Complete $d$-Band Dispersion Relation in Sodium Cobaltates

We utilize fine-tuned polarization selection coupled with excitation-energy variation of photoelectron signal to image the complete $d$-band dispersion relation in sodium cobaltates. A hybridization gap anticrossing is observed along the Brillouin zone corner and the full quasiparticle band is found to emerge as a many-body entity lacking a pure orbital polarization. At low dopings, the quasiparticle bandwidth (Fermion scale, many-body $E_F\sim 0.25\mathrm{eV}$) is found to be smaller than most known oxide metals. The low-lying density of states is found to be in agreement with bulk-sensitive thermodynamic measurements for nonmagnetic dopings where the 2D Luttinger theorem is also observed to be satisfied.

Figure 1

Sample characterization: (a) $\mathrm{Na}\mathrm{\text{−}}2p$ core level signal is monitored during the experiment. Inset shows the LEED image exhibiting a clear sixfold symmetry of the in situ cleaved surface. No ruthenatelike surface reconstruction is observed. (b) No significant shift of the oxygen states are observed at the surface. Experimental configurations employed: Polarization of light is parallel (c) or perpendicular (d) to the plane defined by the sample and the orientation of the detector slits.

Figure 2

Doping evolution of $d$-band complex: Electron band structures of $t_{2g}$ complex measured along $\Gamma \mathrm{\text{−}}K$ (top and middle rows) and $\Gamma \mathrm{\text{−}}M$ (bottom row) for doping levels of (a) $x=0.35$, (b) $x=0.48$, and (c) $x=0.75$. Each panel includes energy distribution curves (left) and corresponding secondary derivative image plot (right). The bands are marked using red dots. Bands along $\Gamma \mathrm{\text{−}}K$ exhibit strong photon energy dependence. Data for two photon energies are shown: $hv=35\mathrm{eV}$ (top) and $hv=60\mathrm{eV}$ (middle). No significant photon energy dependence of bands were found along the $\Gamma \mathrm{\text{−}}M$ cut. The extended flat feature (unmarked) past the Fermi crossing in image plots for low dopings are due to rapid drop of background. (d) A high-resolution Fermi surface shows that the $k$-space cuts $\Gamma \mathrm{\text{−}}K$ and $\Gamma \mathrm{\text{−}}M$. (e) A cartoon view of hole doping.

Figure 3

Experiment versus theory: Comparison of (a) experimental band structures for $x=0.35$, $x=0.48$, and $x=0.75$ with (b) $\mathrm{LDA}+\mathrm{\text{large-}}U$ calculation [17] and (c) LAPW calculations [14]. (d) For sodium- or potassium-ordered samples, $x=1/2$, the $e_g^{\prime }$ band is always below the Fermi level ($\sim 250\mathrm{meV}$) at the temperature below or above insulating transition points.

Figure 4

Polarization dependence: The $t_{2g}$ complex measured along the $\Gamma \mathrm{\text{−}}M$ and $\Gamma \mathrm{\text{−}}K$ cuts under $P$ geometry (a),(c) and $S$ geometry (b),(d). Red and blue (light and gray) dots trace the $e_g^{\prime }$ and $a_{1g}$ bands, respectively. Geometries are defined in Figs. 1c, 1d.

Figure 5

Effect of hydration near $x=1/3$: (a),(b) Fermi crossings are observed at 15 K in the hydrated ${\mathrm{Na}}_x{\mathrm{CoO}}_2·y{\mathrm{H}}_2\mathrm{O}$ compound near $x=0.35$ while “$y$” varies from 0.7 to 1.4 (see text). (c) Quasiparticle bandwidth (many body $E_F$) is estimated from a complete dispersion relation measurement.