Citation
Physics-based thermal impedance models for the simulation of self-heating in semiconductor devices and circuits

Material Information

Title:
Physics-based thermal impedance models for the simulation of self-heating in semiconductor devices and circuits
Creator:
Brodsky, Jonathan Scott, 1969- ( Dissertant )
Fox, Robert M. ( Thesis advisor )
Law, Mark E. ( Reviewer )
Fossum, Jerry G. ( Reviewer )
Eisenstadt, William R. ( Reviewer )
Harris, John G. ( Reviewer )
Hsu, Chen-Chi ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1997
Language:
English
Physical Description:
ix, 257 leaves : ill. ; 29 cm.

Subjects

Subjects / Keywords:
Electric current ( jstor )
Electric potential ( jstor )
Equivalent circuits ( jstor )
Modeling ( jstor )
Oxides ( jstor )
Silicon ( jstor )
Simulations ( jstor )
Thermal resistance ( jstor )
Three dimensional modeling ( jstor )
Transistors ( jstor )
Dissertations, Academic -- Electrical and Computer Engineering -- UF ( lcsh )
Electrical and Computer Engineering thesis, Ph. D ( lcsh )
Semiconductors -- Heat treatment ( lcsh )
Semiconductors -- Thermal properties ( lcsh )
Genre:
bibliography ( marcgt )
theses ( marcgt )
federal government publication ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
Inherent in the operation of semiconductor devices is self-heating, an increase in operating temperature due to a device's own power dissipation. The magnitude of the self-heating effect can be quantified by the value of the thermal impedance, which describes the dynamic response of the device temperature to variations in device power. The thermal impedance is determined primarily by material properties and device structure. The implication of the self-heating effect is that the change in temperature can alter the operating characteristics of a device, which in turn, can affect circuit performance. The primary focus of this dissertation is the development of physics-based models for the thermal impedances of semiconductor devices. Models for the thermal impedances of bipolar and field-effect transistors, on both bulk and silicon-on-insulator (SOI) substrates, are presented. All of the thermal impedance models were derived from the time-dependent heat conduction equation, resulting in compact analytic expressions for the thermal impedances. The physical nature of the thermal impedance models allows them to scale with the device structure and material properties, and they successfully reproduce results from both measurements and three-dimensional finite-element simulations. A circuit model for thermal coupling between transistors in a common substrate is also presented. The coupling model was used in conjunction with the bulk bipolar thermal impedance model to extract a lumped electrothermal model for multiple-emitter bipolar transistors. The secondary objective of this work is the provision of an approach for incorporating these models into circuit simulators. It has been shown that the thermal impedance models can be represented by thermal equivalent circuits made up of resistors and capacitors, making them suitable for efficient circuit simulation. The computer program TIPP (Thermal Impedance Pre-Processor) is introduced. TIPP was developed to provide circuit simulators with convenient algorithms for generating thermal equivalent circuits. TIPP can calculate the component values for thermal equivalent circuits from either physical models or measured data, and is easily modified to interface with different circuit simulators.
Thesis:
Thesis (Ph. D.)--University of Florida, 1997.
Bibliography:
Includes bibliographical references (leaves 247-256).
Additional Physical Form:
Also available on World Wide Web
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Jonathan Scott Brodsky.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
This item is a work of the U.S. federal government and not protected by copyright pursuant to 17 U.S.C. §105.
Resource Identifier:
028004815 ( AlephBibNum )
37823695 ( OCLC )
ALP2249 ( NOTIS )
22109162 ( ALEPH )

Downloads

This item has the following downloads:


Full Text







PHYSICS-BASED THERMAL IMPEDANCE MODELS FOR THE SIMULATION OF
SELF-HEATING IN SEMICONDUCTOR DEVICES AND CIRCUITS






















By

JONATHAN SCOTT BRODSKY


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1997








This work is dedicated to my parents,

Lawrence and Jeraldine,

my brother Matthew and sister Alexandra.












ACKNOWLEDGEMENTS


First, I wish to express my deepest gratitude to my advisor Dr. Robert M.

Fox. His constant support and patient guidance provided a clear path for my research.

It is both a pleasure and a privilege to have worked with Dr. Fox. I thank Drs. Mark

E. Law and Jerry G. Fossum for extending their expertise and help to my unending

questions. I would also like to thank Drs. William R. Eisenstadt, John G. Harris and

Chen-Chi Hsu for their willingness to serve on my supervisory committee. I am also

very grateful to Mary Turner for all of her help throughout my graduate career.

I would like to acknowledge and thank the Semiconductor Research

Corporation (SRC) for the financial support that made this research possible. I am

also grateful to Dr. Surya Veeraraghavan for his guidance and friendship during my

internship at Motorola.

I would like to thank the "TCAD elders", and now my friends, Keith

Green, Dongwook Suh, Ping-Chin Yeh, Haeseok Cho, Chih-Chuan Lin, Ming-Chang

Liang and Scott Miller, for helping me get comfortable in my new surroundings and

setting the standard of excellence.

I am also grateful to my good friends/workmates/"happy hour buddies"

Srinath Krishnan, Samir Chaudhry, David Zweidinger, Omer Dokumaci, Ming-Yeh

Chuang, Dukhyun Chang, Susan Earles, Hernan Rueda, Glenn Workman and Meng-






Hsueh Chiang, for all of the enlightening discussions, the Friday lunch tradition and

the weekend adventures.

There is a very special group of individuals who have my admiration and

love for the friendships they have given me. I would like to thank my best friends

Douglas Weiser, Martin Weiss, Stephen Cea, Edward Cometz and Peter Lynch.

I can not completely express the role my family has played in my life and

in the completion of this dissertation. For simple words seem to diminish their

unconditional and unending love and support. I owe everything I have, everything I

have done and everything I am, to my family. I give my deepest love to my parents,

Jeraldine and Lawrence Brodsky, my brother Matthew and my sister Alexandra.

Finally, I am grateful to all of the wonderful friends that I met in

Gainesville for making this period of my life truly enjoyable.














TABLE OF CONTENTS


ACKNOWLEDGEMENTS .................................


. . . . . . . iii


ABSTRACT ........................................................ viii

1 INTRODUCTION ................................................ 1


1.1 Self-Heating Effects in Semiconductor Devices
1.1.1 Bipolar Transistors ..................
1.1.2 Field-Effect Transistors ..............
1.2 Self-Heating Effects in Semiconductor Circuits
1.2.1 Small-Signal Circuit Performance ......
1.2.2 Large-Signal Circuit Performance ......
1.3 Self-Heating Effects in Parameter Extraction ..
1.4 The Simulation of Self-Heating Effects .......
1.5 Thermal Equivalent Circuits. ...............


. . . . . . . . . . . .
......................3
......................6
. . . . . . . . . . . 8
. . . . . . . . . . . 8
..................... 11
. . . . . . . . . . . 12
. . . . . . . . . . . 13


1.6 The Need for Physics-Based Thermal Impedance Models ..
1.7 O organization ............... ......................


. . . . . . 16
........ .. 20
........... 25


2 A THREE-DIMENSIONAL THERMAL IMPEDANCE MODEL FOR
JUNCTION-ISOLATED BIPOLAR TRANSISTORS. .................... 27


2.1 Introduction .......................................


..........27


2.2 Derivation of the Single-Emitter BJT/HBT Thermal Impedance Model... 28
2.2.1 Modification for Finite Wafer Thickness ................... .. 39
2.2.2 Effects of Interconnect Metallization on the Thermal Impedance ... 43
2.2.3 A Model for the Thermal Impedance of the Emitter Interconnect... 49
2.2.4 Effects of Isolation Structures on the Thermal Impedance ......... 52
2.3 Verification of the Single-Emitter Thermal Impedance Model .......... 61
2.4 Derivation of the Multiple-Emitter BJT/HBT Thermal Impedance Model. 66
2.5 Verification of the Multiple-Emitter Thermal Impedance Model ........ 72
2.6 Summary................... ............................... 76

3 A CIRCUIT MODEL FOR THERMAL COUPLING AND A LUMPED
ELECTROTHERMAL MODEL FOR BULK MULTIPLE-EMITTER BIPOLAR
TRANSISTORS ............... .............................. 77

3.1 Introduction ................................................77
3.2 A Circuit Model for Thermal Coupling ............................ 80









3.3 A Lumped Electrothermal Model for Multiple-Emitter BJT/HBT's ...... 86
3.3.1 A Review of Base-Current Thermometry. ................... .. 90
3.3.2 Generation of the Lumped Electrothermal Model ............... 91
3.4 Verification of the Lumped Electrothermal Model ................... 96
3.5 Summary ............ ....... ....... ........................ 101

4 A THREE-DIMENSIONAL THERMAL IMPEDANCE MODEL FOR
VERTICAL BIPOLAR TRANSISTORS FABRICATED WITH FULL
DIELECTRIC ISOLATION ....................................... 103

4.1 Introduction ........................ ........... ....... .. 103
4.2 Derivation of the DIBJT Thermal Impedance Model ................ 106
4.2.1 Derivation of the Buried-Oxide Heat-Transfer Coefficient ....... 116
4.2.2 Derivation of the Trench Heat-Transfer Coefficient ........... 120
4.2.3 Effects of Interconnect Metallization on the Thermal Impedance . 126
4.2.4 A Model for the Thermal Impedance of the Emitter Interconnect. 129
4.3 Verification of the DIBJT Thermal Impedance Model ............... 132
4.4 Derivation of a Compact DIBJT Thermal Resistance Model .......... 134
4.5 Verification of the DIBJT Thermal Resistance Model ............... 146
4.6 Summary................................................... 149

5 A THREE-DIMENSIONAL THERMAL IMPEDANCE MODEL FOR BULK
METAL-OXIDE-SEMICONDUCTOR FIELD-EFFECT TRANSISTORS .. 150

5.1 Introduction ........................... ..................... 150
5.2 Derivation of the Bulk MOSFET Thermal Impedance Model .......... 153
5.2.1 The Linear Source Thermal Impedance ................... .. 161
5.2.2 The Saturated Source Thermal Impedance ................. .. 164
5.2.3 Effects of the Device Interconnects on the Thermal Impedance ... 165
5.2.4 Effects of Isolation Structures on the Thermal Impedance ........ 171
5.3 Verification of the Bulk MOSFET Thermal Impedance Model ........ 176
5.4 Summary.................. ............................... 182

6 A QUASI-THREE-DIMENSIONAL THERMAL IMPEDANCE MODEL FOR
SILICON-ON-INSULATOR METAL-OXIDE-SEMICONDUCTOR FIELD-
EFFECT TRANSISTORS ........................................ 184

6.1 Introduction ................... ........................... 184
6.2 Derivation of the SOI MOSFET Thermal Resistance Model .......... 187
6.3 Verification of the SOI MOSFET Thermal Resistance Model ......... 202
6.4 Derivation of the SOI MOSFET Thermal Impedance Model .......... 204
6.5 Verification of the SOI MOSFET Thermal Impedance Model ......... 212
6.6 Summary................................................ 216

7 THE THERMAL IMPEDANCE PRE-PROCESSOR: TIPP .............. 219






7.1 Introduction ............................................... 219
7.2 A Description of TIPP ........................................ 220
7.3 Generation of Thermal Equivalent Circuits ........................ 225
7.3.1 Approximation of the Thermal Equivalent Poles/Time Constants. 226
7.3.2 Calculation of the Thermal Equivalent Components ............ 230
7.4 The Interface Between TIPP and Circuit Simulators ................. 232
7.5 Summary................................................... 234

8 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK... 238

8.1 Conclusions ............................................... 238
8.2 Recommendations for Future Work. .......................... 240
8.2.1 The Temperature Dependence of the Thermal Conductivity ...... 240
8.2.2 Models for Thermal Effects Due to Advanced Isolation ......... 242
8.2.3 A Model for Thermal Coupling in SOI MOSFET Circuits ....... 243

REFERENCES ....................................................... 247

BIOGRAPHICAL SKETCH .......................................... . 257












Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


PHYSICS-BASED THERMAL IMPEDANCE MODELS FOR THE SIMULATION OF
SELF-HEATING IN SEMICONDUCTOR DEVICES AND CIRCUITS

By

Jonathan Scott Brodsky

August 1997

Chairman: Robert M. Fox
Major Department: Electrical and Computer Engineering



Inherent in the operation of semiconductor devices is self-heating, an

increase in operating temperature due to a device's own power dissipation. The

magnitude of the self-heating effect can be quantified by the value of the thermal

impedance, which describes the dynamic response of the device temperature to

variations in device power. The thermal impedance is determined primarily by

material properties and device structure. The implication of the self-heating effect is

that the change in temperature can alter the operating characteristics of a device,

which in turn, can affect circuit performance.

The primary focus of this dissertation is the development of physics-based

models for the thermal impedances of semiconductor devices. Models for the thermal

impedances of bipolar and field-effect transistors, on both bulk and silicon-on-






insulator (SOI) substrates, are presented. All of the thermal impedance models were

derived from the time-dependent heat conduction equation, resulting in compact

analytic expressions for the thermal impedances. The physical nature of the thermal

impedance models allows them to scale with the device structure and material

properties, and they successfully reproduce results from both measurements and

three-dimensional finite-element simulations. A circuit model for thermal coupling

between transistors in a common substrate is also presented. The coupling model was

used in conjunction with the bulk bipolar thermal impedance model to extract a

lumped electrothermal model for multiple-emitter bipolar transistors.

The secondary objective of this work is the provision of an approach for

incorporating these models into circuit simulators. It has been shown that the thermal

impedance models can be represented by thermal equivalent circuits made up of

resistors and capacitors, making them suitable for efficient circuit simulation. The

computer program TIPP (Thermal Impedance Pre-Processor) is introduced. TIPP

was developed to provide circuit simulators with convenient algorithms for

generating thermal equivalent circuits. TIPP can calculate the component values for

thermal equivalent circuits from either physical models or measured data, and is

easily modified to interface with different circuit simulators.












CHAPTER I
INTRODUCTION



1.1 Self-Heating Effects in Semiconductor Devices


The physical properties of the materials used to fabricate semiconductor

transistors depend on temperature. Therefore, the operating characteristics of a

transistor (e.g. electrical currents and potentials), which are determined by the

material properties, are also temperature dependent. The temperature at which a

transistor operates is determined by the temperature of the surrounding environment

(referred to as the "local ambient temperature") and the power dissipated in the

device (referred to as the "self-heating effect"). Therefore, the time-dependent

temperature of a transistor can be expressed as


t
T(t) = Tamb + P(t')hTH(t-t')dt', (1.1)
0


where hTH is the thermal impulse response and P is the instantaneous power. The

second term on the right-hand side of (1.1) represents the temperature rise in the

device


AT(t) = P hTH, (1.2)


where 0 is the convolution operator. The temperature rise can also be expressed in







the frequency domain as


AT(t) = - [ZTH(S) P(s)], (1.3)


where -1 represents the inverse Laplace transform and ZTH(s) is the thermal

impedance. The thermal impedance of a transistor describes the dynamic response of

the device temperature to variations in device power, and is determined primarily by

the material properties and the structure of the device. The transient thermal

impedance can be defined as



ZTH(t) = -1 ZTH(s), (1.4)


which represents the normalized thermal step response.

Since the power dissipation in (1.1) is determined by the operating

characteristics of a transistor, it depends on temperature such that


P = P(T) = Idev(T) Vdev(T), (1.5)


where Idev(T) and Vdev(T) represent general currents and potentials within a given

device, respectively. Consequently, there is feedback between the thermal and

electrical operation of the device. Whereas the transistor temperature is usually

assumed to be constant, the electrothermal coupling implied by (1.1) and (1.5) shows

that the temperature actually varies with the device operation. Thus, to fully

characterize the operation of semiconductor transistors, both the electrical and

thermal behavior should be determined.







1.1.1 Bipolar Transistors


In the forward-active mode, the operating characteristics of bipolar

junction and heterojunction transistors (BJT's and HBT's) are controlled by the

injection and diffusion of minority carriers in the base region. For an npn transistor,

electrons are injected across the forward-biased base/emitter junction, causing an

exponential increase in the minority carriers in the base. The electrons diffuse across

the base and are swept into the collector by the reversed-biased base/collector

junction. For a fixed base/emitter voltage, assuming negligible recombination in the

quasi-neutral base, the collector current can be expressed as


2 qVBE'
Ic(T) ni (T) exp EkT (1.6)


where ni is the intrinsic carrier concentration, q is the electron charge, k is

Boltzman's constant and T is temperature. The overall temperature dependence of

(1.6) is dominated by the relation between the intrinsic carrier concentration and

temperature, given by


2 "-Eg
n (T) = N. N, exp.--. ) (1.7)


where Eg is the semiconductor band-gap energy and Nc and Nv are the effective

density of states in the conduction and valence bands, respectively. The junction

voltage is always less than the band-gap and therefore, an increase in temperature

causes an exponential increase of minority carriers in the base, resulting in an

increase in collector current. Since the collector current is a significant component







of the power dissipation in a BJT, self-heating results in a regenerative feedback

between the collector current and the temperature of the device. This positive

feedback can lead to the destructive phenomenon of thermal runaway in BJT's

[Shu90].

For fixed base current, the collector current can be expressed as


Ic(T) = P(T) IB, (1.8)


where P(T) is the common-emitter current gain. For moderate injection levels, the

current gain can be approximated by the ratio of the electrons injected into the base

to the holes back-injected from the base to the emitter. This ratio, and hence p(T),

are typically high since the emitter usually has a higher doping level than the base.

Due to heavy-doping effects in the emitter, the emitter band-gap is typically less than

that in the base so that


P(Y) NDE -AEgN
(T) NB expAE ), (1.9)
NAB V k!T

where AEg is the band-gap difference between the emitter and base, and NDE and

NAB are the doping concentrations in the emitter and base, respectively. As shown

by (1.9), the current gain is greater at higher temperatures; consequently, the

collector current is, again, an increasing function of temperature. The rate of increase

with temperature in this case, however, is not as significant as that for a device biased

with a fixed base voltage. Therefore, the self-heating effect is not as substantial in

BJT's driven by a fixed base current.







HBT's are bipolar devices that use band-gap engineering in either the

emitter or base region to improve the current gain over homojunction BJT's. The

resulting band-gap in the emitter is wider than that in the base, so that the potential

barrier induced by the band-gap discontinuity effectively impedes the injection of

carriers from the base to the emitter. When biased with a fixed base voltage, the

temperature dependence of an HBT is similar to that of a standard BJT. However,

when an HBT is driven with a fixed base current, the temperature dependence of the

collector current is quite different than that of a BJT. While the collector current in

this case can still be determined from (1.8), the common-emitter current gain is now

expressed as


NDE (AEg'
P1(T) exp (1.10)
NAB kT


As a result of the band-gap being wider in the emitter than in the base, the sign of the

exponential argument is now positive. Therefore, as opposed to a standard BJT, the

current gain and the collector current decrease with increasing temperature. As a

result, self-heating in HBT's can lead to the non-catastrophic failure mechanism

known as current collapse [Sei93].

To reduce the effects of parasitic resistances and current-crowding, large

bipolar devices are commonly fabricated using multiple devices connected in parallel

[Shu90]. Multiple-emitter devices, both BJT's and HBT's, are capable of operating

at high frequencies under high power densities [Win67, Mar93, Liu95b]. However,

multiple-emitter devices suffer from more complex self-heating effects due to the

thermal interactions among neighboring devices. The thermal coupling leads to







lateral temperature gradients across the device, resulting in the inner emitters

operating at higher temperatures. Due to the positive feedback between junction

temperature and junction current, the inner devices carry more current than those at

the outer extremes. As the current density in the inner emitters increases, the self-

heating effect in these devices accelerates. The premature activation of thermal

runaway in BJT's and current collapse in HBT's is attributed to this thermal

instability inherent in multiple-emitter devices [Win67, Liu93, Kag94, Lio94,

Lio96].



1.1.2 Field-Effect Transistors


For Metal-Oxide-Semiconductor Field-Effect Transistors (MOSFET's)

operating in strong inversion, the current characteristics are determined by the drift

current of carriers in the inverted channel region. For small drain voltages, a

MOSFET operates in the linear region where the carrier velocity depends on the

longitudinal electric field in the channel. In the linear region, the drain current can

be approximated as



ID(T) oc (T). -VGs -Vt(T) DS, (.1)


where 1p(T) is the carrier mobility, Vt(T) is the threshold voltage, and VGS and VDS

are the applied voltages between the gate and source and drain and source,

respectively. At higher drain voltages, the electric field at the drain end of the

channel is large enough to cause the carrier velocity to saturate. In the saturation







region, the drain current can be expressed as


ID(T) Qc(VGs, VDS, st(T)) vsat(T), (1.12)


where Qc is the channel charge and vsat(T) is the saturated carrier velocity.

The overall temperature dependence of (1.11) and (1.12) are dominated

by the sensitivity of the carrier mobility to changes in temperature. Due to increased

lattice scattering at higher temperatures, mobility decreases as temperature increases.

The reduction in mobility leads to a decrease in drain current, which implies that the

drain current of a MOSFET is a decreasing function of temperature. At high power

dissipation levels, the self-heating effect can cause the drain current to drop below

the ambient temperature value. In such cases, the output conductance becomes

negative, and the device exhibits a negative dynamic resistance (NDR) [Sha83].

MOSFET's fabricated on silicon-on-insulator (SOI) substrates have

temperature dependence that are similar to those of their bulk counterparts, though

the effects of self-heating can be enhanced due to the low thermal conductivity of the

insulating layers. For non-fully depleted (NFD) SOI MOSFET's, however, floating-

body effects further complicate the thermal effects [Wor97]. Impact-ionization-

induced floating-body effects are known to cause the kink, or increase in drain

current, in NFD SOI MOSFET's. The kink is affected by self-heating in two ways.

First, at elevated temperatures, the onset of the impact-ionization is retarded.

Second, an increase in recombination in the quasi-neutral body reduces the

threshold-voltage shift caused by the impact-ionization. Therefore, in addition to a

reduction in drain current due to mobility effects, self-heating also reduces the







current in NFD SOI MOSFET's through temperature-dependent floating-body

effects.




1.2 Self-Heating Effects in Semiconductor Circuits


Since the operating characteristics of transistors are affected by

temperature, the integrated circuits that depend on these transistors will also be

affected by changes in temperature. In modern digital circuits, the high switching

speeds of the transistors, the relatively slow time constants associated with the

temperature response and the low static power dissipation, all help reduce the

instantaneous temperature rise. Consequently, self-heating effects are typically

negligible in digital circuits. On the other hand, analog circuit applications

commonly have significant power dissipation and can operate at frequencies which

are comparable to the thermal time-constants. Therefore, analog circuits are

generally more prone to self-heating effects.



1.2.1 Small-Signal Circuit Performance


The effect of self-heating on small-signal BJT characteristics was derived

by Mueller and investigated in bipolar circuits by Fox et al. [Mue64, Fox93b]. The

two-port small-signal admittance parameters, in the presence of self-heating, were

shown to be


YmnE + DmZTHImIn (1.13)
mn 1 DmZTHP







where ymnE are the admittance parameters neglecting self-heating, and Dm

represents the variation of the current Im with temperature. The denominator of

(1.13) establishes the sensitivity of the admittance parameters to power, and has a

significant impact as DmZTHP approaches unity. The effect of the denominator

generally becomes important only at high power dissipation. However, as the thermal

impedance increases (i.e. due to device scaling), the power level that defines the

threshold for self-heating effects will decrease. The second term in the numerator of

(1.13) shows that the effect of self-heating on the admittance parameters is also

proportional to the operating currents. For yll and y21 the self-heating term in the

numerator is small so that yll =YlE and y21 =Y21E. However, the effect of self-

heating can be substantial in the numerators of yl2 and y22, even at moderate current

levels. The thermal effects on these parameters can result in a coupling between the

collector output admittance and the impedance of the base-driving source. Also, as

shown in Figure 1.1, there can be a significant reduction in the voltage gain of BJT

amplifiers.

The small-signal performance of analog MOSFET circuits can also be

affected by self-heating. For moderate power levels, the thermal effects are similar

to those in bipolar circuits. However, as mentioned previously, the drain current of a

MOSFET decreases with increasing temperature, and significant self-heating can

induce NDR. The effect of a negative output conductance can be investigated by

examining the voltage gain of a MOSFET amplifier. As shown by Fox and Brodsky

[Fox93a], if the devices in the amplifier enter a region of negative output

conductance, the gain of the amplifier changes polarity. For an inverting amplifier,








































102 104 106 108
Frequency, (Hz)


1010


The effect of self-heating on the small-signal gain of a BJT differential
amplifier. The data was simulated using a version of SPICE, which was
modified to account for dynamic variations in temperature [Zwe97],
and the thermal impedance model for bipolar transistors presented in
Chapter Two.


300






200






100






0


Figure 1.1







self-heating effects can therefore cause the gain to become non-inverting, resulting

in hysteresis in the amplifier's output characteristics.



1.2.2 Large-Signal Circuit Performance


The effects of self-heating on the large-signal operation of analog bipolar

circuits was investigated by Fox et al. [Fox93b]. The types of circuits that are

sensitive to thermal effects are typically those that depend on the precise control of

BJT characteristics. For example, the mismatch in the reference and output currents

of a current mirror can be increased due to self-heating-induced differences in the

operating conditions of the transistors. Translinear circuits and band-gap voltage

references can also be affected by self-heating due to their strong dependence on the

thermal voltage. Thus, neglecting self-heating can result in significant discrepancies

between the ideal and actual operation of these types of circuits. The large-signal

transient operation of analog circuits is also affected by self-heating. The long time

constants of the thermal characteristics can effectively slow down the electrical

response of a circuit. Fox et al. showed that the five-percent settling time of a Gilbert

multiplier increased by over an order of magnitude due to self-heating [Fox93b].

While the errors caused by self-heating can be reduced by careful circuit design, they

can not be completely eliminated.







1.3 Self-Heating Effects in Parameter Extraction


The extractions necessary to determine the behavioral characteristics of a

semiconductor transistor are often performed at bias levels that cause moderate- to

high-power dissipation. Typically, the parameters that are extracted are assumed to

correspond to the ambient temperature at which the measurements are carried out.

However, at significant power levels, self-heating will cause a temperature rise in the

device. Neglecting the temperature rise that can occur during the measurements can

lead to erroneous results [Zwe97]. For example, the Early voltage, VA, of a BJT is

commonly extracted from the slope of the IC -VCE characteristics in the linear region

of operation. If self-heating is significant, the slope of the output curves depends on

the source that is driving the base [Fox93b]. Therefore, the exact meaning of the

value extracted for VA would be ambiguous unless the thermal effects were taken

into account.

Various methods have been proposed for removing the effects of self-

heating from parameter extraction. One approach augments a standard extraction

routine with measurements designed to determine the thermal characteristics of the

device. The full set of parameters can then be input to a global optimization routine

to generate electrical-only parameters that are independent of self-heating [Zwe97].

Other techniques attempt to directly remove the effects of self-heating from the

parameter extraction measurements by making the temperature rise in the given

device negligible. The temperature rise can be minimized by performing the

extractions in low-power regions, or by using complex high-speed measurements

[Tu94, Jen95]. Since the device is not allowed to heat, the resulting device parameter






set would be approximately devoid of self-heating effects, and would essentially

correspond to the given device operating under isothermal conditions. Consequently,

the resulting electrical parameters would only pertain to device operation for low-

power or high-speed circuit applications, and would not convey the proper device

characteristics for applications that experience substantial self-heating effects. Thus,

for a set of electrical-only parameters to correctly represent the characteristics of a

device, over a wide range of operating conditions and biases, it should be augmented

by additional parameters that describe the thermal attributes of the device.




1.4 The Simulation of Self-Heating Effects


As shown in the previous sections, the operating characteristics of both

individual transistors and circuits depend on temperature. Due to self-heating, the

effective operating temperature depends on power dissipation and can therefore vary

under different operating conditions.

By solving the time-dependent heat conduction equation and energy

balance equations for electrons and holes, numerical device simulators can model

phenomena associated with dynamic self-heating in individual transistors [Lia94].

While this approach is invaluable for examining the detailed physics that govern the

operation of semiconductor devices, it is impractical for simulating all but the

simplest of circuits. Therefore, to investigate the effects of dynamic self-heating on

a broad range of circuits, a more efficient simulation approach is necessary.







The standard version of most circuit simulators such as Berkeley SPICE

[Nag75] and HSPICE [Hsp92], treat temperature as a static global parameter. This

has two significant implications. All of the semiconductor devices in a simulation

operate at the same temperature, and that temperature remains constant throughout

the simulation. Due to these constraints, a circuit simulator may not accurately

represent the physical operation of a circuit, where spatial and temporal variations of

the temperature can cause each device to operate at its own local temperature. To

account for the temperature dependence of a circuit's operation, circuit simulators

should be capable of independently tracking the dynamic temperature of each device

in the circuit.

A common approach for creating an electrothermal circuit simulator

(ETCS) uses the concept of the thermal impedance and the analogy between

electrodynamics and heat flow to account for dynamic temperature variations. This

approach allows temperature to be represented as an electrical potential and power

as an electrical current [Lee96, Zwe97]; therefore, the local operating temperature of

a device can be thought of as simply another "bias" condition. To facilitate the

temperature "bias" condition, an external node is added to a given compact device

model [McA92, Fos95, Lee96, Zwe97]; such a configuration is shown in Figure 1.2.

Attached to this node, internal to the device model, is a controlled current source that

represents the instantaneous power dissipation. The parameter set for the device

model should be modified to include the correct temperature dependence. When the

modified device model is used for a circuit simulation, a thermal impedance (and, in

some case, a voltage source to represent the reference ambient temperature) can be


























T0p(t)


p(t)


I


Tamb


A generalized schematic showing a common method for modifying a
compact device model to include temperature as a variable. The dashed
box outlines the new model with the added temperature node; DEVICE
represents the original electrical-only model.


IL
I


Figure 1.2


____







attached to the new external node. Therefore, the voltage generated at this additional

node represents the local temperature of the device. The electrical-only device model

is first solved at the ambient temperature; this solution results in an initial guess for

the device power dissipation. This power is then used to calculate the temperature

rise in the device. Once the approximate local operating temperature is calculated, it

is used to update the temperature-dependent model parameters, which are used to

recalculate the electrical bias potentials and currents of the device model. This

procedure is repeated until self-consistent solutions for the temperature and electrical

biases are reached. Thus, an effective operating temperature can be independently

calculated for each device in a simulation, and that temperature can now vary with

the operating point.




1.5 Thermal Equivalent Circuits


The data that quantify the thermal impedance of a transistor are typically

in the form of discrete data points for the temperature rise, normalized to a unit-step

increase in power dissipation, versus time or frequency. In such a format, the thermal

impedance data are not readily accessible by an ETCS. While data in a tabular format

can be used without much complexity for DC and AC simulations, an inefficient

convolution computation would be required to use the data for transient simulations.

Therefore, a representation for the thermal impedance is needed that both accurately

models the physical data and can be easily incorporated into an ETCS for efficient

DC, AC and transient simulations.







In an ETCS, when a current representing the power dissipation in a

transistor is applied to the thermal impedance, ZTH, the resulting voltage represents

the temperature rise in that transistor. By invoking the analogy between

electrodynamics and heat flow, the thermal impedance can be represented as an

electrical impedance. Common representations for the electrical impedance circuits

are shown in Figure 1.3. The resistances and capacitances that comprise the

impedance effectively represent the lumped three-dimensional thermal resistance

and heat capacity of the semiconductor device structure. Therefore, the overall

electrical network can be referred to as a thermal equivalent circuit. The values for

the individual elements of a thermal equivalent circuit can easily be determined by

numerically fitting the circuit to existing thermal impedance data. Thermal

equivalent circuits are directly applicable for DC and AC electrothermal simulations

since, in such cases, the voltage drop across the network is simply equal to the

product of the current and the network resistance or impedance. In addition, such

networks inherently provide an efficient method for effecting the necessary transient

convolution.

As will be shown in the subsequent chapters of this dissertation, the

fundamental nature of heat flow is that of a distributed system. The dynamic

temperature rise in a device due to self-heating can occur over three or more decades

of time or frequency. A single time constant associated with a simple exponential

function can not represent the distributed behavior of self-heating. Consequently, the

network response of the single-pole thermal equivalent circuits which have been

proposed in previous works [McA92, Bau93, Lee93, Tu94], will not accurately












rth2


Cthl Cth2 Cth3
I I


(a)





rthl rth2 rth3
"--- -I ......

I I IAA


Cthl


Cth2


Cth3


Thermal equivalent circuits used to represent a thermal impedance for
circuit simulation: a) Cauer network representation; b) Foster network
representation.


Figure 1.3


rthl


rth3







model the dynamic thermal impedance. Thermal equivalent circuits consisting of

cascaded resistor/capacitor stages, as exemplified in Figure 1.3, effectively provide

a distributed network response, and therefore allow a more accurate representation

of a dynamic thermal impedance [Bro93].

In the work by Szekely and Van Bien [Sze88], the Foster circuit

(Figure 1.3b) was shown to be an invalid representation of a discretized thermal

network. This point is valid in the context of numerical simulations (e.g. finite

difference or finite element) where the transistor structure is modeled by a

distributed thermal network. In that case, the node-to-node capacitances of the Foster

network do not have physical meaning and the Cauer network would be the proper

physical discretization of the given thermal domain. However, in this dissertation,

the thermal equivalent circuit is simply a numerical representation of a thermal

impedance, and the validity of its format is moot. Yet, for the purpose of representing

a lumped thermal impedance in an ETCS, the Foster network form offers an

important advantage over the Cauer form: the time constants associated with a given

Foster network are independent of any surrounding circuit elements. This

characteristic is beneficial when individual thermal equivalent circuits must be

connected to model different components of a transistor structure or the thermal

interactions between transistors. Therefore, the Foster network form will be assumed

for any thermal equivalent circuits within this dissertation.







1.6 The Need for Physics-Based Thermal Impedance Models


In the previous two sections, the concept of the thermal impedance is

adopted to model the temperature rise in a transistor as a function of that device's

power dissipation. For the purpose of circuit simulation, the thermal impedance can

be represented by a network of resistances and capacitances that effectively represent

the lumped thermal characteristics of a transistor. To successfully synthesize a

thermal equivalent circuit, tangible data for the thermal impedance are necessary.

One approach to obtain the thermal impedance of a transistor is to extract

it from measurements [Lee95, Zwe96]. While this empirical approach provides

accurate temperature information, such measurements are somewhat difficult, for

several reasons. To begin with, thermal measurements are very time-consuming. The

extraction procedure is generally divided into two steps, the first of which dominates

the total measurement time. This step is required to calibrate the relation between the

temperature and the physical characteristic that is being used to monitor the

temperature (e.g. the base current and drain current in bipolar and field-effect

transistors, respectively). The calibration is performed at multiple ambient

temperatures at DC and is thus limited by the long time constants associated with

steady-state heat flow. To make such thermal measurements requires special

measurement equipment such as a thermal wafer chuck or oven to accurately control

the temperature of the devices being measured. Finally, the results of any such

extraction are limited to the specific device being measured. Thus, the entire

procedure would have to be repeated for each transistor structure and transistor type

of interest.







Another approach, which avoids the inherent complexities of thermal

measurements, is to derive the thermal impedance of a transistor from the physical

equations that govern the temperature and heat flow in the device. Physical thermal

modeling is desirable because it can give the temperature behavior as a function of

the device structure and material properties alone; therefore, the effects of device

technology scaling on the thermal impedance can be predicted. The requirements of

accurate physical modeling (e.g. multi-dimensional numerical simulations) tend to

conflict with the needs for simplicity and efficiency in circuit simulation. However,

a thermal impedance model does not need to be absolutely accurate to provide

reasonable results within an ETCS. Therefore, by using certain heuristic

assumptions, compact physical models for the thermal impedance, suitable for

efficient simulation, can be derived. It is important, though, that the correlation

between the accuracy of the thermal impedance models and the accuracy of the

simulated electrical characteristics of a semiconductor device in the presence of self-

heating be understood.

The sensitivity of a given electrical parameter, X, of a semiconductor

device to the thermal resistance can be defined as


RTH RTH aX
sx X T (1.14)
X X .RTH


As an example, since the output current of a device is very important for

characterizing performance, (1.14) can be used to determine the sensitivity of the

collector and drain currents of BJT's and MOSFET's, respectively. Using (1.1) in the

steady-state limit and (1.6) with (1.14), the sensitivity of the collector current of a







BJT is expressed as


RTH q. (Vg- VBE) .RTH P
S (1.15)
c k (RTH P + T)2


where Vg is the semiconductor band-gap voltage. A similar expression for the

sensitivity of the drain current of a MOSFET can be determined using the current

equation given by Fox and Brodsky [Fox93a], which results in


SRT RTH. p P RTH. p
SI= -a + 1 (1.16)
1 To To


where a is typically between 1.5 and 1.8 (assuming that the temperature dependence

of the drain current is dominated by the temperature sensitivity of the carrier

mobility). The expected level of error in simulated output currents can be

approximated by the product of the sensitivity and the anticipated error in the thermal

impedance model. Therefore, as shown by (1.15) and (1.16), the relation between the

accuracy of the thermal impedance models and the accuracy of the calculated

electrical parameters depends on the power dissipation and the sensitivity of the

electrical parameters to temperature. Consequently, the level of accuracy of a

thermal impedance model is more critical for devices with electrical characteristics

that are highly sensitive to temperature (e.g. BJT's as opposed to MOSFET's). At

low power dissipation levels, where the temperature rise is small compared to the

ambient temperature, the error in the thermal impedance model will not directly

correspond to the error in the calculated operating temperature. For a temperature

rise of twenty degrees, the sensitivities of the BJT collector current (VBE = 0.8) and







MOSFET drain current are 0.7 and -0.1, respectively. In such a case, the error in the

electrical characteristics will tend to be lower than the error in the thermal

impedance. Whereas at large power dissipation levels, the temperature rise can be

much larger than the ambient temperature, and the error in the thermal impedance

model will directly correspond to the error in the calculated operating temperature

(for a temperature rise of one-hundred degrees, the respective BJT and MOSFET

current sensitivities are 2 and -0.4); in which case, large errors in the calculated

electrical characteristics can result. Figure 1.4 shows an example of BJT

characteristics simulated assuming a 20% error in the thermal resistance model; the

simulations were performed using the modified version of SPICE created by Lee

[Lee96]. The data clearly shows that for larger temperature rises, the error in the

calculated current (due to errors in the thermal impedance model) increases.

The motivation behind this dissertation is the development of compact

thermal impedance models for semiconductor transistors. These models can provide

a reasonably accurate representation of the dynamic temperature response within a

device; more importantly, since the models depend mainly on the physical structure

of a device, they can correctly anticipate the effects of technology scaling on the

thermal behavior. Physics-based thermal impedance models allow an ETCS to

predict dynamic self-heating effects in circuits and can also provide more accurate

electrical parameter extraction. In addition, when the thermal impedance models are

coupled with physics-based compact device models, the combination provides an

efficient tool for studying self-heating in semiconductor transistors.




















40




S30


S20



o
U 10 -




0
0Figure 1.4














Figure 1.4


1 2 3 4 5
Collector-Emitter Voltage, VCE (V)


Simulated output characteristics of a BJT, assuming that
RTH = 1000 C/W, for VBE = 0.80, 0.85, 0.90 and 0.95 V. The
simulations are repeated assuming a 20% error in the thermal
resistance model, so that RTH = 800 oC/W.







1.7 Organization


Chapter Two presents a physics-based model for the thermal impedance of

bulk junction-isolated bipolar transistors. The model is derived by solving the three-

dimensional time-dependent heat conduction equation in the substrate. The ability of

the model to represent bulk BJT/HBT's with either LOCOS or trench isolation is

investigated. To account for multiple-emitter bipolar transistors, the thermal

impedance model is extended to represent multiple heat sources. The accuracy of the

model is evaluated using measurements and three-dimensional finite-element

simulations.

Chapter Three describes a circuit network for modeling thermal

interactions between devices located in the same substrate. The network is developed

for the specific application of multiple-emitter bipolar devices, but is shown to be

valid for general cross-substrate thermal coupling in circuits. A method for

improving the simulation efficiency of a multiple-emitter BJT/HBT electrothermal

model, using a lumped thermal impedance model, is presented. The validity of the

lumped modeling approach is supported with comparisons to the full electrothermal

model.

Chapter Four presents a predictive scalable model for the thermal

impedance of BJT's with full dielectric isolation. The model is derived by solving

the three-dimensional time-dependent heat conduction equation in the substrate

accounting for the buried oxide and trench isolation. In the limit of steady-state heat

conduction, the thermal impedance model is simplified, resulting in a closed-form







model of the thermal resistance. The accuracy of both models is evaluated using

three-dimensional finite-element simulations and measurements.

Chapter Five describes a physics-based model for the thermal impedance

of bulk MOSFET's. The model is derived by solving the three-dimensional time-

dependent heat conduction equation in the substrate. The effects of the device

interconnects and isolation structures, such as LOCOS and trenches, on the thermal

impedance are investigated. The accuracy of the model is evaluated using

measurements and three-dimensional finite-element simulations.

Chapter Six presents a predictive scalable model for the thermal

impedance of SOI MOSFET's. The model is initially derived for steady-state heat

conduction by coupling separate one-dimensional heat conduction analyses in the

silicon film and interconnects. The derivation is then carried out for the case of time-

dependent heat conduction, resulting in a model for the dynamic thermal impedance.

The accuracy of both models is evaluated using three-dimensional finite-element

simulations and measurements.

Chapter Seven describes a computer program developed to facilitate

thermal modeling in circuit simulation. The program, referred to as the Thermal

Impedance Pre-Processor (TIPP), functions as a framework for obtaining the

component values of thermal equivalent circuits from the thermal impedance models

presented in Chapters Two through Six.

Chapter Eight concludes the dissertation with a summary of the

accomplishments of this work and suggestions for future modeling efforts.













CHAPTER 2
A THREE-DIMENSIONAL THERMAL IMPEDANCE MODEL FOR JUNCTION-
ISOLATED BIPOLAR TRANSISTORS



2.1 Introduction


The models derived in this chapter provide closed-form physical solutions

for predicting the thermal impedances for single- and multiple-emitter bipolar

junction (BJT) and heterojunction bipolar (HBT) transistors, based solely on device

geometry and material properties. These models can predict both steady-state and

dynamic self-heating due to the semiconductor substrate. Previous works in this area

provided values for the thermal impedance of BJT's or HBT's, but were either

limited by assumptions or relied on non-predictive measurement techniques. For

example, the thermal impedance model derived by Fox and Lee [Fox91a] is limited

to single-emitter devices. The analyses in other works only provide models for the

steady-state thermal resistance [Lio93, Bau94, Daw94, Lio94, Lio96]. Some authors

have used measurement techniques to extract either the steady-state thermal

resistance or simple one-pole approximations for the thermal impedance [Bau93,

Liu93, Daw94, Liu95a, Liu95b]; in either case, the results do not provide a complete

picture of self-heating and are not predictive.

The thermal analysis by Joy and Schlig [Joy70] serves as the foundation

for deriving of thermal impedance model, and the first part of this chapter re-







examines this work to provide a clear background for modifications made to the

model later in the chapter. The thermal impedance model developed by Joy and

Schlig was derived for single-emitter, junction-isolated BJT's operating in the

forward-active region. A diagram of a typical junction-isolated npn BJT is shown in

Figure 2.1. In this chapter, the basic model is modified to account for variations in

substrate thickness. The effects of interconnect metallization and different isolation

technologies on the thermal impedance, and thus on the performance of the model for

advanced device structures, are investigated. The single-emitter thermal impedance

model is finally extended to account for BJT/HBT's with multiple emitter fingers.




2.2 Derivation of the Single-Emitter BJT/HBT Thermal Impedance Model


For the derivation of the single-emitter bulk BJT/HBT thermal impedance

model, the semiconductor substrate is represented by a homogeneous semi-infinite

half-space with an adiabatic top surface (no heat transfer perpendicular to the

surface). The back side of the substrate is assumed to be held at a constant

temperature, To. Since the substrate material is assumed to be homogeneous, the

model most directly applies to junction-isolated transistors. The effects of other

types of isolation structures used in bulk technologies, such as recessed LOCOS

(local oxidation of silicon) or back-filled trenches, on the thermal response are not

taken into account. Figure 2.2 illustrates the simplified device geometry assumed for

the model derivation; the diagram focuses on the "electrically active" portion of the

device that lies directly beneath the emitter stripe, which has a width W and length





















Collector


p-substrate


Cross-section of a typical junction-isolated bipolar junction transistor
(BJT).


Figure 2.1


Emitter


Base




















(dT/dz) = 0


__-------- -- r_ ----


The simplified device geometry used to define the solution domain for
the bulk, single-emitter BJT/HBT thermal impedance model. The
substrate is represented by a semi-infinite half-space with an adiabatic
surface (the dotted lines). The emitter stripe has a width W and length
L. The heat source (the rectangular volume) is displaced a distance D
below the surface of the device, equivalent to the depth of the base/
collector junction. The heat source has a thickness H which
approximates the base/collector space-charge region (SCR).


D



H


Figure 2.2







L. The imbedded heat source represents the base/collector space-charge region

(SCR), which is further represented by a rectangular volume with a thickness, H. The

heat generated in this region is assumed to be due to uniform power dissipation. This

assumption is reasonable for devices in the forward-active region of operation prior

to any high current effects, as the current distribution in the intrinsic device will be

approximately uniform. The electric field gradient in the base/collector SCR can also

be neglected since it does not greatly affect the thermal impedance model. The heat

source is displaced beneath the surface of the substrate by a distance D, assumed to

be the depth of the base/collector junction. Thus, any encroachment of the base/

collector SCR into the base region is neglected (which is reasonable since the base

typically has a higher doping than the collector).

Representing the substrate as a semi-infinite medium presumes that the

back-side and the lateral edges do not influence the thermal response of the device.

Neglecting the effects of the back side of the substrate on the thermal response is

reasonable since a typical wafer is about 1000 times thicker than the heat source.

Neglecting the effects of the lateral boundaries requires that the device be located

sufficiently far from the substrate edges; the work by Fox et al. [Fox93b] suggests

that this assumption is valid for any device that is at least a distance 5/W L from

any lateral edge. The surface of the substrate is assumed to be the only boundary that

affects the thermal response of the device and it is considered to be adiabatic; thus,

conduction through the interconnects and conduction/convection from the surface

are neglected. Ignoring thermal energy transport from the substrate surface is

supported by the work of Berger and Chai and Goodson et al. [Ber91, Goo95];







however, they were mainly concerned with transport via convection to a surrounding

gas (namely air). Nonetheless, for the regions of the device covered by oxide, it is

unclear whether there is substantial heat conduction to this overlying oxide. From the

analysis of Goodson et al. [Goo95], the device-to-oxide thermal conductance is of the

order of G = 4rk, which corresponds to an isothermal disk of radius r on the

boundary of a semi-infinite medium of thermal conductivity k. Approximating the

radius as J(WL)/7t and using the room-temperature thermal conductivity of SiO2,

the device-to-oxide thermal conductance for a typical device is on the order of

1 x 10-6 (W/OC). Comparably, the device-to-substrate thermal conductance is on the

order of 1 x 10-3 (W/C), showing that the majority of heat will flow through the

substrate.

The temperature rise at any point within the device can be described by the

nonhomogeneous three-dimensional heat conduction equation



V2AT(x, y, z, t) + g(x, y, z, t) I DAT(x, y, z, t) (2.1)
k a at


and the boundary conditions


AT(- y, z, t) = 0 (2.2)


AT(x, +oo, z, t) = 0 (2.3)


aAT(x, y, z, t) 0 (2.4)
z=o







AT(x, y, o, t) = 0, (2.5)


where AT is the temperature rise above the local ambient (AT = T-To), g is the

internal energy generation density, k is the thermal conductivity, a is the thermal

diffusivity (a = k/(p cp) where p is the density and cp is the specific heat) and t

is time. Typical values for the material properties are given in Table 2.1.



Table 2.1 Semiconductor material properties


Source: [Mul77]



Equation (2.1) assumes that the thermal conductivity is independent of temperature

and position. Neglecting the temperature dependence of the thermal conductivity is

reasonable for a moderate temperature rise, where the temperature rise will vary

linearly with power dissipation. However, for large temperature excursions, the

value of the thermal conductivity can vary significantly; the thermal conductivities

of Si and GaAs will vary from their room-temperature values by more than 20%

above 355 and 390 K, respectively [Gao89]. For such large temperature excursions,

the linear relation between temperature rise and power will not be valid. However,

the temperature dependence of the thermal conductivity can be accounted for by


Parameter Si GaAs

k (W cm- K-1) 1.412 0.455

p (g cm-3) 2.328 5.316

Cp (J g-1 K-1) 0.70 0.35






using the Kirchoff transformation [Joy75], as discussed in Chapter Eight. Neglecting

the spatial dependence of the thermal conductivity implies that the effect of dopant

atoms is ignored. In the works by Weber and Gmelin and Goodson et al. [Web91,

Goo95], the thermal conductivity of doped silicon (up to lx1018 and 1.7x1019 dopant

atoms cm-3) above 300 K is shown to differ only slightly from that of intrinsic

silicon. Since the majority of the substrate is typically low-doped semiconductor

material, neglecting the doping effects on the thermal conductivity is reasonable.

With the initial thermal conditions within the substrate specified as


AT(x, y, z, 0) = 0, (2.6)


the solution to (2.1) can be expressed in the form


t
AT G(x, y, z, t) dt G(x, y, z, tx', y ,t')g(x', y, z, t')dv' (2.7)
t'=0 V


where



1 (x-(x) (y)-y)
G(x,y,z, tx',y',zz',t') = 1 __(exp - exp -(
8[to(t-t')]3/2 L4(t t )J 4(t -t)

exp -(-z') + exp -( (2.8)
4a(t -j t) 4(t t )(2


is the Green's function for the given boundary-value problem [Ozi93]. Equation (2.8)

is the solution to







V2G + gpi (x x')8(y y')8(z z')(t t) 1 G(2.9)
k a at


for the boundary and initial conditions given by (2.2) through (2.6), and physically

represents the temperature at point (x, y, z) at time t, due to an instantaneous point

source, gp (W s), of unit strength at point (x', y', z') at time t .

To account for the heat-generation volume (V = W L H), (2.8) is

substituted into (2.7) and integrated over the base/collector SCR, resulting in


t
SP(t') -( L/2 + x ( L/2-x x
AT(x, y, z, t)= P(t erf( L/2x +erf
S8pcV \,4ac(t t) 4c((t t')
t=-0
r W/2+y W/2-y
[erf-( W/2+ + erfW/2
L V4a(t- t') ,4a(t t')
S[erf z+D+H erf( -D-z_
L erf)+erf
L 4a(t t4) a4(t t))
( z-D D+H-z
+ef erf +erf( D+Hzdt' (2.10)
-/4a(t t) J4c(t t )


where g(x', y', z', t') = g(t') = P(t')/V, since the power dissipation is assumed to be

uniform. Equation (2.10) represents the temperature response at any point in the

device at time t due to a change in continuous power dissipation in the base/collector

SCR. Assuming a step increase in power at t' = 0 (P(t') = P U(t) ) and expressing

the temperature rise as


AT(t) = ZTH(t) P (2.11)


yields the transient thermal impedance






r 1 r (L/2 +x+ erfL/2-x)]
ZTH(x, z, t) = J8p- erf-2 + erf( -

/2+y y /2-y y
S[erf (W/ + erf /2-Y

[erf( z+D+H ) -D- z
erf + + erfy--4

+erfz- +erf(+H-z dt (2.12)



where the t value of the thermal impedance corresponds to the thermal

spreading resistance RTH.

Equation (2.12) represents the temperature rise at any point in the device

normalized to a unit-step increase in power dissipation. For circuit simulation, a

single temperature is needed to represent the effective operating temperature of the

device. Fox and Lee [Fox91b] showed that the thermal impedance model evaluated

at a surface corner of the emitter (x = L/2, y = W/2, z = 0) agreed well with

measurements of the thermal spreading resistance RTH; substituting these

coordinates into (2.12) gives the following expression for the thermal impedance


1 L (W (D+H D
ZTH e) = J4 rf er erf I -erf(D I Idt (2.13)
t ,4pcV 4t 4at \F4at 4atJ


In this form, the thermal impedance model has four geometric input

parameters. Of the four, three (W, L and D) are determined directly by the device

layout. However, the fourth parameter, H, depends on the operating bias of the

device. The thickness of the base/collector SCR, H, can be estimated using the







depletion approximation; assuming a one-sided step junction with uniform doping on

each side gives



2 Eg 0- (VR + bi)
H = -R (2.14)
q Nepi


with



kB T N
Ybi n i (2.15)




where Esi is the dielectric constant of silicon, Eo is the permittivity of free space, VR

is the reverse bias voltage on the base/collector junction, q is electronic charge, Nepi

is the doping level in the epi-collector, kB is Boltzman's constant, T is temperature,

Nb is the doping level in the base, and ni is the intrinsic carrier concentration in

silicon. fbi is the built-in potential of the base/collector junction. Equation (2.14)

shows that the thermal impedance depends on the bias of the base/collector junction,

and therefore can change during device operation. However, the square-root

dependence of H on the base/collector voltage, is relatively weak. Figure 2.3

illustrates the variation of the modeled thermal resistance with changes in the

thickness of the base/collector SCR. The three data points plotted for each simulated

device correspond to reverse bias base/collector voltages of 5, 10 and 20 V. The

largest variation is observed for the smallest device, which shows a 25% change in

its thermal resistance going from VR = 0 V to 20 V. The larger devices show a

weaker dependence on H and have no more than a 15% change in thermal resistance

















30






20






10





0-
150












Figure 2.3


200 250 300 350 400
Variation of H (%)










Simulations showing the effect of variations in the thickness of the
base/collector space-charge region on the thermal impedance model
(evaluated at steady-state) for different geometry BJTs. For each
device, D = 0.35 m, Nepi = 1 x 1016 cm-3 and N = 1.5 x 1018 cm3.
The y-axis corresponds to the variation between the model evaluated
at VR = 0 V and the model evaluated at VR equal to 5, 10 and 20 V.







going from VR = 0 V to 20 V. However, at high base/collector biases, the maximum

value of H becomes effectively independent of bias. As shown in Figure 2.1, typical

bipolar technologies use a heavily-doped buried layer to reduce collector resistance.

At high base/collector biases, the low-doped epi-collector region depletes down to

the buried layer; consequently, the maximum value of H should be properly limited

to the thickness of the epi-collector.



2.2.1 Modification for Finite Wafer Thickness


As previously derived, the thermal impedance model for single-emitter

bulk BJT/HBT's represents the substrate as a semi-infinite half-space. This

representation assumes that the back side of the substrate does not affect the thermal

response of the device. In general, this is reasonable since the base-collector junction

is usually within 1 pim of the substrate surface, and a typical wafer is between 350 to

800 pim thick. However, wafers are commonly back-lapped to improve thermal

performance, and substrate thicknesses of 75, 80 and 100 p.m have been reported by

a number of authors [Kag94, Mar93, Liu95a]. As the wafer thickness is reduced, the

substrate can no longer be approximated by a semi-infinite medium and the effects

of the back-side boundary must be taken into account.

The three-dimensional Green's function in the rectangular coordinate

system can be represented by the product of three one-dimensional Green's functions


G(x, y, z, tx', y', z', t') = Gx(x, t|x', t') Gy(y, t|y', t') Gz(z, t|z', t').


(2.16)







The lateral boundaries are still assumed to extend infinitely and their effects on the

thermal response are neglected; thus, the Green's function solutions in both the x and

y directions remain unchanged. In the z-direction, however, the substrate is now

assumed to have a finite thickness Dsub. The top surface of the substrate is still

assumed to be adiabatic. The bottom surface of the substrate is assumed to be at a

constant temperature, T(Dsub) = T0, so that the temperature rise at this surface is

defined by AT(Dsub) = T(Dsub) T = 0. These boundary conditions define the

new Green's function for the z-direction, which is given by



2 2
G,(z, tz', t') = exp[-ac ip(t-t')] Du cos(rpz) cos(7lpz) (2.17)
sub
p=l


where lp is the set of eigenvalues for the boundary-value problem and are given by

the positive roots of


cos(TipDsub) = 0. (2.18)



Equation (2.18) is solved when the argument of the cosine equates to odd multiples

of t/2. Using equations (2.17), (2.16), (2.11) and (2.7), and then integrating over the

base/collector SCR, assuming a unit-step increase in power at t = 0, gives the

following expression for the thermal impedance at any point in the device







Sdtr L/2+ x L/2-x
ZTH(x, y, t) = ] erf --- + erf
t 4pcV 4ot ) 4a t
F (W/2+y\ (W/2-yi
erf W/2 + erf(/2 y

2cos(rOpz)exp(-arpt)
= pf 71PDS~b
p=1 pDsub
{sin[qlp(D + H)]- sin[T[pD]} (2.19)


where V = W- L H and


(2p 1) (2.20)
4 = 2D (2.20)
2sub


Evaluating (2.19) at the coordinates (x = L/2, y = W/2, z = 0) to give a single

effective operating temperature, results in the following expression for the thermal

impedance


7 dt (4L W ( )
ZTH(t) = erf( L rf(W
TH M 4pcV F4 (-t 4att

2
S2exp(-a0rl t)
S p Dsub { sin [rp(D+ H)]- sin [lpD]} (2.21)
p= sub


Since (2.21) is derived from the physical heat conduction equation, it can

be used to anticipate the effects of substrate scaling on the thermal impedance.

Figure 2.4 illustrates equation (2.21) evaluated at various values for Dsub. The

results show that the thermal resistance decreases as the substrate thickness is

reduced, which agrees with the trend predicted by Hattori et al. using a three-

dimensional numerical simulator [Hat95]. Figure 2.4 also shows that (2.13) provides




















675


650


625


600


575


550


525


500


400


500


Simulations showing the effect of substrate thickness on the thermal
impedance model (evaluated at steady-state). The model accounting
for finite substrate thickness is compared to the model assuming
infinite substrate thickness. The device specifications are L = 4 9m,
W = 1 jim, D = 0.35 pm, H = 0.35 im.


200 300
Substrate Thickness, Dsub (Pm)


Figure 2.4


0-.-







I 0G OModel w/finite Dsub
Model w/infinite Dsub



-I I I







an accurate prediction of the thermal resistance over most of the range of substrate

thicknesses: only when the substrate thickness is significantly reduced (< 100 itm),

is there a large deviation between the two models.



2.2.2 Effects of Interconnect Metallization on the Thermal Impedance


For the derivation of the BJT/HBT thermal impedance model, the surface

of the substrate is assumed to be adiabatic. In actual devices, portions of the base,

collector and emitter regions are in direct contact with the metallization used to

electrically connect different devices on a chip. Since the metallization typically has

a high thermal conductivity, it is possible that the heat conduction via the

interconnects significantly influences the thermal response of a device. Therefore,

the validity of such an assumption should be investigated.

Three-dimensional (3-D) finite-element (FE) thermal simulations of a

bipolar transistor, using the ANSYS software package [Ans96], were performed to

examine the effects of the interconnect metallization on the thermal impedance. To

simplify the FE model, the device was considered to be symmetric in both lateral

directions; therefore, only one quarter of the device was simulated. The bottom and

exterior sides of the substrate were assumed to be at a fixed ambient temperature. The

top and side surfaces of the interconnects, as well as the top surface of the inter-layer

dielectric, were assumed to be adiabatic. The FE simulations tend to overpredict the

heat conduction through the interconnects since any contact resistances at the

material interfaces were neglected. The assumed symmetry of the device implies that

the base and collector metallization are equidistant from the emitter. In typical







devices, the collector contact is offset a greater distance from the emitter than the

base contact. Typical ranges for these offsets are 0.5 to 10 urn between the base and

emitter contacts, and 2.5 to 25 Lim between the collector and emitter contacts

[Gra93]. While the FE model does not exactly represent any actual device structure,

it can provide an estimate for the significance of the heat flow through the

interconnects as a function of their distance from the active device.

Figure 2.5a shows the FE model for bipolar devices with full

metallization. Steady-state thermal simulations were run for various interconnect

spacings; this spacing corresponds to the edge-to-edge distance between the emitter

and base/collector interconnects. Simulations were also run of the same structure

with the base/collector interconnect removed. The results of the two groups of

simulations were compared to determine the effect of the base and collector

interconnects on the thermal resistance. Figure 2.5b shows the results of the

comparison between the FE simulations. The data clearly shows that the effect of the

base/collector interconnect metallization is small and decreases as the interconnects

are moved away from the active device area. The collector interconnect has less of

an effect on the thermal impedance than the base interconnect, due to the larger

distance between the collector contact and the active device. In any case, the effect

of either the base or collector interconnect should be negligible compared to the

influence of the emitter metallization.

To determine the extent of the effect of the emitter interconnect on the

thermal impedance, steady-state thermal simulations were run for different devices

with only the emitter metallization in contact with the device. Figure 2.6a shows the




























2.5



2.0
r
c3

S1.5

E

- 1.0
r-
o
i 0.5



0.0


Interconnect Spacing (tm)


Figure 2.5


ANSYS simulations showing the effect of emitter, base and collector
interconnects on the thermal resistance. The device specifications are
D = 0.2 pim and H = 0.35 jIm, the interconnect width Wmet = 2 pmr
and thickness diet = 0.9 [tm, and the inter-layer dielectric thickness
dox = 0.7 gim: a) The finite-element model simulated with ANSYS; b)
the variation between the thermal resistance accounting for emitter,
base and collector interconnects and the thermal resistance accounting
for only the emitter interconnect, plotted as a function of the spacing
between emitter and base/collector interconnects.














































50 100


50 100
Variation of Parameter (%)

(b)


200


250


ANSYS simulations showing the effect of the emitter interconnect on
the thermal resistance for variations in different technology
characteristics. The specifications for the nominal device are
L = 4 pm, W = 1 pm, D = 0.35 pm, H = 0.35 pm, dmet = 0.9 Im,
dox = 0.9 pm and Wmet = 2 [m: a) The finite-element model
simulated with ANSYS; b) the variation between the thermal
resistance accounting for the emitter interconnect and the thermal
resistance neglecting the emitter interconnect. The variation plotted on
the x-axis corresponds to the deviation of each structure parameter
from its nominal value.


30


25
.)
U

S20


S15
.C
H
.E 10
c

5- 5

0
0


Figure 2.6


nominal
SOdmet
a dox
OD
AL
Wmet






o0 A
AA
a/


^- ^







FE model for devices with only the emitter metallization. The same devices were also

simulated with the emitter interconnect removed. The simulations were performed by

independently varying each structure parameter of the FE device model. Reasonable

values were chosen for each parameter to represent a nominal device design; each

parameter was varied about the nominal value to represent a reasonable range of

technology scaling. Figure 2.6b shows the results of the FE simulations. The emitter

metallization becomes a more effective path for heat evacuation as the thickness,

dmet, and the width, Wmet, of the interconnect increase and as the thickness, dox, of

the dielectric layer between the substrate and the interconnect decreases. The data

also show that the effect of the emitter interconnect increases as the depth of the base/

collector junction is decreased (the heat source is moved closer to the surface) and

as the length of the emitter is decreased.

Transient thermal simulations of the FE model in Figure 2.6a were used to

examine the effect of the emitter interconnect on the transient thermal response.

ANSYS was used to simulate the structure with and without the emitter interconnect

in contact with the device; the results are shown in Figure 2.7. The thermal responses

for the device with and without the interconnect match until significant heat reaches

the surface of the device. The time for heat to reach the surface of the device can be

approximated as the square of the distance D divided by the thermal diffusivity of

the substrate material. The resulting time is approximately 0.44 nanoseconds, which

agrees with the FE simulations. The adiabatic boundary condition of the device

without the interconnect predicts a larger response since the heat is completely

reflected once it reaches the surface. The device with the emitter metallization, which




















1.5





U
0

-
1.0






S0.5


[--



0.0
1C














Figure 2.7


10-10 10-8 10-6 10-4 10-2
Time (sec)












ANSYS simulations showing the effect of the emitter interconnect on
the transient thermal impedance. The specifications for the device
structure are L = 2 pm, W = 1 im, D = 0.2 jim, H = 0.35 gim,
dmet = 0.9 gm, dox = 0.7 gm and Wmet = 2 im.







acts as a separate heat sink path, has a reduced thermal impedance and an effectively

slower thermal response.

Based on the results of the 3-D FE simulations, neglecting the base and

collector interconnect metallization in the model derivation is reasonable since it

only slightly affects the thermal impedance of a device. The emitter interconnect,

however, has a greater influence on both the steady-state and transient thermal

responses. The effect on the thermal resistance will be more significant for devices

with small-geometry emitters and shallow base regions, where the transient thermal

response will mainly be affected for large devices with substantial contact structures.

In either case, equations (2.13) and (2.21) will tend to over-predict both the steady-

state thermal resistance and the transient rise of the thermal impedance.



2.2.3 A Model for the Thermal Impedance of the Emitter Interconnect


As shown in the previous section, the assumption that the top surface of

the device is adiabatic neglects heat flow in the emitter interconnect and results in a

thermal impedance model that over-estimates the transient temperature rise in a

bipolar device. To model the effects of the emitter interconnect on the overall

thermal impedance, both the thermal resistance and thermal capacitance of the

metallization need to be considered.

The thermal resistance of the emitter metallization is derived by assuming

that the interconnect can be represented by a one-dimensional cooling fin, so that the

temperature rise at any point xmet along the interconnect,

ATmet(xmet) = Tmet(xmet) To, can be approximated by







m2ATmet 2
2 -mmetATmet 0.
aXmet


(2.22)


The second term on the left-hand side of (2.22) accounts for heat conduction through

the underlying oxide as the heat travels along the interconnect, where


1 kmetdmet
mmet hmet


(2.23)


is the characteristic thermal length in the interconnect and


h = kox
met d
ox


(2.24)


is the heat transfer coefficient from the interconnect to the substrate. The material

properties for the emitter interconnect are given in Table 2.2.




Table 2.2 Emitter interconnect material properties


Source: [Ozi93]
* Assumed to be aluminum


Property Definition Value

kmet Thermal conductivity 2.39 (W cm-I K-)


Pmet Density 2.7 (g cm-3)

Cpmet Specific Heat 0.9 (J g' K-)







Approximating the temperature in the interconnect using a one-dimensional equation

implies that the temperature gradients in the vertical and lateral directions within the

emitter interconnect are negligible. The validity of such an assumption can be

evaluated using the Biot number, which corresponds to the ratio of the internal and

external thermal resistances of a given object [Ozi93]. If the Biot number for the

interconnect is much less than unity, then the interconnect can be approximated as a

one-dimensional thermal medium. The vertical and lateral Biot numbers for the

emitter interconnect are given by Bvmet = hmetdmet/kmet and

BLmet = hmetWmetetdmetmet), respectively. For most practical metallization

geometries, the Biot numbers are much less than one and the cooling-fin model is an

accurate representation of the emitter interconnect.

Assuming that the temperature rise in the interconnect at the emitter

contact is equal to the effective operating temperature of the device, and that the

temperature rise approaches zero far from the contact, the thermal resistance of the

emitter interconnect can be expressed as



RTHmet [kmetmmetmtet]et (2.25)


The thermal capacitance of the emitter metallization can be approximated as


CTHmet PmetCpmetVmet. (2.26)


The volume of the metallization is Vmet = W L met, where 5met represents the

effective length of the interconnect structure. The parameter 5met should be

evaluated to include the volume of the contact and interconnect metallization but can







also be extracted from transient thermal measurements or numerical simulations.

Once the thermal resistance and thermal capacitance have been calculated, the

transient thermal impedance of the emitter interconnect can be approximated by



ZTHmet(t) = RTHmet[l exp -t (2.27)
1 .Tmet '


where Tmet = RTHmetCTHmet. The overall thermal impedance of a bipolar device can

now be represented by the parallel combination of two thermal impedances, such that

effectively


ZTHdev(s) ZTHmet(S)
ZTH(S) = (2.28)
ZTHdev(s) + ZTHmet(S)'


where ZTHdev(s) is determined from the transient thermal impedance given by either

(2.13) or (2.21).



2.2.4 Effects of Isolation Structures on the Thermal Impedance


While junction-isolated technologies are still used, the drive to increase

packing density, improve lateral isolation and increase device operating speeds has

led to the development of newer isolation technologies for VLSI bipolar applications.

The advanced isolation technologies typically used in bulk bipolar fabrication are

recessed LOCOS (local oxidation of silicon) and U-groove [Wol90, Gra93];

Figure 2.8 illustrates examples of bipolar devices fabricated with these isolation

techniques. Since advanced isolation structures typically use low conductivity










Collector


p-substrate




(a)


Base Emitter Collector



Oxide p \ Oxide
INE 1 N


N-,


p-substrate


Cross-sections of typical BJT's fabricated with advanced isolation
technologies: a) Recessed LOCOS; b) U-groove isolation.


Figure 2.8


Emitter


Base







materials like SiO2, the thermal impedance of a device using such isolation tends to

be higher than that of its junction-isolated counterpart. The bulk BJT/HBT thermal

impedance model treats the substrate as a homogeneous material; therefore, it is

unclear whether the thermal impedance model is applicable to devices which are

fabricated with advanced isolation.

Three-dimensional (3-D) finite-element (FE) thermal simulations, using

ANSYS, were performed to examine the effects of advanced isolation structures on

the thermal impedance of bipolar transistors. Two FE models were developed to

separately investigate the effects of recessed LOCOS and U-groove isolation. To

simplify the FE models, the device was considered to be symmetric in both lateral

directions, so that only one quarter of the device was simulated. The bottom and

exterior sides of the substrate were assumed to be at a fixed ambient temperature. The

top surface of the device was assumed to be adiabatic. Due to the assumed symmetry,

the active device region was surrounded on all sides by the isolation structure, which

was at uniform distance from each side of the emitter. As shown by the illustrations

in Figure 2.8, the distance between the isolation structure and the intrinsic device is

not uniform on all sides of the device. Typical values for the distance between the

emitter and the isolation--for the portions of the isolation structure immediately

surrounding the emitter--are in the range from 0.3 ptm to 0.8 p.m for advanced bipolar

devices [Del91, Klo93, Yam93, Pru94]. The distance between the emitter and the

portion of the isolation structure on the far side of the collector contact is generally

larger, typically two to four microns [Del91, Klo93]. For both FE models, the side-

walls of the isolation structures were perpendicular to the top surface of the substrate.







For typical U-groove isolation, the trench is created by anisotropic etching and the

side-walls are nearly perpendicular to the surface. However, actual LOCOS

structures have tapered edges (see Figure 2.8) that get progressively thinner toward

the active device. To determine the implications of the model's non-physicality, two-

dimensional (2-D) FE simulations were run for various angles (30 to 90 degrees)

between the substrate surface and the side-wall of the isolation. The simulations

showed that the temperature rise increased as the angle increased; thus, the 3-D

LOCOS model should show a larger effect than that of an actual isolation structure.

While the finite-element models do not truly represent the physical device layout,

they allow an order-of-magnitude estimate for the effects of the isolation structures

on the thermal impedance.

Steady-state thermal simulations were run for various device-isolation

spacings, corresponding to the edge-to-edge distance between the emitter and the

isolation structure. For the U-groove isolation model, this spacing is the distance

between the emitter and the edge of the surface LOCOS; the actual trench is assumed

to be an additional 0.5 pm away from the edge of the LOCOS [Del91, Yam93].

Simulations were also run for the same devices with the isolation structures removed.

LOCOS isolation is formed by selectively oxidizing regions of the

semiconductor substrate in a dry or wet oxygen-rich ambient. For bipolar

technologies, the resulting Si02 structures are typically no more than one micron

thick, since the growth of thicker oxides is impractical [Wol90, Gra93]. Figure 2.9a

illustrates the FE model for bipolar devices with recessed LOCOS isolation. The

oxide was assumed to be fully recessed beneath the top surface of the substrate and

















































0 0.5 1.0 1.5 2.0 2.5 3.0 3
Device-Oxide Spacing (Lim)


ANSYS simulations showing the effect of recessed LOCOS isolation
on the thermal resistance. The device specifications are L = 2 inm,
W = 1 rim, D = 0.35 jim and H = 0.35 jim: a) The finite-element
model simulated with ANSYS; b) the variation between the thermal
resistance accounting for the isolation and the thermal resistance
assuming a homogeneous substrate, plotted as a function of the edge-
to-edge spacing between the emitter and the isolation.


-Odfox = 1.0 lOm


30





20
C;

E


.2 10



O
.0


0


Figure 2.9


I 1 I 1 1 1 I I I







had a thickness, dfox, of one micron. Figure 2.9b compares the FE simulations with

and without the isolation. The effect of the LOCOS can be significant at small

device-isolation spacings, but decreases as the isolation is moved away from the

active device region. In Figure 2.8a, the portions of the LOCOS structure close to the

emitter have the largest effect on the thermal response, since they directly restrict the

lateral heat flow away from the device. A number of manufacturers are using thinner

standard or semi-recessed LOCOS (0.3 to 0.6 gim) combined with junction isolation

to reduce fabrication times and improve compatibility with existing MOS

technologies [Klo93], [Pru94]. The thinner oxides have a smaller effect on the

thermal resistance, and therefore, the FE simulations can be considered worst-case.

U-groove isolation differs from LOCOS in that trenches are etched

directly into the substrate and then back-filled with oxide and polysilicon. The depth

of the trench, dtr, is typically on the order of 3 pim [Yam93, Ona95], but has been as

large as 5 ptm [Del91]; the width of the trench is generally in a range from 0.6 to

1.5 pim [Del91, Yam93, Ona95, Shi96]. U-groove trenches will typically have a

surface LOCOS layer, but the thickness of this layer is usually no greater than 0.1 to

0.15 jim [Yam93], since the isolation is mainly achieved by the trench. Figure 2.10a

shows the FE model for devices with U-groove isolation. The thickness of each fill

layer (dtrox for oxide and dpoly for polysilicon) in the U-groove was assumed to be

uniform. Figure 2.10b compares the FE simulations with and without the isolation.

The effect of the U-groove isolation on the thermal resistance is greater for small

device-isolation spacings than in LOCOS due to the larger depth of the trench.















































0.5 1.0 1.5 2.0 2.5
Device-Isolation Spacing


3.0
(pm)


3.5 4.0 4.5


Figure 2.10


ANSYS simulations showing the effect of U-groove isolation on the
thermal resistance. The device specifications are L = 2 tm,
W = 1 p-m, D = 0.35 pim and H = 0.35 p.m. The U-groove
specifications are dtr = 3.5 Jm, dtrox = 0.38 pm and dply = 0.75 p-m:
a) The finite-element model simulated with ANSYS; b) the variation
between the thermal resistance accounting for the isolation and the
thermal resistance assuming a homogeneous substrate, plotted as a
function of the edge-to-edge spacing between the emitter and the U-
groove isolation.


60


-50


, 40
CZ
E 30
.-

.5 20
0
' 10
c>


IIIIIIIII







However, these results represent the worst case, since in an actual device the active

region is not immediately flanked by the U-groove on all sides.

Transient thermal simulations of the FE models shown in Figure 2.9a and

Figure 2.10a were used to examine the effects of both LOCOS and U-groove

isolation on the transient thermal response. ANSYS was used to simulate the device

with and without the isolation structures; the results are shown in Figure 2.11. As the

heat travels laterally and reaches the edges of the isolation structure, the response

accounting for the isolation begins to deviate from the response without the isolation.

The time for the heat to reach the edges of the isolation structures can be

approximated as the square of the device-oxide separation (1 jtm) divided by the

thermal diffusivity of the substrate material; the resulting time is on the order of ten

nanoseconds, which agrees with the simulations of both the LOCOS and U-groove

isolation. The oxide used in the isolation structures restricts the lateral flow of heat

away from the device and in both cases results in a larger temperature rise.

Based on the results of the 3-D FE simulations, advanced isolation

structures such as recessed LOCOS and U-groove can considerably increase the

thermal impedance of bipolar devices. In most cases, the bulk bipolar model will tend

to under-predict both the steady-state and transient thermal response of devices

fabricated with oxide-based isolation structures. The error in the model will be the

greatest for advanced, highly-scaled devices fabricated with deep trench isolation.

























10-2


Time (sec)

(a)


10-2


Time (sec)


Figure 2.11


ANSYS simulations showing the effect of advanced isolation
structures on the transient thermal impedance. The device-isolation
spacing is one micron. The specifications for the device structure are
L = 2 rim, W = 1 rim, D = 0.35 ptm and H = 0.35 gim: a) Recessed
LOCOS or BOX isolation with dfox = 1.0 prm; b) U-groove isolation
with dtr = 3.5 pLm, dtrox = 0.38 lim and dpoly = 0.75 pm.







2.3 Verification of the Single-Emitter Thermal Impedance Model


To verify the thermal impedance model, three-dimensional (3-D) finite-

element (FE) simulations of a junction-isolated BJT were performed using ANSYS.

Interconnect metallization was neglected and the substrate was assumed to be

homogeneous silicon with the bulk properties given in Table 2.1. The FE simulations

were evaluated at the surface corner of the emitter and compared to the single-emitter

thermal impedance model given by equation (2.21); the results are shown in

Figure 2.12. The analytic model agrees closely, for both the steady-state and the

transient, with the 3-D FE simulations for both device geometries. The predicted

values for the steady-state thermal resistance agree within twelve percent of the FE

simulations. The error can be partially attributed to numerical error associated with

the FE mesh.

The model was also compared to measured thermal impedances. The

measured data were extracted using the base-thermometry technique developed by

Zweidinger et al. [Zwe96]. Figure 2.13, Figure 2.14 and Figure 2.15 compare the

measured and simulated data for the transient thermal impedance of Harris HBC bulk

BJTs. The thermal impedance model does a good job of predicting the steady-state

thermal resistance, with no more than a 20% error between the model and the

measurements. The model given by (2.13) tends to overpredict the transient

response. As shown with the ANSYS simulations in Figure 2.7, this discrepancy can

be attributed to the model's neglect of the emitter metallization. When the thermal

impedance of the emitter interconnect is accounted for, with 5met = 50 pim extracted







TEMPERATURE (C):


I I


27
27,431
27.863
28,294
28.725
29,157
29.588
30,019
30.451
30,882


Time (sec)


(b)

Figure 2.12 The transient thermal impedance simulated with ANSYS and
calculated with the bulk, single-emitter model: a) The finite-element
model simulated with ANSYS for P = 1.8 mW; b) comparison of
thermal impedances simulated with ANSYS (symbols) and the thermal
impedance model (lines).


1150

950

750

550

350

150

-50
1



























10-"
Time (sec)


~'50
U
0
E40

~30


- 20
E



0


10-2


10-2


10-u
Time (sec)


Figure 2.13


Measured and
Harris HBC
L = 100 tm.


simulated data for the transient thermal impedance of
bulk BJT's with W = 2 m: a) L = 30 m; b)







300


220



140



60


10-6
Time (sec)


-20


10-2


Time (sec)


Figure 2.14


Measured and simulated data for the transient thermal impedance of
Harris HBC bulk BJT's with W = 3 4im: a) L = 10 itm; b) L = 30 pim.








260

220

180

140

100

60

20


10-2


Time (sec)


10-2


Time (sec)


Figure 2.15


Measured and simulated data for the transient thermal impedance of
Harris HBC bulk BJT's with W = 5 ptm: a) L = 10 pnm; b) L = 30 pm.







from the measurements, the model provides a more accurate representation of the

transient thermal response.




2.4 Derivation of the Multiple-Emitter BJT/HBT Thermal Impedance Model


The thermal impedance model for bulk MEBJT/MEHBT's is an extension

of the single-emitter model. A multiple-emitter device consists of single-emitter

devices placed adjacent to each other along their lengths. Since there are multiple

devices (referred to as "emitter fingers") that are thermally coupled through the

substrate operating in close proximity, the temperature rise in each emitter finger is

affected not only by its own power dissipation, but also by the power dissipated by

its neighbors. The heat conduction equation, (2.1), is linear, so superposition can be

used to calculate the total temperature rise in the device. The equation for the

temperature rise can then be manipulated to provide expressions for the effective

temperature rise in each individual finger.

Figure 2.16 illustrates the simplified multiple-emitter device geometry

assumed for the model derivation. The substrate is represented by a homogeneous

semi-infinite half-space with an adiabatic top surface with multiple imbedded heat

sources. The emitter fingers are assumed to be uniform in size and shape, with width

W and length L. Each finger has a corresponding heat source, due to an assumed

uniform power generation in the base/collector SCR; each heat source has a thickness

H and is displaced a distance D below the surface of the substrate. As with the

single-emitter model, D is assumed to equal the depth of the base/collector junction




















W K-"1


Figure 2.16


The simplified device geometry used to define the solution domain for
the bulk, multiple-emitter BJT/HBT thermal impedance model. The
substrate is represented by a semi-infinite half-plane with an adiabatic
surface. Each emitter finger has a width W and length L and each heat
source (the rectangular volumes) is displaced a distance D below the
surface of the device and has a thickness H. The distance D is
equivalent to the depth of the base/collector junction and the thickness
H is approximated by the thickness of the base/collector SCR. The
emitter fingers are uniformly spaced with and edge-to-edge separation
distance S.







and H can be calculated using equations (2.14) and (2.15). The edge-to-edge

separation, S, between adjacent fingers is assumed to be uniform.

The Green's function technique can be employed to find the temperature

rise within the device. By applying superposition, the solution is expressed as the

sum of the Green's function solutions for the multiple heat sources


n t
AT(x, y, z, t) = J dt' G(x, y, z, tx', y', z', t')g(x', y', z, t')dv'. (2.29)
1 t =0 V


where G(x, y, z, t|x', y', z', t') is given by equation (2.8). The summation accounts for

the integration over the different spatial coordinates of each heat source. To clarify

the derivation, certain conventions and definitions can be established. The origin for

the coordinate system is fixed at the center of the left-most finger at the surface of

the device. A device is considered to have a total of n emitter fingers and a reference

order is established with the fingers numbered sequentially starting from the left-

most finger. The character j, where j = 1 -> n, is used to reference a specific emitter

finger. The i-th neighbor (where i = i -> n 1) of a given emitter finger, EFj, is

defined as a finger situated an edge-to-edge distance [iS + (i 1)W] away on either

side. Using equations (2.8) and (2.29), the temperature rise at any point in the device-

-assuming a step increase in power at t = 0 for each finger--is given by







n ^P v dt .(L/2 + x erf(L/2 x
AT(x, y, z,t) = 8V[e[ () erf J

S[erf( -(2j-3)W/2 -(j- 1)S
+ erf(-y+(2j-1)W/2+(j- 1)S']

/-(x-ti
[r z+D+H D-z
erf +D+H + erf(--z

(z-D-) D+H-zcl
+ erf +erf (2.30)



which accounts for n heat sources, one for each emitter finger EFj.

Equation (2.30) can be manipulated to provide the temperature rise in each

emitter finger. As with the single-emitter model, the temperature rise in each finger

is represented by a single effective value. To simplify the derivation, symmetry is

assumed such that the distance from the effective-temperature point of finger EFj to

the heat source of its i -th neighbor, is the same as the distance from the effective-

temperature point of the i-th neighbor to the heat source of EFj. This symmetry is

attained only for the coordinates (x, y = [j I ][S + L], z). When (2.30) is evaluated

at each of these points, the model is reciprocal and the effective temperature rise in

each emitter finger, ATEFj, can be expressed as








ATEFI ZS ZCI ... ZC(n- 1) P
ATEF2 ZC Zs ... ZC(n 2) P2




ATEFn ZC(n-l) ZC(n 2) S Pn (2.31)


where Zs is referred to as a self impedance and ZCi is the i -th coupling impedance.

The system of equations given by (2.31) shows that the temperature rise

in each finger is determined by the power dissipation in its own heat source and by

the power dissipated in the (n 1) neighboring heat sources. The self impedance is

given by



v- dt (L/2 + x. (L/2 x W/2
Zs(x, z, t) = erf + erf erf
I t 4pcV -at at J --t
j=It

z[erf(+D+H + -D-z
+ erf + erf -
/4-c4ct /4at -

erf + erf (D - (2.32)



and accounts for the portion of the temperature rise in a finger due to that finger's

own power dissipation. The i-th coupling impedance can be expressed as







Sf dt r fL/2 +x erfL/2-x
Zc(x, z, t) = -- [erf,-- +erf 4-
j=lt
cJ It 8pcV ( J4xt ) ( /4xt
rf (W/2 + i(W+S)) (W/2 i(W + S))+
e F4at /4- -ct
erf --++H + erf( ->z
( 14at ) \ /4Tt )
+ erf + e D+rH-z (2.33)
,4(t^ J4axt )]


and accounts for the portion of the temperature rise in a finger due to the power

dissipated by its i-th neighbor. Thus, for the assumed symmetry, the thermal

impedance of a device with n emitter fingers can be described with a single self

impedance and (n 1) coupling impedances.

Equations (2.32) and (2.33) should be evaluated at a single point to

represent the temperature rise of each finger by a single effective value. To keep the

MEBJT/MEHBT model as similar as possible to the single-emitter model, the points

x = L/2 and z = 0 are used, giving


r ( L ^ CW/2 > (D+ H> D 'Ni
(t = erf rf W2 erf erf dt (2.34)
Zs(t) = 2pcV 4t 4tA) 4(t y4 dt
t 24 pV At f4 (xat J4at /at

and


ZCi(t) = p erf i
Zc 4pcV t4t cv t
[erf (W/2 + i(W + S)) ef(W/2 i(W +S))
erf -4et (2.35)4t
erfD+H-erf( D -dt (2.35)
L 4at ) 14xt







as the final expressions for the self and coupling impedances. Therefore, the overall

self-heating on the scale of the entire device can be described by the self-heating and

thermal interactions on the smaller scale of each individual emitter finger.

Accounting for a finite wafer thickness can be important for multiple-

emitter devices since the coupling impedances decrease as the wafer thickness is

reduced [Daw94], [Hat95]. The multiple-emitter model can be modified in a similar

fashion as that for the single-emitter model; by simply using equations (2.17) and

(2.20) in place of the Gz(z, tlz', t') in equation (2.29), the expressions for Zs and Zci

will now account for a finite wafer thickness. Since the multiple-emitter thermal

impedance model is simply an extension of the single-emitter model, the effects of

interconnect metallization and advanced isolation technologies are not taken into

account. Neglecting these portions of the overall device structure is assumed to affect

the multiple-emitter model in the same manner, and to the same extent, as to which

it affects the single-emitter model.




2.5 Verification of the Multiple-Emitter Thermal Impedance Model


Two-dimensional (2-D) finite-element (FE) simulations of a junction-

isolated, three-finger BJT were performed using ANSYS to verify the multiple-

emitter thermal impedance model. Two-dimensional FE simulations were used

instead of 3-D simulations due to limitations of the available version of ANSYS. The

validity of comparing 2-D FE simulations to a 3-D analytic model is established by

evaluating the thermal impedance model derived for both two and three dimensions.







Figure 2.17a compares the results, which show that the 3-D the 2-D models converge

for long devices. The difference in the predicted thermal resistance values decreases

from 22% to less than 1% as the length of the device is increased from 50 pim to

800 ptm. Consequently, the 2-D FE simulations can verify the 3-D thermal

impedance model evaluated for devices with long emitters. The 2-D FE simulations

do not verify the model for shorter devices where the heat flow becomes three-

dimensional. However, the verification of the single-emitter thermal impedance

model for 3-D heat flow can be assumed to also verify the MEBJT model. This

assumption is reasonable since the physics that describe the single-emitter model

also apply to the MEBJT thermal impedance model.

The physical device was assumed to be symmetric so that the FE model

represented only half of the device. Figure 2.17b shows an illustration of the FE

model simulated with ANSYS. Interconnect metallization was neglected and the

substrate was assumed to be homogeneous silicon with the bulk properties given in

Table 2.1. The bottom and exterior side of the substrate were held at a constant

ambient temperature while the top surface and the interior side were assumed to be

adiabatic. The FE simulations were compared to the multiple-emitter thermal

impedance model given by equations (2.34) and (2.35), accounting for a finite

substrate thickness; the results are shown in Figure 2.18. The analytic model agrees

well, in both the steady-state and transient, with the 3-D FE simulations of the self

and coupling impedances. The predicted values for the steady-state thermal

resistance agree within three percent of the FE results, which is within the expected

error of the model and the numerical simulations.




74



250

S G-O2D Model: L = 50 pm
U 200 --03D Model: L = 50 pm
0. 3---- 2D Model: L = 200 gm O -
S E--- 3D Model: L = 200 m
A--A2D Model: L = 800 pm
150 -A3D Model: L = 800 gm


S100 -
E
I / ,^-- --B-- --B-
50 --

0 3 ,,, .---_---------------- -----
0f --Q--s-- -.^- '--A-""-----------
10-12 10-10 10-8 10-6 10-4 10-2 100
Time (sec)

(a)


TEMPERATURE (oC):
27
i 29,072
31.145
33,217
35.289
S1-I 37,361
39.434
41,506
43.578
45,651
(b)


Figure 2.17 Two-dimensional heat flow in multiple-emitter bipolar transistors: a)
A comparison of the three-dimensional thermal impedance model to a
two-dimensional model for W = 1 im, D = 0.5 pim and H = 0.5 grm;
b) the finite-element model simulated with ANSYS for P = 500 pW.








15 r ,
OZs: 2D ANSYS Al A A AA
[ OZcl: 2D ANSYS
UA
o AZs+Zc2: 2D ANSYS 0 0 0 00

10 -- Zs: Model
S --- ZCI: Model
a -- Zs+Zc2: Model


E 5



0 /

10-12 10-10 10-8 10-6 104 10-2
Time (sec)

(a)

8
_ _0_ _A A_ A4a
OZs: 2D ANSYS O O O O GD
oZci: 2D ANSYS
o 6 AZs+Zc2: 2D ANSYS
S Zs: Model
T - Zci: Model
S-- ZS+Zc2: Model
Q 4





r/

0 ii
1012 10- 0 10- 10-6 10-4 10-2
Time (sec)

(b)

Figure 2.18 The transient thermal impedance simulated with ANSYS and
calculated with the bulk, multiple-emitter model for L = 500 pm and
S = 1 (im: a) W= 1 (im, D =0.2 pLm, H =0.3 im; b) W = 3 pm,
D = 0.5 tim, H = 0.5 pm.







2.6 Summary


A thermal impedance model for bulk single-emitter BJT/HBT's was

presented and then extended for devices with multiple emitter fingers. The model

was shown to agree reasonably well with three-dimensional finite-element

simulations and measurements of junction-isolated devices. The effects of

interconnect metallization and advanced isolation technologies on the thermal

impedance were investigated; a simple model for the thermal impedance of the

emitter interconnect was demonstrated. The results suggest that the model can be

expected to provide reasonable predictions for the thermal impedance of junction-

isolated devices. However, for highly-scaled devices, the effects of advanced

isolation can be significant and the accuracy of the model will decline. Methods for

modeling the effects of isolation structures are proposed in Chapter Eight.













CHAPTER 3
A CIRCUIT MODEL FOR THERMAL COUPLING AND A LUMPED
ELECTROTHERMAL MODEL FOR BULK MULTIPLE-EMITTER BIPOLAR
TRANSISTORS



3.1 Introduction


Due to increased interest in the role of thermal effects in device and circuit

operation, especially for silicon-on-insulator (SOI) and heterojunction technologies,

circuit simulators and compact device models have been modified to account for the

dynamic temperature response within a device [McA92, Fox93b, Fos95]. Most of the

implementations have been applied to the case of self-heating, where a device's

effective operating temperature (EOT) depends on its power dissipation only. In

many circuits and some devices, such as multiple-emitter bipolar transistors, a

number of devices can operate in close proximity. Under such conditions, the EOT

of a device is no longer determined solely by its own power dissipation but also

depends on the operation of its neighbors. Therefore, not only must circuit simulators

(and compact device models) be able to model dynamic self-heating, they must also

be able to model the dynamic thermal coupling between individual devices or

portions of one device.

An approach for modeling cross-chip thermal coupling using a circuit

simulator was described by Fukahori and Gray [Fuk76]. The thermal coupling

between devices in an arbitrary circuit was modeled using a finite difference







technique. The semiconductor substrate was represented by a three-dimensional

numerical mesh with equivalent thermal resistances and capacitances. The electrical

elements in the circuit (transistors, etc.) were represented by their standard compact

circuit models. The values for the lumped thermal components were calculated by

discretizing the heat conduction equation using a finite difference approximation. In

[Mar93], a similar approach was presented and applied to the simulation of multiple-

emitter HBT's. In this case, two-port theory was used to generate a finite two-dimensional

resistance network that represented steady-state heat conduction in the substrate. For both

applications, the resulting circuit admittance matrix contained elements corresponding to

the electrical circuit and also the thermal elements. The Newton-Raphson-like iteration

scheme of the modified circuit simulators was then used to solve the coupled electrothermal

problem. To simulate both inter-device thermal coupling and self-heating using this

method, a large number of thermal nodes is required; therefore, this approach can

drastically increase simulation time.

Chapter One described a common method for using thermal impedances

to efficiently model self-heating in circuit simulators. A logical progression would

be to expand this method to account for thermal coupling between devices. Such an

approach can provide a more efficient alternative to the semi-numerical methods

mentioned above. This technique was applied by Moinian et al. for modeling cross-

substrate thermal coupling in bipolar circuits [Moi94], and by Baureis for modeling

multiple-emitter HBT's [Bau94]. However, their circuit implementations did not

correctly represent the thermal interactions between (or within) the devices. The

shortcoming of the coupling model used in these works is discussed in this chapter.







A circuit model is then presented which correctly models thermal coupling and is

compatible with the self-heating circuit model described in Chapter One.

Once a valid circuit model for thermal coupling has been developed, it can

be used with the multiple-emitter thermal impedance model to perform both steady-

state and dynamic electrothermal simulations of multiple-emitter BJT/HBT's. The

multiple-emitter thermal impedance model expresses the self-heating of an entire

device as the sum of the thermal actions and interactions of the individual emitter

fingers. The thermal model structure requires that a single multiple-emitter device be

represented by multiple compact device models. While this configuration allows for

the examination of the EOT of each finger in a device, for devices with a large

number of emitter fingers, the overall electrothermal network can become complex

enough to make moderate- to large-scale circuit simulations impractical. To make the

multiple-emitter electrothermal model more suitable for circuit simulation, its

complexity can be reduced by representing the overall thermal response of the device

by a lumped thermal impedance. The lumped thermal impedance is generated by

applying the measurement approach developed by Zweidinger et al. to the simulation

of the complete electrothermal model [Zwe96]. The thermal impedance extraction

technique is briefly reviewed in this chapter. The lumped model generation

procedure is then described and the results are compared to the complete

electrothermal model.







3.2 A Circuit Model for Thermal Coupling


The EOT of any device in a system of n thermally coupled devices (e.g. a

multiple-emitter bipolar transistor with n emitter fingers) can be expressed as


TDEV1(t) = [ATI(t) + AT2(t) + + ATin(t)] +Tamb
TDEV2(t) = [AT21(t) + AT2(t) + + AT2n(t)] +Tamb




TDEVn(t) = [ATnl(t) + ATn2(t) + + AT(t)] + Tamb (3.1)


where ATj(t) = 1 [Zsj P(s)] and ATji(t) = 1 [Zcji Pi(s)]. The impedance Zsj

is the self impedance of device j, and Zcji represents the coupling impedance

between device j and device i. (In general, it is not necessary for Zcji to equal Zcij .)

The modifications described in Chapter One allow circuit simulators to model self-

heating; therefore, the EOT of each device in a simulation is calculated by


TDEVj(t) = ATj(t) + Tamb (3.2)


and is independent of its neighbors. The obvious way to expand this technique to

account for inter-device thermal coupling would be to simply tie together the

temperature nodes of individual devices using coupling impedances; this approach

was used by Baureis and Moinian et al. [Bau94, Moi94]. Figure 3.1 shows an

example of such a thermal coupling network for two devices, where

ZC = ZC12 = ZC21. Unfortunately, simple analysis of the circuit in Figure 3.1

shows that it does not correctly model the expression in (3.1). For example, analyzing






















TDEV1(t)


? P2(t)






Tamb Tamb


















A thermal coupling circuit model for two devices. The temperature
nodes of the two thermally coupled devices are connected using a
thermal coupling impedance.


Figure 3.1


TDEV2(t)







the circuit in the steady-state limit gives the following expression for the EOT of

device 1



Rs I(R + RS2) RslRs2T
TDEV P,+P + Ta m. (3.3)
TDEV (Rc + Rsi + RS2) (Rc + RsI + RS) 2 amb



A similar expression can be derived for the EOT of device 2. The problem with this

network formulation is that when individual temperature nodes are connected

through an impedance path, the entire network becomes distributed among the

coupled devices. The self impedances and coupling impedances, as derived, are not

defined to be distributed elements. In a more simplistic view, the network in

Figure 3.1 does not properly constrain the paths of the respective device power-

currents. The power-current of a given device is divided between its own self

impedance and the rest of the network. The portion of that device's power flowing

through its neighbor's self impedance has no physical meaning. As a result, the

voltages generated at the temperature nodes do not correspond to the correct device

temperatures.

To develop a correct circuit representation of (3.1) and avoid the

shortcomings of the aforementioned coupling technique, control sources can be

utilized in a thermal coupling network group composed of two sub-networks. Each

device in a group of n thermally coupled devices has its own network group.

Figure 3.2 demonstrates how a thermal coupling network group works. Sub-network

A attaches directly to the temperature node of device 1. The current-controlled

current source (F1) in sub-network B has unity gain and is controlled by the current




UJ-)


TDEV(t)


E12(t)



E13(t)


sub-network A


Eln(t)


sub-network B


A new circuit representation of thermal coupling which is compatible
with the self-heating circuit model. Sub-network A is attached to the
temperature node of a device and represents the total temperature rise
in that device. Sub-network B is used to calculate the thermal coupling
between devices. The voltage-controlled voltage sources (E]j)
represent the individual portions of the temperature rise in device 1 due
to the other devices in the circuit. The current-controlled current
source (F,) models the power dissipation in device 1.


Fl(t).


Figure 3.2







flowing through the voltage source Vpl in sub-network A. In this example, Vpl is

also used to set the reference ambient temperature. The voltage drop across each

coupling impedance (Zcil) in sub-network B corresponds to the portion of the

temperature rise in each device i due to the power dissipation of device 1. The

voltage-controlled voltage sources in sub-network A each have unity gain and are

used to couple the voltage drops from each sub-network B of the other devices, back

to device 1. For example, the value of the voltage source E12 is equivalently

EI2-AT12(t), where ZC12 is part of sub-network B of device 2. Therefore, the

voltage generated at the terminal of sub-network A corresponds to the EOT of device

1, and is given by the following expression



TDEVI(t) = ATI(t) +ATI2(t) + + ATn(t) +Tamb. (3.4)


Similar expressions can be obtained for the EOT's of the other (n 1) devices in the

circuit since they each have similar thermal networks.

The thermal coupling model is demonstrated by simulating a five-finger

HBT using a version of SPICE 2G.6 modified to model self-heating [Zwe97]. The

device characteristics are simulated with and without the thermal coupling between

emitter fingers. Figure 3.3 shows the results of the electrothermal simulations. When

accounting for the thermal coupling, the current collapse phenomenon commonly

observed in HBT's can be simulated [Liu93, Sei93, Liu95b]. Figure 3.3b illustrates

how the outer fingers shut down as the middle finger begins to carry all of the current.

The importance of modeling the thermal coupling is established by the fact that the

collapse phenomenon is not reproduced when the simulations only account for self-








60


50

E
u 40


t 30
U

2 20

U


Collector-Emitter Voltage, VCE (V)

(a)


40



30
E
u

r
b 20

u

10
U


Figure 3.3


4 6
Collector-Emitter Voltage, VCE (V)


Simulated current characteristics of a five-finger HBT with
AE = 20 x 2 tm2 for each emitter finger: a) The collector current as a
function of collector-emitter voltage for fixed base currents of 1.0, 2.0
and 3.0 mA; b) the collector current distribution in the device with full
thermal coupling for Ig = 3.0 mA.


---- ---- -------------
-------------



----------------





Full Thermal Coupling
---. Self-Heating Only
l ,I ,l iIl


I I I







heating. Multiple-emitter devices provide just one application for the thermal

coupling model. It can be used on a larger scale for simulating thermal interactions

within circuits, and it can be used on a smaller scale. By dividing a single device (or

each finger of a device) into multiple sub-cells, the thermal coupling model could be

used to simulate the temperature distribution within a device and phenomena such as

current constriction [Koe94].




3.3 A Lumped Electrothermal Model for Multiple-Emitter BJT/HBT's


Used together with a compact device model of either a BJT or HBT, the

multiple-emitter thermal impedance model and the thermal coupling network form a

complete electrothermal model suitable for DC, AC and transient device/circuit

simulation. This type of electrothermal model is generally more efficient for circuit

simulation than either finite difference or finite element techniques; however, it can

be quite complex for devices with a large number of fingers and/or fingers with a

large number of sub-cells. In such a case, simulating moderate- to large-size circuits

could become impractical. The complexity of the electrothermal device model can be

reduced by using a lumped modeling methodology. The measurement technique

described by Zweidinger et al., referred to in this work as base-current thermometry,

can extract the thermal impedance of a bipolar transistor using the temperature

dependence of the base current [Zwe96]. By applying this extraction technique to the

simulation of the complete electrothermal device model, a more compact lumped

electrothermal can be produced. The lumped model implicitly contains all the details







of the thermal actions and interactions described by the complete electrothermal

model, but with less complexity.

To present a clear discussion of the lumped electrothermal model

generation methodology, a few definitions and conventions will be established. Due

to the thermal interactions between fingers in a multiple-emitter device a lateral

temperature gradient will exist across the device. Therefore, the current distribution

among the fingers may not be uniform since the hotter fingers will carry a larger

amount of current. As power dissipation increases, the lateral temperature gradient

also increases and eventually the device will become unstable and enter either

thermal runaway (BJT's) or current collapse (HBT's). Prior to the onset of thermal

instability, the lateral thermal gradient is small and the current distribution among

the fingers is approximately uniform. When the device reaches the point of thermal

instability, the fingers no longer operate under similar bias conditions and the current

no longer divides evenly among the fingers. Therefore, before a device becomes

thermally unstable, it is defined to be in the uniform operating regime; and, once the

device becomes unstable, it is defined to be in the nonuniform operating regime.

In the uniform operating regime, the EOT of the device varies linearly

with the power and the complete electrothermal model can be represented by a single

lumped device model and lumped thermal impedance, ZTHL. The circuit

representation for the uniform lumped model is shown in Figure 3.4a; the emitter

area of the lumped device model is equal to the total emitter area of the multiple-

emitter device. As a device becomes thermally unstable the cooler fingers begin to

turn off, leaving the hottest finger to conduct all of the current; the temperature-














Tamb


Tamb


Circuit representations of the lumped multiple-emitter BJT/HBT
electrothermal model: a) For the uniform operating regime; b) for the
nonuniform operating regime.


Figure 3.4







power relation becomes nonlinear and the uniform lumped model will not accurately

model the device characteristics. To model this shut-down mechanism, the

nonuniform lumped model, which is shown in Figure 3.4b, uses two lumped device

models and four lumped thermal impedances to represent the entire device. Device

QIF is used to represent the hottest emitter finger. In a device with an odd number of

fingers, the hottest finger will be the middle finger. If a device has an even number

of emitter fingers, due to process variation, the hottest finger will be one of the inner

most fingers. For consistency, in either case the hottest finger will be referred to as

the middle finger. The other device model, QoF, represents the remaining outer

emitter fingers. The emitter area of QIF is equal to that of a single emitter finger and

the emitter area of QOF is equivalent to the sum of the emitter areas of the outer

fingers. The lumped thermal impedances ZSg and Zso model the self impedances of

the middle finger and the outer fingers, respectively. In the case of Zso, the

impedance represents the effective temperature rise in the lumped outer fingers due

only to their power dissipation. The lumped coupling impedance Zclo models the

temperature rise in the middle finger due to the power dissipation in the lumped outer

fingers. The reciprocal coupling impedance Zco1, corresponds to the effective

temperature rise in the lumped outer fingers due to the power generation of the

middle finger.

Thermally triggered instability in bipolar devices can lead to circuit

failure and even catastrophic device failure. Typically, this region of operation is

avoided in circuit design. Therefore, in most cases, the uniform lumped model should

be appropriate for most applications. However, if the effects of thermal instability on







device/circuit operation need to be investigated the nonuniform model should be

used.



3.3.1 A Review of Base-Current Thermometry


Base-current thermometry uses the base current as a thermometer to

extract the thermal impedance of a bipolar transistor [Zwe96]. The technique was

developed for measurement-based extraction but can be applied to the simulation of

compact device models as long as the models' temperature dependence are

physically valid.

The first step of the procedure is to determine the dependence of the base

current on temperature. The response of the base current to changes in temperature

is represented by the fractional temperature coefficient, defined as



TCF(I) I (3.5)



By biasing a device in the common-emitter configuration (avoiding impact-

ionization), and separately varying the collector voltage and the ambient

temperature, the thermal resistance of the device can be extracted Since the base-

collector conductance is typically negligible, any changes in the base current during

the measurements are due solely to the change in operating temperature. Therefore,

once the self-heating effects are accounted for, the fractional temperature coefficient

can be determined from the measured base current variations.







The second step of the procedure is to extract the transient thermal

impedance. The collector and base currents of the device are monitored for a step in

the collector voltage. The transient change in temperature can be expressed as


IB(t) IB(0)
AT(t) = (3.6)
IBTCF(IB)


where Ig is the median value of the base current for the transient. The temperature

change is then normalized by the magnitude of the power step, giving the following

equation for the thermal impedance


ZTH(t) AT(t) (3.7)
AP


3.3.2 Generation of the Lumped Electrothermal Model


The first step in the lumped model generation is to extract the temperature

coefficient of the base current by performing DC SPICE simulations at different

ambient temperatures. The .TEMP control card is used to set the ambient

temperature; temperature steps between 4 and 10 degrees are sufficient, where a

geometric mean can be used to average TCF(IB) over temperature to correct for

nonlinearities. The base voltage should be selected for the desired operating point

and for each temperature setting, the collector voltage should be swept over a range

in the forward active region. The range of collector voltages should be large enough

to produce a linear increase in base current. Examples of the resulting base current

characteristics are shown in Figure 3.5. The base and collector current values should




Full Text

PAGE 1

PHYSICS-BASED THERMAL IMPEDANCE MODELS FOR THE SIMULATION OF SELF-HEATING IN SEMICONDUCTOR DEVICES AND CIRCUITS By JONATHAN SCOTT BRODSKY A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1997

PAGE 2

This work is dedicated to my parents, Lawrence and Jeraldine, my brother Matthew and sister Alexandra.

PAGE 3

ACKNOWLEDGEMENTS First, I wish to express my deepest gratitude to my advisor Dr. Robert M. Fox. His constant support and patient guidance provided a clear path for my research. It is both a pleasure and a privilege to have worked with Dr. Fox. I thank Drs. Mark E. Law and Jerry G. Fossum for extending their expertise and help to my unending questions. I would also like to thank Drs. William R. Eisenstadt, John G. Harris and Chen-Chi Hsu for their willingness to serve on my supervisory committee. I am also very grateful to Mary Turner for all of her help throughout my graduate career. I would like to acknowledge and thank the Semiconductor Research Corporation (SRC) for the financial support that made this research possible. I am also grateful to Dr. Surya Veeraraghavan for his guidance and friendship during my internship at Motorola. I would like to thank the "TCAD elders", and now my friends, Keith Green, Dongwook Suh, Ping-Chin Yeh, Haeseok Cho, Chih-Chuan Lin, Ming-Chang Liang and Scott Miller, for helping me get comfortable in my new surroundings and setting the standard of excellence. I am also grateful to my good friends/workmates/"happy hour buddies" Srinath Krishnan, Samir Chaudhry, David Zweidinger, Omer Dokumaci, Ming-Yeh Chuang, Dukhyun Chang, Susan Earles, Hernan Rueda, Glenn Workman and Meng111

PAGE 4

Hsueh Chiang, for all of the enlightening discussions, the Friday lunch tradition and the weekend adventures. There is a very special group of individuals who have my admiration and love for the friendships they have given me. I would like to thank my best friends Douglas Weiser, Martin Weiss, Stephen Cea, Edward Cometz and Peter Lynch. I can not completely express the role my family has played in my life and in the completion of this dissertation. For simple words seem to diminish their unconditional and unending love and support. I owe everything I have, everything I have done and everything I am, to my family. I give my deepest love to my parents, Jeraldine and Lawrence Brodsky, my brother Matthew and my sister Alexandra. Finally, I am grateful to all of the wonderful friends that I met in Gainesville for making this period of my life truly enjoyable. IV

PAGE 5

TABLE OF CONTENTS ACKNOWLEDGEMENTS iii ABSTRACT viii 1 INTRODUCTION 1 1 . 1 Self-Heating Effects in Semiconductor Devices 1 1.1.1 Bipolar Transistors 3 1.1.2 Field-Effect Transistors 6 1.2 Self-Heating Effects in Semiconductor Circuits 8 1.2.1 Small-Signal Circuit Performance 8 1 .2.2 Large-Signal Circuit Performance 11 1 .3 Self-Heating Effects in Parameter Extraction 12 1.4 The Simulation of Self-Heating Effects 13 1.5 Thermal Equivalent Circuits 16 1.6 The Need for Physics-Based Thermal Impedance Models 20 1 .7 Organization 25 2 A THREE-DIMENSIONAL THERMAL IMPEDANCE MODEL FOR JUNCTION-ISOLATED BIPOLAR TRANSISTORS 27 2. 1 Introduction 27 2.2 Derivation of the Single-Emitter BJT/HBT Thermal Impedance Model. . . 28 2.2. 1 Modification for Finite Wafer Thickness 39 2.2.2 Effects of Interconnect Metallization on the Thermal Impedance ... 43 2.2.3 A Model for the Thermal Impedance of the Emitter Interconnect ... 49 2.2.4 Effects of Isolation Structures on the Thermal Impedance 52 2.3 Verification of the Single-Emitter Thermal Impedance Model 61 2.4 Derivation of the Multiple-Emitter BJT/HBT Thermal Impedance Model . 66 2.5 Verification of the Multiple-Emitter Thermal Impedance Model 72 2.6 Summary 76 3 A CIRCUIT MODEL FOR THERMAL COUPLING AND A LUMPED ELECTROTHERMAL MODEL FOR BULK MULTIPLE-EMITTER BIPOLAR TRANSISTORS 77 3.1 Introduction 77 3.2 A Circuit Model for Thermal Coupling 80

PAGE 6

3.3 A Lumped Electrothermal Model for Multiple-Emitter BJT/HBT's 86 3.3.1 A Review of Base-Current Thermometry 90 3.3.2 Generation of the Lumped Electrothermal Model 91 3.4 Verification of the Lumped Electrothermal Model 96 3.5 Summary 101 A THREE-DIMENSIONAL THERMAL IMPEDANCE MODEL FOR VERTICAL BIPOLAR TRANSISTORS FABRICATED WITH FULL DIELECTRIC ISOLATION 103 4. 1 Introduction 103 4.2 Derivation of the DIBJT Thermal Impedance Model 106 4.2.1 Derivation of the Buried-Oxide Heat-Transfer Coefficient 1 16 4.2.2 Derivation of the Trench Heat-Transfer Coefficient 120 4.2.3 Effects of Interconnect Metallization on the Thermal Impedance . . 126 4.2.4 A Model for the Thermal Impedance of the Emitter Interconnect . . 129 4.3 Verification of the DIBJT Thermal Impedance Model 132 4.4 Derivation of a Compact DIBJT Thermal Resistance Model 134 4.5 Verification of the DIBJT Thermal Resistance Model 146 4.6 Summary 149 A THREE-DIMENSIONAL THERMAL IMPEDANCE MODEL FOR BULK METAL-OXIDE-SEMICONDUCTOR FIELD-EFFECT TRANSISTORS ..150 5. 1 Introduction 1 50 5.2 Derivation of the Bulk MOSFET Thermal Impedance Model 153 5.2.1 The Linear Source Thermal Impedance 161 5.2.2 The Saturated Source Thermal Impedance 164 5.2.3 Effects of the Device Interconnects on the Thermal Impedance ... 165 5.2.4 Effects of Isolation Structures on the Thermal Impedance 171 5.3 Verification of the Bulk MOSFET Thermal Impedance Model 176 5.4 Summary 182 A QUASI-THREE-DIMENSIONAL THERMAL IMPEDANCE MODEL FOR SILICON-ON-INSULATOR METAL-OXIDE-SEMICONDUCTOR FIELDEFFECT TRANSISTORS 184 6. 1 Introduction 1 84 6.2 Derivation of the SOI MOSFET Thermal Resistance Model 187 6.3 Verification of the SOI MOSFET Thermal Resistance Model 202 6.4 Derivation of the SOI MOSFET Thermal Impedance Model 204 6.5 Verification of the SOI MOSFET Thermal Impedance Model 212 6.6 Summary 216 THE THERMAL IMPEDANCE PRE-PROCESSOR: TIPP 219 VI

PAGE 7

7. 1 Introduction 219 7.2 A Description of TIPP 220 7.3 Generation of Thermal Equivalent Circuits 225 7.3.1 Approximation of the Thermal Equivalent Poles/Time Constants. . 226 7.3.2 Calculation of the Thermal Equivalent Components 230 7.4 The Interface Between TIPP and Circuit Simulators 232 7.5 Summary 234 8 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK ... 238 8. 1 Conclusions 238 8.2 Recommendations for Future Work 240 8.2.1 The Temperature Dependence of the Thermal Conductivity 240 8.2.2 Models for Thermal Effects Due to Advanced Isolation 242 8.2.3 A Model for Thermal Coupling in SOI MOSFET Circuits 243 REFERENCES 247 BIOGRAPHICAL SKETCH 257 vn

PAGE 8

Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PHYSICS-BASED THERMAL IMPEDANCE MODELS FOR THE SIMULATION OF SELF-HEATING IN SEMICONDUCTOR DEVICES AND CIRCUITS By Jonathan Scott Brodsky August 1997 Chairman: Robert M. Fox Major Department: Electrical and Computer Engineering Inherent in the operation of semiconductor devices is self-heating, an increase in operating temperature due to a device's own power dissipation. The magnitude of the self-heating effect can be quantified by the value of the thermal impedance, which describes the dynamic response of the device temperature to variations in device power. The thermal impedance is determined primarily by material properties and device structure. The implication of the self-heating effect is that the change in temperature can alter the operating characteristics of a device, which in turn, can affect circuit performance. The primary focus of this dissertation is the development of physics-based models for the thermal impedances of semiconductor devices. Models for the thermal impedances of bipolar and field-effect transistors, on both bulk and silicon-onVlll

PAGE 9

insulator (SOI) substrates, are presented. All of the thermal impedance models were derived from the time-dependent heat conduction equation, resulting in compact analytic expressions for the thermal impedances. The physical nature of the thermal impedance models allows them to scale with the device structure and material properties, and they successfully reproduce results from both measurements and three-dimensional finite-element simulations. A circuit model for thermal coupling between transistors in a common substrate is also presented. The coupling model was used in conjunction with the bulk bipolar thermal impedance model to extract a lumped electrothermal model for multiple-emitter bipolar transistors. The secondary objective of this work is the provision of an approach for incorporating these models into circuit simulators. It has been shown that the thermal impedance models can be represented by thermal equivalent circuits made up of resistors and capacitors, making them suitable for efficient circuit simulation. The computer program TIPP (Thermal Impedance Pre-Processor) is introduced. TIPP was developed to provide circuit simulators with convenient algorithms for generating thermal equivalent circuits. TIPP can calculate the component values for thermal equivalent circuits from either physical models or measured data, and is easily modified to interface with different circuit simulators. IX

PAGE 10

CHAPTER 1 INTRODUCTION 1.1 Self-Heating Effects in Semiconductor Devices The physical properties of the materials used to fabricate semiconductor transistors depend on temperature. Therefore, the operating characteristics of a transistor (e.g. electrical currents and potentials), which are determined by the material properties, are also temperature dependent. The temperature at which a transistor operates is determined by the temperature of the surrounding environment (referred to as the "local ambient temperature") and the power dissipated in the device (referred to as the "self-heating effect"). Therefore, the time-dependent temperature of a transistor can be expressed as t T(0 = T amb + JP(t')h TH (t-t')dt\ (1.1) o where h TH is the thermal impulse response and P is the instantaneous power. The second term on the right-hand side of (1.1) represents the temperature rise in the device AT(t) = P®h TH , (1.2) where ® is the convolution operator. The temperature rise can also be expressed in

PAGE 11

the frequency domain as AT(t) =r 1 [Z TH (s)-P(s)], (1.3) where £" ] represents the inverse Laplace transform and Z TH (s) is the thermal impedance. The thermal impedance of a transistor describes the dynamic response of the device temperature to variations in device power, and is determined primarily by the material properties and the structure of the device. The transient thermal impedance can be defined as Z TH (t) =£"' th' S ' (1.4) which represents the normalized thermal step response. Since the power dissipation in (1.1) is determined by the operating characteristics of a transistor, it depends on temperature such that P = P(T) = I dev (T)-V dev (T), (1.5) where I dev (T) and V dev (T) represent general currents and potentials within a given device, respectively. Consequently, there is feedback between the thermal and electrical operation of the device. Whereas the transistor temperature is usually assumed to be constant, the electrothermal coupling implied by ( 1 . 1 ) and ( 1 .5) shows that the temperature actually varies with the device operation. Thus, to fully characterize the operation of semiconductor transistors, both the electrical and thermal behavior should be determined.

PAGE 12

1.1.1 Bipolar Transistors In the forward-active mode, the operating characteristics of bipolar junction and heterojunction transistors (BJT's and HBT's) are controlled by the injection and diffusion of minority carriers in the base region. For an npn transistor, electrons are injected across the forward-biased base/emitter junction, causing an exponential increase in the minority carriers in the base. The electrons diffuse across the base and are swept into the collector by the reversed-biased base/collector junction. For a fixed base/emitter voltage, assuming negligible recombination in the quasi-neutral base, the collector current can be expressed as I c (T)-nf(T)-ex P Ppj^], (1.6) where n, is the intrinsic carrier concentration, q is the electron charge, k is Boltzman's constant and T is temperature. The overall temperature dependence of (1.6) is dominated by the relation between the intrinsic carrier concentration and temperature, given by n?(T) = N c -N v -exp^J, (1.7) where E is the semiconductor band-gap energy and N c and N v are the effective density of states in the conduction and valence bands, respectively. The junction voltage is always less than the band-gap and therefore, an increase in temperature causes an exponential increase of minority carriers in the base, resulting in an increase in collector current. Since the collector current is a significant component

PAGE 13

of the power dissipation in a BJT, self-heating results in a regenerative feedback between the collector current and the temperature of the device. This positive feedback can lead to the destructive phenomenon of thermal runaway in BJT's [Shu90]. For fixed base current, the collector current can be expressed as I C (T) = P(T)I B , (1.8) where (3(T) is the common-emitter current gain. For moderate injection levels, the current gain can be approximated by the ratio of the electrons injected into the base to the holes back-injected from the base to the emitter. This ratio, and hence P(T), are typically high since the emitter usually has a higher doping level than the base. Due to heavy-doping effects in the emitter, the emitter band-gap is typically less than that in the base so that p(T) ~N7B exp M' (l9) where AE g is the band-gap difference between the emitter and base, and N DE and N AB are the doping concentrations in the emitter and base, respectively. As shown by (1.9), the current gain is greater at higher temperatures; consequently, the collector current is, again, an increasing function of temperature. The rate of increase with temperature in this case, however, is not as significant as that for a device biased with a fixed base voltage. Therefore, the self-heating effect is not as substantial in BJT's driven by a fixed base current.

PAGE 14

HBT's are bipolar devices that use band-gap engineering in either the emitter or base region to improve the current gain over homojunction BJT's. The resulting band-gap in the emitter is wider than that in the base, so that the potential barrier induced by the band-gap discontinuity effectively impedes the injection of carriers from the base to the emitter. When biased with a fixed base voltage, the temperature dependence of an HBT is similar to that of a standard BJT. However, when an HBT is driven with a fixed base current, the temperature dependence of the collector current is quite different than that of a BJT. While the collector current in this case can still be determined from ( 1 .8), the common-emitter current gain is now expressed as p p Ut 8 I< iio > As a result of the band-gap being wider in the emitter than in the base, the sign of the exponential argument is now positive. Therefore, as opposed to a standard BJT, the current gain and the collector current decrease with increasing temperature. As a result, self-heating in HBT's can lead to the non-catastrophic failure mechanism known as current collapse [Sei93]. To reduce the effects of parasitic resistances and current-crowding, large bipolar devices are commonly fabricated using multiple devices connected in parallel [Shu90]. Multiple-emitter devices, both BJT's and HBT's, are capable of operating at high frequencies under high power densities [Win67, Mar93, Liu95b]. However, multiple-emitter devices suffer from more complex self-heating effects due to the thermal interactions among neighboring devices. The thermal coupling leads to

PAGE 15

lateral temperature gradients across the device, resulting in the inner emitters operating at higher temperatures. Due to the positive feedback between junction temperature and junction current, the inner devices carry more current than those at the outer extremes. As the current density in the inner emitters increases, the selfheating effect in these devices accelerates. The premature activation of thermal runaway in BJT's and current collapse in HBT's is attributed to this thermal instability inherent in multiple-emitter devices [Win67, Liu93, Kag94, Lio94, Lio96]. 1.1.2 Field-Effect Transistors For Metal-Oxide-Semiconductor Field-Effect Transistors (MOSFET's) operating in strong inversion, the current characteristics are determined by the drift current of carriers in the inverted channel region. For small drain voltages, a MOSFET operates in the linear region where the carrier velocity depends on the longitudinal electric field in the channel. In the linear region, the drain current can be approximated as I D (T)^(T)-(v GS -V t (T)-^]v DS , (1.11) where |i(T) is the carrier mobility, V t (T) is the threshold voltage, and V GS and V DS are the applied voltages between the gate and source and drain and source, respectively. At higher drain voltages, the electric field at the drain end of the channel is large enough to cause the carrier velocity to saturate. In the saturation

PAGE 16

region, the drain current can be expressed as I D (T)ocQ c (V GS ,V DS ,^(T))-v sat (T), (1.12) where Q c is the channel charge and v sat (T) is the saturated carrier velocity. The overall temperature dependences of (1.1 1) and (1.12) are dominated by the sensitivity of the carrier mobility to changes in temperature. Due to increased lattice scattering at higher temperatures, mobility decreases as temperature increases. The reduction in mobility leads to a decrease in drain current, which implies that the drain current of a MOSFET is a decreasing function of temperature. At high power dissipation levels, the self-heating effect can cause the drain current to drop below the ambient temperature value. In such cases, the output conductance becomes negative, and the device exhibits a negative dynamic resistance (NDR) [Sha83]. MOSFET's fabricated on silicon-on-insulator (SOI) substrates have temperature dependences that are similar to those of their bulk counterparts, though the effects of self-heating can be enhanced due to the low thermal conductivity of the insulating layers. For non-fully depleted (NFD) SOI MOSFET's, however, floatingbody effects further complicate the thermal effects [Wor97]. Impact-ionizationinduced floating-body effects are known to cause the kink, or increase in drain current, in NFD SOI MOSFET's. The kink is affected by self-heating in two ways. First, at elevated temperatures, the onset of the impact-ionization is retarded. Second, an increase in recombination in the quasi-neutral body reduces the threshold-voltage shift caused by the impact-ionization. Therefore, in addition to a reduction in drain current due to mobility effects, self-heating also reduces the

PAGE 17

current in NFD SOI MOSFET's through temperature-dependent floating-body effects. 1.2 Self-Heating Effects in Semiconductor Circuits Since the operating characteristics of transistors are affected by temperature, the integrated circuits that depend on these transistors will also be affected by changes in temperature. In modern digital circuits, the high switching speeds of the transistors, the relatively slow time constants associated with the temperature response and the low static power dissipation, all help reduce the instantaneous temperature rise. Consequently, self-heating effects are typically negligible in digital circuits. On the other hand, analog circuit applications commonly have significant power dissipation and can operate at frequencies which are comparable to the thermal time-constants. Therefore, analog circuits are generally more prone to self-heating effects. 1.2.1 Small-Signal Circuit Performance The effect of self-heating on small-signal BJT characteristics was derived by Mueller and investigated in bipolar circuits by Fox et al. [Mue64, Fox93b]. The two-port small-signal admittance parameters, in the presence of self-heating, were shown to be ymnE + D m Z TH I m I n ,, ,ox ^ran — i pv v p ' * ' 1 ~ u m^lH r

PAGE 18

where y mnE are the admittance parameters neglecting self-heating, and D m represents the variation of the current I m with temperature. The denominator of (1.13) establishes the sensitivity of the admittance parameters to power, and has a significant impact as D m Z TH P approaches unity. The effect of the denominator generally becomes important only at high power dissipation. However, as the thermal impedance increases (i.e. due to device scaling), the power level that defines the threshold for self-heating effects will decrease. The second term in the numerator of (1.13) shows that the effect of self-heating on the admittance parameters is also proportional to the operating currents. For y n and y 21 the self-heating term in the numerator is small so that y,, =Yhe and y 21 =y2iEHowever, the effect of selfheating can be substantial in the numerators of y p and y 22 , even at moderate current levels. The thermal effects on these parameters can result in a coupling between the collector output admittance and the impedance of the base-driving source. Also, as shown in Figure 1.1, there can be a significant reduction in the voltage gain of BJT amplifiers. The small-signal performance of analog MOSFET circuits can also be affected by self-heating. For moderate power levels, the thermal effects are similar to those in bipolar circuits. However, as mentioned previously, the drain current of a MOSFET decreases with increasing temperature, and significant self-heating can induce NDR. The effect of a negative output conductance can be investigated by examining the voltage gain of a MOSFET amplifier. As shown by Fox and Brodsky [Fox93a], if the devices in the amplifier enter a region of negative output conductance, the gain of the amplifier changes polarity. For an inverting amplifier,

PAGE 19

10 300 > g a c
PAGE 20

11 self-heating effects can therefore cause the gain to become non-inverting, resulting in hysteresis in the amplifier's output characteristics. 1.2.2 Large-Signal Circuit Performance The effects of self-heating on the large-signal operation of analog bipolar circuits was investigated by Fox et al. [Fox93b]. The types of circuits that are sensitive to thermal effects are typically those that depend on the precise control of BJT characteristics. For example, the mismatch in the reference and output currents of a current mirror can be increased due to self-heating-induced differences in the operating conditions of the transistors. Translinear circuits and band-gap voltage references can also be affected by self-heating due to their strong dependence on the thermal voltage. Thus, neglecting self-heating can result in significant discrepancies between the ideal and actual operation of these types of circuits. The large-signal transient operation of analog circuits is also affected by self-heating. The long time constants of the thermal characteristics can effectively slow down the electrical response of a circuit. Fox et al. showed that the five-percent settling time of a Gilbert multiplier increased by over an order of magnitude due to self-heating [Fox93b]. While the errors caused by self-heating can be reduced by careful circuit design, they can not be completely eliminated.

PAGE 21

12 1.3 Self-Heating Effects in Parameter Extraction The extractions necessary to determine the behavioral characteristics of a semiconductor transistor are often performed at bias levels that cause moderateto high-power dissipation. Typically, the parameters that are extracted are assumed to correspond to the ambient temperature at which the measurements are carried out. However, at significant power levels, self-heating will cause a temperature rise in the device. Neglecting the temperature rise that can occur during the measurements can lead to erroneous results [Zwe97]. For example, the Early voltage, V A , of a BJT is commonly extracted from the slope of the I C -V CE characteristics in the linear region of operation. If self-heating is significant, the slope of the output curves depends on the source that is driving the base [Fox93b]. Therefore, the exact meaning of the value extracted for V A would be ambiguous unless the thermal effects were taken into account. Various methods have been proposed for removing the effects of selfheating from parameter extraction. One approach augments a standard extraction routine with measurements designed to determine the thermal characteristics of the device. The full set of parameters can then be input to a global optimization routine to generate electrical-only parameters that are independent of self-heating [Zwe97]. Other techniques attempt to directly remove the effects of self-heating from the parameter extraction measurements by making the temperature rise in the given device negligible. The temperature rise can be minimized by performing the extractions in low-power regions, or by using complex high-speed measurements [Tu94, Jen95]. Since the device is not allowed to heat, the resulting device parameter

PAGE 22

13 set would be approximately devoid of self-heating effects, and would essentially correspond to the given device operating under isothermal conditions. Consequently, the resulting electrical parameters would only pertain to device operation for lowpower or high-speed circuit applications, and would not convey the proper device characteristics for applications that experience substantial self-heating effects. Thus, for a set of electrical-only parameters to correctly represent the characteristics of a device, over a wide range of operating conditions and biases, it should be augmented by additional parameters that describe the thermal attributes of the device. 1.4 The Simulation of Self-Heating Effects As shown in the previous sections, the operating characteristics of both individual transistors and circuits depend on temperature. Due to self-heating, the effective operating temperature depends on power dissipation and can therefore vary under different operating conditions. By solving the time-dependent heat conduction equation and energy balance equations for electrons and holes, numerical device simulators can model phenomena associated with dynamic self-heating in individual transistors [Lia94]. While this approach is invaluable for examining the detailed physics that govern the operation of semiconductor devices, it is impractical for simulating all but the simplest of circuits. Therefore, to investigate the effects of dynamic self-heating on a broad range of circuits, a more efficient simulation approach is necessary.

PAGE 23

14 The standard version of most circuit simulators such as Berkeley SPICE [Nag75] and HSPICE [Hsp92], treat temperature as a static global parameter. This has two significant implications. All of the semiconductor devices in a simulation operate at the same temperature, and that temperature remains constant throughout the simulation. Due to these constraints, a circuit simulator may not accurately represent the physical operation of a circuit, where spatial and temporal variations of the temperature can cause each device to operate at its own local temperature. To account for the temperature dependence of a circuit's operation, circuit simulators should be capable of independently tracking the dynamic temperature of each device in the circuit. A common approach for creating an electrothermal circuit simulator (ETCS) uses the concept of the thermal impedance and the analogy between electrodynamics and heat flow to account for dynamic temperature variations. This approach allows temperature to be represented as an electrical potential and power as an electrical current [Lee96, Zwe97]; therefore, the local operating temperature of a device can be thought of as simply another "bias" condition. To facilitate the temperature "bias" condition, an external node is added to a given compact device model [McA92, Fos95, Lee96, Zwe97]; such a configuration is shown in Figure 1.2. Attached to this node, internal to the device model, is a controlled current source that represents the instantaneous power dissipation. The parameter set for the device model should be modified to include the correct temperature dependences. When the modified device model is used for a circuit simulation, a thermal impedance (and, in some case, a voltage source to represent the reference ambient temperature) can be

PAGE 24

15 T op (t) P(t) -o amb Figure 1.2 A generalized schematic showing a common method for modifying a compact device model to include temperature as a variable. The dashed box outlines the new model with the added temperature node; DEVICE represents the original electrical-only model.

PAGE 25

16 attached to the new external node. Therefore, the voltage generated at this additional node represents the local temperature of the device. The electrical-only device model is first solved at the ambient temperature; this solution results in an initial guess for the device power dissipation. This power is then used to calculate the temperature rise in the device. Once the approximate local operating temperature is calculated, it is used to update the temperature-dependent model parameters, which are used to recalculate the electrical bias potentials and currents of the device model. This procedure is repeated until self-consistent solutions for the temperature and electrical biases are reached. Thus, an effective operating temperature can be independently calculated for each device in a simulation, and that temperature can now vary with the operating point. 1.5 Thermal Equivalent Circuits The data that quantify the thermal impedance of a transistor are typically in the form of discrete data points for the temperature rise, normalized to a unit-step increase in power dissipation, versus time or frequency. In such a format, the thermal impedance data are not readily accessible by an ETCS. While data in a tabular format can be used without much complexity for DC and AC simulations, an inefficient convolution computation would be required to use the data for transient simulations. Therefore, a representation for the thermal impedance is needed that both accurately models the physical data and can be easily incorporated into an ETCS for efficient DC, AC and transient simulations.

PAGE 26

17 In an ETCS, when a current representing the power dissipation in a transistor is applied to the thermal impedance, Z TH , the resulting voltage represents the temperature rise in that transistor. By invoking the analogy between electrodynamics and heat flow, the thermal impedance can be represented as an electrical impedance. Common representations for the electrical impedance circuits are shown in Figure 1.3. The resistances and capacitances that comprise the impedance effectively represent the lumped three-dimensional thermal resistance and heat capacity of the semiconductor device structure. Therefore, the overall electrical network can be referred to as a thermal equivalent circuit. The values for the individual elements of a thermal equivalent circuit can easily be determined by numerically fitting the circuit to existing thermal impedance data. Thermal equivalent circuits are directly applicable for DC and AC electrothermal simulations since, in such cases, the voltage drop across the network is simply equal to the product of the current and the network resistance or impedance. In addition, such networks inherently provide an efficient method for effecting the necessary transient convolution. As will be shown in the subsequent chapters of this dissertation, the fundamental nature of heat flow is that of a distributed system. The dynamic temperature rise in a device due to self-heating can occur over three or more decades of time or frequency. A single time constant associated with a simple exponential function can not represent the distributed behavior of self-heating. Consequently, the network response of the single-pole thermal equivalent circuits which have been proposed in previous works [McA92, Bau93, Lee93, Tu94], will not accurately

PAGE 27

18 r thl r th2 r th3 o — VvV-t — WV— ? — VW c thi Cth2 c th3 (a) r thl r th2 r th3 rYAn rAVn r^VWi c thi c th2 (b) c th3 Figure 1.3 Thermal equivalent circuits used to represent a thermal impedance for circuit simulation: a) Cauer network representation; b) Foster network representation.

PAGE 28

19 model the dynamic thermal impedance. Thermal equivalent circuits consisting of cascaded resistor/capacitor stages, as exemplified in Figure 1.3, effectively provide a distributed network response, and therefore allow a more accurate representation of a dynamic thermal impedance [Bro93]. In the work by Szekely and Van Bien [Sze88], the Foster circuit (Figure 1.3b) was shown to be an invalid representation of a discretized thermal network. This point is valid in the context of numerical simulations (e.g. finite difference or finite element) where the transistor structure is modeled by a distributed thermal network. In that case, the node-to-node capacitances of the Foster network do not have physical meaning and the Cauer network would be the proper physical discretization of the given thermal domain. However, in this dissertation, the thermal equivalent circuit is simply a numerical representation of a thermal impedance, and the validity of its format is moot. Yet, for the purpose of representing a lumped thermal impedance in an ETCS, the Foster network form offers an important advantage over the Cauer form: the time constants associated with a given Foster network are independent of any surrounding circuit elements. This characteristic is beneficial when individual thermal equivalent circuits must be connected to model different components of a transistor structure or the thermal interactions between transistors. Therefore, the Foster network form will be assumed for any thermal equivalent circuits within this dissertation.

PAGE 29

20 1.6 The Need for Physics-Based Thermal Impedance Models In the previous two sections, the concept of the thermal impedance is adopted to model the temperature rise in a transistor as a function of that device's power dissipation. For the purpose of circuit simulation, the thermal impedance can be represented by a network of resistances and capacitances that effectively represent the lumped thermal characteristics of a transistor. To successfully synthesize a thermal equivalent circuit, tangible data for the thermal impedance are necessary. One approach to obtain the thermal impedance of a transistor is to extract it from measurements [Lee95, Zwe96]. While this empirical approach provides accurate temperature information, such measurements are somewhat difficult, for several reasons. To begin with, thermal measurements are very time-consuming. The extraction procedure is generally divided into two steps, the first of which dominates the total measurement time. This step is required to calibrate the relation between the temperature and the physical characteristic that is being used to monitor the temperature (e.g. the base current and drain current in bipolar and field-effect transistors, respectively). The calibration is performed at multiple ambient temperatures at DC and is thus limited by the long time constants associated with steady-state heat flow. To make such thermal measurements requires special measurement equipment such as a thermal wafer chuck or oven to accurately control the temperature of the devices being measured. Finally, the results of any such extraction are limited to the specific device being measured. Thus, the entire procedure would have to be repeated for each transistor structure and transistor type of interest.

PAGE 30

21 Another approach, which avoids the inherent complexities of thermal measurements, is to derive the thermal impedance of a transistor from the physical equations that govern the temperature and heat flow in the device. Physical thermal modeling is desirable because it can give the temperature behavior as a function of the device structure and material properties alone; therefore, the effects of device technology scaling on the thermal impedance can be predicted. The requirements of accurate physical modeling (e.g. multi-dimensional numerical simulations) tend to conflict with the needs for simplicity and efficiency in circuit simulation. However, a thermal impedance model does not need to be absolutely accurate to provide reasonable results within an ETCS. Therefore, by using certain heuristic assumptions, compact physical models for the thermal impedance, suitable for efficient simulation, can be derived. It is important, though, that the correlation between the accuracy of the thermal impedance models and the accuracy of the simulated electrical characteristics of a semiconductor device in the presence of selfheating be understood. The sensitivity of a given electrical parameter, X, of a semiconductor device to the thermal resistance can be defined as ,Rth_^jh _ax X ' dR s -TH = _m _^~_ ( j j 4) TH As an example, since the output current of a device is very important for characterizing performance, (1.14) can be used to determine the sensitivity of the collector and drain currents of BJT's and MOSFET's, respectively. Using (1.1) in the steady-state limit and (1.6) with (1.14), the sensitivity of the collector current of a

PAGE 31

22 BJT is expressed as „R TH q-(Vg-v B E)-RTH-P ^I j — . (1-15) k(R TH P + T ) 2 where V g is the semiconductor band-gap voltage. A similar expression for the sensitivity of the drain current of a MOSFET can be determined using the current equation given by Fox and Brodsky [Fox93a], which results in SR TH I = -oc 'd R TH ' P " • T i R TH P To (1.16) where a is typically between 1.5 and 1.8 (assuming that the temperature dependence of the drain current is dominated by the temperature sensitivity of the carrier mobility). The expected level of error in simulated output currents can be approximated by the product of the sensitivity and the anticipated error in the thermal impedance model. Therefore, as shown by ( 1 . 1 5) and ( 1 . 1 6), the relation between the accuracy of the thermal impedance models and the accuracy of the calculated electrical parameters depends on the power dissipation and the sensitivity of the electrical parameters to temperature. Consequently, the level of accuracy of a thermal impedance model is more critical for devices with electrical characteristics that are highly sensitive to temperature (e.g. BJT's as opposed to MOSFET's). At low power dissipation levels, where the temperature rise is small compared to the ambient temperature, the error in the thermal impedance model will not directly correspond to the error in the calculated operating temperature. For a temperature rise of twenty degrees, the sensitivities of the BJT collector current (V BE = 0.8) and

PAGE 32

23 MOSFET drain current are 0.7 and -0. 1 , respectively. In such a case, the error in the electrical characteristics will tend to be lower than the error in the thermal impedance. Whereas at large power dissipation levels, the temperature rise can be much larger than the ambient temperature, and the error in the thermal impedance model will directly correspond to the error in the calculated operating temperature (for a temperature rise of one-hundred degrees, the respective BJT and MOSFET current sensitivities are 2 and -0.4); in which case, large errors in the calculated electrical characteristics can result. Figure 1.4 shows an example of BJT characteristics simulated assuming a 20% error in the thermal resistance model; the simulations were performed using the modified version of SPICE created by Lee [Lee96]. The data clearly shows that for larger temperature rises, the error in the calculated current (due to errors in the thermal impedance model) increases. The motivation behind this dissertation is the development of compact thermal impedance models for semiconductor transistors. These models can provide a reasonably accurate representation of the dynamic temperature response within a device; more importantly, since the models depend mainly on the physical structure of a device, they can correctly anticipate the effects of technology scaling on the thermal behavior. Physics-based thermal impedance models allow an ETCS to predict dynamic self-heating effects in circuits and can also provide more accurate electrical parameter extraction. In addition, when the thermal impedance models are coupled with physics-based compact device models, the combination provides an efficient tool for studying self-heating in semiconductor transistors.

PAGE 33

24 40 30 c o fc 20 3 U o "o U R TH = 1000 °C/W R TH = 800 °C/W 12 3 4 Collector-Emitter Voltage, V CE (V) Figure 1.4 Simulated output characteristics of a BJT, assuming that R TH = 1000 °C/W, for V BE =0.80, 0.85, 0.90 and 0.95 V. The simulations are repeated assuming a 20% error in the thermal resistance model, so that R TH = 800 °C/W.

PAGE 34

25 1.7 Organization Chapter Two presents a physics-based model for the thermal impedance of bulk junction-isolated bipolar transistors. The model is derived by solving the threedimensional time-dependent heat conduction equation in the substrate. The ability of the model to represent bulk BJT/HBT's with either LOCOS or trench isolation is investigated. To account for multiple-emitter bipolar transistors, the thermal impedance model is extended to represent multiple heat sources. The accuracy of the model is evaluated using measurements and three-dimensional finite-element simulations. Chapter Three describes a circuit network for modeling thermal interactions between devices located in the same substrate. The network is developed for the specific application of multiple-emitter bipolar devices, but is shown to be valid for general cross-substrate thermal coupling in circuits. A method for improving the simulation efficiency of a multiple-emitter BJT/HBT electrothermal model, using a lumped thermal impedance model, is presented. The validity of the lumped modeling approach is supported with comparisons to the full electrothermal model. Chapter Four presents a predictive scalable model for the thermal impedance of BJT's with full dielectric isolation. The model is derived by solving the three-dimensional time-dependent heat conduction equation in the substrate accounting for the buried oxide and trench isolation. In the limit of steady-state heat conduction, the thermal impedance model is simplified, resulting in a closed-form

PAGE 35

26 model of the thermal resistance. The accuracy of both models is evaluated using three-dimensional finite-element simulations and measurements. Chapter Five describes a physics-based model for the thermal impedance of bulk. MOSFET's. The model is derived by solving the three-dimensional timedependent heat conduction equation in the substrate. The effects of the device interconnects and isolation structures, such as LOCOS and trenches, on the thermal impedance are investigated. The accuracy of the model is evaluated using measurements and three-dimensional finite-element simulations. Chapter Six presents a predictive scalable model for the thermal impedance of SOI MOSFET's. The model is initially derived for steady-state heat conduction by coupling separate one-dimensional heat conduction analyses in the silicon film and interconnects. The derivation is then carried out for the case of timedependent heat conduction, resulting in a model for the dynamic thermal impedance. The accuracy of both models is evaluated using three-dimensional finite-element simulations and measurements. Chapter Seven describes a computer program developed to facilitate thermal modeling in circuit simulation. The program, referred to as the Thermal Impedance Pre-Processor (TIPP), functions as a framework for obtaining the component values of thermal equivalent circuits from the thermal impedance models presented in Chapters Two through Six. Chapter Eight concludes the dissertation with a summary of the accomplishments of this work and suggestions for future modeling efforts.

PAGE 36

CHAPTER 2 A THREE-DIMENSIONAL THERMAL IMPEDANCE MODEL FOR JUNCTIONISOLATED BIPOLAR TRANSISTORS 2.1 Introduction The models derived in this chapter provide closed-form physical solutions for predicting the thermal impedances for singleand multiple-emitter bipolar junction (BJT) and heterojunction bipolar (HBT) transistors, based solely on device geometry and material properties. These models can predict both steady-state and dynamic self-heating due to the semiconductor substrate. Previous works in this area provided values for the thermal impedance of BJT's or HBT's, but were either limited by assumptions or relied on non-predictive measurement techniques. For example, the thermal impedance model derived by Fox and Lee [Fox91a] is limited to single-emitter devices. The analyses in other works only provide models for the steady-state thermal resistance [Lio93, Bau94, Daw94, Lio94, Lio96]. Some authors have used measurement techniques to extract either the steady-state thermal resistance or simple one-pole approximations for the thermal impedance [Bau93, Liu93, Daw94, Liu95a, Liu95b]; in either case, the results do not provide a complete picture of self-heating and are not predictive. The thermal analysis by Joy and Schlig [Joy70] serves as the foundation for deriving of thermal impedance model, and the first part of this chapter re27

PAGE 37

28 examines this work to provide a clear background for modifications made to the model later in the chapter. The thermal impedance model developed by Joy and Schlig was derived for single-emitter, junction-isolated BJT's operating in the forward-active region. A diagram of a typical junction-isolated npn BJT is shown in Figure 2.1. In this chapter, the basic model is modified to account for variations in substrate thickness. The effects of interconnect metallization and different isolation technologies on the thermal impedance, and thus on the performance of the model for advanced device structures, are investigated. The single-emitter thermal impedance model is finally extended to account for BJT/HBT's with multiple emitter fingers. 2.2 Derivation of the Single-Emitter BJT/HBT Thermal Impedance Model For the derivation of the single-emitter bulk BJT/HBT thermal impedance model, the semiconductor substrate is represented by a homogeneous semi-infinite half-space with an adiabatic top surface (no heat transfer perpendicular to the surface). The back side of the substrate is assumed to be held at a constant temperature, T . Since the substrate material is assumed to be homogeneous, the model most directly applies to junction-isolated transistors. The effects of other types of isolation structures used in bulk technologies, such as recessed LOCOS (local oxidation of silicon) or back-filled trenches, on the thermal response are not taken into account. Figure 2.2 illustrates the simplified device geometry assumed for the model derivation; the diagram focuses on the "electrically active" portion of the device that lies directly beneath the emitter stripe, which has a width W and length

PAGE 38

29 Oxide Collector O JZZ Emitter Z2 e Base O I " " ; TI 2 UZ IT" ZL Oxide 7 n n+ p-substrate Figure 2. 1 Cross-section of a typical junction-isolated bipolar junction transistor (BJT).

PAGE 39

30 (dT/dz) = D H ^ Figure 2.2 The simplified device geometry used to define the solution domain for the bulk, single-emitter BJT/HBT thermal impedance model. The substrate is represented by a semi-infinite half-space with an adiabatic surface (the dotted lines). The emitter stripe has a width W and length L. The heat source (the rectangular volume) is displaced a distance D below the surface of the device, equivalent to the depth of the base/ collector junction. The heat source has a thickness H which approximates the base/collector space-charge region (SCR).

PAGE 40

31 L. The imbedded heat source represents the base/collector space-charge region (SCR), which is further represented by a rectangular volume with a thickness, H . The heat generated in this region is assumed to be due to uniform power dissipation. This assumption is reasonable for devices in the forward-active region of operation prior to any high current effects, as the current distribution in the intrinsic device will be approximately uniform. The electric field gradient in the base/collector SCR can also be neglected since it does not greatly affect the thermal impedance model. The heat source is displaced beneath the surface of the substrate by a distance D, assumed to be the depth of the base/collector junction. Thus, any encroachment of the base/ collector SCR into the base region is neglected (which is reasonable since the base typically has a higher doping than the collector). Representing the substrate as a semi-infinite medium presumes that the back-side and the lateral edges do not influence the thermal response of the device. Neglecting the effects of the back side of the substrate on the thermal response is reasonable since a typical wafer is about 1000 times thicker than the heat source. Neglecting the effects of the lateral boundaries requires that the device be located sufficiently far from the substrate edges; the work by Fox et al. [Fox93b] suggests that this assumption is valid for any device that is at least a distance 5jW L from any lateral edge. The surface of the substrate is assumed to be the only boundary that affects the thermal response of the device and it is considered to be adiabatic; thus, conduction through the interconnects and conduction/convection from the surface are neglected. Ignoring thermal energy transport from the substrate surface is supported by the work of Berger and Chai and Goodson et al. [Ber91, Goo95];

PAGE 41

32 however, they were mainly concerned with transport via convection to a surrounding gas (namely air). Nonetheless, for the regions of the device covered by oxide, it is unclear whether there is substantial heat conduction to this overlying oxide. From the analysis of Goodson et al. [Goo95], the device-to-oxide thermal conductance is of the order of G = 4rk, which corresponds to an isothermal disk of radius r on the boundary of a semi-infinite medium of thermal conductivity k. Approximating the radius as */(WL)/7C and using the room-temperature thermal conductivity of Si0 2 , the device-to-oxide thermal conductance for a typical device is on the order of 1 x 10" 6 (W/°C). Comparably, the device-to-substrate thermal conductance is on the order of 1 x 10" 3 (W/°C), showing that the majority of heat will flow through the substrate. The temperature rise at any point within the device can be described by the nonhomogeneous three-dimensional heat conduction equation V2AT(x,y,z,t) + fc^Al) = l 3AT(xy,z,t) v J ' k a dt and the boundary conditions AT(±<», y, z, t) = (2.2) AT(x,±oo, z ,t) = (2.3) 3AT(x, y, z, t) dz = (2.4) z =

PAGE 42

33 AT(x,y,oo, t ) = 0, (2.5) where AT is the temperature rise above the local ambient (AT = T-T ), g is the internal energy generation density, k is the thermal conductivity, a is the thermal diffusivity (a = k/(p • c ) where p is the density and c is the specific heat) and t is time. Typical values for the material properties are given in Table 2.1. Table 2.1 Semiconductor material properties Parameter

PAGE 43

34 using the Kirchoff transformation [Joy75], as discussed in Chapter Eight. Neglecting the spatial dependence of the thermal conductivity implies that the effect of dopant atoms is ignored. In the works by Weber and Gmelin and Goodson et al. [Web91, Goo95], the thermal conductivity of doped silicon (up to lxl0 18 and 1.7xl0 19 dopant atoms cm" 3 ) above 300 K is shown to differ only slightly from that of intrinsic silicon. Since the majority of the substrate is typically low-doped semiconductor material, neglecting the doping effects on the thermal conductivity is reasonable. With the initial thermal conditions within the substrate specified as AT(x,y,z,0) = 0, (2.6) the solution to (2.1) can be expressed in the form AT(x,y,z,t) = 5f J dt'jG(x,y,z,t|x',y',z',t')g(x',y',z' > t')dv' (2.7) t' = o v where G(x, y, z, t|x', y', z, t) = exp 8[7CCC(t-t')] L4cc(t-t)J -(x-x) exp 2-(y-y') _4a(t-t')_ exp -(z-z') 2 _4a(t-t')_ + exp -(z + z) 4a(t-t')_ (2.8) is the Green's function for the given boundary-value problem [Ozi93]. Equation (2.8) is the solution to

PAGE 44

35 gj,-5(x-x')5(y-y')5(z-z')5(t-t') I §9 a 9t (2.9) for the boundary and initial conditions given by (2.2) through (2.6), and physically represents the temperature at point (x, y, z) at time t, due to an instantaneous point source, g' (W s), of unit strength at point (x', y', z) at time t' . To account for the heat-generation volume (V = W • L • H), (2.8) is substituted into (2.7) and integrated over the base/collector SCR, resulting in AT(x, y, z, t) = j" P(t') t' = o 8pcV H K L/2 + x ^ J L/2-x erf . + erf V4a(t-tV V4a(t-t') W/2 + y + erf V4a(t-tV V4a(t-t') W/2-y + erf D-z V4a(t-t') + erf z + D + H */4a(t-t') z-D V4a(t-tV V4a(t-tV. D + H-z \1 , . + erf -=== dt (2.10) where g(x, y, z, t) = g(t') = P(t')/V , since the power dissipation is assumed to be uniform. Equation (2.10) represents the temperature response at any point in the device at time t due to a change in continuous power dissipation in the base/collector SCR. Assuming a step increase in power at t' = ( P(t') = P • U(t) ) and expressing the temperature rise as AT(t) = Z TH (t)P (2.11) yields the transient thermal impedance

PAGE 45

36 Z TH (x, y,z, t) = J" 8pcV crj / L/2 + x A | J L/2 x at /J erf AV^2 + yV erf AV^2-y v V4at / v 74at fl 'z + D + m ,f-D-z erfl z= — 1+ erf V4at + erf| , +erf 74at^ 74at D + H-z 74at dt (2.12) where the t — > oo value of the thermal impedance corresponds to the thermal spreading resistance R TH . Equation (2.12) represents the temperature rise at any point in the device normalized to a unit-step increase in power dissipation. For circuit simulation, a single temperature is needed to represent the effective operating temperature of the device. Fox and Lee [Fox91b] showed that the thermal impedance model evaluated at a surface corner of the emitter (x = L/2, y = W/2, z = 0) agreed well with measurements of the thermal spreading resistance R TH ; substituting these coordinates into (2.12) gives the following expression for the thermal impedance J TH <" I : erf ^U^=)U^M)-crf( ! ;4pcV Kj4mJ WSatA VJ4 KXt 74 at dt (2.13) In this form, the thermal impedance model has four geometric input parameters. Of the four, three (W, L and D) are determined directly by the device layout. However, the fourth parameter, H, depends on the operating bias of the device. The thickness of the base/collector SCR, H, can be estimated using the

PAGE 46

37 depletion approximation; assuming a one-sided step junction with uniform doping on each side gives 2 • e Si • e • (V R + • bi ) H = Sl ° XT R ^, (2.14) V qN e P i with k R T ^bi = -Tln ' N b • N ep , 2 (2.15) where £ Si is the dielectric constant of silicon, e is the permittivity of free space, V R is the reverse bias voltage on the base/collector junction, q is electronic charge, N is the doping level in the epi-collector, k B is Boltzman's constant, T is temperature, N b is the doping level in the base, and nj is the intrinsic carrier concentration in silicon. Yjjj is the built-in potential of the base/collector junction. Equation (2.14) shows that the thermal impedance depends on the bias of the base/collector junction, and therefore can change during device operation. However, the square-root dependence of H on the base/collector voltage, is relatively weak. Figure 2.3 illustrates the variation of the modeled thermal resistance with changes in the thickness of the base/collector SCR. The three data points plotted for each simulated device correspond to reverse bias base/collector voltages of 5, 10 and 20 V. The largest variation is observed for the smallest device, which shows a 25% change in its thermal resistance going from V R = V to 20 V. The larger devices show a weaker dependence on H and have no more than a 15% change in thermal resistance

PAGE 47

38 £ 20 x H c o oj 'C > 200 250 300 Variation of H (%) 350 400 Figure 2.3 Simulations showing the effect of variations in the thickness of the base/collector space-charge region on the thermal impedance model (evaluated at steady-state) for different geometry BJTs. For each device, D = 0.35 urn, N epi = 1 x 10 16 cm" 3 and N b = 1.5 x 10 18 cm" 3 . The y-axis corresponds to the variation between the model evaluated at V R = V and the model evaluated at V R equal to 5, 10 and 20 V.

PAGE 48

39 going from V R = V to 20 V. However, at high base/collector biases, the maximum value of H becomes effectively independent of bias. As shown in Figure 2. 1 , typical bipolar technologies use a heavily-doped buried layer to reduce collector resistance. At high base/collector biases, the low-doped epi-collector region depletes down to the buried layer; consequently, the maximum value of H should be properly limited to the thickness of the epi-collector. 2.2.1 Modification for Finite Wafer Thickness As previously derived, the thermal impedance model for single-emitter bulk BJT/HBT's represents the substrate as a semi-infinite half-space. This representation assumes that the back side of the substrate does not affect the thermal response of the device. In general, this is reasonable since the base-collector junction is usually within 1 |0.m of the substrate surface, and a typical wafer is between 350 to 800 |im thick. However, wafers are commonly back-lapped to improve thermal performance, and substrate thicknesses of 75, 80 and 100 |nm have been reported by a number of authors [Kag94, Mar93, Liu95a]. As the wafer thickness is reduced, the substrate can no longer be approximated by a semi-infinite medium and the effects of the back-side boundary must be taken into account. The three-dimensional Green's function in the rectangular coordinate system can be represented by the product of three one-dimensional Green's functions G(x, y, z, t|x', y', z, t) = G x (x, t|x', t') • G (y, t|y', t') • G z (z, t|z', t'). (2.16)

PAGE 49

40 The lateral boundaries are still assumed to extend infinitely and their effects on the thermal response are neglected; thus, the Green's function solutions in both the x and y directions remain unchanged. In the z -direction, however, the substrate is now assumed to have a finite thickness D sub . The top surface of the substrate is still assumed to be adiabatic. The bottom surface of the substrate is assumed to be at a constant temperature, T(D sub ) = T , so that the temperature rise at this surface is defined by AT(D sub ) = T(D sub )-T = 0. These boundary conditions define the new Green's function for the z-direction, which is given by G z (z,t|z',t') = £ exp[-a-Tip(t-t')]— • cos(Ti p z) • cos(Ti p z') (2.17) P = i sub where r\ is the set of eigenvalues for the boundary-value problem and are given by the positive roots of cos(Ti p D sub ) = 0. (2.18) Equation (2.18) is solved when the argument of the cosine equates to odd multiples of 7i/2. Using equations (2.17), (2.16), (2.1 1) and (2.7), and then integrating over the base/collector SCR, assuming a unit-step increase in power at t' = 0, gives the following expression for the thermal impedance at any point in the device

PAGE 50

41 fL/2 + \\ JL/2-x ft >\ f Ql r L/Z + X J\^/lWu,) = {4^K-^rJ +erf i^si 74at / v 74at i. ~ 2cos(r| p z)exp(-ar|pt) •{sin[Ti p (D + H)]-sin[ri p D]} (2.19) where V = W L H and n p = &£^ (2.20, Evaluating (2.19) at the coordinates (x = L/2, y = W/2, z = 0) to give a single effective operating temperature, results in the following expression for the thermal impedance Z TH (t) = f-^-erff-kArf. " 2exp(-arTt) • X n D (sin[T 1p (D + H)]-sin[T 1p D]}, (2<21) , lp sub Since (2.21) is derived from the physical heat conduction equation, it can be used to anticipate the effects of substrate scaling on the thermal impedance. Figure 2.4 illustrates equation (2.21) evaluated at various values for D sub . The results show that the thermal resistance decreases as the substrate thickness is reduced, which agrees with the trend predicted by Hattori et al. using a threedimensional numerical simulator [Hat95]. Figure 2.4 also shows that (2.13) provides

PAGE 51

42 675 650 " 625 X

PAGE 52

43 an accurate prediction of the thermal resistance over most of the range of substrate thicknesses; only when the substrate thickness is significantly reduced (< 100 (J.m), is there a large deviation between the two models. 2.2.2 Effects of Interconnect Metallization on the Thermal Impedance For the derivation of the BJT/HBT thermal impedance model, the surface of the substrate is assumed to be adiabatic. In actual devices, portions of the base, collector and emitter regions are in direct contact with the metallization used to electrically connect different devices on a chip. Since the metallization typically has a high thermal conductivity, it is possible that the heat conduction via the interconnects significantly influences the thermal response of a device. Therefore, the validity of such an assumption should be investigated. Three-dimensional (3-D) finite-element (FE) thermal simulations of a bipolar transistor, using the ANSYS software package [Ans96], were performed to examine the effects of the interconnect metallization on the thermal impedance. To simplify the FE model, the device was considered to be symmetric in both lateral directions; therefore, only one quarter of the device was simulated. The bottom and exterior sides of the substrate were assumed to be at a fixed ambient temperature. The top and side surfaces of the interconnects, as well as the top surface of the inter-layer dielectric, were assumed to be adiabatic. The FE simulations tend to overpredict the heat conduction through the interconnects since any contact resistances at the material interfaces were neglected. The assumed symmetry of the device implies that the base and collector metallization are equidistant from the emitter. In typical

PAGE 53

44 devices, the collector contact is offset a greater distance from the emitter than the base contact. Typical ranges for these offsets are 0.5 to 10 |Lim between the base and emitter contacts, and 2.5 to 25 |um between the collector and emitter contacts [Gra93]. While the FE model does not exactly represent any actual device structure, it can provide an estimate for the significance of the heat flow through the interconnects as a function of their distance from the active device. Figure 2.5a shows the FE model for bipolar devices with full metallization. Steady-state thermal simulations were run for various interconnect spacings; this spacing corresponds to the edge-to-edge distance between the emitter and base/collector interconnects. Simulations were also run of the same structure with the base/collector interconnect removed. The results of the two groups of simulations were compared to determine the effect of the base and collector interconnects on the thermal resistance. Figure 2.5b shows the results of the comparison between the FE simulations. The data clearly shows that the effect of the base/collector interconnect metallization is small and decreases as the interconnects are moved away from the active device area. The collector interconnect has less of an effect on the thermal impedance than the base interconnect, due to the larger distance between the collector contact and the active device. In any case, the effect of either the base or collector interconnect should be negligible compared to the influence of the emitter metallization. To determine the extent of the effect of the emitter interconnect on the thermal impedance, steady-state thermal simulations were run for different devices with only the emitter metallization in contact with the device. Figure 2.6a shows the

PAGE 54

45 (a) 2.5 8 20 Pi 15 03 0> _c c o 4— > > 1.0 0.5 0.0 9 — OL = 4 urn, W = 1 (im 3 — QL = 4 (im, W= 1.8 \Lm -AL = 8 |lm, W = 1 fim 0.5 2.5 3.5 Interconnect Spacing (fim) (b) 5.5 Figure 2.5 ANSYS simulations showing the effect of emitter, base and collector interconnects on the thermal resistance. The device specifications are D = 0.2 (im and H = 0.35 fim, the interconnect width W met = 2 |^m and thickness d met = 0.9 (im, and the inter-layer dielectric thickness d ox = 0.7 (im: a) The finite-element model simulated with ANSYS; b) the variation between the thermal resistance accounting for emitter, base and collector interconnects and the thermal resistance accounting for only the emitter interconnect, plotted as a function of the spacing between emitter and base/collector interconnects.

PAGE 55

46 (a) 30 £

PAGE 56

47 FE model for devices with only the emitter metallization. The same devices were also simulated with the emitter interconnect removed. The simulations were performed by independently varying each structure parameter of the FE device model. Reasonable values were chosen for each parameter to represent a nominal device design; each parameter was varied about the nominal value to represent a reasonable range of technology scaling. Figure 2.6b shows the results of the FE simulations. The emitter metallization becomes a more effective path for heat evacuation as the thickness, d met , and the width, W met , of the interconnect increase and as the thickness, d ox , of the dielectric layer between the substrate and the interconnect decreases. The data also show that the effect of the emitter interconnect increases as the depth of the base/ collector junction is decreased (the heat source is moved closer to the surface) and as the length of the emitter is decreased. Transient thermal simulations of the FE model in Figure 2.6a were used to examine the effect of the emitter interconnect on the transient thermal response. ANSYS was used to simulate the structure with and without the emitter interconnect in contact with the device; the results are shown in Figure 2.7. The thermal responses for the device with and without the interconnect match until significant heat reaches the surface of the device. The time for heat to reach the surface of the device can be approximated as the square of the distance D divided by the thermal diffusivity of the substrate material. The resulting time is approximately 0.44 nanoseconds, which agrees with the FE simulations. The adiabatic boundary condition of the device without the interconnect predicts a larger response since the heat is completely reflected once it reaches the surface. The device with the emitter metallization, which

PAGE 57

48 1.5 U o g 1.0 o
PAGE 58

49 acts as a separate heat sink path, has a reduced thermal impedance and an effectively slower thermal response. Based on the results of the 3-D FE simulations, neglecting the base and collector interconnect metallization in the model derivation is reasonable since it only slightly affects the thermal impedance of a device. The emitter interconnect, however, has a greater influence on both the steady-state and transient thermal responses. The effect on the thermal resistance will be more significant for devices with small-geometry emitters and shallow base regions, where the transient thermal response will mainly be affected for large devices with substantial contact structures. In either case, equations (2.13) and (2.21) will tend to over-predict both the steadystate thermal resistance and the transient rise of the thermal impedance. 2.2.3 A Model for the Thermal Impedance of the Emitter Interconnect As shown in the previous section, the assumption that the top surface of the device is adiabatic neglects heat flow in the emitter interconnect and results in a thermal impedance model that over-estimates the transient temperature rise in a bipolar device. To model the effects of the emitter interconnect on the overall thermal impedance, both the thermal resistance and thermal capacitance of the metallization need to be considered. The thermal resistance of the emitter metallization is derived by assuming that the interconnect can be represented by a one-dimensional cooling fin, so that the temperature rise at any point x met along the interconnect, AT me t( x me t ) = T met (x met ) T , can be approximated by

PAGE 59

50 a 2 AT 3x met 2 . ry, _ m met AT met = ° (2.22) met The second term on the left-hand side of (2.22) accounts for heat conduction through the underlying oxide as the heat travels along the interconnect, where 1 k rl met u met in met met (2.23) is the characteristic thermal length in the interconnect and met (2.24) is the heat transfer coefficient from the interconnect to the substrate. The material properties for the emitter interconnect are given in Table 2.2. Table 2.2 Emitter interconnect* material properties Property met Pmet 'pmet Definition Thermal conductivity Density Specific Heat Source: [Ozi93] * Assumed to be aluminum Value 2.39 (W cm" 1 K" 1 ) 2.7 (g cm" 3 ) 0.9 (J g" 1 ^ 1 )

PAGE 60

51 Approximating the temperature in the interconnect using a one-dimensional equation implies that the temperature gradients in the vertical and lateral directions within the emitter interconnect are negligible. The validity of such an assumption can be evaluated using the Biot number, which corresponds to the ratio of the internal and external thermal resistances of a given object [Ozi93]. If the Biot number for the interconnect is much less than unity, then the interconnect can be approximated as a one-dimensional thermal medium. The vertical and lateral Biot numbers for the emitter interconnect are given by B Vmet = h met d met /k met and B Lmet = n met w met / ( k met d met)' respectively. For most practical metallization geometries, the Biot numbers are much less than one and the cooling-fin model is an accurate representation of the emitter interconnect. Assuming that the temperature rise in the interconnect at the emitter contact is equal to the effective operating temperature of the device, and that the temperature rise approaches zero far from the contact, the thermal resistance of the emitter interconnect can be expressed as R THmet = ^met"^^!^^*] -1 • (2-25) The thermal capacitance of the emitter metallization can be approximated as ^THmet = Pmet c pmet* met (2.26) The volume of the metallization is V met = W L 5 met , where 8 met represents the effective length of the interconnect structure. The parameter 8 met should be evaluated to include the volume of the contact and interconnect metallization but can

PAGE 61

52 also be extracted from transient thermal measurements or numerical simulations. Once the thermal resistance and thermal capacitance have been calculated, the transient thermal impedance of the emitter interconnect can be approximated by Z THmet( t ) R THr ='['" exp -t 'met'-' (2.27) where T met = RTHmet^-THmet • ^ ne overa ll thermal impedance of a bipolar device can now be represented by the parallel combination of two thermal impedances, such that effectively ^th( s ) — Z-THdev( S ) ' Z THmet (s) Z THdev( S ) + Z THmet( S ) (2.28) where Z THdev (s) is determined from the transient thermal impedance given by either (2.13) or (2.21). 2.2.4 Effects of Isolation Structures on the Thermal Impedance While junction-isolated technologies are still used, the drive to increase packing density, improve lateral isolation and increase device operating speeds has led to the development of newer isolation technologies for VLSI bipolar applications. The advanced isolation technologies typically used in bulk bipolar fabrication are recessed LOCOS (local oxidation of silicon) and U-groove [Wol90, Gra93]; Figure 2.8 illustrates examples of bipolar devices fabricated with these isolation techniques. Since advanced isolation structures typically use low conductivity

PAGE 62

53 Collector p-substrate (a) Emitter o Collector p-substrate (b) Figure 2.8 Cross-sections of typical BJT's fabricated with advanced isolation technologies: a) Recessed LOCOS; b) U-groove isolation.

PAGE 63

54 materials like Si0 2 , the thermal impedance of a device using such isolation tends to be higher than that of its junction-isolated counterpart. The bulk BJT/HBT thermal impedance model treats the substrate as a homogeneous material; therefore, it is unclear whether the thermal impedance model is applicable to devices which are fabricated with advanced isolation. Three-dimensional (3-D) finite-element (FE) thermal simulations, using ANSYS, were performed to examine the effects of advanced isolation structures on the thermal impedance of bipolar transistors. Two FE models were developed to separately investigate the effects of recessed LOCOS and U-groove isolation. To simplify the FE models, the device was considered to be symmetric in both lateral directions, so that only one quarter of the device was simulated. The bottom and exterior sides of the substrate were assumed to be at a fixed ambient temperature. The top surface of the device was assumed to be adiabatic. Due to the assumed symmetry, the active device region was surrounded on all sides by the isolation structure, which was at uniform distance from each side of the emitter. As shown by the illustrations in Figure 2.8, the distance between the isolation structure and the intrinsic device is not uniform on all sides of the device. Typical values for the distance between the emitter and the isolation— for the portions of the isolation structure immediately surrounding the emitter— are in the range from 0.3 fim to 0.8 |im for advanced bipolar devices [Del91, Klo93, Yam93, Pru94]. The distance between the emitter and the portion of the isolation structure on the far side of the collector contact is generally larger, typically two to four microns [Del91, Klo93]. For both FE models, the sidewalls of the isolation structures were perpendicular to the top surface of the substrate.

PAGE 64

55 For typical U-groove isolation, the trench is created by anisotropic etching and the side-walls are nearly perpendicular to the surface. However, actual LOCOS structures have tapered edges (see Figure 2.8) that get progressively thinner toward the active device. To determine the implications of the model's non-physicality, twodimensional (2-D) FE simulations were run for various angles (30 to 90 degrees) between the substrate surface and the side-wall of the isolation. The simulations showed that the temperature rise increased as the angle increased; thus, the 3-D LOCOS model should show a larger effect than that of an actual isolation structure. While the finite-element models do not truly represent the physical device layout, they allow an order-of-magnitude estimate for the effects of the isolation structures on the thermal impedance. Steady-state thermal simulations were run for various device-isolation spacings, corresponding to the edge-to-edge distance between the emitter and the isolation structure. For the U-groove isolation model, this spacing is the distance between the emitter and the edge of the surface LOCOS; the actual trench is assumed to be an additional 0.5 |im away from the edge of the LOCOS [Del91, Yam93]. Simulations were also run for the same devices with the isolation structures removed. LOCOS isolation is formed by selectively oxidizing regions of the semiconductor substrate in a dry or wet oxygen-rich ambient. For bipolar technologies, the resulting Si0 2 structures are typically no more than one micron thick, since the growth of thicker oxides is impractical [Wol90, Gra93]. Figure 2.9a illustrates the FE model for bipolar devices with recessed LOCOS isolation. The oxide was assumed to be fully recessed beneath the top surface of the substrate and

PAGE 65

56 (a) 1.0 1.5 2.0 2.5 Device-Oxide Spacing (|im) (b) 3.0 3.5 Figure 2.9 ANSYS simulations showing the effect of recessed LOCOS isolation on the thermal resistance. The device specifications are L = 2 u.m, W = 1 |im, D = 0.35 u\m and H = 0.35 |im: a) The finite-element model simulated with ANSYS; b) the variation between the thermal resistance accounting for the isolation and the thermal resistance assuming a homogeneous substrate, plotted as a function of the edgeto-edge spacing between the emitter and the isolation.

PAGE 66

57 had a thickness, d tox , of one micron. Figure 2.9b compares the FE simulations with and without the isolation. The effect of the LOCOS can be significant at small device-isolation spacings, but decreases as the isolation is moved away from the active device region. In Figure 2.8a, the portions of the LOCOS structure close to the emitter have the largest effect on the thermal response, since they directly restrict the lateral heat flow away from the device. A number of manufacturers are using thinner standard or semi-recessed LOCOS (0.3 to 0.6 pm) combined with junction isolation to reduce fabrication times and improve compatibility with existing MOS technologies [Klo93], [Pru94]. The thinner oxides have a smaller effect on the thermal resistance, and therefore, the FE simulations can be considered worst-case. U-groove isolation differs from LOCOS in that trenches are etched directly into the substrate and then back-filled with oxide and polysilicon. The depth of the trench, d tr , is typically on the order of 3 pm [Yam93, Ona95], but has been as large as 5 pm [Del91]; the width of the trench is generally in a range from 0.6 to 1.5 pm [Del91, Yam93, Ona95, Shi96]. U-groove trenches will typically have a surface LOCOS layer, but the thickness of this layer is usually no greater than 0. 1 to 0.15 pm [Yam93], since the isolation is mainly achieved by the trench. Figure 2.10a shows the FE model for devices with U-groove isolation. The thickness of each fill layer ( d trox for oxide and d , for polysilicon) in the U-groove was assumed to be uniform. Figure 2.10b compares the FE simulations with and without the isolation. The effect of the U-groove isolation on the thermal resistance is greater for small device-isolation spacings than in LOCOS due to the larger depth of the trench.

PAGE 67

58 (a) 1.0 1.5 2.0 2.5 3.0 Device-Isolation Spacing (urn) (b) 3.5 4.0 4.5 Figure 2.10 ANSYS simulations showing the effect of U-groove isolation on the thermal resistance. The device specifications are L = 2 urn, W = 1 urn, D = 0.35 urn and H = 0.35 urn. The U-groove specifications are d tr = 3.5 am, d trox = 0.38 urn and d poly = 0.75 urn: a) The finite-element model simulated with ANSYS; b) the variation between the thermal resistance accounting for the isolation and the thermal resistance assuming a homogeneous substrate, plotted as a function of the edge-to-edge spacing between the emitter and the Ugroove isolation.

PAGE 68

59 However, these results represent the worst case, since in an actual device the active region is not immediately flanked by the U-groove on all sides. Transient thermal simulations of the FE models shown in Figure 2.9a and Figure 2.10a were used to examine the effects of both LOCOS and U-groove isolation on the transient thermal response. ANSYS was used to simulate the device with and without the isolation structures; the results are shown in Figure 2. 1 1 . As the heat travels laterally and reaches the edges of the isolation structure, the response accounting for the isolation begins to deviate from the response without the isolation. The time for the heat to reach the edges of the isolation structures can be approximated as the square of the device-oxide separation (1 Jim) divided by the thermal diffusivity of the substrate material; the resulting time is on the order often nanoseconds, which agrees with the simulations of both the LOCOS and U-groove isolation. The oxide used in the isolation structures restricts the lateral flow of heat away from the device and in both cases results in a larger temperature rise. Based on the results of the 3-D FE simulations, advanced isolation structures such as recessed LOCOS and U-groove can considerably increase the thermal impedance of bipolar devices. In most cases, the bulk bipolar model will tend to under-predict both the steady-state and transient thermal response of devices fabricated with oxide-based isolation structures. The error in the model will be the greatest for advanced, highly-scaled devices fabricated with deep trench isolation.

PAGE 69

60 2.0 1.5 U a 8 i.o c T3 U E 2 0.5 c3 E t> u H © — ©Without Isolation Q — QWith Isolation 0.04 10" 1 10 -4 10" £ u 3 o c 3 O 2 ~ 1 rt ©—©Without Isolation Q — QWith Isolation ?*r 10 -B10" 10" Figure 2. 11 ANSYS simulations showing the effect of advanced isolation structures on the transient thermal impedance. The device-isolation spacing is one micron. The specifications for the device structure are L = 2 |im, W = 1 (im, D = 0.35 urn and H = 0.35 |im: a) Recessed LOCOS or BOX isolation with d fox = 1.0 (im; b) U-groove isolation with d tr = 3.5 u\m, d lrox = 0.38 |im and d poly = 0.75 |im.

PAGE 70

61 2.3 Verification of the Single-Emitter Thermal Impedance Model To verify the thermal impedance model, three-dimensional (3-D) finiteelement (FE) simulations of a junction-isolated BJT were performed using ANSYS. Interconnect metallization was neglected and the substrate was assumed to be homogeneous silicon with the bulk properties given in Table 2. 1 . The FE simulations were evaluated at the surface corner of the emitter and compared to the single-emitter thermal impedance model given by equation (2.21); the results are shown in Figure 2.12. The analytic model agrees closely, for both the steady-state and the transient, with the 3-D FE simulations for both device geometries. The predicted values for the steady-state thermal resistance agree within twelve percent of the FE simulations. The error can be partially attributed to numerical error associated with the FE mesh. The model was also compared to measured thermal impedances. The measured data were extracted using the base-thermometry technique developed by Zweidinger et al. [Zwe96]. Figure 2.13, Figure 2.14 and Figure 2.15 compare the measured and simulated data for the transient thermal impedance of Harris HBC bulk BJTs. The thermal impedance model does a good job of predicting the steady-state thermal resistance, with no more than a 20% error between the model and the measurements. The model given by (2.13) tends to overpredict the transient response. As shown with the ANSYS simulations in Figure 2.7, this discrepancy can be attributed to the model's neglect of the emitter metallization. When the thermal impedance of the emitter interconnect is accounted for, with 8 met = 50 |im extracted

PAGE 71

62 TEMPERATURE (°C): C3 KXXXXXXXJt WTTTTTT ESI |:x;:x::x::x:l k ;;;; ;;;w;.';;,;l 27 27,431 27.863 28,294 28.725 29,157 29,588 30,019 30.451 30,882 (a) 1150 P 950 ^ 750 -

PAGE 72

63 U u c -a C3 180 130 80 30 -20 Measurement 0----0 Model: Without Emitter Int -•-Model: With Emitter Int. 10 10 -.^@::~ 10" .••-£!}' 10" u X c -o o n 60 50 40 30 20 10 -10 Measurement ©-•-OModel: Without Emitter Int. ---Model: With Emitter Int. 10 -10 10" Figure 2.13 Measured and simulated data for the transient thermal impedance of Harris HBC bulk BJT's with W = 2 ^m: a) L = 30 urn; b) L = 100 u.m.

PAGE 73

64 U 300 220 g 140 03 U s 03 U 60 -20 10 10 & \ ,::::::::& — :: = : = © = Measurement iL 0----OModel: Without Emitter Int. -Model: With Emitter Int. io10" b Time (sec) (a) 10 -4 io140 u o o c 03
PAGE 74

65 260 220 Measurement ©---OModel: Without Emitter Int. ---Model: With Emitter Int. 8 :::: = © = :@::::

PAGE 75

66 from the measurements, the model provides a more accurate representation of the transient thermal response. 2.4 Derivation of the Multiple-Emitter BJT/HBT Thermal Impedance Model The thermal impedance model for bulk MEBJT/MEHBT's is an extension of the single-emitter model. A multiple-emitter device consists of single-emitter devices placed adjacent to each other along their lengths. Since there are multiple devices (referred to as "emitter fingers") that are thermally coupled through the substrate operating in close proximity, the temperature rise in each emitter finger is affected not only by its own power dissipation, but also by the power dissipated by its neighbors. The heat conduction equation, (2.1), is linear, so superposition can be used to calculate the total temperature rise in the device. The equation for the temperature rise can then be manipulated to provide expressions for the effective temperature rise in each individual finger. Figure 2.16 illustrates the simplified multiple-emitter device geometry assumed for the model derivation. The substrate is represented by a homogeneous semi-infinite half-space with an adiabatic top surface with multiple imbedded heat sources. The emitter fingers are assumed to be uniform in size and shape, with width W and length L. Each finger has a corresponding heat source, due to an assumed uniform power generation in the base/collector SCR; each heat source has a thickness H and is displaced a distance D below the surface of the substrate. As with the single-emitter model, D is assumed to equal the depth of the base/collector junction

PAGE 76

67 Figure 2.16 The simplified device geometry used to define the solution domain for the bulk, multiple-emitter BJT/HBT thermal impedance model. The substrate is represented by a semi-infinite half-plane with an adiabatic surface. Each emitter finger has a width W and length L and each heat source (the rectangular volumes) is displaced a distance D below the surface of the device and has a thickness H. The distance D is equivalent to the depth of the base/collector junction and the thickness H is approximated by the thickness of the base/collector SCR. The emitter fingers are uniformly spaced with and edge-to-edge separation distance S .

PAGE 77

68 and H can be calculated using equations (2.14) and (2.15). The edge-to-edge separation, S, between adjacent fingers is assumed to be uniform. The Green's function technique can be employed to find the temperature rise within the device. By applying superposition, the solution is expressed as the sum of the Green's function solutions for the multiple heat sources n t AT(x,y,z,t) = X? j dt'JG(x,y,z,t|x',y',z',t')g(x',y',z',t')dv'. (2.29) 1 .' = v where G(x, y, z, t|x', y', z', t) is given by equation (2.8). The summation accounts for the integration over the different spatial coordinates of each heat source. To clarify the derivation, certain conventions and definitions can be established. The origin for the coordinate system is fixed at the center of the left-most finger at the surface of the device. A device is considered to have a total of n emitter fingers and a reference order is established with the fingers numbered sequentially starting from the leftmost finger. The character j , where j = 1 — > n , is used to reference a specific emitter finger. The i-th neighbor (where i = 1 — »n1) of a given emitter finger, EFj , is defined as a finger situated an edge-to-edge distance [iS + (i 1)W] away on either side. Using equations (2.8) and (2.29), the temperature rise at any point in the device-assuming a step increase in power at t' = for each finger— is given by

PAGE 78

69 AT(x, y, z, t) = 'L/2 x ^ r *jdt r JL/2 + x\ /L/2 . r crf ry-(2j-3)w/2-(j-i)s + erf 74at -y + (2j-l)W/2 + (j-l)S 74at ,,'z + D + H^ /-D-z erf| == — |+ erf V4at + erf I Z , )+erf V4atV 74at D + H-z 74at (2.30) which accounts for n heat sources, one for each emitter finger EFj . Equation (2.30) can be manipulated to provide the temperature rise in each emitter finger. As with the single-emitter model, the temperature rise in each finger is represented by a single effective value. To simplify the derivation, symmetry is assumed such that the distance from the effective-temperature point of finger EFj to the heat source of its i -th neighbor, is the same as the distance from the effectivetemperature point of the i-th neighbor to the heat source of EFj . This symmetry is attained only for the coordinates (x, y = [j 1 ][S + L], z). When (2.30) is evaluated at each of these points, the model is reciprocal and the effective temperature rise in each emitter finger, AT EF : , can be expressed as

PAGE 79

70 AT AT EFl EF2 AT EFn J C1 J C1 ... z ... z _ Z C(n-l) Z C(n-2) C(n-l)

PAGE 80

71 Z Ci (x, z, t) ^ J8pcVL ,, L/2 + x^| JL/2-x erf — ==^ +erf j = it 74at [ erf l f(W/2 + i(W + S)) 74at 74at i + y(w/2-i(w + s)) ^/4at n 'i±D±HVerff^z V4at / V V4at D + H-z + erf[ Z , l+erf 74at 74at (2.33) and accounts for the portion of the temperature rise in a finger due to the power dissipated by its i-th neighbor. Thus, for the assumed symmetry, the thermal impedance of a device with n emitter fingers can be described with a single self impedance and (n1) coupling impedances. Equations (2.32) and (2.33) should be evaluated at a single point to represent the temperature rise of each finger by a single effective value. To keep the MEBJT/MEHBT model as similar as possible to the single-emitter model, the points x = L/2 and z = are used, giving z s (t) = J 1 :erf D + H r_j^W— y erfl 2pcV \j4atJ V4atA ^ J4a erf D 74at dt (2.34) and Z Ci (t) = MpcVL 4pcVL £r I74at [ erf f(W/2 + i(W + S))V erf [ (W/2-i(W + S)) 74at erf[ D±H)_er/ D V4at (2.35)

PAGE 81

72 as the final expressions for the self and coupling impedances. Therefore, the overall self-heating on the scale of the entire device can be described by the self-heating and thermal interactions on the smaller scale of each individual emitter finger. Accounting for a finite wafer thickness can be important for multipleemitter devices since the coupling impedances decrease as the wafer thickness is reduced [Daw94], [Hat95]. The multiple-emitter model can be modified in a similar fashion as that for the single-emitter model; by simply using equations (2.17) and (2.20) in place of the G z (z, t|z', t') in equation (2.29), the expressions for Z s and Z Cj will now account for a finite wafer thickness. Since the multiple-emitter thermal impedance model is simply an extension of the single-emitter model, the effects of interconnect metallization and advanced isolation technologies are not taken into account. Neglecting these portions of the overall device structure is assumed to affect the multiple-emitter model in the same manner, and to the same extent, as to which it affects the single-emitter model. 2.5 Verification of the Multiple-Emitter Thermal Impedance Model Two-dimensional (2-D) finite-element (FE) simulations of a junctionisolated, three-finger BJT were performed using ANSYS to verify the multipleemitter thermal impedance model. Two-dimensional FE simulations were used instead of 3-D simulations due to limitations of the available version of ANSYS. The validity of comparing 2-D FE simulations to a 3-D analytic model is established by evaluating the thermal impedance model derived for both two and three dimensions.

PAGE 82

73 Figure 2.17a compares the results, which show that the 3-D the 2-D models converge for long devices. The difference in the predicted thermal resistance values decreases from 22% to less than 1% as the length of the device is increased from 50 |im to 800 (im. Consequently, the 2-D FE simulations can verify the 3-D thermal impedance model evaluated for devices with long emitters. The 2-D FE simulations do not verify the model for shorter devices where the heat flow becomes threedimensional. However, the verification of the single-emitter thermal impedance model for 3-D heat flow can be assumed to also verify the MEBJT model. This assumption is reasonable since the physics that describe the single-emitter model also apply to the MEBJT thermal impedance model. The physical device was assumed to be symmetric so that the FE model represented only half of the device. Figure 2.17b shows an illustration of the FE model simulated with ANSYS. Interconnect metallization was neglected and the substrate was assumed to be homogeneous silicon with the bulk properties given in Table 2.1. The bottom and exterior side of the substrate were held at a constant ambient temperature while the top surface and the interior side were assumed to be adiabatic. The FE simulations were compared to the multiple-emitter thermal impedance model given by equations (2.34) and (2.35), accounting for a finite substrate thickness; the results are shown in Figure 2.18. The analytic model agrees well, in both the steady-state and transient, with the 3-D FE simulations of the self and coupling impedances. The predicted values for the steady-state thermal resistance agree within three percent of the FE results, which is within the expected error of the model and the numerical simulations.

PAGE 83

74 U o 250 200 x O -a Oh rt o 50 100 50 O — ©2D Model: L = 50 am O--03D Model: L = 50 urn a — a 2D Model: L = 200 fxm Qa 3D Model: L = 200 Jim A — A2D Model: L = 800 |Llm A-A3D Model: L = 800 ^m O 10 12 -W £Ji 10 -10 -fi , .-O --';-.g;--A .-ET' e — e — e — ejO — €>--o--o--o,a* — At-a^g--S ---&--B---B -J .£ S .-^ * r --.2!^ t --.A--^!v^10" 10" 6 Time (sec) (a) 10" 10" 10 ( TEMPERATURE (°C): 27 29,072 31,145 33.217 35.289 37,361 39,434 41,506 43.578 45,651 (b) Figure 2.17 Two-dimensional heat flow in multiple-emitter bipolar transistors: a) A comparison of the three-dimensional thermal impedance model to a two-dimensional model for W = 1 |im, D = 0.5 \xm and H = 0.5 |im; b) the finite-element model simulated with ANSYS for P = 500 uW.

PAGE 84

75 15 U o o c -a E 5 "c5 1 1 OZ s : 2D ANSYS Z C1 : 2D ANSYS AZ S +Z C2 : 2D ANSYS Z s : Model

PAGE 85

76 2.6 Summary A thermal impedance model for bulk single-emitter BJT/HBT's was presented and then extended for devices with multiple emitter fingers. The model was shown to agree reasonably well with three-dimensional finite-element simulations and measurements of junction-isolated devices. The effects of interconnect metallization and advanced isolation technologies on the thermal impedance were investigated; a simple model for the thermal impedance of the emitter interconnect was demonstrated. The results suggest that the model can be expected to provide reasonable predictions for the thermal impedance of junctionisolated devices. However, for highly-scaled devices, the effects of advanced isolation can be significant and the accuracy of the model will decline. Methods for modeling the effects of isolation structures are proposed in Chapter Eight.

PAGE 86

CHAPTER 3 A CIRCUIT MODEL FOR THERMAL COUPLING AND A LUMPED ELECTROTHERMAL MODEL FOR BULK MULTIPLE-EMITTER BIPOLAR TRANSISTORS 3.1 Introduction Due to increased interest in the role of thermal effects in device and circuit operation, especially for silicon-on-insulator (SOI) and heterojunction technologies, circuit simulators and compact device models have been modified to account for the dynamic temperature response within a device [McA92, Fox93b, Fos95]. Most of the implementations have been applied to the case of self-heating, where a device's effective operating temperature (EOT) depends on its power dissipation only. In many circuits and some devices, such as multiple-emitter bipolar transistors, a number of devices can operate in close proximity. Under such conditions, the EOT of a device is no longer determined solely by its own power dissipation but also depends on the operation of its neighbors. Therefore, not only must circuit simulators (and compact device models) be able to model dynamic self-heating, they must also be able to model the dynamic thermal coupling between individual devices or portions of one device. An approach for modeling cross-chip thermal coupling using a circuit simulator was described by Fukahori and Gray [Fuk76]. The thermal coupling between devices in an arbitrary circuit was modeled using a finite difference 77

PAGE 87

78 technique. The semiconductor substrate was represented by a three-dimensional numerical mesh with equivalent thermal resistances and capacitances. The electrical elements in the circuit (transistors, etc.) were represented by their standard compact circuit models. The values for the lumped thermal components were calculated by discretizing the heat conduction equation using a finite difference approximation. In [Mar93], a similar approach was presented and applied to the simulation of multipleemitter HBT's. In this case, two-port theory was used to generate a finite two-dimensional resistance network that represented steady-state heat conduction in the substrate. For both applications, the resulting circuit admittance matrix contained elements corresponding to the electrical circuit and also the thermal elements. The Newton-Raphson-like iteration scheme of the modified circuit simulators was then used to solve the coupled electrothermal problem. To simulate both inter-device thermal coupling and self-heating using this method, a large number of thermal nodes is required; therefore, this approach can drastically increase simulation time. Chapter One described a common method for using thermal impedances to efficiently model self-heating in circuit simulators. A logical progression would be to expand this method to account for thermal coupling between devices. Such an approach can provide a more efficient alternative to the semi-numerical methods mentioned above. This technique was applied by Moinian et al. for modeling crosssubstrate thermal coupling in bipolar circuits [Moi94], and by Baureis for modeling multiple-emitter HBT's [Bau94]. However, their circuit implementations did not correctly represent the thermal interactions between (or within) the devices. The shortcoming of the coupling model used in these works is discussed in this chapter.

PAGE 88

79 A circuit model is then presented which correctly models thermal coupling and is compatible with the self-heating circuit model described in Chapter One. Once a valid circuit model for thermal coupling has been developed, it can be used with the multiple-emitter thermal impedance model to perform both steadystate and dynamic electrothermal simulations of multiple-emitter BJT/HBT's. The multiple-emitter thermal impedance model expresses the self-heating of an entire device as the sum of the thermal actions and interactions of the individual emitter fingers. The thermal model structure requires that a single multiple-emitter device be represented by multiple compact device models. While this configuration allows for the examination of the EOT of each finger in a device, for devices with a large number of emitter fingers, the overall electrothermal network can become complex enough to make moderateto large-scale circuit simulations impractical. To make the multiple-emitter electrothermal model more suitable for circuit simulation, its complexity can be reduced by representing the overall thermal response of the device by a lumped thermal impedance. The lumped thermal impedance is generated by applying the measurement approach developed by Zweidinger et al. to the simulation of the complete electrothermal model [Zwe96]. The thermal impedance extraction technique is briefly reviewed in this chapter. The lumped model generation procedure is then described and the results are compared to the complete electrothermal model.

PAGE 89

80 3.2 A Circuit Model for Thermal Coupling The EOT of any device in a system of n thermally coupled devices (e.g. a multiple-emitter bipolar transistor with n emitter fingers) can be expressed as T DEV1 (t) = [AT 1 (t)+AT 12 (t) + --+AT ln (t)]+T amb T DEV2 (t) = [AT 21 (t) + AT 2 (t) + ---+AT 2n (t)]+T amb T D EVn(t) = [AT nI (t) + AT n2 (t) + ---+AT n (t)]+T amb (3.1) where ATj(t) = £' ] [Z Sj • Pj(s)] and AT^t) = £ _1 [Z Cji • Pj(s)] . The impedance Z Sj is the self impedance of device j , and Z Cji represents the coupling impedance between device j and device i . (In general, it is not necessary for Z Cji to equal Z cj : .) The modifications described in Chapter One allow circuit simulators to model selfheating; therefore, the EOT of each device in a simulation is calculated by T DEVj (t) = ATjW+T^ (3.2) and is independent of its neighbors. The obvious way to expand this technique to account for inter-device thermal coupling would be to simply tie together the temperature nodes of individual devices using coupling impedances; this approach was used by Baureis and Moinian et al. [Bau94, Moi94]. Figure 3.1 shows an example of such a thermal coupling network for two devices, where Z c = Z C12 = Z C21 . Unfortunately, simple analysis of the circuit in Figure 3.1 shows that it does not correctly model the expression in (3.1). For example, analyzing

PAGE 90

T DEVl(t) Pl(t) ambl T DEV2(t) P 2 (t) Figure 3.1 A thermal coupling circuit model for two devices. The temperature nodes of the two thermally coupled devices are connected using a thermal coupling impedance.

PAGE 91

82 the circuit in the steady-state limit gives the following expression for the EOT of device 1 T M L *> l p + M a p +T ex -x\ 1deV! " (R c + R sl + R S2 ) Hl+ (R c + R SI+ R S2 )^ 2+lamb (3 ' J) A similar expression can be derived for the EOT of device 2. The problem with this network formulation is that when individual temperature nodes are connected through an impedance path, the entire network becomes distributed among the coupled devices. The self impedances and coupling impedances, as derived, are not defined to be distributed elements. In a more simplistic view, the network in Figure 3.1 does not properly constrain the paths of the respective device powercurrents. The power-current of a given device is divided between its own self impedance and the rest of the network. The portion of that device's power flowing through its neighbor's self impedance has no physical meaning. As a result, the voltages generated at the temperature nodes do not correspond to the correct device temperatures. To develop a correct circuit representation of (3.1) and avoid the shortcomings of the aforementioned coupling technique, control sources can be utilized in a thermal coupling network group composed of two sub-networks. Each device in a group of n thermally coupled devices has its own network group. Figure 3.2 demonstrates how a thermal coupling network group works. Sub-network A attaches directly to the temperature node of device 1. The current-controlled current source (Fj ) in sub-network B has unity gain and is controlled by the current

PAGE 92

TT E 12 (t) Ei 3 (t) sub-network A Em(t)

PAGE 93

84 flowing through the voltage source V, in sub-network A. In this example, V , is also used to set the reference ambient temperature. The voltage drop across each coupling impedance (Z Cil ) in sub-network B corresponds to the portion of the temperature rise in each device i due to the power dissipation of device 1. The voltage-controlled voltage sources in sub-network A each have unity gain and are used to couple the voltage drops from each sub-network B of the other devices, back to device 1. For example, the value of the voltage source E 12 is equivalently E 12 = AT 12 (t), where Z C12 is part of sub-network B of device 2. Therefore, the voltage generated at the terminal of sub-network A corresponds to the EOT of device 1, and is given by the following expression T DEV1 (t) = AT 1 (t) + AT 12 (t) + "-+AT ln (t) + T amb . (3.4) Similar expressions can be obtained for the EOT's of the other (n 1 ) devices in the circuit since they each have similar thermal networks. The thermal coupling model is demonstrated by simulating a five-finger HBT using a version of SPICE 2G.6 modified to model self-heating [Zwe97]. The device characteristics are simulated with and without the thermal coupling between emitter fingers. Figure 3.3 shows the results of the electrothermal simulations. When accounting for the thermal coupling, the current collapse phenomenon commonly observed in HBT's can be simulated [Liu93, Sei93, Liu95b]. Figure 3.3b illustrates how the outer fingers shut down as the middle finger begins to carry all of the current. The importance of modeling the thermal coupling is established by the fact that the collapse phenomenon is not reproduced when the simulations only account for self-

PAGE 94

85 4 6 Collector-Emitter Voltage, V CE (V) (a) 40 30 c
PAGE 95

86 heating. Multiple-emitter devices provide just one application for the thermal coupling model. It can be used on a larger scale for simulating thermal interactions within circuits, and it can be used on a smaller scale. By dividing a single device (or each finger of a device) into multiple sub-cells, the thermal coupling model could be used to simulate the temperature distribution within a device and phenomena such as current constriction [Koe94]. 3.3 A Lumped Electrothermal Model for Multiple-Emitter BJT/HBT's Used together with a compact device model of either a BJT or HBT, the multiple-emitter thermal impedance model and the thermal coupling network form a complete electrothermal model suitable for DC, AC and transient device/circuit simulation. This type of electrothermal model is generally more efficient for circuit simulation than either finite difference or finite element techniques; however, it can be quite complex for devices with a large number of fingers and/or fingers with a large number of sub-cells. In such a case, simulating moderateto large-size circuits could become impractical. The complexity of the electrothermal device model can be reduced by using a lumped modeling methodology. The measurement technique described by Zweidinger et al., referred to in this work as base-current thermometry, can extract the thermal impedance of a bipolar transistor using the temperature dependence of the base current [Zwe96]. By applying this extraction technique to the simulation of the complete electrothermal device model, a more compact lumped electrothermal can be produced. The lumped model implicitly contains all the details

PAGE 96

87 of the thermal actions and interactions described by the complete electrothermal model, but with less complexity. To present a clear discussion of the lumped electrothermal model generation methodology, a few definitions and conventions will be established. Due to the thermal interactions between fingers in a multiple-emitter device a lateral temperature gradient will exist across the device. Therefore, the current distribution among the fingers may not be uniform since the hotter fingers will carry a larger amount of current. As power dissipation increases, the lateral temperature gradient also increases and eventually the device will become unstable and enter either thermal runaway (BJT's) or current collapse (HBT's). Prior to the onset of thermal instability, the lateral thermal gradient is small and the current distribution among the fingers is approximately uniform. When the device reaches the point of thermal instability, the fingers no longer operate under similar bias conditions and the current no longer divides evenly among the fingers. Therefore, before a device becomes thermally unstable, it is defined to be in the uniform operating regime; and, once the device becomes unstable, it is defined to be in the nonuniform operating regime. In the uniform operating regime, the EOT of the device varies linearly with the power and the complete electrothermal model can be represented by a single lumped device model and lumped thermal impedance, Z THL . The circuit representation for the uniform lumped model is shown in Figure 3.4a; the emitter area of the lumped device model is equal to the total emitter area of the multipleemitter device. As a device becomes thermally unstable the cooler fingers begin to turn off, leaving the hottest finger to conduct all of the current; the temperature-

PAGE 97

88 B Oco k Ed 'op 'THL V o l amb V (a) B 0oc V QlF k Qof k EO 10 p IF ami) V OI p OF ami) V J LOI J LIO V V (b) Figure 3.4 Circuit representations of the lumped multiple-emitter BJT/HBT electrothermal model: a) For the uniform operating regime; b) for the nonuniform operating regime.

PAGE 98

89 power relation becomes nonlinear and the uniform lumped model will not accurately model the device characteristics. To model this shut-down mechanism, the nonuniform lumped model, which is shown in Figure 3.4b, uses two lumped device models and four lumped thermal impedances to represent the entire device. Device Q IF is used to represent the hottest emitter finger. In a device with an odd number of fingers, the hottest finger will be the middle finger. If a device has an even number of emitter fingers, due to process variation, the hottest finger will be one of the inner most fingers. For consistency, in either case the hottest finger will be referred to as the middle finger. The other device model, Q OF , represents the remaining outer emitter fingers. The emitter area of Q IF is equal to that of a single emitter finger and the emitter area of Q OF is equivalent to the sum of the emitter areas of the outer fingers. The lumped thermal impedances Z SI and Z s0 model the self impedances of the middle finger and the outer fingers, respectively. In the case of Z so , the impedance represents the effective temperature rise in the lumped outer fingers due only to their power dissipation. The lumped coupling impedance Z CIO models the temperature rise in the middle finger due to the power dissipation in the lumped outer fingers. The reciprocal coupling impedance Z COI , corresponds to the effective temperature rise in the lumped outer fingers due to the power generation of the middle finger. Thermally triggered instability in bipolar devices can lead to circuit failure and even catastrophic device failure. Typically, this region of operation is avoided in circuit design. Therefore, in most cases, the uniform lumped model should be appropriate for most applications. However, if the effects of thermal instability on

PAGE 99

90 device/circuit operation need to be investigated the nonuniform model should be used. 3.3.1 A Review of Base-Current Thermometry Base-current thermometry uses the base current as a thermometer to extract the thermal impedance of a bipolar transistor [Zwe96]. The technique was developed for measurement-based extraction but can be applied to the simulation of compact device models as long as the models' temperature dependences are physically valid. The first step of the procedure is to determine the dependence of the base current on temperature. The response of the base current to changes in temperature is represented by the fractional temperature coefficient, defined as 1 dI B T c F a B )^af(3-5) By biasing a device in the common-emitter configuration (avoiding impactionization), and separately varying the collector voltage and the ambient temperature, the thermal resistance of the device can be extracted Since the basecollector conductance is typically negligible, any changes in the base current during the measurements are due solely to the change in operating temperature. Therefore, once the self-heating effects are accounted for, the fractional temperature coefficient can be determined from the measured base current variations.

PAGE 100

91 The second step of the procedure is to extract the transient thermal impedance. The collector and base currents of the device are monitored for a step in the collector voltage. The transient change in temperature can be expressed as I R (t)-I R (0) where I B is the median value of the base current for the transient. The temperature change is then normalized by the magnitude of the power step, giving the following equation for the thermal impedance Z TH (t) = ^T(3-7) 3.3.2 Generation of the Lumped Electrothermal Model The first step in the lumped model generation is to extract the temperature coefficient of the base current by performing DC SPICE simulations at different ambient temperatures. The .TEMP control card is used to set the ambient temperature; temperature steps between 4 and 10 degrees are sufficient, where a geometric mean can be used to average TC F (I R ) over temperature to correct for nonlinearities. The base voltage should be selected for the desired operating point and for each temperature setting, the collector voltage should be swept over a range in the forward active region. The range of collector voltages should be large enough to produce a linear increase in base current. Examples of the resulting base current characteristics are shown in Figure 3.5. The base and collector current values should

PAGE 101

92 350 300 < 5 250' c g rj 200 u a PQ 150 100 C38.0 -a bQ— ©T amb = 27 °C B—eT amb = 3i°c A — AT amb = 33 °C ^"7T amb = 35 °C ^ V V ^ -A A A A-jSsl ~~A-B B-a b-e8.2 8.4 8.6 Collector-Emitter Voltage, V CE (V) (a) 8.8 -B9.0 134.4 8.2 8.4 8.6 Collector-Emitter Voltage, V CE (V) (b) 8.8 9.0 Figure 3.5 Simulated base current versus collector-emitter voltage: a) For varying ambient temperatures; b) for T amb = 27 °C.

PAGE 102

93 be stored at each bias point and used to calculate TC F (I B ) . For the uniform model, this process should be performed once, using the net base and collector currents of the complete device. For the nonuniform model, the procedure should also performed just once. The temperature rise in the middle finger, corresponding to either Z SI or Z C10 , can be taken directly from the multiple-emitter thermal impedance model, so an extraction is not necessary. Therefore, only the temperature coefficient of the lumped outer fingers should be needed. The coupling impedances from the middle finger to each of the outer fingers should be turned off and the net currents of only the lumped outer fingers are used to calculate TC F (I B ) . The second step of the procedure is to generate the transient thermal response. The transistor should be set in the common-emitter configuration with the base voltage set to the value used to extract TC F (I B ). The collector voltage should be stepped between two bias points in the forward-active region. The combination of the selected base voltage and collector voltage step-size should set the current level such that a significant base current response results from the step in power. The risetime of the voltage step should be faster than the shortest expected thermal time constant and the step length should be long enough to allow the current response to reach steady-state. The base current should be recorded during the transient as well as the collector current values at the start and end of the step. Figure 3.6a shows an example of the transient base current response. Using (3.6) and the TC F (I B ) from the first step, the transient base current response can be converted into the transient temperature response. The magnitude of the power step can then used to normalize the temperature response, resulting in the transient thermal impedance; an example

PAGE 103

94

PAGE 104

95 is shown in Figure 3.6b. For the uniform lumped model, this procedure should be performed once, using the net transient base current response to calculate the lumped thermal impedance, Z THL . For the nonuniform lumped model, the procedure should only be performed for the extraction of Z so and Z COI from the net base current of the lumped outer fingers. When extracting Z so , the coupling impedances from the middle finger to the outer fingers should be turned off. The calculated temperature response should be normalized by the power dissipated in the lumped outer fingers. When extracting Z COI , the self impedances and the coupling impedances between each of the outer fingers should be turned off. The calculated temperature response should be normalized by the power dissipated in the middle finger. The self impedance Z SI in the lumped model corresponds to the self-impedance of a single finger, which can be calculated directly using the multiple-emitter thermal impedance model. The lumped coupling impedance Z CIO can also be calculated directly from the multiple-emitter thermal impedance model; for a device with an odd number of fingers (n-l)/2 i = 1 and for a device with an even number of fingers Z cio (n _ 1} (n-2)/2 Zc(n/2) + 2 JL ^Ci i = 1 (3.9) where n is the total number of emitter fingers in the device.

PAGE 105

96 3.4 Verification of the Lumped Electrothermal Model To test the accuracy of the lumped electrothermal models, they are compared to the full electrothermal models using DC, AC and transient SPICE simulations. The following simulations of homojunction bipolar transistors use the QBBJT model in a modified version of MMSPICE [Jeo89, Lee96]. The HBT simulations were performed using a modified version of SPICE 2G.6 [Zwe97]. Figure 3.7 shows the simulated DC current characteristics for a ten-finger BJT. The device remains in the uniform operation regime for the simulated biases, and the lumped models produce nearly identical results. The discrepancies between the lumped and full electrothermal models are the greatest when the device is biased with a constant base current, in which case the error is less than 3%. When the device is biased with a fixed base voltage, the error between the lumped models and the full electrothermal models is no greater than 0.5%. Figure 3.8 shows the simulated DC current characteristics for a five-finger HBT. When the device is biased with a fixed base voltage, it remains in the uniform operation regime and the results are similar to those for the homojunction bipolar device. Both lumped models produce similar results and match the full electrothermal model to within 3% error. However, when the device is biased with a constant base current, it becomes thermally unstable and goes into current collapse. Under these operating conditions, the outer fingers shut down, leaving the middle finger to conduct all the current. The uniform lumped model does not show the collapse phenomenon which results in a 17% error at the highest bias point. The nonuniform lumped model, however, does a good job of representing the full electrothermal model, producing no more than 2% error.

PAGE 106

97 10 9 < 8 5 7 u 3 6 c t 5 3 u o o u o U 4 3 2 1 OFull Electrothermal Model • Uniform Lumped Model Nonuniform Lumped Model -e e e e e e e e e2 3 Collector-Emitter Voltage, V CE (V) (a) 2 3 Collector-Emitter Voltage, V CE (V) (b) Figure 3.7 Simulated DC current characteristics for a ten-finger BJT with A E = 20 x 1.6 |im 2 for each emitter finger: a) The collector current as a function of collector emitter voltage for V BE = 0.75, 0.775 and 0.80 V; b) the collector current as a function of collector emitter voltage for I B = 10, 20 and 30 |iA.

PAGE 107

98 2.5 2.0 u

PAGE 108

99 The small-signal and transient performance of the lumped models were tested by simulating a cascode amplifier composed of two five-finger BJT's. The results of the simulations are shown in Figure 3.9. The lumped models produce almost identical results (the curves appear on top of each other) and agree closely with the full electrothermal model. The error in the small-signal gain is less than 1% and is due to errors in the DC operating point. Errors in the magnitude of the transient output waveform are less than 1%. For both the small-signal and transient simulations, the lumped models correctly reflect the frequencyand time-domain responses. The lumped models match the unity-gain frequency and phase response to within 4% of the full electrothermal model. The benefit of the lumped models is that they have fewer components and less complexity than the full electrothermal model. The reduction in complexity results in models that are more efficient to simulate, at the cost of some accuracy. One portion of the increased simulation time of the full electrothermal model is due directly to the added components used for the thermal model. The other part is due to the increase in the number of iterations required to reach convergence in the presence of thermal feedback. Thus, the nonuniform lumped model does not offer as much of a speed enhancement over the full electrothermal model as the uniform lumped model. The total job times for simulations shown in this section (performed on a SPARCstation 5) are given in Table 3.1.

PAGE 109

100 200 c >" w 3 O > 150 o o o o o o c "3 O u bo a -4-» "o > I 100 50 10 L 10.6 3.6 2.6 o o o OFull Electrothermal Model — Uniform Lumped Model — Nonuniform Lumped Model 10^ 10 4 10° Frequency (Hz) (a) 10 10 12 OFull Electrothermal Model Uniform Lumped Model — Nonuniform Lumped Model 100 200 Time (nanosec) (b) 300 Figure 3.9 Small-signal and transient simulations of a cascode amplifier using five-finger BJT's with A E = 20 x 1.6 uim 2 for each emitter finger: a) Small-signal voltage gain; b) the output voltage resulting from a 40 mV peak-to-peak sinewave input at 10 MHz.

PAGE 110

101 Table 3.1 Total simulation time

PAGE 111

102 multiple-emitter device model was reduced by extracting a lumped thermal response. In most cases, a multiple-emitter device can be represented by a single compact device model and a lumped thermal impedance. The lumped model offers a significant speed increase over the full electrothermal model, resulting in more efficient circuit simulations.

PAGE 112

CHAPTER 4 A THREE-DIMENSIONAL THERMAL IMPEDANCE MODEL FOR VERTICAL BIPOLAR TRANSISTORS FABRICATED WITH FULL DIELECTRIC ISOLATION 4.1 Introduction Bipolar junction transistors fabricated using full dielectric isolation (DIBJT's) offer many advantages over those fabricated with junction isolation in bulk wafer technologies. The parasitic capacitances and leakage currents from the collector to the substrate and from the collector to the junction-isolation implant are reduced with full dielectric isolation, which enhances device speed [Dav92]. The area taken up by the lateral dielectric isolation is typically smaller than the area of the diffused junction-isolation implant; therefore, full dielectric isolation is a means to increase the transistor packing density [Jer93]. Finally, full dielectric isolation also negates latchup, and improves radiation hardness [Gan92]. Full dielectric isolation of vertical BJT's can be achieved by using siliconon-insulator (SOI) substrates with trench isolation. Figure 4.1 shows a diagram of a typical npn DIBJT. Direct wafer bonding (DWB) has become a common way to fabricate SOI substrates for bipolar technologies since it produces high-quality, defect-free SOI films [Nis91, Dav92, Fei92, Jer93, Nak95]. The DWB technique thermally bonds one semiconductor wafer to the oxidized surface of another wafer. One wafer serves as the substrate and the other wafer is used for the actual device 103

PAGE 113

104 Base Collector O Buried Oxide Substrate Figure 4.1 Cross-section of a typical bipolar transistor fabricated with full dielectric isolation (DIBJT).

PAGE 114

105 fabrication. The oxide layer interposed between the two wafers becomes the buried isolation layer and is typically 0.4 to 2.0 (im thick [Dav92, Fei92, Nak95]. The device-fabrication wafer is thinned using chemical/mechanical polishing (CMP). One disadvantage of DWB is that the resulting silicon film can have large variation in thickness [Nis91]. Once the SOI wafer is prepared, the BJT fabrication process can follow the typical steps used for bulk, trench-isolated devices. The back-fill for the trench isolation is usually formed with chemical vapor deposition (CVD) oxide (0.1 to 1 .0 urn thick) or a combination of CVD oxide and polysilicon (0.5 to 2.0 p:m thick) [Nis91, Fei92, Nak95]. The device region enclosed by the trench isolation is referred to as the "tub," and the region surrounding the trenches is referred to as the "exterior silicon." Typical tub thicknesses range from 1.5 to 10 Jim [Nis91, Fei92, Nak95]. A major disadvantage of full dielectric isolation is an increase in selfheating. The oxide used in the trench and buried isolation has a low thermal conductivity and impedes the flow of heat away from the device, resulting in higher operating temperatures. The thermal resistance of a DIBJT can be three times larger than that of its bulk counterpart [Gan92]. Previous methods for determining the thermal resistance of DIBJT's have relied on measurement-based extraction or complex numerical techniques such as finite-element solutions [Nis91, Gan92]. Both of these approaches have been limited to steady-state operation and do not provide insight into the dynamic variation of temperature. Furthermore, neither approach is practical for use in circuit simulation: temperature measurements are complicated and are not predictive; finite-element solutions are predictive but can require large amounts of computing time.

PAGE 115

106 This chapter details the derivation of a physics-based model for the dynamic thermal impedance of DIBJT's operating in the forward active region. The effects of interconnect metallization on the thermal impedance are then investigated. The model is compared to three-dimensional finite-element simulations and measurements for verification. The model derivation is then simplified for the limiting case of steady-state heat conduction, resulting in a single closed-form equation for the thermal resistance of DIBJT's. The limitations of the thermal resistance model are shown with comparisons to measurements and to the full thermal impedance model. 4.2 Derivation of the DIBJT Thermal Impedance Model For the derivation of the DIBJT thermal impedance model, the silicon tub is represented by a homogeneous finite medium with an adiabatic top surface (no heat transfer perpendicular to the surface). The interface between the buried oxide and the substrate is assumed to be at a uniform temperature, T . Figure 4.2 shows a threedimensional cross-section of the simplified device geometry assumed for the model derivation. The definitions of the model parameters are given in Table 4.1. The imbedded heat source represents the base/collector space-charge region (SCR), and is modeled by a rectangular volume. The thickness of the base/collector SCR can be estimated using the depletion approximation as given in Chapter Two. The heat generated in this region is assumed to be due to uniform power dissipation. As verified for the bulk BJT thermal impedance model, this assumption is reasonable for

PAGE 116

107 'box V -IK 'trox 'poly Figure 4.2 Cross-section of the simplified device geometry used to define the solution domain for the DIBJT thermal impedance model. The silicon tub is represented by a homogeneous finite medium with an adiabatic top surface. The model parameters are defined in Table 4.1.

PAGE 117

108 Table 4.1 DIBJT thermal impedance model parameters Parameter

PAGE 118

109 the forward-active region before the onset of high-current effects. The heat source is displaced beneath the surface of the wafer by a distance assumed to be the depth of the base/collector junction. Any encroachment of the base/collector SCR into the base region is neglected since the base typically has a higher doping than the epi collector. The width and length of the silicon tub are assumed to scale directly with the width and length of the emitter stripe by the relations W tub = W + C W1+ C W2 (4.1) and L tu b = L + C L , (4.2) where C W1 , C w7 and C L are constants that depend on the fabrication process. Since the tub material is assumed to be homogeneous, the model neglects the effects of the LOCOS isolation that is used to cap the trench structure and to separate the base and emitter from the collector contact implant. This assumption is reasonable since the LOCOS is typically shallow semi-recessed LOCOS. The adiabatic boundary condition at the top surface of the device dictates that conduction through the interconnects and conduction/convection from the surface are neglected. This assumption is valid for the device regions that lie under thick silicon dioxide, since the thermal conductance from the device to the overlying oxide is approximately two to three orders of magnitude smaller than the device-to-substrate conductance. The effects of conduction via the interconnect metallization are examined later in the chapter.

PAGE 119

110 The temperature rise at any point within the tub can be described by the nonhomogeneous three-dimensional heat conduction equation V2 AT tub (x,y,z,t) + I^i) = ± ^,^0 k sl «si dt (4.3) and the following boundary conditions 3AT tub (x, y, z, t) -H tr (t)-AT tub (x,y,z,t) = x = (4.4) r ^AT tub (x, y, z, t) H tr (t)-AT tub (x,y,z,t)l = x = L,, (4.5) f aAT tub (x, y, z, t) dy H tr (t)AT tub (x,y,z,t)] = y = (4.6) p 3AT tub (x,y,z,t) 3y -H tr (t)AT tub (x, y, z, t) = y = w lub (4.7) aAT tub (x, y, z, t) 3z + H b0X (t)-AT tub (x,y,z,t) = z = (4.8) 3AT tub (x, y, z, t) = o z = d lub (4.9) where AT tub is the temperature rise above the local reference temperature (AT tub = T tub -T ) , g is the internal energy generation density, k is thermal

PAGE 120

Ill conductivity, a is thermal diffusivity (a = k/(p • c ) where p is density and c is specific heat) and t is time. Typical values for the material properties are given in Table 4.2. The terms H tr (t) and H box (t) are normalized heat-transfer coefficients that model the time-dependent heat flow through the trench and buried oxide, respectively. Equations (4.3) through (4.9) assume that the thermal conductivity is independent of temperature and position. For the tub material, the variation of k with temperature can be accounted for with the Kirchoff transformation detailed in Chapter Eight. However, the thermal impedance of a DIBJT is mainly determined by the isolation structures, and the thermal conductivity of silicon dioxide varies by less than 13% from 303 to 433 K [Goo95]. Also, as shown in Chapter Two, the thermal conductivity remains approximately constant over a wide impurity doping range, and any spatial dependence of the thermal conductivity can be neglected. With the initial temperature rise of the device specified as AT tub (x,y,z,0) = 0, (4.10) the solution to (4.3) can be expressed in the form t YT tub (x, y, z, t) = -p J dt'JG(x, y, z, t|x', y\ z, t')g(x', y', z', t')dv' (4. 1 1 ) SI f = o v where

PAGE 121

112 Table 4.2 Material Properties Property

PAGE 122

113 G(x, y, z, t|x', y', z, t') = I exp[-a si P^(t-t')]^i-^X(P m ,x)X(P m ,x') Iexp[-a s ^(t-t')]^Y( Yn) y)Y(y n ,y') S exp[-a si Ti p (t 1')]^— ^Z(Ti p> z)Z(Ti p , z') r> = 1 (4.12) is the Green's function for the given boundary-value problem [Ozi93]. The expressions for the eigenfunctions are determined by the boundary conditions for each direction. For the xand y -directions X((3 m ,x) = P m cos(P m x) + H tr (t)-sin(P m x) (4.13) Y(Y n , x) = y n cos(Y n x) + H tr (t) • sin(Y n x) (4.14) N(P m ) = 2[W tub {p^ + H t 2 r (t)} + 2H tr (t)r' (4.15) = 2[L tub {Yn + Hf r (t)} + 2H tr (t)r' N(Y n ) (4.16) where the eigenvalues are determined by the positive roots of the following transcendental equations , ,R T . 2 Pn,H tr (t) tan (P m L .ub) = ~ 2 — (4.17)

PAGE 123

14 2y n H tr (t) tan(Y n W tub )= 2 " (4.18) For the z -direction Z(r>,z) = cos[Ti p (d tub -z)] (4.19) 2[Tl' + Hj 0X (t)] N( V d tub [TiJ + Hj ox (t)] + H box (t) (4.20) where the eigenvalues are determined by the positive roots of the following transcendental equation Tl p tan(Ti p d tub ) = H box (t). (4.21) Equation (4.12) physically represents the temperature rise at any point (x, y, z) in the tub at time t, due to an instantaneous point source at point (x', y', z) at time t'. To account for the heat-generation volume (V = WLH), (4.12) is integrated over the base/collector SCR. Assuming a unit step increase in power dissipation at time t = and expressing the temperature rise in the tub as AT tub (t) = Z TH (t) P (4.22) yields the transient thermal impedance

PAGE 124

115 Z TH (x, y, z, t) = — L_f p c VJ ,2. n 1 I expI-a si P;t] i ^5 X(P m ,x)X(P in ) X eX Pta siY^N(— j Y (Yn'y) Y (Yn) S ex p[a s^p t ]j^n Z(T1 P' z)Z(r| p ) Lp=l N(Ti p ) dt (4.23) where X(p_) = 2sin P m L" cos (U^ C, +L + H tr (t) sin P, C L + L (4.24) Y(y n ) = 2sin rr n wcos Yn C W1 + W H tr (t) . + sin Y n Y„C W l + ~2 (4.25) Z(Ti p ) = -^-{sin[r| p (D + H)]-sin[Ti p D]}. Id (4.26) To represent the temperature rise in the device by a single effective value, (4.23) should be evaluated at a single point. The coordinates for surface corner of the emitter, (x = C L /2, y = C W1 , z = d tub ), are substituted into (4.13), (4.14) and (4.19) to keep the model consistent with the bulk BJT thermal impedance model. As shown later in the chapter, the model evaluated at those coordinates agrees well with measured values of the steady-state thermal resistance. Most of the parameters for the DIBJT thermal impedance model are fixed by the geometry of the device structure. However, the parameter H depends on the

PAGE 125

116 electrical bias of the device and can change with operating conditions. As shown in Chapter Two, the base/collector SCR thickness, and hence the thermal impedance model, depends only moderately on the bias of the base/collector junction. For the DIBJT thermal impedance model, this dependence is weaker than that for the bulk BJT model since the dielectric isolation primarily determines the thermal impedance. The variation in the predicted thermal resistance of a DIBJT due to changes in H is approximately one-half to one-third that of a bulk BJT. The variation will increase as the thickness of the dielectric isolation is decreased or the tub scaling constants are increased; however, for most practical DIBJT structures, the dependence of the thermal impedance on bias can be neglected. The DIBJT thermal impedance model can be extended to account for BJT's with multiple emitter fingers by integrating (4.12) over each base/collector SCR. Using an analysis similar to that in Chapter Two for bulk MEBJT's, expressions for the self and coupling impedances can be derived. 4.2.1 Derivation of the Buried-Oxide Heat-Transfer Coefficient The normalized heat-transfer coefficient of the buried oxide, H box (t) , describes the time-dependent heat conduction through the buried oxide to the substrate. The heat flux through the buried oxide is assumed to be predominantly one-dimensional (1-D). This assumption is valid for most of the tub area (A tub = W tub L tub ) but is questionable at the edges of the tub, where the heat flows out laterally under the trench. However, analogous to field fringing effects in parallel-plate capacitors, the proportion of lateral heat flow becomes smaller as the

PAGE 126

117 tub area becomes larger. Figure 4.3 shows the results of ANSYS simulations of the heat flow through the buried oxide. The data corresponds to the z -component of the heat flux vector in the buried oxide, normalized by the heat flux vector magnitude (referred to as the "flux ratio"). The plots clearly show that the heat flux in the buried oxide is predominantly 1-D, even at the interface between the trench and the buried oxide. The temperature rise in the buried oxide, AT 0X = T ox -T , can be described by the heat conduction equation a 2AT box = i 3AT box (427) dz 2 ' « ox 3t The following boundary conditions are imposed at the interface between the tub and the buried oxide 3AT 0X dAT tub k ^ = k ^ (4 28) AT 0X = AT tub (4.29) and at the interface between the buried oxide and the substrate AT 0X = 0. (4.30) Using the variable substitution

PAGE 127

118 Tub Area, W tub L tub (iim 2 ) (b) Figure 4.3 The ratio of the z -component of the heat flux vector to the magnitude of the total heat flux vector in the buried oxide with d box = 1 u.m. L = 2 |im and W = 0.5 |im: a) For C W] = C W2 = C L /2 = 5 |im; b) the portion of the tub area where the flux ratio is unity (signifying complete 1-D heat flow) for C W1 = C w2 = C L /2 = 3, 4, and 5 (im.

PAGE 128

119 S V^T 1 ' (4.31) (4.27) is transformed into the following ordinary differential equation d AT ox | ^dAT c < 2 d£ = 0. (4.32) Equation (4.32) has a general solution of the form AT ox = C l erf |^) +C 2 (4.33; where c, and c 2 are arbitrary constants. The equation that describes the flux at the interface between the tub and the buried oxide can be derived by substituting (4.31) into (4.33) and then applying the boundary conditions given by (4.28) through (4.30). The resulting expression 3AT tub k-ox^Ttub S1 dz J™^> erf ( d box | (4.34) when rearranged into the form given by (4.8), yields the normalized heat-transfer coefficient for the buried oxide Hox(t) = k siV™W erf box M fiv^hj (4.35)

PAGE 129

120 4.2.2 Derivation of the Trench Heat-Transfer Coefficient The normalized heat-transfer coefficient of the trench isolation, H tr (t) , describes the time-dependent heat conduction through the trench structure and exterior silicon to the substrate. To simplify the analysis, the thickness of the trench is assumed to be uniform along the z -direction and the composite trench structure is represented by a single material with the lumped thermal properties [Man90] H k k , u tr*oxV,1y . 36 K tr " oh k +d k ( ' zu trox K poly + u poly K ox ^trPtr c ptr 2d trox p ox c pox + d poly p poly c ppoly (4.37) d tr = 2d trox + d poly (4.38) k tr Ptr C ptr The heat flux through the trench is assumed to be predominantly one-dimensional ( 1D). This assumption is valid for most of the trench area ( A tr = W tub d tub or L tub d tub ), but is questionable at the corners of the trench structure and at the edges between the trench walls and the buried oxide. Figure 4.4 shows the results of ANSYS simulations of the heat flow through the trench. The data corresponds to either the x or the y -component of the heat flux vector in the trench which is normalized by the heat flux vector magnitude (referred to as the "flux ratio"). The plots clearly show

PAGE 130

121 Z-Direction (um X-Direction (um) Figure 4.4 The ratio of the y -component of the heat flux vector to the magnitude of the total heat flux vector in the trench for half of the trench wall. The device parameters are L = 2 u.m, C L = 10 |im, d [rox = 0.25 |im and poly = 1.6 u.m.

PAGE 131

122 that the heat flux in the trench is predominantly 1-D, even at the corners of the trench and at the interface between the trench and the buried oxide. The temperature rise in the trench, AT tr = T tr -T , can be described by the heat conduction equation 3 2 AT, r i 3AT tr -5?= ^,-ir • < 4 40 ' where n is either the xor y -direction. The following boundary conditions are imposed at the interface between the tub and the trench dAT tr 3AT fnh A T tr = AT tub (4.42) and at the interface between the trench and the exterior silicon 3AT tr 3AT sj AT .r = ^T sl . (4.44) The boundary conditions given by (4.43) and (4.44) require a solution for the temperature rise in the exterior silicon, AT si = T si -T . The temperature in the exterior silicon can be derived by assuming that the heat flow is primarily in the directions normal to the trench walls. Therefore, the temperature rise in the exterior

PAGE 132

123 silicon is assumed to be one-dimensional. The validity of such an assumption can be evaluated using the Biot number, which corresponds to the ratio of the internal and external thermal resistances of a given object [Ozi93]. If the Biot number for an object is much less than unity, then it can be approximated by a one-dimensional thermal medium. The vertical Biot number for the exterior silicon is B Vsi = (k ox d tub )/(k si d box ), so that for d tub « 100 d box the vertical temperature gradient can be neglected. The extent of the temperature gradients in the directions parallel to the trench walls is also investigated by simulating the heat flux in the exterior silicon at the trench/exterior silicon interface. Figure 4.5 shows the results of one such ANSYS simulation. For most of the trench area, the heat flux is primarily perpendicular to the trench walls. However, the one-dimensional assumption does break down around the corners of the trench where the heat flow becomes more twodimensional. Therefore, the one-dimensional model will tend to slightly under predict the heat transfer through the trench. The exterior silicon can be divided into four regions surrounding the trench; each region is modeled as a one-dimensional cooling fin. To approximately account for the lateral spread of heat around the corners of the trench, each cooling fin is assumed to have an increasing cross-sectional area. The temperature rise in the each exterior silicon cooling fin can be described by a 2 AT si i 3AT si 2 i 3AT S1 + 7 „ . S -m'AT,: = *-2. (4.45) dn 2 (n + C n ) 3n S1 s > a si 9t

PAGE 133

124 Z-Direction (um) X-Direction (um) Figure 4.5 The ratio of the y -component of the heat flux vector to the magnitude of the total heat flux vector in the exterior silicon at the trench/exterior silicon interface. The device parameters are L = 2 |im, C L = 10 |J.m, d trox = ° 25 ^ m and d poly = 16 ^ m -

PAGE 134

125 The constant C n is (L tub + 2d tr )/(2tan0 L ) for the x -direction and (W tub + 2d tr )/(2tan8 w ) for the y -direction, where W and G L are the thermal spreading angles for the cooling fins which are assumed to be 45° [Hir93]. The third term on the left-hand side of (4.45) accounts for heat lost from the exterior silicon due to conduction through the buried oxide, where d tub m si V H ox(t) (4.46) is the characteristic thermal length in the exterior silicon. Equation (4.45) does not have a simple closed-form solution. However, a solution to the steady-state form of (4.45) exists and, by using the time-dependent characteristic thermal length, can be used to approximate the temperature rise in the exterior silicon as AT S] = c 1 K [m sj (t)-(n + C n )], (4.47) where c, is an arbitrary constant and K ; is the modified Bessel function of the second kind of order i . The equation that describes the flux at the interface between the tub and the trench is derived by solving (4.40) using (4.31), (4.41) through (4.44), and (4.47). The resulting solution, when arranged into the form given by (4.4), yields the normalized heat-transfer coefficient for the trench

PAGE 135

126 H, r (t) = k tr m si K,(m si C n ) . (4.48) k si m siV™V ' erf ( d tr ^ V 4 ^ K l( m si C n) + k tr eX P v 4« tr t y ^ ( m si C n) 4.2.3 Effects of Interconnect Metallization on the Thermal Impedance For the derivation of the DIBJT thermal impedance model, the surface of the device is assumed to be adiabatic, and therefore conduction through the interconnect metallization is neglected. Since the metallization typically has a high thermal conductivity, it is possible that the heat conduction via the interconnects significantly influences the thermal impedance. Therefore, the validity of such an assumption should be investigated. Based on the finite-element (FE) simulations performed for bulk BJT's (Chapter Two), the heat conduction through base and collector interconnects is typically negligible and only the emitter interconnect is assumed to affect the thermal response of the device. To determine the extent of the effect of the emitter interconnect on the thermal impedance, both steady-state and transient threedimensional (3-D) FE simulations of a DIBJT structure were performed using ANSYS. The FE model represented a simplified device structure and was constructed using similar assumptions and boundary conditions to those detailed in Chapter Two. Figure 4.6a shows the FE model used for the thermal simulations. The simulations were carried out with and without the emitter interconnect present, while independently varying each structure parameter of the FE device model.

PAGE 136

127 (a) 30 ^

PAGE 137

128 The results of the steady-state FE simulations are shown in Figure 4.6b. As with the bulk BJT's simulated in Chapter Two, the emitter interconnect enhances the heat flow away from the active device area, resulting in lower thermal resistance values. As was the case with the bulk BJT FE model simulated in Chapter Two, the effectiveness of the interconnect as a thermal conductance path increases as its crosssectional area increases or as the interconnect is effectively moved closer to the heat source. However, the influence of the emitter interconnect on the thermal resistance is more significant for the DIBJT's than for the bulk BJT's. This effect is due to the lower thermal conductance to the substrate of DIBJT's as compared to the bulk devices. Therefore, the conductance through the emitter interconnect becomes a larger component of the total thermal conductance of the device. This trend is illustrated by the data in Figure 4.6b, which shows that as the conductance to the substrate is increased by enlarging the tub or thinning the isolation oxides, the effect of the emitter interconnect on the thermal resistance is reduced. Since the effect of the emitter interconnect is increased due to the dielectric isolation, it is reasonable to suggest that the effects of the base and collector interconnects on the thermal resistance will also increase. Therefore, the assumption that the base and collector metallization is negligible becomes questionable. However, the increase in the effects of the base and collector interconnects should not be any greater than that for the emitter interconnect. Consequently, even for a two-fold increase in the influence of the base and collector metallization in a bulk technology, the heat conduction through the base and collector interconnects will still be negligible.

PAGE 138

129 Transient thermal simulations were performed to examine the effect of the emitter interconnect on the dynamic temperature response. ANSYS was used to simulate the structure in Figure 4.6a for a step increase in power dissipation, both with and without the emitter interconnect in contact with the device. Figure 4.7 shows the results of the transient FE simulations. The observed trend in the transient temperature rise is similar to that for the bulk bipolar devices in Chapter Two. The additional heat capacity of the emitter metallization effectively slows the temperature rise in the device. The extent of the metallization's effect on the dynamic temperature response is directly related to the effective volume of the interconnect structure, and will be more pronounced for large devices with substantial contact area. Therefore, the model that neglects the conduction through the interconnect by assuming an adiabatic surface will produce the quickest possible temperature rise. 4.2.4 A Model for the Thermal Impedance of the Emitter Interconnect As detailed in Chapter Two, the effect of the emitter interconnect on the thermal response can be modeled by an additional thermal impedance. This thermal impedance is derived by calculating both the thermal resistance and thermal capacitance of the emitter metallization. The emitter interconnect is represented by a one-dimensional cooling fin, such that the thermal resistance can be expressed as R T Hmet = [kmetrnmetWmetdme.r 1 (4.49)

PAGE 139

130 U u c J^ 2 C3 £ sH O O With Emitter Interconnect -O Without Emitter Interconnect 06 10 12 10 tf-B-B — eo 10"° 10 Time (sec) -6 If) -4 ioFigure 4.7 ANSYS simulations showing the effect of the emitter interconnect on the thermal impedance. The simulated FE device model used the nominal structure parameters listed in the caption of Figure 4.6.

PAGE 140

131 where in met k d /'"met^met met (4.50) is the characteristic thermal length in the interconnect and k ox k si met k si( d ox + d box) + k ox d tub (4.51) is the heat-transfer coefficient from the interconnect to the substrate. The material properties for the emitter interconnect are given in Table 4.3. Table 4.3 Emitter interconnect* material properties Property

PAGE 141

132 Therefore, the transient thermal impedance of the emitter interconnect can be approximated as ZTHmetW = R TU me t[ l ~ ex p(; ^met (4.53) where x met = RTHmet^THmet The overa U thermal impedance of a bipolar device can now be represented by the parallel combination of two thermal impedances, such that effectively Z-THdev( S ) • Z THmet (s) ^THdev( s ) + ZxHmet( s ) Zth( s ) = ^ ,_. . J ,-x . (4-54) where Z THdev (s) is determined from the transient thermal impedance given by (4.23). 4.3 Verification of the DIB JT Thermal Impedance Model To verify the thermal impedance model, three-dimensional (3-D) finiteelement (FE) simulations of a DIBJT were performed using ANSYS. The FE model represented a simplified device structure and was constructed using similar assumptions and boundary conditions to those detailed in Chapter Two. In addition, all interconnect metallization was neglected and the tub region was assumed to be composed of homogeneous silicon with the bulk properties given in Table 4.2. The FE simulations were evaluated at the surface corner of the emitter and compared to (4.23); the results are shown in Figure 4.8. The DIBJT thermal impedance model

PAGE 142

133 Figure 4.8 The transient thermal impedance simulated with ANSYS and calculated with the DIBJT model for d tub = 1.5 Jim, d box = 0.5 |im, d trox = 0.13 fxm and d poly = 0.5 (im: a) L = 2 fim and W = 0.5 (im; b) L = 3 |im and W = 0.7 |j.m

PAGE 143

134 agrees closely, for both the transient and steady-state, with the 3-D FE simulation results. The largest error, which is approximately 17%, occurs in steady-state and can be partially attributed to numerical error associated with limitations of the FE mesh. The model was also compared to measured thermal impedance data extracted by Zweidinger et al. using a base-current thermometry technique [Zwe96]. Figure 4.9, Figure 4. 10 and Figure 4. 1 1 compare the measured and simulated data for the transient thermal impedances of Harris UHF DIBJT's. The thermal impedance model does a good job of predicting the steady-state thermal resistance, with no more than 1 1% error between the model and the measurements. However, the model given by (4.23) tends to exaggerate the transient response. This discrepancy can be attributed to the model's neglect of the emitter interconnect metallization. Equation (4.23) predicts the quickest temperature response in the device since it does not account for the additional heat capacity of the metallization. However, when the thermal impedance of the emitter interconnect is accounted for, with 5 met = 100 (im extracted from the measurements, the model provides a more accurate representation of the transient temperature response. 4.4 Derivation of a Compact DIBJT Thermal Resistance Model For the limiting case of steady-state thermal conduction, the complexity of the DIBJT thermal impedance model can be reduced, resulting in a single closedform expression for the thermal resistance. The approach for the derivation of the thermal resistance model is adopted from the analysis by Goodson and Flik for SOI

PAGE 144

135 300 250 U o 200 tf o c

PAGE 145

136 200 U °^ 150 § 100 5 -o o I rt 0) 50 Ofi10" Measurement ©---©Model: Without Emitter Int [•}--•Model: With Emitter Int. -Qi--i-r-y. 150 U o X O -o U On 100 E 50 E su H OS 10 -9 Measurement ©•OModel: Without Emitter Int. a — Model: With Emitter Int. ~QFigure 4.10 Measured and simulated data for the transient thermal impedance of Harris UHF DIBJT's with W = 3 |im: a) L = 90 fim; b) L = 1 10 urn.

PAGE 146

137 100 75 U I u « 50 T3 U I | 25 H Ofr 10" Measurement G>---OModel: Without Emitter Int. B----E3 Model: With Emitter Int. &• .-©• =ei v3 100 & 75 (J g 50 -a u I 03 JS 25 -i — r | I I I I I I | 'T T T I T I 0610' Measurement 0—-0 Model: Without Emitter Int B— €1 Model: With Emitter Int. ~e-3 Figure 4.11 Measured and simulated data for the transient thermal impedance of Harris UHF DIBJT's with W = 3 urn: a) L = 170 urn; b) L = 210 urn.

PAGE 147

138 MOSFET's [Goo92]. The model represents a solitary DIBJT device with three parallel conductance paths that carry heat away from the tub. These heat-flow paths are illustrated in Figure 4.12 and are modeled by a system of coupled onedimensional differential equations. The derivation of the thermal resistance model is divided into separate regions that correspond to the different thermal conductance paths. Different subscripts are used to denote the equations and variables that apply to the different regions; the subscripts tub, e and si refer to the tub region, emitter interconnect and exterior silicon region, respectively. Figure 4. 13 shows the simplified device structure that defines the geometry for the model derivation. The definitions of the model parameters are listed in Table 4.4. The defining assumption for the DIBJT thermal resistance model, and the main departure from the thermal impedance model, is that the entire tub is considered to be a heat source, with uniform power dissipation equal to the actual device power P, at a single uniform temperature, T tub . Since the thermal conductivity of the tub silicon is much greater than that of the insulating oxide, the thermal conductance of the tub is larger than that of the trenches or the buried oxide, and the thermal resistance of the device is mainly determined by the isolation structures. This assumption is supported by both ANSYS simulations and the MEDICI simulation in the work by Ganci et al. [Gan92]. Figure 4.14 shows the ANSYS results, which illustrate that the temperature gradient in the tub is smaller than the gradients across the isolation structures. However, the validity of this assumption declines as the dimensions of the tub, A tub = W tub L tub , become much greater than those of the active device, A dev = W • L, or the thickness of the insulating oxide layers (buried

PAGE 148

139 Tub temperature, T tub Q Along emitter interconnect Substrate temperature, T Figure 4.12 Illustration of the three parallel conductance paths for heat to travel from the device tub to the semiconductor substrate.

PAGE 149

140 V///////A Tub Buried Oxide Substrate 'tub 'box si K tub y S i o w (a) 'tub (b) T o *S1 'poly 'trox T W £ Figure 4. 1 3 The simplified device geometry used to define the solution domain for the DIBJT thermal resistance model. The model parameters are defined in Table 4.4: a) Cross-sectional view; b) top view.

PAGE 150

141 Table 4.4 DIBJT thermal resistance model parameters Parameter

PAGE 151

142 Y-direction (urn Y-direction (um) Z-direction (um) (b) Figure 4. 14 ANSYS simulation results showing the temperature gradient in the tub and across the isolation structures: a) For A tub /A dev = 9; b) for Atut/Adev 1°-

PAGE 152

143 or trench) is reduced. For such conditions, the conductance of the tub is reduced and the temperature gradients within the tub become more significant, so that the model will tend to underestimate the thermal resistance. The boundary conditions and the model parameters for the thermal resistance model, for the most part, are the same as those for the thermal impedance model. The interface between the buried oxide and the substrate is assumed to be at a uniform temperature T , and the device is assumed to cool solely through the substrate. The width and length of the tub region are assumed to scale directly with the width and length of the emitter stripe; one change from the thermal impedance model is that the scaling constants C W1 and C W2 have been lumped together so that W tub = W + C w . (4.55) The top surface of the device is considered adiabatic so that heat conduction through the overlying oxide layers is neglected. Based on the finite-element simulations in Chapter Two, only the emitter interconnect is assumed to affect the thermal resistance. However, since the tub is assumed to be at a uniform temperature, the effects of the base and collector metallization can be incorporated using a similar analysis as shown below. The emitter interconnect is still treated as a one-dimensional cooling fin. Therefore, the temperature rise along the interconnect can be described by the following differential equation

PAGE 153

144 a 2 AT e 2 e -m e AT e = (4.56) 5x e and the boundary conditions AT e | =AT mb (4.57, ATI = 0, (4.58) where AT e (x e ) = T e (x e )-T . The assumption that the temperature gradient in the emitter interconnect is primarily one-dimensional was validated in Chapter Two. The width of the emitter interconnect is assumed to scale directly with the width of the emitter stripe W e = W + C e , (4.59) where C e depends on the fabrication process. The characteristic thermal length and the heat-transfer coefficient for the interconnect are given by (4.50) and (4.51), respectively. The heat lost from the tub through the trench is governed by the steadystate heat-transfer coefficient of the trench. The temperature gradient in the trench is assumed to be one-dimensional and the exterior silicon is modeled by four onedimensional cooling fins where for steady-state, the characteristic thermal length is k-sjdtub m s, V h ox (4.60)

PAGE 154

145 and the buried-oxide heat-transfer coefficient is given by h ox = I ox (4.61) box The flux at the interface between the tub and the trench can be expressed as 3AT tr (4.62) Equation (4.62) can be solved for the heat-transfer coefficient of the trench, h tr , using the steady-state form of (4.40), equations (4.41) through (4.44) and (4.47), resulting in k tr k S1 m S1 K l( m s, C n) "tr d tr k si m si K l( m s, C n) + k tr K o( m si C n) (4.63) The thermal conductance paths are coupled through the following power conservation equation 3AT " k e W e d e JW + hox W tubL t ub AT tub + 2 M,ub( W tub + L t ub)AT tub = P . (4.64) x =0 Equation (4.64) can be solved in the form AT tub = R TH P (4.65: resulting in the following expression for the thermal resistance

PAGE 155

146 r th = [k e W e d e m e + h 0X W tub L tub + 2h tr d tub (W tub y y + L tub Y x )] -l (4.66) k si m s, K l r™.i( W tub + 2d tr)2tan0 Yv w h tr K rm si (w tub + 2d tr )2tan0 w + k si m si K l r m si( W tub + 2d tr)2tan9 w (4.67) k si m si K l Tx = m si( L tub + 2d tr) 2tan0, htr^o r m si( L tub + 2d tr)2 tan 6, + k si m si K l m si( L tub + 2d tr> 2tan6, (4.68) The first term on the right-hand side of (4.66) corresponds to the heat flow out through the emitter interconnect; the second term corresponds to the heat flow out through the buried oxide, and the third term corresponds to the heat flow out through the trench. 4.5 Verification of the DIBJT Thermal Resistance Model The accuracy of the thermal resistance model is tested with comparisons to measured steady-state thermal resistances and to the thermal impedance model; the results are shown in Figure 4.15. The thermal resistance model displays the proper trends with emitter length and tub scaling but as A tub /A dev increases, the accuracy of the model declines. For the DIBJT's with A tub /A dev > 30, the error between the thermal resistance model and the measured data is in excess of 20%. This trend is further illustrated by examining the thermal resistance of Harris

PAGE 156

147 400 U 300 1 200 CO "S3 JS H 100 O Measurement Zj H Model R TH Model 50 100 150 Emitter Length, L (|J.m) 200 250 Figure 4.15 The steady-state thermal resistance extracted from measurements and calculated using both the thermal impedance and thermal resistance models; W = 3 |im.

PAGE 157

148 "Cooling-Zone" DIBJT's. These devices were fabricated in larger tubs to reduce the thermal resistance. Table 4.5 shows a comparison of the measured and predicted thermal resistances for two "Cooling-Zone" devices. Table 4.5 Thermal resistance (°C/W) of Harris "Cooling-Zone" DIBJT's* L (Jim)

PAGE 158

149 4.6 Summary Due to the low thermal conductivity of the silicon dioxide used for the trench and buried oxide isolation, the thermal impedance of DIBJT's can be larger than that of their bulk counterparts. Therefore, self-heating effects can be enhanced and it is important to have a physical model that can predict the dynamic temperature rise. This chapter presented a thermal impedance model for DIBJT's. As with the bulk BJT/HBT's, the device interconnects can affect the thermal impedance. Neglecting the heat capacity of the emitter metallization results in a predicted thermal impedance that exaggerates the transient temperature response. Therefore, the emitter interconnect thermal impedance model, developed in Chapter Two, was utilized in this chapter. The DIBJT thermal impedance model was shown to agree reasonably well with both three-dimensional finite-element simulations and measurements. Finally, in the limit of steady-sate heat conduction, the thermal impedance model was simplified to provide a single, closed-form expression for the thermal resistance.

PAGE 159

CHAPTER 5 A THREE-DIMENSIONAL THERMAL IMPEDANCE MODEL FOR BULK METAL-OXIDE-SEMICONDUCTOR FIELD-EFFECT TRANSISTORS 5.1 Introduction Due to the development of complementary metal-oxide-semiconductor technologies (CMOS), the bulk MOS field-effect transistor (MOSFET) has become the primary device used in semiconductor circuits. Figure 5.1 illustrates the crosssection of a typical bulk MOSFET structure. The MOSFET output current is not as strongly dependent on temperature as the current of bipolar transistors. The current's sensitivity to changes in operating temperature is mainly due to variations in the carrier mobility. A significant change in output current will be observed only for temperature variations of tens of degrees. Based on the thermal resistances of typical devices, such temperature excursions will only occur at high power levels. However, as MOSFET's are aggressively scaled towards tenth-micron channel lengths, thermal impedances are likely to increase and self-heating effects could be enhanced. Previous works in this area have provided methods for calculating the thermal impedance of bulk MOSFET's; however, these approaches were limited by inadequate derivations. In the work by Schutz et al. [Sch84], the two-dimensional steady-state temperature distribution in a MOSFET was solved using a finite difference discretization of the heat conduction equation. The solution was 150

PAGE 160

51 Source p-substrate Figure 5.1 Cross-section of a typical bulk metal-oxide-semiconductor field-effect transistor (MOSFET).

PAGE 161

152 calculated for a thin layer about the channel, where the boundary conditions were set by an effective one-dimensional (1-D) thermal resistance representing conduction in the substrate. The idea of the effective substrate thermal resistance was extended by Hirsch et al. [Hir93] into a quasi-three-dimensional analysis by coupling parallel 1D analyses. While the above-mentioned techniques offer insight into steady-state heat conduction in bulk MOSFET's, they are not capable of determining the dynamic temperature response. A model for the dynamic thermal impedance of bulk MOSFET's was developed by Sharma and Ramanathan [Sha83]. However, this model was derived by neglecting the temperature variation along the width of the channel and the variation of the electric field along the length of the channel. Both of these assumptions can lead to significant errors in the predicted temperature rise within the channel region. The model derived in this chapter is based on an extension of the model developed by Sharma and Ramanathan [Sha83], and provides a closed-form physical solution for the transient thermal impedance of bulk MOSFET's. The first part of this chapter details the derivation of the model and the improvements made to the existing work. The effects of LOCOS and shallow trench isolation and the drain, gate and source interconnects on the thermal impedance are then investigated using threedimensional finite-element simulations. Finally, the accuracy of the thermal impedance model is evaluated using three-dimensional finite-element simulations and measurements.

PAGE 162

153 5.2 Derivation of the Bulk MOSFET Thermal Impedance Model For the derivation of the bulk MOSFET thermal impedance model, the silicon substrate is represented by a homogeneous semi-infinite half-plane with an adiabatic top surface (no heat transfer perpendicular to the surface). The back side of the substrate is assumed to be held at a constant temperature, T . Figure 5.2 illustrates the simplified device geometry assumed for the model derivation; the diagram focuses on the electrically active portion of the device around the drain, source and channel, which has a width W and length L. The heat source represents the power generated in the channel, which is assumed to be uniform along the channel width. Representing the substrate as a semi-infinite medium neglects the influence of the back-side and the lateral edges of the wafer, as well as any LOCOS or trench isolation structures, on the thermal response of the device. The ramifications of neglecting the isolation structures are investigated later in the chapter. The surface of the wafer is assumed to be the only boundary that affects the thermal response of the device and it is considered to be adiabatic. With respect to the assumptions concerning the back-side and lateral edges of the substrate, the bulk MOSFET model is identical to the bulk BJT model and the validation in Chapter Two applies. Assuming that the surface of the device is adiabatic implies that conduction through the source, gate and drain interconnects and conduction through overlying oxide layers are neglected. As shown in Chapter Two, the thermal conductance from the device through the overlying oxide is approximately two to three orders of

PAGE 163

154 (dT/dz) = Figure 5.2 The simplified device geometry used to define the solution domain for the bulk, MOSFET thermal impedance model. The substrate is represented by a semi-infinite half-plane with an adiabatic surface (the dashed lines). Heat is generated due to the power dissipation in the channel, which has a width W and length L.

PAGE 164

155 magnitude smaller than the device-to-substrate conductance. Therefore, neglecting thermal energy transport from the field regions (the regions covered with thick oxide layers) of the device is reasonable. However, the device interconnects can have high thermal conductivities, and can provide effective paths for heat flow from the portions of the device they contact. The consequences of neglecting the interconnects are investigated later in the chapter. The heat source that represents the power dissipation in the channel depends on the region of operation and is considered to be a superposition of two individual heat sources. One source models the power dissipation in the fielddependent-velocity portion of the channel (referred to as the linear source). The other heat source models the power dissipation in the portion of the channel where the carrier velocity is saturated (referred to as the saturated source). The total temperature rise in the device can be expressed as a sum of the individual components due to the separate heat sources, and is given by AT(t) = Z THss (t)-P ss + Z THls (t)-P ls , (5.1) where Z THss (t) and Z THls (t) are the transient thermal impedances of the saturated and linear sources, respectively. When a MOSFET is operating in the saturation region, the power dissipation associated with the saturated source is approximately P„ = (V fU -V fk JI ds , (5.2) and for the linear source

PAGE 165

156 P ls = V dss I ds , (5.3) where V dss is the drain-source voltage at the onset of the saturation region. For a device operating in the linear region, the effect of the saturated source is removed P ss = (5.4) and the linear source accounts for the total power dissipation p is = V ds I ds(5.5) The original model derived by Sharma and Ramanathan assumed that the temperature gradient along the width of a MOSFET was negligible [Sha83]. However, ANSYS simulations of a MOSFET structure have shown that the temperature variations across the width of a device, for aspect ratios (W/L) greater than ten, can be in excess of 10% to 17%. Therefore, the two-dimensional analysis of the original model is extended into three dimensions. The temperature rise at any point within the device can be described by the nonhomogeneous three-dimensional heat conduction equation V2AT(x, y, z, t) + g (x 'y> Z ' C) = l^T(x,y,z,t) k a dt and the following boundary conditions AT(±oo, y , z, t) = (5.7) AT(x, ±oo, z , t) = (5.8)

PAGE 166

157 3AT(x, y, z, t) dz = (5.9) z = AT(x,y,oo,t) = 0, (5.10) where AT is the temperature rise above the local ambient (AT = T-T ), g is the internal energy generation density, k is the thermal conductivity, a is the thermal diffusivity (a = k/(p • c ) where p is the density and c is the specific heat) and t is time. Typical values for the material properties of bulk silicon are given in Table 5.1. Table 5.1 Material properties of silicon Property

PAGE 167

158 accounted for using the Kirchoff transformation described in Chapter Eight. Neglecting the spatial dependence of the thermal conductivity implies that the effect of dopant atoms is ignored. In the highly doped source and drain regions, the thermal conductivity can be significantly lower than the bulk value given in Table 5.1. However, these regions are relatively small and the majority of the device substrate is relatively low-doped; therefore, as shown in Chapter Two, the average thermal conductivity of the substrate will not differ greatly from the intrinsic value. With the initial thermal conditions in the device specified as AT(x,y,z,0) = 0, (5.11) the solution to (5.6) can be expressed in the form AT(x,y, z,t) = ^ J dt'|G(x,y,z,t|x',y',z',t')g(x',y',z',t')dv' (5.12) t' = o where G(x, y, z, t|x', y', z', t') = 1 8[7COC(t-t')] jexp 3/2 exp (z-z) _4a(t-t'). ~(x-x') 2 4a(t-t')_ + exp exp -(y-y) _4a(t-t ) -(z + z) 4oc(t-t'). (5.13) is the Green's function for the given boundary-value problem [Ozi93]. Equation (5.13) physically represents the temperature at point (x, y, z) at time t, due to an instantaneous point source, g' (W s), of unit strength at point (x', y', z') at time t' .

PAGE 168

159 To account for a finite substrate thickness, (5.13) can be modified in a similar fashion as detailed in Chapter Two for the bulk bipolar model. The resulting expression for the Green's function is G(x, y, z, t|x', y', z', t') = 1

PAGE 169

160 rectangular sheet source of length L, and the saturated source is modeled by a line source at the drain/channel junction. To produce a physically consistent model, a more rigorous analysis was attempted that models the saturated source as a rectangular sheet with a length equal to that of the velocity-saturated portion of the channel. The length of the saturated portion of the channel can be derived using the analysis by Green [Gre93], giving L L e = l c • In ^ c {( V d S -V dss ) + J(V ds -V dss ) 2 + 4(^ 1 ^ (5.16) where l c is the characteristic length of the surface potential. While (5.16) provides a more physical characterization for the length of the saturated source, two problems are associated with its implementation. First, equation (5.16) is strongly biasdependent. Thus, the thermal impedance of the saturated source would need to be calculated as a function of device operation, rendering the thermal impedance model incompatible with the ETCS modification described in Chapter One. Second, the physical accuracy gained by using (5.16) did not translate to a significant improvement in the overall accuracy of the thermal impedance model. Therefore, the improvements offered by the use of (5.16) did not justify the complexity of its implementation, and the original approach used by Sharma and Ramanathan [Sha83] was retained.

PAGE 170

161 5.2.1 The Linear Source Thermal Impedance In the original model developed by Sharma and Ramanathan, the potential in the field-dependent-velocity portion of the channel was assumed to vary linearly along the length of the channel [Sha83]. This linear-potential assumption does not accurately represent the actual electric field in the channel. Due to the fielddependent velocity, the surface potential varies parabolically with position along the length of the channel, such that ^s(y) = Is 2 VJ V 2 (5.17) where V ls = V ds for the linear region of operation and V ls = V dss for the saturated region of operation. Therefore, using the relation P = I • E , the energy generation per unit-area can be expressed more accurately as gisCy'1 ') = 2 ' P ls (t j WL 2 (5.18) The temperature rise at any point (x, y, z) in the channel due to the linear source is calculated by substituting (5.13) and (5.18) into (5.12), and then integrating over the width and length of the channel. Assuming a step increase in power dissipation at t' = ( P ls (t) = P, s • U(t) ) and expressing the temperature rise as AT ls (t) = Z THls (t) • P ls , (5.19)

PAGE 171

162 yields the transient thermal impedance of the linear source V2-x ,pcWL 2 77rat 74at V jAax n i y\j -(IW^' + y eif| ^> erf (^3} -Ge>* • ,5 20) Equation (5.20) represents the temperature rise at any point in the device normalized to a unit-step increase in power dissipation in the linear source. To account for a finite substrate thickness, (5.14) can be used in place of (5.13) in equation (5.12). The effect of accounting for the parabolic variation in the surface potential is illustrated in Figure 5.3, where (5.20) is compared to a thermal impedance model derived using the uniform-field assumption of Sharma and Ramanathan [Sha83]. Since assuming a uniform electric field does not accurately model the increased power dissipation near the drain, the original model significantly under-estimates the steady-state thermal resistance in that region. For circuit simulation, a single value is needed to represent the effective operating temperature of the device. Therefore, the thermal impedance model given by (5.20) should be evaluated at a single point. Appropriate values for coordinates can be determined by tuning the temperature rise predicted by (5. 1 ) to measured data in a region of steady-state heat flow. Based on the extracted temperature data shown later in the chapter, the optimum point for the model evaluation was found to be (x = 0, y =0.95L,z = 0).

PAGE 172

163 ! 4.5 4.0 U 3.5 x <£ 3.0 u c 25 'la *§ 2.0 H 1.5 .0 G — OE = E(y) Q — BE = Constant 0.0 0.5 1.0 1.5 Channel Length, L (|J.m) 2.0 Figure 5.3 Comparison of thermal resistance calculated assuming a uniform electric field, E, to the thermal resistance calculated using (5.18), for W = 2 urn. In both cases, the thermal resistance model was evaluated at (x = 0, y = L, z = 0).

PAGE 173

164 5.2.2 The Saturated Source Thermal Impedance The heat source that represents the power dissipation in the velocitysaturated portion of the channel is modeled by a line source located at the surface of the drain/channel junction. The energy generation per unit width can therefore be expressed as g«(t ) = P„(t') W (5.21) The temperature rise at any point (x, y, z) in the channel due to the saturated source is calculated by substituting (5.13) and (5.21) into (5.12), and then integrating over the width of the channel. Assuming a step increase in power dissipation at t' = ( P ss (t) = P ss • U(t) ) and expressing the temperature rise as A T ss (t) = Z THss (t)-P, (5.22) yields the transient thermal impedance of the saturated source Z THss( X 'y' Z '0 = J 4pcWjiat /W/2 + x erf — -—v 74at + erf /2-x 4at exp (-.o^W^v V 4at exp teJ' dt (5.23) Equation (5.23) represents the temperature rise at any point in the device normalized to a unit-step increase in power dissipation in the saturated source. To account for a finite substrate thickness, (5.14) can be used in place of (5.13) in equation (5. 1 2). The thermal impedance model given by (5.23) should be evaluated at a single point to

PAGE 174

165 provide an effective temperature rise suitable for circuit simulation. The optimum point for the evaluation of (5.23) is the same as for the linear source and is given by ( X = 0, y = 0.95 L, z = 0). 5.2.3 Effects of the Device Interconnects on the Thermal Impedance For the derivation of the bulk MOSFET thermal impedance model, the surface of the substrate is assumed to be adiabatic. In actual devices, however, the drain, source and gate are in direct contact with interconnects that are used to electrically connect different devices on a chip. For the drain and source, the interconnects are typically fabricated using aluminum metallization, where the gate is usually contacted with polysilicon. The interconnects represent additional conductance paths which can enhance the heat transfer away from a device. Since the model neglects any such heat conduction, there is a question as to whether or not the heat loss via the interconnects significantly influences the thermal response of a device. Three-dimensional (3-D) finite-element (FE) thermal simulations of a bulk MOSFET structure, using the ANSYS software package, were performed to examine the effects of the device interconnects on the thermal impedance. Two FE models were developed to separately investigate the effects of the drain/source interconnects and the gate interconnect; these models represented simplified device structures and were constructed using similar assumptions and boundary conditions to those detailed in Chapter Two. The FE simulations tend to over predict the heat conduction through the interconnects since the device, contacts and interconnects

PAGE 175

166 were considered to be in perfect thermal contact, so that any contact resistances at the material interfaces were neglected. While the FE models do not truly represent and actual device structure, they serve as an order-of-magnitude estimate for the effects of the interconnects on the thermal response. Figure 5.4a shows the FE model used for investigating the effects of the drain/source interconnect metallization on the thermal resistance. Steady-state thermal simulations were run with the interconnects present and with the interconnects removed. The results of the two groups of simulations were compared to determine the extent of the effect of the drain/source interconnects. The simulations were performed at various channel-to-contact spacings (L d ) , for a fixed interconnect thickness (d met ) and a fixed oxide thickness (d ox ) between the interconnects and the substrate; the results are shown in Figure 5.4b. Overall, the effect of the drain/source interconnect metallization is to reduce the thermal resistance of a device; though, for the simulated devices, this effect is relatively minor. However, the data does illustrates how the drain/source interconnects become more effective thermal conductance paths as they are moved closer to the channel, or as their cross-sectional area is increased. Therefore, the effects of the drain/source interconnects on the thermal resistance of highly-scaled MOSFET's could be more significant. The effects of the gate interconnect on the thermal resistance of bulk MOSFET's are examined by performing steady-state thermal simulations with the FE model shown in Figure 5.5a, both with and without the interconnect present. The gate material was assumed to be heavily-doped polysilicon such that the width of the

PAGE 176

167 (a) 0.0 0.2 0.4 0.6 0.8 L d (Jim) 1.0 1.2 1.4 (b) Figure 5.4 ANSYS simulations showing the effect of the drain/source interconnects on the thermal resistance, for d met = 0.7 |im and d ox = 0.7 |im: a) The finite-element model simulated with ANSYS; b) the variation between the thermal resistance accounting for the drain/ source interconnects and the thermal resistance neglecting the drain/ source interconnects, plotted as a function of the edge-to-edge distance between the channel and the contact openings.

PAGE 177

168 (a) 3 5 7 Channel Width, W (|im) (b) Figure 5.5 ANSYS simulations showing the effect of the gate interconnect, assumed to be heavily-doped polysilicon, on the thermal resistance, for d g = 0.3 |im, d gox = 10 nm and d ox = 0.15 |im: a) The finite-element model simulated with ANSYS; b) the variation between the thermal resistance accounting for the gate interconnect and the thermal resistance neglecting the gate interconnect, plotted as a function of the channel width.

PAGE 178

169 interconnect was equal to the length of the channel. The simulations were run at various channel widths, for both shortand long-channel devices with fixed interconnect (d ) and gate-oxide (d ) thicknesses; the results are shown in Figure 5.5b. As with the drain/source interconnects, the effect of the gate interconnect is a reduction in the device thermal resistance. The trends in the data show that this effect decreases for devices with large channel widths; and, at a fixed channel width, the reduction in the thermal resistance is more significant for longer channel lengths. However, due to the relatively low thermal conductivity of the gate oxide and heavily-doped polysilicon, the extent of the effect is only moderate. Consequently, the use of silicided gate materials and ultra-thin gate oxides, which effectively enhance the thermal conductance of the gate interconnect, could result in a more substantial reduction in the overall device thermal resistance. Transient thermal simulations of the FE models in Figure 5.4a and Figure 5.5a were performed to examine the effect of the device interconnects on the dynamic temperature response. Figure 5.6 shows the results of the simulations for a step-increase in power dissipation. For the simulation with the drain/source interconnects, the thermal response does not deviate from that of a device without interconnects until the heat generated in the channel reaches the contacts. At this time, the additional heat capacity of the interconnect metallization effectively reduces the thermal response. The time it takes for the heat to reach the drain/source contacts can be approximated as the square of the distance L d divided by the thermal diffusivity of the substrate material. For the simulated device structure, the resulting time is approximately one nanosecond, which agrees with the FE simulations. The

PAGE 179

170 3.5 -e-e — b O O With Drain/Source Interconnects Q — Q Without Drain/Source Interconnects Time (sec) (a) 1010 -4 10 -2 1.2 ! u io X o o 1 0.5 S 1 0.2 H o.o 6 10 -12 -e — o ODD Q — OWith Gate Interconnect Q — Q Without Gate Interconnect Time (sec) (b) 0" 10" 10" Figure 5.6 ANS YS simulations showing the effects of the device interconnects on the transient thermal impedance: a) For the drain and source interconnects with W = 2 (im, L = 0.35 urn, d ox = 0.7 |im, d met=°7 l Im ' w m = 5 7 ^m and L d = 1.2]im; b) for the gate interconnect with W = 4 |im, L = 2 \im, d ox =0.15 |im, d gox = 10 nm and d = 0.3 (im.

PAGE 180

171 gate interconnect affects the thermal response almost instantaneously since it is located directly above the channel. As with the drain/source interconnects, the gate interconnect represents an additional conductance path for heat to travel away from the channel, and its heat capacity reduces the thermal response of a device. However, since the thermal conductivity of heavily-doped polysilicon is relatively low, the effect of the gate interconnect on the dynamic temperature response is not substantial. Based on the results of the 3-D FE simulations, neglecting the effects of the device interconnects in the model derivation should not result in substantial errors for larger MOSFET' s. However, the effects of the interconnects on the thermal impedance can become more severe for highly-scaled devices. In such a case, the thermal impedance model will tend to over-predict both the steady-state thermal resistance and the transient rise of the thermal impedance. 5.2.4 Effects of Isolation Structures on the Thermal Impedance As shown in Figure 5.1, a bulk MOSFET is typically surrounded by a region of silicon-dioxide which is used to electrically isolate it from neighboring devices. Such isolation is especially important for reducing latchup in CMOS circuits. The isolation technologies commonly used in bulk MOS fabrication are semi-recessed LOCOS (local oxidation of silicon) and shallow trench (e.g. BOX isolation) [Wol90]. Since the isolation structures use Si0 2 , which has a low thermal conductivity, they will naturally impede the flow of heat away from a device. The bulk MOSFET thermal impedance represents the substrate as a homogeneous

PAGE 181

172 material, and therefore neglects the effects associated with oxide isolation. To understand the limitations associated with this assumption, the effects of isolation structures on the thermal response should be investigated. Three-dimensional (3-D) finite-element (FE) thermal simulations, using ANSYS, were performed to examine the effects of isolation structures on the thermal impedance of bulk MOS transistors. A single FE model was developed to investigate the effects of both LOCOS and shallow-trench isolation. The full thickness (d fox ) of the isolation oxide was assumed to be recessed beneath the surface of the substrate. The model represented a simplified device structure and was constructed using similar assumptions and boundary conditions to those detailed in Chapter Two. The side-walls of the isolation structures were assumed to be perpendicular to the top surface of the substrate, and were directly adjacent to the active device region. The implications of assuming vertical isolation walls are detailed in Chapter Two, where it is shown that the isolation structure model should over estimate the effects of the isolation. Therefore, while the finite-element model does not truly represent the physical device layout, it provides an order-of-magnitude estimate for the effects of the isolation structures on the thermal impedance. Figure 5.7a shows the FE model used for investigating the effects of the isolation structures on the thermal resistance. Steady-state thermal simulations were run for various device-isolation spacings (L dox ) ; this spacing corresponds to the distance between the channel and the edge of the isolation structure, along the length of the device. Simulations were also run for the same devices with the isolation structures removed; Figure 5.7b compares the results of the FE simulations. The

PAGE 182

173 (a) 20 u o c .2 15 © — OW = 2nm B-aW = 4|im AW = 8 |im 0.8 1.0 1.2 1.4 L dox (urn) (b) 1.6 1.8 2.0 Figure 5.7 ANSYS simulations showing the effect of LOCOS or shallow trench isolation on the thermal resistance. The device specifications are L = 0.35 |im and d fox = 0.4 |im: a) The finite-element model simulated with ANSYS; b) the variation between the thermal resistance accounting for the isolation and the thermal resistance assuming a homogeneous substrate, plotted as a function of the distance between the channel and the isolation (along the length of the device).

PAGE 183

174 oxide used in the isolation structures impedes the lateral flow of heat away from the device, resulting in an increased temperature rise over the device simulated with no isolation. The simulation data shows that this effect can be significant and increases for small-geometry devices with small device-isolation spacings. The increase in temperature rise can be reduced by decreasing the depth of the isolation structures or by moving the isolation away from the channel. Transient thermal simulations of the FE model shown in Figure 5.7a were performed to examine the effects of both LOCOS and shallow-trench isolation on the transient thermal impedance. ANSYS was used to simulate the device with and without the isolation structure present; the results are shown in Figure 5.8. The transient temperature response is affected in the same manner as the steady-state response, and as the heat travels laterally and reaches the edges of the isolation structure, the temperature rise accounting for the isolation begins to increase over the response without the isolation. Based on the results of the 3-D FE simulations, neglecting the effects of the isolation structures used in bulk MOSFET technologies can be considered reasonable for larger devices. However, for highly scaled devices where the effective device-isolation separation is small, the model will tend to considerably underpredict both the steady-state thermal resistance and the transient rise of the thermal impedance.

PAGE 184

175 E -e-e-e -B— B ©With Isolation •Q Without Isolation 0"° Time (sec) -6 10 -4 10" Figure 5.8 ANSYS simulations showing the effect of LOCOS or shallow trench isolation on the transient thermal impedance. The specifications for the device structure are W = 2 |im, L = 0.35 |im, d fox = 0.4 |im, L dox = 1-2 Jim.

PAGE 185

176 5.3 Verification of the Bulk MOSFET Thermal Impedance Model To verify the ability of the thermal impedance model to predict the steadystate thermal resistance of bulk MOSFET's, the model was compared to measured temperature data. The temperature rise in the measured devices was predicted by evaluating the model equations (5.20) and (5.23) as t — > °° and then applying them to (5.1). The measured data were extracted using a gate-resistance thermometry technique [Goo95]. Figure 5.9, Figure 5.10 and Figure 5.1 1 show the comparisons between the predicted and measured values for the temperature rise. The thermal impedance model accurately predicts the steady-state temperature rise, and hence the thermal resistance, for both shortand relatively long-channel MOSFET's; the average error between the model and the measurements was 1 1% and 3% for the onevolt and two-volt gate biases, respectively. The thermal impedance model was also compared to three-dimensional (3D) transient finite-element (FE) simulations performed using ANSYS. verification. Two FE models were developed to separately verify the linear and saturated source transient impedances. To simplify the FE simulations, the MOSFET structures were assumed to be symmetric about the y-axis (see Figure 5.2); therefore, only one-half of a device was simulated. The device interconnects and device isolation were neglected and the substrate was assumed to be silicon with the bulk properties given in Table 5.1. The FE simulations were compared to the transient thermal impedance models given by (5.20) and (5.23), using the modification for finite substrate thickness; the results are shown in Figure 5.12 and Figure 5.13. The models for both the linear and saturated sources accurately replicate the transient and steady-state

PAGE 186

177 1 2 Power (mW) (a) U o B 3 — E H Power (mW) (b) Figure 5.9 Temperature rise, extracted using gate resistance thermometry, in Motorola bulk MOSFET's with W = 120 |im and L = 0.65 |im. The drain voltage was swept from 0.0 to 2.0 V: a) 1.0 V on the gate; b) 2.0 V on the gate.

PAGE 187

178 2 3 Power (mW) (a) 20 Power (mW) (b) 30 40 Figure 5.10 Temperature rise, extracted using gate resistance thermometry, in Motorola bulk MOSFET's with W = 120 (xm and L = 0.45 ^m. The drain voltage was swept from 0.0 to 2.0 V: a) 1.0 V on the gate; b) 2.0 V on the gate.

PAGE 188

179 U u e 3 k. u a. E (U H Power (mW) (a) 20 30 Power (mW) (b) 50 Figure 5.11 Temperature rise, extracted using gate resistance thermometry, in Motorola bulk MOSFET's with W = 120 [tm and L = 0.35 urn. The drain voltage was swept from 0.0 to 2.0 V: a) 1.0 V on the gate; b) 2.0 V on the gate.

PAGE 189

180 2.0 o o o r> o o n msrb OANSYS Model 1010" 3.5 0.06O 10 -12 10 10 -i — 000000 oonn> OANSYS — Model 1010" Time (sec) (b) 10" 10" Figure 5.12 The transient thermal impedance of the linear source simulated with ANSYS and calculated with the model: a) For W = 2 |im, L = 0.65 urn; b) for W = 1 urn, L = 0.35 urn.

PAGE 190

181 4.5 £ 4.0 E p 3.5 ? 3.0 X t$ 2.5 | 2.0 r & 1 5 1 1.0 J 0.5 > H o.o ©e 10 12 O O O O O O CXED OANSYS — Model v4 10 -2 10 I I — I I 1 1 III -I — J O O O O O O OOGO OANSYS — Model 0"° ioTime (sec) (b) 10 -4 10" Figure 5.13 The transient thermal impedance of the saturated source simulated with ANSYS and calculated with the model: a) For W = 2 fim, L = 0.65 |im; b) for W = 1 |im, L = 0.35 urn.

PAGE 191

182 temperature response; for the device structures that were simulated, the error between the FE and analytical models is no greater than 13%. Since the parameters for the bulk MOSFET thermal impedance model are based solely on the device geometry, the model can be applied to predict the expected trends for self-heating in highly-scaled devices. Figure 5.14 shows the thermal resistance, of both the linear and saturated sources, plotted as a function of channel length. The model predicts that as bulk MOSFET's are scaled beyond quarter-micron channel lengths, a substantial increase in the thermal resistance can be expected. Therefore, self-heating could become a significant problem in future generation devices and circuits. 5.4 Summary A thermal impedance model for bulk MOSFET's was derived. The model was shown to agree reasonably well with three-dimensional finite-element simulations and measurements of bulk devices. Since the model does not account for interconnect metallization and either LOCOS or trench isolation, the effects of neglecting such structures were investigated. The results suggest that the model can be expected to provide reasonable predictions for the thermal impedance of larger bulk devices. However, for highly-scaled devices, the effects of the drain, gate and source interconnects and isolation structures can be significant and the accuracy of the model will decline. Methods for modeling the effects of isolation structures are proposed in Chapter Eight.

PAGE 192

183 100 G — ©Linear Source Q — Q Saturated Source -B b=e= 3 2.0 Channel Length, L (|im) Figure 5.14 Thermal resistance calculated using the model evaluated at steadystate, plotted as a function of channel length for W/L = 2.

PAGE 193

CHAPTER 6 A QUASI-THREE-DIMENSIONAL THERMAL IMPEDANCE MODEL FOR SILICON-ON-INSULATOR METAL-OXIDE-SEMICONDUCTOR FIELDEFFECT TRANSISTORS 6.1 Introduction Metal-oxide-semiconductor field-effect-transistors (MOSFET's) fabricated using silicon-on-insulator (SOI) substrates offer many advantages over MOSFET's fabricated using bulk wafer technologies. Device speed can be enhanced due to the reduction of the parasitic capacitances to the substrate [Tu94, Goo95]. The improved device isolation prevents latchup in complementary MOSFET (CMOS) circuits since the buried oxide layer eliminates the parasitic bipolar devices between transistors [Wol90]. Short-channel effects can also be reduced due to the limited vertical depletion depth imposed by the finite silicon film thickness [Wol90]. Finally, the use of SOI substrates improves radiation hardness and can increase the device packing density [Wol90]. SOI MOSFET's are commonly fabricated using SIMOX (separation by implanted oxygen) SOI wafers. Figure 6.1 shows a diagram of a typical SOI MOSFET. SIMOX wafers are created by implanting oxygen ions into a silicon substrate. The implant is followed by a high-temperature anneal in an inert ambient to form silicon dioxide (Si0 2 ) [Wol90]. The oxygen is implanted far enough into the silicon such that the Si0 2 is buried beneath the wafer surface; typical values for the 184

PAGE 194

185 r Interconnects Y////Z&,

PAGE 195

186 thickness of the buried oxide layer range from 0.2 to 0.5 fim [Tu94, Goo95, Lee95]. Once the buried oxide layer is formed, the surface silicon layer is used for the fabrication of the MOSFET's, which can follow the standard process steps used for bulk devices. Typical values for the thickness of the silicon film range from 0.04 to 0.2 urn [Tu94, Goo95, Lee95]. Due to the low thermal conductivity of the buried oxide layer, MOSFET's fabricated on SOI wafers exhibit enhanced self-heating effects [Ber91, Che95, Jom95a]. Empirical extraction techniques using gate resistance thermometry or output conductance measurements have been developed to characterize the steadystate thermal resistance of SOI MOSFET's [Goo95, Lee95, Ten95]. The dynamic thermal impedance has also been extracted using output admittance or transient drain current measurements [Cav93, Cav95, Lee95]. The shortcomings of these methods are that they require complicated measurements or the fabrication of special test structures, and they do not provide predictive values for the thermal resistance or impedance. Physics-based models have generally been limited to the steady-state thermal resistance [Ber91, Goo92, Che95, Jom95b]. A physical dynamic electrothermal model for SOI MOSFET's was developed by Bielefeld et al. [Bie95]. However, this model is solved using complex numerical techniques making it inefficient for circuit simulation. This chapter details the derivation of a compact physics-based model for the dynamic thermal impedance of SOI MOSFET's. The thermal impedance model is an extension of a modified version of the steady-state thermal resistance model presented by Goodson and Flik [Goo92]. The work by Goodson and Flik provides an

PAGE 196

187 accurate model for the thermal resistance of SOI MOSFET's for which the width of the drain/source interconnect metallization approximately equals the width of the device, and the device interconnects play a significant role in heat evacuation. However, for many analog and some digital applications, the width of the device is greater than the metallization width, and the heat flow becomes dominated by the silicon film. While the model developed by Goodson and Flik allows separate input parameters for each width, it does not accurately model heat flow for the case where the device width is greater than the width of the metallization. The first part of this chapter re-examines the Goodson and Flik model and presents a modification to correct for the case where the device width is greater than the width of the drain/ source interconnects. The impact of the modification is examined through comparisons to the original model and measurements. The second part of the chapter then details the derivation of the dynamic thermal impedance model. The accuracy of the dynamic model is supported with three-dimensional finite-element simulations and measurements. 6.2 Derivation of the SOI MOSFET Thermal Resistance Model The thermal resistance model represents a solitary SOI MOSFET with three parallel conductance paths that carry heat away from the channel. These heatflow paths are illustrated in Figure 6.2 and are modeled by a system of coupled onedimensional differential equations. The derivation of the thermal resistance model is divided into separate regions that correspond to the different thermal conductance

PAGE 197

188

PAGE 198

189 paths. Different subscripts are used to denote the equations and variables that apply to the different regions; the subscript f is used as a generic identifier, and the subscripts c, g, d, m and ct apply to the channel, gate interconnect, drain/source regions of the silicon film, drain/source interconnects and the drain/source contact regions, respectively. Figure 6.3 shows the simplified device structure that defines the geometry for the model derivation. The definitions of the model parameters are listed in Table 6.1. The channel region is modeled as a heat source with uniform power dissipation and is assumed to be at a uniform temperature T c . In reality, the heat generation rate in the channel has a strong spatial dependence due to variations in the surface potential along the channel. However, the isothermal channel approximation was shown to be valid for predicting the average channel temperature, giving results typically within 10% error [Goo95]. The interface between the buried oxide and the silicon substrate is assumed to be at a uniform temperature T , and the device is assumed to cool solely through the substrate. As shown in Chapter Two, the thermal conductance to the overlying oxide layers is generally much smaller than the conductance to the substrate; therefore, heat loss through the overlying oxide layers can be neglected. The gate interconnect, the drain and source, and their metal interconnects are treated as one-dimensional cooling fins that carry heat away from the channel. The differential equations that describe the temperature rise along the respective fins are given by

PAGE 199

190 source interconnect buried oxide *gate interconnect Figure 6.3 The simplified device geometry used to define the solution domain for the SOI MOSFET thermal resistance model. The model parameters are defined in Table 6.1.

PAGE 200

191 Table 6.1 SOI MOSFET thermal resistance model parameters Parameter

PAGE 201

192 ^I4^m ^AT d (x d ) = 0, (6.1) 9x d 5 2 AT m (x m ) m r^-
PAGE 202

193 h t = -p (6.7) a fo is the heat transfer coefficient from the fin to the substrate. The heat transfer coefficient is derived by solving the one-dimensional, steady-state heat conduction equation in the underlying oxide layers. Values for the necessary material properties are listed in Table 6.2. The variable d fo corresponds to the total oxide thickness between a fin and the substrate, so that d do = d box' (6.8) d mo = d box + d fox + d LTO' (6-9) and d go = d box + d fox • (6.10) Assuming that the temperature along each fin is one-dimensional, neglects the temperature gradients in the vertical and lateral directions within each fin. The validity of this assumption can be evaluated using the Biot number, which corresponds to the ratio of the internal and external thermal resistances for a given fin [Ozi93]. If the Biot number is much less than unity, then the fin can be approximated by a one-dimensional thermal medium. The vertical Biot number for a fin is expressed as B vt = h f d f /k f . For B vf to equal 0. 1 , the thickness of the fin (d f ) would have to be approximately 5, 12 and 2 times greater than d fo for the drain/ source, drain/source interconnects and the gate interconnect, respectively.

PAGE 203

194 Table 6.2 Material properties Property

PAGE 204

195 Consequently, for typical SOI MOSFET geometries, the vertical temperature gradients in the fins can be neglected. The lateral Biot number for a fin is given by 2 B Lt = h f WjV(k f d t ) . For the drain/source interconnects, assuming they are routed in the first metal layer, B Lm approaches unity when W m = \3jd^; for practical metallization geometries, the lateral temperature gradient can be neglected. For a highly-doped polysilicon gate interconnect, the relatively low thermal conductivity can lead to more significant lateral temperature gradients. However, these gradients should be substantial only for channel lengths greater than 1 .2 fim. Due to the effects of the drain/source contacts and interconnects, the heat flow in the silicon film is more complex and the lateral Biot number is not valid for these regions. ANSYS simulations of the drain/source silicon film regions are used to evaluate the nature of the heat flow; the results are shown in Figure 6.4. The simulations show that for low thermal conductances to the substrate (i.e thicker buried oxides), the effect of the drain/source interconnects can lead to significant lateral temperature gradients in the silicon film. However, the model accurately predicts the average temperature distribution in the drain/source silicon film, showing that the one-dimensional approximation is reasonable for these regions. The temperature rise of the silicon film at the edge of the channel (x d = L d ) and of the gate interconnect at x = are given by AT d (x d )| = AT (x e ) 1 + W„d g-gox d„k k k W T g g o d -l = AT C . (6.11) x g =

PAGE 205

196 90 70 U -Bd d = 0.2 |im d box = °5 ^ m o © ANSYS: Edge of Channel a b ANSYS: Middle of Channel Model 0.0 d d = 0.04 fim dbox = 0.2 Jim 0.1 0.2 Distance Along Drain/Source Region, x d (|im) (a) 0.3 U o ANSYS: Edge of Channel -a ANSYS: Middle of Channel Model d d = 0.04 nm d box = 02 ^ m 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Distance Along Drain/Source Region, x d (|im) (b) Figure 6.4 Temperature along drain/source regions as calculated by ANSYS and predicted by the model for an SOI MOSFET with W = 12 u.m and W m = 2 |im (center contact). The ANSYS data is evaluated at both the center and the edge of the channel, to show the extent of the lateral temperature gradient: a) For L d = 0.3 u.m; b) for L d = 1.2 fim.

PAGE 206

197 where k is the thermal conductivity of the gate oxide. The multiplier for the gate interconnect temperature accounts for the thermal resistance of the gate oxide by assuming r£2* = o, (6.12) 8x AT I = AT C (6.13) g x =0 and AT I = AT (x )| . (6.14) e x = d s s x = Goodson and Flik assumed that the thermal resistance of the gate oxide was negligible since d is typically very small [Goo92]. However, (6.1 1) shows that the effect of the gate oxide thermal resistance can become more significant if k decreases or k increases; both conditions are probable in advanced SOI MOSFET's due to phonon boundary scattering in the gate oxide and the use of silicides for the gate interconnect material. The effect of the gate oxide can be removed by setting d = 0, in which case, the model reverts to the original model developed by Goodson and Flik [Goo92]. At the interface between the silicon film and the drain/source interconnect metallization (x d = x m = 0), the temperatures of the silicon and the metal are assumed to be equal and the heat flows are equated, giving the boundary conditions

PAGE 207

198 AT m ( x J| n = AT d( x d) x m = x d = (6.15) k m W m d m • ^T m (x m ) 3x„ k d Wd d x m = ^AT d (x d )n 3xj = 0. (6.16) Equation (6.16) is the boundary condition derived by Goodson and Flik, and is valid when W = W m [Goo92]. However, since the thermal analysis for the drain/source silicon film region only extends to the edge of the interconnect contact, when W > W m (6.16) neglects heat flow from the portions of the silicon film under the contact. Figure 6.5 illustrates the portions of the silicon film that are neglected by (6.16). The heat flow from these portions of the silicon film through the buried oxide is proportional to W-W m ; thus, the error introduced by neglecting this heat flow increases as W becomes greater than W m . To account for heat flow from the portions of the drain/source silicon film under the contact, (6.16) is modified. Due to the high thermal conductivity of the metal contact, the silicon film under the contact is assumed to be at a uniform temperature. For large devices where the contact area is divided into multiple contact windows, significant lateral temperature gradients can develop. ANSYS simulations are used to examine the heat flow in the silicon region under the contact; the results are shown in Figure 6.6. Even for conditions that cause considerable lateral heat flow, the model gives an accurate account of the average temperature in the silicon film under the contact. Therefore, the uniform temperature assumption can be considered appropriate. To account for the neglected conduction, an additional heatflow term is added to (6.16), so that the modified boundary condition becomes

PAGE 208

199 Figure 6.5 The simplified SOI MOSFET structure for the case where W > W m . The cross-hatched regions represent the portions of the drain/source silicon film where heat flow is neglected by the Goodson and Flik model [Goo92].

PAGE 209

200 175 U o U — 3
PAGE 210

201 -k m W m d m

PAGE 211

202 and arg2 = exp(-m d L d )(m d k d d d W-h d (W-W m )L ct -m m k m d m W m ). (6.22) For the case where W < W m , the expression for the thermal resistance is equivalent to the Goodson and Flik model [Goo92] and argl = exp(m d L d )(m d k d d d W + m m k m d m W m ) (6.23) arg2 = exp(-m d L d )(m d k d d d W-m m k m d m W m ). (6.24) The differential equations that describe the temperature rise in each cooling fin neglect the temperature dependences of the material thermal conductivities. However, the measured data plotted in Figure 6.7 show that the temperature dependence of the SOI MOSFET thermal resistance is relatively weak. For the devices used in the measurements, the extracted thermal resistances vary by less than 3% over a 90 °C temperature range. Therefore, it is assumed that the temperature dependences of the thermal conductivities can be neglected. 6.3 Verification of the SOI MOSFET Thermal Resistance Model The accuracy of the SOI MOSFET thermal resistance model derived by Goodson and Flik has been verified [Goo92, Goo95]. The model was shown to agree with thermal resistance measurements of devices with W
PAGE 212

203 880 C 860 U o X & 840 o o 8 820 "c3 H 800 780 i !

PAGE 213

204 MOSFET's with W = 120 fxm and W m = 2 pm were measured using a four-point, gate resistance thermometry technique [Goo95]. Figure 6.8 compares both models to the measured thermal resistance values. By neglecting the heat flow out of the contact regions, the model developed by Goodson and Flik [Goo92] tends to overestimate the thermal resistance, in this case by as much as 40%. With modification to account for the heat flow out of the drain/source contact regions implemented, the model provides a much more accurate estimation of the thermal resistance. For the given devices, the error is reduced to no more than three percent. 6.4 Derivation of the SOI MOSFET Thermal Impedance Model The thermal impedance model for SOI MOSFET's is derived by extending the steady-state thermal resistance model into the time domain. The time-dependent form of the differential cooling-fin equation can be expressed as 9 2 AT f 2 i 3AT f 3x f ^-[m f (t)]AT f= -^t(6.25) where a t = k f /(p f c pf ) is the thermal diffusivity of the fin material. Values for the additional material properties needed to calculate the thermal diffusivities are listed in Table 6.3. In the time domain, (6.25) does not have a simple closed-form solution. However, a simple solution to (6.25) can be found in the frequency domain using the Laplace transform.

PAGE 214

205 1300 1200 U 1100 c 1 1000 f15 -C 900 800 700 ' 1 ' 1

PAGE 215

206 Table 6.3 Material properties Property

PAGE 216

207 In the frequency domain, the differential equations (6.1), (6.2) and (6.3) become 9 3"0 d (x d , s) [ 2 s 1 1 [m d (s)] 2 + -f e d (x d ,s) = (6.26) 3x d a d. a 2 e m (x m ,s) f 2 s 1 —^f [m m (s)] 2 + -l9 m (x m ,s) = (6.27) 3x m I u m J a^e s (x e , s) 2 s g 2 g [m g (s)] 2 + A 9 g (x g ,s) = (6.28) dx' [ a g) where f (x f , s) = £ [AT f (x f , t)] is the Laplace transform of the time-domain temperature rise and the initial condition in each fin is specified as AT f (x f , 0) = [Ozi93]. Examining (6.26), (6.27) and (6.28), shows that the differential cooling fin equation can be expressed in the same form for either the steady-state or the frequency domain. Therefore, the frequency-domain differential cooling-fin equations have closed-form general solutions given as f (x f , s) = c,exp(Y f x f )+c 2 exp(-Y f x f ) (6.29) where Y f = [m f (s)] 2 + A (6.30) CXf

PAGE 217

208 and Cj and c 2 are arbitrary constants that depend on the boundary conditions. The use of the Laplace transform requires that the temperature rise remains a linear function of the power dissipation over the desired region of operation. Since the thermal resistance only depends weakly on device bias and temperature, this assumption is valid for typical device operation. As mentioned in the first part of this chapter, the characteristic thermal length, l/m f , of a cooling fin accounts for heat conduction through the underlying oxide layers. During a thermal transient, the characteristic thermal length will vary, l/m f = l/m f (s) , due to the time dependence of the heat transfer in the underlying oxide. Using the steady-state heat transfer coefficient to calculate the characteristic thermal length would imply that the heat flux in the buried oxide instantaneously adjusts to a steady-state value. Therefore, the heat transfer coefficient for the fin, h f (s) , becomes a function of frequency and should be derived from a dynamic analysis of the heat flux in the underlying oxide. Assuming the heat flow in the oxide is predominantly one-dimensional, the temperature rise in the underlying oxide is described by a 2 AT (n,t) i 9AT (n,t) where AT (n, t) = T (n, t) T , a o is the thermal diffusivity of silicon dioxide and n is the direction of heat flow. Taking the Laplace transform of (6.31) results in a 2 e (n, s) s -£5— -e o (n,s) = 0. (6.32)

PAGE 218

209 where the initial temperature rise in the oxide is assumed to be AT (n, 0) = 0. The fin and the underlying oxide are assumed to be in perfect thermal contact, so that at the interface between the fin and the oxide e (s) = e f (s) (6.33) The interface between the buried oxide and the substrate is still assumed to be at the temperature T , so at that interface e n (s) = o. (6.34) The heat flux at the interface between the fin and the oxide can be derived from the solution to (6.32) for the given boundary conditions, and is given by 30 (n, s) I ' /— d fo 1 exp -2 / — d V a/cl fo 6 f (s) . (6.35) Therefore, the dynamic heat transfer coefficient can be expressed as h f (s) = k o JIcc
PAGE 219

210 and e m (x d ,s)| =0. (6.38) The boundary condition at the edge of the channel becomes 0d( x d> s )| , =r goxl 6(x s)| =6 c (s) (6.39) where r l+exp(-2y gox d gox ) | k g y g W g d g [l-exp(-2 Yg0X d g0X )] g° xl 2exp(Ygox d gox ) 2k goxYgox W-Lexp(Ygox d gox ) and 'V^gox (6.40) (6.41) Equation (6.40) is derived by solving (6.32) in the gate oxide for the boundary conditions implied by (6.13) and (6.14). The boundary conditions that define the temperature and heat flow at the interface between the drain/source regions and the interconnect metallization become e m( x m> s )| =9d( x d> s )| n (6-42) and

PAGE 220

21 = k m W m d m 3x m -k d Wd d + h d (s)[W-WJL ct 9 d (x d ,s)| ^d( x d^)l x d = 0. (6.43) x d = The Laplace transform of the power conservation equation that couples the conductance paths is p(s) = 2k d Wd d [ae d (x d , s> 3x, k g w g d g X.. — L,. •36 g (x g> s)i 3x„ • r gox2 x g = o (6.44) + h d (s)WLe c (s) where 2exp(-y d ) gox2 [exp(-2y d )+l]^eoxTeox^ ^ k 2 Y 2 d e W [^(^YaoX^Ox)1 ] g g g (6.45) and p(s) = £[p(t)] is the Laplace transform of the instantaneous power dissipation in the channel. Equation (6.44) can be solved in the form AT C = Z TH (s)p(s), (6.46) resulting in the following expression for the dynamic thermal impedance ^th' s ) — -2(arg l-arg2)k d y d d d W -r-1 argl + arg2 + k g Tg d g W g r gox2 + h d (s)W • L (6.47) where

PAGE 221

212 argl = exp[L d Y d ]{k d Y d d d W + h d (s)[W-W m ]L ct + k m y m d m W m } (6.48) and arg2 = exp[-L d y d ]{k d y d d d W-h d (s)[W-W m ]L ct -k m y m d m W m }. (6.49) 6.5 Verification of the SOI MOSFET Thermal Impedance Model To verify the thermal impedance model, three-dimensional (3-D) finiteelement (FE) simulations of an SOI MOSFET were performed using ANSYS. The FE model represented a simplified device structure and was constructed using similar assumptions and boundary conditions to those detailed in Chapter Two. The model was further simplified by neglecting the gate interconnect. Figure 6.9a shows the FE model used to represent the SOI MOSFET structure in ANSYS. The thermal impedance was extracted from the FE simulations by averaging the temperature rise over the channel region. The ANSYS results are compared to (6.47) with the term corresponding to conduction in the gate interconnect set to zero. The comparisons are illustrated in Figure 6.9b, Figure 6.10, and Figure 6.1 1. The SOI MOSFET thermal impedance model displays reasonable trends with the scaling of the silicon film thickness, the buried oxide thickness, and both channel width and length. The model does a good job of reproducing both the transient and steady-state temperature responses produced by the 3-D FE simulations. For the device structures that were simulated, the error between the model and ANSYS did not exceed 14%. As is evident from the comparisons in Figure 6.10, the discrepancy between the model and the FE simulations is greatest during the latter half of the transient. This trend can be

PAGE 222

213 (a) o 12 10 o e a -a Cm a OANSYS — Model S 8 6 2 10 -13 -e-Q o oo -5 10" Figure 6.9 The transient thermal impedance simulated with ANSYS and calculated with the SOI MOSFET thermal impedance model: a) The finite-element model simulated with ANSYS; b) comparison of the thermal impedance simulated with ANSYS and the model for W = 8 Jim, L = 0.35 |im, L d = 0.6 ^m, d d = 0.15 |im, d box = 0.4 urn, W m = 2 ixm and d m = 0.9 |im.

PAGE 223

214 15 U o S 10 X 0) o c cd T3
PAGE 224

215 o O
PAGE 225

216 attributed to the lack of modeling of the contact structure itself. The model effectively assumes that the thermal resistance and capacitance of the contact structure is negligible. For devices with a large contact area, this assumption is valid. However, for devices with a relatively small contact area, the effectiveness of the drain/source interconnects as heat conductance paths is diminished. The temperature in such devices will rise quicker and to a greater magnitude than predicted by the model. The extent of the discrepancy can be reduced by tuning the width (and/or thickness) of the drain/source interconnects to reduce their effect on the thermal impedance. Equation (6.47), in the complete form, was also compared to measured thermal impedance data extracted by Lee and Fox using transient drain current measurements [Lee95]. Figure 6.12 compares the measured and simulated data for a transient thermal impedance of an SOI MOSFET fabricated at Texas Instruments. Again, the model accurately predicts both the steady-state and transient temperature response. Part of the discrepancy at the earlier times is due to the finite rise time of the pulse used in the measurements. The model, as calculated for Figure 6.12, represents the temperature response to an ideal step in power dissipation, and will show a faster rise than the measurements. 6.6 Summary Due to the low thermal conductivity of the silicon dioxide used for the buried oxide isolation, the thermal impedance of SOI MOSFET's is typically larger

PAGE 226

15 U o 10 x U e a -a u aM 5 15 xi 217 -i r 1 — i — i i i i I Measurement Model _i i i ' I i t i i i I 10 -9 icr love I0~ 10" Time (sec) Figure 6.12 Measured and simulated data for the transient thermal impedance of a Texas Instruments SOI MOSFET with W = 2.4 (im, L = 1.8 urn, d d = 0.17 |im and d box = 0.33 u.m.

PAGE 227

218 than that of their bulk counterparts. Therefore, self-heating effects can be enhanced and it is important to have a physical model that can predict the dynamic temperature rise. This chapter presented a physical thermal impedance model for SOI MOSFET's. The model was shown to agree reasonably well with both threedimensional finite-element simulations and measurements.

PAGE 228

CHAPTER 7 THE THERMAL IMPEDANCE PRE-PROCESSOR: TIPP 7.1 Introduction The most common way to incorporate dynamic thermal effects into circuit simulators, using a thermal impedance to model the effective temperature rise in a transistor, is described in Chapter One. The analogy between electrical and thermal behavior can be used to represent the thermal impedance with a thermal equivalent circuit consisting of discrete resistances and capacitances. Consequently, thermal equivalent circuits provide a suitable means for efficient DC, AC and transient electrothermal circuit simulation. For steady-state heat conduction, there is a simple correspondence between the thermal impedance and the thermal equivalent circuit: the sum of the resistance components in the thermal equivalent circuit is equal to the thermal spreading resistance. However, for a full dynamic thermal response, the correlation between the thermal impedance data and the component values of the thermal equivalent circuit is more complex. Thus, there is a need for a consistent and efficient approach for generating the component values from dynamic thermal impedance data. This chapter presents a computer program that implements a systematic approach for calculating the component values of the thermal equivalent circuit. The 219

PAGE 229

220 general operation of the program is discussed briefly. The numerical algorithms used to generate the thermal equivalent component values from thermal impedance data are examined in some detail. Examples of calculated thermal equivalent circuits are presented and a method for interfacing the program with electrothermal circuit simulators (ETCS's) is discussed. 7.2 A Description of TIPP The Thermal Impedance Pre-Processor (TIPP) was developed to facilitate the use of ETCS's for modeling self-heating effects in semiconductor circuits. TIPP is a stand-alone software package for the specific purpose of providing an ETCS with access to thermal impedance models and the component values for their thermal equivalent circuits. The evolution of TIPP as a pre-circuit-simulation processor serves three purposes. First, the complexities of modifying an ETCS program to incorporate a thermal impedance model are avoided. The modifications would be specific to each circuit simulator and each thermal impedance model; therefore, the modifications would have to be repeated for each implementation. Second, the numerical burden of calculating the component values for the thermal equivalent circuit is removed from the circuit simulator. Third, TIPP was designed to provide an open frame-work for the generation of physics-based thermal impedance models and their thermal equivalent circuits; and therefore, gives a modifiable platform which can be accessed by any ETCS. TIPP is written in a modular format using the C programming language. The program structure allows for the easy addition of new

PAGE 230

221 thermal impedance models or new numerical algorithms. At the present time, TIPP contains the physics-based models that are derived in the preceding chapters. While physics-based models offer the predictive capabilities that are useful in a development environment, they can not always provide the necessary accuracy required by a design environment. In such cases, the precision of optimized empirical data is essential. The numerical methods that TIPP uses to generate the thermal equivalent circuit are independent of the data source; therefore, TIPP can use either measured thermal impedance data or the predictive physical models to generate the component values for the thermal equivalent circuit. The output from TIPP is in the form of a netlist which contains the components for any necessary thermal equivalent circuit(s). The generated thermal equivalent netlist can then be incorporated into an existing circuit file in preparation for an electrothermal simulation. The TIPP program structure is simple and was designed to provide results without drastically increasing the complexity of circuit simulation. TIPP is invoked from the command line using the parameters in Table 7.1. Table 7.1 TIPP command-line parameters Name Description filename -sp -sa Name of file that contains input for TIPP Option flag to instruct TIPP to use SPICE format for output file (filename. sp) Option flag to instruct TIPP to use SABER format for output file(s) (filename. sin)

PAGE 231

222 The basic operation of TIPP is outlined by the flowchart shown in Figure 7.1. The input for TIPP is contained in a text file that is composed of a set of instruction parameters, which are defined in Table 7.2. Table 7.2 TIPP input parameters Name

PAGE 232

223 Parse TIPP input file Approximate time constants Generate thermal response from model Calculate resistance components Calculate capacitance components Figure 7.1 A flowchart illustrating the general operation of the TIPP software package

PAGE 233

224 haddix a 0EV soimos DATA phys NP 5 W 8 L 0.35 LD 0. 6 LCT . 6 LM 10 WM 2 WG 1 DD 0.1 DM 0. 9 DG 0.3 DBOX 0.4 DOX 1 Figure 7.2 Example of a TIPP input file for an SOI MOSFET. The first three lines contain the TIPP-specific parameters for the simulation. The other parameters are used to describe the geometry of the transistor.

PAGE 234

225 7.3 Generation of Thermal Equivalent Circuits In Chapter One, the representation of a thermal impedance using circuit elements was discussed, and for this work, a Foster network was chosen to model the normalized temperature response. The components for the thermal equivalent circuit are generated by numerically fitting the multiple-pole network response to thermal impedance data; which TIPP assumes is in the form of a normalized (units of °C/W) transient step response or a normalized frequency response. Therefore, the network response of the thermal equivalent circuit must be derived in both the frequency and time domain. The voltage generated across the thermal equivalent circuit can be expressed as V(s) = I(s) • Z TH (s) , (7.1) where V(s) represents the effective temperature rise of the device and I(s) represents the power dissipation. The normalized temperature response for an impulse in the power, where the Laplace transform of the impulse function is £[5(t)] = 1 , is simply equivalent to the impedance of the thermal equivalent circuit. Thus, for an n -stage Foster network, the thermal impedance can be expressed as rj , , r thl r th2 r thn ,~ ~, Z TH (s) = — + — + •••+— , (7.2) 1H 1+s/cO] + s/co 2 + s/co n where r th| is the j -th thermal resistance component, ^r th j = R TH is tne thermal spreading resistance and CO: is the j-th pole. Equation (7.1) yields the temperature

PAGE 235

226 response to a unit step function, U(t) , when I(s) = 1 /s . Therefore, the time-domain thermal step response can be derived by multiplying (7.2) by 1/s and taking the inverse Laplace transform. The resulting expression is Z TH (t) r thl P U" + r th2 expl — \x 2 + --+r thn expg (7.3) where x, is the i -th time constant and co ; = 1/T: . j J j j 7.3.1 Approximation of the Thermal Equivalent Poles/Time Constants The first step in generating the thermal equivalent circuit is to calculate the pole or time constant associated with each stage of the thermal equivalent circuit. The initial approach used to calculate each pole/time constant was nonlinear optimization. The efficiency of the optimization algorithm was strongly dependent on the number of poles/time constants and the accuracy of the initial guess for the poles/time constants. However, the accuracy of the thermal equivalent circuit fit was found to be relatively insensitive to the exact location of the poles/time constants. Figure 7.3 shows an example of the variation in the fit accuracy that can occur when the locations of the poles/time constants are shifted. The variation in the fit accuracy was typically less than five percent for a ten percent change in any pole/time constant, and was never greater than ten percent. The results are typical for each thermal impedance model over most practical device geometries. Consequently, the insensitivity of the fit accuracy to the pole/time constant location can be exploited to develop a heuristic approximation for calculating the location.

PAGE 236

227 ^ o Z D o c C3 l— > < G — 0+10% Variation in Each x ; B — Q-10% Variation in Each X; Number of Poles Figure 7.3 Example of the variation in the accuracy of a thermal equivalent circuit due to changes in the location each time constant. The bulk bipolar model is used to generate the thermal impedance data. The time constants of the thermal equivalent circuit are independently varied by ten percent in either direction. The difference norm between the thermal impedance data (TID) and the thermal equivalent circuit (TEC), /s /(TID 1 -TEC,) 2 + (TID 2 -TEC 2 ) 2 +---+(TID m -TEC m ) 2 , is calculated for m discrete data points. The variation in the difference norm is averaged over each time constant.

PAGE 237

228 The algorithm for approximating the poles/time constants employs a binary search procedure that "scans" the thermal impedance magnitude for specific points. Figure 7.4 shows a graphical illustration of the algorithm when used to approximate five poles of a thermal equivalent circuit from a frequency-domain thermal impedance. For each stage of the thermal equivalent circuit, a single point is found which corresponds to a certain percentage, < Pj < 1 , of the impedance maximum, max. The corresponding frequencies/times at which these points occur are used to approximate the poles/time constants. The value used for each percentage, p j? was determined from the results of the original optimization algorithm performed on the bulk bipolar thermal impedance model. The model was evaluated for a wide range of device geometries and the final value for each percentage was taken as the average over this range. Though the values for the percentages were calculated for a specific model, they tend to be independent of the thermal impedance data source and provide accurate fits in most cases. The aforementioned approach for determining the location of the poles/ time constants for a thermal equivalent circuit is simple and more efficient than numerical optimization. However, this technique is not completely robust, and could produce erroneous results if there is sufficient noise is present in the thermal impedance data. Notwithstanding, the open program structure of TIPP would allow the use of possibly more robust techniques such as moment matching [Pil94].

PAGE 238

229 max 000000000 Pj*max p 2 *max U o O O E 15 p 3 *max H p 4 *max p 5 *max Thermal Impedance Data Multiple-Pole Fit CO] co 2 co 3 co 4 Frequency (Hz) €^) Figure 7.4 Illustration of the algorithm that TIPP uses to approximate the poles/ time constants for a thermal equivalent circuit. In this example, a fivepole thermal equivalent circuit is fit to a frequency-domain thermal impedance. The magnitude of the thermal impedance is scanned for specific points that correspond to certain percentages, pj, of the impedance maximum, max. The frequencies at which these points occur are used to approximate the poles, 0);, of the thermal equivalent circuit.

PAGE 239

230 7.3.2 Calculation of the Thermal Equivalent Components After the necessary poles/time constants are predicted, the resistance and capacitance components of the thermal equivalent circuit can be calculated. Equations (7.2) and (7.3) show that the response of the thermal equivalent circuit is linearly proportional to the resistance components. Thus, the resistance components can be efficiently derived by minimizing the discrete least-squares error between the thermal impedance data and the response of the Foster network. A constraint is placed upon the thermal resistance components such that for an n -stage thermal equivalent circuit n = ssresp-(r,+r 2 +---+r (n _ 1) ) , (7.4) where ssresp is the steady-state value of the thermal impedance data; therefore, the system of unknowns is reduced from n to (n1) independent variables. The leastsquares approach along with (7.4) yields the following system of linear equations A, A. A A 12 22 L l(n-1) L 2(n-1) A (n1)1 A (n-1)2 ••• A (n-l)(n-l)

PAGE 240

231 V= I exp|^)-exp(^ ' l f _t > exp — exp n k /J (7.6) and r,= I "^HKv, • < respj ssresp 1 exp -tjYl (7.7) when using (7.3) to fit a normalized thermal step response, resp, with m discrete data points. If resp corresponds to a normalized thermal frequency response, equation (7.2) is used to fit the data so that V = I i = 1 1 +Sj/a)j 1 + Sj/(JL) n l+s/co k L+s/to n (7.8) and r,= I i = 1 1 + Sj/Cflj + s/co n respj ssresp • l+s/co n (7.9) The n resistance components can be calculated by solving (7.5) and (7.4). Since the resulting coefficient matrix is never larger than 4x4, Gaussian elimination and back-substitution are used to solve the system of linear equations. After the resistance components are determined, the capacitance components can be calculated using the following equation

PAGE 241

232 c thj = V r thj = l/ (°>j r thj)(7 10) Examples of complete thermal equivalent circuit fits are shown in Figure 7.5. The thermal equivalent circuits show good agreement with the thermal impedance data generated from both measurement and a physics-based model. The plots also illustrate how the distributed nature of the thermal impedance is more accurately represented by thermal equivalent circuits using multiple poles. 7.4 The Interface Between TIPP and Circuit Simulators The last operation of the TIPP program is to output the calculated resistance and capacitance components of a thermal equivalent circuit. Since TIPP is intended to function as a companion to any given ETCS, the format of the output should be consistent with the syntax of that simulator. Therefore, the interface that TIPP uses to communicate with an ETCS should be flexible and convenient. The role of a pre-processor was chosen so that TIPP's own program structure would be independent of ETCS program syntax. While this approach is not as immediate as directly coupling TIPP to an ETCS, it avoids the unwieldy modifications, to both circuit simulator and TIPP, that would be required to implement a direct interface. Since it is simple to modify or extend the TIPP program alone, its output can be constructed in a format that is compatible with any ETCS. As shown in Table 7.1, TIPP's output can be formatted for either a SPICE-type or the SABER [Sab95] circuit simulator. This output is written to a text file, or group of text files, that can easily be incorporated into an existing circuit file. If a specific

PAGE 242

150 U o X if o T3 1) E 100 E 50 c3 1) -C H 10 -12 " ' 233 — i — i i n i i f "O" -e — e — eOThermal Impedance Model -1-PoleTh. Eq. Circuit — 3-Pole Th. Eq. Circuit Time (sec) (a) 0' 10" 10 L 150 1 1 — i i i i i i — I 1 1 — i — rrrrr* u o X u o -a
PAGE 243

234 output format is not selected, TIPP writes a simple list of the components for each thermal equivalent circuit. For SPICE-type circuit simulators, TIPP writes a single output file (filename. sp) that contains a subcircuit netlist for each thermal equivalent circuit. For the MEBJT/MEHBT thermal impedance model, the output file will also contain a subcircuit for the complete device. The MEBJT/MEHBT subcircuit consists of the necessary transistor cards and elements to model each finger and each thermalcoupling network. For the SABER circuit simulator, TIPP writes one output file (filename. sin) for each generated thermal equivalent circuit. Each file contains the corresponding netlist in the proper SABER template format. If the MEBJT/MEHBT thermal impedance model is used to generate the thermal equivalent circuits, a separate output file is created for the MEBJT/MEHBT electrothermal model. As with the SPICE-formatted output, the MEBJT/MEHBT template contains the netlists for the necessary thermal-coupling networks. The formats used to generate the SPICE and SABER output files are shown in Figure 7.6 and Figure 7.7, respectively. If a circuit simulator does not use either of the mentioned formats, the TIPP program can be easily extended to provide the required output format. 7.5 Summary In order to investigate thermal effects in circuits and devices, circuit simulators have been modified to account for dynamic temperature variations. The temperature response of a device can be modeled by its thermal impedance, which

PAGE 244

235 * Subcircuit for Thermal Equivalent Circuit .subckt 1 (np+ 1) Z rthl 1 2 cthl 1 2 rth(np) np (np + 1) cth(np) np (np+ 1) .ends Z * Subcircuit for MEBJT/MEHBT .subckt 12 3 4 MEBJT * Thermal Coupling Networks xsl os (os + 1) ZS ecl2(os+l) (os + 2) ncp ncn 1 eclnf (os + nf 1 ) (os + nf) ncp ncn vsl (os + nf) fpl (os + nf + 1) vsl 1 xc21 (os + nf + 1) (os + nf + 2) ZC1 xcnfl (os + 2nf-l) ZC(nf-l) .ends MEBJT Figure 7.6 Template for TIPP output in SPICE format, where os is an arbitrary off-set for the node numbering.

PAGE 245

236 # Template for Thermal Equivalent Circuit template Z 1 (np+ 1) { r.rthl 1 2 = value c.cthl 1 2 = value r.rth(np) np (np + 1 ) = value c.cth(np) np (np+l)= value } # Template for MEBJT template MEBJT 12 3 4 { # Thermal Coupling Networks ZS.zsl os (os + 1) vcvs.tcl2 ncp ncn (os+1) (os + 2)=k=l vcvs.tclnf ncp ncn (os + nf-1) (os + nf)=k v.vsl (os + nf) 0=0 cccs.pl i(v.vsl) (os + nf + 1) = k = 1 ZCl.zc21 (os + nf +1) (os + nf + 2) ZC(nf-l).zcnfl (os + 2nf-l) } Figure 7.7 Template for TIPP output in SABER format, where os is an arbitrary off-set for the node numbering.

PAGE 246

237 can be derived from physical models or extracted from measurements. Multiple-pole thermal equivalent circuits provide an efficient method for incorporating thermal impedance data into a circuit simulation. The TIPP computer program was developed as a general frame-work for thermal impedance modeling. TIPP generates thermal equivalent circuits using its internal physics-based models or imported measured data. The flexibility of the TIPP program structure allows new thermal impedance models to be added easily and simple interfacing with electrothermal circuit simulators.

PAGE 247

CHAPTER 8 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK 8.1 Conclusions The primary goal of this dissertation was to develop physical models that could predict the thermal impedance of semiconductor devices and also remain efficient enough to be used for circuit simulation. Four thermal impedance models were presented for different device structures, all of which satisfy the abovementioned conditions. Each model is derived from the physical heat conduction equation, and is therefore capable of accurately predicting the thermal impedance based solely on material properties and device geometry. In addition, the models are in the form of compact analytical expressions which produce results quickly, therefore offering a more efficient alternative to costly numerical thermal simulations. The development of thermal impedance models for BJT's, on both bulk and SOI substrates, was detailed in Chapters Two and Four. The bulk BJT model, originally developed by Joy and Schlig [Joy70], was extended to account for finite substrate thickness and devices with multiple emitters. The DIBJT model was derived to depict the complete transient temperature response and was then simplified to produce an additional model for the steady-state thermal resistance. Both models were augmented to account for the thermal resistance and heat capacity 238

PAGE 248

239 of the emitter interconnect metallization. In addition, three-dimensional finiteelement simulations were used to show that the effects of advanced dielectric isolation on the thermal impedance of bulk BJT's can be significant for highly-scaled transistors. In Chapter Three, a circuit model for thermal coupling between devices in a common substrate was derived. The circuit representations were developed to facilitate the implementation of the multiple-emitter BJT thermal impedance model in circuit simulation; however, they are also applicable to the general case of thermal coupling between arbitrary devices. The thermal coupling model was then used to develop an extraction methodology for an efficient, lumped electrothermal model for multiple-emitter BJT/HBT's. Thermal impedance models for MOSFET's, on both bulk and SOI substrates, were described in Chapters Five and Six. The bulk MOSFET model improved on the work by Sharma and Ramanathan [Sha83] by accounting for crosswidth temperature gradients and the linear variation of the electric field along the length of the channel. Three-dimensional finite-element simulations were used to investigate the effects of device interconnects and isolation on the thermal impedance of bulk devices. The SOI MOSFET model was based on a modified version of the steady-state thermal resistance model developed by Goodson and Flik [Goo92]. The modified steady-state model was extended for time-dependent heat conduction to produce the dynamic thermal impedance model. In Chapter Seven, the Thermal Impedance Pre-Processor (TIPP) was introduced. The TIPP software program was developed to provide efficient and

PAGE 249

240 systematic algorithms for generating thermal impedance models and the thermal equivalent circuits necessary for electrothermal circuit simulations. Finally, a number of the accomplishments that were the result of this research, have been documented in various journal and conference publications. These papers are cited, as follows, in the reference list at the end of this dissertation [Bro93, Bro97, Fox93a, Zwe95a, Zwe95b, Zwe96]. 8.2 Recommendations for Future Work The thermal impedance models presented in this dissertation can provide accurate predictions when used within the limits of their derivations. However, the robustness of the models, over a wider range of device structure types and sizes, can be enhanced further. Two areas are presented for future research that could improve the accuracy of the models. In addition, a third research topic concerning thermal coupling in SOI MOSFET circuits is suggested. 8.2.1 The Temperature Dependence of the Thermal Conductivity The heat conduction equation used to derive the thermal impedance models in this dissertation assumes that the thermal conductivity is constant, and therefore independent of temperature. For small to moderate temperature excursions, this assumption is valid; however, for a large temperature rise (50-100 °C) above the reference ambient, the thermal conductivity can vary by as much as 31% [Gao89]. Such large temperature excursions in transistors are common in high-power

PAGE 250

241 applications and therefore, the accuracy of the thermal impedance models could be improved by accounting for the temperature dependence of the thermal conductivity. The variation of the thermal conductivity with temperature makes the heat conduction equation non-linear, such that V.k(T)VT + g = p-c p |, (8.1) where the temperature dependence of the specific heat is assumed to be relatively weak compared to that of the thermal conductivity. The Kirchoff transformation [Joy75] linearizes (8.1) by using the following variable transformation T(t) = T + l f k(T)dT, (8.2) 1 o where k is the thermal conductivity at the reference ambient temperature, T . The relation between the linear and non-linear temperature variables can be examined by multiplying both sides of (8.2) by k , taking the gradient and then the divergence, resulting in k V 2 8 = V.k(T)VT = p-c p |j--g. (8.3) Consequently, if the temperature dependence of the thermal conductivity is known and can be expressed as an integrable equation, (8.2) can be used to correct the temperature calculated with the linear heat conduction equation.

PAGE 251

242 The correction to the estimated temperature, accounting for the temperature dependence of the thermal conductivity, can be implemented in an electrothermal circuit simulator by adjusting the voltage generated at the temperature node. Preliminary attempts at such a modification have shown that the approach is viable. However, the implementation must de handled carefully to ensure consistency between the thermal and electrical solutions, and to reduce possible convergence problems. It is suggested that further research be conducted to ascertain the optimum approach for incorporating the Kirchoff transformation into an electrothermal circuit simulator. 8.2.2 Models for Thermal Effects Due to Advanced Isolation The thermal impedance models for bulk BJT's and MOSFET's described in Chapters Two and Five, neglect the effects of the oxide isolation structures (e.g. LOCOS or trench) used to electrically isolate devices in a common substrate. Due to their low thermal conductivity, these isolation structures can significantly increase the thermal impedance of highly-scaled devices. Consequently, there can be substantial errors between the results predicted by the models and actual observed thermal impedances. Therefore, the bulk models should be modified to improve their accuracy for highly-scaled devices that are fabricated with advanced isolation structures. By implementing the Green's function technique that is detailed in Chapter Four, both the bulk BJT and bulk MOSFET thermal impedance models can be modified to account for isolation structures. Adapting this analysis to the bulk

PAGE 252

243 models would require that the physical device be divided into two domains, which can be thought of as intrinsic and extrinsic thermal regions. The intrinsic thermal domain can be approximated by a rectangular volume with boundaries defined at the surface of the substrate, at the depth of the isolation structure and at the interfaces between the intrinsic electrical device and the "side-walls" of the isolation structure. The extrinsic thermal domain would represent the exterior portions of the substrate that surround the isolation structure. The three-dimensional heat conduction equation would be solved within the intrinsic thermal domain. The thermal boundary conditions would be derived from the nature of the thermal conduction in the extrinsic thermal domain. The difficulty in this implementation, lies in the derivation of the boundary conditions. It is suggested that the ANSYS finite-element solver be employed to extensively simulate isolated device structures. From these simulations, reasonable assumptions can be formulated to simplify the heat conduction analysis in the exterior thermal domain for different isolation structures. Once the thermal analysis is simplified, boundary conditions for the intrinsic thermal domain can be deduced. 8.2.3 A Model for Thermal Coupling in SOI MOSFET Circuits For MOSFET's fabricated in a common substrate using a bulk technology, thermal coupling between devices is due primarily to heat transport through the substrate. In such cases, the thermal impedances that describe the thermal interactions between devices can be modeled using the analysis similar to that in Chapter Two for MEBJT's. Due to the insulating properties of the buried and field

PAGE 253

244 oxides, MOSFET's fabricated on an SOI substrate might be considered thermally isolated from one another. However, recent work by Tenbroek et al. [Ten96] showed that there can be thermal coupling between MOSFET's on a common SOI substrate. Since SOI devices are effectively isolated within the substrate, the dominant mechanism of thermal coupling between MOSFET's is the heat conduction through the device interconnects. The effects of thermal coupling in an SOI MOSFET current mirror are illustrated in Figure 8.1; the circuit shown in the inset is modeled after the current mirrors measured by Tenbroek et al. [Ten96] and was simulated in SOISPICE [Fos95]. The reference device and device Ml are assumed to be located in close proximity of one another such that there is significant thermal coupling. Device M2 is assumed to be located a relatively large distance away from the other devices, and is therefore thermally isolated. The drain voltage of M2 is fixed while the drain voltage of Ml is allowed to vary. The drain current of device M2 should be equivalent to, or mirror, the drain current of the reference device and should be independent of V D1 . However, as the simulation shows, the drain current of M2 does depend on the drain voltage V D] . This phenomenon can be explained by the thermal coupling between Ml and the reference device. As the power in Ml increases, the temperature in the reference device will increase. Since the drain current of the reference device is fixed, the gate voltage will have to increase to counter the effects of the reduced carrier mobility caused by the increase in temperature. Therefore, the gate voltage of M2 will also increase, resulting in an increase in drain current.

PAGE 254

245 840 830 820 Q

PAGE 255

246 A preliminary model for thermal-coupling impedance between SOI MOSFET's, which was used for the simulation in Figure 8.1, has been developed. However, the model is not complete. Twoand three-dimensional ANSYS simulations, used in conjunction with measurements of the test structures shown by Tenbroek et al. [Ten96], are suggested to further the development and validate the accuracy of the thermal-coupling impedance model. Once the model is complete, it can be used with a physics-based circuit simulator, such as SOISPICE [Fos95], to investigate the effect of thermal coupling in a wide range of SOI circuits.

PAGE 256

REFERENCES [Ans96] ANSYS User's Manual, Revision 5.3, SAS IP, Inc., Houston, PA, June 1996. [Bau93] Peter Baureis and Dieter Seitzer, "Parameter Extraction for HBT's Temperature Dependent Large Signal Equivalent Circuit Model," in Proceedings GaAs IC Symposium, San Jose, CA, pp. 263-266, 1993. [Bau94] Peter Baureis, "Electrothermal Modeling of Multi-Emitter Heterojunction Bipolar Transistors (HBTs)," in Proceedings International Workshop on Integrated Nonlinear Microwave and Millimeter Circuits, Duisburg, Germany, pp. 145-148, 1994. [Ber9 1 ] Michael Berger and Zhiqin Chai, "Estimation of Heat Transfer in SOIMOSFET's," IEEE Transactions on Electron Devices, vol. 38, no. 4, pp. 871-875, April 1991. [Bie95] Juergen Bielefeld, Geor Pelz, Hans Bernd Abel and Gunter Zimmer, "Dynamic SPICE-Simulation of the Electrothermal Behavior of SOI MOSFET's," IEEE Transactions on Electron Devices, vol. 42, no. 1 1, pp. 1968-1974, November 1995. [Bro93] Jonathan S. Brodsky, David T. Zweidinger and Robert M. Fox, "Physics-Based Multiple-Pole Models for BJT Self-Heating," in Proceedings IEEE Bipolar Circuits and Technology Meeting, Minneapolis, MN, pp. 249 -252, 1993. [Bro97] Jonathan S. Brodsky, Robert M. Fox, David T. Zweidinger and Surya Veeraraghavan, "A Physics-Based, Dynamic Thermal Impedance Model for SOI MOSFET's," IEEE Transactions on Electron Devices, vol. 44, no. 6, pp. 957-964, June 1997. 247

PAGE 257

248 [Cav93] Anthony L. Caviglia and Agis A. Iliadis, "Linear Dynamic SelfHeating in SOI MOSFET's," IEEE Electron Device Letters, vol. 14, no. 3, pp. 133-135, March 1993. [Cav95] Anthony Caviglia and Agis A. Iliadis, "A Large-Signal Model for SOI MOSFET's Including Dynamic Self-Heating Effects," in Proceedings IEEE International SOI Conference, Tucson, AZ, pp. 16-17, October 1995. [Che95] Yu-Guang Chen, Shyh-Yih Ma, James B. Kuo, Zhiping Yu and Robert W. Dutton, "An Analytical Drain Current Model Considering Both electron and Lattice Temperatures Simultaneously for Deep Submicron Ultrathin SOI NMOS Devices with Self-Heating," IEEE Transactions on Electron Devices, vol. 42, no. 5, pp. 899-906, May 1995. [Dav92] C. Davis, G. Bajor, J. Butler, T. Crandell, J. Delgado, T. Jung, Y. Khajeh-Noori, B. Lomenick, V. Milam, H. Nicolay, S. Richmond and T. Rivoli, "UHF-1: A High Speed Complementary Bipolar Analog Process on SOI," in Proceedings IEEE Bipolar Circuits and Technology Meeting, Minneapolis, MN, pp. 260-263, 1992. [Daw94] Dale E. Dawson, "Thermal Modeling, Measurements and Design Considerations of GaAs Microwave Devices," in Proceedings GaAs IC Symposium, Philadelphia, PA, pp. 285-290, 1994. [Del91] V. dela Torre, J. Foerstner, B. Lojek, K. Sakamoto, S. L. Sundaram, N. Tracht, B. Vasquez and P. Zdebel, "MOSAIC V A Very High Performance Bipolar Technology," in Proceedings IEEE Bipolar Circuits and Technology Meeting, Minneapolis, MN, pp. 21-24, 1991. [Fei92] S. Feindt, J-J. J. Hajjar, J. Lapham and D. Buss, "XFCB: A High Speed Complementary Bipolar Process on Bonded SOI," in Proceedings IEEE Bipolar Circuits and Technology Meeting, Minneapolis, MN, pp. 264-267, 1992. [Fos95] J. G. Fossum, R. M. Fox and D. T. Zweidinger, "SOI-SPICE-4 (Version 4.1) Silicon-On-Insulator MOSFET SPICE with SelfHeating," Programmer's Reference Manual, University of Florida Department of Electrical and Computer Engineering, 1995.

PAGE 258

249 [Fox91a] Robert M. Fox and Sang-Gug Lee, "Predictive Modeling of Thermal Effects in BJTs," in Proceedings IEEE Bipolar Circuits and Technology Meeting, Minneapolis, MN, pp. 89-92, 1991. [Fox91b] Robert M. Fox and Sang-Gug Lee, "Scalable Small-Signal Model for BJT Self-Heating, " IEEE Electron Device Letters, vol. 12, no. 12, pp. 649-651, December 1991. [Fox93a] R. M. Fox and J. S. Brodsky, "Effects of Self-Heating-Induced Negative Output Conductance in SOI Circuits," in Proceedings IEEE International SOI Conference, Palm Springs, CA, pp. 152-153, 1993. [Fox93b] Robert M. Fox, Sang-Gug Lee and David T. Zweidinger, "The Effects of BJT Self-Heating on Circuit Behavior," IEEE Journal of Solid-State Circuits, vol. 28, no. 6, pp. 678-685, June 1993. [Fuk76] Kiyoshi Fukahori and Paul R. Gray, "Computer Simulation of Integrated Circuits in the Presence of Electrothermal Interaction," IEEE Journal of Solid-State Circuits, vol. SC-1 1, no. 6, pp. 834-846, December 1976. [Gan92] P. R. Ganci, J-J. J. Hajjar, T. Clark, P. Humphries, J. Lapham and D. Buss, "Self-Heating in High Performance Bipolar Transistors Fabricated on SOI Substrates," in Technical Digest IEEE International Electron Devices Meeting, San Francisco, CA, pp. 417-420, 1992. [Gao89] Guang-Bo Gao, Ming-Zhu Wang, Xiang Gui and Hadis Morkoc, "Thermal Design Studies of High-Power Heterojunction Bipolar Transistors," IEEE Transactions on Electron Devices, vol. 36, no. 5, pp. 854-863, May 1989. [Goo92] K. E. Goodson and M. I. Flik, "Effect of Microscale Thermal Conduction on the Packing Limit of Silicon-On-Insulator Electronic Devices," IEEE Transactions on Components, Hybrids, and Manufacturing Technology, vol. 15, no. 5, pp.7 15-722, October 1992. [Goo95] K. E. Goodson, M. I. Flik, L. T. Su and D. A. Antoniadis, "Prediction and Measurement of Temperature Fields in Silicon-On-Insulator Electronic Circuits," ASME Journal of Heat Transfer, vol. 117, pp. 574-581, August 1995.

PAGE 259

250 [Gra93] Paul R. Gray and Robert G. Meyer, Analysis and Design of Analog Integrated Circuits , Third Edition, John Wiley & Sons, Inc., New York, 1993. [Gre93] Keith R. Green, A Model of the Short-Channel Metal-OxideSemiconductor Field-Effect Transistor for Pragmatic Mixed-Mode Device/Circuit Simulation , Ph. D. Dissertation, University of Florida, August 1993. [Hat95] Ryo Hattori, Teruyuki Shimura, Manabu Kato, Takuji Sonoda and Saburo Takamiya, "Three-Dimensional Modeling of Thermal Flow in Multi-Finger High Power HBTs," in Proceedings IEEE MTT-S International Microwave Symposium, Orlando, FL, vol. 2, pp. 461464, 1995. [Hir93] I. Hirsch, E. Berman and N. Haik, "Thermal Resistance Evaluation in 3-D Thermal Simulation of MOSFET Transistors," Solid-State Electronics, vol. 36, no. 1, pp. 106-108, January 1993. [Hsp92] HSPICE User's Manual, Version H92, Meta-Software, Inc., Campbell, CA, 1992. [Jen95] K. A. Jenkins and J. Y. -C. Sun, "Measurement of I-V Curves of Silicon-on-Insulator (SOI) MOSFET's Without Self-Heating," IEEE Electron Device Letters, vol. 16, no. 4, pp. 145-147, April 1995. [Jeo89] Hanggeun Jeong and Jerry G. Fossum, 'A Charge-Based Large-Signal Bipolar Transistor Model for Device and Circuit Simulation," IEEE Transactions on Electron Devices, vol. 36, no. 1, pp. 124-131, January 1989. [Jer93] R. C. Jerome, I. R. C. Post, P. G. Travnicek, G. M. Wodek, K. E. Huffstater and D. R. Williams, "ACUTE: A High Performance Analog Complementary Polysilicon Emitter Bipolar Technology Utilizing SOI/Trench Full Dielectric Isolation," in Proceedings IEEE International SOI Conference, Palm Springs, CA, pp. 100-101, 1993.

PAGE 260

251 [Jom95a] J. Jomaah, G. Ghibaudo and F. Balestra, "Analysis and Modeling of Self-Heating Effects in Thin-Film SOI MOSFETs as a Function of Temperature," Solid-State Electronics, vol. 38, no. 3, pp. 615-618, March 1995. [Jom95b] J. Jomaah, G. Ghibaudo, F. Balestra and J. L. Pelloie, "Impact of SelfHeating on the Design of SOI Devices Versus Temperature," in Proceedings IEEE International SOI Conference, Tucson, AZ, pp. 1 14115, October 1995. [Joy70] Richard C. Joy and E. S. Schlig, "Thermal Properties of Very Fast Transistors, IEEE Transactions on Electron Devices, vol. ED17, no. 8, pp. 586-594, August 1970. [Joy75] W. B. Joyce. "Thermal Resistance of Heat Sinks with Temperature Dependent Conductivity," Solid-State Electronics, vol. 18, pp. 321322, 1975. [Kag94] A Kager, J. J. Liou, L. L. Liou and C. Huang, "A Semi-Numerical Model for Multi-Emitter Finger AlGaAs/GaAs HBTs," Solid-State Electronics, vol. 37, no. 11, pp. 1825-1832, November 1994. [Klo93] H. Klose, R. Lachner, K. R. Schon, R. Mahnkopf, K. H. Malek, M. Kerber, H. Braun, A. v. Felde, J. Popp, O. Cohrs, E. Bertagnolli and P. Sehrig, "B6HF: A 0.8 Micron 25 GHz/25 ps Bipolar Technology for Mobile Radio and Ultra Fast Data Link IC-Products," in Proceedings IEEE Bipolar Circuits and Technology Meeting, Minneapolis, MN, pp. 125-127, 1993 [Koe94] Eric Koenig, Jurgen Schneider, Ulrich Seiler and Uwe Erben, "Thermally Induced Current Constriction in III-V Heterojunction Bipolar Transistors," in Proceedings IEEE Bipolar Circuits and Technology Meeting, Minneapolis, MN, pp. 127-130, 1994. [Lee93] M. S. L. Lee, W. Redman-White, B. M. Tenbroek and M. Robinson, "Modeling of Thin Film SOI Devices for Circuit Simulation Including Per-Instance Dynamic Self-Heating Effects," in Proceedings IEEE International SOI Conference, Palm Springs, CA, pp. 150-151, 1993.

PAGE 261

252 [Lee95] T. -Y. Lee and R. M. Fox, "Extraction of Thermal Resistance for FullyDepleted SOI MOSFET's," in Proceedings IEEE International SOI Conference, Tucson, AZ, pp. 78-79, October 1995. [Lee96] Tzung-Yin Lee, Model Enhancement and Parameter Extraction for the MMSPICE/OBBJT Model . Ph. D. Dissertation, University of Florida, December 1996. [Lia94] Minchang Liang and Mark E. Law, "Influence of Lattice Self-Heating and Hot-Carrier Transport on Device Performance," IEEE Transactions on Electron Devices, vol. 41, no. 12, pp. 2391-2398, December 1994. [Lio94] J. J. Liou, L. L. Liou and C. I. Huang, "Analytical Model for the AlGaAs/GaAs Multiemitter Finger HBT Including Self-Heating and Thermal Coupling Effects," IEE Proc. Circuits, Devices and Systems, vol, 141, no. 6, pp. 469-475, December 1994. [Lio93] Lee L. Liou, John L. Ebel and Chen I. Huang, "Thermal Effects on the Characteristics of AlGaAs/GaAs Heterojunction Bipolar Transistors Using Two-Dimensional Numerical Simulation," IEEE Transactions on Electron Devices, vol. 40, no. 1, pp. 35-43, January 1993 [Lio96] L. L. Liou, B. Bayraktaroglu and C. I. Huang, "Theoretical Thermal Runaway Analysis of Heterojunction Bipolar Transistors: Junction Temperature Rise Threshold," Solid-State Electronics, vol. 39, no. 1, pp. 165-172, January 1996. [Liu93] William Liu, Steve Nelson, Darrell G. Hill and AH Khatibzadeh, "Current Gain Collapse in Microwave Multifinger Heterojunction Bipolar Transistors Operated at Very High Power Densities," IEEE Transactions on Electron Devices, vol. 40, no. 11, pp. 1917-1927, November 1993. [Liu95a] William Liu and Ayca Yuksel, "Measurement of Junction Temperature of an AlGaAs/GaAs Heterojunction Bipolar Transistor Operating at Large Power Densities," IEEE Transactions on Electron Devices, vol. 42, no. 2, pp. 358-360, February 1995.

PAGE 262

253 [Liu95b] William Liu, "The Interdependence Between the Collapse Phenomena and the Avalanche Breakdown in AlGaAs/GaAs Power Heterojunction Bipolar Transistors," IEEE Transactions on Electron Devices, vol. 42, no. 4, pp. 591-597, April 1995. [Man90] A. M. Mansanares, A. C. Bento, H. Vargas, N. F. Leite and L. C. M. Miranda, "Photoacoustic Measurement of the Thermal Properties of Two-Layer Systems," Physical Review B, vol. 42, no. 7, pp. 44774486, September 1990. [Mar93] Antoine Marty, Thierry Camps, Josiane Tasselli, David L. Pulfrey and Jean Pierre Bailbe, "A Self-Consistent DC-AC Two-Dimensional Electrothermal Model for GaAlAs/GaAs Microwave Power HBT's," IEEE Transactions on Electron Devices, vol. 40, no. 7, pp. 1202-1210, July 1993. [McA92] Colin C. McAndrew, "A Complete and Consistent Electrical/Thermal HBT Model," in Proceedings IEEE Bipolar Circuits and Technology Meeting, Minneapolis, MN, pp. 200-203, 1992. [Moi94] Shahriar Moinian, Peter Feldmann and Robert C. Melville, "ChipLevel Electro-Thermal Simulation of Bipolar Transistor Circuits," in Proceedings IEEE Bipolar/BiCMOS Circuits and Technology Meeting, Minneapolis, MN, pp. 123-126, 1994. [Mue64] O. Mueller, "Internal Thermal Feedback in Four-Poles Especially in Transistors," Proceedings of the IEEE, vol. 52, pp. 924-930, August 1964. [Mul77] Richard S. Muller and Theodore I. Kamins, Device Electronics for Integrated Circuits , John Wiley & Sons, Inc., New York, 1977. [Nag75] L. W. Nagel, "SPICE2: A Computer Program to Simulate Semiconductor Circuits," Electron. Res. Lab. Memo. ERL-M520, University of California, Berkeley, 1975. [Nak95] Tohru Nakamura and Hirotaka Nishizawa, "Recent Progress in Bipolar Transistor Technology," IEEE Transactions on Electron Devices, vol. 42, no. 3, pp. 390-398, March 1995.

PAGE 263

254 [Nis91] H. Nishizawa, S. Azuma, T. Yoshitake, K. Yamada, T. Ikeda, H. Masuda and A. Anzai, "A Fully Si0 2 -Isolated Self-Aligned SOIBipolar Transistor for VLSIs," in Proceedings IEEE Bipolar Circuits and Technology Meeting, Minneapolis, MN, pp. 53-58, 1991. [Ona95] Takahiro Onai, Eiji Ohue, Yohji Idei, Masamichi Tanabe, Hiromi Shimamoto, Katsuyoshi Washio and Tohru Nakamura, "Self-Aligned Complementary Bipolar Technology for Low-Power Dissipation and Ultra-High-Speed LSI's," IEEE Transactions on Electron Devices, vol. 42, no. 3, pp. 413-417, March 1995. [Ozi93] M. Necati Ozisik, Heat Conduction . Second Edition, John Wiley & Sons, Inc., New York, 1993. [Pil94] Lawrence Pillage and Ronald A. Rohrer, "The Essence of AWE," IEEE Circuits & Systems Magazine, pp. 12-19, September 1994. [Pru94] A. Pruijmboom, C. E. Timmering and J. J. E. M. Hageraats, "18 ps ECL-Gate Delay in Laterally Scaled 30 GHz Bipolar Transistors," in Technical Digest IEEE International Electron Devices Meeting, San Francisco, CA, pp. 825-828, 1994. [Sab95] SaberGuide and SaberScope Manual, Version 4.0, Analogy, Inc., Beaverton, OR, December 1995. [Sch84] A. Schutz, S. Selberherr and H. W. Potzl, "Temperature Distribution and Power Dissipation in MOSFETs," Solid-State Electronics, vol. 27, no. 4, pp. 394-395, April 1994. [Sei93] Ulrich Seiler, Eric Koenig, Peter Narozny and Heinrich Dambkes, "Thermally Triggered Collapse of Collector Current in Power Heterojunction Bipolar Transistors," in Proceedings IEEE Bipolar Circuits and Technology Meeting, Minneapolis, MN, pp. 257-260, 1993. [Sha83] D. K. Sharma and K. V. Ramanathan, "Modeling Thermal Effects on MOS I-V Characteristics," IEEE Electron Device Letters, vol. EDL-4, no. 10, pp. 362-364, October 1983.

PAGE 264

255 [Shi96] Takeo Shiba, Yoichi Tamaki, Takahiro Onai, Yukihiro Kiyota, Tokuo Kure and Tohru Nakamura, "A Very Small Bipolar Transistor Technology with Sidewall Polycide Base Electrode for ECL-CMOS LSI's," IEEE Transactions on Electron Devices, vol. 43, no. 9, pp. 1357-1363, September 1996. [Shu90] Michael Shur, Physics of Semiconductor Devices . Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1990. [Sze88] Vladimir Szekely and Tran Van Bien, "Fine Structure of Heat Flow Path in Semiconductor Devices: A Measurement and Identification Method," Solid-State Electronics, vol. 31, no. 9, pp. 1363-1368, September 1988. [Ten95] B. M. Tenbroek, W. Redman-White, M. S. L. Lee and M. J. Uren, "Comparison of SOI MOSFET Self-Heating Measurements by Gate Resistance Thermometry and Small-Signal Drain Admittance Extraction," in Proceedings IEEE International SOI Conference, Tucson, AZ, pp. 48-49, October 1995. [Ten96] Bernard M. Tenbroek, William Redman-White, Michael S. L. Lee, R. John T. Bunyan, Michael Uren and Kevin M. Brunson, "Characterization of Layout Dependent Thermal Coupling in SOI CMOS Current Mirrors," IEEE Transactions on Electron Devices, vol. 43, no. 12, pp. 2227-2232, December 1996. [Tu94] Robert Tu, Clement Wann, Joseph King, Ping Ko and Chenming Hu, "SOI MOSFET Modeling Using an AC Conductance Technique to Determine Heating," in Proceedings IEEE International SOI Conference, Nantucket Island, MA, pp. 21-22, October 1994. [Web91] L. Weber and E. Gmelin, "Transport Properties of Silicon," Applied Physics A, 53, pp. 136-140, 1991. [Win67] Richard H. Winkler, "Thermal Properties of High-Power Transistors," IEEE Transactions on Electron Devices, vol. ED-14, no. 5, pp. 260264, May 1967. [Wol90] Stanley Wolf, Silicon Processing for the VLSI Era. Volume 2: Process Integration , Lattice Press, Sunset Beach, California, 1990.

PAGE 265

256 [Wor97] Glenn Workman, Jerry Fossum, Srinath Krishnan and Mario Pelella, "Physical Modeling of Temperature Dependences of SOI CMOS Devices and Circuits Including Self-Heating," submitted to IEEE Transactions on Electron Devices. [Yam93] Tad Yamaguchi, Sudarsan Uppili, Galen Kawamoto, June Lee and Shaun Simpkins, "Process and Device Optimization of A 30-GHz fT Submicrometer Double Poly-Si Bipolar Technology," in Proceedings IEEE Bipolar Circuits and Technology Meeting, Minneapolis, MN, pp. 136-139, 1993. [Zwe95a] D. T. Zweidinger, J. S. Brodsky and R. M. Fox, "A Physical Thermal Resistance Model for Vertical BJTs on SOI," in Proceedings IEEE International SOI Conference, Tucson, AZ, pp. 84-85, October 1995. [Zwe95b] D. T. Zweidinger, R. M. Fox, J. S. Brodsky, T. Jung and S. -G. Lee, "Extraction of Thermal Parameters for Bipolar Circuit Simulation," in Proceedings IEEE Bipolar/BiCMOS Circuits and Technology Meeting, Minneapolis, MN, pp. 78-81, October 1995. [Zwe96] David T. Zweidinger, Robert M. Fox, Jonathan S. Brodsky, Taewon Jung and Sang-Gug Lee, "Thermal Impedance Extraction for Bipolar Transistors," IEEE Transactions on Electron Devices, vol. 43, no. 2, pp. 342-346, February 1996. [Zwe97] David T. Zweidinger, Modeling of Transistor Self-Heating for Circuit Simulation , Ph. D. Dissertation, University of Florida, August 1997.

PAGE 266

BIOGRAPHICAL SKETCH Jonathan Brodsky was born in Philadelphia, Pennsylvania, on August 3, 1969. In 1991, he received the Bachelor of Science degree in electrical engineering from Lafayette College, Easton, Pennsylvania. He received the Master of Science degree from the University of Florida, Gainesville, in 1993, and in the same year began working towards the Doctor of Philosophy degree in electrical engineering. During the summer of 1995, he worked at the Motorola Advanced Products Research and Development Laboratory, Austin, Texas, where he helped characterize self-heating in SOI MOSFET's. Upon graduation he will join Texas Instruments, Inc. in Dallas, Texas, as a member of the Mixed Signal Products group working in the area of ESD protection. He is a member of The Institute of Electrical and Electronics Engineers. 257

PAGE 267

I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Robert M. Fox, Chairman Associate Professor of Electrical and Computer Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Mark E. Law Professor of Electrical and Computer Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. William R. Eisenstadt Associate Professor of Electrical and Computer Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. i*A/W^ Jcyyl G. Harris Assistant Professor of Electrical and Computer Engineering

PAGE 268

I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Chen-Chi Hsu Professor of Aerospace Engineering, Mechanics and Engineering Science This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 1997 a Winfred M. Phillips Dean, College of Engineering Karen A. Holbrook Dean, Graduate School