Magnetization induced dynamics of a Josephson junction coupled to a nanomagnet

TL;DR

This paper analyzes how a nanomagnet coupled to a Josephson junction influences its superconducting current, revealing Shapiro-like steps in the I-V characteristics under various magnetic field conditions and extending the analysis to include dissipation effects.

Contribution

The study provides an analytic perturbative solution for the coupled JJ-nanomagnet system under arbitrary magnetic fields and explores the impact of dissipation on magnetization-induced Shapiro steps.

Findings

Existence of Shapiro-like steps in I-V characteristics under magnetic fields.

Magnetic field rotation and AC drive affect the Shapiro steps.

Dissipation modifies the magnetization-induced Shapiro steps.

Abstract

We study the superconducting current of a Josephson junction (JJ) coupled to an external nanomagnet driven by a time dependent magnetic field both without and in the presence of an external AC drive. We provide an analytic, albeit perturbative, solution for the Landau-Lifshitz (LL) equations governing the coupled JJ-nanomagnet system in the presence of a magnetic field with arbitrary time-dependence oriented along the easy axis of the nanomagnet's magnetization and in the limit of weak dimensionless coupling $\epsilon_0$ between the JJ and the nanomagnet. We show the existence of Shapiro-like steps in the I-V characteristics of the JJ subjected to a voltage bias for a constant or periodically varying magnetic field and explore the effect of rotation of the magnetic field and the presence of an external AC drive on these steps. We support our analytic results with exact numerical solution of the LL equations. We also extend our results to dissipative nanomagnets by providing a perturbative solution to the Landau-Lifshitz-Gilbert (LLG) equations for weak dissipation. We study the fate of magnetization-induced Shapiro steps in the presence of dissipation both from our analytical results and via numerical solution of the coupled LLG equations. We discuss experiments which can test our theory.

Figures

Tables

Table 1. Table 1: Tabulated values of n 1 0 superscript subscript 𝑛 1 0 n_{1}^{0} and n 3 0 superscript subscript 𝑛 3 0 n_{3}^{0} for all the peaks that appear in Fig.  7 (b) at specific ω 1 subscript 𝜔 1 \omega_{1} values listed above. Note that n 2 0 superscript subscript 𝑛 2 0 n_{2}^{0} does not appear in the table since the peaks correspond to ω B = 0 subscript 𝜔 𝐵 0 \omega_{B}=0 so that their position are independent of n 2 0 superscript subscript 𝑛 2 0 n_{2}^{0} (Eq.  39 ).

${\omega }_1$	$n_1^0$	$n_3^0$	${\omega }_1$	$n_1^0$	$n_3^0$
1.2	0	1	0.3	0	4
1	-1	1	0.28	1	5
0.8	2	2	0.24	0	5
0.6	0	2	0.2	-3	3
0.4	-2	2	0.2	0	6
0.4	0	3	0.08	0	15

Equations

$$E$$

$$E_1$$

$$\gamma (t)$$

$${\gamma }_0(t)$$

$${\gamma }_1(t)$$

$$\vec{A}(\vec{r},t)$$

$${\gamma }_1(t)$$

$$k_0$$

$$\frac{d\vec{M}}{dt}$$

$${\vec{B}}_{\mathrm{eff}}$$

$$\vec{m}$$

$${\omega }_B(t)$$

$$B(t)$$

$$\frac{d\varphi }{d\tau }$$

$$\frac{d\theta }{d\tau }$$

$$\theta (\tau )$$

$$z(\tau )$$

$$\frac{d\delta \varphi }{d\tau }$$

$$\frac{d\delta \theta }{d\tau }$$

$$\delta \theta (\tau )$$

$$\delta \varphi (\tau )$$

$$I_s=I_c\sin \left[{\gamma }_0(\tau )-k_0\sin (\varphi (\tau ))\sin (\theta (\tau ))\right]$$

$$\displaystyle\left(\begin{array}[]{c}m^{\prime}_{x}\\ m^{\prime}_{y}\\ m^{\prime}_{z}\end{array}\right)$$

$$m_x^{\prime }$$

$$\frac{d{\theta }^{\prime }}{d\tau }$$

$$\frac{d{\varphi }^{\prime }}{d\tau }$$

$${\omega }_B^{\prime }(\tau )$$

$${\theta }^{\prime }(\tau )$$

$$\delta \theta (\tau )$$

$$\delta \varphi (\tau )$$

$$I_s$$

$${\omega }_0=n^0{\omega }_c.$$

$$\delta \theta (\tau )$$

$$\delta \varphi (\tau )$$

$$I_s$$

$$I_s^{\mathrm{periodic}}$$

$${\omega }_0-n_2^0{\omega }_1-n_1^0{\omega }_2=0.$$

$$I_s^{\mathrm{periodic}}$$

Full text

I Introduction

The physics of Josephson junctions (JJs) has been the subject of intense theoretical and experimental endeavor for decades likharev1 . The interest in the physics of such JJs has received renewed attention in recent years in the context of Majorana modes in unconventional superconductors kit1 ; kwon1 ; jay1 ; gil1 . Indeed, it has been theoretically predicted kit1 ; kwon1 and experimentally observed rokhin1 that such junctions may serve as a test bed for detection of Majorana end modes in unconventional superconductors. It has been shown that the presence of such end modes lead to fractional Josephson effect kit1 ; kwon1 and results in the absence of odd Shapiro steps shapiro1 when such junctions are subjected to an external AC drive evenlit ; kiril1 ; commentsub .

Recently molecular nanomagnets have been studied as potential candidates for qubit realization owing to their long magnetization relaxation times at low temperatures time1 . Such a realization is expected to play a central role in several aspects of quantum information processing loss1 and spintronics using molecular magnets ref1 ; ref2 . These systems have potential for high-density information storage and are also excellent examples of finite-size spin systems which are promising test-beds for addressing several phenomena in quantum dynamics viz. quantum-tunneling of the magnetization tunnel , quantum information qi , entanglement ent1 to name a few. The study of the spin dynamics of the nanomagnets is a crucial aspect of all such studies. One way to probe such dynamics is to investigate the spin response in bulk magnets using inelastic neutron scattering and subsequent finite-size extrapolation to obtain the inelastic neutron scattering spectra for a single molecule spin1 . Other, more direct, methods include determination of the real-space dynamical two-spin correlations in high-quality crystals of nanomagnets spin2 and transport measurements through nanomagnets spin3 . Another probe of such magnetization dynamics, which we shall focus on in the present study, involves interaction of the nanomagnet with a JJ; the modulation of the Josephson current and the Shapiro steps in the current-voltage (I-V) characteristics is known to bear signature of the nature of the spin precession of the nanomagnet spin4 .

The physics of such a JJ-nanomagnet system has therefore received significant attention both theoretically and experimentally. It has been theoretically studied in Refs. kulik1, ; bula1, , where the effects of spin-flip of the nanomagnet on the Josephson current was charted out. More recently, several theoretical studies have been carried out on a variety of aspects of such systems including effect of superconducting correlations on the spin dynamics of the nanomagnet nuss1 , the influence of spin-orbit coupling of a single spin on the Josephson current nazarov1 , and the effect of deposition of single magnetic molecules on superconducting quantum interference devices(SQUIDs) made out of such junctions delan1 . Another interesting phenomenon which has been widely studied in this context is magnetization switching cai1 ; buzhdin1 ; yury1 which constitutes magnetization reversal of the nanomagnet by a externally driven JJ. In addition, the possibility of a Josephson current to induce exchange interaction between ferromagnetic layers has been studied in Ref. xav1, . Furthermore, the dynamics of both JJs with misaligned ferromagnetic layers and those coupled to single or multiple ferromagnetic layers have also been studied numerically linder1 . It has also been shown that the presence of a single classical spin of a molecular magnet precessing under the action of a constant magnetic field coupled to a JJ may lead to generation of finite spin current whose polarization axis rotates with same frequency as the classical spin holm1 . Such theoretical studies were complemented by experimental work on these systems wernsdorfer1 . More recently, magnetization reversal of a single spin using a JJ subjected to a static field and a weak linearly polarized microwave radiation has been demonstrated in Ref. thirion1, . The possibility of the presence of Shapiro-like steps in the I-V characteristics of such coupled JJ-nanomagnet for constant applied magnetic field has also been pointed out in Ref. cai1, . However, to the best of our knowledge, most of these studies do not provide any analytic treatment of the coupled JJ-nanomagnet system even at a classical level where they are known to be governed by the Landau-Lifshitz-Gilbert (LLG) equations gil1 . Moreover, the current-voltage (I-V) characteristics of a JJ in the presence of a nanomagnet with time-dependent magnetic fields and in the presence of external AC drive has not been studied systematically so far.

In this work we study a JJ coupled to a nanomagnet with a fixed easy-axis anisotropy direction (chosen to be $\hat{y}$) in the presence of an arbitrary time dependent external magnetic field along $\hat{y}$. For nanomagnets with weak anisotropy, we find an analytic perturbative solution to the coupled Landau-Lifshitz (LL) equations in the limit of weak coupling between the nanomagnet and the JJ. Using this solution, we show that a finite DC component of the supercurrent, leading to Shapiro-like steps in the I-V characteristics of a voltage-biased JJ, can occur, in the absence of any external radiation, for either a constant or a periodically time varying magnetic field. Our theoretical analysis provides exact analytic results for the position of such steps. We study the stability of these steps against change in the direction of the applied magnetic field and increase of the dimensionless coupling strength $\epsilon_{0}$ between the JJ and the nanomagnet. We also provide a detailed analysis of the fate of this phenomenon in the presence of an external AC drive and demonstrate that the presence of such a drive leads to several new fractions (ratio between the applied DC voltage and the drive frequency) at which the supercurrent develops a finite DC component leading to Shapiro steps in the I-V characteristics. We support our analytical results with numerical study of the systems which allows exact, albeit numerical, investigation of the dynamics of the coupled JJ and nanomagnet system. We also extend our study to systems with dissipation via perturbative analytic and exact numerical solution of the coupled LLG equations and study the behavior of the steps with increasing dissipation. Finally, we discuss experiments which may test our theory, discuss the significance of our results in context of junctions of unconventional superconductors hosting Majorana end modes, and point out the distinction between voltage and current biased junctions in the present context. We note that the analytic solution of the LLG equations that we present here as well as the presence of additional, more robust, Shapiro steps for periodically time-varying magnetic fields have not been reported so far; thus our work may serve as an ingredient for development of new detectors for magnetization of a nanomagnet in coupled JJ-nanomagnet systems.

The plan of the rest of this work is as follows. In Sec. II, we provide the analytic solution of the LL equations governing the coupled nanomagnet and JJ. This is followed by an analogous solution for LLG equations describing the coupled JJ-nanomagnet system in the presence of dissipation in Sec. II.3. Next, in Sec. III, we back the analytical results with exact numerics and discuss details of Shapiro-step like features in the I-V of the JJ for constant or periodic magnetic field. Finally, we chart out our main results, discuss experiments which can test our theory, and conclude in Sec. IV.

II Formalism and analytical solution

In this section, we obtain analytic solution to the LL equations for the weakly coupled JJ-nanomagnet system. We shall sketch the general derivation of our result for arbitrary time-dependent magnetic field in Sec. II.1 and then apply these results to demonstrate the existence of Shapiro-like steps for constant and periodic magnetic fields in Sec. II.2. The extension of these results for dissipative magnets will be charted out in Sec. II.3.

II.1 Perturbative solution

The coupled JJ-nanomagnet system is schematically shown in Fig. 1. In what follows we consider a JJ along $\hat{x}$ and choose the easy axis of the nanomagnet along $\hat{y}$; the radius vector $\vec{r}$ between the nanomagnet and the JJ thus lies in the $x-y$ plane as shown in Fig. 1. The energy functional governing the JJ and the nanomagnet is given by cai1

[TABLE]

where $K>0$ denotes the magnetization anisotropy constant, $\vec{B}(t)\parallel{\hat{y}}$ is the external magnetic field which can have arbitrary time dependence, and $E_{J}$ is the Josephson energy of the junction. The phase difference $\gamma$ across the junction is given by

[TABLE]

where $\gamma_{00}$ is the intrinsic DC phase of the JJ, $\gamma_{0}$ is the phase generated by the external voltage, $V_{0}(t)=V_{0}g(t)$ is the applied voltage across it, $\omega_{0}^{\prime}=2eV_{0}/\hbar$ is the Josephson frequency of the junction, $g(t)$ is a dimensionless function specifying the time dependence of the applied voltage, $\Phi_{0}=hc/2e$ is the flux quantum, $h=2\pi\hbar$ with $\hbar$ being the Planck constant, $e$ is the charge of an electron, and $c$ is the speed of light. The vector potential ${\vec{A}(\vec{r})}$ is given by

[TABLE]

Note that in our chosen geometry, as shown in Fig. 1, $\vec{d}l\parallel{\hat{x}}$ and $\vec{r}$ lies in the $x-y$ plane, so that

[TABLE]

where the geometrical factor $k_{0}$ can be tuned by tuning the distance $a$ between the JJ with the nanomagnet (Fig. 1). Moreover, in this geometry, the orbital effect of the magnetic field do not affect the phase of the JJ since $\vec{d}l\cdot\vec{A}_{B}\sim\vec{d}l\cdot(\vec{B}\times\vec{r})=0$. In this geometry, the LL equations for the nanomagnet reads

[TABLE]

where $B(M_{y})=KM_{y}+B(t)$ and $\gamma_{g}$ is the gyromagnetic ratio comment1 . These LL equations are to be solved along with the constraint of constant $|\vec{M}|$; in what follows we shall set $|\vec{M}|=M_{0}$. We note that Eq. 5 do not include dissipation which shall be treated in Sec. II.3. Thus the solutions obtained in this section can be treated as limiting case of very weakly dissipating nanomagnets. We also note that our analysis do not take into account the change in $I_{s}$ arising from the spin-flip scattering induced by the coupling of the JJ with the nanomagnet balat1 ; holm1 . This can be justified by the fact that in our geometry, the nanomagnet does not reside atop the junction and thus we expect the spin-flip scattering matrix elements to be small. Further, even with a significant contribution from spin-flip scattering, such effects become important when the Larmor frequency of the magnetization $\omega_{L}\geq\Delta_{0}/\hbar$ holm1 which is not the regime that we focus on. This issue is discussed further in Sec. IV.

Eq. 5 represents a set of non-linear equations which, in most cases, need to be solved numerically. Here we identify a limit in which these equations admits an analytic, albeit perturbative, solution for arbitrary $B(t)$. To this end we define the following dimensionless quantities

[TABLE]

where $f(t)$ is a dimensionless function specifying the time dependence of the magnetic field, $\omega_{0}$ is the dimensionless Josephson frequency (scaled with the frequency associated with the magnetic field $B_{1}$), and $B_{1}$ is the amplitude of the external magnetic field. In what follows we shall seek perturbative solution for $\vec{m}$ in the weak coupling and weak anisotropy limit (for which $\epsilon_{0},\,KM_{0}/B_{1}\ll 1$ and $k_{0}\leq 1$) to first order in $\epsilon_{0}$ and $K$. In terms of the scaled variables, the LL equations (Eq. 5) can be written in terms of $\theta$ and $\phi$ as

[TABLE]

with the initial condition $\phi(\tau=0)=0$ and $\theta(\tau=0)=\theta_{0}$. We note that the choice of this initial condition for $\theta$ and $\phi$ amounts to choosing the initial magnetization of the nanomagnet in the $x-y$ plane: $\vec{M}=(M_{1},M_{2},0)$ where $\cos\theta_{0}=M_{2}/M_{0}$, and $M_{1}^{2}+M_{2}^{2}=M_{0}^{2}$. We choose $\theta_{0}$ such that $\cot\theta_{0}<1$ and the perturbative solutions that we present remains valid as long as $\epsilon_{0}\cot(\theta)\ll 1$. We have checked that this limit is satisfied in all our numerical simulations described in Sec. III.

The perturbative solutions of Eq. 7 can be obtained by writing

[TABLE]

where $\delta\theta(\tau)$ and $\delta\phi(\tau)$ satisfies, to first order in $\epsilon_{0}$ and $K$ [i.e., neglecting terms ${\rm O}(\epsilon_{0}^{2})$, ${\rm O}(K\epsilon_{0})$ and ${\rm O}(k_{0}\epsilon_{0})$],

[TABLE]

The solution of Eq. 9 is straightforward and can be written as

[TABLE]

Eqs. 8, 9, and 10 constitute the central result of this work. These equations describe the dynamics of a nanomagnet in the presence of weak coupling with a JJ. We note that in obtaining these results, we have neglected the normal state resistance of the JJ which can be safely done for tunnel junctions or for weak links with large resistance and small capacitance cai1 . We also note that the domain of validity of these solutions require $\delta\theta(\tau),\delta\phi(\tau)\leq z(\tau)$ at all times; we shall discuss this domain in the context of specific drives in Sec. II.2. We now use these solutions to study the behavior of the supercurrent of the JJ given by

[TABLE]

for several possible magnetic field profiles. Here $I_{c}$ is the critical current of the JJ. Although Eq. 11 yields $I_{s}$ for any magnetic field profile, in what follows we shall concentrate on constant and periodically varying magnetic fields since they allow for Shapiro-step like features in the I-V characteristics of a voltage biased JJ.

Before ending this subsection, we note that the solutions for $\vec{M}$ is stable against small fluctuations of the direction of the applied magnetic field. To see this, we write the external magnetic field $\vec{B}$ is applied in an arbitrary direction in the $x-y$ plane: $\vec{B}=B_{1}f(t)(\sin(\alpha_{0}),\cos(\alpha_{0}),0)$ with $K\alpha_{0}\ll K$. Next, we move to a rotated coordinate frame for which the magnetization $\vec{m}^{\prime}$ is related to $\vec{m}$ by

[TABLE]

We proceed by using the parametrization $\vec{m}^{\prime}=(\sin\theta^{\prime}\cos\phi^{\prime},\cos\theta^{\prime},\sin\theta^{\prime}\sin\phi^{\prime})$. In this representation, the initial values of $\vec{m}^{\prime}$ are given by

[TABLE]

where $\theta_{0}$ and $\phi_{0}=0$ depicts the initial condition for $\vec{m}$. Next, repeating the same algebraic steps as outlined earlier in the section, one finds that the equations governing $\theta^{\prime}$ and $\phi^{\prime}$ are given by

[TABLE]

Note that the analytic solution to Eq. 24 can only be obtained when terms ${\rm O}(K\alpha_{0})$ can be neglected. In this case, the perturbative solution to Eq. 24 can be obtained in the same way as done before in this section. The final result is

[TABLE]

The behavior of these solutions shall be checked against exact numerics in Sec. III.

II.2 Constant and Periodically varying magnetic fields

In this section, we apply our perturbative results on constant and periodically time-varying magnetic fields for which the I-V characteristics of the JJ may have Shapiro-like steps. While this effect has been discussed, using a somewhat different geometry, in Ref. cai1, for constant magnetic field, we demonstrate its presence for periodic magnetic fields.

Constant magnetic field: This case was studied in Ref. cai1, . For an external constant voltage, $g(t)=1$ and one has $\gamma_{0}=\omega_{0}\tau+\gamma_{00}$, where $\gamma_{00}$ is the intrinsic phase difference across the JJ at $t=0$. Further, in this case, $f(t)=1$, and $z(\tau)=\omega_{c}\tau$ where $\omega_{c}=1+KM_{2}/B_{1}$. Thus the supercurrent to the leading order and for $\epsilon_{0},K\ll 1$, is given by

[TABLE]

which indicates the presence of a finite DC component of $I_{s}$ leading to Shapiro steps in the I-V characteristics of the JJ-nanomagnet system at

[TABLE]

To study the stability of these steps we consider the solution to ${\rm O}(\epsilon_{0})$. For constant magnetic field, the ${\rm O}(\epsilon_{0})$ correction to $\vec{M}$ can be obtained from Eq. 10 which, after some straightforward algebra, yields for $z(\tau)=\omega_{c}\tau$

[TABLE]

We note that for $n=n_{\pm}=n^{0}\mp 1$, both $\delta\theta$ and $\delta\phi$ grows linearly in time. These terms turn out to be the most important corrections to the zeroth order solution near $\omega_{0}=n^{0}\omega_{c}$ and leads to the destabilization of the steps as $\epsilon_{0}$ increases. We also note that such terms restrict validity of the perturbative expansion up to a finite time $T_{p}$ so that $\epsilon_{0}\cot(\theta_{0})J_{n_{\pm}}(k\sin(\theta_{0}))T_{p}\sim 1$; we shall discuss this in more details while comparing our perturbative results with exact numerics in Sec. III. The supercurrent to first order in $\epsilon_{0}$ and $K$ is thus given by

[TABLE]

The behavior of the DC component of $I_{s}$ in the presence of these corrections is charted out in Sec. III.

Periodic Magnetic fields: In this case, we choose a periodic magnetic field so that $f(\tau)=\cos(\omega_{1}\tau)$, where $\omega_{1}$ is the external drive frequency measured in units of $\gamma_{g}B_{1}$ commentmag . For this choice, one has $z(\tau)=\omega_{2}\tau+\sin(\omega_{1}\tau)/\omega_{1}$, where $\omega_{2}=\gamma_{g}KM_{2}/B_{1}$. Thus the zeroth order solution for the supercurrent $I_{s}^{\rm periodic}$ reads

[TABLE]

We note that this solution admits a finite DC component of $I_{s}^{\rm periodic}$ and hence Shapiro-like steps for $(n_{1},n_{2})=(n_{1}^{0},n_{2}^{0})$ for which

[TABLE]

The amplitude of these peaks depend on product of two Bessel functions unlike the ones found for constant magnetic field cai1 ; moreover, the condition for their occurrence depends on two distinct integers which allows the peaks to occur in the absence of any DC voltage across the junction. The condition for occurrence of such peaks are given by $\omega_{0}=0$ and $\omega_{2}=n_{2}^{0}\omega_{1}/n_{1}^{0}$; they provide examples of Shapiro steps without any voltage bias across a JJ and have no analog in their constant magnetic field counterparts.

The first order corrections to these solutions can be obtained in a manner analogous to one used for constant magnetic field. The final result is given by

[TABLE]

where $\delta\theta_{p}$ and $\delta\phi_{p}$ are given by

[TABLE]

We note that the main contribution to the zeroth order results again comes from terms linear in time which occurs for

[TABLE]

for $s=\pm 1$. The perturbation theory thus remain valid for $\tau\leq T_{p}^{{}^{\prime}}$ so that

[TABLE]

The behavior of the DC component of $I_{s}^{\rm periodic}$ as a function of $\epsilon_{0}$, as obtained from Eq. II.2 is discussed and compared to exact numerics in Sec. III.

External AC drive: Next, we consider the behavior of $I_{s}$ in the presence of both an external magnetic field and an AC field of amplitude $A$ and frequency $\omega_{A}$ so that $\gamma_{0}(t)=\gamma_{00}+\omega_{0}\tau+A\sin(\omega_{A}\tau)/\omega_{A}$. First we consider a constant magnetic field for which $f(\tau)=1$. In this case, using Eqs. 8 and 11, one obtains, to zeroth order in $\epsilon_{0}$

[TABLE]

A finite DC component of $I_{s}$ leading to Shapiro like steps thus appear in the I-V characteristics for integers $n_{1}^{0},n_{2}^{0}$ which satisfies

[TABLE]

The condition of occurrence of these peaks mimics those for periodic magnetic field in the absence of external AC drive and the peak amplitude depends on the product of two Bessel functions. We note that the resultant Shapiro steps may occur for low AC drive frequencies and thus could, in principle, be amenable to easier experimental realization.

Next, we consider a periodically varying magnetic field in the presence of external radiation. In this case, one has, $f(\tau)=\omega_{2}+\cos(\omega_{1}\tau)$. Using Eq. 8, one has $z(\tau)=\omega_{2}\tau+\sin(\omega_{1}\tau)/\omega_{1}$ which leads to (Eq. 11)

[TABLE]

Thus the presence of the steps now occurs for a set of integers $(n_{1}^{0},n_{2}^{0},n_{3}^{0})$ which satisfies

[TABLE]

The perturbative $O(\epsilon_{0})$ corrections to the above solutions can be carried out in similar manner to that outlined above.

II.3 Dissipative nanomagnets

In this section, we include the dissipative (Gilbert) term in the LLG equations to model dissipative nanomagnets and seek a solution to these equations in the limit weak dissipation and weak coupling between the JJ and the nanomagnets. The resultant LLG equations are given by

[TABLE]

where $\eta$ is a dimensionless constant specifying the strength of the dissipative term. Following the same parametrization as in Eqs. 6, we find that one can express the LLG equations, in terms of $\theta$ and $\phi$, as

[TABLE]

where we have neglected terms $O(\epsilon_{0}\eta)$ and ${\rm O}(\eta^{2})$. We note that the effect of the dissipative term manifests itself in $\theta$ but not in $\phi$; this fact can be understood as a consequence of the fact that to leading order $\vec{M}\times(\vec{M}\times\vec{B}_{\rm eff})$ lies along $\hat{y}$ and hence only effects the dynamics of $M_{y}$ which depend only on $\theta$. For small $\epsilon_{0}$ and $\eta$, Eq. 41 therefore admits a perturbative solution

[TABLE]

where $z(\tau)$, $\delta\theta(\tau)$, and $\delta\phi(\tau)$ are given by Eq. 8 and we have neglected terms ${\rm O}(\epsilon_{0}\eta)$. The supercurrent, in the presence of the dissipative term is given by

[TABLE]

The fate of the DC component of $I_{s}$ leading to Shapiro-like steps in the presence of the dissipative term shall be checked numerically in Sec. III for both periodic and constant applied magnetic fields.

III Numerical results

In this section, we analyze the coupled JJ-nanomagnet system both without and in the presence of dissipation and compare these results, wherever applicable, to the theoretical results obtained in Sec. II.1. In what follows, we focus on cases of constant or periodically varying magnetic fields since Shapiro-step like features are expected to appear in the I-V characteristics of the JJ only for these protocols. The LLG equations for magnetization solved numerically to generate the data for the plots are given by

[TABLE]

where

[TABLE]

In these equations $f(\tau)=1$ for constant and $f(\tau)=\cos(\omega_{1}\tau)$ for the periodically varying magnetic fields, $\alpha_{0}=0$ indicates an applied magnetic field along $\hat{y}$, we have set $\theta_{0}=\pi/3$ and $\gamma_{00}=\pi/2$ for all simulations, and $K^{\prime}=KM_{0}/B_{1}$. Note that Eq. 44 reduces to the usual LL equations for $\eta=0$. The supercurrent is then computed using the values of $m_{z}$ obtained from Eq. 44: $I_{s}=I_{c}\sin(\gamma_{0}(\tau)-k_{0}m_{z})$.

To compare the theoretical results with exact numerics, we first compare the values $I_{s}(\tau)/I_{c}$ computed theoretically (Eq. II.2) with exact numerical result. For comparing the two results, we have fixed the external voltage $\omega_{0}=\omega_{B}$ which leads to a Shapiro step in the I-V characteristics of the JJ with $n^{0}=1$. As discussed in Sec. II.2, one expects one of the perturbative terms to grow linearly in time in this case; the presence of this linear term is expected to invalidate the perturbative theoretical results for $\tau>T^{\prime}\sim\epsilon_{0}^{-1}$. In Fig. 2(a), we show the comparison between theoretical and numerical values of $I_{s}(\tau)/I_{c}$ at $\tau>2\times 10^{4}$ for $\epsilon_{0}=10^{-4}$; we find that the numerical and analytical values differ by less than $5\%$ even at late times ($t\simeq 2T^{\prime}$). In Fig. 2(b), we plot $T^{\prime}$, which is the minimum time at which the deviation between theoretical and numerical values of $I_{s}(\tau)/I_{c}$ reaches $1\%$ near the peak position, as a function of $\epsilon_{0}$; the result shows the expected decrease of $T^{\prime}\sim 1/\epsilon_{0}$ as $\epsilon_{0}$ increases. In Fig. 3(a), we carry out a similar comparison for dissipative nanomagnets with $\eta=0.0001$; we find that $T^{\prime}$ decreases with $\epsilon_{0}$ in a qualitatively similar manner to the non-dissipative case. However, we note that the value of $T^{\prime}$ with finite $\eta$ (Fig. 3(a)) is larger than its $\eta=0$ counterpart (Fig. 2(b)); this feature is a consequence of opposite signs of the correction terms due to $\epsilon_{0}$ (Eq. II.2) and $\eta$ (Eq. 42). For small $\eta$ and $\epsilon_{0}$, these corrections tend to mutually cancel leading to a better stability of the zeroth order result which results in higher value of $T^{\prime}$. Finally, in Fig. 3(b), we plot $T^{\prime}_{p}$ for a periodically varying magnetic field with $\omega_{1}=1.2$. As expected from Eq. 35, $T^{\prime}_{p}\sim 100\epsilon_{0}^{-1}\gg\epsilon_{0}^{-1}$ which implies much better stability for the Shapiro-like steps for periodic magnetic field compared to their constant field counterparts.

Next, we study the presence of a finite DC component of $I_{s}$ in the case of a constant applied magnetic field along $\hat{y}$ ($f(\tau)=1$ and $\alpha_{0}=0$) in the absence of dissipation ($\eta=0$) and external AC voltage ($\gamma_{0}(\tau)=\omega_{0}\tau$). The results of our study is shown in Fig. 4 where we plot $I_{s}^{\rm DC}/I_{c}$, with $I_{s}^{\rm DC}$ given by

[TABLE]

as a function of $\omega_{0}$ for a fixed constant $\omega_{B}$. Here $T_{\rm max}=40,000$ represents the maximum time up to which we average $I_{s}(\tau)$. Note that $I_{s}(\tau)$ is chosen so that increasing it any further does not lead to a change in the peak height for $\epsilon_{0}=0$. As shown in Fig. 4(a), (b) and (c), we find that for $\epsilon_{0}\ll 1$, $I_{s}^{\rm DC}$ shows sharp peaks at $\omega_{0}=\omega_{B},2\omega_{B}$ corresponding to $n^{0}=1,2$ in Eq. 27; the position of this peaks match exactly with our theoretical results. However, the peak heights turn out to be smaller than that predicted by theory and they rapidly decrease with increasing $\epsilon_{0}$. This mismatch between theoretical and numerical results is a consequence of the linearly growing perturbative terms $\sim\epsilon_{0}$ in expression for $\delta\theta(\tau)$ and $\delta\phi(\tau)$ (Eq. II.2) which invalidate the theoretical result for $T^{\prime}\sim\epsilon_{0}^{-1}$. Thus for constant magnetic field and moderate $\epsilon_{0}>0.01$, the step-like feature predicted in Eq. 27 disappears. In Fig. 4(d), we study the behavior of the peak with variation of $\alpha_{0}$. We find that the height of the peak increases with $\alpha_{0}$ for small $\alpha_{0}$ in accordance with the theoretical prediction of Sec. II.1. For larger $\alpha_{0}>\alpha_{0}^{\rm max}$, the peak height starts to decrease and the peak height becomes almost half of its maximum for $\alpha_{0}=\pi/2$ when $\vec{B}\parallel\hat{x}$.

Next, we study the characteristics of the peaks in $I^{\rm DC}_{s}$ for periodically varying magnetic field for which $f(\tau)=\cos(\omega_{1}\tau)$. In Figs. 5(a), (b), and (c), we plot $I^{\rm DC}/I_{c}$ a function of $\omega_{1}$ for a fixed $\omega_{0}=1.2$, $\alpha_{0}=\omega_{2}=\eta=0$, and for several values of $\epsilon_{0}$. We find that the position of the peaks corresponds to integer values of $n_{2}^{0}$ (as indicated in the caption of Fig. 5) in complete accordance with Eq. 31 with $\omega_{2}=0$. Moreover, in contrast to the constant magnetic field case, the peaks of $I^{\rm DC}_{s}$ are much more stable against increasing $\epsilon_{0}$. This features of the peaks can be understood as follows. For periodic magnetic field with $\omega_{2}=0$, the zeroth order solution is given by $z(\tau)=\sin(\omega_{1}\tau)/\omega_{1}$; thus the perturbative terms $\delta\theta(\tau)$ and $\delta\phi(\tau)$ (Eq. 33) involve product of Bessel functions. This renders the effective perturbative parameter to be $\epsilon_{0}^{\rm eff}\simeq\epsilon_{0}J_{n_{1}^{s}}(\frac{1}{\omega_{1}})J_{n_{2}^{s}}(k_{0}\sin(\theta_{0}))J_{n_{3}^{s}}(\frac{n_{2}^{0}}{\omega_{1}})$ (Eqs, 34 and 35). Consequently, the effect of the perturbative correction to the weak coupling solution is drastically reduced in this case leading to a better stability of peak height with increasing $\epsilon_{0}$. Thus periodic magnetic fields are expected to lead to enhanced stability of Shapiro steps compared to their constant field counterparts. Finally in Fig. 5 (d), we show the variation of the peak height of $I^{\rm DC}_{s}$ as a

function of $\alpha_{0}$. We again find similar non-monotonic behavior of the peak height as a function of $\alpha_{0}$; the reason for this is similar to that already discussed in the context of constant magnetic field case. However, in the present case, the correction terms are much smaller and the peak height is accurately predicted by the zeroth order perturbative results: $I^{\rm DC}_{s}/I_{c}\sim 2J_{n_{1}^{0}}(k_{0}\sin(\theta_{0}-\alpha_{0}))J_{n_{2}^{0}}(n_{1}^{0}/\omega_{1})$. This is most easily checked by noting that the peak height vanishes for $\alpha_{0}=\theta_{0}=\pi/3$ for which $J_{n_{1}^{0}}(0)=\delta_{n_{1}^{0}0}$ leading to vanishing of the peak for $n_{2}^{0}=1$.

Next, we study the behavior of the system in the presence of an applied AC field of amplitude $A$ and frequency $\omega_{A}$. In the presence of such a field $\omega(\tau)=\omega_{0}+A\sin(\omega_{A}\tau)/\omega_{A}$. In Fig. 6(a), we show the behavior of the peaks of $I^{\rm DC}_{s}$ as a function of $\omega_{0}$ for a fixed $\omega_{A}=0.2$ and $A=0.1$ in the presence of a constant magnetic field. The peaks in $I^{\rm DC}_{s}$ occur at $\omega_{0}=0.4,0.6,0.8,1$ (from left to right); each of these correspond to two sets of $(n_{1}^{0},n_{2}^{0})=(3,1)\,{\rm and}\,(-2,0),\,(2,1)\,{\rm and}\,(-3,0),\,(1,1)\,{\rm and}\,(-4,0)$ and $(0,1)\,{\rm and}\,(-5,0)$ respectively as predicted in Eq. 37. In Fig. 6(b), we investigate the behavior for $I^{\rm DC}_{s}$ for a periodically varying magnetic field as a function of $\omega_{1}$ for $\omega_{0}=1.2$ and for same amplitude and frequency of the AC field. We find several peaks in $I^{\rm DC}_{s}$; each of these peaks corresponds to a fixed set of integers $(n_{1}^{0},n_{3}^{0})$ (Eq. 39 with $\omega_{2}=0$) as shown in Table 1.

Next, we study the effect of dissipation on these peaks by plotting $I^{\rm DC}_{s}$ as a function of $\omega_{0}$ in Fig. 7(a) (for constant magnetic field) and as a function of $\omega_{1}$ in Fig. 8(a) (periodic magnetic field) for $\eta=0.0001$. As seen in both cases, the position of the peaks remain same as that for $\eta=0$ in accordance with the analysis of Sec. II.3. The variation of the peak height as a function of $\log\epsilon_{0}$ and $\log\eta$ is shown in Figs. 7(b) for a constant magnetic field. We find that the maximal peak-height occur along a line in the $\epsilon_{0}-\eta$ plane. This can be seen more clearly by plotting $I^{\rm DC}_{s}/I_{c}$ as a function of $\eta$ for a fixed $\epsilon_{0}$ as shown in Fig. 9(a); the figure displays a clear peak in $I_{s}^{\rm DC}$ at $\epsilon_{0}\simeq\eta$. This can be understood from Eq. 42 and II.2 as follows. For the constant magnetic field, $z(\tau)=\omega_{c}\tau$; consequently for small $\eta$, the correction to the zeroth order solution from the dissipative term varies linearly with $\eta$ (Eq. 42)

[TABLE]

where the ellipsis indicate higher order term in $\eta$. This correction has opposite sign to the $\tau$-linear correction terms (terms corresponding to $n=n^{0}\mp 1$ in Eq. II.2) arising due to a finite $\epsilon_{0}$ in $\delta\theta(\tau)$. The corrections from $\eta$ and $\epsilon_{0}$ with opposite signs cancel along some specific line $\epsilon_{0}-\eta$ plane leading to enhanced better stability of the zeroth order solution and hence enhanced peak height. We note that the angle of this line depends on details of the relative magnitude of the correction terms. Thus we find that the presence of dissipation in a nanomagnet may lead to enhancement of the Shapiro-like steps for constant magnetic fields.

In contrast, as shown in 8(b), the peak height is almost independent of $\eta$ for small $\eta$ for periodically varying magnetic field. This can also be clearly seen from Fig. 9(b) where $I_{s}^{\rm DC}$ is shown to be indepedent of $\eta$ for small $\eta$ at fixed $\epsilon_{0}$. For such fields, $z(\tau)=\omega_{2}\tau+\sin(\omega_{1}\tau)/\omega_{1}$, where $\omega_{2}=\gamma_{g}KM_{2}/B_{1}\ll\omega_{1}$ for our choice of parameters. In this case, one can write, for small $\eta$

[TABLE]

where the ellipsis indicate higher order term in $\eta$. Thus the correction term is bounded and provides an oscillatory contribution to $\theta(\tau)$. For small $\eta$, it is insignificant compared to the correction term from $\epsilon_{0}$ and hence the peak height stays almost independent of $\eta$. Thus we find that the role of dissipation is minimal for small $\eta$ in case of periodically varying magnetic fields. The oscillatory variation of the peak height as a function of $\epsilon_{0}$ for a fixed $\eta$ can be traced to its dependence on product of three Bessel functions as can be seen from Eq. 33.

Finally, we briefly study the effect of increasing $T_{\rm max}$ in our numerical study. The relevance of this lies in the fact that for any finite $\epsilon_{0}$ and $\eta$, our analytical results hold till $\tau\sim T^{\prime}$ (constant magnetic field) or $\tau\sim T^{\prime}_{p}$ (periodic magnetic field) while the DC signal receives contribution from all $T$. Thus it is necessary to ensure that these deviations do not lead to qualitatively different results for the DC response. To this end, we plot the height of the peak value of $I_{s}^{DC}$ as a function of $1/T_{\rm max}$ in Fig. 10. We find from Fig. 10(a) that for constant magnetic field, the peak height indeed extrapolates to zero indicating that the Shapiro steps will be destabilized due to perturbative corrections if $I_{s}$ is averaged over very long time. However, we note from Fig. 10(c), $I_{s}^{\rm DC}$ could retain a non-zero value in the presence of a finite dissipation parameter $\eta$. This could be understood since the effect of damping, as shown in Fig. 8, negates that of $\epsilon_{0}$ on the peak value of $I_{s}^{\rm DC}$. Furthermore, from Figs. 10(b) and (d), we note that for the periodic magnetic fields the extrapolated value of $I_{s}^{\rm DC}$ is a finite which is lead to finite Shapiro steps in the I-V characteristics of theses JJs. Thus we expect that the Shapiro-step like features in the I-V characteristics of the JJ to be much more stable for periodically varying magnetic fields.

IV Discussion

In this work, we have studied a coupled JJ-nanomagnet system and analyzed the behavior of the supercurrent in the JJ in the presence of a finite coupling to the nanomagnet. We have provided a perturbative analytical solution to both the LL and the LLG equations governing the magnetization dynamics of the nanomagnet for arbitrary time-dependent magnetic field applied along the easy axis of the nanomagnet in the presence of weak coupling to the JJ and for weak dissipation. We have estimated the regime of validity of our perturbative results. We note here that whereas we have mainly focussed on the dynamics of the critical current of the junction in the presence of the nanomagnet in this work, the dynamics of the nanomagnet itself may also have interesting features buzhdin1 ; yury1 ; we leave this issue for future work.

Using these results, we have studied the behavior of the supercurrent of the JJ for constant and periodically varying magnetic fields. The reason for choice of such magnetic fields are that they are the only ones which lead to a fixed DC component of $I_{s}(t)$ which in turn leads to Shapiro step-like features in the I-V characteristics of a voltage-biased JJ. We note that while such features are known for constant magnetic field from earlier works cai1 ; buzhdin1 , the presence of such peaks in $I_{s}^{\rm DC}$ has not been theoretically reported for periodically time-varying magnetic fields. Moreover, we show, both from our analytical results and by performing exact numerics which supports these results, that the peaks in $I_{s}^{\rm DC}$ for periodically varying magnetic field are much more robust against increase of both $\epsilon_{0}$ and $\eta$ compared to their constant field counterparts; we therefore expect such peaks to be more experimentally accessible. We have also studied the behavior of such JJ-nanomagnet systems in the presence of external AC voltage. The presence of such a voltage leads to more peaks in $I_{s}^{\rm DC}$ whose positions are accurately predicted by our theoretical analysis. We note that our analysis, which is carried out at zero temperature, is expected to be valid at low temperature where $k_{B}T\ll\Delta_{0},g\mu_{B}B$ so that the the presence of thermal noise can be neglected. However, we point out that the effect of such noise term in our formalism can be addressed by adding a (white) noise term in the Gilbert equations following standard procedure noise1 .

Our analysis could be easily extended to unconventional superconductors hosting Majorana end states. For these junctions, the current-voltage relation is $4\pi$ periodic and given by $I_{s}=I_{c}\sin(\gamma(t)/2)$ kit1 ; kwon1 ; kiril1 . An analysis using this I-V relation immediately reveals that the Shapiro steps will be present for $\omega_{0}=2n^{0}\omega_{c}$ for constant magnetic field (Eq. 27) and $\omega_{0}=2(n_{2}^{0}\omega_{1}+n_{1}^{0}\omega_{2})$ for periodic magnetic fields (Eq. 31). The additional factor of $2$ is a consequence of $4\pi$ periodicity mentioned above. Thus coupling such JJs with Majorana end modes to nanomagnets in the presence of a magnetic field may lead to new experimental signatures of such end modes.

The experimental verification our work would involve preparing a voltage biased JJ-nanomagnet system with sufficiently small values of $\epsilon_{0}$. The current in such a junction, assuming a resistive junction, is given by

[TABLE]

where $V_{0}$ is the bias voltage, $R$ is the resistance of the junction and $\phi(t)=2eV_{0}t/\hbar=\omega_{0}t$. Thus the DC component of the current will show additional spikes when Eqs. 27 (constant magnetic field) or 31 (periodic magnetic field) is satisfied. We note that it is essential to have a voltage bias to observe these steps. This can be seen from the fact that for current-biased junctions, the phase $\phi(t)$ is not locked to a fixed value of $\omega_{0}t$ but has to be obtained from the solution of

[TABLE]

where $I$ is the bias current. Thus $\phi$ becomes a function of $m_{z}$; we have checked both using perturbative analytic methods kiril1 and exact numerics that in this case no steps exist. This situation is to be contrasted to the case of standard Shapiro steps induced via external radiation of amplitude $A$ where steps can be shown to exist for both current and voltage biases likharev1 ; kiril1 . Thus the present system would require a voltage-biased junction for observation of Shapiro steps.

For experimental realization of such a system, this we envisage a 2D thin film superconducting junction in the $x-y$ plane coupled to the nanomagnet as shown in Fig. 1. We note that value of Josephson energy and superconducting gap in a typical niobium film are $E_{J}\sim 40$K and $\Delta_{0}\sim 3$ meV respectively. Thus for a typical magnetic field $\simeq 100$ Gauss, for which $\gamma_{g}B_{1}=0.28$GHz, one could estimate an $\epsilon_{0}\simeq 0.0005$ for $k_{0}\simeq 0.1$. The Larmor frequency $\omega_{L}$ associated with such magnetic field would be order of GHz while the spin-flip processes responsible for any change in the Josephson current in niobium junctions would be valid only if $\omega_{L}\geq\Delta_{0}/\hbar\sim 4.5$THz conforming the regime for our perturbative analysis to hold. Moreover the critical current in such junctions is $I_{c}=2eE_{J}/\hbar\simeq 1.5\mu$A. Thus the peaks in DC would correspond to $I_{s}^{\rm DC}\simeq 10$nA which is well within detection capability of standard experiments. Typically such peaks would require a voltage bias of $0.1\mu$V which is also well within present experimental capability. We also note here that such experiments should also be possible with 1D junctions which have been prepared experimentally in recent times using nanowires with spin-orbit coupling rokhin1 .

To conclude, we have provided a perturbative analytic results for supercurrent of a coupled JJ-nanomagnet system in the limit of weak coupling between them and in the presence of a time dependent field applied to the system. Using this analytic result and exact numerical solution of the LL and the LLG equations, we predict existence of peaks in $I_{s}^{\rm DC}$ for both constant and periodic magnetic fields which are expected to provide Shapiro-like steps in the I-V characteristics of a voltage-biased JJ without the presence of external AC drive. We have analyzed the effect of finite dissipation of the nanomagnet and the presence of external AC drive on these peaks and discussed experiments which can test our theory.

Acknowledgement: KS acknowledges DST/RFBR grant INT/RUS/RFBR/P-249. RG acknowledges SPM fellowship from CSIR for support. The reported study was partially funded by RFBR according to the research projects 15-29-01217 and 16-52-45011, India.