Distinguishing between s+id and s+is pairing symmetries in multiband superconductors through spontaneous magnetization pattern induced by a defect

Distinguishing between and pairing symmetries in multiband superconductors through spontaneous magnetization pattern induced by a defect

Shi-Zeng Lin Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA    Saurabh Maiti Department of Physics, University of Florida, Gainesville, FL-32611, USA    Andrey Chubukov Department of Physics, University of Minnesota, Minneapolis, Minnesota 55455, USA
September 23, 2019

The symmetry of the pairing state in iron pnictide superconductor is still controversial. At optimal doping (), it is very likely -wave, but for there are experimental and theoretical arguments for both -wave and -wave. Depending on the choice for , intermediate and states have been proposed for intermediate doping . In both states, the time reversal symmetry is broken and a spontaneous magnetization is allowed. In this work we study a spontaneous magnetization induced by a nonmagnetic defect in the and states by using a perturbation theory and numerical calculations for the Ginzburg-Landau free energy functional. We show that the angular dependence of the magnetization is distinct in these two states due to the difference in symmetry properties of the order parameters. Our results indicate a possible way to distinguish between the and pairing symmetries in multi-band superconductors.

74.20.Rp, 74.20.Mn, 74.25.Jb, 76.75.+i

I Introduction

The knowledge of the pairing symmetry of an unconventional superconductor is the first step to elucidate the pairing mechanism. In iron pnictide and iron-chalcogenide superconductors (FeSCs) the identification of the pairing symmetry is complicated by the following reasons. First, the family of iron-based superconductors is large, and it is not certain that the pairing symmetry is the same in all FeSCs. The two major candidates are Mazin et al. (2008); Kuroki et al. (2008); Parker et al. (2008); Chubukov et al. (2008); Seo et al. (2008) and Kuroki et al. (2008); Seo et al. (2008); Chen et al. (2009); Platt C. and R. (2013) states. At optimal doping state is favorable, but at larger hole doping several theoretical work suggested that and are almost degenerate Graser et al. (2009); Maiti et al. (2011); Platt C. and R. (2013). Second, FeSCs are multi-band superconductors with several hole and electron Fermi pockets. The phase of gap changes by between hole and electron pockets and the phase of gap changes by between electron pockets. In general, however, multi-band superconductors allow more complex superconducting states with the phase differences of the gaps on different Fermi pockets as fractions of .

These more complex superconducting states, even with pure wave symmetry, can be understood by noticing that the pairing symmetry originates from an interband repulsive interaction between superconducting condensates on hole and electron pockets. The intra-pocket repulsion favors a phase shift in the superconducting order parameter. When three or more bands are present, the inter-band repulsion leads to frustration, which is resolved by choosing the phases of the gaps to maximize superconducting condensation energy. For example, for three identical pockets with equal repulsive inter-band interaction, the best outcome is the difference between the phases of the three gaps, much like spin configuration resolves frustration in an XY antiferromagnet on a triangular lattice. And just like there, the phase change can go in ingredients clockwise or anticlockwise. The choice breaks the time reversal symmetry in addition to the overall phase symmetry. Agterberg et al. (1999); Stanev and Tes̆anović (2010); Hu and Wang (2012); Bojesen et al. (2013) In a generic state of this kind, the order parameter made out of at a given Fermi pocket, and its complex conjugate are not related by a global phase rotation, i.e., . Below we label such -wave state as .

The transition from a time-symmetry preserving -wave state (e.g., ) to an state is a continuous phase transition. Near the transition, a collective out-of-phase oscillation of (a Leggett mode) becomes soft. Lin and Hu (2012a); Stanev (2012); Lin (2014) Inside an state, the breaking of a discrete time reversal symmetry gives rise to a new kind of a phase soliton between the superconducting domains and Garaud et al. (2011); Lin and Hu (2012b); Garaud and Babaev (2014) It has been proposed that one may detect the pairing symmetry based these properties. Lin and Hu (2012a, b); Garaud and Babaev (2014); Marciani et al. (2013); Burnell et al. (2010); Maiti and Chubukov (2013)

FeSCs are promising candidates for an state. The material is particularly interesting in this regard. It is magnetic at small and superconducting at larger . Its electronic structure for not close to consists of three hole pockets and two electron pockets. Near optimal doping, it is very likely that superconducting interaction is mediated by spin fluctuations. The magnetic order has the momentum equal to the distance between the centra of hole and of electron pockets, and fluctuations of this order clearly favor state without breaking the time-reversal symmetry and phase shift between the gaps on hole and electron pockets. This is consistent with angle-resolved photoemission spectroscopy (ARPES) experiments, which near found a fully gapped superconducting state with little variation of the gaps along the pockets. Ding et al. (2008); Nakayama et al. (2011); Khasanov et al. (2009); Luo et al. (2009); Christianson et al. (2008). At , however, the situation is different: electron pockets disappear and only the hole pockets remain, according to the ARPES measurements. Sato et al. (2009); Okazaki et al. (2012) If the pairing symmetry remains -wave all the way to , as some ARPES experiments suggested based on the measurements on the gaps on hole pockets Okazaki et al. (2012), then the gap must change sign between inner and middle hole pockets, where the gaps are the largest Okazaki et al. (2012). The sign-changing -wave gap structure was found in random phase approximation based theoretical studies for  Maiti et al. (2011). If one analyses how to connect state near optimal doping, with equal sign of the gaps on the hole pockets, and -wave state at with opposite signs of the gaps on the two smallest hole pockets, one finds Maiti and Chubukov (2013); Marciani et al. (2013) that the evolution is continuous near the line, with the gap on one of hole pockets vanishing and re-appearing with a different sign as approaches 1, but necessary involves an intermediate state at , when it is easier to change the phase of the order parameter rather than its amplitude. In this intermediate state, the phases of the gaps on the two smallest hole pockets differ by a fraction of , i.e., the intermediate state is a realization of superconductivity.

Another suggestion, based on measurements of thermal conductivity and resistivity  Reid et al. (2012a, b); Tafti et al. (2013, 2014) and theoretical calculations using functional renormalization group Platt C. and R. (2013), is that the pairing symmetry at is a -wave (). This is also generally consistent with random phase approximation based calculations Maiti et al. (2011), which found that -wave and couplings are almost degenerate. If so, the system must evolve from an superconductor at to a wave superconductor at . Calculations show Thomale et al. (2011); Platt et al. (2009) that this evolution goes via an intermediate phase in which both -wave and -wave components are present, and the phase shift between the two is , i.e, the state is . This is another state which breaks time-reversal pairing symmetry.

In this work we discuss whether it is possible to distinguish between and states in experiments. Both states break time reversal symmetry and allow a spontaneous magnetization to develop. In a homogenous system, magnetization does not develop because a spontaneous current circulates in the band space and is not coupled to a gauge field. However, in the presence of nonmagnetic defects, a spontaneous current can be induced around the defect in superconductors with either or pairing symmetries Lee et al. (2009); Lin and Hu (2012b); Maiti et al. (2015); Garaud and Babaev (2014). Because the state breaks the lattice rotation symmetry, while the preserves it, the profile of the induced spontaneous magnetization are different for and states. This suggests a possible way to experimentally distinguish between the two pairing symmetries.

Below we report the results of our study on the spontaneous magnetization in and superconductors induced by nonmagnetic defects with different shapes. We extend the previous work Lee et al. (2009); Maiti et al. (2015) by developing a self consistent treatment for the magnetization based on a phenomenological Ginzburg-Landau free energy functional. Our results show that one can differentiate between the and states by measuring the magnetization pattern induced by nonmagnetic defects. We first present the symmetry argument to obtain the magnetization profile. We then present perturbative calculations of induced magnetization for a weak defect potential. We next compare the perturbative calculations with the numerical results obtained by minimizing the Ginzburg-Landau free energy functional. Finally, we compare our approach and the results to those in previous work.

Ii Model and perturbative calculations

ii.1 Symmetry analysis

Before going into detailed calculations, let us first perform a symmetry analysis. The state has a full rotational symmetry. For a circular defect, if a supercurrent was induced, it could only flow inward or outward, as sketched in Fig. 1 (a). This would violate the current conservation , hence no supercurrent (and no spontaneous magnetization) is allowed in this case. Consider next a square defect. Because and directions are equivalent, to conserve the supercurrent, the direction of the supercurrent along zone diagonals should be opposite to that in the and the directions. That is if the current flows inward in the and the directions, the current must flow outward in the diagonal direction, and vice versa. As a result, the induced magnetization, measured as a function of the angle with respect to, say, direction, must display a four-fold oscillation [Fig. 1 (b)].

The state breaks the rotation symmetry and is invariant under the combination of the rotation and time reversal operation. In this situation, a supercurrent and a spontaneous magnetization do emerge, even if a defect is circular. For a circular or square defect, if the current in the direction flows inward, then the current in the direction must flow outward, and vice versa. The current conservation at the center of a defect is satisfied automatically. The magnetization pattern, generated by a supercurrent, displays a two-fold oscillation as a function of an angle [Fig. 1 (c) and (d)]. This simple analysis shows that one can indeed differentiate between an pairing state and an pairing state by analyzing the pattern of a spontaneous magnetization induced around a defect.

Figure 1: (color online) Schematic view of the spontaneous supercurrent and magnetization induced by a circular and square defect in the and states based on the symmetry analysis. The arrows are the direction of supercurrent and denotes the direction of the magnetization field perpendicular to the superconducting plane.

ii.2 Ginzburg-Landau free energy

We next calculate a spontaneous magnetization induced by a defect using Ginzburg-Landau theory in two dimensions. We consider and states separately.

ii.2.1 state

Like we said, an state emerges when there are three (or more) Fermi pockets, due to frustration when inter-pocket interactions are repulsive. Stanev and Tes̆anović (2010); Marciani et al. (2013); Maiti and Chubukov (2013) However, recent work Garaud et al. (2016) has demonstrated that one can simplify the analysis of state by reducing the three-pocket model to an effective two-pocket model in which time reversal symmetry breaking is explicitly imposed. In this approach, which we follow, the Ginzburg-Landau free energy functional for an state is, up to terms of quartic order in :


where the free energy density for each component is ()


and the coupling between and is described by


where and is the flux quantum. Because of competition between bilinear and bi-quadratic terms (the and the terms), the phase difference between and can be any value and the resulting state generally can be termed as state [for analogous consideration in coexistence state of superconductivity and magnetism, see Ref. Hinojosa et al., 2014]. To stabilize the state, we set and choose the phase shift between and to be . The coupling between the bands at the bilinear level is via the and terms. The term may be safely set to zero, as in any case it can be eliminated by an appropriate rotation in space. This procedure changes the values of and , but does not introduce new terms. For case the bilinear terms are not allowed by symmetry.

A nonmagnetic defect is modeled by changing . A defect is considered as weak if the defect potential . The variation of the order parameter , where is the order parameters in the absence of defects, can be found from the minimization of the Ginzburg-Landau functional to linear order in . The minimization obviously yields , . In a state with broken time-reversal symmetry an amplitude fluctuation and a phase fluctuation are coupled. As the consequence, a vector potential also fluctuates. Fluctuations of are gapped by Anderson-Higgs mechanism with the gap larger than that of collective excitations of and , at least near the onset of a state which breaks time reversal symmetry. Lin and Hu (2012a); Stanev (2012) To calculate and to the linear order in , we fix the gauge by choosing the global phases such that when . The resulting equations for and are presented in the Appendix. We assume that the defect potential has the angular dependence in the form and consider different integer . The angular dependence of and follows that of .

Minimizing next with respect to , we obtain the Ampere’s law


with the supercurrent


The condition for current conservation, , is satisfied automatically by Eq. (4). Taking of both sides of Eq. (4) and using , which is valid as long as no vortices are present, we obtain


where is the London penetration depth (up to corrections of order ). The induced magnetization is of second order in the defect potential and it is directed perpendicular the 2D superconductor plane.

For a model of a superconductor with identical and , , and , the magnetization is absent because the right-hand side of Eq. (II.2.1) vanishes. In a more generic case, using and , we obtain the magnetization field






The unit vector is along the direction, and is the Bessel function of the first kind. We see that a nonmagnetic defect with an -fold angular variation induces a magnetization which displays a -fold oscillation: . For a centrosymmetric defect potential , the induced magnetic field vanishes.

ii.2.2 state

We proceed to study the magnetization pattern from a defect in an state. Let’s denote an -wave component by and a -wave component by . The Ginzburg-Landau free energy functional is still given by Eqs. (1) and (2), but the coupling term is different and is given by Lee et al. (2009)


For the state, is invariant under a rotation of a reference frame by and subsequent the time reversal operation, as both operations change the sign of a -wave component . The term in Eq. (II.2.1) is not invariant under the change of the sign of and is not allowed. Note also that the mixed gradient terms in the and the directions must have different signs. We show below that, because of the mixed gradient term in Eq. (II.2.2), the magnetization is nonzero already to linear order in a defect potential.

Figure 2: (color online) Spontaneous magnetization induced by a weak defect with -fold angular dependence in the and states, obtained by numerical minimization of the Ginzburg-Landau free energy functional. The magnetization only has the component perpendicular to the superconducting plane. Here , , , and . We consider a weak defect with and .
Figure 3: (color online) Maximal magnetization field induced by a square defect in the and states. Symbols are numerical results and lines are guide to eyes. The parameters are the same as those in Fig. 2 but with a different strength of the defect potential.
Figure 4: (color online) Spontaneous magnetization induced by a square defect in the and states. The magnetization only has the component perpendicular to the superconducting plane. Here , , , and . The strength of the defect potential is and .

To linear order in and , the supercurrent is


where and are unit vectors in and directions, respectively. The magnetic field is determined by


We see that scales linearly with and, hence, with the defect potential . For , the magnetic field behaves as


The angular dependence is obviously different from that in an state [see Eq. (7)]. The right-hand side of Eq. (12) is presented in explicit form in Eq. (A), and the functions are obtained by substituting (A) into (12) and solving the differential equation.

Iii Numerical calculations

To complement the analytical analysis we now perform numerical minimization of using time-dependent Ginzburg-Landau equations


where is the diffusion constant, is the normal state conductivity, and is the electric potential.

We model a defect at a point by setting for and otherwise. The results of the calculation of the induced magnetization for various are displayed in Fig. 2. For , there is no spontaneous magnetization in an state, while in a state the magnetization is non-zero and displays a two-fold angular variation. For , the magnetization in an state displays a -fold variation, consistent with the result of the linear response theory. In an state the induced magnetization predominantly displays the -fold variation for for and the -fold variation for .

In Fig. 3 we show the amplitude of the induced magnetization as a function of the strength of the defect potential . The amplitude increases linearly with for an state and quadratically in an state, in full agreement with the analytical results.

We also analyzed a square defect with if and otherwise (see Fig. 4). The induced magnetization emerges at the corners of the square defect. It has dominant 2-fold (4-fold) angular dependence for an () state. The magnetization pattern in an state has rotation symmetry, while in an state it is invariant under the combination of rotation and time reversal operation.

Iv Relation to previous work

Let us compare our results to the previous work. For a circular defect, our results are consistent with those in Ref. Lee et al., 2009 for an state and in Ref. Maiti et al., 2015 for an state. In Ref. Maiti et al., 2015 it was argued that, to linear order in defect potential, for an superconductor with isotropic impurities, no spontaneous supercurrent (and thus no induced magnetization) appears unless the symmetry and the time reversal symmetry are both broken. In the present work we extend that result and show that an induced magnetization can also appear in an superconductor even if rotation symmetry is preserved: this requires the defect to break rotational symmetry and perturbative calculations to second order in the defect potential. If the rotation symmetry is explicitly broken by a defect, an induced magnetization appears already at the first order in a defect potential.Maiti et al. (2015)

V Summary

To summarize, in this paper we have studied the emergence of a spontaneous magnetization induced by a nonmagnetic defect in a superconductor with either an or an pairing symmetry. We found that the angular dependence of a magnetization depends on the shape of a defect potential and is different for and states. For a weak defect, an induced magnetization for an state is linear in defect potential, while for an state it is quadratic. For a defect with an -fold angular variation, the magnetization displays a -fold angular variation for an state and -fold angular variation for an state. Our results show that and pairing symmetries can be distinguished experimentally by measuring angular dependences of the magnetization patterns.

The induced magnetization can be measured by imaging method, such as magnetic force microscope and superconducting quantum interference device, and muon-spin relaxation experiments. The shapes of the defects can be controlled by focused ion beam milling. For atomic defects, a superconducting order parameter in a spatially isotropic state is suppressed in a circular region with a radius of the order of superconducting coherence length. As a result, no spontaneous magnetization is induced by atomic defects in an state. Because of this, we believe that the fact that a spontaneous magnetization has not been detected in zero-field muon-spin relaxation studies of polycrystalline samples of with in Ref. Mahyari et al., 2014 does not actually rule out pairing symmetry. We hope that future experiments with controlled shape of the defects will be able to resolve the issue whether superconductivity in breaks time-reversal symmetry near and, if yes, whether the superconducting state is or .

The authors are indebted to James Sauls and Filip Ronning for helpful discussions. The work by SZL was carried out under the auspices of the U.S. DOE contract No. DE-AC52-06NA25396 through the LDRD program. The work by AVC was supported by the Office of Basic Energy Sciences, U.S. Department of Energy, under award DE-SC0014402. AVC acknowledges with thanks the hospitality of the Center for Non-linear Studies, LANL, where he was 2015-2016 Ulam Scholar.

Appendix A Calculations of the order parameters

Here we calculate the variations of the superconducting order parameters in the presence of a weak defect. For the state, the Ginzburg-Landau equations after minimizing in Eq. (1) with respect to is


where we have set for simplicity. Here we require to stabilize the state. We have also neglected the variation of the gauge field by setting because the fluctuations of have the gap of the superconducting gap, while the fluctuations of have a smaller gap in the vicinity of the time reversal symmetry breaking transition. Lin and Hu (2012a); Stanev (2012) For a weak defect potential with , the change of the order parameter is


and similarly for the second component. By separating the real and imaginary parts, we obtain four linear equations for and . We assume in the polar coordinate. The Fourier transform is , where is the Bessel function of the first kind. In the momentum space, we need to replace in Eq. (A). By solving Eq. (A) we find and . By taking the inverse Fourier transform, we obtain and similarly for . For identical two band superconductors with , , and , we have


The spontaneous magnetization vanishes according to Eq. (II.2.1). Therefore magnetization appears only in asymmetric two band superconductors.

For the state, we need to replace in the mixed gradient term in Eqs. (16), (17) and (A). In the limit , we have . For , and , but , we obtain


Denote , we have


From Eq. (12), we obtain the angular dependence of in Eq. (13).


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