We study a long topological Josephson junction with a ferromagnetic strip between two superconductors. The low-energy theory exhibits a non-local in time and space interaction between chiral Majorana fermions, mediated by the magnonic excitations in the ferromagnet. While short ranged interactions turn out to be irrelevant by power counting, we show that sufficiently strong and long-ranged interactions may induce a $\mathbb{Z}_2$-symmetry breaking. This spontaneous breaking leads to a tilting of the magnetization perpendicular to the Majorana propagation direction and the opening of a fermionic gap (Majorana mass). It is analogous to the Peierls instability in the commensurate Fröhlich model and reflects the nontrivial interplay between Majorana modes and magnetization dynamics. Within a Gaussian fluctuation analysis, we estimate critical values for the temporal and spatial non-locality of the interaction, beyond which the symmetry breaking is stable at zero temperature -- despite the effective one-dimensionality of the model. We conclude that non-locality, i.e., the stiffness of the magnetization in space and time, stabilizes the symmetry breaking. In the stabilized regime, we expect the current-phase relation to exhibit an experimentally accessible discontinuous jump. At nonzero temperatures, as usual in the 1D Ising model, the long-range order is destroyed by solitonic excitations, which in our case carry each a Majorana zero mode. In order to estimate the correlation length, we investigate the solitons within a self-consistent mean-field approach.
We explore a spin Josephson effect in a system of two ferromagnets coupled by a tunnel junction formed of 2D time-reversal invariant topological insulators. In analogy with the more commonly studied instance of the Josephson effect for charge in superconductors, we investigate properties of the phase-coherent {\it spin} current resulting from the misalignment of the in-plane magnetization angles of the two ferromagnets. We show that the topological insulating barrier offers the exciting prospect of hosting a {\it fractional} spin Josephson effect mediated by bound states at the ferromagnet-top…
We theoretically investigate the magnetization inside a normal metal containing the Rashba spin-orbit interaction (RSOI) induced by the proximity effect in an s-wave superconductor/normal metal/ferromagnetic metal/s-wave superconductor (S/N/F/S) Josephson junction. By solving the linearized Usadel equation taking account of the RSOI,we find that the direction of the magnetization induced by the proximity effect in N can be reversed by tuning the RSOI.Moreover, we also find that the direction of the magnetization inside N can be reversed by changing the superconducting phase difference, i.e., J…
In this work we present a new method to calculate the classical properties of magnetic nanoparticles. Based on the Bethe-Peierls (pair) approximation, we developed a simple system of equations for the classical magnetization of spins at any position within the nanoparticle. The nearest neightbor pair correlations are treated exactly for Ising spins, and the method can be generalized for various lattice symmetries. The master equation is solved for the Glauber dynamics (single-spin-flip) in order to obtain the time evolution of the magnetization. The capabilities of the model are demonstrated t…
The prospect of combining the dissipationless nature of superconducting currents with the spin-polarization of magnetic materials is interesting with respect to exploring superconducting analogues of topics in spintronics. In order to accomplish this aim, it is pivotal to understand how spin-supercurrents interact dynamically with magnetization textures. We investigate the appearance of a spin-supercurrent and the resulting magnetization dynamics in a textured magnetic Josephson current by using three experimentally relevant models: i) a S/F/S junction with spin-active interfaces, ii) a S/F1/F…
We investigate Josephson junctions on the surface of a three-dimensional topological insulator in planar, step, and edge geometries. The elliptical nature of the Dirac cone representing the side surface states of the topological insulator results in a scaling factor in the Josephson current in a step junction as compared to the planar junction. In edge junctions, the contribution of the Andreev bound states to the Josephson current vanishes due to spin-momentum locking of the surface states. Furthermore, we consider a junction with a ferromagnetic insulator between the superconducting regions.…
The Josephson current in a diffusive superconductor/ferromagnet/superconductor junction with precessing magnetization is calculated within the quasiclassical theory of superconductivity. When the junction is phase-biased, a stationary current (without a.c. component) can flow through it despite the non-equilibrium condition. A large critical current is predicted due to a dynamically induced long range triplet proximity effect. Such effect could be observed in a conventional hybrid device close to the ferromagnetic resonance.
Strong dependence of the Josephson energy on the magnetization orientation in Josephson junctions with ferromagnetic interlayers and spin-orbit coupling opens a way to control magnetization by Josephson current or Josephson phase. Here we investigate the perspectives of magnetization control in superconductor/ferromagnet/superconductor (S/F/S) Josephson junctions on the surface of a 3D topological insulator hosting Dirac quasiparticles. Due to the spin-momentum locking of these Dirac quasiparticles a strong dependence of the Josephson current-phase relation on the magnetization orientation is …
We discuss the effects of thermal noise on the magnetic response of a lateral ferromagnetic Josephson junction with spin-orbit coupling and out-of-plane magnetization. The direction of the magnetic moment in the ferromagnetic layer can be inverted by using controlled current pulses. This phenomenon is due to the magnetoelectric effect that couples the flowing charge current and the magnetization of the ferromagnet. We investigate the magnetization reversal effect versus intrinsic parameters of the ferromagnet, such as the Gilbert damping and strength of the spin-orbit coupling. We estimate the…
The review is devoted to the fundamental aspects and characteristic features of the magnetoelectric effects, reported in the literature on Josephson junctions (JJs). The main focus of the review is on the manifestations of the direct and inverse magnetoelectric effects in various types of Josephson systems. They provide a coupling of the magnetization in superconductor/ferromagnet/superconductor JJs to the Josephson current. The direct magnetoelectric effect is a driving force of spin torques acting on the ferromagnet inside the JJ. Therefore it is of key importance for the electrical control …
Josephson current in SFcFS junctions with arbitrary transparency of the constriction (c) is investigated. The emphasis is done on the analysis of the supercurrent dependencies on the misorientation angle $θ$ between the in-plane magnetizations of diffusive ferromagnetic layers (F). It is found that the current-phase relation $I(ϕ)$ may be radically modified with the $θ$ variation: the harmonic $I_{1}\sin ϕ$ vanishes for definite value of $θ$ provided for identical orientation of the magnetizations ($θ=0$) the junction is in the $"π"$ state. The Josephson current may exhibit a nonmonotonic depe…
We investigate the interplay between ferromagnetic and superconducting order at the edge of a quantum spin Hall insulator. Using complementary analytical and self-consistent numerical approaches, we study a ferromagnetic Josephson junction and show how the direct coupling between magnetism and the superconducting U(1) phase gives rise to several unusual phenomena which distinguishes the present system from its non-topological equivalent. In particular, we demonstrate how the anomalous current-phase relation triggers supercurrent-induced magnetization dynamics and also study the spatial localiz…
Josephson junctions based on three-dimensional topological insulators offer intriguing possibilities to realize unconventional $p$-wave pairing and Majorana modes. Here, we provide a detailed study of the effect of a uniform magnetization in the normal region: We show how the interplay between the spin-momentum locking of the topological insulator and an in-plane magnetization parallel to the direction of phase bias leads to an asymmetry of the Andreev spectrum with respect to transverse momenta. If sufficiently large, this asymmetry induces a transition from a regime of gapless, counterpropag…
We study the Josephson current through a ferromagnetic trilayer, both in the diffusive and clean limits. For colinear (parallel or antiparallel) magnetizations in the layers, the Josephson current is small due to short range proximity effect in superconductor/ferromagnet structures. For non colinear magnetizations, we determine the conditions for the Josephson current to be dominated by another contribution originating from long range triplet proximity effect.
We study supercurrent and proximity vortices in a Josephson junction made of disordered surface states of a three-dimensional topological insulator with a proximity induced in-plane helical magnetization. In a regime where the rotation period of helical magnetization is larger than the junction width, we find supercurrent 0-π crossovers as a function of junction thickness, magnetization strength, and parameters inherent to the helical modulation and surface states. The supercurrent reversals are associated with proximity induced vortices, nucleated along the junction width, where the number of…
We theoretically study the magnetization inside a normal metal induced in an s-wave superconductor/ferromagnetic metal/normal metal/ferromagnetic metal/s-wave superconductor (S/F1/N/F2/S) Josephson junction. Using quasiclassical Green's function method, we show that the magnetization becomes finite inside N. The origin of this magnetization is due to odd-frequency spin-triplet Cooper pairs formed by electrons of equal and opposite spins, which are induced by proximity effect in the S/F1/N/F2/S junction. We find that the magnetization M(d,q) in N can be decomposed into two parts, M(d,q)=MI(d)+M…
We study the effect of coupling between the superconducting current and magnetization in the superconductor/ferromagnet/superconductor Josephson junction under an applied circularly polarized magnetic field. Manifestation of ferromagnetic resonance in the frequency dependence of the amplitude of the magnetization and the average critical current density is demonstrated numerically. The IV-characteristics show subharmonic steps that form devil's staircases, following a continued fraction algorithm. The origin of the found steps is related to the effect of the magnetization dynamics on the phase…
We theoretically study the magnetism induced by the proximity effect in the normal metal of ferromagnetic Josephson junction composed of two $s$-wave superconductors separated by ferromagnetic metal/normal metal/ferromagnetic metal junction (${S}/{F}/{N}/{F}/{S}$ junction). We calculate the magnetization in the $N$ by solving the Eilenberger equation. We show that the magnetization arises in the ${N}$ when the product of anomalous Green's functions of the spin-triplet even-frequency odd-parity Cooper pair and spin-singlet odd-frequency odd-parity Cooper pair in the ${N}$ has a finite value. Th…
It has been predicted theoretically that an unconventional odd-frequency spin-triplet component of superconducting order parameter can be induced in multilayered ferromagnetic structures with non-collinear magnetization. In this work we study experimentally nano-scale devices, in which a ferromagnetic spin valve is embedded into a Josephson junction. We demonstrate two ways of in-situ analysis of such Josephson spin valves: via magnetoresistance measurements and via in-situ magnetometry based on flux quantization in the junction. We observe that supercurrent through the device depends on the r…
In the past year, several groups have observed evidence for long-range spin-triplet supercurrent in Josephson junctions containing ferromagnetic (F) materials. In our work, the spin-triplet pair correlations are created by non-collinear magnetizations between a central Co/Ru/Co "synthetic antiferromagnet" (SAF) and two outer thin F layers. Here we present data showing that the spin-triplet supercurrent is enhanced up to 20 times after our samples are subject to a large in-plane magnetizing field. This surprising result can be explained if the Co/Ru/Co SAF undergoes a "spin-flop" transition, wh…
We present a theoretical study of the Josephson coupling of two s-wave superconductors which are connected through a diffusive contact consisting of noncollinear ferromagnetic domains. First, we consider a contact with two domains with magnetization vectors misoriented by an angle $θ$. Using the quantum circuit theory, we find that in addition to the charge supercurrent, a spin supercurrent, which is even in $ϕ$ and odd in $θ$, with a spin polarization normal to the magnetization vectors flows between the domains. Furthermore, with asymmetric insulating barriers at the interfaces of the juncti…
We present an exhaustive study of the coherent heat transport through superconductor-ferromagnet(S-F) Josephson junctions including a spin-filter (I$_{sf}$) tunneling barrier. By using the quasiclassical Keldysh Green's function technique we derive a general expression for the heat current flowing through a S/F/I$_{sf}$/F/S junction and analyze the dependence of the thermal conductance on the spin-filter efficiency, the phase difference between the superconductors and the magnetization direction of the ferromagnetic layers. In the case of non-collinear magnetizations we show explicitly the con…
The simultaneous co-existence and gate-tuneability of the superconducting (SC), magnetic and topological orders in magic angle twisted bilayer graphene (MATBG) open up entirely new possibilities for the creation of complex hybrid Josephson junctions (JJ). Here we report on the creation of gate-defined, magnetic Josephson junctions in MATBG, where the weak link is gate-tuned close to the correlated state at a moiré filling factor of ν=-2. A highly unconventional Fraunhofer pattern emerges, which is phase-shifted and asymmetric with respect to the current and magnetic field directions, and shows…
Superconducting materials with non-centrosymmetric lattices lacking the space inversion symmetry are known to exhibit a variety of interesting parity-breaking phenomena, including the anomalous Josephson effect. Here we consider a Josephson junction consisting of two non-centrosymmetric superconductors (NCSs) connected by a uniaxial ferromagnet, and demonstrate that it exhibits a direct analog of the Chiral Magnetic Effect observed in Dirac and Weyl semimetals. We propose to use this "Chiral Magnetic Josephson junction" (CMJ junction) as an element of a qubit with a Hamiltonian tunable by the …
We study the Josephson effect in a superconductor--ferromagnet--superconductor (SFS) junction with ferromagnetic domains of non-collinear magnetization. As a model for our study we consider a diffusive junction with two ferromagnetic domains along the junction. The superconductor is assumed to be close to the critical temperature $T_c$, and the linearized Usadel equations predict a sinusoidal current-phase relation. We find analytically the critical current as a function of domain lengths and of the angle between the orientations of their magnetizations. As a function of those parameters, the …
We consider a hybrid structure consisting of superconducting or normal leads with a combined ferromagnet-3D topological insulator interlayer. We compare responses of a Josephson junction and a normal junction to magnetic texture dynamics. In both cases the electromotive force resulting from the magnetization dynamics generates a voltage between the junction leads. For an open circuit this voltage is the same for normal and superconducting leads and allows for electrical detection of magnetization dynamics and a structure of a given magnetic texture. However, under the applied current the elect…
We propose the phase dynamics induced by spin waves in a superconductor/ferromagnet/superconductor (SC/FM/SC) Josephson junction. The resistively shunted junction (RSJ) model, which describes the dynamics of superconducting phase difference, is extended to include the spin wave excitation by ferromagnetic resonance (FMR) using the gauge invariant phase difference between two s-wave superconductors. The current-voltage characteristics show step structures when the magnetization in FM is driven by tuning the microwave frequency to FMR in the SC/FM/SC junction. The result presents a new route to …
We investigate Josephson and proximity effects on the surface of a topological insulator on which superconductors and a ferromagnet are deposited. The superconducting regions are described by the conventional BCS Hamiltonian, rather than the superconducting Dirac Hamiltonian. Junction interfaces are assumed to be dirty. We obtain analytical expressions of the Josephson current and the proximity-induced anomalous Green's function on the topological insulator. The dependence of the Josephson effect on the junction length, the temperature, the chemical potential and the magnetization is discussed…
We show that injecting nonequilibrium spins into the superconducting leads strongly enhances the stationary Josephson current through a superconductor-ferromagnet-superconductor junction. The resulting long-range super-current through a ferromagnet is carried by triplet Cooper pairs that are formed in s-wave superconductors by the combined effects of spin injection and exchange interaction. We quantify the exchange interaction in terms of Landau Fermi-liquid factors. The magnitude and direction of the long-range Josephson current can be manipulated by varying the angles of the injected polariz…
We investigate anomalous Hall effect in a magnet coupled to a triplet superconductor under phase gradient. It is found that the anomalous Hall supercurrent arises from non-trivial structure of the magnetization. The magnetic structure manifested in the Hall supercurrent is characterized by even order terms of the exchange coupling, essentially different from that discussed in the context of anomalous Hall effect, reflecting the disspationless nature of supercurrent. We also discuss a possible candidate for magnetic structure to verify our prediction.
We study two different superconductor-ferromagnet (S/F) structures. We consider first a Josephson junction which consists of two S/F bilayers separated by an insulating layer. We show that for an antiparallel alignment of the magnetization in the two F layers the Josephson critical current $I_c$ increases with increasing exchange field $h$. The second system we consider is a S/F structure with a local inhomogeneity of the magnetization in the ferromagnet near the S/F interface. Due to the proximity effect not only a singlet but also a triplet component of the superconducting condensate is indu…
The order parameter of superconducting pairs penetrating an inhomogeneous magnetic material can acquire a long range triplet component (LRTC) with non-zero spin projection. This state has been predicted and generated recently in proximity systems and Josephson junctions. We show using an analytically derived domain wall of an exchange spring how the LRTC emerges and can be tuned with the twisting of the magnetization. We also introduce a new kind of Josephson current reversal, the triplet $0-π$ transition, that can be observed in one and the same system either by tuning the domain wall or by v…
We consider a superconductor-two dimensional topological insulator- superconductor junction (S-2DTI-S) and study how the 2π- and 4π-periodic Josephson currents are affected by the electron-electron interaction. In the long-junction limit the supercurrent can by evaluated by modeling the system as a helical Luttinger liquid coupled to superconducting reservoirs. After having introduced bosonization in the presence of the parity constraint we turn to consider the limit of perfect and poor interfaces. For transparent interfaces, where perfect Andreev reflections occur at the boundaries, the Josep…
We study theoretically charge transport properties of normal metal (N) / ferromagnet insulator (FI) / superconductor (S) junction and S/FI/S junction formed on the surface of three-dimensional topological insulator (TI), where chiral Majorana mode (CMM) exists at FI/S interface. We find that CMM generated in N/FI/S and S/FI/S junctions are very sensitively controlled by the direction of the magnetization ${\bm m}$ in FI region. Especially, the current-phase relation of Josephson current in S/FI/S junctions has a phase shift neither 0 nor $π$, which can be tuned continuously by the component of…
We study the Josephson effect and pairing correlations in SFFS junctions that consist of conventional superconductors (S) connected through two metallic monodomain ferromagnets (F) with transparent and spin inactive interfaces. We solve the Eilenberger equations for arbitrary relative orientation of magnetizations of the two F layers in the clean limit and for moderate disorder in ferromagnets. Spatial variation of pair amplitudes, singlet $f_s$, and odd in frequency triplet $f_{t0}$ and $f_{t1}$, with 0 and $\pm1$ spin projections, as well as the Josephson current-phase relations are calculat…
We consider a SFS Josephson junction made of two superconductors S and a multidomain ferromagnet F with an in-plane magnetization. We assume that the neighboring domains of the ferromagnet are separated by Neel domain walls. An odd-frequency triplet long-range component of superconducting correlations arises in the domain walls and spreads into the domains over a long distance of the order $ξ_T = \sqrt{D / 2 πT}$, where $D$ is the diffusion coefficient (dirty limit is implied). We calculate the contribution of this component to the Josephson current in the situation when conventional short-ran…
Josephson junctions in two-dimensional topological insulators with embedded magnetic domains can host a number of topological phases, in particular, Jackiw-Rebbi solitons and Majorana zero modes. These different non-trivial phases appear in such junctions for multiple-domain magnetic islands, showing a rich multi-gap structure. Features related to the interplay between superconductivity and magnetism in these systems cannot be easily discerned looking at behavior of the Andreev spectrum and the concomitant dc Josephson effect. Instead, the thermal conductance is very sensitive to the nature of…
Ballistic Josephson junctions are studied in the presence of a spin-splitting field and spin-orbit coupling. A generic expression for the quasi-classical Green's function is obtained and with its help we analyze several aspects of the proximity effect between a spin-textured normal metal (N) and singlet superconductors (S). In particular, we show that the density of states may show a zero-energy peak which is a generic consequence of the spin-dependent couplings in heterostructures. In addition we also obtain the spin current and the induced magnetic moment in a SNS structure and discuss possi…
We investigate SFF structures (S-superconductor, F feromagnetic metal) with noncollinear magnetizations of F films with arbitrary transparency of FF interface. We show the existence of phase slips both at SF and FF interfaces which manifest themselves in the anomalous dependence of the spin-triplet correlations on misorientation angle between magnetization vectors in the F-layers. We discuss how these effects can be observed in experiments with Josephson $π$-junctions.
