Received 23 August 2014; accepted 27 November 2015; published 30 November 2015
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1. Introduction
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.
2. Formulation of the Problem
Our model of MgB2 by a Hamiltonian has two bands, labeled
and
, which hybridize through an inter-site hopping term, then the Hamiltonian reads
(1)
where
(2)
(3)
(4)
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]
(5)
(6)
(7)
We can write the similar equations for
band. By using the Gorkov equations in superconducting state:
(8)
(9)
where
is the gap energy in
band and is determined by
(10)
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
(11)
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
(12)
The real and imaginary parts of the optical conductivity are given by
(13)
(14)
where
is a positive infinitesimal. The response kernel is given by the current-current correlation function as [16]
(15)
where V is the volume of the system and the current expression in the case of noninteracting particle is given by
(16)
By using Equation (16), Equation (15) can be written as
(17)
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]
(18)
Then in the isotropic case we obtain the Mattis-Bardeen formula from Equations (13) and (14):
(19)
(20)
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.
3. Numerical Results
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]
(21)
The anisotropy of d-wave gap considered in this paper is
(22)
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:
(23)
(24)
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
(25)
(26)
,
,
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.
4. Conclusion
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.
NOTES
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*Corresponding author.