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Superconductor-To-Metal Transition in Overdoped Cuprates ✉ Zi-Xiang Li1,2, Steven A

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Superconductor-To-Metal Transition in Overdoped Cuprates ✉ Zi-Xiang Li1,2, Steven A www.nature.com/npjquantmats ARTICLE OPEN Superconductor-to-metal transition in overdoped cuprates ✉ Zi-Xiang Li1,2, Steven A. Kivelson3 and Dung-Hai Lee 1,2 We present a theoretical framework for understanding the behavior of the normal and superconducting states of overdoped cuprate high temperature superconductors in the vicinity of the doping-tuned quantum superconductor-to-metal transition. The key ingredients on which we focus are d-wave pairing, a flat antinodal dispersion, and disorder. Even for homogeneous disorder, these lead to effectively granular superconducting correlations and a superconducting transition temperature determined in large part by the superfluid stiffness rather than the pairing scale. npj Quantum Materials (2021) 6:36 ; https://doi.org/10.1038/s41535-021-00335-4 INTRODUCTION superconductor–metal transition. We are aware that our model For over three decades, research on the cuprate superconductivity does not capture various quantitative aspects of the actual primarily focused on the underdoped and optimally doped region materials. In the next two paragraphs, we present the physical of the phase diagram. Here, it is now widely accepted that Tc is not picture underlying this work. set by the scale of Cooper pairing (as in BCS theory), but is instead A sketch of the Fermi surface of an overdoped cuprate is shown largely determined by the onset of phase coherence (i.e., by the in Fig. 1; it is shown as being hole-like, although in some cuprates 1,2 1234567890():,; superfluid density) . Phenomena such as the pseudogap, the Fermi surface passes through a Lifshitz transition at a doping p p intertwined orders, and strange metal behavior remain the focus concentration, Lif < smt, in which case it would be electron- 14,15 of considerable research today. In contrast, it is commonly like . The neighborhood of the van-Hove points—which we believed that the physics of the overdoped cuprates is more will refer to as the antinodal regions—is also the portion of conventional. For example, angle-resolved photoemission spectro- the Fermi surface farthest from the gap nodes in the d-wave scopy (ARPES) shows a large untruncated Fermi surface in the superconducting state and so is where the gap is largest. As we normal state with reasonably well-defined quasi-particle peaks, will discuss, the fact that the Fermi surface passes near the van- and a superconducting gap that decreases with increasing Hove point, meaning that the Fermi velocity is small in the 3 fi doping, more or less in tandem with Tc . Moreover, in at least antinodal regions, plays a signi cant role in the results we obtain; one material4, quantum oscillations, of the sort expected on the whether it is electron or hole-like is relatively less important. basis of band-theory, have been documented. In considering the effects of disorder, the scattering between Thus, it was a surprise that recent penetration depth measure- antinodal regions (indicated by the orange line in the figure) is ments5,6 on crystalline LSCO films suggest that the super- particularly important, as it is pair-breaking. Such antinode to conductivity in the overdoped cuprates is also limited by the antinode scattering is apparent in STM quasi-particle interference onset of phase coherence. Consistent with this result, recent measurements in both non-superconducting16 and superconduct- ARPES measurements7 of overdoped Bi2212 found spectroscopic ing17 overdoped Bi2201. In particular, in ref. 17, it is shown that at evidence that Cooper pairs are already formed at temperatures voltages corresponding to the antinodal gap energy, the Fourier about 30% higher than Tc. Adding to the puzzle, recent optical transform of the local density of states exhibits a broad maximum 8 q = π π conductivity measurements showed that below Tc a large fraction at momentum ( , ). Moreover, from the coherence factor, it is of the normal-state Drude weight remains uncondensed. This is inferred that this scattering occurs between momentum regions consistent with earlier specific heat measurements which show a having opposite signs of the gap function17. T-linear term that persists to the lowest temperatures, T ≪ Tc, with When the Cooper pair coherence length is comparable to the a magnitude that is a substantial fraction of its normal-state correlation length of the disorder potential, the prior mentioned value9,10. A possibly related observation11–13 from scanning pair-breaking causes the pair-field amplitude to be spatially tunneling microscopy (STM) is that, at least up to moderate levels heterogeneous18,19. The superfluid stiffness is large in regions of overdoping, a spectroscopic gap persists in isolated patches up with high pair-field amplitude, whereas the stiffness is low where fi to temperatures well above Tc, so that the normal-state electronic the pair- eld amplitude is small. This is reminiscent of a granular structure is suggestive of superconducting grains embedded in superconductor. The small stiffness in the inter-granular regions a normal metal matrix. Other than the STM results (for which causes the averaged zero-temperature superfluid density to be the relevant data do not exist at very high overdoping), these low, hence superconducting phase fluctuations (both classical and phenomena become increasingly dramatic as the doped hole quantum) are enhanced. In the metallic regions the Cooper concentration, p, approaches the critical value, psmt, at which pairing instability is inhibited since (a) the repulsive interactions the superconductor-to-metal transition occurs at the overdoped between electrons force the average of the superconducting order end of the superconducting dome. parameter to be zero around the Fermi surface, and (b) a sign- The primary goals of this paper are to present a simple changing order parameter is suppressed by disorder scattering. theoretical model that captures what we believe to be the essence The unpaired electrons in the inter-granular regions give rise of the above phenomena, and to explain the cause of the to a substantial uncondensed Drude component in the optical 1Department of Physics, University of California, Berkeley, CA, USA. 2Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA. 3Department of ✉ Physics, Stanford University, Stanford, CA, USA. email: [email protected] Published in partnership with Nanjing University Z.-X. Li et al. 2 Fig. 2 Plot of the density of doped holes as a function of impurity concentration. The disorder strength is fixed with w = 1. different impurity configurations in order to compute the Fig. 1 Model Fermi surface. Antinode to antinode scattering configuration averages of physical observables; typically, we induced by the disorder potential is indicated by the yellow arrow. average over 64 distinct impurity configurations but we average over 128 configurations when the impurity concentration is large T fi conductivity and to a residual -linear term in the speci c heat. A and the superconducting pairing is highly inhomogeneous. superconductor-to-metal transition occurs when the supercon- We solve the model by self-consistent BCS mean-field fi 18 ducting islands grow suf ciently sparse . approximation. However, the disorder scattering is treated exactly. Note that the normal-state transport is dominated by the nodal 1234567890():,; Since H lacks translation invariance, the self-consistency equations quasiparticles, which are known to be less affected by impurity 20,21 need to be solved numerically. Since our calculation does not scattering . In the rest of the paper, we present results capture the thermal and quantum fluctuations, we focus primarily corroborating the physical picture presented above. on zero temperature and on doping sufficiently away from the quantum critical doping of the superconductor–metal transition. RESULTS The carrier concentration is controlled by the chemical potential μ and the impurity concentration n . Sometimes we fix μ while The model imp changing nimp to reach the desired carrier (hole) concentration. In The model we use to describe the superconducting state contains the case where we want to study the effect of disorder at a fixed hopping terms, interaction terms, and disorder potential terms. carrier density, we tune μ while varying nimp to achieve the desired The Hamiltonian is X X carrier density. To avoid the complex issues of the pseudogap, H ¼À t ðcy c þ h:c:Þþ ðw À μÞcy c þ H intertwined orders, and strange metals, we take our lowest carrier ij iσ j;σ i iσ i;σ int (1) i;j;σ i;σ concentration to be slightly larger than optimal doping. Thus except in section “Role of the antinodal dispersion”, the impurity t i j where ij is the hopping integral between sites and on a square concentration is measured relative to that present at optimal lattice which we take (to produce a cuprate-like Fermi surface) to doping. For example, in Fig. 2 we show the hole concentration p be tij = 1 between nearest-neighbor sites, tij = −0.35 between as a function of nimp at fixed μ with w = 1. second-neighbor sites, and tij = 0 for all further neighbors. To compare different pairing symmetries, we consider two different The mean-field solution forms of Hint: (1) As a model of a d-wave superconductor (relevant to the cuprates) we adopt a model with a nearest-neighbor The local value of the gap parameter, Δij, that enters the mean- field equations, which we will refer to as the pair field, is given by antiferromagnetic Heisenberg exchange interaction, Hint = J∑〈ij〉Si ⋅ Sj. (2) As a model of an s-wave superconductor,P we consider an the product of the pairing interaction times the expectation value y y s Δ attractive Hubbard interaction, Hint ¼ÀU ici"ci"ci#ci#.Wefix the of the pair annihilation operator. In the -wave case, is site strength of the pairing interactions to J = 0.8 and U = 1.35, diagonal, Δj ≡ Δjj = U〈cj↑cj↓〉 while for the d-wave case, Δij = J〈ci↑cj↓ respectively, so that the two superconducting gap scales in the + cj↑ci↓〉, where i, j are any pair of nearest-neighbor sites.
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