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Enhancement of superconductivity by local inhomogeneity Ivar Martin,1 Daniel Podolsky,2 and Steven A. Kivelson3 1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545 2Department of Physics, University of California, Berkeley, CA 94720 3Department of Physics, Stanford University, Stanford, CA 94305 5 (Dated: Printed February2, 2008) 0 0 We study the effect of inhomogeneity of the paring interaction or the background potential on 2 the superconducting transition temperature, Tc. In the weak coupling BCS regime, we find that n inhomogeneitywhichisincommensuratewiththeFermisurfacenestingvectorsenhancesTc relative a to its value for the uniform system. For a fixed modulation strength we find that the highest Tc J is reached when the characteristic modulation length scale is of the order of the superconducting coherence length. 7 2 ] Many strongly correlated superconductors, and in achieved when the characteristic length-scale of the in- n particular high-temperature superconducting (HTSC) homogeneities, L, is large, in which case the transition o cuprates,exhibitinhomogeneouselectronicand/orstruc- temperature is that of the regions with the highest lo- c - turalphasesatthenanoscale[1,2,3]. Thecoexistenceof cal Tc. Upon including the effects of phase fluctuations r HTSC and inhomogeneity suggests that the underlying we find that T is maximized when L is comparable to p c u inhomogeneitiescouldbeatleastpartiallyresponsiblefor the superconducting coherence length ξ vF/Tc. The ∼ s thehighvalueofthesuperconductingtransitiontempera- increase of the transition temperature occurs at the ex- . t ture. EmeryandoneofusproposedthatHTSCisrelated pense of the superfluid density, which is reduced in in- a to frustrated electronic phase separation, commonly ex- homogeneous superconductors relative to their homoge- m pectedinstronglycorrelatedsystems[4]. Theseideasfor neous counterparts. - d inhomogeneous superconductivity have been further de- Inhomogeneous pairing: Mean-Field treatment. As a n velopedinthe contextofstripes[5,6]. Itisimportantto first example we consider a Hubbard model with an in- o distinguish these and related scenarios for superconduc- homogeneous attractive potential U(r)>0, c tivity creation or enhancement by inhomogeneity from [ the conventional weak-coupling coexistence of supercon- H = H0+HU (1) v1 ductivity and various density waves [7, 8, 9, 10]. In the 0 = ξkc†kσckσ latter case, the density wave order inevitably suppresses H 9 kσ X superconductivity due to the competition for the Fermi 5 = U(r)n (r)n (r), 6 surface electrons. HU − ↑ ↓ r 1 X It is therefore important to understand the nature 0 5 of the interplay between superconductivity and inhomo- whereξk =ǫk−µandnσ(r)=c†σ(r)cσ(r)istheoccupa- 0 geneities. A complete description of the interplay is tion number of electrons of spin σ at position r. Within / clearly impossible. However, in the case in which the this model, our goalis to understand whether for a fixed t a characteristic energy scale responsible for the formation average pairing strengthU(r), a uniform or non-uniform m of the inhomogeneity is much larger than the supercon- U(r) yields a higher transition temperature, Tc. In the - ductingenergyscale(the gap∆),andwheretheresidual weakcouplinglimit,wecanderivetheBCSconditionfor d interactions are weak, a description based on BCS the- theonsetofsuperconductivityfromtheHamiltonian(1), n o ory should be reliable. The purpose of this work is to c study the effect of such imposed inhomogeneity on su- ddp ∆ = U(q p)K(p)∆ , (2) v: perconductivity within the BCS framework. The origin q Z (2π)d − p i ofthe inhomogeneitycouldbe eitherelectronic,asinthe X where ∆ = U(k) c c , U(k) frustrated phase separation scenario, or structural, that q kp h q/2−k/2+p↑ q/2−k/2−p↓i r is the Fourier transform of the pairing interaction, and is caused by local lattice distortions or non-uniform car- a P K(p)isthepairingkernel. Thepairingkerneldependson rier concentration due to doping irregularities. We will temperatureT andthemean-field(MF)superconducting assume that these structures do not cause Fermi sur- transition is defined by the temperature at which the face nesting either due to the lack of periodicity (e.g. integral equation has a non-trivial solution. The kernel random doping profile) or due to the periodicity being can be calculated from the normal state electron Green incommensurate with the nested momentum transfers functions [11] (e.g. frustrated phase separation, or stripes). Under these conditions we generically find that inhomogeneity 2γω enhances the global superconducting transition temper- K(p) N ln D Θ(ω v p), (3) ature, Tc. At the mean-field level, the maximum Tc is ≈ f "π T2+(vfp)2# D−| f | p 2 whereN isthedensityofstatesattheFermisurface,v f f is the Fermi velocity, T is temperature, and lnγ 0.577 ≈ is Euler’s constant. Here we also introduced an explicit high-energy cut-off for the attraction, ω . For T v p D f ≫ this expression reduces to the well known homogeneous result, K N ln[2γω /(πT)]. f D ∼ The modulation of the pairing interactionleads to the mixing between Cooper pairs with different center-of- mass momenta. For simplicity we first assume a harmonic modulation (Q 2π/L) of the pairing, U(r) = U¯ + U cos(Q r). ≡In this case the integral equa- Q · tion (2) reduces to a system of linear equations ∆ = n U¯Kn∆n+(UQ/2)[Kn−1∆n−1+Kn+1∆n+1] nm∆m, ≡M where ∆n ∆(nQ+q0) and Kn K(nQ+q0). The ≡ ≡ “parent” momentum q0 defines the minimal momentum FIG. 1: Critical temperature for the inhomogeneous neg- of a Cooper pair in the connected family ∆(nQ+q0). ative U Hubbard model with coupling U(x) = U¯ + The paring instability occurs at the temperature Tc, UQcos(Qx). The thick line denotes the mean-field re- suchthatthelargesteigenvalueofmatrix isequalto1. sult, where Tc,a = (2γ/π)ωDexp[−1/NfU¯] and Tc,h = Intheuniformcase,U =0,thisconditionMisU¯K(0)=1. (2γ/π)ωDexp[−1/Nf(U¯ +|UQ|)]. The dashed line shows Q the critical temperature once phase fluctuations of the or- We will now prove that the mean-field transition tem- der parameter are included. For Qξ ≪ 1, the super- perature is greater than in the uniform case, U = 0. Q conductivity is first established locally in regions where Consider the q0 = 0 family and without loss of gener- U(x) is large, but macroscopic phase coherence is achieved ality, take U > 0. Since all of the matrix elements of at a lower temperature, bounded from below by T = Q c,l are non-negative, by Perron’s theorem [12], the max- (2γ/π)ωDexp[−1/Nf(U¯ −|UQ|)]. M imal eigenvalue is a positive number that is larger than any diagonal matrix element, including U¯K(0). Thus, Q,sodoesη. ForvFQ>ωD,Cooperpairscannolonger scatteroffthequicklyoscillatingcouplinglandscape,and generically,the superconducting onset temperature T is c we recoverthe critical temperature for the homogeneous increased whenever U =0. Q 6 case. This result can be understood from an analogy with Small Q limit. In the limit Qξ 1, the a quantum mechanical particle in a tight-binding chain. ≪ Defining a new variable Λ = U¯K ∆ , the BCS condi- global MF transition temperature is determined by n n n tiontakesasimple symmetric form,Λ =(1/U¯K )Λ the regions with the strongest pairing interaction, (UQ/2U¯)(Λn+1+Λn−1). The “hoppinng” term dnelocna−l- Tc ≈(2γ/π)ωDexp[−1/Nf(U¯ +|UQ|)]. The deviations from this result due to finite Q and the effects of the izes the particle and thus reduces the “energy”below its minimal on-site value (1/U¯K0). Clearly, this leads to a phase fluctuations are discussed below. Electron density modulation. Before going further, let relative increase of T . c Large Q limit. In this limit, the quickly oscillating us in parallel consider the case of homogeneous coupling U(r) = U¯, with inhomogeneity caused by a background coupling is ineffective at mixing different modes, so that potential variation. In the simplest case of the harmonic the off-diagonal terms in are rapidly decaying with M modulation, the additional contribution to the Hamilto- n. We are then justified in keeping only a small portion nian (1) is of the matrix surrounding the n = 0 term. The lowest ordercorrectiontothehomogeneousresultisobtainedby =ρ c† c cosQ r . (5) ρ iσ iσ i considering couplings between ∆0 and ∆±1. The largest H · iσ eigenvalue in this case is X Itiseasytoseethatforparticle-holesymmetricdensityof states(DOS),thelinearinρcontributionstotheBCSin- λmax = U2¯ K0+K1+s(K0−K1)2+ 2K0KU¯21UQ2, svtaalubeilsitoyfcQon,dthiteiomnoedquulaattiioonnsavcatnsisahsaidselnotwiclyalvlya.ryFionrgssmhaifltl   in the local chemical potential with the amplitude pro- Given the separation of energy scales, T v Q ω , portionalto ρ. Thus,weonlygetalinearinρcontribu- c f D ≪ ≪ | | we obtain Tc by solving λmax =1, tion for asymmetric DOS, N(ǫ)= Nf +N′(ǫ ǫF). We − then find that BCS equations areidentical to the case of 2γ T = ω exp[ 1/N (U¯ +η)], (4) inhomogeneous pairing interaction with the modulation c D f π − strength awnhderseinηce=KU1Q∼2Klo1g/[[ω2(D1/−vfQU¯]Kd1e)]c.reaWsehsilweiηthisinpcroesaitsiivneg, UQeff =−U¯NN′fρ. (6) 3 Ginzburg-Landau analysis (Qξ 1). Wenowconsider More importantly, it is easy to see that in the limit ≪ the generalcase of slow variationof the pairing strength of small Qξ, the order parameter is exponentially sup- and/orbackgroundpotential. TheGinzburg-Landaufree pressed in the region of smaller pairing interaction rela- energy functional in the presence of inhomogeneity is tive to its value at the peak, F = drdr′K(r r′)∆(r)∆(r′)+ dr∆(r)2 ∆min ∼∆maxexp −A (ξQ)−1 , (11) − − U(r) Z Z This, in turn, implies that ph(cid:8)ase fluctuati(cid:9)ons, which we β +α drρ(r)∆(r)2+ dr∆(r)4, (7) discuss below, can reduce the global phase coherence 2 temperature significantly below TMF. Z Z c The expression of Eq. (11) is only valid at the MF Here we assumed that the order parameter remains real transition temperature, where ∆ is infinitesimal. To de- even in the presence of inhomogeneity. We include both termine ∆ below TMF we need to solve the non-linear thecouplingofthesuperconductingorderparametertoa c Eq. (9). In a d-dimensional superconductor, with ar- densitywave,aswellasthe inhomogeneityofthepairing bitrary smooth variation of U(r), the boundary of the interaction. For smallamplitude modulation ofthe pairing interaction, U(r)=U¯ +δU(r) with δU(r) <U¯, the “classically forbidden” region, g(r) > 0, is a (d − 1)- | | dimensionalsurface. Hence,neartheboundarytheprob- two mechanisms are formally equivalent. For particle- lem is essentially one-dimensional, and in the g(r) > 0 hole asymmetric DOS, from the above considerations, α = U¯N′/N . For simplicity, we only consider the region we can apply the standard WKB approxima- f − tion to solve the linearized Eq. (9). The prefactor is inhomogeneous U(r) case here. The pair susceptibility fixed by matching the WKB solution to the intermedi- kernel is givenby Eq. (3). In the long wave length limit, ate asymptotic at the boundary x0, which can be ob- 2γω tained by solving the full Eq. (9) in a linear potential K(r r′)=δ(r r′)N ln D +ξ2 2 , (8) − − f πT ∇ g(x) = g′(x0)(x x0). We then find that in the partic- (cid:20) (cid:21) − ular case of harmonic modulation discussed above, the where ξ = vF/T. Computing the variation of Eq. (7) order parameter distance d away from the boundary is withrespecttotheorderparameter,wefindtheequation approximately for a stationary solution ∆(r), ∆(d) T AQξexp A Qξ(d/ξ)3/2 . (12) β ∼ − ξ2 2∆(r)+g(r)∆(r)+ ∆(r)3 =0, (9) p h p i − ∇ Nf This expressionis obtained assuming g′(x0) A2Q, and ≈ g(r)= 1 ln2γωD. therefore valid only for the temperatures sufficiently be- N U(r) − πT lowTMF TMF. SolongasT TMF (TMF istheuni- f formcT of∼a smysatxem with pairing≫strmeningthUm¯in U ), the c Q Asafirststep,wedeterminetheinhomogeneousmean- −| | distancefromthe turningpointtotheminimumpointof field (MF) transition temperature for the pairing inter- ∆isd L. Notice,thatthisexpressiondependsontem- action U(r) = U¯ +U cos(Q r), with Qξ 1. Close ∼ Q perature not only explicitly, but also through ξ =v /T. · ≪ f to the MF transition, the cubic terms in Eq. (9) can Phase fluctuation effects (still with Qξ 1). A con- be neglected. Expanding in U /U¯, and transforming ≪ Q sequence of the large spatial variations in the mean-field Eq. (9) to Fourier space, we obtain a system of equa- ∆(r) is that fluctuation effects are severe where ∆(r) tions connecting ∆(k) and ∆(k Q). For small Q, is small. Of these, the most important fluctuations are ± ∆(k Q) ∆(k), and after expanding up to the sec- thermal fluctuations in the phase of the order parame- ± ≈ ond order in Q, we obtain ter,i.e. where∆(r)= ∆ (r)eiθ where∆ (r)isthe MF MF | | solution of Eq. (9), and θ(r) is assumed to be a slowly 1 −gmax∆(k)=(ξk)2∆(k)− 2A2Q2∂k2∆(k). (10) varyingfunctionofr. The free energycostofsuchphase fluctuations can then be readily computed Here g denotes g(r) evaluated at U(r) = U¯ + U max Q | | and A = U /(N U¯2) (note that there is no explicit F = drJ(r)( θ)2 (13) Q f θ ∇ constraint on the value of parameter A since it is a ra- Z p tio of two small numbers). The MF transition temper- where the local superfluid stiffness is ature is determined by the smallest eigenvalue of the J(r)=N ξ2 ∆ (r)2. In general, the phase or- f MF | | differential operator on the rhs. This eigenvalue corre- dering temperature estimated using this as the effective sponds tothe “groundstate energy”ofaharmonicoscil- Hamiltonian is reduced from the mean-field transition lator, ξQA/√2. The corresponding transition tempera- temperature (i.e. the temperature at which J(r) ture TMF = TMF exp ξQA/√2 , is only slightly less vanishes), but by an amount that depends on dimen- c max − than the transitiontemperature,TMF for a systemwith sionality, and on the spatial arrangement of the regions a homogeneous pairing(cid:0) interaction(cid:1)maUx = U¯ + U . of suppressed stiffness. max Q | | 4 For concreteness, we consider the case of a two- its value for the uniform zero center-of-mass momentum dimensional superconductor. At finite temperature, pairing. For a fixed modulation depth we find that the no true long-range order is possible [13]. However, highestT isreachedwhenthe modulationwavelengthis c at T < T , binding of topological excitations into oftheorderofthesuperconductingcoherencelength. For KT vortex-antivortex pairs leads to a state with quasi- shorterwavelengths,thesuperconductorcannottakead- long range order, which has a non-zero superfluid vantage of the locally favorable conditions, while for the stiffness[14]. While for homogeneous BCS supercon- longer wavelengths, the global superconductivity is sup- ductors in 2D, the difference between MF and the pressed due to the phase fluctuations on the weak links, Kosterlitz-Thouless(KT)transitiontemperaturesistiny, where the amplitude of the order parameter is signifi- (TMF T )/TMF TMF/T (where T is the Fermi cantly reduced. Although explicitly derived for s-wave c − KT c ∼ c F F temperature), for inhomogeneous superconductors, the superconductors, similar results will also apply to un- suppression of T , is generally much larger. For a conventional superconductors in the presence of smooth KT smoothrandomdistribution of J(r), anestimate of T (onthe1/k lengthscale)inhomogeneities. Clearlyover- KT f can be made based on the effective superfluid density, simplified, the presented picture bears resemblance to the high-temperature superconducting cuprates, where T J(r) [1/J(r)]−1 J J (14) considerableexperimentalevidence[15]indicatesthatthe KT min max ∼q ≈ maximum Tc occurs at a crossover between a regime This expression has a particularly trapnsparent meaning where T is controlled by the pairing scale and where c fortheunidirectional“striped-like”variationofU(r)that it is a phase ordering transition. we treated explicitly when solving the mean-field equa- We thank G. Blumberg, S. Chakravarty, G. Ortiz, M. tions, above. There, Jmax corresponds to the stiffness Stepanov, and L. Gruvitz for useful discussions. We alongthestripesandJmin –perpendiculartothestripes. alsoacknowledgehospitalityofAspenCenterforPhysics The corresponding anisotropic XY model directly leads wherepartofthisworkwascompleted. IMandDPwere to the result Eq. (14). In this case, we find supported by the US DoE. SAK was supported by NSF grant No. DMR 0421960. T T f TMF∆ (T ). KT ∼ T2 c min KT KT Together with Eq. (12) for ∆ (T ) ∆(L,T ), [1] J. E. Hoffman et al., Science 295, 466 (2002); T. min KT KT this equation implicitly defines T . Wit∼h logarithmic Hanaguri et al.,Nature430, 1001 (2004). KT [2] C. Howald et al.,Phys. Rev.B 67, 014533(2003). accuracy we find that T min(TMF,v Q/A). KT ∼ c f [3] Vershinin et al.,Science 303, 1995 (2004). In any case, baring certain artificial geometries, it is [4] S. A. Kivelson and V. J. Emery, in Strongly Corre- clear that for a long wave length modulation, ξQ 1, lated Electronic Materials: The Los Alamos Symposium ≪ the Kosterlitz-Thouless temperature TKT is exponen- 1993 ed. by K. S. Bedell, Z. Wang, B. E. Meltzer, A.V. tially lower than the MF transition temperature; on the Balatsky, and E. Abrahams (Addison-Wesley, Redding, otherhand,formodulationswithξQ 1,thephasefluc- 1994). tuation region is very narrow and T ∼ TMF. In this [5] E. Arrigoni, E. Fradkin, and S. A. Kivelson, Phys. Rev. KT ≈ c B 69, 214519 (2004). regime, the mean-field superconducting temperature is [6] E. W. Carlson, V. J. Emery, S. A. Kivelson, and D. Or- still exponentially enhanced relative to its value in the gad,inThePhysicsofConventional andUnconventional uniform state with the same average paring interaction Superconductors, edited by K. H. Bennemann and J. B. strength, U¯. For even faster modulation, ξQ 1, the Ketterson (Springer-Verlag, Berlin, 2002). ≫ MF transition temperature drops since the pairings in- [7] K. Machida, T. Koyama, and T. Matsubara, Phys. Rev. teraction modulation averagesout on the length scale of B 23, 99 (1981). ξ. This trend is presented qualitatively in Fig. 1. [8] M. Vojta and S. Sachdev, Phys. Rev. Lett. 83, 3916 (1999). For a “dirty” superconductor with a mean free path [9] I. Martin, G. Ortiz, A. V. Balatsky, and A. R. Bishop, shorterthanthecleancoherencelength,ℓ=vFτ <ξ,the Int.J. of Mod. Phys., 14, 3567 (2000). effect ofphase fluctuations canbe estimatedin the same [10] S. Chakravarty, R. B. Laughlin, D. K. Morr, C. Nayak, way as in the clean limit, with minor changes (involving Phys. Rev.B 63, 094503 (2001). prefactors) but with the coherence length redefined as [11] A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, ξ =√ξℓ. Methods of Quantum Field Theory in Statistical Physics d Summary. We studied the effect of nanoscale inhomo- (Dover, New York,1975). [12] R.Bellman, Introduction to Matrix Analysis(SocforIn- geneity on the superconducting transition temperature, dustrial & Applied Math, 1997). Tc. We considered two possible kinds of inhomogeneity: [13] N.D.MerminandH.Wagner,Phys.Rev.Lett.22,1133 the modulations of the paring strength and of the back- (1966). ground potential. In the weak coupling BCS regime, we [14] J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 find that inhomogeneity which is incommensurate with (1973). theFermisurfacenestingvectorsenhancesT relativeto [15] V.J.Emery and S.A.Kivelson, Nature 374 434 (1995). c