Complex state induced by impurities in multiband superconductors
We study the role of impurities in a two-band superconductor, and elucidate the nature of the recently predicted transition from state to state induced by interband impurity scattering. Using a Ginzburg-Landau theory, derived from microscopic equations, we demonstrate that close to this transition is necessarily a direct one, but deeper in the superconducting state an intermediate complex state appears. This state has a distinct order parameter, which breaks the time-reversal symmetry, and is separated from the and states by continuous phase transitions. Based on our results, we suggest a phase diagram for systems with weak repulsive interband pairing, and discuss its relevance to iron-based superconductors.
It has been long recognized that nonmagnetic impurities strongly influence properties of multiband superconductorsSungJPCS67 (); Muzikar (); GM1 (); Kulic (); Mishonov (); Gurevich (), especially in the case of an order parameter with sign change between different bands ( state)Muzikar (); Senga (); Bang (); Vorontsov (). Recently, it has been pointed out that impurities-induced interband scattering can continuously change the order parameter of a two-band superconductor from to stateGolubov (); Golubov2 (); GM2 () . This is particularly relevant for iron-based superconductorsLaOFeAs (); Paglione (), most of which are believed to be in some form of the state, see recent reviewsReviews1 (); Reviews2 ().
As we demonstrate in this Letter, the -to- transformation may follow a nontrivial scenario, and occur via an intermediate complex state at which a finite phase shift develops between the gap parameters in the two bands. We derive the simplest possible two-band Ginzburg-Landau (GL) free energy of the system from microscopic theory, and show that the presence of interband impurity scattering has important consequences for the different possible order parameters the theory can support. In the case of repulsive interband pairing we indeed observe the to transitioncrossover () with increasing the degree of disorder. We demonstrate that the transition is necessarily a direct one only close to the critical line; deeper in the superconducting state the state gives way to an intrinsically complex order parameter (which can be thought as an state), and only then to a pure state. This complex state breaks time-reversal symmetry and is separated from the other two superconducting states by continuous phase transitions. We discuss the reason and conditions for the appearance of this state. Based on our results, we propose the phase diagram shown in Fig. 1 for two-band superconductors with weak repulsive interband coupling.
We consider a system of two parabolic bands, with partial and total densities of states (DOS) , , and respectively. The pairing interactions are described by coupling matrix , with . In the superconducting state there are two gap parameters and , which are assumed to be complex constants for each band . The relative phase is a gauge-invariant quantity, and it is or in the or states respectively. The presence of impurities introduces scattering rates parametrized by , where are the band indices. For the interband terms () we can write , with , where and are the impurities’ concentration and potential respectively. On general grounds, point defects, such as atomic substitutions or vacancies, can scatter carriers with large momentum change and therefore are expected to give comparable intraband and interband scattering rates. In the case of the iron-based superconductors this was indeed confirmed by the first-principles calculations Kemper ().
Close to the critical temperature the free energy can be expanded in powers of and . (Although GL theory has been generalized to the case of multicomponent order parameters without impuritiesTilley (); ZD (), the proper justification of this multiband extension is a matter of ongoing debateAEK (); Babaev1 (); Kogan1 (); Shanenko (); GLnote ().) In the presence of impurities this can be done systematically, starting from the Usadel equations(Gurevich, ; AEK, ). The resulting GL free energy up to quartic in terms can be written as
We present the derivation of from the microscopic theory, and give exact expressions for its coefficients in the Supplemental Material(SuplMat, ). If the gap parameters are uniform in space and constant within each band, the intraband impurity scattering rate drops out of the theory completely, as a direct consequence of the Anderson theoremAnderson (). In contrast, the interband terms play an important role. The first two terms look similar to the standard GL theory
but with and modified by the presence of impuritiesSuplMat (). combines the electromagnetic field contribution, and the derivative terms that couple and to the electromagnetic vector-potential. For the rest of this paper we assume no field and uniform order parameter, so . The third term in couples and , and without impurities it is . In the presence of interband scattering processes, however, becomes more complicated:
We can see that the presence of impurities introduces several new quartic interband terms in the GL theory(Ng, ). In the limit becomes proportional to and all other coefficients in Eq. (3) vanish. As a consequence, for a clean system the only possible solutions for are and , and which one minimizes is determined by the sign of . When impurities are present, this is not necessarily true any more, and other solutions are possible, due to the term – it can destabilize the and states, provided is positiveTanaka1 (). Thus, the dirty two-band superconductor can have quite rich phase diagram.
The critical temperature at a given disorder strength is determined by the quadratic terms in Eq. (1). The equation for derived in the Supplemental MaterialSuplMat () takes the form , with being the identity matrix, and
We have defined , , and
where is a high-energy cut-off (e.g., the Debye frequency). In the clean limit, , this equation gives transition temperature , where is the largest eigenvalue of the -matrix. Note that the interband impurity scattering processes are always pair-breaking (unless ), and suppress , in contrast with the intraband scattering, which has disappeared.
In general, the dependence has to be found numerically but the extreme dirty limit can be analyzed analytically. Depending on , there are two qualitatively different regimes. If interband pairing is attractive, or negative but weak (i.e., when is positive) no amount of disorder can completely suppress the superconductivity. In this case the critical temperature in the extreme dirty limit can be obtainedSuplMat ():
However, if the interband pairing is repulsive and strong, such that is negative, there is a critical amount of disorder which brings down to zero, in analogy with the Abrikosov-Gor’kov theoryAG (). Numerical calculation of for the different regimes are shown in Fig. 2. We see that for some systems, after the initial drop in from its clean limit , the critical temperature saturates and stays finite in the limit . The reason is that the impurity scattering gradually averages the two gaps, and the closer they get to each other, the less effective the pair-breaking from the impurities is; thus the superconductivity can survive even in the extremely dirty regime (in that limit ). The second regime is also easy to understand – if the sign change between the gaps is necessary for the existence of superconductivity (i.e., if the repulsive interband pairing interactions dominate) then the averaging produced by impurities completely suppresses the order parameter. Note that although our results are broadly consistent with the ones obtained in Ref. Golubov, , our Eq. (4) somewhat disagrees with the dirty limit derived there, since in our expression the effective coupling constant is rather than
For the rest of this Letter we concentrate on systems with positive and repulsive interband pairing – as we will see, these are the systems with the most interesting phase diagram. We turn to the coefficient of the Josephson-like term , and its evolution with . The role of is to couple the gaps, guaranteeing that they appear simultaneously, and close to its sign fixes the relative phase of and . In the presence of impurity scattering it is
with . In the clean limit , , and, as a result, is temperature independent, and can only be or . For finite , however, becomes function of both disorder strength and temperature, and can even change its sign. This has important consequences for the order parameter. Negative leads to the state in the clean limit. However, the second term in Eq. (5) is negative, and for strong disorder it can overcome the term. If is not completely suppressed (i.e., if the intraband pairing dominates), this sign change of means a transition from to state at the line Golubov (). This happens at temperature SuplMat (). At this point the bands are effectively decoupled, and one of them stays normal. At smaller disorder strength the system condenses in the state, while at larger disorder strength it goes into the state.
Below the critical line the quartic terms in the theory become important. Let us consider a system with slightly higher than (meaning that immediately below it is in the state). If is positive then is non-zero solely because of its coupling to through . In the vicinity of we can keep only the linear in terms in the equation (while keeping the cubic in terms), and at the side we get:
It is clear that equation defines a line in the space, originating from , and separating the from the regions. On this line the bands are decoupled and is zero. If, for a fixed , given system has slightly higher than , with decreasing the temperature it will cross the line, and will change its sign. We demonstrate this in Fig. 3. At this - transition point the second band becomes normal again (remember that we are assuming that is still positive). Note however, that neither of the gap parameters have any singularity at this point; in thermodynamic sense this is a crossover, rather than a real phase transition.
What happens if, with decreasing the temperature, the system gets close to the point before the - transition occurs? It can be easily shown that on the line the solution becomes unstable, and non-zero and purely imaginary appears when turns negative. Since is now a superconducting gap in its own right, we have to keep all cubic terms in the equations. More generally, apart from the always-present and solutions, can now take nontrivial values. From the condition we obtain for the equation:
This solution represents a distinct, intrinsically complex superconducting state. The physical picture behind it is simple; instead of changing the relative sign of the gaps by taking one of them through zero, there is alternative, more elegant way – continuous evolution of from to . This intermediate superconducting state can be understood as a linear combination (with complex coefficients) of the two “real” order parameters and . More physically, this means that the fluctuations in the densities of the two condensates (which are induced by fixing the phases) are not in-phase, as in , and not in anti-phase, as in the , but have some nontrivial time shift. One of the modes is lagging the other, and as a consequence the time-reversal symmetry is spontaneously broken (as it should in such intrinsically complex state). It is also easy to understand why such state appears at finite temperature below ; close to the critical line only the state exists. For the state to condense within the state has to turn negative, and only then the complex admixture of and becomes possible. This strongly suggests the necessary condition for the existence of such complex state – the presence of two attractive superconducting channels at the same temperature (which means that has to be positive).
By minimizing the GL free energy, we demonstrate that this solution is indeed realized, as illustrated in Fig. 4. The order parameter starts as () at the critical temperature. However, at some finite temperature below deviates from the solution, and superconducting state is no longer pure , but an intrinsically complex state. According to our model, the time-reversal symmetry breaking state is separated from the both “real” order parameters (which preserve the symmetry) by lines of continuous phase transitions.
Similar complex states in one-band systems ( states)sid1 (); sid2 (); sid3 (); sid4 () and in three-band systems ( states) Agterberg (); VS (); Tanaka2 (); Hu (); Dias (); Orlova (); Maiti (); Babaev2 () have attracted recently a lot of attention. There are some similarities in the underlying physics between these states and state discussed here. As in the case, in our model the complex state appears as a way of avoiding the appearance of non-superconducting parts of the Fermi surface (either the nodes of the -wave state, or an entire band in our model). The similarity with the three-band model is that in both cases the complex order parameter admixes two superconducting states in the trivial representation. Our impurity-induced complex state is also somewhat similar to the surface complex state predicted in the case of strong interband reflection at the boundaryBobkovi ().
We summarize our findings in the phase diagram presented in Fig. 1. Strictly speaking, our results are valid only in the region of applicability of the extended GL theory. To observe the complex state in this region we had to keep and quite close. In the case they are not close the complex state is realized at temperatures significantly lower than and has to be treated within the full microscopic theory. Nevertheless, using analogy with the physics and the phase diagrams discussed in Refs. VS, ; Maiti, we make two conjectures: i) the state is present if the system has to crossover, even if it’s not observable in the GL region; ii) this state extends down to , without any significant modifications. Confirming or rejecting these conjecture is an important direction for future work.
What do our results imply for the iron-based superconductors? Recently a roughly universal complete suppression of was reported for several FeAs-122 compoundsKirshenbaum (). This suggests that these materials are in the state with strong interband pairing, and thus no complex state is expected there. On the other hand, substantial variations in the effects of different impurities in similar 122 systems were observed in Ref. Li, . Also a very recent study of suppression in iron chalcogenidesInabe () showed a non-universal behavior; with some of the curves showing which initially decreases, but eventually saturates, as expected for the to transition. Although more studies are needed, it is already clear that these materials are surprisingly diverse in their normal and superconducting state properties, so it is entirely possible that the state can be induced by impurities (for example, by systematically irradiating a sample) in some of them.
In conclusion, we studied the role of impurities in a two-band superconductor. We derived a Ginzburg-Landau theory to describe the system, and we showed that the interband impurity scattering has a significant impact on the theory. Due to the impurities-induced term in the theory a complex order parameter may appear between the and states.
This work was supported by by UChicago Argonne, LLC, operator of Argonne National Laboratory, a U.S. Department of Energy Office of Science laboratory, operated under contract No. DE-AC02-06CH11357, and by the Center for Emergent Superconductivity, a DOE Energy Frontier Research Center, Grant No. DE-AC0298CH1088.
Appendix A Supplemental Material
We start our derivation of the GL free energy from the Usadel equations for the quasiclassical Greens functions and Gurevich (). We only study uniform states so these functions reduce to and . In the two-band case the equations have the form:
where are the band indices and is implied. Notice that we are treating the impurities in the Born approximation. We do not expect going beyond that approximation to qualitatively change our result.
These equations have to to supplemented by the self-consistency equations for the gap parameters and :
and normalization condition
To derive the GL equations we solve Eqs. (8) for and , and expand the solutions in powers of and . To do this we also have to expand ’s:
where is the zero-th order approximation:
Next order corrections are unwieldy, but straightforward to obtain. For we get:
Inserting , we get an expression for which is of order . If we define
the self-consistency equations give:
with . Expressing via and , we get two equation for the two gap parameters up to . They are identical to the equations obtained by varying the GL free energy with respect to . Collecting all the terms, multiplying by the density of states , and using the notation introduced in the main text, we get:
The coefficients are defined as follows:
The sums for all coefficients can be carried out, and closed-form analytic results can be obtained. Unfortunately, these results are complicated combinations of polygamma functions (digamma function and its derivatives), and since they do not provide any further insight into the problem, we will not show them.
For a fixed coupling constants matrix and disorder strength all coefficients are functions of temperature. The sign change of and drives the superconducting transition, and the sign change of drives the -to- crossover. Close to the quartic coefficients are only weakly temperature dependent, and, with the exception of , are all positive. In addition, tends to be the smallest.
As emphasized in the main text, in the limit all quartic coefficients that couple and vanish, and we recover the clean two-band GL theory. For non-zero , however, we have to use the full free energy .
with for . These equations can be represented in the form of the matrix equation used in the main text,
where the matrix is given by
Here we have used the relation , and defined , with
The quantity can be expresses via the digamma function as with .
Eq. (14) can also be used to derive an analytic formula for in the extreme dirty limit. We rewrite this equation in somewhat different form, more convenient for analytical analysis. The sum can be represented as , where is the largest eigenvalue of , which determines the clean-limit transition temperature, . This allows us to represent the matrix as
Multiplying both sides of the matrix equation (14) with and using , we obtain
Introducing notation , where is the Kronecker delta, we can cast this in an equivalent form:
General equation for is determined by vanishing of the determinant for this linear system which gives
In the dirty limit, , we can use the asympotics of , with . In this case we obtain from Eq. (16)
In the extreme dirty case corresponding to condition , we obtain for the limiting value of transition temperature, ,
This result actually can be obtained directly from Eq. (15) if we take in the left-hand side. Using the definition of , the above result for can be rewritten in somewhat more transparent form
The quantity in the right-hand side represents the band-average of the inverse coupling constant .
Now let us derive the formula for shown in the main text. Remember that is defined as the point at which coefficient vanishes. This happenes at:
Using this condition in the general formula given above we get:
and combining this with the clean limit expression gives the formula in the main text
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