Effects of Pair-Hopping Coupling on Properties of Multi-Band Iron-Based Superconductors

A variety of superconducting materials exhibit multi-band behavior in a form of multicomponent Fermi surfaces. By using a two-band model with a pair hopping, we explain how the interband coupling affects the physical properties of multi-band superconductors. We determine the temperature dependence of the superconducting gap and the specific heat, which strongly diverge from the BCS-type behavior. The anisotropic gap for the system with the mixed gap symmetry is found. Additionally, the spectral function and density of states are significantly modified by the inter-orbital interactions. The results obtained for different symmetries of the order parameter are in a good agreement with the experimental findings for the iron-based superconductors and other multi-band systems.

1. Introduction

The discovery of superconductivity in multi-band materials opened a period of intensive studies of these systems [1, 2]. Superconductivity in such multi-band systems was first considered within the BCS-type formulation by Moskalenko [3] and by Suhl et al. [4]. As a consequence of multi-band properties one can observe several order parameters of different magnitude [5, 6]. The multi-band nature of superconductivity was experimentally observed in many compounds such as NbSe₂ [7], YNi₂B₂C [8], fullerities A₃C₆₀ [9], MgB₂ [10], and high-temperature iron-based systems [11, 12]. In spite of numerous experimental and theoretical studies, the role of the interplay between order parameters in different bands is not well-understood and requires further studies.

Theoretical description of iron-based materials remains a challenge and the nature of pairing interactions in these compounds is still under debate [13–16]. In a standard formulation, the general two-particle on-site interaction is given by intra- and inter-orbital Hubbard repulsion, Hund's exchange, and pair hopping [17]. These interactions are generated automatically in the multiorbital models with two-body interactions using a Hubbard-type approach restricted to intrasite processes [18–20]. The pairing interaction responsible for the occurrence of superconductivity arises probably from an exchange of spin, orbital, or charge fluctuations. In a case of spin fluctuations the pairing vertex can be calculated using the fluctuation exchange approximation (FLEX) [21].

The difficulties with a correct description of iron-based superconductors are connected with a relatively large number of bands emerging at the Fermi surface [5]. Generally, the Fermi surfaces and characteristic band structure of the iron-based superconductors are the consequences of the layered structure of these compounds. In many cases the Fermi surface consists of hole- and electron-like pockets near the Γ and M points of the first Brillouin zone, respectively. Moreover, the Fermi surface strongly depends on doping [22, 23]. It can lead to the Lifshitz transitions induced by doping [24, 25], pressure [26], or external magnetic field [27]. It also modifies the physical properties of supeconductors [28], such as the pairing symmetry. Therefore, the development and investigation of various microscopic models is essential to explain the normal-state and superconducting properties, which can capture the main features of electronic structure near the Fermi level [29].

The conventional multi-band BCS-type and iron-based superconductors differ also in their symmetry of the order parameter. In conventional systems, the order parameter has the same sign on the Fermi pockets, what is observed e.g., in MgB₂ [6]. However, in nearly magnetic Fe-based layered systems mediated by antiferromagnetic spin fluctuations, the s_± symmetry with a sign reversal of the order parameter between different Fermi surface sheets can be favored [13, 30]. As a consequence, the unconventional properties are observed experimentally [31], e.g., in the measurements of energy gap or specific heat. These observations can provide the information about the effective interactions in each band and the symmetry of the order parameter. Therefore, it is important to investigate how different properties of the multi-band systems are affected by these microscopic quantities.

For the reasons mentioned above, in the present paper we investigate a two-band model of iron-based superconductors. We assume the coupling between bands in a form of the pair hopping interaction. The main objective of these studies is to explain how the inter-band coupling and the symmetry of order parameters influence the basic superconducting properties. For different values of the model parameters we predict and discuss the behavior of energy gap, specific heat, electron density of states and spectral function. The rest of the paper is organized as follows. In section 2, we present the model under consideration and the method of calculation. Section 3 is devoted to the numerical results and their discussion. Finally, a summary is included in section 4.

2. Model and Method

For a description of iron-based materials (in the absence of superconductivity) we choose a minimal two-orbital model [32], taking into account only two degenerated d_xz and d_yz orbitals in a case of the unit cell with one Fe atom. This choice is sufficient to describe the low-energy states near the Fermi level [29]. In practice, due to two Fe atoms in the unit cell, the model describes four bands in the case of the folded first Brillouin zone. The model Hamiltonian is given in a general form:

$\begin{array}{ll}{H}_0={\displaystyle \sum _{k\sigma }}{\displaystyle \sum _{\alpha \beta }}{T}_{k\sigma }^{\alpha \beta }{c}_{\alpha k\sigma }^{†}{c}_{\beta k\sigma },& (1)\end{array}$

where $c_{\alpha k\sigma }^{†}$ (c_αkσ) is creation (annihilation) operator of an electron in orbital α with momentum k and spin σ, whereas $T_{k\sigma }^{\alpha \beta }=T_k^{\alpha \beta }-\mu {\delta }_{\alpha \beta }$ describes the kinetic energy of a particle changing the orbital from β to α and they are explicitly given by

$\begin{array}{l}{T}_k^{11}=-2(t_{1}cos{k}_x+t_{2}cos{k}_y)-4t_{3}cos{k}_{x}cos{k}_y,\\ T_k^{12}=T_k^{21}=-4t_{4}sin{k}_{x}sin{k}_y,\\ T_k^{22}=-2(t_{2}cos{k}_x+t_{1}cos{k}_y)-4t_{3}cos{k}_{x}cos{k}_y.\end{array}$

The hopping integrals (t₁, t₂, t₃, t₄) = (−1.0, 1.3, −0.85, −0.85) are given in energy units of |t₁|. The chemical potential is set as μ = 1.54|t₁|. The band structure is given by diagonalization of the above Hamiltonian. Thus, one gets

$\begin{array}{ll}{H}_0^{\prime }={\displaystyle \sum _{\epsilon k\sigma }}(E_{\epsilon k\sigma }-\mu )d_{\epsilon k\sigma }^{†}{d}_{\epsilon k\sigma },& (2)\end{array}$

where E_εkσ are eigenvalues of the Hamiltonian [Equation (1)] given as

$\begin{array}{l}{E}_{\pm k\sigma }=\frac{T_k^{11}+T_k^{22}}{2}\pm \sqrt{{\left(\frac{T_k^{11}-T_k^{22}}{2}\right)}^2+{(T_k^{12})}^2},\end{array}$

where bands are labelled by index ±. This model reproduces the characteristic Fermi surfaces of iron-based superconductors [5, 22, 23] in a case of two Fe-ions in the unit cell – for the folded first Brillouin zone, shown in Figure 1B, it contains two hole-like and two electron-like pockets around the Γ' and M' points, respectively.

FIGURE 1

Figure 1. The Fermi surface given by the model used for a case of one (A) and two (B) Fe atoms in unit cell. Blue (red) line corresponds to the Fermi surface associated with the band ε labelled by 1 (2). In both panels the gray area shows parts of the unfolded and folded first Brouillon zone (FBZ), respectively. We also indicate the high-symmetry points of the FBZ: Γ = (0, 0) and M=(π, π) (or Γ' and M' for a case of folded FBZ).

We should notice, that the minimal two-band model used by Raghu et al. [32] concerns only d_xz and d_yz orbitals. In some situations, additional d_xy orbital should be also included, what leads in natural way to three-orbital models [33, 34]. Inclusion of additional orbitlals can be reasonable due to some particular lattice symmetries [35, 36]. Moreover, in some crucial cases, the tight binding models based on exact DFT calculations should also be used [37–39]. More detailed description, discussion and comparison of different types of tight binding models can be found in the review by Fernandes and Chubukov [29].

2.1. Superconducting State

A realistic description of the interactions in iron–based superconductors contains all possible two-body on-site interactions between electrons in Fe(3d) orbitals (description in the real space) [19, 40]. In a general case, we can distinguish the intra- and interorbital Coulomb repulsions as well as the Hund's rule exchange and the pair hopping term [14, 29]. However, due to the spin fluctuations an interplay between these interaction terms can lead to an effective paring as a source of superconductivity [13]. For a sake of simplicity, in our calculation, we assume a phenomenological form of the interaction in the momentum space [41, 42], where the superconductivity is introduced by a BCS-like term of the following form:

$\begin{array}{l}{H}_{BCS}={\displaystyle \sum _{\epsilon k}}{U}_{\epsilon }{d}_{\epsilon ,-k\downarrow }{d}_{\epsilon k\uparrow }{d}_{\epsilon k\uparrow }^{†}{d}_{\epsilon ,-k\downarrow }^{†}\\ +{\displaystyle \sum _{\epsilon \ne \vartheta ,k}}J{d}_{\epsilon ,-k\downarrow }{d}_{\epsilon k\uparrow }{d}_{\vartheta ,k\uparrow }^{†}{d}_{\vartheta ,-k\downarrow }^{†},& (3)\end{array}$

where U_ε < 0 denotes an effective pairing potential in the band ε (intra-band pairing), and in a general case it can be different in both bands. The pair hopping J couples both the bands (inter-band interaction). Values of U_ε and J can be associated with the coupling constants in the Ginzburg-Landau theory [43, 44]. In the mean-field approximation one gets:

$\begin{array}{l}{H}_{BCS}^{MF}={\displaystyle \sum _{\epsilon k}}{U}_{\epsilon }\left({\chi }_{\epsilon k}{d}_{\epsilon k\uparrow }^{†}{d}_{\epsilon ,-k\downarrow }^{†}+H.c.-|{\chi }_{\epsilon k}{|}^2\right)\\ +{\displaystyle \sum _{\epsilon \ne \vartheta ,k}}J\left({\chi }_{\epsilon k}{d}_{\vartheta ,k\uparrow }^{†}{d}_{\vartheta ,-k\downarrow }^{†}+H.c.-2\text{Re}\left[{\chi }_{1k}{\chi }_{2k}^{*}\right]\right),& (4)\end{array}$

where χ_εk = χ_εη_ε(k) is the superconducting order parameter (SOP) in the band ε. Here, χ_ε denotes the amplitude of the SOP in a given band, whereas η_ε(k) are form factors describing the gap symmetry, i.e., dependence of the SOP on the momentum [42].

The relation describing the symmetry of the superconducting order parameter in the momentum space is a consequence of the paring interactions in the real space existing in the system [14]. By Fourier transforming of these interactions one can find the form factor η(k) corresponding to a given symmetry. In the standard situations the form factors are given as:

$\eta (k)=\{\begin{array}{ll}1\hfill & \text{ for}s-wave,\hfill \\ 2\left(\cos (k_x)+\cos (k_y)\right)\hfill & \text{ for}{s}_{x^2+y^2}-wave,\hfill \\ 4\cos (k_x)\cos (k_y)\hfill & \text{ for}{s}_{x^2{y}^2}-(s_{\pm })-wave,\hfill \\ 2\left(\cos (k_x)-\cos (k_y)\right)\hfill & \text{ for}{d}_{x^2-y^2}-wave,\hfill \\ 4\sin (k_x)\sin (k_y)\hfill & \text{ for}{d}_{x^2{y}^2}-wave.\hfill \end{array}$

The s${}_{x^2{y}^2}$-wave symmetry for the iron-based superconductors is denoted by s_± [13]. The form factors η(k) for different symmetries and for two dimensional momentum space are schematically shown in Figure 2. For symmetries other than s-wave, one can notice a change of a sign of η (blue and red shadows in the figure correspond to negative and positive values of η, respectively). Moreover, one can find momenta, where η = 0, which are called the nodal lines. The mutual relation between η and the shape of the Fermi surface (given in our case by the model) leads to an occurrence of the nodal lines in the superconducting gap at the Fermi surface [e.g., Figures 2A,C,D].

FIGURE 2

Figure 2. The comparison of different symmetries of the superconducting order parameter (in the momentum space k_x × k_y): (A) s${}_{x^2+y^2}$-wave, (B) s${}_{x^2{y}^2}$- (s_±)-wave, (C) d${}_{x^2-y^2}$-wave, (D) d${}_{x^2{y}^2}$-wave. Blue and red regions denote negative and positive values of the form factor η(k) for a given symmetry, whereas nodal lines are white. The Fermi surfaces of the iron-based superconductors in the model used are also shown - cyan and green lines correspond to the electron-like and hole-like pockets, respectively.

The basic possible symmetries of the SOP are mentioned above, while the mixed symmetries of the gap are discussed in section 3.5 (cf. also section 3.4). We should also notice, that in cases investigated in the present work, values of U_ε and J, as well as the ratio between them, are chosen in such a way to have different ratios between amplitudes of the order parameter χ_ε in different bands ε (cf. Table 1 or Figures 3, 4).

TABLE 1

Table 1. Values of the pairing interaction U_ε for chosen gaps Δ_ε at T = 0 and fitted values of α in Equation (13), describing temperature dependence of the superconducting gap in the absence of the pair hopping coupling (J = 0).

FIGURE 3

Figure 3. Superconducting energy gap as a function of temperature T for different values of the pair hopping J (scale on the left, J = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, respectively). Solid (dashed) lines denote the gaps in 1st (2nd) band. Results for (Δ₁; Δ₂) equal (0.2;0.8) and (0.8;0.2) are presented in (A,B), respectively (cf. Table 1).

FIGURE 4

Figure 4. The same as in Figure 3, but for (Δ₁; Δ₂) equal (A) (0.5;2.0) and (B) (2.0;0.5) (cf. Table 1).

The full Hamiltonian $H=H_0^{\prime }+H_{BCS}^{MF}$ in the Nambu notation is rewritten in the form:

$\begin{array}{ll}H={\displaystyle \sum _{\epsilon k}}{\Phi }_{\epsilon k}^{†}{ℍ}_{\epsilon k}{\Phi }_{\epsilon k}+\text{const.},& (5)\end{array}$

with

$\begin{array}{ll}{ℍ}_{\epsilon k}=\left(\begin{array}{cc}{E}_{\epsilon k\uparrow }-\mu & U_{\epsilon }{\chi }_{\epsilon k}\\ U_{\epsilon }{\chi }_{\epsilon k}^{*}& -E_{\epsilon ,-k\downarrow }+\mu \end{array}\right),& (6)\end{array}$

where ${\Phi }_{\epsilon k}^{†}=(d_{\epsilon k\uparrow }^{†}{d}_{\epsilon ,-k\uparrow })$ are the Nambu spinors. The Hamiltonian (Equation 5) can be diagonalized by the unitary Bogoliubov transformation:

$\begin{array}{ll}\left(\begin{array}{c}{\gamma }_{\epsilon k,+}^{†}\\ {\gamma }_{\epsilon k,-}^{†}\end{array}\right)=\left(\begin{array}{cc}{u}_{\epsilon k}& v_{\epsilon k}\\ -v_{\epsilon k}^{*}& u_{\epsilon k}\end{array}\right)\left(\begin{array}{c}{d}_{\epsilon k\uparrow }^{†}\\ d_{\epsilon k\downarrow }\end{array}\right).& (7)\end{array}$

Then, the full Hamitonian H can be rewritten in a diagonal form: $H=\sum _{\epsilon k\alpha }{{E}}_{\epsilon k\alpha }{\gamma }_{\epsilon k\alpha }^{†}{\gamma }_{\epsilon k\alpha }+\mathrm{\text{const.}}$, where the quasiparticle spectrum in the superconducting state takes a BCS-like form:

$\begin{array}{ll}{{E}}_{\epsilon k\pm }=\pm \sqrt{{(E_{\epsilon k\sigma }-\mu )}^2+|{\chi }_{\epsilon k}^{eff}{|}^2},& (8)\end{array}$

with the transformation's coefficients:

$\begin{array}{ll}{u}_{\epsilon k}^2=1-v_{\epsilon k}^2=\frac{1}{2}\left(1+\frac{E_{\epsilon k\uparrow }-\mu }{{{E}}_{\epsilon k+}}\right).& (9)\end{array}$

One needs to be aware that the non-zero pair hopping coupling J between both bands leads to an effective SOP ${\chi }_{\epsilon k}^{eff}$ in a band ε = 1, 2 (last term in Equation 8) given by:

$\begin{array}{ll}{\chi }_{1(2)k}^{eff}=U_{1(2)}{\chi }_{1(2)k}+J{\chi }_{2(1)k}.& (10)\end{array}$

Theoretical description of the interaction in the momentum space is formally given by the form factor η_ε(k). As we know, one of the most important experimental manifestation of the superconductivity is the energy gap in the density of state (DOS), typically defined as a distance between two coherence peaks. Because only the effective SOPs ${\chi }_{\epsilon k}^{eff}$ at the Fermi surface (FS) have a real impact on the gap in the DOS, the superconducting gap 2Δ_ε depends on a maximal value of ${\chi }_{\epsilon k}^{eff}$ for the momentum k at the FS of the ε band:

$\begin{array}{ll}{\Delta }_{\epsilon }={\forall }_{k\in \text{FS}\epsilon }\text{ max}\left|{\chi }_{\epsilon k}^{eff}\right|.& (11)\end{array}$

We will show numerically that this definition is in a good agreement with the DOS properties (cf. sections 3.3, 3.4).

Finally, the grand canonical potential is given by Ω≡−k_BTln (Tr[exp(−H/k_BT)]) and

$\begin{array}{l}\Omega =-k_{B}T{\displaystyle \sum _{\epsilon k\alpha }}log\left(1+exp\frac{-{{E}}_{\epsilon k\alpha }}{k_{B}T}\right)\\ +{\displaystyle \sum _k}\left(-{\displaystyle \sum _{\epsilon }}{U}_{\epsilon }|{\chi }_{\epsilon k}{|}^2-2J\text{Re}\left[{\chi }_{1k}{\chi }_{2k}^{*}\right]\right),& (12)\end{array}$

whereas the equilibrium values of the variational parameters χ_ε at a given temperature T are defined by the global minimum of Ω.

3. Numerical Results and Discussion

The ground state of the system can be obtained from a minimization of the grand canonical potential Ω (Equation 12) with respect to the SOP amplitude χ_ε in both bands ε, at fixed values of other parameters. The calculations have been performed in the momentum space using a square lattice grid k_x × k_y = 300 × 300 and periodic boundary conditions, with the help of the graphic processor unit (GPU) acceleration described in [45]. Such a dense k-point grid strongly reduces the finite-size effects [46]. In the next sections, we present and discuss the temperature dependence of the gaps (section 3.1) and specific heat (section 3.2). In these sections we chose the s_±-wave gap symmetry in both bands, what corresponds to the standard assumption for the iron-based superconductors. Additionally, for the chosen model parameters we study the gap anisotropy and spectral functions for the cases of the same and different gap symmetry in each band (sections 3.3, 3.4, respectively).

3.1. The Temperature Dependence of the Effective Superconducting Gaps

We start from the description of the temperature dependence of the effective superconducting gap in two cases of different values of gaps in each band (the specific values of model parameters are given in Table 1). We compare results for the fixed ratio between gaps in both bands obtained in two cases: (i) of the relatively small gaps (weak coupling) and (ii) large gaps (strong coupling). The chosen ratio corresponds to the typical relation between gaps in iron-based superconductors (examples of those materials will be described in the subsequent paragraphs). Numerical results for these cases are shown in Figures 3, 4, respectively. Here, we assume the s_±-wave symmetry of the superconducting gap in both bands. This assumption corresponds to a typical situation considered in the iron-based materials.

In the absence of coupling between the bands (J = 0) we can find two independent transition temperatures. The temperature dependencies of the superconducting gap Δ_ε(T) in each band show a standard BCS behavior as expected from the mean-field theory (cf. the curves for J = 0 in Figures 3, 4). In this case, the gap value as a function of temperature can be interpolated as [47]:

$\begin{array}{ll}\frac{\Delta (T)}{\Delta (T=0)}\simeq tanh\left(\alpha \sqrt{1-\frac{T}{T_c}}\right),& (13)\end{array}$

where α is a constant (α = 1.74 in the weak coupling limit, i.e., in the BCS theory) [48]. The fitted values of α for the cases considered in this work are also collected in Table 1. For the first band one observes an increasing deviation from the BCS value with increasing U₁, while in the second band the value of α does change only slightly, even for large U₂. We should have in mind that the BCS formula (13) has been defined for the isotropic s-wave superconductivity, while presented numerical results correspond to the s_±-wave gap symmetry in both bands, what can be the source of the disagreement with the BCS value of α. Δ_ε(T) and T_c are associated not only with the pairing interaction U_ε and symmetry of the SOP [given a priori by the form factor η_ε(k)] but also with the band filling n_ε. It should be noted that the bands of considered model have different widths (half-widths: D₁≃1.5|t₁|, D₂≃6|t₁|, D₂/D₁≃4 [32, 49]). Different filling in both bands and band widths can also influence the value of α.

For a non-zero inter-band coupling (J ≠ 0) only a single transition from the superconducting to normal phase occurs in the system. At the transition temperature T_c both gaps Δ₁ and Δ₂ as well as the SOP amplitudes χ₁ and χ₂ go to zero continuously. T_c is rather close to the value of the transition temperature for the band with the larger gap. The lower transition temperature (connected with the band with the smaller gap) disappears. In the case of J ≠ 0, a deflection of Δ_ε(T) from its typical BCS-like temperature dependence (13) is clearly visible (cf. also [4, 50]). The deflection is larger in the band with the smaller gap, while in the band with the larger gap it is almost not noticeable (apart from the fact that it is increased by J ≠ 0)—compare Figures 3, 4. Notice also that Δ_ε(T) (in the band with the small gap) changes its curvature at the intermediate temperatures (if J is relatively small—cf. Figures 3A,B), whereas in the other band Δ_ε(T) shows its typical BCS behavior. Additionally, we notice a relatively small increase of the critical temperature with increasing J. This change is more pronounced when J is strongly enlarged.

Similar deflection of the gap from the BCS-like shape has been observed experimentally for, e.g., 111 [51], 122 [52], and 1111 [53] families of the iron-based superconductors (in a more realistic three band model, there is the pair of leading bands with similar but unresolved gaps coupled by spin fluctuations and the weakly coupled third band [54, 55]) as well as in other multi-band superconductors such as PuCoGa₅ [56], (Li_1−xFe_x)OHFeSe [57], LnOFeAs [58], Sn(Pb)Mo₆S₈ [59], or classical two-band MgB₂ [60–63]. In a case of MgB₂ one has detected also a similar character of the interplay between intra- and inter-band quantities, which has been observed in a temperature dependence of the critical Josephson currents from one band to the other of the MgB₂–insulator–MgB₂ tunnel junctions [61, 64], or in a more general case of the two–band Josephson junction [65].

3.2. Specific Heat

Expanding the approach described in [66], we find numerically the specific heat for the multi-band system with the equilibrium values of the variational parameters χ_ε [67]. From the grand canonical potential Ω(χ_ε) (Equation 12), we determine the entropy as S = −dΩ/dT and the specific heat as C = −T∂²Ω/∂T² (at fixed temperature T). Thus, the entropy can be formally rewritten as:

$\begin{array}{ll}S=-\left[{\left(\frac{\partial \Omega }{\partial T}\right)}_e+{\displaystyle \sum _{\epsilon }}{\left(\frac{\partial \Omega }{\partial {\chi }_{\epsilon }}\right)}_e\frac{\partial {\chi }_{\epsilon }}{\partial T}\right],& (14)\end{array}$

where the subscript e labels the equilibrium values of the SOP amplitudes χ_ε. Because the system is in the state corresponding to the global minimum of Ω, we have ∂Ω/∂χ_ε|_e = 0 for any band ε. As a consequence, we obtain:

$\begin{array}{ll}S={\displaystyle \sum _{\epsilon k\alpha }}{\left[\frac{{{E}}_{\epsilon k\alpha }}{T}f({{E}}_{\epsilon k\alpha })+k_{B}ln\left(1+exp\left(\frac{-{{E}}_{\epsilon k\alpha }}{k_{B}T}\right)\right)\right]}_e,& (15)\end{array}$

where f(ω) = 1/(1+exp(ω/k_BT)) is the Fermi-Dirac distribution. In a similar way one can find the specific heat as $C=-T{\partial }^2\Omega /\partial T^2{|}_e$. However, one needs to remember that the SOP depends on temperature in a non-trivial way, as it has been described in the previous section.

The numerical results of the specific heat C are shown in Figure 5 (similarly as previously we assume the s_±-wave symmetry of the superconducting gap in both bands). In each of the presented cases, in the absence of pair hopping coupling J = 0 (dashed thin black lines) one gets two finite jumps of C, what is a consequence of the existence of two phase transitions from the superconducting to normal phase in each band at two different T_c, separately.

FIGURE 5

Figure 5. Temperature dependence of the superconducting gap Δ_ε (blue lines) and the specific heat C (red lines). Solid (dashed) blue lines show superconducting gap Δ_ε in the first (second) band. For a comparison also C for J = 0 is shown (dashed black line). Results for different values of pair hopping coupling |J| = 0.01, 0.1 (as labeled) in two cases of (Δ₁; Δ₂): (0.2;0.8) and (0.8;0.2) (A–D, respectively; cf. Table 1).

Non-zero J leads to a deflection with respect to the non-coupled case, which is relatively well visible near the first phase transition, i.e., that at lower temperature in the non-coupled case. Moreover, this deflection is greater for J larger than the average pairing in the system, what is well visible in Figures 5B,D. For $J\ll \sqrt{U_1{U}_2}$ (Figures 5A,C), the deflection is well visible only near the first phase transition in the non-coupled case (at lower temperature), whereas at temperatures near the phase transition, where the superconductivity vanishes in the system, results for J = 0 and J ≠ 0 are not distinguishable. Obviously the heat capacity in the normal phase is not dependent on J. Similar behavior has been described theoretically using α model e.g., in MgB₂ [68–70], where interactions are described by the electron-phonon coupling.

The behavior of heat capacity observed here was found experimentally in many multi-band materials like iron-based superconductors from 11 [71, 72], 111 [73–75], or 122 family [52, 76–87], MgB₂—classical two band superconductor [88] and many others (e.g., 2H-Pd_xTaSe₂ [89], Lu₂Fe₃Si₅ [90] or NbS₂ [91]). One needs to stress that the unconventional superconductors (with the gap symmetry other than s-wave) is characterized by a different dependence of C vs. T/T_c.

3.3. The Spectral Functions for Iron-Based Superconductors (Realistic Case of Both Bands of s^±-Wave Symmetry)

At the beginning, let us start from a definition of the spectral function, as a tool to study superconducting systems. This theoretical quantity is important from experimental point of view due to its correspondence to the angle-resolved photoemission spectroscopy (ARPES) results [92]. Additionally, it is worthwhile to mention that the shape of the gap at the Fermi surface has an important influence on the spectral function and the density of states in the superconducting state.

Firstly, we define the Green function in the form Ĝ(ω) = 1/(ω−H), and $G_{\epsilon k\sigma }(\omega )=\langle c_{\epsilon k\sigma }\left|Ĝ(\omega )\right|c_{\epsilon k\sigma }^{†}\rangle$. Formally, it can be rewritten in the form:

$\begin{array}{ll}{G}_{\epsilon k\sigma }(\omega )={(\omega -{ℍ}_{\epsilon k})}^{-1},& (16)\end{array}$

where matrix ℍ_εk is given by Equation (6). Then we define the spectral function as ${{A}}_k(\omega )=-1/\pi \sum _{\epsilon \sigma }\mathrm{\text{Im}}{G}_{\epsilon k\sigma }(\omega +i{0}^{+})$ and the DOS as $\rho (\omega )=\sum _k{{A}}_k(\omega )$. Using an expression (Equation 7) for electron annihilation and creation operators in the language of the Bogoliubov quasiparticle operators we can find [92, 93]:

$\begin{array}{ll}{{A}}_k(\omega )={\displaystyle \sum _{\epsilon \tau }}\left[|u_{\epsilon k}{|}^2\delta (\omega -{{E}}_{\epsilon k\tau })+|v_{\epsilon k}{|}^2\delta (\omega +{{E}}_{\epsilon k\tau })\right],& (17)\end{array}$

where δ(ω) is the Dirac delta function. Numerically δ(ω) is approximated by ζ/[π(ω² + ζ²)], with the damping parameter ζ taken as 0.01. Coherence factors of the Bogoliubov quasiparticles $|u_k{|}^2$ and $|v_k{|}^2$ are nontrivial functions of the SOP (Equation 9).

In the absence of the interactions (i.e., U₁ = U₂ = 0 and J = 0), the spectral function reproduces the non-interacting band structure of the system. It consists of two branches forming the FSs around X and Y points (the first band) and around Γ and M points (the second band). In our case the non-interacting spectral functions (Figure 6B) are in a good agreement with the band structure of the mentioned model [32, 49]. As a consequence of the absence of superconductivity, we do not observe any gap at the Fermi level (Figure 6A). The total DOS is a sum of the DOSs in the bands and this result well agrees with the previous studies [49, 94].

FIGURE 6

Figure 6. The density of states ρ(ω) (A) and spectral function ${{A}}_k(\omega )$ (B) along high symmetry directions (show in inset) in the absence of superconductivity.

The temperature dependence of the gap and specific heat presented in previous sections (sections 3.1, 3.2, respectively), do not changes qualitatively for other gap symmetries. Here, we will discuss in detail the case when gaps in both bands have the s_±-wave symmetry. Similarly to the previous analyses, we have chosen the s_±-wave gap symmetry, which is a characteristic feature of many iron-based superconductors.

The results for the spectral functions are shown in Figure 7. As a consequence of nonzero pairing interaction, which leads to particle-hole mixing [92, 95], we observe typical BCS behavior of the band branches around the Fermi level. In the DOS we can distinguish coherent peaks corresponding to the superconducting gaps Δ₁ and Δ₂. This feature is indeed independent of the chosen model parameters (cf. upper and lower panels). As it was written previously in section 3.1, increasing J leads to larger values of the gaps. This effect of J ≠ 0 is also present for non-isotropic nodeless gaps (what is also well visible in Figure 8).

FIGURE 7

Figure 7. Density of states ρ(ω) [left panels: (A,C)] and spectral function ${{A}}_k(\omega )$ along the high symmetry directions [right panels: (B,D)] in the presence of superconductivity with the s^±-wave symmetry in both bands. Results for (Δ₁; Δ₂) equal to (0.2;0.5) [the top panels: (A,B)] and (0.5;0.2) [the bottom panels: (C,D)] at T = 0 and in the absence of the pair hopping coupling (J = 0). The dashed red lines corresponds to the results in the absence of superconductivity presented in Figure 6.

FIGURE 8

Figure 8. The momentum dependence of the superconducting gap (at the Fermi level) in the first (around X and Y points—red lines) and second (around Γ and M points—green lines) band. Results for s_±-wave symmetry in both bands (and for (Δ₁; Δ₂) equal to (0.2;0.2) at T = 0) for different values of the pair hopping coupling: J/|t₁| = 0.0 (A), J/|t₁| = 0.1 (B), and J/|t₁| = 0.5 (C).

The numerical results of gap anisotropy (i.e., the momentum dependence of the energy gaps at the Fermi level) are shown in Figure 8. In the absence of the pair hopping (J = 0), in both bands one can observe the anisotropic gaps (Figure 8A). Increasing J leads only to the modification of the gap values in both bands (Figures 8B,C) without changing the “symmetry” in both bands.

3.4. Spectral Functions in the Case of Bands With Different Gap Symmetry (s- and d${}_{x^2{y}^2}$-Wave Symmetries)

Theoretically, we can assume that a few gaps with different symmetries can exist in a superconducting material. As a consequence, for J ≠ 0 one can find interesting features of the DOS and the untypical anisotropies of the gaps at the Fermi surface. Now, we will discuss the case when the gap in the first band is of s-wave symmetry and the gap in the second band is of d${}_{x^2{y}^2}$-wave symmetry (J ≠ 0 mixes the gap symmetries). Here we do not consider the s_±-wave symmetry, for better description of effects of one band with strongly anisotropic gap on the second band with constant gap. The gap in the band with d${}_{x^2{y}^2}$-wave symmetry exhibits nodal points and thus the mutual relations between nodal and nodeless gaps are investigated.

The results of spectral function are shown in Figure 9. First, we will discuss the case in the absence of pair hopping coupling (top panels). In the first band we can find well visible gap structure, while in a second band as a consequence of the nodal lines, in some regions along the high symmetry directions, we can find places without the gap (shown by red arrows in Figure 9B). It has a direct impact on the DOS (Figure 9A)—we can find a conventional s-wave gap structure [96] and second non-empty gap with a characteristic ${V}$-shape (similar like in the case of d-wave high temperature superconductors [97–99]).

FIGURE 9

Figure 9. Density of states ρ(ω) [left panels: (A,C)] and spectral function ${{A}}_k(\omega )$ along the high symmetry directions [right panels: (B,D)] in the presence of superconductivity. In the first (second) band the superconducting gap is of the s-wave (d${}_{x^2{y}^2}$-wave) symmetry. Results for (Δ₁; Δ₂) equal to (0.8;0.2) at T = 0 in the absence [(A,B), J = 0] and the presence [(C,D), J/t₁ = 0.5] of the pair hopping coupling J. The dashed red lines corresponds to the results in the absence of superconductivity presented in Figure 6. Red arrows show places, where the coupling J between bands opens additional gaps [cf. (B,D)].

As we have shown in section 3.3, increasing J leads to the change of the gap shape in the momentum space. For the chosen symmetry (conventional s-wave symmetry and d-type symmetry with nodal lines), the gap opening is well visible along the nodal lines (shown by red arrows in Figure 9D). As a consequence, we can observe also modification of the gap structures in the DOS—initially non-empty small gap is modified by the part of the isotropic symmetry and it leads to opening of the conventional gap (marked by the gray are in Figure 9C).

The momentum dependence of the superconducting gap for the considered case are shown in Figure 10. For J = 0, in the first band the gap is isotropic (red lines), while in the second band the gap is anisotropic (green lines) with the nodal lines (Figure 10A). Similarly as in the previous case, increasing J leads to the modification of the gap values. However, due to the different symmetries, the gaps change also their shapes (Figures 10B,C). Due to J ≠ 0 initially isotropic gap in the first band is modified by the anisotropic gap in the second band and vice versa. This is particularly well visible for the second band (green lines), where the influence of J results in vanishing of the gap in some parts of the FS (shown by arrow in Figure 10C).

FIGURE 10

Figure 10. The same as in Figure 8 for two different symmetries of the gap: the s-wave and d${}_{x^2{y}^2}$-wave symmetries in the first and second band, respectively. Results for (Δ₁; Δ₂) equal to (0.2;0.2) at T = 0 for different values of the pair hopping coupling: J/|t₁| = 0.0 (A), J/|t₁| = 0.1 (B), and J/|t₁| = 0.5 (C).

Mutual interplay between two gaps with different symmetries can be important for a realization of a gap anisotropy in multi-band materials like iron-based superconductors that have been observed in these compounds [100–104]. In these materials the results similar to those shown in Figure 10 have been observed experimentally. Moreover, the results presented here reveal the situation when one band has a nodal line in the gap structure. Results, similar to those presented for anisotropic gaps, have been also found in other theoretical studies using the fluctuation exchange approximation (FLEX) [105–110] or ab initio calculations [111]. However, it is possible to change a nodal gap into nodless one by disorder [112].

3.5. Additional Remarks and Comments

One should be aware that the structure of the DOS between the coherence peaks can be different in cases of nodal and nodeless symmetry [113, 114]. In the case of the s-wave symmetry, we observe a full-gaped structure. Any additional anisotropy of the gap symmetry (e.g., in a case of the s_±-wave) leads to emergence of additional structure around the coherence peaks. Finally, for the symmetry with nodal lines we observe the (mentioned) ${V}$-shape DOS structure. From a shape of the in-gap states in the DOS that can be experimentally observed by, e.g., the scanning tunneling microscopy (STM) [115], one can deduce the information about the gap symmetry. Similarly like in the cuprates [116], the gap in the presence of disorder strongly depends on a impurity configuration. This effect can be also observed in the iron-based materials and can lead to the modification of the double-gap structure of the DOS [113, 117].

The experimental results indicate the existence of the two anisotropic s-wave gaps in 11 [118–120] or 122 family compounds [113, 120–122], (Li_1−xFe_x)OHFeSe [57], SmFeAsO_0.9F_0.1 [123]. However, the interplay of coherence peaks in the DOS shows an existence of some kind of interaction between the bands [124], what can be visible as the disappearance of the double-gap structure of the DOS. Similar behavior can be also observed in MgB₂ [125, 126], where the interplay between two (conventional) gaps in the DOS occurs.

We should also have in mind the possibility of mixing gap symmetries other than that investigated in section 3.4, e.g., like s+id symmetry. This scenario was firstly proposed in a context of the cuprates by Ruckenstein et al. [127] and by Kotliar [128]. In such a case, due to a mutual competition between two different symmetries like s and d, for some range of parameters a state with form factor η(k) = η_s(k)+αη_d(k) can be stable [129], where η_i(k) are the form factors and α defines the ratio between amplitudes of the order parameters with symmetry s and d. This competition can be expected in Ba_1−xK_xFe₂As₂, where doping leads to change of the gap symmetry. In such a case, in some range of parameters the mixing gap symmetry s+is_± can be expected [130, 131]. Recent experiments based on the muon spin rotation (μSR) measurements suggest a realization of this state in Ba_1−xK_xFe₂As₂, around x~0.7 [132]. Similar measurements suggest also a realization of s+d symmetry in CsCa₂Fe₄As₄F₂ [133] and KCa₂Fe₄As₄F [134].

The possibility of the realization of a state with the mixing symmetry was suggest in the system with time-reversal symmetry-breaking [107, 135, 136]. This is possible due to lowering spontaneously the fourfold rotational symmetry C4 to C2 symmetry, e.g., by a transition from tetragonal to orthorhombic phase [137]. This transition can be also associated with nematicity of the system [138].

Concluding, competition between two different types of the gap symmetry can lead to a state with mixed symmetry gap. In a context of our study, a role of the interband interaction is important [139], which can lead to an effective modification of the momentum-dependent gap value (cf. section 3.4). In our case, the interplay between different types of the symmetries at Γ- and M-centered FSs leads to a modification of the gap value dependently on J strength.

4. Summary

The multi-band nature of many superconducting materials make them very interesting for experimental and theoretical studies. A mutual influence between bands leads to the unconventional behavior of various physical properties. In this paper, using the two-band model of the iron-based superconductors we have studied a role of the pair-hopping coupling between bands on the physical properties such as superconducting gap, specific heat, spectral function and density of states. We show that the arbitrary small but finite pair-hopping coupling between the bands strongly influence the temperature dependence of superconducting gap and heat capacity, which significantly deflect from the BCS-type behavior. Also the spectral function and electron density of states demonstrate the unconventional nature of superconductivity in the two-band system. These results can be helpful to obtain the information about the symmetry of the gap [72, 140, 141] and the relations between effective interactions in the multi-band systems. The unconventional behavior found in the present work can be observed in many experiments and are in a good agreement with the predictions of the Ginzburg-Landau theory [44]. More recently, similar results have been also found in the case of the two-band superfluid system in the BCS-BEC crossover regime [142].

Data Availability Statement

The datasets generated for this study are available on request to the corresponding author.

Author Contributions

AP initialized the project and performed numerical calculation. AP and KK prepared the first version of the manuscript. All authors analyzed and discussed the results and contributed to the final version of the manuscript.

Funding

This work was supported by the National Science Centre (NCN, Poland) under grants OPUS no. 2017/25/B/ST3/02586 (AP and PP) and SONATINA no. 2017/24/C/ST3/00276 (KK). AP and KK appreciate also founding in the frame of scholarships of the Minister of Science and Higher Education (Poland) for outstanding young scientists (2019 edition, nos. 818/STYP/14/2019 and 821/STYP/14/2019, respectively).

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

The authors are thankful to S.-L. Drechsler for very fruitful discussions and comments.

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