Introduction to Solid State Physics · Problem 9

FREE ELECTRON FERMI GAS — Charles Kittel

Physics — Solid State · static-magnetoconductivity-tensor

Problem

Static magnetoconductivity tensor. For the drift velocity theory of (51), show that the static current density can be written in matrix form as

\begin{pmatrix} j_x \\ j_y \\ j_z \end{pmatrix} = \frac{\sigma_0}{1 + (\omega_c\tau)^2} \begin{pmatrix} 1 & -\omega_c\tau & 0 \\ \omega_c\tau & 1 & 0 \\ 0 & 0 & 1 + (\omega_c\tau)^2 \end{pmatrix} \begin{pmatrix} E_x \\ E_y \\ E_z \end{pmatrix} \;. \tag{64}

In the high magnetic field limit of

\omega_c\tau \gg 1

, show that

\sigma_{yx} = nec/B = -\sigma_{xy} \;. \tag{65}

In this limit

\sigma_{xx} = 0

, to order

1/\omega_c\tau

. The quantity

\sigma_{yx}

is called the Hall conductivity.