The phase of the de Haas–van Alphen oscillations, the Berry phase, and band-contact lines in metals
We point out that measurements of a phase of the de Haas–van Alphen oscillations can give information on a degeneracy of electron-energy bands in a metal even though this degeneracy occurs far away from its Fermi level. As an illustration of this statement, the published experimental data on the d...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Цитувати: | The phase of the de Haas–van Alphen oscillations, the Berry phase, and band-contact lines in metals / G.P. Mikitik, Yu.V. Sharlai // Физика низких температур. — 2007. — Т. 33, № 5. — С. 586-590. — Бібліогр.: 29 назв. — англ. |
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irk-123456789-1278162017-12-29T03:03:15Z The phase of the de Haas–van Alphen oscillations, the Berry phase, and band-contact lines in metals Mikitik, G.P. Sharlai, Yu.V. Электpонные свойства металлов и сплавов We point out that measurements of a phase of the de Haas–van Alphen oscillations can give information on a degeneracy of electron-energy bands in a metal even though this degeneracy occurs far away from its Fermi level. As an illustration of this statement, the published experimental data on the de Haas–van Alphen effect in LaRhIn₅, graphite, and zinc are discussed. 2007 Article The phase of the de Haas–van Alphen oscillations, the Berry phase, and band-contact lines in metals / G.P. Mikitik, Yu.V. Sharlai // Физика низких температур. — 2007. — Т. 33, № 5. — С. 586-590. — Бібліогр.: 29 назв. — англ. 0132-6414 PACS: 71.18.+y, 03.65.Vf http://dspace.nbuv.gov.ua/handle/123456789/127816 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Электpонные свойства металлов и сплавов Электpонные свойства металлов и сплавов |
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Электpонные свойства металлов и сплавов Электpонные свойства металлов и сплавов Mikitik, G.P. Sharlai, Yu.V. The phase of the de Haas–van Alphen oscillations, the Berry phase, and band-contact lines in metals Физика низких температур |
| description |
We point out that measurements of a phase of the de Haas–van Alphen oscillations can give information
on a degeneracy of electron-energy bands in a metal even though this degeneracy occurs far away from its
Fermi level. As an illustration of this statement, the published experimental data on the de Haas–van Alphen
effect in LaRhIn₅, graphite, and zinc are discussed. |
| format |
Article |
| author |
Mikitik, G.P. Sharlai, Yu.V. |
| author_facet |
Mikitik, G.P. Sharlai, Yu.V. |
| author_sort |
Mikitik, G.P. |
| title |
The phase of the de Haas–van Alphen oscillations, the Berry phase, and band-contact lines in metals |
| title_short |
The phase of the de Haas–van Alphen oscillations, the Berry phase, and band-contact lines in metals |
| title_full |
The phase of the de Haas–van Alphen oscillations, the Berry phase, and band-contact lines in metals |
| title_fullStr |
The phase of the de Haas–van Alphen oscillations, the Berry phase, and band-contact lines in metals |
| title_full_unstemmed |
The phase of the de Haas–van Alphen oscillations, the Berry phase, and band-contact lines in metals |
| title_sort |
phase of the de haas–van alphen oscillations, the berry phase, and band-contact lines in metals |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2007 |
| topic_facet |
Электpонные свойства металлов и сплавов |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/127816 |
| citation_txt |
The phase of the de Haas–van Alphen oscillations, the Berry phase, and band-contact lines in metals / G.P. Mikitik, Yu.V. Sharlai // Физика низких температур. — 2007. — Т. 33, № 5. — С. 586-590. — Бібліогр.: 29 назв. — англ. |
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Физика низких температур |
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2025-07-09T07:47:41Z |
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| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, No. 5, p. 586–590
The phase of the de Haas–van Alphen oscillations, the
Berry phase, and band-contact lines in metals
G.P. Mikitik and Yu.V. Sharlai
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: mikitik@ilt.kharkov.ua
Received June 15, 2006
We point out that measurements of a phase of the de Haas–van Alphen oscillations can give information
on a degeneracy of electron-energy bands in a metal even though this degeneracy occurs far away from its
Fermi level. As an illustration of this statement, the published experimental data on the de Haas–van Alphen
effect in LaRhIn5, graphite, and zinc are discussed.
PACS: 71.18.+y Fermi surface: calculations and measurements; effective mass, g factor;
03.65.Vf Phases: geometric; dynamic or topological.
Keywords: Berry phase, de Haas–van Alphen oscillations, degeneracy of electron-energy bands.
As it is well known [1], a frequency of de Haas–van
Alphen oscillations, or of other oscillation effects in the
magnetic field, gives an extremal cross-section area Sex of
the Fermi surface of a metal. On the other hand, a phase of
these oscillations is expressed via the constant � of the
semiclassical quantization rule for the energy of a Bloch
electron in the magnetic field H [2]. However, this phase
is commonly ignored in experiments since it is generally
believed [1,3] that � is equal to 1 2/ in absence of the mag-
netic breakdown, and hence the phase does not contain
any essential information on the electron-band structures
of a metal. In this paper we call attention to the fact that �
can differ from its usual value 1 2/ if a band-contact line
penetrates the extremal cross section. Thus, measurement
of the phase of the de Haas–van Alphen oscillations can
provide an unique information on the band degeneracy
even though this degeneracy occurs far away from the
Fermi level.
In the magnetic field a semiclassical electron orbit in
the space of wave vectors k (i.e., in the Brillouin zone of a
metal) is the intersection of the constant-energy surface,
�( )k � const, with the plane, k H � const, where k H is the
component of k along the magnetic field H. In the case of
the closed orbit the quantization condition for energy lev-
els of the electron looks like [1,2]:
S k
eH
c
nH( , ) ( )�
�
�� �
2
�
, (1)
where S is the cross-sectional area of the closed orbit in
the k space; n is a large integer (n � 0); e is the absolute
value of the electron charge. If the magnetic breakdown
is negligible for the orbit under study, the constant � has
the universal value [4]:
� �
1
2
. (2)
It is this value that is commonly used in describing oscil-
lation phenomena in metals [1]. For simplicity, in Eq. (1)
we neglect the spin of the electron. Effects of the spin
will be briefly discussed at the end of the paper.
In recent years the concept of the so-called Berry phase
[5] has attracted considerable attention thanks to its fun-
damental origin, see, e.g., Refs. 6, 7 and citation therein.
According to Berry, if a Hamiltonian of a quantum system
depends on parameters, and if the parameters undergo adi-
abatic changes so that they eventually return to their origi-
nal values, the wave function of the system can acquire
the so-called geometrical phase in addition to the familiar
dynamical one. This additional phase (the Berry phase)
differs from zero when the trajectory � of the system in
the parameter space is located near a point at which the
states of the system are degenerate [5]. In analyzing this
situation, Berry assumed that the Hamiltonian of the sys-
tem is a Hermitian matrix which is linear in deviations of
the parameters from the point, and he presented his final
result in the pictorial form. He found that such the point
© G.P. Mikitik and Yu.V. Sharlai , 2007
can be considered as a «monopole» in the parameter space
when the geometrical phase is calculated. In other words,
the point «generates» a field which coincides in the form
with that of the monopole, and the flux of this Berry field
through the contour � gives the geometrical phase of the
system.
Berry’s result is applicable to the semiclassical elec-
tron trajectories in crystals, with the Brillouin zone play-
ing the role of the parameter space [8]. However, in crys-
tals with the inversion symmetry and a weak spin–orbit
interaction, the Berry phase of the electrons has the spe-
cific features [9] which are due to the fact that the electron
states are invariant under the simultaneous inversion of
time and spatial coordinates. This invariance permits one
to transform the Hermitian Hamiltonian of the electron
into the real form for any point of the Brillouin zone. As a
consequence, the character of the energy-band degener-
acy differs from that considered by Berry. Now the elec-
tron energy bands � l ( )k contact along lines in the Bril-
louin zone rather than at points, and the lines need not be
symmetry axes [10]. In other words, Berry’s monopoles
in the k space disappear. In this context, it is frequently
implied that the Berry phase is equal to zero for electron
orbits in such crystals, and that a nonzero Berry phase can
occur only in crystals with sufficiently strong spin–orbit
interaction (in this case, the nonzero Berry phase is only
the other interpretation of a nonzero orbital g factor of the
electron). However, as it was shown in our paper [9], a
nonzero Berry phase also exists in crystals in which this
interaction is negligible. We showed that the above-men-
tioned band-contact lines play the role of infinitely thin
«solenoids» which generate the Berry field with the flux
��, and the situation similar to the Aharonov–Bohm ef-
fect [11] occurs: Although this field is zero outside the so-
lenoids, but if the electron orbit surrounds a contact line
of its band with some other band, the flux threads the or-
bit, and the electron acquires the Berry phase �B � �
when it moves around this line. It is clear that in this case
the Berry phase does not depend on the shape and the size
of the electron orbit but is specified only by its topological
characteristics (there is a linking of the orbit with the
band-contact line or not).
The Berry phase of the electron modifies the constant �
in the semiclassical quantization rule. The constant � is
now given by the formula [9]:
�
�
�
1
2 2
�B . (3)
The meaning of formula (3) is the following: When the
electron makes a complete circuit in its orbit, the change
of the phase of its wave function consists of the usual
semiclassical part �cS eH� , the shift
� associated with
the so-called turning points of the orbit where the semi-
classical approximation fails, and the Berry phase. Equat-
ing this change to 2�n, one arrives at Eqs. (1), (3). Thus,
when the electron orbit links to the band-contact line, one
obtains � � 0 (the values � � 0 and � �1are equivalent) in-
stead of the usual value � � �1 2.
As was mentioned above, the constant � specifies the
phase of the de Haas–van Alphen oscillations. For exam-
ple, the first harmonic of the de Haas–van Alphen oscilla-
tions of the magnetic susceptibility has the form [2],
��
�
cos
H
��
�
�
�
�
� , (4)
where � �� ��cS eex ( )2 , Sexis some extremal cross sec-
tion of the Fermi surface of a metal in k H , a positive
is
the amplitude of this first harmonic, and is its phase
which is given by
�� ��
�2 (5)
with � �� � � 4 for a minimum and maximum cross-sec-
tion Sex, respectively (and � � 0 in the case of a two-di-
mensional Fermi surface [12]). The phase (together
with the frequency) can be found by the Fourier analysis
of the magnetic-susceptibility oscillations [12]. Thus,
measurements of the phase for the first harmonic enable
one to find �.
If at low temperatures many harmonics contribute to
the oscillating part M of the magnetization, � can be found
even without the Fourier analysis. In this case it is possi-
ble to take the summation over the harmonics in the
Lifshitz–Kosevich formula for M. Using results of
Ref. 13, we then arrive at
M
e
c
H
S
m
S
k
M
H
�
�
�
�
�
�
�
�
� �
� �
� �
�
�
3 2 1 2
5 2 3 2
1 2
2
2
1 2
�
�
ex
| *|
~
, (6)
where m* is the cyclotron mass of the electron, and
~
,M
c S
eH
�
�
�
�
�
�
�
�
�
��
�
�
��
�
�
1
2 2
� ex (7)
in the case of the maximum cross section Sex, and
~
,M
c S
eH
�
�
�
�
�
�
�
�
�
��
�
�
��
�
�
1
2
1
2
� ex (8)
for the case of the minimum Sex. Here ( , )
�1 2 z is the
generalized Riemann zeta function [13], and {z} denotes
the fractional part of the number z. Equations (7), (8) de-
scribe the saw-tooth oscillations with sharp peaks at the
magnetic fields H n ,
e
c
H
S
nn
�
� �
ex
2
1
�
�( ) , (9)
which correspond to the crossing of the nth Landau level
with the Fermi energy. Note that for the maximum Sex
the peaks are directed upwards, while for the minimum
The phase of the de Haas–van Alphen oscillations, the Berry phase, and band-contact lines in metals
Fizika Nizkikh Temperatur, 2007, v. 33, No. 5 587
Sex downwards. In this situation it is possible to find � us-
ing the procedure of Shoenberg [1]. Plotting measured
1 � H n versus n and extrapolating this straight line to the
origin of the coordinates, one obtains �.
Summing up the above considerations, we can state
that measurements of a phase of the de Haas–van Alphen
oscillations give � for the appropriate electron-energy
band and hence enable one to detect those contact lines of
this band which penetrate the extremal cross section of the
Fermi surface. We now illustrate this statement by several
examples.
In the recent experimental investigation [14] of the de
Haas–van Alphen effect in LaRhIn 5, the oscillations of
magnetization associated with a small cross-section of the
Fermi surface of this metal were detected. Authors of that
paper attributed these oscillations to a small electron po-
cket of the Fermi surface. The experimental data [14] ob-
tained at a low temperature (1.5 K) reveal the sharp peaks
in the magnetization of LaRhIn 5when the magnetic field
H is parallel to the [001] direction of this tetragonal com-
pound, Fig. 1. The analysis of the peak positions gives
� � 0 [15], see inset (a) in Fig. 1, and we conclude that the
oscillations in the magnetization result from some small
group of charge carriers near the band-contact line. The
downward peaks mean that we deal with the situation
shown in the second inset of Fig. 1, i.e., with a small
«neck» containing a band-contact line rather than with a
small electron pocket.
The second example concerns graphite. The electronic
spectrum of graphite is described by the Slonzewski–
Weiss–McClure model [16,17], and values of the main
parameters of this model were found sufficiently accu-
rately from the analysis of various experimental data; see,
e.g., the review of Brandt et al. [18] and references
therein. The Fermi surface of graphite consists of elon-
gated pockets enclosing the edge HKH of its Brillouin
zone, see Fig. 2. These pockets are formed by the two ma-
jority groups of electrons (e) and holes (h) which are lo-
cated near the points K and H of the Brillouin zone, re-
spectively. The electron majority is formed by electrons
of the third band � 3( )k , while the hole majority belongs to
the second band � 2( )k .
As it is well known [18], in the edge HKH of the
Brillouin zone of graphite the bands � 2( )k and � 3( )k are
degenerate, and in a small vicinity of the edge these bands
split linearly in a deviation of the wave vector k from the
edge. In other words, the edge is the band-contact line,
and one might expect to find � � 0 for the orbits surro-
unding this axis. However, the experimental data of
Refs. 19, 20 show that at the magnetic field H directed
along the HKH axis, one has the usual value � � �1 2 for
the maximum cross sections of the electron and hole
majorities. Thus, we conclude that an even number of
the band-contact lines have to penetrate these cross sec-
tions (although each line adds 0.5 to �, but within the
semiclassical approximation � is defined up to an integer).
The recent analysis of the Slonzewski–Weiss–McClure
model [21] does show that near the edge HKH there are
three additional contact lines of the same bands, and thus,
both these cross sections are penetrated by the four
band-contact lines, see Fig. 2.
588 Fizika Nizkikh Temperatur, 2007, v. 33, No. 5
G.P. Mikitik and Yu.V. Sharlai
0.8
0 1 2 3 4 5
n
1
/H
n
a
b
H
M
ag
n
et
iz
at
io
n
0.2
0.1
0
–0.1
0 10 20 30
H, T
Fig. 1. The experimental data [14] on the magnetization of
LaRhIn5 (dots). The inset (a) shows the dependence of the ex-
perimental values of 1 � Hn on n. This dependence gives � � 0.
The inset (b) shows the Fermi surface of the small group in
LaRhIn5, and the band-contact line (the dash-dot line).
K
H
ky
K
P0
P0 H
H
�
h
e
Sh
Se
kz
Fig. 2. Sketch of the Fermi surface (a half of it) and of the
band contact lines in graphite. The accidental contact of the
bands �2( )k and �3( )k occurs along the solid lines, while the
dashed lines mark the accidental contact of the bands �3( )k
and �4( )k . The same bands are in contact along the HKH axis
due to the symmetry of the crystal. All the lines merge at the
point P0 that is very close to the point H (HP HK0 001! . ).
Shown are also the maximum cross sections of the electron
(S e) and hole (Sh) majorities for the magnetic field along the
HKH axis. On the right, the Brillouin zone of graphite is
shown.
Before considering the third example, let us discuss the
effect of the spin–orbit interaction on the quantization
rule and the band degeneracy [but we shall still neglect the
direct spin contribution to Eq. (1)]. With this interaction,
the semiclassical quantization rule (1) is modifies as fol-
lows [1]:
S k
eH
c
nz( , )�
�
�� � ��
�
�
�
�
�
2 1
2�
, (10)
where � � �gm m* 4 , m* and m are the cyclotron and the
electron masses, and g is the so-called orbital g factor of
the electron orbit. The spin–orbit interaction also lifts the
accidental degeneracy of the bands (which is not due to
the symmetry of the crystal). But if this interaction is not
too strong so that the spin–orbit splitting of the two de-
generate bands is essentially smaller than energy gaps on
the electron trajectory between the band under study and
other bands of the metal, the concept of the band-contact
line is still valid approximately. As it was shown in our
paper [22], if the semiclassical electron orbit in the mag-
netic field surrounds such a split band-contact line, one
has g m m! �2 *, � ! �1 2, and formula (10) is equivalent
to Eq. (1) with � � 0. In other words, expression (3) is ro-
bust to «switching on» the spin–orbit interaction. We em-
phasize that in this situation the g factor is large even for
a very weak spin-orbit interaction, and this result is the
equivalent description of the nonzero Berry phase (in-
stead of � � 0).
We now discuss the third example, the electron orbits
on the so-called needle of zinc. The Brillouin zone of zinc
coincides with that of graphite, see Fig. 2, and the needle
is located in the vicinity of the point K. Near this point the
electron-band structure of zinc is described by the
Bennett–Falicov model [23]. According to this model,
there are three electron-energy bands which are close to
each other in the vicinity of the point K. The electrons of
the third (uppermost) band just form the needle. One of
the two energy gaps at the point K is formed by the
spin–orbit interaction, while the other gap is caused by the
crystal potential. Both these gaps are of the same order
and small ("1–3 mRy).
In the framework of this model two sets of its parame-
ters were found [24] that reproduce the known experimen-
tal data on the de Haas–van Alphen effect in zinc [25–27].
These two sets correspond to the energy spectra that differ
in number of the split band-contact lines [28]. The first set
leads to a single contact line of the third and second bands,
and this line coincides with the HKH axis, while the sec-
ond set gives the four contact lines of the same bands, see
the insets in Fig. 3. If the Fermi level is considerably
above the edge of the third band, the energy gaps on the
electron trajectory are large, and according to the above
result for �, this � is completely determined by number of
the band-contact lines: For the case of the single line,� is a
half-integer, while for the case of the four lines, one has
an integer �, and these values of � differ by 3 2� . These
considerations agree with Fig. 3 where we show the de-
pendence of � on calculated in Ref. 28 for the above-
mentioned two sets of the parameters at magnetic fields
when the magnetic breakdown is negligible. Note that un-
der the normal conditions the Fermi energy of zinc is not
far from the edge of the third band, the g factor is thus af-
fected by the spin–orbit interaction, and � differs from the
limiting values. However, the data of Fig. 3 suggest that
investigations [29] of the pressure dependence of � can, in
principle, enable one to distinguish between the two sets
of the parameters, and thus to find the number of the
band-contact lines inside the needle of zinc.
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The phase of the de Haas–van Alphen oscillations, the Berry phase, and band-contact lines in metals
Fizika Nizkikh Temperatur, 2007, v. 33, No. 5 589
1.5
1.0
0.5
0
ky
ky
kx
kx
, Ry
0.001 0.01 0.1 1
|
|�
Fig. 3. The dependence of the quantity � # �gm m* 4 on the
Fermi energy measured from the edge of the third band
whose electrons forms the needle in zinc. Here g and m* are
the g factor and the cyclotron mass of the extremal orbit on the
needle when the magnetic field is along the kz axis (see
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band-contact lines in these two cases. The circles mark the val-
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590 Fizika Nizkikh Temperatur, 2007, v. 33, No. 5
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