We compute the spectral function $A(\mathbf{k},\omega )$ of a model two-dimensional high-temperature superconductor at both zero and finite temperatures $T$. The model consists of a two-dimensional BCS Hamiltonian with $d$-wave symmetry, which has a spatially varying, thermally fluctuating, complex gap $\Delta$. Thermal fluctuations are governed by a Ginzburg-Landau free energy functional. We assume that an areal fraction $c_{\beta }$ of the superconductor has a large $\Delta$ ($\beta$ regions), while the rest has a smaller $\Delta$ ($\alpha$ regions), both of which are randomly distributed in space. We find that $A(\mathbf{k},\omega )$ is most strongly affected by inhomogeneity near the point $\mathbf{k}=(\pi ,0)$ (and the symmetry-related points). For $c_{\beta }\simeq 0.5$, $A(\mathbf{k},\omega )$ exhibits two double peaks (at positive and negative energies) near this $k$ point if the difference between ${\Delta }_{\alpha }$ and ${\Delta }_{\beta }$ is sufficiently large in comparison to the hopping integral; otherwise, it has only two broadened single peaks. The strength of the inhomogeneity required to produce a split spectral function peak suggests that inhomogeneity is unlikely to be the cause of a second branch in the dispersion relation, such as has been reported in underdoped LSCO. Thermal fluctuations also affect $A(\mathbf{k},\omega )$ most strongly near $\mathbf{k}=(\pi ,0)$. Typically, peaks that are sharp at $T=0$ become reduced in height, broadened, and shifted toward lower energies with increasing $T$; the spectral weight near $\mathbf{k}=(\pi ,0)$ becomes substantial at zero energy for $T$ greater than the phase-ordering temperature.

• Received 15 April 2007

DOI:https://doi.org/10.1103/PhysRevB.77.014515