Nonlocal optical properties in periodic lattice of graphene layers

Ruey-Lin Chern; Dezhuan Han

doi:10.1364/OE.22.004817

1. Introduction

Graphene is a single layer of carbon atoms packed into a honeycomb lattice, which exhibits extraordinary electronic transport properties as strong ambipolar electric field effect [1], massless Dirac Fermions [2], and half-integer quantum Hall effect [3]. It is considered the thinnest material that combines aspects of semiconductors and metals with ultrahigh electron mobility [4] and superior thermal conductivity [5]. These properties are exploited in a variety of applications as transparent electrodes [6], ultrafast photodetectors and lasers [7, 8], broadband optical modulators [9], and even metamaterials [10, 11].

The optical properties of a single layer graphene are mainly determined by its sheet conductivity. A universal optical conductance is found for undoped graphene and, interestingly, is determined by the fine structure constant [12–15]. By applying a gate voltage or doping in graphene, the optical properties can be modified due to a shift in the onset of interband transition [16, 17]. In particular, plasmons relations are identified in graphene layers and graphene plasmons provide a suitable alternative to metal plasmons [18–22]. Compared to the latter, graphene plasmons exhibit much tighter field confinement and relatively longer propagation distances, with the advantage of being highly tunable through, for instance, the electrostatic gating [23].

If a number of decoupled graphene layers are arranged as a periodic lattice [24–26], with the period much less than the wavelength, the collection of layers can be regarded as a medium with the optical properties characterized by its effective parameters. Considering the effective medium to be nonmagnetic (the effective permeability being unity) as a viewpoint, the effective permittivity is likely to be spatially dispersive or nonlocal, as in the crystals with excitations [27, 28]. This is true even when the wavelength is much greater than the characteristic length (e.g. lattice period) of the medium [29, 30], although the nonlocal effects are usually not important at long wavelengths. In a periodic structure, the nonlocal effects arise from the coupling of fields between neighboring unit cells and tend to be significant when the resonance (e.g. surface plasmon) occurs. These features are present in metal layers [31, 32] and expected to appear as well in graphene layers.

In the present study, we investigate the nonlocal optical properties in the periodic lattice of graphene layers, with emphasis on the concept of effective medium. Based on a nonlocal effective medium model, the effective permittivity tensor for the graphene layers is derived and expressed in approximate formulas that show explicit dependence on the wave vector. Strong nonlocal effects are indicated by the Lorentzian character of the effective permittivity and are manifest on the emergence of additional wave and the occurrence of negative refraction in the graphene layers. These nonlocal properties are attributed to the excitation of plasmon polaritons, in the form of nonlocal resonance, on the graphene surface that strongly modulate the dispersion of the lattice.

2. Effective medium model

The basic idea of the effective medium model in this study is to extract the effective parameters of a structure from its dispersion relations. This is achieved when analytical relations are available for the structure. A simple way to extract the effective parameters is by expanding the dispersion relations with respect to the wave number [32, 33]. Nonlocal effective parameters, in particular, can be approximately obtained when the expansion order is larger than two, the order for a local medium. For periodic structures, the effective medium model is valid provided that the period is much less than the wavelength.

2.1. Dispersion relations

Consider a periodic lattice of graphene layers with surface conductivity σ and period a embedded in a background with dielectric constant ε, as schematically shown in Fig. 1. Let the wave vector lie on the xz plane, that is, k = (k_x, 0, k_z), without loss of generality. With the time-harmonic dependence e^−iωt, the dispersion relations of the graphene layers are given by (see Appendix for details)

cos (k_{x} a) = cos (q a) - \frac{i σ q}{2 ω ε_{0} ε} sin (q a),

cos (k_{x} a) = cos (q a) - \frac{i σ ω μ_{0}}{2 q} sin (q a),

for TM and TE polarizations, respectively, where

q = \sqrt{ε k_{0}^{2} - k_{z}^{2}}

with k₀ = ω/c. Here, TM refers to transverse magnetic, where E = (E_x, 0, E_z) and H = (0, H_y, 0), and TE to transverse electric, where E = (0, E_y, 0) and H = (H_x, 0, H_z). The relations (1)–(2) are equivalent to those obtained by the transfer matrix method [24, 26].

Fig. 1 Schematic diagram of the periodic lattice of graphene layers with period a embedded in a background with dielectric constant ε. In this study, a = 0.1 h̄c/μ (≈ 98.9 nm) with μ = 0.2 eV and ε = 1.5 will be used as the parameters. A small area of the graphene feature is shown on the top layer for illustration.

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At sufficiently low temperature, kT ≪ μ, where k is the Boltzmann constant, T is the temperature, and μ is the chemical potential of graphene, the surface conductivity of graphene is given, within the random phase approximation, by the following relation [34–38]:

\frac{σ}{ε_{0} c} = 4 α \frac{i}{Ω} + π α [θ (Ω - 2) + \frac{i}{π} ln | \frac{Ω - 2}{Ω + 2} |],

where Ω = h̄ω/μ is the dimensionless frequency, h̄ is the reduced Planck constant, α ≈ 1/137 is the fine structure constant, and θ(x) is the Heaviside step function. The first and second terms on the right side of Eq. (3) stem from the intraband and interband contributions, respectively. The intraband conductivity is of the Drude-like form as in the metal. The imaginary part of σ, however, become negative for Ω > Ω^* ≈ 1.667 due to the interband contribution [38]. For Ω > 2, σ is no longer purely imaginary, leading to the absorption of light in graphene.

2.2. Effective permittivity tensor

Assume that the lattice period a is much less than the wavelength λ, so that the graphene layers can be regarded as an effective medium, characterized by the dispersion relations of a homogeneous yet anisotropic medium as

\frac{k_{x}^{2}}{ε_{z}^{eff}} + \frac{k_{z}^{2}}{ε_{x}^{eff}} = k_{0}^{2},

k_{x}^{2} + k_{z}^{2} = ε_{y}^{eff} k_{0}^{2}

for TM and TE polarizations, respectively, where

ε_{x}^{eff}

,

ε_{y}^{eff}

, and

ε_{z}^{eff}

are components of the effective permittivity tensor. By expanding the dispersion relations of the graphene layers [Eqs. (1) and (2)] with respect to k₀, k_x, and k_z up to fourth order, and rearranging the expanding terms according to the forms in Eqs. (4) and (5), we arrive at the approximate formulas for the effective permittivity components as

ε_{z}^{eff} = \frac{ε_{z}^{0} - \frac{γ}{12} k_{0}^{2} a^{2}}{1 - \frac{1}{12} k_{x}^{2} a^{2}}, ε_{x}^{eff} = \frac{ε (1 - \frac{γ}{12 ε_{z}^{0}} k_{0}^{2} a^{2})}{1 - \frac{γ}{6 ε_{z}^{0}} (k_{0}^{2} a^{2} - \frac{1}{2 ε} k_{z}^{2} a^{2})},

ε_{y}^{eff} = ε_{y}^{0} (1 + \frac{1}{6} k_{z}^{2} a^{2}) - \frac{δ}{6} ε^{2} k_{0}^{2} a^{2} + \frac{a^{2}}{12 k_{0}^{2}} (k_{x}^{4} - k_{z}^{4}),

where

ε_{z}^{0} = ε_{y}^{0} = ε (1 + δ)

, γ = ε² (1 + 2δ), δ = iσ̃/(εk₀a), and σ̃ = σ/(ε₀c).

In this study, we choose a = 0.1 h̄c/μ (≈ 98.9 nm) with μ = 0.2 eV and ε = 1.5 to be the geometric and material parameters. The wavelength, λ = 2πh̄c/(μΩ), corresponds to the interband transition (Ω = 2) is around 3.1 μm. The working wavelength is therefore much greater than the lattice period up to the interband transition, which fulfills the necessary condition for the effective medium to be valid. Theoretically, the expansion is valid when the wave vector component is small. In practice, the effective permittivity tensor based on this expansion successfully recovers the dispersion relations even when the wave vector component is no longer small if the expansion order is suitably increased.

The in-plane effective permittivity component $ε_{z}^{eff}$ [cf. Eq. (6)], which is located in the plane where the wave vector lies, depends on the wave vector component k_x, indicating the nonlocal nature of the graphene layers. The nonlocal effect, however, is weak, as |k_x| is bound by π/a in a periodic lattice. It is shown in Fig. 2(a) that $ε_{z}^{eff}$ (blue solid line) differs not much from $ε_{z}^{0}$ (blue dashed line). Here, $ε_{z}^{0}$ (or $ε_{y}^{0}$ ) is considered the quasistatic effective parameters along the parallel (to graphene surface) direction. This parameter exhibits the Drude-like behavior as in the metal, with an effective plasma frequency Ω₀ determined by the zero of $ε_{z}^{0}$ , that is, $ε_{z}^{0} (Ω_{0}) = 0$ . Using Eq. (3) in $ε_{z}^{0}$ , we have

Ω_{0} \approx 2 {(1 + \frac{ε \tilde{a}}{α})}^{- 1 / 2},

where ã = μa/(h̄c) is the dimensionless lattice period. Here, Ω₀ is very close to the zero of

ε_{z}^{eff}

and Eq. (8) also serves as a good approximation to

ε_{z}^{eff} (Ω_{0}) = 0

, as shown in Fig. 2(b).

Fig. 2 (a) In-plane effective permittivity components $ε_{z}^{eff}$ and $ε_{x}^{eff}$ as the functions of Ω for the same periodic lattice of graphene layers as in Fig. 1, where Ω = h̄ω/μ, K_x = k_xa, K_z = k_za, and ã = μa/(h̄c). (b) Effective plasma frequency Ω₀ and its approximate formula [Eq. (8)] as the functions of ã.

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Another in-plane component $ε_{x}^{eff}$ (green solid line), on the other hand, distinctly deviates from its quasistatic parameter, $ε_{x}^{0} = ε$ (green dashed line). In particular, there exists a pole frequency Ω_p, where $ε_{x}^{eff} \approx \pm \infty$ . Around Ω_p, $ε_{x}^{eff}$ exhibits a Lorentzian resonance feature [cf. Fig. 2(a)]. Using Eq. (3) in $ε_{x}^{eff}$ , we have at small K_z

Ω_{p} \approx 2 {[1 + \frac{ε \tilde{a} (12 + K_{z}^{2})}{2 α (6 + K_{z}^{2})}]}^{- 1 / 2},

where K_z = k_za is the dimensionless wave number. The pole frequency Ω_p is very close to Ω₀ at K_z = 0 and gradually moves toward higher frequencies as K_z increases (also shown in Fig. 6). As there is no constraint for K_z, the nonlocal effects due to

ε_{x}^{eff}

are expected to be strong in the graphene layers, as in the lattice of metal layers [32].

The out-of-plane effective permittivity component $ε_{y}^{eff}$ , which is perpendicular to the plane where the wave vector lies, depends on both k_x and k_z, as shown in Fig. 3. Note that $ε_{y}^{eff}$ and $ε_{z}^{eff}$ are no longer equal as in the quasistatic case, although the layered structure makes no difference between the y and z directions in the geometry. The medium properties, therefore, depend on the polarization. Near K_x = K_z = 0 (where K_x = k_xa), $ε_{y}^{eff}$ is characterized by the effective plasma frequency Ω₀ and the critical frequency Ω^* ≈ 1.667 [38]. Below Ω₀, $ε_{y}^{eff}$ is negative and Ω₀ also serves as the cutoff frequency of the graphene layers for TE polarization [cf. Eq. (5)]. Above Ω^*, where the imaginary part of σ is negative [38], $ε_{y}^{eff}$ is larger than the background dielectric constant. At K_x = 0, $ε_{y}^{eff}$ basically decreases with K_z [Fig. 3(a)], while at K_z = 0, $ε_{y}^{eff}$ increases with K_x [Fig. 3(b)]. Compared to $ε_{x}^{eff}$ , the nonlocal effect due to $ε_{y}^{eff}$ is weaker.

Fig. 3 Out-of-plane effective permittivity component $ε_{y}^{eff}$ for the same periodic lattice of graphene layers as in Fig. 1, where (a) K_x = 0 and (b) K_z = 0. Red and green lines correspond to $ε_{y}^{eff} = 0$ and $ε_{y}^{eff} = ε$ , respectively.

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Note also that there are no first-order k_x and k_z terms in all the effective permittivity components. This is a consequence of inversion symmetry of the underlying structure, an invariance property of a system when the coordinates are inverted [39, 40].

3. Nonlocal optical properties

The optical properties of the periodic lattice of graphene layers are closely related to its dispersion characteristics. In the present problem, the nonlocal effects, in particular, are manifest on the emergence of additional wave, the occurrence of negative refraction, and the excitation of graphene plasmons. These features are well characterized by the nonlocal effective parameters of the graphene layers derived in the preceding section.

3.1. Dispersion characteristics

Figure 4 shows the equifrequency surfaces of the dispersion relations for the same graphene layers as in Fig. 1. For TM polarization [Fig. 4(a)], the dispersion surface consists of two parts: the upper surface is the photonic mode, largely conformed to the light cone: $Ω \tilde{a} = \sqrt{(K_{z}^{2} + K_{x}^{2}) / ε}$ ; the lower surface is the polaritonic mode, arising from the mixing of light wave with the excitations (i.e. plasmons) in the graphene layers. The two modes intersect at a single point on either side of the half space (K_z > 0 or K_z < 0). Along the plane of constant K_x ≠ 0, the polaritonic band forms an anticrossing (avoided crossing) scheme with the photonic band, as shown in Fig. 5(a), indicating the existence of couplings between the two modes.

Fig. 4 Equifrequency surfaces of (a) TM and (b) TE dispersion relations for the same periodic lattice of graphene layers as in Fig. 1. Black circle in (b) is the section of light cone at Ω = 2.

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Fig. 5 (a) TM frequency bands at K_x = 0, 0.03 and (b) TE frequency band at K_x = 0 for the same periodic lattice of graphene layers as in Fig. 1. Insets in (a) are typical magnetic field patterns of photonic mode (P mode) and polaritonic mode (PL mode). Red dot in (b) corresponds to the critical frequency Ω^* ≈ 1.667 that separates P mode and N mode.

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The photonic and polaritonic bands, however, cross on the plane of K_x = 0 [dashed lines in Fig. 5(a)], which is considered a degeneracy due to symmetry. In this situation, the polaritonic mode possesses a completely different symmetry than the photonic mode. The former is a surface-like mode with odd symmetry (along the x direction) in the magnetic field, whereas the latter is a bulk mode with even symmetry [see the insets in Fig. 5(a)]. Here, the surface-like mode is not a surface mode in the usual sense. It is the surface part of the mode that exists in the bulk of the medium [41]. Notice that the crossing point for the photonic and polaritonic bands corresponds to the zero of $ε_{x}^{eff}$ , across which the Lorentzian resonance reverses the order of maximum and minimum. Therefore, the band crossing occurs when the numerator and denominator of $ε_{x}^{eff}$ become zero simultaneously. When K_x ≠ 0, the symmetries in the two modes are no longer completely different. A certain similar symmetry allows for the mode interaction and gives rise to the anticrossing scheme [42].

For TE polarization, the dispersion surface is approximately conformed to the light cone (similar to the photonic mode for TM polarization), but with a cutoff at the bottom, as in the waveguide or cavity mode. The cutoff frequency is very close to Ω₀ [cf. Eq. (8)], which does not scale like $1 / (\sqrt{ε} a)$ as in the case of periodic lattice of thin metal films [43], although the graphene conductivity exhibits the Drude-like character below the interband transition [cf. Eq. (3)]. A distinct feature is observed in the frequency range Ω^* ≈ 1.667 < Ω < 2, where the dispersion surface is located outside the light cone [cf. Fig. 4(b)], indicating that light wave in the graphene layers becomes a slow wave. In this range, the imaginary part of graphene conductivity becomes negative [cf. Eq. (3)], which is considered a very different property than the metal with a positive imaginary part of the conductivity. This wave has been identified as a new electromagnetic mode [denoted by N mode in Fig. 5(b)] in the two-dimensional electron gas for TE wave [38] that does not appear in metal. At the critical frequency Ω^*, the TE frequency band coincides with the light line [indicated by red dot in Fig. 5(b)].

For Ω > 2, the surface conductivity of graphene σ is no longer purely imaginary [cf. Eq. (3)], the dispersion relations of the graphene layers [Eqs. (1)–(2)] do not allow for real frequency with real wave vectors. The corresponding eigenmode is thus a virtual mode with finite lifetime. In this situation, light absorption occurs in the graphene layers.

3.2. Additional wave

The nonlocal optical properties of the periodic lattice of graphene layers come largely from the polaritonic mode. On the one hand, the polaritonic band at K_x = 0 coincides well with the pole frequency Ω_p of the effective permittivity component $ε_{x}^{eff}$ , as shown in Fig. 6. At larger values of K_z, however, a higher order of expansion on the dispersion relation [Eq. (1)] is necessary to deliver a more accurate formula for $ε_{x}^{eff}$ . The polaritonic band, or the permittivity pole, varies with the wave vector component K_z over a wide range of frequency from Ω₀ to Ω^*. This is considered an important feature that characterizes the nonlocal optical properties in graphene layers, as in the case of insulating crystals with excitons [44].

On the other hand, the nonlocal properties associated with the polaritonic band give rise to additional wave. For a wave of frequency Ω, there may exists two K_z components in the graphene layers for a given K_x: a smaller one with the photonic mode and a larger one with the polaritonic mode. This corresponds to the fact that there are two mechanisms of propagating energy in a nonlocal medium: one electromagnetic and one mechanical [44]. As a result, Maxwell’s boundary conditions are insufficient to solve the reflection and transmission coefficients for a nonlocal medium. The additional boundary condition is therefore needed to complete the problem [45].

Fig. 6 Pole frequency Ω_p for $ε_{x}^{eff}$ and its approximate formula [Eq. (9)] as the functions of K_z for the same periodic lattice of graphene layers as in Fig. 1. Black solid lines are photonic and polaritonic modes.

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3.3. Negative refraction

The nonlocal nature of the polaritonic mode and the associated additional wave results in strong modulation of fields, leading to negative refraction in the graphene layers. In a nonlocal medium with the permittivity tensor ε_ij, the time-averaged Poynting vector is written as [46]:

S = \frac{1}{2} Re [E \times H^{*}] - \frac{ω}{4} \frac{\partial ε_{i j}}{\partial k} E_{i} E_{j}^{*} .

The energy transport is modified by an extra term related to the change rate of the permittivity tensor with respect to the wave vector, which is also termed as the mechanical Poynting vector [47]. The energy flow may deviate much from the electromagnetic Poynting vector when the nonlocal effect is significant. Consider a wave incident from vacuum onto the graphene layers with the interface located on the xy plane. The dispersion curves on the wave vector domain, along with the Poynting and wave vectors, are plotted in Fig. 7. Above the frequency Ω₀ ≈ 0.4306, the dispersion curves exhibit strong modulations and separate into two parts [Fig. 7(a)]. The inner part belongs to the photonic mode and becomes flattened in shape. The outer part belongs to the polaritonic mode and orients in a manner that for a wave incident from vacuum with a certain angle of incidence, the Poynting vector in the medium orients toward the different side of the interface normal (the z axis) than the wave vector, which indicates the occurrence of negative refraction.

Fig. 7 Dispersion curves on the wave vector domain at (a) Ω = 0.435 and (b) Ω = 0.35 for the same periodic lattice of graphene layers as in Fig. 1. Black and gray contours are equifrequency curves for vacuum and graphene layers, respectively. Dashed lines indicate the continuity of K_x at the interface.

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If is worthy of noting that the negative refraction in the underlying structure differs from that in the left-handed metamaterial with a negative refractive index [48], which is accompanied by a backward wave (K_z < 0). Here, the graphene layers are highly anisotropic and the negative refraction occurs as a forward wave (K_z > 0). The negative refraction, however, is different from that in an ordinary (local) anisotropic medium with a hyperbolic dispersion relation, where the normal (to interface) permittivity component $ε_{z}^{eff} < 0$ and the tangential component $ε_{x}^{eff} > 0$ [49]. In Fig. 7(a), where Ω = 0.435 is slightly above Ω₀, the negative refraction in graphene layers occurs when $ε_{z}^{eff} > 0$ and $ε_{x}^{eff} < 0$ (see the insets). In a local medium, the same condition would result in a backward wave with ordinary (positive) refraction.

The above feature can be explained by the parabolic-like dispersion associated with the polaritonic mode (or additional wave), which is characterized by a quartic curve (in the xy plane) [32]:

\frac{x^{2} (1 - ε x^{2})}{a^{2}} + \frac{y^{2}}{b^{2}} = 1,

where a and b are constants, and ε is a small positive number. The parabolic-like character of the quartic curve comes from the x⁴ term at larger x, corresponding to the wave vector dependence of the effective permittivity in the present problem. At smaller x, on the other hand, the dispersion is dominant by the elliptic-like character. In Fig. 7(b), where Ω = 0.35 is well below Ω₀, the nonlocal effect is weak and the negative refraction is basically similar to that in a local medium (see the insets for

ε_{z}^{eff}

and

ε_{x}^{eff}

).

3.4. Graphene plasmons

The nonlocal optical properties discussed in the preceding subsections are pertaining to the resonance with varying frequency on the wave vector domain. The physical origin of the nonlocal resonance can be identified as the excitation of plasmons on the graphene surface in two aspects. First, the field patterns on the polaritonic band depicts a typical feature of surface wave. The contours of magnetic field H_y, overlaid with the electric field vectors (E_z, E_x), are plotted in Fig. 8(a) for Ω = 0.6. Both fields decay exponentially away from the graphene surface and become more evanescent as the frequency increases [Fig. 8(b)]. This wave, however, is not a surface mode in the usual sense. It is rather the surface part of a bulk mode that exists in the medium [41]. This feature manifests the couplings of fields between the graphene layers [50]. Across the graphene surface, H_y is discontinuous (and antisymmetric) by the amount of surface current J_s = σE [Fig. 8(b)]. Note that this wave is similar the bonding mode in the periodic lattice of thin metal films [32], with a symmetric alignment of surface charges along the interface. The graphene indeed can be treated as a one-atom-layer thick metal as the chemical potential μ ≠ 0 (away from the Dirac point). The crucial difference is that the graphene layers can only support the bonding mode due to the effective zero thickness for the two-dimensional electronic system, while both the bonding and antibonding modes exist in the layers of metal films with finite thickness.

Fig. 8 Magnetic field (H_y) (a) on the xz plane and (b) along the x axis for the surface mode at the polaritonic band for the same periodic lattice of graphene layers as in Fig. 1, where K_x = 0, K_z = 3.919, and Ω = 0.6. In (a), red and green colors correspond to positive and negative values of H_y, respectively, gray arrows are the electric field vectors (E_z, E_x), and black lines are the locations of graphene layers. In (b), H_y is normalized by its maximum value (at the graphene surface). The profile for Ω = 0.9 is also shown for comparison.

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Second, the surface-like wave on the polaritonic band has points of resemblance to the plasmons in a single graphene layer, characterized by the relation [18, 19, 34, 36]:

\frac{\sqrt{k_{z}^{2} - ε k_{0}^{2}}}{ε} = \frac{2 i k_{0}}{\tilde{σ}} .

It is shown in Fig. 9 that the polaritonic band of the graphene layers approaches toward the plasmon dispersion of a single graphene layer as k_z increases, where k_z = K_z/a. The two curves almost coincide at sufficiently large k_z. There is, however, a discrepancy at small k_z, where the plasmon dispersion of the single layer (red dashed line) scales as

\sqrt{k_{z}}

and is approximated, using Eq. (3), as

Ω = \sqrt{\frac{2 α β k_{z}}{ε}},

where β = h̄c/μ. Nevertheless, the polaritonic band of the graphene layers has a cutoff, owing to the interaction of fields between adjacent layers. At small k_z, the fields spread between the graphene layers and tend to be tightly bound on the graphene surface as k_z increases. At large k_z, where the retardation effects are not important (

Ω ≪ β k_{z} / \sqrt{ε}

), both dispersion curves approach asymptotically to the critical frequency Ω^* ≈ 1.667, at which σ = 0 [cf. Eq. (3)]. Beyond Ω^*, σ < 0 and the graphene plasmons corresponding to TM modes no longer exist [38].

Fig. 9 Plasmon relation of a single graphene layer with the same chemical potential and dielectric background as in Fig. 1. Photonic and polaritonic modes for the periodic lattice of graphene layers are shown for comparison.

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4. Concluding remarks

In conclusion, we have investigated the nonlocal optical properties in the periodic lattice of graphene layers from the view point of effective medium. Approximate formulas of the effective permittivity tensor derived from the nonlocal effective medium model show explicit dependence on the wave vector, indicating the nonlocal nature of the graphene layers. Strong nonlocal effects are found in a wide frequency range (from the effective plasma frequency Ω₀ to the critical frequency Ω^*) and manifest on the emergence of additional wave and the occurrence of negative refraction. These nonlocal properties are attributed to the excitation of graphene plasmons and are well characterized by the nonlocal effective permittivity for the graphene layers.

Appendix

Let the graphene surface locate at x = 0. For TM polarization, the y component of magnetic field in a unit cell (ξ − a < x < ξ, ξ > 0) is given by

H_{y} (x) = {\begin{matrix} A e^{i q x} + B e^{- i q x}, & 0 < x < ξ \\ C e^{i q x} + D e^{- i q x}, & ξ - a < x < 0 \end{matrix}

where

q = \sqrt{ε k_{0}^{2} - k_{z}^{2}}

and A, B, C, and D are coefficients to be determined. The z component of the corresponding electric field is determined by Ampere-Maxwell’s law: ∇ × H = −iωε₀εE as

E_{z} (x) = {\begin{matrix} - \frac{q}{ω ε_{0} ε} (A e^{i q x} - B e^{- i q x}), & 0 < x < ξ \\ - \frac{q}{ω ε_{0} ε} (C e^{i q x} - D e^{- i q x}), & ξ - a < x < 0 \end{matrix}

The boundary conditions at the graphene surface are given by

H_{y} (0_{+}) - σ E_{z} (0_{+}) = H_{y} (0_{-}),

E_{z} (0_{+}) = E_{z} (0_{-}),

where the second term on the left side of Eq. (16) accounts for the surface current. At the unit cell boundary, the fields satisfy

H_{y} (ξ) = e^{i k_{x} a} H_{y} (ξ - a),

E_{z} (ξ) = e^{i k_{x} a} E_{z} (ξ - a) .

The nontrivial solutions for A, B, C, and D require that

| \begin{matrix} 1 + \frac{σ q}{ω ε_{0} ε} & 1 - \frac{σ q}{ω ε_{0} ε} & - 1 & - 1 \\ 1 & - 1 & - 1 & 1 \\ e^{i q ξ} & e^{- i q ξ} & - e^{i k_{x} a} e^{i q (ξ - a)} & - e^{i k_{x} a} e^{- i q (ξ - a)} \\ e^{i q ξ} & - e^{- i q ξ} & - e^{i k_{x} a} e^{i q (ξ - a)} & e^{i k_{x} a} e^{- i q (ξ - a)} \end{matrix} | = 0,

which is simplified to the equation:

cos (k_{x} a) = cos (q a) - \frac{i σ q}{2 ω ε_{0} ε} sin (q a) .

This is the dispersion relation o the periodic lattice of graphene layers for TM polarization. Similar procedures can be performed for TE polarization and give rise to the corresponding dispersion relation:

cos (k_{x} a) = cos (q a) - \frac{i σ ω μ_{0}}{2 q} sin (q a) .

Acknowledgments

The authors thank Prof. C. T. Chan and Prof. Z. Q. Zhang at Hong Kong University of Science and Technology for helpful discussion. This work was supported in part by National Science Council of the Republic of China under Contract No. NSC 102-2221-E-002-202-MY3 and by the National Natural Science Foundation of China under Grant No. 11304038.

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

2. K. Novoselov, A. K. Geim, S. Morozov, D. Jiang, M. K. I. Grigorieva, S. Dubonos, and A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005). [CrossRef] [PubMed]

3. Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201–204 (2005). [CrossRef] [PubMed]

4. K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146, 351–355 (2008). [CrossRef]

5. 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]

6. K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, and B. H. Hong, “Large-scale pattern growth of graphene films for stretchable transparent electrodes,” Nature 457, 706–710 (2009). [CrossRef] [PubMed]

7. F. Xia, T. Mueller, Y.-m. Lin, A. Valdes-Garcia, and P. Avouris, “Ultrafast graphene photodetector,” Nat. Nanotechnol. 4, 839–843 (2009). [CrossRef] [PubMed]

8. Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. C. Ferrari, “Graphene mode-locked ultrafast laser,” ACS Nano 4, 803–810 (2010). [CrossRef] [PubMed]

9. M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474, 64–67 (2011). [CrossRef] [PubMed]

10. P.-Y. Chen and A. Alù, “Atomically thin surface cloak using graphene mnolayers,” ACS Nano 5, 5855–5863 (2011). [CrossRef] [PubMed]

11. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011). [CrossRef] [PubMed]

12. R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science 320, 1308 (2008). [CrossRef] [PubMed]

13. K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. 101, 196405 (2008). [CrossRef] [PubMed]

14. A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, “Universal optical conductance of graphite,” Phys. Rev. Lett. 100, 117401 (2008). [CrossRef] [PubMed]

15. T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B 78, 085432 (2008). [CrossRef]

16. Z. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys. 4, 532–535 (2008). [CrossRef]

17. 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 (2008). [CrossRef] [PubMed]

18. Y. Liu, R. F. Willis, K. V. Emtsev, and T. Seyller, “Plasmon dispersion and damping in electrically isolated two-dimensional charge sheets,” Phys. Rev. B 78, 201403 (2008). [CrossRef]

19. M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009). [CrossRef]

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

21. A. Y. Nikitin, F. Guinea, F. J. García-Vidal, and L. Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B 84, 161407 (2011). [CrossRef]

22. T. Stauber and G. Gómez-Santos, “Plasmons and near-field amplification in double-layer graphene,” Phys. Rev. B 85, 075410 (2012). [CrossRef]

23. F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11, 3370–3377 (2011). [CrossRef] [PubMed]

24. L. A. Falkovsky and S. S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76, 153410 (2007). [CrossRef]

25. R. E. V. Profumo, M. Polini, R. Asgari, R. Fazio, and A. H. MacDonald, “Electron-electron interactions in decoupled graphene layers,” Phys. Rev. B 82, 085443 (2010). [CrossRef]

26. T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys.: Condens. Matter 25, 215301 (2013).

27. V. L. Ginzburg, A. A. Rukhadze, and V. P. Silin, “The electrodynamics of crystals and the theory of excitons,” J. Phys. Chem. Solids 23, 85–97 (1962). [CrossRef]

28. R. M. Hornreich and S. Shtrikman, “Theory of gyrotropic birefringence,” Phys. Rev. 171, 1065–1074 (1968). [CrossRef]

29. P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B 67, 113103 (2003). [CrossRef]

30. C. R. Simovski and P. A. Belov, “Low-frequency spatial dispersion in wire media,” Phys. Rev. E 70, 046616 (2004). [CrossRef]

31. A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012). [CrossRef]

32. R.-L. Chern, “Spatial dispersion and nonlocal effective permittivity for periodic layered metamaterials,” Opt. Express 21, 16514–16527 (2013). [CrossRef] [PubMed]

33. J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007). [CrossRef]

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

35. V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Unusual Microwave Response of Dirac Quasiparticles in Graphene,” Phys. Rev. Lett. 96, 256802 (2006). [CrossRef] [PubMed]

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

37. L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B 56, 281–284 (2007). [CrossRef]

38. S. A. Mikhailov and K. Ziegler, “New electromagnetic mode in graphene,” Phys. Rev. Lett. 99, 016803 (2007). [CrossRef] [PubMed]

39. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevski, Electrodynamics of Continuous Media, 2 (Butterworth-Heinenan, 1984).

40. V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons (Springer-Verlag, 1984). [CrossRef]

41. G. D. Mahan and G. Obermair, “Polaritons at surfaces,” Phys. Rev. 183, 834–841 (1969). [CrossRef]

42. L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-relativistic Theory, 3 (Butterworth-Heinenan, 1977).

43. X. Xu, Y. Xi, D. Han, X. Liu, J. Zi, and Z. Zhu, “Effective plasma frequency in one-dimensional metallic-dielectric photonic crystals,” Appl. Phys.Lett. 86, 091112 (2005).

44. J. J. Hopfield and D. G. Thomas, “Theoretical and experimental effects of spatial dispersion on the optical properties of crystals,” Phys. Rev. 132, 563–572 (1963). [CrossRef]

45. S. I. Pekar, “Theory of electromagnetic waves in a crystal with excitons,” J. Phys. Chem. Solids 5, 11–22 (1958). [CrossRef]

46. V. M. Agranovich and Y. N. Gartstein, “Spatial dispersion and negative refraction of light,” Phys. Usp. 49, 1029 (2006). [CrossRef]

47. M. F. Bishop and A. A. Maradudin, “Energy flow in a semi-infinite spatially dispersive absorbing dielectric,” Phys. Rev. B 14, 3384–3393 (1976). [CrossRef]

48. S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in Media with a Negative Refractive Index,” Phys. Rev. Lett. 90, 107402 (2003). [CrossRef] [PubMed]

49. P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. 37, 259–263 (2003). [CrossRef]

50. B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109, 073901 (2012). [CrossRef] [PubMed]

Nonlocal optical properties in periodic lattice of graphene layers

Abstract

1. Introduction

2. Effective medium model

2.1. Dispersion relations

2.2. Effective permittivity tensor

3. Nonlocal optical properties

3.1. Dispersion characteristics

3.2. Additional wave

3.3. Negative refraction

3.4. Graphene plasmons

4. Concluding remarks

Appendix

Acknowledgments

References and links

Cited By

Figures (9)

Equations (22)

Optics Express