Vacuum space-charge effects in solid-state photoemission

Solid-state photoemission spectroscopy relies to a large part on pulsed photon sources: third-generation synchrotron-radiation sources and ultrafast laser systems in particular. Especially when the photon pulses are intense, Coulombic repulsion between the emitted electrons will be a limiting factor for photoemission experiments aiming at highest energy and angle resolutions. In the present work, the propagation of the photoelectron cloud to the detector is studied with a full $N$-body numerical simulation. The influence of various parameters, in particular number of electrons per pulse, source size, pulse duration, kinetic-energy and emission-angle distributions as well as presence of mirror charges in the sample, is investigated in detail. Previous experimental results obtained with various picosecond and femtosecond light sources are successfully reproduced and the general resolution limits of solid-state photoemission using pulsed photon sources are explored. The results are potentially important for the design and interpretation of photoemission experiments with next-generation light sources, such as free-electron lasers and high-harmonic generation sources.

Figure 1

Schematic of a solid-state photoemission experiment employing a pulsed photon source. A light pulse with energy $\hslash {\omega }_0$ and duration ${\tau }_0$ impinges on a surface area of characteristic width $d_0$ and kicks out photoelectrons. In the numerical simulation, the photoelectrons are divided into cloud electrons (gray dots) and test electrons (black dots), both having specified initial kinetic energy $\left(E_{\text{kin}}\right)$ and emission angle $\left({\vartheta }_c,{\vartheta }_t\right)$ distributions depending on the source parameters and sample properties. For the test electrons, a sharp Gaussian energy distribution (mean position $E_i$ and width $\Delta E_i$) is typically selected. The gray energy areas in the energy spectrum depict simple energy distributions for the cloud electrons. Positive mirror charges in the bulk of the sample (white dots) are generally included in the simulation. The propagation of the electrons toward the detector is influenced by mutual Coulomb interaction. After about 1 ns, when the total Coulomb energy has become negligible, the final energy spectrum of the test electrons is obtained. In general, it will be shifted and broadened with respect to the original spectrum $\left(E_f\ne E_i,\Delta E_f>\Delta E_i\right)$.

Figure 2

Influence of various simulation parameters on a Gaussian test electron spectrum with mean position $E_i=30\text{eV}$ and $\Delta E_i=50\text{meV}$ FWHM [spectrum (a)]. The basic parameters of the numerical simulation, leading to spectrum (b), are number of cloud electrons per pulse $N_c={10}^4$, spot size diameter $d_0=200\mu \text{m}$, pulse duration ${\tau }_0=50\text{fs}$, homogeneous energy and isotropic emission-angle distribution of the cloud electrons $\left(E_c=0–30\text{eV},{\vartheta }_c=0°–90°\right)$, normal test electron emission $\left({\vartheta }_t=0°\right)$, mutual Coulomb interactions between cloud electrons, and interactions with mirror charges included. Changing individual parameters results in the following spectra: (c) $E_i=15\text{eV}$; (d) ${\tau }_0=200\text{ps}$, and $N_c=2\times {10}^4$; (e) ${\tau }_0=50\text{ps}$; (f) ${\vartheta }_t=45°$; (g) no Coulomb interaction between cloud electrons; (h)–(j) Gaussian cloud electron spectra with 50 meV FWHM and mean energies of 1, 30, and 900 eV; (k) no mirror charges; (l) ${\vartheta }_c=0°–45°$; and (m) $d_0=50\mu \text{m}$ and Gaussian cloud electron spectrum with 50 meV FWHM and mean energy 30 eV.

Figure 3

Space-charge effects on the photoemission spectrum of a polycrystalline gold sample under irradiation with ${\tau }_0=60\text{ps},\hslash {\omega }_0=34\text{eV}$ photon pulses. (a) Typical energy distribution curve with the Fermi cutoff at 29.38 eV (taken from Ref. 9). (b) Comparison between simulated and measured (Ref. 9) Fermi edge shifts and broadenings in the range of $\left(0–2000\right)e^{-}$ per pulse. Note the different scales of the vertical axes. Power-law fits to the simulated energy shift and broadening serve as guides for the eyes $\left(\Delta E_{\text{shift}}\propto N_c^{1.11},\Delta E_{\text{broad}}\propto N_c^{0.83}\right)$. The spot size is $0.43\times 0.3{\text{mm}}^2$ and the test electrons are emitted with an angle of $45°$. In the simulation, a cosine distribution of the cloud electron emission angles is used.

Figure 4

Space-charge effect on the two-photon-photoemission spectrum of a Cu(111) surface under irradiation with intense ${\tau }_0=40\text{fs},\hslash {\omega }_0=3.1\text{eV}$ photon pulses. (a) Typical energy distribution curves of the Shockley surface state for emission angles of ${\vartheta }_t=0°$ (solid curve) and ${\vartheta }_t=15°$ (dotted curve), respectively (taken from Ref. 40). In the case of normal emission, the surface-state photoemission peak (light gray) is located at a kinetic energy of 0.912 eV and sits on a characteristic secondary electron background (dark gray). (b) Comparison between the simulated and measured (Ref. 40) energy broadenings of the surface-state peak in the range of $\left(0–80000\right)e^{-}$ per pulse. The simulated broadening shows a power-law behavior $\Delta E_{\text{broad}}\propto N_c^{0.5}$ (solid line). The spot size is $1.15\times 0.9{\text{mm}}^2$ and the test electrons are emitted in the normal direction. In the simulation, an angular distribution of the cloud electrons is assumed that accounts for the parabolic energy-momentum dispersion relation. The inset illustrates the distribution of surface-state (light gray) and secondary electrons (dark gray) in momentum space.

Figure 5

Space-charge effects on a $\text{W}4f$ core-level spectrum under irradiation with ${\tau }_0=40\text{fs},\hslash {\omega }_0=118.5\text{eV}$ photon pulses. (a) Typical energy distribution curve with photoemission peaks at kinetic energies of $\sim 87$ and $\sim 89\text{eV}$ (taken from Ref. 21). (b) Comparison between simulated and measured (Ref. 21) peak shifts and broadenings in the range of $\left(0–200000\right)e^{-}$ per pulse. Note the different scales of the axes for the simulated (bottom axis) and measured data (top axis). Power-law fits to the simulated energy shift and broadening serve as guides for the eyes $\left(\Delta E_{\text{shift}}\propto N_c^{0.98},\Delta E_{\text{broad}}\propto N_c^{0.73}\right)$. The spot size is $0.27\times 0.4{\text{mm}}^2$. In the simulation, a cosine distribution of the cloud electron emission angles is used and the test electron acceptance angle is set to $13°$.

Figure 6

Simulated energy shift (gray data points) and energy broadening (black data points) as functions of the cloud electron density $N_c/d_0$ for a situation similar to core-level photoemission spectroscopy. The error bars result from a systematic variation of the light spot diameter $d_0$ (0.1, 0.4, and 1 mm), mean energy $E_i$ (1, 10, 100, and 1000 eV), and pulse duration ${\tau }_0$ (20 fs, 1 ps, and 50 ps). In the simulation, identical initial Gaussian energy distributions for cloud and test electrons with 100 meV FWHM are used. The cloud electrons are emitted isotropically over the hemisphere; the test electrons are emitted in the normal direction. The solid lines represent power-law fits to the simulated data $\left[\Delta E_{\text{shift}}\propto {\left(\frac{N_c}{d_0}\right)}^{0.98},\Delta E_{\text{broad}}\propto {\left(\frac{N_c}{d_0}\right)}^{0.98}\right]$. The dashed line corresponds to relation (1) (Ref. 37); the dotted lines plotted over the area highlighted in gray are calculated from relation (2) for the different mean energies $E_i$ (Refs. 25, 40). The inset illustrates the space-charge effects on an energy spectrum.

Figure 7

(a) Simulated energy shift (gray data points) and energy broadening (black data points) as functions of the cloud electron density $N_c/d_0$ for a situation corresponding to angle-resolved valence-band spectroscopy. The error bars result from a systematic variation of the light spot diameter $d_0$ (0.04, 0.1, and 0.4 mm), mean energy $E_i$ (1, 10, and 100 eV), and pulse duration ${\tau }_0$ (10 fs and 10 ps). In the simulation, an initial Gaussian test electron energy spectrum with mean energy $E_i$ and 5 meV FWHM is used sitting on top of a rectangular distribution of the cloud electron energies in the interval $0–E_i$ (see inset). The cloud electrons are emitted isotropically over the hemisphere; the test electrons are emitted in the normal direction. The solid lines represent power-law fits to the simulated data $\left[\Delta E_{\text{shift}}\propto {\left(\frac{N_c}{d_0}\right)}^{1.02},\Delta E_{\text{broad}}\propto {\left(\frac{N_c}{d_0}\right)}^{0.98}\right]$. The dashed line corresponds to relation (1) (Ref. 37). (b) Simulated momentum broadening as a function of $N_c/\left(d_0\sqrt{E_i}\right)$. The same parameter sets as in (a) are used, except that the mean energies $E_i$ are now 1, 25, and 100 eV. The inset illustrates the space-charge-induced divergence of the electron distribution in real space.