Theoretical study of multiband Superconductivity and Enhanced Tc in Rhenium Beryllium (ReBe₂₂) superconductor

Multi component superconductivity is realized in multiband superconductors when an inter band pairing interaction is considerably weaker than the intra band interactions [1], where the weak inter band pairing interaction is a necessary condition to understand the two-component superconductivity in multiband systems.

Following the experimental discovery by K Onnes (1911) [2] tremendous works have been done in searching new features and discovery of improved materials for spintronic applications. The BCS theory of superconductivity based on electron-phonon interaction [3] has long been well established later and is now the standard theory for superconductivity. Decades after publication of this theory, the problem of superconductivity in multiband systems was studied theoretically by Suhl et al [4] and independently by Moskalenko [5]. In such a system, two or more bands of itinerant electrons overlap, allowing both inter band and intra band electron-phonon scattering processes. Moreover, it was the discovery of superconductivity in MgB₂ [6] that generated intense interest in the topic because of its relatively high T_C of 39 K, initiated a search in related Boron compounds. The Fermi surface of this compound consists of different bands, each of which has a distinct value of the superconducting gap [7, 8]. Although multi-gap superconductivity had been sofar discussed theoretically [4, 5] the importance of the concept of multi-band, had been proposed recently[9].

Two-band model was first introduced by Kondo [10]. This model is assumed to consist of two basically independent Fermi bands, which differ in their properties. In particular, the coupling strength and thus the energy gaps should be distinctly different for the two bands. Moreover, different Fermi velocities, anisotropies, gap symmetries, impurity scattering rates, densities of states, etc are possible. The two bands are connected by inter band coupling and the system should have only one T_C , one upper critical field (Bc₂), and other common properties [11, 12] because of the interaction between the two bands.

Multiband superconductivity plays an important role in the properties of many emerging systems such as in iron-based superconductors, and gives rise to a rich variety of phenomena [13]. The first observation of superconductivity in the second lightest elemental metal Beryllium (Be) has low transition temperature based on the prediction of Chapnick's empirical theory [14]. This superconducting state caused by spin–orbit coupling in which the energy gap to be known was the result of the very low temperature degeneracy between 2s and 2p conduction bands. Pure Beryllium shows high-frequency lattice vibrations and the expected value of its coupling parameter is 0.21 and the critical transition temperature is 0.026 K [15] as in the BCS theory.

Analogues to Beryllium, the lightest of all elements, hydrogen would become a high-temperature superconductor within the framework of BCS theory [16] as Ashcroft proposed in 1968. The high transition temperature derives in part from hydrogen the uniquely high vibration of phonon frequencies associated with it. Some forty years after the prediction of monatomic metallic hydrogen, Ashcroft [17, 18] proposed that hydrogen-rich metallic compounds are also candidates for high temperature superconductivity. Recently, these ideas of the discovery of several hydrogen-rich compounds at high temperature superconductors have been shown [19–22]. Our group has also studied the phenomena in MgH₃ and MgH₄ superconductors employing DFT technique to show the enhancement of the coupling parameter with pressure. This lead to the possibility of suggesting a new stable superconducting MgH₄ at a T_C of 280K under pressure of 301 GPa [23, 24].

In the same fashion, even though the work that has been done on alloys of Be (Z = 4) is less, the potential for high T_C values in Be compounds has been shown practically [25, 26]. Beryllium tends to form Be-rich compounds like ReBe₂₂, but unlike the case of the super hydrides, these low-Z rich compounds can often be synthesized at ambient pressure. Several Be-rich compounds are found to be superconductors at ambient pressure (Be₁₃U [27], Be₁₃Th [28], Be₁₃Lu [28], and ReBe₂₂), while several others have yet to be reported to be superconducting like Be₁₃La, Be₁₃Y, Be₁₃Re. Among these, ReBe₂₂ adopts a cubic ZrZn₂₂-type structure with space group Fdm³ (No. 227) and Z = 8 formula units per cell. Its lattice parameter is 11.5574 Å, as determined from the XRD pattern [26] consistent with the previously reported value with a highest critical temperatures, by far larger than T_C of elemental Be [29].

On the other hand, high pressure study of the low Z rich superconductors of ReBe₂₂ have shown that the superconducting critical temperature is suppressed by pressure to at least 30GPa. Computational estimates based on electronic density of states and phonon calculations suggest that T_C will continue to be monotonically suppressed at higher pressures, and the calculations and measurements on the study also indicate that lattice stiffening overcomes electronic effects, leading to the observed decrease in and T_C with pressure [30].

Rhenium Beryllium (ReBe₂₂) is a fully-gapped superconductor which is more consistent with a multi gap- rather than a single-gap superconductor. These features are further supported by the density functional calculation that shows multiple dispersive bands crossing the Fermi energy. These evidences indicate that ReBe₂₂ is a multi band superconductor [26]. It is interesting to consider whether higher T_C values can be achieved such that, what enhances its T_C in Be rich compounds using theoretical analysis.

This article focuses on the theoretical study of superconducting state of ReBe₂₂, based on two band model employing double time temperature dependent Green function technique. The study would look in to the possibility for a remarkable enhancement in critical transition temperature with respect to significant changes in the coupling parameters. Finally the results would be summarized and briefly discussed.

The recently observed multi gap nature of Rhenium Beryllium (ReBe₂₂) is a type II multi band superconductor with multi gap features [26] out of which we focus on the two band case. The model Hamiltonian used to describe the system is given by the following BCS type equation:

$$\begin{eqnarray}&&H={H}_{1}+{H}_{2}+{H}_{12}\end{eqnarray} \tag{ 1 }$$

where

$$\begin{eqnarray}&&{H}_{1}=\displaystyle \sum _{k,\sigma }{\varepsilon }_{1}(k){\hat{a}}_{k,\sigma }^{\dagger }{\hat{a}}_{k,\sigma }-\displaystyle \sum _{k,k^{\prime} }{V}_{1}(k,k^{\prime} ){\hat{a}}_{k\uparrow }^{\dagger }{\hat{a}}_{-k\downarrow }^{\dagger }{\hat{a}}_{-k^{\prime} \downarrow }{\hat{a}}_{k^{\prime} \uparrow }\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{H}_{2}=\displaystyle \sum _{k,\sigma }{\varepsilon }_{2}(k){\hat{b}}_{k,\sigma }^{\dagger }{\hat{b}}_{k,\sigma }-\displaystyle \sum _{k,k^{\prime} }{V}_{2}(k,k^{\prime} ){\hat{b}}_{k\uparrow }^{\dagger }{\hat{b}}_{-k\downarrow }^{\dagger }{\hat{b}}_{-k^{\prime} \downarrow }{\hat{b}}_{k^{\prime} \uparrow }\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{H}_{12}=-\displaystyle \sum _{k,k^{\prime} }{V}_{12}(k,k^{\prime} )[{\hat{a}}_{k\uparrow }^{+}{\hat{a}}_{-k\downarrow }^{+}{\hat{b}}_{-k^{\prime} \downarrow }{\hat{b}}_{k^{\prime} \uparrow }+{\hat{b}}_{k\uparrow }^{\dagger }{\hat{b}}_{-k\downarrow }^{\dagger }{\hat{a}}_{-k^{\prime} \downarrow }{\hat{a}}_{k^{\prime} \uparrow }]\end{eqnarray} \tag{ 4 }$$

In the Hamiltonian ε_i in H₁ and H₂ represents energy difference ${\xi }_{{k}_{i}}-{\mu }_{i}$ where ${\xi }_{{k}_{i}}$ and μ_i denote the kinetic energies and the chemical potentials of the two bands, respectively, where i = 1, 2. ${\hat{a}}_{{k}_{1}\sigma }^{+}({\hat{a}}_{{k}_{1}\sigma })$ are the creation (annihilation) operators of an electron having the wave vector (k) and spin σ in the first band and ${\hat{b}}_{{k}_{2}\sigma }^{\dagger }({\hat{b}}_{{k}_{2}\sigma })$ are the creation (annihilation) operators of electron-electron pairing having wave vector (k) and spin σ in the second band. The term H₁₂ in equation (4) contains the effective attractive inter band pairing potentials given by ${V}_{12}(k,k^{\prime} )$ and ${V}_{21}(k,k^{\prime} )$. Assuming pairing interactions, V₁₂ = V₂₁, stemming from inter band pair scattering.

Using the standard definitions for the order parameters equation (1) can be written as,

$$\begin{eqnarray}\begin{array}{rcl}H & = & \displaystyle \sum _{k,\sigma }{\varepsilon }_{1}(k){\hat{a}}_{k,\sigma }^{\dagger }{\hat{a}}_{k,\sigma }-{{\rm{\Delta }}}_{1}\displaystyle \sum _{k,k^{\prime} }({\hat{a}}_{k\uparrow }^{\dagger }{\hat{a}}_{-k\downarrow }^{\dagger }+{\hat{a}}_{-k^{\prime} \downarrow }{\hat{a}}_{k^{\prime} \uparrow })+\displaystyle \sum _{k,\sigma }{\varepsilon }_{2}{\hat{b}}_{k\sigma }^{\dagger }{\hat{b}}_{k\sigma }-{{\rm{\Delta }}}_{2}\displaystyle \sum _{k,k^{\prime} }({\hat{b}}_{k\uparrow }^{\dagger }{\hat{b}}_{-k\downarrow }^{\dagger }+{\hat{b}}_{-k^{\prime} \downarrow }{\hat{b}}_{k^{\prime} \uparrow })\\ & & -{{\rm{\Delta }}}_{12}\displaystyle \sum _{k,k^{\prime} }({\hat{a}}_{k\uparrow }^{\dagger }{\hat{a}}_{-k\downarrow }^{\dagger }+{\hat{b}}_{-k^{\prime} \downarrow }{\hat{b}}_{k^{\prime} \uparrow })-{{\rm{\Delta }}}_{21}\displaystyle \sum _{k,k^{\prime} }({\hat{b}}_{k\uparrow }^{\dagger }{\hat{b}}_{-k\downarrow }^{\dagger }+{\hat{a}}_{-k^{\prime} \downarrow }{\hat{a}}_{k^{\prime} \uparrow })\end{array}\end{eqnarray} \tag{ 5 }$$

where ${{\rm{\Delta }}}_{1}={\sum }_{k,k^{\prime} }{V}_{1}(k,k^{\prime} )\langle {\hat{a}}_{k\uparrow }^{\dagger }{\hat{a}}_{-k\downarrow }^{\dagger }\rangle$ which is equal to its own complex conjugate and ${{\rm{\Delta }}}_{2}={\sum }_{k,k^{\prime} }{V}_{2}(k,k^{\prime} )\langle {\hat{b}}_{k\uparrow }^{\dagger }{\hat{b}}_{-k\downarrow }^{\dagger }\rangle$ in a similar manner, are the superconducting order parameters in the 1^st and 2^nd intra bands, respectively, and Δ₁₂ and Δ₂₁ are that in the interband interactions where Δ₁₂ = Δ₂₁.

To investigate the temperature dependence of superconducting order parameter on both intra- and inter-bands, the retarded double-time dependent Green's function [31] technique is used based on the concept of quantum field theory. Assuming uniform wave propagation in both bands throughout, with k₁ = k₂ taken as k, this last equation is used for H in the equation of motion, equation (6), to determine the order parameter Δ and T_C for the paired states, hence,

$$\begin{eqnarray}&&\omega \langle \langle {\hat{a}}_{k\uparrow }^{\dagger };{\hat{a}}_{-k\downarrow }^{\dagger }\rangle \rangle =\langle [{\hat{a}}_{k\uparrow }^{\dagger },{\hat{a}}_{-k\downarrow }^{\dagger }]\rangle +\langle \langle [{\hat{a}}_{k\uparrow }^{\dagger },H];{\hat{a}}_{-k\downarrow }^{\dagger }\rangle \rangle ;\end{eqnarray} \tag{ 6 }$$

where ≪... ≫ is indicating the Green function and <... > is the thermodynamic average, and ω is that later used as the Fourier Transform frequency of the time dependent Green function which eventually is related to Matsubara frequency.

Substituting equation (5) in to (6), and applying the random phase approximation (RPA), to decouple the higher order terms,we obtain

$$\begin{eqnarray}&&\langle \langle {\hat{a}}_{k\uparrow }^{\dagger };{\hat{a}}_{-k\downarrow }^{\dagger }\rangle \rangle =\displaystyle \frac{-({{\rm{\Delta }}}_{1}+{{\rm{\Delta }}}_{12})}{2\pi [{\omega }^{2}-{\varepsilon }_{1}{\left(k\right)}^{2}-{\left({{\rm{\Delta }}}_{1}+{{\rm{\Delta }}}_{12}\right)}^{2}]}.\end{eqnarray} \tag{ 7 }$$

The two bands of spin singlet superconductor of Rhenium Beryllium (ReBe₂₂) are connected by interband coupling due to the pairing of free electrons forming cooper pairs [32]. In this, a standard model Hamiltonian equation (1) is considered to find Δ, λ and T_C .

3.1. Determination of superconducting order parameter Δ₁, λ_ep1 and T_C in the two bands

3.1.1. In the first band

It has been assumed that there is a Fermi sea below the superconducting order parameter filled with cooper pairs. This order parameter is a representative of the energy required to break the pairs and promote the electrons to the state above the gap [33].

When the interband coupling is ignored the superconducting energy gap and the coupling parameter can be calculated for each band separately. In the first band Δ₁ can be related to the Green function as

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{1}=\displaystyle \frac{{V}_{1}}{\beta }\displaystyle \sum _{n,k}\langle \langle {\hat{a}}_{k\uparrow }^{\dagger };{\hat{a}}_{-k\downarrow }^{\dagger }\rangle \rangle \end{eqnarray} \tag{ 8 }$$

where V₁ is the pairing potential, β⁻¹ = κ_B T, κ_B is the Boltzmann's constant and Δ₁ is the superconducting order parameter. Changing ω → i ω_n which is known as Mitsubara frequency, so that equation (7) becomes,

$$\begin{eqnarray}&&\langle \langle {\hat{a}}_{k\uparrow }^{\dagger };{\hat{a}}_{-k\downarrow }^{\dagger }\rangle \rangle =\displaystyle \frac{-({{\rm{\Delta }}}_{1}+{{\rm{\Delta }}}_{12})}{2\pi [{\omega }_{n}^{2}-{\varepsilon }_{1}{\left(k\right)}^{2}-{\left({{\rm{\Delta }}}_{1}+{{\rm{\Delta }}}_{12}\right)}^{2}]}\end{eqnarray} \tag{ 9 }$$

From Matsubara's frequency, let

$$\begin{eqnarray}&&{\omega }_{n}=\displaystyle \frac{(2n+1)\pi }{\beta }\end{eqnarray} \tag{ 10 }$$

Substituting equation (10) into equation (9),

$$\begin{eqnarray}&&\langle \langle {\hat{a}}_{k\uparrow }^{\dagger };{\hat{a}}_{-k\downarrow }^{\dagger }\rangle \rangle ={\beta }^{2}\displaystyle \frac{{{\rm{\Delta }}}_{1}+{{\rm{\Delta }}}_{12}}{2\pi [{\left(2n+1\right)}^{2}{\pi }^{2}+{E}_{1k}^{2}{\beta }^{2}]}\end{eqnarray} \tag{ 11 }$$

where ${E}_{1k}^{2}=({\varepsilon }_{1}{\left(k\right)}^{2}+{\left({{\rm{\Delta }}}_{1}+{{\rm{\Delta }}}_{12}\right)}^{2})$. Combining equation (8) and equation (11)

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{1}=\displaystyle \sum _{n,k}{V}_{1}\beta \displaystyle \frac{{{\rm{\Delta }}}_{1}+{{\rm{\Delta }}}_{12}}{2\pi [{\left(2n+1\right)}^{2}{\pi }_{n}^{2}+{E}_{1k}^{2}{\beta }^{2}]}\end{eqnarray} \tag{ 12 }$$

Attractive interaction is effective for the region −ℏ ω_D < ε < ℏ ω_D . By introducing the density of state N(0) at the Fermi level (E_f ) and assuming that the density of states does not vary over this integral and changing the summation over k into an integral as:

$$\begin{eqnarray}&&\displaystyle \sum _{k}\to \int {d}^{3}k={\int }_{-{E}_{f}}^{\infty }d\varepsilon N(\varepsilon )\end{eqnarray} \tag{ 13 }$$

and using the relation

$$\begin{eqnarray}&&\displaystyle \frac{1}{2x}\tanh \Space{0ex}{2.3ex}{0ex}(\displaystyle \frac{x}{2}\Space{0ex}{2.3ex}{0ex})=\displaystyle \sum _{k,n}\displaystyle \frac{1}{{\left(2n+1\right)}^{2}{\pi }^{2}+{x}^{2}}\end{eqnarray} \tag{ 14 }$$

where x = β E, leads equation (12) into

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Delta }}}_{1} & = & {{\rm{\Delta }}}_{1}{V}_{1}{N}_{1}(0)\times {\int }_{0}^{{\hslash }{\omega }_{D}}\displaystyle \frac{d{\varepsilon }_{1}}{{\left({\varepsilon }_{1}^{2}+{{\rm{\Delta }}}^{2}\right)}^{1/2}}\tanh \left(\displaystyle \frac{\beta {\left({\varepsilon }_{1}^{2}+{{\rm{\Delta }}}^{2}\right)}^{1/2}}{2}\right)\\ & & +{V}_{12}{N}_{2}(0){{\rm{\Delta }}}_{2}\times {\int }_{0}^{{\hslash }{\omega }_{D}}\displaystyle \frac{d{\varepsilon }_{1}}{{\left({\varepsilon }_{1}^{2}+{{\rm{\Delta }}}^{2}\right)}^{1/2}}\tanh \left(\displaystyle \frac{\beta {\left({\varepsilon }_{1}^{2}+{{\rm{\Delta }}}^{2}\right)}^{1/2}}{2}\right)\end{array}\end{eqnarray} \tag{ 15 }$$

where N₁(0) and N₂(0) are density of states for band one and band two at the fermi level, with a cutoff energies ω₁ and ω₂, respectively.

From equation (15) the superconducting order parameter and the coupling parameter of the first band are obtained, where ${\rm{\Delta }}=\sqrt{{{\rm{\Delta }}}_{1}^{2}+{{\rm{\Delta }}}_{12}^{2}}$. As β → ∞ , it is straight forward to show the gap parameter, Δ(0) in the ground state to be

$$\begin{eqnarray}&&{\rm{\Delta }}(0)\simeq 2{\hslash }{\omega }_{D}\exp \left(-\displaystyle \frac{1}{N{\left(0\right)}_{1}{V}_{1}}\right)\end{eqnarray} \tag{ 16 }$$

where $N{\left(0\right)}_{1}{V}_{1}$ is very small and Δ(0) = Δ₁(0) + Δ₁₂(0). in the first band.

By considering the intra band in equation (15) we obtain the coupling parameter of the first band as:

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\lambda }_{{ep}1}}=\displaystyle \frac{\beta }{2}{\int }_{0}^{{\hslash }{\omega }_{D}}\displaystyle \frac{\tanh (\tfrac{\beta {E}_{1k}}{2})}{\tfrac{\beta {E}_{1k}}{2}}{{dE}}_{1k}\end{eqnarray} \tag{ 17 }$$

where λ_ep1 = V₁ N₁(0) is the electron-phonon coupling parameter in the first band.

Assuming that Δ ≪ E_1k , and after couples of steps the superconducting transition temperature, T_C would be,

$$\begin{eqnarray}&&{T}_{C}=1.14{{\rm{\Theta }}}_{D}{e}^{-\tfrac{1}{{\lambda }_{{ep}1}}}\end{eqnarray} \tag{ 18 }$$

where and ${{\rm{\Theta }}}_{D}=\tfrac{{\hslash }{\omega }_{D}}{{\kappa }_{B}}$ which is a Debye temperature.

3.1.2. In the second band

By the same procedure the superconducting order parameter in the second band can be obtained as:

$$\begin{eqnarray}&&{\rm{\Delta }}(0)^{\prime} \simeq 2{\hslash }{\omega }_{D}\exp (-\displaystyle \frac{1}{{N}_{2}(0){V}_{2}})\end{eqnarray} \tag{ 19 }$$

where ${\rm{\Delta }}(0)^{\prime} =\sqrt{{{\rm{\Delta }}}_{2}^{2}+{{\rm{\Delta }}}_{12}^{2}}$, V₂ is the pairing potential in the second band. Hence,

$$\begin{eqnarray}&&{T}_{C}=1.14{{\rm{\Theta }}}_{D}{e}^{-\tfrac{1}{{\lambda }_{{ep}2}}}\end{eqnarray} \tag{ 20 }$$

where λ_ep2 = V₂ N₂(0) is the electron-phonon coupling parameter in the band.

3.2. The electron-phonon coupling parameter of the inter band, λ₁₂ interaction

The presence of interband pair hopping makes multiband superconductor different from the single band superconductor. We shall take the mediator to be phonons as in BCS case, because pairs are composed of electrons from the two bands. To calculate the superconducting transition temperature T_C the order parameter Δ can be written in matrix form as [34]:

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{i}=\displaystyle \sum _{j}{V}_{{ij}}G\left({{\rm{\Delta }}}_{j}\right){{\rm{\Delta }}}_{j}\end{eqnarray} \tag{ 21 }$$

The $G^{\prime} s$ are defined as

$$\begin{eqnarray}&&G({{\rm{\Delta }}}_{1})={N}_{1}(0){\int }_{0}^{{\hslash }{\omega }_{D}}\displaystyle \frac{d{\varepsilon }_{1}}{\sqrt{{E}_{1}^{2}-{{\rm{\Delta }}}_{1}^{2}}}\tanh \displaystyle \frac{\beta }{2}{\left({E}_{1}^{2}+{{\rm{\Delta }}}_{1}^{2}\right)}^{1/2}\end{eqnarray} \tag{ 22 }$$

Similarly,

$$\begin{eqnarray}&&G({{\rm{\Delta }}}_{2})={N}_{2}(0){\int }_{0}^{{\hslash }{\omega }_{D}}\displaystyle \frac{d{\varepsilon }_{2}}{\sqrt{{E}_{2}^{2}-{{\rm{\Delta }}}_{2}^{2}}}\tanh \displaystyle \frac{\beta }{2}{\left({E}_{2}^{2}+{{\rm{\Delta }}}_{2}^{2}\right)}^{1/2}\end{eqnarray} \tag{ 23 }$$

The G functions relate Δ₁ and Δ₂ as follows:

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Delta }}}_{1} & = & {V}_{1}G({{\rm{\Delta }}}_{1}){{\rm{\Delta }}}_{1}+{V}_{12}G({{\rm{\Delta }}}_{2}){{\rm{\Delta }}}_{2}\\ {{\rm{\Delta }}}_{2} & = & {V}_{2}G({{\rm{\Delta }}}_{2}){{\rm{\Delta }}}_{2}+{V}_{21}G({{\rm{\Delta }}}_{1}){{\rm{\Delta }}}_{12}\end{array}\end{eqnarray} \tag{ 24 }$$

With the assumption that the pairing interaction in the interband is larger than the intra band interaction, V₁ ≡ V₂ ≪ V₁₂, negligibly, so that [35]

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Delta }}}_{1} & = & {V}_{12}G({{\rm{\Delta }}}_{2}){{\rm{\Delta }}}_{2}\\ {{\rm{\Delta }}}_{2} & = & {V}_{21}G({{\rm{\Delta }}}_{1}){{\rm{\Delta }}}_{1}\end{array}\end{eqnarray} \tag{ 25 }$$

Substituting equation (22) and (23) into equation (25) for G(Δ₁) and G(Δ₂) would give

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{1}={V}_{12}{N}_{2}(0){{\rm{\Delta }}}_{2}{\int }_{0}^{{\hslash }{\omega }_{D}}\displaystyle \frac{d{\varepsilon }_{2}}{\sqrt{{E}_{2}^{2}-{{\rm{\Delta }}}_{2}^{2}}}\tanh \displaystyle \frac{\beta }{2}{\left({E}_{2}^{2}+{{\rm{\Delta }}}_{2}^{2}\right)}^{1/2}\end{eqnarray} \tag{ 26 }$$

and

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{2}={V}_{21}{N}_{1}(0){{\rm{\Delta }}}_{1}{\int }_{0}^{{\hslash }{\omega }_{D}}\displaystyle \frac{d{\varepsilon }_{1}}{\sqrt{{E}_{1}^{2}-{{\rm{\Delta }}}_{1}^{2}}}\tanh \displaystyle \frac{\beta }{2}{\left({E}_{1}^{2}+{{\rm{\Delta }}}_{1}^{2}\right)}^{1/2}\end{eqnarray} \tag{ 27 }$$

Considering the same case Δ₁(T_C ) = Δ₂(T_C ) = 0, at T = T_C , equation (26) and (27) becomes, respectively

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{1}={V}_{12}{N}_{2}(0){{\rm{\Delta }}}_{2}{\int }_{0}^{{\hslash }{\omega }_{D}}\displaystyle \frac{d{\varepsilon }_{2}}{{E}_{2}}\tanh \displaystyle \frac{\beta {E}_{2}}{2}\end{eqnarray} \tag{ 28 }$$

and

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{2}={V}_{21}{N}_{1}(0){{\rm{\Delta }}}_{1}{\int }_{0}^{{\hslash }{\omega }_{D}}\displaystyle \frac{d{\varepsilon }_{1}}{{E}_{1}}\tanh \displaystyle \frac{\beta }{2}{E}_{1}\end{eqnarray} \tag{ 29 }$$

Dividing both sides of equations (28) by Δ₂ and (29) by Δ₁ respectively and multiplying gives,

$$\begin{eqnarray}\begin{array}{rcl}1 & = & {V}_{12}{V}_{21}{N}_{1}(0){N}_{2}(0){\int }_{0}^{{\hslash }{\omega }_{D}}\displaystyle \frac{d{\varepsilon }_{1}}{{E}_{1}}\tanh \left(\displaystyle \frac{\beta {E}_{1}}{2}\right)\\ & & \times {\int }_{0}^{{\hslash }{\omega }_{D}}\displaystyle \frac{d{\varepsilon }_{2}}{{E}_{2}}\tanh \left(\displaystyle \frac{\beta {E}_{2}}{2}\right)\end{array}\end{eqnarray} \tag{ 30 }$$

Since V₁₂ = V₂₁, and assuming the two band energies are almost equal E₁ ≃ E₂ denoted by E ≃ ε which is a function of the wave vector, k where $E=\sqrt{{\varepsilon }^{2}+{{\rm{\Delta }}}^{2}}$ so that ε_1k ≃ ε_2k ≃ ε, then equation (30) simplify to

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\lambda }_{12}}={\int }_{0}^{{\hslash }{\omega }_{D}}\displaystyle \frac{d\varepsilon }{E}\tanh \left(\displaystyle \frac{\beta E}{2}\right)\end{eqnarray} \tag{ 31 }$$

where the interaction coupling parameter, ${\lambda }_{12}={V}_{12}\sqrt{{N}_{1}(0){N}_{2}(0)}$. Therefore, interband pairs are energetically favored with our assumption of identical electron energies in the two bands near the Fermi surface[36].

Integrating equation (31) yields

$$\begin{eqnarray}&&{T}_{C}=1.14{{\rm{\Theta }}}_{D}\exp \Space{0ex}{2.3ex}{0ex}(-\displaystyle \frac{1}{{\lambda }_{12}}\Space{0ex}{2.3ex}{0ex})\end{eqnarray} \tag{ 32 }$$

When intra band fluctuations are ignored, i.e. V₁ = V₂ = 0, the superconducting transition temperature is only induced by the interband interaction strength, V₁₂. Figure 1 shows that superconducting transition temperature T_C can be induced by interband coupling parameter the two bands being switched off, hence support the superconducting transition temperature T_C rise in the system.

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For the first band the entire coupling parameter is computed from the superconducting order parameter using equation (15)

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{1}={{\rm{\Delta }}}_{1}{V}_{1}{N}_{1}(0){\int }_{0}^{{\hslash }{\omega }_{D}}\displaystyle \frac{d\varepsilon }{E}\tanh \Space{0ex}{2.3ex}{0ex}(\displaystyle \frac{\beta E}{2}\Space{0ex}{2.3ex}{0ex})+{{\rm{\Delta }}}_{2}{V}_{12}{N}_{2}(0){\int }_{0}^{{\hslash }{\omega }_{D}}\displaystyle \frac{d\varepsilon }{E}\tanh \Space{0ex}{2.3ex}{0ex}(\displaystyle \frac{\beta E^{\prime} }{2}\Space{0ex}{2.3ex}{0ex})\end{eqnarray} \tag{ 33 }$$

Substituting equation (29) for Δ₂ into equation (33), leads to the entire coupling constant;

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\bar{\lambda }}_{1}}={\int }_{0}^{{\hslash }{\omega }_{D}}\displaystyle \frac{d\varepsilon }{\sqrt{{\varepsilon }^{2}+{{\rm{\Delta }}}_{1}^{2}}}\tanh \Space{0ex}{2.3ex}{0ex}(\displaystyle \frac{\beta \sqrt{{\varepsilon }^{2}+{{\rm{\Delta }}}_{1}^{2}}}{2}\Space{0ex}{2.3ex}{0ex})\end{eqnarray} \tag{ 34 }$$

where $\tfrac{1}{{\bar{\lambda }}_{1}}=\tfrac{1}{{\lambda }_{1}+{\lambda }_{12}}$.

One can also show that the temperature dependence of reduced superconducting order parameter in the first band to be,

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{\Delta }}}_{1}(T)}{{{\rm{\Delta }}}_{1}(0)}={\left(1-\displaystyle \frac{T}{1.14{{\rm{\Theta }}}_{D}\exp [-\tfrac{1}{{\lambda }_{1}+{\lambda }_{12}}]}\right)}^{1/2}\end{eqnarray} \tag{ 35 }$$

and illustrate as in figure 2. In this regard Θ_D = 35.02 K and λ_el−p = 0.64 are estimated based on T_C = 9.4 K considered from Shang et al [26] experimental findings for ReBe₂₂. It is demonstrated that, due to the increase in the electron-phonon coupling the superconductor ordering parameter get lifted up suggesting the raising of the superconducting critical temperature.

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In this case the coupling parameter in the second band is neglected. Here we concentrate on a system which consists of two bands, one may be superconducting while the second is not and existence of λ₁₂ is due to the induced superconductivity perhaps by the first band. Due to this reason, the coupling parameter in the first band is influenced further. This could also affect the superconducting order parameter in the entire system where by the flow of cooper pairs from different bands at the fermi surface could be hindered gradually.

For the second band, the total coupling parameter is computed from the superconducting order parameter in it in analogy to equation (15) as,

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Delta }}}_{2} & = & {{\rm{\Delta }}}_{2}{V}_{1}{N}_{2}(0)\times {\int }_{0}^{{\hslash }{\omega }_{D}}\displaystyle \frac{d{\varepsilon }_{2}}{{\left({\varepsilon }_{2}^{2}+{{\rm{\Delta }}}_{2}^{2}\right)}^{1/2}}\tanh \left(\displaystyle \frac{\beta {\left({\varepsilon }_{2}^{2}+{{\rm{\Delta }}}_{2}^{2}\right)}^{1/2}}{2}\right)\\ & & +{V}_{21}{N}_{1}(0){{\rm{\Delta }}}_{1}\times {\int }_{0}^{{\hslash }{\omega }_{D}}\displaystyle \frac{d{\varepsilon }_{2}}{{\left({\varepsilon }_{2}^{2}+{{\rm{\Delta }}}_{2}^{2}\right)}^{1/2}}\tanh \left(\displaystyle \frac{\beta {\left({\varepsilon }_{2}^{2}+{{\rm{\Delta }}}_{2}^{2}\right)}^{1/2}}{2}\right)\end{array}\end{eqnarray} \tag{ 36 }$$

which can be shown to be,

$$\begin{eqnarray}\begin{array}{rcl}1 & = & {V}_{2}{N}_{2}(0){\int }_{0}^{{\hslash }{\omega }_{D}}\displaystyle \frac{d\varepsilon }{\sqrt{{\varepsilon }^{2}+{{\rm{\Delta }}}_{1}^{2}}}\tanh \Space{0ex}{2.3ex}{0ex}(\displaystyle \frac{\beta \sqrt{{\varepsilon }^{2}+{{\rm{\Delta }}}_{1}^{2}}}{2}\Space{0ex}{2.3ex}{0ex})+{V}_{12}\sqrt{{N}_{1}(0){N}_{2}(0)}\\ & & \times {\int }_{0}^{{\hslash }{\omega }_{D}}\displaystyle \frac{d\varepsilon }{\sqrt{{\varepsilon }^{2}+{{\rm{\Delta }}}_{1}^{2}}}\tanh \Space{0ex}{2.3ex}{0ex}(\displaystyle \frac{\beta \sqrt{{\varepsilon }^{2}+{{\rm{\Delta }}}_{1}^{2}}}{2}\Space{0ex}{2.3ex}{0ex})\end{array}\end{eqnarray} \tag{ 37 }$$

hence, the entire coupling parameter in the second band becomes,

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\bar{\lambda }}_{2}}=\displaystyle \frac{1}{{\lambda }_{2}+{\lambda }_{12}}\end{eqnarray} \tag{ 38 }$$

One can plot for the reduced superconducting order parameter Δ₂(T)/Δ₂(0) in the second band and could obtain similar illustration as in figure 2, in which the difference could be size dependence [26].

The entire coupling parameter ${\bar{\lambda }}^{-1}$ is achieved by adding equations (34) and (38), which is the sum of the entire coupling parameters in the band one and band two; hence, the equation for T_C can be shown to be,

$$\begin{eqnarray}&&{T}_{C}=1.14{{\rm{\Theta }}}_{D}\exp \Space{0ex}{2.3ex}{0ex}[-\displaystyle \frac{1}{\bar{\lambda }}\Space{0ex}{2.3ex}{0ex}]\end{eqnarray} \tag{ 39 }$$

Figure 3 is plotted to describe the effect of the inter band coupling parameter, λ₁₂ on T_C , following equation (39) through the entire coupling parameter $\bar{\lambda }$ by the variation of the intraband coupling parameter values in the two bands. It insures that as the intra band coupling parameter strength increases, the critical superconducting transition temperature T_C show sudden rise. This is due to the increase in the density n_s of superconducting electrons of Rhenium Beryllium (ReBe₂₂) at the Fermi level [26] in the two bands. By assuming that the density of states at the two bands are identical (N₁(0) = N₂(0)), raises the coupling parameters of the two bands on the contrary to that in reference [30]. This is because the electron energies in the two bands would come to be equal near the Fermi surface [36] and strengthen the coupling parameters, hence the $\bar{\lambda }$ favors enhancement of T_C of the system.

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Figure 4 is plotted to show the interdependence of superconducting critical temperature on the coupling parameter in the inter band at different values of λ in the first band ignoring the second intra band interaction. This shows that the increase in intra band coupling parameter raises the superconducting critical temperature with the variation of inter band coupling parameter.

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Moreover, the on site couplings raises the averages of the first band coupling (λ₁), the second band coupling (λ₂) and the coupling of interaction term (λ₁₂).

In this article, the superconductivity of Rhenium Beryllium is theoretically studied using two band model Hamiltonian. Green function formalism is employed based on quantum field theory to explain the model for the expression of superconducting order parameter and T_C in the system. We observed that the enhancement of the critical transition temperature T_C is due to the increment of the coupling constants emerged from the two bands perhaps due to the the increase of the density of states near/at the fermi surface in the bands. The effect of the inter band coupling parameter, λ₁₂ on T_C , through the entire coupling parameter $\bar{\lambda }$ by the variation of the intraband coupling parameter values in the two bands is plotted and discussed. It insures that as the intra band coupling parameter increase, T_C show sudden rise. Hence, the entire coupling parameter favors the T_C enhancement for the system. This finding demonstrates that a mixed order parameter system symmetry enhances the critical temperature of ReBe₂₂ superconductor.

All data that support the findings of this study are included within the article (and any supplementary files).