Walter Schirmacher

On the theory of hopping conductivity in disordered systems

Starting with the linearised master equation, the authors present a first-principles theory of conductivity for disordered systems. The theory is valid for all situations to which the master equation description applies. The path summation and configurational average are evaluated by generalising relations obtained from exactly soluble models. Both symmetric and asymmetric energy-dependent transition frequencies are considered. In the latter case they are able to define an energy-dependent conductivity from which it is possible to evaluate the thermopower. All electronic transport properties, including the frequency-dependent conductivity can be evaluated self-consistently, the only input parameter being the density of states. Numerical results for the DC conductivity and thermopower are presented using several model density-of-states functions. For random statistics, the results are in complete agreement with percolation theory for low densities (temperature). The theory is exact in the high-density (temperature) limit.

Theory of Hopping and Multiple-Trapping Transport in Disordered Systems

We present a general theory to describe equilibrium as well as nonequilibrium transport properties of systems in which the carriers perform an incoherent motion that can be described by means of a set of master equations. This includes hopping as well as trapping in the localized energy region of amorphous or perturbed crystalline semiconductors. Employing the mathematical analogy between the master equations and the tight binding problem we develop approximation schemes using methods of many-particle physics to derive expressions for the averaged propagator of the carriers and the conductivity tensor. The calculated conductivity and Hall conductivity of hopping systems compare extremely well to computer simulations over the whole range of frequency, density, and temperature. We are able to derive expressions for dispersive transport in hopping as well as trapping systems that contain the results of earlier theories of Scher, Montroll and Noolandi, Schmidlin as special cases and establish criteria for the occurrence of dispersive transport in such systems. We find that in principle hopping can lead to dispersive transport if the times and densities are very low, but actual experimental data are more easily explained in terms of multiple trapping.

Heterogeneous shear elasticity of glasses: the origin of the boson peak

The local elasticity of glasses is known to be inhomogeneous on a microscopic scale compared to that of crystalline materials. Their vibrational spectrum strongly deviates from that expected from Debyes elasticity theory: The density of states deviates from Debyes law, the sound velocity shows a negative dispersion in the boson-peak frequency regime and there is a strong increase of the sound attenuation near the boson-peak frequency. By comparing a mean-field theory of shear-elastic heterogeneity with a large-scale simulation of a soft-sphere glass we demonstrate that the observed anomalies in glasses are caused by elastic heterogeneity. By observing that the macroscopic bulk modulus is frequency independent we show that the boson-peak-related vibrational anomalies are predominantly due to the spatially fluctuating microscopic shear stresses. It is demonstrated that the boson-peak arises from the steep increase of the sound attenuation at a frequency which marks the transition from wave-like excitations to disorder-dominated ones.

Theory of electronic hopping transport in disordered materials

Abstract A microscopic theory is presented for d.c. and a.c. conductivity and dispersive carrier propagation in disordered systems where the transport is due to hopping between localized states. The microscopic rate equations (which are interpreted as random-walk equations for single-particle diffusion) are solved directly by standard Green function techniques. Exact expressions are given for the frequency-dependent diffusion coefficient D(ω) and the transient current I(t) in an infinite polymer chain and a Bethe lattice. These systems exhibit dispersive-like characteristics as a consequence of structural disorder. Approximate expressions for these quantities in an R-hopping system are derived from a decoupling approximation. If unrenormalized perturbation theory is used (which means that back-and-forth hopping effects are not correlated) our results become identical to the continuous-time random-walk theory of Scher and Lax (1973), and Scher and Montroll (1975). With the help of our renormalized expressi...

Microscopic theory of dispersive transport in disordered semiconductors

Abstract A theory of dispersive transport based on the microscopic master equation is presented. The theory agrees with previous approaches and unifies them but is much more general. By means of the two-site effective medium approximation of Movaghar et al. we derive a generalised master equation for the averaged propagator of the carriers the kernel of which can be calculated directly from the microscopic transfer rates and distribution functions. We give analytic expressions for the transient current i(t) including the conditions for the transition from dispersive to Gaussian transport for three relevant hopping models. The influence of multiple trapping is treated by means of the coherent potential approximation. We find the same results for trapping with an exponentiak trap depth distribution and fixed-range hopping over energy barriers with an exponential barrier height distribution.

Acoustic dynamics of network-forming glasses at mesoscopic wavelengths

The lack of long-range structural order in amorphous solids induces well known thermodynamic anomalies, which are the manifestation of distinct peculiarities in the vibrational spectrum. Although the impact of such anomalies vanishes in the long wavelength, elastic continuum limit, it dominates at length scales comparable to interatomic distances, implying an intermediate transition regime still poorly understood. Here we report a study of such mesoscopic domains by means of a broadband version of picosecond photo-acoustics, developed to coherently generate and detect hypersonic sound waves in the sub-THz region with unprecedented sampling efficiency. We identify a temperature-dependent fractal v3/2 frequency behaviour of the sound attenuation, pointing to the presence of marginally stable regions and a transition between the two above mentioned limits. The essential features of this behaviour are captured by a theoretical approach based on random spatial variation of the shear modulus, including anharmonic interactions.

Anomalous Diffusion of Hydrogen in Amorphous Metals.

Analysing the quasi-elastic neutron scattering from 8 at.% hydrogen in the metallic glass Ni24Zr76, we find that the hydrogen motion can be described in terms of anomalous diffusion. This process is analogous to that which leads to a strong frequency dependence of the conductivity in fast ionic conducting glasses. The data can be well described by an effective medium calculation based on a model with a broad distribution of activation energies. The same model can also successfully be applied to describe the anomalous temperature dependence of proton spin relaxation in amorphous metals.

Theory of phonon-controlled conductivity in high-resistivity conductors

A recently proposed theory for the behaviour of a zero-temperature electron gas in an environment allowing for random scattering and random tunnelling is extended by treating the conductor phase with an electron-phonon coupling taken into account. The conductivity is expressed in terms of two frequency- and temperature-dependent kernels for which self-consistency equations are derived. Solution of these equations in the DC limit shows that phonon-controlled tunnelling processes dominate the conductivity if the electronic mean free path approaches the De Broglie wavelength. The resulting competition between scattering and tunnelling gives rises to anomalies of the temperature-dependent resistivity which are frequently observed in high-resistivity conductors as, e.g. liquid and amorphous alloys or A15 compounds.

Elastic torsion effects in magnetic nanoparticle diblock-copolymer structures.

Magnetic properties of thin composite films, consisting of non-interacting polystyrene-coated γ-Fe(2)O(3) (maghemite) nanoparticles embedded into polystyrene-block-polyisoprene P(S-b-I) diblock-copolymer films are investigated. Different particle concentrations, ranging from 0.7 to 43 wt%, have been used. The magnetization measured as a function of external field and temperature shows typical features of anisotropic superparamagnets including a hysteresis at low temperatures and blocking phenomena. However, the data cannot be reconciled with the unmodified Stoner-Wohlfarth-Néel theory. Applying an appropriate generalization we find evidence for either an elastic torque being exerted on the nanoparticles by the field or a broad distribution of anisotropy constants.

Random walk in disordered hopping systems

Abstract An exact expression is derived for the averaged frequency dependent diffusion coefficient on a disordered Cayley Tree for symmetric hopping. The equivalent conductance network formula of Stinchcome is recovered in the d.c. limit. A frequency dependent effective medium approximation is proposed. The validity of single site approximation as implicitly used in the work of Scher and Lax is in the d.c. limit closely related to the problem of identifying averaged conductiveness with inverse averaged resistivities.