Oscillations of magnetization in topological line-node semimetals
We theoretically investigate the phase of the de Haas–van Alphen oscillations in topological line-node semimetals. In these semimetals the chemical potential of charge carriers can essentially depend on the magnetic field, and this dependence changes the phase of the oscillations as compared to the...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1761682021-02-04T01:30:31Z Oscillations of magnetization in topological line-node semimetals Mikitik, G.P. Sharlai, Yu.V. Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова We theoretically investigate the phase of the de Haas–van Alphen oscillations in topological line-node semimetals. In these semimetals the chemical potential of charge carriers can essentially depend on the magnetic field, and this dependence changes the phase of the oscillations as compared to the phase in a three-dimensional metal with a band-contact line. Our results elucidate recent experimental data on the Berry phase for certain electron orbits in ZrSiS, ZrSiTe, and ZrSiSe. 2018 Article Oscillations of magnetization in topological line-node semimetals / G.P. Mikitik, Yu.V. Sharlai// Физика низких температур. — 2018. — Т. 44, № 6. — С. 727-733. — Бібліогр.: 45 назв. — англ. 0132-6414 PACS: 71.20.–b, 75.20.–g, 71.30.+h http://dspace.nbuv.gov.ua/handle/123456789/176168 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова |
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We theoretically investigate the phase of the de Haas–van Alphen oscillations in topological line-node semimetals. In these semimetals the chemical potential of charge carriers can essentially depend on the magnetic field, and this dependence changes the phase of the oscillations as compared to the phase in a three-dimensional metal with a band-contact line. Our results elucidate recent experimental data on the Berry phase for certain electron orbits in ZrSiS, ZrSiTe, and ZrSiSe. |
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Article |
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Mikitik, G.P. Sharlai, Yu.V. |
| author_facet |
Mikitik, G.P. Sharlai, Yu.V. |
| author_sort |
Mikitik, G.P. |
| title |
Oscillations of magnetization in topological line-node semimetals |
| title_short |
Oscillations of magnetization in topological line-node semimetals |
| title_full |
Oscillations of magnetization in topological line-node semimetals |
| title_fullStr |
Oscillations of magnetization in topological line-node semimetals |
| title_full_unstemmed |
Oscillations of magnetization in topological line-node semimetals |
| title_sort |
oscillations of magnetization in topological line-node semimetals |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2018 |
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Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова |
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http://dspace.nbuv.gov.ua/handle/123456789/176168 |
| citation_txt |
Oscillations of magnetization in topological line-node semimetals / G.P. Mikitik, Yu.V. Sharlai// Физика низких температур. — 2018. — Т. 44, № 6. — С. 727-733. — Бібліогр.: 45 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT mikitikgp oscillationsofmagnetizationintopologicallinenodesemimetals AT sharlaiyuv oscillationsofmagnetizationintopologicallinenodesemimetals |
| first_indexed |
2025-07-15T13:50:45Z |
| last_indexed |
2025-07-15T13:50:45Z |
| _version_ |
1837721139854442496 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6, pp. 727–733
Oscillations of magnetization in topological line-node
semimetals
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 Nauky Ave., Kharkiv 61103, Ukraine
E-mail: mikitik@ilt.kharkov.ua
Received December 26, 2017, published online April 25, 2018
We theoretically investigate the phase of the de Haas–van Alphen oscillations in topological line-node semi-
metals. In these semimetals the chemical potential of charge carriers can essentially depend on the magnetic
field, and this dependence changes the phase of the oscillations as compared to the phase in a three-dimensional
metal with a band-contact line. Our results elucidate recent experimental data on the Berry phase for certain elec-
tron orbits in ZrSiS, ZrSiTe, and ZrSiSe.
PACS: 71.20.–b Electron density of states and band structure of crystalline solids;
75.20.–g Diamagnetism, paramagnetism, and superparamagnetism;
71.30.+h Metal-insulator transitions and other electronic transitions.
Keywords: topological line-node semimetals, de Haas–van Alphen oscillations.
1. Introduction
In recent years much attention has been given to the
topological line-node semimetals in which the conduction
and valence bands touch along lines in the Brillouin zone
and disperse linearly in directions perpendicular to these
lines [1–16]. It is necessary to emphasize that the contact of
the electron energy bands along the lines is the widespread
phenomenon in crystals [7,17–19]. For example, such con-
tacts of the bands occur even in many simple metals, and
graphite [20], beryllium [21], aluminium [22], LaRhIn5 [23]
are among them. However, the degeneracy energy of the
bands, dε , is not constant along such lines, and this energy
dε varies in the interval from its minimum minε to its max-
imum maxε values. A crystal with the band-contact line
can be named the topological semimetal if the difference
max min 2ε − ε ≡ ∆ is sufficiently small and if the chemical
potential ζ of the electrons does not lie far away from the
mean energy 0
max min( ) / 2dε ≡ ε + ε of the line.
Rhombohedral graphite [3,24,25], Ca3P2 [6] and CaAgP [9],
Cu3NZn and Cu3NPd [7,8], ZrSiS [11,12,16], ZrSiTe [13],
alkaline-earth germanides and silicides [14], PbTaSe2 [15]
are examples of the line-node semimetals.
The magnetization of electrons in a crystal with the
band-contact line characterized by large ∆ was theoretical-
ly investigated many years ago [26,27], and it was found
that the magnetic susceptibility of the electrons exhibits a
giant anomaly when ζ approaches one of the energies minε
or maxε which correspond to the points of the electron topo-
logical transitions of 3½ kind [18]. In the topological sem-
imetals the interval 2∆ is small, the critical energies minε
and maxε are close to each other, and the character of the
anomaly in the susceptibility changes. The magnetic sus-
ceptibility in the case of the line-node semimetals was con-
sidered for weak magnetic fields in Ref. 28 and for arbi-
trary magnetic fields in Refs. 29, 30.
It is well known [31,32] that at low temperatures the
magnetization of electrons in metals exhibits the de Haas–
van Alphen oscillations. These oscillations are described
by a periodic function of ex / ( ) 2cS e H − πγ where exS is
the extremal cross-section area of the Fermi surface, and γ
is the constant in the semiclassical quantization rule. This
γ is expressed in terms of the Berry phase BΦ for the elec-
tron orbit in the extremal cross section (see Eq. (5) below).
The characteristic feature of the topological line-node sem-
imetals is that the de Haas–van Alphen oscillations are
shifted in phase [30] as compared to the case of metals for
which the band-contact lines are absent, and = 0BΦ ,
= 1/ 2γ . The shift is due to the Berry phase π for electron
orbits surrounding the band-contact line [33]. Recently, the
de Haas–van Alphen [34,35], Shubnikov–de Haas [36–39],
and thermoelectric power [40] oscillations in magnetic
fields were experimentally investigated in the line-node
semimetals ZrSiS, ZrSiTe, and ZrSiSe, and intermediate
© G.P. Mikitik and Yu.V. Sharlai, 2018
G.P. Mikitik and Yu.V. Sharlai
values of the Berry phase (other than 0 and π) were ob-
tained for a number of the electron orbits. In this paper we
suggest an explanation of these unusual values of BΦ de-
tected in the experiments.
Our explanation is based on the following considerations:
Due to small values of ∆, the dispersion relation for the elec-
trons near the band-contact line is similar to the dispersion
relation in layered metals [30]. It is known [41–44] that in
such metals placed in the magnetic field H a crossover from
the three-dimensional electron spectrum to the quasi-two-
dimensional one occurs with increasing H . In the case of
the quasi-two-dimensional spectrum a dependence of the
chemical potential on the magnetic field is strong [32], and
this dependence changes the phase shift of the oscillations.
We show that in the crossover region of the magnetic fields
and in the region of the quasi-two-dimensional spectrum the
shift can differ from π and 0, simulating the case of the Ber-
ry phase deviating from these values.
2. Formulas for magnetization
To clarify the essence of the matter, we consider the sim-
plest band-contact line, assuming that it has the shape of a
straight line in the quasi-momentum space p, and that the
electron dispersion relation in the vicinity of the contact line
of the two bands “c” and “v” has the form:
, 3 ,
2 2 2
, 11 1 22 2
= ( ) ,
= ,
c v d c v
c v
p E
E b p b p
ε ε ±
+
(1)
where the 3p axis coincides with the line; 3( )d pε de-
scribes a dependence of the degeneracy energy along the
line (the maxε and minε mentioned above are the maximum
and minimum values of the function 3( )d pε ); 11b and 22b
are positive constants specifying the Dirac spectrum in the
directions perpendicular to the line. Below we also use the
simplest approximation for the periodic function 3( )d pε ,
3 3
3
2
( ) = cos = cos ,d
p p d
p
L
π ε ∆ ∆
(2)
where = 2 /L dπ is the length of the line in the Brillouin
zone, and d is the appropriate size of the unit cell of the crys-
tal. Besides, we neglect the electron spin (but take into ac-
count the two-fold degeneracy of the electron energy bands in
spin in the formulas below), and consider the case of the zero
temperature T and of the magnetic field H parallel to the line.
The Fermi surface corresponding to the dispersion rela-
tion (2) is a corrugated cylinder when the chemical potential
ζ lies outside the interval from minε to maxε . If ζ is inside
the interval, the Fermi surface has a self-intersecting shape,
and at min=ζ ε or maxε the electron topological transitions
of 3½ kind occur [18].
If the magnetic field H is directed along the line, the
electron spectrum corresponding to the Hamiltonian (1)
has the form [27]:
1/2
, 3 3( ) = ( ) ,l
c v d
e Hp p l
c
α ε ε ±
1/2
3 11 22= ( ) = 2( )p b bα α , (3)
where l is a non-negative integer ( = 0l , 1,…), with the
single Landau subband = 0l being shared between the
branches “c” and “v”. Interestingly, even for 1l , the
spectrum (3) exactly coincides with that obtained from the
semiclassical quantization rule,
3
2( , ) = ( ),l
e HS p l
c
π
ε + γ
(4)
where 2
3 3( , ) = 2 [ ( )] /dS p pε π ε − ε α is the area of the
cross section of the isoenergetic surface by the plane per-
pendicular to the magnetic field and passing through the
point with the coordinate 3p , the constant γ is expressed in
term of the Berry phase BΦ for the appropriate electron
orbit [33]:
1= ,
2 2
BΦ
γ −
π
(5)
and =BΦ π in our case of the orbit surrounding the band-
contact line.
The magnetization of electrons in a line-node semimetal
was calculated in Ref. 30 at an arbitrary shape of its band-
contact line. Using this result, we obtain the following ex-
pressions for the magnetization 3M directed along the
band-contact line being considered:
3/2
1/2 1/2
3 32
0
1( , ) ( ),
2
LeM H H dp K u
c
ζ = α
π ∫
(6)
where the integration is carried out over this line;
3 1 1( ) = ( ,[ ] 1) ([ ] ),
2 2 2
K u u u uζ − + + + (7)
( , )s aζ is the Hurwitz zeta function,
2
3 3[ ( )] ( , )
= = ,
2
d p c cS p
u
e H e H
ζ − ε ζ
α π
(8)
and [ ]u is the integer part of u .
In the topological semimetals, charge carriers (electron
and holes) are located near the band-contact line, and their
chemical potential ζ generally depends on the magnetic
field, = ( )Hζ ζ . This dependence can be derived from the
condition that the charge-carrier density ( , )n Hζ does not
vary with increasing H ,
0 0( , ) = ( ),n H nζ ζ (9)
where 0n and 0ζ are the density and the chemical potential
at = 0H . At = 0T the densities 0 0( )n ζ and ( , )n Hζ are
described by the following expressions [30]:
728 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
Oscillations of magnetization in topological line-node semimetals
2
0 0 3 0 3 0 32 3
0
1( ) = ( ( )) ( ( )),
2
L
d dn dp p pζ ζ − ε σ ζ − ε
π α ∫
(10)
3 32 2
0
1( , ) = ( [ ]) ( ( )),
22
L
d
eHn H dp u p
c
ζ + σ ζ − ε
π ∫
(11)
where ( ) = 1xσ if > 0x , and 1− if < 0x . On calculating
( )Hζ with Eqs. (9)–(11), one can find the magnetization as
a function of 0n or 0ζ , inserting ( )Hζ into Eqs. (6)–(8).
In analyzing the effect of ( )Hζ on the phase of the de
Haas–van Alphen oscillations, we shall plot the so-called
Landau-level fan diagrams [32] commonly used in treat-
ments of the experimental data. At = 0T the periodic in
1/ H magnetization of electrons in metals exhibits singu-
larities (sharp maxima or minima [45]) when the lower or
upper edge of the lth Landau subband crosses the Fermi
level ζ . In the semiclassical limit ( 1l ) and under the
assumption that ζ is independent of H , such crossings
occur at the magnetic fields lH determined by Eq. (4):
ex
1 2= ( ),
( )l
e l
H cS
π
+ γ
ζ
where ex ( )S ζ is the minimum or maximum value of
3( , )S pζ with respect to 3p . Thus, if the positions lH of
the singularities are known, the constant γ can be found
with the Landau-level fan diagram: Plotting the Landau-
level index l versus 1/ lH and continuing the obtained
straight line up to the intersection of this line with the l
axis in which (1/ ) = 0lH , the coordinate −γ of the inter-
section enables one to obtain γ : =γ γ. It is important to
emphasize that if ζ lies in the energy region where the
Dirac spectrum occurs, one can use lH with 1l in the
construction of the fan diagrams since, as was mentioned
above, the semiclassical spectrum resulting from formu-
la (4) coincides with the exact one given by Eq. (3) even at
small l. Note also that this procedure of determining γ ,
which characterizes the phase of the de Haas–van Alphen
oscillations, is applicable to the case when ζ depends on
H , but as shall be demonstrated below, γ thus extracted
does not generally coincide with the constant γ specifying
the semiclassical quantization rule.
3. Discussion
The quantity u defined by Eq. (8) changes along the
nodal line between its minimal minu and maximal maxu
values which correspond to the minimal minS and maximal
maxS values of 3( , )S pζ with respect to 3p . In the case of
sufficiently weak magnetic fields when minu , maxu ,
max min 1u u− , i.e., when
max min max min
2 , , ,e H S S S S
c
π
−
(12)
the Landau subbands , 3( )l
c v pε with different l overlap as
in three-dimensional metals. According to Ref. 30, in this
case formula (6) reduces to the well-known
expression [32,42,44] describing the de Haas–van Alphen
oscillations in a three-dimensional metal but with = 0γ .
For 3( )d pε given by Eq. (2), this expression is a superposi-
tion of two periodic functions determined by the two
extremal cross-section areas min 3= ( , = 0)S S pζ and
max =S 3( , = / 2)S p L= ζ . The dependence ( )Hζ is suffi-
ciently weak in this three-dimensional case and practically
has no effect on the oscillations [32].
Consider now stronger magnetic fields than in the case of
inequalities (12). If | |ζ ∆ , the difference max minu u− be-
comes less than unity when minu and maxu are still large. In
this situation, the spectrum (2) transforms into the spectrum
of a quasi-two-dimensional electron system since the differ-
ent Landau subbands , 3( )l
c v pε do not overlap, and they look
like broadened Landau levels for which the spacing between
the nearest Landau subbands in the vicinity of the Fermi
level is larger than their width 2∆. At max min 1u u− , the
quantity u is practically independent of 3p running the line,
3( , ) ( )S p Sζ ≈ ζ , the corrugation of the Fermi surface be-
comes unimportant, and formula (6) is simplified as follows:
3/2
1/2 1/2
3 1/2
1( , ) ( ),eM H H K u
cd
ζ ≈ α
π
(13)
where
2 ( )= ,
2
c cSu
e H e H
ζ ζ
≈
α π
(14)
and the function ( )K u at large u has the form [30]:
=1
sin(2 )( ) ( [ ] 0.5) = .
2 2 l
u u luK u u u
l
∞ π
≈ − − −
π ∑ (15)
Equations (13)–(15) describe saw-tooth oscillations of
3M with changing 1/ H , and they coincide with the appro-
priate expression [42–44] for a two-dimensional metal with
the Dirac spectrum. A refined analysis of Eq. (6) at
| |ζ ∆ and maxu , min 1u gives
2 2
3 02
=1
| | 1 ( )sin 2 4 ,
2 l
e c cM l J l
l e H e Hcd
∞ ζ ζ +∆ ζ∆ ≈− π π α α π
∑
(16)
where 0 ( )J x is the Bessel function for which one has
0 ( ) 1J x ≈ at 1x and 1/2
0 ( ) (2 / ) cos( / 4)J x x x≈ π − π at
1x . Formula (16) agrees with the appropriate expression
of Refs. 42, 44 and reproduces both Eqs. (13)–(15) at
max min = 4 / 1u u c e H− ζ∆ α and the formula for the de
Haas–van Alphen oscillations in three-dimensional metals
with = 0γ at max min 1u u− .
The crossover from the three-dimensional electron spec-
trum to the quasi-two-dimensional one occurs at the magnet-
ic field crH defined by the condition max min 1u u− ,
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6 729
G.P. Mikitik and Yu.V. Sharlai
max min
cr
( ) 4 | |= .
2
c S S cH
e e
− ζ ∆
π α
(17)
For crH H , the spacing H∆ε between the Landau
subbands in the vicinity of the Fermi level becomes compa-
rable with their width 2∆. Thus, the quasi-two-dimensional
regime of the oscillations takes place in the interval of the
magnetic fields cr 1< <H H H where 1H is the field of the
ultra-quantum limit,
2
1
( ) = ,
2
cS cH
e e
ζ ζ
π α
(18)
at which H∆ε reaches ζ , and the oscillations of the magnet-
ization disappears. When H changes in this interval, the
chemical potential ( )Hζ moves together with one of the
Landau subbands, and then, at a certain value of H , it jumps
from this subband to the neighboring one [32], Fig. 1. This
strong dependence ( )Hζ noticeably changes the shape of
the de Haas–van Alphen oscillations and can mask the cor-
rect values of γ (and of the Berry phase) when γ is found
with the Landau-level fan diagram. Indeed, the jumps can
occur at the fields lH for which ( )n ζ in Eq. (11) becomes
independent of ζ . This situation is realized when [ ]u in the
right-hand side of Eq. (11) is one and the same integer along
the whole line. Let us denote this integer as ( 1)l − . Then,
Eq. (9) takes the form:
2 2
0 0
1 1= .
22 ( )l
eL l
H cn
−
π ζ
(19)
These lH also mark the singularities in the magnetization
since immediately above and below lH the edges of the
Landau subband touch the chemical potential, Fig. 1. It fol-
lows from equation (19) that the dependence of 1/ lH on l is
a straight line that intersects the l axis at = = 1/ 2l −γ , i.e.,
the Landau-level fan diagram plotted with the fields lH
looks like in the case when = 1/ 2γ and the Berry phase BΦ
is equal to zero. (The value of γ is defined up to an integer,
and so = 1/ 2γ − and = 1/ 2γ are equivalent.) However, in
reality one has =BΦ π , = 0γ , and the phase shift γ extract-
ed from the oscillations in the quasi-two-dimensional regime
does not permit one to find BΦ since γ ≠ γ now. The fore-
going considerations are illustrated in Fig. 1 for which
0 / = 20ζ ∆ and max 0 cr 1 cr( ) / (2 ) / 5cS e H H Hζ π .
Consider now the case when ζ is of the order of ∆. In this
situation 1 crH H , and the crossover occurs near the ultra-
quantum limit. Since, as was explained above, we have = 0γ
and 1/ 2− in the three-dimensional and in the quasi-two-
dimensional regimes of the oscillations, respectively, one may
expect that γ takes intermediate values if this quantity is
found in the crossover region. In Fig. 2 we show the de Haas–
van Alphen oscillations of 3M calculated with Eq. (6) at
0 / = 3ζ ∆ when max 0 cr 1 cr( ) / (2 ) / 1cS e H H Hζ π . In
this situation we find = 0.23 0.04γ − ± if the H -dependence
of ζ is taken into account. Note that a relatively small con-
tribution of the minimal cross section min max= / 4S S into
the oscillations is also visible in the figure. This minimal
cross section slightly affects the maxima in 3M associated
with maxS , and so we find = 0.02 0.01γ ± even for the oscil-
lations calculated at fixed 0=ζ ζ .
In Fig. 3 we present the dependence of γ on 0ζ . In the
construction of this figure, values of γ have been obtained,
using the first four sharp maxima in the calculated functions
3(1/ )M H . The small jump in γ at 0 / 15ζ ∆ ≈ is due to that
at 0 / > 15ζ ∆ all the four maxima in 3M are determined by
the jumps in ζ , whereas at 0 / < 15ζ ∆ the chemical potential
is continues for a part of these maxima, cf. Figs. 1 and 2. At
0 / 1ζ ∆ ≈ the jumps in ζ disappear completely. For 0 / < 3ζ ∆
one has max min/ > 4S S , i.e., the ratio of the oscillation
Fig. 1. (a) The magnetization 3M , Eq. (6), versus 1 / H at fixed
chemical potential 0= = 20ζ ζ ∆ (the dashed line) and at ( )Hζ
shown in the lower panel (the solid line). Here max 0( ) =S ζ
2
02 ( ) /= π ζ + ∆ α , and max 0 cr( ) / (2 ) 5cS e Hζ π . The inset
depicts the Landau-level indexes l versus max 0( ) / (2 )lcS e Hζ π ;
1 / lH are the positions of the maxima of the functions 3(1 / )M H
shown by the solid and dashed curves in the main plot. (b) The
dependence of ζ on 1 / H calculated with Eqs. (9)–(11) at
0 = 20ζ ∆ . We also mark the Landau subbands by the dark back-
ground, and the short and long dashes indicate the lower and the
upper edges of these subbands, respectively.
730 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
Oscillations of magnetization in topological line-node semimetals
periods corresponding to the minimal and maximal cross
sections is larger than 4, and the effect of the low-frequency
oscillations on the first four peaks in 3M decreases. For this
reason the break appears in the dependence 0( )γ ζ at
0 / 3ζ ∆ ≈ . In other words, this break as well as the jump at
0 / 15ζ ∆ ≈ are caused by the relatively small number of the
peaks in 3M used in our plotting the Landau-level fan dia-
grams.
Consider now the situation when apart from the charge
carriers located near the nodal line under study, there is an
additional electron group in the semimetal. In particular,
this situation occurs in ZrSiS. For simplicity, we shall ne-
glect the quantization of electron energy in the magnetic
fields for the charge carriers of this additional group. Note
that this simplifying assumption is easily realized even at
low temperatures if the cyclotron mass of these carries is
essentially larger than the cyclotron mass of electrons near
the nodal line. Then, with the additional electron group,
equation (9) is modified as follows:
0
0 0 0
0
( )
( , ) ( ) = ( ),
dN
n H n
d
ζ
ζ + ζ − ζ ζ
ζ
(20)
where 0 0( ) /dN dζ ζ is the density of the electron states of
the additional group at the Fermi level. We shall specify
this density of the states by the formula
0 0 0
0 0
( ) ( )
= ,
dN n
d
ζ ζ
λ
ζ ζ
(21)
where λ is the dimensionless parameter. If 0,λ → we return
to the case of the single electron group located near the nod-
al line. As was shown above, in this case one always has
= 1/ 2γ − for the oscillations in the quasi-two-dimensional
regime. If 1,λ we arrive at the case of the constant chem-
ical potential which is stabilized by the large additional elec-
tron group. In this case = = 0γ γ in the quasi-two-
dimensional regime. Hence, it is reasonable to expect that γ
will take intermediate values if 1.λ In Fig. 4, we show the
oscillation of 3M in the quasi-two-dimensional regime for
three values of the parameter = 0,λ 1, ∞. It is seen that the
phase of the oscillations at = 1λ indeed has a value which
lies between the values corresponding to the other two cases.
Thus, the intermediate values of γ can be found not only in
the crossover region but also in the quasi-two-dimensional
regime of the oscillations if there is an addition group of
charge carriers in the semimetal.
4. Conclusions
We theoretically investigate the phase of the de Haas–
van Alphen oscillations in topological line-node semimetals,
using the simple model for their nodal lines, Eqs. (1) and (2).
There are two regimes of the oscillations. These regimes are
determined by the relation between spacing H∆ε separating
the Landau subbands in the vicinity of the Fermi level and
the width of these subbands, 2∆, resulting from the disper-
sion of the degeneracy energy along the nodal line.
Fig. 2. (a) The magnetization 3M , Eq. (6), versus 1 / H at fixed
chemical potential 0= = 3ζ ζ ∆ (the dashed line) and at ( )Hζ
shown in the lower panel (the solid line). Here max 0( ) /cS ζ
cr/(2 ) 1e Hπ . (b) The dependence of ζ on 1 / H calculated
with Eqs. (9)–(11) at 0 = 3ζ ∆ . All the notations and the inset are
similar to Fig. 1. The intercept of the solid straight line in the
inset gives = 0.23 0.04−γ ± .
Fig. 3. Dependence of γ on 0ζ . The values of γ have been found
with the Landau-level fan diagrams plotted using the first four
sharp peaks in the functions 3(1 / ).M H
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6 731
G.P. Mikitik and Yu.V. Sharlai
For not-too-strong magnetic fields when 2H∆ε ∆ , the
three-dimensional regime of the oscillations occurs, the de-
pendence of the chemical potential ζ on the magnetic field H
is weak, and the constant γ defining the phase of the oscilla-
tions coincides with the constant γ in the semiclassical quan-
tization rule, i.e., =γ γ . Since the Berry phase =BΦ π and,
according to Eq. (5), = 0γ for the electron orbits surrounding
the nodal lines, one can detect these lines, measuring the
phase of the de Haas–van Alphen oscillations in this case.
With increasing magnetic fields, the spacing between the
Landau subband becomes larger than 2∆, and the quasi-two-
dimensional regime of the oscillations takes place. In this
regime the dependence of ζ on H is strong. This depend-
ence changes the shape and the phase of the oscillations, and
γ ≠ γ in this regime. At low temperatures ( )HT ∆ε we
find that | |= 1/ 2γ for the extremal cross sections for which
= 0γ . Thus, the results of the phase measurements will imi-
tate the case = 1/ 2γ and will not permit one to find the true
values of γ and of the Berry phase.
Due to the experimental data of Refs. 34–40, the special
attention in our paper is given to the situations in which val-
ues of | |γ can be intermediate between 0 and 1/2. We show
that these situations can occur in the region of the magnetic
fields where the crossover from the three-dimensional re-
gime to the quasi-two-dimensional one takes place, and in
the quasi-two-dimensional regime if there is an additional
group of charge carriers in the semimetal. In these cases,
measurements of γ (the phase of the oscillations) provide
information on the electron energy spectrum of the semi-
metal, see Fig. 3, rather than on the Berry phase of the elec-
tron orbits.
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1. Introduction
2. Formulas for magnetization
3. Discussion
4. Conclusions
|