Phase-Tunable Josephson Thermal Router - Nano Letters (ACS

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Phase-tunable Josephson thermal router GiulianoFrancesco Timossi, Antonio Fornieri, Federico Paolucci, Claudio Puglia, and Francesco Giazotto Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b04906 • Publication Date (Web): 14 Feb 2018 Downloaded from http://pubs.acs.org on February 15, 2018

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Phase-tunable Josephson thermal router Giuliano Francesco Timossi,

∗,†

Puglia,

†,‡

†,¶

Antonio Fornieri,

†

Federico Paolucci,

Claudio

∗,†

and Francesco Giazotto

†NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, Piazza S. Silvestro 12, I-56127 Pisa, Italy ‡Dipartimento di Fisica dell’Università di Pisa, Largo Pontecorvo 3, I-56127 Pisa, Italy ¶Present address: Center for Quantum Devices and Station Q Copenhagen, Niels Bohr Institute, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark. E-mail: [email protected]; [email protected]

Abstract A fundamental aspect of electronics is the ability to distribute a charge current among different terminals. On the other hand, despite the great interest in dissipation, storage and conversion of heat in solid state structures, the control of thermal currents at the nanoscale is still in its infancy. Here, we show the experimental realization of a phase-tunable thermal router able to control the spatial distribution of an incoming heat current, thus providing the possibility of tuning the electronic temperatures of two output terminals. This ability is obtained thanks to a direct current superconducting quantum interference device (dc SQUID), which can tune the coherent component of the electronic heat currents flowing through its Josephson junctions. By varying the external magnetic flux and the bath temperature, the SQUID allows us to regulate the size and the direction of the thermal gradient between two drain electrodes. Our results offer new opportunities for all microcircuits requiring an accurate energy management, including electronic coolers, quantum information architectures and thermal logic components.

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Since the first studies of thermodynamics, heat transport has been a crucial element for the understanding of any thermal system. Quantum mechanics has introduced new appealing ingredients for the manipulation of heat currents, such as the long-range coherence of the superconducting condensate. 1,2 The latter has been exploited by phase-coherent caloritronics, a young field of nanoscience, to realize Josephson heat interferometers, 3–6 which can control electronic thermal currents as a function of the external magnetic flux. So far, only one output temperature has been modulated, while multi-terminal devices that allow to distribute the heat flux among different reservoirs are still missing. Here, we report the experimental realization of a phase-tunable thermal router able to control the heat transferred between two terminals residing at different temperatures. Thanks to the Josephson effect, our structure allows to regulate the thermal gradient between the output electrodes until reaching its inversion. Together with interferometers, 3,4 heat diodes 7,8 and thermal memories, 9,10 the thermal router represents a fundamental step towards the thermal conversion of non-linear electronic devices, 2 and the realization of caloritronic logic components. 11,12 A thermal router is a device that allows to direct an incoming heat flow to a (drain) terminal of choice 13 (see Fig. 1a, b). As shown in Fig. 1c, we can define two working regimes: in the splitting regime, one of the drain temperatures is always higher than the other, while in the swapping regime the output thermal gradient is inverted as a function of an external modulating parameter, i.e., the knob represented in Fig. 1a and 1b. This knob can be efficiently embodied by a magnetic flux Φ threading a temperature-biased direct-current superconducting quantum interference device (dc SQUID). 3 The latter consists of a loop composed by two superconductors S1 and S2 residing at electronic temperatures TS1 and TS2 , respectively, and connected by two Josephson tunnel junctions (JJs) in parallel, labelled ‘A’ and ‘B’. If we impose a temperature gradient across the SQUID by raising TS1 above TS2 , a steady state electronic heat current flows through the interferometer: 14 A (TS1 , TS2 ) (1 + r) JSQUID (TS1 , TS2 , Φ) = Jqp s A − Jint (TS1 , TS2 )

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2πΦ , Φ0

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Swapping 0,0

0,5

 

1,0

Figure 1: Basic model of the Josephson thermal router. a. Schematic representation of the phase-tunable Josephson thermal router. The heat current coming from the source electrode can be directed by varying the external magnetic flux. When Φ = 0 the heat current mainly flows to drain 1 and raises its temperature above that of drain 2. b. On the contrary, when Φ = (n + 1/2) Φ0 the heat current mainly flows to drain 2, where n is an integer number and Φ0 is the flux quantum. This periodicity can be implemented by using a dc SQUID as routing element. c. Magnetic flux dependence of the drain temperatures. The router can work in two different regimes, depending on the bath temperature: in the splitting regime, one of the output temperature is always higher than the other, while in the swapping regime the temperature gradient can be inverted. A and J A are the incoherent quasi particle contribution and the phase-coherent component where Jqp int

of the heat current flowing through junction ‘A’, respectively. The second term originates from energy-carrying tunneling processes involving creation and destruction of Cooper pairs on both sides of the JJs 15,16 (see Methods). Consequently, it depends on the phase difference between the superconducting condensates across the JJs and can therefore be modulated by the magnetic flux piercing the SQUID loop. 17 Finally, in Eq. (1) the parameter r = RA /RB accounts for the asymmetry of the normal-state resistances RA and RB of the interferometer JJs, whereas Φ0 ' 2 × 10−15 Wb is the superconducting flux quantum. In the following, we experimentally demonstrate that a temperature-biased dc SQUID can work

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S2

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Figure 2: Structure of the Josephson thermal router. a. Pseudo-colour scanning electron micrograph of the device. Dashed regions indicate the junction areas, i.e., the overlap of two materials separated by a thin oxide layer. The source electrode (red), is coupled to four superconducting probes acting as thermometers and heaters. The source is also coupled with the upper branch of the SQUID S1 (orange) from which the two superconducting islands with anti-phase temperature behaviour spread out, S3 and S2 (light green and light purple, respectively). Each island is connected to its respective normal metal electrodes D1 (dark green) and D2 (dark purple). A pair of superconducting probes for each drain is used to read the temperature. b. Thermal model describing the main heat exchange processes in the structure. Similar colours between the two images a and b have been chosen. Every spring represents a thermal interaction among the components. The colour gradient indicates the heat current direction from red to blue (if Tsource >TBath ). Here the knob (referred to Fig. 1a,b) is the core part between the source and drains. The entering heat flow in S1 is divided between the two superconductors S3 and S2 , and reaches at the end D1 and D2 , respectively. as the directional core of a Josephson thermal router able to control the thermal gradients between two drain electrodes in a superconducting hybrid circuit. The router operation is regulated by two main control parameters: the magnetic flux Φ threading the SQUID and the temperature of the lattice phonons. In our case, the latter are fully thermalized with the substrate phonons residing at the bath temperature, Tbath , thanks to the vanishing Kapitza resistance between thin metallic films and the substrate at low temperatures. 3–7,18,19 As we explained previously, Φ can modulate Jint according to Eq. (1), while instead the average value of the oscillating drain temperatures is mainly controlled by Tbath . As we shall argue, by increasing the value of Tbath , we can tune the

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Figure 3: Thermal behaviour of the router. a. Drain temperatures TD1 (thick lines) and TD2 (thin lines) vs. the magnetic flux Φ piercing the loop of the SQUID for a fixed injected power Jin = 412 pW and different bath temperatures Tbath . It is possible to distinguish between the splitting regime (for Tbath . 300 mK) and the swapping regime (for Tbath & 300 mK). b. TD1 and TD2 vs. Φ for Tbath = 350 mK and different values of Jin . c. Total swing of the drain temperatures as function of Jin for different values of Tbath . The curve are obtained from measurements as those shown in Figs. 3a, b. The coloured regions indicate the amplitudes of the thermal oscillations at a given bath temperature. Four different values of Tbath are shown : 200 mK, 300 mK, 350 mK, and 400 mK. working regime of the thermal router, which can operate as a splitter or as a swapper.

Results and discussion The realization of the thermal router is shown in Fig. 2a and 2b. The structure was fabricated by electron-beam lithography and three-angle shadow-mask evaporation of metals with in situ oxidation (see Methods). Aluminium (with a critical temperature Tc ' 1.55K) was used for all the superconducting parts, and Al0.98 Mn0.02 was used for the normal metal (N) electrodes. 4,5,7,20 The device consists of a N source electrode tunnel-coupled to the superconducting island S1 , forming the upper branch of the SQUID. As mentioned in the introduction, the electrode S1 is connected to the lower branch of the SQUID S2 by means of the JJs ‘A’ and ‘B’. Moreover, S1 is tunnel5


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coupled to the superconducting island S3 , which is necessary to reach the swapping regime, as we shall explain later. S3 and S2 , in turn, are tunnel-coupled to the N drains D1 and D2 , respectively (see Fig. 2a). A N electrode acting as a cooling fin was tunnel coupled to S2 (with a normal state resistance RF ' 1 kΩ) in order to maintain a suitable thermal gradient across the SQUID . 7,18 For the electrical characterization, two superconducting probes P1 and P2 14,17 were tunnel-coupled to S1 and S2 , respectively, with normal state resistances RP1 ' 480 Ω and RP2 ' 320 Ω. The JJ between S1 and S3 (RS3 ' 3 kΩ) was designed to obtain thermal decoupling of D1 from S1 . The normal state resistances of the source and drains junctions are RSource ' 4.6 kΩ, RD1 ' 1.5 kΩ and RD2 ' 1.5 kΩ, respectively. Finally, source and drain electrodes are tunnel-coupled to superconducting probes (see Fig. 2a) realizing NIS junctions (normal state resistance of ∼ 20 kΩ) used as thermometers and Joule heaters. 18 To extract the SQUID parameters, we measured the magnetic interference pattern of its Josephson critical current via a conventional four-wire technique, through P1 and P2 probes. From the fit of the data we obtained the junction ‘A’ normal-state resistance RA ' 650 Ω and the asymmetry parameter r ' 0.95. 3,16,21 On the other hand, thermal measurements were performed using the set-up sketched in Fig. 2a. By injecting a Joule power Jin into the source, we can raise its electronic temperature TSource above Tbath . In this way, we can increase the electronic temperature TS1 of S1 and generate a thermal gradient across the SQUID at the core of the thermal router. This assumption is expected to hold thanks to the cooling fin connected to S2 . Indeed, the cooling fin extends into a large volume which provides a good thermalization with Tbath and represents an efficient channel to reduce TS2 below TS1 . Therefore, a finite heat current JSQUID flows through the SQUID and can be modulated by the magnetic flux Φ piercing its loop. In particular, if we set TSource and Tbath to fixed values, JSQUID is minimized when Φ = n Φ0 (being n an integer) [see Eq. (1)] leading to a maximum of TS1 . For Φ = n Φ0 /2, instead, JSQUID is maximized and TS1 has a local minimum. As a consequence, the heat current flowing out of S1 towards S3 is maximized for Φ = n Φ0 , while the heat current flowing out of S2 results to be minimized. The opposite situation is obtained when Φ = n Φ0 /2. Even though heat currents are not directly detectable, the described behaviour is reflected by the tem-

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Figure 4: Data fit and working regimes. a. Magnetic flux modulations of TD1 and TD2 at two values of Tbath (200mK and 400mK) for Jin = 63 pW. Solid lines represent the experimental data, while dashed lines are obtained from the thermal model of our structure (see Main Text). b. Perδ TD1 +δ TD2 as a function of Tbath for Jin = 63 pW (upper panel) and formance parameter S = 2(hT D2 i−hTD1 i) Jin ' 10.5 nW (lower panel). S describes the working regime of the router, which operates as a splitter (light blue area) for S < 1 or as a swapper (grey area) for S > 1. The insets show the behaviour of R = hTD1 i/hTD2 i as a function of Tbath for Jin = 63 pW and Jin ' 10.5 nW. peratures TD1 and TD2 of the drain electrodes connected to the different branches of the SQUID, which are expected to oscillate periodically with opposite phase dependence. Moreover, the superconducting island S3 is introduced between S1 and D1 in order to reduce the difference between TD1 and TD2 and reach the swapping regime. The thermal behaviour of the D1 and D2 as a function of the magnetic flux is shown in Fig. 3a and 3b for different values of Tbath and Jin , respectively. As anticipated, the drain temperatures manifestly demonstrate an opposite dependence on the magnetic flux: the minima and maxima of the curves corresponding to different drains are inverted. At low bath temperature, the oscillations do not overlap, so the thermal router is operating in the splitting regime. As we increase the value of Tbath , the average temperatures of the drain electrodes hTD1 i and hTD2 i tend to approach each other, leading to overlapping modulations for Tbath & 300 mK. This behaviour is summarized by Fig. 3c, which displays the total swing amplitudes of TD1 and TD2 as a function of Jin for different

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bath temperatures. The injected power Jin also plays an important role, since it can raise hTD2 i more efficiently than hTD1 i. This is due to the stronger coupling of S2 to S1 as compared to that between S1 and S3 . In the splitting regime, we obtain a maximum separation between TD1 and TD2 of ∼40 mK for Jin ' 10.4 nW at Tbath = 200 mK, while in the swapping regime a maximum temperature inversion of ∼4 mK is obtained for Jin ' 6.6 nW at Tbath = 400 mK. To account for our observations we have elaborated a thermal model describing our structure (sketched in Fig. 2b). In this model, we take into account the predominant mechanisms of energy exchange, i.e, the electronic heat currents flowing through SIS, SIN or NIS junctions and the electron-phonon coupling in the normal metal electrodes (see Methods). For any given TSource , Tbath and Φ, the steady-state electronic temperatures of every part of the device can be determined by solving the following system of energy-balance equations:  −Jin + JSource + JSource,e−ph = 0        −JSource + JP1 + JSQUID + JS3 = 0         −JSQUID + JP2 + JFin + JD2 = 0   −JS3 + JD1 = 0        −JD1 − JThermo − J1 + JD1,e−ph = 0       −JD2 − JThermo − J2 + JD2,e−ph = 0.

(2a) (2b) (2c) (2d) (2e) (2f)

These equations account for the detailed thermal budget of each electrode of the structure by equating the sum of all incoming and outgoing heat currents. In particular, Eq. (2a) refers to the thermal budget in the source, Eq. (2b) to S1 , Eq. (2c) to S2 , Eq. (2d) to S3 , and finally Eqs. (2e) and Eq. (2f) describe the energy exchange of D1 and D2 , respectively. In the energy-balance equations, JP1,2 represent the heat currents released from S1 and S2 to the superconducting probes P1 and P2 , JS3 is the heat flux between S1 and S3 , whereas JFin accounts for the power delivered by S2 to the cooling fin. Moreover, JD1,2 are the heat currents flowing into D1,2 , JN,e−ph , where N=Source,D1,D2 are the heat losses due to the electron-phonon coupling in the electrodes and, finally, JThermo consider the energy released by D1 and D2 to the superconducting thermometers. Each term of the energy8


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balance equations is detailed in the Methods. In the model, we neglect photon-mediated thermal transport owing to poor impedance matching between the source and drains electrodes, 22–25 as well as electron-phonon coupling in all the superconducting parts of the structure due to their reduced volume and low Tbath . 26 The model provides a good agreement with the data for both drains at different bath temperature and different injected power, as shown in Fig.4a for Jin = 63pW. Three parameters were extracted from the fit. The first is the resistance of the probe P2 (RP2 = 320Ω), which is not directly obtainable from the electrical measurements (see Methods). The other two parameters are the parasitic heat currents J1,2 injected by the thermometers of the drains, which remain on the order of the femtowatt (in particular J1 varies from 1 fW to 8 fW and J2 ∼ −10 fW). The origin of these parasitic powers may be thermal noise and fluctuations in the thermometers. 22 In a device with equal losses in both drains, we would get a perfect correspondence between JD1,2 and TD1,2 . In order to characterize the performance of our thermal router, we define two parameters outlining its working regime: the ratio between the average drain temperatures R = hTD1 i/hTD2 i, and the ratio between the sum of the amplitudes of the average drain temperature swings and the difference between the drain average temperatures S =

δ TD1 +δ TD2 2(hTD2 i−hTD1 i) .

As shown in the insets of Fig. 4b,

R displays the gradual approach of the drain average temperatures as Tbath is increased, which is not influenced by the injected power Jin . On the other hand, S takes into account the overlap of the thermal oscillations of the two drains. By definition S is lower than 1 if the router works in the splitting regime, while it is greater than 1 if the router is in the swapping regime. The behaviour of S as function of Tbath for two different values of Jin is shown in Fig. 4b. The injected power strongly affects S, influencing the transition point between the different working regimes of the thermal router. This is also clear from Fig.3c: at higher injected power the router reaches the swapping regime at larger values in Tbath . We shall finally discuss the temperatures modulation speed which could be achieved with the device. In previous works 9 it has been shown that the slowest relaxation mechanism is the electronphonon scattering, characterized by the relaxation time τ required by quasiparticles in the super-

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conductors to thermalize with lattice phonons. In particular, at ∼ 400 mK τ −1 is about 1 MHz for Al, whereas at lower Tbath it is drastically reduced owing to increased electron-phonon relaxation time. 27 We stress that in order to obtain a better performance, it is possible to work at higher bath temperatures with other materials characterize by a larger superconducting gap, such as Nb.

Conclusions In summary, we have realized a phase-tunable Josephson thermal router able to control with very high accuracy the heat transferred among two terminals residing at different temperatures. The router provides high versatility of use, allowing to work in different regimes, easily accessible by varying the bath temperature. In the splitting regime, we obtain a maximum separation of the average drain temperatures of ∼ 40 mK, whereas in the swapping regime, our device can invert the thermal gradient between two reservoirs, with a maximum inversion of ∼ 4 mK. The latter regime may be useful for mesoscopic thermal machines. 28,29 Indeed, the possibility of locally inverting the direction of the thermal current between two reservoirs may be used to mimic a thermodynamic cycle in which the reservoir’s temperature is time dependent. The device is a proof of concept which can be easily improved by using a double loop SQUID 5 to achieve a finer control, or a 0 − π thermal JJ 6 to increase the amount of heat that can be controlled at the core of the structure. Furthermore, in order to improve the efficiency of the device is possible to reduce the losses both through the thermometer and through the lattice phonons in the normal metal electrodes. We wish to stress that our thermal router is a modular element, and it can be easily extended to multiple output terminals. Indeed the output of a thermal router can be used as the input for the next one, thus multiplying the output terminals. Finally, one of the fundamental requests of thermal logic, 11 namely switching thermal signals among different channels, is answered by the phase-tunable thermal router.

Supporting Information Supporting information on sample fabrication, measurements and electrical-thermal model.

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Acknowledgements We thank S. Gasparinetti for useful discussion. We acknowledge the MIUR-FIRB2013–Project Coca (grant no. RBFR1379UX), the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 615187 - COMANCHE and the European Union (FP7/2007-2013)/REA grant agreement no. 630925 – COHEAT for partial financial support. The work of F.P. is funded by Tuscany Region under the FARFAS 2014 project SCIADRO.

Author contributions G.T. and F.P. fabricated the samples. G.T. and C.P. performed the measurements. G.T. and C.P. analysed the data and carried out the simulations with input from A.F. and F.G. G.T., A.F. and F.G. conceived the experiment. All the authors discussed the results and their implications equally at all stages, and wrote the manuscript.

Competing Financial Interest The authors declare no competing financial interest.

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References

References (1) Maki, K. & Griffin, A. Entropy transport between two superconductors by electron tunneling. Phys. Rev. Lett. 15, 921-923 (1965). (2) Fornieri, A. & Giazotto, F. Towards phase-coherent caloritronics in superconducting circuits Nat. Nanotechnol 12 944-952 (2017). (3) Giazotto, F. & Martínez-Pérez, M. J. The Josephson heat interferometer. Nature 492, 401-405 (2012). (4) Martínez-Pérez, M. J. & Giazotto, F. A quantum diffractor for thermal flux. Nat. Commun. 5, 3579 (2014). (5) Fornieri, A., Blanc, C., Bosisio, R., D’Ambrosio, S. & Giazotto, F. Nanoscale phase engineering of thermal transport with a Josephson heat modulator. Nat. Nanotechnol. 11, 258-262 (2016). (6) Fornieri, A., Timossi, G., Virtanen, P., Solinas, P. & Giazotto, F. 0-π phase-controllable thermal Josephson junction. Nat. Nanotechnol. 12, 425-429 (2017). (7) Martínez-Pérez, M. J., Fornieri, A. & Giazotto, F. Rectification of electronic heat current by a hybrid thermal diode. Nat. Nanotechnol. 10, 303-307 (2015). (8) Martínez-Pérez, M. J. & Giazotto, F. Efficient phase-tunable Josephson thermal rectifier. Appl. Phys. Lett. 102, 182602 (2013). (9) Guarcello, C., Solinas, P., Braggio, A., Di Ventra, M. & Giazotto, F. A Josephson thermal memory Phys. Rev. Applied 9, 014021 (2018)

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(10) Fornieri, A., Timossi, G., Bosisio, R., Solinas, P. & Giazotto, F. Negative differential thermal conductance and heat amplification in superconducting hybrid devices. Phys. Rev. B 93, 134508 (2016). (11) Paolucci, F., Marchegiani, G., Strambini, E. & Giazotto, F. Phase-tunable thermal logic: computation with heat. arXiv:1709.08609 (2017). (12) Li, N., Ren, J., Wang, L., Zhang, G., Hänggi, P. & Li, B. Phononics: manipulating heat flow with electronic analogs and beyond. Rev. Mod. Phys. 84, 1045-1066 (2012). (13) Bosisio, R., Valentini, S., Mazza, F., Benenti, G., Fazio, R., Giovannetti, V. & Taddei, F. A magnetic thermal switch for heat management at the nanoscale. Phys. Rev. B 91, 205420 (2015) (14) Giazotto, F. & Martínez-Pérez, M. J. Phase-controlled superconducting heat-flux quantum modulator. Appl. Phys. Lett. 101, 102601 (2012). (15) Guttman, G. D., Nathanson, B., Ben-Jacob, E. & Bergman, D. J. Phase-dependent thermal transport in Josephson junctions. Phys. Rev. B 55, 3849-3855 (1997). (16) A. Barone and G. Paternò, Physics and Applications of the Josephson Effect (Wiley, New York, 1982). (17) Tinkham, M. Introduction to Superconductivity (McGraw-Hill, 1996) and references therein. (18) Giazotto, F., Heikkilä, T. T., Luukanen, A., Savin, A. M. & Pekola, J. P. Opportunities for mesoscopics in thermometry and refrigeration: Physics and applications. Rev. Mod. Phys. 78, 217-274 (2006). (19) Wellstood, F. C., Urbina, C. & Clarke, J. Hot-electron effects in metals. Phys. Rev. B 49, 5942-5955 (1994). (20) Taskinen, L. J. & Maaasilta, I. J. Improving the performance of hot-electron bolometers and solid state coolers with disorderd alloys. Appl.Phys.Lett. 89, 143511 (2016). 13


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(21) Clarke, J. & Braginski, A. I. (eds) The SQUID Handbook (Wiley-VCH, 2004). (22) Meschke, M., Guichard, W. & Pekola, J. P. Single-mode heat conduction by photons. Nature 444, 187-190 (2006). (23) Bosisio, R., Solinas, P., Braggio, A. & Giazotto, F. Photonic heat conduction in Josephsoncoupled Bardeen-Cooper-Schrieffer superconductors. Phys. Rev. B 93, 144512 (2016). (24) Timofeev, A. V., Helle, M., Meschke, M., Möttönen, M. & Pekola, J. P. Electronic refrigeration at the quantum limit. Phys. Rev. Lett. 102, 200801 (2009). (25) Pascal, L. M. A., Courtois, H. & Hekking, F. W. J. Circuit approach to photonic heat transport. Phys. Rev, B 83, 125113 (2011). (26) Timofeev, A. V.,Pascual García, C., Kopnin, N. B., Savin, A. M., Meschke, M., Giazotto, F.& Pekola, J. P. Recombination-limited energy relaxation in a Bardeen-Cooper-Schrieffer superconductor.Phys. Rev. Lett. 102, 017003 (2009). (27) Kaplan, S. B., Chi, C. C., Langenberg, D. N., Chang, J. J., Jafarey, S. & Scalapino, D. J., Quasiparticle and phonon lifetime in superconductor Phys. Rev. B 14, 4854 (1976) (28) Kosloff, R. & Levy, A. Quantum Heat Engines and Refrigerators: Continuous Devices. Annu. Rev. Phys. Chem.65, 365-393 (2014). (29) Benenti, G., Casati, G., Saito, K. & Whitney, R. S. Fundamental aspects of steady-state conversion of heat to work at the nanoscale. Physics Reports, 694, 1-124 (2017)

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Nano Letters

Graphical TOC Entry

SQUID Source

Drain1

F Drain2 5 mm

15


Nano Letters

a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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b

c

T

Splitting

Drain1 Drain2

Swapping 0,0

0,5

 


1,0

a

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V

Heater 1

SINIS Thermometer

V

SQUID S1

Source

S3

B

Drain1

F

A P1

S2

Drain2

5mm Cooling Fin

b

S3, TS3

P2

Drain1, TD1

SQUID

Jin Source, TSource

Bath, Tbath

S1, TS1

F Bath, Tbath

S2, TS2


Bath, Tbath

SINIS Thermometer V

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Nano Letters

SINIS Thermometer

Drain2, TD2 Bath, Tbath

a

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T bat h(mK )

360

Tbath (mK)

330

Jin= 412pW

350

200 300 350

300 270

420

400 300

350

200

300

Jin (nW)

Tbath = 350mK

0.063 0.412 10.411

390

360 -3

400

450

b TD1,TD2(mK)

c

390

T (mK )

TD1,TD2(mK)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Nano Letters

-2

-1

0



1

2

250

200 0 3 ACS Paragon Plus Environment

D1 D2 5

Jin(nW)

10

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T (mK )

T bat h= 400mK

1.00

R

3

410

S

2

Jin= 63 pW

0.95 0.90

408

200

300

400

Swapping

T bat h(mK ) 1

406

Splitting 0

D1 D2

270

1.00

3

R

T (mK )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

b

412

260

S

a

Nano Letters

2

Jin= 10.5 nW

0.95 0.90

200

400

Swapping

T bat h(mK )

250

T bat h= 200mK

1

240 -2

300

Splitting -1

0



1

2

0


200

300

T bat h(mK )

400

Phase-Tunable Josephson Thermal Router - Nano Letters (ACS

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