Introduction to Solid State Physics · Problem 7

FREE ELECTRON FERMI GAS — Charles Kittel

Physics — Solid State · dynamic-magnetoconductivity-tensor-for-free-electrons

Problem

Dynamic magnetoconductivity tensor for free electrons. A metal with a concentration nn of free electrons of charge −e-e is in a static magnetic field Bz^B\hat{z}. The electric current density in the xyxy plane is related to the electric field by jx=σxxEx+σxyEy  ;jy=σyzEx+σyyEy  .j_x = \sigma_{xx}E_x + \sigma_{xy}E_y \;; \qquad j_y = \sigma_{yz}E_x + \sigma_{yy}E_y \;. Assume that the frequency ω≫ωc\omega \gg \omega_c and ω≫1/τ\omega \gg 1/\tau, where ωc=eB/mc\omega_c = eB/mc and τ\tau is the collision time. (a) Solve the drift velocity equation (51) to find the components of the magnetoconductivity tensor: σxx=σyy=iωp2/4πω  ;σxy=−σyx=ωcωp2/4πω2  ,\sigma_{xx} = \sigma_{yy} = i\omega_p^2/4\pi\omega \;; \qquad \sigma_{xy} = -\sigma_{yx} = \omega_c\omega_p^2/4\pi\omega^2 \;, where ωp2=4πne2/m\omega_p^2 = 4\pi ne^2/m. (b) Note from a Maxwell equation that the dielectric function tensor of the medium is related to the conductivity tensor as ϵ=1+i(4π/ω)σ\boldsymbol{\epsilon} = \mathbf{1} + i(4\pi/\omega)\boldsymbol{\sigma}. Consider an electromagnetic wave with wavevector k=kz^\mathbf{k} = k\hat{z}. Show that the dispersion relation for this wave in the medium is c2k2=ω2−ωp2±ωcωp2/ω  .(63)c^2k^2 = \omega^2 - \omega_p^2 \pm \omega_c\omega_p^2/\omega \;. \tag{63} At a given frequency there are two modes of propagation with different wavevectors and different velocities. The two modes correspond to circularly polarized waves. Because a linearly polarized wave can be decomposed into two circularly polarized waves, it follows that the plane of polarization of a linearly polarized wave will be rotated by the magnetic field.

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