Effect of interband interactions of phonon and charge fluctuation on superconducting parameters of MgB2

TL;DR

This paper investigates how interband interactions involving phonons and charge fluctuations influence the superconducting properties of MgB2 using a two-band model with various pairing interactions.

Contribution

It introduces a two-band model incorporating multiple interactions to analyze their effects on MgB2's superconductivity, emphasizing the role of interband contributions.

Findings

Estimated transition temperature and isotope effect exponent.

Highlighted the significance of interband interactions.

Showed the impact of charge fluctuations on superconductivity.

Abstract

We have investigated the effect of unconventional pairing mechanism in MgB2 using a two-band model within the framework of Bogoliubov-Valatin formalism. The approach incorporates the intra- band s-wave interaction in the s- and p-bands, as well as interband s-wave interaction between them. The analysis assumes the pairing interaction matrix comprises of attractive electron-phonon, charge fluctuation and repulsive electron-electron (Coulomb) interactions to account for superconductivity in MgB2. The model is used to estimate the transition temperature and isotope effect exponent as well as to elucidate the importance of interband contributions in the superconductivity of the system.

Equations

$$H=\sum _{iks}{ϵ}_{ik}{c}_{iks}^{†}{c}_{iks}+\sum _{ik{k}^{\prime }}{V}_{ik{k}^{\prime }}{c}_{i{k}^{\prime }\uparrow }^{†}{c}_{i-k^{\prime }\downarrow }^{†}{c}_{i-k\downarrow }{c}_{ik\uparrow }$$

$$+\sum _{k{k}^{\prime }}{V}_{\pi \sigma k{k}^{\prime }}\left(c_{\pi k\uparrow }^{†}{c}_{\pi -k\downarrow }^{†}{c}_{\sigma -k^{\prime }\downarrow }{c}_{\sigma k^{\prime }\uparrow }+c_{\sigma k\uparrow }^{†}{c}_{\sigma -k\downarrow }^{†}{c}_{\pi -k^{\prime }\downarrow }{c}_{\pi k^{\prime }\uparrow }\right)$$

$${\Delta }_{\pi k}=-\sum _{k^{\prime }}{V}_{\pi k{k}^{\prime }}\frac{{\Delta }_{\pi k^{\prime }}}{2{ϵ}_{\pi k}}\left(1-2f\left({ϵ}_{\pi k^{\prime }}\right)\right)-\sum _{k^{\prime }}{V}_{\pi \sigma k{k}^{\prime }}\frac{{\Delta }_{\sigma k}}{2{ϵ}_{\sigma k^{\prime }}}\left(1-2f\left({ϵ}_{\sigma k^{\prime }}\right)\right)$$

$${\Delta }_{\sigma k}=-\sum _{k^{\prime }}{V}_{\sigma k{k}^{\prime }}\frac{{\Delta }_{\sigma k^{\prime }}}{2{ϵ}_{\sigma k}}\left(1-2f\left({ϵ}_{\sigma k^{\prime }}\right)\right)-\sum _{k^{\prime }}{V}_{\pi \sigma k{k}^{\prime }}\frac{{\Delta }_{\pi k}}{2{ϵ}_{\pi k^{\prime }}}\left(1-2f\left({ϵ}_{\pi k^{\prime }}\right)\right)$$

$$V_{ikk^{\prime}}=\left\{\begin{array}[]{lll}-V_{iph},&if&\mbox{$\omega\leq\omega_{ph}$}\\ V_{ic},&if&\mbox{$\omega_{ph}\leq\omega\leq\omega_{c}$}\\ -V_{ipl},&if&\mbox{$\omega_{c}\leq\omega\leq\omega_{pl}$}\\ 0&if&\omega>\omega_{pl}\end{array}\right.$$

$${\Delta }_{\pi kj}=-N(0)\int \left(-V_{\pi ph}+V_{\pi c}-V_{\pi pl}\right)\frac{{\Delta }_{\pi kj}}{2{ϵ}_{\pi k^{\prime }}}\left(1-2f\left({ϵ}_{\pi k^{\prime }}\right)\right)d{ϵ}_{\pi k^{\prime }}$$

$$-N(0)\int \left(-V_{\pi \sigma ph}+V_{\pi \sigma c}-V_{\pi \sigma pl}\right)\frac{{\Delta }_{\sigma kj}}{2{ϵ}_{\sigma k^{\prime }}}\left(1-2f\left({ϵ}_{\sigma k^{\prime }}\right)\right)d{ϵ}_{\sigma k^{\prime }}$$

$${\Delta }_{\sigma kj}=-N(0)\int \left(-V_{\sigma ph}+V_{\sigma c}-V_{\sigma pl}\right)\frac{{\Delta }_{\sigma kj}}{2{ϵ}_{\sigma k^{\prime }}}\left(1-2f\left({ϵ}_{\sigma k^{\prime }}\right)\right)d{ϵ}_{\sigma k^{\prime }}$$

$$-N(0)\int \left(-V_{\sigma \pi ph}+V_{\pi c}-V_{\sigma \pi pl}\right)\frac{{\Delta }_{\pi kj}}{2{ϵ}_{\pi k^{\prime }}}\left(1-2f\left({ϵ}_{\pi k^{\prime }}\right)\right)d{ϵ}_{\pi k^{\prime }}$$

$${\Delta }_{\pi kph}=N(0)V_{\pi ph}{\int }_{-{\omega }_{ph}}^{{\omega }_{ph}}{\Delta }_{\pi kph}\frac{(1-2f\left({ϵ}_{\pi k^{\prime }}\right)}{2{ϵ}_{\pi k}}d{ϵ}_{\pi k^{\prime }}$$

$$+N(0)V_{\pi \sigma ph}{\int }_{-{\omega }_{ph}}^{{\omega }_{ph}}{\Delta }_{\sigma kph}\frac{(1-2f({ϵ}_{\sigma k^{\prime }}))}{2{ϵ}_{\sigma k^{\prime }}}.$$

$$(1-{\lambda }_{\pi }{Z}_1){\Delta }_{\pi ph}-{\lambda }_{\pi \sigma }{Z}_1{\Delta }_{\sigma ph}-{\Delta }_{\pi c}=0$$

$${\Delta }_{\pi kc}=-N(0)V_{\pi c}\left({\int }_{-{\omega }_c}^{-{\omega }_{ph}}{\Delta }_{\pi kc}+{\int }_{-{\omega }_{ph}}^{{\omega }_{ph}}{\Delta }_{\pi kph}+{\int }_{{\omega }_{ph}}^{{\omega }_c}{\Delta }_{\pi kc}\right)\frac{(1-2f({ϵ}_{\pi k^{\prime }}))}{2{ϵ}_{\pi k^{\prime }}}d{ϵ}_{\pi k^{\prime }}$$

$$-N(0)V_{\pi \sigma c}\left({\int }_{-{\omega }_c}^{-{\omega }_{ph}}{\Delta }_{\sigma kc}+{\int }_{-{\omega }_{ph}}^{{\omega }_{ph}}{\Delta }_{\sigma kph}+{\int }_{{\omega }_{ph}}^{{\omega }_c}{\Delta }_{\sigma kc}\right)\frac{(1-2f({ϵ}_{\sigma k^{\prime }}))}{2{ϵ}_{\sigma k^{\prime }}}d{ϵ}_{\sigma k^{\prime }}.$$

$$(1+{\mu }_{\pi }{Z}_2){\Delta }_{\pi c}+{\mu }_{\pi }{Z}_1{\Delta }_{\pi ph}+{\mu }_{\pi \sigma }{Z}_1{\Delta }_{\sigma ph}+{\mu }_{\pi \sigma }{Z}_2{\Delta }_{\sigma c}-{\Delta }_{\pi pl}=0$$

$${\Delta }_{\pi kpl}=N(0)V_{\pi pl}\left({\int }_{-{\omega }_{pl}}^{-{\omega }_c}{\Delta }_{\pi kpl}+{\int }_{-{\omega }_c}^{-{\omega }_{ph}}{\Delta }_{\pi kc}+{\int }_{-{\omega }_{ph}}^{{\omega }_{ph}}{\Delta }_{\pi kph}+{\int }_{{\omega }_{ph}}^{{\omega }_c}{\Delta }_{\pi kc}+{\int }_{{\omega }_c}^{{\omega }_{pl}}{\Delta }_{\pi pl}\right)\frac{(1-2f({ϵ}_{\pi k^{\prime }}))}{2{ϵ}_{\pi k^{\prime }}}d{ϵ}_{\sigma k^{\prime }}$$

$$-N(0)V_{\pi \sigma pl}\left({\int }_{-{\omega }_{pl}}^{-{\omega }_c}{\Delta }_{\sigma kpl}+{\int }_{-{\omega }_c}^{-{\omega }_{ph}}{\Delta }_{\sigma kc}+{\int }_{-{\omega }_{ph}}^{{\omega }_{ph}}{\Delta }_{\sigma kph}+{\int }_{{\omega }_{ph}}^{{\omega }_c}{\Delta }_{\sigma kc}+{\int }_{{\omega }_c}^{{\omega }_{pl}}{\Delta }_{\sigma kpl}\right)\frac{(1-2f({ϵ}_{\sigma k^{\prime }}))}{2{ϵ}_{\sigma k^{\prime }}}d{ϵ}_{\sigma k^{\prime }}.$$

$$(1-{\lambda }_{\pi pl}{Z}_3){\Delta }_{\pi pl}-{\lambda }_{\pi pl}{Z}_1{\Delta }_{\pi ph}-{\lambda }_{\pi pl}{Z}_2{\Delta }_{\pi c}-{\lambda }_{\pi \sigma ph}{Z}_1{\Delta }_{\sigma ph}-{\lambda }_{\pi \sigma pl}{Z}_2{\Delta }_{\sigma c}-{\lambda }_{\pi \sigma pl}{Z}_3{\Delta }_{\sigma pl}=0$$

$$\left(\begin{array}[]{cccccc}-1&1-\lambda_{\pi}Z_{1}&0&0&-\lambda_{\pi\sigma}Z_{1}&0\\ 1+\mu_{\pi}Z_{2}&\mu_{\pi}Z_{1}&-1&\mu_{\pi\sigma}Z_{2}&\mu_{\pi\sigma}Z_{1}&0\\ -\lambda_{\pi pl}Z_{2}&-\lambda_{\pi pl}Z_{1}&1-\lambda_{\pi pl}Z_{3}&-\lambda_{\pi\sigma pl}Z_{2}&-\lambda_{\pi\sigma pl}Z_{1}&-\lambda_{\pi\sigma pl}Z_{3}\\ 0&-\lambda_{\pi\sigma}Z_{1}&0&-1&1-\lambda_{\sigma}Z_{1}&0\\ \mu_{\pi\sigma}Z_{2}&\mu_{\pi\sigma}Z_{1}&0&1+\mu_{\sigma}Z_{2}&\mu_{\sigma}Z_{1}&-1\\ -\lambda_{\pi\sigma pl}Z_{2}&-\lambda_{\pi\sigma pl}Z_{1}&-\lambda_{\pi\sigma pl}Z_{3}&-\lambda_{\sigma pl}Z_{2}&-\lambda_{\sigma pl}Z_{1}&1-\lambda_{\sigma pl}Z_{3}\end{array}\right)\left(\begin{array}[]{c}\Delta_{\pi kc}\\ \Delta_{\pi kph}\\ \Delta_{\pi kpl}\\ \Delta_{\sigma kc}\\ \Delta_{\sigma kph}\\ \Delta_{\sigma kpl}\end{array}\right)=0$$

$$\left(\begin{array}[]{ccc}-1&1-(\lambda+\lambda_{\pi\sigma})Z_{1}&0\\ 1+(\mu+\mu_{\pi\sigma})Z_{2}&(\mu+\mu_{\pi\sigma})Z_{1}&-1\\ -(\lambda_{pl}+\lambda_{\pi\sigma pl})Z_{2}&-(\lambda_{pl}+\lambda_{\pi\sigma pl})Z_{1}&1-(\lambda_{pl}+\lambda_{\pi\sigma pl})Z_{3}\end{array}\right)\left(\begin{array}[]{c}\Delta_{0}\\ \Delta_{\alpha}\\ \Delta_{\theta}\end{array}\right)=0$$

$${\Delta }_{\pi kc}={\Delta }_{\sigma kc}\equiv {\Delta }_0;{\Delta }_{\pi kph}={\Delta }_{\sigma kph}\equiv {\Delta }_{\alpha };{\Delta }_{\pi kpl}={\Delta }_{\sigma kpl}\equiv {\Delta }_{\theta };$$

$${\lambda }_{\pi }={\lambda }_{\sigma }\equiv \lambda ;{\lambda }_{\pi pl}={\lambda }_{\sigma pl}\equiv {\lambda }_{pl};{\mu }_{\pi }={\mu }_{\sigma }\equiv \mu .$$

$$T_c=1.14{\omega }_{ph}\exp \{-\frac{1}{{\lambda }_{eff}}\}$$

$${\lambda }_{eff}=(\lambda +{\lambda }_{\pi \sigma })-\frac{1}{Z_2+\frac{1}{(\mu +{\mu }_{\pi \sigma })-{\lambda }_{\pi \sigma pl}^{\ast }}}$$

$${\lambda }_{\pi \sigma pl}^{\ast }=\frac{{\lambda }_{pl}+{\lambda }_{\pi \sigma pl}}{1-({\lambda }_{pl}+{\lambda }_{\pi \sigma pl})Z_3}.$$

$$\beta =-\frac{d\ln T_c}{d\ln M}.$$

$$\beta =\frac{1}{2}\left\{1-{\left\{\frac{1}{{\lambda }_{eff}\left\{Z_2+\frac{1}{(\mu +{\mu }_{\pi \sigma })-{\lambda }_{\pi \sigma pl}^{\ast }}\right\}}\right\}}^2\right\}$$

Full text

Published version: Supercond. Sci. Technol. 23, 045031 (2010).

I Introduction

The discovery of superconductivity in magnesium diboride, MgB2, at a transition temperature, $T_{c}\sim$ 39 K Nagamatsu has attracted a lot of studies both theoretically and experimentally in the field of condensed matter physics. This intermetallic material was associated to conventional superconductors based on pairing by electron-phonon coupling but its remarkable high-$T_{c}$ Vinod2007 cannot be explained using the conventional (Bardeen-Copper-Schrieffer (BCS)) theory BCS . Also, the experimentally Budko2001 ; Hinks2001 observed shift in the isotope exponent reveals the importance of phonon contribution to the $T_{c}$.

Electronic bands structure calculation Belashchenko2001 ; Kortus2001 predicted the existence of 2D covalent in-plane ($\sigma$-band) and 3D metallic type inter-layer ($\pi$-band) conducting bands at the Fermi level, EF for this system. The $\sigma$-band is strongly coupled with phonons within the honeycombed boron layer by electrons while electrons in the $\pi$-band can support only weak coupling.

The multi-band gap nature of superconductivity in MgB2 was first predicted theoretically by Liu et al. Liu2001 and have been observed in various experiments; such as, tunneling spectroscopy Iavarone2002 ; Giubileo2001 , point-contact spectroscopy Szabo2001 , specific heat measurements Bouquet1 and magneto-Raman spectroscopy Blumberg2007 . The two-band nature of superconductivity in MgB2 is already well established Xi2008 and the BCS mechanism have been supported by photoemission spectroscopy Takahashi2001 and scanning tunneling microscopy Karapetrov2001 among others. Also, the specific heat analysis performed by Wälte et al. Walte2006 have revealed the weak interband coupling scenario in MgB2 and they show that the data are best explain within the two-band BCS theory of superconductivity.

Choi et al. Choi2002 used the Eliashberg formalism to show that the $\sigma$-bonding states possess an average energy gap of 6.8 meV while the $\pi$-states have weak pairs with an average energy gap of 1.8 meV. Pickett Pickett2002 explained the double gap as having two transition temperatures, one at 45 K ($\Delta_{\sigma}$) and the other at 15 K ($\Delta_{\pi}$), which compromise to results in a transition temperature (for the bulk material) of 39 K Nagamatsu . Putti et al. Putti2008 , Nicol and Carbotte Nicol2005 provided a good illustration on how the two-bands integrate to results into a single $T_{c}$ due to strong interband processes. That is, the two-gaps merge to one isotropic BCS gap Mazin2003 and possess one critical temperature as a result of strong interband scattering Liu2001 ; Putti2008 ; Nicol2005 ; Ferrando2007 . However, the merging of the two bands does not result in the superconducting properties of a one-band superconductor because of the presence of strong anisotropy coupling as well as both intra- and inter-band scattering that affect the superconducting properties Nicol2005 ; Putti2008 .

In MgB2, the boron isotope exponent, $\beta$(B) is only significant while the Mg isotope exponent, $\beta$(Mg) is small but non-zero. Budḱo et al. Budko2001 measured $\beta$(B) $=0.26$ and the same value was recently obtained by Brotto et al. Brotto2008 . Hinks et al. Hinks2001 measured $\beta$(B) to be 0.30 and $\beta$(Mg) to be 0.02, resulting in a total isotope exponent, $\beta$, equals 0.32 for MgB2. Örd et al. Ord2002 calculated $\beta=0.34$ using electron-phonon and Coulomb interactions in the $\sigma$-bands as well as the interband scattering of intraband pairs within a two-band model. Choi et al. Choi2003 have calculated $\beta$(B) $=0.32$ and $\beta$(Mg) $=0.03$ using isotropic Eliashberg theory. Also, Calandra Calandra2007 group found that inclusion of strong electron-phonon coupling and Coulomb repulsion effect in one-band or two-bands Migdal-Eliashberg approach suppress the isotope exponent of boron, $\beta$(B), to a value of about 0.4 - 0.45. Studies have suggested that the low isotope effect is mainly due to phonon anharmonicity Choi2002PRB and interband Coulomb repulsion Ord2002 ; Abah2009 .

Despite the fact that the isotope effect experiment supports the phonon mediated BCS type superconductor, the observed small value of $\beta$ cannot be explained by the conventional BCS theory BCS . Also, large amount of effort devoted to the study of this phenomenon have not provided a clear understanding of the reduced isotope effect Calandra2007 and the pairing mechanism Vinod2007 . Among the nonconventional models that have been proposed for explaining the superconductivity in MgB2 are electronic mechanism mediated by collective excitations pairing Zhukov2001 ; Sharapov2002 , bipolarons Alexandrov2001 , and the electronic resonance-valence-bond (RVB) model Baskaran2002 .

Ku et al. Ku2002 showed that the presence of collective excitations, consisting of coherent charge fluctuation between Mg and B sheets affects the optical properties of the material. Shortly after discovery, electron energy-loss spectroscopy (EELS) Keast2001 ; Yu2001 suggested that a special electronic contribution may affect the superconductivity of MgB2. First principle calculation predicted that the excitations arise from interband transitions in MgB2 Zhukov2001 . Combined EELS and ab initio calculations of the plasmon structure in MgB2 have shown a peak at 2.4 eV and several peaks for higher energies Keast2005 . Optical experiment Guritanu2006 revealed that the intense peak is due to the transition from the $\sigma$- to the $\pi$-band. The optical study went further to suggest that this collective electronic modes affect the color of MgB2. That is, polarization of light is bluish silver for E$\|$ab and yellow for E$\|$c. Another optical experiment suggested that the multi-band picture is necessary to understand the optical spectra of MgB2 Kakeshita2006 . First principles calculations of the excitations spectra in MgB2 involving all the three symmetry momentum directions have confirmed the existence of the long-lived collective excitations and its presence in the region of optical frequencies strongly affects the optical properties Balassis2008 . Recently, Silkin et al. Silkin2009 demonstrated that the unknown long-lived collective mode corresponds to coherent charge fluctuations between the boron $\sigma$ and $\pi$ band ($\sigma\pi$ mode) as well as having a periodic sine-like dispersion for energies below 0.5 eV. Varshney and Nagar Varshney2007 have employed a model involving collective charge fluctuation, within the Eliashberg formalism, to calculate some superconducting parameters of MgB2 without the inclusion of interband interactions.

The present study is motivated by the predictions of collective coherent charge fluctuation from first principles calculations Zhukov2001 ; Ku2002 ; Balassis2008 ; Silkin2009 , EELS Keast2001 ; Yu2001 and optical Guritanu2006 ; Kakeshita2006 experiments as well as the fact that MgB2 is conclusively agreed to be a two-band BCS type superconductor with exceptional high $T_{c}$ (see reviews, Ref. Vinod2007 ; Xi2008 ). Therefore, we employ the two-band BCS model within the Bogoliubov-Valatin formalism and naively assume that the pairing interaction matrix comprises attractive electron-phonon, repulsive Coulomb and attractive electron-plasmon interactions to study the superconducting parameters of MgB2. This is to elucidate the role of charge fluctuation on the system and the interband contribution to the superconducting properties of MgB2. This kind of pairing mechanism have been used by Tewari and Gumber Tewari within one-band BCS model to study the effect of plasmons on the yttrium and lanthanum based superconductors. The paper is organized as follows; in section 2, we present the model. Then, section 3 will be devoted to results and analysis. Finally, section 4 is the conclusion.

II Model

In accord with the original fomulation by Suhl et al. Suhl1959 , on two-band superconductivity, and more recently, studies on two-band model of MgB2 superconductor Abah2009 ; Kristoffel2003 ; Udoms2005 , the effective Hamiltonian of the system can be written as;

[TABLE]

where $\epsilon_{ik}$ are kinetic energies of the two $\left(i=\pi,\sigma\right)$bands measured relative to the Fermi level, $k(k^{\prime})$ is the Bloch wave vector, $V_{ikk^{\prime}}$ are the intraband potential matrices, s is spin index, $\uparrow$ or $\downarrow$, $c^{\dagger}_{iks}$ ( $c_{iks}$) are the creation (annihilation) operators, for $i^{th}$ band, and $V_{\pi\sigma kk^{\prime}}$ is the interband interaction. Employing the standard Bogoliubov-Valatin transformation Bogoliubov ; Valatin in equation (1), the linearized gap equations (Tc equation) can be written as

[TABLE]

[TABLE]

where $\Delta_{ik}$ are the gap parameters for two bands, $\pi$ and $\sigma$, $f\left(\epsilon_{ik}\right)$ are the Fermi-Dirac occupation number for quasiparticle state to energies $\epsilon_{ik}$ above the Fermi level.

In the present model, we shall assume the pairing interaction matrix $(V_{ikk^{\prime}})$ are made up of contributions from the attractive electron-phonon (Vph), repulsive electron-electron(Coulombic) (Vc) and electron-plasmon interaction (Vpl). Therefore;

[TABLE]

where $\omega_{ph}$ and $\omega_{pl}$ are the cut-off frquencies for the electron-phonon and electron-plasmon (collective excitation), respectively; and $\omega_{c}$ is that for the electron-electron interaction which corresponds to the Fermi energy of the system. Also, $\omega_{pl}>\omega_{c}>\omega_{ph}$ and $i=(\pi,\sigma$ and $\pi\sigma\,(\sigma\pi))$.

Employing equation (4) in equations (2) and (3) and replacing the summation over $k^{\prime}$ by integration over energy, $\epsilon_{ik^{\prime}}$ (using $k_{B}=\hbar=1$), we obtain

[TABLE]

[TABLE]

where j = ph, c, and pl.

We solve the resulting equations following standard procedure Su1990 ; Okoye1998 which involves separating the phonon and non-phonon parts. From equation (5) for the $\pi$-band, we obtained three equations each representing the electron-phonon, Coulombic and electron-plasmon parts as follows: The electron-phonon part;

[TABLE]

Integrating and re-arranging equation (7) gives

[TABLE]

where $Z_{1}=\ln\left(\frac{1.14\omega_{ph}}{T_{c}}\right),\lambda_{\pi}=N(0)V_{\pi ph},\lambda_{\pi\sigma}=N(0)V_{\pi\sigma ph}$ and $\Delta_{\pi c}$ is integration constant.

The electronic part is given by

[TABLE]

This yields,

[TABLE]

where $Z_{2}=\ln\left(\frac{\omega_{c}}{\omega_{ph}}\right),\,\mu_{\pi}=N(0)V_{\pi c}$ and $\mu_{\pi\sigma}=N(0)V_{\pi\sigma}$.

Finally, the electron-plasmon part gives

[TABLE]

Then, we obtain

[TABLE]

where $Z_{3}=\ln\left(\frac{\omega_{pl}}{\omega_{c}}\right),\,\lambda_{\pi pl}=N(0)V_{\pi pl}$ and $\lambda_{\pi\sigma pl}=N(0)V_{\pi\sigma pl}$.

Similarly, for the $\sigma$-band, (equation 6), we obtain another set of three homogeneous equations corresponding to $\Delta_{\sigma ph},\,\Delta_{\sigma c}$ and $\Delta_{\sigma pl}$ respectively. Thus, we can write the resulting six homomogenous equations in matrix form as;

[TABLE]

where, $\lambda_{\sigma}=N(0)V_{\sigma ph},\hskip 14.22636pt\mu_{\sigma}=N(0)V_{\pi c}$ and $\hskip 14.22636pt\lambda_{\sigma}=N(0)V_{\sigma pl}$.

For non-trivial solutions, the determinant of the 6 $\times$ 6 matrix must vanish. To proceed, following the theory of multiband superconductivity Suhl1959 , we consider the case when both $\sigma$- and $\pi$- bands vanish simultaneously at superconducting transition temperature. For MgB2, this vanishing of the gaps is due to interband phonon coupling between the $\sigma$- and $\pi$- bands Putti2008 ; Nicol2005 ; Golubov2002 . Therefore, equation (13) reduces to;

[TABLE]

where

[TABLE]

Solving the determinant of the 3 $\times$ 3 matrix, we obtain an expression for the transition temperature as;

[TABLE]

where the effective coupling parameter,

[TABLE]

and the renormalized electron-plasmon contribution parameter,

[TABLE]

Equation (16) is the expression for the transition temperature. The well known McMillan McMillan1968 expresion in two-band and two-square-well can be recovered from equation (16) if we ignore the contribution of electron-plasmon ($\lambda_{pl}=0=\lambda_{\pi\sigma pl}$). Also, the one-band BCS model result is recovered if we neglected the electron-plasmon contributions as well as all interband contributions.

We proceed to test the suitability of the model in account for the phonon contribution to pairing by exploring the isotope effect. The isotope exponent ($\beta$) can be derived from the expression for $T_{c}$ given in equation (16). The isotope effect exponent, $\beta$ is given by

[TABLE]

where $T_{c}$ is given by equation (16) and M is the ionic mass.

Taking Tc $\propto$ M*-β* and recalling that $\omega_{ph}$ $\propto$ M${}^{-\frac{1}{2}}$ while $\omega_{pl}$ $\propto$ M0, we obtain

[TABLE]

Equation (20) is the expression for the isotope exponent ($\beta$) obtained using a two-band BCS gap equations and assuming that the electron-phonon, repulsive Coulomb and electron-plasmon (charge fluctuation) interacting mechanism are simultaneously present in the system. It can be easily seen that the presence of interband contributions influences the value of $\beta$.

III Results and Discussion

To understand the effect of the assumed pairing mechanism in MgB2, we proceed to estimate the transition temperature and isotope effect exponent using the available data in the literature. At this point, we have to point out that the repulsive Coulomb coupling parameter is associated to the renormalized Coulomb pseudopotential, $\mu^{\ast}=\frac{\mu}{1+\mu Z_{2}}$ which depends on the scaling factor Z${}_{2}=\ln(\omega_{c}/\omega_{ph})$. Although, this can be seen as a crude approximation but we are encouraged by the common believe that the effects of Coulomb screening are drastically reduced by retardation effects due to the different energy scales for electrons and phonons. The effect of Coulombic repulsion is an important concept which is mainly fixed as an adjustable parameter in first principle calculations. A good account of this have been described in Ref. Varshney2007 .

In the present analysis, the electron-phonon and the electron-plasmon cut-off frequency are $\omega_{ph}=812$ K Goncharov2001 and $\omega_{pl}=2.2$ eV Zhukov2001 respectively, while the frequency of Coulomb repulsive parameter, $\omega_{c}=750$ meV Nicol2005 . Assuming an arbitrary total intraband coupling (comprising of electron-phonon and electron-plasmon coupling) $\lambda+\lambda_{pl}=0.45$, we estimated the transition temperature, based on equation (16) as 38.80 K for intra- (inter-)band Coulomb pseudopotential, $\mu^{\ast}(\mu^{\ast}_{\pi\sigma})=0.18\,(0.07)$ and interband parameters of $\lambda_{\pi\sigma pl}=0.09$ and $\lambda_{\pi\sigma}=0.08$.

Figure (1) shows the variation of $T_{c}$ with $\lambda_{\pi\sigma pl}$ for the above set of parameters and three different values of $\mu^{\ast}$, that is 0.17, 0.18, and 0.21. This assumed $\mu^{\ast}$ compares well with the Golubov et al. Golubov2002 values of $\mu^{\ast}_{\pi\pi}=0.17$ and $\mu^{\ast}_{\sigma\sigma}=0.21$. It can be observed that $T_{c}$ increases with $\lambda_{\pi\sigma pl}$ but decreases with $\mu^{\ast}$. Also, $T_{c}\simeq 38.80$ K when $\lambda_{\pi\sigma pl}=0.09$ for $\mu^{\ast}=0.18$, $\mu^{\ast}_{\pi\sigma}=0.07$ and $\lambda_{\pi\sigma}=0.08$. And the effective interband coupling, $\lambda_{\pi\sigma}+\lambda_{\pi\sigma pl}=0.08+0.09=0.17$. Also in figure (1), we can see that $\lambda_{\pi\sigma pl}=0$ corresponds to $T_{c}\sim 20$ K (for $\mu^{\ast}=0.18$), which clearly shows the important of interband interaction in the high transition temperature of the MgB2.

The variation of $T_{c}$ with the interband electron-phonon interaction is shown in figure 2. The lower values of $\lambda_{\pi\sigma}$ yield unphysical value of $T_{c}$ irrespective of $\mu^{\ast}$. The effect of electron-phonon coupling strength on the renormalized repulsive Coulomb parameter, $\mu^{\ast}$ is consistent with the conventional superconductor. This shows that contribution of interband interactions of phonon and plasmon is crucial in high-$T_{c}$ of MgB2.

We proceed to analyze the isotope effect of the superconducting MgB2 based on equation (20). Employing the same parameters used to estimate the transition temperature, $T_{c}\simeq 38.80$ K, we obtain an exponent, $\beta\simeq 0.43$. This is in good aggrement with the theoretical calculated values by Choi et al. Choi2003 and Calandra et al. Calandra2007 but not close to the experimental measured values by Bud’ko et al. Budko2001 and Hinks et al. Hinks2001 .

Figure (3) shows the variation of isotope effect exponent, $\beta$ with the interband electron-plasmon interaction, $\lambda_{\pi\sigma pl}$ for a set of parameter . We can see that $\beta$ increase with $\lambda_{\pi\sigma pl}$ but decrease for increasing Coulomb pseudopotential. The variation of $\beta$ with $\lambda_{\pi\sigma}$ for various $\mu^{\ast}$ is shown in figure 4. Both the interband contributions of phonon and collective excitations increase with the isotope exponent and tend to saturate at BCS predicted value of 0.5 for conventional superconductors.

IV Conclusion

We have formulated a two-band model within the Bogoliubov-Valatin Bogoliubov ; Valatin formalism and by incorporating the effect of the collective excitation in the system, we estimated the transition temperature, $T_{c}\simeq 38.80$ K and the isotope effect exponent, $\beta\simeq 0.43$. Our analysis shows that the inclusion of electron-phonon and electron-plasmon enhances the transition temperature of a superconducting MgB2. Although, the model fails to account for the experimental observed value of $\beta$, it gives evidence that pairing mechanism in MgB2 is not purely phonon and the effect of interband contributions are not negligible. Thus, more work is needed both theoretical and experimentally to understand the effect of this non-phonon mechanism. This is because only isotope effect cannot be used to assign the form of pairing mechanism in a superconducting material and the reduced isotope effect exponent of superconducting MgB2 is still unclear Brotto2008 .

Acknowledgements.

We thank O. Umeh for reading the manuscript.