Free-electron decoherence: Theory and applications

Abstract

Electron microscopy relies on the spatial coherence of electron beams to generate atomic-scale images using interference and diffraction, which can be degraded by inelastic scattering processes that induce decoherence. Here, we present a theoretical study of decoherence arising from the electromagnetic interaction of free electrons with bulk materials and planar surfaces. We show that bulk plasmons dominate decoherence in Al and Au, while electronic excitations above the band gap, supplemented by weaker coupling to phononic and guided modes, are the primary channels in ionic insulators such as LiF. A thermal population of electromagnetic modes leads to a divergence in the energy-loss probability at low frequencies, which in turn produces a pronounced temperature dependence. We show that this effect can be exploited for nanoscale thermometry, predicting that optimized energy-filtered holography enables $\sim0.1\%$ changes in fringe visibility for physically viable temperature variations in metals. Through these results, we establish a unified theoretical framework to describe free-electron decoherence in the bulk and surfaces of arbitrary materials.

Figures (5)

• Figure 1: Free-electron decoherence and interferometry. (a) An incident electron prepared in an arbitrary state corresponding to a density matrix $\rho_i({\bf r},{\bf r}')$ interacts with a specimen and evolves according to Eq. (\ref{['evol']}), acquiring an elastic phase $\chi$ and undergoing a loss of coherence (probability $P$) associated with inelastic processes. (b) We consider an electron prepared in a two-path superposition (interpath distance $d_\perp$, Gaussian path radius $w_0$) such that each path excites the specimen with a different amplitude. Electron paths are then recombined at a plane $z_o$ upon reflection on electrostatic mirrors and propagation over a distance $d_\parallel$. A microscope consisting of two lenses with focal lengths $f_1$ and $f_2$ maps the $z_o$ plane onto an enlarged image at a plane $z_i$, where an electron detector records the magnified interference fringes. The distance $d_1$ from $z_o$ to lens 1 determines the distance $d_2$ from $z_i$ to lens 2 (see Appendix \ref{['apdxprop']}), while the magnification factor is fixed by $g=f_2/f_1$. (c) Interference patterns formed with 200 keV electrons under the configuration of panel (b) as a function of lateral position $x$ for $g=5000$, $d_\parallel/d_\perp=10^4$, $w_0=200$ nm, and two different values of the interpath decoherence probability $P_{AB}$ with $\chi_{AB}=0$.
• Figure 2: Bulk contribution to electron decoherence. (a) An electron prepared in a two-path superposition propagates along a distance $L$ inside the bulk of a material characterized by a nonlocal dielectric tensor $\epsilon(q,\omega)$. The two paths are separated by a distance $d_\perp$. (b) Frequency-resolved decoherence probability $P(\omega)$ for a $200$ keV electron under the configuration of panel (a) when the material is Au, Al, or LiF at room temperature ($T=300$ K). We consider $d_\perp=1$$\mu$m (solid curves) and $d_\perp\to\infty$ (dotted curves, coinciding with the EELS probability for a single-path electron). (c) Comparison between the decoherence probability obtained for Al with inclusion of retardation [taken from (b)] and the nonretarded calculation (i.e., taking $c\to\infty$). (d) Temperature dependence of the frequency-integrated decoherence probability $P$ for $d_\perp=0.3$$\mu$m (solid curves) and $d_\perp=30$$\mu$m (dashed curves). The half-collection angle is set to $\varphi_{\rm out}=10$ mrad throughout this work.
• Figure 3: Surface contribution to electron decoherence: Parallel e-beam configuration. (a) An electron prepared in a two-path superposition propagates along a distance $L$ parallel to the planar surface of a semi-infinite medium of local permittivity $\epsilon(\omega)$. One of the electron paths is inside the material, and the other is in the surrounding vacuum. The path--surface separations are $d_1$ and $d_0$, respectively. (b) Surface contribution to the frequency-resolved decoherence probability $P^{{\rm surf}}(\omega)$ for a $200$ keV electron under the configuration of panel (a) when the material is Au, Al, or LiF at room temperature ($T=300$ K). We set $d_0=d_1=0.5~\mu$m. As $P^{{\rm surf}}$ becomes negative for some frequencies, we plot both $P^{\rm surf}$ (solid curves) and $-P^{{\rm surf}}$ (broken curves). The broken black line indicates a $\omega^{-1}$ slope. (c) Decoherence probability $P$ (frequency-integrated bulk+surface) for LiF at $T = 300$ K as a function of $d_0$ and $d_1$. (d) Partial decoherence probability $P_{\rm cutoff}$ obtained by limiting the spectral integral to $\hbar\omega\le1$ eV under the same conditions as in panel (c).
• Figure 4: Surface contribution to electron decoherence: Perpendicular e-beam configuration.(a) An electron prepared in a superposition of two paths separated by a distance $d_\perp$ traverses a homogeneous film (thickness $L$) made of a material of local permittivity $\epsilon(\omega)$. We consider normal e-beam incidence relative to the film. (b) Frequency-resolved decoherence probability at $T=300$ K for a $200$ keV electron under the configuration of panel (a) when the material is Au, Al, or LiF. We set $d_\perp=1\,\mu$m (solid curves) and $d_\perp\to\infty$ (broken curves, coinciding with the EELS probability). (c) Comparison of retarded and nonretarded regimes for the frequency-resolved decoherence probability in an Al film. (d) Full surface contribution to the frequency-resolved decoherence probability $P^{\rm surf}(\omega)$ (solid curves) compared with the partial contribution to this quantity arising from guided modes (dashed curves). (e) Ratio $P^{\rm surf}/P^{\rm bulk}$ of the frequency-integrated surface to bulk decoherence probabilities as a function of $d_\perp$ for different values of the temperature. (f) Temperature dependence of the decoherence probability for $d_\perp = 10\,{\rm \mu m}$.
• Figure 5: Temperature sensing through decoherence. (a) A normally incident e-beam prepared in a two-path superposition intersects a thin film placed at a temperature $T$. The transmitted e-beam is energy- and space-resolved at a detector, where the two paths are recombined. The spectrally resolved interference is used to measure the temperature of the sample. (b) Energy-filtered temperature sensitivity normalized to the corresponding fringe visibility as a function of energy-loss cutoff for $200$ keV electrons traversing Al, Au, or LiF films of thickness $L=100$ nm at $T=300$ K or $T=1000$ K. We set the path separation to $d_\perp = 50\,{\rm \mu m}$. The dotted line indicates the bulk plasma frequency of Al. (c) Temperature dependence of the energy-filtered temperature sensitivity for a $T$-dependent optimum energy-loss cutoff $\hbar\omega_{\rm opt}(T)$ (see inset). (d) Fringe visibility under the conditions of panel (c). The dotted line indicates $T=300$ K.