Two-component Superconductors Research Articles

Electronic and superconducting properties of CoSi[formula omitted] films on silicon — An unconventional superconductor with technological potential

We report the observation of unusual normal-state electronic conduction properties and superconducting characteristics of high-quality CoSi2/Si films grown on silicon Si(100) and Si(111) substrates. A good understanding of these features shall help to address the underlying physics of the unconventional pairing symmetry recently discovered in transparent CoSi2/TiSi2 heterojunctions (Chiu et al., 2021; Chiu et al., 2023), where CoSi2/Si is a superconductor with a superconducting transition temperature Tc≃ (1.1–1.5) K, dependent on its dimensions, and TiSi2 is a normal metal. In CoSi2/Si films, we find a pronounced positive magnetoresistance caused by the weak-antilocalization effect, indicating a strong Rashba spin–orbit coupling (SOC). This SOC generates two-component superconductivity in CoSi2/TiSi2 heterojunctions. The CoSi2/Si films are stable under ambient conditions and have ultralow 1/f noise. Moreover, they can be patterned via the standard lithography techniques, which might be of considerable practical value for future scalable superconducting and quantum device fabrication.

Two-component nematic superconductivity in 4Hb-TaS2

Most superconductors have an isotropic, single component order parameter and are well described by the standard (BCS) theory for superconductivity. Unconventional, multiple-component superconductors are exceptionally rare and are much less understood. Here, we combine scanning tunneling microscopy and angle-resolved macroscopic transport for studying the candidate chiral superconductor, 4Hb-TaS2. We reveal quasi-periodic one-dimensional modulations in the tunneling conductance accompanied by two-fold symmetric superconducting critical field. The strong modulation of the in-plane critical field, Hc2, points to a nematic, unconventional order parameter. However, the imaged vortex core is isotropic at low temperatures. We suggest a model that reconciles this apparent discrepancy and takes into account previously observed spontaneous time-reversal symmetry breaking at low temperatures. The model describes a competition between a dominating chiral superconducting order parameter and a nematic one. The latter emerges close to the normal phase. Our results strongly support the existence of two-component superconductivity in 4Hb-TaS2 and can provide valuable insights into other systems with coexistent charge order and superconductivity.

Cascade of vestigial orders in two-component superconductors: Nematic, ferromagnetic, s -wave charge- 4e , and d -wave charge- 4e states

Electronically ordered states that break multiple symmetries can melt in multiple stages, similarly to liquid crystals. In a partially melted phase, known as vestigial phase, a bilinear made out of combinations of the multiple components of the primary order parameter condenses. Multicomponent superconductors are thus natural candidates for vestigial order since they break both the $\text{U}(1)$-gauge and also time-reversal or lattice symmetries. Here, we use group theory to classify all possible real-valued and complex-valued bilinears of a generic two-component superconductor on a tetragonal or hexagonal lattice. While the more widely investigated real-valued bilinears correspond to vestigial nematic or ferromagnetic order, the little explored complex-valued bilinears correspond to a vestigial charge-$4e$ condensate, which itself can have an underlying $s$-wave, ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave, or ${d}_{xy}$-wave symmetry. To properly describe the fluctuating regime of the superconducting Ginzburg-Landau action and thus access these competing vestigial phases, we employ both a large-$N$ and a variational method. We show that while vestigial order can be understood as a weak-coupling effect in the large-$N$ approach, it is akin to a moderate-coupling effect in the variational method. Despite these distinctions, both methods yield similar results in wide regions of the parameter space spanned by the quartic Landau coefficients. Specifically, we find that the nematic and ferromagnetic phases are the leading vestigial instabilities, whereas the various types of charge-$4e$ order are attractive albeit subleading vestigial channels. The only exception is for the hexagonal case, in which the nematic and $s$-wave charge-$4e$ vestigial states are degenerate. We discuss the limitations of our approach, as well as the implications of our results for the realization of exotic charge-$4e$ states in material candidates.

Interaction between spin and Abrikosov vortices in doped topological insulators

In the topological superconductor with the nematic superconductivity in $E_u$ representation, it is possible to have different types of vortices. One is associated with the vorticity in the particle-hole space and corresponds to the Abrikosov vortex. Another type corresponds to the vorticity in the spin space and is called spin vortex. We study the interaction of the Abrikosov vortex with the spin vortices. We derive the free energy of the sample with the Abrikosov and the strain-induced spin vortices using the Ginzburg-Landau approach for the two-component superconducting order parameter. We calculate the critical strain at which the spin vortex is formed. We show that the spin vortex and the Abrikosov vortex attract to each other and, as a result, they have a common core. We show that there are no zero-energy states (Majorana fermions) localized near the common vortex core of the Abrikosov vortex and the spin vortex of any type. Possible experimental realization is discussed.

Absence of Ginzburg-Landau mechanism for vestigial order in the normal phase above a two-component superconductor

A two-component superconductor may hypothetically support a vestigial order phase above its superconducting transition temperature, with rotational or time-reversal symmetry spontaneously broken while remaining nonsuperconducting. This has been suggested as an explanation for the observed normal state nematicity of the nematic superconductor ${M}_{x}{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$. We examine the condition for this vestigial order to occur within Ginzburg-Landau theory with order parameter fluctuations, on both the nematic and chiral sides of the theory. Contrary to prior theoretical results, we rule out a large portion of parameter space for possible vestigial order. We argue that very extreme anisotropy is one prerequisite for the formation of a stable vestigial phase via this mechanism, which is likely not met in real materials.

Tuning interfacial two-component superconductivity in CoSi2/TiSi2 heterojunctions via TiSi2 diffusivity.

We report the observation of enhanced interfacial two-component superconductivity possessing a dominant triplet component in nonmagnetic CoSi2/TiSi2 superconductor/normal-metal planar heterojunctions. This is accomplished through the detection of odd-frequency spin-triplet even-parity Cooper pairs in the diffusive normal-metal component of T-shaped proximity junctions. We show that by modifying the diffusivity of the normal-metal part, the transition temperature enhancement can be tuned by a factor of up to 2.3 while the upper critical field increases by up to a factor of 20. Our data suggest that the C49 phase of TiSi2, which is stabilized in confined geometries, underlies this enhancement. These findings are addressed via a Ginzburg-Landau model and the quasi-classical theory. We also relate our findings to the enigmatic 3-K phase reported in Sr2 RuO4.

Josephson junctions of two-dimensional time-reversal invariant superconductors: Signatures of the topological phase

We determine the current-phase relation (CPR) of two-terminal configurations of Josephson junctions containing two-dimensional (2D) time-reversal invariant topological superconductors (TRITOPS), including TRITOPS-TRITOPS, as well as junctions between topological and non-topological superconductors (TRITOPS-S). We focus on long junctions for which several channels intervene in the tunneling coupling through the junction. We present a description of the topological edge modes for different TRITOPS models including $p$-wave pairing and the combination of $s$-wave pairing with spin-orbit coupling. We derive effective low-energy Hamiltonians to describe the Josephson junction, which can be solved analytically to explain the contribution of the edge states to the Josephson current as a function of the phase bias. We find that edge-modes yield singular corrections to the CPR for both junction types. The primary effects occur for the response of the Majorana zero-modes at half-flux quantum phase $\phi\approx \pi$ in TRITOPS-TRITOPS junctions and for integer flux quantum phase $\phi \approx 0$ for TRITOPS-S junctions, respectively. The former effect is particularly strong two-component nematic superconductors. The latter effect leads to a spontaneously broken time-reversal symmetry in the TRITOPS-S junction and to a breakdown of the bulk-boundary correspondence.

Role of electromagnetic gauge-field fluctuations in the selection between chiral and nematic superconductivity

Motivated by the observation of nematic superconductivity in several systems, we revisit the problem of the leading pairing instability of two-component unconventional superconductors on the triangular lattice -- such as $(p_{x},\,p_{y})$-wave and $(d_{x^{2}-y^{2}},\,d_{xy})$-wave. Such a system has two possible superconducting states: the chiral state (e.g. $p+ip$ or $d+id$), which breaks time-reversal symmetry, and the nematic state (e.g. $p+p$ or $d+d$), which breaks the threefold rotational symmetry of the lattice. Weak-coupling calculations generally favor the chiral over the nematic superconducting state, raising the question of what mechanism can stabilize the latter. Here, we show that the electromagnetic field fluctuations can play a crucial role in selecting between these two states. Specifically, we derive and analyze the effective free energy for the two-component superconducting order parameter after integrating out the gauge-field fluctuations, which is formally justified if the spatial order parameter fluctuations can be neglected. A non-analytic cubic term arises, as in the case of a conventional $s$-wave superconductor. However, unlike the latter, the cubic term depends on the relative phase and on the relative amplitudes between the two order parameter components, in such a way that it generally favors the nematic state. This result is a direct consequence of the fact that the stiffness of the superconducting order parameter is not isotropic. Competition with the quartic term, which favors the chiral state, leads to a renormalized phase diagram in which the nematic state displaces the chiral state over a wide region in the parameter space. We analyze the stability of the fluctuation-induced nematic phase, generalize our results to tetragonal lattices, and discuss their applicability to candidate nematic superconductors, including twisted bilayer graphene.

Why Does Maximum Tc Occur at the Crossover From Weak to Strong Electron–phonon Coupling in High-temperature Superconductors?

In cuprate superconductors, a pronounced maximum of superconducting \({T}_{c}^{max}\) is observed in compounds that have an in-plane Cu–O distance \({a}_{Cu-O}\) close to \(\sim 1.92\) angstroms. On the other hand, direct measurements of the electron–phonon coupling \(\lambda \langle {\omega }^{2}\rangle\) as a function of \({a}_{Cu-O}\) show a clear linear correlation, implying that \({T}_{c}^{max}\) is a strongly non-linear function of \(\lambda \langle {\omega }^{2}\rangle\). Conventional superconductivity theories based on the electron–phonon interaction predict a monotonic dependence of \({T}_{c}^{max}\) on \(\lambda \langle {\omega }^{2}\rangle\), which makes them incompatible with the observed behavior. The observed crossover behavior as a function of \(\lambda \langle {\omega }^{2}\rangle\) suggests that \({T}_{c}^{max}\) occurs at the crossover from weak to strong coupling, which is also associated with the onset of carrier localization. A coexistence, with a dynamical exchange of localized and itinerant carriers in a two-component superconductivity scenario are in agreement with the observed anomalous behavior and are suggested to be the key to understanding the mechanism for achieving high \({T}_{c}^{max}\).

Effective method of forming and detecting a fractional magnetic flux quantum

We developed a minimal method of forming and detecting a fractional magnetic flux quantum in a two-component superconductor fabricated using an ultra-thin niobium bilayer that sandwiches a Josephson barrier. The fractional quantum created by cooling a bilayer disk in the magnetic field survived after turning off the magnetic field. The survived quantum was confirmed by the change in the critical current of a superconducting quantum interference device (SQUID) placed on the bilayer disk. By replacing a flux qubit with the SQUID, we can use the fractional flux quantum as the flux bias for the flux qubit. This method enables the investigation of fractional quantization and the related topology.

High-sensitivity heat-capacity measurements on Sr2RuO4 under uniaxial pressure

A key question regarding the unconventional superconductivity of [Formula: see text] remains whether the order parameter is single- or two-component. Under a hypothesis of two-component superconductivity, uniaxial pressure is expected to lift their degeneracy, resulting in a split transition. The most direct and fundamental probe of a split transition is heat capacity. Here, we report measurement of heat capacity of samples subject to large and highly homogeneous uniaxial pressure. We place an upper limit on the heat-capacity signature of any second transition of a few percent of that of the primary superconducting transition. The normalized jump in heat capacity, [Formula: see text], grows smoothly as a function of uniaxial pressure, favoring order parameters which are allowed to maximize in the same part of the Brillouin zone as the well-studied van Hove singularity. Thanks to the high precision of our measurements, these findings place stringent constraints on theories of the superconductivity of [Formula: see text].

Sudden generation of anomalous anti-vortex states in a two-component superconductor

We studied the influences of the inclusion of different geometrical defects (circle, triangle, and square) with different Ginzburg–Landau parameters [Formula: see text] on the vortex state of a mesoscopic superconducting square immersed in an external applied magnetic field. We calculated the magnetization, vorticity, and density of Cooper pairs for this system, solving the time-dependent Ginzburg–Landau equations. We found a novel and interesting behavior of the vorticity [Formula: see text] at low magnetic fields: a spontaneous generation of anti-vortices due to the breaking inversion symmetry.

Stable Hopf-Skyrme topological excitations in the superconducting state

At large scales, magnetostatics of superconductors is described by a massive vector field theory: the London model. The magnetic field cannot penetrate into the bulk unless quantum vortices are formed. These are topological excitations characterized by an invariant: the phase winding. The London model dictates that loops of such vortices are not stable because the kinetic energy of superflow and the magnetic energy are smaller, the smaller vortex loops are. We demonstrate that in two-component superconductors, under certain conditions, such as the proximity to pair-density-wave instabilities, the hydromagnetostatics of the superconducting state and topological excitation changes dramatically: the excitations acquire the form of stable vortex loops and knots characterized by the different topological invariant: the Hopf index and hence termed hopfions. This demonstrates that magnetic properties in a superconducting state can be dramatically different from those of a London's massive vector field theory.

Nematic Skyrmions in Odd-Parity Superconductors.

We study topological excitations in two-component nematic superconductors, with a particular focus on Cu_{x}Bi_{2}Se_{3} as a candidate material. We find that the lowest-energy topological excitations are coreless vortices: a bound state of two spatially separated half-quantum vortices. These objects are nematic Skyrmions, since they are characterized by an additional topological charge. The inter-Skyrmion forces are dipolar in this model, i.e., attractive for certain relative orientations of the Skyrmions, hence forming multi-Skyrmion bound states.

Identification of nematic superconductivity from the upper critical field

Recent nuclear magnetic resonance and specific heat measurements have provided concurring evidence of spontaneously broken rotational symmetry in the superconducting state of the doped topological insulator Cu$_x$Bi$_2$Se$_3$. This suggests that the pairing symmetry corresponds to a two-dimensional representation of the $D_{3d}$ crystal point group, and that Cu$_x$Bi$_2$Se$_3$ is a nematic superconductor. In this work, we present a comprehensive study of the upper critical field $H_{c2}$ of nematic superconductors within Ginzburg-Landau (GL) theory. Contrary to typical GL theories which have an emergent U(1) rotational symmetry obscuring the discrete symmetry of the crystal, the theory of two-component superconductors in trigonal crystals reflects the true $D_{3d}$ crystal symmetry. This has direct implications for the upper critical field. First, $H_{c2}$ of trigonal superconductors exhibits a sixfold anisotropy in the basal plane. Second, when the degeneracy of the two components is lifted by, e.g., uniaxial strain, $H_{c2}$ exhibits a twofold anisotropy with characteristic angle and temperature dependence. Our thorough study shows that measurement of the upper critical field is a direct method of detecting nematic superconductivity, which is directly applicable to recently-discovered trigonal superconductors Cu$_x$Bi$_2$Se$_3$, Sr$_x$Bi$_2$Se$_3$, Nb$_x$Bi$_2$Se$_3$, and Tl$_x$Bi$_2$Se$_3$.

Different electronic charges in two-component superconductor by coherent state

Recently, the different electronic charges, which are related to the different coupling constants with magnetic field, in the two-component superconductor have been studied in the frame of Ginzburg–Landau theory. In order to study the electronic charges in detail we suggest the wave function in the two-component superconductor to be in the coherent state. We find the different electronic charges exist not only in the coherent state but also in the incoherent state. But the ratio of the different charges in the coherent state is different from the ratio in the incoherence. The expressions of the coupling constants are given directly based on the coherence effects. We also discuss the winding number in such a system.

Nambu-Goldstone-Leggett modes in multi-condensate superconductors

AbstractMulti-gap superconductors exhibit interesting properties. In an N-gap superconductor, we have in general U(1)N phase invariance. This multiple-phase invariance is partially or totally spontaneously broken in a superconductor. The Nambu-Goldstone modes, as well as Higgs modes, are important and will play an important role in multi-condensate superconductors. The additional phase invariance leads to a new quantum phase, with help of frustrated Josephson effects, such as the time-reversal symmetry breaking, the emergence of massless modes and fractionally quantized-flux vortices. There is a possibility that half-flux vortices exist in two-component superconductors in a magnetic field. The half-quantum flux vortex can be interpreted as a monopole, and two half-flux vortices form a bound state connected by a domain wall. There is an interesting analogy between quarks and fractionally quantized-flux vortices in superconductors.

Variational method applied to two-component Ginzburg-Landau theory

In this paper, we apply a variational method to two-component superconductors, as in the MgB2 materials, using the two-component Ginzburg-Landau (GL) theory. We expand the order parameter in a series of eigenfunctions containing one or two terms in each component. We also assume azimuthal symmetry to the set of eigenfunctions used in the mathematical procedure. The extension of the GL theory to two components leads to the quantization of the magnetic flux in fractions of ϕ0. We consider two kinds of component interaction potentials: Γ1|ΨI|2|ΨII|2 and \documentclass[12pt]{minimal}\begin{document}$\Gamma _2(\Psi _I^*\Psi _{II}+\Psi _I\Psi _{II}^*)$\end{document}Γ2(ΨI*ΨII+ΨIΨII*). The simplicity of the method allows one to implement it in a broad range of physical systems, such as hybrid magnetic-superconducting mesoscopic systems, texturized thin films, metallic hydrogen superfluid, and mesoscopic superconductors near inhomogeneous magnetic fields, simply by replacing the vector potential by its corresponding expression. As an example, we apply our results to a disk of radius R and thickness t.

Band filling effects on coherence lengths and penetration depth in the two-orbital negative-U Hubbard model of superconductivity

The two-orbital superconducting state is modeled by on-site intra-orbital negative-U Hubbard correlations together with inter-orbital pair-transfer interactions. The critical temperature is mainly governed by intra-orbital attractive interactions and it can pass through an additional maximum as a function of band filling. For the certain number of electrons the clear interband proximity effect is observable in the superconducting state of the band with a smaller gap. The influence of band fillings and orbital site energies on the temperature dependencies of two-component superconductivity coherence lengths and magnetic field penetration depth is analyzed. The presence of proximity effect is probably reflected in the relative temperature behaviour of characteristic lengths.

Semi-Meissner state and nonpairwise intervortex interactions in type-1.5 superconductors

We demonstrate existence of non-pairwise interaction forces between vortices in multicomponent and layered superconducting systems. That is, in contrast to most common models, the interactions in a group of such vortices is not a universal superposition of Coulomb or Yukawa forces. Next we consider the properties of vortex clusters in Semi-Meissner state of type-1.5 two-component superconductors. We show that under certain condition non-pairwise forces can contribute to formation of very complex vortex states in type-1.5 regimes.