Estimation of the on-site Coulomb potential and covalent state in ${\mathrm{La}}_2{\mathrm{CuO}}_4$ by muon spin rotation and density functional theory calculations

The on-site Coulomb potential, $U$, and the covalent state of electronic orbitals play key roles for the Cooper pair symmetry and exotic electromagnetic properties of high-$T_{\mathrm{c}}$ superconducting cuprates. In this paper, we demonstrate a way to determine the value of $U$ and present the whole picture of the covalent state of Cu spins in the mother system of the La-based high-$T_{\mathrm{c}}$ superconducting cuprate, ${\mathrm{La}}_2{\mathrm{CuO}}_4$, by combining the muon spin rotation $\left(\mu \mathrm{SR}\right)$ and the density functional theory (DFT) calculation. We reveal local deformations of the ${\mathrm{CuO}}_6$ octahedron followed by changes in Cu-spin distributions caused by the injected muon. Adjusting the DFT and $\mu \mathrm{SR}$ results, $U$ and the minimum charge-transfer energy between the upper Hubbard band and the O $2p$ band were optimized to be 4.87(4) and 1.24(1) eV, respectively.

Figure 1

Initial condition of the Cu-spin structure for the present DFT calculation study. Blue, red, and green marks are Cu spins, O, and La atoms, respectively. Cu spins form the AF spin alignment with the spin direction along to the $b$ axis within the ${\mathrm{CuO}}_2$ plane. This spin structure is the same as that determined from the neutron-scattering experiment [18].

Figure 2

Temperature dependence of the magnetic susceptibility of the ${\mathrm{La}}_2{\mathrm{CuO}}_4$ single crystal. The magnetic field of 0.5 T was applied perpendicular to the ${\mathrm{CuO}}_2$ plane. The black arrow shows the antiferromagnetic transition temperature $T_{\mathrm{N}}$ of 309 K. The red solid line indicates the best-fit result by using the Curie-Weiss law below 100 K.

Figure 3

(a) $\mu \mathrm{SR}$ time spectrum measured at 1.7 K on the ${\mathrm{La}}_2{\mathrm{CuO}}_4$ single crystal. The solid line is the best-fit results by using Eq. (1) with $i$=1,2,3. (b) Fourier spectrum of the muon spin precession. Solid lines show the best-fit results obtained by using the Lorentzian function. Insets show Fourier spectra of the additional two components.

Figure 4

(a) Three-dimensional map of the Cu-spin density in the ${\mathrm{CuO}}_6$ octahedron of ${\mathrm{La}}_2{\mathrm{CuO}}_4$ obtained from our $\mathrm{DFT}+U$ calculations. The Cu-spin density expands to the in-plane O and reverses the spin direction at the other side centering the nuclear position of O as shown by the dark blue color. This means that the neighboring Cu-spin component which has the opposite spin direction as shown in Fig. 1 flows into the same in-plane O and cancels the total spin component. The $a_0$ in the density unit is the Bohr radius. (b) Electrostatic potential calculation results. The red area is the isosurface showing the energy level of 1.07 eV higher from the minimum potential. The energy level of the isosurface was chosen to make the position of ${\mathrm{M}3}_{\mathrm{DFT}}$ clearly visible. Gray balls indicate three local-minimum potential positions as candidates for initial muon stopped positions.

Figure 5

Final stable muon positions for (a) ${\mathrm{M}1}_{\mathrm{DFT}}$, (b) ${\mathrm{M}2}_{\mathrm{DFT}}$, and (c) ${\mathrm{M}3}_{\mathrm{DFT}}$, respectively, obtained from $\mathrm{DFT}+U$ including the muon using the $4\times 4\times 2$ supercell. Gray and black balls in each panel indicate the initial and final position of the muon after the relaxation, respectively.

Figure 6

Cu-spin density distribution estimated by $\mathrm{DFT}+U$ including the muon using the $4\times 4\times 2$ supercell with the muon at (a) ${\mathrm{M}1}_{\mathrm{DFT}}$, (b) ${\mathrm{M}2}_{\mathrm{DFT}}$, and (c) ${\mathrm{M}3}_{\mathrm{DFT}}$, respectively. Black balls in each panel indicate the final position of the muon after the relaxation where the muon's density has the maximum. The $a_0$ in the density unit is the Bohr radius.

Figure 7

Zero-point vibration motion of the muon itself around the minimum electrostatic potential for ${\mathrm{M}1}_{\mathrm{DFT}}$, ${\mathrm{M}2}_{\mathrm{DFT}}$, and ${\mathrm{M}3}_{\mathrm{DFT}}$. These are views of cross sections perpendicular to the ${\mathrm{CuO}}_2$ plane including the local minimum potential point.

Figure 8

Optimization of $U$ in terms of the difference in internal fields obtained by $\mu \mathrm{SR}$ and $\mathrm{DFT}+U+\mu$ calculations, varying $U$ from 2 to 8 eV. The solid line is the best-fit result by using the Gaussian function. $\mathrm{\Delta }{H}_{\mathrm{M}\mathit{i}}$ is the difference between $H_{\mu \mathrm{SR}}^{\mathrm{M}\mathit{i}}$ and $H_{\mathrm{DFT}}^{\mathrm{M}\mathit{i}}$ ($i$=1,2,3) as described in Eq. (5). ${\sigma }_{\mathit{i}}$ is the fitting-error value of the internal field at each $H_{\mu \mathrm{SR}}^{\mathrm{M}\mathit{i}}$.

Figure 9

ZF-$\mu \mathrm{SR}$ time spectrum observed at 1.7 K with the simulated line by using internal fields which were estimated from the current DFT calculations. The black solid line is the trace of the simulation. The same values for $A_i$, ${\varphi }_i$, and ${\lambda }_i$ listed in Table 1 were used, replacing $H_{\mu \mathrm{SR}}^{\mathrm{M}\mathit{i}}$ with $H_{\mathrm{DFT}}^{\mathrm{M}\mathit{i}}$ to draw the black solid line.