Boltzmann equation for fluctuation Cooper pairs in Lawrence-Doniach model. Possible out-of-plane negative differential conductivity Todor M. Mishonov∗ and Yana G. Maneva† Department of Theoretical Physics, Faculty of Physics, University of Sofia St. Kliment Ohridski, 7 5 J. Bourchier Boulevard, BG-1164 Sofia, Bulgaria 0 (Dated: February 5, 2008) 0 2 The differential conductivity for the out-of-plane transport in layered cuprates is calculated for n Lawrence-Doniach model in the framework of time-dependent Ginzburg-Landau (TDGL) theory. a TheTDGLequationforthesuperconductingorderparameterissolved inthepresenceofLangevin J externalnoise,describingthebirthoffluctuationCooperpairs. TheTDGLcorrelator ofthesuper- 3 conducting order parameter is calculated in momentum representation and it is shown that the so 2 definednumberofparticlesobeystheBoltzmannequation. Thefluctuationconductivityisgivenby an integral over theJosephson phase θ of theparticles distribution, depending on that phase, n(θ) ] and their velocity v(θ). It is demonstrated that in case of overcooling under Tc thetransition from n normal to superconducting phase while reducing the external electric field runs via annulation of o thedifferential conductivity. The presented results can be used for analysis and experimental data c processing of measurements of differential conductivity and fluctuation current in strong electric - r andmagneticfields. Thepossibleusageofanegativedifferentialconductivity(NDC)forgeneration p of THz oscillations is shortly discussed. u s PACSnumbers: 74.40.+k,74.20.De,74.25.Fy,74.72.-h . t a m I. INTRODUCTION insuperconductorsandthereviewarticle(andreferences - therein)2whichisespeciallydevotedtoGaussianfluctua- d n tions in layeredsuperconductors. In addition, the recent o Fluctuation phenomena in high-temperature cuprates experimentalinvestigationsoncut-offeffects3 andreduc- c (CuO2) are greatly pronounced because of the short tionofparaconductivitywithincreaseoftheelectricfield [ lengthofcoherence. Thenumberofdegreesoffreedomin in cuprates4 are also recommended. thesample’svolumeis /ξ2 (0)ξ (0).Thefluctuationsin 2 V ab c Thepurposeofthecurrentworkistodemonstratethat v strongly anisotropic superconductors are approximately negative differential conductivity (NDC) can be reached 4 two-dimensionalinlayersatadistancesfromeachother inthespecificnonequilibriumconditionsofasupercooled 9 and the number of fluctuation modes is /ξ2 (0)s. A re- 4 duction of the dimension, as a rule, leaVdsatbo enhance- belowTc superconductorinnormalstateinexternalelec- 4 ment of the fluctuation effects. The paraconductivity of tric field Ec. This NDC can find possible applications in 0 the usage of layered cuprates as active media in THz Aslamazov-Larkin(AL), when to the normal conductiv- 6 generators. We derived explicit formulas for differential ity we have to add the contribution of the fluctuation 0 conductivityinoutofplanedirectionandforthecurrent / Cooper pairs is the most thoroughly investigated fluc- at tuation phenomenon. The AL conductivity is easier to incaseofexternalmagneticfieldBz.Inthespecialcaseof evanescentmagneticfieldourresultsagreewiththework m be observed if the normal phase conductivity is small. by Puica and Lang.5 Our TDGL result for the momen- Thatisthecaseofastronglydisorderedconventionalsu- - tumdistributionofthefluctuationCooperpairsobeythe d perconductor or high-temperature cuprates, containing n CuO planesasamainstructuraldetail. Theamplitudes Boltzmann equation. The new results for the current jz o 2 canbe usedforthe experimentaldata processingandin- of the electron hopping between CuO layers are small, c 2 terpretation of the experimental results for out-of-plane which leads to intensive diminution of the conductivity : v in the so called dielectric z-direction, perpendicular to fluctuation conductivity for samples in the geometry of i plane capacitor with a layered superconductor between X the CuO2 layers. As a consequenceone could apply con- theplates. Specialattentionispaidtotheanalysisoffor- siderable electric fields that will not result in significant r mulas for the differential conductivity in different phys- a heatingofthesample. Thesmallheatinggivestheoppor- ical conditions. How to reach easily the regime of high tunitytoinvestigatehowthestrongelectricfield’snonlin- frequency oscillations is briefly considered. ear effects influence the fluctuation conductivity σ. Such investigations could reveal new details in the kinetics of Forthis sakethe time dependence ofthe orderparam- the superconducting order parameter. They could spec- eteris estimatedunderLangevinapproachandits corre- ify the parameters of time-dependent Ginzburg-Landau latorincoincidingtimeargumentsiscalculated. Wehave (TDGL) theory which describes, with acceptable accu- demonstratedthatthiscorrelatorincaseofhomogeneous racy, the fluctuation phenomena in almost all supercon- electric field obeys the master equation. In this way ductors. We recommend the monograph by Larkin and BoltzmannequationforthefluctuationCooperpairshas Varlamov1forageneralreviewonfluctuationphenomena been derived, which solution gives the momentum dis- 2 tribution of the superconducting order parameter. The applied to an uniaxial crystal (m = m m ). Con- a b c ≪ electriccurrentisrepresentedasanintegraloverthisdis- versely, when the coherence length in perpendicular di- tribution and the velocity. The derived formula for the rection ξ (0) is much smaller than the distance between c current could be generalized for a self-consistent treat- thelayerswehaveasystemoftwo-dimensionalsupercon- ment of the interactionbetween the fluctuations. Such a ductors with a Josephson coupling. possibility has been discussed in short. We have shown The gauge invariant Josephson’s phase that when T < T the phase transition to the supercon- c p s P e∗A (t) ducting state passes through the critical point of zero z z z θ(t)= = − ( π,+π) (5) differentialconductivity σ =0.This qualitative result ~ ~/s ∈ − diff is one of the important consequences predicted by the actuallyisthekineticmomentumoftheCooperpairmea- theory which can be described by the deduced formula sured in units, connected to the lattice constant in z di- forthe currentj (ǫ, ,h)asafunctionofthe dimension- z less electric field =PeE τ s/~, dimensionless magnetic rection s. For the sake of simplicity from now on we z 0 field h = 2πξ2 (0P)B /Φ and the reduced temperature are going to omit the z index, which would be under- ab z 0 stood by the context. For the relaxation time τ in the ǫ ln(T/T ) (T T )/T Eq. (65). Further decrease 0 c c c ≡ ≈ − TDGL equation, the Bardeen-Cooper-Schrieffer (BCS) oftheelectricfieldEintheregimeT <T couldoriginate c theory gives eitherelectricoscillationsorleadtoformationofdomain structure, but for sure the homogeneous phase loses sta- ~ 8T c bility and a new physics takes place. What we do is to = . (6) 2τ π offersystematicalmeasurementsofthegenerationofhar- 0 monics for investigation the role of strong electric fields. This result for the microscopic theory is extremely ro- The theoretical analysis is completed in the framework bust. The same value we have for dirty bulk supercon- of the TDGL theory applied to the Lawrence-Doniach ductors with isotropic gap and clean two-dimensional d- (LD) model for a layered superconductor, introduced in type superconductors.1 Experimentalresearchesin high- the next section. T cupratesconfirmedthisvaluewithintheexperimental c accuracy. For a discussion of different methods for de- termination of the relaxation time τ , and references of 0 II. MODEL relevant experimental works see Sec. 4.2 and Sec. 4.3 of the review Ref. [2]. That is why TDGL theory does Our starting point is the TDGL equation, which in not need modifications when applied for d-wave layered momentum space takes the form cuprates. TheGLtheoryisapplicableforsmallbyabsolutevalue [ε(P e∗A)+a0ǫ]ψp(t)= 2a0τ0[dtψp(t) ζp(t)], (1) reducedtemperature ǫ 1,whereaswearegoingtouse − − − | |≪ the negative values of this parameter for a description where the argument of the kinetic energy is the gauge of an overcooled superconductor plugged in an external invariant kinetic momentum electricfield. FortheLangevintermintheTDGLtheory p(t)=P e∗A(t), (2) we have a white noise correlator − T ctainsotnhiectoinmeeP, a=ndcoτn0sitsisthaecTonDsGerLvirneglaqxuaatniotnitytifmore.aTfrheee hζp∗1(t1)ζp2(t2)i= a0τ0δp1,p2δ(t1−t2). (7) particle moving in an external electromagnetic field. In Analyzing bulk properties of the material we formally LD model the energy spectrum is given by assumeperiodicboundaryconditionsforagreatdistance L s, so that the quasi-momentum takes the discrete ε(p)= p2x+p2y + a0r(1 cosθ). (3) for≫m 2m 2 − ab 2π~ π~ π~ Here we have used standard notations for the time- pz = (integer) , + . (8) L ∈ − s s dependent vector potential of the homogeneous electric (cid:18) (cid:19) field perpendicular to the layers in our analysis Ez(t) = Let us introduce convenient for this task dimensionless dtAz(t),thechargeoftheCooperpairs e∗ =2e,their variables: u=t/τ for the dimensionless time and − | | | | effective mass in the plane of the layers for the uniaxial superconductors which we study m , the effective mass ε(p) r ab ω(k,θ)= =k2+k2+ (1 cosθ) (9) for motion in c-direction mc and the GL parameter a0 x y 2 − ~2 ~2 a0r ~2 for the dimensionless kinetic energy, where a = = , = . (4) 0 2m ξ2 (0) 2m ξ2(0) 2 m s2 ab ab c c c p ξ (0) p ξ (0) p ξ (0) x a y b z c k = , k = , k = , For high values of the LD parameter r = (2ξ (0)/s)2 x ~ y ~ z ~ c their model simply describes the anisotropic GL theory, θ = 2k /√r, (10) z 3 are the components of the dimensionless wave-vector An elementary integration of the upper equation (17) in and the Josephson’s phase. For the dimensionless noise the limit u , using the trigonometric relation ζ¯∗(u) τ ζ (t) the correlator looks like →∞ p ≡ 0 p sin(φ+2 u˜) sinφ=2cos(φ+ u˜)sin( u˜) P − P P T ζ¯∗(u )ζ¯ (u ) =n δ(u u ), n = . (11) gives h p 1 p 2 i T 1− 2 T a 0 n(θ) ∞ = du˜ For the sake of simplicity let us first consider the n one-dimensional case of hopping of fluctuation current T Z0 r r throughachainofJosephsonjunctions. ThentheTDGL exp ǫ+ u˜+ sin( u˜)cos(θ u˜) .(20) × − 2 2 P −P equation in terms of dimensionless variables reads h (cid:16) (cid:17) P i Here we have turned to a new integration variable d ψ (u)= ν(u)ψ (u)+ζ¯(u), (12) u˜ = u u , see Eq. (17), which describes the birth of u q q q 2 − − fluctuation Cooper pairs in the moment τ u˜ before the where q = P ξ (0)/~ is the conserving dimension- 0 z c time of observation t=τ u. In such a case the integrant less canonical momentum in z direction, A¯(u) = 0 gives the age distribution of fluctuation Cooper pairs. e∗A(u)ξ (0)/~ is the dimensionless potential momen- c The periodical dependence sin(θ +2 u˜) on time u˜ is a − tum corresponding to the dimensionless electric field P reflectionoftheBlochoscillationsofthefreechargedpar- f(u)=e∗E(t)τ ξ (0)/~=d A¯(u). (13) ticles which move in a periodical potential and constant 0 c u electric field. The dimensionless decay rate is given by In case of weak electric fields sin( u˜) u˜ we obtain P ≈P the equilibrium distribution density function 2ν(u) ω(q+A¯(u))+ǫ (14) ≡ n n¯(θ)= T . (21) and the kinetic dimensionless momentum for the one- r (1 cosθ)+ǫ 2 − dimensional task we are solving is The reconstruction of the plane components of the mo- k =q+A¯(u). (15) mentum leads to the general formula of Rayleigh-Jeans z for the equilibrium distribution of the particles T T III. DERBIVOALTTIZOMNAANNNDESQOULAUTTIIOONN OF THE n¯p = p22xm+apb2y + m~2c2s2 1−cosp~zs +a0ǫ = ε(p)−µ. (22) (cid:0) (cid:1) TheTDGLequationEq.(12)isalinearordinarydiffer- Hereby formally we may identify the chemical potential entialequationandone caneasilycheckthatits solution with the Landau parameter µ = a0ǫ. In this sense the − has the form critical temperature Tc sets the point of annulation of the chemical potential and fixes the beginning of Bose- u u2 ψ (u) = ζ¯(u )exp ν(u )du u EinsteincondensationofthefluctuationCooperpairs. It q q 2 1 1 2 (cid:26)Zu2=0 (cid:20)Zu1=0 (cid:21) may be easily checked that the momentum distribution u of the particles obeys the master equation of Boltzmann +ψ (0) exp ν(u )du . (16) q 3 3 } (cid:20)−Zu3=0 (cid:21) dtnP(t)= nP(t)−n¯P−e∗A(t) = nP(t) + nT , The averaging of the superconductor’s order parameter − τP−e∗A −τP−e∗A τ0 (23) overthe noisegives the distributionof the mean number wherethedecayrateisproportionaltothekineticenergy of particles with respect to the canonical momentum measured from the chemical potential n (u) = ψ∗(u)ψ (u) q h q u q i u 1 = 2 ν(P e∗A(t))= ε(p)+a0ǫ. (24) = n du exp 2 du ν(u ) τP−e∗A τ0 − a0τ0 T 2 − 1 1 Z0 (cid:20) Zu2 (cid:21) In such a way we derived the Boltzmann equation for u fluctuation Coper pairs for out-of-plane transport of a +n (u=0)exp 2 ν(u )du . (17) q − 3 3 layered superconductor. The argument of the distribu- (cid:20) Z0 (cid:21) tion and decay rate is the time-dependent kinetic mo- In case of constantelectric field the vector-potentialand mentum. The direct derivation of Boltzmann equation the decay rate take the form directly from the TDGL theory is too long and contains r a lot of technical details.6 A¯(u)=fu, 2ν(u)= [1 cos(φ+2 u)]+ǫ, (18) 2 − P In case of zero electric field the relaxation rate of the order parameter with zero value of the wave vector runs where we have introduced the notations through a typical slowing down P ≡ √fr = eEz~τ0s, φ≡ P~zs = √2r q =const. (19) τ(ǫ)=τp=0 = τǫ0. (25) 4 This is the characteristic time of “drying” at ǫ > 0 of higher dimensions. The density of the current the spatially homogeneous Bose condensate with an or- n(p ) der parameter ψ(r) = const. If we consider the distri- j = e∗v(p ) z (30) z z bution depending on the kinetic momentum we obtain L from TDGL theory the standard form of the Boltzmann Xpz equation derived in 1872 is a product of the particles’ charge e∗, their density n(p )/L and velocity z ∂ n(p,t)+e∗E(t) ∂ n(p,t) t p · ∂ε ~ n(p,t) n¯(p) n(p,t) n v(p )= = sinθ. (31) = − = + T . (26) z ∂p m s − τ(p) − τ(p) τ z c 0 This formula is a special case of the general procedure In our dimensionless variables for the static case this for derivation of the current density in the framework of equation reads the GL’s theory (Ref. [9], sec. 115; Ref. [10], sec. 45) r 2Pdθn(θ)=− 2(1−cosθ)+ǫ n(θ)+nT, (27) j(r)= δ ψ∗(r)ε( i~ e∗A(r))ψ(r)d3x, (32) h i −δA(r) − ∇− which solution is Eq. (20). Z InsuchawaywederivedtheCooperpairStoss-integral where after the functional integration the vector poten- in the framework of the TDGL theory, the result is for- tial is put to be spatially homogeneous and the order mally an energy dependent τ-approximationwith a con- parameter takes the form of a plane wave stant birth rate in momentum space n n¯(p) n ψ(r)= peip·r/~, np = ψ(r)2d3x, = d3x. τ(p) = τT . (28) r V Z | | V Z (33) 0 In the one-dimensional case, imagine current along a This integral describes collisions between normal parti- chain of Josephson junctions, if we use the relation cles and creation of Cooper pairs and back process of the decay of the condensate particles. We wish to point L π dθ = dp =L , (34) out that the TDGL equation is a diffusion-like equa- 2π~ z 2πs tion which does not lead to quasi-classical dynamics of Xpz Z Z−π quasi-particles.7 This shows that Boltzmann equation the substitution of the Boltzmann equation’s solution hasbroaderapplicabilitythanargumentsusedinhistext- Eq. (20) in the general formula for the current Eq. (30) book presentations. gives Let us now turn back to the analysis of the distri- bution function. In the continual limit when the co- e∗rT π ∞ r j(1D) = sinθdθ exp ǫ+ u˜ herence length significantly exceeds the lattice constant z 4π~ − 2 ξc(0) s or in other words when the LD parameter is r Z−π Z0 n (cid:16) (cid:17) big en≫ough r 1 we have to consider small Josephson’s + sin( u)cos(θ u˜) du˜. (35) ≫ 2 P −P phase θ 1andweakfields u˜ 1.Thedistribution P o over th|e|k≪inetic momentum E|qP. (2|0≪) then becomes7,8 Here we have taken into account that θ = pzs/~ and a n =T, see Eq. (5) and Eq. (11). The averaging with n(k )=n ∞exp (ǫ+k2)u˜+k fu˜2 1f2u˜3 du˜. re0spTect to the Josephson’s phase θ can be expressed by z T − z z − 3 the modified Bessel functions11 Z0 (cid:26) (cid:27) (29) π dθ While in Refs. 7,8 are analyzed bulk superconductors in I (z)=( 1)m cosmθe−zcosθ =i−mJ (iz). m m − 2π the present work we consider a layered superconductor Z−π (36) withJosephsoncouplingbetweenthe layersforwhichwe Thus the formula for the current reads apply LD model. The last formula may be derived di- rectly from the Boltzmann equation applied in case of e∗rT ∞ rsin( u˜) pveasrtaibgaotliecddtihspeemrsoiomneεn(tpuzm) ≈dispt2zr/ib2umtico.nAhfteerrewinethhaeveneixnt- jz(1D) = 2~ Z0 du˜e−(ǫ+r/2)u˜sin(Pu˜)I1(cid:18) 2PP (cid:19). (37) section we will obtain a general formula for the density Inordertocalculatetheout-of-planecomponentofthe of the electrical current. currentj inthe LDmodelwehavetoconsiderthe com- z plementary decay rate excited by the kinetic energy in the xy plane, i.e ab CuO plane. For this reasonformula IV. FLUCTUATION CURRENT 2 Eq. (14) has to be modified as follows Letusstartouranalysiswiththeone-dimensionalcase ε(p)+a (ǫ) r 0 2ν = = (1 cosθ)+η+ǫ, (38) as the results may be easily extrapolated for the case of a 2 − 0 5 where according to Eq. (10) AtfirstfromEq.(20)wecalculatetheonedimensional density of the Cooper pairs p2 +p2 η ≡ 2xmabay0 =kx2+ky2 (39) n(1D) = π dθ n(θ). (45) 2πs is the dimensionless in-plane kinetic energy limited by Z−π a cut-off parameter c; for more details see, for example, Analogously to the calculation of the current Eqs. (30) Ref. 2. We have to sum over the kinetic momentums in and (34) now we have the ab-plane n ∞ rsin( u˜) n(1D)(ǫ¯, )= T du˜e−(ǫ¯+r/2)u˜I P . dpxdpyf p2x+p2y = 1 cdηf(η). (40) P s Z0 0(cid:18) 2P (cid:19) ZZ (2π~)2 2maba0! 4πξa2b(0)Z0 The volumedensitycanbe obtainedvia summatio(n46o)f This additional integration with respect to the two- thisone-dimensionaldensitywithrespecttothe in-plane dimensional degrees of freedom taken into account in degrees of freedom Eq. (37) gives the replacement dp dp p2 +p2 e−ǫu˜ e−ǫu˜ ce−ηu˜dη = e−ǫu˜(1−ecu˜). (41) n(3D)(ǫ¯)=Z Z (2πx~)2yn(1D) ǫ¯+ 2xm∗abay0!, (47) → 4πξ2 (0) 4πξ2 (0)u˜ ab Z0 ab cf. the analogoustransition between 2D and 3D case for Thenthe expressionforthe fluctuationcurrenttakesthe in-plane conductivity oflayeredsuperconductors,Ref. 2, final form Sec. 2.3. The kinetic energy cut-off, Ref. 2, Eq. (11) erT ∞ 1 e−cu˜ j (ǫ¯, ) = du˜e−(ǫ¯+r/2)u˜ − p2 +p2 z P 4π~ξ2 (0) u˜ x y <a c (48) ab Zu˜=0 2m∗ 0 rsin( u˜) ab sin( u˜)I P , (42) × P 1 2 leads to the same replacement as in Eq. (41) and for the (cid:18) P (cid:19) 3D density we get where ¯ǫ means the self-consistent treatment of ǫ param- eter, which will be analyzed in the next section. n ∞ 1 e−cu˜ n(3D)(ǫ¯, )= T du˜ − P 4πsξ2 (0) u˜ ab Zu˜=0 rsin V. MAXWELL-HARTREE SELF-CONSISTENT e−(ǫ¯+r/2)u˜I P . (49) 0 APPROXIMATION × 2 (cid:18) P (cid:19) Additionally we may represent the nonlinear parame- Up to this moment we have neglected the influence ter b in terms of the Ginzburg-Landauparameter κ = GL of the nonlinear term in the GL theory. The coefficient λ (0)/ξ (0) and the flux quantum Φ , see Ref. 2, standing in front of this term participates, for instance, ab ab 0 Eq. (165) in the equation for the equilibrium value of the order parameter above Tc π~κ 2 GL b=2µ . (50) [a0ǫ+bψ 2]ψ =0. (43) 0(cid:18)Φ0m∗ab(cid:19) | | The idea of the self-consistent approximation (SCA) is We also introduce the dimensionless Ginzburg number to replace the density of particles with its value aver- for a layered superconductor (cf. Ref. 2, Eq. (168)) agedoverallfluctuationmodes. Insuchawayweobtain a self-consistent equation for the renormalized reduced ǫ (T)=2πµ T λab(0) 2 = 2µ0κ2GLe2ξa2b(0)T. (51) temperature, see Ref. 2, Sec. 3.3 Gi 0s (cid:18) Φ0 (cid:19) π~2s T b In terms of the so introduced notations the self- ǫ¯=ln + n(3D)(ǫ¯). (44) T a consistent equation for the renormalized temperature c 0 reads The idea is coming from the Maxwell treatment of the ringofSaturn–thefirstworkoncollectivephenomenain ¯ǫ=ǫ+ǫ ∞du˜1−e−cu˜e−(ǫ¯+r/2)u˜I rsin(Pu˜) , physics. Maxwellconcludedin1856thatthe ringcannot Gi u˜ 0 2 Z0 (cid:18) P (cid:19) be a rigid object, but consists of “indefinite number of (52) unconnectedparticles”andeachofthemismovinginthe in agreement with the results of Puica and Lang Ref.5, averaged gravitational field of the others. Analogously Eqs. (11) and (17), where the temperature-dependent the fluctuation density of Cooper pairs renormalizes its Ginzburg number ǫ (T) is denoted by gT. The solution Gi chemical potential as shown in Eq. (44). ¯ǫ of equation (52) has to be substituted as an argument 6 in the formula for the current Eq. (42). This agreement In particular, in the exponential dependence, which we demonstrates that it is possible to derive in the begin- haveinEq.(41),wehavetosumupthelimitedgeometric ning an 1-dimensional (1D) formula for the Josephson progression chain and later on to generalize this approach to a 3- dimensional (3D) case of layered superconductor. In the c Nc−1 next section we will analyze the modification of this re- dηe−ηu˜ 2h e−(2n+1)hu˜, (59) → sult in case of out-of-plane external magnetic field. Z0 n=0 X which more easily canbe representedas a subtractionof two infinite ones VI. INFLUENCE OF A PERPENDICULAR MAGNETIC FIELD. MAGNETOCONDUCTIVITY Nc−1 ∞ ∞ = . (60) − A. General case nX=0 nX=0 nX=Nc In this way the summation over the in-plane degrees of Letus consideranexternalmagnetic fieldalsoapplied freedom simply reduces to the shift perpendicularlytotheCuO planes. Inthiscasethevari- 2 ables can be separated and the task for the calculation c h of the current reduces to the 1D case Eq. (37), which dηe−ηu˜ 1 e−cu˜ . (61) → sinhhu˜ − we have already considered. For the two-dimensional Z0 (cid:0) (cid:1) movement the magnetic field arouses equidistant oscil- Then the substitution described in Eq. (41) in the pres- lator spectrum ence of external magnetic field takes the form p 1 e∗B 2ma∗b →~ωc n+ 2 , ωc = m∗z. (53) e−ǫu˜ e−ǫu˜ ce−ηu˜dη e−ǫu˜(1−ecu˜)h . (62) ab (cid:18) (cid:19) ab → 4πξ2 (0) → 4πξ2 (0)sinhhu˜ ab Z0 ab The degenerationrate is determined by the Landausub- zone capacityB/Φ , whereΦ =2π~/e∗ is the magnetic This is the recipe with which one may switch from the 0 0 alreadysolvedone-dimensionaltaskEq.(37)tothethree- flux quantum. For parametrization of the GL theory it dimensional. Comparisonwith the caseofzero magnetic is convenient to introduce the linear extrapolated upper field shows that we may use the 3D formulas Eqs. (52) critical field, represented via the coherence length in the and (42), where we only have to make the substitution ab-plane 1 h B (T) Φ c2 0 . (63) B (0) T = . (54) c2 ≡− c dT 2πξ2 (0) u˜ → sinh(hu˜) (cid:12)Tc ab (cid:12) (cid:12) In this way, in agreement with Ref. 5, Eqs. (13) and Forfacilitationwewillworkwit(cid:12)hthedimensionlessmag- (14), we derive the final formulas for the self-consistent netic field reduced temperature B ~ω z c h = , (55) ∞ (1 e−cu˜)h rsin( u˜) ≡ Bc2(0) 2a0 ǫ¯=ǫ+ǫ du˜ − e−(ǫ¯+r/2)u˜I P Gi 0 sinhhu˜ 2 Z0 (cid:18) P (cid:19) withthehelpofwhichthedimensionlessspectrumquan- (64) tizes as andthe perpendicular to the CuO planes density of the 2 current η (2n+1)h, n=0, 1, 2, 3, ... (56) → e2ξ2(0)E ∞ (1 e−cu˜)h j (ǫ¯, ,h) = c z du˜ − e−(ǫ¯+r/2)u˜ The maximalkinetic energyis reachedatsome big num- z P 16~sξ2 (0) sinhhu˜ ab Zu˜=0 ber N , which cuts off the summation over the Landau c sin( u˜) rsin( u˜) levels P I P , (65) 1 × 2 P (cid:18) P (cid:19) ~ω N =a c, 2hN =c. (57) c c 0 c where we have used the relation The influence of the magnetic field reduces to the re- erT e2ξ2(0)E placement of the integrationover the kinetic energy to a P = c z . (66) 4π~ξ (0) 16~sξ2 (0) restricted summation over the Landau levels ab ab This explicit formula is one of the new results of the c Nc−1 dηf(η) ∆η f((2n+1)h). (58) present work. Let us now analyze the magnetoconduc- Z0 → n=0 tivity in some characteristic particular examples. X 7 B. Weak magnetic fields Inthecaseofstrongmagneticfieldsitismoreappropriate to use the 1D formula Eq. (37), where we have to add a Let us consider the application of the upper formulas summation over the Landau levels, taking into account incaseofweakmagneticfieldh ǫ¯.Ifweusethe well- the degeneration and the cut off ≪| | known summation of Euler-MacLaurin (cf. with Ref. 2, ∞ B Eq. (23)) we have j(3D)(ǫ¯, ,h) j(1D)(ǫ¯ +2nh, ). (73) z P ≈ Φ z h P 0 x 1 7 31 nX=0 =1 x2+ x4 x6+... (67) sinhx − 6 360 − 15120 In close vicinity of the upper critical field ǫ¯ h in the | |≪ summation the share of the lowest Landau level domi- For weak magnetic fields we may use the approximation nates over the others and its contribution gives h 1 1 h2u˜. (68) e∗rTB ∞ rsin( u˜) sinhhu˜ ≈ u˜ − 6 j(3D) du˜e−(ǫ¯h+r/2)u˜sin( u˜)I P . z ≈ 2~Φ P 1 2 0 Z0 (cid:18) P (cid:19) The substitution of this decay into Eq. (65) leads to the (74) specialexpressionforthecurrentinthe presenceofweak Similar procedure applied to the number of particles’ magnetic fields density Eq. (46), see also Eq. (52), j (ǫ¯, ,h) = j (ǫ¯, ) bB z P z P ǫ¯h =ǫh+ n(1D)(ǫ¯h, ) (75) erTh2 ∞ a0Φ0 P du˜e−(ǫ¯+r/2)u˜(1 e−cu˜)u˜ −24π~ξ2 (0) − ab Zu˜=0 determines the equation for the self-consistent reduced rsin( u˜) temperature sin( u˜)I P . (69) 1 × P 2 (cid:18) P (cid:19) ∞ rsin( u˜) ǫ¯ =ǫ +2hǫ du˜e−(ǫ¯h+r/2)u˜I P . (76) h h Gi 0 The coefficient in front of the integral over the reducing 2 Z0 (cid:18) P (cid:19) term may be expressed via the coherent lengths and the lattice constant as follows Closetotheuppercriticalmagneticfieldthefluctuations are practically one-dimensional and such a reduction of erTh2 2πeTξ2(0) ξ2 (0)B 2 the space dimensionality significantly amplifies the fluc- = c ab z . (70) 24π~ξ2 (0) 3~s2 Φ tuation phenomena. ab (cid:18) 0 (cid:19) Nextwe aregoingto deriveanexplicit formulafor the The experimental data fitting with this formula could differential conductivity in the most general case of LD leadto more precisespecification ofξ (0) for the investi- modelandanalyzeitintheparticularcasesofbothweak c gatedsample. TheprocedureEq.(68)appliedtoEq.(64) and strong magnetic fields. The physical conditions of endsinaself-consistentequationforthereducedtemper- possible occurrence of negative differential conductivity ature in case of weak magnetic fields will be discussed. h2 ∞ ǫ¯(h) ǫ ǫ du˜ e−(ǫ¯+r/2)u˜(1 e−cu˜) ≈ − Gi 6 − VII. DIFFERENTIAL CONDUCTIVITY. Z0 POSSIBLE NDC rsin u˜ u˜I P . (71) 0 × 2 (cid:18) P (cid:19) Thenon-lineareffectsovertheconductivityσzz(Ez) ≡ In the next subsection we will analyze the case of values jz/Ez could be best understood in the investigation of of the magnetic field close to the phase curve B (T), the differential conductivity c2 wheretheinfluenceofthefluctuationpairsismostvividly dj expressed. σ (E ) z . (77) diff z ≡ dE z In order to derive it we have to differentiate the gen- C. Strong fields close to Bc2(T) eralformulaforthe currentinanexternalmagnetic field Eq. (65) and use the well-known properties of the modi- Close to the phase curve it is more convenient to ac- fied Bessel functions count the reduced temperature in regards of the phase transition temperature, which is a function of the ex- dIm(z) 2 =I (z)+I (z) (78) ternal magnetic field T (B). Comparison with Eq. (54) dz m+1 m−1 c2 reads which can be found in any special functions manual, for T T (B) exampleseeRef.11orRef.12andreferencestherein. We c2 ǫ = − =ǫ+h, ǫ h. (72) h T | h|≪ turn to differentiation with respect to the dimensionless c 8 electricfield ,takingintoaccountthevalueofthedecay Of course the calculated derivative (∂j /∂E ) does z z ǫ¯ P rate τ that follows from the BCS theory not describe the whole effect. We have to take into ac- 0 ∂ esτ π~ count also (∂¯ǫ/∂Ez)ǫ and the increase of the sample’s 0 (BCS) P = , τ = . (79) temperature above the ambient temperature ∂E ~ 0 16T z c (T T )= j E , (86) FromEq.(65)itisstraightforwardthattheelectricfield- Rtherm − amb V z z dependentpartinthedifferentialconductivityisgivenby due to the thermal resistance. The “chemical” reaction the expression of pairing of normal charge carriers e+e e∗ creates → alsoadecreasingofthenumberofnormalchargecarriers, ∂ rsin( u˜) sin( u˜)I P and in the Drude formula, for example, we have to take 1 ∂ P 2 P (cid:20) (cid:18) P (cid:19)(cid:21) into account1 the density of state’s corrections (DOS) rs˜ σ =(n 2n(ǫ¯, ))e2τ /m . = ( u˜c˜ s˜)(I +I )+u˜c˜I , (80) norm e norm e 4 2 P − 0 2 1 Al those e−ffects arPe however smaller compared to the P where for the sake of brevity we have introduced the no- dominating Aslamazov-Larkinconductivity which quali- tations tatively can describe the appearance of new physical ef- fects such as NDC, for example. rsin( u˜) s˜=sin( u˜), c˜=cos( u˜), I =I P . In order to better understand the nature of NDC in m m P P 2 (cid:18) P (cid:19) supercooling regime in the next section we will analyze (81) the Green functions of the Boltzmann equation. Having inmind the considerationsaboveit canbe easily shown that the general formula for the differential con- ductivity takes the final form VIII. GREEN FUNCTIONS OF THE e2ξ2(0) ∞ (1 e−cu˜)h BOLTZMANN EQUATION σ (ǫ¯, ,h)= c du˜e−(ǫ¯+r/2)u˜ − diff P 16~sξ2 (0) sinhhu˜ ab Zu˜=0 NowletuspaintsomeCooperpairsingreen inorderto rs˜ ( u˜c˜ s˜)(I +I )+u˜c˜I . (82) tracetheirmotioni.e. inordertoanalyzetheinfluenceof × 4 2 P − 0 2 1 the random noise on the momentum distribution of the (cid:20) P (cid:21) This is the equation we are going to use in order to ana- Cooperpairswewillreplacetheconstantincometermin lyze the presenceof negativedifferentialconductivity for theStoss-integraln0/τ0withaδ-functioninhomogeneous given samples. Hereof we may derive the expressions for term in Eqs. (23) and (26) σdiff in the special cases of weak and strong magnetic n0 2π~ fields. δ(t)δ(p p0). (87) τ → L − 0 Intheabsenceofexternalmagneticfieldthelimith → For simplicity we will consider 1D case. 0 gives h/sinhh 1 and we obtain → Before the moment of creation there are no Cooper σ (ǫ¯, )= e2ξc2(0) ∞ du˜e−(ǫ¯+r/2)u˜1−e−cu˜ pairsnp(t<0)=0.Thenimmediatelyaftertheinfluence diff P 16~sξ2 (0) u˜ we have δ-like initial distribution ab Zu˜=0 rs˜ 2π~ × 4 2 (Pu˜c˜−s˜)(I0+I2)+u˜c˜I1 ,(83) np(t=+0)=δp,p0 = L δ(p−p0). (88) (cid:20) P (cid:21) This distribution is normalized to have one Cooper pair which may be derived straight from Eq. (42). in the initial moment For strong magnetic fields we take into account only the contribution of the lowest Landau level and the dif- dp (t)= n (t)=L n (t), (+0)=1 (89) ferentiation of Eq. (74) reads N p 2π~ p N p Z X πe2ξ2(0)B ∞ σ (ǫ¯ , ) = c z du˜e−(ǫ¯h+r/2)u˜ (84) born by thermal fluctuations; pictorially speaking “by diff h P 4~sΦ0 Z0 the sea foam.” According to the separation of variables rs˜ in the Boltzmann equation Eq.(23) in every momentum ( u˜c˜ s˜)(I +I )+u˜c˜I . × 4 2 P − 0 2 1 point the evolution of the distribution function is inde- (cid:20) P (cid:21) pendent If we want to return to the one-dimensional differen- u tial conductivity for strong magnetic fields close to the n (u=t/τ )=n (0)exp 2ν(u )du . (90) p 0 p 3 3 phasecurveweonlyhavetoomitthenumberofmagnetic − (cid:26) Z0 (cid:27) threads B /Φ in the upper equation, which gives z 0 The substitution here of the decay rate from Eq. (18) πe2ξ2(0) ∞ gives σ(1D)(ǫ¯, )= c du˜e−(ǫ¯+r/2)u˜ (85) diff P 4~s (u) r rsin( u) Z0 N =exp ǫ+ u+ P cos(θ u) . rs˜ (0) − 2 2 −P × 4 2 (Pu˜c˜−s˜)(I0+I2)+u˜c˜I1 . N (cid:26) (cid:16) (cid:17) P ((cid:27)91) (cid:20) P (cid:21) 9 It is instructive to consider the 3D limit case of ǫ r; Decreasing of the electric field increases the maximal | |≪ formallyonecanconsidertheunrealisticforthecuprates size of the Cooper pair droplets and the electric current case of r 1. For small angles which is proportional to the number of particles in the ≫ droplets. This precursor of the Bose condensation cre- 2sk f ates the NDC. Our preliminary numerical calculations θ = 1, = , u 1 (92) √r ≪ P √r P ≪ for YBa Cu O (Y:123) taking the parameters from 2 3 7−δ Ref. 4: s = 1.17 nm, ξ (0) = 1.2 nm, ξ (0) = 0.14 nm, ab c the Taylor expansion of the trigonometric functions in (ren) (norm) T = 92.01 K, T = 86.9 K, κ = 70, 1/σ = Eq. (91) gives c c GL zz 5mΩcm, c = 0.5 demonstrated existence of NDC; this givesthehopethatNDCcanbeexperimentallyobserved (u) 1 N =exp (ǫ+q2)u+qfu2 f2u3 . (93) for out-of-plane transport in Y:123. (0) − − 3 N (cid:26) (cid:27) The simplest example is to consider zero initial momen- tumq =ξ (0)p /~=0.Inthesupercoolingregimewhere IX. ANALYSIS, DISCUSSION AND c 0 CONCLUSION ǫ < 0 in the beginning the number of Cooper pairs in- creases. It is a typical lasing process – increasing of the number of coherent Bose particles in one mode; imagine Let us start with the formal discussion. In case of big a dropofrainincreasingby the condensationina humid enough overcooling ǫ+r/2 < 0, when ǫ < 0 and ǫ > r | | atmosphere. Finally,however,the 1f2u3termdominates the integral for the current Eq. (42) is divergent. This 3 intheargumentofthe exponentand ( )=0;onecan means that the electric field cannot prevent the appear- say that the droplet is evaporated byNth∞e big kinetic en- anceofasuperconducting condensation,asis the caseof ergylikeameteoritefallingintheearthatmosphere. The volume superconductors. The reason for this is that the droplets are not smeared during their lifetime one-dimensionalzoneintheenergyspectrumEq.(3)has a finite width a r, in contrast to the LD model, which 0 2π~ (u) shall be obtained if we take the continuallimit of r 1. n (t)= N δ(p p ). (94) ≫ p L (0) − 0 However,iftheLDparameterissmallenoughwehaveto N takeintoaccounttheinteractionbetweenthefluctuation They have just a drift in the kinetic momentum space. modes, described by the nonlinear term in the GL the- This drift related to the electric current is analogous to ory. These effects become important when the reduced the rain fall created by the earth acceleration. temperatureǫiscomparabletotheGinzburgnumberfor layered superconductors 2π~ (u) n(p,t)= N δ(p p e∗Et). (95) 0 1 (0) − − ǫ (T )= , (100) V N Gi c 4πξ2 (0)s∆C ab InupperGreen functionsolutionoftheBoltzmannequa- tion Eq. (26) we have recoveredthe 3D variables. where(seeRef.2,Sec.3.3,Eq.169)∆C isthejumpofthe Nowwecanunderstandqualitativelythemechanismof heatcapacity,extrapolatedfromthecriticalbehaviourof creationof NDC for small electric fields in a supercooled the heat capacity C(T) as a fitting parameter, or can be superconductor. Let us take for illustration p = 0 case evaluated by the electrodynamic properties 0 in Eq. (93). For ǫ < 0, h = 0, and ǫ r this function has a maximum7 | | ≪ 1 Φ0 2 T ∆C = . (101) c 8π2µ λ (0)ξ (0) (u ) 2( ǫ)3/2 0 (cid:18) ab ab (cid:19) max N =exp − (96) (0) 3 f In the Maxwell-HartreeSCA the totalvolume density of N (cid:26) (cid:27) thefluctuationCooperpairsleadstoaneffectiveheating at (see Ref. 2, Sec. 3.3, Eqs. (164) and (173)). As a whole, inclusionof the SCA does not change the qualitative be- ( ǫ) havior of the obtained results. In overcooled supercon- u = − . (97) max f ductors ǫ<0, for example, we expect the appearance of p a negative differential conductivity (NDC) with the de- Themaximalincreaseofthe“weight”oftheBosedroplet creasing of the constant electric field. Whether this will (u )/ (0) can reach big values for small enough N max N lead to formation of a domain structure of normal and electric fields superconducting layers or to existence of electric oscilla- tions is a question of further analysis. Our theory, how- f ( ǫ)3/2, (98) ≪ − ever, predicts that this phase transition when T < Tc will take place via annulation of the differential conduc- or tivity. In this respect the layered superconductors give e∗Eτ(ǫ)ξ(ǫ)/~ 1, τ(ǫ)= τ0, ξ(ǫ)= ξ(0) . (99) an advantage as, because of the strong anisotropy, the ≪ ǫ ǫ1/2 normal conductivity perpendicular to the layers is too | | | | 10 small. Thus the sample will not heat intensively, which happen in the area below the annulation of the differen- givestheopportunitytobeobservedthenonlineareffects tialconductivity. Therealizationofthesimplestpossible ofastrongelectricfieldoverthefluctuationconductivity. scenario of creation of NDC will open the technological These nonlinear effects can be observed as an amplifica- perspective to create a new type of high-frequency oscil- tion ofthe currentharmonics generationas we reachthe lators operating in the THz gap.13 While in Ref. 13 is critical region. Systematical investigation of these har- analyzedthe case ofbulk superconductorsinthe present monics willleadto specificationofthe parametersinthe work we demonstrated that for electric currents applied TDGL theory and to clarification of the kinetics of the in the “dielectric”c-directionthe dissipated powercould superconducting order parameter. Here we wouldlike to be orders of magnitude smaller and perhaps Josephson consider the possible application of our theory for dif- coupled superconductors can realize the first technical ferent layered cuprates. For instance, for the extremely applications of NDC in superconductors; it was one of anisotropicBi Sr CaCu O (Bi:2212)the value of the the motivations to perform the present work. 2 2 2 8+x LDparameterr isexceptionallysmall. Theheatingfrom Few words we wish to add concerning the theoretical the electric field applied in the “dielectric” z-direction is notions used in the present work. It seems very strange very weak, but the area where current harmonics can be that Boltzmann kinetic approach has still limited usage observedisverynarrowaswell. Becauseofthistheirob- inthephysicsofparaconductivityasitwasforthenormal servation would require samples of extraordinary quan- conductivity hundred years ago. Just Green functions tity. Wider and easier for observationwould be the non- of the Boltzmann equation give the transparent picture linearregionforthecuprateswithamoderateanisotropy what is happening in homogeneous electric fields. Sepa- like Y:123. For most favorable we consider the cuprate rationof the variables in optical gauge with zero electric Ta2Sr2CuO6 (Ta:2201)whereacoherentFermisurfaceis potential ϕ = 0 reduces the kinetic equation to an ordi- observed. Whenthe anisotropyissmallandr parameter nary differential equation. Returning to the kinetic mo- bigger, the half-width of the z-zone 21a0r = ~2/m2cs2 is mentum space gives the drift of the fluctuation Cooper bigger and then the electric field Ez causes more signif- pairs created by the electric field. icant increase of the kinetic energy of the Cooper pairs Let us continue the discussion of physics in supercool- 1a r(1 cosp s/~). Because of this, for instance, the 2 0 − z ing regime. Near to the annulation point of the differen- overcoolingregionwhenǫ<0and =0wouldbewider P 6 tial conductivity the paraconductivity is not a small ex- and more easily observed. Exactly, in such more easily cess perturbation but will become comparable with the reachedbytheexperimentconditions,wehopetoobserve fluctuation conductivity. In the suggested experiment of the annulation of the total differential conductivity. For applied DC electric field and small AC voltage one can this purpose submicron high qualitative films of Ta:2201 observe in principle the AC component of the fluctua- are necessary to be used. The investigation of the har- tion magnetization M . This physical effect is described z monics generation above T is a useful beginning for the c by magnetoelectric susceptibility χ = ∂M /∂E ; this ME z z systematicalresearchofthenonlineareffectsofthestrong theory will be the subject of a future research. Measure- electric field over the fluctuation current in z-direction. ments ofmagnetizationandparaconductivityin oneand The experiment which we advocate is principally very the same sample can lead to exact determination of the simple, but according to the best of our knowledge not time constant τ0. This is the most important parame- doneyet. Thefluctuationconductivityhavetobeinvesti- ter of TDGL theory which reveals in part the nature of gatedinavoltage biased circuit;aconstant(DC)voltage superconductivityandpairingmechanism. Theπ/8mul- hastobeapplied. AsignificantlysmallerACvoltagehas tiplier in Eq. (6) is only weak coupling BCS result, but also to be applied to the sample. The DC current gives aswealreadymentioned,forreferencesofrelevantworks the conductivity and a measurement with a smaller AC seeSec4.2ofthereview,2theexperimentalaccuracynow current gives the differential conductivity. The circuit isnotenoughtoobservereliablyanydeviationsfromthis shouldbesimilartothedevicesinventedforinvestigation BCS π/8 value. There is no consensus that BCS theory of the NDC in tunnel diodes. The DC voltage has to be is directly applicable for high-Tc cuprates and one might addedtothenormalphaseaboveTc andthesimplestex- expect some correction multiplier τrel to the life time of perimentwhichwesuggestistomeasurethetemperature theorderofone. Therelativelifetimecangiveimportant dependence of the AC current (differential conductivity) information for the pairing mechanisms in cuprates. asafunctionofthetemperature. Thetheoryreliablypre- From qualitatively point the numerical value of τ is 0 dictsannulationofthedifferentialconductivityatcooling not so important because only fixes the scale for the below T . What exactly will happen at further cooling is graphical presentation of the experimental data. In the c difficulttobepredictedexperimentally: itcouldbeNDC chosen dimensionless units the current voltage curves ifthespacehomogeneityisconserved,creationoflayered should be universal and the most important property domain structure of normal and superconducting layers of the current voltage curves is the predicted annula- or complicated dynamic phenomena. Different instabil- tionofthedifferentialconductivityatsupercoolingbelow ities can be triggered by small perturbations including T . Demonstration of this annulation and possible NDC c thenatureofcontactsandchemicaltreatmentofthesur- inout-of-planetransportcantriggersignificanttechnical faces. One thing is sure: indispensably, new physics will applications13 and we hope that experimental search of