Journal ArticleDOI
Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields
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In this paper, an effective single-band Hamiltonian representing a crystal electron in a uniform magnetic field is constructed from the tight-binding form of a Bloch band by replacing the operator of the Schr\"odinger equation with a matrix method, and the graph of the spectrum over a wide range of "rational" fields is plotted.Abstract:
An effective single-band Hamiltonian representing a crystal electron in a uniform magnetic field is constructed from the tight-binding form of a Bloch band by replacing $\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ by the operator $\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}\ensuremath{-}\frac{e\stackrel{\ensuremath{\rightarrow}}{A}}{c}$. The resultant Schr\"odinger equation becomes a finite-difference equation whose eigenvalues can be computed by a matrix method. The magnetic flux which passes through a lattice cell, divided by a flux quantum, yields a dimensionless parameter whose rationality or irrationality highly influences the nature of the computed spectrum. The graph of the spectrum over a wide range of "rational" fields is plotted. A recursive structure is discovered in the graph, which enables a number of theorems to be proven, bearing particularly on the question of continuity. The recursive structure is not unlike that predicted by Azbel', using a continued fraction for the dimensionless parameter. An iterative algorithm for deriving the clustering pattern of the magnetic subbands is given, which follows from the recursive structure. From this algorithm, the nature of the spectrum at an "irrational" field can be deduced; it is seen to be an uncountable but measure-zero set of points (a Cantor set). Despite these-features, it is shown that the graph is continuous as the magnetic field varies. It is also shown how a spectrum with simplified properties can be derived from the rigorously derived spectrum, by introducing a spread in the field values. This spectrum satisfies all the intuitively desirable properties of a spectrum. The spectrum here presented is shown to agree with that predicted by A. Rauh in a completely different model for crystal electrons in a magnetic field. A new type of magnetic "superlattice" is introduced, constructed so that its unit cell intercepts precisely one quantum of flux. It is shown that this cell represents the periodicity of solutions of the difference equation. It is also shown how this superlattice allows the determination of the wave function at nonlattice sites. Evidence is offered that the wave functions belonging to irrational fields are everywhere defined and are continuous in this model, whereas those belonging to rational fields are only defined on a discrete set of points. A method for investigating these predictions experimentally is sketched.read more
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References
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TL;DR: In this paper, the diamagnetische Suszeptibilitat of freien Elektronen, ihre Beeinflussung durch die Zusammenstose and das magnetische Verhalten gebundener ElektRONen untersucht.
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Single Band Motion of Conduction Electrons in a Uniform Magnetic Field
TL;DR: In this paper, the effect of a uniform magnetic field on the conduction band of metal was investigated, using as model the tight-binding approximation for a simple cubic crystal, and the normally discrete magnetic levels pertaining to free electrons were shown to be non-uniformly spaced and broadened as a result of the lattice forces.
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The Effect of a Magnetic Field on Electrons in a Periodic Potential
TL;DR: In this article, a theorem due to Wannier for treating the motion of electrons in a perturbed periodic field is generalized to include the effect of a slowly varying magnetic field.
Journal ArticleDOI
Bloch Electrons in a Magnetic Field
TL;DR: In this article, a formalism was used to simplify Kohn's derivation of an effective Hamiltonian for arbitrary symmetry and with the inclusion of spin-orbit effects and for bands with degeneracies.