Energy losses and transition radiation in graphene traversed by a fast charged particle under oblique incidence

We perform fully relativistic calculations of the energy loss channels for a charged particle traversing a single layer of graphene under oblique incidence in a setting pertinent to a scanning transmission electron microscope (STEM), where we distinguish between the energy deposited in graphene in the form of electronic excitations (Ohmic loss) and the energy emitted in the far field in the form of transition radiation (TR). Our formulation of the problem uses a definition of two in-plane, dielectric functions of graphene, which describe the longitudinal and transverse excitation processes that contribute separately to those two energy loss channels. Using several models for the electric conductivity of graphene as the input in those dielectric functions enables us to discuss the effects of oblique incidence on several processes in a broad range of frequencies, from the terahertz (THz) to the ultraviolet (UV). In particular, at the THz frequencies, we demonstrate that the nonlocal effect in the graphene's conductivity is not important in the retarded regime, and we show that the longitudinal and transverse contributions to the emitted TR spectra exhibit strongly anisotropic angular patterns that are readily distinguishable in a cathodoluminescence measurement in a STEM. Moreover, we explore the possibility of exciting the so-called transverse mode in the optical response of graphene at the mid-infrared (MIR) range of frequencies by means of a fast charged particle under oblique incidence. Finally, we demonstrate that, aside from the usual high-energy peaks in the longitudinal contribution to the Ohmic energy loss in the MIR to the UV frequency range, there may arise strongly directional features in the in-plane distribution of the transverse contribution to the Ohmic energy loss for an oblique trajectory, which could be possibly observed via momentum- and angle-resolved electron energy loss spectroscopy of graphene in STEM.

Figure 1

Geometry of the problem consisting of a 2D graphene sheet placed in vacuum and traversed by a fast charged particle under oblique incidence.

Figure 2

Normalized integrated energy loss density of the external charged particle ${\overline{P}}_{\mathrm{ext}}=P_{\mathrm{ext}}/P_c$ with $P_c=4/\left(\pi {\mathcal{E}}_F\right)$ where ${\mathcal{E}}_F$ is the Fermi energy of graphene, for two angles of incidence: (a) ${\theta }_0=0$ and (b) ${\theta }_0={60}^{\circ }$, and for several external particle speeds $\beta$, versus the reduced frequency $\overline{\omega }=\omega /{\omega }_c$. Also shown are, in each panel, two separate contributions of longitudinal (dashed curves) and transverse (dotted curves) dielectric functions to the external loss. Nonlocal effects of the graphene's conductivity model on the loss spectra is depicted by the comparison of the Lovat's (thick lines) and Drude (thin lines) models. The reduced damping rate is fixed at $\overline{\gamma }=\gamma /{\omega }_c=0.05$.

Figure 3

Normalized integrated probability densities for (a) Ohmic energy loss ${\overline{P}}_{\mathrm{Ohm}}\left(\overline{\omega }\right)=P_{\mathrm{Ohm}}/P_c$ and (b) radiative energy loss ${\overline{P}}_{\mathrm{rad}}\left(\overline{\omega }\right)=P_{\mathrm{rad}}/P_c$ for the charged particle speed of $\beta =0.01$ (the nonretarded case) and for several angles of incidence. The differences between the Lovat's (thick lines) and Drude (thin lines) models, as well as the relative weights of the longitudinal (dashed lines) and transverse (dotted lines) contributions to the Ohmic and radiative integrated densities in a nonretarded limit, are also shown. The reduced damping rate is fixed at $\overline{\gamma }=0.05$.

Figure 4

Normalized integrated probability densities for (a) Ohmic energy loss ${\overline{P}}_{\mathrm{Ohm}}\left(\overline{\omega }\right)$, (b) radiative energy loss ${\overline{P}}_{\mathrm{rad}}\left(\overline{\omega }\right)$, and (c) external energy loss ${\overline{P}}_{\mathrm{ext}}\left(\overline{\omega }\right)={\overline{P}}_{\mathrm{Ohm}}\left(\overline{\omega }\right)+{\overline{P}}_{\mathrm{rad}}\left(\overline{\omega }\right)$ for the charged particle speed of $\beta =0.5$ and for several angles of incidence. The differences between the Lovat's (thick lines) and Drude (thin lines) models, as well as the decomposition of the longitudinal (dashed lines) and transverse (dotted lines) contributions to those integrated densities are shown. Also included are, in (a), calculations for the Ohmic energy loss density at high frequencies using the approximation for ${\overline{P}}_{\left|\right|}$ given in Eq. (42) for ${\theta }_0\gtrsim {60}^{\circ }$ (dashed–double-dotted curves). The reduced damping rate is fixed at $\overline{\gamma }=0.05$.

Figure 5

Angular distribution of the spectral density for TR in reduced units $\overline{\mathcal{S}}(\theta ,\varphi ,\overline{\omega })=\mathcal{S}/{\mathcal{S}}_c$ with ${\mathcal{S}}_c={\left(Ze\right)}^2/c$ for the external particle with the speed $\beta =0.5$ and the angle of incidence of ${\theta }_0={60}^{\circ }$, and for three different frequencies $\overline{\omega }=0.5$, 1, and 5. In the left, middle, and right columns, we show the longitudinal contribution, transverse contribution, and the total angular radiative spectra, respectively. The reduced damping rate is fixed at $\overline{\gamma }=0.05$ and the Lovat's model has been used for the conductivity tensor.

Figure 6

The cross section of the angular distribution of the spectral density for TR in the plane of incidence $\overline{\mathcal{S}}(\theta ,\varphi =0,\overline{\omega })$ for an external particle with the speed $\beta =0.5$ and three angles of incidence (a) ${\theta }_0={15}^{\circ }$, (b) ${\theta }_0={45}^{\circ }$, and (c) ${\theta }_0={85}^{\circ }$ for several frequencies. These panels represent the total angular distribution, which is identical to the longitudinal contribution. The spectra are calculated employing Lovat's conductivity model for graphene where the reduced damping rate is fixed at $\overline{\gamma }=0.05$.

Figure 7

Integrated probability density for the Ohmic energy loss $P_{\mathrm{Ohm}}$ of a graphene sheet described by the optical conductivity model, along with its decomposition into the longitudinal (dashed lines) and transverse (dotted lines) contributions, for a charged particle at the speed $\beta =0.5$. (a) Represents the results for different angles of incidence with fixed $\overline{\gamma }=0.05$ and (b) shows the investigation of the effects of varying the damping rate $\overline{\gamma }$ which includes the probability density for the excitations of the LDPP and TDPP modes. The inset of each panel depicts a closer focus at the range of $228\lesssim \overline{\omega }\lesssim 274$ where the excitation of the transverse mode occurs.

Figure 8

(a) Integrated probability density of the external energy loss $P_{\mathrm{ext}}\left(\omega \right)$ as a function of the energy $\omega$ (with $\hslash =1$) lost by an incident electron with the speed $\beta =0.5$ for a broad range of incident angles. Shown in the inset of (a) is the corresponding energy loss due to the emitted TR, $P_{\mathrm{rad}}\left(\omega \right)$. Since the magnitude of the radiation energy loss is very much smaller than $P_{\mathrm{ext}}\left(\omega \right)$, practically $P_{\mathrm{ext}}\left(\omega \right)\approx P_{\mathrm{Ohm}}\left(\omega \right)$. (b) Integrated probability density of the energy loss of the external particle $P_{\mathrm{ext}}$ (solid lines) as well as the longitudinal (dashed lines) and transverse (dotted lines) contributions to that density for several incident angles. For both panels, the eHD conductivity model for graphene, with the parameters fixed at $n_{\pi }^0=38{\mathrm{nm}}^{-2}$, $n_{\sigma }^0=115{\mathrm{nm}}^{-2}$, ${\omega }_{\pi r}=4.19$ eV, ${\omega }_{\sigma r}=14.15$ eV, ${\gamma }_{\pi }=2.04$ eV, ${\gamma }_{\sigma }=2.178$ eV, and ${\omega }_{*}=3.54\mathrm{eV}$, is used.

Figure 9

A decomposition of (a) the Ohmic and (b) the radiative energy losses into their respective longitudinal (dashed lines) and transverse (dotted lines) contributions for the incident angle ${\theta }_0={45}^{\circ }$. For both panels, the eHD conductivity model for graphene, with the parameters fixed at $n_{\pi }^0=38{\mathrm{nm}}^{-2}$, $n_{\sigma }^0=115{\mathrm{nm}}^{-2}$, ${\omega }_{\pi r}=4.19$ eV, ${\omega }_{\sigma r}=14.15$ eV, ${\gamma }_{\pi }=2.04$ eV, ${\gamma }_{\sigma }=2.178$ eV, and ${\omega }_{*}=3.54\mathrm{eV}$, is used.

Figure 10

Joint probability densities for the longitudinal (left column) and transverse (middle column) contributions to the Ohmic energy loss, along with their sum $F_{\mathrm{Ohm}}(k,\varphi ,\omega )$ (right column), which practically represents the total energy loss density of the external charged particle, given the smallness of the radiative energy loss. Results are shown as functions of the wave number $k$ and the energy loss $\omega$, in the direction of $\varphi =\pi /4$ with respect to the $x$ axis, i.e., the (projected) direction of motion for an external particle having the speed $\beta =0.5$ and for three angles of incidence relative to the $z$ axis: ${\theta }_0=0$ (top row), ${\theta }_0={45}^{\circ }$ (middle row), and ${\theta }_0={75}^{\circ }$ (bottom row). For all panels, the eHD conductivity model for graphene, with the parameters fixed at $n_{\pi }^0=38{\mathrm{nm}}^{-2}$, $n_{\sigma }^0=115{\mathrm{nm}}^{-2}$, ${\omega }_{\pi r}=4.19$ eV, ${\omega }_{\sigma r}=14.15$ eV, ${\gamma }_{\pi }=2.04$ eV, ${\gamma }_{\sigma }=2.178$ eV, and ${\omega }_{*}=3.54\mathrm{eV}$, is used. Also shown are the nonrelativistic dispersion relations of the $\pi$ (solid orange line) and $\pi +\sigma$ (dash-dotted purple line) plasmons for single-layer graphene [51], and the light line by the dashed red line. Here, $k_{\max }=0.1{\mathrm{nm}}^{-1}$.