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1 Powered by TCPDF ( This is an eletroni reprint of the original artile. This reprint may differ from the original in pagination and typographi detail. Giazotto, F.; Heikkilä, Tero; Taddei, F.; Fazio, Rosario; Pekola,.P.; Beltram, F. Tailoring osephson Coupling through superondutivity-indued nonequilibrium Published in: PHYSICAL REVIEW LETTERS DOI:.3/PhysRevLett Published: 29/3/24 Doument Version Publisher's PDF, also known as Version of reord Please ite the original version: Giazotto, F., Heikkilä, T., Taddei, F., Fazio, R., Pekola,., & Beltram, F. (24). Tailoring osephson Coupling through superondutivity-indued nonequilibrium. PHYSICAL REVIEW LETTERS, 92(3), -4. [37. DOI:.3/PhysRevLett This material is proteted by opyright and other intelletual property rights, and dupliation or sale of all or part of any of the repository olletions is not permitted, exept that material may be dupliated by you for your researh use or eduational purposes in eletroni or print form. You must obtain permission for any other use. Eletroni or print opies may not be offered, whether for sale or otherwise to anyone who is not an authorised user.

2 Tailoring osephson Coupling through Superondutivity-Indued Nonequilibrium F. Giazotto,, * T. T. Heik kilä, 2 F. Taddei, Rosario Fazio,. P. Pekola, 2 and F. Beltram NEST-INFM & Suola Normale Superiore, I-5626 Pisa, Italy 2 Low Temperature Laboratory, Helsinki University of Tehnology, P.O. Box 22, FIN-25 HUT, Finland (Reeived 4 November 23; published 29 Marh 24) The distintive quasipartile distribution existing under nonequilibrium in a superondutorinsulator-normal metal-insulator-superondutor mesosopi line is proposed as a novel tool to ontrol the superurrent intensity in a long osephson weak link. We present a desription of this system in the framework of the diffusive-limit quasilassial Green-funtion theory and take into aount the effets of inelasti sattering with arbitrary strength. Superurrent enhanement and suppression, inluding a marked transition to a juntion, are striking features leading to a fully tunable struture. DOI:.3/PhysRevLett PACS numbers: r, r, b, Nonequilibrium effets in mesosopi superonduting iruits have been reeiving rekindled attention during the last few years [. The art of ontrolling osephson oupling in superondutor-normal metalsuperondutor (SNS) weak links is at present in the spotlight: a reent breakthrough in mesosopi superondutivity is indeed represented by the SNS transistor, where superurrent suppression as well as its sign reversal ( transition) were demonstrated [2,3. This was ahieved by driving the quasipartile distribution in the weak link far from equilibrium [4 6 through external voltage terminals, viz., normal reservoirs. Suh behavior relies on the two-step shape of the quasipartile nonequilibrium distribution, typial of diffusive mesosopi wires and experimentally observed by Pothier and oworkers [7. The purpose of this Letter is to demonstrate that it is possible to tailor the quasipartile distribution through superondutivity-indued nonequilibrium in order to implement a unique lass of superonduting transistors. This an be ahieved when mesosopi ontrol lines are onneted to superonduting reservoirs through tunnel barriers (I), realizing a superondutor-insulator-normal metal-insulator-superondutor (SINIS) hannel. The peuliar quasipartile distribution in the N region, originating from biasing the S terminals, allows one to aess several regimes, from superurrent enhanement with respet to equilibrium to a large amplitude of the transition passing through a steep superurrent suppression. These features are aompanied by a large urrent gain (up to some 5 in the region of larger input impedane) and redued dissipation. The ultimate operating frequenies available open the way to the exploitation of this sheme for the implementation of ultrafast urrent amplifiers. The investigated mesosopi struture (see Fig. ) onsists of a long diffusive weak link of length L muh larger than the superonduting oherene length ( ) oriented along the x diretion. This defines the SNS juntion of ross setion A. The superonduting terminals belonging to the SNS juntion, labeled S (3 and 4), are kept at zero potential. The SINIS ontrol line is oriented along the y diretion and onsists of a normal wire, of length L C and ross setion A C, onneted through idential tunnel juntions of resistane R T to two superonduting reservoirs S C ( and 2), biased at opposite voltages V C =2. The superonduting gaps of S and S C ( and C ) are in general different. The superurrent I flowing aross the SNS juntion is given by [5,6 I V C A Z de f E; V el C f E; V C Im j E ; () and depends on the quasipartile distribution funtion f E.InEq.(), is the normal-state ondutivity whih determines the normal-state resistane of the juntion aording to R N L = A. The distribution funtion f redues to the equilibrium Fermi distribution when V C. The energy-dependent spetral superurrent [8,9, Im j E, an be alulated by solving the Usadel equations [. Following the parametrization of the Green funtions given in Ref. [8, these equations in the N region an be written FIG. (olor). Sheme of the osephson transistor. The superurrent I (along the white dashed line) is tuned by applying a bias V C aross the SINIS symmetri line onneted to the enter of the weak link. All normal wires are assumed quasione-dimensional =4=92(3)=37(4)$ The Amerian Physial Soiety 37-

3 j E sinh x x j E ; hd@ 2 x 2iE sinh hd (2) x 2 sinh 2 ; where D is the diffusion oeffiient and E is the energy relative to the hemial potential in S. x; E and x; E are in general omplex funtions. For perfetly transmissive ontats, the boundary onditions at the S N interfaes redue to artanh =E and =2 in the reservoirs S, where is the phase differene between the superondutors. As required by Eq. (), we must determine the atual quasipartile distribution in the N region of the SINIS struture. This is ontrolled by voltage (V C ) and temperature and by the amount of inelasti sattering in the ontrol line. In the ase of a short ontrol wire with no inelasti interations, the quasipartile distribution, aording to Ref. [, is given by ei R N /E Th (III) ev C / C (T) (II) f (-E)-f (E) T C =.2T (II) (a) φ = π/ (I) f (-E)- f (E) (I) (III) T/T C f (-E)- f (E) 3 2 f E; V C N F N 2 F 2 ; (3) N N 2 where N ;2 N SC E ev C =2 and F ;2 F E ev C =2. The former are the BCS densities of states in the reservoirs S C (labeled and 2 in Fig. ). F E is the Fermi funtion at lattie temperature T [2. In this ase Eqs. () and (3) yield the dimensionless transistor output harateristis shown in Fig. 2(a). The latter plots the superurrent I vs ontrol bias V C at different temperatures for a long juntion (i.e., E Th, where E Th hd=l 2 is the Thouless energy of the SNS juntion, as this is the limit where the superurrent spetrum varies strongly with energy). We assumed =2, T C =T :2, where T C are the ritial temperatures of the superondutors S C and L suh that =E Th 3. At the lowest temperatures, inreasing V C leads to a large superurrent enhanement with respet to equilibrium slightly below V C 2 C T =e VC T (region I in Fig 2). Further inrease of bias leads to a transition (region II) and finally to a deay for larger voltages [4. This behavior is explained in Figs. 2(b) 2(d), where the spetral superurrent (solid line) is plotted together with f E f E (dash-dotted line) for values of V C and T orresponding to regions I, II, and III, respetively. Hathed areas represent the integral of their produt, i.e., the superurrent I of Eq. (). In partiular, region I orresponds to the ooling regime where hot quasipartiles are extrated from the normal metal [,5. The origin of the transition in region II is illustrated by Fig. 2(), where the negative ontribution to the integral is shown. We remark that the intensity of the superurrent inversion is very signifiant. It reahes about 6% of the maximum value of I at V C VC T in the whole temperature range, nearly doubling the -state value of the (b) E/E Th Z e 2 fn R T F f E N 2 F 2 f E g C F () 2 4 E/E Th (d) E/E Th FIG. 2. (a) Superurrent vs ontrol voltage V C at different temperatures (T) for =2 and T C :2 T (see text). Bias regions labeled (I), (II), and (III) indiate superurrent enhanement due to quasipartile ooling, high-voltage state and low-voltage state in the high-temperature regime, respetively. These are qualitatively explained in (b), (), and (d), where hathed areas represent the ontribution to superurrent arising in suh bias ranges (see text). superurrent as ompared to the ase of an all-normal ontrol hannel [5,6. In the high-temperature regime (T=T C * :6), when the equilibrium ritial urrent is vanishing, the superurrent first undergoes a low-bias transition (region III in Fig. 2), then enters regions I and II. This reover of the superurrent from vanishingly small values at equilibrium is again the onsequene of the peuliar shape of f [see Fig. 2(d). Notably, the superurrent enhanement around VC T remains pronouned even at the highest temperatures, so that I attains values largely exeeding 5% of the juntion maximum superurrent. This demonstrates the full tunability of the superurrent through nonequilibrium effets indued by the superonduting ontrol lines. We remark that this is a unique feature stemming from the superondutivityindued nonequilibrium population in the weak link. The length L C of the SINIS ontrol line an be additionally varied to ontrol the superurrent by hanging the effetive strength of inelasti sattering in the N region. For R T R C L C = A C, the distribution funtion f E in the N region is essentially y independent and we have d!d"! I!; "; E : (4)

4 Here F is the normal-metal density of states at the Fermi I energy, C is the volume of the N region, and I is the net!; "; E f "! f E! f " f E : ollision rate at energy E. At low temperatures, the most (7) relevant sattering mehanism is eletron-eletron sattering [6 and we an neglet the effet of eletronphonon sattering. Then [7,7, Eletron-eletron interation is either due to diret Coulomb sattering [8,9 or mediated by magneti impurities [6. Below, we onentrate on the former but I!; "; E I in!; "; E I out!; "; E ; (5) the latter would yield a similar qualitative behavior. From the alulation of the sreened Coulomb interation in the diffusive hannel, it follows [8 that and 3=2 p for a quasi-one-dimensional wire and D=2 h 3=2 F A C [9. We note that C is the most I in!; "; E f " f E f "! f E! ; relevant energy sale to desribe the distribution funtion for different voltages V C. It is thus useful to replae!! (6)!= C and "! "= C in order to obtain a dimensionless equation. Multiplying Eq. (4) by e 2 R T C F, we obtain Z N F f E N 2 f E F 2 K oll d!d"! 3=2 I!; "; E ; (8) p p where K oll R T =R C L 2 C =D C q 2 RT =R K C =E C Th, R K h=2e 2,andE C Th hd=l2 C. In the absene of eletron-eletron interation (K oll ), Eq. (3) is reovered. The influene of inelasti sattering on I is shown in Fig. 3, whih displays the ritial urrent of a long juntion at T : T C for several values of K oll. Here I is obtained by numerially solving Eq. (8). The effet of eletron-eletron interation is to strongly suppress the state and to widen the peak around VC.The transition vanishes for K oll, but the I enhanement due to quasipartile ooling still persists in the limit of even larger inelasti sattering [2. The disappearane of the state an be understood by looking at the right inset of Fig. 3, whih learly shows how f (alulated at ev C 2:5 C ) gradually relaxes from nonequilibrium towards a Fermi funtion upon inreasing K oll. The left inset shows how f (evaluated at ev C :5 C ) sharpens, thus enhaning I, by inreasing K oll. This effet follows from the fat that inelasti interations redistribute the oupation of quasipartile levels in the N region, thus inreasing the oupation at higher energy. As a onsequene, higher-energy exitations are more effetively removed by tunneling, even for biases well below and not only around V C (as in the ase of K oll ). At the same time, superurrent reovery at high temperature is gradually weakened upon enhaning K oll. Notably, these alulations show that a rather large amount of inelasti sattering is neessary to weaken and ompletely suppress the state. For example, using Al=Al 2 O 3 =Cu as materials omposing the SINIS line, K oll orresponds to use a fairly long ontrol line with L C 2:3 m [2. Changing the ratio T C =T shifts the I response along the V C axis, the shape of the harateristis being P (W) (A 2 Hz - ) T C /T..2.4 T =.T T = 3 Ω (a) T C =.2T T =.T T = 3 Ω G I ( 2 ) -2-4 G I ( 5 ) (b) () FIG. 3. Superurrent vs V C for various K oll with T : T C and T C :2 T. Insets show the distribution funtion at ev C :5 C (left) and ev C 2:5 C (right) alulated for the same K oll values. FIG. 4. (a) Power dissipated in the SINIS line vs V C alulated for various ratios T C =T and T :T. (b) Noise power S vs V C alulated for the same parameters as in (a). () Differential urrent gain G I vs V C for T C =T :2. The inset shows G I in the high-bias region. In all these alulations we set K oll and T 9:26 K (Nb)

5 virtually independent of T C. This translates into a different magnitude of ontrol voltages V C and power dissipation P I C V C, where I C is the ontrol urrent aross the SINIS hannel. The funtion P V C is plotted in Fig. 4(a) for some ratios T C =T at T : T, assuming R T 3 and T 9:26 K. The impat of C in ontrolling power dissipation is easily reognized. These effets learly indiate that C is the ondition to be fulfilled in order to minimize P. In pratie, the power dissipation for V C >VC onstitutes an experimental problem as this energy needs to be arried out from the reservoirs. In a similar way the noise properties of the system are sensitive to the different T C =T ratios. Assuming that the noise through one juntion is essentially unorrelated from the noise through the other, it follows that the input noise power S in the ontrol line an be expressed as S R T Z den ff E F F f E g: S V C from Eq. (9) is shown in Fig. 4(b) for the same parameters of Fig. 4(a). For example, for T C =T : (orresponding roughly to the ombination Al/Nb), P obtains values of the order of a few 5 W and S of some 3 A 2 Hz in the ooling regime, while these values are enhaned, respetively, to few tens of 2 W and 26 A 2 Hz for biases around the transition. In light of the possible use of this operational priniple for devie implementation, let us omment on the available gain and swithing times. Input and output voltages are of the order of C =e and E Th =e, respetively, so that it seems hard to ahieve voltage gain. On the other hand, differential urrent gain G I di =di C di =dv C di C =dv C an be very large. For V C >VC a simple estimate gives G I E Th = C R T =R N, meaning that with realisti ratios R T =R N ( 3 ), G I an exeed 2. G I V C alulated for T C =T :2 is plotted in Fig. 4() (the inset shows the gain in the -state region). This alulation reveals that G I an reah huge values, with some 5 for V C <VC [22 and several 2 in the opposite regime. Remarkably, gain is almost unhanged also in the presene of weak inelasti sattering (i.e., K oll ). The same holds for P and S. As far as power gain is onerned, the osephson juntion has to be operated in the dissipative regime in order to get out power. An estimate for the differential power gain gives G P dp =dp E Th = C G I 3 4 for V C <VC and for V C >VC. The highest operating frequeny of the transistor is limited by the smallest energy in the system: min h f C; ;E C Th ;E Th ;h R TC g; where C is the tunnel juntion apaitane. For an optimized devie, working frequenies of the order of Hz an be experimentally ahieved in the high-voltage regime V C >VC.ForV C <VC, onversely, the response is slower (9) (somewhat below 9 Hz), owing to the long disharging time through the juntions. We thank M. H. Devoret, K. K. Likharev, F. Pierre, L. Roshier, A. M. Savin, and V. Semenov for helpful disussions. This work was supported in part by MIUR under the FIRB Projet No. RBNEFSWY and by the EU (RTN-Nanosale Dynamis). *Eletroni address: giazotto@sns.it [ See, for example, Theory of Nonequilibrium Superondutivity, edited by N. B. Kopnin (Clarendon Press, Oxford, 2). [2.. A. Baselmans, A. F. Morpurgo, B.. van Wees, and T. M. Klapwijk, Nature (London) 397, 43 (999);. Huang et al., Phys. Rev. B 66, 257 (22); R. Shaikhaidarov et al., ibid. 62, R4649 (2). [3.. A. Baselmans, T. T. Heikkilä, B.. van Wees, and T. M. Klapwijk, Phys. Rev. Lett. 89, 272 (22). [4 A. F. Volkov, Phys. Rev. Lett. 74, 473 (995). [5 F. K. Wilhelm, G. Shön, and A. D. Zaikin, Phys. Rev. Lett. 8, 682 (998). [6 S.-K. Yip, Phys. Rev. B 58, 583 (998). [7 H. Pothier et al., Phys. Rev. Lett. 79, 349 (997). [8 See W. Belzig et al., Superlatties Mirostrut. 25, 25 (999), and referenes therein. [9 T. T. Heikkilä,. Särkkä, and F. K. Wilhelm, Phys. Rev. B 66, 8453 (22). [ K. D. Usadel, Phys. Rev. Lett. 25, 57 (97). [ D. R. Heslinga and T. M. Klapwijk, Phys. Rev. B 47, 557 (993). [2 At low ontrol voltages there is a region of energies where N N 2. We assume that there the (otherwise weak) oupling to phonons makes the distributions at those energies equal to the equilibrium Fermi distribution. A more detailed disussion and another type of oupling is given in [3. [3. P. Pekola et al., Phys. Rev. Lett. 92, 5684 (24). [4 A similar behavior was predited by.. A. Baselmans, Ph. D. thesis, University of Groningen (22). [5 M. M. Leivo,. P. Pekola, and D.V. Averin, Appl. Phys. Lett. 68, 996 (996). [6 A. Anthore, F. Pierre, H. Pothier, and D. Esteve, Phys. Rev. Lett. 9, 7686 (23). [7 K. E. Nagaev, Phys. Rev. B 52, 474 (995). [8 B. L. Altshuler and A. G. Aronov, Zh. Eksp. Teor. Fiz. 75, 6 (978) [Sov. Phys. ETP 48, 82 ( 978). [9 A. Kamenev and A. Andreev, Phys. Rev. B 6, 228 (999). [2 F. Giazotto et al., Appl. Phys. Lett. 83, 2877 (23). [2 This is straightforward assuming as typial parameters R T 3, D :2 m 2 =s, and C 2 ev: [22 In this alulation we hose to inlude depairing by a phenomenologial but realisti parameter 4 C [3. Its omission would lead to extremely higher G I values. For V C VC and low temperature, determines the differential resistane R R T C = of the SINIS line. Qualitatively, the large G I stems from the fat R R N