January 1968

JOURNAL OF THE OPTICAL SOCIETY OF AMEIRICA VOLUME 58, NUMBER 1 JANUARY 1968

Optical Properties of Tin Telluride in the Visible and Infrared Regions

R. B. SCHOOLAR AND J. R. DIXONU. S. Naval Ordnance Laboratory, While Oak, Silver Spring, Maryland 20910

(Received 11 May 1967)

The optical constants of p-type tin telluride at room temperature have been determined as functionsof carrier concentration over the spectral range from 0.1 to 3.8 eV. The indices of refraction and absorptioncoefficients were obtained from analysis of normal reflectance and transmittance measurements on epitaxialfilms ranging in carrier concentration from 3.6X 101" to 6.8X 1020 cm-3 . At energies greater than approxi-mately 1.0 eV the optical dispersion is found to arise primarily from bound carriers. In contrast, the disper-sion at energies less than about 0.4 eV is in excellent agreement with that calculated on the basis of classicalfree-carrier dispersion. The fundamental absorption edge is characterized by a large, Burstein-type shiftproduced by changes of carrier concentration. Bound-carrier indices of refraction nBc are found to beunusually large and carrier-concentration dependent. A Kramers-Kronig analysis gives values of nBC ingood agreement with experiment for energies less than 2 eV. Extrapolation of our experimental values ofnBC to zero energy using the Kramers-Kronig relation yielded values of the optical dielectric constante.. This quantity is found to be carrier-concentration dependent, ranging in value from 38 to 46 for thecarrier concentrations studied here. It is shown that this variation in e, is primarily due to the Bursteinshift of the fundamental absorption edge.INDEX HEADINGS: Refractive index; Absorption; Tin telluride; Infrared.

TIN telluride is a IV-VI compound which crystal-Tlizes in the NaCl structure. Single-phase SnTeexists over a wide range of compositions and alwaysexhibits p-type conduction with carrier concentrationsranging from 4X 1019 to 2X 1021 cm73 .1-5 It has beenshown that this behavior results from Sn vacancies

' R. F. Brebrick, J. Phys. Chem. Solids 24, 27 (1963).2 J. Umeda, M. Jeong, and T. Okada, J. Appl. Phys. (Japan)

IL, 277 (1962).

which give rise to doubly ionized acceptor levels.6

Studies of the thermal and electrical properties of SnTe3 R. Mazelsky and M. S. Lubell, Nonsloiciionmetric Compounds

(American Chemical Society, Washington, D. C., 1963), Series39, p. 210.

4 J. A. Kafalas, R. F. Brebrick, and A. J. Strauss, Appl. Phys.Letters 4, 93 (1964).

5 R. F. Brebrick and A. J. Strauss, J. Phys. Chem. Solids 41,197 (1964).

" B. B. Houston, R. S. Allgaier, J. Babiskin, and P. G. Sieben-mann, Bull. Am. Phys. Soc. 9, 60 (1964).

R. B. SCHOOLAR AND J. R. DIXON

indicate that it is a semiconductor with a complexband structure. 7 '-10

Optical studies of such a material would normally beexpected to provide a useful and relatively easy methodof obtaining information about the band structure. Suchstudies on SnTe, however, are complicated by the strongabsorption existing throughout the visible and infraredspectral regions. This absorption is due in part to thehigh concentration of free carriers characteristic of bulkmaterial now available. As a consequence, extremelythin samples are required for optical transmittance mea-surements. We have demonstrated in previous workthat thin single-crystal films, grown epitaxially onheated rocksalt substrates, are of great value for opticalmeasurements. For example, by measuring the re-flectance and transmittance of a 1-,4-thick film of SnTe,the energy dependence of the optical absorption co-efficient in the fundamental absorption-edge region hasbeen determined." A particularly appealing character-istic of such films is that their carrier concentrations canbe easily controlled over a wide range by simple heattreatment procedures.' This arises because the anneal-ing time for such thin samples is many orders of magni-tude less than that required for a bulk sample. Thischaracteristic has been exploited in an experimentalstudy of the carrier-concentration dependence of thefundamental absorption edge.'3 The study was prelimi-nary in nature, involving only two carrier concentra-tions, but it clearly established the existence of a largeBurstein-type shift of the edge. Similar but somewhatmore complete work on this subject has recently beendescribed by Finkenrath and Kohler."4

The work reported here represents an extension ofthe research described above. The optical properties often films ranging in carrier concentration from 3.6X 1019to 6.8X 1020 cm-3 have been studied in detail over the

spectral regions from 0.1 to 3.8 eV. Reflectance andtransmittance spectra of these films have been analyzedto determine the absorption coefficients and refractiveindices of SnTe as functions of carrier concentration.

7-A. Sager and R. C. Miller, in Proceedings of the InternationalConference on Semiconductor Physics, Exeter, 1962 (The Instituteof Physics and the Physical Society, London, 1962), p. 653.

8 R. S. Allgaier and B. B. Houston, in Proceedings of the Inter-national Conference on Semiconductor Physics, Exeter, 1962 (TheInstitute of Physics and the Physical Society, London, 1962),p. 172.

9 R. F. Brebrick and A. J. Strauss, Phys. Rev. 131, 104 (1963)."J. Richard Burke, Jr., R. S. Allgaier, B. B. Houston, Jr.,

J. Babiskin, and P. G. Siebenmann, Phys. Rev. Letters 14,360 (1965).

"t E. G. Bylander, J. R. Dixon, H. R. Riedl, and R. B. Schoolar,Phys. Rev. 138, A864 (1965).

12 H. R. Riedl, R. B. Schoolar, and Bland Houston, Solid StateCommun. 4, 399 (1966).

'3 R. B. Schoolar, H. R. Riedl, and J. R. Dixon, Solid StateCommun. 4, 423 (1966).

11 H. Finkenrath and H. Kohler, Phys. Letters 23, 437 (1966);24A, 261 (1967).

We analyze our results in terms of free-carrier andbound-carrier dispersion mechanismis.

I. EXPERIMENTAL

A. Samples

The techniques that we have employed for preparingand heat treating epitaxial films of SnTe have beendescribed previously.' 2 Therefore, only a brief review ofthe procedures will be presented here. A freshly cleavedrocksalt substrate is placed in contact with a heaterblock which is in a bell-jar evaporator. The bell jar isis evacuated to a pressure of IX 1I-' torr and the sub-strate is heated to 300TC. Pulverized crystalline SnTeis evaporated from a quartz furnace onto the heatedsubstrate. The deposition rate is adjusted to 130 A/minand is monitored by a Sloan film-thickness monitor.After a film of desired thickness has been deposited, thefurnace is turned off, and the substrate is cooled toroom temperature. Electrical, x-ray, and optical dataindicate that the films are high quality, single crys-tals.'2",5 The as-grown films generally have a carrierconcentration of 1.4X 1020 cm-3 . This carrier concentra-tion was changed by heat treatment using the tech-niques described in Ref. 12. It has been established thatthe heat-treated films retain bulk transport properties.Analysis of back-reflection Laue photographs reveal noapparent degradation of crystal quality; in appearance,the films remain flat, mirror like, and free of pinholes.

B. Optical Apparatus and Measurement Techniques

The reflectance R and transmittance T spectra ofthe SnTe films were measured using three separatespectrometers.

A Perkin-Elmer model 21 double-beam spectro-photometer, equipped with rocksalt optics, was used tomake preliminary measurements of R and T over thespectral range from 2 to 15 g. This instrument wasequipped with a Perkin-Elmer reflectance attachmentwhich could be easily removed for transmittancemeasurements.

A Perkin-Elmer model 112, single-beam spectrom-eter with additional exit optics was employed for more-accurate measurements of reflectance and transmittanceof several films in the spectral range from 1 to 15 ,. Themodifications to this apparatus have been describedpreviously.'0 This spectrometer was equipped witheither a LiF or NaCl prism, globar source, and thermo-couple detector. An aluminized mirror was used as areflectivity standard in this system.

Another Perkin-Elmer model 112 was used for mea-surements in the 1.0 to 4.0 eV region. It was equippedwith a quartz prism, tungsten source, and phototubedetectors. This system was used to measure R and T of

n J. N. Zemel, J. D. Jensen, and R. B. Schoolar, Phys. Rev.140, A330 (1965).

lo R. B. Schoolar and J. R. Dixon, Phys. Rev. 137, A667 (1965).

Vol. 58

January 1968 OPTICAL PROPERTIES OF

70

z

60

z 40

Z 30

1 2 3 4 5 6 7 8 9 10 11

WAVELENGTH (MICRONS)

FIG. 1. The reflectance and transmittance spectra of an as-grown SnTe epitaxial film (solid curves) and the reflectivityspectra obtained from a thick, chemically polished bulk sample(open circles). The plasma-reflectance minimum occurs at

XR = 6.1 u indicating a carrier concentration of 1.4X 1020 cm-3 . Thebulk sample was chosen so as to have the same carrier concentra-tion as the film.

two thin, homogeneous samples having thicknesses of600 and 2000 A. Such thin samples were required be-cause of the strong optical absorption in this spectralrange. Additional exit optics were designed to measureT and the absolute value of R2. These modificationshave also been described previously." 7

All of the samples studied with the single-beamspectrometers were checked for pinholes and surfaceirregularities. A "sample in-sample out" procedurewas used for taking the data. The error of R and T isestimated to be less than z±3%.

II. RESULTS AND DISCUSSION

A. Reflectance and Transmittance Spectra

Typical reflectance and transmittance spectra of anas-grown film are shown in Fig. 1. The film thicknesswas determined from analysis of the interferencefringes. Such spectra were also used to determine thereal carrier concentration P from the position of thereflectance minimum, XR. For all of our films, the mini-mum values of reflectance and reflectivity occur at thesame wavelength; and the relationship between thewavelength of minimum reflectivity and carrier con-centration has been established.18 We estimate that thecarrier concentrations determined in this way are ac-curate to within ±10%. The reflectance minimum inFig. 1 occurs at approximately 6.1 A, indicating that thefilm has a carrier concentration of 1.4X 1020 cm-3 . Thehomogeneity of the films was checked by measuring thereflectance of different areas of the films on both thefront and back surfaces. Variations were less thanz±3% for all of the samples studied.

17 Paul R. Wessel, Phys. Rev. 153, 836 (1967).Is H. R. Riedl, J. R. Dixon, and R. B. Schoolar, Solid State

Commun. 2,323 (1965); H. R. Riedl and R. B. Schoolar, Bull. Am.Phys. Soc. 11, 348 (1966); Phys. Rev. 162, 692 (1967).

100

90

d 80z1 70

~;60- 5000

z< 40Uz 30

i 20

10

00 1 2 3 4 5 6 7 8 9 10 11

WAVELENGTH (MICRONS)

12 13 14 15

FIG. 2. The reflectance and transmittance spectra of two SnTeepitaxial films of different carrier concentrations. The wavelengthsof the plasma reflectance minima XR are 2.9 and 10.1 a, correspond-ing to carrier concentrations of 6.8X 1020 and 3.6X 1019 cm-3 , re-spectively. The corresponding film thicknesses were 0.77 and 1.9,u.

Typical R and T data for two samples havingcarrier concentrations different from the as-grown filmsare shown in Fig. 2. The striking difference betweenthese spectra is due primarily to the carrier-concentra-tion dependence of the optical constants in this spectralregion. Analysis of such data to obtain the index of re-fraction Uf and extinction coefficient kf of the film iscomplicated by the constructive and destructive inter-ference fringes and by the influence of the film backing.Relations which take these factors into account havebeen given by Hall and Ferguson1 9 and by Heavens.2 0

These lengthy and rather complicated equations willnot be restated here. We will simply indicate that thefunctional forms are

T= f (nfkfngjXl)

_R= g(nf ,kf ,na,X,t).and

(1)

(2)

The wavelength X is a known variable, and the index ofrefraction of the NaCl substrate n, is given in theliterature. The film thickness t is determined as de-scribed in the next section. Thus, Eqs. (1) and (2) canbe used to determine ff and kf from experimental valuesof R and T. Unfortunately, these equations cannot besolved explicitly for ff and kf. For this reason, solutionswere obtained by a computer iterative-search procedure.

In general, there can be more than one set of solu-tions (nf,kf) of the film equations for a given R and T.This point has been discussed in detail by Grant andPaul21 and Wessel."7 Because of this, additional infor-mation is required about the optical properties in orderto choose among these sets. Estimates of ff and kfbased upon classical free-carrier dispersion theory were

19 J. F. Hall, Jr., and W. F. C. Ferguson, J. Opt. Soc. Am. 45,714 (1955).

20 0. S. Heavens, Optical Properties of Thin Solid Films (Aca-demic Press Inc., New York, 1955), Ch. 4, p. 58.

2"Paul M. Grant and William Paul, J. Appl. Phys. 37, 3110(1966); Paul M. Grant, Bull. Am. Phys. Soc. 10, 546 (1965).

SnTe IN VISIBLE AND ir

AND J. R. DIXON

sufficient to eliminate the extraneous roots in the spec-tral range from 0.1 to 1.0 eV. At higher energies theroots were chosen so as to be consistent with the resultsof a Kramers-Kronig analysis.

B. Determination of Film Thickness

Conventional methods of determining film thicknesst by following visible fringes across the step formed atthe film-substrate boundary were complicated by therelatively rough cleavage plane of the substrate. Thistype of measurement was accurate only to within-15% for films of the thicknesses studied here. How-ever, this measurement of I was accurate enough toenable us to make estimates of kf from the averagereflectance and transmittance spectra of the films.22

The results of these estimates indicate that in regionswhere we observed interference fringes

kf<< (if- 1).

This condition made it possible to use a more accuratemethod of determining film thickness, based uponanalysis of interference fringes. The equations used inthis analysis have been derived previously and are pre-sented in a convenient form by Hall and Ferguson.19

They show that the positions of interference-fringe ex-trema are given by

inXn= 4n1 f (\)t, 1f= 1, 2, 3, * * - (3)

where in is even for a transmittance maximum and oddfor a minimum, X," is the wavelength of an extremum,and nf (Xm) is the refractive index of the film evaluatedat Xm. We have determined t by using this relation alongwith experimental values for in and nf, obtained asdescribed below.

The index ntf was determined over the limited regionwhere interference fringes were observed from mea-surements of the reflectance. The index is related to thereflectivity Raf at the air-film interface by

Rwq = (1f- 1)1/ (,If+ 1)2 (4)or

Itgf= (I +Rnf 1)/ (I - R~f (5)

It follows from Hall and Ferguson's Eq. (2), that Rgf isgiven by the average observed reflectance of a filmthrough a region of constructive and destructive inter-ference provided that the amplitudes of the fringes aresmall relative to the total reflectance. This is confirmedby the data points in Fig. 1 which represent values ofreflectivity R. f as measured on a thick, chemicallypolished, bulk sample of SnTe. The bulk sample waschosen so as to have the same carrier concentration asthe film whose spectrum is shown in the figure. It canbe seen that these experimental values of Raf are goodrepresentations of the average of the fringes. Our results

22 T. S. Moss, Optical Properties of Semni-Conductors (AcademicPress Inc., New York, 1959), Ch. 1, p. 14.

z0U

x

z

4.

2,

UI

,0

.0-

6.02.0 3.0 4.0 5.0WAVELENGTH (MICRONS)

FiG. 3. The index of refraction uf of SnTe having a carrier con-centration of 1.4X 1020 cm 3 . The solid curve represents nf deter-mined from bulk reflectivity or average film-reflectance data. Thesymbols indicate values of nif calculated from the positions ofinterference fringes as described in the text. They apply to threefilms of thicknesses 1.3 ju(E), 1.4 A(0), and 1.9 .t(A). The dashedcurves were calculated from interference-fringe data of ourthickest sample, assuming that the correct order of interference litwas mistaken by plus or minus one fringe (ln+2 or Ca-2).

for ,f (X) determined in this manner for SnTe having acarrier concentration of 1.4X 1020 cm-3 are presented inFig. 3 as the solid line.

The order of interference was obtained by takingthe ratio of at least two consecutive even-order fringesgiven by

/ (,il+ 2) = \X+211f (\n)/X Ilj (X,,+2). (6)

In all cases low-order fringes were observed and,consequently, m was determined without difficulty.

The accuracy of t determined from Eq. (3) and thesevalues of Of and m is estimated to be i44%. This pro-cedure for evaluating t was repeated on a number ofheat-treated films. No change of the thickness was ob-served to result from the heat treatment.

The symbols in Fig. 3 represent values of nf calcu-lated from the higher-order fringes using Eq. (3) andour values of t and m for three films having differentthicknesses but the same carrier concentration of1.4X 1020 cm-3 . The excellent agreement between thedata points and the solid curve confirms our assertionthat the correct orders of interference were determinedfor the three films. If an error had been made in ourchoice of in, a large discrepancy would have resulted.This is illustrated in Fig. 3 by the dashed curves whichwere calculated for the thickest film, assuming that thecorrect order of interference was mistaken by onefringe. For thinner films, the corresponding discrepan-cies are even larger.

C. Index of Refraction

The carrier-concentraition dependence of lf *in thenear-infrared spectral region was determined from ananalysis of the interference-fringe patterns. This fringemethod was preferred in this spectral region over themore general technique based upon R and T measurf,-

Vol. 58122 R. B. SCHOOLAR

6.

January1968 OPTICAL PROPERTIES OF SnTe IN VISIBLE AND ir

ments, because it yielded more accurate results. Ourresults for four different carrier concentrations are pre-sented in Fig. 4. These concentrations were 3.6X10's,1.4X1020, 2.9X1020, and 6.8X1020 cm-3 applying in se-quence from the highest to lowest curves. Six films wereinvolved in this study, as indicated in the figure; theircarrier concentrations were varied from the as-grownvalue by heat treatment. Many of the points were cal-culated by observing the shifts of fringes of a given filmas the carrier concentration was altered by relativelysmall increments. Examples of such points are indicatedin the figure by the arrows. The experimental error ofthe absolute values is estimated to be ±3%. The rela-tive error is considerably smaller, being only ± 1%.

The large change of nf with carrier concentration isattributed to the variations of free-carrier dispersionand to the carrier-concentration dependence of theoptical dielectric constant oO. This assertion is basedupon the excellent agreement between our values of nfand the solid curves A, B, C, and D at wavelengthsgreater than -3 ,4. These curves were calculated on thebasis of a classical free-carrier dispersion model. Theparameters which we have used in our calculations aregiven in Table I. These values were obtained fromstudies of the optical properties of single-crystal bulk

TABLE I. Parameters used to calculate the classical free-carriercurves of Figs. 4 and 5.

Calcu-lated ta bcurves P (cm-,) III)nIma (cm2YVe-) a

.. 111 . ___ - X

A 3.6X 1019 0.074 700 49B 1.4X 1020 0.120 195 45C 2.9X 1020 0.130 134 43D 6.8X1020 0.145 84 40

a These values taken from Ref. 18.b 'flhe optical mobility oopt is related to the damping coefficient 'YFC

by .apt =e/YFcm..

material"5 having carrier concentrations correspondingto those of our films. The excellent agreement is con-sidered as impressive evidence for the bulk-like natureof our films.

The calculations involved the determination of thecomplex dielectric constant e, which is given, in gen-eral, by

e= eBc+ eFc, (7)where iBc and iFC are the bound-carrier and free-carrier components. In spectral regions where there isno bound-carrier absorption, iBc becomes independentof wavelength and is referred to as the optical dielectricconstant eo. In such a region, the classical free-carrierdispersion is described by

47rPei 1 ( YFC)

e 17, = .OO-- (8)m b W'+Y'FC'

In this expression, mn is the electric-susceptibility free-carrier mass, -YFC is the free-carrier damping coefficient,

z0U

0z

9

8

7

6

5

432

0 1 2 3 4 5 6 7 8 9

WAVELENGTH (MICRONS)

FIG. 4. The index of refraction nf applying to SnTe havingcarrier concentrations of 3.6X 109, 1.4X1020, 2.9X1020, and6.8X1020cm-3. These carrier concentrations apply in sequencefrom the highest to the lowest curves. The symbols representvalues of nf determined from the interference-fringe patterns of sixfilms as explained in the text. The thicknesses were 0.77,u (tilted A),1.2 (V), 1.3,u(o), 1.

4M(0), 1.4u(i), and 1.9 u(A). The solidcurves A, B, C, and D were calculated on the basis of the classicalfree-carrier model of optical dispersion and the bulk parameterslisted in Table I. Curves A', B', and D' represent bound-carrierdispersion. They were determined from a Kramers-Kronig analysisas explained in Sec. IIE.

and co is the angular frequency of the exciting radiation.The calculated curves for the index of refraction plottedin Fig. 4 were determined from Eq. (8) and the relation

f2= [(e+ (612+622)11/2, (9)

where E1 and 62 are the real and imaginary com-ponents of e.

In the short-wavelength region, the bound-carrierterm in Eq. (7) is expected to become dominant as wellas energy dependent. This is illustrated in Fig. 4 by thesolid curves A', B', D' which represent the wavelengthdependence of ff due to bound carriers. These curveswere calculated for three different carrier concentra-tions, using a Kramers-Kronig-type dispersion relationand our absorption data. Details of the calculations aredescribed in Sec. IIE. Figure 4 shows that in thespectral region shown, free carriers dominate the dis-persion mechanisms of SnTe at wavelengths greaterthan -3 g. At shorter wavelengths our values for lf ap-proach those associated with bound carriers.

D. Absorption Coefficient

The absorption coefficients a of SnTe were determinedfrom our experimental values of k, obtained from analy-sis of R and T data as explained in Sec. IIA. The rela-tion between a and k is

a = 47rk/X. (10)

In the spectral region where interference fringes werestrong, the computed values of a generally showedslight oscillations with the periodicity of the fringes. Wehave not established the origin of these apparent oscilla-tions. They could be the result of a slight inhomogeneityof thickness of the film. We have corrected for theseoscillations by assuming that the true value of a is theaverage of the oscillating values. The maximum error

124

0o6

z-0Uz0or

10v° 1o

0

R. B. SCHOOLAR AND J. R. DIXON

1.0 2.0

PHOTON ENERGY (eV.)

FIIG. 5. The absorption coefficient applying to four carrier con-centrations of SnTe given in Table I. The solid curves A, B, C,and D were calculated on the basis of a classical free-carrier modeland the bulk parameters listed in Table 1. The solid curve from1.2 to 3.0 eV represents the results of Cardona and Greenaway(Ref. 23). The dashed curves represent the bound-carrier compo-nents of the total absorption determined as described in the text.

arising from this assumption is expected to bevery small.

Our results for a applying to four different carrierconcentrations of SnTe are presented in Fig. 5. Atphoton energies greater than 1.0 eV, the absorptionspectrum is characteristic of bound carriers and agreeswith the results of Cardona and Greenawav.yY The ab-sorption spectrum at lower photon energies becomescharacteristic of free carriers. This is illustrated by theexcellent agreement between our data and curvesA, B, C, and D calculated on the basis of the classicalfree-carrier model and the parameters in Table l. Thecalculations were based upon Eqs. (8) and (10) andthe relation

k2= [-e+ (e+- e-92)x)J/2. (1 1)As in the case of the index of refraction, the agreementusing the bulk parameters is excellent.

In the intermediate spectral region, between 0.2 and0.8 eV, strong mixing of the absorption mechanismsoccurs. The bound-carrier absorption anc, representedby the dashed curves in the figure was obtained bvapplying the relation

Ec= e- ire, (12)

which is a rearrangement of Eq. (7). The classical free-carrier values of iFC, calculated as described above,were used in the evaluation. The rapid rise of aBC ischaracteristic of a fundamental absorption edge. Theshift of this edge to lower photon energies with decreas-ing carrier concentration is attributed to the Bursteineffect; i.e., the onset of optical transitions between theconduction and valence bands is determined by theposition of the Fermi level and, consequently, by the

23 Manuel Cardona and D. L. Greenaway, Phys. Rev. 133,A1685 (1964).

Vol. 58

X carrier concentration. Such shifts have been observedpreviously in SnTe.Y3,'1

The optical absorption in the plateau spectral regionabove the fundamental absorption edge is more thanan order of magnitude greater than it is for the relatedcompounds, PbS, PbSe, and PbTe.2 4 This is regardedas further evidence for the complex band structure be-lieved to be associated with this material. Because of theundefined complexity, we are unable to analyze ourdata to determine the separation of the conduction-and valence-band extrema or a density-of-states mass.However, our results should serve as a useful check on

3.0 band models proposed for SnTe in the future.

E. Extended Bound-Carrier Spectra andKramers-Kronig Analysis

Our experimental values of nric and kBc, extendingto 3.8 eV, are indicated by the filled symbols in Fig. 6.They were determined from R and T measurements, asdescribed in Sec. IIA. The unfilled symbols representbound-carrier values obtained by correcting the experi-mental quantities for free-carrier dispersion in regionsof overlap, as described by Eq. (12). No points areshown for 11BC in the spectral region from 0.8 to 1.9 eV,because the scatter in our results was unacceptablylarge. This is believed to be related to the very greatuncertainty of Of as determined by R and T measure-ments in regions where n2~ 1+k2 . This point has beendiscussed in detail by GCrant and Paul2 ' and Wessel.17

The low-energy values of slync are of considerableinterest. In this range sniC is unusually large and carrier-concentration dependent. We have attempted to de-termine whether or not the magnitude and carrier-con-centration dependences of nBC in this region are con-sistent with the nature of observed kBC spectra. Ouranalysis was based upon the Kramers-Kronig disper-sion relation given by25

2 rnBC(E)- 1 =- kBC(E')[E'2-A2}]-'E'dE', (13)

7r

which relates nnc and kB. It follows from this equa-tion that the complete k33,(E') spectrum specifies thevalue of flisy at all energies, E.

We have evaluated Eq. (13), using our values of kasCout to 3 eV and those of Cardona and Greenawav2 3

from 3 eV to 20 eV. Values of kBc beyond 20 eV havenot been reported. Such values are expected to con-tribute onlv a small, constant amount to the calculatedvalue of nlic(E) for E<2.0 eV. The evaluation of theintegral over the range of known k uc wag carried outnumerically on a computer.

The values of )Inc obtained using the kBC spectrumapplying to our film having P= 1.4X 1020 cm-' are found

21 W. W. Scanlon, Solid State PiVsics (Academic Press inc.New York, 1959), Vol. IX, p. 115.

F5 Frank Stern, Phys. Rev. 133, A1653 (1964).

January 1968 OPTIrCAL PROPERTIES OF

9

8

7

6

5

4

R'

wi

2 -

0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6PHOTON ENERGY (eV.)

FIG. 6. The bound-carrier indices of refraction nBc and extinc-tion coefficients kBC for three carrier concentrations of SnTe. Thecarrier concentrations were 3.6X 1019, 1.4X 1020, and 6.8X 1020cm-3 , and are represented by the triangles, circles, and squares,respectively. The filled symbols are values determined from anal-ysis of R and T as explained in Sec. IIA. The unfilled symbolsare experimental values corrected for free-carrier effects. Thevalues of nBc for E<0.8 eV were determined from interferencefringes and are uncertain to within 43%. The error of their rela-tive values is, however, considerably less than this. All otherpoints have an estimated uncertainty of 4-10%. The solid curvesA', B', and D' were calculated using the Kramers-Kronig relation,as explained in the text.

to be within 10% of the observed values at all energiesless than 2.0 eV. Thus, the observed magnitude of nscis essentially accounted for by the kBc spectrum from 0to 20 eV. By using a very small constant, 0.50, to repre-sent the integral beyond 20 eV, our results become thoserepresented by curve B' in Fig. 6. They are in excellentagreement with the observed values of n1C, both inmagnitude and energy dependence.

The origin of the carrier-concentration dependence ofnBC was also studied using Eq. (13). We have assumedthat the kBc spectrum is carrier-concentration depen-dent only in the region of the fundamental absorptionedge, as shown in Fig. 6. Using the values of kBc as-sociated with the low and high carrier concentrationsamples, we obtained the n1C spectra shown as calcu-lated curves A' and D'. The agreement between curveD' and the observed values is excellent; only a smalldiscrepancy is associated with curve A'. These resultsindicate that the observed carrier-concentration de-pendence of nBc arises primarily from the Burstein-type shift of the fundamental absorption edge.

A useful by-product of the Kramers-Kronig disper-sion analysis described is that it yields a value of theoptical index of refraction nO. By definition, nO, is equalto nBc(E) evaluated at E= 0. The results of the disper-sion analysis provide a basis for extrapolating our ex-

SnTe IN VISIBLE AND ir 125

I10, , I I , , , , I

"I.-

perimental results to zero energy and, consequently, forthe determination of nit, The values determined in thisway are 6.8±0.2, 6.4±0.2, and 6.2±0.2 for carrierconcentrations of 3.6X 10's, 1.4X 1020, and 6.8X 1020cm-3 , respectively. The optical indices of refraction forthe two highest carrier concentrations were derived fromthe intercepts of curves B' and D', as they are shown inFig. 6. In contrast, the value associated with the lowestcarrier concentration was not obtained from curve A',but from a similar curve shifted so as to be in betteragreement with the observed values of the correspond-ing nBc(E). The shift was effected by simply increasingthe constant representing the dispersion integral atE'>20 eV, from 0.50 to 0.66. By so doing, excellentagreement was obtained between the calculated andobserved results. We should not conclude from this thatpart of the carrier concentration dependence of nBC isnecessarily due to variations in kBc at E'> 20 eV, sincevariations at lower energies could also be representedby such a constant in some cases. The optical index ofrefraction is the square root of the more commonlyquoted optical dielectric constant E. Values for thelatter quantity are 4643, 42i3, and 38±t3 applyingfrom the lowest to highest carrier concentrations, respec-tively. These values are in good agreement with thoseobtained from studies of the bulk material.'4"18

Another useful by-product of the dispersion analysisis that it provides reliable values of nBC in the regionfrom 0.8 to 1.9 eV where we were not able to obtainthem experimentally.

III. SUMMARY

The optical dispersion of SnTe in the spectral regionstudied here arises primarily from bound and freecarriers, each dominating, respectively, in the high- andlow-energy regions. Free-carrier dispersion is well de-scribed by classical theory. The bound-carrier indices ofrefraction are unusually large and carrier-concentrationdependent in their low-energy range. Both of thesecharacteristics are consistent with our results of aKramers-Kronig analysis based upon the absorptionspectrum. The optical dielectric constant is also largeand carrier-concentration dependent. This dependencearises primarily from a Burstein-type shift of thefundamental absorption edge.

IV. ACKNOWLEDGMENTS

We wish to thank our associate H. R. Riedl forstimulating discussions and suggestions during thecourse of the work.