Momentum-space electronic structures and charge orders of the high-temperature superconductors Ca${}_{2-x}$Na${}_x$CuO${}_2$Cl${}_2$ and Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$

We study the electronic structure of Ca${}_{2-x}$Na${}_x$CuO${}_2$Cl${}_2$ and Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$ samples in a wide range of doping, using angle-resolved photoemission spectroscopy, with emphasis on the Fermi surface (FS) in the near antinodal region. The “nesting wave vector,” i.e., the wave vector that connects two nearly flat pieces of the FS in the antinodal region, reveals a universal monotonic decrease in magnitude as a function of doping. Comparing our results to the charge order recently observed by scanning tunneling spectroscopy (STS), we conclude that the FS nesting and the charge order pattern seen in STS do not have a direct relationship. Therefore, the charge order likely arises due to strong-correlation physics rather than FS nesting physics.

Figure 1

(Color online) Doping evolution of the Fermi energy intensity maps for (a) UD18, (b) UD21, (c) OP27, and (d) OD20 Na-CCOC samples. The energy integration window $[-10\text{meV},+10\text{meV}]$ centered at $E_F$ was used. These maps are obtained by symmetrizing the original data with respect to the $(0,0)-(\pi ,\pi )$ line. The Fermi momentum values determined by the maximum of the MDC at $E_F$ are marked by open circles (green). The solid lines (gray) show fits to the Fermi surface of the tight-binding band theory (Ref. 28). In the color map, dark (black) means high intensity and pale (yellow) means low intensity. The insets of (a) and (d) are discussed later (“Luttinger sum rule”).

Figure 2

(Color online) (a) EDCs in the antinodal region [see (b) for the $\vec{k}$ information], as a function of doping for Na-CCOC. (b) $\vec{k}$ space cuts A,B corresponding to the data of (d) and (c), respectively, and the approximate $\vec{k}$ value (point) corresponding to the data of (a). (c), (d) MDCs as a function of the binding energy for UD21 (c) and OP27 (d). Red dashed lines are fit curves (see text) and the vertical dashed lines (blue) correspond to the peak positions at $\omega =0$, i.e., at $E_F$. The curves are separated vertically for easy view, and the binding energy values for adjacent curves differ by 10 meV. The inset of (c) redisplays the $E_F$ data, emphasizing the clear two-peak structure. The two peaks for each curve are marked by triangular symbols, each of which indicates both the peak position and its uncertainty (horizontal size) as determined by the MDC fit procedure.

Figure 3

(Color online) (a) “Fermi surfaces” as a function of doping for Na-CCOC, corresponding to the data of Fig. 1. The arrows indicate the nesting vectors. (b) FS nesting vectors (open symbols) and STS wave vectors (filled symbols). The vertical dashed line (red) indicates the position of $T_{c,\mathit{max}}$.

Figure 4

(Color online) (a)–(c) Normal-state Fermi energy intensity maps of Bi2212. White lines show fits to the Fermi surface of the tight-binding band theory (Ref. 28). (d) The FS nesting vectors and the STS charge order wave vectors for Bi2212, from this work and from the literature (Refs. 10, 11, 12, 13, 14, 15, 31, 32). (e) Similar plot for Bi2201 samples. All data are from the literature (Refs. 17, 31, 33). The formats of (d) or (e) are similar to those of Fig. 3b. (f), (g) Summary of all ARPES nesting vectors (f) and STS charge order wave vectors (g), showing their overall contrasting behaviors.