PH YSICAL RE VIE% B VOLUME 17, NUMBER 2 j, 5 JANUARY 1978 Analytical wave functions for crystalline electrons in a magnetic field~ A. Rauh Fachbereich Physik, Universita't, D-8400 Regensburg, West Germany S. R. Salinas and L. C. Menezes Instituto de Fisica, Universidade de Sao Paulo, Caixa Postal 20516, Sao Paulo, Brasil (Received 6 July 1977), We examine the properties of the wave functions which are related to the fine structure of the Landau levels of crystalline electrons. Within a recently proposed formulation we are able to calculate explicit wave functions for a special periodic potential and a certain rational field. The wave functions turn out to be highly singular in terms, of the degeneracy parameter k, of the Landau free-electron functions. We show how the singularities can be removed to obtain physically reasonable results. In addition, we discuss transformation properties of the wave functions with respect to rotations of the reference frame connected to the free-electron Landau functions. Also we come to an agreement with Brown and Meisel on the degeneracy of the diamagnetic fine-structure levels. I. INTRODUCTION In the presence of a constant magnetic field the calculated band structure of crystalline electrons exhibits a peculiar fine structure which is dis- played for example by the beautiful graphs in the work of Hofstadter. ' However, this fine structure has not yet been observed in experiments. The dlamagnetlc effects of metals and semiconductors seem to be adequately described within the WKB- type scheme of Lifshitz and Kosevitch, ' provided magnetic-breakdown conditions are not met. Ae experiment in which the diamagnetic fine structure may show up has been suggested in the literature. ' One of the motivations for our work is the at- tempt to settle a controversy about the degeneracy of the diamagnetic fine-structure levels. ' ~ ' Let us briefly indicate the main points of the contro- versy. The degeneracy predicted by the magnetic translation group is finite for the so-called rational magnetic fields and infinite otherwise. ~' Of course, the true degeneracy cannot be smaller than that predicted by the invariance group which is con- sidered. Very often there are additional degenera- cies due to further symmetries or to an accidental degeneracy. It has been claimed' that under cer- tain conditions the degeneracy related to the trans- lational symmetry is infinite for all magnetic fields, including rational fields. This result has been questioned by Brown and Meisel. 4 In the present paper we come to an agreement with Brown and Meisel4: the additional degeneracy which has been previously claimed' is indeed an artifact. In order to clear up this situation completely, we perform an analytic calculation of the wave func- tions for some special cases. We also clarify the nature of the wave functions represented in the basis of the Landau free-elec- I tron functions. It turns out that the wave functions we obtain for a crystalline electron in a magnetic field depend on the degeneracy parameter k„of the Landau functions in a highly singular fashion; they are initially defined only in the sense of a distribution in k, . In the paper of Rauh, ' this pathology of the wave functions is to be blamed for the misleading conclusions on the degeneracy from correct mathematical expressions. As an impor- tant result of this paper we show how the singulari- ties in the wavefunctions can be removed. Thus it is demonstrated that the method of Landau func- tions introduced by Rauh' leads to mell-defined physical wave functions. In addition, we discuss some transformation properties of the wave func- tions with respect to rotations of the reference frame connected to the Landau functions. In order to be able to calculate the wave func- tions analytically, we adopt a special periodic po- tential V and a special magnetic field H. The po- tential V has the form~, io V(x, y) = 2v, (eos& x + eosay), where x and y refer to Cartesian coordinates. The special field H is rational and given by H = (g'/2m) (s c/e) z, where z is a unit vector perpendicular to the x-y plane. This kind of field assures the commutation of all magnetic translation operators, as it will become clear in the following sections. In this case the degeneracy controversy occurs in its sharpest form: no degeneracy at all is predicted by the magnetic translation group, while in the energy-matrix formalism of Rauh' an infinite de- generacy is supposed to be found. Therefore, we consider this special case as a crucial test ex- ample for the degeneracy question. In Sec. VI we 591