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Electronic band structure and optical characteristics of quantum- size CdTe crystals in glass films

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Electronic band structure and optical characteristics of quantum- size CdTe crystals in glass films
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Potter, Barrett G., 1963-
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vii, 252 leaves : ill., photos ; 29 cm.

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Absorption spectra ( jstor )
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Crystallites ( jstor )
Crystals ( jstor )
Electronics ( jstor )
Electrons ( jstor )
Energy ( jstor )
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Ions ( jstor )
Semiconductors ( jstor )
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Thesis (Ph. D.)--University of Florida, 1991.
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Includes bibliographical references (leaves 239-251).
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Typescript.
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Vita.
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by Barrett G. Potter.

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ELECTRONIC BAND STRUCTURE AND OPTICAL CHARACTERISTICS
OF QUANTUM-SIZE CdTe CRYSTALS IN GLASS FILMS




















By

BARRETT G. POTTER, JR.


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

UNIVERSITY OF FLORIDA


1991
















ACKNOWLEDGMENTS


During the course of performing the research leading to the

present work, I have had the distinct pleasure and honor of interacting

with many fine individuals, both within the University of Florida and,

on occasion of collaborative work, outside the state. Contact with

these people will remain in my mind as one of the most valued facets of

my educational and personal experience during graduate school,

contributing immeasurably to my professional and personal development.

I would first like to express my deepest appreciation and

gratitude to my academic advisor, Dr. Joseph H. Simmons, who took me

into his group six years ago and allowed me to work on some truly

exciting research. His continued encouragement, direction and

assistance through the past years have been indispensable. In building

on the foundation we have begun here, I hope that I will continue to

find science and its pursuit as exciting and fulfilling as I have found

it to be under the guidance of Dr. Simmons. To Kate Simmons, a gifted

scientist and close friend, I also wish to express my thanks for a great

deal of assistance and direction in scientific matters and more.

Much gratitude must also be accorded to my committee, Drs.

Holloway, DeHoff, Jones and Kumar, who have all found time to aid me

with helpful discussions and suggestions concerning the work. Special

thanks are due to Dr. G.I. Stegeman and his research group now based at










the University of Central Florida, Center for Research in Electro-Optics

and Lasers. Their ready acceptance of me coupled with their enduring

patience while I was involved with optical research within their group,

made for an extremely productive and enjoyable period of research.

To attempt to individually thank all my friends and colleagues for

their ever-present support would certainly take numerous pages of text.

I, however, wish to take this opportunity to acknowledge their help and

reassurance. Additionally, many thanks are due to my parents for their

continued faith in my abilities and support during my graduate career.

I also wish to express my appreciation to AT&T Bell Laboratories

for a generous doctoral fellowship.

To close, I want to thank my best friend and wife, Kelly, for her

support, guidance and understanding during the research and writing of

this dissertation. I only hope that I can adequately return the favor

when she writes her dissertation!


















TABLE OF CONTENTS


ACKNOWLEDGMENTS .


ABSTRACT

CHAPTERS


I INTRODUCTION .

Overview .
Statement of Problem .

II BACKGROUND ..............

Fabrication Techniques .
Condensed Phase Methods .
Gas Phase Methods .
CdTe: Structural and Optical Properties
General Considerations .
Band Structure .
Quantum Confinement .
Theoretical Considerations .
Prior Experimental Results .


III EXPERIMENTAL TECHNIQUE .

Sample Production: RF-Magnetron Sputter Deposit
General Considerations .
Dual Source Sputtering Technique .
Samples Fabricated .
Sample Analysis .
Film Thickness .
Microstructure .
Optical Properties .

IV RESULTS AND DISCUSSION .

Thin Film Microstructure .
Film Thickness .
Transmission Electron Microscopy .
X-Ray Diffraction .
Optical Characterization .
Discussion .
Microstructural Attributes .
Identification of Absorption Features .


. 81


ion


. 100

. 100
. 100
. 101
. 116
. 119
. 123
. 123
. 127


. vi


. 1


. 8


. 1
. 6










Crystal Size Dependent Optical Behavior .... .134
Effects of Crystal Size Distribution .. .158
X-Ray Photoelectron Spectroscopy ... 186
Waveguide Applications ... 202

V CONCLUSIONS ... .207

VI FUTURE WORK 212

APPENDICES

A. QuickBasic Interface Program to Control Leybold-Heraeus
Film Thickness Monitor ... 218

B. QuickBasic Program to Control Stepper Motor Motion
During Sequential Sputtering Deposition ... 229

C. QuickBasic Program to Calculate Planar Waveguide
Properties ... .232

D. QuickBasic Program Used to Perform Numerical
Integration ... .. .236

REFERENCES ... 239

BIOGRAPHICAL SKETCH ... ..... .252
















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


ELECTRONIC BAND STRUCTURE AND OPTICAL CHARACTERISTICS
OF QUANTUM-SIZE CdTe CRYSTALS IN GLASS FILMS

by

Barrett G. Potter, Jr.

August, 1991

Chairman: Dr. Joseph H. Simmons
Major Department: Materials Science and Engineering


Low-dimensional semiconductor structures now occupy a position of

central importance with regard to the understanding and application of

the basic physics of quantum confinement. Isolated II-VI semiconductor

crystals embedded in transparent, insulating matrices represent a

convenient medium for the study ofquantum-size effects on the

electronic and optical properties of compound semiconductors. The

present study simultaneously examines finite crystal size-related shifts

in the energies of optical transitions originating from states located

at two different critical points of the zincblende Brillouin zone of

CdTe. Using a versatile, dual source, RF sputtering technique, CdTe-

glass composite thin films have been produced possessing average crystal

sizes ranging from 24 to 125 A in films containing 5 vol% semiconductor

as determined by cross-sectional, transmission electron microscopy.

Previously unattainable control over such microstructural










characteristics as volume fraction and crystalline phase distribution

throughout the matrix have been demonstrated using the sequential

sputtering process.

Analysis of quantum-size induced transition energy shifts,

monitored by optical absorption, indicates the persistence of

significant Coulomb interactions between carriers at the r-point of CdTe

in crystallite sizes 0.3 times the size of the bulk exciton. L-point

transition energy shifts support the existence of two-dimensional bound

electron-hole pair states whose center-of-mass motion is confined within

the potential well. The influence of finite crystal size distribution

width on the interpretation of quantum confinement effects in these

materials was also analyzed using a numerical integration technique.

Findings substantiate the relative dominance of inhomogeneous broadening

effects over homogeneous broadening in determining the observed

absorption lineshape of the polydisperse collection of crystallites.

This does not, however, explain an apparent saturation of the energy

shifts exhibited by both r- and L-point transitions at very small sizes.

Such behavior is likely due to an increasing dominance of interface-

related modifications to the bulk semiconductor crystalline and

electronic structure.

Insight into the chemical state of the semiconductor constituents

is obtained using x-ray photoelectron spectroscopy with particular

emphasis on the Te 3d5/2 photoelectron peak. Spectrum components of

this energy region, associated with the oxide, semiconductor and matrix,

are monitored with heat-treatment of the films.

















CHAPTER I
INTRODUCTION



Overview

Analysis of material structures in which one or more real space

dimensions are restricted, i.e. are on the order of the bulk crystal

lattice spacing, contributes to an increased fundamental understanding

of bulk material properties and their evolution from constituent atomic

and molecular units. In the limit in which the spatial extent of the

material is truncated in all three dimensions, a wide variety of

materials, including metallic alloys, semiconductors and metal-oxides

have been examined. Interest in these atomic clusters stems from their

novel physical properties which can deviate markedly from that of the

corresponding bulk material, exhibiting intermediate or hybrid behavior

between molecular-like and bulk-like properties. Major efforts in

determining cluster structure and its relationship to the resulting

thermodynamic and chemical behavior, for example, have yielded

information helpful in understanding the reaction energetic and

important catalytic effects exhibited by some lower-dimensional systems

[1,2]. Films or ultrafine-grained bulk materials composed of these

clusters, i.e. nanophase or cluster-assembled materials, allow the

exploitation of the cluster-derived properties in a technologically

relevant manner. Bulk "nanocrystalline" ceramics composed of yttria-

stabilized zirconia and titania have already been produced, exhibiting

1








2

enhanced sintering dynamics and ductility over submicron-grained samples

of the same composition [3]. These and other projected applications

must involve the development of controlled fabrication techniques and

processes allowing the formation and manipulation of clusters possessing

well-defined sizes and structures to effectively utilize the structure-

property relationships of interest. The present work deals with a

distinct subset of the broad field of cluster materials and phenomena.

In the area of compound semiconductor physics, a great deal of research

effort has recently been focused on the size-dependent electronic and

optical properties of very small III-V and II-VI semiconductor

structures. As will be discussed in greater detail, when charge carrier

motion is restricted within the semiconductor by an electric potential

energy step, a distortion of the bulk electronic energy levels and a

concomitant deviation in the optical properties from the bulk

semiconductor behavior results. Drastic reduction in extent of the

crystal lattice eventually results in a breakdown of the bulk lattice

symmetry and a corresponding tendency to molecular-like electronic and

vibrational behavior. Within the present discussion, "intermediate"

sized clusters are examined. This size range is characterized by bulk-

like lattice symmetry and phonon properties while exhibiting hybrid

electronic and optical properties. This "quantum size effect"

represents a fundamental size-dependent optical property which can have

important ramifications upon the application of these materials in the

field of optics and electronics.

Low-dimensional semiconductor structures now occupy a position of

central importance with regard to the understanding and application of

the basic physics of quantum-size materials. Using molecular beam








3

epitaxy (MBE), III-V semiconductor multiple quantum wells (MQW),

exhibiting pronounced one-dimensional carrier confinement effects have

been grown with finely tuned structural and compositional variation,

thus allowing the electronic band structure of the semiconductor to be

artificially modified according to the projected device application.

Lateral patterning, using various ion and chemical etching techniques,

enables the fabrication of structures of even lower dimensionality, e.g.

quantum wires and dots [4]. Recently, this highly versatile deposition

technique has also been employed in the fabrication of well structures

based on II-VI semiconductor compositions [5].

From the standpoint of three-dimensional confinement effects.in

semiconductors, another class of materials has also received a great

deal of attention stemming originally from their significant optical

nonlinearity. Isolated II-VI semiconductor crystals embdeded in

insulating matrices represent a convenient medium for the study of

quantum-size effects on the electronic and optical properties of

compound semiconductors. Systematic study of quantum confinement in the

optical, structural and vibrational properties of these materials has

been pursued to elucidate the predominant physical mechanisms at work.

Various optical spectroscopies, including absorption, photoluminescence,

and resonant Raman, provide a means with which to examine the effect of

crystal size and structure on the optical transitions exhibited and,

consequently, the underlying electronic energy states present. Under

high incident electromagnetic field intensities, resonant excitation

produces significant carrier population redistribution within these

allowed energy levels of the isolated, quantum-size semiconductor

crystals. Such effects yield intensity-dependent changes in the optical










constants of the solid, which, under certain conditions, are greater in

magnitude and speed than the same phenomena in the corresponding bulk

semiconductor crystal. These and other behaviors, including a

significant electro-optic effect under an applied dc field, can be

,employed in the production of devices based on the semiconductor

crystallites as an active medium. Thus, careful examination of the

microstructural development and linear optical activity of quantum size

semiconductor crystals can be used to complement nonlinear optical

studies, allowing a more thorough evaluation of the materials for

device-related applications.

Previous studies of the three-dimensional confinement effects in

these materials have utilized several synthesis techniques to produce

nanometer-sized crystallites in a transparent, suspending matrix.

Examples include arrested precipitation in aqueous or organic solvents

[6] and liquid or gas phase reaction of semiconductor components within

polymer films [7], porous glasses [8], zeolites [9] or sol-gel derived

monoliths [10]. One widely studied group is the melt-derived

semiconductor-doped glasses (SDG) or semiconductor-glass composites [11-

14]. The general availability of these materials in the form of sharp-

cutoff optical filters, coupled with the discovery of a significant

resonant optical nonlinearity associated with the semiconductor phase,

has resulted in a concentrated effort to understand the influence of

finite crystal size on the evolution of the bulk electronic energy band

structure and its corresponding effect on the nonlinear optical

properties of the semiconductor.

The relative ease with which these materials may be fabricated is

balanced by the limited control afforded over such microstructure








5

parameters as semiconductor composition and stoichiometry, matrix

composition, relative volume fraction of semiconductor, width of the

crystal size distribution (which results in a significant inhomogeneous

broadening of the quantum confined optical transitions), and the

distribution of crystalline phase throughout the matrix volume. In the

case of melt-derived composites, for example, redox behavior and

solubility of the semiconductor components at elevated temperatures and

the characteristic broadening of the crystallite size distribution with

thermal treatment time [14] represent fundamental obstacles, to the

production of composites exhibiting narrow crystal size distributions

and large volume fractions of semiconductor.

The limitations imposed by the synthesis methods used to fabricate

these microstructures necessarily restrict the range of materials-

related parameters which can be examined in the context of their effects

on the resulting linear and nonlinear optical properties of the

composite. Control over matrix composition, for example, may provide a

means to investigate the nature of the semiconductor-matrix interface,

presently believed to play an important role in the magnitude and

dynamics of the linear and nonlinear optical behavior exhibited by the

material.

It is thus of interest to explore other avenues to the production

of compound semiconductor clusters within isolating matrices. Greater

freedom in the incorporation of a variety of microstructural and

chemical attributes into such composites will ultimately lead to an

improved understanding of their effect on the resulting optical

behavior.









6

Recently, several groups have begun fabrication initiatives

involving the R.F.-sputter deposition of semiconductor-glass composite

thin films. This technique promises to allow greater microstructural

control in these composites as well as an increased range of possible

semiconductor and matrix chemical identity. The investigation of this

approach in the production of CdTe-glass films and the analysis of

quantum-size related variation in their optical absorption is the topic

of the present work.



Statement of Problem

The present study represents a preliminary investigation of CdTe-

glass composite thin films fabricated using a sequential R.F.-sputtering

technique. As has been briefly introduced in the previous section, the

fabrication of such materials has generally been difficult to control,

with a limited range of microstructural attributes attainable. This

sputtering technique offers distinct advantages over more conventional

synthesis methods, allowing independent control of semiconductor volume

fraction and average crystallite size within a geometry directly

applicable to waveguide based measurements and applications.

It is of interest to first demonstrate the feasibility of using

this technique to produce composites of varying microstructural

characteristics and to then examine the effect of the microstructure on

the resulting linear optical absorption behavior. In addition,

waveguide fabrication and characterization can be performed on the films

without further processing, allowing their evaluation for future device-

related investigations. As will be discussed in subsequent sections,

quantum confinement effects in the CdTe crystallites should dominate the








7

observed optical properties. Analysis of the crystal size dependent

optical behavior in the context of current quantum confinement

treatments should yield insight into the mechanisms prevalent.

Particular attention will be paid to the effect of a finite crystal size

distribution upon the observed absorption spectral shape and its

corresponding impact on the interpretation of quantum confinement in the

semiconductor; a computer aided, numerical integration technique will

enable a representative, single-crystallite absorption lineshape to be

inhomogeneously broadened by the experimentally observed crystallite

size distribution.

Before a complete discussion of the experimental procedure and

data is undertaken, it is prudent to begin with an examination of the

current state of quantum confinement research in semiconductor optical

and electronic properties. General models and treatment of three-

dimensional finite size effects will be discussed, illustrated with data

and conclusions collected from the literature dealing predominantly with

II-VI semiconductor structures. An introduction to the various

fabrication techniques used to produce the microstructures of interest

in this study and to the general optical and electronic properties of

bulk CdTe will precede this theoretical discussion.

















CHAPTER II

BACKGROUND



Fabrication Techniques

One of the goals of the present work is the successful production

of II-VI semiconductor-glass composites using an R.F.-sputtering

approach. This technique promises to provide greater control over

microstructural parameters not addressed by other methods. It is

therefore of interest to examine these efforts in the context of past

research.

The introduction briefly mentioned the use of molecular beam

epitaxy (MBE) in the production of quantum-size III-V and II-VI

semiconductor structures. This multiple source, ultra-high vacuum

deposition process enables monolayer-resolution control over the growth

of epitaxial layers of semiconductor possessing a wide variety of

compositions. Synthesis of structures to very close tolerances is

routinely achieved, making MBE the technique of choice in the production

of III-V semiconductor-based devices. Independent control of structure

size and chemical composition enable electronic energy band structures

and hence the corresponding electronic and optical properties to be

designed and implemented. Molecular beam epitaxy is thus used to

produce the starting material used in the fabrication of such solid

state optical devices as lasers, diodes, electro-optic modulators, and









9

waveguide-based switches (e.g. nonlinear directional couplers) all of

which impact a wide range of integrated electronic and optical

applications.

Quantum confinement in MBE-grown materials is readily analyzed in

terms of models to be discussed in the next section. Epitaxial

boundaries between regions of different dielectric constant represent a

potential energy step of well-characterized depth and width. In the

case of ion or chemically etched structures, side wall damage results in

sources for nonradiative recombination centers and optical losses in the

case of waveguiding structures. The problem can be, however, addressed

through annealling and regrowth techniques.

These characteristics contrast the less easily controlled

microstructures that are the topic of this work. As has been mentioned,

the optical and electronic properties of quantum size crystals in

amorphous media can be dominated, in some cases, by interface-related

effects. The boundary conditions defining the semiconductor in the

matrix are in reality very complex. The interface is necessarily

composed of various structural defects including strained and dangling

bonds, vacancies, segregation effects, and adsorbents, thus making a

complete analysis of confinement difficult.

As will be discussed, the size of the confining structures in

these materials is also only partially controlled, generally through

thermal treatments. Diffusion-controlled growth processes result in a

wide distribution of microstructure sizes leading to significant

inhomogeneous broadening in the optical response of the composite.

Nevertheless, the synthesis techniques examined in this section

remain a viable alternative to a more elaborate and expensive MBE









10

system, enabling a wide range of chemical compositions and structures to

be produced using standard condensed and gas phase reactions and

procedures. The discussion of cluster fabrication will be, in most

cases, limited to JI-VI semiconductor materials. Two classes of

fabrication procedures will be examined: condensed and gas phase

methods.



Condensed Phase Methods

Colloidal solutions. A large body of work presently exists in

which quantum size effects in compound semiconductors suspended in

liquid media have been examined. An in-depth review of the field is

inappropriate for the present introductory discussion. However, an

attempt will be made to deliver a brief overview of the large number of

variations on accepted preparation techniques. Generally, the samples

consist of nanometer-size crystallites suspended in a dilute aqueous or

organic solution. The clusters are formed by reaction between

appropriate ions obtained from dissociated salts or organometallics.

Interest in these materials originally stemmed from the photocatalytic

properties of such colloidal particles [1,8]. Colloidal semiconductors

in solution are also widely utilized as photosensitizers in which

optically excited carriers may react with molecular species adsorbed

onto the particle surface [15]. Thus, an understanding of the size

dependence of electronic energy levels within colloidal particles is of

fundamental importance.

Proper choice of precursors, reaction media and conditions enable

a wide range of different chemical compounds to be grown in colloid

form. Within the present topic, such compound semiconductors as CdS









11

[16,17], CdSe [18,19], ZnS [18,20], ZnSe [18], Cd3As2 [21], and PbS [22]

have been produced in a variety of solvents, including water,

acetonitrile (methyl-cyanide), 2-propanol, methanol or mixtures of

these.

The synthesis method involves the reaction of free ions to form

the covalently bonded structures characterizing the compound

semiconductors of this work by the general reaction: N(A+ + B') -

(AB),. Steigerwald and Brus [23], in a lucid introduction to this

field, note that the formation of internal crystallinity implies that

the reaction is, in fact, complex. Ionic dissolution and readsorption

after the initial bonding of the ions to the growing cluster or severe

restrictions on the allowed bonding site geometries must be invoked to

explain the relatively rapid development of a bulk-like crystalline

structure with colloid growth.

Standard colloid preparation techniques have been documented by

numerous researchers in the field [1,8,15-23]. In the case of cadmium-

based colloids, raw materials supplying the cationic species include

nitrates, chlorides, perchlorates, and sulphates. Ammonium and sodium

salts of the chalcogenide generally furnish the anion. Variation in

this general procedure has also involved the introduction of gaseous

hydrogen chalcogenides, e.g. H2S and H2Se, into cation-containing

solutions [17,22,24,25].

Initial studies utilizing an aqueous-based solution [15] produced

CdS crystallites as small as 35 A. However, this colloidal population

is inherently unstable, exhibiting a marked increase in average particle

size and broadening of the distribution with time, the result of an

Ostwald ripening process. The coarsening behavior necessarily limits









12

the use of these types of samples in a size-dependent study of optical

properties. The ripening process, reviewed elsewhere [26,27], is a

thermodynamically driven, diffusion-limited process in which surface

curvature dependent considerations dictate the stability of particles of

a given size. Smaller, unstable particles dissolve and contribute ions

and multiatomic fragments to larger more stable crystals, thus

increasing the average size of the colloid ensemble with time.

Several approaches have been developed in an attempt to improve

the stability of newly precipitated semiconductor crystals in solution.

Some success has been achieved by altering or changing the solvent.

Acetonitrile has been observed to significantly slow the growth of

colloid particles when used as the suspending/reaction media. The

effect is attributed to the low dielectric constant of the liquid which

decreases the stability of dissolved ions in solution thus slowing the

dissolution of semiconductor crystallites [15]. A gradual decrease in

the average particle size is also observed in water/acetonitrile

mixtures as the concentration of organic solvent is increased [24].

Partial success in slowing the aging process is attributed to increased

Ph levels in aqueous solutions, again the result of decreased solubility

of the metal sulfides in a basic environment [15,28].

Another avenue for increasing colloid stability involves

restriction of diffusional motion of reactants between the growing

particles in solution. The most straightforward method is through a

reduction in the solution temperature, thus resulting in a greater

viscosity. This has in fact resulted in smaller average crystallite

sizes of both CdSe and ZnSe, when compared to room temperature

reactions, embedded in isopropanol solutions held at -80*C [18].









13

Similar improvements have also been observed in CdS colloids [21].

Cryogenic glasses composed of 2-propanol at less than 200 K containing

CdS colloids have also been examined [29]. In a broad sense, the

temperature-dependent viscosity approach is the primary method for

crystallite size control in molten silicate glasses to be discussed

later.

The use of microheterogeneous reaction media has been employed to

isolate nucleation sites and to restrict the transport of reactants to

the growing particles. Several specific techniques exist including the

development of vesicles [30,31] and reverse micelles [32,33] both

involving the production of isolated domains of solution through the

addition of surfactant molecules possessing both hydrophillic and

hydrophobic functional groups [23]. In the case of reverse micelle

microemulsions, a small amount of water is emulsified in a hydrocarbon

solvent using a surfactant. Sodium dioctylsulfosuccinate (AOT) is a

commonly used surfactant for the production of CdS in such structures

[33]. The addition of surfactant results in isolated water pools

surrounded by surfactant molecules suspended in the organic solvent.

The size of the water pool is dependent upon the water to surfactant

ratio, generally denoted as W in the literature. Colloid growth occurs

within these micelles when separate emulsions containing the cation and

anion are mixed. Thus, variation in the W parameter can be used to

influence the final size of the colloids. A significant increase in the

average CdS crystallite size has, in fact, been observed by Petit et al.

[33] with an increase in W.

In addition to blocking kinetic mechanisms for growth, ligands

bonded to the surface of the growing particles effectively passivates










the colloid, halting further incorporation of ions. Several sources

have been used to yield these molecular groups including a

stryrene/maleic anhydride copolymer (to stabilize CdS) [15], polyacrylic

acid (CdS), dextran sulfate (In2S3) [34], and sodium hexametaphosphate

(CdS) [28,35].

Steigerwald et al. [36] report a second method of surface

passivation using organometallic polymer groups to render CdSe and CdTe

clusters unreactive. In the case of CdSe, the procedure begins with

inverse micelle encapsulated CdSe particles formed using Se(TMS)2 (TMS -

trimethylsilyl) as a reactant precursor. Substitution of the original

precursor by a similar compound with a phenyl group replacing one TMS,

i.e. Ph(TMS)Se (phenyl(trimethylsilyl)selenium), results in the

incorporation of Se into the semiconductor lattice with the phenyl group

strongly bonded to the Se atom [23]. The crystallite is passivated or

"capped" with the weakly bonded surfactant of the micelle replaced by a

strong Se-C bond [23]. The capped clusters possess extreme stability,

enabling their isolation and removal from the solution, forming a waxy

solid which can be redissolved in suitable solvents (dependent on the

organic cap identity) with no agglomeration of clusters. Recently

Olshavky et al. [37] report the fabrication of GaAs capped clusters by

the above method.

Glasses. Semiconductor-glass composites or semiconductor-doped

glasses were actually first examined from the standpoint of finite

crystal size effects as early as 1954 [38]. More recently, this group

of materials has received a great deal of attention stemming from their

significant resonant electronic optical nonlinearity first reported by

Jain and Lind [39] in optical phase conjugation studies of the material.









15

The relative ease of fabrication coupled with their widespread

availability as commercial filter glasses enabled many examinations of

nonlinear optical behavior in these materials to be pursued [40-44]. It

was discovered early on that crystallite sizes in the commercial

glasses were larger than that expected to produce significant quantum

confinement effects in the semiconductor. Further research [14,45,46]

then concentrated on the fabrication and microstructural evolution of

these glasses in an attempt to understand the concepts important in the

controlled development of quantum-size crystals and their influence on

the optical properties of the composite.

Semiconductor-doped glasses typically consist of II-VI

semiconductor microcrystallites dispersed in a Zn-crown silicate glass

matrix. More recently, CdS [47] and CdSe [48] crystallites have been

grown in a germanate glass matrix. After melting and annealing, the

semiconductor constituents are bound into the glass atomic structure.

Upon heating to 500-8000C (dependent on the matrix and semiconductor

identity), however, crystals nucleate and grow from this supersaturated

matrix producing an optical absorption edge indicative of the crystal

composition and size. The kinetics of the phase transformation have

been examined by a number of researchers [14,45,46] who have shown it to

consist of a nucleation and growth period followed by a diffusion-

limited coarsening regime when the degree of supersaturation becomes too

low to favor the formation of new nuclei. This latter behavior is

typified by a cube root heat-treatment time dependence of the average

crystal size and a continually broadening size distribution with heat-

treatment time.









16

In addition to this inherently broad crystal size distribution

(which can produce significant inhomogeneous broadening in the quantum

confined states) other problems accompany the conventional melting route

to the production of these samples. Substitutional solid solution of

other glass constituents into the crystallizing semiconductor can

produce an absorption edge shift due to composition alloying. Such a

shift could be construed to be of a size-dependent nature. During

melting and subsequent thermal treatment to precipitate the

semiconductor phase, oxidation of semiconductor elements results in a

significant loss of material from the glass. This problem becomes more

prevalent for compounds with heavier anions that exhibit lower band gap

energies and where the oxidizing tendency of the anion increases. The

limited solubility of the semiconductor components in the glass also

prevents high loading of the composite with typical volume fractions of

semiconductor crystals in the glass remaining at approximately 0.001.

Despite the above limitations, careful control of matrix

composition and thermal treatment have enabled several groups [14,45,46]

to separate artifact effects from quantum-size related variation in

optical properties, allowing theoretical evaluation of the materials.

Further discussion of these findings will be presented in the next

chapter.

Porous Media. Introduction of nanometer-sized semiconductor

crystallites into the interstices of a porous solid is an intriguing

hybrid of the colloid and glass melting approaches previously discussed.

With a solid, transparent matrix to support the crystallites, optical

and microstructural examinations are greatly aided. In addition,

chemical approaches to crystallite production avoid the high temperature









17

conditions necessary in melting methods. Control of pore size and

distribution in the matrix would also allow the distribution and size of

the semiconductor phase to be defined. Several different materials have

been employed as supporting structures including: leached, phase-

separated borosilicate glass (e.g. Vycor), silica sol-gel monoliths,

polymers, and zeolites.

Reaction of free ions within a porous glass matrix is generally

accomplished by introducing a solution (aqueous or organic solvent-

based) containing the cation species into the structure by immersion.

The sample is either dried, reprecipitating the cationic salt in the

glass [8] or is allowed to remain saturated [49] before subsequent

immersion in the anion-containing solution. In the case of II-VI

semiconductors, cadmium salts, e.g. CdC12 or Cd(NO3)2, are employed as

they dissociate readily yielding free ions. Ammonium, sodium-based, or

organic compounds furnish the anions for the reaction, e.g. Na2S or

selenourea would contribute sulphur and selenium ions, respectively.

In contrast to the impregnation route above, Potter and Simmons

[50], employing an ion-exchange process developed by Simmons et al.

[51], successfully utilized a two step process to form both CdS and CdSe

crystallites in a leached, phase-separated borosilicate glass matrix.

Samples of porous glass were first placed in either a NaOH aqueous

solution saturated with CdC12 or Cd(N03)2 to exchange Cd2+ ions into the

surface of the glass pores. After rinsing and drying, the samples were

then immersed in either Na2S or selenourea solutions to produce CdS or

CdSe crystallites, respectively. Significant quantum-confined shifts in

the optical absorption of the crystals were observed. Consolidation of

the glass matrix around the crystals was attempted unsuccessfully due to









18

significant oxidation losses of semiconductor at the elevated

temperatures necessary to densify the glass. As is the case for

techniques involving a wet chemical process, this approach was limited

in the ability to vary microstructure size and volume fraction of

semiconductor.

Borrelli and Luong [52] successfully produced a number of

semiconducting compounds in the II-VI, III-V and IV-VI group within

porous glass using an impregnation technique. Cadmium sulfide, CdSe,

PbS, PbSe, MoS2, and GaAs crystallites were formed by the reaction of

the previously introduced metal cation and gaseous hydrogen

chalcogenide. In addition to the gas-phase path, selenium-containing

compounds were obtained via a photoinitiated process in which N,N-

dimethyl selenourea is co-impregnated into the glass with the metal.

Upon exposure to the ultra-violet, 308 nm line of an excimer laser, the

selenium compound decomposed enabling its reaction with the metal.

Masking of the sample prior to exposure allowed patterning of the areas

to contain the semiconductor crystallites.

Polymers have also been employed as matrices for the support of

semiconductor crystallites. Wang and Mahler [7] and Hilinski et al.

[53] synthesized CdS in Nafion thin films. After cleaning in dilute

nitric acid and washing, Cd2+ ions are absorbed into the polymer from a

cadmium acetate solution. Again, gas phase reaction of the dried sample

with H2S yielded CdS crystallites with an average size of 55 A and

distribution width of 20% of the average. Lead sulfide crystallites

were grown in an E-MAA polymer film by Wang et al. [54]. In this

process, however, the lead ions were ion exchanged into the polymer

structure prior to hydrogen sulfide exposure. In addition, modification











of the as-reacted microstructure was achieved by thermal treatments

employed to vary the average crystal size of the semiconductor.

Use of the matrix to define the spatial extent and distribution of

the newly formed semiconductor phase is especially well demonstrated by

work performed by Wang and Herron [9] and Herron et al. [55] in which

CdS clusters are produced within alumino-silicate zeolite structures.

The zeolite exists in a powder form and, after a cadmium ion-exchange

process, the slurry is dried and calcined. The semiconductor is formed

during exposure of the dried, ion-exchanged powder to H2S. Discrete

(CdS, 0)4 clusters were formed at low loading density and were found to

exhibit a room temperature absorption edge at 290 nm (bulk CdS 514

nm). Interconnection between these clusters, through bonding between

cages in the zeolite, results in imperfect "superclusters" with an

absorption edge of 350 nm. This technique thus allows the three

dimensional structure and electronic properties of the semiconductor to

be controlled by using different zeolites as the matrix. These

researchers report, however, that the samples produced were extremely

sensitive to moisture which aided in the migration and aggregation of

semiconductor crystals to form larger structural units [55] within the

zeolite powder.

In an innovative application of sol-gel processing, cadmium-doped

sol-gel silica monoliths were fabricated by Nogami et al. [10]. The

dried samples were given a 500 *C, 2 hour heat-treatment to partially

consolidate the structure. The glass is then placed into an evacuated

glass tube into which hydrogen sulfide gas is introduced. Variation in

the time of H2S exposure results in the successive red-shift of the

observed optical absorption edge due to the formation and growth of CdS









20

particles. Although not a parameter of study in the reported work, it

is expected that variation in the consolidation time and temperature of

the precursor porous glass monolith would result in a controlled change

in the pore size and distribution, thus exercising further control over

the final crystallite size and morphology.



Gas Phase Methods

Soot. In a rather limited fabrication approach, quantum-size

effects in resonant Raman scattering from ZnTe crystals were examined by

Hayashi et al. [56] in samples produced by evaporating polycrystalline

ZnTe crystals under Ar gas. The microcrystals took the form of soot-

like deposits collected on pieces of Si wafer. Optical and structural

examinations were performed directly from these samples. Average

particle size variation was obtained through changes in the gas pressure

and heater temperature used during evaporation.

Vacuum deposition. With the exception of the semiconductor-

polymer materials, the fabrication techniques discussed above generally

result in bulk samples or powders. From a technological standpoint,

further processing must be performed to make the sample more compatible

with device-related applications. For example, channel waveguiding

structures have been produced, using potassium and cesium ion-exchange,

in melt-derived semiconductor-doped glasses allowing the patterning of

switching device geometries [57,58].

Molecular beam epitaxy represents a specialized form of vacuum

evaporation to produce thin, epitaxial semiconducting films. The

technique is widely used and was discussed earlier. In line with the

present class of materials, the technique has also been used to produce









21

GaAs crystallites within a porous glass matrix [59]. Electron

microscopy shows the crystals to be spherical and have a nearly bulk-

like crystal structure.

Recently, a number of research groups have investigated the use of

R.F.-sputtering to produce glass films containing quantum-sized

semiconductor crystals. The goal has been to mimic the microstructural

attributes of the previously discussed melt-derived material within a

thin film geometry, directly applicable to waveguide-based studies. A

more complete introduction to the sputtering process follows in a

subsequent section. Presently, past use of this well established

technique in the production of semiconductor-glass composite films will

be discussed.

One of the first attempts to introduce a semiconducting phase into

a glassy matrix was made by Clausen [60]. Using a sequential sputtering

method, employing independent sources for the matrix and semiconductor,

a CdS-BK-7 glass film was deposited onto silica substrates. Films

containing 5, 50, and 100 vol% CdS were produced and subjected to post-

deposition thermal treatments at 650*C. Heat-treatments of the 5 and 50

vol% samples resulted in the precipitation of various phases; the

identification of bulk CdS diffraction patterns was achieved in the high

volume fraction samples with relatively large, 700 A, crystals

observed using transmission electron microscopy. Prior to heat-

treatment the samples exhibited diffuse diffraction features and were

extremely electron beam sensitive in the TEM, phase-separating in

minutes upon exposure [60]. In the 100 vol% samples, Clausen [60] also

demonstrated the ability to produce polycrystalline CdS films of high

crystallographic quality and excellent stoichiometry as shown in









22

measurements of excitonic features whose energetic positions closely

match those of single crystal CdS platelets.

Jerominek et al. [61] produced pure CdS crystals and mixed

composition crystallites of CdSSe1_. in silicate glass films using a

single sputtering source and a variety of target materials. Films were

deposited onto room temperature soda-lime silicate glass or Corning 7059

glass substrates in a 2 Mt Ar atmosphere. The first target employed

consisted of a disk made from commercially available Schott GG495

filter-glass, containing the mixed CdS/Se crystals. Other targets were

composite in nature consisting of a bulk glass disk with a fused silica

crucible, containing CdS powder, placed at its center. The bulk glass

was either the Schott filter glass or Corning 7059, an undoped glass.

Development of a crystalline phase during heat-treatment of the films

was monitored using Raman spectroscopy and optical absorption.

Waveguiding properties of these planar structures were investigated

using prism coupling. Refractive indices for films produced using

varying R.F. powers during deposition were determined from coupling

angle. It was found that the index of the film increased with R.F.

power, presumably due to an increased fraction of semiconducting phase

in the glass matrix. Incident power-dependent coupling into the planar

waveguides was also observed.

Quantum-size related shifts in the optical transitions of CdTe,

CdSe, and GaAs have been reported by Tsunetomo et al. [62] and Nasu et

al. [63] in R.F.-sputtered thin films also produced using single,

composite targets. In this case, the semiconductor is introduced in the

form of polycrystalline chips placed on the surface of a fused silica

target. The average size of the resulting crystallites, as determined









23

by X-ray diffraction and TEM, was found to vary with the sputtering

power used, the substrate temperature and the relative surface area of

semiconductor and glass on the target.

Picard [64],' at the National Optics Institute in Quebec, Canada,

has also recently reported the fabrication GaAs-doped glass films using

a single source R.F.-sputtering technique. Results were preliminary but

XPS measurements suggested the existence of additional As-based species,

e.g. AlAs, which could have contributed to observed shifts in the

optical absorption exhibited.

The R.F.-sputtering technique offers several advantages over the

previously discussed fabrication methods, making it a very versatile

technique for the production of these microstructures. As was mentioned

in the Overview, limitations imposed by the synthesis techniques

commonly used restrict the analysis of confinement effects and other

properties of the composite under variation of such material parameters

as volume fraction, average crystallite size, matrix and semiconductor

identity and phase distribution throughout the body of the composite.

All the specific R.F.-sputtering approaches offer some access to these

variables, and can actually enable a greater variable range to be

addressed than is possible with the previously discussed synthesis

methods. The present study concentrates on the dual source sputtering

technique of Clausen [60] and its application in the production of CdTe-

glass composite films. As will be demonstrated, a two-source approach

can offer additional advantages in the independent control of several

deposition parameters that are linked in techniques utilizing composite

targets.







24

CdTe: Structural and Optical Properties

General Considerations

Cadmium Telluride is a member of the IIB-VIA compound

semiconductor family which includes diatomic compounds composed of Cd,

Zn (group IIB) and S, Se and Te (group VIA). The cadmium-tellurium

effective bond ionicity is 3% (where 0% is fully convalent and 100% is

ionic) [65] resulting in a largely covalent bond character. The

interatomic bonds are formed from sp3-hybridized orbitals, thus

determining the tetrahedrally coordinated crystal structures typically

observed in this material. Cadmium telluride exhibits a cubic,

zincblende bulk crystal structure, the diatomic analog to the Si diamond

lattice (see Figure 1). The noncentrosymmetric structure incorporates 4

CdTe molecules in a unit cell with each Cd (Te) surrounded by four Te

(Cd) atoms at the corners of a regular tetrahedron. The lattice

constant is 6.481 A [66]. Small energetic differences between

zincblende and the hexagonal, wurtzite, crystal lattices, result in the

observation of this structure in other compounds of the II-VI group,

e.g. CdS, and CdSe. In general, the zincblende phase is more stable for

greater molecular weights and higher bond ionicities [65]. In thin

films of CdTe produced using various vacuum deposition techniques,

however, control of deposition conditions can result in a multiphase

structure consisting of both the hexagonal and cubic components [66,67].

The static and optical frequency dielectric constants of CdTe

range from approximately 10.4 to 7.2, respectively with the refractive

index equal to 2.70 at 2.5 pm [66]. Rowe et al. [68] determined phonon

dispersion relations using neutron inelastic scattering. Longitudinal

and transverse optical phonon frequencies at the r (k 0) critical






























































A view of the zincblende crystal structure, typical of
bulk CdTe [65].


Figure 1.








26

point of the Brillouin zone are found to be 5.08 and 4.20 THz,

respectively.

As will be discussed presently, cubic (zincblende) CdTe is a

direct-gap semiconductor with a fundamental absorption edge of 1.49 eV

at room temperature. Early interest in the material stemmed from this

long wavelength absorptive behavior and its impact on solar energy

conversion. Other applications involve its use as infrared-transmitting

laser windows. Cadmium telluride also possesses a relatively high

electro-optic coefficient when compared to GaAs and other III-V

semiconductors useful from the standpoint of signal modulation device

applications. As early as 1970, the Franz-Keldysh effect was used to

modulate radiation in the near infrared [69].

Extrinsic effects, due to intentional doping of the semiconductor,

also enable its incorporation into heterojunction structures useful both

from an optical and electronic device perspective. Several groups of

substitutional atoms give rise to both n and p-type doping in the

material up to relatively high levels [66]. Members of the Group V (P,

As and Sb), Group IB (Cu, Ag, and Au) and Group IA (Na and Li) have been

shown to furnish shallow acceptor levels within the band gap of CdTe

while substituting for Cd in the crystal structure. Group III (Al, Ga,

and In) which substitute for Cd and Group VII (Cl, Br, and I),

substituting for Te, result in n-type doping [66]. It is important to

note, in addition, that the CdTe also exhibits native defect centers,

i.e. point defects in the crystal structure involving vacancies and

interstitial atoms, which also result in shallow donor and acceptor







27

levels. Such defects have been utilized to obtain very low carrier

concentrations in impurity-doped material through self-compensation

[70].

Recently, several research groups [71-73] have focused on the use

of the photorefractive effect in CdTe doped with In or V which produce

deep traps within the band gap. These trap states furnish photoexcited

carriers and also provide a mechanism for the trapping of diffusing

carriers at ionized centers. This effect, coupled with the high

electro-optic coefficient of the semiconductor and its modest

photoconductivity, have enabled the observation of photorefractive beam

coupling with relatively high gain at frequencies resonant with the trap

state energies, generally at 1.5 [73] and 1.06 pm [71]. These phenomena

have also been utilized to produce power limiting and self-switching in

CdTe [72].



Band Structure

The Brillouin zone of CdTe, and other zincblende structure

semiconductors, has been examined by several authors either through

calculation [74-77] or optical spectroscopy [78-83]. A calculated

energy band diagram is depicted in Figure 2. The band structure

contains several critical points identified by locations in k-space at

which the derivative of the energy separation between the valence and

conduction band is zero. (This condition originates from the

incorporation of the density of states into the e2 expression obtained

by an evaluation of transition probabilities between a single filled and

unfilled energy level using first order, time dependent perturbation

theory [84].)































































The calculated energy band structure for
zincblende CdTe [79].


Figure 2.







29

Generally, these critical points correspond to different symmetry

directions in the Brillouin zone. Interband transitions at these points

give rise to structure in the reflectivity and absorption spectra of the

bulk material. Critical points in the band structure are designated Mi,

where i is the number of reduced mass components that are less than zero

[85], thus corresponding to the local energy band shape at that point in

k-space. Two critical point transitions occurring in CdTe are of

particular interest in the present investigation and will be discussed

here.

The lowest energy direct transition occurs between the degenerate

rF-symmetry valence bands and the lowest (r6) conduction band at the

center of the Brillouin zone, i.e. the r-point. This Mo critical point

occurs at k 0. Transitions at the r point produce the fundamental

absorption edge of the semiconductor occurring at 1.49 to 1.53 eV (300K)

or at approximately 1.60 eV (2K) [66]. In CdTe, the next observable

interband absorptions occur at El 3.24 to 3.38 eV (300K) and EI + Ai -

3.84 to 3.96 eV (300K) [67], between the spin-orbit split valence bands

(separated by the spin-orbit splitting, A,) and the lowest conduction

band at the L-point. This is an M1 critical point located at the zone

boundary along the [111] direction in inverse space [67]. Some

uncertainty, however, does exist regarding this identification in the

literature. While Cardona and Harbeke [86] also attribute these

features to the L-point, Cardona and Greenaway [83] and Chadi [79], for

example, subsequently identify the transitions as occurring at the A

point (at ka/n 0.43, 0.43, 0.43) in accordance with a pseudopotential

band structure calculation which predicts a significantly smaller







30

oscillator strength for the L-point transition than is observed in

reflectivity measurements.

At the zone center, the effective masses are largely isotropic

in this cubic semiconductor. The electron and hole effective masses are

found to be 0.1 in and 0.4 m, [87], respectively. Effective mass

tensors for carriers at the L-point have only been partially evaluated,

however. The carrier masses at the L-point, as alluded to in the last

paragraph, are anisotropic, i.e. the effective mass value varies with

carrier propagation direction in the crystal lattice. In this case, the

longitudinal (along the [111] direction) reduced mass component is

negative with an absolute value whose magnitude is large when compared

to the smaller, positive transverse masses [85]. The transverse masses

can, in fact be calculated using expressions adapted by Cardona and

Greenaway [82] to II-VI semiconductors from similar equations developed

by Ehrenreich [88] for the III-Vs using k-p perturbation theory:


mO E
m =1+ p
m eE +








m= = -1 + E )
m, 2(E, + A,)

Typical effective masses for the electron, light and heavy hole at the

L-point are calculated to be approximately 0.14, 0.45, and 0.56mo,

respectively, using typical transition energies observed in the present

study.







31

Coulomb interactions between photoexcited carriers can occur at

all direct interband transitions, producing, in some cases, large

distortions in the dielectric response function derived from a one

electron treatment of excitation within the solid. As will be

discussed, at M0 critical points interparticle correlation results in

sharp peaks below the fundamental absorption energy and an enhancement

in the absorption above the edge. The sharper features correspond to

the formation of correlated electron-hole pairs, i.e. excitons.

Although somewhat controversial, Coulomb effects are also predicted to

give rise to enhancements in the imaginary portion of the dielectric

function at Mi critical points [75,84,85]. The pair state in this case

is sometimes referred to as a "saddle-point" or "hyperbolic" exciton,

hence describing the nature of the energy surface in k-space [84,89].

Both M2 and M3-type points exhibit a less prominent absorption strength

under the influence of an electron-hole interaction [84].

Excitonic effects at the M0 point and their influence on the

resulting absorption behavior have been theoretically treated in a

number of reviews [90,91,92]. A brief overview of the pertinent points

follows.

Within the present discussion, i.e. direct transitions within a

low band gap, high dielectric constant semiconductor, the appropriate

description for the weakly interacting electron-hole pair formed is that

of Wannier [90]. In this case, the spatial extent of the pair is large

compared to the lattice constant of the crystal. The delocalized state

can be described, therefore, within the effective mass approximation,

incorporating the long range periodicity of the crystal structure. The

charged carriers are assumed to move within an average background










potential possessing effective masses ultimately linked to the lattice

symmetry and the resulting band structure. The energy bands are assumed

to be parabolic in k-space and are isotropic.

In contrast, interacting carriers within a low dielectric

constant, insulating material are generally localized on one or a small

number of lattice sites, polarizing the material around them. In this

case, the pair is designated a Frenkel-type exciton. An atomic or

molecular tight binding approach [90] is then used to formulate a

description of the state. Eigenstates and energies are evaluated using

a Hamiltonian incorporating interactions from the surrounding nuclei and

electrons. Thus, the approach requires detailed knowledge of the local

structure of the system. The excitonic wavefunctions are consequently

built from localized, atomic-like wavefunctions, contrasting the above

mentioned Wannier-type exciton formed from the free carrier or Bloch

wavefunctions.

A description of intermediate electron-hole interaction strength

effects can be approached from either of the above points. In the

Frenkel model, the excitation is taken to be "localized" on a larger

number of sites, resulting in an increased number of localized

wavefunctions contributing to the treatment [84]. The approach is

therefore limited in the ability to handle the matrices involved [93].

Conversely, the Wannier approach must be modified to include a wider

wavevector (k) range, to describe a more limited real space dimension.

The exciton is therefore no longer associated with a particular point in

k-space [84]. The model would eventually break down when k-space

regions involving nonparabolic energy surfaces become involved [93].







33

As was mentioned earlier, the Wannier exciton description of

electron-hole interaction is most applicable to the CdTe system. Within

the effective mass approximation, the eigenfunctions describing the

motion of the electron and hole can be obtained through solution of

Schroedinger's wave equation:



S2mV 2V V,2-+




The Hamiltonian used contains the kinetic and Coulomb interaction

energy operators of the electron and hole. The equation is identical to

that commonly used to describe the hydrogen atom with proper

substitutions made for the mass and dielectric constant terms.

Solution of the equation is generally achieved through separation

of the problem into center-of-mass and relative coordinates. This

results in a wavefunction consisting of a free particle, plane wave

component (describing the center-of-mass motion) and a hydrogenic

component describing the internal, orbital motion of the electron and

hole about their common center of mass. The eigenenergies for the

state, relative to the top of the valence band, are thus


a4 h2K2
E= E 4 +----
2he2n2 2M



where


mm











M = + mh

a dielectric constant
K center-of-mass wavevector
n radial quantum number
p excitonic reduced mass
M excitonic translational mass.


Figure 3 depicts the energy spectrum for the exciton in terms of the

excitonic wavevector, K. The second term denotes the excitonic

"binding" energy, i.e. the energy necessary to ionize the exciton into

free particles. The absorption spectrum thus exhibits a series of

discrete lines whose energy separation from the bottom of band edge

continuum varies as a function of 1/n2. The n 1 state defines the

Rydberg energy of the exciton.

The excitonic radius or spatial extent of the electron and hole

orbit can be calculated as

=e

112
The exciton radius may also be expressed in terms of the angular

momentum and frequency of the pair:



a= 4'" (n-11



where nh angular momentum and a angular frequency.

The "critical radius," a,,it corresponds to the frequency of the

lowest optical phonon branch of the lattice, wo. If the exciton radius

is less than acrit then the pair revolves at frequencies greater than the

Wo and the high frequency dielectric constant must be used when

examining the interaction between the charge carriers and the material.

















E


p ~ 'K
0


The energy spectrum found for a Wannier exciton
assuming parabolic, isotropic energy bands [94].


nS.
n 2


Figure 3.








36

For a > acrit, the static dielectric constant may be used [93].

Calculations involving the suitable parameters for CdTe at the r-point

show that the Rydberg energy of the exciton is 10 meV as compared to

the LO-phonon energy of 21 meV [95]. In other words, the excitonic

radius of 75 A is greater than the critical radius determined from the

LO-phonon frequency, i.e. 67 A. Thus, the use of the static

dielectric constant is justified in this semiconductor as is the case

for many semiconductors [93].

The above treatment for excitonic effects adequately explains the

energetic positions of free exciton structure in band edge absorption

spectra. Coulomb effects on the absorption strength exhibited were

examined theoretically by Elliott [96]. Evaluation of transition

probabilities using the Wannier, hydrogenic wavefunction shows that the

strength of the excitonic transition varies with the inverse cube of the

exciton radius, i.e. is proportional to i/volume of the excited state.

The absorption strength also follows a 1/n3 dependence, where n is the

excitonic radial quantum number. Within the continuum portion of the

spectrum, the absorption coefficient does not fall to zero at the band

edge as it does in the case of noninteracting particles. At the edge,

the absorption falls to a finite value, an enhancement over the

noninteracting case, with the above excitonic lines falling to lower

energies within the gap. The overall effect of Coulomb interaction can

be envisioned, therefore, as a reduction in the free particle energy

states of the system, shifting the entire absorption spectrum to lower

energies with the lowest lying electrons in the band forming the

exciton complexes [97]. Figure 4 shows a schematic of the predicted e2

behavior (the imaginary portion of the complex dielectric function) at













































Eo+5R


ENERGY


A typical M, critical point absorption-edge spectrum
with (solid lines) and without (dashed line) electron-
hole interaction. The inset shows the effects of
broadening [84].


I -




_ ^ -______----


Eo-R Eo Eo+R


Figure 4.








38

the M0 critical point with and without electron-hole interaction.

Of importance in the interpretation of the absorption spectra of

excitonic resonances are broadening effects which can significantly

influence the shape of the absorption structure (see Figure 4). Thus,

the excitonic structures are not lines but broadened peaks, at times so

broad that higher order transitions are not observable. Several

mechanisms have been determined to lead to broadening in the optical

response of semiconductors, e.g. phonon interactions and impurities or

defects in the lattice. Further discussion of this topic will be

pursued in a subsequent section due to its direct applicability to the

investigation of absorption structure in polydisperse, quantum-sized

crystallite ensembles.

At this point in the discussion, it is of interest to discuss

deviations from the above theory for the development of Wannier-type

excitons. The theory employs the effective mass approximation and

assumes a parabolic, isotropic dispersion of electron and hole moment

in k-space. Semiconductors possessing a zincblende or diamond structure

exhibit a very complex, 6-fold degenerate (including spin) valence band

structure at k 0, as shown in Figure 2. The valence band is spin-

orbit split into four-fold degenerate (Pa) and two-fold degenerate (r7)

parts. The hole bands are not isotropic and possess off-diagonal

elements in the Hamiltonian describing the particle that can cause a

mixing or splitting of states which would compose the resulting exciton

complex [98]. Treatment of this property, as will be discussed in the

next section, can be important in the interpretation of quantum size

effects in such tetrahedrally bonded semiconductors, including CdTe.










Baldereschi and Lipari [98] present a calculation in which the

anisotropic components of the valence band hole Hamiltonian are treated

as a perturbation on the Hamiltonian describing the Coulomb interaction

between the electron and the isotropic part of the hole. The total 6X6

Hamiltonian describing the exciton is expressed in terms of two 6X6

matrices, one containing the isotropic, orthogonal components of the

holes and the other describing the anisotropy of the valence bands.

Coupling between states associated with the anisotropic components of

the hole is only observed by Baldereschi and Lipari in second-order

perturbation theory.

Results of the treatment show that coupling to anisotropic

components of the hole bands does distort the simple, parabolic band

representation of excitons formed from the two spin-orbit split valence

bands. The effect is observed in corrections to the binding energies

exhibited by the excitons formed from the two spin-orbit split valence

bands [98]. The magnitude of the corrections is governed by a coupling

parameter, 4, which is in turn a function of valence band shape

parameters (Luttinger parameters). Due to the relatively large

effective masses which characterize the anistropic components of the

hole states, the coupling parameter remains small, substantiating the

validity of the perturbation approach. In general, the degree of

coupling also decreases as the spin-orbit splitting increases. The

authors note that the upper and lower valence bands are implicitly

assumed to be characterized by the same isotropic mass, defined using

the relevant Luttinger band shape parameter [98]. Thus, apart from

small perturbations in the hole energy level positions due to band

anisotropy, the Bohr radii and Rydberg energies of the excitons formed








40

from these hole bands should be the same. Confinement-related shifts in

the hole energies should, thus, also be equivalent. As will be

discussed later, this has in fact been observed in small ZnSe

crystallites in which the bulk spin orbit energy separation between the

respective Is states associated with each hole band is retained even in

crystallites approximately 20 A in diameter [99].



Quantum Confinement

Theoretical Considerations

The previous section examined electronic band structure properties

necessarily the result of collective interactions between atoms arranged

in a crystal structure. Examination of delocalized electronic motion

within the periodic boundary conditions imposed by the lattice symmetry

yields a dispersion of allowed carrier eigenenergies throughout k-space

represented schematically by band structures, an example of which was

given in Figure 2.

The evaluation of distorting effects on the Bloch-type carrier

wavefunctions due to a truncation in the real-space extent of the

crystal structure and the eventual decomposition of the corresponding

bands into more discrete, molecular-like orbital states as the number of

participating units in the structure decreases, can be an extremely

difficult endeavor. The decreasing validity of extrapolations made from

bulk crystal properties as the crystal size is reduced, coupled with

largely uncontrollable fabrication-related deviations from more easily-.

handled theoretical assumptions, both contribute to the complexity of

the problem.








41

The present discussion will attempt to furnish a review of various

treatments used in the interpretation of finite crystal size effects on

optical and electronic properties of largely covalent, compound

semiconductors.

Theoretical interpretations of the effects of crystal size on the

electronic energy level structure and the ensuing optical behavior of a

material can be separated into two main crystal size regimes, dependent

on real- and inverse- (wavevector) space structural characteristics

exhibited by the atomic ensemble within each size range. These size

ranges are denoted by small clusters and larger crystals by the present

author.

Small clusters. Quantum confinement studies enable an examination

of the development of bulk crystalline properties with the addition of

atoms to a growing cluster. In very small clusters the material

exhibits atomic structures which tend to minimize the energy of the

system as a whole. A spherical cluster morphology is generally assumed

thus minimizing the surface area to volume ratio. The atomic structure

is more close-packed in nature with a significant population of strained

bonds. In this size regime, the bulk structure of the corresponding

crystal has not been attained and the electronic and optical properties

of the cluster are molecular-like in nature. The typical approach in

the investigation of these clusters and their corresponding electronic

energy level structures involves the use of quantum mechanical

calculations within molecular orbital (MO) theory.

Generally, MO theory involves the solution of the molecular

Schroedinger equation containing a total Hamiltonian which includes

individual contributions from each particle in the system [100,101] and








42

their interactions. A "total" molecular wavefunction is then determined

from which the energy states and real-space distribution of electronic

probabilities can be found. Differences in specific methods used to

solve this equation are rooted in varied approximations used to obtain

tractable Hamiltonians to describe the system. Two general

approximations are made initially. First, the motion of nuclei is

neglected when compared to the high kinetic energy of the electrons,

thus reducing the nuclear kinetic energy to zero. This is known as the

adiabatic or Born-Oppenheimer approximation [102] and is generally

considered to be a reasonable simplification owing to the large

difference in mass between the two particles. The second initial

approximation assumes that as the electron moves within the molecule, it

perceives an average potential generated by all other electrons. Thus,

the electron motion is calculated neglecting the correlation between

wavefunctions describing the individual electrons present, i.e.

electrons are considered to be independent particles (opposite spin

types are not correlated) This assumption is known as either the

Hartree-Fock (HF) or the self-consistent field theory (SCF) and involves

the use of a Slater determinant to express the overall molecular

wavefunction in terms of its component one-electron functions [100,103].

Solution of the problem even after these significant

simplifications is still very difficult, involving iterative solution of

the modified Schroedinger equations (HF equations) to make the electron-

electron potential terms self-consistent. Further assumptions in which

the one electron wavefunctions of the Slater determinant are expanded in

terms of a suitable set of orthonormal basis functions is termed the

algebraic approximation [103]. The set is usually finite with the







43

understanding that-the use of an infinite series will converge the

wavefunction to the original HF solution. Atomic orbitals can be used

as the basis set in the calculations, resulting in the commonly used

method of linear combination of atomic orbitals (LCAO). The initial

molecular wavefunction can then be used to determine the Hamiltonian

describing the system. This is then compared with the original test

Hamiltonian and any discrepancy is reduced through an iterative

procedure.

Calculations utilizing the above approximations can be described

as "ab initio" in that accuracy of the results can be successively

improved through lifting of the approximations until all parameters of

the system are explicitly defined and incorporated. Conversely, semi-

empirical methods have been developed for specific systems, including

experimentally determined parameters such as bond angle or energy.

Zero-differential overlap methods modify or neglect the overlap of

any but nearest neighbor bonding orbitals. Several variations of

differential overlap methods exist characterized by the degree to which

the overlap is neglected: complete neglect of differential overlap

(CNDO), intermediate neglect of differential overlap (INDO), modified

INDO (MINDO), and neglect of diatomic differential overlap (NDDO) [101].

Each method was developed for the calculation of different physical

properties and hence has been optimized to achieve this end through the

use of parameters (incorporated in the Hamiltonian) known to result in

reasonable agreement with experimentally determined properties. The

CNDO and INDO methods, for example, are used to ascertain such electron-

related properties as energy level spectra and charge distributions










while MINDO is directed toward the calculation of such chemical

properties as dissociation energy and molecular structure [103].

The Huckel MO method is also a semi-empirical calculation based on

several assumptions. Originally developed to examine organic

structures, the treatment involves only unsaturated, x-bonding orbital

contributions, i.e. there exists only one electron per atom in the

structure. A Hamiltonian is formed based on the ionization energy from

the p, orbital (ai) and x-bond energy (Ei) of carbon. Additional, non-

carbon atoms are included in the calculation by modifying their

respective energies using heteroatom parameters, arrived at empirically

[101]. Once obtained, the Hamiltonian is diagonalized to produce the

eigenenergies and the eigenvectors of the electrons in the structure.

Thus, not only can the transition energies be calculated but also the

respective probabilities that an electron will be associated with any

one atom can be found.

The approaches briefly discussed above, whether ab initio or semi-

empirical, are limited in their ability to handle structures involving

more than a few tens of atoms due to calculation time constraints. A

rule of thumb estimate says that calculation time for HMO, semiempirical

and ab initio methods goes up with the squared, cubed, and quadrupled

power of the number of atoms involved in the calculation [104]. Recent

corraborative results to experimental observables have however been

achieved, using semiempirical methods, for a number of vibrational and

electronic properties exhibited in the siloxyl structures of sol-gel

silica with controlled water content [105].

Large crystalB. At some point in the growth of the cluster, the

bulk crystal structure becomes energetically favorable and the material








45

begins to assume a bulk lattice symmetry and can be considered an

excised fragment taken from a bulk crystal. The point at which this

cross-over occurs is material dependent and is the focus of several

studies, most utilizing quantum chemistry approaches such as those

above. In fact, it has been shown that as many as 1000 silicon atoms

may be necessary to stabilize the bulk diamond structure in silicon

[106].

In particles of this type, however, bulk electronic and optical

properties are still not attained; the material exhibits a "hybrid"

molecular/bulk-like character. Treatments used to evaluate the effect

of finite size (to be discussed presently) generally involve the use of

known, bulk material properties within the effective mass approximation

(EMA). Thus, within the two size ranges introduced here, the problem of

quantum size effects is approached from the two possible extremes of

behavior, i.e. within a first principles, atomic and molecular bonding

methodology and from the standpoint of an already existing symmetric

array of atoms whose structural attributes are extrapolated into the

quantum size regime.

The evolution of a bulk-like electronic energy spectrum from

molecular orbitals is perhaps best illustrated through the use of

discussions by Steigerwald and Brus [23] and Bawendi et al. [107]. This

simple treatment, reviewed below, presents an intuitive link between the

more fundamental MO approach discussed above and the EMA-based, confined

particle model which will be introduced presently.

In a linear, conjugated polyene (carbon chain), the Huckel MO

model can be used to describe the nearest neighbor x bonding with the

addition of increasing numbers of participating atoms. If the %-bonds







46
are modelled as a chain of one-electron, one orbital atoms separated by

a distance, a, the one electron eigenenergies for the system can be

found to be (neglecting bond overlap)


E(s,N) = e + 2pcos N1 s = 1,23,...,N



E(s,N) = e + 2pcos[ka ] where k, = si
(N+1)a



and E is the exchange integral between adjacent orbitals, i.e. is a

measure of the bond strength [23]. Thus as N, the number of atoms,

approaches infinity, the wavenumber, k, becomes essentially a continuous

variable as do the allowed energies of the system, i.e. a band is

formed. For a finite number of atoms, various stable orbitals are

selected from this continuum, the energy spacing between them decreasing

with the addition of more atoms.

If ka < 1, the second equation above can be expanded to

yield [23]



E(k) = (e + 2p) + 2 = Eo x pa2
2 2L

where L (N + l)a.

Comparing this with the quantum mechanical result for the problem of a

particle or mass, m, confined within a one-dimensional infinite

potential well of length, L,


R2 t2n2
E 2L2 M
n ~2L2 m







47

Thus, the N-atom chain eigenenergies near the lower band edge (k,

< 1) are equivalent to the confined particle representation with


=
2pa2

It should also be noted that the kga < 1 condition in effect

denotes an effective mass approximation [23]. This condition implies

that the electron wavelength is much greater than the atomic separation,

making the line of atomic (nuclear) potentials essentially continuous

with respect to the electronic motion. The electron therefore moves in

an effectively constant potential with an effective mass determined by

the underlying periodicity (a) and bonding behavior (8) of the atomic

structure. As will be discussed in the next few paragraphs, this 1/a2

dependence of electronic energy with size is the general result obtained

in confinement theory based on an EMA model.

Quantum confinement effects within the compound semiconductors of

interest here typically have been examined within samples exhibiting

bulk-like crystalline structure as measured using X-ray or electron

diffraction techniques. These samples contain crystals of average size

greater than approximately 30 to 40 A thus containing oyer 1000 atoms.

Calculations involving large numbers of atoms have recently been

performed by Lippens and Lannoo [108,109] using a semi-empirical, tight

binding approach utilizing the known symmetry of the bulk crystal

lattice to choose appropriate atomic orbitals (or linear combinations of

them) to be used as a basis set approximating the true cluster

wavefunction. Only interactions between first nearest neighbors is

included and parameters are used to optimize the calculation so as to

obtain the correct bulk band structure. Calculation time constraints








48

are alleviated through the use of a recursion method which reduces the

number of matrix elements which must be stored during diagonalization of

the multi-atom Hamiltonian to obtain the eigenenergies of the system

[108]. Calculations involving up to 1000 to 2000 atoms have been

completed for such semiconductors as CdS, ZnS, and CdSe.

In addition to numerical calculations, the majority of theoretical

treatments used to interpret energy shifts in the optical spectra of

crystals in this size range have utilized assumptions linked to the bulk

structural attributes of the semiconductor, i.e. have employed the

effective mass approximation.

The general approach to the problem again involves the solution of

the Schroedinger equation for the Hamiltonian containing both the

kinetic and interparticle potential energies of a photoexcited electron

and hole whose masses are obtained from bulk band structure calculations

or experiment:



S2m, 2m, h ]r-r




Of primary importance in the successful application of this method is

the correct interpretation of interparticle Coulomb forces and the

pertinent choice of boundary conditions employed in the solution of the

above equation. Differences between treatments in the EMA thus

typically involve these considerations. Although the system can be

solved in any dimensionality (using the appropriate coordinate system),

the present discussion deals with three-dimensional structures usually

exhibiting nearly spherical symmetry. These types of semiconductor

structures have been designated quantum dots by some researchers [110].







49

Efros and Efros [111], in solving the above equation, divide the

problem into 3 size regimes depending upon the relative radius of the

confining sphere, a, and the Bohr radius of the carriers, a,(h) (given by

h2X/(m.(hoe2)). The Efros and Efros treatment represents the most

fundamental approach to this particular problem, assuming that the

energy bands describing the carriers are parabolic and symmetric in k-

space and that the wavefunctions are reduced to zero at the surface of

semiconducting sphere, i.e. the potential energy well in which the

carriers are confined is infinite. The differences between the size

regimes chosen lie in the relative magnitude of the kinetic energy of

localization imparted to each of the carriers by the infinite well

boundaries (proportional to 1/R2) and the interparticle Coulomb

interaction (proportional to 1/R) in this case assumed to be the

classical electrostatic attraction screened by the dielectric constant

of the semiconducting material. The two extremes of size effectively

represent two extremes of quantum confinement behavior.

In the smallest size regime, a < ah < a,, it is assumed that the

magnitude of the Coulomb interaction is small enough that it can be

neglected in comparison to the kinetic energy of either carrier. This

is termed the single-particle confinement regime. The solution to the

problem is then the solution to the quantum mechanical confinement of

noninteracting particles confined in a spherical, infinite potential

well. The resulting wavefunction for the confined particles in a well

of radius, a is given by










J,4Q1(k4,
4,<(r,,p) = Y,(.p) 2 2
2

yielding the familiar electron (e) and hole (h) quantized energy levels,


32x2n2
2m,ha2

for the 1 0 or s-like angular momentum states.

This result can be interpreted as the decomposition of previously

continuous energy bands into discrete single particle states as the size

of the crystallite is reduced. As will be discussed later, this "sub-

banding" effect has been reported in II-VI semiconductor-doped glasses

and colloids exhibiting crystallite sizes larger than the Efros and

Efros cutoff for this size range.

The optical absorption threshold is then evaluated for the lowest

allowed transition between the electron and hole levels, employing

selection rules Al 0 and An 0 [111]. Assuming that the free

particle energies for the electron and hole are defined by the lowest

state in the conduction band and highest point in the valence band,

respectively, the absorption threshold energy for the transition between

the Is levels (n-l, 1-0) of the individual carrier energy spectra is


E= E+
2pa2

where Es the bulk band gap of the semiconductor at k 0 and p is the

reduced mass of the electron-hole pair (exciton).

Within the largest size regime, ah < a, < a, the Coulomb

interaction is assumed to result in the formation of a correlated







51

electron-hole pair'state, the exciton. The threshold or lowest excited

state absorption energy is then found to be


h2X2
E= E E +
2Ma2

where E, again corresponds to the bulk band gap energy, E,. bulk

excitonic binding energy (n 1) and M m, + mn the translational

mass of the exciton. Thus, the eigenenergy includes terms corresponding

to wavefunctions describing both the relative motion of the electron and

hole (E.,) and the motion of the electron-hole center of mass. In this

case, only the translational motion of the correlated pair is affected

by the presence of the potential boundaries.

Thus the two size regimes are characterized by a shift in the

absorption edge energy which is inversely proportional to the confining

sphere radius squared. The actual magnitude of the shift is governed by

an effective mass value ranging from the reduced to the translational

mass of the bulk exciton in the semiconductor. For the smallest size

range, the reduced mass indicates that the carrier motion is completely

uncorrelated by a Coulomb effect. At the other size extreme, the

Coulomb interaction results in the formation of an exciton whose center

of mass motion appears to be confined in the structure as if it were a

quasi-particle whose mass is that of the translational exciton mass. As

a result, the mass parameter observed to govern the energy shift with

size in a particular system can be used as a sensitive measure of the

relative correlation of excited carriers through the Coulomb

interaction, assuming the Efros and Efros assumptions mentioned above

are valid.







52

The intermediate size regime deliniated by Efros and Efros, ah < a

< a,, is handled through the use of an adiabatic approximation in which

the kinetic energy of the hole may be neglected when compared to that of

the lighter electron. The hole is then found to move within the

spherically symmetric potential created by the electron, behaving as an

isotropic, three-dimensional harmonic oscillator. As a result, the

absorption spectrum is characterized by line shifts which are mainly

determined by the electron mass and the corresponding state shifts with

sphere size. Peaks in the spectrum are further divided into smaller

lines corresponding to quantized, equally spaced energy levels

characterizing the oscillating hole in the electron potential. The

energies expected for the lowest excited state are given by



E, = E, + X2r 2 + 2~1(2t + F + )
2mn,a2 ea 2

where the last two terms correspond to the harmonic oscillator energies

for the hole.

The treatment of Efros and Efros given above is a valuable tool to

determine the general behavior of carriers confined in small

semiconductor structures. Due to its general nature and the assumptions

made in its formulation, accurate interpretation of experimental data

has necessitated the inclusion of several deviations from the

assumptions made in addition to the investigation of a more continuous

variability in the degree of Coulomb interaction between the carriers.

Although some overlap can occur, three main areas of interest can be

deliniated, each of which must be investigated to fully evaluate the

effect of quantum size in these structures.







53

a. Coulomb Interactions: Due to the relatively small hole Bohr

diameter in the semiconductors of this study, the smallest size regime

of Efros and Efros [111] has not, in general, been attained in these

composite materials. As a result, even the smallest crystallites

studied thus far have fallen into the intermediate size range. Although

oscillations presumably have been observed by Ekimov et al. [112], which

can be explained using the adiabatic treatment discussed earlier,

confining structures within this intermediate size range have been

examined using a number of approaches including variational principles

in which the degree of electron and hole correlation can be made size

dependent. The ultimate goal of these studies is to model the

transition between largely uncorrelated particle behavior which is

moderated by the Coulomb interaction and the translational, excitonic

mass confinement expected to evolve as the size of the crystals are

increased and the relative contributions of the kinetic and Coulomb

terms in the Hamiltonian change.

The influence of intercarrier correlation effects, in a similar

manner to the effects discussed in the previous section upon the bulk

allowed energy states, generate a number distortions in the optical

behavior of a purely noninteracting particle system.

Brus [6,113], in one of the first treatments of the problem,

employs a perturbation calculation in an attempt to model the effect of

both interparticle Coulomb interaction and medium polarization on the

single particle states within small crystallites. The treatment is thus

only valid for crystal sizes where the single particle kinetic energies

dominate the effect of Coulomb attraction. Brus develops size dependent

polarization terms which describe the loss of dielectricc solvation







54

energy" as the dielectric material present to screen the charges of the

confined particles is reduced with sphere size. These terms are

incorporated into the Hamiltonian for the charged particles such that

the interparticle potential is given as


e2
KR1,RW 4 = + P(R1) + P(R2 + Pm(R1,R)


where 2 is the dielectric constant inside the sphere. As the radius of

the confining sphere increases, these polarization terms approach zero

and the remaining Hamiltonian is just that discussed earlier in the

development of Wannier exciton theory.

The resulting energy dependence for the lowest excited state, i.e.

the Is (hole level) to Is (electron level) transition, then becomes


E~ = -ZI 1 +- 1.8 + smaller terms
2R2 m, mh e2R

where the first term is equivalent to the uncorrelated particle energy

derived by Efros and Efros, the second term corresponds to the Coulomb

attraction between the carriers, and the third term (often neglected in

subsequent calculations and treatments) represents the loss of

dielectric solvation energy [113].

Proceeding along the lines of Brus, Nair et al. [114] also treats

the relatively small influence of Coulomb interaction in small

semiconductor crystallites (with R < 1.5ab, ab excitonic Bohr radius)

as a perturbation on the single particle confinement behavior

illustrated by Efros and Efros. Linear combinations of the single

particle state wavefunctions (given above) were used to form a basis set

from which the Hamiltonian for the wave equation above was obtained.







55

Due to limitations in the matrix size which could be diagonalized, only

particles less that 1.5ab could be handled. Larger spheres were

evaluated using a variational calculation involving the product of the

aforementioned single particle wavefunctions and an exponential term

describing the Coulomb interaction to form the trial wavefunction of the

ensemble which satisfied the infinite well boundary conditions of the

problem. Suitable choice of the three variational parameters can be

shown to yield the expected asymptotic behaviors. Minimization of the

total energy expectation value was achieved to determine the

eigenenergies for a given sphere radius. Again, because this treatment

forms its basis from free particle behavior, it tends to inaccurately

predict behavior at larger crystal sizes where the Coulomb interaction

is an increasingly dominant factor.

Kayanuma [115,116] reproduces both Brus' result and the excitonic

confinement behavior of Efros and Efros when examining the asymptotic

limits of strong and weak confinement, respectively, of charged carriers

within a spherical, infinite potential well of radius, R, neglecting the

surface polarization effects included by Brus [113]. A variational

method is used to investigate the transition size range between one

extreme and the other. The procedure involves a two-parameter

optimization of the lowest eigenenergy of the system in which the

wavefunction describing the two particle arrangement is expanded in a

series of basis sets including two polynomials, each modelling the true

single-particle functions, and an exponential factor, describing the

size-dependent interparticle forces. This is similar to the procedure

followed by Nair et al. [114] above. The use of polynomials simplifies

the numerical calculation used to evaluate the eigenenergies which are







56

then minimized through variation in a parameter incorporated in the

exponential.

Kayanuma [116] finds that the behavior of the confined particles

can also be classified into three size regimes, although these ranges

differ from those -of Efros and Efros. In this treatment, excitonic

confinement is predicted to occur in the range R/ab z 4, where ab was

defined earlier as the bulk exciton radius. Subsequent numerical

calculation shows that a term involving the finite size of the exciton

must also be included which is dependent on a parameter equal to the

ratio of the hole and electron masses. The total confining volume must

therefore exclude the volume inaccessible to the exciton center of mass.

This "dead layer" is a function of the relative magnitude of the hole

and electron masses and corresponds to the strong deformation of the

carrier pair relative motion necessary to enable the center of mass to

approach the potential boundary. The resulting lowest excited state

energy for the weak confinement limit is then given as



E = E E, + 2
2M [R-n(o)a,]


where q(o) increases with a, the ratio of hole to electron effective

mass in the semiconductor. This is a direct result of the increased

spatial extension of the relative electron and hole motion about the

center of mass when the mass ratio becomes larger. With a decrease in

the hole mass, this particle attains a greater degree of mobility,

effectively allowing greater deformation of the two-particle unit and a

closer approach of the center of mass to the edge of the crystallite.

When q is multiplied by the excitonic Bohr radius, ab, the product is a

measure of the dead layer discussed above.







57

Conversely, individual particle behavior exists for R/ab 2.

Similar to Brus, the energy of the is state is


h2X2 e2
E= E, + 1.786-- 0.248E,
S2pR2 eR

where in this case the third term is described by Kayanuma [116] as a

remnant of the excitonic effect.

The numerical calculation bridges the size range between the two

extremes. Graphical representation of the Kayanuma calculation as well

as other confinement theories discussed in this section will be

presented in the context of experimental data in the next section.

In addition to an examination of the effect of quantum size and

the influence of the Coulomb interaction between the confined carriers

on the energetic positions of optical transitions exhibited by them,

Kayanuma [116] also follows the evolution of the two behavioral extremes

through investigation of the redistribution of oscillator strength

throughout the allowed higher order transitions of the ensemble. Using

the wavefunctions calculated as a function of size, Kayanuma determines

the oscillator strength of the jth level per unit volume adapting a

formula developed by Henry and Nassau [117]:



ftx v


where f,. the Is bulk exciton state oscillator strength and V volume

of the confining sphere. As the crystal size is reduced, transitions

between all s-like orbital states become allowed because of a breakdown

in the translational invariance of the structure [116]. In fact

calculations by Bawendi et al. [107] show that inclusion of the Coulomb







58

term results in the is excited state having approximately 1% character

of both Ip-lp and ls-2s transitions. Kayanuma [116] finds that for a

microcrystal of volume, V, in the region R/ab s 2, the ground state

oscillator strength is inversely proportional to the volume of the

semiconductor sphere behaving as




f 41R

For R/ab k 4, the oscillator strength is distributed within the center-

of-mass motion states which become more and more closely spaced as the

radius increases. The asymptotic result is the collection of oscillator

strength into the familiar hydrogenic, Rydberg series of the bulk

exciton. The Is bulk excitonic state is then seen to decompose and

evolve into the Is-ls free particle transition in the small sphere limit

[110].

Finite Coulomb effects have also been used to explain induced

absorption features exhibited by CdSe and CdTe quantum crystallites in

femtosecond pump-probe measurements performed by Peyghambarian et al.

[118] and Esch et al. [119], respectively. The lifting of selection

rules due to the existence of a photo-excited electron-hole pair is

hypothesized to result in previously unallowed transitions during the

subsequent absorption of a second photon.

The overall results, then, of the introduction of Coulomb

interaction into the treatment of quantum-confined, noninteracting

particles is a general redistribution of oscillator strengths, a slight

lifting of optical transition selection rules, and the more readily

observable red-shift in the single-particle states expected from a basic

confined particle representation.







59

b. Band Structure Considerations: All of the above treatments

are based on the effective mass approximation (EMA) utilizing parabolic

energy bands which provides a framework from within which to address the

problem of carrier confinement in crystals exhibiting bulk-like crystal

attributes. In general, at large crystal sizes, this is a good

approximation in that the crystal size is many times the lattice

constant and transitions can still be considered to occur at a single

point in k-space. Substantial deviations from this assumption however,

can occur in sufficiently small structures. The complicated nature of

the actual band structure must also be included.

As has been shown in the previous section, semiconductor band

structures are actually complicated functions of the wavevector, k. In

the case of tetrahedrally coordinated crystal structures, several

valence bands exist at the center of the Brillouin zone which may be

degenerate or separated from each other by quantum mechanical spin-orbit

splitting [87]. Of significance from the standpoint of the present

discussion is the anisotropy of the band surfaces with orientation in k-

space. The carrier effective masses are defined by [120]


f-1 1 d2E
%2 dkdk,

The effective carrier mass is therefore described by a tensor.

In a crystallite of diameter, D, the localization of a particle in

real space can be depicted by the summation of Bloch-like, plane wave

wavefunctions to form a wavepacket describing the particle. The more

localized the particle, the wider the wavevector range which much be

used in the formation of the wavepacket describing it [121]. Generally,

the result is that a finite k-space region, approximately equal to








60

abs(k) s w/D, is involved in describing the transitions occurring

between the single particle states bounding the forbidden band of the

semiconductor. In the case of real semiconductors possessing energy

surfaces which deviate from parabolicity for finite k, a complete

analysis of confinement must include the anisotropic components of the

carrier masses, individual elements of which may be of different sign

and magnitude. This complicates the otherwise straightforward use of a

scalar mass in the treatments discussed earlier. In principle, Wang et

al. [54] notes that the problem can be solved by using the full E(k)

curve, determined by a band structure calculation, to numerically

integrate the expectation value of the transition energies at the

required points of k-space using the corresponding electron and hole

wavefunctions.

Following the treatment of Baldereschi and Lipari [98], Chestnoy

et al. [99] examine the effect of anisotropic, degenerate valence bands

when examining higher order transitions they find to occur in zincblende

ZnSe. As discussed earlier, contributions from the three valence bands

at k 0 can lead to mixing and splitting of hole states leading to

previously unallowed transitions. Anisotropic components of the hole

Hamiltonian result in distortions of the parabolic band model assumed to

hold within the effective mass approximation. Chestnoy et al. do not

find that the anisotropic hole mass Hamiltonian (introduced in the last

section) substantially affects the zero order state transitions (Is) in

first order perturbation theory (due to the relatively large magnitudes

of the anisotropic masses), they do report a weak coupling in second

order perturbation theory to the Is transition involving the split-off

valence band. They also report an increased coupling to the lowest








61

order states when the spin orbit splitting of the valence bands at k 0

becomes smaller [99]. In approximately 20 A ZnSe crystallites, the

spin-orbit splitting (0.43 eV) of the valence bands of the zincblende

semiconductor is apparently retained while in ZnS crystallites the

separation between the two Is hole states increases to 0.22 eV, a marked

change over the bulk value of 0.07 eV. This effect indicates the

increased coupling of anisotropic to isotropic hole states predicted by

Baldereschi and Lipari [98] in structures exhibiting a small spin-orbit

splitting, i.e. the quantum confined shifts of the two hole states in

ZnS are governed by different effective mass parameters.

A theoretical treatment of electronic structures in three-

dimensionally confined semiconductors exhibiting these complex valence

bands was performed by Xia [122]. Using the analysis of Baldereschi and

Lipari [123] in the limit of strong spin-orbit splitting between the

valence bands, the hole energy spectrum is calculated for the excited

states of all six valence bands as a function of a parameter describing

the anisotropy of the hole bands (Luttinger parameter). Xia notes that

the variation is not monotonic with variation in the hole shape

parameter as would have been expected in the case of the parabolic band

model. With the hole wavefunctions in hand, Xia also calculates the

expected transition probabilities and "excitonic" binding energies

(defined as the difference between the Coulombically correlated

electron-hole state energy and the energies of the corresponding

electron and hole eigenenergies) using the Coulomb interaction derived

by Brus [6]. Significant weight is found for transitions between the

electron Is and the r8 hole second excited state as well as additional








62

previously unallowed transitions due to the mixing of d-like symmetry

hole terms [122].

Anisotropic carrier masses also increase the sensitivity of the

confinement effect to the geometry of the confining surface, i.e. the

crystal shape. Needles or platelets oriented along specific

crystallographic directions will necessarily induce broadening of the

pertinent k-space region along preferred directions in the Brillouin

zone. Thus, an anisotropic incorporation of mass elements must be

included in an interpretation of effects in these materials, in addition

to the macroscopic, geometry-related changes in the boundary conditions

used in solving the Schroedinger equation. This effect is also of

importance as it represents one mechanism of inhomogeneous broadening of

observed transition energies for the crystallite ensemble.

Wang et al. [54] uses an approximation of the true band shape in

PbS (a IV-VI semiconductor), in the vicinity of the L-point, when

examining deviations from the EMA treatment exhibited by the fundamental

absorption of small particles in polymer films. In this case, the

energy surface is approximated to be isotropic, described by a

hyperbolic function which approaches a parabolic dependence with k near

the L-point. Improved fit to experimental data is obtained for

crystallite average sizes down to approximately 30 A. The net effect of

incorporating this synthetic band shape is the reduction in the Is

transition energy, exhibited as the crystal size is reduced, below that

predicted by the EMA which assumes parabolic energy bands. This can be

intuitively understood from the standpoint of the corresponding mass

value defined by the smaller curvature of the hyperbolic-shaped band.

Looking at the transition energy shift expressions developed earlier for








63

the single particle confinement size regime, the reduced curvature of

the proposed band shape would result in a greater effective mass

parameter in the numerator, thus resulting in a smaller energy shift for

the same crystallite size when compared to the parabolic model.

It can be readily seen from the above discussion that inclusion of

actual band structure parameters for the semiconductor under study

results in a substantial increase in the complexity of an analysis of

quantum confinement in very small crystallites.

c. Interfacial effects: In the previous sections, the

semiconductor lattice is assumed to be truncated without structural

relaxation or the influence of adjacent matrix constituents. The high

curvature, high surface area, interfacial region between the

semiconductor and matrix will be characterized by a locally high free

energy. Unbound orbitals (dangling bonds) and defect states (e.g.

interstitials, vacancies), due to a rearrangement of the bulk

crystalline structure to reduce this high energy condition, will all

contribute localized energy levels to the allowed energy structure

characterizing the internal volume of the crystal. Interactions with

the matrix itself in the form of adsorbed molecules or bridging bonds

between the semiconductor and matrix constituents will also serve to

distort the "pristine" quantum confined energy characteristics of the

crystal.

Indeed, as the crystallite size decreases, a greater proportion of

the particle can be considered "surface" resulting in optical and

electronic properties poorly described using treatments based on the

bulk structural properties of the material. Such surface or trap

states at the semiconductor/glass interface in the materials of the








64

present work have been hypothesized to be the origin of a wide variation

in the magnitude and dynamics of both linear and nonlinear optical

properties. Such effects include the observation of a photochemical-

like darkening of these materials when exposed to high intensity laser

radiation tuned above the band edge of the crystals [124-129] and the

extremely rapid recombination times exhibited in time-resolved

photoluminescence and high intensity pump-probe

measurements.[118,119,130-137] In Figure 5, Brus [121] schematically

depicts the influence of these surface related states on the developing

energy level structure of a small semiconductor crystal.

At small crystal sizes, the crystallite morphology itself can also

have a drastic influence on the electronic structure. Valence bands of

low band gap semiconductors, while still degenerate in spherical

crystals, would split, characterized by different confinement energies

and optical selection rules [110]. This is widely observed in one-

dimensionally confined multiple quantum well structures in GaAs.

One of the most fundamental repercussions on quantum confinement

caused by the existence of a boundary between regions of different

dielectric constant is the distortion of energy bands over a

characteristic length, the result of Fermi level matching at the

boundary to achieve thermodynamic equilibrium. From the standpoint of

the above theoretical discussions, this invalidates the assumption of an

infinite potential energy well depth in which carriers are confined. In

addition, the potential energy well profile can no longer be considered

step-like.

The fundamental effect of a finite potential energy step between

the matrix and semiconductor results in a finite probability that the






















BULK SEMICONDUCTOR


*-4
CONDUCTION BAND .'

--SHALLOW TRAP---"
S- DEEP TRAP.-* ..."




VALENCE BAND
*.._


7
1


DELOCALZED
MOLECULAR
OR81TALS


DEEP TRAP


-SURFACE STATE


CLUSTER DIAMETER


Energy band schematic depicting the effect of
crystallite size reduction on the electronic
structure of a semiconductor material [121].


Figure 5.


CLUSTER








66

charge carriers actually exist in the matrix. This result can be

partially understood using the more simple example of the interaction

between a travelling wave packet and a one dimensional potential energy

step. Solving for these semi-infinite boundary conditions assuring

wavefunction continuity across the boundary, the wavefunction for the

particle no longer goes to zero at the interface. On the low potential

side of the step, the wavefunction is still oscillatory. The solution

to the wave equation within the high potential region, however, results

in an exponentially decaying function. Thus, there is a quantum-

mechanical tunnellingg" of the particle into the matrix, effectively

increasing the size of the confining structure. The depth of

penetration is a function of the effective mass of the particle, the

height of the potential energy step and the profile of the step. The

analysis of finite potential well depth in a real material is thus a

very complex problem.

Depending on the band structures of the matrix and semiconductor

and the real space structural characteristics of the boundary separating

them, the photoexcited electron and hole initially formed within the

semiconductor will each perceive a different potential boundary step.

Each particle will therefore have a tendency to penetrate the matrix to

varying degrees. Schmitt-Rink et al. [110] note that this effect can

lead to a lifting of optical transition selection rules which allow

transitions only between states of the same quantum number. The finite

probability of higher order transitions in which the final energy states

of the photoexcited particles are even greater than the lowest excited

state also implies a lower potential energy step defining the

semiconductor surface and an increased tendency for matrix penetration.










Given that the carrier existence in the matrix is characterized by a

finite probability, the dynamic properties of the carriers in the matrix

will not be determined by the semiconductor lattice but by the local

atomic and molecular orbital structure of the matrix or, depending on

the penetration depth, the interface region itself. Thus, for example,

the effective mass for the charge carrier should change while "outside"

the semiconductor.

The influence of surface-related considerations is a major barrier

in the comparison of experimental data taken in different materials;

interface structures will necessarily vary between samples employing

glass and those using an organic solvent or heterogeneous reaction

media. Taking this point to the extreme, the carrier behavior in the

boundary material could in reality be used as the asymptotic limit of

the confinement effect. Contrary to the basic theoretical discussions

presented, the energy levels are not expected to diverge with an inverse

radius dependence. Eventually, as alluded to earlier, the molecular-

like limit of behavior should be reached with the precise attributes

heavily influenced, and perhaps dominated, by the neighboring atomic

orbital characteristics of the matrix. Experimental evidence for this

effect was found by Wang and Herron [138] who noted a modification of

optical properties exhibited by lead and bismuth iodide colloids through

an interaction with the surrounding molecules in aqueous or acetonitrile

solutions.

Brus [6] outlines a treatment for handling carrier penetration

outside a spherical crystallite in water when the carrier mass varies

with position. In this treatment the eigenstates of a partially

confined particle are developed from known single particle solutions for








68

r < R (the radius of the sphere) and an exponentially decaying function

for r > R. The potential function is step-like at the boundary, R, the

sphere radius. In addition to continuity of wavefunctions at R, Brus

also equates the charge flux moving across this boundary in either

direction. The eigenenergy for the particle is similar in form to the

energies found for the infinite well problem with the Bessel function

root replaced by a quantization parameter dependent on the relative mass

of the particle inside and outside the sphere and on the depth and width

of the well. Using pertinent electron affinity values for water and for

CdS and InSb, Brus calculates that the kinetic energy for an electron

confined in a 120 A diameter InSb crystallite in water would be 0.17 eV,

only 25Z of its value in the case of an infinite well. Additionally,

Brus finds that the charge density distribution for the lighter mass,

0.015 mn, InSb electron is actually at its maximum at the surface of an

80 A diameter crystal. This redistribution of the electronic charge

density is not so drastic in the case of CdS where the electron mass is

greater, 0.19 m.. Thus, as the depth of the potential well is reduced,

interactions with the crystallite-matrix interface are expected to be

important, especially for carriers exhibiting very small effective

masses.

Although knowledge of the respective band structures can yield the

height of the electron or hole potential energy steps (to the first

approximation), the addition of defects, such as those mentioned above

would also distort the predicted transition region energy structure

through the incorporation of localized states which could behave as

carrier traps. These attributes make for a very complicated potential

profile between the two materials, in turn increasing the difficulty in








69

solving the wave equation to describe particle motion in the structure.

An interesting possibility in attempting to deal with this problem might

be to model the potential energy function at the boundary within the WKB

(Wentzel, Kramers, and Brillouin) approximation [102].



Prior ExDerimental Results

Examination of quantum confinement effects on the electronic

energy level structures of finite sized semiconductor crystallites

typically involves measurement of optical transition energies, using

either optical absorption or photoluminescence as the predominant tool.

Precise determination of energy positions in the spectra obtained is

often difficult due to significant homogeneous and inhomogeneous

broadening mechanisms at work in the ensemble of crystallites. This

topic will be discussed further in a later section. In addition, the

determination of the microstructure size is often difficult, most often

involving the direct examination and identification of semiconductor

size, morphology and crystallinity using transmission electron

microscopy and electron diffraction. Small angle X-ray scattering may

also yield size information although its use requires the assumption of

a particular crystal morphology. X-ray diffraction can yield both size

and crystallinity information.

It is generally of interest to investigate the functional

dependence of various transition energies with the average particle size

of the ensemble. In light of the theoretical discussion above, looking

at the energy behavior as a function of size should yield information

allowing a more educated assessment of the predominant physical

mechanisms at work. Depending on the fabrication technique employed to










obtain the necessary microstructures, a variation in the average

particle size can, however, be restricted. Examples of such techniques

are the liquid-based, colloidal precipitation methods introduced above.

As has been previously stated, Brus [6,113,139,140], Rossetti et

al. [15,16], Chestnoy et al. [99], and Steigerwald et al. [36] have

grown colloidal II-VI semiconductor particles (e.g. CdS, CdSe, ZnS,

ZnSe) in a variety of aqueous and organic solvents using temperature,

solvent identity or heterogeneous reaction media to control and

stabilize the growing particles. This work is characterized by extreme

attention to theoretical detail in the analysis of optical absorption

and photoexcited response data and represents some of the most coherent

studies in the field. In general, the average crystallite size,

determined using TEM, is smaller than the excitonic Bohr radius, thus

exhibiting optical absorption spectra containing at times several

oscillations in the frequency range above the absorption onset due to

transitions between sub-banded, nearly free particle energy states.

In both aqueous and acetonitrile based CdS colloids, Brus [139]

observes a shift in the CdS absorption edge from its bulk value in 100 A

crystallites (2.4 eV) to approximately 3.18 eV in 20 A diameter

particles. Accompanying this shift is the appearance of an unresolved

peak at 360 nm which Brus ascribes to a Is-type transition in the

spherical potential energy well. This shift is corroborated by resonant

Raman studies in these colloids in which the frequency range exhibiting

the scattering peaks is red-shifted with colloid aging, i.e. crystal

size increase. Detected LO-phonon harmonics present at 395, 416, and

448 nm excitations in 45 A colloids are seen only with incident

radiation at 463-480 nm in 125 A particles. Rossetti et al. [16]










attribute this to the quantum-shifted Is transition which acts as the

intermediate state for the behavior. The shifts observed are

interpreted within Brus' approach in which the Coulomb interaction is

included in a variational calculation on the single particle state

functions. Similar effects were also observed in the absorption spectra

of CdS particles prepared via other methods in various liquid phase

media by other researchers [35,141,32,30].

Chestnoy et al. [99] also utilize a sub-banded interpretation of

optical absorption features they observe in approx. 20-30A diameter

crystallites of ZnS and ZnSe [140]. The identification of the two broad

peaks exhibited in the crystallites has already been discussed in terms

of the degenerate valence band structure of the zincblende

semiconductors. Using a variety of synthesis procedures involving

ethanol, methanol, and isopropanol at reduced and room temperature, a

wide range of resulting optical absorption is observed. In ZnSe

produced in isopropanol, the lowest excited state absorption is shifted

as much as 1.3 eV from the bulk value of 2.58 eV. A room temperature

preparation in an ethylene glycol/water mixture however exhibits a shift

of only 0.6 eV, indicating larger crystallites. Similar tendencies are

also exhibited by CdSe and ZnS prepared via the same methods. Precise

determination of the crystallite size is however not possible due to

aggregation of colloidal particles in TEM analyses precluding a clean

measurement of size. The crystallites are estimated to range from 20-50

A in size however, based on the size of lattice plane fragments

observed.

These studies, and others involving the precipitation of

crystallites within low viscosity, homogeneous or heterogeneous phase








72

matrices, typically do not allow precise control over the final average

size of the crystal ensemble. Exceptions would include the work of, for

example, Steigerwald et al. [36], Petit et al. [33] and Lianos and

Thomas [32]. Here, the semiconductor is precipitated within inverse

micelle structures. Some control of the final particle size is

demonstrated through variation in the ratio of water to surfactant added

to form the micelles thus changing their diameter. Such controlled

variation in size is however limited with Steigerwald et al. [36]

reporting a range of size from 17 to 45 A in CdSe. Wang et al. [54]

achieved a wide average particle size ranging from approx. 13 to 125 A

in PbS embedded in E-MAA polymer films. Control of both the Pb2+

concentration introduced in the initial ion-exchange process and the

subsequent thermal treatments to which the films were subjected allowed

this wide variation.

Within the context of the preceding theoretical discussion, it is

of interest to examine the behavior of quantum confinement-induced

shifts in the optical transition energies as a function of size over a

wide range of crystal sizes. Most of the theoretical treatments predict

specific trends with crystal size which will vary depending on the

relative size of the confining structure and the Bohr radii of either

the free carriers themselves or the bulk excitonic Bohr radii of the

Coulombically bound carriers in the semiconductor. Effective control in

the synthesis of crystallites possessing a wide range of average

microstructure sizes would allow the experimental testing of the

predicted tendencies.

Semiconductor-glass composite fabrication was briefly discussed in

the first section of this chapter. The diffusive precipitation of








73

crystallites from a supersaturated silicate glass matrix has been well-

characterized by several groups, with wide variation in the average

crystallite size adequately demonstrated for a number of II-VI materials

through the use of suitable thermal treatments. This material thus

represents a convenient, easily handled medium in which to study the

effects of finite size on the electronic energy bands of semiconductors.

In 1954, Kuwabara [38] examined a red-shift in the absorption edge

of CdS-containing silicate glass with increasing heat-treatment. Unable

to obtain microstructural data, e.g. crystal structure or size, he

tentatively assigns the effect to a size-dependent lattice constant in

the crystals although he does recognize the possibility that the effect

could be tied to crystal size.

In recent years, this class of material has been examined by

several groups in an attempt to investigate size dependent effects on

the electronic structure of the precipitated crystallites. Co-workers

of Efros and Efros were some of the first researchers to examine both

the precipitation kinetics and quantum confinement behavior of the

material. In this case, work involved glasses doped with a variety of

II-VI and I-VII materials.

Ekimov and Onushchenko [12] first observed confinement effects on

the translational behavior of excitons formed from both spin-orbit split

valence bands and the conduction band of CuCl. With a bulk Bohr exciton

radius of approx. 8 A in this material, the center-of-mass confinement

predicted by Efros and Efros was expected to be exhibited in the 15 to

1000 A particles produced. (The crystallite sizes in this case being

determined by small angle X-ray scattering.) Indeed, 1/R2 behavior was

exhibited by two transitions attributed to exciton formation from the









74

upper, doubly degenerate r7 (Z3 transition) and lower, quadruply

degenerate Pr (Z1,2 transition) valence bands. Deviation from expected

magnitude of the behavior, based on a simple parabolic band model for

the degenerate hole bands, was later explained by Ekimov et al. [142]

using a similar treatment to that provided by Baldereschi and Lipari

[98]. The identical hole Hamiltonian containing the anisotropic

components of the valence bands was used to determine the dispersion of

the excitonic energy allowing for nonparabolicity. Ekimov et al. [142]

determine that an increase in the momentum of the exciton causes a

considerable increase in the translational mass formed using a hole in

the spin-orbit split-off band, r7, depending on the degree of

nonparabolicity. This would then predict a deviation from a linear

dependence of the transition energy with 1/R2 at smaller crystallite

radii and higher energy. After full treatment of the problem, Ekimov et

al. found that the Z3 and Z1,2 transition should be governed by different

effective translational masses dependent upon the pertinent Luttinger

band shape parameters for CuCl, with the Z3 transition, involving the

upper r7 valence band, closely approximating parabolic band behavior due

to the relatively low anisotropy exhibited. Close agreement to

experimental data was obtained for both transitions. In fact, by

fitting the data and using the appropriate band shape parameters, the

"light" and "heavy" hole components of the r8 band were determined by

the researchers for the first time [142].

Addressing the intermediate size regime of Efros and Efros, Ekimov

and Onushchenko [143] evaluated finite size effects on the position of

the lowest excited state absorption edge (Is transition) and two higher

energy peaks (attributed to the Ip and ld transitions) in CdS-doped








75

glasses containing crystals ranging from R 15 to 400 A. In this case,

the slopes of the l/R2 plots for each transition at larger sizes defined

an effective mass in agreement with literature values for the electronic

mass in CdS when the contribution due to a finite crystal size

distribution is included in the energy shift expression developed by

Efros and Efros [111]. Here, the size distribution was taken to be that

derived by Lifshitz and Slyozov [26] in their treatment of diffusive

decomposition of a supersaturated solid solution in the limit of small

volume fraction of precipitate. By numerically integrating the

expression for the transition probability, including this distribution

function and assuming that the homogeneous linewidth for a single

particle could be approximated as a delta function, Efros and Efros

found that the energy of the lowest excited state must be corrected by a

constant, multiplicative term: 0.67 for the large crystal size limit and

0.71 for the small size range. Use of the latter term for CdS

crystallites in the intermediate size regime by Ekimov et al. [144],

however, led to their close agreement with literature electron mass

values. The validity of this analysis is somewhat in question.

Significant deviation below the linear dependence with 1/R2 was seen for

all three transitions at smaller crystal sizes with a greater effect

observed for the higher energy transitions. The omission of the

influence of Coulomb interaction between carriers, nonparabolicity of

the conduction band and the finite depth of the potential well was

hypothesized to account for the deviation [144].

Further research by Ekimov et al. [112,145] in CdS crystallites 15

to 30A in size appears to substantiate the intermediate size behavior

predicted by Efros and Efros in which the hole is taken to oscillate as










a three-dimensional harmonic oscillator within the adiabatic potential

created by the lighter, faster moving electron. The energy spacing of

small features superimposed on the electron free particle transitions

observed in absorption measurements taken at T 4.2K is in good

agreement with the energy eigenvalues calculated for the oscillating

hole.

In other research, Warnock and Awschalom [146,147] examined peak

energy shifts in photoluminescence spectra taken from three different

Corning commercial glass compositions containing mixed CdS/Se crystals

of size ranging from 94 to 133 A. These shifts are taken by them to be

the result of quantum confinement effects. The assumption was therefore

made that the crystals within each sample had the same composition and

that it was invariant with heat-treatment. Subsequent work by Borrelli

et al. [11] and Potter and Simmons [14], however, suggest that this

conclusion is erroneous. Work on identical compositions by Borrelli et

al. [11] revealed that the crystallite sizes in these glasses are

significantly larger than would be appropriate to achieve energy state

shifts of the order observed by Warnock and Awschalom [146,147].

Furthermore, x-ray diffraction data presented by Borrelli et al. [11]

suggest that the peak shifts observed by the previous researchers

originate from crystallite stoichiometry differences between the samples

of different glass compositions. Potter and Simmons [14] also noted

that uncontrolled stoichiometric changes are likely in Schott, mixed

crystal composition glasses where additional heat-treatment results in a

red-shifted optical absorption edge without a corresponding increase in

crystal size. Fuyu and Parker [46] also report x-ray diffraction








77

results indicating a substantial change in the S/Se ratio with heat-

treatment time and temperature in CdS/Se-doped glass compositions.

The conclusion of such findings supports the use of pure,

nonalloyed crystal compositions in the study of quantum confinement.

Thus, careful attention must be paid to the base glass formulation to

avoid incorporation of constituents into the growing crystal lattice and

anomalous shifts in the optical behavior.

To eliminate the competing crystal stoichiometry effect from the

confinement-induced shift in optical properties, Borrelli et al. [11]

performed tests on glass composites doped only with CdS or CdSe. Sub-

banding type oscillations were observed in the optical absorption

spectra of both compositions subjected to heat-treatments of 550 to 700

*C, for 0.5 to 4 hours. The incorporation of Zn into the base glass

resulted in anomalous blue shifts in the absorption edge of the CdS-

glass composite subjected to longer thermal treatments. In the CdSe

glass, TEM analyses showed the crystal sizes to range from 30 to 79 A

after 30 minutes at 600, 650 and 700"C, respectively. Noting the

influence of a finite particle size distribution, and the uncertainty in

literature values for the carrier masses and dielectric constant, the

absorption edge shifts in the CdSe crystals were not used to

quantitatively verify Brus' equation. However, good agreement with the

predicted energy eigenvalues characterized by consecutive roots of the

spherical Bessel function was observed in the energy ratio of adjacent

absorption peaks. Thus the single-particle state transition model was

substantiated.

Potter and Simmons [14] examined both the growth kinetics and

quantum confinement induced changes in optical behavior exhibited by a










CdS-doped glass containing neither Zn nor Se additions. In this case,

the researchers are assured an examination of finite size effects free

from compositional artifacts. Extraction replica TEM analysis was

performed on samples subjected to heat-treatments at 800"C for 5 minutes

to 11 hours. The average particle size development with the cube root

of the heat-treatment time and the size distributions measured support

the use of the Lifshitz and Slyozov model for coarsening. Both optical

absorption and photoluminescence showed distinct quantum size shifts of

as much as 70 meV in crystal sizes ranging from 60 to 425 A in size.

The energy of the optical absorption edge was found to vary linearly

with the expected radius parameter. The effective mass governing the

shift with size is found to agree within 10% of the literature value for

the excitonic translational mass. Thus, good agreement is found, in

crystallites larger than the CdS exciton diameter of 60 A, with the

Efros and Efros excitonic confinement behavior. Relative peak height

changes, exhibited by LO-phonon harmonics in resonant Raman experiments,

with shifts in the position of the underlying photoluminescence peak,

lead to the identification of the PL state as being excitonic in nature.

Free excitons in the bulk semiconductor had previously been found to

serve as an intermediate state in the resonant Raman process [148].

Continuing experiments in the mixed crystal system, Shinojima et

al. [149] also examined both microstructural and optical behavior

development with heat-treatment of CdS/Se doped glasses. Crystallite

sizes range from 30 to 200 A were produced with the crystallite

stoichiometry measured in each sample using the ratio of CdS to CdSe LO-

phonon peak intensities in Raman scattering. The stoichiometry is found

to be CdS0.12Seo.s8 independent of heat-treatment time or temperature, in









79

contradiction to the aforementioned studies. Two peaks in the

absorption spectra are observed and their energy separation found to be

independent of crystal size (as measured by TEM). This is in agreement

with their identification as sub-banded transitions occurring between

the lowest conduction band and the two spin-orbit split valence bands,

similar to the behavior seen in ZnSe colloids by Chestnoy et al. [99].

Shinojima et al., however, found that the effective mass which

characterized the lowest peak energy variation with crystal size was

greater than the reduced mass of the bulk exciton. This is in

contradiction to their incorrect assertion that reduced mass behavior is

expected in this size regime by the Efros and Efros interpretation. The

researchers thus conclude that in the crystal size range 30 to several

hundred angstroms, the effective mass approximation is no longer valid.

Although the effect of a finite potential well is also mentioned, no

attempt is made to investigate substantial Coulomb interaction between

the carriers which is possible in these structures. In fact, the

effective mass they calculate from the data, 0.46 mn is intermediate

between the reduced (0.1 mn) and translational (0.59 m0) masses

determined by them for the measured stoichiometry. A reasonable result

for crystallite sizes falling either in the size range below or slightly

above the excitonic Bohr diameter.

Germanate base glasses containing CdS were examined by Arai et al.

[47]. The crystals were found to grow according to the predicted t1/3

dependence to sizes ranging from 50 to 250 A. Although a 1/R2 type

dependence was observed in the lowest optical absorption feature energy

with crystal radius, no attempt was made to obtain quantitative








80

information from the data due to difficulty in assigning precise energy

positions to the very broad features observed.

















CHAPTER III

EXPERIMENTAL TECHNIQUE



Sample Production: R.F.-Magnetron Sputter Deposition

General Considerations

The R.F.-magnetron sputtering technique is a special case of the

more basic process of cathodic sputtering. The following discussion is

drawn from information in References 150 to 152 which contain full

reviews of the phenomena involved.

In the sputtering process, a plasma, generally composed of a noble

gas, e.g. Ar, Xe, or Kr, is produced between electrodes under an applied

voltage. The process occurs in a vacuum chamber which can be

backfilled, after initial evacuation, with the plasma-forming gas. When

a voltage is applied, the negative electrode (cathode) is bombarded by

gas ions which are produced in the plasma usually through collisions

with secondary electrons ejected from the electrode surface on previous

ion impacts. Materials placed on the cathode are therefore subject to

this bombardment. The plasma is said to be self-sustaining when the

number of secondary electrons generated by ion impact is sufficient to

create enough ions to in turn produce the same number of electrons. In

practice, the ratio of the number of impacting ions to secondary

electrons evolved is greater than one. Therefore, to maintain the

plasma, electrons accelerated away from the cathode must each ionize








82

approximately 10 ions. This requirement can be attained by the

variation of several deposition parameters which influence the energy

and mobility of the electrons and ions in the plasma, mainly the

potential applied to the electrodes and the sputtering gas pressure.

The ionization energy of Ar is approximately 15.75 eV, thus

representing the minimum applied voltage necessary to sustain the

discharge. Increasing the applied voltage insures that the electrons

accelerated through the potential have the opportunity to collide with

numerous neutral gas atoms before losing their energy. If the gas

pressure however is too low, the mean free path of the electron is too

long to interact with many gas atoms before collision with components of

the vacuum chamber, e.g. the walls, or the anode. A high gas pressure

however, generally reduces the energy of the Ar ions and sputtered

species from the cathode through collisions in the gas phase, resulting

in lower kinetic energies at the substrate and corresponding effects on

the morphology of the growing film. Higher gas pressures also

necessarily increase the tendency for impurity incorporation into the

films.

One method to increase the electron-Ar atom interaction while

still maintaining a lower pressure is to confine the ejected secondary

electrons with a magnetic field. The electron is subjected to the

Lorentz force imposed by specifically configured magnetic fields

associated with the cathode, causing it to travel in helical patterns

along the field lines, increasing its path in the plasma region.

Typically the field lines are designed to be parallel with the target

(cathode) surface thus increasing the production of Ar+ directly above

the material to be sputtered. Increased sputtering in this region









83

typically results in well-defined erosion areas on the target. Use of

the magnetron process in the dc sputtering of metallic targets can

reduce the pressures required from 15 75 mT to 0.5 to 10 mT. The

magnetic field presents other advantages including reduced heating of

the substrates due to electron impingement and the confinement of the

plasma itself in the area of the target.

The dc sputtering process is limited in the range of materials

that can be sputtered. An insulating target attached to the cathode

would quench the discharge process due to surface charge accumulation

under the action of the applied voltage. This effect can be

circumvented using ac voltages with frequencies in the megahertz range,

i.e. radio frequencies (R.F.). Under the application of an R.F.

potential, the relative mobility of the electrons and Ar ions in the

plasma, coupled with the requirement that the net current through the

insulator be zero, results in a negative self-biasing of the insulator

surface which serves as an accelerating potential for the Ar ions.

Thus, targets capacitively coupled to the R.F. voltage on either

electrode will develop a self-bias and will be subjected to ion

bombardment. Sputtering from a single electrode is possible if that

electrode is capacitively coupled to the voltage (either the target is

an insulator or uses an insulator to isolate a conductive target) and

the other is directly coupled to the voltage source. The relative

voltage obtained on the capacitively coupled electrode with respect to

the plasma potential is related to that on the directly coupled

electrode by















where Vc(d) and Ac(d) are the potential developed on the capacitively

(directly) coupled electrode and its area, respectively. As a result,

sputtering of the capacitively coupled electrode is insured by making

the area of the directly coupled electrode as large as possible compared

to the former. In general, this is accomplished by making the directly

coupled electrode the system ground, including the vacuum chamber walls

and fixtures.

Whether a dc or R.F. voltage is used, the end result is the

bombardment of the cathode by energetic gas ions produced in the plasma.

In fact, any surface placed in a plasma will obtain a net negative

potential below its surroundings and must also be considered prone to

the same sputtering effects as the target. The sputtering phenomenon is

an extremely complex process, resulting in a number of effects as the

kinetic energy of the ions is dissipated in the collision with the

target. In addition to production of secondary electrons, necessary to

sustain the discharge process, ionized or neutral target atoms and

molecules are liberated from the surface through a momentum transfer

process. The sputter yield, S, defined as the number of atoms or

molecules ejected per incident ion, is therefore related to the bond

strength of the target atoms, the relative masses of the incident and

target atoms and the energy of the bombarding ions. A general

expression for the sputter yield is given by


4m/n, E, 8(,
(, + m U lm,
(Min+ m), U mi









85

where m(it) is the mass of the incident or target particle, Ei is the

incident particle energy and U is the heat of sublimation of the target

material. The second term in the expression is equal to the fraction of

kinetic energy transferred to the target atoms by the incident ion.

Thus, a maximum energy transfer is predicted when the masses of the

participating species are of the same order of magnitude. The last term

effectively limits the growth of S as the incident energy is increased.

The sputter yield increases up to approximately Ei 1-2 keV and then

drops off due to this term.

Neutral particles emitted during ion bombardment are the primary

source for film deposition. In addition, secondary ions, reflected

incident particles and photons may also be produced. Negative secondary

ions are not expected to be a large fraction of the emitted population

but will be accelerated by the applied potential toward the anode,

which, depending on the specific arrangement of the sputtering system,

could impinge on the substrates. Of nominally greater significance,

dependent upon the target material and the energy of the incident ions,

is the reflection of bombarding ions which have been neutralized. Such

particles are not influenced by the electric fields present and

therefore can represent another source of substrate bombardment.

Photons in the ultra-violet and x-ray range may be detrimental due to

radiation damage to substrate surfaces. Of primary interest, from the

standpoint of practical utilization of a sputtering technique, are

further interactions between the target and ions. Even at low

sputtering ion energies significant implantation of ions into the target

can occur, resulting in the energy dissipation being put into heating

the target. Thus, depending on the thermal properties of the target









86

material, attention must be paid to target cooling to avoid cracking.

Temperature changes during sputtering and defects caused by implanted

species can ultimately result in substantial enhancement of diffusional

processes in multi-component targets leading to local variation in

target stoichiometry. In addition, the sputter yield observed in multi-

component targets is dependent upon the identity of the constituents

present, the identity of the matrix in which the constituents exist, the

surface structure of the target, and the microstructure (e.g. grain

size) of the target material.

It should be evident from the above discussion that the sputter

deposition process is indeed fundamentally complicated. Interdependence

of several deposition parameters upon the sputtering conditions attained

and the resulting film chemical and physical properties frequently

require that the deposition conditions for a particular process be

tailored to insure high quality, reproducible results.



Dual Source Sputtering Technique

In the present study, an R.F.-magnetron sputtering process was

employed to produce CdTe-glass composite thin films by independent

sputtering of semiconductor and glass from separate sources. The

sputtering system used was designed by Dr. E.M. Clausen to produce pure

CdS polycrystalline films and later to form CdS-glass composites. The

deposition chamber was custom built by MDC Vacuum Corp. (Haywood, CA)

and consists of a rectangular, 1/2 inch thick stainless steel plate box

measuring 12" X 18" X 18". The Viton 0-ring sealed chamber is pumped

using a 330 1/sec turbomolecular pump (Balzers, Hudson, NH) backed by a

300 1/min two-stage mechanical pump (Sargent Welch, Skokie, IL). A









87

molecular sieve trap (MDC Vacuum Corp., Haywood, CA) is incorporated

into the foreline, preventing oil backstreaming from the mechanical

pump. Chamber pressures were monitored from atmosphere to approximately

1xl0-6 T using a Leybold-Heraeus model CM 330 Combitron (Leybold-

Heraeus, East Syracuse, NY) to control both the thermocouple and Penning

discharge gauge heads employed. Base pressures in the 10'6 T range were

typically attained in the above system. Sputtering was achieved by

backfilling the evacuated chamber with high-purity Argon gas (99.9995%).

The process pressure used during sputtering was held at approximately 3

mT through the use of a mass flow controller (Matheson Gas Products,

Norcross, GA) and adjustment of the conductance allowed through a

throttle valve. A gas flow rate of 20 seem was used.

Two planar magnetron sputter guns (US Guns, Campbell, CA) serve as

separate sputter sources for the semiconductor and glass. Both guns

were controlled by R.F. power supplies (Eratron, Campbell, CA) operating

at 13.57 MHz with a maximum output power of 600 W. Power from the

supplies was coupled to the guns using an auto-load match tuning network

to match the impedance of the plasma produced with that of the power

supply, minimizing reflected R.F. signal. A common oscillator (Eratron,

Campbell, CA) was also employed to phase match the two plasmas.

Two-inch diameter targets were mounted in each of the sputter

guns. The semiconductor source employed a 1/8" thick, hot-pressed, high

purity (99.999%) CdTe target obtained from Cerac (Milwaukee, WI). Pyrex

7740 glass was obtained in the form of a 1/16" X 2" disc for use in the

other sputter source. The 1/16" thickness enabled the glass to be

sputtered at reasonable rates without cracking due to thermal stresses.








88

Film thickness was monitored during the deposition process only on

the CdTe side of the chamber using a IC-6000 deposition controller

(Leybold-Heraeus, East Syracuse, NY). The system utilized a water-

cooled, quartz crystal oscillator sensor head placed near the aperture

of the baffle plate over the CdTe source. To aid in deposition

parameter control and recording, an interactive interfacing program (in

QuickBasic) was written to control the IC-6000 though the RS-232 port of

a Zenith personal computer. The program allowed film deposition

conditions to be reviewed, altered and downloaded to the printer for

hard copy or to floppy disk prior to process initiation. In addition,

the deposition could be started from the keyboard with the computer

continuously monitoring and recording deposition conditions during the

process, saving this data to floppy disk. A print out of this program

is reproduced in Appendix A. No in situ thickness monitoring was

available for the glass source.

In both cases, deposition rate calibrations were performed in

which an array of glass cover slip masks were attached to a silica

substrate held over each source. After deposition for a known time and

at several R.F. powers, the films were removed from the system. Using a

diamond stylus profilometer (Sloan Technology Corp., Santa Barbara, CA)

the film thicknesses could then be measured at several points over the

entire sample surface, yielding both the average film thickness and its

variability over the deposited area. Figure 6 presents a representative

deposition rate calibration curve for Pyrex glass targets supplied by

Esco (Oak Ridge, NJ). The majority of composite films produced employed

a 1 A/sec deposition rate at each target. This rate corresponds to a


















SI I I I I I


4 5 6 7 8
POWER DENSITY (W/cm2)


Representative deposition rate calibration curve
obtained for the Pyrex sputtering targets used in the
production of composite films. (Target thickness -
1/16", diameter 2".)


1.3

1.2

S1.1

1

0.9
z
0 0.8
E
0 0.7
LU
Sf


Figure 6.








90

power density of approximately 7.4 W/cm2 and 0.89 W/cm2 at the glass and

semiconductor sources, respectively.

It is known that the thickness uniformity in planar magnetron

sputtering is sensitive to the geometry of the target and substrate

placement [150]. As can be is seen in Figure 7, the substrates are

located approximately 3.2" directly above the sputtering targets. The

ratio between the radius of the most heavily eroded region of the target

(0.6") and the distance separating the substrates from the target is

approximately 0.2. Given this limiting situation, Glang's work [153]

suggests that near uniform deposition rates across the substrate should

be expected if the ratio of the substrate radius to separation distance

is less than approximately 0.1. (This calculation assumes that

sputtered atoms follow straight line trajectories to the substrate and

that the standard cos(8) distribution of sputtered material holds, where

0 is the take-off angle for the sputtered species taken from the target

normal.) The substrates are kept within a 3 cm diameter region in the

center of the substrate platter during deposition thus some thickness

variation is expected across the substrates. Subsequent profilometer

measurements on calibration samples for both the glass and CdTe sources

exhibit a variation in thickness of approximately 10 % of the average.

Figure 7 contains a simplified schematic of the sputtering chamber

and the components used to produce the sequentially sputtered composite

films. The chamber interior is divided by stainless steel baffle plates

into three main volumes, two containing the sputter gun sources

supplying the semiconductor and glass and an upper region in which

circular substrate support platters are rotated about a common axis

enabling their exposure to each of the sources. Flux from the sources

























A. R.F. Magnetron Sputter Gun
B. Halogen Lamp for RTA
C. Shutter
D. Film Thickness Monitor
E. Substrate
F. Substrate Platter
G. Thermocouple Feedthrough
H. Stepping Motor
I. Voltage Control
J. Rotation Feedthrough
K. Baffle Plate


Schematic depicting the deposition chamber used in
the production of sequentially-sputtered, CdTe-glass
thin films.


Figure 7.









92

reaches the substrates through apertures in the horizontal baffle plate

and can be blocked by manually operated shutters. Modifications in the

system allowed substrate temperatures to be monitored using type K

(chromel-alumel) thermocouples and in situ thermal treatments to be made

with a 100 W substrate heating lamp (Kurt Lesker, Inc., Clairton, PA)

controlled by a variable voltage source. No attempt was made to control

the substrate temperatures during deposition. Substrate temperatures,

however, were observed to increase to approximately 55*C while the films

were deposited.

A computer program was written within the QuickBasic environment

to control the motion of a 200 step/revolution stepping motor during

deposition (see Appendix B). The motor rotated the central post onto

which the sample platter arms were connected thus alternately exposing

the substrates to each of the sputter sources. Proper programming

allowed any overall composition to be produced in the final film based

on the relative exposure time of the substrates to each of the sources

and their deposition rates. In an attempt to improve film thickness

uniformity, the platters were made to oscillate back and forth over each

aperture during the programmed exposure time. This procedure yielded an

apparent increase in the film thickness uniformity by factor of 1.5 to 2

as measured by profilometry.

Fused silica substrates, generally employed in the production of

films for optical analysis, were cleaned using a multistep process.

Substrates were first cut to the desired dimensions and then swabbed

with a soap solution and rinsed with deionized water. After ultrasonic

cleaning for approximately 2 minutes in deionized water, the substrates

were then rinsed and sonicated in acetone. Another acetone rinse








93

followed by a methanol rinse and sonication completed the next step.

After a final methanol rinse, the substrates were attached to the

aluminum substrate platters with tungsten spring clips and spin dried

while attached to the platter. The platters were then placed in the

deposition system for pump-down. This procedure typically produced the

cleanest substrate surfaces observed in the less than clean-room

conditions available.



Samples Fabricated

Sequentially sputtered films were deposited to a number of

different substrates depending on their anticipated use. The majority

of films were deposited to fused silica slides (Esco, Oak Ridge, NJ),

enabling their optical evaluation. The characteristically low

refractive index of the substrate insured the planar waveguiding

behavior required of the films for some studies. For the purposes of

TEM analyses, very thin films (700 to 1000 A thickness) were deposited

to single crystal, cleaved NaC1 substrates or Formvar/C-coated copper

grids. Films ranging in thickness from 0.5 to 4.5 pm were produced for

optical analysis.

The deposition system discussed above allowed two sets of films to

be produced simultaneously on each of the substrate support platters.

Thus, the film compositions on either platter were complementary, i.e.

if a 10 vol% CdTe film was produced on the right platter, a 90 vol% film

was automatically formed on the left. As a result, a relatively wide

range of film compositions were produced in the study. Films containing

0, 0.5, 5, 10, 20 and 30 vol % CdTe and their complements were produced.

The overall semiconductor content was altered by changing the relative




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