Abstract

Terahertz plasmons and magnetoplasmons propagating along electrically and chemically doped graphene p-n junctions are investigated. It is shown that such junctions support non-reciprocal magnetoplasmonic modes which get concentrated at the middle of the junction in one direction and split away from the middle of the junction in the other direction under the application of an external static magnetic field. This phenomenon follows from the combined effects of circular birefringence and carrier density non-uniformity. It can be exploited for the realization of plasmonic isolators.

© 2013 OSA

1. Introduction

The linear band structure, tunability and ambipolarity of graphene [1,2] have recently opened up new horizons in the area of plasmonics. These fundamental properties lead to unique plasmonic phenomena, such as the existence of both TE and TM plasmons [37] and voltage tunable plasmonic modes [811]. These modes have been recently investigated towards the realization of various enhanced plasmonic devices [3, 7, 10, 1216].

Using electrical gating, one can modify and tune the charge profile to generate useful modes on graphene structures. For instance, one can obtain regions with opposite carrier types on a graphene strip by applying a tangential transverse electric field across it so as to create a p-n junction. In [9], it was theoretically shown at zero temperature that such a junction supports localized plasmonic and magnetoplasmonic modes with the magnetoplasmonic modes existing only for one direction of propagation (the modes in the other direction being cut off) in the limit of a very high magnetic bias field.

Allowing non-zero temperatures leads to the excitation of thermal carriers, which induce non-zero conductivity at the center of the junction and hence affects the electromagnetic field distribution across the strip. Moreover, allowing arbitrary magnetic bias fields leads to the existence of a circular birefringence regime where magnetoplasmonic modes exist in the two propagation directions but exhibit very different distributions, one being concentrated at the center of the junction and the other one being split away from it. This paper reveals these non-reciprocal phenomena and proposes isolator devices based on them.

The magnetoplasmonic modes are computed by numerically solving Maxwell equations for the graphene strip with exact non-uniform carrier densities in the Kubo conductivity, which properly captures the quantized Landau regime prevailing near the center of the junction. Two types of doping are considered and compared: doping by a transverse electric field and chemical doping, represented in Figs. 1(a) and 1(b), respectively. The corresponding structures are analyzed with the 2D finite difference frequency domain technique (FDFD) [17], where graphene is modeled as a zero-thickness conductive sheet with a conductivity tensor given by the Kubo formula [18, 19]. Adaptive mesh refinement is used around the edges and small features for better accuracy.

 

Fig. 1 (a) Electrically and (b) chemically doped graphene p-n junctions.

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2. Magnetoplasmon energy concentration and splitting in graphene p-n junctions

2.1. Electrically doped graphene

First consider the electrically doped graphene structure [Fig. 1(a)]. A transverse electric field applied tangentially to the graphene strip creates a non-uniform carrier density and therefore a non-uniform conductivity across the strip. The net charge profile can be found by solving the integral equation

w/2w/2ρ(x,y)G(x,y;x,y)dxE0x=0,
with
w/2xw/2,y=0,y=0,
for the net charge density ρ (x′, y′), where G(x,y;x,y)=12πε0ln(xx)2+(yy)2 is the 2D free-space Green function for the Poisson equation and E0 is the applied electrostatic field. The resulting net carrier density, nnet = np, where n and p are the electron and hole densities, respectively, is plotted in Fig. 2 along with the electric potential for a graphene strip of width w = 50 μm. Different carrier types symmetrically appear at the opposite sides of the strip, which results in the formation of a p-n junction at the center of the strip.

 

Fig. 2 Net carrier density and electric potential for a graphene strip doped with an electric field; w = 50 μm and E0 = 108 V/m.

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The chemical potential (μc) and the electron and hole densities are found from nnet by numerically solving the equation

nnet=0fd(E,μc)N(E)dE0[1fd(E,μc)]N(E)dE,
where fd(E, μc) is the Fermi-Dirac distribution and N(E)=2|E|πh¯2vf2 is the density of energy states. The integrals represent the electron and hole densities (n and p, respectively). The carrier densities are plotted in Fig. 3 while the corresponding chemical potential and Kubo conductivity are plotted in Fig. 4. Note that at the center of the strip, despite the zero net carrier density, a significant amount of thermally excited electrons and holes are present, which leads to the significant conductivity observed in Fig. 4. For simplicity, we assumed an energy independent scattering time of τ = 0.1 ps. Moreover, we assumed that the graphene has less than 1010 cm−2 of unintentional doping fluctuations.

 

Fig. 3 Thermally excited electron and hole densities at room temperature for a graphene strip doped with an electric field; w = 50 μm, E0 = 108 V/m, T = 300 K.

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Fig. 4 Chemical potential and Kubo conductivity for a graphene strip doped with an electric field; w = 50 μm and E0 = 108 V/m, B0 = 0 T, τ = 0.1 ps, T = 300 K, f = 1 THz.

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The graphene strip is next simulated as a conductive sheet in the FDFD resolution of Maxwell equations in order to compute the plasmonic and magnetoplasmonic modes. The slow-wave factor and loss for the plasmon modes propagating along the structure are shown in Fig. 5, where kz and k0 are the propagation constant in the z direction and the free space wave number, respectively. The structure supports a plasmon mode localized at the p-n junction, an infinite number of bulk modes (only the first four are shown here) and two edge modes (not shown here). The losses for the first few modes are shown in Fig. 5(b). Since the carrier density is low near the center of the graphene strip, the p-n junction mode has higher loss than the other modes. The farther the mode is from the center, the lower its loss is. Note that the p-n junction mode has a longitudinal current component that is maximal at the center. Since essentially no carriers are present at the center at zero temperature, this mode is not supported at zero temperature.

 

Fig. 5 Slow-wave factor and loss for a graphene strip biased by an electric field; w = 50 μm, E0 = 108 V/m, B0 = 0 T, τ = 0.1 ps, T = 300 K. The p-n junction mode is represented in red. The insets show the electric field patterns.

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In the presence of a magnetic biasing field, the lowest magnetoplasmon mode exhibits particularly interesting non-reciprocal properties. If the field is sufficiently high, it propagates only in one direction, as shown in the dispersion diagram of Fig. 6 for B0 = 0.1 T. As the magnetic field is increased, the forward mode concentrates at the center, whereas its energy splits away from the center in the backward direction, as shown in Fig. 6. Therefore, if the strip is excited at its center, in the forward direction, the mode whose energy is localized at the junction is excited, whereas no modes are excited in the backward direction, as there is no mode with energy at the center. However, the source beam will need to be highly confined to avoid exciting the backward mode. Note that the modes exhibit non-commensurate field patterns, with energy being squeezed near the center, as a result of the non-uniform conductivity. In the chemically doped p-n junction, to be studied next, it will be shown that the higher order modes exhibit commensurate resonances with higher contrast between the field patterns of the p-n junction mode in the forward and backward direction.

 

Fig. 6 Dispersion curves for a magnetically biased graphene strip biased by an electric field; w = 50 μm, E0 = 108 V/m, B0 = 0.1 T, τ = 0.1 ps, T = 300 K. The p-n junction mode is represented in red. The grey area represents the light cone.

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2.2. Chemically doped graphene

The chemically doped p-n junction is shown in Fig. 1(b). This structure is composed of two chemically doped graphene strips with opposite polarities forming a p-n junction and separated by a nano-gap isolating electron and hole carriers. As the structure of Fig. 1(a), this structure supports a plasmonic mode localized at the middle of the p-n junction. The dispersion curves for the non-biased structure is shown in Fig. 7. The structure supports two edge modes, an infinite number of bulk modes, and the p-n junction mode, plotted in red. The unbiased structure has symmetric dispersion for the forward and backward propagation directions. However, as the magnetic bias is applied, time reversal symmetry is broken and the p-n junction mode exhibits different properties for the forward and backward directions. The dispersion curves for the structure of Fig. 1(b) under magnetic bias is plotted in Fig. 8 for a magnetic bias of B0 = 1 T. It is seen that the mode propagating at the junction exhibits very different properties for the forward and backward directions. In the forward direction, this mode is concentrated at the center, whereas in the backward direction it has very little energy at the center.

 

Fig. 7 Slow-wave factor and loss for the structure in Fig. 1(b) with no magnetic bias; w = 100 μm, s = 10 nm, n = p = 1013 cm−2, B0 = 0 T, τ = 0.1 ps, T = 300 K. The p-n junction mode is represented in red.

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Fig. 8 Slow-wave factor and loss for the magnetoplasmonic isolator in Fig. 1(b); w = 100 μm, s = 10 nm, n = p = 1013 cm−2, B0 = 1 T, τ = 0.1 ps, T = 300 K. The p-n junction mode is represented in red. The grey area represents the light cone.

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It should be noted that the electrically doped and chemically doped graphene junction structures presented in this section are isolators in the sense that propagation is supported along one direction only but not in the sense of conventional isolators [20]. In the latter case, the power sent in the stop direction is dissipated in the devices whereas in the former case it is reflected by the device. Another point to notice is that, although we are considering modal excitation of a multimode structure, the time reversal symmetry breaking is inherent to our structure in the presence of all the modes, due to the odd nature of the magnetic biasing field under time reversal symmetry. Therefore, the proposed isolator represents a true optical isolator in the sense of [21].

3. Phenomenological explanation

The non-reciprocal property of the magnetically biased structures of Fig. 1(a) and Fig. 1(b), results from the circular birefringence for the two propagation directions. The electric field pattern of the p-n junction mode in the plane of graphene is shown in Fig. 9 for the unbiased structure of Fig. 1(b). As the wave propagates along the strip in the forward direction, point R on the right strip sees a clock-wise rotating electric field, and point L on the left strip sees a counter clock-wise rotating electric field. In magnetically biased graphene, which is characterized by a conductivity tensor σ̿ = σd(x̂x̂ + ẑẑ) +σo(x̂ẑẑx̂), where σd and σo are the diagonal and off-diagonal conductivities, respectively, the right and left-hand circularly polarized waves see different scalar conductivities, σd + o and σdo, respectively. On the other hand, when a magnetic field is applied, the p-doped and n-doped strips have off-diagonal conductivities with opposite signs. As a result, for the forward propagation both strips see the conductivity σdo while for the backward direction the conductivity seen by the wave is σd + o. Therefore, the mode sees different media for different propagation directions, corresponding to different dispersions, as observed in Fig. 8. In the forward direction, the p-n junction mode sees a conductivity with a slightly higher imaginary part and is therefore more concentrated. In the opposite direction, the imaginary part of the conductivity is slightly decreased and the mode becomes less localized. The evolution of mode 1 with increasing magnetic field is plotted in Fig. 10 for the forward and backward directions. As the magnetic field is increased, in the forward direction the mode gradually becomes more concentrated on the p-n junction, whereas in the backward direction it moves away from the center.

 

Fig. 9 Electric field pattern in the plane of graphene for the structure of Fig. 1(b) with B0 = 0 T. Points R on the right strip sees a clock-wise rotating electric field in the forward direction and a counter clock-wise rotating electric field in the backward direction as the wave propagates. Point L on the left strip sees an oppositely rotating electric field to point R in each direction.

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Fig. 10 Evolution of the electric field pattern of the mode propagating at the p-n junction (mode 1) for the forward and backward propagations as the magnetic field is increased.

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4. Conclusions

The energy concentration and splitting effect allows the realization of non-reciprocal plasmonic devices such as isolators. If the structure is excited at the center, in the forward direction the p-n junction mode is excited, whereas in the backward direction there is only negligible coupling to the backward mode which has little energy at the center. The electrically doped isolator will be very lossy however, as the carrier density is low at the center (the loss is shown in Fig. 6(b)). In a chemically doped graphene p-n junctions the loss can be mitigated by appropriate doping (the loss is shown in Fig. 8(b) for n = p = 1013).

References and links

1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 22306, 666–669 (2004). [CrossRef]  

2. A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nature Materials 6, 183–191 (2007). [CrossRef]   [PubMed]  

3. A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photon. 6, 7490 758 (2012). [CrossRef]  

4. S. A. Mikhailov and K. Ziegler, “New Electromagnetic Mode in Graphene,” Phys. Rev. Lett. 99, 016 803 (2007). [CrossRef]  

5. G. W. Hanson, “Dyadic Greens functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103, 064 302 (2008). [CrossRef]  

6. I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Letters 12, 2470–2474 (2012). [CrossRef]   [PubMed]  

7. A. Vakil and N. Engheta, “Transformation Optics Using Graphene,” Science 332, 1291–1294 (2011). [CrossRef]   [PubMed]  

8. P. G. Silvestrov and K. B. Efetov, “Charge accumulation at the boundaries of a graphene strip induced by a gate voltage: Electrostatic approach,” Phys. Rev. B 77, 155 436 (2008). [CrossRef]  

9. E. G. Mishchenko, A. V. Shytov, and P. G. Silvestrov, “Guided Plasmons in Graphene p-n Junctions,” Phys. Rev. Lett. 104, 156 806 (2010). [CrossRef]  

10. S. Thongrattanasiri, I. Silveiro, and F. J. G. de Abajo, “Plasmons in electrostatically doped graphene,” Appl. Phys. Lett. 100, 201 105 (2012). [CrossRef]  

11. I. Petković, F. I. B. Williams, K. Bennaceur, F. Portier, P. Roche, and D. C. Glattli, “Carrier Drift Velocity and Edge Magnetoplasmons in Graphene,” Phys. Rev. Lett. 110, 016 801 (2013). [CrossRef]  

12. T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun. 2, 458 (2011). [CrossRef]   [PubMed]  

13. T. Mueller, F. Xia, and P. Avouris, “Graphene photodetectors for high-speed optical communications,” Nat. Photon. 4, 297–301 (2010). [CrossRef]  

14. N. M. Gabor, J. C. W. Song, Q. Ma, N. L. Nair, T. Taychatanapat, K. Watanabe, T. Taniguchi, L. S. Levitov, and P. Jarillo-Herrero, “Hot carrierassisted intrinsic photoresponse in graphene,” Science 334, 648–652 (2011). [CrossRef]   [PubMed]  

15. N. Chamanara, D. Sounas, and C. Caloz, “Non-reciprocal magnetoplasmon graphene coupler,” Opt. Express 21, 11248–11256 (2013). [CrossRef]   [PubMed]  

16. D. L. Sounas and C. Caloz, “Edge surface modes in magnetically biased chemically doped graphene strips,” Appl. Phys. Lett. 99, 231 902:13 (2011). [CrossRef]  

17. Y. Zhao, K. Wu, and K. M. Cheng, “A Compact 2-D Full-Wave Finite-Difference Frequency-Domain Method for General Guided Wave Structures,” IEEE Trans. Microwave Theory Tech. 50, 1844–1848 (2002). [CrossRef]  

18. V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” Journal of Physics: Condensed Matter 19, 026 222 (2007). [CrossRef]  

19. V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “On the universal ac optical background in graphene,” New Journal of Physics 11, 095 013 (2009). [CrossRef]  

20. D. M. Pozar, Microwave engineering(Danvers, MA: Wiley, 2005), 3rd edn.

21. D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, and H. Renner, “What is and what is not an optical isolator,” Nature Photonics 7, 579–582 (2013). [CrossRef]  

References

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  1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science22 306, 666–669 (2004).
    [Crossref]
  2. A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nature Materials 6, 183–191 (2007).
    [Crossref] [PubMed]
  3. A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photon. 6, 7490 758 (2012).
    [Crossref]
  4. S. A. Mikhailov and K. Ziegler, “New Electromagnetic Mode in Graphene,” Phys. Rev. Lett. 99, 016 803 (2007).
    [Crossref]
  5. G. W. Hanson, “Dyadic Greens functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103, 064 302 (2008).
    [Crossref]
  6. I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Letters 12, 2470–2474 (2012).
    [Crossref] [PubMed]
  7. A. Vakil and N. Engheta, “Transformation Optics Using Graphene,” Science 332, 1291–1294 (2011).
    [Crossref] [PubMed]
  8. P. G. Silvestrov and K. B. Efetov, “Charge accumulation at the boundaries of a graphene strip induced by a gate voltage: Electrostatic approach,” Phys. Rev. B 77, 155 436 (2008).
    [Crossref]
  9. E. G. Mishchenko, A. V. Shytov, and P. G. Silvestrov, “Guided Plasmons in Graphene p-n Junctions,” Phys. Rev. Lett. 104, 156 806 (2010).
    [Crossref]
  10. S. Thongrattanasiri, I. Silveiro, and F. J. G. de Abajo, “Plasmons in electrostatically doped graphene,” Appl. Phys. Lett. 100, 201 105 (2012).
    [Crossref]
  11. I. Petković, F. I. B. Williams, K. Bennaceur, F. Portier, P. Roche, and D. C. Glattli, “Carrier Drift Velocity and Edge Magnetoplasmons in Graphene,” Phys. Rev. Lett. 110, 016 801 (2013).
    [Crossref]
  12. T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun. 2, 458 (2011).
    [Crossref] [PubMed]
  13. T. Mueller, F. Xia, and P. Avouris, “Graphene photodetectors for high-speed optical communications,” Nat. Photon. 4, 297–301 (2010).
    [Crossref]
  14. N. M. Gabor, J. C. W. Song, Q. Ma, N. L. Nair, T. Taychatanapat, K. Watanabe, T. Taniguchi, L. S. Levitov, and P. Jarillo-Herrero, “Hot carrierassisted intrinsic photoresponse in graphene,” Science 334, 648–652 (2011).
    [Crossref] [PubMed]
  15. N. Chamanara, D. Sounas, and C. Caloz, “Non-reciprocal magnetoplasmon graphene coupler,” Opt. Express 21, 11248–11256 (2013).
    [Crossref] [PubMed]
  16. D. L. Sounas and C. Caloz, “Edge surface modes in magnetically biased chemically doped graphene strips,” Appl. Phys. Lett. 99, 231 902:13 (2011).
    [Crossref]
  17. Y. Zhao, K. Wu, and K. M. Cheng, “A Compact 2-D Full-Wave Finite-Difference Frequency-Domain Method for General Guided Wave Structures,” IEEE Trans. Microwave Theory Tech. 50, 1844–1848 (2002).
    [Crossref]
  18. V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” Journal of Physics: Condensed Matter 19, 026 222 (2007).
    [Crossref]
  19. V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “On the universal ac optical background in graphene,” New Journal of Physics 11, 095 013 (2009).
    [Crossref]
  20. D. M. Pozar, Microwave engineering(Danvers, MA: Wiley, 2005), 3rd edn.
  21. D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, and H. Renner, “What is and what is not an optical isolator,” Nature Photonics 7, 579–582 (2013).
    [Crossref]

2013 (3)

I. Petković, F. I. B. Williams, K. Bennaceur, F. Portier, P. Roche, and D. C. Glattli, “Carrier Drift Velocity and Edge Magnetoplasmons in Graphene,” Phys. Rev. Lett. 110, 016 801 (2013).
[Crossref]

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, and H. Renner, “What is and what is not an optical isolator,” Nature Photonics 7, 579–582 (2013).
[Crossref]

N. Chamanara, D. Sounas, and C. Caloz, “Non-reciprocal magnetoplasmon graphene coupler,” Opt. Express 21, 11248–11256 (2013).
[Crossref] [PubMed]

2012 (3)

S. Thongrattanasiri, I. Silveiro, and F. J. G. de Abajo, “Plasmons in electrostatically doped graphene,” Appl. Phys. Lett. 100, 201 105 (2012).
[Crossref]

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photon. 6, 7490 758 (2012).
[Crossref]

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Letters 12, 2470–2474 (2012).
[Crossref] [PubMed]

2011 (4)

A. Vakil and N. Engheta, “Transformation Optics Using Graphene,” Science 332, 1291–1294 (2011).
[Crossref] [PubMed]

T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun. 2, 458 (2011).
[Crossref] [PubMed]

N. M. Gabor, J. C. W. Song, Q. Ma, N. L. Nair, T. Taychatanapat, K. Watanabe, T. Taniguchi, L. S. Levitov, and P. Jarillo-Herrero, “Hot carrierassisted intrinsic photoresponse in graphene,” Science 334, 648–652 (2011).
[Crossref] [PubMed]

D. L. Sounas and C. Caloz, “Edge surface modes in magnetically biased chemically doped graphene strips,” Appl. Phys. Lett. 99, 231 902:13 (2011).
[Crossref]

2010 (2)

T. Mueller, F. Xia, and P. Avouris, “Graphene photodetectors for high-speed optical communications,” Nat. Photon. 4, 297–301 (2010).
[Crossref]

E. G. Mishchenko, A. V. Shytov, and P. G. Silvestrov, “Guided Plasmons in Graphene p-n Junctions,” Phys. Rev. Lett. 104, 156 806 (2010).
[Crossref]

2009 (1)

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “On the universal ac optical background in graphene,” New Journal of Physics 11, 095 013 (2009).
[Crossref]

2008 (2)

G. W. Hanson, “Dyadic Greens functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103, 064 302 (2008).
[Crossref]

P. G. Silvestrov and K. B. Efetov, “Charge accumulation at the boundaries of a graphene strip induced by a gate voltage: Electrostatic approach,” Phys. Rev. B 77, 155 436 (2008).
[Crossref]

2007 (3)

S. A. Mikhailov and K. Ziegler, “New Electromagnetic Mode in Graphene,” Phys. Rev. Lett. 99, 016 803 (2007).
[Crossref]

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nature Materials 6, 183–191 (2007).
[Crossref] [PubMed]

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” Journal of Physics: Condensed Matter 19, 026 222 (2007).
[Crossref]

2004 (1)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science22 306, 666–669 (2004).
[Crossref]

2002 (1)

Y. Zhao, K. Wu, and K. M. Cheng, “A Compact 2-D Full-Wave Finite-Difference Frequency-Domain Method for General Guided Wave Structures,” IEEE Trans. Microwave Theory Tech. 50, 1844–1848 (2002).
[Crossref]

Avouris, P.

T. Mueller, F. Xia, and P. Avouris, “Graphene photodetectors for high-speed optical communications,” Nat. Photon. 4, 297–301 (2010).
[Crossref]

Bennaceur, K.

I. Petković, F. I. B. Williams, K. Bennaceur, F. Portier, P. Roche, and D. C. Glattli, “Carrier Drift Velocity and Edge Magnetoplasmons in Graphene,” Phys. Rev. Lett. 110, 016 801 (2013).
[Crossref]

Britnell, L.

T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun. 2, 458 (2011).
[Crossref] [PubMed]

Caloz, C.

N. Chamanara, D. Sounas, and C. Caloz, “Non-reciprocal magnetoplasmon graphene coupler,” Opt. Express 21, 11248–11256 (2013).
[Crossref] [PubMed]

D. L. Sounas and C. Caloz, “Edge surface modes in magnetically biased chemically doped graphene strips,” Appl. Phys. Lett. 99, 231 902:13 (2011).
[Crossref]

Carbotte, J. P.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “On the universal ac optical background in graphene,” New Journal of Physics 11, 095 013 (2009).
[Crossref]

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” Journal of Physics: Condensed Matter 19, 026 222 (2007).
[Crossref]

Chamanara, N.

Chen, J.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Letters 12, 2470–2474 (2012).
[Crossref] [PubMed]

Cheng, K. M.

Y. Zhao, K. Wu, and K. M. Cheng, “A Compact 2-D Full-Wave Finite-Difference Frequency-Domain Method for General Guided Wave Structures,” IEEE Trans. Microwave Theory Tech. 50, 1844–1848 (2002).
[Crossref]

Crassee, I.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Letters 12, 2470–2474 (2012).
[Crossref] [PubMed]

de Abajo, F. J. G.

S. Thongrattanasiri, I. Silveiro, and F. J. G. de Abajo, “Plasmons in electrostatically doped graphene,” Appl. Phys. Lett. 100, 201 105 (2012).
[Crossref]

Dubonos, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science22 306, 666–669 (2004).
[Crossref]

Echtermeyer, T.

T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun. 2, 458 (2011).
[Crossref] [PubMed]

Efetov, K. B.

P. G. Silvestrov and K. B. Efetov, “Charge accumulation at the boundaries of a graphene strip induced by a gate voltage: Electrostatic approach,” Phys. Rev. B 77, 155 436 (2008).
[Crossref]

Eich, M.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, and H. Renner, “What is and what is not an optical isolator,” Nature Photonics 7, 579–582 (2013).
[Crossref]

Engheta, N.

A. Vakil and N. Engheta, “Transformation Optics Using Graphene,” Science 332, 1291–1294 (2011).
[Crossref] [PubMed]

Fan, S.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, and H. Renner, “What is and what is not an optical isolator,” Nature Photonics 7, 579–582 (2013).
[Crossref]

Ferrari, A.

T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun. 2, 458 (2011).
[Crossref] [PubMed]

Firsov, A. A.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science22 306, 666–669 (2004).
[Crossref]

Freude, W.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, and H. Renner, “What is and what is not an optical isolator,” Nature Photonics 7, 579–582 (2013).
[Crossref]

Gabor, N. M.

N. M. Gabor, J. C. W. Song, Q. Ma, N. L. Nair, T. Taychatanapat, K. Watanabe, T. Taniguchi, L. S. Levitov, and P. Jarillo-Herrero, “Hot carrierassisted intrinsic photoresponse in graphene,” Science 334, 648–652 (2011).
[Crossref] [PubMed]

Gaponenko, I.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Letters 12, 2470–2474 (2012).
[Crossref] [PubMed]

Geim, A.

T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun. 2, 458 (2011).
[Crossref] [PubMed]

Geim, A. K.

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nature Materials 6, 183–191 (2007).
[Crossref] [PubMed]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science22 306, 666–669 (2004).
[Crossref]

Glattli, D. C.

I. Petković, F. I. B. Williams, K. Bennaceur, F. Portier, P. Roche, and D. C. Glattli, “Carrier Drift Velocity and Edge Magnetoplasmons in Graphene,” Phys. Rev. Lett. 110, 016 801 (2013).
[Crossref]

Gorbachev, R.

T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun. 2, 458 (2011).
[Crossref] [PubMed]

Grigorenko, A.

T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun. 2, 458 (2011).
[Crossref] [PubMed]

Grigorenko, A. N.

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photon. 6, 7490 758 (2012).
[Crossref]

Grigorieva, I. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science22 306, 666–669 (2004).
[Crossref]

Gusynin, V. P.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “On the universal ac optical background in graphene,” New Journal of Physics 11, 095 013 (2009).
[Crossref]

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” Journal of Physics: Condensed Matter 19, 026 222 (2007).
[Crossref]

Hanson, G. W.

G. W. Hanson, “Dyadic Greens functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103, 064 302 (2008).
[Crossref]

Jalas, D.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, and H. Renner, “What is and what is not an optical isolator,” Nature Photonics 7, 579–582 (2013).
[Crossref]

Jarillo-Herrero, P.

N. M. Gabor, J. C. W. Song, Q. Ma, N. L. Nair, T. Taychatanapat, K. Watanabe, T. Taniguchi, L. S. Levitov, and P. Jarillo-Herrero, “Hot carrierassisted intrinsic photoresponse in graphene,” Science 334, 648–652 (2011).
[Crossref] [PubMed]

Jasnos, P.

T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun. 2, 458 (2011).
[Crossref] [PubMed]

Jiang, D.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science22 306, 666–669 (2004).
[Crossref]

Kuzmenko, A. B.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Letters 12, 2470–2474 (2012).
[Crossref] [PubMed]

Levitov, L. S.

N. M. Gabor, J. C. W. Song, Q. Ma, N. L. Nair, T. Taychatanapat, K. Watanabe, T. Taniguchi, L. S. Levitov, and P. Jarillo-Herrero, “Hot carrierassisted intrinsic photoresponse in graphene,” Science 334, 648–652 (2011).
[Crossref] [PubMed]

Lombardo, A.

T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun. 2, 458 (2011).
[Crossref] [PubMed]

Ma, Q.

N. M. Gabor, J. C. W. Song, Q. Ma, N. L. Nair, T. Taychatanapat, K. Watanabe, T. Taniguchi, L. S. Levitov, and P. Jarillo-Herrero, “Hot carrierassisted intrinsic photoresponse in graphene,” Science 334, 648–652 (2011).
[Crossref] [PubMed]

Mikhailov, S. A.

S. A. Mikhailov and K. Ziegler, “New Electromagnetic Mode in Graphene,” Phys. Rev. Lett. 99, 016 803 (2007).
[Crossref]

Mishchenko, E. G.

E. G. Mishchenko, A. V. Shytov, and P. G. Silvestrov, “Guided Plasmons in Graphene p-n Junctions,” Phys. Rev. Lett. 104, 156 806 (2010).
[Crossref]

Morozov, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science22 306, 666–669 (2004).
[Crossref]

Mueller, T.

T. Mueller, F. Xia, and P. Avouris, “Graphene photodetectors for high-speed optical communications,” Nat. Photon. 4, 297–301 (2010).
[Crossref]

Nair, N. L.

N. M. Gabor, J. C. W. Song, Q. Ma, N. L. Nair, T. Taychatanapat, K. Watanabe, T. Taniguchi, L. S. Levitov, and P. Jarillo-Herrero, “Hot carrierassisted intrinsic photoresponse in graphene,” Science 334, 648–652 (2011).
[Crossref] [PubMed]

Novoselov, K.

T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun. 2, 458 (2011).
[Crossref] [PubMed]

Novoselov, K. S.

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photon. 6, 7490 758 (2012).
[Crossref]

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nature Materials 6, 183–191 (2007).
[Crossref] [PubMed]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science22 306, 666–669 (2004).
[Crossref]

Orlita, M.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Letters 12, 2470–2474 (2012).
[Crossref] [PubMed]

Ostler, M.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Letters 12, 2470–2474 (2012).
[Crossref] [PubMed]

Petkovic, I.

I. Petković, F. I. B. Williams, K. Bennaceur, F. Portier, P. Roche, and D. C. Glattli, “Carrier Drift Velocity and Edge Magnetoplasmons in Graphene,” Phys. Rev. Lett. 110, 016 801 (2013).
[Crossref]

Petrov, A.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, and H. Renner, “What is and what is not an optical isolator,” Nature Photonics 7, 579–582 (2013).
[Crossref]

Polini, M.

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photon. 6, 7490 758 (2012).
[Crossref]

Portier, F.

I. Petković, F. I. B. Williams, K. Bennaceur, F. Portier, P. Roche, and D. C. Glattli, “Carrier Drift Velocity and Edge Magnetoplasmons in Graphene,” Phys. Rev. Lett. 110, 016 801 (2013).
[Crossref]

Potemski, M.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Letters 12, 2470–2474 (2012).
[Crossref] [PubMed]

Pozar, D. M.

D. M. Pozar, Microwave engineering(Danvers, MA: Wiley, 2005), 3rd edn.

Renner, H.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, and H. Renner, “What is and what is not an optical isolator,” Nature Photonics 7, 579–582 (2013).
[Crossref]

Roche, P.

I. Petković, F. I. B. Williams, K. Bennaceur, F. Portier, P. Roche, and D. C. Glattli, “Carrier Drift Velocity and Edge Magnetoplasmons in Graphene,” Phys. Rev. Lett. 110, 016 801 (2013).
[Crossref]

Seyller, T.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Letters 12, 2470–2474 (2012).
[Crossref] [PubMed]

Sharapov, S. G.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “On the universal ac optical background in graphene,” New Journal of Physics 11, 095 013 (2009).
[Crossref]

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” Journal of Physics: Condensed Matter 19, 026 222 (2007).
[Crossref]

Shytov, A. V.

E. G. Mishchenko, A. V. Shytov, and P. G. Silvestrov, “Guided Plasmons in Graphene p-n Junctions,” Phys. Rev. Lett. 104, 156 806 (2010).
[Crossref]

Silveiro, I.

S. Thongrattanasiri, I. Silveiro, and F. J. G. de Abajo, “Plasmons in electrostatically doped graphene,” Appl. Phys. Lett. 100, 201 105 (2012).
[Crossref]

Silvestrov, P. G.

E. G. Mishchenko, A. V. Shytov, and P. G. Silvestrov, “Guided Plasmons in Graphene p-n Junctions,” Phys. Rev. Lett. 104, 156 806 (2010).
[Crossref]

P. G. Silvestrov and K. B. Efetov, “Charge accumulation at the boundaries of a graphene strip induced by a gate voltage: Electrostatic approach,” Phys. Rev. B 77, 155 436 (2008).
[Crossref]

Song, J. C. W.

N. M. Gabor, J. C. W. Song, Q. Ma, N. L. Nair, T. Taychatanapat, K. Watanabe, T. Taniguchi, L. S. Levitov, and P. Jarillo-Herrero, “Hot carrierassisted intrinsic photoresponse in graphene,” Science 334, 648–652 (2011).
[Crossref] [PubMed]

Sounas, D.

Sounas, D. L.

D. L. Sounas and C. Caloz, “Edge surface modes in magnetically biased chemically doped graphene strips,” Appl. Phys. Lett. 99, 231 902:13 (2011).
[Crossref]

Taniguchi, T.

N. M. Gabor, J. C. W. Song, Q. Ma, N. L. Nair, T. Taychatanapat, K. Watanabe, T. Taniguchi, L. S. Levitov, and P. Jarillo-Herrero, “Hot carrierassisted intrinsic photoresponse in graphene,” Science 334, 648–652 (2011).
[Crossref] [PubMed]

Taychatanapat, T.

N. M. Gabor, J. C. W. Song, Q. Ma, N. L. Nair, T. Taychatanapat, K. Watanabe, T. Taniguchi, L. S. Levitov, and P. Jarillo-Herrero, “Hot carrierassisted intrinsic photoresponse in graphene,” Science 334, 648–652 (2011).
[Crossref] [PubMed]

Thongrattanasiri, S.

S. Thongrattanasiri, I. Silveiro, and F. J. G. de Abajo, “Plasmons in electrostatically doped graphene,” Appl. Phys. Lett. 100, 201 105 (2012).
[Crossref]

Vakil, A.

A. Vakil and N. Engheta, “Transformation Optics Using Graphene,” Science 332, 1291–1294 (2011).
[Crossref] [PubMed]

Walter, A. L.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Letters 12, 2470–2474 (2012).
[Crossref] [PubMed]

Watanabe, K.

N. M. Gabor, J. C. W. Song, Q. Ma, N. L. Nair, T. Taychatanapat, K. Watanabe, T. Taniguchi, L. S. Levitov, and P. Jarillo-Herrero, “Hot carrierassisted intrinsic photoresponse in graphene,” Science 334, 648–652 (2011).
[Crossref] [PubMed]

Williams, F. I. B.

I. Petković, F. I. B. Williams, K. Bennaceur, F. Portier, P. Roche, and D. C. Glattli, “Carrier Drift Velocity and Edge Magnetoplasmons in Graphene,” Phys. Rev. Lett. 110, 016 801 (2013).
[Crossref]

Wu, K.

Y. Zhao, K. Wu, and K. M. Cheng, “A Compact 2-D Full-Wave Finite-Difference Frequency-Domain Method for General Guided Wave Structures,” IEEE Trans. Microwave Theory Tech. 50, 1844–1848 (2002).
[Crossref]

Xia, F.

T. Mueller, F. Xia, and P. Avouris, “Graphene photodetectors for high-speed optical communications,” Nat. Photon. 4, 297–301 (2010).
[Crossref]

Yu, Z.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, and H. Renner, “What is and what is not an optical isolator,” Nature Photonics 7, 579–582 (2013).
[Crossref]

Zhang, Y.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science22 306, 666–669 (2004).
[Crossref]

Zhao, Y.

Y. Zhao, K. Wu, and K. M. Cheng, “A Compact 2-D Full-Wave Finite-Difference Frequency-Domain Method for General Guided Wave Structures,” IEEE Trans. Microwave Theory Tech. 50, 1844–1848 (2002).
[Crossref]

Ziegler, K.

S. A. Mikhailov and K. Ziegler, “New Electromagnetic Mode in Graphene,” Phys. Rev. Lett. 99, 016 803 (2007).
[Crossref]

Appl. Phys. Lett. (2)

S. Thongrattanasiri, I. Silveiro, and F. J. G. de Abajo, “Plasmons in electrostatically doped graphene,” Appl. Phys. Lett. 100, 201 105 (2012).
[Crossref]

D. L. Sounas and C. Caloz, “Edge surface modes in magnetically biased chemically doped graphene strips,” Appl. Phys. Lett. 99, 231 902:13 (2011).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

Y. Zhao, K. Wu, and K. M. Cheng, “A Compact 2-D Full-Wave Finite-Difference Frequency-Domain Method for General Guided Wave Structures,” IEEE Trans. Microwave Theory Tech. 50, 1844–1848 (2002).
[Crossref]

J. Appl. Phys. (1)

G. W. Hanson, “Dyadic Greens functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103, 064 302 (2008).
[Crossref]

Journal of Physics: Condensed Matter (1)

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” Journal of Physics: Condensed Matter 19, 026 222 (2007).
[Crossref]

Nano Letters (1)

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Letters 12, 2470–2474 (2012).
[Crossref] [PubMed]

Nat. Commun. (1)

T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun. 2, 458 (2011).
[Crossref] [PubMed]

Nat. Photon. (2)

T. Mueller, F. Xia, and P. Avouris, “Graphene photodetectors for high-speed optical communications,” Nat. Photon. 4, 297–301 (2010).
[Crossref]

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photon. 6, 7490 758 (2012).
[Crossref]

Nature Materials (1)

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nature Materials 6, 183–191 (2007).
[Crossref] [PubMed]

Nature Photonics (1)

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, and H. Renner, “What is and what is not an optical isolator,” Nature Photonics 7, 579–582 (2013).
[Crossref]

New Journal of Physics (1)

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “On the universal ac optical background in graphene,” New Journal of Physics 11, 095 013 (2009).
[Crossref]

Opt. Express (1)

Phys. Rev. B (1)

P. G. Silvestrov and K. B. Efetov, “Charge accumulation at the boundaries of a graphene strip induced by a gate voltage: Electrostatic approach,” Phys. Rev. B 77, 155 436 (2008).
[Crossref]

Phys. Rev. Lett. (3)

E. G. Mishchenko, A. V. Shytov, and P. G. Silvestrov, “Guided Plasmons in Graphene p-n Junctions,” Phys. Rev. Lett. 104, 156 806 (2010).
[Crossref]

S. A. Mikhailov and K. Ziegler, “New Electromagnetic Mode in Graphene,” Phys. Rev. Lett. 99, 016 803 (2007).
[Crossref]

I. Petković, F. I. B. Williams, K. Bennaceur, F. Portier, P. Roche, and D. C. Glattli, “Carrier Drift Velocity and Edge Magnetoplasmons in Graphene,” Phys. Rev. Lett. 110, 016 801 (2013).
[Crossref]

Science (3)

N. M. Gabor, J. C. W. Song, Q. Ma, N. L. Nair, T. Taychatanapat, K. Watanabe, T. Taniguchi, L. S. Levitov, and P. Jarillo-Herrero, “Hot carrierassisted intrinsic photoresponse in graphene,” Science 334, 648–652 (2011).
[Crossref] [PubMed]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science22 306, 666–669 (2004).
[Crossref]

A. Vakil and N. Engheta, “Transformation Optics Using Graphene,” Science 332, 1291–1294 (2011).
[Crossref] [PubMed]

Other (1)

D. M. Pozar, Microwave engineering(Danvers, MA: Wiley, 2005), 3rd edn.

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Figures (10)

Fig. 1
Fig. 1

(a) Electrically and (b) chemically doped graphene p-n junctions.

Fig. 2
Fig. 2

Net carrier density and electric potential for a graphene strip doped with an electric field; w = 50 μm and E0 = 108 V/m.

Fig. 3
Fig. 3

Thermally excited electron and hole densities at room temperature for a graphene strip doped with an electric field; w = 50 μm, E0 = 108 V/m, T = 300 K.

Fig. 4
Fig. 4

Chemical potential and Kubo conductivity for a graphene strip doped with an electric field; w = 50 μm and E0 = 108 V/m, B0 = 0 T, τ = 0.1 ps, T = 300 K, f = 1 THz.

Fig. 5
Fig. 5

Slow-wave factor and loss for a graphene strip biased by an electric field; w = 50 μm, E0 = 108 V/m, B0 = 0 T, τ = 0.1 ps, T = 300 K. The p-n junction mode is represented in red. The insets show the electric field patterns.

Fig. 6
Fig. 6

Dispersion curves for a magnetically biased graphene strip biased by an electric field; w = 50 μm, E0 = 108 V/m, B0 = 0.1 T, τ = 0.1 ps, T = 300 K. The p-n junction mode is represented in red. The grey area represents the light cone.

Fig. 7
Fig. 7

Slow-wave factor and loss for the structure in Fig. 1(b) with no magnetic bias; w = 100 μm, s = 10 nm, n = p = 1013 cm−2, B0 = 0 T, τ = 0.1 ps, T = 300 K. The p-n junction mode is represented in red.

Fig. 8
Fig. 8

Slow-wave factor and loss for the magnetoplasmonic isolator in Fig. 1(b); w = 100 μm, s = 10 nm, n = p = 1013 cm−2, B0 = 1 T, τ = 0.1 ps, T = 300 K. The p-n junction mode is represented in red. The grey area represents the light cone.

Fig. 9
Fig. 9

Electric field pattern in the plane of graphene for the structure of Fig. 1(b) with B0 = 0 T. Points R on the right strip sees a clock-wise rotating electric field in the forward direction and a counter clock-wise rotating electric field in the backward direction as the wave propagates. Point L on the left strip sees an oppositely rotating electric field to point R in each direction.

Fig. 10
Fig. 10

Evolution of the electric field pattern of the mode propagating at the p-n junction (mode 1) for the forward and backward propagations as the magnetic field is increased.

Equations (3)

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w / 2 w / 2 ρ ( x , y ) G ( x , y ; x , y ) d x E 0 x = 0 ,
w / 2 x w / 2 , y = 0 , y = 0 ,
n net = 0 f d ( E , μ c ) N ( E ) d E 0 [ 1 f d ( E , μ c ) ] N ( E ) d E ,

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