Supercurrent-Induced Magnetization Dynamics
SSupercurrent-Induced Magnetization Dynamics
Jacob Linder and Takehito Yokoyama Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway Department of Physics, Tokyo Institute of Technology,2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan (Dated: November 8, 2018)We investigate supercurrent-induced magnetization dynamics in a Josephson junction with two misalignedferromagnetic layers, and demonstrate a variety of effects by solving numerically the Landau-Lifshitz-Gilbertequation. In particular, we demonstrate the possibility to obtain supercurrent-induced magnetization switch-ing for an experimentally feasible set of parameters, and clarify the favorable condition for the realization ofmagnetization reversal. These results constitute a superconducting analogue to conventional current-inducedmagnetization dynamics and indicate how spin-triplet supercurrents may be utilized for practical purposes inspintronics.
PACS numbers: 74.45.+c
Introduction . The interplay between superconducting andferromagnetic order is presently generating much interest ina variety of research communities [1]. Besides the obviousinterest from a fundamental physics point of view, a majorpart of the allure of superconductor | ferromagnet (S | F) hy-brid structures is the prospect of combining the spin-polarizedcharge carriers present in ferromagnets with the dissipation-less flow of a current offered by the superconducting envi-ronment. By tailoring the desired properties of a hybrid S | Fsystem on a nanometer scale, this interplay opens up new per-spectives within spin-polarized transport.Closely related to the transport of spin is the phenomenonof current-induced magnetization dynamics [2]. The gen-eral principle is that a spin-polarized current injected into aferromagnetic layer can act upon the magnetization of thatlayer via a torque and thus induce magnetization dynamics[3]. Previous works in this field have considered mainly non-equilibrium spin accumulation via quasiparticle spin-injectionfrom a ferromagnet into a superconductor [4–8]. The conceptof supercurrent-induced magnetization dynamics suggests aninteresting venue for combining the seemingly disparate fieldsof superconductivity and spintronics. However, magnetiza-tion dynamics in a Josephson junction has so far been dis-cussed only in a handful of works [9–13]. In particular, itwas demonstrated in Ref. [9] how a supercurrent would in-duce an equilibrium exchange interaction between two non-collinear ferromagnets in an S | F | N | F | S junction (N stands fornormal metal). Taking into account the fact that a Joseph-son current flowing through such an inhomogeneous magne-tization profile will have a spin-triplet contribution [1], suchan interaction implies that it should be possible to generate supercurrent-induced magnetization dynamics in this type ofjunction. This would constitute a superconducting analogueto magnetization dynamics in a conventional spin-valve setup.To this date, this remains unexplored in the literature.Motivated by this, we study in this Letter for the first timethe magnetization dynamics of a multilayer ferromagneticJosephson junction when a current bias is applied, based onthe Landau-Lifshitz-Gilbert (LLG) equation [14]. Our main
FIG. 1: (Color online) The proposed experimental structure. Twoferromagnetic layers separated by a normal spacer are sandwichedbetween two s -wave superconductors. The magnetization directionsmay be non-aligned and tuned by means of an external field, as longas the exchange coupling between the ferromagnets is sufficiently re-duced by the thickness of the normal spacer. By current-biasing thissystem one may generate a supercurrent which induces magnetiza-tion dynamics. idea is to utilize the spin-triplet nature of the Josephson cur-rent, which very recently has been observed experimentally[18–20], in order to induce a torque on the magnetic order pa-rameter and thus generate magnetization dynamics. The ex-perimentally relevant setup is shown in Fig. 1: two ferromag-nets with different coersive fields are separated by a normalspacer and sandwiched between two conventional s -wave su-perconductors. The coersive fields are such that the magneticorder parameter is hard in one layer, while soft in the other. a r X i v : . [ c ond - m a t . s up r- c on ] J un τ M ag n e t i z a t i o n c o m p o n e n t s (a) m y ( τ ) m z ( τ ) τ (b) τ M ag n e t i z a t i o n c o m p o n e n t s (c) −1 0 1−101 0 5 10 15 20 25 30 35 40 45 50−1−0.8−0.6−0.4−0.200.20.40.60.81 τ (d) −1 0 1−101 m x ( τ ) m y ( τ ) m z ( τ ) FIG. 2: (Color online) The time-evolution of the normalized magnetization components m j . Here, we have set ζ =
290 with an initial anglemisalignment θ / π = .
1. We have considered the weak damping regime α (cid:28) ω = . α = .
05 while in(b) ω = . α = .
05. In the lower row, we have considered stronger Gilbert damping and set (c) ω = . α = . ω = . α = .
5. The insets display a parametric plot of the ˆ yyy - and ˆ zzz -components of the magnzetization with the red circles indicating the point τ = τ ∈ [ , ] . By application of an external field
HHH ext , it is thus possible totune the relative orientation of the local magnetization fields inthe two layers. When this junction is current-biased, a super-current flows without resistance up to a critical strength. Thesupercurrent strongly modifies the equilibrium exchange in-teraction between the ferromagnetic layers and should thus beexpected to result in supercurrent-induced magnetization dy-namics. We investigate this by solving numerically the LLGequation, which provides the time-evolution of the magneti-zation components in the soft magnetic layer. As we shallsee, a number of interesting opportunities arise in terms of thetorque exerted on the soft layer by the Josephson current. Weproceed by first establishing the theoretical framework used inthe forthcoming analysis and then present our main results.
Theory . The magnetization dynamics of the Josephsonjunction is governed by the LLG equation, which in the free magnetic layer reads: ∂ mmm L ∂ t = − γ mmm L × HHH eff + α (cid:16) mmm L × ∂ mmm L ∂ t (cid:17) , (1)where γ is the gyromagnetic ratio and α is the damping con-stant. The effective magnetic field is obtained from the freeenergy functional via the relation HHH eff = − ( δ F / δ mmm L ) / ( V M ) where V is the unit volume and M is the magnitude of themagnetization. Note that only transverse magnetization dy-namics are described by the LLG equation due to the cross-terms coupling to mmm L , meaning that | mmm L | = F S = A ∆ λ F cos φ [ J ( mmm L · mmm R ) + J ( mmm L · mmm R ) − J ] , (2)where A is the unit area, λ F is the Fermi wavelength, and ∆ is the gap magnitude. The parameters J i are analogues tothe quadratic and biquadratic coupling constants for a mag-netic exchange interaction. In addition to Eq. (2), one shouldalso include the anisotropy contribution F M to the free energywhich provides an effective field HHH M eff = ( Km y / M ) ˆ yyy where K is the anisotropy constant. We have here assumed that theanisotropy axis (cid:107) ˆ yyy . After some algebra, one arrives at the finalform of the LLG equation in our system: ∂ mmm L ∂ t = mmm L × (cid:16) − γ Km y M ˆ yyy + mmm R γ∆ cos φ ( t ) dM λ F [ J + J ( mmm L · mmm R )]+ α ∂ mmm L ∂ t (cid:17) (3)where φ ( t ) = ω J t and d is the thickness of the ferromagneticlayer. Importantly, we note that the anisotropy contributionproportional to K was not included in the effective field usedin previous works [9]. This contribution is nevertheless essen-tial since the supercurrent-induced torque must overcome theanisotropy contribution in order to switch the magnetizationorientation. In what follows, we will present a full numeri-cal solution of this equation to investigate the supercurrent-induced magnetization dynamics. To this end, we first estab-lish experimentally relevant values of the parameters in Eq.(3). The above equation may be cast into a dimensionlessform by introducing ω F = γ K / M , τ = ω F t , ω = ω J / ω F ,and ζ = ∆ / ( Kd λ F ) . Here, ω F is the ferromagnetic reso-nance frequency and ω J is the Josephson frequency. Employ-ing a realistic estimate [16] for transmission probabilities inthe F | N | F part of the system, one finds [9] that Eq. (2) ac-counts well for the Josephson current when J = .
007 and J = . K (cid:39) × − K ˚A − . Moreover, we set ∆ = λ F = d =
10 nm as standard values [17] forthe hybrid structure under consideration. The Josephson fre-quency ω J is typically of order GHz, but may be tuned ex-perimentally. We will therefore consider several choices of ω = ω J / ω F and the damping constant α to model a variety ofexperimentally accessible scenarios. As long as the require-ment ¯ h ω J (cid:28) T c is fulfilled, i.e. the time-dependent part is asmall perturbation, one may consider φ ( t ) as a time-dependentexternal potential in the static expression for the free energy[13]. Since typically ω J ∼ µ V, this condition is easily met.The supercurrent-induced magnetization dynamics come intoplay when the local magnetizations { mmm L , mmm R } are misalignedwith an angle θ = θ ( t = ) to begin with, and vanishes when θ = { , π } . The magnetization in the right (hard) layer isfixed at mmm R = ˆ yyy . Results and Discussion.
Using the parameters discussedabove, we now proceed to investigate the resulting magneti-zation dynamics for both weak damping ( α (cid:28) ) and more considerable damping ( α ∼ ) . Choosing a small initial an-gle of misalignment θ / π = .
1, the numerical solution ofthe LLG-equation is shown in Fig. 2. The upper row showsthe weak-damping regime for two different choices of the fre-quency ratio ( ω < ω > mmm L (cid:39) ˆ yyy . The oscillations eventually dieout and the magnetization orientation of the two layers sat-urates in a parallel configuration. This happens on a fastertime-scale in (b) due to the larger value of ω used comparedto (a). Consider now the case of stronger Gilbert dampingshown in the lower row of Fig. 2. The a.c. Josephson current-induced torque now induces more ”violent” oscillations andis able to rotate the magnetization orientation by π . Whenthe frequency is large enough, exemplified in (d) by ω = . full magnetization reversal is achieved and maintained. Ona larger time-scale, the same behavior is seen for lower fre-quencies. In general, we observe in our numerical simula-tions that the main difference between the weak and strongGilbert damping regime is that the magnetization dynamics isperiodic and oscillating in the former case, whereas it tendsto saturate to a fixed orientation in the latter case. This isreasonable as the damping effectively leads to a more rapiddecay of the oscillating magnetization dynamics induced bythe a.c. current. The results in Fig. 2 suggest that it is pos-sible to generate supercurrent-induced magnetization rever-sal in the setup shown in Fig. 1. This phenomenon shouldbe intimately linked with the spin-triplet correlations presentin the non-collinear setup considered here since spin-singletcorrelations cannot carry torque. Recently, controllable long-range triplet supercurrents have been experimentally observedin Josephson junctions with strong inhomogeneous ferromag-netic layers [18–20]. In the present setup, torque carried bythe supercurrent does not become long-ranged [21] (an addi-tional source of triplet correlations would be required for thatpurpose [22]) whereas it is still due to spin-triplet, thus me-diating magnetic correlations between the two ferromagneticlayers.We proceed by investigating the role of the anisotropy pref-erence in the soft magnetic layer. This is modelled by theterm proportional to K in Eq. (3). The parameter ζ effectivelymodels the relative weight of the anisotropy energy and theJosephson energy related to the presence of superconductiv-ity. In Fig. 3(a), we focus on the behavior of the magnetiza-tion component m y . We plot its time-evolution for increasingvalues of ζ . It is seen that supercurrent-induced magnetiza-tion reversal occurs for ζ > ζ c with a critical value ζ c (forthe present parameters, we find ζ c (cid:39) ζ becomes larger. In Fig. 3(b)-(d), we give a phase-diagramfor the magnetization switching by plotting m y ( t → ∞ ) in the α − ζ , ζ − ω , and α − ω plane. In general, the results indi-cate that the torque generated by the a.c. Josephson effectcan reverse the magnetization orientation when the Gilbert τ m y ( τ ) (a)
200 300 4000.10.20.30.40.5 ζ (b) α (d) ω α (c) ω ζ ζ =P-regime(switched)AP-regime FIG. 3: (Color online) (a) The time-evolution of the normalized mag-netization component m y . Here, we have set ω = . α = . θ / π = .
1. Above a criticalvalue of ζ , permanent switching occurs from mmm L (cid:107) ˆ yyy to mmm L (cid:107) ( − ˆ yyy ) .The phase-diagram for magnetization reversal [from the parallel (P)configuration mmm L (cid:107) mmm R to the anti-parallel (AP) configuration mmm L (cid:107) ( − mmm R ) ] is shown in (b-d). A contour-plot is given for m y ( t → ∞ ) with (b) ω = .
5, (c) α = .
5, and (d) ζ = damping is non-neglible, the anisotropy contribution is suf-ficiently weak and Josephson frequency is sufficiently small.The phase-diagram in (b) and (d) shows that the magnetizationcan be trapped in the switched direction when the damping α is sufficiently large, in agreement with the results in Fig. 2.It should also be noted that the switching becomes less viableas ω increases as seen from (c) and (d). When ω J (cid:29) ω F , thetorque from the Josephson current oscillates very fast com-pared to the characteristic motion of the magnetization. In thisway, the magnetization would experience an averaged torqueover many oscillations, which results in small effect due topartial cancellation of the net torque. Conclusion.
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