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

Behavior of the Electronic Dielectric Constant in Covalent and Ionic MaterialsS. H. Wemple and M. DiDomenico, Jr.

Bel/ Telephone Laboratories Incorporated, Murray Hill, Nege Jersey 07974(Received 30 July 1970)

Refractive-index dispersion data below the interband absorption edge in more than 100 widelydifferent solids and liquids are analyzed using a single-effective-oscillator fit of the formn 1=E&Ep/(Ep 5 ct) ) wherekcois thephotonenergy, Epis the single oscillator energy, andEz is the dispersion energy. The parameter E, which is a measure of the strength of inter-band optical transitions, is found to obey the simple empirical relationship E& =PN, Z~N~, whereN~ is the coordination number of the cation nearest neighbor to the anion, Z, is the formalchemical valency of the anion, N, is the effective number of valence electrons per anion(usually N~=s), and P is essentially two-valued, taking on the "ionic" value P; =0.26+0. 04 eVfor halides and most oxides, and the "covalent" value P, =0.37 +0.05 eV for the tetrahedrallybonded A B zinc-blende- and diamond-type structures, as well as for scheelite-structureoxides and some iodates and carbonates. Wurtzite-structure crystals form a transitional groupbetween ionic and covalent crystal classes. Experimentally, it is also found that Ez does notdepend significantly on either the bandgap or the volume density of valence electrons. Theexperimental results are related to the fundamental &2 spectrum via appropriately definedmoment integrals. It is found, using relationships between momentintegrals, thatforaparticu-larly simple choice of a model &2 spectrum, viz. , constant optical-frequency conductivity withhigh- and low-frequency cutoffs, the bandgap parameter E~ in the high-frequency sum ruleintroduced by Hopfield provides the connection between the single-oscillator parameters (Ep,Ez) and the Phillips static-dielectric-constant parameters (E~, Se&), i.e. , {h~&) =E,EandE& = E+Ep, Finally, it is suggested that the observed dependence of Ed on coordination numberand valency implies that an understanding of refractive-index behavior may lie in a localizedmolecular theory of optical transitions.

I. INTRODUCTION

The fundamental electronic excitation spectrumof a substance is generally described in terms of afrequency-dependent complex electronic dielectricconstant e((d) = a(((d)+te2((d). Either the real part(&&) or the imaginary part (ez) contains all the de-sired response information, since causality argu-ments relate the real and imaginary parts via thewell-known Kramers-Kronig (K-K) relations, i.e. ,

2 ~ (d e2((d )'E(((d) 1 = (P &p g d(dg (d (d

0

2(d " e(((d') 1i2 a

0

where g denotes the principal part. In materialsexhibiting a bandgap, the real part in the region oftransparency below the gap is related to optical ab-sorption above the gap by

&(((d) 1=n ((d) 1=(P',2 2 d(d(d e2((d )(dt (2)

where &, is the threshold frequency, and we haveidentified a& with the square of the refractive indexn. The frequency & is assumed to lie above all lat-tice vibrational modes, i.e. , only electronic ex-citations are being considered. In the language ofE(l. (2), a theory of the refractive index would en-

tail computation of &~((d) followed by integration overall frequencies. Although a formal perturbationtheory procedure for computing &3((d) exists withinthe framework of the one-electron band theory ofsolids, such computations require integration overthe full Brillouin zone as well as over all frequen-cies, so that important physical quantities tend tobe obscured by the computational details of theanaylsis. Furthermore, several adjustable param-eters are generally introduced into energy-bandcalculations, and in the more ionic materials exci-tonic effects are difficult to account for quantita-tively. It is useful, therefore, to approximate thegeneral theoretical expressions for e, ((d) in waysthat display explicitly certain physically meaningfulparameters. These parameters depend on the par-ticular approximation being made. Phillips' andVan Vechten, ' for example, make use of the Pennmodel description of the static electronicdielectric constant to define an "average energygap" E~, whereas we use a single-oscillatordescription of the frequency-dependent dielectricconstant to define a "dispersion-energy" parameterE~ . These parameters are useful because they turnout to obey remarkably simple empirical rules in largegroups of materials. Although these rules are quitedifferent in detail, one common feature is the over-whelming evidence that both crystal structure andionicity influence the refractive-index behavior ofsolids in ways that can be simply described. This

1338

BEHAVIOR OF THE ELECTRONIC DIELECTRIC CONSTANT 1339

Using time-dependent perturbation theory, thefollowing formal expression for the frequency (+)dependence of the real part of the electronic dielec-tric constant can be derived:

e, ((o) =1+, Q d k,. " p.7T m ] g (0]) -(d (3)

Here e and m are, respectively, the electroniccharge and mass. The sum extends over all bandsi and j such that i& j, and the integral extends overthe volume of the Brillouin zone (BZ). The inter-band oscillator strength for polarization directionn is given by f,&(k). We shall consider two ap-proximations to Eq. (3) which contain parametersthat can be measured experimentally. The firstis the zero frequency or static-electronic dielec-

experimental observation should be contrasted withexisting theories for which any simple connectionbetween the final result (i. e. , the dielectric con-stant) and crystal structure or ionicity tends to beobscured by the analysis.

In the present article, our purpose is to expandthe analysis of refractive-index behavior reportedpreviously. We show how the single-oscillatordescription is a "natural" approximation to the di-electric response function and compare in detail thesingle-oscillator and Penn-model parametrizationwith emphasis on the relationships between momentsof the e, (ar) spectrum and the model parameters.Tables of experimental oscillator parameters arepresented for nonmetallic ionic and covalent semi-conductors and insulators, magnetic insulators, andliquids. These tables enable us to extend the em-pirical rule reported previously, connecting thedispersion energy Ewith coordination number,valency, and ionicity, to a wider range of mater-ials.

Finally, we discuss the implications of the ex-perimental results on our understanding of elec-tronic structure and chemical bonding in solids andliquids. Within the framework of the constant con-ductivity model, it is shown that the dispersion en-ergy E~ is simply related to &,(v) at high frequenciesabove the e2(v) cutoff via the f sum rule and theHopfield sum rule. As a result, E is related tothe charge distribution within each unit cell and is,thus, a quantity closely related to chemical bonding.The observed simple dependence on coordinationnumber and chemical valency suggests further thatnearest-neighbor atomiclike quantities strongly in-fluence the electronic optical properties of mater-ials, and that an understanding of these propertiesas well as bonding may lie within a riearly localizedmolecular theory.

II. SINGLE-OSCILLATOR DESCRIPTION OFELECTRONIC DIELECTRIC CONSTANT

tric constant, and the second is the frequency dependence of the dielectric constant in the regionof transparency, i. e. , ~& v;&.

Penn has shown that the static dielectric con-stant of a semiconductor can be computed using anisotropic free-electron model containing a singleenergy gap E~ . Apart from a factor of order unity, wecan obtain his result from Eq. (3)by letting Ke&& = E, ,where E is an average energy gap. Equation (3)then becomes

2g2 f~l(0)=1+ 8 8 & d'kflJ(k)

7F m g $ g gg(4)

Making use of the f sum rule and noting thatZ f d k=4mn,

el(&)=I+&p+ I a s) ~ (8)Equation (8) has the same form as the classicalKramers-Heisenberg dispersion formula for anassembly of weakly interacting atoms. In Eq. (8),f is the electric-dipole oscillator strength associ-ated with transitions at frequency &. The sum-mation over oscillators vcan be sensibly approxi-mated for ~ & ~by isolating the first (strong) oscil-lator f,/(u&l + ) and combining the remaining terms

where n is the effective density of valence elec-trons, Eq. (4) reduces to

e, (0) = 1+ (If(u~) /Eg . (8)Here &u~ = 4vnea/m is the plasma frequency of thevalence electrons. Phillips and Van Vechten' haveshown that the average energy gap E~ can be sensi-bly decomposed into homoyolar and heteropolar partsEand C obeying the quadrature relation E = E+Cand that new and useful scales of ionicity and elec-tronegativity can be defined within the frameworkof the single-gap dielectric model. Furthermore,these authors have also shown that a remarkablywide range of experimental observations in manymaterials can be correlated using a "small number"of adjustable parameters associated with E~. '

To calculate the frequency dependence of the di-electric constant we note that for a single group ofvalence and conduction bands Eq. (3) can be rewrit-ten as

el(+) 1+ Q g 24ve~ f,"(k)5%0 g (O,k -(0where 0 is the volume of the crystal, and c and vdenote conduction and valence bands. If we approxi-mate the important interband transitions in the BZby individual oscillators and recognize that eachvalence electron contributes one such oscillator,Eq. (I) may be approximated by

1340 S. H. WEMP I E AND M. DiDOMENICO, JH.

l.36l.32

l.28

I 24

l.20I

I. I 6

I. I 2

I,08I.04

I I I I I I I

4 8 l2 I6 20 24 28 32I/X (MICRONS)

FIG. 1. Plot of refrective-index factor (n 1) 'versus ~- for NaF.

I.OO0

in the form

(f /& ) (1+ & /&, )Combining these higher-order contributions with thefirst-resonant oscillator and retaining. terms to or-der co then yields the single-oscillator approxima-tion

e, ((u) -1= F/[E,'(her)'], (9)

n (sr) 1 =E~E0/[ED- (Nv) ] . (10)Experimental verification of Eq. (10) can be ob-

tained by plotting 1/(n 1) versus va ( or A. ~). Theresulting straight line then yields values of the pa-rameters Ea and E~ . Some typical results are shownin Figs. 1-3for NaF, BazNaNb, O, and CdS. Thesecurves illustrate the general features observed ina total of over 100 materials investigated to date.At long wavelengths, a positive curvature deviationfrom linearity is usually observed due to the nega-

where the two parameters E0 and E are relatedstraightforwardly to all the fand &u in Eq. (8).Equation (9) provides a two-parameter approxima-tion at low energies co & co, to, the theoretical resultexpressed by Eq. (3). The usefulness of Eq. (9)depends, of course, on affirmative answers to thefollowing two questions: (i) Do real solids obey thesingle-oscillator approximation with "reasonable"accuracy; and (ii) do the experimentally observedvalues of the parameters E0 and E (or some com-bination) provide new insights into the optical pro-perties of matter'P In an earlier paper, 3 we assert-ed that a positive answer could be given to the firstquestion for more than 50 widely different ionic andcovalent nonmetallic crystals, and we showed ex-perimentally that a special combination of param-eters (E~ =F/E0) obeyed an extraordinarily simpleempirical relation for this same large group ofmaterials. In terms of the dispersion energy E,Eq. (9) can be rewritten in the form

tive contribution of lattice vibrations to the refrac-tive index. At short wavelengths, a negative cur-vature deviation is sometimes observed due to theproximity of the band edge or excitonic absorption.The largest deviations occur when strong excitonpeaks are present below the interband edge, as inCdS (see Fig. 3). However, in all materials studied(both solids and liquids), a sufficiently extendedregion of linearity is observed to allow unambiguousexperimental determination of the two parametersE0 and E, . In the worst case (CdS), linearity ex-tends over a factor of 4 in co, and the quantity(n -1) ' is depressed approximately 15% below thelinear extrapolation close to the absorption (exci-ton) threshold.

In addition to the single-oscillator parametriza-tion given by Eq. (10), many other curve-fittingforms involving three or more parameters havebeen used in the literature to describe refractive-index dispersion data. In general, no physical signif-icance has been attached to the parameters, andthe expressions serve primarily as interpolationformulas. Our justification for using Eq. (10),apart from its experimental validity, is the a pos-teriori argument that the parameters obtained havefundamental physical significance.

0.26

0.25

0.24

0.23

0.22

0.2 I

0.20

O. I 9

O. I8

O. I 70I I I

3 4 5I/X (MICRONS)

FIG. 2. Plot of refractive-index factor (g 1) ~versus A, ~ for Ba&waNb&0&5.

III. RELATIONSHIP OF PARAMETERS TOFUNDAMENTAL e2(u) SPECTRUM

A simple connection between the single-oscillatorparameters E~ and E~ and the ez(v) spectrum can beobtained by equating Eq. (10) with Eq. (1) and com-paring terms in an expansion in powers of & . Theresulting relationships can be compactly expressedin terms of moments of the e2(e) spectrum. We de-fine the Hh moment of the optical spectrum by therelation

BEHAVIOR OF THE ELECTRONIC DIELECTRIC CONSTANT 1341

where E=- S~, and E, is the absorption thresholdenergy. It should be noted that the definition ofmoments given here differs somewhat from the def-inition used in Ref. 3. The parameters Ep and E~are then given by

and

Eo M g/M q

E=M g/M q .

(12)

The oscillator energy Ep is independent of the scaleof E& and is consequently an "average" energy gap,whereas Edepends on the scale of &&, and thusserves as an interband strength parameter. Sincethe -1 and -3 moments are involved in computationof Ep and E, the && spectrum is weighted mostheavily near the interband absorption threshold.It is instructive to compare Eqs. (12) and (13) withcorresponding expressions for the quantities Sco~and E, entering into the Phillips-dielectric model.Using the f-sum-rule integral and the K-K relation,it is easy to show that

and

(a~,)'= M, ,

Eg Mg/M g .

(14)

0.24

0.25

0.22

0.2 I

0.20I

O. I9

O. I 8

O. I7

O. I 6

0000

O. I5

O. I 40I I

2I/X (MICRONS)

FIG. 3. Plot of refractive-index factor (n 1)"versus ~ for CdS.

The Phillips "bandgap" E~ is, thus, simply theratio of the +1 to the -1 moments, and consequentlyweights the e& spectrum at higher energies moreheavily than the corresponding expression for Epgiven by Eq. (12). Interband transition strengthsare described by the plasma energy M& in thismodel, whereas the single-oscillator model usesthe quantity M, /M

~

to describe transition strengths.Hopfield' has recently derived a new sum rule re-

lating optical properties to the charge distribution

within a unit cell. He introduces the energy-gapparameter E, defined by the moment relation

E, = Mg/Mg, (16)

A third parametrization of refractive-index be-havior that has gained wide acceptance is theClausius-Mossotti local-field polarizability model.This description introduces physicaDy appealingquantities (i. e. , the Lorentz local-field factor I'and the electronic polarizability o!), but thesequantities are not separately measureabl, in gen-eral, nor can they be computed from fundamentaltheories. In terms of ~ and I", the zero-frequencyrefractive index can be expressed in the form

n~(0) 1 = 4mNQ n (1 N Q I', n, ), (18)where a; and I'& are the electronic polarizabilitiesand local-field factors for the ith atom in each unitcell, and N is the volume density of unit cells. Fora cubic array of isolated ions, I'= ~~r, and Eq. (18)reduces to the well-known Lorentz-Lorenz form

(n' I)/(n'+ 2) =~

mNuo,

where ao=g& o, & . In most solids neither the assump-tion of cubic symmetry nor isolated ions is valid.In the covalent limit, extreme overlap between near-est-neighbor wave functions gives 1"-0. ' TheLorentz factor is, thus, clearly related in some wayto ionicity as well as structure. There is, however,no fundamental theory that allows quantitative de-termination of I' in solids that are not strongly ion-ic. For this reason a value for 1" is often assumed,and the total polaribility ap is then determined ex-perimentally from Eq. (18). Such a procedure isuseful provided polarizabilities are additive andprovided they have essentially the same value foreach ion when placed in a variety of crystalline andchemical environments. This turns out to be limitedto certain families of ionic crystals, so that Eq.(18), apart perhaps from giving useful qualitativeinsights, does not yield parameters that can alwaysbe unambiguously measured nor are the parame-ters necessarily applicable to large groups of crys-tals. It is of interest to note in passing that a sin-

and shows that this energy is related to an integralof the product of the Laplacian of the bare crystalpotential and the fluctuation in the electron density.The symmetry in the expressions for Ep, E, , and E, isevident in Eqs. (12), (15), and (16), as is the in-equality

E &E &EOther "average" energy gaps can be defined by thegeneral relation E=M/Malthough thus faronly the gaps Ep, E~, and E, have been found useful.IV. OTHER PARAMETRIZATIONS OF REFRACTIVE INDEX

1342 S. H. WEMP LE AND M. D iDOM E NI CO, JR.gle-oscillator description of no in the Lorentz-Lorenz expression [see Eq. (19)] cannot be distin-guished experimentally from a single-oscillator de-scription of n -1.

Another parametrization of refractive-index datathat is sometimes used involves normalization ofvarious functions of the refractive index [e.g. , n-1or (n' - I)/(n + 2)] by the density. Although a varie-ty of empricial rules are observed to apply to groupsof complex materials, it is difficult to ascribe phys-ical significance to the resulting parameters. Thereader is referred to the book by Batsonov for adetailed presentation of this "refractometry" ap-proach.

V. EXPERIMENTAL RESULTS

Using available refractive-index-dispersion data,we have computed the single-oscillator parametersEo and E, for over 100 solids and liquids. The re-sults, listed in Tables I-IV are grouped into thefollowing four categories: Table I, nonmagneticcrystals containing a single anion species; TableII, nonmagnetic crystals containing anion radicals;Table III, magnetic crystals; and Table IV, liquids.For brevity we have included values of Eo and Efor only one direction of light polarization in uni-axial and biaxial crystals. In most cases, E isfound to be very nearly isotropic.

A. Table I Nonmagnetic Crystals Containing aSingle Anion Species

The crystals in Table I are grouped into separatesubgroups according to the formal anion valency(Z,), the coordination number of the nearest-neigh-bor cation (N, ), and the effective number of valenceelectrons per anion (N, ). We take N, = 8in all com--pounds containing only filled s-p valence bands. Thiswould include, in the tetrahedrally coordinated A"8' "covalent compounds, all 8 electrons in the directedsP hybridized orbitals. An argument for takingN, =- 6 in ionic crystals is given in Sec. VII, althoughwe have not done so in the tabulations. As dis-cussed below and in Ref. 3, some complicating fea-tures occur in the thalium halides (N, = 10) and inthe noble metal salts (N, =18).

We now turn to the values of E listed in Table Iand make the following pertinent observations.

(a) The parameter E~ in ten six-coordinated al-kali halides (excluding LiF) has very nearly thesame value of 12. 6+ 1.4 eV, although the oscillatorenergy Eo varies by a factor of 2 and the unit cellvolume varies by a factor of 4. The value of E forLiF appears to fall slightly outside the listed limits.

(b) In the eight-coordinated CsCl- and CaF2-typestructures, E=16.2+ 1 eV. This value is verynearly a factor 6 times that observed in the six-coordinated structures. A simple proportionalitybetween E~ and N, is thus implied by this result.

Note in particular the values of E~ for six- andeight-coordinated Cs Cl.

(c) The parameter E in 15 six-coordinated ox-ides in which Eo varies by a factor of 2. 5 is 24. 7a 2. 8 eV. This factor-of-two difference between six-coordinated oxides and halides suggests the possibil-ity of a simple proportionality between Eand Z', .

(d) The parameter E~ in five four-coordinatedoxides (17.Va 0. 6 eV) is about 6 times its value inthe six-coordinated oxides, thus lending further sup-port to the view that E is approximately proportion-al to N, .

(e) The four scheelite-structure oxides (N, = 4)listed in Table I have E=22.3+1.0 eV. This valueis considerably larger than that noted above forother four-coordinated oxides (E~ = 17.7+ 0. 6 eV).As a result, the scheelite crystals do not appear tofall within the simple framework of a proportion-ality between Eand N, . It should be noted thatother nonscheelite crystals containing MoO4 tetra-hedra have values of E in line with a proportion-ality between E~ and N, [e.g. , E, = 18.4 eV inGdz(Mo04)3 and Tb2(MoO, ), ]. There are at least twoexplanations for the unusual behavior of scheelites.First, it is possible that the divalent cations (Ca,Pb, Sr), which are essentially eight-coordinated,contribute somewhat to optical transitions and raiseEabove its four-coordinated value. Second, thetetrahedra in the scheelite structure may be morecovalently bonded than in nonscheelites. We willreturn to this point below.

(f) The parameter Ehas the value 26+ 1 eV infour II-IV compounds (Z, = 2) having the zinc-blendestructure. In the three III-V compounds (Z, = 8), Eis approximately a factor ', larger, and in the1V-IV compounds (Z, = 4), E~ is approximately afactor of 2 larger. There thus appears to be atleast an approximate proportionality between EandZ, even in the covalent tetrahedrally coordinatedbinary compounds. We have excluded small-band-gap and impure materials from this listing becausefree carrier and donor (or acceptor) photo-ionizationabsorption can contribute significantly to the appar-ent dispersion of the refractive index, and thus toE. Such effects may in fact influence the listedvalue of E~ for Ge.

(g) The parameter E is about 21 eV in theeight-coordinated thalium halides. This value isapproximately a factor '8 times that observed in theisostructural alkali halides. We attribute this dif-ference to the extra 6s electrons associated withthe Tl' ion. Thus, for the thalium salts we takeN, = 10 and assume that E is proportional to N, .

(h) The noble-metal salt AgCl crystallizes in theNaCl-type structure, yet the observed value of Eis larger by nearly a factor of 2, i. e. , Ehas thevalue expected for a six-coordinated oxide. Simi-larly, E in four-coordinated CuC1 is the same as

BEHAVIOR OF THE ELECTRONIC DIEL EC TRIC CONSTANT 1343

TABLE I. Dispersion parameters for nonmagneticcrystals containing a single anion species. Crystal

TABLE I. (continued)E()(eV) Eq(eV) P (eV)

CrystalNaCl typePr, =6, Z, =i,LiFNaFKFNaClKC1RbC1Cs ClKBrRbBrKIRbI

N, =S)17.11514, 81P.310.510.410.69.29.17.77.7

14.911.312.313.612.312.214.012.412.112.812.1

0.310.240.260.280.260.250.290.260.250.270.25

E (eV) E(eV) P(eV) Oxides(Nc 4& Za 2& Ne 8

ZnOSi02LiGa02Gd, (Moo4),Tb2(Moo4) 3

6.413.69.58.38.1

9.158.265.48.6

Oxides (scheelite structure)(N, =4, Z, =2, N, =S)CaWO4CaMo04PbMo04SrMo04

17.118.318.118.518.4

23.323.022. 621.3

0.270.290.280.290.29

0.360.360.350.33

CsCl type(N~=8, Z~=1, N~=S)Cs ClCsBrCsI

CsC1 type(N =8, Z =1, N =10)T1ClTlBr

CaF2 type(N~=S, Z~=i, N~=S)CaF,BaF2

Noble-metal salts(Z.=i, N, =iS)AgC1 (N =6)CuC1 (N =4)

10.69.47.5

5.85.3

15.713.8

7.48.3

17.117.015.2

20. 621,7

15.915.9

2218.6

0.270.270.24

0.260.27

0.250.25

0.200.26

Zinc blende(N~ = 4, Z~ = 2, N~ = 8)ZnSZnSeZnTeCdTe

Zinc BlendeQr, =4, Z.=3, N, =S)

GaPGaAsZnGeI'~

Diamond type(N~=4, Z~=4, N~=S)CP SiCSiGe

6.365.544.344.13

4.463.554. 04

10.97.64.02.7

26.127.027.025.7

36.033.535.2

49.7424441

0.410.420.420.40

0.380.350.37

0.390.330.350.32

Wurtzites(N~=4, ZfI=2, N~=S)ZnOCdsCdSeZnS

Oxides(N, =6, Z, =2, N, =S)MgOCaoA1203Y3A150)2Te02SrTi03BaTi03KTa03KTao 6)Nbp 3503LiTa03LiNb03Ba2NaNbgO(gTi02MgA1, 204ZnWO4

6.44.94.06.15

11.39.9

13.411.1t. 245.685. 636.506.177.496.656. 195.24

12.17.46

17,120.420.625. 2

2222. 627. 525.423.223.724. 023.723.426, 125.924. 425.723.326.0

0.270.320.320.39

0.230.240.290.260.240.250. 250.250.250.270.270.260.270.270.27

observed in four-coordinated oxides (ZnO and SiOq).We attribute the occurrence of anomalously largevalues of E~ in these noble-metal salts to contribu-tions of the filled d band to interband transitions,and thus take N, =18.

(i) The parameter &~ in four II-VI wurtzite-structure crystals varies between 17.1 (ZnO) and25. 2 eV (ZnS). We will comment on this observa-tion below.

As suggested in Ref. 3, the above observationslead us to conclude that the quantity

P = E/N~Z, N, eV (2o)

P& = 0. 26 + 0. 04 eV, (21)and taking on the "covalent" value P, in the zinc-

has very nearly the same numerical value in largegroups of crystals containing a single anion species.In particular, we find from Table I that P is essen-tially two-valued, taking on the "ionic" value P, forhalides and most oxides (scheelites being an excep-tion as noted above), i. e. ,

1344 S. H. WEMP I.E AND M. DiDOMENICO, JR.blende, scheelite, and diamond-type structures,~. e. ,

TABLE II. Dispersion parameters for nonmagneticcrystals containing anion radicals.

P, =O. 37+ 0. 05 eV. (22) Crystal Nc Ep {eV) E(eV) P(eV)

The wurtzite-structure crystals appear to rangebetween the ionic and covalent extremes with ZnOfalling at the ionic limit and ZnS at the covalentlimit. Thus, it would appear based solely on ex-perimental values of E that large groups of crys-tals containing a single anion species can be groupedinto distinct ionic and covalent classes with wurtzitecrystals forming a transitional group between theseextremes. This rather remarkable result has alsobeen noted within the ionicity scale of Phillips andVan Vechten. Furthermore, Kurtin et al. ' deducedthe existence of distinct ionic and covalent crystalclasses based on observations of barrier energiesat metal-semiconductor interfaces, excitonstrengths, and the relative importance of "direct"and "nondirect" transitions. These authors alsofind that wurtzite-structure crystals form a transi-tional group between covalent and ionic crystalclasses.

Turning to the oxides, it is of interest that all thesix-coordinated oxides have the ionic value of P = P;,whereas the four-coordinated oxides seem to beeither ionic or covalent (scheelites). Although it ispossible that the four-coordinated oxides, in analogywith the wurtzites, fall near the ionic-covalent tran-sition with scheelites on the covalent side, this ex-planation cannot be distinguished from that suggestedabove attributing the larger E~ value in scheelitesto the influence of the divalent cation-oxygen bond.

B. Table II Nonmagnetic Crystals ContainingAmon Radicals

In Table II we list values of Ep, E, and P for arepresentative sample of nonmagnetic crystals con-taining XO anion radicals. If we continue to defineP by Eq. (20) and take for N, the coordination num-ber of the nearest-neighbor X ion (e. g. , Cl, C, P,I), the values of P shown in Table II are obtained.Although no general rules can be deduced becauseof the limited number of crystals listed and the re-stricted wavelength range of the data, the resultsshown in Table II are clearly reminiscent of thosetabulated in Table I. Thus, the nitrates, phos-phates, sulphates, and chlorates appear to have theionic P value (P, = 0. 26' 0. 04 eV), whereas the io-dates and carbonates have the covalent value (P,=0. 31+0.05). It is of interest that the single per-iodate listed (KIO4) appears to fall into the ionicclass. These experimental results suggest that thesimple empirical rule expressed by Eqs. (20)-(22)may apply to many, if not most, crystalline solidscontaining anion radicals, and that even in thesematerials coordination number and ionicity aresimply related to refractive-index dispersion.

KH2PO4NH4H2PO4AlPOg

12.8 1612.2 1613.9 18

0.250.250.28

KNO3NaNO, (l)'NaNO3(ll)'Ba(NO, ),K)SO4Rb)SO4LiKSO4

NaC104RbC104NH4C 104

Rb2SeO4KIO4

HIO3LiIO3SrCO~CaCO,

13.58.5

159.6

13.813.815.812.414.513.612.29.17.68.34

15.210.9

10.312.111.513.416;517.514.015.416.415.416.714

212019.818.5

0.220.250. 240.280;260.270.220.240.260.240.260.220.440.420, 410.39

We include for NaNO& data for light polarized parallel(II) and perpendicular (j.) to the uniaxial c axis. Notethat Ez is very nearly isotropic despite a large anisotropyln Ep,

Eu&o &A'o2 8E~ E~ E E0

(23)

where P, , Eo applies to the f- d transitions and Eo,Ep applies to the s, p - d transitions. It is straight-forward to combine terms in Eq. (23) and arrive atthe following expressions for the equivalent single-oscillator parameters Ep and E:

C. Table III Magnetic Crystals

In Table III we show values of Ep and E for themagnetic europium chalcogenides. The value of Ein EuO (9 eV) is much lower than in isostructuralMgO (22 eV). Furthermore, there is a, monotonicincrease in E in going from the oxide to the tellu-ride. These results fall outside the framework ofTables I and II; however, they can be easily under-stood by taking into account the 4f7-4f (5d) transi-tions as well as transitions between the s-p valenceband and the d-like conduction band. Transitionsoriginating from the 4f electrons are responsiblefor the observed bandgap and contribute substantial-ly to dispersion of the refractive index. The higher-energy transitions originating in the s-p valenceband are the major contributors to the long wave-length refractive index. The above statements canbe made quantitative by using a two-oscillator de-scription of the refractive-index dispersion, i. e. ,

BEHAVIOR OF THE ELECTRONIC DIELECTRIC CONSTANT 1345

TABLE III. Dispersion parameters for magneticcrystals.

CrystalEuOEuSEuSeEu Te

Ep (eV)2.464 04. 34.3

E(eV)9

141821

and

-, 1+ (E,/E, )(E,/E, )1 (E./E. )(E,/E, )

~a "s I&+(&.I&.)(&D&d]')1 + (E/E~) (Eo/Eo)

(24)

(25)

To give a numerical example, we take the EuO val-ues for Zo and E listed in Table III, assume thatto =1.9 eV from the photoemission experiments ofEastman et al. , ' and assume further that E=25 eVbased on the Evalues observed in the nonmagneticsix-coordinated oxides. We then find, using Eqs.(24) and (25), that E=l.5 eV, and ED=10 eV. Sim-ilar calculations for EuS yield E=2 eV and ED =8eV, and for EuSe, 2, =2 eV and ED=V eV. The cal-culated values of Eo are similar to those observedin nonmagnetic six-coordinated crystals (see TableI). Thus, the refractive-index behavior of thesemagnetic materials containing f electrons can beunderstood as arising from strong interband (p- d)transitions at high energy (E~ =25 eV) and muchweaker f- d transitions (E~ =2 eV) occurring nearthe absorption threshold at lower photon energies.

D. Table IV Liquids

We show in Table IV experimental values of Eoand E for several liquids. All of the listed Eval-ues except for CH2I2 fall in the range E=10+ 2 eV.The relatively low refractive indices observed inthese liquids, when compared with many solids, arethus a consequence of weak optical transitionstrengths as measured by the dispersion energy E.A connection between Eand the coordination num-ber is, of course, complicated in liquids by uncer-tainties as to the short-range order. If we makethe reasonable assumption that molecules remainintact in the liquid so that nearest-neighbor coordi-nation is a valid concept, we then find for CC14(taking N, =8, Z, = 1, N, =4) that P=0. 38 eV, andwe find for CS, (taking N, = 8, Z, = 2, N, = 2) thatP = 0. 34 eV. Both liquids thus display the P valuefound in covalent solids. A complication occurs forwater where the proton lies between two oxygenatoms at a distance 1 A from the nearest neighborand 1.7 A from the next-nearest neighbor. Forthis situation, the appropriate coordination numberis not clearly defined but certainly falls between1 and 2. By taking Ne-8, Za 2 we find P= 0. 31 eV

for N, = 2 and P = 0. 62 eV for N, = 1, thus bracketingthe crystalline covalent value of P, = 0. 37 + 0. 05 eV.For the organic liquids included in Table IV, it isnot clear that N, and Z, can be sensibly defined soas to provide a useful ordering of the data. In sum-mary, the refractive-index behavior of some in-organic liquids appears to fit well into the frame-work observed for inorganic solids, although morerefractive-index data are clearly required to drawfirm conclusions.

VI. EMPIRICAL RULES

TABLE IV. Dispersion parameters for liquids.Liquid

H2OCCl4CS2C6H6 (benzene)C6H5OH (phenol)C2H5OH (ethyl alcohol)CH3OH (methyl alcohol)C,I-I(pentane)C&H&N (chirolin)CH2I2

E()(eV)13.011.26, 98.98.6

10.512.510.67.47.7

E(eV)9.9

11.210.710.511.110.79.38.3

11.114.5

The refractive-index dispersion behavior of morethan 100 solids and liquids, summarized in TablesI-IV, falls into a remarkably simple pattern.Apart from some liquids for which nearest-neighborcoordination number may be a poorly defined con-cept, the dispersion energy E~ obeys the empiricalrelations given by Eqs. (20)-(22). Wurtzite-struc-ture crystals are found to form a transitional groupbetween distinct ionic and covalent crystal classes(four-coordinated oxides may also form a transi-tional group). Equation (20) expresses the strikingresult that the influence of structure, chemistry,and ionicity on Eis simply related to the associateddiscreet quantities NZand p. We emphasizethat the dependence on anion valency is not limitedto ionic materials, but holds also for the tetrahe-drally cpprdinated A 8 cpvalent cpmppunds.We emphasize also that Eq. (20) contains the fol-lowing very important implicit observations: (i)E is independent of the absorption threshold E,(bandgap) to within the indicated error limits of+15%,' and (ii) E~ is independent of the lattice con-stant (or density) to within these same error limits.The first observation imposes constraints on theform of the &2(&o) spectra as discussed in Sec. VII.The second observation places the dispersion energyE in a separate category from all other knownparametrizations of ref ractive-index behavior.For example, the volume density of valence elec-trons is central to the Phillips description (via theplasma energy), the Claussius-Mossotti model, andall ref ractometry approaches. Because the volume

1346 S. H. WEMPLE AND M. DiDOMENICO, JR.that automatically forces E to be independent ofEviz. , constant-optical-frequency conductivitywith high- and low-frequency cutoffs:

b

QUIVALENT OSCILLATOR E~,=47ISo, E, &&E&&~=0, E&E, and E& bE, .

(30)

I

I Eo/Eg 2E/Eg

FIG. 4. Model dielectric functions &2 and 0. together~ith equivalent single oscillator for b =4.

n (0) 1 = E/Eo pN, Z,N, /Eo . (26)The oscillator energy Eo is, to a fair approxima-tion, related empirically to the lowest direct band-gap E, by

Eo=&. 5 E (27)

density is not required, Eq. (20) provides a usefulbasis for predicting and understanding the magni-tudes of refractive indices in new materials. Forexample, the long-wavelength value n(0) is given by

e, (E) = E g (E/E, ), (31)where g(E/E, ) is an arbitrary function of the nor-malized photon energy. The constant-conductivitymodel, for example, requires that g= ah'( E/E) '.Equation (31) imposes the implicit constraint thatthe area under the e2(E) spectrum for a given classof materials is a constant independent of E, .Aslaksen has also pointed out that an E2 spectrumapproximated by a series of ~ functions, i. e. ,

e~=Q n, 6(E a,E,), a, ~1 (32)

ln Kq. (30), h is an initially arbitrary bandwidthparameter, and o is the optical conductivity. Themodel spectrum described by Eq. (30) is shown inFig. 4 for b=4. Also shown is the single oscillatorthat produces equivalent dispersion of the refractiveindex for E & E, .

It has been pointed out recently by Aslaksen' thatthe constant-conductivity model is the simplestmember of a general class of a& spectra obeying therequirement that Ebe independent of E, . Theform proposed by Aslaksen is

so that for ionic compoundsn (0) 1 =N, Z,N, /6E, ,

and for covalent compoundsn (0) 1=N,Z,N,/4E) .

(28)

(29)

(33)satisfies the constraint, in which case

2 8 3naua

For the constant-conductivity model, it has beenshown3 that

Equations (28) and (29) provide a more widely ap-plicable connection between bandgaps and refractiveindices than that given by Moss's rule' and revealexplicitly the importance of coordination number,valency, and ionicity. It is of interest to note thatEqs. (20) and (27), together with Eq. (10), providea general means for estimating both the magnitudeand dispersion of the refractive index in terms ofthe lowest direct bandgap.

VII. DIELECTRIC MODELS

As noted above, E is almost independent of thethreshold energy E, . This res ~lt imposes con-straints on the E2 spectrum via the. aoment integralscontained in Eq. (13). We now consider variousmodel a, spectra that satisfy this constraint andcompare these spectra with those observed experi-mentally. Comparison will then be made betweenthe model spectrum derived from dispersion dataand the Penn model a& spectrum derived theoreti-cally'3 using a simplified single-gap model of theisotropic three -dimensional electron gas.

In Ref. 3 we proposed a simple model spectrum

E~= 8v3 ho(b 1)/(1+ b+ h')'i (34)It is clear that refractive-index dispersion dataalone cannot distinguish between these or other di-electric models obeying Eq. (31).

To make a sensible choice we must examine thea& spectra of real materials. When this is done wefind that many of the available spectra appear to havethe following common features: (a) There is a con-tinuous background which increases towards thebandgap and is governed approximately by the ex-pression ez=4vo/u&; (b) relatively sharp structureis superimposed on this constant-conductivity back-ground; and (c) the spectra cut off rapidly at highfrequencies. The structure is related to interbandcritical points and possibly excitons in ionic materi-als. Several typical E~ spectra are shown in Figs.5-8. Figures 5 and 6 give data for the ionic crys-tals NaC1' and MgO, ' while Figs. V and 8 give datafor the covalent crystals Si' and GaP. ' In eachfigure the approximate constant conductivity back-ground is indicated by a dashed line, and the averageoscillator position Eo is shown by an arrow. The

BEHAVIOR OF THE ELECTRONIC DIELECTRIC CONSTANT 13476

Na Cl

0 I10 l5fi(ev)

I

20

FIG. 6. Experimental &2 spectrum for MgO takenfrom Ref. 16 together with approximate constant-con-ductivity background and average energy gaps Eo andE

06I I

IO I2t)~ (ev)

I

l4I

l6

FIG. 5. Experimental &2 spectrum for NaCl takenfrom Ref. 15 together with approximate constant-con-ductivity background and average energy gaps Eo and E~.

Phillips bandgap parameter E~ is also indicated.Based on these and other similar, though limited,data we suggest that a reasonable model &~ spectrumfor many materials consists of a superposition of aconstant-conductivity background [Eq. (30)] andcritical point and/or excitonic structure [Eq. (3&)].This raises the question as to whether the backgroundor the structure determines the value of E. Clearlythe structure-related quantities (n, , a() are not ex-pected to be the same for all members of a crystalgroup having the same values of NZand N, . Forexample, among the oxides having N, = 6 the tita-nates, niobates, and tantalates have d-like conduc-tion bands, whereas s-like bands are found in Mgo,CaO, and A1203 . Furthermore, sharp excitonsare observed in MgO while excitons are not observedin the niobates and tantalates. Also, the oxygenanion coordination number is 2 in all the listed tita-nates, niobates, and tantalates, while in MgO andCaO the oxygen coordination number is 6. Thissame argument applies also to CsC1- and CaF&-typehalides (1V', = 8) in whichthe halide anion coordinationnumbers are 8 and 4, respectively. Figures 5-8clearly show that the background is the major con-tributor to c& near the bandgap, and from the mo-ment integrals in Eq. (13), it is this portion of the tespectrum that makes the largest contribution to E~ .We therefore suggest, to within +15%%u() limits, thatthe behavior of the refractive index below the inter-

40

50

20

Ip

02 5

ti~ (ev)FIG. 7. Experimental &2 spectrum for Si taken from

Ref. 17 together with approximate constant-conductivitybackground and average energy gaps Eo and E~.

band edge is determined primarily by the approxi-mate constant-conductivity background and that thecritical-point (and/or excitonic) structure plays aminor role. This tentative conclusion may be anoversimplification for such covalent materials asSi and Ge (cf. Fig. V).

It is of interest to note that a constant-conduc-tivity background with superimposed structure ispredicted by theoretical arguments. The expres-sion for the electronic conductivity is given byo(cu) = (e I/8v m) J d )t f,(k)5[8,(k) -8(k) K&d],

(35)where f,is the oscillator strength for transitionsbetween valence and conduction bands, 8, and 8are the conduction- and valence-band energies, andthe integral extends over the Brillouin zone. Interms of an average oscillator strength f~ and jointdensity-of-states function p(&u), Eq. (35) becomes

1348 S. H. WEMP I E AND M. DiDOME NICO, JR.

o/N, Z,N, = 80+ 12 (0 cm) ' . (33)Our choice of a sharp upper-frequency cutoff de-scribed by the parameter b is, of course, unreal-istic, although a rapid decrease in o is expected athigh frequencies because of exhaustion of the inter-band transitions and the form of the crystallinepseudopotential. Rather than distinguish betweenionic and covalent crystals solely on the basis ofbandwidth alone, we now postulate an alternativemodel spectrum in which the bandwidth is fixed andthe conductivity, i. e. , f,C in Eq. (3V), is differentin ionic and covalent crystal classes. Based onavailable experimental data in large numbers ofcrystals we take 5 =2. 5. Combining Eqs. (20)-(22)with (34) we obtain

30

25 GOP

20

15

10

4~ (ev)I

10 15

FIG. 8. Experimental && spectrum for GaP taken fromRef. 18 together with approximate constant-conductivity

background and average energy gaps Eo and E~.

o((u) = (ne'ff/2m) f,p((u) . (35)By expanding E,(k) E(k) in Taylor series in theneighborhood of critical points, it can be shown thatEq. (36) reduces to

m) &OO + ~42&ti 4& Otnts+~si44424 ~(3V)

where |"is a constant related to transitions through-out the zone. The product f C is, thus, a measureof the background conductivity.

The simplified model spectrum described by Eq.(30) contains an unknown bandwidth parameter bIn Ref. 3, we postulated that the normalized conduc-tivity (o/NQ, N, ) was the same in all ionic and co-valent materials. As a consequence we derived avalue b, = 3.4 for covalent materials and b, = 2. 1 forionic materials. Using Eqs. (20) and (34), we thenfound a "universal" value of 0 given by

o,/N, Z,N, = 70+ 10 (0 cm) ', ionico,/ N,Z,N, = 100+ 15 (0 cm), covalent.

Equation (30) then yields the relations@2= (0. 5+0. 07)N,Z,N, /E, ionice2 = (0. 75 + 0. 12)N,Z,N, /E, covalent,

(39)(40)

(41)(42)

where E= 8& in units of electron volts, and E,5E

Equations (39)-(42) describe a simplified approxi-mate dielectric model that correctly predicts boththe magnitude and dispersion of refractive indicesin more than 100 widely different substances. Theonly distinction between ionic and covalent materialsusing this constant-bandwidth model is the 50%%uo-larger conductivity (or transition strength) observedin the covalent class. Crystal structure enters onlythrough the nearest-neighbor coordination numberNchemistry through the anion valency Zandvalence-band structure through the equivalent num-ber of electrons per anion N, . To within the +15%error limits, the model spectrum does not dependon the lattice constant or volume density. It shouldbe pointed out that the choice of N, =8 for filled s-pvalence bands assumes that the s-p splitting is suf-ficiently small so that the bands derived from theatomic s-like anion orbitals can contribute to inter-band transitions over the range of energies of im-portance in determining Eo and E~ . In the moreionic materials, energy-band computations" 'suggest that the s-like bands may lie so far belowthe p-like bands that this assumption is invalid, inwhich case it might be more appropriate to chooseN, =6. In covalently bonded materials having sphybridized orbitals, all eight valence electrons con-tribute to the bonding. This observation could ac-count for the approximately 50%%uo lower transitionstrength observed in the ionic materials and wouldmake Eqs. (40) and (42) apply to all materials ionicas well as covalent.

We now turn to a comparison between the sim-plified dielectric model presented above and thec& spectrum derived theoretically by Bardasis andHone' based on the Penn model. The Penn-modelc& spectrum is given to a good approximation by'3

]6e2( ) 3 g (h )4 [(g )2 E2]1/ s 2g

(43)e2 -0, h(o &E~

where o,'0= 0. 529 A is the Bohr radius, and k~ andE&are the Fermi wave number and energy, respec-tively, of a free-electron gas having an electrondensity equal to that of the valence electrons. Thespectrum is shown in Fig. 9 for Ge and comparedwith both the observed spectrum and the constantcr model. It is clear that the Penn spectrum bears

BEHAVIOR OF THE ELECTRONIC DIELECTRIC CONSTANT 1349

little resemblance to either the observed spectrumor the constant-o model.

5p(r) g 'V, (r) d'r,mNu

all sp~e(44)

where N is the number of electrons per unit cell,5p(r) is the variation of the electron density in the

VIII. DISCUSSION

We have shown that a single oscillator having ad-justable strength and position accurately describesthe dispersion of the refractive index in more than100 ionic and covalent materials (solids and liquids).The utility of such a simple result for large num-bers of very different materials lies in the empiri-cal rule obeyed by the dispersion energy parameterE. Through this quantity we find that ionicity,coordination number, and chemical valency playcentral roles in determining the behavior of refrac-tive indices, and, as a consequence, electro-optic,nonlinear-optical, and photoelastic effects. ' '0 Thedispersion energy properly normalizes the interac-tion potential describing these optical effects and,thus, directly influences their magnitudes.

Before discussing whether or not the empiricalrules on E fall within the framework of existingtheories, we now ask if there is some connectionbetween the various parameters introduced by dif-ferent authors to describe optical properties (viz. ,Eo, E~, S&d&, Eand E,). We have shown that E~obeys interesting empiricisms, while Phillips andVan Vechten' find widely applicable empiricismsassociated with E, . The other quantities of inter-est obey the f sum rule (if&a~) and the Hopfield sumrule~ (E,). The latter sum rule, Eq. (16), is givenby

E,= Ms/M, = 3E,(b + 5+ 1),E~=M(/M ( E,b,E() M, /M -s E,Sb /(I) + 5+1),

(45)(46)(47)

while the interband-strength parameters are givenby

(k&d~)'= 8 (ho)E, (b -1), (48)E~ = 192(ho) (b 1)s/(bs+ b+ 1). (49)

Hopfield has pointed out that these strength param-eters are interrelated via the E, gap [Eq. (45)] by

(5(u(, ) =End . (50)It is also clear from Eqs. (45)-(47) that the three

energy gaps are interrelated by

crystal about the average, and V,(r) is the bareatomic-crystal potential. Physically, E, is the nat-ural frequency of the electronic charge clouds,treated as a rigid-charge density, vibrating againstthe atomic cores, whereas v~ is the natural frequen-cy of vibration of the free-valence-electron gas.We note that the sum-rule parameters h~~ and E,describe the magnitude and dispersion of e&(&0) athigh frequencies above the cutoff of the &2 spec-trum. Thus, according to Sec. III, the Ms and M,moments determine the high-frequency behavior of

On the other hand, at low frequencies, belowthe bandgap, the magnitude and dispersion of &, ((d)are determined by the M

~and I 3 moments. Be-

cause different moments are involved, there is nogeneral connection between the high- and low-fre-quency behavior of e&((d). A simple connection isobtained, however, if we choose the specific model&, spectrum given by Eq. (80). For this constant-conductivity model the following expressions givethe three energy gaps of interest;

50 E -EE (51)

40

30

20E

IO

Ge

4~ CONSTANT 0

3 4(ev)

PENNMODEL

We emphasize that Eqs. (50) and (51) are indepen-dent of all the parameters of the constant-conduc-tivity model (a, b, E,). Within the framework of thismodel, then, the Hopfield gap Eas given by thesum rule Eq. (44), provides the connection betweenthe Phillips parameters and the oscillator param-eters Eo and E~. In Fig. 10 we show our model E&spectrum for b = 2. 5 and indicate the positions of thethree energy gaps Eo, E~, and Eas well as theplasma energy h(d~. The e, (&u) spectra at high andlow frequencies are shown by the dashed lines, andthe moments which determine these spectra are al-so indicated in parentheses. From the moment ex-pressions for the quantities in Eqs. (50) or (51), itcan be shown that

FIG. 9. Experimental &2 spectrum for Ge comparedwith results of Penn and constant-conductivity models.The energy gaps Eo and E~ are also indicated.

E=M(/M s M, /Ms . (52)Equation (52) holds for the constant-conductivity

1350 S. H. WEMP LE AND M. D iDOME NICO, JR.

CA

X

K

CO

N

'4l

(V,P,)7)r~(M ()

o g o

2EIEt

(M),M~)y C((m) l~

4

FIG, 10. Schematic illustration of the high- and low-frequency behavior of e& (dashed lines) compared withthe constant-conductivity model c2 spectrum (solid line).The moments determining e& at high and low frequenciesare indicated in parentheses.

E, =37(n, &10 )/N, Z, eV, (53)where n, is the volume density of anions, and wehave taken P = 0. 37 eV. It is not as yet clear, how-ever, how Eq. (53) is contained within the Hopfieldsum rule expressed by Eq. (44), although the formof Eq. (53) suggests that atomiclike nearest-neigh-bor quantities are of major importance.

Shaw" has recently introduced a "dispersion en-ergy" analogous to E into the Phillips-Van Vechtendescription by defining the quantity

(E~) = (If(dp) /E~= M,M ( . (54)For the constant-conductivity model, Eqs. (46) and(48) then yield the relation

(E') = 192 (So) (b 1) /3b.Comparison with Eq. (49) shows that

(E~/E~) =3b/(b + b+1)1.

(55)

For a typical value of b=2. 5, E'=1.14E~. Shawfinds that both Eand E' correlate in a simple waywith the Phillips ionicity f, = C 2/E ~, i. e. , the dis-

model but is not generally valid for Aslaksen's'form of the e, spectrum given by Eq. (31).

Either Eq. (50) or (51) can be used to determineexperimental values of E, using known values of E,or Av~ and Ep or Ey Because of the empirical ruleon E~ given by Eq. (20) and the proportionality be-tween (h&o~) and the volume density of electrons,Eq. (50) provides a simple expression for E i. e. ,

persion energies are approximately proportional to(1 -f, ). According to Eqs. (49) and (55) this resultimplies, as pointed out by Shaw, that

o cc(l, f,.), (56)that is, the conductivity decreases continuously withincreasing ionicity. Thus far there do not appear tobe clear-cut experimental or theoretical argumentsthat unambiguously support either the ionicity-re-lated conductivity given by Eq. (56) or the two-val-ued form given by Eqs. (39) and (40).

In terms of the average oscillator strength f(v)and joint density of states p(v) Eqs. (20) and (36)yield the relation

o ~fpo: PN, Z,N, , (57)

ACKNOWLEDGMENTS

We wish to thank J. R. Brews and K. K. Thornberfor helping to clarify several points in the manu-script, and acknowledge useful discussions withR. W. Shaw, Jr. We also wish to thank J. C.Phillips for reading the manuscript.

independent of the volume density of electrons towithin +15%. To our knowledge, none of the existingtheories of optical properties explains the experi-mental observations described by Eq. (57). In nocase do coordination number and valency enter insuch a simple way, and in no case does the volumedensity of electrons fail to be an important quantity.In existing theories, as well as in the Phillips-VanVechten model, the volume density enters naturallyvia the f sum rule (M, moment). Only the M, andM 3 moments determine E, so that the higher-energyinterband transitions that inQuence M& have little in-fluence on E~ . However, within the confines of theconstant-conductivity model Eq. (52) relates E, tothe sum-rule moments M3 and M& . As a result,the dispersion energy may depend upon the detailedcharge distribution within each unit cell via Eq. (44)and, consequently, would then be closely related tochemical bonding. The dependence on N, and Z,suggests further that nearest-neighbor atomiclikequantities are of major importance, and that an un-derstanding of the electronic optical properties ofmaterials and their relationship to chemical bondingmay lie within a nearly localized orbital theory.

~J. C. Phillips, Phys. Rev. Letters 20, 550 (1968);J. C. Phillips and J. A. Van Vechten, ibid. 22, 705(1969);J. A. Van Vechten, Phys. Rev. 182, 891 (1969).

D. R. Penn, Phys. Rev. 128, 2093 (1962).S. H. Wemple and M. DiDomenico, Jr. , Phys. Rev.

Letters 23, 1156 (1969).4J. J. Hopfield, Phys. Rev. 8 2, 973 (1970).5J. C. Philips, Rev. Mod. Phys. 42, 317 (1970).

See, for example, J. R. Tessman, A. H. Kahn, and

W. Shockley, Phys. Rev. 92, 890 (1953); and I. M.Boswarva, Phys. Rev. B 1, 1698 (1970).

P. Noizieres and D. Pines, Phys. Rev. 113, 1254(1959).

8S. S. Batsanov, Refractometry agd Chemical Structure(Consultants Bureau, New York, 1961).

~S. Kurtin, T. C. McGill, and C. A. Mead, Phys. Rev.Letters 22, 1433 (1969).

D. E. Eastman, F. Holtzberg, and S. Methfessel,

BEHAVIOR OF THE ELECTRONIC DIELECTRIC CONSTANT 1351Phys. Rev. Letters 23, 226 (1969).

I'D. Eisenberg and W. Kauzmann, Structure andProperties of Water (Oxford U. P. , Oxford, England,1969).

T. S. Moss, Optical Properties of Semiconductors(Butterworths, London, 1961).

S. Bardasis and D. Hone, Phys. Rev. 153, 849 (1967).I4E. W. Aslaksen, Phys. Rev. Letters 24, 767 (1970).

C. Y. Fong and Rf. L. Cohen, Phys. Rev. 185, 1168(1968).

C. Y. Fong, W. Saslow, and M. L. Cohen, Phys.

Rev. 168, 992 (1968).~~D. Brust, Phys. Rev. 134, A1337 (1964).

H. R. Philipp and H. Ehrenreich, Phys. Rev. 129,1550 (1963).

~BS. H. Wemple and M. DiDomenico, Jr. , AppliedSolid State Science, edited by R. Wolfe (Academic, NewYork, to be published), Vol. III.

S. H. Wemple and M. DiDomenico, Jr. , Phys. Rev.B 1, 193 (1970).

R. W. Shaw, Jr. , Phys. Rev. Letters ~25 818 (1970).

PHYSICAL REVIEW B VOLUME 3, NUMBER 4 15 F EBRUARY 1971

Far-Infrared Properties of Lattice Resonant Modes.V. Second-Order Stark Effect*

B. P. ClaymanI-aboratory of Atomic and Solid State Physics, Cornell University, Ithaca, Negro York 14850

Department of Physics, Simon I"raser University, Burnaby 2, British Columbia, Canadaand

R. D. Kirbyt and A. J. Sieverslaboratory of Atomic and Solid State Physics, Cornell University, Ithaca, Nese York 14850

(Received 8 September 1970)Small electric-field-induced frequency shifts have been observed for resonant modes assoc-

iated with three defect systems. For NaI: Cl, the shifts have been used to measure the quar-tic anharmonic terms of the interionic potential of the impurity ion. For KBr: Li', the quarticanharmonic terms are found to be very small, and an harmonic potential which includes a cen-tral barrier with the barrier height less than the zero-point energy of the oscillator is requiredto explain the experimental results. For NaCl: Cu', only an "on-center" resonant-mode con-figuration is consistent with the experimental results.

I. INTRODUCTION

The response of lattice resonant modes in alkali-halide crystals to an external dc electric field isa sensitive probe of the local impurity environment.For a harmonic-oscillator resonant mode associ-ated with an "on-center" defect, an applied electricfield shifts all energy levels by the same amount,and no change in the far-infrared absorption fre-quency is to be expected. For paraelectric im-purities, whose far-infrared properties are stronglymodified by the tunneling motion of the "off-center"impurity ion, giant electric field effects have beenobserved. ' We have measured small electric-field-induced frequency shifts associated with three"on-center" defect systems: NaI: NaC1, KBr:LiBr,and NaCl: CuCl. Because these experiments com-plement previous far-infrared measurements in-corporating other perturbations, ~' some definitefeatures of the anharmonic potentials which bindthese impurities can now be resolved.

Of the three lattice-defect systems, the largest

electric-field-induced frequency shifts have beenobserved in NaI:Cl . The resonant-mode frequencyshifts have been used to determine the cubic sym-metry of the defect site and to measure the quar-tic anharmonic terms of the interionic potential ofthe impurity ion. A preliminary description ofthese findings has been given earlier.

The much smaller electric-field-induced shifts'observed for KBr:Li' are signif icant in that theyeliminate the possibility that quartic anharmonicterms play an important role in the local potentialof this defect. Both the small electric field shiftand the large isotope shift previously observed areexplained with an harmonic potential containing acentral barrier. The barrier is small comparedto the zero-point energy of the resonant mode,hence the impurity still appears to occupy the nor-mal equilibrium lattice site.

The null electric field effects observed forNaC1: Cu' are consistent with an "on-center" res-onant-mode configuration. For this case, the exactshape of the potential has not been determined.