Josephson junction based on highly disordered superconductor/low-resistive normal metal bilayer
aa r X i v : . [ c ond - m a t . s up r- c on ] F e b Josephson junction based on highly disorderedsuperconductor/low-resistive normal metal bilayer
P M Marychev and D Yu Vodolazov
Institute for Physics of Microstructures, Russian Academy of Sciences, NizhnyNovgorod, 603950 RussiaE-mail: [email protected]
Abstract.
We calculate current-phase relation (CPR) of a SN-S-SN Josephsonjunction based on a variable thickness SN bilayer composed of highly disorderedsuperconductor (S) and low-resistive normal metal (N) with proximity inducedsuperconductivity. In case when the thickness of S,N layers and length of S constrictionis about of superconducting coherence length the CPR is single-valued, could be closeto sinusoidal one and the product I c R n can reach ∆(0) / | e | ( I c is the critical currentof the junction, R n is its normal-state resistance, ∆(0) is the superconductor gap ofsingle S layer at zero temperature). We argue that the normal layer should providegood heat removal from S constriction and there is range of parameters when current-voltage characteristic is not hysteretic and I c R n is relatively large. Keywords : normal metal-superconductor bilayer, Josephson junction, Joule heating osephson junction based on superconductor/low-resistive normal metal bilayer
1. Introduction
Various technological applications of Josephson junctions (AC voltage standard[1], rapid single-quantum logic [2], SQUID-magnetometers [3] and particle detectors[4]) require to have nonhysteretic current-voltage characteristic (IVC). Tunnelsuperconductor-insulator-superconductor (S-I-S) junctions is characterized by smallcritical current densities and hysteretic IVC (the latter is related with large capacitanceof the insulator layer) which restricts their applicability. S-N-S and S-S’-S junctions(where N is a normal metal and S’ is the geometric constriction or superconductorwith smaller critical current) have small capacitance of the weak link but IV curves arehysteretic due to Joule dissipation [3, 5–7].The current-phase relation (CPR) of these types of the Josephson junctions is oftendifferent from the sinusoidal form I = I c sin ϕ where I c is the junction critical currentand ϕ is the phase difference between electrodes [8]. Specific form of the CPR dependson the junction parameters and temperature [9]. In the case of the weak link made ofpure superconductor or normal metal (having mean free path ℓ much larger than thecoherence length of the electrode ξ and the coherence length of the weak link ξ ), theCPR transform from the sinusoidal one at temperature close to critical temperature ofelectrodes T c to the saw-toothed shape with the maximum at ϕ = π at T ≪ T c . Indirty S-S’-S junctions ( ℓ ≪ ξ ) the CPR with decreasing temperature can change fromthe sinusoidal one to the quite different multi-valued relation. In the latter case themaximum is attained at ϕ > π and two values of current correspond to a fixed value of ϕ . Similar multi-valued CPR is typical for the weak link in the form of superconductingbridge whose length is much larger than the superconducting coherence length ξ ( T ).In the case of short bridges (whose length is smaller than the bridge coherence length ξ ) the CPR remains single-valued at all temperatures but it is sinusoidal one only attemperature close to critical temperature and in the case of sufficiently small ratio ξ /ξ .For technological applications important characteristic is the characteristic voltage V c = I c R n where R n is the normal-state resistance of the junction. On the one side, tohave large V c one needs S-N-S or S-S’-S junction with high-resistive N or S’ layer. On theother side, in these junctions IVC becomes hysteretic below certain temperature whichis associated with Joule heating in the weak link ( ∼ I c V c ∼ I c R n ) and the formation at I > I c of the stable region of suppressed superconductivity (so called ’hot spot’) [3,5–7].Therefore, the eliminating of the thermal hysteresis without sacrificing the voltage V c is important and nontrivial problem. One solution is a normal metal shunt either ontop of the junction [10] or in parallel to it [11]. However, in this case the resistanceand the position of the shunt play important role and they can lead to reduction of thejunction characteristics because of the proximity effect or very small shunt resistance.In the work [12] it was proposed to use as the Josephson junction the variable thicknessSN-N-SN bilayer where the superconducting layer was partially (or entirely) etched by afocused ion beam. Sufficiently thick normal-metal layer act as a heat sink which providesnonhysteretic current-voltage characteristic even at low temperatures. But the increase osephson junction based on superconductor/low-resistive normal metal bilayer R n and, hence, smaller V c .In our work we calculate current phase relation for recently proposed variablethickness SN-S-SN Josephson junction based on thin dirty superconductor with largenormal state resistivity ρ S & µ Ω · cm and thin normal metal layer with low ρ N & µ Ω · cm [13]. In [14] it has been demonstrated theoretically and experimentallythat in such a bilayer the superconducting current mainly flows in N layer (due toproximity induced superconductivity and ρ S /ρ N ≫
1) and the critical current of SNbilayer may exceed the critical current of single S layer if thicknesses of S and N layersare about of superconducting coherence length. Below we show that in comparison withSN-N-SN junction the critical current density could be about of depairing current densityof S layer, which makes it possible to have I c R n ∼ ∆(0) / | e | . Due to large diffusioncoefficient D N and small minigap in N layer the heat could be effectively removed fromthe junction area and current-voltage characteristic could be not hysteretic. Besides,because of D N ≫ D S current-phase relation could be single-valued at all temperatuesand close to sinusoidal one at temperature near the critical temperature of bilayer.
2. Model
The model system consists of SN bilayer strip with length L made ofsuperconducting film with thickness d S and normal-metal film with thickness d N . Atthe center of bilayer there is a constriction with length a and thickness d c where N layerand partially S layer are removed (see figure 1). We assume that in our system thecurrent flows in the x direction and in the y direction the system is uniform. To find thecurrent-phase relation of such SN-S-SN Josephson junction at all temperatures below T c we solve two-dimensional Usadel equation for quasiclassical normal g and anomalous f Green functions. With the angle parametrization g = cos Θ and f = sin Θ exp( iφ ) thisequation in different layers can be written as SN z x La d N d S d c I y Figure 1: Sketch of SN-S-SN Josephson junction based on variable thickness SN strip. ~ D S (cid:18) ∂ Θ S ∂x + ∂ Θ S ∂z (cid:19) − (cid:18) ~ ω n + ~ D S q cos Θ S (cid:19) sin Θ S +∆ cos Θ S = 0 , (1) osephson junction based on superconductor/low-resistive normal metal bilayer ~ D N (cid:18) ∂ Θ N ∂x + ∂ Θ N ∂z (cid:19) − (cid:18) ~ ω n + ~ D N q cos Θ N (cid:19) sin Θ N = 0 , (2)where subscripts S and N refer to superconducting and normal layer, respectively.Here ~ ω n = πk B T (2 n + 1) are the Matsubara frequencies ( n is an integer number), q = ∇ φ = ( q x , q z ) is the quantity that is proportional to supervelocity v s , φ is the phaseof superconducting order parameter. ∆ is the magnitude of order parameter whichshould satisfy to the self-consistency equation∆ ln (cid:18) TT c (cid:19) = 2 πk B T X ω n > (cid:18) sin Θ S − ∆ ~ ω n (cid:19) , (3)where T c is the critical temperature of the single S layer. We assume that ∆ is nonzeroonly in the S layer because of absence of attractive phonon mediated electron-electroncoupling in the N layer. Equations (1),(2) are supplemented by the Kupriyanov-Lukichevboundary conditions [15] between layers D S d Θ S dz (cid:12)(cid:12)(cid:12)(cid:12) z = d S − = D N d Θ N dz (cid:12)(cid:12)(cid:12)(cid:12) z = d S +0 . (4)In the model we assume transparent interface between N and S layers which leadsto continuity of Θ on the NS boundary. At boundaries of the system with the vacuumwe use d Θ /dn = 0.To find the phase distribution φ the equations (1) – (3) are supplemented by two-dimensional equationdiv j s = 0 , (5)where j s is the superconducting current density, which is determined by the followingexpression j s = 2 πk B Teρ q X ω n > sin Θ , (6)where ρ is the residual resistivity of the corresponding layer. At the SN-interface we usethe boundary condition similar to ((4)) and for the interfaces with the vacuum we use dφ/dn = 0. At the system ends the rigid boundary conditions are imposed φ (0 , z ) = − δφ/ , φ ( L, z ) = δφ/ , (7)where δφ is the fixed phase difference between the system ends. One should differs itwith the phase drop near the junction which we define as ϕ = δφ − kL, (8)where k = q x ( x = 0) is far from the constriction (in similar way ϕ is defined in [16, 17]).The value of k is found from self-consisting solution of (1) – (3),(5).In numerical calculations we use dimensionless units. The magnitude of the orderparameter is normalized by k B T c = ∆(0) / .
76, length is in units of ξ c = p ~ D S /k B T c ≃ . ξ (0) ( ξ (0) = p ~ D S / ∆(0) is the superconducting coherence length at T = 0) andcurrent is in units of depairing current I dep of superconductor at T = 0. osephson junction based on superconductor/low-resistive normal metal bilayer δφ . When the self-consistency is achieved (we stop calculations when maximalrelative change of ∆ between consequent iterations is less than 10 − ) the Green functionsare used to calculate j s and the supercurrent per unit of width I s I s = d S + d N Z j sx ( x = 0) dz. (9)We also compare calculated CPR with the current-phase relation for 1D S’-S-S’system with large ratio of diffusion coefficients D S ′ /D S ≫ a ). To calculate it we use 1D Usadel equation.
3. Current-phase relation of SN-S-SN Josephson junction
The dependence I s ( q ) in SN bilayer may have one or two maxima depending onvalue of d S (see figure (2)) or d N (see figure 3(a) in [14]). The maxima at small q is connected with suppression of proximity induced superconductivity in N layer at q > q c ∼ / √ D N while the second maxima at q = q c ∼ / √ D S ≫ q c comes fromsuppression of superconductivity in S layer when q > q c . Large difference in q c and q c leads to larger phase concentration in S constriction (see figure 1) in comparison withthe variable thickness strip (or Dayem bridge) made of the same material and havingthe similar geometrical parameters. Because of that for relatively thin S layers the CPRis single-valued (see figure 3 (a)) which is not easy to achieve for Dayem bridge [18].For relatively large d S there is noticeable contribution to total supercurrent from Slayer which means smaller current (phase) concentration in constriction like in ordinaryDayem bridge and CPR becomes multi-valued (see figure 3(a) for d S = 2 , ξ c ).In some respect studied Josephson junction resembles Josephson junction basedon S’-S-S’system composed of two superconductors S and S’ having D S ′ ≫ D S and thesame thicknesses d S = d S ′ [16,19,20]. Josephson junction based on this quasi 1D systemhas single-valued CPR which tends to the sinusoidal shape with increasing temperature.In figure 3 (b) we compare CPR calculated for 1D S’-S-S’ and 2D SN-S-SN systems.Since in 1D model there is no suppression of T c by N layer, in calculations we use ratio T /T c which corresponds to ratio T /T c of 2D SN structure. Visible differences betweenCPRs calculated using different models could be related with transversal inhomogeneitynear the S constriction in the 2D case.We have studied evolution of CPR of SN-S-SN Josephson junction by varyingdifferent parameters. In figure 4(a) we demonstrate that with increase of thetemperature the current phase relation becomes closer to sinusoidal one which is typicalfor S’-S-S’ junctions [20] and it is related with increase of the temperature-dependentcoherence length ξ ( T ). Effect of different d N is shown in figure 4(b). An increase in d N leads to slight shift of maximum of I s ( ϕ ) to the left and decrease of I c which areexplained by lowering of T c of SN bilayer for thicker N layers. Lower I c means smaller osephson junction based on superconductor/low-resistive normal metal bilayer q c2 d s =3 c d s =2 c d s =1.5 c d s = c single S-film I s / I d e p ( ) q c q c1 d N = c T=0.2T c0 S / N =150 Figure 2: Dependence of the superconducting current I s flowing along SN bilayer on q for different d S . Solid line shows the dependence I s on q for the single S strip. Dashedlines show the critical values of q . Current is normalized by the depairing current I dep of the single S strip with thickness d S at T = 0. I c R n but how we discuss below large d N provides better cooling of S constriction andnonhysteretic IV curves.An increase of the weak-link length a leads to the shift of the maximum of I s ( ϕ )to the right (see figure 4(c)) as it is typical for ordinary variable thickness Josephsonjunctions. Interestingly, that contrary to that junctions the I c increases in SN-S-SNsystem. This result is explained by lower value of superconducting order parameter inSN banks in comparison with ∆ in S constriction at I s = 0. With increasing a thesupercondcuting order parameter in constriction increases and I c increases too.And finally figure 4(c)) illustrates that three-fold decrease of ratio ρ S /ρ N does notchange current-phase relation drastically. Both the critical current and shape of CPRvary a little.
4. Effect of Joule heating in SN-S-SN junctions
The absence of hysteresis in current-voltage characteristic is important for devicesbased on Josephson junctions. The hysteresis in Dayem, variable thickness, S’-S-S’ orS-N-S junctions is mainly caused by the temperature rise in the weak-link region in the osephson junction based on superconductor/low-resistive normal metal bilayer (b) I s / I d e p ( ) d S =3 c d S =2 c d S =1.5 c d S = c a=0.5 c d N = c d c =0.5 c T=0.2T c0S / N =150 (a) a=0.5 cS / N = S / S’ =150 I s / I c Figure 3: (a) Current-phase relation of SN-S-SN Josephson junction at different d S .Current is normalized by the depairing current I dep of the single S strip with thickness d c at T = 0. The junction parameters are shown in the figure. (b) Comparison ofcurrent-phase relations calculated on the basis of 1D and 2D models. For 2D casethe parameters are following: d S = d N = ξ c , d c = 0 . ξ c , T = 0 . T c . In the 1Dcase temperature T = 0 . T c corresponds to T = 0 . T c , where T c = 0 . T c is criticaltemperature of SN bilayer with chosen parameters. The superconducting current isnormalized by critical current of Josephson junction.resistive state due to Joule heating and the formation of hot spot [3, 6, 7]. Local heatproduction should be large in SN-S-SN junction due to large critical current densitywhich is about of the depairing current density of the superconductor. But as we showbelow the presence of relatively thick N layer with large diffusion coefficient providesefficient cooling of constriction.To estimate the increase of temperature in the resistive state we use twotemperature (2T) model [21, 22] for SN-S-SN junction. We suppose that electron T e = T + δT e and phonon T p = T + δT p temperatures are near the substrate temperature δT e , δT e ≪ T and do not vary along the thickness. Because of inverse proximity effectthe gap in relatively thin S layer ( d S . . ξ c ) is suppressed in comparison with single Slayer, which permits heat diffusion from N to S layer in SN banks. In S constriction beingin the resistive state at I > I c the superconducting order parameter is also suppressed. osephson junction based on superconductor/low-resistive normal metal bilayer (c)(a) a =0.5 c d S =1.25 c d N =3 c d c =0.5 cS / N =150 T=0.1T c0 T=0.2T c0 T=0.3T c0 I s / I d e p ( ) (b) a =0.5 c d S =1.25 c d c =0.5 c T=0.3T c0S / N =150 I s / I d e p ( ) d N = c d N = c d N =3 c d s = c d n = c d c = c S / N =150 a= c ,T=0.1T c a= c ,T=0.1T c a= c ,T=0.2T c a= c ,T=0.2T c I s / I d e p ( ) (d) a=0.5 c d S = c d N = c d c =0.5 c T=0.2T c0 I s / I d e p ( ) S / N =150 S / N =50 Figure 4: Variation of current-phase relation of SN-S-SN junction with change of:(a) temperature; (b) thickness of N layer d N ; (c) length of constriction a ; (d) ratio ofresistivities. Current is normalized by the depairing current I dep of the superconductingstrip with thickness d c at T = 0.It allows us to use normal state heat conductivity both in SN and S regions in heatconductance equation for calculation of δT e . In our model Joule dissipation is takeninto account only in S constriction, because in SN bilayer it is considerably lower due tomuch lower resistivity and lower current density. Because of small length of constrictionand large difference in diffusion coefficients and thicknesses in constriction and bankswe can neglect heat flow to the phonons and substrate in constriction (main cooling ofjunction comes from diffusion of hot electrons to SN banks). In SN bilayer D N ≫ D S andheat diffusion occurs mainly along N layer. With above assumptions we have followingequation for δT e d δT e dx + ρ S ( j c ) /κ S = 0 , | x | ≤ a/ , (10) d δT e dx − δT e λ T = 0 , | x | ≥ a/ , where κ S = 2 D S N (0) k B T / osephson junction based on superconductor/low-resistive normal metal bilayer N (0) is the one spin density of states on the Fermi level, λ T = p D N τ (cid:18) T c T (cid:19) / s π (1 + β )720 ζ (5) (11)is the healing length, β = [ γτ esc ζ (5) T / [ τ π T c ], ζ (5) ≃ . τ esc is the escape timeof nonequilibrium phonons to substrate, γ = 8 π C e ( T c ) /C p ( T c ) is the ratio of electronand phonon heat capacities at T = T c and τ determines the strength of electron-phonon inelastic scattering in S and N layers (see equations (4,6) in [22]). For τ weuse the smallest time for S and N materials due to assumed good transfer of electronsbetween S and N layers and their small thickness. On the boundary between S and SNregions we use continuity of the electron temperature ( δT e | a/ − = δT e | a/ ) and heatflux ( d c D S δT e dx | a/ − = d N D N δT e dx | a/ ).Using (10) and above boundary conditions we find maximal temperature increasein the constriction δT maxe T = 0 . (cid:18) aξ c (cid:19) (cid:18) T c T (cid:19) (cid:18) I c I dep (0) (cid:19) (cid:18) D S d c D N d N λ T a + 1 (cid:19) (12)In following estimations we use parameters of NbN (S layer) and Cu (N layer): T c = 10 K, D S = 0 . /s, ρ S = 200 µ Ω · cm, D N = 40 cm /s, ρ N = 2 µ Ω · cm, τ = 1ns (theoretical estimation for NbN is taken from [22]), ξ c = 6 . γ = 9, d S = 1 . ξ c , d N = 2 ξ c , τ esc = 4( d N + d S ) /u ≃
41 ps ( u = 2 · cm /s is a mean speed of sound), T /T c = 0 . T c /T c = 0 . a = 0 . ξ c , d c = 0 . ξ c . With these parameters β ≃ . I c ≃ . I dep (0) (see figure 4(b)) and δT maxe /T ∼ .
24 is small, thanks to D N ≫ D S and d N ≫ d c .
5. Discussion
We use Usadel model to calculate current-phase relation of SN-S-SN Josephsonjunction based on high-resistive superconductor and low-resistive normal metal. In[14] from comparison of the experiment and theory it was concluded that Usadelmodel underestimates proximity induced superconductivity in N layer and overestimatesinverse proximity effect in S layer in NbN/Al, NbN/Ag and MoN/Ag bilayers. Namely,the suppression of critical temperature of SN bilayer is smaller while change in magneticfield penetration depth of SN bilayer is larger than Usadel model predicts. Therefore,present results should be considered only as a route for possible experimental realizationof SN-S-SN Josephson junction. They demonstrate that the thickness of S layer shouldnot exceed ∼ . ξ c , otherwise current-phase relation is not single-valued for reasonablelength and thickness of S constriction. The thickness of N layer should not be too small(small d N leads to large overheating) and not too large (the larger d N leads to lower T c and smaller I c at fixed substrate temperature). osephson junction based on superconductor/low-resistive normal metal bilayer V c = I c R n = ∆(0) | e | aξ c I c I dep (0) , (13)can reach 0 . / | e | at low temperature ( T = 0 . T c ) and a = ξ c (see figure 4(c))due to use of superconductor in constriction area, instead of normal metal as in [12].In case of NbN with T c = 10 K one may have V c = 0 .
75 mV but according to (12) δT maxe will be larger than T at these parameters. However there is a hope, that criticaltemperature of real SN bilayer is higher than Usadel model predicts (see discussionabove) and therefore large I c could be reached at higher operating temperature T /T c ,leading to drastic reduction of δT maxe (see (12)).The SN-S-SN junctions made of NbN/Al bilayer have been fabricated recently [13]and indications of Josephson effect (the presence of Shapiro steps and Fraunhofer likedependence of critical current on the magnetic field) have been observed. But due tonot optimized parameters ( d S = d c ∼
15 nm ∼ . ξ c , d N ∼
29 nm ∼ . ξ c , a = 20nm ∼ . ξ c ) the IV curves were hysteretic already at temperature close to criticalone and width of Shapiro steps did not follow the theoretical expectations [13]. Moderntechnology allows to make constriction with length about 5 nm with help of helium beam,which is smaller than ξ c in NbN. Successful implementation of this method could lead tocreation of low temperature nano-scale Josephson junction or their arrays. For exampleSN-S-SN junctions can be promising to use in programmable voltage standards [1] wherelarge value of V c allows to reduce the number of junctions and to use Shapiro steps oforder higher than one. Nonhysteretic current-voltage characteristics with large V c at lowtemperatures enables to use these structures for various low-temperature applications,e.g., particle detectors [4].
6. Conclusion
In conclusion, we have calculated current phase relation of Josephson junction basedon variable thickness SN-S-SN strip, where S is dirty superconductor with large normalstate resistivity and N is low resistive normal metal. We find the range of parameterswhen CPR is single-valued, close to sinusoidal one and product I c R n . ∆(0) / | e | . Ourestimations demonstrate that relatively thick N layer serves as effective heat-conductorproviding weak overheating and nonhysteretic current-voltage characteristic of SN-S-SNJosephson junction. Acknowledgments
P. M. M. acknowledges support from Russian Scientific Foundation (project No.20-42-04415) and D. Yu. V. acknowledges support from the Foundation for theAdvancement of Theoretical Physics and Mathematics BASIS (program No. 18-1-2-64-2). osephson junction based on superconductor/low-resistive normal metal bilayer References [1] Benz S P 1995
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