1.Introduction

The possibility of realization of anisotropic superconductivity with triplet pairing has been subject of interest since the 1970s, when the discovery of superfluity was found in the fermionic isotope ³He, with p -wave Cooper pairs of atoms and a critical transition temperature of about 1 mK ¹. In 1994, Y. Maeno ² reported that strontium ruthenate (Sr₂RuO₄), with a critical temperature of 1.5 K, could possess a superconducting gap with p -wave symmetry. In this system, the critical temperature of the superconducting state cannot be tuned by changing the electron density or concentration as in cuprate superconductors such as La_2-xSr_xCuO₄³, where an optimal doping of holes xop≈0.15 from half filling leads to a maximum critical temperature Tc-max≈40 K. Nevertheless, Sr₂RuO₄ (SRO) and La_2-xSr_xCuO₄ (LSCO) have similar tetragonal crystal structures but, in the former, the planes where the particles flow without any resistance are RuO₂, whereas in the latter are CuO₂². Although Ru atoms may play an analogous role for electronic pairing as Cu in LSCO, these systems SRO and LSCO have very distinct superconducting properties; in particular, the former could have p -wave Cooper pairs and the latter has a d-wave superconducting gap. It has been suggested that in the underdoping regime, for 𝑑-wave superconductors, the antiferromagnetic state could be a precursor of the superconducting state ⁴, whereas the ferromagnetic state would be appropriate for 𝑝-wave superconductors, as possibly occurs in the family of superconductors compounds with uranium (U), such as Ute₂, UGe₂, URhGe₂ and UCoGe ⁵, with critical temperatures around 1.5 K. A model which accounts for both p- and 𝑑-wave superconductivity, within the same theoretical framework, is the generalized Hubbard model (GHM) ^6-8, where nearest and next-nearest-neighbor correlated hoppings (bond-charge interactions) can lead to the formation of Cooper pairs with different symmetries. For SRO an infinitesimal distortion of the RuO₂ square lattice is also considered. This distortion leads to an asymmetry between the second-neighbor correlated hoppings along the two diagonals of the former square lattice, which give rise to two 𝑝-wave superconducting states of different energies ⁶. In previous works, we have studied the dependence of the p-wave ground-state critical temperature and p-wave gap amplitude on the electron density (n) and the electron-electron interaction parameters within the generalized Hubbard model, maintaining the ratio between first- (t) and second-neighbor (t´) single-electron hoppings constant ^6,7,13. Also, we have studied the temperature dependence of the p-wave gap amplitude for a fixed set of Hamiltonian parameters which allows the calculation of the temperature behavior of the electronic specific heat, which has shown a good agreement with that experimentally obtained for SRO ¹³. However, in a square lattice the ratio t´/t determines the energy at which the van Hove singularities occur, that in turn affect the optimal electron density where the maximum T _c (n) is attained for a given set of electron-electron interaction parameters. Hence, in this work, in a similar manner as was recently done for d- and s*-wave superconductors ^9,10, the superconducting critical temperature (T _c ) versus the electron density (n) and the ratio t´/t is calculated for a set of electron-electron interaction parameters. In particular, for the optimal electron density (n _op ), i.e., that n where the critical temperature attains its maximum T _c,max with the other Hamiltonian parameters fixed, the corresponding ground state energy was determined for different values of t´. For a given set of electron-electron interaction parameters, the results show that the largest T _c,max occurs at low electron densities, and the minimum of T _c,max can be encountered close to half filling (n = 1). Moreover, there could be a competition between s*-, d- and p-wave superconducting states around half filling. Finally, this study suggest that the values of the optimal doping (n _op ) and second-neighbor hopping (t´) that lead to a critical temperature Tc≈1.5 K, which matches with that of the SRO, are found close to have filling (nop∼1) and for t´< 0, in agreement with first-principles studies performed for SRO ¹¹. Finally, an analysis of the corresponding Fermi surfaces and the single-particle excitation energy were performed.

2. The model

We start from a single-band generalized Hubbard model on-site (U) and nearest-neighbor (V) Coulombic interactions, first- (t) and second-neighbor (t´) hopping parameters, and first- (Δt) and second-neighbor (Δt₃) correlated hopping interactions ^6,7

where ni=ni,↑+ni,↓, ni,σ=ci,σ†ci,σ, and ci,σ†ci,σ, is the creation (annihilation) operator with spin σ=↓ or ↑ at site i. < i,j > and << i,j >> denote nearest- and the next-nearest neighbour sites, respectively. In the case of SRO, a surface distortion of the RuO₂ square lattice has been observed by using X-ray diffraction ¹². Hence, we considered an infinitesimal distortion of the square lattice that leads to an asymmetry between the second-neighbor hopping (t´) and correlated hopping (Δt₃) along the two square diagonals. The former undisturbed second-neighbour hopping (t´) and second-neighbor correlated hopping (Δt₃) of a non-distorted square lattice respectively become t'+χijδ and Δt3+χijδ3, where χij=+1 if second-neighbour sites i and j are along the x^-y^ direction, and χij=-1 if they are along the x^+y^ direction. Here δ and δ₃ characterize the degree of asymmetry between the mentioned hoppings along the two square diagonals. The expressions for the model parameters are given in Table I in terms of Wannier functions [φ(r-Ri)] centred at lattice site R _i.

The real-space Hamiltonian ((1)) can be transformed into reciprocal space by performing a Fourier transform of the electron operators ck,σ†=(1/Ns)∑jexp⁡(ik⋅Rj)cjσ†. After a mean-field Hartree decoupling, the reciprocal-space Hamiltonian can be written as:

where the Fourier transform of the electron-electron interaction for electrons with the same spin component is given by:

With

where α is the lattice parameter and Δt3±=Δt3±δ3. Notice that the real-space term Δt3+χijδ3 in Eq. (1) leads, after the Fourier transform, to Δt3+ and Δt3- depending on the value of χij(±1). Also, εMF(k) is the mean-field dispersion relation and N_s is the total number of lattice sites. In this case, εMF(k) is given by:

With

where n is the electron density and t'±=t'±δ. Notice that the mean-field single-particle hoppings t _MF and t'MF,±, depend on the electron density (n) and correlated hoppings (Δt,Δt3), in particular on the distorted values of the second-neighbor correlated hopping (Δt3±=Δt3±δ3).

In general, the BCS coupled integral equations can be written as ^6,7:

and

whose solution gives the chemical potential (μ) and the amplitude of the superconducting gap for given n and T. For the particular case of an interaction Wk,k’,q given by Eq.(3), Eq. (11) and (12) admit a solution for the superconducting gap (Δ(k)) of the form:

known as p-wave superconducting gap, where Δp is the temperature-dependent superconducting gap amplitude and the sign ± indicates the direction of the real-space distortion. It is worth mentioning that, in general, the gap of a superconducting state formed by triplet Cooper pairs could have another symmetry different from p-wave, such as f-wave symmetry, and the possible symmetries depend on the form of the electron-electron interaction potential Wk,k’,q.

By substituting (13) in (11) and (12) we obtain:

where the sums over the first Brillouin zone (1BZ), defined as

-πa,πa⊗-πa,πa

have been transformed into integrals. Here E(k) is the quasi-particle energy given by:

It is important to mention that the chiral p-wave gap given by

also satisfies (11) and (12) with Wk,k’,q given by (3), where the second term in the square root of (16) is

instead of

However, in this work, the chiral case will not be considered because it does not seem to successfully apply to SRO ¹³.

The critical temperature T_c can be determined from (14) and (15) considering that Δp(T=Tc)=0. In this case, these equations transform into:

Here, for given Hamiltonian interaction parameters and n, the two-coupled equations determine the values of T _c and μ. Moreover, the ground state energy (E_g) per site is given by ¹⁴:

which, in the case of a p-symmetry superconducting gap Eq. (13) can be simplified to:

It is important to mention that for p-wave superconductors, U plays a similar role in the superconducting gap equation as for d-wave ones, i.e., it does not affect the shape of εMF, and only modifies the electron self-energy (E_MF) shifting the numerical value of the superconducting chemical potential (μ) without changing the superconducting critical temperature. Therefore U can be taken equal to zero. Moreover, to obtain a solution for the superconducting gap equation ⁷, (V-4δ3) should be negative and therefore the condition V<4δ3 must to be satisfied. In order to keep a minimum set of parameters but considering that a p-wave superconducting gap can be generated from δ₃, V will be set to zero too. It is worth mentioning that models which consider a negative V can lead to triplet pairing in frustrated lattices ¹⁵, however, this term would also lead to a phase separated state, where electrons double occupy sites over a macroscopic region of the lattice, which would strongly compete with the superconducting state ¹⁶.

3.Results

Figure 1a) shows the critical temperature T _c of p-wave superconducting states as a function of the electron concentration n and the second-neighbor hopping parameter t´ for Δt=0.5|t|, Δt3=δ=0.05|t|, and δ3=0.08|t|. Likewise, 1b) shows the critical temperature versus the electronic density (n) for a particular value of -t'/t=-0.15. The value of n, where the maximum critical temperature (T _c-max ) is attained for each value of t´/t is called the optimal electron density n _op .

Figure 1 a) Critical temperature (T _c ) versus electronic density (n) and t´/t for p-wave superconductors with U = V = 0, Δt = 0:5|t|, δ = Δ t3 = 0:05|t|, and δ₃ = 0:08|t| (red circles), with t = -1 eV. The dark gray circles corresponds depicts the maximum Tc for each value of t´/t. b) Critical temperature (T _c ) versus electronic density (n) for a p-wave superconductor with t´/t = ¡0:15 and the same interaction parameters as in Fig. 1a). 

The p-channel maximum critical temperature (T _c-max ) versus the optimal electronic density (n _op ) and t´/t, for the same interaction parameters Δt and Δt₃ can be obtained from Fig. 1a). These curves are shown in Fig. 2a) for different values of δ₃. The projections of these curves Tc-max(nop,t'/t) on the (nop,t'/t) plane are presented in Fig. 2b). For a given δ₃, a global maximum of T _c-max can be found for t´> 0 (solid magenta triangles) within of the intervals nop-max∈[0.08,0.32] and t'∈[0.47,0.43]. The latter interval is very narrow. On the other hand, for a given δ₃, minimum values of T _c-max can be found for t´< 0. These minima are located within the intervals nop∈[0.863,0.921] and t'∈[-0.190,-0.105] (open magenta triangles). The values of n _op and t´ where the global maxima and minima of T _c-max occur, for each δ₃, are shown in Table II.

Figure 2 a) Maximum superconducting critical temperature (T _c-max ) versus the optimal electronic density (n _op ) and -t´ for U = V = 0, Δt = 0:5|t|, δ = Δt₃ = 0:05|t|, and δ₃ 2 [0:06|t|; 0:1|t|] (colors lines). b) Projections of the curves depicted in Fig. 2a) on the plane (n _op ;-t´/t). The magenta solid (open) triangles correspond to maxima (minima) of T _c-max . 

It is important to underline that for t´< 0, n _op is close to half filling and, in this region, T _c-max abruptly diminishes. Moreover, as shown below, in this zone p-wave superconducting states compete with extended s-wave (s*) and d-wave ones.

It is also worth mentioning that close to (nop,-t'/t)=(0.868,-0.57), n _op is almost the same for all δ₃. Figure 3 shows T _c-max versus δ₃ at this point.

Figure 3 Maximum critical temperature (T _c-max ) as a function of δ₃ for (n _op ;-t´/t) = (0:868;-0:57). 

Moreover, when δ₃ = 0.045 eV, the maximum critical temperature (T _c-max ) of the state (nop,-t'/t)=(0.868,-0.57) agrees with the critical temperature of SRO, i.e., Tc≈1.5 K. Other points that satisfy Tc-max≈1.5 K with different values of δ₃ are summarized in Table III. Two cases correspond to t´> 0: (nop,-t'/t)=(0.684,0.115) with δ₃ = 0.07 eV; and (nop,-t'/t)=(0.586,0.210) with δ₃ = 0.06 eV. The other cases correspond to t´< 0. Taken into account that the distortion of the square lattice is assumed to be very small, it is expected that the parameter δ₃ should be very small. Hence, according with the results of this work, the smaller value of δ₃ satisfying Tc-max≈1.5 K, corresponds to δ₃ = 0.045 eV, which is attained at (nop,-t'/t)=(0.868,-0.57). However, it has been estimated, by using first-principles calculations ¹¹, that the first- and second-neighbor hoppings of SRO on the RuO₂ planes satisfy -t'/t=-0.3 and then, the states (nop,-t'/t)=(0.960,-0.26) with δ₃ = 0.08 and (1.027, -0.385) with δ₃ = 0.07 eV, both of which also have small values of δ₃ and optimal electron concentrations (dopings) close to half-filling, are good candidates to be the experimentally observed p-wave ground state of SRO.

The p-wave gap amplitude (Δp-max) and the ground state energy (E_g) of the systems depicted in Fig. 2a) are shown in Fig. 4a) and 4b), respectively. Notice that, although there is a global minimum of T _c-max and Δp-max, and a local minimum of E_g, they are not located at the same (nop,-t'/t) point as occurs for d-wave superconducting states ⁹.

Figure 4 a) The maximum gap amplitude (∆_max) for the same p-wave superconductors as in Fig. 2. b). Ground state energy (E _g ) of the systems depicted in Fig. 4a). 

Figures 5a)-d) show the Fermi surfaces (FS) corresponding to the states listed in Table III. Likewise, 6a)-d) show the corresponding single-particle excitation energies (Δ₀). This energy is defined as the minimum energy necessary to break a Cooper pair when the momentum k of the pairing electrons are along the angle θ=ky/kx. Notice that the FS are symmetric with respect to the k _x + k _y diagonal. It is also important to point out that the states (nop=0.586, t'=0.2|t| eV), (nop=0.684, δ₃ = 0.07 eV), (δ3=0.06, δ₃ = 0.08 eV) whose single-particle excitation gaps are depicted in Fig. 6a)-6c), Δ0≈2Δp at the antinode. However, for the other cases, the FS is more elongated along the k _x + k _y diagonal, and the relation Δ0≈2Δp is not fulfilled at the antinode. In the case (nop=0.868,t'=-0.570|t| δ₃ = 0.045eV), the FS is disconnected, see Fig. 5d).

Figure 5 Fermi surface of p-wave superconducting states with a) (n _op = 0:815, t´ = -0:045|t|, δ₃ = 0:08 eV), b) (n _op = 0:684, t´ = 0:115|t|, δ₃ = 0:07 eV), c) (n _op = 0:586, t´ = 0:2|t|, δ₃ = 0:06 eV), and d) (n _op = 0:868, t´ = -0:570|t|, δ₃ = 0:045 eV). 

Figure 6 Single-particle excitation energies (∆₀) for the same systems as Figs. 5a)-d), respectively 

Besides p-wave superconducting ground states, the first- (Δt) and second-neighbor (Δt₃) correlated hoppings can also induce extended s-wave (s*) and d-wave superconducting states ⁷, where the respective gaps (order parameters) are given by Δs*(k)=Δs+Δs*[cos(kxa)+cos(kya)] and Δd(k)=Δd(cos[kxa]-cos[kya]). To compare the behavior of p-wave superconducting systems with those with extended s- and d-wave superconducting gaps, Fig. 7a) shows those points (nop,-t'/t) where the critical temperature is maximum, for s*- (black line), p- (purple line), and d-wave (gray line) superconductors. The depicted d- and s*-wave states have Δt=0.5|t| and Δt3=0.05|t| ^9,10, whereas the p-wave ones have, in addition, δ = 0.05 eV and δ₃ = 0.06 eV. In each curve, the red open and solid circles correspond to those states where T _c-max , and E_g, respectively attain their minimum values. Figure 7b) shows the corresponding ground state energies for the same systems as in Fig. 7a). Notice that, at low densities, around the point (nop,-t'/t)=(0.14,0.455), all the curves coincide, and the ordering from lowest to highest value of E_g correspond to s*-, p- and d-wave symmetries. However, for higher densities, the maximum critical temperatures for the different superconducting symmetries are located in different regions, and can be distinguished two crossing points; one around (nop,-t'/t)=(0.260,0.415), where the s* channel curve crosses with the p-channel one, and other at (nop,-t'/t)=(0.459,0.310), where the d-channel crosses with the p-channel one. Close to these points it is expected that the different possible superconducting symmetries could coexist, if they had similar ground state energies.

Figure 7 a) (n _op ;-t/t) states where the maximum critical temperature (T _c-max ) is attained for s*- (black line), p- (purple line), and d-wave (gray line) superconducting states with ∆t = 0:5|t|, and ∆t₃ = 0:05|t|. The red open and solid circles correspond to minimum of T _c-max , and ground state energy (E _g ), respectively. b) Ground state energies (E _g ) of the superconducting states depicted in Fig. 7a). 

Table IV summarizes the superconducting physical properties of the mentioned points of possible coexistence.

4.Conclusions

For the first time, by means of the generalized Hubbard model, we have studied the variation of the maximum critical temperature (T _c-max ), the gap amplitude (Δ _p ) and the ground state energy (E_g) of p-wave superconducting ground states in a square lattice within the (nop,t'/t) space for fixed values of the electron-electron interaction parameters, where n _op denotes the optimal electron density where the maximum T_c is attained for a given value of the second-neighbor hopping (t´). In contrast to previous studies, here the effects of different values of t´ on the superconducting properties were analyzed. The ground state energies (E_g) were obtained for all optimal electron concentrations (n _op ) where the critical temperature (T_c) is maximum, for systems with given values of the second-neighbor hopping (t´) and δ₃ and with Δt = 0.5 eV, δ = Δt₃ = 0.05 eV. It is important to emphasize that it is not possible to define a supremum value for the set of maximum critical temperatures (T _c-max ), as occurs for 𝑑-wave superconducting states ⁹, where the minimum ground state energy is found for that state with the minimum value of T _c-max . Moreover, the results presented in this work suggest that the p-wave superconducting ground states that reproduce ecuacion K, are found close to half filling (nop∼1), and should posses t ´< 0, in agreement with first-principles calculations. In addition, in this parameter zone the Fermi surface is close and the single-particle excitation energy [ Δ₀(θ)] possess a clear p-wave pattern. Moreover, within this zone, the p-wave ground state energies are lower than the surrounding d- and s*-wave superconducting states. In contrast to previous works ^6,7,13, this study indicates that the Hamiltonian-parameter zone where stable p-wave superconductivity with low critical temperatures can be found embraces positive and negative values of t´, and thus other superconducting materials whose pairing symmetry has not been established can be studied under the framework considered in this work. For example the iron-based superconductors, where the K-doping of Sr_1-xK-xFe−2As₂, modifies the former lattice and electronic structure ¹⁷. In addition, we can also investigate compounds such as RuSr_2-xBa_xGdCu₂O₈ which contains both Ru-O and Cu-O planes, and could have superconducting gaps with different symmetries ¹⁸.