Nonvanishing Subgap Photocurrent as a Probe of Lifetime Effects

For semiconductors and insulators, it is commonly believed that in-gap transitions into nonlocalized states are smoothly suppressed in the clean limit; i.e., at zero temperature, their contribution vanishes due to the unavailability of states. We present a novel type of subgap response which shows that this intuition does not generalize beyond linear response. Namely, we find that the dc current due to the bulk photovoltaic effect can be finite and mostly temperature independent in an allowed window of subgap transitions. We expect that a moderate range of excitation energies lies between the bulk energy gap and the mobility edge where this effect is observable. Using a simplified relaxation time model for the band broadening, we find the subgap dc current to be temperature independent for noninteracting systems but temperature dependent for strongly interacting systems. Thus, the subgap response may be used to distinguish whether a state is single-particle localized or many-body localized.

Figure 1

Main elements of the proposal. (a) Schematic of the processes leading to a photocurrent below the gap. The photon with subgap energy creates an excitation in the region between the mobility edge and the lower end of the conduction band. (b) Current response up to second order. The incident light of frequency $\omega$ creates a linear response current $I^{(1)}$ at frequency $\omega$. At second order, a photovoltaic current ($\omega =0$) is observed. (c) Sketch of the antiferromagnetic honeycomb lattice with next-nearest neighbor hoppings between sublattices $A$ and $B$ which is used as a model system.

Figure 2

Modulus of the nonlinear conductivity as a function of the frequency for three different relaxation rates and three ratios $\gamma /\mathrm{\Gamma }$ when TRS is broken. The gap is at $\mathrm{\Delta }=0.4t$. Above the gap, the conductivity increases with increasing lifetime; below the gap, the conductivity is generically independent of the lifetime and depends only on the ratio $\gamma /\mathrm{\Gamma }$. The exception to this behavior is the fine-tuned value $\gamma /\mathrm{\Gamma }=2$. The gray dashed lines show schematically how taking localization physics into account will modify the result: It leads to an exponential decay of the current below the mobility edge $E_{\mu }$. The parameters used in the Hamiltonian Eq. (4) are $m_1=0.4t$, $m_2=0$, and $\delta =0.2t$.

Figure 3

Modulus of the nonlinear conductivity as a function of the frequency for three different relaxation rates, but now for a system that preserves TRS. Above the gap, the conductivity is independent of the lifetime. It is always independent of the ratio $\gamma /\mathrm{\Gamma }$. The figure does not include any effects of localization. The parameters are chosen such that the gap is again at $\mathrm{\Delta }=0.4t$. They are $m_1=0$, $m_2=0.4t$, and $\delta =0.2t$.