aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Dynamic effects in double graphene-layer structures with inter-layerresonant-tunneling negative conductivity
V Ryzhii , , A Satou , , T Otsuji , , M Ryzhii , , V Mitin and M S Shur Research Institute for Electrical Communication,Tohoku University, Sendai 980-8577, Japan Department of Computer Science and Engineering,University of Aizu, Aizu-Wakamatsu 965-8580, Japan Japan Science and Technology Agency,CREST, Tokyo 107-0075, Japan Department of Electrical Engineering,University at Buffalo, Buffalo,New York 1460-1920, USA Department of Electrical, Electronics,and Systems Engineering and Physics,Applied Physics, and Astronomy,Rensselaer Polytechnic Institute,Troy, New York 12180, USA.
We study the dynamic effects in the double graphene-layer (GL) structures with the resonant-tunneling (RT) and the negative differential inter-GL conductivity. Using the developed model,which accounts for the excitation of self-consistent oscillations of the electron and hole densities andthe ac electric field between GLs (plasma oscillations), we calculate the admittance of the double-GL RT structures as a function of the signal frequency and applied voltages, and the spectrumand increment/decrement of plasma oscillations. Our results show that the electron-hole plasma inthe double-GL RT structures with realistic parameters is stable with respect to the self-excitationof plasma oscillations and aperiodic perturbations. The stability of the electron-hole plasma atthe bias voltages corresponding to the inter-GL RT and strong nonlinearity of the RT current-voltage characteristics enable using the double-GL RT structures for detection of teraherz (THz)radiation. The excitation of plasma oscillations by the incoming THz radiation can result in a sharpresonant dependence of detector responsivity on radiation frequency and the bias voltage. Due to astrong nonlinearity of the current-voltage characteristics of the double-GL structures at RT and theresonant excitation of plasma oscillations, the maximum responsivity, R maxV , can markedly exceedthe values (10 − ) V/W at room temperature. I. INTRODUCTION
As demonstrated recently [1–4], double graphene-layer(GL) structures with the inter-GL layers forming rela-tively narrow and low energy barriers for electrons orholes can be effectively used in novel devices. The excita-tion of plasma oscillations in these structures, i.e., the ex-citation of spatio-temporal variations of the electron andhole densities in GLs and the spatio-temporal variationsof the self-consistent electric field between GLs, by in-coming terahertz (THz) radiation, modulated optical ra-diation, or ultra-short optical pulses provides additionalfunctional opportunities. In particular, the double-GLstructures with the inter-GL tunneling or thermionic con-ductance can be used in the resonant THz detectors andphotomixers [5, 6]. As was recently discussed [7–9] andrealized experimentally [10], the inter-GL resonant tun-neling (RT) in the double-GL structures, with the banddiagrams properly aligned by the applied voltage, leadsto inter-GL negative differential conductivity (NDC) andenables novel transistor designs with the multi-valuedcurrent-voltage characteristics. A strong nonlinearity ofthe inter-GL current-voltage characteristics at the volt-ages near tunneling resonance can be used in double-GL- based frequency multipliers [8], detectors [5], and othermicrowave and THz devices. However, NDC might, inprinciple, result in the instability of stationary states,modifying or even harming normal operation of double-GL transistors and two-terminal devices.In this paper, we consider the double-GL structureswith tunneling transparent barrier layers exhibiting RTand NDC. Using the proposed model, we calculate the de-vice admittance and demonstrate that a steady-state cur-rent flow in these structures is stable with respect to theself-excitation of plasma oscillations and aperiodic per-turbations despite NDC. The incoming electromagneticradiation, particularly, THz radiation can result in an ef-fective resonant excitation of plasma oscillations, whichcan be used for the THz detection. We calculate the rec-tified current and the responsivity of the THz detectorsbased on the double-GL RT structures and show, thatdue to a strong nonlinearity of the double-GL current-voltage characteristics and the possibility of the resonantexcitation of plasma oscillations by incoming THz radia-tion, the structures under consideration can serve as THzdetectors exhibiting very high responsivity. V / d B a r r i e r l a y e r - V / 2n - G Lp - G L V / 2 d B a r r i e r l a y e r - V / 2G L V g / 2 - V g / 2G L W g W g G a t eG a t e ( a ) ( b ) e V D B a r r i e r ( c ) e V B a r r i e r ( d )
FIG. 1: Schematic views of (a) a doped double-GL structure and (b) a gated double-GL structure with ”electrical” doping,(c) band diagram at V < V t and (b) at V = V t . Horizontal arrows correspond to inter-GL non-resonant and RT transitions II. MODEL EQUATIONS
We consider the double-GL structures either withchemically doped GLs (figure 1(a)) or with undoped GLsbut sandwiched between two highly conducting gates (fig-ure 1(b)). In the first case, one of the GLs is doped bydonors, whereas the other one is doped by acceptors, sothat the electron and hole sheet densities in the perti-nent GLs are equal tothe dopant density (donors andacceptors) Σ i . In the second case, the voltage V g appliedbetween the gates, induces the electrons and holes withthe sheet charge densities ± e Σ i ∝ V g /W g , where W g isthe thickness of the layers separating GLs and the gates(see figure 1(b)).Each GL is supplied by an Ohmic side contact (left-side contact to lower GL and right-side contact to upperGL) between which the bias voltage V is applied. Thisvoltage affects the electron and hole steady-state densi-ties Σ : Σ = Σ i + κ V π ed . (1)Here e = | e | is the value of the electron charge, κ is thedielectric constant, and d is the spacing between GL (i.e.,the thickness of the inter-GL barrier). Figures 1(c) and1(d) show the band diagrams of the double-GL structuresunder consideration at V < V t and V = V t , respectively.The energy difference, ∆, between the Dirac points inGLs is determined by the following equation (see figure1(c)): ∆ = 2 ε F − eV , (2)where ε F is the Fermi energy of electrons and holes inthe pertinent GL. In practically interesting cases, theelectron and hole systems in GLs are degenerated evenat room temperatures. This occurs when Σ i is suffi-ciently large in comparison with the equilibrium den-sity Σ T : Σ i ≫ Σ T = πk B T / ~ v W , where ~ and k B are the Planck and Boltzmann constants, respectively, v W ≃ cm/s is the characteristic velocity of electronsand holes in GLs, and T is the temperature. In such acase, ε F = ~ v W s π (cid:18) Σ i + κ V π ed (cid:19) , (3)Using equations (1) and (2) and the alignment condition∆ = 0, we obtain the following formula for the alignmentvoltage: eV t = κ ~ v W e d (cid:18) s π e d Σ i κ ~ v W (cid:19) & ~ v W p π Σ i = 2 ε F i . (4)A difference between the quantities eV t and 2 ε F i is dueto the quantum capacitance effect [11].The local value of the inter-GL resonant-tunneling cur-rent density as a function of the bias voltage V is givenby [7, 8] j t = j maxt exp (cid:20) − (cid:18) V − V t ∆ V t (cid:19) (cid:21) , (5)where j maxt is the peak value of the current density and∆ V t = 2 √ π ~ v W /el determines the peak width, l is thecoherence length (the characteristic size of the orderedareas in GLs).Using equation (5) , at small signal variations of thelocal potential difference between GLs ( δϕ + − δϕ − ) , thevariation of the RT current density can be presented as δj t ≃ σ t ( δϕ + − δϕ − ) + β t ( δϕ + − δϕ − ) . (6)Here σ t = ( dj t /dV ) | V = V is the differential tunnelingconductivity and β t = 12 ( d j t /dV ) | V = V determines thenonlinearity of the RT current-voltage characteristics: σ t = − j maxt ∆ V t (cid:18) V − V t ∆ V t (cid:19) exp (cid:20) − (cid:18) V − V t ∆ V t (cid:19) (cid:21) , (7) β t = 2 j maxt (∆ V t ) (cid:20) (cid:18) V − V t ∆ V t (cid:19) − (cid:21) exp (cid:20) − (cid:18) V − V t ∆ V t (cid:19) (cid:21) . (8)As follows from equation (7), σ t changes its sign at V = V t and becomes negative when V > V t . The maximumvalue of | σ t | , which is achieved at V − V t = ± ∆ V t / √ | σ t | = √ e − / ( j maxt / ∆ V t ). Equation (8) forthe value β t at V = V t yields β = − j maxt / (∆ V t ) .The spatial distributions of δϕ + ( x ) and δϕ − ( x ) in theGL plane (along the axis x ) can be found from linearizedhydrodynamic equations (adopted for the energy spec-tra of the electrons and holes in GLs [12]) coupled withthe Poisson equation in the gradual channel approxi-mation [13]. We limit our treatment to the in-phaseperturbations of the electron and hole densities. Forsuch perturbations, the self-consistent ac electric field δE = ( δϕ + − δϕ − ) /d ) is located between GLs. The differ-ence in the local values of the ac potentials of the upperand lower GLs causes the inter-GL current, which caneither increase or decrease with varying ( δϕ + − δϕ − ) de-pending on the value of V − V t . For the out-off-phase per-turbations of the electron and hole densities, the ac elec-tric field is located mainly outside the double-GL struc-ture, so that the inter-GL tunneling is insignificant andno effects associated with NDC can be expected.Calculating the ac potentials, one can neglect the non-linear component of the ac inter-GL current. i.e., thesecond term in the right-hand side of equation (6) (see,however, Sec. IV) and searching for the ac potential inthe following form: δϕ ± = δϕ ± ( x ) exp( − iωt ), where ω is the complex signal frequency. In this case, the systemof equations in question can be reduced to the equationsfor the ac components of the potential at the frequency ω [3, 4]: d δϕ + dx + ω ( ω + iν ) s ( δϕ + − δϕ − ) = − iδj ( ω + iν ) 4 π dκ s , (9) d δϕ − dx + ω ( ω + iν ) s ( δϕ − − δϕ + ) = iδj ( ω + iν ) 4 π dκ s . (10)Here ν is the collision frequency of electrons and holesin GLs with impurities and acoustic phonons and s isthe characteristic velocity of plasma waves in double-GLstructures. Since electrons and holes belong to differ-ent GLs separated by a relatively high barrier, we havedisregarded the electron-hole scattering and, hence, theeffect of mutual electron-hole drag [12]. The character-istic velocity s in the double-GL structures (similar tothat in the two-dimensional electron or hole channelsin the standard semiconductors with metal gates) is de-termined by the net dc electron and hole densities Σ and the inter-GL layer thickness d [14–17]. In double-GL structures with the degenerate electron-hole plasma, s = p π e Σ d/κ m , where m ∝ √ Σ is the ”fictitious”mass of electrons and holes in GLs. This implies that s ∝ Σ / [14]. The value of s in the GL structures un-der consideration can be fairly high, always exceedingthe characteristic velocity of electrons and holes in GLs v W [14, 18]. Considering equations (6), (9), and (10), weobtain Normalized frequency, ω/Ω -2-101234 A d m i tt an c e , a . u . Im( Y ω ) ν t / Ω = -0.01Q = 4 π Re( Y ω ) Re( Y ω )Im( Y ω ) Normalized frequency, ω/Ω -0.2-0.10.00.10.20.3 A d m i tt an c e , a . u . Im( Y ω ) ν t / Ω = -0.01Q = 0.4 π Re( Y ω ) Re( Y ω )Im( Y ω ) FIG. 2: Real and imaginary parts, Re Y ω and Im Y ω ofthe admittance versus signal frequency ω normalized by theplasma frequency Ω for double-GL RT structures with differ-ent plasma oscillation quality factor Q = Ω /ν ( ν t / Ω = − . d δϕ + dx + ( ω + iν t )( ω + iν ) s ( δϕ + − δϕ − ) = 0 , (11) d δϕ − dx + ( ω + iν t )( ω + iν ) s ( δϕ − − δϕ + ) = 0 . (12)Here ν t = 4 πσ t d/k is the characteristic frequency of theinter-GL tunneling. At V = V t , σ t = 0 and ν t = 0, whileat V & V t , σ t < ν t < V = V + δV ω , where δV ω is thesmall signal voltage component, one can use the followingboundary conditions for equations (11) and (12): δϕ ± | x = ± L = ± δV ω − iω t ) , dδϕ ± dx (cid:12)(cid:12)(cid:12)(cid:12) x = ∓ L = 0 . (13)The latter boundary conditions reflect the fact that theelectron and hole currents are equal to zero at the dis-connected edges of GLs (at x = − L in the upper GL andat x = L in the lower GL), while the difference of the acpotentials δ V ω can generally be nonzero.Solving equations (11) and (12) with boundary condi-tions (13), we obtain δϕ + − δϕ − = δV ω (cid:18) cos γ ω xγ ω sin γ ω L cos γ ω Lγ ω sin γ ω L − L (cid:19) , (14)where γ ω = p ω + iν t )( ω + iν ) /s . III. ADMITTANCE OF THE DOUBLE-GLRESONANT-TUNNELING STRUCTURES
First, we calculate the small-signal admittance, of thedouble-GL RT structures, Y ω = δJ ω /δV ω , where δJ ω = H (cid:18) − i κω π d + σ t (cid:19) Z L − L dx ( δϕ + − δϕ − ) (15)is the net ac current, including the displacement currentand H is the width of the double-GL structure in thedirection perpendicular to the currents. Using equations(14) and (15), we find Y ω = − i (cid:18) κ HL π d (cid:19) ( ω + iν t )[ γ ω L (cot γ ω − γ ω L )]= − i (cid:18) κ HL π d (cid:19)s(cid:18) ω + iν t ω + iν (cid:19) Ω(cot γ ω − γ ω L ) . (16)Here we have introduce the plasma frequency Ω =( π s/ √ L ), so that γ ω L = π p ( ω + iν t )( ω + iν ) / Ω. Inthe limit ω = 0, equation (16) yields Y = 2 HLσ t ifΩ ≫ ν . This implies that in this case the dc admit-tance is determined by the inter-GL conductivity. WhenΩ ≪ | ν t | ν , we find Y = ( κ HL/ π d )(Ω /πν ) ∝ σ ,where σ ∝ ( e Σ /mν ) stands for the dc conductivity ofGLs.Figure 2 shows the frequency dependences of the realand imaginary parts of the admittance calculated usingequation (16) for different values of the plasma oscilla-tion quality factor Ω /ν ( Q = 4 π and Q = 0 . π ) and ν t / Ω = − .
01. As seen, Re ( Y ω ) is negative in a nar-row range of small frequencies. This is due to NDC at( V − V T ) & ∆ V t associated with RT. However, at highersignal frequencies, Re ( Y ω ) is positive. One can also seethat Im ( Y ω ) does not change its sign in the frequency re-gion, where Re ( Y ω ) <
0. This implies that the electron-hole plasma is stable, at least for the chosen parameters.Indeed, the stability of stationary state of the electron-hole plasma at given bias voltage V is determined usingthe condition Z ω = Y − ω = 0. Considering the latter, weobtain the following condition for the damped or growing of perturbations of the electron and hole densities, i.e.,the following dispersion equation for plasma oscillations:cot γ ω L − γ ω L = 0 . (17)Equation (17) determines the complex frequency ω = ω ′ + iω ′′ , where ω ′ is real value of possible plasma os-cillations and ω ′′ is their damping/growth rate. For thestructures with the characteristic plasma frequency Ω ≫ ν, | ν t | , equation (17) yields ω ′ = ω n , ( n = 0 , , , , .. ) and ω ′′ = Γ, where ω ≃ . π Ω , ω n ≃ n Ω + Ω π n , Γ ≃ − ν + ν t . (18)If Ω ≪ ν , we find ω ′ = 0 from equation (17)Γ ≃ − Ω π ν − ν t . (19)The plasma frequency can vary in a wide range de-pending on the structure length 2 L . In particular, setting s = (2 − × cm/s and 2 L = 1 µ m (as in Ref. [10]),one obtains Ω / π ≃ . − . ω ≃ . − . L = 10 µ m,one can get Ω / π ≃ . − .
28 THz.As follows from equations (18) and (19), the plasmainstability (the increment, i.e,the growth rate Γ >
0) ispossible if ν t < − ν in the structures with a high qualityfactor Q = Ω /ν ≫ ν t < − Ω /π ν in the structureswith Q ≪
1. In the former case, plasma oscillations withthe frequencies ω n can self-excite, while in latter case,the growth of the perturbations of the electron and holedensities is aperiodic, which could potentially result inthe domain formation. In such situations, the stationarycurrent flow between GLs could be unstable. However,in real double-GL structures the value of the differen-tial inter-GL RT conductivity is not sufficiently large toprovide the condition ν t < − ν . To estimate the realvalue of ν t , we assume that j maxt = (5 −
30) A/cm [8–10] and l = 100 nm, so that V t ≃
30 mV. Thisyields | σ t | = (143 − . If d = 4 nm and κ = 4 (hBN four atomic layers thick barrier), one ob-tains | ν t | ≃ (1 . − . × s − . As seen, at realis-tic ν = (10 − ) s − , the value | ν t | ≪ ν . This isin contrast to the double-barrier RT devices based onInGaAs-AlAs, where the frequency | ν t | can be ratherhigh, being of the order of or even exceeding the elec-tron collision frequency. This is due to a very high peakcurrent density and modest width of the tunneling reso-nance ∆ V t in the double-barrier RT diodes. This can leadto the instability of the stationary current with respectto the self-excitation of plasma oscillations [19]. For in-stance, in one of the best resonant-tunneling diode [20], κ = 12, | σ t | = 3 . × S/cm and d = 31 nm, so that ν t ≃ s − ν t < − Ω /π ν can be satisfied if the collision frequency Frequency, ω /2 π (THz)10 -2 -1 N o r m a li z ed r e s pon s i v i t y , | R V / R V m a x | Ω/2π = 1.0 THz ν = 0.5 x s -1 x s -1 x s -1 Frequency, ω /2 π (THz)10 -2 -1 N o r m a li z ed r e s pon s i v i t y , | R V / R V m a x | Ω/2π = 2.0 THz ν = 0.5 x s -1 x s -1 x s -1 Frequency, ω /2 π (THz)10 -2 -1 N o r m a li z ed r e s pon s i v i t y , | R V / R V m a x | Ω/2π = 3.0 THz ν = 0.5 x s -1 x s -1 x s -1 FIG. 3: Normalized responsivity R V /R V max at versus signal frequency ω for different values of the electron and hole collisionfrequency ν and plasma frequency Ω. ν is large and, hence, the mobility of electrons and holesis low. Indeed, setting Ω / π = 0 . − .
28 THz and ν = (1 − × s − , the latter inequality needs | ν t | > (1 . − , × s − . The latter condition is notmet for the double-GL structures considered recently [8–10]. IV. RESONANT DETECTION OF RADIATION
The ac potential δV ω between the contacts to GLs canarise due incoming electromagnetic radiation received byan antenna. This results in the excitation of plasma os-cillations in the double-GL structure described by equa-tion (14). The ac potential drop ( δϕ + − δϕ − ) causes notonly the linear component of the tunneling current δJ ω but also the rectified dc current, δJ , associated with thenonlinear (quadratic) component. The rectified ac cur-rent is given by the following formula: δJ = H Z L − L dxβ t | δϕ + − δϕ − | , (20)This rectified component of the inter-GL RT currentcan be used for detection of THz signals. Using equa-tions (14) and (20), we obtain the following formulae forthe rectified current δJ and the detector Volt-Watt re-sponsivity R V δJ = β t ( δV ω ) LH | cos γ ω L − γ ω L sin γ ω L | Z − dξ | cos( γ ω Lξ ) | . (21) R V = R V R − dξ | cos( γ ω Lξ ) | | cos γ ω L − γ ω L sin γ ω L | . (22)Here the characteristic responsivity R V is given by R V = 2 πcG (cid:18) β t σ t (cid:19) = 2 πcG (cid:18) β t V j t (cid:19) , (23) where c is the speed of light in vacuum, G ≃ . σ t = j t /V and j T are the inter-GLdc conductivity and current density, respectively. Con-sidering that at V = V t (when the resonant-tunnelingcurrent exhibits a maximum) above expression yields β t = j maxt / (∆ V t ) , we arrive at the following formula R V max = − πcG (cid:20) V (∆ V t ) (cid:21) . (24)Equation (23) provides the frequency dependence simi-lar to the responsivity R nV of the double-GL detectorsusing the nonlinearity of the inter-GL relatively smoothcurrent-voltage characteristic with the tunneling assistedby electron scattering (non-resonant tunneling detector)considered by us recently [5]. However, there are twodistinctions (apart from the difference in the R V sign).First, the absolute value of R V is much larger than R nV .This is due to a significantly larger value of | β t | in theRT detectors associated with a high and sharp RT peak.Indeed, considering equation (24) and the pertinent equa-tion in [5], one can arrive at R V max R nV ≃ (cid:18) V ∆ V t (cid:19) . (25)Setting ∆ V t = 30 −
90 mV and V = 1000 mV, we obtain R V /R nV ≃ − . At the above parameters, assumingthat j t = j maxt , one obtains R V ≃ (1 . − . × V/W.Second, the responsivity peaks of the inter-GL RT de-tector width is characterized by the collision frequency ν (because ν t = 0), but in the detectors using non-resonanttunneling the width of the peaks is determined by ( ν + ν t )with ν t >
0. The possibility to achieve very high valuesof the characteristic responsivity R V is connected witha large value of β t and, hence, small values of the widthof tunneling resonance ∆ V t . One can assume that thelatter quantity weakly depends on the temperature [8],so that the responsivity can be very high even at roomtemperature.As follows from equation (22), the frequency depen-dence of the responsivity exhibits sharp maxima at the Voltage swing, ( V - V t )/ ∆ V t N o r m a li z ed r e s pon s i v i t y , | R V / R V m a x | Ω /2 π = 3.0 THz ν = 0.5 x s -1 ω /2 π = 0.82 THz 3.35 THz3.27 THz3.19 THz FIG. 4: Dependence of normalized responsivity R V /R V ver-sus bias voltage swing ( V − V t ) / ∆ V t .n at different signal fre-quency ω near the zeroth plasma resonance (upper panel) andnear the first plasma resonance (lower panel). plasma resonant frequencies ω = ω n , where the fre-quencies ω n are given by equation (16). The widths ofthe peaks are determined by the parameter Γ ∝ ν (atthe RT resonance ν t = 0). At the plasma resonances, | R V | ≫ R V . Thus, very high values of the responsiv-ity of the detectors in question can be achieved due tocombining of the tunneling and plasma resonances.Since the resonant plasma frequencies fall into the THzrange, the detector under consideration can be particu-larly useful for the resonant detection of THz radiation.Figure 3 demonstrates examples of the frequency de-pendences of the responsivity calculated using equation(22) for the double-GL RT structures with different val-ues of the electron and hole collision frequency ν anddifferent values of the plasma frequency Ω. The ob-tained dependences of the responsivity versus signal fre-quency exhibit several resonant peaks associated with theplasma oscillations. The highest peaks correspond to thezeroth resonances at the frequency ω < Ω, while theother resonances correspond to multiples of the plasmafrequency Ω (see equation (18)). One can see that R V > R V max not only at the zeroth plasma resonance,but also a higher resonances. The number of such reso-nances depends on the quality factor Q . The responsivityis very high due RT with sharp maximum at the cur-rent voltage-characteristics even at the moderate qualityfactors. However, it is much higher at the pronouncedplasma resonances.The high values of the responsivity with the frequencycharacteristics of figure 3 are associated with the com-bination of the tunneling and plasma resonances. Thedeviation from the tunneling resonance leads to loweringof the responsivity. If V = V t , the factor R V in equation (22) becomes smaller than R V max : R V = R V max (cid:20) (cid:18) V − V t ∆ V t (cid:19) − (cid:21) × exp (cid:20) − (cid:18) V − V t ∆ V t (cid:19) (cid:21) . (26)Figure 4 shows the dependences of the normalizedresponsivity, R V /R V max on the voltage swing ( V − V t ) / ∆ V t calculated using equation (26) for different sig-nal frequencies in the vicinity of the zeroth and firstplasma resonances. The plasma frequency at V = V t waschosen to be Ω / π = 3 . V is given by Ω = Ω [1 + ( V − V t ) /V ] / , where V = 4 π ed/κ Σ i . At κ = 4, d = 4 nm,and Σ i = (1 − × cm − , V ≃ −
900 mV. Weset V / ∆ V t = 6 and ν = 0 . × s − . As seen fromfigure 4, an increase in the absolute value of the voltageswing | V − V t | / ∆ V t results in a marked drop of the re-sponsivity. It is also seen that detuning of the plasmaresonance leads to a significant decrease in the respon-sivity (compare the curves for the plasma resonances at ω/ π ≃ .
82 THz and ω/ π ≃ .
27 THz with those cor-responding to a detuning δω/ π = ± .
08 THz. Slightlydifferent maximum values of the responsivity shifted fromthe plasma resonances are due to a small asymmetry ofthe resonant peaks. Different positions of these maximaare associated with the dependence of the plasma fre-quency on V .The excitation of plasma oscillation by electromagneticsignals can be used not only for the resonant reinforce-ment of the rectified current (i.e., the detector respon-sivity), but also for a more effective generation of higherharmonics [8]. V. CONCLUSIONS
In summary, we considered the dynamic behaviorof the double-GL RT structures. We calculated thefrequency-dependent admittance and the responsivity ofthe double-GL RT structures to the incoming signals asfunctions of the structural parameters, bias voltages, andfrequency. It was demonstrated that the stationary statesof the electron-hole plasma are stable with respect to theself-excitation of plasma oscillations and aperiodic per-turbations for the structures with realistic parameters.As shown, the responsivity exhibits sharp resonant max-ima corresponding to the excitation of plasma modes byincoming electromagnetic radiation. The plasma oscil-lations and the pertinent responsivity peaks are in theTHz range. The responsivity of the double-GL RT de-tectors operating at room temperature can exhibit veryhigh values markedly exceeding (10 − ) V/W. Acknowledgments
The work was supported by the Japan Science andTechnology Agency and the Japan Society for Promotion of Science, PIRE TeraNano Program, NSF, USA, and theArmy Research Laboratory under ARL MSME Alliance,USA. [1] Liu M, Yin X, Ulin-Avila E, Geng B, Zentgraf T, Ju L,Wang F and Zhang X 2011 Nature ...[7] Feenstra R M, Jena D, and Gu G 2012 J. Appl. Phys Physics of Semiconductor Devices (Engle-wood Cliffs, NJ: Prentice-Hall)[14] Ryzhii V, Satou A and Otsuji T 2007 J. Appl. Phys. L923[19] Ryzhii V and Shur M 2001 Jpn J. Appl. Phys.5