Terahertz plasmonic resonances in coplanar graphene nanoribbon structures

We analyze plasmonic oscillations in the coplanar graphene nanoribbon (GNR) structures induced by the applied terahertz (THz) signals and calculate the GNR impedance. The plasmonic oscillations in the CNR structures are associated with the electron and hole inductances and the lateral inter-CNR capacitance. A relatively low inter-GNR capacitance enables the resonant excitation of the THz plasmonic oscillations in the CNR structures with long GNRs. The GNR structures under consideration can be used in different THz devices as the resonant structures incorporated in THz detectors, THz sources using resonant-tunneling diodes, photomixers, and surface acoustic wave sensors.

In the present paper, we explore the structures comprising two co-planar GNRs on a dielectric substrate focusing on their terahertz applications.In such structures the plasmonic wave number is determined by the GNR length, which is much longer than their widths.As a result, the perpendicular plasmonic modes are not excited in the frequency range under consideration (the THz range).
It is assumed that the bias dc voltage V G and the THz signal voltage δV ω with the frequency ω are applied between the GNR edges.The bias voltage results in the formation of the two-dimensional electron and hole systems, 2DES and 2DHS.The signal voltage can be produced by the impinging electromagnetic radiation and received by an antenna integrated with the GNR structure.The THz signals excite the plasmonic standing waves (plasmonic oscillations) in the GNRs.Examining the plasmonic properties of the co-planar GNRs, we show that with a relatively low geometrical inter-GNR capacitance, the plasmonic resonances can be in the THz range even for fairly long GNRs.This property of the coplanar GNRs, as well as their lateral periodic arrays, can find applications in different devices using the THz plasmonic resonant structures.

II. EQUATIONS OF THE MODEL
Considering the co-planar GNR structures, we assume two types of signal voltage bias, P-GNR and S-GNR, as shown in Fig. 1(a) and Fig. 1(b).The lengths and widths of the GNRs are 2H and 2L G , respectively, whereas the spacing between the GNRs is equal to 2L being much smaller than other geometrical parameters (L G , L ≪ H ).
The GNR potentials are ϕ(z, t) ± = δϕ ± ω (z)e −iωt , where the signs "+" and "-" correspond to the GNRs to which the voltages of different polarities are applied.The displacement current density between the GNRs and the continuity equations for the electrons and holes governing their transport along the GNRs can be presented as (see Appendix) Here and are the GNR ac conductivity and the geometrical inter-GNR capacitance, respectively, where σ G = e 2 µ/π 2 ν is the dc Drude conductivity of a GNR with the carrier Fermi energy µ and the carrier collision frequency ν (scattering by impurities and acoustic phonons), κ = (κ S +1)/2 and κ S are the effective dielectric constant and the substrate dielectric constant (the dielectric constant of the media above the structure is assumed to be unity), the factor reflects the GNR blade-like shape [30] (see also Refs.[31][32][33][34][35]) with parameter a = L G /L.The electron and hole Fermi energy in the pertinent GNR is determined by the bias voltage , where v W ≃ 10 8 cm/s is the characteristic electron and hole velocity in graphene, Σ G the steady-state electron and hole density, and is the Planck constant.
Using Eqs. ( 1) -(3), we arrive at the following equations governing the potentials δϕ ± ω : with s = 2e 2 µ L G /π C 2 being the characteristic velocity of the plasmonic wave along the GNRs.Depending on the signal voltage application, the boundary conditions for Eqs. ( 5) and ( 6) are: for the situation corresponding to the device corresponding to structure Fig 1(a) P-GNR, and for the device shown in Fig. 1(b) -interdigital GNR connection S-GNR.
Figure 2 shows the dependences of the plasmonic frequency Ω/2π on the bias voltage V G for different GNR geometries calculated using Eq. ( 13) with Eq. ( 4).It is assumed that κ S = 4, H = 10 µm, and L G /L = 2.In line with Eq. ( 12), an increase in the bias voltage V G leads to an increase in the characteristic plasmonic frequency Ω due to the carrier density Σ G and the carrier Fermi energy µ rise when V G increases.The plasmonic frequency Ω is sensitive to the geometrical parameters L G and L, which determine the carrier density and the GNR ac conductivity.The specifics of the GNR structures under consideration lead to relatively small inter-GNR capacitance and elevated values of the plasmonic wave velocity s.Consequently, rather high values of plasmonic frequency Ω (in the THz range) can be realized at fairly large lengths of the GNRs.

IV. GNR STRUCTURE ADMITTANCE AND IMPEDANCE
The net inter-GNR displacement current is given by with the current density given by Eq. ( 9) or Eq.(10).
For the admittance of GNR structure under consideration Y ω = δJ ω δV ω , we obtain (for the P-and S-devices, respectively) The impedances of the P-GNRs and S-GNRs are given by At low signal frequencies (ω ≪ ν, Ω) the admittances and impedances of both GNR structures tend to zero and infinity, respectively, provided the inter-GNR leakage current is insignificant.In the range of elevated frequencies comparable with the characteristic plasmonic frequency Ω, the admittances and impedances exhibit the oscillations associated with the plasmonic response of the carriers in the GNRs.Figures 3 and 4 show the P-GNRs and S-GNRs spectral characteristics.
The frequency dependences of the S-GNR admittance shown in Fig. 4 differ from those for the P-GNRs.This is due to different conditions of the plasmonic resonances.Indeed, considering the second Eq. ( 15), one can see that the maxima of Re Y S ω correspond to resonant frequencies ω = ω S n , which satisfy the condition cot γ ω S n = γ ω S n .This condition yields: ω S 0 ≃ 1.72 Ω/π ("soft" mode) and ω S n ≃ 2Ω(n + 1/n π 2 ) with n = 1, 2, ....As seen from Figs. 4(a and 4(b), the positions of the maximum of Re Y S ω and the point, where Im Y S ω = 0 are close to ω S 0 /2π ≃ 0.4 (for Ω/2π = 0.72 THz) and to ω S 0 /2π ≃ 0.4THz (for Ω/2π = 1.09THz).As can be derived using the second Eq. ( 15), the height of the Re Y S ω maxima is proportional to (4Ω/πν)/[2+(πω S n /2Ω) 2 ] = Q/(2+π 2 n 2 ), demonstrating a substantial decrease with increasing plasmonic index n and the fact that the peak height is generally lower than the pertinent peaks of the P-GNR admittance.Such a dependence of Re Y S ω on n is clearly seen in Fig. 4(a).In particular, the ratio Re It is interesting that in both P-and S-GNRs the Re Z P ω and Z S ω exhibit maxima at approximately the same frequencies having approximately the same height (compare Figs. 3(c) and 4(c).
One notes that the resonant frequencies at which Im Y P ω , Im Y S ω , Im Z ω P , and Im Z S ω equal to zero correspond to very low values of Re Z P ω and Re Z S ω .The sharpness of the resonant peaks substantially de-pends on the carrier collision frequency ν.Their height is proportional to the plasmonic oscillations quality factor Q ∝ Ω/ν ∝ M , where M is the carrier mobility in the GNRs.A decrease in the mobility and, hence, in the quality factor, leads to the pertinent lowering of the resonant peak heights.This implies that the realization of the GNR structures exhibiting pronounced plasmonic resonances requires using sufficiently perfect GNRs or lowering the operation temperature down from room temperature leading to a substantial increase in the carrier mobility.

V. DISCUSSION AND COMMENTS
The structures based on the vertically stacked separately contacted GLs and GNRs with the hBN, WS ers, and others [36][37][38][39][40][41][42][43][44][45].The resonant plasmonic effects in such structures can lead to a substantial enhancement of device characteristics.The resonant plasmonic response in the lateral co-planar structures with separately contacted GNRs, considered above, is akin to that in the vertically stacked multiple-GL structures [40].However, as predicted above, the lateral coplanar GNR structures can exhibit a pronounced plasmonic response controlled by the bias voltage even when the GNRs are relatively long (about several micrometers).This is because the plasmonic frequency in such structures is determined by the inter-GNR geometrical capacitance.This capacitance can be much smaller than the capacitance of the vertically stacked GLs.Indeed, the geometrical capacitance (per unit of the GNR length) of the lateral structure with two coplanar GNRs considered above is given by C = (κ/2π 2 )c, whereas the geometrical capacitance of the vertically stacked GNRs of the same lateral sizes separated by the spacing W is equal Comparing the plasmonic frequencies of co-planar and vertically stacked GNRs, Ω and Ω V , and assuming that GNR lengths are equal to 2H and 2H V , respectively, we obtain 1/2 (18) if the carrier Fermi energy µ (carrier density) in the GNR structures is the same, and 1/4 (19) for equal bias voltages, V G , applied between the GNRs.Considering that in the devices with the vertically stacked GNRs, the thickness of the inter-GNR dielectric layer W is relatively small and setting 2L G = 0.75 µm, L = 0.375 µm (c ≃ 2.36), W = (0.01 − 0.03) µm, and κ S = 4, we find (Ω/Ω V )| µ ≃ (4.1 − 7.1)H V /H.This implies that in the coplanar GNR structures with the same plasmonic frequency can be realized at much longer GNRs (H ≫ H V ).
As has been shown previously, the plasmonic resonances in the structures with the arrays of several vertically stacked GNRs (GNR stacks) can be more pronounced [17,18,27].The results obtained above can be applied for the lateral arrays of vertically stacked GNRs provided the proper renormalization of the characteristic plasmonic frequency Ω given by Eq. ( 12).In the case when the GNR stack comprises K non-Bernal stacked GNRs, the electron and hole density in each GNR is equal to Σ G,K = Cκ V G /4π e K. Hence, for the carrier Fermi energy µ K , which determines the GNR Drude conductivity, we obtain µ K ∝ Σ G,K ∝ 1/ √ K. Accounting for that the net GNR stack conductivity is proportional to K, for the renormalized plasmonic frequency Ω K we find Ω K ∝ √ K.
In principle, the plasmonic waves along the lateral depleted channel formed in the graphene p-n-junction predicted previously [46] resemble those considered above.However, there is a substantial difference between the plasmonic waves associated with the oscillating charges localized in the isolated GNRs, considered by us, and the plasmonic waves associated with the residual charges in the depleted area in graphene lateral p-n junctions.
The voltage-controlled coplanar GNR plasmonic resonators can be used in the THz detectors exploiting the signal rectification in p-i-n diodes and the THz radiation sources based on the resonant-tunneling diodes.In the first case, an enhanced real part of the GNR structures admittance at the plasmonic resonances promotes the rectified current amplification, which leads to an elevated detector responsivity.In the second case, low values of the real part of the GNR resonant impedance, can support the THz oscillation self-excitation and radiation emission.
Using periodic coplanar S-GNR structures with multiple GNR pairs can simplify the problem matching these structures with the external circuits (e.g., with a load resistance or a THz antenna).These structures might be also used for the THz photomixers exploiting the plasmonic resonances, GNR optical transparency, and strong light absorption in underlying layers, as well as for the acoustic and photoacoustic transducers.The resonant response associated with the excitation of plasmonic modes along the GNRs might result in a substantial increase in the impinging THz radiation.This can be used for the performance enhancement of the bolometric THz detectors exploiting the carrier heating, in particular, their nonuniform heating (see, for example, [47][48][49][50]).

CONCLUSIONS
We explored the plasmonic response of the lateral coplanar GNR structures to the impinging radiation.The pertinent resonant frequencies are determined by the plasmonic frequencies, which can fall into the THz range in structures with fairly long GNRs (about several micrometers).Due to the sensitivity of the plasmonic frequency to the bias voltage, the resonant response can be effectively voltage-controlled.The coplanar GNR structures and their version with multiple interdigital GNRs can serve as the resonant cavities for different passive and active THz devices, including THz detectors, photomixers using the interband transitions in the structure substrate (photoconductive antenna), oscillators with the diodes exhibiting negative dynamic conductivity (for example, resonant-tunneling diodes), and the acoustic transducers.

FIG. 1 .
FIG. 1. Schematic top view of the co-planar GNR structures with different types of voltage bias.

FIG. 3 .
FIG. 3. Frequency dependences of (a, b)) the real part Re Y P ω , and the imaginary part Im Z P ω of the P-GNR admittance and (c, d) the real part Re Z P ω , and the imaginary part Im Z P ω of the P-GNR impedance for different plasmonic frequencies Ω/2π ( ν = 1.0 ps −1 , a = LG/L = 2).