Reconstruction of the ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ Fermi surface

The effects of structural supermodulation with the period $\lambda \approx 26$ Å along the $b$ axis of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ have been observed in photoemission studies from the early days as the presence of “diffraction replicas” of the intrinsic electronic structure. Although predicted to affect the electronic structure of the Cu-O plane, the influence of supermodulation potential on Cu-O electrons has never been observed in photoemission. In the present study, we clearly see the effects on the ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ electronic structure; we observe a hybridization of the intrinsic bands with the supermodulation replica bands in the form of avoided crossings and a corresponding reconstruction of the Fermi surface. We estimate the hybridization gap, $2{\mathrm{\Delta }}_h\sim 25$ meV, in the slightly underdoped samples. The hybridization weakens with doping and the anticrossing can no longer be resolved in strongly overdoped samples. In contrast, the “shadow” replica, shifted by $(\pi ,\pi )$, is found not to hybridize with the original bands within our detection limits.

Figure 1

Reconstructed Fermi surface of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$. Constant energy contour at $E=-5$ meV of ARPES intensity from slightly underdoped (UD87) Bi2212, taken in the superconducting state ($T=25$ K). The black arrows indicate the reconstruction due to the structural supermodulation in the form of avoided crossings between the original FS and supermodulation replicas. The red arrow indicates the quasi-1D segments formed by reconstruction. The dashed arrows indicate the coordinate axis used in Fig. 2, with the origin at the nodal point. The map was recorded with the analyzer slit in the $k_R$ direction, by rotating the sample in the $k_T$ direction in steps of $0.5^{\circ }$. The blue curves represent the tight-binding fit of the experimental FS.

Figure 2

Reconstructed electronic structure of Bi2212. (a)–(d) Constant energy contours of ARPES intensity near the nodal region in the $\mathrm{\Gamma }\text{−}X$ quadrant of the Brillouin zone at energies as indicated. (e) The ARPES spectrum taken along the momentum line indicated in panel (d). (f)–(i) The ARPES spectra taken along the four-momentum lines nearly tangential to the nodal segment of the FS as indicated in panel (a). The top (bottom) spectrum corresponds to the left (right) momentum line in (a). The momenta are measured from the nodal Fermi wave vector. All the spectra were taken with the analyzer slit in the $k_R$ direction, while the mapping in the $k_T$ direction was done by rotating the sample in steps of $1/3$ of a degree. (j) The energy distribution curve integrated over the momentum region between the two dotted lines in panel (g) for UD87 and OD91. The arrow indicates the magnitude of the hybridization gap in the UD87 sample, $2{\mathrm{\Delta }}_h\approx 25$ meV.

Figure 3

Hybridization gap and density of states. The energy distribution curve from Fig. 2 (red) and several other EDCs obtained by integrating the spectral intensity over the $k$ regions as indicated in the inset, with the corresponding colors.

Figure 4

The $(\pi ,\pi )$ replica. (a) The constant energy contour of ARPES intensity from the OD91 sample recorded at $E=-26$ meV in the superconducting state ($T=25$ K). (b),(c) ARPES spectra showing the dispersion of the states along the lines indicated in (a). The $(\pi ,\pi )$ replica does not hybridize with the supermodulation replica (line 1) or with the original band (line 2). The raw intensities are shown as functions of the emission angle. The mapping was done with the analyzer slit in the $x$ direction, by rotating the sample in the $y$ direction in steps of $1^{\circ }$. (d) EDCs taken at the points on the original FS as indicated in (a).