1. Introduction

Plasmonics aims to combine the advantages of nanometer scale electronics with the high operating frequency (terahertz and beyond) of photonics [1]. 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, $q=\frac{i\kappa \omega }{2\pi \sigma }$, 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.

 

Fig. 1 A normally incident plasmon is partially reflected and transmitted by a 1D junction of conductivity σ different from the background value σ₀. The strength of reflection is determined by two types of effects, the capacitive coupling of the two edges of the leads (represented by the + and − electric charges) and the cavity resonances in the strip. The field profiles of the first few resonant modes (white solid and dashed curves) are calculated for the case of a narrow junction with an infinite conductivity contrast between the gap and the leads.

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Fig. 2 (a) Equivalent circuit for the system. The leads have impedance Z₀, while the junction J has different representations depending on its width a and conductivity σ. (b) For wide junctions J is represented by two interfaces and a waveguide of length 2a. The waveguide can be replaced by a T-junction, while the interface I consists of two phase shifters and an ideal transformer. The right interface I_R has reversed number of coils and signs of θ compared to I_L. (c) For narrow junctions J is a capacitor in parallel with a LC network that describes the cavity resonances. (d) In the dc limit, a ≪ λ, the LC network reduces to a single inductance L₀.

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2. Results

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 [19] to be

 

Fig. 3 (a) Reflectance of a “wide” junction, a = λ₀ without damping. (b) Similar quantities for damping γ = 0.05. (c) Reflectance of a “narrow” junction, a = 0.01λ₀ without damping. Blue curves are numerical results, dotted red curves are from the F-P formula Eq. (3), and green curves are from Eq. (15). (d) Similar quantities for γ = 0.05.

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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) [24]. However, the plasmonic wave impedance of a 2DEG, Z₀ = π/ωκ, was found to be independent of the sheet conductivity [25] (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 $Z_j=Z_0\frac{{\sigma }_0}{\sigma }$ 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 = −iZ_j tan(qa − θ/2) and Z₂ = iZ_j csc(2qa − θ) [24]. 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 E_i = E₀e^iq₀xx̂ with the asymptotic form of the scattered field E_s ≃ −rE₀e^−iq₀xx̂ at large distances from the junction. The scattered field E_s can be completely determined by the total field E inside the junction via the Green’s function G,

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 $E\simeq \frac{{\sigma }_0}{\sigma }{E}_0$. This yields the reflection coefficient

 

Fig. 4 Reflectance of a narrow vacuum gap in a lossless sheet. The curve labeled "analyt." is computed using Eq. (10) and R = |r|².

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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,

The reflection coefficient can be found by substituting the field Eq. (13) into Eq. (12), then using the general form of r [Eq. (8)] to obtain

Equivalent circuits are again helpful in understanding Eq. (15). Consider the impedance of a 2a-long piece of plasmonic waveguide Z = −4iZ₀ tan qa [24], which can be expanded into the resonant form $Z^{-1}={\left(-4i{Z}_{0}qa\right)}^{-1}+i\frac{qa}{2Z_0}{\sum }_n{\left(q^2{a}^2-{\pi }^2{n}^2\right)}^{-1}$. 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 $Z^{-1}={\left(-i\omega L_0\right)}^{-1}+i\omega {\sum }_n{\left({\omega }^2{L}_n-C_n^{-1}\right)}^{-1}$, which yields L₀ = 2ia/σω, $L_n=\frac{1}{2}{L}_0$, and C_n = qaκ/2π³n². The load impedance of the junction, including the capacitor, the LC network and the right half-plane is then Z_L = ZZ_c/(Z + Z_c) + Z₀, which yields the reflection coefficient

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 Z_C = i/ωC and $Z_{L_0}=-i\omega L_0=\frac{2a}{\sigma }$, the later resembling the usual dc formula for resistance. When the inverse impedances, i.e., admittances $Z_C^{-1}$ and $Z_{L_0}^{-1}$ cancel, the total impedance of the junction diverges and the reflectance is unity. We call this situation the open-circuit resonance. It occurs when

 

Fig. 5 (a) False color plot of the reflectance of the junction for γ = 0.05. (b) Schematic diagram of the location of the open-circuit and the F-P resonances. The red curve is predicted by Eq. (18).

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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) Excess power absoption ΔP̄ for a = 0.01λ₀ and γ = 0.05. The power absorbed by the leads (blue) is proportional to the reflectance R. The power absorbed by the junction (red) has a maximum at the open-circuit and the F-P resonances. (b) Excess power absoption spectrum for a junction of width 2a = 1000 nm in graphene on hBN. The background conductivity is assumed to be Drude like, ${\sigma }_0=\frac{i{D}_0}{\pi \left(\omega +i\nu \right)}$ with ν = 3 cm⁻¹. The Drude weight $D_0=\frac{e^2}{\pi {\hslash }^2}$ μ is calculated at chemical potential μ = 0.12eV, the conductivity ratio is D/D₀ = 0.03. At low frequencies ω ∼ ν the conductivity is dominated by damping, resulting in a negative ΔP̄_l, i.e., the absorption in the leads is smaller than the background value.

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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 ΔP̄_j 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 $\text{Re}\sigma \propto \frac{\nu }{{\omega }^2+{\nu }^2}$ 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 E_s ≃ −re^iq₀|x|, we obtain

3. Discussion

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-T_c 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 T_c locally using focused ion beams or by etching and reducing the number of layers [33]. 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 ω_c = 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 $\omega \simeq \gamma {\omega }_c\sqrt{{(1+2\kappa /{∊}_{\infty }ql)}^{-1}}$ [34]. 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.

4. Conclusion

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 [24] 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, that

(22)$$P=\frac{1}{2}\text{Re}\left(v^{*}i\right).$$

In our case the propagating wave is a plasmon. The plasmon travelling in the x-direction can be described by the field amplitudes

(23)$$E_x=e^{i{q}_{0}x}{e}^{-q_0\left|z\right|},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{E}_z=i{E}_x,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{H}_y=\frac{2\pi }{c}{\sigma }_0{E}_x,$$

which decay exponentially away from the plane, see Fig. 7(a). The power transmitted by the plasmon (per unit length in y) is given by the integrated Poynting vector,

(24)$$P=\frac{1}{2}\text{Re}{\int }_{-\infty }^{\infty }dz\frac{c}{4\pi }{H}_y^{*}{E}_z=\frac{1}{4}\text{Re}\left(j^{*}\varphi \right).$$

Here ϕ is the electric potential on the plane,

(25)$$\varphi ={\int }_0^{\infty }dz{E}_z,$$

and j = σ₀E_x is the current density. In this work we define our effective current to be equal to the physical current, i = j. This choice leads to a simple form of the current conservation in our equivalent circuits and fixes the effective voltage v to be one-half of the physical potential:

(26)$$v=\frac{1}{2}\varphi ,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=j.$$

The characteristic impedance of the sheet is then

(27)$$Z_0=\frac{1}{2}\frac{\varphi }{j}=\frac{\pi }{\omega \kappa }.$$

One further application of this result is to a wide but finite strip. If this strip has the unperturbed sheet conductivity σ = σ₀ and the total length $2a\gg q_0^{-1}$ in the x-direction, then it is equivalent to a T-junction shown in Fig. 1(b) of the main text. The horizontal legs have the impedance [24]

(28)$$Z_1=-i{Z}_0\text{tan}{q}_{0}a$$

each and the vertical leg has the impedance

(29)$$Z_2=i{Z}_0\text{csc}{q}_{0}a.$$

However, using Eqs. (28) and (29) for a narrow strip $2a\ll q_0^{-1}$ 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,

(30)$$Z_C=\frac{1}{-i\omega C}.$$

Notably, it has no additional factor of one-half, which one would naively guess based on 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 ϕ_i = e^iq₀x, 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).

 

Fig. 7 (a) Distribution of E_z and H_y outside the sheet. (b) Schematic diagram of the potential distribution for the case of a narrow vacuum gap. The large distance oscillations have amplitude V, while the potential right next to the gap oscillates with amplitude V/2.

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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 by

(31)$$Z\simeq -4i{Z}_{0}qa=-4i\frac{\pi }{\omega \kappa }\frac{i\omega \kappa }{2\pi \sigma }a=\frac{2a}{\sigma },\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}qa\ll 1,$$

which is the usual formula for dc resistance of a conductor, similar to Eq. (30) being the standard formula for a capacitor.

Appendix B. Reflection coefficient of general 1D inhomogeneities

In our problem the plasmon wave ϕ_i = e^iq₀x impinges upon a 1D inhomogeneity in the conductivity of the sheet localized around the y-axis, parametrized as

(32)$$g(x)=\frac{\mathrm{\Delta }\sigma (x)}{{\sigma }_0},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{\Delta }\sigma (x)\equiv \sigma (x)-{\sigma }_0.$$

The scattered electric potential ψ from the inhomogeneity can be written in terms of the total electric potential ϕ as (cf. Supplementary Information of [26])

(33)$$\psi (x)=\frac{1}{q_0}(G_1*V_1)*\left[-{\partial }_{x}g(x){\partial }_x\varphi (x)\right],$$

where * denotes convolution and V₁ is the Coulomb kernel with the Fourier transform Ṽ₁ = 1/|k|. The Green’s function $G_1(x)={\int }_{-\infty }^{\infty }\frac{dk}{2\pi }{e}^{ikx}{∊}^{-1}(k)$ is the 1D Fourier transform of the dielectric function ∊ given by

(34)$$G_1(x)=-i{q}_0{e}^{i{q}_0\left|x\right|}-\frac{q_0}{2\pi }\left[e^{i{q}_0\left|x\right|}{E}_1(i{q}_0\left|x\right|)+e^{-i{q}_0\left|x\right|}{E}_1\left(-i{q}_0\left|x\right|\right)\right],$$

where the exponential integral $E_1(z)={\int }_z^{\infty }dt\hspace{0.17em}{t}^{-1}{e}^{-t}$ has a branch cut on the negative real axis. The 2D dielectric function ∊(k) = 1 − |k|/q₀, which encodes the sheet conductivity σ₀ through $q_0=\frac{i\kappa \omega }{2\pi {\sigma }_0}$, is frequency and wavevector-dependent and determines the response of the sheet to an external potential [26], ϕ̃(k) =ϕ̃_ext(k)/∊(k). Equation (33) can be further simplified in momentum space in terms of the total electric field E = −∂_xϕ and the scattered field E_s = −∂_xψ,

(35)$${\tilde{E}}_s(k)=-ik\tilde{\psi }(k)=\left(\frac{1}{∊(k)}-1\right)\left(\widetilde{gE}\right)(k),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left(\widetilde{gE}\right)=\frac{1}{2\pi }\tilde{g}*\tilde{E}.$$

In the real space this reads,

(36)$$E_s(x)={\int }_{-\infty }^{\infty }d{x}^{\prime }\left(G_1(x-x^{\prime })-\delta (x-x^{\prime })\right)g(x^{\prime })E(x^{\prime }).$$

This is the equation for the scattered field used in the main text. It indicates the scattered field E_s can be determined by the conductivity profile g(x) and the local field E(x) around the inhomogeneity using the Green’s function G₁.

The reflection coefficient r of the normally incident plasmon wave can be obtained by analyzing the scattered field E_s at large negative x, E_s ≃ −rE₀e^−iq₀x, where E₀ is the amplitude of the incident field E_i = E₀e^iq₀x. From Eq. (35), the long range behavior of E_s is determined by the pole of ∊⁻¹(k) at k = −q₀. Taking the residue, we get

(37)$$r=\frac{i{q}_0}{E_0}{\int }_{-\infty }^{\infty }dx\hspace{0.17em}g(x)E(x)e^{i{q}_{0}x}.$$

This formula is exact and can be used for any conductivity profile g(x). The value of r can be obtained numerically by calculating the field E(x) from Eq. (36). The transmission coefficient t can be similarly obtained by taking the residue of the pole at k = q₀,

(38)$$t=1-\frac{i{q}_0}{E_0}{\int }_{-\infty }^{\infty }dx\hspace{0.17em}g(x)E(x)e^{-i{q}_{0}x}.$$

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,

(39)$$j={\sigma }_0{E}_0=\sigma (x)E(x),$$

so that $E(x)=\frac{{\sigma }_0}{\sigma (x)}{E}_0{e}^{i{q}_{0}x}$ and

(40)$$r=i{q}_0{\int }_{-\infty }^{\infty }dx\frac{\sigma (x)-{\sigma }_0}{\sigma (x)}{e}^{2i{q}_{0}x}=i{q}_0{\int }_{-\infty }^{\infty }dx\frac{g(x)}{g(x)+1}{e}^{2i{q}_{0}x}.$$

Appendix C. Reflection coefficient of narrow junctions

In this section we consider a particular type of conductivity profile,

(41)$$g(x)=g\mathrm{\Theta }\left(a-\left|x\right|\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g=\frac{\mathrm{\Delta }\sigma }{{\sigma }_0}=\frac{\sigma -{\sigma }_0}{{\sigma }_0},$$

which describes a junction that has a constant conductivity σ and a width 2a. Further, we assume the width is narrow compared to the background plasmon wavelength, $a\ll q_0^{-1}$. In this limit, the leads can be considered perfect metals of effectively infinite conductivity. This allows us to make simplifications and obtain analytical expressions for the reflection coefficient r.

C.1. Vacuum gap

Analytical solution of the field distribution E_vac for the special case of a vacuum gap, σ = 0 or g =−1, is well known [35],

(42)$$E_{\text{vac}}(x)=\mathit{VF}(x),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}F(x)=\frac{1}{\pi }\frac{\mathrm{\Theta }\left(a-\left|x\right|\right)}{\sqrt{a^2-x^2}},$$

where V is the voltage difference across the junction,

(43)$$V={\int }_{-a}^{a}dx\hspace{0.17em}E(x).$$

Substituting E_vac(x) into Eq. (36), we get

(44)$$\begin{array}{ll}{E}_i(x)\simeq E_0\hfill & =V{\int }_{-a}^{a}d{x}^{\prime }\hspace{0.17em}{G}_1(x-x^{\prime })F(x)\hfill \\ \hfill & =V\frac{q_0}{\pi }\left[c+\text{log}\left(\frac{q_{0}a}{2}\right)-i\pi \right],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-a<x<a.\hfill \end{array}$$

Here we used the small-distance approximation for the Green’s function,

(45)$$G_1(x)\simeq \frac{q_0}{\pi }\left[c+\text{log}(q_0\left|x\right|)-i\pi \right]+𝒪\left(q_0\left|x\right|\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{q}_0\left|x\right|\ll 1,$$

where c ≃ 0.577 is the Euler-Mascheroni constant, and the table integral

(46)$${\int }_{-a}^{a}d{x}^{\prime }\frac{\text{log}\left|x-x^{\prime }\right|}{\pi \sqrt{a^2-x^{\prime 2}}}=\text{log}\frac{a}{2}.$$

(It can be evaluated by the change of variables x′ = a cos θ′) Identifying the capacitance of the junction

(47)$$C=\frac{\kappa }{2{\pi }^2}\text{log}\frac{2e^{-c}}{q_{0}a},$$

and the reflection coefficient for a narrow junction [Eq. (37) with e^iq₀x→ 1]

(48)$$r\simeq \frac{i{q}_0}{E_0}gV,$$

our Eq. (44) yields the reflection coefficient of a narrow vacuum gap,

(49)$$r=\frac{i\pi }{\text{log}\left(\frac{2}{q_{0}a}\right)-c+i\pi }=\frac{i\kappa }{2\pi C+i\kappa }.$$

Note that as the gap gets wider, r approaches unity. This can be understood by considering an infinitely wide gap where the current j(x) is completely reflected and is zero at the edge, so that the current reflection coefficient r_j = −r = −1.

C.2. General cases

We add E_i to both sides of Eq. (36) to get

(50)$$E_i(x)\simeq E_0=-\left[G(x)*gE(x)\right]+(1+g)E(x),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-a<x<a.$$

We multiply this by σ₀F(x), where F(x) is given by Eq. (42), and integrate it from x = −a to a,

(51)$${\sigma }_0{E}_0=-{\sigma }_{0}gV\frac{q_0}{\pi }\left[c+\text{log}\left(\frac{q_{0}a}{2}\right)-i\pi \right]+{\sigma }_0(1+g){\int }_{-a}^{a}dxE(x)F(x).$$

We arrive at the equation (mentioned in the main text)

(52)$${\sigma }_0{E}_0(1-r)=-i\omega V{C}^{\prime }+{\int }_{-a}^{a}dx\hspace{0.17em}\sigma E(x)F(x),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{C}^{\prime }\equiv -gC,$$

which represents the conservation of charge. Note that this equation has the correct limiting behavior, reproducing the reflection coefficient for the vacuum gap when g = −1 and σ = 0, and yielding r = 0 when g = 0 and σ = σ₀.

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,

(53)$$E(x)=d_0+\sum _{n=1}^{\infty }{d}_n{E}_n(x),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-a<x<a,$$

which is an expansion into the eigenmodes E_n of the system when there is no external field. The eigenmodes E_n have the following properties,

(54)$$E_n\left(\left|x\right|>a\right)=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{V}_n={\int }_{-a}^{a}dx\hspace{0.17em}{E}_n(x)=0,$$

as the system now consists of a junction with nonzero conductivity σ_n and two sheets of infinite conductivity. The eigenvalue equation governing E_n(x) can be derived from Eq. (50) using the substitutions g = −1 and 1 + g = σ_n/σ₀ and invoking Eqs. (46) and (54),

(55)$$E_n(x)=A-\frac{q_n}{\pi }{\int }_{-\infty }^{\infty }d{x}^{\prime }\text{log}\frac{\left|x-x^{\prime }\right|}{L}{E}_n(x^{\prime }),$$

where q_n = iκω/2πσ_n, the constant L is arbitrary, and the constant A can be obtained by multiplying both sides of Eq. (55) by F(x) and integrating over x,

(56)$$A={\int }_{-a}^a{E}_n(x)F(x).$$

Since the kernel in Eq. (55) is self-adjoint, the eigenmodes E_n are orthogonal,

(57)$${\int }_{-a}^{a}dx\hspace{0.17em}{E}_n(x)E_m(x)=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n\ne m,$$

and the eigenvalues q_n are real. Eq. (55) can be solved numerically to obtain the eigenmodes. As shown in Fig. 8, they have the asymptotic forms

(58)$$\frac{E_n}{E_0}\simeq \text{cos}\left(q_{n}x-\frac{n\pi }{2}\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{q}_n\simeq \frac{\pi }{2a}\left(n+\frac{1}{4}\right),$$

as n is increased.

 

Fig. 8 (a) Eigenvalues q_n and (b)–(f) the first five eigenmodes f_n of a junction in a perfect metal sheet. The analytical formula is Eq. (58), while the numerical results are obtained by solving Eq. (55).

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Next, we determine the coefficients d_n for the fields E_n. Under an external field the voltage across the junction V ≠ 0 but all V_n = 0. Thus V must be accounted for by d₀, i.e.,

(59)$$d_0=\frac{V}{2a}.$$

To find d_n, we rewrite Eq. (50) using Eqs. (53) and (55) as

(60)$$\text{log}\frac{\left|x\right|}{L}*h(x)=0,$$

with a solution

(61)$$h(x)=V\left[\frac{1}{2a}-F(x)\right]+\sum _{n=1}^{\infty }\left(1+\frac{g+1}{g}\frac{q_n}{q_0}\right)d_n{E}_n=0.$$

Here Eq. (46) was used to convert constants into the convolution,

(62)$$1=\frac{1}{\text{log}(a/2)}F(x)*\text{log}\frac{\left|x\right|}{L},$$

and Eq. (51) was used in the form

(63)$$E_0-(g+1){\int }_{-a}^{a}dx\hspace{0.17em}E(x)F(x)=-g\frac{q_0}{\pi }V\text{log}\frac{a}{2}.$$

Using the orthogonality of the eigenmodes, we find

(64)$$d_n=\frac{V}{1+\frac{g+1}{g}\frac{q_n}{q_0}}\frac{{\int }_{-a}^{a}dx\hspace{0.17em}{E}_n(x)F(x)}{{\int }_{-a}^{a}dx\hspace{0.17em}{E}_n^2(x)}.$$

Note that in the limit of a vacuum gap where g =−1, d_n reduces to coefficients of the expansion of E_vac = VF(x) in the E_n basis, as expected. In the manuscript, the dimensionless expansion coefficients b₀ and b_n are defined as

(65)$$b_0=\frac{d_0}{(V/2a)}=1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{b}_n=d_n\frac{E_0}{(V/2a)}=\frac{2a}{1+\frac{g+1}{g}\frac{q_n}{q_0}}\frac{{\int }_{-a}^{a}dx\hspace{0.17em}{f}_n(x)F(x)}{{\int }_{-a}^{a}dx\hspace{0.17em}{f}_n^2(x)},$$

and the field is expanded as

(66)$$E(x)=\frac{V}{2a}\left(1+\sum _{n=1}^{\infty }{b}_n{f}_n\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{f}_n=\frac{E_n}{E_0}.$$

C.3. Analytic approximation

Having found the coefficients d_n, the expression for the reflection coefficient can now be written from Eq. (51) using Eq. (48),

(67)$$r^{-1}=1-\frac{2\pi i}{\kappa }C-\frac{i}{q_0}\frac{g+1}{g}\left\{\frac{1}{2a}+\sum _{n=1}^{\infty }\frac{1}{1+\frac{g+1}{g}\frac{q_n}{q_0}}\frac{{\left[{\int }_{-a}^{a}dx\hspace{0.17em}{f}_n(x)F(x)\right]}^2}{{\int }_{-a}^{a}dx\hspace{0.17em}{f}_n^2(x)}\right\}.$$

An analytical expression for r can be found using the approximation Eq. (58), so that

(68)$${\int }_{-a}^{a}dx\hspace{0.17em}{f}_n^2(x)=a\left[1+\frac{1}{\sqrt{2}\pi \left(n+\frac{1}{4}\right)}\right],$$

and

(69)$$\begin{array}{ll}{\left[{\int }_{-a}^{a}dx\hspace{0.17em}{f}_n(x)F(x)\right]}^2\hfill & =0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n\hspace{0.17em}\hspace{0.17em}\text{odd}\hfill \\ \hfill & =J_0^2(q_{n}a)\simeq \frac{2+\sqrt{2}}{{\pi }^2}\frac{1}{n+\frac{1}{4}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n\gg 1\hspace{0.17em}\hspace{0.17em}\text{even},\hfill \end{array}$$

where J₀(z) is the Bessel function of the first kind. The summation is then

(70)$$\sum _{n=1}^{\infty }\frac{1}{1+\frac{g+1}{g}\frac{q_n}{q_0}}\frac{{\left[{\int }_{-a}^{a}dx\hspace{0.17em}{f}_n(x)F(x)\right]}^2}{{\int }_{-a}^{a}dx\hspace{0.17em}{f}_n^2(x)}\simeq \frac{2q_0}{\pi }\frac{g}{g+1}\frac{2+\sqrt{2}}{{\pi }^2}\sum _{n=1}^{\infty }\frac{1}{(2n+\alpha )(2n+\beta )},$$

where $\alpha =\frac{2q_{0}a}{\pi }\frac{g}{g+1}+\frac{1}{4}$ and $\beta =\frac{1}{\sqrt{2}\pi }+\frac{1}{4}$. Using the identity

(71)$$\sum _{n=1}^{\infty }\frac{1}{(2n+\alpha )(2n+\beta )}=\frac{\mathrm{\Psi }\left(1+\frac{\alpha }{2}\right)-\mathrm{\Psi }\left(1+\frac{\beta }{2}\right)}{2(\alpha -\beta )},$$

where Ψ(z) = Γ′(z)/Γ(z) is the digamma function, the reflection coefficient is

(72)$$\begin{array}{l}{r}^{-1}\simeq 1-\frac{2\pi i}{\kappa }C-\frac{i}{2q_{0}a}\frac{g+1}{g}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-i\frac{\sqrt{2}+1}{{\pi }^2}\frac{2(g+1)}{2\sqrt{2}{q}_{0}ag-(g+1)}\left[\mathrm{\Psi }\left(\frac{9}{8}+\frac{q_{0}a}{\pi }\frac{g}{g+1}\right)-\mathrm{\Psi }\left(\frac{9}{8}+\frac{\sqrt{2}}{4\pi }\right)\right],\end{array}$$

in the limits q₀a ≪ 1 and n ≫ 1.

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 profile

(73)$${\sigma }_{\text{sm}}\left(\left|x\right|<a\right)={\text{sin}}^2\left(\frac{\pi }{2a}\left|x\right|\right)+\frac{\sigma }{{\sigma }_0}{\text{cos}}^2\left(\frac{\pi }{2a}\left|x\right|\right),$$

where σ now denotes the lowest value of conductivity occuring at x = 0. As shown in Fig. 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.

 

Fig. 9 (a) Reflectance of a junction with a smooth conductivity profile for a = 0.1λ₀ and γ = 0. (b) Similar plot at γ = 0.05. (Inset) The conductivity profile σ_sm.

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Funding

Department of Energy (DOE) (DE-SC0012592, DE FG02 84ER45118); Office of Naval Research (ONR) (N00014-15-1-2671); Semiconductor Research Corporation (SRC) (2701.002).

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