Anomalous Josephson effect

In conventional superconductor/insulator/superconductor JJs, the current–phase dependence near the critical temperature is sinusoidal, I(φ) = I_c sin φ; however, with decreasing temperature, the contribution of higher harmonics ∼ I_n sin (nφ) can be observed, but the current–phase dependence remains antisymmetric, I(–φ) = –I(φ) [4]. When symmetry is broken with respect to time reversal, a more general dependence I(φ) = I₀sin (φ + φ₀) arises, as pointed out in Josephson's paper [67]. Such a general dependence is also predicted in JJ with unconventional superconductors [41, 43, 44].

Buzdin [6], using the phenomenological Ginzburg–Landau (GL) equations, showed that a JJ with a magnetic normal metal as a weak coupling with a Rashba-type spin-orbit interaction has a specific nonsinusoidal current–phase dependence. The ground state of such a junction is characterized by a finite phase difference φ₀, which is proportional to the magnitude of the spin-orbit interaction and the exchange energy in the magnetic metal. As a result, a direct coupling is realized between the magnetic moment and the Josephson current, and the corresponding JJs are called φ₀ junctions. The thickness of the metal layer in such a junction determines the magnitude of the phase shift, which may be of interest for superconducting spintronics. It should be noted that there is a difference with the case of a JJ with a dominant second harmonic, where there is also a phase shift through the junction at a negative shift of the second harmonic, but there is no coupling between the magnetic exchange field and the superconducting phase. The anomalous properties of the φ₀ junction are associated with the features of the superconducting proximity effect in a magnetic metal with broken inversion symmetry.

The special nature of the electronic spectrum in materials with broken inversion symmetry arises due to Rashba-type spin-orbit coupling [68, 69] α (σ × p)n, where n is the unit vector along the gradient of the asymmetric potential, and the parameter α describes its value. This type of interaction, taking into account the exchange field h acting on the electron spin, leads to the following GL free energy density [70, 71]:

where ψ is the superconducting order parameter, D_i = –i∂_i – 2eA_i , a and b are the GL coefficients, and γ is the gyromagnetic ratio. The special nature of superconductivity in a material with broken inversion symmetry is described by the last term in (1) with the coefficient ∼ α.

Neglecting the orbital effect in geometry, where the n and h vectors are mutually perpendicular and perpendicular to the current direction (the x-axis, see Fig. 1), and also neglecting the nonlinear terms, the GL equation is reduced to the form [6]

whose solution has the form

where . The expression for the superconducting current in the limit of a long junction is written as

i.e., .

A similar dependence can be obtained up to |Δ|² at temperatures close to the critical one T_c if we use the expression for the superconducting current in terms of the anomalous Green's functions f_ij (v, r):

where N(0) is the density of states at the Fermi level. Anomalous Green's functions f_ij (v, r) are determined from the Eilenberger equations [72]. Under the condition L > ν/h, the main contribution to (4) comes from directions with |ν_x | ≲ ν and the formula for the current takes the simple form

Here,

and, in the absence of spin-orbit interaction (α = 0), this expression coincides with the corresponding expression for the 2D SFS junction obtained in [7]. Comparing (5) with the formula for j(φ) derived from the Ginzburg–Landau theory (3), we can see that the phase shift φ₀ = 4αhL/ν² in both cases is proportional to the spin-orbit interaction and the product hL. On the other hand, the critical current in (5) oscillates with L, changing its sign. This is typical for SFS junctions with a strong exchange field h ≫ T_c [3]. Such oscillations are absent in the GL approximation (3), since it is valid for h ≲ T_c; otherwise, the gradient terms in (1) change signs and it becomes necessary to take higher derivatives into account. Such a modified GL functional indeed qualitatively describes the oscillations of the superconducting order parameter under the proximity effect in the S/F structure [3].

As shown in [6], in the 1D weak coupling model (single-channel approximation), the current–phase dependence is obtained similarly to (5):

Above are the results in the pure limit (ballistic mode). In diffusion mode, a convenient approach is provided by the Usadel equations [73] for Green's functions integrated over the Fermi surface: F_ij (r) = 〈f_ij (v, r)〉. The superconducting current in this case can also be represented as j (φ) = j₀sin (φ + φ₀). Consequently, φ₀ junction formation based on materials with broken inversion symmetry is a fairly common phenomenon that can be observed in both 'clean' and 'dirty' limits.

In the φ₀ Josephson junction, the phase shift is proportional to the magnetic moment perpendicular to the gradient of the asymmetric spin-orbit potential [6], which makes it possible to control the internal magnetic moment of the ferromagnetic layer using the superconducting phase difference, that is, the Josephson current. The spin dynamics of the SFS junction has been intensively studied recently, demonstrating a number of unique properties. We note the pioneering research [74], in which the narrowing of the ferromagnetic resonance below the superconducting transition temperature in Nb/Ni₈₀Fe₂₀ was observed. The dynamics of a single spin in the Josephson junction was studied theoretically in papers [75–78], the dynamically induced triplet proximity effect in the SFS was demonstrated in [79, 80], and the properties of junctions with several ferromagnetic layers with different magnetizations were discussed in [31, 32].

Investigations of the SFS φ₀ junction in the low frequency regime ℏω_J ≪ T_c (where ω_J = 2eV/ℏ is the Josephson frequency) [7] using a quasi-static approach to the superconducting subsystem, in contrast to the case analyzed in [79, 80], led to the conclusion that the superconducting current can produce a strong orientational effect on the magnetic moment of the ferromagnetic layer. More interestingly, an alternating Josephson current, when there is DC voltage V at the φ₀ junction, causes magnetic precession, which can be controlled by the appearance of higher harmonics in the CPR, as well as DC components in the superconducting current. In certain regimes, a complete magnetization reversal can be observed, and in the case of a strong coupling between the magnetic and superconducting subsystems, complex nonlinear dynamic regimes arise [7].

To demonstrate the unusual properties of the φ₀ junction, the authors in Ref. [7] consider the case of magnetic anisotropy of an easy-axis ferromagnet (Fig. 2). Both the easy axis and the gradient of the asymmetric spin-orbit potential n are directed along the z-axis. It was assumed that suitable materials for the intermediate layer F could be MnSi or FeGe. In these systems, the absence of an inversion center is associated with the crystal structure, but the origin of symmetry with broken inversion can be due to external factors, as in the case near the surface of a thin F film. We note that study [7] did not take into account the magnetic induction, which in the xy plane is negligibly small for a thin layer F, while the demagnetization coefficient cancels internal induction along the z-axis (N = 1). The coupling between the subsystems F and S, due to the orbital effect, was studied in Ref. [81] and, as it turned out, is very weak and quadratic in the magnetic moment M when the flow M through the layer F is small compared to the flux quantum Φ₀ = h/2e.

For I < I_c, the total energy of the φ₀ junction is given by [82]

where the superconducting part is

In the ballistic limit, an estimate of the characteristic Josephson energy E_J = Φ₀ I_c/2π leads to Φ₀ I_c/S∼ with ℓ = 4hL/ℏν_F, where S, L, and h are the section, length, and exchange field in layer F, respectively [3]. The phase shift is defined as

where the parameter ν_SO/ν_F characterizes the relative intensity of the spin-orbit interaction [6]. It is assumed, that ν_SO/ν_F ∼ 0.1.

The contribution of the magnetic energy is reduced to the anisotropy energy

where K is the anisotropy constant and is the volume of the F-layer.

Naturally, one can expect that the most interesting situation corresponds to the case when the magnetic anisotropy energy does not greatly exceed the Josephson energy. Measurements [83] on permalloy with very weak anisotropy imply K ∼ 4 × 10⁻⁵ K Å ⁻³. On the other hand, the typical value of L in an SFS junction is L ∼ 10 and sin ℓ/ℓ ∼ 1. Then, the ratio of the Josephson energy to the magnetic energy will be E_J/E_M ∼ 100 at T_c ∼ 10 K. Naturally, in a more realistic case of stronger anisotropy, this ratio will be smaller, but one can expect a wide variety of regimes from E_J/E_M ⪡ 1 to E_J/E_M ⪢ 1.

The shift of the superconducting phase difference φ and the precession of the magnetic moment M_y = M₀ sin θ (where θ is the angle between the z-axis and the M direction) are determined from the minimum energy condition ∂_φ E_tot = ∂_φ0 E_tot = 0, which results in

This means that the superconducting current forces the rotation of the magnetic moment M_y in the plane yz. Therefore, for small angles, the dependence θ (I) is linear. In principle, the Γ parameter can be greater than one. In this case, if the condition I/I_c ⩾ 1/Γ is satisfied, the magnetic moment will be oriented along the y-axis. Therefore, the application of DC superconducting current changes the direction of magnetization, while applying an AC superconducting current to a φ₀ junction can generate magnetic moment precession.

It was noted in [7] that, when the spin-orbit potential gradient is directed along y (perpendicular to the easy axis z), then φ₀ = ℓ (ν_SO/ν_F) cos θ. The total energy (7) has two minima θ = (0, π), and the degeneracy between them is lifted when a current is applied. However, an energy barrier exists for a transition from one minimum to another. This barrier can disappear if Γ > 1 and the current is large enough, I > I_c/Γ. In this mode, the superconducting current will cause the magnetization to switch between one stable configuration, θ = 0, and another, θ = π. This corresponds to switching between the +φ₀ and –φ₀ states. Reading the state of the φ₀ junction can be easily done if it is part of some SQUID-like circuit (φ₀ junction causes the diffraction pattern to shift by φ₀) [7].

The JJ in the given voltage mode and, accordingly, the AC Josephson effect provide an ideal tool for studying magnetic dynamics in the φ₀ junction. In this case, the superconducting phase changes with time as φ (t) = ω_J t. With ℏ ω_J ⪡ T_c, you can use a static value for JJ energy (7), considering φ (t) to be an external potential. In Josephson junctions with a thin ferromagnetic layer, the superconducting phase difference and the F-layer magnetization are two coupled dynamic variables. The system of equations describing the dynamics of these variables is formed from the Landau–Lifshitz–Gilbert (LLG) [84] equation and the Josephson relations for the phase difference and current. In particular, the dynamics of the magnetization of the system is described by the LLG equation with an effective field depending on the phase difference:

where γ is the gyromagnetic ratio, α is the phenomenological dissipation parameter, φ is the phase difference between superconductors along the junction, M₀ = |M|, , K is the anisotropy constant, is the volume of the F-layer, r = lν_SO/ν_F is the spin-orbit interaction parameter, ν_SO/ν_F characterizes the intensity of spin-orbit interaction, ν_F is the Fermi velocity, l = 4hL/(ℏ ν_F), L is the length of the F layer, and h denotes the exchange field in the ferromagnetic layer. The complete system of equations used in numerical calculations, in normalized units, takes the form

where β_c = 2eI_c CR²/ℏ is the McCumber parameter, m_i = M_i /M₀ for i = x,y,z, and ω_F = ω_F/ω_c with ferromagnetic resonance frequency ω_F = γK/M₀ and characteristic frequency ω_c = 2eRI_c/ℏ. Here, time is normalized to , external current I is normalized to I_c, and voltage V, to V_c = I_c R.

Usually, this system of equations is solved numerically by the Runge–Kutta or Gauss–Legendre methods, as a result of which m_i (t), V(t), and φ (t) are determined as functions of time and external current I. After the averaging procedure [85, 86], the CVC is calculated for fixed system parameters [35]. Below are a number of results of modeling the properties of the φ₀ junction based on system of equations (13).

Ferromagnetic resonance (FMR) is one of the main phenomena that occurs in the SFS structure when the Josephson frequency approaches that of the ferromagnet eigenmode. To demonstrate the manifestation of FMR in the φ₀ junction along the CVC, in Refs [35, 87] the maximum and minimum values of the magnetization components were determined, in particular, and , calculated at each value of the bias current. FMR also appears in the dependence of the average superconducting current as a function of bias current. The manifestation of FMR is shown in Fig. 3, which shows the CVC of the φ₀ Josephson junction, which demonstrates specific behavior in the vicinity where the Josephson frequency coincides with the ferromagnetic one, i.e., in the region of FMR [87]. The dependence of the average value of the superconducting current on the value of the bias current also demonstrates a manifestation of FMR in the form of a maximum at I = 0.6 (see Fig. 3b). FMR manifests itself clearly in the dependence of the maximum value of the oscillation amplitude m_y on voltage, which is shown in the inset to Fig. 3b. The above dependences reflect the mutual influence of the Josephson current and magnetization precession in the ferromagnetic layer in the φ₀ junction.

Depending on the magnitude of the spin-orbit coupling, the manifestation of FMR in the CVC can be quite significant, as shown in Fig. 4, which shows parts of the CVC of the φ₀ junction for G = 0.1, α = 0.1, ω_F = 0.5 for different values of the spin-orbit interaction parameter.

Based on the presented results, it can be noted that a change in the parameters of the Josephson junction and the ferromagnetic layer in a system with damping can lead to a fairly strong coupling between the superconducting current and magnetization. The contribution of the superconducting DC current manifests itself here as a deviation of the CVC from a linear dependence in the resonant region. Note that the observed feature in the CVC in the resonance region actually reflects the appearance of a resonant branch, which is emphasized by the appearance of the corresponding hysteresis at r = 0.7 and 1. With an increase in the spin-orbit coupling parameter, the rate of increase in the amplitude of the magnetic moment increases, and, accordingly, the length of the resonant branch in the CVC increases. The mechanism of the appearance of this branch is similar to the mechanism in shunted Josephson junctions at parallel resonance [88, 89], as well as the appearance of a resonant branch in the CVC of a two-terminal SQUID [90].

The influence of the spin-orbit interaction on the resonant character of the dependence of on voltage, shown in Fig. 5 for various values of the spin-orbit interaction parameter, can serve as a theoretical justification for developing an experimental method to determine the intensity of spin-orbit coupling in noncentrosymmetric materials.

Note that, in the equation of the RCSJ (Resistively Capacitance Shunted Junction) model (the fourth equation in system (13)), which describes the dynamics of the φ₀ junction, the phase difference φ is replaced by φ –rφ₀ to preserve the gauge invariance. Taking it into account leads to an additional term r dm_y /dt in the equation of the RCSJ model, which was neglected in [7, 35]. Figure 5b shows the results without this term. As can be seen, its contribution does not change the qualitative picture of the described phenomenon for small values of the spin-orbit coupling parameter r. Up to values of the order of r = 0.5, the dependences of on V practically coincide in both cases. Figure 5c compares the results for r = 0.3.

The dynamics of the system can be studied analytically in the ℏ ω_J ⪡ T_c approximation, i.e., when the energy of the Josephson junction and the magnitude of the superconducting current are determined by a fixed Josephson frequency ω_J [7, 87], also neglecting the displacement current. In this case, the Josephson phase φ can be replaced by ω_J t, which means the choice of one point on the CVC of the junction. At a fixed voltage in the case without dissipation (α = 0) in the 'weak coupling' mode G ⪡ 1 (i.e., the Josephson energy E_J is small compared to the magnetic energy E_M), the last two equations in (13) lead to a linear time dependence of the phase difference φ = Vt (Josephson junction with voltage). At the chosen normalization V = ω_J, so φ = ω_J t. If the other components satisfy the conditions m_x , m_y ⪡ 1, then equations (13) can be linearized:

and the corresponding solutions are

Thus, the magnetic moment precesses around the z-axis. The precessing magnetic moment affects the φ₀ junction current:

where it is taken into account that . Thus, in addition to the oscillations of the first harmonic, the current contains contributions from higher harmonics. The amplitude of the harmonics increases near resonance and changes sign when ω_J = ω_F. Thus, monitoring the second harmonic of the current oscillations will make it possible to monitor the dynamics of the magnetic system.

An important role in the dynamics of the system under consideration is played by dissipation, the inclusion of which leads to a constant contribution to the Josephson current. Near the resonance ω_J ≈ ω_F, the linearization conditions leading to equations (15) are violated, and allowance for dissipation becomes necessary. In this case [7], the linearization of the LLG equation in system (13), taking into account m_z ≈ 1 and neglecting the quadratic terms m_x and m_y , leads to

The corresponding expression for m_y in the presence of dissipation takes the form

where

Thus, m_y exhibits resonance with dissipation when the Josephson frequency is tuned to the ferromagnetic frequency (ω_J → ω_F). In addition, dissipation results in phase oscillations m_y (t) (term proportional to cos (ω_J t) in equation (18)). As a result, the superconducting current

contains a time independent (DC) component:

The presence of this DC contribution shows that the Gilbert damping plays an important role in the dynamics of the φ₀ junction. This contribution depends on the value of the spin-orbit interaction r and the ratio of the Josephson energy to the magnetic one G, and is absent at α = 0.

On the other hand, the presence of a given constant Josephson current at a constant voltage V applied to the junction means the presence of a dissipative regime, which can be easily detected experimentally. The appearance of a direct current peak near resonance resembles the appearance of a Shapiro step in the Josephson junctions in an external electromagnetic field. Note that the presence of the second harmonic in I(t) in Eqn (20) should also lead to half-integer Shapiro steps on the CVC of φ₀ junctions [7, 91].

Figure 6 shows the dependence of the maximum amplitude on voltage, calculated on the basis of system of equations (13) and the analytical dependence m_y (ω_J) according to formula (18), in addition to the dependence of the superconducting current I_s on voltage, calculated on the basis of system of equations (13) and the analytical dependence I₀(ω_J) according to formula (21). As can be seen, the numerical and analytical results are in good agreement with each other. We emphasize that numerical calculations do not use any approximations, as opposed to analytical ones (where the weak coupling mode is used and the case m_x , m_y ⪡ 1 is considered). This manifests itself in characteristic features at V ≈0.25 and V ≈0.16 in the numerically simulated dependence , which reflects the occurrence of ferromagnetic resonance harmonics at ω_J = ω_F/2 and ω_J = ω_F/3.

It is expected that the impact of external microwave radiation with frequency ω_R on the φ₀ junction will lead to a number of new interesting phenomena. It was noted in [7] that, in addition to integer Shapiro steps at ω_J = nω_R, half-integer steps will appear in the CVC. Second, the microwave magnetic field can also generate an additional magnetic precession with a frequency of ω_R. Depending on the parameters of the φ₀ junction and the microwave radiation amplitude, the main precession mechanism can be associated either with the Josephson current or with microwave radiation. In the latter case, the spin-orbit coupling can significantly affect the width of the Shapiro steps. Therefore, one can expect its sharp increase at frequencies near the ferromagnetic resonance. In the event that the influence of radiation and Josephson current on the precession of the magnetic moment are comparable, a rather complex regime can be observed. In the case of a ferromagnet with weak in-plane anisotropy, the detailed dynamics of the magnetic precession can change dramatically. Note that a detailed study of these phenomena has not yet been carried out.

Of interest is the 'strong coupling' limit Γ ⪢ 1 (but r ⪡ 1), which can also be considered analytically [7]. In this case, m_y ≈ 0, and the solution of the LLG equations leads to

which are magnetization reversal equations; a complete flip occurs at Γ/ω > π/2. Strictly speaking, these solutions are not exact oscillatory functions in the sense that m_z (t) rotates counterclockwise around the center of the sphere, and then rotates and returns to the position m_z (t = 0) = 1 clockwise like a pendulum in a spherical potential.

One of the interesting results of numerical simulations of the ferromagnet magnetic moment dynamics along the CVC of the φ₀ junction is the discovery of a rather simple precession of the magnetic moment in some current intervals, leading to specific trajectories in the m_y – m_x , m_z – m_x , and m_z – m_y planes [35]. In this case, the spin-orbit interaction exerts a strong influence on the appearance of such intervals. The transformation of the dependence in the region of ferromagnetic resonance with a change in the spin-orbit interaction parameter is shown in Fig. 7.

With an increase in parameter r, along with the appearance of chaotic dynamics of the magnetization, regular regions appear in the dependence , denoted as R_i in Fig. 7d. These regions are characterized by specific trajectories, such as an apple (b), sickle (d), mushroom (e), fish (g), and moon (h) in the m_y – m_x , m_z – m_x , and m_z – m_y planes shown in Fig. 8 at different values of the bias current.

The transformation of trajectories with a change in the bias current is extremely interesting, and its experimental detection would contribute to the discovery of a new direction in the study of the properties of the φ₀ junction. As already noted, external electromagnetic radiation leads to a number of new effects; for example, it can fix the type of structure in the current range corresponding to the Shapiro step, and a change in the radiation amplitude can cause certain transformations of the magnetic precession, in particular, the transformation of a left mushroom into a right one [35].

A particle moving simultaneously in a constant field and in a field oscillating at a high frequency exhibits unusual behavior [92, 93]. In particular, in a pendulum with a vibrating suspension point, an external sinusoidal force can invert the stability position of the pendulum. Kapitza gave an analytical explanation of the causes of stability by introducing fast and slow motion variables. By averaging the classical equations of motion over fast oscillations, Kapitza found that the upper position of the pendulum becomes stable at sufficiently large perturbation amplitudes, while the lower one turns out to be unstable. This pioneering study marked the beginning of the field of vibrational mechanics, while the Kapitza method is used to describe periodic processes in various physical systems (see [94, 95] and references therein). In nonlinear control theory, the Kapitza pendulum is used as an example of a parametric oscillator that demonstrates the concept of dynamic stabilization.

The properties of a mechanical pendulum with an oscillating suspension point, in particular, the inversion of the stability position of the pendulum, the stabilization of new equilibrium positions [92], manifest themselves in the φ₀ Josephson junction, in which the reorientation of the easy axis of the ferromagnet occurs by changing the critical current of the Josephson junction and the spin-orbit interaction in the ferromagnet [96]. An example of such behavior is presented in Fig. 9, where it is shown that the magnetization component m_z in the φ₀ junction, which is an easy axis, with the corresponding values of the parameters indicated in the figure, vanishes, while m_y becomes equal to one. The influence of parameter G, which determines the ratio of the Josephson energy to the magnetic one in the φ₀ junction, is shown in Fig. 9c. As the value of G increases, the mean value of m_y , relative to which oscillations of a given magnetization component occur, approaches unity.

Since the magnitude of the magnetization depends on the magnitude of the spin-orbit interaction, the results obtained may contribute to the development of new methods for determining the magnitude of the spin-orbit interaction in ferromagnetic metals.

The use of the φ₀ Josephson junction can make it possible to detect macroscopic quantum tunneling and quantum oscillations of the magnetic moment by measuring the alternating voltage at the junction, and the rate of magnetic tunneling in the φ₀ junction can be controlled by a superconducting current [8]. Below, following [8, 97], we discuss the main results leading to this conclusion.

At I = 0, the equilibrium state of the φ₀ junction corresponds to two opposite orientations M (for example, along the y-axis) with an energy barrier between them equal to U₀ = (1/2)K_∥ V. A decrease in the barrier value to zero under the action of a current can lead to switching between these states [6, 7]. Of interest is quantum switching M at a finite barrier.

The equations of motion for φ and M in this case have the form

where C and R are the capacitance and resistance of the junction, respectively, and

is the effective field acting on the magnetic moment. To implement quantum tunneling, the junction must be sufficiently small; therefore, the capacitance can be neglected.

Quantum tunneling M is implemented as an instanton solution of equations (23) and (24), which at I = 0 for M = M₀(sin θ cos ϕ,sin θ sin ϕ,cos θ) has the form [97–99]

where ω₀ = [ω_∥ (ω_∥ + ω_⊥)]^1/2, λ = ω_∥/ω_⊥, ω_{∥, ⊥} = 2γ K_∥,⊥/(M₀ V). The instanton switches magnetization at τ = –∞ to at τ = +∞.

The interaction of the magnetic moment with the superconducting order parameter renormalizes the level splitting upon tunneling,

where Δ₀ is the splitting at I = 0, and the decoherence rate provided by the finite junction resistance R is given by

At in the state M with orientation along the y-axis, oscillations M_y and φ occur:

Allowance for damping leads to a decrease in the intensity of quantum voltage oscillations at the φ₀ junction with the corresponding quality factor

For φ₀ ∼ 0.1, the estimates give Q ∼ 0.1R(Ω), which is a fairly high value for a dielectric ferromagnetic layer.

Oscillations φ lead to voltage oscillations at the junction:

At φ₀ ∼ 0.1 and Δ_eff ∼ 0.1 K, the initial (t = 0) amplitude of the alternating voltage will be of the order of 1 μV, and the frequency of Δ_eff/(2πℏ) will be of the order of 1 GHz.

With a strong enough interaction,

tunneling is frozen: Δ_eff = 0.

Thus, the following main conclusions can be drawn [8]. First, the interaction between the magnetic moment and the superconducting order parameter in the φ₀ junction renormalizes the tunneling splitting in a way that can be accurately calculated and measured. The second point is that the φ₀ Josephson junction makes it possible to detect quantum tunneling and quantum oscillations of the magnetic moment by measuring the voltage across the junction. The third point is that the φ₀ junction allows us to control the rate of magnetic tunneling of the superconducting current through the junction. Note that the exact form of the spin-orbit interaction (Rashba, Dresselhaus, or others) is important for the specific dependence φ₀(M); the rest is determined by the symmetry of the magnetic anisotropy (crystal field).

The foregoing suggests an alternative approach to detecting coherent quantum spin oscillations compared to probing Rabi oscillations using the electron spin resonance method [80]. Such experiments are rather difficult, as they are carried out with small samples at low temperatures in order to freeze the superparamagnetic behavior. However, magnetic tunneling in a nanoparticle was studied on the basis of JJ at millikelvin temperatures [100]. In the φ₀ junction, the coupling of the magnetic moment to the Josephson dynamics is strong enough that it provides an interesting new tool for studying magnetic tunneling.

Another type of anomalous phase shift occurs in a superconductor/ferromagnet/superconductor JJ (Fig. 10) in the presence of moving domain walls [101]. Such systems, in the presence of magnetization dynamics, become dissipative in nature and, in principle, cannot support a superconducting current of any magnitude due to the voltage generated by the magnetization precession. The situation is analogous to type II superconductors, in which the mixed state is resistive, since, at an arbitrarily small value of the electric current, vortex motion occurs, which generates an electric field, leading to resistance and ohmic losses. The precession of magnetization in the SFS structure, created by supercurrent, necessarily generates an electric field and ohmic losses, similar to the motion of Abrikosov vortices (flux-flow regime). The difference is that, in the case of a magnetic system, the dynamics of the magnetic order parameter is responsible for the emerging electric field and ohmic losses in the superconducting state due to Gilbert dissipation.

In [101], the SFS junction was considered, in which the coupled dynamics of the magnetization M and the Josephson phase difference φ are determined by the system of equations

Equation (33) represents a generalized RSJ (Resistively Shunted Junction) model with a nonequilibrium current–phase relation with an anomalous phase shift φ₀ M determined by the spin-orbit interaction and magnetic texture. The LLG equation (34) contains the spin torque T = (γ/M)(J_s∇) M + (2γ/M)(M × B_j ) J_{s, j} due to the current, where the first term is due to the spin current J_s and the second term is due to the spin-orbit interaction determined by the spin vector B_j = (B_xj ,B_yj ,B_zj ) corresponding to the jth spatial component of the spin-orbit interaction tensor B_{i j} .

The anomalous phase shift φ₀ M is expressed as

where Z = Z^m + Z^SO.

The term is due to the spin-orbit interaction, with B_ij describing a linear spin-orbit coupling of the general form . It is assumed that . Z^m is nonzero only for a noncoplanar magnetic structure, and, in the case under consideration, Z^m = 0.

Equation (33) is quite general and is applicable to a wide class of Josephson systems that exhibit an anomalous phase shift. In the case of a Néel domain wall and Rashba spin-orbit interaction, , and the anomalous phase shift is determined by the expression

The results obtained are valid for |d/2 ± x₀| ⪢ d_w, i.e., when the domain wall is separated from the interface between the superconductor and the ferromagnet.

Figure 11 shows the CVC at rest and stationary moving domain wall. In fact, V(t) was determined by the derivative of .

The anomalous Josephson effect can arise in S/AF/S structures with an antiferromagnet in the presence of Rashba spin-orbit interaction [102]. A diagram of such a system is shown in Fig. 12.

In [102], junctions were considered both with an uncompensated magnetic moment (in the figure, the AA junction, in which there are A-type atoms at both interfaces with the superconductor) and with a fully compensated moment, the AB junction. It was shown that the presence of an uncompensated magnetic moment at the S/AF boundary leads to an anomalous phase shift, which strongly depends on the magnitude of the spin-orbit coupling. One of the most interesting results is the strong dependence of the anomalous phase shift on the orientation of the Néel vector with respect to the S/AF interface. The uncompensated magnetic moment at the interface does not require the expenditure of antiferromagnetic exchange energy, in contrast to the uncompensated moment in the bulk of the antiferromagnet. In this regard, by analogy with ferromagnetic systems [101], the presence of an anomalous phase shift in Josephson systems makes it possible to detect electrically and to control the dynamics of the Néel vector by a superconducting current.

The dependence of the anomalous phase shift on the magnitude of the local magnetization m, presented in Fig. 13, shows that, as m increases, the system transitions from state 0 to state π, and the transition occurs through a wide region of intermediate states φ₀. It can also be seen that φ₀ is a strongly nonlinear function of m, which contrasts sharply with the available theoretical and experimental results on the anomalous phase shift in Josephson junctions with low-dimensional ferromagnetic layers or in an in-plane magnetic field in the presence of Rashba spin-orbit interaction [6, 26, 28, 29]. This is also a direct consequence of the fact that, for antiferromagnets, the manifestation of the magnetoelectric effect is determined not by the magnitude of the sublattice magnetization m but by the uncompensated magnetic moment, which is rather small and can lead to large values of the anomalous phase only when the system is close to the 0 – π transition. Here, we also see the ambiguous behavior of the anomalous phase shift with stable and metastable branches.

The anomalous phase shift exhibits a strong dependence on the angle α between the magnetization of the A-site m and the boundary. Figure 14 shows the phase shift φ₀ as a function of the m_z -component of the magnetization at site A. The Néel vector rotates in the plane x,z. When the Néel vector component along the boundary vanishes, φ₀ = 0. The nature of the dependence of the anomalous phase shift on the orientation of the Néel vector is dictated by the Lifshitz-type term: the symmetry of the tensor is determined by the symmetry underlying the spin-orbit coupling. In the considered case of Rashba spin-orbit interaction, the only nonzero elements of are . As a result, the anomalous phase shift, which is the phase difference along the x-axis, can only be associated with m_z , i.e., the anomalous phase shift is proportional to m_z , at least for small m_z , when the linear approximation is performed. In principle, this dependence of the anomalous phase shift on the orientation of the Néel vector opens up a new direction for studying the prospects for controlling the Néel vector using a superconducting current.

Systems with multiple conducting channels provide a unique opportunity to design devices with tunable transport properties on the quantum length scale. One of the promising implementations of such devices is based on localized electronic states arising, for example, on the surface of a topological insulator [103] or at the edges of graphene nanoribbons [104] and various types of nanowires [105–107]. The physics of charge transfer through these states seems to be extremely rich due to the strong spin-orbit coupling, the large anisotropic g factor, and a number of other properties. The physics of edge states associated with bulk superconducting banks [104, 105] makes it possible to create a new type of Josephson devices with controlled current-phase dependences [3, 6]. Here, favorable conditions arise for observing Majorana fermions [108].

In [109], the appearance of a one-dimensional φ₀ junction was shown in the framework of the Bogoliubov–de Gennes formalism for the case of a nonquadratic electronic spectrum (it is sufficient to take into account nonquadratic corrections to the spectrum near the bottom of the band). The authors studied magnetotransport phenomena in a Josephson system containing several conducting channels simulating edge states localized, for example, on the surface of a single nanowire, taking into account strong spin-orbit and Zeeman interactions. This model, illustrated in Fig. 15, allows us to describe both the orbital and spin mechanisms of the influence of the magnetic field, as well as the nontrivial ground state of the Josephson junction with a nonzero superconducting phase difference. The Zeeman interaction creates spatial oscillations of the wave function of the Cooper pair on the scale ℏν_F/gμ_B H (similar to those observed in superconductor/ferromagnet structures [3]), which lead to magnetic oscillations of the critical current with a characteristic period ℏν_F/gμ_B L, where L is the channel length. The orbital effect causes a standard phase enhancement of ∼ 2πHS/Φ₀ (Φ₀ = πℏc/|e| is the flux quantum) in the electron wave function, similar to that arising in the Aharonov–Bohm effect. Here, S is the area bounded by a pair of interfering trajectories projected onto a plane perpendicular to the magnetic field. In this case, the interfering quantum-mechanical amplitudes cause magnetic oscillations in the total transfer amplitude with a period of 2Φ₀/S. Andreev reflection at the boundaries of superconductors can double the effective charge in the oscillation period [110]. The authors of [109] showed that, in the general case, the resulting critical current oscillates with competing periods of 2Φ₀/S and Φ₀/S. This physical picture is modified in the presence of the spin-orbit coupling, which is responsible for the dependence of the Fermi velocity on the spin projection and momentum direction. Such a specific dependence creates a spontaneous Josephson phase difference [6, 11, 12] and can cause a significant renormalization of the indicated oscillation periods.

The current–phase dependence was determined on the basis of the following equation [111]:

where, to calculate the energy of quasiparticle excitations ε, the Bogoliubov–de Gennes equation

was solved with the Hamiltonian of an insulated wire, which in the absence of a magnetic field had the form

Here, is the x projection of the momentum, ξ(p) is the energy of an electron in an insulated wire, and μ is the chemical potential. The term describes the Rashba spin-orbit interaction arising due to inversion symmetry breaking in the y direction [68], the operator is the 2 × 2 identity matrix in the channel subspace, and the potential describes the scattering at the superconductor-nanowire interface. The magnetic field is taken into account by the Zeeman term in (39) and by replacing with with gauge A_x (y) = –Hy.

For a short junction , only the intragap Andreev states contribute to the Josephson current, which leads to four positive intragap energy levels:

where n numbers the channels. As a result, the current-phase dependence (37) at T ⪢ Δ_n takes the form

Here, is the critical current of the nth channel at H = 0, the magnetic flux ϕ creates squid-like oscillations I_c, the term cos (γ_nH ), which depends on the constants , describes the oscillatory dependence I_c due to the Zeeman interaction, similar to the dependence in the SFS structure [3]. The term describes the formation of the φ₀ junction due to the spin-orbit interaction [6].

In [48], the possibility of realizing AJE in diffuse superconductor/ferromagnet/superconductor junctions was studied. It was shown that the conditions for observing this effect are a noncoplanar distribution of the magnetization and violation of the invariance of the superconducting current upon inversion of the magnetization. This symmetry is inherent in the widely used semiclassical approximation, and taking it into account leads to the absence of an anomalous superconducting current. In diffuse systems, it can be eliminated if the spin-dependent boundary conditions for the semiclassical equations at the superconductor/ferromagnet interface are taken into account. Using this procedure, the authors determined the ideal experimental conditions for increasing the anomalous Josephson current.

The anomalous current obtained in [48] demonstrates fast oscillations, depending on the thickness of the ferromagnet. These oscillations are the result of Fabry–Perot interference of electron waves reflected at S/F and F/F interfaces.

In diffuse SFS structures used in experiments [56–59, 61, 112], scattering by impurities makes the directions of electron propagation random; therefore, suppression of a rapidly oscillating anomalous current can be expected. Semiclassical studies of diffuse Josephson junctions with various noncoplanar structures, including helical [62], magnetic vortices [63], and skyrmions [64], have not shown AJE. On the contrary, in studies devoted to diffuse systems with half-metallic elements [15, 50] and junctions between magnetic superconductors with spin filters [54, 55], a finite anomalous current is predicted.

It was shown in [48] that AJE can appear in any diffuse SFS system with a noncoplanar magnetization texture under fairly general conditions. Figure 16 shows typical non-coplanar three-layer SFS systems leading to AJE. The reason why anomalous currents were not detected in previous studies on diffuse SFS systems is related to the additional symmetry of the magnetization inversion I(φ, M) = I(φ, –M), which was taken into account in the semiclassical approximation [72, 73] with respect to the original Hamiltonian [48].

The Josephson effect provides a fundamental feature of phase-coherent transport through mesoscopic samples with time reversal symmetry [4].

In Ref. [113], the authors calculated the equilibrium Josephson current through a multilevel quantum dot with Rashba or Dresselhaus spin-orbit coupling. The critical current can change radically with a change in the spin-orbit coupling parameter, which, in an external magnetic field, leads to oscillatory dependences on the Datta–Dass-type coupling parameter.

A new impetus to the study of mesoscopic and nanoscale Josephson junctions was given by the experimental demonstration of a gate-tunable Josephson current through junctions with several electronic levels (quantum dots) in various systems, in particular, in InAs-nanowires [114–116], carbon nanotubes [117], and 2D electron gas in semiconductors [118, 119].

Mesoscopic systems based on ordinary s-wave superconductors form a new class of systems with spontaneously broken symmetry with respect to time reversal and, therefore, exhibit anomalous supercurrents. In [11], a long ballistic one-dimensional Rashba quantum wire was studied, where I_a ≠ 0 was determined by the Zeeman effect and the difference between the velocities of electrons moving to the right and to the left. The obtained value α and the value I_a ∝ α⁴ turned out to be very small, practically unrealizable for the experiment. The anomalous Josephson effect was investigated by numerical simulation for a multichannel spin-polarized quantum point contact [12]. In Ref. [36], the authors calculated the Josephson current through a typical phase-coherent mesoscopic system (arbitrary multilevel quantum dot). Figure 17 shows the transfer of a Cooper pair through a two-level quantum dot and indicates the contributions to the matrix element during the forward and reverse tunneling processes. It was found that the necessary conditions for I_a ≠ 0 are the presence of a spin-orbit coupling and a suitably oriented Zeeman field. In addition, the quantum dot must be a chiral conductor.

The presence of a strong spin-orbit interaction in narrow-gap InSb and InAs semiconductors [120–122], which makes it possible to electrically control the spin, makes them natural materials for the realization of the anomalous Josephson effect [123].

For conduction electrons in direct-gap semiconductors, the spin-orbit interaction is expressed as

where V (r) is the external potential and σ indicates the electron spin s = σ/2. For an external electric field , the replacement V(r) = ez in expression (42) leads to the Rashba interaction:

Here, the coupling constant α = eλ is tunable by an electric field or gate voltage.

The diagram of the Josephson junction and a brief description of the model used in [9] are shown in Fig. 18, and the results in the case of one impurity, in Fig. 19.

In [9], the anomalous Josephson effect was studied numerically using the tight coupling model for a nanowire in the case of a short junction in the presence of a magnetic field. The energy levels E_n of the Andreev bound states were calculated numerically as functions of the phase difference φ between the superconductors; the DC Josephson current was estimated from the Andreev levels. I(–φ) = –I(φ) ratio violation has been demonstrated. It was also shown that the anomalous Josephson current and the direction dependence of the critical current are qualitatively the same as in the single scatterer model, and that the spin-dependent mixing channel plays an important role in the presence of spin-orbit interaction.

It is well known that the magnetic flux Φ through a superconducting loop placed in an external magnetic field is quantized, i.e., is equal to an integer number of flux quanta Φ = nΦ₀. The possibility of transitions between states with different vortex structures (different n) makes it possible to implement multiply connected superconducting systems, such as flux qubits [124, 125], ultrasensitive magnetic field detectors [126], and Josephson generators of a sequence of sharp uniformly distributed voltage pulses [127]. Flow quantization also naturally implies the use of a superconducting loop as a topologically protected memory cell in fast single-quantum logic devices (RSFQ) [128].

However, as the loop size R decreases at a fixed value of the magnetic flux, the field value increases as R⁻², which becomes a serious obstacle to the miniaturization of flux devices. Switching without the use of an external magnetic field can be implemented in a superconducting loop containing a Josephson junction with built-in magnetic order and/or broken inversion symmetry [1, 3, 4, 129]. Unlike conventional Josephson systems, such junctions maintain a nonzero phase difference in the ground state. The φ₀ junctions included in the superconducting circuit play the role of a phase battery in the creation of a spontaneous electric current, which corresponds to the magnetic flux Φ = (φ₀/2π)Φ₀ through the circuit [58, 130, 131]. It can be expected that control of the phase of the ground state of the φ₀ junction, for example, by voltage or radiation, should effectively rearrange the magnetic flux and switch the system between states with a different number of vortices without applying an external magnetic field.

However, in most existing φ₀ junctions, the phase change is limited (δφ₀ <2 π) and the corresponding flow Φ < Φ₀ cannot change the number of vortices n through the superconducting circuit. Indeed, Josephson junctions with a ferromagnetic layer between superconducting banks support only π-states with φ₀ = π [56, 57, 132], which allow the design of environmentally decoupled qubits [58], but are not suitable for changing the loop vorticity. A variable F-layer thickness (see, for example, [133] and references therein), as well as the presence of an Abrikosov vortex or a pair of current injectors in one of the superconductors [134, 135], can lead to a second-order phase transition with a spontaneous phase φ₀ varying from zero to π (so δ φ₀ < π). The situation looks more promising for SFS compounds with broken inversion symmetry, where the spin-orbit interaction creates an arbitrary spontaneous phase φ₀ ∝ ν sin θ (θ is the angle between the exchange field in the ferromagnet and the direction of the broken inversion symmetry), while the constant ν characterizes the intensity of the spin-orbit interaction and exchange field [6, 12, 27, 36, 50]. However, the ν parameter is usually small, which limits the φ₀ variation.

It was shown in [136] that a ψ Josephson junction with a half-metallic weak link integrated into a superconducting circuit (Fig. 20) makes it possible to pump a magnetic flux penetrating the loop.

In such a compound, the phase of the ground state ψ is determined by the mutual orientation of the magnetic moments in two ferromagnets (Fig. 21) surrounding the half-metal. The precession of the magnetic moment in one of the ferromagnets, controlled, for example, by microwave radiation, leads to the accumulation of phase ψ and subsequent switching between states with different numbers of vortices. The proposed flux pumping mechanism does not require the application of voltage or an external magnetic field, which makes it possible to design electrically decoupled memory cells for superconducting spintronics.

Appropriate selection of the driving frequency Ω makes it possible to realize controlled switching between different stable states φ_n , an example of which between states n = 0 and n = 1 is shown in Fig. 22.

In Ref. [137], a thermal analogue of the anomalous Josephson effect in an SFS structure with a noncoplanar magnetic texture is predicted. It is shown that the thermal current through the junction has a phase-sensitive interference component proportional to cos (θ – θ₀), where θ is the Josephson phase difference and θ₀ is the texture-dependent phase shift.

In a three-layer magnetic structure with a tunnel barrier with a spin filter, the value of θ₀ is determined by the spin chirality of the magnetic configuration and can be considered a direct manifestation of energy transfer involving spin-triplet Cooper pairs. In the case of an ideal spin filter, it is shown that the phase shift is resistant to spin relaxation caused by spin-orbit scattering. Possible applications of the relation between heat flux and magnetic precession are discussed.

As mentioned above, in recent years, much attention has been paid to phase-coherent caloritronics in hybrid superconducting structures [138]. The mechanism of phase-sensitive heat transfer is based on the thermal analog of the Josephson effect [139–141], which occurs in a system consisting of two superconductors S₁ and S₂ separated by a weak link and at temperatures T₁ and T₂, respectively. A nonzero temperature shift (for definiteness, it is assumed that T₁ > T₂) generates a constant heat flux from S₁ and S₂, which is expressed in terms of thermal CPR:

where θ is the phase difference between the superconducting electrodes. Here, the first term on the right-hand side is the usual quasi-particle thermal current, and the second one describes the contribution of energy transfer involving Cooper pairs. In accordance with the Onsager symmetry, the thermal current is invariant under time reversal, since the phase-coherent term in equation (44) does not change with phase inversion: .

The mutual influence of heat transfer and the Josephson effect have been experimentally studied starting from observations of thermoelectric effects in superconductor/normal metal/superconductor junctions [142]. Recently, the presence of coherent thermal currents (44) was confirmed in experiments using Josephson thermal interferometry in tunnel junctions [138, 143]. Subsequently, a number of applications have been proposed, including thermal interferometers [138, 143, 144], transistors [145], phase-sensitive ferromagnetic Josephson valves [146], and topological Andreev coupled state probes [147].

The direction in equation (44) can be controlled experimentally, providing the implementation of the 0 – π thermal Josephson junction [148].

Paper [137] reported on the possibility of obtaining a generalized thermal CPR in the form

where, in contrast to equation (44), there is an arbitrary phase shift θ₀.

This effect takes place in systems with broken time reversal symmetry and chiral symmetry, such as SFS junctions with noncoplanar magnetic texture or spin-orbit interaction. It can be considered a thermal analogue of the anomalous Josephson effect characterized by the generalized CPR:

Here, I_c is the critical current, and φ₀ is an arbitrary phase shift, which in the general case differs from the phase shift in the generalized thermal CPR: θ₀ ≠ φ₀.

Paper [137] demonstrates a thermal CPR with a phase shift (45) using the example of a Josephson spin valve [149], which contains three noncoplanar magnetic vectors (Fig. 23). It consists of two ferromagnetic layers with exchange fields h_{1, 2} interacting with superconducting electrodes separated by a spin filter barrier with magnetic polarization directed along m. Figure 24 shows the phase shifts in the CPR with φ₀ and θ₀ for two barrier spin filter efficiencies P. Recently, the spin filter effect was demonstrated in SF structures based on ferromagnetic insulators of europium chalcogenides [150, 151] and GdN tunnel barriers [152]. The role of external contacts F_{1, 2} is to induce effective exchange fields in superconducting electrodes. In the case of metallic ferromagnets, this can be achieved on the basis of the inverse proximity effect [153, 154]. Alternatively, F_{1, 2} can be ferromagnetic insulators and induce an effective exchange field in S_{1, 2} as a result of spin-mixing scattering of conduction electrons [129].

A distinctive feature of a topological insulator is edge or surface states. In the two-dimensional case, helical modes appear at the edge of the sample, i.e., a pair of one-dimensional modes coupled by time reversal symmetry and propagating in opposite directions for opposite pseudospins. A 2D chiral Dirac fermion on the surface of a strong TI (an odd number of chiral Dirac fermions on the surface) is protected by the topology and the presence of a volume bandgap. Such systems are interesting objects for the search for 2D phonon or exciton superconductivity (see references in [155]). Fu and Kane [156] predicted the appearance of chiral Majorana fermions as bound Andreev states at the interface of an insulator (FI) and an ordinary superconductor (S) having dispersion along the interface.

The charge transfer phenomena in normal metal/ferromagnetic insulator/superconductor structures and S/FI/S with a chiral Majorana mode on a 3D surface of TI were studied by Tanaka et al. [155]. Emergence of the chiral Majorana mode can be controlled with high sensitivity by the direction of the magnetization m in FI. The phase shift can be continuously tuned with the component m perpendicular to the interface. Control of the Andreev reflection and the Josephson current by means of the Majorana mode opens unique opportunities for superconducting spintronics.

Figure 25 shows a contour diagram of the energy level of the chiral Majorana mode E_J (CMM) as a function of θ and φ for m_x = 0, 0.4m_z , and –0.4m_z and the resulting Josephson current in junctions, showing the change in CPR with the direction of magnetization in a ferromagnetic insulator.

The calculation of the Josephson current, taking into account the chiral Majorana mode and magnetization in FI, carried out in [155], leads to the expression

where R_N is the resistance in the normal state, δ = m_xd/ν_F is the phase shift, T is the temperature, and σ_N is the junction transparency in the normal state.

The Josephson current is practically independent of m_y , because its contribution to I is compensated by Majorana modes with opposite chiralities. On the other hand, m_x significantly affects the value I as an effective vector potential, which directly enters the Josephson phase φ. The absence of a small factor e/c, which reduces the connection with the magnetic field, greatly facilitates the restructuring of the CPR. It is remarkable that, in this model, the anomalous CPR arises by changing the magnetization vector only, without using unconventional pairing mechanisms.

One of the variants of the φ₀ Josephson junction based on the topology of the barrier is the junction formed by the contact of two superconductors through the spiral edge states of a quantum spin-Hall insulator (QSHI) [25]. The influence of an external magnetic field on the CPR of such a junction leads to the fact that, as a result of the Zeeman effect, along the axis of spin quantization of the edges of the helix AJE arises. The phase shift φ₀ is associated with so-called helical superconductivity, which is the result of the mutual influence of the Zeeman effect and spin-orbit coupling.

Helical superconductivity due to edge states leads to current

where Θ is the Heaviside function and h is the applied magnetic field. The current I (φ = 0) flows in a short JJ, L ⪡ ξ with ξ = ν_F/Δ₀, at zero phase difference. Thus, proximitized superconductivity brings the system into an excited state, which leads to the anomalous Josephson effect. The anomalous current increases in proportion to the applied magnetic field h at h < Δ₀, and then decreases as at h ⪢ Δ₀. Note the fact that Δ₀ is an induced gap, which also implies the validity of the condition h > Δ₀ as long as the gap Δ₀ is sufficiently small compared to the intrinsic gap of the superconductors forming the junction.

Figure 26 shows the phase dependence of the current for various values of the applied magnetic field h, as well as the anomalous Josephson current at φ = 0 as a function of h.

The spatial separation of the two spirals is important. AJE occurs only if the connections at the edges are of unequal length, as shown schematically in Fig. 27a, where the Josephson current is carried by edge states. The anisotropy of the gyromagnetic tensor should make it possible to observe the effect with a magnetic field in the plane. The effect is stable with respect to a small shift between the applied field and the spin quantization axis. The dependence of the anomalous Josephson current on the magnetic field h at φ₀ = 0 for various temperatures is shown in Fig. 27b. We note an additional phase shift between the two edges due to the orbital effect of the field [157–159].

Thus, the basis of the anomalous Josephson effect in S/QSHI/S systems is the helical nature of edge states QSHI exposed to a magnetic field. The resulting anomalous superconducting current flowing at zero phase difference between the two superconducting terminals is rebuilt by the magnetic field. The fields in the superconductor and in the contact region, which contribute to the superconducting current, feel the helical nature of the edge states in the corresponding parts of the system. Analyzing the contributions of both edges, the required direction of the magnetic field, and the stability of the effect with respect to the final chemical potential and the misorientation of the magnetic field and the spin quantization axis, Dolcini et al. [25] proposed methods for experimental observation of the effect using hybrid structures based on available implementations. The CPR was determined as a function of the external magnetic field and the contact length. It is expected that AJE will be pronounced in junctions based on nanowires with strong spin-orbit interaction [160, 161] in the topological regime.

The connection between the electric charge and spin polarization during equilibrium and nonequilibrium electric transport through a two-dimensional Josephson junction containing disordered surface channels of a three-dimensional topological insulator can serve as the basis for the appearance of AJE [40]. In this case, the Edelstein effect is more pronounced in equilibrium than in a conventional material with spin-orbit coupling.

Using the semiclassical technique of Keldysh, Bobkova et al. [40] demonstrated the formation of a φ₀ junction resulting from the modulation of quasiparticle injection into the junction, the scheme of which is shown in Fig. 28. The ground state of the system corresponds to zero superconducting current, realized at a nonzero phase difference φ = φ₀ = –4d_Ih/α. The anomalous phase shift is proportional to the voltage difference V between the superconductors and the normal injection electrode, which determines the injection rate of quasiparticles, making it possible to switch the φ₀-state of the JJ on and off experimentally by controlling the injection of the quasiparticle flow.

The strong dependence of the Josephson energy on the orientation of the magnetization in Josephson junctions with ferromagnetic interlayers and spin-orbit coupling opens up new possibilities for controlling the magnetization of a ferromagnet by a superconducting current (Josephson phase). Such a dependence arises in Josephson SFS junctions on the surface of a three-dimensional topological insulator containing Dirac quasiparticles.

Dirac quasiparticles exhibit complete spin-momentum locking: the electron spin always forms a right angle with its own momentum, which leads to a pronounced dependence of the CPR on the magnetization direction [16, 155, 163]. At present, great progress has been made in the experimental implementation of hybrid structures F/TI. In particular, alloying with transition metal elements, such as Cr or V, has been used to introduce a ferromagnetic order into TI. Another way to introduce a ferromagnetic order into TI, which has been successfully implemented experimentally, is based on the coupling of a nonmagnetic TI with a magnetic insulator with a high T_c to create a strong exchange interaction in surface states through the proximity effect. Such a structure was studied by Nashaat et al. [162], and it was shown that the phase shift of the anomalous ground state of the φ₀ junction on the surface of a 3D TI is proportional to the magnetization component in the plane perpendicular to the direction of the superconducting current [16, 163]. An anomalous phase shift causes precession of the magnetization, similarly to the case of a system with spin-orbit coupling. However, for the system under consideration, the absolute value of the critical current highly depends on the orientation of the magnetization, namely, on the magnetization component in the plane along the current direction. This dependence in the regime with a given voltage can lead to splitting of the easy axis of the ferromagnet and stabilization of the fourfold degenerate ferromagnetic state, which contrasts sharply with the usual twofold degenerate ferromagnetic state of the easy axis.

Figure 29 shows a diagram of the described system, in which two ordinary s-wave superconductors and a ferromagnet deposited on a 3D TI surface form a JJ. The calculation procedure consists of calculating the CPR based on the formalism of semiclassical Green's functions. For the case with a given voltage, the electric current through the junction comprises two parts: the Josephson current j_s and the normal j_n. The Josephson current is associated with the presence of nonzero anomalous Green's functions in the interlayer and manifests itself in equilibrium. In the low applied voltage mode eV/(k_B T_c)⪡ 1, the deviation of the distribution function from equilibrium is small and can be ignored in the calculation of the Josephson current. The result is the following final expression for the Josephson current:

where ε_n = πT(2n + 1). At high temperatures, T ≈ T_c ⪢ Δ, the lowest Matsubara frequency makes the main contribution to the current, and expression (50) is simplified:

where j_b = eν_F N_F Δ²/(π^2T ). A similar expression was obtained for Dirac materials by Linder et al. [164]. The normal current is due to the deviation of the distribution function from equilibrium. However, for the system under consideration, assuming that the ferromagnet is metallic, practically the entire normal current flows through the ferromagnet, since in real experimental setups the TI resistance must be much larger than the resistance of the ferromagnet. As for the Josephson current, it is greatly suppressed inside the ferromagnetic layer, since the exchange field there is usually much larger than the induced exchange field h_eff in the TI surface layer. Therefore, the current flows through the TI surface states.

The dynamics of the magnetization of a ferromagnet is described in terms of the LLG equation. The electric current flowing through the TI surface states induces a spin-orbital torque [165–168] due to the presence of a strong coupling between the quasi-particle spin and the momentum. In principle, if ferromagnetism and spin-orbit coupling coexist spatially, then this torque is determined by the total electric current flowing through the system. However, in the case under consideration, only the superconducting current flows through the TI surface states, where spin-momentum synchronization takes place. Therefore, only this superconducting current creates a torque acting on the magnetization. A normal current flows through a homogeneous ferromagnet in which there is no spin-orbit coupling.

The torque caused by the superconducting current can be taken into account as an additional contribution to the effective field, which has the form

where m = M/M_s, is the length of the JJ, Γ = φ₀ j_bSr/2πKV_F is proportional to the ratio of the Josephson and magnetic energy, r = 2dh_eff/ν_F, Ω_J = 2eV is the Josephson frequency, and H_F = Ω_F/γ = K/M_s.

Figure 30 shows the evolution of the magnetization m obtained from the numerical solution of the LLG equation. It can be seen that, under different initial conditions, the system goes into four different stable states: two with a positive component m_y and ±m_x (shown in panels a and c) and two with ±m_x and a negative component m_y (panels b and d).

The system can switch to these states spontaneously under the influence of noise acting on states with ± m_z . The results are presented in Fig. 31 showing all four possible end states.

A similar approach to studying the dynamics of magnetization in voltage-shifted junctions has already been applied to systems with spin-orbit coupling in the intermediate layer [7, 96]. The qualitative difference between the system based on TI surface states and these investigations is that, in the case under consideration, the critical current demonstrates a strong dependence on the x component of the magnetization. Previously, it was considered independent of the direction of magnetization. This dependency results in a nonzero value of H_{eff, x} ∼ m_x for small values of m_x , which means that the easy axis y can become unstable in a junction in a state with voltage or bias current, while this axis is always stable if the critical current does not depend on the direction of magnetization. Moreover, for the system, there is no difference between the ±m_x -components of the magnetization. This leads to the remarkable fact that, in a controlled system, the easy axis does not reorient itself, retaining two stable directions of magnetization, as was already obtained earlier, but splits, demonstrating four stable directions of magnetization.

Within the framework of this structure, it is assumed that a number of characteristic phenomena will be observed [100], in particular, the appearance of Shapiro-like steps in the CVC of the JJ, created by the NM precession, the reversal of the magnetic moment when a time-varying voltage is applied to the JJ, and Rabi oscillations of the quantum spin induced by applied constant voltage. The intensity of the interaction between the NM and the JJ is determined by the parameter = E_J/E_B, which describes the ratio of the Josephson energy to the magnetic one, which is, in order of magnitude, the ratio of the magnetic field generated by the tunneling current to the effective field B_eff acting on the magnetic moment due to the magnetic anisotropy and the applied external field. Estimates show that the possibility of observing the first Shapiro step at (as well as the peak at V₀ = 0.5 due to nonlinearity) looks quite realistic [100]. In this case, the width of the first step decreases linearly with decreasing , the width of the second harmonic is proportional to ², and so on.

A remarkable property of the system is that, despite the weakness of the field generated by the tunneling current, for a certain time dependence of the applied voltage, an effective pumping of spin excitations into a nanomagnet and a reversal of its magnetic moment can occur. The reversal is realized under the condition > α, while the realistic value of the damping parameter is α = 0.01 [170]. Parameter determines the number of cycles in the precession of the magnetic moment leading to a reversal, which is proportional to 1/. For = 0.05, the required time to reversal is close to , which for ω_g ∼ 10¹¹ s⁻¹ is ∼ 10 ns. Smaller would require a slower voltage V₀ versus time, which is not a problem for such an experiment [100]. However, a smaller will require less dissipation due to the > α condition. In addition, the smaller is, the more sensitive the time evolution of the magnetic moment to the time dependence of the voltage. A small change in this dependence is sufficient to return to the original state after the reversal. Below is an example of studying the reversal of the magnetic moment in this system.

Rabi oscillations of the quantum spin in the JJ/NM system, induced by an applied constant voltage, are determined not by parameter but by the ratio of the Zeeman interaction of the spin with the tunnel current field and tunnel splitting Δ [100]. They strongly depend on the applied voltage. The greatest effect occurs when V₀ satisfies one of the resonant conditions eV₀ = (m/n) Δ, where m and n are integers. With such a resonant behavior, the probability of finding a spin in the 'up' or 'down' state is very different from the nonresonant case, which indicates the fundamental possibility of electromagnetic control of the JJ/NM qubit by means of an applied voltage.

The JJ/NM model was considered in [100]. The phase difference φ = φ₀ + φ_A is determined by the applied voltage V₀(t) (dφ₀/dt = 2eV₀(t)/ℏ) and the voltage V_A = (ℏ/2e) × (dφ_A /dt), which is the electromotive force induced in the junction by the time dependent magnetic field generated by the rotating magnetic moment of the nanomagnet. The vector potential is determined by the sum A = A_B + A_M of the vector potential of the external field A_B = 1/2(B × r) and the vector potential A_M = (μ₀/4π)(M × r)/r³ created by the magnetic field of the nanomagnet. The derivative dφ/dt is proportional to the total voltage drop across the junction:

Here, E is the electric field, and integration is carried out from one end of the weak link to the other.

Thus, in the JJ/NM system, the phase shift in the expression for the superconducting current

arises due to taking into account the magnetic field created by the rotating magnetic moment of the nanomagnet.

The dynamics of the magnetic moment is determined by the LLG equation:

where Γ_g is the gyromagnetic ratio for M, α is the Gilbert damping, and

is the effective field acting on M. The last term in this expression is equal to the magnetic field B_J, created by the tunnel current I = I_c sin φ at the location of the nanomagnet.

In normalized units, the LLG equation takes the form

where m_i = M_i /M_s are the normalized components of the magnetic moment, M_s is the saturation magnetic moment, Ω_F = ω_F/ω_c is the normalized ferromagnetic resonance frequency, ω_c = 2eRI_c/ℏ, I_c is the critical current,

Φ₀ is the magnetic flux quantum, m is the absolute value of magnetic moment, and α is the Gilbert damping parameter. Here, time t is normalized to , voltage V is normalized to ℏω_c/(2e). The components of the effective magnetic field h_i in normalized values are determined by the expressions [100]

where = G k, G = _J/K_an ν, ν is the volume of the nanomagnet, and K_an is the magnetic anisotropy constant. The components of the effective magnetic field are normalized to H_F = ω_F/γ_g. Equations (59), (60), together with the equations of the RSJ model, form the basis for studying the dynamics and current–voltage characteristics of the JJ/NM system.

Josephson oscillations in the junction excite the precession of the magnetic moment of the nanomagnet, which leads to FMR when the precession frequency becomes equal to the natural frequency of the magnetic system Ω_F. In [169], to describe the resonance, system of equations (59) was solved by the Gauss–Legendre method, as a result of which the time dependences of the magnetic moment components were determined, and the maximum amplitude of oscillations of the magnetic moment components in the time domain was calculated for each given voltage value.

Figure 33a shows the results of calculations of the maximal amplitude of oscillations as a function of voltage V on the JJ at Ω_F = 0.5 and two values of the damping parameter α = 0.001 and 0.3. In the chosen normalization, V = Ω_J; therefore, at a voltage corresponding to the frequency of Josephson oscillations Ω_J = 0.5, an FMR peak is observed. For , the result is qualitatively the same. An increase in damping leads to an increase in the width of the resonance and its shift towards lower frequencies, which is shown in Fig. 33a at α = 0.3. The positions of the peaks at low damping are in good agreement with the frequency values following from the analytical formulas obtained from the linearization of the LLG equations [169]. In particular, if the deviation of the magnetic moment from the equilibrium direction due to interaction with the Josephson current is small, i.e., G < 1 and , then the resonant frequency is given by

where

So, at G = 0.3, k = 0.01, and Ω_F = 0.5, for α = 0.001, the resonant frequency is Ω_res ≈ 0.5, and, for α = 0.3, Ω_res ≈ 0.45, which is quite close to the values obtained numerically (Fig. 33a).

The width of the resonance depends on the Gilbert damping parameter α, the ratio of the Josephson energy to the energy of the nanomagnet G, and coupling parameter k. Figure 33b demonstrates the influence of parameter G on the properties of ferromagnetic resonance. As G increases, a decrease in the resonant frequency and asymmetry of the resonant peak with respect to Ω_J = Ω_F are observed. Analytic expressions in this case give Ω_res ≈0.492 at α = 0.1, G = 0.1, k = 0.01, and Ω_F = 0.5. However, at G = 3π, analytical calculations lead to overestimated values, which means that higher order terms must be taken into account at G ⪢ 1. Thus, the deviation m_z in resonance at certain values of G, k, and α can be quite strong and manifest itself under experimental conditions [169].

Another interesting result that manifests itself in the JJ/NM system is the demonstration of the reorientation of the easy axis of the nanomagnet with an increase in the ratio of the Josephson energy to the magnetic one, i.e., a peculiar manifestation of the properties of the Kapitza pendulum in the Josephson junction/nanomagnet system [169]. Figure 34 shows the dynamics of the magnetic moment component m_z (t) for different values of parameter G. We emphasize that at the initial moment of time the magnetic moment is directed along the easy axis (the y-axis, see Fig. 32). We see that for small values of G the time dependence of the m_z (t) component reaches a certain constant value. As G increases, this dependence changes significantly, and at G = 3π, m_z (t) oscillating, tends to unity, i.e., m_y goes to zero. Thus, the easy axis of the nanomagnet is reoriented. In intermediate states, the magnetic moment of the nanomagnet is oriented between the axes y and z, and the reorientation time decreases with increasing G.

The dynamics of m_z at various values of the Josephson frequency Ω_J is shown in Fig. 35. For small Ω_J, the m_z component precesses near a certain fixed value, while for large Ω_J, oscillating, it goes to unity [169].

It is known that the position of stable equilibrium of a pendulum changes if its suspension point oscillates with a high frequency [93]. The ratio of the Josephson energy to the magnetic energy (G) corresponds to the amplitude of the variable force in the problem of the Kapitza pendulum, which should contribute to the reorientation of the easy axis of the ferromagnet. The nature of the increase in the average value of m_z depending on the ratio of the Josephson energy to the magnetic one is shown in Fig. 36, which also demonstrates an analogy with Kapitza's pendulum. A similar behavior is observed with increasing coupling parameter k of the Josephson and magnetic subsystems.

Until now, we have considered the effect of Josephson oscillations on the dynamics of the magnetic moment of a nanomagnet. Let us now briefly describe the reverse effect, i.e., the influence of the dynamics of the magnetic moment on the CVC of the Josephson junction [5]. The calculation of the CVC is carried out for the junction in the biased current case. In this case, within the framework of the RCSJ model [82], the system of equations has the following form:

where φ is the phase difference in the Josephson junction, and β_c is the McCumber parameter. In addition, to calculate the CVC in system of equations (59), in the expression for the effective field (60), it is necessary to replace Vt with φ.

Figure 37 shows the results of calculating the CVC of a JJ with and without a nanomagnet (CVC of the SIS junction) together with the dependence of the maximum component m_z on voltage [169]. The CVC of a JJ with a nanomagnet demonstrates the feature indicated by the arrow, which is absent in the case of a JJ without a nanomagnet. The voltage position of this feature corresponds to the position of the m_z resonant peak. Thus, the nanomagnet precession manifests itself in the CVC of the JJ, which can serve as a method for controlling its dynamics. Note that justification for the parameter values used and their correspondence to the experimental conditions is given in [100, 171], which implies the possibility of an experimental study of the observed effects.

As unexplored properties of the JJ/NM system, the appearance of chaotic states under periodic action on the system should be noted [172]. Chaos in the JJ under external electromagnetic radiation, which is modeled by adding the term A sin (ωt) to the bias current (ω is the frequency, A is the radiation amplitude), was considered in detail in [173]. It is assumed that, in the system under study, the precession of the magnetic moment under superconducting oscillations in the JJ can also lead to the appearance of chaotic states. Research in this area has not yet been carried out, but undoubtedly it is important for the practical applications of these systems.

The possibility of magnetization reversal in the φ₀ Josephson junction by a current pulse was demonstrated in [34]. It was shown that the reversal of the magnetic moment is extremely sensitive to the values of the system parameters. In view of the sufficient complexity of the system under consideration, the question of the possibility of predicting a complete reversal for given parameters of the system and the current pulse remained open until recently. Below, we discuss an analytical criterion for reversal, which makes it possible to do this for certain parameters of the φ₀ junction and the current pulse.

The scheme of the considered φ₀ junction is shown in Fig. 38. The easy axis of the ferromagnetic layer is directed along the z-axis, which also coincides with the direction of the spin-orbit potential gradient. The magnetic moment component m_y is related to the Josephson current directed along the x-axis.

The dynamics of the magnetic moment of the system under consideration is described by the LLG equation [34], for which the effective field H_eff depends on the Josephson phase difference φ:

where γ is the gyromagnetic ratio, α is the Gilbert dissipation, M₀ = |M|, is the ratio of Josephson energy to magnetic anisotropy energy, K is the anisotropy constant, is the ferromagnetic layer volume, and r is the spin-orbit interaction parameter.

In dimensionless quantities, the system of equations is written as

where the magnetic moment components m_i are normalized to M₀, i = x, y, z, and H_i are the effective field components normalized to K/M₀, which are determined by the expressions

In system of equations (65), the time is normalized to (where ω_F = γK/M₀ is the ferromagnetic resonance frequency). The equation for the phase difference is written in the framework of the resistive model [82], where, for simplicity, a JJ with a small capacitance C (R² C/L_J ⪡ 1, L_J is the inductance of the Josephson junction, R is its resistance in the normal state), i.e., displacement current, is not taken into account. In this case, the expression for the electric current I through the Josephson junction, normalized to the critical current I_c, is written as

where w = V_F/(I_cR ) = ω_F/ω_R , V_F = ℏω_F/(2e), and ω_R = 2eI_cR /ℏ. It should be noted that the results in [7, 34] were obtained under the assumption that the term r(dm_y /dt) is small. It is shown below that taking it into account does not really lead to qualitative changes, but it is necessary to maintain the gauge invariance of the equations used [178].

In (67), a rectangular current pulse with amplitude A_s and duration Δt,

and with initial conditions

was used.

The calculations were carried out using an implicit scheme based on the two-step Gauss–Legendre method [180]. This approach provided higher accuracy (fourth order O(h⁴) ≈ 10⁻⁸) and stability than the Runge–Kutta method. System of equations (65) was solved numerically together with Eqn (67) using (68) with initial conditions (69). In all calculations, w = 1 was assumed; the values of the remaining parameters are indicated in the captions to the corresponding figures.

An example of transient dynamics for a reversal m_z with residual oscillations is shown in Fig. 39a, and the dynamics of the components of the magnetic moment, phase difference, and superconducting current are shown in Fig. 39b. As can be seen, in the transition region, the phase difference changes from 0 to 2π and, accordingly, the superconducting current changes its direction twice. This is followed by an interval with damping of the oscillations of the superconducting current.

In Fig. 39b, the characteristic moments of time are indicated by vertical dashed lines. Line 1 corresponds to the phase difference π/2 and indicates the maximum superconducting current I_s. Line 1 ', which corresponds to the maximum of m_y and m_z = 0, has a slight offset from line 1. This fact demonstrates that, in general, the time dependence features of m_x and m_y do not coincide with the features on I_s (t), i.e., there is a delay in the response of the magnetic moment to changes in the superconducting current. Another characteristic point corresponds to φ = π. At this point in time, line 2 crosses points I_s = 0, m_y = 0 and the minimum of m_z . At the time when φ = 3π/2, line 3 crosses the minimum of I_s. When the pulse is turned off, a superconducting current flows through the resistance, exhibiting damped oscillations and causing residual oscillations in the magnetic moment components. Note also that the end time of the pulse (t = 28) does not actually appear immediately in the dynamics of m_y (and m_x not shown here). They demonstrate a continuous transition to damped oscillating behavior.

The data of Fig. 39b indicate a direct way to determine the magnitude of the spin-orbit coupling in the junction by means of the estimate r. For this, we note that can be determined up to an initial time-independent constant φ₀₀ as the voltage V(t) across the junction changes. In addition, maxima and minima of I_s occur at times t_max and t_min (Fig. 39b), for which

Determining φ₀₀ from these equations, one can obtain

which makes it possible in principle to determine r from magnetization m_y at the maximum and minimum supercurrent and voltage V at the junction. We emphasize that, for the experimental implementation of the proposed method, it is necessary to distinguish between the magnetization values at times of the order of 10⁻¹⁰ – 10⁻⁹ s. At present, studying the dynamics of magnetization with such time resolution is a rather difficult task. To experimentally determine the spin-orbit coupling constant r, it is more convenient to vary the parameters of the current pulse I (t) and study the switching threshold of the magnetic moment.

The dynamics of the system in the form of magnetization trajectories in the m_y – m_x and m_z – m_x planes during the transition time interval for the same momentum and JJ parameters at α = 0 is shown in Fig. 39. It can be seen that the magnetic moment performs a spiral rotation, approaching the state with m_z = –1 after turning off the electric current pulse. The figures clearly show the features of the dynamics near the points B, A', and Q and the damped oscillations of the magnetization components (Fig. 39b, d). Point B in Fig. 39c corresponds to the transition from an increase in the absolute value m_x to its decrease and back at point A'. The behavior of the magnetic system turns out to be quite sensitive to the parameters of the electric current pulse and the JJ. Paper [34] shows different magnetization reversal protocols when changing the parameters A_s, G, and r.

It is interesting to compare the effect of a rectangular pulse with a Gaussian one of the form

where σ denotes the full width at half maximum of the pulse, and A is its maximum amplitude at t = t₀. An example of the reversal of the magnetic moment in this case is shown in Fig. 40, which displays the dynamics of m_z at r = 0.1, G = 10, A_s = 5, σ = 2 at low dissipation α = 0.01 .

In this case, the magnetization reversal occurs more smoothly than in the case of a rectangular pulse.

The study of the magnetization dynamics of the φ₀ junction led to the discovery of a periodicity in the occurrence of intervals of reversal of the magnetic moment with a change in the parameters of the JJ and the current pulse. Figure 41 shows examples of the dynamics of the magnetic moment m_z for two values of the ratio of the Josephson energy to the magnetic one: G = 9 (curve 1) and G = 8 (curve 2), as well as the applied current pulse I_pulse (curve 3). In the first case (G = 9), a reversal of the magnetic moment is observed, while in the second (G = 8), it is absent, which reflects the dependence of the implementation of the reversal on the chosen values of the system parameters. The influence of the model parameters and the current pulse on the reversal of the magnetic moment in the φ₀ Josephson junction was discussed in Refs [34, 179]. However, the study of the possibility of predicting the reversal and determining the intervals of parameters at which the reversal of the magnetic moment occurs have not been carried out until recently.

To determine the intervals of G in which the reversal occurs, the time dependence of the magnetic moment was calculated in [181] for values of G from 1 to 130 with step ΔG = 1 for values of α from 0.01 to 0.5 with step Δα = 0.001. The value of the spin-orbit coupling parameter was assumed to be r = 0.1. For each pair of values (α, G), systems of equations (65), (67), and (68) were solved by the Gauss–Legendre method with step h = 0.01 in the interval t ∈ [0,T_max], T_max = 200. For t = T_max, the inequality |m_z + 1| ⩽ 0.0001 was checked in order to fix the realization of the reversal of the magnetic moment. In the case of its implementation, the corresponding values of α and G were selected and stored. These data, presented in Fig. 42, indicate a periodic dependence in the implementation of the reversal of the magnetic moment with increasing G. Let us emphasize some features in the manifestation of this dependence, in particular, the absence of a reversal at small G and the shift of intervals along G with increasing α. In this case, an increase in the width of these intervals along the G-axis is observed.

Paper [181] also presents the results of calculating the implementation of the reversal of the magnetic moment on the plane (G, r), where a periodicity appears in the occurrence of reversal intervals with a change in G. An increase in the spin-orbit coupling parameter r leads to a shift of the domain with a reversal to the region of small G with a simultaneous decrease in its width.

As mentioned above, the easy axis of a ferromagnet is directed along the z-axis and has two stable states, m_z = ±1. The current pulse causes the magnetic moment to oscillate. The critical value for the reversal is its value at the end of the pulse, which is determined by the parameters of the φ₀ junction and the parameters of the pulse. Paper [181] presents the time dependences of the magnetic moment component m_z for different values of the spin-orbit coupling parameter r, the Gilbert damping parameter α, and the ratio of the Josephson to the magnetic energy G for the first and second bands. If the value of m_z turns out to be close to zero or negative, then the presence of Gilbert damping ensures that m_z tends to –1. The periodicity in the occurrence of reversal intervals can be explained by assuming a periodic dependence of the m_z component on the parameters of the model used. Section 6.3 presents analytical studies of the magnetic moment reversal in Josephson structures with an anomalous phase shift.

As already noted, one of the main problems in superconducting electronics is the creation of a memory element [174, 175, 182] with low energy dissipation. Various versions of such devices have been considered, including those based on φ₀ Josephson junctions [66, 182–184].

The DC component of the superconducting current applied to the SFS φ₀ junction can have a strong orientational effect on the ferromagnetic magnetic moment in the layer [34]. Guarcello and Bergeret in [66] indicated the possibility of using the SFS φ₀ junction as an element of cryogenic memory based on the pulse switching proposed in [34]. In such a scheme, a bit of information is associated with the direction of the magnetic moment along or against the direction of the easy axis of the ferromagnetic layer. The recording is carried out as an inversion of the magnetic moment by a current pulse, and the readout is carried out by detecting the magnetic flux by a SQUID inductively coupled to the φ₀ junction. The stability of the current-induced magnetization reversal to thermal fluctuations was also studied, and a method was proposed for breaking the coupling between the Josephson phase and the magnetization dynamics by tuning the intensity of the Rashba spin-orbit interaction by the gate voltage. We emphasize that, in all the studies mentioned above, magnetization reversal was studied only numerically.

As noted above, the physics of switching in the φ₀ junction is determined by the LLG equation (65) and the model equation of RSJ (67). An interesting feature of this system in the case of strong dissipation is the decoupling of the equations, i.e., equation (67) for Φ = φ – φ₀ = φ – rm_y turns out to be unrelated to the LLG equation (65). This makes it possible to find an analytical solution for Φ and construct a theory of magnetization reversal for certain values of the junction and current pulse parameters.

Thus, at the period of the current pulse action t₀ ⩽ t ⩽ t₀ + δt, the equation for Φ has the form

with the initial condition Φ (t = t₀) = 0. At A_s < 1, this gives

where defines the time scale of approaching a constant value Φ. Formula (73) allows finding sin Φ during the current pulse t₀ ⩽ t ⩽ t₀ + δt. In the second time interval t ⩾ t₀ + δt, when the impulse is disabled (I_p(t) = 0),

This expression exhibits an exponential decay towards zero with a τ₁ ∼ w time scale. Here, tan [Φ (t₀ + δt)/2] is determined by equation (73).

The theory is based on the following basic assumptions: (1) for small w and for momenta with A_s ≠ 1, the wdΦ/dt term in equation (72) can be neglected, which implies the relation I_p(t) = sin Φ; (2) the condition Gr ⪢ 1, which does not imply G ⪡ 1 for w ⪡ 1 (see [7]), so that G can vary over a wide range, from G ⪡ 1 to G ∼ 100 ⪢ 1; (3) the smallness of the Gilbert damping α ⪡ 1 [185–187]. Under these conditions, the LLG equation during the period of the current pulse can be written in the form

The tight coupling limit Gr ⪢ 1 (but r ⪡ 1) can be considered analytically [7]. In this case, m_y (t) ≈ 0, and for the applicability of the method, we must put GrI_p(t) ⪢ 1. Then, in polar coordinates ρ and ϕ, we get m_x = ρ sin ϕ, m_z = ρ cos ϕ, and . Thus:

As can be seen from (73), after the current pulse is turned off, sin Φ quickly drops to 0 due to the condition w ⪡ 1. In this time interval, the magnetization dynamics is determined only by magnetic anisotropy and Gilbert damping, which causes the magnetization to align along the easy axis [96].

As follows from (76), magnetization reversal occurs at

where δt is the pulse duration.

The results of analytical calculations, together with the results of the numerical solution of the complete system of equations, are shown in Fig. 43 for rectangular pulse I_p(t) = A_s[θ (t – t₀) – θ (t – t₀ – δt)] with A_s = 0.5 for δt₁ = 1 and δt₂ = 3.

In the case of δt = 1, criterion (77) gives cos (GrA_s δt₁) = 0.28 > 0, so there is no reversal, while, for δt₂ = 3, it turns out to be cos (GrA_s δt₁) = –0.76 < 0 and remagnetization is observed. It can be seen that numerical solution (76) represented by the blue squares coincides with the analytical solution represented by the green solid curve. When the pulse is disabled, damping prevents any deviation from the easy m_z = ±1 axis. This is shown in the insets to Fig. 43. It should be noted that the magnetization reversal is affected not by the shape of the current pulse but only by the integral over the pulse duration [65].

According to (77), the magnetization reversal in plane r – G under the current pulse action I_p(t) = A_s[θ (t – t₀) – θ (t – t₀ – δt)] leads to hyperbolic regions at

for n = 0, ±1,..., while the most efficient reversal occurs when condition

is satisfied, i.e., at GrA_s δt = π + 2πn.

In the case of a rectangular current pulse in the weak damping regime and small w, Eqn (78) gives hyperbolic curves for different n. From a physical point of view, they are curves of constant amplitude for the force in the LLG equation (64). In such a situation, the magnetic moment aligns in the m_z = –1 direction immediately after the pulse is turned off, and the appropriate time scale is determined only by the pulse duration and not by the Gilbert damping. This helps to optimize the pulse width to achieve a fast reversal. It can be seen from (79) that the minimum time for the reversal is realized at n = 0, i.e.,

The described situation is shown in Fig. 44 for G = 100, r = 0.1, α = 0.005, w = 0.01, A_s = 0.5, and δt_eff = 0.628, resulting in a remagnetization time of δt_rev ≈0.6 × 10⁻¹⁰ s for typical ω_F ∼ 10 GHz. This time is two orders of magnitude shorter than that calculated in [34].

Figure 45 compares the numerical results with the analytical ones obtained on the basis of equations (78) and (79). The almost perfect agreement between numerical and analytical calculations emphasizes the validity of the theory for the selected system parameters. Comparing both results, we can conclude that, in fact, term (72), which makes the equations gauge-invariant, only slightly shifts the remagnetization regions. But, on the other hand, the gauge-invariant form of the equations makes it possible to consider Eqn (72) analytically.

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In quantum dots, the breaking of both symmetries can be achieved by a combination of an external magnetic field and spin-orbit interaction [36, 37, 113]. The finite Zeeman splitting for spin 'up' and spin 'down' electrons breaks symmetry under time reversal. On the other hand, chiral symmetry breaking arises under the mutual influence of the spin-orbit interaction and the magnetic field, when the orbital inside the quantum dot changes during tunneling. Szombati et al. [27] report the first experimental implementation of the φ₀ junction based on a nanowire quantum dot, where it was demonstrated that the phase shift φ₀ can be controlled using an electrostatic gate. The tunneling of an electron through a quantum dot with two orbitals that mix during the spin-orbit interaction was considered. When the orbital does not change, the contributions from both processes cancel each other out. When the orbital changes during tunneling, due to the mutual influence of the spin-orbit interaction and the magnetic field, compensation does not occur. In this case, an additional phase factor is acquired, which depends on the tunneling direction and is different for the left and right tunneling processes. As a consequence, the two processes cannot cancel each other, and this results in chiral symmetry breaking. The magnitude of the phase shift φ₀ depends on the intensity of the spin-orbit interaction and the magnitude of the magnetic field in the plane. The results of observing a continuous phase change in the φ₀ Josephson junction in a magnetic field in the plane were presented. In contrast to the case B_in-plane = 0, the phase shift of the voltage oscillations in the flow is tuned using the gate voltage.

An anomalous phase shift φ₀ can be obtained in systems with both a Zeeman field and the Rashba spin-orbit coupling term H_R = (α/ℏ)(p × e_z )σ in the Hamiltonian [6, 39], where α is the Rashba coefficient, e_z is the direction of the Rashba electric field, and σ is the vector of Pauli matrices describing the spin. Physically, these terms lead to spin-induced dephasing of the superconducting wave function.

The anomalous phase shift is associated with the inverse Edelstein effect observed in metals or semiconductors with a strong spin-orbit coupling. While the Edelstein effect generates a spin polarization in response to an electric field [190], the inverse Edelstein effect [191], also called the spin-galvanic effect, generates a charge current by nonequilibrium spin polarization. These two magnetoelectric effects also arise in superconductors as a consequence of a Lifshitz-type term in the free energy [26, 192]. Thus, in a superconductor with a strong Rashba coupling, the Zeeman field induces an additional term in the supercurrent. In Josephson junctions, this term leads to an anomalous phase shift in the CPR [39].

An anomalous phase shift φ₀ can be detected experimentally in a Josephson interferometer (JI) by measuring the CPR. Anomalous phase shifts have recently been identified in a JI created from a parallel combination of a normal 0 and π junction [27], which breaks the parity symmetry.

Due to the large values of the g factor (g = 19.5) [193] and the Rashba coefficient, Bi₂Se₃ is a promising candidate for observing the anomalous Josephson effect due to the mutual influence of the Zeeman field and the spin-orbit interaction.

As described in detail in [26, 39], the amplitude of the anomalous phase depends on the amplitude of the Rashba coefficient α, the transparency of interfaces, spin relaxation terms such as the Dyakonov–Perel coefficient, and whether the junction is in the ballistic or diffusion mode. It is predicted that for small values of α the anomalous phase is proportional to α³; for large values, it should be proportional to α.

In the ballistic regime [6] and for large α, the anomalous phase shift is given as φ₀ = 4E_Z αL/(ℏν_F)² for a magnetic field of magnitude B perpendicular to the Rashba electric field, where E_Z = (1/2)gμ_B B is the Zeeman energy, L is the distance between the superconductors, and ν_F is the Fermi velocity of the barrier material. For the spin-split Rashba conduction band with α = 0.4 eV Å, ν_F = 3.2 × 10⁵ m s⁻¹, and junction length L = 150 nm, the magnetic field B = 100 mT generates an anomalous phase φ₀ ≃ 0.01 π, while for the Dirac states with ν_F = 4.5 × 10⁵ m s⁻¹ the value φ₀ ≃ 0.005π [28].

In the diffusion regime, the expected anomalous phase shift was calculated in [39]. For weak α with highly transparent interfaces and ignoring spin relaxation, the anomalous phase shift is determined by the relation

where τ = 0.13 ps is the elastic scattering time, is the diffusion constant, and m* = 0.25m_e is the effective electron mass [194].

To test these theoretical predictions, a JJ and a Josephson interferometer were prepared from Bi₂Se₃ in [28]. According to measurements of the relative phase shift between two JIs with different orientations of the JJs relative to the magnetic field in the plane, an anomalous phase shift was determined corresponding to formula (81). The results are presented in Figs 46 and 47 and in the table.

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In the absence of disorder, the anomalous phase shift induced by Rashba spin-orbit coupling can only be generated by an in-plane magnetic field B_y . The stress map shown in Fig. 46b shows the oscillations of the critical current of the two devices as a function of the magnetic field B. The critical current of both devices oscillates due to the perpendicular component of the magnetic field B_z = B sin θ. The oscillation frequency can be changed by mechanically tilting the sample, i.e., by changing the angle θ between the plane containing the superconducting loop and the magnetic field. Due to the anomalous phase shift, the frequency of the anomalous device is greater than the reference.

Figure 47 compares the JI frequencies as functions of the angle θ. Since the critical current of a large JI decreases with increasing magnetic field, this leads to a decrease in the background set for both devices, shown in the figure with solid lines. As can be seen, the anomalous device is characterized by a higher frequency of oscillations than the control one. The two JIs are in-phase in a weak magnetic field and out of phase in a strong magnetic field, with the anomalous phase shift reaching φ₀ ≃ π for B ≃ 100 mT. In Figure 47c, the ratio of the oscillation frequencies is shown as a function of the angle θ. Without generating an anomalous phase shift, this ratio should be constant and equal to the surface ratio S/S_ref ≃ 1. It was found that this ratio diverges as 1/θ for small θ in accordance with the equation

Curve fitting equation (82) determines the spin-orbit coupling coefficient α.

The table shows the anomalous phase shifts obtained at B ≃ 100 mT, as well as theoretical values. The first three columns show the anomalous phase shifts extracted from the last nodes of the critical current oscillations shown in Fig. 47 for three curves taken at different angles θ. The last three columns show the calculated anomalous phase shifts in the ballistic mode for Rashba splitting conduction states, Dirac states, and in the diffusion mode for Rashba splitting conduction states. In theoretical calculations, the values α = 0.4 eV Å were used for conduction states with Rashba splitting and α = 3 eV Å for Dirac states.

As noted, AJE has been demonstrated in JJs with InSb nanowires in the quantum dot geometry [27] and more recently in JJs using Bi₂Se₃ [28]. In a JJ with a quantum dot, the phase shift can be changed by the gate voltage, but is limited by the geometry and supports only a few modes; therefore, small critical currents are realized in the JJ. Based on the topological insulator Bi₂Se₃, it is possible to implement a planar φ₀ junction, but in this case the phase shift cannot be tuned by the gate voltage [29, 177]. Heterostructures formed by InAs and epitaxial superconducting Al [195] are promising not only for mesoscopic superconductivity [196] but also for the realization of topological superconductivity and Majorana fermions [197]. This is due to the fact that the induced superconducting gap Δ_ind in InAs can be as large as in Al [198], and InAs has a large g-factor and spin-orbit coupling. As a consequence, a JJ fabricated on this platform can have a high critical current and high transparency [177, 199]. In addition, it is possible to control the magnitude of the spin-orbit coupling by changing the InAs density through an external gate [200].

The possibility of changing the magnitude of the anomalous phase shift in a JJ formed on the basis of InAs and Al is due to the possibility of varying the intensity of the spin-orbit coupling through an external gate [29]. The observation of a finite phase shift φ₀ indicates a relationship among the phase difference of superconductors, electric current, and spin in these heterostructures. The phase shift for different in-plane magnetic fields and gate voltages is shown in Fig. 48, and the evolution of the phase shift is shown in Fig. 49.

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The value of φ₀ is proportional to the Zeeman energy and turns out to be much larger than the given theoretical estimates. Most likely, this is due to the fact that such scaling is valid for a long junction with several channels, which is not directly related to the system under study.

The implementation of large values of the phase shift φ₀ and its tuning are very important for applications in superconducting spintronics, where large spin gradients can be used to create a phase battery [1]. This opens up the possibility of generating spin gradients in a controlled manner via Josephson currents or phase shift. The possibility of achieving a large value φ₀ in heterostructures InAs/Al and the fact that it strongly depends on the InAs density are directly related to efforts to realize topological superconducting states [201, 202].

A phase battery is a quantum device that provides a continuous (constant) phase shift for the wave function of a quantum circuit and is a key element for quantum technologies based on quantum coherence. In [30], the first experimental implementation of a phase battery in a hybrid superconducting circuit is reported. It consists of an n-doped InAs nanowire with unpaired spin surface states proximitized by aluminum superconducting contacts. The ferromagnetic polarization of unpaired spin states is effectively converted into a constant phase shift φ₀ along the wire, which leads to the anomalous Josephson effect [6, 39]. By applying an external magnetic field in the plane, a continuous change in φ₀ is achieved, which allows the battery to be charged and discharged. The joint action of the spin-orbit coupling and the exchange interaction breaks the phase rigidity of the system, causing a strong coupling among the charge, spin, and phase. This relationship opens up broad prospects for topological quantum technologies [1].

Side hybrid JJs made from materials with strong spin-orbit interaction [27] or topological insulators [28] are ideal options for creating φ₀ junctions. The lateral arrangement breaks the symmetry of the inversion and provides a natural polar axis that is perpendicular to the current direction. Moreover, the spin polarization of electrons s, induced either by a Zeeman field or by the exchange interaction with ordered magnetic impurities, breaks the time reversal symmetry. As a result, the structure acquires a toroidal symmetry described by the anapole moment , retaining the magnetoelectric effects in the presence of the spin-orbit interaction. In this case, the anomalous shifts φ₀ are controlled by a Lifshitz-type invariant in the free energy (F_L), which can be composed from the product of the anapole moment and the superfluid velocity [39]:

where f (α, h) is the odd intensity function of the Rashba spin-orbit interaction α, h is the exchange or Zeeman field, n_h is the unit vector pointing in the direction of the latter, and v_s is the superfluid velocity of Cooper pairs flowing in the JJ. The scalar triple product defines vector symmetries φ₀, while the magnitude of the shift depends on specific microscopic details of the sample, as well as macroscopic quantities such as temperature.

As shown in Fig. 50, the phase battery consists of a JJ made on the basis of an InAs nanowire enclosed between two Al superconducting poles. The supercurrent and, hence, v_s flow along the nanowire (x-direction) orthogonally to the Rashba spin-orbit interaction vector indicating the plane of the substrate (z-direction). In the same nanowire, surface oxides or defects generate unpaired spins that behave like magnetic impurities (indicated by yellow arrows in Fig. 50) that can be polarized along the y-direction to provide an exchange interaction h in this direction. This leads to the final triple product in equation (83) and, hence, to an anomalous phase shift φ₀. Due to the ferromagnetic order of the unpaired spins, the phase shift persists even in the absence of an applied magnetic field. The phase battery converts the ferromagnetic order into a quantum phase shift. It was shown in [30] that the shift φ₀ can also be controlled by the Zeeman interaction with an external magnetic field B. By scanning the shifts φ₀ in all directions of the plane magnetic field, the authors showed the geometric origin of the anomalous phase described by Eqn (83).

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An important problem for the memory element is effects due to inevitable thermal fluctuations. In [66], an exhaustive analysis of the noise in the φ₀ junction dynamics was presented, taking into account the influence of stochastic thermal fluctuations. The current-induced magnetic bistability makes it possible to determine two well-distinguishable logical states and to investigate the stability of such a memory against noise effects. A sensing scheme based on a current controlled SQUID (Fig. 51) has been proposed in which no additional magnetic flux is required to set the optimum operating point. In addition, the intriguing possibility of effective screening of the memory state by voltage gating in a device formed by a ferromagnetic layer with a linear momentum term of the Dresselhaus spin-orbit coupling was discussed.

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In Ref. [66], a nonvolatile memory element based on a lateral ferromagnetic Josephson junction with spin-orbit coupling and out-of-plane magnetization is proposed. The interaction between the magnetic moment and the exchange field of a ferromagnet leads to a magnetoelectric effect, which couples the electric current through the junction and magnetization, which makes it possible to switch the direction of the magnetic moment in the ferromagnet by a current pulse. The two memory states are encoded in the direction of out-of-plane magnetization. In order to determine the optimal operating temperature for a memory element, the influence of noise on the average stationary magnetization was studied, taking into account thermal fluctuations that affect both the Josephson phase and the dynamics of the magnetic moment. The switching process is studied depending on the parameters of the ferromagnet, such as the Gilbert damping and the spin-orbit coupling intensity, and a non-destructive readout scheme based on the DC SQUID is proposed. In addition, in [66], a method was analyzed for protecting the memory state from external disturbances by voltage gating in systems with Rashba and Dresselhaus type linear spin-orbit coupling in the momentum.