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

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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]

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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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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).
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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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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, 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]

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).
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Wong, T. J.

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).
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Wu, J. 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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Wu, J.-S.

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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Xiong, L.

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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Yablonovitch, E.

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).
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Yan, J.

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).
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Yngvesson, S. K.

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).
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Zettl, A.

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).
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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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