The quantum mechanical B-dependence

The quantum mechanical B-dependence of τ also leads to weak-field oscillations in ρ||, with the same periodicity as the oscillations in ρ⊥ discussed earlier, but of much smaller amplitude and shifted in phase (see Fig. 37, where a maximum in the experimental σ|| around 0.3T lines up with a minimum in ρ⊥). These small antiphase oscillations in ρ|| were explained by Vasilopoulos and Peeters227 and by Gerhardts and Zhang259 as resulting from oscillations in τ due to the oscillatory Landau bandwidth. The Landau levels En = (n − 12 )¯hωc broaden into a band of finite width in a periodic potential.261 This Landau band is described by a dispersion law En(k), where the wave number k is related to the classical orbit center (X, Y ) by k = Y eB/¯h (cf. the similar relation in Section III.A). The classical guiding-center-drift resonance can also be explained in these quantum mechanical terms, if one so desires, by noticing that the bandwidth of the Landau levels is proportional to the root-mean-square average of vdrift = dEn(k)/¯hdk. A maximal bandwidth thus corresponds to a maximal guiding center drift and, hence, to a maximal ρ⊥. A maximum in the bandwidth also implies a minimum in the density of states at the Fermi level and, hence, a maximum in τ [Eq. (1.29)]. A maximal bandwidth thus corresponds to a minimal ρ||, whereas the B-dependence of τ can safely by neglected for the oscillations in ρ⊥ (which are dominated by the classicalguiding-center-drift resonance).