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 ï¿½1M, 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 . ï¿½

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 ï¿½ 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]ï¿½ 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 ï¿½ 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) ï¿½

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 ï¿½

(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 ï¿½

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 - ï¿½n(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 ï¿½

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 ï¿½ 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 + ï¿½n(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 ï¿½(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 ï¿½ 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 ï¿½ 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 ï¿½ 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) ï¿½ (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 - ï¿½n

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 ï¿½

(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) ï¿½ (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

ï¿½ 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.