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 lightâmatter interaction. In a conceptual breakthrough over 80 years old, Äerenkov showed how charged particles emit shockwaves of light when moving faster than the phase velocity of light in a medium. To modern eyes, the Äerenkov effect offers a direct and ultrafast energy conversion scheme from charge particles to photons. The requirement for relativistic particles, however, makes Äerenkov 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 Äerenkov emission, giving a versatile, tunable and ultrafast conversion mechanism from electrical signal to plasmonic excitation.
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Introduction
Achieving ultrafast conversion of electrical to optical signals at the nanoscale using plasmonics1,2 is a long-standing goal, owing to its potential to revolutionize electronics and allow ultrafast communication and signal processing. Given their strong field confinement, plasmonic systems combine the benefits of high frequencies (1014â1015âHz) with those of small spatial scales, thus avoiding the limitation of conventional photonic systems. However, the realization of plasmonic sources that are electrically pumped, power efficient and compatible with current device fabrication processes (for example, complementary metal-oxide semiconductors (CMOS)), is a formidable challenge. In recent years, several groups have demonstrated the potential of surface plasmons as a platform for strong and ultrafast lightâmatter interaction3,4,5,6. Grapheneâs tunability and strong field confinement7,8,9,10 have motivated proposals for the use of graphene plasmons (GPs)7,11,12,13 in electrically pumped plasmonic sources14 and in the conversion of electrical energy into luminescence15,16,17.
Here we show that under proper conditions charge carriers propagating inside graphene can efficiently excite GPs, through a two-dimensional (2D) Äerenkov emission process. Graphene provides a platform in which the flow of charge alone is sufficient for Äerenkov radiation, eliminating the need for accelerated charge particles in vacuum chambers and opening up an opportunity for the study of the Äerenkov effect (ÄE) and its applications, especially as a novel plasmonic source. Unlike other types of plasmon excitations, the 2D ÄE manifests as a plasmonic shock wave, analogous to the conventional ÄE that creates shockwaves in a three-dimensional (3D) medium. On a quantum mechanical level, this shockwave is reflected in the wavefunction of a single GP emitted from a single hot carrier.
Results
The mechanism of the graphene ÄE
The mechanism of 2D ÄE is enabled by two fundamental characteristics of graphene. On one hand, hot charge carriers moving with high velocities (up to the Fermi velocity νFâ106âmâsâ1) are considered possible, even in relatively large sheets of graphene (10âμm and more18). On the other hand, plasmons in graphene can have an exceptionally slow phase velocity, down to a few hundred times slower than the speed of light7,9,19. Consequently, velocity matching between charge carriers and plasmons is possible, enabling the emission of GPs from electrical excitations (hot carriers) at very high rates. This paves the way to devices using the ÄE on the nanoscale, a prospect made even more attractive by the dynamic tunability of the Fermi level of graphene. For a wide range of parameters, the emission rate of GPs is significantly higher than the rates previously found for photons or phonons7,20,21, suggesting that taking advantage of the ÄE enables near-perfect energy conversion from electrical energy to plasmons. Of course, it is impossible to reach perfect conversion due to other competing decay processes as discussed below. We further show that, contrary to expectations, plasmons can be created at energies above 2EFâthus exceeding energies attainable by photon emissionâresulting in a plasmon spectrum that extends from terahertz to near-infrared frequencies and possibly into the visible range. Furthermore, we show that tuning the Fermi energy by external voltage can control the parameters (direction and frequency) of enhanced emission. This tunability also reveals regimes of backward GP emission and regimes of forward GP emission with low angular spread, emphasizing the uniqueness of ÄE from hot carriers flowing in graphene. Surprisingly, we find that GP emission can also result from intraband transitions that are made possible by plasmonic losses. These kinds of transitions can become significant and might help explain several phenomena observed in graphene devices, such as current saturation22, high frequency radiation spectrum from graphene17,23 and the black body radiation spectrum that seems to relate to extraordinary high electron temperatures24.
Recent studies25,26,27, which focus on cases of classical free charge particles moving outside graphene, have revealed strong Äerenkov-related GP emission resulting from the charge particleâplasmon coupling. In contrast, in this work we focus on the study of charge carriers inside graphene, as illustrated in Fig. 1a. For this purpose, we develop a quantum theory of ÄE in graphene. As we shall see, our analysis of this system gives rise to a variety of novel Äerenkov-induced plasmonic phenomena.
The conventional threshold of the ÄE in either 2D or 3D (v>vp) may seem unattainable for charge carriers in graphene, because they are limited by the Fermi velocity vâ¤vF, which is smaller than the GP phase velocity vF<vp, as shown by the random phase approximation calculations19,28. However, we show that quantum effects come into play, to enable these charge carriers to exceed the actual ÄE threshold. Specifically, the actual ÄE threshold for free electrons is shifted from its classically predicted value by the quantum recoil of electrons on photon emission29,30. Because of this shift, the actual ÄE velocity threshold can in fact lie below the velocity of charge carriers in graphene, contrary to the conventional predictions. At the core of the modification of the quantum ÄE is the linearity of the charge carrier energyâmomentum relation (Dirac cone). Consequently, a careful choice of parameters (for example, Fermi energy or hot carrier energy) allows the ÄE threshold to be attainedâresulting in significant enhancements and high efficiencies of energy conversion from electrical to plasmonic excitation.
The quantum theory of the ÄE from hot carriers
The quantum ÄE can be described as a spontaneous emission process of a charge carrier emitting into GPs, calculated by Fermiâs golden rule29,30. In our case, the matrix elements must be obtained from the lightâmatter interaction term in the graphene Hamiltonian, illustrated by a diagram such as Fig. 1b. To model the GPs, we use the random phase approximation19,28,31,32, combined with a frequency-dependent phenomenological lifetime19 that takes into account additional loss mechanisms such as optical phonons and scattering from impurities in the sample (assuming graphene mobility of μ=2,000âcm2âVâ1âsâ1). This approach has been shown to give good agreement with experimental results8,12,13,33,34. The graphene sheet is in the yz plane and the charge carrier is moving in the z direction (Fig. 1a). For the case of low-loss GPs, the calculation reduces to the following integral (Lossy GPs are described later in this workâequation (6)).
where is the matrix element, S is the surface area used for normalization, qe is the electric charge, É0 is the vacuum permittivity, [SP] is the spinorâpolarization coupling term and is the GP dispersion-based energy normalization term35 (, using the group velocity vg=âÏ/âq). Supplementary Notes 1 and 2 elaborate on these definitions and describe the normalization (respectively). The GP momentum q=(qy, qz) satisfies , with the phase velocity vp=vp(Ï) or vp(q) obtained from the plasmon dispersion relation as vp=Ï/q. The momenta of the incoming (outgoing) charge carrier ki=(kiy, kiz) (kf=(kfy, kfz)) correspond to energies according to the conical momentum-energy relation . The charge velocity is v=Ek/|â k|, which equals a constant (vF). The only approximation in equation (2), henceforth, comes from the standard assumption of high GP confinement (free space wavelength/GP wavelength much larger than 1)19. Substituting equation (1) into equation (2) we obtain (denoting ):
where is the fine-structure constant, c is the speed of light and is the relative substrate permittivity obtained by averaging the permittivity on both sides of the graphene. We assume for all the figures. As material dispersion is neglected, all spectral features are uniquely attributed to the GP dispersion and its interaction with charge carriers, and not to any frequency dependence of the dielectrics. We further define the angle Ï for the outgoing charge and θ for the GP, both relative to the z axis, which is the direction of the incoming charge. This notation allows us to simplify the [SP] for charge carriers inside graphene to |SP|2=cos2(θâÏ/2) or |SP|2=sin2(θâÏ/2) for intraband or interband transitions, respectively. The emission is restricted to two angles θ=Â±Î¸Ä , as reflected in the delta function in equation (3) (a clear signature of the ÄE) and so we simplify the rate of emission to (see Supplementary Notes 3 and 4 for the complete derivation):
We note in passing that by setting ââ0 in the above expressions we recover the classical Äerenkov angle cos (θÄ)=νp/ν and the classical 2D ÄE, which can also be obtained from a purely classical electromagnetic calculation. However, although charge particles outside of graphene satisfy âÏ<<Ei, making the classical approximation almost always exact29, the charges flowing inside graphene can have much lower energies, because they are massless. Consequently, the introduced â terms in equations (4) and (5) modifies the conventional velocity threshold significantly, allowing ÄE to occur for lower charge velocities. For example, although the conventional ÄE requires charge velocity above the GP phase velocity (v>vp), equation (4) allows ÄE below it and specifically requires the velocity of charge carriers in graphene (v=vF) to reside between . Physically, the latter case involves interband transitions made possible when graphene is properly doped: when the charge carriers are hot electrons (holes), interband ÄE requires negatively (positively) doped graphene. Figures 2 and 3 demonstrate this interband ÄE that indeed occurs for charge velocities below the conventional velocity threshold. More generally, the inequalities can be satisfied in two spectral windows simultaneously for the same charge carrier, owing to the frequency dependence of the GP phase velocity (shown by the intersection of the red curve with the blue regime in Fig. 2d). Moreover, part of the radiation (or even most of it, as in Fig. 2) can be emitted backward, which is considered impossible for ÄE in conventional materials36,37. Several spectral cutoffs appear in Figs 2c, 3c and 4c, as seen by the range of non-vanishing blue spectrum. These can be found by substituting θÄ=0 in equation (4), leading to âÏcutoff=2Ei/(1±vF/vp), exactly matching the points where the red curves in Figs 2d, 3d and 4d cross the border of the blue regime. The upper most frequency cutoff marked by the thick orange line in Figs 2, 3, 4 occurs at âÏ=Ei+EF due to the interband transition being limited by the Fermi sea of excited states. Since Ei can be larger than EF, this implies that GP emission from electrical excitation can be more energetic than photon emission from a similar process (that is limited already by âÏâ²2EF). Finite temperature will broaden all cutoffs by the expected FermiâDirac distribution. However, for most frequencies, the GP losses are a more significant source of broadening.
Including plasmonic losses into the graphene ÄE
To incorporate the GP losses (as we do in all the figures), we modify the matrix elements calculation by including the imaginary part of the GP wavevector qI=qI(Ï), derived independently for each point of the GP dispersion curve19. This is equivalent to replacing the delta functions in equation (3) by Lorentzians with 1/γ width, defining γ(Ï)=qR(Ï)/qI(Ï). The calculation can be done partly analytically (Supplementary Note 3) yielding:
The immediate effect of the GP losses is the broadening of the spectral features as shown in Figs 2c, 3c and 4c. Still, the complete analytic theory of equations (4) and (5) matches very well with the exact graphene ÄE. For example, regimes of enhanced emission in Figs 2b and 3b match with the blue dashed curves marked according to the Äerenkov angle formula of equation (4) (see Supplementary Note 4 for its derivation in the lossless limit). The presence of GP loss also opens up a regime of quasi-ÄE that takes place when the charge velocity is very close to the Äerenkov threshold but does not exceed it. The addition of Lorentzian broadening then closes the gap, creating significant non-zero matrix elements that can lead to intraband GP emission (Fig. 4). This GP emission occurs even for hot electrons (holes) in positively (negatively) doped graphene, with the only change in Fig. 4 being that the upper frequency cutoff is instead shifted to âÏâ¤EiâEF (eliminating all interband transitions). The dip in the spectrum at the boundary between interband and intraband transitions (Fig. 4c) follows from the charge carriersâ density of states being zero at the tip of the Dirac cone.
The interband ÄE in Fig. 4 shows the possibility of emission of relatively high-frequency GPs, even reaching near-infrared and visible frequencies. These are interband transitions as in Figs 2 and 3, thus limited by âÏâ¤Ei+EF. Such a limit can get to a few eVs, because Ei is controlled externally by the mechanism creating the hot carriers (for example, pân junction, tunnelling current in a heterostructure, scanning tunneling microscope (STM) tip, ballistic transport in graphene with high drainâsource voltage and photoexcitation). GPs at near-infrared frequencies have already been shown to exist through direct and indirect experimental evidence38,39,40,41. The only fundamental limitation is the energy at which the graphene dispersion ceases to be conical (â¼1âeV from the Dirac point42). Even then, our equations are only modified by changing the dispersion relations of the charge carrier and the GP, and therefore the graphene ÄE should appear for Ei as high as (approximately) 3âeV43. The equations we presented are still valid, as they are written for a general dispersion relation, with vp(Ï) and γ(Ï) as parameters; thus, the basic predictions of the equations and the ÄE features we describe will continue to hold regardless of the precise plasmon dispersion. For example, a recent paper44 suggests an alternative way of calculating GP dispersion, giving smaller GP phase velocities at high frequenciesâthis will lead to more efficient GP emission and another intraband regime that can occur without being mediated by the GP loss.
Opportunities for experimental observation of the ÄE in graphene
There exist several possible avenues for observing the quantum ÄE in GPs, having to do with schemes for exciting hot carriers. For example, apart from photoexcitation, hot carriers have been excited from tunnelling current in a heterostructure45 and by a biased STM tip38. These methods can be used to generate hot carriers with narrow energy distributions but broad angular spreads. To show that many of the ÄE features discussed in this study can be observed experimentally even with a broad angular distribution of hot carriers, Figs 2c and 3c plot the spectral distribution of the ÄE averaged over the angular degree of freedom. Notably, the spectra contain clear signatures of the ÄE (for example, the two distinct spectral peaks in Fig. 2c). This confirms that even with non-directional hot carriers plasmons with the distinctive attributes of Äerenkov radiation can be created and observed with established experimental means.
To generate hot carriers that are also directional, a possible method would be to apply a strong drainâsource voltage across a graphene pân junction46 or in other graphene devices with ballistic transport. This could be achieved by using concepts of graphene electron optics47, where recent experimental demonstrations have shown controlled and directional ballistic motion of hot carriers by specially designed gates and/or an external magnetic field47,48,49. The generation of directional hot carriers would facilitate the measurement of even more features of the ÄE, such as the GP Äerenkov angle (as shown in Figs 2b, 3b and 4b). Yet, another intriguing approach for observing additional features of the graphene ÄE could be exciting the hot carriers and measuring the generated ÄE with the photon-induced near-field electron microscopy (PINEM)50,51, which might allow the visualization of the temporal dynamics of the Äerenkov emission. This approach is especially exciting, as the temporal dynamics of the ÄE is expected to appear in the form of a plasmonic shockwave (as the conventional ÄE appears as a shockwave of light).
If the hot carriers have a broad distribution in both energy and direction, the spectral features will be partially smoothened but still present a clear signature of the graphene ÄE. Hot carriers with a wide energy distribution (instead of a single Ei), for instance, may be generated from a tunnelling current or pân junction. The ÄE spectrum corresponding to an arbitrary hot carrier excitation energy distribution is readily computed by integrating over a weighted distribution of ÄE spectra for monoenergetic hot carriers. An example is presented in Fig. 5, where we assume the hot carriers have an exponential energy distribution (multiplied by the appropriate electron energy of states). This scenario can be readily created in a heterostructure45 having a sheet of graphene isolated from another conductor (possibly graphene as well), or by biasing an STM38 brought near the graphene surface. In both cases, the hot carriers tunnel through a potential barrier; thus, their distribution can be controlled by modifying the barrier thickness or the bias voltage. Figure 5a,b show that the spectral features are only partially smoothened and still present a clear signature of the graphene ÄE. To give a rough estimate for energy conversion efficiency, it is noteworthy that the average GP energies in Fig. 5a,b are 0.93EF and 1.1EF, respectively. Dividing by max(Ei)+EF (upper limit on the energy invested into a hot carrier), we get 77% and 78% (respectively) energy conversion efficiency. Here we assume that not more than one plasmon is emitted from each hot carrierâthe efficiency is higher when GP emission from intraband trasitions is significant, as then multiple emissions from a single hot carrier are probable (as in Fig. 4).
Future studies could directly generalize our approach to regimes of higher rates of excitation of hot carriers, by solving for the steady-state occupation probability. Here we assume the hot carriers are sparse enough so that they relax to the Fermi sea after their GP emission without creating a significant occupation that would exclude the transitions of other hot carriers. Altogether, higher excitation rates are therefore expected to cause gradual saturation of the GP emission and thus reduce the conversion efficiency. Importantly, the conversion efficiency remains high even when the carriersâ energy distribution is broad, as implied by the high ÄE rate of emission for the representative values of Ei studied here (Figs 2, 3, 4 and 5 all show rates on the order of ÎÏâ¼1). Emission rates on the order of ÎÏâ¼1 are two to three orders of magnitude higher than those found in the conventional ÄE in a 3D homogeneous dielectric, . For example, in a bulk silicon nitride having , we find values ranging from ÎÏ=0.0013 for v=0.6c to ÎÏ=0.0055 for vâc. In graphene, the high ÄE emission rate over a broad range of Ei owes itself to the low phase velocity and high confinement of GPs over a wide frequency range, which are properties unique to 2D plasmons.
Notably, the ÄE emission of GPs can be coupled out as free-space photons by creating a grating or nanoribbonsâfabricated in the graphene, in the substrate, or in a layer above it (see, for example, refs 39, 52, 53, 54)âwith two arbitrarily chosen examples marked by the green dots in Figs 3b and 4b. Careful design of the coupling mechanism can restrict the emission to pre-defined frequencies and angles, with further optimization needed for efficient coupling. This clearly indicates that the GP emission, although usually considered as merely a virtual process, can be, in fact, completely real in some regimes, with the very tangible consequences of light emission in terahertz, infrared or possibly visible frequencies. Such novel sources of light could have promising applications due to grapheneâs dynamic tunability and small footprint (due to the small scale of GPs). Moreover, near-perfect conversion efficiency of electrical energy into photonic energy might be achievable due to the ÄE emission rate dominating all other scattering processes. Lastly, unlike plasmonic materials such as silver and gold, graphene is especially exciting in this context, as it is CMOS compatible. Still, further research is needed in the design of gratings and/or cavities to minimize losses in the GP-to-photon conversion.
Discussion
The high efficiency of GP emission through electrical excitation makes it very promising as a plasmonic source. The efficiency of coupling to the plasmons from free space, in contrast, is very low due to momentum mismatch (the GP momentum is two orders of magnitude higher than the momentum of a photon of the same frequency in free space). For regular plasmons, this mismatch is typically overcome using a grating (nanoribbons are also a viable option in graphene); however, this results in relatively low coupling efficiency. In general, the efficiency of out-coupling is similar to the efficiency of in-coupling, for example, 2% has been shown in the best cases of ref. 39. Yet, another approach to solve the momentum mismatch challenge is by using the nonlinear optical response of graphene55, for which coupling efficiencies approaching 10â5 have been reported. The prospect of attaining energy conversion efficiencies on the order of 77â78%, as in the example above, thus makes the quantum ÄE from electrical excitation of hot carriers an exciting candidate for GP generation.
Furthermore, electrical excitation is interesting in itself due to its direct generation of radiation, without the need to use another radiation source (as in coupling lasers from free space). The unavoidable disadvantage of this scheme lies in the broad spectral range of the generated plasmons, compared with the highly monochromatic spectrum possible when using an external laser.
The hot carrier lifetime due to GP emission is defined by the inverse of the total rate of GP emission (integrating equation (5) or equation (6)) and can therefore be exceptionally short (down to a few femtosecond). Such short lifetimes are in general agreement with previous research on the subject (for example, see refs 20, 21, 56) that have shown electronâelectron scattering as the dominant cooling process of hot carriers, unless hot carriers of relatively high energies (Eiâ2EF and above) are involved. In this latter case, one expects single-particle excitations to prevail over the contribution of the plasmonic resonances57. This is also in agreement with the fact that plasmons with high energies and momenta (in the electron-hole continuum, pink areas in Figs 2, 3, 4) are very lossy7,19. Additional factors that keep the ÄE from attaining near-perfect conversion efficiency include other scattering processes such as acoustic and optical phonon scattering. Owing to the relatively long lifetime from acoustic phonon scattering (hundreds of femtoseconds to several picoseconds7), any deterioration due to this effect is not likely to be significant. Scattering by optical phonons is more significant for hot carriers above 0.2âeV, but its contribution is still about an order of magnitude smaller in our regime of interest20. Further research is necessary to optimize the emission process, including considerations of multi-plasmon emission and other higher-order effects.
Interestingly, the high rates of GP emission also conform to research of the reverse processâof plasmons enhancing and controlling the emission of hot carriersâthat is also found to be particularly strong in graphene58,59. This might reveal unexplored relations between ÄE and other novel ideas of graphene-based radiation sources that are based on different physical principles60,61,62,63,64.
It is also worth noting that Äerenkov-like plasmon excitations from hot carriers can be found in other condensed matter systems such as a 2D electron gas at the interface of semiconductors. Long before the discovery of graphene, such systems have demonstrated very high Fermi velocities (even higher than grapheneâs), while also supporting meV plasmons that can have slow phase velocities, partly due to the higher refractive indices possible in such low frequencies65. The ÄE coupling, therefore, should not be unique to graphene. In many cases66,67, the coupling of hot carriers to bulk plasmons is even considered as part of the self-energy of the carriers, although the plasmons are then considered as virtual particles in the process. Nonetheless, graphene offers a unique opportunity where the Äerenkov velocity matching can occur at relatively high frequencies, with plasmons that have relatively low losses. Crucially, these differences are what makes the efficiency of the graphene ÄE so high. Continued research into other 2D materials (for example, 2D silver and Beryllium7,68,69,70) may lead to materials with higher frequency, lower loss and higher confinement (lower phase velocity) than GPs. The prospect of higher frequency plasmons is especially exciting, as the ÄE radiation intensity increases with frequency (explaining the bluish colour of conventional ÄE).
We conclude with some very intriguing yet at this stage admittedly very speculative comments. Effects associated with the highly efficient emission of interband GPs and the unexpected emission of intraband GPs predicted by our quantum ÄE theory may have already manifested themselves in current graphene experiments, even in ones that do not involve any optical measurement, such as transistor-based graphene devices22,45. For example, such GP emission could be a contributing factor to the effect of current saturation observed in graphene devices22, as large sourceâdrain voltages can take graphene out of equilibrium and create hot carriers. When these hot carriers cross the energy threshold for significant GPs emission they lose energy abruptly, causing a sudden increase in resistivity. As another example, our graphene ÄE might play a role in explaining the surprisingly high frequency of emitted light from graphene17,23,24, as GPs can couple out as light emission by surface roughness, impurities and so on. This hypothesis is encouraged by reports17 in which the measurements show characteristics typical of ÄE variants, such as threshold values and power scaling behaviour that do not fit simple black body models. If our theory is indeed applicable here, then the extremely high-temperature estimates of the electron gas would need to be modified, to account for the contribution of GP emission in the high-frequency range of the observed spectrum. This would imply a lower black body radiation spectrum and thus lower graphene temperatures than otherwise expected24. Finally, as the GP energy can be higher than both Ei and EF, the ÄE could form part of the explanation for the observed frequency up-conversion23, especially given that multi-plasmon effects are expected due to the high rate of the emission process. Of course, future detailed studies of the systems will be needed to verify the ÄE connections proposed here.
Data availability
The data supporting the findings of this study are available within the article and its Supplementary Information File.
Additional information
How to cite this article: Kaminer, I. et al. Efficient plasmonic emission by the quantum Äerenkov effect from hot carriers in graphene. Nat. Commun. 7:11880 doi: 10.1038/ncomms11880 (2016).
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Acknowledgements
We thank Professor L. Levitov and Professor Y.D. Chong for helpful discussions, and Dr Marinko Jablan for useful advice. The work was supported by the US Army Research Laboratory and the US Army Research Office through the Institute for Soldier Nanotechnologies (contract number W911NF-13-D-0001). H.B. acknowledges support from the QuantiXLie and G2D Center of Excellence. The research of L.J.W. was supported by the Science and Engineering Research Council (SERC; grant number 1426500054) of the Agency for Science, Technology and Research (A*STAR), Singapore. J.J.L. was supported in part by the MRSEC Program of the National Science Foundation under award number DMRâ1419807. The research of I.K. was partially supported by the Seventh Framework Programme of the European Research Council (FP7âMarie Curie IOF) under grant agreement number 328853âMCâBSiCS.
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Kaminer, I., Katan, Y., Buljan, H. et al. Efficient plasmonic emission by the quantum Äerenkov effect from hot carriers in graphene. Nat Commun 7, ncomms11880 (2016). https://doi.org/10.1038/ncomms11880
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DOI: https://doi.org/10.1038/ncomms11880
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