1.4.4 Parametrization of the Electron Coulomb Interaction

The contribution of the electron-electron Coulomb repulsion to the overall electron-electron interaction in a metal is not well understood. The topic will not be reviewed in detail here, but a short introduction to the typical parametrization is given below following a mainly qualitative discussion of the interaction.

First, notice that the electron-electron Coulomb interaction (at least for independent electrons) is clearly repulsive, as their charge is of the same sign. This is in contrast to the electron-phonon interaction which can be effectively attractive. The electron-phonon interaction in metals (Eq. 1.4) is a model for the electrostatic (screening) interaction between the system of ions and conduction electrons. In typical metals, the screening response of the lattice is much slower than the response of the electron system. This is just the Adiabatic Approximation which, according to Migdal's Theorem, applies in the limit where the Debye energy, E_D, is much smaller than the Fermi Energy. Typically this holds because the ion masses are much larger than the electronic mass. The ``retarded'' response of the lattice causes the electron interaction to aqcuire energy dependence. In particular, for electrons differing in energy by more than E_D, the interaction is negligible, and for electrons close in energy, the interaction can be strong and can become negative, i.e. attractive.

To make a complete model for the net interaction between electrons in a metal, one must also consider the effect of (self)screening of the electrons. The effect of the Coulomb repulsion is to introduce a halo of positive charge around an electron. Screening of this kind modifies the Coulomb interaction from the unscreened form (Eq. 1.1) to

where $\epsilon$ is the effective dynamic dielectric response function of the medium which we consider here to be just the conduction electrons. The effect of screening is always to limit the (infinite) range of the interaction. The co-ordinate space screened interaction will fall off exponentially with a characteristic screening length scale, e.g. the Thomas-Fermi length $r_{TF} = (4\pi e^2 g(0))^{-1}$ which is typically less than 1Å. The screening is complete for length scales large in comparison to r_TF, i.e. for wavevectors much smaller than $k_{TF} \sim r_{TF}^{-1}$. The limited range of the screened interaction leads naturally to simplified local models, such as the Hubbard model. In this kind of model the electron-electron repulsion is parametrized by a single number which is essentially the repulsive energy cost of bringing two electrons close together (i.e. onto the same site or separated by a distance less than the screening length). The magnitude of this energy, which is conventionally denoted U, depends on the full screened electron-electron interaction including both the electron-phonon and electron-electron terms. An upper bound for U can be estimated by simply calculating the unscreened Coulomb energy required to bring two electrons to some minimal distance. The effect of screening by the two media (the positive lattice and the negative electrons) is to reduce this energy. An appropriate dimensionless measure of U is

where g(0) is the density of states per energy per spin. A simple treatment of screening (e.g. see [53]) leads to a renormalization of this energy to the conventional parameter known as the ``Coulomb Pseudopotential'', $\mu^*$ (e.g. Eq. 1.9).

Thus, in the limit where $E_F \gg E_D$, $\mu^* = [\ln{(E_F/E_D)}]^{-1}$,and thus for all typical metals $\mu^* \approx 0.1$.In the theory leading to this result, the electron-electron repulsion is only added after the electron-phonon interaction. Recently, there has been a theoretical attempt to treat these interactions on a more equal footing[54]. These authors find that at low electron density (such as in molecular metals), there are significant deviations from the McMillan equation (Eq. 1.9) / $\mu^*$ result.