The discovery of MgB₂ superconductor [1] at relatively high temperature has appealed much attention in theoretical and applied condensed matter physics. This material has been known as the first superconductor which has two energy gaps at the Fermi surface: 1) in the two dimensional band (σ) and 2) three dimensional band (π) [2] [3] . The inter-band scattering between them is negligible. To explore the mechanism of superconductivity in this material, it is crucial to determine the symmetry of the superconducting order parameter which governs the behavior of quasiparticle excitations below .

There have been several studies to detect the MgB₂ gaps. The isotope effect of boron has suggested that MgB₂ is a BCS-type superconductor [4] and the high is realized through strong electron-phonon coupling with light boron mass. Several studies have shown two different superconducting gaps [5] [6] : a gap much smaller than the expected BCS value and another is comparable to the BCS value given by . Their ratio is estimated to be using several experiments. The two-gap model is shown to consistently describe the optical conductivity and thermodynamic properties of MgB₂ [7] -[9] . However, there is no general agreement whether MgB₂ is an s-wave BCS type superconductor or not. In conventional s-wave superconductors, there is no quasiparticle excitation at low energies and the thermodynamic and transport coefficients decay exponentially at low temperatures. In this superconductors, the deviation of penetration depth from its zero temperature value exhibits activated behavior [10] i.e. (we set through the paper), reflecting the isotropic BCS energy gap at the Fermi surface. In contrast, in unconventional superconductors with gap nodes, such as in high- oxides, power law behaviors are expected in thermodynamic and transport coefficients at low temperatures [11] .

Pronin et al. measurements [12] show that the low temperature dependence of penetration depth of MgB₂ film has a behavior. This disagreement with BCS calculations could be caused by an additional absorption. Also, theoretical calculations of A. A. Golubov et al. [13] and A. Brinkman et al. [14] show that the penetration depth is well described by two band model.

Kaindl et al. [15] measured both components of complex conductivity of MgB₂ film as a function of frequency for different temperatures. They compared their results with conventional superconductors and concluded that their results were inconsistent with BCS calculations. This disagreement with BCS calculations could be caused by an additional absorption.

In this paper we introduce the new view of the frequency dependence of optical properties of MgB₂ . Numerical calculations of frequency dependence of optical conductivities are carried out by proposing different kinds of energy gaps. We show that the optical conductivities are well described by a two-band superconductor model with different anisotropies in k-space. First, we conclude that the single-gap model is insufficient to understand consistently the optical behaviors. Then, it will be shown that the two-gap model with different symmetries in k-space is sufficient to understand optical properties. In this model the larger gap approximately follows of ordinary usual BCS-like curve and the smaller gap deviates from the usual BCS-like behavior and is similar to a d-wave energy gap. Both gaps are expected to close at the same transition temperature.

Our model of MgB₂ by a Hamiltonian has two bands, labeled and, which hybridize through an inter-site hopping term, then the Hamiltonian reads

where

Here, c and d are referred to and bands with creation and annihilation operators, c, , d, respectively, and is the quasi-particle energy with respect to Fermi energy. The pairing potentials and act intra-band and is the inter-band interaction dominated by multi-phonon processes. We define the green function for the -band as [16]

We can write the similar equations for band. By using the Gorkov equations in superconducting state:

where is the gap energy in band and is determined by

We assume that the hybridization between and bands is negligible, and then the last term in Equations (1) and (10) can be ignored. In this case two parts of the Hamiltonian (and) are independent. Therefore, the and bands has the similar relations and we omit the indexes c and d.

Optical conductivity describes the linear response of a material, which is exposed to an electromagnetic field. This field induces shielding currents

where, is the phonon energy, is the Fourier transform of the covariant vector potential and is the response kernel which depends only on the properties of the material. It can be expressed in terms of quasiparticle propagators and once this is known, the optical conductivity follows

The real and imaginary parts of the optical conductivity are given by

where is a positive infinitesimal. The response kernel is given by the current-current correlation function as [16]

where V is the volume of the system and the current expression in the case of noninteracting particle is given by

By using Equation (16), Equation (15) can be written as

where and.

Here, we consider thin film satisfying for the film thickness d and the coherence length. In these cases we can regard and as independent variables. Since we can do Abrikosov’s replacement [17]

Then in the isotropic case we obtain the Mattis-Bardeen formula from Equations (13) and (14):

where is the real part of the conductivity for normal state and is the density of

states that generalized to, where the bracket indicates the average over the Fermi

surface.

Now, we present the numerical solutions of complex conductivity of MgB₂ film in the frequency range for different temperatures. We use the temperature dependence of energy gaps as [18]

The anisotropy of d-wave gap considered in this paper is

Here, θ is the angular deviation of from the given node direction in the basal plan. The parameter a determines the anisotropy. We have chosen, , and so that the theoretical curves for two-band model at lowest frequency match the experimental values of (solid squares curve of Figure 3 in Ref. [15] ). For, the average over the Fermi surface in Equation (19) for is given by:

where, and is the elliptic

integral of the first kind.

In Figure 1 and Figure 2 we show our numerical results for the real and imaginary parts of optical conductivity as a function of frequency for and. The solid and dotted curves represent the real and imaginary parts of optical conductivity for s-wave and d-wave gaps separately. These curves do not fit the experimental results of Kaindl et al. [15] , which is shown in the Figure 3 of their paper. The d-wave curve of real part of conductivity is bigger than s-wave curve at same temperature. Thus the main contribution to the optical absorption comes from band. However, within the single-gap model, it is difficult to understand the optical behaviors measured by experimental method of Kaindl et al. [15] .

Here, a two-band model with different anisotropies is investigated. We assume that the hybridization between and bands is negligible so that the optical conductivities are given by

, , and are the weighting factors with and, which determines the contributions from and bands. The open circle curves in Figure 1 and Figure 2 indicate optical conductivities using the present two-band anisotropic model. For, these curves are in good agreement with experimental result of Kaindl et al.

In this curves, the best fit to the experimental data are obtained if we assign the ratio of the weights of the band to that of the -band as and, which approximately agrees with band structure [19] and complex conductivity [20] calculations, respectively. These weights show that the main contribution to the optical conductivities comes from the three dimensional band. The open circle, solid and dotted curves in Figure 3 and Figure 4 are calculated for, and respectively. These curves are in good agreement with Kaindl et al. [15] measurements.

By using Green’s function method and linear response theory we have calculated the frequency dependence of the real and imaginary parts of optical conductivity of MgB₂ film in the framework of two-band theory. We have shown that a single-gap model is insufficient to describe the optical behaviors, but the two-band model with different symmetries can explain the experimental results consistently. Also, we have shown that the electrons in band have greater contribution in the optical and transport behaviors than do electrons in the band. We have considered that the optical conductivities are a weighted sum of the continuation from each band and the interaction between them is negligible.

AdelShojaei,MohammadMoarrefi-Romeileh,AsadollahJoata-Bayrami, (2015) Frequency Dependence of Optical Conductivity in MgB₂ Superconductor. World Journal of Condensed Matter Physics,05,353-360. doi: 10.4236/wjcmp.2015.54036