PROPERTIES OF NOISE AND CHARGE TRANSPORT

IN LAYERED ELECTRONIC MATERIALS

By

SAID 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 disser-

tation.

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 ....................... o ....... .. ... 101

6.1 Introduction.......................................101

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 CURRENT-

VOLTAGE, 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 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

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 um) 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 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

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.

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 1 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 (I) GB,

where T is the pulse length, T-1 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 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-1. 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

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 CRT

TEK 7D20 1 Dlat

Drive

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 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

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-l.

r''''''" ''''''"- -- ------I'' ''

I I

A

CCD

Output

Port

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

- 1k j

Table II-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-10MS 5I/ 1024 400 KHZ 400 KHz TIDIV

TIM/DIV TIM/DIV

Real Time 50 100

Real Time 50MS-500.S 50 1024 400 KHz 400 KHz 10T

(RD) TIM/DIV TIM/DIV

Extended 40 80 *

Real Time 200pS-2PS TIM/DIV 820 TIM/DIV 400 KHz 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 04t

defined in terms of Fourier series as10

x(t) = k akexp(j2nfkt) (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

N-1

ak NAt Atx(nAt)exp(-j2rkAfnAt) (3.2)

n=0

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

k

component fk = N- kAf. It is clear that if al is the complex

conjugate of ak,

ak = a. (3.3)

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

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

The ensemble average of x2 is found to be equal to

S= a exp(j4rfkt) + akexp(-j4fkt) + 2aka_k (3.5)

Since the Fourier coefficients ak have an arbitrary phase,

a2 = a2 = 0. Hence,

N-1 N-1

x2 = 2aka = -2 1 (At)2 x(n)x(m) exp(j2wfk(m-n)At) .

(NAt)2 n=0 m=O

(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 M, we can write

M{L

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 N-l N+M

k N2 1I 1 (At)2 x(n)x(m) exp(j2wfk(m-n)At) .

(NAt)2 n=0 m=n-M

(3.7)

We define a new variable s = m n. Then

2 N-1 M

xk 2- 1 (At)2 x(n)x(n+s) exp(j2ifksAt) (3.8)

(NAt)2 n=0 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 2 At x(n)x(n+s) exp(j2lfksAt) (3.9)

NAt s--H

s=-M

Since x(n)x(n+s) = 0 for s>JMI 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 2N I At x(n)x(n+s) exp(j2rfksAt) (3.10)

xi t k

The spectral intensity of x(t) defined by the discretized Wiener-

Khintchine theorem is given by

Sx(f) = 2 1 x(n)x(n+s) exp(j2nfksAt)At (3.11)

1 1

Since Af = = is defined as the frequency interval between

adjacent fks, the spectral intensity of the input signal can be written

as

k 21ak 2

Sf ) 1 '( f i ak (3-12)

Sx(fk) A Af (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 Ot<(T, with a frequency response given

by

g(f) T in((4.1)

rfT

The function g(t) is often called a "rectangular (uniform) time

window". The effective noise bandwidth is

00sinfTI2 1

Bff = g( df fT df (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

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 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. 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.

__ _ ____ ____ _

I

t

9

1;

rl

r

I

r

r

Fold Over

20 -10

81012 20

F (KHz)

(a)

Fold Over

F (K H z)

(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.

-2

I

-1-

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[(SDUT SA)R2 + SV] (7.1)

M2 = GB[(SMMY + + SL)R2 + S (7.2)

M3 = GB[(SDUM + S )R + SA] (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

M1 M3

SI DUT [ M3]SL + 4kT Re(Y) (7.4)

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

V

SR

I

II

I

I I

i I

_ _~R __ ____ _

----- M P.. ... .

Fig. 11-7. Equivalent input noise scheme.

22

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

DUT adinittance.

CHAPTER III

CURRENT-VOLTAGE AND IMPEDANCE CHARACTERISTICS

OF SCL FLOW IN a-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, 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 Hysell13

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 < Ti, where T is the dielectric

23

relaxation time of the unexcited specimen, and Ti 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 polytypisml,2 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....

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

To

5.048A

5.048A

4.396

To

10.46

I. 460

SILICON

CARBON

ABCABCABCA

.3C

ABCABCAB

2H

ABCABCA

4H

(a)

Fig. III-1.

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

i,

ABCABCA

6H

(b)

ABCABCABCA

15R

Fig. III-i.

T

15.12

_I

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)

(0)

(33)

(323232)

(22)

Sequence of

Inequivalent Layer

h

c

hcc

hcchc

he

Table III-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(r-~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-24 dependence was found between 300K and 800K. This was attributed

to acoustic and intervalley scattering. A relationship of the form

V6H < 15R < 4H< 13 3C 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 m0 and 4.4 mO, while

a value of (1.0 A 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 cm3 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 dependence 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.

Table III-3

A summary of electron mobilities, ionization energies

of nitrogen donor (E ), and effective masses of electrons

in n-type SiC (ND x 10 cm-3) [27] at room temperature

Mobility

Polytype (cm2/volt-sec.) ED(meV) m* /mI m*/m0

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-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 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 e = 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

5.7m

Width:.825 mm

C-axis

Gol d/ A

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.

POLYTYPE I

POLYTYPE

POLYTYPE 2

E

VI

EV2

Fig. 111-3.

Energy band structure representing the situation of a strongly compensated

polytype (polytype i) between low-resistive polytypes 1 and 2.

leads to complications in the quadratic regimes of the characteristic.

Then J scales with 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 1/n 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

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 = nO/n(x) = e n0pE(x)/J (3.1)

v = e3n03l2V(x)/Ee J2 (3.2)

w = e2n02px/e 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 = g0JELI/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:

= qpnE 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 + nt -N + N) (3.6a)

dx e 0t D A

and for donor type traps

dE q +

VE n + nd N + NA) (3.6b)

0

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 + A

zero, we have for acceptor type traps n + n = N N and for

N D A

0 0 + -

donor type traps n + nt = ND N + N Thus (3.6a) and (3.6b)

read also

dE q [(n-nO) + (nn) (n-n)] +n-n) (3.7)

dx 0 0 t

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

c t

equation is

Ynt = Bn(Nt-nt) (3.8)

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

all rates quasi-bimolecular by writing y Bn 31 where n1 is the

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

nlnt n(Nt-nt) or

n-

n n 1 (3.9)

t t

which a fortiori holds with superscripts zero added.

With Fermi-Dirac statistics

o (C- 0 )/kT

no = Nt/[g -le F

t t

+ 1] ,

(3.10)

((,6- 6 )/kT

no N e F c (3.11)

c

we find33

( ( ()/kT

nI = (Nc/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

n + n

t 1 + n1/n

(3.13)

For n we have the quasi-Fermi level description n = N

exp[( 6F(X)- 6c)/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 n1/n >> 1, and (3.13) yields

n

P n = n = constant << 1

n+n nt N

t nt t

(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

CONDUCTION

F(Xl )=Sc-KT

Fig. III-4.

I I ." --- -(X -. + KT

I I

I II

VALENCE BAND

REGION IREGIONI REGION I REGION

I I nz

X(J) X (J) X3(J) L

n=ni=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)

p = n ++ 1 (M-G regime) .

n + N-

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

Sg 0gE

dx = dE -

qnc I + gOE '

where

go qp(n0+nO)A = qn0pA ;

t

(3.16)

(3.15)

g0/L is the conductance of the unexcited specimen. For later use we

also introduce the dielectric relaxation time

T = ee /qnOi .

02

(3.17)

Equation (3.15) is immediately integrated to yield, with boundary

condition E(0) = 0:

(3.18)

x qn [gE Ian( + )]

qn g :

In particular, evaluating this at x = L and defining

(3.19)

a = goIELI/I gEL/I

we find

qncg0L 1

I= -C0 a + 2.n(1--)

(3.20)

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

1 L2 = 0

Sqncg0

[gov +

qn c

dEI g0E

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

(3.15). Let g0/I = 8 .

Then, noticing

B E 1

S+ BE dE = [BE n(l+E)] ,

1 + SE

we find from integration by parts

dE 1- nE

dE E n(+BE)

1 + BE

= [BEL n(l+BEL)]An(l+BEL) Y

(3.23)

E

Y =

0

EL

=f

0

E E n(l+SE)

dE

1 + BE

[ fi 1 1+ dAn(l+BE) 1

dE 1 1 -1d dn(+E) an(l+BE)]

1 + BE a dE

1 [n(1+) 2 .

SEL (+E) [n(l+EL)I2

Together with (3.23) and setting

EL = -a this yields

L

SE ) n(-) [n(-a)]2 .

X = EL n(I-) EL + tn(-c) [Zn(-a)]2

L L 20 2

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

(3.25)

The

result is

g0E

+I-)]

(3.21)

(3.22)

EL

0x

0

where

(3.24)

see 2 2

SL2 V + [(l-a)An(l-a) + a (Zn(l-a))2]} (3.26)

2 nc qn

Solving for V and using (3.20) this yields

qncL2 qncL2 (1-a) n(l-a) + a [An(1-a)]2

V 2 ~ (3.27)

2E0 c0 [a + an(l-a)]2

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 L (l-a)an(l-a) + a [n(l-a)]2

-= lim { -- [a + Zn(l-a) ] + L

S +l 2g0 g0 a + Zn(l-a)

^L a2 + An(l-a) + a

= --lim -- (3.28)

go a+l a + .n(l-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 sOa2

1 eqn0a2 (3.29)

I 2qnc 0L

qn L2 4qn L2

V +---- 3 (3.30)

EE 2 3a) 3e0a

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

arrive at

I = ePAV2/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 = eeAV2/L3 (3.32)

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

low-voltage quadratic regime occurs at

V 8 qn0L2 (3.33)

X 9 peec

from which

p = 1.18 x 10- n0L2/e1 (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 << n. 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 = qNtL2/2e0 (3.35)

from which

Nt 1. x 106 VTFL/L2 (3.36)

4. Experimental I-V characteristics

In Figs. 111-5 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,

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.

-1i

10

i-23

-3

I0

O.._

E 10

-5

-6

I0

-7

10 .

103 102 Io

V (volts)

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

T= 2 96 K

2

IONV

1.0 10

10'

IaV

IaV

102 10' 1.0 10

V (volts)

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

T=200K

laV/

H

I I

-7

10o,

12

10

-2

10

-3

10

K

Icav

1 Ic0V

101

-6

-7

10 .

I0" I0 2 10-' 1.0 10 102

V (volts)

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

47

-2

10

T=52.6 K

-4

10-

E i&

i3

Is I

10-

-67.

10 lc aV

-8

103 102 o10'

V(volts)

10

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

lav2

cVV2

10

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:

VTFL, VX, and R ohmicc 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 observations 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 n0 is quite small, indicative for

the insulating nature of the polytype involved. A plot of log no vs

1000/T, given in Fig. III-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 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 mO. We notice, however, that in what follows we need log Nc,

which is not too sensitive to the choice of m**. Thus 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.

n (m3)

C6-

= OmeV

1016 -- -

10 .A ..

1000

T(K)

Equilibrium free carrier density nO vs. 1000/T.

2-

I7

5.

Fig. III-9.

(c- E) (meV)

C F

100

200 300

T(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 (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 Cc- t was computed from the

equation

SgNt (6 -C t)/kT

S N --e e (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 III-3. Comparing 6c t with 6 -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

Table 111-4

Results Obtained

T(K) 296

VTFL(V) 2.80

Vx(V) 0.4

R(0) 2.9K

2(m2/V-sec)

nO(m-3)

N(m-B3) 1.23x1019

Nc(m-3) 3.0x1025

(6-C 0(eV)

l/p(eq. 3.27)

l/p(obs.)

fc- t(meV)

8.!

6.!

2.:

250 200

1.50 1.60

0.27 0.26

6.2K 6.7K

0.035 0.023

58xi016 1.38xl017

58x1018 7.01x1018

30xl025 1.64x1025

424 325

10.6 6.36

7.2 4.6

361 274

167

1.50

0.22

8K

0.020

1.33x1017

6.57x1018

1.25x1025

268

5.58

4.0

223

125

1.30

0.19

14K

0.0175

8.68xl016

5.70xl018

8.10x1024

200

7.39

5.8

167

100

1.10

0.125

23K

0.0134'

6.9x1016

4.82xl018

5.80x1024

160

6.12

4.8

130

77

0.70

0.12

40K

0.0106

5.02x1016

3.07x1018

3.92x1024

122

8.07

6.7

103

62.5

0.70

0.12

50K

0.007

6.08x1016

3.07x1018

2.87x1024

95

6.66

4.4

81

52.6

0.75

0.09

100K

0.006

3.55x1016

3.29x1018

2.12x1024

S81

8.56

7.6

67

8.

6.'.

2.3

I

the ohmic and low-voltage quadratic regimes:

ZL = r(I)F3(X,ea)/F3(1,O,a)

where

1 dvdv du 1/u-a x)/x -au

0 (1-av) v

Here r(I) is the low-frequency differential

0 = jwa, a as in Section 3, while

T and T2 being trapping-detrapping times. In

can be shown to lead to the simple result,

resistance djlV/dl,

X = (1+JmT2)/(1+juT),

the ohmic regime, (5.1)

L 1

ZL= go 1 + jwt

Note that

e0

qnO

qn 9

0A L

L- = RC ,

L qn0pA

so we have the usual result

z R

L 1 + j RC

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

(5.3)

(5.4)

(5.5)

(5.6)

.(5.1)

(5.2)

ZL = r(I)F4 (, F)/Fq(1,0) ,

where ( = jwt* t* being the drift time, and where

1 1 (X-1)/x

F4(x,) = f dv f du v() e(5P)( ) (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 limpedancel2 plots at 77K are given in Fig. III-11. We

notice that all curves can be represented by a form

r(I)

Z = r()) (5.8)

"L 1 + jwr(I)C(I)

A plot of C(I) vs V is shown in Fig. 1-12. We notice that C(I)

increases, though not as fast as r(I) decreases, (r V-) 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 M 1.1 x 10-6

C\Jc

C~_

Nu

S(Hz 106

f (Hz)

Fig. III-11. Impedancel2 vs. frequency for various voltages at 77K.

lOt

UL

CQ

77

T

2 3 71

V(

2 3

71.0

2 3

volts

Fig. III-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 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 ni n1 for that set of traps, where ni is the

density of injected carriers and n1 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 quasi-

Fermi 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 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-

recombination 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.,3 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

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) = qunE + H(x,t) (2.1)

dE

S- (q/eO) (n + nt ne) (2.2)

an(x,t) 1nn Jn(N ) J- J(x,t)

at t n(Nt q ax + Y(x,t) (2.3)

ant(x,t)

3t = 8n1nt + On(Nt-nt) y(x,t) (2.4)

Here n is the free electron density, nt the trapped density, ne is

equilibrium charge, n1 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. Since the total

current (including displacement current) Jt = J + ee08E/at 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, n0 = , etc.

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

SdE

z(x,x',W) u(x-x') I d0 1

X dx 1+0 g0E0(x')

I /E () + g0 (1-X)/ 0 + gE(x) j9/

I/EO(X) + g0 0 + go0o(x')

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

p = no/(n0+nt) go = qpncA ,

1 + j2WT2

1 + jwr /

T1 1/B(Nt-nt0) T2 = 1/0(n1+n0) 1/T = l/T1 + 1/2 .

(2.5)

(2.6)

(2.7)

(2.8)

The noise source for trapping is given by35

S (xx') = 4ABnlnt6(x-x') K(x)S(x-x')

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

LL 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 = Jf dx dx' z(x,x',w)

L 0

(2.11)

the current noise spectrum is then SAV(L)/1ZI2 The following

result, valid in both the ohmic and low-voltage quadratic regimes, was

obtained after a rather involved analysis

T2 1

SAI(w) = 4qpIOIVI0 -2(X,,a)

L2TI 1 + W2T2

1

(2.12)

where e = j ta

Section 3, see

sufficiently low

#(x,e,a) =

, a = g0IEO(L) /I0 (same parameter as in Part I,

eq. (3.19)), while 4 is normalized to be unity at

frequencies. It is given by

F2(X,6,a)/F2(1,0,a)

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

(2.13)

with

F2(x,6,a) =

v2dv 1 du 1/u a (-X)/X1 u 6/ 2

fv -1 ( ) (2.14)

(1-av)3 v av

F3(x,8,a) =

vdv 1 du 1/u a (1-x)/x au 6/X

(1-av)2 v v l- a 1 av

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

regime t is complex. For an "insulator-trapping device" (1/T << 1/T)

the function t rises beyond wl = 1/Tq to a new plateau which is 20/9

times the low-frequency plateau. It then falls off beyond

(2.15)

S2 = 1/T The roll-off is complex. For a "semiconductor-trapping

device" (1/T << /T ) the function 0 has not yet been fully studied.

However, we believe that it will only have a mild structure between

,1 = 1/T and w2 = 1/Ta ; 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), S' is a number

between 0.9 and 1.0 throughout both regimes. Further, uIV01/L2 is of

the order of the transit time t see references 30, 34 and 40, while

T/TI is the statistical factor

T 2 t tO

n T1 T1 + T2 Nt nt +n + n (2.16)

Thus the low-frequency plateau is also ~ 4 ql0 (T/t*)(/n) ,

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, /n + 0 As we noted above, if the trapping time

constants i are far enough apart, the multiple trap spectrum is

approximately a sum of Lorentzians with appropriate statistical factors,

i.e.,

4quI0oV0' 1

S AI() =i Ti I + 2 (KXi,,a) (2.17)

L2 i li 1 + T2, i

with

n Nti ntiO + ni + 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

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

velocity,

1/Ti = i[(Nti-nti,) + n,i + nO] (2.19)

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

nti0 << Nti. We noted before (Table 111-4, chapter III) that n0

(thermal equilibrium value) is of order 1010 cm-3 while Nti is of order

1012 cm-3. The ratio N /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,

L g- )IkT

T e (2.20)

i 0 N

i c

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 nTi 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-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 I2 in all cases.

From the data we determined the time constants involved from the

best fit. These Tis are plotted vs 1000/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.

*5

OS

T=296 K

\ ,

S.

- S

T=250K

-23

IO0

-24

I0

10

106

104

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.

S1(A/Hz)

-18

-19

10

-20

10

-21

-19

10

-21

10

10' 102 I I

f(Hz)

Fig. IV-2.

Current

and T =

lines:

noise spectral density in ohmic regime for T = 200K

175K. Circles o and 0 : measured data. Full

resolution into Lorentzians.

-20

10

-2f

10

-23

10

-24

10

-25

10

-26

10

1C

105

T=IOOK

a 0 M

an a

T= 25K

f(Hz)

Fig. IV-3.

Current

and T =

lines:

noise spectral density in ohmic regime for T = 125K

100K. Circles 0 and squares: measured data. Full

resolution into Lorentzians.

S 1(2/Hz) io

-2O

10.

16

-23

10

21

0

li3

io5

S (A/Hz)

T= 62.5 K

* *

T=77K

101

-26

10' *1(

Fig. IV-4. Current

and T =

lines:

-2.5

noise spectral density in ohmic regime for T = 77K

62.5K. Circles o and 0 : measured data. Full

resolution into Lorentzians.

-22

10

-23

10

-24

10

21"

8

10 I I I II I

3 5 7 9 II 13 15

IOOOQT(K)

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

5 7

9 II

I000/T(K)

Fig. IV-6.

Relative plateau values S /12 of the Lorentzian spectra spectra as a function

of 1000/T.

-10

C 10

CO

-12

I0

-14

3

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 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 /12 = 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

72

off occurs at TO = 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 11I-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 106A ohmic regime

3 x 10-6A < I < 3 x 104A 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 ) 6.62xl0-15 9.8xl0-15 2.40xl0-18 1.27xl0-17 2.5xl0-17 3.05x10-17 4.87x10-17

Table IV-2

Comparison between the trap activation energies as found

from the I-V curves (Table III-4) and noise

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

meV

(noise -- 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 C.

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 0lOV0 see eq. (2.17),

modified by the factor /n0 of eq. (2.18). Thus, in Fig. IV-11 we

plotted S AR/IO 11V 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., S IJV101 ,

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

n + Nti so that (An2>/n0 + 0 see eq. (2.18). Thus the behavior

plotted in Fig. IV-11 is in most respects as expected.

T=77K

-a

(/)

I=717xIO6A

Id

I0 10

f(Hz)

-6

Fig. IV-7. Current spectral density at 77K and I = 7.17 x 10- A

ohmicc regime).

-20

10.

-23

10

jc4

N~

10o

10-

(I)

T=77

.

*

0

S

0

0.

0

0

K

I=2.5 xIOA

f(Hz)

-4

Fig. IV-8. Current spectral density at 77K and I = 2.5 x 10 A (low-

voltage quadratic SCL regime).

0 *

T=77K

*

09

*\ 0

\0

-21

i i

1=3.74 x10o A

-23

10 0 1 1 LfI

f(Hz)

Fig. IV-9. Current spectral density at 77K and I = 3.74 x 10-

(TFL regime).

-18

0

-19

0

C()

T(S)

T=77 K

0

2j

0di

O 7

10i6

105(

I(A)

104

Fig. IV-10.

Observed time constants of the Lorentzian spectra at 77K as

a function of current I0.

ohmic--l-- -ow volt. quad.*.-I-- .TFL --

I

10

I0

10

16n

id,5L

Fig. IV-11.

T=77K

ohmic4-Ilow Volt. quad. -*- TFL *

S1(A)

Normalized current spectral density S R/Io lvo as a

function of current IO.

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 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.

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 + H(x,t) (2.1)

dE

d-E -(q/e )(n + n) (2.2)

an(x,t) 1n aJ(x,t)

t t t nt) q x + (x,t) (2.3)

ant(x,t)

t -8n1nt + 3n(Nt nt) T Y(x,t) (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, 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. 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, n, = , etc.

The equations describing the DC part are related to the quasi-Fermi

potential (4 n) and the electrostatic potential (Xc) as

Qe_ + (2.5)

eE\ ^ + nt0 0<)

aI

--= V.(Aq nnO(x)Vn(x) = 0 (2.6)

The density of the trapped electrons is related to the quasi-Fermi

potential $Fn(x) and the trap potential Xt(x) by Fermi-Dirac statistics,

i.e.,

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

where g is the electron spin degeneracy and jq(x 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) = Ncexp(-q( n(x) Xc(x))/kT) (2.8)

where N 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 nn and nn+

contacts and thus

83

+

n = N (3.1)

Consequently, the Fermi potential at few Debye lengths away from the

electron injecting contact (cathode) is given by

X kT DN

'Fn Xc q- n

c

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

obtain for Fn

Fn

kT

Fn q n (N) (3.2)

cFn q

The Fermi potential few Debye lengths away from the electron extracting

contact (anode) is given by

N+

kT

n= VD -q (N ) (3.3)

c

where VD is the applied voltage.

The coupled system of partial differential equations

F-q(n(x) Xc(x))

Fl(X Fn) c2(x) + (Nexp( kT

-1 -Et -q(c(x) n())

+ NT/[I + g exp(- )exp( kT n

(3.4)

and

-q(n((x) Xc(x))

F2(XC,Fn) = V[(exp( kT ) V7Fn(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

-q.0 n(x)

since V[exp( T ) n] indicates a poorly scaled, first-order

derivative due to the variation of OFn(x). Hence we define a new

variable

-q bF

U = exp(- n). (3.6)

The variable U also has the advantage of having much greater variation

than the variable Fn. So equation (3.5) becomes

qxc(x)

F2(X,U) = 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

/

qXc/kT qx /kT

Xc [e VU](K+1), [e Vu]K'

V[e [ Vu] + ) K'] (3.8)

kT [(K + 1)' K']Ax

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+1' Ax

and

V, u(K) u(K-1)

VuK' = u Ax

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

C

d ex q(x)

dx exp[ MkT

q

kT

qxc(x) dx (x)

exp[ kT i .

After rearranging (3.11), we get from an integration

K.Ax -qxc(x)

f exp( kT

(K-1)Ax

qX (x)

[exp( kT )]dx

KAx dx (x)

I dx dx = x(K) c(K-1)

(K-1)Ax x

(3.12)

3

According to the mean value theorem for integrals, there exists a value

K',K-1

KAx

(K-l)Ax

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

S

(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 can be discritized

kT KAx -qX (x) qX (x)

f exp( ) d [exp( )dx

q (K-1)Ax kT kT

-qX KAx q (x)

= exp( ~T) [-- exp( kT )]dx (3.14)

K' (K-1)Ax

or

e -qX(K') xc(K) X,(K-1)

exp[ T ] (3.15)

kT e qXc(K) qx(K-1) (3.15)

exp] exp[ kT

A similar approach gives us an expression for

qxc(K+1)'

exp( kT

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

FK1 K(m+l) K ,UK + 6UKm)

S*-6Xc -F 1(Xc cu)

(3.16)

3FK

2 6uK(m+l) K + K(m+l) ,K)

aT- U = -F2 (c c XU

K(m+1) K(m+1)

where we solve for xK(ml) and UK(m) with K denoting the

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

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

location of (K+1)' and K' is unimportant and does not affect the value

of Xc.

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 Xc(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-1 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

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.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

T=77 K

/ 2

laV

S2

ieV

V(v)

I-V characteristics at T = 77K. The dots indicate the

measurements. The solid line represents the results of

computer calculations.

(A)

i7

10

Fig. V-1.

Linear Regime

I=6X 10 -7A

1021:

S1020

t

E

% 1 01 Trap

-- C----arrier

S\Concentratio n,

,10 nt(x) ,

'- 1018-

101

Carrier

Concentration

106 n(x)

1010 5 10 15 20

L(pm)

Fig. V-2. Carrier concentration profile of the trap level and the

conduction band in the linear regime (I = 6 x 10-7A).

Linear Regime

I = 6 x 107 A

T=77K

V

Conduction

----- Fermi Level

-rap Level

n+ a n

\

\\

\

\

Band

lI

SI I I

0 5 10

L (um)

15 20

Energy band diagram in the linear regime (I = 6 x 10-7A) 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.

0.08

.02 H

-0.02 k

EF

/E

-I-~

ET

-0.04

-0.06

-0.08

0.10

-0.12 -

-0.14

Fig. V-3.

. .. _

0.06

0.04 -

10-2

10-3

10

-4

10

10-6

-7

10

10-8

-9

10

T=77K

Linear Regime

1=6 x10 7A

+ +

\ 'drift \

Potential

Minimum drift

-._^--+--1-C-7- ----*

Idiffusion-'-

-- n+ --n ------4 n+ -*

0 5 10 15 20

L(Um)

Current diagram in the linear regime (I = 6 x 10-7A) 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.

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. 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

field occur at n+n contacts due to sudden change in carrier

concentration, Fig. V-1l.

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 (IaV2).

Linear Regime

1=6X10 -7A

1.0-

.81-

-.2

-.6

Fig. V-5.

- n+" n n--

I I I .

-5 0 5 10 15 20

L(pm)

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

at 77K.

Low Voltage Quadratic

I = 3.92 x10-5 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.