Electronic properties with and without electron-phonon coupling

To decent approximation, electronic properties P of solids have a temperature dependence of the type $\Delta $P(T) $= \quad \Sigma $ (dP/d$\omega _{\mathrm{i}})$[n$_{\mathrm{i}}$(T)$+$1/2], where $\omega_{\mathrm{i}}$ is the frequency of the i$^{\mathrm{th}}$ vibrational normal mode, and n$_{\mathrm{i}}$ is the Bose-Einstein equilibrium occupation of the mode. The coupling constant (dP/d$\omega_{\mathrm{i}})$ comes from electron-phonon interactions. At T$=$0, the ``1/2'' gives the zero-point electron-phonon renormalization of the property P, and at T\textgreater $\Theta_{\mathrm{D}}$, the total shift $\Delta $P becomes linear in T, extrapolating toward $\Delta $P$=$0 at T$=$0. This form of T-dependence arises from the adiabatic or Born-Oppenheimer approximation, where electrons essentially ``don't notice'' the time-dependence of thermal lattice fluctuations. In other words, the leading order theory for P is $\Delta $P(T) $= \quad \Sigma $ (d$^{\mathrm{2}}$P/du$_{\mathrm{i}}$du$_{\mathrm{j}})$\textless u$_{\mathrm{i}}$u$_{\mathrm{j}}$\textgreater , responding to the thermal average mean square lattice displacement, as if it were static. There are two situations where non-adiabatic effects alter things. (1) In metals at low T, the thermal smearing k$_{\mathrm{B}}$T of the sharp Fermi edge gets small ($\omega_{\mathrm{i\thinspace }}$\textless \textless k$_{\mathrm{B}}$T). Then non-analyticity of k-integrals requires phonon energy to be included in perturbative denominators. (2) In insulators with polar phonons, Froehlich polaron effects enter, and k-integrals diverge unless phonon energies are kept. Most non-adiabatic effects become unimportant by room temperature, but the low T consequences can be very interesting (e.g. superconductivity.) This talk will discuss the confusing history and predict some future developments in this field.