Past exam of the condensed matter physics major option course of the University of Oxford / 2019 / 2 / c

In a parabolic conduction valley, $\mathbf{v}=\hslash M^{-1}\mathbf{k}$. The Lorentz force on an electron gives $M\dot{\mathbf{v}}=-e\mathbf{v}\times \mathbf{B}=e\mathbf{B}\times \mathbf{v}$. In the principal axes of the stated effective-mass tensor,Call the matrix on the right $A$. Its characteristic polynomial isOne mode has zero frequency, corresponding to constant drift along the field; the other two have eigenvalues $\pm i\omega$. ThusMore generally the ellipsoidal cyclotron mass is $m_{\mathrm{cr}}^2=\det M/({\hat{\mathbf{B}}}^{T}M\hat{\mathbf{B}})$. The tensor axes must be rotated for the differently oriented valleys.