Three-dimensional Superconductors Research Articles

Topological superconductors with spin-triplet pairings and Majorana Fermi arcs

Abstract We construct a three-dimensional topological superconductor Bogoliubov-de Gennes(BdG) Hamiltonian with the normal state being a three dimensional topological insulator. By introducing inter-orbital spin-triplet pairings term △3, there are topological Majorana nodes in the bulk, and they are connected by Majorana Fermi arcs on the surface, similar to the case of Weyl semimetal. Furthermore, by adding an inversion breaking term to the normal state, momentum independent pairing terms with different parities can coexist in the BdG Hamiltonian, which creates more Majorana modes similar to Andreev bound states and a richer phase diagram.

Fully gapped pairing state in spin-triplet superconductor UTe2.

The recently discovered superconductor UTe2 is a promising candidate for spin-triplet superconductors, but the symmetry of the superconducting order parameter remains highly controversial. Here, we determine the superconducting gap structure by the thermal conductivity of ultraclean UTe2 single crystals. We find that the a-axis thermal conductivity divided by temperature κ/T in zero-temperature limit is vanishingly small for both magnetic field H‖a and H‖c axes up to H/Hc2 ∼ 0.2, demonstrating the absence of nodes around the a axis contrary to the previous belief. The present results, combined with the reduction of nuclear magnetic resonance Knight shift, indicate that the superconducting order parameter belongs to the isotropic Au representation with a fully gapped pairing state, analogous to the B phase of superfluid 3He. These findings reveal that UTe2 is likely to be a long-sought three-dimensional strong topological superconductor, hosting helical Majorana surface states on any crystal plane.

Oscillating paramagnetic Meissner effect and Berezinskii-Kosterlitz-Thouless transition in underdoped Bi2Sr2CaCu2O8+δ.

Superconducting phase transitions in two dimensions lie beyond the description of the Ginzburg-Landau symmetry-breaking paradigm for three-dimensional superconductors. They are Berezinskii-Kosterlitz-Thouless (BKT) transitions of paired-electron condensate driven by the unbinding of topological excitations, i.e. vortices. The recently discovered monolayers of layered high-transition-temperature ([Formula: see text]) cuprate superconductor Bi2Sr2CaCu2O8+δ (Bi2212) meant that this 2D superconductor promised to be ideal for the study of unconventional superconductivity. But inhomogeneity posed challenges for distinguishing BKT physics from charge correlations in this material. Here, we utilize the phase sensitivity of scanning superconducting quantum interference device microscopy susceptometry to image the local magnetic response of underdoped Bi2212 from the monolayer to the bulk throughout its phase transition. The monolayer segregates into domains with independent phases at elevated temperatures below [Formula: see text]. Within a single domain, we find that the susceptibility oscillates with flux between diamagnetism and paramagnetism in a Fraunhofer-like pattern up to [Formula: see text]. The finite modulation period, as well as the broadening of the peaks when approaching [Formula: see text] from below, suggests well-defined vortices that are increasingly screened by the dissociation of vortex-antivortex plasma through a BKT transition. In the multilayers, the susceptibility oscillation differs in a small temperature regime below [Formula: see text], consistent with a dimensional crossover led by interlayer coupling. Serving as strong evidence for BKT transition in the bulk, we observe a sharp jump in phase stiffness and paramagnetism at small fields just below [Formula: see text]. These results unify the superconducting phase transitions from the monolayer to the bulk underdoped Bi2212, and can be collectively referred to as the BKT transition with interlayer coupling.

Gradual emergence of superconductivity in underdoped La2−xSrxCuO4

We present triple-axis neutron scattering studies of low-energy magnetic fluctuations in strongly underdoped La$_{2-x}$Sr$_{x}$CuO$_{4}$ with $x=0.05$, $0.06$ and $0.07$, providing quantitative evidence for a direct competition between these fluctuations and superconductivity. At dopings $x=0.06$ and $x=0.07$, three-dimensional superconductivity is found, while only a very weak signature of two-dimensional superconductivity residing in the CuO$_2$ planes is detectable for $x=0.05$. We find a surprising suppression of the low-energy fluctuations by an external magnetic field at all three dopings. This implies that the response of two-dimensional superconductivity to a magnetic field is similar to that of a bulk superconductor. Our results provide direct evidence of a very gradual onset of superconductivity in cuprates.

Plasmons in three-dimensional superconductors

We study the plasma branch of an homogeneous three-dimensional electron gas in an $s$-wave superconducting state. Although a sum rule guarantees that the departure of the plasma branch always coincides with the plasma frequency ${\ensuremath{\omega}}_{p}$, the dispersion and lifetime of plasmons is strongly affected by the presence of the pair condensate, especially when ${\ensuremath{\omega}}_{p}$ is close to the pair-breaking threshold $2\mathrm{\ensuremath{\Delta}}$. When ${\ensuremath{\omega}}_{p}$ is between $1.7\mathrm{\ensuremath{\Delta}}$ and $2\mathrm{\ensuremath{\Delta}}$, the level repulsion is strong enough to give the plasma branch an anomalous, downward dispersion and a dispersion minimum strictly lower than ${\ensuremath{\omega}}_{p}$. Then for ${\ensuremath{\omega}}_{p}>2\mathrm{\ensuremath{\Delta}}$, plasmons damp out in pair-breaking excitations, acquiring a small damping rate at zero temperature, which we compute in a non-perturbative way. Finally, the density-density response function displays a resonance near $2\mathrm{\ensuremath{\Delta}}$ (not to be confused with the amplitude mode), which can beat with the main plasma resonance, and subsists for ${\ensuremath{\omega}}_{p}$ large compared to $\mathrm{\ensuremath{\Delta}}$, thereby distinguishing charged from neutral condensates.

Vortex supercurrent inversion by frequency-symmetry conversion of Cooper pairs

We theoretically demonstrate that the vortex supercurrent can be reversed by odd-frequency Cooper pairs accompanied by surface Andreev bound states. The surface of a three-dimensional superconductor pierced by a flux quantum is considered. We compare the vortex supercurrents near the surface of the spin-singlet $s$-wave and spin-triplet $p_z$-wave superconductors using quasiclassical Eilenberger theory, where the surface is perpendicular to the $z$ direction. We demonstrate that the vortex supercurrent near the surface of a $p_z$-wave superconductor is reversed compared to those far from the surface, whereas that of an $s$-wave superconductor is not. The splitting of the zero-energy states caused by the interference of the surface Andreev bound states and Caroli-de Gennes-Matricon modes is demonstrated.

Fourfold Symmetric Superconductivity in Spinel Oxide LiTi2O4(001) Thin Films.

The charge frustration with the mixed-valence state inherent to LiTi2O4, which is found to be the only oxide superconductor with spinel structure, is the impetus for paying special attention to unveil the underlying intriguing superconducting properties. Here, we report a pronounced fourfold rotational symmetry of the superconductivity in high-quality single-crystalline LiTi2O4(001) thin films. Both the magnetoresistivity and upper critical field under an applied magnetic field manifest striking fourfold oscillations deep inside the superconducting state, whereas the anisotropy vanishes in the normal state, demonstrating that it is an intrinsic property of the superconducting phase. We attribute this behavior to the unconventional d-wave superconducting Cooper pairs with the irreducible representation of Eg protected by the Oh point group in cubic LiTi2O4. Our findings show the nontrivial character of the pairing interaction in a three-dimensional spinel oxide superconductor.

Topological classification of intrinsic three-dimensional superconductors using anomalous surface construction

Intrinsic topological superconductors have protected gapless Majorana modes, bound and/or propagating, at the natural boundaries of the sample, without requiring field, defect, or heterostructure. We establish the complete classification/construction of intrinsic topological superconductors jointly protected by point-group and time-reversal symmetries in three dimensions. This is obtained from enumerating distinct ways for stacking $n$th-order irreducible building blocks, minimal anomalous surface states of $n$th-order topological superconductors. Particularly, our method provides a unified description of possible surface anomalies away from high-symmetry points/lines in terms of the homotopy group of the surface mass field.

Anomalous gapped boundaries between surface topological orders in higher-order topological insulators and superconductors with inversion symmetry

We show that the gapless boundary signatures---namely, chiral/helical hinge modes or localized zero modes---of three-dimensional higher-order topological insulators and superconductors with inversion symmetry can be gapped without symmetry breaking upon the introduction of non-Abelian surface topological order. In each case, the fractionalization pattern that appears on the surface is ``anomalous'' in the sense that it can be made consistent with symmetry only on the surface of a three-dimensional higher-order insulator/superconductor. Our results show that the interacting manifestation of higher-order topology is the appearance of ``anomalous gapped boundaries'' between distinct topological orders whose quasiparticles are related by inversion, possibly in conjunction with other protecting symmetries such as time-reversal symmetry and charge conservation.

Prevalence of trivial zero-energy subgap states in nonuniform helical spin chains on the surface of superconductors

Helical spin chains, consisting of magnetic (ad-)atoms, on the surface of bulk superconductors are predicted to host Majorana bound states (MBSs) at the ends of the chain. Here, we investigate the prevalence of trivial zero-energy bound states in these helical spin chain systems. First, we show that the Hamiltonian of a helical spin chain on a superconductor can be mapped to an effective Hamiltonian reminiscent of a semiconductor nanowire with strong Rashba spin-orbit coupling. In particular, we show that a varying rotation rate between neighbouring magnetic moments maps to smooth non-uniform potentials in the effective nanowire Hamiltonian. Previously it has been found that trivial zero-energy states are abundant in nanowire systems with smooth potentials. Therefore, we perform an extensive search for zero-energy bound states in helical spin chain systems with varying rotation rates. Although bound states with near zero-energy do exist for certain dimensionalities and rotation profiles, we find that zero-energy bound states are far less prevalent than in semiconductor nanowire systems with equivalent non-uniformities. In particular, utilising varying rotation rates, we do not find zero-energy bound states in the most experimentally relevant setup consisting of a one-dimensional helical spin chain on the surface of a three-dimensional superconductor, even for profiles that produce near zero-energy states in equivalent one- and two- dimensional systems. Although our findings do not rule them out, the much reduced prevalence of zero-energy bound states in long non-uniform helical spin chains compared with equivalent semi-conductor nanowires, as well as the ability to measure states locally via STM, should reduce the experimental barrier to identifying MBSs in such systems.

Dephasing-enhanced Majorana zero modes in two-dimensional and three-dimensional higher-order topological superconductors

The 1D Kitaev model in the topological phase, with open boundary conditions, hosts strong Majorana zero modes. These are fermion parity-odd operators that almost commute with the Hamiltonian and manifest in long coherence times for edge degrees of freedom. We obtain higher-dimensional counterparts of such Majorana operators by explicitly computing their closed form expressions in models describing 2D and 3D higher-order superconductors. Due to the existence of such strong Majorana zero modes, the coherence time of the infinite temperature autocorrelation function of the corner Majorana operators in these models diverges with the linear system size. In the presence of a certain class of orbital-selective dissipative dynamics, the coherence times of half of the corner Majorana operators is enhanced, while the time correlations corresponding to the remaining corner Majoranas decay much faster as compared with the unitary case. We numerically demonstrate robustness of the coherence times to the presence of disorder.

Observing the Influence of Reduced Dimensionality on Fermionic Superfluids.

Understanding the origins of unconventional superconductivity has been a major focus of condensed matter physics for many decades. While many questions remain unanswered, experiments have found the highest critical temperatures in layered two-dimensional materials. However, to what extent the remarkable stability of these strongly correlated 2D superfluids is affected by their reduced dimensionality is still an open question. Here, we use dilute gases of ultracold fermionic atoms as a model system to directly observe the influence of dimensionality on the stability of strongly interacting fermionic superfluids. We find that the superfluid gap follows the same universal function of the interaction strength regardless of dimensionality, which suggests that there is no inherent difference in the stability of two- and three-dimensional fermionic superfluids. Finally, we compare our data to results from solid state systems and find a similar relation between the interaction strength and the gap for a wide range of two- and three-dimensional superconductors.

Anisotropy of 2H-NbSe2 in the superconducting and charge density wave states

Anisotropy is an important feature of layered materials, and a large anisotropy is usually related to the two-dimensional characteristics. We investigated the anisotropy of the layered transition metal dicalcogenide 2H-NbSe2 in the superconducting and charge density wave (CDW) states using magnetotransport measurements. In the superconducting state, the normalized is independent of the thickness of 2H-NbSe2, while increases significantly with decreasing thickness, where H p is the Pauli limiting magnetic field and and are the upper critical fields in the c and ab directions, respectively. It is found that the superconducting anisotropy parameter increases with reduction in the thickness of 2H-NbSe2. In the CDW state, the angular (θ) dependence of magnetoresistance, R(H,θ) scales with , which decreases with increasing temperature and disappears at about 40 K. It is found that the CDW anisotropy parameter γ CDW is much larger than the effective mass anisotropy but does not change a lot for ultrathin and bulk samples. Our results suggest the existence of three-dimensional superconductivity and quasi-two dimensional CDWs in bulk 2H-NbSe2.

Hot-lines topology and the fate of the spin resonance mode in three-dimensional unconventional superconductors

In the quasi-two-dimensional (quasi-2D) copper- and iron-based superconductors, the onset of superconductivity is accompanied by a prominent peak in the magnetic spectrum at momenta close to the wave-vector of the nearby antiferromagnetic state. Such a peak is well described in terms of a spin resonance mode, i.e., a spin-1 exciton theoretically predicted for quasi-2D superconductors with a sign-changing gap. The same theories, however, indicate that such a resonance mode should be absent in a three-dimensional (3D) system with a spherical Fermi surface. This raises the question of the fate of the spin resonance mode in layered unconventional superconductors that are not strongly anisotropic, such as certain heavy-fermion compounds and potentially the newly discovered nickelate superconductor NdNiO$_2$. Here, we use the random-phase-approximation to calculate the dynamical spin susceptibility of 3D superconductors with a $d_{x^2-y^2}$-wave gap symmetry and corrugated cylindrical-like Fermi surfaces. By varying the out-of-plane hopping anisotropy $t_z/t$, we demonstrate that the appearance of a spin resonance mode is determined by the topology of the hot lines -- i.e. lines on the Fermi surface that are connected by the magnetic wave-vector. For an in-plane antiferromagnetic wave-vector, the hot lines undergo a topological transition from open lines to closed loops at a critical $t_z/t$ value. The closed hot lines cross the nodal superconducting lines, making the spin resonance mode overdamped and incoherent. In contrast, for an out-of-plane antiferromagnetic wave-vector, the hot lines remain open and the spin resonance mode remains sharp. We discuss the experimental implications of our results for the out-of-plane dispersion of the spin resonance mode and, more generally, for inelastic neutron scattering experiments on unconventional superconductors.

Vortex Matter in a Two-Band SQUID-Shaped Superconducting film

In the present work, we studied the magnetization, vorticity, Cooper pairs density, and the spatial distribution of the local magnetic field in a three-dimensional superconductor with a SQUID geometry (a square with a central hole connected to the outside vacuum through a very thin slit). Our investigation was carried out in both the Meissner-Ochsenfeld and the Abrikosov state solving the two-band Ginzburg-Landau equations considering a Josephson coupling between the bands. We found a non-monotonic vortex behavior and the respective generation of vortex clusters due to the Josephson coupling used between condensates.

Dimensionality of the Superconductivity in the Transition Metal Pnictide WP.

We report theoretical and experimental results on the transition metal pnictide WP. The theoretical outcomes based on tight-binding calculations and density functional theory indicate that WP is a three-dimensional superconductor with an anisotropic electronic structure and nonsymmorphic symmetries. On the other hand, magnetoresistance experimental data and the analysis of superconducting fluctuations of the conductivity in external magnetic field indicate a weakly anisotropic three-dimensional superconducting phase.

Driving force on flowing quantum vortices in type-II superconductors with finite Ginzburg-Landau parameter

The origin of the driving force on quantum vortices in superconductors has long been discussed. We investigate the origin of this force using the momentum flux tensor $\mathsc{P}$, Maxwell stress tensor $\mathsc{T}$, and numerical solutions for a flowing rectilinear vortex in the time-dependent Ginzburg-Landau (TDGL) theory for three-dimensional superconductors with finite Ginzburg-Landau parameter $\ensuremath{\kappa}$ and the Maxwell equations. We calculate the hydrodynamic force ${\mathbit{F}}_{\mathrm{hydro}}(C)$ and magnetic Lorentz force ${\mathbit{F}}_{\mathrm{mag}}(C)$ respectively using the contour integral of $\mathsc{P}$ and $\mathsc{T}$ along a closed path that winds around the vortex line. The calculations show that neither ${\mathbit{F}}_{\mathrm{hydro}}(C)$ nor ${\mathbit{F}}_{\mathrm{mag}}(C)$ reaches the full magnitude of the driving force. However, when the path $C$ is farther than the penetration depth from the vortex line and hence the energy dissipation is negligible on $C$, the sum of the two forces becomes independent of the choice of $C$ and accounts for the full magnitude of the driving force on the vortex. We demonstrate the applicability of this result to a flowing vortex described in the generalized or modified version of the TDGL equation and to a pinned vortex. We then discuss the driving force on the Pearl vortex in two-dimensional superconductors and a curved vortex line in three-dimensional superconductors. We propose an experiment that locally probes the magnetic field with a pinned vortex to verify our results that the contribution of the magnetic pressure (Lorentz force) to the total driving force on the vortex is less than half.

Three-dimensional peridynamic modeling of crack initiation and propagation in bulk superconductor during field cooling magnetization

Bulk high-temperature superconductors exhibit outstanding electromagnetic properties and are capable of trapping very large magnetic fields. However, bulk superconductors are subjected to a large Lorentz force during field cooling magnetization (FCM), which can cause crack initiation and propagation. Superconducting performance is then limited by the damage to the bulk. In this paper, we study the mechanical behavior and brittle damage of a three-dimensional (3D) bulk GdBCO superconductor during FCM. Firstly, the distribution of electromagnetic field in the bulk is obtained based on the H-formulation with a finite element model. Then, a 3D bond-based peridynamic (PD) model is used to simulate the mechanical behavior and possible brittle damage to the bulk under a given electromagnetic force. The initiation and propagation of cracks can be predicted using the 3D bond-based PD model. The effect of residual stress due to the presence of a reinforcement ring during the cooling process is also discussed. The results show that mechanical stability can be improved by reinforcing the sample.

Novel vortex structures in a three-dimensional superconductor under a helical magnetic field from a chiral helimagnet

We have investigated vortex structures in three-dimensional superconductors under a helical magnetic field from a chiral helimagnet numerically. In order to obtain vortex structures, we solve the three-dimensional Ginzburg–Landau equations with the finite element method. A spatial distribution of the helical magnetic field corresponds to a spatial distribution of magnetic moments in the chiral helimagnet. Then, the magnetic field is parallel to a yz-plane and helical rotation along a helical axis (x-axis). An arrangement of the vortex is different from the spatial oscillation in each component of the magnetic field. In this calculation, two-types of vortex structures appear. One is an appearance of vortices perpendicular to a xy-plane and a zx-plane of the superconductor. The other arrangement is that vortices are tilted toward the helical axis, although the magnetic field along the helical axis is zero. The arrangement of the tilted vortex depends on a direction of a rotation in the helical magnetic field.

Unconventional Majorana fermions on the surface of topological superconductors protected by rotational symmetry

Topological superconductors are exotic gapped phases of matter hosting Majorana mid-gap states on their boundaries. In conventional three-dimensional topological superconductors, Majorana in-gap states appear in the form of spin-1/2 fermions with a quasirelativistic dispersion relation. Here, we show that unconventional Majorana states can emerge on the surface of three-dimensional topological superconductors protected by rotational symmetry. The unconventional Majorana surface states are classified into three different categories: a spin-$s$ Majorana fermion with $(2s+1)$-fold degeneracy $(s\ensuremath{\ge}3/2)$, a Majorana Fermi line carrying two distinct topological charges, and a quartet of spin-1/2 Majorana fermions related by fourfold rotational symmetry. The spectral properties of the first two types, which go beyond conventional spin-1/2 fermions, are unique to topological superconductors and have no counterparts in topological insulators. We show that unconventional Majorana surface states can be obtained in the superconducting phase of doped ${Z}_{2}$ topological insulators or Dirac semimetals with rotational symmetry.