Graphene as a Tunable Anisotropic or Isotropic Plasmonic Metasurface
Supplemental Material for: Graphene as a Tunable Anisotropic or Isotropic Plasmonic Metasurface Paloma A. Huidobro,∗ Matthias Kraft, Stefan A. Maier, and John B. Pendry Imperial College London, Department of Physics, The Blackett Laboratory, London SW7 2AZ, UK E-mail: [email protected]
∗
To whom correspondence should be addressed
1
Graphene’s conductivity Within the random phase approximation, the conductivity of graphene is written as a sum of intraband and interband contributions, σ = σintra + σinter , as follows 1
σintra σinter
1 2ie2 t ln 2 cosh , = h ¯ π [Ω + iΓ] 2t e2 1 1 Ω−2 i (Ω + 2)2 = + arctan − ln . 4¯ h 2 π 2t 2π (Ω − 2)2 + (2t)2
(1) (2)
where Ω = h ¯ ω/µ and t = kB T /µ are frequency and temperature normalized to the chemical potential, respectively. The normalized damping term is Γ = h ¯ /(µτ ), where τ = mµ/vF2 is the carriers’ scattering time (m is the mobility and vF the Fermi velocity).
(a)
μ = 0.65 eV, m =104 cm2/(Vs) α = 76 GHz/Ω, β = 1.5THz
(b) μ = 0.65 eV, m =1.6x103 cm2/(Vs) α = 76 GHz/Ω, β = 9.8THz
f (THz)
Figure S1: Graphene’s conductivity for the RPA model (solid lines) and for the Drude approximation (dashed lines). The conductivity for two sets of parameters used in the paper are plotted, with a mobilities of m = 104 cm2 /(V·s) (upper panel) and m = 1.6 × 103 cm2 /(V·s) (lower panel).
At low frequencies and high doping values, h ¯ ω µ, the intraband transitions dominate 2
and the conductivity can be written as follows,
σ(ω) ≈ σintra (ω) ≈
i 2ie2 t 1 e2 µ = . 2 h ¯ π [Ω + iΓ] 2t π¯h ω + i/τ
(3)
Defining α = e2 µ/π¯h2 and γg = 1/τ we arrive to Eq. 1 in the main text,
σg =
α γg − 2πif
(4)
This Drude-form conductivity is a very good approximation at low frequencies, where absorption in graphene comes only from scattering losses, γg . This can be appreciated in Fig. S1, where the Drude model is compared to the full RPA conductivity for a high value of the chemical potential (µ = 0.65 eV) and for two different values of the mobility, m = 104 cm2 /(V·s) as for instance in Ref. 2 (τ = 0.7 ps, upper panel), and m = 1.6 × 103 cm2 /(V·s). Note that this last data set corresponds to the experimental data in Ref. 3 , and was also used in the theoretical study in Ref. 4 (τ = 0.1 ps, lower panel).
Transformation optics applied to graphene metasurfaces Conductivity modulation In Fig. S2 we sketch a structure to generate the periodically modulated doping in a graphene sheet. The graphene is placed on top of a dielectric spacer (not shown for clarity) that lies on top of a wafer with a periodically structured surface. This surface can be metallic or dielectric, and even a highly resistive material transparent at the appropriate frequency range could be used. The electrostatic potential generated by the structured plane reproduces the periodic patterning of the film. By placing the graphene at distance above the plane larger than the corrugation period, the shape of the electrostatic potential will smooth out to a sinusoidal, as the potential will be given by a fourier series and higher order modes decay as exp(−gx). Note that this plane does not affect the near field on the graphene as the plasmons
3
σg
a=2πγ
z y
x
Figure S2: Periodically patterned plane to generate a periodic conductivity modulation in the graphene.
are confined to the sheet at distances much less than the distance to the corrugated plane. Transformation optics applied to graphene with conductivity modulation The application of transformation optics to a modulated graphene sheet is detailed in Ref. 5. Here we only reproduce its main details needed for completeness of the main text. The conformal transformation, given by Eq. (3) of the main text maps a flat and homogeneously doped graphene sheet into a flat graphene with periodic conductivity modulation σ(x) that depends on the modulation strength, w0 , and whose period is fixed by γ. A Fourier expansion of the transformation shows that the conductivity retrieved from Eq. (3) is in fact a sinusoidal one to a very good approximation, see Eq. (2). The value of modulation parameters {a0 , a1 } in Eq. (2) for different values of w0 in Eq. (3) is given in Table 1 Table 1: Values for {a0 , a1 } in Eq. (2) of the main text for different parameters of the modulation strength, w0 . w0 a0 a1
1.0
1.5
2.0
2.5
3.0
1.1048 1.2513 1.4973 1.9035 2.6237 0.4696 0.7538 1.1144 1.6196 2.4225
Graphene with 1D conductivity modulation Figure S3 presents results for the same metasurface considered in Figs. 2 and 3 of the main text, but for a graphene with lower mobilities, m = 1.6 × 103 cm2 /(V·s), as reported in 4
μ = 0.65 eV, γg = 9.8 THz
γg = 19.6 THz
γg = 9.8 THz
w0 = 2.5
w0 = 2.5
γ = 400 nm 0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1 0
0
Re(σ)
Re(σ)
Im(σ)
Im(σ)
Figure S3: Metasurface with 1D conductivity modulation for more lossy graphene.
experimental works. Compared to the plots shown in the main text, when γg = 1.5 THz is increased to γg = 1.5 THz, the peaks broaden and a stronger modulation is needed to suppress transmittance. However, all the main conclusions derived from our work are still valid: all resonant peaks appear at the same frequency and there is a peak splitting away from normal incidence. It is interesting to note that the dependence of the effective conductivity on the angle of incidence can be virtually removed by considering a larger collision frequency,
Transmittance/Absorption
γg = 1.5 THz
γg = 9.8 THz
1 0.8 w0 = 1
0.6
w0 = 1.5 w0 = 2
0.4
p-polarization μ = 0.65 eV 2πγ = 1.25μm
w0 = 1 w0 = 1.5 w0 = 2 w0 = 2.5
w0 = 2.5
0.2 0 27
w0 = 3
w0 = 3
28
29
30
31
32
33 27
f (THz)
28
29
30
31
32
33
f (THz)
Figure S4: Metasurface with 2D modulation: transmittance and absorption for the lower frequency mode. Two different loss parameters are considered (left and right panel).
5
γg = 19.6 THz, as the line width of each mode is increased while their separation, fixed by our transformation optics recipe, is maintained.
Graphene with 2D conductivity modulation Here we complement the results of the graphene metasurface with 2D conductivity modulation. First, we justify the value of the modulation strength used in the main text, w0 = 2.5. From the simulation results presented in Fig. S4 we see that the value w0 = 2.5 minimises transmittance as well as maximises absorption on resonance (T ≈ 0.3 and Q = 0.5). Increasing the losses has the effect of broadening the resonances (see right-hand side panel). In Fig. S5 we provide a full spectrum for the metasurface considered in Fig. 4 of the main text. Panel (a) shows transmittance and absorbance, panel (b) reflectance, and panel (c) displays the field patterns at each resonance peak. From panel (a) it is evident that the strongest resonance is provided by the two lower energy modes, that correspond to the splitting of the electrical dipole peak, as confirmed from the field patterns in panel (c), first and second columns. Then, peaks 3 and 5 and peaks 4 and 6 are the split pairs of multipolar peaks. The reflectance spectrum features a Fano shape for all the peaks, related to the fact that all the modes arise from the interaction between the localised dipoles and plasmon polariton sustained by the continuous graphene sheet. Figure S6 complements Fig. 5 in the main text by showing the spectrum and effective conductivity of the metasurface for graphene with conductivity parameters taken from the experimental work. 3 The effect of increasing scattering losses in the graphene is to broaden the resonance peaks and reduce their height. On the other hand, in this case the retrieved effective conductivity is virtually independent of the incidence angle, θ, the azhimutal angle, φ, or the polarisation. Finally, in Fig. S7 we show results for the same graphene metasurface but with a deeply 6
(a)
(b)
Transmittance/Absorbance
1
Reflectance
p-polarization μ = 0.65 eV 2πγ = 1.25μm θ = 0, Φ = 0
0.2 0.8 0.15 0.6
2
1
0.1
0.4
3
0.2
4
0.05
6
5
0 30
35
40
45
0
50
30
35
40
f (THz)
1
(c)
45
50
f (THz)
2
3
4
5
6
Ex max
Ey
0
min
Hz
Figure S5: Metasurface with 2D modulation: transmittance (blue) and absorption (red) under p-polarized incident light for the same parameters as in Fig. 4 of the main text. (a) Reflectance and absorbance. (b) Reflectance. (c) Electric field patterns at all resonance peaks. The colour scale was chosen to reveal the symmetry patterns of the fields and has a different range for each plot. In all cases red (blue) corresponds to the maximum (minimum) field value.
subwavelength unit cell. In order to compare with the results discussed in Ref. 2 for graphene nano islands, we choose a unit cell size of size 2πγ = 314 nm and a chemical potential of µ = 0.4 eV while we keep the value of the mobility used in our main text and in that work, m = 104 cm2 /(V·s). For the Drude formula given by Eq. (1) of the main text, this set of parameters corresponds to α = 45 THz and γg = 2 THz. In panel (a) we see the same splitting between the two dipole peaks discussed as for the metasurface with 2πγ = 1.25 µm, accompanied by similar Fano shapes in the reflectance spectrum. On the other hand, panel (b) presents a zoom of the spectrum of the lower energy mode including different angles of incidence, which shows a virtually perfect agreement in the retrieved conductivities
7
p-polarization
s-polarization
Trans / Abs
w0 = 2.5 μ = 0.65 eV 2πγ = 1.25μm γg = 9.8 THz
Φ=0
θ=0
Figure S6: Metasurface with 2D conductivity modulation for lossy graphene. All the parameters are the same as those for Fig. 5 of the main text except γg , which changes from 1.5 THz to 9.8 THz. Both polarisations are shown (p – left column–, s –right column–). The first row shows transmittance and absorbance, and the second and third rows show effective conductivities at fixed φ and fixed θ, respectively.
8
(bottom inset). In this case, λ0 ∼ 20 × 2πγ. Lastly, the higher energy peak, see panel (c), is characterised by an effective conductivity that is more affected by the change of incidence angle. Note that while only p-polarisation is shown, given all the results in this paper, we
(a)
Transmittance/Absorption
expect the same response for s-polarisation also in this case. 1
0.8
p-polarization
(b) Peak 1
(c) Peak 2
μ = 0.4 eV 2πγ = 314 nm
0.6 0.4 0.2
0 0.06
T Q Ex
Ey
Reflectance
0.05
Ex
Ey
0.04
Hz
0.03
Hz
0.02 0.01 0
Re(σeff) Im(σeff)
Figure S7: Graphene with 2D conductivity modulation in the deeply subwavelength regime.
It is worth comparing the last results with a graphene metasurface where isolated islands are used as building blocks. Based on the structure considered in Ref. 2, we consider a square array of graphene nano islands but with two islands per unit cell, see Fig. S8. We take a similar periodicity as for the metasurface in Fig. S7, 2πγ = 200 nm in this case, and chose a disk diameter of 60 nm, such that this structure is resonant at similar frequencies than that studied in Fig. S6. As a difference with the metasurfaces based on modulated considered here, in this case there is no splitting of the electrical dipole mode since the islands are not connected and there is not a continuum to support a propagating mode. On the other hand, the performance in terms of transmission suppression and absorbance enhancement is quite similar for the two devices, both of them showing a virtually perfect onmidirectionality for these deep subwavelength unit cells. 9
Transmittance/Re ectance
Nanoislands 1
0.6 0.4 0.2
p-polarization μ = 0.4 eV 2πγ = 200 nm D = 60 nm
Hx
Ey
Hy
Ez
Hz
0 0.5
0.4
Absorption
Ex
0.8
0.3 0.2 0.1 10 0
Re ectance
0
10 -2 10 -4 10 -6 10 -8
0
10
20
30
40
50
60
70
f (THz)
Figure S8: Square array of graphene nanoislands with two islands per unit cell.
References 1. Wunsch, B.; Stauber, T.; Sols, F.; Guinea, F. Dynamical Polarization of Graphene at Finite Doping. New J. Phys. 2006, 8, 318–318. 2. Thongrattanasiri, S.; Koppens, F. H. L.; Garc´ıa De Abajo, F. J. Complete Optical Absorption in Periodically Patterned Graphene. Phys. Rev. Lett. 2012, 108, 047401. 3. Yan, H.; Li, X.; Chandra, B.; Tulevski, G.; Wu, Y.; Freitag, M.; Zhu, W.; Avouris, P.; Xia, F. Tunable Infrared Plasmonic Devices Using Graphene/Insulator Stacks. Nat. Nanotechnol. 2012, 7, 330–334. 4. Fan, Y.; Shen, N.-H.; Koschny, T.; Soukoulis, C. M. Tunable Terahertz Meta-Surface with Graphene Cut-Wires. ACS Photonics 2015, 2, 151–156. 5. Huidobro, P. A.; Kraft, M.; Kun, R.; Maier, S. A.; Pendry, J. B. Graphene, Plasmons and Transformation Optics. J. Opt. 2016, 18, 44024. 10
∗
To whom correspondence should be addressed
1
Graphene’s conductivity Within the random phase approximation, the conductivity of graphene is written as a sum of intraband and interband contributions, σ = σintra + σinter , as follows 1
σintra σinter
1 2ie2 t ln 2 cosh , = h ¯ π [Ω + iΓ] 2t e2 1 1 Ω−2 i (Ω + 2)2 = + arctan − ln . 4¯ h 2 π 2t 2π (Ω − 2)2 + (2t)2
(1) (2)
where Ω = h ¯ ω/µ and t = kB T /µ are frequency and temperature normalized to the chemical potential, respectively. The normalized damping term is Γ = h ¯ /(µτ ), where τ = mµ/vF2 is the carriers’ scattering time (m is the mobility and vF the Fermi velocity).
(a)
μ = 0.65 eV, m =104 cm2/(Vs) α = 76 GHz/Ω, β = 1.5THz
(b) μ = 0.65 eV, m =1.6x103 cm2/(Vs) α = 76 GHz/Ω, β = 9.8THz
f (THz)
Figure S1: Graphene’s conductivity for the RPA model (solid lines) and for the Drude approximation (dashed lines). The conductivity for two sets of parameters used in the paper are plotted, with a mobilities of m = 104 cm2 /(V·s) (upper panel) and m = 1.6 × 103 cm2 /(V·s) (lower panel).
At low frequencies and high doping values, h ¯ ω µ, the intraband transitions dominate 2
and the conductivity can be written as follows,
σ(ω) ≈ σintra (ω) ≈
i 2ie2 t 1 e2 µ = . 2 h ¯ π [Ω + iΓ] 2t π¯h ω + i/τ
(3)
Defining α = e2 µ/π¯h2 and γg = 1/τ we arrive to Eq. 1 in the main text,
σg =
α γg − 2πif
(4)
This Drude-form conductivity is a very good approximation at low frequencies, where absorption in graphene comes only from scattering losses, γg . This can be appreciated in Fig. S1, where the Drude model is compared to the full RPA conductivity for a high value of the chemical potential (µ = 0.65 eV) and for two different values of the mobility, m = 104 cm2 /(V·s) as for instance in Ref. 2 (τ = 0.7 ps, upper panel), and m = 1.6 × 103 cm2 /(V·s). Note that this last data set corresponds to the experimental data in Ref. 3 , and was also used in the theoretical study in Ref. 4 (τ = 0.1 ps, lower panel).
Transformation optics applied to graphene metasurfaces Conductivity modulation In Fig. S2 we sketch a structure to generate the periodically modulated doping in a graphene sheet. The graphene is placed on top of a dielectric spacer (not shown for clarity) that lies on top of a wafer with a periodically structured surface. This surface can be metallic or dielectric, and even a highly resistive material transparent at the appropriate frequency range could be used. The electrostatic potential generated by the structured plane reproduces the periodic patterning of the film. By placing the graphene at distance above the plane larger than the corrugation period, the shape of the electrostatic potential will smooth out to a sinusoidal, as the potential will be given by a fourier series and higher order modes decay as exp(−gx). Note that this plane does not affect the near field on the graphene as the plasmons
3
σg
a=2πγ
z y
x
Figure S2: Periodically patterned plane to generate a periodic conductivity modulation in the graphene.
are confined to the sheet at distances much less than the distance to the corrugated plane. Transformation optics applied to graphene with conductivity modulation The application of transformation optics to a modulated graphene sheet is detailed in Ref. 5. Here we only reproduce its main details needed for completeness of the main text. The conformal transformation, given by Eq. (3) of the main text maps a flat and homogeneously doped graphene sheet into a flat graphene with periodic conductivity modulation σ(x) that depends on the modulation strength, w0 , and whose period is fixed by γ. A Fourier expansion of the transformation shows that the conductivity retrieved from Eq. (3) is in fact a sinusoidal one to a very good approximation, see Eq. (2). The value of modulation parameters {a0 , a1 } in Eq. (2) for different values of w0 in Eq. (3) is given in Table 1 Table 1: Values for {a0 , a1 } in Eq. (2) of the main text for different parameters of the modulation strength, w0 . w0 a0 a1
1.0
1.5
2.0
2.5
3.0
1.1048 1.2513 1.4973 1.9035 2.6237 0.4696 0.7538 1.1144 1.6196 2.4225
Graphene with 1D conductivity modulation Figure S3 presents results for the same metasurface considered in Figs. 2 and 3 of the main text, but for a graphene with lower mobilities, m = 1.6 × 103 cm2 /(V·s), as reported in 4
μ = 0.65 eV, γg = 9.8 THz
γg = 19.6 THz
γg = 9.8 THz
w0 = 2.5
w0 = 2.5
γ = 400 nm 0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1 0
0
Re(σ)
Re(σ)
Im(σ)
Im(σ)
Figure S3: Metasurface with 1D conductivity modulation for more lossy graphene.
experimental works. Compared to the plots shown in the main text, when γg = 1.5 THz is increased to γg = 1.5 THz, the peaks broaden and a stronger modulation is needed to suppress transmittance. However, all the main conclusions derived from our work are still valid: all resonant peaks appear at the same frequency and there is a peak splitting away from normal incidence. It is interesting to note that the dependence of the effective conductivity on the angle of incidence can be virtually removed by considering a larger collision frequency,
Transmittance/Absorption
γg = 1.5 THz
γg = 9.8 THz
1 0.8 w0 = 1
0.6
w0 = 1.5 w0 = 2
0.4
p-polarization μ = 0.65 eV 2πγ = 1.25μm
w0 = 1 w0 = 1.5 w0 = 2 w0 = 2.5
w0 = 2.5
0.2 0 27
w0 = 3
w0 = 3
28
29
30
31
32
33 27
f (THz)
28
29
30
31
32
33
f (THz)
Figure S4: Metasurface with 2D modulation: transmittance and absorption for the lower frequency mode. Two different loss parameters are considered (left and right panel).
5
γg = 19.6 THz, as the line width of each mode is increased while their separation, fixed by our transformation optics recipe, is maintained.
Graphene with 2D conductivity modulation Here we complement the results of the graphene metasurface with 2D conductivity modulation. First, we justify the value of the modulation strength used in the main text, w0 = 2.5. From the simulation results presented in Fig. S4 we see that the value w0 = 2.5 minimises transmittance as well as maximises absorption on resonance (T ≈ 0.3 and Q = 0.5). Increasing the losses has the effect of broadening the resonances (see right-hand side panel). In Fig. S5 we provide a full spectrum for the metasurface considered in Fig. 4 of the main text. Panel (a) shows transmittance and absorbance, panel (b) reflectance, and panel (c) displays the field patterns at each resonance peak. From panel (a) it is evident that the strongest resonance is provided by the two lower energy modes, that correspond to the splitting of the electrical dipole peak, as confirmed from the field patterns in panel (c), first and second columns. Then, peaks 3 and 5 and peaks 4 and 6 are the split pairs of multipolar peaks. The reflectance spectrum features a Fano shape for all the peaks, related to the fact that all the modes arise from the interaction between the localised dipoles and plasmon polariton sustained by the continuous graphene sheet. Figure S6 complements Fig. 5 in the main text by showing the spectrum and effective conductivity of the metasurface for graphene with conductivity parameters taken from the experimental work. 3 The effect of increasing scattering losses in the graphene is to broaden the resonance peaks and reduce their height. On the other hand, in this case the retrieved effective conductivity is virtually independent of the incidence angle, θ, the azhimutal angle, φ, or the polarisation. Finally, in Fig. S7 we show results for the same graphene metasurface but with a deeply 6
(a)
(b)
Transmittance/Absorbance
1
Reflectance
p-polarization μ = 0.65 eV 2πγ = 1.25μm θ = 0, Φ = 0
0.2 0.8 0.15 0.6
2
1
0.1
0.4
3
0.2
4
0.05
6
5
0 30
35
40
45
0
50
30
35
40
f (THz)
1
(c)
45
50
f (THz)
2
3
4
5
6
Ex max
Ey
0
min
Hz
Figure S5: Metasurface with 2D modulation: transmittance (blue) and absorption (red) under p-polarized incident light for the same parameters as in Fig. 4 of the main text. (a) Reflectance and absorbance. (b) Reflectance. (c) Electric field patterns at all resonance peaks. The colour scale was chosen to reveal the symmetry patterns of the fields and has a different range for each plot. In all cases red (blue) corresponds to the maximum (minimum) field value.
subwavelength unit cell. In order to compare with the results discussed in Ref. 2 for graphene nano islands, we choose a unit cell size of size 2πγ = 314 nm and a chemical potential of µ = 0.4 eV while we keep the value of the mobility used in our main text and in that work, m = 104 cm2 /(V·s). For the Drude formula given by Eq. (1) of the main text, this set of parameters corresponds to α = 45 THz and γg = 2 THz. In panel (a) we see the same splitting between the two dipole peaks discussed as for the metasurface with 2πγ = 1.25 µm, accompanied by similar Fano shapes in the reflectance spectrum. On the other hand, panel (b) presents a zoom of the spectrum of the lower energy mode including different angles of incidence, which shows a virtually perfect agreement in the retrieved conductivities
7
p-polarization
s-polarization
Trans / Abs
w0 = 2.5 μ = 0.65 eV 2πγ = 1.25μm γg = 9.8 THz
Φ=0
θ=0
Figure S6: Metasurface with 2D conductivity modulation for lossy graphene. All the parameters are the same as those for Fig. 5 of the main text except γg , which changes from 1.5 THz to 9.8 THz. Both polarisations are shown (p – left column–, s –right column–). The first row shows transmittance and absorbance, and the second and third rows show effective conductivities at fixed φ and fixed θ, respectively.
8
(bottom inset). In this case, λ0 ∼ 20 × 2πγ. Lastly, the higher energy peak, see panel (c), is characterised by an effective conductivity that is more affected by the change of incidence angle. Note that while only p-polarisation is shown, given all the results in this paper, we
(a)
Transmittance/Absorption
expect the same response for s-polarisation also in this case. 1
0.8
p-polarization
(b) Peak 1
(c) Peak 2
μ = 0.4 eV 2πγ = 314 nm
0.6 0.4 0.2
0 0.06
T Q Ex
Ey
Reflectance
0.05
Ex
Ey
0.04
Hz
0.03
Hz
0.02 0.01 0
Re(σeff) Im(σeff)
Figure S7: Graphene with 2D conductivity modulation in the deeply subwavelength regime.
It is worth comparing the last results with a graphene metasurface where isolated islands are used as building blocks. Based on the structure considered in Ref. 2, we consider a square array of graphene nano islands but with two islands per unit cell, see Fig. S8. We take a similar periodicity as for the metasurface in Fig. S7, 2πγ = 200 nm in this case, and chose a disk diameter of 60 nm, such that this structure is resonant at similar frequencies than that studied in Fig. S6. As a difference with the metasurfaces based on modulated considered here, in this case there is no splitting of the electrical dipole mode since the islands are not connected and there is not a continuum to support a propagating mode. On the other hand, the performance in terms of transmission suppression and absorbance enhancement is quite similar for the two devices, both of them showing a virtually perfect onmidirectionality for these deep subwavelength unit cells. 9
Transmittance/Re ectance
Nanoislands 1
0.6 0.4 0.2
p-polarization μ = 0.4 eV 2πγ = 200 nm D = 60 nm
Hx
Ey
Hy
Ez
Hz
0 0.5
0.4
Absorption
Ex
0.8
0.3 0.2 0.1 10 0
Re ectance
0
10 -2 10 -4 10 -6 10 -8
0
10
20
30
40
50
60
70
f (THz)
Figure S8: Square array of graphene nanoislands with two islands per unit cell.
References 1. Wunsch, B.; Stauber, T.; Sols, F.; Guinea, F. Dynamical Polarization of Graphene at Finite Doping. New J. Phys. 2006, 8, 318–318. 2. Thongrattanasiri, S.; Koppens, F. H. L.; Garc´ıa De Abajo, F. J. Complete Optical Absorption in Periodically Patterned Graphene. Phys. Rev. Lett. 2012, 108, 047401. 3. Yan, H.; Li, X.; Chandra, B.; Tulevski, G.; Wu, Y.; Freitag, M.; Zhu, W.; Avouris, P.; Xia, F. Tunable Infrared Plasmonic Devices Using Graphene/Insulator Stacks. Nat. Nanotechnol. 2012, 7, 330–334. 4. Fan, Y.; Shen, N.-H.; Koschny, T.; Soukoulis, C. M. Tunable Terahertz Meta-Surface with Graphene Cut-Wires. ACS Photonics 2015, 2, 151–156. 5. Huidobro, P. A.; Kraft, M.; Kun, R.; Maier, S. A.; Pendry, J. B. Graphene, Plasmons and Transformation Optics. J. Opt. 2016, 18, 44024. 10
