Calculating the conductivity of periodic systems in the presence of scattering is a formidable challenge because scattering couples the many degrees of freedom in the system. The most broadly employed resolution of this dilemma is the semiclassical treatment given by Boltzmann transport theory, which simplifies the coupling to render the problem tractable. In this approximation, all scattering is assumed to be incoherent. While this is usually acceptable in the limit of weak scattering, it is not valid for strong scattering. The conductivity derived within Boltzmann transport theory also only describes intraband motion of electrons, leaving open the question of how the conductivity is affected when multiple bands are present near the Fermi level.
Alternatively to the semiclassical approach of Boltzmann transport theory, the Kubo formula gives the exact conductivity. However, because it requires the system's exact eigenstates and their energies, it may only be applied to microscopic systems which are small enough to be solved numerically. The behavior of such small systems does not mimic that of macroscopic systems, especially in the direct-current (dc) limit of the conductivity. In particular, the dc conductivity of a finite supercell is either zero with open boundary conditions or infinite with periodic boundary conditions. Only with a small numerical broadening does the Kubo formula yield a finite dc conductivity, but strong dependence on the broadening parameter obscures the true dc conductivity. Hence the precision of this approach is limited by the size of a supercell one can diagonalize, making it impractical for the study of realistic macroscopic systems.
In this dissertation, we derive a method to calculate the conductivity of macroscopic multiband systems which is valid for any scattering strength. The conductivity is a dynamical relaxation function, so to calculate it, we must solve the dynamics of operators in the system. We use a Mori projection on the equation of motion for the operators, which separates time evolution of operators into two contributions, one due to the pristine system and a memory term due to scattering. This yields a system of generalized Langevin equations for the relaxation functions, which we solve by approximating the memory term by a single memory function. This memory function describes the relaxation of the slowest collective variables, which dominate long-time dynamics.
With this result, we derive a formula for the conductivity of the full system with scattering in terms of the conductivity of the pristine system without scattering and a complex memory function, Ïƒ(Ï‰)=Ïƒâ°(Ï‰+M(Ï‰)). The imaginary part of the memory function describes the broadening of transitions by scattering at low frequencies, and its real part provides a frequency shift. This formula is independent of the nature of the scattering, such as by disorder, the Coulomb interaction, or phonons, is independent of the method used to evaluate the memory function, and is valid for all scattering strengths.
Our formula for the conductivity includes not only intraband motion, but also the interband contributions which are not described in standard Boltzmann transport theory. Although the interband contributions vanish for zero scattering, they grow with scattering as the bands are broadened, so that this term is important for multiband systems with moderate scattering. Note that multiband systems can refer not only to complex materials, but also simple materials studied through large supercells, where the density of bands is increased through the folded-zone scheme. The importance of the interband contribution is most pronounced when the Fermi level is in a gap, where the intraband contribution vanishes and only interband contributions to the conductivity are present. Surprisingly, we find that at low to moderate scattering strength, scattering increases the dc conductivity by broadening the bands into the gap, an effect that cannot be described by standard Boltzmann transport theory.
By comparing the intraband contribution in our formula for the conductivity to that given by standard Boltzmann transport theory, we show that in the zero-frequency limit the memory function parallels the inverse relaxation time. Hence, this formula can be implemented into existing Boltzmann transport theory codes with the memory function replaced by the inverse relaxation time. This simple extension would make existing Boltzmann transport valid for multiband systems in the weak-scattering limit.
To address the strong-scattering limit, we must calculate the memory function consistently with its formalism. Specifying the case in which scattering is due to disorder, we derive a self-consistent equation for the memory function. In the zero-frequency limit, the self-consistent equation becomes an analytic formula for the memory function. This formula shows that a finite level of disorder, the zero-frequency memory function diverges, which corresponds to vanishing conductivity, i.e., a metal-to-insulator transition. Taking the limit of large M(0), we obtain an analytic formula for the critical disorder.
The only inputs for the formulae described above are the density response function of the pristine system and a potential representing the disorder. The former is readily available in ab initio codes and the latter is easily obtained analytically, e.g., a screened Coulomb or a Gaussian potential. Hence, all our equations are suitable for implementation into ab initio code, and we have implemented them into the density functional theory code exciting. For comparison to previous work, we use a disorder potential which is the continuous analogue of Anderson disorder, and evaluate the zero-frequency memory function and dc conductivity for sodium, chosen as a nearly-free-electron gas. We find that disordered sodium undergoes a metal-to-insulator transition at a finite level of disorder. We compare our results to Boltzmann transport theory and to existing memory function calculations for the free-electron gas. For low disorder, where the approximations of Boltzmann transport theory are valid, the two theories agree, while for strong disorder only the memory function captures the metal-to-insulator transition.