Superconductivity in cuprates with square-planar coordinated copper driven by hole carriers

The phase diagram of cuprate superconductors comprises cuprates with octahedral and/or pyramidal coordinated copper as well as square-planar coordinated copper to represent hole- and electron-doped cuprates, respectively. In hole-doped cuprates, superconductivity is induced by doping, resulting in the well-known superconducting dome in the phase diagram. Recently, electron-doped cuprates, e.g., Pr_2−xCe_xCuO₄, have regained interest since several reports have shown that the superconductivity in electron-doped cuprates is induced by annealing rather than doping, and the CuO₂ plane itself is a metal.^1–3) The annealing process is known to control the presence or absence of apical oxygen.^4,5) It has been known that the electronic properties of the electron-doped cuprates [Nd₂CuO₄ type (T') and infinite-layer (IL) type, Fig. 1] are highly sensitive to the off-stoichiometric oxygen residing at the apical position.⁶⁾ The presence or absence of apical oxygen at a level of 0.04 per formula unit holds the decisive vote on the dramatic difference (insulator or superconducting metal) observed by transport measurements.⁷⁾

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Regarding the electronic structure, angle-resolved photoemission spectroscopy (ARPES) has shown that the Fermi surface is rather simple (Fig. 1), evolving around the X point of the Brillouin zone.^8,9) ARPES measurements on as-grown and annealed Pr_1.85Ce_0.15CuO₄ show strong evidence that the annealing process neither changes the band filling nor the global shape of the band dispersion. Only the spectral weight distribution in the momentum space shows a marked change,¹⁰⁾ which leads to the marked change observed by transport measurements. The annealing-dependent antiferromagnetic-induced folding observed by ARPES between the first and second Brillouin zones (Fig. 1) is likely to be the sole trigger for the observed metal-insulator transition. Theoretically, electronic structure calculations thoroughly scrutinized the class of electron-doped high-temperature superconducting materials.¹¹⁾ However, certain aspects of electronic correlations driven by the presence or absence of apical oxygen have not been taken into account.¹²⁾

Fermi surface mapping by ARPES suggests a holelike Fermi surface, a hole pocket centered at X, which, according to semiclassical theory, contains hole charge carriers. However, reported Hall data for the Nd₂CuO₄-type superconductors, such as Nd_1.85Ce_0.15CuO₄, are highly contradictory to each other. Although some reports have shown that the Hall coefficient is positive¹³⁾ as the Fermi surface mapping by ARPES suggests, others found a negative Hall coefficient, and thus claim that they are electron-doped superconductors.¹⁴⁾ In the case of IL-type superconductors, Hall data have been very limited owing to the difficulty in the preparation of single-crystalline samples. In this letter, we present transport data of high-quality, single-crystal thin-film samples, and we shed light into the somewhat murky correlations between the Hall coefficient and the underlying Fermi surface.

High-quality, epitaxial, c-axis-oriented single-crystalline films of IL Sr_0.90La_0.10CuO₂ (SLCO), T'-Pr_1.85Ce_0.15CuO₄ (PCCO), T'-Pr₂CuO₄ (PCO), and T'-La₂CuO₄ (LCO) were grown by molecular beam epitaxy (MBE) onto (110) DyScO₃, (001) SrTiO₃, and (110) PrScO₃ substrates. PCO films have been annealed by a two-step annealing procedure described recently¹⁾ in order to induce superconductivity. PCCO,¹⁵⁾ SLCO,¹⁶⁾ and LCO films have been annealed in situ for the induction of superconductivity. These thin films were characterized by high-resolution reciprocal space mapping. Photolithography and ion milling techniques were used to pattern the films into a standard six-probe Hall bar. Resistivity and Hall-effect measurements were carried out in a cryostat equipped with a 14 T magnet. The magnetic field was aligned perpendicular to the ab-plane of the films. A superconducting quantum interference device was used to probe the magnetic flux expulsion from these superconducting films.¹⁷⁾ We determined Hall coefficient data by performing symmetric magnetic field sweeps (−14 → +14 T) at constant temperatures between 1.8 and 400 K and at a DC current density of 1 kA cm⁻². Throughout all of our Hall voltage measurements, the Hall voltage is linear to the applied magnetic field. Below the superconducting transition, the Hall coefficient was determined from Hall voltage data taken above the upper critical magnetic field. This limits the accessible Hall data at low temperatures for SLCO and T'-LCO since these films are grown onto DyScO₃ and PrScO₃ substrates, respectively. In rare-earth scandates such as DyScO₃, high magnetic fields (>5 T) induce spontaneous ordering of magnetic moments,¹⁸⁾ resulting literally in an explosion of the substrate. Figure 2 shows the temperature-dependent Hall coefficients of SLCO [Fig. 2(a)], PCCO [Fig. 2(b)], PCO [Fig. 2(c)], and T'-LCO [Fig. 2(d)] samples. The lowest common multiple of these superconductors is that copper is square-planar coordinated. The fact that the Hall data plotted (Fig. 2) are taken from superconducting cuprates with two different crystal structures (IL and T'), different doping levels (optimal and zero), and different rare-earth elements (Pr and La) highlights two common trends. At high (>300 K) and low (<100 K) temperatures, the Hall coefficient is positive. In the intermediate-temperature range of 100–300 K, the Hall coefficient may take either sign. A variation in the sign of the Hall coefficient may point to marked changes in overall conduction behavior, but rapid variations in the temperature dependence of resistivity (Fig. 3) are absent. Such behavior indicates that holelike and electron-like bands are in competition and that these bands stabilize different electronic ground states that are commonly found in systems of correlated electrons. At low temperatures, the four superconductors have in common a rapid increase in Hall coefficient. Again, such a strong variation in Hall coefficient may result in a strong variation in electronic conductance but that is not observed (Fig. 3). Instead, for all cuprate superconductors discussed here, the resistivity below 40 K remains remarkably temperature-independent. In Fig. 3, we also added the resistivity behavior under magnetic fields. With increasing magnetic field strength, the superconducting transition is systematically shifted towards lower temperatures owing to the pair breaking. Although such behavior is expected for a type-II superconductor, the temperature dependence of the Hall coefficients shown here is in contrast to other reports and is associated with the well-known material issue of electron-doped cuprate superconductors. In contrast to other reports, we tailored the annealing conditions to be optimal for every doping level.¹⁾ Actually, the fact that superconductivity in cuprates with square-planar coordinated copper can only be induced by annealing^4,19) has far reaching implications on the underlying electron correlations. In particular, for optimal doping levels of Ce, annealing is necessary, although with less rigid thermodynamic constraints. Such lax annealing constraints allow that even under less-than-optimal annealing conditions, a superconducting transition appears for PCCO, although at a somewhat lower temperature (Fig. 4). In contrast to optimally annealed PCCO (Fig. 3), the temperature dependence of the resistivity shows an upturn at low temperatures in randomly annealed PCCO (Fig. 4). The upturn in resistivity with decreasing temperature is commonly seen and reported in many publications. In Fig. 4, the temperature dependence of the Hall coefficient is also plotted for PCCO prepared under random annealing conditions. Although such a sample still shows superconductivity, the strong increase in Hall coefficient disappears and the overall Hall coefficient remains negative over the entire temperature range. This indicates that the topology of its Fermi surface has been severely altered, although superconductivity is induced. Again, a negative Hall coefficient has been widely reported for electron-doped cuprate superconductors.^14,20)

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Unlike the situation in hole-type superconductors, e.g., La_2−xSr_xCuO₄ and Bi₂Sr₂Ca_n−1Cu_n+1O_2n+4+δ, the transport properties of cuprates with square-planar coordinated copper might be explained using a two-band model where electrons and holes coexist. Despite its inability to embrace electronic correlations found in strongly correlated systems, the two-band model is widely used. In optimally annealed samples (Fig. 2), R_H shows a sign change as a function of temperature, which has also been reported for some superconducting ceramic samples.¹³⁾ To elucidate the sign of the relevant superconducting carriers, a qualitative understanding of the trend of the Hall coefficient is required. The sensitivity of its sign to temperature is largely determined by two competing bands, i.e., a hole band coexists with an electron band. Noteworthy is that a change in the sign of R_H observed using a one-band model has been demonstrated by Kontani et al.,^21,22) introducing vertex corrections in a marginal Fermi liquid. At this point, however, we interpret our measurement data using a two-band model. The resistivity data seem to show only temperature-independent disorder-potential scattering present below 40 K. However, the persistent increase in Hall coefficient particularly for T < 100 K indicates the existence of a different temperature-dependent scattering mechanism that operates down to 1.8 K. The second mechanism affects a second band of carriers. Quite generally, R_H of a two-band system is given by

$$\begin{equation} R_{\text{H}}=\frac{\sigma_{\text{H}1}+\sigma_{\text{H}2}}{B(\sigma_{1}+\sigma_{2})^{2}}, \end{equation} \tag{ 1 }$$

where σ_i and σ_Hi are respectively the diagonal and Hall conductivity elements of band i. For definiteness, band 1 (2) is electron-like (holelike), i.e., σ_H1 < 0, and σ_H2 > 0. It is possible to account for the major features of R_H and ρ without making specific assumptions about either band. σ_Hi is the product of σ_i tan ϑ_Hi (ϑ_Hi is the Hall angle); thus, its temperature dependence follows that of the relaxation time τ_i raised to some positive integer p.²³⁾ Matthiessen's rule for both bands implies

$$\begin{equation} \frac{1}{\tau_{i}}=\frac{1}{\tau_{i}^{0}}+\frac{1}{\tau_{i}^{\text{in}}}\ (i=1,2), \end{equation} \tag{ 2a }$$

where $1/\tau _{i}^{0}$ and $1/\tau _{i}^{\text{in}}$ are the scattering rates due to potential scattering caused by disorder and inelastic processes, respectively.²²⁾ The corresponding temperature-independent and temperature-dependent parts of the resistivity will be called $\rho _{i}^{0}$ and $\rho _{i}^{\text{in}}$, respectively, i.e., Eq. (2a) is equivalent to

$$\begin{equation} \frac{1}{\sigma_{i}}=\rho_{i}^{0}+\rho_{i}^{\text{in}}. \end{equation} \tag{ 2b }$$

In the low-temperature regime (T < 40 K), R_H is observed to be positive and to increase steeply with decreasing T, but the total resistivity ρ = 1/(σ₁ + σ₂) remains nominally T-independent. Rewriting Eq. (1) as R_HB/ρ² = σ_H2 − |σ_H1|, we see that σ_H2 must also increase steeply, but |σ_H1| may be either T-independent or may increase (less steeply than σ_H2). The second option would require both τ₁ and τ₂ to increase with increasing T, in contradiction with the observed constancy of ρ. Therefore, the only possibility is that τ₁ is T-independent in this regime, and that σ₂ dominates σ₁ (such that ρ is hardly affected by the increase in $\tau _{2}^{\text{in}}$). Thus, the two-band model is consistent with the measurements provided:

$$\begin{align} \sigma_{\text{H}2}&\gg|\sigma_{\text{H}1}|, \end{align} \tag{ 3a }$$

$$\begin{align} \sigma_{1}&\ll\sigma_{2}\quad(T<40\,\text{K}). \end{align} \tag{ 3b }$$

Actually, a less restrictive inequality suffices in place of Eq. (3b). We only need the temperature-dependent part of 1/σ₂ to be small compared with the sum of σ₁ and σ₂, i.e., $\rho _{2}^{\text{in}} \ll ( \rho _{2}^{0} )^{2}( \sigma _{1} + \sigma _{2} )$. Adopting Eq. (3), we may understand the data on R_H and ρ as follows. A holelike Fermi surface coexists with a band of electrons. At T > 40 K, the conductivity of the electron band dominates that of the hole pocket. However, the Hall conductivities (determined by the respective mean free paths l_i) are comparable at high T values. Thus, R_H at 300 K is small and sensitive to the relative magnitudes of l₁ and l₂, i.e., R_H may be of either sign. When T decreases below 40 K, the scattering rate of the electron band saturates because disorder scattering dominates ($1/\tau _{1} \sim 1/\tau _{1}^{0}$), whereas that of the holes continues to decrease to 1.8 K ($1/\tau _{2} \sim 1/\tau _{2}^{\text{in}}$). As a result, the Hall current of the holes dominates that of the electrons,²⁴⁾ and R_H increases to large positive values [Eq. (1)]. We discuss the electron-like band (band 1) first. Since σ₁ dominates the total temperature dependence of conductivity (T > 40 K), the behavior of 1/τ₁ may be inferred from ρ. The RRR in cuprates with square-planar coordinated copper is highly variable (4 for SLCO to 15 for T'-LCO); thus, the scattering rate $1/\tau _{1}^{0}$ is sensitive to the degree of disorder present. Although ρ becomes T-independent only below 40 K, a deviation from the high-temperature linear behavior is observable at a temperature as high as 250 K. The ρ versus T profile is qualitatively different from that in cuprates with octahedral or pyramidal coordinated copper, although similar to what is seen in conventional highly disordered metals. The infrared reflectivity is known to be dominated by the electron band. It is worth mentioning that the Drude model explains the reflectivity profiles of PCCO²⁵⁾ much better than, for example, YBa₂Cu₃O₇.²⁶⁾ The plasma frequency derived from those reflectivity spectra was found to be 19700 cm⁻¹ or an electron density of $4.4 \times 10^{21}m_{1}^{*}/m_{0}$ cm⁻³, where $m_{1}^{*}/m_{0}$ is the effective mass ratio in the electron band. Hence, both ρ and the reflectivity data suggest that the electron band comprises weakly correlated quasiparticles that behave much like carriers in a conventional disordered metal.

The hole band is much more interesting. By the analysis, its inelastic scattering rate is observable down to 1.8 K. T-dependent resistivity and R_H may be observed down to rather low T values in conventional metals, e.g., 6 K in ultrahigh-purity Bi metals with mobilities in the range of 5 × 10⁷ cm² V⁻¹ s⁻¹.²⁷⁾ For comparison, the effective Hall mobility for Sr_0.90La_0.10CuO₂ μ_H = R_H/ρ varies from 2.5 cm² V⁻¹ s⁻¹ at 300 K to 3500 cm² V⁻¹ s⁻¹ at 1.8 K. What is remarkable here is that the electron-band resistivity shows that a high degree of disorder exists, yet the hole band displays a scattering rate 1/τ₂ that remains T-dependent at 1.8 K. The disparity between 1/τ₁ and 1/τ₂ within the same sample presents clear evidence that the hole quasiparticle experiences an inelastic scattering mechanism qualitatively different from those of the electrons. The origin of the hole inelastic scattering is unlikely to be due to phonons (otherwise, 1/τ₂ should also saturate to a constant value, similarly to 1/τ₁). The absence of saturation at low T values is similar to the resistivity profile observed in Bi₂Sr₂Ca_n−1Cu_n+1O_2n+4+δ and suggests that the holes may, in fact, be the carriers responsible for superconductivity in cuprates with square-planar coordinated copper. The electron band may become superconducting owing to its proximity with the holes, but this is of secondary importance. These results present evidence against the case for n-type superconductivity as the commonly reported negative Hall coefficient is induced by electron pockets.

In all optimally annealed samples studied, R_H is found to be positive below 100 K. The most interesting feature of R_H is its persistent T dependence down to 1.8 K, despite the saturation of the ρ versus T curve below 40 K. Kontani et al.²¹⁾ pointed out that in nearly antiferromagnetic Fermi liquids, a Curie-like behavior of the Hall coefficient is expected. However, in stark contrast to Hall coefficient data reported in Ref. 21, the Hall coefficient is positive for the electron-doped cuprate superconductors. Nonetheless, there exists a band of negatively charged carriers that dominate the normal-state resistivity. The resistivity profile is dominated by this band and shows that the negatively charged carriers in the band are quite conventional in behavior. Using a two-band model that has broad validity, we account for the general features of R_H and ρ. The low-temperature Hall effect reveals the existence of a holelike Fermi surface as confirmed by our analysis using a two-band model. We show that the temperature dependence of R_H, which persists to 1.8 K, reflects the anomalous inelastic-scattering mechanism of the holes. Thus, the fundamentally distinct scattering mechanisms in the two bands explain the different temperature scales observed in the transport quantities. The Hall results provide direct evidence that an inelastic scattering mechanism involving the holes remains observable to 1.8 K in cuprates with square-planar coordinated copper. The similarity between the behavior of the hole-scattering rate and that in hole-doped superconductors suggests that the holes, in fact, may be driving the superconducting transition. Our comparison of the Hall coefficients of electron-doped PCCO and undoped PCO endorses the results determined by ARPES measurements as they clearly point towards a shrinkage of the Fermi surface.²⁸⁾ One would expect that the free carrier concentrations diminish with increasing doping but that is in contrast to both the resistivity and Hall coefficient results. The Hall coefficients decrease with increasing Ce doping, consistent with a direct increase in carrier density. In contrast to the hole band, the assignment of the electron band responsible for σ₁ is not very straightforward. The large antiferromagnetically induced band, which has an electron character, can be ruled out because this band is known to be triggered by the apical oxygen, which substantially reduces both τ₁ and τ₂. It is worth recalling the fact that doping is not responsible for the induction of superconductivity, but annealing is. During the annealing procedure, apex oxygen sites are evacuated whereby the system is converted from an antiferromagnetic insulator into a superconductor. Therefore, superconductivity is found in the vicinity of antiferromagnetic order. Although no long-range order is detected, short-range antiferromagnetic fluctuations may call for vertex corrections of the Hall coefficient. It was shown that in marginal antiferromagnetic Fermi liquids, those vertex corrections are accountable for the sign change of the Hall coefficient.^21,22) Although other electron pockets have been inferred by high-energy ARPES measurements,²⁹⁾ their existence itself and correspondence to the transport characteristics are far from being understood and demand further investigations.

Acknowledgments