• TABLE OF CONTENTS
HIDE
 Title Page
 Acknowledgement
 Table of Contents
 Abstract
 Introduction
 Modeling of multi-dimensional...
 Non-quasi-static modeling of BJT...
 Analytic accounting for carrier...
 MMSPICE-2 development
 Summary and suggestions for future...
 Appendices
 References
 Biographical sketch














Title: Assessment and modeling of non-quasi-static, non-local, and multi-dimensional effects in advanced bipolar junction transistors /
CITATION PDF VIEWER THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00097383/00001
 Material Information
Title: Assessment and modeling of non-quasi-static, non-local, and multi-dimensional effects in advanced bipolar junction transistors /
Physical Description: vii, 161 leaves : ill. ; 28 cm.
Language: English
Creator: Jin, Joohyun, 1958-
Publication Date: 1992
Copyright Date: 1992
 Subjects
Subject: Electrical Engineering thesis Ph. D
Dissertations, Academic -- Electrical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis (Ph. D.)--University of Florida, 1992.
Bibliography: Includes bibliographical references (leaves 156-160).
Additional Physical Form: Also available on World Wide Web
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by Joohyun Jin.
 Record Information
Bibliographic ID: UF00097383
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 001796795
oclc - 27483477
notis - AJM0508

Downloads

This item has the following downloads:

PDF ( 4 MBs ) ( PDF )


Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
        Page iii
    Table of Contents
        Page iv
        Page v
    Abstract
        Page vi
        Page vii
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
    Modeling of multi-dimensional currents
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
    Non-quasi-static modeling of BJT current crowding
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
    Analytic accounting for carrier velocity overshoot
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
        Page 101
        Page 102
        Page 103
        Page 104
        Page 105
        Page 106
        Page 107
        Page 108
        Page 109
        Page 110
    MMSPICE-2 development
        Page 111
        Page 112
        Page 113
        Page 114
        Page 115
        Page 116
        Page 117
        Page 118
        Page 119
        Page 120
        Page 121
        Page 122
        Page 123
        Page 124
        Page 125
        Page 126
        Page 127
        Page 128
        Page 129
        Page 130
        Page 131
        Page 132
        Page 133
        Page 134
        Page 135
        Page 136
        Page 137
        Page 138
        Page 139
        Page 140
        Page 141
        Page 142
        Page 143
        Page 144
    Summary and suggestions for future work
        Page 145
        Page 146
        Page 147
    Appendices
        Page 148
        Page 149
        Page 150
        Page 151
        Page 152
        Page 153
        Page 154
        Page 155
    References
        Page 156
        Page 157
        Page 158
        Page 159
        Page 160
    Biographical sketch
        Page 161
        Page 162
        Page 163
        Page 164
Full Text











ASSESSMENT AND MODELING OF NON-QUASI-STATIC,
NON-LOCAL, AND MULTI-DIMENSIONAL EFFECTS IN
ADVANCED BIPOLAR JUNCTION TRANSISTORS















By

JOOHYUN JIN


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


1992












ACKNOWLEDGEMENTS


I would like to express my sincere gratitude to my

advisor, Jerry G. Fossum, for giving me an opportunity to

work as one of his privileged graduate students on

interesting research topics. Without his devoted guidance,

encouragement, concern, support and patience, this work could

not have reached fruition. My interaction with him has been

a most gratifying learning experience.

I also would like to thank the other members of my

supervisory committee, Professors Dorothea E. Burk, Mark E.

Law, Sheng S. Li, and Timothy J. Anderson, for their

willingness to serve on my committee.

I am also indebted to numerous people I have interacted

with during my stay in Gainesville. First I am grateful to

Mr. D. FitzPatrick for his help in the MMSPICE software

development. Thanks are also extended to many of my

colleagues who helped me through technical discussions or by

cheering me up in difficult times. I cannot mention all of

them, but I should mention Drs. H. Jeong, Y. Kim, J. Choi,

and Messrs. H. J. Cho, S. Lee, H. S. Cho, G. Hong, K. Green,

D. Suh, P. Yeh, M. Liang, D. Apte, S. Krishnan. My deepest

gratitude goes to my parents and sisters Hyesook and Minjung








for their endless love and encouragement throughout the years

of my graduate study. Last but not least, I thank the Lord

for His guidance in my life. I also acknowledge the

financial support of the Semiconductor Research Corporation

and Samsung Semiconductor & Telecommunication Co. Ltd.


iii












TABLE OF CONTENTS


Page


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

ABSTRACT .. . .............................................

CHAPTERS

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

2 MODELING OF MULTI-DIMENSIONAL CURRENTS................

2.1 Introduction .................................... ..
2.2 Model Development..................................
2.2.1 Experimental Characterization.................
2.2.2 Analytic Model ................................
2.3 Simulations and Verification ......................
2.4 Summary.......................................... ..

3 NON-QUASI-STATIC MODELING OF BJT CURRENT CROWDING.....

3.1 Introduction .................................... ..
3.2 Model Development..................................
3.2.1 Switch-on Case ................................
3.2.2 Switch-off Case................................
3.3 NQS Model Implementation..........................
3.4 Simulations ......................................
3.5 Summary.......................................... ..

4 ANALYTIC ACCOUNTING FOR CARRIER VELOCITY OVERSHOOT....


.1 Introduction..............................
.2 Model Development........................
4.2.1 Velocity Overshoot....................
4.2.2 Velocity Relaxation...................
4.2.3 Effective Saturated Drift Velocity...
4.2.3.1 Junction SCR......................
4.2.3.2 Current-induced SCR...............
4.2.3.3 Special case .....................
.3 Comparisons with Energy Transport Model..
.4 Implementation...........................
.5 Simulations ..............................







4 .6 Summary ......................................... 109

5 MMSPICE-2 DEVELOPMENT................................. 111

5.1 Introduction ...................................... 111
5.2 New Features.......................................... 112
5.2.1 Multi-dimensional Currents..................... 112
5.2.2 Current Crowding............................... 113
5.2.3 Velocity Overshoot ............................ 115
5.2.4 Extrinsic Collector-base Capacitance.......... 115
5.2.5 Substrate Capacitance ......................... 119
5.3 Parameter Evaluation............................... 119
5.4 Model Implementation .............................. 122
5.4.1 Subroutine Modifications....................... 122
5.4.1.1 Subroutine MODCHK.......................... 122
5.4.1.2 Subroutine QBBJT........................... 123
5.4.1.3 Subroutine QBCT ........................... 127
5.4.2 Subroutine Additions .......................... 127
5.4.2.1 Subroutine CROWD......................... 127
5.4.2.2 Subroutine OVERSHOOT ...................... 129
5.5 Demonstration .. .................................. 131
5.6 Summary............................................... 144

6 SUMMARY AND SUGGESTIONS FOR FUTURE WORK................ 145

APPENDICES

A EVALUATION OF JSEO, nEB, JEOP AND nEBP ................... 148
B DISCUSSION ON JQ...................................... 150
C LIMITING JEO(eff) IN THE SWITCH-OFF SIMULATION.......... 152
D VALIDITY OF THE DEPLETION APPROXIMATION............... 154

REFERENCES .................. ................................. 156

BIOGRAPHICAL SKETCH........................................ 161












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

ASSESSMENT AND MODELING OF NON-QUASI-STATIC,
NON-LOCAL, AND MULTI-DIMENSIONAL EFFECTS IN
ADVANCED BIPOLAR JUNCTION TRANSISTORS

By

JOOHYUN JIN

August 1992



Chairman: Dr. J. G. Fossum
Major Department: Electrical Engineering

This dissertation is concerned with assessment,

modeling, and simulation of non-quasi-static (NQS), non-

local, and multi-dimensional effects in advanced bipolar

junction transistors. A simple analytic model for the

sidewall injection of the base current, which is shown to be

the most important multi-dimensional component in scaled

devices, is developed based on the separation of the base

current into internal and peripheral components. Simulation

results for typical test BJTs with various emitter geometries

are compared against corresponding measurements to support

the model. A novel NQS model for transient current crowding

in advanced BJTs is developed for circuit simulation. The

new model, implemented based on a novel use of the previous

time-step solution in the current time-step analysis,







characterizes a time-dependent effective bias on the emitter-

base junction in a semi-numerical analysis, accounting for

base conductivity modulation and the NQS nature of the

crowding. The (dc) debiasing effect, which is important in

analog circuits, is inherently accounted for as well. An

analytic model for electron velocity overshoot resulting from

non-local transport in advanced silicon-based BJTs is

developed. The model, which characterizes an effective

saturated drift velocity, larger than the classical value

because of overshoot, is intended for circuit simulation.

The model uses an augmented drift-velocity formalism that

involves a length coefficient derived via Monte Carlo

analysis. The associated velocity relaxation is

characterized phenomenologically to be consistent with

overshoot analysis. The developed charge-based models are

implemented in MMSPICE-2, a semi-numerical mixed-mode

device/circuit simulator, such that users may activate any

combination of the new features by option. The resulting

hierarchical tool, along with the parasitic charge

(capacitance) models included to enhance the usefulness of

the simulator, could indeed enable predictive yet

computationally efficient mixed-mode simulations for bipolar

(and BiCMOS) VLSI technology/manufacturing CAD. Utility of

MMSPICE-2 is demonstrated by transient simulations of ECL

circuits and devices.


vii












CHAPTER 1
INTRODUCTION



In recent years, advances in process technology have led

to the realization of high-performance bipolar junction

transistors (BJTs). While continual improvement in the

lithographic capability allows the lateral dimensions to be

reduced, scaling down the BJT requires a coordinated change

in both the lateral dimensions and vertical profile to

achieve proper device operation and to improve the intrinsic

device speed. Furthermore, in order to reduce the extrinsic

portion of the bipolar device so that circuit performance can

be more closely tied to the intrinsic device performance,

various self-alignment schemes using polysilicon as base and

emitter contacts have been developed. They all have a

similar structure (see Fig. 1.1), and generally provide much

improved performance over the conventional BJT structure via

a reduction in base-collector junction area and base

resistance.

Despite the impressive progress made in bipolar

technology, computer simulation tools, which are essential to

the optimization of device and circuit designs for the

technology, have not kept pace with it. In integrated























I I Polysilicon

- Nitride


////// Oxide

FisTTT1s +ln*


Fig. 1.1 Cross section of an advanced bipolar junction
transistor fabricated by double-polysilicon
process.








circuit development and manufacturing today, a technology CAD

(TCAD) system is essential for exploring alternative designs

and evaluating various trade-offs without time-consuming and

costly fabrications.

An effective TCAD system requires integrated, physics-

based tools for predictive process, device, and (small-scale)

circuit simulation. Computational efficiency is desirable

and indeed essential if the TCAD system is to be used in

manufacturing CAD involving statistical simulation.

Conventional TCAD systems comprise robust, numerical process

and device simulators which drive optimization of empirical

device model parameters for circuit simulation. This

optimization can miss parametric correlations, and hence the

integrated system, although CPU-intensive, could yield

nonunique (erroneous) predictions.

Numerical mixed-mode device/circuit simulation would

obviate this deficiency, but with a high cost of computation

time. Alternatively, improvement of the TCAD system can

possibly be afforded by incorporation of semi-numerical

device models into the circuit simulator which have physical

parameters that relate directly to the device structure. The

resulting tool is an application-specific, computationally

efficient mixed-mode simulator that can easily be integrated

with the process simulator by a program that evaluates the

model parameters from the doping profile. The MMSPICE








[Jeo90] is such a simulator, which is integrated with SUPREM

[SUP88] by a parameter-extraction program, SUMM [Gre90].

The model development for MMSPICE has emphasized the

advanced BJTs. A physical, one-dimensional charge-based

model [Jeo89] has been developed and implemented. High-

current effects, impact ionization, and non-reciprocal

(trans)capacitances are physically accounted for in the semi-

numerical model. This model is sufficient for many

applications, but more work is needed to enhance the

usefulness of MMSPICE.

In most advanced BJTs, the lateral dimension of the

emitter has become the same order of magnitude as the

emitter-base junction depth. Thus, multi-dimensional current

effects in the peripheral region of the junction are expected

to play a significant role in device performance.

Especially, the variation in common-emitter forward current

gain 3 with geometric shape and size is troublesome to IC

designers [Hwa87]. Hence, some accounting of peripheral

currents is needed for circuit simulation.

High-current effects (e.g., quasi-saturation and base

widening, or pushout) are physically accounted for in the

MMSPICE model, but emitter current crowding, caused by

lateral voltage drops in the intrinsic base region, has not

yet been considered. Today's advanced (scaled) BJTs commonly

operate at high current density, and hence transient base




5


current can be much greater than the steady-state current;

this clearly implies the non-quasi-static (NQS) nature of

transient current crowding [Ham88] Therefore, it can be

significant even though dc crowding may be insignificant

[Tan85].

In semiconductor devices where the electric field

increases rapidly over distances comparable to the energy-

relaxation mean free path, carrier velocity can overshoot the

value corresponding to the local field because the carrier

(kinetic) energy, which controls the collision time and hence

limits the velocity, lags the field and remains relatively

small [Ruc72]. This non-local effect on electron transport

has been recognized as significant in MOSFETs and MESFETs for

years, and now has become important in scaled BJTs [Lee89,

Cra90]. Recent work [Fus92] has indicated that the velocity

overshoot in scaled BJTs can be beneficial, and must be

accounted for in the device and circuit design. However, the

effect has not yet been accounted for in any existing circuit

simulators, and indeed is missing in many device simulators

because of the implied computational intensiveness.

For bipolar integrated circuits, reducing parasitic

capacitances is one of the key issues for speed enhancements.

The extrinsic collector-base junction capacitance (charge)

has a predominant effect on the circuit performance because

the extrinsic base region is not reduced in proportion as the








intrinsic device is scaled down. The collector-substrate

capacitance (charge) is also important.

This dissertation addresses these problems; it is

concerned with the development and implementation of new

models to account for the aforementioned effects in the

advanced BJTs. This work will enable not only truly

predictive, scalable BJT simulations, but also

computationally efficient (semi-numerical) mixed-mode

device/circuit simulations for bipolar TCAD. The major

contributions made in this work are as follows:

(1) modeling of multi-dimensional current effects, based on

the separation of the current into internal and

peripheral components;

(2) development of an NQS transient current-crowding model,

based on a novel use of the previous time-step solution

in the current time-step analysis;

(3) development of an analytic model for electron velocity

overshoot resulting from non-local transport in advanced

silicon-based BJTs;

(4) implementation of the new models, including both the

extrinsic collector-base and collector-substrate

capacitances (charges), in MMSPICE to create MMSPICE-2.

In Chapter 2, a simple analytic way of accounting for

multi-dimensional current effects is described. The approach

is based on the separation of the current into areal and








peripheral components. For high VBE, an effective junction

bias (described in Chapter 3) is necessarily defined to

account for the emitter debiasing (a.k.a. crowding) effect.

The model is supported by experimental results of test BJTs

having varied emitter geometries.

In Chapter 3, a new NQS model for transient current

crowding is presented. The model, which characterizes a

time-dependent effective bias on the emitter-base junction in

a semi-numerical analysis, follows the previous work by

Hauser [Hau64], but physically accounts for base conductivity

modulation and the NQS nature of the crowding. The novel

modeling/implementation is based on the use of the previous

time-step solution in the current time-step analysis, which

in fact could enable general accounting of NQS effects in

semi-numerical mixed-mode device/circuit simulation. The

tool is supported by numerical simulations of advanced BJTs

using PISCES [PIS84].

In Chapter 4, an analytic model for electron velocity

overshoot in advanced BJTs is presented. The model, which

characterizes an effective saturated drift velocity in the

collector space-charge regions, is intended for circuit

simulation. The model uses an augmented drift-velocity

formalism that involves a length coefficient derived from

Monte Carlo simulations. The associated relaxation of the

carrier velocity is characterized phenomenologically to be









consistent with the overshoot analysis. Demonstrative

simulation results are presented to assess the significance

of the electron velocity overshoot in advanced bipolar and

BiCMOS technologies, and to support model.

The developed charge-based models are implemented into

MMSPICE-2 so that users may activate any combination of the

new features by option. This hierarchical tool is discussed

in Chapter 5. Representative simulations are presented, with

descriptions of the new parameters.

In Chapter 6, the main accomplishments of this

dissertation are summarized, and future research areas are

suggested.













CHAPTER 2
MODELING OF MULTI-DIMENSIONAL CURRENTS



2.1 Introduction



For bipolar integrated circuits, reducing parasitic

effects and achieving shallow profiles are two of the key

issues in improving performance. Many self-aligned bipolar

technologies have been developed to achieve low parasitic

capacitance and low base resistivity. They all have a

similar device structure using polysilicon as base and

emitter contacts. In the scaled structure, the distance

between base and emitter contacts is greatly reduced as

determined by the boot-shaped sidewall spacer (see Fig. 1.1).

The lateral dimensions of the device have also been scaled

down; for example, the emitter width of today's most advanced

transistors has become the same order of magnitude as the

emitter-base junction depth. Thus, multi-dimensional effects

in the peripheral region of the junction can play a

significant role in device performance [Hur87].

For digital applications, a most predominant multi-

dimensional effect is the lateral injection of significant

base current along the emitter sidewall, which is controlled

by the morphology of the link region [Li88]. One simple way









to reduce this sidewall current component is to increase the

width of the spacer [Dej88, Saw88]. However, many desirable

features of the device depend on the limitation of the spacer

width. For example, as the spacer width increases, the base

resistance and parasitic capacitances increase. Also, the

emitter-collector punchthrough current increases due to

insufficient extrinsic-intrinsic base overlap in the emitter

periphery [Chu87, Saw88], while an increase in the extrinsic-

intrinsic base overlap results in excessive perimeter

tunneling current [Sto83] and hence reduced emitter-base

breakdown voltage. Thus, the control of spacer thickness is

vital to the performance of the device.

The peripheral component of the base current does not

modulate the collector current, and is therefore a parasitic

that degrades the dc current gain 3 in proportion to the

ratio of its magnitude relative to that of the areal

component. Hence, P is degraded more as the perimeter-to-

area ratio (PE/AE) increases. This implies that the sidewall

effect can be an obstacle for down-scaling the emitter size

[Hwa87, Dej88] Therefore, some accounting of peripheral

currents for a given process is needed for a circuit

simulator, e.g., MMSPICE, which actually gives an extra

degree of freedom to the IC designer [Ver87].

In Section 2.2, a simple model based on measurements is

presented to account for the peripheral currents in the








advanced BJT structure. This model, combined with the

current-crowding analysis described in Chapter 3, will be the

basis for a more predictive and scalable BJT model for

MMSPICE. In Section 2.3, experimental results of test BJTs

having varied emitter geometries are presented to support our

formalism. In fact, interpretation of these results requires

the crowding model of Chapter 3, which was hence developed in

conjunction with the work described in this chapter.



2.2 Model Development



2.2.1 Experimental Characterization



For digital applications, the most important peripheral

current is the sidewall component of the base current.

However, the peripheral component of the collector current is

not significant compared with the areal component, provided

the extrinsic base is well-linked with the intrinsic base

[Li88].

This fact is also supported by our own measurements of

representative (advanced) BJTs provided by Dr. D. Verret of

Texas Instruments. The lateral geometries of the test

devices are described in Table 2.1; LE and WE are the

effective (or actual) length and width of the emitter, and PE

(=2LE+2WE) and AE (=LEWE) are the perimeter and area















TABLE 2.1
LATERAL EMITTER GEOMETRIES OF TEST DEVICES



LE [1Lm] WE [im] PE/AE [1.m-1]

9.2 5.2 0.60

9.2 4.2 0.69

9.2 3.2 0.84

9.2 1.7 1.39

9.2 1.2 1.88

9.2 0.7 3.08

9.2 0.45 4.66








respectively. The spacer width of these devices is estimated

to be 0.4j.m. Fig. 2.1 shows the base (JB) and collector (Jc)

current densities versus PE/AE for the devices with LE fixed

at LE=9.2RLm when VBE=0.4 or 0.7V. Since JC is almost constant

regardless of PE/AE as well as VBE and VBC, we infer that the

peripheral collector current can be neglected at least for

relatively low VBE. On the contrary, JB clearly increases

with PE/AE, obviously implying a significant lateral-injection

component. We note that this parasitic effect becomes more

significant as VBE is reduced, which we believe reveals that

the peripheral base current is due to the recombination of

excess carriers in the peripheral junction space-charge-

region (SCR) near or at the oxide-silicon interface.

The lateral injection can be understood better if the

peripheral component of base current is quantified.

Empirically, the total base current IB can be separated into

areal and peripheral parts as follows [Rei84]:



IB = IBA + IBP

_VBE VBEv
SCAA e xp- 1 + CPPE exp--- 1 (2.1)




where CA, nA, Cp, and np are (process-dependent) empirical

constants, which can easily be evaluated using the basic

experimental method discussed in Appendix A. In (2.1), the



















E





0
-"l
13

Oa


10-10




10-11
10


U j I I I I I I I I I I I I I III I
0 1 2 3 4 5
PE/AE [1/um]

(b)


E


m


o
rn


10-6




10-7



10-87
1-8


PE/AE [1/um]


Fig. 2.1 Base and collector current densities versus PE/AE
for devices with LE=9.2tm: (a) VBE=O .4V; (b)
VBE=0.7V.


6 6 6 0




Sa




A VBC= -3.0V
O VBC= O.OV


6 6 6 6



S VBC = -3.0V
0 VBC = O.OV



0









voltage drop across the extrinsic base resistance is

neglected for low-current conditions.

Based on this formalism, it is possible to calculate the

contribution of the peripheral current to the total base

current. Doing this for the devices previously characterized

yields in Fig. 2.2 IBP/IB versus PE/AE for VBC=O.OV. As

discussed before, the peripheral base component increases

with PE/AE. For example, when VBE=0.7V and PE/AE=0.60/Llm

(actually, this is equivalent to the device with WE=5.21m),

IBP is only 16% of the total base current, but it increases to

50% when PE/AE=3.1/tm (i.e., WE=0.7 m). For reduced VBE, the

effect of lateral injection becomes more significant in

accord with our previously stated recognition; when VBE=0.4V,

the mentioned ratios are changed to 47% and 82% respectively.

Our other simulations and measurements show that the

peripheral collector current evaluated via this methodology

is about 10% of the total collector current on the average.



2.2.2 Analytic Model



With this insight, we can extend the MMSPICE BJT model

to account for the peripheral base region, at least to first

order. The extended model is restricted to include only the

lateral injection of the base current, which has been shown

to be the most important multi-dimensional effect in modeling














I I 1 1 1I I 1 1I I 1I r 11 11


1-



0.8



0.6



0.4



0.2



0-


1 2 3 4 5
PE/AE [1/um]


Fig. 2.2 Simulated IBP/IB versus PE/AE for the devices used
in Fig. 2.1.


VBE =0.7V
BE-








advanced BJTs. Based on the insight derived from the

measurements, we add only a peripheral component of base

current to the existing BJT routine in MMSPICE. This

additional component is proportional to the emitter perimeter

PE, and represents peripheral SCR recombination near the

surface. The peripheral base current IBp can be expressed as



IBP = JEOPPE exp VBE 1 (2.2)
L \nEBpVT



where JEOP and nEBP represent the peripheral saturation

current density (per unit length) and the peripheral emission

coefficient respectively. The sidewall injection effect

could also be dependent on the emitter junction depth, but we

assume that this dependence is implicitly included in the

above formalism.

In a dc case, the predominant components of the areal

base current are typically back-injection current from the

base to the emitter and the recombination current at the

(emitter-base) junction SCR. (Recombination in the quasi-

neutral base and the epi collector is neglected here since it

is typically insignificant in advanced BJTs.) Hence, the

total base current IB can be expressed as


IB = IBA + IBP








= JEoAE [expB 1 + JSEOAE [exp( VBE
T nEBV


+ JEOPPE [exp V1E (2.3)
L \nEBpVT J


where JEO is the (areal) emitter saturation current density,

and JSEO and nEB are (areal) SCR saturation current density

and SCR emission coefficient respectively.

Although (2.3) is sufficient for many operating ranges,

it is necessary to examine whether it is valid for high-

current operation where additional effects are significant.
In this case the actual (peripheral) junction bias V'BE cannot

be approximated as the terminal voltage VBE; V'BE is

considerably less than VBE since the voltage drops across the

extrinsic base and emitter resistances are no longer

negligible. Furthermore, the areal component is degraded by

the lateral voltage drops in the intrinsic base region. In

fact, interpretation of data necessitated the current-

crowding modeling described in Chapter 3. Hence we modify

(2.3):


IB = JEOAE expBE(e) 1 + JSEOAE [exp(V ) -
VT nEBV E


+ JEOPPE exp-VIB E 1 (2.4)
L \ngBpVTl I








where VBE(eff) is defined (in Chapter 3) as the effective bias

on the emitter-base junction to account for the debiasing

(a.k.a. current crowding) in terms of the actual (peripheral)

bias V'BE. Note that in (2.4), the peripheral current term is

not threatened by the current crowding because the peripheral

junction voltage is always fixed at V'BE. Although the

debiasing effect was classically characterized by Hauser

[Hau64], his treatment is inadequate for advanced BJTs

because it neglects conductivity modulation of the base. On

the contrary, the concept of the effective bias can account

for the high-current effects via the charge-based BJT model

[Jeo89]. When the debiasing effect is significant, the

effective bias is of course less than the actual junction

bias V'BE. (In this case, V'BE is also significantly less

than VBE.) Otherwise, VBE(eff) would be almost the same as

V'BE. This effective bias is derived from the quasi-three-

dimensional crowding analysis, which involves a coupling of

the vertical and lateral carrier-transport analyses in the

base region. Details are described in Chapter 3.

Fig. 2.3 illustrates (V'BE-VBE(eff))/VT versus WE

predicted by the debiasing analysis for typical advanced

devices with LE=9.211m. When VBE=0.7V, the debiasing effect

is, as expected, negligible resulting in VBE(eff)=V'BE=VBE

regardless of WE and VBc. However it becomes noticeable for

higher VBE and especially for greater WE, due to the increased
















0.8




0.6

- -

S0.4 -

w
m






0-




-0.2


I t H I ~ ~ I I I I i I I I I I 1 1


VBC=-3.0V


VBC=O.OV


VBC=O.O or -3.0V
.. .-.-- .---.-----------------------


SI . . . . I I
0 1 2 3 4 5 6

WE [urn]


Fig. 2.3 Simulated (V'BE-VBE(eff))/VT versus WE for typical
advanced BJTs with LE=9.21m.


I . I I I I I I i I , I I I I I I I I I I I ,








voltage drops in the intrinsic base region. The debiasing

effect also becomes more important with increasing reverse

bias on the base-collector junction because the base

resistivity increases correspondingly. For contemporary

scaled BJTs however, it is not significant [Tan85]; for

WE=2jm at VBE=0.9V and VBC=-3.0V, the voltage difference

between the actual and effective bias is about 20% of the

thermal voltage.



2.3 Simulations and Verification



The test devices, representative of the advanced bipolar

technology, were used to verify the model. The devices, from

Texas Instruments, were fabricated using a double-polysilicon

process in conjunction with a sidewall spacer technique,

which enables a self-aligned submicrometer emitter structure.

In order to identify significant multi-dimensional effects,

transistors with different PE/AE (see Table 2.1) were

measured.

Simulations were done with MMSPICE-2, which includes the

peripheral base current [eq. (2.2)] and the current-crowding

model as described in Chapter 3. At first, the model

parameters associated with the lateral injection were

extracted as described in Appendix A. Then, with no

additional parameter extraction, all BJTs were simulated with








reasonably good accuracy simply by scaling AE.

Simulated Ic/WE and IB/WE compare quite well with the

corresponding measurements in Fig. 2.4(a) when VBE=0.4V and

VBC=O.OV. Note that the lateral injection effect on the base

current becomes significant as WE is scaled down; IB/WE

increases because the ratio of the peripheral to the areal

component increases. However, the contribution of the

peripheral collector current is negligible for each device.

Note that if IBP had not been accounted for, IB/WE would have

been predicted to be a constant, since the voltage drops

across the extrinsic resistances are negligible for each

device at this bias point. For the corresponding 0 shown in

Fig. 2.4(b), the simulations are excellent. As expected, P

is reduced with decreasing WE. Although 3-degradation is an

obstacle for down-scaling WE, we expect that our first-order

accounting of the lateral injection could give an extra

degree of freedom to the circuit designer.

The peripheral collector current is still negligible

when VBE is increased to 0.7V, as shown in Fig. 2.5(a).

Still, the sidewall injection of the base current, although

not as significant as in the low-current region, is important

especially for devices with small WE. The simulations are

good, although there is a small discrepancy between the

measured and predicted values of IB/WE for submicron devices.

Indeed this discrepancy seems to be inevitable because the


















10-100
LUi


S1011

10 11


140-

120-

100

an

60-

40-

20-


0 1 2 3 4 5 6
WE [um]

(b)
L''~' , I , I , I , I I ,


A VBC= -3.0V
O VBC= 0.0V
-- Simulation















1 2 3 4 5


WE [um]




Fig. 2.4 Measured and simulated IC/WE, IB/WE in (a) and P in
(b) for the test BJTs with LE=9.2.m for VBE=0.4V.


O Measurement (VBC=O.OV)

Simulation


00n o n 0







0










(a)
Ir1 1, 1 1, 1


10-5




10-6




10-7




10-8
0


O
0



~ VBC= -3.0V
0 VBC= O.OV
-- Simulation




0 1 2 3 4 5 6

WE [um]


Fig. 2.5 Measured and simulated IC/WE, IB/WE in (a) and P in
(b) for the test BJTs with LE=9.2pm for VBE=0.7V.


2 3 4
WE [um]


Oo o o n O




0 Measurement (VBC=O.OV)
Simulation


0


I


r r a I I I I i s r r r I


f i l l








IBp-related model parameters were evaluated from the devices

operating in the low current region; according to (2.3), the

PE-dependent term would become negligible with increasing VBE.

However, our model seems adequate, as implied by the

corresponding 3 results in Fig. 2.5(b).

For VBE=0.9V in Fig. 2.6(a), the simulations are also

reasonably good. We note that IC/WE and IB/WE decrease with

increasing WE, not because the lateral injection becomes less

significant as in Figs. 2.4 and 2.5, but because both the

debiasing of the internal junction and high-current-induced

voltage drops across the extrinsic resistances, including

base resistance, increase with WE. From the figure however,

we can infer that the voltage drops, which become greater for

large devices due to the increased terminal currents, are

most dominant. The effect of current crowding on P is well

illustrated in Fig. 2.6(b); of course, the better simulations

obtain with debiasing accounted for. However the debiasing

seems to be insignificant for contemporary scaled devices, as

discussed before. Our other simulations show that for

devices with WE>LE, the debiasing effect is almost the same

for each device, since the predominant base current flow

under the rectangular emitter is laterally along the shorter

emitter dimension (LE in this case).

















10-3


0 1 2 3 4 5 6
WE [um]

(b)
. . I . . eI I I I * si, ,


140-

120-

100-

80-

60-

40-

20-


2 3 4


5 6


WE [um]




Fig. 2.6 Measured and simulated IC/WE, IB/WE in (a) and 3 in
(b) for the test BJTs with LE=9.2lm for VBE=0.9V.


0



------- ------



O Measurement (VBC=0.(
- - w/o Crowding
S-- w/ Crowding
0


VBC = -3.0V
O VBC = O.OV
-- w/ Crowding
- - - w/o Crowding






6p









2.4 Summary



A simple analytic model for the lateral injection of

base current, which is shown to be the most predominant

multi-dimensional current effect in advanced BJTs, has been

developed by separating the base current into internal and

peripheral components. The model is intended for (digital)

circuit simulation and has been implemented in MMSPICE-2.

For high VBE, the effective bias (see Chapter 3) on the

emitter-base junction is defined to account for the debiasing

effect. The tool is well supported by experimental results

of test BJTs having varied emitter geometries, despite the

fact that the simulation for each device was done by scaling

only AE for a given parameter set. Therefore, this lateral

injection model, combined with the current-crowding analysis,

can be the basis for more predictive and scalable BJT

simulation for TCAD.

For analog circuit simulations, more precision is

usually required. In this case, it is possible to analyze

more physically the multi-dimensional effects by cascading a

second (peripheral) BJT to the intrinsic one, each

represented by the one-dimensional BJT model in MMSPICE; the

composite transistor is also useful to account for the

parasitics associated with the extrinsic base region as well

as the lateral injection effect, for example in RF IC design









applications [Jaf92].

From the measurement and simulation results for

contemporary BJTs, the following conclusions were reached:

(1) The lateral injection of the base current becomes more

significant with decreasing VBE, which reveals that the

nature of this perimeter effect is recombination at the

peripheral junction SCR near the oxide-silicon interface.

(2) The peripheral component of collector current is

typically negligible.

(3) In high-current regions, the voltage drops across the

extrinsic resistances are most predominant, and the dc

debiasing effect seems to be negligible for contemporary

BJTs.













CHAPTER 3
NON-QUASI-STATIC MODELING OF BJT CURRENT CROWDING



3.1 Introduction



In contemporary digital circuits containing advanced

(scaled) BJTs, high transient base current can be much

greater than the steady-state current; this clearly implies

the non-quasi-static (NQS) nature of transient current

crowding. (We generally define an NQS effect in the time [or

ac-frequency] domain as one that cannot be inferred nor

characterized from steady-state [dc] conditions.) Hence it

can be significant even though dc crowding may be

insignificant [Tan85]. The classical treatment of emitter

current crowding by Hauser [Hau64], although useful, is

inadequate for advanced BJTs because it neglects conductivity

modulation of the base, which can occur because of high

injection and/or base widening, and because it assumes

steady-state or quasi-static conditions. In fact, transient

current crowding is NQS, as well as being dependent on the

base conductivity modulation [Ham88].

There has been some modeling done addressing the NQS

nature of current crowding, but generally involving

distributed lumped-model representations of the base region.








Indeed NQS effects can be physically accounted for by

cascading a sufficient number of elemental quasi-static

models, but computational efficiency must be sacrificed. Rey

[Rey69] used a more novel approach to model ac crowding and

derived a frequency-dependent base impedance for an

equivalent-circuit model.

In this chapter we extend the one-dimensional BJT model

in MMSPICE-1 to account for three-dimensional transient

current crowding in advanced, self-aligned devices which have

peripheral base contacts. The formalism includes a novel

methodology for semi-numerically modeling general NQS effects

in transient device/circuit simulation. The new model

characterizes a time-dependent effective bias on the emitter-

base junction for each Newton-Raphson iteration of the

circuit nodal analysis at each time-step. The semi-numerical

analysis follows Hauser, but physically accounts for base

conductivity modulation and the NQS nature of the crowding.

The latter extension is effected by the novel

modeling/implementation that involves the use of the previous

time-step solution in the current time-step analysis. The

model naturally accounts for dc crowding as well, which is

important in analog circuits, and which was needed in Chapter

2 to interpret the multi-dimensional current measurements in

the BJT. It does not require a lumped intrinsic base

resistance [Jo90], which is commonly used in BJT circuit









models.

The NQS model, implemented in MMSPICE-2, enables a semi-

numerical mixed-mode device/circuit simulation capability for

application-specific TCAD. The tool is supported by

numerical simulations of advanced BJT structures using PISCES

[PIS84]. It is used to clarify the nature of the added (NQS)

delay due to current crowding in switch-on and switch-off

transients in representative BJT inverting circuits, and it

reveals the significance of transient crowding even in

submicron devices.



3.2 Model Development



The intrinsic base of the advanced (self-aligned) BJT is

surrounded by a high-conductivity extrinsic base. Hence the

predominant base current flow under a rectangular emitter is

along the shorter emitter dimension (WE); this is assumed in

our (quasi-three-dimensional) crowding analysis. Consider a

section of the base of an npn BJT as shown in Fig. 3.1, where

WE is shorter than the emitter length LE. For transient

conditions at a point in time, let iB(y) be the lateral base

current which causes the crowding in the emitter-base

junction. Then, the emitter-base junction voltage v(y) can

be expressed as








EMITTER


I -Y


y=O


y=WE/2


Fig. 3.1


Cross section of the advanced (symmetrical)
bipolar junction transistor. Wb(eff) is the
widened (due to possible quasi-saturation) base
width.


BASE










v(y) = v(O) dv




= VBE iB(y)dRBi




= VBE iB(y)pdy (3.1)




where VBE is the peripheral junction voltage and p is the

specific base resistivity,


dRBi 1 WE
-- (3.2)
dy 2pqLpLEWb(eff) 2jLp(QBB + QQNR)



In (3.2), p represents an average hole density at y, which we

assume can be represented in terms of the total hole charge

(QBB+QQNR) in the quasi-neutral base (possibly widened to

Wb(eff) due to quasi-saturation); QBB, the hole charge in the

metallurgical base region, and QQNR, the hole charge in the

widened base region, both integrated over the emitter area AE

as well as over the base width, are characterized in the one-

dimensional model [Jeo89] This assumption in (3.2) is

consistent with a quasi-two-dimensional analysis (to be

described) which links the one-dimensional ambipolar








transport to the lateral hole flow. Implicit in the

assumption is a neglect of lateral hole diffusion, which

indeed is typically small compared to the lateral drift

current when crowding is significant. The model deficiency

resulting from this neglect will be shown to be

inconsequential later. The hole mobility at y is also

approximated by an average value Lp which is reasonably

estimated from common sources. Note that the factor of 2 in

the denominator of (3.2) accounts for the symmetry of the

transistor obvious in Fig. 3.1.

For transient excitation, the main components of the

intrinsic base current iB are typically hole current back-

injected from the base to the emitter (IBE) and majority-hole

charging/discharging current (dQBE/dt) Note that QBE

includes components of (QBB+QQNR) communicating with the

emitter [Jeo89]. It comprises space charge (e.g., junction

depletion charge) as well as quasi-neutral-region charge in

the intrinsic device structure. Generally, IBE(y) can be

expressed as



IBE(y) = IBE(O) 2JEoLEexp ) l dy (3.3)




where JEO is the (constant) emitter saturation current

density. We assume that the y-dependence of dQBE(y)/dt, at a








particular point in time, may be similarly expressed as


m+l m+l1
dQBE (y) dQBE (0) j v(y)
BE = B 2J L exp v ( 1dy
dt dt f Q VT 1 (3.4)



where JQ(t) is a transient (time-dependent) counterpart to

JEO. Implicit in (3.4) is an idea that JQ can be estimated

from the previous time-step (t=tm) solution for dQBE/dt for

use in the current time-step (t=tm+l) analysis as follows:


dQmE (0)
Ja+l = dt
SLEWf BE(eff) (3.5)

VT



where VBE(eff) is an NQS effective bias on the emitter-base

junction defined (see (3.7)) to account for the current

crowding (see the discussion in the Appendix B). So our

model, when implemented based on the previous time-step

solution, accounts for transient crowding non-quasi-

statically. The approximation in (3.5) is viable even for

fast transients because of the automatically controlled time-

step reduction in the simulator, which is needed to ensure

acceptable truncation error and convergence of the time-point

solution.

With (3.3) and (3.4), the intrinsic base current iB(y)


is written as









dQBE(y)
iB(y) = IBE(Y) + dQBE
dt


d (0y
= IBE(0) + dQBE(0 2(JEo + JLE expP 1 dy
dt f VT




= B(O) 2JEO(eff)L exPv -1dy (3.6)




where the time-dependent JEO(eff) is defined as the sum of JEO

and JQ(t) To facilitate an analytic accounting for the

crowding (reflected by the integral in (3.6)), we define

VBE(eff) based on the total intrinsic base current:


VBE (eff) 1
iB(0) = LEWEJEO(eff) exp V 1 (3.7)




Note that (3.7) is consistent with (3.5).

Now, following Hauser's classical analysis [Hau64], we

differentiate (3.6) combined with (3.1) to get



S= -2JEO(eff)L exp-(VBE iB(y)pdy) 1 (3.8)
Dy VT



This integral-differential equation for iB (y) may be

transformed into a closed-form second-order differential








equation by differentiating it. This differentiation, with


v(y)
ex >>[ 1 (3.9)
VT



for all values of y, which is generally valid for problems of

interest, yields


2 iB P iB
+ --B- (3.10)
ay2 VT ay



For transient crowding, (3.10) has two different types of

solution depending on the sign of BiB/ay. We consider the two

cases separately.



3.2.1 Switch-on Case



When the BJT is switched-on, iB>O tends to cause

peripheral-emitter current crowding, as in dc crowding

[Tan85]. In this case, aiB/By is negative, and the solution

of (3.10) is



iB(y) = A tan- _B- B (3.11)
L2V \ B j



where A and B are arbitrary constants of integration. The

constants can be evaluated from the boundary conditions of








the problem. For the structure shown in Fig. 3.1, we have

due to the symmetry


iWE ,
X21


which gives B=WE/2. Then from (3.11),


iB(y) = A taz( 2y
W \


(3.12)


(3.13)


where z=ApWE/(4VT) Hence, the total base current is


iB(0) = A tan(z)


(3.14)


which is equated to (3.7) to characterize vBE(eff)-

Using (3.13) in (3.1) and doing the integration yields


v(y) = VBE 2VT n-csz(l 2y/WE)
cosz


(3.15)


Note for this case that


v(W) = VBE + 2VT In(cosz) < VBE = v(0)



Now using (3.15) in the integration in (3.6), with the

boundary condition (3.12), yields another expression for the








total base current:


BE sinz COSZ
iB(O) = LEWEJEO(eff)exp -- sin (3.16)




With (3.7),(3.14), and (3.16) we now have a set of three

nonlinear equations in three unknowns (vBE(eff) A, and

iB(O)), which can be numerically solved by the iterative

Newton-Raphson method. An interesting relationship is an

expression relating VBE(eff) to VBE. This is obtained by

equating (3.7) to (3.16):


xVBE(ef f)) V=BE sinz COSZ
exp = exp (3.17)
VT VZ z



for exp [BE(eff)/VT]>>i. Note that vBE(eff) is always less

than VBE for the switch-on case since (sinz cosz/z) is less

than unity.

The accounting for dc crowding in the model is inherent

in the switch-on analysis described above. For the dc case,

JQ=O and JEO(eff)JEO in (3.16)



3.2.2 Switch-off Case



For the switch-off case, iB<0 tends to cause central-

emitter current crowding. The analysis is very similar to







that for switch-on, except that now aiB/ay is positive.

Actually this condition does not obtain instantaneously when

the BJT is abruptly turned off from an on-state. A very fast

transient occurs during which holes diffuse out of the

intrinsic base periphery to support the central-emitter

crowding that ultimately controls the predominant switch-off

transient. Our model presented below is invalid during this

fast transient since it neglects lateral diffusion flow.

However this brief invalidity is typically inconsequential

with regard to simulating the predominant transient. Note

that the fast (diffusion) transient is governed by a lateral

quasi-neutral base transit time for minority electrons; it is

proportional to (WE/2)2/Dn where Dn is an average diffusion

constant for electrons.

With the same boundary condition (3.12), the solution of

(3.10) with iiB/ay>0 is



iB(y) = -A tanhz1 2y (3.18)




So, the total base current is now



iB(O) = -A tanh(z) (3.19)



Once again we define the NQS effective bias vBE(eff) by (3.7),

in which JEO(eff) is now negative because predominant









discharging current flows in this case. Following the steps

in the switch-on analysis, we get


v(y) = VBE + 2VT in[cosh{z( S 2y/z (3.20)
cosh{z(1 2y/WE)

Note here that


v- =- VBE + 2VT In(coshz) > VBE = v(0)




The total base current can now be derived, analogously to

(3.16), as


VBE COshz sinhz
iB(O) = LEWEJEO(eff)exp(T coshz sinhz
T (3.21)



Once again we have a system of three nonlinear

equations, (3.7), (3.19), and (3.21), that define VBE(eff)

semi-numerically via iterative solution. Another interesting

relationship between VBE and vBE(eff) is obtained from (3.7)

and (3.21):


VxBE(eff) exVBEi coshz sinhz
exp = exp-------- (3.22)
VT VT z



for exp[vBE(eff)/VT>>1 Note that vBE(eff) is always greater

than VBE in the switch-off case since (coshz sinhz/z) is

greater than unity.








We note that the switch-off analysis described above has

no solution for extremely large negative JEO(eff), which tends

to obtain when the discharging current dQBE/dt (viz., JQ in

(3.5)) becomes too large compared with the dc current IBE*

This condition is non-physical, and reflects the deficiency

of our model during the initial fast (diffusion) transient

discussed previously.

The no-solution problem can be avoided by limiting

JEO(eff). Such limitation results in a solution, albeit

invalid, that most importantly carries the simulation through

the fast transient to the most significant lateral-drift-

controlled switch-off transient. So, for each iteration at

each time-step, we calculate a hypothetical maximum absolute

value of JEO(eff) for which the system of equations is

solvable, and then compare it with the actual JEO(eff); the

smaller value is used for the analysis. Details are given in

the Appendix C. This hypothetical limit for JEO(eff) is, as

expected, used only at the very beginning of the switch-off

transient, where the model is non-physical anyway, and indeed

is insignificant with regard to the predominant transient.



3.3 NOS Model Implementation



Our novel NQS modeling/implementation in MMSPICE-2 of

the BJT current crowding involves a coupling of the vertical








and lateral carrier-transport analyses in the base region.

For the npn device, the analysis of the two-dimensional hole

flow semi-numerically defines vBE(eff) for each Newton-Raphson

iteration of the circuit nodal analysis at each time step.

The implemented transient-crowding model algorithm is

flowcharted in Fig. 3.2. The calculation of JQ from the

previous time-step solution for use in the current time-step

is done only in the first iteration at each time step, and

the value is used for all subsequent iterations. With the

terminal biases VBE and VBC passed in from the nodal analysis,

the one-dimensional model routine in MMSPICE solves the

ambipolar transport, accounting for constant extrinsic

terminal resistances, and characterizes the base charge in

both the metallurgical (QBB) and widened (QQNR) base regions.

These charges define the specific base resistivity (p) for the

current time-step analysis, which is needed in the solution

of the hole transport to derive a new VBE(eff). As discussed

in Section 3.2, this derivation requires a Newton-Raphson

iterative solution because of nonlinearities due to the

conductivity modulation.

Note in Fig. 3.2 that vBE(eff) is not iteratively coupled

to the one-dimensional model solution; that is, p is not

updated to correspond with vBE(eff) VBE. Although this one-

pass derivation of vBE(eff) using p(VBE) might seem incomplete,

it is proper. A complete iterative solution, which would


























































Fig. 3.2


Flowchart of the MMSPICE-implemented transient
current crowding analysis, for every iteration at
each time step.









require an outer Newton-like loop in the algorithm, would be

non-physical. The reason is that in the switch-on case where

VBE(eff) is less than vBE, the smaller VBE(eff) in the one-

dimensional model would not adequately account for possible

high-current effects at the periphery, and that in the

switch-off case where VBE(eff) is greater than VBE, the larger

VBE(eff) in the one-dimensional model would tend to diminish

the central crowding effects by implying a smaller p.

With VBE(eff), the one-dimensional MMSPICE model routine

is called again to obtain the nominal bias-point solution.

Since the model is semi-numerical, analytic derivatives of

the currents and charges cannot be given explicitly. Thus,

numerical (divided-difference) approximations are used to

evaluate (trans-)conductances and (trans-)capacitances for

use in the subsequent nodal analysis. In order to do that,

the model routine is called twice more with perturbed values

of vBE(eff) and vBC as indicated in Fig. 3.2. The admittance

matrix is then loaded, and ordinary circuit nodal analysis

follows.



3.4 Simulations



Examples of transient simulations using MMSPICE-2 are

presented in this section. One circuit chosen for simulation

is a single-transistor inverter shown in Fig. 3.3, with no













Vcc=2 V


RCC=200 Q


-0 OUT


RBB=100


Fig. 3.3


A single transistor inverter circuit. The base
terminal is driven with a voltage pulse that is
delayed by 200ps and then ramped up (down) from
0.4V (0.9V) to 0.9V (0.4V) at a rate of 0.1V/ps.









load on the output. The assumed BJT model parameters

characterize a typical advanced device structure with

WE=1.2tm. The peak base doping density is 1.5x1018cm-3 and

the metallurgical base width is 0.15p.m. For the switch-on

transient, the NQS nature of the transient current crowding

is well illustrated in Fig. 3.4 where the simulated time-

dependent JQ, defined in (3.5), is compared with JEO. Note

that JQ is several orders of magnitude greater than JEO at the

moment the device is switched-on. It decreases monotonically

with time and finally becomes less than JEO only when the

device nears steady state. In the switch-off case, JQ is

negative, and its magnitude is not so large as for the

switch-on case. This is due to the exp[vBE(eff)/VT] term in

the denominator of (3.5), which is large when the device is

switched off.

For the complete switch-on/switch-off cycle, Fig. 3.5

contrasts the simulated vBE(eff) with VBE in time, accounting

for constant extrinsic/external base resistance, which is

reflected by the discrepancies between VBE and the input

voltage vin. The moment the device is switched-on, vBE(eff)

becomes, as mentioned earlier, less than VBE due to the high

transient base current-induced crowding, but then increases

steadily with time to a value that corresponds to dc

crowding, which is relatively insignificant. For the switch-

off transient, vBE(eff) is greater than vBE, but the difference





48








I .I , I I , I I I


0 110-10


210-10


3 10-10 410-10

Time [sec]


51I 610 I 710
510-10 610-10 710-10


Fig. 3.4


Simulated JQ versus time in the switch-on case.
JEO is the emitter saturation current density.


102
10



10 -


-2
10


i
10-6


10-8


0
0
0
o
o -





o
o
J 00o

- - --E O 0 0 0 00 0 0
O000


-4 .


. . .


. . l ' '














1.2



1



LU 0.8


oa
" 0.6

LU


0.4



0.2


0 210-10 410-10 61010 81010


1 10-9


Time [sec]


Fig. 3.5


Simulated VBE(eff) versus time for the complete
switch-on/switch-off cycle. The input pulse and
the actual (peripheral) base-emitter junction
voltage are shown for comparison.








is not so noticeable as for the switch-on case. These

results suggest that the central-emitter current crowding

during a switch-off transient is much less significant than

the peripheral-emitter crowding during a switch-on transient.

This can be attributed to the level of base conductivity

modulation (reflected by p) at the initial stages of the

respective transients.

Fig. 3.6 shows the output voltage characteristics of the

inverter simulated with (MMSPICE-2) and without (MMSPICE-1)

the current crowding accounted for. In accord with

conclusions drawn from Fig. 3.5, the result of the switch-on

transient crowding is a substantively slower response, while

the added delay is insignificant for the switch-off

transient. Other simulations show that accounting for only

quasi-static crowding (due to JEO in (3.6)) yields an output

voltage characteristic which is virtually identical to that

predicted by the simulation in Fig. 3.6 for which crowding

was completely neglected.

Predicted switch-on delays of the single transistor

inverter versus WE, with the emitter area fixed

(AE=LExWE=9.2x2.0m2), and with the emitter area scaled with

WE, are plotted in Fig. 3.7. The emitter width WE was varied

using the values 0.1, 0.4, 1.2 and 2.Opm. The delay was

defined as the time for the output current to reach 50% of

its final (high) value. The effect of the crowding is made







































210-10 410-10 610-10 810-10

Time [sec]


Fig. 3.6


Output voltage characteristics of the single
transistor inverter simulated with (MMSPICE-2) and
without (MMSPICE-1) the transient current crowding
accounted for.


1 10-9














I I i I i I


' ' I '.. I .1 I I ' ' I

0 0.5 1 1.5 2 2.E
WE [um]


Fig. 3.7


Predicted switch-on delays of the single
transistor inverter versus WE, with the emitter
area fixed (AE=9.2x2.0lm2), and with the emitter
area scaled with WE.


MMSPICE-2, AE fixed
- - MMSPICE-1,AE fixed
[ MMSPICE-2, AE scaled
- f- MMSPICE-1, AE scaled


200




160




-0 120

-

a 80 -


40




0-


.... I


- ) ---------- -O









apparent by including in the figure delays predicted by one-

dimensional (MMSPICE-1) simulations. For the switch-on

transient, the results, consistent with previous work

[Tan85], show that peripheral-emitter crowding causes an

added delay, one that tends to become insignificant only when

WE is reduced to deep-submicron values [Ham88]. Note in Fig.

3.7 that when the emitter area is scaled with WE, the delay

is more sensitive to WE. The reason of course is that, in

addition to the crowding effect, the amount of charge that

must be stored in the BJT varies with WE. Other simulations

show that the relative importance of the crowding varies

inversely with the extrinsic (plus external) base resistance.

Results of switch-off simulations with varying WE show

that the added delay due to central-emitter crowding is

negligible, at least for WE<2p~m. Indeed the simulations

predict that the reduced delay of a scaled (WE and AE) device

is due predominantly to the reduced charge storage in the

BJT.

The effect of the emitter length LE on the current

crowding is reflected in Fig. 3.8, which shows normalized

predicted switch-on delays versus WE for devices with AE fixed

at 9.2x2.Om2 or 3.2x2.Om2. Note that for a fixed WE, the

crowding effect on the delay diminishes with increasing LE.

This is due to the decreasing specific resistivity p in

(3.2).



































0 0.5 1 1.5 2 2.5


WE [um]


Fig. 3.8


Predicted normalized switch-on delays versus WE,
with fixed AE, for devices with different LE.









The influence of the nominal base resistivity, viz., the

Gummel number, on the added switch-on delay due to crowding

is revealed in Fig. 3.9 where predicted normalized delays are

plotted versus WE (with fixed AE=9.2x2 0m2) for three

different metallurgical base widths WBM. The peak base doping

density was fixed at 1.5xl018cm-3. The plots show how the

transient crowding becomes more significant as WBM is scaled

down, independent of the increasing current gain of the BJT

since there is no load on the inverter (Fig. 3.3).

In order to verify our model, two-dimensional numerical

simulations of the nominal BJT inverter were performed using

PISCES [PIS84], the results of which for varying WE are shown

in Fig. 3.10. In these switch-on and switch-off simulations,

the actual emitter length was fixed at lLm because the output

currents of PISCES are always normalized by the length

perpendicular to the simulated structure. Also, the values

of WE used for the plots are the effective emitter widths,

which are about 0.2pLm wider than the poly-emitter windows

because of lateral diffusion. The contact resistances at the

collector and base terminals were specified to include the

external resistances in the inverter circuit. Included in

Fig. 3.10 are corresponding MMSPICE device/circuit

simulations, with LE=1lm. In the switch-on case, the

transient current crowding is significant and is faithfully

predicted by MMSPICE-2, as contrasted by the inaccurate















,, ,I I ~ I I I _


S1 11 1 1 I I 1 I 1

0 0.5 1 1.5 2 2.5

WE [um]


Fig. 3.9


Predicted normalized switch-on delays versus WE,
with fixed AE, for devices with different WBM.


1 i I I I I I I


1.2


1 -



0.8



0.6



0.4



0.2


--- WBM=0.20um
- - WBM=0.15um
-WBM=0.10um


" '


' '














140


120 -


100


80-


60 -





20 -


0


0- I


0.5


WE [um]


Fig. 3.10


PISCES simulations of the switch-on and switch-off
delays versus (effective) WE of the single
transistor inverter, with corresponding MMSPICE
simulations. LE=1~m for all simulations.


--0 -PISCES [On]
--- MMSPICE-2 [On]
- O- MMSPICE-1 [On]
-- -PISCES [Off]
-- MMSPICE-2 [Off]
- j- MMSPICE-1 [Off]








G-


---- -8
p^ /


. . . . . . . . .








MMSPICE-1 simulations which are also shown. Some discrepancy

in the submicron region is apparent. This could be due to a

parasitic peripheral-region transistor unaccounted for in

MMSPICE-2 simulations; or possibly to slightly different

physical model parameters, e.g., mobility, assumed by PISCES

and MMSPICE-2. In the switch-off case, the crowding is seen

to be insignificant as implied previously. It can be

inferred then that the reduction of switch-off delay of a

scaled device is primarily caused by the reduced charge

storage rather than the diminished crowding in the BJT.

Additional verification of the NQS crowding formalism in

MMSPICE-2 is provided in Fig. 3.11 where switch-on transient

collector currents predicted by PISCES, MMSPICE-2, and

MMSPICE-1 are plotted. These currents were taken from the

WE=l 4.m simulations of Fig. 3.10. Note the good

correspondence in time between the PISCES and MMSPICE-2

currents, which are separated from the MMSPICE-1 current by a

significant (added NQS) delay.

In MMSPICE-1, a semi-empirical accounting for current

crowding can be effected by using a parameter which defines

the intrinsic base resistance as a function of the current-

dependent charge. Although the parameter could account for

the current crowding for given device dimension, it is not

applicable to other device dimensions since the parameter is

neither scalable nor predictable. Hence it cannot yield a














10-2



1-3


10-7


0 110-10 210-10 310-10


Time [sec]


Fig. 3.11


Predicted switch-on transient collector currents
taken from the PISCES, MMSPICE-2, and MMSPICE-1
simulations of Fig. 3.10 for WE=1.4(Jm.


410-10








trend like Fig. 3.10.

Finally, to emphasize the mixed-mode NQS simulation

capability of MMSPICE-2, transient simulations of an ECL

inverter stage, the basic building block of high-speed

digital circuits, were done. Fig. 3.12 shows the circuit

diagram; the four nominal BJTs have WE=1.2pm. The output

voltage waveforms of the circuit predicted with and without

(via MMSPICE-1) current crowding are plotted in Fig. 3.13.

The effect of the NQS current crowding is apparent; the

propagation delay is increased by almost 50%.



3.5 Summary



A novel NQS model for transient current crowding in

advanced BJTs has been developed. The new model, based on

the use of the previous time-step solution in the current

time-step analysis, characterizes a time-dependent effective

bias on the emitter-base junction for each circuit nodal

iteration at each time-step in a semi-numerical analysis

following Hauser [Hau64], but physically accounting for base

conductivity modulation and the NQS nature of the crowding.

The NQS model, implemented in MMSPICE-2, enables a semi-

numerical, scalable, mixed-mode device/circuit simulation

capability for application-specific TCAD. The tool is

supported by numerical simulations of advanced BJT structures






















Vin O




-2.5 V















Fig. 3.12 A
f


GND






Vout










-5.2 V


n advanced-technology ECL inverter circuit. The
our BJTs have LE/WE=9.22lm/l.2j2m.














-0.5




-0.7




-0.9




-1.1




-1.3




-1.5


0 1 1010 21010 31010 41010


Time [sec]


Fig. 3.13


Switching waveforms of the ECL inverter circuit
simulated with (MMSPICE-2) and without (MMSPICE-1)
the transient current crowding accounted for.


510-10









using PISCES. From the simulations of a representative BJT

inverter circuit, the following conclusions were reached.

(1) For the switch-on transient, peripheral-emitter crowding

causes an added delay, and tends to become insignificant

only when WE is scaled to deep-submicron values.

(2) For the switch-off transient, the added delay due to

central-emitter crowding is negligible, at least for

WE<2pm. Indeed the reduced delay of a scaled (WE and AE)

device is due predominantly to the reduced charge storage

in the BJT.

We note that the novel modeling/implementation involving

use of the previous time-step solution to update the model

for the current time-step analysis could be a viable means of

accounting for general NQS behavior in semi-numerical

transient device/circuit simulation. Such behavior must

indeed be modeled to enable truly predictive mixed-mode

simulation for TCAD.












CHAPTER 4
ANALYTIC ACCOUNTING FOR CARRIER VELOCITY OVERSHOOT



4.1 Introduction



In advanced silicon-based bipolar technology, the

vertical as well as the lateral dimensions of the BJT are

being scaled to deep-submicron values. Consequently, very

high electric fields and field gradients are not uncommon in

the scaled device. When the field increases rapidly over

distances comparable to the energy-relaxation mean free path,

carrier velocity can overshoot the value corresponding to the

local electric field. This enhanced transport occurs because

the carrier (kinetic) energy, which controls the collision

time and hence limits the velocity, lags the field and

remains relatively small [Ruc72]. Such a non-local effect

has been recognized as significant in MOSFETs and MESFETs for

years, but only now is its significance in advanced bipolar

transistors (BJTs) becoming an issue [Lee89, Cra90].

Recent work [Fus92] has indicated that velocity

overshoot in scaled silicon BJTs can be beneficial, and must

be accounted for in the device and circuit design. The

effect, however, has not yet been physically accounted for in

any circuit simulator. Indeed, this phenomenon is not









accounted for in most device simulators because of the

implied computational intensiveness. The conventional drift-

diffusion current equation used in ordinary circuit and

device simulators does not account for the non-local effect

of an inhomogeneous electric field on the carrier velocity.

It is based on the assumption that the drift velocity is a

function of the local electric field, and ignores the actual

dependence (of mobility) on carrier energy.

Non-local effects on carrier transport have been

accounted for using different analyses, but with severe

restrictions because of the accuracy/computational efficiency

trade-off. Hence these analyses--which include rigorous

Monte Carlo statistical treatments [Lee89], less complex

solutions of the hydrodynamic equations involving the

solution of the moments of Boltzmann transport equation

(i.e., a set of equations describing conservation of particle

number, momentum, and energy solved in conjunction with

Poisson's equation) [Blo70], and even simpler solutions of

the energy transport equations which, with some assumptions,

can be derived from the hydrodynamic model [Bor91]--have

limited utility for device simulation and virtually no use

for circuit simulation. Alternatively, the so-called

augmented drift-diffusion (ADD) transport model [Tho82],

which retains most of the efficiency of the drift-diffusion

equation but uses additional analytic terms to account for









the non-local effects, has been proposed as a way of

efficiently extending the utility of drift/diffusion-based

tools for scaled technologies.

In Section 4.2, a simple but physical analytic model for

first-order accounting of the electron velocity overshoot in

advanced silicon-based BJT "circuit simulation" is presented.

The model, which characterizes the non-local electron

velocity in the high-field collector space-charge regions

(SCRs), is shown to be identical to the ADD formalism when

the electron diffusion is negligible. The associated

velocity relaxation, which is not accounted for in the ADD

model, is characterized phenomenologically to be consistent

with the overshoot analysis. In Section 4.3, the comparison

of our model with the energy transport analysis is presented.

In Section 4.4, the implementation of the model in MMSPICE is

discussed. In the last section, device and circuit

simulation results are presented to assess the significance

of the electron velocity overshoot in advanced silicon

bipolar and BiCMOS technologies, and to support the model.

This is the first time that a non-local effect has been

explicitly accounted for in a circuit simulator.









4.2 Model Development



4.2.1 Velocity Overshoot



When the randomly moving conduction-band electrons in a

semiconductor encounter an electric field, they experience an

increase in average (drift) velocity, and an increase in

average kinetic energy which however tends to lag the drift

velocity [Ruc72]. When the kinetic energy is important

(i.e., when the electrons are not in thermal balance with the

lattice), a phenomenological force acting on the electrons

can be expressed in one dimension as


qE(eff) d (EC- W (4.1)
dx



where Ec and W are the (average) potential and kinetic

energies of the electrons respectively. Note that Ec and W

in (4.1) are "correlated" in accord with electron flow. When

Wt is small (=3kT/2 where T is the lattice temperature),

E(eff) is the actual field, E, proportional to dEc/dx as it is

classically expressed.

Ballistic transport of the electrons, driven by E, would

result in unlimited 7W. However the electrons in a crystal

lattice frequently collide with impurities and phonons, the

result of which is to randomize their motion and limit their









(average) drift velocity, v, and hence their momentum.

Effectively the collisions give rise to a retarding force

proportional to the velocity, as characterized by the balance

of momentum [Shu81]:


m*d = qE(eff) m* v (4.2)
dt T(71



where m* is the effective mass of conduction (sub-)band

electrons and T(7'4 is an energy-dependent momentum relaxation

time. Combining (4.1) and (4.2) yields


*dv dEc dW v
m- + -- m-
dt dx dx (74



= qE + m (4.3)




For dc or quasi-static analysis, dv/dt=0 in (4.3) and



v m -qE + (4.4)




Note that when dW/dx is negligible, (4.4) becomes a well-

known equation defining the electron mobility 1(7'4 (=1v/El):



( q (4O (4.5)
m










The mobility is expressed as a function of 4' to emphasize

that it depends more on the local carrier energy than on the

local electric field. Using (4.5) in (4.4) with the chain

rule for differentiation gives



LIq dlEl E dxH


= vo(E) + L(E) dE (4.6)
L E dxJ (4.6)



where vo(E) is the conventional drift velocity defined by the

local field, and L(E)E(d'W/dlE )/q is a phenomenological

length coefficient [Pri88], which describes to first-order

the non-local effect of the electric field gradient on v.

For L(E)#0, a large dE/dx in (4.6) implies a possibly

significant velocity overshoot, Ivi>|vo(E) in accord with

the more rigorous physics underlying the electron transport.

Note that (4.6) is identical with the ADD formalism [Tho82]

when the diffusion of carriers is negligible [Kan91]. The

field gradient in (4.6) was substituted with the quasi-Fermi

level by other authors [Kiz89], to avoid inappropriate

overshoot corrections in the presence of built-in electric

field. However this would not be important in real

applications, since the simulation of the equilibrium

condition is not needed in most cases.








The length coefficient has been characterized via Monte

Carlo analysis [Art88] by several investigators. However the

results show some quantitative differences, possibly because

of the different transport parameters and band structures

used. Recently, Chen et al [Che91] derived an analytic

formula for L(E), but its utility is subject to uncertainties

in the evaluation of some model parameters. Hence we suggest

a simplified piecewise-linear representation of L(E) for

electrons in silicon at room temperature, based on Artaki's

Monte Carlo simulations [Art88], which is illustrated in Fig.

4.1. In fact, L(E) can be negative for low |E|, although the

velocity undershoot thereby implied by (4.6) is generally not

significant [Lun90] and will be neglected here.

Equation (4.5) implies that the classical mobility

decreases with increasing electric field since the electrons

gain kinetic energy which reduces the average (scattering)

time between collisions. When the velocity imparted to an

electron by the applied field is much less than the random

thermal velocity, T is however insensitive to E, implying a

linear v(E) dependence: vo=-p1oE where 4o is the low-field

mobility. At high fields however, the drift velocity becomes

comparable to the random thermal velocity, and T is reduced.

The drift velocity (magnitude) in this case, in the absence

of a high gradient of E, approaches a limiting (saturated)














1 10-5 I I I I


810-6


610-6-
0 O
410-6O O


210-6-


0 O
0 O Artaki's Work
-210-6 Our Model


-410-6
0 20 40 60 80 100 120 140
Electric Field Magnitude [KV/cm]


Fig. 4.1 The length coefficient versus electric field
(magnitude) for silicon at room temperature. The
points were derived from Monte Carlo simulations
[Art88], and the piecewise-linear approximation is
used in our model.








value vs (=107cm/sec in silicon at room temperature), which

can be empirically expressed as the product of o0 and a

critical electric field (magnitude) Es defining the onset of

velocity saturation: vs=oEs.

Hence depending on the magnitude of the electric field

in a region with dlEl/dx > 0, the magnitude of the carrier

drift velocity in (4.6) can be expressed as


v|, = olEI [1 + L(E) dE] = olEI for IE| < Es (4.7)
L E dxJ

and
= vs + L(E) dE] for IE| > Es. (4.8)
L E dxJ



The typical value of Es for electrons in silicon at room

temperature is less than 30KV/cm, and for IEI
vanishes as shown in Fig. 4.1. Hence as indicated in (4.7),

Ivl=ol IEl for this case, in accord with the conventional

characterization. This simplification means that the

velocity overshoot characterization is needed only when

IEI>Es as in (4.8), and that otherwise the conventional

drift-diffusion formalism with (4.7) is still applicable even

though dlEl/dx is high.









4.2.2 Velocity Relaxation



The analytic velocity overshoot characterization in

(4.8) is strictly valid only when the magnitude of the

electric field is increasing in the drift current direction.

It would yield no overshoot when diE|/dx = 0 or an undershoot

when dlEl/dx < 0, independent of the history of the

transport, and hence is non-physical for these cases. For

example, a hot (high-'4t electron entering such a region where

dlEl/dx is not positive must travel a few mean free paths to

reach the velocity corresponding to the local field, and

hence would experience velocity overshoot. This relaxation

can be neglected for MOSFETs and MESFETs because the only

significant non-local effects occur under the gate where

electrons are accelerated to the drain by a high field with

dlEl/dx > 0 [Kiz89, Kan91]. However for the BJT, which

contains significant (space-charge) regions with diE|/dx < 0

adjacent to those with dlEi/dx > 0, the velocity relaxation

following overshoot must be simulated. Details on various

types of SCRs will be presented in next section.

To understand the velocity relaxation in the advanced

BJT, consider a mental experiment. Fig. 4.2 shows the

possible relaxation of the drift velocity in the collector

side of the base-collector junction SCR where ]El is














v(x)



v(0)
v(O) Case 1

Case 3
vs .----------------

Case 2




Edge of SCR











Fig. 4.2 Possible distributions of the drift velocity when
IEl is decreasing with distance. Note that the
electric field magnitude at the edge of SCR is
assumed to be Es.









decreasing with distance (see Fig. 4.3(a)). Note that the

electric field magnitude at the (nebulous) edge of the SCR is

implicitly assumed to be Es [Jeo89] Normally when a hot

electron leaves a high-field region, its velocity will

decrease with distance due to the scattering by which it

transfers its energy to the lattice (see Case 1 in Fig. 4.2).

The relaxation however becomes somewhat different when the

width of the SCR gets smaller. At a glance, it seems likely

that the velocity would not decrease very much from its value

at the junction because of the reduced scattering. But

actually this tendency would be compensated by the velocity

undershoot tendency [Lun90], which obtains when the electric

field is decreasing very rapidly. The kinetic energy

responds to fields more slowly than does the carrier

velocity; hence immediately after the high- to low-field

transition, the carrier's kinetic energy is still high, and

thus its mobility is lower than that corresponding to thermal

balance between the carrier and the lattice. After the

electron has dissipated its excess energy, it would then have

the velocity vs (see Case 2). This is supported by the fact

that L(E) in (4.8) is 0 at the edge of the SCR because |El is

assumed to be Es. Taking these two conflicting phenomena

into consideration, we assume that the velocity would decay

monotonically with distance and finally reach vs at the edge

of the SCR (as described by Case 3).








Based on this insight, we use a phenomenological

representation of the velocity relaxation in an SCR where

dlEl/dx < 0 by simplifying (4.2) to


dv dv v
dt dx t



or


dv 1 ,_ v (4.9)
dx T s



where s is an average mean free path for velocity relaxation.

The solution of (4.9) is



v(x) = v(0) exp(-x/s) (4.10)



where v(0) is the velocity at the point where IEl is maximum

in the SCR. Since the velocity must be continuous, v(0) is

derived from the analysis of the velocity (overshoot) in the

adjacent region where dlEl/dx > 0. To estimate s, we assume

as discussed above that the carrier velocity reaches vs at

the edge of the SCR. (This assumption is consistent with a

common designation of an SCR [Jeo89].) Thus


s WRR (4.11)
InV(O)
Vs,











where WRR is the width of the relaxation region.



4.2.3 Effective Saturated Drift Velocity



To this point, we have modeled the hot-electron velocity

in an SCR using either the length coefficient or the

scattering mean free path, depending on the sign of diE|/dx.

To facilitate the implementation (discussed later) of the

model into the bipolar device/circuit simulator MMSPICE, we

define now an effective saturated drift velocity vs(eff) based

on the actual transit time of electrons in the SCR being

analyzed:



dt = -WSCR (4.12)
IC I- C v(x) Vs(eff)
WSCR WsCR




where v(x) is given by (4.8) or (4.10), and WSCR is the width

of the SCR in which IEl is greater than Es.

For the advanced BJT, different operating conditions are

distinguished by the charge conditions [Jeo89] in the

epitaxial collector region, as reflected in Fig. 4.3. The

electric field distributions shown are determined by the bias

on the base-collector junction and the collector current.










-Wvs 0 Wscc


E(x)


0 WQNR


WEPI WBL


E(x)


WEPI


E(x)


Fig. 4.3 Electric field distributions in a base-collector
junction SCR (a), and a current-induced SCR (b)
associated with non-ohmic quasi-saturation, i.e.,
base pushout. When either SCR expands, the entire
epi layer can become space-charged (c).








Fig. 4.3(a) represents the conventional junction SCR at the

base-collector junction under low-current conditions. For

high-current conditions, when non-ohmic quasi-saturation

(base pushout) prevails, the current-induced SCR exists in

the epi-collector as denoted in Fig. 4.3(b). Note that the

electric field is assumed to be -Es at the edge of the

collector-side SCR in both cases; this assumption in fact

defines the SCRs [Jeo89] When either SCR expands, the

entire epi layer can become space-charged, as shown in Fig.

4.3(c). We must consider the three SCR types in the BJT

separately.



4.2.3.1 Junction SCR



When the SCR exists across the base-collector junction,

as shown in Fig. 4.3(a), (4.12) applied to it yields


f0 fWscc
dx + dx Wvs + Wscc (4.13)
V(x) v(x) Vs(eff)



where Wvs and Wscc are the widths of the base and collector

sides of the SCR respectively. The carrier velocity v(x) is

evaluated depending on the sign of dIE|/dx. In the base

side, the velocity is characterized via (4.8), using the








depletion approximation coupled to a first-pass (vs(eff)->Vs)

MMSPICE simulation to describe E(x) and Wvs:


dE -q [NA(x) + n]
dx E


= - NA(X)



- NAO exp-( x + WBM) (4.14)
C LWBM



where the assumed exponential doping profile is consistent

with the base-transport analysis of the BJT model [Jeo89] in

MMSPICE; WBM is the metallurgical base width. Thus


(4.15)
E(x) qWBM NAO exp +x +W C (4.15)
EM [WBM



The integration constant C can be easily evaluated from the

electric field at the junction (x=0), which is available from

the output of the BJT model routine in MMSPICE. E(x) and

dE/dx are then substituted into (4.8) to give v(x) for the

first integral in (4.13). The validity of using the

depletion approximation here will be discussed in Appendix D.

In the collector side, (4.10) is used directly for the

second integral in (4.13), with v(0) being equated to that

derived from the analysis of base side. Both integrals in








(4.13) are evaluated by a numerical method to give vs(eff).

Strictly, the value of Es in the base side tends to be

greater than that in the collector side because the electron
mobility (Io) in the base is lower due to the higher doping

concentration. However because the (compensated) doping is

generally not known precisely and because this variation in

Es is only a second-order effect, we neglect it.


4.2.3.2 Current-induced SCR


When the current-induced SCR exists, as illustrated in

Fig. 4.3(b), (4.12) applied to it yields


WEP dx WEPI WQNR (4.16)
d-- ,W, -(4.16)
v(x) Vs(eff)



where WQNR is the extended width of the pushed-out (quasi-

neutral) base region. The transit time across the portion of

the SCR in the adjacent buried layer of the BJT structure is

neglected since the heavy doping there implies only a
negligibly thin depletion-region width, WBL-WEPI.

From the first-pass MMSPICE simulation [Jeo89], the
electric field in the SCR and WQNR are obtained in accord with


dE An
dx E











C \qAvs I



where NEPI is the doping concentration of the epi-collector

layer and An is the excess electron density in the SCR, which

is assumed to be spatially constant since the current Ic is

constant. From (4.17),


E(x) = q Ic N (x WQNR) Es (4.18)




Equations (4.17) and (4.18) are substituted into (4.8) to

yield v(x), and vs(eff) is evaluated from (4.16). In the

vicinity of the boundary between the SCR and the quasi-

neutral region in the epi-collector, the electric field

gradient is very large. However this transitional region can

be ignored because the length coefficient is, as shown in

Fig. 4.1, assumed to be 0 when IEI


4.2.3.3 Special case



In previous sections, the velocity overshoot effect was

characterized via regional analyses depending on the sign of

the field gradient. There is a special case for the BJT

however where the overshoot effect would not be properly

accounted for in this manner. This is the case where the epi









layer is completely space-charged, and the magnitude of the

electric field is still increasing with distance due to non-

ohmic quasi-saturation, as shown in Fig. 4.3(c). (Note that

when the entire epi layer is space-charged, but |El is

decreasing with distance, the overshoot analysis for the

junction SCR is still applicable.)

According to our formalism, the same overshoot analysis

would be applied in the collector side as in the base side.

Of course, this is adequate if the field gradient is

relatively large. When the electric field is increasing

slightly however, the direct application of our model would

tend to exaggerate the overshoot effect since velocity

relaxation is ignored. In fact, the carrier velocity would

decrease with distance in the epi-collector. In order to

cope with this deficiency of our formalism, we empirically

combine the overshoot model with the relaxation model for

this case as follows:



v(X) = Vrei(X) + ov (x) exp -- + Voffset (4.19)

f dx)



where vrel(x) and vov(x) are the velocity distributions

characterized by the relaxation and overshoot models

respectively, and f is an empirical weighting factor. When

the field gradient is very small, (4.19) reduces to (4.10),









implying that velocity relaxation would be predominant in the

collector side. When the gradient becomes large, v(x) is

given as the sum of vrel(x) and vov(x) with the empirical

factor chosen to ensure a smooth transition from velocity

relaxation to velocity overshoot. The offset velocity,

offset in (4.19) is used to make the velocity at the junction

continuous.



4.3 Comparisons with Energy Transport Model



One way to characterize the velocity overshoot effect is

to solve the energy transport equation [Bor91]. Such a

solution can provide support for our simple analytic model.

In this section, we will numerically solve the energy

transport equation Goldsman et al presented [Gol88], and

contrast it with our model. By assuming the electron energy

as entirely thermal, they derived the steady-state momentum

equation from the Boltzmann transport equation as


S= (w) qE 2 dw 2w dn (4.20)
m* 3 dx 3n dxx


1 3 3
where w is the average electron energy (= m*v2 + kTe = kTe

where Te is the electron temperature), Tp(w) is the energy-

dependent momentum relaxation time, and n is the electron

concentration. Combining (4.20) (with dn/dx=0) with the









steady-state energy equation, Goldsman et al derived an

equation for average electron energy that includes the effect

of velocity overshoot:


dw 21 qE 9 40 m* (-o) + 1/2 (4.21)
dx 20 20 [9 Tpw qE2]



where w (w) is the energy relaxation time, and wo is the

thermal energy of the lattice (=3kT/2).

In order to solve these equations, both Tp and Tw must be

known as functions of the electron energy. Although Goldsman

et al evaluated the relaxation times by Monte Carlo

simulations in homogeneous fields, we use simple functions to

empirically approximate the parameters they derived:


Tp(w) = co + c and (4.22)
w



Tw(w) = do + d1w + d2w2 + d3w3 (4.23)



where Cn and dn denote empirical constants. In Fig. 4.4, the

discrete points represent the momentum and energy relaxation

times Goldsman et al have derived, and the solid lines which

best fit the data are given by (4.22) and (4.23). Then the

energy dependent carrier velocity can be numerically

evaluated from (4.20) and (4.21), since those equations are a

function of the single variable w.























0.08



0.06



0.04



0.02



0


0 0.1


Fig. 4.4


0.3



m
r3
CD

0.28 <
CD


o
-I
0.26 3
CD
"0
Zcn




0.24


0.2 0.3 0.4 0.5 0.6 0.7 0.8

Average Electron Energy [eV]


Momentum and energy relaxation times as functions
of energy.









For comparisons, we evaluated the velocity distributions

for the typical advanced BJT, when the junction or the

current-induced SCR exists, using our model and that of

Goldsman et al. Fig. 4.5(a) shows the predicted velocity

distributions in the junction SCR when VBE=0.7V and VBC=O.OV

are applied to the terminals of the device. (For the

effective mass of conduction sub-band electron, m*=0.26mo was

used, where mo is the rest mass [Mu189]. The electric field

used as inputs for both the models was available from the

output of MMSPICE.) As described before, our overshoot

analysis is done when the magnitude of the electric field is

increasing (x<0). In accord with our piecewise-linear L(E)

model, the carrier velocity reaches its peak value Vpeak when

the length coefficient is at its maximum value at E=50KV/cm

(see Fig. 4.1). Note that the location of peak is about the

same as that predicted by the energy transport model. When

reverse bias is applied on the base-collector junction (VBC=

-2.0V), peak increases as shown in Fig. 4.5(b), because the

gradient of the electric field also increases. Figs. 4.6(a)

and (b) illustrate the velocity distributions in the current-

induced SCR (for VBE=1.OV, VBC=O.OV or VBC=-2.0V) .

We note in the above figures that our model predicts a

higher peak overshoot velocity than that yielded by either

the energy transport model or Monte Carlo simulations (not





88



(a)
3.5 107 i I ,I I I 140

7-
3.0 10 Our model 120
S- -- Energy transport
7
2.5 10- 100

7
Q 2.0 10 80
-. -


7 <-
1. -510A A60 |

1.0 10 - 40

5.0 106- 20



-0.1 0 0.1 0.2 0.3 0.4
Distance from B-C junction [um]

(b)
3.5107 I I I i 140

S3.0 Our model
3.0 10 120
S- - - Energy transport
7 !
2.5 10- 100
N 10

0) 7
U 2.010 -80 x

7 <
S1.510 6

1.0 107 ------- -- -40
I ----- \_

5.0106 20



-0.1 0 0.1 0.2 0.3 0.4
Distance from B-C junction [urn]




Fig. 4.5 Drift velocity and electric field in junction SCR:
(a) VBE=0.7V and VBC=O.OV; (b) VBE=0.7V and VBC=
-2.0V.













2.0 107





1.5 107





1.0 107





5.0 106


0






2.0 107


1.5 107





1.0 107





5.0 106


, I I I I ,I


0 0.1 0.2 0.3 0.4 0.5

Distance from B-C junction [urn]

(b)
.I.I I I .1 1 1 1 1 1 1 1 1 I ,


-l r I I I I I I I I I I I~l 1 I I I I r


- 200



- 150




- 100 0



-50


250


- 200



m
- 150 "




- 100 3



-50


Distance from B-C junction [um]


Fig. 4.6 Drift velocity and electric field in current-
induced SCR: (a) VBE=I.OV and VBC=0.OV; (b) VBE=1.0V
and VBC=-2.0V.


-- Our model
- - - Energy transport

















/"


-- Our model
- - - Energy transport






-I

I" /
/
/

/
I
/
/


. . I I, , I








shown in the figures) [Prof. M. Lundstrom of Purdue

University, private communication, 1991]. This discrepancy

could mean that the length coefficient [Art88] which we used

might be erroneous. Indeed when the high-IEI saturation

value of L(E) in Fig. 4.1 is reduced from 4.5x10-6cm to

2.0x10-6cm, which has been suggested [Art88], the MMSPICE-

predicted velocity overshoot is in better agreement with that

predicted by the energy transport and Monte Carlo analyses.

This uncertainty in L(E) can be attributed to the different

set of transport parameters used. However we stress that the

terminal characteristics of advanced BJTs predicted by our

model, which will be shown later, agree quite well with

results [Fus92] of measurements and numerical simulations

based on a hydrodynamic model for energy transport.

Conversely then, we note that the energy transport model has

several uncertainties as well. It is based on several

equivocal assumptions. For example, it assumes that the

electron energy is entirely thermal. Also, the results

depend on the degree of the energy transport equation, and

there are still some uncertainties in the evaluation of the

model parameters such as m*, Tp, and T,. Monte Carlo analysis

is not unequivocal either. For example, detailed and

accurate information about the numerous scattering parameters

as well as needed details of the energy-band structure are

lacking. With these deficiencies then, our model is









reasonable for first-order accounting of the electron

velocity overshoot in circuit simulation, which has never

been done before.



4.4 Implementation



The implementation of the electron velocity overshoot

model in MMSPICE is based on a single iteration of the

existing (conventional) model routine [Jeo89] for the (n+pnn+)

BJT, as illustrated in Fig. 4.7. The analysis is done for

each iteration of the circuit nodal analysis at each time

step. With VBE and VBC passed in from the nodal analysis, the

(one-dimensional) BJT model routine, which assumes a

saturated drift velocity vs (no overshoot) in the high-E epi-

collector SCRs, is called to solve the conventional ambipolar

transport, and characterize E(x) Thus unlike empirical

circuit models, the MMSPICE BJT model is susceptible to an

extension to account for the augmented non-local carrier

velocity distribution. From the predicted E(x), combined

with the length coefficient L(E), the carrier velocity is

evaluated depending on the SCR type (see Fig. 4.3). Then

from v[E(x)], the effective saturated drift velocity,

vs(eff)>Vs, is evaluated as described in Section 4.2.3.

























































Fig. 4.7 Algorithm for implementation of velocity overshoot
model in MMSPICE.









Once Vs(eff) is characterized, Es is correspondingly

updated as well to Es(eff)yVs(eff)/Io, which is higher than the

preliminary value. Fig. 4.8 illustrates the resulting

velocity-field model in the epi-collector SCRs used in

MMSPICE, which we believe is suitable for first-order

accounting of the velocity overshoot in circuit simulation.

Note that the v(E) slope (i.e., the low-field mobility p0) is

not changed. Hence the solution obtained in regions where

IEI
physically appropriate.

With vs(eff) and Es(eff), the MMSPICE BJT model routine is

called once more to effect the first-order accounting for the

non-local transport in the predicted device currents and

charges. The accounting for velocity overshoot, which is

done here in a circuit simulator for the first time, is

computationally efficient, and enables representative mixed-

mode simulation for advanced bipolar technologies as we now

demonstrate.



4.5 Simulations



In this section, MMSPICE device and circuit simulation

results are presented to assess significance of the velocity

overshoot effects in advanced silicon-based bipolar




University of Florida Home Page
© 2004 - 2010 University of Florida George A. Smathers Libraries.
All rights reserved.

Acceptable Use, Copyright, and Disclaimer Statement
Last updated October 10, 2010 - - mvs