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Ginzburg-Landau theory of Fulde-Ferrell-Larkin-Ovchinnikov states in unconventional singlet superconductors with impurities

We derive the Ginzburg-Landau theory of unconventional singlet superconductors in the presence of a Zeeman field and impurities, in order to examine the resulting Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phases. We show that the behavior of the FFLO phases in unconventional superconductors in the presence of impurities is qualitatively different from that found for s-wave superconductors.


I. INTRODUCTION
In 1964 Fulde and Ferrell 1 , and Larkin and Ovchinnikov 2 demonstrated that a superconducting state with an order parameter that oscillates spatially may be stabilized by a large applied magnetic field or an internal exchange field. Such a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state was subsequently shown to be readily destroyed by impurities 3 and has never been observed in conventional low-T c superconductors. The question of observing an FFLO phase in an unconventional superconductor has only been addressed more recently. In particular organic, heavy fermion, and high T c superconductors appear to be promising candidates for such states [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] . These new classes of superconductors are believed to provide conditions that are favorable to the formation of FFLO state, because many of them are (i) strongly type II superconductors so that the upper critical field H c2 can easily approach the Pauli paramagnetic limit; and (ii) layered compounds so that when a magnetic field is applied parallel to the conducting plane, the orbital effect is minimal, and the Zeeman effect (which is the driving force for the formation of FFLO state) dominates the physics. Indeed, some experimental indications of the existence of the FFLO state have been reported 4,11,13,19 . All of these materials have been argued to be unconventional superconductors and in this way differ from the case originally considered by Fulde, Ferrell, Larkin, and Ovchinnikov. Motivated by this possibility we have derived the Ginzburg-Landau (GL) free energy functional for unconventional superconductors in the presence of Zeeman splitting and impurity potential, and use it to study the possible FFLO phases. This is not a commonly used approach of examining the FFLO phase. To our knowledge it has previously only been discussed in the context of clean s-wave superconductors by Buzdin and Kachkachi 17 . However, as we show below, it represents a very powerful approach to study the FFLO phase since the simplicity of the resulting theory allows complexities such as non s-wave pairing, impurities, and even strong-coupling effects (though this is not done here) to be included. The stability of the various superconducting phases and to some degree, the topology of the superconducting phase diagram can be examined within this approach. These results help to clarify the nature of the FFLO phase and also can be used as a guide for a theory extended to all temperatures and magnetic fields.
Using a functional integral formalism, we derive the GL free energy functional for single component singlet superconductors in the presence of impurities and Zeeman fields in the weak coupling limit. The resultant GL free energy is valid near the second order normal to superconductor phase transition line in the (T, H) plane. This line will be denoted by [T, H(T )]. The resulting instability to the FFLO phase appears readily within this approach due to the change of sign of the gradient term κ|∇Ψ| 2 along the line [T, H(T )]. The point at which the coefficient κ changes sign [denoted [T * , H(T * )]] is a tricritical point. At this point the normal, uniform superconducting, and FFLO phases all meet (see Fig. 1). An intriguing feature of the weak-coupling clean limit is that the fourth order uniform term β|Ψ| 4 also changes sign at the tricritical point. It is this term that determines the form of the FFLO phase. In particular, for a given momentum q both solutions Ψ ∼ e iq·r and Ψ ∼ e −iq·r are degenerate superconducting states at the normal to superconductor instability. The fourth order term breaks this degeneracy and selects either a cos(q · r) (LO phase) or a e iq·r (FF phase) type order parameter. Since the magnitude of the order parameter is spatially uniform for the FF phase and vanishes at lines in real space for a LO phase, a negative β stabilizes the LO phase. Note that the complete GL free energy in this case requires inclusion of terms of the form |Ψ| 6 and |Ψ| 2 |∇Ψ| 2 to be bounded. These considerations have appeared in the work of Buzdin and Kachkachi 17 for conventional s-wave superconductors in the clean limit and are shown here to remain true for unconventional superconductors.
We further extend these considerations to include the effect of impurities. It is found that impurities suppress the FFLO phase for both conventional and unconventional superconductors. However, we also find that impurities lead to qualitatively different (T, H) phase diagrams for conventional (s-wave) and unconventional (non s-wave) superconductors (see Figs. 2 and 3). This difference is most easily understood by looking at the coefficients κ and β. For conventional superconductors it is known that β is unchanged by non-magnetic impurities (this is a consequence of Anderson's theorem) while κ is changed. The point on the [T, H(T )] line that κ changes sign is pushed to lower temperatures with impurities (which illustrates that impurities suppress the FFLO phase). Consequently, the normal to FFLO transition in the clean limit is replaced by a first order normal to uniform superconducting transition. For unconventional superconductors impurities change both the κ and β coefficients, due to the absence of Anderson's theorem. It is found that the points on the [T, H(T )] line that κ = 0 and β = 0 move to lower temperatures with increasing impurity concentration. The point β = 0 is more rapidly suppressed than the point κ = 0. This implies and that the initial instability into the FFLO phase is into a FF phase (Ψ ∼ e iq·r ) as opposed to the LO phase (Ψ ∼ cos q · r) that is typically encountered. We are not aware of any other report of the stability of a FF phase. The results here also indicate that the first order normal to uniform superconductor phase transition does not occur for unconventional superconductors.

Consider the HamiltonianĤ
where T (x) represents the kinetic energy and takes the form −∇ 2 /(2m) − ǫ F for a free electron and more generally takes the form T (x) = ǫ(k = i∇) for a band with dispersion ǫ(k) measured from the Fermi energy ǫ F , U (x) is the disorder potential and satisfies U (x) = 0 and , n i is the concentration of impurities, and µ = gµ B /2 is the magnetic moment of the electron. We will primarily be interested in singlet pairing in the ground state; so we neglect interactions between electrons with the same spin. The pairing interaction (V int ) is taken to have the separable formV f (k) describes the gap dependence on the Fermi surface and is defined to satisfy k |f k | 2 = 1. It is also understood that a cutoff exists in momentum space so that only electrons that are close enough to the Fermi surface interact with each other. After taking the appropriate Fourier transforms the interaction in real space takes the form V is the volume of the system (note this definition of f (x) implies dx|f (x)| 2 = 1). One may also describe the system using an Euclidean action in terms of Grassman variables: where S 0 is the action for free electrons, and τ is the imaginary time. The partition function is We now decouple the quartic term in S by introducing a pair of Hubbard-Stratonovich fields ∆(R, τ ) and ∆(R, τ ), which will become the superconducting order parameter: With this decoupling, the fermionic action becomes quadratic, and can be integrated out, after which we obtain an effective action in terms of the order parameter ∆(R, τ ): where The mean-field solution corresponds to the saddle point of S e [∆, ∆]: where ∆s stands for quantum and thermal averaging in the presence of the pairing field ∆ s (R). Here we have assumed a static saddle point so that ∆ s has no τ dependence. The functional integral formalism can be used to derive the effective Ginzburg-Landau free energy, in the vicinity of the second order normal to uniform superconductor transition line [T, H(T )]. This has been done for the short-range attractive interactions (that gives rise to s-wave pairing) 24 . Our starting point is the effective action, Eq. (7). Near [T, H(T )], we may make two simplifications: i) We may neglect the τ dependence of ∆ as we expect the thermal fluctuations to dominate the quantum fluctuations; ii) We may expand S e in powers of ∆. The quadratic terms take the form where The quartic term takes the form where We will need the sixth order term as well, whose explicit expression (that involves the product of six Green's functions) is not included here. The above quadratic and quartic terms apply for a particular impurity configuration. We will average over impurity distributions when calculating the form of the free energy. We assume that the gap function that appears above corresponds to the gap function averaged over impurities and that we can ignore impurity induced correlations in the gap function and between the gap function and the single particle Greens functions. To proceed further the impurity averaged correlation functions GG and GGGG must be calculated. We determine these within the Born approximation and much of the derivation follows that of Werthamer for conventional s-wave superconductors 25 . The impurity averaged normal Greens functions are G 0,σ (k; iω n ) = 1 iω n + i 1 2τ sgnω n − ǫ k + 2σµB (14) where 1 2τ = Γ = πn i W N (0) and σ is 1/2 (−1/2) for ↑ (↓). Consider the average Q(x 1 , y 1 ; x 2 , y 2 ; iω n ) = G 0,↓ (x 1 , y 1 ; iω n )G 0,↑ (x 2 , y 2 ; −iω n ) imp = Q(x 1 − y 2 , y 1 − y 2 , x 2 − y 2 ; iω n ) due to translational invariance. Summing the usual ladder diagrams shown in 25 gives the self-consistent solution This can be solved after taking the Fourier transforms with respect to R 1 , R 2 , and R 3 Substituting this result into Eq. 11 gives Note that the form of the vertex corrections found here is not the same as for conventional s-wave superconductors. In particular, when deriving the terms up to second order in the gradients, the vertex corrections vanish for unconventional superconductors. However, for higher order gradient terms, the vertex corrections are not zero. The impurity averaged correlation function that appears in δS 4 is less straightforward to calculate. For the terms in the free energy that are fourth order in the order parameter we consider the only up to second order in the gradients of the order parameter. In this case the non-zero diagrams have the same form are those that contribute in the s-wave case 25 .
After performing the appropriate Taylor series expansions the following GL free energy for hexagonal and square lattices is found [this expression is valid only for non s-wave superconductors ( f k = 0)] the coefficients are where . Theδ term does not appear for a hexagonal lattice. For an orthorhombic lattice the following terms also appear in the free energy The coefficientsκ,μ, andη are given by κ, µ, η, with v 2 ⊥ replaced by (v 2 x − v 2 y ) respectively and the coefficient δ 2 is given by . The free energy is the main result of this paper.

III. S-WAVE SUPERCONDUCTORS
Here we review the known results about the FFLO phase for a conventional s-wave superconductor and show how it arises from the free energy. As mentioned in the Introduction the only two coefficients that are required to study the instability from the normal phase into the FFLO phase near the tricritical point are κ and β. For a conventional s-wave superconductor (for which f k is a constant) these can easily be determined by following Werthamer's derivation 25 (note that the above free energy does not apply here since f k = 0): The coefficient β does not depend upon the impurity concentration in agreement with Anderson's theorem. The second order normal to uniform superconductor phase line [T, H(T )] is given by α(Γ = 0) = 0 and is shown in Fig. 4. Note that once κ < 0 or β < 0 the phase line [T, H(T )] no longer denotes the true normal to superconductor phase line. Numerical evaluation of κ s and β show that in the clean limit both κ and β vanish at T = 0.56T c . This point is the tricritical point. The phase diagram is shown in Fig. 1. When impurities are added numerical evaluations show κ vanishes at a lower temperature than β vanishes (note that the presence of impurities does not change β). In this case there is a first order superconducting transition to a homogeneous phase for T ≤ 0.56T c . Once β < 0 then the normal to superconductor instability line is no longer given by Fig. 4. To determine whether there exists an FFLO phase for some arbitrary impurity concentration requires a calculation that goes beyond the GL free energy presented here. This is because the G.L. theory is only valid close to the transition at T = 0.56T c (note that the G.L. theory can be used to study the transition into the FFLO phase if the impurity concentration is small enough). The calculations of Bulaevskii and Guseinov for layered superconductors indicate that there exists no FFLO phase at T = 0 when Γ/T c > 0.6 26 . The qualitative phase diagram in the presence of impurities is shown in Fig. 2.

IV. UNCONVENTIONAL SUPERCONDUCTORS
The last Section demonstrated that for T near 0.56T c the G.L. theory accurately reproduced the phase diagram for conventional superconductors. Here we apply the same approach to unconventional superconductors where it turns out the G.L. theory is more powerful. This arises because the transition from the normal state to the superconducting state is second order for all fields and impurity concentrations for unconventional superconductors. For conventional superconductors this transition is sometimes first order, which limits the applicability of G.L. theory. For example, the G.L. theory for unconventional superconductors can give the maximum impurity concentration that allows the FFLO phase to exist; it was not able to do this for s-wave superconductors. As a concrete example we study a d-wave superconductor (f k ∝ k 2 x −k 2 y ) with a cylindrical Fermi surface. Choosing some other f k will not change the qualitative form of the phase diagrams (provided f k = 0). The clean limit phase diagram is qualitatively the same as that for the s-wave case. In fact, the clean limit theory indicates that the FFLO phase appears for T < 0.56T c independent of the order parameter symmetry. When impurities are added the main conclusion is that κ vanishes at a higher temperature than β does when these quantities are evaluated on the phase boundary [T, H(T )] (see Fig. 5 and Fig. 6 for the temperature evolution of κ and β). This implies that there is no first order transition from the normal state to a uniform superconducting state, but rather a second order transition into a FFLO phase. The temperature at which κ = 0 gives the maximum temperature that allows the existence of the FFLO phase (we call this temperature T F ). This is in sharp contrast to what happens in s-wave superconductors, where a first-order phase boundary separates the normal and uniform superconducting state for temperatures just above the point at which β vanishes. Fig. 7 shows how T F varies as the impurity concentration is increased (note that impurities also suppress T c , hence we plot T F /T c as a function of T c /T c0 since the latter is an experimentally measurable quantity). This figure indicates that the FFLO phase survives a considerable impurity concentration, only for Γ/T c ≥ 0.6 does the FFLO phase cease to exist (superconductivity is destroyed when Γ/T c ≥ 0.88).
The structure of the FFLO phase can also be addressed within the G.L. theory in the neighborhood of tricritical point (given by κ = 0 on the phase line [T, H(T )]). To do this we compare the free energy for three different phases: (1) ∆ 1 = ∆ 0 (uniform phase), (2) ∆ 2 = ∆ 0 e iqr (FF phase), and (3)∆ 3 = ∆ 0 cos(q · r) (LO phase). We note here that terms O(∆ 6 ) do not need to be included in the calculations done here because the terms O(∆ 4 ) are positive. Also the minimization with respect to the orientation of q for ∆ 2 and ∆ 3 implies that q is oriented along the nodes for all impurity concentrations. This agrees with earlier calculations that go beyond the GL theory in the clean limit 18,15 . After minimizing with respect to q the resulting free energies are where β 2 = β − 3κη δ and β 3 = 3 2 β − 5κη 2δ (where we have used µ = 8η). It is clear that when κ = 0, F 1 = F 2 < F 3 which implies that α = 0 and κ = 0 gives the tricritical point where the normal, uniform superconducting, and FF superconducting phases meet. Note that the FF phase is stable while the LO phase is not at the tricritical point since β > 0 (the FF phase is never stable in the s-wave case). For temperatures below that of the tricritical point the phase transition from the normal phase into the FFLO phase is given by α = κ 2 4δ . Intriguingly, along this phase line β changes sign. This implies a first order transition between the FF and the LO phase. This can be seen by comparing F 2 and F 3 which is equivalent to comparing β 2 and β 3 . If β = 0 and κ < 0 then 0 < β 3 < β 2 which implies F 3 < F 2 implying that there is a phase transition between the FF and the LO phases. For the singlet superconductors considered here the FF phase should exhibit a spin current.
A detailed calculation was carried out for an impurity concentration for which T c /T c0 = 0.573 (where T c0 is the transition temperature with no impurities present). Fig. 8 shows the resulting phase diagram calculated within the G.L. theory. Note that G.L. theory in this case gives a reasonable description of the phase diagram in this case because it is valid along the entire normal to superconducting phase transition boundary. For other impurity concentrations the phase diagram is similar. If multiple scattering from the impurities becomes important (e.g. going beyond the Born approximation to a T -matrix treatment), then it is found that the region of the phase diagram where the FF phase appears is decreased 27 .

V. CONCLUSIONS
In conclusion we have derived the GL free energy for singlet superconductors in the presence of a Zeeman field and non-magnetic impurities. This free energy was used to examine the resulting phase diagram. It was shown that the phase diagrams for unconventional superconductors and conventional superconductors are qualitatively different in the presence of impurities. In particular the first order normal to uniform superconductor phase transition that exists for conventional superconductors does not exist for unconventional superconductors. Also, for unconventional superconductors impurities induce a change in the structure of the FFLO phase. In the clean limit the FFLO phase is described by an order parameter of the form cos(qr) (LO) while impurities stabilize a e iqr (FF) type order parameter.  when there are no impurities present. This is the boundary on which κ and β are determined. Note that for T ≤ 0.56Tc this phase boundary will not coincide with the actual normal to superconducting phase boundary (the transition will either be to a non-uniform (FFLO) phase or will be first order).