Penn Model Research Articles

Frequency dependence of the photoelastic coefficients of silicon

We have measured the frequency dependence of the individual photoelastic coefficients of silicon using a simplified acousto-optic technique. We define the photoelastic tensor $q$ by $\ensuremath{\Delta}{\ensuremath{\epsilon}}_{\mathrm{ij}}={q}_{\mathrm{ijkl}}{e}_{\mathrm{kl}}$, where $e$ is the particle displacement gradient tensor and $\ensuremath{\Delta}\ensuremath{\epsilon}$ is the strain-induced change in the optical dielectric tensor. We find ${q}_{1111}(3.39 \ensuremath{\mu}m)=+13.0\ifmmode\pm\else\textpm\fi{}0.6$, ${q}_{1111}(1.15 \ensuremath{\mu}m)=+15.7\ifmmode\pm\else\textpm\fi{}1.1$, ${q}_{1122}(3.39 \ensuremath{\mu}m)=\ensuremath{-}2.41\ifmmode\pm\else\textpm\fi{}0.1$, and ${q}_{1122}(1.15 \ensuremath{\mu}m)=\ensuremath{-}1.45\ifmmode\pm\else\textpm\fi{}0.1$. The extrapolated long-wavelength limit of the average photoelastic coefficient agrees well with our previous estimate for the frequency-independent Phillips---Van Vechten model. However, within a proper frequency-dependent Penn model we have shown that the oscillator strength does not vary as ${r}^{\ensuremath{-}3}$. Further we have demonstrated that the dispersion energy ${E}_{d}$ in the Wemple-DiDomenico model is proportional to ${r}^{\ensuremath{-}1.9\ifmmode\pm\else\textpm\fi{}0.8}$ and not volume independent as would be expected from the model. It is concluded that whereas a simple single-gap model works well to describe the low-frequency dispersion in the dielectric constant of silicon, it is incapable of describing the dispersion in the photoelastic tensor.

Screened-exchange plus Coulomb-hole correlated Hartree-Fock energy bands for LiF

Green's-function techniques have been used to derive screened-exchange plus Coulomb-hole correlation corrections to Hartree-Fock energy bands. These correlation corrections or energy shifts consist of a statically screened exchange term which is state dependent, and a Coulomb-hole term which is constant in our diagonal approximation for the screening (dielectric) function. These energy shifts raise the occupied bands and lower the conduction bands with a resulting decrease in energy differences. The calculation was done for LiF at general points in the first zone using linear-combination-of-atomic-orbitals Hartree-Fock energy bands and the Penn-model dielectric function. A change in.the band gap at $\ensuremath{\Gamma}$ of 5.0 eV was obtained compared with the experimental value of 9.3 eV. The calculation was also done using the random-phase-approximation (RPA) dielectric function and it was found that the diagonal part of the RPA gives less than half of the correlation obtained with the Penn model.

Correlated Hartree-Fock energy bands for diamond

The screened-exchange-plus---Coulomb-hole method has been used to add correlation to Hartree-Fock energy bands. These correlation corrections, or energy shifts, are state dependent due to the screened-exchange term. The Coulomb-hole term is constant throughout the zone in our diagonal approximation for the screening (dielectric) function. The energy shifts raise the occupied bands and lower the conduction bands with a resulting decrease in energy differences. The calculation has been done for diamond using linear combination of atomic orbitals Hartree-Fock bands and the Penn-model dielectric function. The energy shifts, which tend to flatten the bands, were computed at general points in the first zone. Values of 5.6 and 7.6 eV were obtained for the indirect and direct band gaps respectively, both of which are in close agreement with experiment. The diagonal part of the random-phase approximation was also used in the calculation and it was found to produce about two-thirds of the correlation obtained with the Penn model.

Improved calculations of the complex dielectric constant of semiconductors

Expressions for the real, static part ${\ensuremath{\epsilon}}_{1}(0)$ and the imaginary part ${\ensuremath{\epsilon}}_{2}(\ensuremath{\omega})$ of the dielectric constant of semiconductors in the long-wavelength limit are obtained using the isotropic nearly-free-electron band approximation (Penn model). Earlier calculations of these functions do not satisfy the Kramers-Kronig relations and yield an excessively large result for the $f$-sum rule. The corrected expressions eliminate these inconsistencies. Values of the energy gap between the bonding and antibonding states are obtained for diamond, silicon, and germanium, respectively. ${\ensuremath{\epsilon}}_{1}(\ensuremath{\omega})$ is obtained from ${\ensuremath{\epsilon}}_{2}(\ensuremath{\omega})$ through the use of the Kramers-Kronig relation. The theoretical curves for ${\ensuremath{\epsilon}}_{1}(\ensuremath{\omega})$ and ${\ensuremath{\epsilon}}_{2}(\ensuremath{\omega})$ are compared with experimental results.

Dielectric function of one-dimensional peirls insulator

We use Penn's model to calculate the complex dielectric functional and reflectivity of a quasi-one-dimensional insulator. A brief comparison with KCP is made.

Three-dimensional model for surface states

A detailed theoretical discussion of the surface bands in the Penn model is presented. It is found that a surface band always exists; another surface band may exist, if the gap is not too small and well below the vacuum level, but it plunges always in the valence band.

Dielectric Function of a Model Semiconductor

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Pressure Dependence of the Refractive Index of Amorphous Lone-Pair Semiconductors

The pressure dependence of the refractive index ($\frac{\mathrm{dn}}{\mathrm{dP}}$) of several amorphous chalcogenide semiconductors is reported. $\frac{\mathrm{dn}}{\mathrm{dP}}$ is positive for materials containing group-VI elements in twofold coordination (lone-pair semiconductors), whereas it is negative for tetrahedral semiconductors. Furthermore, $\frac{\mathrm{dn}}{\mathrm{dP}}$ is the same for the amorphous and crystalline phases of a given semiconductor. Although a one-oscillator (Penn) model adequately predicts $\frac{\mathrm{dn}}{\mathrm{dP}}$ for tetrahedral semiconductors, a similar approach does not predict the positive $\frac{\mathrm{dn}}{\mathrm{dP}}$ of lone-pair semiconductors. Local-field corrections appear to be the cause of the positive $\frac{\mathrm{dn}}{\mathrm{dP}}$. The Lorenz-Lorentz description of the local field predicts $\frac{\mathrm{dn}}{\mathrm{dP}}$ in agreement with experiment for most molecular lone-pair materials. It fails, however, for materials containing large concentrations of group-IV elements. The limitations of the local-field correction in describing $\frac{\mathrm{dn}}{\mathrm{dP}}$ of partially molecular semiconductors is discussed.

Optical properties of amorphous III–V compounds. I. Experiment

AbstractFrom reflectivity measurements ε1 and ε2, the real and imaginary part of the dielectric constant, of the amorphous III‐V compounds InSb, InAs, InP, GaSb, GaAs, and GaP are determined in the fundamental absorption region up to 12 eV. Compared with the corresponding curves of the crystals, the spectra of the disordered materials are smeared out and shifted to lower energy. This shift leads to an increase of ε1(0). In the energy region of the E2‐peaks a selective decrease of ε2 is observed. The results for ε2 are briefly discussed in terms of structural properties, and a modified Penn model is applied to the spectral dependence of ε2.

The effective mass and q dependent dielectric function of simple semiconductors

The Penn dielectric function is often used in the theory of semiconductors. It is shown that although the Penn model is reasonable as regards the electronic spectrum it considerably overestimates the interband matrix element. A correction factor is determined by group theoretical considerations and is shown to give agreement with an appropriate effective mass.

The microscopic dielectric function and charge density in germanium

The microscopic dielectric function epsilon (q) of germanium was calculated using a pseudopotential scheme. Fifteen bands and 23040 points were included in the sampling of the Brillouin zone 14.67+or-0.4 was obtained from the simple Penn model. The Fourier components of the charge density and the histogram of the density of states are also presented. The comparison with experiment gives good agreement.

Temperature Coefficient of the Refractive Index of Diamond- and Zinc-Blende-Type Semiconductors

We have calculated the temperature coefficient of the long-wavelength refractive index of several group-IV and III-V semiconductors, using the Penn model for the electronic contribution to the dielectric constant. The isotropic band gap of this model is identified with the band gap at the $X$ point of the Brillouin zone, which can be simply expressed in terms of pseudopotential coefficients. The explicit temperature dependence of this gap is calculated by applying to these pseudopotential coefficients the appropriate Debye-Waller factors. The thermal expansion effect is obtained in the manner suggested recently by Van Vechten. Good agreement between the calculated and the observed temperature dependence of the long-wavelength refractive index is found.

The microscopic dielectric function in silicon and diamond

The microscopic dielectric function epsilon (q) of diamond and silicon was calculated using a pseudopotential scheme. Sufficient plane waves were included to obtain energy convergence to within 0.1 eV. Four valence, seven conduction bands, and 11520 points were used in the sampling of the Brillouin zone. Good agreement with experiment was obtained for the static limit epsilon (0). The effect of the higher conduction bands was investigated and it was shown that it cannot be neglected. The calculated values of epsilon (q) for q>or=q (Fermi) are smaller than those predicted from the simple Penn model. The covalent 222 contribution to the valence charge density in Si was calculated as 0.219 electrons/atom, in agreement with the experimental value 0.22.

Optical Properties and Electronic Density of States.

The fundamental absorption spectrum of a solid yields information about critical points in the optical density of states. This information can be used to adjust parameters of the band structure. Once the adjusted band structure is known, the optical properties and the density of states can be generated by numerical integration. We review in this paper the parametrization techniques used for obtaining band structures suitable for density of states calculations. The calculated optical constants are compared with experimental results. The energy derivative of these optical constants is discussed in connection with results of modulated reflectance measurements. It is also shown that information about density of empty states can be obtained from optical experiments involving excitation from deep core levels to the conduction band. A detailed comparison of the calculated one-electron optical line shapes with experiment reveals deviations which can be interpreted as exciton effects. The accumulating experimental evidence pointing in this direction is reviewed together with the existing theory of these effects. A number of simple models for the complicated interband density of states of an insulator have been proposed. We review in particular the Penn model, which can be used to account for response functions at zero frequency, and the parabolic model, which can be used to account for the dispersion of response functions in the immediate vicinity of the fundamental absorption edge.

Electronic band structure and covalency in diamond-type semiconductors

The band structure of a diamond-type semiconductor near the energy gap can be described simply in terms of the Jones zone. The energy gap is nearly constant over the Jones-zone faces except close to the edges and corners. This accounts for the success of the spherically symmetric Penn model of the dielectric constant . It also explains the qualitative shape of the optical absorption spectrum, in particular the occurrence of the first peak. Specifically covalent effects arise from the fact that the ν(111) Fourier component of the pseudopotential is relatively large and has to be treated beyond lowest order of perturbation theory. Simple algebraic formulae give the principal energy gap and the position of the minimum in the conduction band along ΓΔX, which agree with the large computer calculations to within about 25%. This picture accounts in a simple approximate way for the variation with pressure of the dielectric constant and of the position of the main peak in the optical absorption spectrum. Finally a microscopic calculation of the extra charge Qb located in the covalent bond gives agreement with Phillips' conjecture that Qb = 2-1.

Microscopic Dielectric Function of a Model Semiconductor

Penn's model of an isotropic semiconductor is used to calculate $\ensuremath{\epsilon}(q)$, the diagonal part of the static micro-scopic dielectric function. Penn's results have been extended beyond $q={K}_{F}$. This region is important in developing a quantum theory of lattice vibrations of diamond and zinc-blende crystals. Explicit results are given for the range $0<q<4{K}_{F}$.

Covalent Bond in Crystals. I. Elements of a Structural Theory

An a posteriori theory is developed for the structural energy of covalent crystals. The microscopic theory is based on ionic pseudopotentials and valence dielectric screening. The theory explains the difference between empirical pseudopotential form factors derived from the optical spectra of semiconductors and the metallic form factors calculated from free-ion term values by Animalu and Heine. A byproduct of the theory, which utilizes Penn's model isotropic semiconductor dielectric function, is a relation between the covalent bonding charge and the macroscopic dielectric constant. In self-consistent form the theory is an example of the "bootstrap" approach, applied here to treat the effect of covalent bonding on the ground-state energy of the valence electron gas. It is argued that the axiomatic character of the covalent theory is to be expected on symmetry grounds, and it is shown that the theory is superior to a nonlinear multiple-scattering theory based on the free-electron dielectric function. The extension of the theory to III---V and II---VI semiconductors is described briefly. The theory may be used to calculate elastic and macroscopic dielectric properties of covalent crystals starting only from ionic pseudopotential form factors.

Infrared Dielectric Constant and Ultraviolet Optical Properties of Solids with Diamond, Zinc Blende, Wurtzite, and Rocksalt Structure

The room-temperature optical constants of InP, CdTe, and ZnTe are calculated for photon energies up to 20 eV by the Kramers-Kronig analysis of the normal-incidence reflection spectra. Data at 77°K are also presented for ZnTe and CdTe. At these temperatures, asymmetries in some of the peaks in ε2, of the type attributed by Phillips to resonant exciton effects interacting with a continuous background, are seen. The position of the E2 peaks (due to transitions at X) is listed for materials of the diamond-zinc blende family, as observed in the spectrum of R (the reflection coefficient), ε1, ε2 (real and imaginary parts of the dielectric constant) n, k (real and imaginary parts of the refractive index), and the absorption coefficient α. We have also listed the various peaks corresponding to E2 peaks of zinc blende for materials with wurtzite structure. The infrared dielectric constant ε∞ is calculated for these materials from the position of E2 on the base of Penn's model for an isotropic free electron gas with an energy gap. Good agreement with the experimental values of ε∞ is obtained. The model is generalized to treat PbS, PbSe, PbTe, and SnTe. Good agreement between calculated and experimental values of ε∞ is also obtained for these materials.