t JOU RN AL O F RESEAR CH of th e National Bureau of Sta ndards -A. Ph ysics and Chemistry j Vol. 74 A, No.2, March- April 1970

7 Optical Properties and Electronic * 1 ( Manuel Cardona2 > Brown University, Providence, Rhode Island and DESY, Homburg, Germany

(October 10, 1969)

T he fundame nta l absorption spectrum of a so lid yie lds information about critical points in the opti· cal density of states. This informati on can be used to adjust parameters of the band structure. Once the adju sted band structure is known , the optical prope rties and the de nsity of states can be generated by nume ri cal integration. We revi ew in this paper the para metrization techniques used for obtaining band structures suitable for de nsity of s tates calculations. The calculated optical constants are compared with experim enta l results. The e ne rgy derivative of these opti cal constants is di scussed in connection with results of modulated re fl ecta nce measure ments. It is also shown that information about de ns ity of e mpty s tates can be obtained fro m optical experime nt s in vo lving excit ati on from dee p core le ve ls to the conduction band. A deta il ed comparison of the calc ul ate d one·e lectron opti cal line s hapes with experime nt reveals deviations whi ch can be interpre ted as effects. The acc umulating ex pe rime nt al evide nce poin t· in g in this direction is reviewed togethe r with the ex isting theo ry of these effects. l A number of simple mode ls for the co mpli cated interband de nsit y of states of a n in s ul ator have been proposed. We revi e w in pa rti cu lar the Pe nn mode l, whi ch can be used to account for res po nse functions at zero frequency, and t he paraboli c model, whi ch can be used to account for the di s pers ion of I response fun ction s in th e immedi ate vi cinity of the fundamental absorption e dge. T Key words: C ritical points; de nsity of states; d ielectric constant; modulated re Aectance; opti cal absorption.

1. Optical Properties and One- Density oscillator s tre ngth tensor F ef is related to th e matrix of States ele ments of p through F e/= 2 < flp le > < el p lf > w eI- I. The Bloc h fun c ti ons are normalized over unit The opti cal behavior of semi conductors and ins ula· volume. Degenera te statisti cs has been assumed in e q tors in the near infrared, visible, and ultraviolet is (1) and spatial dispersion effects have been neglected. determined by electronic interband transitions. An ad· It is customary to take the slowly varying oscillator ditional intraband or free electron contributio n to the strength out of the integral sign in eq (1) and thus write: optical properties has to be considered for me tals. We shall di scuss here the relationship between the inter­ band contribution and the density of states. The inter· (2) band contribution to the imaginary part of the dielectric constant can be written as (in atomic units, Ii = 1, m = I ,e= I): where F is an average oscillator strength and Net the (1) combined optical density of states. Structure in E;(W) (eq (1)) appears in the neighborhood of critical points, where V k Wef= O. Such critical points where w eFwe-WI is the difference in e nergy be tween can be localized in a small region of k s pace or can ex· the empty bands (e) and the filled bands (E). The spin tend over large portions of the over multiplicity must be included explicitly in eq (1). The which filled and e mpty bands are parallel (sometimes only nearly parallel). Once the critical points which cor· *i\n invit ed paper P"(,SCll lc{j £lIt he 3d .Materials Research Symposium, Electronic Density respond to observed optical structure are ide ntified in of Sr.(lt.es. Novernber 3- 6. 1969. Gaithersburg, Md. 'S upported by the National Science Fou ndati on and the Army Researc h Office, Durham. terms of the band structure through various de vious :! John Simon Guggenheim Foundation Fellow. and sometimes dubious arguments, their energies can 253 372-505 0 - 70 • 6 be used to adjust parameters of semiempirical band ing oscillator strength and thus the integral of eq (1) is structure calculations. obtained. Four different parametric techniques of calculating The real part of the dielectric constant E,. can be ob­ band structures have been used for this purpose: the tained from E i by using the Kramers-Kronig relations. empirical pseudopotential method (EPM) [1], the k· p It is also possible to obtain E,. and E; simultaneously by method [2], the Fourier expansion technique (FE) [3], calculating the integral: and the adjustable orthogonalized plane waves method (AOPW) [4]. Once reasonably reliable band structures are known (3) it is important to calculate from them the imaginary part of the dielectric constant Ei(W) and to compare it with experimental results so as to confirm or disprove (~ with YJ small and positive. For YJ ~ + 0 the imaginary the initial tentative assignment of critical points and part of eq (3) coincides with eq (1). Equation (3) can be thus the accuracy of the band structures. Rich struc­ evaluated with a Monte Carlo technique. Points are ture is obtained in both experimental and calculated generated at random in k space within the Brillouin spectra and hence a rather stringent test of the accura­ zone and the average value of the integrand for these cy of the available theoretical band structure is in prin­ points calculated. The process can be interrupted when (: ciple possible. reasonable convergence as a function of the number of In order to calculate numerically the integral of eq (1) random points is achieved [7,8] . it is necessary to sample eigenvalues and eigenfunc­ We show in figure 1 the results of a calculation of Ei tions at a large number of points in the Brillouin zone. from the k . p band structure of InAs with the method The amount of computer time required for solving the of Gilat and Raubenheimer [6]. The band structure band structure problem with first-principles methods problem, including spin-orbit effects, was solved at (OPW, APW, KKR) at a general point of the Brillouin about 200 points of the reduced zone (1/48 of the BZ). zone makes such methods impractical for evaluating eq We have indicated in this figure the of the t? (1). The parametric methods (EPM, k· p, FE, but not critical points (or of the approximate regions of space) AOPW) require only the diagonalization of a small where the structure in Ei originates. The experimental X matrix (typically 30 30) and hence it is possible to Ei spectrum, as obtained from the Kramers-Kronig anal­ sample the band structure at about 1000 points with ysis of the normal incidence reflectivity [9] , is also only a few hours of computer time. Cubic materials, in shown. The agreement between calculated and experi- _< particular those with Td , 0, and 0" point groups, are mental spectra is good, with regards to both position simple in this respect: symmetry reduces the sampling and strength of the observed structure, with the excep- required for the evaluation of eq (1) to only 1/48 of the

Brillouin zone. Hexagonal and tetragonal materials 25 have relatively larger irreducible zones and hence a -- THEORY larger number of sampling points is necessary if the In As resolution of the calculation is not to suffer. Once the 20 --- EXP. band structure problem has been solved for all points of a reasonably tight regular mesh, the bands and 15 matrix elements at arbitrary points can be obtained by 3 means of linear or quadratic interpolation. 1 The method of Gilat and coworkers [5] has become 10 rather popular for the numerical evaluation of eq (1) [ 4,6]. In the case of a cubic material the Brillouin zone is divided into a cubi c mesh and the band structure problem solved at the center of these cubes (sometimes a finer mesh is generated by quadratic interpolation 2.0 4.0 6 ,0 from the coarser mesh [6]). Within each cube of the ENERGY leV) ) mesh the bands are linearily interpolated and approxi­ FIGURE 1. Imaginary part of the dielectric constant of InAs as mated by their tangent planes. The areas of constant 'I calculated from the k· p method (--) [6J and as determined I energy plane within each cube corresponding to a given experimentally (.... J [91· The group theoretical symmetry assign· We! are added after multiplying them by the correspond- ments were made with the help of the calculated isoenergy plots. 254 tion of the pOS]tlOn of the E2 peak. This is to be at- non-local pseudopotential calculation with 14 adjusta­ i tributed to an improper assignment of the E2 peak when ble parameters [11]. The di screpancy between experi­ the 6 adjustable band structure parameters were deter­ mental and calculated c urves at hi gh energy, a common mined. The E2 peak had been attributed, followin g the feature of many zincblende-type materials [12] , has tradition, to an X critical point while it is actually due two on gms: the measured reflectivity should be low > to an extended region of k space centered around the because of in creased diffuse refl ectance at small U points [8]. It s hould be a si mple matter to readjust wavelengths while the calculated one should be high > th e band structure parameters to lower the energy of because of the finite number of bands included in the the calculated E2 peak by about 0.5 e V; in view of the calculation. In this region where EI'-1 is small, the con­ large amount of computer time required to recalculate tribution to E,. of transitions not included should lower th e energy bands this has not been done. The structure the calculated reflectivity. calculated around 6 eV, due mostly to spin-orbit During the past few years a lot of activity has been splitting of the L3 levels, has not yet been observed devoted to the measurement and analysis of differential experime ntally. reflection spectra obtained with modulation techniques The conventional experime ntal determination of Ei [1 3-15]. The wavelength (or energy) derivative from normal incidence reflection data [9] suffers from s pectra [14] should permit an accurate analysis of the 1- consid erable inaccuracy: to the experimental error line shapes of the spectra of figures 1 and 2. We show > produced by possible improper s urface treatment and in figure 3 the temperature modulated reflecti on spec­ contamination one has to add the uncertainty in the trum (thermoreflectance) of GaSb [15]: it has been high-e nergy extrapolation of th e experime ntal data show n that for th e III-V materials [15] this spectrum required for the Kramers-Kronig analysis. Some of is very similar to th e photon energy derivative spec­ " these difficulties are avoided by comparing the trum, difficult to obtain experimentally. The cor­ calculated re fl ectivity spectra (obtained from E with responding photon energy derivative spectrum ob­ Fresnel's equation) with the experimental results. This tained from th e calculation of figure 2 is also shown in is done in figure 2 for GaSb: th e experim ental data [10] figure 3. The calculated and experime ntal shapes of the :=' have not been Kramers-Kronig analyzed because of the E I , EI + ~I peaks show di screpancies of the type at­ small range of the energy scale. Two calculated spectra tributed in secti on 2 to exciton interaction. Derivative have been plotted in this figure: one obtained from the spectra for other germanium- a nd zincblende-type k . p band structure [6] and the other obtain ed from a materi als have been calculated by Walter and Cohen [12] and by Higginbotham [16] . 09 ----.----.------, The methods to calcul ate band s tructures from first Go Sb principles, without or with only a few adjustable parameters (o ne [17] or three [4]) have recently 0 8 - - PSEUDO POTE N TI A L -- k P achieved consi derable success. However the calcula- •••• EXPFRIMEN T (29 7°K ) 0 .7 -- THERMOREF. BOOK R - - CALCULATED Go Sb 1'\ 06 1.0 II

~E, I I II 0.5 'I (\ I I 1\ 0.5 'I E21 I \ •0 / I E~ ~ I \ 1\ I 0 4 x / I / E: I \ r 0:: / ('\j\../ I \ I "- ,./ ...... 0:: I - 0 /

and Raubenheimer together with experimental results 0.7 [20J. Comparison with other experimental results for ~ the germanium family suggests that the high-energy IE ~ /2\ end of the measured spectrum is too low, probably due 0.6 ./' \ to surface imperfections in the delicate crystals, grown r from mercury solution, which were used for this experi­ R / EI ment. .. 0.5 The k . p fitting procedure has also been applied to a ••• Eo first principles relativistic APW calculation of the band structure of PbTe by Buss and Parada [7J. Figure 5 0.4 shows the reflectivity of PbTe obtained by this method CALCU LAT IO N CA R DONA 8 GREENAWAY ZEMEL , J EN SEN 8 0.3 SCHOOLAR < 0.6

o 2 3 4 5 eV

(I FIGURE 5. R eflectivity of PbTe calculated from the A PW-k· p band >- 0.5 I- I \-McELROY'S structure [71 , compared with experimental results [21]. ;; I \ DATA I­ U W / \ ....J H G LJ.. I \ W a:: 0.4 \IJH)' \ -, E '--1.../ \ 10'

/ \ z / / o / f-- 6 / (LO.5xIO / ' ...... , \ Ef / 0:: / CA LCULAT ION 0.3 / o / CARDO NA 80 GREENAWAY (/) / / \-ll ro / SCANLON / « / V 2 .0 3.0 4.0 5.0 ENERGY (eV) 2 3 4 < eV FIGURE 4. Reflectivity of gray tin calculated from afirst principles OPW band structure fitted with the k · p method [1 9]. Also FIGURE 6. Absorption coefficient PbTe calculated from the APW­ experimental results [20]. k . p band structure, compared with experimental results [21 ,221. 256 We have so far di scussed optical constants for cubic F e! must be removed. As an example we show in figure material s. While calculations for materials with lower 8 the individual density of states of the 3 highest symmetry require more computer time, one has the valence bands (s ix including s pin) and the 3 lowest con­ extra reward of being able to predict the experimentall y duction bands of gray tin [19] . Direct information observed . Figure 7 shows the two principal about the individual density of states can be obtained ;> components of Ei for trigonal Se as calculated by San­ by a number of methods di scussed in this conference. > drock [24] from the pseudopotential band structure. We mention , in particular, optical techniques involving The similarity be tween calculated and experi me ntal transiti ons from deep core levels to the conduction results [25] , also shown in figure 7, is especially re­ band or from the valence band to temporarily e mpty > markable in vi ew of the method used to determine the core levels (soft x-ray emission) [27]. If the sometimes l pseudopotential parameters: they were determined questionable assumption of constant matrix elements from the pseudopotential parameters required to fit the is made, the corresponding spectra represent the con­ optical structure of ZnSe. Only a small adjustment was duction (for absorption spectra) and the valence (for performed so as to bring the calculated fundamental emission spectra) density of states because of the small gap (1.4 e V) into agreement with the experimental one width of the core bands. We show in figure 9 the densi- (2.0 e V). The di electri c constant of antimony (trigonal) " for the ordin ary and the extraordinary ray has also been - - VALENCE BANDS -- · I· - CO NDU CT ION BA NDS - · ( calculated by a similar procedure [26]. 1.0 The reasonable agreement obtained between experi­ mental and calculated optical cons tants suggests the E use of the corresponding band structure to determine o o the individual de nsity of states D(w): the main work, '- > that of diagonalizing the Hamiltonian at a large number Q) of points, has already been done. The programs ;;; 05 Q) r required to calculate individual density of states are o very similar to those used for the evaluation of eq (1): '" We! must be replaced by the single band energies and

30 EN ERGY (eV)

FIGURE 8. Individual density of states for gray tin, obtained /rom the OPW-k· p band structure [/ 9]. The top of the valence band is 25 at aev' The lowest valence band is not included. fl\£ EJ.c 20 ------0 calculated IGel ------0 calculated IGa Sb) --_ Ej (.\l2 measured o Ei Ii 15 [\ o . 4·~ ': u ~ . , r" I f I \ E o d VI I. l , I \ ..... , OJ~ 10 J \ \. I)' \, 3 ,I" _...-/ 0.1 ' ..... / " t 5 0.1

2 3 4 5 6 ~-~-~-~--~-~-~---~ O 4 8 eV - eV-

F IGURE 7. Inwginary part of the dielectric constant of trigonal FIGURE 9. Conduction density of siates calculate for Ce [4 ] and for selenium for both principal directions of polarization of the electric CaSb [6ltogether with thefnnction E iW 2 obtainedfrom. experimental field vector E as calculated from the pseudopotentiaJ band struc­ data in the vacuum. uv [28] (the horizontal scale for the EiW2 cnrve ture (histograms) [24 ] and as determined experimentally [25 1. has been shifted by 29.5 eV). 257 ty of states of the conduction band of Ge calculated by spectra are explained by the one-electron theory. The Herman, et aL. [4] and the corresponding density of exciton interaction is responsible, at most, for small states for GaSb as obtained by the k· p method [6]. details concerning the observed line shapes. It is The densities of states for both materials are very simi· generally accepted [31,32] that the exciton interaction lar because of the similarity of their band structures. suppresses structure in the neighborhood of M3 critical We also show in figure 9 the quantity EiW2 obtained by points: the Coulomb attraction with negative reduced Feuerbacher et al. [28] for Ge in the region of the M4 ,5 masses is equivalent to a repulsion with positive edge. The origin of energies has been shifted so as to masses. Such a repulsion smooths out critical point make a comparison with the conduction density of structure: no M3 critical point has been conclusively states possible: EiW2 should be proportional to D(w) identified in the experimental spectra. The El and El + under the assumption of constant matrix elements of p. ~, critical points of figures 1-3 are of the M, variety. While the rich structure of the calculated density of Hence the line shape of the corresponding Ei spectrum < states is not seen in the EiW2 curve, this curve is should be characterized by a steep low-energy side and reproduced quite well if the density of states is a broader high-energy side. Figure 10 shows the shape broadened so as to remove the fine structure. The of the E, peak observed at low temperature by Marple required lifetime broadening of about 1 eV is not un· and Ehrenreich [33] and by Cardona [34]. In order to reasonable for the M4 ,5 transitions. Using eq (2) with Nd avoid effects due to the overlap of the El and the E, + replaced by the conduction density of states we obtain ~, peaks it has been assumed that they have exactly the an average oscillator strength at the maximum of EiW2 F same shape but shifted by 0.55 eV. The contribution of = 0.15. This oscillator strength corresponds to the 20 4d per unit cell and hence it should be divided o 0 0 8 o by 20 to obtain the average oscillator strength per d­ o . CdTe band. If one reasons that the transitions from 10 of the /-". / I. 20' d bands to a given conduction band are forbidden / I· .. / I· because of the spin flip involved while transitions from 6 . / I: 5 of these 10 bands are forbidden or nearly forbidden by / 10 / ~ parity, one finds for the average o~cillator strength of / r each one of the 5 allowed bands F = 0.03, which cor­ 4 ,,/ \ /_- THEORY (KANE) \ = responds to a matrix element of p 0.13 (in atomic -- MARPLE ~. units): this value is quite reasonable in view of the fact • • • • CARDONA \ . \ ·0 -_ that the typical valence-conduction matrix element is 2 \ '-'•• ,.7 0.6. The small value of this matrix element explains ...... why the d core electrons are negligible in the k . P 3.3 3.4 3.5 3.6 analysis of the valence and conduction masses. eV

FIGURE 10. Contribution of the EI gap to E; in CdTe as measured 2. Exciton Effects at low temperatures by Marple and Ehrenreich [33] and by Cardona [34]. Also calculation by Kane [32) using the adiabatic approxima- We have devoted section 1 to a comparison of experi­ tion. mental optical spectra with calculations based on the only El has been extracted from the measured Ei one-electron band structure. Exciton effects, i. e. the spectrum and displayed in figure 10. It is clear from final state Coulomb interaction between the excited this figure that the E, peak is steeper at high energies electron and the hole left behind, are known to modify than at low energies, against the expectations for an Ml substantially the fundamental edge of peak. Also in figure 10 we show the results of a calcula- l and insulators [29]. Exciton-modified interband spec- tion by Kane [32] of the effect of Coulomb interaction tra seem also to occur in at interband edges on the E, line shape for CdTe, using the effective mass which have the final state on the Fermi surface [30]. approximation. The solution of the effective mass Experimental evidence for these effects is reported at Hamiltonian with non-positive-definite mass is made this conference in the paper by Kunz et al. easier by the fact that the negative mass (along the A We shall now discuss the question of exciton effects direction) has a magnitude much larger (about ten above the fundamental edge of insulators and semicon- times) than the two equal positive masses. It is possible ductors with special emphasis on the zincblende fami- to use the adiabatic approximation [31], i. e., to solve iy. A, mentioned in ,ection 1 the ",0" featme, of the.c ;he two·dime"ional hydrogen pmblem w;th thJ 25 third coordinate as a parameter and then solve the ONE ELECT RON KOS TER-SLATER adiabatic equation for the third coordinate. The agree· EXCITO N ment between the calculated and the experimental line s hapes of fi gure 10 is excellent. Attempts have been made to calculate the dielectric \, constant including exciton interactions at an arbitrary point of k space, independently of the stringent restric- > tions of the effective mass approximation [35,36] . Such calculation is possible if one truncates the Coulomb in­ teraction between electron and hole Wannier packets to extend to a finite number of neighboring cells. The ( extreme and simplest case of a a·function (Koster­

I Slater) interaction can be solved by hand [31,35] and I gives around an MJ critical point the shapes of EI' and Ei s hown in figure 11: for an Mi critical point the Koster­ Slater interaction mixes the Mi one·electron line shape l with the Mi+ l • The hi gh energy side of the Ei peak . becomes steeper , in agreement with fi gure 10. The line shape observed for the E I- E, + ~I peaks in the reflec- ( tivity spectrum is composed almost additively of the Ei and EI' line s hape: at the energies of these peaks dR/dEi F IGU RE 11 . Modification in €,. and €; introduced by the Koster­ and dR/dEl' are almost equal. We also s how in figure 11 Slater excitoll illteraction ill th.e neighborhood of all M, critical the line shapes expected for the refl ectivity spectra of point. Also, effect on the reflectivity under the ass umption of an the EI-EI + ~I peaks and for the corresponding dif- equal contribution of LI.E ,. and t1€; to the reflectivity line shope. r ferential spectra (dR/dw). We s how in figure 12 the photon energy derivative spectrum of these peaks in ) HgTe [37] : the o bserved line shapes di sagree with 2 .0 Hg Te those expected from the one-electron theory (equal 77°K 1.0 positive and negative peaks) but agree with those pre­

dicted in the presence of a Koster-Slater interaction 0 '"0 ( (fig. 11). Similar results have been found for other x l zincblende-type materials [ 37] . - -10 0:: "- 0:: - 20 3. Simplified Models for the Density of States

-3 .0 As seen in secti on 1 the optical density of states, and E., + 6, :> thus the dielectric constant, is a complicated function

of frequency and its calculation requires lengthy nu­ 2.2 2.4 2.6 2 .8 3 .0 3.2 merical computation. For some purposes, however, it eV can be approximated by simple functions. In the vicini- ~ ty of a critical point of the Mi vari ety, for instance, the FIGURE 12. Photon energy derivative spectrum of the reflectivity of HgTe in the neighborhood of the E, andE, + t1, structure [37]. singular behavior of the dielectric constant can be ap­ ~ proximated by: which most of the optical density of states is concen­ trated. The corresponding transitions occur over a large ECC i l' +' (w - W y ) 1/2 + co nstant (4) region of the BZ, close to its boundaries. In order to represent this fact, Penn [38] suggested the model of if exciton effects are neglected. Exciton interaction can a non-physical spherical BZ with an isotropic gap at its lbe included, within the Koster-Slater model, by mul­ boundaries. The complex energy bands of the material tiplying eq (4) by a factor eicJ> with 4> small and are then replaced by those of a free electron with an > positive. isotropic Wg at the boundary of a spheri cal BZ. This As shown in figure 1, Ei for the zincblende-type gap should occur in the vicinity of the E2 optical struc­ materials has a strong peak (E 2) in the neighborhood of ture. While this model represents rather poorly the rich 259 structure of Ei (fig. 1), it is expected that it should give Equation (5) yields for q = 0 the electronic contribu­ a good picture of E,· at zero frequency. The reshuffling tion to the static dielectric constant: of density of states involved in the case of the isotropic model should not affect E,.(W = 0) very much because of Wp - Wp - Eo=l+ § (- )" = 1+ ()"- (6) the large energy denominators which appear in eq (3) Wy Wy for W = 0: the lowest gap wo, usually much smaller than W" , accounts only for a very small fraction of the optical The experimental values of Eo agree reasonably well density of states. Penn obtained with this model the with the results of eq (6) using for Wg the energy of the static dielectric constant for a finite wavevector q. The E2 peak [34]. Equation 6 has gained recent interest as result can be approximated by the analytic expression the basis of Phillips and Van Vechten's theory of [38]: covalent bonding [41,42,43]. These authors use eq (6) and the experimental values of Eo to define the average ; ' ) WI) - { W'" q ? }_.)- dw = o, q) = l+ (- ) § 1+--§ 1/- (5) gap Wy. With this gap and the corresponding gap of the Wy Wy k", isoelectronic group IV material they can interpret a wide range of properties such as [42] , with binding energy [43] , energies of interband critical points [41], non-linear susceptibilities [44J , etc. As an example we discuss the hydrostatic pressure (i. e. In eq (5) Wp is the frequency obtained for the volume) dependence of Eo for germanium and . density of valence electrons and w'" and k", the cor­ According to Van Vechten [41], Wy for C, Ge, Si, and responding free electron Fermi energy and wave a-Sn is proportional to (ao) - 2, 5 where ao is the lattice number. The dimensionless quantity § is usually close constant. If one makes the assumption that this law to one. gives also the change in Wy with lattice constant for a Figure 13 shows eq (5) for Si compared with the exact given material when hydrostatic stress is applied one results of the Penn model [39]. These results are obvi­ can calculate the volume dependence of Eo [41]. ously independent of the direction of q. A s mall depen­ Neglecting the one in eq (6), a valid approximation for dence on this direction is found from a complete pseu­ Ge and Si, one finds: dopotential calculation by N ara [40] (see also fig. 13). The function E(O,q) is of interest for the treatment of .ldEo=2(dln Wp din W y ) dielectric screening. Eo dV dV dV

= 2 [0.83 - 0.50] = 0.66 (7) - Srinivasan's Result --- Nara's Result 12 Along [Ill] Equation (7) explains the sign and the small magnitude ---Penn's Interpolation \ Formula observed for (I/Eo)(dEo/dV). The experimental values of 10 \ . this quantity are 1.0 for Ge and 0.6 for Si [41,45]. \\ According to eq (6) the average gap Wg determines e:IO,ql \\ 8 \ \ SILICON the electronic dielectric constant for W = o. As the c \ \ lowest gap Wo is approached (wo ~ Wy usually), E,. I \ '\ 6 \ . exhibits strong dispersion. This dispersion is due, in the \ \ spirit of eq (3), to the density of states in the vicinity of " ". woo For the purpose of calculating the dispersion of E,. 4 , " "',. " ~ immediately below wo, the density of states can be ap- ' '.... ' :::::::.,. " proximated by that of parabolic bands with a reduced mass equal to the reduced mass f.L at woo These bands are assumed to extend to infinity in k space: the O~~~~~~~~~~~.-~ 0.1 0.3 0.5 O} 09 1.1 unphysical contribution to Er for Ik I ~ 00 should be ql KF small for W ~ W o, because of the large energy denomina­ tors of eq (3). We thy.s obtain for a cubic material the FIGURE 13. Static dielectric constant E( 0 , q ) obtained by Srinivasan [39 1for Si with the Penn model compared with the interpolation following contribution of the Wo gap to the scalar dielec­ formula of eq (5) and with the results of a pseudopotential calcu­ tric constant below Wo (under the assumption of a con­ lationbyNara [40Jforq along (lll). stant matrix element of p equal to P ) [46]: 260 ~E,. = 2(2fL)3/2 wg/2 i P i2f(wlwo) = C~f(wlwo) with (8) o f(x) = 2- (l +x) 1/2_ (I-X) 1/2.

Equation (8) represents quite well the behavior of Er GaAs [1001 STRESS \ immediately below Wo for the lead chalcogenides [47] and a number of other semiconductors [48]. As an ex­ ample we show in figure 14 the observed dispersion of E,. below Wo at room temperature [49] together with a fit based on eq (8) [48]. For the sake of completeness we have included in the fittiftg equations not only the effect - 2 I _~ _ ~ L-_~_~ 0.4 0 .8 1.2 of Wo (Eo) but also that of its spin-orbit-split mate Eo + ~o ° eV (also represented by an expression similar to eq (8)), the FIGURE 15. Piezobire}i-ingence in CaAs for an extensive stress along dispersion due to the EI and EI ~I gaps, and that due + (J 00) (room temperature). The circles are experimental points. The to the main Wg gap assuming Wg == E2 • Thus the fitting line is a fit based on the model of eq (5) [481. equation with three adjustable parameters C~, C;, C~ is [48]: Ce, CaAs [48J, and other IIl-V semiconductors ;> [50,51]: uniaxial stress splits th e top valence band state ([8) and a birefringence in the contribution of Eo to Er results because of the selection rules for transi­ tions from the split bands. The main contribution to this piezobirefringence is expected to be proportional where: to f'(x), which diverges like (w- wo) - 1/2 for W ---c> woo W Such behavior can be seen in the experime ntal results Wos=Wo+~o, Xos==-, W as (circles) of figure 15 obtained for CaAs at room tem­ perature. Included in this fi gure is the corresponding W W XIS=-, x·,=- WI., - W" fit based on the model of eq (9) [481. The long wavelength, non-dispersive contribution to x:! h(xd=l +i· the piezobirefringence of fi gure 15 can be interpre ted, at least qualitatively, in terms of the Penn model of eqs The fitting values of C~ (6.602) and C; (2.791) are in (6) and (7). Equation (7) yields two contributions to the qualitative agreement with those calculated from the c hange in Er one due to the change in plasma frequency band parameters [48]. (i. e. carrier de nsity) with stress and th e other due to The parabolic model density of states can also be the c hange in the isotropic gap. The first contribution used to interpret the strong dispersion in the piezo­ should not exist for a pure shear stress. For a hydro­ birefringence observed near the lowest direct gap of ;, static stress the second contribution can be written in tensor form as: GaAs 13

1 l (10) -- ~E'o= 5e 12 Eo

II where e is the strain tensor. We postulate that eq (8) remains valid for pure sheer stress. This crude generalization has a clear physical meaning in terms of 10 L-___ L-_L-_~_~1 _~ ___L-_L_~ o 0.4 0 .8 1.2 1.6 the Penn model. The spherical BZ becomes ellipsoidal eV under a sheer stress and the at an arbitrary point of the BZ boundary kv becomes anisotropic. The FIGUHE 14. Experimental results for E,· in GaAs below the funda· mental edge at room temperature [491 (circles) and fitted curve gap at kv is assumed to become larger as kv becomes based on a model density of states. larger (kF is the distance be tween atomic planes per- 261 pendicular to kl'). Equation (8) gives the right sign for where the one·dimensional electro·optic function GI ('>7) the long wavelength contribution to the piezobirefrin­ is given by [54]: gence of figure 15 but a magnitude about five times GI (->7) = 21TAi(y/)Bi(y/) -H(y/)y/-1/2 larger. The agreement becomes better if the contribu­ (14) , tion of the Eo edge to the long wavelength piezobirefrin· gence, of opposite sign to that predicted by eq (10), is subtracted from the experimental results. In eq (14) H (y/) is the unit step function and fL . the re­

duced mass of the Penn model, given by J.-t= wll2kl')- 2. 4. Third Order Susceptibility and Model The Fermi momentum of the valence electrons Density of States is related to the plasma frequency Wp through It has been recently suggested [52] that the third k[<'3 = (3/4)1TWp2. The limit for ?if --70 in eq (14) is easily found using order susceptibi}jty of Ge, Si, and GaAs at long the asymptotic expansions of Ai (y/) and Bi ('ry) for wavelengths is related to the Franz·Keldysh effect (i. e. y/--7+oo [55]. By subsequently performing the limit the intraband coupling by the field) of interband critical for W--70 one finds: points [53]. We discuss now the Franz-Keldysh con· tribution of the Wg , Eo, and EI gaps to X~~)II' (15) 4.1. Average Gap Wy Or, for ease of evaluation, with X~~)II in e.s.u., and the In the spirit of Penn's model [38] we represent the energies in e V: long· wavelength dielectric constant by eq (6) with .'F = 1. The corresponding imaginary part of the dielec· tric constant is, for W > Wg [47]:

We list in table I the values of Wg,~) and Eo - 1 for Ge, Si and GaAs. The values of X(3) calculated with eq (11) 1111 with (16) are then listed in table II. This table shows agree­ ment in sign and magnitude between the values of X(:3) IlIl The Franz-Keldysh effect for the one-dimensional abo predicted from Wg and the experimental ones. An in- ' sorption edge of eq (11) can be expressed in terms of crease in the polarizability with field (X(3) 0) is to be 1111 > the Airy functions Ai and Bi [54]. One must mention, expected for the Franz-Keldysh effect since the in- however, that the isotropic gap problem in the presence traband coupling by the electric field produces a of an electric field would only be completely equivalent decrease in the energy gap. to the one·dimensional problem if the field experienced We shall consider now the contribution to X(11113) of the by every electron were along the direction of the cor­ interband coupling by the electric field across the responding k. The fact that the field ~ is the same for isotropic gap %. This coupling produces an increase in all electrons, regardless of k, can be taken into account energy gap, and thus its contribution to xBL is nega­ by using an average field: tive. This contribution to Xl~)11 is readily found from eq (6): (12)

where f3 is the angle between k and '6'. The long­ wavelength expression for X~~)II thus is [54]: =_ 3(En-l) (31T) 2/:3 w1:l 51T 4 w~ (17)

In eq (17) we have made use of the second-order pertur· bation expression: (13)

2k2 dWg 1 < vlrlc > 12 -'-' d(fP ) = 2 Wg uV (18) 262 Equation (18) is in agree men t with the results of ref. [44]. Comparison of this equation with eq (15) shows of /-to and Wo = Eo + ~o for Ce, Si and CaAs are li sted in that the magnitude of the interband contribution to Xml table I. Using these numbers, eq (20) yields th e values differs from that of the intraband contribution by a fac­ of x\~L (w = 0) listed in table II. While this contribution tor of order wr/ (Wy in atomic units). It is therefore al- TABLE 1. Values of the parameters required for the evaluation (> most two orders of magnitude smaller and hence of the Franz-Keldysh co ntributions to X<,'~)II ' Frequ.e ncies in ~ negligible. e V, 30 in Bohr radii, 1-'-0 in units of the free electron. mass. p' has been. taken equal to 0.4 for all materials. 4.2. Lowest Gap Eo l We use for the contribution of an isotropic Mo critical Ce Si CaAs /' point to the real part of the dielectric constant the result of eq (8). A calculation similar to that performed W g 4.3 4.8 5.2 , above yields for an Mo critical point the followin g Franz­ Wp 15.5 17.35 15.5 Keldysh contribution to X\:~)ll [52]: f o - 1 11 15 10 1-'- 0 0.03 0.04 0.05

'J I / ') .J Wo 0.9 4 1.5 A (:1) - 006P-fLO-(1 185(W)- 10.7 10.3 10.7 LlXllll-' --,,-.,- + . - + . . .) (19) ~ w ~- W u w, 2.2 3.3 3.0

or, transforming IlXm I to e .s.u. and Wo to e V (P 2 and fLo are left in atomic units for ease of co mputation): TABLE 11. Contribution of th e vorious Franz- Keldysh eileCl.s discllssed here 10 x\'ll, and Xm,·I II IInit s of 10- 11' e.s.u. Also, experimental values of th e bOllnd carrier con trilJlltions to xl::, and xW, ·

10' , E" Ell Experi - contribu- co nt ribu- co ntribu- lll enl 57 ti on tion tion

Xl:)) 1 1\ 1 0.26 0.67 0.20 1.0 We have included in eqs (19) a nd (2 0) the first term in Ce XC)) 0.26 0.67 0.26 1.5 the di spersion of IlXWII since it may be possible to ob­ UU

Xl :!) serve it experime ntally in s mall materials. 1 111 0.22 0.00 0.27 0.06 This di spersion is give n exactl y by the function: Si x"U(:" 0.22 0.00 0 ..3 6 0.08

X( :lI 2 1111 0. 12 0.087 0.045 0. 12 W (( W ) - 2 . .', ( W ) - 2. 5 ) (21) w ~ 1 + Wo + 1 - W o - 2W;;-2 ..; CaAs Xuul:!) 0.1 2 0.087 0.060 0.10 :> Equation (21) is not immediately valid for the Eo edge because of the degeneracy of the vale nce band. How­ is zero for Si and is not excessive in CaAs (it may, there­ ever , one can apply it to the Eo edge if one neglects the fore, be assumed as included in the average gap calcu­ field coupling between degene rate valence bands and lation given above), it is dominant in Ce. In first approx­ ~ uses appropriate average values of p2 and fLo. Each one imation it may be added to the average gap calculation: "7 of the three valence bands can be assumed to have a excellent agreement with the experimental results is mass equal to three times the conduction band mass then found. and a corresponding ma trix element equal to tp2 [48]. For InAs, with Wo= 0.5 eV and fL = 0.02, we find from He nce eq (20) must be used with the matrix element p 2 eq (20) IlXWll (w = 0) = 7 . 10- 10 , which is of the order of and /-to=;f m e if the three valence bands are to be in­ the free-carrier contribution for the samples with the cluded. The s pin-orbit splitting Ilo of the valence band lowest electron concentrations measured (N = 2 . 10 16 / is ta ke n into account, if Ilo ~ E o, by replacing W o by its cm- 3 ) [57] . This is contrary to the state ment found in average value Eo + ~o. Since p 2 is almost the same for the literature that for these carrier concentrati ons in all materials of the germanium family, we can replace InAs XWll is dominated by the free-carrier contribution it by a typical value = 0.4 (in atomic units). The values [56,57,58]. 263 X(3) W = The interband contribution of Eo to 1111 for 0 is (xmg > Xm I) observed for Ge and Si, but not for GaAs easily obtained from the expression (see eq (8)): [57]. The Franz-Keldysh contribution of EI to X(3) as 1111 found by the procedure sketched above, is: (2fLo P / ~ p2 . ~Er(W=O)= W - ·l / 2 (22) 2 y A (3) -052 p2 UXIIII-' -5 (24) aowl If one assumes that the repulsion produced by the field affects fLo in the manner predicted by the k . p expres­ or, with WI in e V and X\~)l1 in e.s. U.: sion with constant matrix elements of P(fLo ex Wg), the corresponding interband contribution to Xml vanishes. ~X · (3) = 2 7 . 10- 8 p2 1111' 5 (25) If, on the other hand, one assumes fLo to be field inde­ aowl pendent one also finds a negative contribution to X\1)11 " about two orders of magnitude smaller than the Franz­ (ao in Bohr radii and Pz in atomic units.) Keldysh contribution, and hence negligible. The matrix element P should have approximately the < same value as for the Eo gap. In order to take care of the spin-orbit splitting ~I of EI we substitute WI by EI 4.3. EI Critical Points + ~d2. The approximate values of ao, and WI for Ge, Si .-' and GaAs are listed in table 1. The values calculated for The EI optical structure is usually attributed to MI < X(3) y(3) critical points along the {Ill} directions. While M1 the Franz-Keldysh contribution to EI to 1III and ',{;ga critical points are known to yield no contribution to are listed in table II. While the calculated anisotropy Xml [52], there are Mo critical points of the same sym­ has, for Ge and Si, the sign observed experimentally, its metry slightly below the M1 critical points. This com­ magnitude is far too small to explain the experimental bination of Mo and MI critical_points with a very large anisotropy, especially after the Eo and the E" longitudinal mass, can actually be approximated by contributions are added. There is a possibility that the two-dimensional minima [ 48J . The contribution of one EI contribution of eq (25) may have been underesti­ of these two-dimensional mlDlma to the long­ mated. Exciton quenching effects [59,60J , not included wavelength dielectric constant is (we assume four, and in our calculation, may increase this contribution. not eight equivalent {Ill} directions): We cannot offer even a qualitative explanation of the sign of the X(3) anisotropy observed for GaAs. It would be interesting to determine, through measurements of ~Er (23) other III- V or II-VI compounds, whether it is con- ~ nected with the lack of inversion symmetry in these where ao is the lattice constant and Pz the appropriate materials. square matrix element. We have tried to calculate the The interband contribution of the L edges can be evaluated in a manner analogous to that used for the Eo Franz-Keldysh contribution to X(3)1111 of these two-dimen- sional critical points in a way similar to that used above, and the Wg gaps. We also find that this contribution is but we have run into difficulties when evaluating the negative and, typically, two orders of magnitude below limits of the two-dimensional electro·optic functions for the Franz-Keldysh contribution. YJ ~ + 00. In view of this we have instead evaluated the effect of the three-dimensional Mo critical points, with S. Acknowledgments the longitudinal effective mass replaced by the value I am indebted to Drs. Buss, Kane, Phillips, Van required to give at long wavelengths a contribution to V echten, and As pnes for sending pre prints of their E,. equal to that in eq (23). work, prior to publication and to the staff of DESY for Under these conditions, and because of the large lon­ their hospitality. gitudinal mass, only fields transverse to the critical point axis contribute to XmJ' When summing the con­ tributions of the four equivalent valleys, it is found that 6. 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