Phase transitions to dipolar clusters and charge density waves in high Tc superconductors
PPhase transitions to dipolar clusters and charge density waves in high T c superconductors M. Saarela a , and F. V. Kusmartsev b a Department of Physics, P.O. Box 3000, FIN-90014, University of Oulu, Finland b Department of Physics, Loughborough University, LE11 3TU, UK
Abstract
We show that doping of hole charge carriers leads to formation of electric dipolar clusters in cuprates. They are created by many-body interactions between the dopant ion outside and holes inside the CuO planes. Because of the two-fold degeneracy holes inthe CuO plane cluster into four-particles resonance valence bond plaquettes bound with dopant ions. Such dipoles may order intocharge-density waves (CDW) or stripes or form a disordered state depending on doping and temperature. The lowest energy of theordered system corresponds to a local anti-ferroelectric ordering. The mobility of individual disordered dipoles is very low at lowtemperatures and they prefer first to bind into dipole-dipole pairs. Electromagnetic radiation interacts strongly with electric dipolesand when the sample is subjected to it the mobility changes significantly. This leads to a fractal growth of dipolar clusters. Theexistence of electric dipoles and CDW induce two phase transitions with increasing temperature, melting of the ordered state anddisappearance of the dipolar state. Ferroelectricity at low doping is a natural consequence of such dipole moments. We developa theory based on two-level systems and dipole-dipole interaction to explain the behavior of the polarization as a function oftemperature and electric field.
Keywords: high Tc superconductors, charge density wave, pseudogap, dipolar cluster
PACS:
1. Introduction
Since the discovery of the hole-doped cuprate superconduc-tors a great deal of e ff ort has been devoted to understand theircomplex behaviour. Recently guided migration of doped oxy-gen atoms in La CuO + y (LCO) has been successfully demon-strated with nano-scale synchrotron radiation scanning X-raydi ff raction in the temperature range 180 K < T <
330 K. [1, 2, 3]Originally randomly distributed dopant atoms in the LaO planeorder into clusters after many hours of illumination. Moreover,these ordered nano grains grow and form a fractal structure thatextends through the sample. The cluster growth looks as if thereis a continuous phase transition (or a critical state) where thereare two competing long-range orders(LRO). There exists a po-tential barrier separating minima associated with these ordersand it takes a time for the dopant oxygens to self-organize andreach the new stable ordered phase. Probably because of thiscriticality the cluster sizes observed have a scale free distribu-tion. The superconducting critical temperature depends on thetype of the cluster order. There are two critical temperatures ofthe clusters LRO observed. It was found that the higher tran-sition temperature is associated with the better order where thecritical percolation through ordered oxygens takes place. Thesefundamental discoveries [1, 2, 3] pose questions: Why does theradiation increase mobility of oxygen ions and why do they or-der? Is this order critical in increasing the onset temperature tosuperconductivity?The inhomogeneity of cuprate superconductors coupled withdopant atoms is well documented besides LCO also in othermaterials and in di ff erent experiments using scanning tunneling spectroscopy [4, 5, 6, 7, 8], photoemission spectra (ARPES) [9]– suggesting that the oxygen dopant-induced states are mixedwith Cu atoms – and measurements of dielectric constant [10]etc.Very recently a surprising discovery of ferroelectricity alsoin La CuO + y was reported at exceptionally low oxygendoping.[11, 12] To describe this e ff ect the Ginsburg-Landautheory of the ferroelectricity, where the existence of themagneto-electric coupling has been assumed, was developed.[13] Other models based on polaron fromation and vortex-antivortex pairs have been suggested. Here we propose thatit is more natural to assume that the ferroelectricity arises dueto electric dipoles caused by the Coulomb interaction of thedopant ions.Debate between two competing orders and preformed pairsof superconductivity in the pseudogap region focuses mainlyin the interpretation of ARPES measurements on the nature ofgap functions and Fermi arcs. [14, 15, 16, 17, 18, 19] Very re-cent experiments found evidences of sharp phase transitions attemperatures between the superconducting critical temperatureT c and the pseudogap temperature at T*. [20, 21, 22, 23] Atzero temperature inside the superconducting dome two phasetransitions are discovered by studying the behavior of the gapfunction as a function of doping. [24] Also the temperaturedependence of resistivity changes from quadratic to linear be-havior inside the pseudogap region with increasing doping andtemperature. [25, 26] Optically stimulated ultrafast changes inthe charge-density wave correlations have been studied by fem-tosecond resonant x-ray di ff raction indicating that charge order- Preprint submitted to Physica C October 12, 2018 a r X i v : . [ c ond - m a t . s t r- e l ] A ug ng and superconductivity are competing orders.[27] However,in spite of many attempts there is still no clear physical mech-anism of the pseudogap and why there are phase transition be-tween the pseudogap temperature T ∗ and T c .Important experimental findings for our model come froma peculiar temperature dependence of the Hall coe ffi cient, R H ,measured in Sr doped La − x Sr x CuO (LSCO) [28, 29]. Theseresults have been analyzed within a two-band model [30]. Itwas argued that the charge carrier density, in that case the holedensity, can be written as the sum of two components n h ( x , T ) = n ( x ) + n e − ∆ ( x ) / k B T . (1)The first term depends on doping x and is independent of tem-perature T . The second term is of activation type contributionwith a doping dependent activation energy ∆ ( x ) multiplied by aconstant n . The analysis [30] suggested that at small doping,0 . < x < .
08, and below the room temperature each dopantatom creates one hole and hence n ( x ) = x . At higher tem-peratures the hole density increases rapidly and for each x thattemperature behavior was well fitted with the activation typeexponential component in Eq. (1). The constant value n = . n ≈ ∆ ( x )was left as an open question. At higher doping 0 . < x < . x when the temperatureincreases from 0K to 50K the carrier density decreases. Thissuggests that the charge density can fluctuate between the twobands.In this paper we propose a microscopic approach [33] wherebound states appear because dopant atoms ( e. g. Sr + are re-placing the La + ) and therewith inducing extra negative chargesinto spacer layers. At low doping these negative charges areable to bind holes. The key ingredient in binding holes intodopant atom is the near degeneracy of the d-state bands with x − y and z symmetry in the CuO plane. The Jahn-Tellerdistortions in cuprates, such as La CuO , elongate the oxygenoctahedron (green in Fig.1) surrounding the central Cu ion andsplit the degeneracy of the associated e g orbitals of the Cu d -state. This happens in the highest partially occupied x − y - orbital, which together with the doubly occupied O p x , p y -orbitals form a strong covalent bonding. Their straight hy-bridization gives rise to the bonding, non-bonding and half-filled anti-bonding bands and predicts a good metal, in sharpcontrast with experiments finding a large charge gap. The fail-ure of the band theory indicates a strong Coulomb interaction(in particular, on site Hubbard U ), which may exceed well thebandwidth of the tight-binding, anti-bonding band.If we take the parent compound as La CuO then two chargecarriers (holes) put into the Cu x − y orbital would experiencea large energy penalty, ( U ∼ eV ). Since on site Hubbard U is much larger than the energy separation between the Cu x − y and O p − orbitals, which defines the charge transfer gap ∆ CT ∼ − eV , and this in turn is much larger than the hybridi-sation integral, t dp ∼ . meV [34, 35, 36], single electrons Figure 1: Parts of the La CuO crystal structure with one dopant impurity (top-most red sphere) are shown in two panels. Green octahedra have oxygens inall corners and Cu atom at the center (purple sphere). The apex oxygens (redspheres) and La atoms (blue spheres) form one of the two (LaO) + spacer lay-ers, which separates the (CuO ) − planes. In the upper panel we show howdopant impurity atom with its negative charge pushes the apex oxygen downfrom the top of light red octahedron to the top of the green octahedron. Thisrestores the e g symmetry of the Cu orbitals (the anti-Jahn-Teller e ff ect), whichreleases electrons from Cu atoms. Note, that there original ionic states on Cu + have small positive charge transfer gap ∆ CT . But the formal valence state Cu + ,which should be created by doping, have large negative ∆ CT <
0. This meansthat the states should be rather represented as Cu + → Cu + L , (see, the Ref.[31] for a detail), where the doped holes would go not so much to the d-shellsbut rather to the oxygens, creating oxygen holes (though the quantum num-bers of the respective states would be the same as those of Cu + ). In the lowerpanel we show schematically how electrons are redistributed. Thus, each Cu(precisely saying the Cu atom with surrounding oxygens) can ”loose” e ff ec-tively one electron. Two of them are captured by the impurity oxygen to fillits p-shell. This extra charge repels nearest neighbour Cu electrons and alto-gether five holes are revealed. The degeneracy allows four holes to bind intothe ground state of the impurity charge 2- and form a dipolar cluster. are localized on Cu sites on the Cu x − y orbitals forming theCu + states. Their spins are anti-ferromagnetically aligned (viathe super-exchange interaction that involves virtual hopping tothe neighbouring O p - orbitals) to create the anti-ferromagneticMott insulator[37].Since these original ionic states on Cu + have positive chargetransfer gap ∆ CT , which is small relatively to U , the formal va-lence state Cu + , which with doping would already have largenegative ∆ CT < + → Cu + L , (see, the Ref. [31] for details).There doped holes would go not so much to the d-shells butrather to the oxygens, creating oxygen holes (though the quan-tum numbers of the respective states would be the same as thoseof Cu + ). In these Zhang-Rice singlets the hole is distributedbetween the Cu + and four neighbouring oxygens. Having thisin mind to make it short below we call these Zhang-Rice sin-glets simply saying as Cu + states.In our approach we consider this Mott state and take into ac-2ount that cuprates, like La CuO , are also layered ionic crys-tals. The (CuO ) − layer is negatively charged and two spacerlayers (La O ) + are positively charged. Oxygen atoms forman octahedral cell around Cu + and the apex oxygen atoms con-taining two extra electrons filling the p-shell are in the LaO-layer as depicted in Fig. 1. In the parent anti-ferromagnetic in-sulator compound the degeneracy of cubic e g states is removedby the Jahn-Teller e ff ect, which elongates the oxygens octahe-dron. Copper, on the other hand, is a transition metal and it caneasily give out one electron and transfer into Cu + → Cu + L sharing the electron with neighbouring octahedron oxygens andforming the Zhang-Rice singlet.Dopant atoms settle in the LaO spacer layer substituting Laatoms like in the case of Sr doping or intercalated like oxygenatoms. Oxygen atoms are small in size and thus mobile. Sur-prisingly that these atoms play an important role in the vortextrapping[32]. They easily fill their p-shell with two electronsand become ionized. When this happens the negative impuritycharge pushes the negatively charged apex oxygen down (see,Fig.1). This apex atom displacement restores the e g symmetryand helps to remove one electron from neighbouring Cu-ionsto form Zhang-Rice singlet (the anti-Jahn-Teller e ff ect). Atlarge enough doping holes created in this way in Cu sites forma charge carrying hole (”Zhang-Rice singlet”) band in the CuOplane. Many complexes containing specific transition-metalcentral ions with special valency show this e ff ect. [39] Ev-ery new hole appearing during hole-doping in the parent com-pound leads to strong frustration of the original antiferromag-netic state. This leads to rapid suppression of antiferromag-netism in La − x Sr x CuO even by small hole concentration.
2. Charge Density Waves composed of resonance plaque-ttes and electrical dipolar moments
Four holes bound by dopant atoms due to correlations [40]form resonance plaquettes in a manner similar to the resonancevalence bond described by Anderson[37]. These plaquettes to-gether with dopant atoms induce electrical dipolar moments.There may also arise a second nearly flat band[38].As discussed above in hole doped cuprates dopant atomsare located in spacer layers between CuO planes and chargecarriers are confined within CuO planes. Dipolar clusters ex-ist in a broad range of doping from deeply under-doped up toover-doped region. They disappear in over-doped region dueto screening associated with the increasing hole density. Inthe region of the optimal doping plaquettes are decoupled fromdopand atoms and become mobile. Then they are playing a keyrole in the Planckian dissipation observed in all cuprates[41].The dipole-dipole interaction between clusters has a strong di-rectional dependence. It is attractive when parallel dipolespointing to the same direction are on top of each other and re-pulsive when they are side by side. Note that with this anti-Jahn-Teller e ff ect associated with the squeezingof the oxygen octahedra, the hole state has a two fold degeneracy that is inaddition of the conventional Kramers degeneracy. Dopant oxygen atoms are mobile although their mobility isvery low at low temperatures. Then at temperatures below 80K, at very low doping and random distribution of dopant atomsit is likely that dipoles remain in random, glassy order. Mobilityrises with temperature and when the sample is subjected to elec-tromagnetic (X-ray) radiation. The radiation interacts stronglywith electric dipoles as well as doped oxygen impurities. Theycan be excited by irradiation and moved to new positions as-sociated with the minimum of the total energy. That is howirradiation increases the e ff ective mobility of dopant atoms andordering of dipoles. As the result they reorganize themselveswith the help of the dipole-dipole interaction and after waitinglong enough, as it was done in Refs. [1, 2], a short-ranged orderarises.The interaction between dipolar clusters is short-ranged andvaries as a function of separation distance with the 1 / r tail.Clusters move towards the minimum energy configuration, but,because of low mobility, nucleation can start at di ff erent partsof the sample and lead to patches of di ff erent directional stripyorder with a fractal structure like in classical dipolar systems.The electron-phonon interaction will further enhance this order-ing tendency leading to charge density waves[42, 43, 44].Any isolated charge associated with an impurity or a chargefluctuation in LaO spacer layer is inducing the anti-Jahn-Tellershift of the apex O ion[33, 40] and therewith creates a reso-nance plaquette in CuO plane. To describe these resonance pla-quettes we developed the many-body variational theory wherefour holes are trapped by the impurity [33, 45, 46, 47] andderived a proper energetic description of resonance plaquettestates, which reproduces experimental pseudogap temperaturewithin the two-fluid model [30, 29].
3. Many-body theory of the dipole moment formation
In order to describe properties of a single dipole we need tosolve a quantum many-body problem where holes interact vialong-ranged Coulomb interaction among themselves and withthe isolated charge associated with an impurity inducing theanti-Jahn-Teller shift of the apex O ion [33, 40, 48]. In factthe many body interaction results in the bound state of the im-purity in the LaO plane with holes in the CuO plane. We cal-culate below the binding energy of holes forming the dipoleas a function of doping. To make a quantitative description ofsuch a state it is su ffi cient to use a continuum approximation,where mobile charged holes are confined to a CuO plane. Theire ff ective mass is determined by the band structure of the ma-terial with the use of the e ff ective mass or the (kp-) method[49]. For a simplicity, one may start with a three band Hub-bard model[31, 50] where the e ff ective holes mass, m ∼ t − pd .The charge neutrality is implemented by the inert background,called the jellium model[33]. Holes are allowed in general tomove in a 2D-plane and the impurity is set at a given distance c away from the plane. Thus the impurity serves as a nucleationcenter of the dipole moment pointing to c-direction.The Hamiltonian of such a system is the sum of two terms, H = H h + H I which includes the mutual Coulomb interaction3nd kinetic energy of holes, H h and the apex O ion interactionwith holes, H I , H h = − N (cid:88) i = (cid:126) m ∇ i + N (cid:88) i , j = , i (cid:44) j e π(cid:15) | r i − r j | H I = − (cid:126) M ∇ − N (cid:88) i = e π(cid:15) (cid:112) | r − r i | + c . (2)The number of mobile holes is N , they have the mass m , charge | e | and position r i . The impurity is placed at r , its charge is − | e | and kinetic energy is controlled by the mass M . For alocalized impurity we let the mass grow to infinity. The strengthof the Coulomb interaction depends on the dielectric constant ε . It is convenient to use the atomic units where all distancesare given in units of r = r s r B and energies in Rydbergs. Theparameter r s = / √ π n r B is defined by the density n of holesand the Bohr radius r B .At low doping in the under-doped region the hole gas is di-lute, but very strongly correlated. We assume that the holeband is two-fold degenerate, because the mechanism to producethe conducting hole band relies on the anti-Jahn-Teller typeshift in almost degenerate energy levels. Under such circum-stances, in strongly correlated regime, the di ff erence betweenthe fermionic and bosonic gases is of minor importance. Thecluster formation takes place in both cases although in slightlydi ff erent regions of r s values [46, 47]. The bosonic part ofthe ground-state wave function which contains correlations be-tween holes and impurities is chosen in the form of the Jastrow-type variational ansatz[51] Ψ = e (cid:80) Ni , j = u hh ( | r i , r j | ) Ψ I = e (cid:80) Ni = u It ( | r i , r | ) e (cid:80) Ni , j = u In ( | r i , r | ) Ψ (3)We have extended the conventional, many-body variational the-ory to the case where four holes are trapped by the impurity.[33]The wave function includes now the product of three com-ponents, hole-hole (hh) correlations, impurity-trapped holes(It) correlations and impurity non-trapped holes (In) correla-tions. The correlation functions u hh ( | r i , r j | )), u It ( | r i , r | )) and u In ( | r i , r | )) are determined by minimizing the total energy E ofthe hole gas and the chemical potential µ of the impurity. E = (cid:104) Ψ | H h | Ψ (cid:105)(cid:104) Ψ | Ψ (cid:105) µ = (cid:104) Ψ I | H I | Ψ I (cid:105)(cid:104) Ψ I | Ψ I (cid:105) − (cid:104) Ψ | H h | Ψ (cid:105)(cid:104) Ψ | Ψ (cid:105) − E bin , (4)where the last term is the sum of binding energies 4 E bin of fourtrapped holes. The maximum number of trapped holes in thesame state is determined by the Pauli principle. It is twice thedegeneracy factor of the hole band, which we set equal to two.The fact that four holes are bound by the charge -2 | e | is causedby strong many-body correlations. As a result of the variationalcalculation we get the total and binding energies, chemical po-tential and the hole-hole and hole-impurity distribution func-tions. For details we refer to Ref. [33].In Fig. 2 we show the binding energy E bin as a function of r s parameter. The binding energy vanishes when r s < . -3-2.5-2-1.5-1-0.5 0 2 4 6 8 10 12 -0.6-0.5-0.4-0.3-0.2-0.1 0 E b i n [ R y ] E b i n [ e V ] r s Figure 2: The calculated binding energy of the hole cluster E bin ( x ) (solid bluecurve) identified as the activation energy and fitted to the Hall coe ffi cient mea-surements [29, 52] (red plus signs). saturates very slowly to the single hole limit when r s → ∞ . Atzero temperature the number of holes in CuO layers is equalto x , the number of dopant atoms, but when the density is highenough, r s < .
5, then four more holes per dopant atom arereleased. The maximum number of holes is one per Copper siteand that is reached when x = / = .
2. This is clearly seen byCooper et al. [25] as a change in the temperature dependenceof the resistivity at x ≈ .
19, which is identified as a quantumcritical point. In the Gorkov-Teitelbaum two-fluid model[30]the activation energy ∆ ( x ) vanishes also at x ≈ . ffi cient data [29] we haverefitted them with the formula, n h = x (1 + ∗ f ( x )) + − f ( x )) e − ∆ ( x ) / (2 k B T ) (5)The function f ( x ) shown in Fig. 3 gives the fraction of holesfreed from the clusters already at zero temperature. In an idealsystem f ( x ) should be a step function, zero at low doping andone when all holes are released from the trap. At zero temper-ature that happens at x = . n h =
1. That is the phase transition from dipolar clusterstate to homogeneous metallic state. In practice high T c super-conductors are inhomogeneous. There are high and low densityregions and that is seen as a smooth behavior in the fitted f ( x ).In the high density regions, r s < .
5, the binding energy ofclusters vanishes and all four bound holes per cluster are thenfree to carry the charge and out of the activation process. In thefit we have averaged the low temperature fluctuations between0K < T < x < . f ( x ) vanishes and then increases up to 0.6 atthe highest doping, x = f ( x ) grow fasterwith increasing doping reaching unity when x = .
2, whichindicates that all holes are freed.The fitted activation energies ∆ ( x ) shown in Fig. 2 by redsymbols can be compared with our binding energies − E bin ( r s )after the conversion of our Rydberg unit into electron volts and4 f ( x ) x Figure 3: The function f ( x ) from Eq. (5) fitted to the Hall coe ffi cient data byOno el al. [29] at values of doping show by blue stars. The solid line is just toguide the eye.
200 400 600 800 1000 0 0.05 0.1 0.15 0.2 0.25 T * [ K ] x Figure 4: The pseudogap temperature calculated from Eq. (7) is comparedwith the crossover temperature of the resistivity curves (open squares)[[54]], thetemperature of the maximum magnetic susceptibility (stars) [[55] ] and (solidsquares) [ [56]], and with the activation energy (plus-signs)[[30]]. r s parameter scale into the doping scale x . In our notation thedensity of holes n h = /π ( r s r B ) = x /π r where r Cu ≈ . r s = . E bin ( x ) vanishes with the valueof doping x = r B ≈ . = (cid:15) =
21 and thee ff ective hole mass m = . m e in units of the electron mass m e .The dielectric constant found here is in reasonable agreementwith the value (cid:15) =
29 determined from Chen et al. experimentsin LSCO [53].We identify the pseudogap temperature with the temperaturewhen all trapped holes are released. Then x = e − ∆ ( x ) / (2 k B T ∗ ) , (6) g I (r) r/r Figure 5: The radial distribution functions of holes in the CuO layer around theimpurity charge − | e | in the LaO layer . The impurity is located at the origin andthe peak around it is huge in this scale. The highest peak is at the lowest densitywhen r s =
12 with the height 780. The curves from left to right are calculated at r s =
12, 10,8,6,4 and 2.5, respectively. All distributions are normalized to unityat r → ∞ . The deep minima at r < ff ective interaction as a function of separationdistance r and multiplied by r calculated at r s = and the density of holes n h = x . That is the phase transi-tion line in temperature doping plane when dipolar clusters meltinto the bad metal state. This transition is not sharp because ofthe density fluctuations, and one enters the Fermi liquid regimewith regions of clusters. That would be some kind of an emul-sion state. In Fig. 4 we show the calculated pseudogap temper-ature T ∗ ( x ) = E bin ( x ) / (2 k B log( x )) (7)and find a good agreement with di ff erent experimental values.Doping dependence of T ∗ ( x ) is dominated by the binding en-ergy because log( x ) is a slowly varying function in the range0 . < x < . g I ( r ) around the origin. At the highest density5here the activation energy vanishes ( r s = . r = .
1Å and x = .
22) the radius of the bound state wave function barelycovers the nearest neighbour Cu atoms, since the distance be-tween Cu atoms is about 3.8Å. When the hole density decreasesthe range of wave function extends further covering the wholeneighbouring octahedron, see, the Fig. 1 and the yellow circlein the Fig. 7. As pointed out earlier the Pauli principle allowsonly four particles to be bound in the lowest state and that iswhy one of the five covered holes (the maximum number ofholes) is free to act as a charge carrier. That is consistent withthe experimental fact that the number of holes is equal to thenumber of dopant atoms at low doping and the maximum holedensity is reached when all five holes per dopant atom are re-leased.Finally in Fig. 6 we show the e ff ective interaction betweenthe impurity and holes. At short distances it is attractive, butmany body e ff ects strongly over-screen attraction at intermedi-ate ranges leading to bound clusters.Four bound, positively charged holes over-screen the impu-rity ion charge -2 | e | . It means that two electrons are expelledfrom the octahedron as circled out in Fig. 1. They are attractedto the positively charged LaO-layer and get loosely trapped byfour apex oxygens squeezed to their symmetry positions in thesame way as the impurity ion does. That is a new source ofelectron pairs, which will form a very flat band together withthe bound holes in the CuO plane. The binding energy of holesto the electron charges is roughly the same as to the impuritycharge and that is why at low doping the factor in front of theactivation contribution to the charge carriers is independent of x in Eqs. (1) and (5). This phenomenon is called Mottness.In Refs. [57, 58], however, a di ff erent microscopic mechanismfor the Mottness accounting for spectral weight transfer exper-iments has been suggested.The positively charged clusters in the CuO-plane expel sur-rounding holes further away and reveal the background elec-trons. In our jellium approximation this shows up in the holedistribution function g I ( r ) as a deep minimum around the cen-tral hole peak extending to 2 r (see, Fig. 5). At infinity the dis-tribution is normalized to one, which accounts for the fact thatthe charge of holes compensates totally the background chargeand the integral of g I ( r ) −
4. The dipolar clusters
The cluster of four holes in the CuO plane self-bound dueto many-body e ff ects to the excess charge created in the LaOplane around the apex oxygen pushed by a dopant ion (the anti-Jahn-Teller e ff ect) has a strong dipolar moment and low mo-bility. With increasing doping these clusters form a classicaldipolar gas. The potential energy of N dipoles is determinedby the impurity charges q LaO = − | e | located in LaO layers,each of which creates the opposite sign charge cluster in a CuOlayer q CuO = | e | . For simplicity, we assume that these chargesin both layers are point like, heavy objects and calculate their Figure 7: A schematic top view of the charged hole cluster in CuO plane ofLa CuO noted by yellow disk. At the center of the cluster there is the Cu atomsurrounded by four oxygen and other Cu atoms with the d and p-state orbitalsshown respectively. Inside the cluster there are four holes trapped. Outside -double occupied states (Mottness) arising to keep the electro-neutrality. Coulomb potential energy classically V = π(cid:15) N (cid:88) i (cid:44) j (cid:88) α,β q α q β | r α ( i ) − r β ( j ) | . (8)It depends on the charges and dielectric constant (cid:15) , but most im-portantly it depends on the orientation of clusters. The Greekindices refer to the location of the charge, either in LaO or CuOlayer, and r α ( i ) to the position of the charge in that layer. Inthe calculated examples we have required the following regularorder. A bound dipole pair is separated by a distance d pointingto the b direction in the a , b -plane. The pairs are then separatedby distances R a and R b in a and b directions, respectively. Themotion of holes is strictly two-dimensional in the CuO-layers,but dopant atoms have some flexibility at high temperatures to c R b d (cid:68) c Figure 8: Ordered structure of dipolar clusters in the layered superconductors inthe bc-plane. Four-hole clusters (red) are in the CuO plane and i-O ions (blue)in LaO planes. The separation distance between LaO and CuO layers is c / d and the distance between dipole-dipole pairs is R b .
20 40 60 80 R b R a R c Figure 9: Regularly ordered clusters in 3D. The red circles determine the CuOplane. The i-O (blue circles) can be below or above that plane. The minimumenergy in this configuration is reached when the dipole-dipole pairs are sepa-rated by d = di ff use between two LaO layers. The octahedron can then bepolarized either from above or below as shown in Fig. 8. Withthese assumptions the potential energy of two neighboring, op-positely oriented dipoles (d-d interaction) associated with thesame LaO layer as a function of their separation distance d isthen V dd ( d ) = v ( d , ∆ c ) + v ( d , c / − v ( d , c / + ∆ c /
2) (9)where v ( d , x ) = e / (4 π(cid:15) √ d + x ), c = .
13Å is the sizeof the unit cell in c-direction and ∆ c ≈ c / = .
3Å is theseparation distance of the impurity charge centers in c-directionas shown in Fig. 8. V dd ( d ) has the minimum at d = . (cid:15) =
21 taken from our activation energy fit shown in Fig. 2.This defines the temperature scale in the self organization ofthe ordered structures seen in the experiments [1, 2].The growth of nano grains under soft X-ray radiation [1, 2]takes place in the temperature range, 330K > T >
180 K, wellabove the transition to superconducting state, but below thepseudogap where dipolar clusters exist. The d-d interactiondominates the growth process. First appear bound d-d dimers inthe a-b plane and they arrange themselves into anti-ferroelectricstripes as shown in Fig. 9. The growth into larger clusters slowsdown the mobility and therefore patches of high density dipolarclusters can appear in di ff erent places of the sample. At highenough temperature depending on doping or hole density di-rectional order melts and stripes disappear indicating a secondphase transition inside the pseudogap region.In Fig. 10 we show the contour plot of the potential energyin ( R a , R b ) - plane for eight layers of 10 ×
10 dipoles in a layerorganized in a regular order like in Figs. 8 and 9. The dipole-dipole separation distance d =
13 Å, which gives the minimumenergy in this configuration. The energy surface has local min-ima and the growth process may not find the absolute minimumvery easily. Fig. 11 then shows the potential energy per dipoleas a function R b , the d-d pair separation distance. We have fixed d =
13 Å as above and R a = R b (cid:64) Å (cid:68) R a (cid:64) Å (cid:68) Figure 10: The contour plot of the potential energy per dipole in the ( R a , R b )-plane of regularly ordered clusters shown in Figs. 8 and 9. The third parameterin the potential energy of this ordered structure is the dipole-dipole separationdistance d in the b direction, which has a fixed value d =
13Å obtained by mini-mizing the potential energy.
10 20 30 40 50 (cid:45) (cid:45) (cid:45) R b (cid:64) Å (cid:68) V (cid:64) m e V (cid:68) Figure 11: The potential energy per dipole of regularly ordered clusters shownin Figs. 8 and 9 as a function of separation distance between dipole-dipolepairs in b -direction R b . Two other parameters have fixed values, d =
13Å and R a = at R b = ff erent,nearly degenerate dipole orientations which grow slower andslower together but without the underlying, long-ranged crystalstructure. Instead the system may favor the fractal like growth.
5. Ferroelectricity at extremely low doping
At extremely low doping, x ≈ . − x Sr x CuO andLa CuO − y ferroelectricity has been discovered experimentally[11, 12]. Measurements are done well inside the antiferro-magnetic insulator phase and charge carrier holes are all local-ized. The dipolar structure between a dopant atom and clusteredholes in CuO plane is strongly bound. In such dilute systemdistance r between dopant atoms and, consequently, dipoles islarge, r ≈ / r and be-comes very weak. Experiments in LCO [1, 2] suggest that thegrowth process of macroscopic dipolar structure under radia-tion takes place layer by layer. Thus in a very dilute systemwe may consider clustering of dipoles in separated layers. Asdiscussed above dopant ions are located in spacer layers, pushdown apex oxygens and bind holes in CuO plane, say belowthe spacer layer. The dipole points to c-direction. The interac-tion with a neighboring dipole pointing to the same direction isrepulsive. In order to gain energy dipoles must tilt away from c-direction more than the critical angle θ = π/ − arccos(1 / √ (3)).By tilting one increases the length of a dipole and the strengthof the dipole moment, but looses in binding energy.Experiments [11, 12] show that components of polarizationin c- and ab-directions are roughly equal, which is consistentwith the above estimate of the tilting angle. We can calculatethe dipole-dipole binding energy q / (4 π(cid:15) ) d / r (1 − ( θ )) = . meV (1 − θ )), where (cid:15) = (cid:15) , the length of dipole d =
3Å and dipole separation distance r = d and d under theexternal electric fields E and E acting on them are − ( d · E )and − ( d · E ), see Fig. 12. Therefore with each dopant ion wemay introduce a partition function Z = exp β ( d · E ) + exp β ( d · E ) (10)where β = / k B T . The polarization associated with such adopant ion P i can be estimated with the equation P i = d exp β ( d · E ) / Z + d exp β ( d · E ) / Z (11)Taking into account the staggered character of the intrinsicelectrical field, we may introduce the fields E = E int + E and E = − E int + E . Assuming that d = − d = d i we can write po-larization in the following form, P i = d i tanh( β E · d i ) (12)In the regime of low doping total polarization P of the crystalconsists of contributions from all dopants ions. At very lowtemperatures P = n i < P i > where n i is the density of dopantions.Next we develop a pseudo-spin formalism of this two-levelsystem. The energy di ff erence 2 E · d i = h corresponds to an CuO ________________________________________________________________LaO_________________________________________________________________LaO _________________________________________________________________CuO ________________________________________________________________ + + + + + ++ + + + + +- - - - - -- - - - - - E E _ d d Figure 12: There is an intrinsic electric field between the LaO layers and CuO2layers, It is noted as E and E . These fields form a staggered layered structureover the crystal. The two states of the dipoles formed by a dopant atom locatedin the spacer layer, LaO. The dipole may be located in the upper layer betweenLaO and CuO2, and its value is equal to d or in the lower layer, below LaOwhere its value will be equal to d . Thus each dopant atom may be a sources oftwo states of a dipole. e ff ective field h . Without an interaction between dipoles ther-modynamics of our model is simply equivalent to a standardbehavior of non-interacting magnetic moments in an externalfield h . Therefore, the average ”magnetization”, which is herethe polarization, associated with the pseudo-spins < d i · E > isdescribed by a standard Brillouin function for the spin S = / ff ective Hamiltonian describing interacting dipoles inthe staggered electric field E ( z ), may be written in the followingform: H e f f = (cid:88) i j J ij d i ⊗ d j − (cid:88) i E · d i (13)where the second term describes the interaction with staggeredelectric field as discussed above. The first term of the Hamilto-nian is the complex interaction between dipoles and staggeredelectrical field associated with the charged layers CuO − andLaO + . It is represented with the use of the tensor J αβ i j , whichincludes also the direct dipole-dipole interaction. In our situa-tions of an interacting two level system shown in Fig. 12 thee ff ective pseudo-spin Hamiltonian becomes H e f f = (cid:88) < i j > J i j τ zi τ zj − h (cid:88) i τ zi (14)here we assume that there is an interaction between nearesttwo level systems in the sandwich of CuO − and LaO + pairsof layers. Note that we have introduced here the notation τ zi ≡ d i · E / | d i · E | . One can easily show that although thecoupling is a long-ranged in the first approximation we cantake into account a nearest-neighbor interaction which can beboth repulsive, J >
1, i.e. an antiferromagnetic interactionor attractive, J <
1, i.e. a ferromagnetic interaction between8seudospins τ . In accordance with the qualitative considera-tions presented above (the state τ z = + τ z = − ff erentsign, but usually the nn interactions dominate, and this is whatwe will assume further on.With this assumption we can reduce our model to an antifer-romagnetic or ferromagnetic Ising model with nn coupling J in a parallel field. In this case the standard mean-field equationfor the total magnetization takes the form: τ = < τ > = tanh h − J z τ T (15)( z is the number of nearest neighbours), from which we candetermine the temperature dependence of τ and consequentlythe total polarisation of our system as P = < d i > n dip .It is convenient to rewrite Eq. (15) as τ = tanh ˜ ∆ + J z (1 − τ ) T , (16)where ˜ ∆ = h − J z τ (0) = h − J z is the renormalized initial( T =
0) splitting of these two states. If we would take this split-ting due to h to be constant (i.e. if we ignore the second termin the argument of Eq. (16), we would get the conventionaltemperature dependence of τ (Brillouin function) and, conse-quently, of the polarisation, which at low temperature would beexponential in temperature: τ ( T ) = − exp (cid:32) − ˜ ∆ T (cid:33) , (17)Eq. (15) can be solved numerically. It contains two pa-rameters h and J z , which can be fit to experiments. Resultsare shown in Figs. 13 and 14 where the experimental data istaken from Fig. 2 of Ref. [11]. The value of h = J z = − . P z to J z = − P ab .The interaction strength is consistent with the estimate of thedipole-dipole interaction given above. The over all strength ofthe polarization is fitted to the experimental value at T = P z (0) = − and P ab (0) = − . Thesevalues are defined by the total number of dipoles created in thesamples. The figures show an excellent agreement between thedeveloped theory and existing experiments. It is then clear thatthe critical temperature of the ferroelectric phase transition isdetermined by the dipole-dipole interaction at very low dop-ings, x . P as a function of applied external field E at the tempera-ture T = K . Again we have a very good fit of our theory to theexperiments of Ref. [11] for for the LCSO samples with verysmall doping x = . Pab on applied electric field E , seethe blue curve (the developed theory) and red dots (the experi-mental data from the Ref. [11] are well described by a standardBrillouin function for the spin S = /
2, which coincides with (cid:72) K (cid:76) P z (cid:72) n C c m (cid:45) (cid:76) Figure 13: Out-of-plane P z electric polarization as a function of temperature T .Solid line corresponds to the theoretical calculation performed in the frameworkof the developed model, when the value of dipole interaction is chosen to be J z = − .
9K and h = . (cid:72) K (cid:76) P a b (cid:72) n C c m (cid:45) (cid:76) Figure 14: In-plane P ab electric polarization as a function of temperature.Solid line corresponds to the theoretical calculation performed in the frameworkof the developed model, when the value of dipole interaction is chosen to be J z = − .
0K and h = . the derived Eq. (12). However the similar dependence for thez-component of the polarisation, P z ( E ) shows already small de-viation between the theory and experiments, that indicates thatthe dipole-dipole interaction should be here properly taken intoaccount.
6. The resonance plaquettes at optimal doping are viewedas Planckian dissipators
For free hole current carriers each dipole is a source of strongscattering. The scattering becomes stronger when the dopingincreases and reaches its maximum at the optimal doping. Toestimate the scattering time τ we have to look into the detailedstructure of individual dipole clusters and their evolution withdoping. Here each dipole is formed by a cluster of four holeslocated in the CuO plane and created by the many-body over-screening of the Coulomb attraction of these holes to the impu-rity or electron polarons located in the LaO spacer layer. Thewhole process of the cluster formation is accompanied by thesqueezing of the neighbouring oxygen octahedron and creationof heavy polarons located in LaO spacer. That process of the9 (cid:45) (cid:45) (cid:45) (cid:72) kV cm (cid:45) P (cid:72) n C c m (cid:45) (cid:76) Figure 15: Polarization as a function of applied electric field (P-E) curves. Thedata for P ab (red solid circles) and for P c (black solid circles) were measuredat 2K and presented in the Ref. [11]. Blue and orange solid lines are calculatedwith the use of the derived Eq. (12). Here in the plot to fit the data the saturated P z was taken to be equal to P z =
42 nC cm − and P ab = − oxygen octahedron squeezing is known as anti-Jahn-Teller ef-fect and in turn it may provide four holes for the cluster and thesinglet electron pair for the flat band as shown in Fig. 1. Therest doped free Zhang-Rice holes form a strongly-correlatedquantum liquid. The many-body interaction of these holes inthe strongly-correlated regime provides the over-screening ofthe charged impurities and polarons located in the LaO layerleads to the cluster formation. It is important to note that evenwithout impurities the fluctuation associated with the squeezingof the oxygen octahedron locally, originated in such stronglycorrelated liquid may lead to formation of a similar clusterof four holes bound to polarons. However, such states aremetastable and their structure depends on the doping level.With increasing temperature clusters become metastable.The life time of such a fluctuating cluster, τ li f e ∼ (cid:126) / E bin ( x ),depends on its binding energy at a given doping x . It is maxi-mal near optimal doping where x ∼ .
2. This e ff ectively meansthat in the region of the optimal doping there are continuousquantum fluctuations associated with formation of these reso-nance plaquettes consiting of four holes. In other words theyare decoupled from dopant impurities and become mobile.Four holes clusters behave as quasi-classical objects movingalong in a nearly flat band. Obviously such continuous fluc-tuations play huge role in the scattering of the remaining stillvery mobile free holes. As classical objects the thermal energyof clusters is E kin ∼ k B T . The motion of free current carri-ers through such a clustered media is very dissipative. Freeholes can be individually trapped by cluster fluctuations andturn all their energy irreversibly into heat. Using the Heisen-berg uncertainty principle we can estimate the scattering timeof free holes as the time they leave in the cluster fluctuation, E kin τ ∼ k B T τ ≥ (cid:126) /
2. Such a process takes a characteristic(relaxation) time, τ = τ Planck = (cid:126) / k B T , (18)which is the shortest possible relaxation time named as the Planck time[41].
7. Quantum Criticality and comparison with experiments
The resistivity of the mixture of two components – free cur-rent carrier holes and four hole clusters – can be described byDrude formula, R = m / n h ( x ) e τ , where m is the e ff ective massof holes and n h ( x ) is their density as a function of doping x .Near the optimal doping x ∼ . R = mk B Tn h e (cid:126) = α ( x ) T . (19)In the underdoped region the binding energy of clusters in-creases with decreasing doping or increasing r s as shown inFig. 2. Trapping of free holes into clusters becomes less proba-ble and their transport becomes more Fermi liquid like with T term in the resistivity.We are now in position to make a comparison with experi-ments on LSCO [25, 26], where in the region of optimal dopingthe T- linear dependence of resistivity has been in detail inves-tigated. The value of the e ff ective mass m = . m e and the holedensity n h ( x ) are determined above from the comparison withthe Hall coe ffi cient measurements [29]. Inserting these num-bers into Eq. (19) we get the T-linear coe ffi cient. At x = . α = . µ Ω cm / K and at x = . α = . µ Ω cm / K. Thesevalues are within the error bar of the experimental results[25],which is surprising for such an order of magnitude estimate.When x < .
18 the increasing binding energy of clusters di-minishes the T-linear behavior and increses the normal Fermiliquid contribution as also seen in experiments [25].These results confirm that the above theoretical calculationsof the pseudogap (see, Fig.2) associated with the formationof the four holes clusters - the resonance plaquettes providenot only qualitative but also quantitative description of the un-derdoped and optimally doped cuprates, That also anticipatesformation of anti-ferroelectric fractal structures which are allmelted at optimally doping where a quantum phase transitionarises.
8. Discussion and Conclusions
We have shown that in LSCO and other related compoundsCoulomb interaction between charge carriers and doped ions isvery important. The strong electronic correlations tightly con-nected with anti- Jahn-Teller lattice distortions lead to boundstates of four holes forming a resonance plaquette. The for-mation of the plaquettes located in the CuO plane is associatedwith the over-screening of the Coulomb interaction. In the un-derdoped region they are bound with dopant atoms creating acluster which has a dipole moment due to the separation of theplaquette and dopant impurity having opposite charges.Recently Poccia et al. [3] proposed a mysterious secondgrowth mechanism, which they have studied at T = + , which may be, in general, decoupled.Such excitonic mechanism of the local lattice defect forma-tion and distortions has been observed experimentally in Refs.[60, 61], indeed. When such a vacancy will approach to anotheri-O, they may annihilate. The creation of such virtual or shorttime living pairs increases significantly the mobility of the i-O’sand form as special pattern of the local lattice distortions, notedas Q3 in the Ref. [62, 3]. The pairs located in the LaO planeform dipoles lying in the plane that change the structure of theclusters. In fact there due to the presence of these extra cou-pled and decoupled i-O-V pairs the mobility of i-O’s increase.This leads to a faster formation of the new type of clusters dif-ferent from the ones described in the Refs. [1, 2, 63]. Theadditional dipoles associated with the new pairs created withX-ray illumination lead to a new period of the stripe orderingin these new type of clusters noted as Q3 in the Ref. [59]. Be-sides the excitons generated the LLD-Q3 may arise only in thedeeply underdoped case as excitons at all. The probability tosuch excitons with X-ray illumination very fast vanishes withthe doping. This explains why the Q3 clusters are arising onlyin the underdoped cuprates.Possibly, also the oxygen atoms may be displaced from theiroriginal positions in tetragons. They take one or two electronswith them, act as ions and leave behind holes, which are re-sponsible of the superconducting state when the temperatureis lowered below the critical temperature. These ions are notlifted into the LaO layer and are more mobile. However, nega-tive ions form again bound states with holes and dipoles orientin the growth process as in the case of i-O and form fractal clus-ters. The only di ff erence between these two mechanisms is thelocation of the oxygen ions and shows up as slightly di ff erentstripe order of the nano grains.Recently, the nature of oxygen dopant-induced states foundat − . eV in Bi Sr CuO + x by measuring ARPES spectra in awide photon energy range has been investigated.[9] The foundresonance profile of the corresponding nondispersive peak in-dicates an unexpected mixing with Cu states. The A g symme-try of the peak suggests that the oxygen dopant-induced statesare mixed with Cu through the O dopant -O apex z -Cu 3 d z − r channel. [9]Experiments indicate that the doped cuprates have a mix-ture of characters. To see this, we start with an undoped par-ent compound such as La CuO . This material is an insulatorwith a charge-transfer gap of ∼ planes as a consequence of strong on- site Coulomb repulsion between electrons in the same Cu 3d x − y orbital. The e ff ective magnetic interaction is well char-acterized by the superexchange mechanism, and the magneticexcitation spectrum is described quite well by spin-wave theorywith nearest-neighbor superexchange energy.[64]Thus, the dipolar clusters appear when dopant oxygen ionsin LaO layers trap charge carriers in CuO planes. The densityof dipole moments increases with doping and vanishes at thepseudogap temperature. The binding energy of holes into thesedipolar clusters defines the pseudogap, which we identify alsoas the activation energy of the two fluid model used by Gorkovand Teitelbaum. [30, 29, 57] Dipole moments have a strong in-teraction with electromagnetic radiation and subjected to suchradiation dipoles become mobile. Due to the dipole-dipole at-traction existing for some dipole orientations and repulsion forother orientations they may form clusters. This finding explainsthe recent observation of the self-organization of mobile oxygendopant ions in LCO by Bianconi et al. [1, 2]. Under irradia-tion oxygen ions resting in the LaO layer as well as dipoles getexcited and are forced to move towards the energetically mostfavourable positions.Ferroelectricity in La CuO + y and La − x Sr x CuO was re-ported at exceptionally low doping.[11, 12]. We have shownthat the dipole formation and its reduction to the two-level sys-tem under external electric explains the behavior of the ferro-electric polarization as a function of temperature and electricfield. It could also be responsible of the peculiar behavior ofthe dielectric constant [10]. The resonance plaquettes describedin this paper may play a very important role in the mechanismof high temperature superconductivity. Their presence in thevicinity of the hole band edge may lead to shape resonances asdescribed recently by Bianconi[65], that can be an origin of theCooper pairing in cuprates. Our results strongly suggest thatmany phenomena in cuprates such as pseudogap formation andhigh temperature superconductivity are interaction driven andtherefore these materials belong to a new class of holographicsuperconductors[66].
9. Acknowledgements
We thank Antonio Bianconi, Lev Gorkov, Danya Khomskii,Christos Panagopoulos, Montu Saxena, Grisha Teitelbaum andJan Zaanen for useful discussions. We also thank NanoSc-CostAction MP1201 for financial support.
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