Landau Level Spectroscopy

778 Pages·1991·10.942 MB·English

by  G. LANDWEHR and E.I. RASHBA (Eds.)

MODERN PROBLEMS IN
CONDENSED MATTER SCIENCES
Volume 27.1
V.M. AGRANOVICH
Moscow, USSR
A.A. M A R A D U D IN
Irvine, California, USA
Advisory editorial board
F. Abeles, Paris, France
F. Bassani, Pisa, Italy
N. Bloembergen, Cambridge, MA, USA
E. Burstein, Philadelphia, PA, USA
I.L. Fabelinskii, Moscow, USSR
P. Fulde, Stuttgart, FRG
M.D. Galanin, Moscow, USSR
V.L. Ginzburg, Moscow, USSR
H. Haken, Stuttgart, FRG
R.M. Hochstrasser, Philadelphia, PA, USA
LP. Ipatova, Leningrad, USSR
A.A. Kaplyanskii, Leningrad, USSR
L.V. Keldysh, Moscow, USSR
R. Kubo, Tokyo, Japan
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Yu.A. Ossipyan, Moscow, USSR
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J.M. Rowell, Red Bank, NJ, USA
NORTH-HOLLAND
AMSTERDAM · OXFORD · NEW YORK · TOKYO
LANDAU LEVEL SPECTROSCOPY
Volume editors
G. L A N D W E HR
Wiirzburg, Germany
E.I. R A S H BA
Moscow, USSR
1991
NORTH-HOLLAND
AMSTERDAM · OXFORD · NEW YORK · TOKYO
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MODERN PROBLEMS IN CONDENSED MATTER SCIENCES
Vol. 1. SURFACE POLARITONS
V.M. Agranovich and D.L. Mills, editors
Vol. 2. EXCITONS
E.I. Rashba and M.D. Sturge, editors
Vol. 3. ELECTRONIC EXCITATION ENERGY TRANSFER IN
CONDENSED MATTER
V.M. Agranovich and M.D. Galanin
Vol. 4. SPECTROSCOPY AND EXCITATION DYNAMICS OF
CONDENSED MOLECULAR SYSTEMS
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H.Z. Cummins and A.P. Levanyuk, editors
Vol. 6. ELECTRON-HOLE DROPLETS IN SEMICONDUCTORS
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V.L. Gurevich
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SEMICONDUCTORS
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N.B. Brandt, S.M. Chudinov and Ya.G. Ponomarev
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MODERN PROBLEMS IN CONDENSED MATTER SCIENCES
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Vol. 25. CHARGE DENSITY WAVES IN SOLIDS
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Vol. 26. HELIUM THREE
W.P. Halperin and L.P. Pitaevskii, editors
Vol. 27. LANDAU LEVEL SPECTROSCOPY
G. Landwehr and E.I. Rashba, editors
In preparation
HOPPING TRANSPORT IN SOLIDS
B. Shklovskii and M. Pollak, editors
NONLINEAR SURFACE ELECTROMAGNETIC
PHENOMENA
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MESOSCOPIC PHENOMENA IN SOLIDS
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ELECTRONIC PHASE TRANSITIONS
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ELASTIC STRAIN FIELDS AND DISLOCATION MOBILITY
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Oh, how many of them there
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IX
L. D. Landau(1908-1968)
Introduction
by
G. Landwehr
and
E.L Rashba
After quantum mechanics was developed in the mid-twenties, the new theoret­
ical concept was subsequently applied to problems of solid state physics. Felix
Bloch was the first to address the problem of free electrons in a periodic
potential in 1928. Two years later L.D. Landau, a young theoretician from
Leningrad, worked out in Cambridge the quantum theory of diamagnetism in
metals (Landau 1930). Up to then it had been tacitly assumed that the magnetic
properties of electrons in metals were determined by their spin and by the electron
binding in atoms. According to a theorem by Bohr and van Leeuwen, based on
classical physics, it was argued that free electrons did not contribute to the
susceptibility because a magnetic field did not change the velocity and
consequently the energy of the electrons. Landau showed that this approach was
inadequate. By solving the Schrodinger equation, incorporating a magnetic field
by a vector potential in a gauge which we now call the Landau gauge, he showed
that the motion of the electron perpendicular to the magnetic field is quantized.
The quantization is coupled with a change in the density of states and results in a
non-zero diamagnetic susceptibility.
Landau performed the calculation for a range of magnetic fields and
temperatures in which the difference in energy between two subsequent
magnetic sub-bands hw is small compared with the thermal energy /cT, with
ω = eB/m (h = Planck's constant/271, ω = cyclotron frequency, Β = magnetic
field, m = electron mass). This condition is not satisfied at low temperatures and
high magnetic fields. Landau noted that under these circumstances no linear
dependence of the magnetic moment on Β could be expected and that a strong
periodic variation in Β should occur. He concluded that it should hardly be
possible to observe the periodic effects experimentally, because they would be
XI
xii G. Landwehr and E.I. Rashba
averaged out due to inhomogeneities of the magnetic field. This estimate was
based on the electron rest mass. Referring to Bloch's theoretical work (Bloch
1928) Landau noted that his calculations should, in principle, also be valid for
crystal electrons although the quantitative application of the results should not
be possible.
At the same time at the University of Leiden, the susceptibility and the
magnetoresistance of bismuth single crystals were investigated at temperatures
between 14 and 20 K. De Haas and van Alphen studied the susceptibilities in
magnetic fields up to 15 kG and found an oscillatory behaviour in the high-field
range (De Haas and van Alphen 1930). In the same year, Shubnikov and de
Haas found an oscillatory magnetoresistance (Shubnikov and de Haas 1930).
The new experimental findings showed that Landau's estimates about the
observability of the periodic variations in the susceptibility had been too
pessimistic. It was Rudolf Peierls, then working with a Rockefeller fellowship as
a guest of Enrico Fermi in Rome, who made the first detailed calculations of the
oscillatory susceptibility, which we now call the de Haas-van Alphen effect
(Peierls 1933). Due to the quantum effects in the regime hco > kT (high magnetic
fields, low temperatures) it is not possible to extend the rather general method
used by Landau for the calculations into the low-field range. Therefore, a model
calculation was performed assuming that the number of electrons was so small
that only the lowest quantized energy bands were occupied. In order to empha­
size the essential features of the quantization caused by a magnetic field, Peierls
discussed a two-dimensional model for Τ = 0, which he considered as physically
meaningless, but instructive. It is interesting to note that nowadays we have
been able to realize semiconductor heterostructures, which really behave like
two-dimensional systems.
Peierls showed that oscillatory behaviour of the susceptibility, which is
periodic in l/B, is expected not only in two dimensions, but also for three-
dimensional systems. In order to allow a comparison with the experiments by
de Haas and van Alphen, he performed rather tedious calculations for finite
temperatures. It turned out that there was qualitative agreement between theory
and experiment. Peierls recognized that bismuth has a particular band structure
with a very small but anisotropic effective mass. The carrier concentration he
estimated was about two orders of magnitude too small. However, in sub­
sequent years the band structure of bismuth was studied in some detail and
reasonable agreement between theory and experiment was obtained (Mott and
Jones 1936).
Further work by Shoenberg (1939) on the de Haas-van Alphen effect revealed
a wealth of detailed information about the electronic band structure of bismuth
close to the conduction-band edges. The experiments by Shoenberg were very
successful, because they were performed at liquid-helium temperatures, whereas
the original measurements had been done with liquid hydrogen.
Whereas a theory was at hand for the interpretation of the oscillatory