PHYSICAL RE VIE W B VOLUME 8, NUMBER 4 15 AUGUST 1973

Electronic Band Structure and Optical Properties of PbTe, PbSe, and PbS~

S. E. Kohn, P. Y. Yu, ' Y. Petroff, ~ Y. R. Shen, Y. Tsang, and M. L. CohenDepartment of Physics, University of California

and Inorganic Afaterials Research Division, Lawrence Berkeley Laboratory, Berkeley, California 94720(Received 5 September 1972)

By using the techniques of wavelength-modulation spectroscopy, we have measured the reflectivityR (co) and the derivative of the reflectivity, (1/R)d R/d co, for PbTe, PbSe, and PbS from 1.5 «6eV. We have independently calculated the band structure, frequency-dependent dielectric function, density of states,and derivative reflectivity spectrum for all three compounds by the empirical pseudopotential method.

I. INTRODUCTION

The narrow- gap IV-VI semiconductors havebeen the object of extensive research over thepast few years. ~ Compounds of Pb, Sn, and Gehave been studied, and a great deal is known aboutmany of them. Particular emphasis has beenplaced on studies of the band-edge properties andthe fundamental gap. The location of the gap inthe Brillouin zone, 4' the temperature and pressuredependence, 6' and the alloy dependence of thefundamental gap have been studied extensively.Higher-energy gaps have also been explored, butless is known about these gaps, and few experi-mental measurements have been made at higherenergies.

In this paper we will present experimental op-tical data and theoretical band-structure calcula-tions for the lead chalcogenides PbTe, PbSe, andPbS. Several band-structure calculations of thesecompounds are available in the literature using theaugmented-plane-wave (APW) method, ' theKorringa-Kohn-Rostoker (KKR) method, ' theorthogonalized-plane-wave (OPW) method, ' andtwo versions of the pseudopotential method. '"All of these calculations have a minimum directgap at the L point in the Brillouin zone and manyother features in common. There are differences,however, of the order of 1 eV for the energy bandsaway from the fundamental gap. These differenceslead to different interpretations of the optical spec-trum. These interpretations can be tested by cal-culating the optical constants in the visible andultraviolet regions from the various band-structurecalculations, and then comparing the results withthe measured optical spectra. Some calculationsof this kind exist for the IV-VI semiconduc-tors, ' ' but not all of the above band-structurecalculations have been tested in this way.

Until recently, the only optical data in the vis-ible and ultraviolet spectral regions available forcomparison to theoretical calculations have beenthe near-normal incidence ref lectivity measure-ments of Cardona and Greenaway, '6 and of Belle. '

Since these experiments were performed, deriva-tive spectroscopic techniques have become verypopular. Aspnes and Cardona have measured theelectroreflectance of all three lead salts. In theirexperiment, the electric field inside the sampleswas considerably distorted from the ideal square-wave shape because of the high dielectric constantof the compounds. This made the usual inter-pretation of the electroreflectance data more dif-ficult. Nishino et ul. ' have measured the elec-troreflectance and thermoreflectance of PbSe,showing some differences between these measure-ments. We have measured the wavelength-mod-ulated reflectance of PbTe, PbSe, and PbS froml. 5 to 6 eV at temperatures ranging from 5 to300'K. The only previously published wave-length-modulation ref lectivity measurements onany of these compounds is the work of Korn,Welkowsky, and Braunstein on the Pb„Sn& „Te alloysystem.

We have also made empirical-pseudopotential-method (EPM) calculations of the band structureand ez(&u) spectrum of PbSe and PbS. In addition,by Kramers-Kronig transformations of the theoret-ical az(ur) spectrum, the ref lectivity derivativespectrum (I/R)dR/du& is calculated for PbTe,PbSe, and PbS, and compared with the experi-mental measurements. These results confirm thecritical-point assignment for the ez((o) spectrum ofPbTe made previously by Cohen and Tsang. Byusing the method of Gilat and Dolling, ' we havealso calculated the density of states for all threecompounds.

II. EXPERIMENTAL PROCEDURE

The theory of wavelength-modulation spectro-scopy and a description of our experimental appa-ratus has been given elsewhere. A double-beamoptical system is used to normalize the ref lectivityand to cancel the background contributions to thederivative signal. The system is operated in aconfiguration that measures R(X) and dR/RdX si-multaneously. These results are then convertedto R(&u) and dR/Rd&u by a relatively simple corn

1477

1478 AND CQHE NKOHN, YU, PE TTROF F SHE N, TSANG,

—501.0-

PbS5K

100

0

I (gg -Q5—

)%

h

-1.0-

I

6I

4 52 3ENERGY (eV)

FIG. 1. Refj.ec xvx zl t ' . and derivative re ectxvi ofd tted curve is the theoretzcaPbTe at 5'K. The o e

(1/R)dR/dv spectra calculate yted b EPM.

2 3 4 5ENERGY (eV)

3. Reflectivity and derivative ref lecti ''vi of PbS~5at 5'K. The dotted curve is the core '

spectra calculated by EPM.

I

Pbse5'K

IOO

I

6I I

4 52 3ENERGY(eV)

. and derivative ref lectxvt. ofThe dotted curve is the theore ic

(1/R)dR/der spectra calculated by E

uter manipulation.wereSam les of all three compounds studied wer

i . All of the samples used werefl t .~.....d;...d'i le crystals.

cleaved to prododuce a clean a suThe PbS crys-

relatively small, an e0f PbTe and PbSe were rela ' y, erf ct. Therefore, the a-cleavages were not as pe ec .

t' 'ty measurementsma nitude of their re ec ivy.

h th relative measure-may not bebe correct, thoug e r1 Once inside thements are stroll yver acceptab e.

uousl maintainedthe samples were continuous y mDewar, e sT rature control wasptmos here. Temper

ed b flowing helium gas pas e ss ec '

d always reproducible,s ectra thus obtained are w

t mination has no observ-indicating that surface contamina ibl ffect on our spectra.

We have also investigate e efield on the structures in our spectra y

thmeasuring two PbS pPbS samples wx- 10 and- 10' cm, respective y.el . We haveth t of thfound no deferencece between e spe

5 08 eV whichtwo samples exceptt the structure at . ethan theerature dependence anhas a stronger temper

V. Thuss as we will show in Sec.onclude that, excep orwe are led to conc

of PbS is notour derivative spectrumaffected by the surface field. e s

1 true for the other two lead salts althoughwe n to rform the samewe did not have the crystals o pe

in PbS. The same conclusionmeasurements as inct ofined if one estimates the effec

surface field on the wavelength-m a eof the lead s s ualt sing an expression recently o-tained by Seraphin and Aspnes.

III. EXPERIMENTAL RESULTS

easured ref lectivity spec-The experimentally meas ' ' e-ra R,u&) and derivative ref lectivity spectra

S d'Rde for PbTe, PbSe, anofi s. 1-3 along wx'th theoretical calculationsF g

pere were measured atpresented here

ectrum of PbTe agrees' e well with the results of Korn et al. at

The derivative spectrum oectra theare striking y simi

tion of Cardonad E structures (in the notationand Greenaway'~) are well reso ve .1 ed. It is of par-ticular ininterest to no ein t the fine structure in the

1 th existence ofderivative spectra, particular y ethe Eks on the low-energy side of e

simila i the pect apeak. These peaks are very simi ar inQ p ~

ierformed to determine &,(&o), a, &v,tra was pe or

nd d& d~. The resu ts o1 f these calculations5. A description ofp g ~in Fi s. 4 and

ith those of Korn et a .The results agree well wx

of PbTe. A quick glance at the t2(&cr)

rahl more structure than the ea "reveals considera y mntif i criti-tra and is therefore useful in i en yi

oin '1 of the optical spectrum.oints and finer detai s od f-m "-,-'-- .1Critical points obtaine rom

re listed in Table I, along with experx-Th ds of revious wor

o wa s. First, crt xc ptermined in two ways.dR/Rd&oref lectivity were determinined from the

in the Kramers-Kronig analy-spectra. However, an e

IO-

SI 0

IO

t5(b)

IO-

IV. CALCULATION

The EPM has ncnv been miTh en widely applied ten mz' o many

ments {extracted ifor many clem tpotential form f tac ox's

c culations ax'and-structure ale x'om different EPM

ese existinge ' pseudopotential) are now availabl .e.

for Pb, 8e, and 8ial form factors V

n q respectlvelyq 1n

smooth curve issm s drawn through

PbS

es and can be px'o

ts of Pb8e andop-

61 n f PbS are akn s of P18e and

s. The

Follomin th

8ors are sli htl

erne„

ps. Th8 choice1 S8V-

ln8 assumption that

c gapathe correspond-

e ref lectivit se 'y spectrum in Pbg

e same regions of thm e sam ' ee, vrhich have been anal-

Pb Te5'K

l5 E2

30

E LE C T RONIC BA ND STRUCTURE AE AND OP T ICAL

I l

PROPERTIES

30

1 ~ e

"IO- 50

-l5

-20

-25I

4

- l5

I

ENERGY {AO

-75

IO

3I

ENERGY {eV)6

FIGG. 4. ~g(co) aalu ~ 4r) o(M) 0 e eq RQicl (0)

g analysis of the

0

QJ 000 "0

I

3ENERGY {eV)

sis of the experimental datahift d Th fos . re ore, a second liatre ore of critical-point

rom the dest'd~ spectra is

e that, in electrore-

s. A close corn& peak for all three

data andcompax'iaon of oue 4 d~ cojnfirms this result.

PbS5 K —50

0

-l503 4

FIG. 5

ENERGY {eV)5 6'5

. 5. d~~jdco and A

d~ SPeotra.Dig," SGa,lysis Of Qle

1480 KOHN, YU, PETROFF, SHEN, TSANG, AND COHEN

TABLE I. Critical-pointenergies of (a) PbTe, {b) PbSe, and (c) PbS. The second columncontains critical points derived from experimentally measured (1/R)dR/dry. t;(w) stands forweak. ] The third column has critical points from the F2/d~ spectra obtained by Kramers-Kronig analysis. The fourth column contains the results of EPM calculations.

Other experimentaldata

o. 19o'

1.24

2.18

3.S"

(a) PbTe~1 dR d&2

Rdu dM

2. 16{w)d2.25 2.12.3e"2.56(w)~ 2.3

3.47 3.34.83(w) 4.95.9 5.9

Theory

0.19

1.071.27

2.002.03

2.783.1

Transition

1.{5-6)z(s -v)I {5-v)

go

11.2'

0. 16S'(b) I bSe

0.172

Volume effectstransitions near X

Mp

1.591 54c1 78c

2.9S"3 123.3c

8c

4.s'4, 6c

7.1'

4.47 4.40

1.97(w) 2.02.18(w) 2.1S

2 ' 84(vr)d 2.653.12' 2.8

1 ~ 61.92.3

2.72.9

4.24.3

Mp

Mg

Mo

z(s -v)x{5-6)

a{4-6)z(4-v)

9.lb

12.Sb

Volume effectstxansitions near I" and X

0. 286

1.85b c

1.98 2.1

0.287

1.61.9

Mo

Mo

M)

L, (s -6)I.(s -v)z(s -6)

3.49b3.Vc

4.o'4, 6c

5 27

8.1'

2.46

3.3S'

5.08

5.75(w)

5.05.2

4.925 ~ 1

3.03.3 2.943.9(w) 3.5

Mg

M2

z(s -v)z{s-6)

z(4-v)~{4-e)

9.8'Volume effect

txansitions near F and X

Reference 4."Reference 16.'Reference 18.

Structure in (1/R) dR/des due to structure in e&.'Reference 26.

ELECTRONIC BAND STRUCTURE AND OPTICAL PROPERTIES. . . 1481

0.1 TABLE II. Interpolated and final set of pseudopoten-tial form factors used for the calculation of the bandstructures of PbSe and PbS.

—0.2—

—0.3

v'(4)

v'(8)

v'0.2)

V4(3)

V+(11)

PbSe

Interpolated(Ry)

—0.2334

—0.0171

0.0097

0 ' 0657

0.0154

Finaladjusted

(Ry)

—0.2377

—0.0171

0.0097

0.0735

0.01224

Interpolated(Ry)

—0.256

—0.0116

0.0242

0.091

0.0154

Finaladjusted

(Ry)

—0.2532

—0.0181

0.0242

0.0859

0.01038

q(k ')

FIG. 6. Pseudopotential form factors for Pb. Thepoints are derived from PbTe data (Refs. 23 and 24).

yzed previously. 4 In Table II are listed the start-ing set of form factors for PbS and PbSe interpo-lated and scaled from the Pb, S, and Se poten-tials, as discussed above. This table also con-tains the final set used in calculating the bandstructure and optical properties. The startingand final sets of form factors are remarkablyclose. The adjustment from the former to thelatter set is minimal.

Spin-orbit coupling effects are included in theEPM calculation. The symmetric and antisym-metric spin-orbit parameters X» X&

' in eachmaterial are chosen so that the ratio Xz/X„corre-sponds to that of the constituent outermost coreatomic spin- orbit splitting.

The resultant band structure for PbSe and PbS

are shown in Figs. 8 and 9, respectively. Theminimum direct gap E~ is at the L point of theBrillouin zone with symmetry L6 for the top va-lence band and symmetry L6 for the bottom con-duction band, as in PbTe. The fundamental gapsare 0. 172 and 0. 287 eV, respectively, for PbSeand PbS. From studies of interband magnetoab-sorption, the corresponding experimentally mea-sured values are 0. 165+ 0. 005 and 0. 286+ 0. 003eV. The over-all features of the band structuresfor PbSe and PbS are similar and resemble thatfor PbTe (Fig. 10).

An intrinsic property of an EPM calculation isthat the high-energy bands crowd together as a re-sult of the truncation of the Hamiltonian matrix.This is illustrated by the PbS conduction bands at

In a nonrelativistic calculation, the conduction-band symmetry at I' for the lead salts would be I'15,I'25, and I'&2, counting upward in energy from theforbidden gap. The spin-orbit coupling splits theI'» into I", and I'8, I'25 toI'and I'&, and I'» to r,.

0.1

a.

—0.1—

ac —0.2—

—0.3—

FIG. 7. Pseudopotentialform factors for (a) Se and

(b) S (Refs. 23 and 24).

—0.4— nS

cIS

—0.5

I

5 1

q (A'j

1482 PE TROF F, 8 HE I, TSANQ, AND C OHE N

'Io—

Fs'

r;r„

' Ks'

~ Kg

Ks~

+Xj, Xe

x.I

Xe

Xy

FsI

Fs

r,

, L4,g

'L

FIG. 8. Energy-bandstructure of PbSe.

re—d

—24—r,'

r

J--r

K K

XjXe

Xe

J-X4

X

AL4,$

Le

) Le

L

In Pb8, the levels arising from F35 and ~&3 crowdtogether as they decrease in energy away from, and

drop below, the ~8 level of I'». This same effectof the crowding of the higher-energy bands is seen'

in the optical spectrum where the energies of thetheoretical high-energy structure (E+ EI, E~) de-viate somewhat from the measured data.

The imaginary part of the frequency-dependentdielectric function ex(co) has the form

8ex(&u)= » Z ~ 5(E,(k)-E„(k)-$(o)3~ co

10-Ks&

,Le

Le

Fs' pr.'r;

xs)'Ks

Xq

e

2IC5I4a 0

LLI

—2Fs

Fe' i—6

&Ksi

& Ks»

XeX7

, r;,4

Lq

+Le

4.$Lt

Le/

FIG. 9. Energy-band .

structure of PbS.

Le

L

ELECTRONIC BAND STRUCTURE AND OPTICAL PROPERTIES. . .where up, „and up„are the periodic parts of the va-lence- and conduction-band Bloch wave functions atk and E„(k) and E,(k) are the energies of thesestates. In the previous calculation of ca(w) forPbTe, ' these wave functions and energies are ob-tained by diagonalizing a pseudopotential Hamilton-ian at 3M mesh points in the irreducible part of thefcc cubic Brillouin zone. Approximately 3&&10 ran-dom sampling points are then chosen using a MonteCarlo method. The energy eigenvalues and dipolematrix elements associated with each point arechosen by linear interpolation between the 356 meshpoints. The prominent structure in the e2(&o) origi-nate from Van Hove singularities at critical pointswhere VQE,(k)]=0. This can be more clearly seenif we write the k integration in Eq. (1) in terms ofan integration over a surface S, corresponding to aconstant interband energy difference E,(k) —E„(k).Assuming the dipole matrix to be a constant,

dS

I V;[E,(k) —E„(k)]1

'

The expression on the right-hand side in Eq. (2) isthe joint density of states.

Gilat and Dolling ~ developed a different methodof ca1culation first applied to a computation of thephonon density of states for a cubic crystal. ~' Thismethod proved to be of much higher resolution,speed, and accuracy in comparison with traditionalrandom-point sampling methods. This method alsoinvolves dividing the irreducible section of the firstBrillouin zone into a cubic mesh, but instead of tak-ing discrete random points within the mesh, theconstant frequency surfaces inside every smallcube are approximated by a set of parallel planes[the directions of the planes are determined by

&pE (k) at the mesh points] and may be continuouslyintegrated. Since the spectra contain two impor-tant factors, the density of states and the transi-tion probability or dipole matrix element, thesehave to be known accurately when comparing withexperimentally measured spectra. Gilat aod co-workers obtain such accuracy for the density ofstates in Refs. 21 and 2'7 by treating the matrixelement of each cube as a constant. This impliesthat fine details of the computed spectrum couldbe significantly distorted. The computation of thematrix elements was later taken to the same de-gree of accuracy as the density of states by Gilatand Kam. ~

Along with the experimental derivative reflec-tivity data of the present work, we employ theGilat scheme ~' in computing the e2(~) for PbTe,PbSe, and PbS. The pseudo- Hamiltonian is diago-nalized at 308 k points within the irreducible partof the Brillouin zone. The mesh chosen is slightlydifferent from the former calculation for PbTe 4

in that here the mesh points are shifted away fromsymmetry directions to avoid degeneracy, a nec-essary condition of the Gilat scheme. For thiscalculation, the dipole matrix element inside eachmesh is taken to be a constant equal to the valueat the mesh point. This is done for two reasons.First of all, previous experience with PbTe indi-cates the usefulness of even the joint density ofstates in determining the origin of the main struc-ture in the ref lectivity. Second, a first-order per-turbation calculation on the dipole matrix elementat the mesh point to allow interpolation betweenpoints requires far more computer time than doinga perturbation calculation on the energy. There-fore, as a first attempt, we have taken the matrixelements as constant in each small cubic mesh.

12

S+

r+~s'r,'r;—'

4

0—r;r-i

4

x4

K5

x'":K5 4

X"XK5 7

X-X

K5

K5

r;r4—

I IG. 10. Energy-bandstructure of PbTe (a repro-duction from Ref. 4).

r'— 5 x+

54

X+

x

V. DISCUSSION

The final theoretical results are presented inFigs. 1-3 and the theoretical dR/Rd~ spectra areshown as dashed lines in Figs. 1-3, together withthe experimental spectra. The corresponding bandstructures for PbSe, PbS, and PbTe are given inFigs. 8-10, respectively. In Fig. 11, the corre-sponding &2(&u) spectra are shown. They agreewell with the ez(&o) (Fig. 4) obtained from the ex-perimental data.

The peaks in e~(&o) are typically shifted down inenergy from the reQectivity spectrum by about0. 3 eV. The purpose of the theoretical and exper-imental e2(&o) curves (Figs. 11 and 4) is then todisplay the position of the main peaks and shoul-ders, whereas the experimental and theoreticalderivative reflectivity spectra (Figs. 1-3) are de-signed to display the fine structure on the mainones. In Table I are listed our critical-point as-signments for PbTe, PbSe, and PbS using theEPM calculations, and experimental values ofcritical points obtained by us and by others'6' 8

for the E;E6 structures. Note that the critical-point assignments for all three materials are al-most identical. The values of E4, E» and Ez inthe theoretical E& all gxoup together and are shifteddawn in energy, owing to the crowding of higher-energy conduction bands previously discussed.

To obtain the theoretical dR/Rd to spectra, &,(&o)

was calculated from the theoretical em(v) byKramers-Kronig transformation using an appropri-ate tail function for frequencies above u, „&,&f.

he respective +cucorr for PbTe PbSe anare 8. 5, 9.0, and 9. 5 eV.

In the following discussion for the results of thePbTe, PbSe, and PbS, the notations E0-E8 of Car-dona and Greenaway' wiQ be used freely since theoptical spectra for the three materials are indeedqualitatively similar and discussion of a qualitativenature is applicable to all three cases.

(i) I'bTe [Figs 1and l. l(a)]. The solid curve inFig. 11(a) is the previous theoretical cz(&o) calcula-tion obtained from a random-point sampling. Thedotted curve is the present calculation using theGilat sampling scheme. Note how the criticalpoints are accentuated in the present (Gilat)scheme. The approximation that dipole matrixelements are constant within each cube in the pres-ent scheme shows up most obviously at the bumpsat 0. 3, 0. 6, and 1.2 eV. The previous random-point sampling scheme, which gives less resolutionthan the present scheme when structure arises fromcritical-point singularities, does take account ofthe interpolation of dipole matrix elements withinthe cubic mesh. So in case of fine structures aris-ing mainly from volume effects, such as those at0. 3, 0. 6, and 1.2 eV in PbTe, the random-point

50 lI I

~5I&). Z,

I',40—

30

PbTe

20

Transition near X

0 I I I I

2 3 4 5 6 7 8 9 10

Energy (eV)

50

40—~5~ 7

3ee

%l

20

10ransitions near I', X

0 I I I I—

I I I

2 3 4 5 6 7 8 9 10

Energy (eV)

50

(c}

40—

ebs

30—3

eeIU

20—

10— Transihonsnear I, X

0 1 2 5 6 7 8 9 10

Energy (eV)

FIG. 11. Theoretical imaginary part of the frequency-dependent dielectric function c2(~) for (a) PbTe, (b)PbSe, and (c) PbS. In case of PbTe the two theoreticalcurves correspond to bvo different sampling schemes:the Gilat scheme (broken curve) and the random-pointsampling scheme (solid curve). The 4-eV peak in PbSis artificial and disappears when linear interpolation forthe matrix elements is used in this energy range (dottedline). For details, see text.

sampling scheme probably portrays the actual &~(~)better than the Gilat scheme. Otherwise, theagreement for the energy positions and the generalprofile between the experimental and theoretical&,(~) using the Gilat scheme is good. The theoreti-

AND OPTICAL PROPERTIES. . .cal derivative ref leetivity spectrum derived frome,(~) is extremely sensitive to any slight variationof the slope as well as the main peaks and shoul-ders in the ez(&o). A comparison of the theoretical(1/R)dR/der with experiment in Fig. 1 is thereforeencouraging. The theoretical dR/Rd &u would re-semble the experimental curve at 3.35 eV if the E,shoulder in the theoretical ez(&u) were slightly flat-ter. The experimental dR/Rd&u curve changesslope from negative to positive at 5. 4 eV due to anupward turn of slope in e,(&o) at energies just belowthe E4(6. 3 eV as measured from ref lectivity) struc-ture. The theoretical E4 value is at 7. 1 eV, andthus the theoretical dR/Rd&o continues to have anegative slope at 5. 4 eV. These results seem toindicate that our previous critical-point assign-ments for PbTe were essentially correct.

(ii) PbSe [Figs 2an.d 11(b)]. One essential fea-ture that differentiates PbSe from PbTe in the ex-perimental dR/Rd(u spectrum (Fig. 2) is the broadstructure with small peaks between 2. 0 and 3. 1 eVin the dR/Rd~ spectrum. The corresponding struc-ture in PbTe (Fig. 1) is much narrower, having awidth of only 0. 5 eV. This ean be understood fromthe theoretical critical-point analysis (Table I) andthe theoretical e~(&o) curves [Figs. 11(a) and 11(b)].For PbSe, there are critical points at 1.9 and2. 3 eV, both with moderate dipole matrix ele-ments. The main peak E~ is composed of two crit-ical points with large dipole matrix elements at2. 7 and 2. 9 eV. This cluster of critical pointswith increasing dipole matrix elements gives riseto the gentle rising E~ peak. In PbTe, the criticalpoints with moderate dipole matrix elements occurat 1.07 and 1.27 eV while those with large dipolematrix elements are at 2. 00 and 2. 02 eV. Thusin PbTe, critical points with moderate matrix ele-ments are more widely separated from those with

large matrix elements than in PbSe. Furthermore,the two critical points with large matrix elementsin PbTe occur at almost the same energy. Theseresult in a sharper and narrower Ea peak in PbTethan in the case of PbSe.

The small structure on the main peaks arisesfrom varying dipole matrix elements for differentregions of k space. As can be seen by the abovearguments, both the position of the critical pointsand the dipole matrix elements are important togive the optical spectrum. This point will be evenmore apparent when discussing PbS. The struc-ture near the 4. 4-eV region in the experimentaldR/Rd&u (Fig. 2) is again due to the Es shoulder inea(&u) and the rise in slope at 6. 1 eV again arisesbecause of the E4 structure at '7. 1 eV.

(iii) PbS [Eigs. 3 and 11(c)]. The theoreticale2(v) in Fig. 11(c) shows that in addition to theE~-E6 structure, there is an extra bump between3. 5 and 4. 1 eV. No critical point is found in this

energy region. A close examination of all the en-ergy gapa that may contribute to e2(~) in this en-ergy range uncovers the following: (a) the contri-bution to this part of ea(~) arises from the sameband transitions in neighboring cubic mesh, i.e. ,volume effect (each mesh covers an energy levelof about 0. 2-0. 3 eV), and (b) the dipole matrix ele-ments of these neighboring mesh points vary by afactor of 2 or 3. Recall that the matrix element ineach cubic mesh is taken to be constant and equalto the value at the mesh point, whereas the energyis allowed to vary continuously within the mesh.Therefore in the continuous energy range of above6.6 eV, the dipole matrix elexnent jumps discon-tinuously by about a factor of 2 to 3 at every 0. 2or 0. 3 eV interval.

If one now uses linear interpolation for the ma-trix el.ernents 1n the continuous energy range ofthese particular neighboring cubic mesh points inquestion, the resultant sz(&o) between 3. 5 and 4. 1 eVwill follow the dotted line in Fig 11(.c). In the re-gion of the spurious bump, there remains now onlysome smooth structure, which does show up whentransformed to the theoretical (I/R)dR/dv inFig. 3. This structure deviates considerably inshape from the experiment (I/R)dR/d&o in the re-gion from 3. 5 to 4. 5 eV. The Gilat scheme ' wouldbe necessary if one were interested in obtaining anundistorted spectrum and retaining the qualitativefeatures of the fine structures of the optical spec-trum, because it takes into account the interpola-tion of the matrix elements within each cubic mesh.The distortion of the fine structure is especiallyimportant when the structure arises from volumeeffects. In PbSe and PbTe, the E3 critical pointsare close 1n energy to the Ea erit1cal points. Thesingularities help to mask the distortion of finestructures arising from the approximation imposedon the dipole matrix elements. %8 are less fortu-nate in PbS in that E3 critical points are muchhigher in energy than the Ez critical points, allow-ing volume effects to dominate in between. Thestructure between 2 and 3. 5 eV in the experimental(I/R)dR/d~ (Fig. 3) is even broader than the cor-responding one in PbSe. This is again a result ofthe clustering of critical points at 1.6, 1.9, 2. 94,and 3. 5 eV. Note the two large matrix elementcritical points that give rise to the Ez peak (2. 94and 3. 5 eV) are separate by 0. 56 eV. This givesrise to a very broad E& peak.

In measuring the ref lectivity spectrum of PbSas a function of temperature, we observe a pro-nounced sharpening in the fine structure at 5. 08 eVas the temperature decreases. This sharpeningis observable even between liquid-nitrogen andliquid-helium temperatures. This is shown inFig. 12. No other structure in the spectra of allthree compounds has a similar behavior. The

1486

PbSzero-ener-energy reference. The ener—12 to+ 12 eV. Thlence bands and fi

is energy ra eng covers four va-s an five conduction bands.

pmore than 13 eVn e ge. Since these band

dt o ld~ ~

e results in explainin' 'pa iyone would antici te

so be useful in stud insion experiments. " u yang photoemis-

Wavelength-modulation s ctrocapable of detectin mane ecting many fine structures '

thxn e op-

(o.u.)-+0

+

I—/

ggK —+

0

—42

KOHN, YU, PETRO FF, SHEN T SANG, AND COHE N

I

-41

K 40 Xo

-39—38

4 5I

ENERGY (eV)6

FIG. 12. Temperature dependencereflectivity (1/R)

p ence of the derivativedR/d~ of PbS around th e E3 structure.

0

CIO

I I t I I I I I 1

PbTe

sample used in this measurement was n t

-e ect measurements. Ho

we repeated the ms. However, when

e measurement on an n-twith a carrier conc tncen ration of - 101

n- ype sample

obtained exactly they e same spectra as thesample, except this 5. 0came less prono

is . -eV structure , which be-nounce at low tern r

not show appreci blperature, and did

ecla e change as the

what was observed fThis result si, sl.medlar to

cates that thve or theE e& peaks of Ge, indi-

es structure is not sime strong tern eratu

together with the dep ture dependence

e dependence on the carrseems to suggest th t

carrier density

e citonic effect. Accordia the structure

(),atZ and b, dh

ue o&

and M2 critical--poxnt transitions

model h, an ence based on the Toyozawayperbolic excitons'tons can exist in the E

PbSto ' t thgainst the idea of

, xn our case, the excitonsi th hi h d'g conduction bands ma

smaller effective di 1'

n've &electric constantxe ectric constant E .

carriers tend toant E„. The free

en o screen the attractive '

b tw 1 t nd

plain why the excitns a holes, and this

exes onic effect decreacarrier concentration in

eases with higher

Forn ra ion in our observation.

or completeness, we also res1 1 tio ofth dd PbSi Fi . 13. i

e density of states

e valence-band edge i takis en to be the

0 I

-12 -10 -8 -6I

-6 -4 -2 0 2

Energy (eVj

2 I I

(b)

4 6 8 10 12

PbSe

0~1CIO

0—12 -10 -8 -6 -4

(c)

-2 0 2

Energy (eVj

6 8 10 12

I I I

Cl

0

PbS

0 1

CIO

0—12 -10 2 4 6 8 10

FIG. 13. DensEnergy (eVj

12

ensity of states (numberper primitive cell) for a) PbTe

er of states per eV(b) PbSe, and (c) PbS.

E LE CTRONIC BAND STRUC TURE AND OPTICAL PROPE RT IE S. . . 1487

To obtain e,(&u) and e2(&u), we need to know boththe absolute ref lectivity R(&u) and the correspond-ing phase angle 8(ur). ' These two quantities arerelated by the Kramers-Kronig relation

8( )A. ( d, lnR(X')7r] (X')'-~'

This can be transformed into the form

(Al)

(A2)

which is more appropriate for our own case, sincewe measure d lnR/dX with greater accuracy thanR. This method of calculation has the disadvan-tage that the function ln l(X+ V)/(X- X') I does notgo to zero as fast as I/(X"- X ) as X' becomes

tical spectrum of PbTe, PbSe, and PbS. Theagreement of the experimental and theoretical(I/R) dR/d~ and e,(&o) is quite good. Based uponthese results, we conclude that the EPM givesband structures of the lead chalcogenides that areessentially correct throughout the Brillouin zone.With these band structures, we have gained con-fidence in a more thorough understanding of theoptical properties of the lead chalcogenides. Thecomputational techniques may be improved bychanging the approximation used in handling thedipole matrix elements. If this is done, we be-lieve that the band-structure calculations are cap-able of giving not only the main structure but alsothe very fine structure in the optical ref lectivityspectra.

ACKNOWLEDGMENTS

One of us (Y. W. Tsang) would like to thankJohn Joannopoulos for helpful discussions on theGilat scheme of computation. We would also liketo thank M. Mulrey for assistance in prepariigsome of the graphs.

APPENDIX: KRAMERS-KRONIG ANALYSIS

very different from X. Hence, the calculation isa little more sensitive to the extrapolation condi-tions used for the low- and high-energy limits ofthe integral.

By differentiating Eq. (A2) with respect to X,we have

d8 I I, d lnR, InR(X')(A3)

The derivatives of && and &2 can then be calculatedfrom d InR/dX and d8/d),

In evaluating the integrals for 8 and d8/dX, itis necessary to make truncating approximationsfor the low- and high-energy limits of the inte-grals. It is well known that the ref lectivity ofsemiconductors becomes approximately constantat very low energies. It is common to take R(X)as constant for long wavelengths when computing8 via Eq. (Al). ~~ We have made a similar approx-imation d InR/d&u = 0 for long wavelengths. Also,structure in d lnR/dX far away from the long-wave-length limit of Eqs. (A2) and (A3) tend to oscillatearound zero, giving small contributions to the in-tegrals. At very high energies, it is expected,from a Drude type of dielectric function, that theref lectivity should decrease roughly as ~ 4.

Therefore, we approximated a functional depen-dence for R of (d ", where A is an adjustable pa-rameter around 4. To make the calculations lesssensitive to A, the room-temperature reflectivitydata of Cardona and Greenaway from 6 to 20 eVwere used in computing the integrals. The re-sults are very insensitive to the choice of A used.Varying A by over a factor of 2 does not changethe shape of the curves appreciably, but doeschange the over-all magnitudes somewhat. Theabsolute magnitudes of the curves are also some-what in error because the absolute magnitude af(I/R) dR/dX is a little uncertain, although this doesnot have a very strong effect in the Kramers-Kronig analysis.

~Work supported in part by the U. S. Atomic EnergyCo~mission and the National Science Foundation, Grant No.GP 13632, Al.

«IBM Postdoctoral Fellow.~On leave from the University of Paris.'M. L. Cohen, Y. Tung, and P. B. Allen, J. Phys. (Paris) Suppl.

29, C4-163 (1968).The Physics of Semimetals and Narrow Gap Semiconductors:Proceedings of the Conference held at Dallas, Texas, 20-21March 1970, edited by D. L. Carter and R. T. Bates(Pergamon, New York, 1971).

Proceedings of the Conference on the Physics of IV-VICompounds and Alloys, Philadelphia, 1972 (unpublished).

Y. W. Tung and M. L. Cohen, Phys. Rev. 180, 823 (1969);Phys. Rev. B 2, 1216 (1970); Y. W. Tung and M. L. Cohen,Phys. Lett. A 29, 236 (1969).

Y. W. Tsang and M. L. Cohen, Solid State Commun. 9, 261(1971).

6Y. W. Tsang and M. L. Cohen, Phys. Rev. B3, 1254(1971).

'M. L. Cohen and Y. W. Tsang, Ref. 2, p. 303.L. E. Johnson, J. B. Conklin, Jr., and G. W. Pratt, Jr., Phys.Rev. Lett. 11, 538 (1963); J. B. Conklin, Jr., L. E. Johnson,and G. W. Pratt, Jr., Phys. Rev. 137, A1282 (1965).

L. G. Ferreira, Phys. Rev. 137, A1601 (1965).' S. Rabii, Phys. Rev. 167, 801 (1968);S. Rabii, Phys. Rev.

173, 918 (1968)."H. Overhof and V. Rossler, Phys. Status Solidi 37, 691

(1970)."F. Herman, R. L. Kortum, I. Ortenburger, and J. P. Van

Dyke, J. Phys. (Paris) Suppl. 29, C4-62 (1968).' P. J. Lin and L. Kleinman, Phys. Rev. 142, 478 (1966).'4See Ref. 1."D. D. Buss and N. J. Parada, Phys. Rev. B 1, 2692 (1970).' M. Cardona and D. L. Greenaway, Phys. Rev. 133, A1685

(1964).

KOHN, YU, PETE, OFF, SHEN, TSANG, AND COHEN

"M. L. Belle, Fiz. Tverd. Tela 5, 3282 (1963) [Sov. Phys. -SolidState 5, 2401 (1964)]; Fiz. Tverd. Tela 7, 606 (1965} [Sov.Phys. -Solid State 7, 480 (1965}].

' D. E. Aspnes and M. Cardona, Phys. Rev. 173, 714 (1968}.' T. Nishino, H. Ogawa, and Y. Hamaka, J. Phys. Soc. Jap.

30, 1113 (1971).D. M. Korn, M. Welkowsky, and R. Braunstein, Sohd StateCommun. 9, 2001 (1971); D. M. Korn and R. Braunstein,Phys. Rev. B 5, 4837 (1972).

"G. Gilat and G. Dolling, Phys. Lett. 8, 304 (1964}."R. R. L Zucca, Ph.D. thesis (University of California,

Berkeley, 1970) (unpublished).~3B.O. Seraphin and D. E.Aspnes, Phys. Rev. B6, 3158 (1972).'"M. L. Cohen and V. Heine, in Solid State Physics, edited by

H. Ehrenreich, F. Seitz, and D. Turnbull (Academic, NewYork, 1970), Vol. 24. The data points in Figs. 6, 7(a), and 7(b)are taken directly from Tables 15 and 16, pages 212 and 215.

"Y. W. Tsang, Ph.D thesis (University of California, Berkely,1970) (unpublished).

"D. L. Mitchell, E. D. Palik, and J. N. Zemel, in Physics ofSemiconductors, Paris, 1964, edited by M. Hulin (Academic,New York, 1964), p. 325.G. Gilat and L J. Rankenheimer, Phys. Rev. 144, 390(1966).

G. Gilat and Z. Kam, Phys. Rev. Lett. 22, 715 (1969).' J. P. Walters and M. L. Cohen, Phys. Rev. 183, 763 (1969).' Y. Petroff, S. E. Kohn, and Y. R. Shen, in FirstInternational Conference on Modulation Spectroscopy, Tucson,Arizona, November, 1972 (unpublished).

'Y. Toyozawa, M. Inoue, M. Okazaki, and E. Hanamura, J.Phys. Soc. Jap. Suppl. 21, 133 (1967).

"A very good agreement with the theoretical valence-banddensity of states has been obtained by photoemissionmeasurements on PbTe."R. Pinchaux, Y. Petrol, D.Dagneaux, and M. Balkanski, in Conference on Physics ofIV-VI Compounds and Alloys, Philadelphia, 24-25 March1972 (unpublished); and W. E. Spicer and G. J. Lapeyre,Phys. Rev. 139, A565 (1965). After this work has beencompleted, x-ray photoemission [F. R. McFeely, S.Kowalczyk, L. Ley, R. A. Pollak, and D. A. Shirley, Phys.Rev. B (to be published)] and uv photoemission PVI. Cardona,D. W. Langer, N. J. Shevchik, and J. Tejeda (unpublished)]have been reported and in both cases good agreement wasfound in all three lead salts between the experimentallydetermined valence-band density of states and those shown inFigs. 13(a)—13(c).

"M. Cardona, in Optical Properties of Solids, edited by S.Nudelman and S. S. Mitra (Plenum, New York, 1969),p. 59.

PHYSICAL REVIEW 8 VOLUME 8, NUMBER 4 15 AUGUST 1973

Spontaneous and Stimulated Luminescence in CdS and ZnS Excited by MultiyhotonOptical Pumying

I. M. Catalano, A. Cingolani, and A, MinafraIstituto di Fisica, Cenfro Studie Applicazioni in Tecnologie Avanzate, Bari, Italy

(Received 12 February 1973)

Spontaneous and stimulated emission at 85'K in CdS and ZnS single crystals is reported. The excitingsource was a ruby laser and its second harmonic. The two-photon spectrum of CdS shows stimulationeffects in the phonon-assisted excitonic lines at a pumping level of about 10~6 (photons/cm~)/sec. Thesame effect is observed in cubic ZnS with the second-harmonic light. A three-photon excitedspontaneous lu~&acescence is also obtained in this last sample. The results are in good agreement withthe theoretical predictions of multiphoton-absorption theory.

I. INTRODUCTION

Several authors have reported spontaneous andstimulated emission in II-VI compounds. Themechanism responsible for most of the lasing tran-sitions is reported4 to involve bound excitons andacoustical-phonon-assisted transitions, particu-larly at very low temperatures (4. 2 'K) and in thepresence of impurity centers. At bquid-nitrogentemperature, line-shape measurements' in thespontaneous-emission spectrum have shown that inpure Cd8, CdSe, and ZnQ, the laser transitionsinvolve the E,„,-LQ line, i.e. , the longitudinaloptical-phonon-assisted emission of free excitons.

Other recombination mechanisms which can givela, ser action in II-VI compounds are the exciton-exciton2 interaction and the excitonic-moleculerecombination. This last effect is difficult to ob-serve experimentally owing to the small bindingenergy of the excitonic molecule, while the first

one may become prominent at very low tempera-tures. Generally, the laser emission has been ob-tained by electron-beam pumping, ~ ~ by linear op-tical pumping with ultraviolet laser light, 6 and bybvo-photon excitation with a ruby laser.

In the present work we report stimulated emis-sion in pure CdS and ZnS samples at a temperatureof 85'K under two-photon optical pumping, andthree-photon excited spontaneous luminescence inZn8 at the same tempera, ture.

II. BASIC THEORY

The fundamental energy gap E~ of CdS and Zn8(cubic) are, at V7'K, 2. 5 and 3. |i eV; the photonenergy of the ruby laser is I~, = 1.78 eV and thatof its second harmonic is 1~2= 3.56 eV.

The nonlinear cross section y„ for an n-photonprocess is defined by

+St} g+ Itt