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The possibility of magnetization resonant control in a Josephson superconductor-ferromagnet-superconductor $\varphi_{0}$ junction shunted by an $LC$ circuit is demonstrated. As a result of the resonance of Josephson oscillations with oscillations in the circuit, a time-independent superconducting current arises in the junction. Due to the coupling of the Josephson phase and the magnetization of the ferromagnetic layer, the resulting superconducting current leads to a deviation of the easy axis from its initial position and to a precession of the magnetization around the tilted axis. We show that the tilt value increases with the increasing spin-orbit interaction and the Josephson to magnetic energy ratio. An analytical expression for the magnetization tilt is obtained, which agrees well with the results of numerical calculations. The emerging possibility of resonant control of magnetization in a shunted $\varphi_{0}$ junction can be used in the development of novel technologies in the field of superconducting electronics and spintronics.

Usually, overdamped Josephson junctions do not exhibit chaotic behavior in their phase dynamics, either because the phase space dimension is less than three (as in the case of a single overdamped ac-driven junction) or due to the general tendency of systems to become less chaotic with increasing dissipation (as in the case of coupled overdamped junctions). Here we consider the so-called $\varphi_0$ superconductor/ferromagnet/superconductor Josephson junction in which the current flowing through the junction may induce magnetization dynamics in the ferromagnetic interlayer. We find that due to the induced magnetization dynamics, even in the overdamped limit, i.e. for a junction without capacitance, the junction may exhibit chaos and hysteresis that in some cases leads to multiple branches in its current-voltage characteristics. We also show that pulsed current signals can be used to switch between the different voltage states, even in the presence of added thermal noise. Such switching could be used in cryogenic memory components., Comment: 17 pages, 12 figures

Periodic structure of the magnetization reversal (MR) domains is studied within the superconductor-ferromagnetic-superconductor $\phi_0$-junction model. The model is described by the Cauchy problem for the system of nonlinear ordinary equations which is numerically solved by means of the 2-step Gauss-Legendre method. Two versions of parallel implementation on the basis of MPI and OpenMP techniques have been developed. Efficiency of both versions is confirmed by test calculations. An effect of frequency of ferromagnetic resonance on the configuration of MR domains has been investigated., Comment: The XXVI International Scientific Conference of Young Scientists and Specialists (AYSS-2022), JINR, Dubna

We demonstrate unusual features of phase dynamics, IV-characteristics and magnetization dynamics of the $\varphi_0$ Josephson junction at small values of spin-orbit interaction, ratio of Josephson to magnetic energy and Gilbert damping. In particular, an anomalous shift of the ferromagnetic resonance frequency with an increase of Gilbert damping is found. The ferromagnetic resonance curves show the Duffing oscillator behaviour, reflecting the nonlinear nature of Landau-Lifshitz-Gilbert (LLG) equation. Based on the numerical analysis of each term in LLG equation we obtained an approximated equation demonstrated both damping effect and Duffing oscillator features. The resulting Duffing equation incorporates the Gilbert damping in a special way across the dissipative term and the restoring force. A resonance method for the determination of spin-orbit interaction in noncentrosymmetric materials which play the role of barrier in $\varphi_0$ junctions is proposed.

We study a magnetization reversal by an electric current pulse in a superconductor/insulating ferromagnet/superconductor Josephson junction placed on top of a 3D topological insulator. It is demonstrated that such a system is perspective for low-dissipative spintronics because of the strong spin-momentum locking in the TI surface states. This property provides an ideally strong coupling between the orbital and spin degrees of freedom thus giving a possibility of efficient reversal of the magnetic moment by current pulse with amplitude lower than the critical current, that results in strongly reduced energy dissipation. The underlying physical mechanism of the reversal is discussed. The influence of the magnetic anisotropy on the controllability of the reversal by the pulse duration is investigated. In addition, a way of a simultaneous electrical detection of the reversal is proposed., Comment: 8 pages, 6 figures

Appearance and origin of resonance phenomena have been studied in the annular system of underdamped Josephson junctions. If no fluxon is trapped in the system, dynamics is governed by the motion of fluxon-antifluxon pairs, while if trapped fluxons are present, they can move solely but also simultaneously with the pairs. Locking between the rotating excitations (fluxons and antifluxons) and the Josephson frequency leads to the appearance of zero field steps in the current-voltage characteristics, which can further exhibit branching due to resonance between the rotating excitations and plasma oscillations in their tale. The number of zero field steps and their branching are strongly determined by the total number of excitations present in the system. High resolution analysis further reveals not only some interesting properties of zero field steps, but also shows that the current-voltage characteristics is determined not only by the number, but also by the type of excitations, i.e., whether the dynamics is governed only by the motion of fluxon-antifluxon pairs or the trapped fluxons, or they move simultaneously in the system.

Examination of an annular system of underdamped Josephson junctions in the presence of external radiation showed that the ability of the system to lock with some external radiation was determined not only by the number but also by the type of rotating excitations (fluxons or antifluxons). Shapiro steps can be observed in the current-voltage characteristics only in the system with trapped fluxons or in the system with fluxon-antifluxon pairs. If the trapped fluxons circulate simultaneously with fluxon-antifluxon pairs, there are no Shapiro steps regardless of the amplitude or frequency of the applied external radiation.

The $\varphi_0$ Josephson junctions formed by ordinary superconductors and a magnetic non-centrosymmetric interlayer are studied. We derive an analytical solution for the magnetization dynamics induced by an arbitrary current pulse and formulate the criteria for magnetization reversal. Using the obtained results, the form and duration of the current pulse are optimized. The agreement between analytical and numerical investigations is reached in the case of a large product of the ratio Josephson to magnetic energy, strength of spin-orbit interaction and a minimal value of the current pulse. The obtained results allow one to predict magnetization reversal at the chosen system parameters.

The magnetization reversal in ${\varphi_0}$-Josephson junction with direct coupling between magnetic moment and Josephson current has been studied. By adding pulse signal, the dynamics of magnetic moment components have been simulated and the full magnetization reversal at different parameters of the junction has been demonstrated. We obtain a detailed pictures representing the intervals of the damping parameter $\alpha$, Josephson to magnetic energy relation $G$ and spin-orbit coupling parameter $r$ with full magnetization reversal. A periodicity in the appearance of magnetization reversal intervals with increase in Josephson to magnetic energy relation is found. The obtained results might be used in different fields of superconducting spintronics.

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 realized. It is demonstrated that this can lead to splitting of the ferromagnet's easy-axis in the voltage driven regime. We show that such a splitting can lead to stabilization of an unconventional four-fold degenerate ferromagnetic state.

The Josephson $\varphi_0$ junctions with the current-phase relation $I = I_c \sin (\varphi-\varphi_0)$, where the phase shift $\varphi_0$ is proportional to the magnetic moment perpendicular to the gradient of the asymmetric spin-orbit potential, demonstrate a number of unique features important for superconducting spintronics and modern informational technologies. Here we show that a current sweep along IV-characteristic of the $\varphi_0$ junction may lead to regular magnetization dynamics with a series of specific phase trajectories. The origin of these trajectories is related to a direct coupling between the magnetic moment and the Josephson oscillations in these junctions, and ferromagnetic resonance when Josephson frequency coincides with the ferromagnetic one. We demonstrate that an external electromagnetic field can control the dynamics of magnetic moment within a current interval corresponding to a Shapiro step and produce topological transformation of specific precession trajectories. We demonstrate the appearance of the DC component of superconducting current and clarify its role in the transformation of IV-characteristics in the resonance region. Good agreement between numerical and analytical results has been found in the ferromagnetic resonance region. The presented results might be used for developing novel resonance methods of determination of the spin-orbit coupling parameter in the non-centrosymmetric materials. We discuss experiments which can test our results.

We study theoretically a dynamics of $\varphi_0$ junction with direct coupling between magnetic moment and Josephson current which shows features close to Kapitza pendulum. We have found that starting with oscillations along $z$-axis, the character of magnetization dynamics changes crucially and stable position of magnetic moment $\vec m$ is realized between $z-$ and $y$-axes depending on parameters of the system. Changes in critical current and spin-orbit interaction lead to the different stability regions for magnetization. An excellent agreement between analytical and numerical results is obtained for low values of the Josephson to magnetic energy ratio.

The peculiarities of the phase dynamics of stacks of coupled Josephson junctions with topologically trivial and nontrivial barriers have been investigated numerically and their comparative analysis is carried out. The effect of coupling and dissipation parameters on the parametric resonance in the breakpoint region is shown. It is found that the dependence of the breakpoint voltage on the dissipation parameter demonstrates a minimum due to the changing of frequency of longitudinal plasma wave. We have shown that in case of the stack with nontrivial barriers the observed minimum shifts along the $\beta$ to the $\sqrt{2}$ in comparison with trivial barriers stack. We assume that the found features may be used for the experimental determination of Majorana fermions in the stack of JJs with the nontrivial barriers.

We study magnetization reversal in a $\varphi_0$ Josephson junction with direct coupling between magnetic moment and Josephson current. Our simulations of magnetic moment dynamics show that by applying an electric current pulse, we can realize the full magnetization reversal. We propose different protocols of full magnetization reversal based on the variation of the Josephson junction and pulse parameters, particularly, electric current pulse amplitude, damping of magnetization and spin-orbit interaction. We discuss experiments which can probe the magnetization reversal in $\varphi_0$-junctions.

We study the phase dynamics and IV--characteristics of DC--SQUIDs consisting of Josephson junctions with topologically nontrivial barriers, which cause the appearance of Majorana bound state. Its comparative analysis with the trivial case is performed. The influence of external electromagnetic radiation is considered and the analysis of the amplitude dependence of the Shapiro step widths is performed. We have shown that in nontrivial case the width of even harmonic of Shapiro step is larger than width of odd harmonic. In the presence of external dc magnetic field a beating state is realized in the DC--SQUID, which leads to a resonance branch in the IV--curve. We show that in the presence of resonance branch the maximum width of Shapiro step and periods of its amplitude dependence are decreased in comparison of situation without resonance in both trivial and nontrivial cases. We demonstrate that in the presence of the resonance branch the chaotic behavior of IV--curve is reduced.

We calculate the Josephson current between two one-dimensional (1D) nanowires oriented along $x$ with proximity induced $s$-wave superconducting pairing and separated by a narrow dielectric barrier in the presence of both Rashba spin-orbit interaction (SOI) characterized by strength $\alpha$ and Zeeman fields ($h$ along $\hat z$ and ${\bf B}$ in the $x-y$ plane). We formulate a general method for computing the Andreev bound states energy which allows us to obtain analytical expressions for the energy of these states in several asymptotic cases. We find that in the absence of the magnetic fields the energy gap between the Andreev bound states decreases with increasing Rashba SOI constant leading eventually to touching of the levels. In the absence of Rashba SOI, the Andreev bound states depend on the magnetic fields and display oscillatory behavior with orientational angle of B leading to magneto-Josephson effect. We also present analytic expressions for the dc Josephson current charting out their dependence on ${\bf B}$, $h$, and $\alpha$. We demonstrate the existence of finite spin-Josephson current in these junctions in the presence of external magnetic fields and provide analytic expressions for its dependence on $\alpha$, $\bf B$ and $h$. Finally, we study the AC Josephson effect in the presence of the SOI (for $|{\bf B}|=h=0$) and an external radiation and show that the width of the resulting Shapiro steps in such a system can be tuned by varying $\alpha$. We discuss experiments which can test our theoretical results.

We present the results of numerical study of the phase dynamics of the DC-SQUID with topologically trivial and nontrivial barriers. In our calculations we take into account two components of supperconducting current, Cooper pairs ($2\pi$-periodic) and Majorana fermions ($4\pi$-periodic) currents. Magnetic field dependence of return current is presented. The qualitative behavior of this dependence is explained. We show that in case of two component superconducting current the periodicity of magnetic field dependence of return current displaced by Cooper pairs and Majorana fermion ratio over the magnetic field. This effect makes possible the experimental determination of ratio of Cooper pairs and Majorana fermions currents.

We have investigated the phase dynamics and IV-characteristics of shunted Josephson junctions coupled to an $LC$ circuit. When the Josephson frequency $\omega_J$ is close to the eigen frequency $\omega_{rc}$ of the coupled resonance circuit, the IV-characteristic demonstrates an additional $rc$-branch. We have investigated the features of the $rc$-branch and of the superconducting current component for different values of the resonance frequency. It has been found that the maximal value of the superconducting current depends on the resonance frequency and that it increases when the end point of the $rc$-branch approaches the critical current. The dependence of the maximal superconducting current on the resonance frequency at different values of the dissipation parameter is peculiar, for the optimized maximum appears to be independent of the system parameters within $1\%$., Comment: 8 pages, 8 figures

Two parallel stacks of coupled Josephson junctions are investigated to clarify the physics of transitions between the rotating and oscillating states and their effect on the IV-characteristics of the system. The detailed study of phase dynamics and bias dependence of the superconducting and diffusion currents allows one to explain all features of simulated IV-characteristics and demonstrate the correspondence in their behavior. The coupling between JJ in the stacks leads to the branching of IV-characteristics and a decrease in the hysteretic region. The crucial role of the diffusion current in the formation of the IV-characteristic of the parallel stacks of coupled Josephson junctions is demonstrated. We discuss the effect of symmetry in a number of junctions in the stacks and show a decrease of the branching in the symmetrical stacks. The observed effects might be useful for development of superconducting electronic devices based on intrinsic Josephson junctions.

The effect of external radiation on the phase dynamics of Josephson junctions shunted by an LC circuit is examined. It is shown that additional resonant circuit (rc) branches appear in the current voltage characteristics of the junctions when the Josephson frequency is equal to the frequency of the formed resonance circuit. We show that the amplitude dependence of the Shapiro step width crucially changes when the Shapiro step is on the rc-branch in comparison to the case of Josephson junction without shunt. The experimental implementation of these effects might give very important advantages for existing methods and technologies., Comment: 4 pages, 4 figures

The effect of electromagnetic wave irradiation on the phase dynamics of intrinsic Josephson junctions in high temperature superconductors is investigated. We predict three novel effects by variation of the radiation amplitude and frequency: changing of the longitudinal plasma wavelength at parametric resonance; double resonance of the Josephson oscillations with radiation and longitudinal plasma wave; charging of superconducting layers in the current interval corresponding to the Shapiro step. The "bump" structure in IVC recently observed experimentally is demonstrated. We also observe ragged Shapiro steps at double resonance., Comment: 5 pages, 7 figures; cond-mat.supr-con