Origin of the T c enhancement in heterostructure cuprate superconductors
aa r X i v : . [ c ond - m a t . s up r- c on ] S e p Origin of the T c enhancement in heterostructure cuprate superconductors Doron L. Bergman and T. Pereg-Barnea
Department of Physics, California Institute of Technology,1200 E. California Blvd, MC114-36, Pasadena, CA 91125 (Dated: November 19, 2018)Recent experiments on heterostructures composed of two or more films of cuprate superconductorsof different oxygen doping levels[1, 2] have shown a remarkable T c enhancement (up to 50%) relativeto single compound films. We provide here a simple explanation of the enhancement which arisesnaturally from a collection of experimental works. We show that the enhancement could be causedby a structural change in the lattice, namely an increase in the distance of the apical oxygen fromthe copper-oxygen plane. This increase modifies the effective off-site interaction in the plane whichin turn enhances the d-wave superconductivity order parameter. To illustrate this point we studythe extended Hubbard model using the fluctuation exchange approximation. In 2008 Yuli at al. [1] presented a novel technique to en-hance the superconducting transition temperature (T c )of LSCO films. The enhancement was achieved in a het-erostructure of two thin layers of the same cuprate super-conductor with different carrier density in the two layers.The experimental setup used La − x Sr x CuO (LSCO)structures where the top layer had x = 0 .
35 (overdoped)and in the bottom layer x ranged from the underdopedto the overdoped. In all measured heterostructures T c was higher in the heterostructure than that of a singlecompound LSCO film with the same doping x as thebottom layer. The largest enhancement occurred around x = 0 .
12 where T c of the heterostructure was 32K, about50% larger than T c of the single compound film.In another experiment Gozar et al. [2] created het-erostructures with high T c from layers of metallic LSCO(overdoped) and either insulating LCO (the parent com-pound) or superconducting LCO (in this experiment thecompound was LaCuO δ with δ = 0 for the insulator).These intriguing results present a possible new direc-tion in the exploration of high temperature superconduc-tors. At the moment, the phenomenon of T c enhance-ment is far from understood and seems to depend cru-cially on materials and growth method[3]. A few ideashave been put forth regarding the mechanism by whichthe enhancement occurs. These ideas are closely relatedto the different point of views on the origin of the pseu-dogap and its relation to the superconducting phase.For example, if one takes the point of view that thepseudogap is related to an order parameter which com-petes with superconductivity then the tendency to de-velop such an order reduces T c . Therefore, if in the het-erostructure this tendency is suppressed superconductiv-ity will prevail to higher temperatures. Some support forthis point of view is provided by the experiment[1] in thefollowing way. The largest enhancement occurs close to1/8 filling. At this filling, the single compound films havea dip in the T c vs. x curve. The dip is believed to bedue to charge stripes which are favorable at this filling.In the heterostructures no such dip is observed.Another point of view of the pseudogap physics is that in the underdoped side of the superconducting dome, T c is restricted by phase fluctuations while pairing persistsup to higher energy scale, possibly of the order of thepseudogap temperature, T ∗ . Taking this point of view, ithas been suggested by Yuli et al. [1] and theoretically ex-plored by Berg et al. [4] and by Goren and Altman[5], thatin the heterostructure, enhancement may occur if phasefluctuations are suppressed compared to the single com-pound film. To suppress the fluctuations the superfluiddensity should be increased, making vortices more costlyin energy. Higher superfluid density can be achieved byhigher density of states at the Fermi level which can beprovided by a metallic layer. This view is supported bythe fact that the enhancement only occurs on the under-doped side of the superconducting dome.Both of the directions above are interesting and deservefurther investigation. In this work, however, we point ata different direction which does not necessarily discrimi-nate between the two scenarios above but may lead theway to a microscopic description of the phenomenon. Weare guided by both experimental evidence and relevantnumerical findings. We first assume that the cause for theT c enhancement resides within the copper-oxygen plane.This starting point should be contrasted with approachesthat include a bilayer interface of the two materials andrely on strong hopping between the layers[4, 5]. Unfor-tunately, the inter-layer hopping strength in the cupratematerials may be too weak to explain such strong vari-ations in T c . A quantitative measure of the interlayerhopping strength has been seen, for example, in transportproperties[6, 7] and optical conductivity[8]. There is noevidence to suggest the heterostructures have strongerinter-plane coupling than that of the single compoundfilm.What is then the difference between the single com-pound film and the heterostructure? One difference maybe the electronic density. The amount of doping x in thebottom layer of the heterostructure may be different fromthat of the starting material due to charge migration be-tween the two layers. This leads to a ’self doping’ effectwhich may change T c . This effect alone, however, cannotaccount for the large T c enhancement in the experiment.Doping alone can not yield a transition temperature thatis significantly higher than that of the single compoundfilm at optimal doping.Another difference is structural; atoms move andbonds stretch/shrink to relieve strain resulting from thetwo layer mismatch. A recent experiment by Zhou etal. [9] provides evidence that structural changes are inti-mately related to the T c enhancement. In these exper-iments heterostructures of the parent compound (LCO)and overdoped metallic LSCO are fabricated. These het-erostructures also display superconductivity with an en-hanced T c relative to the single film. In order to de-tect where in the heterostructure superconductivity oc-curs, the experimental group introduce zinc impuritiesto the sample, layer by layer[10]. In-plane zinc impu-rities are known to suppress T c in the cuprates by afactor of about 2. In the heterostructure their effectwas significant only when introduced to one specific layerwhich resides 1 layer above the interface, on the insulat-ing/superconducting LCO side. This leads to the conclu-sion that only one layer is responsible for the heterostruc-ture’s high T c . In addition, the apical oxygen distancefrom each layer ( d ) was measured by x-ray scattering[9]and was found to vary significantly in the heterostruc-ture, between d ≈ . A and d ≈ . A . In contrast,both the insulator (LCO) and metal (LSCO) have a bulkapical distance of d ≈ . A (as pointed out in Ref. 9and measured in Ref. 11). Therefore, the heterostructureachieves apical oxygen distances in doped LSCO, that arenot reached in bulk samples. We believe that this latticerearrangement is responsible for the T c enhancement andmotivate this point of view using a microscopic model be-low.In order to discuss superconductivity in the cuprateswithout speculating on the pairing mechanism and theorigin of the pseudogap we explore the extended Hub-bard model. This model contains both the hopping pa-rameters that are relevant to the cuprates and can befit to the band structure as it is observed in experiment(e.g. ARPES[12]) and the strongly correlated nature ofthis electronic system. The Hamiltonian reads H = − X ij,σ t ij c † iσ c jσ + U X i n i ↑ n i ↓ + 12 V X h ij i n i n j (1)where the indices i, j go over the lattice sites with angularbrackets denoting nearest neighbors and σ = ↑↓ is thespin label. The number of electrons at site i is denotedby n i = n i ↑ + n i ↓ = P σ c † iσ c iσ .Numerical analysis of this model leads to d-wavesuperconductivity[13]. In order to see how this comesabout one has to go beyond simple mean field and in-clude strong spin and charge fluctuations. In this modelthe onsite repulsion U promotes d-wave superconductiv-ity while the off-site repulsion V suppresses it. This V ( meV ) − V e ff / U e ff FIG. 1: The effective attractive off-site interaction V eff as afunction of the bare repulsive, off-site, interaction V in unitsof the on-site repulsion U . In the calculation the values t =488 meV and U = 2 . eV were used. The (bare) repulsive V axis is in units of meV . The values taken for the barerepulsive V range from V = 0 to its expected value for bulkLCO as found in Ref. 15. has been seen, for example, in the fluctuation exchange(FLEX) approximation[13]. In this approximation spinand charge fluctuations are taken into account (to allorders in perturbation theory) in the renormalized in-teraction vertex function. This vertex together with theelectronic Green’s function is used in the Eliashberg[14]theory to determine the strength of pairing. The analysisshows that when the off-site interaction V is increasedd-wave pairing is suppressed. If V is larger than somecritical value a charge density wave becomes more ener-getically favorable than superconductivity.The connection between the off-site interaction termand the apical oxygen distance from the Cu-O plane hasbeen recently explored from first principles by Yin andKu[15] and by Weber et al. [16]. The conclusion of thesestudies is that the distance of the apical oxygen fromthe layer is inversely related to the bare V . For mate-rials with larger apical distance the off-site repulsion issmaller. Other terms in the Hamiltonian, however, arerather insensitive to this distance. The result is a d-wave superconductor with larger amplitude. This is themain point of the paper - the apical oxygen distances inthe heterostructure are significantly larger than in bulkLSCO, in particular at the superconducting layer. Thiscauses a reduction of the off-site repulsion V and super-conductivity is therefore enhanced.In order to relate these findings to the experimentalsetup we investigate the Hamiltonian in Eq. 1 as a modelfor the highest T c layer close to the interface. We startwith repulsive interactions both on-site ( U ) and off-site( V ) and vary the strength of V to capture the effect of − V eff /U eff ∆ /tS/t −5 FIG. 2: The mean field result for d-wave superconductivityorder parameter (squares) and Ne´el order (diamonds) as afunction of the ratio between the off-site interaction V andthe on-site interaction U . The inset shows a zoom-in on therelevant interaction ratio for the superconducting order pa-rameter ∆ . the apical oxygen distance. As a first step we use theFLEX approximation to include the Fermionic responsein the effective interaction strength. The renormalizedinteractions U eff and V eff are presented in Fig. 1. Notethat the effective off-site interaction is attractive . Usingrenormalized interaction coefficients is crucial, since su-perconductivity requires pairing channel.An intuitive understanding of the mechanism bywhich off-site interaction is related to superconductiv-ity emerges from the FLEX approximation. We startwith a repulsive on-site and off-site interaction in the ex-tended Hubbard model, as appropriate for this system.However, when considering both interactions together inthe vertex function the total off-site channel is attrac-tive. This can be seen even at the level of the standardHubbard model (without off-site interaction). The Hub-bard model close to half filling was studied by Scalapinoin Ref. [17] where it was found that the on-site repul-sive interaction U leads to an effective off-site attraction.This is the result of proximity to the antiferromagneticphase at half filling which is due to double hopping pro-cesses of the order of t /U . Away from half filling thereis no long range Ne´el order but the susceptibility is stillpeaked around ( π, π ). The contribution of such a struc-ture to the interaction vertex is, again, strongest at thispoint in momentum space which leads to a large attrac-tion on nearest neighbor sites. This can be viewed aslarge Friedel oscillations. Electrons on nearest neighborsites take advantage of these oscillations through pairingin the d-wave channel.In order to determine T c realistically one should usethe renormalized vertex in a full Eliashberg calculation as was done by Onari et al. [13] In the present work we cal-culate effective interaction parameters using the FLEXapproximation, and turn to mean field theory in orderto sketch the qualitative effect of the interaction on theorder parameter. In the effective model the on-site in-teraction U eff is repulsive while the off-site interaction V eff is attractive. The bare repulsion, and the inducedeffective attraction combine to give an effective attractive off-site interaction and a renormalized on-site repulsion.When the apical oxygen distance is increased, the barerepulsion is decreased, and the overall effective interac-tion is more strongly attractive. We show the mean-fieldresult for the above effective Hubbard model in Fig 2. Inthis figure, as the ratio between the magnitude of the at-tractive V eff and the on-site interaction U eff is increasedd-wave superconductivity is enhanced and the Ne´el or-der (or spin density wave) is suppressed. This result isobtained by the usual mean field decomposition of theinteraction terms in the superconducting and antiferro-magnetic channels. The order parameters are found byminimizing the mean-field free energy. F = − X k ∈ RBZ [ E + E ] − N U (¯ n − S ) − N V ∆ E = s ( U S ) + ( V ∆ k ) + ǫ k + ǫ k + Q ǫ k + ǫ k + Q ) ξE = s ( U S ) + ( V ∆ k ) + ǫ k + ǫ k + Q − ( ǫ k + ǫ k + Q ) ξξ = s ( U S ) + (cid:18) ǫ k − ǫ k + Q (cid:19) (2)where RBZ stands for the sum over the reduced Brillouinzone, ǫ k = − t (cos( k x ) + cos( k y )) − µ + U ¯ n is the banddispersion which includes a chemical potential shift U ¯ n ,∆ k = ∆ χ k with χ k = 2(cos( k x ) − cos( k y )) is the gapfunction, S is the Ne´el order parameter which folds theBrillouin zone with the wavevector Q = ( π, π ) and N isthe number of sites. We have used an attractive potentialto derive these equations and write | V eff | to emphasizethis point. U N P k ∈ RBZ h ǫ k + ǫ k + Q ) / ξE + − ( ǫ k + ǫ k + Q ) / ξE i = 1 − V N P k ∈ RBZ χ k h E + E i = 1 − N P k ∈ RBZ (cid:20) ǫ k + ǫ k + Q + ξE + ǫ k + ǫ k + Q − ξE (cid:21) = ¯ n (3)where the third equation has only a trivial solution(¯ n = 0) at half filling. We must point out the numbers wefind from our simple mean field treatment in Fig. 2, sug-gest the superconductor is significant only if | V eff | U eff > . .
08 for thisratio. However, in a full Eliashberg treatment even asmall attractive off-site potential (like that of Fig. 1) cancause the superconducting T c to be non-zero[13]. Wetherefore conclude that our simplistic analysis gives thecorrect qualitative results, while leaving something to bedesired in terms of quantitative results.Before our conclusions, let us briefly discuss the possi-bility that phase fluctuations are suppressing T c on theunderdoped side. In the models above only the ampli-tude of the d-wave pairing order parameter is considered.However, there is a lot of experimental evidence thatphase fluctuations are important in the pseudogap phase.Naturally, the question arises whether phase fluctuationsare reducing T c from the temperature in which pairs areformed (which should be associated with a higher en-ergy scale, possibly as high as T ∗ ) or pairing and phasecoherence appear at the same temperature (T c ) whilethe higher energy pseudogap is due to a competing orderparameter. To shed more light on this issue it wouldbe helpful to study the pseudogap regime of the het-erostructure and measure T ∗ . We propose to perform thefollowing experiments. Though experimentally challeng-ing, it may be possible to measure the density of states(DOS) in the heterostructure and determine its pseudo-gap temperature T ∗ . If we adopt the point of view thatT ∗ is due to a competing order, unrelated to supercon-ductivity we expect this temperature to be lower in theheterostructure than in the single compound films. Thiswill reflect the competition; when the competing orderis suppressed by a larger apical oxygen distance, d-wavesuperconductivity is enhanced. If, on the other hand T ∗ is the temperature at which pairing begins it should behigher in the heterostructure since T c and T ∗ are related.Another important measurement is that of the Nernst ef-fect which has provided evidence for phase fluctuationsin the past[18] together with a diamagnetism probe[19].It is important to determine whether the phase fluctua-tions are enhanced or suppressed in the heterostructurerelative to the single compound film.In addition, we would like to propose a third experi-ment which would directly address the role of the apicaloxygen in the T c enhancement. Applying tensile stressalong the c-axis will act to reduce the apical oxygen dis-tance. This, according to the scenario we present hereshould increase the bare off-site repulsion and reduce T c .To summarize, we provide a simple explanation tothe enhancement of the superconducting transition tem-perature, T c , in LSCO heterostructures compared withheterogenous, single compound films, as seen in recentexperiments[1, 2]. We have analyzed the extended Hub-bard model in the FLEX approximation and used aheuristic mean-field analysis to asses the effect of theapical oxygen distance on order formation. We con-clude that the off-site nearest neighbor interaction hasa tremendous effect on T c in d-wave superconductors. Inparticular, when this interaction becomes more negative (in other words when it’s bare repulsive component is re-duced) the transition temperature increases. This inter-action has been found to be inversely related to the apicaloxygen distance from the copper oxygen planes[15, 16].It has also been seen that the apical oxygen distance islarger in the heterostructure, compared to the single com-pound film[9]. We conclude that it is this structural dif-ference between the heterostructure and the single com-pound film that is responsible for the enhancement ofT c .The authors wish to acknowledge useful discussionswith O. Millo, N. Lindner, G. Refael and N.-C. Yeh.TPB acknowledges funding from the Research Corpora-tion Cottrell Fellowship and DLB was supported by theSherman Fairchild Foundation. [1] O.Yuli, I. Asulin, O. Millo, D. Orgad, L. Iomin and G.Koren, Phys. Rev. Lett. , 057005 (2008).[2] A. Gozar, G. Logvenov, L. Fitting Kourkoutis, A. T.Bollinger, L. A. Giannuzzi, D. A. Muller and I. Bozovic,Nature , 782 (2008).[3] G. Koren and O. Millo, Phys. Rev. B , 134516 (2010).[4] E. Berg, D. Orgad and S. A. Kivelson, Phys. Rev. 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