r. .'¡ r l- 4/;.,i '<'r/ot r\l_ ¿3 , o't LHlY. è.

\/^ .-rJ\n'14,¡.1-o'

OPTÏCAI PROPERTIES OF TITÏN

VACUI]M DEPOSTTED

SEMICONDUCTTNG FILMS.

BY

ROBIN ERIC DENTON

B.Sc. HorlB. (l¿etaiAe)

A TÎIESTS

SI.IBMTTTED FOR THE DEGREE OF

DOCTOR OF PHTLOSOFHY

rN lfrE

DEPIiRTMENT OF PHYSICS

IJNT\IERSITY OF ADELAIDE

DECEI{BER L?TI,

TABLE OF CO}ITEIITS

SUMMARY

DECLATIAÎION

ACKNOI,f LEDGEMENTS

CJ{APTER 1 IN'IRODUCTIONNotationMeasurements on BuIk MaterialsUse of fhin FihnsMethod.s of Determining the Optical Constantsof Ttrin Absorbing Fil¡rs4.1 Polarimetric Method.sI+,2 Schopperts Methocll+.3 Spectrophotometry at llon-norrnal fncid.encel+.1+ Photometric Llethod.s at Normal Tncid.enceAirns of the Present Inveetigation

2 EXPERIMENTAI APPATATUSSpectrophotometer for l,{easuring Reflectanceand Trs.nsmíttance

Generel DescriptionLight SourceOptical SysternMeasurements of Refleetance and TransÌnittanceSpecimen Hold-er

2,!,6 Refractive Ind.ex of and. Absorption in the Substrstcs2,2 Beam Detection

2.2.L Light-sensitive Detectors2.2.2 Arnplifico,tion and. Rectification of the lleam Signal

2,3 Preparation of ¡'i1¡rsEvaporation of Ger"¡mniumSubstratesSubstrate CleaningPr.rmping UnitSubstrate Support and Oven

Z,)+ Measr,rre¡îent and. Calculation of Filn Ttrickness2.1+,I Optical System for Measuring the hlavelengths

of the Interferenee Orders2.\.2 Ifigh Reflectence Coo,tings2,1+.3 Caleul-ation of FiI¡: Thickncss

CÎ{APÎER 3 CAICULATION OF I'IIE OPTTCAL CONST.^,NTS 0F A TIII}IASSOFBTNG STNGLE-LAYER FTLM

3.1 fntz'od.uction3,2 Ttre Equations for the Optieal Const¿mts

3,2,L Reflectance and. Transmittance Equations3,2,2 Corrected Transmittanee liquation

3.3 Solution of the Reflectance and Trensnittance Ectruations

1.l_I,21.3L.4

PAGE

J-L

13131l+

$16ITITTT1B191920202I2IIL

23c22L+

I¿

3h

h

5,6

7B

1ol.1.1o

!,,

gHAPTER

2.rì^

2.f.12.r,22.1. 32.I ,'l+2,r,5

2.3.f2,3,2?..3.32.3.)+2'3'5

2T2.7

282B2930

T/IBLE OF CONTENTS ( cont. )

Ttre Equations (r+n)/r ana (r-n)/rSolution of the Equations (f+n)/f a¡ra (r-n)/rfor ni and. k1

3.5.1 CalcuLation of n1 and h1CalcuJ-ation of the Error in the Solutionslhe Effects of Fifn Ttrickness on the Calculationf n1 end k1

The H¡lothetic¿ìJ- tr'ilxlCalculation of n1 and k1 for the HypotheticaJ. Fil¡nFll-n Thickness Too LargeFiltr Ttrickne..sri Tr:o SmallEffect ofi Previous lilorkAccurate Dctemination bf the Optical Filn TliiekneesEffects of Errors in R and T on the Caleula'1.,ed.F'1J-m thLclçness

3.7.8 Very ihin I¡íl-ins3.8 Approximate Filn ltríckness3,9 Deteniination of the Optical Constants fron the Measured.

Transmittance and. Reflectance from the Back Surfact'.3"10 Alternative Denivation of the Equations for

(ItR)/t ana (rtnt ) /rCT.I.APTER I+ SUN¡'¡,CE LATERS ON ]ÍTTN FILI\ß

]+.1 fntrod.uction\,2 Reflecta¡rce ancL Transmitta¡rce Equations for a

Tr,ro-Iayer Filnh.3 The Hypothetíeal Filns)+.l+ Effect of Using the Single-Iayer Equations

l+.l+.1 Systern 1l+.h.2 System 2

4.5 Solving the I'r¡o-layer Ecluations[.6 Application to Rea]" Fihns

\.6.1 Examples for Gerrna¡riu¡r and. Selenium\,6,2 Effects of Errors in R and T on the Calculated-

Filn and. Surface Layer Thicknesses)+,T Explicit Expressions for (ftn)/f

l+.7.L Exact Expressions for the Case k1=k3=0. Systenr 1l+,7,2 Exact E:cpressions for the Case kr=t<t=¡. System 2)+.7,3 Error Änalysis fcr the Case kt=kj*O. Systen 1

l+.8 Use of the Single-lr.yer Equations ior-a 'fbo*leyer Filrnl+.8.1 System 1. First Layer Very Ttrinl+.8.2 Systen 2. Second Layer Very fhin

]+,9 Choice of fncorreet Refractive Index for f irst Leyerl+.10 More Than One Very Ttrin Layer on the Surface!.11 Surface Layers on Gernanir¡:n Filns

l+.11.1 Surface Topograph..¡ of Gernaniun Films\.1-1.2 Treatment of the Surface Layer

)+.12 Surface Layers on Selcnir:m Films

PAGE

3.l+3,5

33

6

7

32

33353T

3939l+o

hohrl+f,+Z

o3,73.73,73,73.73,73,7

123l+

,6

T

52,+55,,,65T596o

\zh3l+l+

\¡

l+t

l+9

l+e

6t62.

636',t+

6\a6j66676g

T2T3T,+

76

7r

üA3LE OF CONTENTS (cont.)

ll"13 Conclusions

CH/IPÎER 5 OPTTCAI, COIISÎANIE 0I' SELEI{IUM FILI\4S

,J Introduction,.2 Incorrect Solutions

,,2,L lrlissing Solutions5,3 Reflectanee and Trans¡nittance of Selenium FiLns,,1+ Corrected. Filn Ttrickness,,5 Refraetive fndex of Seleniur,t Films

,,5,L Errors in the Ca1eulated. $ol-utions5,5.2 Comparison r,¡ith Previous trüork

5¿6 Absorption Inclex of Selenium Films5rT Absorption Coefficient of Selenir¡n Fituas5.8 Absorption in Seleniun Fihns

PAGE

929292

9\979T99oc)

100

111

111113111+

1r5

rf7

77

T9T97980BzB3BhB,B686Bt8?

CTlAPTER

6.r6,26"3

6.7

CIIAPTER

T.I

6.1+6,56,6

6 optTCAI PROPERTTES OF GERT4AIITUM FIL},ßIntrod.uctionFozrîs of Genranium FílmsPrevious Colculations of the Optieal Constantsof Germanir:m FitnsReflectonee and. Transmittance of Gerrranium FilnsCalculated. Fi1re and. Oxid.e T.hicknessesOptical Constants of Gernanium Films.1 Amorphous Filns,2 Very Thin .Anorphous Fi1¡ns.3 Fil¡ts Annealed at 35oo0 and Films Deposited-

on Substrates at 3rOoC6,6|l+ opticJ õonstants of Fiþns A:rneal-ed et 65ooc

Absorption Processes in Gemra¡riurq FilasBasic TLreoryDirect TransitionsIndireet TransitionsAbsorption in Crystalline Gerrnanir¡n FilmsAbscrption in Ä:norphous FilmsLong l,Iavelength Absorption in Gezno,niu¡n Filrns

7 CONCLUSIONSOn the Ðeter"minetion of the Optical Constantsof thin Fil-nsOn the Optieal Consta:rts of Selenir:¡n FilnsOn the Optical Constents of Gerrna¡riu¡n l-ilmsFuture Work

6,66,66.6

101IO2102IO2lOh106ro71081.10

6,7.L6,7 ,?_

6.7.36,7.\6,7,'6,7.6

7,27.37.l+

ÄPPENDTX A PARTTA-L DnnïVATTV!,lg OF (rtn) /r FOn rHnSrNGLE-LAYEIì EQUAllrONs

AppriNDrx B DERTVATTON oF (l.tn)/r ¿no (rtnl )/t rnoi'l tuuFOFJ{ULAE C,]TVEIü BY ASELES 120

APPENDIX

APPENDIX

t4Þ._1,_u ,o¡' C9$SIL (cgrt J,

c pARTrÁt DERTVATTVES 0F (ltn)/r rOn mnTIIO-I"AYER EQUATIONS SYSTEM 1 (t<1=tr=6)

D MEÀSURED RTFLECTAI\ICES AIüD TR.ANSMITTANTCES AI{DCATCUI,A,TED REFRACTIVE AÑD ABSORPTTON INDTCESOF \,:[IREOUS SEI,ENIUM FTLI\4S

PAGE

L23

137

150

T2B

APPENDTX E MEAÉURED REFLECTANCES ÁT{D IAA}TSMITTANCES A}TD

CATCUI,ATED REFRACTTVE AND ABSORPTTON INDICESOF GERI\4ANÎUM TIT¡4S

RM'EREI{CES

SUMMANY

TLre thesis begins with a brief introduction to the problem of

cleterrnining the optical constants of thin films, with a critical

review of the various techniques for <i.ete::rnining these parameters

in a¡r effort to arrive at a suitable method. applicable to the

absorption ed.ge of semicond.ucting fitns. It is shor¡n that the

measurement of normal ineid.ence refl-eetance end. transnittance is

potentially the most aecurate method..

In Chapter 2 is cl.escribed. a speetrophotometer which wos designed

to enabLe the absolute refl.eetences and. trensr¡ittanees of thin filns

to be measured. with a¡r aceuraey of tO,OO2, The thieknesses of the

fil¡rs were measured by the method. of Fringes of Equal Chrornatic Ord.er,

and. the apparatus and necessa,ry computations for this ere also

described. in this chapter.

Ttre proble¡n in the past with nornal incidence reflectance end.

trans¡dttance measurements Iay in the enorrnous eornputations necessary

to extract the optical consta¡rts, since rnultiple-beam interference

effects must be taken into account resulting in conplicated. equations

for the refleetance and transrnitta¡tce. In Chapter 3 is d.escribecl a

mrrch sirrplified. ¡rethod for the exact d,eternination of the refraetive

ind.ex and. absorption index of thin fílms from measured. normal

ineidence reflectcr¡rces and transmittances. Ttre ¡aethod. requires

níni¡ral- programming and. the computing timc is at least a factor of 10

faster thon previously reportecl. Beceuse of the perioclic nature of

the reflectance and. tra^rìsníttance equations, nultipJ-e solutions for

the opticaL conste¡rts eccur, and. an analysis of hypotlietical fil¡ls is

made in a^r¡ effort to 6.etezmj.ne which crf the eolutions is correct,

An unambiguous choice can only be ¡rnrl.e if rnoasurements ere têliêlr over

a witle wavelength range, and. it is shown how the behavior¡r of the

calculatecl solutions Ìrnd.er smal-l changes in filn thickness enables

the thickness to be tieter.nínetl with precision without recourse to

its explicit measurêr€îtr

A najor attvantage of nonnal incid.ence measurements ig that the

effects of the surface are mininized. Ilowever it is shown that

certa,in behaviour of the ce,lculated solutions is aseociate¿ wittl

oxitle layers or irregularities on the surface of the fil¡ls. This

behavior:r is d.escribecl in Chapter l+, anct it is sholrn how the effects

of the surface nay be eliminated. by consid.ering a two-layer film.

If the surface layer is of known refractive index, it is ehown how

the optical constants of the uncterlying filn antL the filn ancl

surface-layer thicknesses may be accurately d.eterrnined from the

measurect refleetance a¡rcl transuittanee of the system.

The above enalyses revealed that a previous worker, in his

cletermination of the optica.l constants of selenir¡m fil¡ls, chose

incorrect solutions from the nultiple values that existed. IIis

ehoice of the wrong aolutions arose prirnarily from using incorrect

filn thickness values, In Chapter 5 the optieal constants of

selenium filns are recalcuÌatecl fro¡r his data, and the results are

interpreted in te¡as of a qualitative semieonclucting bond mod.el.

One of the most elctensively stucliecl semiconductors is gerrnanium,

yet the reeu.Its obtainecl by various authors for the optical eonstants

of thin filns of this naterial shoty ft ld.e discrepancies. It was

appropriate therefore to apply the theory developed in chapters 3

and. l+ to d.ete¡lrine accuratcly the optical constents of germarrirrm

films. [he results of this investi6o+,íon ore presented- in Chapter

6. Anorphoue films are considered. in greater d'etail, but fil-ns with

various degrees of crystalline perfection are investigated' for the

purpoee of eomparigon. A model for the observe¿l absorption in

germaniun fiLns is discussed. Id.eas for ft¡ture work conclude the

thesis.

pu0,LA3ATI0.I\T

flris thesís contains no ¡natêrial whích Ïras been acceptecl

for the aÌ¡afd of arry other degree or diplona ln any University ædr

to the beàt of tbe authortE knoùIed.ge and' beliefr eonta-ins no

naterial previousl]¡ publiBheô or written by another pergont except

where due reference is mad.e in the text of the thesis.

(n. oPrtlol)

ACKN Olv'LED GElvlEl[TS

The author gratefi:J-Iy acknowled.ges the fol I owing people and

organisations for their aSsista¡ce during the course of this work'

¡lr. T. Spurr of the Physics Departnent workshop for the construction

of the spectrophotometer.

!fr. A. Ewart for his vã,luable technical assistsnee.

tr4r. S. tr{eIIs for photographing the d.iagrems.

I{rs. K. Possinghatr for typing the thesis.

Dr. D. McCoy a¡¡d Messrs. R" Goocltrin and T. McDonnell for the many

thought Provoking discussions.

Íhe Ractio Reseaych Board. Of Australia for fina¡rcing the project.

The University of Ad,elaid.e for the ar,¡ard of a University Research

Grent (1968).

Ttre Connonwealth Department of Eclucation And Seienee for a Post-

graduate Award' (]-969-7t),

Finally the author wishes to thank his Supervisor, Dr. S.G. Tomlin,

for his guictance encl etinulating critical diseussions during the

course of this work.

o9¿t

61.

''d-l nCHAPTE]R 1.

INTRODUCTION

A feature contrnon to all semieonductors is the repict increase Ín

optical absorption which occurs over a, smel1 energy range when the

absorbed. radiation has Bn enerry roughly equal to the enerry gap of

the semiconductor. Itris is called. the absorption ed.ge of the material,

and. is due to the onset of aÞsorTtion in r¡hich eleetrons are raised

from the valence band across the forbidcten gap to the conduction bancl.

A study of the position of this edge and any structure it may have

will yielct infomation about the energy gap ond. the properties of

those el-ectron states just above it at the conduction ba^nd ed.ge a^nd.

belorr it at the valence ba¡rd. ed,ge. ntris tytrre of inforuration is

useful as it is these states elose to the eonduction and valence band

etLges which d.eter¡rine the electrical properties of the semiconductor.

Consequently, measurement and. analysis of the absorption ed.ge spectrum

of a semicond.uctor is an inportant feature in the study of its

properties.

2.

f. 1 urATr9N.

The optical properties of a materiel may be described- by its

complex refractive index, IT. The signifieance of N is t;nat' a plane

electromagnetic we,ve of frequency v and TÍavelength À propogated'

through the material in the x- direction has the fo::øt

y = Aexp{zniv(t-Nx/c)} r't'r

c being the velocity of light ín free ep&ce'

N may be written in real and' inaginary parts in the forr

N=n-ik I'L?

where n is calIed. the refractive ind.ex and. k the absorption index'

fheee tno quantities constitute the so-called optieal constants of

the med.iumo In te:ms of n arrd' k, equation I'1'1 becomes

y = A erç{-2nvkx/c} exp{eniv(t-nx/c)}

In tems of the wavelength À

v = A exp{-zntcx/r} exp{zni(ct-nx)/À} l'r'3

fhe first term of equation 1.1.3 represents the attenua'tion of the

anplitud.e of -the electromagnetic lüave on passing through the medium'

The square of the factor, exp(-lrnkx/l), tn.refore represents the

attenuation in i.ntensity of the w&vê¡ The factor K clefíned' by the

relation

r = \nk/À 1'1'h

ie temed. the abeorption coefficient of the material'

Determination of the optical constants over a wid'e wavelength

ransemaythereforebeerpectedtoyield,valuableinformationon

absorption processes in materia'ls'

3.

I. 2 ME.ASUREMENTS ON BULK MAIERIALS.

Measurements of the refractive indices of br¡lk transparent

materials present no serious d.ifficufties. lltre cffect on a transnitted-

light bea,rn by slight surface clisorders, for exa¡¡p1e, fron polishing

or from surface fiLus, is generally negligible if the bea¡r traverses

a total distance in the medir:m of the orrler of millimetres.

Trtre study of absorbing materials, particularly where the

absorption is 1arge, presents a much more difficrrlt problem.

Measurements usually entail examination of light reflected- from the

specimen surface. Since the tlepth of penetration of the light wave

may rvel-I be much less than a. visible vavelengthr it is apparent that

the behaviour exhibited. by a specimen depends almost entirely on the

state of the surface which may be conpletely unrepresentative of the

st¿te of the interior of the specinen. Polishing usually produces a

surfaee gtructure (SiefUy layer) in whieh the nornal erystal structure

of the materiaf is consid.erably disordered.. the presence of a thin

film remaining after polishing, or fonned in so¡re other way, will

also alter the meosured. values. These problems have been overcome

by cleoving the specimen, vhen possible, and. tal-.ing me&surements of

the cLeaved surface. The absorption ind.ex is usually found from a

succesgive series of meagrrrements on a crystal v¡hich is progressivcly

thinned. Ttris procedure ofben requires specinlised grind.ing and

etching techniques.

In ad.tlition, if rÌea,surements are taken at other than norrnaf

incitlence, the polarised. state of the incid.ent and reflected waves

¡rust be known and. taken into account. Sueh measure¡aents are not

easily made.

l+.

1. 3 USE OF TIÍIN FILI\4S.

It was thought originalJ-y that the optical properties of

absorbing bulk materiats could. be stud.ied through the use of thin

v€rcuun ttepositecl fitns. The aclvantages of filns are that they ean be

mad.e much more r:nifom in thickness and. nany tiraes thinner than the

bulk material, peroitting transnission measurenents to be mad'e even

when the absorption index is large. Ítle whole d.epth of the filn is

sa.ropled, ¡dnimising the effect of any surfaee irregularities.

It has been subsecluently found that the optical properties of

films ofben depend.-^d. on the fil:n thickness and method. of preparation,

and were different from those which had been fountl- for the bulk

material. Instead of an aid. in the d.etemination of the properties

of butk rnaterial, the study of thin films has beco¡tie a fielcl of

research in its own right.

Although vacuum depositer] fil-ms are easy to prepare for optical

measurenents the anount of work which has been d.one on them has not

been extensive. Tlris is because the courputation involved. in

calculating n and. k is much nore laborious than for bulk speeimens.

Once the film thichness bccomes comparable r¡ith the r+avelength, the

effect of multiple bean interference must be includ.ed in any reduction

of data. lüorkers in the past have for.¡nd this problem so fo:nrid.able

that n and k have usual-ly been deterrrined from approxiuate ec¿uations.

1. I+ I\MTIÍODS OF DII1ERMINTNG OPTICAL CONSTANTS OF fiTTN

-ABSORBING FILN{S.

Extensive reviews of varior:-s nethod.s of clete::roiníng the optical

5,

pafffiteters of thin trarrdparent or absorbing fiL¡rrs have been given by

Rouard. and Bousquef (Ð65) and Abe1es (1963). In this section a

brief critical sumary of the various ¡rêthod.s of d.eterrniníng the

opticat constants of absorbing filns is given. Since the present

investigation entailed tlie exa¡rination of the absorpticn ed.ge of tlrin

film senicorrd.uctors, a suitable nethod. must enabl-e n and. k to be

detennined. over a wid.e wavelength range¡ from the transparent region

to the rnoderately high absorbing region.

1, l+. 1 POLARIMETRTC METI{ODS.

There o.re a nr¡mber of variations of this experinent involving

cither the measurencnt of the absofute intensity of the reflected'

components of a plane potarired. bea¡r of 1ight, or tÏre relative

phases and. intensi'ries.

Measurenents ere carried. out at non-DormBl incidence so that

problems arise in measuring occurately the a,ngle a¡rd. intensities of

the reflected componerrts.

l,{ost of the results obtained. using polarimetric methocLs are

inconsistent. The grenular Surface of filns as revealerL by electron

m:icroscopy and the inhonogeneity and. the existence of tre¡rsition

layers shom by Bousquet (tg>l) meke ít exceptionaf for the theoret-

iee,t conclitions on r¡hich the polarimetric methods ore bs,sed to exist'

1. \. 2 SCHOPPERîS METI{OD.

Schopper (tgSZ) shcv.rd. that it r^tas possible to eompute n, k

anrl the fil-m thickness, d, as firnctions of the eomplcx e.uplitud'es of

reflectance from the air end. substrate sid.es of the film and

6,

trons¡n:ittarlee, using eertsin approxiraatíons. This method. is

excellent in theory but it ínvoives six measurements, of whieh three

affe mes"suxements of phase chânges, these being d.ifficult to make

â,ccurately.

OnIy four of the quantities c8tl be measured" without ambíquity

and with reasonable accuraey.

1. \. 3 SPECTBOPHOTOMETRY AT NON-NORMAr, INCIDENCE.

This nethod. involves the measurement of the reflected. and

transüitted. intensities of radiation incident obliquely on the fi1m.

As vith polarirnetric methods the surface of the film will strongly

influence the results. Apart from surface irregularities the

possibility of anisotropy must also be consideredo Sehopper (tg52)

has noted. that if thin fil-¡ns produced. by theraral evaporatic¡n are

anisotropic they probably take the forrn of r:niaxial crystals with

the optic axis perpendicular to the plene of the film surface. It

is assumecl here that the d.ei:osition occurs at nonnal inciC.ence,

so that there is no privilegecl d.irection in the surface of the fil:n.

Tf the fil¡o behaves like a uniaxial crystal with its axis normal to

the plane, the anisotropy wou]d. affect all measurenents mad.e at

oblique incidence.

On the other hanå, onisotropy cannot affeet neasutements mad.e at

nonnal incidence. Rouard (tgS6) tras stressed. that in these eircr¡n-

stances norm&I ineid.ence metlrods are cfearly superior.

,|,

1. l+. )+ PUOfOWtnfC AT }IOTII"TAÍ, cE.

There are several variatione of this technique.

(f). Measurement of the absolute reflected. intensity at no:mal

incidence from the eubstrate siae (nl) ena air side (n) or the filn

together with the absclute trensmission (T).

Tlris method. wor¡ld. require an extensive arnount of computation.

tr\¡rthermore in the low absorbing region the two reffeeteci intensities

would. be nearly equal, resulting in poorly d.efined curve intersections

in the calculation of the optice,l constants'

(Z). Measurement of the absol-ute refleeted. intensity at nortal

inciclence from the substrate side and- air sid.e of tÌTe filn together

l¡ith the fil-n thickness.

Conputation for this m()thod. would. be less cxtensive than for

(f ) since the fil-u thiclcness cloes not enter into the equations es all

unknown para,roeter. However this nethod. is again unsuitable except

in the highly absorbing region. The optical constants, n and. k, are

d.eterudned, by plotting n os a function of k for the measured. values

of R a¡rd. RI, the interscction of theee t¡¡o curves giving the solution.

In the low abeorption re6ion these tvo eurves exhibit near tangential

intersections, giving riee to large errors in the calculated.

qusntities for sma1l errors in the measured quantities.

fhue bcth method.s (r) a"nd. (Z) are unsuitabl-e for the present

investigation where lÍe are particr:larly intereeted. in the absorption

ed.ge "

(S). Ioleaeurement of the abgolute reflected intensity at normal

ineiaience from the air (or substrate) siae of the film anct measurement

BO

of the abeolute tranemitted. intensity at norrnal incidence, together

witt¡ ttre tifn tnicknegE.

This nethod. was eonsicl-ered. potentially the most aceurate.

lvleagufements at no:mAl incidence mininise thè eff,ects of surface

irregularities and transmiseion üreasurements minimise them further

still since the whole thickness of the filü is involved' Un1ike R

and. Rt, deterrnining n and k from R (or RI) a¡rd T doeg not suffer

from unstable curve intersections in the transparent or low absorption

region, except in limited I'avefength regions, these rcgions d.epentting

on the thickness of the filn.

1.5 AI}4S OF TITE PNNSENT INVESTTGATION.

Ihe original ain of this investigation was to d'etennine

accurotely the optical constants of a number of semiconducting filns

near the fi,¡nda¡rental a,bsorption ed.ge, from measurements of the nomal

incid.ence reflectance and. transmittance of the filns. Initially it

was proposed to study germaniurn fi1¡rs as these filns can be preparecl'

with various degrees of crystalline perfection.

In C'hapter 2 Ls d.escribed. the d.esign and construction of

a,pparå,tus for the mea,surement of the absolute reflectance end

trenenittance of thin filns. Also in this ehapter is described' the

apparatus ancl consequent calcrrlations for the d.eternination of fífun

thickness b1' nultiple bea,m interferometry. Apparatus for the

preparation of vacuu¡n d.epositeð germaniunc filns is also clescribed.

Ihe accurate calculation of the optica-l constei¡rts from the

measured. reflectance, tranismitta¡rce and. thiekness of thin filns is

9.

1.hen consid.ered.. Until now the computation necessary for determining

n a¡rcl k was lengthy a^nd extensive. In Chapter 3 a much simplified'

rnethod- is d.escriberl Vhere n nnd k are cLctermined, not frorn B and. Tt

but from the firnctions (l+n)/fl ana (f-n)/f. Although a hi¿5h speed

com¡-ruter is still required, the progra.nøiing is mininaf and- computing

tirne at least a factor of ten less than previously t'epor.'uetl.

rt soon became apparent that a thorough investigation of the

catculation of n and. k fronr R and' 'I r'ras needed-. This arose because

ofton several soLutions satisfied. the reflectance a¡rd transnittance

equetione (f+n¡/f ana (f-n)/T, and en unanbiguous choice of the

correct solution could not be ttad.e. This investigatic,n is also

d.escribed in Ch.r,pter 3, firstiy ín relation to hypothetieal fil¡os

so that all the properties of the film are knor,¡n exactlyr and

seconday to show horç the rcsults from this analysis could be applied.

to real filns. A eonsequence of this investigation ie the d'evelop'

ment of a method r+herebynprovid.ed. the film is sufficiently thj.ck to

extribit at teast one turning point in the reflectance and. trans-

mittance curves, the opticcl constants tcgether with the thiekness

of the filn nay be detc::.rinecl r,¡ith precision.

Other anomaJ-ies in the calculatecl solutions are shown in Chapter

l+ to te due to oxid.e layers ancl structural írregularities on the

surface of the films. In many cases, and in particular gerrnanirm,

the oxid.e is transparent in the region of the absorption ed.ge of the

r,md.erlyíng fiIm. A technique is d.eveloped in this chaptcr wherebyt

provir1ecL the oxid.e layer is transparent and. its refractive ind'ex

known, the opticaf constsnts of the und-erlying fihn ancl the fiha

10.

and. oxide thicknesses may be aceurately determined" 'Itre analysis

is applicable to any two-layer system provided one layer is trans-

parent ancl ie of knoçn refractive index trhile the second. Iayer is

absorbing over at least a part of the wavelenglh range of measure-

ments. The two systems: transparent film - absorbing fil¡t - substrate

and absorbing fi]ln - transparent film - substrate ore shom to be

read.ily distinguishable. It is also shom that, under certain

eonditions, surface irregularities may be treated in a siníl-ar

matr¡ner.

Tkrere is sone d.oubt reported in the literature as to the

accura,cy of measuring film thickness by nultip]e beam interferometry

(Rouard antl Bousquet, 1965). The analyses given in Chepters 3 and' l+

have enabled. the thicknesses of the films to be d'eterminett without

recourse to their explicit measurement. It was found. that in the

case of seleniu:n and germeniwr filns the measured film thicknesses

agreed r¡ithin the experimental eTÏor to those colculated.o thus

dispelling fears that the measured fih thickness nay be seriously

]-l1 êffOTr

Ítre a^nalyses also revealed that previous calculations of the

optical- eonstanta of selenirrn fifuns (Ca¡rybel-l , :-.968) were ineorrect,

ond. in Chapter ! these quantities are recaleulated- from the d'ata of

Calnpbe1l. Íhe results are then briefly discussed''

In Chapter 6 tne optical consta¡rts of gerrnaniurn fiLms are

presented. l,norphous films are considered in greater detail, but

fil¡ns with various d.egrees of crystalline perfection are considered'

for the purpose of conparison. A nod.el to explain the observed

11.

absorption is then Presented.

Ideas for fr¡ture work conclud'e the thesis.

L2,

CIIAPTER 2,

EXPERIMEN'IAL APPARAIUS

2. 1 SPECTROPHOTOMETER FOR I'EA,SURTNG NEFI..T]CTANCE AND

A spectrophotometer for mes,suring the trans¡rittance s"nd-

absolute reffectance of thin filns Ì{as d.esigned. by the author aJrd

¡uilt in the Physics Department worhshop. A photograph of the

apparatus is shorøn in Figure 2.1. A nr:mber of ty¡res of systens

trere consid.ered. before the prcsent one was d-esigned end' most of the

systenatic errors associated. vith previous instruments have been

elirninated or rninimized.

lüith the existing spe':ctrophotometer (Campbel1, f96B) ttre

measurement of reflectance recluired' a calibratect rcference miriort

which was.bed.ious to ca.librater ffid any charrge in its reflectance

bet'ween neasurements was r:nknolrn.

An elaborate spectrophotcmeter has been buil-t by Bennett and'

Koehler (fg6O) which has the advantages that a calibrated mirror is

not required., and the light bear:r position on the detector is not

affected. by changes in the dispersion of the substrate at d-ifferent

vavelengths. However the positions of the specimen for measuring

the reffectance and transmittance require the specimen hold'er to

be mor¡nted. in different positions. As the present system was

required to be operatec under vacuum l¡ith the specimen cooled to

near liquid. air temperatures, if d.esired., such movement woufd- have

bcen ùifficult to achieve"

The Strong - type reflectometer (Kuhn and Inli1sonrlp50) appeared

to be the most suitable instrunent and. the present systcm was designed'

Þ t

o

GI

c a a v

q ry

0 ei q

46, a

oG

n¡ H'

r_c o N)

Þ"

I { ,t j

þ-

13.

on this basis. The disadvantage of this type is that the change in

ùispersion of the substrate at d-ifferent vavelengths causes the

Iight bean position on the d.etector to alter. In the strong - type

refLectometer this was partially compensated for by enploying an

integrating sphere. The present system r¡s,s d.esignerl eo that this

error could. be elininated. OnIy tran6verse movement of the specimen

hold.er \¡as required. which is read.ily achieved under vEì,cllüú.

2. !. 1 GE"¡IERAL DESCRIPTION.

The measurements of reflectanee and transmittence were made at

near normal ineid.ence, the angle of incidence bein6¡ about 50 which

resulted. in a negligible ê1.1.or¡ tight frorn a Hilger-ïratts prisn

monochrouator wo,s mechenically choppefand., after reflection or

trensmission by the specimen, the intensity of the ehopped- bean was

measured by a light-sensitive d.etector. The square Ì¡ave signal from

the d.etector was nrnplified., rectifiedr sxnoothed' and' recorded on a

<Ligita1 voltmeter. fhe square of the reflectance and transmittance

wa,s measured. which results in greate? &ccllr&cJrr

2.L.2 LIGHT SOIIRCE.

Tnne light source consisted. of a 1oo - watt quartz - iodicle

IÊmF powerecl by a 12 volt i[.c. rep¡ulated. supply. A concave mirror

was positionec. so that the image of the lanp could. be focussed' on

the entrance slit of the monochrcm&tcr' Af'ber an initial va:m - up

period. the intensity of the light source was constant.

* the ctropping frequency was IOO Hertz.

1l+

2,L,3 OPTÏCAI, S TEM.

Ttre optÍcal system is shonn diagranatically in Fipçure 2.2.

tight fron thc monochrornator, M, was nrechonically ehopped. ond. passecl

through atljustable horízontal a¡rd vertieal slits to l-i¡ni'b the spread

of the bea¡n. Coneave mirrors MI a¡rtt Ì42 focusged the beem onto the

plane mirror M3. After reflection from M3, mirrors Ml+ anci M5

focussed the bea¡r onto a light sensitive detector D. Ttre nirror

system vas encloaed. in a vaeuum cha¡nber so that experiments could. be

earried out at Iot pressures and temperatures if required.

Ttre concave ¡rirrors were housed. in nritted. ahutiniu¡n bloeks.

Each block'was attached. to its support by three spring-loaded position-

ing aerews to a mount to aseist in the alignment of the opties.

L¿tera1 movement of the blocks was possitrle, and. each nirror nount

could. be rotated. external-ly via rotating trotr - ring seals. t?re light

beem was positioned. on that part of the d,etector whieh 6;ave moximurn

output sipgral. At each r.ravelength the bea.r:r position coul-d. be altered.

read-ily, by a slight rotation of mirror M3 or Mh, to give manimum

output signal fro¡r the detector, ensUring the bea¡a fe11 on the sarne

part of tbe detector a¡rd. consequently elininating the error that

arises when the bean position alters due to different d.ispersion in

the substrate at ðifferent wavelengths.

For the purpose of ¡neasu-ring reflectance (section 2.1.h), mirror

M3 could. be rotatcd externei-Iy about a perpendicular axis. Ttris

resul-ted in the sane eirror being used. for the reference and specinen

siggrals, elirrinati^g (a) ttre errrrr thaÈ eculd. arise due to d.ifferent

reflectivities if two clífferent mirrors were used., and. (t) ttre neea

AOJUSTAôlE HOtlzoNl^LANO VERIICAT SLITS

ðEÀA CHO¡FER

t\A4

M

DIAGFAM CF THE OtrTICAL SYSTEM

Figure 2"2

MONOCH ROMAIOR

M

D

2

DE'TECIOR

15,

to ealibrate a reference mirroro

Al-1 the mirror mormts were equipped. with planetary drive geaÎs

connected to worm a¡c1 wheel gears so tþat fine ad.justr¡ents could be

na,d.e.

2. I. l+ MEASUREMENIS OF REFLECT.êJ,ICE Af'lD TRANSMITTAI\ICE.

For measuring reflecta,nce, the referehce signal was obtained-

by allowing tTre bee¡n to traverse the system as shown in Figure 2.3'ã1

where the incid.ent bea¡o was reflected by M3. Ttre specimen vas then

xc.oved into the path of the bea¡n as shown in Figure 2.3.br and mirror

M3 rotated. l8oo to the position shown, eueh that the bea¡l was

reflected. firstly from the speeimenr then M3 and. again from the

specini:.en, and. the corresponding siSna1 recorded'.

Let Io be the incid.ent intensity of the light beam and' Rlt

R2, R3r Rl+r R5, RF the reflecta,nces of mirrors I[Lr M2, M3, Ml+t M5

and. the filn F respectively. Ttre intensity of the reference bea¡:r is

thus:

Io. RI. R2. R3. R\. R5.

onri. that of the bee,m afber reflection by the film is

ro. RI. R2. RF. R3. RF. Rh. R5

the ratio of the two signals is thus RFz, the square of the absoLute

reflectance of the film.

For measuring transmitta¡rce, the reference signal r'I8,s obtained'

by allowing the bean to traverse the system ag shor¡n in Figure

203.c, where the incid.ent bea¡r passefl through an uncoal,ed' half of

the substrate, r¡as reflectcrl'by M3, and. again passed through the

(fl) r:rerlrcr RErrEcrroN P^lH

(Ç) nrrtrrNcr rßaNsMrssioN ¡ÂrH

,l-¡lrlIt,l

(þ) t,.nt RåtrtcrtoN PArñ

($) r'." rR^NsMrssroN PAIH

FÍgurre 2"3

T6,

uncoated- substrate. TLre specimen we,s then üoved- to the position

shorùn in Figure 2.3.¿[ so that the beam passed. through the coated

half of the substra,te, Îüas reflected. by M3 and. again passed' through

the coated. substrate, anil the corresponding signal recorded' ff

Íi and. TF are the transn:ittefices of the substrate g¡ICt filn respectively

the reference íntensitY is

Io, Rl. R2. TS. R3. TS. Rh. R5

and the inteneity after transmission through the filn antt substrate is

IO. TLL. R2. TF. TS. R3. TS. TF. Rh' R'

The ratio of the two signa,ls thus gives TF2r the Ðquare of the

transmittan¡ce of the film.

2.r.5 SPECIMT]N IIOLDER.

The copper specimen hold.er Ï'as attached. to the litt of the

vacuum chanber (see Figure 2.\). The specimen cot¡lcl be moved. into

the positions shovn in Figure 2.3 by means of a rotating seal

tocated in the licl. The tilt of the specimen hold'er could also be

adtjusted. externally and. the lid itsþ]f cor:Lct be rotatetl with respect

to the charnber to assiet in the alignment'

A liquid. air tank was positionecl above the specimen holtler, ancl

contact to the hold.er was achieved. by copper band's on either sicle of

e copper block welcled. to the base of the to¡:k. The sorrple could'

then be cooled. to near liquid. air temperature if desirecl.

A second liquicl air tank was attached. to the lid- for use as a

r¡ater vapour trap. For measurements at l-or¡ tenperatures, liquict air

vas first pJ-aced. in this tank to remove any water vapour from the

IT,

evacuated. chamber before the speeimen was coofedr cnsuring no ice fil¡r

fo:med. on the specimen as it wo,s cooleil.

Insul-ated. feetl-througlrs were velded in the 1id. to enabLe

el-ectrica,I nea8urements to be taken if ttrev were reguired.

20 !, 6 n¡rnAcTI1rE INDEX OF AND A3SORPÎION IN ÎIiE SÜBSTNATES.

Tt¡e refractive ind.ex of the quartz-homosil substrates was

supplied. by the manufacturer, and the experinental d.ete::rrination

previously carried out in this laboratory (Canpbell, Ioe. cit.)

agreed. vith that supplied. by the ma¡rufacturer'

Conparíson of the trarlsmittances of substrates of, different

thickneeses showed. negligitle absofption over the wavelength Tange

2000L - 2.1+U. Beyond. 2.\U the homosil quartz gradually became

absorbin8, ï-ith a strong abso4rtion pe8Ï at ahout 3.0u .

2. 2 BEA},I DETECTTON.

2. 2, T LIGHT - SENS TTI1Æ DE1tsCTORS.

For the wavelength range 55o9l - 2'l+u, a lead sulphi¿e

photoconcluctive ce11 was *sed, a,nd. for ttre range 25008 - 55008

a Philips 1P2B photonultiplier u8,s used. Ttre gernonium fil¡ns vere

strongly absorbing before the cross-ovêI point of 55OOf, thue for

these filns only the leed. sulphid,e cell was required..

TLre sensitivity of the lead sulphid.e celI changed- with

temperature, and as a consequence the signal from it d.rifted- as the

enbient temperature altered. Ttris problem was overcome by nounting

the cell in an a}:niniwr housing to whieh vas attached' a Peltier

18.

battery. Fou-r diod.es 'were morrnted- in the housing as temperature

sensors. fhe fon¡ard bias vol-t*ge across ihe diodes decreases with

increasing temperature and. this was used. as a monitor to control the

current through the Peltier battery maintaining the temperature

to +o.o! degreeg c. A circuit ôiagralr of the temperature controller

and. Pcltier eell supply is shovn in Figure 2.5. If the temperature

of the housing vag such that the for'ç¡ard voltage acroÊs the d'iod'es

rose above a pred.eterrined. vafue (set by the variable voltage VI)o

the comparator would. turn transistor T1 on, shutting off the current

to the Peltier battery. Likevise ir tne forv¡ard voltage across the

diod.es fell bel-ow the sane value, the comparator lrould turn transistor

T1 off, thus al_lowing current to flow throu6h the battery.

Ehis arrangement had the arlvantaSe that the detector could' be

coolerl considerably, thus increasing its sensitivity. The operating

temperature w8,s determined by the setting of tbe variable voltage

supply V}.

2.2. 2 AMPLTFTCATIOIf AND RECTIFICATTON OF TIIE BEAI'4 SIGNAI,.

A block diagram of the amplifier and reetifier is shor'¡n in

Figure 2.6. The output impeda.nce of the lead- sulphid'e cel1 was

approximatelylOmegohmsandconsequentlyohighinputimpedarrce

amplifier was required. The square wave signal from the d-etector

'¿ro.s capacitively coupled to a bootstrapped voltage follower with an

input impedance of the r:rd-er of lO11 chms. The r:utput of the voltage

forrower .wos capacitively coupled to a Fairchild uA?09 integrated

circuit operational amplifier witrr fixed. gains of 10, 20, ancl 1O0'

.48

7V

TEMPÊPATUFE CONTSOLLEtrAND

PELTIEFì CELL SUPPLY I

5Vt-+

2.2x

10x

4¡ ANlí0f5'6v

1rdr€R^ruR! sErsrNc Dlocis î 6

6

10

óv t^M¡suFPLÌ oN /otF

rNo rc^l oÍ

644

-3.OV

BAX13

723

3.9 x

Figuie 2.5

bhq

CELL

LMß02 DVM

AI.f PLIFIER RECTIFIEN,

II,ITEGRATOR F,ECORDER

A.MPLIFIEFI ANÐ FIECTIFIERVOLT,\G=

FOLLOY{EF.

Fl,xeC oains= tgll-Õ- 10, 20; IOo

Hl-oh c on¡non-moderejection o unityçafn arnplifier

.Froure'2-"O

l9,

Ttre e,nrplified. signal was rectifíed. by enploying a d-iotle-bricl¿1e

in ttre feeübaek loop of an operatioi:.e-t amplifier. Ttrc'output from the

d.iod.e-brid.ge wae eoupled. to a gain 1 operational amplifier d'esigned to

give high cotmon-mode rejection. 'rhis rsas requirec. as the output of

the brid.ge floats at a potential of about one volt. This method of

rectification reduces non-linearities in the d'iodes to negligiTrle

l-eveIs.

The rectified. signal l¡as integrated for one second using a

capacitor in parallel witlr the feedba,ck resistor around an operational

anrplifier, on the front-end. of r¿hich was a National Instrunents LM302

integrated. circuit voltage follower to reduce therrnal drifts ' The

smoothed. signal was d-ispiayed' on a d'igital voltmeter'

The linearity of the enplifier and rectifier was tested- by

attenuating an ínput signal by known anounts using a Merconi attenuatort

¿xrc1 was for.¡nd. to be better than I part in lOOo for input signals from

3OuV to 300nV.

After an initial war.rn-up period, the noise level of the lampt

detector a,nd. amplifier was less than O'5 millivolts'

2.3 PREPARATION OF FILIIS.

2.3.r EVAPOAA TION OF GER]\4ANIUM.

The ger.manir:n fitns Ìfere prepared. by evaporation in vêct'lo¡ TÌre

spectroscopieally pure geflnaniun was supplied' by Nel¡ Metals and-

Chemicals Ltdl. , England. Ge::ina¡riwr lumps were placed' in a tungsten

conical basket. The evaporation rate r¡as controtled, by varying the

current f]-owing tlrrough +,he basket. An electromagnetieally operated

24,

shutter was positioned above the source to enable the genaanium to be

outga.ssed prior to evaporation onto the substrate.

2, 3" 2 STßSTRATES.

For the Ineg.surement of reflecta¡ce and. transmittance, the

germanium filns were deposited. on scratch free, opticatly flat quarbz

weaged prepared to specífication by the Scientific and Optical

Laboratories of Australia, Adelaide. Each wedge mee.sured- 2rt x 1]rt snd

was cut ín half to forrn a thielc and a thin wetige. The films were

cleposited. on a pôrt of tbe thieker half. TLre reason for using wedges

was tiro-foltl. Firstly it is difficult and expensive to obtain plane

pa,r&11e1-sid.ed. substrates of sufficient aceurecy to accurately take

€,ccou¡t of nrultiple-beam interferenee effects within theÌn. Seeondly

by using rÍedges, the beam reflected from the back surface passes aw8'y

from the d.etector so that the rrultiple reflections in the substrate do

not have to be taken into account, consequently sinrplif\ring the

reflectance and. transmittance eo¡ations .

2,3.3An

obtain a

it. Ttre

The

SUBSTFIATE CLEANING.

aciC. solution, and then washed in clean chromic acid. solution at BOoC

tu:tiL the acid forrned en even film over the substrate. Ttrey were

subsequently rinsed. in d.oubly d.istiileC vater, dried. in a strea¡r of

extremely clean surfaeed substrate is necessary in order to

pin-hoIe free film and to ensure that the fiki rllilI adhere to

following clearring procedure was for.:nd. to be sufficient.

substrates were first cle¡¡red. in warm concentrated chromic

* The wedge angle was 30.

2I,

dry air and. finaLly placed. in the dark space betr'reen tl¡o eleetrodes

in a vacuum cha¡rber anct ion bonbard.ed for three to five minutes.

The chromic acia solution was preparett by d.issolvíng chronic

oxide in weter (approxinately 100 gns. of chromie oxitte in 100 mf. of

water) and ad.cling concentrated sulphuric acid.. At first a vhite

precipitate formed.. tr\rrther sulphuric acid was ad.d-ed until- the

precipit ate d.issol-vr.;tl.

To remove ger-nanium films, the substrates were washed in aqua

regia.

2, 3, l+ _PUMPII{G u$.rT..

Ttre purnping r:nit consisted. of a 6rr oil diffusion pr:mp and a

single-stage rotary backing pump. A phosphorous pentoxid-e vapour

trap was inserbed. betueen the diffusion and backing prxllps to remove

water vapour, ancl a liquid. air trap was mountecl between the tliffusion

prunp e,nd. chanber to improve the evaporating pressure and to prevent

conte¡nination of the evaporating equipnent by oiJ- va,pour. Ttre

pressure cluring evaporation tr&s less than 1O-5 torr.

2,3.5 .

[he eubstrates had to be maintained. at ternperatures between

roon tenperature anit ?OOoC. It çill be shown in the next chapter

that the uniformity of the fil-rns is an inportant factor in the

aecurate d.etersination of the optieal constants of them. It l¡as

therefore necessary to rotate the substrate during evs,poration in

order to obtain s, more r:nifonn film. A d.iagram of the substrate

22,

support anal oven is shorrn in Figure 2.7.

Ttre oven consisted of a silica cylindur ¡þout vhicÌr was wound'

molybd,entun heating wire. A stainless steel jacket was inserted. inside

the cylintter to prevent contenination from the oven, md.a water-

coq-rled. copper jacket was mounted around thc oven" ftrc substrate

support ças constructed- of stainless steel, md the shafb of the

support could. be adjustccl to enable the d.istance frorn the souree to

the substrate to be varied.. fhe support r.¡as milLed. tc receive the

ved.ges which were hefd in i:lace by tungsten cIips.

The substrates were hcated by rad.iation from thc oven, the

tcrapcrature of which was corrtrolled- by the heating current. Afber

about two hours the substratc reached. the temperaturc of tlte oven.

Ttrc oven tenperature r.ras rneasured as a function of thc heating current

r,¡ith a chromel-ah¡neI thernoeoulrle, so that if a particrrrar temperature

was desired., the appropriote hcating current could Ì:e rletermined. I,Iith

this arrangement the tenper.eture v&s aecurete to within about 10Ø which

was sufficient for the nrescni purpose. The rotation of the substrate

was obtained- by d.irect d.rive from a small electric motor mounted. ebove

the oven. Íhe motor was shieldecl from the high tempa::ature of the

oven by a thick copper d.isc boltecl to the water-coolcd. jacket.

2, I+ W¡SUN¡N¡NNT AND CAT,CULATTON OF FTLM ,IHICÍIItrESS.

Ttre method. of Frin¡1es of Equal Chronatie Order (fäCO) viewed in

rcfleetion (Tol-ansky, 196C) rsas u.sed for the ne&surcment of fi1¡r

thiclcness. À review of thc a*¡ailable technigues ond the rlrguments for

usin¡1 this method have 'been given by Ca.mpbell (1967) .rncl trl' McCoy

.+

<___MOÌOR AND GEÀR8OX

COPPER HÊÁI SHIEID

ÂDJUSfAðI-E SHATf

<_coPPIR W,\tER-COOtEÞ,AC KÊT

MICA INSUI.AI ION

ÌUNGSfEN HEÀIING WIR€wour'tD oN vllRoslt

SIÀINI.E55 STEEI. JACKEf

5U B SIRAIE

SHUTIER

rU¡.]CSIEN çONICÀT 8A5KEI

CfVËN,A.ND SUBSTFìA.TE HDI-DEFI

Figure 2-.7

23,

(1966).

ftre apparatus for measuring film thickness i¡as built in this

laboratory (Mccoy, Ioe. cit.) and. wiIL only be briefly described

here.

2.\.1 OPTICAL SYSTEI,I FOR MEASURING ffTE WAVELET'TGTHS OF ÎIIE

INTERFERSNCE ORDERS.

TLre optical systen ie shown schematically in Figure 2.8.

the light from a carbon arc lo.p was eollimated' by a system of lenses

and was incid.ent on a half-reflecting half-transnitting mirror' lllre

light reflected. from the film r,¡as collected antl focussed onto the

entra¡rce slit of a sPectrometeroo

the nearest IA.

fhe wavelength coul-d. be measurecl to

2. l+. 2 HIGTT REFI,ECTANCE COATINGS.

Forwelldefined'interferencefringes'ahighreflectance

silver coating is necessary. A scratch-free optical flat, used' as

a reference surface, lfas coatecl with a silver layer with a reflectance

of approximately 954ø each time a thickness measurement was made'

The germaniun films lfere prepared' with a sharp step' ancl an

opaque silver layer vas deposited. over this vhen thickness ltrê8'sllfê-

ments vere made. Tolansky has shovn that a thin silver layer

accurately eontours the surface over çhich it is evaporated'

Thereferencesurfacew€Lsctampedoverthecoatedfilmtoforu

a Fabry-Perot interferometer' fhe existence of a step causes the

fringesofequalchromaticord.ertobedisplaced-.Measurementof

MItrROH

PtrC]JECTICJ N LEN

OBJEC-TIVE LENI-|A.I-I=- SI LVEFED MI FìTìÜËI

F=2.gtt

F : o'9tr

CC]LI.-IMATING ANt]COT'JDENS ING L-E NSES

F= 1'5lr

f-.trOM A.Fl(]SOURCE

FILI\4

OPTICAL SJYSTËM FOFI MEASURINCS FIL_M TIIICI<NESS

F-i.;ure Z,l\

2\,

the wavelengths of the fringes before and afber the step provides a

nethod of accurately d.etermi.ning the hcigJrt of the step, i.e. the

geometrical- fiL:n thickness.

2. I+, 3 CAT,CUI,ATTON OF FTLM TIIICKNESS.

Ttre thickness of each fil:n was d.eterriained at several points

across the fiIn, and the average fil¡r thickness together with an

estinate of the non-uniformity of the film calculated.. f?re calcul-

ation procedure used. is d"escri.bed in d.etail- by McCcy (loc. cit.)

and- Carnpbell (Ioc. cit.). îre prob-lcm with the technique of FtrlCO

is to account for phase changes on refleetion. llhen the fringe

system is viewed, in reflection, only the phase effects of the over-

coatíng silver layer need. be consid.ered.

Frorn the d.erivation of the Airy equatíon r¡hich includes phase

cha.nges on reflectíon, the conclition for a ¡rinimum in the reflected

intensity is given to a high degree of accuracy by (Bennett, 1961+):

NÀ = 2nt - (¡/2r) {gr(r) + ß2(À)} 2.\.1

wherc N is an integer, n is the refractive index of the material

betveen the two reflecting surfaces which forrl the Fabry-Perot

interferometer, t is the gcometrical separation of thc plates and

ßr(f) and ß¿(f) are the absol,ute phose chonges on refl-ee'bion at the

two reflecting surfaces.

For attjaeent fringes at Ài and. À2 (11>12), the Nth. mininun

in the reflected fringe systern is given by (Bennett, J-96ì+)

rI+1 = x2/(^r-Àz) 2,1+,2

For the fringe pattern on each sj-d.e of the step, tho equntions

25.

NÀ.Ë 2t - (l/e¡){er(,r') + gr(},)i 2.1+.3

NÀr = e(t+ot) - (¡t /pr) {er(x') + ßz(l')} 2.h.\

follow from equation 2.\.1 l¡here tr and Àr are the wavelengths of

correÊpond.ing ninima and n for air is taken as I.

For high reflectance fil¡rs d.eposited- simulta^neously in vacuot

the phase change on reflection at the two refleeting surfaces will

be very nearly the sanen Thenr to a good' approximation,

gr(r)=ßz(r)=P(r)

Bennett (Loc. cit.) has shown that Ê(f) tor silver fiLns is a

slowly varying firr¡ction of À and may be approxinated to a straight

line of the folro

g(r) = n(c+riÀ) 2,\,,

Hence

NÀ =Zt-cÀ-clÀ2 2,\,6

A plot of N), against tr is no longer linear, but is rnodified' by the

tern in À2. lvlcOoy (loc. cit.) encl Canpbell (loc. cit.) have shown

that íf a straight line is fítted through ttte (ÀrNÀ) points, the

maximl¡m deviation from the parabolic fit over the visible re'nge iso

less tha¡¡ t25Ã and is independent of both N and' 6t.

For a linear fit through the (f,rlL) points, the most convenient

equations for each set of points are

Nili=mÀ. + c 2'l+'7

N.l.=mÀ.*c+ôt 2.\.8J.'J J

where N., is the integer closest to the value given by equationI

2.\.3. ftre two lines of slope m are required to be para].lel and

the pararneters are calcr¡l-ated. by ninimising the sta¡ad'ard d'eviation of

26,

each set of points from the corresponding straight 1ine. Using these

equations, the fil¡l thicknees ie calculated as one of the para^netèrs.

Cerrpbell (Ioc. cit.) and. McCoy (Ioc, cit.) have programned the

computer to calcul-ate the average fil¡n thiekness from the measurements

at a number of points across the film, together with an estinate Of

the non-unifo:rnity of the fil-n catcrrlated from the staridard. d.eviation

of the thickness at each poínt fron the a1¡erage thickness. they have

shown that, at least for carl:on a¡rd seleníurn films, the uniformity of

the filn is generally the limiting factor in the aecuraey of the

calculations.

27,

CHAPTER 3.

OF OPTIC/I.L OFA TFIIN ASSOBSTNG

SINGI,E - LAYER FI ['I'I

3.L INTRODUClÏON

fhc erlultions for the reflectance, R, a'nd transnitta'ncet Tt of

light at norral incid.ence on a thin absorbing fil-m on a transparent

substrate are eomplicated., and. the exact solution for the refractive

index, n, ond. absorption index, k, from them inr¡olves extensive

conrputation (Mccoyl- ]:966¡ 0anr¡.,be11, 1968). Consialeration of the

expressions for (f + n)/t an¿ (f - n)/T resultect in simpler equations

which were more reaclilY golved.

A progra,rn has been written for a d,igitat- computer to calcul-ate n

and k from measured. values of R anct T, and uses B, colcul-ation

proceôure euitable for data from any thin fil^n. The progra,m has been

used to calcutate the optical conetants of germa^nium filrrrs, seleniu:L

fil¡rs and carbon filns. The eoLutions from the data for the earbon

fil$s agree exactly with thoee calcu-Iatett by McCoy (loc. cit.)'

In the calculation of the optieal constants, multiple solutions

cs,n occurr md the solutions for the refraetive índex are markeclly

affected. by smal1 changes in film thickness. Ttte behaviour of the

solutions under such changes is d.iscussed, ffid the resuLts of this

investígation show that the ehoice of the correct solutions d'epend's

critieally on fíIt'n thickness, üd use can be made of this in d'ete::srin-

ing accurately the optics,l fiha thickness '

28¿

2 MIE EOUATIONS a

3 . 2, L REFI,E CE AND TRAI{SMITTAT\ICE ONS.

The equations for the reflectsJrce c¡¡rd transmitte¡rce of a

single absorbing fil-n on a transparent substrate are (Heavenst 1955)

(gr2*trr2)exp(2o1) + (gz2+:nz2)exp(-2o1) + a cos2y1 + B sin2y1ff=

exp(2o1) + (g]+1at2)(ez2+;¡z2)e:ç(-Zor) + C cos2y1 + D sin2y1

3.2,I

{ (r+er )z + n}}{ (r**r )2 + :n22I

exp(2o1) + (gr2+hrz)(ezz+hz2)exp(-2u1) + c cos211 + D sin2y1

3,2,2

.02-nt2-kt2 nLZ-nr2+kr?8z

where

gI=(ns+n1)2 + t<J

2nsk1

(n1+n2)2 + t<J

-2n2k-¡hr= }l2'-

(ni+n2)z + kJ(ne+n1)2 + kJ

t=2(grg2+:n¡:n2) B=z(gilnz-ez}.t)

C=2(BtEz-h1h2) D=2(grhz+ezlnt)

or = 2nktdt/À Y1 = 2nn1d1/À

and.

ng is the refractive index of the surround.ing med.ium (nO = I for air) '

n1 is the refraetive index of the film.

n2 is the refractive ind.ex of the substrate.

k1 is the absorption index of the fi1m.

d.1 is the thickness of tlre fil¡r.

À is the wavelength of thc ineident light'

29,

3.212 IRANSI\Ef T^{rNCE TTON.

Equation 3.2.2 for the transnitta¡rce is the elçreEsion for the

inteneity trefisnítteil into the med.iun of refractive ind'ex II2r T-]xis

equatíon has to be corrected for the case of a substrate of finite

thickness where the transnitted. intensity passes through the substrate

and. is measured. in the mediu¡r of refraetive index rg r For the vucin:c"

shaped. subetrates used., the bearn reflected. from the baek face of the

subetrate passed out of the substrate away fron tbe incident beam and

hence produced. no multiple bea¡r effect in the substrate' The

transnitte¡rce of the half of the ued.ge with the fiJm deposited. on

it was comparecl to tha,t of the blsnk half'

Fromthediagranbelowrifloistheincid.entintensityrthen

the reference intensity I* is given by

f* = Iot2

rrhere t ie the . transmission coefficient of the substrate-

air interfs,ce.

The specimen intensitY I, is

I, = IOtb

rr.here T is the transmittance given by equation 3,2,2, Thus the

ncasured transnittanee ratio I is

I^T¡ =-.Ì=-rnt

(n2+no ) 2ñ-mI _ a¡

\n2ng1.G. 3.2,3

30.

To

+FILM

rot <-SUBSTItAtF'>

Iott r tt0

S

3. 3 SOLUTION OF lTiE RF]FLECTANCE A-IID TNANSMITTAITCE EQUATTONS .

Equations 3,2,I ancl 3.2.2 cannot be separated tO give explicit

expressions for n1 and k1 e the alternative is to solve the equations

graphically. Bennett and Booty' (tg66) used a trial and error process

in which n1 and. k1 were altered in increments and' the values of n1

end. trl stored for which the conputed values of R ond T were ne&rest

to the measured va-lues. Snafler increments Were then used fOr n1

and. k1 arrd the process repeated wltil the measured' and calculated

values of R ancl T were within the d.esired. degree of accuracy. ft¡is

procegs has a number of d.isad.velrtages. Firstly, approximate solutions

to the equations have to be inserted as the starting point for the

calculation, these being obtained. from the bulk naterial. Seconcllyt

Mccoy (Ioc. cit.) has found. the number of coutputatione involved. in

find.ing a solution to the equ$tions can become excessive uncler certain

conditions. Ihircllyn the eErotions often had more tha¡r one solution,

either of which rras s,cceptable et the wavelength at vhich the

calcr¡lation was mad.e. Alternatively no solution may exist and this

31..

posaibility was not taken i¡to accor:nt'. Fou:*h\r, in the paper

quotetl there vas no egtimate of tbe erro/ in the solutione arising

f?om experi¡¡ental êrrore r

Nilsson (fg68) recognisett the exietence of nultiple solutions

a,ntl attemptecl to golve the reflectance a¡rd transnittance equations

in the following mannerl Ttre equationg for R ancl T were rewritten

in the inplieit fo¡ms:

F(Rrn1rk1) = 0

G( lrn¡ ¡k1 ) = 0

Approxinate values of n1 antl k1 gaVe eertain values of F and G.

Ttrrougþ the correspontiing points in the F¡n1¡k1 âDd' G¡n1tkl systemt

tangent planea were laicl, [tre coamon point of those planes in the

plane F = G = O waa then taken aB a Eecontl appro:cimation, ancl the

process repeatecl rx¡ti] the equation system nas eatisfiecL.

However again with this ¡nethott only one soluticn was d.etermined.,

althougþ it ghor¡Ict be possible to ealculate the other soLutions

using appropriate initial values of n1 antl k1. Nilsson gave an ad'

hoc methodt for cteteroining the initial- values of n1 antt k1 uncler eertain

conclitions. He ad¡aittefl that over Bome spectral regions the resultE

obtainect by this metbod. were uncertain. No estinate of the error in

the solutions wae given although he realieecl that the error clepended

on the curve intersections.

Mccoy (Ioc. cit.) ancl. Carrpbell (Ioc. cit.) solved. the equations

graphically with the aicl of a ttigital conputer. lttris was achievea

by aJ-lowing nI to talce o, range of va,lueer aIId for eaeh nlt calculat-

ing the values of k1 which satisfied the experinentally dletenninecl

32,

valuee of reflecte¡rce ond transr,rittance. I[?te8e loei of const8nt

reflectance and constar'it transxdttanee were ptottecl as n1 against

k1¡ the solutions for n1 and' k1 being the points of intersection of

the two loci. The large nunber of types of eurve intersections

which cor¡J-d. occuf necessitated extensive progrnmmi¡¡g and- requirecl

considerable conputing tirne, but nevertheless exact solutiOns 'were

obtained., and. all- the solutions in a given yÞnge of n1 were deterrnineð'

An estimate of the error in the solution at each uavelength was also

eel-cuLated., involving the nunerical differentiation of the reflectanee

and. tra¡rsrnitta¡ree equations .

3. l+ $TE ES.IIATIONS (r+n) /t AND (l--R) /T.

The denominators of the right-hantt sid.es of equations 3'2'1 a¡rd'

3,2,2 are id.entical exeept for the factor ng in equation 3'2'2' By

consiclering the expression" (r+n)/r ana (r-n)/t, the following

sinpler equations can be tterivedl (ton'tin, 1968):

1+R I(rr92+n1 z+l'f) {(n¡z+nrz+t12 ) cosh2al

rF \nsn2(n12+k12)

+ 2n1n2sinh2qrl

+ (ns2-n1 2-kJ ){(n12-n2 z+^t2)cos2.¡1 - 2n2k1si*2Yt}J

I-R I

2n2 (nr 2+kr2 ) þ, { (rrt*r, z2+',.:2)sinh2o1 + 2n¡n2cosh2a1 }

3.l+.1

T

+ kr{ (ny2-n22+} 12)sin21¡ + 2n2k1cos2'¡1i

3,\,2

33.

For the vedge-shapecl substrates used., the measured. quantities

are R and. I.

t+R=

T

t-R

Thus from equations 3.2,3t 3,+.I and 3,4f.2t

(ns2+n1 z+kf ) {(n12+n22+ky2 )cosh2o1(n12+k12)(ne+n2)2

+ 2n1n2sinh2o1]

+ (ne2-n1z-:nf ){("r 2-nz2+:nt2)cos2y1 - 2n2k1sin2yr}

1

3. \.3

T # þr{(r,t'*n z2+kt2)sinh2o'i + 2n1n2cosh2o1}

+ krt (n¡2-n22+k1.2)sin2y1 + 2n2k1.o"zy1)]

3.4. h

3.5 SOLUTION OF ÎTIE EOUATTONS (1+R \/ T ÆTD (r-n) /r FoR nr AND kr .

Equations 3.\.3 a,ncl 3.\.h aga;in cannot be separatecL to give

explicit expressions for n1 and k1. Ttrey þave been solved. in a

similar menner to that d.escribed. ín section 3.3 by Canpbell, with less

extensive computation due to a decrease in the number of mocles of

curve interseetiong. Tt¡e behavior.lr of these equations is much

sinpler the¡r for R a¡rd. T (or[) as can be seen in Figure 3.I where

nor:m.alised contour eurves of R, T, (f+n)/f ana (1-R) fI, are plotted

as functions of n1 and k1 for rr2 = 1'5 and d1/tr = o'\' By plotting

curves of constant (ftn)/f(orf) in the first quadrant cf the nt-kt

plane for a series of vafues of cI1 /X, ít was found that ec¡uations

3.h.3 and 3.1+,\ were sinE¡le-valuetl in k1 for a given n1' Since

ecluation 3.l¡.1+ is read.ily di.fferentiated with respect to k1r the

solutions to the equations nay be d.eternined as follows:

t*+l*

{'I,:..)Ç

l

3\.

Rerriting equ.ations 3.1+.3 and 3.h.\

( n s2{.a 1 2+k

1 2 ) { (n1 2+n

22 +lrt2) coehZo ¡ + 2nrnzsinh2cl }

+ (ns2-n12-tr2) { (n12-n22+k12 )eos2.¡1 - 2n2k1sin21¡}

- (n12+rr12)(ns+n2,t[*, J = o

2ngn1 {(n12+n22+k12)sinb2c1 + 2n1n2cosh2a1 }

+ 2nsk¡ {(n¡z-n2z+k12)sin2y1 + 2n2:r.reos2Yt}

- (n12+r¡2)(ner,n2)tFfJ = o

3.5.1

3.5.2

IÊt

f1(n1rkt) = left-hanct siate of equation 3.5.1-

f2(n1rkt) = lefb-hancl side of equation 3,5,2

The solution is thue that (ntrkt) tor which

f1(nrrkr) = f2(n1¡k1) = o

For a given n1r eguation 3.5.2 cal¡ bè solved. for k, using

Newbonrs methocl (see for exanple, Nationa.l Physical LaboratorTr 1961)'

If ktf0ì is an approxinate solution of equation 3.5,2 lor a given n1¡

a better approximation is given bY

kr rtì = ¡, f 0ì - f2(n1rkr r0ì ) /f zt (n1rk1 r0ì )

where

f2r(n1rkr) = t'hr)

â1

= 2n0{2nt (k1+an2y1 )sinh2cr1 + 2\t(n12+nr2+x12)cosh2c1

+ (nr2-nz2+3r.12)ein2y¡ + hn2kl{cos2y1 - (n6+n2)zt$lf }

For the present investígation it vas sufficient to set kr f0ì to zero'

35,

ttius þ suecessive approxinations, iie.

kI Él = kt ln-l'ì ¡f2(n1rk, ln-'rì ¡

f2 t (n1rk, fn-'rt ,

k1 can be forlnd. to any preÞet Lilltit, for a given n1.

For the present inveati¿4ation the limit wà,s sueh t':nrlxb f 2(tt rkr ) was

'¡rithin 0.0001 of zero.

?. E. r cALCIILATION 0F nr -ANÐjI.

n1 and k1 which satisf! both equations 3.5.1 and 3,5,2 may be

d.ctermined as followe: n1 is set to a lower linrit ancl the value of

k1 which satisfies equation 3,r,2 is ca-lculated as ôescribed. above.

The set value of n1 and. caiculated. k1 are substítuted' in ttre equation

for f1(n1rk1). If f1(nrrkr) = o, then this (trrkr) satisfies both

equations 3.5.1 ancl 3.5.2 and hence eonstitutes a solution.

If f1(nt rkr ) # o these vo,lues of n1rk1 and f 1(n1¡k1) are stored

in the computer, an incrcment is ad.d.ed. to n1, ffid the process repeatecl.

T'he sign of the new value o:f f 1(nr rki ) is comparecl çith that of the

s.bored value, ild if identical, the stored values of n1¡k1 and

f1(ntrk1) are rejected. and. the ncw values stored. A fi¡r'bher increment

iÊ add.ed. to n1 end the procerlure eontinued until a change of sign

oceurs betrreen corresponrlíng val-ues of f 1(n1rk1) , which then inplies

that a solution exists betweerr these two values of n1, 'i'Ìris can be

seen from Figure 3.2,a, where f1(n1¡h1) is plotted as a fì:r¡ction oft-R

nrfor a given dt/À and. -¡- .

Let n, fIì and n1 12ì ¡s the vatues of n1 correspondj.ng to the

vnLues of f 1(tf rkr ) bctçeen .*trich there occurred o ch:ìflBe of sign'

fr ( n,,k,) 0

5

4

3

2

1

0

0

-1

-¿

-3

-4 I 2 3 4 5 6

ft1

f

fP

f ,(n,,k,) {,(n¡,k,) o

n¡o) n lßl fì l(2ì

b )

-t-

I

II

II

-lI

II

------t-- - -\II

I\.1

(c)

Figure 3,2

ntl'l np) nter

36.

Let r, 11\ = f1(n1 11\¡k1) an,t ft12ì = fr(nrf2ìrk1). There are tvo

c&ses to coneid'er corregpohôíng to f1 lIì ¡sft* negativc or positive'

Coneid.er the caee where ft fll ís negatiVet An approxinAtÍon to n1

cen be found by linear intcrpolatíon betreen n, flì and' nI f2ì '

Fron Figures 3.2.b e¡rd. J,2.c¡ an s,pproximate solution is n, 13ì

whcre, from sinilar trianglest

n, 13ì = ¡, f2ì -{ n, rzt-n, llì }r, rzt

3,r,3fl f2ì-fr flì

k1 is calculateil as before for this new value of n1r and the

corresponding 11 13ì = fr(nr 13ì rk1) calcr¡Lated.. If f1 13ì<6 ¡¡6tt

ft fll an¿ n, fIì are replaced by fI f3l and n, f3ì ond. er¿uation 3.5.3

again applied. rf fl f3ì>6, then f, f2ì and nt f2ì are replaced' by

fi f 3ì snd n, f 3ì and. ecluotion 3.5.3 again applied'. By continuing

this process until Bome prêBct linit (e.g.lft fet l.C.OO1), the valuee

n, f 3ì snd. correspond.ing k1 íire gooci approximations to the correct

solutions.

For the case where ft rrl is initially positivc, thc procedure

errtL equations are id.enticat to the above ir (nr 11ì ,¡, f Il ) and'

(n, fzt ,ft f2l ) are interchr-urgcd.. Thus n1 and k1 can be fould. to any

required. degree of aceurÊcy by inposing appropriate li¡nits.

.trlthough for a given n1, there is only one positive k1 nhich

ss,tj.sfies equation 3,5.2, there may exist more thon one set of n1

and. k1 which sotisf! both ecluations 3.5.1 a¡rd. 3,5.2. Consequentlyt

flrrther increments must be ad.decl to n1r until- some preset l-iulit is

reached., to d.ete¡'nine all posoíble solutions.

37,

3. 6 CATCUTATIO}I OF ITIE ÉRROR II{ THE SOLUTION.

Along with each value of n1 and k1 calculated, m estimate of th'e

maximum err6r in each, due to err6rs in Rr¡¡tl1 and. n2¡¡f&S evaluated

as follows:

ÏÍe may write (f+n)/f. anA (1-R)/l in the functional form

(f+n) /f = F(ni rk1rc11¡n2) 3'6'1-

(r-n) /r = G(nr rk1¡d1¡n2) 3'6'2

Teking total derivatives

¿¡n = **, dnr + H, *r * #, aar + ffr, dn2 3.6,3

3.6. h

3.6,5

3.6,6

¿c = #r dnr + **, *, * å$, dtil + #, *,

From equationa 3.6.3 a¡rd. 3.6.1+

dkr = * [ **.-elr - ds.#, * i#,.'3å, - å*,.åå,]uu,

. {å*,.f*, - #,.#,1*, ]

dnr =i [*.tfi, - u*.å*, * {**,.*å, -å*,.#,}uu'

* {**,.#, - *f,,.**,}*, J

where

3,6.7

Although explicit expressions for n1 sJnd k1 eannot be'r found.t

they may be written in the firnctional- fom

llt = rlt(Frcrd.1rn2) 3'6'8

kr = kt (FrGrdl rn2) 3,6,9

-âGàFãGâF'=Er'81 -.ãã'r'E,.

38.

Sinilarly

Equating coefficiente of equotions 3,6,5 a¡rcl 3.6.6 r¡ittr those of

equations 3.6.10 arld 3.6.11

clnr =;Ët.* *#t.dc +*å1.uu, *#,r].*t

crkr =*ät.* *€ät.dG +#t.uu, *#r,|.*,

àfl râF

-Qtr--IÐraG J âkl

ân, rfar aG aG ar ìããi =ïl..Er'Fr -Er'frrj

ân, rlar aG aG aF Iñ;= ï[ãE'ã-n2 -E''ãñrJ

âk, 1 ãGi-¡ =AF J Ðn1

ãK, I âF

-¡=AG J ân1

àk, rfac âF aF âG Iããi =ïl.Er'ããI -Er'ErJ

âk. rfac âF aF ðG ìã'; =ïL*r'82 - *t'ft'rj

anr=lþ.orl .lP.ool .l #.¡arl .l#.4*.1

akr =[F.orl .l #,0.1 *l *i .oa, l*l # .a', I

3.6. r0

3.6 . 11

3,6,I2

3,6,I3

3.6.1\

3,6,I5

3,6,16

3,6,I7

3.6.18

3,6,I9

3.6.20

aG

akI=-J I

Hence an estinate of the maximr'¡n error in n1 and' k1 is given by

3,6,21

39,

where the partiaL derivatives of n1 and k¡ are given by equations

3,6,!2 to 3.6.Igi AF a.nd. AG are found from the experimental errors in

R and. f ¡ Âd1 is the estimate of tbe error in the fiLrt thickness, and.

An2 is the error in n2. ILtc partiat tterivatives of F and. G are given

in Appendix A. Íhese firet ord.er fotmulae are valid. provicied J is

not too smalI.

3.7 TTIE EFFECTS OF TTITCKNESS ON Í5TE TION OF n AND K

fhe effects of variations in fil¡r thickness on the ealculated

values of n1 considered in this section B,re fol the situation where

D1>ng¡n2r ttris is general-ly the case for a semiconducting fí1m

supported. on a transparent substrate. It will- be shovn that an

r¡nd.erestinate or overestine,te of the fil¡l thickness produces quite

ùistinct resufts.

Although not considered in this thesis, the cases 116<n1<n2

and. n1<ng give òifferent behoviour but again und.eyestime;tes or

Õverestine,tes of the film thickness prod.uce quite distinct results.

3. 7 . 1 T}IE HYPOTHETI FILM.

In order to detemine the behaviour of the optical consta¡rts

of a film r¡nd.er small changes in film thickness, a hypothetieal fil¡r

'rilas forautated so that its properties were knor¡n exactly. Ttre film

w&s a,sstuted to have properties approxinatin6 those of gennanium.

TLre refractive inclex, il1¡ Ìr€ì,s set equal to l+.0 over the

wavelength ra,:rge Q.5V - 2.5Y

Ttre absorption ind.ex vo.ried. as O,25/)\ (f in ¡r)

\0.

Film thickness was set eq.ue.l to 2OOOfi''

fhe reftectance and trensr,ritta¡rce for this film vere calculatecl

using equations 3,2r! and 3.2.2 (eorrected- using ecluation 3'2'3) ' The

reflectance ond transrnittance curves are shOvn in Figure 313'ai and'¡

o

vhen conpared with thoöe obtained for a gemaniun filmt 1800Ã thick,

(Figure 3.3.b) it can been seen that the hypothetical- filn is not

far d.ifferent from the real situation'

2nÒJ. I a ç CAICULATION 0F n r AIÙD kr FOR TIIE IiTPOIÍÍETICAL FTLM.

The calculatecl reflectanee a,nd transmittance for the hypcthetical

fil¡,r were used ag ôata for the caleuLation of n1 s,nd' k1 *sing the

method. d.escribed. above. ILte calcuLated nt - curve is shown in Figure

3.h.€ro In various parts of the wevelength ra^nge, nultipte solutions

exist, a^lthough at each point the exact value of l+.0 is o solution'

some eolutiong are close togettrer a¡rd an iterative process, 8,e usecl

by Bennett and Booty (tg66), assuming initiutly bulk values nay

read,ily converge to an incorrect solution'

The incorrect solutions are seen to forn loops intersecting the

correct curve. Írhe nurnber and position of these loops d'epend,s on the

fil¡n thickness. l\8 the thickness increasesr more loops appea'r' If

the film is very thin no such loops will appee,r and. only the correct

sol-utions will be dete:mined. Ûris situation will be d'iscussed later'

3. T. 3 FILM TfiI CKNESS TOO LARGE.

lJsing the s¿rme reflectance and transnittenee d.ata as in 3,7 '2,ô

n1 Bnc. k1 were calculated, assr¡ming a fiJ.rn thickness of 21004 insteacl

T

(a)rYPorHrrrc^! tr!rd : 2000 A.U-

\A/AVELENGI'II IN MICFÌONIS

\A/AVELENGTH iN MICFIONS

l--

oz

tr

t-

oz

tr

T

T

T

(þ)crnnrttuu rrrxd : ls00 A,tr"

Figure 3.3

5

\t

(a)o

d= 2000 A

5

XLLJ

Õ7__

LLJ

t-c)

É.l!LUG

4

3

2

2"4 2.2

oA

2,0 1.8

WAVELT NGTt-I

1'4

M ICRONS

2 1,0f'6IN

'B

B

.)

XLljôz

LL¡

ìF(-)

ÉLL!É

5

4

(b)

d= 21002

5

2'0 1'û

WAVEIENGTI-I

1'6 1'4 1'2

IN M ICRONSf.0 62.4

><t!o

=

LIJ

;O

É.LL!ga

3

(c)

cl = 190d Å

,).(\ î.fì 1'Ê

\¡,i ¡i\,r[i ING ifl ir'J

Figure

1.4 1'2

IVlICRONS

3.4

2

.]. .4 t'¿ 1,0 e

\1.

of the correct value of 2oOO1.. T'e e¿Ieu'e;ted. nl - eurve is shovn

in Figure 3.h.b. In Bome lrave]-engÈh regions no solutions for nI were

found., ¿md a continuoue clispersion curve eannot be drawn' Parts of

thc correct curve have shifbed donn and uerged. with the lor'ier parts of

the loops to forrn isolatecl isloJxds '

3..r.\ FTLM 1'flICIC{ESS TOO SMALL,

Againusingthesa¡rrereflectaneeandtransnittcurcedataas

above, n1 an¿ k1 were carculatcc- assuming a firn thickness of 19001''

TÌre ca1culaterL n1 - cuxve is sh'ou"n in Figure 3'\'c' Altho'ugh a

continuous d.ispersion curve cfft be dram, it is vastly irrcoruect'

Various regions of the correct solutions have shifbecl up to fom' !:ope

with parts of the incorreet solutions. flhe remaining regions hal'e

shifted. dom anC mergeC with tl¡e incorrect solutions ti¡ forrr a

continuous d.ispersion eur'¡e with the appearance of vnlleys. Ttnis

curve is also nr:lti-valued at some points '

3. T. 5 EFFECT ON PR¡\rIOUS hfORK.

canpbell (loc. cit.)n in his calculations of the optical

constarrts of setenir:¡n filns, founcl that use cf the measurcd film

thickness produced. curvcs sinilar to that shc-tm in Figrre 3'l+'b' He

recognised this as being re.La1;erj. to too lorge a value for the filrn

thickness being uged. in ealculations. He aceordin6ly recuced' ttre fifun

thickness but failed to reco¡Erise the behaviour of nn '"mrlcrcstinate in

tbe fil-u thíckness, üd curves si¡ailar to that shown in fi¡lure 3"\'c

were ossumed, correct. Tt¡us el-f the refraetive ind-ex ancl absorption

)+2,

curvee caleulated. by him are incorrect'

Etre reason vhy the measurecl fil¡r thiekness correspond'ed' to the

situation of an overestimate in this quantity wirl be consid'ered in

chapter 4, TÌre optícaI constantg of selenium fiLns were recalculated'

from the d.ata of cenpbell a¡rd the results of this 8,re presented' in

Chapter 5.

3. '.l, 6 ACCURATE DETERMINATI OF ISIE OPTICAL FILM TITTCKNtr]SS.

Ttre behaviour of the refractive ind-ex solutions as a function of

wavetength is quite d.istinct for fil-n thicknesses whieh are either too

small or too large. Tttis characteristi-c bchavior:r is reailily tletected'

for. fil¡n thickness variations as smal1- as O,5/", Hence ad-justnents can

be mad.e to the filJt thickness r:ntil the correet behaviour oecurs, thus

giving the correct optical eonstants together wíth a¡r accurate value

of the optical fil-n thickness '

ERRORS IN R .I\ND T ON 1TTE FIL}{ UIICKNESS.3, 7, 'l EFFECTS CF

From our calculations on Ìrypothetical fil-ns it was apparent

that obtaining the correct behaviour for the refractive indcx solutions

was vêrîr critical on the choice of the correct fil^n thickness'

Beforea,nyconctusionscouldbedrawnastotheabgoluteaccuracyof

the fi1¡l thickness detendned by this nethotl it was necessary to

determine the sensitivity of obtaining the correct refractive ind'ex

betra'riour to errors in R oncl T.

Ttris was most suitably detersined. from the use of the h¡rpothetical

fil-ns. TLre caf culated. reflectB,nces and tronsmitta¡rees for hypothetical

43.

fi.tns of TaryiÈg thicknesE¡es were elteredL in tl¡e four follorrring ways:

(r). R+o.oo3 T+o'oo3

(e). R-o.OO3 T-0'003

(s). R+o.oo3 T-o'oo3

(4). R-o.oo3 1-o'oo3

Cases (f) e¡ra (Z) clid. not affeCt the cs,leulated fílm thickness to

any d.etectabLe extent. For case (3) the filn thickness had. to be

reduced by 0.5/, vní]:e in case (l+) it had to be increased by A,5/o'

Proviôed. one has a uniform fil¡n of sufficient thickness to enable

the above method. to be used., end provid.ed that R and T can be measured

to about ! o.oo3, then the optical thickness of the film may be

d.etemined. with considerable precision.

3.7.8 VERY TTIIN FTL¡{S.

If the thickness of the fi]l.a is less than the order of 5008,

maxima and minina in the R and. T curves are no longer apparent in

the wavelength range of meagurenents. For such fil¡ls nultiple

solutions tlo not occllr¡ Variations in filn thickness for these fihis

therefore d,o not e:chibit the behaviour d.iseussed above, and. the ¡rethod'

d.eseribed. in section 3.?,6 cannot be used to detertine the filn

thickness.

fn order to be eertain that the filn thickness is correct, and

.3orrespond.ingly the optical constants, it is preferable to use films

thick enough to exhibit nultiple solutions. For geroanium this

correspond.s to 6OOl ot grea;ber. Fron the d.iscussion of surfece layers

in the next chapter, the existenee of nultiple solutions provides

l+h.

va-l¡able info:metioa¡ e.nð enbancea the desirability to use sufficient-

Iy thiclr. filns such that nultiple solutions occlllt¡

?. 8 APPROXIMATE FILM TJTIICKNEÊS.

If two consecutive turning points of the (f+n)/f vs. tr cu:t¡e

exists in the infra-reci (i.e. awa¡¡ fron the absorltion ed'ge)¡ and the

long-1¡r¿yulength refractive index is knorn, an approxinote v¡ilue for

tlre fiLn thickness nay be fowrtt as follolrs:

In the infra-rettr kl = O, hence' from equation 3'l+'3

t ort (ns+n2 )2

â{ (ITF) /r,} = -----h?Tdl- (n12-r) (nt2-nzz) sin2y1aÀ À2n2n1 (ns+n2)2

Ttrning points occur wlren

â{ (r+R) /r} oa^

sin2y1 = 0

znnl ¿h /I , henec trrning points occur whcn

\nn1,11/À = mîr m an intcger

Í.e. when

Now y1

1.eo d.1 3.?.1

Let n correspond. to thc turning point at wavelength À1, then (m+1)

corrcsponcls to tbe consecutive turning point at wavelength À2 '

r¡1Ftt

Hence

cl1 3.7.2

4r.

Therefore

mÀt (n+I) ¡,

)t2m=-

À t-Àz

Substituting in equation 3,7.2

trr trz

3,7,3

3. ?.1+d1 4n1( À1-À2

where 11 atrd 1,2 are the wavelengths of consecutive turning points in

the infra-red, and n1 is the long-wavelength refractive index'

3. 9 DETERI{INATTON OF ITTE OPTTCAL C ONSTA}ÏTS FROM TTIE MEASURED

AI$D REFLECTANCE FROM rÍiE BACK STMFACE.

An alternative nethod of d.etemining the optical constsfits of a

thin filn d.epositett on a tra¡rsparent substrate is from the transnittance

and" reflecta,r¡ce from the baek surface vhere the incidlent bean passes

through the substrate and is reflectecl by the fil:n'

fhe reflectance from tho back surface is given by (Heavenst

Ioc. cit. ):

(ezz+nzz )ery(2or ) + (g12+trr2)exp(-eo1) + A cos2'¡1 - B sin2'¡'1 tRl=

e:ç(2or ) + (gr2+hr 2)(ez2+hz2)exp(-aai ) + C cos211 + D sin2'¡1

where the s¡mrbolg have tbe same meaning as in section 3'2'!'

Following the nethod of romlin (Ioe. cit.), much sinpler equations

can be derived. by consictering the expressions (ftnl)/t.

f There ig e¡r incorrect sign in the numerator of thc forurulagiven by Heavens.

,l+ u.

+ = [#] [tt t (ez2+,^z2)]{exp(eor) t (er2+hr2)exp(-zer}

+ (Ctl)eos2y1 + (ote)sin2y1

{(r*ur)2 + hrz}{(r+g2)2 + ,21

ctA=l+grgz or

DtB=l+gthe or

(r+er)2 + hf =

-l+rrtn,

\ezhr

\nn2

(ìrs+n1)2 + kr2

\(n12+n1n z+kt2)2 + l+nz2utz'

i(n1*trr)2 + :' JIz(.0-.r )2' + kJ

i(re+tr1)2 + kr2Ì

\n1n2

( r+ez )2 + lnz2 =

912+hr2 =

Z(nf +n22+tr12 )::'=-2tY--2\ - '-.'_=_._--__-

' (nr +n2)2 + kr2or

(n1+n2)2 + kr2

exp(ecr ) t (gr2+hr2)exp(-za1 )

{ (ns+n1)2 + kflu*p(2o1) t {(ns-tt1)z + kflexp(zo1)

(ns+n1)2 + kf

Substituting the above in the cquations for (ltRl)/T antl siurplifling

we obtain

I+Rl-r' 1 (6¡2+n"2+k12) i(nq2+ntz+t12)coen2o1hn6n2(n12+k12) t

+ 2n¡n1sinh2o1]

+ (nr 2-n22+kr2) { (n92-n1z-i¡-t|) cos2li + 2nsklsit2yr }

J' B.t

)+7.

f-Rl = IT ene(n12it12) ["r

{ ("02*" t'*kr') sinh2ol + 2nsn1cosh2o1}

+ kr{(n12-ne2+k12)sin2vr * 2nsklcos2vr}

3.8.2

For the wed.ge-shapetl substrates used:

1 (nf +n22+kr 2 ) { (no 2+n, 2+k1 2 ) cosh2ol

(n12+k12 ) (n2+ne )2

+ 2npn1sinh2a1)

+ (n12-n2z+ll"t2) i (ne2-n12-kt2)cos2y1 + 2n¡klsin2yrI

3. 8.3

]-+RI-T-

f-Rt = Znz

[n1{(ne2+n tl+kf)sinh2o1r (nr z+kf) (n2+n6 )2 l.

+ 2ngn1cosh2a1)

+ kr{ (ni2-ns2+k12)sinzyl + 2neklcoszvr}l

3. B .l+

Itre method of solving equationg 3.8.3 snd 3.8.h is itlentical

to that enployed. for solving equations 3.\.3 srid 3.1+.4. The

multiple soLutíons that occur are slightly different in magnitud-e

to those obtained. from equations 3.\.3 arld 3.\.b, with the exception

of the correct solutions which are id.entical. The effects of filn

thickness are id.entical to those ctiscussed in section 3.7.

3. ro AITERNAT]VE DERIVATTON OF lTfE ESUATIONS FOR (lrR)/T AND (trnl)/r.

A sinpler cierivation of the equations (ftn)/f ana (ftnl)/f can

be achieved by coneidering thc forrnulae for R, Rl a¡rcl T given by

48.

Ábeles (fg6S) which are written in a different form to those given by

Heavens (loc. cit.). Ítris d.erivation is given in Append.ix B. I'Ie

have retained. the nomenclature of Heavens in this chapter for

consistency with the two-layer equations consid.ered. in the next chapter.

l+9.

CHAPT}IR I+.

SURFACE LAYERS ON THIN FILMS

l+. l- INTRODUCTfON.

Althougþ accurate measurements vere taken of the thieknesses

of the seleni!:m fitms, it was found. that the meagured' value used in

the calculation of n and. k, correspond.ed. to the situation of too large

a film thickness being used. This behaviour was also apparent with the

gemaniun films which were d.eposited. on hot gubstrates.

Electron microscope exarnina,tions of the surface of vitreous

selenium fiLns (canpbell, 1968) <lid" not indicate any definite surfa,ce

rougbness which woul-d. cause some scattering of the incid.ent beal¡,o

Hovever macroscopic r:ndulations d.id exist which vor¡l-d. mean that the

beesr trans¡aitted. througb the film traversed. different thicknesses

of the fil-!x.

Tn the present investigation of genaanium fiins, sueh r.¡nd.ulations

were reðucecl consid.erably d-ue to the use of a rotating substrate

during the evaporation of the fil¡ns. Tktus the non-uniforoíty of the

films ùid nct appear to explain the necessity to reduce the measured'

filn thicknesso

Carrpbell has also suggested. that overcoating layers may give a

meagured. thickness value less tha¡l the geometrical height of the stcp

in the fil:n d.ue to the tend.ency of the overeoating naterial to

nigrate tor.¡ards the uncoated. portion of the substrate. Ttrus the

presence of the overcoating silver layer in the measurement of the

fil-m thickness is r:nlikeIy to resrr-l-t in an overestinate of its

thickness.

5O

ftre second. problen encountered. was the non-closure of the

refractive ind.ex curve over. parts of the vavelength range. By

closure is meant the re¡roval of the island or vafley t¡le apFear.ances

of the ealcufatecl solutions as vas discussed in the previs¿g chaptert

and. the production of a snooth single-valued d.ispersion curve. This

is denonstrated. in Figure h.L where the refractive index is plottecl

fcr a selenir:rn filn, anci a geruranir:m fifun deposited on a 35OoC

substrateo llhen the filn thickness rvas adjusted. to give the correc't

behaviour in the Icng wavelength region, then the short lravelength

region beha.¡iour eorrespond-ecl to that cf tco smalf a value of filn

thickness. Likevise vhcn thc thickrless was increased. such that

correct behaviour wos obtained,. in the short wavelength re¿{ion, the

long wavelength region exliibited. the behaviour of too large a value

of thickness. No filnr thicicncss existed whieli would. close the curve

over the entire vavelenpçth rr:,nge.

Both of these problems were solved by consid.ering the existence

of a thin oxid.e layer, or sorie other Iayer, cn the surfecc cf the

fil-i':. potter (1961+) in hi-s calculations of the optical consta¡rts of

genranir.:nr correeted for en oxirLe layer 501 tU.et. Dash (f95\) fras

shom. that at least a menclayer and prcbably nc're is fomred very

quickly on an a,norphous or crystalline film of genraniu:r. Recent

r+r:rk by a number of authors in surface stete scienee support the

find.ings of Dash (see for exr:ra;rle, Henzlcr, 1971). llrus evidence

cæists which inùicates that such oxid.e layers d.o exist for ge:rnanium.

A scarch of the l-iteratur+ rlid- nct reveal a¡ry work on l+hcther oxid-c

Iayers also forned on seLeniwü. lIowever, postulating the existence

SELË:NIUM

Xujo1

ul

l-0trtLUJ

tr

WAVELENGTH IN î/IICFIONS.4

XuJo4

IU

l-0

trILtxg

GEtrlMANIUM

t.{\A/AVELENGTH

1.2l'8

Figure 4" L

IIJ MICFIONS.8

5l-.

of such a lo.yer solved both nroblems mentioned. above and is good

evidence that selenir¡¡r oxid.e, or ó, compound of eeleniurn with a

sinrilar refractive ind.ex, does for¡n on the surface of selenium fiI¡rs"

Papazian (fgf6) has found. that the oxid.e forned on germaniun is

usually hexagonal Ge02 vhich he found. had. a strong absorption bancl at

Z\!ia"A. Garino - Canina (fg:$, 1959) and Cohen anC S¡rith (fgf8) found

sinil-ar results. Thus in the wavelength range of the present Írêâsüre-

ments, the gennanium oxid.e layer is non-absorbing.

ltre oxid.e of selenium was also assulred. to be transparent over

the wavelength raJrge of measurements. A literature survel/ did. not

reveal any previous optical ahsorptj.on meesurements cln this compound.

Ttre errcr introd.uced. by pcssible absorption in the oxid.e layer not

bein6 accounted. for vould be very small since this layer is very thin.

Thus the problerc reduccd to considering the systen: tronsparent-

film - absorbing filn - transparent substrate. Although l{È &re

concerned prinarily with gemor¡ir:rc a,¡:d seleniu¡l and their corresponcting

cxidcs, the analyses presented in this chopter are applieabfe tc any

such systen. Ttre systert: absorbing fil.ut - transparent filin -trcnaparent substrate is also considered (for reasons discussed in

section h.tt) and it lrill- be shown that the tvo systens nay be

read.ily d.ifferentiateC. We thcn proceed. to solve the two*layer sys'ben

rn¡here the refractive ind"ex of the transparent layer is knol'n.

Provided. the system is thick cnough to exhibit at leest one turning

point in the refl-ectance anC. transmittance eurves, it is sholrn huw the

optical constants of the absorbing fiJ-n together witlr the thicknesses

of both the absorbing eind. transparent layers may be accurately

,2,

d.etemined. from the measure¡nent of the refleete¡ce ancl transmittance

of the system"

I+" 2 REFLECTIUICE AISD TRANSIUITT.A]üCE FOR A TIIO LAYER FILM.

Consid.er the systern sholrn in Figure l+,2

ng is the refractive inclex of airr

n1 - ik1 is the coinplex refractive index

thickness d1.

n2 - í1r2 is the cornp-lex refractive index

thickness d,2.

n3 - ik3 is the cornplex refractive ind.ex

Fron ïIeavens (].)>S)z

no2-nt2-:¡-tZ

vhere

of the first layer of

of the seeoncl layer of

of the substrate'

2n9k1g1 hr=

(ne+n1)z + kf

nLZ-n22+!-J-:l'z2

(ns+n1)2 + kf

82

83=

(n1+n2¡z + (trr+t2)2

p2 = edlcosJl

t2 = s-ot(g2cosy1 + h2sinyl)

o1 = 2nk1d1/)t

PtZ=PZ+ 8It2 -htuZ

tI2 = t2 + gyÞ2 - h1Ç.2

nz2-ng?+}.zZ-:r.g2

h2 = 2(n1kr - n2k1)

(n1+n2)2 + (t<t+6r)2

qz = e0lsinyl

u2 = 6-01 (h2eosy1 - g2sinyl )

y1 = 2rn1d1/À

QLZ=q2+}:¡t2+ßtuz

ü12=u2+h1p2+Bl8Z

Z(n2lr.s - n3k2)h3=

(n2+n3 )2 + (kz+ts ) 2(n2+n312 + (tcr+tr3)2

u2 = 2rk2d.2/1, y, = 2nn2d2/),,

,AIR Í'l 6

OXIDE Í11 - ikr 1

FIEM ['!2 - I $qe

SUBSTRATL= rl3- ãks

T\A/O-LAYER SYSTËM

Figure 4.2

d

d

)(

2,

53,

p3 = eo2cosyz q.3 = ed2siny2

t3 = s-d2(g3"osy, + h3siny2) us = e-d2(h3eosy, - e3siny2)

12 = edl(g2eo"y, - h2sinyl) sz = eol(hrcosyl + grsinyl)

v2 = s-dlcogyl w2 = -s-o1slnyl

l.Lz = Tz + gI:uz - htwz slz = s2 + hlv2 + Etwz

.rtz = v2 + 8rT2 - htsz w!2 = wt + htrt + 8f z

P13 = Pf z!¡ - gtzo.g + rl2t3 - st2u3

Q13 = QrzPS + PtzQg * s¡rt3 + r12u3

tI3 = tfZPg - utZQ¡ + vtrte - wtZu3

ü13 = utZPg + t12Qa + l¡I2t3 + vt2u3

113 = (r+gr)(l+ez)(r+sr) - hzhs(l+er) - hghr (t+ez) - hrhz(r+g,¡

rrr3 = ht(1+g2)(r+gr) + hz(r+ge)(l+er) + hs(t+er)(r+gr) - rrtnrn,

ftren the refleetance and tra¡rsurittance of the ctout¡1e-Iayer system

are given by

R = 1-¡r2 + urr2

h.2.1prs2 + qls2

ng lt t2 I mtg2l+ n2,2T=

ng pI g2 * qtz2

As in the single-layer caÉee if a wed.ge-shaped transparent

substrate is used, the measurcrl tranrlnittance ratio, I, is refated-

toTby

r = (ng+no)2 . 1 \.2.3

\n3n9

5l+ .

l+. 3 TT{E HYPOIHETTCAL FIL}ß.

As in the d.iscussion of the effects of film thickness ín

section 3.?, the t+ro-layer system was best analysed' by consid'eringt

first of all, hy¡rotbetical fil¡rs. Íhe transparent atrd. absorbing films

had the following proPerties:

Tra,nsparent film refrÐ,ctive index = 2,O

Absorbing fil¡¡ refractive inrLex E l+.0

Absofbing film absorption index = 0,25/^3 (r in p)

Tre"nsparent substrate refractive index = 1.h5

Ttre properties of the absorbing film approximate those of gemanir:m

ancl the refractivc index of the transparent layer approximates that

of gerrnanium d.ioxid.e.

The two systems nentioned above are d.esignated thus:

System 1 : Traneparent filn - absorbing filn - transparent substrate

System 2 : Absorbing film - transparent film - transparent substrate

The reflectanees and tronsnittanees of both systems for various

transparent and. absorbing fil-n thicknesses were caJ-culated from

equations \,2,I and. l+,2,2, TTre refLecta¡rce and transmittance curves

so obtained are shown in Figure l+.3 for Systen 1a¡rd- Figure l+'l+ for

system 2. It can'be secn that the transparent layer has the greater

infLuence in System I lrhere the rroet sigrrifican'b differences occr¡r in

the short wavelength re6ion of the reflectanee eurves, particularly

rrith the thicker absorbing fil¡rs, and. become more pronounced' as the

transparent layer thickness is increaserl. The transparent layer has

Ð,1esser influence on Systeu 2, particutarly in the short wavelength

ÌeglOII r

- dr:aÂ.u

..... dt = tooru.

T

FlSlÉM I d¡' rool.u

l-

!a

F

\^/AVELÊNGT}{ IN MÉRONS

WAVELENGTH IN MICFIONS

IN MI:fIONS

F

-T

syslEM I d¡ =rû¡o^.u

A,..,...p'-'*-'=t ':'-

--dt=s¡.u..-.dt=zoo¡.u'- - - dr=roolu.

SYSÎEM 'l dr. zoeo ¡¡,

T

-¡h= ¡ 4-s.. . dr=¡ooÀu--- dr-røl.u

tI'

F igure 4.3

ß

-d2.aN" . d¡.ou3Fl[I r dt, ¡o u

wavEttÍclt rfl HrqRcìs

w^vÉttðGÌH lx ÀlrcRoils

WAVEIT?IGTH II urcRofl s

sYsrtH 2 dts urÀu

I

-d2='""dr.-'- d¿=h0Ma!0Àu

t¡[

t-..!'.:'.;,.

Figr:re 4.4

55.

J+. 4 EFFIrcr OF USING THE STNGI,E - I.AYTìR EQUATIONS,

\. l+. 1 SYSTEM 1.

Consider a hypothetical film vith absorbin6¡ layer thicknessqo

2ooo.,1 and transparent layer thickness IOA. If the single-1ayer

equations are notrr used to d.eten,ine the refractive index and. absorptiono

index for the above film assuming a thickness of 20104, r,¡hich i,¡ou1d. be

the value obtained. if tne thickness waË rlÌcasured. directly, the solutions

for the refractive ind.ex so obtained. are shol¡n in Figure l+.5,a, One

can see that this is the behar/íour exhibited. when too large a film

thickness is uSed. in the calculation. lühen the fi1¡r thickness iso

reduced. to 201!4, the refractive index eurve shovn in Figure h.5.b is

ol¡tained. llith the exception of the short davelength rcgion, where

there is a fatl in the refractive index, the values obtoined. are

negligibly d.ifferent fron the exact t¡afue of l+,0.

fhus a transparent oxid-e 1a¡rs¡ vitl explain the need to reduee

the neasured. fil:n thickness in order to obtain the correct behaviour.

Consid.er now the case where tire absorbing fi1rû thiekness is 2OOOÎ

obut the transparent layer is 2OOÃ thiek. Figure \,6,a shovs the

calculated refractive ind.ex eolutions vhen a thickness of eaOoi is

used ín the single-Ioyer equationso When the thickness is reduced

oto 2O5OÃ the curve shown in Figure l+.6.b is obtained. Ttre curve rs

closed in the long vavelength regíon, but not in the short wavelength

regíon, anct no adjustrnent of the fil¡r thiekness wilt gi.ve a curve

which is closed over the entire wavelength range, In the figure

shorrn, the short wavelength region has the anpe&rance of too small a

v¿1ue for the fitrn thickness. If the thickness is inereased so that

(alo

d=2050 A

.6

WAVËLENG'TH IN1.2

MICRONS

xo=

tsoÉ

É

xoz

ìÞ-()

É

E

t.6ã0

rb:

od=2015 A

WAVELENGTH IN MICRONS

Figure 4"5

5

XLL¡o

=IJJ

ìFO

É.LÙJÉ

4

3

2

2.2 1,8 f'6

WAVELENGTH

1.4 1'2

IN MICRONS

1.0

1.0 oo

I ,6

5

xLUôz

LU

IFO

É.u-LUÉ.

4

3

2

2'0 1.8 1'6

WAVELENGTH IN

1.4 1.2

MICRONS

62.4 2.2

Effect of using the sÍngle-layer equati.ons fora 2000 A.U. f íÍnn r¡ith an oxide layer 200 A.U. thick.

Fígure 4"6

-o(a)d=2200 A

o(b) d=2050 A

56.

cl-osure occurs in the short wavelength region then the long wavelength

region has the a,ppeara,ïtce of too large a fil¡r thickness.

Ttre effect of a transparent oxide layer on a hy¡rothetical film

vith opticol congtants approxir:rating those of selenium showed id'ent-

íca} behs,viour, except that the effects l¡ere more pronounced'. Ttris

is d.ne to the refractive index of selenirrm oxid.e (t,76) being compar-

able to that of selenium (Z)+l) r¿hereas the refractive ind'ex of

ge:rcanium oxid.e is 2.0 ut¡ire that of gemanir:m is \'o'

TÌrus the presence of a trensparent oxicle layer on the surface

of the fih wi1f produce both the problems discussed in section 4'1'

\. \. z SYSTFJ'I 2.o

For the case where the absorbing layer is 20004 thick and. theo

transparent layer 50Ã thick, the effect of using the single-layero

eguations assr,uring a thickness of 2O5OA is sirdl-ar to that for

system 1. Íhat is the refractive indcx solutions have the appearance

of too }arge a value of fiLtìl thickness being used' in the calculations'

If the thickness is reduced. to 2015f. ,n"rr, as vith System 1, the

sol-utions are negligibly different from the value h.O, exeept in the

sbort wavelength region r¿here there is a elight rise ín the refractive

index (conpared. to a faII in System 1).o

For the case'where the thickness of the absorbing layer is 2oo0A

obut that of the transparent tayer is 2OOA, the fil¡r thickness again

appears too large íf tne single-layer equations are used. assuming a

thickness of 22OOi, (Figure l4.?.a). If the thickness is reduced toU

2O5OA, the refractive ind.ex sofutions shown in Figure l+.7.b are

Þ

RE

FR

AC

TIV

E

IND

EX

RE

FR

AC

TIV

E I

ND

EX

€ z o { z = õ Ð o z

ryl

ll. to H o .Þ \¡

€ Þ c) ¡ I = o - z

,7,

obtained. the curve is clogeil in the long wovelengbh region while

the short vavelength region has the ieland. ty¡re appearance character-

istic of using too large a value of fíJrn thickness, ff the thickness

is reduced. to give closure in the short wavelengÈh region, the long

wavelength region has the appearanee of too small a value of fifun

thickness bein6 used in the calcuJ-o.tions.

ÍÌrat is, the behaviour is just the opposíte to that for System 1'

Thus, for mod.erate transparent layer thicknesses it is possible to

d.:'.stinguish between the two systems.

Systen I characterised ttre behaviour observed for the eelenium

and. geroaniun films. For very thin oxid.e layers, the results obtaíned-

using the single-layer equations are negligibly different fron the

correct valuee of the absorbin6¡ fil-ul, provid.ed. the total film thick-

ness is reduced in ord.er to obtain a closed refractive ind.ex cürvê¡

For moderate oxid.e thieknesses the use of the single-Is,yer equations

no 1on6çer produces accurate results. It vas therefore necessary to

attempt to solve the two-layer equations.

)+. 5 SOLVING TTTE TT'IO - LAYER ESUATTONS.

Equations l+.2.I and h,2o2 can be solved for the refractive index

of tr1re absorbing film, the absorption ind.ex of the absorbing fiL:n and-

the transparent e¡d absorbing filn thicknesses, from the measured'

values of R and. T and. given the refractive ind.ices of the transparent

layer end. substrate.

consid.er system ] vhere ni is the refractive ind.ex of the

transparent layer of thickness d1¡ n2 and k2 are the refractive and

58.

absorptíon ind.ices of the absorbing filn of thiekness d2'

From equations l+.2of a¡rd \,2,2

1+R n¡ (ptz2 +qrg2 +ttzz +u132)

T n3 (trs2+n132)

t - R n6 (prg2 + qr32 - ttz2 - tis2)

l+,2,3

l+.2.h

T n3 (rrs2+14132)

As with the single-layer equations, these tvo funetions are

single-valued. in k2 given thc other paraneters. If l¡e clefine the

following functions:

F1 (n2 ¡k2 ) n¡ (Þr g2+'q.r 32+t r 32+ur 32) - n3 (11 32+ml 32 ) i (r+n) /t)

F2(n2-rk2) n¡ (pr 32+c{r z2-ttgZ'-ur 32) - n3 (1132+mt g2) { (:-n) /r}

then the solutions are those n2 and. k2 thich satisflr

F1 (n2 ,k2 ) F2 (n2 ¡k2 ) 0

sinul-taneously, given the other paroneters'

F2(n2rk2) is no longer readily differentiabLe with respect to k2 as

rras the correspond.ing function in the single-layer caÉQr For a

given n2 there ís only one k2 which satisfies F2(n2¡k2) = O'

Consequentially for a given n2¡ k2 which satisfios F2(n2rkZ) = O was

found. by allowing Kz to start from some preset value and increnenting

it until a change of eign occurred in Ir2 (n2rll'2) ' An iterative

interpolation was then used to find. the intersection poirrt.

Tfhe set value of n2 ond calculeted. k2 'were then substitutecl into

the equation F1 (n2ek2), n2 w&s then incremented and the process

repeated. u¡rtil a chenge of sign occurred ín F1(n2;k2-), md again an

59,

iterative interpolation was used to d'etermine the intersection point'

Fwther increments were attd.eil Lo n2, until some upper limit was

reached, to d.etennine alt the possible solutions'

TLre above rnethod. vas used. to caleulate n2 anci k2 for the hypo-

thetical fil-n arith treneparcnt and. absorbing J-ayer thicknesses ofoO

hgOA erra 2OOOA respectively. 'Ihe results for n2 âre shown in

Figure \.8. At each wavelength the correct sol'ution (ì+.0) was founcl,

end. again nultiple solutions occurred as with the oingllc-Iayer film'

It was also found. that only the correct values of the tvo fil:n

thicknesses would result in closure of the refraetive index curve

over the entire wavelength ran6¡e.

For Systen 2, where n1 and. k1 are the::efractive and absorption

ind.ices of the absorbing fil¡r of thickness d-1 anit n2 is the refractive

index of the transparent fi]jt of thickness d2¡ gíven the other

parametergr trl anrj. k1 may be forrnd. in exactJ-y the salnc manner as ÌIas

n2 and k2 for system 1. Again only the correct values of the tvo

fil¡a thicknesses vou1d. result in complete closure of the refraetive

ind.ex cu.flrê r

l+.6 -A?PLrcATroN TO RII.AL FILI\4S.

In practice it is not possible to ueasure explicitly both the

transparent and. absorbing fim tnicknesses ar¡d' one therefore has two

equations, (ftn)/T, and" three unknowns, the absorbing fíln refractive

and. absorption ind.ices arìd. eitl¡er d1 or d2 (ttre total filn thicknesst

d.r+d.2r maï be measured. expl.ícitty). Ilowever the refre.c'cive index

eur.ve will nOt cloge unlcss accurate values of d.1 a':rd d'2 are usecl, and

¡ f r ¡ r ¡ ¡ r ¡ ! r ¡ ¡ i ¡ r i r ¡ I' t È t t' t rlt'f ¡ "

t f t ¡ ¡' \ l' I ¡ ¡ ¡ I r t t t t I ¡ I ¡

1rI

I Calculated solutions for ahypothetical tt'torlayer film with

dr = 400 A..U.

dr = 2000 A,U.

I

xuo

U¿t--()ôlUJ

l. .8WAVE TH IN

Figure 4"8

NS

60.

this fact may be used. to d.etemrine both d1 and. d2. If the total film

thickness is knovn exactly, tlren d1 and. d-2 may be altered, such that

df+d.2 remains constant, until closure oceurs, thus giving the optical

consta,nts of the absorbing fi.lnr together with the trarisparcnt film

thickness and absorbing film thickness.

It is not even necessery to knory d.1+c12 exactly. An approximate

value for cl1 and. d.2 may be estimated. frorn the solutions obtained from

the single-Iayer equations. l'hat thickness which gives closure in the

long wavelength region is slightly greater tha.n the absorbing fil-n

thiekness, and. the extent of the non-closure gives a good, ind-ieation

of the transparent layer thickness. t'hese values may then be substit-

uted. in the two-l-ayer ecluations and. odjusted. vithin sma1l linits '.mtil

closure of the refractive index curve occurs. llre rnethod. adopted here

was to increase the transparent layer thickness to close up the short

r,ravelength region, ¡¡hile at *,he sane tine redueing the absorbing layer

thickness to retain closure in the long wavelength region. After gome

experience eomplete closure coufd be achieved. in about four or five

steps.

h. 6. l- TTilMPLES FOR GERI'4ANTUM AND SELENTUM.

As mentioned e¿¡.rlier, the behaviour of t?re germanium and'

seleniun fíIms were characterised. by Syster.r 1. The refractive ind.ex

curves for the seleniun and. ge:meniu¡a films discussed. in section \.f,

obtained. frolr the C.ouble layer eguations (Systen 1) are shown in

Figure l+.9. The values of absorbing filrn and. oxid.e layer thicknesses

required. to give closure of the eurves are shown on the figures

xoz

ts()É

SELENIUM

d, = d$t5 A"U.dr= 930!5 A.U.drnou.* 942t49 A'U'

.4

WAVELENG IN MICRO

xôz

:tsoÉ

É

GERMAl'i!UM

d, = t00!5 A.U.dr= It4$rb A.U.d.þos.= I209J65 A.U.

WAVEi-TIIGTH IN MICAONS1.8

Figure 4,9

6r.

together rrith the measured total thicknesses.

It ca¡r be seen that the inclusion of an oxid.e layer enables

closure to be achieved over the entire wavelength range. Also the

calculatecl total filn thicknesses are in agreement, within experiroen-

tal error, vith the measured vaLues.

\,6,2 EIry'ECT OF ERRORS IN R AND T ON TTIE CALCUI,ATED FILM AND

SURFACE - LAYER fiÍICKNESSES.

From our calcuLations on hypothetical films a^nd. ge:meniun fiJ-ms,

the closure of the refractive inctex euryes appeared to be very

seneitive fr:nctiong of fi]¡n and surface-layer thicknessee. As with

the single-layer calculatíons in the previous chapter, it Ìfas necesss'ry

to d.etermine the effects of errorg in R and' T on the eafculated

thicknesses.

Ttris was again most suitably deternined. from the use of hy¡rothet-

ical fi1ms. The calculated. reflectances and transmittances for films

of varying thicknesses 1Íere aLtered in the four following I'fays:

(r) R+o.oo3 T+o.oo3

(z) R-o.oo3 T-o.oo3

(E) R+o.oo3 T-o.oo3

(4) R-o.oo3 T+o.oo3

Cases (1) anA (2) did liot affect the filn or sr¡rface-Iayer

thicknesses. For case (:) ttre fílm thickness had. to be reduced

by 0.5/" vhíLe in case (h) it had to be increased by O,5'iÅ, in both

cases the surface layer thickness remaining u¡raltered.

62,

l+. 7 EKPLÏ crr Ð(PREssroNS Fon (ItR)/T.

one drawback in solving the d.ouble-layer equations is the

reJ-atively large computing time recluired, being about twenty tines

that requirett for the single-layer equations. lhis increase in tine

is brougþt about largely by having to increnent k, for a given nt

until it satisfies F2(nrk) = O instead. of using Ner'rbonts nethod' as in

thc single-Iayer case.

Folloving the success of explicit expressions for (ftn)/T for the

single-layer film which afford.ed. a large reduction in cornputing time,

TourLin (fgff) has derivefl explicit e:çressions for (ftn)/T for two

absorbing layers on an absorbíng substrate. From Tomlín we have:

l+R (n¡2+n12+kr2)Fr + (ns2-n12-uf)pz

l6nsn 3 (n1 2+k, 2 ) (n22 +:r22)

Fr = {(n1+n2)z + (tr+k2)2-}{A cosha(o1+sr) t e sinh2(oi+o2)}

+ { (n1-n2)2 + (kr-k)z\{A cos}r2(o1-o2) - s sinh2(o1-c2) }

+ 2(n12-nr2+kr2-l<r2)coeh2o1(C cos2y2 + D sin2y2)

+ l+(n1k2-n2k1)sinh2o1(C sinZy2 - D cos2y2)

Fz_ = {(n1+n2)2 + (ki +x2)2}{c cosz(yrnz) + o sin2(v1+v2)}

+ {(n1-n2)z + (ki-u)z\{c cos2(ytt) - D sin2(vrrr)}

+ 2(n12-î22+k.2-kz2)cos211(A eosh2oz + B sinh2c2)

+ l+(n1k2-n2k1)sin2y1(R sintrZc2 + B coshZo2)

4.7.1T

where

A.=n22 +n32 *kZ2 +kg2

C=nz?-n32+kz?-ks2

B=Z(nzns+k2k3)

D = 2(nzk - n3k2 )

and.

63,

I-R\,7 ,2

rvhere

Gr = {(n1+n2)2 + (trr+t<r-)'}{a sinh2(o1+qr) + B cosh2(o,1+c2)}

+ {(n1-n2)2 + (tcr-xz)2}{a sintrz(o1-42) - s eosh2(qr-cz)}

+ 2(nf-nz2*r2;k22)sinh2o1(C cos2y2 + O sin2y2)

+ l+(n1k2-n2k1)cosh?or1(C sina12 - D cos2y2)

Gz = {(n1+n2)2 + (t1+uù21{c sine(yrIz) - D cos2(vr*ve)}

+ {(n1-n2)2 + (kr-t z)2}{c sina (yryz) + o cos2('¡1-v2)}

+ 2(nr2-n r2+k¡2Jx22)sin2y1(A cosh2o 2 + B sinh2a2)

- \(n1k2-n2k1)cos2y1(a sinnecrz + B cosh2o2)

Ttre symbols have their meaning given in section l+.2.

h.7.1 EXACT EXPRESSIONS FOR $TE CASE k' = ka = O. SYSTEI'{ L.

For the caee where the first layer and. substrate are transparentt

the following expressions can be derived' frorn equations l+.?.1 and \'T'2¿

1+R =

(no2+nt2)rrr+(no2-nrz)¡'rt [.?.3

T 16nsn3n12 (n22+k22)

r.¡hefe

F I I = 2 (n¡?' +n22 +U22 ) [ (n22 +n 3z +t<22 ) eosin?:a2 * 2n2n 3 s í n}t?a2l

+ 2(ni2-n z2-k22){ (n22-n3z+:nz2)cos?y2 - 2n3k2sin2y2l

F2r = { (o1+rr, )2 + kzz} { (n22-n3z'+|l.zz)cos2(yr+yz ) - 2n3k2sin2(vr+vz) }

+ i (n1-n2 )2 + :rzz] { (tt22-tt, z+}'.zl) cos2 (v r-vz ) + ?-n3k2sin2 (v r-vt ) }

T

n1G1 + k102

Bn3(n1 z+k.2)rr,22*t*2\

6\.

+ 2(n12-n r"-tr,)cos211 {(n22+nrz'+x22)cosh2o2 * 2n2n3sinh2o2}

+ \nlk2sinavr i(n22+n32+k22)sinhza 2 t 2î2nscosh2o2i

11-R

T 2n3(n22+k22) þr{ (tr'* n32+,'22) sinh2a2 + 2n2n3cosh2c2}

+ kz{ (n22-ns2+u22)sín2y2 + 2n3h2co"2Yr}l

)+.7.\

Equation l+.?.\ is identical to the equation for a singte film on

a transparent substrate, and is thus read.ily differentiated with

respect to k2. consecluentially the metl¡od, for solving the two-layer

equations, where the first layer is Ùransparent and is of knom

refractive ind.ex, is id.entical to that employed for the single-layer

equations, For a given n2, k2 may be solvcd by Newbonrs method from

equation l¡.7.1+ instead of the increnental rcethod discussed- in section

\,r, Ttris resul-ted. in a reduction of eomputing time by a factor of

ten.

l+, '1, 2 EXACT Ð(PRESSIONS FOR TTIE CASE KE = ke = 0o SYSTEM 2.

From equations )+.?.f and \,'(,2 for the case k2 = k3 = 0 we

obtain:

1+R (ns2+n12+kr2 )Frz + (ns2-n1 z-trf)F 22_ \.7 ,'

Bngn22n3 (n12+k12)

Ft2 = (n2-2+n32) (nf+n22+t12)eosh2c1 + l+n2n3sinh2c1

+ (n22-n32) {(n12-n zl+krz)cosh2o1 eosZy2 - 2n2k1sinh2a1 sin2y2}

T

where

6l+.a

Fz2 = (nzz-ns2){(o12+nrz+kt?)cos211 cos2'¡2 - 2n1n2sin211 sin2y2}

* (n22+n32) (n¡2-n22+l<f)cos2y1 - hn22n3k1sin2y1

ît?tz + k¡G22 \.7,6\n3n22(nt2+t12)

where

Gtz = (n¡2+n22+E¡2) (n22+n32 ) sinh2oll + l+n1n22n3cosh2o1

+ (n22-n32) { (n12-n ,2+k¡2)sinh2o1 eos2y2 - 2n2k1cosh2o1 sin212}

Gzz = (nzl-nz2) { (n12+n, z+xtz) sin2y1 cos2^¡2 + 2n1n2cos2'¡1 sin2y2}

+ (n12-n2z+kf) (n22+n32)sin2y1 + l+n22n3k1eoË2y1

Althougþ not as sinple as equation \.f .h, ecluation h.7,6 may

again be readily differentiated. with respect to k1 ' Tlrus the method

for solving the two-layer equations, 'where the second' layer is

transparent and. is of knorrn refraetive index, is the ss,ne as for the

previous case with a consequent reduetion in computing timc'

l+o T. 3 ERROR AI{ALYSÏS FOR THE CASE K 1 - Â? = 0. SYSTm/i 1.

Since the two-layer equations (Systen 1) had' to bc employed for

the selenium and germanirun films, it was d.esirable to calcul-ate the

errors in the calculated refractive index and absorption index

solrrtions due to errors in the other parameters. The method. is

id.entical to that employed. for the single-1ayer equations. It was

founcl that the error in the bubstrate refractive index contributed' a

ne¿E1igible a¡oount to the total erl'or in the calculated solutions'

fhus this variable was eliminated and was replacecl by the oxid'e

thickness. The errors in the absorbing filn and- oxidc fil:n thicknesses

RI

65,

vere estinated. from the accur&cy l¡ith which the refractive ind'ex

curve could. be cfosed over the entífe 'n¡avelength range' The error in

the oxid.e layer refractive ind.ex vas assunèd' to be zero. Ttre partia'l

derivatives for the two-Iayer ectruations (Syste¡l 1) are given in

Append.ix C.

h. B USE OF 1TITI SINGLE-LAYER EQUATTONS FOR A ÎWO-LAYER FTLM.

In generaL there is no single-Iayer film equivalent to a t\'¡o-

layer filnr even if both layerÉt are transparent ' Hor'¡ever for the

special case r¡here one of the layers is very thin, as is generally

the case for oxid.e layers, then in the transparent region it can be

shom that the single-Iayer equations may be used provicLed' the total

film thickness is reduced.

we d.efine a very thin film as one whose thickness is sueh that

Y<0.1

Tlre two-l-ayer equations, for the case k1 - k2 k3 = 0, then

become:

1+R F ll.8.1T 16n¡n3n12n22

where

p = ( n e 2 +n 72 ) lz (n¡2 +n 22 ) (n2z +n 32 ) + p- (n ¡2 -n22) (n z2 -n z2 ) cos 2"¡ 2 ]

+ ( ns 2-n1 2 ) ((n22 -n32 ) { ( n1 +n 2) 2 cosl(y r +vz ) + ( n 1-n 2)

2 cosl( v r-vz ) }

+ Z(nf -n22) (n22+n32 ) cosaYt)

66,

\. B. I SYSTEÀ{ 1 : L\YER VERY TTÍTN.

For a single fil:n with refraetive index n2 and. thickness d2 on

a substrate with refractive index n3r tben

L+R IE:

-- l(ns2+nrz) (n22+n32) + (no2-n22) (n22-nrz)cos2lz!

T l+ngn3n2z \.g.2

T.f \2 is increased to \z + ^(O vith 19 << 1 then

I+R I=

-:

{(nçz+nr2) (nr2+n32)T \nsn3n22

+ (ns2-n2 2) (nzz-nzz) cos?(vz+vo ) Ì

L

{(n62+nrz) (nrz+nsz)l+n6n3n22

+ (ns2-n22) (nzz-ns2) (cos2^¡2 - 2yesín212 ) i

h.8. 3

From equation h.8.1 vith Yt << 1 we have

F = 2(n02+n12) {(n12+nr2 ) (n22+n32) + (ni 2-n22) (nzz-ntz )cos2y2}

+ (ne2-ni2) { (n1+n2 )2(nzz-nzz) (eos2y2 - 2.¡¡sín2y2)

+ (n1-n2 )z (nzz-nt?) ftos2y 2 + 2y ¡sín2"¡2)

+ e(nr'-nr')(n22+ry2)l

1cê¡

Hence

L+R

F o 4nr2 (ng2+n22) (n22t-n32) + l+n12 (no2-nzz) (n22-nt2) eos2y2

- l+n1n2 (no2-nr2 ) (rz2-nt2 )211sin2y2

T

I;..--.-.; l(n22+nrz) (ns2+n22) + (tto2-n22) (n22-n32 ) cos2y24n6n3n2¿

- 2Yrff (noz-nf) (nz2-nz2)sinavr] \'B'l+

Conpariug equation \.8,h r+i'Uh equation l+.8.3r t?re condition for

equivalence becomes

yo n2 (noz-nf)

n1 (n62-n22)

67,

1¡€¡

;

do,

d.1

(no2-n¡2)

(noz-n22)

vhere d.g is the equival-ent thickness of the thin first layer (of

refractive index n1 and thickness a1) if it is considered' to have a

refractivc index n2.o

For exa,urple if nl = 2.0, n2 = \'or dr = 1004, tro = 1, theno

dO = 20Ã. ftrat is, the timeasuredtr film thickness hes to be reduced

oUy BOã.. Ttris supports the computations in seetion h.h rvtrerc a

red.uction in total filn thickncss ga,ve a, cfosed refractive ind.ex eurve

in the infra'redo

\. B. 2 SYSTH{ 2 : VERY fiIrN SECOND LAYER.

For a single fil-n of refractive ind.ex n1 s.nd thickness d1 on a

substrate with refractive index n3.

L+R I= " i(ne2+nr2)(nt2+n32) +(ne2-n12)(n12-n32)cos2y1]

T l+n¡n3n 12

If yt is increased to Yr + Yo uith y9 << 1, then

L+R I

-

{ (ne2+rt12 ) (n12+n32)hn6n 3n1?

'r (ne2-n12) (nr2-n¡2) (cos2v1 - 2yesin2yr ) lT

\.8.5

68.

From equation \.8.1 with \z << 1 we have

I+R 1

-

o : i(ns2+nr2) (n12+n32) + (no2-n12) (n12-n32)eos2Yr

T hn¡n 3n1z

- å1t"0'-n¡2) (n22-n32)2v2sLtt2vr l l+"8'6

Conparing equation l+.8.6 with equation 4'B'5 the conditj'on for

ecluivalence is

Yo =

rt1 (n22-n32)

\2. n2 (nt2-n32)

ioe.

do

d,2

(nrz-n32)

(n12-n32 )

rshere d"g is the equivalent thickness of the thin leyer if it is

consideretl to have refractive index n1 'o

For example, if nl = h'O¡ n2 = 2'ot dz = lOoAr tr3 = I'Jt then

odO = t5Ã. Ttrat is, the ltueasureiltt fi]-m thickness would. have to be

reduced ¡V S:l it tfre single-layer equations were used, which is agaín

in a6¡reement witlr the eomputationg.

ft is aleo Iúorth noting that if the refractive ind-ex of the very

thin layer wa,s greater than that of the thick Iayer, then for both

systems the total filïr thickness t¡ould have to be inerea'sed to alfow

the use of the single-layer equations. Tkris also shows that the

surface layers on the germanium and seleniun tilns have a refraetive

ind.ex lees than that of the gc:rnani¡m or seleniu$ respectively, in

agrcement rr¡ith the original ossuinptions 8,s to the nature of these

layers.

69,

l+, 9 CHOICE OF REFRACTT VE TNDTX( FOR TIIE Ist. LAYER.

Provitted the first layer is very thin and transparentr it will

be shown tirat alr incorrect choiee of its iefraetive index will only

result ih 8, diffefence in its calcuLated trricrness, and' will not

affeot the resul.ts of the underlying filb'

For the csse of â, verxr thin lst. layer, i'e' y1 << lt ol << ft

Tornlin (rgtf) has derivecl the following expressions from equations

\.7.1 a¡rcl \ ,7 ,2,

I+R F

where

1-R

-

=

T \nsn3 (nrz+x2z)

F = (n92+n22+k 22){(n22+nr2+kr2+lx32) coslnãa2

+ 2(n2n3+krt3 ) sinh2azl

+ (no2-nz2-kzr)U n22-n32+k ,'-kg') eos}y2 + 2 (n2k3-n3k2) sin212)

+ et, [{rrr(no2-n12+t12)

+ 2n1n2k1}{ (n22+nr2+kz2+:xs2)sinh2a2

+ 2(n2n3+k2k3) cosh2c2]

{n2 ( ns 2-rr, 2 +t<. 1

2 ) - 2n 1k 1h2 } { ( n2z-", 2 +inz2-;r s2) s in2y2

- 2(n283-n 3k2 ) "osa12 ]J

)+.9.1

l+.9 ,2G

T 2n3kt22+]K22)

= n2l(n22+n 92+;622 +k 32) sírú72øz + 2 (n2n3+k2k3 ) coshZo2 ]

+ k21|n22-n32+r22-iy.32) sin2y 2 - 2(n2k3-n 3k2 ) eos?^¡ 2l

+ 2n1k1tr ((n22+rr32+l'22+:xg2) cosh2o2 + 2(n2n3+k2k3)sinh2o2

+ (n22 -n 32 +]nr2 -k z2 ) eos}y 2 + 2 (n2k 3-n 3h2 ) sin2y 2)

where

G

70.

where

t1 = 2rð'¡/)t

It is seen that these equations are those for a single absorbin6

layeronanabsorbingsubstrg,teplusthead'd.itionofacorrection

tema in t1.

For the cage where kt = k3 = o (transparent first layer and'

substrate), tfre correction term for (1-R) /T becomes u eTo whil-e that

tor (1+R)/T becomeg:

z(nsl-n¡2lt 1 [r.r1 (n22+n32+k22) sinh2o 2 + 2n2nseosh2o2i

- nz{(nzz-nr2+k2?)sin2v2 + 2n3k2co"2Yr}J l+'9'3

ûre expression between the curly brackets is ind-ependent of n1 and d1'

Ihereforetobeequivalenttoathinfirstlayerofrefractive

ind.ex n1r and. thickness d1t ¡ the cond'ition for equivalcnce of (f-n)/T

is always ¡net r¡hile that for (f+n)/T is, from equation I+'9'3

(n62-n12 )t1 = (no2-tr t 2)t1 ?

l-¡€o

t 1 tl1 no2-n¡'2

tlt dtt n02-nt29O

for exampr_e if n0 = fr nr = 2, trIr = 3r dr = looAr then d'1t = JJA'

fhat isr a very thin firet 1aycr of thickness rool and refraetíveo

ind_ex 2.0 has the sarne effect o.s a layer 3TÃ thick with refractive

index 3.0.

so provid.ed. the first layer is very thin and- transparent we may

s,ssume an incorrect va,Iue for its refractive ind'ex and' still- elirnineJe

its effect when d.ete:mining the optical properties of the underlying

TI

film using the tt¡o-Iayer equatíons'

h. to MOR!] $TAN ONE VERY UÍTN LAYT,]Iì ON TIIE SURFACE.

l{e cnn show that two very thín transparent layers are equival-

ent to a single transParent laYer'

Consider two very thin transparent layers with refractive

indicesnland'n2ofthicknessestlland,d.2respectivelyonanabsorb-

ing substrate of refractive index n3 and absorption ind'ex lca'

Equation \.9.1 with k1 - k2 = 0 becomes:

l"+R

For 12 << 1 equation l+.10'1 bccomes

1+R 1

=\ngn3n22 \

+ (nsz-n2z¡ l(n"z-n32;r.32) cos2y2 + 2nrk3si nzy z\

- zt,yn2(no2-rrr2) { (n22-n3 2Jl'z2)sín2y2 - 1nzkscoszyz!

T

l+ .10.I

p (ns2.+n 32+k 3 2 ) + 2k 3 | (nsz -nrz ) zt r+ (noz-n t

z ) zt t i

- (no2-or 2) (n22-n32-y,ez)\t rtrJT \ngn3

Since t1 << I and- 12 << 1, r*e may neglect ternrs in t1t2' Hence

I+R I (r f z(r,e2+r,32+kgz) + 2k3{ (no2-nzz)et2+(ns2-n1

T hnsn3 \'z)ztrll

l+.10.2

For :: single v.:r-rr tlrin tra.nsparent fil¡:. of refractive index n and

thickness d on an abeorbing substrate vith optical constants n3 and' k3:

}+R 1

= r

{z(ne2+n32+ksz) + 2k3(ns2-n2)et} h'ro'3T \ngn3

72,

For two very thin transpa-rent layers on aJl absorbj-ng substra,te to be

equivalent to a single transparent fil-n on oJr absorbing substrotet the

conditions for (f-n) Æ to be equivalent are a]1¿û,yg met ond from

equations Ll.1O.2 and. ì+.10.3 the cond.itions for (t+n)/t to te

equivalent are thus

(noz-nz)t = (no2-nzz)tz + (ns2-n12)t1

or

(n¡2-n2)a = (noz-nz2)dz + (nq2-n12)41 \.10.ho

êogr if cL1 - d.2 !OA, n1 = 2¡ î2 = 3r trg = 1r theno

(n2-1)d = 5504

€rgo ifn=2r a=r8¡lo

D=3¡ ð,=69A.

Ttrat is, two very thin non-absorbing layers with refractive indiceso

2.0 end 3.0 end. thickness JOÃ each, have the sane effect as a single

fil-¡r vith refractive inclex 2.0 end thickness fOSi, or a single fil-mo

with refractive index 3.0 and thickness 694.

I+. ]1 SURFACE LAYERS ON GERI4ANTUM FIL}4S.

ftre caleulated. oxide thicknesses on the gennanir.rm films wereo

estirnated. to be aecurate to within about t5A (assuning the ehoice of

refractive ind.ex was correct ) . Amorphous geruaniun fikns a"nd. fil-ns

cieposited. on room temperature substrates &nd subsequentl'¡ :;mealcc

shoved oxid.e layer thiclcnesses 1ess tfran 101, which is of the sane

order of magnitud.e as has been previously reported.

However fihls cieposited. on substrates at 35OoC evJribited. apparent

oxide layer thicknesses of the order of 1OOl., which is much larger

73.

than that found. for the obher films' One possibre explanation for this

was that the oxid-e may have forrred on thc hot substratc during

d.eposition and not on the top gurfaee of the film' lkris ¡ossibility

was precluded. since tlre ane'lyses presented in this chapter shol¡ed'

thot the layer was d'efinitely on the surface'

Acluetotheanswertothisproblenwasfoundfromther'rorlcof

Sloopeand.TilleT(::962)r¡hostud.ied.thefor¡nationcond.itionsand

structure of thin epitaxial geroanium films on single-crîrstaI

substrates. They forrrd that til-rns d"eposited. on calcilm flouriðe

substrates below 3OOoC and above 55OoC were smooth' whereas' between

3OOoC and ,5ooc, the fil-ns were wrinkled. Although quartz substrates

wereemploy'ed.inthepresentinvestigationitwaspossib}ethata

simiLar effect existed. Electron microscope examinations of the

surfacesofthegermaniumfilmsr'¡erethereforeundertaken.

l+. fl. I STJRFACE TOPOGRAPHY OF GERMANIUM FILÌ'{S'

surface replicas of ge:zranir.r¡n fil¡s deposited' on room iernperature

substrates,filmsdepositedonroomtemperaturesubstratesBnd

subsequenttyannealedrand'filnsd"eposited'onsubstratespreheatedto

35OoCr trere prepared. and stud'ied with the aid- of an electron micro-

scope. Figure \.10.a is a micrograph of the surface of a filn

d.epositect on a room temperature substrate, a.nd is al-so typical of

thosethatwereer¡nealed.f.tresurfaeeisgenerallyquites¡nooth

with oceassionar crater-rike d.epressions. Figure l+.ro.b is an

electron micrograph of the surface of a film depositedon a substrate

which rtas prehe¡;bed. to 35OoC' TLre surface of the filÏ is seen to be

tç

tII

t

II ¡ I

I

T

I I

Ilì I

t

fttå

\f;. Ù

Itrirl

i

i

I

I

T!+.

oolrrrinkleil. ftre wrinkles aye of the crd.e:' of 1OOA rliarneter and 10004

long. ff we consider the surface as being composed of serni-cylinders

9Orooà in üia.meter and lOOoà long, then the ratio of the surface of the

fiJm conparecl to that íf it was perfectly flat is about 1.6, in

exce¡-ent agreement with that dete¡nined 'by l{aeDonald' (f967) ' He

ínvestigated" the absorption of gases on germenium films deposited' on

quartz-crysta] oscillators and deduced. the above ratio for films

d.eposited. on 350oC substrates.

\.1I.2 TREATMI]NT OF THE SURFACE LAYER.

It is apparent that the large surface layer calculated. for the

gerranium films deposited. on 35OoC substrates is not geznaniurn oxid'e,

but has arisen from the wrinkled. nature of the surface vhich procluces

the sa¡ne effect &a an oxide layer. Although not an oxide, we can still

treat thc surface as a separote layer and r,re vil1 show thatr to a

good. approximation, it may be treeted as tre^nsparent so {het the

analyses presented. in this chapter in eIi¡rinating the effects of the

surface layer on the properties of the underlyine filnû are still

valid..

Treating the surface of the filnrr as a separate layerr we nay

appfy Schopperrs theory (Sctropper t I95I; l.leave.ns, 1955) which has been

appJ-ied. successfully to aceount for the anomalous optical behaviour

observed in very thin metat fil¡rso In the símpIe case the surface

layer may be consid.ered as a filn consisting of eJ-lipsoid'al particles

of the same size and axial ratio. Ihe ratio of the average fil-n

thickness to that cled.uced fz'on the mass per unit area on the assumption

75,

of butk density ir y. If Ne is the observed. 1¡alue of the complex

refractive ind.eX for such o fil¡n (i.e. Ne = n" - i ku l¡her:e nu and'

ku are the observed refractíve and. ebsorption ind.iees), and- N the

conplex refractive ind.ex of the bulk material, then

N2-.1y(Ne2-1) = -=-' (1¡2-r)r+r b.1I.1

where f is a firnction (Davidrs function) of the axial ratio of the

ellipsoid.e whose variatíon is seen in Figure l+'11'

Thus frorn equaticn l+.11.1,

I ' ^ i f,--ô -\ ìr

Ne = [r + (n2-r)¡v{(nz-r)r+r}lz \'11'2t)

Ttris was the most convenient fonn of the equation as the computer wa|,s

capable of executing cornptex aríthnetic'

Althougrr it was not possible to measure accuratel-y y ernd' f,

they .were estimated. frorn the electron nicrographs to be of the order

of f.5 for y,and o.I for f. using these vcfues and. apptying equation

l+.II.2 to values of the optical constants approxinating those of

bulk ge:manium prociuced the results shown in Figure l+'I2'

It can be seen that the effeetive refractive ind.ex of this layer

is less tha¡r that of the bulk ge:rraniun in agreement with the

abservecl results. TLre velue of n" in this case is 2,?- wtlíe]n is close

to the value which was originally assumed for this layer. The

absorption edge of the surface layer is shifbed towards sborter

wavelengths. For our range of measurements, the inclusion of this

absorption in a very thin surface layer d.id. not alter the reflectance

and tra,nsnittance of the two-layer systen outsid.e the experimental

5

ßL.¿

ry

î

(!

¿-Õf-*c--)

¿*:)LL

Ç)

?

ñ

t 'E 236(æ)AXI/-',t- RATIC)

Ir Í.t¡ u:r:e ¿î." Jl

l'le

Application of SchöPPer's theorY

to a lrypothetical .film

52.0

1'6

Þ€Øov-{at

1.2 >

zÇmf

B

xLrlô

=

LLI

1-CJ

ÉLLtr,d

3

2

1

0

4

0

2.s 1.8 1.6

WAVELENGTI-1

1'4 1.2

.Mrc RoN.$

1.0oo 62'4 2.2

IN

Figure 4,Lz

76.

errors in these c¿uantities. Thus for fhe pre-oent purpose the surface

le.yer may bc considered. transparent i^¡ithout introdu-ci-ng a s-'lgnifi-

c:i;.rt error.

Al'clrc.,ugh the treatrnent of the surface layer given above is

sinplified.D it nevertheless demonstrates rvhy the postulation of a

transparent oxid.e layer on the surface of the gerrnanium fil¡rs

produced excellent closure of the refraetive index curves and gave

good ogreement betwcen the measured and ealculated filn thicknesses'

To a good approxir:e"tion, thereforet lre may treat the surfaee of the

filr,r as o transparent layer with a refractive ind'ex of 2'0' We have

shown in section )+.9 that an error in the choice of the refractive

índ.ex of the transparent layer will only result in a difference in

its calcul-ated. thickness and wilt not affect the results of the

r:nd.erlying film. Furthermore if gerrranium cxid.e is also present on

the surface.r^re have shown in section \.10 that two or mcre very thin

transparent layers on the surface may be treated effectively as a

singlelay+;1.withno¡neasurableeffectonthequantitiescalculated.

for the und.erlYing film.

l+. 12 sTjRFACE LAYEBS ON SELETIIUM FIL}4S.

Íheca]-culated.surfacelayerthicknessesfortheseleniumfil¡rs

varied between LOI anA f251, these vafues being esti¡nated' to within

about ZOl, fæ each fil¡,r. fhe -large e].ror connared' to that åeterminecl

for the gerrnanír:m fitms r:rost probably arises fron the non-uniforr:ity

of t¡ese firns as a rotating substrate vas not employed. during the

d.eposition of the fiIms.

77.

Ttre v.alues obtained anpear rather larSçe for an oxid-e layer,

bub no other me&surementS to compe,re l¡ith coukL be fou¡o. ín 'bl:e

.i--r";r:rar;ure. Campbell (fq6S) reported. that there was no significant

sul'f ¿ce t.oughness for these films, end. that they vere generrr,lly

sncoth with isolated crysteJ-line patches. I^lhether surface irregul-

arities of the ord.er of )+Ol - 1201. 'were present the author r+as not

in a position to ileterrnine. If the large appa:rent oxid.e layer is in

fact d-ue to surfaee irregularities then the surfaee may be treated' in

a si¡lilar nanner to the surface of the gerr:naniun fi}ns in the previous

section, with the consequent eliraination of the effects of the

surface on the calculated quantities for the underlying filn.

\.13 coNcLUSroNS.

'Ihe existence of surface layers aceounts read.ily for the need to

reduce the neasured- film thickness in ord.er to obtain the correet

type of refractive ind,ex curve from the sin.qle layer equations, and

also the non-closure of the refraetive ind.ex curve when these equations

are used.. For very +,hin tra¡rsparent layers, the resul-ts obtained'

using the single layer eguations r¿ith recLuced. fil:r thickness are

neglig-i,i,.l.y d.ifferent fron the exact results. For moderate surface

l.).ers it becones necesse,ry to use the d,ouble layer equations in

orcler to cbtain a closed. refractive index cürveo

As vith the single-layer ecluations where one can use the

property of cfosure to obtain an accurate value of the fih thichness,

so with the two-layer case one ce,n use the property of closure over

the entire wavelength range to obtain accurnte values of fifun and

.r8.

transparent tayer thicknessee along rrith accura,:te values of the optical

consta¡'its of the film. Tt¡e s,bsolute aecuracy of the trar¡sparent

Ir:;-er thickness is dependent on the accuracy with which its refractive

inårr:,¡ í.; l-r¡owno

Ttre property of closure leads to the possibility that a fil:n

may be coated. with a thin layer of trensparent material of known

refractive ind.ex. Itre cptical constants of the und'erlying film and'

its thickness may then be calcr:lated from the measured. reflectance

and transmittance of the two-layer systen. 'itris nethod. night prove

ueeful for the investigation of the optieal properties of subst¿mees

which are higþIy reactive in air.

79,

CIIAPTER 5.

OPTICAL CONSTANTS OF SELENIUIvI FILMS.

5. 1 INTRODUCTÏON.

Íltre calcu-l-ation of the optical eonstants of thin vitreous

se:l_=niun filils, from the measured- reflecto¡rce and transmittance of

the fil¡;rs, using exact theory uas atternpted by Canpbell (f968)'

Inad.equate infor.mation on the behaviour of the c¿rlculation proeeclure

led to incorrect solutions being chosen from the nrultiple solutions

that existed. Tkre reason for the incorreet choice is diseussed't

and" the optical constants of selenium fi.lms are recalculated' on thc

basis of tbe knowledge of the behaviour of the caiculation proeess

d.etemined. in the previous two chapters. Ítre effects of surfaee

layers are al-so accountec. for. Tlre absorption found in these filns

is then briefly d.iseussed in terms of a o-ualitative semi-eonc'ucting -

bond ¡rodef .

5,2 TNCORRECT SOLUTIONS.

The argument put forç¡ard by canpbel-l- for the selection of his

sol_utions was that at certain r¡averengths only one solution for the

refractive inclex n (and. absorption index k) exísted. flhis solution

rnust therefore have becn the ccrrect solution and' conseguently the

correct d.ispersion curve must pass through this point' For exarnplet

in Figure 5.I one of his rcfr¿lctive ind.ex curves is reprodueed''

At the wavelength O.B ¡r the only solution for¡nd.'v'as n = 1'l+, and it

L,:sNou 3l!/\¡

9- L.

T'ç a¡nõTE

NI HIgN]]3AVM8' 6'0r l.t ¿'l

L

Ðm-t'l

aÐ(.>o=m

2(fmx

E

?

t

IIT-t ¡

ïI

I

x

T

T

5.

xx.

'¡-.t-L -t¡ x

I

\*¡¡¡rr

ï

T

ï

ff'E t

r

tr-I' 0"f,l

-T iI

fL;!:l

SEIITTT

TT

I lT

80.

wes assr¡xxed that the curve nust pass through this point and as a

result the curve shown as a dashed Line lras obtaineil.

It tri11 be gll'.,liil that, given accurate experimental datar more

then one solution d.oes exist at these wavelengbhs, but their existence

may be missed in the calculation rnethod.. In the present caee the

nissing of solutions s¡as enhanced by experinental errors in the

mes,sr:recL reflectance and. transnittance, by the presence of surface

layerø on the fiIns, ffid particularly by errors in the fih'I thickness

values used. in the calculations.

5 O 2' ! MISSING SOLUTTONS'

The reason why scme sofutions raay not be found. at particuler

wavelengths c$n be clearly dernonstrated by consid.ering the cafculation

for the hypothetical film d.iscussed. in section 3.7. ftte caleulated.

refractive inclex solutions for this film are shol'7n again in Figure

5.2.

Initially it was for¡nd. tho,t at wavelength 1.6U only the solution

n = 1o5 i¡as reproduced., the correct sol-ution n = h.O being omitted..

Ttre reason for this was as follows¡

rn Figure 5.3.e the right-hand" sid.e of equation 3.4.1 (i.e.

(f+n)/T) is plotted as a function of n for vavelength 1.5u (for eaeh

n, k was for.¡nd euch that (nrk) satisfied the equation for

(1-R)/T). Ítre measured value of the lefb-hand" side of equation

3"1+,1 is includ.ed.. Ttrree solutions exist.

(f). n = 1.76

(z), ¡ = J.\\

{

5

iI

I.

I

tf

I ¡ l¡ I ¡ ¡ a . t a \. t a x u * t t .. x¡t¡x¡ff{, t1¡¡x

xUJô

=14,IF-o

þ-U

t

16-8

WAVELENGTH IN

Figure 5;2MICRONS

+

f n324

+

+

4

n

+

,4t1

A

I 2 3 4 1 ?- 3 4

n

(f+n),/T as a f unctÍon of n neal the critícaI wavelength'

Figure 5.3

n

81.

(g). n = \.Oo

Figure 5.3.b showe the si¡oilar eurve for À = 1.55u.

Sol-ution (e) rras Tnoved closer to sol-ution (3).

Figure 5.3.c is the eurre for l, = 1.65u.

Solution (z) fras now movecl to the other side of solution (3).

Fi6çure 5.3.d is the curve for À = 1.60u. Solution (1) still

exists, but solutions (e) and (3) trave merged to give two coincicent

soLrrtions at the correct value of l+.0.

TLre measured value of (f+n)/1, represented by the straight line

parallel- to the n-axis, is tangentiaL to the (f+n)/T vs. r cllrVê &t

this point. It is this solution that was previously missed.

Unobtainable accuracy in the experimental data and filn thickness is

necessary to reproduce this sofution. An error of I part in tO4 in

the reflectance or transmittance will cause a separation of the tangent

at this point from the correct vafue of tbe order of 0.01, and. any

çul-oulation procedure gearching for rfzeroestr less than this at the

particrrlar navelength of concern will ¡niss the solution. Ttre

separation is greatly inereased, and exists for a wider wavelen6çbh

range, if incorrect fil¡r thickness values &re uged.. Às shown in

Ctrapter l+ ttre presence of surface layers wiIl olso eause this behaviour

if one attempts to use the single-layer equations in plaee of the

two-Iayer equations.

Although Canpbell employccl a clifferent caleulation proced.ure,

a sinil-ar effect existed. In his method., loci of eonstcrnt reflecta¡rce

and consta¡rt transmittance were plottecl as n against k, the solutions

for n antt k being the points of interoection of the loci. A typical

82,

curve ie shown in Figure 5o[.a. At the critical çavelengths the loci

touch at a point ae shown in Figure 5'l+'b' rt was these points that

were missed. in his calcul-ations.

,,3 AIVD TTANCE OF SELENIUM FIL¡,6.

Tbe val-ues of R and T for eight vitreous seLenir:m films,

measured previor;sly in this lc,"boratory by Ca^npbe3-I, are tabulated'

in A1:;and.ix Ð. The curves for four of the filns are shown in Figure

5,5. MultipJ-e bea¡r interference effects are apparent in all but the

thinnest filn a¡rd. become more pronounced as the fil¡n thickness

inereases. The transmittance of the fil¡m rapid.Iy falLs to zero st

short vavelengbhs, correspond.ing to the regj.on of strong absorption,

fhese fil-us were sti1l strongly absorbing at 0.21p.

Ca,mpbeJ-l suggests tha+, no nrultiple bearn interferenee effect

oceurs in film 1. (Figure 5.5.e) as it is near the liniting thiekness

forwhich the film is a continuous layer. However the number of

maxima and nrini¡ra observcil in the reflectance and tra¡rsmittance

curves d.epends primarily on ttro ratio d./À, the value of d for this

fíl-m being such that nul,tiple bean effects are not ,seerr for the ltave-

length range consid.ered.. In fact this was the only fil-¡n for vhich

Ce,rrpbell obtained results which were in qualitative B,greement with

those for the bulk material, but he d.ismissed. these results as being

meaningless with the suggestion that this filn nay not be a continuous

layer.

Tïn

(a)

(b)

R

In

[t*

------+

T

R

Ëigure 5"4

{

ô!

wavELaNGlts lt MlcRots

wÄvEttN6la ril MlcRofls

.a

w^vE!tNGrH l{ MICROtS

Figure 5.5

83.

f . l+ CORRECTT]D FTI,I4 fiiICKNESS .

Table !.1 gives the rneasured values of the thickncsses of the

eight films, the values assuned by Campbe11, and the fiLm and surface

layer thicknesses ealcuLated. by the method. d.eseribed in Chapter l+.

The surface layer Ìras assuîed. to be SeO2 vhieh has a vefractive index

of I.76. With the exeeption of films 2 and 3 the total of the

celc11l-ate¿ filn and oxicle thicknesses agree rernarkabfy well with the

mea.sl':ied. thicknesses. Ttris would. ind.icate that our assumption of the

first layer being selenirllr d.ioxid.e is possible, although a wrink'fed

surfoce as found for gemaniu¡l r,rilt give a sirnilar effect. As shown

in the previous ehaptern regardless of the exact nature of the surface

layer, provid.ed. it is thin a¡d. near trnnsparent its effect on the

r:nderlying fiJm is sti11 eliminated.

It is possible that the difference in meesured anC. calculated

thickness values for fil¡rs 2 ernd. 3 nay be the resul-ts of experi¡rental

errors. Sel-enir¡n reacts vith silver, anil so an inert layer uriust be

d.epositcd. over the filn befo::e the silver -;,n order to ma,Ìte thickness

Êteasrrïemerrtse Some material may migrate tovards part of the substrate

not eoate¿ r,¡itn seleniun. This would result in a snaller measured.

fitn thickness and. it is possible that this has occurred with these

two fi]¡rs. Alternatively the nature of the surface for these fi1¡rs

nay be such that our choice of its effective ind.ex (t,76) nay be tocr

smaII, resulting in a larger calculated. surface layes thickness.

Table 5.1 also gives the aecuracy witl: whieh the filn ond surface

layer thicknesses could be calculated. The accuracy is p;enerally less

than for the gerrn,rniun fil-urs (Chapter 6), probably rlue to non-uniformit-

8l+.

ies in the fiI¡rs as a rotating substrate was not employed as in the

case of tÌ:e gernanium films.

520lro

68etro

7l+zt5

930t5

101215

r-600130

25AO!5O

2700150

T.A3LE 5.1

ER). ) REFRACTIVE TNDEX OF SELE}]]UM FILMS.

The caLcufated. refractive ind.ex sofutions for the eight vitreous

sel-enium fil¡rs in the range of n = 1 to n = l+ are plotted. in Appendix

D. The values for one of the films are shown in Figure !.6. Ilultiple

solutions occur bccause of the period.ic nature of the reflcctance ¿u:d

transnittence equations, the ¡reriodicity being deternined' by the

parameter Y wbich is affected.by the ratio d/À. As d. increaseßr rÌore

solutions are c¡:,lculatcd' in. a 6;iven rall,ge of n.

For the thicker fihts thcre vere sma1l r¡aveleng$h ranges where

l-o

2.

Ja

)+.

5,

6.

7,

B,

538

T2\

?BT

9\2

10u

T'82

2380

2567

l+OtfO

I25lIO

r2o!5

I+5!,

I+5t5

L20!20

Bo12o

L20!20

j60!20

Bo?tao

B6etro

975!LO

1057110

l-720!50

7SBO|TO

2B2O!lO

FILPI

NIjMBI]R

CA¡{PBELLISFÏLMTHTCIC.IESS

A )(o

(a)

TIIICIilIESS

CALC. FTLI\Í CAI,C.OXIDtr]

lrfrclflùBssoA)(

MEASURND

FILT4TTTICIff{ESS

538!?-'

7zl+!\3

7Brt2g

g\2!\9

ro}9t32

r732t26

2680r?3

2867!57

o(.t)

1'OTALFTLM+OX]DE

o(A)

5

4

3

xl¿lo

=U

l--(J

É.IUJÉ.

2

1,8 4I1.6 1'4 1'2

WAVELENGTH IN-/h'ìarilYls \ fìJ. ¿

1.0

MIC RON S

.6

Example of the calcuriatecl refractlve lndexsoiu'bions for a selenlum fili:r

sE3

III

I

I

II

'IhI

85.

the correct val-ues l¡ere not found. Íhe reason for this has been

discussed in seetion ),2" Afthough these solutions could have been

found by sea::ching fr:r the turning points in the 1r+n)/T vsr ft cllÏV€e

the increase in cornputation and. eorre6ponding computin6 tirne vas

consiclered, unnecessary sinee the problem oecurred only for smal1

regions in the infra-red where there is littIe change in the refractive

anrl nhsorption ind,ices.

5. 5, r IRRORS rN lÍIE CALCUL/\TED SQLUTIONS.

Íhe errors in the cal-culated, sotutíons werc¡ detertined by the

method cleserihed. in Chapter h, and are ind.icated on the curves by

verticoL bars. The error in R ond. T vas estimated. by Canpbell to be

O.OOZ5. T'Lte errors ín the surface layer and film thicknesses Ì¡ere

estimated. from the B,ccuracy vith ',¡hicÌ¡ the refractive ind'ex curve

could be closed over thc entire wavelength range, md are given in

Table 1.1.

Near the regions where the correet curve intersects with the

incor:lect loops, the ealculated errors are seen to be much larger than

elsewhere. fltesc regions eorrespond. to the wavelength range where

the curve (f+n)/t vs. n ie nea.r tangential to the straight line,

paraIIel to thc n axis, representing the e.rxperimental value of

(f+n)/f. fn these circrrmstances a sm¡r1l errLl in (f+n)/l nakes a

large error in n. It is possible that the first order fo:rrulae for

the errors are inadequate in these regions, since the expression for

the Jaeobian (equetion 3,6.7r page 37) tend.s to zero in these re¿Jions.

fhe smoothness of the cor.nputed curve suggests that the errors may be

86,

overestimated. in these regions.

5. 5. 2 COMPAATSON t{TTg PREVîQUS T¡IORK.

In Figure J,'l the average refractive index curve for the

vitreous seleniu¡l films is plotted. as a function of wavelength

together with the results of llood (rgoe) for thin sputtered filJns

ancl Dor.¡d. (fglf) which were obtained. from the bulk material. There is

seen .Lo be close agreement between the present results 8nd those of

these authors.

5, 6 ABSORPTION TNDÐ( OF SEI,ENIUM FILlvl'S.

Tfhe absorption ind.ex curves obtained. for the sel-eniull films

are ptotted. in Append.ix D. The correct solutions for one of the

fitns, together r,¡ith the calculated errors in the sofutionst are

plotted. in Figure 5.8. The incorrect solutions, correspond'ing to the

incorrect refractive ind.ex values r 'were in general close to the

correct vafues and. are consequently not shown on the graphs.

TÌre large errorB assoeiated. with certain criticel regions in the

refractive ind.ex curves are not apparent in 'the absorption ind'ex

curves. l,Ie have found. that, although the rrultiple solutions for the

rcfractive index at a given wa,ve1en6$h nay be lrid-ely separated, the

corresponcling absorption ind.ex solutions are not widely diffcrent'

Thus although the calculatecl errors in n at the critical points ma'y

be large¡ they do not necessarily 6ive rise to large errors in k

in these regions.

.f DOWD - ôU!( AÂlrilÀl

O wcroD - !PUlltl90 lltMs

-

rîEs:tsÌ REsuLlS

,i

xUJô

=t ¡.1

F()

rU

1.6WAVELENGTH IN MICROITS

Figure 5.'î

l'0

I

B

7

6

5

,:

3

XUo

z

;o_

É.

o)co

2

I'8 ¡'o 1'4 1.2 1'0

WAVELENGTH IN MICROI'IS

;¡iU:U:e 5 ^t

I 60

IÏl

E::arlpie of the celcuiaÌ;ed abscrption inde,'<çc--Lut",5-lcn-o fo'r a sei-eç-it.iin iil-m

lI

I'i{¡

i,t)l

I

SE3

BT.

5. 7 ABSORPTTON COEFFTCTENT 0rsEI,ENTUT/I FILMS.

The al¡sorption coefficient, d'efined' b¡r the relation

K = l+nk/À

is plotted as a f\rnetion of pboton energy in Figure 9'9 fot a typical

seteniun film, together wittr the curve of Seimsen and Fenton (tg6l)

which is representativer of previoue 't'¡ork. ftte ¡rain difference is in

thrr l.ov enerßy region vhere the present filns exhibit much stronger

absc:.'¡rtion tha¡r have previously been reported' Tlris result is

consistent vith the conclusiong of calrpbell r,¡here he has shovn that

approximate equetions are not suitr:,bl-e for the wavelen¡"'th range where

Kisnotverylargene.rrd.thatinrportanti.nfornrationeoncernin¡q

absorptionprocesBeswittrinthefi]¡rsmaynotbeapporentfromthe

approximate oPtical constants.

5. B ABSORPTION IN SELENTUM FILMS.

From the theory of 1if,;ht obsorption in semiconC.uetors (see, for

exa¡npleD R.A. Sïûith in ttserqiconductorstt ) IgSg) ttre absorption near

the edge maY be rePresented' bY

EnK., (u-¡g)ß 5'B'1

whcre E is the photon encTgy, n the refractj-ve índex r:f the material'

ß- tht-. absorption coefficient and. Eg the forbi<ld.en enerfîr Ãap. For

direct transiticns p hns tire value å and fcr phonon-assistcd indireet

transitions it has the velue 2. For forbid.d-en direct flIrcl" indirect

transitions $ has the vaLrres 312 anc.- 5/2 respectively'

Ecluation 5,8 J may be riorÜ conveniently v¡ritten

E-Es o (¡:nrc) r/g '

'B '2t

I5(J

z

l-zr!

9LuLlloc)

z.9t-o-do(n@

106

10

1----¡

+_1

+-2

I srrrst¡¡ ANo rENtoN

I rnrstxr ßtsulr5 - rltr't sE2

0

4.

PHOTON ENERGY

F ígure

IN ELECTRON - VOLTS

5.9

t0

BB.

I

A plot '¡t (¡ntr) t/ß t'" ' E for: various values of S would' therefore

be expected. to reveal the absorption ¡nechanisms occurring'

For the sefeniun filrcs the best value of p was found' to be

I.Oo ancl ín Figure 5"10, IijnK ie ¡Iotted' as a functi-on of photon

ener&r on a linear scale fo:: one of the selenium films. It is seen

to consist of two straight lines. ft¡is beharriour was typical of all

thc seleniun films, with thc extrapolateci line intersecting the

eÍle;: ,j¡'axis betrn¡een 2.18eV euid 2.2!eV, an¿ the chan¡-ge in s1'ope of the

ljne occurring betweetr 2.)OcV ancl 2'63eV"

Tkrusforthesefenitunfilnrsghasthevalueof]-whichisnot

reprcsentative of any of the -r,bove sta¡rd.ard transitions " It is

recognised. that it is doubtful whether selenium ean be dcseribed" b1r

thc theory of metalLic conduction 6iving rise to an enerm¡ band

norlcI. The wel-l d-efinecl cncr¿y ve'Jue of the sharp increasc in

absorption ind.icates that band strueture itself has ¿rn applícation

to selcniurn, wlrich ap?e¡j.ïs to l¡e a,n B,rgument for aJl 1'pproacÌt to semi-

concluction which does not require a period'ic lattice'

Mooser and Pears on (1916, Lg58r 1960), by a quali'bative argument

based. on the chenical bonding between atoms, deduceÖ an encrgy levef

structure for hexagonal selcnium. TLre struct'¿re of the hexagonal

form consists of spiral chains with the seleniu'-.n atoms arranged- at

tire corne:rs a,nd. in the centre of a hexa5¡on. 'rhe chain d-i::ection is

*uhe c-arcis of the crystal. Every thi:ld. atom in a chain completes

onerevc]-utionofthespiral.Thcdistanceofanatomtothetwo

nearest neighbours in thc seme chain is 2.32L an¿ tl:e d-j'sta'ce too

the four next-neB,rest neipþ'f.rours in aôjaeent chains is 3.h6'4.

EnK as a function of Photonenergy on a linear scalefor a selenium film

d

/

/

I

:¿c

]JJ

2,2PHOTON

2.4

EN ERGY IN2.6

ELECT RON-VOLTS?0

Fígure 5.10

30

89.

ftre spirø1 ehain structure is preserved. in the vitreous forrt but the

chains are rand.omlY oriented.

1'he selenium atom has six outer electrons (u2pa) above five

completed. shells. The bonding between nearest neighbours in the

same chain involves the s and. p orbitals and. feads to allo¡'red' and'

forbid.d"en energf ¡'.evels, ftre oecurrence of resonating bonds between

next-nearest neighbours ín ad.jacent chains suggested' that the forbid'd'en

gap :L= bridged by a band. of low state density overlapping the valence

band.. The resonating bond requires that either or both of the

unpaired. p electrons is promoted to the higher ô orbital'

Mooser and- Pearson suggest that the overlapping band' does not

exist in vítreous selenium filns because the random arr&ngenent of

the chains d.estroys all resonances. This conclusion wog ded'ueed

from the increased dista¡rce to next-nearest neighbours in vitreous

selenir:m. ltris, at fi.rst sig¡t, cppears inconsistent with the

present measurements r,¡hich ind-icate that consid'erable long waveleirgth

absorption exists, particularly in the thinner fi1ms" However, if

the residual long wavelength absorption is plotted. as a function of

fil¡r thickness (trigure 5.11) a ri.efinite trend is observed' As the

fil:n thickness increases so the resid.ual absorption d.ecreases in an

exponential mannero

,Ihe chain structure of selenium grolrs upward-s frorn the substra'be

vith a:norphous naterial separating the chains (Campbellr 1968) '

As the thickness increaseg the chain structure becomes better developed'

Krebs and Schullze - Gebhardt (lrg,5) have suggested' that chains B're

o

IOO to 1OOO atoms long which represents a length between 23OA and'

I

=Oôlo

z.

zotro_

ot¡)(Ð

J

f9</,L¡J

î00tFILM THIC(NESS IN

FiE'.lre 5. LI

2000ANGSTROM UNITS

3C000

90.gO

23OO;.. For filme fess than about 1OOOÃ thick most of the chains üil1

be und.erdeveloped. Open-end.ed. chains wiLf abound. in the materialt

p€Ìrticularly near the fi}n surface. Carnpbell has attributed tlie lov

enerry absorption mechanism to the ease with vhich electrons &rc

excited. at the open ends of the che,ins. As the thickness incre&sest

anr1 the chains become better d.eveloped, this absorption meehanism

wil-I clecre&ser in agreement with the present results"

'f:e change in slope of tbc line in the EnK vs. E cltrve corresponds

tr: the energ,r at which photoeonductivity in the amorphous fil¡ns

rises sharply (Campbell, 1oc. cit.). Moss (tgSg) suggests that the

clifference in photon enerÉ3r for absorption and photoconduction

arises from potential- barriers withín the rnaterial vhich results from

the besic structure of selenium. Tt is asgurneC. that r'iithin inclivid.-

ual chains of seLeniu¡l atoms carriers may move freely, but that to

t:..aveL to nei6Lrbouring chains it is necessary to cross potential

barriers, It is suggested. that these barriers are responsible for +;he

dj.fferenee in photon enerry between the rise in absorptiorr zurd photo-

cond.uction. It is also sugqcsted that the barrier hei¡:'ht correspond.s

tc the difference in energ-y betveen tlrc oecuÏrenee of thc change in

sLqpe of the EnK vs. E curve rrnd. the energy at vhich the ex'brapolatcd

line intersects the energy anis. For our neo,surements this is

O"3e'/ to O.heV. Fro¡r the C.ifference ín photon energl/ lcvcfs betveen

ecluivalent photocontLuction B,nà absorption levefs, Mcss (toc. cit.)

estirrates the barrier height as'v O.6eV. It is sugeesteC' that our

nethocl efford,Ë 8, rìrore accurs"tc estirnate. The change in slope of the

EnK vs. II curve is assccintccl with the eha"nge in tlre clensity of

91.

a^llowed states when suffícient enerÐr beeomes available for the

electrons to eross the potential barriers. The position of the

change in slope ca¡r be d.ete::mine¿ vitn reagonable aecuracy.

ÎÏre above qualitative arguments aceor¡nt readily for the observecl

absorption in tbin vitreous sel.eniurn films. Until fi¡rther

quantitive analysee are available to d.eternine whether the above

mechanism l,fill resuLt in a Linear dependence of the absorption with

energlij it is not possible to e,ssess further the validity of the

model.

aìô)L.

CHAPTER 6,

OPTTCAL PROPERITIES OT GERMANTUTVI FIL}4S.

6,T

ConsiderabLe d.iscrepancies exist in tbe published. values of the

optical cons.bants of germanir¡¡l fiIns. No tl.oubt, in many c&ses t

accur,-ite measurements have been taken of the reflectance ancl/or

transmittance of these filns but, as seen from the d.iscussions in the

previous chepters, these measu.remcnts must be followed' by accurate

calculations using exact theory to obtain accurate and meaningflrl

resufts.

Abriefsì"ut.. o,ryofneagurementsongerm&niumfilmsuptothe

present time is 6çiven. Ttre resrrfts of ou-r me&surements and eafeul-

ations are then presented-¡ md a model to explain the observed'

absorption in the filns is d.iscussecl'

6. z FoRI,ts oF GERMANTUM FrLl,ß.

Evaporation conditions for obtaining gertna^nium films with

varying d.egrees of crystalline perfcction, from highl-y disord.ered'

amorphous filme to relatively well-oriented' epita>cial fil-ns, are veII

known (Kurov, Vosilev and. Kosaganova, Ig62e I963t K:,'.rovr Semiletov

arrd Pinsker, L957¡ Donoven and Ashley, 1p6\) '

Filns cteposited. at room temperature on fused' quartz substrates

are highly disordered. and. bave an apperently anorphous structuret as

sho.wn by X-ray powder diffraction measurements on metcrial seraped

93,

from the quartz flats (Donovan and' Ash1ey' 1oe' cit')'

Fil¡Ts deposite<]' on quartz' substrates at 325oC anå higher are

polycq¡stalline, vhile films erraporated' onto substrates betveen 6Z'oC

endT25oC are polycrystalline and- higJrty oriented, the principal

planes being tne (roo) ana (]1o) planes as shor¡n by electron d'iffrac-

tion (Donovan and- Asbley, loc' cit')'

Filnsdeposited.onsingle-crystalelectropolisherl.gerrranium

su.ostrates 'raintainecl at Soooc are shor+n by electron d.iffraetion

measurements to be epitaxial. Ttrat is, electron d'iffraetion patterns

of these films exhi-bit a spot strueture ennd sharp Kiehuchi lines

sirnilar to the patterns obtained. from electropol-ished' single crystals

(Donovan and' Àsh1ey, Ioe' ci*") '

Ge'bbie(:1952)fou¡iðthatarrnealinghisfilmsafterd.epositíon

for several hours at tenperatures up to 525oC produced- an electron

diffraction pattern of fine Debye - scherrer - HulI rin¡4s charac-

teristic of the polycrystal-line state'

Cardona and Harbeke (rg6¡) have deposited' epitaxial films on

cLeaved. calcium flnuriðe substrates at a temperature of 60ooC'

Refleetion efectron rliffraction patterns clearly showed their epit-

exial behaviour.

Forthepresentinvestigation,amorphousfilns}Ìereprepare<lby

evaporatingBermeniunontoroorntenperaturequartzgubstrates.Po}y-

crystallinefilmswerepreparedei.therbyevaporationontoquartz

substrates at 35OoC or by d'epcsition onto room tempernture quartz

substratesandsubsequentanneatingat35oocforseveralhours.ffhe

polycrystallinebutlriShtyoriented.filmsvereprepareôbyd.epositing

9l+.

geruanium onto room temperature quartz eubstrates âJxd subsequently

ennealing them at 65OoC for severe-l hours.

6. 3 PREVIOUS CALCIII"ATT ONS OF TÍIE OPTICA], CONS TAN'IS OF GERI{ANIUM

rIil4s.

Brattain ond Briggs (fg\g) caleulated the optieal- eonsta.nts of

thíclc polycrystall-ine fil¡as from transmission measurements. In the

ínÍra,^y'ed- region, where the absorption is 1ow, the refraetive ind-ex

was cletezmined from the cond.itions for transmiesion marcima and minima.

Ítrat is,

ma¡cimr:¡r trons¡aission occurs when 2 nd. = m À

ninimr¡n transmission occurs when 2 nd- = å(am+r)f

where n ie the refractive i,ndex of the film, À the wavclengtht m the

orri.cr of interferenee, and d the thiclcness of the fiI¡l $hich was

detenninect by weighing.

For the absorbing region they used. a wedge of gerrnaniur d.eposíted.

on a quartz substrate. The thickness versus d.ista¡rce l'ras d.etennined'

and. then trarrsmission versus distanee. Using approximate equations,

end. neglecting the quartz-aír interface, they calculated' n a¡¡d. k at

several wavelengths. For the hi6h1y absorbing region it was aser¡ned.

that the transmission was an exponential functíon of the thickness

whereby the logarith¡r of the transnissic¡n versus thiekness should.

give a eurve which is a straight J-ine with a slope which d.etertnines k.o

Lr:kes (fgeO) has detemined n in the wavelength range 35004 -oo

2.5U and k in the region 35OOA - 78OOA from measured. values of the

refleetonce ancl transroittâûc€¡ To calculate k they used the ¡rethod. of

95.

Brattain aJlrì. Briggs, emp.l oyin6 filnis of rlifferent thicknesseB instead'

of B. 'nredr4e-shelped filrt.o

The refractive index vos calculated-n in the region 35004 --

8oooL, from the reflectance using the rels'tion

(to-.)2 + k2f=

(no+n) 2- + k2

where n6 is the refractive ind.ex of air. Trhis is the equation for

the refl-ectence from en air - to - metal interface, anrl is only vatid'

in the higþIy absorbinq region. Greater than 9Oool, o **u assumed'

to be zero and. n d.eterr:rined. fron the transmission noxin¡r ¿encl mini::n'a'

Grant anct paul (rg6l+) dete:minecl n and h for an epitexial filn9oo

ZSOX tfrick, in the regíon 2OO0i( - 6OOOÃ, fror:a measurements of the

nortrral incirience reflcctance and. transni'rtance of the fil-xt' 'Ihe film

thickness was measured. by infra-reú. transmissivity. fttey dc not

give cLetails of their calculetion prr:ced.ure, only to say n and k were

deteznined" from .the theoretica.t expressions for R and' T throuEh a

Newton - Raphscn iteraticn using a high-speed computer'

Tauc et e,l (rg6h,:,966, 1970) calculated- the optical eonsta.:n'ts

of anorphous and. polycrysto,lline filns for thr,' vavelen¡r,,th range

SOOOI - 3U from rneagurerl valtles of nozrnal. inciclcnce reflecte¡rce ¿md

transnitt8.l1c€. T]re refractive and absorption indices were rLete::nined'

from the for::ruL¿¿e given Ì-r¡r Meyer (fg:O), but no rietails c'f the

calculation methoð lrere gi,vcn. They found. the opticar properties of

the polycrystalline layers vere practically id-enticaJ- tc' those of

single crxrsta,ls. Tkrey also reported significa¡t long wavelength

absorptíon in the fi}"¿s, in agreement with the theoretieal prerlictions

96,

of Gubanov (t965),

ÏtraLes, Lovitt and HiLl (fg6f) have d"etermined n and k for

asrorphous ond. polycrystalline films in the region IU - 5U. ftrey

d.eterndned n in û, simil€ùf manner to Brattain ancl Brig¿3s. For the

absorption index, the observed tfonsnisslon we,s comParer1 with that

ca,lculated for pairs of n enit k until the two agrced.. However a

lar¿;4c nu¡nber of sueh pairs exist and it is d.ifficult to assess the

vnli,Jí-'.;y of their results.

Donovan, Spicer and Bennett (tgíg) invest:L¡4ateC. the a.bsorption

in amorphous films. Fron neosrlrements of refleetance, tro,ns¡nittance

n¡:d. fil¡c thickness they calc:ulated the absorption cocfficient of the

fiL:ns using on iterative methocj <Lescribed by Bennett and. Booty (loc"

cí*,, ) . ftris methocl of celcul.r'r,tin6 n and k was shown in Chapter 3

tr¡ be doubtful- as it may c(-ìnvcrge on the wrong solutione, ¡ortie-

ularltrr where the nrultiple solutions are elose together.

Ttrey were particularly i.nterestetl. in dete::nining r¡hether long -

wavclength e.bsorpti6n existecl in the filns as had. becn reported.

earlier (nauc, 1oc. cit.). lhey found no evidenee fr:r absorption

in the infra-recl beyond- the absorption eclge, in agrecment with their

photoenission experirnents .

Tkre results of the pbove a.uthors are sunnlsrized. in Firju::e 6.1.

fher.c a,re seen to be r,riùe vari.e.tions betlreen the't¡aricus authors,

shoving tbe need fr:r a rûorc: í,rceurate and reliabl-e cleterminstion of

the r:ptical constents of ¡"temaniurn films.

Nc previous worker has bcen able to talce accou¡t of any oxid.e

or surface layers which mciy fo:m on the surface of the fil¡ns, nor have

5

13

rltl.8 6

6

xUo

=t¡J

F

É.EU

Þ

5

4

5f !iAlt^rN

^ND !trGGs

2 w¡tr5 ¡r At - ?o[TcRYsrA!(tflÉ

3 truç - ¡6¿*t"orté w¡t¡s tr ¡I - Auoßråous

5 truc - rotvcevs'¡tLrrt$ toxts

7 or¡pt ¡po ¡¡ur

0

lr1.2 1.0

WAVILENGTH IN MICRONS

Figure 6.1-a

zIt-o-0¿ou)

' 2.0 1.8 1.6 1.4

WAVÊ LE NGT H IN

FÍgure 6.t"b

1.2

MICRON S

1.0 .8 .6

10 5

I

I

C)

=t--z.vl9LLLLt!o()

zo;ô-ÉoU)lo

103

01

DONOVAN, SPICEß Å BEI{NET¡.

PI.IOTON ENERGY IN E I.ECTRON- VOLTS

Figure 6.Ì"c

10

6

firru€* 3

2

1

5'4

50

4.6

4+-

&E

n

ü

R ESULTS OF T,AUC

E¡,{ERGy * eV

Figure ó" J.d

97.

exact calculations been perfonned. Potter (fg¿í[), in his cal-cu]-ations

on single crystals, arbitrarily eopected for an oxid.e layer lol

thick"

The results presented. below constitute the most aecurate

d.eternination of the optical constants of gennaniurn fíIns thus far

presented.

6" )+ FIËLECÏANCE AND TNANS¡4ITTANCE OF GEP'rr{A}TIUM FILMS.

The reflectance end. trsflsmittance o1 12 gerrnanium films,

neasurr¡d. witn tne reflectometer d.escribed in Chapter 2, are given

in Appendix E. The c\rves for four of the fil¡rs are plotted. in

Fígure 6.2. As with the selenium fi1¡rs multiple bearn interference

effects are apparent, particularly in the thicker fihnso TLre estimated"

eïr:ors in the ¡neasured. values of R and. T are t0.002"

6.5 CALCULATED TILM AND OXTDE THICKItrIiSgtr]S.

Ttre film and surfaee layer thicknesses of the germanium films,

calculated. by the method d.escri't¡ed- in Chapter l+o are tabulated in

Table 6.1. The exceptions are the first three films which were too

thin for this method to be applicable" Thej.r'thicknesses Ρere meesured

explicitty by the rnethocl of fringes of ectrual chromatie ord-er d"escribed

in Chapter 2. flae thicknesses of some of the thicker films were

also meosurcrd explicitly, the calcul-aterl total thicknesses a¡grecing

well- with the measureC, values. After a few films the practic,e of

measuring the thickness of the thicker films r^¡a¡ discontinued.

For many of the filns, the closure of the refractive index curve

w^YE!tr6tH lt Mrcnoìs

t6la rN M¡CFOXS

à

WÀVEL¡NGIB Iil rcRct9

til MICRONS

/^ts].grire ii, z

98,

was densitive to about five angstrom unite in both fifun and' surface

layer thicknesses. Ttris repregents about one monolayer of substs'nce

whieh appes,rB strange since the non-uniformity of the fil¡rs would'

exceed this value. It is suggested. that the calculated' thicknesses

a;re accurate averages taken over the erea of the light bean on the

fil:ns.

GE].

GEÍ?

GE3

Annealed. at 35OoC

Deposi-,,ed. on 35OoCsubstrate.

tor3

0r2

0t5

or5

1015

0r5

997!3

1O9Bt10

1130t5

Il+25!2o

1T50t20

9151)+o

810

Room Temp.Substratc

GE5

ll

il

lt

!l

It

It

il

il

il

il

il

il

ll

il

il

It

ll

r

Í

n

ll

G][?

GEB

It

l-55!5

1o0r5

015

7t5!5

855!5

1072r10

lOO?r6

109Bt12

1130r10

rl+25f.25

r760!?5

915th5

Bg5!27

rTl+stl+6

11h7r35

10T2t

2l+0120

25OtzO

5LA!25

r020r23

BT0r10

955tl0

EVAPORATICIN

CONÐITIONS.NO.

FIIM d 1 CAJ,C .

o(r) (r)

(ar+42)d2 CALC.

o

cAl,c.o}T

MEASIIRED d.

oA(

Et2 Anneoled. at 65OoC

fuAtìLE 6.L.

99

6.6 OPTIC.A], CONSTA]ÍTS OF GER¡4A.T\TTUM FILVIS.

The calculated. refractive and. absorption indices for al-I of the

geraranir.rn fil-ms investigated az'e plotted' in Append.ix E" Ttre solutions

for one of the filns are plotted in Figure 6.3. Trhe behaviour of the

solutions are sinilar to those for selenium. Ttre behaviour of the

errors is seen to be sinil-ar to those caleulated. for the seleniun

fil-¡as. As with the seleniurn films only the correet solutions for the

absorption inclex have been plotted..

6. 6, ]- AMORPHOUS FTLMS.

Films witn tnickness greater than about l+ool proauced. similar

results r md the rcfractive ind.ex and. absorption ind.ex curves for these

fil¡rs are plotted. together i.n Fi.gure 6..l+. Tnne srnall variations from

filn to film are not surprising since one woul-d. expect the optical

properties of a film to d.epend on the structure of the filn. fn turnt

the structure of evaporated, filns d.epenrl on a nunber of cond.itions

such as substrate temperature du:ring and af'ter evaporationt evaporation

rate, partial pressures of the residuat €ls,ses in the vacuuln charubert

source temperature, etc. These conditions change slightly fro¡r film

to fifun resulting in snal-t va,rietir:ns in the optical properties of

the fil-ns.

Fron these results however, reasr:n¿rb]¡r aecurate avera.ges for

the refractive and e,þsorption ind.ices can be dravn and sre shc¡r¡n in

Figure 6.!. These values have been usecl in the preparation of

nultitayer thin filn interference fil-ters with exceflent agreer:nent

between predicteri. and experimental resul-ts (Warcl, 19TI) '

xo

=u=tsoú

G

3

ã0 t.ô 1.4

WAVELENGTII+2

I N MICRONS

'g

.?

xä.0

=

.a

z9tsoÉoo

t.8 t.6WAVËLENGI[I

12tN t¡lckoNS

1.0

the Çêì.culatedir i.g r:re

Eoiult.io¡rs for a Ge film"O.J

IIIIITI JiII IIIiIf III IIIfI IIIII III II

e--

GÉ4

)lì"H"sca Ie

L.H.sca Ie

6

t'5

t'3

7.2

.z

Exaraple of

xôz

Ëoæ

ÍRefractive irrdex cL¡rves of tlieamorphous germaníum f ilnis

l.{WAVELE NGT}.I

xoz

z9FÀ-

IN MICRONS

VJAVELENGTI-I IN MICROì.IS

þ'iqrrre 6 "4.

1.0

'9

.8

4fnrna,{rrl

z.or1><

xl+,o

z,o.5Þô-É.o.4.J'6

.3

,2

.1

21.2

.6I2,ît 1'8 1.6

WAVELENGTH

Fig'.lre 6"5

MICRONSIN

Average namorphous

and k curves forgermanium filns

j

n

100.

6,6,2 VERY 1IIIN .AIIORPHOUS FIU/IS.

Films Gel (z\01 tr,;.cX) e,nd Ge2 (2501 thick) had a much lower

refractivc ind.ex tho¡. the thicker arnorphous filns, md also exhibited'

on apparent shifb in the absorption ed.ge, the shift being towards the

short ,¡avelength end. the optical constants of film GeI are compared

to those of a thicker filn in Figure 6.5, Ttris behaviour is readily

explainetL on the basis of Schopperrs theor.l,' (section \.lLÌ), From

ecluation l+,IL,L,

r ìlrir = [{t

- y(nn2-t)i(r-r) llL - vr(lru2-r) }jã 6'6'a

For values r:f y = 1.02 and' f = 0'012, the results of fil¡r Gel-

are brought into near eEìreenent with those of the thicher fi1m, as

cen be seen in Figure 6.6" Considering the simplificotions involveô,

the agreenent is excell-ent "

The srna-lI value of f indicates that the ma,jor axis of the

ellipsoitl.s, parellel- to the substrate, is nuch larger then the rninor

ançis. Thet is, the fifuir consists of long islands. This picture is

consistent with stud.ies of the kinetics of the fornation of thin

evaporated fil-ns (Johnsonn 196r), which show that initiatty some ¿toms

fo:m nuclei on the substrate about which the fil¡r grotrs, at first in

the forn of islands. These isl-and.s grow until eventuslly the

d.eposition is unifcrrm over the entire substrate. Stud.ies on cadmium

sr:Iphid.e fil¡ns in this laboratory (Goodwin, 19T1) shotr that for these

1,irns the d-eposition does not bercotle wrifor:n until- a thickness ofo

ebout 3ooA. Electrc,.n nicroscope exor,':.inaticns of selenium rihns by

Ca.npbe11 (loc. cit.) showed that very thin filrns lrere discontinuous

and consisted of islands.

xo=

FoE

É

WAVELENGTH IN MICRONS

WAVEIENGTH IN MICRONS

Applic.¡¡tion of scÌ1oppel:r s theory to a very thin f ilrn"

Figure 6.6

z

zo;:cÉo@

01 .

Ttrus the posbulation of very thin gemaniun fil-ns consisting of

ellipsoic1s, although sinplified. by the assunption that they are/,9

jìl

¡'\1'ì\the

sane size, is therefore reasonable ar¡d in agreernent with experinental

observations.

6,6,3 FTL}'{S ANNEÁIED AT 35OOC ATüÐ FILI\4S DEPOSTTED ON SU]JSTRATES

AT 35ooc.

Ttre filns deposited. on room temperature substrates and sub-

sequently a¡rnea1ed. at 35OoC for several horrrs cxhibited optical

constants sirnilar to thc,,se r,¡hich I'ere not anne¿llecl. Ttre optieal

constants of films deposited on substrates preheated. to 3rOoC

showed similar behaviour to the very thin ornorphcus films¡ that is,

the refractive ind.ex r,¡as smaller and the absorption edge appeared

to have shifbetl to shcrter wavelengths.

It appears that these films have a d.ifferent strueture to those

deposited. on roon tenperature substrates, this structure arising in

some w€,,y frcm the presence of a hot substrate during the evaporation.

lle have already seen that the surfeces of these films are vrj.nkled

which is not ihe case for the ot,her methods of fil-m ¡reparation lle

have usedo lfrere is a need. for further experirnents on these films

before any conclusions can be drawn, ¡articula,rIy in relation to thc

kineties of the fomiation of these fil-::¡s and. tireir subsequent struc'bure.

Apart frcm this differenee, the optieal constants of ernorphous

and polycrystalline filns exhibit a sirnilar dependence on w&velength.

Tlrese results are consistent with the theory of absorption in

geruenium fil-ms to be presented. in section 6,7,

LOz,

6. 6, l+ optrc¡:, CONSTATÍTS OF FILI{S AIINE/ILED Ar 65ooc.

The optical constants of a gennanium film annealeô at 65OoC

6,re comparecl to those of an enorp¡ous fitrn in Figure 6.T. Íhe refrac-

tíve index of the hieþ te¡eperature fil¡t is slightly legs than that for

the a.nrorphous filn. Ttrere is a significe¡rt d'ifference in the absorp;

tion indices of the two fil¡os. Ïnitially both fihes show sí-Iûilar

behavicur, but at a l¡avelength of about 1.05u the higlh temperature

filn absorption index departs markeùLy from that of the anorphous

fil-n. Ttre significance of this will ¡e diseussed in section 6,7,

6,7 ABSORPTION PROCESSES TN GERMANIUM F1L}4S.

The theory of absorption proeesses in erystalline naterial-s is

treated. in d.etail in a nu¡rber of books (see, for exalrpl-e, Smith in

tVave Mechanics of Crysta1l:Lne Sol-id.srtt 1961). A brief treatment is

presented and. this is then extended to the case of amorphous gerrnanir'ur

films.

6.7 . 1 BASIC IIIEORY.

TLre interaction between ttre incid.ent electro-magnetic rad-iation

and. the electrons in a semicondueting crystal is d.escribed by the

Ha¡riltonian

H - =9 '-"1 6,7,Lrao In

where A is the veetor potential of the efectromagnetic field' and'

p the momentum operato::.

For rad.iation of r'rave vecto" I"oa and- angurar frequency û)

5

200c 6 5ooC

ÐñîÐ

fm

-2çñx

nooc

xUoz 6500c

z9F-ùÉ

co

kI

,.6.8

0WAVELENGTH IN ¡JIICRONS

Figure 6 "7

l-03.

I = Asdcos(ot - Iroa'f ) 6'7 '2

vhere o is a u¡rit vector in the direction of polarisation of the

radietion.

Equation 6.7 ,L maY then r¡ritten

Hrad.

6,7,3

The first and- seconcl part of equation 6'7 '3 represent the

enrission and absorption of a photon respectively. Íhus the transition

probability for absorption mey be cieteruined by applying time d'epencl*

ent perburbation theory to the second. part. Integrating over a renge

of frequencies for the incid.ent rad.iation (rand.on phase approxination)

gives

wif * l.i l"*ni(Irra'r)o'pl f' ô (8. -Ef-bh!) 6,7,1+

where lI., is the transition probability per u¡rit time between the two

enere$r fevels E' and' Er. lho total enerry of frequency o absr:rbed

per unit volt¡¡ne per unit tinc is then

P(o)=lItut-nr)]Iir 6,"(,5

where the srrnnation is over afl initial and finot stâtes per unit

volr¡ne. If S is the poynti-ng vector, then the absorption coefficient

K is given by

eAo

ñ [*n{r(.rt-5rou'l) } * exp{-i(d-Irad'") }J :'P

l(=

cil' It_

If

<i I expi (5rna'r) o'p I tt

2o(n.-ur-Tro) 6,7,6

10b

where C is o constant.

The absorption can therefore be for.¡nd. by sr:mning over all non-

zero matrix elements between the initial anct finat states 'ilr'fl,

for which the energr is equal to the photon energy licr;.

tauc (t965) has sho'wn that if the natrix elenent changes only

slightly for all the possible transitions then it nay ìre put in front

of the surrnation e'd the sr:rmnation replacecl ¡V g(E), where e(¡)dg is

the nu¡a,ber of possible tronsitions per rrnit volume in which the

trcnsition enerry lies in the ronge E to E+dE. lhe firnetion e(g) is

knol¡n as the joint d.ensity of states function. Equation 6.7.6 then

beeonres

K(o¡) = #lurrl2s(n) 6,7.7

where

Mir <i lexpi (1"*¿.I)î.f lr'

6. T. 2 DIRECT TF.AI{SITTO}TS .

Ttre matrix elenent Mt, nalr be evaluated as follows:

Mir = -infu,f,i*{expi (Iracl':)1'liU,' av

Fhere úi ana V, are the wave functions of the initial and final

gta+,es respectively. For a periodic lattice these ney bc l¡ritten in

the fa,nilier Bloch forn

ûrr(rrk) = urr(1'f)expi(k'r)

where Urr(1r!) has the period-icity of the lattice'

Ttrus

Mif = -injuur*(:,:r)*rrp(-iIi.r).:o-1'{i(lJ"oa'r)u.Y}ur(r,r'r)exP(ilr'1) dv

105.

where 5i *U |, are the wave veetors of the initiel and' final electron

states respectivelY.

Mir -ih Iuuro{*'Yur * i(o'kr)ur}expi{(5, * 5r.,a - t<r)'r} av

6.7,8

Using the period^icity of U. anrì. U,,

Mir = -inf O(r,5i,Ir) av [expi(k, 5ra¿ - Ii)'l¡'*rii J

cell,

6.7,9

I{here R. is the vector to the "i-th ceIl'-J

fhe sunmation in equation 6,7,9 nay be shown to be zero unless either

5r * I"o.a - Ii = o or ,n2n 6 '7 'Lo

where l' i" a reciproca.t labtice veetor. |"u.u is very snaJ'f for

optieal vavelengths ent therefore the seconð cond'ition does not

aÞ?ltrrr thus:

k._1

i oeo k, - ko 6'7 ''-'--r -f

This requirernent therefore se'i'ereLy li¡níts the nu¡nl¡er of initial

and. final states which can eontribute to the absorption for a given

encrg$¡ of separation hul. such tre.nsitions are known as direct

transitions where the wave vector of the electron is conserved' during

the transition. If Ec. is mcasured. up frcm the bottour ofl thc concluc-

tion banrL a¡rct E d.own from the top of the volence band thenv

Itr¡

k-rad5t

E +E +Egcv

The joint clensity of gtates function is given by

6,7 ,r2

B(E) = ec(Ec) + err(u.r) 6'7 'L3

If the enerry Öependence for the valence ancl eond.uction bands ean be

repreeented. by the usual k2 approxirnation near the band. ed'ge, then

¡¡2. ¡1s2

E = and. E-- =-c 2n* v 2m--*--C 1I

vhere m * antl m * are the effective masses of electrons and' holes"---- - -c v

respectively. Using equation 6.?.11+ ond' the fact that k before

transition is equal to k a,ftcr transition, gives

s(E) (n - u*)12c

106.

6,7.L\

6,7 .Lj

Hence

Ice.

6.7. 3

K(E) c # Itirlt (n - nr)'t

2

1

2nr¡nK(E) (u-s)'øCË

INDTRECT TRANSTTTONS.

In a periodic structure the conttitions given by equation 6.7'I1

can be relarced. by conserving momentr:m r,¡ith an electron-phonon

interaction at the sarie time ae the transition' The transition

probability is calculated by first considering that the electron

me.kes a vertical transition to a virtual state r'¡here the lifetime is

short and enerry is not congervetl. Electron-phonon scattering then

takes place so that both energy and momentum a.re finally conserved''

Fip¡re 6.8.b is a d.iagra¡nmat,ic representation of the transition.

For these phonon-o,seistetl transitions

CONDUCT ION BAND CONDUCTION BAND

ELECTRON -PHOTON

INTÉRACTION

VALENCE BAND

ETECTRCN.PHONON SCATIERETECTRON-PHOTON

iNTERACTION

VALENCE BAND

(a) NIR ECT T RAN S IT I ON (b) TNDiRECT TRANSITION

Figure 6.8

107.

c(E) * (Tro-tut ttirn¡)2

rçhere EUi is the ind.ircct band--gap and- ,ph tht phonon frcq.uency' Since

ñrph is very small, thcn to a good approximation

rrr¡nK(E) * (n-urt)' (>'7'16

6, T, \ ABSORPT]ON TN CRYSTALLINE GERMANTUM FILI!4S.

Band structure cnlculations on crystalline germanium (Herman,

L95\¡ Ig55) shor.r that its vaJ-ence rurd" cond.uction bends ',re represented

by Figure 6.8.b. From equo,tions 6,T,I, and' 6 'T'16, the absorption in

a crysta1 nay be rePresented. bY

E-E_ * {rrrrrr(n)}BI

where B -- 2 for d.ireet transitions and' B = I for ind'irect transitions'

fr plot of ih.rnK(n)Ìß, ¿r,s a ftmction of photon enerey E, for various

values of ß nay therefore bc expected to reveol the a.i:sorption

processes in the naterirJ.

For germanir.m fj.l¡rs vh:i-eh had been annea-Ied at 65OoC, the

va.lnc of Ê r,¡as I to. photon encrgies from O'?5eV to I'J-gV, an¿ S ças 2

for photon energies fram l.1cV to l-.5eV. Tttis can bc' scen in Figure1

6.9 ,¡Ïrere {non1ç(E)}t *n {n.rnli(n)i2 trave been plottcd a¡;aiirst E.

The initial absorp-bion in these filns is thus of the form given

fo:: i.nd.irect .transitions, where the mini¡aum in thc eond'rrction bo.nd'

and maximu:n in the valence ba,nti. corresponcl to d'ifferent vp'Iues of the

electron wave vecto" 5. flhe energy separation between these two

points is o.75ev. T'tr.e nbsorpti.on then chonges tc the forrn Siven for

,lirect trarrsiticlns, where a nininum in the conduetion bu'nrL correspond's

o

ô

o

o

o

o

c

ce

o

ø

o

o

ca

o

o

o

6

5

80

70

40

20

10

0

rn

xP

x

c(o0

o

x

IE 3

R.LI. scale

CTRON - VOL

L.H.sca Ie

(sntr)7t .na (enr<)2 u,functíons of photonenergy for a hÍghlYorierìted polycrYsta Ilinegerma nir.¡rn f ihn

0

0

PIIOTON ENER(ìY IN EL

Fi"gure 6 (i

60

l-o8.

to the ninimr¡m in the valence band. for the se¡ne k. Ik¡e vertical

enerey gap in this case is about 1.1eV.

lhus the band. structure ancl absorption processes in the Ìrighly

oriented. polycrystalline fitxrs are basically wraltered, from those

found for single crysta1 gerta,nirrm, The main difference is in the

values of the threshold.s for the onset of índ.irect and. d.irecb transit-

ions, those for single crystal gerua,nium being O,62e',1 and" O.B1eV

respectively (Dash and Ner,¡man ¡ f955), compared. to O.75eV and l.IeV

for the fil-ms.

6.7. , .ASSORP'IION TN .AMONPHOUS FII,T{S.

lnitiauS' the absorption in the a,norphous ge:manium films

follows that of the highly oriented polycrystalline fil-ng. fhat is

from 0,72eY to about l.leV the al:sorption follows the fonn given for

ind.irect transitions. From 1.}eV to about 1.\cV the absorptíon also1

folLows the se,ne fort, as is shown in Figure 6.10 where {n.,rnt(E)}ã

h,as been plotted as a fUnction of E for the er,morphous filns. It ís

seen to consist of two straight lines with a change in slope at ln1eVt

and.intersectingtheenerg¡ra:cisatO.T2eV.Tttisbehaviourhasalso

been observed. by lauc (loc. cit.).

Ttrus either the bend structure, or absorption process or both

have changetl for anorplrous gennanium. Theoretical calculations on

anorphous semiconductors by Gubeurov (toc. eit.) indicate that the

essential features of the band structure remain unaltered, except for

a possible change in the magnitud.es of the enerry SaFsr which would.

meen thot the absorption process has alteretl.

I 000

800

600

l:<lclu->

400

2 {i0

,4 .8

PHOTON ENERGY ELECTRON: VOLTSf.0

IN

,o

ol/

(EnK)'2 as a function of photonenergy f,or amorpho¡.¡s germaníurn films o

I7t

i6

Á.þ

.O

,4f,f

P

,f¿'

Fiçure 6"L0

fo9.

Ttre d.erivation of the joint density of states function, g(E) I

for d-irect transitions rntas evaluated using all initial and' final

states for Which l.oth energy and mo¡rentUm wcre conserved. The con-

servation of mo¡rentum given by equation 6.?.11 depends upon the

electron wave fr:nction being representecl by a Bloeh Ì'¡aver and. the

regular periodicity of the lattice allowing the separation of the

integral in equation 6.7.8 into two partsr (f) an integral over the

g¡rit ceLl an¿ (2) the sr.gn over all r:nit ce]Is in the crystal. In

general it nay be eaid that translational invariance wou1d. lead- to

conservation of momentum.

In an amorphous structure where either of l¡oth of these

cond.itions do not applÍr there would. be a refucation on the cond-ition

which makes the matrix element Don-zero. TlIe evaluation of g(E)

must then be mad.e using all initiol ond final- states which are

separated. by an enerry T1o. Equation 6,7,L3 may therefore be repleced'

by the integral:

e(or)s ø"(n")sv(Ev) dE., 6.T,rT

where

E = fio¡-E -iicgv1

tt gc *ru Bv are consid.ered to be proporbional to 82, then integrat-

ing equation 6.7.11 gives (Mccoy, L96r)

e(o) ,. (Trar - uu)2 6.T.18

ïhis expression is of the s¿ìiiìe foro as thnt given for phonon aidecl

indirect transitions, exeeìpt for .the omission of the term htonn for

110,

the phonorl êll€rgre Hence

ätonK(E) (¡-u )2 6,7,L9g

in agreement with the observed results.

Figure 6.11 is a diagre,unatic representati.on of the transitions

which occur in aroorphous ge:maniun.

6, T, 6 LONG I,IAVELEI{GTTI ABSORPÎIoN IN GERMÁJVTUM FILI¡XS.

Althou6h the calculated errors in the solutions for the absorp-

tion index are relatively large in the long wavelength region, it is

apparent that the absorption falle to very small values in this

region. Ttris result is in agreement with the photoenrission experim-

ents of Donovan et al (Ioc. cit.). Ta.ue (l-oc. eit.) on the other

hand reported. significa¡rt long wavelength absorption in the germanium

filrns he investigated. fhe error analysis used in the present

cal-cul-otions showed. that 8.n er?or of tO.OO1 in R and- I resulted' in en

error in the absorption ind.ex of about t0.003. Thus a s¡rall error

in R or T uiË result in a significant error in the calculated

absorption coefficient in the long wavelength region. It is suggested

that, the absorption observed. by Tauc in this region may have resulted

from smaLl- errors in the measured. reflectance anö/or transmitt&TlCeo

Es

C. B.

V.B

hw-Eg

lDirecttl'rañsit ion s

-1!,

hw - Ect

Eg''= '72 eV E!, = 1'1 e\r

1."'i.cirrse 6 " 11

111

CHAPTER.f

CONC].USIONS.

?. 1 ON TETE DETERMINATÏON OF TTIE OPTTCAL CONSTANTS 0F UIIN I'IL¡/6.

lduch of this thesis has been concernecl with the cal.culation of

the optical consluants of thin fifuns. For the stutly of the absorption

ed.ge of semicond.uctor films, the mea.surenent of ncr:rnal incidence

refl-ectance and. transmittance is clearly thc most reliable method-

for deternining these parameters, as other method.s d"epend. eritically

on the properties of the surface of the fj.J:.o. Even son thi.s method

has not been applied extensively to the study of thin films because

of the conplicated n.ature of the reflectance and trensmitta^nce

equations. Analybical solutions for the refractive ind.ex and

absorption index are no 1on6er possi.ble and. one nust resort to

nr.merieaf method.s to eal-culate these quanti+.ics. Ttre situation is

further complicated. by the occurrence of multiple sol-utionsn and.

this has probabÌy been the most formid.able problem in the past sinee

often an incorrect sofution nay be nj.stuken for the correet va1ue.

The nunerical solution of the refl.ectance ¿-md transmittance

equotions itself has in the past presented a major problcm beeause of

their conplicated behaviour,

lfe have shor¿n that the functions (ftn)/T exhibit a rnuch sinpler

behaviour, and ve have d.escribed. in d.etail the caicul-ation of n and.

k from thern. Given R, T and the film thickness d., i+, is no'lr a

simple mr+tter to caleula-çe n and k.

The protilem of rlultiple sofu'fions requircd careful anatysis end.

.11_2.

wå,s best exaTrlined by consi'lering hypothetical filns ' Tle have shown

that an una,nrbiguous choice of the correct solution can only be rna'le

if measurements are taken over a, sufficiently wid.e wavelength rartge.

lle have also seen that tbe fil.:o thickness is a critical para,rneter

and that the behaviour of the cafcul-a.ted solutíons under small

changes in filn thickness is quite d.istinct for thicknesses which

are either too sna1l or too Iarge, leading to a method. r^¡here'by the

film thickness may be calculated. along r¿itn the optical constantsn

without recourse to its explicit measurement.

Ttre inability to find a sing1e filn thickness which woul-d enable

the refractive ind.ex curve to be closed over the entire waveJ-ength

range, assuming R and T were ¿ccurate, .lv'&s shown to be ðue to either

en oxide layer or an irregular surface. Provided the layer is very

thin and. near transperent over the wavelength raxrge of mensurements,

its effects on the colculation of the optical constants of the

ud.erlying filn'r may be eliminated. by consid.ering a tvo-layer system.

fhe equations for the refl-ectance a,::d. transmittance of a tr,¡o-

layer system are more eomplieated. than those for a sin651e-Iayer,

but aga,in the funetions(ftn) /I are much simpler. For the special

case r,¡here one of the layers is transparent and. of known refractive

index, we have described how the optieal- constants of the absorbing

fiim, together with the 'bl.riclcnesses of the transparent and absorbing

Iayers r may be accurately C.eternrined. fron the meetsurcd. reflcetance

and. tronsmittance of the system. Although r,¡e 'were coneerned primar-

ity vitn accor¡nting for the su.rface, the theory devcloperl i.o quite

general a¡¡d. is applicable to any two-Iayer system where onc of the

113.

layers is transpârêrlt.

7,2 ON T'}IE OPTICAL CONSTANTS OF SEI,ENITN4 FTLT'IS.

TLre analyses of the behaviour of the calculatecl solutions for

the optical consta¡rts of thin filns revealed. that campbell, in his

calculation of the optical consta¡rts of vitreous selenir¡m films,

chose incorrect solutions from those that existed. lüe therefore

recalculated the optical constants of these films from his refleetance

and transmittance data. The resufts obtained- for thc refractive

ind.ex of these fiLns are in quantitative agreement with previous

vork on the bulk material and on sputtered films'

Ttre absorption in the filns, although in qualitative agrecment

with other authorsl results in the highly absorbing regibn, depart

fron previous results in the low absorption region where the present

filns exhibit considerable absorption, particularly in the thinner

films. As the thickness of the filn increased the long wavelength

absorption clecreased, and the absorption mechanieru was attributed"

to the cleveloping chain structure of the vitreous selenium.

A plot of the absorption cocfficient as a function of photon

enel.ry .was folxÌc1 to consist of two straight finesf with the change

of slope occrrrring at the enersr value of the onset of photoconcluct-

ivity. The results are consistent with the view that within

inctividual chains or rings carriers are free to move, but to travel

to neighbouring chains or rings it is necessal"Jr to cross potential

barriers.

* see Figure 5.10.

111+.

7,3 ON TIÍE OPTICAI, CONSTATITS OF GERMANIUM FTLMS.

ltre wide discrepancies in tire publislied. values of the optical

eonstants of ge::rraniiun fi1¡rs ind.icated, thre need for a¡r accurate

deterrninat,ion of ther:o Fihls with various degrees of crystalline

perfection were stud.ied. with amorphous filns beíng studied' in greater

d.etail.

Tlre optical constants of filns a.nnealed at 65ooc, that is

polycrystal-line and. highly oriented. films, pfoduced. qualitatively

sinilar results to those found. for single-crystal gemanium' From

o"75eV to I.IeV the obsorption in these fil-ms followed. the 1al'¡ for

indirect transitions, Greater than l-.leV the absorption folfowed

the 1a.w for ùirect transitions. T'he main difference between thesc

films and the bu-l-k material is the tirreshold energies for the onset

of indirect and d.irect transitions, being O.62eV a¡d O.BLeV respec'b-

ively for the bulk nateríal, a^nd O.75eV and l.leV foilbhe fiLns'

tlee optical constants of fitns d-eposi.ted on room temperature

substr¿tes, that is arnorphous films, Ìfere in clualitative agreement

with the results of Tauc, with the exception of the long-wavelength

region beyond. the absorption edge. fhat is, the absorpticn ed-¡¡e in

these films foll-or.¡ed. the lefr for ind.ireet transitions fron 0'l2eV

to l.l.eV, and. foll-owed the same lar¡ brrt with a change in slope for

energies greater than 1ofeV. The results are consistent wittr the

vi-ew th¿Lt the esoen'r,ial features of the banC. picture for amorphous

gennaniun are basieally unalterecl from that of crysteilj-ne germanium,

anct that the rand.on structure of the anorphous state removes -vhe

cond.itions for the conserveJion of nomentum, givin¡¡ riso to an

f15.

absorption law similar to ind.irect trensitions '

The long wavelength absorption in the ger:naniu¡l fil¡rs was found

to fall to very smalf values, in agreement with the photo-cmission

measurelnents of Donovan et al.

Polycrystalline films, prep&red by d.eposition onto room teirçcr-

ature substrates with subseguent annealing at 35OoC for several

hours, prod.uceC. sinil-ar results to the anorphous fihns, indicating

ttrat the size of tl:re rendomly oriented crystallites does not have

a significe¡rt influence on the absorption proeesses"

The applieation of schopperts theorxr to very thin arnorphous

fil¡rs brought the results for these fil-ns into near a¿¡reenent rrith

those of the thicker filnrs. fne vie¡¡ that these films consi.st of

Iong islands is consistent with thc kinetics of thin fil¡r forrnation.

Filns d.eposited. on substrates at 35OoC sho'nIed sinila"r properties

to the very thin filns, The surfaee of these films wns found to bc

wrinkl-ed. in o.greement uith Sloope and. TilÌer, but tkrerc is a need for

further wcrk on the structure of these filns before arry conelusions

cen be d.Tavn.

7. \ FUTURE I,IORK.

The theory involved. ín the catcul-ation of the optic¿tl consta"nts

of thin fitns from no::¡inl- incid.ence reflectence i,mc1 tr¿lnsr,:ittancc

rucasurements was Oevelopcd. in Chapters 3 ond" \, rrnd thcn applicd-

successfully to the d.eter.uiin¿l,tion of n and Ic of selcnirm and

gcni,anir:m films, The tÌieory Las paved the way for the accurerte

deternrination of n and. k for thin semiconducting fihas and- vork

116,

is alreacty proceed.ing in this laboratory on the optical constants

of silicon and. group 11-Vl- semiconduetorgo

Surface layers and. inho¡nogeneous surfaces Ïrave preoentecl

problens in the past, but now theee can be read.ily aecounted for¿

It should now be possible to dete:mine accurately the optical constente

of thin fil¡rs of a large number of se¡riconcluctors.

IL?.

Ai,-rEND:1-i( "Éi

DERIVATI\rujs oF (lt B) /r rqrr EE- SINGL"F-:I,åIEE-IÌQUATI0,N$.'::{^/.i --Ê .

t+R 1

T (ns+n2) 2(¡r2+¡t2)

t-R

(ne 2+n, 2+kr 2 ) { ( n1 2+nr2+kl 2 ) cosh2ql

{ 2n1n2sinh2o1}

+ (ng2-o1 2.'f) l(n¡z-nrz+t12 ) cos2'¡1 - 2n2k1si"2Y, lJ

2non1 { (n12+n r2+k¡2)sin}r2a¡ t 2n1n2eosh2c¡ }

+ kr{(n12-n22+k12)ain2vt + 2n2k1cos?vr}r (ns+n2) 2(¡12+k12)

fr-RìITJa

--=

l+n

âdr ¡(n's+n2) 2(¡12+k12)

^ Ll*nì"[TJ 2

ã = (rirt**r2) (no+n2) 3

þr {ro t*rr, 2+r1 2 ) { ( n1 z+nrz+t'1 2 ) sinh2ar

+ 2n1n2eosh2c1)

- n1(ns2- nt!-Etl){(n1 2-nr2+lt2)sin21¡ + 2n2k1co'zYtiJ

(ns2+n1 2+kr 2 ) { (oso2-nr 2-k r

2 ) cosh2el

* n1(ng-n2)sinhZal]

+ (n¡2-n1 z-kt2) {t r (nz-nq)sin2vr- (nen2+n12+r'12)"o"zv1}J

^ fr*Rl'[TJ Iânr (ns+n2)2(¡¡r2¡¡r2)2 [rr, {

( n ¡ 2+k

1 2 ) z-rLozrLzz\cos}r2o

1

+ zn2l ( n, 2+t 1

2 ) (n 9

2+n1 2+k r

2 ) - 2n r 2n0 2

Ì s inh2a 1

* f 1 rrnrr,rno 2k

1 2- ( n 1

2+k 1

2 ) (ns 2-n 1 2-k, 2 ) ( n I

2-n22+k r2)

Ix 21 tÌ" in2y

118.

+ z{ (ne2*nr2-kr 2) {nf ¡-2-,n2(n¡2'+txt2)Zo,y}

-, 11,1 (n1 2 nbt2) ( n1 il^'rl:.!rh1 i ) ) ccs2"¡ 1

âkr (n6+n2)2(rlr?'+6r2)2 Iit", z+l.tz) (ro2- t'i:.i?4ur?-) (rrr2'u,, ?2" r'?)-t

- l+n1n2it¡ 2k1 ] sinh2o' 1

+ zl(2n2"¡r+k1 ) (ni 2+tr2 ) (ne2+n1 2+t<1 2)

-*o 2kr (n¡2+n22+l-12 ) ] cosh2o 1

+ znz{ (n12+k12) 2 n no2(kr2-nr2)}sinz^¡1

+ 2k1 {ns2n22-(n12+k12) 2}cosaYll

t'.FlG

-=

l_

. 8rrn9n1k1

( n ¡2 +n 22 +k1 2- ) c o sh2 o 1

ãdr l(ns+n2) 2(nl+k72)

+ 2n1n2s!nh2o1 + (n¡z-n2z+kf)cos2'¡1 - 2n2k1sin2'¡1

\ro

ðnz (n12+k12) (n6+n2) 3 [r, r

2- ( rrs-r,2 ) cosh2o 1

+ n1 (nsn2-nr2-kl ?-) sinh2ol - k1 (nsn2+n12+k12-) sin2yi

+ kr2(ns-n2)"nuzv1J

2no

{ (n1 z+t rz ) (n}+n22+t1 2 )

ânr (n6+n2)2(nr2avt2)2

- 2nr2n22 sintr2ol + )+n1n2k1zeosh2a1

+ { (n12+trr2) (r,r2-n zà*ktz)¿q - \n1n2k12 eos2)y¡

+ 2n2k1{rrr' - 2at(n12+kr2) }sir,zyrl

119

?n¡i (n1 2+r'

12 ) (n1 2+nr2 +k-rì) 2'r tâkr (ns+n2 )2 (,rr 2+1r.t2)z

.- [rr 2.nzkt]cosh2c1 + 2n1n2{ (rr12+h12)n-, y-n2k1isirrÌr2nr.

+ { (n12+tr J)z - n22'(n¡z-ki2)}sineyl + l+n2kr::t1.2-er:rs2'y1

r20,

DERIVATÏCIJ] CF

AllI\ÌjiIX B.

ltR l:IfÐ .1.''.R1 FnoM T}IE FOF.Ì',ljÏ,AE Cilirri ,3Ï ¿,I..',;,li'i'-î **- -fl'--

ff=

Rl=

m-

*¡u2o1.r-s6:.u:-:lot+2rcos2y1 + A s sin2y1

bde2dl+ace-2d1+2tcos2y1 + 2 u sin2y1

cde2ol+abe-2o1+2rcos2y1 - 2 s sin2y1

bd.e2o1+ace-241+2teos211 + 2 u sin2y1

L6nsn2(nr2+k12 )

I

!6nsn2(n12+k12 )

b(dra)"2o1 * c(atd.)e-2ol + 2(ttr)cos2y1 + 2(urs)sin2yi

= 2(nt2+no2+kl2) or \ngn1

= 2(no2-nt2-kr2) (nr2+:K¡2-n22) or Bn6n2k12

= 4nzkr(n12+k12-ns2) or hngtl (n¡2+u¡z-n22)

f1-RlITJ

bde2ol+ace-2o1+2tcos2y1 + 2 u sin2y1

where

! = (rr*.ro)2+ tr2 (n1tn2)2 + k3

r = (ns2+nz2)(nf*xrz) - (n12+t<12)2 - no2nz2 çl+nsn2k¡2

I = rot (nz+no ) (n12+tr12tnsn2)

ot = 2nkrtlr/À yt = 2rnrclr/^

bc

Ttrus

1tR-rwhere

!'=

dta

t+r

uts

Íhus for

F

F = 2(n12+n¡2+k12) {(r1+rr2)2 + kr2ie2dI

T2I,

IIence

1+RT

+ 2(ni z+ns2+y.¡2){(n1*n2) t- -," kt2-}e*2o1

+ l+(irg2-n rl^;,;rz)(ni2.r-x,2 -nzz)cos2^¡1 rr fn/r:¡(n12+lc12'"nç2):;i.n2y1

= )+ ( n r 2 +ne2-.:tç Lz) { ( " 1, ^ r'

*k r') cosh2c 1 + 2n Ln zs inir;ì,-r 1 }

+ 4 ( n s 2-n L2-k f ) {(n ¡2 -tr rz+t 1

?' ) cos2y 1 - 2n2k 1 e :'-n'Íli 1 }

(n12+n92+kr2 ) { ("r 2+n22+k12 )cosh2a1

+ 2n1n2sinh2o1]

1

+

\ngn2(n12+k12)

(ns2-n12-rr2) { (nrz- nz2+kt2)cos2y1 - 2nz^k1uinzv1}j

IT

1-R

1-RT

\nen1{{(n1+n2 )2 + l f}"tot

- {(t1-n2)2 + k1lu-2or}J-6nsn2 (n

1 2+N¡2)

+ 16n gn2k ¡2 eos2y ¡ + Bn gk 1 ( n 1

2+k 1 2- n22 ) sín?\ t

2n2(n12+kL2) [n1 { (rr12on z2+l.t2) sinh2al + 2n¡n2eosh2o1 }

* kr { (n12+tr2-n22)sín2y1 + 2n2k1cos2yi }

I

Similarly

ËRl IG whereT t6nsn2(n12+k12)

G = d(btc)*2ot + a(ctb)u-2ol

+ 2(tt")cos2y1 + 2(u-.'s)sin2Y1

btc = 2(n¡2+n22+k12) or 4n1n2

Hence for t+ntT

c = z(rL]-+n22+l'y2){t(r,r+.r,0)2 + kt2'ie2d1 + {(r,1-ro)2 + kr2}e-2d1}

I22.

+ h ( nq 2-n t 2-k

t 2 ) ( n r

2 4-y r;| *'" r"l l c,rs2y 1 t C',.r sì,; 1 ( rr i

2+ii 1

2*,n2.2 ) sin2Y t

= l+(nt 2+n22 {* ¡2) { (n12+ng2+X12) cosh2ol + 2::-:i1sinh2a1}

-r l+(nr2+h 12-*n22J {(t92-n.¡ z-kJ )cos2y1 + Zztç";:i-,1.,:2y1}

Therefore

Ì+RI, =T

I

hn9n2 (n12+k12)(n¡2 +n22 +kr 2 ) i ( tt 1

2+ng 2+k 1

2 ) cosh2c 1

+ 2n9nlsinhzo1i

+ (n12+k12-n22) { (ns2-r,1z-kf)cos2y1 + 2nsklsinzyr}

Ll-Rl-1i- l+n1n2{{ (o1+rro ) 2 + kr2}e2or

- { (n1-ns ) 2+kr2}e-2dr }16n6n2 (n12+k12)

l-Rl-r

+ 16ngn2k12cos2y1 + Bn2k1 (n12+k12-n62)sin211

2no(nt2+kr2)

1

þr{{r, 2+ns2+k¡2)sinh2o1 + 2nen1cosh2o1i

+ kr{ (n12+kt2-ns2)sin2y1 + 2nek1cos2y1}

]-23,

A})J?fl1TDTX C '

PAETT II J-.]IË.IVATIVES oF (]rR)/r ron TI{E l'!üO*LAy}lr TIONS

SYSTEM r. (kr=ka=O)

1-R

1 F where

l+nd2 /À

2n"

(n6+n3 )z(n22+k22)(Asính2o2 + Bcosh2o, + Csín2y, + Dcos2y2)

A = nz2(n22-n32+k22) * kz2(n22+nn2+k22)

B = l+nz_ntkz2

C = 2n3k2{n2n3-2clz(n rz+X2z)l

t = 2n0 |.rrr{ (nrt*r, 3z+kzz)sinh2a2 + 2n2n3cosh2a2}

(ns+n312(lr.r2+k22) \

+ kz{(nr2-n32+k 12)sinllz + 2n3k2eo"2Y2}J

T

1+R

F

\n1 2 (nr2+t22 ) (ne+n3 ) 2

= (ns2+n12) [a(r,r2*r,

z2*kz2) { (n22+n3 z+ixz2) cosh2o2 + 2n2n3sinh2o2}

+ 2 ( n 1 2-n z2 -lo zz) { (n 22 -n sz +u 22 ) cas2y 2 - 2n 3k2s i"tr, }

J

+ (n¡2-n1,) [{trr,*rr

r)' * kr2}{ (n22-n32+k22)cos2(vr+vz)

- 2n3k2sin2(vr+1z) Ì

+ {(n1-n2)z + kzz!lkr22-n32+u22)cos2(1 r^rz) + 2n3k2'sina(vr-vz)}

+ 2 ( n r 2. -nzí- -k z2) coa 21 1 | 6 22 +n 12 +u 22)cosh2o 2 + 2nrn 3

s inh2o 2 )

+ hn 1k2s in2y r { ( n22+n, 2+k2 2 ) sinh2o, + 2n2n gcosirao2 } J

l:2I+,

D = 2d2(n z2,rkz?) (n22-n32'+kz') - \n2n3u22

fr-nlal. r ,] ! c.É--+r-. (asintr2a, + Bcosh2a, + csín2\2 + DtÒs?yz)l2n6

oJ: .t

vhere

(nq+n3 )z(nrz+t<rz)2

I\ = pn2n3{2.¡2k22+Ur2) - nekzi

B = 2\z(n"2+x2z) (n22+n32+kzz) - I+n22n3}*2

c = nz? (nr2-n32+:l.22) + kz2 (n22+n32+k22)

D = \nzZn3kz

(asintrZo, + Bcosh2o, + Csín2Y2 + DcosZyz )

Ðdz

r¡here

,A = 2n2n3

B = nz2+n r'*]KzZ

c = -2n3k2

D = n22-rLZ2*kZ2

a

âdr

I

6 \n¡2(nr2+t<22) (n6+n3) 2iAsintrzo2 + Bcosh2a2

+ Caín?y2 + Dcos2y2 + Esin2(y1+y2) + Fcos2(y1+y2)

+ Gsin2("¡ryù + Ilcos2(yryz) +;sineyl + Kcos2yl]

Bnn¡n2k2

tr(ne+n3 )z(nrz+x2z)

fr-nl[-rJ 0

!25.

ìiìì.efe

}¡here

A = hn3(n62+n1r)I:Kr'(n12-+kr-2) - nzz(ni2-t 22)]

B = )+nz(n62-+n12)l(n22ntt.z2)z - nr?n32Ì

C * Z( no 2 +n 1 2 ) { l+n

1 2-n

2n sk2- g ( n2 2+k 22 ) (n ¡2 -n r'-v' r' ) ( n 22'n 32 +k r'' ) I

2 ) 2 | -a su rE ( n 22 +u rz ) ( n rz -n r' -E z'- )

- g(n 22 +u 12 ) (n 22 -x't 32 +k z2 ) { ( n I +n, )

*ortlJ

F = 2(no2-or2) [(rrr*rrz

)1rr.22+t<r2)z * n1n2n32] -nyn32k22

G=(,,02-,,12)[(,,22+x22)(nz2-nsz+_;ïil;lÏ":'^,ï!r',,i'.-"t'

H = p(r,02-rrr2l Itrrr-nr){nin2n32

-(n22+ll'r2)2} * n¡ns2r<22

r=-Bnrn3k2(ns2-n12)in2n3r,o"nrlr"î'r:'r::-::;;"]l*.r,

K = -b (nol-ntz, þr{

(nr2+k"2)2 + nfna2}cosh2o2

+ n3{n1 2(nrz-krz) * kzz(nrz+urz)}sinrrzc2]

{asinh2o2 + Bcosh2cr2 + Csín2y2- \n12(ne+n )2(n22+]x22)z

+ Dcos2y2 + Esin2(yr+y2) + Fcos2(y1+^¡2) + Gsín2(yr-Vz)

+ Iicos2(y ryù + Jsin2Yl + KcoszYl}

A. = Z(ne2+n12) {e(n22+t 22) (ny2+nr2+k22) (n22+n32+kr') - \n12n2n3t2}

B = l+ (n62+n1 z) {t2t (n22+kr2)2-n}nz2} + n2n3g (nr2+k22) (n}+nz2+1r-zz) I

c = Lng(ns2+ni 2)lntrz - (nzz-kz2) (nf-nz2-rtzz)I

Lr = l+kz(ne2+n1') {"t znj2 (n22+k22)zl

l2

1

E = 2n3(no2-nr2){(n12+zn.¡n2) (l.,2-nz2) - (nr2+t< r')'I

F = Zr'z(n¡2--:r121[{tnr*rr, )2+kz2] + (nr2'+lrr 2)(.n2.2' - '::''::rz))

G = 2nî (noz-*nrr){(n22+xr2)2- + (n12-2n1n2)fu22'ur2)l

TI = p;6z(r,02-.rr2) { (n2z+urz)2 - nrn32(Zn2+n1) i

.l = \(no2-rr2) [rrnru(rr,

z+'xz2) { (n2z+nr 2+kzz)cosh2o2 + l+nrn3sinlnàa2l

+ 2n 1 n2n 3

(n zz -k zz) cosir2o,2 + n r { (n 22 +n 32 +k rz ) (n rz +k 22 )

- 2nr2kr2}sinrracr.2J(

K = p(no2-nr 2) [ø("rz*tz2

) (n, 2-nr2-kz2) { (n22+nrz+kzz ) cosh2a2

+' 2n2n3si ntr2ø2]¡ - l*tt 2n2n3k2-sinh2a2

-. zyzl(n2z +r,r2) 2 + nyznaz l cosrrzo2l

].:26

{Asinfrao2 + Bcosh2d2 + Csin2y2 + Dcos?^¡2'Í

where

Àn12 (nr2+k 22) (ns+n3)2

+ Esin2 (yt\z) + Fcos2(yr+yz) + csin2(yr-yz) + IIeos2(yr-vz)

+ Jsin2yl + Kcos2Yl

Ix = 2kz(ns2+n12 ) (n¡z+nrz+v2?) (nr2+n32+k22)

B = \nzntk2 (n62+n12) (n¡z+n2z+kzz)

C = -Zn 2 ( n t2 -n zz -k 22 ) (n 92 +n ¡2 ) ( n 22 -n 12 +tr 22 )

D = -hn2n3k2 (n62+n12 ) (n12-n z2-xz2)

E = -r.z(no2-rr2){(n1 +n2)2 * kzz!(nr2-ns2+u22)

F = -pn2n3k2(n62-n12){ (n1+n)z + xrzl

G = r1z(no2-nt2) (nzz-nz2+:x22) { (n1-n2 )z + rr22l

H = -2nzn3k2(n6 2-nL2)

{ (n1-n2 )2 + }'zzl

J = l+ntkZ 2 { (n22 +nr2 *lr,r2) cosh2o2 + 2n2n3sin}r1ct2l

K = Z1'z (n ¡2-n22 -k z2 ) { (n22 +n 32 +k 22) s inh2o2 * 2n2n 3 e oshzo2 }

T2T,

'[oPJ =r (n0 2-n

12 )

i^tslne (y1¡tz) + scos2(y1+y2)--.ì-d; Àrr (n,2'rk22) (n¡+n3) 2

r Cein2(vr-yz) + DcoÊz(vr-vz) + Esín211 + Fcos2yl]

uhere

A = -t(n1+n2)2 *:l.z2!(n22-n32+k'22)

B = -2n3k2{(n1+n2)2 *:Kzz!

c = -(nz2-n32+:x22) {(n1-t2)z * lxzzl

D = 2n3kz{(n1-n2)2 *kzz!

E = -Z (nJ -o z2-}'22 ) { (n22 +n rz +k 22 )cosh2o2 + 2n2n 3 s i n}laa2]l

f = l+nikz {(n22+nrz+k22)sinh1ø2 + 2n2n3codn2a2l

128.

APPENDTX DO

TMASIJRED ]JIì¡LECTANCES AIüD TA.AAISMITTÆTCES Ä}ID CA-LCULA.TII'D REFBACTTVE

AND AltbuilPTION TNDICES OF VITREOUS SELENITJM FTL¡iIS.

r29.

o\o t roa

FILM SEl.

ð.2 L.i '

*

r(u)

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t_oA

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2.000f'750L.5001.2501.000

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r

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WAVELEIIGTII IN MICROÌ{S

1,4 1.2 t'0

WAVETENGTTI IN MICRONS

't '6 -1

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131.

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FÎLM SE3.

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2.0001.'r501,5001.2501.000

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2"5002.1+OO

2.3002,2502,2002.1002.0001.9001.800f.7501.7O01.6001.5001. )+00

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WÀVEtENGTI{ IN MICRONS

f0

.e

.t

.1

'¡

x!tôv

zoF&ao.çô

wAvELENGTH 1N MlcRoNsf6

.t

133.

d1

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.35f

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.308

r À(u) R

LJ\jtit

FTLM ÍJflTJ

ð,2o

= l-012 t 5Á.

r À(u) R rr(u) R

1.500r,25Ol_.000

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. ?30,720.710,700,69o.6Bo,6To

"66o,6jo.6ho6so6¿o6ro6oo

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,5rO.51+o.530,r2O.510.500. \90.lrBo.l+70.\60. \50. \40. l+30

. l+20

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f

I

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WAVELÊNGTH IT¡ MICRONS

l.û

10

.g

õ.¡ô

=

.?

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2oËc&oØ6

WAVELENGTH III MICRONS

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tli

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6\s6+Z6l+r6l+r6\z

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.900

.BT,

. B5o

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.820

.810

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" 730,T20.710.70C1

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t.gg3.9tl+.91+\,969,978,987,9Bg,990,990,ggr.g82,9TI,957,936.911.886. 86r.832.803,777,7\2.710.680

.6jo

.6ho

.630,6ao.610.600,19o.5BO,570,160,5ro.5\o,530,52O.510.500.!90. l+Bo

.4To

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. \50

.l+Lo

R

,352.381.Lo\,)+23.l+37.l+U+. h\o,l+23,389.334.258,196,16T,169.185,202,LLT.zzï,237,zl+3.z\9,252

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ñ120 r 20A

o16oo t l+i;i

r(u)

I\L2

r(u) R r r(u) R r

2.000l,9ro1.900t"B5o1.Boo1.7501.700t,6501.6001.5501.5001. \50l".l+OC)

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.351

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- sE8

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tt

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I

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t.8WAVELEN6IH IN MICRONS

10

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l.oôz

sE0zoËÀqroøo-

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.t

1,01.2,6.t

WAVELENGT}I IN MICRO'{St.ô

135.

cl1o

Bo t eon

.l'-,T,:Y*Æl'

d2

.218

.255,292,327,355.3TT.383.38?.390.3Bg.385.378.371.360.31+T,326.302,273.21+1.202.I5T

o25OO !

'O,4.

,690.6Bo.6To,660.6jo.6ho,630.620.610.600.590.580,5TO,j6o,550.5hO,530,r2O.510.500.490

r(u) R

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r À(u) R r À(u) R

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WAVELE N GT H IN MICRONS

1.4 1'2 1.0

wavEtf.NGT[l lN n'! lcRo¡ls

f0

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1.0 '8 .0 ,4

136.

SEB

r(u) R

rlt

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1.1501.1001.0501", ooo,97',9rO,92'.900. B9o. BBO

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d.2

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.700

l-.300r,2r01.200

r r(u) R

o120 r 2oA

o27OO t 50Ä

r(u) R t

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.820

.B1o

. Boo,790.780,770,T60,Tro.71+o,730,720

5

1

I

xôz_

IÈoc

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zo;ÀÉo@

t0

-9

.t

sE8

t.6 t.{ 1-2

WAVETENGTI.I IN MICRONS

t'a 1.2 t'0

WÀVELENGTH IN I/ICRONS

x-o'z

.1

.t

.5

IIfI

I!

I

IsE8

tltrrrII\i{rI

Ìrt

ttlr

i

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t.E t.6 '6 .5 .¡

137.

APPENDTX E.

MEASURI|D

AND .ASSORPTION TNDTCES OF GERMANIUM FTI,I\'F.

TTANCES AND CATCUI,ATED

138.

FTLM GEl.

di

R

.lh?

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2.o2 .6762. OT ,6T82.OO .679L,gg .6Bt1.gB ,68zr,gT ,682r,96 .68l+l.gj .68l+t.gh ,685t.g3 .686r,g2 .6861.91 .6881.go .688r. 89 ,689t.BB ,6901, BT .69rr,86 .692r. B5 .692r.Bl+ ,693r.. 83 ,693r,B2 ,693I.{Jr ,69\l. Bo ,69,1,79 ,6961.78 ,696T.TT ,697t.76 ,697L,Ti ,697f .'f l+ ,697r,73 .698L.Tz ,698i.?f ,698r.To ,698r.69 .6981.68 ,698I,6'T ,697t,(,6 ,69'(r,6'2 .697r.6\ .696t,63 "696L,6n .695L,6r ,695r,6ct ,691+L.59 .6931.58 ,692.r,j'( .69r

r r(u) R r r(u) R I

,335.33\.333.332.331.330.329.328,32'r,326.32',321+,323.322

ADD

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r,56r.5,1' 5)+

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Ie J11.311.30L,29T,2.8T,2Tr,2.6r.25i,zI+1."23r,22I,2T1.201.19r.181.17r.r6T.I51.1\1.f3I"L21.11

,69t,689.688,686.68,.693.68t.68o.678.67i,673.67r.668,66,.662,6i9.6j,,65l-,6\T.6l+3,639.631+,629,6zj,620.6tl+.6o8,602,595.5BB.580,572,56¿,55\,,\,'531+,523,5L2.\99. \BB. hTI+

. )+6t

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.00,99.98,97,96.95.9\,93.92.91,go.B9.BB.BT.86.85.B\.93"Bz.BI.Bo,79.?B.fl.'f 6,75

"Th,73,72.TL,70,69.68,6'(,66

.386

.368

"352.331+,3IT.301.285.269,253.2hl-,2?-7.2.r5.20\,]-9,. t-BB. IB2.TT9,T7T.1TB.r8o. fB5.r92.20I,2L2,225" 2\o

.255,273,292.310.329.316.36\.380,395.\oB.l+I9. \28. l+3tr

.l+38

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.3f0 I

.31r I

.311 1

.3r2 l_

,3l-,2 1.312 1.313 I.311+ l-.315 I.316 I.317 1.318.319.32r,322.,3ù+ß26.328.330.332,335.338.3\]-.3\3.3\6.3\9¡ 5)1ß56.360.363ß67.371,375,3Tg.38h.3BB,393.3gB.l+03. l+o8

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1'4WAVELENGTH IN

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150.

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