Abstract

Using Maxwell’s equations for the incoming and outgoing electromagnetic field, in interaction with a metallic arm-chair graphene nanoribbon (AGNR), and the relationship between the density-density response function and the conductivity, we study surface plasmons (SPs) in a AGNR following the Lindhard, random-phase approximation (RPA), and Hubbard approaches. For transverse magnetic (TM) modes we obtain analytical dispersion relations (DRs) valid for qkF and assess their width dependence. In all approaches we include screening. In the long-wavelength limit q → 0 there is a small but noticeable difference between the DRs of the three approaches. In this limit the respective, scattering-free conductivities differ drastically from those obtained when scattering by impurities is included. We demonstrate that the SP field is proportional to the square of the quality factor Q. The reflection amplitude shows that metallic AGNRs do not support Brewster angles. In addition, AGNRs do not support transverse electric (TE) SPs.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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2016 (3)

Z. Sun, A. Martinez, and F. Wang, “Optical modulators with 2D layered materials,” Nature Photonics 10, 227–238 (2016).
[Crossref]

Y. Wang and D. R. Andersen, “First-principles study of the terahertz third-order nonlinear response of metallic armchair graphene nanoribbons,” Phys. Rev. B 93, 235430 (2016).
[Crossref]

S. Xiao, X. Zhu, B.H. Li, and N. A. Mortensen, “Graphene-plasmon polaritons: From fundamental properties to potential applications,” Front. Phys. 11, 117801 (2016).
[Crossref]

2015 (4)

Y. Zhong, S. D. Malagari, T. Hamilton, and D. Wasserman, “Review of mid-infrared plasmonic materials,” J. Nanophotonics 9, 093791 (2015).
[Crossref]

A. A. Shylau, S. M. Badalyan, F. M. Peeters, and A. P. Jauho, “Electron polarization function and plasmons in metallic armchair graphene nanoribbons,” Phys. Rev. B 91, 205444 (2015).
[Crossref]

S. Jang, E. Hwang, Y. Lee, S. Lee, and J. H. Cho, “Multifunctional Graphene Optoelectronic Devices Capable of Detecting and Storing Photonic Signals,” Nano Lett. 15, 2542–2547 (2015).
[Crossref] [PubMed]

Y. Zhao and Y. Zhu, “Graphene-based hybrid films for plasmonic sensing,” Nanoscale.,  7, 14561–14576 (2015).
[Crossref] [PubMed]

2014 (7)

T. Low and P. Avouris, “Graphene Plasmonics for Terahertz to Mid-Infrared Applications,” ACS Nano. 8, 1086–1101 (2014).
[Crossref] [PubMed]

M. Bagheri and M. Bahrami, “Plasmons in spatially separated double-layer graphene nanoribbons,” J. Appl. Phys. 115, 174301 (2014).
[Crossref]

T. Stauber, “Plasmonics in Dirac systems: from graphene to topological insulators,” J. Phys.: Cond. Matter. 26, 123201 (2014).

G. Z. Magda, X. Jin, I. Hagymasi, P. Vancso, Z. Osvath, P. Nemes-Incze, C. Hwang, L. P. Biro, and L. Tapaszto, “Room-temperature magnetic order on zigzag edges of narrow graphene nanoribbons,” Nature 514, 608–611 (2014)
[Crossref] [PubMed]

F. J. G. Abajo, “Graphene Plasmonics: Challenges and Opportunities,” ACS Photonics. 1, 135–152 (2014).
[Crossref]

D. N. Basov, M. M. Fogler, A. Lanzara, Feng Wang, and Y. Zhang, “Colloquium: Graphene spectroscopy,” Rev. Mod. Phys. 86, 959 (2014).
[Crossref]

A. Politano and G. Chiarello, “Plasmon modes in graphene: status and prospect,” Nanoscale 6, 10927–10940 (2014).
[Crossref] [PubMed]

2013 (5)

W. Wang and J. M. Kinaret, “Plasmons in graphene nanoribbons: Interband transitions and nonlocal effects,” Phys. Rev. B 87, 195424 (2013).
[Crossref]

J. H. Strait, P. Nene, W. M. Chan, C. Manolatou, S. Tiwari, and F. Rana, “Confined plasmons in graphene microstructures: Experiments and theory,” Phys. Rev. B 87, 241410 (2013).
[Crossref]

C. E. P. Villegas, M. R. S. Tavares, G. Q. Hai, and P. Vasilopoulos, “Plasmon modes and screening in double metallic armchair graphene nanoribbons,” Phys. Rev. B 88, 165426 (2013).
[Crossref]

D. R. Andersen and H. Raza, “Collective modes of massive Dirac fermions in armchair graphene nanoribbons,” J. Phys. Condens. Matter 25, 045303 (2013).
[Crossref]

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative Plasmonic Materials: Beyond Gold and Silver,” Advanced Materials 25, 3264–3294 (2013).
[Crossref] [PubMed]

2012 (4)

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nature Photonics 6, 737–748 (2012).
[Crossref]

Q. Bao and K. P. Loh, “Graphene Photonics, Plasmonics, and Broadband Optoelectronic Devices,” ACS Nano. 2, 3677–3694 (2012).
[Crossref]

S. A. Mikhailov and D. Beba, “Nonlinear broadening of the plasmon linewidth in a graphene stripe,” New Journal of Physics. 14, 115024 (2012).
[Crossref]

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

2011 (3)

V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, “Electron-Electron Interactions in Graphene: Current Status and Perspectives,” Rev. Mod. Phys. 84, 1067 (2011).
[Crossref]

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nature Nanotechnology 6, 630–634 (2011).
[Crossref] [PubMed]

G. Seol and J. Guo, “Bandgap opening in boron nitride confined armchair graphene nanoribbon,” Appl. Phys. Lett. 98, 143107 (2011).
[Crossref]

2010 (1)

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nature Photonics 4, 83–91 (2010).
[Crossref]

2007 (1)

L. Brey and H. A. Fertig, “Elementary electronic excitations in graphene nanoribbons,” Phys. Rev. B 75, 125434 (2007).
[Crossref]

2006 (3)

Y-W Son, M. L. Cohen, and S. G. Louie, “Energy Gaps in Graphene Nanoribbons,” Phys. Rev. Lett. 97, 216803 (2006).
[Crossref] [PubMed]

E. Ozbay, “Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions,” Science 13, 189–193 (2006).
[Crossref]

B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8, 318 (2006).
[Crossref]

2001 (1)

M. S. Kushwaha, “Plasmons and magnetoplasmons in semiconductor heterostructures,” Surf. Sci. Reports. 41, 1–416 (2001).
[Crossref]

Abajo, F. J. G.

F. J. G. Abajo, “Graphene Plasmonics: Challenges and Opportunities,” ACS Photonics. 1, 135–152 (2014).
[Crossref]

Andersen, D. R.

Y. Wang and D. R. Andersen, “First-principles study of the terahertz third-order nonlinear response of metallic armchair graphene nanoribbons,” Phys. Rev. B 93, 235430 (2016).
[Crossref]

D. R. Andersen and H. Raza, “Collective modes of massive Dirac fermions in armchair graphene nanoribbons,” J. Phys. Condens. Matter 25, 045303 (2013).
[Crossref]

Avouris, P.

T. Low and P. Avouris, “Graphene Plasmonics for Terahertz to Mid-Infrared Applications,” ACS Nano. 8, 1086–1101 (2014).
[Crossref] [PubMed]

Badalyan, S. M.

A. A. Shylau, S. M. Badalyan, F. M. Peeters, and A. P. Jauho, “Electron polarization function and plasmons in metallic armchair graphene nanoribbons,” Phys. Rev. B 91, 205444 (2015).
[Crossref]

Bagheri, M.

M. Bagheri and M. Bahrami, “Plasmons in spatially separated double-layer graphene nanoribbons,” J. Appl. Phys. 115, 174301 (2014).
[Crossref]

Bahrami, M.

M. Bagheri and M. Bahrami, “Plasmons in spatially separated double-layer graphene nanoribbons,” J. Appl. Phys. 115, 174301 (2014).
[Crossref]

Bao, Q.

Q. Bao and K. P. Loh, “Graphene Photonics, Plasmonics, and Broadband Optoelectronic Devices,” ACS Nano. 2, 3677–3694 (2012).
[Crossref]

Basov, D. N.

D. N. Basov, M. M. Fogler, A. Lanzara, Feng Wang, and Y. Zhang, “Colloquium: Graphene spectroscopy,” Rev. Mod. Phys. 86, 959 (2014).
[Crossref]

Beba, D.

S. A. Mikhailov and D. Beba, “Nonlinear broadening of the plasmon linewidth in a graphene stripe,” New Journal of Physics. 14, 115024 (2012).
[Crossref]

Bechtel, H. A.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nature Nanotechnology 6, 630–634 (2011).
[Crossref] [PubMed]

Biro, L. P.

G. Z. Magda, X. Jin, I. Hagymasi, P. Vancso, Z. Osvath, P. Nemes-Incze, C. Hwang, L. P. Biro, and L. Tapaszto, “Room-temperature magnetic order on zigzag edges of narrow graphene nanoribbons,” Nature 514, 608–611 (2014)
[Crossref] [PubMed]

Boltasseva, A.

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative Plasmonic Materials: Beyond Gold and Silver,” Advanced Materials 25, 3264–3294 (2013).
[Crossref] [PubMed]

Bozhevolnyi, S. I.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nature Photonics 4, 83–91 (2010).
[Crossref]

Brey, L.

L. Brey and H. A. Fertig, “Elementary electronic excitations in graphene nanoribbons,” Phys. Rev. B 75, 125434 (2007).
[Crossref]

Bruus, H.

H. Bruus and K. Flensberg, Introduction to Many-Body Quantum Theory in Condensed Matter Physics (Oxford University, 2004).

Castro Neto, A. H.

V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, “Electron-Electron Interactions in Graphene: Current Status and Perspectives,” Rev. Mod. Phys. 84, 1067 (2011).
[Crossref]

Chan, W. M.

J. H. Strait, P. Nene, W. M. Chan, C. Manolatou, S. Tiwari, and F. Rana, “Confined plasmons in graphene microstructures: Experiments and theory,” Phys. Rev. B 87, 241410 (2013).
[Crossref]

Chiarello, G.

A. Politano and G. Chiarello, “Plasmon modes in graphene: status and prospect,” Nanoscale 6, 10927–10940 (2014).
[Crossref] [PubMed]

Cho, J. H.

S. Jang, E. Hwang, Y. Lee, S. Lee, and J. H. Cho, “Multifunctional Graphene Optoelectronic Devices Capable of Detecting and Storing Photonic Signals,” Nano Lett. 15, 2542–2547 (2015).
[Crossref] [PubMed]

Cohen, M. L.

Y-W Son, M. L. Cohen, and S. G. Louie, “Energy Gaps in Graphene Nanoribbons,” Phys. Rev. Lett. 97, 216803 (2006).
[Crossref] [PubMed]

Fertig, H. A.

L. Brey and H. A. Fertig, “Elementary electronic excitations in graphene nanoribbons,” Phys. Rev. B 75, 125434 (2007).
[Crossref]

Flensberg, K.

H. Bruus and K. Flensberg, Introduction to Many-Body Quantum Theory in Condensed Matter Physics (Oxford University, 2004).

Fogler, M. M.

D. N. Basov, M. M. Fogler, A. Lanzara, Feng Wang, and Y. Zhang, “Colloquium: Graphene spectroscopy,” Rev. Mod. Phys. 86, 959 (2014).
[Crossref]

Geng, B.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nature Nanotechnology 6, 630–634 (2011).
[Crossref] [PubMed]

Girit, C.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nature Nanotechnology 6, 630–634 (2011).
[Crossref] [PubMed]

Giuliani, G. F.

G. F. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid (Cambridge University, 2005).
[Crossref]

Gradshteyn,

Gradshteyn and Ryzhik, Tables of Integrals, Series, and Products (Academic, 2014).

Gramotnev, D. K.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nature Photonics 4, 83–91 (2010).
[Crossref]

Grigorenko, A. N.

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

Guinea, F.

V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, “Electron-Electron Interactions in Graphene: Current Status and Perspectives,” Rev. Mod. Phys. 84, 1067 (2011).
[Crossref]

B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8, 318 (2006).
[Crossref]

Guo, J.

G. Seol and J. Guo, “Bandgap opening in boron nitride confined armchair graphene nanoribbon,” Appl. Phys. Lett. 98, 143107 (2011).
[Crossref]

Hagymasi, I.

G. Z. Magda, X. Jin, I. Hagymasi, P. Vancso, Z. Osvath, P. Nemes-Incze, C. Hwang, L. P. Biro, and L. Tapaszto, “Room-temperature magnetic order on zigzag edges of narrow graphene nanoribbons,” Nature 514, 608–611 (2014)
[Crossref] [PubMed]

Hai, G. Q.

C. E. P. Villegas, M. R. S. Tavares, G. Q. Hai, and P. Vasilopoulos, “Plasmon modes and screening in double metallic armchair graphene nanoribbons,” Phys. Rev. B 88, 165426 (2013).
[Crossref]

Hamilton, T.

Y. Zhong, S. D. Malagari, T. Hamilton, and D. Wasserman, “Review of mid-infrared plasmonic materials,” J. Nanophotonics 9, 093791 (2015).
[Crossref]

Hao, Z.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nature Nanotechnology 6, 630–634 (2011).
[Crossref] [PubMed]

Horng, J.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nature Nanotechnology 6, 630–634 (2011).
[Crossref] [PubMed]

Hwang, C.

G. Z. Magda, X. Jin, I. Hagymasi, P. Vancso, Z. Osvath, P. Nemes-Incze, C. Hwang, L. P. Biro, and L. Tapaszto, “Room-temperature magnetic order on zigzag edges of narrow graphene nanoribbons,” Nature 514, 608–611 (2014)
[Crossref] [PubMed]

Hwang, E.

S. Jang, E. Hwang, Y. Lee, S. Lee, and J. H. Cho, “Multifunctional Graphene Optoelectronic Devices Capable of Detecting and Storing Photonic Signals,” Nano Lett. 15, 2542–2547 (2015).
[Crossref] [PubMed]

Jang, S.

S. Jang, E. Hwang, Y. Lee, S. Lee, and J. H. Cho, “Multifunctional Graphene Optoelectronic Devices Capable of Detecting and Storing Photonic Signals,” Nano Lett. 15, 2542–2547 (2015).
[Crossref] [PubMed]

Jauho, A. P.

A. A. Shylau, S. M. Badalyan, F. M. Peeters, and A. P. Jauho, “Electron polarization function and plasmons in metallic armchair graphene nanoribbons,” Phys. Rev. B 91, 205444 (2015).
[Crossref]

Jin, X.

G. Z. Magda, X. Jin, I. Hagymasi, P. Vancso, Z. Osvath, P. Nemes-Incze, C. Hwang, L. P. Biro, and L. Tapaszto, “Room-temperature magnetic order on zigzag edges of narrow graphene nanoribbons,” Nature 514, 608–611 (2014)
[Crossref] [PubMed]

Ju, L.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nature Nanotechnology 6, 630–634 (2011).
[Crossref] [PubMed]

Kauranen, M.

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nature Photonics 6, 737–748 (2012).
[Crossref]

Kinaret, J. M.

W. Wang and J. M. Kinaret, “Plasmons in graphene nanoribbons: Interband transitions and nonlocal effects,” Phys. Rev. B 87, 195424 (2013).
[Crossref]

Kotov, V. N.

V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, “Electron-Electron Interactions in Graphene: Current Status and Perspectives,” Rev. Mod. Phys. 84, 1067 (2011).
[Crossref]

Kushwaha, M. S.

M. S. Kushwaha, “Plasmons and magnetoplasmons in semiconductor heterostructures,” Surf. Sci. Reports. 41, 1–416 (2001).
[Crossref]

Lanzara, A.

D. N. Basov, M. M. Fogler, A. Lanzara, Feng Wang, and Y. Zhang, “Colloquium: Graphene spectroscopy,” Rev. Mod. Phys. 86, 959 (2014).
[Crossref]

Lee, S.

S. Jang, E. Hwang, Y. Lee, S. Lee, and J. H. Cho, “Multifunctional Graphene Optoelectronic Devices Capable of Detecting and Storing Photonic Signals,” Nano Lett. 15, 2542–2547 (2015).
[Crossref] [PubMed]

Lee, Y.

S. Jang, E. Hwang, Y. Lee, S. Lee, and J. H. Cho, “Multifunctional Graphene Optoelectronic Devices Capable of Detecting and Storing Photonic Signals,” Nano Lett. 15, 2542–2547 (2015).
[Crossref] [PubMed]

Li, B.H.

S. Xiao, X. Zhu, B.H. Li, and N. A. Mortensen, “Graphene-plasmon polaritons: From fundamental properties to potential applications,” Front. Phys. 11, 117801 (2016).
[Crossref]

Liang, X.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nature Nanotechnology 6, 630–634 (2011).
[Crossref] [PubMed]

Loh, K. P.

Q. Bao and K. P. Loh, “Graphene Photonics, Plasmonics, and Broadband Optoelectronic Devices,” ACS Nano. 2, 3677–3694 (2012).
[Crossref]

Louie, S. G.

Y-W Son, M. L. Cohen, and S. G. Louie, “Energy Gaps in Graphene Nanoribbons,” Phys. Rev. Lett. 97, 216803 (2006).
[Crossref] [PubMed]

Low, T.

T. Low and P. Avouris, “Graphene Plasmonics for Terahertz to Mid-Infrared Applications,” ACS Nano. 8, 1086–1101 (2014).
[Crossref] [PubMed]

Magda, G. Z.

G. Z. Magda, X. Jin, I. Hagymasi, P. Vancso, Z. Osvath, P. Nemes-Incze, C. Hwang, L. P. Biro, and L. Tapaszto, “Room-temperature magnetic order on zigzag edges of narrow graphene nanoribbons,” Nature 514, 608–611 (2014)
[Crossref] [PubMed]

Maier, S. A.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

Malagari, S. D.

Y. Zhong, S. D. Malagari, T. Hamilton, and D. Wasserman, “Review of mid-infrared plasmonic materials,” J. Nanophotonics 9, 093791 (2015).
[Crossref]

Manolatou, C.

J. H. Strait, P. Nene, W. M. Chan, C. Manolatou, S. Tiwari, and F. Rana, “Confined plasmons in graphene microstructures: Experiments and theory,” Phys. Rev. B 87, 241410 (2013).
[Crossref]

Martin, M.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nature Nanotechnology 6, 630–634 (2011).
[Crossref] [PubMed]

Martinez, A.

Z. Sun, A. Martinez, and F. Wang, “Optical modulators with 2D layered materials,” Nature Photonics 10, 227–238 (2016).
[Crossref]

Mikhailov, S. A.

S. A. Mikhailov and D. Beba, “Nonlinear broadening of the plasmon linewidth in a graphene stripe,” New Journal of Physics. 14, 115024 (2012).
[Crossref]

Mortensen, N. A.

S. Xiao, X. Zhu, B.H. Li, and N. A. Mortensen, “Graphene-plasmon polaritons: From fundamental properties to potential applications,” Front. Phys. 11, 117801 (2016).
[Crossref]

Naik, G. V.

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative Plasmonic Materials: Beyond Gold and Silver,” Advanced Materials 25, 3264–3294 (2013).
[Crossref] [PubMed]

Nemes-Incze, P.

G. Z. Magda, X. Jin, I. Hagymasi, P. Vancso, Z. Osvath, P. Nemes-Incze, C. Hwang, L. P. Biro, and L. Tapaszto, “Room-temperature magnetic order on zigzag edges of narrow graphene nanoribbons,” Nature 514, 608–611 (2014)
[Crossref] [PubMed]

Nene, P.

J. H. Strait, P. Nene, W. M. Chan, C. Manolatou, S. Tiwari, and F. Rana, “Confined plasmons in graphene microstructures: Experiments and theory,” Phys. Rev. B 87, 241410 (2013).
[Crossref]

Novoselov, K. S.

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

Osvath, Z.

G. Z. Magda, X. Jin, I. Hagymasi, P. Vancso, Z. Osvath, P. Nemes-Incze, C. Hwang, L. P. Biro, and L. Tapaszto, “Room-temperature magnetic order on zigzag edges of narrow graphene nanoribbons,” Nature 514, 608–611 (2014)
[Crossref] [PubMed]

Ozbay, E.

E. Ozbay, “Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions,” Science 13, 189–193 (2006).
[Crossref]

Peeters, F. M.

A. A. Shylau, S. M. Badalyan, F. M. Peeters, and A. P. Jauho, “Electron polarization function and plasmons in metallic armchair graphene nanoribbons,” Phys. Rev. B 91, 205444 (2015).
[Crossref]

Pereira, V. M.

V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, “Electron-Electron Interactions in Graphene: Current Status and Perspectives,” Rev. Mod. Phys. 84, 1067 (2011).
[Crossref]

Polini, M.

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

Politano, A.

A. Politano and G. Chiarello, “Plasmon modes in graphene: status and prospect,” Nanoscale 6, 10927–10940 (2014).
[Crossref] [PubMed]

Rana, F.

J. H. Strait, P. Nene, W. M. Chan, C. Manolatou, S. Tiwari, and F. Rana, “Confined plasmons in graphene microstructures: Experiments and theory,” Phys. Rev. B 87, 241410 (2013).
[Crossref]

Raza, H.

D. R. Andersen and H. Raza, “Collective modes of massive Dirac fermions in armchair graphene nanoribbons,” J. Phys. Condens. Matter 25, 045303 (2013).
[Crossref]

Ryzhik,

Gradshteyn and Ryzhik, Tables of Integrals, Series, and Products (Academic, 2014).

Seol, G.

G. Seol and J. Guo, “Bandgap opening in boron nitride confined armchair graphene nanoribbon,” Appl. Phys. Lett. 98, 143107 (2011).
[Crossref]

Shalaev, V. M.

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative Plasmonic Materials: Beyond Gold and Silver,” Advanced Materials 25, 3264–3294 (2013).
[Crossref] [PubMed]

Shen, Y. R.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nature Nanotechnology 6, 630–634 (2011).
[Crossref] [PubMed]

Shylau, A. A.

A. A. Shylau, S. M. Badalyan, F. M. Peeters, and A. P. Jauho, “Electron polarization function and plasmons in metallic armchair graphene nanoribbons,” Phys. Rev. B 91, 205444 (2015).
[Crossref]

Sols, F.

B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8, 318 (2006).
[Crossref]

Son, Y-W

Y-W Son, M. L. Cohen, and S. G. Louie, “Energy Gaps in Graphene Nanoribbons,” Phys. Rev. Lett. 97, 216803 (2006).
[Crossref] [PubMed]

Stauber, T.

T. Stauber, “Plasmonics in Dirac systems: from graphene to topological insulators,” J. Phys.: Cond. Matter. 26, 123201 (2014).

B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8, 318 (2006).
[Crossref]

Strait, J. H.

J. H. Strait, P. Nene, W. M. Chan, C. Manolatou, S. Tiwari, and F. Rana, “Confined plasmons in graphene microstructures: Experiments and theory,” Phys. Rev. B 87, 241410 (2013).
[Crossref]

Sun, Z.

Z. Sun, A. Martinez, and F. Wang, “Optical modulators with 2D layered materials,” Nature Photonics 10, 227–238 (2016).
[Crossref]

Tapaszto, L.

G. Z. Magda, X. Jin, I. Hagymasi, P. Vancso, Z. Osvath, P. Nemes-Incze, C. Hwang, L. P. Biro, and L. Tapaszto, “Room-temperature magnetic order on zigzag edges of narrow graphene nanoribbons,” Nature 514, 608–611 (2014)
[Crossref] [PubMed]

Tas, M.

M. Tas, Dielectric Formulation of One-Dimensional Electron Gas (Wiley, 2004).

Tavares, M. R. S.

C. E. P. Villegas, M. R. S. Tavares, G. Q. Hai, and P. Vasilopoulos, “Plasmon modes and screening in double metallic armchair graphene nanoribbons,” Phys. Rev. B 88, 165426 (2013).
[Crossref]

Tiwari, S.

J. H. Strait, P. Nene, W. M. Chan, C. Manolatou, S. Tiwari, and F. Rana, “Confined plasmons in graphene microstructures: Experiments and theory,” Phys. Rev. B 87, 241410 (2013).
[Crossref]

Uchoa, B.

V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, “Electron-Electron Interactions in Graphene: Current Status and Perspectives,” Rev. Mod. Phys. 84, 1067 (2011).
[Crossref]

Vancso, P.

G. Z. Magda, X. Jin, I. Hagymasi, P. Vancso, Z. Osvath, P. Nemes-Incze, C. Hwang, L. P. Biro, and L. Tapaszto, “Room-temperature magnetic order on zigzag edges of narrow graphene nanoribbons,” Nature 514, 608–611 (2014)
[Crossref] [PubMed]

Vasilopoulos, P.

C. E. P. Villegas, M. R. S. Tavares, G. Q. Hai, and P. Vasilopoulos, “Plasmon modes and screening in double metallic armchair graphene nanoribbons,” Phys. Rev. B 88, 165426 (2013).
[Crossref]

Vignale, G.

G. F. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid (Cambridge University, 2005).
[Crossref]

Villegas, C. E. P.

C. E. P. Villegas, M. R. S. Tavares, G. Q. Hai, and P. Vasilopoulos, “Plasmon modes and screening in double metallic armchair graphene nanoribbons,” Phys. Rev. B 88, 165426 (2013).
[Crossref]

Wang, F.

Z. Sun, A. Martinez, and F. Wang, “Optical modulators with 2D layered materials,” Nature Photonics 10, 227–238 (2016).
[Crossref]

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nature Nanotechnology 6, 630–634 (2011).
[Crossref] [PubMed]

Wang, Feng

D. N. Basov, M. M. Fogler, A. Lanzara, Feng Wang, and Y. Zhang, “Colloquium: Graphene spectroscopy,” Rev. Mod. Phys. 86, 959 (2014).
[Crossref]

Wang, W.

W. Wang and J. M. Kinaret, “Plasmons in graphene nanoribbons: Interband transitions and nonlocal effects,” Phys. Rev. B 87, 195424 (2013).
[Crossref]

Wang, Y.

Y. Wang and D. R. Andersen, “First-principles study of the terahertz third-order nonlinear response of metallic armchair graphene nanoribbons,” Phys. Rev. B 93, 235430 (2016).
[Crossref]

Wasserman, D.

Y. Zhong, S. D. Malagari, T. Hamilton, and D. Wasserman, “Review of mid-infrared plasmonic materials,” J. Nanophotonics 9, 093791 (2015).
[Crossref]

Wunsch, B.

B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8, 318 (2006).
[Crossref]

Xiao, S.

S. Xiao, X. Zhu, B.H. Li, and N. A. Mortensen, “Graphene-plasmon polaritons: From fundamental properties to potential applications,” Front. Phys. 11, 117801 (2016).
[Crossref]

Zayats, A. V.

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nature Photonics 6, 737–748 (2012).
[Crossref]

Zettl, A.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nature Nanotechnology 6, 630–634 (2011).
[Crossref] [PubMed]

Zhang, Y.

D. N. Basov, M. M. Fogler, A. Lanzara, Feng Wang, and Y. Zhang, “Colloquium: Graphene spectroscopy,” Rev. Mod. Phys. 86, 959 (2014).
[Crossref]

Zhao, Y.

Y. Zhao and Y. Zhu, “Graphene-based hybrid films for plasmonic sensing,” Nanoscale.,  7, 14561–14576 (2015).
[Crossref] [PubMed]

Zhong, Y.

Y. Zhong, S. D. Malagari, T. Hamilton, and D. Wasserman, “Review of mid-infrared plasmonic materials,” J. Nanophotonics 9, 093791 (2015).
[Crossref]

Zhu, X.

S. Xiao, X. Zhu, B.H. Li, and N. A. Mortensen, “Graphene-plasmon polaritons: From fundamental properties to potential applications,” Front. Phys. 11, 117801 (2016).
[Crossref]

Zhu, Y.

Y. Zhao and Y. Zhu, “Graphene-based hybrid films for plasmonic sensing,” Nanoscale.,  7, 14561–14576 (2015).
[Crossref] [PubMed]

ACS Nano. (2)

T. Low and P. Avouris, “Graphene Plasmonics for Terahertz to Mid-Infrared Applications,” ACS Nano. 8, 1086–1101 (2014).
[Crossref] [PubMed]

Q. Bao and K. P. Loh, “Graphene Photonics, Plasmonics, and Broadband Optoelectronic Devices,” ACS Nano. 2, 3677–3694 (2012).
[Crossref]

ACS Photonics. (1)

F. J. G. Abajo, “Graphene Plasmonics: Challenges and Opportunities,” ACS Photonics. 1, 135–152 (2014).
[Crossref]

Advanced Materials (1)

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative Plasmonic Materials: Beyond Gold and Silver,” Advanced Materials 25, 3264–3294 (2013).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

G. Seol and J. Guo, “Bandgap opening in boron nitride confined armchair graphene nanoribbon,” Appl. Phys. Lett. 98, 143107 (2011).
[Crossref]

Front. Phys. (1)

S. Xiao, X. Zhu, B.H. Li, and N. A. Mortensen, “Graphene-plasmon polaritons: From fundamental properties to potential applications,” Front. Phys. 11, 117801 (2016).
[Crossref]

J. Appl. Phys. (1)

M. Bagheri and M. Bahrami, “Plasmons in spatially separated double-layer graphene nanoribbons,” J. Appl. Phys. 115, 174301 (2014).
[Crossref]

J. Nanophotonics (1)

Y. Zhong, S. D. Malagari, T. Hamilton, and D. Wasserman, “Review of mid-infrared plasmonic materials,” J. Nanophotonics 9, 093791 (2015).
[Crossref]

J. Phys. Condens. Matter (1)

D. R. Andersen and H. Raza, “Collective modes of massive Dirac fermions in armchair graphene nanoribbons,” J. Phys. Condens. Matter 25, 045303 (2013).
[Crossref]

J. Phys.: Cond. Matter. (1)

T. Stauber, “Plasmonics in Dirac systems: from graphene to topological insulators,” J. Phys.: Cond. Matter. 26, 123201 (2014).

Nano Lett. (1)

S. Jang, E. Hwang, Y. Lee, S. Lee, and J. H. Cho, “Multifunctional Graphene Optoelectronic Devices Capable of Detecting and Storing Photonic Signals,” Nano Lett. 15, 2542–2547 (2015).
[Crossref] [PubMed]

Nanoscale (1)

A. Politano and G. Chiarello, “Plasmon modes in graphene: status and prospect,” Nanoscale 6, 10927–10940 (2014).
[Crossref] [PubMed]

Nanoscale. (1)

Y. Zhao and Y. Zhu, “Graphene-based hybrid films for plasmonic sensing,” Nanoscale.,  7, 14561–14576 (2015).
[Crossref] [PubMed]

Nature (1)

G. Z. Magda, X. Jin, I. Hagymasi, P. Vancso, Z. Osvath, P. Nemes-Incze, C. Hwang, L. P. Biro, and L. Tapaszto, “Room-temperature magnetic order on zigzag edges of narrow graphene nanoribbons,” Nature 514, 608–611 (2014)
[Crossref] [PubMed]

Nature Nanotechnology (1)

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nature Nanotechnology 6, 630–634 (2011).
[Crossref] [PubMed]

Nature Photonics (4)

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nature Photonics 6, 737–748 (2012).
[Crossref]

Z. Sun, A. Martinez, and F. Wang, “Optical modulators with 2D layered materials,” Nature Photonics 10, 227–238 (2016).
[Crossref]

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nature Photonics 4, 83–91 (2010).
[Crossref]

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

New J. Phys. (1)

B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8, 318 (2006).
[Crossref]

New Journal of Physics. (1)

S. A. Mikhailov and D. Beba, “Nonlinear broadening of the plasmon linewidth in a graphene stripe,” New Journal of Physics. 14, 115024 (2012).
[Crossref]

Phys. Rev. B (6)

W. Wang and J. M. Kinaret, “Plasmons in graphene nanoribbons: Interband transitions and nonlocal effects,” Phys. Rev. B 87, 195424 (2013).
[Crossref]

J. H. Strait, P. Nene, W. M. Chan, C. Manolatou, S. Tiwari, and F. Rana, “Confined plasmons in graphene microstructures: Experiments and theory,” Phys. Rev. B 87, 241410 (2013).
[Crossref]

A. A. Shylau, S. M. Badalyan, F. M. Peeters, and A. P. Jauho, “Electron polarization function and plasmons in metallic armchair graphene nanoribbons,” Phys. Rev. B 91, 205444 (2015).
[Crossref]

L. Brey and H. A. Fertig, “Elementary electronic excitations in graphene nanoribbons,” Phys. Rev. B 75, 125434 (2007).
[Crossref]

C. E. P. Villegas, M. R. S. Tavares, G. Q. Hai, and P. Vasilopoulos, “Plasmon modes and screening in double metallic armchair graphene nanoribbons,” Phys. Rev. B 88, 165426 (2013).
[Crossref]

Y. Wang and D. R. Andersen, “First-principles study of the terahertz third-order nonlinear response of metallic armchair graphene nanoribbons,” Phys. Rev. B 93, 235430 (2016).
[Crossref]

Phys. Rev. Lett. (1)

Y-W Son, M. L. Cohen, and S. G. Louie, “Energy Gaps in Graphene Nanoribbons,” Phys. Rev. Lett. 97, 216803 (2006).
[Crossref] [PubMed]

Rev. Mod. Phys. (2)

V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, “Electron-Electron Interactions in Graphene: Current Status and Perspectives,” Rev. Mod. Phys. 84, 1067 (2011).
[Crossref]

D. N. Basov, M. M. Fogler, A. Lanzara, Feng Wang, and Y. Zhang, “Colloquium: Graphene spectroscopy,” Rev. Mod. Phys. 86, 959 (2014).
[Crossref]

Science (1)

E. Ozbay, “Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions,” Science 13, 189–193 (2006).
[Crossref]

Surf. Sci. Reports. (1)

M. S. Kushwaha, “Plasmons and magnetoplasmons in semiconductor heterostructures,” Surf. Sci. Reports. 41, 1–416 (2001).
[Crossref]

Other (5)

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

H. Bruus and K. Flensberg, Introduction to Many-Body Quantum Theory in Condensed Matter Physics (Oxford University, 2004).

Gradshteyn and Ryzhik, Tables of Integrals, Series, and Products (Academic, 2014).

M. Tas, Dielectric Formulation of One-Dimensional Electron Gas (Wiley, 2004).

G. F. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid (Cambridge University, 2005).
[Crossref]

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Figures (11)

Fig. 1
Fig. 1 Geometry of a AGNR.
Fig. 2
Fig. 2 A AGNR at the interface of two media with permittivities ε1 and ε2.
Fig. 3
Fig. 3 Matrix element of the screened potential vs q/kF in (a) a AGNR for N = 14, EF = 0.1 eV, and 2D graphene in (b) the RPA and (c) Thomas-Fermi approaches.
Fig. 4
Fig. 4 (a) TM Lindhard and (b) Hubbard SP DRs in a metallic AGNR with n1 = 2, n2 = 1, and EF = 0.1 eV. From top to bottom the AGNR width is N = 5, 8, 20.
Fig. 5
Fig. 5 TM Lindhard, RPA, and Hubbard SP dispersions in a AGNR for EF = 0.1 eV and width N = 8 for (a) q/kF ≤ 1 and (b) long wavelength limit, q/kF → 0.
Fig. 6
Fig. 6 (ω, q) contour plot of the RPA, Hubbard, and Lindhard conductivities of a metallic AGNR for N = 14 and EF = 0.1 eV.
Fig. 7
Fig. 7 Cross sections of Fig. 6 for (a) q/kF = 0.18 and (b) ħω/EF = 0.48.
Fig. 8
Fig. 8 Hubbard conductivity for N = 8, 14, 20 and EF = 0.1 eV. Panel (a) is for fixed ħω/EF = 0.48 and panel (b) for fixed q/kF = 0.18.
Fig. 9
Fig. 9 (ω, q) amplitude contour plot of the RPA, Hubbard, and Lindhard conductivities of a metallic AGNR, in the long-wavelength limit, for N = 14 and EF = 0.1 eV. The top panels are for γ′ = 0, the middle ones for γ′ = 0.001, and the bottom panels for γ′ = 0.03.
Fig. 10
Fig. 10 (ω, q) contour plot of −πħvFIm χRPA for a metallic AGNR with N = 14 and EF = 0.1 eV. The first row of panels is for an unscreened potential with γ′ = 0.001, 0.005, and 0.009., the second row for a screened one with γ′ = 0.001 and k′s = 0.001, 0.01, 0.1..
Fig. 11
Fig. 11 (a) RPA Reflection coefficient for N = 14, EF = 0.1 eV, n1 = 2 and n2 = 1 in the absence of screening and scattering. (b) The solid red curve is a cross section of (a) for ħω/EF = 0.5 and the dot-dashed blue one the result for a 2D substrate.

Equations (50)

Equations on this page are rendered with MathJax. Learn more.

1 E 1 2 E 2 = ρ s ,
E 1 E 2 = 0 .
t ( k , ω ) = 2 [ n 1 n 2 + sin θ T sin θ I + sin θ T n 2 ε 0 c σ ( k , ω ) ] 1
r ( k , ω ) = 1 sin θ T sin θ I t ( k , ω ) ;
σ ( q , ω ) = ( i N e 2 ω / q 2 ) χ ( q , ω ) ,
q i ω ( 1 + 2 ) / σ ( q , ω ) .
χ 0 ( q , y , ω ) = 2 W h v F q 2 v F 2 q 2 ω 2 .
χ Lin ( q , ω ) = χ 0 ( q , ω ) = 0 W χ 0 ( q , y , ω ) d y ,
χ RPA ( q , ω ) = χ 0 ( q , ω ) 1 V ( q ) χ 0 ( q , ω ) ,
V ( q ) = 2 e 2 0 0 1 0 1 K 0 [ δ λ | ( y y ) | ] d y d y ,
χ Hub ( q , ω ) = χ 0 ( q , ω ) 1 V ( q ) [ 1 G ( q ) ] χ 0 ( q , ω ) ,
G ( q ) = V ( ( q 2 + k F 2 ) 1 / 2 ) / 2 V ( q ) .
ω / E F = λ 2 + ζ λ , Lindhard ,
ω / E F = λ 2 ( 1 + λ β λ ) + ζ λ , RPA ,
ω / E F = λ 2 ( 1 + λ γ λ ) + ζ λ . Hubbard
ζ = 2 e 2 / [ π 3 ( 1 + 2 ) a c c E F ] ( N / ( N + 1 ) )
β λ = 2 e 2 v F 0 0 1 0 1 K 0 [ δ λ | ( y y ) | ] d y d y ,
χ i m 0 ( q , ω ) = ( 1 i ω τ ) χ 0 ( q , ω + i γ ) 1 i ω τ + [ χ 0 ( q , ω + i γ ) / χ 0 ( q , 0 ) 1 ] ,
λ = [ P 1 + ζ 2 P 2 / 2 S ] 1 / 2 .
λ = [ P 4 + P 5 / 2 P 3 ] 1 / 2 ,
Im χ RPA = Im χ imp 0 [ 1 V ( q ) χ imp 0 ] 2 + [ V ( q ) Im χ imp 0 ] 2
R ( θ , ω ) = ( 1 δ θ , θ B ) | r | 2 ,
t ~ 4 Q 2 / r n ,
E SP ~ 4 Q 2 E I / r n ,
2 ( E I cos θ I + E R cos θ R ) 1 E T cos θ T = ρ s ,
( E I sin θ I E R sin θ R ) E T sin θ T = 0 .
ω ρ s ( q , ω ) + k x J x ( q , ω ) = 0 ,
J x ( q , ω ) = σ ( q , ω ) E x ( q , ω ) ,
ρ s ( q , ω ) = [ σ ( q , ω ) / ω ] k T E T sin 2 θ T / 2 .
( 1 + r ) cos θ I = ( 1 + [ σ ( q , ω ) / ω ] k T sin θ T ) t cos θ T / 2 ,
( 1 r ) sin θ I = t sin θ T .
ρ s ( x , t ) = i = 1 N ρ ( x , y , z , t ) δ ( y y i ) δ ( z ) d y d z = N ρ ( x , t ) ,
E ( k , ω ) = ϕ ( k , ω ) = i k ϕ ext ( k , ω ) ,
ρ ( k , ω ) = e 2 χ ( k , ω ) ϕ ext ( k , ω ) .
n 1 n 2 + k 2 z n 2 k 1 z n 1 + k 2 z n 1 n 2 ε 0 σ ( k , ω ) ω ,
k j z = i | k x | [ 1 ( k j / k x ) 2 ] 1 / 2 i | k x | ,
| k x | = [ k R x 2 + k I x 2 ] 1 / 2 = k R x [ 1 + Q 2 ] 1 / 2 ,
t ( k , ω ) ~ 4 Q 2 / r n ,
H 2 H 1 = k f × n ^ ,
B 2 B 1 = 0 .
1 r = t ( n 2 n 1 + σ y y n 1 0 sin θ T ) sin θ T sin θ I ,
1 + r = t .
t = 2 [ 1 + ( n 2 n 1 + σ y y n 1 0 sin θ T ) sin θ T sin θ I ] 1 .
λ 2 ζ 2 S = ν 2 ( S λ 2 ) 2 + γ 2 ( 2 ν 2 λ 2 ) 2 ,
P 1 = [ 3 γ 2 ν 2 + S / 2 ζ 2 + ν 4 ] / S
P 2 = S 2 + 4 ν 2 ( S + 2 γ 2 ) [ γ 2 ( S 2 γ 2 ) / ζ 2 + S ] / ζ 2 .
λ 2 ζ 2 S = ( S λ 2 ) 2 + γ 2 [ 2 ν 2 λ 2 ( 1 + β λ ) ] 2 .
P 3 = [ S ( 1 + β λ 2 ) + 2 β λ γ 2 ( 1 + ν 2 ) ] / ζ 2 ,
P 4 = [ ν 2 ( S + 2 γ 2 ) ( 1 + β λ ) + S ζ 2 ] / 2 ζ 2 ,
P 5 = 4 P 4 2 4 P 3 ν 2 ( S 2 + 4 γ 2 ν 2 ) / ζ 2 ,

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