We present a comprehensive study of the reflection of normally incident plasmon waves from a low-conductivity 1D junction in a 2D conductive sheet. Rigorous analytical results are derived in the limits of wide and narrow junctions. Two types of phenomena determine the reflectance, the cavity resonances within the junction and the capacitive coupling between the leads. The resonances give rise to alternating strong and weak reflection but are vulnerable to plasmonic damping. The capacitive coupling, which is immune to damping, induces a near perfect plasmon reflection in junctions narrower than 1/10 of the plasmon wavelength. Our results are important for infrared 2D plasmonic circuits utilizing slot antennas, split gates or nanowire gates. They are also relevant for the implementation of nanoscale terahertz detectors, where optimal light absorption coincides with the maximal junction reflectance.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Plasmonics aims to combine the advantages of nanometer scale electronics with the high operating frequency (terahertz and beyond) of photonics . A promising platform for plasmonics is graphene, which features high confinement, wide range of operating frequencies ω, long lifetimes, and tunability [2,3]. Recent experiments demonstrated long plasmon propagation distance and high quality factor for graphene [4–6], making 2D plasmonic circuitry feasible. An important basic element in a plasmonic circuit is a switch that has a small size and a large on-off ratio. Numerous numerical studies of the interaction of plasmons with 1D obstacles has been done in search of such a device [7–15]. It has been shown that a narrow 1D junction of low conductivity in an otherwise uniform 2D conductive sheet can potentially serve as a plasmonic switch [13,16]. This is a surprising result as deeply subwavelength obstacles usually cannot impede the propagation of a wave. In this work we explain the physical principles behind the plasmonic interaction with this type of inhomogeneities and provide analytical solutions for the reflection coefficient, which can be simply understood in terms of equivalent circuits. The anomalous complete reflection is revealed to have the same origin as the stop-band behavior exploited in LC-loaded radio frequency waveguides or various metamaterial structures.
Plasmons propagating in a uniform conducting film or a two-dimensional electron gas (2DEG) have a momentum q that is inversely proportional to the (frequency-dependent) sheet conductivity σ, , where κ is the dielectric function of the environment exterior to the sheet. A local variation in the conductivity causes a change in q and thus acts as a scatterer for plasmons. Controlled conductivity variations can be realized in graphene which has a conductivity σ determined by its chemical potential. Using patterned electric gates, the chemical potential can be tuned locally. We consider an idealized case where the sheet conductivity has a piecewise constant 1D profile with a value σ inside a strip of width 2a and another value σ0 in the semi-infinite leads on both sides (Fig. 1). In practice, the width of the junction is determined by the geometry of the gate and can be as narrow as a few nanometers for a nanowire or nanotube gate [17,18]. Two types of phenomena govern the propagation of plasmons across such a junction, the cavity resonances inside the junction and the capacitive coupling between the leads. Depending on how the width (2a) of the junction compares to the plasmon wavelength in the leads (λ0 = 2π/Re q0) and in the junction (λ), the strength of these phenomena varies. For wide junctions λ0 ≪ a, the reflectivity is determined chiefly by the cavity resonances while the capacitive coupling is negligible. The resonances give rise to alternating maxima and minima in the reflectance, which are, however, quickly suppressed by plasmonic damping. For narrow junctions λ0 ≫ a, the capacitive coupling dominates. This regime can be further divided into two, depending on how a compares to λ. When a ≫ λ, the cavity resonances are present but have their amplitudes modified by the capacitive coupling. When a ≪ λ, instead of the oscillating resonant fields, the junction has a constant electric field like a capacitor and becomes a parallel LC circuit in the dc limit [Fig. 2(d)]. A near perfect plasmon reflection occurs at a specific conductivity σ where the impedance of the LC circuit diverges. As there is no wave propagation in the junction in this limit, the anomalously strong reflection is robust against plasmonic damping.
2.1. Wide junctions
A wide junction contains two interfaces that partially reflect and transmit plasmons but otherwise do not interact with each other, i.e., it is a plasmonic cavity. Using the analytical Wiener-Hopf method, the reflection coefficient r for a normally incident plasmon wave from the left at the left (L) and the right (R) interfaces was found in  to be20,21], is given by 22,23]. We define the reflection coefficient r to be the prefactor in the asymptotic form of the scattered potential ϕs ≃ re−iq0x at large negative x for an incident plasmon potential ϕ0 = eiq0x. (Our definition differs in the overall sign from  where r is the reflection coefficient for the current.) The total reflection coefficient from the left interface of the junction is then found from the usual Fabry-Pérot (F-P) formula to be Equation (3) yields a reflectance that has local minima at resonances, ϕ = nπ+θ, and local maxima at anti-resonances, , where n is a positive integer. These alternating maxima and minima can be seen in Fig. 3(a), where an example of the reflectance R = |r|2 for a lossless junction of width a = λ0 is shown. Here we have parametrized the plasmon momentum as with the dimensionless damping factor defined as γ = Im q/Req. Under these circumstances, the junction acts as a plasmonic switch with a high on-off ratio, tuned by the conductivity contrast σ0/σ (or alternatively, by varying the width a). However, the resonances are quickly suppressed by plasmonic damping as shown in Fig. 3(b), removing the switching behavior and rendering wide junctions less desirable for nanoplasmonic applications. Note that we assume the same damping in the leads and the junction, γ0 = γ, in this work, so that the conductivity ratio σ0/σ is real. A higher damping in the junction than in the leads further diminishes the resonances but does not alter the results qualitatively. Note also that we neglect the screening of plasmons by the gates forming the junction. This effect is quite small for graphene plasmonic structures operating in mid-IR with typical parameters λ0 ∼ 100 nm, d ∼ 300 nm, where d is the distance between the 2D sheet and the gate. The screening would become important, however, if the plasmon confinement length λ0/2π exceeds d, which is the case in the THz domain. In the first approximation, the screening effectively replaces the long-range Coulomb interaction by a short-range one, so that the propagation of the plasmons is described by a wave equation with a position-dependent λ0. In such a limit, the theoretical problem becomes trivial. Both the reflection phase shift and the capacitive coupling of the two sides of the junction can be neglected, so that the simplest version of the Fabry-Perot formula (with θ = 0) [20,21] applies.
A wide junction can be described with an equivalent circuit. The reflection and transmission coefficients described in Eq. (1) is what one would expect at the interface of two waveguides with impedances σ0 and σ (up to the phase factor in r) . However, the plasmonic wave impedance of a 2DEG, Z0 = π/ωκ, was found to be independent of the sheet conductivity  (see Appendix A.) To retain the analogy to conventional waveguides, we treat the interface as a composite object consisting of an ideal transformer and two phase shifters [Fig. 2(b)]. The transformer effectively rescales the junction impedance into while the complementary phase shifters provide the requisite phase factor e±iθ to the reflection coefficients. The junction itself can be replaced by a piece of waveguide of the effective length 2a − (θ/q), which is equivalent to a T-junction with impedances Z1 = −iZj tan(qa − θ/2) and Z2 = iZj csc(2qa − θ) . The total reflection coefficient from the left interface [coinciding with Eq. (3)] can then be found directly from the standard waveguide-theory formula,
2.2. Narrow junctions
The F-P formula (3) remains numerically accurate for junctions as narrow as a ≃ λ0. However, as the two leads get closer, they start to couple capacitively. The system resembles a leaky capacitor. The capacitive coupling becomes dominant when the junction is narrow. The cavity resonances remain but their amplitude gets strongly modified. This can be seen in Fig. 3(c), where the reflectance for a narrow junction a = 0.01λ0 deviates greatly from the F-P formula but retains the same resonant locations. In the following we present key intermediate steps in calculating the analytical form of this modified reflectance, leaving the detailed derivation to the Appendix.
We start by discussing the general form of r for a narrow junction. The reflection coefficient r can be found by comparing the incident in-plane plasmon field Ei = E0eiq0xx̂ with the asymptotic form of the scattered field Es ≃ −rE0e−iq0xx̂ at large distances from the junction. The scattered field Es can be completely determined by the total field E inside the junction via the Green’s function G,Appendix B), Eq. (6), one arrives at the relation Eq. (7) can be simplified and written in terms of the voltage drop across the junction ,
Equation (8) has simple solutions in limiting cases. In the perturbative limit σ ≲ σ0, the current density j(x) is approximately constant across the junction, which implies that the field inside the junction is . This yields the reflection coefficient26]. The reflection coefficient depends linearly on the strength of the perturbation Δσ, as expected. In fact, the perturbative expression is also valid for any σ > σ0 as long as the resulting reflection coefficient is small, |r| ≪ 1. However, it is clearly inadequate as σ is decreased toward zero, i.e., the case of a vacuum gap, where the perturbative r diverges [Eq. (9)]. For such a vacuum gap, the current j is stopped and completely reflected by the gap, i.e., the two edges of the junction act as a (non-leaky) capacitor. By considering the charge conservation equation on either edge of the capacitor, ji + jr = σ0E0(1 − r) = Q̇ = −iωVC, the reflection coefficient for the vacuum gap can be found: Equation (10) is in good agreement with our numerical result (Fig. 4) and with . It can also be simply explained by the equivalent circuit for the system, a resistor-capacitor-resistor series with impedances Z0, Zc = i/ωC, and Z0. The circuit yields the load impedance ZL = Zc + Z0, so that r = Zc/(Zc + 2Z0), matching Eq. (10).
For intermediate conductivities 0 < σ < σ0, the capacitive component is diminished compared to the vacuum gap, but the junction can now host plasmonic resonances. This is seen in the modified charge conservation equation,27]. The field in the junction can be expanded in the basis of resonant eigenmodes 27] are inaccurate), but they quickly approach as n is increased [7,27]. The first three eigenmodes fn(x), n = 1, 2, 3, are plotted in Fig. 1 and also in Appendix C.2.Eq. (15). These terms vary smoothly with σ and a, giving the resonances a Fano shape and causing r to deviate significantly from the F-P formula, as shown in Fig. 3(c) and (d). Another distinction from wide junctions is that even and odd resonances become very different in strength. Numerical results show that the odd resonances are more narrow and less strong than the even ones. This can be seen more clearly in the analytical formula for r calculated under the approximation Eqs. (14) and (15) yielding bn = 0 for an odd fn(x). The residual presence of the odd modes in the numerical results is due to the incident field Ei = E0eiq0x containing both even and odd components. Except for this difference, the analytical approximation agrees well with the numerical result, and the agreement gets better at larger n where the approximation for fn becomes more accurate.
Equivalent circuits are again helpful in understanding Eq. (15). Consider the impedance of a 2a-long piece of plasmonic waveguide Z = −4iZ0 tan qa , which can be expanded into the resonant form . This inverse impedance has the same structure as a network of parallel LC circuits [Fig. 2(c)], which is in fact how resonant cavities are conventionally described. The impedance of the LC network is , which yields L0 = 2ia/σω, , and Cn = qaκ/2π3n2. The load impedance of the junction, including the capacitor, the LC network and the right half-plane is then ZL = ZZc/(Z + Zc) + Z0, which yields the reflection coefficientEq. (15) (under the limit σ ≪ σ0). The strength of the resonances does not match Eq. (15) but this is expected, as the effect of capacitive coupling on the resonant modes was not accounted for. To correctly describe the junction Ln and Cn will have to be modified, which is effectively accomplished by Eq. (14). The resonant condition qa = πn is however qualitatively correct (apart from the phase shift θ) and predicts the disappearance of the odd modes.
As in the case of wide junctions, the high-order resonances quickly diminish in the presence of plasmonic damping [Fig. 3(d)], so these resonances are not robust enough for plasmonics applications. However, there is a prominent peak in the reflectance that occurs apart from the resonances and is persistent under damping, making it a promising candidate for controlling a compact plasmonic switch. The physical origin of this peak is as follows. If the field in the junction is approximately uniform, the junction reduces to a parallel LC circuit [Fig. 2(d)] with impedances ZC = i/ωC and , the later resembling the usual dc formula for resistance. When the inverse impedances, i.e., admittances and cancel, the total impedance of the junction diverges and the reflectance is unity. We call this situation the open-circuit resonance. It occurs whenFig. 5.
2.3. Power absorption
Low-conductivity junctions based on narrow slots in metallic films or 2DEGs under split-gates have also been proposed as basic units of terahertz detectors. They function by converting photons into plasmons, which are then detected as a dc current either through nonlinear optical effects [28,29] or thermoelectric effects [30–32]. The thermocurrent generated by the plasmons is proportional to the total power P absorbed by the system. Here we show that P is related to the plasmon reflectance R of the junction, so that the analytical formula Eq. (15) can be used to predict the detection efficiency. The photon to plasmon conversion can be understood from our formulas above since the external terahertz field effectively replaces the field of the incident plasmon wave. For a narrow gap both fields are approximately constant over the junction, so that they launch plasmons from the junction with the same efficiency.
To quantify the extra absorbed power when the junction is present compared to a uniform 2DEG sheet, we define the dimensionless excess power absorption,Fig. 6(a), ΔP̄l is very similar to the reflectance shown in Fig. 3(d). The resonances occur at the same locations but the odd modes disappear, due to the incident field being even in x.
A similar resonant structure is found when the frequency ω instead of the junction conductivity σ is varied, as shown in Fig. 6(b). Varying ω changes the background wavelength λ0 and thus the ratio a/λ0. The enhanced ΔP̄j at low frequencies is due to our assumption of a Drude-like conductivity, σ = iD/π(ω + iν) where the damping rate ν is taken to be constant, so that has a maximum at ω = 0. The Drude weight D is assumed to be proportional to the carrier density and independent of frequency. At low frequencies using the approximations E ≃ V/2a and Es ≃ −reiq0|x|, we obtain
As previously mentioned, in the THz range where the plasmon wavelength λ0 is often larger than the distance d between the 2D sheet and the gate, the effects of screening cannot be neglected in general. However, if the junction is narrow enough such that a ≪ d, the capacitive coupling that gave rise to the absorption peak still occur. Screening effectively replaces the long range cutoff 1/q0 by d in the capacitance, Eq. (11), causing a small shift in the peak absorption frequency.
Besides electrostatically gated graphene or 2DEGs, tunable conductivity profile needed for the plasmon-reflecting 1D junction can be also realized with layered high-Tc superconductors by varying the temperature. Plasmon reflection in such a system would also yield information about local optical conductivity, which can be difficult to measure by other means. If the conductivity contrast is large, which is the case of our primary interest, the system becomes a weak link, i.e., a Josephson junction. Such a junction can be fabricated by lowering the Tc locally using focused ion beams or by etching and reducing the number of layers . An estimate of the required parameters are as follows. Assume a BCS-like superconductor film has a thickness l = 10 nm, interlayer Josephson plasmon frequency ωc = 50 cm−1, lattice dielectric constant ∊∞ = 27, anisotropy γ = 17.5, and is placed on a strontium titanate (STO) substrate. The low-frequency plasmon dispersion is then . At a frequency ω = 0.25 THz, the effective dielectric constant of STO is κ ≃ 1000, and so the plasmon wavelength λ0 ≃ 10 μm. If the conductivity inside the central region is σ = 0.1σ0, then the strongest reflection/optimal THz detection occurs for a junction of width 2a ∼ 200 nm, which is quite practical.
We presented a theory of plasmonic interaction with a conductivity dip and showed that a narrow junction is a minimalistic yet robust plasmonic switch. Our analytical results and the associated physical insights into the plasmonic interaction with a 1D defect may be useful for the design of nanoscale plasmonic reflectors as well as terahertz detectors.
Appendix A. Definition of impedances in the equivalent circuits
There is always some arbitrariness  in the value of the characteristic impedance Z0 = v/i of a waveguide, which derives from the freedom of choosing the definition of “voltage” v and “current” i of a propagating wave. The only constraint is that the product of the two is proportional to the power P transmitted by such a wave, more precisely, thatFig. 7(a). The power transmitted by the plasmon (per unit length in y) is given by the integrated Poynting vector, Fig. 1(b) of the main text. The horizontal legs have the impedance 
However, using Eqs. (28) and (29) for a narrow strip would be wrong because in that case the electric and magnetic fields have a much more complicated distribution than what is sketched in Fig. 7(a). Nevertheless, a simple result is obtained for a narrow vacuum gap, σ = 0, whose interaction with the plasmons is governed by the capacitive coupling of the two leads. As shown in the main text, the correct plasmon reflection coefficient is obtained if we assume that the impedance of this type of junction is given by the usual circuit formula for the capacitor,Eq. (26). The reason for its absence is the difference between the “near-field” potential V and the “far-field” potential ϕ. Potential ϕ in our definiton of the impedance refers to the asymptotic amplitude of the plasmon waves. Our formula for the the reflection coefficient [Eq. (48) below] indicates that for the incident wave ϕi = eiq0x, the scattered potential at large distances oscillates with an amplitude V. On the other hand, since V is also the potential difference across the gap, its amplitude just outside the gap oscillates with an amplitude V/2, as shown in Fig. 7(b). This factor-of-two difference between the far-field and near-field amplitudes cancels the factor of one-half in the definition of the effective voltage v = ϕ/2, leading to the standard form of Eq. (30).
The same cancellation occurs also if the short strip has a nonzero sheet conductivity σ. In other words, the correct result for the reflection coefficient is obtained if impedances of the T-junction Z1 and Z2 are multiplied by the factor of two. As far as the vertical leg is concerned, this modification has virtually no effect because it already has a large impedance Z2 ≫ Z1, and so can be cut from the circuit. But the two horizontal legs now combine to the total impedance of Z ≃ 4Z1 = −4iZ0 tan qa, as mentioned in the main text. The last expression can be further approximated byEq. (30) being the standard formula for a capacitor.
Appendix B. Reflection coefficient of general 1D inhomogeneities
In our problem the plasmon wave ϕi = eiq0x impinges upon a 1D inhomogeneity in the conductivity of the sheet localized around the y-axis, parametrized as26]) 26], ϕ̃(k) =ϕ̃ext(k)/∊(k). Equation (33) can be further simplified in momentum space in terms of the total electric field E = −∂xϕ and the scattered field Es = −∂xψ,
The reflection coefficient r of the normally incident plasmon wave can be obtained by analyzing the scattered field Es at large negative x, Es ≃ −rE0e−iq0x, where E0 is the amplitude of the incident field Ei = E0eiq0x. From Eq. (35), the long range behavior of Es is determined by the pole of ∊−1(k) at k = −q0. Taking the residue, we getEq. (36). The transmission coefficient t can be similarly obtained by taking the residue of the pole at k = q0,
An analytical expression for E(x) (and thus r) can be found in the perturbative case, r ≪ 1, where the current density is approximately constant across the inhomogeneity,
Appendix C. Reflection coefficient of narrow junctions
In this section we consider a particular type of conductivity profile,
C.1. Vacuum gap
Analytical solution of the field distribution Evac for the special case of a vacuum gap, σ = 0 or g =−1, is well known ,Eq. (36), we get Eq. (37) with eiq0x→ 1] Eq. (44) yields the reflection coefficient of a narrow vacuum gap,
C.2. General cases
We add Ei to both sides of Eq. (36) to getEq. (42), and integrate it from x = −a to a,
Equation (51) can be used to find the reflection coefficient r if the field E(x) were known. We use the following ansatz for the field,Eq. (50) using the substitutions g = −1 and 1 + g = σn/σ0 and invoking Eqs. (46) and (54), Eq. (55) by F(x) and integrating over x, Eq. (55) is self-adjoint, the eigenmodes En are orthogonal, Eq. (55) can be solved numerically to obtain the eigenmodes. As shown in Fig. 8, they have the asymptotic forms
Next, we determine the coefficients dn for the fields En. Under an external field the voltage across the junction V ≠ 0 but all Vn = 0. Thus V must be accounted for by d0, i.e.,Eq. (50) using Eqs. (53) and (55) as Eq. (46) was used to convert constants into the convolution, Eq. (51) was used in the form
C.3. Analytic approximationEq. (58), so that
Appendix D. Smooth conductivity profiles
In our model we assume sharp conductivity changes at the edges of the junction. This may be difficult to achieve in experiment, or yield unphysical results in theory. Here we show that the physics for a sharply-changing conductivity profile remains qualitatively the same for a smoothly-varying one. To demonstrate, we calculated numerically the reflectance of a smooth conductivity profileFig. 9, the reflectance retains the same features – capacitive open-circuit resonance and Fano-shaped cavity resonances with weak odd modes. The open-circuit resonance is still persistent under damping, while the cavity resonances are less sharp due to a larger |t′| for smooth profiles. Note that to get an anti-resonance at the same location, σ0/σ ≈ 10, the junction width had to be increased by an order of magnitude, from a/λ0 = 0.01 to 0.1.
Department of Energy (DOE) (DE-SC0012592, DE FG02 84ER45118); Office of Naval Research (ONR) (N00014-15-1-2671); Semiconductor Research Corporation (SRC) (2701.002).
References and links
2. A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6, 749–758 (2012). [CrossRef]
3. Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012). [CrossRef] [PubMed]
4. A. Woessner, M. B. Lundeberg, Y. Gao, A. Principi, P. Alonso-González, M. Carrega, K. Watanabe, T. Taniguchi, G. Vignale, M. Polini, J. Hone, R. Hillenbrand, and F. H. L. Koppens, “Highly confined low-loss plasmons in graphene-boron nitride heterostructures,” Nat. Mater. 14, 421–425 (2014). [CrossRef] [PubMed]
5. G. X. Ni, L. Wang, M. D. Goldflam, M. Wagner, Z. Fei, A. S. McLeod, M. K. Liu, F. Keilmann, B. Özyilmaz, A. H. C. Neto, J. Hone, M. M. Fogler, and D. N. Basov, “Ultrafast optical switching of infrared plasmon polaritons in high-mobility graphene,” Nat. Photonics 10, 244–247 (2016). [CrossRef]
6. G. X. Ni, A. S. McLeod, Z. Sun, L. Wang, L. Xiong, K. W. Post, S. S. Sunku, B.-Y. Jiang, J. Hone, C. R. Dean, M. M. Fogler, and D. N. Basov, “Fundamental limits to graphene plasmonics,” Nature 557, 530–533 (2018). [CrossRef]
7. V. Ryzhii, A. Satou, I. Khmyrova, A. Chaplik, and M. S. Shur, “Plasma oscillations in a slot diode structure with a two-dimensional electron channel,” J. Appl. Phys. 96, 7625–7628 (2004). [CrossRef]
8. V. V. Popov, G. M. Tsymbalov, M. S. Shur, and W. Knap, “The resonant terahertz response of a slot diode with a two-dimensional electron channel,” Semiconductors 39, 142–146 (2005). [CrossRef]
9. A. Satou, V. Ryzhii, and A. Chaplik, “Plasma oscillations in two-dimensional electron channel with nonideally conducting side contacts,” J. Appl. Phys. 98, 034502 (2005). [CrossRef]
11. V. V. Popov, O. V. Polishchuk, and S. A. Nikitov, “Electromagnetic renormalization of the plasmon spectrum in a laterally screened two-dimensional electron system,” JETP Lett. 95, 85–90 (2012). [CrossRef]
12. J. Chen, M. L. Nesterov, A. Y. Nikitin, S. Thongrattanasiri, P. Alonso-González, T. M. Slipchenko, F. Speck, M. Ostler, T. Seyller, I. Crassee, F. H. L. Koppens, L. Martin-Moreno, F. J. García de Abajo, A. B. Kuzmenko, and R. Hillenbrand, “Strong plasmon reflection at nanometer-size gaps in monolayer graphene on SiC,” Nano Lett. 13, 6210–6215 (2013). [CrossRef] [PubMed]
14. J. Polanco, R. M. Fitzgerald, and A. A. Maradudin, “Scattering of surface plasmon polaritons by one-dimensional surface defects,” Phys. Rev. B 87, 155417 (2013). [CrossRef]
15. A. Woessner, Y. Gao, I. Torre, M. B. Lundeberg, C. Tan, K. Watanabe, T. Taniguchi, R. Hillenbrand, J. Hone, M. Polini, and F. H. L. Koppens, “Electrical 2π phase control of infrared light in a 350-nm footprint using graphene plasmons,” Nat. Photonics 11, 421–424 (2017). [CrossRef]
17. B.-Y. Jiang and M. M. Fogler, “Electronic response of graphene to linelike charge perturbations,” Phys. Rev. B 91, 235422 (2015). [CrossRef]
18. B.-Y. Jiang, G. Ni, C. Pan, Z. Fei, B. Cheng, C. Lau, M. Bockrath, D. Basov, and M. Fogler, “Tunable plasmonic reflection by bound 1D electron states in a 2D Dirac metal,” Phys. Rev. Lett. 117, 086801 (2016). [CrossRef] [PubMed]
19. B. Rejaei and A. Khavasi, “Scattering of surface plasmons on graphene by a discontinuity in surface conductivity,” J. Opt. 17, 075002 (2015). [CrossRef]
20. G. C. Dyer, G. R. Aizin, S. Preu, N. Q. Vinh, S. J. Allen, J. L. Reno, and E. A. Shaner, “Inducing an incipient terahertz finite plasmonic crystal in coupled two dimensional plasmonic cavities,” Phys. Rev. Lett. 109, 126803 (2012). [CrossRef] [PubMed]
21. G. C. Dyer, G. R. Aizin, S. J. Allen, A. D. Grine, D. Bethke, J. L. Reno, and E. A. Shaner, “Induced transparency by coupling of Tamm and defect states in tunable terahertz plasmonic crystals,” Nat. Photonics 7, 925–930 (2013). [CrossRef]
22. A. Y. Nikitin, T. Low, and L. Martin-Moreno, “Anomalous reflection phase of graphene plasmons and its influence on resonators,” Phys. Rev. B 90, 041407 (2014). [CrossRef]
23. J.-H. Kang, S. Wang, Z. Shi, W. Zhao, E. Yablonovitch, and F. Wang, “Goos-hänchen shift and even-odd peak oscillations in edge-reflections of surface polaritons in atomically thin crystals,” Nano Lett. 17, 1768–1774 (2017). [CrossRef] [PubMed]
24. C. G. Montgomery, R. H. Dicke, and E. M. Purcell, Principles of Microwave Circuits, vol. 8 of MIT Radiation Laboratory Series (McGraw-Hill, 1948).
25. S. Farajollahi, S. AbdollahRamezani, K. Arik, B. Rejae, and A. Khavasi, “Circuit model for plasmons on graphene with one-dimensional conductivity profile,” IEEE Photon. Technol. Lett. 28, 355–358 (2016). [CrossRef]
26. Z. Fei, A. S. Rodin, W. Gannett, S. Dai, W. Regan, M. Wagner, M. K. Liu, A. S. McLeod, G. Dominguez, M. Thiemens, A. H. C. Neto, F. Keilmann, A. Zettl, R. Hillenbrand, M. M. Fogler, and D. N. Basov, “Electronic and plasmonic phenomena at graphene grain boundaries,” Nat. Nanotech. 8, 821–825 (2013). [CrossRef]
27. V. Ryzhii, A. Satou, and M. S. Shur, “Admittance of a slot diode with a two-dimensional electron channel,” J. Appl. Phys. 93, 10041–10045 (2003). [CrossRef]
28. M. Dyakonov and M. Shur, “Detection, mixing, and frequency multiplication of terahertz radiation by two-dimensional electronic fluid,” IEEE Trans. Electron Dev. 43, 380–387 (1996). [CrossRef]
29. W. Knap, M. Dyakonov, D. Coquillat, F. Teppe, N. Dyakonova, J. Łusakowski, K. Karpierz, M. Sakowicz, G. Valusis, D. Seliuta, I. Kasalynas, A. El Fatimy, Y. M. Meziani, and T. Otsuji, “Field effect transistors for terahertz detection: Physics and first imaging applications,” J. Infrared Millim. Terahertz Waves 30, 1319–1337 (2009).
31. M. B. Lundeberg, Y. Gao, A. Woessner, C. Tan, P. Alonso-Gonzalez, K. Watanabe, T. Taniguchi, J. Hone, R. Hillenbrand, and F. H. L. Koppens, “Thermoelectric detection and imaging of propagating graphene plasmons,” Nat. Mater. 16, 204–207 (2017). [CrossRef]
32. A. Woessner, R. Parret, D. Davydovskaya, Y. Gao, J.-S. Wu, M. B. Lundeberg, S. Nanot, P. Alonso-González, K. Watanabe, T. Taniguchi, R. Hillenbrand, M. M. Fogler, J. Hone, and F. H. L. Koppens, “Electrical detection of hyperbolic phonon-polaritons in heterostructures of graphene and boron nitride,” NPJ 2D Mater. Appl. 1, 25 (2017). [CrossRef]
33. S. A. Cybart, E. Y. Cho, T. J. Wong, B. H. Wehlin, M. K. Ma, C. Huynh, and R. C. Dynes, “Nano Josephson superconducting tunnel junctions in YBa2Cu3O7−δ directly patterned with a focused helium ion beam,” Nat. Nanotech. 10, 598–602 (2015). [CrossRef]
34. H. T. Stinson, J. S. Wu, B. Y. Jiang, Z. Fei, A. S. Rodin, B. C. Chapler, A. S. McLeod, A. Castro Neto, Y. S. Lee, M. M. Fogler, and D. N. Basov, “Infrared nanospectroscopy and imaging of collective superfluid excitations in anisotropic superconductors,” Phys. Rev. B 90, 014502 (2014). [CrossRef]
35. W. Smythe, Static and Dynamic Electricity, 2nd ed. (McGraw-Hill, 1950).