Maintaining the local temperature below the critical value in thermally out of equilibrium superconducting wires
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J u l Maintaining the local temperature below the critical value in thermally out ofequilibrium superconducting wires
Y. Dubi and M. Di Ventra
Department of Physics, University of California San Diego, La Jolla, California 92093-0319, USA
A generalized theory of open quantum systems combined with mean-field theory is used to studya superconducting wire in contact with thermal baths at different temperatures. It is shown that,depending on the temperature of the colder bath, the temperature of the hotter bath can greatlyexceed the equilibrium critical temperature, and still the local temperature in the wire is maintainedbelow the critical temperature and hence the wire remains in the superconducting state. The effectsof contact areas and disorder are studied. Finally, an experimental setup is suggested to test ourpredictions.
PACS numbers: 72.15.Jf,73.63.Rt,65.80.+n
Ever since the discovery of superconductivity, fabricat-ing a superconducting (SC) wire that conducts electricitywithout dissipation at room temperature has been a ma-jor goal of modern condensed matter physics. However,since the discovery of High- T c superconductors [1] (whichhave T c as high as ∼ T c . Recently, several suggestions have been put forwardto increase T c by fabricating nanoclusters [3, 4] or layer-ing of the SC material [5–7].In this paper we consider a different route which mayallow a substantial increase in T c for small SC wires,based on local cooling. To this aim we study a SC wirein contact with two different heat baths, held at differenttemperatures. We consider at first a wire in contact withtwo heat baths at its edges, held at different temperatures T L (left bath) and T R (right bath), with T R > T eqc > T L ,where T eqc is the equilibrium critical temperature (up-per panel of Fig. 1(a)). We find that depending on thevalue of T L , the critical value T R,c (defined as the max-imum value of the temperature T R of the hot bath atwhich the wire is still SC) can be much larger than T eqc ,which implies that the local temperature in the wire ismaintained below T eqc although the average of T L and T R exceeds it. We then study the effect of different cou-plings to the baths and of disorder on the above result.We find the dependence of T R,c on the coupling to thedifferent baths, and demonstrate that, in agreement withAnderson’s theorem [8], weak disorder does not change T R,c by much. Strong disorder leads to the breakdown ofthe SC state near the right (hot) bath, and to a spatialdependence of the SC order parameter.Normal metallic wires in contact with two heat bathsat the edges were recently studied in detail, in the contextof heat flow in such systems [9–11]. One of the main con-clusions of Ref. [10] was that in a system which is not dif-fusive, one cannot define a non-equilibrium temperatureas the average of the temperatures of the different baths.Rather, it is the energy distribution function (DF) of the two baths which are averaged (a similar observation wasverified experimentally for short wires, when interactionsare relatively unimportant [12], a situation which is alsolikely to hold for the quasi-particles in SC wires). Thisobservation will allow us to provide an analytic expres-sion for the critical temperature which shows excellentfit with our numerical calculations, and provides a directprediction which may be tested experimentally.The method we use is a generalization of theBogoliubov-De Gennes (BdG) mean-field theory [13]to non-equilibrium. The starting point is the tight-binding BdG Hamiltonian on a square lattice (withlattice constant a =1), H BdG = P i,σ ( ǫ i − µ ) c † iσ c iσ − t P h i,j i ,σ c † iσ c jσ + P i (cid:16) ∆ i c † i ↑ c † i ↓ + h.c. (cid:17) , where c † i,σ cre-ates an electron in the i -th lattice site with spin σ , t is the hopping integral ( t =1 serves as the energy scalehereafter), ǫ i are random on-site energies drawn froma uniform distribution U [ − W/ , W/
2] (hence W is thestrength of disorder, with W = 0 representing a cleansystem), µ is the chemical potential and ∆ i is the SC or-der parameter in the i -th lattice site. The order parame-ter is to be determined self-consistently on every site via∆ i = − U h c i ↓ c i ↑ i , where U > h·i stands for a statis-tical average. In this paper we treat only s-wave super-conductors, but the formalism can easily be extended toaccount for other kinds of symmetry.Since the BdG Hamiltonian is quadratic, it can beexactly diagonalized to describe the quasi-particle ex-citations γ nσ . To diagonalize it, one performs a Bo-goliubov transformation [13] for the electron operators, c † iσ = P n (cid:0) u n ( i ) γ † nσ + σv ∗ n ( i ) γ n ¯ σ (cid:1) , where u n ( i ) and v n ( i )are quasi-particle and quasi-hole wave-functions, respec-tively. Requiring that this transformation diagonalizesthe Hamiltonian yields a set of eigenvalue equations forthe quasiparticle (QP) wave functions [13], (cid:18) ˆ ξ ˆ∆ˆ∆ ∗ − ˆ ξ ∗ (cid:19) (cid:18) u n ( i ) v n ( i ) (cid:19) = E n (cid:18) u n ( i ) v n ( i ) (cid:19) , (1)where ˆ ξu n ( i ) = − t P h i,j, i u n ( j ) + ( ǫ i − µ + U n i / u n ( i )(here n i is the local electron density), and ˆ∆ u n ( i ) =∆ i u n ( i ). From Eq. (1) one obtains the QP wave func-tions u n ( i ) and v n ( i ) and the energies E n . In equilibrium,this procedure results in a closed self-consistent set ofequations for the local SC order parameter and density,which were recently used to study, e.g., effects of disorderand magnetic fields in two dimensional superconductors[14–16].Since the QP excitations are non-interacting Fermions,they can be treated by the formalism of Refs. [9–11],which is aimed at studying such particles out of equilib-rium. To generalize the method of Refs. [9–11] to theQP excitations, we define a single-particle density ma-trix ˆ ρ , with matrix elements ρ nn ′ = h γ † n ↑ γ n ′ ↑ i (note thatthe particle-hole symmetry allows one to treat only theup-spin excitations). The master equation for ˆ ρ is ofthe Lindblad form [17] (setting ~ = k B = 1 hereafter)˙ ρ = − i [ H , ρ ] + L L [ ρ ] + L R [ ρ ] , where H is the diagonalmatrix of energies E n and L (L , R) [ ρ ] = X nn ′ (cid:16) − n V (L , R) † nn ′ V (L , R) nn ′ , ρ o + V (L , R) nn ′ ρV (L , R) † nn ′ (cid:17) (2)describe environment-induced inelastic transitions be-tween different single-particle states. The V-operatorsin Eq. (2) take on a local form [10] V (L , R) nn ′ = q Γ (L , R) nn ′ f (L , R) D ( E n ) γ † n ↑ γ n ′ ↑ Γ (L , R) nn ′ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ X r i ∈ S L , R ( u n ( r i ) u ∗ n ′ ( r i ) + v n ( r i ) v ∗ n ′ ( r i )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3)where S L , R are the contact area of the left(right) heat bath with the sample, f (L , R) D ( E n ) =1 / (1 + exp( E n /T L , R )) is the Fermi distribution and Γ is some constant scattering rate. (We take Γ =0.1,changing Γ does not alter the results presented above.)Note that the V -operators operate on the QPs andnot on the electron operators, since the Fermi functionis defined for the occupation of the QPs (to put itdifferently, the part of the electrons which is in thesuperfluid phase does not feel scattering from the baths,only the QPs do).Once the V-operators are evaluated, the master equa-tion is solved in the asymptotic time limit (i.e., ˙ ρ = 0)and a solution for ρ is obtained. From this solution, thelocal density and order parameter are evaluated via theself-consistency condition, which reads∆ i = U X n u n ( i ) v ∗ n ( i )(1 − ρ nn ) n i = 2 X n (cid:0) | u n ( i ) | ρ nn + | v n ( i ) | (1 − ρ nn ) (cid:1) . (4)With ∆ i and n i determined, the whole procedure (i.e.,the solution of the BdG equations, the evaluation of the V-operators and the solution of the master equation) isrepeated until ∆ i and n i no longer change (within thenumerical tolerance of < − ). We point that this treat-ment is of a mean-field type, and as such neglects statis-tical fluctuations in the Hamiltonian [18] or phase-slips[19]. As we will show below, in the ideal case one candefine an effective temperature of the wire, and hence insuch a case one can follow the usual treatment of phase-slips in SC wires, but using the effective temperatureas input. Phase fluctuations are unlikely to change thelocal effective temperature, since these are excitationsthat do not carry heat (as opposed to QP excitations).In the disordered case, such a treatment cannot work asthe temperature becomes space-dependent and one needsa different formalism to treat phase-fluctuations in thepresence of a temperature gradient. Such a theory is be-yond the scope of the present work and will be the subjectof future studies.We begin by presenting the averaged order parameter¯∆ = N P i ∆ i , where N is the total number of latticesites. We consider the geometry shown in the side panelof Fig. 1(a), where the SC wire (gray area) is connectedto the thermal baths only at its edges. The numericalparameters are as follows. The wire dimensions are 100 × , U = 2 , W = 0 (clean system) and the density is heldat n = 0 .
875 (i.e., n electrons per site on average, and thechemical potential is chosen self-consistently to maintainthis filling). We define T L by its ratio with the criticaltemperature at equilibrium T eqc , T L /T eqc = γ < T R for differ-ent values of γ , γ = 0 .
05, (top curve), 0 . , ..., γ , T R,c can greatly exceed T eqc (marked by a solid arrow). In the inset of Fig. 1(a) weplot the local order parameter as a function of positionalong the wire (averaged over the transverse direction),at γ = 0 .
05 for different values of T R = 0 . , . , . .
185 (in units of t ). We find that although there aretwo different temperatures at the edges, the order param-eter is practically uniform along the wire, in agreementwith the results of Refs. [10, 11].In Fig. 1(b) we study a somewhat different (and per-haps more realistic) situation, in which the heat bathsare in contact with the wire not only at the edges butover some area (see side panel of Fig. 1(b)). We definethe parameter α (1 − α ) to be the ratio between the con-tact area of the right (left) heat bath and the area of thewhole wire, such that α = 1 stands for a system in fullcontact only with the right heat bath. In Fig. 1(b), ¯∆is plotted as a function of T R (at γ = 0 . α , α = 0 .
1, (top curve), 0 . , ..., α = 1 curve is the equilibrium curve (with T eqc marked by an arrow). As seen, for different valuesof α , T R,c may again exceed T eqc . Also in this case wefound that both ∆ and the local temperature (calculatedfor a similar one-dimensional geometry, with the methodof Refs. [10, 11]) are uniform in space (not shown). Thisemphasizes the fact that the temperature is defined notby the local baths, but rather by the (inelastic) scatteringbetween states, which in the clean case span the entiresystem. D D position g=0.05g=1 T R a=0.1a=0.2a=1 a=0.3 T R T L T c (eq) T R T R T L T c (eq) T = R T = R T = R T = R (a) (b) FIG. 1: (color online) (a) Main panel: ¯∆ as a function of T R for different values of γ . At small values of γ , T R cangreatly exceed the equilibrium critical temperature (indicatedby a solid arrow) without destroying superconductivity in thewire. Inset: position dependence of the order parameter fordifferent values of T R (at γ = 0 . γ = 0 .
083 and different values of α (describing thecontact area of the right heat bath with the sample). Upperpanel: the geometries considered in (a) and (b). If one could simply define a local temperature whichgradually shifts from T L to T R , then one would expectthat T R could not exceed T eqc and that the order pa-rameter would not be uniform in space. In order to ex-plain our findings, we recall that one of the main resultsof Refs. [10, 11] is that in the ballistic limit the tem-perature is uniform, and a non-equilibrium DF develops,which is the average of the two DFs of the left and rightbaths. In the case represented in Fig. 1(b), we find thata weighted average between the DFs of the left and rightbaths develops, the weight being α . This result, alongwith the observation that ∆ i is uniform in space, allowsus to find an analytical expression for the effective T c asfollows. In the equilibrium theory of superconductivity[13], the critical temperature T eqc is determined by thegap equation N U = R ω D /T eqc x (1 − f ( x )) dx , where N is the density of states at the Fermi energy, ω D is the Debye frequency and f ( x ) is the DF. From theabove discussion, in the non-equilibrium case we have f ( x ) = αf (R) D ( x ) + (1 − α ) f (L) D ( x ). The resulting equa-tion for T R,c then reads1 N U = α Z ω D /T R,c x (1 − f ( x )) dx ++ 1 − α Z ω D /T L x (1 − f ( x )) dx . (5)These integrals may be evaluated exactly, and with T L /T eqc = γ we find T R,c T eqc = γ − /α . (6)In Fig. 2 we plot T R,c /T eqc as a function of γ (Fig. 2(a))and of α (Fig. 2(b)), taken from the data of Fig. 1(a)and(b), respectively. The solid line corresponds toEq. (6) for the two cases, with the corresponding param-eters taken from the numerical calculation. The agree-ment between Eq. (6) and the numerical results confirmsthat indeed the DF is a weighted average of the DFs ofthe two baths. One can now use Eq. (6) to estimate theeffective T R,c of real materials. As a practical use, it isadvantageous to raise T R,c above the freezing point ofliquid Nitrogen, ∼ K . For example, consider a desiredworking temperature of ∼ K , deposited on a SC wire with T eqc ∼ T R,c to = 78K.Perhaps a more intriguing possibility is the enhance-ment of T R,c to room temperature. For a wire madeof the newly-found Iron compound [20–22] ( T eqc ∼ α = 0 . T L = 7 .
5K would drive T R,c above room temperature.For nano-scale wires made of a high- T c material, the fab-rication of which was recently demonstrated [23], taking T eqc ∼ T L = 20K and coverage of α = 0 . T R,c above room temperature. Ofcourse, other effects (e.g. phonon scattering, phase fluc-tuations etc.) might become very important in such highvalue of T R and inhibit the zero-resistance state. T R,c T c(eq) a g , g=0.083 T R,c T c(eq) g g , a=1/2 numerics numerics (a) (b) FIG. 2: (a) The ratio T R,c /T eqc as a function of γ , taken fromthe data of Fig. 1(a). The points correspond to the numericaldata and the solid line is Eq. (6) with α = 0 .
5. (b) T R,c /T eqc as a function of α , taken from the data of Fig. 1(b). Thepoints correspond to the numerical data and the solid line isEq. (6) with γ = 0 . Next we turn to study the effect of disorder. InRef. [11] it was shown that the form of the non-equilibrium DF is robust against disorder, but thatthe local temperature profile changes from a constant-temperature to a position-dependent profile. In Fig. 3we plot the order parameter ∆ i as a function of positionalong the wire for T R = 1 . T eqc ∼ . T L = 0 .
02 and dimensions 50 ×
10, fordifferent values of disorder, W = 0 (dark curve), 0 . , ..., T L ) is lower thanthe effective temperature of the clean sample, and hencethe rise in the value of ∆ with increasing disorder nearthe left edge. The distance from the right edge at which∆ vanishes indicates the ”thermal length” where the lo-cal temperature is close to T R [11]. We also note thatalready for relatively small values of disorder ( W ≈ . W ≈ .
4) does not di-rectly correspond to the onset of localization [11, 24]. position D W=4 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
HeaterInsulating layer xxxxxxxx contacts xxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx tunnel junction xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
S C w ire
W=0
FIG. 3: Position dependence of the order parameter ∆, cal-culated with increasing values of disorder, from the clean case W = 0 (dark line) to strongly disordered case W = 4 (lightgray line), averaged over 500 realizations of disorder. For thisexample T L = 0 . T eqc = 0 . T R = 1 .
5. Inset: Sug-gested experimental setup. An insulating layer is depositedon top of a SC wire with etched contacts, and a heater coilis placed on top of it, to generate local heating. The resis-tance is then monitored along the other contacts (which maybe either 4-terminal contacts or tunnel junctions).
In order to test our predictions, we suggest the experi-mental setup shown in the inset of Fig. 3. It consists of aSC wire with etched contacts (either regular 4-terminalcontacts, or made as tunnel junction, aimed at measuringthe local density of states at that location). On top ofone of the contacts an insulating layer is deposited, andon top of it a heater coil is set. The temperature belowthe heater can be calibrated by measuring the resistance between the contacts beneath the heater when the wireis in the normal state. Then the whole device is cooleddown, and the resistance through the other contacts ismeasured as a function of the current that passes throughthe heater (i.e., the local temperature beneath it). In auniform system, the resistances of all the contacts shouldvanish if the system is SC, and have finite values once thelocal temperature beneath the heater rises above T R,c . Ifthe system is disordered, the different contacts shouldexhibit a gradual transition to a normal state.The length-scale that determines the onset of a tem-perature gradient is in this case the QP mean-free path(or diffusion length)[24]. While it can be estimated forlow T c metals, for high- T c materials it is unknown (al-though it is suspected to be small, based on their poorconduction in the normal state). By controlling the dis-tances between the contacts our proposed experiment canthus serve to determine this length, by relating it to thelength-scale at which a temperature gradient develops.In this work we have neglected the effect of phononscattering, and assumed the effective electron-electroninteraction is not appreciably changed by temperature,supported by the fact that the Debye temperature ismuch larger than T c . However, considering our geometry,electron-phonon (e-ph) interaction effects may play a sig-nificant role as T R reaches the Debye temperature. Thisis probably more important in disordered wires (for cleanwires we expect that, on equal grounds, phonons willalso acquire a uniform temperature). The e-ph interac-tion may induce inelastic electron transitions, which willreduce the inelastic mean-free path. Since the temper-ature profile is sensitive to the inelastic mean-free path[24], this effect may be seen in our suggested experiment.To conclude, we point out that in recent years therehave been tremendous advances in fabricating micro-refrigerators, based on the thermo-electric Peltier effect,and which can locally cool down their environment sub-stantially [25, 26]. Since the efficiency of thermo-electricmaterials is likely to increase in the future [27], one canconceive a device composed of a (relatively) high- T c ma-terial, on top of which are embedded a series of micro-refrigerators, covering an area of the material and cool-ing it enough for it to operate at a temperature whichexceeds its T c , a possibility that may allow for integrat-ing superconducting wires as circuit elements in variousdevices.We are grateful to A. Sharoni for valuable discussions.This research was funded by DOE under grant DE-FG02-05ER46204. [1] A. Bednorz and K. A. Mueller, Z. Phys. B. , 189(1986).[2] V. Z. Kresin, S. A. Wolf,and Yu. N. Ovchinnikov, Phys. Rep. , 347 (1997).[3] V. Z. Kresin and Yu. N. Ovchinnikov, Phys. Rev. B ,024514 (2006).[4] B. Cao et al. , J. Supercond. Nov. Mag. , 163 (2008).[5] O. Yuli, I. Asulin, O. Millo and D. Orgad, Phys. Rev.Lett. , 057005 (2008).[6] E. Berg, D. Orgad and S. A. Kivelson, Phys. Rev. B ,094509 (2008),[7] S. Okamoto and T. A. Maier, Phys. Rev. Lett. ,156401 (2008).[8] P. W. Anderson, J. Phys. Chem. Solids , 26 (1959).[9] Yu. V. Pershin, Y. Dubi, and M. Di Ventra, Phys. Rev.B , 054302 (2008).[10] Y. Dubi, and M. Di Ventra, Nano Letters , 97 (2009).[11] Y. Dubi and M. Di Ventra Phys. Rev. B , 115415(2009).[12] H. Pothier et al. , Phys. Rev. Lett. , 3490 (1997).[13] P. G. De Gennes, Superconductivity in Metals and Alloys (Benjamin, New York, 1966).[14] A. Ghosal, M. Randeria and N. Trivedi, Phys. Rev. Lett. , 3940 (1998). [15] Y. Dubi, Y. Meir and Y. Avishai, Nature , 876 (2007).[16] Y. Dubi, Y. Meir and Y. Avishai, Phys. Rev. B ,024502 (2008).[17] G. Lindblad, Commun. Math. Phys. , 119 (1976).[18] R. D’Agosta and M. Di Ventra, Phys. Rev. B , 165105(2008).[19] see, e.g., A. Zharov et al. , Phys. Rev. Lett. , 197005(2007); D. S. Golubev and A. D. Zaikin, Phys. Rev. B bf78, 144502 (2008) and references therein.[20] Y. Kamihara et al. , J. Am. Chem. Soc. , 10012(2006).[21] H. Takahashi et al. , Nature , 376 (2008).[22] P. M. Grant, Nature , 1000 (2008).[23] K. Xu, and J. R. Heath, Nano Letters , 3845 (2008).[24] Y. Dubi and M. Di Ventra, Phys. Rev. E , 1613 (2006).[26] I. Chowdhury et al. , Nature Nanotechnology , 235(2008).[27] A. Majumdar, Science303