A cryogenic memory element based on an anomalous Josephson junction

C. Guarcello1 and F.S. Bergeret1, 2

1Centro de Física de Materiales, Centro Mixto CSIC-UPV/EHU,Paseo Manuel de Lardizabal 5, 20018 San Sebastián, Spain

2Donostia International Physics Center, Paseo Manuel de Lardizabal 4, 20018 San Sebastián, Spain(Dated: September 27, 2019)

We propose a non-volatile memory element based on a lateral ferromagnetic Josephson junction with spin-orbit coupling and out-of-plane magnetization. The interplay between the latter and the intrinsic exchangefield of the ferromagnet leads to a magnetoelectric effect that couples the charge current through the junctionand its magnetization, such that by applying a current pulse the direction of the magnetic moment in F canbe switched. The two memory states are encoded in the direction of the out-of-plane magnetization. Withthe aim to determine the optimal working temperature for the memory element, we explore the noise-inducedeffects on the averaged stationary magnetization by taking into account thermal fluctuations affecting both theJosephson phase and the magnetic moment dynamics. We investigate the switching process as a function ofintrinsic parameters of the ferromagnet, such as the Gilbert damping and strength of the spin-orbit coupling, andproposed a non-destructive readout scheme based on a dc-SQUID. Additionally, we analyze a way to protect thememory state from external perturbations by voltage gating in systems with a both linear-in-momentum Rashbaand Dresselhaus spin-orbit coupling.

I. INTRODUCTION

Superconducting electronics is suggested as playing an im-portant role in the search of ultra-low-power computers [1–5].One of the key challenges towards this objective is the fabrica-tion of a reliable and scalable cryogenic memory architecture.Superconductor-ferromagnet-superconductor (SFS) junctionsare promising structures suggested for such memories [6–15].Indeed, the interplay between the intrinsic exchange field andthe induced superconductivity in the ferromagnet leads to theso-called π-junction, i.e., a Josephson junction exhibiting anintrinsic π-phase shift in its ground state. Vertical ferromag-netic multilayer structures are being used as Josephson mag-netic memories. The two logic states of these memories usu-ally correspond to states with different relative orientation ofmagnetic layers, that in turn determines whether the junctionis in the 0- or π-state. Readout schemes are commonly basedon distinguish between resistive and non-resistive states.

Here, we suggest an alternative cryogenic memory elementbased on a ferromagnetic anomalous Josephson junction, usu-ally called ϕ0-junction [16]. It consists of a SFS Josephsonjunction with a Rashba-like spin-orbit coupling (SOC). Itsground state corresponds to a finite phase shift in its current-phase-relation 0 < ϕ0 < π . Such anomalous phase has beenrecently detected experimentally in hybrid Josephson junc-tions fabricated with the topological insulator Bi2Se3 [17] andAl/InAs heterostructures [18]. Both materials have strongspin-orbit coupling and in those experiments, time reversal isbroken by an external magnetic field that acts as a Zeemanfield. In our proposal, the memory element is a Josephsonjunction with a ferromagnetic link, thus in principle time-reversal is broken intrinsically by the exchange field. Asdemonstrated theoretically, in these junctions the magneti-zation of F can be controlled by passing an electric currentthrough the device [19–24]. We propose to use such a junc-tion as a memory element with the information encoded in themagnetization direction of the F layer. An important issue forthe memory element is the effects stemming from the unavoid-

able thermal fluctuations on both phase and magnetization dy-namics. Therefore we present an exhaustive analysis of thenoisy dynamics of a current-biased SFS Josephson junction,see Fig. 1, considering the influence of stochastic thermal fluc-tuations. Indeed, we explore this current-induced magneticbistability, in order to define two well-distinguishable logicstates, and investigate the robustness of such memory againstnoise-induced effects. We also suggest a suitable non-invasivereadout scheme based on a current-controlled superconduct-ing quantum interference device (SQUID), which does notrequire an additional magnetic flux to set the optimal oper-ating point. Moreover, we discuss the intriguing possibilityof effectively shielding the memory state by voltage gating,in a device formed by a ferromagnetic layer with a linear-in-momentum Dresselhaus spin-orbit coupling term.

The work is organized as follows: In Sec. II, we presentthe theoretical model used to describe the time evolution ofboth the magnetic moment and the Josephson phase of acurrent-driven SFS junction. In Sec. III we discuss the mag-

SSF x y

zIbias

Mz

Imax

Mz

FIG. 1. S/F/S Josephson junction driven by a rectangular bias currentpulses, Ibias, with amplitude Imax. The z-component of the magneti-zation, Mz, is the observable used to define the logic memory states0 and 1.

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netic configuration of the junction after applying a currentpulse. Specifically we determine its stationary magnetiza-tion, as a function of the Gilbert damping parameter and theSOC strength. In Sec. IV, we focus on the effect of stochas-tic thermal fluctuations on both the phase and the magnetiza-tion dynamics. In Sec. V, we explore how a generic Rashba-Dresselhaus type SOC can affect the stationary magnetization.In Sec. VI, we discuss a feasible readout scheme based on adc-SQUID magnetometer. In Sec. VII we present our conclu-sions.

II. THE MODEL

We consider a similar setup as the one studied in Ref. 20. Itconsists of a SFS junction in which the ferromagnet is a thinfilm with an out-of-plane magnetic anisotropy and a Rashba-like SOC, see Fig. 1.

Because of the interplay between the exchange field and theSOC, the current-phase relation of the SFS Josephson junc-tion reads Iϕ = Ic sin(ϕ −ϕ0), where Ic is the critical currentof the junction, ϕ is the Josephson phase difference, and ϕ0is the so-called anomalous phase shift. The latter depends onseveral parameters of the system, such us the Rashba coeffi-cient α [25, 26], the transparency of S/F interfaces, the spinrelaxation, and the disorder degree of the junction. The ex-act dependence on these parameters is here not crucial, butthe geometric one. If we consider a two-dimensional SOCwith momenta in the plane of the F film, and the charge cur-rent flows in x-direction, then the phase shift ϕ0 is propor-tional to the y-component of the magnetic moment accordingto [16, 20, 27, 28]

ϕ0 = rMy

M, (1)

where M =√

M2x +M2

y +M2z is the modulus of the magneti-

zation vector, and the parameter r quantifies the SOC strengthand contains the α-dependence. The specific dependence ofr on the SOC will be discussed in Sec. IV, when we considerthe coexistence of both Rashba and Dresselhaus SOC contri-butions and their impact on the magnetization dynamics.

As discussed in Refs. 20, 22, 24, and 29, Eq. (1) estab-lishes a direct coupling between the magnetic moment andthe supercurrent. The time evolution of the magnetization canbe described in terms of the Landau–Lifshitz–Gilbert (LLG)equation [30, 31]

dMdτ

=−grM×Heff +γ

M

(M× dM

dτ

), (2)

where gr denotes the gyromagnetic ratio. The first term onthe right-hand side describes the precession motion aroundHeff, whereas the second term describes the dissipation, whichis accounted by the phenomenological dimensionless Gilbertdamping parameter γ . The i-th component of the effectivefield Heff, with i = x,y,z, can be calculated as [32]

Heff,i =−1Ω

∂F

∂Mi, (3)

where Ω is the volume of the F layer, and F is the free energyof the junction, which can be written as

F =−EJϕIbias +Es(ϕ,ϕ0)+EM. (4)

Here, EJ = Φ0Ic/(2π) with Φ0 being the flux quantum, Ibias isthe external current in units of Ic, Es(ϕ,ϕ0) = EJ [1−cos(ϕ−

ϕ0)], and EM =−K Ω

2

(MzM

)2is the magnetic energy depend-

ing on the anisotropy constant K . The ratio between the en-ergy scales of the system will be indicated by the parameterε = EJ/(K Ω).

The effective magnetic field calculated from Eq. (3) reads

Heff =K

M[εr sin(ϕ− rmy) y+mzz] , (5)

with mx,y,z = Mx,y,z/M.Since the normalized components of the magnetization sat-

isfy the condition m2x + m2

y + m2z = 1, it is convenient to

write the LLG equations in spherical coordinates [33], so thatthe normalized components of the magnetization can be ex-pressed in terms of the polar and azimuthal angles θ and φ

as

mx(τ) = sinθ(τ)cosφ(τ)

my(τ) = sinθ(τ)sinφ(τ) (6)mz(τ) = cosθ(τ).

By normalizing the time to the inverse of the ferromagneticresonance frequency ωF = grK /M, that is t = ωF τ , the LLGequations in spherical coordinates reduce to the following twocoupled equations [33]

dθ

dt=

11+ γ2

(Heff,φ + γHeff,θ

)(7)

sinθdφ

dt=

11+ γ2

(γHeff,φ − Heff,θ

), (8)

where the θ and φ components of the normalized effectivefield are defined as

Heff,θ = εr sin(ϕ− rmy)cosθ sinφ −mz sinθ (9)

Heff,φ = εr sin(ϕ− rmy)cosφ . (10)

Next, we discuss how the Josephson phase ϕ(t) of a mag-netic junction responds to a driving bias current. The dy-namics of an overdamped SFS Josephson junction can bedescribed through the modified resistively shunted junction(RSJ) model [34, 35] generalized to include the anomalousphase shift ϕ0 = rmy. The phase dynamics is described by theequation:

dϕ

dt= ω [Ibias(t)− sin(ϕ− rmy)]+ r

dmy

dt, (11)

where the time is still normalized to the inverse of the ferro-magnetic resonance frequency and ω = ωJ/ωF , with ωJ =2πIcR/Φ0 being the characteristic frequency of the junc-tion [34] (R is the normal-state resistance of the device).

3

mzst

-1.0

-0.5

0

0.5

1.0

mzst

-1.0

-0.5

0

0.5

1.0

(a)

(d)

(b)

(e)

(c)

(f)

FIG. 2. Stationary magnetization, mstz , as a function of r and γ , at

different values of the amplitude Imax of the current pulse, in theabsence of noise fluctuations. The other parameters are: ε = 10,ω = 1, σ = 5, and mz(t = 0) = +1.

Last term in the right-hand side of Eq. (11) stems from thetime derivative of the anomalous phase, ϕ0, cf. Eq. (1), andwas ignored in Refs. 20, 22, and 24, but has to be included inorder to preserve gauge-invariance [35].

We assume that the SFS junction is driven by a rectangularcurrent pulse, Ibias, centered at tc:

Ibias(t) =

Imax, tc−σ ≤ t ≤ tc +σ

0, elsewhere. (12)

Here, σ is the width and Imax is the intensity, in units of Ic,of the pulse, so that a value Imax > 1 indicates a bias currentlarger than the critical value.

Because of the magnetoelectric effect in a ϕ0-junction, thecharge current induces an in-plane magnetic moment [27, 28,36–39], which in turn acts as a torque on the out-of plane mag-netization of the F layer and eventually leads to its switch-ing [20, 22].

In the next sections we search for an optimal combinationof system parameters to induce the magnetization reversal.Specifically, we explore the response of the magnetization byvarying γ and r in suitable ranges, whereas the energy andtimescales ratios ε and ω , are fixed. The energy ratio ε rangesfrom ε ∼ 100 [20] in systems with weak magnetic anisotropy,to ε ∼ 1 for stronger anisotropy [40]. In our calculation wechoose an intermediate value ε = 10. The typical ferromagnetresonance frequency is ωF ' 10 GHz, while the characteristicJosephson frequency, usually of the order of gigahertz, maybe tuned experimentally. Therefore we choose ω = 1. Aslong as the injected bias current is below the critical value, theresults discussed in this work are only weakly affected by thevalue of ω . In contrast, if Ibias > 1 the magnetic switchingwould become more unlikely as ω increases. In particular,for ω 1 the torque exerted by the Josephson current oscil-lates very fast, in comparison with the timescale of the mag-netization [21]. This means that the magnetization would ex-perience an effective torque averaged over many oscillations,

-1.0-0.50.00.51.0

mzst

r = 0.1 - γ = 0.25

Imax = 0.8

-1.0-0.50.00.51.0

mzst

Imax = 0.9

-1.0-0.50.00.51.0

mzst

Imax = 1.

-1.0-0.50.00.51.0

mzst

Imax = 1.1

-1.0-0.50.00.51.0

mzst

Imax = 1.2

-1.0-0.50.00.51.0

mzst

Imax = 1.3

-1.0-0.50.00.51.0

mzst

Imax = 1.4

0 10 20 30 40 50-1.0-0.50.00.51.0

σ

mzst

Imax = 1.5

FIG. 3. Stationary magnetization, mstz , as a function of the current

pulse width, σ , at different values of the amplitude Imax ∈ [0.8−1.5],in the absence of noise fluctuations. The other parameters are: r =0.1, γ = 0.25, ε = 10, ω = 1, and mz(t = 0) = +1.

which results in a small contribution due to a partial cancella-tion of the net torque.

At t = 0 we assume that the F magnetization points towardsthe z-direction, that is M = (0,0,1). With this initial valueswe solve Eqs. (7)-(8) and (11) self-consistently for differentvalues of the parameters. From the solution we determine themagnetization direction after the current pulse.

III. THE DETERMINISTIC ANALYSIS

We first neglect the effect of thermal noise, as in Refs. 22and 24, and explore the magnetic switching of the junction.

The overall behavior of the stationary magnetization mstz ,

namely, the value of mz at the time t = tmax = 100, as afunction of both the Gilbert damping parameter, γ , and theSOC strength, r, at different current pulse intensities Imax andσ = 5, is summarized in Fig. 2. We note that each contour plotis characterized by a dark fringes pattern, namely, we observeregions of the (γ,r) parametric space in which the magneti-zation reversal systematically occurs, i.e., in which mst

z =−1.This means, for instance, that by increasing r at a fixed γ weobserve a sequence of mst

z =+1 and mstz =−1 values.

For small enough r values, the magnetization reversal effectis absent. Interestingly, the pattern of dark fringes evidently

4

changes for Imax > 1, so that for Imax = 1.6 the dark fringesmerge together in large areas of (γ,r) values in which magne-tization reversal takes place.

With the aim of selecting a driving pulse suitable for amemory application, we observe that the magnetization rever-sal effect should be sufficiently robust against small changesof both the current pulse intensity and the duration of thepulse. In Fig. 3 we show the stationary magnetization as afunction of the pulse width σ , by changing the pulse am-plitude Imax. We have chosen the parameters r = 0.1 andγ = 0.25. We observe that for a bias current below the crit-ical value, the need for an accurate current-width regulation issignificantly relaxed, since the magnetization reversal defini-tively occurs for any width above a specific value. Instead, forcurrent amplitudes higher than the critical value, the station-ary magnetization versus σ is highly scattered between thetwo possible values, mst

z = ±1, so that even a slight changein the pulse width may lead to a non-switching situation. Tounderstand this behavior, we observe that, for a bias currenthigher than Ic, a pulse sufficiently long can make the junctionto switch to the resistive state, so that the Josephson phaserapidly evolves and a voltage drop across the device appears,since V ∝

dϕ

dt [34]. In this case, the steady magnetizationstrongly depends on the dynamical state of the system whenthe current pulse is switched off. Moreover, the higher the biascurrent is, the faster the Josephson phase evolve and more pro-nounced will be the σ -dependence of mst

z . In view of a mem-ory application, a current pulse smaller than the critical valueis therefore recommended, in order to make the magnetizationreversal unrelated on the pulse width σ . For these reasons inthe subsequent analysis we set Imax = 0.9 and σ = 5 and fo-cus on the thermal effect on the phase and the magnetizationdynamics.

IV. EFFECTS OF NOISE

In this section, we focus on the noisy dynamics of the junc-tion, specifically on how it affects the magnetization reversal.The temperature can significantly influence the time evolutionof the system, eventually inducing unwanted magnetizationflip or preventing a stable magnetization reversal. Therefore,we consider stochastic thermal fluctuations in both the phaseand the magnetic moment dynamics. We start discussing theeffect of a thermal noise source only on the phase within theRSJ model. In the second part of this section we include alsothe thermal noise on the magnetization dynamics.

A. Thermal current effects on the RSJ model

The phase dynamics can be directly perturbed by thermalfluctuations accounted by adding to the RSJ model, Eq. (11),a Langevin Gaussianly distributed, delta correlated stochas-tic term, Ith(t). This “thermal current” has the usual white-noise statistical properties that, in normalized units, can be

FIG. 4. (a) Stationary magnetization, mstz , as a function of r and

γ . (b) Average stationary magnetization, mstz , as a function of r and

γ , at DI = 0.01, calculated by averaging over Nexp = 100 indepen-dent numerical repetitions. (c) Average stationary magnetization,mst

z , as a function of the thermal current intensity, DI, at γ = 0.25and r = 0.1 [namely, the (γ,r)-values highlighted with a red circlein panel (b)], calculated by averaging over Nexp = 1000 independentnumerical repetitions. The inset shows the normalized temperaturescorresponding to the noise intensities DI. For all panels Imax = 0.9and σ = 5, whereas the values of other parameters are the same usedto obtain Fig. 2.

expressed as [34, 41, 42]

〈Ith(t)〉= 0 (13)⟨Ith(t)Ith(t ′)

⟩= 2DIδ

(t− t ′

). (14)

Here, we introduced the dimensionless amplitude of thermalcurrent fluctuations defined as

DI =kBT

RωF

I2c

=1ω

kBTEJ

. (15)

For example, if ω = 1 we obtain DI ∼ 0.04 TIc

µAK , so that, for

instance, a junction with Ic = 1 µA at T = 250 mK is affectedby a thermal fluctuation of intensity DI ∼ 10−2.

By taking into account the noise contribution, Eq. (11) be-comes

dϕ

dt= ω [Ibias(t)− sin(ϕ− rmy)+ Ith(t)]+ r

dmy

dt. (16)

In Fig. 4 we compare the current-induced magnetization re-versal obtained without and with accounting of the noise ef-fects, see panel (a) and (b) respectively. We set the intensityof the current pulse Imax = 0.9 and its width σ = 5.

5

In Fig. 4(a) we show the behavior of mstz as a function of

r and γ in the deterministic case, namely, in the absence ofnoise, DI = 0. Here, we observe a contour plot composed bymany narrow dark fringes in which mst

z =−1, see Fig. 4(a).The situation drastically changes if we include the thermal

noise. In this case we focus on the average stationary magne-tization, mst

z , which is computed by averaging the stationarymagnetization over Nexp = 100 independent numerical runs.The behavior of mst

z as a function of r and γ for DI = 0.01 isillustrated in Fig. 4(b). At small values of r the magnetiza-tion reversal is still absent, whereas noise mostly affects theregions with large r where the averaged value of the magne-tization is mainly distributed around zero. Nevertheless, onecan still identify dark regions in which magnetization switch-ing takes place. With a red circle we highlight in Fig. 4(b) theregion around the point (γ,r) = (0.25,0.1) where the mag-netization takes the largest negative average magnetizationmst

z ' −1. In other words, the region with the most robustswitching.

By increasing the noise intensity the switching process issuppressed, as shown in Fig. 4(c), at the optimal values r = 0.1and γ = 0.25 in Fig. 4(b). In Fig. 4(c) the value of mst

z isthe average over Nexp = 1000 independent numerical repeti-tions. In the inset we show the normalized temperatures cor-responding to the noise intensities DI, calculated by assuminga junction with a temperature-dependent critical current andIc = 100 µA at low temperatures [43]. From this figure, onesees that mst

z '−1 only for DI . 0.01, that is for T . 0.75Tc.For higher noise intensities both the average magnetizationand the error bar increase, approaching a zero magnetizationaverage only for DI & 0.3.

In Fig. 5 we explore the time evolution of the different ob-servables with and without thermal noise. Specifically, weshow the response of the junction with γ = 0.25 and r = 0.1to a current pulse with amplitude Imax = 0.9 depicted in panel(a). In the absence of noise, DI = 0, we plot in Fig. 5(b)the time evolution of the phase and the supercurrent, and inFig. 5(c) the different components of the magnetic moment.During the current pulse, i.e., the yellow shaded region, thephase first increases, and then it goes to zero when the pulseis turned off, see Fig. 5(b). To understand the phase behav-ior, we observe that, in the washboard-like picture [34], thetilting imposed by the bias current Imax = 0.9 is not enoughfor allowing the “particle” to overcome the nearest potentialbarrier and switch the system to the finite voltage “running”state. Instead, the phase-particle remains confined within apotential minimum, so that when the current is turned off, theslope of the washboard potential goes again to zero and thephase restores its initial position, i.e., ϕ → 0.

We observe that the larger the bias current pulse is, thehigher is the washboard potential slope, and therefore forImax > 1 the greater the speed of the phase particle, so thatit can take a longer time to restore the initial position afterthe current pulse is switched off. Moreover, a large bias cur-rent pulse may also longer switching times. Hence, a currentImax < 1 is, in general, more advantageous for a memory ap-plication.

In Fig. 5(c) we show how all components of the magne-

0 10 20 30 400.00.20.40.60.81.0

I bias

φ/π

Iφ

0.00.20.40.60.8

mxmymz

0 10 20 30 40-1.0

-0.5

0.0

0.5

1.0

mx,my,mz

-0.20.00.20.40.60.81.0

DI =0.05

0 10 20 30 40-1.0

-0.5

0.0

0.5

1.0

t

mx,my,mz

DI =0.05

(b)

(c)

(d)

(a)

(e)

FIG. 5. Current pulse (a) and following time evolution of phase andJosephson current, see panel (b), and magnetization components, seepanel (c), in the absence of noise and including a thermal currentcontribution with amplitude DI = 0.05, see panels (e) and (d). Thevalues of other parameters are Imax = 0.9, σ = 5, r = 0.1, and γ =0.25. The legends in panels (b) and (c) refer also to panels (d) and(e), respectively.

tization are induced by the current pulse. Whereas mx andmy are generated during the current pulse, and they undergo adamped oscillations around zero when the current is switchedoff, the z-component, after a transient regime, flips definitivelyto the value mz = −1. From this figure we can also estimatethe switching time tSW ' 10, as the time mz roughly takes toapproach the value −1 after switching off the current.

The scenario described so far essentially persists also in thestochastic case, as shown in Figs. 5(d-e) for DI = 0.05. There-fore, at the temperature that we are considering, the overallbehavior is still quite similar to the one obtained in the ab-sence of noise. In fact, the z-component of the magnetizationflips again to the value−1, while the x and y components tendto oscillate around zero, without, however, vanishing defini-tively.

The magnetization switch can be achieved in a short timescale, by passing through the junction a sequence of current

6

FIG. 6. Time evolution of the magnetization mz, in response to a se-quence of three current pulses shown in the top panel, in the presenceof a thermal current noise with amplitude DI = 0.05. The values ofthe other parameters are: Imax = 0.9, σ = 5, r = 0.1, and γ = 0.25.

pulses, as it is shown in Fig. 6. In the bottom panel we showthe time evolution of the magnetization mz, when the junctionis excited by the three subsequent current pulses presentedin the top panel, in the presence of a thermal current noisewith amplitude DI = 0.05. In response to each current pulse,mz follows first a transient regime, and then, as the current isswitched off, it approaches the steady value with an oppositesign.

B. Effect of thermal noise on the magnetization dynamics

Thermal noise also affects directly the magnetization dy-namics [44–49] via a stochastic field Hth, a sort of “thermalfield”, which is added to the effective magnetic field term inEq. (2), as done in Ref. [50]. Inclusion of the thermal noise inEq. (2) leads to [33]

dMdτ

=−grM× (Heff +Hth)+γ

M

(M× dM

dτ

). (17)

This stochastic differential equation has to be solved numeri-cally by a stochastic integration prescription by keeping themodulus of the magnetic moment constant during the timeevolution (see Ref. 33 and references therein). For this pur-pose it is again convenient to write the equations in sphericalcoordinates, see Eq. (6), so that the stochastic LLG equationreads [33, 51]:

dθ

dt=

11+ γ2

[Heff,φ + Hth,φ + γ

(Heff,θ + Hth,θ

)](18)

sinθdφ

dt=

11+ γ2

[γ

(Heff,φ+Hth,φ

)−Heff,θ−Hth,θ

], (19)

where

Hth,θ = Hth,x cosθ cosφ+Hth,y cosθ sinφ−Hth,z sinθ (20)

Hth,φ =−Hth,x sinφ+Hth,y cosφ . (21)

The normalized field, Hth = (M/K )Hth is assumed to be a

10-4 0.001 0.01 0.1 1

-1.0

-0.5

0.0

0.5

1.0

DI

mzst

r = 0.1 - γ = 0.25 - Nexp = 1000

FIG. 7. Average stationary magnetization, mstz , as a function of the

noise intensity, DI, calculated by taking into account both the thermalcurrent and the thermal field noise contribution, and by averagingover Nexp = 1000 independent numerical repetitions. The values ofother parameters are Imax = 0.9, σ = 5, r = 0.1, and γ = 0.25.

Gaussianly distributed random field with the following statis-tical features ⟨

Hth,i(t)⟩= 0 (22)⟨

Hth,i(t)Hth,i(t ′)⟩= 2DHδ

(t− t ′

), (23)

where i = x,y,z and

DH =

(γ

MkBT|gr|Ω

)(MK

)2

ωF = γkBTK Ω

(24)

is the dimensionless amplitude of thermal field fluctuations.In all previous equations the time is still normalized to theinverse of ωF .

Interestingly, by recalling the definition of the parameterε = EJ/(K Ω), from Eqs. (15) and (24) we can easily obtainthe following relation between the normalized thermal noiseintensities

DH = (γ εω)DI. (25)

Thus, by changing the magnetization energy, the Gilbertdamping parameter, or the magnetic resonance frequency wecan effectively modify the relative strength of the two noisemechanisms. This means that one could optimize the systemparameters in such a way to make, for instance, the impact ofthe thermal field negligible with respect to the thermal current.This allows us to study the effects produced by these noisesources independently. In the following, even if we explicitlywrite only the value of DI, we are taking into account boththermal current and thermal field independent noise sources,which amplitudes are related by Eq. (25).

The overall effect of both the thermal current and field ispresented in Fig. 7, where we show the behavior of mst

z , calcu-lated by averaging over Nexp = 1000 independent numericalruns, at different values of the noise intensity DI, and by set-ting Imax = 0.9, σ = 5, γ = 0.25, and r = 0.1. We observe thatthe average magnetization remains close to the value mst

z '−1

7

FIG. 8. Time evolution of phase and Josephson current (a) and mag-netization components (b) as the system is excited by the currentpulse in Fig. 5. Here we are taking into account both a thermal cur-rent and a thermal field contribution, with noise intensity DI = 0.005.The values of other parameters are Imax = 0.9, σ = 5, r = 0.1, andγ = 0.25.

only for DI . 0.003, that is for T . 0.58Tc, see the insetof Fig. 4(c). For larger values of DI, mst

z approaches zeroand hence the magnetization reversal probability is reduced,Fig. 4(c). In view of the memory application, one should, inprinciple, carefully choose the F layer and its characteristics(such as its volume or the Gilbert damping parameter) in or-der to make the thermal field effect as small as possible. Theaim is to reduce the thermal field intensity in order to increasethe working temperature suitable for a memory application,e.g., through a lower Gilbert damping or a larger F volume,according to Eq. (25).

The time evolution of ϕ , Iϕ , and mi (with i = x,y,z), asthe junction dynamics if affected by both a thermal currentand a thermal field, for r = 0.1, γ = 0.25, and DI = 0.005, isshown in Fig. 8. Here, we consider again the system excitedby a current pulse with intensity Imax = 0.9, as that one shownin Fig. 5(a). We observe that all noisy curves still resemblein shape the deterministic evolution presented in Figs. 5(b)-(c). The value tSW ' 10 is a quite good estimation for theswitching time of the device also in this noisy case.

V. RASHBA-DRESSELHAUS SOC

In all previous analysis it was assumed a pure Rashba SOC.However, the theory of ϕ0-junctions can be generalized forany linear-in-momentum SOC [27, 28], by using the SU(2)-covariant formulation [28], where the SOC is described interms of a SU(2) vector potential A . For a 2D SOC withboth Rashba and Dresselhaus contributions one obtains Ax =−ασ y + βσ x and Ay = ασ x− βσ y (here, α and β are theRashba and Dresselhaus coefficients and σ x and σ y are thefirst two Pauli matrices).

The appearance of the anomalous phase is related to theexistence of a finite Liftshitz invariant term in the free en-ergy [16, 52–54] which is proportional to Ti∂iϕ , where Ti

is the i-th component of a polar vector which is odd undertime reversal, ∂i is the i-the derivative of the superconduct-ing phase, and the sum over repeated indices is implied hereand below. For the particular junction geometry sketched inFig. 1 the supercurrent, and hence the phase gradient, is fi-nite in x-direction. Thus, according to Eq.(5.17) of Ref. 27,the anomalous phase can be written in the following compactform [27, 28]:

ϕ0 = rβ(βmx +my). (26)

Here, we defined the SOC coefficients ratio, β = β/α , andthe parameter r

β= r(1− β 2), with r depending this time on

both α and β . In the absence of the Dresselhaus SOC, thatis when β = 0 and r

β→ r, we recover Eq. (1). If both con-

tributions are similar in magnitude, i.e., when β → 1, sincer

β→ 0 the phase shift vanishes, i.e., ϕ0 → 0. This is a very

interesting situation that we explore in this section. In fact,whereas the Dresselhaus contribution is due to the breakingof crystal inversion symmetries, the Rashba SOC stems fromstructural broken symmetry and therefore can be controlled bya gate voltage [55, 56]. In other words, a voltage gate can con-trol the ratio β between Dresselhaus and Rashba coefficients,and hence the phase shift and the supercurrent flow, accord-ing to Eq. (26). Specifically, by tuning α such that β ' 1 onecan fully decouple the phase and magnetic moment dynamics.Such a process can be eventually used to protect the memorystate in one of the storage elements of a distributed architec-ture.

We provide next a quantitative analysis of this situation, sothat by taking into account the generic ϕ0, Eq. (26), into theexpression for the effective field, Eq. (5) becomes

Heff =K

M

β εr

βsin[ϕ− r

β

(βmx +my

)]x+

εrβ

sin[ϕ− r

β

(βmx +my

)]y+mzz

. (27)

The θ and φ components of the normalized effective field tobe included in LLG Eqs. (7)-(8), read

Heff,θ= β εrβ

sin[ϕ− r

β

(βmx +my

)]cosθ cosφ (28)

+εrβ

sin[ϕ−r

β

(βmx +my

)]cosθ sinφ−mz sinθ

Heff,φ=−β εrβ

sin[ϕ− r

β

(βmx +my

)]sinφ (29)

+εrβ

sin[ϕ− r

β

(βmx +my

)]cosφ ,

whereas the RSJ equation becomes

dϕ

dt=ω

Ibias(t)− sin

[ϕ− r

β

(βmx +my

)](30)

+rβ

(β

dmx

dt+

dmy

dt

).

The behavior of the stationary magnetization as a function ofr and γ , at different values of β ∈ [0− 1] is shown in Fig. 9.

8

mzst

-1.0

-0.5

0

0.5

1.0

mzst

-1.0

-0.5

0

0.5

1.0

(a)

(d)

(b)

(e)

(c)

(f)

FIG. 9. Stationary magnetization, mstz , as a function of r and γ , at

different values of the relative Dresselhaus coefficient β = β/α , inthe absence of noise fluctuations, by imposing Imax = 0.9 and σ = 5.

The current pulse intensity and width are chosen equal toImax = 0.9 and σ = 5, respectively. As expected from the dis-cussion above, the region where no magnetization switchingoccurs, bright color in Fig. 9, increases by increasing β to-wards 1. For β = 1, the ϕ0 behavior is fully suppressed, cf.Eq. (26), and hence no magnetization switching takes place,despite the current pulse flowing through the junction. Inter-estingly, we note that for intermediate values of β , the area ofthe switching fringes, i.e., where mst

z =−1, increases consid-erably.

In summary of this section, by tuning α = β one can decou-ple the magnetic behavior from the Josephson dynamics andfreeze the memory state in order to protect it from externalcurrent pulses and other perturbations. The major challengein this regard is to find materials with a sizable magnetic mo-ment and tunable by means of a gate voltage.

VI. THE MEMORY READ-OUT

As discussed in previous section, the writing operation ofthe proposed memory element can be performed by excitingthe junction with controlled current pulses. We could envis-age an array of ϕ0-junction-based memory elements, each oneeventually having its own current line so that it can be writtenby sending individual current inputs. Alternatively, exploitingthe tuning of the SOC discussed in previous section, one couldcontrol locally, via individual gates at each junction, severalmemory elements connected in series to a common currentline. In this way one could selectively write via a commoncurrent pulse only a specific set of memory units.

The read-out of the memory state can be non-destructivelyperformed by direct measurement of the magnetization statethrough a dc-SQUID inductively coupled to the ϕ0-junction.A SQUID is essentially a magnetic flux detector [57], whichcan be employed to measure with a very high sensitivity any

VV

IbiasMz

IreadΦµ

IbiasMz

Iread(a) (b) Φµ

SQ SQ

FIG. 10. SQUID-based memory readout and cartoon showing thecritical current interference pattern of the SQUID, in the cases of bothpositive and negative orientation along the z-axis of the magneticmoment, see panel (a) and (b), respectively.

physical quantity that can be converted in a magnetic flux [58].We suggest a SQUID sensor along the lines of the read-

out scheme implemented in Ref. 13 for a π-junction mem-ory. Our scheme is based on an asymmetric inductive dc-SQUID, which consists of a superconducting ring with a non-negligible total inductance, L, with two Josephson junctionswith different critical currents, i.e., Ic,1 6= Ic,2 (here, we are ne-glecting for simplicity any asymmetry in the ring inductance).With such asymmetric SQUID, one can avoid the use of an ad-ditional magnetic flux to adjust the working point of the devicein a high sensitivity position of the ISQ

c −Φ characteristics,where ISQ

c is the SQUID critical current and Φ is the magneticflux threading the loop. In fact, such asymmetric dc-SQUIDshows non-negligible screening and asymmetry parameters,that is βL = 2π

Φ0L2 (Ic,1 + Ic,2) 6= 0 and αI =

Ic,1−Ic,2Ic,1+Ic,2

6= 0, re-

spectively. In this case, the ISQc maximum is not centered in

Φ = 0, but shifted from zero by ∆Φ, see Fig. 10, where [57]2π∆Φ = Φ0βLαI . Accordingly, our readout SQUID demon-strates a high sensitivity point of the ISQ

c −Φ characteristicsalso in Φ = 0, that is in the absence of an external magneticflux.

We assume that the unbiased critical current, ISQc (Φ = 0),

lies in the positive branch of the critical current diffraction

pattern of the SQUID, that is dISQc

dΦ

∣∣∣Φ=0

> 0, just like in thecase shown in Fig. 10. Thus, the magnetic moment mz = +1generates a positive magnetic flux Φ=+Φµ through the loop,and gives a critical current higher than the unbiased value, i.e.,I+c = ISQ

c (+Φµ)> Ic(0), see the red dashed line in Fig. 10(a).Conversely, if mz = −1 the magnetic flux through the loopis negative, i.e., Φ = −Φµ , and the critical current is lowerthan the unbiased value, I−c = ISQ

c (−Φµ)< Ic(0), see the bluedashed line in Fig. 10(b).

The SQUID readout loop is then sensed by passing a bit-read current with intensity Iread = ISQ

c (0) through the device,see the orange dashed line in Fig. 10, that is a current whichlays in between I−c and I+c . In this way, if the magnetic mo-ment points in the negative z-direction, which encodes the

9

“1” logic state of the memory, the bit-read current makes theSQUID to switch to the voltage state, since Iread > I−c . Con-sequently, a voltage drop appears across the readout SQUIDin response to the bit-read current. For the opposite magneticmoment orientation, which encodes the “0” logic state, theSQUID critical current is larger than the bit-read current, thatis Iread < I+c , so that the SQUID remains in the superconduct-ing, zero-voltage state.

To provide an estimate of the dimensions of a SQUID loopsuitable to detect the magnetic moment reversal we considerthe simple case of a circular readout SQUID with radius aand a magnetic moment oriented along the z-axis and placedon the symmetry axis of the SQUID at a height z′ above thex− y plane. In this case, the total magnetic flux through theloop can be estimated as [58, 59] Φµ = µ0Mz

2 a2/(

a2 + z′2)3/2.

This flux is maximum at z′ = 0 where it reads Φµ = µ0Mzd ,

with d = 2a being the loop diameter. If we assume a satura-tion magnetization [60] M = 8×105 A/m and a volume Ω =(10× 100× 100) nm3 = 10−22 m3, we get a total magneticmoment Mz = 8× 10−17 Am2 (that is Mz = 8.6× 106 µB,with µB being the Bohr magneton), so that the total magneticflux reads Φµ = 32π10−24d−1 Wb m. Thus, for a SQUID withdiameter' 0.5 µm we obtain a measurable flux Φµ 'Φ0/10,(see vertical dot-dashed lines in Fig. 10).

VII. CONCLUSIONS

In conclusion, we propose a non-volatile superconductingmemory based on a bistable magnetic behavior of a current-biased ϕ0-junction, that is a superconductor-ferromagnet-superconductor Josephson junction with a Rashba-like spin-orbit coupling (SOC). The memory state is encoded in themagnetization direction of the ferromagnetic layer, which canbe switched via controlled current pulses. We explore the ro-bustness of the current-induced magnetization reversal againstthermal fluctuations, in order to find the optimal working tem-perature at which the magnetization switching induced by acurrent pulse is stable. We also suggest a way of decouplingthe Josephson phase and the magnetization dynamics, by tun-ing the Rashba SOC strength via a gate voltage.

Finally, we discuss a suitable non-destructive readoutscheme based on a dc-SQUID inductively coupled to the ϕ0-junction. The suggested readout scheme is exclusively con-trolled by current pulses, and no additional magnetic flux isneeded.

ACKNOWLEDGMENTS

This work was supported by the Horizon Research and In-novation Program under Grant Agreement No. 800923 (SU-PERTED) and the Spanish Ministerio de Economía, Industriay Competitividad (MINEICO) under Project No. FIS2017-82804-P.

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