We present a gauge-invariant theory of the electromagnetic response of a chiral px+ipy superconductor in the clean limit. Due to the spontaneously broken time-reversal symmetry, the effective action of the system contains an anomalous term not present in conventional superconductors. As a result, the electromagnetic charge and current responses contain anomalous terms, which depend explicitly on the chirality of the superconducting order parameter. These terms lead to a number of unusual effects, such as coupling of the transverse currents to the collective plasma oscillations and a possibility of inducing the charge density by the magnetic field perpendicular to the conducting planes. We calculate the antisymmetric part of the conductivity tensor (the intrinsic Hall conductivity) and show that it depends on the wave vector of the electromagnetic field. We also show that the Mermin-Muzikar magnetization current and the Hall conductivity are strongly suppressed at high frequencies. Finally, we discuss implications of the theory to the experiments in Sr2RuO4.
In this paper, the px+ipy topological superconductor with vortex superlattice is studied. We found that there exist mid-gap energy bands induced by the vortex superlattice and the mid-gap energy bands have nontrivial topological properties including the gapless edge states and non-zero winding number. An topological anisotropic tight-binding Majorana lattice model is proposed to describe the mid-gap states.
We calculate frequency and temperature dependence of the anomalous ac Hall conductivity induced by impurity scattering in a chiral px+ipy superconductor, such as Sr2RuO4, with spontaneous time-reversal-symmetry breaking in the absence of an external magnetic field. We consider two models of disorder, Gaussian and non-Gaussian, characterized by the second and third moments of the random impurity potential, respectively. Within both models, we find that the anomalous Hall conductivity has a finite real value at zero frequency, exhibits singularities at the threshold of photon absorption across t…
Directly observing a zero energy Majorana state in the vortex core of a chiral superconductor by tunneling spectroscopy requires energy resolution better than the spacing between core states $Δ^2/eF$. We show that nevertheless, its existence can be decisively tested by comparing the temperature broadened tunneling conductance of a vortex with that of an antivortex even at temperatures $T >> Δ^2/eF$.
To investigate Majorana fermionic excitations of a $p_x+ip_y$ superconductor, the Bogoliubov-de-Gennes equation is solved on a sphere for two cases: (i) a vortex-antivortex pair at opposite poles and (ii) an edge near the south pole and an antivortex at the north pole. The vortex cores support a state of two Majorana fermions, the energy of which decreases exponentially with the radius of the sphere, independently of a moderate disorder potential. The tunneling conductance of an electron into the superconductor near the position of a vortex is computed for finite temperature, and is compared t…
We study the Josephson effect between chiral p-wave superconductor / diffusive normal metal (DN) / chiral p-wave superconductor (CP/DN/CP) junctions using quasiclassical Green's function formalism with proper boundary conditions. The px+ipy-wave symmetry of superconducting order parameter is chosen which is believed to be a pairing state in Sr2RuO4. It is shown that the Cooper pairs induced in DN have an odd-frequency spin-triplet s-wave symmetry, where pair amplitude is an odd function of Matsubara frequency. Despite the peculiar symmetry properties of the Cooper pairs, the behavior of the Jo…
Superconductors connected to normal metallic electrodes at the nanoscale provide a potential source of non-locally entangled electron pairs. Such states would arise from Cooper pairs splitting into two electrons with opposite spins tunnelling into different leads. In an actual system the detection of these processes is hindered by the elastic transmission of individual electrons between the leads, yielding an opposite contribution to the non-local conductance. Here we show that electromagnetic excitations on the superconductor can play an important role in altering the balance between these tw…
We show that a moving vortex lattice, as it comes to a crystal edge, radiates into a free space the harmonics of the washboard frequency, $ω_0=2πv/a$, up to a superconducting gap, $Δ/\hbar$. Here $v$ is the velocity of the vortex lattice and $a$ is the intervortex spacing. We compute radiation power and show that this effect can be used for generation of terahertz radiation and for characterization of moving vortex lattices.
A vortex in a model spinless px+ipy superconductor induces two Majorana fermions (MFs), one in the core and the other at the sample edge. In the present work, we show that edge MF can be generated, fused, transported, and braided easily by tuning gate voltages at point-like constriction junctions. Solving the time-dependent Bogoliubov-de Gennes equation, we demonstrate that the braiding of edge MFs obeys the non-Abelian statistics. The present setup is therefore a promising implementation for topological quantum computation, and has the advantage of easy manipulation and simple device structur…
Starting from the generalised London equations, which include a gravitomagnetic term, the gravitational and the electromagnetic properties of superconductors are derived. A phenomenological synthesis of those properties is proposed.
Within the framework of the kinetic energy driven superconducting mechanism, the doping dependence of the electromagnetic response in the electron-doped cuprate superconductors is studied. It is shown that although there is an electron-hole asymmetry in the phase diagram, the electromagnetic response in the electron-doped cuprate superconductors is similar to that observed in the hole-doped cuprate superconductors. The superfluid density depends linearly on temperature, except for the strong deviation from the linear characteristics at the extremely low temperatures.
Recent theoretical and experimental studies demonstrated that an electromagnetic metamaterial approach is capable of drastically increasing the critical temperature Tc of composite metamaterial superconductors. This progress was achieved by engineering the frequency-dependent dielectric response function of the metamaterial. Here we further extend this approach by introducing the concept of hybrid acousto-electromagnetic hyperbolic metamaterial superconductors, which are projected to exhibit strongly enhanced superconducting properties due to artificially engineered broadband divergency of the…
Motivated by the recent controversy surrounding the Kerr effect measurements in strontium ruthenate \cite{xia:167002}, we examine the electromagnetic response of a clean chiral p-wave superconductor. When the contributions of the collective modes are accounted for, the Hall response in a clean chiral superconductor is smaller by several orders of magnitude than previous theoretical predictions and is too small to explain the experiment. We also uncover some unusual features of the collective modes of a chiral superconductor, namely, that they are not purely longitudinal and couple to external …
The interaction between radiation and superconductors is explored in this paper. In particular, the calculation of a plane standing wave scattered by an infinite cylindrical superconductor is performed by solving the Helmholtz equation in cylindrical coordinates. Numerical results computed up to $\mathcal{O}(77)$ of Bessel functions are presented for different wavelengths showing the appearance of a diffraction pattern.
We derive the current response to the linearly polarized electromagnetic field with finite frequency and wave vector incident normally on the specular surface of a clean nonconventional superconductor with orbital spontaneous magnetization parallel to the crystal axis and perpendicular to the crystal surface. The result includes the usual part known from the theory of conventional superconductivity and as well the magneto-optical term typical for the superconductors with spontaneous time reversal breaking. As an application of the basic current-field relation we consider the Kerr effect for th…
We reveal that due to the presence of disorder oscillations of the order parameter amplitude called the Higgs mode can be effectively excited by the external electromagnetic radiation in usual BCS superconductors. This mechanism works for superconductors with both isotropic s-wave and anisotropic, such as d-wave, pairings. The non-linear response in the presence of impurities is captured by the quasiclassical formalism. We demonstrate that analytical solutions of the Eilenberger equation with impurity collision integral and external field drive coincide with the exact summation of ladder impur…
We investigate theoretically the shielding of the electromagnetic field fluctuations by s-wave and d-wave superconductors within the framework of macroscopic quantum electrodynamics. The spin flip lifetime is evaluated above a niobium and a bismuth strontium calcium copper oxide (BSCCO) surface, and the screening effect is studied as a function of the thickness of the superconducting layer. Further, we study the different temperature dependence of the atomic spin relaxation above the two superconductors.
Recently a generic model has been proposed for the single-particle properties of gapless superconductors with simultaneously present pair-conserving and pair-breaking impurity scattering (the so-called Dynes superconductors). Here we calculate the optical conductivity of the Dynes superconductors. Our approach is applicable for all disorder strengths from the clean up to the dirty limit and for all relative ratios of the two types of scattering, nevertheless the complexity of our description is equivalent to that of the widely used Mattis-Bardeen theory. We identify two optical fingerprints of…
The electromagnetic properties of a Fermi liquid superconductor with singular Landau parameter $F_{1s}\to-1$, corresponding to a state with marginal spin-charge separation (MSCS) is analyzed in this paper. We show that the MSCS state describes a strongly phase-disordered superconductor which is a diamagnetic metal with Drude-like optical conductivity. The phase diagram and electromagnetic properties of the MSCS state is found to be in qualitative agreement with what is observed in the pseudo-gap phase of (underdoped) High-$T_c$ cuprates. We predicted that vortices with {\em un-quantized} magne…
Effects of order parameter collective modes on electromagnetic response are studied for a clean spin-triplet superconductor with $k_x\pm ik_y$ orbital symmetry, which has been proposed as a candidate pairing symmetry for Sr$_2$RuO$_4$. It is shown that the $k_x \pm ik_y$ superconductor has characteristic massive collective modes analogous to the clapping mode in the A-phase of superfluid $^3$He. We discuss the contribution from the collective modes to ultrasound attenuation and electromagnetic absorption. We show that in the electromagnetic absorption spectrum the clapping mode gives rise to a…
We present results of a microwave surface impedance study of the heavy fermion superconductor UBe13. We clearly observe an absorption peak whose frequency- and temperature-dependence scales with the BCS gap function. Resonant absorption into a collective mode, with energy approximately proportional to the superconducting gap, is proposed as a possible explantation.
The spread of the Cooper pairs into the ferromagnet in proximity coupled superconductor - ferromagnet (SF) structures is shown to cause a strong inverse electromagnetic phenomenon, namely, the long-range transfer of the magnetic field from the ferromagnet to the superconductor. Contrary to the previously investigated inverse proximity effect resulting from the spin polarization of superconducting surface layer, the characteristic length of the above inverse electrodynamic effect is of the order of the London penetration depth, which usually much larger than the superconducting coherence length…
We consider here a layered superconductor subject to an externally applied moderately-strong electromagnetic field. We predict hysteretic jumps in the dependence of the surface reactance of the superconductor on the amplitude of the incident electromagnetic wave. This very unusual nonlinear phenomenon can be observed in thin superconducting slabs at not very strong ac amplitudes, if the frequency of the irradiating field is close to the Josephson plasma frequency. Using the set of coupled sine-Gordon equations, we derive the expression for the phase shift of the reflected wave and obtain the c…
We study theoretically the space-time evolution of the thermal and electromagnetic perturbation in a superconductor with a nonlinear current-voltage characteristics in the flux creep regime. On the basis of a linear analysis of a set of differential equations describing small perturbations of temperature and electromagnetic field, it is found that under some conditions a branching instability may occur in a superconductor sample.
The nano-scale intrinsic Josephson junctions in highly anisotropic cuprate superconductors have potential for generation of terahertz electromagnetic waves. When the thickness of a superconductor sample is much smaller than the wavelength of electromagnetic waves in vacuum, the superconductor renders itself as a cavity. Unlike conventional lasers, the presence of the cavity does not guarantee a coherent emission because of the internal degree of freedom of the superconductivity phase in long junctions. We study the excitation of terahertz wave by solitons in a stack of intrinsic Josephson junc…
We theoretically study the transition radiation of bulk and surface electromagnetic waves by an electron crossing an interface between a layered superconductor and an isotropic dielectric. We assume that the direction of the electron motion and the orientation of the superconducting layers are perpendicular to the interface. We derive the analytical expressions for the strongly anisotropic radiation fields and for the time-integrated energy fluxes of bulk and oblique surface electromagnetic waves (OSWs). We show that the OSWs with frequencies close to the Josephson plasma frequency $ω_J$ provi…
A multilayered structure with a single superconductor layer and a single insulator layer formed on a bulk superconductor is studied. General formulae for the vortex-penetration field of the superconductor layer and the magnetic field on the bulk superconductor, which is shielded by the superconductor and insulator layers, are derived with a rigorous calculation of the magnetic field attenuation in the multilayered structure. The achievable peak surface field depends on the thickness and its material of the superconductor layer, the thickness of the insulator layer and material of the bulk supe…
We show that in superlattices with alternating superconducting (S) and ferromagnetic (F) layers the spontaneous magnetic field induced in the superconducting layers due to the electromagnetic proximity effect becomes dramatically enhanced compared to the previously studied S/F bilayers. The effect reveals itself for the in-plane orientation of the magnetic moments both for ferromagnetic and anti-ferromagnetic ordering of the moments in the F layers. In the finite size samples the magnetic field decays from the sample surface towards the bulk of the structure, and the decay length strongly depe…
It is shown that color-superconducting quark matter, where quarks of the same flavor form Cooper pairs with spin one, exhibits an electromagnetic Meissner effect. This is in contrast to spin-zero color superconductors where Cooper pairs consist of quarks with different flavors.
We determine the position and shape of the melting line in a layered superconductor taking the electromagnetic coupling between layers into account. In the limit of vanishing Josephson coupling we obtain a new generic reentrant low-field melting line. Finite Josephson coupling pushes the melting line to higher temperatures and fields and a new line shape $B_{\rm m} \propto (1-T/T_c)^{3/2}$ is found. We construct the low-field phase diagram including melting and decoupling lines and discuss various experiments in the light of our new results.
The electromagnetic response of the CuO2-planes is calculated within a strong coupling theory using model tight binding bands and momentum dependent pairing interactions representing spin fluctuations and phonon exchange. The superconducting state resulting from these interactions has d-wave symmetry. With phonon exchange included the order parameter amplitude grows rapidly below Tc at elevated frequencies which leads to improved agreement with the observed temperature dependence of the penetration depth. Good agreement between theory and experiment can only be achieved if it is assumed that t…
We revisit the importance of collective-mode fluctuations and gauge invariance in the electromagnetic response of superconducting systems. In particular, we show that order-parameter fluctuations, gapless or not, have no contribution to the Meissner effect in both $s$- and $p$-wave superconductors. More generally, we extend this result to uniform and nonuniform superfluids with no external wavevector scale. To facilitate this analysis, we formulate a path-integral-based matrix methodology for computing the electromagnetic response of fermionic fluids in the presence of concomitantly fluctuatin…
Left-handed materials usually are realized in artificial subwavelength structures. Here we show that some anisotropic superconductors, such as $\mathrm{Bi_2Sr_2CaCu_2O_{8+δ}}$, $\mathrm{YBa_2Cu_xO_y}$ and $\mathrm{La_{2-x}Sr_xCuO_4}$, are intrinsic left-handed materials. The condition is that the plasma frequency in the $c$ axis, $ω_c$, and in the $ab$ plane, $ω_{ab}$, and the operating frequency, $ω$, satisfy $ω_c<ω<ω_{ab}$. In addition $ω$ should be smaller than the superconducting energy gap to sustain superconductivity. We study the reflection and transmission of electromagnetic waves, and…
It has been proposed that the effective Hamiltonian describing high T_c superconductivity in cuprate materials has an approximate SO(5) symmetry relating the superconducting (SC) and antiferromagnetic (AF) phases of these systems. We show that robust consequences of this proposal are potentially large optical conductivities and Raman scattering rates in the AF phase, due to the electromagnetic response of the doubly-charged pseudo Goldstone bosons which must exist there. This provides strong constraints on the properties of the bosons, such as their mass gap and velocity.
The nonlinear dynamics of thermal and electromagnetic perturbations in the vortex state of type II superconductors is analyzed with account of dissipation and dispersion effects. A theoretical analysis shows that nonlinear thermal and electromagnetic dissipative waves (structures) may be formed under certain conditions on the sample surface. The structures possess a finite-amplitude and propagate at a constant velocity. The possibility of experimental observation of thermomagnetic shock waves is briefly discussed.
The interrelation between the condensation energy and the optical sum rules has been investigated. It has been shown that the so called 'partial' sum rule violation is related mainly to a temperature dependence of the relaxation rate rather than to the appearance of superconductivity itself. Moreover, we demonstrate that the experimental data on the temperature dependence of the optical sum rule can be explained rather well by an account of strong electron-phonon interaction.
Perspectives on Magnetic/Superconductor Hybrid Systems: Long-range Electromagnetic Phenomena Induced by Proximity Effect and Interfacial Spin-orbit Coupling
Institute for Physics of Microstructures, Russian Academy of Sciences（俄罗斯科学院微观物理研究所）；Moscow Institute of Physics and Technology (National Research University)（莫斯科物理技术学院（国立研究型大学））；Institute for Computer Science and Mathematical Modeling, Sechenov First Moscow State Medical University（谢切诺夫第一莫斯科国立医科大学计算机科学与数学建模研究所）；University Bordeaux, LOMA UMR-CNRS 5798（波尔多大学，LOMA CNRS联合研究实验室）
It is well known that a covariant Lagrangian for relativistic charged particles can lead to a vanishing Hamiltonian. Alternatively, it is shown that using a "space+time" Lagrangian leads to a new canonical momentum and minimal coupling rule that describes the coupling of both electromagnetic and gravitational fields to a relativistic charged particle. Discrepancies between Hamiltonians obtained by various authors are resolved. The canonical momentum leads to a new form of the London equations and London gauge. Using the linearized Einstein field equation in harmonic coordinates, and a non-rela…
In recent years, the $H$ formulation of Maxwell's equation has become the de facto standard for simulating the time-dependent electromagnetic behavior of superconducting applications with commercial software. However, there are cases where other formulations are desirable, for example for modeling superconducting turns in electrical machines or situations where the superconductor is better described by the critical state than by a power-law resistivity. In order to accurately and efficiently handle those situations, here we consider two published approaches based on the magnetic vector potenti…
The study of the electromagnetic response in cuprate superconductors plays a crucial role in the understanding of the essential physics of these materials. Here the doping dependence of the electromagnetic response in cuprate superconductors is studied within the kinetic-energy driven superconducting mechanism. The kernel of the response function is evaluated based on the linear response approximation for a purely transverse vector potential, and can be broken up into its diamagnetic and paramagnetic parts. In particular, this paramagnetic part exactly cancels the corresponding diamagnetic par…
A theoretical study of small thermal and electromagnetic perturbations in type-II superconductors is studied in the flux creep regime with a power-law voltage-current characteristics. The space-time evolution of temperature and electric field was calculated using the heat diffusion equation, coupled with Maxwell's equations and material law, assuming that heat diffusion is small that the magnetic diffusion. An explicit blow-up-like solution is obtained, which describes the time and space evolution of the electromagnetic field and temperature due to flux jump instabilities.
We investigate the response of a stack of intrinsic Josephson junctions (IJJs) to terahertz (THz) electromagnetic (EM) irradiation. A significant amplification of the EM wave can be achieved by the IJJs stack when the incident frequency equals to one of the cavity frequencies. The irradiation excites pi phase kinks in the junctions, which stimulate the cavity resonance when the bias voltage is tuned. A large amount of dc energy is then pumped into the Josephson plasma oscillation, and the incident wave gets amplified. From the profound current step in IV characteristics induced at the cavity r…
We use the quasiclassical theory of superconductivity to calculate the response of the currents in the core of a vortex to an alternating electric field. We consider relatively clean superconductors with a mean free path larger than the coherence length (but still not in the 'superclean limit'). The current response of the core is dominated by the bound states of Caroli, de Gennes and Matricon, and differs significantly from the response obtained from the model of a ``normal core'' introduced by Bardeen and Stephen. The model of Bardeen and Stephen describes the limit of a dirty core and fails…
The new rare-earth arsenate superconductors are layered, low carrier density compounds with many similarities to the high-Tc cuprates. An important question is whether they also exhibit weak-coupling across randomly oriented grain-boundaries. In this work we show considerable evidence for such weak-coupling by study of the dependence of magnetization in bulk and powdered samples. Bulk sample magnetization curves show very little hysteresis while remanent magnetization shows almost no sample size dependence, even after powdering. We conclude that these samples exhibit substantial electromagneti…
We investigate the macroscopic effects of charge density waves (CDW) and superconductivity in layered superconducting systems with broken lattice inversion symmetry (allowing for piezoelectricity) such as two dimensional (2D) transition metal dichalcogenides (TMD). We work with the low temperature time dependent Ginzburg-Landau theory and study the coupling of lattice distortions and low energy CDW collective modes to the superconducting order parameter in the presence of electromagnetic fields. We show that superconductivity and piezoelectricity can coexist in these singular metals. Furthermo…
We report numerical calculations of a dynamic pairbreaking current density $J_d$ and a critical superfluid velocity $v_d$ in a nonequilibrium superconductor carrying a uniform, large-amplitude ac current density $J(t)=J_a\sinΩt$ with $Ω$ well below the gap frequency $Ω\ll Δ_0/\hbar$. The dependencies $J_d(Ω,T)$ and $v_d(Ω,T)$ near the critical temperature $T_c$ were calculated from either the full time-dependent nonequilibrium equations for a dirty s-wave superconductor and the time-dependent Ginzburg-Landau (TDGL) equations for a gapped superconductor, taking into account the GL relaxation ti…
Within the framework of the kinetic energy driven superconductivity, the electromagnetic response in cuprate superconductors is studied in the linear response approach. The kernel of the response function is evaluated and employed to calculate the local magnetic field profile, the magnetic field penetration depth, and the superfluid density, based on the specular reflection model for a purely transverse vector potential. It is shown that the low temperature magnetic field profile follows an exponential decay at the surface, while the magnetic field penetration depth depends linearly on tempera…
A nonlinear dynamics of the thermal and electromagnetic instability of critical state in type II superconductors has been analysed taking into account effects of dissipation and dispersion. An existance of a nonlinear running waves describing a final stage of evolution of the thermomagnetic instability in superconductors is demonstrated analitically.
We report on the superconducting properties of In doped SnTe which has recently been explored as a topological crystalline superconductor. Single crystals of Sn0.5In0.5Te have been synthesized by modified Bridgman method. Resistivity measurement performed in the range 1.6K to 300K shows metallic normal state with onset of superconducting transition at Tc = 4.5K. Bulk superconductivity has also been confirmed by DC magnetization, AC susceptibility and rf penetration depth measurements. The zero temperature upper critical field, lower critical field, coherence length, penetration depth are estim…
Structural, electronic, and magnetic properties of undoped and aliovalent-ion (Ca, F)-doped LaOFeP, which undergo superconducting transitions at transition temperatures (Tc) 4-7 K [Kamihara et al, J. Am. Chem. Soc. 128, 10012, (2006)], were investigated. Tc of the samples varied from 2.4 to 5.5 K in the undoped samples, and was increased up to ~7 K by Ca- and F-doping. The Tc increases are correlated with a decrease in the lattice volume. LaOFeP exhibits paramagnetism in the normal conducting state. Photoemission spectroscopy combined with first-principle band calculations clarified that Fe 3d…
We calculate the response of a vortex core in a layered superconductor to {\em ac} electromagnetic fields with frequencies $ω\lsim 2Δ/\hbar$. In this frequency range the response is dominated by order parameter collective modes which are coupled to the vortex-core bound states of Caroli, de Gennes and Matricon. Our calculations show that a vortex core has a more complex and richer dynamics than predicted by previous theories. The {\em ac} field drives an oscillating, nearly homogeneous supercurrent in the direction of the electric field, superimposed with a dissipative current flow which has a…
Many large-scale applications require electromagnetic modelling with extensive numerical computations, such as magnets or 3-dimensional (3D) objects like transposed conductors or motors and generators. Therefore, it is necessary to develop computationally time-efficient but still accurate numerical methods. This article develops a general variational formalism for any ${\bf E}({\bf J})$ relation and applies it to model coated-conductor coils containing up to thousands of turns, taking magnetization currents fully into account. The variational principle, valid for any 3D situation, restricts th…
Nonlinear dynamics of the thermal and electromagnetic instabilities of the mixed state in type II superconductors has been analysed taking into account the effect of dissipation and dispersion. The existence of nonlinear running waves describing the final stage of evolution of the thermomagnetic instability in superconductors is demonstrated analytically.
