B K Ridley

The electron-phonon interaction in quasi-two-dimensional semiconductor quantum-well structures

Approximate analytic expressions are obtained for the scattering rates and momentum-relaxation rates of an electron in a quasi-two-dimensional quantum well interacting with acoustic, optical and intervalley modes via the deformation potential and with longitudinal optical modes via the polar interaction. These analytic expressions are obtained using a momentum-conservation approximation. The threshold for optical phonon emission, unlike the case in the bulk, is abrupt. All scattering rates are energy-independent and are inversely proportional to L, the thickness of the well. The momentum-relaxation rate associated with the absorption of polar optical phonons, on the other hand, proves to be proportional to L. These properties are shown to lead to a negative differential resistance for pure polar mode scattering, and to the existence of a runaway field for deformation-potential scattering. The self-energy associated with the emission of polar optical phonons at absolute zero is shown to be divergent unless the polar interaction is screened, and some consequences of this for laser and other optical processes are pointed out. The description of scattering by perturbation theory breaks down in very narrow wells.

Reconciliation of the Conwell-Weisskopf and Brooks-Herring formulae for charged-impurity scattering in semiconductors: Third-body interference

The divergence at small scattering angles for unscreened charged-impurity scattering may be removed by including the probability that a closer scattering centre does not exist. This introduces an exponential function like the screening factor, which allows a straightforward bridging to be made between non-screening and screening situations. An expression for the mobility which encompasses the Conwell-Weisskopf and Brooks-Herring results is derived.

A model for the interpretation of measurements of photoionisation and capture cross sections associated with deep-level impurities

A model of a deep-level impurity centre involving a highly localised wavefunction is suggested as a convenient basis on which to interpret experimental results for photoionisation cross section and multiphonon capture rate. The model is probably the simplest one which incorporates all the basic features of a deep-level centre; size and symmetry, charge, and degree of lattice coupling, without being committed to an effective-mass approximation. Its application to measurements of photoionisation allows the lattice coupling strength and the zero-phonon optical threshold to be obtained. It is pointed out that a substantial part of the observed temperature dependence of threshold is caused solely by thermal broadening and a method of estimating the magnitudes of the effect is given. Only measurements in the neighbourhood of the threshold are considered to be useful, since far above threshold the role of band structure is too great. The effect of charge on the dependence of photoionisation is shown to be relatively unimportant. Charge is important in determining the temperature dependence of capture rate at low temperatures. Explicit expressions are given for the thermally broadened photoionisation cross section and for the multiphonon process capture rate in this simple model.

The photoionisation cross section of deep-level impurities in semiconductors

A simple analytic expression for the photoionisation cross section of deep-level impurities in semiconductors is obtained in a form that is explicitly related to the electron-scattering strength of the impurity. It also includes the dependence on size and charge, and the character of the localised-state Bloch functions. All of these quantities affect the spectral dependence immediately above threshold. The model introduces a simple billiard-ball wavefunction and the shape functions near threshold are derived, for neutral and charged (plus or minus) centres which are either donor-like or acceptor-like. The effect of phonon coupling is discussed in the context of a single-frequency model. At low temperatures the spectral shape near the shoulder of the curve is determined by the charge, and the effect of phonon coupling is to raise the apparent threshold.

A comparison of simple theoretical models for the photoionisation of impurities in semiconductors

A comparison of two simple theoretical models, the quantum-defect (QD) model with unit normalisation, and a new billiard-ball model (BB) recently proposed by Ridley (see ibid., vol.13, p.2015, 1980), is effected by relating to a third model which is based upon approximating the true impurity potential to a square-well core and, if the centre is charged, a Coulombic tail. This third model, which requires numerical computation, goes over to the QD or BB models under specific conditions, and so acts as a kind of compromise model. The photoionisation cross section for various depths of level, and for neutral, attractive and repulsive centres, is calculated for each model and conditions for the applicability of the QD and BB models are obtained. It is shown that the QD model predicts spectral dependences well, but underestimates magnitude because of incorrect normalisation. The QD model is expected to be reasonably good for shallow and moderately deep levels, provided the centres are not repulsive, but for very deep levels, and for all levels associated with repulsive centres, the BB model is expected to be more useful.

Impurity scattering of electrons in non-degenerate semiconductors

A unified approach to the theory of impurity scattering in non-degenerate semiconductors is presented using a simple model for the scattering potential, and partial-wave techniques. The theory describes both charged and neutral impurity scattering and distinguishes between sign of charge. It also allows a novel extension to be made to resonance (or Breit-Wigner) scattering. Third-body screening is taken to provide a bridge between the Brooks-Herring and Conwell-Weisskopf formulae, and its effect on neutral impurity scattering is examined for both Erginsoy and Sclar cross sections. The relationship between scattering and the depth of a bound state is exhibited using quantum-defect wavefunctions. Resonance scattering, if the resonance is sharp, has only a weak effect on the mobility, but a large effect on rH, the ratio of Hall and drift mobilities. The latter reaches the magnitude 3.7 for sharp resonance. The ratio rH is therefore a sensitive detector of resonance scattering.

On the origin of slow domains in GaAs:O

Measurements between 90K and 200K of the field outside the domain, the domain velocity, and other entities associated with slow domains in n-type GaAs:O (2 Omega cm at 300K) are shown to point unambiguously to the origin of the associated negative differential resistance. This origin is an impurity-barrier mechanism connected with trapping at Gaussian-broadened complex of levels peaking near 0.05 eV below the conduction band. The natures of the barrier and centre are unknown. It is shown that if the barrier is taken to be unscreened Coulombic originating in a negative charge of magnitude Ze then the measured activation energy (0.077 eV) entails Z=10, which appears unacceptably high.

Photoconductivity in n-type GaAs:O associated with the deep level at 0.4 eV

The photoconductivity of GaAs:O (2 Omega cm at 300K) has been studied between 90-200K using a pulsed HF gas laser, in addition to conventional steady-state and chopped-light techniques. The substantial power available has been exploited to study photoresponse up to saturation level. Besides confirming the existence of a broadened trap distribution of width 160 meV below the conduction band, the authors have determined for the 0.4 eV centre the density NT, the volume capture rate cn, the absolute photoionisation cross section sigma n, the effective radius of the centre rT*, the Huang-Rhys factor S, the Franck-Condon shift dFC, the optical ionisation energy ET and its dependence on temperature. They are NT=1.4*1014 cm-3, cn=1.3*10-8 cm3 s-1 at 100K, sigma n=7.1*10-18 cm2 at 100K and at 0.4682 eV, rT*=11 AA, S=3.5, dFC=0.11 eV, ET=0.420-7*10-5 TeV. The spectral dependence of sigma n agrees with a new model by Ridley (1980) rather than with the Lucovsky model, and this new model is exploited in the derivation of the above quantities. It is pointed out that the Franck-Condon shift may be more a property of the host crystal than of the defect.

Photoconductivity, electron traps and level-broadening in GaAs:O

The steady state and transient photoconductivity of n-type GaAs doped with oxygen (2 Omega cm at 300K) was measured between 90 and 150K as a function of the intensity of filtered (1-3 mu m) light. Besides two deep levels which provided photoexcited electrons, discrete levels at 0.23 eV (5*1014 cm-3) and at 0.15 eV (3*1014 cm-3) below the conduction band were found. The main discovery was of a complex of traps at 0.05 eV which exhibited level-broadening. The broadening could be shown to possess a Gaussian form, and to have a half-width of about 10 meV. The capture cross-section associated with the 0.05 eV complex was 1*10-19 cm2 at 100K rising with temperature, with an activation energy of 0.077 eV with about 10% error. Confirmation of shallow trapping with this temperature dependence was obtained from measurements of photoconductivity produced by a pulsed CO2-laser at 10.6 mu m. Rough values of the capture cross-sections for the other centres were also obtained.

The distribution function of photoexcited electrons in semiconductors subject to laser illumination

The steady-state form of the distribution function of electrons in the conduction band of a semiconductor, when the electrons are photoexcited from an impurity level by one or two monochromatic laser beams, is analysed for non-polar acoustic and for both non-polar and polar optical phonon scattering. A single laser line does not alter the Maxwellian distribution, but inverts the population near the band edge. Two laser lines induce non-Maxwellian elements into the distribution function whose character depends upon the laser bandwidths relative to the bandwidth for acoustic phonon scattering. Small bandwidths give rise to spikes in the distribution, but bandwidths comparable to the range of frequencies involved in acoustic phonon scattering may produce, in the absence of strong optical phonon scattering, profound deviations from the Maxwellian form. Numerical solutions of the Boltzmann equation for a few illustrative cases are presented.