Solitonic Josephson Thermal Transport

Solitonic Josephson thermal transport

Claudio Guarcello,1, 2, ∗ Paolo Solinas,3 Alessandro Braggio,1 and Francesco Giazotto1 1NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, Piazza San Silvestro 12, I-56127 Pisa, Italy 2Radiophysics Department, Lobachevsky State University, Gagarin Avenue 23, 603950 Nizhni Novgorod, Russia 3SPIN-CNR, Via Dodecaneso 33, 16146 Genova, Italy (Dated: March 22, 2018) We explore the coherent thermal transport sustained by solitons through a long Josephson junction, as a thermal gradient across the system is established. We observe that a soliton causes the heat current through the system to increase. Correspondingly, the junction warms up in correspondence of the soliton, with temperature peaks up to, e.g., approximately 56 mK for a realistic Nb-based proposed setup at a bath temperature Tbath = 4.2 K. The thermal effects on the dynamics of the soliton are also discussed. Markedly, this system inherits the topological robustness of the solitons. In view of these results, the proposed device can effectively find an application as a superconducting thermal router in which the thermal transport can be locally mastered through solitonic excitations, which positions can be externally controlled through a magnetic field and a bias current.

I. INTRODUCTION manipulation of heat currents in mesoscopic superconducting devices. Here, the aim is to design and realize The physics of coherent excitations has relevant im- thermal components able to master the energy transfer plications in the field of condensed matter. Such co- with a high degree of accuracy. In this regard, we pro- herent objects emerge in several extended systems and pose to lay the foundation of a new branch of fast co- are usually characterized by remarkable particle-like fea- herent caloritronics based on solitons, with the end to tures. In the past decades, these notions played a crucial build up new devices exploiting this highly-controllable, role for understanding various issues in different areas of “phase-coherent” thermal flux. Specifically, the feasibility the physics of continuous and discrete systems [1, 2]. A of using a LJJ as a thermal router [36], in which ther- Josephson junction (JJ) is a model system to appreciate mal transport can be locally handled through solitonic coherent excitations, and, specifically, a superconductor- excitations, is very promising. insulator-superconductor (SIS) long JJ (LJJ) is the pro- After the earlier prediction in 1965 by Maki and totypal solid-state environment to explore the dynamics Griffin [37], only recently phase-coherent thermal trans- of a peculiar kind of solitary waves, called soliton [3, 4]. port in temperature-biased Josephson devices has been These excitations give rise to readily measurable physical confirmed experimentally in several interferometer-like phenomena, such as step structures in the I-V character- structures [38–43]. The thermal modulation induced by istic of LJJs and microwaves radiation emission. More- the external magnetic field was demonstrated in super- over, a soliton has a clear physical meaning in the LJJ conducting quantum-interference devices (SQUID) [38, framework, since it carries a quantum of magnetic flux, 39] and short JJs [40, 41]. Furthermore, in LJJs the induced by a supercurrent loop surrounding it, with the heat current diffraction patterns in the presence of an local magnetic field perpendicularly oriented with respect in-plane external magnetic field have been discussed the- to the junction length [5]. Thus, solitons in the context of oretically [44]. However, until now no efforts have been LJJs are usually referred to as fluxons or Josephson vor- addressed to explore how thermal transport across a LJJ tices. Measured for the first time more than 40 years is influenced by solitons eventually set along it. Nonethe- ago [6, 7], LJJs are still nowadays an active research less, it has been demonstrated theoretically that the pres- field [8–26]. Indeed, the fact that a single topologically ence of a fluxon threading a temperature-biased inductive protected excitation, i.e., a flux quantum, can be moved SQUID modifies thermal transport and affects the steady and controlled by bias currents, created by the magnetic temperatures of a floating electrode of the device [45, 46]. field, manipulated through shape engineering [9, 27–30], Similarly, we demonstrate theoretically that a fluxon ar- or pinned by inhomogeneities [31, 32], naturally stimu- ranged within a LJJ locally affects, in a fast timescale, lated a profusion of ideas and applications.

arXiv:1712.00259v3 [cond-mat.mes-hall] 21 Mar 2018 the thermal evolution of the system, and, at the same Practically, several electric and magnetic features con- time, we discuss how the temperature gradient aﬀects cerning solitons in LJJs were comprehensively hitherto the soliton dynamics. Finally, being solitons, namely, re- explored, but little is known about the soliton-sustained markably stable and robust objects [47], at the core of coherent thermal transport through a temperature- its operation, this system provides an intrinsic topologi- biased junction. This issue falls into the emerging ﬁeld cal protection on thermal transport. of coherent caloritronics [33–35], which deals with the The paper is organized as follows. In Sec. II, the the- oretical background used to describe the phase evolution of a magnetically-driven LJJ is discussed. In Sec. III, ∗ [email protected] the thermal balance equation and the heat currents are 2

y (a) y ϕ (b) Pheat 2π soliton Hz(t) Hz(t) d Pin 0 D2 z Pe-ph x L W x

FIG. 1. a, A superconductor-insulator-superconductor (SIS) rectangular long Josephson junction (LJJ) excited by an external in-plane magnetic field Hz(t). The length and the width of the junction are L λJ and W λJ , respectively, where λJ is the Josephson penetration depth. Moreover, the thickness D2 λJ of the electrode S2 is indicated. A soliton within the junction, corresponding to a 2π-twist of the phase ϕ, is represented. Ti is the temperature of the superconductor Si and d is the insulating layer thickness. b, Thermal model of the device, as the thermal contact with a phonon bath is taken into account. The heat current, Pin, flowing through the junctions depends on the temperatures and the solitons eventually set along the system. Pe−ph represents the coupling between quasiparticles in S2 and the lattice phonons residing at Tbath, whereas Pheat denotes the power injected into S1 through heating probes in order to impose a fixed quasiparticle temperature T1. The arrows indicate the direction of heat currents for T1 > T2 > Tbath.

introduced. In Sec. IV, the evolution of the temperature a long and narrow junction, we assume that W λJ of the ﬂoating electrode is studied, as a thermal gradi- and L λ , where we introduced the length scale q J ent across the system is taken into account. In Sec. V, λ = Φ0 1 called Josephson penetration depth, conclusions are drawn. J 2πµ0 tdJc where Jc = Ic/A is the critical current area density. Then, in normalized units, the linear dimensions of the

junction read L = L/λJ 1 and W = W/λJ 1. II. PHASE DYNAMICS The electrodynamics of a LJJ is usually described by a partial differential equation for the order parameter In Fig. 1a long and narrow SIS Josephson junction, in phase difference ϕ, namely, the perturbed sine-Gordon the so-called overlap geometry, formed by two supercon- (SG) equation, that in the normalized units x = x/λJ ducting electrodes S and S separated by a thin layer of q e 1 2 2π Ic and t = ωpt, with ωp = being the Josephson insulating material with thickness d is represented. We e Φ0 C consider an extended junction with both the length and plasma frequency [48], reads [48, 49] the width larger than d (namely, W, L d). In the ge- ∂2ϕ(x, t) ∂2ϕ(x, t) ∂ϕ(x, t) ometry depicted in Fig. 1, the junction area A = WL e e − e e − sin ϕ x, t = α e e . (1) 2 e e extends in the xz-plane, the electric bias current is even- ∂xe ∂et2 ∂et tually flowing in the y direction, and the external mag- The boundary conditions of this equation takes into ac- netic field is applied in the z direction. The thickness count the normalized external magnetic field H(t) = of each superconducting electrode is assumed larger than 2π Φ µ tdλJ H(t) the London penetration depth λL,i of the electrodes ma- 0 0 terial. Since the applied field penetrates the supercon- dϕ(0, t) dϕ(L, t) ducting electrodes up to a thickness given by the London = = H(t). (2) penetration depth, an effective magnetic thickness of the dxe dxe −1 junction td = λL,1 + λL,2 + d can be defined. If λL,i In Eq. (1), α = (ωpRC) is the damping parameter are larger than the thickness of the electrodes Di, the (with R and C being the total normal resistance and effective magnetic thickness has to be replaced by t˜d = capacitance of the JJ). λL,1 tanh (D1/2λL,1) + λL,2 tanh (D2/2λL,2) + d [40, 41]. The SG equation admits topologically stable In the presence of an external in-plane magnetic field travelling-wave solutions, called solitons [3, 4], cor- H(r, t) = (0, 0, −H(t) bz), the phase ϕ, namely, the phase responding to 2π-twists of the phase (see Fig. 2). For difference between the wavefunctions describing the carri- the unperturbed SG equation, i.e., α = 0 in Eq. (1), ers in the superconducting electrodes, changes according solitons have the simple analytical expression [48] 2π to ∂ϕ(x, t)/∂x = µ0tdH(t) [48], where Φ0 = h/2e ' Φ0 2 × 10−15Wb is the magnetic flux quantum (with e and     xe − xe0 − uet  h being the electron charge and the Planck constant, ϕ(x − uet) = 4 arctan exp ± √  , (3) e 1 − u2 respectively), and µ0 is the vacuum permeability. For   3 where the sign ± is the polarity of the soliton and u the sake of simplicity, we assume that the electrode S1 is the soliton speed normalized to the Swihart’s veloc- resides at a fixed temperature T1, which is maintained by ity [48], namely, the largest group propagation velocity of the good thermal contact with heating probes. The elec- the linear electromagnetic waves in long junctions. The trode S2 is in thermal contact also with a phonon bath moving soliton corresponds to a time variations of the at temperature Tbath ≤ T2 < T1. phase, which generates a local voltage drop according to A characteristic length scale for the thermalization in V (x, t) = Φ0/(2π)ϕ ˙(x, t). the diffusive regime can√ be estimated as the inelastic scat- 2 For the numerical simulation of the soliton dynamics, tering length ìn = Dτs, where D = σN /(e NF ) is the we modelled the normalized external magnetic field H(t) diffusion constant (with σN and NF being the electrical as a Gaussian pulse exciting the junction end in x = 0. conductivity in the normal state and the density of states Accordingly, the boundary conditions become at the Fermi energy, respectively) and τs is the recombination quasiparticle lifetime [54]. For Nb at 4.2 K, one dϕ(0, t) dϕ(L, t) = H(t) and = 0. (4) obtains ìn ∼ 0.3 µm, namely, a value well below the di- dx dx mension of a soliton, ` λ , since λ 6 µm for the e e in J J & device considered here below. When only the length of For simplicity, in our model, i.e., Eq. (1), both the ∂ϕ S2 is much larger than ìn, i.e., L ìn (namely, the so- terms β 2 [3, 49] (with β = ωpLP /RP , where LP = called quasiequilibrium limit [33]), the electrode S can ∂xe ∂et 2 µ0td/W and RP represents scattering of quasiparticles be modelled as a one-dimensional diffusive superconduc- ∂H in the superconducting surface layers) and ∆c [3, 50] tor at a temperature varying along L. ∂xe (with ∆c being a coupling constant) are not included. For the sake of readability, hereafter we will adopt in These terms account for the dissipation due to the sur- equations the abbreviated notation in which the x and face resistance of the superconducting electrodes and for t dependences are left implicit, namely, T2 = T2(x, t), the spatial gradient of the magnetic field along the junc- ϕ = ϕ(x, t), and V = V (x, t). Then, the evolution of the tion, respectively. We neglect these contributes since we temperature T2 is given by the time-dependent diffusion are interested only to look the interplay between a soliton equation and the thermal effects resulting from its presence along the system as a temperature gradient across the junction d dT dT κ(T ) 2 + P (T ,T , ϕ) = c (T ) 2 , (5) is imposed. In this regard, also the specific mechanism dx 2 dx tot 1 2 v 2 dt used to excite a soliton is not so relevant. In fact, in the place of a moving soliton generated by a magnetic pulse, where the rhs represents the variations of the internal we can alternatively design the local control of thermal energy density of the system, and the lhs terms indicate flux through configurations of steady solitons excited in the spatial heat diffusion, taking into account the inho- specific points of the junction via a slowly-varying ex- mogeneous electronic heat conductivity, κ(T2), and the ternal magnetic drive applied to both edges of the de- total heat flux density in the system, namely, vice [44]. In this manner, the positions of the solitons are directly dependent on the boundary conditions. Anyway, Ptot (T1,T2, ϕ) = Pin (T1,T2, ϕ, V ) − Pe−ph,2 (T2,Tbath) . we observe that, still in this case, a dynamical treatment (6) is crucial for the realistic description of the manipulation This term consists of the incoming, i.e., Pin (T1,T2, ϕ, V ), of the system, and it leads to peculiar results, such us the and outgoing, i.e., Pe−ph,2 (T2,Tbath), thermal power hysteresis and the trapping of fluxons [44]. Alternatively, densities in S2. We stress that the phase dynamics is es- in an annular geometry [51], i.e., a “closed” LJJ folded sential, through Pin, to determine the heat flows and the back into itself in which solitons move undisturbed, i.e., temperature evolution. Therefore, both Eqs. (1) and (5) without interaction with borders, fluxons can be excited have to be solved numerically self-consistently to thor- at will [52, 53], allowing highly-controlled soliton dynam- oughly explore the thermal behaviour of the system. ics. In Eq. (6), the heat current density Pin(T1,T2, ϕ, V ) Below, we will briefly discuss also the possibility to flowing from S1 to S2 is control the soliton position by an applied bias current. This feasibility adds an external control knob, making Pin(T1,T2, ϕ, V ) =Pqp(T1,T2,V ) − cos ϕ Pcos(T1,T2,V ) this device more interesting for practical applications. + sin ϕ Psin(T1,T2,V ), (7) and contains the interplay between Cooper pairs and III. THERMAL EFFECTS quasiparticles in tunneling through a JJ predicted by Maki and Griffin [37]. In fact, Pqp is the heat flux den- The aim of this section is to explore the thermal flux sity carried by quasiparticles and represents an incoher- through the junction, as a soliton is set and a tempera- ent flow of energy through the junction from the hot to ture gradient across the junction is imposed. Specifically, the cold electrode [33, 37, 55]. Instead, the “anomalous” we observe the evolution of the temperature T2(x, t), terms Psin and Pcos determine the phase-dependent part which depends on all the energy local relaxation mech- of the heat current originating from the energy-carrying anisms occurring in the electrode S2 (see Fig. 1b). For tunneling processes involving, respectively, Cooper pairs 4 and recombination/destruction of Cooper pairs on both quasi-particle and the anomalous heat current densities, sides of the junction. In the adiabatic regime [56], the Pqp, Pcos, and Psin read, respectively, [37, 56, 57]

1 Z ∞ Pqp(T1,T2,V ) = 2 dεN1(ε − eV, T1)N2(ε, T2)(ε − eV )[f(ε − eV, T1) − f(ε, T2)], (8) e RaD2 −∞ Z ∞ 1 ∆1(T1)∆2(T2) Pcos(T1,T2,V ) = 2 dεN1(ε − eV, T1)N2(ε, T2) [f(ε − eV, T1) − f(ε, T2)], (9) e RaD2 −∞ ε ZZ ∞ " # eV ∆1(T1)∆2(T2) 1 − f(E1,T1) − f(E2,T2) f(E1,T1) − f(E2,T2) Psin(T1,T2,V ) = d1d2 + (10), 2 2 2 2 2 2 2 2πe RaD2 −∞ E2 (E1 + E2) − e V (E1 − E2) − e V

where Ra = RA is the resistance per area of the junc- phonons are very well thermalized with the substrate q 2 2 E/kB T that resides at T , thanks to the vanishing Kapitza tion, Ej = + ∆j(Tj) , f(E,T ) = 1/ 1 + e bath j resistance between thin metallic films and the substrate is the Fermi distribution function, and Nj (ε, T ) = at low temperatures [33, 60]. Re √ ε+iγj is the reduced superconducting Going forward in the description of the terms in 2 2 (ε+iγj ) −∆j (T ) dS(T ) Eq. (5), cv(T ) = T dT is the volume-specific heat ca- density of state, with ∆j (Tj) and γj being the BCS en- pacity, with S(T ) being the electronic entropy density of ergy gap and the Dynes broadening parameter [58] of the the superconductor S [61, 62] j-th electrode, respectively. 2 Interestingly, if we calculate the values of the heat cur- Z ∞ S(T ) = −4kBNF dεN2(ε, T ) × (12) rent density Pin(T1,T2, ϕ) in the presence of a steady 0 unperturbed soliton, described by Eq. (3) for u = 0, × {[1 − f(ε, T )] log [1 − f(ε, T )] + f(ε, T ) log f(ε, T )} . an enhancement of Pin just in correspondence of the soliton is observed (see Fig. 2 assuming for simplicity In Eq. (5), κ(T2) is the electronic heat conductivity, given an homogeneous temperature profile with T1 = 7 K by [43] and Tbath = 4.2 K). Correspondingly, in the presence of a thermal gradient, we expect in the station- n h ∆(T2) io Z ∞ cos2 Im arctanh ary regime a soliton to induce a local warming-up in S . σN 2 ε+iγ2 2 κ(T2) = dεε . 2 2 2 The peaked shape of Pin shown in Fig. 2 results from 2e kBT2 −∞ cosh ε the ϕ-dependence of the anomalous contribute P in 2kB T2 cos (13) Eq. (7) (notably, the anomalous term P vanishes in sin In order to comprehensively account all the thermal the stationary case, i.e., ϕ˙ = 0). In fact, the coefficient effects, we observe also that the temperature affects − cos ϕ, that multiplies the P term, tends to −1 for cos both the effective magnetic thickness t (T ,T ) and the ϕ → {0, 2π}, and it is +1 for ϕ = π, namely, in cor- d 1 2 respondence of the center of the soliton. Nevertheless, the quasiparticle contribute Pqp represents a positive off- 2π set that makes Pin still positive, so that the total heat current flows however from the hot to the cold reservoir. In Eq. (6), the energy exchange between electrons and

phonons in the superconductor is accounted by Pe−ph,2, φ π which reads [59] Z ∞ Z ∞ −Σ 2 Pe−ph,2 = 5 dEE dεε sign(ε)ME,E+ε 96ζ(5)kB −∞ −∞ 0 ( ε 0 5 10 15 20 × coth [F(E,T2) − F(E + ε, T2)] x~ 2kBTbath ) FIG. 2. Phase proﬁle ϕ (left vertical scale, black line) − F(E,T2)F(E + ε, T2) + 1 , (11) and the heat power density Pin(T1,T2, ϕ) (in units of 2 2 ∆2(0)/(e RaD2)), see Eq. (7) (right vertical scale, orange where F (ε, T2) = tanh (ε/2kBT2), ME,E0 = line), for T1 = 7 K and T2 = 4.2 K, as a function of the 0 2 0 Ni(E,T2)Ni(E ,T2) 1 − ∆ (T2)/(EE ) , Σ is the normalized position xe, when a steady unperturbed soliton, electron-phonon coupling constant, and ζ is the Rie- see Eq. (3) for u = 0, is located in the midpoint of a junction mann zeta function. We are assuming that the lattice with normalized length L = 20. 5

10 0.7 T2=4.2K xs 0.6

) 9

μ m 0.5 ( 8 α

J 0.4 λ 7 0.3

5 6 7 8 9

T1 (K)

FIG. 3. Josephson penetration length λJ (left vertical scale, blu line) and damping parameter α (right vertical scale, red line) as a function of the temperature of the hot electrode T1, for T2 = 4.2 K, for a Nb-based LJJ with values of the junction parameters discussed in the main text.

Josephson critical current Ic(T1,T2), which varies with FIG. 4. Phase evolution as a function of the position x and the temperatures according to the generalized Ambe- the time t, for T1 = 7 K and Tbath = 4.2 K. A soliton gaokar and Baratoﬀ formula [63–65] magnetically excited in x = 0 shifts along the junction. Cor- respondingly, the Josephson phase ϕ undergoes a 2π step (see ∞ 1 Z n red lines). The phase values and the position xs of the soli- I (T ,T ) = f(ε, T )Re [F (ε)] Im [F (ε)] c 1 2 2eR 1 1 2 ton, which is marked by a black dashed line, are highlighted −∞ in the contour plot underneath the main graph.

+f(ε, T2)Re [F2(ε)] Im [F1(ε)] dε , (14) thermally isolated, so that boundary conditions of Eq. (5) .q read ∂T2 = 0. The choice of the initial temperature where F (ε) = ∆ (T ) (ε + iγ )2 − ∆2 (T ). Accord- ∂x x=0,L j j j j j j of the electrode S is not essential for our discussion, since ingly, both the Josephson penetration depth λ and the 2 J we will assume to excite a soliton only when T reaches damping parameter α vary with the temperatures, see 2 a steady value T2,s in-between Tbath and T1. Fig. 3. Since the soliton width depends on λJ , this thermal dependence affects both the dynamics and the shape of the soliton and, then, the temperature profile along the junction. IV. RESULTS The feasibility to affect the soliton dynamics by locally We consider an Nb/AlOx/Nb SIS LJJ characterized heating the system is the cornerstone of the low tem- 2 by a resistance per area Ra = 50 Ω µm and a specific perature scanning electron microscopy (LTSEM) [66–69]. 2 This techniques was proved to be a powerful experimental capacitance Cs = 50 fF/µm . The linear dimensions of the device are L = 150 µm, W = 0.5 µm, D2 = 0.1 µm, tool for investigating fluxon dynamics in Josephson de- 0 vices. The main idea behind this technique is to locally and d = 1nm. For the Nb electrode, we assume λL = 6 −1 −1 9 −3 −5 heat a small area (∼ µm) of the junction by a narrow 80nm, σN = 6.7 × 10 Ω m , Σ = 3 × 10 Wm K , 47 −1 −3 electron beam. The generated hot spot acts as a small NF = 10 J m , ∆1(0) = ∆2(0) = ∆ = 1.764kBTc, thermal perturbation with the aim to drastically locally with Tc = 9.2 K being the common critical temperature −4 increase the effective dissipation coefficient. This process of the superconductors, and γ1 = γ2 = 10 ∆. results in a change of the I-V characteristic of the de- Here we focus on the simplest case in which we magnet- vice. By gradually scanning the electron beam along the ically excite a soliton which then moves along the junc- junction surface and measuring the voltage, an “image” of tion as the friction affecting its dynamics stops it. The the dynamical state of the LJJ can be produced. Alterna- resulting standing soliton is stable and, if it is far enough tively, in our work we discuss a sort of thermal imaging of to the junction edges and in absence of further pertur- a magnetically excited soliton, through the temperature bations, definitively remains in this position. Then, to profile of the floating electrode of the device. model this situation, the “left”, i.e., in x = 0, junction In Fig. 3, we assume a fixed T2, since, in the small edge is excited by a Gaussian magnetic pulse, with nor- range of variation of T2 that we will discuss, the effect of malized amplitude Hmax = 8.5 and width σ = 1 (in units µ0 Φ0 −1 this temperature on λ and α is vanishingly small, and of and ωp , respectively), which induces a soliton J 2π tdλJ then can be neglected. moving rightward along the junction. The width and the Finally, we assume that the electrode S2 is initially velocity of the generated soliton directly depend on the at T2(x, 0) = Tbath ∀x ∈ [0,L], and that its ends are temperatures of the system through λJ and α, respec- 6

T1=7K Tbath=4.2K

(a) (b)

FIG. 5. a, Heat current Pin(T1,T2, ϕ, V ) flowing from S1 to S2, see Eq. (7). b, Evolution of the temperature T2(x, t) of S2. In both panels, the soliton is magnetically excited to the left end, i.e., x = 0, after ∼ 2 ns. At this time, the superconducting electrode S2 is already fully thermalized at the steady temperature T2,s ∼ 4.23 K. Then, in correspondence of the induced soliton, we observe a clear enhancement of both Pin and T2. In panel (b), the phase values ϕ(x, t) and the position of the soliton, which is marked by a black dashed line, are highlighted in the contour plot underneath the main graph. For both panels, T1 = 7 K, Tbath = 4.2 K, and the junction is initially at the temperature T2(x, 0) = Tbath ∀x ∈ [0 − L]. tively. In fact, the higher the temperatures the larger ized at the steady “unperturbed” (i.e., unaffected by ex- −1/2 citations) temperature T2,s ' 4.23 K. Interestingly, the both λJ and α, since both are proportional to Ic (see Fig. 3). Therefore, by increasing the temperatures, the soliton induces a local intense warming-up in S2, with a steady maximum temperature T2,Max ' 4.29 K. soliton enlarges and slows down, since both λJ and α increase. This shows that the manipulation of the thermal We observe that, as the soliton sets in xs, the tem- profile along the junction can be also eventually used to perature enhances exponentially approaching its steady modify the soliton dynamics [70]. value, see Fig. 5b. The thermal response time can be We impose a thermal gradient across the system, estimate as the characteristic time of the exponential specifically, the bath resides at Tbath = 4.2 K, and S1 evolution by which the temperature approaches its sta- is at a temperature T1 = 7 K kept fixed throughout the tionary value. Then, from Fig. 5b we deduce the value computation. The electronic temperature T2(x, t) of the τth ∼ 0.25 ns. Markedly, a quite good estimate of this electrode S2 is the key quantity to master the thermal thermal response time results also in a linear response route across the junction, since it floats and can be driven regime, namely, by first order expanding the heat cur- by controlling the soliton along the system. rent terms in Eq. (5). In fact, by following the same The evolution of the Josephson phase ϕ(x, t) in the procedure developed in Ref. [46], we obtain a thermal presence of a magnetically excited soliton is shown in switching time τsw ' 0.1 ns. Fig. 4. In this figure, a rightwards moving soliton (which The role of the temperature T1 is illustrated in Fig. 6, corresponds to a 2π step of the phase along the junc- where T2(x, τ) is calculated at τ = 10 ns at a few values tion) at different instants is outlined by red lines, whereas of T1 and Tbath = 4.2 K. By rising T1, the temperature a dashed line in the contour plot underneath the main peak shifts leftwards and becomes wider, just because graph marks the soliton position. As expected, due to the soliton slows down and enlarges, as a consequence of the friction (which is accounted by a value of the damp- the parameter variations discussed in Fig. 3. Interest- −1 ing parameter α = (ωpRC) ' 0.3) the soliton sets in ingly, the T2 modulation amplitude, δT2 = T2,Max −T2,s, xs ∼ 74.8 µm and definitively stays in this position. defined as the difference between the maximum and the We observe that in correspondence of the soliton, the minimum values of T2(x, τ) along the junction at a fixed heat flux Pin clearly enhances (see Fig. 5a). Specifically, time τ, behaves nonmonotonically by varying T1 (see the the steady value of the heat current in correspondence of inset of Fig. 6). In fact, δT2 is vanishing for low T1’s the soliton is Pin ∼ 1.1 µW, whereas it is Pin ∼ 0.3 µW (specifically, for T1 = Tbath there is no thermal gradient elsewhere. across the system). It then increases up to δT2 ∼ 56 mK Finally, the behaviour of the temperature T2(x, t) re- for T1 = 7 K, and it finally reduces again for T1 → Tc, flects the behavior of the thermal flux Pin, as it is shown due to the temperature-induced suppression of the energy in Fig. 5b. In this case, the soliton is excited after gaps in the superconductors. ∼ 2ns, namely, as the whole electrode S2 is thermal- The physical effect we have described here can 7

ing “cold” electrode of the junction, as the temperature T1 of the “hot” electrode is kept fixed and the thermal contact with a phonon bath is taken into account. Specif- ically, in correspondence of a magnetically excited soliton we observe a clear enhancement of the heat current Pin flowing through the junction. Correspondingly, a soliton-induced temperature peak occurs, with height up to δT2 ∼ 56 mK in a realistic Nb-based proposed setup. Finally, the physical properties of the device depend on the evolution of the superconducting order parameter along the junction, and, hence, on the dynamics of solitons which can be accurately controlled by external magnetic field, bias current, and shape engineering. This flexibility will allow to suggest new caloritronics applications enabling, for instance, the handling of the local thermal transport in specific points of the junction, i.e., FIG. 6. Temperature T2(x, τ) at τ = 10 ns for a few val- a solitonic thermal router. The analysis shows also the ues of T1. In the inset, the T2 modulation amplitude, δT2, possibility to affect the solitonic properties by manipulat- as a function of T is shown. The bath temperature is 1 ing the thermal profile, increasing the possible interplay Tbath = 4.2 K and the junction is initially at the tempera- between thermal and solitonic dynamics. Additionally, ture T2(x, 0) = Tbath ∀x ∈ [0 − L]. the solitonic nature of the system ensures the protection against environmental disturbances and a highly- promptly find an application as a Josephson thermal controllable, unaffected by noise, heat flow. The results router [36]. Specifically, we can design to direct through obtained will clarify the interplay between solitons and a soliton the heat to a superconducting finger electrode, caloritronics at nanoscale, paving the way to the realization of new coherent devices based on the soliton- attached for instance in xs, in order to selectively warm it up. Additionally, this idea can be improved further by sustained thermal transport. including an external electric bias current across the junc- Moreover, this device could represent the link between two recent proposals concerning a Josephson based tion. In fact, a bias current density, Jb, acts on the soliton with a Lorentz force, F = J × Φ (with the direction phase-tunable thermal logic [71] and a logic using fluxons L b 0 in LJJs [72]. of Φ0 depending on the polarity of the soliton). So, in the presence of an external bias current, according to the The suggested systems could be implemented by stan- perturbational approach [5], a soliton drifts with a veloc- dard nanofabrication techniques through the setup used, for instance, for the short JJs-based thermal diffrac- .q 2 ity approximately given by ud = 1 1 + [4α/ (πγ)] [4], tor [41]. The modulations of the temperature of the with γ = Jb/Jc. Specifically, for a low bias current drain “cold” electrode is usually obtained by realizing a πγ Josephson junction with a large superconducting elec- ud ' 4α . This allows us to actively control the dynamics and the final position of the soliton and, thus, the trode, which temperature is blocked at a fixed value, and local temperature of the electrode. Therefore, a multi- a small electrode with a small thermal capacity. In this terminal device allowing to distribute the heat among way, the heat transferred significantly affects the temper- several reservoirs can be conceived, in which we can se- ature of the latter electrode, which is then measured. lect which terminal to heat by shifting through the bias current the soliton along the junction. Clearly, the time dependent approach we illustrated so far is indispens- ACKNOWLEDGMENTS able to accurately describe the dynamical temperature response when the soliton moves from a finger to the next C.G. and F.G. acknowledges the European Research one, and then to properly master the operating principles Council under the European Union’s Seventh Frame- of a multi-terminal device. work Program (FP7/2007-2013)/ERC Grant agreement No. 615187-COMANCHE for partial financial support. P.S. has received funding from the European Union V. CONCLUSIONS FP7/2007-2013 under REA Grant agreement No. 630925 – COHEAT and from MIUR-FIRB2013 – Project Coca In conclusion, we have discussed the phase-coherent (Grant No. RBFR1379UX). A.B. acknowledges the Ital- thermal transport in a temperature-biased LJJ, where ian’s MIUR-FIRB 2012 via the HybridNanoDev project the thermal conduction across the system can be con- under Grant no. RBFR1236VV and CNR-CONICET trolled through solitonic excitations. Specifically, we cooperation programme “Energy conversion in quantum analyse the evolution of the temperature T2 of the float- nanoscale hybrid devices”. 8

[1] A. Scott, M.P. Sørensen, and P.L. Christiansen, Non- [18] D. Valenti, C. Guarcello, and B. Spagnolo, “Switching linear Science: Emergence and Dynamics of Coherent times in long-overlap Josephson junctions subject to ther- Structures, Oxford applied and engineering mathemat- mal fluctuations and non-Gaussian noise sources,” Phys. ics (Oxford University Press, 1999). Rev. B 89, 214510 (2014). [2] T. Dauxois and M. Peyrard, Physics of Solitons (Cam- [19] I. I. Soloviev, N. V. Klenov, A. L. Pankratov, L. S. Revin, bridge University Press, 2006). E. Il’ichev, and L. S. Kuzmin, “Soliton scattering as a [3] R. D. Parmentier, “Solitons and long Josephson junc- measurement tool for weak signals,” Phys. Rev. B 92, tions,” in The New Superconducting Electronics, edited 014516 (2015). by Harold Weinstock and Richard W. Ralston (Springer [20] C. Guarcello, D. Valenti, A. Carollo, and B. Spagnolo, Netherlands, Dordrecht, 1993) pp. 221–248. “Stabilization effects of dichotomous noise on the life- [4] A. V. Ustinov, “Solitons in Josephson junctions,” Physica time of the superconducting state in a long Josephson D 123, 315–329 (1998). junction,” Entropy 17, 2862 (2015). [5] D. W. McLaughlin and A. C. Scott, “Perturbation anal- [21] A. Vettoliere, C. Granata, and R. Monaco, “Long ysis of fluxon dynamics,” Phys. Rev. A 18, 1652–1680 Josephson junction in ultralow-noise magnetometer con- (1978). figuration,” IEEE Trans. Magn. 51, 1–4 (2015). [6] A. C. Scott and W. J. Johnson, “Internal flux motion in [22] A. L. Pankratov, K. G. Fedorov, M. Salerno, S. V. Shi- large Josephson junctions,” Appl. Phys. Lett. 14, 316– tov, and A. V. Ustinov, “Nonreciprocal transmission of 318 (1969). microwaves through a long Josephson junction,” Phys. [7] T.A. Fulton and R.C. Dynes, “Single vortex propagation Rev. B 92, 104501 (2015). in Josephson tunnel junctions,” Solid State Commun. 12, [23] C. Guarcello, D. Valenti, A. Carollo, and B. Spagnolo, 57 – 61 (1973). “Effects of Lévy noise on the dynamics of sine-Gordon [8] S. Ooi, Sergey Savel’ev, M. B. Gaifullin, T. Mochiku, solitons in long Josephson junctions,” J. Stat. Mech.: K. Hirata, and Franco Nori, “Nonlinear nanodevices us- Theory Exp. 2016, 054012 (2016). ing magnetic flux quanta,” Phys. Rev. Lett. 99, 207003 [24] I A Golovchanskiy, N N Abramov, V S Stolyarov, (2007). O V Emelyanova, A A Golubov, A V Ustinov, and [9] D. R. Gulevich and F.V. Kusmartsev, “New phenomena V V Ryazanov, “Ferromagnetic resonance with long in long Josephson junctions,” Supercond. Sci. Technol. Josephson junction,” Supercond. Sci. Technol. 30, 054005 20, S60 (2007). (2017). [10] S. Anders, M.G. Blamire, F.-Im. Buchholz, D.-G. Crété, [25] C. Guarcello, P. Solinas, M. Di Ventra, and F. Giazotto, R. Cristiano, P. Febvre, L. Fritzsch, A. Herr, E. Il’ichev, “Solitonic Josephson-based meminductive systems,” Sci. J. Kohlmann, J. Kunert, H.-G. Meyer, J. Niemeyer, Rep. 7, 46736 (2017). T. Ortlepp, H. Rogalla, T. Schurig, M. Siegel, R. Stolz, [26] D. Hill, S. K. Kim, and Y. Tserkovnyak, “Easy-plane E. Tarte, H.J.M. ter Brake, H. Toepfer, J.-C. Villegier, magnetic strip as a long Josephson junction,” arXiv A.M. Zagoskin, and A.B. Zorin, “European roadmap preprint arXiv:1802.04229 (2018). on superconductive electronics–status and perspectives,” [27] E. Goldobin, A. Sterck, and D. Koelle, “Josephson vortex Physica C: Superconductivity 470, 2079 – 2126 (2010). in a ratchet potential: Theory,” Phys. Rev. E 63, 031111 [11] K. K. Likharev, “Superconductor digital electronics,” (2001). Physica (Amsterdam) 482C, 6 – 18 (2012). [28] G. Carapella, G. Costabile, N. Martucciello, M. Cirillo, [12] R. Monaco, “Magnetic sensors based on long Josephson R. Latempa, A. Polcari, and G. Filatrella, “Experimen- tunnel junctions,” Supercond. Sci. Technol. 25, 115011 tal realization of a relativistic fluxon ratchet,” Physica C (2012). 382, 337 – 341 (2002). [13] C. Guarcello, D. Valenti, G. Augello, and B. Spagnolo, [29] G. Carapella, N. Martucciello, and G. Costabile, “Ex- “The role of non-Gaussian sources in the transient dy- perimental investigation of flux motion in exponentially namics of long Josephson junctions,” Acta Phys. Pol. B shaped Josephson junctions,” Phys. Rev. B 66, 134531 44, 997–1005 (2013). (2002). [14] R. Monaco, C. Granata, R. Russo, and A. Vettoliere, [30] T. Dobrowolski, “Kink motion in a curved Josephson “Ultra-low-noise magnetic sensing with long Josephson junction,” Phys. Rev. E 79, 046601 (2009). tunnel junctions,” Supercond. Sci. Technol. 26, 125005 [31] A. V. Ustinov, “Solitons in long Josephson junctions with (2013). inhomogeneities,” in Nonlinear Superconductive Electron- [15] C. Granata, A. Vettoliere, and R. Monaco, “Noise perfor- ics and Josephson Devices, edited by Giovanni Costa- mance of superconductive magnetometers based on long bile, Sergio Pagano, Niels Falsig Pedersen, and Maurizio Josephson tunnel junctions,” Supercond. Sci. Technol. Russo (Springer US, Boston, MA, 1991) pp. 315–336. 27, 095003 (2014). [32] R. Fehrenbacher, V. B. Geshkenbein, and G. Blatter, [16] J. Cuevas-Maraver, P.G. Kevrekidis, and F. Williams, “Pinning phenomena and critical currents in disordered The sine-Gordon Model and its Applications: From Pen- long Josephson junctions,” Phys. Rev. B 45, 5450–5467 dula and Josephson Junctions to Gravity and High- (1992). Energy Physics, Nonlinear Systems and Complexity [33] F. Giazotto, T. T. Heikkilä, A. Luukanen, A. M. Savin, (Springer International Publishing, 2014). and Jukka P. Pekola, “Opportunities for mesoscopics [17] V. P. Koshelets, “Sub-terahertz sound excitation and de- in thermometry and refrigeration: Physics and applica- tection by a long Josephson junction,” Supercond. Sci. tions,” Rev. Mod. Phys. 78, 217–274 (2006). Technol. 27, 065010 (2014). [34] M. J. Martínez-Pérez, P. Solinas, and F. Giazotto, “Co- 9

herent caloritronics in Josephson-based nanocircuits,” J. ducting tunnel junctions,” Phys. Lett. A 235, 281 – 284 Low Temp. Phys. 175, 813–837 (2014). (1997). [35] A. Fornieri and F. Giazotto, “Towards phase-coherent [56] D. Golubev, T. Faivre, and J. P. Pekola, “Heat transport caloritronics in superconducting circuits,” Nat. Nan- through a Josephson junction,” Phys. Rev. B 87, 094522 otechnology 12, 944–952 (2017). (2013). [36] G. Timossi, A. Fornieri, F. Paolucci, C. Puglia, and [57] P. Virtanen, P. Solinas, and F. Giazotto, “Spectral repre- F. Giazotto, “Phase-tunable Josephson thermal router,” sentation of the heat current in a driven Josephson junc- arXiv preprint arXiv:1710.04606 (2017), [Nano Lett. (to tion,” Phys. Rev. B 95, 144512 (2017). be published). DOI:10.1021/acs.nanolett.7b04906]. [58] R. C. Dynes, V. Narayanamurti, and J. P. Garno, “Di- [37] K. Maki and A. Griffin, “Entropy transport between two rect measurement of quasiparticle-lifetime broadening in superconductors by electron tunneling,” Phys. Rev. Lett. a strong-coupled superconductor,” Phys. Rev. Lett. 41, 15, 921–923 (1965). 1509–1512 (1978). [38] F. Giazotto and M. J. Martínez-Pérez, “Phase-controlled [59] A. V. Timofeev, C. P. García, N. B. Kopnin, A. M. superconducting heat-flux quantum modulator,” Appl. Savin, M. Meschke, F. Giazotto, and J. P. Pekola, Phys. Lett. 101, 102601 (2012). “Recombination-limited energy relaxation in a Bardeen- [39] F. Giazotto and M. J. Martínez-Pérez, “The Josephson Cooper-Schrieffer superconductor,” Phys. Rev. Lett. 102, heat interferometer,” Nature 492, 401–405 (2012). 017003 (2009). [40] F. Giazotto, M. J. Martínez-Pérez, and P. Solinas, “Co- [60] F. C. Wellstood, C. Urbina, and John Clarke, “Hot- herent diffraction of thermal currents in Josephson tunnel electron effects in metals,” Phys. Rev. B 49, 5942–5955 junctions,” Phys. Rev. B 88, 094506 (2013). (1994). [41] M. J. Martínez-Pérez and F. Giazotto, “A quantum [61] H. Rabani, F. Taddei, O. Bourgeois, R. Fazio, and diffractor for thermal flux,” Nat. Commun. 5, 3579 F. Giazotto, “Phase-dependent electronic specific heat (2014). of mesoscopic Josephson junctions,” Phys. Rev. B 78, [42] A. Fornieri, C. Blanc, R. Bosisio, S. D’Ambrosio, and 012503 (2008). F. Giazotto, “Nanoscale phase engineering of thermal [62] P. Solinas, R. Bosisio, and F. Giazotto, “Microwave transport with a Josephson heat modulator,” Nat. Nan- quantum refrigeration based on the Josephson effect,” otechnology 11, 258–262 (2016). Phys. Rev. B 93, 224521 (2016). [43] A. Fornieri, G. Timossi, P. Virtanen, P. Solinas, and [63] F. Giazotto and J. P. Pekola, “Josephson tunnel junction F. Giazotto, “0–π phase-controllable thermal Josephson controlled by quasiparticle injection,” J. Appl. Phys. 97, junction,” Nat. Nanotechnology 12, 425–429 (2017). 023908 (2005). [44] C. Guarcello, F. Giazotto, and P. Solinas, “Coherent [64] S. Tirelli, A. M. Savin, C. P. Garcia, J. P. Pekola, F. Bel- diffraction of thermal currents in long Josephson tunnel tram, and F. Giazotto, “Manipulation and generation junctions,” Phys. Rev. B 94, 054522 (2016). of supercurrent in out-of-equilibrium Josephson tunnel [45] C. Guarcello, P. Solinas, M. Di Ventra, and F. Giazotto, nanojunctions,” Phys. Rev. Lett. 101, 077004 (2008). “Hysteretic superconducting heat-flux quantum modula- [65] R. Bosisio, P. Solinas, A. Braggio, and F. Gia- tor,” Phys. Rev. Applied 7, 044021 (2017). zotto, “Photonic heat conduction in Josephson-coupled [46] C. Guarcello, P. Solinas, A. Braggio, M. Di Ventra, and Bardeen-Cooper-Schrieffer superconductors,” Phys. Rev. F. Giazotto, “Josephson thermal memory,” Phys. Rev. B 93, 144512 (2016). Applied 9, 014021 (2018). [66] S. G. Lachenmann, T. Doderer, R. P. Huebener, [47] A.R. Bishop, J.A. Krumhansl, and S.E. Trullinger, “Soli- D. Quenter, J. Niemeyer, and R. Pöpel, “Spatially re- tons in condensed matter: A paradigm,” Physica D 1, 1 solved study of the dynamics of Josephson tunnel junc- – 44 (1980). tions,” Phys. Rev. B 48, 3295–3303 (1993). [48] A. Barone and G. Paternò, Physics and Applications of [67] R. Gross and D. Koelle, “Low temperature scanning the Josephson Effect (Wiley, New York, 1982). electron microscopy of superconducting thin films and [49] P. S. Lomdahl, O. H. Soerensen, and P. L. Christiansen, Josephson junctions,” Reports on Progress in Physics 57, “Soliton excitations in Josephson tunnel junctions,” Phys. 651 (1994). Rev. B 25, 5737–5748 (1982). [68] B. A. Malomed and A. V. Ustinov, “Analysis of testing [50] N. Grønbech-Jensen, “Zero-voltage states in ac-driven the single-fluxon dynamics in a long Josephson junction long Josephson junctions,” Phys. Rev. B 45, 7315–7320 by a dissipative spot,” Phys. Rev. B 49, 13024–13029 (1992). (1994). [51] A. Davidson, B. Dueholm, and N. F. Pedersen, “Experi- [69] T. Doderer, “Microscopic imaging of Josephson junction ments on soliton motion in annular Josephson junctions,” dynamics,” International Journal of Modern Physics B , J. Appl. Phys. 60, 1447–1454 (1986). 11, 1979–2042 (1997). [52] A. V. Ustinov, T. Doderer, R. P. Huebener, N. F. Ped- [70] V. M. Krasnov, V. A. Oboznov, and N. F. Pedersen, ersen, B. Mayer, and V. A. Oboznov, “Dynamics of “Fluxon dynamics in long Josephson junctions in the sine-Gordon solitons in the annular Josephson junction,” presence of a temperature gradient or spatial nonunifor- Phys. Rev. Lett. 69, 1815–1818 (1992). mity,” Phys. Rev. B 55, 14486–14498 (1997). [53] A. V. Ustinov, “Fluxon insertion into annular Josephson [71] F. Paolucci, G. Marchegiani, E. Strambini, and F. Gi- junctions,” Appl. Phys. Lett. 80, 3153–3155 (2002). azotto, “Phase-tunable thermal logic: computation with [54] S. B. Kaplan, C. C. Chi, D. N. Langenberg, J. J. Chang, heat,” arXiv preprint arXiv:1709.08609 (2017). S. Jafarey, and D. J. Scalapino, “Quasiparticle and [72] W. Wustmann and K. D. Osborn, “Reversible fluxon phonon lifetimes in superconductors,” Phys. Rev. B 14, logic: Topological particles allow gates beyond the stan- 4854–4873 (1976). dard adiabatic limit,” arXiv preprint arXiv:1711.04339 [55] B. Frank and W. Krech, “Electronic cooling in supercon- (2018).