Thermal flux-flow regime in long Josephson tunnel junctions
Claudio Guarcello, Paolo Solinas, Francesco Giazotto, Alessandro Braggio
TThermal flux-flow regime in long Josephson tunneljunctions
Claudio Guarcello , Paolo Solinas , Francesco Giazotto ,Alessandro Braggio Centro de Fisica de Materiales (CFM-MPC), Centro Mixto CSIC-UPV/EHU,20018 Donostia-San Sebastian, Basque Country, Spain SPIN-CNR, Via Dodecaneso 33, 16146 Genova, Italy NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, Piazza SanSilvestro 12, I-56127 Pisa, ItalyE-mail: [email protected]
Abstract.
We study thermal transport induced by soliton dynamics in a longJosephson tunnel junction operating in the flux-flow regime. A thermal bias across thejunction is established by imposing the superconducting electrodes to reside at differenttemperatures, when solitons flow along the junction. Here, we consider the effect ofboth a bias current and an external magnetic field on the thermal evolution of thedevice. In the flux-flow regime, a chain of magnetically-excited solitons rapidly movesalong the junction driven by the bias current. We explore the range of bias currenttriggering the flux-flow regime at fixed values of magnetic field, and the stationarytemperature distribution in this operation mode. We evidence a steady multi-peakedtemperature profile which reflects on the average soliton distribution along the junction.Finally, we analyse also how the friction affecting the soliton dynamics influences thethermal evolution of the system. a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug hermal FF regime in a LJJ
1. Introduction
The possibility of mastering the local temperature of a long Josephson junction (LJJ)by acting on solitonic excitations is scientifically intriguing and it has an applicativepotential in fast heat mastering. Since a soliton is a magnetic flux quantum surroundedby a loop of dissipationless supercurrents, how can it affect thermal transport throughthe system? In this regard, it was recently discussed theoretically [1] that in a longJosephson tunnel junction a steady localized 2 π -twist of the phase, that is a soliton [2–7], is able to locally affect the quasiparticle heat-current flowing through a junctionformed by superconducting electrodes residing at different temperatures. In this case,the emerging temperature modulation in correspondence of a soliton is not ascribed toa direct Cooper pairs contribution, but it is a local phase-dependent modulation of heatcarried by quasiparticles flowing from the hot to the cold electrode. In fact, after theearlier theoretical prediction that heat transport can depend on the Josephson phasedifference [8–12], this phenomenon was recently confirmed experimentally in severaltemperature-biased Josephson devices [13]. This phenomenon gives, for instance, thecapability to control the temperature of the system via an external magnetic field, as itwas demonstrated, both theoretically and experimentally, in heat interferometers [14–17] and quantum diffractors [18–20] of heat currents. These examples fall within theso-called phase-dependent caloritronics [13, 21] from which many temperature-basednovel devices were recently conceived [22–31]. In a long Josephson tunnel junction, i.e.,a junction in which one dimension is longer than the Josephson penetration length [61],the externally applied magnetic field can penetrate the junction in the form of fluxons.This kind of excitations can be controlled and handled in different ways, for instancethey can be moved by a bias current, or created by a magnetic field or a dissipativehotspot [32, 33], pinned by inhomogeneities [34, 35], and also manipulated throughshape engineering [36–39]. Additionally, it was recently understood that solitons caninduce thermal effects in a temperature-biased junction, so that applications as thermalrouter [1, 40] and heat oscillator [41] have been suggested.In this paper we give a further insight in the research field of phase-dependentcaloritronics based on LJJs. In fact, here we explore the effects of both an externalmagnetic field and a bias current on thermal transport through a temperature-biasedLJJ. In particular, we investigate the so-called flux-flow regime [42–44], that is the casein which solitons in the form of fluxons are continuously magnetically excited from oneedge of the junction and then forced to shift towards the opposite junction edge under theaction of a bias current. In this operation mode, we observe some peculiar thermal effectsdepending on the dynamical state of solitons excited along the system. In particular,despite solitons move very rapidly, we observe temperature rises inhomogeneously inspecific points of the system. We study also how friction affecting the phase dynamicsinfluences the stationary temperature distribution.The number of applications in different fields based on LJJs is still nowadaysgrowing [45–52], not to mention that ones in which LJJs are used in both flux-flow hermal FF regime in a LJJ xyz S T S T H ext H ext 𝐼 𝑏 P e-ph T >T ( x ) >T bath P e-ph P in 𝐼 𝑏 I b Φ F P e-ph L Figure 1.
A tunnel LJJ driven by both an external in-plane magnetic field, H ext ( t ),and a homogeneously distributed bias current, I b . The temperature T i of each electrode S i is also indicated. A chain of solitons drifting along the junction under the action of I b is depicted. The incoming, i.e., P in ( T , T , ϕ, V ), and outgoing, i.e., P e -ph ( T , T bath ),thermal powers in S are also represented, for T > T ( x ) > T bath . In the inset: biascurrent-induced Lorentz force, F L ∝ I b × Φ , acting on a soliton [where the directionof Φ depends on the polarity, σ , of the soliton, see Eq. (2)]. regime and oscillator [53–57]. How a homogeneous temperature gradient applied alonga LJJ (namely, from one edge of the junction to the other) affects soliton dynamics wasearlier studied both theoretically and experimentally [58–60], but the soliton-sustainedthermal transport as a temperature gradient is imposed across the system (namely, asthe electrodes forming the junction reside at different temperatures) was exclusivelyaddressed recently [1, 40, 41]. In this regard, both the heat oscillator application [41]and the thermal router [1] readily lend themself to a further advance in the flux-flowoperation mode.This paper is organised as follows. In the next section both the sine-Gordonequation and thermal model are presented. In section 3 the theoretical results, includingthe temperature dynamics as a function of the bias current, the external magneticfield, and the damping parameter, are shown and analysed. Finally, in section 4 theconclusions are drawn.
2. The Model
The system is driven by both an external magnetic field, H ext ( t ), applied in both sides ofthe device and a homogeneous bias current, I b , flowing through the junction, see Fig. 1.The behavior of a long and narrow Josephson tunnel junction depends on the dynamicsof the Josephson phase ϕ , which can be described by the perturbed sine-Gordon (SG)equation [61] ∂ ϕ (cid:0)(cid:101) x, (cid:101) t (cid:1) ∂ (cid:101) x − ∂ ϕ (cid:0)(cid:101) x, (cid:101) t (cid:1) ∂ (cid:101) t − sin (cid:2) ϕ (cid:0)(cid:101) x, (cid:101) t (cid:1)(cid:3) = α ∂ϕ (cid:0)(cid:101) x, (cid:101) t (cid:1) ∂ (cid:101) t + (cid:101) I b . (1) hermal FF regime in a LJJ (cid:101) x = x/λ J and (cid:101) t = ω p t , respectively, where λ J = (cid:113) Φ πµ t d J c is the Josephson penetration depthand ω p = (cid:113) π Φ J c C is the Josephson plasma frequency. Here, we introduced thecritical current area density J c = I c / ( L × W ) (where L and W are the length andthe width of the junction, respectively) and the effective magnetic thickness [18, 19] t d = λ L, tanh ( d / λ L, ) + λ L, tanh ( d / λ L, ) + d (where λ L,i and d i are the Londonpenetration depth and the thickness of the electrode S i , respectively, and d is theinsulating layer thickness). Moreover, µ is the vacuum permeability and Φ = h/ e (cid:39) × − Wb is the magnetic flux quantum, with e and h being the electron chargeand the Planck constant, respectively. We point out that we make use in this work ofa notation in which a tilde over a letter labels a dimensionless normalized quantity, sothat, for instance, the term (cid:101) I b = I b /I c in Eq. (1) indicates the normalized bias currentflowing through the junction. The friction in the phase dynamics is accounted by thedamping parameter α = 1 / ( ω p R a C ), with R a and C being the normal-state resistanceper area and the specific capacitance of the junction, respectively [62]. For simplicity,in the SG model we neglect the surface losses in the electrodes ‡ .The Josephson penetration depth λ J represents the main length-scale in our system,so that the junction is called “long” when its length and width, in units of λ J , read (cid:101) L = L/λ J (cid:29) (cid:102) W = W/λ J (cid:28)
1, respectively. Moreover, λ J roughly indicates alsothe width of a soliton [5, 43]. This is a 2 π -twists of the phase that in the LJJ frameworkhas a clear physical meaning, since it carries a quantum of magnetic flux, induced by asupercurrent loop surrounding it, with the local magnetic field perpendicularly orientedwith respect to the junction length. Thus, solitons in the context of LJJs are alsoindicated as fluxons or Josephson vortices . In the unperturbed case, i.e., Eq. (1) withno drive and dissipation, a moving soliton has the simple analytical expression [61] ϕ (cid:0)(cid:101) x − (cid:101) u (cid:101) t (cid:1) = 4 arctan (cid:34) exp (cid:32) σ (cid:101) x − (cid:101) x − (cid:101) u (cid:101) t √ − (cid:101) u (cid:33)(cid:35) , (2)where σ = ± − signindicates an antisoliton) and (cid:101) u is the speed of the soliton, given in units of the Swihartsvelocity ¯ c = λ J ω p [61]. The latter is the phase velocity of electromagnetic wavespropagating in the junction, and can approach the values ¯ c ∼ − m / s in high-quality tunnel LJJs. The velocity-dependent factor in Eq. (2) represents the relativisticcontraction of the soliton when its velocity approaches the maximum speed [4]. This isthe consequence of Lorentz invariance of the unperturbed SG equation [61].An external magnetic field, H ext , affects the phase dynamics, since it is accounted ‡ For moderate fluxon velocities, the effective damping term including both the quasiparticle tunnelingand the surface current losses can be accounted by α eff = α + β/ β being the parameterquantifying the surface losses of the junction. hermal FF regime in a LJJ dϕ (0 , t ) d (cid:101) x = dϕ ( (cid:101) L, t ) d (cid:101) x = 2 H ext H c, = (cid:101) H. (3)The coefficient H c, = Φ πµ t d λ J is called the first critical field of a LJJ [44]. So, for magneticfields H ext exceeding this critical value, that means for (cid:101) H ≥ (cid:101) H thr with (cid:101) H thr = 2, solitonspenetrate the junction also in the absence of an applied bias current.When a bias current is flowing through the system the situation changes, since itexerts a Lorentz force, F L ∝ I b × Φ , on a soliton, see the inset of Fig. 1, with thedirection of Φ depending on the polarity of the soliton, see Eq. (2). Thus, in thepresence of an external bias current the soliton is forced to shift along the junction. Inthe case of several solitons excited in the system, the chain of solitons moves againstthe damping forces under the action of the Lorentz force exerted by the bias current.Reaching an edge of the junction, the fluxon leaves the system, while a new fluxon entersthe junction from the opposite edge. This operation mode is called flux-flow regime [42–44]. In this state, the magnetic flux penetrates effectively the junction in the form offluxons.In Ref. [1], it was demonstrated that the phase distribution along a LJJ affectsthermal transport through the system, when a temperature bias is imposed. In thiswork we investigate the time evolution of the temperature T of the floating electrode S by changing both the bias current flowing through the system and the applied magneticfield. Specifically, we assume to work with a JJ in which the electrode S is kept at afixed temperature T , while S has a floating temperature T . This can be accomplishedby optimizing the volumes of the electrodes. For the sake of readability, hereafter wewill use an abbreviated notation in which x and t dependences are left implicit, namely, T = T ( x, t ), ϕ = ϕ ( x, t ), and V = V ( x, t ). A characteristic length scale for thethermalization in the diffusive regime can be estimated as the inelastic scattering length (cid:96) in = √D τ s , where D = σ N / ( e N F ) is the diffusion constant (with σ N and N F beingthe electrical conductivity in the normal state and the density of states at the Fermienergy, respectively) and τ s is the quasiparticle scattering lifetime [63]. In Ref. [1], thevalue (cid:96) in (cid:39) . µ m was estimated for a Nb lead at 4 . L (cid:29) (cid:96) in , the electrode S canbe modelled as a one-dimensional diffusive superconductor at a temperature varyingalong x direction [1]. In this case the evolution of the temperature T is given by thetime-dependent diffusion equation [1]dd x (cid:20) κ ( T ) d T d x (cid:21) + P in ( T , T , ϕ, V ) − P e -ph ( T , T bath ) = c v ( T ) d T d t . (4)Here, the rhs represents the variations of internal energy density of S and the lhs termsindicate the spatial heat diffusion, taking into account the inhomogeneous electronic heatconductivity, κ ( T ), and both the phase-dependent incoming, i.e., P in ( T , T , ϕ, V ), and hermal FF regime in a LJJ P e -ph ( T , T bath ), thermal power densities in S . The phase-dependentthermal power density flowing from S to S reads P in ( T , T , ϕ, V ) = P qp ( T , T , V ) − cos ϕ P cos ( T , T , V ) . (5)In the adiabatic regime [64], that is when the voltage drop is smaller than the relevantenergy scales in the system, eV (cid:28) min { k B T , k B T , ∆ ( T ) , ∆ ( T ) } , the contributions P qp and P cos can be written as P qp ( T , T , V ) = 1 e R a d (cid:90) ∞−∞ dε N ( ε − eV, T ) N ( ε, T )( ε − eV )[ f ( ε − eV, T ) − f ( ε, T )] , (6) P cos ( T , T , V ) = 1 e R a d (cid:90) ∞−∞ dε N ( ε − eV, T ) N ( ε, T ) × ∆ ( T )∆ ( T ) ε [ f ( ε − eV, T ) − f ( ε, T )] , (7)where f ( E, T ) is the Fermi distribution function and N j ( ε, T ) = (cid:12)(cid:12)(cid:12)(cid:12) Re (cid:20) ε + iγ j √ ( ε + iγ j ) − ∆ j ( T ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) is the reduced superconducting density of state, with ∆ j ( T j ) and γ j being the BCSenergy gap and the Dynes broadening parameter [65] of the j -th electrode, respectively § .Equation (6) describes heat power density carried by quasiparticles, namely, it is anincoherent flow of energy through the junction from the hot to the cold electrode [8, 66].Instead, Eq. (7) represents the phase-dependent part of heat transport originating fromthe energy-carrying tunneling processes involving recombination/destruction of Cooperpairs on both sides of the junction [8, 66]. As we are going to discuss later morespecifically, this term is responsible for the localized temperature modulation in thepresence of a soliton.At this point we make a step backward to clarify better what we mean whenwe refer to heat transport due to a soliton. First of all, a soliton in a LJJ carries amagnetic flux quantum which is generated by a circulating supercurrent loop [5]. Thesedissipationaless superconducting currents give no contribute in the thermal dynamics weare going to discuss. Furthermore, the energy transport in a thermal-biased JJ includesalso a phase-dependent term due to energy-carrying tunneling processes involvingdirectly Cooper pairs [64]. However, since it is a purely reactive contribution [67],in writing the thermal balance equation, see Eq. (4), we have to neglect it since it doesnot contribute to the average heat flux, which determines the stationary temperatureprofile T . So, when we mention “soliton-induced” thermal effects, we are still dealingwith temperature variations produced by heat carried by quasiparticles flowing throughthe junction. This “heat current” is due to the imposed temperature gradient, i.e.,the electrodes have to reside at different temperatures, but depends on the phasedifference according to Eq. (5), as it was recently demonstrated in many caloritronics § We observe that the width d of the electrode S appears in Eqs. (6)-(7) since we wrote the thermalbalance equation, see Eq. (4), in terms of volume power densities. hermal FF regime in a LJJ π phase twist,can locally affects thermal transport and, therefore, the temperature of the junction.The term P e -ph in Eq. (4) represents the energy exchange, per unit volume, betweenelectrons and phonons in the superconductor and reads [68] P e -ph = − Σ96 ζ (5) k B (cid:90) ∞−∞ dEE (cid:90) ∞−∞ dεε sign( ε ) M E,E + ε (cid:40) coth (cid:18) ε k B T bath (cid:19) × (cid:104) F ( E, T ) − F ( E + ε, T ) (cid:105) − F ( E, T ) F ( E + ε, T ) + 1 (cid:41) , (8)where F ( ε, T ) = tanh ( ε/ k B T ), M E,E (cid:48) = N i ( E, T ) N i ( E (cid:48) , T ) [1 − ∆ ( T ) / ( EE (cid:48) )], Σis the electron-phonon coupling constant, and ζ is the Riemann zeta function. Here, weare assuming that the lattice phonons of the superconductor are very well thermalizedwith the substrate that resides at T bath , thanks to the vanishing Kapitza resistancebetween thin metallic films and the substrate at low temperatures [66].Finally, in Eq. (4), c v ( T ) = T d S ( T )d T is the volume-specific heat capacity, with S ( T )being the electronic entropy density of S [25] S ( T ) = − k B N F (cid:90) ∞ dε N ( ε, T ) { [1 − f ( ε, T )] ln [1 − f ( ε, T )] + f ( ε, T ) ln f ( ε, T ) } , (9)and κ ( T ) is the electronic heat conductivity [69] κ ( T ) = σ N e k B T (cid:90) ∞−∞ d εε cos (cid:110) Im (cid:104) arctanh (cid:16) ∆( T ) ε + iγ (cid:17)(cid:105)(cid:111) cosh (cid:16) ε k B T (cid:17) . (10)For gaining insight in thermal transport through the junction, it only remains toinclude in Eq. (4) the specific phase ϕ ( x, t ) for a LJJ given by solving numericallyEqs. (1) and (3), with initial conditions ϕ ( (cid:101) x,
0) = dϕ ( (cid:101) x, /d (cid:101) t = 0 ∀ (cid:101) x ∈ [0 − (cid:101) L ].
3. Numerical results
In the present study, we consider an Nb/AlO x /Nb tunnel LJJ characterized by a normalresistance per area R a = 50 Ω µ m and a specific capacitance C = 50 f F/µ m . Thelinear dimensions ( L × W × d ) of the junction are set to (150 × . × . µ m and d = 1 nm is the thickness of the insulating layer.For Nb electrodes, we assume λ L = 80 nm, σ N = 6 . × Ω − m − , Σ =3 × Wm − K − , N F = 10 J − m − , ∆ (0) = ∆ (0) = ∆ = 1 . k B T c (with T c = 9 . γ = γ = 10 − ∆.We impose a thermal gradient across the system, specifically, the bath resides at T bath = 4 . S resides at a temperature T = 7 K kept fixed throughout thecomputation. This value of the temperature T assures the maximal soliton-inducedheating in S , for a bath residing at T bath = 4 . hermal FF regime in a LJJ Figure 2. (a) and (c), Temperature evolution for a fixed bias current (cid:101) I b = 0 .
2, at (cid:101) H = 1 .
75 and (cid:101) H = 2, respectively. (b) and (e), Soliton number, N , and mean voltagedrop, V mean , as a function of the bias current (cid:101) I b at (cid:101) H = 1 .
75 and (cid:101) H = 2, respectively.(c) and (f), Stationary temperature profile T st2 ( x ) as a function of (cid:101) I b , at (cid:101) H = 1 .
75 and (cid:101) H = 2, respectively. then adjusting the temperature T . However, we stress that reducing the workingtemperatures could lead to a significantly longer thermal response time [17].The electronic temperature T ( x, t ) of the electrode S is the key quantity to masterthermal transport across the junction, since it can modulate under the influence of boththe external magnetic field and the bias current. By taking into account the specifictemperature-dependence in both t d ( T , T ) and J c ( T , T ) [70, 71], we can estimate thevalues of the parameters of the system: λ J (cid:39) . µ m, ω p (cid:39) . H c, (cid:39) . α (cid:39) .
3. According to this λ J value, the length of the junction in normalized unitsreads (cid:101) L = L/λ J (cid:39)
21, while the value of the damping parameter α (cid:39) . T inthe range of values that we are going to discuss.The evolution of the temperature T at fixed values of bias current (cid:101) I b = 0 . (cid:101) H = 1 .
75 is shown in Fig. 2(a). Here we are assuming to switch on theexternal magnetic field only when T has reached a steady value T s between T bath and T . We observe that the temperature of S locally increase at the left junction edge, x = 0, and is double-peaked close to the right junction edge, x = L . This temperature hermal FF regime in a LJJ (cid:101) H < (cid:101) H thr (where (cid:101) H thr = 2), the imposed bias current is high enough to induce a flux-flow regime.The bias-current conditions giving a flux-flow regime can be grasped by studying thenumber of solitons and the mean voltage drop across the junction. The number ofsolitons N excited along the junction can be roughly evaluated through the quantity [72] N ( t ) = (cid:22) ϕ ( L, t ) − ϕ (0 , t )2 π (cid:23) , (11)where (cid:98) ... (cid:99) stands for the integer part of the argument. The mean voltage across thejunction can be estimated as V mean ( t ) = 1 L (cid:90) L Φ π d ϕ ( x, t )d t dx, (12)according to the a.c. Josephson relation [61].In Fig. 2(b) we show the behaviour of both the number of solitons (left vertical scale,blue line) and the mean voltage drop (right vertical scale, red line) by quasi-adiabatically,i.e., very slowly, increasing the bias current, at a fixed external magnetic field (cid:101) H = 1 . Meissner state [72], that isthe fluxon-free state, corresponding to N = 0. Instead, when (cid:101) I b (cid:38) .
156 a flux-flowregime is established. In this case, we obtain
N >
0, that is solitons fill the junctionmoving from the left towards the right edge of the device driven by the current. In thisregime a non-zero mean voltage drop appears, so that the larger the bias current, thehigher the speed of soliton and therefore the larger V mean .Then, despite the fast soliton dynamics, the flux-flow regime triggered by the biascurrent results in a peculiar temperature profile along the junction. In Fig. 2(a), wealso highlight with a black solid curve the stationary temperature profile, T st2 ( x ), sincewe are going to show shortly how this stationary profile modifies as the bias currentchanges. In fact, Fig. 2(c) is drawn collecting several stationary temperature profiles T st2 ( x ) by changing the bias current, at (cid:101) H = 1 .
75. In the flux-flow regime, that is for (cid:101) I b (cid:38) . (cid:101) H = (cid:101) H thr .In Fig. 2(d) we show the time evolution of the temperature profile for (cid:101) I b = 0 . (cid:101) H = 2. Now the situation is somewhat different with respect to what we shownpreviously when we set an under-threshold magnetic field value. In fact, now whenthe magnetic field is switched on several temperature peaks come into being along the hermal FF regime in a LJJ x ( μ m ) I b = T st ( K ) H ~ ~ Figure 3.
Stationary temperature T st2 as a function of x , at different values of magneticfield and for a fixed (cid:101) I b = 0 . junction. The analysis of the number of solitons N and the mean voltage peak V mean as a function of (cid:101) I b shown in Fig. 2(e), reveals that there is no Meissner state in thiscase, since already for (cid:101) I b = 0 two solitons populate the junction, i.e., N = 2. Then,by slightly increasing the bias current, solitons are pushed rightwards becoming moretightly-packed. At a certain point, a new soliton enters from the left edge. Each timethat a new soliton enters, the phase configuration abruptly changes and a peak in themean voltage drop appears. Finally, for (cid:101) I b (cid:38) .
049 the flux-flow mode begins and a non-negligible mean voltage drop definitively appears. The stationary temperature profile, T st2 ( x ), as a function of the bias current evidences two different regimes, see Fig. 2(f).For (cid:101) I b < . (cid:101) I b (cid:38) . T is well outlined in Fig. 3,where we show the stationary temperature profile T st2 ( x ), by changing the intensity ofthe underthreshold magnetic field, at a fixed (cid:101) I b = 0 .
2. At (cid:101) H = 1 .
68 the temperaturedistribution is asymmetric, but there are no temperature peaks along the junction, sincethe system is in the Meissner state. Instead, by increasing further the magnetic field,the system goes into the flux-flow regime and some temperature peaks appear, so thatthe stronger the magnetic field, the greater the number of temperature peaks.Although the timescale of the solitonic evolution is generally shorter than thetimescale of thermal relaxation processes (cid:107) , the results discussed so far show that soliton-induced thermal effects emerge also in the flux-flow regime, that is when a chain of (cid:107)
The Swihart velocity can be of the order of ¯ c ∼ m / s. Thus, a soliton moving at a speed equal,for instance, to 0 . c takes approximatively 7 ps to cover a distance roughly equal to the length scale ofthe system ∼ λ J (cid:39) µ m. Conversely, in a Nb-junction the estimated thermal response time is of theorder of a fraction of nanosecond [29]. hermal FF regime in a LJJ Figure 4.
Time averages of the normalized local magnetic field (a), see Eq. (14), andthe cosine of the phase (b), see Eq. (15), as a function of x , for (cid:101) H = 1 .
75 and (cid:101) I b = 0 . solitons rapidly moves along the junction. To understand the physical origin of thisbehaviour, we define two quantities giving information on both the soliton position andthe distribution of the phase-dependent component of heat current flowing through thesystem. The soliton configurations are well depicted by the space derivative of the phase, ∂ϕ ( x,t ) ∂x , since it is proportional to the local magnetic field according to the relation [61] H in ( x, t ) = H c, λ J ∂ϕ ( x, t ) ∂x . (13)Thus, to understand the thermal response in the flux-flow regime, we can define thetime average of the normalized local magnetic field along the junction as (cid:68) (cid:101) H in ( x, t ) (cid:69) t = 1 T p (cid:90) t + T p t ∂ϕ ( x, t ) ∂x dt, (14)where t is the time at which flux-flow starts and T p is a much longer time than the typicaltimescale of the solitonic evolution. The (cid:68) (cid:101) H in ( x, t ) (cid:69) t profile obtained for (cid:101) H = 1 .
75 and (cid:101) I b = 0 .
16 is shown in Fig. 4(a). We note that this function is peaked at x = 0 andalso that it significantly enhances close to the right junction edge. This picture confirmsthat, in the range of bias current and magnetic field under investigation, during the timeevolution some solitons are preferentially localized in the right part of the junction. Infact, during the flux-flow evolution, for a magnetic field close to the critical value andfor a low enough bias current, the dynamics we observed is composed by a soliton chain“surfing” on a standing solitons background ¶ . This standing solitons configuration isresponsible for the peaked temperature profile close to the right junction edge discussedso far.Since the peaked behaviour of the temperature profile along the junction can bemainly ascribed to the phase-dependent contribution in Eq. (5), namely, to the term − cos ϕ P cos , it can be useful to define the quantity ζ ( x ) = − T p (cid:90) t + T p t cos [ ϕ ( x, t )] dt, (15) ¶ A similar surfing soliton dynamics was also previously discussed investigating the behaviour of SGchiral soliton lattice in helimagnetic structures in the presence of a magnetic field [73]. hermal FF regime in a LJJ x ( μ m ) I b = H = T st ( K ) ~ α ~ Figure 5.
Stationary temperature T st2 as a function of x , at fixed values of (cid:101) I b = 0 . (cid:101) H = 1 .
75, by varying the damping parameter α ∈ [0 . − which behaviour as a function of x , for (cid:101) H = 1 .
75 and (cid:101) I b = 0 .
16, is shown in Fig.4(b). Bycomparing this curve with results in Figs. (2)(a) and (c), the response of the temperature T reflects the behavior of the function ζ ( x ), which is also clearly double-peaked nearthe right edge of the junction.Finally, since the temperature reached locally depends on dynamical aspects ofthe soliton evolution, we investigate how the damping affecting the phase dynamicsinfluences thermal transport. As a friction parameter, α opposes the variations of ϕ .Then, we present in Fig. 5 the stationary temperature profile as a function of α , at fixedvalues of the bias current and the magnetic field, (cid:101) I b = 0 . (cid:101) H = 1 .
75, respectively. Weobserve that in the range of α values taken into account, the overall thermal behaviour isonly slightly modified by a change in the damping parameter. By increasing α the shapeof the two temperature peaks becomes clearer, and the small ripples in the temperatureprofiles, probably due to leftward retro-reflected phase components, tend to fade. Forhigher damping values (i.e., α (cid:29) hermal FF regime in a LJJ
4. Conclusions
In this paper we study thermal transport in a temperature-biased LJJ operating in theflux-flow regime. Specifically, the electrodes forming the junction reside at differenttemperatures and the device is driven by both a bias current, (cid:101) I b , and a magnetic field, (cid:101) H . The magnetic field used is close to the threshold value, (cid:101) H thr = 2, the latter beingthe magnetic field, in normalized units, above which solitons enter a junction also inthe absence of a bias current. In this case the fluxon density along the junction is notvery high and we can suppose to highlight soliton-induced well-localized thermal effectsin the system.We observe that, by increasing the bias current, as soon as the junction enters inthe flux-flow mode, the temperature profile along the junction modifies. Despite in thisregime solitons rapidly move along the junction, the temperature tends to rise just inspecific points, depending mainly on the value of the applied magnetic field. In fact,by increasing further the magnetic field a multi-peaked structure in the temperatureemerges. Specifically, we compare thermal evolutions obtained by setting the normalizedmagnetic field to an underthreshold value (cid:101) H = 1 .
75 and to (cid:101) H = (cid:101) H thr = 2. We studyalso the number of fluxons and the mean voltage drop across the junction as a functionof the bias current, to highlight the (cid:101) I b values triggering the flux-flow regime.Finally, we investigate how the friction affecting the phase dynamics enters intoplay in the thermal evolution of the system. Specifically, we study how a change in thedamping parameter α influences the stationary temperature profiles along the junction.Our findings are important for understanding the interplay between solitondynamics and thermal evolution in a LJJ, in those conditions in which, despite theextremely rapid time evolution of the Josephson phase should suggest the absenceof relevant thermal effects, we are still able to highlight the emergence of intriguingphenomena. Markedly, we observe that, starting from a homogeneous system, we finallyobserve some inhomogeneously-distributed temperature profiles. The initial symmetryof the system is indeed broken by the applied bias current, which forces the soliton tomove in a specific direction. Since thermal effects that we discussed are independent onthe polarity of the magnetically-excited soliton, similar temperature configurations canbe obtained by inverting both the bias current flowing direction and the polarity of thesolitons, the latter depending on the direction of the in-plane external magnetic field.
5. Acknowledgments
We thank G. Filatrella for the useful discussions. A.B. and F.G. acknowledge theEuropean Research Council under the European Union’s Seventh Framework Program(FP7/2007-2013)/ERC Grant agreement No. 615187-COMANCHE and the TuscanyRegion under the PAR FAS 2007-2013, FAR-FAS 2014 call, project SCIADRO, forfinancial support. A.B. acknowledges the CNR-CONICET cooperation programme“Energy conversion in quantum nanoscale hybrid devices” and the Royal Society though
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