10 - Electron transport properties in periodic systems (I)

Summary

Prologue

Electron transport properties can be investigated by measuring the response of conduction electrons to a temperature gradient or to external fields such as an electric field, a magnetic field, or a combination of these applied to a specimen. In this chapter, we study basic transport properties with subsequent derivation of the Boltzmann transport equation. The electrical conductivity is then formulated in the light of the Boltzmann transport equation. The temperature-dependent electrical resistivity expression known as the Baym resistivity formula is derived by taking into account the electron–phonon interaction and is applied to obtain the well-known Bloch–Grüneisen law for a crystal metal. The remaining transport properties including the thermal conductivity, the thermoelectric power, the Hall effect and the magnetoresistance will be discussed in Chapter 11.

The Drude theory for electrical conductivity

In Chapter 2, we learned that electrons on the Fermi surface in a metal like Na carry a Fermi wave number k_F of the order of a few 10 nm^-1, which is converted to a Fermi velocity v_F of approximately 10⁶ m/s through the relation v_F=ħk_F/m. In spite of such high velocities of electrons on the Fermi surface, no electrical current can flow, unless an electric field is applied. The reason for this is that the Fermi surface is always symmetric with respect to the origin in reciprocal space and that there always exists an electron with -v_F for the electron with v_F. Obviously, a current flows only when the applied field displaces the Fermi sphere from the origin and its symmetrical geometry breaks.