Distinction of Electron Dispersion in Time-Resolved Photoemission Spectroscopy

While recent experiments provided compelling evidence for an intricate dependence of attosecond photoemission-time delays on the solid’s electronic band structure, the extent to which electronic transport and dispersion in solids can be imaged in time-resolved photoelectron (PE) spectra remains poorly understood. Emphasizing the distinction between photoemission time delays measured with two-photon, two-color interferometric spectroscopy, and transport times, we demonstrate how the effect of energy dispersion in the solid on photoemission delays can, in principle, be observed in interferometric photoemission. We reveal analytically a scaling relation between the PE transport time in the solid and the observable photoemission delay and confirm this relation in numerical simulations for a model system. We trace photoemission delays to the phase difference the PE accumulates inside the solid and, in particular, predict negative photoemission delays. Based on these findings, we suggest a novel time-domain interferometric solid-state energy-momentum-dispersion imaging method.

Figure 1

Scaling factor $\alpha (R_{2q})$ in Eq. (1) as a function of the effective mass ratio $R_{2q}$ of PEs released by successive odd HHs of order $2q\pm 1$ of the XUV APT. Approximate results according to Eq. (5) and exact results for final PE energies ${ϵ}_{2q+1}=5{\omega }_{\mathrm{IR}}$ (diamonds and solid blue solid line), $10{\omega }_{\mathrm{IR}}$ (squares and dashed blue line), and $20{\omega }_{\mathrm{IR}}$ (circles and dotted blue line). The vertical dotted line indicates equal effective masses with an enlarged view in the inset.

Figure 2

Model system used to derive the scaling relation in Eqs. (1), (3), and (4). PE dispersion is tunable by adjusting the well depth (barrier height) $V_0$. Case A depicts free and case B dispersive propagation with effective mass $m_{e,2q+1}$ in a slab of width $d_W$ [Eq. (6)].

Figure 3

(a) Simulated photoemission-time delays $t_{pe,50}$ for the model system in Fig. 2 as a function of the transport-region length $d_W$. The transport region is modeled by a potential well of depth $V_0=-5\mathrm{eV}$ (open circles), free propagation ($V_0=0$, open squares), or a barrier of height $V_0=5\mathrm{eV}$ (open diamonds). Corresponding analytical results from Eqs. (1), (3), and (4) are given by the blue dash-dotted, magenta dashed, and red solid lines. (b)–(d) Quantum-mechanically simulated PE spectra on a logarithmic yield scale versus the APT-IR pulse delay $\tau$ for $d_W=20$ and (b) $V_0=5\mathrm{eV}$, (c) 0, and (d) $-5\mathrm{eV}$. Vertical red lines in (b)–(d) indicate photoemission delay changes relative to $\tau =0$.

Figure 4

Photoemission time delays $\delta {\tau }_{a_s}$ (red circles), for $\mathrm{Mg}\text{−}2p$ core-level PEs traveling over one lattice constant $a_s=4.923$ inside solid Mg, extracted from the calculated total $\tau$-dependent SB-50 yield of PEs from 15 lattice sites as a function of the effective mass ratio $R_{50}=m_{e,49}/m_{e,51}$, where $m_{e,51}\equiv 1$. The blue line shows the analytical prediction by Eq. (8).