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Hai-Yao Deng, [Katsunori Wakabayashi](https://orcid.org/0000-0002-9147-9939)

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[Retardation effects on plasma waves in graphene, topological insulators, and quantum wires](https://mdr.nims.go.jp/datasets/79e26eb6-842a-40c9-a651-4bd75c89115d)

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Retardation effects on plasma waves in graphene, topological insulators and quantum wires Hai-Yao Deng1,2 and Katsunori Wakabayashi1,2∗ 1International Center for Materials Nanoarchitechtonics (WPI-MANA), National Institute for Materials Science (NIMS), Namiki 1-1, Tsukuba 305-0044, Japan and 2Department of Nanotechnology for Sustainable Energy, School of Science and Technology, Kwansei Gakuin University, Gakuen 2-1, Sanda 669-1337, Japan (Dated: Jan. 06, 2015) Retardation effects (REs) are known to cause a crossover from linear to sublinear behaviors in the dispersion relation of two-dimensional (2D) plasma waves at long wave lengths. In the present work, we systematically analyze REs on plasma waves in both 2D and one-dimensional (1D) electron gases and clarify the experimental conditions for observing the crossover. We show that graphene is the only material that might gratify the conditions with current technology. Although it closely imitates graphene, the surface of three-dimensional (3D) topological insulators is found not suitable for this purpose due to strong electronic scattering by intrinsic disorder. We also show that the crossover, though theoretically possible, virtually never occur in 1D plasma. In addition, the relevance of our results to teraherz plasmonics is discussed. I. INTRODUCTION Electrons in solids can undergo plasma oscillations due to long-range Coulomb forces. The fundamental plasma theory was accomplished long ago.1 In three dimension (3D), plasma frequency is nearly independent of the wave number q.2 How- ever, in two dimension (2D) it possesses a sublinear dispersion relation,3–6 ω2D p ∼ √q, which would derive unphysical su- perluminal transmission of wave packets at sufficiently small q ∼ q∗, viz. the group velocity v2D g = ∂qω 2D p ∼ 1/ √ q would exceed the speed of light c.7 Retardation effects (REs, result- ing from the finite speed of light) critically modify the disper- sion and eventually the dispersion becomes linear,7 ω2D p ∼ cq for q ≪ q∗. Experimentally studying the crossover around q∗ turns out to be difficult, partly because q∗ is typically mi- nuscule. Simple estimate shows that q∗ ∼ 1 Km−1 for 2D electron gas precipitated on the surface of liquid helium (elec- tron density n0 ∼ 107cm−2) or q∗ ∼ 1 dm−1 for that formed in semiconductor heterostructures (n0 ∼ 1011cm−2). Therefore, q∗−1 is generally in excess of available sample dimensions. The crossover phenomenon has thence not been thoroughly analyzed in existing studies. The interest in 2D plasma has recently been boosted by the advent of graphene,8,9 which is an atomically thin sheet of carbon atoms, and 3D topological insulators (TIs) that har- bor gapless energy bands on their surfaces.10,11 These materi- als have one common feature in their electronic structure:12,13 near the Fermi level εF their charge carriers behave as 2D massless Dirac fermions characterized by conical energy- momentum dispersion, ε(p⃗) = vF p. Here vF ≈ 106ms−1 de- notes the Fermi velocity and p = \|p⃗\| with p⃗ being the momen- tum. This feature makes them very attractive for applications in surface plasmonics, which is poised as a viable path to- ward nanophotonics and terahertz (THZ) technology.14–16 In the latter, 2D plasma has been gaining a progressively signif- icant status as it might provide a means of attaining efficient coupling with THZ electromagnetic waves and thereby an av- enue to manipulate them. Due to a fundamental mismatch be- tween the wavelength of THZ waves and that of plasma at the same frequency, such coupling is generically weak and plas- monic crystals are required to enhance it.17 However, if it is x y z z r = (x,y) (a) (b) λ  =  2π /q λ  = &#32;2 π /q + \- r0 FIG. 1. (a) A sheet of electron gas is placed in the x − y plane, in which plasma waves are incited to propagate along x direction. (b) Sketch of a quantum wire with lateral dimension ∼ r0. Plasma waves are propagating along the wire (z direction). Throughout the paper, we use x⃗ = (x, y, z) and reserve r⃗ = (x, y) for points in the x− y plane. Colors schematically encode charge density. engineered into the regime where q \< q∗, the plasma will have the same dispersion relation as electromagnetic waves and are expected to show strong light-matter interaction.18 Therefore, the wavelength mismatch can be mitigated and the need for artificial structuring may be largely dispensed with. In this paper we study REs on plasma in graphene, TIs and quantum wires such as graphene nanoribbons (GNRs). Unlike previous work,18,19 we systematically clarify the crossover scenario by two parameters q∗ and ω∗ = cq∗. Their depen- dence on the electronic structure is then established. Practi- cal requirements for observing the crossover are clarified. We find that in order to observe the crossover, not only the sam- ple dimension must be sufficiently bigger than q∗−1 but also sufficiently clean so that the electronic scattering rate τ−1 is much lower than ω∗. Checking the conditions against com- mon 2D electron gases, we find graphene to be the only sys- tem that might meet both conditions. Although a TI surface closely imitates graphene, the small bulk band gap (∼ 0.3eV) and the relatively strong electronic scattering due to intrinsic disorder makes it inferior. Moreover, we show that REs are negligibly small in realistic quantum wires and can thence be2 ignored under ambient circumstances. Estimate shows that high-quality and large-area graphene might be used for de- tecting THZ waves. In the next section, we present the framework and de- rive the generic expressions for 2D. Similar results for one- dimension (1D) systems are given in Appendix A. The REs- driven crossover phenomenon in graphene, TI and quantum wires are scrutinized in detail in Sec. III, where effects due to the presence of dielectrics are also briefly discussed. In Sec. IV, we analyze electronic scattering effects on observing the crossover. We discuss our results and conclude the paper in Sec. V. II. FORMALISM Plasma are self-sustained long wavelength charge density waves. To establish the equation of motions for the charge density (assuming neutrality on the whole), which we desig- nate by n(x⃗, t), let us begin with the continuity equation ∂tn(x⃗, t) + ∂x⃗ · j⃗(x⃗, t) = 0, (1) which locally embodies the conservation of total charge en- tailed by gauge symmetry. Here j⃗(x⃗, t) stands for the elec- trical current density while x⃗ = (x, y, z) represents a point in space and t an instant in time. Next we relate j⃗(x⃗, t) to local fields via an appropriate response function. For our purpose, we only consider weak electric fields E⃗(x⃗, t). In the linear re- sponse regime we can write ji(x⃗, t) = ∑ j ∫ dx⃗′ ∫ dt′ σi j 0 (x⃗, x⃗′, t − t′) E j(x⃗′, t′), (2) where the indice are i, j = x, y, z and σ0 denotes the conduc- tivity tensor of the system without taking into account elec- tromagnetic interactions. In addition, according to Maxwell’s theory we cast E⃗(x⃗, t) = −∂x⃗V(x⃗, t) − 1 c ∂tA⃗(x⃗, t), (3) where V(x⃗, t) and A⃗(x⃗, t) represent the scalar and vector po- tential fields, respectively. c ≈ 3 × 108ms−1 is the speed of light in vacuum. In the Lorenz gauge the Fourier components of V(x⃗, t), A⃗(x⃗, t), n(x⃗, t) and j⃗(x⃗, t), denoted by Vω(x⃗), A⃗ω(x⃗), nω(x⃗) and j⃗ω(x⃗) respectively, satisfy the following equations20∂2 x⃗ + ω2 c2  Vω(x⃗) A⃗ω(x⃗)  = −4π  nω(x⃗) c−1 j⃗ω(x⃗)  . (4) Equations (1) - (4) complete the semi-classical (i.e., treating the electromagnetic fields classically) description of plasma waves. They only involve quantites that exist irrespective of the underlying microscopic dynamics. The bare conductivity tensor σi j 0 serves as the single input, which can be either cal- culated or just phenomenological. This description reduces to the random phase approximation (RPA)1,4,21 in the static limit c → ∞.22 Or it boils downs to the hydrodynamical ap- proach7,23,24 if one assumes a Drude-Boltzmann model25 for σ i j 0 . Hitherto we have not included any dielectrics in the equa- tions. Their effects will be briefly discussed in Sec.III. Throughout this paper we confine ourselves to long wave- length responses (which is relevant for plasma) of isotropic systems, so that σ0 becomes a local scalar, viz. σ i j 0 (x⃗, x⃗′, τ) = σ0(τ) δi j δ(x⃗ − x⃗′). (5) Here σ0(τ) (together with its Fourier transform σω0 ) will be specified in later sections. Let us apply those equations to a homogeneous 2D system lying in the plane z = 0 as depicted in Fig. 1 (a). We seek plane wave solutions for n(x⃗, t) and j⃗(x⃗, t). Without loss of generality we may put the wave vector as q⃗ = (q, 0), thereby decoupling jy from both jx and n. Then it suffices to consider j⃗ = ( j, 0), i.e. jy ≡ 0 and hence Ay = 0 by Eq. (4) so that A⃗ = (A, 0). Accordingly, we write s(x⃗, t) = sqω ei(qx−ωt) δ(z), s = n, j/c. (6) From the continuity equation (1) we infer that jqω = (ω/q) nqω. (7) The linearity of Eq. (4) dictates that ψω(x⃗) = ψqω(z) ei(qx−ωt), where ψ = V, A. Using (4) we get( κ2 − ∂2 z ) ψqω(z) = 4πsqω δ(z), (8) where κ2 = q2 − ω2 c2 . Solving this equation and using (7) we arrive ( Vqω(z) Aqω(z) ) = 2π  1 ω cq  nqω e−κ\|z\| κ−1. (9) We require κ to be real positive, κ = √ q2 − ω2c−2 with qc \> ω, so that no radiation emanates from the plasma. Combining Eqs. (3) and (9), we see that the y-component of the local electric field vanishes. At z = 0 the x-component, denoted by E(⃗r, t), is given by E(x, y, t) = Eqω ei(qx−ωt), Eqω = κ2 iq Vqω = 2π i κ q nqω, (10) which together with (2) and (5) yields jqω = σω0 Eqω = 2π i κ q σω0 nqω, (11) Equating (11) with (7) leads to ω + 2πi κ σω0 = 0, (12) which will be used to determine the dispersion relation ω2D p (q) for 2D plasma. We can derive a similar equation as (12) for determining the dispersion relation of plasma waves in quantum wires sketched in Fig. 1 (b). Referring the calculation details to Ap- pendix A, here we only quote the result. Let the wave with wave number q propagate along a wire of radius r0. Then the desired equation reads ω + 2iσω0 κ 2K0(κr0) = 0, (13) where K0(z) is the modified Bessel function.3 TABLE I. Expressions for the factor F involved in the Drude- Boltzmann conductivity, σω 0 = i F ω (dissipationless limit). We use ε(p) to denote the energy-momentum relation of the system under question, n0 is the average electron density, m the effective mass, vF the Fermi velocity, εF the Fermi level, N(ε) density of states per unit area, e the elementary charge and ℏ (h = 2πℏ) the reduced Planck constant. ε(p) Parabolic (2D) Linear (2D) Parabolic (1D) Linear (1D) F n0e2 m e2v2 F N(εF ) 2 n0e2 m e2vF h III. CHARACTERIZING THE CROSSOVER Anticipating that REs become pronounced only at very small q and ω, we may assume the Drude-Boltzmann model for the bare conductivity. Namely, we put σω0 = Fτ 1 − iωτ , (14) where F is a prefactor that depends on the particulars of the system (see Table I) and τ is the transport relaxation time.25 For 2D systems, it is convenient to express ω and τ−1 in units of ω∗ = 2πF/c and q of q∗ = ω∗/c. Then we can rewrite Eq. (12) as follows ω + i √ q2 − ω2 τ 1 − iωτ = 0, (15) which depends on a single parameter τ. Its solution Ω2D p = ω2D p −iγ2D p /2 gives the plasma frequencyω2D p and the damping rate γ2D p . In the limit τ → ∞, γ2D p vanishes and ω2D p is given by (after recovering the units) ω2D p = ω ∗ √√√√√√1 2  √ 1 + 4 ( q q∗ )2 − 1 , for τ→ ∞, (16) Finite but large τ will make negative corrections of the order τ−2 to ω2D p . There is a crossover to occur around q∗. Actually, we find ω2D p ≈ ω∗ √ q/q∗, if q ≫ q∗, ω∗ (q/q∗) = cq, if q ≪ q∗. (17) The associated vg goes as q−1/2 for large q but saturates at c for small q. The crossover is clear in Fig. 2, where ω2D p is plotted versus q. The value of q∗ varies significantly from one system to an- other. For systems with quadratic energy-momentum relation, ε( p⃗) = p⃗2 2m , where m denotes the effective electronic mass, we have F = n0e2 m with n0 being the average electron density. Ex- pressing n0 = n∗0 × 1016cm−2 and m = m∗ × 9.1× 10−31Kg, we obtain ω∗ = 2πn0e2 mc ≈ n∗0 2m∗ × 1015Hz, q∗ ≈ n∗0 6m∗ × 107m−1. (18) For electrons precipitated on the surface of liquid helium, on which the 2D plasma dispersion \[upper line in Eq. (17)\] was first verified5,26, one has n∗0 ∼ 10−9 and m∗ ∼ 1. Thus, ω∗ ∼ 0.5MHz and q∗ ∼ 2Km−1, which is too small a value to be experimentally achieved. To observe the crossover, a sample area much larger than q∗−2 ∼ 0.25km2 is demanded. For electrons accumulated at the interface of a semiconductor heterojunction like AlGaAs/GaAs, typically n∗0 ∼ 10−5 and m∗ ∼ 0.1.27 Therefore, ω∗ ∼ 50GHz and q∗ ∼ 1.67cm−1, which are still very tiny for realization. The sample area re- quired far surpasses 2.78dm2. These numerics show that REs are generically negligible in conventional 2D electron gases. We proceed to reason that graphene has a disparate story. This material in its pristine form has a conduction and a va- lence band near the Fermi level εF , which touch conically at the corners of the first Brillouin zone.28,29 Relative to the neutrality point (chosen to be ε = 0), the electronic energy- momentum relation is nearly linear: ε( p⃗) = vF ∣∣∣p⃗∣∣∣, where vF ≈ 106ms−1. For εF close to the neutrality point, inter- band transitions prevail at all frequencies and hence Drude- Boltzmann theory \[Eq. (14)\] is not applicable. Nevertheless, far away from that point interband transitions are greatly sup- pressed at frequencies below 2εF/ℏ. Under such circum- stances, we may still use Eq. (14). As shown in Ref.30, for graphene F = e2v2 F N(εF)/2, where N(εF) = √ πn0 hvF denotes the density of states per unit area per valley per spin. Now we find ω∗ = πe2v2 F N(εF) c ≈ √ n∗0 × 6.5 × 1013Hz, q∗ ≈ √ n∗0 × 2.1 × 105m−1. (19) Current doping technology can commonly achieve n∗0 ∼ 10−4 in graphene,31 for which ω∗ ∼ 0.65THz and q∗ ∼ 2.1mm−1. Good quality single crystal monolayer graphene sheet of sizes ∼ 1cm2 can hopefully be fabricated using chemical vapor de- position method.32 Therefore, observation of the REs-driven crossover is fairly plausible in graphene. To complete the analysis, let us examine TIs, which have recently caused a great sensation. TIs possess surfaces which each features a pair of conically dispersing energy bands crossing Fermi level, in close analogy with graphene.11 The same ε(p⃗) applies here as for graphene, but with vF ∼ 6 × 105ms−1. The lacking of valley and spin degeneracy leads to N(εF) = √ πn0 2hvF . As a result, q∗ ∼ √ n∗0 × 5 × 104m−1 becomes much smaller. Additionally, There is an upper bound on n∗0 be- cause of the existence of the bulk bandgap, which is ∼ 0.3eV for Bi2Se3, the most extensively studied TI. The Fermi level must be well inside this gap to avoid strong mixing with bulk bands. Namely, εF = ℏvF √ 4πn0 ≈ 10 √ n∗0eV \< 0.1eV, which gives n∗0 \< 10−4. This fixes q∗ \< 0.5mm−1, corresponding to a surface area no less than 10mm2. The observed carrier mobil- ity in TIs is much lower than that in graphene, due to intrin- sic dopant disorder.33,34 This means relatively smaller τ and hence stronger plasma damping, which makes low frequency (relevant to REs) observation especially difficult (see the next section). Moreover, the fabrication of good quality TIs with desired surface area is currently formiddable. The coupling4 q/q\* 10 -3 10 -2 10 -1 10 0 10 1 ω /ω \* 10 -3 10 -2 10 -1 10 0 10 1 Eq. (16) light line ω = q 1/2 q = q\* FIG. 2. 2D plasma dispersion relation ω2D p (q). The red solid curve is given by Eq. (16), which reduces to the green solid curve for q \> q∗ but becomes degenerate with the light line (dashed curve) for q \< q∗. The crossover is controlled by the single parameter q∗, which varies significantly from one system to another. In units of ω∗ = cq∗ and q∗, ω2D p assumes a form that is regardless of the materials. between parallel surfaces is likely to erase the crossover phe- nomenon.7 Graphene has an additional merit that it can be inserted be- tween dielectrics ϵ with relative ease. The main effects of the dielectrics consist of reducing c to c/ √ ϵ(qd) and sub- sequently increasing q∗ to ϵ(q∗d)q∗. Here d is the separa- tion of graphene layer from the dielectric layer and ϵ(qd) = ϵ cosh(qd)/2, which becomes ϵ at long wavelengths.35 This further decreases the desired sample sizes and then lowers the barrier of observing the REs-driven crossover. For example, taking the dielectric to be HfO2 with ϵ ≈ 22 and n∗0 ∼ 10−3, we expect q∗ ∼ 0.12µm−1 and a sample area of ∼ 100µm2, which can already be obtained using current fabrication meth- ods.32,36 Let us also mention hexagonal boron nitride (hBN) on this occasion. Although it has a smaller ϵ ≈ 4, hBN is capable of preserving the high mobility of graphene.37 As we will discuss in the next section, high mobility is also crucial in observing the crossover. We note that the surface of TIs is sandwiched between the bulk (ϵ ≈ 100 for Bi2Se3 34\) and the vacuum. This gives an effective dielectric constant of ≈ 50 and thus q∗ ≤ 0.025µm−1. For quantum wires, an analogous but more subtle reason- ing has been presented in Appendix A, where it is shown that REs are not discernible at all. The predicted crossover occurs at q∗ ∼ r−1 0 e−(c/c∗), which is practically zero in the systems known to us. Here c∗ = √ 2F, with F given in Table I. IV. DAMPING EFFECTS We have so far focused on the limit τ → ∞. In reality, τ is finite and the plasma spectrum can be modified. To gain some insights, let us first consider 2D plasma in the extreme that ωτ ≫ 1, whereby (ωτ)−1 serves as a small parameter for perturbative treatment. Let us write Ω2D p = ω2D p + δω 2D p − iγ2D p /2, where ω2D p is prescribed by Eq. (16), and substitute it in (15). To the first order in τ−1, we find δω2D p ≈ 0 and γ2D p ≈ 1 τ 1 1 + 1 2 ( ω∗ ω2D p )2 ≈  τ−1, if q ≫ q∗, 0.7τ−1, if q ∼ q∗, 0, if q ≪ q∗. (20) Note that γ2D p can be directly measured by e.g. electron en- ergy loss spectroscopy (EELS).1 The plasma shows up as a peak in EELS at fixed q. The width of this peak gives γ2D p . From Eq. (20), we infer that the peak will get broadened as q increases. In the other extreme where ωτ ≪ 1, plasma becomes heav- ily damped and we expect to see a central peak (relaxational mode) in EELS for not-so-big q. Actually, to the first order in ωτ, which is treated as a small parameter, Eq. (15) becomes ω + iω∗τ √ c2q2 − ω2 ≈ 0, (21) yielding ω2D p ≈ 0 and γ2D p ≈ 2 ω∗τ qc. This shows that the central peak gets narrower when q decreases. For sufficiently big q, the central peak disappears. The overall profile of EELS can be numerically assessed. For this purpose, we resort to the fact that the l.h.s. of Eq. (12) \[or equivalently (15)\] corresponds to the dynamic dielectric function ϵ(ω, q) of the electron gas. EELS is then mainly de- termined as Im ( − 1 ϵ(ω,q) ) , which is a positive-definite function in ω. In Fig. 3 we plot it for representative values of q and τ. In line with the analytical expressions, we see that (i) the plasma dispersion is linear for q \< q∗ but becomes sublin- ear for q \> q∗; (ii) damping effects are more pronounced for higher q and (iii) for very heavy damping (ω∗τ \< 1) a central peak appears for q around q∗. To unambiguously observe the crossover, it is necessary that ω∗τ \> 1. Let us compare the EELS for the 2D systems scrutinized in Sec. III. For systems with quadratic energy-momentum dis- persion, we can relate τ to the electronic mobility µ = µ∗ cm2 V·s , i.e. τ = (m/e)µ. By Eq. (18), ω∗τ ≈ 0.28 × n∗0 µ ∗, which does not rely on m∗. For electrons living on the surface of Helium at very low temperatures, µ∗ ∼ 105 and we estimate that ω∗τ ∼ 0.003, which is much smaller than unity. Hence, in the vicinity of q∗, we expect to see a central peak. The ex- periment performed in e.g. Ref.5 looked at q ≫ q∗ and the authors were able to see the dispersion of ω2D p . For electrons accomodated in high quality AlGaAs/GaAs heterojunctions, it is realistic to take µ∗ ∼ 107 and then ω∗τ ∼ 28. Therefore, even for q ∼ q∗ we expect to see univocal plasma dispersion. Unfortunately, available sample sizes are still way below what is desired. For graphene, τ may be conveniently estimated us- ing the value of electronic mean free path le and vF , namely, τ = le/vF . By Eq. (19), this leads to ω∗τ ≈ 65 × √ n∗0 × l∗e, with le = l∗e µm. Good graphene samples possess le ∼ 10µm, which puts ω∗τ ∼ 6.5. The best TIs so far have le that is only a few percent of that of graphene. In this respect, graphene is5 1 2 3 4 5 q /q\* 0 0 1 2 3 ω /  ω \* γ p 2D light line &#32;ω  τ = 5.0\* (a) q /q\* ω /  ω \* 1 2 3 4 50 0 1 2 3 light line &#32;ω  τ = 0.8\* (b) ω /  ω\* 0 2 E E L S &#32;( 2 D ) 0 4 8 12 16 20 1 (c) &#32;ω  τ = 0.1\* q = 0.5q\* 1\.5q\* q = 200q\*   (ω/10, x20) 0 1 2 0 1 2 FIG. 3. Electron energy loss spectra (EELS) evaluated by Im− ( ϵ−1(ω, q) ) , with ϵ(ω, q) given by the left hand side of Eq. (15). EELS is useful in unveiling the effects of electronic scattering (scattering rate = τ−1) on plasma dispersion ω2D p (q) and life time, which is the inverse of the broadening factor γ2D p (q). The profile of EELS critically depends on the value of ω∗τ. As shown in panel (a), if ω∗τ \> 1, the plasma disperses according to ω2D p with γ2D p increasing as q increases. However, as displayed in (b), if ω∗τ \< 1, a nondispersive central peak is seen for q around q∗. In (c), we plot frequency scan for three values of q. For q ∼ q∗ (curves with q = 0.5q∗ and q = 1.5q∗), central peaks are revealed, as in (b). For very large q (e.g. q = 200q∗), very weak dispersion remains. For the sake of comparison, the line labelled q/q∗ = 200 has been multiplied by 20 and ω has been rescaled by a factor 10. To observe the crossover phenomenon, it is necessary that ω∗τ \> 1. most suitable for studying the REs-driven crossover in plasma dispersion. Apart from EELS, optical techniques can also be used to extract the same information. For example, one may study the reflection (R) and transmission (T ) of normally incident radi- ation on the sample. From TR−1, one can infer the dielectric function ϵ(ω, q) and then plasma properties. V. DISCUSSIONS AND CONCLUSIONS In summary, we have derived two criteria that must be ful- filled in order to observe the REs-driven crossover in 2D and 1D plasma dispersion relations. Firstly, the system size must be much larger than q∗−1. Secondly, the system must be suffi- ciently pure so that the electronic scattering rate is much lower than ω∗ = cq∗. The criteria have been systematically checked against common 2D and 1D electron gases. We synthesize the results in Table II. We found that the crossover virtually never occurs in 1D. Among 2D systems, to our knowledge the only candidate that can hopefully meet both criteria seems to be graphene. In actual setup, we propose to sandwitch graphene between two dielectrics, so as to further increase q∗ and reduce the min- imum flake size. To this end, hBN may be used. Although its dielectric constant is smaller than HfO2, hBN is able to pre- serve the high electronic mobility of graphene. The minimum sample area required is then estimated of ∼ 0.01mm2. Note that gating structure and bilayer coupling must be avoided, as it will have dramatic impact on plasma dispersion7,23 and could erase the desired phenomenon. Our estimate shows that the crossover frequency ω∗ for typ- ical graphene samples falls within the THZ regime, whereby the plasma dispersion follows closely that of THZ electromag- netic waves. Thus, the wavelength mismatch problem that has plagued the THZ community for years may be bypassed by optimizing graphene. The present study is expected to be use- ful in the design of THZ detectors. Let us note that the use of Drude-Boltzmann formula for σω0 \[Eq. (14)\] does not undermine our results. This is because the scenario under question occurs on the scale of q∗−1, which is much larger than atomic dimensions. The characteristic fre- quency ω∗ is tiny in comparison with energy bandgap (or 2εF in case of graphene). Quantum effects play a minor role under such circumstances: the conductivity calculated via quantum mechanical perturbative theory actually reduces to Eq. (14). Another concern pertains to coupling with optical phonons arising from the substrate supporting the 2D sample. Ex- periments38 have demonstrated strong coupling around q ∼ 1nm−1 between graphene and silicon carbide substrate. How- ever, we do not expect this coupling to be significant in the regime of q ∼ q∗. In this regime, the phonon frequency (which is highest at q = 0) is much higher than that (lowest at q = 0) of plasmon. This frequency mismatch effectively decouples them at small q, in accord with experiments.38 ACKNOWLEDGMENTS HYD acknowledges useful discussions with K. Florian. This work is supported by JSPS fellowship and KAKENHI (Project Nos. 15K13507, 25107005) from MEXT of Japan.6 TABLE II. Estimated order of magnitudes of several parameters which are relevant for observing the REs-driven crossover in 2D plasma dispersion relation ω2D p (q). Four systems are listed: electrons precipitated on the surface of liquid helium, AlGaAs/GaAs heterojunction, graphene and surface of 3D topological insulators (TIs). The parameters considered are: average electron density n0 = n∗0 × 1016cm−2, electronic effective mass m = m∗ × 9.1 × 10−31Kg, ω∗ = 2πF/c and q∗ = ω∗/c as well as ω∗τ. Here c ≈ 3 × 1010cm · s−1 denotes the speed of light in vacuum, F is given in Table I and τ is the electronic relaxation time. Two conditions must be gratified to observe the crossover: (i) sample dimension≫ q∗−1 and (ii) ω∗τ \> 1. For graphene and TIs, m does not exist, as indicated by a hyphen in the entry. System Helium surface26 AlGaAs/GaAs39 Graphene14,36,40–42 TIs33 n∗0 10−9 10−5 10−4 10−4 m∗ 1 0.1 - - ω∗ 1 MHz 10 GHz 1 THz 0.1 THz q∗ 1 Km−1 1 cm−1 1 mm−1 0.3 mm−1 ω∗τ 0.003 28 6.5 0.01 a a For best quality graphene µ∗ ∼ 105 while for best TIs µ∗ ∼ 103, which underlies this value. Here µ = µ∗ cm2 V·s denotes electronic mobility. Appendix A: Quantum wires Plasma oscillations in quantum wires were optically de- tected over two decades ago43,44 and have since been inten- sively studied.45,46 Here we nalyze the REs-driven crossover in quasi-1D systems including graphene nanoribbons. We imagine a homogeneous 1D system lying along the z- axis, as portrayed in Fig. 1 (b). For convenience, we shall write r⃗ = (x, y). In analogy with Eq. (6), we make the follow- ing ansatz s(x⃗, t) = sqω ei(qz−ωt) δ2(⃗r), s = n, j/c. (A1) Here j is the z-component of j⃗. The continuity equation (7) still holds, whereas Eq. (8) is modified to be( κ2 − ∂2 r⃗ ) ψqω (⃗r) = 4πsqω δ 2 (⃗r), (A2) where we have defined ψ(x⃗, t) = ψqω(⃗r)ei(qz−ωt) with ψ = V, A. Here A denotes the z-component of A⃗. κ = √ q2 − ω2/c2 re- mains the same as before. The solution to Eq. (A2) is given by ψqω (⃗r) = 2 K0(κr) sqω, (A3) where r = ∣∣∣⃗r∣∣∣ and K0(z) denotes the zeroth order mod- ified Bessel function of second kind. Again Aqω(⃗r) = (ω/qc)Vqω (⃗r). The z-component of the local electric field is obtained as follows E(x⃗, t) = Eqω (⃗r) ei(qz−ωt), Eqω(⃗r) = 2 κ2 iq K0(κr) nqω. (A4) The limit r → 0 is ill-defined for Eqω (⃗r). Instead of r → 0, we shall take r = r0, where r0 signifies the fact that in reality any 1D system has a lateral dimention of order r0 ∼ 10−10m. 47 Henceforth, from Eqs. (2) and (5) we have jqω = 2σω0 κ2 iq K0(κr0) nqω = ω q nqω, (A5) where the second equality comes from the continuity equa- tion. Finally we obtain ω + 2i σω0 κ 2 K0(κr0) = 0. (A6) This equation determines the dispersion relation ω1D p (q) for 1D plasma. The bare conductivity is given by Eq. (14), where the factor F is to be specified later. Introducing Q = ω c and expressing all wave numbers (including both q and Q) in units of r−1 0 , we can recast Eq. (A6) in the following form (neglecting damping effects, viz. τ→ ∞) ( q2 − Q2 ) K0 (√ q2 − Q2 ) = λ Q2, λ = ( c c∗ )2 , c∗ = √ 2F, (A7) which depends on a single parameter λ. The static limit has c → ∞ and hence λ → ∞, where Q is of the order λ−1/2. In case q ≫ Q, one can drop the Q2 on the l.h.s. in (A7) to get Q ≈ (c∗/c) q √ K0(q), which would produce a singular group velocity vg ≈ c∗ √ K0(q) at small q. On other hand, for q lying close to zero, we may make the ansatz that Q ≈ Vg q, where Vg = vg/c can be analytically evaluated. Approximat- ing K0(z) ≈ ln(2/z) − γ (here γ ≈ 0.577 denotes the Euler constant) and differetiating Eq. (A7), we find 2λQVg + q ( 1 + ln(q2 − Q2) ) = 0. (A8) Now using (A7) to rewrite ln(q2 − Q2) ≈ 2λQ2 Q2−q2 and plugging it in this equation, we obtain 2λV2 g + V2 g − 1 = 0, which yields Vg = √ √ 1 + 8λ − 1 4λ ≈ 1, if λ ≪ 1, (2λ)− 1 4 , if λ ≫ 1. (A9) As required, Vg \< 1 always. Piecing everything together, we see that there must exist a crossover to occur around some wave number q∗, above which vg/c ∼ (c∗/c) √ ln q−1 while below it vg/c ∼ Vg. Obviously, we have (c∗/c) √ ln q∗−1 ∼ Vg. After recovering the units, it gives q∗ ∼ r−1 0 e − ( Vgc c∗ )2 ≈ r−1 0 e− 0.71 × √ λ, (A10) where the second equality follows from the fact that λ gener- ally far exceeds unity. Now we see that 1D plasma dispersion7 (ω /c )r 0 λ = 10 Eq. (A7) upper  line of Eq. (A11) light line lower line of Eq. (A11) 10 0 10 -4 10 -6 10 -8 10 0 10 -2 10 -4 10 -6 10 -8 q = q\* q r 0 10 -2 10 -3 10 \-2 FIG. 4. 1D plasma dispersion relation, ω1D p (q). In contrast to ω2D p , ω1D p depends sensitively on the paramter λ = (c/c∗)2, where c∗ de- pends on the materials. The red solid curve is given by Eq. (A7). It goes atop the green solid curve \[upper line of Eq. (A11)\] for q \> q∗ but collapses to the solid black line \[lower line of Eq. (A11)\] for q \< q∗. Here q∗ ∼ exp ( − √ λ ) drastically relies on λ, which makes quantum quasi-1D electron gases essentially immune from retarda- tion effects (REs). behaves asymptotically as follows ω1D p ≈ c∗ q √ K0(r0q), if q ≫ q∗, (2λ)− 1/4 c q, if q ≪ q∗, (A11) which is confirmed in Fig. 4, where ω1D p is plotted versus q. Now we discuss the value of q∗ in a variety of quasi-1D electron gases. We first consider systems bearing quadratic energy-momentum relation for charge carriers, ε(p) = p2 2m , for which F = n0e2 m the same as for 2D. Such systems in- clude quantum wires that may be fabricated out of conven- tional semiconductor heterostructures44, semiconducting car- bon nanotubes48 and semiconducting armchair GNRs (aG- NRs).49 To facilitate the analysis, let us write n0 = n∗0 × 108cm−1 and m = m∗ × 9.1 × 10−31Kg. Then we have c∗ ≈ 2.3 √ n∗0 m∗ 106ms−1, λ ≈ 1.7 × 104 × m∗ n∗0 . (A12) Taking r0 ∼ 10−8m as typically realized, we get q∗ ∼ 108m−1 × exp − 92 × √ m∗ n∗0  . (A13) In AlGaAs/GaAs quantum wires, commonly n∗0 ∼ 10−3 and m∗ ∼ 0.1. Therefore, the as-estimated q∗ is vanishingly small and the crossover never happens for such parameters. How- ever, considering that q∗ is extremely sensitive to m∗ n∗0 due to the exponetial relation, it might not be completely out of question to attain the observation. Actually, by increasing n∗0 to 0.1 and simultanesouly diminishing m∗ to 0.001, we can increase q∗ to 105m−1, which corresponds to a length of 10µm that is in- credibly smaller than typical value. Hitherto we have not been aware of any materials which meet the demand. Next we look at systems with linear dispersion, ε(p) = ℏvF p. In this catagory one finds metallic aGNRs, 2D topo- logical insulators50 and perhaps the so-called Su-Schrieffer- Heeger chains (such as polyacetylene, when Peierls instabil- ity is weak or suppressed by e.g. doping).51 Now we have F = e2h−1vF (see Appendix). The corresponding c∗ and λ are given by c∗ ≈ 0.05 × √c vF , λ ≈ 430 × c vF . (A14) Since these systems have relatively smaller lateral dimension, we may take r0 ∼ 10−10m. This gives q∗ ∼ 1010m−1 × exp −15 × √ c vF  , (A15) which does not hinge on carrier concentration. The magnitude of vF is primarily set by the electronic hopping energy t ∼ 1eV, the lattice constant a ∼ 10−10m and the Planck constant ℏ, namely vF ∼ t a ℏ ∼ 105ms−1 by dimensional analysis. Thus, q∗ is too small to be realized in these systems. Finally it might be worthwhile to mention zigzag GNRs (zGNRs).49 zGNRs feature a pair of partially flat subbands which touch at the boundaries of the first Brillouin zone. Thus, they are always metallic. For very narrow zGNRs with only these subbands partially filled, we find F = e2h−1v(pF), where v(pF) denotes the Fermi velocity. Eqs. (A14) and (A15) remain valid. 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