Magnetic Breakdown in Zinc and Its Alloys as Seen in the de Haas-van Alphen Effect

The amplitude of the longest-period de Haas-van Alphen oscillation in zinc has been measured as a function of magnetic field strength and field direction in pure zinc and dilute alloys. The field dependence, which is anomalous at high fields, was fitted by the Lifshitz-Kosevich (LK) expression modified to include magnetic breakdown, using a correction factor due to Pippard. This expression fits the data well for field directions up to 70° from [0001], and gives accurate values of the breakdown field parameter $H_0$. Values of $H_0$ obtained for pure zinc by this method range from 2.2 kG (H ∥ [0001]) to 10 kG (H 70° from [0001]), with fields in either the ($10\overline{1}0$) or ($11\overline{2}0$) plane. These results are interpreted as evidence that the lattice band gap is essentially constant over the central third of the needle, the angular dependence of $H_0$ being due to the velocity term in Blount's expression for $H_0$. Values of the Dingle scattering temperature ($X=1.5\mathrm{}°$K in this sample) show no angular dependence; the scattering relaxation time is therefore isotropic on the needle. This result disagrees with previous measurements, but the conflict is shown to be due to the persistence of magnetic breakdown to very low fields, which invalidates the usual graphical procedure for evaluating $X$. In amplitude measurements on impure systems (0.14% Cu, 0.21% Cu, 0.11% Al, 0.008% Mn), magnetic breakdown is far less apparent than in pure Zn. High values of $H_0$ (>5 kG) result from a fit to the breakdown-corrected LK expression. Arguments about band-gap changes are unable to explain this result satisfactorily. The increase in $H_0$ correlates with the decrease in the measured scattering relaxation time. It is suggested that the Pippard breakdown correction, based on a coupled phase-coherent network of orbits, is not appropriate in the presence of strong impurity scattering. An alternative breakdown correction is derived, appropriate to the random-scattering limit, using a generalization of the Dingle scattering factor due to Brailsford and a breakdown-scattering probability suggested by Condon. It is shown that this results in a high-field amplitude qualitatively similar to that observed in the alloys, without changing the value of $H_0$ (obtained from pure zinc where the Pippard approach is valid). Further measurements and theoretical calculations are suggested.