The Theory of High-Temperature Superconductivity in Many-band Systems. MgB2
The main stages of development of the theory of superconducting systems with overlapping energy bands are formulated. The main references of the classical papers of the author of this theory, Prof. V.A. Moskalenko, and his coworkers are listed. The list also includes papers related to high-temperatu...
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| Cite this: | The Theory of High-Temperature Superconductivity in Many-band Systems. MgB2 / M.E. Palistrant, L.Z. Kon // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 44-54. — Бібліогр.: 107 назв. — англ. |
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| citation_txt | The Theory of High-Temperature Superconductivity in Many-band Systems. MgB2 / M.E. Palistrant, L.Z. Kon // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 44-54. — Бібліогр.: 107 назв. — англ. |
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| description | The main stages of development of the theory of superconducting systems with overlapping energy bands are formulated. The main references of the classical papers of the author of this theory, Prof. V.A. Moskalenko, and his coworkers are listed. The list also includes papers related to high-temperature superconductivity. Some peculiarities of the two-band model which gives qualitatively new results in comparison with the usual one-band model, are enumerated. The application of the two-band model to the description of thermodynamical properties of compound MgB2 is also discussed. The references covering our research of the kinetic properties of superconductors with overlapping energy bands are provided. In particular, we present the Ginzburg–Landau (GL) equations for the two-band superconductors doped with impurities and the results on the influence of impurities on the energy gap, as well as those concerning the dynamical properties of twoband superconductors.
Сформульовано основнi етапи розвитку теорiї надпровiдних систем iз зонами, що перекриваються. Наведено основнi посилання на класичнi роботи автора цiєї теорiї, професора В.Л. Москаленко, i його спiвробiтникiв разом iз роботами з високотемпературної надпровiдностi. Вiдзначено особливостi двозонної моделi, яка дає якiсно новi результати порiвняно зi звичайною однозонною. Обговорено застосування двозонної моделi для опису термодинамiчних властивостей сполуки MgB2. Дано огляд наших дослiджень кiнетики надпровiдникiв iз зонами, що перекриваються. Зокрема, наведено рiвняння Гiнзбурга–Ландау для двозонних надпровiдникiв з домiшками та результати впливу домiшок на ширину забороненої зони. Розглянуто динамiчнi властивостi двозонних надпровiдникiв.
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SOLID MATTER
44 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
THE THEORY OF HIGH-TEMPERATURE
SUPERCONDUCTIVITY IN MANY-BAND SYSTEMS. MgB2
M.E. PALISTRANT, L.Z. KON
Institute of Applied Physics, Kishinev
(5, Str. Academiei, MD 2028, Republic Moldova; e-mail: mepalistrant@ yandex. ru )
PACS 74.25.Bt, 74.25.Ha
c©2010
The main stages of development of the theory of superconduct-
ing systems with overlapping energy bands are formulated. The
main references of the classical papers of the author of this the-
ory, Prof. V.A. Moskalenko, and his coworkers are listed. The list
also includes papers related to high-temperature superconductiv-
ity. Some peculiarities of the two-band model which gives qualita-
tively new results in comparison with the usual one-band model,
are enumerated. The application of the two-band model to the
description of thermodynamical properties of compound MgB2 is
also discussed. The references covering our research of the kinetic
properties of superconductors with overlapping energy bands are
provided. In particular, we present the Ginzburg–Landau (GL)
equations for the two-band superconductors doped with impuri-
ties and the results on the influence of impurities on the energy
gap, as well as those concerning the dynamical properties of two-
band superconductors.
1. Two-Band High-Temperature
Superconductivity
In this work, we partially touch upon that distant heroic
time, when the mechanism of such amazing effect as su-
perconductivity was discovered (the year of 1957).This
problem had been solved as a result of the long-time and
intense theoretical and experimental researches (it had
taken almost 50 years), and it had appeared as the most
important cryogenics discovery. The significant results
in the construction of the theory of superconductivity
belong to the great physicist and mathematician N.N.
Bogoliubov [1–4], whose centenary we celebrate nowa-
days. The great contribution to the development of
the theory was given by his disciples: S.V. Tyablikov,
D.N. Zubarev, D.V. Shirkov, and other eminent scien-
tists. Along with the development of the theory of super-
conductivity in the isotropic systems, Bogoliubov came
up with the idea of the anisotropic properties of super-
conducting systems accounting for the need to describe
real metals. The first work in this direction, where the
anisotropy appears in the band structure of the consid-
ered system, belongs to V. Moskalenko.
The model of a superconductor with the overlapping
of energy bands on the Fermi surface was proposed by
V. Moskalenko [5] and some later also by H. Suhl et al.
[6] about 50 years ago.
This model considers the anisotropy of energy bands
and describes the superconducting properties of transi-
tion metals that have two or more groups of electrons
belonging to different energy bands on the Fermi sur-
face.
The main assumption of the model is the formation of
Cooper pairs of electrons inside one energy band and the
transition of a pair as the whole entity from one energy
band to another one. This results in the appearance
of the intraband Vnn and the interband Vnm(m 6= n;
n, m = 1, 2) electronic interactions, which leads to the
additional attraction of electrons. This, in its turn, fa-
vors an increase of the superconducting transition tem-
perature. This is equivalent to the diagonal approxima-
tion over band indices, and two order parameters Δ11
and Δ22 appear in the two-band model.
Having made these assumptions, V. Moskalenko and
his co-workers have carried out the investigations of the
thermodynamic and electromagnetic properties of many-
band superconductors. A few books and a lot of articles
on this problem have been published (see, e.g., [7–16]
and references therein, as well as [17, 18]).
It had appeared a new scientific trend of studies of
the multiband superconductors’ properties. Along with
the Moldavian theoretical physicists, a lot of scientists
in different countries worked in this direction (see, e.g.,
THE THEORY OF HIGH-TEMPERATURE SUPERCONDUCTIVITY
[19–22] and references in [15]). It should be noted that
the most valuable contribution to the development of the
theory of superconductivity in the systems, where two
and more energy bands overlap on the Fermi surface,
belongs to the Moldavian scientists.
At the time preceding the discovery of high-Tc super-
conductivity, the theory of superconductivity with over-
lapping energy bands had been developed in order to
describe the physical characteristics in the systems with
heavy fermions [23, 24]. The discovery of high-Tc super-
conductivity [25] has aroused the attention to this prob-
lem again, which is explained by the overlapping of two
or more energy bands on the Fermi surface in the high-
Tc superconductor materials. Such a conclusion results
from the band structure of oxide ceramics [26, 27]. This
can be understood as well, having considered a layered
structure of these systems.
The Hamiltonian that describes the system with n lay-
ers and takes interactions of electrons inside each layer
and between layers into account can be transformed
without difficulty into a Hamiltonian that describes n
band systems [28–30]. This transformation has been
done by the diagonalization and the introduction of the
symmetric and anti-symmetric states of electrons.
Since the discovery of high-Tc superconductivity, a
great number of theoretical studies of the many-band
model [5] regarding the high-Tc materials have been
made.
An increase in the number of energy bands on the
Fermi surface leads to an increase in the total density of
electron states and to an additional interband electron-
electron interaction which favors the superconducting
state. This interaction destroys the validity of the uni-
versal BCS relations, and the thermodynamic charac-
teristics depend significantly on the properties of the
anisotropic system.
Series of experimental studies have been performed
there as well. They discovered the presence of two energy
gaps in the energy spectrum of electrons and proved, in
its turn, the appearance of singularities caused by the
overlapping of energy bands on the Fermi surface in the
energy spectrum of electrons (see, e.g., [31]).
Nowadays, the researchers have concluded that the ex-
periment in the oxide metals does not confirm the pres-
ence of two energy gaps. This may be a result of many
difficulties of the high-Tc superconductivity and, in par-
ticular, of a disorder in the system (great concentrations
of impurity or oxygen deficiency). In this case, despite
the presence of two order parameters, only a single en-
ergy gap is found in two-band superconductors at high
concentrations of impurity [13]. There has appeared the
averaged order parameter Δ̄ = N1Δ1 +N2Δ2
N1 +N2
that de-
termines the averaged energy at T = 0 [32]
ES − EN
V
= −1
2
Δ̄2(N1 +N2),
where Ni is the density of electron states on the ith fold
of the Fermi surface.
At great concentrations of an impurity, the jump of
the heat capacity at the critical point T = Tc in a two-
band system has the form [13]
CS − CN
CN
=
8π2
7βc
N1 +N2
ζ(3)
.
These expressions coincide formally with the case of
the pure one-band model with the only difference that
the quantity βc = (kTc)−1 is determined on the two-
band background and contains all electron-electron in-
teraction couplings (the intraband and interband ones).
In addition to this, the density of electron states is re-
placed by the total density of both bands. Note that,
in the two-band systems with the great concentration
of a non-magnetic impurity, the equation of Ginzburg–
Landau is reduced to the one-band equation for the wave
function Δ(x) that is the gravity center of wave functions
Δ1(x) and Δ2(x). In this way, we have to take into ac-
count that the quantity Tc and the parameter κ of this
one-band equation are determined on the two-band ba-
sis.
Therefore, the overlapping of energy bands on the
Fermi surface contributes essentially to the physical
quantities even at a strong disorder in the system.
The conclusion about the absence of two gaps in the
electron energy spectrum (because the experiment does
not confirm it) or even omitting the contribution of both
energy bands that overlap on the Fermi surface to the
thermodynamic and electromagnetic characteristics of
the oxide metals was earlier stated. More detailed ex-
perimental studies should be done both at the high and
low densities of charge carriers in order to decrease the
disorder of a two-band system.
Note also that Moskalenko’s model [5], as well as its
generalizations for the arbitrary densities of charge car-
riers (including the very low densities), is an isotropic
one. So, the model assumes the singlet pairing of Cooper
pairs. There is no difficulty to construct a theoretical
two-band model with the d-pairing of electrons, as it
has been done for the two-band superconductors with
non-magnetic impurity [33].
The consideration of the overlapping of energy bands
leads not only to the quantitative difference of the results
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 45
M.E. PALISTRANT, L.Z.KON
from the case of the one-band superconductor, but, in
some cases, to qualitatively new results. For example:
(1) In the two-band superconductors, the high tem-
peratures of the superconducting transition are possible
not only in the case of the attractive interaction be-
tween electrons, but even in the case of the repulsive
one (λnm < 0, n,m = 1 − 2), but when the relation
λ11λ22 − λ12λ21 < 0 is fulfilled.
(2) In the impurity two-band superconductors, for ex-
ample, the Anderson theorem is violated at Δ1 6= Δ2,
and the dependence of thermodynamic quantities on the
concentration of a non-magnetic impurity due to the in-
terband scattering of electrons on the impurity atoms is
observed.
(3) In the two-band superconductors, the collective
oscillations of the exciton-like Leggett mode caused by
fluctuations of the phase of order parameters from differ-
ent bands are observed. In the three-band systems, and
in the two-band ones with account of the Cooper pairs
of electrons from different energy bands, that is reduced
to the effective three-band model, the number of such
oscillatory modes can be two.
(4) Using the two-band model and assuming the mod-
erate values of coupling constants, one can obtain high
values of Tc, the two energy gaps 2Δ1/Tc > 3.5 and
2Δ2/Tc < 3.5, large values of negative dlnTc/dlnV (V is
the volume), positive curvature of the upper critical field
near the transition temperature, etc. [34–38]. Further-
more, in the two-band model, it is possible to describe
a decrease of Tc with increase of the system disorder
[39–41].
(5) The location of the Fermi level that can be changed
by doping plays an important role in the determination
of thermodynamic and magnetic properties of a two-
band superconductor. Having assumed a non-phonon
pairing mechanism of superconductivity, as well as the
phonon mechanism in the many-band systems with low-
ered densities of charge carriers, the account of the sin-
gularities mentioned above is very crucial. A particular
interest is attached to the possibility to obtain a bell-
shaped dependence of Tc and a jump of the heat capac-
ity (CS − CN )/CN at the point T = Tc on the carrier
density [42–44]. In the three-band model, it is possible
to obtain a “step” in the dependence of Tc on the carrier
density [45, 46]. The investigation of the properties of
superconductors with energy bands on the Fermi surface
and the electronic topological transitions, was reviewed
in [47]. This review contains the classical results of the
problem.
An increase in the number of energy bands on the
Fermi surface increases the overall density of electron
states and leads to the onset of an additional interband
electron-electron interaction that contributes to the on-
set of superconductivity. This interaction violates the
universal BCS relations and leads to the substantial de-
pendence of a number of the physical characteristics on
properties of the anisotropic system [12, 13, 15, 47–49].
Note that the experimental confirmation of two-band
superconductivity has been made long before the discov-
ery of the high-Tc superconductivity. For example, there
have been studied the tunnel characteristics of SrTiO3:
NbIn in [50], and two order parameters have been found.
The appearance of two energy bands that overlap on the
Fermi surface in this compound is likely to be caused by
the great value of the dielectric constant of strontium
titanite. This results in the weak interband scattering of
electrons on the ionized donors, and even their high con-
centration does not initiate the transition of the system
to the one-band case. When the carrier concentration is
gradually increased by adding niobium, the consequent
filling of the first and second energy bands takes place.
The discovery of superconductivity in the intermetallic
borocarbide nitrides was the important whirling point in
superconductivity. These compounds possess interesting
superconducting and magnetic properties. The more de-
tailed information about the discovery of diverse new in-
termetallic compounds and their properties can be found
in [51].
2. Superconductivity in compound MgB2 and
in the Systems with Reduced Density of
Charge Carriers
The discovery of superconductivity in the intermetallic
compound MgB2 was even more intriguing [52]. The
main physical properties and singularities are as follows
(see, e.g., [53] and [54]):
(1) The high temperature of the superconducting tran-
sition, Tc ∼ 39 K [52].
(2) The interaction responsible for the formation of
superconducting pairs is caused by the exchange of
phonons [55, 56], and the symmetry of Cooper pairs is
of the s-wave type [57].
(3) The average phonon frequencies are 2–3 times as high
as those for classical superconductors Nb2Sn [58, 59], the
mass renormalization factor (1 + λ) is small, the Som-
merfeld’s coefficient γn and the condensation energy also
appear to be inconsistent with superconductivity near
∼ 40 K.
Researchers have concluded that the superconductiv-
ity in MgB2 cannot be understood from the isotropic
one-band BCS model. The anisotropy of the system is
46 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
THE THEORY OF HIGH-TEMPERATURE SUPERCONDUCTIVITY
the very key factor here that is revealed in the anisotropy
of energy bands (the overlapping of energy bands on the
Fermi surface) and in the anisotropy of the order param-
eter, i.e. its dependence on the momentum direction.
As was mentioned above, the properties of the
isotropic two-band (multiband) superconductors have
been studied long before the discovery of MgB2 com-
pounds by V. Moskalenko and his co-workers. These
studies showed that the thermodynamic and electromag-
netic properties of a two-band superconductor are quali-
tatively different from those of a one-band superconduc-
tor.
Another field of studies in the BCS theory is an ac-
count for an anisotropy of the electron-phonon interac-
tion [60, 66]. As shown in the works mentioned above,
the band anisotropy, as well as the anisotropy of the
matrix element of the electron-phonon interaction, de-
creases the relative jump of the electron heat capacity
(CS − CN )/CN at T = Tc in comparison with the value
1.43, which is specific of the isotropic case.
Works [62–64] have developed the method taking
the existence of a greater number of bands and the
anisotropy of energy gaps in each band into consider-
ation in the calculation of the electron heat capacity.
We did not intend to make a thorough analysis of
the works mentioned above. Note only that the clear
picture with regard for the band structure, the topol-
ogy of the Fermi surface, the values of densities of
electron states, the averaged velocities of electrons on
the Fermi surface, and other characteristics necessary
for consistency of theory with experiment has been ob-
tained. In MgB2, two energy gaps Δ1(0) = 6.8 meV
(2Δ(0)/kBTc = 4.0); Δ2(0) = 1.8 meV (2Δ2(0)/kBTc =
1.06) have been experimentally determined [65]. As a
result, some singularities of thermodynamic properties
have been observed in these compounds. The shoulder-
type anomaly, for example, appears in the tempera-
ture dependence of the heat capacity around 0.25Tc
as well as (CS − CN )/CN ) ≈ 0.8 at T = Tc [59],
which is consistent with the theory [5, 9, 64]. Another
anomaly observed experimentally is the positive curva-
ture of the upper critical magnetic field Hc2(T ) near
the temperature of superconducting transition (see the-
oretical studies [47, 65]). Note also the breakdown of
the Anderson theorem in the two-band system with a
non-magnetic impurity due to the interband scattering
of electrons on the impurity, which leads to a disorder
in the system [7, 8, 13]. These and other anomalies in
MgB2 can be understood only by considering the over-
lapping of energy bands on the Fermi surface. The two-
band model describes qualitatively the main singular-
ities in the behavior of the physical characteristics of
MgB2.
A possibility to describe superconductivity on the ba-
sis of the two-band model in other compounds cannot be
excluded. This fact validates the following generaliza-
tions and the development of superconductivity theory
with regard for the overlapping of energy bands.
In all the above-mentioned works, the two-band model
can be used to describe the properties of the supercon-
ductors, for which the relation µ � Tc is satisfied (µ
is the chemical potential). This description is made
in the diagonal approximation over the band indices
[7, 10].
In systems with low carrier densities, however, the re-
lation µ � Tc does not hold. Therefore, it is necessary
to develop the theory of superconductivity for the two-
band systems without constraints on the Fermi energy.
We consider simultaneously two possible superconduc-
tivity mechanisms – the phonon and electron ones. The
substantial dependence of the chemical potential µ on
the order parameter in the superconducting phase is an
inherent feature of the systems with low carrier densi-
ties. These circumstances have been noted in many pa-
pers, and the feasibility of experimental observation of
these anomalies in the temperature dependence of the
chemical potential was first suggested in [66]. It was
shown there with the BCS model as an example that
the µ(T ) curve has an experimentally observable bend
at the point T = Tc. Below, we show that, in the two-
band case, this effect is enhanced by the presence of
two or four order parameters (Δnm; m = 1, 2), and is
manifested at µ values more easily observed in experi-
ment [67].
In the works cited above, the investigations were car-
ried out by the Cooper pairing scenario. In the sys-
tems with small carrier concentrations, the bound states
may arise following a decrease in the carrier concentra-
tion, and the transition to the Bose condensate of lo-
calized pairs with a finite bound energy may occur (the
Schafroth scenario [68]). The possibility of such a transi-
tion in the one-band systems was discussed in a number
of works (see, e.g., [67–74]).
As was shown in [69, 72], a change of the sign of
the chemical potential with decrease in the carrier con-
centration corresponds to the transition from the BCS
to Schafroth scenario. The condensation of localized
pairs occurs at the concentrations of carriers, for which
µ ≤ 0.
We present the theory of superconductivity of the two-
band systems that is valid at any carrier density and con-
siders all possible pairing of electrons due to the intra-
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 47
M.E. PALISTRANT, L.Z.KON
band and interband interactions by the Cooper pairing
scenario. The critical temperature Tc, chemical poten-
tial µ, and heat capacity (CS −CN ) at the point T = Tc
as functions of the carrier density are shown in [47, 67,
75]. This theory is used to describe the properties of
MgB2-based compounds, where magnesium and boron
are replaced by different elements of the Periodic ta-
ble. The works [76,77] are designed for the self-consistent
discussion, in the mean-field approximation, of the sys-
tem of equations for the order parameters Δn and µ
at T = 0. There have been revealed the influence of
the overlapping of energy bands on these quantities and
the carrier concentration, at which the system experi-
ences the transition from the Cooper pairing (µ > 0)
to the Schafroth (µ < 0) scenario. The equation for
the binding energy εb of the two-particle state is also
obtained, and the relation between εb and µ is estab-
lished.
The path integral method as applied to the two-band
model is also developed, and, on this basis, the pro-
cedure for the transition from the Fermi to Bose ele-
mentary excitations at T = 0 is given in [75, 78, 79].
The Bose system condensation temperature Tk is also
determined. The theory of superconductivity in two-
band non-adiabatic systems with strong electron corre-
lations in the linear approximation over non-adiabaticity
is built [80].
The superconducting ordering in the systems with
two characteristic features – the small concentration of
charge carriers and the overlapping of energy bands on
the Fermi surface – is investigated.
Along with the above-described phenomena related
to the overlapping of energy bands on the Fermi sur-
face, a very interesting one has to be examined – the
appearance of collective oscillations due to the phase
fluctuations of the order parameters of different bands.
Fistly, this phenomenon was researched in the theo-
retical work by Leggett [74]. According to its na-
ture, it can appear only in the systems with two or
more energy bands on the Fermi surface. Our follow-
ing results on collective oscillations in three-dimensional
systems and in the systems with reduced dimension-
ality develop the Leggett’s researches, by considering
two or more energy bands, within the phonon and
non-phonon superconducting mechanisms, supposing a
reduced and a weak carrier concentration until the
transition from BCS state (µ > 0) to the Schafroth
state (µ < 0). The collective exciton-type modes dif-
fer quantitatively in different systems and are deter-
mined by physical features of the examined systems
[81–85].
3. Thermodynamic and Magnetic Properties of
Doped Compound MgB2
It should be noted that the above-discussed two-band
model [5] was found to be very fruitful, since it had ex-
plained a lot of abnormal physical properties of super-
conducting anisotropic systems and had given a quite
good accordance with the experimental data.
Let’s give an example of the determination of ther-
modynamic and magnetic properties in MgB2, when Mg
and B are replaced by other chemical elements.
a. Thermodynamic properties of doped
compound MgB2
The theory of two-band superconductors with variable
or small density of charge carriers [42, 33, 28] can de-
scribe the behavior of thermodynamic quantities such
as Tc,Δ1,Δ2, (CS − CN )/CN at T = Tc as functions of
the chemical potential µ or the charge carrier density in
MgB2.
To this end, it is necessary to do the following:
(1) Start from the system of equations for the BCS-
type order parameters Δn (n = 1, 2) for the two-band
model with the electron-phonon interaction constants
λnm corresponding to the strong electron–phonon cou-
pling renormalized in the two-band model and to the
presence of the Coulomb interaction µ∗nm [86, 87].
(2) Add an equation that determines the chemical po-
tential to the system of equations for the quantities
Δn. This addition is necessary for systems with the low
charge carrier density µ ∼ Δn. The system MgB2 is not
such a system, compound because µ = µ0 ≈ 0.74 eV
for a pure substance; that is, µ � Δn. However, the
additional equation would be introduced due to a spe-
cific band structure of the considered system: the up-
per boundary of the σ-band, which is responsible for
superconductivity in MgB2, is situated in the vicinity of
µ0. This circumstance plays the decisive role in the de-
pendence of thermodynamic quantities on the parameter
µ varying at the substitution of magnesium and boron
atoms by chemical elements of another valence.
(3) Consider the overlapping of the two-dimensional σ-
and three-dimensional π-bands on the Fermi surface.
The dependence of thermodynamic quantities on the
chemical potential µ should be built with regard for
its proximity to µ0 ≈ 0.74 eV for pure MgB2. From
the experimental data for the renormalized constant of
the electron-phonon interaction, we obtain λ11 = 0.302,
λ22 = 0.135, λ12 = 0.04, and λ21 = 0.038.
48 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
THE THEORY OF HIGH-TEMPERATURE SUPERCONDUCTIVITY
(4) Introduce the relative charge carrier density δ =
(µ − µ0)/µ0 that coincides with the corresponding
value which is calculated with regard for the valence
of the elements constituting compounds Mg1−xLixB2,
Mg1−xCuxB2, Mg0.8Li0.2B2−xCx, Mg0.95Cu0.05B2−xCx,
and MgB2−xCx at different values of x. The depen-
dences of the Tc values on δ built in this way allow us
to compare our theoretical results with the experimental
data (see, e.g., [88]). As follows from our calculations,
the doping of MgB2 with electrons (δ > 0) leads to a
decrease of the critical temperature Tc (MgB2−xCx and
Mg0.95Cu0.05B2−xCx). But, at the doping with holes
(δ < 0), the value of Tc (Mg1−xLixB2) does not vary
with the parameter δ. This scheme does not cover com-
pound Mg0.8]Li0.2B2−xCx, in which Tc takes the value
39.4 (that corresponds to MgB2) at δ = −0.02 and de-
creases with the hole density.
Our theory takes the occupation of energy bands
(variation of the chemical potential µ) into account,
as well as the scattering of charge carriers on the im-
purity potential [8, 38], when carbon atoms are in-
troduced into the layered structure instead of boron
atoms responsible for superconductivity. In view of these
two mechanisms, we obtain the dependence which ade-
quately describes the experimental data (MgB2−xCx and
Mg0.95Cu0.05B2−xCx). The doping with holes (δ < 0)
(Mg1−xLixB2 and Mg1−xCuxB2) does not change the
Tc value, because the impurity is not introduced into
the layer responsible for superconductivity, and lithium
and copper introduced instead of magnesium cause only
the variation of the effective valence of boron.
A decrease of Tc in Mg0.8Li0.2B2−xCx is due to the
scattering of electrons on the impurity potential of car-
bon atoms. This jump of the electron heat capac-
ity (CS − CN )/CN at T = Tc is very small (0.8) at
δ = 0 that corresponds to MgB2 without impurity. This
small value is due to the overlapping of energy bands
on the Fermi surface. As this overlapping decreases
(to the right or to the left from this point), the value
of this jump increases and becomes equal to 1.43 (at
δ ≈ 0.06, e.g.) that corresponds to the one-band sys-
tem. These estimations have been realized with regard
for the effect of occupation of energy bands only, with-
out consideration of the impurity scattering. The ob-
tained results correctly reflect the transition from the
two-band system to the one-band one and are in qual-
itative agreement with the experimental data (as for
details, see [86, 87]). The above-given experimental
and theoretical results describe well enough the depen-
dences [88, 89] which were observed during the experi-
ment.
b. The upper critical fields H
(ab)
c2 and H
(c)
c2 in
intermetallic compound MgB2
The experimental investigations of the magnetic prop-
erties of MgB2 show the bright appearance of an
anisotropy of the upper critical field Hc2 [90]. The up-
per critical fieldH(ab)
c2 , which corresponds to the external
magnetic field in the plane (ab), exceeds H(c)
c2 with the
magnetic field along the c-axis by several times.
We pose the problems to build the microscopic the-
ory of the upper critical field Hc2 of a pure anisotropic
two-band superconductor applicable on the whole tem-
perature interval 0 < T < Tc, to describe the pat-
tern of the Hc2 behavior as a function of the temper-
ature in MgB2, to determine the curvature of the upper
critical field H
(ab)
c2 and H
(c)
c2 close to the temperature
of superconducting transition, and to reveal then the
anisotropy of the temperature dependence of the coeffi-
cient γH = H
(ab)
c2 /H
(c)
c2 . We determine also the influence
of the mechanism of occupation of energy bands on Tc
and Hab
c2 , when the system is doped with electrons or
holes. We note that the researches of two-band systems
are based on the microscopic approximation of the the-
ory of superconductivity [91–93].
Herewith, the following peculiarities of the MgB2 band
structure are taken into account: the mutual arrange-
ment of energy bands, presence of the overlapping of
the two-dimensional σ-band and the three-dimensional
π-band, and differences of the topologies of Fermi surface
cavities of the bands under consideration.
The values of the above-given magnetic fields are de-
termined on the base of the Ginzburg–Landau equations
for a two-band system. In this case, we apply the method
of Maki and Tsuzuki extended to the two-band case [92]
with the account for MgB2 compound band structure
peculiarities (as for details, see [94–96]). This method
allows one to obtain the analytic solutions for the critical
fields H(ab)
c2 and H(c)
c2 in the low-temperature range (T �
Tc) and near the critical temperature (Tc − T � Tc).
The account for the anisotropy results in an anomaly
of physical characteristics of compound MgB2. At the
same time, the proposed method allows one to consider
both a pure anisotropic two-band superconductor and
intermetallic compound MgB2 with the Mg and B atoms
replaced by the other elements of the Periodic table. We
now give the results of calculations of the upper crit-
ical fields H(ab)
c2 and H
(c)
c2 which were obtained on the
base of the constructed two-band theory. We use the
following constants of the electron-phonon interaction
which correspond to MgB2: λ11 = 0.302; λ22 = 0.135;
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 49
M.E. PALISTRANT, L.Z.KON
λ12 = 0.04; λ22 = 0.038, and λ = v1
v2
= 0.8 (the ratio of
electron velocities on the different cavities of the Fermi
surface) [86]. We assumed that the chemical potential
in MgB2 without doping amounts to µ0 = 0.74 eV. The
parameter is selected as ε = 0.31. This value gives the
closest approach to the experimental results. The pa-
rameter ε defines by the declination of the σ-band from
two-dimensionality.
We obtain the values of H(ab)
c2 � H
(c)
c2 . This result
corresponds well both to the results of many theoret-
ical works and to the experimental data. The strong
anisotropy of the upper critical field is explained by a
weak dispersion of the electron energy in the axis z di-
rection and by the low value of the average electron ve-
locity on the Fermi surface in this direction. In the case
of Mg and B replacement by the other chemical ele-
ments, which furthers the doping by electrons (increas-
ing the chemical potential), the behavior of the upper
critical field as a function of the temperature is similar
to that in the case of pure MgB2. However, the values
of these quantities decrease in comparison with those
in the case of pure MgB2. The correlation of the su-
perconducting phase transition temperature Tc and the
critical fields takes place with increasing the chemical
potential. An increase in the hole conduction has no
effect on the critical temperature and the upper criti-
cal field. The interrelation of the upper critical tem-
peratures of doped and pure MgB2, as well as the de-
pendence of upper critical fields on the electron density
(the chemical potential µ), is considered in [96]. We
obtain that all the quantities decrease with increase in
the electron density of charge carriers at µ > 0.74 eV
and remain constant at µ < 0.74 eV. Consequently, the
hole doping leaves constant the values of the temper-
ature of the superconducting phase transition and the
upper critical field. The essential dependence of the
anisotropy coefficient γH on the temperature in pure
MgB2 (µ0 = 0.74 eV) and doped MgB2 (µ = 0.76 eV)
was obtained. The above-presented results correspond
well to the experimental data on the magnetic prop-
erties of both pure intermetallic compound MgB2 and
doped by electrons and holes (see [97], e.g.). This
tells about the ability of the two-band model to de-
scribe the properties of real materials and the ability
to calculate the anomalies of physical properties which
were generated by the anisotropy of a system. We
note that the filling of energy bands was concerned as
the main mechanism of action of a substitutional im-
purity. It was assumed that the scattering on the im-
purity potential is weak. The account for the electron
scattering on impurities essentially complicates the re-
sults for the systems, where the impurity scattering is
strong.
4. Some Kinetic Properties of Two-Band
Superconductors
The discovery and the experimental investigation of su-
perconducting properties of MgB2 have attracted a spe-
cial interest to the model with overlapping energy bands.
In some cases, this model is extended to take various
anisotropies of the order parameters into account, as well
as the strong electron-lattice coupling. The two-band
model is used to interpret experimental data on tunnel-
ing, specific heat, electron Raman scattering, thermal
conductivity, penetration depth of a magnetic field, and
other properties of MgB2.
Prof. V.A. Moskalenko and his coworkers from the In-
stitute of Applied Physics of the Academy of Sciences
of Moldova have studied most of these properties on the
basis of the above model. The equilibrium problem is
described by the two-band Hamiltonian [5]. This Hamil-
tonian has been extended to consider the scattering of
electrons by non-magnetic impurities [8]. The results for
pure and doped systems are valid for arbitrary values of
the two-band parameters.
We present here only some qualitatively new kinetic
properties which have been obtained within the model
with overlapping energy bands.
1. The GL system of equations for the two-band model
has been formulated to cover the whole range of param-
eters from the pristine to the dirty limit. On this basis,
the magnetic penetration depth of a superconductor, the
jump of the specific heat per unit cell at the critical tem-
perature, and other properties have been investigated in
[9, 15, 98].
For a high concentration of non-magnetic impurities,
the system of GL equations for the two-band model is
similar to the GL system of the one-band model. How-
ever, the critical temperature Tc and the dimensionless
GL-parameter k in the equations are determined by the
two-band model. In this case, the expression for the
relative jump of the specific heat at Tc coincides with
the corresponding expression for the pristine one-band
model, the density of states being the sum of densities
of two bands, and the dependence of Tc on impurities
being specified by the two band model. The effect of
magnetic impurities is to decrease the relative specific
heat jump and to increase considerably both the param-
eter k and the magnetic field penetration depth in the
superconducting phase.
50 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
THE THEORY OF HIGH-TEMPERATURE SUPERCONDUCTIVITY
2. The influence of impurities (non-magnetic and
paramagnetic) on the thermodynamic properties of two-
band superconductors at zero, close to zero, and at the
critical temperatures has been considered in [11, 16, 99].
It has been found that, due to the interband scattering
of electrons on impurities, the superconducting state in
the two-band model is described by a single-energy gap.
Thus, when one of the densities of states becomes zero,
the other density vanishes too. In particular, the energy
gap in the “dirty” limit for a non-magnetic impurity de-
couples in a product of averages of the order parameters
of individual bands with their densities serving as weight
factors.
3. The non-equilibrium process of charge imbalance in
a two-band superconductor has been investigated by em-
ploying the technique of Keldysh-Green functions. The
kinetic equation, penetration depth of a longitudinal
electric field, and distribution of this field in the super-
conductor are given in [100].
A new mechanism of relaxation of the charge imbal-
ance in non-equilibrium two-band superconductors has
been revealed. This mechanism is due to the inter-band
electron-impurity scattering and leads to a decrease of
the penetration depth of a longitudinal electric field into
the superconductor.
4. A model of a superconductor with two dielec-
tric gaps and two superconducting order parameters
corresponding to two parts of the Fermi surface has
been formulated. The phase diagram obtained for this
model contains an area of coexistence of structural, an-
tiferromagnetic, and superconducting phase transitions
versus the non-magnetic impurity concentration, which
agrees qualitatively with the experimental data on high-
temperature superconductors [101].
5. In [102], the electron Raman scattering in super-
conductors, taking the collective oscillations, Coulomb
screening, and scattering of electrons by non-magnetic
impurities into account, has been studied in the frame-
work of the two-band model.
Two contributions to the scattered light intensity have
been singled out: an additive contribution from each
of the two bands, and a term caused by the interband
transitions of Cooper pairs which exists for an arbitrary
light polarization. Experimentally, this means that the
lowest gap should be active for any light polarization.
6. The propagation of a longitudinal ultrasound in the
one- and two-band models of superconductors at arbi-
trary temperatures has been investigated by considering
the collective oscillations in the presence of nonmagnetic
impurities for an arbitrary mean free path. The effect of
superconductivity and impurities on the relative shift of
the sound velocity turned out to depend strongly on the
choice of a model [103,104].
In particular, we have predicted a more efficient sup-
pression of fluctuations of the superconducting gaps by
impurities within the two-band model, than that within
the one-band model. The two-band model has also al-
lowed us to explain such a spectacular feature of high-Tc
superconductors as an increase of the sound velocity for
all the temperature interval below Tc.
7. For a two-band superconductor, the amplitude of
the multiple electron scattering by nonmagnetic impuri-
ties has no electron-hole symmetry with respect to the
Fermi surface, and this may be the cause of an increase in
the thermoelectric effect in superconductors. As a result,
the temperature dependence of the additional contribu-
tion to the thermoelectric coefficient reaches a maximum
in the region of temperatures T < Tc [105].
8. The collective modes related to phase fluctuations
near Tc have been investigated by assuming the exis-
tence of a two-component neutral superfluid. The equa-
tion for collective modes describes the interference of
two processes: small fluctuations of the relative density
of the condensate of electrons (Leggett-type) and small
fluctuations of the charge imbalance of the electron-hole
branches. This equation is analogous to the well-known
equation in solid state physics describing, e.g., the col-
lective modes of polaritons [106].
The amplitudes of the collective modes of the two-
band model have also been studied. We mention that
these modes in the case of non-identical traditional two-
band superconductors do not occur [107].
5. Conclusion
This work is put forward for the publication with the
purpose to turn scientists’ attention to the information
which is related to the two-band superconductors’ prop-
erties and which was obtained by the Moldavian physi-
cists guided by Prof. V.A. Moskalenko who is the cre-
ator of the multiband superconductivity model. Thus,
we underline the essential contribution of the N.N. Bo-
goliubov’s school to the development of superconductiv-
ity in this direction. We note that our theory contains
the classical results which are related to the essential
distinction of the two-band superconductor properties
from the one-band ones not only in the quantitative
sense, but also in the qualitative sense. These results
had been obtained long before the discovery of HTSC
and superconductivity in MgB2. The theory describes
well enough all kinds of anomalies of physical character-
istics of multiband superconductors (e.g., MgB2). The
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 51
M.E. PALISTRANT, L.Z.KON
large number of noncuprated compounds is obtained
nowadays, such as LaFeAsO1−xFx, Pr[O1−xFx]Fe4As,
CeO1−xFxFeAs, and others, for which, in particular, the
presence of energy bands which overlap on the Fermi
surface is essential. Consequently, the works on the two-
band theory of superconductivity are the base, to some
extent, for the investigation of properties of these new
compounds. We adduced by far not all of the inves-
tigations on the theory of multiband superconductors
in this work. We’d like to express our thanks to all
the Moldavian physicists who manifested the interest in
this problem and who made the contribution to its solu-
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Received 07.07.09
ТЕОРIЯ ВИСОКОТЕМПЕРАТУРНОЇ НАДПРОВIДНОСТI
В БАГАТОЗОННИХ СИСТЕМАХ. MgB2
М.Є. Палiстрант, Л.З. Кон
Р е з ю м е
Сформульовано основнi етапи розвитку теорiї надпровiдних
систем iз зонами, що перекриваються. Наведено основнi по-
силання на класичнi роботи автора цiєї теорiї, професора
В.Л. Москаленко, i його спiвробiтникiв разом iз роботами з
високотемпературної надпровiдностi. Вiдзначено особливостi
двозонної моделi, яка дає якiсно новi результати порiвняно
зi звичайною однозонною. Обговорено застосування двозон-
ної моделi для опису термодинамiчних властивостей сполуки
MgB2. Дано огляд наших дослiджень кiнетики надпровiдни-
кiв iз зонами, що перекриваються. Зокрема, наведено рiвняння
Гiнзбурга–Ландау для двозонних надпровiдникiв з домiшками
та результати впливу домiшок на ширину забороненої зони.
Розглянуто динамiчнi властивостi двозонних надпровiдникiв.
54 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
|
| id | nasplib_isofts_kiev_ua-123456789-13285 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 2071-0194 |
| language | English |
| last_indexed | 2025-12-07T16:36:05Z |
| publishDate | 2010 |
| publisher | Відділення фізики і астрономії НАН України |
| record_format | dspace |
| spelling | Palistrant, M.E. Kon, L.Z. 2010-11-04T09:56:21Z 2010-11-04T09:56:21Z 2010 The Theory of High-Temperature Superconductivity in Many-band Systems. MgB2 / M.E. Palistrant, L.Z. Kon // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 44-54. — Бібліогр.: 107 назв. — англ. 2071-0194 PACS 74.25.Bt, 74.25.Ha https://nasplib.isofts.kiev.ua/handle/123456789/13285 The main stages of development of the theory of superconducting systems with overlapping energy bands are formulated. The main references of the classical papers of the author of this theory, Prof. V.A. Moskalenko, and his coworkers are listed. The list also includes papers related to high-temperature superconductivity. Some peculiarities of the two-band model which gives qualitatively new results in comparison with the usual one-band model, are enumerated. The application of the two-band model to the description of thermodynamical properties of compound MgB2 is also discussed. The references covering our research of the kinetic properties of superconductors with overlapping energy bands are provided. In particular, we present the Ginzburg–Landau (GL) equations for the two-band superconductors doped with impurities and the results on the influence of impurities on the energy gap, as well as those concerning the dynamical properties of twoband superconductors. Сформульовано основнi етапи розвитку теорiї надпровiдних систем iз зонами, що перекриваються. Наведено основнi посилання на класичнi роботи автора цiєї теорiї, професора В.Л. Москаленко, i його спiвробiтникiв разом iз роботами з високотемпературної надпровiдностi. Вiдзначено особливостi двозонної моделi, яка дає якiсно новi результати порiвняно зi звичайною однозонною. Обговорено застосування двозонної моделi для опису термодинамiчних властивостей сполуки MgB2. Дано огляд наших дослiджень кiнетики надпровiдникiв iз зонами, що перекриваються. Зокрема, наведено рiвняння Гiнзбурга–Ландау для двозонних надпровiдникiв з домiшками та результати впливу домiшок на ширину забороненої зони. Розглянуто динамiчнi властивостi двозонних надпровiдникiв. en Відділення фізики і астрономії НАН України Тверде тіло The Theory of High-Temperature Superconductivity in Many-band Systems. MgB2 Теорія високотемпературної надпровідності в багатозонних системах. MgB2 Article published earlier |
| spellingShingle | The Theory of High-Temperature Superconductivity in Many-band Systems. MgB2 Palistrant, M.E. Kon, L.Z. Тверде тіло |
| title | The Theory of High-Temperature Superconductivity in Many-band Systems. MgB2 |
| title_alt | Теорія високотемпературної надпровідності в багатозонних системах. MgB2 |
| title_full | The Theory of High-Temperature Superconductivity in Many-band Systems. MgB2 |
| title_fullStr | The Theory of High-Temperature Superconductivity in Many-band Systems. MgB2 |
| title_full_unstemmed | The Theory of High-Temperature Superconductivity in Many-band Systems. MgB2 |
| title_short | The Theory of High-Temperature Superconductivity in Many-band Systems. MgB2 |
| title_sort | theory of high-temperature superconductivity in many-band systems. mgb2 |
| topic | Тверде тіло |
| topic_facet | Тверде тіло |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/13285 |
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