The de Haas-van Alphen effect

Abstract

The magnetic anisotropy of single crystals of various metals has been studied experimentally particularly at liquid-helium temperatures, and an oscillatory variation with field (de Haas-van Alphen effect) has been discovered in gallium, tin, graphite, antimony, aluminium, cadmium, indium, mercury and thallium. Previously the effect has been found only in bismuth and zinc, and recently it has been found by Verkin, Lazarev & Rudenko also in magnesium and beryllium. After a brief statement of Landau’s theory of the effect and some recent modifications by Dingle, the experimental technique is described and the results for the individual metals are presented. The effect has been studied most thoroughly for gallium, tin, graphite and antimony, and it has been possible to explain the results in considerable detail on the basis of the theory, though some features such as the modulations of the oscillations cannot be fully explained; the theoretical interpretation of the results for the other metals is less complete, mainly because of experimental difficulties specific to each metal which hindered a complete investigation. Comparison with the theory shows that the effect can be explained if it is assumed that only a very small number of free electrons (ranging from 10 -6 to 10 -3 per atom) are effective and that these electrons have effective masses which are small (usually of order of one-tenth of an electron mass) and depend on the direction of the applied magnetic field. The period, amplitude and temperature-dependence of the oscillations vary considerably from one metal to another, depending on the particular values of these parameters. These ‘effective’ electrons are presumably those which overflow at certain places in wave-number space from one Brillouin zone into another, or the ‘holes’ left behind in nearly full zones, and their small effective masses are associated with large curvature of the Fermi surface in these regions. The theory assumes that the relevant parts of the Fermi surface can be represented by ellipsoids, and for some of the metals the form of these ellipsoids can be worked out in detail on the basis of the experimental results. The fact that the de Haas-van Alphen effect has not been found in monovalent metals such as copper, silver and gold up to fields of 15800 G, supports this interpretation, since the Fermi surface in these metals does not cross Brillouin zone boundaries. Although the oscillatory variation of anisotropy was the main object of the investigation, some new data on the steady part of the anisotropy were also obtained, and where a detailed comparison with theory was possible it was found that the free electrons effective in producing the oscillations could account only partly for the observed steady anisotropy. An important feature of the comparison with theory is that in order to explain both the temperature and field variation of the amplitude of the oscillations consistently it is necessary to add to Landau’s formula an exponential ‘ damping factor ’ involving a parameter x which has the dimensions of temperature and is usually of order 1°K. The effect of this ‘damping’ is equivalent to that of raising the temperature by x º K. Dingle has shown that just such a factor is to be expected if broadening of the energy levels due either to collisions or other causes is taken into account. Experiments on the de Haas-van Alphen effect in a series of alloys of tin with mercury and indium support Dingle’s interpretation in showing that the parameter x varies approximately linearly with the reciprocal of the collision time (i.e. with the residual resistance), and the slope of the linear relation gives a reasonable value of the collision time. It is clear, however, that collision broadening alone cannot account for the experimental values of x for pure metals, and other causes of level broadening, such as the effect of the electric field of the crystal lattice, must be invoked.