De Haas-van Alphen Frequencies Research Articles

De Haas-van Alphen Effect in Gallium at High dc Magnetic Fields

Results of low-frequency field-modulation de Haas-van Alphen (DHVA) measurements in gallium single crystals to 80 kG are presented. The data show many sections of a complex Fermi surface (FS). The DHVA frequency and amplitude data near the $c$ axis are consistent with partial magnetic breakdown. DHVA Dingle temperature measurements for the low-frequency oscillation along the $b$ axis, along with line shape of the giant quantum oscillations of Shapira and Lax for the same piece of FS and field orientation, indicate that the Landau levels associated with these carriers are not collision-broadened. Qualitative comparisons are made with recent magneto acoustic and rf size-effect data. The augmented-plane-wave (APW) calculation of Wood and the nearly-free-electron (NFE) model for the gallium FS are compared with experimental results. The NFE model does not agree with the DHVA or other related data, whereas qualitative comparisons are possible with the APW Fermi surface.

The Fermi surface of beryllium

The Fermi surface of beryllium has been determined experimentally by studying the de Haas–van Alphen effect of single crystals in pulsed magnetic fields. The de Haas–van Alphen frequency (proportional to the extremal area of the Fermi surface normal to the field) was measured as a function of field direction. Consideration of the hexagonal symmetry of the Brillouin zone (discussed in the Appendix) shows that only six distinct classes of fre­quency variation with field direction are possible, and these considerations are used to deduce the locations and forms of the various sheets of the Fermi surface. The Fermi surface is found to consist of hole and electron surfaces of equal volume (each containing 0∙162 carrier per atom). The hole surface is somewhat like a coronet, i. e. a ring of six smoothed tetrahedra joined by small necks lying in the central (0001) plane of the first double Brillouin zone, and the electron surface is a set of six roughly ellipsoidal surfaces (cigars) lying on the vertical edges of the second double zone. Detailed shapes and sizes are deduced for the coronet and cigars such that the extremal areas of cross-section are consistent to within 1 % of those obtained from the observed de Haas–van Alphen frequencies. No oscillations of frequency corresponding to the outer (0001) orbit round the coronet were, however, observed; a study of the field dependence of amplitude of the oscillations from the coupled orbit round the cigar shows that this absence can be explained by magnetic breakdown of the {101̄0} band gap. The model described is in good agreement with the predictions of recent band structure calculations, and is consistent with other experimental evidence.

Band Structure and Fermi Surface of Beryllium

Using a self-consistent potential, the energy eigenvalues along the symmetry edges, on the zone faces, and at the equivalent of 5184 general points in the first Brillouin zone were calculated for beryllium by expanding the conduction electron wave functions in a linear combination of 23 orthogonalized plane waves. From this the Fermi energy and density of states were calculated and the Fermi surface constructed. The density of states is in agreement with soft x-ray emission and absorption data and with the experimental low-temperature specific heat coefficient. The Fermi surface consists of three pieces: A region of unoccupied states in the first double zone which resembles a coronet, and two identical pockets of electrons in the second double zone similar in shape to a cigar with a triangular cross section. The de Haas-van Alphen frequencies in $\frac{1}{H}$ predicted from the Fermi surface are in good agreement with those measured experimentally.