Intraband exciton relaxation in a biased lattice with long-range correlated disorder

We study numerically the intraband exciton relaxation in a one-dimensional lattice with scale-free disorder, in the presence of a linear bias. Exciton transport is considered as incoherent hoppings over the eigenstates of the static lattice. The site potential of the unbiased lattice is long-range-correlated with a power-law spectral density $S(k) \sim 1/k^{\alpha}$,$\alpha > 0$. The lattice supports a phase of extended states at the center of the band, provided $\alpha$ is larger than a critical value ${\alpha }_c$ [F. A. B. F. de Moura and M. L. Lyra, Phys. Rev. Lett. $\mathbf{\text{81}}$, 3735 (1998)]. When the bias is applied, the absorption spectrum displays clear signatures of the Wannier-Stark ladder [E. D'{\i}az \emph{et al.}, Phys. Rev. B$\mathbf{\text{73}}$, 172410 (2006)]. We demonstrate that in unbiased lattices and in weakly correlated potentials the decay law is non-exponential. However, the decay is purely exponential when the bias increases and $\alpha$ is large. We relate this exponential decay to the occurrence of the Wannier-Stark ladder in the exciton band.