Table Of ContentMODERN 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
R. Loudon, Colchester, UK
Yu.A. Ossipyan, Moscow, USSR
L.P. Pitaevskii, Moscow, USSR
A.M. Prokhorov, Moscow, USSR
K.K. Rebane, Tallinn, USSR
J.M. Rowell, Red Bank, NJ, USA
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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
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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
V.M. Agranovich and R.M. Hochstrasser, editors
Vol. 5. LIGHT SCATTERING NEAR PHASE TRANSITIONS
H.Z. Cummins and A.P. Levanyuk, editors
Vol. 6. ELECTRON-HOLE DROPLETS IN SEMICONDUCTORS
CD. Jeffries and L.V. Keldysh, editors
Vol. 7. THE DYNAMICAL JAHN-TELLER EFFECT IN
LOCALIZED SYSTEMS
Yu.E. Perlin and M. Wagner, editors
Vol. 8. OPTICAL ORIENTATION
F. Meier and B.P. Zakharchenya, editors
Vol. 9. SURFACE EXCITATIONS
V.M. Agranovich and R. Loudon, editors
Vol. 10. ELECTRON-ELECTRON INTERACTIONS IN
DISORDERED SYSTEMS
A.L. Efros and M. Pollak, editors
Vol. 11. MEDIUM-ENERGY ION REFLECTION FROM SOLIDS
E.S. Mashkova and V.A. Molchanov
Vol. 12. NONEQUILIBRIUM SUPERCONDUCTIVITY
D.N. Langenberg and A.I. Larkin, editors
MODERN PROBLEMS IN CONDENSED MATTER SCIENCES
Vol. 13. PHYSICS OF RADIATION EFFECTS IN CRYSTALS
R.A. Johnson and A.N. Orlov, editors
Vol. 14. INCOMMENSURATE PHASES IN DIELECTRICS
(Two volumes)
R. Blinc and A.P. Levanyuk, editors
Vol. 15. UNITARY TRANSFORMATIONS IN SOLID STATE
PHYSICS
M. Wagner
Vol. 16. NONEQUILIBRIUM PHONONS IN NONMETALLIC
CRYSTALS
W. Eisenmenger and A.A. Kaplyanskii, editors
Vol. 17. SOLITONS
S.E. Trullinger, V.L. Pokrovskii and V.E. Zakharov, editors
Vol. 18. TRANSPORT IN PHONON SYSTEMS
V.L. Gurevich
Vol. 19. CARRIER SCATTERING IN METALS AND
SEMICONDUCTORS
V.F. Gantmakher and LB. Levinson
Vol. 20. SEMIMETALS - 1. GRAPHITE AND ITS COMPOUNDS
N.B. Brandt, S.M. Chudinov and Ya.G. Ponomarev
Vol. 21. SPECTROSCOPY OF SOLIDS CONTAINING RARE
EARTH IONS
A.A. Kaplyanskii and R.M. Macfarlane, editors
Vol. 22. SPIN WAVES AND MAGNETIC EXCITATIONS
(Two volumes)
A.S. Borovik-Romanov and S.K. Sinha, editors
Vol. 23. OPTICAL PROPERTIES OF MIXED CRYSTALS
R.J. Elliott and LP. Ipatova, editors
MODERN PROBLEMS IN CONDENSED MATTER SCIENCES
Vol. 24. THE DIELECTRIC FUNCTION OF CONDENSED
SYSTEMS
L.V. Keldysh, D.A. Kirzhnitz and A.A. Maradudin, editors
Vol. 25. CHARGE DENSITY WAVES IN SOLIDS
L.P. Gor'kov and G. Gruner, editors
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
G. Stegeman and H.E. Ponath, editors
MESOSCOPIC PHENOMENA IN SOLIDS
B.L. Altshuler, R. Webb and P.A. Lee, editors
ELECTRONIC PHASE TRANSITIONS
W. Hanke and Yu. Kopaev, editors
ELASTIC STRAIN FIELDS AND DISLOCATION MOBILITY
V.L. Indenbom and J. Lothe, editors
Oh, how many of them there
are in the fields!
But each flowers in its
own way -
In this is the highest achievement
of a flower!
Matsuo Β as ho
1644-1694
P R E F A CE TO T HE S E R I ES
Our understanding of condensed matter is developing rapidly at the present
time, and the numerous new insights gained in this field define to a significant
degree the face of contemporary science. Furthermore, discoveries made in
this area are shaping present and future technology. This being so, it is clear
that the most important results and directions for future developments can
only be covered by an international group of authors working in cooperation.
"Modern Problems in Condensed Matter Sciences" is a series of con
tributed volumes and monographs on condensed matter science that is pub
lished by North-Holland Physics Publishing, a division of Elsevier Science
Publishers. With the support of a distinguished Advisory Editorial Board,
areas of current interest that have reached a maturity to be reviewed, are
selected for the series. Both Soviet and Western scholars are contributing
to the series, and each contributed volume has, accordingly, two editors.
Monographs, written by either Western or Soviet authors, are also included.
The complete series will provide the most comprehensive coverage available
of condensed matter science.
Another important outcome of the foundation of this series is the emer
gence of a rather interesting and fruitful form of collaboration among scholars
from different countries. We are deeply convinced that such international
collaboration in the spheres of science and art, as well as other socially useful
spheres of human activity, will assist in the establishment of a climate of
confidence and peace.
The publishing house "Nauka" publishes the volumes in the Russian lan
guage. This way the broadest possible readership is ensured.
The General Editors of the Series,
V.M. Agranovich A.A. Maradudin
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
