Electron-beam broadening in electron microscopy by solving the electron transport equation

Scanning transmission electron microscopy (STEM) and scanning electron microscopy (SEM) are prominent techniques for the structural characterization of materials. STEM in particular provides high spatial resolution down to the sub-ångström range. The spatial resolution in STEM and SEM is ultimately limited by the electron-beam diameter provided by the microscope's electron optical system. However, the resolution is frequently degraded by the interaction between electron and matter leading to beam broadening, which depends on the thickness of the analyzed sample. Numerous models are available to calculate beam broadening. However, most of them neglect the energy loss of the electrons and large-angle scattering. These restrictions severely limit the applicability of the approaches for large sample thicknesses in STEM and SEM. In this work, we address beam broadening in a more general way. We numerically solve the electron transport equation without any simplifications, and take into account energy loss along the electron path. For this purpose, we developed the software package CeTE (Computation of electron Transport Equation). We determine beam broadening, energy deposition, and the interaction volume of the scattered electrons in homogeneous matter. The calculated spatial and angular distributions of electrons are not limited to forward scattering and small sample thicknesses. We focus on low electron energies of 30 keV and below, where beam broadening is particularly pronounced. These electron energies are typical for SEM and STEM in scanning electron microscopes.

Figure 1

Beam radii in silicon as a function of sample thickness calculated by the indicated analytical models for different primary electron energies. Energy loss and large-angle scattering are neglected.

Figure 2

Spatial distribution of 20 keV electrons after traveling different path lengths $s$ in silicon. Ellipses centered at the mean penetration depth $\langle z\rangle$ (indicated by dashed-dotted horizontal lines) for $\mathrm{s}=300$, 600, and 900 nm show the distribution of the electrons less than one standard deviation away from the mean displacement.

Figure 3

(a) Mean penetration depth in silicon for electrons of different primary energies as a function of the traveled path length. The dashed-dotted line indicates a depth equal to the path length. (b) Mean cosine of the scattering angle as a function of the traveled path length in silicon for electrons of different primary energies.

Figure 4

Beam radius as a function of penetration depth $\langle z\rangle$ for the Reimer (dashed), Goldstein (dashed-dotted), and MM (continuous lines) formalisms for primary electron energies of 10, 20, and 30 keV indicated by different colors. (a) Beam radius in Si for the small scattering-angle approximation and neglecting energy loss. (b)–(d) Beam radius in Si, Cu, and Pt, respectively, taking into account energy loss along the electron path and without limitation to small scattering angles for the MM.

Figure 5

Time evolution of beam spreading in Cu for (a) 10 keV and (b) 30 keV. The ellipses drawn with centers at different mean penetration depths contain 68% of the scattered electrons.

Figure 6

Energy dissipation in copper as a function of (a) traveled path length $s$ and (b) depth $\langle z\rangle +{\sigma }_z$ below the sample surface for primary electron energies of 10 and 20 keV.

Figure 7

Angular distribution of 20 keV electrons after traveling a path length $s$ of (a) 300 nm, (b) 500 nm, and (c) 700 nm in silicon obtained by the exact solution of the transport equation (continuous line), Cosslett's Gaussian distribution (dotted line), and the solution with the Fokker-Planck approximation (dashed line).