PHYSICAL REVIEW B VOLUME 3, NUMB ER 4 15 F EBRUARY 1971

Wannier Exciton in an Electric Field. II.Electroabsorption in Direct-Band-Gap Solids

Daniel F. BlosseyXerox Research Laboratories, Xerox Square, Rochester, Nese 1'oak, 14603

(Received 11 August 1970)

The electric field and temperature dependence of excitonic electroabsorption is presentedfor direct-band-gap semiconductors and insulators. Each excitonic electroabsorption spectrumis characterized in terms of three energy parameters: (i) R, the binding energy of the exci-ton; (ii) I', the thermal broadening parameter; and (iii) I'8, the electro-optical energy. Therelative magnitude of these three quantities determines which, if any, of the competing forcesdominate. In comparison with the single-particle electroabsorption theory, the excitonicelectroabsorption theory gives several new results. The electric field and temperature de-pendence of g, 4&2, and &e& are displayed. Fitting of the excitonic electroabsorption theoryto lead iodide data gives the values of 2. 58 eV for the direct gap in lead iodide and 0. 21 elec-tron masses for the reduced mass of the electron-hole pair associated with the extrema in thevalence and conduction bands, as compared with previous values of 2. 55 eV and 0.24 electronmasses.

I. INTRODUCTION

Several calculations of electric field effects onoptical absorption by Wannier excitons have beenpublished in recent years. ' All of these calcula-tions have sought a solution to the quantum-me-chanical problem of the hydrogenic atom in an elec-tric field. The fact that there is no known analyticsolution to this problem has led the theorists totry approximate models ' and numerical integra-tion" 6 of the differential equation. A Green's-function solution' which was presented as an exactsolution later proved to be adequate only in thelimit of very high fields. ' The results of themodels, ' unfortunately, do not bear a close enoughresemblance to the real system and only the nu-merical integrations" have given reasonableresults for all values of electric field. A compre-hensive study of electric field effects on opticalabsorption by Wannier excitons was presented ina previous paper, ' hereafter denoted I, in whichfield effects on both bound and continuum stateswere shown. The effect of the electron-hole inter-action on the electric-field- induced oscillations inthe optical density of states was also demonstratedwith the result that the electron-hole interactionenhances these oscillations near an M, -type edge(positive effective masses) and quenches these os-cillations near an Mtpye edge (negative effectivemasses). It was suggested that this effect wouldinhibit the observation of M, -type edges in elec-troreflectance. The most glaring deficiency of allthese calculations' ' is that not one of them hasbeen put in a form such that it can be comparedwith a modulated spectrum, i. e. , an e1,ectroabsorp-tion or an electroreflectance spectrum. ' It is tothis deficiency that we address ourselves in thispaper.

8AND STRVC'TVRF.'

A8SORPTION SPECTRA.'

ABSORPTIONCOEFFICIENT, 0i

PHOTON- INDUCEDELECTRONICTRANSITION

WAVE VECTORLIGHT ENERGY

FIG. 1. Simple two-band model for direct-band-gapsolid and resulting absorption spectrum near the funda-mental edge.

Modulated spectroscopy, particularly electro-absorption (EA) and electroreflectance (ER), hasgenerated a great deal of information about theelectronic band structure of solids. This technique,under the name of electric-field-induced spectro-scopy (EFS), has also been applied to the electronicspectra of gaseous molecules and has generatedextremely well-resolved rotational structure. " Inthis paper, we will concentrate on the EA spectrumnear the fundamental edge of direct-band-gap solids.Figure 1 shows a simple two-band model for a di-rect-band-gap solid and the resulting absorptionspectrum near the direct edge. The exact positionof the edge, as shown in Fig. 1, is somewhat hardto pinpoint. The EA spectrum gives much betterresolution as well as additional information aboutband anisotropy and effective masses. Figure 2

shows the electric-field-induced oscillations in theoptical density of states and the EA spectrum with

its strongest structure about the energy gap E, .This spectrum is measured using standard phase-sensitive techniques. ' The temperature and fielddependence of the excitonic EA spectrum will bedemonstrated later in the paper but first it is useful

1382

WANNIER EXCITQN IN AN ELECTRIC FIELD. II ~ ~ ~ 1383

ABSORPTIOIII SPECTRA:FINITE AND ZERO FIELD

ABSORPTIONCOEFFICIENT, O

NODIJLA TED SPECTR'A:

aR

and then become a true continuum above the gap.The effective Rydberg R is given by

4

A= 2 = —- &&e &&13.6 eV,2 e

(2)

where JLI. is the reduced mass of the electron-holepair, m is the electronic mass, and c is the staticdielectric constant of the solid. The radius (ef-fective Bohr radius} of the exciton is given by

LIGHT ENERGY LIGHT ENERGYa= ~ = —- ~e &&5. 29&&10 cm.g

p, e mFIG. 2. Electric-field-induced oscillations in the opti-cal density of states near the fundamental edge and theresulting electroabsorption spectrum representing thedifference between the finite- and zero-field absorptioncurves.

to review some basic concepts of Wannier excitontheory and electric field effects.

The Wannier exciton is an electron-hole paircreated by photoexcitation of a crystalline solid. 'This type of exciton is a hydrogenic atom typifiedby the effective masses of the electron and hole and

by the dielectric constant of the solid. This modelfor the exciton is applicable to crystals which cangenerate charge carriers upon exposure to light,i. e. , photoconductors. Also, from the formalismof the theory, it is required that the effective sizeof the excitation cover several unit cells of thecrystal. The size of the excitation is related tohow much an electron is shared among the variouslattice sites. For covalently bonded crystals, theelectron is shared equally among many sites and,consequently, the excitation covers many sites;but, for ionic, molecular, and rare-gas solids, theelectronic motion is much more restricted and theexcitations are thereby more localized. Theselocalized excitations are generally called Frenkelexcitons. ' Thus, the Wannier theory for ex-citons is primarily applicable to semiconductingcrystals, ' particularly from group IV of the Pe-riodic Table; many of the more qualitative resultsof this theory are applicable to noncovalently bondedcrystals. 20

The Wannier exciton in the effective-mass ap-proximation is equivalent to the hydrogen atom dif-fering only in the values of the effective masses ofthe electron and hole and the dielectric constant ofthe medium. " The bound states of the electron-hole pair occur in the forbidden gap of semicon-ductors and insulators at energies given by

Ef = Eg Rs

where E~ is the energy gap and R is the effectiveRydberg energy. The intensities of the absorptionlines for these bound levels are proportional ton '. These lines blend into a quasicontinuum forenergies very near but just below the gap energy

(3)

@z = If/ea = (y. /m) && e '&& 2. 59 && 10 V/cm. (5)

6 FINITE

C=0

FIG. 3. Electron-hole interactions with and withoutexternally applied electric field.

Thus, we see that a small effective mass (highlymobile carriers) and a large dielectric constantboth combine to give large radii for the excitonswhich is one of the criteria for validity of the Wan-nier theory.

Since the Coulomb potential is altered by thedielectric constant c, the electron-hole pair in anelectric field sees the combined potential

V(r} = —e /er e8 -~ r, (4)

where r = r, —r„and 8 is the electric field. Thispotential is shown in Fig. 3 for zero and finiteelectric fields. The major effect of the electricfield is to lower the lip of the potential well whichcauses the bound levels to be mixed and broadenedinto a continuum. A secondary effect of the elec-tric field is a slight widening of the Coulomb wellwhich causes a shift of the ls level to lower ener-gies which produces the well-known phenomena ofthe second-order Stark shift. For the electricfield to be capable of ionizing the exciton, it mustprovide at least a potential drop of 1 Ry acrossthe effective Bohr radius; this ionization field 8»is defined as

1384 DANIE L F. BLOSSEY

TABLE I. Energy gaps, exciton binding energies(exciton associated highest-energy split-off valence band),and ionization fields for various semiconductors and

insulators.

Crystal E,(eV) R (me V) 81(10 V/cm)

InSbInAsGeGaSbInPGaAsAlsbCdTeCdseZnTePbI2CdsZnSeZnS

0.23570.3600.8000.8131.291.411.61.606l.84152.3012. 552. 58312.8183.9115

0.5

1.81,41.86.55.17.5

10.015.713.073.029.419.040. 1

0.080.700.551.007.85.7

12316047

460140

75200

In Table I, there is a listing of energy gaps, binding

energies, and ionization fields for various semi-conductors and insulators calculated from tabulatedvalues of effective masses and dielectric con-stants'~ '8; the crystals are ordered by increasingvalue of energy gap.

There are very few published calculations ofelectric field effects on the hydrogenic atom. Forcalculation of Stark shifts and splittings, ' ' per-turbation methods are appropriate if 8/Sz«1. Butfor electric fields on the order of or larger than

8», the electric field dominates the Coulomb po-tential and a nonperturbative solution is needed.The first nonyerturbative approach was proposedby Duke and Alferieff in which they reduced theelectric field plus Coulomb potentials to the ana-lytically solvable model of only the Coulomb po-tential inside a given radius and only the electricfield potential outside the given radius. This mod-el is adequate in the high-field limit but fails topredict correct Stark shifts in the low-field limit.To improve on the results of Duke's model, nu-

merical integrations of the hydrogenic Schrodingerequation have been performed by Ralph, ' Dow and

Redfield, 4 and the author. " Ralph' calculatedthe shifts and broadenings of the 1s hydrogeniclevel and Dow and Redfield4 demonstrated the elec-tric field dependence of the excitonic absorptiontail in the band gap and compared this with Urbach'srule. Enderleinv used a Green's-function approachto solve the problem but one of his assumptionsmade his results valid only in the limit ofg gpss Byg

The field dependence of e~ for excitonic absorp-tion has been calculated in paper I. ' In Secs. IIand III, we will be examining the field and temper-ature dependence of the excitonic EA spectrum. 'This involves calculating a properly normalized

zero-field spectrum as well as a finite-field spec-trum and taking the difference. The dielectric func-tions e2($) and ea(0) will be displayed as functionsof temperature and electric field in Sec. IIA. Thefield-induced changes &&2 and 4&& will be displayedas functions of temperature and electric field in

Secs. IIB and IIC, respectively. Fits to lead io-dide EA data will be shown in Sec. IID and thethermally ionized exciton limit is considered in theAppendix. A discussion of the results is containedin Sec. III.

II, ELECTROABSORPTION

Since the early work (1958) of Franz" and Kel-dysh on photon-assisted tunneling between bands,the theory of EA has progressed a long way. Theseearly theories"' presented an asymptotic formulafor electric-field-induced optical absorption in theforbidden gap and predicted a red shift of the band

edge with field. The first published experimentalwork appeared two years later and showed a field-induced shift to lower energies of an exponentialabsorption edge in CdS. " Similar results appearedtwo years later (1962) for GaAs. '4 The following

year, Tharmalingam" presented a closed formsolution which agreed with the Franz-Keldysh theorybelow the edge and showed electric-field-inducedoscillations in the optical density of states abovethe edge. Similar results were obtained by Cal-laway' using field-dependent electron and holewave functions. " The equivalence of these twotheories was not demonstrated until several yearslater (1968) by Aspnes, Handler, and Blossey'8in which a more general expression for EA wasderived. The electric-field-induced oscillationswere first observed in EA by Frova and Handlerand in ER by Seraphin. Since then a deluge ofexperimental data has appeared in the literature. 'Experimentalists have attempted to fit EA data tothe nonexcitonic EA theory' ' ' with some success.Qualitative but not quantitative agreement could beachieved. In 1966, Hamakawa, Germano, andHandler4 suggested that exciton effects were thedominate cause of discrepancies between EA theoryand experiment. They later showed that only oneof the six possible predictions of the nonexcitonicEA theory was confirmed by experiment. Ad-ditional work on CdS and PbIz, ' in which excitonabsorption is known to dominate the spectra, con-firmed that the nonexcitonic EA theory was inade-quate. Sec. IIA will contain the first display oftheoretical excitonic EA spectra as functions ofelectric field and temperature and a fit to the leadiodide EA data of Perov et al.

A. Zero- and Finite-Field Absorption

It was shown in I that the imaginary part of thedielectric constant ~, may be expressed as

WANNIER EXCITQN IN AN E LECTRIC FIELD. II. ~ ~ 1385

Q (0) =4vS~n '6 —', he& E, (Sa)

y'(0) =2&I(& co"8 )/g j

I~ &E, . (8b)

Below the edge, &f& (0) contains the hydrogenic se-ries with each line having strength 4m' . Nearthe edge, both expressions approach the limitQ (0) = 2w, below the edge as a quasicontinuum andabove the edge as a continuum. Far above theedge, Qa(0) approaches the desired limit of[(If+ —E,)/R]" . For h 40, Qa(0) is expressed interms of parabolic coordinate eigenfunctions f„and g„. as

(6)

for absorption near the direct edge where e is thestatic dielectric constant, p, is an interband di-pole-matrix element, a is the exciton radius, andQa(0) is the density-of-states function. The inter-band dipole-matrix element p. ,„ is defined in termsof an interband momentum-matrix element Pwhere

p, ,„=(e/m(d) P,„=(e/m(d) (q, I(If/c) v

Iy„).

The g, and g„are the Bloch states for the conductionand valence bands, respectively. From Eq. (6),it is evident that the square of the ratio of the tran-sition dipole moment to the exciton dipole momentdetermines the strength of the exciton absorption.The density-of-states function Q (0), which con-tains all electric-field-induced structure, is di-mensionless and is normalized to approach thelimit [(h&u- E,)/R]"' far above the edge. For elec-tric field 8 = 0, the density-of-states function in-cludes the hydrogenic series below the edge andblends into a continuum above the edge. Thus,from I, in the zero-field limit, Q (0) is given by

40

0.0~'.30-

s20.2~

I I I

(o) I"/R%0. I

2/3h0/R*(K/Ct)

40

30—

42

20—

(b) I /R%0. 2

"e/R (6/6t)

IO—

the zero- and finite-field curves will become clearbelow.

To be able to compare theory with experiment,a phenomenological broadening factor I'- kT mustbe introduced. The author tried both Gaussian-and Lorentzian-type broadenings and found that theLorentzian-type correlated best with experimentaldata. c' The Lorentzian broadening of Eqs. (8a) and(8b) can be accomplished analytically, whereas thebroadening of Eq. (9) must be accomplished numer-ically. Upon broadening Eqs. (8a) and (Sb), thetwo equations may be combined into one for thebroadened zero-f ield curve. The temperature andfield dependence of Q'(0) or e, is shown in Fig. 4.Each figure is for a given value of 1'/R and containsthree curves, one for zero field and two for nonzerovalues of h/hz, the numbers on the curves cor-respond to different values of this parameter. Therelationship between the electro-optical energy h8,the exciton binding energy A, the electric field 8,and the ionization field 8, is simply

@8/R = (8/h, )"'.The consistency of normalization between the

zero- and finite-field curves is fully demonstratedby Fig. 4, in that the oscillations above the edgeare centered on the zero-field curve as they shouldbe. This is somewhat remarkable considering thatthe finite-field curves were calculated numericallyfrom Eqs. (9) and (10), whereas the zero-fieldcurves were calculated analytically from Eqs. (8a)

g 1/3Q'(0) =

@2 f„,(0) g„;(0), (9)

p I I p I I I I

0 I 2 3 4 -5-4-3-2-I 0 I 2 3 4 5(hcu -EII )/R (hcu- E(I) /R

where f„and g„. are solutions of the differentialequations

1 d d t 1 I& fx— + -- i + P-'& =0)x dx dx g

(10)where x

' = ~ + 2(~/&~) '"~

P =(h/h, )-"'[(Z- Z,)/R],

IO I I I I I 1 I I I

(G) I /R%0.5

0 p .'] h8/R%(6/Et)

'2

8 ) ( i

(d) I /R I.p~p

and the a& are the allowed eigenvalues as continuousfunctions of P. The solutions of Eq. (10) and thevarious parameter definitions are fully documentedin I and need not be belabored here. All that needbe said is that Eq. (10) is numerically integrableand that computer programs have been written todo this, The consistency of normalization between

0 I I I 0 I I I I

-10-8-6-4 -2 0 2 4 6 8 IP "I6 -8 0 8 16 24(hcs-K )/R0

(hcu-f )/R0

FIG. 4. Temperature and electric field dependence ofE2 as function of (E —EII)/R for (a) I/R=0. 1, 8/@1=0.0,0.2, and 0.5; (b) I'/R=0. 2, 8/I=0. 0, 0.5, and 1.0;(c) &/R=0. 5, 5/$1=0. 0, 1.0, and 2.0; and (d) I'/R=1. 0,g/fr=0. 0, 2.0, and 5.0.

1386 DANIE L F. BLOSSE Y

-0.0

FIELD DEPENDENCE

0 2 — EXCITON GROUND

-OA—

lp—

0—

IJ) c2-IO—

-20—

I I I I I

02 (a) I /R~O. I2/3

0.5 R (E/Et)

I I I I I I I I

(b) I /R 0.25— tIe/R*(E/E2) 2'3

0-ds2

-lp——0.6—

ex]'R

-0.8

I

-4 -3 -2 - I 0 I

(t)QI-E )/R9

I

2 3 4I

-5 -4

I.6

I I I

-3-2 -I 0 I 2 3 4 5(t)QI-E )/R

9

I I I

-1.0-

I I I I I

0.05 O. l

I

0.2

PERTUR8ATIONTHEORY

I I I 1

0.5 I.O 2.0 5.0

0—

A&2-2-

(c) I"/R =0,5tetR (E/E, ) 0.8—

00-IJ)

20.8-

(d) I'/R ~ I.O

FIG. 5. Field dependence of exciton (hydrogenic)ground-s tate energy.

4—-16-

I I I I I I I I I -2.4"10 "8 "6-4-2 0 2 4 6 8 lp(tIQJ-E )/R9

I I I I

-8 0 8 I6 24('%QJ-E )/R

9

and (Sb) with no adjustable scaling factors. Theconsistency of normalization is also demonstratedby conservation of total oscillator strengths (f sumrule). This consistency is especially importantbecause, to calculate the modulated spectrum, onemust take the difference between the finite- andzero-field spectra.

The field dependence of the exciton (hydrogenic)ground-state energy is shown in Fig. 5. The peakfollows the quadratic Stark shift up to a field ofabout 0. 25 81. A minimum energy is obtained inthe 0. 4$,-0. 581 range and the peak moves to higherenergies above 0. 581. For 8/SJ &1, the peakexists only as the first electric-field-induced os-cillation in the optical density of states. The broad-ening due to the electric field is demonstrated inFig. 7 of I.

B. Field-Induced Changes in &2

Several theoretical excitonic EA spectra, repre-sented by ~&2, are shown in Fig. 6. These curveswere evaluated by taking the difference between thefinite- and zero-field spectra of Fig. 4. As thefield increases, the excitonic EA spectrum expandsas (h/$~)'" above the edge with the first negativeoscillation being pinned at S~ = E~ —R if the excitonis not thermally ionized (I' & A). The temperatureand field dependence of amplitudes and energy sep-arations in ~~2 are shown in Figs. 7 and 8. Fromthe field and temperature dependence of these EAparameters, several predictions can be made: (a)ha should be greater than h, except for the case ofboth small field (h/8, & 0. S) and small broadenings(I"/R & 0. 2); (b) JI, and JI, can increase or decrease withfield whereas h, shouldalways increase withfield;(c) for large fields, JIa should approach a constant val-ue if the exciton is not thermallyionized; (d) when

FIG. 6. Temperature and electric field dependenceof &62 as functionof (E-Eg)/R for (a) I'/R=0. 1, 8/@I=0.2 and 0.5; (b) I'/R =0.2, 8/81 ——0.5 and 1.0; (c)I'/R=0. 5, g/@~=1.0 and 2.0; and (d) I'/R=1. 0, b/hl= 2.0 and 5.0.

IO0.1

, ~O,2IO-

~0.$tip

io'I.O

~O.I

IO o.s~hl

Ip-t-I.O

Ip I

1O-' IOo IOI IO2

SISr

)pl

104-1.0

5Ip-I—

O ~l, J J, 102I I I

IO-' IO4 IO' KP IO-I IOo IOI 1028/Sr Kr%r

FIG. 7. Temperature and electric field dependenceof amplitudes in 4g for ~/R=0. 1, 0.2, 0.5, 1.0, 2.0,and 5.0.

the exciton is neither field or thermally ionized,the first negative oscillation is pinnedathco = E, —R;(e) the width ~I increases slower than (8/Sz)a"and is temperature dependent, whereas the widthnE2 increases as (8/SJ)2" and is largely tempera-ture independent. It should be emphasized thatthese results are for nondegenerate bands. Foroverlapping bands, the effects are generally ad-ditive. 48 The above predictions have been borneout qualitatively by experiment. " A quantitativecomparison between lead iodide EA data and the

WANNIER EXCITON IN AN E LECTRIC FIE LD. II. ~ ~ 138V

(f ~f )+P g

o, 4-

w 2-IL

W~sage 0

"21O-I

O.1-Q.S

l

IO0 10 10s~sz

10 10

I I I I I & I10-(cI) I /R s0.1

ht/%6/6f )

0-

I I I I I I I

(b) P/Rs0. 25- I R/3

0.50-

Idec \

cI

l~-5—

10 l I I )/ I I

10-' 10' 10' 10' 10-' KP 10' 10'&&x KIK

FIG. 8. Temperature and electric field dependenceof energy differences in «2 for ~/R = 0.1, 0.2, 0.5, 1.0,2.0, and 5.0.excitonic EA theory is made in Sec. IID.

C. Field-Induced Changes in g l

Using the Kramers-Kronig relationships, ~~,may be calculated from 4e2; 4e, is important sinceboth b c, and Sea contribute to the ER spectrum. Sev-eral «, curves are shown in Fig. 9. These curvesare calculated, assuming R «E„ from the ~~2curves in Fig. 6. The 4e, spectrum expands ap-proximately as (6/h, ) ', as does the he, spectrum,with the first zero below the edge being pinned at

S(d= 8, —R if the exciton is not thermally ionized.The field and temperature dependence of amplitudesand energy separations in 4c, are shown in Figs.10 and 11. From the field and temperature de-pendence of these 4~, parameters, several predic-tions can be made: (a) h, and h, should be about thesamesizewith ha&h, for highfields; (b) h, is smallrelative to hsand h„(c) h, can increase or decreasedepending on field, whereas h2 always should in-crease with field; (d) hs saturates for 8 &8, andsmall broadenings; (e) for small fields (8/hz & 1)or small broadenings (I"/ff & 0. 5) the first zero ispinned at hco=Ec —8 irrespective of field; (f) thewidths &E, and ~2 increase slightly slower than(8/8, )'~' with AE, approaching this limit at highfields; (g) the width ~I is less temperature de-pendent than &E2 at low fields but is more temper-ature dependent at high fields. In many cases theER spectra are largely composed of ~~„ thus, inthese cases, these results can be directly correlatedwith ER data.

D. Comparison with Experiment: Lead Iodide

For comparison between theory and experiment,the lead iodide EA data of Perov et al. was used.This data was chosen in preference to CdS or44

Ge data" because of the lack of overlapping struc-ture, which in the case of CdS complicates the EAspectrum. Also, PbI2 exhibits a stronger excitonpeak than Ge which should provide a good test foran excitonic EA theory. The fit of both the excitonicand nonexcitonic EA theories to experiment isshown in Figs. 12 and 13. The theoretical curvesin these figures were generated using the param-eters I =kT, T= VV'K, a=6. 25, p. =0. 21m, , andthe electric fields 8 = 8 && 10c V/cm and 2. 85 x10'V/cm, respectively. The energy-gap values usedwere 2. 58 and 2. 48 eV for the excitonic and non-excitonic theories, respectively.

At best, the nonexcitonic theory gives a poor fit.

-4 -5 -2 -I 0 I 2 3(%cu-E ) /R

4

I I I I I I I I I

(c) f'/R ~0.52— /6 )2/j

I

0-

-2—

1.6

0.8-

I I I I

(d) r/R- lu&\I I

0.0-de,-0.8—

-10-I « I I

4 -5-4-5 "2 -I 0 I 2 5 4 5(%cs-E4) / R

IO I

O.lI-»lP 0.2~-

0.$~10

l 0'2.0

lQ 5.06«2 I I

lp-' lpo Ol

6/Kz

lp

lQ2

-4-

-10-8 "6 -4-2 0 2 4 6 8 10(%cu - E ) /R4

"16-

I I I I

-16 -8 0 8 16 24(hcu-E )/R4

FIG. 9. Temperature and electric field dependenceof dc& as function of (E Ec)IR for (a) f'IE=0.—1, Elh q

=0.2 and 0.5; (b) ~/R=0. 2, 8/Bz=0. 5 and 1.0; (c)I'/R=0. 5, 8/Br=1. 0 and 2.0; and (d) &/R=1. 0, ~/$1=2.0 and 5.0.

O I-%g

!lp4 lp' lp2

6%z

IO0.2

Lo~~2.0 5

I6 lp" I lpO lplSitz

lp2

FIG. 10. Temperature and electric field dependenceof amplitudes in 4&l for I'/R=0. 1, 0.2, 0.5, 1.0, 2.0,and 5.0.

1388 DANIE L F. BLOSSEY

Io

~ Io

4Jjoo

0

~-4fhE I

4—

~~2—IN

hl oO.l "0,5

2 I I

IO-' IO' IOI

E/E z

Io

~ IO

~AI

IO4

IO

tuality, the electron and hole are not point particlesand can overlap if they are in the same unit cell.This effect causes the potential well to have a bot-tom, and thereby causes deviations from the hydro-genic series. The greatest weight of the fit wasplaced on the first three half-oscillations whichwould cause a difference of phase in the oscillationsabove the edge if the exciton series is not purelyhydrogenic. From Fig. 13, the exciton peak inPbI~ seems to be about 0.02 eV higher than thatwhich would be predicted by the hydrogenic model.

III. DISCUSSION

IO"I I0 ' I

IO-' IO4 IO' IO' IO-' IOo IO' Io~6/Sg

FIG. 11. Temperature and electric fieM dependenceof energy differences in &&~ for ~/8=0. 1, 0.2, 0.5, 1.0,2.0, and 5.0.

There are many points of discrepancy between thenon-excitonic theory and experiment: (a) the calcu-lated energy gap is lower in energy than the firstexciton absorption peak (a ridiculous result); (b)the oscillations are too narrow for the smaller fieldand too wide for the larger field: (c) the oscilla-tion amplitudes are not damped sufficiently abovethe edge; (d) the first negative half-oscillation istoo narrow and deep to fit with the nonexcitonic EAtheory. As is clearly demonstrated by Figs. 12and 13, the excitonic EA theory gives a much bet-ter fit to PbIz EA data than the nonexcitonic EAtheory. The nonexcitonic EA theory is treated inthe Appendix as the thermally ionized limit of theexcitonic EA theory.

Considering the simplicity of the exciton model,the fit of the excitonic EA theory to experiment isquite good. Using these parameters, the ioniza-tion field as defined by Eq. (5) is calculated to be4. 67x10' V/cm in which case h/Sz=0. 17 and 0. 61for the two figures. For comparison between theo-ry and experiment, 42 cur ves for diff erentvalues of electric field ranging from 8/8, = 0. 079 to1000.0 have been calculated and placed in directaccess storage on the computer. The theoreticalcurves in Figs. 11 and 12 correspond to values of8/Sz = 0. 16 and 0. 63, respectively, since these val-ues of 8/8, were the closest of the 42 stored curvesto the actual values of 8/h, . The general featuresof the EA data are reproduced by the theory. Forexample, in the case of the lower field (8 x 104

V/cm), 8, is almost two times h~; whereas, in thecase of the higher field (2. 85x 10' V/cm), h, is lessthan h3. In both cases, the magnitudes of h„h&,and h, are in the right proportions to fit the theoryto the experimental data. The oscillations abovethe edge are not quite in phase which may be due tothe oversimplification of the electron-hole interac-tion as being simply a Coulomb potential, In ac-

t I I I

~~ PbI&, 8.0)fIO V/Cm—EXCITONIC EA THEORY

——NONEXCITONIC EA THEORY

l'I ~I

I

I',

I

It( I'

I l I I

2.44 2.54 2.64 2.74PHOTON ENERGY IN eV

0.8—

0.0

-0.8—

—I.62.34 2.84

FIG. 12. Fit of excitonic and nonexcitonic electroab-sorption theories to lead iodide data of Perov et al.(Ref. 46) for electric field of 8X10 V/cm at 77'K.

If the excitonic and nonexcitonic (single-particle)EA theories are to be compared, there are severalpoints that need clarification. Since the EA spec-trum is the difference between a finite-field spec-trum and a zero-field spectrum, the EA spectrumcontains structure from both spectra. In the caseof the nonexcitonic EA theory, the zero-field spec-trum contributes little structure except at the fun-damental edge where a singularity inc~ occurs. In

contrast, the zero-field exciton spectrum can con-tain many absorption peaks corresponding to thebound exciton states. This structure is absent onlyif the thermal broadening is much greater than theexciton binding energy, in which case the nonexci-tonic EA theory is applicable (see the Appendix).The electric field induces spectral shifts and broad-enings of any structure in the zero-field spectrumand causes electric-field-induced oscillations in theoptical density of states above the edge. The elec-tron-hole interaction enhances these oscillationsnear the edge and causes a phase shift in energythere. The nonexcitonic EA spectrum expands as

whereas only the part of the excitonic EAspectrum above the edge follows this power law.Below the edge, the excitonic EA spectrum is dom-inated by the structure in the zero-field spectrum.The excitonic EA theory introduces an additional

1.6%IO

WANNIER EXCITQN IN AN E LECTRIC FIELD. II. ~ ~ 1389

4x IO I I I

~ ~ ~ Pbl2, 2.85x IO V/cm

EXC I TON I C EA THEORYNONEXCITONIC E A THEORY

~ /r .g.

(1I:('

l, ~ II

2— /~ /

ha, cm"I

Eg

I ( l ( I

2.44 2.54 2s64 2.74 2.84PHOTON ENERGY IN e'V

FIG. 13. Fit of excitonic and nonexcitonic electro-absorption theories to lead iodide data of Perov et al.(Ref. 46) for electric field at 2.85x10~ V/cm at 77'K.Both theoretical curves in this and the previous figureused the parameters I.'=kT, E~=2. 58 eV, and an elec-tron-hole-pair reduced mass p = 0.2lme.

42s34

complication in characterization of the EA spec-trum, i. e. , the single-particle EA theory can becharacterized by the energy parameters I', thethermal broadening energy SO, the electro-opticalenergy, and E~, the energy gap, whereas the exci-tonic EA theory also includes R, the binding energyof the exciton.

In this theory, the electron-hole interaction istreated as a Coulomb potential between point parti-cles. It has already been pointed out in the text thatthis potential tends slightly to overestimate the ex-citon binding energy owing to the fact that the parti-cles are actually charge densitites covering entireunit cells of the crystal and not point particles. Butto make the problem tractable certain simplifyingapproximations must be introduced, such as theCoulomb potential for the electron-hole interactionand effective masses for the band structure. Evenwith these approximations the agreement betweentheory and experiment is quite good. The reducedmass of the electron-hole pair JLI, along with staticdielectric constant e of the solid combine to definethe exciton binding energy R and the exciton radiusa. The exciton radius in lead iodide is about 16 A,thus making it a marginal case for study by Wannierexciton theory which requires that the exciton spanseveral unit cells of the crystal. Even so, a goodfit was obtained. From the best fit to EA data fortwo different values of electric field, the parametersE~=2. 58 eV, 1(L=0. 21m, were obtained for & =6. 25and I' = k T, where T = 77 'K for lead iodide whichare in reasonable agreement with previous valuesof 2. 55 eV and 0. 24m, .

In summary, the electric field and temperaturedependence of excitonic EA is presented in graphicalform for direct band-gap semiconductors and in-sulators. Each excitonic EA spectrum is character-

d2Ai(z) = zAi(z) (12)

whose integral representation is given by'

Ai(s)= —f dss'""" '"'" sos(-', ass+-', s).(13)

0.2I

(a) heq (F8) '

O.I—

"o.l—

-O.2 ~4I I

-2 0(E-E )/t89

0.2I

(b) a, (%8) "'O.l—

"O.l—

-O.2 -4 -2 0(E" gh8

FIG. 14. Electroabsorption spectra for thermallyionized excitons, I'»R: (a) b&& vs (E- E~)/8'~ for ~/S'&= 0.1 and 0.3; (b) 4&~ vs (E- E~)/Se for ~/S& = 0.1 and0.3.

ized in terms of ratios of three parameters R, I',and 58, the magnitude of each relative to one anotherdetermining which, if any, of the competitive forcesdominate. In only one case does the excitonic EAtheory agree with the nonexcitonic theory —theoscillations in the optical density of states above theedge expand as 8 . This theory explains many ofthe features of measured EA spectra which are in-consistent with the nonexcitonic EA theory such astemperature-dependent and field-independent oscil-lations and hitherto anomolous dependencies of theamplitudes of the oscillations on field. 43

APPENDIX: ELECTROABSORPTION BYTHERMALLY IONIZED ELECTRONS

In the limit of large thermal broadening of the ex-citon lines I'» R, the excitonic EA spectrum ap-proaches the nonexcitonic EA spectrum. Thebroadened nonexcitonic EA spectrum may be ex-pressed in terms of complex Airy functions ' which

are Bessel functions of fractional order. Thecomplex Airy function is the analytic continuationof the convergent real-axis solution of the equation

1390 DANIE L F. B LOS SE Y

For completeness, a second solution to Eq. (12) isBi(z) whose integral representation is given by'

Bi(z) = — dse ' '" "~'" sin( —,'W3sz+-,'v)77

p

+ — Qse ss-(1/ 3)s (14)0

The complex functions Ai(z) and Bi(z) are interre-lated by the formula'

Ai(z) +iBi(z) = 2e"~ Ai(ze 'z~ ).

For EA calculations, we have that in the limit ofl»R

P'(0)-(&/8, )' 2v Re[e " 'Ai'(z)Ai'(ze '~ ')

+ze '" 'Ai(z)Ai(ze '"~')](16)

for finite fields and broadenings where Ai'(z) =

(d/dz)Ai(z) and the zero-field limit is given by

where hen is the photon energy, E~ is the energygap, R is the exciton binding energy, and I' is thethermal broadening energy. The complex param-eter z may be expressed intermsofphysicalparam-eters as

z = (h/8, )-"' [(E, -@~)+f1]/ft= [(E, -a~)+ f1]/ge,

where AO is the electro-optical energy. The mod-ulated spectrum &&2 representing the difference be-tween Egs. (16) and (17) is shown in Fig. 14 with itsKramers-Kronig transform ~c, for two values of1/h6. The energy coordinate expands as 8 "andthe first negative oscillation in 6~2 occurs abovethe gap in contrast to the limit l &R, where thefirst negative oscillation is pinned at Aco=E, —R.This theory was used to generate the nonexcitoniccurves in Figs. 12 and 13.

'D. F. Blossey, Phys. Rev. B 2, 3976 (1970).C. B. Duke and M. E. Alferieff, Phys. Rev. 145,

583 (1966).H. I. Ralph, J. Phys. C 1, 378 (1968).J. D. Dow and D. Redfield, Phys, Rev. B 1, 3358

(1970).5D. F. Blossey, Bull. Am. Phys. Soc. 14, 429 (1969).6D. F. Blossey, Ph. D. thesis, University of Illinois,

Urbana, Ill. , 1969 (unpublished).R. Enderlein, Phys. Status Solidi 26, 509 (1968).J. D. Dow, Phys. Status Solidi 34, K71 (1969).

BC. M. Penchina, J. K. Pribram, and J. Sak, Phys.Rev. 188, 1240 (1969).

The term electroabsorption as it is used here refersto the electric-field-modulated spectrum which is thedifference between the finite- and zero-field spectra.Reviews on the various aspects of modulated spectros-copy may be found in M. Cardona, in Solid State Physics,edited by F. Seitz, D. Turnbull, and H. Ehrenreich(Academic, New York, 1969), Suppl. 11; B. O. Seraphin,in Semiconductors and Semimetals, edited by B. K.Willardson and A. C. Beer {Academic, New York, 1970)Vol. VI; D. E. Aspnes and ¹ Bottka, ibid. , Vol. VI;D. F. Blossey and P. Handler, ibid. , Vol. IX.

"D. A. Dows and A. D. Buckingham, J. Mol. Spectry.12, 189 (1964); A. D. Buckingham and D. A. Ramsay,J. Chem. Phys. 42, 3721 (1965); N. J. Bridge, D. A.Haner, and D. A. Dows, ibid. 44, 3128 {1966);48, 4196(1968); D. A. Haner and D. A. Dows, ibid. 49, 601(1968); B. M. Conrad and D. A. Dows, J. Mol. Spectry.32, 276 (1969).

G. H. Wannier, Phys. Rev. 52, 191 (1937).~3G. Dresselhaus, Phys. Chem. Solids 1, 14 (1956).'4R. J. Elliott, Phys. Bev. 108, 1384 (1957).

R. J. Elliott, in Polarons and &xcitons, edited byC. G. Kuper and G. D. Whitfield (Oliver and Boyd,London, 1963), pp. 269 ff.

~6J. Frenkel, Phys. Rev. 37, 17 (1931);37, 12760.931).

' R. E. Peierls, Ann. Physik 13, 905 (1932).'8A. W. Overhauser, Phys. Rev. 101, 1702 (1956).'9A. S, Davydov, Theory of Molecular &xcitons,

translated by M. Kasha and M. Oppenheimer, Jr.

(McGraw-Hill, New York, 1962).R. S. Knox, in Solid State Physics, Theory of &x-

citons, edited "& F. Seitz and D. Turnbull (Academic,

New York, 1963) Suppl. 5, .J. O. Dimmock, in Optical Properties of III-V Com-

pounds, edited by R. K. Willardson and A. C. Beer(Academic, New York, 1966), Vol. 3, pp. 259 ff.

B. A. Smith, Semiconductors (Cambridge U. P. ,Cambridge, England, 1961), pp. 347, 350, 367, and410.

C. Hilsum, in Semiconductors and Semimetals,Physics of III-V Compounds, edited by B. K. Willardsonand A. C. Beer (Academic, New York, 1966), Vol. 1,p. 9.

Reference 21, p. 314.250. Madelung, Physics of III-V Compounds (Wiley,

New York, 1964), p. 101.B. Segall and D. T. F. Marple, in Physics and

Chemistry of II-VI Compounds, edited by M. Aven andJ. S. Prener (Wiley, New York, 1967) pp. 335 and 344.

C. Lanczos, Z. Physik 62, 518 (1930); 68, 204 (1931).H. Rausch von Traubenberg and R. Gebauer, Z.

Physik 54, 307 (1929); 56, 254 (1929); 62, 289 (1930); 71,291 (1931).

BJ. R. Oppenheimer, Phys. Bev. 31, 66 (1928).H. A. Bethe and E. E. Salpeter, Quantum Mechanics

of One and Takeo Electron Atoms (Academic, New York,1957).

'W. Franz, Z. Naturforsch. 13, 484 (1958).32L. V. Keldysh, Zh. Eksperim. i Teor. Fiz. 34, 1138

(1958) [Soviet Phys. JETP 7, 788 {1958)].B. Williams, Phys. Bev. 117, 1487 (1960).

34T. S. Moss, J. Appl. Phys. Suppl. 32, 2136 (1962).3~K. Tharmalingham, Phys. Rev. 130, 2204 (1963).J. Callaway, Phys. Bev. 130, 549 (1963); 134, A898

(1964).3~W. V. Houston, Phys. Rev. 57, 184 (1940); E. ¹

Adams, J. Chem. Phys. 21, 2013 (1953); E. O. Kane,J. Phys. Chem. Solids 12, 181 (1959); P. W. Argyres,Phys. Rev. 126, 13Q6 (1962).

D. E. Aspnes, P. Handler, and D. F. Blossey,Phys. Rev. 166, 921 (1968).

SA. Frova and P. Handler, in Proceedings of the

WANNIER EXCITON IN AN ELECTRIC FIELD ~ 1391

International Conference on the Physics of Semiconductors,Paris, 1964, edited by M. Huleir (Dunod, Paris, 1964),p. 157.

B. O. Seraphin, in Proceedings of the InternationalConference on the Physics ofSemiconductors, Paris,1964, edited by M. Hulin (Dunod, Paris, 1964), p. 165.

4 D. E. Aspnes, Phys. Rev. 147, 554 (1966); 153, 972(1967).

42Y. Hamakawa, F. A. Germano, and P. Handler, J.Phys. Soc. Japan Suppl. 21, 111 (1966).

43Y. Hamakawa, F. A. Germano, and P. Handler,Phys. Rev. 167, 703 (1968).

44B. B. Snavely, Solid State Commun. 4, 561 (1966);Phys. Rev. 167, 730 (1968).

4~C. Gahwiller and G. Harbeke, Phys. Rev. 185, 1141

(1969).4'P. I. Perov, L. A. Avdeeva, and M. I. Elinson,

Fiz. Tverd. Tela 11, 541 (1969) [Soviet Phys. SolidState 11, 438 (1969)].

4~Y. Toyazawa, Progr. Theoret. Phys. (Kyoto) 20,53 (1958). A Lorentzian shape is predicted for weakexciton-lattice coupling and a Gaussian shape for strongcoupling.

48P. Handler, S. Jasperson, and S. Koeppen, Phys.Rev. Letters 23, 1387 (1969).

H. A. Antosiewicz, in Handbook of MathematicalFunctions, edited by M. Abramowitz and I. A. Stegun(U. S. Department of Commerce, Natl. Bur. Std. ,Washington, D. C. , 1964), Appl. Math Ser. 55, 446 ff.

5 D. F. Blossey (unpublished).

PHYSICAL REVIEW B VOLUME 3, NUMBER 4 15 FEBRUARY 1971

Nonpairwise Interactions and Vacancy Formation Energies in Simple Molecular Solids~

S. D. DrugerDePartment of Physics, College of William and Mary, Williamsbmg, Virginia 23185

(Received 10 June 1970)

A method is given for summing all orders of nonpairwise contributions to the van der Waalsinteraction energy of a substitutional impurity in a monatomic molecular crystal using a Lorentzoscillator model as an approximation. The nonpairwise contribution to the energy of removingan atom from the lattice is obtained as a limiting case. Applied to rare-gas solids, the modelsuggests vacancy formation energies reduced from the two-body values, but insufficiently togive complete agreement with experiment.

I. INTRODUCTION

Calculations using two-body model potentialsfor the interactions between rare-gas atoms yieldthird vjrial coefficients in disagreement with ex-periment, and predict vacancy formation energiesin solid argon and krypton about equal to the co-hesive energy per atom, while observed ther-mal vacancy concentrations'6 suggest much smallervalues. Also, the two-body potentials generallyemployed predict stability for the hexagonal close-packed (hcp) solid relative to the face-centeredcubic (fcc), ' contrary to observation. The pos-sibility of explaining these apparent discrepanciesbetween theory and experiment in terms of non-pairwise contributions to the cohesive energies ofrare-gas solids has been widely examined.

Jansen et al. consider nonpairwise interactionsinvolving three-atom electron exchange in theoverlap region; they obtain rather large three-bod;- contributions (25% of the cohesive energy)that decisively stabilize the fcc lattice, reducethe two-body vacancy formation energy by as muchas 47%, '0 and reportedly produce large relaxationsaround vacancies. " Swenberg' ' points out, how-ever, that Jansen's effective one-electron Gaus-sian wave functions give unrealistically large

nearest-neighbor overlaps, and that Gaussianswith more realistic width parameters produce neg-ligible three-body effects in Jansen's theory.Other possible difficulties with Jansen's approachhave also been discussed. '

Nonpairwise contributions to the van der Waalsinteraction energy alone have also been considered.When interatomic overlaps are neglected and amultipole expansion used, the van der Waals in-teractions among rare-gas atoms are found to bepairwise additive to second order in perturbationtheory, "and a three-body "triple-dipole" inter-action arises in third order. " ' The triple-dipoleinteraction favors the fcc structure, but insuf-ficiently to decisively stabilize it, ' and Burton'shows that it decreases the vacancy formationenergy in solid argon, although insufficiently togive agreement with experiment. The triple-dipoleinteraction also reduces the discrepancies betweenobserved and calculated third virial coefficientsin gaseous argon and krypton, ' ' ' and its pos-sible effects on other rare-gas properties havealso been considered. Present's calculationsuggests that this interaction might be much moresignificant than three-body interactions arisingfrom overlap and exchange at normal lattice sepa-ration.