Resistance of High-Temperature Cuprate Superconductors

Cuprate superconductors have many different atoms per unit cell. A large fraction of cells (5-25%) must be modified ("doped") before the material superconducts. Thus it is not surprising that there is little consensus on the superconducting mechanism, despite almost 200,000 papers. Most astonishing is that for the simplest electrical property, the resistance,"despite sustained theoretical efforts over the past two decades, its origin and its relation to the superconducting mechanism remain a profound, unsolved mystery."Currently, model parameters used to fit normal state properties are experiment specific and vary arbitrarily from one doping to the other. Here, we provide a quantitative explanation for the temperature and doping dependence of the resistivity, Hall effect, and magnetoresistance in one self-consistent model by showing that cuprates are intrinsically inhomogeneous with a percolating metallic region and insulating regions. Using simple counting of dopant-induced plaquettes, we show that the superconducting pairing and resistivity are due to phonons.

Since superconductivity requires coherent Cooper pairing of electrons, knowing what couples most strongly to electrons is absolutely necessary for understanding what causes high-temperature cuprate superconductivity. The resistivity, the Hall effect, and the magnetoresistance are fundamentally measurements of the momentum dependent Fermi surface scattering rate, 1/τ (k), that measures the strength of whatever is coupling to electrons. If the origin of the scattering rate and its temperature dependence are not understood, then most likely cuprate superconductivity is not understood either.
The earliest resistivity (ρ) measurements [3] on cuprates found ρ to be approximately linear in temperature over the huge temperature range of 10 < T < 1000 K. A linear T resistivity is characteristic of electron-phonon (or electron-boson) scattering. Yet phonons are not believed to cause ρ, despite the fact that historically, the dominant scattering mechanisms of electrons in metals have been phonons and impurities. There are two reasons for this conclusion: First, at low temperatures, the Bose-Einstein statistics of the phonons reduces the phase space for scattering, leading to ρ(T ) ∼ T 5 (the Bloch-Gruneisen law) [4].
The T 5 scaling should be observable for T < Θ Debye /10 where Θ Debye is the Debye temperature (the characteristic energy of the highest energy phonons). Since Θ Debye ∼ 400 K in cuprates [5], phonon scattering is not compatible with the observed linearity. Magnons are also eliminated because Θ Mag ∼ 1500 K [6].
Second, the magnitude of ρ at high T for some dopings exceeds the Mott-Ioffe-Regel limit (MIR) [7] that occurs when the electron mean free path reaches the shortest Cu−Cu distance (≈ 3.8Å). For cuprates, ρ(T ) should saturate to ρ MIR ∼ 1000 µΩ-cm. Instead, ρ increases linearly right through ρ MIR , leading to the conclusion that the normal state may not even be a typical metal (a non-Fermi liquid).
Both of these conclusions are invalid when the crystal is intrinsically inhomogeneous, as we will show. Briefly, the first point implicitly assumes that phonon momentum is a good quantum number. The second underestimates ρ MIR by overestimating the density of charge carriers. For example, at optimal hole doping of x = 0.16 (we define x to be the number of holes per planar CuO 2 ), we find the fraction 4 × 0.16 = 0.64 of the crystal is metallic leading to a ρ MIR that is (1/0.64) = 1.56 times larger than the conventional estimate. ρ(T ) remains below the larger ρ MIR up to the melting temperature.
Recently, the doping and k-vector dependent scattering rate, 1/τ (x, k), has been extracted from a series of beautiful experiments by Hussey et al. [2,[8][9][10]. In the first set of experiments [8,9], the magnetoresistance (MR) of large single crystals of single layer Tl 2 Ba 2 CuO 6+δ was measured as a function of the direction of an applied 45 Tesla magnetic field for T < 40K and large overdoping. They found that 1/τ (x, k) is the sum of three terms, 1/τ (x, k) = 1/τ 0 (x) + A 1 (x) cos 2 (2ϕ)T + A 2 (x)T 2 , where A 1 (x) and A 2 (x) are hole doping dependent constants and ϕ is the angle between the Cu−O bond direction and the k-vector as shown in the top right corner of figure 6a. The first term, 1/τ 0 (x), is a constant that is sample dependent even at the same doping. The anisotropic linear T term is zero for k-vectors along the diagonal (ϕ = π/4) and largest for k along the Cu−O bond directions (ϕ = 0, π/2).
In an elegant theoretical analysis, Hussey et al. [2,11,12] showed that a scattering rate of the above form explained the resistivity, the Hall effect, and the MR for a large range of dopings, temperatures, and different cuprate materials. The authors suggested that the isotropic T 2 part of the scattering rate arose from electron-electron Coulomb scattering while the origin of the T term is unknown. Finally, the strong doping dependence of A 1 (x) and A 2 (x) was noted, but not explained.
In another experiment, Hussey et al. [2,10] used large magnetic fields to lower the T c (the superconducting transition temperature) of La 2−x Sr x CuO 4 . They found that ρ(T ) was the sum of doping dependent T and T 2 terms at low temperatures plus a sample and doping dependent constant. These results suggested that the 1/τ (x, k) form found for Tl 2 Ba 2 CuO 6+δ was applicable to all dopings and cuprate materials.
The authors also found that the low temperature ρ evolved into linear T at high temperatures, as seen previously [13]. They concluded that the isotropic T 2 term was due to electron-electron Coulomb scattering and Mott-Ioffe-Regel saturation of the scattering rate at high temperatures was reducing this term down to linear T .
Even for a fixed doping value, it is clear that the low and high T forms of 1/τ (k) put strong constraints on theory. An even stronger constraint is explaining the doping dependence of α 1 (0) and α 1 (∞) and their peculiar "crossover" at doping of x ≈ 0.19, as can be seen in figure 1.
In prior publications [14,15], we showed that cuprates are comprised of a percolating metallic region and insulating regions. The metallic region is formed from 4-Cu-site square plaquettes in the CuO 2 planes that may overlap (share a Cu atom). These plaquettes are centered around the dopants (e.g., Sr in La 2−x Sr x CuO 4 ). The insulating regions are comprised of localized d 9 Cu spins that are coupled antiferromagnetically.
By simply counting the volume and surface areas of these two regions as a function of doping, we explained the origin and doping dependence of the pseudogap (PG) [14], the universal room-temperature thermopower, the STM incommensurability, the neutron resonance peak, and the generic cuprate phase diagram [15]. Here, we show that α 1 (0) and α 1 (∞) are also explained by the same intrinsic inhomogeneity (figure 2).
We will show that α 1 (0) is due to phonon scattering inside non-overlapping plaquettes and α 1 (∞) is due to scattering from all phonons. Therefore, α 1 (0) ∝ N 4M and α 1 (∞) ∝ N Cu , where N 4M equals the number of planar Cu sites inside non-overlapping plaquettes and N Cu is the total number of planar Cu sites. Dividing by the total number of charge carriers, N M , leads to α 1 (0) = C(N 4M /N M ) and α 1 (∞) = C(N Cu /N M ). The best fit (see figure 1) is C = 0.904 µΩ-cm/K. Figure 1a has one adjustable parameter, C. Figure 1b is the ratio of α 1 (0)/α 1 (∞) and has zero adjustable parameters. The excellent fit in figure 1 using simple counting is the main result of this paper. Prior to describing our detailed model for the anomalous normal state transport, we summarize the cuprate doping phase diagram. All cuprate superconductors have CuO 2 planes. When undoped, they are insulating Heisenberg spin-1/2 antiferromagnets (AF) with a Néel temperature T N ∼ 200 K and spin-spin coupling J dd ≈ 0.13 eV [6]. The spin is localized on planar Cu atoms in a d 9 configuration with the single unpaired spin in the Cu d x 2 −y 2 orbital, where the x and y axes point along the Cu−O bonds. Doping of more than x ≈ 0.05 holes per planar Cu destroys the bulk Néel AF phase and superconductivity occurs for T < T c where T c is the superconducting transition temperature. Above T c , the material is metallic. When x is increased past ≈ 0.27, superconductivity vanishes (T c = 0).
The highest T c occurs for x ≈ 0.16 [16]. The band structure consists of approximately 2D CuO 2 bands with Cu d x 2 −y 2 and O p σ (p x and p y ) orbital characters that cross the Fermi level [17]. The Fermi surface is hole-like and centered around k = (π/a, π/a) where a is the  In the cuprates, no long-range static distortion compatible with figure 3a or 3b is found.
Thus the coupling between these out-of-plane hole states is strong, leading to a vibronic or dynamical Jahn-Teller state being formed instead. Figure 3c shows an instantaneous configuration of the out-of-plane holes in a single CuO 2 plane. The out-of-plane hole states are shown by the red "dumbbells". Coupling between adjacent plaquettes leads to correlation between the orientations of neighboring dumbbells.
When plaquettes overlap as shown by the green squares in figure 3c, the orbital degeneracy of the out-of-plane holes shown in figures 3a and 3b is removed. Thus overlapping plaquettes have no dynamical Jahn-Teller state.
In figure 3d, the correlation energy between neighboring dynamical Jahn-Teller states is shown for plaquettes inside a single CuO 2 plane, planar , and for plaquettes in adjacent CuO 2 planes, perp . In the Supplement, perp is estimated to be ∼ 3.6 × 10 −5 eV ∼ 0.42 K and planar is on the order 0.01 to 0.1 eV (∼ 100 − 1000 K).
For temperatures greater than ∼ 1 K, there is no correlation between dynamic Jahn-Teller states between planes. In analogy to the "dynamical detuning" proposed for electron transport normal to the CuO 2 planes [19,20], the lack of correlation of dynamical Jahn-Teller states normal to the plane disrupts the phase coherence of phonons between CuO 2 layers. Hence, the phonon modes inside non-overlapping plaquettes become strictly 2D for temperatures greater than perp .
Since momentum is not a good quantum number for the 2D phonon modes inside the region of the non-overlapping plaquettes, these modes have the same character as those in amorphous metals. We show later that these amorphous 2D phonons lead to the lowtemperature linear resistivity (the α 1 (0) term in figure 1). Also, α 1 (0) ∝ N 4M because the number of amorphous 2D phonon modes is proportional to the total number of Cu sites inside the non-overlapping 4-Cu-site plaquettes, N 4M .
The phonon modes occurring "outside" the non-overlapping plaquettes are amorphous and 3D. The number of these modes is proportional to N Cu − N 4M . These modes lead to the T 2 resistivity term, as we show later.
A percolating pathway of plaquettes leads to a delocalized metallic Cu d x 2 −y 2 and O p σ band inside the percolating swath (yellow region in figure 2). The undoped regions (not part of the doped metallic swath) remain localized d 9 Cu spins with AF coupling. Thus cuprates are intrinsically inhomogeneous with a percolating metallic region and insulating AF regions.
Our proposed inhomogeneity appears prima facie to be at odds with the recent observation of quantum oscillations in heavily overdoped Tl-2201 cuprates [21][22][23]. In the Supplement, we estimate the Dingle temperature arising from the inhomogeneity in our model and show that it is compatible with these measurements.
The phonon modes that lead to superconductivity involve the planar displacement of the O atoms at the interface between the metal and AF regions (figures 4 and 5). These longitudinal-optical (LO) O phonon modes are the softened modes seen by neutron scattering [24][25][26] for the superconducting range of dopings for momenta along the (π, 0) and (0, π) directions. They are different from the Jahn-Teller modes (figure 3a,b) that lead to the linear T term in ρ. Figure 6 shows how these modes lead to the observed d-wave (d x 2 −y 2 ) superconducting gap.
To complete the derivation of the results in figure 1, we must show that phonon modes inside the non-overlapping plaquettes produce the anisotropic cos 2 (2ϕ)T scattering rate and that the remaining phonons contribute the T 2 isotropic term. In a typical metal with full translational symmetry, the scattering rate for a metallic electron with momentum k is [4] where the sum is over k on the Fermi surface and q is the phonon momentum. M is the nuclear mass, ω q is the phonon energy, θ is the angle between k and k , and The delta function maintains momentum conservation. The (1 − cos θ) term becomes T dependent at low T because n B (ω) restricts the scattering to small-angles (θ 1).
All phonons can scatter k to k in cuprates because of their intrinsic inhomogeneity. Hence, the (1 − cos θ) term in 1/τ (k) has no T dependence. In this case [27], where λ is summed over all phonon modes and the delta function has disappeared. The number of excited phonon modes at temperature The 2D Jahn-Teller modes in figure 3 have d = 2, leading to 1/τ ∼ T . This linear scattering rate persists for T > perp ∼ 0.4 K. The remaining 3D modes lead to 1/τ ∼ T 2 .
Since the phonon modes are amorphous, 1/τ (k) is independent of k for the 3D T 2 modes.
The 2D Jahn-Teller modes are derived from the vibronic distortions in figure 3. These modes raise the Cu orbital energy for two diagonal Cu atoms in a plaquette and lower the orbital energy for the other two Cu atoms. Hence, their scattering is modulated by an envelope function ≈ cos 2 (2ϕ). Thus 1/τ (k) ∼ cos 2 (2ϕ)T . This modulation of 1/τ (k) does not change the total scattering around the Fermi surface. Instead, it merely redistributes its weight around the surface. Hence, the proportionality constant in front of the expressions for α 1 (0) and α 1 (∞) is the same.
Thus the doping and temperature dependence of the resistivity, Hall effect, and mag-netoresistance of cuprates is due to the intrinsic inhomogeneity of these materials. This inhomogeneity arises from the dopant atoms and leads to a percolating metallic region and insulating antiferromagnetic regions. The percolating metallic region is comprised of 4-Cusite plaquettes that are centered around dopants. The plaquettes may overlap each other.
We show that the ratio of non-overlapping plaquettes to the total number of Cu atoms equals the ratio of the low and high temperature linear T terms in the resistivity, with no adjustable parameters. We calculate this ratio and find very good agreement with experiment. The resistivity is shown to be due to phonon scattering. The special "crossover" doping, x ≈ 0.19, for α 1 (0) and α 1 (∞) occurs at the doping when plaquettes first begin to overlap. We calculate this value to be x = 0.187 with no adjustable parameters. The low doping value of α 1 (0) and the high doping value of α 1 (∞) are found to be exactly the same in our model. This is observed by experiments within the error bars. We find an excellent fit to the doping evolution of α 1 (0) and α 1 (∞) with one adjustable multiplicative constant.
We also provide a physical picture for the origin of the observed softened LO O phonon mode and show that it leads to a d-wave superconducting gap.
We conclude that the anomalous normal state transport in cuprates provides strong evidence for intrinsic inhomogeneity, metallic percolation, phonon superconducting pairing, and phonon normal state scattering.

Methods
We dope plaquettes using a simple model for the Coulomb repulsion of the dopants. Since the dopants reside in the metallic region, the repulsion is screened over a short distance.
Thus we assume plaquettes are randomly doped without overlap. This doping is possible Acknowledgements. The author is grateful to Andres Jaramillo-Botero and Carver A.
Mead for many stimulating discussions.
[31] Phillips, J. C. Self-organized networks and lattice effects in high-temperature superconductors.
Phys. Rev. B 75 (2007).     Figure S1 shows the calculated number of Cu atoms inside the metallic region, N M , and the number of metallic Cu atoms that reside in non-overlapping plaquettes, N 4M . In figure   2 of the main text, the number of Cu sites in the black squares with "4" inside is N 4M and the yellow region of the same figure is N M . The doping evolution of these two numbers determines the low and high T linear coefficients of the resistivity shown in figure 1 (α 1 (0) and α 1 (∞)). Pure density functionals such as local density (LDA) [S1, S2, 17] and gradient-corrected functionals (GGA, PBE, etc) [S3] obtain a metallic ground state for the undoped cuprates rather than an antiferromagnetic insulator because they underestimate the band gap due to a derivative discontinuity of the energy with respect to the number of electrons [S4, S5]. In essence, the LDA and PBE functionals include too much self-Coulomb repulsion.
This repulsion leads to more delocalized electronic states. Removing this extra repulsion is necessary in order to obtain the correct localized antiferromagnetic spin states of the undoped cuprates.
The self-repulsion problem with LDA has been known for a long time [S6]. Very soon after the failure of LDA to obtain the undoped insulating antiferromagnet for cuprates, several approaches were applied to correct the flaws in LDA for La 2 CuO 4 . Using a selfinteraction-corrected method (SIC-LDA) [S6], Svane [S7]  Our ab-initio calculations [S10, 18] were performed using the hybrid density functional, B3LYP. Due to its remarkable success on molecular systems, B3LYP has been the workhorse density functional for molecular chemistry computations for almost 20 years [S11, S12]. For example [S11], B3LYP has a mean absolute deviation (MAD) of 0.13 eV, LDA MAD = 3.94 eV, and PBE MAD = 0.74 eV for the heats of formation, ∆H f , of the 148 molecules in the extended G2 set [S13, S14]. B3LYP has also been found to predict excellent band gaps for carbon nanotubes and binary and ternary semiconductors relevant to photovoltaics and thermoelectrics [S15, S16].
The essential difference between B3LYP and all pure density functionals is that 20% exact Hartree-Fock (HF) exchange is included. B3LYP is called a hybrid functional because it includes exact HF exchange. The HF exchange removes some of the self-Coulomb repulsion of an electron with itself found in pure DFT functionals. A modern viewpoint of the reason for the success of hybrid functionals is that inclusion of some exact Hartree-Fock exchange compensates the error for fractional charges that occur in LDA, PBE, and other pure density functionals [S17]. The downside to using hybrid functionals is they are computationally more expensive than pure density functionals.
Our B3LYP calculations reproduced the experimental 2.0 eV band gap for undoped La 2 CuO 4 and also had very good agreement for the antiferromagnetic spin-spin coupling, J dd = 0.18 eV (experiment is ≈ 0.13 eV) [S10]. We also found substantial out-of-plane apical O p z and Cu d z 2 character near the top of the valence band in agreement with LDA + U and SIC-LDA calculations.
We also performed B3LYP calculations on La 2−x Sr x CuO 4 for x = 0.125, 0.25, and 0.50 with explicit Sr atoms using large supercells [18]. Regardless of the doping value, we always found that the Sr dopant induces a localized hole in an out-of-the-plane orbital that is delocalized over the four-site region surrounding the Sr as shown in figures 3 and S3. This result is in contrast to removing an electron from the planar Cu d x 2 −y 2 /O p σ as predicted by LDA and PBE.
Our calculations found that the apical O's in the doped CuO 6 octahedron are asymmetric anti-Jahn-Teller distorted. In particular, the O atom between the Cu and Sr is displaced 0.24Å while the O atom between the Cu and La is displaced 0.10Å. XAFS measurements [S18] find the apical O displacement in the vicinity of a Sr to be ≈ 0.2Å. Significantly, no samples with lower doping showed quantum oscillations.

Compatibility of Our Proposed Inhomogeneity and the Quantum Oscillations in
Using the doping methodology described in the Methods section of our paper, we find that 4.5% of the planar Cu atoms are localized d 9 spins at 0.26 doping and 0.6% of the planar Cu atoms are localized d 9 spins at 0.31 doping. At these two dopings, only isolated d 9 spins remain in our model. Below ≈ 0.26 doping, some of the d 9 Cu spins neighbor other d 9 Cu spins. These d 9 Cu atoms are no longer isolated.
In his classic book, Shoenberg [S19] states that Dingle temperatures much above 5K smear out quantum oscillations so that they are not resolvable (page 62). Shoenberg also states (page 62 and Table 8.1) that every 1% impurity typically adds 10 − 100 K to the Dingle temperature. Thus it is possible the ≈ 0.6% "impurities" we expect for the 0.31 doped samples may still lead to observable quantum oscillations, but the 4.5% impurities at 0.26 doping appears to be too large to be compatible with experiment.
The flaw in the above simplified analysis is that a localized Cu d 9 "impurity" is not a typical impurity. The charge distribution around a metallic and localized d 9 Cu are virtually identical in the CuO 2 planes in cuprates. For both Cu atoms, the Cu has approximately +2 charge with a hole in the d x 2 −y 2 orbital. The d 9 Cu impurity is not a charged impurity.
Instead, it is a neutral impurity with exactly the same sized valence orbitals as the metallic Cu atom. Since the impurity scattering cross section (and hence the scattering length) scales as the square of the difference in the potential between these two types of Cu atoms, the scattering can be up to two orders of magnitude smaller than a typical impurity, as we show below. Thus 4.5% d 9 Cu impurities may lead to the same amount of scattering as 0.045% atomic substitutions of different atoms. At 10 − 100 K Dingle temperature per 1% impurity, this leads to a Dingle temperature of 0.45 − 4.5 K for the 0.26 doped sample.
We can use the Friedel sum rule [4] to quantify this physical argument. The Friedel sum rule incorporates the response of the metallic electrons around an impurity to screen out the impurity charge distribution and maintain a constant Fermi energy throughout the crystal.
The relevant Cu orbitals near the Fermi level are d x 2 −y 2 , d z 2 , and 4s with symmetries B 1g , A 1g , and A 1g , respectively. The charge difference between the impurity Cu atom and the metallic Cu atom is Z = 0, leading to Z = 0 = 2 π [δ A + δ B ] where δ B is the d x 2 −y 2 phase shift at the Fermi level and δ A is the combined d z 2 and 4s phase shifts at the Fermi level.
Since there is predominantly d x 2 −y 2 character at the Fermi level, the d z 2 and 4s phase shifts should be small, δ A ∼ 0.1.
The Friedel sum rule therefore forces the d x 2 −y 2 phase shift to be small, δ B ∼ 0.1 rather than a δ B ∼ 1 for a "standard" impurity arising from atomic substitution.
Since the scattering cross section scales as sin 2 δ B [4], the scattering cross section may be two orders of magnitude smaller than a typical impurity. Thus we have shown that the scattering length expected from the inhomogeneity of our plaquette model leads to Dingle temperatures from 0.45 − 4.5 K at 0.26 doping and 0.006 − 0.6 K at 0.31 doping.
Rourke et al measure the attenuation of the oscillations due to the Dingle temperature, but they do not quote their Dingle temperature. We can extract their Dingle temperature in two ways.
First, the Dingle temperature, x D , is given by where k B is Boltzmann's constant and τ is the scattering time. Multiplying the top and bottom of the right-hand-side of this equation by the Fermi velocity, v F , we obtain where l 0 is the scattering length. l 0 = 400Å in Rourke et al. v F ≈ 1 − 3 × 10 7 cm/s in cuprates [11]. For v F = 2 × 10 7 cm/s, x D = 6.1 K.
Second, we may use where m therm /m elec is the ratio of the band electron mass measured by the temperature dependence of the oscillation amplitude (∼ 5 in Rourke et al), µ B is the Bohr magneton, F 0 is the oscillation frequency (1.8 × 10 4 Tesla in Rourke et al). This leads to x D = 5.2 K.
Therefore, estimates of the scattering from our plaquette induced inhomogeneity lead to a Dingle temperature compatible with observations.
In our model, below ≈ 0.26 doping, there are d 9 Cu atoms that are no longer isolated.
These Cu atoms form antiferromagnetic regions in the material. We anticipate that the scattering from these "non-isolated" regions will lead to greater scattering than isolated d 9 Cu atoms. This may explain why no quantum oscillations were found for dopings less than ≈ 0.26.
Despite the calculations done here, further work needs to be done to obtain precise numbers for the scattering cross section from an isolated d 9 Cu spin. What we have shown here is that it is most likely that the inhomogeneity we propose in this paper does not contradict the quantum oscillations found in heavily overdoped Tl-2201.

The Debye-Waller Factor and Amorphous/Glassy Metals
For liquid metals, Ziman and Faber [S20, S21] showed that the T dependence of the structure factor, S(K), evaluated at K = 2k F , where k F is the Fermi momentum determines the T dependence of the resistivity. Their resistivity expression is given by [S20, 28], Here v F is the Fermi velocity, t(K) is the scattering matrix element, and Ω is the volume.
The integral for ρ is dominated by the maximum momentum transfer across the Fermi surface, or 2k F . Thus the T dependence of the structure factor, S(2k F , T ), determines the temperature dependence of ρ.
In amorphous/glassy metals, the structure factor is dominated by the "zero-phonon" elastic term that is given by S(K, T ) = S(K)e −2W (K) where S(K) is the static structure factor and W (K) is the Debye-Waller factor arising from the nuclear motion at temperature S23, S24, S25]. Here, K = 2k F . Using S(K, T ) ≈ S(K)[1 − 2W (K)], we conclude that the temperature dependence of the W (K) determines the temperature dependence of ρ(T ).
The Debye-Waller factor is given by W (K) = (1/2) q |K · U q | 2 where the sum is over all phonon modes q and U q is the amplitude for the q mode. U q can be determined from where M is the nuclear mass,hω q is the energy of the phonon mode q, and n q is the Bose-Einstein occupancy of the phonon mode, n q = 1/(e βhωq − 1). From this expression, we find the form of the the Debye-Waller factor to be, The expression for W (K) has exactly the same form that we obtained from the matrix element argument in the main text [27].
The Bose-Einstein occupation, n q , cuts off the integral at ∼ k B T , where k B is Boltzmann's constant. Since ω q = cq, where c is the speed of sound, W (K) ∼ T 2 . For 2D phonon modes, the integral is over d 2 q instead of d 3 q leading to W (K) ∼ T .
For 2D, the Debye-Waller factor diverges as dω/ω ∼ log T as T → 0. In fact, the 2D phonon modes are not strictly 2D for T < perp ∼ 0.4 K. Less than 0.1 K, these modes are 3D leading to a finite value for the Debye-Waller factor. For T > 0.1 K, the Debye-Waller factor is linear in T leading to a linear resistivity.
Estimating the Coupling Energy of Dynamical Jahn-Teller Plaquettes in Neighboring CuO 2 Planes The coupling energy, perp , between two dynamical Jahn-Teller plaquettes in adjacent planes is given by the Coulomb repulsion energy difference between the different orientations of the plaquette hole orbitals. The details of the energy difference expression are shown in Figure S4.
The Thomas-Fermi screening length, λ, is estimated from the expression [4], where e is the electron charge and N (0) is the density of states. There is approximately one metallic electron per unit cell leading to a density of states of approximately one state per eV per unit cell. The volume of the unit cell is, Ω cell = a 2 d, where a = 3.8Å and d = 6.0Å.
Substituting these numbers into the expression shown in figure S4 leads to perp = 3.6 × 10 −5 eV = 0.42 K.

Oxygen Modes Around an Isolated (Red) Plaquette
The red arrows in figure S5 show the softened phonon mode for the O atoms between the AF spins and the delocalized electrons inside the isolated plaquette. In this case, the difference in energy between the doubly occupied Cu site and the planar O orbital energy, x 2 −y 2 + U − p , is large, leading to localization of spins on the Cu. In the right figure, the Cu orbital energy is reduced due to the missing electron in the apical O sites directly above the Cu atoms leading to the neighboring doubly-occupied O p σ electrons delocalizing onto the Cu sites.
When two plaquettes are neighbors, the delocalization occurs over all eight Cu sites. When the doping is large enough that the plaquettes percolate in 3D through the crystal, a "metallic" band is formed in the percolating swath and current can flow from one end of the crystal to the other.
This metallic band carries the current in the normal state and becomes superconducting below T c . Jahn-Teller modes in neighboring CuO 2 planes. The coupling, perp , is given by the difference in Coulomb repulsion energy between the blue and red "dumbbells", (B, R), and the orange and red "dumbbells", (O, R). The 1/R Coulomb repulsion is attenuated by the Thomas-Fermi metallic screening length (dervied in the text to be, λ ≈ 0.69Å). Since the charge in a dumbbell is divided equally between the two ends, a charge of (1/2)e is used for each interaction term (e is the electron charge). The Cu-Cu lattice spacing in the plane is 3.8Å and the distance between planes is 6.0Å.

Cu
We find perp = 3.6 × 10 −5 eV = 0.42 K.  figure 2). Yellow is used here to signify that there is delocalization of the Cu/O d x 2 −y 2 p σ orbitals inside the plaquette. Since this plaquette is not connected to any other plaquette, these electronic states remain localized inside the plaquette. An orbital degeneracy at the Fermi energy of these electronic states leads to the pseudogap [14]. The O atoms that reside between the localized spins in the AF region and the delocalized electronic states inside the plaquettes will distort and the energy of this mode will change. We show the phonon mode by red arrows.