Optical Transistor for an Amplification of Radiation in a Broadband THz Domain
Kristian Hauser A. Villegas, Fedor V. Kusmartsev, Y. Luo, Ivan G. Savenko
OOptical Transistor for an Amplification of Radiation in a Broadband THz Domain
K. H. A. Villegas, F. V. Kusmartsev,
2, 3, ∗ Y. Luo, and I. G. Savenko
1, 4 Center for Theoretical Physics of Complex Systems,Institute for Basic Science (IBS), Daejeon 34126, Korea Micro/Nano Fabrication Laboratory Microsystem and THz Research Center, Chengdu, Sichuan, China Loughborough University,UK A. V. Rzhanov Institute of Semiconductor Physics,Siberian Branch of Russian Academy of Sciences, Novosibirsk 630090, Russia (Dated: March 3, 2020)We propose a new type of optical transistor for a broadband amplification of THz radiation. It ismade of a graphene–superconductor hybrid, where electrons and Cooper pairs couple by Coulombforces. The transistor operates via the propagation of surface plasmons in both layers, and the originof amplification is the quantum capacitance of graphene. It leads to THz waves amplification, thenegative power absorption, and as a result, the system yields positive gain, and the hybrid acts likean optical transistor, operating with the terahertz light. It can, in principle, amplify even a wholespectrum of chaotic signals (or noise), that is required for numerous biological applications.
The growing interest in terahertz (THz) frequencyrange (0.3 to 30 THz) is due to its potential appli-cations in diverse fields such as non-destructive prob-ing in medicine, allowing for non-invasive tumor detec-tion, biosecurity, ultra-high bandwidth wireless commu-nication networks, vehicle control, atmospheric pollu-tion monitoring, inter-satellite communication, and spec-troscopy [1–4]. However, the THz range still remains achallenge for modern technology due to the lack of a com-pact, powerful, and scalable solid state source [5]. Thisproblem is known as the terahertz gap .To ‘close’ this gap from the lower frequencies, one canmention electronic devices with negative differential re-sistance (NDR). For instance, super-lattice electronic de-vices (SLED) generate higher harmonics by means ofNDR [6] and can reach 0.5 THz gap, while the outputpower is less than 0.5 mW [7]. The radiation power ofresonant tunneling diodes (RTDs) [8] is less than 1 µW ,and it further decreases by three orders of magnitude atroom temperature. Also, RTDs suffer from their smallelectron transition times and parasitic capacitance, asso-ciated with the double-barrier structure.The use of layered high-temperature superconductors(HTSC) with intrinsic Josephson junctions, such as BIS-CCO [9–11] can produce radiation with Josephson os-cillations generated by an applied bias voltage [12, 13].Here a tunable emission, from 1 to 11 THz, has been re-cently observed [14]. However, the power output is 1 µ W,which is still inadequate for practical applications.It can be enhanced with the use of Bose-Einsteincondensates [15, 16]. However, such approach requiresa hybridization of several bands with different parity,making the output power small. Quantum cascadelasers (QCL) [17–19] can generate a high-frequency THz-radiation, while transistors [20–23], Gunn diodes [24],and frequency multipliers [25] are approaching the THz-gap from the low-frequency side. The latter covers thewhole THz range while having small power. The general fundamental obstacle of all these THz sources is the smallemission rate (of the order of 10 ms). It can be increasedwith the Purcell effect when the THz source is placedin a cavity [26, 27], however, the quantum efficiency isstill about 1%, and the manufacture of these devices isdifficult.Graphene and carbon nanotubes may serve as highlytunable sources and detectors of THz radiation [28–34], and even in THz lasers [35–40]. In the dual-gategraphene-channel field-effect transistor [41] embeddedinto a cavity resonator [42, 43], one observes spontaneousbroadband light emission in the 0.1-7.6 THz range withthe maximum radiation power of ∼ µ W at the tem-perature 100 K. There are also emerging sources of THz-radiation that can deliver several frequencies at roomtemperature, e.g., multiple harmonic generation in super-lattices [44, 45], frequency difference generation in midinfrared QCLs [46] and THz optical combs [47].Graphene covered with a thin film of colloidal quantumdots has strong photoelectric effect, that provides enor-mous gain for the photodetection (about 10 electronsper photon) [48]; graphene grown on SiC has strong pho-toresponse [49]; and graphene composites can improvesolar cells efficiency [50]. Note, both graphene and super-conductor alone are practically insensitive to light [51].In this Letter, we show that graphene placed in the vicin-ity of a superconductor represents an active media withstrong light-matter coupling. It can operate as an opticaltransistor that amplifies broadband electromagnetic ra-diation. This new feature allows one to use this device forstudying chemical and biological processes or in telecom-munications for encryption-decryption procedures, whereit is important to image the whole spectrum.We consider a system consisting of parallel layers ofgraphene and a superconductor, exposed to an elec-tromagnetic (EM) field incident with the angle θ andlinearly polarized along the x-z plane (p-polarisation), E ( r , t ) = (sin θ, , cos θ ) E e − i ( k ⊥ z + k (cid:107) · r + ωt ) , where k (cid:107) , a r X i v : . [ phy s i c s . a pp - ph ] M a r a ēē superconductorgraphenelight θ pump FIG. 1. System schematic. Graphene coupled with a two-dimensional superconductor by the Coulomb force and con-nected to an electrical pump source (a battery). Figure alsoshows the pump-probe configuration for the THz radiationamplification: The hybrid system is exposed to an externallaser ( pump , depicted by red arrow) and broadband EM fieldat incidence angle θ ( probe , depicted by yellow arrows). Thefrequency of the pump (probe) should be above (below) thesuperconducting gap. Both the optical and electrical pumpcan provide energy for the amplification. ω , and r are the in-plane wave vector of the field, fre-quency, and coordinate, respectively (see Fig. 1). Be-tween graphene layer and superconductor there is a gatevoltage that controls its chemical potential and providesAC power. It may pump the energy in the supercon-ductor with AC bias, larger than the superconductinggap exciting quasiparticles and creating NDR. Alterna-tively, the battery can be replaced by an external laser(pump) with frequency exceeding the superconductinggap (Fig. 1).The electrons in graphene are coupled by the Coulombinteraction, which has the Fourier image given by v k =2 πe /k , where k is in-plane momentum (lying in the x-y plane). The electrons between the two layers are alsoCoulomb-coupled, and the Fourier image of the interlayerinteraction reads u k = 2 πe exp( − ak ) /k , where a ∼ δn kω and Cooper pair density fluctuationsin the superconducting layer δN kω as [54] δn kω = Π kω ( v k δn kω + u k δN kω + W kω ) ,δN kω = P kω ( v k δN kω + u k δn kω + W kω ) , (1)where Π kω = Π Rkω + i Π Ikω and P kω = P Rkω + iP Ikω are thecomplex-valued polarization operators of the grapheneand superconductor, respectively, and W kω = eE /ik isthe Fourier image of the potential energy due to the ex-ternal electric field. From Eqs. (1) we can find the densityfluctuations in graphene n kω and in superconductor N kω as linear functions of applied electric field amplitude E (see Supplemental Material [55] for details). Collectiveplasmonic hybrid modes in graphene and superconductorcan be found from the same system of equations, takinginto account the expressions for the polarization loops of the superconductor [54] and graphene [56, 57]. Substitut-ing the expressions for n kω and N kω into the continuityequations, kj kω = − eωδn kω and kJ kω = − eωδN kω , forgraphene and superconductor, respectively, we can deter-mine the electric currents in each of the layers and theirimpedances Z G and Z SC . The collective modes of thehybrid system are presented in Fig. 2(a) for the undopedand doped graphene cases. The upper mode has a gap2∆ = 2 meV. If in this hybrid a single graphene layeris not interacting with a superconductor, only one modeexists, which is due to the superconductor.The formula for the power absorption or gain reads [58] P ( ω ) = 12 (cid:28) R e (cid:20)(cid:90) d r J ( r , t ) · E ∗ ( r , t ) (cid:21)(cid:29) , (2)where the integration is over the graphene plane, (cid:104)· · ·(cid:105) denotes time-averaging. We normalize the power withthe sample area (cid:82) d = l and the square of the fieldamplitude E to get P ( ω ) = P ( ω ) l E = 12 eωkE R e (cid:2) δn kω (cid:3) . (3)Figure 2(b) shows the dependence of the power absorp-tion on the EM field incidence angle θ calculated with (3),fixing µ = 0 at the Dirac point by gate voltage. All thecurves exhibit critical angles at which the power absorp-tion becomes negative, α <
0. This suggests that theincident angle can be used to switch the amplifier deviceon or off. Furthermore, increasing the frequency of theincident EM wave increases the critical angle.Figure 2(c) shows the power absorption spectrum. Wesee that coupling graphene to the superconductor layerresults in a negative power absorption in THz frequencyrange (solid curves and the shaded regions). There is nonegative absorption region for isolated graphene, wherethe power absorption remains positive for any frequency ν (dashed curves). When the light incidence angle θ in-creases, both the maximal intensity (slightly) and thefrequency range of the negative light absorption increase,see the shaded area in Fig. 2(b,c). Thus, the angle of lightincidence allows us to control the range of light frequen-cies with the negative absorption.To understand the θ -dependence, note that the wavevector of the plasma wave is related to the projection ofthe incident light wave vector on the plane of the sample.Both the angular dependence of the absorption and gainare related to the amplitude of this wave propagatingon the surface. The light incident perpendicular to thegraphene surface can not excite such plasma waves andtherefore, in this case we do not have the gain. However,at large incident angle, there is a reflection of the incidentradiation due to the difference in the refractive index ofthe hybrid and air. Thus we conclude that the most opti-mal effect will be observed at small but nonzero θ . If thesystem is embedded into a cavity resonator, there might Quasiparticle dispersion � � � � � ���������������� ν ( ��� ) (cid:1) ( � � � � �� � � � ) ��� ��� ��� ��� ������������������ θ ( ������� ) (cid:1) ( � � � � �� � � � ) (c) ��� ��� ��� ��� ��� ��������������� � ( ��� ) ω ( � � � ) ���� ���� ���� ���� ���� ���� �������� × �� - � (a) (b) ��� ��� ��� ��� ��� - ������������ ω ( ��� ) × - T - parameter (THz) ( a r b . un it s ) ( a r b . un it s ) T - parameter (degrees) k (meV) ω ( m e V ) FIG. 2. (a) Hybrid collective plasmonic modes for undoped (blue curves) and doped (red curves, µ = 3 . θ [see also Fig. 1] for graphene-superconductor hybrid for frequencies ν = 0 . . . θ = 1 . ◦ (red), θ = 1 . ◦ (green), and θ = 3 . ◦ (blue). Dashed curves show the data corresponding to the isolated graphene case.In (b) and (c) µ = 0; in insets, the effect of temperature T = 0 (red), T = 0 . T c (green), and T = T c (blue) is shown. Theseparation between graphene and superconductor is a = 10 nm. even arise lasing similar to one observed in plasmoniclattices [59, 60] or semiconductor superlattices [43].The mechanism of gain here is similar to one in awaveguide coupled with a superconducting Josephsonjunction [61]. Then, the optical reflectivity of the systemreads Γ = ( Z G − Z SC ) / ( Z G + Z SC ) . Near the frequencyof the plasmon resonance, there is an area of negativedifferential resistance of the superconductor, R SC < X G = 0 and X SC = 0 [61, 62], we findΓ >
1. Note, that graphene can also have NDR (seeSec. III of [55] and [63–65]) as in a graphene transistor,which consists of two graphene layers separated by a BNinsulator [66].The graphene-superconductor junction (Fig. 1) haslarge tunneling resistance. An electron in graphene withenergy below the superconducting gap can tunnel intosuperconductor only due to the Andreev scattering [67].The probability of such tunnelling is small since all elec-trons are paired. Therefore, the resistance of the junc-tion is high. With applied bias voltage above the gap,quasiparticles appear and they can tunnel. As a result,the resistance decreases and NDR arises. The latter canappear even at zero bias when we pump the supercon-ductor with external light with the frequency above thegap. The light excites electron and hole quasiparticlescoexisting with superconducting fluctuations on the sur-face of the superconductor [68]. Then, in addition tothe Andreev scattering, there starts normal tunneling ofquasiparticles into the superconductor. The resistance ofthe junction decreases and the NDR arises. Such mech-anism of NDR can exist only in a highly nonequilibriumexcited state created by the pump.Scattering and de-phasing mechanisms are limiting thegain bandwidth and can flat and eliminate the gain [69].Here the plasmon scattering within each and betweentwo graphene and superconducting layers can not only broaden the width of the optical transition, but also en-able optical gain and absorption to coexist, constitutingthe Wacker-Pereira mechanism of optical gain [69–71], asobserved in QCLs [72, 73]. It provides one of the ex-planations, why α ( ω ) = 1 − Γ( ω ) is negative below theplasmon resonance.From another perspective, graphene separated by di-electric layer (e.g., made of BN, SiO or Ta O ) fromthe superconductor (Nb, Pb, or HTSC) together forma parallel plate capacitor, which capacitance C is givenby C = C plate + C q , where C plate is the classical capaci-tance C plate = (cid:15) A/a , A is the area of the sample and (cid:15) o is the dielectric constant (e.g., (cid:15) o = 3 . ). Thequantum capacitance C q of graphene emerges due to itsconical energy-momentum relation, and it has the form C q = 2 Ae | E F | /π (cid:126) v F [74–76].This parallel plate capacitor is connected to a powersupply that charges the capacitor and provides the en-ergy for the amplification of incident radiation. Theincident light (in particular, its component parallel tothe superconducting surface) induces the fluctuation ofcharge density δn s , which is associated with a travel-ling plasmon wave with the amplitude E s ∼ δn s , where δn s = δn s cos( kx + ωt ) (Fig. 3). This charge densitywave on the surface of the superconductor generates amirror charge wave of the opposite sign in the neutralgraphene layer, being of the same order as the charge fluc-tuations in superconducting layer, i.e. δn G ∼ δn s . Notethat the sign of these charge density fluctuations changeseach half-wavelength of the plasmon wave. These plas-mons have the wavelength larger than a micrometer, andtherefore the charge density for each half-wavelength canbe viewed at as a local temporary graphene doping, mov-ing with the wave. During this half-period, the chargefluctuation corresponds to the local change in the chem-ical potential or the Fermi energy E F , which is directly E G E SC E F E F + superconductorgraphene - ++ ++ + -- --- FIG. 3. Schematic of the mechanism of THz amplification.The incident light induces a collective hybrid plasmon mode.The interplay between this mode and the quantum capaci-tance of graphene amplifies the incident electromagnetic field(see text for details). related to the amplitude of the plasmon wave propagat-ing in graphene E G . Due to the quantum capacitanceof graphene, described by the relation E F ∼ √ n G , thewaves amplitude, E G ∼ E F , is significantly enhanced andit is different from the amplitude of the plasmon wave,propagating on the surface of the superconductor.In order to stimulate radiation of electromagneticwaves in the THz range (similar to photo-conductiveantennas [77]), one has to add a supply of energy. Itcan be done by introducing a pump-probe setup, wherethe energy required for the amplification of the probecomes from the pump. For the pump, we have to ex-pose graphene to an external laser with the frequencyabove the superconducting gap. Or we can apply ACbias voltage with the amplitude larger than the gap. Dueto pump, there forms a state, in which energy is accu-mulated in electronic excitations. To have the pump-probe configuration, we expose the junction to an ad-ditional external light (probe) with the frequency lowerthan the gap. Then, there on the surface will emergesuperconducting charge density fluctuations. Effectivelythey represent coherent waves traveling inside the capac-itor formed by graphene and superconductor. Due tothe graphene quantum capacitance [74–76], the ampli-tude of these electromagnetic waves becomes amplified(by pump), resulting in reflection coefficient larger than1 (as shown above).Note that the graphene–superconductor system pos-sesses many benefits. Both superconducting andgraphene layers have giant mobility and small resistivitygiving rise to minute losses. With increasing temperaturethe superconducting gap decreases, while graphene mo-bility changes a little. Since the radiation amplification occurs in graphene the temperature has a weak effect onthe operation of THz transistor (see insets in Fig. 2(b,c).The working temperature range is similar to one for thestack of the Josephson junctions made of HTSC [9, 10]and it is limited by the critical temperature T c .The radiation power reads P r = (cid:104) V g I (cid:105) , where thevoltage V g and the transverse current I are periodicallychanging in time around the Dirac point. Then, as-suming simple periodic behavior for both I ( t ) and V g ( t )and the graphene-superconductor separation, a =10 nm[82], the maximal outcome power reads (cid:104) I × V g (cid:105) ∼ µ W/cm [81]. Evidently, it can be increased forlarger areas of the surface or employing multilayer hy-brid structures. Conclusions.
We have shown that in a hybridgraphene–superconductor system exposed to an electro-magnetic field of light the absorption coefficient can be-come negative in a certain range of frequencies and ata non-zero angle of incidence. We suggest that the sys-tem can serve as an amplifier of THz radiation. Theessence of the amplification is the quantum capacitanceof graphene, which provides the conversion of the chargedensity wave induced by incident light into emitted radi-ation with much stronger intensity. That is also relatedto the negative differential conductivity of the hybrid,where there is a strong Coulomb coupling of grapheneand superconductor.Such devices are now in strong demand and may becomplementary to quantum cascade lasers. Moreover,the usage of high-temperature superconductors extendsthe range of temperatures required for their operation.The existence of Dirac or Weyl cones in graphene, topo-logical insulators, and Weyl semimetals brings in a newphysical concept called quantum capacitance. Its essenceis in a strong dependence of the Fermi energy on thecharge doping. A weak charge density wave can inducea strong electric field in these materials, allowing us toachieve the amplification of incident electromagnetic ra-diation.The situation is somewhat similar to lasers, where thepumping results in the population inversion. The dif-ference is that here the amplification can occur in abroad frequency range simultaneously, while in lasers it ispinned to a specific resonant frequency. Such amplifica-tion of the broadband spectrum, e.g., for chaotic or noiseradiation, opens exciting opportunities of new types ofmolecular and biological noise spectroscopy, where theresponse of the system can be measured in a broad fre-quency range opening new opportunities in molecularand biological noise spectroscopy [83, 84].
Acknowledgements.
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