Nanophotonics 2019; 8(1): 135–143
Research article
Igor V. Smetanin*, Alexandre Bouhelier and Alexander V. Uskov Coherent surface plasmon amplification through the dissipative instability of 2D direct current https://doi.org/10.1515/nanoph-2018-0090 an electronic drive is a unique asset to enhance the speed Received July 9, 2018; revised October 5, 2018; accepted October 11, of signal processing and decrease energy consumption [3, 2018 4] as well as to provide for electrical sources of propagat- ing surface plasmon polaritons (SPP) [5–8]. The search of Abstract: We propose an original concept for on-chip exci- alternative mechanisms for effective and controllable elec- tation and amplification of surface plasmon polaritons. trical generation of SPP is however plagued by a number Our approach, named nanoresotron, utilizes the collective of fundamental physical and technological issues. The effect of dissipative instability of a 2D direct current flowing quantum efficiency rarely exceeds 10−4, the wavevector in vicinity of a metal surface. The instability arises through mismatch limits the coupling strength and the surface the excitation of self-consistent plasma oscillations and modes are typically attenuated by different loss mecha- results in the creation of a pair of collective surface elec- nisms. Several attempts were made to boost the conver- tromagnetic modes in addition to conventional plasmon sion yield either by optimizing the local density of optical resonances. We derive the dispersion equations for these states [9] or by engineering the energy barrier [10]. Like- modes using self-consistent solutions of Maxwell’s and 2D wise, the inherent losses of the metal were mitigated by hydrodynamics equations. We find that the phase velocities different strategies with various degrees of success includ- of these new collective modes are close to the drift veloc- ing back reflection of radiation leakage [11], coupling SPP ity of 2D electrons. We demonstrate that the slow mode is with gain media [12–14] or shifting altogether to alterna- amplified while the fast mode exhibits absorption. Esti- tive plasmonic materials [15]. mates indicate that very high gain are attainable, which Here, we propose a novel approach to electrically makes the nanoresotron a promising scheme to electrically excite and amplify SPP on-chip in a single process which excite and regenerate surface plasmon polaritons. is based on the dissipative instability of a direct current Keywords: electrical excitation of nanoantenna; self-con- (DC) carried in a 2D nanostructure near the lossy metal sistent 2D plasma oscillation; dissipative instability of DC surface. Our strategy has its origin in conventional plasma current. physics and relativistic electronics and utilizes the general principle of self-consistent excitation of plasma instabili- ties in the stream of electrons resulting in coherent ampli- fication of the electromagnetic wave. Primarily, the idea to 1 Introduction use the dissipative instability for light amplification had been elaborated in the concept of the so-called resotron, The development of a hybrid approach merging an elec- a free-electron laser in which the surface electromagnetic tronic control layer with a plasmonic transport circuitry wave is amplified by high-current relativistic electron beam is a very promising road to drastically downsize on-chip moving above the surface of a resonant absorbing material functionalities [1, 2]. Combining surface plasmons with [16–18]. The amplification effect in this scheme emerges at frequencies within the material’s resonant absorption line, *Corresponding author: Igor V. Smetanin, P. N. Lebedev Physical instability is absent for frequencies at which the material is Institute, Leninsky pr. 53, 119991 Moscow, Russia, transparent. In contradistinction to the microwave resistive- e-mail: [email protected]. wall amplifier [19–21], the resotron is able to operate in the https://orcid.org/0000-0002-0303-4543 optical frequency range and takes advantage of the narrow Laboratoire Interdisciplinaire Carnot de Alexandre Bouhelier: resonance absorption line of the surface material. Being Bourgogne, CNRS-UMR 6303, Université Bourgogne Franche-Comté, 21078 Dijon, France essentially the collective excitation process, the mechanism Alexander V. Uskov: P. N. Lebedev Physical Institute, Leninsky pr. 53, of SPP generation in our concept qualitatively differs from 119991 Moscow, Russia the Cherenkov emission by electron beam moving above
Open Access. © 2018 Igor V. Smetanin et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution- NonCommercial-NoDerivatives 4.0 License. 136 I.V. Smetanin et al.: Coherent surface plasmon amplification through the dissipative instability of 2D
the grating (see, for instance, [22] and references therein). film, ε2 for the dielectric spacer located between the QW and
The necessity of material’s absorption to provide the dis- the metal film (–d < x < 0), ε1 for the superstrate at x > 0, and sipative instability of electron stream makes our concept ε4 for the substrate dielectric medium situated at x < –d–Δ. also qualitatively different from the well-known collective We assume the system invariant in the third dimension. two-stream and Dyakonov-Shur [23] THz-range instabilities A monochromatic wave propagating along the z direc- in ballistic field-effect transistor. tion consists of three field components, either polarized
In the following, we demonstrate that a 2D electric TM (Ez, Ex, Hy) or TE (Hz, Hx, Ey). We are interested in the current circulating in a 2D nanostructure is unstable TM mode because coupling with the DC electric current is when situated in the vicinity of a gold metal film. This enabled due to the non-zero electric field component Ez in instability has a resonance character with the maximum the QW plane. We assume the electromagnetic mode having increment in the vicinity of the conventional SPP reso- the field components (Ez, Ex, Hy) × exp[i(hz–ωt)], where the nance condition (i.e. at Re{εm + ε1} → 0, where εm and ε1 field amplitudes Ez, Ex, Hy are functions of the transverse are the dielectric permittivities of the metal and the sur- co-ordinate x. Our goal is to derive the self-consistent dis- rounding material, respectively) and thus lies in the persion equation relating the mode frequency ω and the optical domain. The instability arises through the excita- longitudinal wavenumber h. The conditions leading for tion of self-consistent 2D plasma oscillations and results the amplification effect are then found when the imaginary in pair of surface modes additional to conventional SPP part of the wavenumber becomes negative Im{h} < 0. solutions. We derive the dispersion equations describing Let us first analyze the simplest case where there is both the conventional SPP modes and the new collective no spacer and the metal film is of infinite thickness, that slow surface modes. The velocities of these new collective is d = 0 and Δ → ∞. Maxwell’s equations result in the fol- modes are close to the drift velocity of 2D electrons, which lowing transverse distributions of the field amplitudes. provides an effective interaction and consequently a high In the ε1 dielectric medium (x > 0), we write Ez = Aexp gain increment. (–q1x), Ex = i(h/q1)Aexp(–q1x), and Hy = iε1(ω/q1c)Aexp(–q1x), 22 22 where qh11=/− εω c is the transverse wavenumber, and A is the amplitude coefficient. The necessary condition Re{q } > 0 for the evanescent solution must be implied. 2 Dispersion equation 1 Within the metal slab (x < 0) we find Ez = Bexp(qmx), E = –i(h/q )Bexp(q x), and H = –iε (ω/q c)Bexp(q x), with Let us consider the simplest two-dimensional configura- x m m y m m m the transverse wavenumber qh22=/− εω22c , and tion shown in Figure 1. It consists of a 2D nanostructure mm Re{q } > 0. The amplitude coefficients A and B are related carrying DC current separated from a metal film of finite m to each other through the boundary conditions at x = 0: thickness Δ by a spacer of thickness d. This nanostructure could be a high-mobility semiconductor heterostructure, a EEzx|==0 zx|,=0 +− 2D graphene monolayer or other 2D material which is char- ()εε1=EExx|(0=− mx)|x 0 =4πδρ (1) acterized by high mobility carriers. Let us suggest for defi- +− niteness this nanostructure be a current carrying quantum Here, δρ is the amplitude of the charge density oscillation well (QW). The dielectric permittivities are ε for the metal m which is excited in the QW’s current-carrying plasma at the frequency ω. The second equation in Equation (1) is the Gauss theorem written in CGS units. Assuming that the holes in QW are sufficiently heavy, we can write down the following Euler hydrodynamics equations for the 2D density n and the velocity v of the cold electron liquid [24]
∂∂nn()v + =0, ∂∂tz ∂∂vvsn2 ∂ e +−vE=e−−z xp[(ihztωγ)]− v (2) ∂∂tznz0 ∂ m Figure 1: Simplified scheme of the nanoresotron. Plasma oscillations in a quantum well (QW) carrying DC current where sv=/2 is the speed of sound, vF is the Fermi veloc- result in surface wave amplification at the metal film. εi, where i = 1 F ity of 2D electron gas, m is the effective mass of 2D elec- to 4 are the permittivities of the media, ε3 = εm, v0 is the drift velocity of the electrons in the QW. trons, γ is the velocity relaxation rate, n0 is the unperturbed I.V. Smetanin et al.: Coherent surface plasmon amplification through the dissipative instability of 2D 137
2D density of electrons. Note, analogous approach has been find that the equation has two pairs of roots. The first pair used to describe localised plasmons near the constriction represents the conventional rapidly decaying plasmon in a point contact geometry [25]. Note also, the magnetic solution which is determined by the left hand side of contribution to the Lorentz force in Equation (2) has been Equation 4 at vanishing DC current. At n0 → 0 we arrive 2 neglected due to its relative smallness ~(v0/c) with respect at the well-known surface plasmon dispersion equation to the longitudinal electric force. In the field of TM electro- ε1 εm magnetic mode of our particular interest, the main compo- + =0 (6) hc22−−εω //22hcεω22 nent of the magnetic contribution vH0 × y is along the x axis 1 m and does not affect the longitudinal motion of 2D electrons.
At sufficiently small field amplitudes, we assume in which at |εm | | ε1 | results in the wavenumbers of coun- ≈± the first-order perturbation n = n0 + δn exp[i(hz − ωt)] and ter-propagating waves hcSPP εω1 /. As both the Fermi v = v0 + δv exp[i(hz − ωt)], v0 is the drift velocity of 2D elec- velocity and the drift velocity are much less than the speed trons providing the DC electric current. From Equation 2, of the plasmon, vvf , 01c /,ε correction to these con- the velocity and density perturbation amplitudes are ventional solutions due to 2D plasma oscillations is small:
the difference between hSSP and the correspondent solu- e hv0 − ω δvi=,A 22 tion of the dispersion equation (5) is δh/h ≡ (h − h )/ m [(hv −−ωω)(hv −−isγ)]h SPP SPP 00 h ≈ (4πn e2/mω2)h ~ 3 × 10−5 assuming for estimate a hn SPP 0 e 0 (3) δni= − A 22 gold-GaAs interface (the effective mass of electrons is m [(hv00−−ωω)(hv −−isγ)]h m = 0.067m0, the high-frequency dielectric constant is ε = 10.89, ħω ≈ 2 eV, and the 2D electron density is As far as δρ = −eδn, the boundary conditions of Equa- 1 n = 1012 cm−2). tion 1 result in the following dispersion relation 0 The presence of DC current leads to the appearance 2 ε επ4/ne m of another pair of solutions with wavenumbers close to 1 + m 0 = 22 (4) qq [(hv −−ωω)(hv −−isγ)]h h → ω/v , which are thus much more slower than conven- 1 m 00 0 tional surface plasmons (Equation 6). Since the transverse
In the general case of a finite spacer width and finite wavenumbers are then close to each other, q1 ≈ qm ≈ h (with thickness of the metal film, doing calculations in the same Re{h} > 0), we can rewrite the dispersion relation as straightforward manner, we find a dispersion equation of 4πne2 the form: ()hv −−ωω()hv −+ihγ = 0 sh22 (7) 00 m()εε+ 1 m qqεεq ε mm44++tanh()φφm 2 1t+ anh( )tanh(ψ) εεε which explicitly highlights the coupling of the electro- εε12qq42mmq4 m + magnetic surface wave to the 2D plasma wave (note that qqq ε qqεε 12m 2 1t++mm44anh(φφ)t+ anh( )tanh(ψ) the right hand side of Equation 7 is squared the eigen qqεε q ε 24mm4 m frequency of 2D plasma sheet oscillations having the 2 4/πne m wavenumber h). Typical momentum relaxation rate for = 0 (5) [(hv −−ωω)(hv −−isγ)]22h electrons in semiconductor QW at room temperature is 00 12 13 −1 γ ~ 10 to 10 s [26], so one can omit it in Equation 7 as far as we are interested in the optical frequencies domain. where the parameters φ = q Δ and ψ = q d are introduced, m 2 Introducing the following dimensionless parameters qh22=/− εω22c and qh22=/− εω22c are the transverse 22 44 Γ = 4π n e2/mωv , ε′ = Re{(ε + ε )−1} and ε″ = Im{(ε + ε )−1} wavenumbers for the spacer and for the substrate dielec- 0 0 1 m 1 m = −Im{ε }/ | ε + ε | 2, the solution to equation (7) is tric. One can easily see that dispersion equation of Equa- m 1 m tion 5 simplifies to Equation 4 in the limit of negligible hv 12+±Γε′ Z R=e0 spacer and infinitely thick metal film; d = 0, Δ→ ∞. ω 2 1(− sv/)2 0 (8) hv 12Γε′′ + Γε′ I=m0 1 ± ω 2 1(− sv/)2 Z 3 DC plasma-supported slow 0 collective surface modes where Z is the positive root of the following biquadratic 4 2 2 2 2 2 equation Z +[4(1 −(s/v0) ) −(2 + Γε′) +(Γε″) ]Z −(Γε″) To reveal the physics of the nanoresotron, let us analyze (2 + Γε′)2 = 0. One can easily find that the amplification the dispersion equation of Equation 4. One can easily effect Im{h} < 0 arises when the drift velocity exceeds the 138 I.V. Smetanin et al.: Coherent surface plasmon amplification through the dissipative instability of 2D
2D sound velocity: v0 > s (note, the defined above para- is GaAs. One can expect the largest amplification incre- meter ε″ is negative, ε″ < 0). In the frequency domain of ments to arise when Re{ε1 + εm} ~ 0, which corresponds negative ε′ < 0, there is an additional restriction on the to the frequency region near ħω ≈ 2 eV. The frequency 2 drift velocity: v0 > −2πn0e ε′/mω. Depending on the choice dependencies of Equation 8 are shown in Figure 2A and of the sign, Equation 8 describes two modes: one of which B. The amplified mode is very slow and has the phase is slow with respect to electron drift velocity, Re{hv0/ω} > 1, velocity ω/Re{h} approximately 3.41 times less than the and the second mode is fast, Re{hv0/ω} < 1. In analogy to drift velocity, which is about three orders of magnitude classical electronic devices, the slow mode is thus able less than the phase velocity of conventional surface to acquire energy from the DC current and thus exhibits plasmon modes. The largest amplitude in the evolu- amplification, while the fast mode decays. tion of Im{hv0/ω} reaches 1.2. The increment is thus 7 −1 Let us make estimates considering an InGaAs QW Im{hv0/ω} × ω/2πv0 ≈ 8.4 × 10 cm , which is indeed a very 12 −2 with a 2D electron density n0 = 10 cm and a gold metal high gain. Two dotted lines in Figure 2A correspond to film. We model dielectric susceptibility of gold with the the dispersion curves of oscillations in recluse current- data [27]. We assume the electron drift velocity v0 equals carrying QW which in the optical frequency range are to the Fermi velocity, which with above electron density determined as ω ≈ h(v0 ± s), the damping coefficient for 7 −1 12 −2 is vF ≈ 4.3 × 10 cm s . At n0 = 10 cm the 2D plasma these oscillations is of the order of γ which we omit in density we find the 2D current density is 6.88 A/cm which the above estimates. For reference, the dispersion char- for QW with the transverse dimensions 2 nm × 100 nm acteristics of the conventional SPP mode which are εε results in the current density 3.44 × 107 A/cm2 and in the determined by the relation hc22=/1 m ω 2 are shown in εε+ total current 68.8 μA. The superstrate dielectric material Figure 2C and D. 1 m
AB4.0 0.2
3.6 0.0
3.2 –0.2
/ ω ) –0.4 / ω ) 0
0 2.8 –0.6 0.7 Im (h v Re (h v –0.8 0.6 –1.0 0.5 –1.2 0.4 1.50 1.75 2.00 2.25 2.50 1.50 1.75 2.00 2.25 2.50 ω (eV) ω (eV)
CD12 12
10 10
8 8
6 6 Im (hc/ ω ) Re (hc/ ω ) 4 4
2 2
0 0 1.50 1.75 2.00 2.25 2.50 1.50 1.75 2.00 2.25 2.50 ω (eV) ω (eV)
Figure 2: The dispersion characteristics of the slow collective dissipative modes.
The modes are either slow with respect to the 2D electron drift velocity (black curves) or fast with respect to v0 (red curves). (A) the normalized (by ω/v0) wavenumber and (B) the instability increment vs frequency. Slow mode exhibit amplification (negative increment) while the fast mode is decaying. Dotted lines in (A) corresopond to oscillations in recluse current-carrying QW. The dispersion characteristic of the conventional SPP mode is shown in (C) and (D), the normalized (by ω/c) wavenumber and the damping coefficient vs frequency, 12 −2 respectively. Calculations are made for an InGaAs quantum well with n0 = 10 cm , a gold film and a drift velocity equals to the Fermi 7 −1 velocity of 2D electron liquid, v0 = vF ≈ 4.3 × 10 cm s . I.V. Smetanin et al.: Coherent surface plasmon amplification through the dissipative instability of 2D 139
To get the above amplification effect one needs the the opposite limiting case, when |εmδ | | ε1 + εm | the dis- drift velocity of the electrons exceeding the speed of persion equation acquires the form sound, v > s. This is a rather hard condition but recent 0 2πεne2 experimental studies demonstrated that such large drift ()hv −−ωω()hv −−isγδ22hh= − 0 m (10) 00 m()εε+ ε velocities might be attainable in 2D materials. In gra- 11m phene monolayers for instance the drift velocity has been The wavenumber and the growth increment for the measured at 7 × 107 cm s−1, exceeding the saturation veloc- amplified mode are ity [28, 29] and approaching the the Fermi velocity [30]. In GaN QWs with the negative differential mobility the 2 hv00vv1 εε10mm′ + ||ε R1e≈−Γδ , drift velocity can be as high as twice the saturation veloc- ω vs− 4 εε+ ε 2 s 0 11||m ity with the peak value of 3.5 × 107 cm s−1 [31]. In InGaAs/ hv vv1 ε′′ γ 00 m 0 GaAs structures, the maximum drift velocity is signifi- Im ≈− 2 Γδ− (11) 7 −1 ωω vs− 42||εε+ s cantly enhanced up to 4 × 10 cm s by increasing the frac- 0 m 1 tion of Indium in the QW itself and the barrier layer [32].
Finally, the nonstationary voltage switch-on transients The limiting spacer thickness dmax is thus determined with very high drift velocities up to 8 × 107 cm s−1 emerg- by the balance between the instability increment and the ing on sub-picosecond time scale in short quasi-ballistic losses due to 2D electron momentum relaxation GaAs, GaInAs, InP structures had been also intensively 2ωΓε′ v ω studied since 80’s [33]. Clearly, the regime of amplifica- ≈ m 0 expdmax 2 (12) vs− ||εε+ s γ tion seems to reach with material quality available. The 0 m 1 second important aspect for an experimental operation of the proposed device requires a strategy to inject a DC Hence, the amplification effect survives at spacer electrical current flowing in the QW layer. The simplified thicknesses of the order of few wavelengths of the scheme depicted in Figure 1 features a multilayer structure amplified mode. The transverse profile of the amplified that can be readily fabricated with current technology. mode is shown in Figure 3 at various spacer thickness Field-effect transistors and semiconducting lasers share a for the case of sufficiently thick metal film. The field similar stacked geometry as in Figure 1 and benefit from amplitudes are normalized by the amplitude of the lon- charge injection strategies. In this context, an interest- gitudinal electric field in QW plane. The electric field ing example is the hybrid plasmonic semiconductor laser is concentrated near the metal surface for small spacer proposed by R. Colombelli and co-workers [34] whereby thicknesses d < 3(Re{h})−1 and the largest field amplitude a metal clad supporting SPP is interacting with an active near the metal surface is attained at d ≈ 2(Re{h})−1. This layer. The technological steps developed in this report could be adapted to accommodate an active material fea- turing a charge mobility sufficiently high to trigger the amplification. The amplification rapidly drops with the increase of the spacer thickness. To demonstrate it, let us assume the spacer material to be the same as the superstrate dielec- tric, ε1 = ε2, and keep the thickness of the metal film suf- ficiently high, tanh(φ) → 1. The dispersion equation now writes
q ε 1t+ m 2 anh(ψ) ε εεqn4/π em2 1 + mm20= (9) qq q ε [(hv −−ωω)(hv −−isγ)]22h 1 m 1t+ 2 m anh(ψ) 00 Figure 3: The transverse profile of the amplified electromagnetic q ε mode, (A) |E | 2 and (B) |E | 2 vs the distance from the metal surface x/l m 2 x z normalized by the wavelength of the amplified mode, l = (Re{h})−1. The profiles are calculated at various spacer thicknesses, d = l Introducing the exponentially small parameter (red), d = 2l (blue), d = 3l (olive) and d = 4l (magenta). Calculations δ = ψ ≈ ε δ ε + ε 1 − tanh( ) 2exp( −2hd), we find for | m | | 1 m | are made at the exact resonance condition correspondent to the that the coupling between the DC current and the metal maximum in Figure 3B. The field amplitudes are normalized by the surface disappears as well as the dissipative instability. In amplitude of the longitudinal electric field in QW plane. 140 I.V. Smetanin et al.: Coherent surface plasmon amplification through the dissipative instability of 2D relative enhancement of the field amplitude is due to the from support of the Région Bourgogne Franche-Comté resonance condition Re{ε2 + εm} → 0, as one can easily see (project EXCELLENCE-APEX). from the relation for the field amplitudes (A15), see the Appendix section. Appendix: Derivation of the dispersion equation 4 Conclusion In this Appendix, we present the detailed derivation of the In conclusion, the nanoresotron presented here is a new dispersion Equation (5). We consider the geometry of the concept device for electrical surface plasmon excitation nanoresotron which is shown in the Figure 1. The scheme and amplification by a DC electric current flowing in adja- consists in the QW carrying DC current and the parallel cent 2D nanostructure. We find that when placed at the metal film of finite thickness Δ. QW is placed in the z–y vicinity of the metal surface, the 2D DC electric current plane at x = 0. Above the QW is the dielectric medium with becomes unstable at optical frequencies close to that of dielectric permittivity ε1, the distance between the QW and conventional SPP resonance, which is the result of the the metal film –d < x < 0 is the spacer region with the die- dissipative instability (excitation of self-consistent 2D lectric permittivity ε2. Below the metal film is the dielectric plasma oscillations). The coupling between the plasma medium with the permittivity ε4, the dielectric permittivity oscillation and the absorbing metal leads to an additional of the metal film is εm. pair of slow collective surface modes. In the simplest It is assumed that the system is infinite in y direc- planar configuration of the nanoresotron, the dispersion tion. The monochromatic electromagnetic TM wave con- equation for these collective modes is derived and we find sists in three components and propagates along the z that the phase velocities of these new collective modes are axis, (Ez, Ex, Hy) × exp(i[hz − ωt]). The Maxwell equations close to the drift velocity of 2D electrons. The slow mode then read as is then amplified while the fast mode is attenuated. Esti- ∂Hy εω mates show that very high increments are attainable, sug- = −iE ∂xcz gesting that a practical realization of the nanoresotron εω HE= may offer a promising alternative to electrically excited yxhc and amplify surface plasmons. The simplified geometry ∂E ω z −−ihEi= H can be extended to excite other types of plasmon-sup- ∂xcxy porting structure e.g. nano-sized optical antennas, and ∂Ex + ihEz =0 (A.1) will be discussed elsewhere. Let us stress that not only 2D ∂x QWs but also the other low-dimensional current-carrying structures, such as quantum wires, carbon nanotubes, Combination of these equations leads to the equation graphene nanoribbons and other topological insulators which determines the transverse wavenumber, are able to provide excitation energy for the nanoresotron. 2 ∂ E ω2 z 2 − ε The extremely important energy efficiency issues, as well 22=hEz (A.2) ∂xc as study of the saturation dynamics in the nanoresotron requires further development of the presented approach Solutions to the Maxwell equations in each the above to include the higher-order perturbations which will be space regions can be thus immediately written in the fol- done elsewhere in the near future. lowing way. At x > 0 we write Acknowledgments: The work was partially funded by the −−qx hh∂E qx European Research Council under the European Commu- E =,Ae 11Ei==− z iAe , zx2 ∂ nity’s Seventh Framework program FP7/2007-2013 Grant q1 xq1 Agreement 306772, the CNRS/RFBR collaborative research ω −qx H =iAε e 1 (A.3) program number 1493(Grant RFBR-17-58-150007), and the y 1 qc 1 COST Action MP1403 “Nanoscale Quantum Optics,” sup- ported by COST (European Cooperation in Science and Here, the transverse wavenumber q1 is determined by 22 22 Technology). A.U. and I.S are thankful to Russian Science the relation qh11=/− εω c and must satisfy the follow-
Foundation (Grant 17-19-01532) for support. A. B. benefited ing necessary condition of evanescent solution Req1 > 0. I.V. Smetanin et al.: Coherent surface plasmon amplification through the dissipative instability of 2D 141
In the spacer region between the QW and the metal At x = −d the boundary conditions are (ψ = q2d) film –d < x < 0 we have DFsinh(/φφ2)+−cosh(/2) =sBCinh(ψψ/2)c+ osh( /2) EB=sinh[qx(/++dC2)]cosh[qx(/+ d 2)], z 22 h −+εφφ h iDm [cosh( /2)sF inh( /2)] Ei=c−+Bqosh[ (/xd2)]s++Cqinh[ (/xd2)], qm x q {}22 2 h −− ω =[iBεψ2 cosh(/2) Csinh(/ψ 2)] (A.9) −+ε ++ q Hiy =c22{}Bqosh[ (/xd2)]sCqinh[ 2 (/xd2)], 2 qc2 2 22 ω qh22= − ε 2 (A.4) Taking into account the relations (A8), one can find c qmε4 Dsinh(/φφ2)++FGcosh(/2) =cosh(φφ)sinh( ), q ε Within the metal film –d–Δ < x < –d the solution is 4 m qmε4 Dcosh(/φφ2)++FGsinh(/2) =cosh(φφ)sinh( ) EDzm=sinh[qx(/++dF∆∆2)]c++osh[qxm(/d + 2)], q ε 4 m h −++ (A.10) Eixm=c{Dqosh[ (/xd∆ 2)] qm +++ ∆ which allows us to get the following relations for the field Fqsinh[(m xd /2)]}, amplitudes B and C inside the spacer ω −++ Hiym=cε∆{Dqosh[ m(/xd 2)] qc ε q q ε m B =c m 2 m 4 osh(φφ)s+ inh( )cosh(ψ/2) +++ ∆ εεqq Fqsinh[(m xd /2)]}, 24mm 2 22 ω q ε − ε (A.5) m 4 qhmm= 2 ++cosh()φφsinh()sinh(/ψ 2) G c ε q4 m q ε C =c osh(φφ)s+ m 4 inh( )cosh(ψ/2) And finally, in the dielectric medium below the metal ε q4 m film, we have the evanescent wave εmq2 qmε4 qx()++dq∆∆h ()xd++ ++cosh()φφsinh()sinh(/ψ 2) G EG=,eE44=,−iGe εεqq zx 24mm (A.11) q4
ω qx()++d ∆ Hi= − ε Ge 4 At the position of the QW, the continuity of the normal y 4 qc 4 component of electric displacement must be replaced by ω2 qh22= − ε (A.6) the Gauss theorem, 44c2 AB=sinh(ψψ/2)c+ C osh( /2) hh with the necessary condition for the evanescent solution εε++ψψπδρ iA12iB( cosh(/2) Csinh(/2))=4 (A.12) Req > 0. qq 4 12 Taking into account the boundary conditions, namely the continuity of the tangential field components and of Relations (A11) allows us to find the normal components of electric displacement, one can q ε relate the field amplitudes B, C, D, F, and G to each other. Bsinh(/ψψ2)++Ccosh(/2) =cosh(φφ)sm 4 inh( ) ε q4 m First, at the boundary x = −d − Δ we have (ϕ = qmΔ) ε q q ε GD=s−+inh(φφ/2)cF osh( /2) cosh()ψφ++m 2 m 4 cosh() sinh()φψsinh()G εεqq hh 24mm −−εε φφ− (A.7) iG4 =[iDm cosh(/2) Fsinh(/2)] ε ε qq mq2 qm 4 4 m Bcosh(/ψψ2)++Csinh(/2) =c osh(φφ)sinh( ) εε 24qqmm which result in the following relations for the amplitudes q ε cosh()ψφ++cosh() m 4 sinh()φψsinh()G D and F (A.13) q4εm qmε4 D =cosh(φφ/2)s+ inh( /2),G ε q4 m which, after substitution into the boundary conditions qmε4 (A12) and the use of relations (3) for the amplitude of the F =cosh(φφ/2)s+ inh( /2) G (A.8) q ε density perturbation, results in the following equality 4 m 142 I.V. Smetanin et al.: Coherent surface plasmon amplification through the dissipative instability of 2D
ε q qqεε m 2 mm44cosh()φφ++sinh()cosh()ψφcosh()+ sinh()φψsinh() εεεεqq q ε 12+ 24mm4 m qqqqεεε q 12cosh()φφ++mm44sinh()cosh()ψφm 2 cosh()+ sinh()φψsinh() εεε qq42mmq4 m 4/πne2 m = 0 (A.14) [(hv −−ωω)(hv −−isγ) 22h ] 00
Dividing both the numerator and the denominator by [10] Uskov AV, Khurgin JB, Protsenko IE, Smetanin IV, Bouhelier A. the product cosh(φ)cosh(ψ), we arrive at the dispersion Excitation of plasmonic nanoantennas by nonresonant and resonant electron tunnelling. Nanoscale 2016;8:14573. Equation (5). [11] Lee T-W, Gray SK. Regenerated surface plasmon polaritons. It is interesting to compare the field amplitudes in the Appl Phys Lett 2005;86:141105. limit of thick metal film when tanh(φ) → 1. One can easily [12] Grandidier J, Colas des Francs G, Massenot S, et al. Gain- find using Equations (11–13) the field amplitudes in the assisted propagation in a plasmonic waveguide at telecom spacer related to the field amplitude in QW are wavelength. Nano Lett 2009;9:2935–9. [13] Handapangoda D, Rukhlenko ID, Premaratne M, Jagadish C. ε q Optimization of gain-assisted waveguiding in metal-dielectric m 2 cosh(/ψψ2)+ sinh(/2) B ε q nanowires. Opt Lett 2010;35:4190–2. = 2 m A ε q [14] Berini P, De Leon I. Surface plasmon-polariton amplifiers and m 2 sinh()ψψ+ cosh() lasers. Nat Photon 2012;6:16–24. ε 2qm [15] West P, Ishii S, Naik G, Emani N, Shalaev V, Boltasseva A. ε q Searching for better plasmonic materials. Laser Photon Rev m 2 sinh(/ψψ2)+ cosh(/2) C ε q 2010;4:795–808. = 2 m (A.15) A ε q [16] Oraevsky AN, Smetanin IV. Intensification of light by an elec- m 2 sinh()ψψ+ cosh() tron beam near an absorbing surface. JETP Lett 1995;62:258. ε q 2 m [17] Oraevsky AN, Smetanin IV. Resotron – a new concept of IR free- electron laser driven by a low-voltage electron beam. Laser Phys 1997;7:155. [18] Smetanin IV. Free-electron lasers driven by supercurrent. Nuc Instrum Meth Phys Res A 1999;429:445–50. References [19] Birdsall CK, Brewer GR, Haeff AV. The resistive-wall amplifier. Proc IRE 1953;41:865–75. [1] Schuller JA, Barnard ES, Cai W, Jun YC, White JS, Brongersma ML. [20] Neil VK, Sessler AM. Longitudinal resistive instabilities of Plasmonics for extreme light concentration and manipulation. intense coasting beams in particle accelerators. Rev Sci Nature Mat 2010;9:193–204. Instrum 1965;36:429–36. [2] Gramotnev DK, Bozhevolnyi SI. Plasmonics beyond the diffrac- [21] Chao AW, Tigner M, eds. Handbook of accelerator physics and tion limit. Nat Photonics 2010;4:83–91. engineering. Singapore, New-Jersey, London, World Scientific, [3] Cai W, White J, Brongersma M. Compact, high-speed and 1998. power-efficient electrooptic plasmonic modulators. Nano Lett [22] Gong S, Hu M, Zhong R, Zhao T, Zhang C, Liu S. Mediated 2009;9:4403–11. coupling of surface plasmon polaritons by a moving electron [4] Haffner C, Heni W, Fedoryshyn Y, et al. All-plasmonic Mach- beam. Opt Express 2017;25:25919–28. Zehnder modulator enabling optical high-speed communication [23] Dyakonov MI, Shur MS. Shallow water analogy for a ballistic at the microscale. Nat Phot 2015;9:525. field effect transistor: new mechanism of plasma wave genera- [5] Neutens P, Lagae L, Borghs G, Van Dorpe P. Electrical excitation tion by dc current. Phys Rev Lett 1993;71:2465–8. of confined surface plasmon polaritons in metallic slot wave- [24] Fetter A. Electrodynamics of a layered electron gas. I. Single guides. Nano Lett 2010;10:1429. layer. Ann Phys 1973;81:367–93. [6] Bharadwaj P, Bouhelier A, Novotny L. Electrical excitation of [25] Bruus H, Flensberg K. Localized plasmons in point contacts. surface plasmons. Phys Rev Lett 2011;106:226802. Semicond Sci Technol 1998;13:A30–2. [7] Rai P, Hartmann N, Berthelot J, et al. Bouhelier A. Electrical exci- [26] Smetanin IV, Vasil’ev PP. Enhanced longitudinal mode spacing tation of surface plasmons by an individual carbon nanotube in blue-violet InGaN semiconductor lasers. Appl Phys Lett transistor. Phys Rev Lett 2013;111:026804. 2012;100:041113. [8] Du W, Wang T, Chu H-S, Nijhuis CA. Highly efficient on- [27] Johnson PB, Christy RW. Optical constants of the noble metals. chip direct electronic-plasmonic transducers. Nat Photon Phys Rev B 1972;6:4370–9. 2017;11:623. [28] Akinwande D, Petrone N, Hone J. Two-dimensional flexible [9] Bigourdan F, Hugonin J-P, Marquier F, Sauvan C, Greffet J-J. nanoelectronics. Nat Commun 2014;5:5678. Nanoantenna for electrical generation of surface plasmon [29] Li Z, Dorgan VE, Serov AY, Pop E. High-field and thermal trans- polaritons. Phys Rev Lett 2016;116:106803. port in graphen. In: Houssa M, Dimoulas A, Molle A, eds. 2D I.V. Smetanin et al.: Coherent surface plasmon amplification through the dissipative instability of 2D 143
materials for nanoelectronics. Boca Raton, FL, CRC Press Taylor [32] Požela K, Požela J, Jucienė V. High-field electron drift velocity in and Francis Group, 2016, 107–38. InGaAs quantum wells. J Basic Appl Phys 2014;3:110–8. [30] Shin HJ, Kim J, Kim S, et al. Unsaturated drift velocity of mon- [33] Shur M. GaAs devices and circuits. New York, Plenum Press, olayer graphene. Nano Lett 2018;18:1575–81. 1987. [31] Dai Y, Yanga L, Chen Q, Wang Y, Hao Y. Enhancement of the [34] Costantini D, Greusard L, Bousseksou A, et al. A performance of GaN IMPATT diodes by negative differential hybrid plasmonic semiconductor laser. Appl Phys Lett mobility. AIP Advances 2016;6:055301. 2013;102:101106.