Vacuum space-charge effects in nano-ARPES

Angle-resolved photoemission spectroscopy (ARPES) provides a powerful tool for probing the spatially averaged electronic structure of condensed matter at high energy and momentum resolution. Current efforts to add spatial resolution by reducing the focal spot size of the incident radiation to the nanometer regime may result in an effective loss of resolution due to the increased Coulomb interaction between the photoelectrons emitted into vacuum. Here, the potential limitations of “nano-ARPES” at pulsed light sources are determined by molecular-dynamics simulations over a wide range of the most relevant parameters: the pulse duration and focal spot size of the incident light as well as the number of emitted electrons per pulse and their mean kinetic energy. The dependence of the space-charge-induced spectral shift and broadening on these parameters is found to be describable by empirical formulas. The simulation results particularly reveal a saturation of vacuum space-charge effects in the limit of small focal spot sizes and suggest that “nano-ARPES” at an effective energy resolution of 5 meV is possible and practical, employing typical synchrotron radiation in the extreme ultraviolet spectral range.

Figure 1

Schematic of photoemission from a solid surface using monochromatic light pulses with an energy $\hslash {\omega }_0$, a duration ${\tau }_0$, and a focal width $d_0$. With decreasing ratio $d_0/{\tau }_0$, the shape of the emitted electron cloud changes from a quasi-2D disk via a 3D sphere to a quasi-1D chain. Within the numerical simulation, the emitted electrons are divided into cloud electrons (gray dots) and test electrons (black dots), both having specified initial kinetic energy and emission angle (${\vartheta }_{c,t}$) distributions. In all simulations presented in this work, the test electrons start with a fixed energy $E_i$, while the cloud electrons are uniformly distributed over the energy interval $[0,E_i]$ (light gray filling). After each simulation run, the space-charge-induced spectral shift and broadening are obtained by fitting a Gaussian to the final test electron energy distribution (dark gray filling). The shift and broadening are defined as the Gaussian mean energy shift $E_f-E_i$ and the Gaussian full width at half maximum $\Delta E_f$, respectively.

Figure 2

Simulated space-charge-induced energy shift and broadening as a function of the spot diameter $d_0$ for a typical ARPES experiment. The error bars represent the standard deviations of repeated simulations. Data for pulse durations of 20 and 100 ps are shown. In the simulations, the test electrons have an initial kinetic energy $E_i=50$ eV and sit on top of a uniform distribution of cloud electron energies in the interval $[0,E_i]$. The number of cloud electrons is $N_c=1000$. The cloud electrons are emitted isotropically over the hemisphere, whereas the test electrons are emitted in the surface normal direction. The dashed horizontal and vertical lines mark plateau energies ($\Delta E_{\max }$) in the quasi-1D regime and the half-way spot diameter ($d_{\mathrm{half}}$) between the quasi-1D and quasi-2D regimes, respectively.

Figure 3

Vacuum space-charge effects in the small spot diameter limit ($d_0=10$ nm). (a)–(c) Maximum energy shift and broadening $\Delta E_{\max }$ as a function (a) of the pulse duration ${\tau }_0$ ($E_i=50$ eV, $N_c=1000$), (b) of the kinetic energy $E_i$ (${\tau }_0=50$ ps, $N_c=1000$), and (c) of the number of electrons per pulse $N_c$ (${\tau }_0=50$ ps, $E_i=50$ eV). (d), (e) Half-way spot diameter $d_{\mathrm{half}}$ as a function (d) of ${\tau }_0$ ($E_i=50$ eV) and (e) of $E_i$ (${\tau }_0=50$ ps) (in both cases: $N_c=5\sqrt{E_i}{\tau }_0$ eV${}^{-1/2}$ps${}^{-1}$). Solid lines represent power-law fits as indicated.

Figure 4

Simulated space-charge-induced energy shift and broadening (filled and open circles) as a function of the spot diameter $d_0$ in comparison to values calculated by formula (2) (solid lines). The circles and lines represent average values for the following parameter ranges: kinetic energy $E_i\in [10,200]$ eV, pulse duration ${\tau }_0\in [10,100]$ ps, and number of electrons per pulse $N_c=5\sqrt{E_i}{\tau }_0$ eV${}^{-1/2}$ps${}^{-1}$. The gray shaded areas indicate the maximum variation of the numerically simulated results.