Physics 361Problem Set 7due November 25, 2009P7.1 Anisotropic mass(Cf. Problem 12.1) As we have seen, the maxima and minima ofan energy band need not be isotropic. Thus the general form ofthe energy near a minimum isε(k) = constant +¯h22(k-k0)·ˆM-1(k-k0),whereˆMis a 3×3 (symmetric) matrix called the mass matrix, andk0is the position ofthe minimum. Ifk0is an energy maximum, the second term is negative. For some metalsand most semiconductors all the occupied or empty states are close enough to maxima orminima thatε(k) is well described by this mass-matrix formula.a) For such a material find the density of statesg(ε) and find the specific heat in termsofˆM. That is, find the specific heat effective massm*of p. 48in terms ofˆM.b) For such materials the crystal momentumkis proportional to the velocityvin thesensev=ˆAkfor some matrixˆA. Using Hamilton’s equations (i.e.,the semiclassicalformalism of Chapter 12), find the proportionality matrixˆAin terms ofˆM.
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