X-ray imaging of dispersive charge modes in a doped Mott insulator near the antiferromagnet/superconductor transition

Momentum-resolved inelastic resonant x-ray scattering is used to map the doping evolution of bulk electronic modes in the doped Mott insulator class ${\text{Nd}}_{2-x}{\text{Ce}}_x{\text{CuO}}_4$. As the doping induced antiferromagnet/superconductor (AFM/SC) transition is approached, we observe an anisotropic redistribution of the spectral weight of collective excitations over a large energy scale along the $\Gamma \to \left(\pi ,\pi \right)$ direction, whereas the modes exhibit broadening $\left(\sim 1\text{eV}\right)$ with relatively little softening along $\Gamma \to \left(\pi ,0\right)$ with respect to the parent Mott state $\left(x=0\right)$. Our study reveals a closing of the charge gap in the vicinity of the zone center even though the mode softening and spectral redistribution involve an unusually large energy scale over the full Brillouin zone. The collective behavior of modes in the vicinity of the AFM/SC critical transition is demonstrated.

Figure 1

(Color online) Incident energy dependence of electronic excitations. (a) Vertical scattering geometry employed with an x-ray field along [001]. [(b) and (c)] X-ray energy dependence of inelastic excitations for ${\text{Nd}}_2{\text{CuO}}_4$ and ${\text{La}}_2{\text{CuO}}_4$. The $Q$ values are chosen to minimize the large quasielastic background. [(d)–(g)] Excitations near the Mott gap are seen to be enhanced near the first absorption peak, not only in the undoped insulator but also in the doped system. A few energy-loss curves corresponding to the data images in (f) and (g) are shown in (h) and (i). The red (dark gray) curves show the fits to the quasielastic data.

Figure 2

(Color online) Doping evolution of charge excitations in NCCO: (a) $x=0$ (parent Mott state), (b) $x=0.09$ (doped AFM), and (c) $x=0.14$ (superconductor). The upper row shows intensity maps in the reduced zone by using a color scheme wherein high intensity is denoted as red (dark gray) and low intensity as blue (black). The black, gray, and white dots are guides to the eye for the dispersion of low-, medium-, and high-energy excitation branches, respectively. The lower row gives a few representative spectra corresponding to the images in the upper row. Absolute momentum values (in units of $\pi /a_o$) are shown attached to various spectra and lie in the fourth BZ along [110] and the second BZ along [100].

Figure 3

(Color online) (a) Spectral curves (raw data) along [110] and [100] for the undoped system [red (dark gray) squares] are compared at various momenta to the corresponding spectra for the doped system [black circles and green (gray) diamonds]. (b) The positions of peaks [marked in Figs. 2a, 2b, 2c] based on the center of gravity method (Refs. 4, 8) of various spectral features (filled symbols) and some of the associated leading edges (open symbols) as a function of momentum and doping are plotted.

Figure 4

(Color online) Top row: Doping evolution of RIXS spectra obtained within a three-band Hubbard model as a function of the energy transfer $\omega$ at momenta $\mathbf{q}$ along $\Gamma$ to $\left(\pi ,\pi \right)$ and $\left(\pi ,0\right)$ directions for three different dopings $x$. The positions in $\omega -\mathbf{q}$ space of various experimentally observed spectral peaks (filled circles) and the leading edge of the low-energy feature (filled diamonds) are superposed. The computed spectra have not been broadened to reflect the experimental resolution in order to highlight spectral features. Bottom row: Comparison of theoretical [blue (black) lines] and experimental [red (gray) lines] spectra. The three columns refer to the three indicated doping levels $x$. Each column includes four different $q$ values as shown; the experimental spectra are at the closest $q$ value measured. The theoretical spectra have been smoothed to reflect experimental broadening (Ref. 22).

Figure 5

(Color online) (a) Schematic temperature-doping $\left(T\text{−}x\right)$ phase diagram of NCCO. (b) The leading-edge gap (filled squares) and peak positions (white diamonds) of the lowest-energy excitation branch in RIXS data at the $\Gamma$ point are compared to antiferromagnetic-correlation gap ${\Delta }_m$ from the present calculation (filled circles) and from SPS (triangles) (Ref. 10) as a function of doping. (c) The excitation modes (branches) are fitted and extrapolated to the $\Gamma$ point (0,0) based on the branch curvatures to extract the data points presented in (b). The peak positions correspond to markings in Figs. 2a, 2b, 2c and the leading edges are extracted from data presented in Fig. 3.