Superconductivity Studied by Solving Ab Initio Low-Energy Effective Hamiltonians for Carrier Doped ${\mathrm{CaCuO}}_2$, ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CuO}}_6$, ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_8$, and ${\mathrm{HgBa}}_2{\mathrm{CuO}}_4$

Understanding the materials dependence together with the universal controlling parameter of superconductivity (SC) in copper oxide superconductors is one of the major challenges in condensed matter physics. Here, we numerically analyze SC by using ab initio low-energy effective Hamiltonians consisting of the antibonding combination of Cu $3d_{x^2-y^2}$ and O $2p_{\sigma }$ orbitals without adjustable parameters. We have performed the state-of-the-art variational Monte Carlo calculations for the four carrier doped cuprates with diverse experimental optimal SC critical temperature $T_c^{\mathrm{opt}}$: ${\mathrm{CaCuO}}_2$ ($T_c^{\mathrm{opt}}\sim 110\mathrm{K}$), ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CuO}}_6$ ($T_c^{\mathrm{opt}}\sim 10–40\mathrm{K}$), ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_8$ ($T_c^{\mathrm{opt}}\sim 85–100\mathrm{K}$), and ${\mathrm{HgBa}}_2{\mathrm{CuO}}_4$ ($T_c^{\mathrm{opt}}\sim 90\mathrm{K}$). Materials and hole doping concentration ($\delta$) dependencies of the SC order parameter $F_{\mathrm{SC}}$ and the competition with spin or charge order show essential and quantitative agreement with the available experiments on the four materials in the following points. (1) In a wide range $0.05\le \delta \le 0.25$, the ground state is commonly the uniform SC state, which is severely competing with the charge or spin stripe and antiferromagnetic states. (2) $F_{\mathrm{SC}}$ at the optimum doping shows amplitude consistent with the superfluid density measured in the muon spin resonance and its dome structure found in $\delta$ dependence shows consistency with that of the SC gap in the tunneling and photoemission measurements. Based on the confirmed materials dependence, we further find insights into the universal SC mechanism. (I) $F_{\mathrm{SC}}$ increases with the ratio $U/|t_1|$ within the available realistic materials, indicating that $U/|t_1|$ is the principal component controlling the strength of the SC in the real materials dependence. Here, $U$ and $t_1$ are the on-site Coulomb repulsion and the nearest neighbor hopping, respectively, in the ab initio Hamiltonians. (II) A universal scaling $T_c^{\mathrm{opt}}\sim 0.16|t_1|F_{\mathrm{SC}}$ holds. (III) SC is enhanced and optimized if $U$ is increased beyond the real available materials, and it is further enhanced when the off-site interaction is reduced, while the presence of the off-site interaction is important to make the SC ground state against other competing states. The present findings provide useful clues for the design of new SC materials with even higher $T_c^{\mathrm{opt}}$.

Popular Summary

High-temperature copper oxide, or cuprate, superconductors have held the record for highest superconducting temperature at ambient pressure since their discovery nearly four decades ago. Understanding the microscopic origin governing the material dependence of the superconducting temperature—which ranges from below 10 K to above 130 K—has been a major challenge, partly because of the strong repulsion between electrons, which necessitates solving difficult quantum many-body problems. Here, we numerically solve the first-principles Hamiltonian—an expression of the system’s energy—of the cuprates and reproduce the detailed materials dependence and distinctions of superconducting properties as well as common properties seen in experiments.

Our computational study, done by a state-of-the-art quantum many-body solver and without adjustable parameters, based on recent rapid development of methodology, has made it possible to uncover the principal component that controls the superconducting amplitude, identified as the ratio of the strength of the electron repulsion to the electron’s kinetic energy. We also propose a scaling formula to predict the optimum critical temperature of superconductivity and explain the detailed real materials dependence quantitatively. This realistic firm basis provides insights into the origin of the emergent attraction between electrons required for Cooper pairing and into long-debated superconducting mechanisms.

This newly established first-principles methodology with predictive power opens a route for designing useful functional materials in strongly correlated electron systems in general and, in particular, designing materials that superconduct closer to room temperature.

Figure 1

SC correlation function $P_d(r)$ for ${\mathrm{CaCuO}}_2$ at different hole dopings. (a) $\delta =0.028$, (b) $\delta =0.045$, (c) $\delta =0.087$, (d) $\delta =0.101$, (e) $\delta =0.125$ (here $L=28$ instead of $L=30$), (f) $\delta =0.167$, (g) $\delta =0.208$, (h) $\delta =0.247$. In each case, we show $P_d(r)$ for the square lattice size $L=24$, 30, 36. For the same distance $r$ we employ the largest value of correlation. We perform the same procedure in later figures. Inset of each panel: size extrapolation of ${\overline{P}}_d(L)$ to the thermodynamic limit ${\overline{P}}_d^{\infty }$, whose numerical value is listed in Table 2. The gray line shows the linear approximation. Statistical errors originating from the Monte Carlo sampling are smaller than the symbol size.

Figure 2

The SC order parameter $F_{\mathrm{SC}}$ as a function of $\delta$ for doped ${\mathrm{CaCuO}}_2$. The gray filled circles show the values of $F_{\mathrm{SC}}(L)$ at $L=24$ square lattice calculated from ${\overline{P}}_d(L)$ shown in Fig. 1 by using $F_{\mathrm{SC}}(L)=\sqrt{{\overline{P}}_d(L)}$. The red squares are the size-extrapolated values $F_{\mathrm{SC}}^{\infty }$ calculated from ${\overline{P}}_d^{\infty }$. Inset shows the corresponding ${\overline{P}}_d(L)$ at $L=24$ and ${\overline{P}}_d^{\infty }$.

Figure 3

(a) Variance extrapolated energies of doped ${\mathrm{CaCuO}}_2$ for various ground-state candidates, SC, charge or spin stripe states C3S3 and C4S8, and AFM state as a function of hole doping $\delta$ on a $L=24$ square lattice. All energies are subtracted by the function $\mathcal{F}(\delta )=-12.76470\times \delta +6.44626$ for better visibility. (b) Energy difference $\mathrm{\Delta }E$ for the variance extrapolated data from (a).

Figure 4

SC correlation function $P_d(r)$ for Hg1201 at $\delta =0.146$, which is close to the experimental optimal doping, at the square lattice size $L=24$, 28, 30, 32, and 36. Error bars are smaller than the symbol size. Inset: size extrapolation of $P_d$ to the thermodynamic limit $L\to \infty$. Because of relatively scattered data we employ the average of the two biggest lattice sizes ($L=32$, 36) for the size extrapolation $L\to \infty$. In fact, the value at the largest sizes is consistent with the systematic $\delta$ dependence observed near $\delta =0.146$ after the size extrapolation (not shown). The error bar at $1/L=0$ is estimated as the biggest difference to $F_{\mathrm{SC}}^{\infty }$ from the given data.

Figure 5

SC correlation function $P_d(r)$ for Bi2212 at $\delta =0.167$ for the square lattice size $L=16$, 24, 30, 36. Inset: size extrapolation of ${\overline{P}}_d$ to the thermodynamic limit. The gray line shows the linear extrapolation to $1/L\to 0$ by using the values for $L=16$, 24, 30, 36. Error bars are smaller than the symbol size.

Figure 6

SC correlation function $P_d(r)$ for Bi2212 at $\delta =0.167$ in the cases of the single- and the explicit two-layer calculations. In the single-layer case a $16\times 16$ square lattice was considered, which corresponds to a $16\times 16\times 2$ lattice in the two-layer case. Error bars are smaller than the symbol size.

Figure 7

SC correlation function $P_d(r)$ for Bi2201 at $\delta =0.167$ at the square lattice sizes $L=24$, 30, 36. Inset: size extrapolation of ${\overline{P}}_d$ to the thermodynamic limit. The gray line shows the linear extrapolation to $1/L\to 0$ by using the values for $L=24$, 30, 36. Error bars are smaller than the symbol size.

Figure 8

Variation of SC order arising from uncertainty of the apical oxygen position via $\alpha =\xi$ scaling for Bi2212 and Bi2212. The realistic range is $0.95\le \alpha =\xi \le 1.1$ for Bi2212 and $0.8\le \alpha =\xi \le 1.0$ for Bi2201. Size extrapolation of ${\overline{P}}_d$ for (a) Bi2212 at $\delta =0.167$ and $\alpha \in \{0.8,0.9,1.0,1.1,1.2\}$, (b) Bi2201 at $\delta =0.167$ and $\alpha \in \{0.8,0.9,1.0\}$. (c) $F_{\mathrm{SC}}^{\infty }$ as a function of $\alpha =\xi$ for Bi2212 (orange) and Bi2201 (blue).

Figure 9

(a) $F_{\mathrm{SC}}$ over scaling $\alpha$ or $\xi$ (effective scaling of $U/|t_1|$ and $V_i/U$) of the SC state on the $L=24$ square lattice and $\delta =0.167$ hole doping. The inset is an enlarged plot around the peak $0.9<\alpha ,\xi <1.5$. For further details, see the main text. (b) Variance extrapolated energy difference $\mathrm{\Delta }E$ between the SC and C4S8 as a function of $\xi$ in the case $\mathcal{H}(\alpha =1.2,\xi )$.

Figure 10

$F_{\mathrm{SC}}^{\infty }$ as a function of $U/|t_1|$ for the four cuprate compounds at $\delta =0.167$ plotted from the list in Tables 3 and 4.

Figure 11

Experimental $T_c^{\mathrm{opt}}$ (black crosses or bars) in comparison to the $T_c=0.16|t_1|F_{\mathrm{SC}}^{\infty }$ scaling for each compound (purple bars). $T_c$ is taken from Tables 3 and 4.

Figure 12

Variance extrapolation for the example of ${\mathrm{CaCuO}}_2$ at $\delta =0.167$ on a $L=24$ lattice. The inset shows an enlarged plot around $(\delta E{)}^2=0$.

Figure 13

Comparison of $F_{\mathrm{SC}}$ between that obtained by using $\delta$ dependent Hamiltonians (red) and the fixed Hamiltonian (black) as in the main text.

Figure 14

Size dependence of the spin structure factor ${10}^3\times S_s({\mathit{q}}_{\max })/N$ at $(\pi ,\pi )$ for (a) the AFM state after optimization at half filling ($L=12$, 16, 24) and $\delta =0.167$ ($L=12$, 18, 24), and for (b) the SC state at $\delta =0.167$ for different lattice sizes ($L=24$, 30, 36).

Figure 15

Size dependence of the charge and spin stripe state after the optimization of variational parameters for different lattice sizes and fillings. For C3S3 (red crosses) $\delta =0.207$ and $L=12$, 24 and for C4S8 (blue dot) $\delta =0.125$ and $L=16$, 24. (a) ${10}^3\times S_s({\mathit{q}}_{\mathrm{SDW}})/N$ vs $1/L$. (b) ${10}^3\times S_c({\mathit{q}}_{\mathrm{CDW}})/N$ vs $1/L^2$. The dashed lines are only to guide the eyes. ${\mathbf{q}}_{\text{SDW}}$ and ${\mathbf{q}}_{\text{CDW}}$ are the wave numbers at the peak of $S_s$ and $S_c$, respectively, for the spin density wave (SDW) and charge density wave (CDW) modulations, respectively.

Figure 16

SC correlation function $P_d(r)$ for modified $t_2$ from the ${\mathrm{CaCuO}}_2$ Hamiltonian for $L=24$ lattice.

Figure 17

SC order parameter $F_{\mathrm{SC}}$ as a function of $\delta$ calculated for three choices of $t_2$ modified from ${\mathrm{CaCuO}}_2$ Hamiltonian.

Figure 18

Energy per site as a function of $\delta$ in the SC state on a $L=24$ square lattice. Total energy $E_{\text{total}}/N$ (a) and $E_{\text{total}}/N-F(\delta )$ (b), as well as on-site Coulomb energy $E_U/N$ (c) and $E_U/N-F(\delta )$ (d), are plotted. Here, a $\delta$-linear function $F(\delta )$ defined below is subtracted in (b) and (d) for better visibility. Note that $F(\delta )$ is different depending on the type of the energy as seen in Table 5. The gray line in the left-hand column indicates $F(\delta )=b_0+b_1\delta$, which is subtracted in (b) and (d). Right-hand column: energies after subtraction of $F(\delta )$ are fitted via a quadratic function $\mathcal{P}(\delta )=c_0+c_1\delta +c_2{\delta }^2$. Fitting parameters are given in Table 5.

Figure 19

Variance extrapolated energies of Bi2212 for various ground-state candidates (SC, charge or spin stripe C3S3 and C4S8, and AFM states) as a function of hole doping $\delta$ on a $L=24$ square lattice. (a) Total energies per site subtracted by $\mathcal{F}(\delta )$. All energies are subtracted by the function $\mathcal{F}(\delta )=-12.34985·\delta +6.36953$. (b) Energy difference $\mathrm{\Delta }E$ for the variance extrapolated data from (a).

Figure 20

Variance extrapolated energies of Bi2201 for various ground-state candidates [SC, charge-spin stripe (C3S3 and C4S8), and AFM state] as a function of hole doping $\delta$ on a $L=24$ square lattice. (a) Total energies per site subtracted by $\mathcal{F}(\delta )$. All energies are subtracted by the function $\mathcal{F}(\delta )=-15.38797·\delta +7.79753$. (b) Energy difference $\mathrm{\Delta }E$ for the variance extrapolated data from (a).

Figure 21

SC order parameter in strong-coupling region for $\delta =0.101$ and 0.167. The lattice size is $L=24$. The dashed lines are fitting $F_{\mathrm{SC}}\propto (|t_1|/U{)}^p$ with $p=0.58$ for $\delta =0.167$ and $p=0.60$ for $\delta =0.101$.

Figure 22

SC order parameter in the space of $\delta$ and $\alpha =(1/8.1)U/|t_1|$ modified from the ab initio ${\mathrm{CaCuO}}_2$. Calculations were performed on $L=24$ lattice.

Figure 23

Momentum distribution $n(\mathit{k})$ in the Brillouin zone for doped ${\mathrm{CaCuO}}_2$ on a $L=24$ square lattice in the SC state. The doping concentration is $\delta =0.028$ for (a),(f); 0.045 for (b),(g); 0.101 for (c),(h); 0.125 for (d),(i); and 0.167 for (e),(j). The lower panels [(f)–(j)] are contour plots.