Deep-level impedance spectroscopy of electronic materials

MISSING IMAGE

Material Information

Title:
Deep-level impedance spectroscopy of electronic materials
Physical Description:
xiv, 221 leaves : ill. ; 29 cm.
Language:
English
Creator:
Jansen, Andrew Norbert, 1964-
Publication Date:

Subjects

Subjects / Keywords:
Impedance spectroscopy   ( lcsh )
Varistors   ( lcsh )
Semiconductors   ( lcsh )
Thin films   ( lcsh )
Chemical Engingeering thesis Ph. D
Dissertations, Academic -- Chemical Engineering -- UF
Genre:
bibliography   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1992.
Bibliography:
Includes bibliographical references (leaves 212-220).
Statement of Responsibility:
by Andrew Norbert Jansen.
General Note:
Typescript.
General Note:
Vita.

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 001862348
oclc - 28922392
notis - AJT6815
System ID:
AA00002091:00001

Full Text









DEEP-LEVEL IMPEDANCE SPECTROSCOPY
OF
ELECTRONIC MATERIALS









By


ANDREW NORBERT JANSEN


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
















ACKNOWLED GEMENTS


The author thanks the following individuals:

to my wife Rosalie for her love and encouragement since we first met,
to my parents Charles and Shirley for their love and guidance throughout
the years,
to Professor Mark E. Orazem for his advice and support during my grad-
uate studies,
to Pankaj Agarwal for his Indian recipes and numerical regression exper-
tise,
to Matthew Esteban, Oliver Moghissi, and Paul Wojcik for their assistance
and friendship,
to Professor Chih-Tang Sah for his discussions and reprints on semicon-
ductor physics and,
to Professors Timothy J. Anderson, Oscar D. Crisalle, and Kevin S. Jones
for their suggestions during the course of this work.


The author thanks Mr.


Darwin Thusius of Sula Technologies for the GaAs sample


and discussions on the DLTS instrumentation.


The author also thanks Dr.


David


Morton and Dr.


Robert Miller of the U.S.


Army


Electronics


Technology and De-


vice Laboratory for providing the ZnS:Mn TFEL panel and Dr.


Raychem Corporation for providing the ZnO varistors.


DARPA


Gary Trost of the


This work was supported by


under the Optoelectronics program of the Florida Initiative in Advanced


11. snlan4.n: n a &a4 .. a1.














TABLE OF CONTENTS


CHAPTERS


INTRODUCTION
BACKGROUND


a a a a a a a a a a S S 0 0 5 a 0 * a S
. . . . . . . . * * *


EXPERIMENTAL METHOD AND INSTRUMENTATION


General Schematic


Electrochemical Impedance Spectroscopy


3.2.1
3.2.2
3.2.3


Theory * *
Signal Generation and Analysis
Experimental Difficulties .


Standard Characterization Techniques


. a a a a .26


3.3.1
3.3.2
3.3.3


Mott-Schottky Profiling . .
Deep-Level Transient Spectroscopy (DLTS)
DLTS Equipment and Parameters .


32


Optical Spectroscopy


DEVELOPMENT OF PROCESS MODEL
4.1 Shockley Read Hall Processes


4.1.1
4.1.2
4.1.3
4.1.4


Equilibrium . .
Non-equilibrium .
Small Signal Analysis
Equivalent Circuit


Surface States


4.2.1
4.2.2


S . a * 0 . .74


Single Energy Level Surface State
Surface Band . .


Complete Equivalent Circuit


4.3.1


j a


a a a S S S S a a a a a a82


Simplifications of the Complete Circuit . .
A 1 0 .1 13% 1 i a/l" < i l I


ACE(NOWI;EDGEMENTS
LIST OF TABLES ................... ..............
LIST OF FIGUIEIES
AIISTRACT









Cuprous Oxide in Alkaline-Chloride Solutions


99


5.4.1
5.4.2


Background
Experiment


* S 9 S 9 9 S 9 . 9 9 9 9 9
* . 9 . S 9 9 9 9 S 9 9 9 9


RESULTS AND DISCUSSION
6.1 Gallium Arsenide


9 9 9 9 . 9 9 S 106
. 9 . 9 . 9 9 5 9 .106


6.1.1
6.1.2


Thermally Stimulated Deep-Level Impedance Spectroscopy
Optically Stimulated Deep-Level Impedance Spectroscopy


. 106
. 112


Importance of Weighting on Regression Results . . . 117


6.3 ZnS:Mn TFEL Panel .. .. .. ..
6.4 ZnO Varistors . . . . .
6.5 Cuprous Oxide in Alkaline-Chloride Solution
6.6 Comparison of Impedance and Admittance.
6.7 Three-Dimensional Representation .
CONCLUSIONS . . . . .
SUGGESTIONS FOR FUTURE WORK .....


S. .. 120


APPENDIX


PROGRAM FOR DLTS INTERFACE


REFERENCES


S9 205


5 .9 .. .. 9. .. .. .9. *212


BIOGRAPHICAL SKETCH .


221


126


200
202













LIST OF TABLES


A comparison of methods used to identify deep-level states.

Comparison of values from DLZS, DLTS, and literature [71].


. . 143


Model variance weighting parameters used in regression of impedance
data. Values were determined by Pankaj Agarwal.. . .













LIST OF FIGURES



Energy band diagram of a n-type semiconductor with a deep-level state.


Experimental setup for Deep-Level Impedance Spectroscopy.


Basic inverting operational amplifier circuit (a) and overly simplified
equivalent circuit of an operational amplifier (b).. . . .

Simplified block diagram for the Solartron 1286. . . .

Simplified block diagram for the PAR 273. . . . .

Equivalent circuit from regression of (a) Solartron 1286 and (b) PAR
273 impedance data. . . . . . .

Comparison of Solartron cell data and equivalent circuit data for the
Solartron and the PAR potentiostats in the form of an impedance plane


plot.


. S S S S S S S S S S S S S S S S S S S S jt


Comparison of Solartron cell data and equivalent circuit data for the
Solartron and the PAR potentiostats in the form of phase angle as a


function of frequency.


. S S S S S S S S S S S S S z 5


Comparison of PAR cell data and equivalent circuit data for the So-
lartron and the PAR potentiostats in the form of an impedance plane
plot. .. . . . . . .


Comparison of PAR cell data and equivalent circuit data for the So-


lartron and


the PAR potentiostats in


the form of phase angle as a


function of frequency.


Frequency dependence of gain and phase shift for control op amp [27].


Compensated


(stable)


uncompensated gain


and phase shift for


control op amp [27] .


Energy


levels as a function


distance for a n-type semiconductor


- t t --- -r .- t -- -- -








Intensity of the focused light as a function of the wavelength for the
410-720 nm order sorting filter with 1200 and 600 grooves/mm diffrac-
tion gratings and the spectrometer exit/entrance slit width as param-
eters. . . . * * * * *


Intensity of the focused light as a function of the wavelength for the
720-1350 nm order sorting filter with the 600 grooves/mm diffraction
grating and the spectrometer exit/entrance slit width as parameters.


Intensity of the focused light as a function of the wavelength for the
1200-2000 nm order sorting filter with the 300 grooves/mm diffraction
grating and the spectrometer exit/entrance slit width as parameters.


3.17 Intensity of the focused light as a function of the wavelength for the
1800-3000 nm order sorting filter with the 300 grooves/mm diffraction
grating and the spectrometer exit/entrance slit width as parameters.


Shockley Read Hall processes of a deep-level state.


Equivalent circuit corresponding to the small signal non-equilibrium
case for a bulk deep-level state. Equations and notation are given in
the text.

Equivalent circuit corresponding to the small signal non-equilibrium
case for a surface state. Equations and notation are given in the text.

The complete non-equilibrium equivalent circuit with surface and bulk
deep-level states for a Schottky barrier on the left side and an Ohmic
contact on the right side with no DC leakage current. . .

The reduced equilibrium equivalent circuit with surface and bulk deep-
level states for a Schottky barrier on the left side and an Ohmic contact
on the right side. .

Simplified equilibrium equivalent circuit with only surface states (a),
and only bulk deep-level states (b) for a Schottky barrier on the left
side and an Ohmic contact on the right side. . .

The reduced equilibrium equivalent circuit used in this work with a
Schottky barrier on the left side and an Ohmic contact on the right side.


Cell configuration for the copper electrode experiments.


Imaginary component of the impedance as a function of the real com-
nonent of the imnedance for freauendes ranrine from 65000 to 1 Hz








Capacitances from regression of simplified equivalent circuit to impe-
dance data as a function of inverse temperature for n-type GaAs with
Schottky contact. .. .. . .. .. .. .. .. ... .


Energy diagram of a shallow donor state (El), a midgap state (E2)
slightly below the Fermi level (EF), and a shallow acceptor state (E3)
for negative half cycle (upper) and positive half cycle (lower) of sinusoid.135


Inverse of resistances from regression of simplified equivalent circuit to
impedance data as a function of inverse temperature for n-type GaAs
with Schottky contact. . . . .


6.6 Equivalent circuit characteristic frequencies from regression of simpli-
fied equivalent circuit to impedance data as a function of inverse tem-
perature for n-type GaAs with Schottky contact. . . 137

6.7 Product of temperature and equivalent circuit resistance as a function
of inverse temperature for n-type GaAs with Schottky contact.. 138

6.8 Product of temperature and equivalent circuit capacitance as a function
of inverse temperature for n-type GaAs with Schottky contact.. 139

6.9 Equivalent circuit characteristic frequencies divided by the square of
temperature as a function of inverse temperature for n-type GaAs with
Schottky contact. .. . .. . . .. .. 140

6.10 DLTS emission rate divide by square of peak temperature as a function
of inverse temperature for n-type GaAs with Schottky contact.. 141


Mott-Schottky plot for n-type GaAs with Schottky contact.


. 142


Imaginary component of
ponent of the impedance
with optical energy as a
contact . .


the impedance as a function of the real com-
for frequencies ranging from 20000 to 4 Hz
parameter for n-type GaAs with Schottky
. . . 144


Dimensionless real component of the impedance as a function of optical
energy with electrical frequency as a parameter for n-type GaAs with
Schottky contact. . . . . . .

Dimensionless imaginary component of the impedance as a function
of optical energy with electrical frequency as a parameter for n-type
GaAs with Schottky contact.. . . . . .

Dimensionless real component of the imnedance as a function of lifht








6.16



6.17


6.18



6.19


6.20


6.21


6.22


Dimensionless imaginary component of the impedance as a function of
light intensity at 1.1 eV with electrical frequency as a parameter for
n-type GaAs with Schottky contact.

Optical cross sections from DLOS experiments as a function of optical
energy, from references [98, 99].

Dimensionless total impedance as a function of optical energy with
electrical frequency as a parameter for n-type GaAs with Schottky
contact.

Dimensionless phase angle as a function of optical energy with electrical
frequency as a parameter for n-type GaAs with Schottky contact.

Resistances from regression of impedance data to equivalent circuit as
a function of optical energy for n-type GaAs with Schottky contact.

Capacitances from regression of impedance data to equivalent circuit
as a function of optical energy for n-type GaAs with Schottky contact.

Characteristic frequencies from regression of impedance data to equiva-
lent circuit as a function of optical energy for n-type GaAs with Schot-
tky contact. .. .


Total system energy as a function of defect position..


Resistances from regression of impedance data to equivalent circuit
with no weighting as a function of optical energy for n-type GaAs
with Schottky contact. . . . . . . 156


Characteristic frequencies from regression of impedance data to equiva-
lent circuit with no weighting as a function of optical energy for n-type
GaAs with Schottky contact. . . . . .


6.26 Resistances from regression of impedance data to equivalent circuit
with proportional weighting as a function of optical energy for n-type
GaAs with Schottky contact. .. . .


Characteristic frequencies from regression of impedance data to equiv-
alent circuit with proportional weighting as a function of optical energy
for n-type GaAs with Schottky contact. . . . .


6.28 Resistances from regression of impedance data to equivalent circuit
with modulus weighting as a function of optical energy for n-type
GaAs with Schottkv contact.. ... .... .....








6.30 Imaginary component of the impedance as a function of the real com-
ponent of the impedance for frequencies ranging from 0.2 to 100 Hz
and with optical energy as a parameter for the ZnS:Mn TFEL panel.

6.31 Dimensionless real component of the impedance as a function of optical
energy with electrical frequency as a parameter for the ZnS:MN TFEL
panel. . . * * * * *

6.32 Dimensionless imaginary component of the impedance as a function of
optical energy with electrical frequency as a parameter for the ZnS:Mn
TFEL panel.


Change in photon-released residual charge as a function of photon
energy for a ZnS:Mn TFEL panel from reference [108]. . . 166

Total impedance as a function of frequency with optical energy as a
parameter for the ZnS:Mn TFEL panel.. . . . 167

Phase angle as a function of frequency with optical energy as a param-
eter for the ZnS:Mn TFEL panel. 168


6.36 Resistances from regression of simplified equivalent circuit to impe-
dance data as a function of inverse temperature for the ZnS:Mn TFEL
panel.


6.37



6.38



6.39


Capacitances from regression of simplified equivalent circuit to impe-
dance data as a function of inverse temperature for the ZnS:Mn TFEL
panel. . . . . * * *

Equivalent circuit characteristic frequencies from regression of simpli-
fied equivalent circuit to impedance data as a function of inverse tem-
perature for the ZnS:Mn TFEL panel.

Imaginary component of the impedance as a function of the real com-
ponent of the impedance for frequencies ranging from 20000 to 0.05 Hz
with optical energy as a parameter for the unaged ZnO varistor.


6.40 Imaginary component of the impedance as a function of the real com-
ponent of the impedance for frequencies ranging from 1500 to 0.2 Hz
with optical energy as a parameter for the aged ZnO varistor. . 173

6.41 Dimensionless real component of the impedance as a function of optical
energy with electrical frequency as a parameter for the aged ZnO varistor. 174
6.42 Dimensninmlpn s ima.oinarv mmnnnpnt nf t.h imp rlann ao a fmnnmn n4r








6.43



6.44



6.45



6.46


Dimensionless real component of the impedance as a function of optical
energy with electrical frequency as a parameter for the unaged ZnO
varistor

Dimensionless imaginary component of the impedance as a function of
optical energy with electrical frequency as a parameter for the unaged
ZnO varistor. .

Resistances from regression of simplified equivalent circuit to impe-
dance data as a function of inverse temperature for the unaged ZnO
varistor.

Capacitances from regression of simplified equivalent circuit to impe-
dance data as a function of inverse temperature for the unaged ZnO
varistor. .


Equivalent circuit characteristic frequencies from regression of simpli-
fled equivalent circuit to impedance data as a function of inverse tem-
perature for the unaged ZnO varistor.. . . . 180

Imaginary component of the impedance as a function of the real com-
ponent of the impedance for frequencies ranging from 1500 to 0.2 Hz
with optical energy as a parameter for the unaged ZnO varistor an-
nealed at 600C.. . . . . 181


Total impedance as a function of frequency under no illumination and
illumination of 1.77 eV for the aged and unaged ZnO varistors annealed
at 600 C. * * .

Phase angle as a function of frequency under no illumination and illu-
mination of 1.77 eV for the aged and unaged ZnO varistors annealed
at 6000C. ..


Imaginary component of the impedance
ponent of the impedance for frequencies
with optical energy near the bandgap as
varistor.


as a function of the real corn-
ranging from 1500 to 0.2 Hz
a parameter for the aged ZnO
S . 0184


Imaginary component of the impedance as a function of the real com-
ponent of the impedance for frequencies ranging from 65000 to 0.1 Hz
with optical energy as a parameter for cuprous oxide (aged in light) in
alkaline-chloride solution. . . . . . 185


6.53 Phase angle as a function of frequency with optical energy as a param-








Dimensionless imaginary component of the impedance as a function
of optical energy with electrical frequency as a parameter for cuprous
oxide (aged in light) in alkaline-chloride solution. . . 188

Black body radiation intensity per wavelength as a function of wave-
length with temperature as a parameter, from Planck distribution law. 189


Imaginary component of the impedance as a function of the real com-
ponent of the impedance for frequencies ranging from 10000 to 0.2 Hz
with optical energy as a parameter for cuprous oxide (aged in dark) in


alkaline-chloride solution.


. . . S .* 4 190


Phase angle as a function of frequency with optical energy as a param-


eter for cuprous oxide (aged in dark) in alkaline-chloride solution.


. 191


Dimensionless real component of the impedance as a function of optical
energy with electrical frequency as a parameter for cuprous oxide (aged
in dark) in alkaline-chloride solution. .. .. ..


Dimensionless imaginary


component of the impedance as a function


of optical energy with electrical frequency as a parameter for cuprous
oxide (aged in dark) in alkaline-chloride solution. . . .

Density of states as a function of energy for a disordered semiconductor


(left) and a crystalline semiconductor with a discrete state (right).


. 194


Imaginary component of the impedance and of the admittance as a
function of electrical frequency with optical energy as a parameter for
the aged ZnO varistor. . . . . . .


Dimensionless real component of the impedance as a function of optical
energy and electrical frequency for n-type GaAs with Schottky contact.


Dimensionless imaginary component of the impedance as a function of
optical energy and electrical frequency for n-type GaAs with Schottky


contact.


Dimensionless real component of the impedance as a function of optical
energy and electrical frequency for the ZnS:Mn electroluminescent panel.198


Dimensionless imaginary component of the impedance as a function of
optical energy and electrical frequency for the ZnS:Mn electrolumines-


cent panel.


.. 199













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


DEEP-LEVEL IMPEDANCE SPECTROSCOPY
OF
ELECTRONIC MATERIALS


By


ANDREW NORBERT JANSEN

December, 1992


Chairman: Prof. Mark E. Orazem
Major Department: Chemical Engineering


Deep-Level Impedance Spectroscopy (DLZS) is suggested to be a new approach


to analyze transitions of deep-level states in electronic materials.


The technique is


based on interpretation of both the real and imaginary components of the impedance


response over a continuous range of electrical frequencies.


The two parameters that


were varied, temperature and optical energy, directly influenced the transition rate

or occupancy of deep-level states.

Transition rates of the deep-level states were determined to have an Arrhenius

relationship with temperature, indicating activation energy controlled reactions. Mea-

surement of the impedance response as a function of temperature is termed Thermally








deep-level state and a carrier band.


This variant of the technique is termed Optically


Stimulated Deep-Level Impedance Spectroscopy (OS-DLZS).

A mathematical model was developed from mass balance equations and Poisson's

equation to aid the analysis of the impedance data. For convenience during the regres-

sion, the model was posed in terms of an equivalent circuit of resistors and capacitors

whose values were related to system properties such as reaction rate constants and


concentrations. The data were regressed to the equivalent circuit through complex

nonlinear regression. The influence of temperature, monochromatic light, and elec-

trical frequency was evident in the impedance data and the regressed resistances and


capacitances.


The results from DLZS (data and regressed parameters) and the well-


accepted Deep-Level Transient Spectroscopy (DLTS) for the EL2 state in GaAs were


found to be in close agreement.


This technique was also used to analyze semicon-


ducting materials such as ZnS:Mn thin film electroluminescent panels, ZnO varistors,

and cuprous oxide on copper in alkaline chloride solutions.













CHAPTER 1

INTRODUCTION



Deep-Level Impedance Spectroscopy (DLZS) is presented here as a new technique

for characterization of electronic transitions in large band-gap semiconductor devices.


This technique involves an analysis of the real and imaginary


components of the


impedance response over a continuous range of applied frequency under the influence

of sub-bandgap illumination and/or temperature.


A history of impedance techniques is presented in Chapter 2.


This chapter deals


only with


techniques that are relevant to


the work presented in


this manuscript.


There are a considerable number of techniques and their variations used to analyze

semiconductors that are described elsewhere in literature.


The experimental theory and application is discussed in Chapter 3.


This discussion


is designed to bring the next investigator up to speed on the experimentation and the


details behind the impedance measurements.


For those who have a good electronics


background, much of this chapter may serve as a review.


Chapter


4, some physical insight is provided as to the nature of electronic


transitions between deep-level states and the carrier bands.


The governing equations


are developed and transformed into a simplified equivalent circuit.

A brief description of the samples used in this work is given in Chapter 5.


samples described are:


n-tvue


GaAs. a ZnS:Mn thin film pwlp.rtrnlmnonpFant+ annl





2

troscopy, and hopefully, provide new insight to the semiconducting nature of these


materials.


Chapter


7 is a summary of the key points in this work, and suggestions


for future work are presented in Chapter 8


The computer program used to interface


the DLTS equipment is included in the Appendix.













CHAPTER 2

BACKGROUND


In contrast to most spectroscopic techniques,


Optically Stimulated Deep-Level


Impedance Spectroscopy (OS-DLZS) can be regarded as encompassing two frequency


domains, one that is electrical and one that is optical.


The potential utility of em-


playing a broad electrical frequency range is consistent with experimental observation

that surface states have the largest influence on the impedance response at low fre-

quencies [1, 2] and that the space charge capacitance is obtained most easily from

high frequency measurements.

The application of sub-bandgap optical excitation of deep-level states was sug-


gested


body


work


describing


Electrochemical


Photocapacitance


Spec-


troscopy (EPS) [3-6].The emphasis on interpretation of both the real and imaginary

components of the impedance was driven by the results of a mathematical model [7-


l0]that


treated


the influence of


deep-level states on


the impedance response of a


semiconductor by solving the equations which govern the physics of the system, e.g.,


Poisson's


equation, conservation equations for electrons and holes,


and homogeneous


and heterogeneous rate expressions for generation and recombination.


The modeling


work suggested that, through use of monochromatic sub-bandgap optical excitation,

the influence of even low concentrations of deep-level electronic states could be seen

on the real Dart of the imnedance measured at low electrical frenllnncies






4

to determine their concentration distribution and the associated rate constants for

electronic transitions. Knowledge of these parameters is essential for the engineering

of many electronic devices.

For example, deep-level states are undesirable when they facilitate electronic tran-


sitions which reduce the efficiency of photovoltaic cells.


In other cases,


the added


reaction pathways for electrons result in desired effects.


Electroluminescent panels,


for example, rely on electronic transitions that result in emission of photons.


The en-


ergy level of the states caused by introduction of dopants determines the color of the

emitted light. Interfacial states are believed to play a key role in electroluminescence,

and commercial development of this technology will hinge upon understanding the

relationship between fabrication techniques and the formation of deep-level states.


Deep-level states also influence the performance of solid-state varistors.


While this


technology is more established, development of new processing recipes will be facili-

tated by understanding how fabrication techniques, composition, and aging processes


influence the energy and location of deep-level states.


Copper in alkaline-chloride


solutions (marine environment) does not suffer from stress corrosion cracking if its


oxide film forms under illumination[ll].


Deep-level states may be involved because


cuprous oxide is a large band gap semiconductor.


The impact of deep-level states can be significant,

are very low by normal chemical standards. Several sta


even in concentrations that


Ltes can be associated with a


chemical species, and such states may also appear as a result of vacancies or other


crystalline defects.


Traditional chemical means of detection, therefore, do not provide


complete identification of


deep-level


electronic states.


The techniques commonly


, r.,-, ,, +a 4, iAa/,.+ ,Aanlnral a+ fa +,,-,A A ^ nk0 ,l,,4n*-, 1 i -,n + -+n.aa, 4.+ o 4a I ,l.n. el,,








The electronic behavior of an electron donor deep-level state with energy


ET is


illustrated in Figure 2.1. An electron can be emitted from the deep-level state to the

conduction band if it acquires energy of (Ec Er). Likewise, transfer of energy equal

to (ET Ev) can excite an electron from the valence band to an ionized deep-level


state,

(ED-


hole emission.


An electron in the deep-level state that acquires energy


Er) can be excited to the donor state.


Under equilibrium conditions,


occupancy of the state is governed by the Fermi-Dirac distribution function.


In the


absence of illumination, the deep-level state will influence the impedance response of

the semiconductor only if the probability of occupancy of the state is close to 1/2,


i.e., if the state energy is close to the Fermi energy.


If, for example, the probability


of occupancy is essentially zero, the unfavorable energetic of the transition prevents


the electrical excitation from moving electrons into the state.


On the other hand, if


the probability of occupancy is essentially unity, the need to satisfy Pauli exclusion

prohibits transfer of electrons into the state. Similar arguments apply for transferring

electrons out of the state.

At an interface with a dissimilar material, a redistribution of charged species oc-

curs which results in the formation of a space charge region in the semiconductor. A

space charge region can be formed at an interface with a metal, another semiconduc-

tor, or an electrolyte. In polycrystalline materials, a space charge region can also be

formed at a grain boundary. Within the space charge region, the electron energy for

the valence band, the conduction band, and the deep-level state can be described as


varying with reference to a fixed Fermi energy.


Thus the probability of occupancy of


the deep-level state is a function of position, and, since the degree of band bending

is determined by the Dotential annlied to the svstemn the der, nf nnnrmanmrv u a








level states.


The energy needed to excite deep-level electronic transitions can also


be provided through illumination. Monochromatic illumination with energy less than

the bandgap energy is preferred because band-to-band transitions can easily obscure

the transitions involving deep-level states.

The distinguishing features of some electrical techniques used to characterize deep-


level states are presented in


Table 1.


Anomalous changes in the space charge ca-


pacitance with applied potential have been attributed to the influence of deep-level


states.


In the absence of deep-level states, Mott-Schottky theory (see for example


references [12, 13]) suggests that the inverse of the square of the space charge capac-


itance be a linear function of applied potential.


The slope in this plot is proportional


the concentration of


carriers and


the intercept is related by a constant to the


flat band potential.


Changes in the slope of the Mott-Schottky plot can be related


to nonuniform distribution of dopants or to potential-dependent ionization of deep-


level states [8].


The presence of deep-level states is suggested if the capacitance is a


function of the electrical frequency at which the capacitance is measured.


Development


electrochemical


photocapacitance


spectroscopy


(EPS)


6]extended analyses based on Mott-Schottky theory by using monochromatic sub-


bandgap illumination to change the occupancy of deep-level states.


Changes in the


space charge capacitance, measured at a fixed electrical frequency (usually around


1000 Hz) and applied potential,


were observed as a function of photon energy.


energy at which a change in capacitance was observed was the energy of an allowed


electronic transition.


This approach can be used to map out the energy structure


of an interface because transitions from


valence band


to the deep-level state


reduced the snace charge canacitance. and transitions from the deen-level state to






7

be very small as compared to that associated with doping species, the sensitivity of

capacitance-based techniques requires precise measurement of the imaginary part of

the impedance response.

Admittance Spectroscopy (AS) is a relatively new technique used to analyze deep-

level states [14-18].In AS, the conductance (and the capacitance) of semiconductor

samples are determined as a function of temperature at selected electrical frequencies.

A plot of the conductance as a function of temperature is observed to have a peak.

This peak temperature corresponds to a condition in which the emission rate of the


trap equals the frequency of the perturbing sinusoid.


An Arrhenius curve results if


the frequency divided by the square of the peak temperature is plotted as a function


of inverse temperature.


The slope of this curve is proportional to the trap energy


level from one of the carrier bands and the intercept is related to the capture cross-


section and trap concentration.


This technique is similar to DLTS (described below)


in that the emission rate is varied via temperature sweeps.


conductance is monitored in AS,


The difference is that the


whereas, a capacitance difference is monitored in


DLTS. Results from AS and DLTS were found to be in close agreement [14-16].


Deep-Level


Transient


Spectroscopy


(DLTS)


perhaps,


dominant


capacitance-based technique used to detect deep-level electronic states.


In DLTS,


one monitors the space charge capacitance of the semiconductor in response to a


pulse in potential.


In addition, the temperature is changed in the experiment to vary


the coefficient of the emission rate.


The capacitance is measured at two different


times following a pulse (in the forward direction); therefore, the transient change in

occupation of a state in response to a step change in potential is observed as a change
4n .n- .*%t.. .a ,.. ..4 +n .t .*^i r, T..f + a.t+ -rrr& +4h ,..; r^ Of ^^l K 0. +..0 4a





8

The experimental part of the technique presented here bears closest resemblance


to Optical Admittance Spectroscopy (OAS) [20,


The major difference is that the


complex impedance response is analyzed in DLZS; whereas, the complex admittance

response is treated in OAS. The distinction between the two experimental techniques


is, perhaps, subtle since impedance is simply the reciprocal of admittance.


difference lies in the data analysis.


The main


As shown later, analysis in terms of impedance


may be more sensitive to deep-level states at low frequencies.

This work addresses a light-enhanced form of electrochemical impedance spec-

troscopy. In this technique, the effect of photonic excitation of electronic transitions

by light at selected wavelengths is detected by impedance spectroscopy applied over a


broad frequency range.


The photonic energy of the light used is less than the bandgap


energy; therefore, any changes in the impedance spectrum with illumination can be


attributed to transitions involving energy levels within the bandgap.


This method


differs from the more commonly used DLTS in that the wavelength of monochromatic

sub-bandgap light is varied to excite electronic transitions at a fixed temperature (e.g.,

room temperature); whereas, in DLTS, temperature is varied to change the occupancy

of the states. A broad range of frequency (with emphasis on lower frequencies) for the

impedance measurements is used instead of measuring an effective capacitance at a


single high frequency.


The use of a broad frequency range is the essential distinction


between this approach and photocapacitance spectroscopy.

The effect of temperature on the impedance response is also analyzed in this work;


this technique is referred to as


Thermally Stimulated Deep-Level Impedance Spec-


troscopy (TS-DLZS). The rate of electronic transitions is dependent on the temper-
nfII.. in a n A' .rr ni,- o nrnnnaarl n o -n fTT TQ -,ar .-n.ll-7 0+i;,,,1 ,,1 TAflT AW. ,.f;f,, -,,,.,,..,-
























hi-


Figure 2.1:


Energy band diagram of a n-type semiconductor with a deep-level state.


-t-












Table 2.1: A comparison of methods used to identify deep-level states.


Technique


Optical Excitation


Thermal Excitation


Electrical Frequency


of Deep-Level States


of Deep-Level States


Range


single value about 1 kHz

single value about 1 kHz

rate windows (1 Hz to 5 kHz)


DLTS3


capacitance measured at 1 MHz


OAS5


single values (1 Hz to 1 MHz)

0.1 Hz to 1 MHz

0.05 Hz to 65 kHz

1 Hz to 65 kHz


OS-DLZS6

TS-DLZS7


1. MS: Mott-Schottky Theory.


Anomalous changes in differential capacitance are ob-


served as a function of applied potential.


2. PS:


Photocapacitance


or Electrochemical


Photocapacitance


Spectroscopy


(EPS).


Changes in differential capacitance are observed as a function of photon energy.


DLTS: Deep-Level Transient Spectroscopy.


capacitance are made


Transient measurements of differential


as a function of temperature.


4. AS: Admittance Spectroscopy. Changes in the admittance are observed as a function
of frequency and temperature.


OAS: Optical Admittance Spectroscopy.
observed as a function of photon energy.


Changes in the admittance spectrum are


OS-DLZS: Optically Stimulated Deep-Level Impedance Spectroscopy:


Changes in


the impedance spectrum are observed as a function of photon energy and applied
potential.

7. TS-DLZS: Thermally Stimulated Deep-Level Impedance Spectroscopy: Changes in












CHAPTER 3


EXPERIMENTAL METHOD


AND


INSTRUMENTATION


This chapter provides a detailed description of the equipment used throughout this


work.


The theoretical principles behind the techniques used are presented to help the


reader understand the functions and limitations of the equipment. It is intended that

this chapter supplement the operation manuals provided by the equipment distribu-


tors rather than replace them.


A general description of the overall setup is given in


the first section to provide an outline for the remainder of the chapter.


The following


sections describe the overall setup in greater detail.


General Schematic


A schematic of the experimental setup is given in Figure 3.1.


The setup can be


divided into two sections, one responsible for the optical frequency and the other for


the electrical frequency.


The light source was a 450


W Xenon lamp able to emit


wavelengths of light greater than 300 nm.


This light was diffracted in a SPEX Model


1681B Spectrometer to produce monochromatic light in accordance with Bragg's Law.


The diffraction grating rulings were 1200, 600, 300 grooves/mm.


The bandpass of the


monochromatic light was set by means of an entrance and exit slit; the bandpass used
was 18 nm unless otherwise noted. Harmonic frequencies were eliminated with order
sorting filters and the lirht intensity was fixed with the use nf nuntlral rPnaitv filtir-





12

A potential bias was maintained across the sample by a PAR 273 potentiostat


or a Solartron 1286 Electrochemical Interface (potentiostat).


This was coupled with


a Solartron


1250


Frequency Response Analyzer (FRA) which applied a sinusoidal


voltage at select frequencies across the sample.


The current was measured by the


potentiostat and the phase relationship between the applied voltage and the measured

current was analyzed by the FRA. The output of the FRA was the real and imaginary

components of the impedance at each measured electrical frequency.


A Sula


Technologies DLTS setup was used to perform the capacitance-voltage


measurements and the DLTS experiments.


The cryostatic chamber and tempera-


ture controller from this setup were also used to perform the temperature controlled

impedance experiments.


Electro chemical Impedance Spectroscopy


The underlying principles


behind impedance spectroscopy is presented first to


better the understanding of the impedance equipment.


basic schematic of the


electronics is then given.


3.2.1


Theory


The best single reference on the theory and application of impedance spectroscopy


is a technical report written by Gabrielli [22].


This section provides a brief overview


of some of the key concepts.

The principle behind impedance spectroscopy is to relate a physical model to ex-


perimental data in the form of real and imaginary components of the impedance.


This






13

an equivalent circuit with real and imaginary components, is in a form that can be

directly compared to the impedance data.

The electrochemical cell, or sample, can be thought of as a black box with two

terminals where the black box consists of a circuit of electronic components men-


tioned above.


The impedance of this circuit is Z(w) = a(w) + jb(w) where


is the


complex impedance that can be related to a real component, a(w), and an imaginary


component, b(w), both being real numbers and having units of Ohms.


The electrical


frequency is given by w with units of radians per second, and


- ,C


All three


impedance values are a function of potential.


The potential dependence is not written


explicitly because in typical impedance analysis the potential is fixed as a parameter

while the frequency is varied.


Let the voltage perturbation applied across


the terminals be represented by a


sinusoid,


V(t) = Vo sinwt


where Vo is the amplitude of the voltage perturbation.


(3.1)


The above equation assumes


that the voltage perturbation is referenced to the steady state potential value.


measured current that flows through the cell as a result of the applied voltage per-

turbation can be represented by a sinusoid,


V0
I(t)= sin(wt + {(w)) + YmAn, sin(mwt ekm(w)) + n(t)


(3.2)


where Z(w) is the total impedance of the cell and (w) is the phase shift between the


applied voltage and the measured current.


If the electrochemical cell was a resistor,


the phase shift would be zero.


If the cell was a capacitor,


the phase shift would


,,


..





14

The current expression above contains more information than is desired; only the

impedance as a function of w is sought, not mw, and noise decreases the confidence


in the impedance data.


The digital frequency response analyzer minimizes harmonic


responses and noise through integration of equation (3.2) multiplied by either sin wt

or cos wt, i.e.,


1
-I,
T


() sin
I(t) sinutdt= o cos 4(W)


(3.3)


1
J T


V0
I(t) cos wtdt = sin (w)
z(tlt


as T


-- 00


(3.4)


where T is the integration time in integer multiples of the sinusoid period, 1.

The integrals that contain the harmonic response are zero when the integration is

over a complete cycle(s) because sinwt, cost, and sin mwt form an orthogonal set


of functions (see Chapter 2 of Haberman [24]).

zero as the integration time increases ( 0).


The noise integral will tend towards

A trade off has to be made between


"noise free"


data (T


S oo) and a practical time length for the experiment (T is


finite), particularly if the system is corroding.


Non-stationary effects would not be


eliminated in the noise integral even as T


--+ 00.


The long integration time or short


integration time in the FRA parameter menu refers to the integral time T.

The impedance is the voltage divided by the current which can be represented as

a complex number, i.e.,


Zcell = Zr,cell + jZ,cell =


+31j'FI
+14ir


(3.5)


where the subscript "cell" has been added to denote that this is the impedance of the


cell (sample).


Because the voltage is the applied signal, phase shift set to zero as a


reference, the imaginary component of the voltage is zero (V.


= 0.


The real Dart of


as II"








The voltages and currents discussed so far occur across the sample.


these values are modified by various gains in the potentiostat


However,


before reaching the


frequency response analyzer. For instance, the voltage amplitude, Vo, from the signal


generator will become KVo where K is the transfer function of the potentiostat.


measured voltage passes through an amplifier of gain Gv before it reaches the FRA.

The amplitude of the voltage signal at the FRA input becomes GvKVo. Likewise, the

current signal becomes GIKVo/Zceu which is converted to a voltage signal by means


of the measuring resistor, R.


The current signal at the FRA input,


in the form of a


voltage, becomes GCRKVo/Zeeru.

The impedance that is actually measured by the FRA is


Zme aa r,menas


Zjmeas -


Gv
-=
GI


Zceli
R


GvKV.


GIj R cos + jGx zV R sin


(3.6)


cos + j sin 6


The relationship between the measured real and imaginary impedance and the cell

real and imaginary impedance is


Z -
r,meaa -


GVl
SR rZ,cell


(3.7)


j,meas


Gvl
G-- -Rcel*
CI RZ,1


(3.8)


The ratio of the voltage and current gains in the potentiostat is designed to be unity


over the frequency range used.


Hence, the measuring resistor is the proportionality


constant between the measured impedance and the cell impedance.





16

without first reading the ZPLOT manual and the operating manuals for the poten-


tiostats [27


28] and the frequency response analyzer [29].


The default parameters


selected by ZPLOT are rarely appropriate to the system being studied.


3.2.2.1


Potentiostat


There are several parameters involved with each potentiostat (Solartron 1286 or


PAR 273)


that must be tailored to each experiment.


The first is the desired cell


configuration.


All potentiostats allow at least three electrodes to be monitored or


controlled; these are the working electrode, the counter electrode, and the reference


electrode.


The working electrode is the electrode of interest (i.e.,


the sample elec-


trode) and its potential is controlled relative to the reference electrode via the counter

electrode. For instance, suppose that the working electrode is desired to be 0.5 volts

more positive than the reference electrode, the potentiostat would adjust the potential

of the counter electrode to maintain this desired potential drop.

The reference electrode potential versus the working electrode potential is mea-


sured with an operational amplifier, also referred to as an "op amp"


It is worthwhile


here to describe briefly the function of the operational amplifier (see for instance

Malvino [30]). A basic inverting operational amplifier circuit is given in Figure 3.2(a)

and the equivalent circuit for an operational amplifier is given in Figure 3.2(b). The


minus sign is used to designate that the input is connected to the minus terminal of

the op amp which means that the output has a phase shift of -180 degrees.


The overall voltage gain of the op amp circuit,


G, is defined as the ratio of the


output voltage, V,, to the input voltage, V1g,


fain. A. is defined as the ratio of V^.. /(V -


= Vt/ ) n


*'D U.


A is larre (


The differential voltage


100. noon and V.., is






17

The op amp varies the output voltage Vo, in response to the input voltage VI,.


The output voltage is feedback through the resistor, Ry,


and thus affects the potential


at the input terminal such that


V1 V2 is small.


Rin is the input impedance of the


op amp and is typically


the op amp.


MO, large enough that essentially no current flows through


There are two main features of an ideal op amp; no potential drop or


current flow across the input terminals.


Thus, a potential or current measurement


can be made without affecting the system being measured.


The overall voltage gain


can be determined from the differential gain equation


(AfVt


-V21 -


Vo,) and the fact that the currents through the two external resistors,


Rf and R,, must be equal because no current flows through the op amp (I, = If = I).

With these two assumptions, the voltage across the op amp can be expressed as


V2 -V =Vn


- IR,


(3.9)


V2 = V + IRf.


(3.10)


Equation (3.9) can be related to the differential voltage gain, A, in order to derive an

explicit current expression, i.e.,


-A(MV


- IR,) =


vou=~I=


Vot + AV1,


(3.11)


AR.


This current expression is substituted into equation (3.10) to eliminate I, i.e.,


V011 + AV
- AV,t ARvut + A R',t
AR,


(3.12)


The last expression is rearranged to yield the overall voltage gain,


TVot
r _


-ARf/R,


(3.13)


1 + A + R/R,






18

This information will be used later in analyzing the current measurement capabilities

of each potentiostat and in determining the proper bandwidth selection to ensure

stability in potentiostatic control of the system.

The PAR 273 potentiostat is capable of applying 100 volts between the counter

and working electrodes just to maintain a fraction of a volt difference between the


reference and working electrodes.


As a safety precaution, the PAR 273 is installed


with a cell enable/disable switch so that the electrode leads can be adjusted without


fear of electrocution.


The Solartron 1286 is capable of applying 20 volts between


the counter and working electrodes and does not have such a safety switch.

Three electrode configurations are used primarily in electrochemical experiments.

For example, one of the systems studied here had a 0.5 M NaC1 (pH 10) electrolytic

solution with a cuprous oxide working electrode, a saturated calomel reference elec-


trode (SCE),


and a platinum counter electrode.


The two cell configuration is used


primarily in solid state systems such as semiconductor devices with


either metal


Schottky barriers or Ohmic contacts. It is extremely important to be mindful of the

correct lead connections when using the two electrode cell configuration; the reference


electrode lead is connected (shorted) to the counter electrode lead.


This permits the


measurement of the working electrode versus the reference electrode and control of


this potential difference via the counter electrode. Never use just the reference or

counter electrode alone versus the working electrode. This would not allow poten-


tiostatic control of the system because the reference electrode can only measure the

potential between the reference and the working electrodes and the counter electrode

can only control the potential between the reference and the working electrodes by
n\ 0 a vi nr ak iT rAn + 1m a .-n-, v + r0 ar+ tunYo an A + 1. 4 M v i nnT al I n> alff*/ A





19

Simplified block diagrams that include only the options relevant to the impedance

work done in this work are presented for the Solartron 1286 (Figure 3.3) and the


PAR 273 (Figure 3.4).


Not surprisingly,


both block diagrams have the same gen-


eral features, with the exception of the current measurement circuit and the second


reference electrode capability of the Solartron 1286. The second reference electrode

permits measurement of the potential difference between two sections of the elec-


trolytic solution.


The counter electrode maintains this potential difference.


Note


that the potential of the working electrode is not measured in this four electrode


configuration.


Thus, all two and three electrode cell configurations require that the


second reference electrode lead be shorted to the working electrode.


3.2.2.2


Current measurement


There are differences in the current measurement circuits of the two potentiostats


that are worth discussing here.


Both potentiostats determine the current passing


through the working electrode with the aid of a high quality measuring resistor in se-


ries with the working electrode.


The current flows through the resistor which creates


a potential drop across the resistor equal to IP.


This potential drop is measured


with an op amp as shown in the block diagrams.

Two differences in the current measuring circuits become evident from a closer


look at the block diagrams.


The first is that


the resistors in the Solartron range


from 0.1 f to 100 AkL and those of the PAR range from 1 f to 10 MSl.


The second


difference is that the Solartron uses an additional op amp to boost the low current


signal (1, 10,


and 100 kQl resistors).


A possible disadvantage of the latter point is


the introduction of the additional on amn circuit in the Solartron.


This circuit micht





20

The larger measuring resistors in the PAR 273 permit low current measurements


with less noise.


This statement is based on the simple relation V


= IR; small I re-


quires a large R to maintain the same potential drop and, hence, potential sensitivity.

Currents in the nanoamp range are observed when the impedance of the cell is large


(108lO),


as was the case for most of the devices analyzed in this work.


This brings


up an important point-how to be certain that the signal being measured actually
reflects the cell alone or the electronic system of the potentiostat as well, especially


if the impedance of the cell is near the limits of detection.


The majority of systems


analyzed can be represented as passive circuit elements such as resistors, capacitors,

and inductors; some of the very same components used to apply and measure the

voltage and current.

A solution to this dilemma was suggested by Dr. Ron Haak of the ALZA Corpora-

tion. He proposed that the impedance data of the cell being analyzed be regressed to

a suitable equivalent circuit and to build this circuit with resistors and capacitors on a


breadboard.


The same impedance scan would be performed on this equivalent circuit


and the result compared with the original cell data.


This test proved to be very fruit-


ful in the case of the ZnS:Mn thin film electroluminescent panel. Both potentiostats

gave reproducible results; however, these results depended on the potentiostat used.

The Solartron showed a change in phase angle from around -90 degrees to around -20

degrees at low frequencies (a 0.2 Hz); whereas, the PAR indicated that the phase

angle remained relatively constant around -90 degrees in the same frequency range.

The Solartron cell data was regressed with ZFIT [31] to a resistor in series with a


resistor and capacitor parallel to each other as shown in Figure 3.5(a).


This equivalent


rcuit wa




21

only predict the qualitative features of the equivalent circuit, even though the original

cell data was obtained with the Solartron.

The PAR 273 cell data was regressed in a like manner to yield the equivalent


circuit shown in Figure 3.5(b).


Note that the capacitor is roughly the same as for


the equivalent circuit above but that the parallel resistor is about 40 times greater.

This resistor was considered to be so large that it could be assumed to have an in-


finite resistance, thus reducing the circuit to a resistor and capacitor in series.


circuit would have a -90 degree phase shift at low frequencies.


This


It was not expected


that the resistor and capacitor in series was a good fit to the original data; however,

it was adequate to test the behavior of each potentiostat to such a highly capaci-


tive system such as the ZnS:Mn panel.


The results from both potentiostats on the


resistor-capacitor series are presented in Figures 3.8 and 3.9.


The PAR was able to


predict the capacitive nature much better than was the Solartron at low frequencies.

Based on these series of experiments, it was decided that the PAR 273 was the ap-


propriate choice in potentiostats to analyze the ZnS:Mn panel.


Similar checks were


performed for the other systems.


3.2.2.3


Stability and bandwidth


This subsection is a brief review of Chapter 9 of the Solartron 1286 operating


manual


A system is considered to be stable if its output is a direct response to


an applied input signal. An unstable system can exhibit spontaneous undriven output

or an output due to a previous input signal. Such a system could result in oscillations

in the potentiostatic control of the system. In terms of operational amplifier theory,





22

a system becomes unstable when the loop gain is greater than or equal to unity and

the phase shift is 0 or 360 degrees.

The loop gain is defined as the product of all the gains that incorporate the loop.

There are four gains in Figures 3.3 and 3.4:

1) GC differential voltage gain of potential control amplifier, A1,


2) GC


- cell gain,


3) G2 differential voltage gain of potential measurement amplifier, A2, and


4) G,


- gain due to resistors R1 and R2 (not shown in Figure 3.4).


The last gain, Gr, is composed of resistors and hence will not impart a phase shift.


It will only attenuate the signal.


G2 has been designed to be frequency independent


with a gain of unity except at very high frequencies.

cell and the gain of the control op amp, GC is the


The cell gain varies from cell to


parameter that can be varied to


ensure stability.

The input signal to amplifier A1 is connected to the minus terminal which means

that the output will experience a phase shift of -180 degrees. If the cell being analyzed

is capacitive, it will also impart a phase shift of -90 degrees for a total of -270 degrees.


The differential voltage gain, G1, is constant at low frequencies.


However, the gain


decreases at some frequency with an additional phase shift of -90 degrees as shown


in Figure 3.10.


The total phase shift becomes -360 degrees which means that the


feedback signal is in phase with the input drive signal.


at the minus terminal


The two signals add together


to form a larger amplitude input signal which the op amp


amplifies with gain A1


to form the output signal.


The output signal is feedback to


the minus terminal where it is added to the in phase input signal and again amplified.
This cyclic behavior will lead to inntahilitv if thp lnnn anin in wrap tha win r nMnil In








The differential voltage gain of A1

occurs away from the -360 degree phase


can be varied such that the unity loop gain

. shift. This point can be graphically demon-


strated with the aid of Figure 3.11 where two G1 gains are shown.


The uncompensated


loop gain has unity gain at -360 degrees,


and hence, is unstable.


With the selection of


a lower bandwidth gain (compensated) for the amplifier A1, the phase shift at unity

loop gain becomes -270 degrees and the system is stable.

The Solartron 1286 has ten ranges (labelled A to J) of bandwidths to choose from


and the PAR 273 has two (labelled "high speed"


and "high stability"). Bandwidths


C, D, and "high stability" have high and low pass frequency filters and, due to the

extra electronic circuits, should only be selected if none of the other bandwidths are


appropriate.


An oscilloscope is used to determine the proper bandwidth with a po-


tential square wave form applied across the cell.


If the bandwidth is too wide, the


square wave will exhibit


'ringing'


which indicates that the differential voltage gain


is too large.


Too narrow of a bandwidth means that the gain is too small and the


resulting wave form will be overdamped.


correct bandwidth choice will yield a


square wave form on the oscilloscope with perhaps a slight overshoot.


3.2.2.4


Frequency response analyzer


The Solartron 1250 Frequency Response Analyzer (FRA) performs two duties: 1)


generates an output wave form and 2) analyzes up to two input wave forms.


The gen-


erated wave form can be a sine, square, or triangle wave at frequencies from 10 p Hz to


65,500 Hz.


These wave forms are synthesized with a 10 MHz crystal oscillator along


with digital processing to provide a frequency accuracy of 1 part in 10,000 [32].


method is reported to be better than the nrevinms nhas-lnrlk mpthnr.l


This


ThP vR A





24

The analyzer section of the FRA can simultaneously measure two input channels

which in the work presented here is the voltage and current (converted to a voltage


signal) from the potentiostat.


The analyzer displays the ratio of the two channels in


either rectangular (a+ jb), polar (r,0), or log polar (log r, 0) coordinates. Integration

time on the analyzer menu specifies the maximum number of cycles to be integrated

if the convergence criteria is not meet. At least 1 cycle is integrated below frequencies


of 655 Hz and 61


cycles above 655 Hz regardless of the number of cycles specified


in the ZPLOT parameter menu.


Only one integration over a cycle is necessary to


eliminate the D.C. and harmonic frequencies of the fundamental frequency desired.

However, integration over more cycles reduces the amount of noise in each channel.


Two convergence criteria are available in the auto integration mode.


The first


is labelled


"short integration time"


which means that the channel measurement is


continued and averaged until the standard deviation of the average does not vary

by more than 10 percent of the average or until the maximum integration time


has been reached as specified above.


"Error 82"


message will occur when the


latter case has occurred; this does not imply that the data point should be discarded,


only that the convergence criteria has not been met.


"long integration time"


The second option is labelled


and is similar to the short integration time except that the


convergence criteria is 1 percent.


input channel.


The convergence criteria can be applied to either


The current channel (Channel 1) should be chosen since this channel


is the response of the voltage signal from the generator and is expected to have the


most noise.


The voltage channel is the controlled channel in the potentiostat and


hence is not expected to be as noisy.


There are five voltage range selections for the channel innmit (f mV


.nn mV


RV








should be measured on the oscilloscope to determine the peak voltage.


Note that


the voltage channel (Channel 2) can be fixed because the applied sine wave remains

relatively constant when the Solartron 1286 or the PAR 273 is used as a potentiostat.

However, the current channel (Channel 1) is best left in the Auto selection because

the current varies several orders of magnitude as the impedance of the cell changes


with frequency.


3.2.3


The reverse is true when in galvanostatic (current) control.


Experimental Difficulties


The large impedance of the materials used in this study made collection of accurate


impedance data difficult at low frequencies. The experimental frequency range was

constrained by an unfavorable signal to noise ratio. The currents measured at low


frequencies were extremely small (in the nanoamp range), and currents smaller than


this were beyond the range of the potentiostat used.


This problem was addressed in


this work by increasing the amplitude for the potential perturbation from the 10 mV


characteristic of electrochemical studies to


ZnO varistors.


to 3 V for the ZnS:Mn panel and the


Reproducibility of the impedance response for different magnitudes


suggested that the system response was linear at even these large amplitudes.


The signal to noise ratio was further improved,


when possible, by increasing the


area of the device tested. For example, the current output of the electroluminescent


panel was increased by 2 orders of magnitude by electrically coupling 100 pixels.


electronic transitions involving deep-level states have time constants that are large


in comparison to the inverse of the lowest measured frequency.


In other words


imaginary impedance did not tend toward zero at the lowest frequency used.


lb .. a S a





26

required between runs for the ZnS:Mn TFEL panel and the unaged varistor; whereas,


a wait time of 1


hour was sufficient for the aged varistor, and a few minutes for


the GaAs semiconductor and the copper/cuprous oxide electrode.


The problem of


memory effects in electroluminescent panels is discussed in references [33, 34, 35].


Standard


Characterization


Techniques


Two standard techniques were used in this work to provide a comparison to elec-


trochemical impedance spectroscopy.


These are Mott-Schottky or C-V profiling and


Deep-Level Transient Spectroscopy (DLTS).


3.3.1


Mott-Schottky Profiling


The Mott-Schottky relationship is used to determine the concentration of shallow


level states in the bulk of a semiconductor.


This relationship can be represented


as [36]


1 2
-
C2 A2

where C is the measured capacitance,


((V


qKeo(ND


- hi)


(3.15)


-NA)


Vbi is the built in potential,


V is the applied


potential, A is the surface area, Keo is the permittivity of the semiconductor, ND


is the donor concentration, and NA is the acceptor concentration.


can be incorporated into this expression


Deep-level states


by considering them to be either donor-


like or acceptor-like in nature.


Assume, for example,


that the deep-level state is


acceptor-like in nature in that it becomes negative when ionized.


Equation (3.15)


then becomes


1


2(Vi V


- kT)
P


(2 nfl\


--~~~~~I A -- t IA





27

The above equation can be expressed explicitly in terms of the net donor concen-

tration through the derivative of equation (3.16) with respect to the applied potential.

The result is


ND -NA-NT= -


Equations (3.16)


(3.17)


A2qKeo d(--i)/dV


(3.17) are implemented easily by plotting the inverse of the


square of the measured capacitance ( ) versus the applied voltage, V


The slope of


this curve is referred to as the Mott-Schottky slope and is used in equation (3.17) to

calculate the net donor or acceptor concentration. If the semiconductor is n-type the

slope will be negative, and for p-type semiconductors the slope is positive.

The capacitance is measured by applying a high frequency sinusoidal voltage per-


turbation about the set applied voltage bias.


This frequency is typically in the MHz


range,


values too high to change the occupation of most deep-level states.


Changes


in the Mott-Schottky slope with frequency as a parameter have been attributed to

interface states or bulk deep-level states in the literature [1,2,37-39].


3.3.2


Deep-Level Transient Spectroscopy (DLTS)


Yau and Sah


[40] first


demonstrated


the influence of


deep-level states on the


capacitance decay of a pulsed semiconductor as a function of temperature in 1971.

This concept was further refined by Lang [19] in 1974 into the present day technique

known as Deep-Level Transient Spectroscopy (DLTS). DLTS has become the most

widely used technique to determine the energy and concentration of deep-level states.

The theory behind DLTS is briefly outlined in this section for a n-type Schottky

barrier, a more involved presentation for other situations is given by Lang [19].





28

charge region of width W, which is a region void of majority carriers (electrons). All


deep-level states with


energy greater than


Fermi energy


level, EFp,


will emit


their electron into the conduction band.


all such states are empty.


Steady state equilibrium is reached when


This is shown in an energy versus distance diagram in


Figure 3.12 (a).


The semiconductor is forward biased for approximately


1 msec.


The majority


carriers are swept into the space charge region at a rate that can be considered to be


instantaneous.


The width of the space charge region decreases to Wf and the energy


level of the deep-level state falls below the Fermi energy level in a portion of the


region Wr


-Wf.


All the states in this portion capture electrons to reach a forward


biased equilibrium.


This situation is shown in Figure 3.12 (b).


These states will not


have time to capture electrons, reach equilibrium, if the pulse is too short in duration.

The semiconductor is again reverse biased which creates a transient space charge


region of width W7, that decays to the reverse biased equilibrium value, Wr.


>w,


because the trapped electrons are neutral species, and hence, more conduction band


electrons vacate in order to maintain the same reverse bias level as


before.


deep-level state energy level is above the Fermi energy level in a portion of the newly


acquired space charge region.


The trapped electrons in this region will be emitted into


the conduction band to reach the reverse biased equilibrium.


The space charge width


will decrease towards the equilibrium width W, as these trapped electrons are emitted


into the conduction band.


This transient case is shown in Figure 3.12 (c).


It is this


transient decay of the space charge region that is monitored in DLTS experiments by

observing the time dependence of capacitance.

The emission rate constant can he extracted from the transient canacitance from








Equation (3.16) can be rearranged into a more appropriate form,


C(t) =


A2qKeo(ND


2(Vu


- NA


- aT(t))


(3.18)


k )
- -'


where it is noted that the capacitance and deep-level state concentration are a func-

tion of time. It is convenient here to define a capacitance difference, i.e.,


AC(t) = C(t)- C,,


A2qK


2(Vb


kT\
- --


ND- NA- n(t)-


ND-NA nT


(3.19)


where Cu, and nT, are the steady state reverse bias capacitance and deep-level con-


centration, respectively.


This equation is made dimensionless upon division by Co,,


AC,) (
Ca.


ND-NA


- nT(t)


/ND


- /WN NA nT.


-NA- rT.


ND -NA


-NA


- nT(t)


-- RT


(3.20)

Equation (3.20) relates the measured transient capacitance to the transient con-


centration of ionized deep-level states.


This is not of the form typically used in DLTS


analysis to estimate the total concentration of states.


Two additional assumptions


are needed; the steady state (reverse bias) concentration of deep-level states is sev-

eral orders of magnitude smaller than the net donor concentration, and the forward


bias pulse is long enough that all of the deep-level states are occupied.

assumption can be expressed as


The former


AC(t) /
Ca.


nr(t)


(3.21)


-NA


A Taylor's series approximation of equation (3.21) about -"
ND--NA


AC(t) 1 n
C,, 2 ND


H 0 yields


r(t)


(3.22)


- a -


-NA








where NrT is the total trap concentration.


This is the relationship used in most DLTS


analysis.

The kinetic expression for the rate of change of trap occupancy is used to link

the concentration,. and hence, the measured capacitance to the desired emission rate


constant.


The transient concentration of trapped electrons can be expressed as


dnT
- = -eJNr + e(NTT NT) + cnN(NTT Nr) + CpPNT
dt


(3.24)


where en is the emission rate constant for electrons, ep is the emission rate constant for

holes, c, is the capture rate constant for electrons, and cp is the capture rate constant


for holes.


The concentration of electrons and holes in the space charge region can be


neglected when the reverse bias is applied. Hence, there are no electrons or holes to

be captured by the deep-level state and equation (3.24) is simplified to


dnT
dnT --NT(e,
dt


+ e,) + eNTT.


(3.25)


Under steady state reverse bias conditions,


NT.
NTT


-p (3.26)
en + ep ,


An electron trap under reverse bias with energy greater than the Fermi energy will


tend to be empty of electrons at steady state, or in other words, NT,


implies from equation (3.26) that e,


0.


This


e,. Equation (3.25) can be solved easily with


the neglect of the hole emission rate constant, the result being


nT(t) = NTTexp(-e-t).


(3.27)


Equations (3.22) and (3.27) are combined to provide the necessary link between





31

As can be seen from this relation, the capacitance decay following the forward bias


to reverse bias step change is a simple exponential in time. Lang [41],


in a preceding


paper, also addresses the


case


in which non-exponential transient decays are present.


He indicated that this case is likely for intermediate and shallow level states.


DLTS


can still be applied to non-exponential transients, however, the concentration of states

cannot be determined via equation (3.23) because an extrapolation to AC(t = 0) is

required [41].

The emission rate constant is given by


1 ErT
e,= a1, -T2exp(-


kT


(3.29)


where an is the capture cross-section of the trap, g is the trap spin degeneracy, and

ET Ec is the energy difference between the deep-level state and the conduction


band.


7, is a temperature independent quantity that is derived in Section 4.3.2 along


with equation (3.29).

The procedure of DLTS can be best illustrated with the aid of Figure 3.13 where

capacitance is shown as a function of both temperature and time. In the actual DLTS

experiment, the complete capacitance decay is not monitored but rather, it is only


determined at two points in time as shown in Figure 3.13.


The difference between


these two capacitance readings is the output signal that is typically connected to the


y-terminal of an


plotter (the x-terminal is connected to the temperature output


signal).


Time tl


is referred to


as the initial delay,


i.e., the time between the start


of the reverse bias (end of the forward bias) and the time of the first capacitance


measurement.


The ratio of tl and t2 is designed to be a constant. It is convenient to


consider the emission rate. e.. as the inverse of a time constant. r.





32

out each DLTS experiment, and hence, the peak capacitance difference is found by


varying the emission rate via the temperature (see equation (3.29)).


The relationship


between the emission rate and the two correlation times can be easily derived. From

equation (3.28) we have


c(t,) c(t2)


oc exp(-entl) exp(-et2).


(3.30)


The derivative of equation (3.30) with respect to e, is zero at the peak capacitance

difference. Setting the derivative of equation (3.30) to zero yields


(3.31)


eT=I( )/( 2).
t2


The goal is to find


the temperature at which equation (3.31) is true,


temperature at which C(tl) C(t2) has its maximum value. A plot of the natural log

of e,/T2 versus the inverse of the peak temperature will be a straight line of slope

(ET Ec)/k and the intercept is proportional to an.


3.3.3


DLTS Equipment and Parameters


The DLTS equipment was purchased through Sula Technologies (Palo Alto, CA)

and consisted of a cryostatic chamber, a Sula Technologies Deep Level Spectrometer,

a Lake Shore Model 805 Temperature Controller (Lake Shore Cryotronics, Inc. of

Westerville, OH), a 7-liter Dewar flask to hold liquid nitrogen, and a variable speed

circulation pump to draw the liquid nitrogen through the cryostatic chamber, along


with


various tubing and electrical connectors.


A roughing pump was supplied by


Pete Axson, a technician for Dr.


Timothy J. Anderson's research group located in


Itt.i 21L .1 'I a ii 4


Ltf j __ _L _~





33

Inside the cryostatic chamber was a thermal block connected to a heater element,

controlled by the temperature controller, and a stainless steel tube through which


liquid nitrogen is drawn.


peratures.


The combination provides a wide range of controllable tern-


The lowest temperature possible is 78K to 84K and the highest practical


temperature is around 420K. A rather large brass block (0.25 inch thick


x 3 inch di-


ameter) was attached to the thermal block by teflon screws with a thin teflon spacer

in between the thermal block and the brass block to provide electrical insulation. The

anode and cathode leads were loose wires; there was no sample mount provided. It

was left to the purchaser to design his or her own sample mount.

Pat Watson, while at Cornell University, suggested that the brass block be replaced

because of a severe temperature lag between the thermal block and the sample due


to the block's


disk (1mm thick


large mass. He recommended replacing the brass block with a sapphire


x 2 cm diameter) that has a thin layer of gold deposited on it.


sapphire has good thermal conductivity and is also an electrical insulator.


One of


the leads was connected to the gold with silver paste and the other lead was soldered

to a needle. Silver paste was used to improve the connection between the gold layer

and the semiconductor metal backing. However, the gold layer became damaged after

repeated pasting and removal of the samples.

It was decided to replace the sapphire disk with a 304 stainless steel plate of similar


dimensions.


block.


The original teflon sheet was placed between the steel and the thermal


Stainless steel was chosen because it does not form an electrically insulating


oxide layer as do iron and most brasses particularly at higher temperatures. However,

it is near impossible to solder to stainless steel. It was necessary to attach one of the

leads to the steel late by drilling a small hole at one corner nf the nlats. hPnrm;n








needle.


The needle was blunted with a honing stone to prevent it from gouging into


the semiconductor.


Vacuum grease was used on both sides of the teflon spacer to


improve the thermal contact.


The needle was mechanically fastened to the thermal


support with a cut and heat-formed


plexi-glass sheet in the shape of a


.The


stainless steel plate was also fixed in place with a plexi-glass block screwed into the

thermal block.

A small copper shell was "riveted" to the stainless steel plate to house the 1000


platinum


temperature sensor used


to measure the sample temperature,


the other


temperature sensor (sensor A) is in the thermal block and is used as the control sensor


by the temperature controller.


With this configuration,


the thermal block sensor


temperature reached 78K at its lowest and the sample sensor temperature reached


84K at its lowest.


Although the temperature controller can maintain a temperature


as high as 800K, it was found from experience that at temperatures above 420K

problems arose due to the coaxial cables shorting out near the stainless steel plate.

This problem was minimized by keeping the cables as far from the thermal block as

possible.

The Sula Technologies Deep Level Spectrometer did not have an IEEE-488 com-

puter interface although the temperature controller did. Initially, a Metrabyte Corpo-

ration (Taunton, MA) analog-to-digital converter was used but was later overloaded in

a non-related experiment. However, a Hewlett-Packard 7090A x-y plotter was avail-

able in our lab capable of analog-to-digital data acquisition with 12 bit resolution.

The interface programs for both the Metrabyte converter and the HP 7090A [42, 43]


were written in Microsoft QuickBASIC 4.5 computer language.


The program for the


HP 7090A is included in the annendir








thorough understanding of Lang's


[19] paper before using the spectrometer.


Several


important relationships will be stressed here. The manual lists two heuristics that

should be obeyed in the choice of front panel parameters. These are:


The period (time between pulses) is greater than or equal to the length


of the pulse plus ten times the initial delay (time tl).


This guarantees that


enough time has been allowed for completion of the correlation process,

i.e., measurement of C(tl) and C(t2).

The period is less than or equal to one hundred times the initial de-


This guarantees that the electronics of the spectrometer have time


to respond to the pulsing in order to make an accurate capacitance mea-

surement.


The ratio of times tl


and t2 is designed to be constant in the spectrometer as


recommended by Lang [19]. For this spectrometer, the rate window, or time constant,

at the peak capacitance difference is


"r "-


1
--= 4.3


where tl is also referred to as the initial delay.


(3.32)


Another important relationship used


to estimated the state concentration is the capacitance difference at the start of the

reverse bias,


AC(t = 0) = 3[C(tl) C(t2)]


evaluated at its peak value.


(3.33)


The steady state capacitance or background capacitance, C,,, is determined with

a 150 mV peak to peak voltage sinusoid at a frequency of 1 MHz applied across the





36

in the Mott-Schottky plots described in an earlier section, however, the forward bias

pulse should be off during these measurements.


Optical Spectroscopy


A SPEX Model 1681B Spectrometer was used to create the monochromatic light.

The spectrometer was chosen over lasers because of the greater range of wavelengths

available (the bandgap of the semiconductors analyzed ranged from 1.4 to 3.6 eV).

Three diffraction gratings were used with rulings of 1200, 600, and 300 grooves/mm


for visible light, near infrared, and infrared, respectively.


The light source was a 450 W


Xenon lamp that emits wavelengths of light greater than 300 nm.


The light beam


exiting the spectrometer passed through: (1) a order sorting filter to eliminate higher

energy harmonics of the fundamental wavelength, (2) a collimating lens to convert the

divergent rays of light to parallel rays, (3) neutral density filters to control the light


intensity, and (4) a focusing lens to focus the beam onto the sample.


The intensity of


the focused light spot was typically 5 to 20 mW/cm2 and the dimensions were 8


mm.

The intensity of the monochromatic beam was dependent on the wavelength of


the light.


This was due to the Xenon light source and the diffraction gratings.


intensity of the focused light as a function of the wavelength are shown for each order

sorting filter in Figures 3.14 to 3.17 with the diffraction grating and the spectrometer


exit/entrance slit width as parameters.


Curves are shown for two diffraction gratings


in Figure 3.14 because the intensity with the 600 grooves/mm grating was larger than

that of the 1200 grooves/mm grating for wavelengths greater than 660 nm.








density filters and, in some cases,


small variations of the exit slit width.


Checks


were made periodically on the accuracy of this method.


There were some indications


that the intensity spectrum is changing as the Xenon lamp ages.


An estimate was


made during the last month of experiments of the error in the intensity value by

comparing the desired intensity with the intensity measured after the appropriate

neutral density filters. It was found that the intensity varied by approximately 22

percent of the desired value. The error should be less for experiments that were


conducted earlier.


The order of experiments was: cuprous oxide in alkaline-chloride


solution, ZnS:Mn electroluminescent panel, n-type GaAs, and ZnO varistors. For the

GaAs sample, the intensity was measured before the impedance experiments for each

wavelength to be used and, thus, should have a much lower error in intensity.

The bandpass of the spectrometer is the band of wavelengths that pass through

the exit slit and is determined from the dispersion (wavelength in nanometers divided

by the slit width in millimeters) and the exit slit width in millimeters. For instance,


1200


grooves/mm grating


(dispersion


of 3.7


nm/mm)


set at


a wavelength of


500 nm and an exist slit width of 5 mm will have a bandpass of 18.5 nm, or in other

words, light of wavelength 490.75 to 509.25 nm will pass through the exit slit. The

dispersion value is considered constant for each diffraction grating and harmonic order


of the wavelength that is selected


The dispersion is 3.7 nm/mm for the 1200


grooves/mm grating, 7.2 nm/mm for the 600 grooves/mm grating, and 14.4 nm/mm

for the 300 grooves/mm grating all at the fundamental wavelength of the grating.


The bandpass is one disadvantage of the spectrometer over a laser.


The exact


location of an optical transition is obscured by the range of wavelengths that exit


& on mf&I ar an afarI A"^ Io ia n rr-tn'rn wi 4-1,hia a eat4 W,4-- r k* .1a a


~hP PnPr~mmcrt~cr*


~.C ~11A *Aj ~nn







the spectrometer is set at


720 nm, light of wavelength


730 nm and its harmonic


wavelength at 365 nm will also pass if the bandpass is large.


The harmonic at 365 nm


is the wavelength of concern because the 410-720 nm order sorting filter begins to

pass light of this wavelength as can be seen from an inspection of Figure 3.14.















COUNTER AND|
REFERENCE
ELECTRODE
LEAD


X" nIcRA


IIC-


SPRING-LOADED
CLAMP
/


/ U.SO


WORKING
ELECTRODE
LEAD


.. .
0**t
..... S
S.....
*5~***
*5*St*
S.....
S....
9.....
9*9e*
*5*


Light


Source


Spectrometer


Black


Order
Sorting
Filter


Lens


Neutral
Density
Filter


Lens


Sample


IiI


Computer


Compudrive


Power
Supply


Potentiostat


v



















out


out


Figure 3.2:


Basic inverting operational amplifier circuit


overly simplified


~I~

















Bias


External
Input

Voltage
Output









Current
Output


High
Measuring
Resistors


Low
Measuring
Resistors




















Bias


External
Input









Current
Output 7


Voltage
Output


IR Comp


CeLL












121


MOhms


835


Ohms


9.23E-9 farads


4000


MOhms


1620 Ohms


b 9.23E-9 farads















-14.0


-7.0


14.0


n ~~107















-90.0


-80.0


-70.0


-60.0


-50.0


-40.0


-30.0


-20.0


1 1 111 I I


Original


Circuit


data


data


S U *


Fi11T~


(Solartron)


(PAR)


o Circuit data (Solartron)


-10.0


0.0


-210
1O


100


10'


Frequency


. i.


, Hz


11 1 1 1 i 1 1i A r i


102


103

















-4.0







-3.0


-2.0


-1.0


2.0 4.0 6.0


Il 3e 107















-90.0






-87.0






-84.0






-81.0






-78.0


-2 10-0 100 101 102


i0a


Frequency,




























log gain















gain: =xl


A1 pole
JJ


GAIN PLOT


PHASE PLOT


--- --


cell pole


phase





180-




270'




-- -360'


log frequency





























IogigainI


phase


Al pole


\\UNCOMPENSATED 180




---- ---- -- 270O

COMPENSATED

S- --3603




gain= rl -

log frequency

cell pole











Metal


Metal


Semiconductor



S------------- EC
------------^ E


pI


- E,
--ET


Semiconductor




--------------- E


- Er,
-E,





Ev


Metal


E-
V I


Semiconductor


SMor
cap


- Er
--ET


lutor this
aocltance


as a
transient


a - I


L
t---,
~4~ EC










































Tpeak



- --l

I I
I
I I
I I
I I
I

I I


Time


C(tl)


- C(t2)


tO





































0+--
300


400 500 600 700


800


WAVELENGTH, nm





1:.._ 1 A. T-- t r --- 1r :.l. --s ..I. ,- .. .nl4. .. ',s *, rTloans4 in, +1,


















































600


700 800 900 1000 1100 1200 1300


1400


WAVELENGTH, nm


r 91ct, T4na,r, at 4tn ,.1 nn in k .%.. -vta pn .4 +h nnroant few


a a


*-l
















Slit Width


Entrance/Exit

9/7 mm
7/5 mm
4/2 mm


1200-2000 OSF


300 gr/mm


5-




1000
1000


I ~ I I I I I I I I


1200


1400


I I--I 1 I II I I I


1600


1800


2000


WAVELENGTH, nm


- -r U


-- '3 IC. T......:2.. nL t tl n C...I AF i: n ..,. .tm An nC 4l\ ,utann*I tr


II


- --














1800-3000 OSF


300 gr/mm


Slit Width


Entrance/Exit


9/7


mm


7/5 mm
4/2 mm


1500


1700


1900


I 1 I
2100


I2300 1
2300


2500


2700


WAVELENGTH, nm




S( r i y T I n hA ...P r P 1 r -- v .- Sr /f +r .rlralonn+h trw +h













CHAPTER


DEVELOPMENT OF PROCESS MODEL



The objective of this chapter is to develop a mathematical model that describes


transport and reaction processes involving deep-level states.


Some physical insight


into the electronic processes of these states is provided in the first section.


In the


following sections, the governing equations for both bulk and surface states are devel-

oped, and the model is posed in terms of an equivalent circuit composed of resistors,


capacitors,


and current generators.


Finally,


various assumptions are made to obtain


a usable circuit.


Shockley Read Hall Processes


The influence of deep-level states or traps on the statistics of electron-hole re-


combination was first described by Shockley and Read [46] and Hall [47].


Deep-level


states, as their name implies, lie close to the middle of the energy band gap of the


semiconductor.


Due to the large energy separation from the valence band and con-


duction band edges, deep-level states are not fully ionized at room temperature. In

contrast, shallow-level states are those considered to be fully ionized at room tem-

perature due to thermal excitation.

The interaction between a deep-level state and electrons and holes can be de-





57

to (Ec Er) by either radiative (photon) or non-radiative (phonon) processes, or a


combination of both.


Process (b) involves a trapped electron being emitted to the


conduction band after receiving an amount of energy equal to (Ec Er) from either

optical or thermal excitation for instance.


Process (c) is the capture of a deep-level state electron


by the valence band.


Valence band electrons are more tightly held to the crystal atom than are the con-


duction band electrons


constants).


, which have a much larger radius of travel (of several lattice


This allows the analogy that conduction band electrons can be considered


to be negative charges floating in a sea of fixed positive nuclei (lattice sites) as in a


metal.


Valence band electrons are held in a tight sphere about a nucleus and can


move only if the neighboring nucleus has an electron vacant site that it can jump


into. Once the electron moves into this vacant site, it leaves a net positive charge

behind. This positive charge will appear to be an entity, a hole, moving opposite


that of the valence band electron.


Thus


, it is feasible to consider process (c) as being


the capture of a valence band hole.


For a deep-level state electron to be captured by


the valence band, it must lose an amount of energy equal to (Er Ev), or likewise,

for a valence band hole to be captured by the trap, it must lose the same amount of

energy.

Process (d) involves the emission of a valence band electron to the deep-level state


after receiving energy (Er Ev).


This can also be thought of as hole emission from


the deep-level state to the valence band since an electron vacant deep-level state has

been filled by a valence band electron, thus leaving a hole in the valence band.

The four processes involving deep-level states can be described in terms of chemi-


cal reactions between two species. holes and electrons.


Elementary reactions proceed





58

concentration of conduction band electrons and the concentration of electron-vacant


deep-level states.


The concentration of conduction band electrons per energy incre-


ment can be expressed as


at energy


energy


f(E)D(E)dE where f(E) is the probability that a state


E will be occupied by an electron and D(E) is the density of states at

The concentration of electron-vacant deep-level states can be expressed


as [1


- fTr(r,t)]Nrr(r) where


fr(r,t) is the probability that a deep-level state will


be occupied by an electron and is a function of position and


time.


- fTr(r,t)]


is the probability that this state will be vacant or occupied by a hole.


The total


concentration of deep-level states is given by NTT(r) which is a function of position

only.


Process (a) becomes dra = c (E)[1


- fT(r, t)]NT(r)f(E)D(E)dE where c,(E) is


called the capture coefficient and is a function of energy. c,(E) can be thought of as


a reaction rate constant.


This expression is integrated over the energy range of the


conduction band (Ec -' oo),


[1 fr(r, t)]NTrr(r)


c4(E)f(E)D(E)dE.


(4.1)


Here, it is convenient to use a mean capture coefficient defined as


c,(E)f(E)D(E)dE


(4.2)


;f f(E)D(E)dE


Process (a) becomes


cn > [1 fT(r, t)lNr(r)n =< c,


(4.3)


where it is noted


that the denominator of


equation(4.2) is equal


to n,


the total


concentration


of conduction


band


electrons.


and pr is the concentration of holes


Ta =


ro


------1








Cn >= vUon


Blakemore [48] suggests that this definition is a carry over from


atomic physics where it was hoped that the capture cross-section would be propor-


tional to the cross-section of an atom (~ 10-16 cm-2

the capture cross-section can range anywhere from


However, he points out that


10-s2


to 10-12cm-2.


Despite


this apparent discrepancy,


the term capture cross-section remains in common use.


Grove [49] considers a,o to be a measure of how close a carrier has to come to a trap

center in order to be captured. Hence, a deep-level state can easily capture an elec-

tron if it has a large capture cross-section and/or if the mean thermal velocity of the

electron is increased, by an increase in the temperature, for example.

Electron emission from the deep-level state, process (b), can be represented as a

reaction between an electron-occupied deep-level state and an electron-vacant con-


duction band site.


The concentration of electrons occupying a deep-level state is given


by fT(r, t)Nrr(r) and the concentration of vacant conduction band sites is given by


[1- f(E)]D(E)dE at an energy increment dE.


These symbols are defined above; note


that [1 f(E)] is the probability that a conduction band site is vacant. An incremen-

tal expression for process (b) becomes: drb = en(E)fT(r,t)NTTrr(r)[1 f(E)]D(E)dE


where e,(E) is the reaction rate constant or the emission constant.


This expression


is integrated over the conduction band energy range,


rb =


fT(r, t)NTT(r)


e,(E)[1 f(E)]D(E)dE.


(4.4)


This rate expression is typically redefined as


(4.5)


where nT is the concentration of electrons occupying deep-level states and e, is defined

as


rb = enfT(r, t)NTT(r)= enrT







e, can be approximated by


en(E)D(E)dE.


(4.7)


Note that as the concentration of free electrons increases, f(E) increases, and there-


fore,


the emission rate constant decreases.


Or in


other words,


the emission rate


decreases as the number of free electrons increases because there are fewer vacant
conduction band sites for the deep-level state to emit its electron into.


Hole capture by the deep-level state can


by treated as a reaction between an


electron occupying a deep-level state and a free hole in the valence band.


centration of trapped electrons is given by


The con-


fT(r,t)Nrr(r) and the concentration of


holes in the valence band is given by [1


- f(E)]D(E)dE.


The incremental rate of


process (c) is dr, = %(E)fT(r,t)NTT(r)[1 -


f(E)]D(E)dE which can be integrated


over the energy range of the valence band to yield


rc =


fT(r, t)NTT(r)


B~
K


P(E)[1 f(E)]D(E)dE.


(4.8)


The hole capture coefficient is averaged similar to the electron capture coefficient,


fE c,(E)[1 -


f_ [1


f(E)]D(E)dE


(4.9)


f(E)]D(E)dE


in which case


rc --4.


> fr(r,t)NTT(r)p =--< Cp


(4.10)


where it is noted that the denominator of equation(4.9) is the concentration of holes


in the valence band.


> is traditionally defined as the product of a hole capture


cross-section, a,, and the mean thermal velocity of a hole, v,.

Hole emission bv the deem-level state is a reaction between an electron-vacant


e, X








rate of hole emission is given as: drd = e,(E)[1 fTr(r,t)]NTT(r)f(E)D(E)dE where

e,(E) is the hole emission constant. This expression is integrated over the valence


band energy range to yield


rd =


[1 fT(r, t)]Nrr(r)


ep(E)f(E)D(E)dE.


(4.11)


-00


This rate expression is redefined as


rd = e[1 fT(r, t)]NTTrr(r) = eprP


(4.12)


where


ep


Ev
La:


e(E)f (E)D(E)dE


IEv
-00


e%(E)D(E)dE


(4.13)


because f(E)


4.1.1


1 in the valence band for the non-degenerate case.


Equilibrium


The above formulations are general in that no equilibrium constraints were placed

on them. Here, relationships are derived based on the conditions of thermal equilib-


rium.


"principle of detailed balance"


will be used.


Shockley [50] quotes John


C. Slater's


description of this principle:


"When a system has reached thermal equi-


librium, it has run down and is no longer changing.


Past and future are alike to


Now imagine that a motion picture is made of the system, showing atoms and


electrons in detail.


This film can be projected backwards in time and since past and


future are alike, the observer will not be able to tell the difference. Now suppose, for

example, the forward-running picture shows on the average (Cap. n) electrons being

captured per unit volume per unit time on traps giving up the energy in the form of

he}at wavpn (nhnnnn s and llmrnn e it shnws (Eml. n') electrtnns hjinur emittAl nnr nnit








to (Cap.


n), forward and backwards running of the film can be distinguished, con-


trary to the assumption that the system is run down.


Thus, the principle of detailed


balance requires that each process and its reverse proceed at equal rates."

With this in mind, process (a) must proceed at the same rate as its reverse, process

(b). On average, a conduction band electron loses energy and is captured by a trap

at the same time a trapped electron gains the same unit of energy and is emitted


to the conduction band.


Likewise, process (c) proceeds at the same rate as process


valence band hole loses energy and is captured by a trap at the same time a


trapped hole gains energy and is emitted to the valence band.


These rate expressions


are related as follows:


ra = rb =--r<


r --= rd ---'<


where the subscript


PTenc cut fll~e


nTePe = Cpe lre


e has been added to denote equilibrium conditions.


(4.14)


(4.15)

These equa-


tions can be solved in terms of ratios by equating f of each relation,
Pre


nTe
pT,


The ratio of the third term to the second term is equal to unity, i.e.,


(4.16)


(4.17)


flepe


The mass-action law states that the product of the electron and hole concentration


is equal to the intrinsic concentration squared (nepe


= n?) under thermal equilibrium.


II lIlk\ III U11 1 1 I~


epeene





63

The rate coefficients can be expressed in terms of an Arrhenius activation energy


through the use of the principle of detailed balance.

rates (a) and (b) at thermal equilibrium [46, 51],


1cPTe


We have from the equality of


[1 el2


(4.19)


Under the assumption of non-degeneracy, the equilibrium concentration of electrons

in the conduction band is given as (Boltzmann's approximation)


EF Ei
nE, = n exp( )


(4.20)


where EF is the Fermi level of the semiconductor, E, is the intrinsic energy level, and


k = 8.6173


x 10 -s is the Boltzmann's constant.


The probability of the deep-level state at energy level ET being occupied by an

electron can be described by a Fermi-Dirac distribution function at equilibrium,


fTe =


1+frexp( t")


(4.21)


where g is the impurity level spin degeneracy [48] which refers to the number of states


having the same energy level Er.


The subscript e has been added to the distribution


function to indicate that this is an equilibrium expression.

to electron occupancy in equation (4.19) becomes


The ratio of hole occupancy


- fTe]


1 Er
= -exp(
g


- E
kT


(4.22)


Equations (4.20) and (4.22) are substituted into equation (4.19) to yield


1 ET
-~ ~ T~ A
= ni-exp(
g


(4.23)


where nl is defined as the concentration of condnr.tinn hand wplrtimnn if thp TFnmi lnal








The ratio pe/


> can be described in a similar manner when process (c)


proceeds at the same rate as process (d) (thermal equilibrium),


fTe
[1 e
[1- fr8]


EF
=gexp(-


- ErT (E
- )ni exp(
kT "


(4.24)


kT


Ei
= gniexp(--


-1


(4.25)


pi is defined as the equilibrium concentration of valence band holes if the Fermi level

coincides with the energy level of the deep-level state, modified by g.


4.1.2


Non-equilibrium


This section considers the carrier concentrations and the deep-level states under


non-equilibrium conditions as developed by Sah [51,


The following transport


equations are of the form


Accumulation


= Flux


Gradient + Net


Rate


Production


on a unit volume basis.


The rate of change of the conduction band electrons can be


represented as


On(r,t)


Cn > n(r, t)pTr(r, t) + e nT(r, t) + G.(r) + gn(r, t).


(4.26)


The rate of change of the valence band holes can be represented as


ap(r, t)


-Q


> p(r, t)nT(r, t) + e,,pr(r, t) + G,(r) + gp(r, t).


(4.27)


The rate of change of the concentration of deep-level state electrons can be repre-

sented as


53].


~,,I, r\





65

The notation in parenthesis has been added to indicate the dependence on position


and time.


The terms G,(r),g,(r,t),G,(r), and gp(r,t) have been introduced to ac-


count for other net generation processes which are separated into steady state and


dynamic terms.


These net generation processes include such contributions as the in-


fluence of other deep-level states, and conduction band to valence band generations

and recombinations. GT(r) and gt(r, t) have been added to allow for optical excitation


of the deep-level states [51


The carrier flux can be attributed to contributions of


drift (migration) and diffusion processes [52] i.e.,


jn = qpnnE + qD,Vn, and


(4.29)


jp = qpppE qDpVp.


(4.30)


These carrier flux expressions would have an additional term to treat any DC leakage


current.


For the present development, it is assumed that DC leakage current is not


present.

The above equations are related to the electrostatic potential via Poisson's equa-

tion:


P = Q[(r, t)


- n (r,t) + (ND(r, t) nD(r,t)) (NA(r,t)


= KeoV


- nA(r, t)) nT(r,t)]


E = -KeoV2


(4.31)


where p is the concentration of charged species, q = 1.602


x 10-19 C is the electronic


charge,


K is the dielectric constant,


= 8.85419


x 10-12 -


is the permittivity


of vacuum,


E is the electric field,


4 is the electrostatic potential,


ND(r,t) is the


donor concentration, n(r, t) is the concentration of un-ionized donors, NA (r, t) is


the concentration of arrentorsn.


nA (rt1 is thei ronrcfntratinn of nn- innimd a.rrentnra








4.1.3


Small Signal Analysis


The above non-equilibrium equations will be analyzed here with the small signal


approach.


are:


There are several assumptions made in the present development which


the perturbation from equilibrium (np


n?) is small such that only the first


two terms of the Taylor series are sufficient, the shallow-level donors and acceptors


are fully ionized


(fD, nA


= 0)


at the


temperatures


analyzed,


the semiconductor


remains non-degenerate such


that the non-equilibrium concentration of electrons,


holes, and trapped electrons can be expressed with a quasi-Fermi energy level for

each (F., F,, FT), and the system is one-dimensional.


The four rate constants are assumed to be equal to their equilibrium values (


- aepc,


>=<


c, > and e/


e-,


This implies that the ratios


> are constant and, in light of equations (4.23) and (4.25),


ET-Ei=


constant.


(4.32)


The quasi-Fermi energy level assumption,


valid for a non-degenerate semicon-


ductor, permits the use of the Einstein relation in the flux equations.


The Einstein


relation is


kT
Di = -- p.

The one-dimensional electron and hole flux equations become


(4.33)


OF,
jn = pun ,x

OFp
jp = pp x


(4.34)


(4.35)


The concentrations can be approximated by the first


two terms of the


Taylor


>,


>--<








where


n(r,t)


=F -Ei-
= niexp( kT ).


(4.37)


The partial derivative with respect to time can be expressed as


9n(r,t)
m It


= exp(


___ Fn


'9K
Ot'


N(r) (OFI
LeT Ot


Ot*


(4.38)


The hole concentration can be represented as


p(r,t)


Op(r, t)


= P(r) + Op(r, t)


(4.39)


where


(Ei
= n"exp(


(4.40)


kT '


Op(r, t)


OF,
-~ ~)


ni E;
k^e xP(
= nc exp(
kfT


=P(r)( e9Ei
tm -A


OF
~ ) .


(4.41)


The trapped electron concentration can be represented as


nT(r, t)


-= NT(r) +


Onr(r,t)


= NT(r)+ nt(r, t)


(4.42)


where


nT(r, t)


NTT(r)
1+lexp()
2 ItT


(4.43)


dnT(r,t)
cu It


NTT(r)
[l+ lexp(-F )


1 ET
T exp(
gkT


-FTr aE,
kT a Ot


OF-
-)a


(4.44)


PT(r) aE,
NTT(r) at


O)F,
--


The trapped hole concentration can be represented as


Opr(r,t) ,


n' /_\ i -- \


p(r,t)


NT(r)


P() +


- li~p aEi
o


r\


In nr\









Opr(r, t)
stIt


Nrr(r)T exp(E --T)
1 + exp(Z-F)


N_ (r) 1 1 ET-F
- 1 -F( ex+pT( ))T
C1+~expQ~r&)J2 kT g


,DE,


I'


OF, ,


I


= PT(r)
-tn Cl- ~


PT(r) aEt OF
NTT(]) t -t
Nnr(r)1 at 01


PTr(r) NT(r) (EO
) = (r)
kT NTT(r) 9t


Ft
at


(4.47)


The results of the above derivations with the substitution of OEt
produced here for clarity


= E E;are re-


n(r,t)


= N(r) + k(F, Ei),


(4.48)


p(r,t)


= P(r) (r)
IT"


(4.49)


nT(r, t)


= NT(r)-


NT(r) Pr(r)
kT NTT(r)'


(4.50)


pr(r,t) = PT(r) +


PT(r) NT(r) ( .
kT NTT(r)


(4.51)


Equations (4.48) to (4.51) are substituted in the non-equilibrium equation (4.26)
to yield


N(r) jF,
kT Di


OEI
ai '


" [N(r) +
q


S(OF OE,)I[V(P, + OF)])
kT F-Eiy+F,


cn > [N(r) + N(OF. 8Ei)][PT(r) +
kT


PT(r) NT(r)
NTT(rDE OF,)]
kT NTV(r)


NT(r) PTr(r)
+en[NT(r) (E (F,)] + Gn(r) + g(r,t)
kT NTr(r)


'("l
q


*(N(r)VF,)-< c


Sf" N(r)VOF, +
q


(4.52)


> N(r)Pr(r) + eNT(r) + G,(r)

N(r)VF.
kT (OF, OE) + g(r,t)
kTf


r -- d A -. n


at at


v


r








NT(r) ,F.
S> N(r)PT(r)Nr(
NTT (r)


-9E, OE-OFt)
Lr )r Im )*


The terms in the curly bracket is the steady state equation and hence can be set equal

to zero, and the last two terms are second order terms which can be dropped since

the Taylor series included only first order terms.

It is advantageous to divide the semiconductor into elements and to assume that

in each element the static quasi-Fermi potentials and the static electron, hole, and


trap concentrations are independent of position.


The dynamic generation terms are


also assumed


to be independent of position within each element.


However,


these


properties can vary from element to element but are constant within each element.

Equation (4.52) reduces to


kcT Ot attiN 0xL2+


cm>N-PT T-
cn > NPT( T )


(4.53)


-[

NT Pr aF,
> NPT + eNT-](


le aI)


Substitution of equations (4.48) to (4.51) into equation (4.27) yields


P(r) (OE OF,
kT t t)= -


Cp(r)+ P(r)(B-F
S[P(r) + (dEi dF,)][V(F,


+ aF,)]


> [P(r) + ((Ei OF,)][NT(r) N
kT keT


PT(r)
Nr(r)(ES OF,)]
NTT (r)


P [P(r)(r)
+ e,[PT(r)+
ST


NT(r) (E aFt)] + G,(r) + g,(r, t).


This equation can be reduced with the subtraction of the steady state equation, the

elimination of second order terms, and with the assumption of position independent

static properties within each element as well as the dynamic generation term to yield


P dI. REni2 82 F.


~Ar.~


- E2


(4.54)










Substitution of equations (4.48) to (4.51) into equation (4.28) yields


NT(r)PT(r) (OF,
kTNTT(r) 9t


OEi)
at


N(r) + N(r)(F"


(4.56)


r + PT rN(r) NEU
PTr + PT(]P) F


> [P(r)+ P(r)(


- aF,,
-rr


-8F
-kT)][NT(r)
kT


- )F,


PT(r) 9OE -9 F,,
- NT(r) ( T )
NTr(r) kT


S+ NT(r) O-Ei
+ e, [PT(r) + PrT (r) ( (
NT (r)


- + GT(r) + Ft(rt).
--)] + GT(r) + g(rt).


This equation is reduced by subtraction of the steady state equation, elimination


of second


order terms,


with


the assumption


of position independent element


properties to yield


NTPT
kTNTT


OF,


NT PT
c > NPTN + eSNT-
NVTT N TT




PT NT ] OF,
> PNT- +epPr- (
NTT NTT


-O~i)
leT


> PNTOF
> PNr( -


- 'E*.
kT )+
LeT


F> N
c, > NPT( "


-OE
kT ) + t(t).


(4.57)


Poisson's equation becomes


CV2(E, + OE,)


EP(r)P(r)(
P(r)+P(r)(a


-OF
T 1')


- N(r)


- N(r)( E O)
kT


+ND(r) NA(r)


,PT(r) (Ei
- NT(r) + NT(r)T(r)
NTr(r)(


- OF,
kT


(4.58)


where it was assumed that the electrostatic potential is equal to the intrinsic energy


level divided by the electronic charge,


-Ei.


The derivative with respect to time


of this equation is more useful here,


1
Keo
q2


atV OE


PO Ei
-kT-


-9F
9P)


N OF,
-kT
kCT


-OEi)+
at


NTPT
kTNTT


OEi
(--


-9F)
).
at


(4.59)


NT (r) ) (aEi
[NT(r)- NT (r)N(
NTT(r)


OE;
a)


,


Cn >[


,] e,


--aEi
)]








4.1.4


Equivalent Circuit


The small signal non-equilibrium equations derived above can be related to an

equivalent circuit composed of common circuit elements such as resistors and capac-


itors.


To do this


, the partial quasi-Fermi and the partial intrinsic energy levels have


to be related to a voltage [51


This can be done by dividing the energy levels by


the electronic charge,


OF
--


OFt
- -


(4.60)


Kirchhoff's current law is applied to each small signal equation.


This law states that


the sum of all the currents entering and exiting a circuit node must be zero. At this

point it is useful to remember that the current through a capacitor is equal to the

capacitance multiplied by the time derivative of the voltage across the capacitor and

the current through a conductor is the conductance multiplied by the voltage drop


across the conductor.


The small signal equation derived above are reproduced below


for clarity,


N OF,, E aEi aaF,
kT( at -t )- -q
kT~ Di Ot q Ozx2


OF, --Ei
cI > NPT( kT )-
k:T


(4.61)


NT
> NPT-N
Nrr


PT OFt
+ eNT -](-


-OaEi
lv)?


= g,


kT(
kT 9t


-[< cP


- -)E
Ot


PT
> PNT -
Nrr


+ NT
+ ePtr-


OF,-9E,
> PNT( kT )
kT


(Oft
3(-


-DaEi
lv)?


(4.62)


= -9p,


NrPT
kTNrr


NT PT
c. > NPr-_ + e.Nr _-+


SF,
Ot


- 'a


BEi)
at


PT NT 1 (OF
> PN-- + ePr-I (-


Clp 020Fp
- P '"
-a 89


--[< Cn


-aEi










1
qK^ Vtd


SP(OF,
kT
kT


- 9Ei+ N8aF,
at IcT'


-oEi
9E)+
at


NrPr
kTNTT


aOF


(4.64)


The next step is to replace the quasi-Fermi energy level perturbations,


with


the corresponding voltages,


-qv,


and multiply the above equations by the electronic


charge, q.


This changes the voltage coefficients into capacitances and conductances.


There are four voltage parameters in equations (4.61) to (4.64),


, Vp, vt,


and vi.


These equations are re-arranged into voltage drops across circuit elements, i.


- vi, vt


-V 2i,Vn1


- vt, and vp


- vt. It is assumed that direct band to band processes


are included in g, and g, and hence, v,


- v, need not be considered here.


To find


the coefficients of the these voltage drops,


it is best to collect all the coefficients of


like voltage and then regroup the voltages with common coefficient for each equation.


The relationship, NTT


= NT + PT is used several times.


Upon doing the above, the equations can be expressed as


q2 Na( V,
lvi


- vi)


22, +
- qpN8-2


cn> NPT(
-T^----- {n


- t)


2 PT


kTNTT


[

> NPT


- enNTr](vt


= -qgn,


(4.65)


q2P O(v,


- vi)


- qpP- P --- +


> PNT
r-- ("


-Vt)


q2NT


kTNTT


ReC


>PNT


-e,Pr](vt -vi)


= qgp,


(4.66)


q2 NTPT (vt


kTNTr


-Vi)


_ q- [<
kTN TT





PNT,
(vP


enz'NP~


-Vt)


- ePrNT+


Cn> NPT(
--^----( n


>PN4


-Vi)


-epPrNT (Vt


= qgt,


-vi)


(4.67)


e., t/v,-vi,


--a~Eli
at


- vi)





73

At this point, the semiconductor is divided into equally spaced elements in order


to treat the second order derivatives.


The second order derivatives are discretized


with the central finite difference method applied to three elements.


The second order


derivatives of each potential in the above governing equations can be expressed as


02Vrk
BA:


Vk+1 21vk + Vk_-


(Az)2


Vk+l Vk


(Aa)2


-ktr-


(4.69)


(Az)2


where the subscript k is used to indicate the element number. It is apparent that the

second derivative in potential can be thought of as a potential drop across a circuit

component at the beginning of the element minus the potential drop across a circuit

component at the end of the element.

We now let


Cn=


q2N
kT'


(4.70)


C, =


(4.71)


q2NT PT
kTNTT


, (4.72)


an= qp N
=(Ax)2'

= (Aaz)2'


(4.73)


(4.74)


(4.75)


Ke,
(Ax)2'


gr < Cn > NPT
kT
q2< Cp > PNT


(4.76)


(4.77)


q2apT


kfTNTT
.2 AL.


[ Ccc


> NPT eNT],


(4.78)


Cr -


CK -


GnT =

GpT =


GnTi =








(4.82)


- qgt


The above conductances have units of current per volt per volume and the capaci-

tances have units of charge per volt per volume.

These conductances and capacitances are substituted in the above equations to

yield


-Vi)


-Unk-,+)+GnT(vn


(4.83)


-vi)


-Vp,k)+Gp,k+i (vp,k-Vp,k+1 )+GpT(Vp


=- p,

(4.84)


- uv)


- (GnTi + G,Ti)(v,


- vi)


- GpT(Vp


- vt)


- GnT(V,


- t)


= iT, (4.85)


S(v,
^p-


- vi)


+cn


8(v,


-vi)


-v.)


+Cr


- Vi,k)


= -CK,k


- Vi,k+1)


+ CK,k+1


(4.86)


The corresponding equivalent circuit is given in Figure 4


and Figure 4.2 are consistent with that derived by Sah [51,


Equations (4.70) to (4.86)

, 52].


Surface States


The treatment of surface states presented here follows that of Sah


Surface


states can be a discrete single energy level defect or a distributed energy level defect


(surface band).


4.2.1


The single energy level surface state is treated first.


Single Energy Level Surface State


cys


-S,,


- Gnk (2t,,kl 2)nk)+ Gnk+l (Vnk


-vt)+GTi(vt--vi)


a(v,
c,-


-- Gpk (Vpk-~l


-- ttt)+ GpTi (tlt t)i)


d(Vik


a(v;,kl








is a boundary condition to Poisson's


Equation in that it relates the first derivative


of the potential to the charge located at


the surface.


The surface concentrations


can be expressed in terms of quasi-Fermi energy levels that are either bulk values

evaluated at the surface (F,, F,.) or newly defined levels associated with the surface


state (FT.).


The concentration of electrons evaluated at the surface is given by


F= n exp(
= niexp(-


- E
kT )


(4.87)


and of holes,


p (t)


=Ein
= ni exp(-


FS)
k:T


(4.88)


The concentration of electrons trapped by the surface state can be expressed as


nTr(t)


(4.89)


1+1 exp(&;E)F


The concentration of surface states that are electron vacant is given by


NTT,


mi.t)


exp( k-)


(4.90)


l+ pexp( C )


where NTT is the total concentration of surface states, Ei, is the intrinsic energy

level evaluated at the surface, ET, is the surface state energy level, and the subscript

s denotes that these are either surface properties or parameters associated with the

surface state.

The Taylor series approximations of the surface concentrations are


n,(t)


= N,


N.


- 9E;,),


(4.91)


p.(t)


P + P
P,+ (T(Ei,


- OF,,),


(4.92)


nfT,(t)


= NT.


NTTB


(8Ei8


(4.93)


n.(t)


NTTI


-- aF,,),





76

The non-equilibrium equations that describe the rate processes between the carrier

bands and the surface state are the time rate of change of surface state occupancy


49fT8(t)


(4.95)


and Gauss'


Law


- Keo^ = qrT,(t)
nf


(4.96)


where it is assumed that the potential gradient on the metal contact is zero.

These equations apply to a two-dimensional plane and hence, are based per unit of


area, cm2


The surface state electron occupancy equation is two-dimensional through-


out because nT, is per surface area, not volume.


units of inverse time.


The emission rate coefficients have


The product of the capture rate coefficients and the respective


carrier concentration have units of time. Both of these expressions are multiplied by

a concentration of surface state, the product of such has units of per time per area.


Upon substitution of the


Taylor series approximations of the surface concentra-


tions, the above governing equations become


NTPTr, (Ft, OE_
kTNTTr, Ot t


pNa( OF
P, NT,(


- 9E;,
kT


OF8 9E)
> NPT,( )
kT


+

NT, PT.+
> NPTrTT + eNT, ~+
NTTs NTT


Nrri (Oft.


PT.
> PNT,
NTTs


- 9Ei,
,-m ) = gt (t)


(4.97)


and the time derivative of Gauss'


Law becomes


1 8 OOEi,
-s K Ox
ao t -:


qNTPT. (dEi,
kTNrT,r


-OF,t
at


(4.98)


where the stePnAv state p1innatinn hna hen elimninatld as well as the srm ndnl nrdsr terms


Cns > n~(t)pTo(t)- ens 1ZTs(t)- < Cpa


> P.(t)nT, (t)+e~pr, (t)+ GT, +gt, (t)






77

can then be considered as simple resistors and capacitors with units of current and

charge per volt per square area.

The governing equations can be expressed as


O(v~.


-v)
vi,


- (GnTis


+ GpT,) (vt,


- v)i.)


- GpTS(vps


- Vt,) GnT,(vns


- ve)


== fl's


(4.99)


- t. 1
v^8)


= CKs


- Vil)


(4.100)


where


q2NTPT.
k TNTTs


(4.101)


(4.102)


GnTs

GuT.
GyT.


(4.103)


q < Cn, > NPT,
lkT
qZ < C > PNT,.


(4.104)


GnTis


q2PTs


kTNTT.


ke n.


> N,PT.


- ensNT.],


(4.105)


GpTiBr


QNT.)


k!TNTT,


> PNT.


- epPT,],


(4.106)


- nt


The corresponding equivalent circuit is given in Figure 4.3

and Figure 4.3 are consistent with that derived by Sah [


4.2.2


(4.107)


!. Equations (4.99) to (4.107)

54].


Surface Band


An equivalent circuit is developed in this section to incorporate a surface deep-


a(v,,


a(vi,





78

before in that the capture and emission rate coefficients are a function of the carrier

band energy level and also the energy between the carrier band and surface band.


Each incremental process is developed first,


then the small signal analysis is applied,


and the equivalent circuit is deduced.


Process (asb)


is the rate of electron


capture


by the surface


band and


can be


expressed as


d2rab


= c.,(E


E,)f (E)D(E)[1


- flb (Esb)]D,b(Ea)ddEdE.


(4.108)


where E8, is the energy level of the surface band increment dEg,


and the subscript


sb is used to indicate a surface band property. All other symbols are consistent with


earlier definitions.


This equation becomes upon integration over the conduction band


dr.,b


> n[1l


-- fab(Ea)]DD,(E,)dEb


(4.109)


where


< Cnab(Eb)


1 o
cn-b >= J
ri Ec


Eb)f(E)D(E)dE


(4.110)


Process (bsb) is the rate of electron emission from the surface band and is given


Pblbb


- f(E)]D(E)f.b(E.,)D.b (E,b)dEdEb.


= -enb(E


(4.111)


This equation becomes upon integration over the conduction band


= enb(Esb)fsb(E,b)Drb(E,a)dE,b


(4.112)


where


enab (Esb)


=- nsb


- f(E)]D(E)dE.


(4.113)


>=<


--- -- -


< C~,b (Elb)


Cnbb (E


Ea)[l


drbsb


enrb(E


,Eirb)[l





79

where the distribution function was assumed to have the same form as equation (4.20)

and equation (4.22) was used to relate the equilibrium electron concentration to the


intrinsic concentration.


This relation will be used in the derivation of the equivalent


circuit.


Process (csb),


the rate of hole capture can be expressed as


(4.115)


and upon integration over the valence band becomes


> P.fSb(ESb)Dab(Eab)dEb


(4.116)


where


< Cpb(Eb)


>=<


1 Ev
-L E


- f(E)]D(E)dE.


(4.117)


Process (dsb) is the rate of hole emission from the surface band and is given by


d'~rd,


= eps(E


, Eb)f(E)D(E)[1 f,b(Eb)]D,a(Eb)dEdE,b.


(4.118)


This equation becomes upon integration over the valence band


drdsb


- eps(E,))[1


- fab(E,b)]Dab(Ea )dE,sb


(4.119)


where


eps (E,a)


- ,.b


Ev
J1o


, E,)f(E)D(E)dE.


(4.120)


The rate of process (csb) equals the rate of process (dsb) at each increment of


surface band energy under thermal equilibrium.


This yields the equilibrium relation


hwptxwv -rta~ wrnnntnn


- f(E)]D(E)fsb(Eab)Dn6(Eb)dEdEb


=< Cp,(Esh)


ddr,~


Cprb(E, E~b)[l


drc~b


Edb )[1


Cpb (E


%,a(E





80

The governing equations at the surface are similar to those of the single energy


level surface state.


The rate of surface band state concentration must be addressed at


a particular energy level and not as an integrated sum of reactions because equilibrium


conditions only apply to individual states at a particular energy level.


Thus the rate


of change of surface band concentration is given by


anab(E.)


= [dr.,b


- drbsb


- dr'es + dra, + g.b( Eb, t)J IlEs


(4.122)


Gauss'


Law can be expressed as


e08alt


=fE


(4.123)


where E, and EI are the upper and lower energy bounds of the surface band, respec-

tively.

The concentrations of electrons and holes, and the surface band distribution func-


tion can be approximated with the first two terms of the


Taylor series expansion.


These expressions are


=N,


N+ ,
+ 8FT


(4.124)


P= +
=P,+ (Eia,


(4.125)


f b(Esb)


= Jfb


- f8&)(OEi.


- OF,,,),


(4.126)


- fAb(Eb)


1
f~b(1


- 9F


- fb)(9Ei,


(4.127)


where fb (without the energy notation) denotes the static distribution function which

is a function of E,.

An additional assumption is necessary in the present case due to the distributed

nature of the surface band; small perturbations in potential are independent of the


f~a (-Eb)Dsb(Eb)dEa


--


- aF,,),


1
fdb(l
LT


1 fb ~








- 8E. )][1


1
- fb + fob(1
kTfa


- fb9)(aEi.


1
- f-(1
-^-fT


- f.b)(9Ei.


- OF,b)]Do,(Eb)


(4.128)


P, + -(AE,


-< pb~lb


- OFo)][fb


1
k f~a(l -f.6)(8Ei,


+epb(Eb) [1


- fb)(OEi.


- 8F,6)]D,6(Eb) + g6(E.6, t)] dEs.


The time derivative of Gauss'


Law can be expressed as


-, all
1 Ot dr


feb (1l


-- f8E 8


OF.
--)Da&(E~a)dE&.


(4.129)


The steady state equation is subtracted from


the above equations and second


order terms are neglected.


The governing equations


become,


upon regrouping the


dynamic energy terms,


1 Fb
fa&(1 feb)D.b(Ea)dEb(


0g.)
at


(4.130)


N,(1 fb)Dsb(Ea)


- 8Fb )


< cn (E,a)


N,(1


- fsDb)2Ds(Esb)


dEoa(OFa


-OaEi.)


< Cpb(E&b)


Pfb Db (Esb)


- epb (Eab) f(1


- fb)D4b(Eb)


dEb (OFb


< Cpb(Eb)


- OFB,) +


P~f~ olatin b


gsb (Ebs, t)dE,b


e, 8 8OEx
at Oax


=qE.


1 Eb)-
k-fa((l fab)(
k:T at


OFs
)D )Db(EIb)dEsb.
dt


(4.131)


These governing equations are related to common circuit components through the


N


- esh(Es}) [fs


- fsb)Dab(Eb)


1
f.a + fbb(l
k:[;r


- ar~b)l Dbli(Elb)


- dFb)]DI)(~Eb)


< cnQ (Esb)


f~b(i
--en~b(E~L)


-aEi~)









- vi)


= CK


8(vi.,


- vil)


(4.133)


where the conductances have units of current per volt per area and the capacitances


have units of charge per volt per area.


These circuit components are defined as


= q2


pfab(1 fsb)D,b(Eb )dEa


(4.134)


(4.135)


N,(1 fb)Dsb (Eb)
kT
P.,f6 D,sb(Es) ,
> --~oEb


(4.136)


(4.137)


< C ,an(Esb)


N,(1


- f, )2 Dsb (Es)


- enaa(Eob) -


(4.138)


P, f,Dsb(Esba)


< Cpsb(Eab)


=Eq
= q ,


- fsb)D^s(Esb)


- ep%(Eob) f,(1


dR,6


(4.139)

(4.140)


9g, (E,, t)dEs.


The corresponding equivalent circuit is identical to the equivalent circuit for the single


energy level surface state given in Figure 4.3.


are consistent with that of Sah



4.3 Co


This circuit and its related equations


implete Equivalent Circuit


The surface state equivalent circuit can be coupled to the bulk state equivalent


circuit by converting the bulk state capacitances and conductances from a


h;aqi to a snrfarpe area ha.in


volume


This is dmnn hv mnltinlvintr thePQs terms hv the width of


9(Va~b


dR.6


< Cnb (Esb)


Gnsb


< Cpb(E~b)


- f~b) Dba(Eda)


G,;,a


Gpisb








current.


The Schottky contact can be treated as a capacitance, Co, connected only


to the intrinsic energy line because electrons and holes cannot flow across a Schottky


contact; only charging and discharging is allowed in an ideal case.


The Ohmic contact


on the right side of the circuit can be treated as a short of all three energy lines because

a potential difference cannot be maintained in a metal contact.


The circuit shown in Figure 4.4 represents


the minimum number of


elements


needed to model a Schottky contact (one element) with surface states (one element).


In the case of a pn junction, at least two bulk state elements are needed.


It can


be assumed that surface states at the Ohmic contacts can be neglected because the


contacts would appear as a short across the surface states.


One of the bulk state ele-


ments would represent a semiconductor with p-type behavior and the other element

would represent n-type behavior.

The use of only one element can be rationalized from an analysis of the deep-


level state terms at various locations in the semiconductor.


The trap capacitance,


Ct, is proportional to the product of the trapped electron concentration, NT, and the


trapped hole concentration, PT.


The trap can be assumed to be completely full or


empty in the bulk of the semiconductor depending on the location of the trap energy


level in relation to the Fermi level.


Therefore the product NTPT is zero and, hence,


the trap capacitance is zero. Likewise, band bending at the surface will make either


NT or PT zero and the trap capacitance zero.


The maximum in the trap capacitance


will occur at the point where the trap energy level, ET, crosses the Fermi energy level


at which point NT = PT.


The product of the trapped electron probability and the


trapped hole probability, fr[1 fT] is its maximum value of one fourth at this point.


Tt in this rPwinn rnf the sPmircnrhdtrnr that thin n1nivalent cirrn1it aAIzdrP.aMPRP


Th u nint





84

either a Schottky contact or pn junction, the equivalent circuit would be represented


by a single conductance term, G, or G,.

Figure 4.4 with a short between v,,, vi,, a


(Replace the Schottky capacitance, Co in


nd vt,). No space charge region would form


, hence, the filling and emptying of traps would not take place. Although surface


and/or bulk states may be present, neither would influence the electrical response.


4.3.1


Simplifications of the Complete Circuit


It was assumed that the experiments performed in this work do not perturb the


semiconductor far from equilibrium sucd

equal the equilibrium Fermi level, EF.


the steady state quasi-Fermi levels


The steady state concentrations are equal


to their equilibrium concentrations with this assumption. As a direct result of this

equilibrium assumption, the transconductance terms, GnTi, GpTi, etc., are zero (rate


of emission equals rate of capture) and can be removed from the circuit.

Additional assumptions can be made if the semiconductor is n-type or p-type.

All of the semiconductor devices studied in this work can be treated as n-type semi-


conductors with Schottky contacts.


The concentration of electrons is many orders


of magnitude greater than the concentration of electrons for this case (pe, e


=- ,


, 1017


a 010).


This permits


the removal of the hole conductance (Gp),


hole capacitance (CG),


ure 4.4.


and the hole recombination conductance (GTi) terms in Fig-


This also has the effect of removing the hole surface state recombination


term (GpTi,) from the circuit even though the concentration of holes at the surface

might be of the same order of magnitude as the bulk concentration of electrons [57].

The resulting equivalent circuit is presented in Figure 4.5.
r.1l* .. -- "^J A i 4 4S1 #. 4 4








series is in parallel to a capacitance.


The conductance-capacitance series can


thought of as a recombination arm and the capacitance as a space charge capacitance.

The space charge capacitance is composed of either the dielectric capacitance and


the electron capacitance in series for the surface state case, or,


capacitance for a bulk state.


only the electron


This space charge capacitance will remain constant


in either case so long as the concentration of deep-level states is several orders of

magnitude smaller than the concentration of electrons due to shallow-level states.

It was assumed throughout the derivation of this circuit that the two dielectric

capacitances were identical because in the finite difference approximation of the sec-


ond derivative of potential,


the element


width,


was assumed to be constant.


This is certainly true for the case of a large number of bulk elements.


However, in


the present


case


only one bulk state element was used and the assumption of equal


sized element widths is not appropriate.


The element width that comprises the space


charge region is very small and of the same order of magnitude as the depletion width,


Whereas, the element width near the Ohmic contact incorporates the bulk of the


semiconductor and is on the order of a millimeter.


capacitance terms,


This implies that the dielectric


which are inversely proportional to the element width, differ by


several orders of magnitude.


The dielectric capacitance near the Schottky contact


is much larger than the dielectric capacitance near the Ohmic contact to the extent

that the latter term can be treated as an open circuit.


governing equations in


this chapter have


been


developed for the case of


one bulk deep-level state and one surface state.


There will be one additional gov-


ering equation that resembles the kinetic trap equation for each additional state


nrefsentt


TheSp will annear as rprrmhinati nn arms in naTallel tn thna glsir.tp l in





86

Based on the above discussion, the final simplified equivalent circuit used to regress


the experimental data is given in Figure 4.7.


This circuit has all the essential features


that were mentioned above, namely, the equivalent circuit for a surface state is iden-


tical to that of a bulk state.


It is possible to distinguish the two types of states by


means of a Mott-Schottky plot (C2 versus applied bias) as discussed in Chapter


A leakage resistor has been added to the equivalent circuit of Figure 4.7 to account


for the non-ideal behavior of the Schottky contact (DC leakage current).


is proportional to the exponential of an activation energy.


This term


The activation energy can


be the barrier height of the Schottky barrier or the energy difference between the

conduction band and a surface state energy level. Leakage resistance was addressed


for a p-n junction in the literature [59, 60].


The leakage resistor may also be the in-


ternal resistor, R1,, of the measuring op amp, especially in cases in which the sample

impedance is of the same order of magnitude as Rn[61]

A similar circuit was derived by van der Ziel [62] in 1959 to account for generation-


recombination noise due to traps in semiconductors.


Van der Ziel [63] also determined


trap activation


energies


by measuring noise resistance as


a function of frequency


and temperature (similar to Admittance Spectroscopy).


Dare-Edwards


et al. [64]


developed a similar circuit by eliminating terms from the governing equations be-

fore developing an equivalent circuit. Nicollian and Goetzberger [55] developed this


circuit


(without leakage resistance) to analyze interface states of metal-insulator-


semiconductor (MIS) capacitors.

The impedance data was regressed to the simplified equivalent circuit by complex


nonlinear regression [65,


This regression method is based on


the Levenberg-


MarrarUt nln,.mnthm ffi7l


AnI altanatu' a tr- iiea this nirtnI-nonr rnurpaarrocnn nnr lrsn