Magnetic Flux Pumping in Superconducting Loop Containing a Josephson Ψ Junction S

Magnetic Flux Pumping in Superconducting Loop Containing a Josephson ψ Junction S. Mironov,1 H. Meng,2, 3, 4 and A. Buzdin2, 5 1)Institute for Physics of Microstructures, Russian Academy of Sciences, GSP-105, 603950 Nizhny Novgorod, Russia 2)University Bordeaux, LOMA UMR-CNRS 5798, F-33405 Talence Cedex, France 3)School of Physics and Telecommunication Engineering, Shaanxi University of Technology, Hanzhong 723001, China 4)Shanghai Key Laboratory of High Temperature Superconductors, Shanghai University, Shanghai 200444, China 5)Sechenov First Moscow State Medical University, Moscow, 119991, Russia We demonstrate that a Josephson junction with a half-metallic weak link integrated into the superconducting loop enables the pumping of magnetic flux piercing the loop. In such junctions the ground state phase ψ is determined by the mutual orientation of magnetic moments in two ferromagnets surrounding the half-metal. Thus, the precession of magnetic moment in one of two ferromagnets controlled, e.g., by the microwave radiation, results in the accumulation of the phase ψ and subsequent switching between the states with different vorticities. The proposed flux pumping mechanism does not require the application of voltage or external magnetic field which enables the design of electrically decoupled memory cells in superconducting spintronics.

When a superconducting loop is put in the external magnetic field the magnetic flux Φ through the loop is B~ quantized being equal to the integer number of the flux 1 quanta Φ0: Φ = nΦ0 . The interplay between the states with different vorticities n enables the implementation of multiply connected superconducting systems as flux qubits2,3, ultra-sensitive magnetic field detectors4,5, gen- erators of ac radiation6 etc. The flux quantization also Controllable Energy naturally suggests using superconducting loop as topo- switching logically protected memory cell in the devices of the rapid single flux quantum (RSFQ) logics7. The straightforward way to switch the loop between the states with different vorticity n requires application of external magnetic flux Φ ∼ Φ . Even for the loops -1 0 1 ext 0 Φ / Φ of the radius R ∼ 0.1 µm the corresponding magnetic 0 field should be ∼ 0.1 T and it grows ∝ R−2 with the de- crease of the loop size which becomes a serious obstacle FIG. 1. Sketch of the superconducting loop containing a for the miniaturization of the flux-based devices. Thus, Josephson ψ junction which enables the switching between the discovery of the physical phenomena which enable the states with different vorticities by periodic driving of the ground state Josephson phase ψ. the controllable switching of the vorticity n without application of external flux is strongly desired. One of promising mechanisms allowing the flux- free switching can be realized in the loops containing different vorticity without application of external mag- a Josephson junction with embedded magnetic order netic field. 8–11 and/or broken inversion symmetry . These systems However, in most existing ϕ0 junctions the phase vari- are known to reveal unusual collective dynamics of mag- ation is limited (δϕ0 < 2π) and the corresponding flux 12–19 netic and superconducting order parameters. In con- Φ < Φ0 cannot change the vorticity n of the supercon-

arXiv:2004.13403v1 [cond-mat.supr-con] 28 Apr 2020 trast to the conventional Josephson systems, such junc- ducting state. Indeed, the junctions with a ferromagnetic tions support the non-zero ground state phase ϕ0 be- (F) layer between the superconducting (S) leads support 25–27 tween the superconducting leads. Being incorporated only π states with ϕ0 = π which allow the design 23 into the loop such ϕ0-junctions play the role of the phase of the environmentally decoupled qubits but are not batteries producing the spontaneous electric current suitable for changing the loop vorticity. The alternat- which corresponds to the magnetic flux Φ = (ϕ0/2π)Φ0 ing F layer thickness (see, e.g., Ref. 28 and references through the loop20–24. One can expect, thus, that con- therein) as well as the presence of the Abrikosov vortex 30–32 trolling the ground state phase ϕ0, e.g., by voltage or or the pair of current injectors in one of S leads may radiation should enable effective tuning of the magnetic produce the second order phase transition with the spon- flux and switching the system between the states with taneous phase ϕ0 gradually changing between zero and 2

FIG. 2. (a) Sketch of the Josephson ψ junction where the weak link consists of a half-metallic layer sandwiched between two ferromagnets. The ground state phase ψ coincides with the angle between the projection of magnetic moments in ferromagnets to the plane perpendicular to the spin quantization axis of half-metal. (b) Sketch of possible experimental setup where circularly polarized radiation controls the rotation of magnetic moment in the F2 layer and subsequent variation of the spontaneous Josephson phase ψ. The superconducting electrodes are covered by the reﬂecting material to minimize heating eﬀects.

π (so that δϕ0 < π). The situation looks more promis- netic field. ing for the the S/F/S junctions with the broken inversion The described phase accumulation in ψ junctions re- symmetry where the spin-orbit coupling (SOC) produces minds the Berry phase effect45 since the full recovery of the arbitrary spontaneous phase ϕ0 ∝ ν sin θ where θ the initial magnetic state after the one precession period is the angle between the exchange field in the F mate- is accompanied by the 2π change of the Josephson ground rial and the direction of the broken inversion symmetry state phase ψ. Note that this phenomenon contrasts to while the constant ν characterizes the strength of SOC the behavior of the ϕ0 junctions where the recovery of the and the exchange field33–37. However, the parameter ν is exchange field direction (the angle θ) unavoidably results typically small which limits the ϕ0 variations. in the recovery of the spontaneous phase ϕ0 ∝ sin θ. In this Letter we unravel the flux-free mechanism of For the experimental realization of the described the remote control of the superconducting state vortic- radiation-stimulated vorticity switching it is convenient ity in the loop by the Josephson ψ junction in which to use the Josephson ψ junctions of the “overlap” ge- the range of the spontaneous phase ψ variation is not ometry (see Fig. 2b). This design provides a convenient limited (see Fig. 1). The weak link of the basic ψ junc- access to the surface of the F2 layer for the circularly tion consists of the layer of half-metal (HM) sandwiched polarized electromagnetic wave. At the same time, the between two ferromagnets (Fig. 2a). If the magnetic superconducting electrodes should be covered by the suit- moments in the three magnetic layers are non-coplanar able reflecting material to prevent parasitic heating ef- the current-phase relation of the junction takes the form fects. The recent experiments have demonstrated the active control of triplet supercurrents in different types I(ϕ) = Ic sin(ϕ − ψ) where the spontaneous phase ψ and 46–50 the critical current I are controlled by the directions of of S/F/S junctions including the Josepshon systems c with half-metallic layers.51,52 which provides the guid- the magnetic moments M1 and M2 in the F1 and F2 layers. Specifically, for the spin quantization axis in the ance for the appropriate choice of materials and their half-metal directed along the z-axis the phase ψ coincides parameters. To demonstrate the controllable switching of vorticity with the angle between the projection of M1 and M2 to the xy-plane while the critical current I ∝ sin α sin α and the magnetic flux pumping we consider the supercon- c 1 2 ducting loop of inductance L interrupted by the Joseph- where α1 (α2) is the angle between M1 (M2) and the z- axis.38–43. The geometry of the ψ junction favorable for son ψ junction characterized by the the current-phase the control of the loop vorticity requires the direction of relation I = Ic sin(ϕ − ψ) where Ic is the critical current (see Fig. 1). The dynamics of the Josephson phase ϕ is M1 to be fixed along the y-axis and the F2 layer to have 1 the easy-plane (xy) magnetic anisotropy. In this case described by the sine-Gordon equation : radiating the F layer with the circularly polarized elec- 2 2 ∂ ϕ 1 ∂ϕ 1 2πIc tromagnetic wave with the magnetic field rotating in the 2 + + ϕ + sin [ϕ − ψ(t)] = 0, (1) ∂t RN C ∂t LC Φ0C xy plane should cause the precession of M2 around the 44 z-axis resulting in the unlimited accumulation of the where C and RN are the capacity and normal state resis- spontaneous phase ψ. If the described ψ junction is inte- tance of the Josephson junction, respectively. Consider- grated in the superconducting loop such phase accumu- ing the case of identical easy-axis anisotropy of magneti- lation gives rise to the switching between the states with zation in both F layers of the ψ junction we assume that different vorticities and subsequent pumping of magnetic in the absence of external driving ψ(t) = 0. To analyze flux through the loop without actual application of mag- the system dynamics it is convenient to introduce the di- 3

p mensionless time τ = ωpt where ωp = 2πIc/Φ0C is the p plasma frequency, the quality factor Q = R 2πIcC/Φ0 2 (a) Stable state sin ϕ and the normalized loop inductance λ = 2πLIc/Φ0. In the absence of external driving the number of equi- 0 librium superconducting states in the loop is controlled Saddle −ϕ/λ by the loop inductance. At small λ the equation sin ϕ = -2 point −ϕ/λ defining the equilibrium states has only one triv- -6 -4 -2 0 2 4 6 ial solution ϕ = 0. When increasing λ at the threshold ϕ/π values λ = λn one gets the sequential emergence of additional pairs of the stable states ϕ = ±ϕn (n = 1, 2, 3...) 2 (b) sdl and the nearby saddle points ϕ = ±ϕn (see Fig. 3a, τ b). The precession of the magnetic moment with the fre- 0 quency ω in one of the F layers produces the subsequent ∂ϕ/∂ growth of the Josephson ground-state phase ψ = ωt and -2 the dynamics of ϕ is controlled by the equation -6 -4 -2 0 2 4 6 ϕ/π ∂2ϕ 1 ∂ϕ 1 + + ϕ + sin(ϕ − Ωτ) = 0, (2) ∂τ 2 Q ∂τ λ 5 (c) Ω = 0.1 where Ω = ω/ωp. Further we analyze the dynamics of the Josephson phase ϕ numerically solving Eq. (2) with the explicit Euler method. 4 The periodic driving with rather small Ω being applied to the system at the initial equilibrium state ϕ = 0 pro- 3 duces almost linear growth of ϕ accompanied by the in- π ϕ/ crease in the magnetic flux through the loop (see Fig. 3c). τ 0 The described flux pumping is limited by the maximal 2

stable Josephson phase, and the further precession of the ∂ϕ/∂ -1 magnetic moment gives rise to the periodic change of ϕ 1 around the maximal stable state ϕn. 0 2 4 The appropriate choice of the driving frequency Ω al- ϕ/π lows the controllable switching between different stable 0 states ϕ . Let us assume that initially the system is in 0 100 200 300 400 500 n τ the stable state ϕ = 0 and the periodic driving is switched on at τ = 0. The solution of Eq. (2) with the initial con- FIG. 3. (a) Stable states (filled red circles) of the system ditions ϕ(0) = 0 and ∂τ ϕ(0) = 0 at large τ relaxes to the periodic oscillations of Josephson phase and, thus, the separated by the saddle points (empty red circles). (b) Phase portrait of the system. Red lines are the separatrices of the oscillations of the magnetic flux through the loop (see saddle points. (c) Dependence ϕ(τ) and the phase portrait Fig. 4a). The corresponding phase portrait on the plane (see inset) under the driving at Ω = 0.1 in the flux pumping (ϕ, ∂τ ϕ) has the form of the closed cycle (see Fig. 4b) regime. In all panels λ = 15 and Q = 1. which position at the phase diagram is controlled by the driving frequency Ω. For the appropriate choice of Ω the phase trajectory crosses one of separatrices and the whole limiting cycle lays between two neighboring sep- such regime unfavorable for the design of the switching aratrices. In this case switching off the driving at any devices. arbitrary moment with 100% probability should result in In Fig. 5 we provide the diagram guiding the choice the further relaxation of the system to the state which of the driving frequency Ω corresponding to the 100% is different from the initial one. Note that the external probability switching. If for the fixed normalized loop driving with another frequency Ω can return the system inductance λ the driving frequency is chosen inside the back to the initial state, so that the switching between filled blue region then the limiting cycle does not cross the states with different n is reversible. The described the separatrices and after the driving is switched off the scenario is illustrated in Fig. 4 where the periodic driv- system would relax to the state ϕn (the corresponding ing with the frequency Ω = 0.7 switches the system from index n is written inside the filled blue region). In con- the state with n = 0 to the state with n = 1 while the trast, the choice of Ω in the intermediate white region further driving with the frequency Ω = 1.3 produces the would produce the cycle crossing the separatrices and, inverse switching. thus, cannot guarantee the controllable switching of the Note that if the limiting cycle crosses one of the sep- state vorticity. aratrices the final state strongly depends on the specific Note that throughout the paper we focused on the peri- moment when the driving is switched off which makes odic variations of the Josephson phase stimulated by lin- 4

3 (a) Ω = 0.7 Ω = 0 Ω = 1.3 Ω = 0 4 λ 30 5 2.5 3 λ 4 n = 1 2 20 2 λ π 1.5 λ 3 ϕ/ 1 λ 10 1 2 0.5 λ n = 0 n = 0 1 0 0 0 0 50 100 150 200 250 0 0.5 1 1.5 τ Ω

1 (b) 1 (c) FIG. 5. Diagram of the switching regimes controlled by the

τ τ driving frequency Ω. Each point inside the ﬁlled blue ar- 0 0 eas guarantees the switching to the stable state ϕn with the

∂ϕ/∂ ∂ϕ/∂ fixed vorticity (the corresponding index n is shown with the -1 -1 red color). For the parameters in the intermediate white re- 0 1 2 3 0 1 2 gions the final state depends on the specific moment when ϕ/π ϕ/π the driving is switched off. The values λn corresponding to the appearance of additional stable states are shown with the horizontal black lines. FIG. 4. (a) Dynamics of the Josephson phase corresponding to the controllable switching from the state n = 0 to the state n = 1 and back. The applied driving frequencies are indicated on top of the figure. The dashed lines indicate the and the quality factor Q ∼ 1. Then from Fig. 5 one sees moments when the driving is switched on and off. (b), (c) that the existence of at least one stable state with n = 1 Corresponding phase portraits of the system under the driving requires λ ∼ 10 which corresponds to the loop inductance at Ω = 0.7 and Ω = 1.3, respectively. We took λ = 15 and −11 L & 10 H (rather typical values for the existing de- Q = 1. vices). The operating frequencies ω then should be of the order of ωp. Comparing required ω values with the fre- 11 −1 quencies ωf ∼ 10 sec of the magnetization precession 57 early growing ψ(t). At the same time, for ω . ωp with the induces, e.g., by the circularly polarized radiation one increase of the quality factor Q the periodic dependence sees that the proposed mechanism of the vorticity switch- ϕ(t) first experiences the series of the period doubling bi- ing can be realized in the currently fabricated RSFQ cir- furcations and then becomes non-periodic which is rather cuits. 53,54 typical for the rf SQUIDs. Exactly such doubling of Obviously, the effects of the finite temperature and the period is responsible for the curvature inversion of noise which were not taken into account should modify the region boundaries in Fig. 5. However, to distinguish the systems dynamics. However, the estimates58 show whether the non-periodic behavior of ϕ(t) is truly chaotic that the effects of temperature-induced fluctuations do or not the further investigation is needed. not result in qualitative changes provided the loop in- We have considered the influence of the magnetic mo- ductance L is small compared to the threshold value 2 −9 ment re-orientation on the Josephson phase. Note that Lc ∼ (Φ0/2) (1/kBT ) ∼ 10 H at T = 4 K. Thus, −11 the inverse effect, when the voltage applied to the ψ junc- for rather small loops of the inductance L & 10 H the tion produces the precession of the magnetic moment, is effects of fluctuation should not be crucial. also possible. In such a case we may expect the magnetic To sum up, we have demonstrated that the integra- moment re-orientation, similar to the situation consid- tion of the Josephson S/F/HM/F/S ψ junction into the ered in Refs. 55 and 56 for the ϕ0 junction. superconducting loop enables the controllable switching Finally, let us estimate the parameters of the super- between the superconducting states with different vortic- conducting loop and the Josephson ψ junction required ities without application of external magnetic flux. The for the realization of the described memory cell. For the peculiar coupling between the precession of the magnetic typical Josephson junctions in the RSFQ devices with the moment in the F layer and the Josephson phase oscil- critical current Ic ∼ 1 µA, the normal state resistance lations in the ψ junction allow the realization of the RN ∼ 100 Ω and the capacitance C ∼ 1 pF the plasma magnetic flux pumping into the loop controlled, e.g., by 10 11 −1 frequency lays in the GHz range (ωp ∼ 10 ÷10 sec ) the microwave radiation. The proposed mechanism may 5

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