Impact ionization processes in the steady state of a driven Mott-insulating layer coupled to metallic leads

We study a simple model of photovoltaic energy harvesting across a Mott-insulating gap consisting of a correlated layer connected to two metallic leads held at different chemical potentials. We address, in particular, the issue of impact ionization, whereby a particle photoexcited to the high-energy part of the upper Hubbard band uses its extra energy to produce a second particle-hole excitation. We find a drastic increase of the photocurrent upon entering the frequency regime where impact ionization is possible. At large values of the Mott gap, where impact ionization is energetically not allowed, we observe a suppression of the current and a piling up of charge in the high-energy part of the upper Hubbard band. Our study is based on a Floquet dynamical mean-field theory treatment of the steady state with the so-called auxiliary master equation approach as impurity solver. We verify that an additional approximation, taking the self-energy diagonal in the Floquet indices, is appropriate for the parameter range we are considering.

Figure 1

A sketch of the energy distribution of the system studied and an illustration of the dominant steady-state photoinduced processes. The dark-blue (light-blue) region describes the full (empty) lead, while the lower and upper Hubbard bands of the central layer are marked in dark- and light-red, respectively. Electromagnetic radiation with energy $\mathrm{\Omega }$ (yellow) initially produces a particle-hole excitation (red wiggly line). For ${\mathrm{\Delta }}_{ss}<\mathrm{\Omega }<{\mathrm{\Delta }}_{ss}+2D$, panel (a) shows an electron coming from the left (full) lead which is photoexcited into the upper Hubbard band at energies such that it can directly escape into the right (empty) lead without further scattering processes. For $\mathrm{\Omega }>2{\mathrm{\Delta }}_{ss}$, panel (b), we have impact ionization: First, an electron-hole pair is created with a high-energy electron via photoabsorption ($e_1^{-},h_1^{+}$ in the figure). Since the energy of the photoexcited electron is incompatible with states in the right lead, it cannot escape the correlated region. Thus, it can only contribute to the current if it can get rid of its excess energy. The simplest, and therefore dominant process is impact ionization, where $e_1^{-}$ scatters with a second electron from the lower band, thereby exciting it over the steady-state gap and creating a second e-h pair $e_2^{-},h_2^{+}$. In the final state, both electrons and holes can now escape into the leads and contribute to the current. Notice that in the present steady-state situation only processes that eventually recover the initial configuration are allowed.

Figure 2

Sketch of the considered lattice system. The leads, Hubbard layer, monochromatic light, and the coupling between the lead and layer are illustrated in blue, red, yellow, and green, respectively.

Figure 3

(a) Time-averaged current density and (b) double occupancy as a function of driving frequency for a situation with a gap larger than the bandwidth so that impact ionization is impossible. Parameters as in $R_a$ in Table 1.

Figure 4

(a) Time-averaged current density (in units of $j_0=e\hslash /t_c{a}^2$) and (b) double occupancy as a function of driving frequency for parameters compatible with impact ionization, $R_b$ in Table 1. The current and the double occupancy show the same behavior in the region of the main peak as expected for impact ionization processes. (c) Time-averaged spectral function $A\left(\omega \right)$ and corresponding filling $A\left(\omega \right)f\left(\omega \right)$ [here, $f\left(\omega \right)$ denotes the nonequilibrium distribution function] for representative driving frequencies for direct excitations ($\mathrm{\Omega }=6$) and impact ionization ($\mathrm{\Omega }=12$).

Figure 5

(a) Time-averaged spectral function and filling at $\mathrm{\Omega }=37$. (b) Time-averaged current density as a function of driving frequency for the two solutions discussed in Sec. 4b3. Parameter set $R_a$ as in Fig. 3.

Figure 6

Test of the FDSA (see Sec. 3d1). Time-averaged current as a function of the effective coupling constant $\alpha \equiv t_c{E}_0/{\mathrm{\Omega }}^2$ for different driving frequencies obtained with IPT. The lines labeled with “FDSA” correspond to data computed with the FDSA, while the label “full” refers to the solution with the full Floquet impurity problem. For $\alpha \lesssim 1/2$, the FDSA approximation is very reliable.

Figure 7

Test of the FDSA introduced in Sec. 3d1. (a) Time-averaged spectral function for different electric field strengths at constant frequency $\mathrm{\Omega }=5$ obtained with IPT. (b) The lower plot shows a zoom onto the upper Hubbard band as indicated by the box in the upper plot. Numerical parameters are the same as in Fig. 6.