Quantization of Coupled Orbits in Metals II. The Two-Dimensional Network, with Special Reference to the Properties of Zinc

A. B. Pippard


In an earlier paper the wave functions and eigenvalues for an electron moving in a magnetic field, and interacting with one component of lattice potential, were analysed in terms of a model of coupled localized orbits. The model is now examined in more detail and shown to be a reasonable approximation to one possible representation of the true wave function. It is then extended to cover the case of a two-dimensional metal, the model now consisting of a network of interlocking orbits, on which an electron can move with specified probability amplitude for making a transition between orbits at any junction point. The problem of periodicity of the structure is discussed carefully, and it is found that the phase changes accompanying gauge transformations assume great importance. It is shown that the magnetic field imposes a periodicity on the network which is not in general compatible with that of the lattice potential, and the consequences are briefly investigated with the conclusion that they are probably observable only with difficulty. A special case, the hexagonal network, is then solved exactly, the magnetic field being chosen to avoid the above-mentioned difficulty of incompatible periodicities. From the solution an energy level diagram is constructed, showing how the free-electron levels are broadened by the lattice potential and, as this is made stronger, reconstruct themselves into the sharp level system predicted by Onsager's semi-classical method. In the intermediate stages of the process the bands contact each other frequently and other types of singularity appear. It is claimed that the structure revealed by this simple model is more elaborate than anything that could be readily derived by a perturbation treatment of the magnetic field. The electrons are able to move as quasi-particles in straight lines in any direction through the lattice, the velocity being derived from the energy level structure by the standard formula $\hslash ^{-1}\nabla _{k}$E. When the bands are at their broadest the velocity is comparable with that of a free electron near the corners of the Brillouin zone. The contribution of the quasi-particles to the conductivity of the metal is evaluated on the assumption that the width of individual bands is rather less than k T, so that much of the rapid variation of conductivity with Fermi energy is smoothed out. The variations that are left are still considerable and have a periodicity determined by the smallest quantized orbits. The results of the theory are applied to the fairly extensive, though not always consistent, observations of oscillatory behaviour in zinc. The anomalous variation with field strength of the de Haas-van Alphen amplitude can be satisfactorily explained if it is assumed that the energy gap across the sides of the Brillouin zone is about 0.027 eV. The vigorous resistance oscillations, attributed by Stark to magnetic breakdown changing some of the hole orbits into electron orbits, are shown to require more than this, though this effect is certainly important and is implied by the theory. It is suggested that the quasi-particles provide the necessary extra mechanism to account for resistance and Hall-effect data, but quantitative comparison is far from satisfactory, and it is concluded that more data and further analysis are probably needed. Stark's proposal that the fine structure of the oscillations are due to spin, with a g-factor of 34, is disputed since it appears that the quasi-particle conductivity possesses the right sort of fine structure to account for the observations.

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