Electrical Transport in Fermi Liquids

Abstract

This report is aimed at answering the following question: “why do real metals oppose the flow of current?” The simple answer is “because electrons can’t be accelerated indefinitely by an external field without loosing energy”. I aim at discussing this question in the framework of Fermi Liquids – however, a large part of the argument only depends on weakly interacting charge carriers, irrespective of them being Landau quasiparticles. The main elements that make Landau quasiparticles unique in the context of conductivity is the fact that they have an intrinsic decay time. As we’ll see, this phenomenon leads to both a \(T^2\) temperature dependence of the resistivity and a form of magnetoresistance.

Physical basis for the resistance of metals

The two key elements for conducting a current are mobile charged particles and an energy dissipation mechanism.1 Since a current is just the motion of charged particles, you need the former by definition. In order to prevent these particles accelerating indefinitely, you also need an energy dissipation mechanism – without it, you have no steady state current for nonzero electric field.

We’ll now derive Ohm’s law starting only from these assumptions, following the argument set by Drude (1900). Assume that we have a uniform density \(n\) of particles, carrying a charge \(q\). These carriers are accelerated by an electric field \(\mathbf{E}\) and scatter via a certain mechanism with a characteristic lifetime \(\tau\). Assuming they have a well defined momentum \(\mathbf{p}\), their equation of motion is:2 \[\frac{d\mathbf{p}}{dt} = - \frac{1}{\tau} \mathbf{p} + q \mathbf{E} \label{eq:eom}\]

The force proportional to \(-\mathbf{p}\) serves to dissipate energy, while the electric field \(E\) serves to accelerate the particle. The steady state velocity \(\mathbf{v}\) is: \[\frac{d\mathbf{p}}{dt} = 0 \Leftrightarrow \mathbf{v} = \frac{q \tau}{m} \mathbf{E} \label{eq:v}\] The current density is given by multiplying the charge density of carriers with their velocity:(Halliday 2008) \[\mathbf{j} = n q \mathbf{v} \label{eq:j}\] Substituting the value of \(\mathbf{v}\) from equation \eqref{eq:v} gives \[\mathbf{j} = \frac{n q^2 \tau}{m} \mathbf{E}\]

This equation is essentially Ohm’s Law: the current density is linearly proportional to the applied field. We can calculate the conductivity tensor from its definition, \[j_\alpha \equiv \sigma_{\alpha\beta} E_\beta\] giving a diagonal conductivity tensor of \[\sigma_{\alpha\beta} = \frac{n q^2 \tau}{m} \delta_{\alpha\beta}\]

This result is intuitive: the conductivity increases with the carrier density and their charge, while it decreases with increasing mass. As long as the charge carriers are weakly interacting, all that is left to do is to calculate \(\tau\) for different scattering mechanisms. We’ll look at this problem in the next section.


  1. We’re discussing here about non-superconducing metals. Superconductivity is a totally different beast.

  2. As long as we only consider small changes in momentum, can also be crystal momentum

Mechanisms of scattering in crystals

The discussion in this chapter is partially inspi