## Abstract

Based on the effective medium model, nonlocal optical properties in periodic lattice of graphene layers with the period much less than the wavelength are investigated. Strong nonlocal effects are found in a broad frequency range for TM polarization, where the effective permittivity tensor exhibits the Lorentzian resonance. The resonance frequency varies with the wave vector and coincides well with the polaritonic mode. Nonlocal features are manifest on the emergence of additional wave and the occurrence of negative refraction. By examining the characters of the eigenmode, the nonlocal optical properties are attributed to the excitation of plasmons on the graphene surfaces.

© 2014 Optical Society of America

## 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*), without loss of generality. With the time-harmonic dependence

_{z}*e*

^{−iωt}, the dispersion relations of the graphene layers are given by (see Appendix for details)

*k*

_{0}=

*ω/c*. Here, TM refers to transverse magnetic, where

**E**= (

*E*, 0,

_{x}*E*) and

_{z}**H**= (0,

*H*, 0), and TE to transverse electric, where

_{y}**E**= (0,

*E*, 0) and

_{y}**H**= (

*H*, 0,

_{x}*H*). The relations (1)–(2) are equivalent to those obtained by the transfer matrix method [24, 26].

_{z}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]:

*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

*k*

_{0},

*k*, and

_{x}*k*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

_{z}*γ*=

*ε*

^{2}(1 + 2

*δ*),

*δ*=

*iσ̃*/(

*εk*

_{0}

*a*), and

*σ̃*=

*σ*/(

*ε*

_{0}

*c*).

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* =
2*πh̄c*/(*μ*Ω),
corresponds to the interband transition (Ω = 2) is around 3.1
*μ*m. 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 ${\epsilon}_{z}^{\text{eff}}$ [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*| is bound by

_{x}*π/a*in a periodic lattice. It is shown in Fig. 2(a) that ${\epsilon}_{z}^{\text{eff}}$ (blue solid line) differs not much from ${\epsilon}_{z}^{0}$ (blue dashed line). Here, ${\epsilon}_{z}^{0}$ (or ${\epsilon}_{y}^{0}$) 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*Ω

_{0}determined by the

*zero*of ${\epsilon}_{z}^{0}$, that is, ${\epsilon}_{z}^{0}({\mathrm{\Omega}}_{0})=0$. Using Eq. (3) in ${\epsilon}_{z}^{0}$, we have

*ã*=

*μa*/(

*h̄c*) is the dimensionless lattice period. Here, Ω

_{0}is very close to the zero of ${\epsilon}_{z}^{\text{eff}}$ and Eq. (8) also serves as a good approximation to ${\epsilon}_{z}^{\text{eff}}({\mathrm{\Omega}}_{0})=0$, as shown in Fig. 2(b).

Another in-plane component ${\epsilon}_{x}^{\text{eff}}$ (green solid line), on the other hand,
distinctly deviates from its quasistatic parameter, ${\epsilon}_{x}^{0}=\epsilon $ (green dashed line). In particular, there
exists a *pole* frequency Ω* _{p}*,
where ${\epsilon}_{x}^{\text{eff}}\approx \pm \infty $. Around Ω

*, ${\epsilon}_{x}^{\text{eff}}$ exhibits a Lorentzian resonance feature [cf. Fig. 2(a)]. Using Eq. (3) in ${\epsilon}_{x}^{\text{eff}}$, we have at small*

_{p}*K*

_{z}*K*=

_{z}*k*is the dimensionless wave number. The pole frequency Ω

_{z}a*is very close to Ω*

_{p}_{0}at

*K*= 0 and gradually moves toward higher frequencies as

_{z}*K*increases (also shown in Fig. 6). As there is no constraint for

_{z}*K*, the nonlocal effects due to ${\epsilon}_{x}^{\text{eff}}$ are expected to be strong in the graphene layers, as in the lattice of metal layers [32].

_{z}The out-of-plane effective permittivity component ${\epsilon}_{y}^{\text{eff}}$, which is perpendicular to the plane where the
wave vector lies, depends on both *k _{x}* and

*k*, as shown in Fig. 3. Note that ${\epsilon}_{y}^{\text{eff}}$ and ${\epsilon}_{z}^{\text{eff}}$ are no longer equal as in the quasistatic case, although the layered structure makes no difference between the

_{z}*y*and

*z*directions in the geometry. The medium properties, therefore, depend on the polarization. Near

*K*=

_{x}*K*= 0 (where

_{z}*K*=

_{x}*k*), ${\epsilon}_{y}^{\text{eff}}$ is characterized by the effective plasma frequency Ω

_{x}a_{0}and the critical frequency Ω

^{*}≈ 1.667 [38]. Below Ω

_{0}, ${\epsilon}_{y}^{\text{eff}}$ is negative and Ω

_{0}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], ${\epsilon}_{y}^{\text{eff}}$ is larger than the background dielectric constant. At

*K*= 0, ${\epsilon}_{y}^{\text{eff}}$ basically decreases with

_{x}*K*[Fig. 3(a)], while at

_{z}*K*= 0, ${\epsilon}_{y}^{\text{eff}}$ increases with

_{z}*K*[Fig. 3(b)]. Compared to ${\epsilon}_{x}^{\text{eff}}$, the nonlocal effect due to ${\epsilon}_{y}^{\text{eff}}$ is weaker.

_{x}Note also that there are no first-order *k _{x}* and

*k*terms in all the effective permittivity components. This is a consequence of

_{z}*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: $\mathrm{\Omega}\tilde{a}=\sqrt{\left({K}_{z}^{2}+{K}_{x}^{2}\right)/\epsilon}$; 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*< 0). Along the plane of constant

_{z}*K*≠ 0, the polaritonic band forms an

_{x}*anticrossing*(avoided crossing) scheme with the photonic band, as shown in Fig. 5(a), indicating the existence of couplings between the two modes.

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 ${\epsilon}_{x}^{\text{eff}}$, across which the Lorentzian resonance reverses the order of maximum and minimum. Therefore, the band crossing occurs when the numerator and denominator of ${\epsilon}_{x}^{\text{eff}}$ become zero simultaneously. When

*K*≠ 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].

_{x}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 Ω_{0} [cf. Eq. (8)], which does not scale like $1/(\sqrt{\epsilon}a)$ 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

*of the effective permittivity component ${\epsilon}_{x}^{\text{eff}}$, as shown in Fig. 6. At larger values of*

_{p}*K*, however, a higher order of expansion on the dispersion relation [Eq. (1)] is necessary to deliver a more accurate formula for ${\epsilon}_{x}^{\text{eff}}$. The polaritonic band, or the permittivity pole, varies with the wave vector component

_{z}*K*over a wide range of frequency from Ω

_{z}_{0}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*: 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

_{x}*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].

#### 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]:

*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 Ω

_{0}≈ 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.

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*> 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 ${\epsilon}_{z}^{\text{eff}}<0$ and the tangential component ${\epsilon}_{x}^{\text{eff}}>0$ [49]. In Fig. 7(a), where Ω = 0.435 is slightly above Ω

_{z}_{0}, the negative refraction in graphene layers occurs when ${\epsilon}_{z}^{\text{eff}}>0$ and ${\epsilon}_{x}^{\text{eff}}<0$ (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]:

*a*and

*b*are constants, and

*ε*is a small positive number. The parabolic-like character of the quartic curve comes from the

*x*

^{4}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 Ω

_{0}, the nonlocal effect is weak and the negative refraction is basically similar to that in a local medium (see the insets for ${\epsilon}_{z}^{\text{eff}}$ and ${\epsilon}_{x}^{\text{eff}}$).

#### 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*), 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,

_{x}*H*is discontinuous (and antisymmetric) by the amount of surface current

_{y}**J**

*=*

_{s}*σ*

**E**[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.

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]:

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*increases, where

_{z}*k*=

_{z}*K*. The two curves almost coincide at sufficiently large

_{z}/a*k*. There is, however, a discrepancy at small

_{z}*k*, where the plasmon dispersion of the single layer (red dashed line) scales as $\sqrt{{k}_{z}}$ and is approximated, using Eq. (3), as where

_{z}*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*, the fields spread between the graphene layers and tend to be tightly bound on the graphene surface as

_{z}*k*increases. At large

_{z}*k*, where the retardation effects are not important ( $\mathrm{\Omega}\ll \beta {k}_{z}/\sqrt{\epsilon}$), both dispersion curves approach asymptotically to the critical frequency Ω

_{z}^{*}≈ 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].

## 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 Ω_{0} 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

*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**= −

*iωε*

_{0}

*ε*

**E**as

*A*,

*B*,

*C*, and

*D*require that

## 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.

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