Imaging studies of hole-electron asymmetry observed in a lightly-doped high-$T_{\rm c}$ superconductor Ca$_{2-x}$Na$_x$CuO$_2$Cl$_2$

We have reported spectroscopic imaging on the electronic state intervening between the Mott insulator and the $d$-wave superconductor [1]. The electronic state with $|E|<100$~meV shows complex spatial modulations with $4a_0\times 4a_0$ ($a_0$: in-plane lattice constant) correlations. Moreover, the tunneling spectra show characteristic hole-electron asymmetry, which is thought to be related to the approach of the Mott insulator [2]. Here we report on new studies of imaging the 'Mottness' through mapping the asymmetry in tunneling spectra associated with high energies $|E|>100$~meV. Spectroscopic mapping of the asymmetry reveals strong $4a_0\times 4a_0$ periodic modulation up to several hundreds meV. However the $4a_0/3\times 4a_0/3$ component, which appears in the Fourier transform of the checkerboard pattern observed at $|E|<100$~meV, is negligle. This spatial modulation of the 'Mottness' may indicate that hole density is modulated with the $4a_0\times 4a_0$ periodicity. This would imply that the $4a_0/3\times 4a_0/3$ periodicity observed at lower energies may arise from umklapp scattering due to the hole density modulation. [1] T. Hanaguri {\it et al}., Nature 430, 1001 (2004). [2] P. W. Anderson and N. P. Ong, cond-mat/0405518.