Pump-probe Auger-electron spectroscopy of Mott insulators

In high-resolution core-valence-valence (CVV) Auger electron spectroscopy from the surface of a solid at thermal equilibrium, the main correlation satellite, visible in the case of strong valence-electron correlations, corresponds to a bound state of the two holes in the final state of the CVV Auger process. We discuss the physical significance of this satellite in nonequilibrium pump-probe Auger spectroscopy by numerical analysis of a single-band Hubbard-type model system, including core states and a continuum of high-energy scattering states. It turns out that the spectrum of the photo-doped system, due to the increased double occupancy, shares features with the equilibrium spectrum at higher fillings. The pumping of doublons can be watched when working with overlapping pulses at short $\mathrm{\Delta }t$. For larger pump-probe delays $\mathrm{\Delta }t$ and on the typical femtosecond timescale for electronic relaxation processes, spectra are hardly $\mathrm{\Delta }t$ dependent, reflecting the high stability of bound two-hole states for strong Hubbard $U$. We argue that taking into account the spatial expansion of single-particle orbitals when these are doubly occupied, as described by the dynamical Hubbard model, produces an oscillation of the barycenter of the satellite as a function of $\mathrm{\Delta }t$. Pump-probe Auger-electron spectroscopy is thus highly sensitive to dynamical screening of the Coulomb interaction.

Figure 1

Sketch of pump-probe Auger electron spectroscopy from a Mott insulator with lower (LHB) and upper Hubbard bands (UHB) separated by the Hubbard interaction $U$. An optical pump pulse drives the system to a nonequilibrium state with an enhanced fraction of double occupancies. The system evolves in time. After time $\mathrm{\Delta }t$, a core hole is created by an x-ray probe pulse. The spectrum of the subsequent Auger decay is characteristic for the local correlated electronic structure at time $\mathrm{\Delta }t$.

Figure 2

Main figure: Total AES cross section as a function of the cutoff time $t_{\text{max}}$ as obtained numerically (dots and dashed lines) compared with the analytical formula (solid lines) given by Eq. (18), valid for the full band $n=2$ and all $U_A$ as well as for small $U_A$ and all band fillings $n$. Additional parameters: $L=10, U=6, U_{\text{cv}}=0, {\tau }_{\text{prb}}=0.1, L_{\text{ch}}=200$ (to describe the continumm, see Appendix pp1). Inset: Convergence of the spectrum for $n=2$ and $U_A=0.1$ as a function of $t_{\text{max}}$ and eventual appearance of finite-size effects (see green line for $t_{\text{max}}=10$). B: Bandlike part. D: Doublon satellite (see text). The energy (and with $\hslash \equiv 1$ also the timescale) is fixed by $T=1$.

Figure 3

Core-hole lifetime ${\tau }_c=1/2\mathrm{\Gamma }$ as a function of $U_A$ extracted from the exponential fit Eq. (23) for $U_A>0.01$ and from the linear fit Eq. (24) for $U_A<0.01$. The fits are done in the range $t_{\text{max}}=1,2,...12$ (see Fig. 2). Note that the lifetime for $n=2$ does not depend on $U_{\text{cv}}$. Other parameters as in Fig. 2.

Figure 4

Total double occupancy after a pump [see Eq. (4)] of duration ${\tau }_{\text{pmp}}=1$ for various values of ${\mathrm{\Omega }}_{\text{pmp}}$ as indicated. Further parameters: $L=10, U=6, {\mathcal{E}}_{\text{pmp}}^{\left(0\right)}=8$, periodic boundary conditions.

Figure 5

Pump-probe Auger spectra of the half-filled Hubbard model for various delay times $\mathrm{\Delta }t$ at $U=6$. Calculations for $L=10$, periodic boundary conditions. Comparison with the equilibrium case for the full band $n=2$ (solid line) and the half-filled band $n=1$ (dashed line) for the same parameters. The upper inset shows the vector potential of the pump $A_{\text{pmp}}\left(t\right)$. The probe starts at $t=0$, i.e., $t_{\mathrm{prb}}=0$. The lower inset is a magnification of the quadruplon peak at small kinetic energies (Q). D denotes the doublon satellite and B is the bandlike part; see Ref. [12] and discussion in the text.

Figure 6

Pump-probe Auger spectra of the half-filled dynamic Hubbard model for various delay times $\mathrm{\Delta }t$ at $U=6$. Calculations for the same parameters as in Fig. 5, for coupling strength $k=2$ and for ${\omega }_0=0.1$ (adiabatic case). Upper inset: The effective Coulomb interaction as a function of time, see Eq. (29). Note: ${\tau }_{\mathrm{prb}}=0$. Lower inset: The barycenter of the spectrum, see Eq. (30), compared with the time-averaged $U_{i,\text{eff}}\left(t\right)$, see Eq. (31).

Figure 7

The same as Fig. 6, but for the antiadiabatic case with ${\omega }_0=1$.

Figure 8

Equilibrium phase diagram of the dynamic Hubbard model. The total double occupancy Eq. (25), see color code on the right, is displayed as a function of the electron-oscillator coupling strength $k$ and the Hubbard interaction $U$. Results for half filling and for $L=10$ sites (periodic boundary conditions). In the CDW phase, the local double occupancy $〈n_{i\uparrow }{n}_{i\downarrow }〉$ alternates between 0 and 1, leading to $d_{\text{tot}}=0.5$.