Anisotropic scattering caused by apical oxygen vacancies in thin films of overdoped high-temperature cuprate superconductors

Introduction.— For a long time, the overdoped cuprate superconductors were believed to be described quite well by the Bardeen-Cooper-Schrieffer (BCS) theory Lee et al. (2006); Keimer et al. (2015). However, in 2016, such a belief was challenged by the measurement of the superfluid density $\rho_{s}$ using mutual inductance technique on a large amount of high quality overdoped La_2-xSr_xCuO₄ (LSCO) films Božović et al. (2016). Two seemingly contradicting results were reported: $\rho_{s}(0)-\rho_{s}(T)\propto T$ and $\rho_{s}(0)\propto T_{c}$ where $\rho_{s}(0)$ is the zero temperature value of $\rho_{s}(T)$ and $T_{c}$ is the transition temperature. Within the conventional BCS theory, the former scaling law indicates the d-wave superconductors are very clean, but the latter is a result of dirty BCS superconductors Abrikosov et al. (1963). This dichotomy regarding the dirtiness/cleanness of the overdoped cuprates motivated exotic theoretical investigations Božović et al. (2019); Zaanen (2016); Herman and Hlubina (2018); Fu et al. (2018); Wang (2018); Goutéraux and Mefford (2020); Phillips et al. (2020); Li et al. (2021).

The superfluid density has also been measured by THz optical conductivity experiment Mahmood et al. (2019), and is quantitatively consistent with the mutual inductance measurement Božović et al. (2016). In the meantime, the Drude fitting of the optical conductivity $\sigma_{1}(\nu)$ indicates the dirty limit Lee-Hone et al. (2017, 2018). So the same dichotomy also exists in optical measurements. Moreover, as yet another puzzle, $\sigma_{1}(\nu\to 0)$ should be identical to, but is fitted to be significantly smaller than the dc conductivity $\sigma_{dc}$ Mahmood et al. (2019). This superficial loss of Drude weight seems to increase with overdoping.

In this Letter, we propose a scenario to resolve all of the above mysteries. We realize that in addition to the conventional isotropic scattering rate $\Gamma_{s}$ Lee-Hone et al. (2017, 2018), the apical oxygen vacancies cause an anisotropic scattering rate $\Gamma_{d}\cos^{2}(2\theta)$, with $\theta$ the azimuthal angle of the quasiparticle momentum relative to the antinodal direction. Since the low-energy nodal quasiparticles are barely affected by $\Gamma_{d}$, they reduce the superfluid density linearly in temperature if in addition $\Gamma_{s}\rightarrow 0$. But the total scattering rate $\Gamma_{\theta}=\Gamma_{s}+\Gamma_{d}\cos^{2}(2\theta)$ determines the typical behavior $\rho_{s}(0)\propto T_{c}$ in the dirty limit. Meanwhile, the strong anisotropy in $\Gamma_{\theta}$ causes a continuous distribution of Lorentzian components in $\sigma(\nu)$, so that $\sigma_{1}(\nu\to 0)$ would be underestimated by extrapolation from finite frequencies if a single isotropic scattering rate were assumed instead Mahmood et al. (2019).

Oxygen vacancies and “cold spot” model.— From both early Torrance et al. (1988); Sato et al. (2000) and recent Kim et al. (2017) studies, apical oxygen vacancies were found to be an important factor to prevent obtaining (high-quality) overdoped LSCO samples. Therefore, the effect of apical oxygen vacancies deserves study in overdoped cuprates carefully. This is another motivation of the present work.

In Fig. 1, we plot a configuration of one apical oxygen atom above the CuO₂-plane (only Cu-sites plotted). Notice that since the carriers have $d_{x^{2}-y^{2}}$-like orbital content Zhang and Rice (1988), there is no coupling between the oxygen and the $d_{x^{2}-y^{2}}$-orbital just below it. Therefore, the leading couplings are with the next nearest neighboring sites, giving the hopping integrals switching signs alternatively, as shown in Fig. 1. First, if there is no oxygen vacancy, the second order process of these out-of-plane hoppings gives an additional band dispersion $-4\kappa(\cos k_{x}-\cos k_{y})^{2}$ where $\kappa=t_{o}^{2}/\varepsilon_{o}$ with $t_{o}$ and $\varepsilon_{o}$ the hopping amplitude and energy level distance between the apical oxygen and $d_{x^{2}-y^{2}}$ orbitals Xiang and Wheatley (1996). Such a dispersion has already enters the carrier band. Next, we consider one oxygen vacancy at the origin. Relative to the uniform background, the vacancy now leads to an additional term $H_{i}=\sum_{\delta\delta^{\prime}\sigma}\kappa f_{\delta}f_{\delta^{\prime}}c_{\delta\sigma}^{\dagger}c_{\delta^{\prime}\sigma}$ where $f_{\delta}=1(-1)$ for $\delta=\pm\hat{x}(\pm\hat{y})$ and $\sigma$ denotes spin, which gives rise to the following scattering Hamiltonian

where $k$ and $k^{\prime}$ are momentums, and $N$ is the number of copper atoms. Since the oxygen vacancies are out-of-plane and only couple to next nearest neighboring $d_{x^{2}-y^{2}}$-orbitals, the scattering strength is anticipated to be small. Under the Born approximation Abrikosov et al. (1963), $H_{i}$ gives a scattering rate $\Gamma(k_{x},k_{y})\propto\Gamma_{d}(\cos k_{x}-\cos k_{y})^{2}/4$. For analytical convenience but without loss of qualitative physics, throughout this work, we further assume circular fermi surface and use wide band approximation, so that the scattering rate is only angle-dependent, i.e.

where we have added the isotropic scattering rate $\Gamma_{s}$ arising from, e.g., in-plane impurities. In the following calculations, $\Gamma_{s}$ and $\Gamma_{d}$ are our model parameters. In fact, this kind of scattering rate has been proposed to explain the transport phenomena in underdoped cuprates, called “cold spot” model Ioffe and Millis (1998), where the $\Gamma_{d}$-term is attributed to the pair fluctuations or interaction effects, which hence is expected to decrease upon doing. But here, in our case, the $\Gamma_{d}$-term is caused by oxygen vacancies and should increase upon doping, according to both early Torrance et al. (1988); Sato et al. (2000) and recent Kim et al. (2017) experiments.

We now look at the effect of the anisotropic scattering $\Gamma_{d}$ on single particle excitations. By choosing the d-wave pairing function $\Delta(\theta)=\Delta_{0}\cos(2\theta)$ and fixing $\Gamma_{s,d}=0.2\Delta_{0}$, we calculate the density of states (DOS) $\rho(\omega)=\int\frac{\mathrm{d}\theta}{2\pi}\rho(\omega,\theta)$, where $\rho(\omega,\theta)$ is the partial DOS (PDOS) given by

where $\mathcal{N}$ is the normal state DOS and $E_{\theta}=\sqrt{\varepsilon^{2}+\Delta(\theta)^{2}}$. $\rho(\omega)$ and $\rho(\omega,\theta)$ are plotted in Fig. 2(a)-(d). In the calculations, we perform the integrals over $\varepsilon$ and $\theta$ using standard numerical integral technique. Fig. 2(a) is the typical behavior of the conventional d-wave superconductor with isotropic scattering rate: zero energy cusp and coherence peak are both smeared out by isotropic scattering rate. Differently, the $\Gamma_{d}$-scattering shown in Fig. 2(b) only suppresses the coherence peak but does not change the zero energy cusp, as a result of the vanishing of scattering rate for nodal quasiparticles. Accordingly, we can divide the energy space into two regions: lower energy “clean” one and higher energy “dirty” one. The above picture is also seen clearly from the PDOS in Fig. 2(d), where the spectral becomes smeared out for high energy quasiparticles (near the antinodes) but remains sharp for low energy ones (near the nodes).

The dirty BCS theory for d-wave superconductors Alloul et al. (2009) is a direct generalization of the Abrikosov-Gor’kov (AG) theory Abrikosov and Gor’kov (1961), since the potential scattering here causes pair-breaking which is similar to the magnetic impurities in s-wave superconductors. The gap $\Delta_{0}$ is determined by the self-consistent equation

where $\lambda$ is the BCS coupling constant, $\omega_{n}=(2n-1)\pi T$ the Matsubara frequency, and $\phi_{\theta}=\cos 2\theta$ the d-wave form factor. ¹¹1Exactly speaking, the frequency summation should be bounded with a soft-cutoff (with the form of a boson propagator) instead of the hard-cutoff used here. However, only the latter can be mapped to the momentum cutoff strategy as in the BCS treatment. In numerical calculations, we use $\lambda=0.3$ and $\Omega=800$, which gives $T_{c0}=1.134\Omega\mathrm{e}^{-2/\lambda}=1.15$ for clean superconductors (with this setup, $0.87T_{c0}$ is taken as the energy unit). By letting $\Delta_{0}\to 0$, we obtain the generalized $T_{c}$-formula

where $\psi(z)$ is the digamma function. The results of $T_{c}$ vs $\Gamma_{s}$ and $\Gamma_{d}$ are shown in Fig. 2(e). A stronger $\Gamma_{d}$ is needed to kill superconductivity as the low energy excitations are less affected. In Fig. 2(f), we show the results of $2\Delta_{0}/T_{c}$. It is interesting to find that the anisotropic scattering drives the ratio farther away from the BCS prediction $4.28$ in the clean limit to values as large as $\sim 10$. In the literature, a large gap-$T_{c}$ ratio is often taken as an indication of strong coupling superconductivity. Here, the anisotropic scattering gives another interpretation.

Superfluid density.— After recognizing the clean-dirty dichotomy caused by oxygen vacancies at the single particle level, we turn to discuss the superfluid density, which can be obtained as Coleman (2015)

where $v_{F}$ is the Fermi velocity. Here, the current vertex correction can be proven to vanish in the non-crossing approximation as a result of the factorized scattering potential (Eq. 1), as shown in the Supplementary Material sm . In Fig. 6(a) and (b), $\rho_{s}$ vs $T$ is plotted for pure $\Gamma_{s}$ and $\Gamma_{d}$ scatterings, respectively. The most obvious difference is that any nonzero $\Gamma_{s}$ causes power law temperature dependence of $\rho_{s}(T)$, but $\Gamma_{d}$ preserves the linear dependence as in the clean limit. This is already anticipated from the DOS feature in Fig. 2(b) because the normal fluid density $\rho_{n}$ is roughly determined by quasiparticles and thus satisfies $\rho_{n}(T)-\rho_{n}(0)\propto T$, leaving the superfluid density satisfying $\rho_{s}(0)-\rho_{s}(T)=\rho_{n}(T)-\rho_{n}(0)\propto T$.

In real materials, both $\Gamma_{s}$ and $\Gamma_{d}$ are expected to coexist. In order to make quantitative comparisons with the experiment, we renormalize $T$ and $\rho_{s}$ by $T_{c}$ and $\rho_{s}(0)$, respectively, as shown in Fig. 6(c). We have superposed the experimental data of all $100$ samples with $T_{c}$ ranging from $41.6$ to $5.1$K Božović et al. (2016), shown as the red dots. Four typical theoretical curves are plotted: $(\Gamma_{s},\Gamma_{d})=(0,0)$, $(0.1,0)$, $(1,0)$, and $(0,1.2)$. As can be seen, almost all the experimental data reside within the region enclosed by the curves of $(0.1,0)$ (green line) and $(0,1.2)$ (blue line), while if $\Gamma_{s}$ is large and dominant, as the $(1,0)$-line shows, the renormalized $\rho_{s}$-$T$ curve bends more significantly, which is at odds with the experimental data. These observations indicate the more important role of $\Gamma_{d}$. As an example, the experimental data (open circles) with $T_{c}=5.1$K are shown in the inset, which is clearly very linear. We can fit the data with pure $\Gamma_{d}=0.75$ (line) quite well.

Another key observation of the experiment Božović et al. (2016) is the linear relationship between $\rho_{s}(0)$ vs $T_{c}$. To see that, we plot our results in Fig. 6(d). Pure $\Gamma_{s}$ and $\Gamma_{d}$ scatterings correspond to blue and red curves, which enclose the physical region (shaded region in the inset), in which $\rho_{s}(0)$ is always roughly proportional to $T_{c}$ although not exactly. Therefore, this is a hallmark of the dirty BCS superconductors regardless of isotropic ($\Gamma_{s}$) or anisotropic ($\Gamma_{d}$) scatterings. Furthermore, we have also fix $\Gamma_{s}=0.05$ and $0.1$ and tune $\Gamma_{d}$ to suppress $T_{c}$. Both of them show the bending behavior near $T_{c}\to 0$, showing much better behavior than the separate scattering cases in view of the experimental data .

Optical conductivity.— From the Kubo formula, the optical conductivity $\sigma_{1}(\nu)$ can be obtained as Hirschfeld et al. (1989)

where $f(\omega)$ is the Fermi distribution function, $G(\omega,\varepsilon,\theta)$ is the Green’s function in the Nambu space: $G^{-1}=(\omega+i\Gamma_{\theta})\sigma_{0}-\varepsilon\sigma_{3}-\Delta_{0}\phi_{\theta}\sigma_{1}$ with $\sigma_{0,1,3}$ the identity and Pauli matrices. By numerical calculations, we obtain the frequency dependence of $\sigma_{1}(\nu)$ in both normal and superconducting states (with $\Delta_{0}=1$ as the energy unit), as shown in Fig. 7(a) and (b), respectively. In the normal state, as $\Gamma_{d}$ increases, $\sigma_{1}(\nu)$ becomes more and more broadened to show long tail behavior. This is an important feature obtained in the “cold spot” model Ioffe and Millis (1998). In the superconducting state, a nonzero $\Gamma_{d}$ breaks the universal conductivity Lee (1993), which applies only in the case of pure $\Gamma_{s}$ scattering as shown in the inset of Fig. 7(b). Furthermore, we renormalize the superconducting state $\sigma_{1}(\nu)$ by the normal state value $\sigma_{1n}(\nu)$ to obtain the results shown in Fig. 7(c). The resulted curves show similar behavior to the DOS at low frequencies: nearly linear $\nu$-dependence if $\Gamma_{d}$ dominates which is quite different from the case of pure $\Gamma_{s}$ scattering (inset) with quadratic $\nu$-dependence. This can be used to examine the types of scattering in future experiments.

The most obvious feature when $\Gamma_{d}$ dominates is the behavior of $\sigma_{1}(\nu)$ is no longer a simple Lorentian-like Drude peak (as in the case of isotropic scattering), since the quasiparticles experience angle dependent scattering rates. In fact, the conductivity should be an integration over a continuous distribution of Lorentzian components. Since the Lorentzian is sharper and higher for smaller scattering rates, the overall line shape of $\sigma_{1}(\nu)$ should be steeper as the frequency approaches zero. Therefore, if the finite frequency data are used to perform the fit to a single Lorentzian-like Drude peak, as in the THz experiment, the extrapolated value of $\sigma_{1}(\nu\to 0)$ will be lower than the actual one, leading to superficial “missing” of the Drude weight. As an example, in Fig. 7(d) we fit the experimental data with $T_{c}=17.5$K (symbols) in both normal (blue) and superconducting (red) states Mahmood et al. (2019). The best fits with only $\Gamma_{s}$ (dashed lines) are found to underestimate $\sigma_{1}(\nu\to 0)$ and the spectral weight, as compared to the result with both $\Gamma_{s}$ and $\Gamma_{d}$ (solid lines). Since the pairing gap $\Delta_{0}$ is not available in the experiment, we take it also as a parameter, and the fitted value is $1.11$THz. More systematic fitting for a series of overdoped samples can be found in the Supplemental Material sm , which supports our main point that apical oxygen vacancy increases with overdoping, leading to larger $\Gamma_{d}$ scattering.

Summary and discussions.— In summary, we have found that the apical oxygen vacancies give rise to an anisotropic scattering rate $\Gamma_{d}\cos^{2}(2\theta)$ which causes a clean-dirty dichotomy for low-high energy excitations. This provides a natural explanation of the anomalous behavior of the superfluid density and THz optical conductivity in the experiments. Therefore, we conclude that the superconducting states of overdoped cuprates are still captured by the BCS theory, as long as the anisotropic scattering rate is adequately considered.

Finally, we make some remarks. (1) In this work, we have omitted many material-dependent details (such as specific tight-binding models, pairing interactions and interaction effects) in order to obtain universal results. The doping dependence enters the problem via the scattering rate $\Gamma_{d}$, which increases with doping. (2) The anisotropic scattering rate has been reported in angle-dependent magnetoresistivity experiments in Nd-LSCO Grissonnanche et al. (2021) and Tl₂Ba₂CuO_6+δ Abdel-Jawad et al. (2006). But to the best of our knowledge, there has not been such report in overdoped LSCO films. A systematic study of the anisotropic scattering rate can check our model and finally answer the question on whether overdoped cuprates are dirty BCS superconductors or not. (3) Our prediction suggests that it is necessary to extend the optical measurements down to even lower frequency (e.g. GHz) in order to obtain accurate behavior of the optical conductivity. (4) About superfluid density, there are two other kinds of scalings reported in literature: Uemura’s law Uemura et al. (1989) $\rho_{s}(0)\propto T_{c}$ and Homes’ law $\rho_{s}(0)\propto\sigma_{dc}T_{c}$ Homes et al. (2004). The former is obtained in underdoped cuprates, where phase fluctuations are very strong and cannot be neglected. Emery and Kivelson (1995). On the other hand, Homes’ law can be explained by dirty BCS superconductors with pair-preserving impurities only Abrikosov et al. (1963); Kogan (2013). Here, in the overdoped LSCO films, the oxygen vacancies are pair-breaking, hence Homes’ law does not apply. (5) Recently, the apical oxygen vacancies have also been observed in optimally doped YBa₂Cu₃O_7-x Hartman et al. (2019). Then, according to our study, $\sigma_{1}(\nu)$ at low frequency should be non-standard Drude-like, which may be consistent with the early microwave experiment Turner et al. (2003). Furthermore, if the apical oxygen vacancies are widely present in different families of cuprates, then the effect of $\Gamma_{d}$ scattering should be considered carefully in future studies.

D. W. thanks Congjun Wu, Qi Zhang, Tao Li, Yuan Wan for helpful discussions during the past few years. D. W. also acknowledges Ivan Bozovic for his encouragement on studying their experiments. This work is supported by National Natural Science Foundation of China (under Grant Nos. 11874205, 11574134, and 12074181), National Key Projects for Research and Development of China (Grant No.2021YFA1400400), Fundamental Research Funds for the Central Universities (Grant No. 020414380185), and Natural Science Foundation of Jiangsu Province (No. BK20200007).

Supplementary Materials

In this supplementary material, we first derive the scattering potential caused by the apical oxygen vacancies and then the anisotropic scattering rate $\Gamma_{d}\cos^{2}(2\theta)$ within the Born approximation. The effects of this scattering rate on some superconducting properties, including density of states, gap, $T_{c}$, superfluid density, and optical conductivity, are derived in a self-contained way. Finally, some further discussions are given, including first principle calculations to determine the position of the oxygen vacancies (apical or planar), the effect of planar oxygen vacancies (if they exist), and the current vertex corrections.