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Solitonic Josephson Thermal Transport

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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 junc- tion, 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 supercon- ducting 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 ', 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 ', 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 = !pt, with !p = 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- @2'(x; t) @2'(x; t) @'(x; t) ometry depicted in Fig. 1, the junction area A = WL e e − e e − sin ' 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- d'(0; t) d'(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 = (!pRC) 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 ', 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 @'(x; t)=@x = µ0tdH(t) [48], where Φ0 = h=2e ' Φ0 2 × 10−15Wb is the magnetic flux quantum (with e and 8 2 39 < xe − xe0 − uet = h being the electron charge and the Planck constant, '(x − uet) = 4 arctan exp 4± p 5 ; (3) e 1 − u2 respectively), and µ0 is the vacuum permeability.
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