Correlation effects in the triangular lattice single-band system ${\mathrm{Li}}_x\mathrm{Nb}{\mathrm{O}}_2$

Superconductivity in hole-doped ${\mathrm{Li}}_x\mathrm{Nb}{\mathrm{O}}_2$ has been reported with $T_c\approx 5\mathrm{K}$ in the range $0.45⩽x<0.8$. The electronic structure is based on a two-dimensional triangular Nb lattice. The strong trigonal crystal field results in a single Nb $d_{z^2}$ band isolated within a wide gap, leading to a single-band triangular lattice system. The isolated, partially filled band has a width $W=1.5\mathrm{eV}$ with dominant second neighbor hopping. To identify possible correlation effects, we apply dynamical mean field theory and quantum Monte Carlo (QMC) using on-site Coulomb repulsion $U=0–3\mathrm{eV}$ and check selected results using determinant QMC. For $U$ as small as $1\mathrm{eV}$, the single particle spectrum displays a robust lower Hubbard band, suggesting the importance of correlation effects in ${\mathrm{Li}}_x\mathrm{Nb}{\mathrm{O}}_2$ even for $U⩽W$. At half-filling $(x=0)$, a Mott transition occurs at $U_c\approx 1.5\mathrm{eV}$. Coupling between the O $A_g$ phonon displacement and correlation effects is assessed.

Figure 1

(Color online) Two-dimensional tight-binding band structure and density of states, using up to third neighbor hopping parameters ($t_1=64\mathrm{meV}$, $t_2=100\mathrm{meV}$, and $t_3=33\mathrm{meV}$), for the single isolated $d_{z^2}$ state in $\mathrm{Li}\mathrm{Nb}{\mathrm{O}}_2$. The dashed horizontal lines indicate the Fermi energy $E_F$ in the rigid band picture for electron fillings $n=1$, $4∕3$, and $5∕3$ from bottom to top. The centroid of the band $E_0$ lies on $18\mathrm{meV}$ below $E_F$ for $n=4∕3$. Note that the lower half of weight is localized within $0.2\mathrm{eV}$ width, only 13% of the total bandwidth.

Figure 2

(Color online) Top: Electron filling $n$ versus effective chemical potential ${\mu }^{*}=\mu -U∕2$, when changing the on-site Coulomb repulsion $U$. The lines and symbols are the results of DMFT and DQMC calculations, respectively. The excellent agreement is partly a consequence of the magnetic frustration of the triangular lattice which diminishes the importance of including intersite correlations (see text). Even for $U=1.0\mathrm{eV}$, the slope changes significantly near half-filling, which is indicated by the dashed horizontal line. Above $U=1.5\mathrm{eV}$, there is a plateau at half-filling. Inset: $n$ vs $U$ plot at ${\mu }^{*}=0.05$, which displays clearly a metal-to-insulator transition completing $(n\to 1)$ near $U=1.5\mathrm{eV}$. Bottom: Compressibility $\kappa \equiv dn∕d\mu$ versus ${\mu }^{*}$ near the critical $U$ regime, showing more clearly where it approaches zero and finally becomes vanishingly small.

Figure 3

(Color online) Determinant QMC result for the spin-spin correlation function for temperatures of $1100\mathrm{K}$ $(\beta =10.9)$ and $750\mathrm{K}$ $(\beta =16)$, versus band filling $n$. $U=1$, and energies are in eV. The fifth neighbor (5nn) is the third neighbor along a straight line from the reference site.

Figure 4

(Color online) Effect of the on-site Coulomb repulsion $U$ (in units of eV) on spectral density, which is obtained from MaxEnt, for $n=1$. At $U=1.5\mathrm{eV}$, a gap opens clearly.

Figure 5

(Color online) Top: Change in the correlation function $⟨m\left(\tau \right)m\left(0\right)⟩$ of local moment $m$ as $U$ increases. Bottom: $U$-dependent local susceptibility ${\chi }_{\mathit{loc}}$ (left) and standard deviation $\sigma$ (right) over $(0,\beta )$ for the correlation function. The $\sigma$ plot shows that variance of the correlation function is strongly reduced by the transition to the insulating phase.

Figure 6

(Color online) Effect of the on-site Coulomb repulsion $U$ (in units of eV) on spectral densities, which are obtained from MaxEnt, for $n=4∕3$ (top) and $5∕3$ (bottom). As expected from the bandwidth, correlated behavior already appears at $U=0.5\mathrm{eV}$. (For details, see text.) Note that the spectral densities are normalized, i.e., $\int A\left(\omega \right)d\omega =1$.

Figure 7

(Color online) Effect of O displacement on spectral densities. O is displaced by $\pm 0.10\mathrm{\AA }$ in the ⟨001⟩ direction. $d_0$ is the optimized position. $d_{-}$ and $d_{+}$ denote contraction and elongation of the Nb–O bond length, respectively. Top: Tight-binding DOS (i.e., $U=0$) for the O displacement. The densities of states are aligned with respect to $E_F$ for $n=5∕3$, set to zero. For each case, the dashed vertical lines denote $E_F$ for $n=1$, $4∕3$, and $5∕3$ (left to right). Bottom: Change in spectral densities, which shows effects of the phonon mode, at $U=2\mathrm{eV}$. The line designations are the same as in the top panel.