Electronic transport properties of materials: the memory function approach

The study of the transport properties is very important for understanding various interactions in electronic systems. These properties such as electrical conductivity, thermal conductivity and thermoelectric coefficients have been widely studied within the Bloch-Boltzmann approach. In this approach, the transport equations are generally solved analytically under the relaxation time approximation (RTA) and in the zero frequency limit. Success of this approach is mostly limited to the zero frequency behavior. It becomes very complicated while investigating the finite frequency behavior of these transport coefficients especially beyond RTA. Thus, one needs an alternative approach which goes beyond RTA and captures the finite frequency features of these coefficients with much ease. This approach is known as the memory function approach. By construction, this formalism is beyond RTA and using this formalism one can calculate the time dependent correlation functions upto any order. It has been used by G�otze and W�olfle (GW) to calculate the dynamical electrical conductivity for metallic electrons. It is successfully applied to study the transport behavior in presence of weak electron-phonon, electron-impurity interactions in metals under the assumption of constant electronic density of states (EDOS). An attempt to extend the GW approach beyond its original assumption of constant EDOS is made here and also we have applied GW approach to a wide variety of transport coefficients (dynamical thermal conductivity, dynamical Seebeck coefficient, etc). Sharapov and Carbotte have also calculated the generalized Drude scattering (GDS) rate for systems with gapped density of states based on Kubo formalism. We reconsider that problem here using the memory function formalism. We show the suppression in GDS due to the presence of gap. We also compare the resulting GDS with that calculated by Sharapov and Carbotte (SC). We find discrepency in the scattering rate using both approaches in the low frequency limit. This is due to the crucial assumption made by SC approach which is not assumed in the memory function approach. We then study the dynamical thermal conductivity of metals within the memory function formalism. Here we introduce the thermal memory functions for the first time and calculate them for the cases of the electron-impurity and electron-phonon interactions. Several new results have been obtained and discussed in various temperature and frequency regimes. In the zero frequency limit, we find that the results are consistent with the results predicted by using the Bloch-Boltzmann approach and are also in accord with the experiments. Furthermore, we also investigate the dynamical behavior of the thermo-electric coefficient, namely Seebeck coefficient. This analysis is done to explore the possibility of obtaining large figure of merit in various materials so that the efficiency of thermoelectric devices can be enhanced. We first confirm that at the zero frequency and in the high temperature case, the results of the Seebeck coefficient are in qualitatively agreement with the experimental findings. We further find that the Seebeck coefficient increases with increasing frequency. This enhancement hints towards a possibility of greater figure of merit if the device is operated at a certain non-zero frequency. We have also applied the memory function approach to other systems such as graphene, a two dimensional system and we investigate the electronic thermal conductivity. In that, we explore the roles of different acoustic phonons, characterized by different dispersion relations. It is found that at the high temperature, the thermal conductivity saturates for all type of phonons. But the longitudinal phonons gives larger contribution to the total thermal conductivity. While at the low temperature, it follows different temperature power law behavior for different type of phonons. We have also found the results at finite frequency regimes which are identical to the case of conventional metals. In the above studies, we performed analytical studies of various transport coefficients that have been done for the weak perturbative interactions by using the memory function approach. However, with the increase in the interaction strength, one needs to go beyond GW approach. In this context, we propose a high frequency expansion of the memory function in term of its various moments. Taking simple example of the electron-impurity interaction for the case of the metal, we calculate the memory function upto the second order moment. It is found that the higher moments contribute more in the low frequency regimes and in the case of large interaction strength. In a nutshell, we extend the GW memory function formalism to various physical situations of interest with encouraging new results in the dynamical regime. While in the dc limit, our results agree with the traditional approaches.by Pankaj BhallaPh.D