R.F. Wallis

Energy levels of an electron in the field of a finite dipole

Abstract A quantum mechanical investigation has been made of the energy levels of an electron moving in the field of two fixed charges with values + e and − e and having a separation R . The Schrodinger equation for this problem is separable in confocal elliptic coordinates. The evaluation of the energy eigenvalues requires the solution of two simultaneous transcendental equations involving continued fractions. The solutions of these transcendental equations have been worked out for the ten lowest bound states over a range of values of R using the IBM 704 electronic computer at the Research Laboratories of the General Motors Corporation. The binding energy of the electron decreases monotonically as R decreases for all states considered. For a given state the electronic binding energy is very small for values of R less than the Bohr radius of the corresponding hydrogenic state obtained by letting R → ∞. Applications of the results to color center problems and to other solid state physics problemsare discussed.

Theory of impurity photo-ionization spectrum of semiconductors in magnetic fields

Abstract The impurity photo-ionization magneto-optic (IPMO) effect is the effect of an external magnetic field on the optical photo-ionization spectrum of impurities in semiconductors. A theory of the IPMO effect has been developed for simple semiconductors. Two cases can be distinguished, depending on whether the impurity ionization energy for zero field is larger or smaller than 1 2 h ω c = h eH/2m ∗ c . In the low-field case, the photo-ionization spectrum consists of a series of rather strong peaks corresponding to transitions from bound levels to the various magnetic sub-bands of the continuum. In the high-field case, the photo-ionization spectrum consists of one, or at most two, strong peaks if the temperature is such that only the lowest bound level is appreciably occupied. Detailed calculations are presented of the high-field absorption spectrum for donor impurities in InSb, based on wave functions similar to those of Yafet, Keyes and Adams .

Absorption spectra of impurities in silicon—I: Group-III acceptors

Abstract The low-temperature absorption spectra of group-III impurities in silicon yield information about the ionization energies and excited states of the impurity centers. The optical values for the ionization energies of boron (0.046 eV) and indium (0.154 eV) are in good agreement with the thermal values, but the optical values for aluminum (0.67 eV) and gallium (0.71 eV) are appreciably larger than the thermal values. Variations in ionization energy among the group-III impurities are found to be accompanied by variations in the character of the excitation and photoionization spectra. The term schemes for the impurities are derived from the position of the excitation bands and the assumption that the 4p level is the same for all of the group-III impurities. Aside from the appearance of fine structure, the p-type states appear to fit a simple hydrogen model. From the positions of the p levels, an effective mass of 0.045 is calculated for holes using the expression for the energy levels of a hydrogen model. Variations in the oscillator strengths of the excitation bands among the group-III impurities are attributed to variations in the position and character of the 1s levels which arise from effects at the impurity atom that cause the simple effective-mass formalism to break down. The factors which should be included in a more complete theoretical treatment of the impurity centers are discussed.

Surface effects on lattice vibrations

Abstract The introduction of a free surface into an otherwise perfect, periodic crystal generally leads to changes in the normal mode frequencies and to changes in the displacement amplitudes and velocities of the atoms or ions. Localized surface modes of vibration may occur in which the displacement amplitudes decrease exponentially from the surface. Certain types of surface modes may interact with infrared radiation. Theoretical studies of the vibrational specific heat have shown that free surfaces should lead to a contribution at low temperatures proportional to the surface area and to the square of the absolute temperature. The mean square amplitudes of vibration of the atoms determine the temperature dependence of the intensities of X-rays or electrons scattered by a crystal. Since the mean square amplitude of an atom decreases as the forces of interaction of that atom with the other atoms increase, the scattering of X-rays or electrons can be used in principle to obtain information about the variation of interatomic forces near the surface of a crystal. In the Mossbauer effect lattice vibrations cause a temperature dependent shift of the resonance line which is proportional to the mean square velocity of the absorbing or emitting atom. The mean square velocity increases as the forces of interaction increase, so that the Mossbauer effect may provide another means of investigating the effect of surfaces on interatomic forces.

Surface Vibrational Modes in Crystal Lattices with Complex Interatomic Interactions

The investigation of surface vibrational modes in crystal lattices is complicated by the necessity for satisfying the free boundary conditions at the surface if the interatomic interactions include next‐nearest neighbor interactions, next‐next‐nearest neighbor interactions, etc. The number of equations specifying the free boundary conditions may then become rather large. A general method is presented in this paper for the investigation of the normal modes of one‐, two‐, and three‐dimensional lattices which are finite or semi‐infinite in one dimension and have free boundary surfaces, assuming interatomic interactions of various ranges. The method is illustrated by calculations of the normal modes for the finite and semi‐infinite diatomic linear chains with nearest and next‐nearest neighbor interactions. Some remarks are made regarding the applications of the mathematical techniques to the evaluation of certain continuant determinants of large order.

Theory of cyclotron-resonance absorption by conduction electrons in indium antimonide

Abstract A semi-classical theory of cyclotron resonance by conduction electrons in InSb is presented. It is assumed that the conduction band is a simple band with a minimum at k = 0, and spin interactions are neglected. Terms up to the fourth order in the expansion of E in powers of k about the band edge are taken into account in a computation of the energy levels and wave functions of an electron in a constant magnetic field, using a simplified form of the effective mass equation of Kjeldaas and Kohn in which warping of the conduction-band energy surfaces is neglected. An expression for the absorption coefficient has been obtained which is accurate to terms linear in the fourth-order energy coefficient. Collision broadening is incorporated, using a single relaxation time. The fourth-order coefficient in the energy results in a splitting of the cyclotron-resonance absorption band into a series of peaks, each broadened towards the low-frequency side. The frequencies of the maxima of these peaks are shifted towards lower values by an amount which is proportional to the magnetic field and to the fourth-order coefficient in the energy and increases with magnetic quantum number. Using the energy-band parameters computed by Kane , the theoretical frequency shift is larger than that observed in infra-red cyclotron-resonance experiments, but it is of the right order of magnitude.

Surface elastic waves in body-centered cubic lattices

Abstract Surface elastic waves in a monatomic body-centered cubic lattice are investigated using a Born-von Karman lattice model. The model includes central force interactions between nearest and next-nearest neighbors and non-central forces corresponding to angular stiffness terms in the strain energy. A dispersion curve for surface waves is obtained, which lies below the dispersion curve for bulk shear waves. In the long wave approximation, the results match those given by the continuum theory of elasticity which predicts nondispersive surface waves.

Note on semiconductor statistics

Abstract The Guggenheim method for treating semiconductor statistics using the grand partition function is applied to multi-level impurities in Ge assuming a model for electron configuration at the impurity based on the present knowledge of the energy band structure of Ge. The results are expressed in terms of effective one-electron energies corresponding to the one-electron levels associated with the impurities. The difference between these effective one-electron energies and actual one-electron energies is emphasized.

Zeeman perturbations on shallow acceptor states in germanium

Abstract Based on the effective mass theoretical approach, the magnetic field splittings of the shallow acceptor states in Ge have been studied. In this calculation, we have used the acceptor wave functions previously obtained by Mendelson and James. It is found that the g -factor for the ground state Γ 8 quartet, so obtained, is in close agreement with the one calculated by Suzuki, Okazaki and Hasegawa using full six component envelope functions. The Zeeman splitting for the excited Γ 6 doublet appears to be quite large. The g -factors for the Γ 7 doublet and other excited states have been calculated. The Zeeman splittings of Raman transitions involving the ground state multiplet and a low-lying excited state multiplet of even parity have also been determined.

Zeeman type magneto-optic studies of energy band structure☆

Abstract The energy levels associated with an energy band in a magnetic field are characterized by the following quantum numbers: h k H , the crystal momentum along the magnetic field direction; l , the Landau magnetic quantum number; and M J the component of the total angular momentum along the magnetic field which is characteristic of the atomic states in the tight binding limit. In the case of degenerate valence bands, the effect of a magnetic field is complicated by degeneracy effects and the levels in a magnetic field are characterized by two or more pairs of ( l , M J ) values. The selection rules, polarization effects, and the character of the absorption spectra of interband transitions in a magnetic field are discussed and illustrated by experimental data for Ge. The purpose of the present paper is to discuss the physics underlying the Zeeman type interband magneto-optic (IMO) effects in semiconductors. The effect of a magnetic field on the electronic energy levels of a semiconductor will be presented and the various optical transitions that can take place between these levels will be discussed. Finally, data on the room temperature IMO effect in Ge will be reported and compared with the theoretical calculations.