Selected Papers of Kun Huang

The following sections are included:

• Contents

• Preface

• Preface by C. N. Yang

• COMMENTARY

The effect on X-ray reflexion of deviations of the atoms from the ideal lattice sites caused by the presence of randomly distributed foreign atoms in a dilute solid solution is investigated quantitatively. The form of the function used to describe these deviations is suggested by the distortions produced in an elastic medium by a number of spherically symmetric point centres of distortion. Hence one is led to two types of effects, exactly analogous to the thermal effects: (i) a weakening of the ordinary interference maxima; (ii) the presence of ‘diffuse maxima’ associated with the ordinary maxima. The change of lattice constant appears naturally in the analysis. It is used to determine the magnitude of the effects (i) and (ii). By applying the theoretical formulae to the solid solutions Au-Cu, we find that it should be possible to detect (i) experimentally; the thermal effect is secondary and cannot mask the distortion effect. But the effect (ii) mixes with the thermal diffuse maxima and is found to be very much smaller at ordinary temperatures. This conclusion is, however, not regarded as general, especially in view of the anisotropic nature of the thermal effect.

An attempt is made to calculate the heat of solution of gold in silver on the basis of-the quantum theory-of metals. These two metals are chosen because they have the same atomic volume, and therefore are the simplest case.

The steps in the argument are as follows : Suppose that a gold atom replaces a silver atom in the lattice. Then, to a certain approximation, one can represent the substitution of the silver ion by a gold ion, in its effect on the electrons, by a “ potential hole ” of depth ΔE and radius r₀. This potential hole will alter the energy of the conduction electrons. To a first approximation the change in energy is just ΔE, which would give zero heat of mixing. A second-order term of order (ΔE)²/E_F always gives a positive heat of mixing ; E_F is here the Fermi energy. This term is calculated exactly by wave-mechanical methods ; it gives 0.69 ev. per atom. The same calculation- shows, however, that there is a concentration of charge in the gold atom in excess of that in the surrounding silver atoms ; this alters the potential in which the electrons move, so that a self-consistent calculation is required to obtain the true energy. For this the labour required would be almost prohibitive; therefore we use instead the Thomas-Fermi method and obtain 0.45 ev. We thus find 0.15 ev. per atom for the heat of solution, which compares well with the observed value 0.13 ev.

With the help of the potential obtained with the Thomas-Fermi method the residual resistance of gold in silver is found to be 0.16 micro ohm cm. for 1% solution. The considerable discrepancy as compared with the experimental value 0.38 seems closely connected with very similar discrepancies found in other theoretical work on temperature resistance of the noble metals.

A scheme has recently been developed to present the general theories in lattice dynamics without specific assumptions about the atomic interactions. The present note aims at clarifying some basic points in using this scheme, as well as giving the correct expressions for the elastic constants (the available results given in previous works are shown to be correct only for central forces in spite of the use of the general scheme). The fact is emphasized that neither the potential energy of a homogeneous deformation from a reference configuration, nor the complete stresses in the configuration can in general be represented in the general scheme. Hence, for instance, the elastic constants cannot be deduced by straightforward means, nor the equilibrium conditions (vanishing stresses) imposed; the latter itself being necessary for the definition of the usual elastic constants.

A different technique is shown to be necessary for discussing such problems; one result obtained is that only five of the stresses in an arbitrarily chosen reference configuration can be explicitly represented, namely, all the anisotropic stresses. After introducing the condition that these stresses should vanish, the expressions for the elastic constants can be obtained, which are, however, to be used subject to the condition that the remaining stress, namely, an isotropic pressure, vanishes. Only when the general theory is applied to a concrete case, can the latter condition be explicitly introduced by the use of the given interaction.

The results are illustrated by the special example of central forces in the last section. The Cauchy relations follow as an incidental result; the assumptions upon which the relations rest are clearly exhibited in the simple derivation.

The following sections are included:

• Summary

• Introduction

• Phenomenological relations in a homogeneously polarized region of the dielectric

• Free vibrations in the infinite lattice

• Forced irrotational motion by charge sources

• Determination of remaining phenomenological constants

• A special microscopic model

• Acknowledgment

• Appendix: Bearing on and relation to previous work

• References

Please refer to full text.

A symmetric treatment of the interaction between the radiation field and a diatomic ionic crystal with optical isotropy is presented; the explicit solutions of the problem can be readily obtained, which describe all the possible modes of vibration of the combined system of the radiation field and the ionic lattice. An unexpected conclusion is that the optical waves immediately below the infra-red dispersion frequency are in fact none other than what are normally considered as the transverse lattice vibrations of long wave-length in the theory of ionic crystals. It is shown that such waves consist mainly of mechanical vibrational energy rather than radiative energy.

A quantitative theory for the shapes of the absorption bands of F-centres is given on the basis of the Franck-Condon principle. Underlying the treatment are two simplifying assumptions: namely, (a) that the lattice can be approximately treated as a dielectric continuum; (6) that in obtaining the vibrational wave functions for the lattice, the effect of the F-centre can be considered as that of a static charge distribution. Under these assumptions, it is shown that the absorption constant as a function of frequency and temperature can be expressed in terms of the Bessel functions with imaginary arguments. The theoretical curves for the absorption constant compare very favourably with the experimental curves for all temperatures.

Also considered in the paper are the probabilities of non-radiative transitions, which are important in connexion with the photo-conductivity observed following light absorption by F-centres. The treatment given differs from the qualitative considerations hitherto in one important aspect, namely, the strength of the coupling between the electron and the lattice is taken into account. The adiabatic wave functions for the F-centre electron required for the discussion are obtained by perturbation methods. The probability for an excited F-centre to return to its ground state by non-radiative transitions is shown to be negligible; similar transitions to the conduction band are, however, important if the excited state is separated from the conduction band by not much more than 0.1 eV. The temperature dependence of such transitions is complicated, but, for a wide range of temperatures, resembles e^−W/kT. Tentative estimates show that the result is consistent with the observed steep drop of the photo-conductive current with temperature.

The symmetry coordinates for the long wave modes of the Cu₂O lattice are derived explicitly; the selection rules for the first order optical effects (absorption and combinatory scattering) are given. It is pointed out that owing to macroscopic field effects degeneracies cannot be ascertained on symmetry grounds alone, thus nine distinct frequencies are obtained instead of seven. A tentative estimate of the frequencies on a simple ionic model has been made, giving for one of the optically active modes λ = 16.5 μ, in close agreement with the observed position of the strongest absorption peak. The tentative nature of such an estimate is stressed, and effects due to electrical polarisation of the ions and possible bond angle distortion forces are briefly discussed.

The aim of the present paper is to point out and give solution to certain unsolved problems of a fundamental nature, which have emerged in the course of development of the theory of nonradiative transitions in the last two decades. It is shown that the so-called Condon approximation involves an inconsistent and consequently impermissible application of the perturbation method, and a clear-cut criterion for correct choice of the non-Condon wave function is established. On this basis, a proof is given to show that within the approximation that consideration of lattice relaxation is restricted to the diagonal electron-phonon interaction and the nondiagonal interaction is considered as a perturbation taken to the first order, the adiabatic approximation and static coupling theories are equivalent, thus apparently divergent developments of nonradiative transition theory are unified. This conclusion, apart from its significance from the point of view of basic theory, will also facilitate practical calculations of transition rates.

Certain contributions to the multiphonon transition theory are reviewed. The topics reviewed include: i) some theoretical work directed towards clearing up certain long standing muddles in the historical development of the nonradiative transition theory; ii) the concept of multi-frequency model and the method of steepest method as the appropriate method for its implementation; iii) statistical frequency distribution law for phonons emitted in a multiphonon transition.

It is shown that the phonons emitted in a multiphonon transition obey a simple distribution law with respect to the phonon energy. It implies that the relative roles of phonons of different frequencies change systematically with the transition energy and affect accordingly the multiphonon transition probability. The prevalent use of single-frequency models, which completely obliterates such effects, can thus he seriously inappropriate. Such inadequacy of single-frequency models is illustrated by comparing results calculated from multi-frequency models and corresponding single-frequency models.

The method of steepest descent applied to the evaluation of multiphonon transition probabilities is discussed, especially in relation to multi-frequency models. It is shown that the saddle points correspond to a most probable distribution for the phonons. The variation of the phonon distribution with physical parameters and its effects on multiphonon transitions are discussed.

In the investigation of multiphonon transitions, single-mode or single-frequency models are widely used. In view of the fact that such oversimplified models can be seriously inadequate, the present work bridges the gap between the complexity of the general formal theory and the simplicity required for concrete applications by introducing the concept of multi-frequency models. That is, the theory is so formulated that a general system can be approximated by multi-frequency models of any degree of elaboration. A statistical thermodynamic formalism is developed for treating such multi-frequency models, which, on the one hand, greatly reduces the labour of calculation with such models and, on the other hand, leads directly to a simple statistical distribution law for numbers of phonons of each frequency participating in a multiphonon transition. Applications of the theory to concrete models lead to certain general conclusions on frequency dispersion effects in multiphonon transitions. The use of the theory is further demonstrated by fully accounting for the paradoxical experimental results reported by Jia and Yen that the isotopic substitution of H by D in CsMn Cl₃· 2H₂O reduces the multiphonon nonradiative transition probability of excited Mn²⁺ ion by more than ten-fold, and yet leaves the corresponding luminescence phonon sideband little changed. In the last section of the paper, the relation between the statistical thermodynamic formalism and existing multiphonon transition theory is elucidated, thereby the theoretical basis of the statistical formalism becomes clearly defined.

For a critical examination of the dielectric continuum model as applied to a superlattice, we have introduced a microscopic model, which takes proper account of the long-range Coulomb interaction and yet permits easy solution for the long-wavelength LO and TO modes. With this model, it is shown that such long-wavelength modes approach different limits depending on directions of propagation relative to the axis of the superlattice. This demonstrates that to treat the confined bulklike modes in superlattices as modes of isolated slabs is inadequate.

In an attempt to establish an equivalent of the Fröhlich interaction in superlattices, we are led to a critical examination of the dielectric continuum model by comparing with a parallel microscopic model. The reason that the usually quoted confined bulklike phonon modes derived from the dielectric continuum model are completely at variance with the results calculated from the microscopic model is explained. Simple rules for obtaining the proper bulklike modes are then set up, which lead to analytical expressions for the modes, which are found to agree closely with numerical results calculated from the microscopic model in the limit of zero dispersion for the bulk LO and TO phonons. They directly furnish expressions for the interaction with charged particles, which can be considered the equivalent to the Fröhlich interaction in superlattices. Phonon dispersion has the effect of mixing the interface modes into the bulklike modes with nearby frequencies. The small number of bulklike modes so affected are no longer confined to one material. The potentials of these modes apparently cannot be described by simple analytical expressions.

A microscopic theory of Raman scattering by optic phonons is worked out systematically, on the basis of recent advances in our knowledge of the electronic and phonon structure of quantum-well systems. With our recently reformulated analytical expressions for the optical modes, explicit expressions for the Raman tensor for the various phonon modes (interfaces as well as bulklike LO and TO modes) are derived, displaying in full the selection rules regarding polarization configuration, phonon parity, and the phonon-scattering mechanisms. As the theoretical results show, certain specific features of quantum-well wave functions are of special importance for a quantitative theory. Thus heavy- and light-hole mixing effects, and the angular momentum state of the excitons, can play a decisive role in determining the predominant scattering channels. These are illustrated by numerically calculated results for various intra- and intersubband scattering channels. Special emphasis has been given to the Fröhlich-interaction-induced scattering, which is dipole allowed in multiple quantum wells owing to the barrier penetrations and heavy- and light-hole mixing.

Results of theoretical investigations on hole subbands in quantum wells and superlattices are reviewed. Topic covered include: hole subband calculation by an expansion method; pseudopotential calculation by a two-step procedure; heavy and light hole mixing and Coulomb energy of excitons; other applications of the expansion method.

An investigation is made into the effect of valence-band coupling on Wannier excitons in GaAs-Ga_1−xAl_xAs quantum wells with well widths ranging from 30 to 200 Å. The results of our calculation show that the effect is twofold. On the one hand, hole-subband nonparabolicity due to mixing of the heavy- (HH) and light-hole (LH) states causes an increase in the binding energies E^ex, of both ground- and excited-state excitons; on the other hand, the different orbital behaviors of the spinor components of the excitonic wave function result in a decreased E^ex of s-state excitons and an increased E^ex of p- and d-state excitons. The former effect dominates in narrower wells and the latter effect dominates in wider wells. The two-band model is a good approximation for calculating E^ex(HH1), but can cause a significant error in calculating E^ex(LH1) in wider wells because of the stronger coupling between exciton states from different subbands.

The main results of a microscopic theory on the resonant Raman profile mediated by exciton states in multiple quantum wells (MQW's) are presented. The theory takes proper account of effects of heavy- and light-hole mixing of the exciton states, and adopts an appropriate expression for electron-phonon interaction. The selection rules on exciton–LO-phonon scattering in MQW's are summarized, which reveals the roles played by intrasubband and intersubband exciton Raman scattering. Numerical calculations show the dependence of exciton-phonon scattering on the well width and the phonon modes. The mechanism for the asymmetry of the incoming and outgoing resonance is discussed.

An electric field is applied to a superlattice in the growth direction. While the phonon parity selection rules break down, the selection rules for scattering mechanisms in various polarization configurations are still in effect for zero field. Numerical estimates show that Raman scattering due to normally parity-forbidden phonon modes can dominate that due to allowed modes in large but realizable electric fields.

The following sections are included:

• List of Scientific Publications