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# Electronic excitation spectrum

The dispersion and the relaxation of electronic states near the Fermi level can be measured by angle-resolved photoemission, ARPES. The general trend is that quasiparticles are poorly defined (strongly damped) at very low dopings, but gradually become better defined with increasing doping. Notably, only part of Fermi surface expected from band structure calculations is visible below optimal doping.

Figures 1a and 1b show a schematic of the nominal Fermi surface for LSCO as predicted by band structure calculations. It is useful to distinguish between different regions of the Fermi surface. The language used is based on the $d$-symmetry pair wave function observed in cuprates. The gap goes to zero (i.e., has a node) along the diagonal direction; the gap maximum occurs in the "anti-nodal" region. Even in the normal state, there are differences in the spectral function measured in these two regions.

The quasiparticle spectral weight in the nodal direction increases with doping in the underdoped regime. Its doping dependence is consistent with the doping dependence of the carrier density obtained by other means (Hall and ac conductivity).
For instance, Fig. 2 shows the spectral function along the nodal (a) and antinodal (b) directions for LSCO with $x$ from zero to 0.15 [1]. Along the nodal direction, a sharp peak is seen near the Fermi level even for $x=0.03$, whereas in the antinodal direction a sharp peak at the Fermi level does not appear until one approaches optimal doping ($x=0.15$). The antinodal spectra tend to show a diffuse, gapped behavior, providing one definition of the pseudogap.

In more detail, Figure 3 shows ARPES measurements of the quasiparticle dispersion of filled states for LSCO with several dopant concentrations [2]. In the undoped system, the spectral weight is diffuse and far below the Fermi energy, $E_{\rm F}$. With doping, a sharp, roughly-linear dispersion develops along the nodal direction [(0,0) to $(\pi,\pi)$]. In the antinodal region, near $(\pi,0)$, a region of fairly flat dispersion just below $E_{\rm F}$ becomes apparent at $x=0.15$, but this feature shifts through $E_{\rm F}$ for $x=0.22$ and above.

The absence of visible dispersion line crossing in the antinodal region makes the difinition of a continuous Fermi surface problematic in underdoped cuprates. The antinodal gap (or pseudogap) persists up to a characteristic temperature $T^*$ (which decreases with doping) [3].
Figure 4 shows ARPES measurements of spectral weight along a quadrant of the Fermi surface for a range of dopings in LSCO [4]. At low doping, the weight is only apparent on a small arc near the nodal point. With doping, the nodal arc expands, until it covers most of the nominal Fermi surface near optimal doping. When the flat, antinodal dispersion moves through $E_{\rm F}$ near $x=0.22$ as seen in Fig. 3, the shape of the Fermi surface changes. For lower doping, the nominal Fermi surface is centered around $(\pi,\pi)$, while for $x=0.22$ and above the Fermi surface is closed around (0,0). This corresponds to a change from a hole-like to an electron-like Fermi surface.