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

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Optical Properties and Electronic Density of States* 1 ( Manuel Cardona2 > Brown University, Providence, Rhode Island and DESY, Homburg, Germany 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 Density of States* 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 exciton 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-Electron 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 Brillouin zone 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 symmetry 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].
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