1. Introduction
The surprising discovery of superconductivity in the novel system MgB2 with Tc = 39 K by Nagamatsu et al. [1] has stimulated new excitement in condensed matter physics. This discovery certainly revived the interest in the field of superconductivity especially in non-oxides, and initiated a search for superconductivity in related boron compounds [2]. Its high critical temperature gives hope for obtaining even higher Tc for similar compounds.
The crystal structure of MgB2 is very simple. It is composed of layers of boron and magnesium, alternating along the c-axis. Each boron layer has a hexagonal lattice similar to that of graphite. The magnesium atoms are arranged between the boron layers in the centers of the hexagons. This has allowed to perform consistent calculations of its electronic structure. Band structure calculations of MgB2 show that there are at least two types of nearly separated bands with two superconducting gaps in the excitation spectrum at the Fermi surface. The first one is a heavy hole band, built up of boron σ orbitals. The second one is the broader band with a smaller effective mass, built up mainly of π boron orbitals [3-7].
It is now well established that MgB2 is an anisotropic two-gap superconductor [4]. The gap ratio 
for the larger gap 
is 7.6. For the smaller gap
, this ratio is around 2.78, so that
. Seemingly, both the energy gaps have s-wave symmetries, the larger gap is highly anisotropic, while the smaller one is either isotropic or slightly anisotropic. The induced character of 
manifests itself in its temperature dependence. The larger energy gap 
occurs in σ-orbital band, while 
in the π-orbital band. For a simplified description, single effective σ- and π-bands can be introduced.
The Fermi surface consists of four sheets: two three dimensional sheets form the π bonding and antibonding bands
, and two nearly cylindrical sheets form the two-dimensional σ-band 
[4,8]. There is a large difference in the electron-phonon coupling on different Fermi surface sheets and this leads to multiband description of superconductivity. The average electronphonon coupling strength is found to have small values [9-11]. Ummarino et al. [12] proposed that MgB2 is a weak coupling two band phononic system where the Coulomb pseudopotential and the interchannel paring mechanism are key terms to interpret the superconducting state. Garland [13] has remarked that Coulomb potential in the d-orbitals of the transition metal reduce the isotope exponent, whereas sp-metals generally shows a nearly full isotope effect. Clearly, like sp metal, for MgB2 the Coulomb effect cannot be considered to explain the reduction of isotope exponent.
It is quite natural to describe a two-gap superconductor by means of a two-band model with interband coupling [14,15]. For MgB2, an approach of such kind is also directly proposed by the nature of the electron spectrum mentioned. There is a number of two band type approaches for superconductivity in MgB2 [16]. We may note that two band models have been exploited in various realizations for high-Tc cuprate superconductivity [16,17].
Liu et al. [4] pointed the role of the electron-phonon interaction between effective σ- and π-bands in the two gap system MgB2. In the present study, we use σ-π interband coupling with a strong σ-interband contribution of electron-phonon and Coulobmic nature. Following Liu et al. [4], the interband interaction is considered to be repulsive (an advantage of two band models) corresponding to electron-electron interaction.
Using two band models, we study the basic MgB2 superconductivity characteristics, specific heat and density of states and compare the theoretical results qualitatively with the available experimental data.
2. The Model Hamiltonian
The model Hamiltonian has the form [18]
(1)
where
(2)
(3)
and
(4)
Here p and d are momentum labels in the π- and σ- bands respectively with energies 
and
, μ is the common chemical potential. Each band has its proper pairing interaction 
and
, while the pair interchange between the two bands is assured by 
term.
We have assumed
, and we define the following quantities
Further we define
(5)
Now 
in Equation (1) read as
Final Hamiltonian can be written as
(6)
We study the Hamiltonian (6) with the Green’s function technique and equation of motion method.
2.1. Green’s Functions
In order to study the physical properties, we define the following normal and anomalous Green’s functions [18- 28]:
(7)
Following equation of motion method, we obtain Green’s functions as follows. In obtaining Green’s functions, we have assumed
and 
and 
Then
(8)
1) Green’s functions for π-band
(9)
(10)
2) Green’s functions for σ-band
(11)
(12)
2.2. The Correlation Functions
Using the following relation [23-27],
(13)
and employing the following identity,
we obtain the correlation functions for the Green’s functions given by Equations (9) and (10) as:
(14)
(15)
where
(16)
and 
are Fermi functions.
Similarly correlation functions for Green’s functions (11) and (12) for σ holes are obtained.
One can define the two superconducting order parameters related to the correlation functions corresponding to Green’s functions 
and 
for π- and σ-bands respectively. In a similar manner electronic specific heat can also be defined related to both π- and σ-bands.
3. Physical Properties
3.1. Superconducting Order Parameters
Gap parameter 
is the superconducting order parameter, which can be determined self consistently from the gap equations
(17)
(18)
In a matrix form, the order parameter for the superconducting state is given by [19]
(19)
where 
is the pairing interaction constant and function G’s are defined as
(20)
(21)
Here 
and 
are density of states for π- and σ-bands respectively at the Fermi level.
There are two superconducting gaps corresponding to π- and σ-bands in this interband model. One can write the equations for superconducting gaps corresponding to π- and σ-bands as follows
(22)
(23)
where 
and 
is pairing interaction for π- and σ- bands respectively, while the pair interchange between the two bands is assured by the 
term. The quantity 
has been supposed to be operative and constant in the energy interval for higher band and lower band, keeping in mind the integration ranges, the gap parameter satisfy the system if the interband interactions are missing, i.e.
, the transition is solely induced by the interband interaction [16] and given by
(24)
Using Equations (24), we can write the simultaneous equation as
(25)
(26)
3.2. Electronic Specific Heat (Ces)
The electronic specific heat per atom of a superconductor is determined from the following relation [3,23-28]1) For π-band
(27)
where 
is the energy of π-band and μ is the common chemical potential.
Substituting 
from (10) and changing the summation over p into an integration by using the relation
, we obtain
(28)
where 
and 
are given by Equation (16).
2) For σ-band Similarly one can write the expression for electronic specific heat 
for σ-band, as
(29)
where 
and 
are
Electronics specific heat for π-band and σ-band are given by Equations (28) and (29) respectively.
3.3. Density of States N (ω)
The density of states is an important function. This helps in the interpretation of several experimental data e.g. many processes that could occur in crystal but are forbidden because they do not conserve energy. Some of them nevertheless take place, if it is possible to correct the energy imbalance by phonon-assisted processes, which will be proportional to 
[25]. For
, the density of states per atom 
is defined as [26,27].
(30)
where, 
is the density of state function for π- band. For σ-band we have
(31)
where 
is one particle Green function for π- and σ-bands, defined by Equations (9) and (11) respectively. We have the Green’s function Equation (9),
(32)
where
. Now solving Equation (32) and using partial fraction method, we obtain
(33)
Now substituting the Green function from Equation (33) in Equation (30) and using the delta function property,
we obtain
(34)
Changing the summation into integration and after simplification, one obtains
(35)
Similarly, for σ-band
(36)
4. Numerical Calculations
Values of various parameters appearing in equations obtained in the previous section are given in Table 1. Using these values, we study the various parameters for the system MgB2.
4.1. Superconducting Order Parameter 
For the study of superconducting order parameter for MgB2 system within two band models, one finds the following situations1) The SC order parameter for π- and σ-bands. Using Equation (25), one can write
and 
Changing the variables as
, 
, and taking
, we obtain Solving Equation (37) numerically, one can study the variation of superconductivity order parameters 
and 
with temperature corresponding π- and σ-bands. The behavior of superconducting order parameters corresponding to π- and σ-bands with temperature is shown in Figure 1.
2) SC order parameter in the presence of both π- and σ-bands.
The superconducting order parameter for combined π- and σ-bands can be studied by taking a simple sum of both the parameters. Taking the sum of order parameters
, one can obtain the values by solving numerically. A comparison of 
with BCS type curve is shown in Figure 2.
4.2. Electronic Specific Heat (Ces)
1) For π-band Using Equation (28) and putting a1, a2 and
, after simplification, we obtain
(38)
Changing the variables as
, 
, and using parameters from Table 1 with taking
, we obtain
(39)
2) For σ-band Similarly, we can write expression for specific heat for σ-band using Equation (29)
(37)
Table 1. Values of various parameters for MgB2 system.
Figure 1. Variation of superconducting order parameter corresponding to π- and σ-bands with temperature [29].
Figure 2. Variation of superconducting order parameter Δ = (Δπ + Δσ) corresponding to π- and σ-bands with temperature.
(40)
The variation electronic specific heat (Ces) with temperature (T) for π- and σ-band is shown in Figure 3. There is a good agreement with experiment data.
4.3. Density of States
Density of states function for the π-band is given by Equation (35). Now using the following values of y, x1 for 
and x2 for
, and taking
, one obtains,
(41)
Similarly, using Equation (36), density of states for σ- band is obtained as
(42)
The above two expressions of density of states function for π- and σ-bands are similar, hence we have evaluated the values with different values of x1 and x2 for π- and σ-bands. The behavior of density of states function for both π- and σ-bands is shown in Figure 4.
5. Discussion and Conclusions
NOTES