Phase solitons in multi-band superconductors with and without time-reversal symmetry

The Josephson-like interband couplings in multi-band superconductivity exhibit degenerate energy minima, which support states with kinks in phase of superconductivity. When the interband couplings in systems of three or more components are frustrated, the time-reversal symmetry (TRS) can be broken, which generates another type of phase kink between the two time-reversal-symmetry breaking (TRSB) pair states. In this work, we focus on these novel states of phase kinks, and investigate their stability, similarity, differences and physical consequences. Main results are summarized as follows: (1) We find a new type of phase slip when the kink becomes unstable. (2) In the kink region, TRS is broken and spontaneous magnetic fields are induced. (3) In superconductors with TRSB, composite topological excitations associated with variations of both superconductivity phase and amplitude can be created by local perturbations, or due to proximity effect between normal metals.

Introduction -It has been known for long time that superconductors with different pairing symmetries in contact with one another can form stable domain structures [1,2]. Properties of domain walls are governed by the pairing symmetries in the domains, thus these heterogeneous systems become vital in understanding the pair symmetry. Meanwhile, there are growing evidences that superconductors may break discrete symmetries in addition to the U(1) (local) gauge symmetry whose loss defines superconductivity [3,4]. Examples include TRSB in some unconventional superconductors [5,6], which results in unusual phenomena such as the appearance of magnetic flux when the superconductivity is perturbed by nonmagnetic impurities [7]. These superconductors can also form stable domain walls between domains of distinct symmetry-breaking states.
The discovery of MgB 2 [8] and iron-pnictide superconductors [9] has opened intense and exciting discussions of multi-band superconductivity in condensed matter physics. In these systems, superconductivity in one band is coupled through interband Josephson coupling to that in another band γ i j ∆ i ∆ j cos(φ i − φ j ), with φ i and ∆ i being the superconductivity phase and amplitude in the i-th band respectively. This gives rise a collective oscillation of the superconductivity phases, known as the Leggett mode [10].
It is interesting to observe that the interband coupling has degenerate energy minima φ i − φ j = 2nπ for γ i j < 0, which supports various topological excitations in the form of phase kinks belonging to the homotopy class π 0 (S 0 ), whereas the well known vortex solution in type II superconductors belongs to the homotopy class π 1 (S 1 ). The existence of the kink solution was first discussed by Tanaka [11] for two-component superconductors, and later it was discussed that phase kinks can be excited in nonequilibrium processes such as current injection [12]. The phase kinks have been observed experimentally in layered aluminium mesocopic rings with two order parameters [13].
In the presence of frustrated interband couplings in super-conductors with three or more components, the system may break the TRS [14][15][16][17][18]. In the TRSB state,Ψ e iθΨ * for any phase θ withΨ ≡ (Ψ 1 , Ψ 2 , ..., Ψ n ) a vector of the complex order parameters. A phase kink may appear between two degenerate statesΨ andΨ * . One thus sees that multi-band superconductors support two types of kink solutions of different origins. In a recent paper by Garaud et. al. [19], composite topological excitations associated with phase kink and vortex in superconductors with TRSB have been found numerically. The stability of these phase kinks however still remains to be investigated.
In the present work, we investigate the stability, similarity, differences and physical consequences of these two types of kink solutions. In superconductors with TRS in bulk, the phase kink breaks TRS at the domain wall, and induces local magnetic flux. Upon elevation of temperatures, kinks become unstable as a consequence of increasing coherence length. At the instability, a phase slippage occurs accompanying a voltage pulse. Contrarily, kinks between TRSB pair states remain stable even with the increasing coherence length. Moreover, in superconductors with TRSB, various types of composite topological excitations associated with the variation of superconductivity phase and amplitude can be created by perturbing the superconductivity locally, such as heating and/or nonmagnetic impurities, and at interface to a normal metal.
Kink solutions -We start from the standard multi-band Ginzburg-Landau (GL) theory with Josephson-like interband couplings [20,21], which is adequate for discussions on physics addressed here where symbols are conventionally defined [22]. Throughout the paper, we use the units = 2e = c = 1. γ l j for l j is the arXiv:1111.3850v2 [cond-mat.supr-con] 16 Jun 2012 interband coupling, which can be either repulsive or attractive depending on the strengths of the Coulomb and electronphonon interactions. The interband repulsion may cause frustration of the superconductivity in different bands and results in TRSB [14][15][16][17][18]. For MgB 2 , the interband coupling is commonly accepted as attractive γ 12 < 0, while for iron-pnictide superconductors, there are growing evidences that some of γ i j are positive and the system favors s± pair symmetry. [23,24] The kinetic energy in Eq. (1) can be rewritten as where ξ j = 1/2m j |a j | and H c j are coherence length and thermodynamic critical field in respective single-band condensates (γ l j = 0). When width of the kink λ k (derived below) is much larger than ξ j , λ k ξ j , the suppression of the amplitude of the order parameters by the phase kink is weak, and the order parameter is approximately constant in space. In this case, we can concentrate on the phase variables of the order parameters.
First we consider phase kink between domains with TRS in one dimension, where we can take the gauge A = 0. The minimal model for this domain structure is of two bands. Since the sign of γ 12 can be gauged away in this case, we consider γ 12 < 0 without loss of generality. We also assume an identical amplitude of order parameter ∆ i = ∆ for simplicity. The variation of the phase difference φ 12 ≡ φ 1 − φ 2 is described by the sine-Gordon equation [11] ∂ 2 x φ 12 + 2γ 12 (m 1 + m 2 ) sin φ 12 = 0, and . The width of the kink is λ k = 1/ −2γ 12 (m 1 + m 2 ), which is temperature independent. The condition that λ k ξ j then becomes |γ l j | α j . A typical phase kink is shown in Fig. 1 (middle). The TRS is broken at the domain wall while reserved in the domains. There are finite phase differences between the right and left domains in both components. Now we consider the kink solution of a superconductor of three or more components with frustrated interband couplings, where TRS is broken in bulk. As a minimal model, we treat a superconductor with three equivalent bands α j = α, γ i j = γ > 0, and m i = m. The two degenerate ground stateŝ Fig.1 (right). For constant amplitudes of order parameters at γ α, the phase kink is described by The potential associated with Eq. (3) V p = cos φ 12 + cos(2φ 12 )/2 has many degenerate minima φ 12,m = ±2π/3 + 2nπ. One can construct kink solution between any pair of the energy minima. Their stability and magnetic response are qualitatively the same. To be specific, we only consider the following kink solution, which can be found analytically using the Bogomolny inequality [25] and the associated energy is In one dimension (1D), there is no supercurrent in the domain wall due to the current conservation ∂ x J s = 0. In higher dimensions, supercurrent and the associated magnetic field are induced at the domain wall as a result of TRSB. We consider a closed domain wall described by either Eq. (2) or Eq. (4) in a 2D superconductor. To investigate the dynamic evolution of the domain wall, we solve the time-dependent GL equation(TDGL) numerically [26] with D j the diffusion constant, σ the normal conductivity, and Φ the electric potential. In simulations, we prepare a closed domain wall with square or rectangular shapes as initial conditions. In order to minimize its energy, the domain wall organizes itself into a circular shape irrespective to its initial shape during the time evolution in simulations. Magnetic fields appear spontaneously at the domain wall with alternating directions, as shown in Fig. 2 (a) and (b). As revealed by numerical simulations, for phase kinks in superconductors with TRS [see Eq. (2)], the induced magnetic field changes polarization in both radial and azimuthal directions as shown in Fig. 2 (a). While for kinks between TRSB pair states [see Eq. (4)], the magnetic field changes polarization only in the azimuthal direction as shown in Fig. 2 (b). One may treat the domain wall at the left semicircle as a phase kink, then the domain wall at the right semicircle is an anti-kink. They attract each other, which causes the whole circular domain wall collapsing, and renders a uniform state. Since the attraction between two domain walls becomes exponentially weak at a large separation, the life time of the domain walls increases with the size of the domain enclosed. This allows for possible experimental detections on the induced magnetic flux after quenching when domain walls are excited by chance.
Stability of the Kink Solution-We proceed to investigate the stability of the kink solution in Eq. (2) taking into account the suppression of amplitude of order parameter by the phase kink. The magnitude of the suppression depends on the ratio of the kink width λ k to the coherence length ξ as briefly mentioned above. As the coherence length increases when temperature is elevated while the width of kink remains almost unchanged, the superconductivity in the domain wall will be greatly depleted. At a threshold value, the phase kink loses its stability, and system evolves into a uniform state. There is a voltage pulse associated with varying magnetic field across the domain wall which is experimentally detectable. This process is a new type of phase slip, different from that in singleband superconductors carrying supercurrent close to the critical one, with the latter one caused by fluctuations. [22] We explicitly consider a superconductor of two identical bands with a phase kink localized at the center of a superconducting wire. We solve numerically the TDGL equations and derive the stable configuration of the superconductivity phase as temperature (namely α) varies. When temperature increases, the amplitude of the superconductivity at the domain wall decreases as depicted in Fig. 3(a). At a threshold α for given value of γ 12 [symbols in Fig. 3(b)], the phase kink becomes unstable and the system evolves into a uniform state, during which a voltage pulse appears. Therefore, the phase kinks in superconductors with TRS are stable only for weak interband couplings.
A superconducting wire with a phase kink can be alternatively considered as a Josephson junction since the superconductivity is suppressed at the domain wall. In the ground state, the phase difference between two domains is finite, thus it is a realization of φ-junction [27], or π-junction [28] if the two bands are identical. When current is injected into the wire, the phase kink is deformed due to the phase gradient created by the injected current. At a threshold current, the phase kink becomes unstable, and the system evolves into a uniform superconducting state, during which a voltage pulse appears, similar to the case with increasing temperature. The threshold current is still much smaller than the depairing current of the uniform state. One may regard the threshold current as a critical current for the present Josephson junction.
We perform numerical calculations on the critical current of a superconducting wire with a phase kink, introducing supercurrent into the system by twisting the phases at the two edges of the wire far away from the phase kink. The kink structure is deformed into the shape depicted in Fig. 3(c) by the current injection. The critical current for the kink state decreases with |γ 12 | as shown in Fig. 3(d), since the kink state gradually loses its stability when |γ 12 | increases as discussed above. At the critical current, we observe a phase slip with a voltage pulse, and finally the system reaches a uniform state.
The phase kink in Eq. (4) between two TRSB pair states of a three-component superconductor is stable because the system takes different states in the left and right domains, and one cannot transform one state to the other by adjusting superconductivity at the domain wall. This kink is thus protected by symmetry and is very different from those with bulk TRS as in Eq. (2), where the states in the left and right domains are essentially the same except for a common phase factor, as shown in Fig. 1 (middle), and the system evolves into a uniform state by rotating the phase of domains globally at the instability of the kinks.
Consequences of TRSB -We have shown that spontaneous magnetic flux and voltage pulse appear during the nonequilibrium evolution of superconductivity phase when a phase kink becomes unstable. Here we discuss possible experimental observations on stable phase kinks at equilibrium. In the presence of phase kink, the TRS is violated at the domain wall, namely φ 1 − φ 2 0 or π. The variation of the superconductivity amplitude is coupled with that of superconductivity phase, which can be checked by expanding the interband coupling term γ i j ∆ i ∆ j cos(φ i − φ j ) to the quadratic order of phase difference. When the superconductivity is suppressed locally by nonmagnetic impurities, proximity effect at sample edge or heating, variations in superconductivity phases are induced, which in turn excites supercurrent and magnetic flux, as confirmed numerically. We study the proximity effect between a superconducting strip and a normal metal when a phase kink is present in the superconductor. In order to describe the proximity effect correctly, a boundary condition between a multi-band superconductor and a normal metals should be formulated. The boundary condition in terms of the Usadel equation has been derived in Ref. [29]. In the framework of phenomenological GL theory, the boundary condition for a single-band superconductor can be generalized straightforwardly to a multi-band one [22] where the off-diagonal coefficient p jk with j k accounts for the interband coupling while the diagonal coefficient j = k represents suppression of superconductivity as a consequence of the leakage of Cooper pairs at the interface. We minimize the GL energy numerically, and the results are presented in Fig. 4 (a). Spontaneous magnetic field is induced at the interface between the normal metal and superconductor at the position of the domain wall, which is strong enough (in Fig.  4(a), H ∼ 10 −5 H c2 ) to be measured experimentally by scanning SQUID, Hall, or magnetic force microscopy. The magnetic field has opposite signs at the two interfaces, leaving a zero integration over the sample.
In TRSB superconductors, stable domain walls associated with the variation of superconductivity phase and amplitude can be created by local perturbations, because the phase is coupled with the amplitude when TRS is violated. In Fig. 4(b), we consider the proximity effect between a threeband superconductor with TRSB and a normal metal. Magnetic fluxes appear at the corners of the superconductor, as- sociated with sharp changes of phase gradients. In Fig. 4(c), we introduce an impurity by modifying α i locally. We see that magnetic flux is induced around the impurity. For superconductors with TRS, no magnetic filed can be induced by the proximity effect or impurities, which implies a possible way to detect the TRSB in experiment.
Discussions -In multi-band superconductors with Josephsonlike interband coupling, the phase kinks as topological excitations can exist because of the multiple degenerate energy minima associated with the interband Josephson coupling. The phase kink suppresses the superconductivity nearby depending on the ratio of the width of the phase kink to the superconducting coherence length, which causes instability when the suppressed superconductivity is insufficient to maintain the phase coherence for the phase kink. The existence of the phase kink does not require breaking of additional symmetry besides U(1). Instead, the presence of phase kink violates the TRS locally. When bulk multi-band superconductors break TRS, a new type of phase kink can be formed between the two TRSB pair states. The topological solutions (phase kinks) in multi-band superconductors discussed in the present work are different from the topological superconductors realized in materials with strong spin-orbit couplings.
In 1D, both phase kink in a superconductor with TRS and that between TRSB pair states are stable. In 2D, because of the attraction between opposite kinks, the domain wall collapses and the system reaches a uniform state. This is in accordance with the Derrick's theorem [30], i.e. for an infinite system, the kink state is only stable in 1D. Kinks can be pinned by the pinning centers where the superfluid densi-ties are small, since the loss of superconductivity condensation energy can be reduced by adapting the domain wall to the pinning centers. This may prevent the domain wall from collapsing and stabilize the kink in 2D. The kink can also be stabilized when vortices are present as discussed by Garaud et al. [19]. Domain walls created by local heating or impurities in superconductors with TRSB are stable in 2D and 3D since they are enforced by external perturbations.
Let us discuss the realization of the phase kink in Eq. (2). In iron pnictide superconductors, interband scatterings are strong [31], and thus the phase kinks are unlikely realized. While for the well-known two-band superconductor MgB 2 and V 3 Si, it is revealed that interband scatterings are weak [32,33], which may allow for the excitation of stable phase kinks in low-temperature region. While for the realization of kink in Eq. (4), we need multi-band superconductors with TRSB. As discussed in Ref. [18], the TRSB state can be achieved by chemical doping in iron pnictide superconductors. Moreover, phase kinks can also be realized in hybrid structures with two superconducting films coupled with Josephson coupling. Recently, Vakaryuk et al. proposed to realize the phase kinks in superconductors with the s± pairing symmetry by exploiting the proximity effect to a conventional s-wave superconductor. [34]