Abstract

We describe theoretically a new long-wave infrared optical modulator based on the characteristics of TM surface plasmons in graphene. Calculations made using a finite-τ random-phase approximation model, of relevant surface plasmon propagation parameters, are presented. We show that the plasmon losses vary as a function of carrier density; for large carrier densities, the interband absorption of the plasmon energy is blocked due to filling of the conduction band states, and for small carrier densities, the plasmon energy is absorbed by interband optical transitions. The carrier density versus plasmon loss curve exhibits a kink at the boundary between these two qualitatively dissimilar absorption mechanisms, corresponding to the intersection between the plasmon dispersion curve and the onset threshold for the interband absorption. The modulator device can be switched between high and low transmission states by varying the carrier density with an applied gate bias voltage. The device is limited in optical frequency to plasmons with photon energies less than the optical phonon energy (200 meV in graphene). An example modulator design for light with vacuum wavelength λ0=10μm is presented. This modulator exhibits a contrast ratio in the transmitted optical power between ON and OFF states of 62  dB. A simple circuit model indicates that the switching speed of the modulator should be limited by the carrier relaxation time (1.35×1013  s).

© 2010 Optical Society of America

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References

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  1. J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1-87 (2007).
    [CrossRef]
  2. A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109-162 (2009).
    [CrossRef]
  3. M. Jablan, H. Buljan, and M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 024535 (2009).
    [CrossRef]
  4. F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320, 206-209 (2009).
    [CrossRef]
  5. G. W. Hanson, “Dyadic Green's functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103, 064302 (2008).
    [CrossRef]
  6. F. Rana, “Graphene terahertz plasmon oscillators,” IEEE Trans. Nanotechnol. 7, 91-99 (2008).
    [CrossRef]
  7. E. H. Hwang and S. Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75, 205418 (2007).
    [CrossRef]
  8. N. D. Mermin, “Lindhard dielectric function in the relaxation-time approximation,” Phys. Rev. B 1, 2362-2363 (1970).
    [CrossRef]
  9. 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 306, 666-669 (2004).
    [CrossRef] [PubMed]
  10. A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett. 8, 902-907 (2008).
    [CrossRef] [PubMed]
  11. Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. 19, 3077-3083 (2009).
    [CrossRef]
  12. H. Zhang, D. Y. Tang, L. M. Zhao, Q. L. Bao, and K. P. Loh, “Large energy mode locking of an erbium-doped fiber laser with atomic layer graphene,” Opt. Express 17, 17630-17635 (2009).
    [CrossRef] [PubMed]

2009

A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109-162 (2009).
[CrossRef]

M. Jablan, H. Buljan, and M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 024535 (2009).
[CrossRef]

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320, 206-209 (2009).
[CrossRef]

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. 19, 3077-3083 (2009).
[CrossRef]

H. Zhang, D. Y. Tang, L. M. Zhao, Q. L. Bao, and K. P. Loh, “Large energy mode locking of an erbium-doped fiber laser with atomic layer graphene,” Opt. Express 17, 17630-17635 (2009).
[CrossRef] [PubMed]

2008

G. W. Hanson, “Dyadic Green's functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103, 064302 (2008).
[CrossRef]

F. Rana, “Graphene terahertz plasmon oscillators,” IEEE Trans. Nanotechnol. 7, 91-99 (2008).
[CrossRef]

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett. 8, 902-907 (2008).
[CrossRef] [PubMed]

2007

E. H. Hwang and S. Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75, 205418 (2007).
[CrossRef]

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1-87 (2007).
[CrossRef]

2004

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 306, 666-669 (2004).
[CrossRef] [PubMed]

1970

N. D. Mermin, “Lindhard dielectric function in the relaxation-time approximation,” Phys. Rev. B 1, 2362-2363 (1970).
[CrossRef]

Balandin, A. A.

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett. 8, 902-907 (2008).
[CrossRef] [PubMed]

Bao, Q.

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. 19, 3077-3083 (2009).
[CrossRef]

Bao, Q. L.

Bao, W.

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett. 8, 902-907 (2008).
[CrossRef] [PubMed]

Buljan, H.

M. Jablan, H. Buljan, and M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 024535 (2009).
[CrossRef]

Calizo, I.

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett. 8, 902-907 (2008).
[CrossRef] [PubMed]

Chulkov, E. V.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1-87 (2007).
[CrossRef]

Crommie, M.

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320, 206-209 (2009).
[CrossRef]

Das Sarma, S.

E. H. Hwang and S. Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75, 205418 (2007).
[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,” Science 306, 666-669 (2004).
[CrossRef] [PubMed]

Echenique, P. M.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1-87 (2007).
[CrossRef]

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,” Science 306, 666-669 (2004).
[CrossRef] [PubMed]

Geim, A. K.

A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109-162 (2009).
[CrossRef]

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 306, 666-669 (2004).
[CrossRef] [PubMed]

Ghosh, S.

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett. 8, 902-907 (2008).
[CrossRef] [PubMed]

Girit, C.

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320, 206-209 (2009).
[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,” Science 306, 666-669 (2004).
[CrossRef] [PubMed]

Guinea, F.

A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109-162 (2009).
[CrossRef]

Hanson, G. W.

G. W. Hanson, “Dyadic Green's functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103, 064302 (2008).
[CrossRef]

Hwang, E. H.

E. H. Hwang and S. Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75, 205418 (2007).
[CrossRef]

Jablan, M.

M. Jablan, H. Buljan, and M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 024535 (2009).
[CrossRef]

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,” Science 306, 666-669 (2004).
[CrossRef] [PubMed]

Lau, C. N.

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett. 8, 902-907 (2008).
[CrossRef] [PubMed]

Loh, K. P.

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. 19, 3077-3083 (2009).
[CrossRef]

H. Zhang, D. Y. Tang, L. M. Zhao, Q. L. Bao, and K. P. Loh, “Large energy mode locking of an erbium-doped fiber laser with atomic layer graphene,” Opt. Express 17, 17630-17635 (2009).
[CrossRef] [PubMed]

Mermin, N. D.

N. D. Mermin, “Lindhard dielectric function in the relaxation-time approximation,” Phys. Rev. B 1, 2362-2363 (1970).
[CrossRef]

Miao, F.

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett. 8, 902-907 (2008).
[CrossRef] [PubMed]

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,” Science 306, 666-669 (2004).
[CrossRef] [PubMed]

Neto, A. H. C.

A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109-162 (2009).
[CrossRef]

Ni, Z.

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. 19, 3077-3083 (2009).
[CrossRef]

Novoselov, K. S.

A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109-162 (2009).
[CrossRef]

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 306, 666-669 (2004).
[CrossRef] [PubMed]

Peres, N. M. R.

A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109-162 (2009).
[CrossRef]

Pitarke, J. M.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1-87 (2007).
[CrossRef]

Rana, F.

F. Rana, “Graphene terahertz plasmon oscillators,” IEEE Trans. Nanotechnol. 7, 91-99 (2008).
[CrossRef]

Shen, Y. R.

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320, 206-209 (2009).
[CrossRef]

Shen, Z. X.

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. 19, 3077-3083 (2009).
[CrossRef]

Silkin, V. M.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1-87 (2007).
[CrossRef]

Soljacic, M.

M. Jablan, H. Buljan, and M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 024535 (2009).
[CrossRef]

Tang, D. Y.

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. 19, 3077-3083 (2009).
[CrossRef]

H. Zhang, D. Y. Tang, L. M. Zhao, Q. L. Bao, and K. P. Loh, “Large energy mode locking of an erbium-doped fiber laser with atomic layer graphene,” Opt. Express 17, 17630-17635 (2009).
[CrossRef] [PubMed]

Teweldebrhan, D.

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett. 8, 902-907 (2008).
[CrossRef] [PubMed]

Tian, C.

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320, 206-209 (2009).
[CrossRef]

Wang, F.

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320, 206-209 (2009).
[CrossRef]

Wang, Y.

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. 19, 3077-3083 (2009).
[CrossRef]

Yan, Y.

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. 19, 3077-3083 (2009).
[CrossRef]

Zettl, A.

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320, 206-209 (2009).
[CrossRef]

Zhang, H.

H. Zhang, D. Y. Tang, L. M. Zhao, Q. L. Bao, and K. P. Loh, “Large energy mode locking of an erbium-doped fiber laser with atomic layer graphene,” Opt. Express 17, 17630-17635 (2009).
[CrossRef] [PubMed]

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. 19, 3077-3083 (2009).
[CrossRef]

Zhang, Y.

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320, 206-209 (2009).
[CrossRef]

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 306, 666-669 (2004).
[CrossRef] [PubMed]

Zhao, L. M.

Adv. Funct. Mater.

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. 19, 3077-3083 (2009).
[CrossRef]

IEEE Trans. Nanotechnol.

F. Rana, “Graphene terahertz plasmon oscillators,” IEEE Trans. Nanotechnol. 7, 91-99 (2008).
[CrossRef]

J. Appl. Phys.

G. W. Hanson, “Dyadic Green's functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103, 064302 (2008).
[CrossRef]

Nano Lett.

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett. 8, 902-907 (2008).
[CrossRef] [PubMed]

Opt. Express

Phys. Rev. B

E. H. Hwang and S. Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75, 205418 (2007).
[CrossRef]

N. D. Mermin, “Lindhard dielectric function in the relaxation-time approximation,” Phys. Rev. B 1, 2362-2363 (1970).
[CrossRef]

M. Jablan, H. Buljan, and M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 024535 (2009).
[CrossRef]

Rep. Prog. Phys.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1-87 (2007).
[CrossRef]

Rev. Mod. Phys.

A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109-162 (2009).
[CrossRef]

Science

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320, 206-209 (2009).
[CrossRef]

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 306, 666-669 (2004).
[CrossRef] [PubMed]

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Figures (7)

Fig. 1
Fig. 1

Band structure of graphene monolayer near the K point at ( k x a 0 / 2 , k y a 0 / 2 ) = ( 2 π / 3 , 0 ) calculated using a tight-binding model. This calculation clearly shows the linear character of the band structure near the K point. The vertical axis of the plot ranges from −1.0 eV to + 1.0   eV and a 0 = 2.46   Å is the lattice constant of the graphene unit cell. The Fermi energy E F for intrinsic graphene is located at E = 0 .

Fig. 2
Fig. 2

Schematic illustration of interband plasmon absorption in graphene near a K point. The transition occurs between an initial state E i in the valence band and a final state E f in the conduction band. For small doping densities corresponding to the Fermi level E F   low , the final state is open and the interband transition is allowed resulting in high-loss plasmon propagation. For large doping densities corresponding to the Fermi level E F   high , the final state is filled and the interband transition is blocked resulting in low-loss plasmon propagation. It should be noted that due to the plasmon momentum enhancement, non-vertical transitions are allowed. For plasmons with energy ω = ( E f E i ) / the RPA theory sums over all allowed transitions q = q ( E f ) q ( E i ) .

Fig. 3
Fig. 3

Device geometry for the plasmon modulator. A gate bias is applied to the graphene monolayer by applying a voltage from the Au contacts to the Si : n + substrate. Plasmon propagation takes place along the graphene monolayer in the z direction as shown, with the field decaying exponentially in the ± x direction.

Fig. 4
Fig. 4

The normalized dispersion relation for TM surface plasmons calculated using a f τ -RPA model is plotted as the solid black curve and is the same for all plasmons. The intersection of the dispersion curve with the dot-dashed line indicates the location of the λ 0 = 10 μ m plasmon for the n = 2.41 × 10 12 cm 2 (low-loss) case; the intersection with the dotted line indicates the location of the n = 1.52 × 10 12 cm 2 (low-loss) case; and the intersection with the dashed line indicates the n = 3.8 × 10 11 cm 2 (high-loss) case. The region below the bold black line corresponds to high intraband absorption and the region above the bold dashed line corresponds to high interband absorption. The bold dot marks the intersection of the plasmon dispersion relation with the interband absorption boundary and corresponds to the inflection point (kink) in the q Im versus n curves of Fig. 6.

Fig. 5
Fig. 5

Plot of TM plasmon parameters near λ 0 = 10 μ m calculated using a f τ -RPA model. Part (a) shows the ratio of the real part of the plasmon wavenumber to the imaginary part of the plasmon wavenumber; part (b) shows the momentum enhancement λ 0 / λ p ; and part (c) shows the normalized group velocity of the plasmon. Curves are shown for three values of the Fermi energy: the solid curves are calculated with E F = 0.170   eV corresponding to a carrier density of 2.41 × 10 12 cm 2 (low-loss near λ 0 = 10 μ m ); the dashed curves are calculated with E F = 0.135   eV corresponding to a carrier density of 1.52 × 10 12 cm 2 (low-loss near λ 0 = 10 μ m ); and the dotted curves are calculated with E F = 0.0675   eV corresponding to a carrier density of 3.8 × 10 11 cm 2 (high-loss near λ 0 = 10 μ m ). For all carrier densities discussed here, a carrier relaxation-time constant of τ = 1.35 × 10 13 as well as dielectric constants of 1 (air) and 4 ( SiO 2 ) for the superstrate and substrate, respectively, are assumed. Further, the Fermi energies and plasmon photon energy are below the optical phonon energy of 200 meV, and thus optical phonon decay mechanisms are not considered in these calculations.

Fig. 6
Fig. 6

Plot of q Im versus carrier density curves for ω = 0.5 (dotted), 1.0 (short-dashed), 1.5 (short dot-dashed), 2.0 (long-dashed), 2.5 (long dot-dashed), and 3.0 (solid) × 10 14   rad / s calculated with the f τ -RPA model. The carrier density bounding the weak and strong interband absorption regions for each frequency is located at the kink in q Im , separating the virtual interband absorption region (larger n) from the real interband absorption region (smaller n). The vertical dotted line indicates the carrier density for E F = 200   meV .

Fig. 7
Fig. 7

Propagation of λ 0 = 10 μ m plasmons with parameters determined by the f τ -RPA model. Part (a) corresponds to a carrier density of n = 2.41 × 10 12 cm 2 (low-loss state) and part (b) corresponds to a carrier density of n = 3.8 × 10 11 cm 2 (high-loss state). Axes in both plots are normalized to q Re and q x calculated for the low-loss state.

Equations (11)

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

E z = A   exp [ i q z q x 1 x ] , E y = 0 , E x = B   exp [ i q z q x 1 x ] ,
x > 0 ,
E z = A   exp [ i q z + q x 2 x ] , E y = 0 , E x = B   exp [ i q z + q x 2 x ] ,
x < 0 ,
ϵ 1 q 2 ϵ 1 ω 2 c 2 + ϵ 2 q 2 ϵ 2 ω 2 c 2 = i σ ( q , ω ) ω ϵ 0
q q x 1 q x 2 = ϵ 0 ϵ 1 + ϵ 2 2 i 2 ω σ ( q , ω ) ,
χ ( q , ω ) = e 2 q 2 Π ( q , ω ) ,
Π ( q , ω ) = g s g v L 2 k s s f s k f s k ω + E s k E s k F s s ( k , k ) ,
χ τ ( q , ω ) = ( 1 + i / ω τ ) χ ( q , ω + i / τ ) 1 + ( i / ω τ ) χ ( q , ω + i / τ ) / χ ( q , 0 ) .
ϵ RPA ( q , ω ) = ϵ 1 + ϵ 2 2 + q 2 ϵ 0 χ τ ( q , ω ) ,
σ RPA ( q , ω ) = i ω χ τ ( q , ω ) .

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