Time-reversal Symmetry Breaking State Research Articles

Proximity induced band gap opening in topological-magnetic heterostructure (Ni80Fe20/p-TlBiSe2/p-Si) under ambient condition

The broken time reversal symmetry states may result in the opening of a band gap in TlBiSe2 leading to several interesting phenomena which are potentially relevant for spintronic applications. In this work, the quantum interference and magnetic proximity effects have been studied in Ni80Fe20/p-TlBiSe2/p-Si (Magnetic/TI) heterostructure using physical vapor deposition technique. Raman analysis shows the symmetry breaking with the appearance of A21u mode. The electrical characteristics are investigated under dark and illumination conditions in the absence as well as in the presence of a magnetic field. The outcomes of the examined device reveal excellent photo response in both forward and reverse bias regions. Interestingly, under a magnetic field, the device shows a reduction in electrical conductivity at ambient conditions due to the crossover of weak localization and separation of weak antilocalization, which are experimentally confirmed by magnetoresistance measurement. Further, the photo response has also been assessed by the transient absorption spectroscopy through analysis of charge transfer and carrier relaxation mechanisms. Our results can be beneficial for quantum computation and further study of topological insulator/ferromagnet heterostructure and topological material based spintronic devices due to high spin orbit coupling along with dissipationless conduction channels at the surface states.

Stability of Mixed-Symmetry Superconducting States with Broken Time-Reversal Symmetry against Lattice Distortions

We examine the stability of mixed-symmetry superconducting states with broken time-reversal symmetry in spatial-symmetry-broken systems, including chiral states, on the basis of the free-energy functional derived in the weak-coupling theory. We consider a generic a_1 + i a_2 wave state, with a_1 and a_2 being different symmetry indices such as (a_1,a_2) = (d,s), (p_x,p_y), and (d,d').The time-reversal symmetry of the mixed-symmetry state with the a_1- and a_2-wave components is broken when the phases of these components differ, and such a state is called the time-reversal-symmetry breaking (TRSB) state. However, their phases are equated by Cooper-pair scattering between these components if it occurs; i.e., when the off-diagonal elements S_{a_1 a_2} = S_{a_2 a_1} of the scattering matrix are nonzero, they destabilize the TRSB state. Hence, it has often been believed that the TRSB state is stable only in systems with a spatial symmetry that guarantees S_{a_1 a_2}=0. We note that, contrary to this belief, the TRSB state can remain stable in systems without the spatial symmetry when the relative phase shifts so that S_{a_1 a_2} = 0 is restored, which results in a distorted TRSB (a_1 + a_2) + i a_2 wave state. Here, note that the restoration of S_{a_1 a_2} = 0 does not imply that the symmetry of the quasi-particle energy E_k is recovered. This study shows that such stabilization of the TRSB state occurs when the distortion is sufficiently small and \Delta_{a_1} \Delta_{a_2} is sufficiently large, where \Delta_a is the amplitude of the a-wave component in the TRSB state in the absence of the distortion. We clarify the manner in which the shift in the relative phase eliminates S_{a_1 a_2} and prove that such a state yields a free-energy minimum. We also propose a formula for the upper bound of the degree of lattice distortion, below which the TRSB state can be stable.

Time-reversal symmetry-breaking in noncentrosymmetric superconductors

In this paper we examine the appearance of time-reversal symmetry-breaking (TRSB) states in the bulk and at the surface of a noncentrosymmetric superconductor. We utilize a Ginzburg–Landau theory, with coefficients derived from a microscopic model of the superconductor. We show that suppression of the triplet order parameter at the surface stabilizes a TRSB state by locally tuning the system into the bulk TRSB phase. We compare these results with those from centrosymmetric systems, and examine the appearance of a magnetization at the surface.

From linear magnetoresistance to parabolic magnetoresistance in Cu and Cr-doped topological insulator Bi2Se3 films

We report on the magnetotransport properties of Cu- and Cr-doped topological insulator Bi2Se3 films grown on silicon (100) substrate by chemical vapor deposition. Upon doping of non-magnetic Cu, Bi2Se3 film's carrier density is reduced and the magnetoresistance changes from a strong positive parabolic magnetoresistance to a linear magnetoresistance. This linear magnetoresistance is similar to the linear magnetoresistance in other gapless topological states of matter and has a strong temperature dependence. In contrast, the magnetoresistance of Cr-doped Bi2Se3 films remains to be parabolic and becomes temperature independent upon increased Cr-doping. These results suggest that the absence of linear magnetoresistance in Cr-doped Bi2Se3 is related to the magnetic dopants that induce time reversal symmetry breaking and surface states gap opening.

Superconductivity with broken time-reversal symmetry in ion-irradiated Ba0.27K0.73Fe2As2 single crystals

Over the last years a lot of theoretical and experimental efforts have been made to find states with broken time reversal symmetry (BTRS) in multi-band superconductors. In particular, it was theoretically proposed that in the Ba$_{1-x}$K$_{x}$Fe$_2$As$_2$ system either an $s+is$ or an $s+id$ BTRS state may exist at high doping levels in a narrow region of the phase diagram. Here we report the observation of an enhanced zero field muon spin relaxation rate below the superconducting transition temperature for a high quality crystalline sample with $x \approx$ 0.73. This indicates that indeed the time reversal symmetry is broken in superconducting Ba$_{1-x}$K$_{x}$Fe$_2$As$_2$ at this doping level.

Polar Kerr effect studies of time reversal symmetry breaking states in heavy fermion superconductors

The connection between chiral superconductivity and topological order has emerged as an active direction in research as more instances of both have been identified in condensed matter systems. With the notable exception of 3He-B, all of the known or suspected chiral – that is to say time-reversal symmetry-breaking (TRSB) – superfluids arise in heavy fermion superconductors, although the vast majority of heavy fermion superconductors preserve time-reversal symmetry. Here we review recent experimental efforts to identify TRSB states in heavy fermion systems via measurement of polar Kerr effect, which is a direct consequence of TRSB.

Pairing of j=3/2 Fermions in Half-Heusler Superconductors.

We theoretically consider the superconductivity of the topological half-Heusler semimetals YPtBi and LuPtBi. We show that pairing occurs between j=3/2 fermion states, which leads to qualitative differences from the conventional theory of pairing between j=1/2 states. In particular, this permits Cooper pairs with quintet or septet total angular momentum, in addition to the usual singlet and triplet states. Purely on-site interactions can generate s-wave quintet time-reversal symmetry-breaking states with topologically nontrivial point or line nodes. These local s-wave quintet pairs reveal themselves as d-wave states in momentum space. Furthermore, due to the broken inversion symmetry in these materials, the s-wave singlet state can mix with a p-wave septet state, again with topologically stable line nodes. Our analysis lays the foundation for understanding the unconventional superconductivity of the half-Heuslers.

Unconventional superconductivity in Y5Rh6Sn18 probed by muon spin relaxation.

Conventional superconductors are robust diamagnets that expel magnetic fields through the Meissner effect. It would therefore be unexpected if a superconducting ground state would support spontaneous magnetics fields. Such broken time-reversal symmetry states have been suggested for the high—temperature superconductors, but their identification remains experimentally controversial. We present magnetization, heat capacity, zero field and transverse field muon spin relaxation experiments on the recently discovered caged type superconductor Y5Rh6Sn18 ( TC= 3.0 K). The electronic heat capacity of Y5Rh6Sn18 shows a T3 dependence below Tc indicating an anisotropic superconducting gap with a point node. This result is in sharp contrast to that observed in the isostructural Lu5Rh6Sn18 which is a strong coupling s—wave superconductor. The temperature dependence of the deduced superfluid in density Y5Rh6Sn18 is consistent with a BCS s—wave gap function, while the zero-field muon spin relaxation measurements strongly evidences unconventional superconductivity through a spontaneous appearance of an internal magnetic field below the superconducting transition temperature, signifying that the superconducting state is categorized by the broken time-reversal symmetry.

Berry phase in cuprate superconductors

Geometrical Berry phase is recognized as having profound implications for the properties of electronic systems. Over the last decade, Berry phase has been essential to our understanding of new materials, including graphene and topological insulators. The Berry phase can be accessed via its contribution to the phase mismatch in quantum oscillation experiments, where electrons accumulate a phase as they traverse closed cyclotron orbits in momentum space. The high-temperature cuprate superconductors are a class of materials where the Berry phase is thus far unknown despite the large body of existing quantum oscillations data. In this report we present a systematic Berry phase analysis of Shubnikov - de Haas measurements on the hole-doped cuprates YBa$_2$Cu$_3$O$_{y}$, YBa$_2$Cu$_4$O$_8$, HgBa$_2$CuO$_{4 + \delta}$, and the electron-doped cuprate Nd$_{2-x}$Ce$_x$CuO$_4$. For the hole-doped materials, a trivial Berry phase of 0 mod $2\pi$ is systematically observed whereas the electron-doped Nd$_{2-x}$Ce$_x$CuO$_4$ exhibits a significant non-zero Berry phase. These observations set constraints on the nature of the high-field normal state of the cuprates and points towards contrasting behaviour between hole-doped and electron-doped materials. We discuss this difference in light of recent developments related to charge density-wave and broken time-reversal symmetry states.

Phase Transition to a Time-Reversal Symmetry-Breaking State in d-Wave Superconducting Films with Rough Surfaces

The normal-superconducting phase transition in d-wave superconducting films is discussed with a focus on the effect of diffuse surface scattering. A specularity parameter S characterizing the boundary condition is introduced for systematic analysis of the surface effect. When S = 1 (the specular limit), the film can exhibit a novel superconductivity that spontaneously breaks time-reversal symmetry, as was shown by Vorontsov [Phys. Rev. Lett. 102, 177001 (2009)]. We find that this superconducting phase is suppressed as the degree of surface roughness increases, i.e., as the specularity S decreases. In particular, it is completely suppressed when S = 0 (the diffusive limit). Those results are explained from the viewpoint of surface odd-frequency pairing.

Time-reversal symmetry breaking state in dirty three-band superconductor

I study the effects of disorder on the superconductivity of a three-band model with repulsive interband pairing. Such a model can support several possible superconducting order parameters, including a complex time-reversal symmetry breaking (TRSB) state. Impurity scattering suppresses the critical temperature of all these states, but the complex state survives, and remains a part of the phase diagram of the model even in the presence of moderate amount of disorder. This means that the TRSB states could be experimentally accessible in multiband materials like iron pnictides and chalcogenides.

Remarks on the Hall Conductivity in Chiral Superconductors: Weak vs. Strong Coupling Approach

We discuss the calculations of the ac Hall conductivity for superconductors with the time reversal symmetry breaking states. In the weak coupling theories these states show vanishing Hall response in one band models, even though one expects otherwise on symmetry grounds. On the other hand, the strong coupling approach based on the anti-de Sitter conformal eld theory correspondence leads to the non-vanishing Hall conductivity. We discuss the possible reasons of the discrepancy. The weak coupling many orbital theory leading to the Hall conductivity with correct temperature dependence is also brie y presented.

Frustration phenomena in Josephson point contacts between single-band and three-band superconductors

Within the formalizm of Usadel equations the Josephson effect in dirty point contacts between single-band and three-band superconductors is investigated. The general expression for the Josephson current, which is valid for arbitrary temperatures, is obtained. We calculate current-phase relations for very low temperature and in the vicinity of the critical temperature. For three-band superconductors with broken time-reversal symmetry (BTRS) point contacts undergo frustration phenomena with different current-phase relations, corresponding to φ-contacts. For three-band superconductors without BTRS we have close to sinusoidal current-phase relations and absence of the frustration, excepting the case of very low temperature, where under certain conditions two ground states of the point contact are realized. Our results can be used as the potential probe for the detection of the possible BTRS state in three-band superconducting systems.

H–T Phase Diagram of Multi-component Superconductors with Frustrated Inter-component Couplings

Multi-band superconductors in which frustrated inter-band couplings yield a time-reversal-symmetry breaking (TRSB) state are investigated. Stability condition for the TRSB state are derived based on the Bardeen-Cooper-Schrieffer (BCS) theory. With the time-dependent Ginzburg-Landau (GL) method, vortex states are investigated first at the vicinity of critical temperature $T_\text{c}$ where the GL theory is valid, and the results are extended to compose the $H$-$T$ phase diagram. When material parameters satisfy the condition that the nucleation field is slightly larger than the thermodynamic field ($H_\text{n} \gtrsim H_\text{tc}$) derived in a previous work (X. Hu and Z. Wang, Phys. Rev. B 85, 064516 (2012)), an unconventional intermediate state characterized by clustering vortices appears. Calculation of interface energy reveals that the clustering vortices are associated with positive interface energy.

Spin liquid phases of alkaline-earth-metal atoms at finite temperature

We study spin liquid phases of spin-5/2 alkaline-earth-metal atoms on a honeycomb lattice at finite temperatures. Our analysis is based on a Gutzwiller projection variational approach recast to a path-integral formalism. In the framework of a saddle-point approximation we determine spin liquid phases with lowest free energy and study their temperature dependence. We identify a critical temperature, where all the spin liquid phases melt and the system goes to the paramagnetic phase. We also study the stability of the saddle-point solutions and show that a time-reversal symmetry breaking state, a so-called chiral spin liquid phase, is realized even at finite temperatures. We also determine the spin structure factor, which, in principle, is an experimentally measurable quantity and is the basic tool to map the spectrum of elementary excitations of the system.

Phase diagram of Josephson junction betweensands±superconductors in the dirty limit

The ${s}_{\ifmmode\pm\else\textpm\fi{}}$ state in which the order parameter has different signs in different bands is a leading candidate for the superconducting state in the iron-based superconductors. We investigate a Josephson junction between $s$ and ${s}_{\ifmmode\pm\else\textpm\fi{}}$ superconductors within microscopic theory. Frustration, caused by interaction of the $s$-wave gap parameter with the opposite-sign gaps of the ${s}_{\ifmmode\pm\else\textpm\fi{}}$ superconductor, leads to nontrivial phase diagram. When the partial Josephson coupling energy between the $s$-wave superconductor and one of the ${s}_{\ifmmode\pm\else\textpm\fi{}}$ bands dominates, $s$-wave gap parameter aligns with the order parameter in this band. In this case, the partial Josephson energies have different signs corresponding to signs of the gap parameters. In the case of strong frustration, corresponding to almost complete compensation of the total Josephson energy, a nontrivial time-reversal-symmetry breaking (TRSB) state realizes. In this state, all gap parameters become essentially complex. As a consequence, this state provides realization for so-called $\ensuremath{\phi}$-junction with finite phase difference in the ground state. The width of the TRSB state region is determined by the second harmonic in Josephson current, $\ensuremath{\propto}$$\mathrm{sin}(2\ensuremath{\phi})$, which appears in the second order with respect to the boundary transparency. Using the microscopic theory, we establish a range of parameters where different states are realized. Our analysis shows insufficiency of the simple phenomenological approach for treatment of this problem.

Anomalous proximity effects at the interface ofs- ands±- superconductors

We study proximity effects close to a boundary between $s$ and ${s}_{\ifmmode\pm\else\textpm\fi{}}$ superconductors. Frustration, caused by interaction of the $s$-wave gap parameter with the opposite-sign gaps of ${s}_{\ifmmode\pm\else\textpm\fi{}}$ superconductor, leads to several anomalous features. In the case of strong frustration a nontrivial time-reversal-symmetry breaking (TRSB) state, with nonzero phase angles between all gap parameters, is possible. In a more typical state, the $s$-wave order parameter is aligned with one of the ${s}_{\ifmmode\pm\else\textpm\fi{}}$ gaps. The other (antialigned) gap induces negative feature in the $s$-wave density of states, which can serve as a fingerprint of the ${s}_{\ifmmode\pm\else\textpm\fi{}}$ state. Another consequence of the frustration is an extended region in the parameter space in which $s$-wave superconductivity is suppressed, despite being in contact with nominally stronger superconductor. This negative proximity effect is always present for the TRSB state, but extends even into the aligned states. We study these effects within a simple microscopic model assuming a dirty limit in all bands, which allows us to model the system in terms of minimum number of the most relevant parameters. The described anomalous features provide a route to establishing the possible ${s}_{\ifmmode\pm\else\textpm\fi{}}$ state in the iron-based superconductors.

Magneto-optical and optical probes of gapped ground states of bilayer graphene

We study the influence of different kinds of gaps in a quasiparticle spectrum on longitudinal and transverse optical conductivities of bilayer graphene. An exact analytical expression for magneto-optical conductivity is derived using a low-energy two-band Hamiltonian. We consider how the layer asymmetry gap caused by a bias electric field and a time-reversal symmetry breaking gap affect the absorption lines. The limit of zero magnetic field is then analyzed for an arbitrary carrier density in the two-band model. For a neutral bilayer graphene, the optical Hall and longitudinal conductivities are calculated exactly in the four-band model with four different gaps and zero magnetic field. It is shown that two different time-reversal symmetry breaking states can be distinguished by analyzing the dependence of the optical Hall conductivity on the energy of photon. These time-reversal symmetry breaking states are expected to be observed experimentally via optical polarization rotation either in the Faraday or Kerr effects. We analyze a possibility of such an experiment for a free-standing graphene, graphene on a thick substrate, and graphene on a double-layer substrate.

Josephson effect between a two-band superconductor withs++ors±pairing symmetry and a conventionals-wave superconductor

In this work, we investigate the Josephson effect between a two-band superconductor either with the $s++$ (two energy gaps have the same sign and are fully gapped) pairing symmetry or $s\pm$ (two energy gaps have $\pi$ phase difference and are fully gapped) pairing symmetry and a conventional s-wave superconductors. The ground state, critical current, plasma modes, flux flow dynamics, and response to external ac electric field, possible soliton solutions are investigated. For junctions with the charge neutrality breaking, we find a new plasma mode for junctions, which gives rise to new resonance peaks in the Josephson flux flow region. Because of the frustrated interaction in junctions with $s\pm$ pairing symmetry, time-reversal-symmetry (TRS) can be broken if the frustration is optimized. In the TRS broken (TRSB) state, there is a non-trivial phase difference between the two Josephson tunnelling channels, which results in a non-trivial interference. Furthermore, we find a novel massless plasma mode at the TRSB transition for junctions with the charge neutrality breaking. In the TRSB state, a spontaneous magnetic flux appears where there is a spatial inhomogeneity in the Josephson coupling, thus provides a possible smoking-gun evidence for the underlying pairing symmetry.

Intrinsic Optical Dichroism in the Chiral Superconducting State ofSr2RuO4

We present an analysis of the Hall conductivity σ(xy)(ω,T) in time reversal symmetry breaking states of exotic superconductors. We find that the dichroic signal is nonzero in systems with interband order parameters. This new intrinsic mechanism may explain the Kerr effect observed in strontium ruthenate and possibly other superconductors. We predict coherence factor effects in the temperature dependence of the imaginary part of the ac Hall conductivity Imσ(xy)(ω,T), which can be tested experimentally.