Fock-Landau quantisation and magnetic oscillations

When an electron is placed within a magnetic field, it will be subjected to a Lorentz force in the directions perpendicular to that of the field, which results in it exhibiting a circular motion along the plane described by those directions. In addition, these radii of the orbits are restricted to discrete values corresponding to given energy levels. This is called 'Fock-Landau' quantisation, and has many applications within condensed matter physics. Metals placed in strong magnetic fields at low temperatures undergo oscillations in the intensity of their magnetisation (the De Haas-Van Alphen or dHvA effect) and conductivity (the Shubnikov-De Haas or SdH effect) as the magnetic field is varied. This phenomenon is caused by the increase in the spacing of the Fock-Landau energy levels as the strength of the field is increased which causes some of them to pass above the Fermi energy, causing them to become unoccupied and therefore changing the density of states of the metal. From the frequencies of these oscillations, we are able to infer properties such as the area of the Fermi surface in the plane perpendicular to the direction of the field that are needed in order to further understand the properties of these metals.

1. L D Landau and E M Lifshitz Quantum Mechanics (Non-relativistic Theory), Third Edition, Pergamon Press (1977)
2. N W Ashcroft and N D Mermin, Solid State Physics, Thomson Learning (1976)
3. D Shoenberg, Magnetic Oscillations in Metals, Cambridge University Press (1984)

We have been researching extensions to the conventional theory of these effects in the following areas

Oscillations in nanowires

The traditional account of the dHvA and SdH effects assumes that the sample is large enough that the boundaries of the metal sample in question have no significant effect on the spacing of the Landau levels. In the case of metallic nanowires, this is no longer true due to their extreme thinness, and so the question of the effect of the system boundaries on the nanowire cannot be ignored. We approach this problem by examining 'soft' boundary conditions -- modelled as a parabolic well -- and 'hard' boundary conditions -- modelled as an infinite potential well -- since the behaviour of the systems with these ideal boundary conditions should be a good guide to the behaviour of the dHvA or SdH effects in real metallic nanowires. So far, we have found important differences between the behaviours predicted for large samples and those predicted for both sets of nanowire boundary condition; this should prove useful in furthering our knowledge regarding the nature of these systems.

Animation of an electron in a magnetic field inside a nanowire

References

1. A S Alexandrov and V V Kabanov, Magnetic quantum oscillations in nanowires, Phys Rev Lett 95, 076601 (2005)
2. A S Alexandrov, V V Kabanov and I O Thomas, Interplay of size and Landau quantisations in the de Haas-van Alphen oscillations of metallic nanowires; Phys Rev B 76, 155417 (2007)

Oscillations in low-dimensional metals

dHvA oscillations in 2 dimensional metals exhibit some interesting qualitative differences from those observed in 3 dimensional metals. One of the most striking is that when one is dealing with two-band metals, one may observe clear differences in the behaviour of systems that are in the canonical potential (fixed number of particles) and those within the grand canonical potential (fixed chemical potential). The chemical potential in the canonical potential is not constant -- in fact, it has an oscillatory component that varies with the strength of the magnetic field. These oscillations interfere with the dHvA frequencies of the bands and result in the appearance of additional 'mixing' frequencies given by the sum and the difference of the band frequencies. These mixing frequencies may be distinguished from the band frequencies by their behaviour as the temperature is raised.

References

1. A S Alexandrov and A M Bratkovsky, de Haas-van Alphen Effect in Canonical and Grand Canonical Multiband Fermi Liquid; Phys Rev Lett 76, 1308 (1996)
2. A S Alexandrov and A M Bratkovsky, New fundamental dHvA frequency in canonical low-dimensional Fermi liquids, Phys Lett A 234, 53 (1997)
3. A S Alexandrov and A M Bratkovsky, Semiclassical theory of magnetic quantum oscillations in a two-dimensional multiband canonical Fermi liquid, Phys Rev B 63, 033105 (2001)

Oscillations in multiband quasi-2d metals

Layered metals such as the organic charge transfer salts where the conductivity in one direction (usually denoted as the z-direction) is significantly worse than in the other two also exhibit interesting behaviour due to their quasi-two dimensional behaviour, which entails the existence of a cylindrical Fermi surface whose radius varies periodically in the direction of the z component of the momentum. For example, in a single band system in the canonical potential, we have predicted the observation of additional frequencies caused by the mixing between the two frequencies corresponding to the smallest and the largest radii of the Fermi surface induced by oscillations in the chemical potential (similar to what is observed in 2 dimensions); however, if the system is in the grand canonical potential no additional frequencies will be observed as the chemical potential is constant. In multiband quasi-2d metals at intermediate values of the magnetic field, we observe dHvA and SdH oscillations with frequencies corresponding to the Fermi surface areas of both bands. In the dHvA effect, mixing frequencies can only arise if the system is in the canonical potential, as one would expect from the 2-dimensional case; however, in the SdH effect, we have predicted that one may also observe mixing in the grand canonical potential due to the contribution of inter-band scattering to the behaviour of the conductivity. The elucidation of these sometimes quite subtle differences in behaviour is important if we are to understand these materials more fully.

References

1. A M Bratkovsky and A S Alexandrov, Angular dependence of nonclassical magnetic quantum oscillations in a quasi-two-dimensional multiband Fermi liquid with impurities, Phys Rev B 65, 035418 (2002)
2. A S Alexandrov and V V Kabanov, Combination quantum oscillations in canonical single-band Fermi liquids, Phys Rev B 76, 233101 (2007)
3. I O Thomas, V V Kabanov and A S Alexandrov, The Shubnikov-de Haas effect in multiband quasi-two dimensional metals, to be published in Phys Rev B

Oscillations in d-wave superconductors

One current area of interest is the unexpected observation of magnetic quantum oscillations in underdoped cuprate superconductors. Conventional interpretations of these effects would seem to indicate that they are consistent with a Fermi energy of around room temperature and a Fermi surface that is of a few percent of the Brillouin zone. This conclusion does not seem to be easily compatible with both the known band structure of cuprates and the conclusion that the normal state is a non-Fermi liquid. Moreover, the observation of these oscillations below the upper critical field casts some doubt on whether they have a normal state origin. We have proposed a possible origin of these oscillations based on the interference between the d-wave modulations of the order parameter of a charged Bose liquid and the vortex lattice, and have also begun to investigate aspects of magnetic quantum oscillations in anti-ferromagnets, which may prove helpful in our exploration of this issue since the parent cuprate compounds of high temperature superconductors are themselves antiferromagnets.

References

1. A S Alexandrov, Theory of quantum magneto-oscillations in underdoped cuprate superconductors, submitted to Phys Rev Lett
2. V V Kabanov and A S Alexandrov, Magnetic quantum oscillations in doped anti-ferromagnetic insulators, submitted to Phys Rev Lett