Central-cell Corrections Research Articles

Instabilities and Δ‐Distributions in Solid State Physics

AbstractIt is not admissible to use Δ‐distributions as potentials, e.g. as central cell corrections, as it has been done by Lucovsky and Kiefer for doped semiconductors considered in effective mass approximation (EMA). Rigorous theorems like Kato's one for the wave function and Kimball's one for the pair distribution function are due to the Δ‐terms contained not appropriate to test the quality of trial functions. The intensity formula for the radiative recombination contains the dipole matrix element having in EMA two instabilities, one arising from a Δ‐term. The other one is connected with the asymptotic decay of the trial functions which also makes the usefulness of the notion of the localization length doubtful.

Residual donors in LEC indium phosphide

We report well-resolved structure in the neutral donor bound exciton “two-electron” transition photoluminescence satellites from selected relatively pure, nominally undoped single crystals of InP grown by the liquid encapsulated Czochralski (LEC) technique. This structure results from transitions at different donor species, adequately resolvable only in spectra recorded at high magnetic fields and low temperatures. Three or four donors appear. Sulphur is usually a significant species, Si is a minor donor, and an unidentified species is a principal donor which has a small negative central cell correction. These data contrast strongly with those for InP layers grown by the halide and metal organic vapour epitaxial techniques in which the dominant residual donors are S and Si, respectively.

Calculation of the first ionization energies of hydrogenic acceptor pairs in silicon

We have calculated ground state energies of neutral and singly ionized diatomic acceptor pairs involving boron, aluminum, gallium, indium, and thallium at the first 13 nearest-neighbor separations in the silicon lattice. Each acceptor is characterized by its own central-cell correction and is treated within the effective-mass approximation. The ionization energies of the neutral or the singly ionized pairs do not conform to the Ohmer–Lang empirical half-sum rule for the ionization energies of the associated acceptor X centers. The results of the calculation might find application in further studies of heavily or double-doped p-type silicon.

Identification of donors in vapor grown indium phosphide

We report well resolved substructure in the ‘‘two electron’’ satellite (TES) near band-gap photoluminescence (PL) of excitons bound to shallow donors in InP, observable in high magnetic fields above ∼3 T. This substructure arises from differences in the central cell corrections to the binding energies of different donor species, resolved here for the first time in PL spectra for InP. Six different donor species are differentiated in undoped vapor phase epitaxial (VPE) layers. The two most persistent contaminants are identified with S and Si from back-doped crystals which yield useful data for uncompensated donor concentrations up to ∼5×1015 cm−3 at ∼9.5 T. The most important donor contaminant in refined halide transport VPE InP is S, but Si is dominant in layers grown with metalorganic sources of In. The TES identifications are supported by similar substructure with appropriately reduced splittings in the ionized donor BE PL. A profile of the Si/S donor concentration ratio as a function of layer thickness is presented, and the ‘‘mole-fraction’’ technique is shown to control the relative proportion of Si donors.

Effect of chemical shift on polarizabilities of donors in semiconductors

Following Hassé's variational method, an analytic expression of the donor polarizability is obtained, including the effect of central-cell correction by means of a semi-empirical short range potential. It is shown that the electrical polarizability is very sensitive to the shape of the impurity potential. Quantitative results are given for donor polarizabilities in Si, GaP and ZnSe and compared to previous theoretical and experimental work.

Excited states of the Zn and C acceptors inAl0.47Ga0.53As

Six excited states of Zn and of a shallow acceptor $X$ in the semiconducting alloy ${\mathrm{Al}}_{0.47}$${\mathrm{Ga}}_{0.53}$As have been determined by means of the selective pair luminescence technique. Their binding energies are compared with the effective-mass theory of Baldereschi and Lipari. For Zn the agreement is only qualitative. The central-cell correction amounts to 14.5 meV. Quantitative agreement is obtained for the shallow acceptor $X$ (no central-cell correction), provided that the Kohn-Luttinger valence-band parameters are slightly adapted: ${\ensuremath{\gamma}}_{1}=4.68$, ${\ensuremath{\gamma}}_{2}=1.29$, ${\ensuremath{\gamma}}_{3}=1.87$, and ${R}_{0}=23.6$ meV. Most probably, $X$ has to be identified with carbon.

Zero-phonon recombination spectra of donor-acceptor pairs in GaP and ZnSe: Model-impurity-potential approach

The influence of central-cell corrections on electron-hole recombination energies of donor-acceptor (DA) pairs in III-V and II-VI semiconductors is investigated. A variational treatment of the DA complex is developed in the framework of the effective-mass theory, including the effects of interaction between the charge carriers and the longitudinal-optical phonons within the static approximation. The effective-mass Hamiltonian of the DA pair, based on Fr\ohlich's continuum theory, allows for central-cell corrections by means of a suitable impurity model potential adjusted to experiment in the case of isolated impurities. The orbital radii of the electron and hole wave function, its departure from spherical symmetry, and the DA-pair energy are given as a function of the donor-acceptor separation $R$. The central-cell correction is found to decrease appreciably at small $R$. The results are compared with previous theoretical work and experiments on zero-phonon spectra of DA pairs in ZnSe and in GaP. It is shown that the central-cell correction cannot be neglected in order to explain the radiative DA-pair recombination energies. The model potential adopted in the present work reproduces the general trend of the $R$-dependent electron-hole recombination energies deduced from DA-pair spectra involving dopants of various depth.

Photoluminescence study of acceptors in AlxGa1−xAs

A photoluminescence study of C, Si, Ge, Be, Mg, and Zn acceptors in AlxGa1−xAs is made. The binding energy of these acceptors is determined from the free-to-bound transitions at 75 K as a function of aluminum composition up to x∼0.4. It is observed that C, Be, and Mg behave like effective mass acceptors in AlxGa1−xAs while Si, Zn, and Ge show positive deviations from the effective mass theory with Ge exhibiting the maximum deviation. This difference in the behavior of the acceptors is suggested to be due to the difference in their central cell corrections in GaAs. The relative merits of these acceptors, in terms of their solubility, ionization energy, and diffusivity are considered. It is concluded that Mg is an attractive acceptor dopant in AlxGa1−xAs grown by liquid phase epitaxy.

K.p perturbation theory for light absorption in semiconductors doped by 3d metals

A theory of light absorption in semiconductors doped by transition metal impurities is constructed. The absorption cross section near the edge of the optical band is calculated by means of k.p perturbation theory within the framework of the Kane model supplemented by the d impurity levels. Special attention is given to the problem of gauge invariance in k.p theory. It is shown that in the case of an impurity state with definite parity the Lucovsky formula is valid when modified by the more detailed account of the central cell corrections. In the substitutional positions without the inversion centre the impurity states possess no definite parity which allows the dipole transitions forbidden by the conventional crystal field theory and causes the linear Stark effect for the t2 levels.

Core excitons at the boronKedge in hexagonal BN

High-resolution measurements of soft-x-ray absorption near the boron $K$ threshold of hexagonal BN display a prominent peak at 192.0 eV. The angular dependence of this absorption for $p$-polarization geometry is consistent with a final state constructed mostly of $\ensuremath{\pi}$-like wave functions. Although higher-energy parts of the spectrum can be approximated by an effective-mass exciton theory, the 192.0-eV peak requires a substantial central-cell correction. This core exciton is thus highly localized in real space and delocalized in $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ space. It seems likely that a mixture of $\ensuremath{\sigma}$- and $\ensuremath{\pi}$-like wave functions is necessary to describe the final state; this could account for the observed line strength at normal incidence which is not zero.

Radiative recombination of donor-acceptor pairs in ZnTe

Lifetime of the first excited state of donor-acceptor pairs has been measured in ZnTe as a function of the donor-acceptor distance. The measured lifetime for the radiative recombinations of donor-acceptor pairs agrees well with the calculated one in which the central cell correction for the 1s state of the acceptors is taken into account. It has been found that the lifetime depends on the impurity concentration. The concentration dependence is discussed in connection with the non-exponential decay of the luminescence observed for more distant donor-acceptor pairs.

High intensity magnetic fields for the identification and study of donors in epitaxial gallium arsenide

High intensity magnetic fields are needed for the far infrared photoconductivity method to be used reliably in the identification of unitentional contaminants in ultra high purity epitaxial GaAs. We show experimental evidence that the inhomogeneous Stark broadening of the 1s→2p (m=−1) transition of the hydrogen-like donor almost disappears as the magnetic field is increased to 20 Tesla. Since the spectral lines also become narrower and the central cell correction (chemical shift) becomes larger, the “signature curve” method of identification permits positive identification of donor species. In particular, the donors Ge and Se have been identified in specimens that were reported to contain carbon and Sn respectively.

Theory of Shallow Donor States in Silicon –Substitutional Donors–

A variational formulation is presented in order to discuss the shallow donor states in Si. The theory is a combined one of the linear combination of atomic orbitals (LCAO) approach and the effective mass approach. It is shown that the central cell corrections play significant roles in 1 s donor states, and that they should be expressed in a symmetry dependent form. The chemical shifts of the donor states are easily obtained. The observed binding energies and Fermi contact hyperfine interaction constants are reasonably explained.

Calculation of the binding energies of doubly ionizable centres in II–VI compounds

AbstractThe ionization energies of a doubly ionizable centre are calculated on the basis of two different models. One of these is the effective mass approximation (i.e. shallow centre approximation) suitably extended so as to simulate central cell corrections, whilst the other, based on the concept of an electronic polaron, is more appropriate to the deeper centre regime. The results obtained provide a parametrization scheme which can be checked by experiment and thus lead to a possible means of identifying such centres.

Electronic structure of the self-trapped exciton in rare gas solids

The Rydberg-like series of the self-trapped exciton R 2 ∗ in rare gas solids (Ar, Kr and Xe) are obtained by solving the effective mass equation which incorporates different corrections, including the central cell correction. The results are in good agreement with the recent transient optical absorption data in which the electron is excited into higher excited states. The origin of the luminescence bands is interpreted by analogy with a similar structure of the self-trapped excitons in alkali halides.

Magneto-optical study of shallow donors in transmutation-doped GaAs

Lineshapes and peak positions of 1 s→2 p −1, donor transitions in epitaxial GaAs samples of relatively low compensation have been studied as functions of magnetic field by use of photoconductivity measurements. Some of these samples were produced by transmutation doping using thermal neutrons—a method which is useful for the controlled introduction of donor impurities in GaAs. Two new effects, tentatively attributed to van der Waals interactions between neutral donor atoms, are observed: (1) although both Se and Ge donors are introduced by thermal neutron transmutation, the Se line is much broader than the Ge line, and (2) deviations from isolated-donor behavior occur in the magnetic field dependence of the chemical shift of the shallowest donor present. The separation of lines from two deeper donors, Ge and Si, verified the simple phenomenological theory of the magnetic field dependence of central cell corrections of isolated donors up to at least 10 T.

Photoionization of group-III acceptors in silicon

A simple model is presented to describe the bound-to-continuum optical absorption process for group-III acceptors in silicon. Hydrogenic continuum states are used and the hydrogenic-ground-state wave function is scaled to account for central-cell corrections. Very good agreement with experiment is obtained along with physical insight into details of the photoionization process.

The location and shape of the conduction band minima in cubic silicon carbide

We have measured the inter-bound state excitation spectrum of the N C donor in cubic β-SiC through the ‘two-electron’ transition satellites observed in the luminescent recombination of excitons bound to neutral N donors. Transitions are seen to p as well as s-like donor states although the transition oscillator strength is derived from interaction with the impurity core since parity is conserved through inter-valley scattering by p-like X phonons. The Zeeman splitting of a luminescence line involving the 2p± donor state yield the electron mass parameter m t = 0.24 ± 0.01 m 0. This and the directly measured energy separations of the 2p 0 and 2p± states yields m t/ m 1 = 0.36 ± 0.01 with the static dielectric constant K = 9.92 ± 0.1. Mutually consistent central cell corrections of 1.1 and 8.4 meV are observed for the 2s(A 1) and 1s(A 1) donor states, the latter being in agreement with a recent estimate from electronic Raman scattering by Gaubis and Colwell. The ionization donor energy of the N C donor, 53.6 ± 0.5 meV is consistent with earlier, less accurate estimates from donor-acceptor pair and free to bound luminescence. There is no evidence for a ‘camel's back’ conduction band structure in cubic SiC, unlike GaP. The two-phonon sidebands of the N C donor exciton luminescence spectrum in SiC can be constructed by X and Г phonons only.

Electron-hole effects on interband transitions between surface states in semiconductors

We present calculations of binding energies and wave functions for surface state excitons. Our use of the effective-mass approximation has been implemented by including central-cell corrections. Binding energies reported by recent experiments are obtained for surface electron and hole masses of order unity.

Surface state excitons in semiconductors

We calculate binding energies and wave-functions of surface state excitons. Our approach is based on the effective mass approximation, and assumes a small but finite surface state penetration depth. Central cell corrections are included. We obtain binding energies in the tenth of eV range, in agreement with recent observations on GaAs (110) and Si (111) 7 × 7, for surface masses of order unity.