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Properties of noise and charge transport in layered electronic materials

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Title:
Properties of noise and charge transport in layered electronic materials
Creator:
Tehrani-Nikoo, Saied, 1960- ( Dissertant )
Van Vliet, C. M. ( Thesis advisor )
Bosman, Gijs ( Thesis advisor )
van der Ziel, A. ( Reviewer )
Hench, L. L. ( Reviewer )
Chenette, E. R. ( Reviewer )
Neugroschel, Arnost ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1985
Language:
English
Physical Description:
vi, 163 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Computer printers ( jstor )
Electric current ( jstor )
Electric fields ( jstor )
Electric potential ( jstor )
Narrative devices ( jstor )
Noise measurement ( jstor )
Noise spectra ( jstor )
Signals ( jstor )
Trucks ( jstor )
Waveforms ( jstor )
Charge transfer devices ( lcsh )
Dissertations, Academic -- Electrical Engineering -- UF
Electrical Engineering thesis Ph. D
Layer structure (Solids) ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
The charge transport mechanism in layered structures, in which both space-charge injection and strong band bending occur, is studied. Samples of nitrogen-doped a-SiC are used as models to test and verify the various aspects of transport theory in short, layered devices. The samples studied consisted of a high resistive (strongly compensated) polytype layer sandwiched between two low resistive polytype layers. This creates an n+nn+--like structure. The low resistive polytypes act as injecting and extracting "contact" for the high resistive polytype, introducing into the latter single-carrier, space-charge-limited (SCL) flow. The samples which we investigated all showed typical SCL I-V characteristics with four clearly discernible regimes for most temperatures between 50K and 300K. The DC characteristics show a succession of trap filling; yet they are characterized by a single trap level at a given temperature with regard to the onset of the trap-filling regime. The energy band profile and the electric field profile in the different regimes of the I-V characteristics are calculated with the aid of a computer. The effect of carrier spillover from the n+ regions into the n region and the importance of the diffusion current are discussed. Theoretical values of the noise in the four different regimes of the I-V characteristics are obtained, using a discrete transfer impedence method (including diffusion), and are compared with the experimental results. Various transport quantities of a-SiC are deducted from the I-V characteristics, the impedence, and the noise. The insight gained from this study is used to unravel some of the problems associated with charge transport in submicros (.47 um) n+pn+ GaAs devices.
Thesis:
Thesis (Ph. D.)--University of Florida, 1985.
Bibliography:
Bibliography: leaves 159-162.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Saied Tehrani-Nikoo.

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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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14632125 ( OCLC )
AEH1124 ( NOTIS )

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PROPERTIES OF NOISE AND CHARGE TRANSPORT
IN LAYERED ELECTRONIC MATERIALS










By

SAIED TEHRANI-NIKOO


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


1985














ACKNOWLEDGMENTS


The author wishes to express his sincere gratitude to Dr. C.M. Van Vliet and Dr. G. Bosman for their research guidance and helpful suggestions, and to Dr. A. van der Ziel for fruitful discussions.

I extend my gratitude to Dr. L.L. Hench for the opportunity to do this research and for his guidance and encouragement.

Finally, the author appreciates the help of his fellow students in the Noise Research Laboratory and Ms. Katie Beard for typing the dissertation.
















TABLE OF CONTENTS


Page

ACKNOWLEDGMENTS . ii

ABSTRACT . v

CHAPTER

I INTRODUCTION . 1

II COMPUTER CONTROLLED NOISE SPECTRUM ANALYZER
EMPLOYING A DIGITAL OSCILLOSCOPE . 3

2.1 Introduction . 3
2.2 Programmable Digital Oscilloscope. 6 2.3 Spectral Intensity of a Random Noise Signal . 10 2.4 Time Window . . . 14 2.5 Aliasing . 15 2.6 Computer Software - . 18 2.7 Noise Measurement Procedure . 20

III CURRENT-VOLTAGE AND IMPEDANCE CHARACTERISTICS OF
SCL FLOW IN a-SiC. 23

3.1 Introduction. 23 3.2 Properties of SiC . 24
3.2.1 Crystallography and band structure . 24 3.2.2 Transport properties . .29 3.2.3 Our samples . 31
3.3 On the Theory of SCL Flow in the
Presence of Traps . 34 3.3.1 Analytical results . 34 3.3.2 Regional approximations. 42
3.4 Experimental I-V Characteristics . 43 3.5 Impedance Measurement . .51

IV ELECTRICAL NOISE OF SCL FLOW IN a-SiC . .57

4.1 Introduction. 57
4.2 Review of the Theory of Trapping Noise
in SCL Flow . . 59
4.3 Experimental Results in the Ohmic Regime . 64 4.4 Discussion of Spectra in the Ohmic Regime . 71 4.5 Noise Spectra in Ohmic and SCL Regimes at 77K . 72










V. COMPUTER CALCULATION OF DC SCL FLOW IN a-SiC . 80

5.1 Introduction . 80 5.2 Theoretical Model . 81 5.3 Computer Simulation. . . . 82 5.4 Theoretical Results . 87

VI COMPUTER CALCULATION OF THE SCL IMPEDANCE
AND NOISE OF a-SiC. . 01

6.1 Introduction.1 01 6.2 Theoretical Model . 102 6.3 Theoretical Results-.106

VII COMPUTER CALCULATIONS OF THE CURRENT-VOLTAGE AND THE NOISE CHARACTERISTICS OF SUBMICRON n+pn+ GaAs DEVICES . 117

7.1 Introduction . 117 7.2 I-V Characteristics. . 118 7.3 Noise Characteristics . 125

VIII CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE STUDY . 132

APPENDIX A COMPUTER PROGRAM FOR TEK 7D20 NOISE
SPECTRUM ANALYZER . 134

APPENDIX B COMPUTER PROGRAM TO CALCULATE THE CURRENTVOLTAGE, THE IMPEDANCE, AND THE NOISE CHARACTERISTICS . 151

REFERENCES . .159

BIOGRAPHICAL SKETCH . 163

















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


PROPERTIES OF NOISE AND CHARGE TRANSPORT IN LAYERED ELECTRONIC MATERIALS

By

Saied Tehrani-Nikoo
June 1985

Chairperson: C.M. Van Vliet
Major Department: Electrical Engineering

The charge transport mechanism in layered structures, in which both space-charge injection and strong band bending occur, is studied. Samples of nitrogen-doped a-SiC are used as models to test and verify the various aspects of transport theory in short, layered devices. The samples studied consisted of a high resistive (strongly compensated) polytype layer sandwiched between two low resistive polytype layers. This creates an n+nn+-like structure. The low resistive polytypes act as injecting and extracting "contacts" for the high resistive polytype, introducing into the latter single-carrier, space-charge-limited (SCL) flow.

The samples which we investigated all showed typical SCL I-V

characteristics with four clearly discernible regimes for most temperatures between 50K and 300K. The DC characteristics show a

succession of trap filling; yet they are characterized by a single trap level at a given temperature with regard to the onset of the trapfilling regime. The energy band profile and the electric field profile










in the different regimes of the I-V characteristics are calculated with the aid of a computer. The effect of carrier spillover from the n+

regions into the n region and the importance of the diffusion current are discussed.

Theoretical values of the noise in the four different regimes of the I-V characteristics are obtained, using a discrete transfer impedance method (including diffusion), and are compared with the experimental results.

Various transport quantities of c-SiC are deducted from the I-V characteristics, the impedance, and the noise.

The insight gained from this study is used to unravel some of the problems associated with charge transport in submicron (.47 urm) n+pn+ GaAs devices.
















CHAPTER I
INTRODUCTION


Recent progress in crystal growth techniques (MBE, MOCVD) has made it possible to fabricate multilayered structures having small layer thicknesses. Charge transport in layered materials is strongly

dependent on doping, stacking sequence, thickness, and energy bandgap of the individual layers. This enables one to grow semiconductor material ideally suited for particular device applications.

Due to the phenomenon of polytypism,1,2 silicon carbide (SiC) is a natural layered semiconductor. This implies that the same chemical

compound of silicon and carbon atoms ,crystallizes into different crystallographic modifications known as polytypes. These polytypes are all similar in the plane perpendicular to the symmetry axis (c-axis), but differ from each other in the direction parallel to the c-axis.

Silicon carbide (SiC) is also known to have one of the largest energy bandgaps (- 3.0 eV) of common semiconductor materials. This

property makes it valuable for high-temperature device applications and blue light-emitting diodes.3-6

The charge transport mechanism in layered structures, in which space-charge injection and strong band bending will occur, is studied, using a-SiC as a model. Various transport quantities of c-SiC are

deducted from the I-V characteristics, the impedance, and the noise in the temperature range of 50 K - 300 K. The insight we gained from this study is used to unravel some of the problems associated with charge transport in submicron (.47 pm) n+pn+ GaAs devices.

i










The organization of the chapters is as follows. In Chapter II, a newly developed computer controlled spectrum analyzer system for noise measurement is presented. A review of the properties of SiC, the

experimental results of I-V and impedance measurements, and a somewhat different version of the standard theory of space-charge limited flow is presented in Chapter III. In Chapter IV, the theory of noise in SCL flow in the presence of traps is reviewed, and experimental data on noise in the ohmic regime and SCL regimes is presented. Chapter V

contains the results of computer calculations of the DC characteristics of c-SiC in particular, and layered structures in general. In Chapter

VI, computer calculations of the impedance and the noise in c-SiC are presented. Chapter VII contains the study of the charge transport in submicron n+pn+ GaAs. Finally, conclusions and recommendations for

further work are given in Chapter VIII.

Appendix A includes the computer program for the TEK 7D20 noise spectrum analyzer. The computer program to simulate the I-V

characteristics, the impedance, and the noise is given in Appendix B.
















CHAPTER II
COMPUTER CONTROLLED NOISE SPECTRUM ANALYZER EMPLOYING A DIGITAL OSCILLOSCOPE


1. Introduction

Substantial improvements in the performance of measurement equipment have been obtained in recent years by using either desk-top computers or built-in microprocessors to control the different functions of a particular instrument. In addition, simple calculations to verify the experimental findings can be done almost instantaneously by these data processors. Significant advances in speed, stability, and accuracy of spectral noise measurements have been achieved by using the computer controlled spectrum analyzer system discussed in this chapter.

Pulsed bias noise measurements, as well as continuous bias noise measurements, can be performed in the frequency range from 10 Hz to 10 MHz. Pulsed bias noise measurements are used to avoid excessive joule heating when a device is pulsed biased at high voltage or current levels.

Nougier et al.7 described an apparatus for pulsed bias noise measurement in the frequency range of 100 MHz - I GHz. Recently,

Whiteside8 developed a similar system for pulsed bias noise measurements between 1 MHz and 22 GHz. Both systems are complicated and gate the noise spectral "on" and "off" synchronously with the bias pulse. The

total output noise power in these systems is proportional to (_) GB, where T is the pulse length, T-I is the pulse repetition rate, and GB is equal to system power gain-bandwidth product. In order to detect small










noise signals, the system power gain and the pulse duty cycle must be relatively large. Additional disadvantages are that a true r.m.s. detector with an extremely large crest factor is required and that if the system is not band limited, the frequency component of the pulse will saturate the power amplifier. The later phenomenon sets a lower limit of frequency that can be measured since the noise signal needs to go through a bandpass filter to eliminate the frequency components of the pulse.

In the system discussed here, the pulsed bias noise measurement is done by collecting the sampled data only when the bias pulse is

applied. The sampled data is then fast Fourier transformed in the same way as is done in the continuous bias noise measurement.

This system can also be used to measure the spectral intensity ofburst noise. The noise spectral intensity is calculated using sampled data collected only during current or voltage bursts.

A block diagram of the system is shown in Fig. II-i. The system

consists of a low-noise amplifier, a set of passive low-pass filters, a digital programmable oscilloscope, and a desk-top computer.

The input signal is amplified with the low-noise amplifier. A lowpass filter is chosen for the desired frequency span which is set by the TIM/DIV knob on the oscilloscope [section 2]. These low-pass filters

are implemented to remove the high-frequency signal components which would appear as low-frequency components when sampled by the digital oscilloscope [section 5]. A finite segment of the discrete time data is then transformed to the computer and is translated into a discrete frequency spectrum using a fast Fourier transform (FFT). The FFT

coefficients are used to obtain a power spectral intensity [section 6].














I.
I I
iCRT


TEK 7D20 D




TEK 4052


Fig. II-i. Block diagram for computer-controlled noise spectrum analyzer.









The final result is stored in the computer for further noise calculations. The noise spectrum can be plotted on the CRT or UP plotter, or it can be stored on the. tape. The magnitude of the spectrum can also be transferred to the printer or the CRT.

In the following sections we describe the various system components and data processing steps in more detail.

2. Programmable Digital Oscilloscope

A sampling of the analog input waveform is obtained by a TEK 7D20 programmable digital oscilloscope (p.d.o.).9 The preamplifier circuitry attenuates the input signal according to the setting of the front panel VOLTS/DIV control. It then amplifies the signal, converts it into a differential signal, and applies it to the charge coupled device (CCD) circuitry (Fig. 11-2).

The charge coupled device contains two analog shift registers which are driven differentially. One register samples the (-) side of the differential input signal, while the other register samples the (+) side of the signal. Triggered by a sampling clock pulse, a sample of the signal is stored in the first cell of the analog shift register. At

subsequent clock pulse triggers, this sample is shifted from one cell to another until it reaches the output amplifier and the analog-to-digital (A/D) converter. The timing and synchronization of the CCD is set by the time-base circuitry.

There are four basic modes of operation for the time-base circuitry: roll, real-time digitizing (RD), extended real-time

digitizing (ERD), and equivalent-time digitizing (ETD). These modes are selected by the TIM/DIV control knob. The characterization of the

different modes is summarized in Table II-1.





- -- ------I
I I


A M
I CD


Output Port


Fig. 11-2. Simplified block diagram of 7D20 digital programmable oscilloscope.


1 k I















Table 11-1
Digitizing Mode Characteristics


Digitizing TIM/DIV Nyquist Number of- CCD Sampling A/D Sampling Memory Access
Mode Range Frequency (Hz) Points/Window Rate Rate Gate Rate

ROLL 20S-lOOMS; 50 1024 400 KHZ 400 KHz 100
TIM/DIV TIM/DIV

Real Time 50MS-500.S 50 1024 400 K z 400 Kz TIM/DIV
(RD) Y-/I TIM/DIV

Extended 40 80 *
Real Time 200PS-2PS TIM/DIV 820 TIM/DIV 400 KIz 400 KHz
(ERD)
*Samples are shifted out of the CCD at a rate of 400 Hz.










In the roll mode, the CCD circuitry continuously samples the input signal at a rate of 400 KHz. The output of the CCD is continuously

digitized by the A/D converter. Selected samples are then stored in a 1K block of waveform memory at a rate determined by the TIM/DIV setting. In this mode, triggering is' not required since the waveform memory is continuously being filled with new waveform information, which in turn is being displayed on the p.d.o.

In the real-time digitizing (RD) mode, the CCD also continuously samples the input signal at a 400 KHz rate. The A/D converter digitizes each sample, and selected samples are stored in a 1K block of the waveform memory at a rate determined by the TIM/DIV setting. In this

mode, upon triggering, the acquisition of waveform samples continues until a complete waveform is stored. Then the acquisition is halted, the time base is reset, and another waveform is acquired. This waveform is stored in the second 1K block of the waveform memory, while the first waveform is being displayed on the scope.

In the extended real-time digitizing (ERD) mode, the two CCD registers sample the input signal continuously at a rate determined by the TIM/DIV setting. Upon receiving a trigger signal, the CCD samples the signal continuously until the waveform stored in the CCD corresponds to the amount of pretrigger or posttrigger desired. The samples are

then shifted out of the CCD at a 400 KHz rate, digitized by the A/D converter, and stored in the waveform memory. Then the time base is

reset, and a second waveform is acquired which is written into another block of memory while the first waveform is being displayed.

In the ERD mode, the dead time between two successively acquired waveforms is equal to the sum of the time-base reset time and the time









needed to transfer the data into the A/D converter, since the data is transferred at a slower rate than the sampling rate.

In, the equivalent-time digitizing (ETD) mode, a limited number of samples from a periodic signal are taken at successive trigger events. These samples are used to reconstruct an accurate composition representation of the waveform in the same way as is done in a

conventional sampling oscilloscope.

Since a noise signal is nonperiodic and might contain correlation times, this triggering mode cannot be used for the noise measurements. The upper frequency limit for noise measurements is set by the extended real-time digitizing mode and is equal to the maximum Nyquist frequency of 20 MHz, which corresponds to a 3 dB system bandwidth of 10 MHz. The digitized samples stored in the waveform memory are read by the computer for fast Fourier transform (FFT) calculations.

3. Spectral Intensity of a Random Noise Signal

A noise signal x(t) measured in the time period 0t4T can be

defined in terms of Fourier series asI0



x(t) = k akexp(J2rfkt) (3.1)
k

where fk = (k = 0, *1, *2, .), and ak is the Fourier coefficient of x(t). We obtain the value of ak using the decimation-in-frequency FFT approach of Sande-Tukey.11,12 The discrete Fourier transform

coefficients ak are defined as



1 N-i
ak = - I Atx(nAt)exp(-j27kAfnAt) (3.2)
n=O










where x(nAt) is the sampled time domain data, N is the total number of samples acquired in the time interval T = AtN, Af is the frequency
1
spacing defined as Af = ,-- and k denotes the frequency
At
component fk = k kAf. It is clear that if al is the complex
k NAt
conjugate of ak,



ak =a*. (3.3)
kk


The Fourier component xk of x(t) having frequency fk is given by



xk = akexp(j2wfkt) + a_kexp(-j2 wfkt). (3.4)



The ensemble average of 2 is found to be equal to


x2=. 2 +x~~lft + a.2 (.5
X a exp(i4rfkt) akexp(-j41fkt) + 2aka_k (3.5)



Since the Fourier coefficients ak have an arbitrary phase,

2 .a2 = 0. Hence,
ak -k

N-1 N-i1___-= 2aka' 2 N- (At)2 x(n)x(m) exp(j21rfk(m-n)At)
iK(NAt)2 n-0 m0O


(3.6)

The magnitude of the terms in eq. (3.6) peaks along the line n = m (Fig. 11-3) and decreases as we go away from this line. If we introduce

a domain of summation along the two lines parallel to the diagonal time (n = m) at a vertical distance of �1M, we can write-

























M 1


Fig. 11-3.


n


Two areas of integration are considered:
(a) square area of integration of side length T;
(b) parallelogram of height 2M. The two areas differ by two large triangles for which the integrand is negligible, and by two small triangles that give a negligible contribution if T >> M.











2 2 N-I N+M
2 (At)2 x(n)x(m) exp(j2lrfk(m-n)At) (NAt)2 n-0 m=n-M
(3.7)

We define a new variable s = m - n. Then


2 N-I M
2K 2 (At)2 x(n)x(n+s) exp(j2rfksAt) . (3.8)
(NAt)2 n=O s=-M


For a stationary process, x(n)x(n+s) is independent of n, and consequently the two summations in eq. (3.8) can be decoupled, resulting in



2 N 2 At x(n)x(n+s) exp(j2rfksAt) (3.9)
x NAt s-Hk


Since x(n)x(n+s) = 0 for s>IM and the two small triangles of side M have a negligible contribution if N>>M, we can change the limits of the summation, such that



2 2 At x(n)x(n+s) exp(j27rfksAt) (3.10)
xi - NAtk



The spectral intensity of x(t) defined by the discretized WienerKhintchine theorem is given by



Sx(fk= 2 1 x(n)x(n+s) exp(j2wfksAt)At . (3.11)


1 1
Since Af = - = - is defined as the frequency interval between
T NAt
adjacent fks, the spectral intensity of the input signal can be written as











Sx(fk) i alkf (3.12)
xk Af Af


4. Time Window

Since samples of x(t) are taken during the time interval T only, this causes the continuous input signal x(t) to be multiplied by a function g(t), where g(t) - 1 for Ot4T, with a frequency response given by



g(f) = T sin(wfT) (4.1)
7rfT


The function g(t) is often called a "rectangular (uniform) time window". The effective noise bandwidth is



B f 00 f 0* sinifTI2 df=1(42
Beff =- -g(f) df _ fT d (42



and is equal to the bandwidth obtained in section 3.

Depending on the type of measurement, different passband filters (windows in the time domain) can be constructed by multiplying the uniform function g(t) with an arbitrary continuous time function. The

spacing between these filters and the width of the individual filter in the frequency domain is determined by the window shape and the sampling rate.

These synthesized filters exhibit a characteristic referred to as "leakage." Leakage occurs when the energy of the signal leaks into the sidelobes of the filter. The simplest filter to examine is the uniform sin x
window with the filter shape of . The displayed spectrum depends
X









on where the selected discrete points.fall. Fig. 11-4 illustrates two distinct possibilities.

Leakage can be minimized by using filters with lower sidelobes (Hanning). However, the trade-off is that the basic filter shape is widened considerably.

5. Aliasing

Before any waveform can undergo digital signal processing, it must be sampled and windowed. The sampling rate determines how well the

waveform is defined and how accurate the discrete representation is.

Nyquist's sampling theorem4 governs the rule for sampling. It

states that the sampling rate must be at least twice the frequency of the highest frequency component of the waveform being sampled. If the

sampling rate is less than twice the highest frequency present, then aliasing will occur.

To better visualize the aliasing problem, we refer to Fig. 11-5, in which the Nyquist frequency is 10 KHz. When the sampled waveform has

frequency components above the Nyquist frequency fN' these components are folded about the Nyquist frequency into the frequency domain between zero' and fN" If they fall outside this domain (see Fig. II-5b), then they fold again around zero Hz and eventually end up between zero and fN Hz.

If the waveform is band limited, the aliasing problem can be avoided by the right choice of sampling rate. However, when the signal is not band limited, as is the case in noise measurements, it is impossible to escape aliasing by adjusting the sampling rate. In this

case, aliasing is prevented by filtering the waveform before it is sampled. The filters limit the high-frequency contents of the waveform to the known and acceptable cutoff frequency.

























(a)


(b)


Fig. 11-4.


(a) Energy leaks into the sidelobes with discrete points falling on top of the sidelobes.
(b) Good approximation to impulse with discrete points falling between the sidelobes.


J . �










Fold Over


-20 -10


(a)


-2


F (KHz) Fold Over


2022 F (KHz)


(b)


Fig. 11-5.


When the sampling rate is 10 KHz: (a) a 12 KHz component is folded down to become an 8 KHz alias;
(b) a 22 KHz component is folded twice and becomes a
2 KHz alias.









In our system, the Nyquist frequencies are lower than the system bandwidth, so the frequencies above the Nyquist frequency fold back into the frequency band of zero to fN Hz. To overcome this problem, a set of low-pass filters with a sharp cutoff are used in front of the digital oscilloscope. The relation between Nyquist frequency and time setting of p.d.o. is given in Table II-i.

6. Computer Software

Special effort has been devoted to making the system both interactive and fast. The flow chart of the computer program is shown in Fig. 11-6. The total number of spectrum averages is set by the operator.

In the extended real-time digitizing (ERD) mode, 820 sample points are transferred into the computer. In the roll and real-time digitizing mode, this number is equal to 1,024. An improvement in speed is

obtained by dividing the data blocks into blocks of 256 points. The FFT performs N log2N operations to obtain the Fourier coefficients of a signal of N sample points. If N is reduced by factor four, the total number of operations is reduced from 10,240 for 1,024 points to 2,048 for 256 points. In addition, for every data transfer from the oscilloscope to the computer, four or three spectra can be obtained.

The reduction in block size effectively shortens the time window and therefore increases the spacing between adjacent, discrete Fourier components by about a factor of 4. For noise measurements, this is no problem since in general the signal is wideband, and its magnitude varies slowly with frequency.

The FFT operation is performed in an ROM pack, and the magnitude of each frequency component is obtained. The signal averaging is done on a

































































Fig. 11-6. Flow chart of the computer program.









point-by-point basis using RMS calculations. Finally, the total spectrum is divided by the square root of the effective noise bandwidth and is displayed on the screen.

7. Noise Measurement Procedure

To determine the current noise of the device under test (DUT) and to eliminate the effects of system gain and bandwidth, three different noise measurements are required.4

First, the noise magnitude is calculated when the DUT is biased

(Ml). Then the DUT is replaced by a noise calibration source and a dummy resistor having the same impedance (M2).

In the third measurement, the calibration source is turned off

(M3). From the equivalent noise scheme of the experimental setup in Fig. 11-7,





Ml =GB[(SU + S i)R2 + SV] (7.1)

M2 = GB[(S~ +s i + SL)R2 + SV] (7.2)

M3 = GB[(S' + S )R2 + S] (7.3)





where GB is equal to the product of gain and bandwidth, and R is equal to the parallel combination of RDUT and RA. From these three equations, it follows that the current noise of the DUT is



SI DUT - M3]S + 4kT Re(Y) (7.4)



where SCAL is the calibration current noise source, k is Boltzmann's

























V











I I




' AMP I


Fig. 11-7. Equivalent input noise scheme.






22


constant, T is absolute temperature, and Re(Y) is the real part of the DUT admittance.
















CHAPTER III
CURRENT-VOLTAGE AND IMPEDANCE CHARACTERISTICS OF SCL FLOW IN a-SiC


I. Introduction

We report here on the electrical properties of samples diced from a nitrogen-doped a-SiC crystal, being light in color and transparent. The crystal had the form of a thin hexagonal platelet, m7 mm across and 1 mm thick. It had well-developed crystal faces normal to the c-axis, but at the edge of the crystal the lamellar structure due to the layering of different polytypes (section 2) was clearly discernible. Samples were diced in the shape of a rectangular bar.

As it turned out, the current-voltage characteristics were indicative of single carrier space-charge-limited (SCL) flow. In fact, it is shown that this material provides a very good example of SCL flow, governed by various types of shallow traps. No prior measurements on

n-type SiC of SCL currents have been reported in the literature to our knowledge, though limited data on p-type SiC exist (Ozarov and Hysel113 and English and Drews4 ). Their results are summarized in Lampert and Mark's excellent monograph on the subject of injection currents.15 The SCL flow in n-type a-SiC results from injection of electrons into a compensated large bandgap polytype, sandwiched between higher conductivity polytypes (for details see Section 2).

Chapter IV concerns the electrical noise of the same samples. It

is shown there that our devices were "semiconductor-trapping devices" (terminology of reference 30), T < T, where T is the dielectric


23










relaxation time of the unexcited specimen, and -i are the various

trapping time constants. The noise shows as many as five trapping

levels, ranging in energy depth (below the conduction band) from 63 meV to 302 meV). The DC characteristics show a succession of trap fillings; yet they are characterized by a single set of traps at a given temperature with regards to the onset of the trap-filling regime. Thus the curves can be synthesized as multiple discrete trap curves, see Lampert and Mark's book, Fig. 2.2b. The noise data supplement the DC data as to the various trap levels involved. As we show in chapter IV, the noise studies on SiC reported here fully confirm and satisfy the theory of noise in SCL flow, in particular as developed in reference 30.

2. Properties of SiC

2.1. Crystallography and band structure. Silicon carbide exhibits the phenomenon of polytypism1,2 in which phe same chemical compound of silicon and carbon atoms crystallizes into different crystallographic modifications known as polytypes. These polytypes are all similar in the plane perpendicular to the symmetry axis (c-axis), but differ from each other in the stacking sequence and cycle in the direction of the symmetry axis. The additional modulation of the main crystal field allows us to regard the more complex polytypes as superlattices, having miniband Brillouin zones.

The nearest neighbor-bonding in all polytypes is tetrahedral, but the second nearest neighbor determines whether we have a cubic or a hexagonal close-packed structure. For the zinc blende structure the regular succession of three alternating pairs is ABCABC., whereas for the wurzite structure we have a succession of two alternating layers ABAB.










The various polytypes are much more complicated than these two basic structures. Since relatively minor variations in the stacking sequence of layer pairs along the symmetry axis can produce many

alternate forms of close-packed structures, polytypism is a general phenomenon affecting a large number of substances with close-packed as well as layered structures.16 The different polytypic modifications, corresponding to different stacking sequences, can still be described by ABC notations. Also, other equivalent notations of polytypic structure have been developed. Figure III-I shows the schematic arrangement of atoms in the (1120) plane along the c-axis of five important polytypic modifications, 3C, 2H, 4H, 15R and 6H. The first number in this

notation denotes the number of layer pairs in one unit cell and the succeeding letter (C,H,R) denotes the Bravais lattice, i.e., cubic, hexagonal, or rhombohedral, respectively. The Bravais lattice is

discernible from the closest adjacent layer pairs. In Table III-1 we

elaborate on the nomenclature for the above five polytypes.17-19 There seem to be no limitations to the occurrence of different polytypes. Over 45 different polytypes of SiC have been discovered by different workers from x-ray investigations.2 However, the above five structures seem to be the basic units of many larger period modifications. The

self-stabilization of a given polytypic structure during crystal growth is influenced by a large number of factors.2

Theoretical calculations of the band structure of the various polytypes have been carried out by several authors.20,21 Since the

large number of atoms per unit cell considerably complicates the

problem, calculations have mainly been done on the two simplest modifications, 3C and 2H. Table 111-2 gives some data for four
















4.396


ABCABCA BCA


.3C


/0

To
5.048A

ABCABCAB


2H


I0 10. 46 A


SILICON

CARBON


ABCABCA

4H


(a)


Fig. I1-I.


Schematic arrangement of Si and C atoms in the [11201 plane of different polytypes of SiC. The solid lines indicate the sequence in the zigzag movement of sublattices along the c-axis.











37. 30


ABCA BCA
6H


(b)


A BCA BCABCA


15R


Fig. III-i.


T
15.12

IL


Continued.
















Table III-1
Nomenclature of the polytypes of silicon carbide


ABC Notation

AB ABC

ABCACB

ABCACBCABACABCB

ABCB


Ramsdell

2H 3C 6H

15R

4H


Zigzag Sequence
(11)

(00)

(33)

(323232)

(22)


Sequence of
Inequivalent Layer

h

c

hcc

hcchc

hc


Table 111-2
A summary of band structure for SiC polytypes


Polytypes 3C 6H 4H 2H

Direct bandgaps [12] (eV) 5.14 4.4 4.6 4.46

Indirect bandgaps (eV)

Experimental values 2.39 [12] 3.0 [15] 3.26 [12] 3.35 [13]

Theoretical values [10] 2.4(r-x) 2.4(--M) 2.8(r-M) 3.35(r-K)










polytypes. We note that all of these polytypes have indirect bandgaps. The maximum of the valence band is located at the r point whereas the minimum of the conduction band is near the zone edge for all polytypes. 22-25

2.2. Transport properties. Barrett and Campbell26 measured the mobility perpendicular to the c-axis for n-type 6H, 15R, and 4H.

Ionized impurity scattering was evident at temperatures up to 300K, and a T-2.4 dependence was found between 300K and 800K. This was attributed to acoustic and intervalley scattering. A relationship of the form

1J6H < 1si5R < 4H < 13C was found. An extensive review paper on mobility in SiC was given by Van Daal.27 He postulated that polar

scattering was the dominant process. A successful fit to the mobility data indicated a hole effective mass of between 3.0 mO and 4.4 mO, while a value of (1.0 � 0.2)m0 was found for the electron effective mass.

A comparative analysis of polytype dependent transport properties was also presented by Lomakina.28 In general, the electrical properties of n-type SiC were found to be polytype dependent. Electron mobility, ionization energy of donors (nitrogen), and effective mass anisotropy are summarized in Table 111-3. These data were obtained with the

nitrogen concentration adjusted to 6 x 1016 cm-3 and at room temperature. The ionization energies of the nitrogen impurity in other

polytypes are 30 meV for 27R, 35 meV for 10H, 40 meV for 21R and 60 meV for 330R.

The temperature dependences of the conductivity in equally doped ptype 4H, 6H, and 15R were all the same. This implies that the valence band of SiC is not dependent on polytypic structure.



























Table 111-3
A summary of electron mobilities, ionization energies of nitrogen donor (E ), and effective masses of electrons in n-type SiC (ND x 1016c-3) [27] at room temperature


Mobility
Polytype (cm2/volt-sec.) ED(meV) m* /M0 m*/mO


4H 700 33 0.19 0.21

15R 500 47 0.27 0.25

6H 330 95 1.3 0.35









2.3. Our samples. In the introduction we mentioned that our devices were rectangular bars, cut out of a fairly large crystal of nitrogen-doped a-SiC. Tungsten films were deposited on both flat sides of the crystal by sputtering, prior to cutting our samples. The contact area was defined by etching out the residual tungsten films in a diluted solution of HF-HN03. Bonding to the tungsten pad was done with a

silver epoxy featuring high electrical and thermal conductivity (Type SCS-I, Lake Shore Cryogenics, Inc.). Fig. 111-2 gives the

geometry obtained.

As indicated in the introduction, space-charge limited flow was observed in all samples, with the current flowing between contacts I and 2 or 1 and 4. The SCL structure was most pronounced, however, if the current was flowing between contacts 1 and 4, with the voltage measured either between 1 and 4 or 2 and 3. This led us to believe that the

layered structure perpendicular to the c-axis contained high resistivity polytype(s) sandwiched between low resistivity polytypes; this is sketched in Fig. 111-3. Note that no notches occur in the polytype independent valence band. The low resistivity polytypes act as injecting (cathode) and extracting (anode) contacts.

Impedance measurements in the ohmic region at 77K (section 5) give a capacitance of 55 pF. It is estimated that half of this is parasitic, so we have CO = 28 pF. With relative dielectric constant s = 10.2 and an area of 4.7 x 10-6m2 (see Fig. 111-3), this yields L = 1.6 x 10-5m for the width of the insulator polytype which governs the SCL flow. Notice that L/Ltotal is only 1.6% where Ltotal is the macroscopic thickness of the crystal.

In principle, there could also be several insulating polytypes in series. Due to the universal scaling law,15 J/Lp = f(V/L2) , this










a I
a a
* - _ I


Width:.825 mm


I Caxis


Gol d/ Alumina


Fig. 111-2.


The device structure mounted on an alumina plate showing the lamellae along the c-axis direction and the arrangements of the tungsten contact areas.







POLYTYPE I


POLYTYPE


POLYT YPE 2


EVI


EV2


Fig. 111-3.


Energy band structure representing the situation of a strongly compensated polytype (polytype i) between low-resistive polytypes I and 2.









leads to complications in the quadratic regimes of the characteristic. Then J scales withN, the number of series connected polytypes in these regimes; the mobility p, as calculated from the Mott-Gurney law in the high voltage quadratic regime, then scales with i/n . Computation

indicated that P becomes unreasonably low if * I. We will therefore

ignore the possibility of series connection altogether and assume henceforth that the electrical behavior observed is governed by a single insulating polytype structure with A = 4.7 x 10-6m2 and L = 1.6 x 10-5m.

3. On the theory of SCL flow in the presence of traps

3.1. Analytic results. The standard mathematical solution of the current equation, the trapping balance equation, and Poisson's equation was given by Lampert.29 It is reviewed in Lampert and Mark's book.15 The full problem has been solved using dimensionless variables u, v, and w, defined as follows:



u = n0/n(x) = e n0tE(x)/J , (3.1)

v = e3n0312V(x)/Ee J2 , (3.2)
0
w - e2n02ix/c0 J . (3.3)



Here no is the equilibrium density of electrons, n(x) the actual

position dependent density after injection, p is the mobility, E(x) the field strength, V(x) the potential, and J the current density. With

these dimensionless variables one obtains a very simple form for

Poisson's equation in the presence of traps. This equation can then

easily be solved. The current is essentially 1/wa and the voltage

Va/wa2, as is apparent from (3.1) - -3.3); the subscript a means

evaluation at the anode. Though the solution is straightforward, it is









rather awkward to extract physical information for the limiting regimes from the solution, as is evident from the discussion in Sections 4.2 and

4.6.2 of reference 15.

For the above reasons we present here a different solution, based on the principles and notation of reference 30. We express I and V

explicitly parametrically in the parameter a = goIELI/I, where go is

the conductance per unit length of the unexcited specimen and EL is the field at the anode. Limiting regimes require a + 1 for ohmic flow

and a + 0 for SCL flow. Thus 0 < a < 1 . We noted before that this parametric presentation is also extremely useful for a computation of the impedance and of the noise.30. Thus, together with reference 30,

this section gives a unified description for DC and AC behavior, as well as for the noise.

The pertinent equations are:



J=qanE E = Ex (3.4)



I=- JA = - AqvnE (3.5)



where we assumed injection of electrons along the positive x-axis, the cathode being at x = 0 and the anode at x = L; notice that J and E are negative quantities (x is a unit vector along the positive x-axis). Poisson's equation for acceptor type traps reads



V.E= dE q (n + n + NA) (3.6a)
S te D0 N A


and for donor type traps










dE q+ N
V.E E -l- (n + nt - Nt - % - (-b
d 0 A)(3.6b)



where ND is the number of ionized donors, NA is the number of ionized acceptors, Nt is the number of traps, n is the number of conduction electrons, and nt is the number of trapped electrons. In the unexcited specimen, i.e., in thermal equilibrium, denoted by a superscript
0 0 +
zero, we have for acceptor type traps n + nt = ND - N and for
o 0 +
donor type traps n + n0 = ND - NA + N*t Thus (3.6a) and (3.6b)

read also


dE q [(n-n0) + (n -n0)] q _ (n+nt-n ) , (3.7)
dx c0 t c


where n = nO + nO is a constant charge. The trapping balance
c t
equation is



Ynt = On(N t-nt) , (3.8)



where 0 is a capture constant and y is an emission constant. We make

all rates quasi-bimolecular by writing y = nI , 31 where nI is the

Shockley-Read quantity32 as we will see shortly. Thus (3.8) yields

nlnt = n(Nt-n) , or



n N nt - 1 (3.9)
t Vt


which a fortiori holds with superscripts zero added.








With Fermi-Dirac statistics


no ZN/[-1e(6&- eF)/kT t t


+i] ,


(3.10)


(,-6 )/kT
no - N e , (3.11)


we find33


( (.-N) nIi =(N c/g) e


(3.12)


thus, apart from the spin degeneracy g, nI is the number of electrons that would be in the conduction band if the Fermi level coincided with t. We now write (3.6) in the form


N
t
n t l +n 1/n


(3.13)


For n we have the quasi-Fermi level description n = N exp[( 6F(X)- (.)/kT]� If the quasi-Fermi level remains below the

trapping level, which happens prior to the trap-filling regime, see Fig. 111-4, regions III and IV, we have n1/n >> 1, and (3.13) yields


n n nl
p n n _ constant << I
n + nt nt Nt


(3.14a)


If, on the other hand, the quasi-Fermi level is above t. as in

Fig. 111-4, regions I and II, nl/n << 1, and nt $ N � This occurs in the trap-filling and asymptotic (Mott-Gurney) regimes. For the

latter regime










CONDUCTIO N
F(Xl )=-c--KT


Fig. 111-4.


SF,(X- KT

I [
I I I
I I

I I I






EGION IREGIONI REGION I REGION
I , lI ,
XI(J) X(J) X4(J) L
nni=Nt n=ni=N ni=no








Schematic energy band, regional approximation diagram for the problem of SCL currents with a single set of shallow (above equilibrium Fermi level) traps. After Lampert and Mark, reference 3, Fig. 4.8.








(3.14b)


= +n . 1 (M-G regime) �


If the first possibility applies, as we assume presently, Poisson's equation can be rewritten with the aid of (3.5) and (3.14a). Thus the pertinent equations become (3.5) and


ee 0 g0E
dx - dE
qn c I + g0E'


where


go M qp(n0+n0)pA = qn0pA
t


(3.16)


(3.15)


go/L is the conductance of the unexcited specimen. For later use we

also introduce the dielectric relaxation time


T = ee0 /qn0i �


(3.17)


Equation (3.15) is immediately integrated to yield, with boundary condition E(O) = 0:


(3.18)


S qc0 [g0E - I.n(l + g0E
qn g :I


In particular, evaluating this at x = L and defining


(3.19)


a , g 0ELI/I = - g0E/I, we find


qn g0L 1
I= -CC a + 2.n(1-a)


(3.20)








Further, integrating (3.18) once more from 0 to L, we obtain


1 L2 = cc0 Tqn cg0


0g v qn -


go 9E Xn (l
dEI + g0E


where we used the standard trick to change dx into dE by means of


(3.15). Let g0/1 - a �


Then, noticing


E 1
1+E dEff [SE -Zn(I+SE)],


we find from integration by parts


1 OE tn(l+OE)
d1 + aE


i [SEL - �n(l+0EL)] Xn(l+aEL) - y (3.23)


E
Y f
0
EL

=f
0


dE SE - Zn(1+OE)
1 + SE


I_ [1 1 d2.n(I+SE) jn(+OE) E + $E S dE


EL - - tn(I+aEL) - I [Xn(l+$EL)I �


Together with (3.23) and setting


OEL = -a , this yields


X = EL n(ic) - EL + n(1-) - 1 [Zn(1-a)]2


When this is substituted into (3.21), equation (3.22) results.


(3.25)


The


result is


+ 90 E


(3.21)


(3.22)


EL
x f
0


where


(3.24)









12 L0 0 (Zn(1-a))2
L2 =- {V + [(O-a)Pn(l-a) + a - - ]I . (3.26)
2qn c qncg0 22



Solving for V and using (3.20) this yields


qncL2 qncL2 (1-a) Xn(l-a) + a - - [2n(
V0 2e0 [a + Xn(l-a)]2 (3.27)


Equations (3.20) and (3.27) are the full solution for the current and the voltage, parametrically expressed in a.

We first consider the ohmic limit, a + 1 � Then,
1
lir { L )(l-a)Zn(l-a) + a- [2En(1-a)]2
fM --- [a + Zn(l-a) ] + L ]
*I l 2go g0 a + Zn(l-a)



1 a2 + �n(1-a) + a
=L li L
S lim y --,L (3.28)
90 a+l a + Zn(1-a) g0


which is Ohm's law.

For the SCL regime we expand in orders of a. Then (3.20) and

(3.27) yield


1 E0a2 (3.29)
I 2qn goL


V= qn L2 4qncL2 (3.30)
e--- 0 - 3 ) 3ce0 a .0



Eliminating a between the two expressions and using again (3.16), we arrive at







I = .- E0 2/L (3.31)



which is the low-voltage quadratic regime. The trap-filled limit is not contained in this analysis, but the asymptotic high-voltage regime gives with (3.14b)


1 1 e0 pAV2/L3 (-2
I =ffi eIIVLL , (3.32)



the famous Mott-Gurney law.33) The transition from the ohmic to the

low-voltage quadratic regime occurs at

8 qn0L2 (3.33)

X pee0


from which



p - 1.18 x 10-8 nL2/ Vx . (3.34)



3.2. Regional approximation. In the regional approximation method the device is divided into four zones, as indicated in Fig. 111-4. Here ni is the injected free carrier density, n - no. In region I ni

is highest; it decreases in the other regions until in region IV ni << no. For details, see Lampert and Mark's book. Poisson's equation can be considerably simplified for each region, according to the appropriate carrier densities' approximations. The regions I - IV are called the perfect insulator region, the trap-filling region, the

semiconductor region, and the ohmic region, respectively. The method is again more lucid by not introducing the dimensionless variables, but






43
using the physical variables I, E, and V. The solutions can be

accomplished as in the previous subsection, but with considerably more ease. This will be shown elsewhere.

We recall that four current regimes are found. The ohmic regime

prevails when region IV nearly fills the entire device. When region IV becomes negligible ("is swept out at the anode") and region III prevails in most of the sample, we obtain the low-voltage quadratic regime. When this regime becomes negligible and region II prevails in most of the sample, we have the TFL (trap-filling-limited) regime. Finally, with

region I taking up most of the sample, the Mott-Gurney regime occurs. We recall the TFL-low voltage quadratic regime transition voltage,



VTFL = qNtL2/20, (3.35)



from which



Nt- 1.1 X 106 eVTFL/L2 . (3.36)



4. Experimental I-V characteristics

In Figs. 111-5 - 111-8 we show some data for 296K, 200K, 77K, and 52.6K. Data were also obtained at 250K, 167K, 125K, 100K, and 62.5K. All characteristics but the one at the highest temperature clearly show the four regimes. The trap-filling limited regime does not give a sharp near-vertical line. We notice that this regime is rather short,

indicating34 that 1/p is not more than an order of magnitude, cf. eqs. (3.35) and (3.37) for the two quadratic regimes. This is also born out by an explicit evaluation of p via eq. (3.31), see below.









-2
Id

-3
to


T=296 K


2
IccVA


ol v
-4




10 1d2 101
V/(volts)


Fig. 111-5. I-V characteristic at 296K.


1.o 10







I0


IaV


IcaV


12 I0-' 1.0 10
V (volts)


Fig. 111-6. I-V characteristic at 200K.


T=200K






JciV/


H
II


Io7


10 2







I10.'


-3
10.


T-77


K4


Ic(V


. -4
o-Io E 2
0 I V2
H -5 101

-6
I0


0.3 1002 100 0
V (volts)


Fig. 111-7. I-V characteristic at 77K.





47
1-2
T=52.6 K


-3I 10

-4
10
0
(o







-8


V/(volts)


Fig. 111-8. I-V characteristic at 52.6K.


lc(V


1.0


aV 2






48
The basic data and the results deduced from them are presented in Table 111-4. The first three rows refer to the experimental data: VTFL, VX, and R (ohmic regime resistance), as observed. The fourth row lists the mobility, computed from the Mott-Gurney asymptotic law. The

values for p are in the same ballpark as those of Table 111-3. We note that p increases as temperature increases, but not as fast as T3/2 (rather - T0"8). Thus the scattering may be a mixture of ionized impurity scattering and polar phonon scattering, the latter coming in near room temperature. This agrees with the observatios of van Daal, dealt with in Section 2.

The values of no, listed in the fifth row, are found from R, p and the geometry factors. We note that no is quite small, indicative for the insulating nature of the polytype involved. A plot of log no vs

10OO/T, given in Fig. 111-9, indicates a very shallow donor, c - d ' 10 meV. Obviously, these donors are ionized at all

temperatures in our measurements.

From the trap-filled-limit transition voltage, VTFL, we computed the number of traps using (3.36). The number of traps per cm3 is only of order 1013 - 1012, indicating that the crystals are quite pure. The statistical weight of the conduction band is calculated in the next row. We assumed here effective masses as for 6H in Table 111-3, with a valley degeneracy factor of three (corresponding to point M in the Brillouin zone). This leads to a density of states effective mass m** = 1.13 m. We notice, however, that in what follows we need log Nc, which is not too sensitive to the choice of m**. Thus c 0 c- F' computed

from eq. (3.11), is quite accurate. Figure III-10 gives the equilibrium Fermi level as a function of T. We notice that the Fermi level position in the forbidden gap is lowered almost linearly with rising T.









n0 (m-3)


C-


,- OmeV


I161 j I
10 .A .


I000
T(K)


Equilibrium free carrier density no vs. 1000/T.


2
d7


5.


Fig.- 111-9.







(- E') (m e V)
C F


I00


200 300 1(K)


Fig. III-10.


Location of equilibrium Fermi level 0 as a function of temperature.


300



200 100


0.










Next we computed 1/p from Vx using eq. (3.33), and we also computed it from the displacement of the two quadratic regimes (1/p obs.). The agreement is very reasonable. As noted above, the values of 1/p turn out to be rather small (making the theoretical assumption p << 1 rather crude). The values of 1/p are not much dependent on temperature, and a log (l/p) vs 1000/T plot reveals no significant slope. This indicates that different traps were responsible for the TFL regime at different temperatures. The pertinent trap depth Cc- 6 was computed from the

equation


I gNt (6c- (t)/kT
P N -- - e 3(4.1)



which follows from (3.14a) and (3.12). The results are entered in the last row of Table 111-4. In chapter IV we will correlate these trap levels with those obtained from the noise measurements. The lowest

trapping levels (81 and 67 meV) may be due to the ionized nitrogen donors of Table 111-3. Comparing 6c- ft with c-C0, we see that the traps which are involved are only a few kT above the Fermi level. They are "shallow" but not very much so. When the temperature decreases,

"shallower" traps are involved. E.g., at 200K the traps involved are 274 meV below c; the traps at 361 meV are "deep" traps at this

temperature and are filled up. They do not show in the I-V characteristics, since none of our I-V curves showed a transition of the ohmic range directly to the TFL range. Probably very low temperatures would be required to observe that behavior.

5. Impedance measurement

The impedance of SCL devices with traps was calculated in previous work by Van Vliet et al., see ref. 20. The following result is valid in















Table 111-4 Results Obtained


T(K) 296

VTFL(V) 2.80

Vx(V) 0.4

R(Q) 2.9K
j(m2/V-sec)

*0 -3
n (m- ) Nt(m-3) 1.23xi019 Nc(m-3) 3.0xlO25

(f- - (eV). l/p(eq. 3.27) 1/p(obs.)

f4-&t-(meV)


8.! 6.!

2.:


250 200

1.50 1.60

0.27 0.26

6.2K 6.7K

0.035 0.023

58xi016 1.38xi017 58xi018 7.01x1018 30xlO25 1.64xi025 424 325

10.6 6.36

7.2 4.6

361 274


167

1.50 0.22

8K

0.020

1.33xl017 6.57x1018 1.25x1025

268

5.58 4.0 223


125 1.30 0.19

14K

0.0175

8.68xi 016 5.70xiO18

8.10xlO24

200 7.39 5.8

167


100

1.10 0.125

23K

0.0134" 6.9x10l6

4.82xIo18 5.80x1024

160 6.12

4.8 130


77

0.70

0.12 40K

0.0106

5.02x1016 3.07xl018

3.92x1024

122 8.07 6.7 103


62.5 0.70

0.12 50K

0.007

6.08x1016 3.07xl018

2.87x1024

95

6.66

4.4 81


52.6 0.75 0.09 100K

0.006

3.55x1016 3.29x 1018 2.12x 1024

81

8.56 7.6

67


8.

6.'.

2.3









the ohmic and low-voltage quadratic regimes:



ZL = r(I)F3(x,c)/F3(1,O,a) where


1 vdv 1 1du riiu.a (l-x)/x rllau6/X

0 (l-av) v


Here r(I) is the low-frequency differential

0 - JWTQ, a as in Section 3, while T and T2 being trapping-detrapping times. In can be shown to lead to the simple result,


resistance dIVI/dI, X = (+JWT 2)I(I+J(T), the ohmic regime, (5.1)


L 1
S0go + jit"


Note that


T0
qnOp


0 L
- = RC
L qn0 pA


so we have the usual result



z R
L 1 + j wRC


In the SCL limit, a+0 , eq. (5.1) reduces to


(5.3)


(5.4) (5.5)


(5.6)


(5.1)


(5.2)


Z L = r(1)F 4(X,)IF 4 (1,0) ,









where jwt* j , t* being the drift time, and where


1 1 (X-l)/x ( px)(v-u)
F4 (x,) = f dv f du v(u) e (5.7)
0 v V


In the SCL regime t* << Ta , so the structure in frequency of (5.5) is
L
pushed to higher frequencies compared to that of the ohmic flow.

Typical jimpedancel2 plots at 77K are given in Fig. III-11. We

notice that all curves can be represented by a form



ZL = r(I) (5.8)
L I + jwr(I)C(l)


A plot of C(I) vs V is shown in Fig. 111-12. We notice that C(I)

increases, though not as fast as r(I) decreases, (r - ) , so

r(I)C(I) decreases with increasing V, indicating higher frequency turnover when we go further into the SCL regime. The equilibrium

capacitance seen is 55 pF. The dielectric-relaxation time of the sample (subtracting parasitic capacitance) is T 1. x 10-6.






























C\J C~J_

N


f(Hz)


Fig. III-11. llmpedancel2 vs. frequency for various voltages at 77K.







l lI0


U
0L Q)


77


T


2 a 7 1


V(


2 3


71.0


235


volts


Fig. 111-12. Apparent capacitance as a function of voltage. (The
parasitic capacitance of wires and connectors outside the
cryostat are subtracted.)













CHAPTER IV
ELECTRICAL NOISE OF SCL FLOW IN a-SiC


1. Introduction

From the I-V characteristics we observed space-charge-limited (SCL) flow in a-silicon carbide, due to the injection of electrons in a strongly compensated polytype sandwiched between low resistive polytypes. The I-V characteristics showed all four regimes pertaining to SCL flow involving shallow traps: ohmic, low-voltage quadratic,

trap-filling limited, and high-voltage quadratic (Mott-Gurney regime). (As usual, "shallow" traps means traps above the equilibrium Fermi level.) It was shown that a number of trapping levels are involved, ranging from 67 meV to 360 meV below the conduction band; the temperature range investigated was 50K - 300K. At a given temperature the onset of the trap-filling limited (TFL) regime occurs when the quasi-Fermi level passes the deepest trapping level which for that temperature has a substantial number of empty traps; computations from the results (Table 111-4) indicated that these are traps which are

- 2 kT above the equilibrium Fermi level. When we go further into the TFL regime, shallower traps are filling up. After all traps are filled, we observe the Mott-Gurney range. In terms of carrier densities, the TFL regime begins when ni $ n1 for that set of traps, where ni is the density of injected carriers and nI the Shockley-Read density for that trap, see chapter III, section 3.

The above picture should be corroborated by noise measurements, since trapping noise exhibits the same features. At a given temperature










the deepest traps which play a role are a few kT away from the quasiFermi level. Shallower traps may be seen, but deeper traps are generally not, since the statistical factor /n0 goes rapidly to

zero for such traps (section 2). Thus, at a given temperature the same range of traps should play a role as in the TFL portion of the DC I-V characteristics. Whereas the latter showed, however, no discernible structure (the presence of multiple traps is only manifest in a more gradual slope of the TFL curve), in the noise we should see discrete Lorentzians for each trap. With this in mind we undertook to measure the noise in the ohmic, low-level quadratic, and TFL regimes of the same SiC specimens as employed in chapter III.

This chapter is divided as follows. Section 2 reviews the theory of trapping noise in SCL flow, as developed by Van Vliet, Friedmann,

Zijlstra, Gisolf, Driedonks and van der Z iel (see references 30 and 34 and the references therein). Fortunately, closed analytical expressions exist in the first two regimes. For the TFL regime no detailed

expressions have been derived as yet, but it is obvious from generationrecombination noise theory35 that the noise should rapidly go. down. We also mention the fact that, as shown in a recent paper by Van Rheenen et al.,36 the theory of single-level trapping noise can be justifiably applied to each Lorentzian in a multiple trap noise spectrum whenever

the time constants are a factor of ten or more apart. In section 3 we present the experimental results in the ohmic regime for temperatures

from 62.5 - 300K. In section 4 we give a discussion of these results. In section 5 we present the spectra at 77K, with applied voltage ranging through the ohmic, low-voltage quadratic and TFL regimes. The MottGurney regime could not be reached; it would require the noise










measurements to be done under pulsed conditions. In this regime there

should be only thermal noise of magnitude 8 kT/ReY.

2. Review of the theory of trapping noise in SCL flow

Noise in SCL flow was first computed by van der Ziel,37, further by Zijlstra and Driedonks,34 Rigaud, Nicolet and Savelli'38 and by Van Vliet, Friedmann, Zijlstra, Gisolf, and van der Ziel30 in an approach based on the transfer impedance method39. We review here the results of reference 30.

The basic equations given in reference 30 are the same as those of chapter III, section 3. However, we need DC, AC, as well as Langevin equations, so the full time dependent equations including noise sources are needed. These equations are



J(x,t) = qpnE + H(x,t) (2.1)


dE
dx -(q/co0) (n + nt - nc) (2.2)


n(x,t) On nt - n(N~nt) I aJ(x,t)
at 1q + y(x,t) (2.3)

ant(x,t)
3n = - Onlnt + On(Nt-nt) - y(x,t) . (2.4)




Here n is the free electron density, nt the trapped density, nc is equilibrium charge, n1 is the Shockley-Read parameter, a is a capture constant, H is the thermal noise source, and y is the trapping noise source. The other symbols have their usual meaning. Since the total

current (including displacement current) Jt = J + ee 0aE/3t is solenoidal, eq. (2.3) is found to be redundant and can be dismissed.









These equations are split into DC and AC parts or in a noise analysis into DC and fluctuating parts, so that n = n0 + An , I - I0 + AI , etc. Note that the suffix zero now denotes DC or average values, no = , etc.

The transfer impedance was found to be from eqs. (2.1) - (2.4)



z(x,x',W) = - u(x-x') I dE0 1
X dx 10 + g0E 0(x')


xI o/E �(x) + go (1-x)/x 1o + goEo(x) iWTj /x
1 0 /E 0(Xt) + 90 (10+ g0E0(x' )


where u(x) is the Heaviside function, and where



P n0/(n0+nt) , go = qpn cA ,


1 + jWT2
1 + jW '0 c


T 1 1/8(Nt-nt0) T2 = 1/8(n1+n0) , 1/T = l/T1 + 1/T2


(2.5)


(2.6) (2.7) (2.8)


The noise source for trapping is given by35


S Y(x,x) = 4An1nt0 (x-x') - K(x)S(x-x')



The terminal noise is computed with z(x,x',w) and K(x):


L L L
SAV(L) = A f f dx dx' f dx" z(x,x",w)z*(x',x",w)K(x")
00 0


(2.9)


(2.10)









and the impedance follows from


LL
ZL = f f dx dx' z(x,x',w)
S00


(2.11)


the current noise spectrum is then SAV(L)/1Z]2 � The following

result, valid in both the ohmic and low-voltage quadratic regimes, was obtained after a rather involved analysis


SAI(w) - 4qpII0V0I TX L2r1 1 + W2T2


(2.12)


where 0 = JwTa Section 3, see sufficiently low



XOS


a = g0IE0(L)I/I0 (same parameter as in Part I, eq. (3.19)), while 4D is normalized to be unity at frequencies. It is given by


F2(X,Q,a)/F2(1,0,a)

IF3(x,,ct) I2/1(F3(1,0,a)12'


(2.13)


with


F2(X,0,c) =


v2dv 1 du (lu - a) (l-x)/x1 I u 6/X,2
(l )3 vf u/ )X[ (2.14)


F 3(x,e,a) =


vdv 1du (/u - ua(<-x)/xri - au6 (1- v)2 v - a a' )


In the ohmic regime f is flat up to wT. 0 1 � However, in the SCL

regime t is complex. For an "insulator-trapping device" (1/Ta << 1/T) the function t rises beyond wI = 1I/T to a new plateau which is 20/9 times the low-frequency plateau. It then falls off beyond


(2.15)









2= /t . The roll-off is complex. For a "semiconductor-trapping device" (l/T << l/T) the function 0 has not yet been fully studied. However, we believe that it will only have a mild structure between WI = i/T and w2 l 1/T a ; for frequencies beyond this it decreases monotonically. Detailed computer studies of the integrals (2.14) and (2.15) are underway.

For our devices, T& being of order 10-5, some trapping times are smaller and some larger than T. Thus both situations discussed

above can occur. As to the other factors in (2.12), ' is a number between 0.9 and 1.0 throughout both regimes. Further, uIV0I/L2 is of

the order of the transit time tt, see references 30, 34 and 40, while T/TI is the statistical factor



_T _ 2 t nto (2.16)
no T1 T1 + T2 N t - nt0 + n, + no


Thus the low-frequency plateau is also - 4 q10 (T/t*)(/n0)

i.e., modified shot noise. Notice that as long as the traps are empty and the injection is low /n0 is finite. When the traps fill up, however, /n0 + 0 � As we noted above, if the trapping time

constants Ti are far enough apart, the multiple trap spectrum is approximately a sum of Lorentzians with appropriate statistical factors, i.e.,


4quI01v0j' T2
SAI(w)= L2 i Tli I + W2T2 (Xi,8,a) , (2.17)


with'










i i ti - n tiO (2.18)
no Tli Nti - nt,0 + n +i + no(





where i is a "partial covariance" due to interactions of the conduction band electrons with traps i. Equations (2.17) and (2.18) are the basis for our experimental results.

We still mention the result for Ti. From (2.8) we obtain with

8 = ai, ai being the cross section and the mean thermal

velocity,



l/Ti a i[(N ti-nti,0 + n1 + no]. (2.19)



In the ohmic regime, no << nli. For shallow traps, moreover,

nti0 << Nti. We noted before (Table 111-4, chapter III) that no

(thermal equilibrium value) is of order 1010 cm-3 while Nti is of order 1012 cm-3. The ratio Nti/no for all temperatures is of order 100. Hence, in order that nl,i [see chapter III, eq. (3.12)] dominates over the term Nt,i, we must be about 4 kT above the Fermi level (e4"6 = 100). Thus, T. is exponential, ( c )IkT
T I - e , (2.20)
i N


in a temperature range where the trap level is 4 kT or more above the Fermi level. Since the Fermi level rises when the temperature decreases (chapter III, Fig. III-10), we must find that in a plot of XnTi vs









1000/T the curve is linear up to a low temperature limit, where the Fermi level becomes as close as 4 kT below the trap level; thereafter, Ti must level off.

3. Experimental results in the ohmic regime

We refer to the geometry of Fig. 111-2. In all cases the current was passed through contacts 1 and 4 while the noise was measured between contacts 2 and 3, in order to avoid contact noise. Noise spectra were measured in the range 1 Hz - 1 MHz for T = 62.5K - 300K. Typical data

are shown in Figs. IV-l - IV-4. Fig. IV-l has spectra at 296K and 250K, Fig. IV-2 lists data for 200K and 175K, Fig. IV-3 lists data for 125K and 100K, while Fig. IV-4 lists data for 77K and 62.5K. All spectra

show three to five Lorentzians. There are probably other Lorentzians at lower frequencies. We note that the 77K and 62.5K curves show a sharp roll-off above 5 KHz (77K) or above 1 KHz (62.5K), with no indication of other Lorentzians coming in. Then these roll-offs give the smallest lifetimes for these temperatures, indicative of the shallowest traps that occur in this sample. In the ohmic regime the noise was found to be proportional to 12 in all cases.
0
From the data we determined the time constants involved from the best fit. These Tis are plotted vs 10001T on a semilog scale in Fig. IV-5, while in Fig. IV-6 we plotted the plateau values normalized by the current squared vs 1000/T, also on a semilog scale. Points

pertaining to the same Lorentzian are connected by straight lines. We notice that a number of nonconnected points occur for temperatures above 200K. These belong to other Lorentzians, which can only be fully determined if measurements far above 300K were made.































































Fig. IV-i.


Current noise spectral density in ohmic regime for T = 296K and T = 250K. Circles o and 0 : measured data. Full lines: resolution into Lorentzians. Dashed line: 1/f approximation. Please note: right vertical axis refers to 296 K curve, left vertical axis refers to 250K curve.










































10 102 IV I
f(Hz)


Fig. IV-2.


Current and T = lines:


noise spectral density in ohmic regime for T = 200K 175K. Circles 0 and 0 : measured data. Full resolution into Lorentzians.


-20
10



-2!
I0







-22
I0



-24
I0

-245 -25
10


-26
I0


105


















,. @TzOOK


a


an a


4

T125K


f(Hz)


Fig. IV-3.


Current and T = lines:


noise spectral density in ohmic regime for T = 125K lOOK. Circles 0 and squares: measured data. Full resolution into Lorentzians.


s1 A Hz) (


-20~




I0.


-23

10.


(f


-2-3

10


1 O:


































































Fig. IV-4. Current
and T =
lines:


noise spectral density in ohmic regime for T = 77K 62.5K. Circles 0 and 0 : measured data. Full resolution into Lorentzians.

















8
I m I II I I I |
3 5 7 9 II 13 15
IOOOT( K)

Fig. IV-5. Observed time constants of the Lorentzian spectra as a function of 1000/T.


























5 7


9 II
IO00/T(K)


Fig. IV-6.


Relative plateau values SA /12 of the Lorentzian spectra spectra as a function of 1O00/T.


v -I0 1, 0

CO


-12
I0


-14
0









4. Discussion of spectra in the ohmic regime

Though the choice of the plateaus of some Lorentzians leaves some leeway, we are quite convinced that there is not a continuous uniform distribution of traps. The latter would lead to a 1/f spectrum, while an exponential distribution would lead to a spectrum 1/f6, with 6

between zero and two (though usually close to one, say 0.8). It should be noticed that a straight line approximation, e.g., to the 296K curve, see dashed line, would cause a number of points to be well off by a factor 1.5 to 2.0. In this respect we should keep in mind that a loglog plot tends to obscure details. However, the accuracy of the

measured points was never less than 10%, so that the structure, where appearing, should be taken at face value. However, the most important clue to the fact that this is not 1/f noise is indicated by the pronounced rise at high frequencies in the 200K curve, and by the sharp roll-off at high frequencies in the 77K and 62.5K curves. Therefore, we believe that all spectra represent trapping noise, i.e., a form of generation-recombination noise. A final indication that this is not 1/f noise comes from the fact that the Hooge constant S /12 = a/fN for

such a process, using the dashed line in Fig. IV-l, would be

a m 0.6 , a value that is orders of magnitude higher than observed

1/f noise in silicon or germanium. Most clearly the nature of the

trapping noise is revealed by Figs. IV-5 and IV-6. In Fig. IV-5 we find a number of straight lines, the slope of which gives the trap depth, see eq. (2.20). We also note that in some cases a horizontal portion of the lifetime appears. This portion should appear at temperatures for which the Fermi level approaches the trap level within 4 kT from below, see section 2. E.g., for the 98 meV trapping curve, the leveling






72

off occurs at T0 = 77K. With kT = 6 meV and the Fermi level being at 122 meV below the conduction band at 77K, see Table 111-4, chapter III, the distance between trap level and Fermi level is indeed 4 kT0 when the leveling off sets in. Similar good agreement is found for the other trapping curves.

From the noise spectra at 77K and 62.5K, it is clear that no traps shallower than 63 meV appear, except perhaps for the ionized donor level of 10 meV, found in chapter III. Generation-recombination noise due to these donors should be observable at very low temperatures.

From the magnitude of the Ti and eq. (2.20), together with the data for Nc given in chapter III and a spin degeneracy g = 2, we computed

the electron capture cross section ai of each trapping level. The

results are shown in Table IV-1. The cross sections are within the

normal range for neutral or negatively charged traps.

In Table VI-2 we have tried to correlate the trap depths as found from the I-V curves (Table 111-4) with the trap depths found from the noise. The correlation of the two sets of data is far from perfect; yet the fact that they range over a similar latitude (31 - 63 meV) lends strong support to the interpretation and consistency of the data in both papers.

5. Noise spectra in ohmic and SCL regimes at 77K

At 77K the noise was measured for currents ranging from 7.7 x 10-7A up to 3.74 x 10-3A. From Fig. 111-7 we deduce that the following ranges occur:

I < 3 x 10-6A ohmic regime

3 x 10-6A < I < 3 x 1O-4A low voltage quadratic regime

3 x 10-4A < I < 8 x 10-3A TFL regime.









Table IV-1
Activation energies and capture cross sections of different
trap levels observed in the noise measurements


C t 302 281 135 121 98 81.5 63
(meV)

o(cm ) 6.62xi0-15 9.8xi0-15 2.40xi0-18 1.27xi0-17 2.5xi0-17 3.05xi0-17 4.87xi0-17





Table IV-2
Comparison between the trap activation energies as found from the I-V curves (Table 111-4) and noise




(f-V)t 361 274 223 167 130 -- 103 81 67
meV


(noise5 302 281 --- 135 121 98 81.5 63
meV






74
These three regimes are covered by these noise measurements. Typical

results are given in Figs. IV-7 - IV-9. The time constants involved in all spectra are nearly the same, see Fig. IV-10. The slight decrease

with current might be due to the modulation by the function .

As to the magnitude, we note that the noise at 1 Hz is of order 10-18 in the ohmic regime, of order 10-15 in the low-voltage quadratic regime, while it goes down again to 10-18 in the TFL regime. Such a

large variation seems at first hand unexplicable. However, the

normalizing factor for all regimes is l1V0j , see eq. (2.17), modified by the factor /n0 of eq. (2.18). Thus, in Fig. IV-l1 we plotted SAIR/I01VOI vs I0, where R is the ohmic resistance of the unexcited specimen, for the four Lorentzians involved. In the first

two regimes the normalized noise is constant, i.e., SI C I01VOj,

with /n0 = N /ni = constant, in excellent agreement with the

theory of Section 2. In the TFL regime there is a sharp drop-off, since nti + N t so that /n0 + 0 , see eq. (2.18). Thus the behavior pl nti 0
plotted in Fig. IV-11 is in most respects as expected.






T-77K


N III

-a
(/)


V=77xIO A


[d


f(Hzo f(Hz)


Fig. IV-7. Current spectral density at 77K and I0 = 7.17 x 10-6 A
(ohmic regime).


-20
)0.


-23
10


jc4






N


I0 10


(I)


T:77



0.


K


=2.5xIOA


f(Hz)


Fig. IV-8. Current spectral density at 77K and I0 = 2.5 x 10-4 A (lowvoltage quadratic SCL regime).







0 *


T-77 K



S
0


I0
100


100


-2-3


S i i

f(Hz)


Fig. IV-9. Current spectral density at 77K and I = 3.74 x 10-3
(TFL regime).


-18
0

-19
0


CIO






(S)


T-77 K


-(5
10


O)7


ic06


I(A)


10 4


Fig. IV-jO.


Observed time constants of the Lorentzian spectra at 77K as a function of current 10.


o h m ic- *_ . lo~w v o lt. q u d - -- T F L ".*- I


0 I5









10


0
10


-d7


Fig. IV-il.


T=77K


ohmic4-low Volt. quad. -- TFL
I (A)


Normalized current spectral density SAIR/Iolvoi as a function of current 1.0















CHAPTER V
COMPUTER CALCULATION OF DC SCL FLOW IN a-SiC


1. Introduction

Many observations on several samples cut from single crystals led to the conclusion that the layered structure which makes up the device contains a highly resistive (strongly compensated) polytype, sandwiched between low resistive polytypes [chapter III and chapter IV]. This

creates an n+nn+-like structure. The low resistive polytypes act as injecting and extracting "contacts" for the high resistive polytype, introducing into the latter single-carrier, space-charge limited flow.

The I-V characteristics measured in the temperature range of 50300K show four different regimes of operation: a low-bias, ohmic

regime; a low-bias, quadratic Mott-Gurney regime (electron traps empty); a fast-rising trap-filling regime; and, finally, a quadratic Mott-Gurney regime (electron traps filled).

In chapter III we used the regional approximation method to explain the presence of these four regimes. The model we used to explain our DC characteristics shows a succession of trap fillings. However, at any

given temperature, a single trap level controls charge transport.

In order to verify the experimental results and the model presented in the preceding chapters and to obtain a better understanding of the charge transport mechanism in a-SiC in particular, and of short n+nn+ devices in general, the appropriate transport equations are solved with the aid of the computer.










We present a detailed description of the theoretical charge transport model and the results of our simulation for the linear regime and the three regimes of SCL flow in our n+nn+ a-SiC samples at T - 77K.

2. Theoretical Model

The basic transport equations, including DC and AC as well as Langevin noise sources, are:

dn

J(x,t) = qunE + qD d- + H(x,t) (2.1)


dE2
d- -(q/ee )(n + n nc) (2.2)



an(x,t) . a n - n(N n l 1J(x,t) + y(x,t) (2.3)
at I t t t q ax

ant(x,t)
at -8nnt + Bn(Nt - nt) 7y(xt) � (2.4)



Here n is the free electron density, nt is the trapped density, nc is the equilibrium charge, n1 is the Shockley-Read parameter, 0 is a capture constant, H is the thermal noise source, and y is the trapping noise source. The other symbols have their usual meaning. These

equations are split into DC and AC parts or in a noise analysis into DC and fluctuating parts, so that n - n0 + An, I - 10 + AI, etc. Note

that the suffix zero now denotes DC or average values, no = , etc. The equations describing the DC part are related to the quasi-Fermi potential (4 n) and the electrostatic potential (Xc) as



V2c q0 (no nt0 -nc) (2.5)

I0

- V.(Aqpnn0x WVn(x)) = 0 (2.6)
ax_ n 0 )~() 26









The density of the trapped electrons is related to the quasi-Fermi potential F(x) and the trap potential X (x) by Fermi-Dirac statistics,
Fn t
i.e.,



nto(x) = Nt/[l + g exp(-(qxt(x) - qtn(x))/kT)] (2.7)



where g is the electron spin degeneracy and jq(xc - xt)I is the trap activation energy. The density of the electrons in the conduction band for a nondegenerate semiconductor is given by Maxwell-Boltzman statistics as



nO(x) = N exp(-q( Dn(x) - X (x))/kT) (2.8)



where Nc is the effective density of states. The two coupled secondorder partial differential equations (i.5) and (2.6) are solved simultaneously with the aid of equations (2.7) and (2.8).



3. Computer simulation

The charge transport equations described in Section 2 are applied to an n+nn+ structure. In order to find a unique solution of the

second-order, differential charge transport equations, we have to specify two boundary conditions. Numerical stability dictates that

these boundary conditions have to be specified at the two opposing n+ contacts (Dirichlet problem). Specification of, for example, the charge concentration and its derivative at only one n+ contact (Riemann problem) leads to numerical instability. Hence we assume that the

device behaves ohmic at a few Debye lengths away from n+n and nn+ contacts and thus






83

nND . (3.1)



Consequently, the Fermi potential at few Debye lengths away from the electron injecting contact (cathode) is given by




'Fn Xc - �n
c


Since we chose Xc = 0 at the n+ cathode, for reference purposes .we obtain for 4D
Fn

T N+
kT n (N) (3.2)
Fn q c


The Fermi potential few Debye lengths away from the electron extracting contact (anode) is given by

kT (._ (33
"Fn = V D - q- Xn ND 33
Dq c


where VD is the applied voltage.

The coupled system of partial differential equations


"t(q(n(X) - Xc(x))
Fl(xF) - V2(X) + kT

NI g -E -q(xC(x) - 4nx)

+ NT/[I + g exp(- )exp( kTn

(3.4)


and



F2(X 'Fn) = (exp( - Xc(x)) VFn(X)] (3.5)










are solved simultaneously using Newton's method41 to get F1 and F2 approximately zero. The boundary conditions are given by equations

(3.2) and (3.3).

The discretization of equation (3.5) is somewhat ill-conditioned
sinceVexpq. n(x) )
since n F VI] indicates a poorly scaled, first-order
kT )vFn
derivative due to the variation of cn (x). Hence we define a new

variable



U = exp(-V . (3.6)



The variable U also has the advantage of having much greater variation than the variable 0. n So equation (3.5) becomes Fn


F2(XcU) = V[exp( kT )VU] . (3.7)

qx (x)

The discretization of V[exp( kT )VU] has been the subject of lengthy discussions in the semiconductor simulation literature since the backward or forward difference does not adequately describe strong varying exponents. We choose an approach similar to that of Bank et al.42 We define point K' between K - 1 and K, and point (K + 1)' between K and K + 1, so that
/


qeq Xc /kT qXc/kT
VuqXc Vu] (K+l), - [e VU]K,
V[e k [(K + 1)' - K']Ax (3.8)


The Fermi potential is a smoothly varying function of position.








Consequently, u = exp[-q n/kT] is


accurately as u(K+l) - u(K) VUK+I' ' u Ax


and


VK, = u(K) - u(K-1) Ax


To find the values for exp[qx c/kT] at (K+1)' and K', we define


.exp[qx (x)
dex~dx kT


q
kT


qx (X) dx (x) exp[ kT ] dx "


After rearranging (3.11), we get from an integration


K'Ax -qXc(x)
f exp( kT
(K-1)Ax


qX (x)
[exp( kT )Jdx


KAx dx (x)
f dx =Xc(K) Xc(K1)
(K-1)Ax


(3.12)


According to the mean value theorem for integrals,3 there exists a value K',K-l

KAx
f
(K-I)Ax


f(x)g(x)dx = f(K')


f
(K-1) Ax


g(x)dx �


(3.13)


Applying the theorem to the left-hand side of eq. (3.12) results in


(3.9)


(3.10)


(3.11)


e
well behaved and canbe discritized










KAx -qX (x) qX (x)
-q f exp( k ) Td [exp( )]dx
q (K-1)Ax


-q KAx d qx x)
= exp([ ) I [-- exp( .- )]dx (3.14)
K' (K-1)Ax


or


exp-qX (K') I xc(K) - Xc (K-i)
kT " e qX c(K) qxc(K-1) � (3.15)
]T kT



A similar approach gives us an expression for


qxc (K+1) '
exp( C )



The Jacobians of F1 and F2 are obtained after the discretization, yielding a system of nonlinear equations. Assuming that the Jacobians

are defined, we can write


aFK K(m+l) KUK + 6UKm
6X- xc fi-F I(XC, + Um


(3.16)


3F K
2 6U K(m+1) = _F(K + 6XK(m+1), UK
8F2 K Kmm) K K(m+l) K


where we solve for 6xK(ml) and 6UK(mnl) , with K denoting the

Newton step and m being the iteration step. Since the Jacobian of F1

and F1 both have the same denominator as given by eq. (3.8), the actual










location of (K+l)' and K' is unimportant and does not affect the value of 6xc"

To obtain a higher convergence rate, Poisson's equation is solved for the electrostatic potential until a total convergence is obtained. The updated values of X c(K) are then substituted into the continuity equation with the U(K) being updated for convergence. The new values

are then substituted back into Poisson's equation, and the procedure continues until a full convergence of the electrostatic potential and the quasi-Fermi potential has been obtained.

4. Theoretical Results

Fig. V-I shows that the computer simulation program explains the measured I-V characteristic at 77K very well. The simulation program

includes only one trap level at this temperature.

The carrier concentrations of the nt regions are assumed to be approximately equal to 5 x 1021m. This value is in the range of

carrier concentrations measured for various polytypes of a-SiC at T - 77K.43 The other transport parameters are taken from Table 111-4. The charge transport mechanism in the four regimes of the I-V characteristics is discussed below.

Linear regime. The overflow of carriers from the n+ regions into the n region plays an important role at low-bias voltage levels. This

large overflow of carriers is due to the large gradient in the carrier concentration at the n+n interface. In the linear regime the traps in the n- region are mainly empty (Fig. V-2), and the trap energy lies above the Fermi level for most of the n- region (Fig. V-3).

The current profile of the sample is shown in Fig. V-4. In this

regime the diffusion plays a dominant role in charge transport. The








-1
1011(A)


-51
10 -


-7
10


Fig. V-i.


T=77 K


/ 2
1 lAV


V(v)


I-V characteristics at T = 77K. The dots indicate the measurements. The solid line represents the results of computer calculations.








Linear Regime
I=6X 10 -7A


1021i


1020

E
� 10,19
%10 iTrap
-- -- - , Carrier
% Concent ratio n,,
od ' n Wx ,
'- 1018-" t ,



1017
Carrier
1W0 6 Concentration
n(x)


1015 1 0 1'5 2'0

L(pm)









Fig. V-2. Carrier concentration profile of the trap level and the
conduction band in the linear regime (I = 6 x 10-7 A).









Linear Regime I = 6 xl0-7A


T=77K


V


Conduction Fermi Level . rap Level n+ _n


Band


I I I


0 5 10
L (um)


15 20


Energy band diagram in the linear regime (I = 6 x 10-7 A) at T = 77K. The solid line represents the conduction band, the dashed line the quasi-Fermi level, and the dot-dashed line the trap energy level.


01.08


.02 F


-0.02 k


EF


-0.04


-0.06


-0.08


0.10


-0.12 1


-0.14


Fig. V-3.


0.06 0.0 4[












10-2



10-3 10



-4
10


1-6


-7
10 10- 8



-9
10


T=77K
Linear Regime
1=6 x10_7A






+ '










II

I diffusion
I \ /
I r" / _. " Potential
Minimum idrft4--. I---n--- ndT~OI -____i I , , I I


0 5 10 15 20
L(m)


Current diagram in the linear regime (I = 6 x 10-7 A) at T = 77K. The solid line represents the drift current and the dashed line the diffusion current. The sign conven tons are given at the bottom of the figure.


Fig. V-4.









field profile, Fig. V-5, shows a large variation at n+n contacts due to an abrupt change in carrier concentration.

Low-voltage quadratic regime. As the biasing voltage increases, the potential minimum shifts towards the n+ cathode contact, Fig. V-6. The Fermi level lies below the trap level for most of the device. Consequently, the traps are mainly empty (Fig. V-7). The width of the

section in the n region dominated by the drift current increases, and the current becomes space charge limited and proportional to the square of the voltage.

Trap-filling regime. In this regime the quasi-Fermi level passes through the trap level as is indicated in Fig. V-8. The traps are being filled (Fig. V-9), and the I-V characteristic shows a fast rise in the current. The drift current dominates current flow in the major part of the n region (Fig. V-10).

The field profile is negative in most of the n region, indicating that drift is dominating the device. Large variations in the electric field occur at n+n contacts due to sudden change in carrier concentration, Fig. V-il.

High-voltage Mott-Gurney regime. In this regime the Fermi level lies completely above the trap level, and almost all of the traps are filled (Fig. V-12). Drift dominates the sample, and the I-V

characteristic shows the quadratic dependence between current and voltage (,eV2).








Linear Regime 1=6X1 0-7A


1.01-


.8-


-.2


-.6


Fig. V-5.


-n+ n n
, , , I I II , I . . . .
-5 0 5 10 15 20
L(pm)




Electric field profile in the linear regime (I = 6 x 10-7 A) at 77K.








Low Voltage Quadratic
I = 3.92 xl0-5 A


-5 0 5 10 15 20


L (jLm)


Energy band diagram in the low-voltage, quadratic regime at 77K. The solid line represents the conduction band, the dashed line the quasi-Fermi level, and the dot-dashed line the trap energy level.


-0.


Fig. V-6.




Full Text
67
Fig. IV-3 Current noise spectral density in ohmic regime for T 125K
and T = 100K. Circles and squares: measured data. Full
lines: resolution into Lorentzians.


9
In the roll mode, the CCD circuitry continuously samples the input
signal at a rate of 400 KHz. The output of the CCD is continuously
digitized by the A/D converter. Selected samples are then stored in a
IK block of waveform memory at a rate determined by the TIM/DIV
setting. In this mode, triggering is' not required since the waveform
memory is continuously being filled with new waveform information, which
in turn is being displayed on the p.d.o.
In the real-time digitizing (RD) mode, the CCD also continuously
samples the input signal at a 400 KHz rate. The A/D converter digitizes
each sample, and selected samples are stored in a IK block of the
waveform memory at a rate determined by the TIM/DIV setting. In this
mode, upon triggering, the acquisition of waveform samples continues
until a complete waveform is stored. Then the acquisition is halted,
the time base is reset, and another waveform is acquired. This waveform
is stored in the second IK block of the waveform memory, while the first
waveform is being displayed on the scope.
In the extended real-time digitizing (ERD) mode, the two CCD
registers sample the input signal continuously at a rate determined by
the TIM/DIV setting. Upon receiving a trigger signal, the CCD samples
the signal continuously until the waveform stored in the CCD corresponds
to the amount of pretrigger or posttrigger desired. The samples are
then shifted out of the CCD at a 400 KHz rate, digitized by the A/D
converter, and stored in the waveform memory. Then the time base is
reset, and a second waveform is acquired which is written into another
block of memory while the first waveform is being displayed.
In the ERD mode, the dead time between two successively acquired
waveforms is equal to the sum of the time-base reset time and the time


146
1020 REM MEASUREMENT WITH DEVICE OFF
15223 DIM Mre2(70)
10240 Mre2=A
0260 GO TO 060
10260 PAGE
10300 REM MEASUREMENT WITH CALIBSRATIQN SOURCE ON
10320 DIM Mm3<70)
10340 Mm3=A
10360 C-0 TO 8060
1038 PAGE
10480 REM MEASUREMENT OF THE NOISE CALIEERATIQN MAGNITUDE
10420 DIM Sc(70)
10448 Sc=A
10468 GO TO 8868
10480 PAGE
18500 PRINT "WHAT IS THE VALUE FOR Real ?
10520 INPUT Real
10548 DIM Mm4(70),Mm5(705,M6(78)
10560 REM CALCULATE THE EXCESS CURRENT NOISE SPECTRUM
18580 Mm4=Mml-Mm2
10600 Mm5=Mm3-Mm2
18620 Mm6=Mm4/Mre5
18648 Mre4-Mm*ac
18660 MmsMm4/(Rcal#Rcal)
10688 A=Mre6
18780 GO TO 8060
18728 REM THIS PART OF THE PROGRAM IS FOR THE HP-FLOTTER
10748 REM BY MR. MIKE TRIPPE.
18760 PAGE
10780 REM SET THE HP PLOTTER
10300 DIM X(A1),Y(A1>
18320 FOR 1=1 TO A
10340 X=A3/70*I
10860 Y(I)=i0*LGT!A(D)
10380 NEXT I
18900 PAGE
10928 DIM Xscaled(Ai),Yscaied(Ai>
10948 N=A1
18960 Xreax=A3
10980 Xmin=l
11000 Xint*1
11828 PRINT "IS THE X-AXIS TO BE LOG OR LIN?"
11048 INPUT Lx$
11068 Yreax=W6
11030 Ymin=W4
11108 Yirvt=W4
11120 Ly$="LIN"
11148 IF L;-:$="LOG" THEN 11130
11168 Xtie=A3/10
11188 Yi:c=(W6-W4)/3
11288 PRINT DO YOU WANT TO USE ANY OF THE FOLLOWING PLOTTING OPTIONS?"
11228 PRINT "PLOTTING SYMBOL, GRID, OR VECTOR (DOT TO DOT).(Y/N)
11240 INPUT Opt:sns$


34
leads to complications in the quadratic regimes of the characteristic.
Then J scales with T)} the number of series connected polytypes in these
regimes; the mobility p, as calculated from the Mott-Gurney law in the
high voltage quadratic regime, then scales with l/fy Computation
indicated that p becomes unreasonably low if 1. We will therefore
ignore the possibility of series connection altogether and assume
henceforth that the electrical behavior observed is governed by a single
-6 2 S
insulating polytype structure with A 4.7 10 m and L 1.6 x 10"Jm.
3. On the theory of SCL flow in the presence of traps
3.1. Analytic results. The standard mathematical solution of the
current equation, the trapping balance equation, and Poisson's equation
was given by Lampert.^ It is reviewed in Lampert and Mark's book.^
The full problem has been solved using dimensionless variables u, v, and
w, defined as follows:
u => n/n(x) e npE(x)/J ,
(3.1)
v e3n03p2V(x)/ee0J2 ,
(3.2)
w e2n02px/eeQj .
(3.3)
Here xP is the equilibrium density of electrons, n(x) the actual
position dependent density after injection, p is the mobility, E(x) the
field strength, V(x) the potential, and J the current density. With
these dimensionless variables one obtains a very simple form for
Poisson's equation in the presence of traps. This equation can then
easily be solved. The current is essentially 1/w and the voltage
a
vfl/wa2, as is apparent from (3.1) (3.3); the subscript a means
evaluation at the anode. Though the solution is straightforward, it is


POLYTYPE I POLYTYPE i
POLY TYPE 2
Fig. III-3. Energy band structure representing the situation of a strongly compensated
polytype (polytype i) between low-resistive polytypes 1 and 2.


T=77 K
Fig. IV-10. Observed time constants of the Lorentzian spectra at 77K as
a function of current Iq.


66
Current noise spectral density in ohmic regime for T = 200K
and T = 175K. Circles o and : measured data. Full
lines: resolution into Lorentzians.
Fig. IV-2.


Energy (eV)
96
TFL 1 = 1.9 x 10"3A
L (^m)
Fig. V-8. Energy band diagram in the trap-filling regime at T = 77K.
The solid line represents the conduction band, and the dot
dashed line the trap energy level.


in the different regimes of the I-V characteristics are calculated with
the aid of a computer. The effect of carrier spillover from the n+
regions into the n region and the importance of the diffusion current
are discussed.
Theoretical values of the noise in the four different regimes of
the I-V characteristics are obtained, using a discrete transfer
impedance method (including diffusion), and are compared with the
experimental results.
Various transport quantities of a-SiC are deducted from the I-V
characteristics, the impedance, and the noise.
The insight gained from this study is used to unravel some of the
problems associated with charge transport in submicron (.47 pm) n+pn+
GaAs devices.
vi


50
K
Fig. III-10. Location of equilibrium Fermi level £ p as a function of
temperature.


Table II1-4
Results Obtained
T(K) 296
250
200
' 167
125
100
77
62.5
52.6
VTf.L(V) 2.80
1.50
1.60
1.50
1.30
1.10
0.70
0.70
0.75
VX(V) 0.4
0.27
0.26
0.22
0.19
0.125
0.12
0.12
0.09
R(fi) 2.9K
6.2K
6.7K
8K
14K
23K
40K
50K
100K
2
pCm^/V-sec)
~ 0.035
0.023
0.020
0.0175
0.0134'
0.0106
0.007
0.006
n(m-3)
8.58xl016
l.38xl017
1.33xl017
8.68xl016
6.9xl016
5.02xl016
6.08x1o16
3.55xl016
Nc(nf3) 1.23xl019
6.58x1o1
7-OlxlO18
6.57xl018
5.70xl018
4.82xl018
3.07xl018
3.07xl018
3.29xl018
Nc(m-3) 3.0xl025
2.30xl025
1.64xl025
1.25xl025
8-lOxlO24
5.8Oxl024
3.92xl024
2.87xl024
2.l2x1024
<5c-6(a.eV).
424
325
268
200
160
122
95
81
l/p(eq. 3.27)
10.6
6.36
5.58
7.39
6.12
8.07
6.66
8.56
l/p(obs.)
7.2
4.6
4.0
5.8
4.8
6.7
4.4
7.6
£c"£'t(eV)
361
274
223
167
130
103
81
67


n n n n n n
153
c. SET AN ABRUPT JUNCTION & LINEAR BAND PROFILE
C
DO 10 I = 1> N1
F< 1 + 1 > = F ; l o- -EXP < < C
V<1 + 1 ) 7 !.R** )
C(1+1)*C(I)+F(1+1)-F{I)
IF i W ) C (1+1 ) = C < 1 +1 ) + 0765
IF(I.EG.45)C(1+1)~C(1 + 1>-.0765
1 CONTINUE
C
C SOLUS POISSON EQUATION
DO 40 J-.Ml
DO 20 K = 2 > N1
XN ( K / L,*,'. )*8KT>
XNT< K >"NT/(1 +.5*ET*)>
IF(K.LT.16.OR.K.GT.45 > GO TO 15
F1=2.*C < K}-C -C(K-1)-Q2*H*H* DF12.-Q2*H*H* *.5*ET*
GO TO 25
C
C SOLVE POISSON EQN* FOR N+ REGION.
-n 1
w
15 XNE=NP
F1"2.*C < K)-C < K+1)-C < K-1)-Q2*H*H*(XN < X)-XNE)
DF1=2.-G2*H*H*(-QKT*XN 25 DC < K)=-Fl/DF1
C < K)=C K)+DC 20 CONTINUE
SOLVE THE CONTINUITY EQN.
DO 30 K=2.N1
AN=>/<2.718**(8KT*C(K))-2.718**
AN2 . C i x + I ) C CK ) ) / {?. 718** >-2.71S**< S3KT*C < K ) ) )
F2 = AN* -V(K-1)>/H-AN2*>/H
DF2-AN/H
DF2-DF2+AN2/H
DF{K > =-F2/DF2
V (K >= V< K 5 +DF < K >
F(K)=<1/QKT>*GL0G vO lONTNUc.
FIND THE CURRENT MAGNITUDE & CHECK FOR CONVERGENCE.
K ~35
ZI* )/<2.71Q**(QKT*C ) )
2 I=-ZI 1.6£-19* < V{K)-V(K-l))*MU*NC*A/H
IF < ABS(< 2 I -ZZI)/ZI>.LT.TX )GO TO 35
ZZi=/!
40 CONTINUE


75
T=77 K
1=717x10 A
-22
10-
-23
10
id icf io3 io'
f(Hz)
Fig. IV-7. Current spectral density at 77K and I = 7.17 x 10_6A
(ohmic regime). 0


154
WRITE THE DATA IN A FILE FOR IMEDENCE
35 WRITEO, iOO)C(N) > ZI
XI*XI*1.5
50 CONTINUE
100 FORMAT<2E11.4)
WRITE < 8 >120)NC MU > ET > H > ZI
120 FORMAT<5E11.4)
WRITE(8 140)NT A > N
140 FORMAT<2E11.4*14)
DO 200 K = 1/N
WRITE(Sr 130)C(K)r XN 130 FORMAT(3E11.4)
200 CONTINUE
STOP
END
PROGRAM


107
Fig. VI-1. |impedanceat T = 77K. Dots: measured data. Solid
line: computer calculation with four trap levels. Dot-
dashed line: computer calculation with a single trap level
(Eact = 103 meV) at V = 30 meV.


nnn nnn
155
FILE
TIM
COMPUTER SIMULATION FOR N+ N N+ SiC .
NOISE AND IMPEDANCE CALCULATION WITH DIFFUSION INCLUDED
DIMENS ION CZ(100 >,XNZ(100)* XNTZ<100),FI ELD(100)*TAU12(100),RU(100 5
DIMENSION TAU11(100) ,TAU22U00) > ERE< 100) XNTZ2( 100) TAU21 (100)
DIMENS ION TAU(100),SI(50 50),XK Z(50 >
COMPLEX 2KAPA100) ,DENQ(100) ,DIEL<100) .ACF(IOO) ,2(50.50)
COMPLEX XNDR(50,50),XND11(50,50),XND12(50,50),XND13(50,50>
COMPLEX XND15(50,50),XND16(50.50),XND17(50.50),AZ(50)
COMPLEX XAK(50,50),XKAC < 50,50),CDKC 50,50),OK(50,50),XC(50)
COMPLEX DIO,SUM2,D102,XNOISE XNOISE1,XNOISE2,XNOISE3,XNO13E4
COMPLEX C(50,50),XND14(50,50),XNOISEX,V < 50,50 >
COMMON C OK Z Y,CDK,XND14,XND15,XND16,XND17 > XND13,XNDI 2,XND11
*X,NDR, XK AC > XAK ,31, ZK APA DENO DI EL, ACF XK Z TAU XNI SE)
NOISE4
READ DATA FROM 1-0 SIMULATION
READ(8 >134>CZ(N),ZXI
WRITE(6,139)CZ(N>,ZXI
WRITE(7,139)CZ(N),ZXI
139 FORMAT (5X "VOLTAGE" El 1,4>5X, CURRENT',
134 FORMAT(2E11.4)
READ (. : 20) XNO, ZMU, ETZ,HZ,2X1
READ(3,140)ZNT,A,N
140 FORMAT(2E11.4,14)
120 FORMAT(5E11.4)
Nl=N-i
T 7 7
GKT*1.6E19/(1.381E23*T)
ZNT2 = 5,OE18
XN2-' 2N;p (- 083*QK T ) ) /2,
EPS I slii, i
E ; i - ' >4L 12
CDIFF(ZMU*EPSI*EPST0>/ . XNlETZ*ZNC/2.0
DO 200 K = 1,N
READ ( 8,1:70;" (K) ,XNZ(K) ,XNTZ(K )
XNTZ2 (K ) =2NV2/ ( 1 ;i -i VX7.' ; ;< ) )
130 FORMAT(3E11.4)
200 CONTINUE
BET A = 2.OE-17
BET A 2 = BE": A* I ,
WRITE(6,1015 BETA
101 FORMAT (1 OX CA PTURE COEFFIOI ENT* .-Ell. 4 )
11
, 4)


83
n N *
(3.1)
Consequently, the Fermi potential at few Debye lengths away from the
electron injecting contact (cathode) is given by
, kT rND>|
Fn Ac q 'N J
n c
Since we chose x 0 at the n+ cathode, for reference purposes we
c
obtain for $
Fn
kT
IL
$ = An () .
Fn q '
n c
(3.2)
The Fermi potential few Debye lengths away from the electron extracting
contact (anode) is given by
% V In (^)
Fn D q 1
(3.3)
where Vp is the applied voltage.
The coupled system of partial differential equations
F_ (X ) V*x (x) + N exp(
1 c Fn c ee v c v
-q(%n(x) Xc(x))
kT
r -1 r(liXC(x) V(X)^n
+ NT/[l + g exp(-^-)exp( ^ )] n
c
(3.4)
and
V*c
$Fn) vt(exp(-
-q(A(x) -
) V%(*)
kT
(3.5)


14
(3.12)
4. Time Window
Since samples of x(t) are taken during the time interval T only,
this causes the continuous input signal x(t) to be multiplied by a
function g(t), where g(t) = 1 for 0 by
(4.1)
The function g(t) is often called a "rectangular (uniform) time
window". The effective noise bandwidth is
!_
T
(4.2)
and is equal to the bandwidth obtained in section 3.
Depending on the type of measurement, different passband filters
(windows in the time domain) can be constructed by multiplying the
uniform function g(t) with an arbitrary continuous time function. The
spacing between these filters and the width of the individual filter in
the frequency domain is determined by the window shape and the sampling
rate.
These synthesized filters exhibit a characteristic referred to as
"leakage." Leakage occurs when the energy of the signal leaks into the
sidelobes of the filter. The simplest filter to examine is the uniform
window with the filter shape of
x
The displayed spectrum depends



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PAGE 170

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PAGE 171

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81
We present a detailed description of the theoretical charge
transport model and the results of our simulation for the linear regime
J. J.
and the three regimes of SCL flow in our n nn a-SiC samples at T = 77K.
2. Theoretical Model
The basic transport equations, including DC and AC as well as
Langevin noise sources, are:
J(x,t) qynE + 3d (2.1)
(2.2)
(2.3)
(2.4)
dE
to '(ci/eeo)(n + V V
3n(x,t) N 1 3J(x,t) ,
at 6nint Sn(Nt V 5 sr*+ r<**t>
3n (x,t)
g- -fra^n + 6n(Nt nt) 7 Y(x>t)
Here n is the free electron density, nt is the trapped density, n£ is
the equilibrium charge, n^ is the Shockley-Read parameter, 8 is a
capture constant, H is the thermal noise source, and Y is the trapping
noise source. The other symbols have their usual meaning. These
equations are split into DC and AC parts or in a noise analysis into DC
and fluctuating parts, so that n < n^ + An, I I + AI, etc. Note
that the suffix zero now denotes DC or average values, tig , etc.
The equations describing the DC part are related to the quasi-Fermi
potential (4^n) and the electrostatic potential (x ) as
V2x =* (n_ + n n )
Ac eeQ v 0 tO c'
3x
(2.5)
V(Aqynn0(x)V^,n(x)) 0 .
(2.6)


131
chapter VI to a single trap level with an activation energy of 70 meV, a
IQ
trap concentration of 10 m and a capture cross section of
-18 o
~ 5 x 10 cm We also apply a trap level indicated in Table
VII-lb. As shown in Fig. VII-9, the noise magnitudes obtained from the
computer calculation are much larger than we experimentally observed.
We should remember that due to the strong bending of the conduction
band and the trap level, the description of G-R noise is complicated.
Because in these devices different traps are responsible for current
fluctuation in different sections of the device, we have also neglected
the possibility of carriers traveling through some part of the device
collision free. In this case, the concept of thermal velocity is no
longer valid, resulting in a position-dependent value for the capture
coefficient 3. This makes, in turn, the trap time-constant position
dependent, creating a smoothing effect on the noise spectrum.
In addition, we neglected hot electron effects, so the computer-
calculated noise spectrum is obviously overestimating the real G-R noise
inside the device because of the overestimation of the mobility.


Fig. IV-8. Current spectral density at 77K and In =2.5 x 10~4A (low-
voltage quadratic SCL regime). u


Fig. II 2. Simplified block diagram of 7D20 digital programmable oscilloscope.


28
Table III-l
Nomenclature of the polytypes of silicon carbide
ABC Notation
Ramsdell
Zigzag Sequence
Sequence of
Inequivalent Layer
AB
2H
(11)
h
ABC
3C
(00)
c
ABCACB
6H
(33)
hcc
ABCACBCABACABCB
15R
(323232)
hcchc
ABCB
4H
(22)
he
A summary of
Table Ill-
band structure
2
for SiC
polytypes
Polytypes
3C
6H
4H
2H
Direct bandgaps [12] (eV)
5.14
4.4
4.6
4.46
Indirect bandgaps (eV)
Experimental values
2.39 [12]
3.0 [15]
3.26 [12]
3.35 [13]
Theoretical values [10]
2.4(T-X)
2.4(TM)
2.8(r-M)
3.35(T-K)


43
using the physical variables I, E, and V. The solutions can be
accomplished as in the previous subsection, but with considerably more
ease. This will be shown elsewhere.
We recall that four current regimes are found. The ohmic regime
prevails when region IV nearly fills the entire device. When region IV
becomes negligible ("is swept out at the anode") and region III prevails
in most of the sample, we obtain the low-voltage quadratic regime. When
this regime becomes negligible and region II prevails in most of the
sample, we have the TFL (trap-filling-limited) regime. Finally, with
region I taking up most of the sample, the Mott-Gurney regime occurs.
We recall the TFL-low voltage quadratic regime transition voltage,
(3.35)
from which
Nt 1.1 x 106 eVm/L2
(3.36)
4. Experimental I-V characteristics
In Figs. III-5 III-8 we show some data for 296K, 200K, 77K, and
52.6K. Data were also obtained at 250K, 167K, 125K, 100K, and 62.5K.
All characteristics but the one at the highest temperature clearly show
the four regimes. The trap-filling limited regime does not give a sharp
near-vertical line. We notice that this regime is rather short,
indicating"^ that 1/p is not more than an order of magnitude, cf. eqs.
(3.35) and (3.37) for the two quadratic regimes. This is also born out
by an explicit evaluation of p via eq. (3.31), see below.


97
TFL Regime
1=1.9X10 ~3A
Fig. V-9.. Carrier concentration profile of the trap level and the
conduction band in the trap-filling regime at T => 77k.


REFERENCES
1. R.C. Marshall, J.W. Faust, Jr. and C.E. Ryan, Silicon Carbide,
University of South Carolina Press, Columbia, S.C. (1974).
2. A.R. Verma and P. Krisna, Polymorphism and Polytypism in Crystals,
Wiley, New York (1966).
3. John Gosch, "Silicon carbide ends long quest for blue light-
emitting diode," Electronics Week, p. 24, (Oct. 8, 1984).
4. G. Ziegler, P. Lanig, D. Theis and C. Weyrich, "Single crystal
growth of SiC substrate material for blue light-emitting diodes,"
IEEE Trans. Electron Devices ED^SO, 4, 277 (1983).
5. L. Hoffman, G. Ziegler, D. Theis and C. Weyrich, "Silicon carbide
blue light-emitting diodes with improved external quantum
efficiency," J. Appl. Phys. 53, 10, 6962 (1982).
6. E. Pettenpaul, W. von Munch and G. Ziegler, "Silicon carbide
devices," Inst. Phys. Conf., Ser. No.. 53, 21-35 (1980).
7. J.P. Nougier, J. Comallonga and M. Rol land, "Pulsed technique for
noise temperature measurement," Journal of Physics E: Scientific
Instruments 7_, 287 (1974).
8. C.F. Whiteside, "Pulsed bias noise measurements on submicron
devices," Master's thesis, University of Florida (1983).
9. "7D20 programmable digitizer," TEK operational manual, Tektronix,
Inc., Oregon (1983).
10. A. van der Ziel, Noise in Measurements, Wiley-Interscience, New
York (1976).
11. A.V. Oppenheim and R.D. Schafer, Digital Signal Processing,
Prentice-Hall, New Jersey (1975).
12. "The FFT fundamentals and concepts," TEK instruction manual,
Tektronix, Inc., Oregon (1975).
13. V. Ozarov and R.E. Hysell, "Space-charge-limited currents in
single-crystal silicon carbide," J. Appl. Phys. 33, 3013 (1962).
14. A.C. English and R.E. Drews, "Space-charge-limited currents in
silicon carbide single crystals," Sci. Elect. 9, 1 (1963).
159


Table II-l
Digitizing Mode Characteristics
Digitizing
Mode
TIM/DIV
Range
Nyquist
Frequency (Hz)
Number of*
Points/Window
CCD Sampling
Rate
A/d Sampling
Rate
Memory Access
Gate Rate
ROLL
20S-100MS
50
1024
400 KHZ
400 KHz
100
TIM/DIV
TIM/DIV
Real Time
50MS-500pS
50
1024
400 KHz
400 KHz
100
(RD)
TIM/DIV
TIM/DIV
Extended
40
80 *
Real Time
(ERD)
200yS-2pS
TIM/DIV
820
TIM/DIV
400 KHz
400 KHz
*Samples are shifted out of the CCD at a rate of 400 Hz


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AUTHOR: Tehrani-Nikoo, Saied
TITLE: Properties of Noise and Charge Transport in Layered Electronic
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PUBLICATION DATE: 1985
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104
AE(x)AE(x')
m 31 // ^dx^Cx^.j^S^Cx^.x^ZfCx^.x',ji)
where the impedance elements are defined in discre
(2.6). To obtain the noise source S_(x ,x ), we
5 n m
di:
:e fora by
fine
C = Ay + BH + C
dY
dx
where A, B, and C are constants, given by eq. (2.4).
density of 5>S^ is then
S A(x )A*(x )S (x ,x ) + B(x )B*(x )S (x
s n m t n m n m H n
dS (x ,x )
+ A(x )C*(x ) ~nr + A*(x )C(x )
n m ax m n
dS
m
dS (x ,x )
+ C(x )C*(x ) 1-3--
n nr dx dx
n m
with
Sv^xn*xra) 4ABn n- 6(x x ) K (x )6(x
T n m it n m Yn, n
(shot noise in the transition rates)
Sw(xnx> ^Aq2n D6(x x ) K (x )6(x
Hnm unm Hnn
(velocity fluctuation noise in the conduction band)
where A in eqs. (2.12) and (2.13) is the device area
find for S
AE(x)AE(x')
From eq.
using only the first add second
(2.11),
m
The noise
x)
tn
(x ,x )
n m
dx
n
x )
m
x )
m
(2.9)
equation
(2.10)
spectral
(2.11)
(2.12)
(2.13)
(2.9) we
terms of


51
Next we computed 1/p from using eq. (3.33), and we also computed
it from the displacement of the two quadratic regimes (1/p obs.). The
agreement is very reasonable. As noted above, the values of 1/p turn
out to be rather small (making the theoretical assumption p 1 rather
crude). The values of 1/ p are not much dependent on temperature, and a
log (1/p) vs 1000/T plot reveals no significant slope. This indicates
that different traps were responsible for the TFL regime at different
temperatures. The pertinent trap depth £, c-was computed from the
equation
1 St
(4.1)
which follows from (3.14a) and (3.12). The results are entered in the
last row of Table III-4. In chapter IV we will correlate these trap
levels with those obtained from the noise measurements. The lowest
trapping levels (81 and 67 meV) may be due to the ionized nitrogen
donors of Table III-3. Comparing £c£t with (S c~C%, we see that the
traps which are involved are only a few kT above the Fermi level. They
are "shallow but not very much so. When the temperature decreases,
"shallower" traps are involved. E.g., at 200K the traps involved are
274 meV below c; the traps at 361 meV are "deep" traps at this
temperature and are filled up. They do not show in the I-V
characteristics, since none of our I-V curves showed a transition of the
ohmic range directly to the TFL range. Probably very low temperatures
would be required to observe that behavior.
5. Impedance measurement
The impedance of SCL devices with traps was calculated in previous
work by Van Vliet et al., see ref. 20. The following result is valid in


125
field is higher than the critical field (~ 3.3 KV/cm for T 77K). At
this point velocity overshoot occurs, and the carriers travel through
the device with few or no collisions.
Even though the physics of the charge transport model outlined
above is basically right, it is not an exact picture. In our computer
calculations we did not consider the transfer of carriers from central
valley to satellite valley, i.e., we neglected the change in the
velocity v a liE of the carriers above the critical field as calculated
with the Monte Carlo simulations,^ as shown in Fig. VII-6.
As the biasing voltage increases, the potential minimum moves
toward the cathode as illustrated in Fig. VII-7. The carrier
concentration profile for this case is shown in Fig. VII-4.
Fig. VII-8 shows the electric field profile. The retarding field
in this case is smaller than it was in. the ohmic regime. At this
current level almost half of the device observes an electric field
higher than the critical field. Velocity overshoot is very likely to
occur. Neglecting the mobility reduction at high fields results in an
overestimation of the current. This can be observed in the results of
the computer calculation of the I-V characteristics (Fig. VII-2).
Finally, the linear regime of the I-V characteristic observed at
high-voltage levels can be due to the velocity overshoot. One should
remember that at these voltage levels most of the device is controlled
by a field higher than the critical field. This enhances the
possibility of the velocity overshoot.
3. Noise Characteristic
The noise measurements^ for various current levels at T 77K are
shown in Fig. VII-9. The frequency dependence of the spectra is about
(1/f)*^ for all current levels.


44
ft
Fig. III-5. I-V characteristic at 296K.


EnergyeV)
90
-5 0 5 10 15 20
L (/im)
Fig. V-3. Energy band diagram in the linear regime (I = 6 10 A) at
T = 77K. The solid line represents the conduction band,
the dashed line the quasi-Fermi level, and the dot-dashed
line the trap energy level.


60
These equations are split into DC and AC parts or in a noise analysis
into DC and fluctuating parts, so that n = n^ + An ,
I Iq + AI etc. Note that the suffix zero now denotes DC or
average values, Hq , etc.
The transfer impedance was found to be from eqs. (2.1) (2.4)
z(x,x',m) = u(x-x')
(2.5)
where u(x) is the Heaviside function, and where
p nQ/(n0+nt)
qpn yA
c
(2.6)
(2.7)
Y sat '
* 1 + j)T
t1 l/3(Nt-nt0) t2 I/S^-HIq) 1/t 1/t1 + 1/t2
(2.8)
The noise source for trapping is given by
35
S^(x,x') = AAgn^n^Six-x') 2 K(x) (2.9)
The terminal noise is computed with z(x,x', S
AV(L)
L L L
A / / dx dx' / dx" z(x,x" ,w)z*(x',x" ,a))K(x") (2.10)
0 0
0


19
Fig. II-6. Flow chart of the computer program


-14
3
5 T~ 9 0 13 15
lOOO/KK)
Fig. IV-6. Relative plateau values S /I2 of the Lorentziau
of 1000/T. 1
spectra spectra as a function


116
O
CD
Fig.
VI-6.'
Normalized current spectral density S^R/Iq | VJ as a
function of voltage Vq. Dashed lines: best fit to the
measured data. Solid lines: results of the computer
calculations.


20
point-by-point basis using RMS calculations. Finally, the total
spectrum is divided by the square root of the effective noise bandwidth
and is displayed on the screen.
7. Noise Measurement Procedure
To determine the current noise of the device under test (DUT) and
to eliminate the effects of system gain and bandwidth, three different
noise measurements are required.^
First, the noise magnitude is calculated when the DUT is biased
(Ml). Then the DUT is replaced by a noise calibration source and a
dummy resistor having the same impedance (M2).
In the third measurement, the calibration source is turned off
(M3). From the equivalent noise scheme of the experimental setup in
Fig. II-7,
(7.1)
(7.2)
(7.3)
where GB is equal to the product of gain and bandwidth, and R is equal
to the parallel combination of and R^. From these three equations,
it follows that the current noise of the DUT is
S
I DUT
Ml M3-i
M2 M3JbCAL
]S_.. + 4kT Re(Y)
(7.4)
where S,
'CAL
is the calibration current noise source, k is Boltzmann's


Energy (eV)
94
Low Voltage Quadratic
I = 3.92 MO"5 A
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
T=7 7K
Potential Minimum
-5 0 5 10 15 20
L <(o.m)
Fig. V-6. Energy band diagram in the low-voltage, quadratic regime at
77K. The solid line represents the conduction band, the
dashed line the quasi-Fermi level, and the dot-dashed line
the trap energy level.


48
The basic data and the results deduced from them are presented in
Table III-4. The first three rows refer to the experimental data:
^TFL* an<* R (ohmic .regime resistance), as observed. The fourth row
lists the mobility, computed from the Mott-Gurney asymptotic law. The
values for p are in the same ballpark as those of Table III-3. We note
that p increases as temperature increases, but not as fast as T^^
(rather ~ T*). Thus the scattering may be a mixture of ionized
impurity scattering and polar phonon scattering, the latter coming in
near room temperature. This agrees with the observatios of van Daal,
dealt with in Section 2.
The values of iP, listed in the fifth row, are found from R, p and
the geometry factors. We note that n is quite small, indicative for
the insulating nature of the poly type involved. A plot of log nQ vs
1000/T, given in Fig. III-9, indicates a very shallow donor,
<£*c 10 meV. Obviously, these donors are ionized at all
temperatures in our measurements.
From the trap-filled-limit transition voltage, V^p^, we computed
the number of traps using (3.36). The number of traps per cm^ is only
of order 10*3 10^-^ indicating that the crystals are quite pure. The
statistical weight of the conduction band is calculated in the next
row. We assumed here effective masses as for 6H in Table III-3, with a
valley degeneracy factor of three (corresponding to point M in the
Brillouin zone). This leads to a density of states effective mass
m** 1.13 mg. We notice, however, that in what follows we need log Nc,
which is not too sensitive to the choice of m**. Thus p, computed
from eq. (3.11), is quite accurate. Figure III-10 gives the equilibrium
Fermi level as a function of T. We notice that the Fermi level position
in the forbidden gap is lowered almost linearly with rising T.


140
2780 INPUT OU:A0,Ai,A3,A,A2,A5,A4
2800 PRINT QU'-"CURVE'?"
2820 WBYTE 3U+4:
2840 BYTE A7
280 RBYTS AA8
2380 RE M CHECK FOR CORRECT DATA TRANSFER
2900 A9=SUM(A)+A7(9)+A7(8)
2920 A9=A9-25*INT(A9/25)
2940 A9=25-A9
290 IF AS3(A8)=A9 THEN 3020
2930 GO TO 2800
3000 REM CHSNGE FROM BINARY TO DECIMAL FORMAT
3020 A=A-l2c
3040 A=A*A5,
30s0 A=A/25
3033 A=A+A4
3100 LET A=A-SUM(A)/820
3120 REM DIVIDE THE SAMPLED DATA INTO THREE BLOCKS
31*0 DIM E(25>,E7(25),E*<256>
3130 FOR Ii4=l TO 25
3133 BI14)=A(Ii4;
3200 B7;Ii4>=A(Ii4+25)
3220 E8(Ii4)=A(i4+5i2)
3240 NEXT I4
3230 RSM CALCULATE THE FFT
3230 CALL "FFT",E
3303 CALL FFT",B?
3320 CALL "FFT'SBS
3340 DIM S1(129),C(129),32(129),E329),A<129)
330 REM FIND THE MAGNITUDE AT EACH FREQ.
3380 CALL POLAR" ,B,B 1 C0
3400 CALL 'POLAR",B7,B2,C,0
3420 CALL POLAR",B8,B3,C,0
3440 REM CALCULATE THE FOURIER COMPONENTS
3460 E1=B1*31
3430 B2=B2*B2
3500 B3=B3*B3
2520 RSM ADD THE SPECTRUM3
3540 A=3i+B2
3560 A=A+B3
3580 S5-A+35
2600 NEXT 115
3220 RSM
3648 REM
3360 RSM
2330 REM
3700 REM
3720 REM
3743 3w=SW
DIVIDE THE SPECTRUM BY THE NOISE BW AND THE NUMB!
OF SAMPLED POINT SQUARE AND ff OF AVERAGES
SI=2*(M AGN1TUDS! A) 2) / (N 2)* i 2*BW
FOR 820 SAMPLES THE TIME WINDOW FOR 256 POINTS
IS EQUAL TO 1.28 TIMES THE TIMS WINDOW
USED FOP 1024 SAMPLE POINTS.
R
3760 A*B5/(Bw#399384>
3780 C3=A*C5
3800 A=C5/I10
3820 VIEWPORT 20,120,15,30


(A2/Hz)
129
Fig. VII-9. Noise calculation at T 77K. Solid lines; measurements
at various current levels. Dashed line: computer
calculation of G-R noise for I = 100 yA with the trap level
given in Table VII1. Dotdashed line: computer
cniculation of G-R noise with the trap concentration of
l(rVJ for I 100 yA.


Widtlv.825 mm
Gold'*
Alumina
Fig. III-2. The device structure mounted on an alumina plate showing the lamellae along
the c-axis direction and the arrangements of the tungsten contact areas.


10
needed to transfer the data into the A/D converter, since the data is
transferred at a slower rate than the sampling rate.
In' the equivalent-time digitizing (ETD) mode, a limited number of
samples from a periodic signal are taken at successive trigger events.
These samples are used to reconstruct an accurate composition
representation of the waveform in the same way as is done in a
conventional sampling oscilloscope.
Since a noise signal is nonperiodic and might contain correlation
times, this triggering mode cannot be used for the noise measurements.
The upper frequency limit for noise measurements is set by the extended
real-time digitizing mode and is equal to the maximum Nyquist frequency
of 20 MHz, which corresponds to a 3 dB system bandwidth of 10 MHz. The
digitized samples stored in the waveform memory are read by the computer
for fast Fourier transform (FFT) calculations.
3. Spectral Intensity of a Random Noise Signal
A noise signal x(t) measured in the time period 0 defined in terms of Fourier series as*
00
x(t) = l afcexp( j2irfkt) (3.1)
k=-
where ^ 0, 1, 2, ), and a^ is the Fourier coefficient of
x(t). We obtain the value of a^ using the decimation-in-frequency FFT
approach of Sande-Tukey.***^ The discrete Fourier transform
coefficients a^ are defined as
N-l
ak 53 t E Atx(nAt)exp(-j2irkAfnAt)
n=0
(3.2)


130
If we assume that the measured noise is due to mobility
fluctuations which can be described by the flooge formula, then
SAI
(3.1)
where Iq is the DC current, f is the frequency, N is the total number of
carriers in the sample contributing to the noise, and is the Hooge
parameter.
For a collision-limited sample with the conduction determined by a
constant mobility, the expression for obtained for the mesa
structure by van der Ziel and Van Vliet^ applies. They find
AI
overall
(f)
a I2
HO
fm2L2
(qpp)
(3.2)
where m is the number of alternating n+ and p layers, p is the total
device resistance, and L is the length of the p-layer. As we observe
from Fig. VII-3, the major variation in the quasi-Fermi level occurs in
the mid-section of the p-layer. From Fig. VII-4 the mid-section is
where the carrier concentration has its smallest magnitude causing the
largest contribution to the 1/f noise. If we assume that the effective
length contributing to the noise is approximately 1/2 of the total
-a -4
length of the p-layer at Iq 10 A, for f = 1 Hz we obtain 10
The Hooge parameters measured^-^ for submicron n+nn+ structures are in
the 10^ range. We conclude that 1/f noise alone cannot explain the
total noise spectral magnitude.
Assuming that generation-recombination (G-R) noise is the dominant
source of noise in the p-layer, we apply the computer model of


119
Fig. VII-1.
Submicron (.47 pm) n pn
GaAs mesa structures.


162
46. J.H. Andrian, "Noise properties of very short GaAs Devices," Ph.D.
dissertation, University of Florida (1985).
47. R.R. Schmidt, G. Bosnian, C.M. van Vliet, L.F. Eastman and H.
Hollis, "Noise in near-ballistic n nn and n pn gallium arsenide
submicron diodes," Solid State Electron. 26, 437 (1983).
48. S.M. Sze, Physics of Semiconductor Devices, Wiley-Interscience, New
York (1981), p. 21.
49. J.G. Ruch and W. Fawcett, "Temperature dependence of the transport
properties of gallium arsenide determined by Monte Carlo method,"
J. Appl. Phys. 41, 3843 (1970).
50. A. van der Ziel and C.M. van Vliet, "Mobility fluctuation 1/f noise
in non-uniform non-linear samples and in mesa structures," Physica
Status Solidi (a) ]!> 453 (1982).
51. P. Das and D.K. Ferry, "Hot electron microwave conductivity of wide
bandgap semiconductors," Solid State Electronics 19, 851 (1976).
52. K. Sasaki, M.M. Rahman and S. Furukawa, "An amorphous SiC H emitter
heterojunction bipolar transistor," IEEE Electron Device Letters
EDL-6, 311 (1985).


35
rather awkward to extract physical information for the limiting regimes
from the solution, as is evident from the discussion in Sections 4.2 and
4.6.2 of reference 15.
For the above reasons we present here a different solution, based
on the principles and notation of reference 30. We express I and V
explicitly parametrically in the parameter a g^E^I/I, where gg is
the conductance per unit length of the unexcited specimen and is the
field at the anode. Limiting regimes require a -* 1 for ohmic flow
and a 0 for SCL flow. Thus 0 < a < 1 We noted before that this
parametric presentation is also extremely useful for a computation of
on
the impedance and of the noise. Thus, together with reference 30,
this section gives a unified description for DC and AC behavior, as well
as for the noise.
The pertinent equations are:
(3.4)
J => qynE E = Ex
iv v iv iv
(3.5)
I = JA = AqynE
where we assumed injection of electrons along the positive x-axis, the
cathode being at x 0 and the anode at x L; notice that J and E are
negative quantities (x is a unit vector along the positive x-axis).
Poisson's equation for acceptor type traps reads
(3.6a)
and for donor type traps


20
2 20
F (KHz)
-22-20
Fold Over
-10 -202
1 0
(b)
Fig. II-5. When the sampling rate is 10 KHz: (a)
component is folded down to become an
(b) a 22 KHz component is folded twice
2 KHz alias.
2 02 2
F (KHz)
a 12 KHz
KHz alias;
and becomes
a


30
Table II1-3
A summary of electron mobilities, ionization energies
of nitrogen donor (Eg), and effective masses of electrons
in n-type SiC (Ng = 6 x 10 cm-^) [27] at room temperature
1
Polytype
Mobility
2
(cm /volt-sec.)
Ep(meV)
m* /mg
m*/mg
4H
700
33
0.19
0.21
15R
500
47
0.27
0.25
6H
330
95
1.3
0.35


63
.
Nti ~ nti,0
li
Nti nti,0 + nli + no
(2.18)
where ^ is a "partial covariance" due to interactions of the
conduction band electrons with traps i. Equations (2.17) and (2.18) are
the basis for our experimental results.
We still mention the result for T^. From (2.8) we obtain
with
8 o^, being the cross section and the mean thermal
velocity,
+ ni,i+ v (2-19)
In the ohmic regime, nQ n^ For shallow traps, moreover,
nti 0 ^e note<* before (Table III-4, chapter III) that n
(thermal equilibrium value) is of order 10^ cm^ while Nt^ is of order
10^ cm^. The ratio N^/n fr temPeratures is of order 100.
Hence, in order that n^ ^ [see chapter III, eq. (3.12)] dominates over
the term Nt we must be about 4 kT above the Fermi level
(e^*6 a 100). Thus, is exponential,
x S
i o N
i c
(6c-^/kT
(2.20)
in a temperature range where the trap level is 4 kT or more above the
Fermi level. Since the Fermi level rises when the temperature decreases
(chapter III, Fig. III-10), we must find that in a plot of lnx^ vs


i4440 Tny=100*(Xints-Plx)/P2x-Pix)
14460 PRINT 95: USING 14480:Tpy}Tny
14480 IMAGE HTL",6D,VD," JJ1
14500 RETURN


109
(1 + I
1/T.
)"1 = r
+ i)-' '1. +
i 1/T2i
1 + jWT
22>-l
1
jwt,
(3.1)
22
where the left-hand side was evaluated for trap two only. Since iiq,
nt2> and n^ are much smaller than Nt2 at low-bias voltages,
1/ 8 Nfc2 and t ^ &(nQ + nl^ The ^mPe<^ance calculation of
the single trap observes a pole at a frequency corresponding to the
reciprocal detrapping time constant u> => l/x^ and later observes a
zero at the reciprocal trap time constant a) 1/x2 which cancels the
effect of the first pole. The impedance finally sees the pole created
by the dielectric time constant x^. If other traps are included,
specially shallow traps with n-^ Nt, then the aforementioned effect
of trap two on the impedance calculation is cancelled and the results
are as shown by the solid lines. The break-point frequency is
controlled by the dielectric time constant.
The magnitude of the impedance is in excellent agreement with our
measurements at all voltage levels. However, the break-point frequency
differs at high-bias voltages. Presently we do not know the exact
source of this discrepancy. Our guess is that at high-voltage levels
the break-point frequency is in the MHz range, so the pulsed bias
impedance measurement was not accurate enough.
The results of the noise calculation in the ohmic regime are shown
in Fig. VI-2. The dots indicate the measured values of the total
spectral current density, whereas the dashed lines represent the noise
contributions of the individual traps. The solid lines give the results
of the computer calculations with drift and diffusion included in the
noise source term (eq. 2.10). The effect of the diffusion part of the


40
Further, integrating (3.18) once more from 0 to L, we obtain
L2 -
ee
Ise0 E-L SoE
qncS0
[Eov +i^ / dE ^ +~r)] <3-21>
c 0 n
where we used the standard trick .to change dx into dE by means of
(3.15). Let gg/I = 0 Then, noticing
, RT? 1
/ rriE dE j [SE to(l+0E)] ,
(3.22)
we find from integration by parts
X / dE An(l+8E)
j [8EL An(l+6EL)]Jln(l+3EL) Y ,
(3.23)
where
gE J2.n( 1+3E)
.aw]
- El -j n(l+8EL) [An(l+8EL)]2 .
(3.24)
Together with (3.23) and setting $E -a this yields
Ju
X ET An(l-a) ~ \ +J *(!-) [n(l-)]2
28
(3.25)
When this is substituted into (3.21), equation (3.22) results. The
result is


148
12320 IF Lv$="LOG" THEN 13940 REM PLOT THE Y-AXIS LOGARITHMICALLY
12340 REM PLOT THE Y-AXIS LINEARLY
12360 Ytics=Ky#Ytic
12330 Nyirrt= 12400 GOSUB 14360 !REM CHANGE THE Y-TIC LENGTH
12420 PRINT S5: USING 12440:Xint*,Piy
12440 IMAGE n;PA%6D'V',D,,,;PD¡YT¡
12460 FOR 1*1 TO Nyint
12430 PRINT 35: USING 12500:Xints,Ply+I*Ytics
12500 IMAGE ¡PA" ,6D,"," ,6D,"; YT;"
12520 NEXT I
12540 REM RETURN HERS FROM ROUTINE WHICH PLOTTED LOG Y-AXIS
12560 PRINT 25: USING 12580:5m$,Xscaled(l),Yscaiedii)
12530 IMAGE ";PU;SMMA,";?A",6D,V,6D
12600 IF Vector*" N" THEN 12640
12620 PRINT 35:"PD;
12640 FOR 1=2 TO N
12660 PRINT 25: USING 12680:Xscaied(I),Yscaled(I)
12630 IMAGE 'LPA^D/VLD,";"
12700 NEXT I
12720 PRINT 25:SM;" !REM CANCEL SYMBOL MODE
12740 GO TO 12920
12760 PRINT "DO YOU DESIRE ANOTHER PLOT?(Y/N)H
12730 INPUT A$
12300 IF A*S"N" THEN 13130
12320 PRINT "WOULD YOU LIKE TO CHANGE ANYTHING ABOUT THE PLGT?"
12340 INPUT A*
12360 IF A*="N" THEN 11500
12330 IF A*=Y'! THEN 10960
12900 GO TO 13130
12920 PRINT "WOULD YOU LIKE TO LABEL THIS PLOT?(Y/N)M
12940 INPUT A$
12960 IF AI="N" THEN 12760
12980 PRINT "ENTER THE STRING, THEN HIT RETURN."
13000 PRINT "TO ESCAPE TYPE CNTRL A, RETURN."
13028 PRINT "LIMIT THE STRING TO 15 CHARACTERS."
13040 PRINT "USE THE MANUAL PLOTTER CONTROLS TO POSITION THE PEN TO THE"
13068 PRINT "BOTTOM LEFT CORNER OF THE FIRST CHARACTER."
13088 INPUT A$
13100 IF A*="A__" THEN 12768
13120 PRINT 25: USING 13140:A$
1314-3 IMAGE ,,;LB"i5A,"C_jPU;"
13160 GO TO 13030
13130 PRINT 35:"3P0J_
13,280 GO TO 7960
12220 END
13240 REM SCALE THE DATA, PREPARE THE PLOT PARAMETERS
13260 U2x*LGT(Xmax)
13230 Ui>;=LGT(Xmin)
12308 Nxdec=U2:-;-Uix
13328 Kxi= 13340 FOR 1=1 TO N
13360 Xscaled(I)=Kxl*:


n(x)& nt(x)(m3)
89
Linear Regime
I = 6 X 1 0 ~7 A
Fig. V-2. Carrier concentration profile of the trap level and the
conduction band in the linear regime (I = 6 x 10 ^A).


4* (o ro
135
REM SOFTWARE FOR THE COMPUTER CONTROLLED SPECTRUM ANALYZER
REM EMPLOYING A DIGITAL OSCILLOSCOPE.
REM MAY 1,1983
REM FOR MORE DETAILS REFER TO CHAPTER II.
10 REM
20 REM THE PROGRAM IS STORED IN FILE 2 IN BINARY FORMAT.
30 REM IN ORDER TO LOAD THE PROGRAM PRESS "AUTO LOAD".
100 FIND 2
120 CALL "BOLD"
130 RUN


6
The final result is stored in the computer for further noise
calculations. The noise spectrum can be plotted on the CRT or HP
plotter, or it can be stored on the- tape. The magnitude of the spectrum
can also be transferred to the printer or the CRT.
In the following sections we describe the various system components
and data processing steps in more detail.
2. Programmable Digital Oscilloscope
A sampling of the analog input waveform is obtained by a TEK 7D20
Q
programmable digital oscilloscope (p.d.o.). The preamplifier circuitry
attenuates the input signal according to the setting of the front panel
VOLTS/DIV control. It then amplifies the signal, converts it into a
differential signal, and applies it to the charge coupled device (CCD)
circuitry (Fig. II-2).
The charge coupled device contains two analog shift registers which
are driven differentially. One register samples the (-) side of the
differential input signal, while the other register samples the (+) side
of the signal. Triggered by a sampling clock pulse, a sample of the
signal is stored in the first cell of the analog shift register. At
subsequent clock pulse triggers, this sample is shifted from one cell to
another until it reaches the output amplifier and the analog-to-digital
(A/D) converter. The timing and synchronization of the CCD is set by
the time-base circuitry.
There are four basic modes of operation for the time-base
circuitry: roll, real-time digitizing (RD), extended real-time
digitizing (ERD), and equivalent-time digitizing (ETD) These modes are
selected by the TIM/DIV control knob. The characterization of the
different modes is summarized in Table II-l.


22
constant, T is absolute temperature, and Re(Y) is the real part of the
DUT admittance.
1


118
computer program outlined in preceding chapters is applied to a .47 pm
GaAs mesa structure device (Fig. VII-1) measured by Schmidt.^ The
results are compared with the I-V and noise measurements at T 77K.
2. I-V Characteristic
The I-V characteristics measured^ at various temperatures indicate
the presence of an ohmic regime, a fast-rising regime, and finally a
linear regime at very high voltage levels, as can be seen in Fig. VII-2
for T 77K.
The results of. the computer calculation without any traps are shown
by a dashed line in Fig. VII-2, using the parameters given in Table
VII-la. The solid line is the results of the computer calculation with
a trap level as given in Table VII-lb. To the best of our knowledge, a
21 -3
trap concentration of ~ 10 m with activation energy close to 70 meV
AO
has not been observed in GaAs before, so we assume that the fast rise
in the I-V characteristics is due to the presence of the space-
charge-limited flow with no traps.
Fig. VII-3 shows the energy band profile in the linear regime of
the I-V characteristic. Due to the large overflow of carriers from the
n+ regions, the short p-layer is inverted to an n layer. The potential
minimum has a large value indicative of the possibility of the space-
charge-limited flow. The carrier concentration profile in the ohmic
regime is given in Fig. VII-4. The carrier concentration shows a large
variation with position inside the p-layer.
The results of the field calculation are shown in Fig. VII-5. From
the injecting contact (cathode) up to ~ 25 pm, the carriers experience
a large retarding electric field. From that point on, the field is
negative and the carriers accelerate until they reach a point where the


16
Fig. II-4. (a) Energy leaks into the sidelobes with
discrete points falling on top of the sidelobes.
(b) Good approximation to impulse with discrete
points falling between the sidelobes.


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
E.R. Chenette
Professor of Electrical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
A. Neugroschel
Professor of Electrical Engineering
This dissertation was submitted to the Graduate Faculty of the
College of Engineering and to the Graduate School, and was accepted as
partial fulfillment of the requirements for the degree of Doctor of
Philosophy.
June 1985
Dean, College of Engineering
Dean, Graduate School


ACKNOWLEDGMENTS
The author wishes to express his sincere gratitude to Dr. C.M.
Van Vliet and Dr. G. Bosman for their research guidance and helpful
suggestions, and to Dr. A. van der Ziel for fruitful discussions.
I extend my gratitude to Dr. L.L. Hench for the opportunity to do
this research and for his guidance and encouragement.
Finally, the author appreciates the help of his fellow students in
the Noise Research Laboratory and Ms. Katie Beard for typing the disser
tation.
ii


65
Fig. IV-1. Current noise spectral density in ohmic regime for T = 296K
and T =* 250K. Circles and o : measured data. Full
lines: resolution into Lorentzians. Dashed line: 1/f
approximation. Please note: right vertical axis refers to
296 K curve, left vertical axis refers to 250K curve.


160
15. M.A. Lampert and P. Mark, Current Injection in Solids, Acad. Press,
New York and London (1970).
16. F.A. Levy, Crystallography and Crystal Chemistry of Materials with
Layered Structures, D. Reidel, Dordrecht, Holland (1976).
17. R.W.G. Wyckoff, Crystal Structure, vol. 1, p. 23, Interscience, New
York and London (1948).
18. W.F. Knippenberg, "Growth phenomena in silicon carbide," Philips
Res. Reports 18, 161 (1963).
19. H. Jagodzinskii, "One-dimensional disorder in crystals and its
influence on x-ray diffraction," Acta Cryst. Camb. 2, 201 (1949).
20. H.C. Junger and W. van Haeringen, "Energy band structures of four
polytypes of silicon carbide calculated with the empirical
pseudopotential method," Phys. Stat. Sol. 37, 709 (1970).
21. L.A. Hemstreet and C.Y. Fong, in Silicon Carbide-1973 (eds. R.C.
Marshall, J.W. Faust, Jr. and C.E. Ryan), Univ. of South Carolina
Press, Columbia, S.C. (1974).
22. W.J. Choyke, D.R. Hamilton and L. Patrick, "Optical properties of
cubic SiC: Luminescence of nitrogen-exciton complexes, and
interband," Phys. Rev. 133, A1163 (1964).
23. L. Patrick, D.R. Hamilton and W.J. Choyke, "Growth luminescence,
selection rules, and lattice sums of SiC with wurtze structure,"
Phys. Rev. 143, 526 (1966).
24. W.J. Choyke and L. Patrick, "Absorption of light in alpha SiC near
the bandedge, Phys. Rev. 105, 1721 (1957).
25. W.J. Choyke and L. Patrick, "Exciton recombination radiation and
phonon spectrum of 6H SiC," Phys. Rev. 127, 1868 (1962).
26* D.C. Barrett and R.B. Campbell, "Electron mobility measurements in
SiC polytypes," J. Appl. Phys. 38, 53 (1967).
27. H.J. Van Daal, "Mobility of charge carriers in silicon carbide,
Philips Res. Rept. Suppl., p. 1 (1965).
28. G.A. Lomakina, "Electrical properties of various polytypes of
silicon carbide," in Silicon Carbide-1973 (eds. R.C. Marshall, J.W.
Faust, Jr. and C.E. Ryan), Univ. of South Carolina Press, Columbia,
S.C. (1974), p. 520.
29. M.A. Lampert, "Simplified theory of space-charge-limited currents
in an insulator with traps," Phys. Rev. 103, 1648 (1956).
30. K.M. van Vliet, A. Friedman, R.J.J. Zijstra, A. Gisolf and A. van
der Ziel, "Noise in single injection diodes II. Applications," J.
Appl. Phys. 46, 1814 (1975).


CHAPTER IV
ELECTRICAL NOISE OF SCL FLOW IN ct-SiC
1. Introduction
From the I-V characteristics we observed space-charge-limited (SCL)
flow in a-sllicon carbide, due to the injection of electrons in a
strongly compensated polytype sandwiched between low resistive
polytypes. The I-V characteristics showed all four regimes pertaining
to SCL flow involving shallow traps: ohmic, low-voltage quadratic,
trap-filling limited, and high-voltage quadratic (Mott-Gurney regime).
(As usual, "shallow traps means traps above the equilibrium Fermi
level.) It was shown that a number of trapping levels are involved,
ranging from 67 meV to 360 meV below the conduction band; the
temperature range investigated was 50K 300K. At a given temperature
the onset of the trap-filling limited (TFL) regime occurs when the
quasi-Fermi level passes the deepest trapping level which for that
temperature has a substantial number of empty traps; computations from
the results (Table III-4) indicated that these are traps which are
~ 2 kT above the equilibrium Fermi level. When we go further into the
TFL regime, shallower traps are filling up. After all traps are filled,
we observe the Mott-Gurney range. In terms of carrier densities, the
TFL regime begins when n^ n^ for that set of traps, where n^ is the
density of injected carriers and n^ the Shockley-Read density for that
trap, see chapter III, section 3.
The above picture should be corroborated by noise measurements,
since trapping noise exhibits the same features. At a given temperature
57


11
where x(nAt) is the sampled time domain data, N is the total number of
samples acquired in the time interval T = AtN, Af is the frequency
spacing defined as Af = and k denotes the frequency
It
component fk = kAf. It is clear that if a£ is the complex
conjugate of a^,
a
-k
*
(3.3)
The Fourier component x^ of x(t) having frequency fk is given by
xk = akexp( j2irfkt) + a_fcexp(-j2irfkt) (3.4)
The ensemble average of x2 is found to be equal to
x£ a2 exp(j4irfkt) + afkexp(-j4irfkt) + 2afca_k (3.5)
Since the Fourier coefficients ak have an arbitrary phase,
a2 =* a2k 0. Hence,
*k 2ak*k
. N-l N-l
^ l l
(NAt) z n=0 df*0
(At)2 x(n)x(m) exp(j2irfk(m-n)At) .
(3.6)
The magnitude of the terms in eq. (3.6) peaks along the line n = m
(Fig. II-3) and decreases as we go away from this line. If we introduce
a domain of summation along the two lines parallel to the diagonal time
(n m) at a vertical distance of M, we can write


127
L(nm)
Fig. VII-7. Energy band diagram in the SCL regime (1 =
T 77K.
35A) at


Table VII-la
Transport parameters used In the computer calculation
without including
any trap level
Nc(m 3)
H(m2/V-sec)
ND(n+ regions)(m 3)
nc(m"3)
2
Area(m )
L( player)(m)
6.1 x 1022
8.0
1.5 x 1023
2.1 x 1021
7.9 x 10~9
.47 x 10-6
i
Table VII-lb
Transport parameters used In addition to Table VII-la
when a trap level is included
nc(m-3)
NT(m3)
Eact(meV>
8(m3/sec)
o(cm 2)
6 x IQ20
2.6 x lo21
71
5 x IQ-17
8.2 x IQ-18
121


77
Fig. IV-9 Current spectral density at 77K and I = 3.74 x in'3
(TFL regime). 0


n(x), nt(x)(m
100
Mott-Gurney Regime
I = 2.17 x 102
L(/x.m)
Fig. V-12. Carrier concentration profile of the trap level and the
conduction band in the Mott-Gurney regime at T 77K.


143
7020 CALL "MAXU,AM2L2
7040 Wi=M
7060 W2=M2
7080 IF Wl=8 THEN 7140
7100 W3=INT(LGT(ABS 7120 Wl=INT(Wl*10"'-W3))*i0-(W3-i)
7140 W2 7160 IF W2O0 THEN 7200
7180 W2=0.139
7200 W3 = 0 -I N T(L GT W2)))
7220 W2=INT(W2*W3+0.5)+i
7240 W2=W2/W3
7260 W2=Wi+3*W2
7230 W3=LGT(ABS(W2>)
7300 IF W3=8 THEN 7420
7328 W4=10*LC-T
7340 W6=18*LGT(M2)
7360 WINDOW i,AiW4,W6
7330 AXIS Ai/10,(W6-W4)/8,i,W4
7400 W5=/8
7420 MOVE 1,W6
7448 PRINT "";V*
7460 REM PRINT THE Y-AXIS
7480 FOR 1=0 TO 8
7500 MOVE 1 *(W6-W4)/8#I+W4
7520 PRINT USING "'','H_H_H_H_H_H_H_"')4D,2D":W4+I*W5
7548 NEXT I
7560 A3= 1 /(A3*MA 1 -1)*2))
7580 W3=INT(LGT(Ai#A3))
7680 A$="
7620 IF W3=8 THEN 7660
7648 A*=STR(10"W3)
7660 MOVE Ai/2*W4
7630 PRINT J_JJ';A$;" "¡HI;" ;Ai;" p/w";
7700 Fre=AS
7720 REM PRINT THE X-AXIS
7740 FOR 1=0 TO 10
7760 MOVE I#(A1-1)/18+1,W4
7730 PRINT USING *,,iH_H_JJ1,,,2D.2D(S,':a*(Al-l)/18*A3+A2)/10-W3
7880 NEXT I
7820 FOR 1=1 TO Ai
7340 Dtz(I)=18*LGT(A(I))
7360 NEXT I
7380 CALL "DISP" ,Ddz
7900 RETURN
7920 F3=108/35
7940 RETURN
7960 SET NOKEY
7938 PRINT 032,21:0*0
8000 PRINT "PRESS RETURN TO CONTINUE";G$;
8820 PRINT 032,21:0,40
8840 INPUT A
3060 PRINT "L_"("MENU"


Ln
TEK 4052
^-S* II-l. Block diagram for computercontrolled noise spectrum analyzer.


2
(NAt)2
N-l
I
n=0
N+-M
I (At)2 x(n)x(m) exp(j2irfk(m-n)At)
m=n-M
We define a new variable s = m n. Then
(3.7)
N-l
l
M
l
(NAt)2 n=0 s=-M
(At)2 x(n)x(n+s) exp(j2irf sAt)
tc
(3.8)
For a stationary process, x(n)x(n+s) is independent of n, and
consequently the two summations in eq. (3.8) can be decoupled, resulting
in
a 1 At x(n)x(n+s) exp( j2irfksAt) (3.9)
s=-M
Since x(n)x(n+s) 0 for s>|m| and the two small triangles of side M
have a negligible contribution if NM, we can change the limits of the
summation, such that
x2
He NAt
00
£ At x(n)x(n+s) exp( j2irfksAt)
(3.10)
The spectral intensity of x(t) defined by the discretized Wiener-
Khintchine theorem is given by
Sx(fk) = 2 l x(n)x(n+s) exp(j2irfksAt)At .
s=-
(3.11)
Since Af = i is defined as the frequency interval between
adjacent f^s, t*ie spectral intensity of the input signal can be written
as


37
With Fermi-Dirac statistics
- Nt/[g e
-i + i] .
n (64- C )/a
n N e
c
(3.10)
(3.11)
we find^
( -6;)/kT
\ (Nc/g) e
(3.12)
thus, apart from the spin degeneracy g, n^ is the number of electrons
that would be in the conduction band if the Fermi level coincided
with t< We now write (3.6) in the form
n
t
N
t
1 + n^/n
(3.13)
For n we have the quasi-Fermi level description n N£
exp[( £ p(x)- ^*c)/kT]. If the quasi-Fermi level remains below the
trapping level, which happens prior to the trap-filling regime, see Fig.
III-4, regions III and IV, we have n^/n 1, and (3.13) yields
n.
p = ~ a = = constant 1 (3.14a)
n + n n_ IT v '
If, on the other hand, the quasi-Fermi level is above t, as in
Fig. III-4, regions I and II, n^/n 1, and n This occurs
in the trap-filling and asymptotic (Mott-Gurney) regimes. For the
latter regime


Ill
noise source term only is shown by the dot-dashed line for trap level
two. The current noise spectral densities for the low-voltage quadratic
regime and trap-filling regime (TFL) are shown in Fig. VI-3 and
Fig. VI-4. The importance of diffusion in the noise source term
decreases as the biasing voltage increases, as can be seen in these
figures.
The parameters of the trap levels observed in the noise spectral
density are given in Table VI-1. The capture cross sections a are
within the normal range for neutral or negatively charged single ion
44
trap centers.
The time constants obtained from the computer calculation are in
good agreement with the measured values. The difference between the
measured and the calculated values of the noise magnitudes, however,
increases with the bias voltage. The experimental circumstances may
explain this discrepancy. The noise measurement is done under
continuous bias conditions, and therefore at high electric fields the
Joule heating of the device may create changes in the electrical charac
teristics. Specially in the low-voltage quadratic and trap-filling
regimes, small changes in the voltage can create a large change in the
current level. In measuring the DC characteristics and the impedance of
a-SiC, this problem was avoided by using a pulsed bias system.
To obtain a better understanding of the variation of the noise with
bias for the different regimes of the I-V characteristics, a plot of the
trap time constant x vs. Vq is shown in Fig. VI-5, where
1/x = g[Nt nfc + n1 + nQ].
(3.1)


49
n (rr3)
T(K)
Fig. III-9. Equilibrium free carrier density n vs. 1000/T.


(e_LU)(X)U
123
Fig. VII-4. Carrier concentration profile at T = 77K. Solid line:
ohmic regime I = 45pA. Dashed line: SCL regime I = .35A.


158
C
n
w
C
C
c
c
c
c
c
CALCULATE THE FIRST MULTI PLICATION OF MATRIX AND NOISE
DO 10 1=2.49
DO 20 J =2.49
DO 30 M=2,49
XNDR < I J ) 7. < I M ; *S I < M, J ) -f-XNDR < I, J )
XND i l X i ~Z(I > M) *XAK < M. J ) + XNDI < I J )
XNDI 2 ( I J ) =-!.:! IN A 11: i ; a, -*C.A i ( Z ( J M ) 5 + HND12 ( I J )
X N D13 ( I J ) = Z (I M) C < M J 5 + X N D13 < I J )
XND14(I,J ) = XKAC (I> M ) *CONJG < Z(J,M))+XND14 ( I > J )
XND15(I.J)=Z(I.M)*CDK(M>J)+XND15 XND 16 ( I, J) CONJQ(C( I.M) ) *CO.N JG (Z ( J > M) >+XNDJ6( I, J)
XND17(I,J)=Z(I> M >*CK< M,J)+XND17{I.J)
30 CONTINUE
20 CONTINUE
10 CONTINUE
CALCULATE THE NOISE FOR EACH TERM
XN0ISE1=0,
XNOISE2*0.
XNOISE3=0.
XNOISE4=0.
DO 40 1=3.48
DO 50 J =3.48
DO 60 M=3.4S
XNO ISE =XNO ISE1+XNDR ( I > M ) *CON J3 (Z < J > M.) >
XNO I Hi- XNt : UE2+XND11 ( I M) ( XND 12 < M J ) -XND 12 (M-1 J ) > /HZ
XNOISE3*XNOI8E3+(XND13 )*XND14 < M,J)/HZ
XNOISE4 *XNOISE4+(XND13I.M)-XND13 )*XND16 < M,J >/HZ-
*{XND17(I,M+1)-2.*XND17(1,Ml+XND17(1,M- ) )*XNB16(M, J)/(KZ*HZ)
60 CONTINUE
50 CONTINUE
40 CONTINUE
121 FORMAT(5X 2E11.4.5X.?E11.4)
FREQ = GMEOA/< 2.*3.14)
Z1 = ( CABS < 8UMZ >)**2.
CALCLATE THE CURRENT NOISE SPECTRUM
XNOISEXNOISEl/Zi
12S FORMAT(5X4(5X.2E11.4))
XNOISEX*(XNOISE2+XN0ISE3+XN0ISE4>/2l
XNOISE*XNOISEi+XNCI SEX
NRITE(7 > 3015 FREQ,21,XNOI SE1,XNCISEX,XNOISE
301 FORMAT <5X,"FRES*.E11.4
*2E -4. fX OR:.- * 2E11
GM£GA=CMEGA*2.
ISO CONTINUE
STOP
END
5X,"IMPED*".Ell
. 4.5X,"7CTAL=".2E
4 5X 11 DIF-
11. 4 5


Fig. VII-6. Monte Carlo simulation of the velocity-field
characteristic for impurity concentration of
10^m-* at T = 77K. From J.G. Ruch and
W. Fawcett, reference 49, Fig. 5.
126


36
dE
VE -7
~ ~ dx
q
ee.
(n + n
Nt -
K +
V
(3.6b)
where N is the number of ionized donors, N. is the number of ionized
U A
acceptors, Nfc is the number of traps, n is the number of conduction
electrons, and nfc is the number of trapped electrons. In the unexcited
specimen, i.e., in thermal equilibrium, denoted by a superscript
zero, we have for acceptor type traps n^ + n^ = N* and for
donor type traps n + n^ = N* N. + N Thus (3.6a) and (3.6b)
read also
dE
dx
where n
equation is
[(n-n) + (nt-nj)] - (n+VV (3.7)
n + n is a constant charge. The trapping balance
ynt gn(Nt-nt) ,
(3.8)
where g is a capture constant and y is an emission constant. We make
all rates quasi-bimolecular by writing y = gn^ ^ where n-^ is the
Shockley-Read quantity^ as we will see shortly. Thus (3.8) yields
n.n n(N -n ) or
It t t
(3.9)
which a fortiori holds with superscripts zero added.


161
31. K.M. van Vliet and J.R. Fassett, in Fluctuation Phenomena in Solids
(ed. R.E. Burgess), Academic Press, New York (1965), p. 267, in
particular Section IV, C3.
32. W. Schokley and W.T. Read, "Statics of the recombinations of holes
and electrons, Phys. Rev. 87, 835 (1952).
33. N.F. Mott and R.W. Gurney, Electronic Processes in Ionic Crystals,
Oxford Univ. Press, London and New York (1940).
34. R.J.J. Zijlstra and F. Driedonks, "Theory of trapping noise of
solid stat single injection diodes, Physica 50, 331 (1970).
35. K.M. van Vliet and J.R. Fassett, in Fluctuation Phenomena in Solids
(ed. R.E. Burgess), Academic Press, New York and London (1965), p.
267.
36. A.D. van Kheenen, G. Bosman and C.M. Van Vliet, "Decomposition of
generation-recombination noise spectra in separate Lorentzians,
Solid State Electronics, in press.
37. A. van der Ziel, "Thermal noise in space-charge-limited diodes,"
Solid State Electr. 9^, 8799 (1966).
38. A. Rigaud, M.A. Nicolet and M. Save Hi, "Noise calculation by the
impedance-field method: Application to single injection," Phys.
Stat. Solidi (a) 18, 531 (1973).
39. K.M. van Vliet, A. Friedmann, R.J.J. Zijlstra, A. Gisolf and A. van
der Ziel, "Noise in single injection diodes I. Survey of methods,"
J. Appl. Phys. 46, 1804 (1975).
40. A. Rigaud and M. Savelli, in Le Bruit de Fond des Composants Actifs
Semiconductors, Colloques Internationaux du Centre National de la
Recherche Scientifique, Toulouse (1971), p. 277.
41. R.L. Burden, J.D. Fairs and A.C. Reynolds, Numerical Analysis, FWS
Publishers, Boston (1981).
42. R.E. Bank, D.J. Rose and W. Fichtner, "Numerical methods for
semiconductor device simulation, IEEE Electron Devices ED-30, 9,
1031 (1983) .
43. A.G. Milnes, Deep Impurities in Semiconductors, Wiley-Interscience,
New York (1973).
44. G.A. Lomakin, in Silicon Carbide-1973 (eds. R.C. Marshall, J.W.
Faust, Jr. and C.E. Ryan), Univ. of South Carolina Press, Columbia,
S.C. (1974).
45. R.R. Schmidt, "Noise and current-voltage characteristics of near-
ballistic GaAs devices," Ph.D. dissertation, University of Florida
(1983).


CHAPTER VI
COMPUTER CALCULATION OF THE SCL IMPEDANCE AND NOISE OF a-SiC
1. Introduction
Expressions for the Impedance and the noise of a two-terminal
device, operating under conditions of SCL flow, were derived by Van
SO
Vliet et al., employing the transfer impedance method. Neglecting the
diffusion term in the current equation, closed analytical expressions
were obtained for the first two regimes, i.e., the ohmic and the low-
voltage quadratic regime.
From our DC computer calculations, however, we conclude that
diffusion plays an important role, specifically at low-voltage bias
levels. Therefore, we include the diffusion term in our present study
and apply the transfer impedance method to obtain values for the
impedance and the noise in the ohmic regime and three SCL regimes.
The basic equations describing charge transport are the same as the
ones we used in the preceding chapters to explain the current-voltage
characteristics. For easy reference these equations are repeated here:
J(x,t) qunE + qD + H(x,t)
(1.1)
dE
d^ - (1.2)
- 6Vt to(Nt -%>-? + t1-3
3n (x,t)
= -fr^n. + Sn(Nt nfc) Y(x,t),
(1.4)
101


CHAPTER VII
COMPUTER CALCULATIONS OF THE CURRENT-VOLTAGE AND NOISE
CHARACTERISTICS OF SUBMICRON n+ pn+ GaAs DEVICES
1. Introduction
Advances in semiconductor device technology, such as MBE, have made
it possible to fabricate devices with thin layer thicknesses. If the
layer thicknesses are smaller than the mean free path of the carriers,
under favorable conditions carriers may cross the layers undergoing few
or no collisions. As a result of this so-called ballistic or near-
ballistic transport, very high-speed devices can be developed.
The I-V and noise measurements on submicron n+nn+ GaAs devices
indicate that ballistic or near-ballistic transport can occur.^-*47
However, the explanation of the charge transport mechanism in submicron
n+pn+ has been the topic of lengthy discussions^^^-47 an clear.
The exact modeling of submicron devices requires a combination of a
Monte Carlo simulation and a computer program similar to the ones
described in the preceding chapters. The Monte Carlo simulation
provides information about the current due to ballistic and near-
ballistic electrons, and the number of electrons in the central and
satellite valleys at each point in the device. The electric field and
energy band profile is then obtained from a computer program similar to
the one described in chapter V.
To shed some light on the possible mechanism that can control the
charge transport in the submicron n pn and p np GaAs devices, the
117


V. COMPUTER CALCULATION OF DC SCL FLOW IN ct-SiC 80
5.1 Introduction 80
5.2 Theoretical Model 81
5.3 Computer Simulation 82
5.4 Theoretical Results 87
VICOMPUTER CALCULATION OF THE SCL IMPEDANCE
AND NOISE OF o-SiC 101
6.1 Introduction 101
6.2 Theoretical Model 102
6.3 Theoretical Results 106
VIICOMPUTER CALCULATIONS OF THE CURRENT-VOLTAGE AND THE NOISE
CHARACTERISTICS OF SUBMICRON n+pn+ GaAs DEVICES 117
7.1 Introduction 117
7.2 I-V Characteristics 118
7.3 Noise Characteristics 125
VIII CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE STUDY 132
APPENDIX A COMPUTER PROGRAM FOR TEK 7D20 NOISE
SPECTRUM ANALYZER 134
APPENDIX B COMPUTER PROGRAM TO CALCULATE THE CURRENT-
VOLTAGE, THE IMPEDANCE, AND THE NOISE
CHARACTERISTICS 151
REFERENCES 159
BIOGRAPHICAL SKETCH 163
iv


106
S
dK (x'")
Y
*",ja>)C(x)KY(x*)].
c*(x' ')Z*(x* ,x''' ja>) }.
(2.18)
The total noise is given by
S5v(L) = // dxdx' S
AE(x)AE(x)
(2.19)
where ^,E(x)AE(x) is comPose<* e9s* (2*16), (2.17) and (2.18). The
current noise spectral is then
(2.20)
3. Theoretical results
The results of the impedance calculation are shown in Fig. VI-1.
The computer calculation includes four trap levels with activation
energies of 120 meV, 103 meV, 85 meV, and 67 meV, as shown in Table
VI-1. The second trap level (Eact = 103 meV) is the one that is used to
explain the I-V characteristics. The low-bias impedance calculation
using only this trap is shown by the dot-dashed line. The low-frequency
magnitude of this single trap is in good agreement with the measurement.
However, there is a break-point frequency corresponding to the detrap
ping time constant t^. The break point is due to the following term
in eq. (2.4) :


CHAPTER III
CURRENT-VOLTAGE AND IMPEDANCE CHARACTERISTICS
OF SCL FLOW IN ot-SiC
1. Introduction
We report here on the electrical properties of samples diced from a
nitrogen-doped a-SiC crystal, being light in color and transparent. The
crystal had the form of a thin hexagonal platelet, *7 mm across and 1 mm
thick. It had well-developed crystal faces normal to the c-axis, but at
the edge of the crystal the lamellar structure due to the layering of
different polytypes (section 2) was clearly discernible. Samples were
diced in the shape of a rectangular bar.
As it turned out, the current-voltage characteristics were
indicative of single carrier space-charge-limited (SCL) flow. In fact,
it is shown that this material provides a very good example of SCL flow,
governed by various types of shallow traps. No prior measurements on
n-type SiC of SCL currents have been reported in the literature to our
knowledge, though limited data on p-type SiC exist (Ozarov and Hysell
and English and Drews^). Their results are summarized in Lampert and
Mark's excellent monograph on the subject of injection currents.^ The
SCL flow in n-type a-SiC results from injection of electrons into a
compensated large bandgap polytype, sandwiched between higher
conductivity polytypes (for details see Section 2).
Chapter IV concerns the electrical noise of the same samples. It
is shown there that our devices were "semiconductor-trapping devices"
(terminology of reference 30), x < t where xn is the dielectric
06 X
23


I (amps)
46
V (volts)
Fig. III-7. I-V characteristic at 77K.
ro


n n n ri n n n n n n n n n n n n n a n n n
152
FUS > NPNNP
: COMPUTER SIMULATION FOR N+ N N+ GaAs AND SiC .
i
REAL*12 DF2 V < 100),F2,DF<100>.F(100),ZZI>AN,AN2,C<100),CC
REA; QKT 2E. EACT A ,TX > XI> 02,H,ET, XN < 00 ) > XNT ( 100 ) DF1 F1
REAL* 12 MU.NC.NP.NT'.XNE.ZI ,DC( ICO)
SET THE GENERAL TRANSPORT PARAMETERS
ZB=TOTAL LENGHT OF N+N N+
N=TOTAL # OF POINTS
NP-CARRIER CONCENTRATION IN N+ REGION
MU-MOBILITY
NCEFFECTIVE DENSITY OF STATE
A-DEVICE AREA
EACT= TRAP ACTIVATION ENERGY
NT-TRAP DENSITY
T-DEVICE TEMPERATURE
GKT-ELECTRON CHARGE< =1) /BQLTZMAN CONST. H=DISTANCE BETWEEN TWO DISCRETIZED POINT
G2-ELECTR0N CHAGE<*>/DEVICE DIELECTRIC CONSTANT(EPSI*EPSIO)
TX-CQNERGANCE ACCURACY
XI-DEVICE CURRENT
Ml-TOTAL # OF CALCULATION IF CONVERSANCE IS NOT ACHIEVED.
ZB-2.667E-5
N = GO
NP-5.0E21
MU-.0106
NC-3.92E24
A-4.7E-6
EACT. 103
NT-5.07E+1S
GKT-150.56
02=1.77E-9
H-ZB/N
TX-5E-4
XI-2.2E-5
Ml = 200
E': -EXP(EACT*GKT)
Nl-N-1
DO 50 11-1,1
SET THE BOUNDARY CONDITIONS
C<1)*0.
F{1) = (1/QKT)*GLGG V < 1 > = 2 V IS**- ( y/< TV F W ))
CC'- ( XI *H ) / ( 1 ,6E-19*A*frfU*NCi


56
Fig. III-12. Apparent capacitance as a function of voltage. (The
parasitic capacitance of wires and connectors outside the
cryostat are subtracted.)


68
fCHz)
Fig. IV-4. Current noise spectral density in ohmic regime for T =* 77K
and T = 62.5K. Circles o and : measured data. Full
lines: resolution into Lorentzians.


74
These three regimes are covered by these noise measurements. Typical
results are given in Figs. IV-7 IV-9. The time constants involved in
all spectra are nearly the same, see Fig. IV-10. The slight decrease
with current might be due to the modulation by the function $.
As to the magnitude, we note that the noise at 1 Hz is of order
101-8 in the ohmic regime, of order 10^ in the low-voltage quadratic
regime, while it goes down again to 10^8 in the TFL regime. Such a
large variation seems at first hand unexplicable. However, the
normalizing factor for all regimes is Iq|Vq| see e modified by the factor /ng of eq. (2.18). Thus, in Fig. IV-11 we
plotted S^.j.R/Iq|Vq| VS *0* w^ere R is the ohmic resistance of the
unexcited specimen, for the four Lorentzians involved. In the first
two regimes the normalized noise is constant, i.e., S^ Iq|Vq|,
with /ng <* N^/n^ = constant, in excellent agreement with the
theory of Section 2. In the TFL regime there is a sharp drop-off, since
n ^ so that /n^ 0 see eq. (2.18). Thus the behavior
plotted in Fig. IV-11 is in most respects as expected.


21
Fig* II-7. Equivalent input noise scheme


Table IV-1
Activation energies and capture cross sections of different
trap
levels observed
In the noise
measurements
c" t
(meV)
302
281
135
121
98 81.5
63
2
o(cm )
6.62xl015
9.8xl0-15
2.40xl018
1.27xl0-17
2.5xl0-17 3.05xl0-17
4.87xl0-17
Table IV-2
Comparison between the trap activation energies as found
from the I-V curves (Table III-4) and noise
(f-V)* 361 274 223 167 130 103 81 67
meV
(noisef 302 281 135 121 98 81.5 63
meV


124
Fig. VII-5. Electric field profile in the ohmic regime (I = 45 uA) at
T = 77K.


142
5960
5980 GOSUB ¡5800
6000 NEXT 1:0
6020 G$=""
6040 GO TO 7960
6060 REM TAPE DIRECTOR2
6080 PRINT J_ TAPE FILES"
6100 PRINT "^FileV Contents J_"
6120 FOR I=F TO F+9
6140 FIND I
6160 X=TYP(0)
6180 GOSUB X+ OF 6260,0320,6380,6460,6520
6200 NEXT I
6220 RETURN
6240 REM TYP<0)=0
6260 PRINT I,""
6230 RETURN
6300 REM TYP(0)=i
6320 PRINT I,"";G*
6340 RETURN
6360 REM TYP(3)=2
6330 INPUT S33:A$
6400 PRINT I,A*
6420 RETURN
6440 REM TYP<0)=3
6460 PRINT I,"";G$
6430 RETURN
6500 REM TYP(0)=4
6520 READ 333:A$
6540 PRINT I,A$
6560 RETURN
6530 REM FILE NUMBER INPUT
6600 PRINT J_Enter file number (0 to exit/";F4;",;F4+Ni-i;"): ";G$;
6620 INPUT F
6640 IF F=0 THEN 6730
6660 IF FF4+Nl THEN 6600
6630 FIND F
6700 IF TYF(0)=4 THEN 6760
6720 PRINT "INVALID FILE SELECT ANOTHER OR ENTER ""0"": ";G*
6740 GO TO 6620
6760 READ Q33:A$,A0,Ai,A3,A6,A2,A5,A4,V$,H$,A,Ii0,Bw
6780 RETURN
6800 11 = 1
6820 PAGE
6840 Al=129
6860 DIM A(Al),Ddz(Al)
6330 I2=A1
6900 KK0=Ii0*12
6920 A2=A3
6940 PRINT "NUMBER OF AVE =,KK0
6960 FRINT "NOISE BW= ",Bw
6930 REM CALCULATE THE MAXIMUM AND THE MINIMUM
7000 CALL MIN",A,Mi,LI


86
kl r,XcW,d r rqXcM,i.
J expl J -T- [exp ( r J Jdx
q (K-l)Ax ax xi
-qX KAx qx (x)
exp(-j~) / [ exp(£- )jdx
L K' (K-l)Ax ax Ki
(3.14)
or
-qXc(K)
exp[ 0 J
XC(K) XC(K-1)
rqXcW1 rqXcCK-l)
6XP * kTJ XpL Zr J
kT
(3.15)
A similar approach gives us an expression for
qX (K+l)'
exp( ^ ) .
The Jacobians of and F2 are obtained after the discretization,
yielding a system of nonlinear equations. Assuming that the Jacobians
are defined, we can write
3XC %
-P^.U* + U*1)
(3.16)
!!i roK(+l) _F2(xK + SxK(-l)joK)
where we solve for and ^(ro+l) with K denoting the
c
Newton step and m being the iteration step. Since the Jacobian of F-^
and F^ both have the same denominator as given by eq. (3.8), the actual


o
T.
15.12 A
i
ABCABCA
6H
Fig. III-l. Continued.
ABCABCABCA
I5R
N5
(b)


53
the ohmic and low-voltage quadratic regimes:
ZL r(I)F3(x,e,a)/F3(l,0,a)
(5.1)
where
F3(X9.a)
\ vdv \ du fl/u-cr/^ fl-au^*
0 (1-av)2 v ll~
(5.2)
Here r(I) is the low-frequency differential resistance d|v|/dl,
0 = jwx^, a as in Section 3, while x 3 (l+jwx^/il+jwx),
t and x^ being trapping-detrapping times. In the ohmic regime, (5.1)
can be shown to lead to the simple result,
L_ 1
s0 1 + j)Ta
(5.3)
Note that
ee
£e0A L
qnli L
qnyA
RC ,
so we have the usual result
R
JL 1 + juRC
In the SCL limit, a-H) eq, (5.1) reduces to
(5.4)
(5.5)
\ = r(I)F4(x,5)/F4(l,0) ,
(5.6)


157
C
C
FIND THE ELEMENTS OF Y-MATRIX
ZKAPA()=ZKAPA(2)
DO 850 K-2,49
Y {K K 1 > = DENO (K ) -C;;: i ZK A P A < K ) +C DIFF* < ZK A PA (K )
Y(K K ) --DIEL(K)-DENO < K)+2.*CDIFF*ZKAPA < K)-CD IFF
-2KAPA(K-15
*(ZKAPA(K)
1-ZKAPA
Y (K i K +1 C.'
50 CONTINUE
C FIND THE INVERSE GF Y-MATRIX (Z)
C
Z (2 > 2 ) = 1 / Y ( 2 ? 2 )
Y < 2 7 3) = Y (2,3 > / Y (2 2)
Y (2 2} 1 .
Z < 3,2)=-2 < 2,2 > *Y < 3,2 5 +Z < 3,2 >
Y(3.2 > = -Y2,2)*Y(3,2)+Y < 3,2)
Y(3 3)=-Y(2,3)*Y(3,2)+Y<3,3 >
DO 550 K=349
Z t t K ) s i
DIV = Y < K +1 K )
Y (K X -1 ) = Y (K K i ) / Y {K K )
Y(K K +1>- Y iX > X>ii/v K > K )
DIV2=Y(K,K)
Y < K K ) .
DO 750 J = 2,K + i
2 (K J >-:? { X > J ) DIV2
Z < X +1 J > --Z < K J>*DIV+Z J)
Y(K +1J)=-Y(K,J> #DIV+Y£X + > J)
750 CONTINUE
550 CONTINUE
DO 250 J = 2 > 48
K-50-J
DIVY(K/K+l)
Y< K K-*- i ) Y : X f I > X +1 > *DIV+Y ( X > K + )
DO 360 1=2,49
2 GAO CONTINUE
250 CONTINUE
C
C CALCULATE THE TOTAL IMPEDANCE
C
3UMZ=0.
DO 800 1=2,49
DO 900 J-2,49
Z< I ' : J)*HZ/A
aUMZ*SUMZ+Z 900 CONTINUE
800 CONTINUE


xml record header identifier oai:www.uflib.ufl.edu.ufdc:UF0008244000001datestamp 2009-02-09setSpec [UFDC_OAI_SET]metadata oai_dc:dc xmlns:oai_dc http:www.openarchives.orgOAI2.0oai_dc xmlns:dc http:purl.orgdcelements1.1 xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.openarchives.orgOAI2.0oai_dc.xsd dc:title Properties of noise and charge transport in layered electronic materialsdc:creator Tehrani-Nikoo, Saieddc:publisher Saied Tehrani-Nikoodc:date 1985dc:type Bookdc:identifier http://www.uflib.ufl.edu/ufdc/?b=UF00082440&v=0000114632125 (oclc)000873795 (alephbibnum)dc:source University of Floridadc:language English


FIELD(kV/cm)
99
TFL Regime
1=1.9X10 ~3A
Fig. V-ll. Electric field profile in the trap-filling regime.


1(A)
98
TFL Regime
I = 1.9 x 10" 3 A
L(/x.m)
Fig. V-10. Current diagram in the trap-filling regime at I = 77K. The
solid line represents the drift current, the dashed line the
diffusion current. The sign conventions are given at the
bottom of the regime.




APPENDIX B
COMPUTER PROGRAM TO CALCULATE THE CURRENT-VOLTAGE
THE IMPEDANCE, AND THE NOISE CHARACTERISTICS
151


88
V(v)
Fig. V-l. I-V characteristics at T 77K. The dots indicate the
measurements. The solid line represents the results of
computer calculations.


92
field profile, Fig. V-5, shows a large variation at n n contacts due to
an abrupt change in carrier concentration.
Low-voltage quadratic regime. As the biasing voltage increases,
t
the potential minimum shifts towards the n+ cathode contact, Fig. V-6.
The Fermi level lies below the trap level for most of the device. Con
sequently, the traps are mainly empty (Fig. V-7). The width of the
section in the n region dominated by the drift current increases, and
the current becomes space charge limited and proportional to the square
of the voltage.
Trap-filling regime. In this regime the quasi-Fermi level passes
through the trap level as is indicated in Fig. V-8. The traps are being
filled (Fig. V-9), and the I-V characteristic shows a fast rise in the
current. The drift current dominates current flow in the major part of
the n region (Fig. V-10).
The field profile is negative in most of the n region, indicating
that drift is dominating the device. Large variations in the electric
J.
field occur at n n contacts due to sudden change in carrier
concentration, Fig. V-ll.
High-voltage Mott-Gumey regime. In this regime the Fermi level
lies completely above the trap level, and almost all of the traps are
filled (Fig. V-12). Drift dominates the sample, and the I-V
characteristic shows the quadratic dependence between current and
voltage (IaV2).


59
measurements to be done under pulsed conditions. In this regime there
should be only thermal noise of magnitude 8 kT/ReY.
2. Review of the theory of trapping noise in SCL flow
Noise in SCL flow was first computed by van der Ziel,^, further by
Zijlstra and Driedonks,^ Rigaud, Nicolet and Savelli'^ and by Van
Vliet, Friedmann, Zijlstra, Gisolf, and van der Ziel^ in an approach
OQ
based on the transfer impedance method We review here the results of
reference 30.
The basic equations given in reference 30 are the same as those of
chapter III, section 3. However, we need DC, AC, as well as Langevin
equations, so the full time dependent equations including noise sources
are needed. These equations are
J(x,t) = qynE + H(x,t)
(2.1)
dE
di=" (q/eV (n + nt" V
(2.2)
8n(x,t)
Bn. n Bn(N -n ) -
1 3J(x,t)
3t Mlt Kv"t **t' q 3x + Y(x,t)
(2.3)
(x,t)
^ Bn^ + Bn(Nt-nt) Y(x,t)
(2.4)
Here n is the free electron density, nt the trapped density, nc is
equilibrium charge, n-^ is the Shockley-Read parameter, B is a capture
constant, H is the thermal noise source, and y is the trapping noise
source. The other symbols have their usual meaning. Since the total
current (including displacement current) J =* J + eGgSE/at is
solenoidal, eq. (2.3) is found to be redundant and can be dismissed.


61
and the impedance follows from
L L
ZT = / / dx dx' z(x,x',w) ; (2.11)
h 0 0
the current noise spectrum is then Sav(L)^Z^2 Tlie fo3-lowinS
result, valid in both the ohmic and low-voltage quadratic regimes, was
obtained after a rather involved analysis
SAI(o>) 4qpl |VJc* (X,0,a) (2-12)
L2T^ 1 + (i)2X2
where 9 jtox^ a = g |Eq(L)|/Iq (same parameter as in Part I,
Section 3, see eq. (3.19)), while $ is normalized to be unity at
sufficiently low frequencies. It is given by
F2(X,e,a)/F2(l,0,a)
F3(X,9,)|2/|(F3(l,0,a)
with
(2.13)
(x.e.) =
F2(x?a)
j v2dv
0 (1-av)3
f H. f^-/u ~ X)/X(l
x ^1/v cr '1
au
0/X
av'
(2.14)
F3(x,9,a)
f vdv \ du <-l/u X^Xfl au^^X
L v 9 X ^1/v a-* ^1 av'
0 (l-av)^ v A
(2.15)
In the ohmic regime $ is flat up to 1 However, in the SCL
regime $ is complex. For an "insulator-trapping device" (1/x^ 1/x)
the function $ rises beyond o>^ = 1/ x^ to a new plateau which is 20/9
times the low-frequency plateau. It then falls off beyond


29
polytypes. We note that all of these polytypes have indirect
bandgaps. The maximum of the valence band is located at the F point
whereas the minimum of the conduction band is near the zone edge for all
polytypes.^2-25
2.2. Transport properties. Barrett and Campbell2** measured the
mobility perpendicular to the c-axis for n-type 6H, 15R, and 4H.
Ionized impurity scattering was evident at temperatures up to 300K, and
a T-2*^ dependence was found between 300K and 800K. This was attributed
to acoustic and intervalley scattering. A relationship of the form
tigH ^ ^lSR ^ y4H ^ ^30 was fun<* An extensive review paper on
mobility in SiC was given by Van Daal.2^ He postulated that polar
scattering was the dominant process. A successful fit to the mobility
data indicated a hole effective mass of between 3.0 niQ and 4.4 mg, while \
a value of (1.0 0.2)mQ was found for the electron effective mass.
A comparative analysis of polytype dependent transport properties
was also presented by Lomakina.2 In general, the electrical properties
of nrtype SiC were found to be polytype dependent. Electron mobility,
ionization energy of donors (nitrogen), and effective mass anisotropy
are summarized in Table III-3. These data were obtained with the
nitrogen concentration adjusted to 6 x 10 cm-J and at room
temperature. The ionization energies of the nitrogen impurity in other
polytypes are 30 meV for 27R, 35 meV for 10H, 40 meV for 21R and 60 meV
for 330R.
The temperature dependences of the conductivity in equally doped p-
type 4H, 6H, and 15R were all the same. This implies that the valence
band of SiC is not dependent on polytypic structure.


136
3840 V$="DBV/SQR(HZ)"
3860 Hl="HZ"
3S80 GOSUB 6800
3900 NEXT 0
3920 G$=""
3940 GO TO 7960
390 REM AVERAGING NOISE SPECTRUM
3980 PAGE
4000 PRINT VAVERAGING THE NOISE SPECTRUM"
4020 PRINT J_V CHOOSE APROPRIATE FILTER AND TIM/DIV SETTING"
4040 PRINT "J
4060 PRINT "J.
4080 PRINT "J
4100 PRINT "J
4120 PRINT "J_",
4140 PRINT "J_",
4160 PRINT "J_",
4180 U=4
TIM/DIV
1 SEC
50 mSEC
5 mSEC
500 mmSEC
20 MMSEC
2 MMSEC
FILTER "
50 Hz "
1 KHz"
10 KHz"
100 KHz "
2 MHz"
20 MHz"
4200 PRINT "JJVNUMBER OF AVERAGES REQUIRED (*12)?"
4220 INPUT Nave
4240 REM FIND THE EFFECTIVE NOISE BANDWIDTH
4260 REM TDIV IS TIM/DIV SETTING.
4280 PRINT 34:"HOR?Tr
4300 INPUT '34:Tdiv
4320 REM THE TIME WINDOW FOR 256 POINTS OUT OF 1024 SAMPLE POINTS
4340 REM IS EQUAL TO 2.5* 4360 Bw=l/(Tdiv*2.5)
4380 Nave=INT(Nave>
4400 DIM A<1024),A?(9),E5<129),C5(i29>
4420 REM READ THEN Sampled DATA From THE Osc IN BINARY FORMAT.
4440 PRINT 3U:"DA ME:1,SNC:BININTE:0FF"
4460 PRINT 3U:"WFM?"
4480 REM A0 IS THE X INCREMENTS IS THE NUMBER OF POINTS.
4500 REM A3 IS THE X MULTIPLIERS IS THE Y-AXIS ZERO
4520 REM A2 IS THE INTERPOLATION FLAG.A5 IS THE X-AXIS UNIT.
4548 REM A4 IS THE Y-AXIS UNIT.
4560 INPUT 3U:A0,AfA3,A,A2,A5,A4
4580 C5=0
4600 PAGE
4620 REM CHECK FOR EXTENDED REAL-TIME DIGITIZING(ERD) MODS.
4640 IF AK1024 THEN 22000
4668 FOR 110=1 TO Nave
4680 B5=8
4700 A=0
4720 FOR 1=1 TO 3
4740 PRINT SU:"WFM?"
4768 INPUT SU:A0A1 >A3A6A2A51A4
4780 REM THE DATA IS READ INTO MEMORY SPACE A.
4800 DIM A(1024),A7<9)
4820 PRINT OU'."CURVE?"
4840 WEYTE 3U+64:
4860 REM A7 IS TWO BYTES REPRESENTING NUMBER OF DATA POINTS*!
4888 REM A IS 3-BIT BYTE OF DATA


102
where the symbols have their usual meaning. Since the total current
gg
J = J + ee is solenoidal, eq. (1.3) is found to be redundant and
L U Ot
can be dismissed. The remaining equations are split into AC and DC
parts. In this chapter we solve for the AC part to obtain the impedance
and the noise.
While the DC characteristics could be explained by a single trap
level, the impedance and the noise measurements show clearly the
presence of more than one trap level a few kT away from the Fermi
level. Hence, in the following AC and noise analysis we consider
multiple trap levels.
2. Theoretical model
The AC part with the Langevin noise terms included becomes, after
linearization,
-AI
A
dAE
dx
d(PEQ) dAn
qnQ jg AE + qyEQAn + qD + H(x,t)
0
(An + l An )
i
(2.1)
(2.2)
3An
ti
~3t SinlAnti + 0iAn(Nti
nti) e.nAnti Y^x.t),
(2.3)
where subscript i indicates different trap levels.. Equations (2.2) -
(2.3)are used to express An and An in terms of AE. Then, after
ti
a Fourier transformation the equation for the current AI reads
iL qn iE ( + 4 )(0 w .[! + l ]-l)
A H 0 dEg 0 dxJV q dx L £ ^'X2i + '
+ + -1 i/tJ\ 3)Cl + l 1/.IV1 + +
(2.4)


64
1000/T the curve is linear up to a low temperature limit, where the
Fermi level becomes as close as 4 kT below the trap level; thereafter,
must level off.
3. Experimental results in the ohmic regime
We refer to the geometry of Fig. III-2. In all cases the current
was passed through contacts 1 and 4 while the noise was measured between
contacts 2 and 3, in order to avoid contact noise. Noise spectra were
measured in the range 1 Hz 1 MHz for T = 62.5K 300K. Typical data
are shown in Figs. IV-1 IV-4. Fig. IV-1 has spectra at 296K and 250K,
Fig. IV-2 lists data for 200K and 175K, Fig. IV-3 lists data for 125K
and 100K, while Fig. IV-4 lists data for 77K and 62.5K. All spectra
show three to five Lorentzians. There are probably other Lorentzians at
lower frequencies. We note that the 77K and 62.5K curves show a sharp
roll-off above 5 KHz (77K) or above 1 KHz (62.5K), with no indication of
other Lorentzians coming in. Then these roll-offs give the smallest
lifetimes for these temperatures, indicative of the shallowest traps
that occur in this sample. In the ohmic regime the noise was found to
be proportional to 1^ in all cases.
From the data we determined the time constants involved from the
best fit. These x.js are plotted vs lOOO'/T on a semilog scale in
Fig. IV-5, while in Fig. IV-6 we plotted the plateau values normalized
by the current squared vs 1000/T, also on a semilog scale. Points
pertaining to the same Lorentzian are connected by straight lines. We
notice that a number of nonconnected points occur for temperatures above
200K. These belong to other Lorentzians, which can only be fully
determined if measurements far above 300K were made.


87
location of (K+l)' and K' is unimportant and does not affect the value
of c
To obtain a higher convergence rate, Poissons equation is solved
for the electrostatic potential until a total convergence is obtained.
The updated values of X£(K) are then substituted into the continuity
equation with the U(K) being updated for convergence. The new values
are then substituted back into Poisson's equation, and the procedure
continues until a full convergence of the electrostatic potential and
the quasi-Fermi potential has been obtained.
4. Theoretical Results
Fig. V-l shows that the computer simulation program explains the
measured I-V characteristic at 77K very well. The simulation program
includes only one trap level at this temperature.
The carrier concentrations of the n* regions are assumed to be
21 -3
approximately equal to 5 x 10 m This value is in the range of
carrier concentrations measured for various polytypes of a-SiC at
T 77K.^ The other transport parameters are taken from Table III-4.
The charge transport mechanism in the four regimes of the I-V
characteristics is discussed below.
Linear regime. The overflow of carriers from the n+ regions into
the n region plays an important role at low-bias voltage levels. This
large overflow of carriers is due to the large gradient in the carrier
concentration at the n+n interface. In the linear regime the traps in
the n- region are mainly empty (Fig. V-2), and the trap energy lies
above the Fermi level for most of the n~ region (Fig. V-3).
The current profile of the sample is shown in Fig. V-4. In this
regime the diffusion plays a dominant role in charge transport. The


47
Fig. III-8. I-V characteristic at 52.6K.


42
g ee0pwAV2/L3
(3.31)
which is the low-voltage quadratic regime. The trap-filled limit is not
contained in this analysis, but the asymptotic high-voltage regime gives
with (3.14b)
I -f eeQpAV2/L3 ,
(3.32)
the famous Mott-Gurney law.'*'*) The transition from the ohmic to the
low-voltage quadratic regime occurs at
v
X 9 peeQ *
from which
(3.33)
p 1.18 x 10"8nL2/eVx (3.34)
3.2. Regional approximation. In the regional approximation method
the device is divided into four zones, as indicated in Fig,. III-4.
Here n^ is the injected free carrier density, n rig. In region I n^
is highest; it decreases in the other regions until in region IV
ni V For details, see Lampert and Mark's book. Poisson's equation
can be considerably simplified for each region, according to the
appropriate carrier densities' approximations. The regions I IV are
called the perfect insulator region, the trap-filling region, the
semiconductor region, and the ohmic region, respectively. The method is
again more lucid by not introducing the dimensionless variables, but


38
Fig. III-4. Schematic energy band, regional approximation diagram for
the problem of SCL currents with a single set of shallow
(above equilibrium Fermi level) traps. After Lampert and
Mark, reference 3, Fig. 4.8.


T(S)
Fig. IV 5. Observed time constants of the Lorentzian
spectra as a function of 1000/T.


62
)2 1/t The roll-off is complex. For a "semiconductor-trapping
device (1/t 1/t^) the function $ has not yet been fully studied.
However, we believe that it will only have a mild structure between
u)^ 1/t and u>2 = 1/t^ fr frequencies beyond this it decreases
monotonically. Detailed computer studies of the integrals (2.14) and
(2.15) are underway.
_e
For our devices, t^ being of order 10 some trapping times are
smaller and some larger than t^ Thus both situations discussed
above can occur. As to the other factors in (2.12), 4' is a number
between 0.9 and 1.0 throughout both regimes. Further, u|Vg|/L2 is of
the order of the transit time t£, see references 30, 34 and 40, while
t/t^ is the statistical factor

T_
T.
Nt \o
T + T
1 2
N n + n. + n.
t tO 1 0
(2.16)
Thus the low-frequency plateau is also ~ 4 qlg (T/t*)(/ng) ,
i.e., modified shot noise. Notice that as long as the traps are empty
and the injection is low /ng is finite. When the traps fill
up, however, /ng 0 As we noted above, if the trapping time
constants t^ are far enough apart, the multiple trap spectrum is
approximately a sum of Lorentzians with appropriate statistical factors,
i.e.,
SAI
4qni0|v0|s'
I
i
1 1 + Ur T
2t2
(x, >0>O
(2.17)
with


n n n n n n n n n
156
CALCULATE the time coastants
DO 150 K-2,50
TAU11 K ) = i / (BETA*(ZNT-XNTZ*.99>)
TAU2 TAU12 TAU22(K 5 = 1./ TAUCK)=7AUi1(K)*7AU2 FIELD(K)=(CZ(K!-CZ < K-1')/HZ
'ERE(K)1.E-19*2MU*XNZ 150 CONTINUE
OMEGA- o
TO ISO ui--i ;>
DO 280 ¡< = 1,30
DO 290 , = i > 5 0
Z(K> J)= 0.
Y < K J ) = 0 ,
XNDRK,J)*0.
XND
::ndi2 j>o.
XND13 K >J)*0.
XND 14 *; 0 .
XND15(K, 0.
XNDI6(K.J)-Q.
XND17 < K J) 0
290 CONTINUE
280 CONTINUE
CALCULATE THE COMPONENT CF THE FOURIER.TRANSFORM ESN,
XKZ DIE-E?SI*£PSIO*OMEGA
C< 1 1 ) - < < 1.6E-19*ZMU/QKT) ( TAU (1 ) /CMPLX < i OMEGA*TAU 1 1 > ) ) >/HZ
DO 350 1=2,50
ZKAPA *1,/ DENO =e?SIO*£PSI*ZMU*FIELD*2KAPA DIEL CALCULATE THE NOISE CURRENT ELEMENTS
XX2*H2*A
AZd ) = < 1 .UE-9*ZMU*FIE;_D( I> *7AU(I ) /CMPLX C i , 0ME3A*7AU d) ))/H2
Cd I) = ( < i SE-1 9*ZMU/QK T ) *7AU ( I ) /CMPLXd ,OMEGA*TAU< I ) ) ) /HZ
XCd ) = Cd d )-Cd-i I- 5
XAK (Id) = (AZd )+XCd >/HZ) *XKZ< I )
SI< : : ; = : +CONJG(XCI)/HZ> )
XKAC(I.I)"MKZ(I)* < CONJGiAZ(I))+CONJG(XC(I)/HZ))
CDK (!.!)= C ( I I) (XKZ < I) -XK2 d-1 ) ) /HZ
CK( I I )=C( I ,I)*XKZ(I)
350 CONTINUE


41
X^S£
[(l-o)in(l-a)
.V
+ a -
(&n(l-a)) ]}
(3.26)
Solving for V and using (3.20) this yields
qncL2 qn L2 (l-a)Jln(l-a) + a y [Jln(l-a)]2
2ee0 e£0 [a + &n(l-a)]2
(3.27)
Equations (3.20) and (3.27) are the full solution for the current and
the voltage, parametrically expressed in a.
We first consider the ohmic limit, a -* 1 Then,
V
I
1M 1 2g
a+1 g
0
[a + &n(l-a)] +
0
(l-a)Xn(l-a) + a y [£n(l-a)]:
a + £n(l-a)
lim
S0 a+1
y a2 + £n(l-a) + a
a + in(l-a)
(3.28)
which is Ohm's law.
For the SCL regime we expand in orders of o. Then (3.20) and
(3.27) yield
1_
I
V
eeoa
2qnc80L
qn Lz ,
c f. I + L.)
( 9 + 3ctj
ee
4qncLzl
3ee0a
(3.29)
(3.30)
Eliminating a between the two expressions and using again (3.16), we
arrive at


133
computer. The computer program described in this dissertation for DC
calculations should be further developed into a two-dimensional program
and applied to three-terminal devices. The effect of degeneracy and
bandgap narrowing should be included also. For small semiconductor
devices the general transport equations do not apply and Monte Carlo
simulations should be used to calculate the electron distribution, while
the electric field and energy band profile are calculated by the
computer program outlined in this dissertation.
The computer calculations of the noise are extremely useful,
especially when analytical expressions become very complicated. The
noise computer program developed in this work should also be expanded to
apply to three-terminal devices. For short devices (submicron devices)
a microscopic model is then required, and the model explained in this
work becomes questionable. An extensive effort should be devoted to the
formation of a method to calculate the noise of submicron devices.


FIELD(k V/cm)
93
Linear Regime
l = 6X 1 0 "7 A
L(ym)
Fig. V-5. Electric field profile in the linear regime (I = (, x 10~^A)
at 77K.


PROPERTIES OF NOISE AND CHARGE TRANSPORT
IN LAYERED ELECTRONIC MATERIALS
By
SAIED TEHRANI-NIKOO
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


CHAPTER I
INTRODUCTION
Recent progress in crystal growth techniques (MBE, MOCVD) has made
it possible to fabricate multilayered structures having small layer
thicknesses. Charge transport in layered materials is strongly
dependent on doping, stacking sequence, thickness, and energy bandgap of
the individual layers. This enables one to grow semiconductor material
ideally suited for particular device applications.
Due to the phenomenon of polytypism, silicon carbide (SiC) is a
natural layered semiconductor. This implies that the same chemical
compound of silicon and carbon atoms .crystallizes into different
crystallographic modifications known as polytypes. These polytypes are
all similar in the plane perpendicular to the symmetry axis (c-axis),
but differ from each other in the direction parallel to the c-axis.
Silicon carbide (SiC) is also known to have one of the largest
energy bandgaps (~ 3.0 eV) of common semiconductor materials. This
property makes it valuable for high-temperature device applications and
blue light-emitting diodes."^-
The charge transport mechanism in layered structures, in which
space-charge injection and strong band bending will occur, is studied,
using a-SiC as a model. Various transport quantities of a-SiC are
deducted from the I-V characteristics, the impedance, and the noise in
the temperature range of 50 K 300 K. The insight we gained from this
study is used to unravel some of the problems associated with charge
transport in submicron (.47 pm) n+pn+ GaAs devices.
1


1(A)
120
Fig. VII-2'. I-V characteristic at T = 77K. Dots: measured data.
Dashed line: computer calculation without any traps.
Solid line: computer calculation with a trap level given
in Table VII-1.


137
REM PROGRAM FOR AVERAGING THE NOISE SPECTRUM
2 REM SET THE USER DEFINABLE KEYS
3 GO TO m
9 GO TO 31350
12 GO TO 3960
24 GO TO 10030
23 GO TO 10130
32 GO TO 10230
3 GO TO 10330
40 GO TO 10480
44 GO TO 9020
43 GO TO 9520
56 PAGE
57 A3=A2
53 GOSUB 6300
59 GO TO 7960
60 PAGE
61 GO TO 1120
64 GO TO 3460
63 GO TO 10760
72 GO TO 730
76 CALL "RATE1'>300,0(2
7? CALL "CMFLAG">0
79 GO TO 7960
80 GO TO 31360
100 PRINT "L_VWPXX10 7D20/4052 SYSTEM
120 PRINT AVERAGING FFT PROGRAM"
140 PRINT SPECIAL EFFORT HAS BEEN GIVEN TO MAKE THE SYSTEM
160 PRINT FRIENDLY.HOWE VER SOME CARE MUST BE TAKEN WHEN USING"
130 PRINT THIS SYSTEM.SOME OF THESE POINTS ARE GIVEN BELOW: "
200 PRINT J DUSE THE APPROPRIATE FILTERS FOR EACH BAND"
2)THE COMMUNICATION SYSTEM IS INITIALLY SET FOR"
THE SCOPE .IF CHANGES ARE MADE IN ORDER TO "
USE THE PRINTER THEY HAVE TO BE RECHANGED "
DURING THE MEASUREMENT."
31MAKE SURE THAT THE FILTERS ARE TERMINATED"
CORRECTLY."
360 REM SET UP SRQ'S
330 F3=130/S5
408 X1*10
420 SET KEY
220 PRINT "J
240 PRINT "
260 PRINT "
230 PRINT "
300 PRINT "J
320 PRINT "
340 F9=10
440 DELETE A,A*,B5
460 Ai=0
430 REM SET ENVIRONMENT FLAGS
500 G$=""
520 REM SET MENU FLAG
540 Fl=-1
560 G$=G_"
580 F4=3
600 F5=13
620 F7=i
640 F8=l
660 F9=0
II


18
In our system, the Nyquist frequencies are lower than the system
bandwidth, so the frequencies above the Nyquist frequency fold back Into
the frequency band of zero to f^ Hz. To overcome this problem, a set of
low-pass filters with a sharp cutoff are used in front of the digital
oscilloscope. The relation between Nyquist frequency and time setting
of p.d.o. is given in Table II-l.
6. Computer Software
Special effort has been devoted to making the system both
interactive and fast. The flow chart of the computer program is shown
in Fig. II-6. The total number of spectrum averages is set by the
operator.
In the extended real-time digitizing (ERD) mode, 820 sample points
are transferred into the computer. In the roll and real-time digitizing
mode, this number is equal to 1,024. An improvement in speed is
obtained by dividing the data blocks into blocks of 256 points. The FFT
performs N log2N operations to obtain the Fourier coefficients of a
signal of N sample points. If N is reduced by factor four, the total
number of operations is reduced from 10,240 for 1,024 points to 2,048
for 256 points. In addition, for every data transfer from the
oscilloscope to the computer, four or three spectra can be obtained.
The reduction in block size effectively shortens the time window
and therefore increases the spacing between adjacent, discrete Fourier
components by about a factor of 4. For noise measurements, this is no
problem since in general the signal is wideband, and its magnitude
varies slowly with frequency.
The FFT operation is performed in an ROM pack, and the magnitude of
each frequency component is obtained. The signal averaging is done on a


84
re solved simultaneously using Newton's method^* to get F^ and F2
approximately zero. The boundary conditions are given by equations
(3.2) and (3.3).
The discretization of equation (3.5) is somewhat ill-conditioned
r rq%(x)^ i
since V[exp[ J^Fn.-' *-n<^cates a poorly scaled, first-order
derivative due to the variation of 4^n(x). Hence we define a new
variable
U exp( y.-)
(3.6)
The variable also has the advantage of having much greater variation
than the variable 4^. So equation (3.5) becomes
F2(Xc,U)
V[exp(
qxc(x)
kT
)VU] .
(3.7)
The discretization of V[exp(~)Vu] bas been the subject of
lengthy discussions in the semiconductor simulation literature since the
backward or forward difference does not adequately describe strong
varying exponents. We choose an approach similar to that of Bank et
al.^2 We define point K' between K 1 and K, and point (K + 1)'
between K and K + 1, so that
/
V[e
r HXc/feT qx /kT
Vu3(K+1)'
[(K + 1)' K']Ax
(3.8)
The Fermi potential is a smoothly varying function of position.


CHAPTER VIII
CONCLUSION AND RECOMMENDATIONS FOR FUTURE STUDY
Charge transport in layered materials is strongly dependent on
doping, stacking sequence, thickness, and energy bandgap of the
individual layers. A typical sample cut from a single crystal of
a-SiC has a lamellar structure due to stacking of different polytypes
along the c-axis. When a high resistive polytype is sandwiched between
low resistive polytypes, space-charge-limited flow will occur. The
samples studied for this dissertation all showed typical SCL I-V
characteristics in the presence of traps. The noise in these samples is
due to trapping noise. SiC presents a good example (one of only a few)
for verification of the complicated theory of trapping noise in the
presence of the space-charge-limited flow.
Our studies on the bulk properties of SiC should be used to develop
new devices. However, the attention should be focused more toward the
processing of SiC. Recent advances have been achieved in fabrication of
single crystal SiC platelets.^ SiC is currently being used to
A
fabricate blue light-emitting diodes. It should also be considered
for the fabrication of high-power microwave devices and sensors capable
fi SI
of performing at very high temperatures (up to T 1000K).
Amorphous SiC has recently been used as an emitter to develop a
52
heterojunction bipolar transistor.
A unique insight into the charge transport in layered materials has
been obtained from analyzing our measurements with the help of a
132


APPENDIX A
COMPUTER PROGRAM FOR TEK 7D20 NOISE
SPECTRUM ANALYZER
134


163
BIOGRAPHICAL SKETCH
Saled Tehrani-Nikoo was born on February 20, 1960, in Tehran,
Iran. He graduated from Kharazmi High School in 1978. In August of
1981 he received the degree of B.Sc. in Electrical Engineering, with
honors, from the University of North Carolina at Charlotte. He began
his graduate study at the University of Florida and obtained an M.E.
degree in December, 1982. In 1983 he joined the Noise Research
Laboratory at the University of Florida as a graduate research
assistant. He is a member of Tau Beta Pi and IEEE.
I
O


138
S0 GQSUB 7960
700 REM WAIT FOR INTERRUPTS (F9=0 => NOT BUSY)
720 F9=0
740 GO TO 700
760 REM END INITIALIZATION
7-30 REM READ SPECTRUM FROM THE TAPE
300 PAGE
820 F9=2
340 GOSUB 7920
360 PRINT 11,"READ SPECTRUM FROM THE TAPE"
880 F=F4
900 DELETE A
920 DIM AA1)
940 GOSUB 033
960 GOSUB 00
930 IF F=3 THEN 1040
1000 GOSUB 7920
1020 PRINT ",Ai
1040 VIEWPORT 15,120,10.90
1060 GO TO 56
1080 RETURN
1100 GO TO 17800
1120 REM SAVE WAVEFORM QN FILE
1140 IF F9O0 THEN 1540
1160 F9=12
1180 GOSUB 7920
1230 PRINT H "/'SAVE WAVEFORM ON FILE"
1220 IF Ai>0 THEN 1280
1240 PRINT J_SORRY NO WAVEFORM DEFINED"
1260 GO TO 1520
1230 FSF4
1300 GOSUB 6080
1320 PRINT J_Enter -file number (0 to exit/";F4;~";F4+Nl-l;">: ";G$;
1343 INPUT F
1360 IF F=3 THEN 1520
1330 IF FF4+N1 THEN 1320
1430 PRINT J_Snter a label tor this tile: ";G*;
1420 INPUT A*
1440 KILL F
1463 FIND F
1480 WRITE 33 3: A $ ,0, A1, A3, A, A2 > A5, A4, V$ ,H$, A ,1:0 ,B w
1530 CLOSE
1520 GOSUB 7960
1540 RETURN
1560 REM COPY SET-UP FROM 7D20 TO TAPE
1580 IF F9O0 THEN 1940
1600 F9-13
1620 GOSUB 7920
1640 PRINT "/'COPY SET-UP FROM 7D20 TO TAPS"
1660 F=F5
1630 GOSUB 6030
1700 PRINT "J_Snter file number (0 to exii/";F5;"~";F5+Ni-l;"): ";G$;


105
SAE(x)AE(x) f dx' [Z(x,x" ,jo>)A(x")A*(x' )K^(|x" )Z*(x' ,x'
+ Z(x,x'' ,jo))B(x' )B*(x' ,)Kh(x")Z*(
The third and fourth terms in eq. (2.11) give
AE(x)AE(x')
II dx'-dx-Z(x,x-,ju)A(x") j^ttT [k|v(x-)(x"-
C*(x")Z*(x',x" ,ju>) + hcj
where hcj is the Hermitean conjugate (x' -* x' x
of the first term. For a Dirac delta function it ho]
/ dx..
(dx")
(dx'
Consequently, the noise contribution of third and ffourth terms
(2.11) can be written as
SAE(x)AE(x') + / dx- {[Z(x,x' ,jo))A(x'')
ITT [Kv(x-)C*(x-)Z*(x',x-,ja>)]} +
dx" 1 Y'
Finally, for the fifth term of equation (2.11)
x ,x- ,ju>) ]
X ju>
ds that
.n
tcj
we obtain,
Y
(2.14)
*-)]
(2.15)
* -jw)
(2.16)
of eq.
(2.17)
after
substitution of S


25
The various polytypes are much more complicated than these two
basic structures. Since relatively minor variations in the stacking
sequence of layer pairs along the symmetry axis can produce many
alternate forms of close-packed structures, polytypism is a general
phenomenon affecting a large number of substances with close-packed as
well as layered structuresThe different polytypic modifications,
corresponding to different stacking sequences, can still be described by
ABC notations. Also, other equivalent notations of polytypic structure
have been developed. Figure III-l shows the schematic arrangement of
atoms in the (1120) plane along the c-axis of five important polytypic
modifications, 3C, 2H, 4H, 15R and 6H. The first number in this
notation denotes the number of layer pairs in one unit cell and the
succeeding letter (C,H,R) denotes the Bravais lattice, i.e., cubic,
hexagonal, or rhombohedral, respectively. The Bravais lattice is
discernible from the closest adjacent layer pairs. In Table III-l we
elaborate on the nomenclature for the above five polytypes.17-19 There
seem to be no limitations to the occurrence of different polytypes.
Over 45 different polytypes of SiC have been discovered by different
workers from x-ray investigations. However, the above five structures
seem to be the basic units of many larger period modifications. The
self-stabilization of a given polytypic structure during crystal growth
2
is influenced by a large number of factors.
Theoretical calculations of the band structure of the various
polytypes have been carried out by several authors. Since the
large number of atoms per unit cell considerably complicates the
problem, calculations have mainly been done on the two simplest
modifications, 3C and 2H. Table III-2 gives some data for four


103
where x^ = 1/6^ n^) x^ = 1/i^^ + nQ),
and 1/x 1/t + 1/x^.
To obtain the impedance, we proceed as follows. We set H s y = 0
and discretize eq. (1.4). Formally, the current density Aj^ can be
related to the field AEa by
p
Aj = Y AE
Ja a3 3
(2.5)
where Aj^ = AC current density at a and AE^ electric field at
3 The inverse of eq. (2.5) defines the impedance operator ZQ^ and
the transfer impedance tensor Z^, i.e.,
AE Z Aj0 A T Z Aj Ax
a op J3 a3 J3
(2.6)
where A is the device area. The total voltage across the device is
given by
Av =* 7 AE Ax. (2.7)
L a
a
Since the current density Aj^ is independent of position 3, the total
device impedance is given by
\ m (2-8)
In order to calculate the noise, we include the effect of diffusion
in the noise term. In general, the noise spectral density of the
30
electric field is defined as


39
P n +\ 1 (M-G re8ime)
(3.14b)
If the first possibility applies, as we assume presently, Poissons
equation can be rewritten with the aid of (3.5) and (3.14a). Thus the
pertinent equations become (3.5) and
ee.
dx dE
g0E
qnc I + g0E
(3.15)
where
g qp(n+n)pA qnpA ;
u t
(3.16)
gg/L is the conductance of the unexcited specimen. For later use we
also introduce the dielectric relaxation time
Tn eeo/qn0y
(3.17)
Equation (3.15) is immediately integrated to yield, with boundary
condition E(0) 0:
sen gnE
[gE I*n(l + )] .
Vo
(3.18)
In particular, evaluating this at x = L and defining
a g0|EL|/l - gQEL/I ,
(3.19)
we find
I
VoL
E£q a + Jln(l-a)
(3.20)


4
noise signals, the system power gain and the pulse duty cycle must be
relatively large. Additional disadvantages are that a true r.m.s.
detector with an extremely large crest factor is required and that if
the system is not band limited, the frequency component of the pulse
will saturate the power amplifier. The later phenomenon sets a lower
limit of frequency that can be measured since the noise signal needs to
go through a bandpass filter to eliminate the frequency components of
the pulse.
In the system discussed here, the pulsed bias noise measurement is
done by collecting the sampled data only when the bias pulse is
applied. The sampled data is then fast Fourier transformed in the same
way as is done in the continuous bias noise measurement.
This system can also be used to measure the spectral intensity of-
burst noise. The noise spectral intensity is calculated using sampled
data collected only during current or voltage bursts.
A block diagram of the system is shown in Fig. II-l. The system
consists of a low-noise amplifier, a set of passive low-pass filters, a
digital programmable oscilloscope, and a desk-top computer.
The input signal is amplified with the low-noise amplifier. A low-
pass filter is chosen for the desired frequency span which is set by the
T1M/DIV knob on the oscilloscope [section 2] These low-pass filters
are implemented to remove the high-frequency signal components which
would appear as low-frequency components when sampled by the digital
\
oscilloscope [section 5]. A finite segment of the discrete time data is
then transformed to the computer and is translated into a discrete
frequency spectrum using a fast Fourier transform (FFT). The FFT
coefficients are used to obtain a power spectral intensity [section 6].


54
where £ jwt* t* being the drift time, and where
F4(X,5) / dv / du v(^)(X >/Xe(5/Px)(v-u) (5.7)
0 v
In the SCL regime t* t so the structure in frequency of (5.5) is
pushed to higher frequencies compared to that of the ohmic flow.
Typical |impedance|^ plots at 77K are given in Fig. III-ll. We
notice that all curves can be represented by a form
, r(I)
'L 1 + jor(I)C(I)
(5.8)
A plot of C(I) vs V is shown in Fig. III-12. We notice that C(I)
increases, though not as fast as r(I) decreases, (r V S so
r(I)C(I) decreases with increasing V, indicating higher frequency
turnover when we go further into the SCL regime. The equilibrium
capacitance seen is 55 pF. The dielectric relaxation time of the sample
*6
(subtracting parasitic capacitance) is i. 1.1 x 10


58
the deepest traps which play a role are a few kT away from the quasi-
Fermi level. Shallower traps may be seen, but deeper traps are
generally not, since the statistical factor /ng goes rapidly to
zero for such traps (section 2). Thus, at a given temperature the same
range of traps should play a role as in the TFL portion of the DC I-V
characteristics. Whereas the latter showed, however, no discernible
structure (the presence of multiple traps is only manifest in a more
gradual slope of the TFL curve), in the noise we should see discrete
Lorentzians for each trap. With this in mind we undertook to measure
the noise in the ohmic, low-level quadratic, and TFL regimes of the same
SiC specimens as employed in chapter III.
This chapter is divided as follows. Section 2 reviews the theory
of trapping noise in SCL flow, as developed by Van Vliet, Friedmann,
Zijlstra, Gisolf, Driedonks and van der Ziel (see references 30 and 34
and the references therein). Fortunately, closed analytical expressions
exist in the first two regimes. For the TFL regime no detailed
expressions have been derived as yet, but it is obvious from generation-
OC
recombination noise theoryJJ that the noise should rapidly go. down. We
also mention the fact that, as shown in a recent paper by Van Rheenen et
al., the theory of single-level trapping noise can be justifiably
applied to each Lorentzian in a multiple trap noise spectrum whenever
the time constants are a factor of ten or more apart. In section 3 we
present the experimental results in the ohmic regime for temperatures
from 62.5 300K. In section 4 we give a discussion of these results.
In section 5 we present the spectra at 77K, with applied voltage ranging
through the ohmic, low-voltage quadratic and TFL regimes. The Mott-
Gurney regime could not be reached; it would require the noise


TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ii
ABSTRACT v
CHAPTER
I INTRODUCTION 1
II COMPUTER CONTROLLED NOISE SPECTRUM ANALYZER
EMPLOYING A DIGITAL OSCILLOSCOPE 3
2.1 Introduction 3
2.2 Programmable Digital Oscilloscope 6
2.3 Spectral Intensity of a Random Noise Signal.. 10
2.4 Time Window 14
2.5 Aliasing 15
2.6 Computer Software ^ 18
2.7 Noise Measurement Procedure 20
III CURRENT-VOLTAGE AND IMPEDANCE CHARACTERISTICS OF
SCL FLOW IN a-SiC 23
3.1 Introduction. 23
3.2 Properties of SiC 24
3.2.1 Crystallography and band structure 24
3.2.2 Transport properties 29
3.2.3 Our samples 31
3.3 On the Theory of SCL Flow in the
Presence of Traps 34
3.3.1 Analytical results 34
3.3.2 Regional approximations 42
3.4 Experimental I-V Characteristics 43
3.5 Impedance Measurement 51
IVELECTRICAL NOISE OF SCL FLOW IN a-SiC 57
4.1 Introduction 57
4.2 Review of the Theory of Trapping Noise
in SCL Flow 59
4.3 Experimental Results in the Ohmic Regime 64
4.4 Discussion of Spectra in the Ohmic Regime..... 71
4.5 Noise Spectra in Ohmic and SCL Regimes at 77K 72
iii


108
TABLE VI-1
Trap parameters used for the computer calculations
Trap #
Nt(m'3)
Eact
g(m3/sec)
a(cm2)
1
121
5
x 1019
2.0
1
o
H
X
3.6
X
00
r4
o
H
2
103
5
x 1018
1.4
X lo"16
2.5
X
10-17
3
85
6
x 1018
3.0
x 1016
5.4
X
10"17
4
67
1
x 1020
3.0
x 10"16
5.4
X
10-17
\
\


2
The organization of the chapters is as follows. In Chapter II, a
newly developed computer controlled spectrum analyzer system for noise
measurement is presented. A review of the properties of SiC, the
experimental results of I-V and impedance measurements, and a somewhat
different version of the standard theory of space-charge limited flow is
presented in Chapter III. In Chapter IV, the theory of noise in SCL
flow in the presence of traps is reviewed, and experimental data on
noise in the ohmic regime and SCL regimes is presented. Chapter V
contains the results of computer calculations of the DC characteristics
of a-SiC in particular, and layered structures in general. In Chapter
VI, computer calculations of the impedance and the noise in a-SiC are
presented. Chapter VII contains the study of the charge transport in
submicron n pn GaAs. Finally, conclusions and recommendations for
further work are given in Chapter VIII.
Appendix A includes the computer program for the TEK 7D20 noise
spectrum analyzer. The computer program to simulate the I-V
characteristics, the impedance, and the noise is given in Appendix B.


S, ( A /Hz)
no
Ohmic Regime
V=30mv
Fig. VI-2. Current noise magnitude in the ohmic regime. Dots :
measured data. Dashed lines: resolution into
Lorentzians. Solid lines: computer calculations. Dot-
dashed lines: the effect of the diffusion part of the noise
source term for trap level two.


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
C.M. Van Vliet, Chairperson
Professor of Electrical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
6. Bosman, CoChairperson
Assistant Professor of Electrical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
A. van der Ziel
Graduate Research Professor
of Electrical Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
L.L. Hench
Professor of Materials Science & Engineering


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
PROPERTIES OF NOISE AND CHARGE TRANSPORT
IN LAYERED ELECTRONIC MATERIALS
By
Saied Tehrani-Nikoo
June 1985
Chairperson: C.M. Van Vliet
Major Department: Electrical Engineering
The charge transport mechanism in layered structures, in which both
space-charge injection and strong band bending occur, is studied.
Samples of nitrogen-doped a-SiC are used as models to test and verify
the various aspects of transport theory in short, layered devices. The
samples studied consisted of a high resistive (strongly compensated)
polytype layer sandwiched between two low resistive polytype layers.
This creates an n+nn+-like structure. The low resistive polytypes act
as injecting and extracting "contacts for the high resistive polytype,
introducing into the latter single-carrier, space-charge-limited (SCL)
flow.
The samples which we investigated all showed typical SCL I-V
characteristics with four clearly discernible regimes for most
temperatures between 50K and 300K. The DC characteristics show a
succession of trap filling; yet they are characterized by a single trap
level at a given temperature with regard to the onset of the trap
filling regime. The energy band profile and the electric field profile
v


139
1720 INPUT F
1740 IF F=0 THEN 1920
1760 IF FF5+Nl THEN 1700
1730 FIND F
13*0 PRINT J_Enter a name for this set-up tile: ";G$;
132*INPUT A$
1340 PRINT 333:A$
1360 PRINT aU:set?"
1330 INPUT QU:A*
1900 PRINT 333:A$
1920 GOSUB 31010
1940 RETURN
I960 REM TAPER IS NOT AVAILABLE WITH THE SYSTEM AT THIS TIME
1930 REM TAPER IS BASICALLY SIMILAR TO USING DIFFERENT
2000 REM TIME WINDOW.
2020 REM TAPER
2040 IF F9O0 THEN 2420
2060 F9=16
2080 PAGE
2100 F3=100/35
2120 PRINT "/'TAPER"
2140 IF F8=i THEN 2200
2160 PRINT "J_SORRY 4*5xR8 REQUIRED FOR THIS FUNCTION"
2180 GO TO 2400
2200 IF A1>0 THEN 2260
2220 PRINT J_SORRY NO WAVEFORM IS DEFINED"
2240 GO TO 2400
2268 PRINT J_Snter percent ot cosine taper desired 0 < 50): ";Gi;
2230 INPUT X
2300 IF X<0 OR X>50 THEN 2260
2320 X=X/180
2340 CALL "TAPER,A,X
2360 VIEWPORT 15,120,10,80
2330 GOSUB 6300
2400 GOSUB 31010
2420 RETURN
2440 PRINT B$;G$;
2460 INPUT A$
2430 A$=SEG 2500 X=0
2520 IF A$="N" OR A$="n" THEN 2530
2540 X=i
2560 IF A$<>"Y" AND A$<>My" THEN 2440
2580 RETURN
2600 Ew=INT(Bw)
2620 REM AVERAGING THE NOISE SPECTRUM
2640 FOR 110=1 TO Nave
2660 B5=0
2680 A=0
2700 FOR Ii5=l TO 4
2720 REM READ DATA FROM OSCILLOSCOPE IN BINARY FORMAT
2740 DIM AC320),A7(9)
2760 PRINT '3U:"WFM?"


82
The density of the trapped electrons is related to the quasi-Fermi
potential $ (x) and the trap potential xt(x) by Fermi-Dirac statistics,
rn
1*6)
nt0(s) ,Nt/11 + 8 1exp(-(qxt(x) q ^(x))/kT) ] (2.7)
where g is the electron spin degeneracy and |q(Xc x£) | is the trap
activation energy. The density of the electrons in the conduction band
for a nondegenerate semiconductor is given by Maxwell-Boltzman
statistics as
nQ(x) = Ncexp(-q( $Fn(x) Xc(x))/kT) (2.8)
where Nc is the effective density of states. The two coupled second-
order partial differential equations (2.5) and (2.6) are solved
simultaneously with the aid of equations (2.7) and (2.8).
3. Computer simulation
The charge transport equations described in Section 2 are applied
to an n nn structure. In order to find a unique solution of the
second-order, differential charge transport equations, we have to
specify two boundary conditions. Numerical stability dictates that
these boundary conditions have to be specified at the two opposing n
contacts (Dirichlet problem). Specification of, for example, the charge
concentration and its derivative at only one n+ contact (Riemann
problem) leads to numerical instability. Hence we assume that the
device behaves ohmic at a few Debye lengths away from n+n and nn+
contacts and thus


I (amps)
45
Fig. III-6 I-V characteristic at 200K.


72
off occurs at Tq 77K. With kT 6 meV and the Fermi level being at
122 meV below the conduction band at 77K, see Table III-4, chapter III,
the distance between trap level and Fermi level is indeed 4 kTg when the
leveling off sets in. Similar good agreement is found for the other
trapping curves.
From the noise spectra at 77K and 62.5K, it is clear that no traps
shallower than 63 meV appear, except perhaps for the ionized donor level
of 10 meV, found in chapter III. Generation-recombination noise due to
these donors should be observable at very low temperatures.
From the magnitude of the and eq. (2.20), together with the data
for given in chapter III and a spin degeneracy g 2, we computed
the electron capture cross section a^ of each trapping level. The
results are shown in Table IV-1. The cross sections are within the
normal range for neutral or negatively charged traps.
In Table VI-2 we have tried to correlate the trap depths as found
from the I-V curves (Table III-4) with the trap depths found from the
noise. The correlation of the two sets of data is far from perfect; yet
the fact that they range over a similar latitude (31 63 meV) lends
strong support to the interpretation and consistency of the data in both
papers.
5. Noise spectra in ohmic and SCL regimes at 77K
At 77K the noise was measured for currents ranging from 7.7 x 10""^A
up to 3.74 x 10 A. From Fig. III-7 we deduce that the following
ranges occur:
I < 3 x 10^A ohmic regime
3 x 10^A < I < 3 x 10*"4A low voltage quadratic regime
3 x 104A < I < 8 x 10-3A TFL regime.


n(x)&n (x)(m _3)
95
Low Voltage Quadratic
1 = 3.9X10 ~5 A
Fig. V-7. Carrier concentration profile of the trap level and the
conduction band in the low-voltage, quadratic regime at 77K.


CHAPTER II
COMPUTER CONTROLLED NOISE SPECTRUM ANALYZER
EMPLOYING A DIGITAL OSCILLOSCOPE
1. Introduction
Substantial improvements in the performance of measurement
equipment have been obtained in recent years by using either desk-top
computers or built-in microprocessors to control the different functions
of a particular instrument. In addition, simple calculations to verify
the experimental findings can be done almost instantaneously by these
data processors. Significant advances in speed, stability, and accuracy
of spectral noise measurements have been achieved by using the computer
controlled spectrum analyzer system discussed in this chapter.
Pulsed bias noise measurements, as well as continuous bias noise
measurements, can be performed in the frequency range from 10 Hz to 10
MHz. Pulsed bias noise measurements are used to avoid excessive joule
heating when a device is pulsed biased at high voltage or current
levels.
Nougier et al.^ described an apparatus for pulsed bias noise
measurement in the frequency range of 100 MHz 1 GHz. Recently,
Q
Whiteside0 developed a similar system for pulsed bias noise measurements
between 1 MHz and 22 GHz. Both systems are complicated and gate the
noise spectral "on and "off" synchronously with the bias pulse. The
total output noise power in these systems is proportional to () GB,
where x is the pulse length, T~^ is the pulse repetition rate, and GB is
equal to system power gain-bandwidth product. In order to detect small
3


Energy(eV)
122
L(nm)
Fig. VII-3. Energy band diagram in the ohmic regime(I = 45 uA) at
T = 77K.


31
2.3. Our samples. In the introduction we mentioned that our
devices were rectangular bars, cut out of a fairly large crystal of
nitrogen-doped o-SiC. Tungsten films were deposited on both flat sides
of the crystal by sputtering, prior to cutting our samples. The contact
area was defined by etching out the residual tungsten films in a diluted
solution of HF-HN03. Bonding to the tungsten pad was done with a
silver epoxy featuring high electrical and thermal conductivity
(Type SCS-1, Lake Shore Cryogenics, Inc.). Fig. III-2 gives the
geometry obtained.
As indicated in the introduction, space-charge limited flow was
observed in all samples, with the current flowing between contacts 1 and
2 or 1 and 4. The SCL structure was most pronounced, however, if the
current was flowing between contacts 1 and 4, with the voltage measured
either between 1 and 4 or 2 and 3. This led us to believe that the
layered structure perpendicular to the c-axis contained high resistivity
polytype(s) sandwiched between low resistivity polytypes; this is
sketched in Fig. III-3. Note that no notches occur in the polytype
independent valence band. The low resistivity polytypes act as
injecting (cathode) and extracting (anode) contacts.
Impedance measurements in the ohmic region at 77K (section 5) give
a capacitance of 55 pF. It is estimated that half of this is parasitic,
so we have Cq => 28 pF. With relative dielectric constant e = 10.2 and
an area of 4.7 x 10~^m^ (see Fig. III-3), this yields L 1.6 x 10~^m
for the width of the insulator polytype which governs the SCL flow.
Notice that W^total is only 1.6% where Ltota-^ is the macroscopic
thickness of the crystal.
In principle, there could also be several insulating polytypes in
series. Due to the universal scaling law,^ J/Lp = f(V/L2) this


85
Consequently, u = exp[-q4^/kT] is well behaved and can be discritized
accurately as
Vn U(K+1) U(K)
"K+1' Ax
(3.9)
and
7,, 3 u(K) ~ u(K-l)
TC' Ax
(3.10)
To find the values for exp[qxc/kT] at (K+l)' and K*, we define
d rqxe(x)i q r^c^T dxc(x>
_ exP[-1Sf-] pHhT-J dx
(3.11)
After rearranging (3.11), we get from an integration
, KAx -qx (x) qx (x)
/ exp( ) [exp( -"- ) jdx
q (K-l)Ax
kT
KAx dx (x)
/
(K-1)AX
dx dx *cW "
(3.12)
According to the mean value theorem for integrals, there exists a value
K',K-1 KAx KAx
/ f(x)g(x)dx = f(K') / g(x)dx .
(K-l)Ax (K-l)Ax
(3.13)
Applying the theorem to the left-hand side of eq. (3.12) results in


112
Low-Voltage Quadratic
V = .68 V
Fig. VI-3. Current noise magnitude in the low-voltage quadratic
regime. Dots: measured data. Dashed lines: resolution
into Lorentzians. Solid lines: computer calculations.
Dot-dashed lines: the effect of the diffusion part of the
noise source term for trap level two.


71
4. Discussion of spectra In the ohmic regime
Though the choice of the plateaus of some Lorentzians leaves some
leeway, we are quite convinced that there is not a continuous uniform
distribution of traps. The latter would lead to a 1/f spectrum, while
S'
an exponential distribution would lead to a spectrum 1/f with
between zero and two (though usually close to one, say 0.8). It should
be noticed that a straight line approximation, e.g., to the 296K curve,
see dashed line, would cause a number of points to be well off by a
factor 1.5 to 2.0. In this respect we should keep in mind that a log-
log plot tends to obscure details. However, the accuracy of the
measured points was never less than 10%, so that the structure, where
appearing, should be taken at face value. However, the most important
clue to the fact that this is not 1/f noise is indicated by the
pronounced rise at high frequencies in the 200K curve, and by the sharp
roll-off at high frequencies in the 77K and 62.5K curves. Therefore, we
believe that all spectra represent trapping noise, i.e., a form of
generation-recombination noise. A final indication that this is not 1/f
noise comes from the fact that the Hooge constant S^/I2 > a/fN for
such a process, using the dashed line in Fig. IV-1, would be
a 0.6 a value that is orders of magnitude higher than observed
1/f noise in silicon or germanium. Most clearly the nature of the
trapping noise is revealed by Figs. IV-5 and IV-6* In Fig. IV-5 we find
a number of straight lines, the slope of which gives the trap depth, see
eq. (2.20). We also note that in some cases a horizontal portion of the
lifetime appears. This portion should appear at temperatures for which
the Fermi level approaches the trap level within 4 kT from below, see
section 2. E.g., for the 98 meV trapping curve, the leveling


ABCABCAB
ABCABCA
3C 2H 4H
(a)
III-!* Schematic arrangement of Si and C atoms in the [1120] plane of different polytypes
of SiG. The solid lines indicate the sequence in the zigzag movement of
sublattices along the c-axis.


10
8
6
4
2
0
-2
-4
-6
-8
10
VII-
128
L(jjm)
Electric field profile in the SCL regime (I = .35A) at
T 77K.


T(sec)
114
Fig. VI-5. Characteristic time constants of the Lorentzian spectra
-as a function of voltage V. Dashed lines: best fit to the
measured data. Solid lines: results of the computer
calculations.


144
3080 PRINT "J_KEY"," FUNCTION J_"
£100 PRINT 3("AVERAGE THE NOISE SPECTRUM"
£120 PRINT 6","STORE DATA FOR DUT ON MEASURE ME NT(Mi)"
£140 PRINT ?","STORE DATA FOR DUT OFF(M2)"
8160 PRINT 8,"STORE DATA FOR CALIBURATION SOURCE ONM3)"
3188 PRINT 9","STORE CALIBURATION SOURCE SPECTRUM(Sv cal)"
8200 PRINT 10"("CALCULATE THE DUT SPECTRUM"
8220 PRINT 11 ","PRINT FFT VALUES ON CRT"
£240 PRINT 12 "("TRANSFER FFT VALUES TO PRINTER"
£260 PRINT 14 /'PLOT FFT ON CRT "
£280 PRINT'" 15 "."COPY THE SFECTRUM ON THE TAPE"
£300 PRINT 16 "("ELIMINATE FREQUENCIES "
£320 PRINT 17 ","PLOT ON HP PLOTTER"
£340 PRINT 18 ","READ WAVEFORM FROM THE TAPE"
8360 PRINT 20 "."RESET"
£380 PRINT 232,21:0,0
£400 PRINT "NEXT KEY ";Gi;
£420 SET KEY
£440 GO TO 700
£460 PAGE
£480 PRINT WHAT RANGE OF FRE. DO YOU WANT TO ELIMINATE?(FMIN,FMA
£500 INPUT JJ.J.I
£520 J0=INT(Jj0/A3)
£548 Ji=INT(Jji/A3>
£560 IF J0>0 THEN £600
£580 J0=i
£600 FOR 10=J8 TO J1
8620 AI0-AL1
£640 NEXT 18
£660 GO TO 56
£680 FOR Jj*i TO M
£700 FOR Ii=i TO N
£720 DIM 55(1024)
£740 Ij=I.+l
£760 IF Aa>1800 THEN £820
£730 L=512
£800 GO TO £840
£820 L=1024
£840 35=0
3860 GO TO 17244
8880 AA/te
8900 NEXT K
8920 NEXT li
8940 GOSUB 6800
£960 NEXT Jj
89S0 GO TO 7968
9000 END
9020 PAGE
9040 PRINT "NUMBER OF DATA POINTS REQUIRED ?"
9060 INPUT No
9080 PRINT L_'FREQUENCY (HZ)
9100 Nal=INTiAi/Moi
9120 FOR 1=1 TO No
M AGNITUDE (DEV)"


147
11260 IF OptionsM"N" THEN 11420
11280 PRINT "WHAT PLOTTING SYMBOL SHALL BE USED?"
11300 INPUT Sm$
11320 PRINT "DO YOU WANT A GRID OR NORMAL TIC-MARKS?iG/N)."
11343 INPUT Ti$
11360 PRINT "DO YOU WANT A LINE TO CONNECT THE DATA POINTS? 11383 INPUT Vector*
11400 GOTO 11530
11420 REM SET DEFAULT VALUES OF PLOTTING OPTIONS
11440 Ti$="N"
11463 Sm*="-r
11433 Vector$="Y"
11500 PRINT 35:"OPJ_"
11520 INPUT S5:Plx,Piy,P2xP2y
11540 GO TO 11628 IREM TEMPORARILY SKIP THE WINDOW COMMAND
11560 PRINT 35: USING li530:i.0i*Fix,i.0*Piy>i.0l*P2x.i*P2y
11530 IMAGE "IW,6D,",,6D,,6D,,h,6D,J
11600 PRINT 25:"TLJ_"
11628 IF Lx 5=" LOG" THEN 13240 !REM LOG SCALE THE X DATA
11640 REM LINEARLY SCALE THE X DATA
11660 U2x=Xmax
11688 Ulx*Xmin
11780 Kx 11728 FOR 1=1 TO N
11740 Xscaied (I)=Kx *(X4I)-U 1 x HP 1 x
11760 NEXT I
11780 Xints=KxXint-Ulx!+Pl¡:
11300 REM CONTINUE, RETURN FROM ROUTINE WHICH LOG SCALED X POINTS
1.1828 IF Ly$="LOG" THEN 13720 IREM LOG SCALE THE Y DATA
11840 REM LINEARLY SCALE THE Y DATA
11860 U2y=Yma;<
11880 Uiy=Ymin
11980 Ky=lP2y -P i y > / (U2y-U i y)
11928 FOR 1=1 TO M
11940 Yscaled(I)=Ky* 11960 NEXT I
11930 Yints=Ky* 12000 REM CONTINUE, RETURN FROM ROUTINE WHICH LOG SCALED THE Y POINTS
12020 IF L>;$="LQG" THEN 13460 iREM PLOT THE X-AXIS LOGARITHMICALLY
12040 REM PLOT THE X AXIS LINEARLY
12060 Xtics=Kx*Xtic
12330 N>;irt= 12108 PRINT 35:"SPi;PU"
12123 GGSUB 14280 iREM CHANGE THE XTIC LENGTH
12148 PRINT 05: USING l2160:Pl;:;Yints
12160 IMAGE ''¡PAil,6D,",",6D,',;!
12183 PRINT 05:";PD;XT;"
12280 FOR 1=1 TO Nxint
12220 PRINT 05: USING 12248:Plx+I*Xtics,Ylnts
12240 IM AGE" ¡PA" ,6D,"," ,6D," ;XT; J_"
12263 NEXT I
12280 PRINT 05:";PU;"
12300 REM CONTINUE, RETURN FROM ROUTINE WHICH PLOTTED LOG X-AXIS


(3)J Z I
55
Fig. III-ll. |Impedancevs. frequency for various voltages at 77K.


149
13380 NEXT I
13400 Xints=k'xl*(LGT:
13420 GO TO 11800
13440 REM PLOT THE LOGARITHMIC X-AXIS
13460 PRINT 05:";SP1;PU;"
13480 GOSUB 14200 !REM CHANGE THE X TIC-LENGTH
13500 PRINT 05: USING 13520:Pix,Yin+.s
13520 IMAGE" ;PA" ,6D,"",6D,H ;PD;XT;
13540 FOR 1=1 TO Nxdec
13580 FOR J=2 TO 10
13580 Xs=-i+LGT 13800 PRINT 05: USING 13620:Xs,Yints
13820 IMAGEh;PA",6D,V,6D";XT;"
13840 NEXT J
13880 NEXT I
13880 PRINT S:";PU;"
13700 GO TO 12300
13720 REM SCALE THE Y DATA, PREPARE FOR PLOTTING
13740 U2y=LGTiYmax)
13780 Uly=LGT(Ymin)
13780 Nydec=U2y-Uly
13800 Xyl=(P2y-?iy)/(U2y-Uly>
13820 FOR 1=1 TO N
13840 Y5caied(I)=Kyl*iLGT!Y!I))-Uiy)+Ply
13880 NEXT I
13888 Yints=Kyl*(LGTiYint)-Uiy)+Piy
13900 GOTO 12000
13920 REM PLOT THE LOGARITHMIC Y-AXIS
13940 PRINT 05:";SP1;PU;"
13980 GOSUB 14380 ¡REM CHANGE THE Y TIC-LENGTH
13980 PRINT 05: USING 14000:Xints,Piy
14000 IMAGE" ;PA" ,8D,",8D," ;PD;YTJ_"
14020 FOR 1=1 TO Nydec
14040 FOR J=2 TO 10
14060 Ys=-l+LGT(J)-Uiy)*Kyi+Ply
14080 PRINT 05: USING 14100:Xints,Ys
14100 IM AGE" ;PA" ,6D,'',8D,"; YT;
14120 NEXT J
14140 NEXT I
14160 PRINT 05:;PU;
14180 GOTO 12540
14208 REM CHANGE THE X TIC-LENGTH
14220 PRINT 05:";TL;"
14240 IF T!$=N" THEN 14348
14260 Tpx=108*(?2y-Yints)/(P2y-Piy)
14280 Tnx=! 00*Yints-P 1 y) / 14300 PRINT 05: USING i4320:TDx,Tnx
14320 IMAGE '7L",6D,V,6D,HJ_
14340 RETURN
14360 REM CHANGE THE Y TIC-LENGTH
14380 PRINT 05:";TL;"
14400 IF T1$="N" THEN 14500
14420 Tpy=100*:)


113
"i5 TFL Regime
x V= 1.5 v
Fig. VI-4. Current noise magnitude in the trap-filling regime. Dots:
measured data. Dashed lines: resolution into
Lorentzians. Solid lines: computer calculations. Dot-
dashed lines: the effect of the diffusion part of the noise
source term for trap level two.


24
relaxation time of the unexcited specimen, and t are the various
trapping time constants. The noise shows as many as five trapping
levels, ranging in energy depth (below the conduction band) from 63 meV
to 302 meV). The DC characteristics show a succession of trap fillings;
yet they are characterized by a single set of traps at a given
temperature with regards to the onset of the trap-filling regime. Thus
the curves can be synthesized as multiple discrete trap curves, see
Lampert and Mark's book, Fig. 2.2b. The noise data supplement the DC
data as to the various trap levels involved. As we show in chapter IV,
the noise studies on SiC reported here fully confirm and satisfy the
theory of noise in SCL flow, in particular as developed in reference 30.
2. Properties of SiC
2.1. Crystallography and band structure. Silicon carbide exhibits
the phenomenon of polytypism^^ in which ,the same chemical compound of
silicon and carbon atoms crystallizes into different crystallographic
modifications known as polytypes. These polytypes are all similar in
the plane perpendicular to the symmetry axis (c-axis), but differ from
each other in the stacking sequence and cycle in the direction of the
symmetry axis. The additional modulation of the main crystal field
allows us to regard the more complex polytypes as superlattices, having
miniband Brillouin zones.
The nearest neighbor-bonding in all polytypes is tetrahedral, but
the second nearest neighbor determines whether we have a cubic or a
hexagonal close-packed structure. For the zinc blende structure the
regular succession of three alternating pairs is ABCABC..., whereas for
the wurzite structure we have a succession of two alternating layers
ABAB...


1(A)
91
Fig. V-4. Current diagram in the linear regime (I = 6 x 10~^A) at
T = 77K. The solid line represents the drift current and the
dashed line the diffusion current. The sign conven tions are
given at the bottom of the figure.


CHAPTER V
COMPUTER CALCULATION OF DC SCL FLOW IN ct-SiC
1. Introduction
Many observations on several samples cut from single crystals led
to the conclusion that the layered structure which makes up the device
contains a highly resistive (strongly compensated) polytype, sandwiched
between low resistive polytypes [chapter III and chapter IV]. This
creates an n+nn+-like structure. The low resistive polytypes act as
injecting and extracting "contacts" for the high resistive polytype,
introducing into the latter single-carrier, space-charge limited flow.
The I-V characteristics measured in the temperature range of 50-
300K show four different regimes of operation: a low-bias, ohmic
regime; a low-bias, quadratic Mott-Gurney regime (electron traps empty);
a fast-rising trap-filling regime; and, finally, a quadratic Mott-Gurney
regime (electron traps filled).
In chapter III we used the regional approximation method to explain
the presence of these four regimes. The model we used to explain our DC
characteristics shows a succession of trap fillings. However, at any
given temperature, a single trap level controls charge transport.
In order to verify the experimental results and the model presented
in the preceding chapters and to obtain a better understanding of the
charge transport mechanism in a-SiC in particular, and of short n+nn+
devices in general, the appropriate transport equations are solved with
the aid of the computer.
80


141
4980 REM AS IS THE 2'S COMPLEMENT OF THE MODULO 256 SUM OF THE
4920 REM PRECEDING BINARY DATA BYTES AND THE BINARY COUNT.
4940 RBYTE A?
4960 RBYTE A,AS
49S0 REM CHECK FOR ERROR IN DATA TRANSFER
5000 A9SUM(A)+A7(9)+A7<8)
5020 A9=A9-256*INT(A9/256)
5040 A9=256-A9
500 IF ABS(A8)=A9 THEN 5120
5080 GO TO 4820
5100 REM CHANGE DATA FROM BINARY TO DECIMAL FORMAT
5120 A=A-128
5140 A=A*A5
510 A=A/25
51S0 A=A+A4
5200 LET A=A-SUM 5220 DIM E6¡25)B7(25)S8(25),B9<256)
5240 REM DIVIDE THE DATA INTO FOUR BLOCKS
5260 FOR Iii=i TO 256
52S8 B6iIii)=AUil)
5300 B7 5320 ES(Ii)=A(Iii+512)
5340 S9 5360 NEXT Ill
5380 REM CALCULATE THE FFT OF EACH BLOCK
5400 CALL FFT" ,E6
5420 CALL 11FFT",B7
5440 CALL "FFT",B8
5460 CALL "FFT"E9
5480 DIM B(129),C(i29),Bl(i29)B2(129)>B3(129)A(i29)
5500 REM CALCULATE THE MAGNITUDE
5520 CALL POLAR" B6,B,C,0
5540 CALL "POLAR",B7,B1,C,0
5560 CALL "POLAR",38,B2,C,0
5580 CALL "POLAR",B9,B3,C,0
5600 REM CALCULATE THE FOURIER COMPONENTS
5620 3=B#S
5640 Bi=Bl*Bi
5660 B2=B2*B2
5680 B3=B3*B3
5700 REM ADD THE NOISE SPECTRUM
5720 A=E1+E2
5740 A=A+B3
5760 A=A+B
5780 B5=A+35
5800 NEXT I
5820 REM REFER TO LINE 23420
5840 A=E5/iEw*399384)
5860 REM ADD THE SPECTRUMS
5880 C5=A+C5
5900 A=C5/Ii0
5920 VIEWPORT 20,120,15,90
5940 V$="DBV/SGR(H2>"


115
The theory .of the noise in SCL flow indicates that the normalizing
2
factor for all regimes is Iq|Vq| modified by ^ where n is the
number of carriers in section Ax (chapter IV).
S6IR
In Fig. VI-6 a plot of =r vs. Vq is shown where R is the
O' O'
ohmic resistance of the unexcited specimen. For the Lorentzians
2
involved, the statistical factor - is
no

T
T,
N.
n.
Nt nt + n0 + nl
(3.2)
Even though there is a difference in the magnitude between the
measured values and the computer calculations, the trend of a reduction
in the noise magnitude at higher bias is found.
In the ohmic regime the charges in nfc and ng with bias are
small, and since nt Nt and ng n^, the noise magnitude and the
time constant changes with bias will be small. As the biasing
increases, the carrier concentration ng increases. For the trap
closest to the Fermi level (Eact 103 meV), the trap carrier concentra
tion nfc will increase rapidly, so the noise magnitude and time
constant will show a rapid decrease. The noise magnitude of the other
traps decreases because of the increase in the carrier concentration ng.


145
9146 K=I*No
9160 Dd=10#LGT(A(K))
9186 J=K*Fre
9260 PRINT J,Dd
9220 NEXT I
9240 B$=" J_DO YOU WISH TO LOOK Af SPECIFIC FREQUENCY (Y/N)?"
9260 GOSUB 2440
9280 IF X=0 THEN 7960
9300 PRINT PRESS KEY 19 TO GET OUT"
9320 PRINT "WHAT ARE THE FREQUENCIES ?"
9346 INPUT Fr
9360 K0=Fr/Fre
9380 Ki=INT!K0)
9400 K2=Kl+i
9420 Ddb=10*LGT(A(K25)
9440 Dda=10*LGT 9460 Dd0=Ddb-KDda-Ddb)*((K2-K0)/(K2-Kl))
9486 PRINT Fr,Dd0
9500 GO TO 9340
9520 PAGE
9540 PRINT V SET THE PRINTER"
9560 CALL "RATE",9600,6,2
9586 CALL "CMFLAG" ,3
9606 PRINT NUMBER OF DATA POINTS REQUIRED ?"
9620 INPUT N0
9640 PRINT Me-." FREQUENCY (HZ) MAGNITUDE 9660 N0i=INT(Ai/N6)
9680 FOR 1=1 TO N0
9700 K=I*N01
9720 Dd=10*LGT(A(K))
9740 J=K*Fre
9760 PRINT 040:J,Dd
9780 NEXT I
9806 5$=" DO YOU WANT TO LOOK AT SPECIFIC FREQUENCY 9820 GOSUB 2440
9848 IF X=0 THEN 7966
9860 PRINT PRESS KEY 19 TO EXIT "
9880 PRINT "WHAT ARE THE FREQUENCIES ?"
9960 INPUT Fr
9920 K0=Fr/Fre
9940 Ki=INT(K0)
9960 K2=Ki+i
9980 Ddb=10*LGT(A(K2))
10000 Dda=10*LGT(A!Kl))
10020 Dd0=Ddb+(Dda-Ddb)*K2-K0)/(K2-Kl))
10640 PRINT O40:Fr,Dd0
10060 GO TO 9900
18080 PAGE
10100 REM MEASUREMENT WITH DEVICE ON
10120 DIM Mm 1(70)
10140 Mmi=A
10160 GO TO 8060
10130 PAGE


15
on where the selected discrete points fall. Fig. II-4 illustrates two
distinct possibilities.
Leakage can be minimized by using filters with lower sidelobes
(Hanning). However, the trade-off is that the basic filter shape is
widened considerably.
5. Aliasing
Before any waveform can undergo digital signal processing, it must
be sampled and windowed. The sampling rate determines how well the
waveform is defined and how accurate the discrete representation is.
Nyquist's sampling theorem^ governs the rule for sampling. It
states that the sampling rate must be at least twice the frequency of
the highest frequency component of the waveform being sampled. If the
sampling rate is less than twice the highest frequency present, then
aliasing will occur.
To better visualize the aliasing problem, we refer to Fig. II-5, in
which the Nyquist frequency is 10 KHz. When the sampled waveform has
frequency components above the Nyquist frequency fN, these components
are folded about the Nyquist frequency into the frequency domain between
zero- and f^. If they fall outside this domain (see Fig. II-5b), then
they fold again around zero Hz and eventually end up between zero and f^
Hz.
If the waveform is band limited, the aliasing problem can be
avoided by the right choice of sampling rate. However, when the signal
is not band limited, as is the case in noise measurements, it is
impossible to escape aliasing by adjusting the sampling rate. In this
case, aliasing is prevented by filtering the waveform before it is
sampled. The filters limit the high-frequency contents of the waveform
to the known and acceptable cutoff frequency.


12
M
Fig. II-3. Two areas of integration are considered:
(a) square area of integration of side length T;
(b) parallelogram of height 2M. The two areas
differ by two large triangles for which the
integrand is negligible, and by two small triangles
that give a negligible contribution if T M.