Efficient plasmonic emission by the quantum Čerenkov effect from hot carriers in graphene

Abstract

Graphene plasmons have been found to be an exciting plasmonic platform, thanks to their high field confinement and low phase velocity, motivating contemporary research to revisit established concepts in light-matter interaction. In a conceptual breakthrough over 80 years old, Čerenkov showed how charged particles emit shockwaves of light when moving faster than the phase velocity of light in a medium. To modern eyes, the Čerenkov effect offers a direct and ultrafast energy conversion scheme from charge particles to photons. The requirement for relativistic particles, however, makes Čerenkov emission inaccessible to most nanoscale electronic and photonic devices. Here we show that graphene plasmons provide the means to overcome this limitation through their low phase velocity and high field confinement. The interaction between the charge carriers flowing inside graphene and the plasmons enables a highly efficient two-dimensional Čerenkov emission, giving a versatile, tunable and ultrafast conversion mechanism from electrical signal to plasmonic excitation.

Figure 1. Illustration of the plasmon emission from charge carriers in graphene via a 2D Čerenkov process.

(a) GP emission in graphene from a hot carrier flowing inside it. The hot carrier (white arrow marking a transparent-blue arch shape) excites GPs that propagate sideways (glowing red-blue bars) along the graphene surface (plotted on the yellow–orange–red substrate). The Čerenkov angle into which the GPs are emitted is denoted by θ (defined between the wiggling red arrows and the z axis, which is the direction of motion of the hot carrier). (b) A diagram describing the GP emission process from a hot carrier in graphene.

Figure 2. GP emission from hot carriers.

(a) Illustration of the possible transitions. The hot carrier (green dot) has a range of potential transitions (red arrows) with distinct final states (green curves and circles), emitting plasmons that satisfy conservation of momentum and energy (corresponding to the height and angle of the red arrows). This way the cone geometry correlates the GP frequency and angle. The projection of these arrows to a 2D plane predicts the in-plane angle θ of the plasmonic emission, matching the (b) map of GP emission rate as a function of frequency and angle, equation (6). We find most of the GP emission around the dashed blue curves that are exactly found by the Čerenkov angle equation (4). (c) Spectrum of the ČE GP emission process, with the red regime marking the area of high losses (as in ref. 19). Black, emission spectrum with GP losses, equation (6). Blue, lossless emission approximation, equation (5). The thick orange line marks the spectral cutoff due to the Fermi sea, beyond which all states are occupied (therefore, ℏω<E_i+E_F). (d) Explaining the GP emission with the quantum ČE. The GP phase velocity is plotted as a red curve, with its thickness presenting the GP loss. The blue-shaded regime shows the range of allowed velocities according to the quantum ČE. We find enhanced GP emission in the frequencies for which the red curve crosses the blue regime, either directly or due to the curve thickness. The vertical dotted red line that crosses both c and d divides between interband to intraband transitions (exactly at ℏω=E_i). At the parameters presented in this figure, there is only negligible intraband transitions (zero spectrum on the left of the dotted line). All figures are presented in normalized units, except for the angle shown in degrees. The hot carrier energy E_i=0.2E_F and n_s=3 × 10¹³ cm⁻² (corresponding to E_F=0.639 eV).

Figure 3. GP emission from hot carriers.

Caption and notations same as in Fig. 2. The green dots in b show the GPs can be coupled out, as light, with each dot's size illustrating the strength of the coupling (in this case, the dots correspond to coupling out through a square lattice with a period of 32 nm in both dimensions). The hot carrier energy E_i=0.4E_F. E_F as in Fig. 2.

Figure 4. GP emission from hot carriers.

Caption and notations same as in Fig. 2. Unlike conventional ČE, most of the emission occurs in the forward direction with a relatively low angular spread as is shown by b. The green dot shows that GPs a particular frequency can be coupled out as light (we assume a grating with period of 3.5 nm). For the parameters used here, ČE emission occurs due to intraband transitions that are becoming allowed by the GP losses, whereas high-frequency (such as ℏω>2E_F) emission occurs due to interband transitions at areas of high GP losses. The hot carrier energy E_i=1.9E_F. E_F as in Fig. 2.

Figure 5. The spectrum of the Graphene ČE: hot carrier excitation energy having a wide versus narrow distribution.

(a) Illustration of the distribution of hot carriers, which is taken to be an exponential multiplied by the linear electron density of states in graphene. The exponential decay is (b) with maximum hot carrier energy of E_i,max=0.2 eV corresponding to Fig. 2, or (c) with maximum hot carrier energy of E_i,max=0.4 eV corresponding to Fig. 3. In both b and c we plot the exact spectrum (integrating equation (6) over the energy distribution) in solid black and the lossless approximation (integrating equation (5) over the energy distribution) in solid blue. The dashed curves are for the respective cases of narrow energy distribution (matching Figs 2c and 3c). The Fermi energy E_F is as in Figs 2, 3, 4.