Unified Origin for the 3D Magnetism and Superconductivity in ${\mathrm{Na}}_x{\mathrm{CoO}}_2$

We analyze the origin of the three-dimensional (3D) magnetism observed in nonhydrated Na-rich ${\mathrm{Na}}_x{\mathrm{CoO}}_2$ within an itinerant spin picture using a 3D Hubbard model. The origin is identified as the 3D nesting between the inner and outer portions of the Fermi surface, which arise due to the local minimum structure of the $a_{1g}$ band at the $\Gamma \mathrm{\text{−}}A$ line. The calculated spin wave dispersion strikingly resembles the neutron scattering result. We argue that this 3D magnetism and the spin fluctuations responsible for superconductivity in the hydrated systems share essentially the same origin.

Figure 1

Upper left: The 3D model in the present study. Upper right: The Brillouin zone. The wave numbers in the parentheses are those in the unfolded BZ scheme. Lower left: The band structure of the present model with $t_2=-0.35$, $t_3=-0.07$, $t_z=-0.15$, and $U=0$. Lower right: The band for $t_2=t_3=0$, $t_z=-0.15$. $k_z=0$, $\pi /2$, $\pi$ are $k_z$ in the unfolded BZ scheme.

Figure 2

The vertical cross section ($k_y=0$) of the FS for various $n$ (upper left corner) with $U=0$. Here we use the unfolded BZ scheme, but for $n=1.60$ we also show the FS in the usual folded BZ scheme by dashed lines. The unfolded BZ edge in the $k_z$ direction is denoted as “$\frac{\pi }{c/2}$” to stress that the unit cell is halved in the $c$ direction. The finite thickness is due to a finite range of energy ($E_F\pm 0.03t_1$) taken in obtaining the FS. The thickness thus represents the density of states or the effective mass. For $n=1.82$, the FS for $t_2=t_3=0$ (dashed line) is shown for comparison. The arrows represent the diameter of the FS.

Figure 3

Upper left: The band filling dependence of $T_{\mathrm{SDW}}$ obtained by FLEX. The inset shows the $T$ dependence of ${\chi }_s(\mathbf{0},0{)}^{-1}$ for $n=1.82$ and $n=1.7$. Upper left: The nesting vector of the FS for $n=1.82$ (left) and the FS in the ordered state ($T=0.015$) obtained by the mean field calculation (right) are shown. We use the extended zone scheme in the ordered state, and the regions $0<|k_z|<\frac{\pi /2}{c/2}$ and $\frac{\pi /2}{c/2}<|k_z|<\frac{\pi }{c/2}$ are the 1st and the 2nd BZ, respectively. Lower left: The FLEX ${\chi }_s(\mathit{q},0)$ for $n=1.82$, $U=6$, and $T=0.05$. Lower right: The calculated in-plane (left) and out-of-plane (right) spin wave dispersion for $n=1.82$, $U=4$, and $T=0.015$. Here the wave vectors are presented in units of the reciprocal lattice primitive vector in the usual folded BZ for clear comparison with Fig. 4 of Ref. 25.

Figure 4

The FS nesting relevant for SC (left) and the 3D magnetism (right).