Role of impact ionization in the thermalization of photoexcited Mott insulators

We study the influence of the pulse energy and fluence on the thermalization of photodoped Mott insulators. If the Mott gap is smaller than the width of the Hubbard bands, the kinetic energy of individual carriers can be large enough to produce additional doublon-hole pairs via a process analogous to impact ionization. The thermalization dynamics, which involves an adjustment of the doublon and hole densities, thus changes as a function of the energy of the photo-doped carriers and exhibits two time scales: a fast relaxation related to the impact ionization of high-energy carriers and a slower time scale associated with higher-order scattering processes. The slow dynamics depends more strongly on the gap size and the photodoping concentration.

Figure 1

Equilibrium spectral functions of the Hubbard model for $\beta =5$ and indicated values of $U$ as calculated by DMFT (with NCA impurity solver).

Figure 2

Relaxation after pulse excitations at different frequencies at an initial inverse temperature $\beta =5$. The amplitude of the pulses is adjusted such that the number of photodoped doublons at $t=12$ (shortly after the pulse) is $0.01$. Top: Time evolution of the normalized doublon density and expected thermal values (horizontal lines) for $U=2.5$ and $3.5$. Bottom: Relaxation times ($c$) obtained by fitting $D\left(t\right)/D(t=12)$ to the function $a+bexp(-t/c)$ in the range $t\in [30,60]$. The extrapolated long time values ($a$) from this fit are indicated by arrows in the upper panels.

Figure 3

Time-resolved photoemission spectra for pulse amplitude 2 and initial inverse temperature $\beta =5$. Left: Results for $U=3$, $\Omega =3.5\pi /2$; initial photodoping concentration $D(t=12)=0.0056$. Right: Results for $U=3.5$, $\Omega =4\pi /2$; initial photodoping concentration $D(t=12)=0.0021$. Colored curves in the upper panels show the nonequilibrium photoemission spectrum $I(\omega ,t)$ for indicated values of $t$, while dashed black curves plot that of the initial equilibrium state. Solid arrows sketch the energy transfers associated with an impact ionization process: left-pointing arrow—kinetic energy loss of a high-energy doublon; right-pointing arrow—excitation of an electron across the gap. Bottom: Time-dependent change in the photoemission spectrum. The (red) arrows indicate the energies ${\omega }_{\mathrm{gain}}=0.85$ and ${\omega }_{\mathrm{loss}}=2.55$ ($U=3$; left) and ${\omega }_{\mathrm{gain}}=1.05$ and ${\omega }_{\mathrm{loss}}=3.1$ ($U=3.5$; right). Shaded (red) areas correspond to the increase (decrease) in low-energy (high-energy) doublons from $t=24$ to $t=42$. For $U=3$, the increase in the number of low-energy doublons is about $2.7$ times as large as the decrease in the number of high-energy doublons, while for $U=3.5$, the corresponding factor is about $2.3$.

Figure 4

Top: Doublon production rate plotted as a function of $D_{\mathrm{th}}-D\left(t\right)$ for different pulse energies and indicated values of $U$. Fits of model (7) to the curves corresponding to the highest pulse energy are shown by dashed lines. Bottom: Time evolution of the normalized doublon population as predicted from the fit to model (7) for $U=3.5$, $\Omega =3.5\frac{\pi }{2}$ ($t_s=15$).

Figure 5

Time evolution of the doublon concentration for $U=3$, $\Omega =3.5\pi /2$, and different pulse amplitudes. Top: Normalized doublon population. Bottom: Relaxation of the doublon concentration to the thermal value. The curves for amplitudes $<6$ are multiplied by an arbitrary factor, to enable a better comparison of the long-time behavior. Dashed lines are fits to model (7) in the time interval $t\in [15,60]$.

Figure 6

Difference between the time-resolved photoemission spectra measured at $t=36$ and $t=24$ for $\Omega =4\pi /2$ and the indicated values of the pulse amplitude. Top: $U=3.5$. Bottom: $U=4$. For better comparison, curves have been normalized such that the maximum difference is 1. For $U=3.5$, the doublon concentration at $t=12$ is $D=0.0065$, 0.0021, and 0.00015 for amplitude 5, 2, and $0.5$, respectively. For $U=4$, the corresponding numbers are $D=0.014$, $0.0043$, and $0.00030$.