Formation of Gapless Fermi Arcs and Fingerprints of Order in the Pseudogap State of Cuprate Superconductors

We use angle-resolved photoemission spectroscopy and a new quantitative approach based on the partial density of states to study properties of seemingly disconnected portions of the Fermi surface (FS) that are present in the pseudogap state of cuprates called Fermi arcs. We find that the normal state FS collapses very abruptly into Fermi arcs at the pseudogap temperature ($T^{*}$). Surprisingly, the length of the Fermi arcs remains constant over an extended temperature range between $T^{*}$ and $T_{\mathrm{pair}}$, consistent with the presence of an ordered state below $T^{*}$. These arcs collapse again at the temperature below which pair formation occurs ($T_{\mathrm{pair}}$) either to a point or a very short arc, whose length is limited by our experimental resolution. The tips of the arcs span between points defining a set of wave vectors in momentum space, which are the fingerprints of the ordered state that causes the pseudogap.

Figure 1

Determination of the partial density of states at $E_F$, $A_{\mathrm{MDC}}(E_F)$, from the ARPES data of optimally doped Bi2201 ($T_c=32\mathrm{K}$). (a) MDCs measured along the nodal direction. The inset shows a dispersion image at the lowest temperature (12 K). (b) The temperature dependence of $A_{\mathrm{MDC}}(E_F)$ estimated from the area of MDCs in (a). The momentum cut and estimated area of MDC are shown in the inset. Statistical error bars are smaller than the size of the symbols, and the main source of noise in the data is stability of the ARPES spectrometer, which can be evaluated by the small scatter of the data points. (c) Same data as in (a), but measured along a momentum cut slightly off the antinode. (d) The temperature dependence of $A_{\mathrm{MDC}}(E_F)$ estimated from the area of MDCs in (c). (e) Symmetrized EDCs at $k_F$ for several temperatures, obtained from the data of (c). The $T_{\mathrm{peak}}$ is defined as the temperature where two peaks in the spectrum merge to one peak at elevated temperatures.

Figure 2

Temperature dependence of the MDC area along the Fermi surface $A_{\mathrm{MDC}}(E_F)$ for optimally doped Bi2201 ($T_c=32\mathrm{K}$) measured at several points on the Fermi surface. $A_{\mathrm{MDC}}(E_F)$ is expressed as a percentage of the MDC area above $T^{*}$. The top left panel corresponds to the antinodal direction. The inset of each panel shows the FS and the location of the cut, normal to the FS. The pseudogap temperature ($T^{*}$) and the temperature above which the arc exists ($T_{\mathrm{pair}}$) are indicated by arrows. $T_c$ is indicated by a dotted green line. $A_{\mathrm{MDC}}(E_F)$ at the nodal point is constant down to lowest temperature and already shown in Fig. 1b.

Figure 3

Temperature dependence of the MDC area along the Fermi surface $A_{\mathrm{MDC}}(E_F)$ for optimally doped Bi2212 with $T_c=93\mathrm{K}$ measured at several points on the Fermi surface similarly to Fig. 2. The pseudogap temperature ($T^{*}$) and the temperature above which the arc exists ($T_{\mathrm{pair}}$) are indicated by arrows. $T_c$ is indicated by a dotted green line. $A_{\mathrm{MDC}}(E_F)$ at the nodal point is constant down to the lowest temperature and shown in Fig. S5 of the Supplemental Material [25].

Figure 4

Momentum dependence of the temperatures at which the gap opens and the length of the Fermi arc is determined from ARPES data and marked by red and blue arrows in Figs. 2, 3 (a) for optimally doped Bi2201 and (b) for optimally doped Bi2212. The length of the arcs remains constant for $T_{\mathrm{pair}}<T<T^{*}$. Below $T_{\mathrm{pair}}$, we can only impose an upper limit on the length of the remaining arc indicated by black arrows. (c)–(e) The Fermi surface for three key temperature ranges. The antiferromagnetic zone boundary is marked with a dashed black line. (f) The Fermi arcs of the Bi2212 and Bi2201 samples for $T_{\mathrm{pair}}<T<T^{*}$ determined from our ARPES data. The key $q$ vectors connecting the tips of the Fermi arcs are indicated with arrows.