Optical conductivity of overdoped cuprate superconductors: application to LSCO
N. R. Lee-Hone V. Mishra D. M. Broun P. J. Hirschfeld





Department of Physics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada

Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

Department of Physics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada
Canadian Institute for Advanced Research, Toronto, ON, MG5 1Z8, Canada


Department of Physics, U. Florida, Gainesville FL 32611


We argue that recent  measurements on both the superfluid density and the optical conductivity of high-quality  films can be understood almost entirely within the theory of disordered BCS d-wave superconductors.  
The large scattering rates deduced from experiments are shown to arise predominantly from weak scatterers, probably the Sr dopants out of the CuO_2 plane, and correspond to significant suppression of T_c relative to a pure reference state with the same doping.  
Our results confirm the “conventional” viewpoint that the overdoped side of the cuprate phase diagram can be viewed as approaching the BCS weak-coupling description of the superconducting state, with significant many-body renormalization of the plasma frequency. They suggest that, while some of the decrease in T_c with overdoping may be due to weakening of the pairing, disorder plays an essential role.



Optical conductivity of  overdoped cuprate superconductors: application to LSCO
    P. J. Hirschfeld
    December 30, 2023
===============================================================================





§ INTRODUCTION


The cuprate phase diagram has been the subject of considerable controversy over the 30 years since the discovery of high-T_c superconductivity.<cit.>  Of the various exotic phases observed, including pseudogap, charge order, etc., most are located on the underdoped side.  The  superconducting phase is thought to be the simplest to understand, particularly on the overdoped side, where in many systems it exists without obvious competing or coexisting orders of other types.     Recently an experiment cast doubt on this simple picture of a “garden variety” BCS d-wave superconductor.  Božović et al.<cit.> measured the  superfluid density of a finely spaced set of high-quality epitaxial grown films of overdoped ,  and showed  that the superfluid density ρ_s and the superconducting transition temperature T_c approached zero together as a function of doping. This by itself contradicts BCS theory, which predicts that the T=0 superfluid density should be simply the carrier density (in appropriate units) independent of T_c in a clean system.  In a dirty superconductor, the superfluid density correlates with T_c,<cit.> but  the T-dependence of the penetration depth deduced from the measurements in Ref. Bozovic:2016ei is nearly linear down to the lowest temperatures, suggesting that the films are in fact largely free of disorder.  Under the assumption that the systems are clean,  Božović et al.<cit.> concluded that the scaling of ρ_s with T_c implies a substantial reduction in superfluid density relative to the nominal carrier density, inconsistent with a BCS description.

Subsequently, two of the present authors, with J. S. Dodge, analyzed the data of Ref. Bozovic:2016ei and reached rather  different conclusions.<cit.>  They pointed out that if the Sr dopants  were treated as weak  (Born limit) scatterers, the lack of a T^2 term in the penetration depth down to the lowest measurement temperatures of Božović et al. could easily be understood.  It has been known for many years that in this limit the quasiparticle states near the d-wave nodes are broadened by disorder, and that this broadening occurs significantly over an energy range that is exponentially small in Γ_N/Δ_0, where Γ_N is the normal state disorder scattering rate and Δ_0 the d-wave gap maximum.  For clean systems, states at energies greater than this scale are largely unaffected, and the penetration depth retains its linear dependence. Even for systems where Γ_N/Δ_0 becomes appreciable, however, the linear-T behavior obtains over a surprisingly large range.<cit.>  Band structure effects can also enhance this quasi-linearity in the same intermediate temperature range.  It was shown in Ref. Lee-Hone:2017 that the data could be fit extremely well over the entire range of doping and temperature using a single choice of disorder parameters and the known doping-dependent Fermi surface measured by ARPES.<cit.>  This comparison is summarized in Fig. <ref>.

The most logical way to distinguish between the disorder scenario of Ref. Lee-Hone:2017 and more exotic explanations is to measure the spectral weight of the condensed and uncondensed carriers directly using optical probes.  Recently, Mahmood et al. have performed terahertz (THz) spectroscopy<cit.> on  films very similar to those used in the Božović et al. superfluid density measurements.<cit.>  As anticipated, they found that  a significant fraction of
the carriers remained uncondensed in a broad Drude-like
peak at low temperatures. They showed consistency with the earlier superfluid density measurements, but argued that the broad uncondensed
spectrum implied a quasiparticle scattering rate too large  to be consistent with the linear-T-penetration depth observed.  Mahmood et al. therefore concluded that their results definitively ruled out a disorder-based explanation of the overdoped data.

In this work, we re-examine the conductivity of a  superconductor in the presence of disorder with a view towards refining the interpretation of Mahmood et al.'s work.  We argue that the weak scattering limit used predominantly in Ref. Lee-Hone:2017 yields a self-energy that grows roughly with energy up to a scale of Δ_0 in the clean limit, becoming roughly constant when the scattering rate is a significant fraction of T_c0, the critical temperature of the pure system, but before superconductivity is fully suppressed.    This implies that the conductivity spectrum has a Drude-like shape even in the superconducting state, for reasonably dirty systems.  This  result is qualitatively different from earlier works on the optical properties of d-wave superconductors, which focused primarily on underdoped to optimally doped systems where strong in-plane defects dominated the scattering.  Using exactly the same model and parameters employed in Ref. Lee-Hone:2017 to describe superfluid density,  we obtain a qualitatively very reasonable fit to the results of Mahmood et al.  Nearly perfect fits can be obtained with slight further fine tuning.  While for each sample, substantial pairbreaking is indeed involved, the Born limit scattering rate parameter Γ_N/T_c0 is still sufficiently small compared to 1 so as to allow the near-linear T dependence of the penetration depth in this limit. 

Our analysis thus implies that a BCS-like disordered d-wave scenario can indeed explain the unusual dependence of the superfluid density on T_c that is observed, and has further implications for the phase diagram as well. Taken at face value, our analysis suggests that  the superconducting dome is suppressed at high doping in  part due to disorder, but also indicates the existence of a reference “pure” T_c0 that is significantly higher than the observed T_c, implying that such critical temperatures might be achieved if the given doping level could be reached without potential scattering.    We discuss how T_c0 might depend on interactions, as well as possible roles of spin fluctuation scattering and forward scattering due to out-of-plane defects. 



§ THEORY






 §.§ Dirty d-wave superconductivity

The so-called “dirty d-wave" theory of cuprate superconductivity is a simple extension of the original field-theoretical formulation of the theory of disordered superconductors by Abrikosov and Gor'kov.<cit.>  This theory was applied for the first time to unconventional (nodal) superconductors  independently by Gor'kov and Kalugin<cit.> and by Rice and Ueda,<cit.> and extended to arbitrary impurity phase shifts by Hirschfeld et al.<cit.> and Schmitt-Rink et al.<cit.>  Impurity effects on the superfluid density of unconventional superconductors were treated within the same formalism by Gross et al.<cit.> for p-wave superconductors, and applied to d-wave systems by Prohammer and Carbotte,<cit.> and Hirschfeld and Goldenfeld.<cit.>
The latter work focused on strong scattering, in an attempt to explain Zn-substituted YBCO penetration depth experiments by Bonn, Hardy and co-workers that had made the case for d-wave superconductivity in cuprates.<cit.>  In the unitarity limit appropriate for the strong Zn scatterer in the CuO_2 plane, the linear density of states N(ω)∼ω in the pure d-wave superconductor was found to give way to a constant residual N(0) ∼γ, leading to a T^2 term in the penetration depth over a range of energies roughly equal to γ. Above this range, in the unitarity limit, states are relatively unaffected by disorder.<cit.>  The success of this early work has led to the impression that disorder always gives rise to immediate, strong asymptotic T^2 dependence in the penetration depth at low temperatures.  That this is not the case in the weak scattering (Born) limit, at least in practical terms  was noted by Hirschfeld et al.,<cit.> and recently reemphasized by Kogan et al.<cit.> and in a work by two of the current authors.<cit.>



We begin with the Nambu space Green's function for a dirty d-wave superconductor, written as 

    G(𝐤,iω_n)=-iω̃_n τ_0 + Δ_ kτ_1 + ξ_ kτ_3/ω̃^2_n + Δ_ k^2 + ξ_ k^2 ,

where Δ_𝐤 is the d-wave superconducting gap, ξ_ k is the single-particle dispersion relative to the Fermi level, τ_i are the Pauli matrices, and ω̃_n is a renormalized Matsubara frequency that, in the self-consistent t-matrix approximation (SCTMA),<cit.> follows 

    ω̃_n    ≡ω̃(ω_n)  = ω_n + i Σ
        =ω_n + πΓ⟨ N_𝐤(ω̃_n) ⟩_FS/c^2 + ⟨ N_𝐤(ω̃_n) ⟩_FS^2 .

Here the particular form of the self energy Σ is
for a single type of scatterer characterized by parameters (Γ,c), where  of the scattering phase shift, Γ is a scattering parameter proportional to the concentration of impurities, and

    N_𝐤(ω̃_n) = ω̃_n/√(ω̃_n^2 + Δ_𝐤^2) .

We have used the definition  ⟨ ... ⟩_FS as a Fermi surface angular average defined by

    ⟨ ...⟩_FS≡1/N_0∫_0^2 π N_ϕ (...) dϕ ,

where 

    N_ϕ = 1/2 π^2 ħ d|k_F|^2/𝐤_F ·𝐯_F

is the angle-resolved density of states on the Fermi surface, N_0 = ∫  N_ϕdϕ is the integrated density of states (including both spin channels), and d is the spacing of the two-dimensional conducting layers, in  taking the value .  The Fermi wavevector,  𝐤_F, and Fermi velocity, 𝐯_F, are both functions of ϕ, the momentum angle on the Fermi surface.

In much of this paper we will consider a specific model that was found in Ref. Lee-Hone:2017 to explain the superfluid density results well.  It is assumed that scattering is determined by a large concentration of Born scatterers (c ≫ 1, presumably the out-of-plane Sr dopants), and a small concentration of unitarity scatterers (c=0, possibly Cu vacancies).  For this model the self-consistency condition reads

    ω̃_n    =   ω_n + Γ_N^B⟨ N_ k(ω̃_n) ⟩_FS + Γ_N^U/⟨ N_ k(ω̃_n) ⟩_FS ,

with Γ_N^U/π = 1 K and Γ_N^B/π =17 K.

Note that for a d-wave order parameter, which averages to zero over the Fermi surface, there is no  renormalization of the gap Δ_𝐤 by nonmagnetic pointlike scatterers.  Impurities are pairbreaking, however, and suppress the gap through the effect of disorder on ω̃_n in the gap equation

    Δ_𝐤 =2 π T N_0 ∑_ω_n > 0^ω_0⟨ V_𝐤,𝐤^'Δ_𝐤^'/√(ω̃_n^2 + Δ_𝐤^'^2)⟩_FS ,

where Δ_𝐤 is the gap parameter at wave-vector 𝐤,  are the fermionic  Matsubara frequencies, V_𝐤,𝐤^' is the  pairing interaction, and ω_0 is a high energy cutoff.
This equation as Δ_ k→ 0 determines the critical temperature T_c, which is suppressed according to the universal Abrikosov–Gor'kov formula<cit.> with pairbreaking parameter twice the normal state single-particle scattering rate, Γ_N=Γ/(1+c^2). 







For concreteness, in what follows we assume a separable pairing interaction V_ k, k' = V_0 Ω_ kΩ_ k' in the  d-wave eigenfunction
Ω_𝐤 defined in the first Brillouin zone of the two-dimensional CuO_2 planes,

    Ω_𝐤∝(cos(k_x a) - cos(k_y a) ) ,

where a is the lattice spacing and Ω_𝐤 is normalized such that ⟨Ω_𝐤^2⟩_FS = 1. 




 §.§ Superfluid density


The superfluid density, ρ_s ≡ 1/λ^2, and optical conductivity are closely related, with the spectral weight available to form the superfluid set by the normal state conductivity, via the Ferrell–Glover–Tinkham sum rule,

    ∫_0^ω_cσ_1(ω) ω = π2ϵ_0 ω_p^2 .

Here ω_p is the in-plane plasma frequency of the conduction electrons, with the cutoff ω_c chosen to capture the Drude weight of the free carriers, leading to 

    ω_p^2 = e^2 N_0/ϵ_0⟨ v_F,x^2⟩_FS .

In the absence of disorder, the entire Drude weight would condense to give a clean-limit zero-temperature superfluid density

    ρ_s00≡1/λ_00^2 = μ_0 ϵ_0 ω_p^2 .

Interaction effects reduce the plasma frequency below its bare value, and will be discussed later in the context of the cuprates.

Expressions for the penetration depth or superfluid density of a d-wave superconductor in the presence of disorder have been given in many places, and were specifically reviewed in Ref. Lee-Hone:2017. We assume, as in most of these works, that nonmagnetic scatterers are pointlike so that impurity vertex corrections to the current–current correlation function 
vanish.<cit.>  Many references use a formulation that explicitly or implicitly assumes a circular Fermi surface for a quasi-2D system like the cuprates.  Here we work with a tight-binding model appropriate for overdoped ,<cit.> which was shown in Ref. Lee-Hone:2017 to be crucial to understanding the T dependence of the superfluid density.

Within our model, where we linearize the electronic structure near the Fermi surface, the finite temperature superfluid density in the presence of disorder is given by

    ρ_s(T) = μ_0 e^2 2 π T N_0∑_ω_n > 0⟨v_F,x^2Δ_𝐤^2/(ω̃_n^2 + Δ_𝐤^2 )^3/2⟩_FS .

At T=0, disorder suppresses ρ_s from the clean-limit value ρ_s00.  In Fig. <ref>, we have reproduced the results of Ref. Lee-Hone:2017 based on Eqs. (<ref>) and (<ref>), and shown that the temperature and doping dependence compares semiquantitatively with the experimental results of Ref. Bozovic:2016ei.




 §.§ Optical conductivity


We now proceed to calculate the real (dissipative) part of the conductivity  σ_1(Ω) within the same framework.  Again, this has been done in many places earlier,<cit.> so we provide only the final expressions, which agree, e.g., with Refs. Hirschfeld:1994,Quinlan:1996.  We  find

    σ_1( Ω )    =    -N_0 e^2/2 Ω∫_-∞^∞dω[ f(ω) -f(ω+Ω) ]×
          ⟨  v_F,x^2  Re{ A_++-A_+-}⟩_FS ,

where

    A_+±   =   Δ_ k^2+ω̃_+ ω̃'_± + Q_+ Q' _±/ Q_+ Q '_±(  Q_+ + Q '_±)
    ω_±   =   ω± i η
    ω'_±   =   ω+Ω± i η
    
    Q_±   =   √(Δ_ k^2-ω̃^2_±)
    
     Q'_±   =   √(Δ_ k^2-ω̃'^2_±) ,

and the renormalized real axis frequencies are ω̃_± (ω) =ω̃_n (iω_n→ω± i η). 
 Here the branch cut for the complex square root function is the negative real axis.
These equations, taken together with the definitions of the renormalized frequencies (<ref>), are sufficient to calculate the optical conductivity at all frequencies and temperatures.   In the normal state limit Δ_ k→ 0, (<ref>) reduces to the Drude conductivity 
 
    σ_1N(Ω) = σ_N0(4Γ_N^2/Ω^2+4 Γ_N^2) ,

where σ_N0 is the DC conductivity  e^2 N_0 ⟨ v_F,x^2 ⟩_FS/(2Γ_N).
.2cm









§ RESULTS



 §.§ Qualitative considerations

            
It is essential to appreciate the unusual and sometimes counterintuitive effects of the different energy dependences of the impurity scattering rate of the d-wave superconductor in the Born and unitarity limits.  In Fig. <ref>(a)-(c), we illustrate how the density of states varies with energy in the two limiting cases, as well as a special mixture of the two that was found in Ref. Lee-Hone:2017 to fit the superfluid density well.

  

    It is well-known that the residual density of states  N(ω→ 0) scales as γ∼√(Γ_N Δ_0) in the unitarity limit, creating a plateau in N(ω) over a range of energies γ (the “impurity band"), at the expense of the coherence peak.  Relatively small scattering rates Γ_N are sufficient to strongly modify these low-energy states.      
            In the clean Born limit, on the other hand, the corresponding density of states N(0) is also finite, but scales with a γ that is exponentially small in Δ_0/Γ_N.<cit.>  Since a scattering resonance is never produced in this limit, all states are modified equally weakly, such that for a  range of small disorder negligible effects are seen on 1-particle spectral quantities.  In Fig. <ref>(d), we compare the dependence of the residual DOS N(0) on Γ_N, proportional to the impurity concentration.   From  Figs. <ref>(a) and (d), it is clear that in the Born limit the scattering rate must reach a very large fraction of T_c0, i.e., a very large fraction of the critical rate to destroy superconductivity altogether, before the density of states is substantially modified over any significant range of energies.
            
            To study the optical conductivity, it is even more important to understand the differences in the energy dependences of the scattering rate in the Born and unitarity limits, shown in Fig. <ref>(a)-(b).  It is easy to see from Eq. (<ref>) that in the unitarity limit c=0, the scattering rate 1/τ≡ 2  Im ω̃_+ varies inversely proportional to N(ω) with the exception of the impurity band region, where the 1/ω divergence is cut off by self-consistency.   On the other hand, 1/τ is proportional to N(ω) in the  Born limit, again cut off at the lowest energies, but in a manner hardly visible in the clean limit.  In the dirty Born limit, the energy dependence of 1/τ becomes generally smeared out over the entire energy range, a fact that will be important for our analysis below.  
            
  
            
            The superfluid density  (<ref>) reflects the density of states directly.  In particular, the nonzero residual density of states N(0) may be shown to lead directly to a T^2 term in the penetration depth <cit.> at sufficiently low T.  This is confined to a temperature region of order T^∗∼γ in the unitarity limit, and if T^∗≪Δ_0 it is possible to observe a crossover linear-T region as well.<cit.>  It is sometimes assumed that the same statement may be applied in the Born limit, with the only difference being the much smaller γ. However, as is clear from Fig. <ref>(a), in the Born limit there is no such separation of energy scales, leading to a quasi-linear T behavior in the penetration depth over the whole range, or, for sufficient disorder, a T^2 behavior over the whole range.<cit.>  As emphasized in Ref. Lee-Hone:2017, the regime of  quasi-linear T behavior in the intermediate T regime extends to rather high disorder values for realistic band structure models appropriate for  underdoped . 
               
            
From these considerations exhibited in Figs. <ref> and <ref>, it is possible to sketch the expected behavior in the conductivity.  There are two additional factors that influence the peculiar form of the spectrum in a cuprate d-wave superconductor, however.  The first is the existence of a unique zero-frequency, zero temperature limit of σ_1(Ω), the so-called “universal” d-wave conductivity  σ_00≡ e^2 N_0 v_F^2 ħ/(2 πΔ_0), 
which is, unlike a normal quasiparticle conductivity, finite in the limit of vanishing disorder.<cit.>  The expression for σ_00 is not strictly independent of disorder, since it depends on the gap magnitude which is itself suppressed within the present theory, such that σ_00 is expected to increase slightly with addition of impurities.  Furthermore, it is enhanced by impurity vertex corrections if the impurity potential has nonzero range,<cit.> but these effects vanish  in our model with pointlike scatterers.

 

The second effect that must be accounted for is inelastic scattering by spin fluctuations.  The relaxation time of nodal quasiparticles that dominate the conductivity at low frequencies may be shown to vary as 1/τ_sf∼ω^3 at T=0 for ω≪ 3Δ_0.<cit.>  This behavior of the relaxation time leads to a scenario for the shape of the conductivity spectrum seen in optimally doped cuprates, as discussed in Ref. Quinlan:1996.  The low energy behavior is dominated by disorder physics in the form of a relatively narrow residual Drude component tied at T=0 to σ_00, while the higher energy physics (“mid-infrared component”) arises from the 
crossover between these low- and high frequency inelastic scattering regimes, and is found to peak at 4Δ_0  in weak coupling theory,<cit.> or more generally at 2Δ_0+Ω_0, where Ω_0 is the frequency of the pairing boson.  This picture is sketched in Fig. <ref>a, and leads to the familiar  conductivity spectrum of optimally doped cuprates, where spectral weight is lost at finite frequencies relative to the normal state in an intermediate frequency region between γ and 2Δ_0+Ω_0.  It should be noted that there are other proposals for the origin of the mid-infrared peak, discussed in Ref. BasovTimusk2005.

The implications of weak scattering for the conductivity spectrum within this scenario have not been systematically explored.  In Figs. <ref>(b) and (c), we show what is anticipated from the discussion of Figs. <ref> and <ref> for disorder scattering in the Born limit. In the clean case, the conductivity must tend to σ_00 at low T, but the effect is essentially invisible due to the exponentially small residual scattering γ.   The disorder-limited  conductivity is then very small and flat out to high frequencies.<cit.>  We have depicted a situation where spin fluctuations scattering is weak, anticipating a loss of low frequency spin fluctuations in the overdoped cuprates under consideration.  Note in this limit one expects the spectral weight loss between T=T_c and T=0 to be frequency dependent, although there is no well-defined region where it occurs, as in the unitarity scattering case.  In the dirty Born limit, on the other hand, the spectrum is completely dominated by disorder.  While it must still approach the value σ_00 as T,Ω→ 0, in absolute units this value is larger due to the suppression of Δ_0.  There is no well-defined residual Drude component, as all states over a wide energy range are broadened (see Fig. <ref>b).  Because the zero-temperature superfluid density is considered to be strongly suppressed in this example, the T=0 conductivity should closely
follow the normal state spectrum.  Again, we have assumed in Fig. <ref>c that spin fluctuations are weak, and may not play a major role.



 §.§ Disorder-limited optical conductivity

We now present evaluations of Eq. (<ref>)  to illustrate explicitly the importance of the impurity phase shift to the conductivity spectrum.  Our specific goal is to compare the predictions of the disorder-based model elaborated in Ref. Lee-Hone:2017, and show that it explains semiquantitatively the results of Mahmood et al. for the THz conductivity of overdoped .  In Fig. <ref> we plot the optical conductivity in the normal state and at low temperatures, which should be compared with the experimental data in  of Ref. Mahmood:2017. Each column shows results for three different dopings, corresponding to critical temperatures T_c=27.5,13.5, and 7 K,   In the first column the impurity parameters and T_c0 are identical to those used in Ref. Lee-Hone:2017.  In all cases the normal state curve has a pure Drude form with physical scattering rate 2Γ_N.  In the dirty Born limit represented here (with small unitarity limit corrections that affect only the very low frequency range),  the superconducting state conductivity spectrum is also surprisingly Drude-like.  That spectral weight has indeed been lost and condensed into a zero-energy δ-function is clear in all cases from the area between the normal state and low temperature curves, but there is no sign of a well-defined low-energy region where scattering states have been removed, as exhibited, e.g.,  in the unitarity limit scenario in Fig. <ref>(a).  As doping increases, T_c0 decreases and the effect of disorder becomes more pronounced, shifting more weight from the condensate to the finite-frequency quasiparticle spectrum.   Despite the simplicity of the disorder-based model considered in Ref. Lee-Hone:2017, the calculated conductivity shows semiquantitatively the same behavior as the THz experiments.  The only essential difference is the decrease of the magnitude of the conductivity as the system is overdoped, not captured here because a fixed disorder strength has been used for all dopings.<cit.>  Note as well that there is no signature of the superconducting gap, in either the model or the experiment, despite the fact that the frequency range covered (2 THz) corresponds to 50 K in temperature units.  This is a particular consequence of the  order parameter, for which disorder (especially in the dirty Born limit) rapidly smooths out sharp gap features in both the density of states and conductivity, as shown in Figs. <ref> and <ref>, respectively.




The doping dependence of the  uncondensed dimensionless spectral weight loss can now be easily calculated by integrating  S_u=2/(πϵ_0 ω_p^2)∫_0^∞σ_1(ω) ω. At the same time, the condensed spectral weight is obtained from the superfluid density via S_δ=ρ_s(T)/(μ_0ϵ_0 ω_p^2), with the Ferrell–Glover–Tinkham sum rule implying that S_u+S_δ=1 at any temperature. In Fig. <ref>, we plot the components of the spectral weight as a function of doping at T=1.6 K, illustrating how the condensed spectral weight indeed vanishes with doping, even as the total carrier density (∼ω_p^2) in the model  remains finite, a signature of the disorder-dominated regime.   In the same figure are plotted the experimental data obtained by Mahmood et al. by fitting a Drude form over the frequency range range from 0.3 to 1.7 THz.  It is easy to see that the model of fixed disorder scattering rate Γ_N considered in Ref. Lee-Hone:2017 (dotted lines) provides a very reasonable fit to the data except for the samples closest to optimal doping, which we discuss further below.  



§ DISCUSSION






 §.§ Renormalized plasma frequency


It is well known in interacting electron systems that plasma frequency and Drude weight are reduced below the bare values calculated from band theory.<cit.>  This arises both from many-body effects that locally flatten the dispersion near the Fermi surface,<cit.> simultaneously transferring  spectral weight to higher frequencies;<cit.> and from Fermi-liquid effects,<cit.> in which residual quasiparticle interactions induce a backflow that partially cancels the current response.
These effects can be accounted for by employing a renormalized plasma frequency.  To gauge the magnitude of the renormalization in overdoped , Fig. <ref> plots ω_p^2 calculated directly from the ARPES-derived tight-binding energy dispersion 
(TB ω_p^2) and compares it to fitted Drude weights obtained from the THz experiments (expt ω_p^2). The calculated ω_p^2 is larger than the experimental value by a factor of 3 to 4, consistent with previous work on the  system.<cit.> Indeed, the authors of the ARPES study from which the  tight-binding band structure was obtained<cit.> point out that their fitted tight-binding dispersion does not capture the electron–phonon kink structure in ϵ_𝐤 that appears near the Fermi energy.<cit.>  The inset of Fig. <ref> plots the overall renormalization of the Drude weight, given by the ratio of the experimental to tight-binding ω_p^2.  While some doping dependence is apparent, for the purposes of our calculation we  simply assume the renormalization takes a doping independent value of 0.3, and have applied this as a prefactor to the conductivities in Fig. <ref>.  In contrast, for the dimensionless spectral weights plotted in Fig. <ref> the renormalization of plasma frequency cancels out, making this measurement a particularly robust context in which to compare theory and experiment. 



 §.§ Clean limit critical temperature parameter T_c0

In this work, for simplicity, we have considered an idealized, weak-coupling BCS theory.  The model corresponds to a situation in which the characteristic frequency of the exchange bosons responsible for Cooper pairing, ω_0, is much greater than the superconducting transition temperature.  This leads to a pairing interaction that is roughly independent of frequency, allowing the combined effects of coupling strength and boson frequency to be captured by a single parameter, the notional clean-limit transition temperature T_c0.  Although beyond the scope of this paper, a more realistic model, e.g., based on spin fluctuations, would result in a boson spectrum spread over a wide range of frequencies.  As discussed in Ref. Millis:1988, low frequency boson spectral weight, at ω≤ T, gives rise to inelastic electron–boson scattering, leading to significant pair breaking in unconventional superconductors.  Inelastic scattering is known to be particularly strong in optimally doped cuprates and so in this regime we expect the T_c0 parameter of the weak-coupling theory to substantially overestimate the T_c of the hypothetical disorder-free reference material that might be obtained, say, by gating.  



There is one region of the phase diagram where the parameters of the weak-coupling model can be checked relatively directly against experiment: near the overdoped critical point at which T_c → 0.  In the zero-temperature limit, inelastic scattering is irrelevant and pair-breaking can be attributed entirely to disorder.  The measured transport scattering rate τ^-1_tr then places a lower bound on the single-particle elastic scattering rate, τ^-1_sp =  2Im{ω̃_+}, where in the normal state Im{ω̃_+}≡Γ_N.  In the THz conductivity study of Mahmood et al.,<cit.> the most overdoped sample (T_c = 7 K) has a normal-state spectrum with Drude width 1.75 THz, corresponding to  in temperature units, placing a lower bound on the normal-state scattering rate such that Γ_N ≥ 42 K.  Equivalently, we can obtain a lower bound for the clean-limit transition temperature by taking the Abrikosov–Gor'kov result for the critical scattering rate of a weak-coupling  superconductor, Γ_c = 0.881 T_c0, from which we infer T_c0≥ 48 K.  Against these experimental bounds, the values assumed in our weak-coupling model, Γ_N = 56.5 K and T_c0(p → p_c) = 64 K, are not unreasonable.



 §.§ Disorder dependence of T_c0 and Γ_N  


In Fig. <ref>, the small discrepancies for the more optimally doped samples between  experimental values of the uncondensed spectral weight compared to the solid line representing the present theory    may reflect one of two things: a) these samples are closer to the clean limit, meaning that the fit performed on the experimental data over a limited intermediate frequency range misses   some weight at the lowest frequencies by extrapolating the Drude form to DC (see, e.g., Fig. <ref>(b)); or b), we have until now neglected  the doping dependence of the scattering rate (à  la Ref. Lee-Hone:2017). The actual sources of disorder and their scattering potentials are not completely characterized,  but it seems reasonable to assume that the small number of unitarity scatterers are Cu vacancies, and that the weak out of plane disorder is a combination of Sr dopants, whose number is reasonably well known, and O vacancies, whose number is poorly known.  To give some idea  of the sensitivity of our results to differing assumptions about these sources, we fix the 
number of unitarity scatterers but consider two additional models where the Born limit contribution, Γ_N^B, varies with doping but is normalized to give the same result as the Ref. Lee-Hone:2017 model at T_c=0.  
In one, Γ_N^B(p) increases linearly with doping by a factor of two over the relevant range, in rough correspondence to what is observed in the terahertz experiments;<cit.> in the second, we vary Γ_N^B(p) so as to give a constant T_c0  vs. hole doping.  The  normal state scattering rates for both
the Born and unitarity scatterers, along with T_c and T_c0, are plotted vs. hole doping in Fig. <ref>.   

The values of T_c0 required to fit the experimental data in the three models of doping dependence are all substantially larger than T_c itself, so this qualitative feature of the weak-coupling theory remains.  It is interesting to note, however, that the fit to the spectral weight loss in Fig. <ref>  improves markedly in the near-optimally doped samples when one includes the doping dependence of scattering.  In fact, the lower scattering rate closer to 
optimal doping suggests that cleaner samples may have a weak  low Ω residual Drude upturn below the experimental observation window.



§ CONCLUSIONS


We have shown that a model of “dirty d-wave" superconductivity used to fit superfluid density data by Bozovic et al.<cit.> on high quality overdoped  films is also capable of fitting most features of the terahertz conductivity measured on similar samples<cit.> with identical parameters.  This is surprising given that scattering rates deduced for these films from Drude fits in the normal state appeared to be
much larger than T_c.   The explanation for the excellent description of two independent aspects of the electromagnetic response requires two assumptions.  First, we have postulated  that the intrinsic critical temperature in the absence of disorder is substantially higher than T_c itself over the entire optimally doped range, and discussed why this may be reasonable.  This ansatz is indirectly supported, in fact, by the recent observation of the persistence of strong magnetic fluctuations to high doping in RIXS.<cit.>   The effective “pure T_c0” if one goes beyond BCS and includes  inelastic scattering in the theory may be substantially lower, but we do not expect that these effects will alter our qualitative conclusion that disorder is playing an essential role in current measurements of the vanishing of T_c on the overdoped side of the cuprate phase diagram.  

The second crucial aspect of the theory that allows for  strong suppression of superfluid density, linear dependence of superfluid density on T_c, and Drude-like conductivity spectra at low temperatures,  is the unusual  role of weak scattering.  Here we have explored the effect  of Born scattering on the d-wave conductivity in some detail, and tried to exhibit some of the more  salient qualitative aspects.  A more complete theory of  the  conductivity spectrum and the overdoped phase diagram may require treatment of forward impurity scattering and a more proper treatment of the dynamics of the pairing bosons.  The semiquantitative success of the present theory suggests strongly, however, that exotic physics beyond Fermi liquid and Eliashberg  theory is probably not  required to understand the data.

 We emphasize that the possibility of a “dirty d-wave” explanation  for the apparently surprising results of Bozovic et al.<cit.> and Mahmood et al.<cit.> does not imply that the overdoped state of the cuprates can be described by a weakly interacting Fermi gas with a d-wave pairing instability.  While the dirty d-wave results fit the superfluid density and conductivity data well, we have also shown that a large renormalization of the plasma frequency by many-body effects, consistent in magnitude with that reported by other authors for this system, is required.  Renormalizations of similar magnitude have been reported for other cuprates for some time.<cit.>  A reliable  theory of the cuprates, even on the overdoped side of the phase diagram, must therefore account simultaneously for these effects on a microscopic basis.   



We acknowledge useful discussions with N. P. Armitage, L. Benfatto,  J. S. Dodge, M. P. Kennett, S. A. Kivelson, F. Mahmood, J. Orenstein, D. J. Scalapino, and C. M. Varma.   We gratefully acknowledge financial support from the Natural Science and Engineering Research Council of Canada, the Canadian Institute for Advanced Research, and the Canadian Foundation for Innovation.  PJH was supported by NSF-DMR-1407502.











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