Understanding electron-doped cuprate superconductors as hole superconductors
aa r X i v : . [ c ond - m a t . s up r- c on ] A p r Understanding electron-doped cuprate superconductors as hole superconductors
J. E. Hirsch a and F. Marsiglio b a Department of Physics, University of California, San Diego, La Jolla, CA 92093-0319 b Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2E1
Since their experimental discovery in 1989, the electron-doped cuprate superconductors have pre-sented both a major challenge and a major opportunity. The major challenge has been to determinewhether these materials are fundamentally different from or essentially similar to their hole-dopedcounterparts; a major opportunity because answering this question would strongly constrain thepossible explanations for what is the essential physics that leads to high temperature superconduc-tivity in the cuprates, which is still not agreed upon. Here we argue that experimental results overthe past 30 years on electron-doped cuprate materials have provided conclusive answers to thesefundamental questions, by establishing that both in hole- and electron-doped cuprates, supercon-ductivity originates in pairing of hole carriers in the same band. We discuss a model to describe thisphysics that is different from the generally accepted ones, and calculate physical observables thatagree with experiment, in particular tunneling characteristics. We argue that our model is simpler,more natural and more compelling than other models. Unlike other models, ours was originallyproposed before rather than after many key experiments were performed.
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I. INTRODUCTION
Shortly after the discovery of high temperature su-perconductivity in cuprate superconductors in 1986 [1],it became clear that the carriers responsible for super-conductivity in these materials were holes [2–7]. Uponchanging the chemical composition of the parent insulat-ing compound so that hole carriers were added to thecopper-oxygen planes, the superconducting T c was foundto increase, go through a maximum and then decrease tozero in the ‘overdoped’ regime [4, 8].Then, on January 26, 1989 it was reported by Tak-agi, Tokura and Uchida [9] that by doping a parent insu-lating material with electrons instead of holes supercon-ductivity also occurred, albeit with a smaller maximumcritical temperature. Initially, experiments appeared toshow very definitely that indeed the charge carriers inthese electron-doped materials were electrons [9–11], asthe title of Ref. [9] claimed: “A superconducting copperoxide compound with electrons as the charge carriers” .The general reaction to this discovery was that it pro-vided evidence for an approximate electron-hole symme-try [12–15]. For example, Art Sleight stated in March1989 [16] “ This symmetry between adding and subtract-ing electrons will have to be reflected in any theory thatexplains high-temperature superconductivity, and existingtheories based on the supposition that there is somethingunique about hole carriers are ‘out the window’ .”Electron-hole symmetry is to be expected if the un-doped parent insulating compound is assumed to be de-scribed by a half-filled band governed by the HubbardHamiltonian [17], which is particle-hole symmetric, anddoping of holes or electrons results in changing the car-rier occupation in this band . This was widely assumedto be the case at that time and continues to be widelyassumed to be the case today.Instead, immediately after the discovery of the
FIG. 1: Schematic depiction of how holes are created by elec-tron doping. The electron added to Cu repels an electronfrom O − to the neighboring Cu , leaving behind a hole inoxygen ( O − ). electron-doped materials we pointed out [18–22] that anatural explanation existed for why hole carriers of thesame nature as the hole carriers in the hole-doped mate-rials [23] would be induced in the electron-doped materi-als, and we predicted that subsequent experiments wouldshow that hole carriers exist and are responsible for su-perconductivity also in the electron-doped materials [18–22]. Figure 1 shows schematically how holes on O = canresult from electron-doping of Cu ++ , we will delve intothe details later. No experimental evidence suggestingneither that hole carriers existed nor that they were re-sponsible for superconductivity in these materials existedat that time.Already very soon thereafter, EELS experiments sug-gested the presence of holes at the oxygen sites inelectron-doped cuprates [24]. That holes participate inthe transport was shown by detailed and extensive mag-netotransport measurements by several different experi-mental groups extending over many years [25–35]. Theseexperimental results and their analysis showed that thereare both electron and hole charge carriers in the electron-doped cuprates, that hole carriers dominate the transportin the regime where electron-doped cuprates become su-perconducting, and that it is the hole carriers that likelydrive superconductivity in these materials [25, 34, 35].However several key questions remain unsettled. Whatis the nature of the electron and hole carriers in theelectron-doped materials? Why, if there are hole carriersin electron-doped cuprates, aren’t there electron carri-ers in hole-doped cuprates? Are the hole carriers in theelectron-doped materials of the same nature as those inthe hole-doped materials? Even if they are, is the pair-ing mechanism the same? We argue that definite answersto these questions whould go a long way towards eluci-dating the origin of superconductivity in both hole- andelectron-doped cuprates.In this paper we elaborate on the simple answers tothese questions that we proposed 30 years ago [18–22],and argue that various experimental results obtained dur-ing these 30 years support our original proposal. We willalso argue that other proposals to explain these questionsare complicated, unnatural and implausible.In a nutshell, our proposal was and is: hole carriersresponsible for superconductivity in both hole-doped andelectron-doped materials reside in a band resulting fromoverlapping oxygen pπ orbitals in the Cu-O plane thatpoint perpendicular to the Cu − O bonds, as shown inFig. 2. This band is full in the undoped case and be-comes slightly less than full both on the hole-doped andthe electron-doped side, for reasons we will explain. Theelectron carriers in the electron-doped cuprates reside inthe Cu − O band formed by the overlapping Cu d x − y and O pσ orbitals pointing along the Cu − O bond. Pro-posals that hole carriers in the hole-doped cuprates residein the O pπ orbitals were also made early on by Goddardet al [36], Stechel and Jennison [37], Birgeneau et al [38]and Ikeda [39]. II. TWO-BAND MODELS
As discussed above, experiments indicate [25–35] thathole carriers exist and are responsible for superconductiv-ity in both electron-doped and hole-doped cuprate super-conductors. Furthermore, these experiments show thatin electron-doped cuprates there is two-band conductionin the normal state, with the other band being electron-like. The question is, where are these carriers? We startby giving a brief overview of three different possibilitiesthat have been proposed: (i) Our two-band model, (ii)two-band t − J model, and (iii) reconstructed Fermi sur-face models. Cu ++ p σ O = p σ p π p π d x -y FIG. 2:
Cu d x − y and oxygen orbitals in the Cu-O planes.In the undoped parent compound the nominal valence is Cu ++ and O = and there is one hole in the filled Cu d or-bital. The O pπ orbitals point perpendicular to the Cu-Obonds, the pσ orbitals parallel. We propose that doped holesreside in a band resulting principally from overlapping O pπ orbitals for both hole- and electron-doped cuprates. (i) Our two-band model The simplest way to have two-band conduction for thesystem shown in Fig. 2 is if one band involves princi-pally the O pσ orbitals hybridized with the Cu d x − y ,which we will call the Cu-O band, and the other bandinvolves principally the O pπ orbitals orbitals pointingperpendicular to the O pσ orbitals in the plane, whichwe will call the O band. For the hole-doped materials wehad proposed [23, 40], before the electron-doped materi-als were discovered, that their high T c could be under-stood as arising from hole carriers in the O band. Fur-thermore within this theoretical framework it is predictedthat superconductivity can only arise from hole carriersin a nearly full band [41–43]. In other words, within thistheoretical framework ‘A superconducting copper oxidecompound with electrons as the charge carriers ’, as an-nounced [9] by Tokura, Takagi and Uchida on January26, 1989, cannot exist. As we now know from experi-ments [25–35], it does not exist, at least to date.There is a simple way to understand why doping withelectrons can create holes in the O band, illustrated inFig. 3, in the hole representation. First, we assume theO pπ energy level for a hole is lower than the O pσ level,in other words it costs less energy to remove an electronfrom the O pπ orbital than from the O pσ orbital. This isplausible for two reasons: first, as pointed out by Birge-neau et al [38] and Goddard and coworkers [36], becausethere is more negative charge near the center of a plaque-tte than along the Cu-O-Cu line, it costs less Coulombenergy to remove an electron (create a hole) from the pπ orbitals that point towards the center of the plaquettethan from the pσ orbitals directed along the Cu-O bond.Second, as discussed in ref. [44], the orbital relaxationeffect that occurs when an electron is removed from the O = ion is stronger if the electron is in the pπ orbital thatis doubly occupied (by electrons) in the undoped casethan if it is in the pσ orbital that is only about 1 . pπ orbital relative to creating it in the pσ orbital. zzzzzzzadd a hole add an electronelectronadd a hole add an electron hole-doped electron-doped Cu ++ O Cu ++ Cu ++ O = Cu ++ Cu ++ O = Cu ++ Cu ++ O Cu ++ Cu + O Cu + add a hole add an electron doped electron doped Cu ++ O Cu ++ Cu ++ O Cu ++ Cu ++ O Cu ++ Cu + O Cu + p σ p π p σ p π p σ p π p σ p π FIG. 3: Illustration of how oxygen hole carriers are createdby both hole and electron doping in high T c cuprates in thehole representation (see text). Arrows denote holes. The dif-ference in the relative locations of the O = and Cu ++ orbitalsin the hole-doped and electron-doped cases arises due to theirdifferent crystal structures, T and T ′ . In the undoped case, there is one hole at each Cu ++ site. For the electron-doped material, we assume thesingle-hole O pπ energy level is lower than the Cu energylevel. Nevertheless in the undoped compound the Cuhole doesn’t ‘fall’ onto the neighboring O = ion becauseof the cost in Coulomb repulsion between neighboringholes at Cu and O sites. Upon electron doping the holeis removed from (electron is added to) a Cu ++ ion, nowthe hole from a neighboring Cu ++ can fall into the O = ion without paying nearest neighbor Coulomb repulsionenergy, as illustrated in the lower right panel of Fig. 3.The net result from adding an electron to a Cu ++ is two Cu + ions and one O − ion, with the hole in the pπ or-bital. In other words, two extra electrons reside in theCu-O band and one hole in the O band.This process can (and will) happen in the electron-doped cuprate materials because of the absence of apicaloxygens in their structure (T ′ structure), that increasesthe electrostatic potential at the Cu ++ site relative to thecase of the hole-doped materials (T structure). Becausethe apical oxygen is relatively closer to the Cu atomsthan to the the O atoms in the plane, the Cu hole energylevel is relatively higher with respect to the O levels inthe T ′ structure (right side in Fig. 3). This facilitates both the electron-doping of the material (it is not possibleto dope electrons in the T structure) and the transfer ofelectrons from O = sites to neighboring Cu ++ sites, thuscreating O holes. A detailed analysis of the energeticsof these processes and the important role of reduction ingetting the carriers to delocalize is given in ref. [45]. In all the other models that have been proposed todescribe the electron-doped and hole-doped cuprate su-perconductors in recent years, the O pπ orbitals are notincluded. It is assumed that the O band is far below theFermi energy and can be ignored. We discussed in Ref.[44] why this may not be so. (ii) Two-band t − J model Another possible model to give rise to two-band su-perconductivity with electrons and holes was suggestedby T. Xiang and coworkers [46]. They proposed that thetwo bands in question are a Zhang-Rice singlet band andthe upper Hubbard band, both originating in the overlapof orbitals
Cud x − y and Opσ shown in Fig. 2 (what wecalled the Cu-O band). The authors argue that both theZhang-Rice singlet band and the upper Hubbard bandshould be described by effective one-band t − J models,or, more accurately, a ‘hybridized two-band t-J model’.Then, this model maps onto a one-band t − U − J model,using approximations that according to the authors ‘maynot be fully satisfied in real materials’. The authors ar-gue that this model gives results for a Fermi surface den-sity map consistent with ARPES observations. However,they don’t explain how this model explains the transportexperiments [25–35] that clearly show electrons and holecarriers in two different bands. (iii) Reconstructed Fermi surface models In these models it is suggested that some kind of trans-lational symmetry breaking with wavevector ( π, π ) dou-bles the unit cell and this gives rise to both electron andhole carriers. Lin and Millis [47] argue that this recon-struction of the Fermi surface occurs below Ce concentra-tion x = x c = 0 .
16 due to antiferromagnetic long-rangeorder, giving rise to electron pockets at (0 , π ) and smallhole pockets at ( π/ , π/ x > x c only a largehole-like Fermi surface exists occupying about half theBrillouin zone. However, Motoyama and coworkers re-port [48] that the antiferromagnetic order disappears al-ready at x = 0 . III. MAGNETOTRANSPORT
Initially, Hall coefficient measurements on electron-doped cuprates yielded a negative Hall coeffcient [11],consistent with the expectation that these were supercon-ductors ‘with electrons as the charge carriers’ [9]. Laterthis changed, when experiments were performed on singlecrystals and thin films.Figure 4 shows measurements of the Hall coefficientversus temperature for a range of doping levels for a typ-ical electron-doped material,
P r − x Ce x CuO [32], and ahole-doped material, ( La − x Sr x ) CuO [54]. Results for T c versus doping are also shown [53, 54]. There is a cleardifference in the behavior of the Hall coefficient R H inboth cases. FIG. 4: Temperature dependence of Hall coefficient and T c versus doping for electron-doped (upper panels) [32, 53] andhole-doped (lower panels) [54] cuprates.. In the hole-doped material, R H is almost temperatureindependent and positive over the entire doping rangewhere superconductivity occurs (up to x = 0 . x = 0 .
125 the positive Hall coef-ficient becomes very small. For doping x = 0 .
15 andabove (not shown), the Hall coefficient is negative andno superconductivity exists [54].These results for the Hall coefficient of hole-dopedcuprates were initially qualitatively interpreted [54] asa crossover from a Mott-Hubbard regime ( R H >
0) atlow doping to a Fermi liquid regime ( R H <
0) at highdoping [55] in a single band model. However contrary toinitial expectations it was found within dynamical meanfield theory that the Hall coefficient of a half-filled Hub-bard model doped with holes is negative for all dopings and given by the bare band structure results [56, 57]. Atfinite temperatures it was reportedly found that the Hallcoefficient can turn positive [57–59]; however the tem-perature and doping dependence does not resemble theexperimental results shown in Fig. 4. In addition, thesetreatments predict that R H ( δ ) = − R H ( − δ ), where δ isthe doping away from half-filling, which is very differentfrom the behavior shown in Fig. 4.Instead, we argue that the results for R H for hole-doped cuprates shown in Fig. 4, namely the near tem-perature independence and the positive value decreasingwith hole doping, are most simply interpreted as resultingfrom doping of a single band from an initial state wherethe band is completely full, as predicted by our model,where the band is the Opπ band. As hole carriers areadded the magnitude of the Hall coefficient decreases asexpected from the simple formula R H ∼ / ( n h ec ). As theband becomes half full the Hall coefficient would changesign from positive to negative in a simple picture. Thechange in sign at high hole doping from positive to nega-tive before the band is half-full can be simply explainedas due to a high scattering rate that would prevent carri-ers from completing closed hole orbits without scattering,which is necessary for the Hall coefficient to be positive.The behavior of the Hall coefficient for electron-dopedcuprates as a function of temperature and doping shownin Fig. 4 is drastically different from a ‘mirror image’of the hole-doped cuprates with a sign change, as wouldbe predicted by a single band model. It shows unmistak-able evidence for two-band conduction, one with electroncarriers and one with hole carriers, with electrons dom-inating at low electron doping and holes dominating athigh electron doping. For an isotropic two band modelwith electron and hole carriers of densities n e and n h theHall coefficient is R H = − n e ec − ( n h /n e )( µ h /µ e ) n h /n e )( µ h /µ e ) (1)and will be negative if the mobility of the hole carriers( µ h ) is much smaller than that of the electron carriers( µ e ). Wang et al [25] and Crusellas et al [27] have arguedthat the hole mobility increases rapidly as the tempera-ture is lowered due to a decrease in the hole scatteringrate, and have fitted the temperature and doping depen-dence of resistivity and Hall coefficient measured in ex-periments using reasonable assumptions for temperature-dependent electron and hole mobilities. Wang et alpointed out [25] the similarity in the temperature depen-dence of the scattering rate of the hole carriers in hole-doped cuprates inferred from the resistivity with that ofthe hole carriers in electron-doped cuprates inferred from R H , suggesting it is the same carriers and the same scat-tering processes. Crusellas et al pointed out [27] thatthe measured hole mobility in electron-doped cupratesis very similar to that observed in hole-doped cuprates.Note also that in a single-band situation a temperature-dependent scattering rate does not result in temperaturedependence of the Hall coefficient.Further evidence for the existence of two-band con-duction in the regime where electron-doped cuprates su-perconduct comes from the sign and magnitude of themagnetoresistance. Already in 1994 Jiang et al pointedout [29] that “we see a remarkable correlation betweenthe occurrence of superconductivity and the appearanceof a positive MR” and that a “signature of two-bandconduction is a positive magnetoresistance”. RecentlyLi et al [35] pointed out that there is a considerable in-crease in the magnitude of the magnetoresistance in theregion where bulk superconductivity is first seen, whichreveals the underlying two bands. Jiang et al [29] as wellas Fournier et al [30] also pointed out that the anoma-lously large Nerst coefficient they observed together withthe measured small thermopower cannot be explained bya single band model and is direct evidence of two-bandconduction with carriers of opposite sign in both bands.Over the years Greene and coworkers have performedextensive measurements [29, 30, 32–34] on Hall coeffi-cient, thermopower, magnetoresistance and Nerst effectin electron-doped cuprates, and carefully analyzed theirdata. They found compelling evidence from these mea-surements for two-band conduction, an electron-bandand a hole-band, and dominant role of hole carriers inthe regime where the materials are superconducting, andin particular that the regime where hole transport beginsto dominate coincides with the onset of superconductiv-ity. We believe that these together with other transportexperiments and analysis [25, 28, 31, 35] have establishedexperimentally that “ in electron doped cuprates holes areresponsible for the superconductivity ” [34]. IV. QUANTUM OSCILLATIONS
In 2007, Shubnikov-de Haas oscillations in underdopedhole-doped cuprates provided clear evidence for the ex-istence of closed small Fermi surface pockets in thosematerials [60, 61]. This was very surprising in viewof the consensus that existed that Fermi liquid con-cepts did not apply to underdoped cuprates, and thatthe Fermi surface consisted of Fermi arcs rather thanclosed surfaces. The frequency of these oscillations was530 T in Y Ba Cu O . [60], corresponding to 1 .
9% ofthe two-dimensional Brillouin zone area, and 660 T in Y Ba Cu O [61, 62] with T c = 80 K , corresponding to2 .
4% of the Brillouin zone. For the first case T c = 57 . K and p = 0 .
10 holes per planar Cu atom, for the secondcase T c = 80 K and p = 0 .
125 [61] or p = 0 .
14 [62] holesper Cu atom. Quantum oscillations have also been ob-served in underdoped
HgBa CuO δ [63, 64], with fre-quency 840 T corresponding to about 3% of the Brillouinzone, or 0.061 carriers per pocket. Several other suchobservations in hole-doped cuprates have been reported[65–67]. Whether these oscillations are due to hole pock-ets or electron pockets or both has been controversial[68, 69] and is unsettled. Quantum oscillations have alsobeen observed in the overdoped regime of T l Ba CuO δ [70], with frequency 18,100T, corresponding to cross-sectional area 65% of the Brillouin zone.Within our model, in hole-doped cuprates holes aredoped into the Opπ band that is initially full, so we wouldexpect small hole orbits around the ( π, π ) point in theBrillouin zone, consistent with the Shubnikov-de Haasoscillations observed. It is not clear how these observa-tions can be consistent with doping a half-filled Hubbardband with holes, since according to what we reviewed inthe previous section the Hall coefficient at low tempera-tures in that model reflects the bare band structure.In electron-doped cuprates, Shubnikov-de Haas oscil-lations first detected in 2009 in
N d − x Ce x CuO [71]showed that small Fermi surface pockets exist both inthe optimally doped and overdoped samples [49, 50, 71].The measured frequency is approximately 300T, corre-sponding to 1 .
1% of the Brillouin zone. As mentionedearlier, it is hypothesized that they originate in recon-struction of the Fermi surface, however no long range an-tiferromagnetic order exists in this doping range [48]. Inaddition, from the reconstructed Fermi surface scenarioone would expect also quantum oscillations from electronpockets with frequency half of that produced by the holepockets, however a single low frequency is observed. In
N d − x Ce x CuO at high doping, both low frequency andhigh frequency ( ∼ kT ) oscillations are detected coex-isting in the range x > .
15 to x = 0 .
17 [49, 50], withthe high frequency oscillations, corresponding to a Fermisurface area of ∼
41% of the Brillouin zone dominating athigh doping, and the low frequency ones at low doping.The authors suggest that the reconstructed Fermi surfaceexists until the point where superconductivity disappearsat x ∼ .
175 and that the high frequency oscillations orig-inate in magnetic breakdown orbits, and they attributethe absence of evidence for electron pockets to damping.They acknowledge that “The mechanism responsible forthe broken translational symmetry is still to be clarified”,and suggest “ ‘hidden’ d-density-wave ordering” as a pos-sibility. Similar low frequency quantum oscillations arefound in
P r − x Ce x CuO ± δ and La − x Ce x CuO ± δ [51].Instead, within our model these observations are sim-ply explained by the existence of a small number of holecarriers in the Opπ band giving rise to the low frequencyoscillations, and electrons doped into the Cu − Opσ bandgiving rise to a large Fermi surface corresponding to amore than half-filled band and corresponding high fre-quency Shubnikov-de Haas oscillations.
V. PHOTOEMISSION
Recent photoemission investigations on electron-dopedcuprates were reviewed by Horio and Fujimori [72]. Inparticular they point out that the electron concentrationinferred from the area of the electron Fermi surface mea-sured by ARPES is significantly larger than the nominal Ce concentration. This was already found long ago byAlp and coworkers [73] via x-ray absorption spectroscopy.It is consistent with our picture that electron doping in-duces hole carriers in the plane, hence by charge conser-vation this generates extra electron carriers in additionto the ones doped.Horio and Fujimori (HF) emphasize the fact that an-nealing in a reducing atmosphere plays an essential rolein giving rise to superconductivity, and point out thatsuperconductivity can arise even in the absence of Cedoping. Contrary to earlier findings, they suggest that re-duction removes oxygen from the rare-earth layers ratherthan from impurity oxygens at the apical sites. This isin agreement with our prediction [45].The photoemission results reviewed by HF show aFermi surface developing around k ∼ ( π, π, π ). Similar results are found in photoe-mission measurements by Song et al [74]. We infer thatall these measurements are only detecting the quasiparti-cles residing in the Cu − Opσ band, that near half-fillingare strongly affected by the Hubbard U and for suffi-cient electron doping evolve into a simple Fermi surfacecentered around ( π, π ). These measurements don’t showthe hole carriers that according to the transport measure-ments must exist in a different band. We expect thosehole carriers to be in the Opπ orbitals forming a smallhole pocket at ( π, π ), both for electron-doped and hole-doped cuprates. We conclude that because of the strongorbital relaxation effects in this purely oxygen band thequasiparticle weight is very small [75, 76] and not visiblein current photoemission experiments.
VI. TUNNELING ASYMMETRY
Tunneling measurements have often been at odds withphotoemission experiments; both types of experimentshave unknown uncertainties because of the difficultieswith surface preparation, etc. as discussed in Ref. [77].Here we wish to focus on tunneling asymmetry, i.e.the difference in coherence peak heights, depending onwhether the sample is negatively or positively biasedwith respect to the tip. For hole-doped cuprates, a largenumber of studies have found that tunneling spectra areasymmetric, with asymmetry of universal sign [78]. Forelectron-doped cuprates, much less tunneling work hasbeen done. We will focus on data from Refs. [79–81],since specific details have been clarified through privatecommunication. In particular in the raw data (Figs. 2and 3 in Ref. [79]) the coherence peak is clearly higherwhen the sample is negatively biased with respect tothe (normal) tunneling tip. The authors report (privatecommunication) that this is universally true for all theirmeasurements. Furthermore, this remains true for bothNd . Ce . CuO − y at optimal doping (subject of Ref.[79]) and for Pr − x Ce x CuO − y as a function of doping,as shown in Ref. [80] (see their Fig. 1) and Ref. [81](Figs. 1 and 2).Tunneling spectra of Jubileo and coworkers [82] on P r − x LaCe x CuO − y also show a clear asymmetry of thesame sign (Figs.1 and 2), and this is also seen in tunnel-ing on P r − x LaCe x CuO by Miyakawa et al [83] (Figs.1 and 3) and Diamant et al [84] (Fig. 1).This tunneling asymmetry is important to confirmsince the hole mechanism of superconductivity [41, 42]predicts an energy-dependent superconducting order pa-rameter that results in a tunneling asymmetry [19] ofuniversal sign, as observed in these experiments. Instead,the RVB model of Anderson and coworkers also predictstunneling asymmetry [85] but of opposite sign for hole-doped and electron-doped cuprates [86]. Thus, estab-lishing the sign of the tunneling asymmetry in electron-doped cuprates can rule out one of these two theories[87].In the theory of hole superconductivity for a singleband, the asymmetry predicted by this model is clear[19]. We will provide further model calculations in thenext section, particularly in light of the fact that thetunneling data given in the different references give noindication of two-band behavior. VII. MODEL CALCULATIONS
We consider a tight binding model for the orbitalsshown in Fig. 2. There are 5 orbitals per unit cell
CuO :one for the Cu atom, and two for each of the oxygens.We denote the d − pσ hopping amplitude by t d , and thedirect hopping amplitudes between oxygen orbitals by t for π − π or σ − σ hopping and t for π − σ hopping. Fol-lowing estimates by McMahan et al [88] and Stechel andJennison [37] we take t = 0 . t = 0 .
35 and t d = 1 . eV . For site energies we take ǫ d = − . ǫ pσ = − . ǫ pπ = − . eV . Because of electrostatics, ǫ pπ is higherthan ǫ pσ . The resulting 5 bands are shown in Fig. 5.The bands of interest for us are bands 4 and 5. Figure6 shows the weight of the atomic orbitals for the Blochstates of these bands. It can be seen that band 5 has sim-ilar content of Cu − d orbital and Opσ orbital. This is theband that is generally considered to be the important onefor both hole-doped and electron-doped cuprates. TheHubbard U opens up a gap in the undoped compound inthis band, rendering the system insulating. From this sin-gle band it is argued that both hole-like and electron-likecarriers can result in the electron-doped cuprates accord-ing to the theories discussed in Sect. II.Instead, in our picture band 4 is the more importantone. As seen in Fig. 6, it is predominantly of Opπ char-acter, particularly as k approaches the ( π, π ) point in theBrillouin zone, when the band becomes full. Band struc-ture calculations predict that it is about 1.5 eV below theFermi energy and remains full under doping. Instead, wehave argued [44] that strong oxygen orbital relaxationmakes it easier to create holes in this band than whatband structure predicts: qualitatively, when one electronis removed from the doubly occupied pπ orbital the re-maining electron orbital shrinks, thus lowering its energy. FIG. 5: Band structure in the Cu − O planes in the Γ − X direction ((0 ,
0) to ( π, π )) from a tight binding calculationwith 5 orbitals per unit cell (see Fig. 2). Parameters usedare t ππ = t σσ ≡ t = 0 . t πσ ≡ t = 0 . t dσ ≡ t d = 1 . ǫ d = − . ǫ pσ = − . ǫ pπ = − . J. E. HIRSCH PHYSICAL REVIEW B , 184515 (2014)FIG. 1. Unit cell in the Cu-O plane with five electronic orbitals:Cu and O ,p There are two oxygen atoms in theunit cell denoted by O us is band 4, which is almost entirely of pπ character at itshighest energies near the point as shown in the lower leftpanel of Fig.The Fermi level corresponds to energy 0. This tight-bindingband structure, extending from 2 to 1 eV, resemblesthe main features of the band structures obtained fromdensity functional calculations [ ]. The Fermi level cuts theCu -O pσ antibonding band that extends from energy6 to 1 eV, hence according to this band structure when thesystem is doped with holes they should occupy this band. Thisis the general consensus. The antibonding oxygen pπ band(band 4) is full, and its top is approximately 2 eV below theFermi level, hence it should remain full and inert when thesystem is doped with holes according to this band structure.There are two problems with this argument, the first one iswell recognized, but the second one is not.The first problem is that the band structure in Fig. doesnot reflect the fact that the undoped system is insulating. Thisis attributed to the strong Coulomb repulsion of electrons inthe Cu orbital, which is argued to open up a gap (Mott-
2. Band structure in the Cu-O planes in the di-to ( π,π )] from a tight-binding calculation withfive orbitals per unit cell (see Fig. ). Parameters used ππ σσ , t πσ , t dσ , ǫ, ǫ pσ = −
5, and pπ = − pσ and pπ for all values of Hubbard gap) when the band is half-filled, corresponding tothe undoped case. Hence, band 5 in Fig. is argued to splitinto two upper and lower Hubbard bands when the system isundoped with the Fermi level in the gap between the two bandsrendering it insulating.There have been various calculations performed using theseideas that take the Mott-Hubbard gap into account [ 14 1724]. The general consensus is that doped holes still go into theCu -O pσ band and are responsible for the transport inthe underdoped through overdoped regimes. However, thesecalculations rely on approximations that are not necessarilywell controlled.The second problem is that the antibonding pπ band (band4 in Fig. ), which comes to about 2 eV below the Fermi levelat the point, is assumed to be rigid. Here we argue thatthis assumption is incorrect and that in reality the energy ofholes doped into this band is raised by several eV by orbitalrelaxation and that as a result doped holes will go into thisband rather than into the Cu-O pσ band. In other words,the energetics of orbital relaxation, not reflected in the bandstructure shown in Fig. , makes it easier to remove electronsfrom band 4 rather than from the band above it (band 5 inFig. ). III. ORBITAL RELAXATION
When an atomic orbital is doubly occupied, its size expands.This is certainly well known from atomic physics [25], butsurprisingly its consequences are not properly taken intoaccount in band-structure calculations nor in the many-bodyHamiltonians commonly used for solids. In a series of paperswe have argued that this effect is essential to understandthe physics of electrons in electronic energy bands that are26 30] and have proposed a new classof model Hamiltonians, “dynamic Hubbard models” to take itinto account. The magnitude of this effect increases as the net
FIG. 6: Weights of the different orbitals in the band statesfor bands 4 and 5.
This local effect is not captured by band structure calcu-lations.Band 4 contains the hole carriers that we believe areresponsible for superconductivity both in hole-doped andelectron-doped cuprates. In the hole-doped cuprates, weargue that when holes are added to the undoped sys-tem they go into this band rather than into band 5. Forelectron-doped cuprates, we argue that adding electronscreates electron carriers in band 5 and through the pro-cess depicted in Fig. 3 also creates holes in band 4. Weexplained how this process works in the
F eAs compoundsin ref. [89] (Fig. 3) and argue that it is the same here.Thus there will be carriers at the Fermi energy both fromband 4 and band 5.We consider a reduced Hamiltonian to describe trans-port and superconductivity in those 2 bands. Following Suhl et al [90] we take H = X kσ ( ǫ ak − µ ) a † kσ a kσ + X kσ ( ǫ dk − ǫ − µ ) d † kσ d kσ + X kk ′ V aakk ′ a † k ↑ a †− k ↓ a − k ′ ↓ a k ′ ↑ + X kk ′ V ddkk ′ d † k ↑ d †− k ↓ d − k ′ ↓ d k ′ ↑ + X kk ′ V adkk ′ (cid:0) a † k ↑ a †− k ↓ d − k ′ ↓ d k ′ ↑ + d † k ↑ d †− k ↓ a − k ′ ↓ a k ′ ↑ (cid:1) . (2)As discussed in Ref. [91] we retain the simplest interbandinteraction, and in what follows adopt a constant inter-band potential: V adkk ′ = V ad ≡ V . We have used a holenotation, so that the a † and d † operators correspond tohole creation operators in the Opπ and Cu − Opσ band,respectively, and similarly for the annihilation operators.We adopt a flat density of states for both bands, eachwith bandwidth D i . The single particle energies are mea-sured from the center of each band, and the Cu − O bandis shifted by an amount ǫ with respect to the O band.The intraband potentials are assumed to have identicalform; we adopt the form from Ref. [41]: V iikk ′ = U i + K i (cid:18) ǫ ik D i / ǫ ik ′ D i / (cid:19) + W i ǫ ik D i / ǫ ik ′ D i / , (3)where U i corresponds to the on-site repulsion, K i themodulated hopping, and W i the nearest-neighbor repul-sion ( i = 1 , a, d in Eq. (2)). Theseinteractions lead to a BCS ground state that is super-conducting, and an (s-wave) order parameter with theform ∆ i ( ǫ ) = ∆ mi (cid:0) c i − ǫD i / (cid:1) , (4)as found previously [41, 91]. Further details are availablein these references.Figure 7 shows T c versus hole concentration in the holeband for two sets of parameters given in the figure cap-tion. We have taken into account the fact that withinour model the bandwidth in the hole band increases withhole concentration [22], i.e. it is not a rigid band model.The behavior shown in Fig. 7 looks similar to the exper-imental results in Fig. 4, top right panel. If we were notto take into account the renormalization of the band-width with hole occupation the range of doping wheresuperconductivity occurs would be about twice as largeas shown in Fig. 7, which would be inconsistent withexperiments.The fact that the T c found in experiments (Fig. 4)starts at finite Ce concentration rather than 0 would re-sult if initially the doped electrons and induced holesremain localized, as discussed in Ref. [45] that also sug-gested an explanation why oxygen reduction is essentialfor hole delocalization.Fig. 8 shows calculated tunneling spectra for our two-band model for parameters corresponding to the solidline in Fig. 7 just to the right of the maximum with FIG. 7: T c (K) versus hole concentration ( n h ) in our twoband model, for parameters D = D = 2 eV, ǫ = 1 eV, V = 0 . U = U = 10 eV, K = W = 0, and: solidline, K = 3 .
64 eV, W = 0, dashed line K = 12 .
92 eV, W = 16 eV. T c ∼ K , together compared with the Shan measure-ments. For this calculation we used a background den-sity of states of the same shape as Shan’s, and as-sumed an intrinsic broadening with Dynes’ parameterΓ = 0 . − . It can be seen that our results lookvery similar to Shan’s data. FIG. 8: Comparison of tunneling measurements of Shan etal [81] for a sample with T c = 25 K with the predictions ofthe model of hole superconductivity for parameters as in Fig.7 (solid line) with n = 0 .
030 and T c = 31 . = 4 . = 0 .
05 meV. Because of the intrinsic broadening thesmaller gap is not visible, even at the lowest temperatureshown, as is the case in the experimental results.
We conclude that our model is compatible with theexperimental tunneling results.
VIII. SUMMARY AND DISCUSSION
In this paper we have argued that a simple two-bandmodel resulting from the orbitals shown in Fig. 2, where pairing originates in hole carriers in the
Opπ orbitals, canexplain in a simple way a large variety of experimentalfindings in electron-doped cuprates obtained over manyyears, as well as in hole-doped cuprates. The basic prin-ciples of the model were proposed before electron-dopedcuprates were even discovered. Our model says that pair-ing of the same hole carriers drives superconductivity inhole-doped and in electron-doped cuprates. The pair-ing mechanism is intimately tied to the hole nature ofthe carriers and gives rise to high T c when holes con-duct through a network of negatively charged ions, as wehave argued is the case in hole-doped cuprates, electron-doped cuprates, M gB , iron pnictides and chalcogenidesand H S [89, 91, 94–96].In contrast, all other theoretical explanations of exper-imental observations in electron-doped cuprates assumethat the only electrons involved are from Cu d x − y or-bitals hybridized with O pσ orbitals. To explain howclear experimental signatures of two-band physics arisefrom such a single band necessitates invoking electroniccorrelations, Fermi surface reconstruction and hidden ex-otic orders, which in our view are complicated and con-trived explanations not supported by experimental ob-servations.In addition to the experiments already discussed, im-portant information about electron-doped cuprates hasbeen recently inferred from the behaviour of the super-fluid density. The upward curvature of superfluid den-sity versus temperature in electron-doped cuprates hasbeen argued to be clear evidence for the presence oftwo types of superfluid carriers [92]. Li et al [35] ana-lyzed experimental results for upper critical field versustemperature and superfluid density versus temperaturein electron-doped cuprates and concluded that they areconsistent with a 2-band model where 25% of the car-riers are hole-like and 75% are electron-like, that thedominant interaction giving rise to pairing is in thehole-like band, and that the interband coupling is small( λ hh >> λ ee , λ eh ∼ λ he ∼ M gB [93]. The superfluid hole densityinferred by Li et al [92] matches the general scaling lawbetween superfluid density and T c proposed by Uemuraand coworkers [97].The physics uncovered by the analysis of Li et al. is inagreement with what the model of hole superconductiv-ity predicts, that was discussed in Sect. VII and in ourearlier work on hole superconductivity in two-band mod-els [89, 91]. In our model, the parameters K , K and V of Sect. VII are proportional to λ hh , λ ee and λ eh ofLi et al [35]. In the Li et al. analysis the fact that holesdrive superconductivity is derived from the experimentalresults. Instead, for us this is a prediction of the model:when there is two-band conduction with electrons andholes at the Fermi energy, it is necessarily the holes thatpair and drive the entire system superconducting [91].Li et al. conclude from their analysis [35] that it“ points to a single underlying hole-related mechanism of superconductivity in the cuprates regardless of nominalcarrier type ”. Dagan and Greene concluded from theiranalysis [34] that “in electron doped cuprates holes areresponsible for the superconductivity” . Already in 1991Wang et al had concluded [25] that “The similarity be-tween the behavior of the hole-scattering rate and thatin earlier “hole” superconductors suggests to us that theholes, in fact, may be driving the superconducting tran-sition in N d − x Ce x CuO − δ ” . These conclusions agreewith what our model has predicted since 1989. To furthersupport this picture it would be important to confirmexperimentally that tunneling asymmetry of the samesign as for hole-doped cuprates is the generic behavior inelectron-doped cuprates, as initial experimental resultsappear to show [79–84]. This would rule out theoriesbased on electron-hole symmetric models [85].There has been conflicting experimental evidence onthe question of the symmetry of the order parameter inelectron-doped cuprates [77, 98]. In the optimally dopedand overdoped regime there is substantial evidence of anodeless order parameter, i.e. of s-wave symmetry [99–101]. Within our model the symmetry of the gap is in-dependent of doping and is the same for electron- andhole-doped cuprates, namely s-wave. We suggest thatthe experimental evidence for non-s-wave superconduc- tivity in the underdoped regimes of electron-doped andin hole-doped cuprates is due to extrinsic factors, for ex-ample the existence of gapless excitations due to strongcorrelations.In summary, our model provides a simple, natural,elegant, unified and compelling explanation for a widevariety of experimental results gathered through inten-sive experimental research and analysis thereof over threedecades [24–35, 49, 50, 71, 72, 74, 77, 79–84, 98]. Wesuggest that the fact that we formulated the model evenbefore electron-doped cuprates were discovered and pre-dicted the conclusions reached through thirty years ofexperimental work argues for the validity of the model todescribe physical reality. Acknowledgments
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