Resistance of High-Temperature Cuprate Superconductors

Cuprate superconductors have many different atoms per unit cell. A large fraction of cells (525%) 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 [1]. 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 [2].” 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. ∗Electronic address: [email protected]; Fax: 626-525-0918. Tel: 626-395-8148. 1 ar X iv :1 30 5. 10 58 v1 [ co nd -m at .s up rco n] 5 M ay 2 01 3 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 < ΘDebye/10 where ΘDebye is the Debye temperature (the characteristic energy of the highest energy phonons). Since ΘDebye ∼ 400 K in cuprates [5], phonon scattering is not compatible with the observed linearity. Magnons are also eliminated because ΘMag ∼ 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 (≈ 3.8 Å). For cuprates, ρ(T ) should saturate to ρMIR ∼ 1000 μΩ-cm. Instead, ρ increases linearly right through ρMIR, 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 ρMIR 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 CuO2), we find the fraction 4×0.16 = 0.64 of the crystal is metallic leading to a ρMIR that is (1/0.64) = 1.56 times larger than the conventional estimate. ρ(T ) remains below the larger ρMIR 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, 8–10]. In the first set of experiments