Reconstructed Fermi Surface of Underdoped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ Cuprate Superconductors

The Fermi surface topologies of underdoped samples of the high-$T_c$ superconductor ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ have been measured with angle resolved photoemission. By examining thermally excited states above the Fermi level, we show that the observed Fermi surfaces in the pseudogap phase are actually components of fully enclosed hole pockets. The spectral weight of these pockets is vanishingly small at the magnetic zone boundary, creating the illusion of Fermi “arcs.” The area of the pockets as measured in this study is consistent with the doping level, and hence carrier density, of the samples measured. Furthermore, the shape and area of the pockets is well reproduced by phenomenological models of the pseudogap phase as a spin liquid.

Figure 1

Fermi surface of underdoped Bi2212 at 140 K ($T_c=65\mathrm{K}$). (a)–(f) Energy distribution curves (EDCs) of raw data taken at cuts 1–6 in (p), and (g)–(l) after the analysis described in the text. The EDC peak positions are marked with triangles. FS crossing points exist in cuts 1–4, but not in cuts 5 and 6. (m) Image plot of cut 2 after Fermi normalization, with the FS crossing (solid circle) and the top of the band (cross) indicated. The ghost FS crossing (open circle) deduced from symmetrization in momentum is also indicated. The dashed line represents the location of the magnetic zone boundary. (n) The same image plot for cut 6 showing no FS crossing. (o) Schematic showing points analyzed in the measured spectra as indicated in the text. (p) Pseudopocket determined for the 65 K sample. Black solid circles indicate the measured FS crossings corresponding to point $A$ in the schematic. Crosses show the measured extremity of the dispersion corresponding to point $B$, and black open circles represent the ghost FS corresponding to point $C$. Red open circles are the FS crossings obtained from symmetrization. The dashed line indicates the large FS by LDA.

Figure 2

(a) The pseudopockets determined for three different doping levels. The black data correspond to the $T_c=65\mathrm{K}$ sample, the blue data correspond to the $T_c=45\mathrm{K}$ sample, and the red data correspond to the nonsuperconducting $T_c=0\mathrm{K}$ sample. The area of the pockets $x_{\mathrm{ARPES}}$ scales with the nominal of doping level $x_n$, as shown in the inset. (b) The Fermi pockets derived from YRZ ansatz with different doping level.

Figure 3

(a) The Fermi surface crossings determined for the $T_c=45\mathrm{K}$ sample at three different temperatures. The triangles indicate measurements at a sample temperature of 140 K, the circles measurements at 90 K, and the diamonds measurements at 60 K. (b) The measured arc lengths in (a) plotted as a function of temperature. We note that rather than cycling the temperatures on the same sample, the data in (a) are measured on different samples cut from the same crystal.

Figure 4

(a)–(d) Spectra intensity measured at four points in the antinodal region as indicated in the schematic (e). The measurements are made with the sample in the normal state at a temperature of 140 K. (f) Intensity cuts through the spectral intensity maps of  (a)–(d) as indicated by the lines in (a)–(d). They are also indicated by the open circles in (e).