Abstract

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

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=iκω2πσ, 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 λ0a, 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 λ0a, the capacitive coupling dominates. This regime can be further divided into two, depending on how a compares to λ. When aλ, 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.

 figure: Fig. 1

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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 figure: Fig. 2

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 IR has reversed number of coils and signs of θ compared to IL. (c) For narrow junctions J is a capacitor in parallel with a LC network that describes the cavity resonances. (d) In the dc limit, aλ, 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

rL=eiθσ0σσ0+σ,rR=eiθσσ0σ+σ0,tL=tR=2σ0σσ0+σ.
The nontrivial reflection phase shift, frequently overlooked in previous studies [20,21], is given by
θ=π42π0dutan1(σσ0u)u2+1,
which approaches π/4 in the limit σ → 0, as in the case where the lead-junction boundary corresponds to a physical termination of the lead [22,23]. We define the reflection coefficient r to be the prefactor in the asymptotic form of the scattered potential ϕsreiq0x at large negative x for an incident plasmon potential ϕ0 = eiq0x. (Our definition differs in the overall sign from [19] 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
rFP1=σ02+σ2σ02σ2+i2σ0σσ02σ2cot(ϕθ),
where ϕ = 2qa is the phase accumulated across the junction. Equation (3) yields a reflectance that has local minima at resonances, ϕ = +θ, and local maxima at anti-resonances, ϕ=(n+12)π+θ, 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 q=2πλ(1+iγ) 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.

 figure: Fig. 3

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 Zj=Z0σ0σ while the complementary phase shifters provide the requisite phase factor e± 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(qaθ/2) and Z2 = iZj csc(2qaθ) [24]. The total reflection coefficient from the left interface [coinciding with Eq. (3)] can then be found directly from the standard waveguide-theory formula,

r=(ZLZ0)/(ZL+Z0),
where the total load impedance, calculated by the rules of parallel and series connections, is ZL=Z1+(Z21+(Z1+Z0)1)1.

2.2. Narrow junctions

The F-P formula (3) remains numerically accurate for junctions as narrow as aλ0. 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 = E0eiq0x with the asymptotic form of the scattered field Es ≃ −rE0eiq0x 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,

Es(|x|>a)=Δσσ0aadxG(xx)E(x),
where Δσσσ0, and the Green’s function G is the 1D Fourier transform of the dielectric function = 1− |q|/q0 of the 2D sheet (See Appendix B),
G(x)=dq2πeiqx1(q).
By analyzing the asymptotic form of Eq. (6), one arrives at the relation
r=iΔσσ0q0E0aadxE(x)eiq0x,
which is valid for junctions of any width. For example, it can be used to evaluate the reflection coefficient numerically. For narrow junctions, Eq. (7) can be simplified and written in terms of the voltage drop across the junction V=aadxE(x),
riΔσσ0q0VE0,(aλ0)
since eiq0x ≈ 1 in the junction.

Equation (8) has simple solutions in limiting cases. In the perturbative limit σσ0, the current density j(x) is approximately constant across the junction, which implies that the field inside the junction is Eσ0σE0. This yields the reflection coefficient

r=2iq0aΔσσ,(|Δσ|σ0)
in agreement with [26]. 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) = = −iωVC, the reflection coefficient for the vacuum gap can be found:
r=iκ2πC+iκ=iπlog2q0ac+iπ,(σ=0,aλ0).
Here the junction capacitance per unit length
C=κ2π2(log2q0ac),c=0.577
can be derived by considering the charge distribution on perfectly conducting leads. The sublinear dependence on a is due to the 2D geometry of the system, where the fringing fields, i.e., fields exterior to the sheet, dominate. Equation (10) is in good agreement with our numerical result (Fig. 4) and with [13]. 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).

 figure: Fig. 4

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,

σ0E0(1r)=iωVC+aadxσE(x)πa2x2,C=Δσσ0C,
where the capacitance acquires a correction factor −Δσ/σ0 which is 1 for the vacuum gap but decreases to 0 for a uniform sheet where σ = σ0. The additional integral represents the Shockley-Ramo image current in the leads induced by the current in the junction [27]. The field in the junction can be expanded in the basis of resonant eigenmodes
E(|x|<a)=V2a(1+n=1bnfn(x)).
The coefficients
bn=2a1+σΔσqnq0aadxfn(x)πa2x2aadxfn2(x)
diverge whenever the plasmon wavevector q inside the junction matches the resonant wavevector qn, and the field E becomes dominated by the n-th resonant mode fn. The wavevector qn of the resonant field is again determined by the resonance condition 2qna = + θ as in the case of wide junctions. For narrow junctions, all resonances occur at σ0/σ ≫ 1, so that θπ/4 and qn=π2a(n+14). The exact forms of the dimensionless resonant fields fn has to be calculated numerically (the simple forms described in [27] are inaccurate), but they quickly approach cos(qnx12nπ) 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.

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

r1=1i2πκCi12q0aσΔσ(1+n=1bnaadxfn(x)πa2x2).
At a resonance, r−1 diverges for zero damping and has a maximum for finite damping, so the reflectance has a local minimum, similar to the case of wide junctions. The main difference from a wide junction comes from the capacitive coupling, which gives rise to the non-resonant terms in 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
fn=cos(qnx12nπ),qn=π2a(n+14).
In this case the odd modes disappear completely due to the integral in 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 [24], which can be expanded into the resonant form Z1=(4iZ0qa)1+iqa2Z0n(q2a2π2n2)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 Z1=(iωL0)1+iωn(ω2LnCn1)1, which yields L0 = 2ia/σω, Ln=12L0, 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 coefficient

r1=1i2πκC+i2qa+iqan=11q2a2π2n2.
The first three terms represent the low q or dc response and match those in Eq. (15) (under the limit σσ0). 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 ZL0=iωL0=2aσ, the later resembling the usual dc formula for resistance. When the inverse impedances, i.e., admittances ZC1 and ZL01 cancel, the total impedance of the junction diverges and the reflectance is unity. We call this situation the open-circuit resonance. It occurs when

σσ0σλ0a=4(log2q0ac).
The open-circuit resonance arises only for narrow junctions because it requires capacitive coupling of the leads. As the junction width increases, the capacitive coupling gradually weakens, and the junction evolves towards a regular F-P resonator, as shown in Fig. 5.

 figure: Fig. 5

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,

ΔP¯=dx(P(x)P0)λ0Pinc
where
P(x)=Reσ(x)|E(x)|2
is the local power absorption per unit area, P0 = Re σ0|E0|2 is the same quantity for a uniform system without a junction, and Pinc=c4π|E0|2 is the incident power per unit area. We further divide Δ into two parts, absorption by the junction (Δj) and by the leads (Δl). The former is determined by the junction conductivity as well as the strength of the resonant fields, while the latter is determined by the strength of plasmon emission. The power absorption of the junction has a peak at each resonance where the resonant field is strong. It is also peaked at the open-circuit resonance, where the field in the junction is approximately constant and proportional to r, EV/2ar, so that the peak in R is mirrored in Δj. The power absorbed by the leads is determined by the scattered plasmon field which quickly approach −reiq0|x| away from the junction. Hence Δl is approximately proportional to the reflectance. As shown in Fig. 6(a), Δl 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.

 figure: Fig. 6

Fig. 6 (a) Excess power absoption Δ 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, σ0=iD0π(ω+iν) with ν = 3 cm−1. The Drude weight D0=e2π2 μ is calculated at chemical potential μ = 0.12eV, the conductivity ratio is D/D0 = 0.03. At low frequencies ων the conductivity is dominated by damping, resulting in a negative Δl, 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 Δj at low frequencies is due to our assumption of a Drude-like conductivity, σ = iD/π(ω + ) where the damping rate ν is taken to be constant, so that Reσνω2+ν2 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 EV/2a and Es ≃ −reiq0|x|, we obtain

ΔP¯j2πκg2D02caν(ω2+ν2)2R,ΔP¯l4πD0cωω2+ν2R,
where c is the speed of light and gD/D0 − 1. Using these expressions, the optimal operating frequency can be found given the device parameters.

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 ad, 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 [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 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 ωγωc(1+2κ/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

P=12Re(v*i).
In our case the propagating wave is a plasmon. The plasmon travelling in the x-direction can be described by the field amplitudes
Ex=eiq0xeq0|z|,Ez=iEx,Hy=2πcσ0Ex,
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,
P=12Redzc4πHy*Ez=14Re(j*ϕ).
Here ϕ is the electric potential on the plane,
ϕ=0dzEz,
and j = σ0Ex 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:
v=12ϕ,i=j.
The characteristic impedance of the sheet is then
Z0=12ϕj=πωκ.
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 2aq01 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]
Z1=iZ0tanq0a
each and the vertical leg has the impedance
Z2=iZ0cscq0a.

However, using Eqs. (28) and (29) for a narrow strip 2aq01 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,

ZC=1iωC.
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 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).

 figure: Fig. 7

Fig. 7 (a) Distribution of Ez and Hy 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 Z2Z1, 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

Z4iZ0qa=4iπωκiωκ2πσa=2aσ,qa1,
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 ϕi = eiq0x impinges upon a 1D inhomogeneity in the conductivity of the sheet localized around the y-axis, parametrized as

g(x)=Δσ(x)σ0,Δσ(x)σ(x)σ0.
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])
ψ(x)=1q0(G1*V1)*[xg(x)xϕ(x)],
where * denotes convolution and V1 is the Coulomb kernel with the Fourier transform 1 = 1/|k|. The Green’s function G1(x)=dk2πeikx1(k) is the 1D Fourier transform of the dielectric function given by
G1(x)=iq0eiq0|x|q02π[eiq0|x|E1(iq0|x|)+eiq0|x|E1(iq0|x|)],
where the exponential integral E1(z)=zdtt1et has a branch cut on the negative real axis. The 2D dielectric function (k) = 1 − |k|/q0, which encodes the sheet conductivity σ0 through q0=iκω2πσ0, is frequency and wavevector-dependent and determines the response of the sheet to an external potential [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,
E˜s(k)=ikψ˜(k)=(1(k)1)(gE˜)(k),(gE˜)=12πg˜*E˜.
In the real space this reads,
Es(x)=dx(G1(xx)δ(xx))g(x)E(x).
This is the equation for the scattered field used in the main text. It indicates the scattered field Es 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 Es at large negative x, Es ≃ −rE0eiq0x, 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 get

r=iq0E0dxg(x)E(x)eiq0x.
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,
t=1iq0E0dxg(x)E(x)eiq0x.

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,

j=σ0E0=σ(x)E(x),
so that E(x)=σ0σ(x)E0eiq0x and
r=iq0dxσ(x)σ0σ(x)e2iq0x=iq0dxg(x)g(x)+1e2iq0x.

Appendix C. Reflection coefficient of narrow junctions

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

g(x)=gΘ(a|x|),g=Δσσ0=σσ0σ0,
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, aq01. 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 Evac for the special case of a vacuum gap, σ = 0 or g =−1, is well known [35],

Evac(x)=VF(x),F(x)=1πΘ(a|x|)a2x2,
where V is the voltage difference across the junction,
V=aadxE(x).
Substituting Evac(x) into Eq. (36), we get
Ei(x)E0=VaadxG1(xx)F(x)=Vq0π[c+log(q0a2)iπ],a<x<a.
Here we used the small-distance approximation for the Green’s function,
G1(x)q0π[c+log(q0|x|)iπ]+𝒪(q0|x|),q0|x|1,
where c ≃ 0.577 is the Euler-Mascheroni constant, and the table integral
aadxlog|xx|πa2x2=loga2.
(It can be evaluated by the change of variables x′ = a cos θ′) Identifying the capacitance of the junction
C=κ2π2log2ecq0a,
and the reflection coefficient for a narrow junction [Eq. (37) with eiq0x→ 1]
riq0E0gV,
our Eq. (44) yields the reflection coefficient of a narrow vacuum gap,
r=iπlog(2q0a)c+iπ=iκ2πC+iκ.
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 rj = −r = −1.

C.2. General cases

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

Ei(x)E0=[G(x)*gE(x)]+(1+g)E(x),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,
σ0E0=σ0gVq0π[c+log(q0a2)iπ]+σ0(1+g)aadxE(x)F(x).
We arrive at the equation (mentioned in the main text)
σ0E0(1r)=iωVC+aadxσE(x)F(x),CgC,
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,

E(x)=d0+n=1dnEn(x),a<x<a,
which is an expansion into the eigenmodes En of the system when there is no external field. The eigenmodes En have the following properties,
En(|x|>a)=0,Vn=aadxEn(x)=0,
as the system now consists of a junction with nonzero conductivity σn and two sheets of infinite conductivity. The eigenvalue equation governing En(x) can be derived from Eq. (50) using the substitutions g = −1 and 1 + g = σn/σ0 and invoking Eqs. (46) and (54),
En(x)=Aqnπdxlog|xx|LEn(x),
where qn = iκω/2πσn, 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,
A=aaEn(x)F(x).
Since the kernel in Eq. (55) is self-adjoint, the eigenmodes En are orthogonal,
aadxEn(x)Em(x)=0,nm,
and the eigenvalues qn are real. Eq. (55) can be solved numerically to obtain the eigenmodes. As shown in Fig. 8, they have the asymptotic forms
EnE0cos(qnxnπ2),qnπ2a(n+14),
as n is increased.

 figure: Fig. 8

Fig. 8 (a) Eigenvalues qn and (b)–(f) the first five eigenmodes fn 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 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.,

d0=V2a.
To find dn, we rewrite Eq. (50) using Eqs. (53) and (55) as
log|x|L*h(x)=0,
with a solution
h(x)=V[12aF(x)]+n=1(1+g+1gqnq0)dnEn=0.
Here Eq. (46) was used to convert constants into the convolution,
1=1log(a/2)F(x)*log|x|L,
and Eq. (51) was used in the form
E0(g+1)aadxE(x)F(x)=gq0πVloga2.
Using the orthogonality of the eigenmodes, we find
dn=V1+g+1gqnq0aadxEn(x)F(x)aadxEn2(x).
Note that in the limit of a vacuum gap where g =−1, dn reduces to coefficients of the expansion of Evac = VF(x) in the En basis, as expected. In the manuscript, the dimensionless expansion coefficients b0 and bn are defined as
b0=d0(V/2a)=1,bn=dnE0(V/2a)=2a1+g+1gqnq0aadxfn(x)F(x)aadxfn2(x),
and the field is expanded as
E(x)=V2a(1+n=1bnfn),fn=EnE0.

C.3. Analytic approximation

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

r1=12πiκCiq0g+1g{12a+n=111+g+1gqnq0[aadxfn(x)F(x)]2aadxfn2(x)}.
An analytical expression for r can be found using the approximation Eq. (58), so that
aadxfn2(x)=a[1+12π(n+14)],
and
[aadxfn(x)F(x)]2=0,nodd=J02(qna)2+2π21n+14,n1even,
where J0(z) is the Bessel function of the first kind. The summation is then
n=111+g+1gqnq0[aadxfn(x)F(x)]2aadxfn2(x)2q0πgg+12+2π2n=11(2n+α)(2n+β),
where α=2q0aπgg+1+14 and β=12π+14. Using the identity
n=11(2n+α)(2n+β)=Ψ(1+α2)Ψ(1+β2)2(αβ),
where Ψ(z) = Γ′(z)/Γ(z) is the digamma function, the reflection coefficient is
r112πiκCi2q0ag+1gi2+1π22(g+1)22q0ag(g+1)[Ψ(98+q0aπgg+1)Ψ(98+24π)],
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

σsm(|x|<a)=sin2(π2a|x|)+σσ0cos2(π2a|x|),
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.

 figure: Fig. 9

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 σsm.

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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).

References and links

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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]  

10. F. Huerkamp, T. A. Leskova, A. A. Maradudin, and B. Baumeier, “The Goos-Hänchen effect for surface plasmon polaritons,” Opt. Express 19, 15483–15489 (2011). [CrossRef]   [PubMed]  

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]  

13. J. L. Garcia-Pomar, A. Y. Nikitin, and L. Martin-Moreno, “Scattering of graphene plasmons by defects in the graphene sheet,” ACS Nano 7, 4988–4994 (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]  

16. J. S. Gómez-Díaz and J. Perruisseau-Carrier, “Graphene-based plasmonic switches at near infrared frequencies,” Opt. Express 21, 15490–15504 (2013). [CrossRef]   [PubMed]  

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]  

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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]  

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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).

References

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  • |

  1. E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
    [Crossref] [PubMed]
  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]
  10. F. Huerkamp, T. A. Leskova, A. A. Maradudin, and B. Baumeier, “The Goos-Hänchen effect for surface plasmon polaritons,” Opt. Express 19, 15483–15489 (2011).
    [Crossref] [PubMed]
  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]
  13. J. L. Garcia-Pomar, A. Y. Nikitin, and L. Martin-Moreno, “Scattering of graphene plasmons by defects in the graphene sheet,” ACS Nano 7, 4988–4994 (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]
  16. J. S. Gómez-Díaz and J. Perruisseau-Carrier, “Graphene-based plasmonic switches at near infrared frequencies,” Opt. Express 21, 15490–15504 (2013).
    [Crossref] [PubMed]
  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]
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2018 (1)

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]

2017 (4)

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]

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]

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]

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]

2016 (3)

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]

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]

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]

2015 (4)

B.-Y. Jiang and M. M. Fogler, “Electronic response of graphene to linelike charge perturbations,” Phys. Rev. B 91, 235422 (2015).
[Crossref]

B. Rejaei and A. Khavasi, “Scattering of surface plasmons on graphene by a discontinuity in surface conductivity,” J. Opt. 17, 075002 (2015).
[Crossref]

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]

J. Tong, M. Muthee, S.-Y. Chen, S. K. Yngvesson, and J. Yan, “Antenna enhanced graphene THz emitter and detector,” Nano Lett. 15, 5295–5301 (2015).
[Crossref] [PubMed]

2014 (3)

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]

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]

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]

2013 (6)

J. S. Gómez-Díaz and J. Perruisseau-Carrier, “Graphene-based plasmonic switches at near infrared frequencies,” Opt. Express 21, 15490–15504 (2013).
[Crossref] [PubMed]

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]

J. L. Garcia-Pomar, A. Y. Nikitin, and L. Martin-Moreno, “Scattering of graphene plasmons by defects in the graphene sheet,” ACS Nano 7, 4988–4994 (2013).
[Crossref] [PubMed]

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]

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]

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]

2012 (4)

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]

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]

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6, 749–758 (2012).
[Crossref]

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]

2011 (1)

2009 (1)

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).

2006 (1)

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[Crossref] [PubMed]

2005 (2)

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]

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]

2004 (1)

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]

2003 (1)

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]

1996 (1)

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]

AbdollahRamezani, S.

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]

Aizin, G. R.

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]

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]

Allen, S. J.

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]

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]

Alonso-Gonzalez, P.

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]

Alonso-González, P.

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]

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]

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]

Andreev, G. O.

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]

Arik, K.

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]

Bao, W.

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]

Basov, D.

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]

Basov, D. N.

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]

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]

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]

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]

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]

Baumeier, B.

Bethke, D.

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]

Bockrath, M.

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]

Carrega, M.

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]

Castro Neto, A.

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]

Chapler, B. C.

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]

Chaplik, A.

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]

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]

Chen, J.

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]

Chen, S.-Y.

J. Tong, M. Muthee, S.-Y. Chen, S. K. Yngvesson, and J. Yan, “Antenna enhanced graphene THz emitter and detector,” Nano Lett. 15, 5295–5301 (2015).
[Crossref] [PubMed]

Cheng, B.

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]

Cho, E. Y.

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]

Coquillat, D.

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).

Crassee, I.

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]

Cybart, S. A.

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]

Dai, S.

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]

Davydovskaya, D.

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]

Dean, C. R.

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]

Dicke, R. H.

C. G. Montgomery, R. H. Dicke, and E. M. Purcell, Principles of Microwave Circuits, vol. 8 of MIT Radiation Laboratory Series (McGraw-Hill, 1948).

Dominguez, G.

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]

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).
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Dyakonov, M.

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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).
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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).
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Huynh, C.

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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).
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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).
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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).
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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).

Keilmann, F.

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).
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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).
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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).
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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).
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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).
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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).
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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).
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Lau, C.

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).
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Lau, C. N.

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).
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Lee, Y. S.

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).
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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).
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[Crossref]

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Figures (9)

Fig. 1
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.
Fig. 2
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 IR has reversed number of coils and signs of θ compared to IL. (c) For narrow junctions J is a capacitor in parallel with a LC network that describes the cavity resonances. (d) In the dc limit, aλ, the LC network reduces to a single inductance L0.
Fig. 3
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.
Fig. 4
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.
Fig. 5
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).
Fig. 6
Fig. 6 (a) Excess power absoption Δ 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, σ 0 = i D 0 π ( ω + i ν ) with ν = 3 cm−1. The Drude weight D 0 = e 2 π 2 μ is calculated at chemical potential μ = 0.12eV, the conductivity ratio is D/D0 = 0.03. At low frequencies ων the conductivity is dominated by damping, resulting in a negative Δl, i.e., the absorption in the leads is smaller than the background value.
Fig. 7
Fig. 7 (a) Distribution of Ez and Hy 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.
Fig. 8
Fig. 8 (a) Eigenvalues qn and (b)–(f) the first five eigenmodes fn of a junction in a perfect metal sheet. The analytical formula is Eq. (58), while the numerical results are obtained by solving Eq. (55).
Fig. 9
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 σsm.

Equations (73)

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r L = e i θ σ 0 σ σ 0 + σ , r R = e i θ σ σ 0 σ + σ 0 , t L = t R = 2 σ 0 σ σ 0 + σ .
θ = π 4 2 π 0 d u tan 1 ( σ σ 0 u ) u 2 + 1 ,
r FP 1 = σ 0 2 + σ 2 σ 0 2 σ 2 + i 2 σ 0 σ σ 0 2 σ 2 cot ( ϕ θ ) ,
r = ( Z L Z 0 ) / ( Z L + Z 0 ) ,
E s ( | x | > a ) = Δ σ σ 0 a a d x G ( x x ) E ( x ) ,
G ( x ) = d q 2 π e i q x 1 ( q ) .
r = i Δ σ σ 0 q 0 E 0 a a d x E ( x ) e i q 0 x ,
r i Δ σ σ 0 q 0 V E 0 , ( a λ 0 )
r = 2 i q 0 a Δ σ σ , ( | Δ σ | σ 0 )
r = i κ 2 π C + i κ = i π log 2 q 0 a c + i π , ( σ = 0 , a λ 0 ) .
C = κ 2 π 2 ( log 2 q 0 a c ) , c = 0.577
σ 0 E 0 ( 1 r ) = i ω V C + a a d x σ E ( x ) π a 2 x 2 , C = Δ σ σ 0 C ,
E ( | x | < a ) = V 2 a ( 1 + n = 1 b n f n ( x ) ) .
b n = 2 a 1 + σ Δ σ q n q 0 a a d x f n ( x ) π a 2 x 2 a a d x f n 2 ( x )
r 1 = 1 i 2 π κ C i 1 2 q 0 a σ Δ σ ( 1 + n = 1 b n a a d x f n ( x ) π a 2 x 2 ) .
f n = cos ( q n x 1 2 n π ) , q n = π 2 a ( n + 1 4 ) .
r 1 = 1 i 2 π κ C + i 2 q a + i q a n = 1 1 q 2 a 2 π 2 n 2 .
σ σ 0 σ λ 0 a = 4 ( log 2 q 0 a c ) .
Δ P ¯ = d x ( P ( x ) P 0 ) λ 0 P inc
P ( x ) = Re σ ( x ) | E ( x ) | 2
Δ P ¯ j 2 π κ g 2 D 0 2 ca ν ( ω 2 + ν 2 ) 2 R , Δ P ¯ l 4 π D 0 c ω ω 2 + ν 2 R ,
P = 1 2 Re ( v * i ) .
E x = e i q 0 x e q 0 | z | , E z = i E x , H y = 2 π c σ 0 E x ,
P = 1 2 Re d z c 4 π H y * E z = 1 4 Re ( j * ϕ ) .
ϕ = 0 d z E z ,
v = 1 2 ϕ , i = j .
Z 0 = 1 2 ϕ j = π ω κ .
Z 1 = i Z 0 tan q 0 a
Z 2 = i Z 0 csc q 0 a .
Z C = 1 i ω C .
Z 4 i Z 0 q a = 4 i π ω κ i ω κ 2 π σ a = 2 a σ , q a 1 ,
g ( x ) = Δ σ ( x ) σ 0 , Δ σ ( x ) σ ( x ) σ 0 .
ψ ( x ) = 1 q 0 ( G 1 * V 1 ) * [ x g ( x ) x ϕ ( x ) ] ,
G 1 ( x ) = i q 0 e i q 0 | x | q 0 2 π [ e i q 0 | x | E 1 ( i q 0 | x | ) + e i q 0 | x | E 1 ( i q 0 | x | ) ] ,
E ˜ s ( k ) = i k ψ ˜ ( k ) = ( 1 ( k ) 1 ) ( g E ˜ ) ( k ) , ( g E ˜ ) = 1 2 π g ˜ * E ˜ .
E s ( x ) = d x ( G 1 ( x x ) δ ( x x ) ) g ( x ) E ( x ) .
r = i q 0 E 0 d x g ( x ) E ( x ) e i q 0 x .
t = 1 i q 0 E 0 d x g ( x ) E ( x ) e i q 0 x .
j = σ 0 E 0 = σ ( x ) E ( x ) ,
r = i q 0 d x σ ( x ) σ 0 σ ( x ) e 2 i q 0 x = i q 0 d x g ( x ) g ( x ) + 1 e 2 i q 0 x .
g ( x ) = g Θ ( a | x | ) , g = Δ σ σ 0 = σ σ 0 σ 0 ,
E vac ( x ) = VF ( x ) , F ( x ) = 1 π Θ ( a | x | ) a 2 x 2 ,
V = a a d x E ( x ) .
E i ( x ) E 0 = V a a d x G 1 ( x x ) F ( x ) = V q 0 π [ c + log ( q 0 a 2 ) i π ] , a < x < a .
G 1 ( x ) q 0 π [ c + log ( q 0 | x | ) i π ] + 𝒪 ( q 0 | x | ) , q 0 | x | 1 ,
a a d x log | x x | π a 2 x 2 = log a 2 .
C = κ 2 π 2 log 2 e c q 0 a ,
r i q 0 E 0 g V ,
r = i π log ( 2 q 0 a ) c + i π = i κ 2 π C + i κ .
E i ( x ) E 0 = [ G ( x ) * g E ( x ) ] + ( 1 + g ) E ( x ) , a < x < a .
σ 0 E 0 = σ 0 g V q 0 π [ c + log ( q 0 a 2 ) i π ] + σ 0 ( 1 + g ) a a d x E ( x ) F ( x ) .
σ 0 E 0 ( 1 r ) = i ω V C + a a d x σ E ( x ) F ( x ) , C g C ,
E ( x ) = d 0 + n = 1 d n E n ( x ) , a < x < a ,
E n ( | x | > a ) = 0 , V n = a a d x E n ( x ) = 0 ,
E n ( x ) = A q n π d x log | x x | L E n ( x ) ,
A = a a E n ( x ) F ( x ) .
a a d x E n ( x ) E m ( x ) = 0 , n m ,
E n E 0 cos ( q n x n π 2 ) , q n π 2 a ( n + 1 4 ) ,
d 0 = V 2 a .
log | x | L * h ( x ) = 0 ,
h ( x ) = V [ 1 2 a F ( x ) ] + n = 1 ( 1 + g + 1 g q n q 0 ) d n E n = 0 .
1 = 1 log ( a / 2 ) F ( x ) * log | x | L ,
E 0 ( g + 1 ) a a d x E ( x ) F ( x ) = g q 0 π V log a 2 .
d n = V 1 + g + 1 g q n q 0 a a d x E n ( x ) F ( x ) a a d x E n 2 ( x ) .
b 0 = d 0 ( V / 2 a ) = 1 , b n = d n E 0 ( V / 2 a ) = 2 a 1 + g + 1 g q n q 0 a a d x f n ( x ) F ( x ) a a d x f n 2 ( x ) ,
E ( x ) = V 2 a ( 1 + n = 1 b n f n ) , f n = E n E 0 .
r 1 = 1 2 π i κ C i q 0 g + 1 g { 1 2 a + n = 1 1 1 + g + 1 g q n q 0 [ a a d x f n ( x ) F ( x ) ] 2 a a d x f n 2 ( x ) } .
a a d x f n 2 ( x ) = a [ 1 + 1 2 π ( n + 1 4 ) ] ,
[ a a d x f n ( x ) F ( x ) ] 2 = 0 , n odd = J 0 2 ( q n a ) 2 + 2 π 2 1 n + 1 4 , n 1 even ,
n = 1 1 1 + g + 1 g q n q 0 [ a a d x f n ( x ) F ( x ) ] 2 a a d x f n 2 ( x ) 2 q 0 π g g + 1 2 + 2 π 2 n = 1 1 ( 2 n + α ) ( 2 n + β ) ,
n = 1 1 ( 2 n + α ) ( 2 n + β ) = Ψ ( 1 + α 2 ) Ψ ( 1 + β 2 ) 2 ( α β ) ,
r 1 1 2 π i κ C i 2 q 0 a g + 1 g i 2 + 1 π 2 2 ( g + 1 ) 2 2 q 0 a g ( g + 1 ) [ Ψ ( 9 8 + q 0 a π g g + 1 ) Ψ ( 9 8 + 2 4 π ) ] ,
σ sm ( | x | < a ) = sin 2 ( π 2 a | x | ) + σ σ 0 cos 2 ( π 2 a | x | ) ,

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