P. N. Butcher

Analytical formulae for d.c. hopping conductivity degenerate hopping in wide bands

Abstract Analytical formulae for d.c. hopping conductivities are derived for degenerate hopping in wide energy bands. The formulae are in good agreement with computer data for two- and three-dimensional systems. For impurity conduction in n-type crystalline Ge they give hopping conductivities within a factor of four of those observed experimentally. The formulae are also used to analyse T 1/4 data in amorphous Ge and T 1/3 data obtained from studies of inversion layers in MOSFET devices. In both cases, reasonable values for the system parameters are deduced.

Analytical formulae for d.c. hopping conductivity

Abstract Analytical formulae for d.c. hopping conductivity are derived from the conductance network analogue of the steady-state rate equations. The formulae provide valid approximations for both high and low site densities and for both degenerate and non-degenerate electron statistics. They are applied to the special case of non-degenerate hopping in narrow energy bands. The formulae are then particularly simple and are in excellent agreement with computer calculations in two and three dimensions. They yield hopping conductivities for n-–type and p–type Ge within about an order of magnitude of the experimental values. The prefactor in the conductivity is proportional to the inverse site density raised to the power (v – d + 1)/d in d dimensions where v is the power of the intersite separation in the prefactor to the hopping rate.

Numerical calculations of a.c. hopping conductivity

Abstract The first direct computer calculations of a.c. conductivity are presented for electrons in three-dimensional, r-percolation hopping systems. Kirchhoffs equations are solved for 1600 randomly distributed sites using constant capacitances and conductances in which the dependence on the intersite separation r has the form r 3/2 exp (– 2αLr). The computer data are compared with the predictions of the pair approximation, corrected by the addition of the known d.c. limit , and with the predictions of the continuous-time random-walk (CTRW) model proposed by Scher and Lax (1973). Good agreement is obtained with the predictions of the corrected pair approximation over the entire frequency range. The CTRW predictions are qualitatively incorrect both at zero frequency and at high frequencies. At intermediate frequencies, however, the CTRW predictions are in excellent numerical agreement with the computer results, for which ∝ωs but with s significantly less than 0.8.

The d.c. conductivity of chalcogenide films due to the correlated barrier hopping mechanism

Abstract Within the pair approximation, the correlated barrier hopping (CBH) mechanism apparently correlates the observed trends in the a.c. conductivities of chalcogenide films, Invoking an extended pair approximation, it is found that this mechanism is consistent in the case of wide band-gap materials in that it gives rise to an insignificant contribution to the observed d.c. conductivity which may then be correctly accounted for by invoking other processes. In the case of small band gap materials, however, this mechanism yields d.c. conductivities orders of magnitude larger than are observed.

Calculation of hopping transport coefficients

Abstract The calculation of d.c. conductivity, a.c. conductivity, magnetoresistance and Hall mobility from the Miller–Abrahams equivalent circuit is reviewed. Analytical formulae are derived by making simple applications of percolation theory. They are shown to be in good agreement with results obtained by direct numerical solution of Kirchhoffs equations for large models.

The effect of temperature-dependent energies on semiconductor thermopower formulae

Abstract It is confirmed that temperature-dependent energies may be allowed for in the standard thermopower formulae for crystalline semiconductors by inserting the energies appropriate to the temperature of the experiment. Recent suggestions that this procedure is incorrect are shown to violate both Onsager symmetry and particle conservation.

The Hall mobility of carriers hopping in an impurity band

Abstract A simple new general formula for the Hall mobility μH is derived. It involves the contribution to the equilibrium hopping rates from Holsteins three-site process and the product of potential differences in the Miller–Abrahams network in zero magnetic field calculated with an electric field oriented in two orthogonal directions. We find that μH is proportional to site density in the r-percolation regime. The exponential behaviour found by previous authors is missing because the percolation exponentials cancel out of the current ratio which determines μH. For P-doped c-Si at 4 K, μH is in the order of 10−1–10−2 cm2 V−1 s−1 and is considerably larger than the drift mobility.

Low-frequency hopping conductivity in amorphous semiconductors

Abstract The excess a.c. conductivity [sgrave]A(ω) = [sgrave](ω) - [sgrave](0) due to hopping processes in a disordered material is usually interpreted in terms of the ω5 law, s< 1, predicted by the pair approximation. However, recent data on [sgrave]A(ω) for amorphous germanium shows a roll-off behaviour as ω → 0; [sgrave]A(ω) falls off faster than ω5 below a characteristic frequency ωc which exhibits a strong temperature dependence. In the present work it has been found that the roll-off is obtained within the extended pair approximation (EPA)—it marks the onset of a.c. loss by subsets of sites contained in the infinite cluster determining [sgrave](0). Decreasing ω through ωc changes s from less than unity to 2. To within a scale factor, the EPA gives a good account of the observed frequency dependence of [sgrave]A(ω) and the temperature dependence of ωc.

Application of the rate-equation formulation of a.c. hopping conductivity to amorphous semiconductors and impurity bands

Abstract The ratio of the imaginary and real parts of the a.c. hopping conductivity is evaluated numerically in the pair approximation for some simple models. The results confirm the analytical formula 0·293 log (ωτ0)−1 where τ0 is a time constant. The application of this formula to experimental data is discussed.

The theory of anomalous carrier pulse propagation in amorphous semiconductors

A simple derivation is given of a macroscopic conservation equation for carriers obeying rate equations and Boltzmann statistics, which hop between sites having both random positions and random energies. The macroscopic conservation equation is identical to that used previously bythe authors to determine the salient features of anomalous carrier pulse propagation. It involves the a.c. mobility and embodies the Einstein relation. These conclusions are contrary to those obtained by some other authors and the reasons for this are discussed.