Recent experiments have shown that the superconducting energy gap in some cuprates is spatially inhomogeneous. Motivated by these experiments, and using exact diagonalization of a model $d$-wave Hamiltonian, combined with Monte Carlo simulations of a Ginzburg-Landau free energy functional, we have calculated the single-particle local density of states LDOS $(\omega ,r)$ of a model high-${\mathrm{T}}_c$ superconductor as a function of temperature. Our calculations include both quenched disorder in the pairing potential and thermal fluctuations in both phase and amplitude of the superconducting gap. Most of our calculations assume two types of superconducting regions: $\alpha$ with a small gap and large superfluid density, and $\beta$ with the opposite. If the $\beta$ regions are randomly embedded in an $\alpha$ host, the LDOS on the $\alpha$ sites still has a sharp coherence peak at $T=0$, but the $\beta$ component does not, in agreement with experiment. An ordered arrangement of $\beta$ regions leads to oscillations in the LDOS as a function of energy. The model leads to a superconducting transition temperature $T_c$ well below the pseudogap temperature $T_{c0}$ and has a spatially varying gap at very low $T$, both consistent with experiments in underdoped Bi2212. Our calculated LDOS $(\omega ,r)$ shows coherence peaks for $T<T_c$, which disappear for $T>T_c$, in agreement with previous work considering phase but not amplitude fluctuations in a homogeneous superconductor. Well above $T_c$, the gap in the LDOS disappears.

• Received 3 July 2006

DOI:https://doi.org/10.1103/PhysRevB.74.174506