Theory for polaritons in graphene photonic crystals in an applied magnetic field

ACCEPTED MANUSCRIPT

Theory for polaritons in graphene photonic crystals in an applied

magnetic field

To cite this article before publication: Manoel S Vasconcelos et al 2019 J. Phys. D: Appl. Phys. in press https://doi.org/10.1088/1361-6463/ab6518

Manuscript version: Accepted Manuscript

Accepted Manuscript is “the version of the article accepted for publication including all changes made as a result of the peer review process, and which may also include the addition to the article by IOP Publishing of a header, an article ID, a cover sheet and/or an ‘Accepted Manuscript’ watermark, but excluding any other editing, typesetting or other changes made by IOP Publishing and/or its licensors” This Accepted Manuscript is © 2019 IOP Publishing Ltd.

During the embargo period (the 12 month period from the publication of the Version of Record of this article), the Accepted Manuscript is fully protected by copyright and cannot be reused or reposted elsewhere.

As the Version of Record of this article is going to be / has been published on a subscription basis, this Accepted Manuscript is available for reuse under a CC BY-NC-ND 3.0 licence after the 12 month embargo period.

After the embargo period, everyone is permitted to use copy and redistribute this article for non-commercial purposes only, provided that they adhere to all the terms of the licence https://creativecommons.org/licences/by-nc-nd/3.0

Although reasonable endeavours have been taken to obtain all necessary permissions from third parties to include their copyrighted content within this article, their full citation and copyright line may not be present in this Accepted Manuscript version. Before using any content from this article, please refer to the Version of Record on IOPscience once published for full citation and copyright details, as permissions will likely be required. All third party content is fully copyright protected, unless specifically stated otherwise in the figure caption in the Version of Record. View the article online for updates and enhancements.

Theory for Polaritons in Graphene Photonic

Crystals in an Applied Magnetic Field

M S Vasconcelos‡ and M G Cottam

Department of Physics and Astronomy, University of Western Ontario, London, Ontario N6A 3K7, Canada

E-mail: mvasconcelos@ect.ufrn.br

Abstract. The dispersion relations for bulk and surface plasmon-polaritons in a semi-infinite 1D photonic crystal interlayered with graphene are calculated in the presence of an applied magnetic field. The results are applied to SiO2as the constituent material in a geometry where the layers are arranged in a periodic array with the same layer thickness. The static magnetic field is applied perpendicular to the plane of layers. Numerical results are presented for the modes in THz range, up to 10 THz, to illustrate the important role of the applied magnetic field on the graphene sheets in modifying the polariton dispersion curves, especially for magnetic fields of the order of 1 T. It is found that the polariton frequencies and band gaps have a sensitive dependence on the electron scattering rate parameter (and hence the applied magnetic field strength) in the graphene sheets. Electromagnetic retardation effects are fully taken into account for the bulk bands, while for the surface modes (which are shown to have a novel non-reciprocal propagation characteristics) it is convenient to focus on the regime where retardation is small.

Keywords: Plasmon-polaritons, Graphene, Magnetic field effects, Photonic crystals, Transfer-matrix methods

Submitted to: J. Phys. D: Appl. Phys.

‡ Permanent address: Escola de Ciˆencias e Tecnologia, Universidade Federal do Rio Grande do Norte, 59078-900, Natal-RN, Brazil 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Accepted Manuscript

1. Introduction

One-dimensional (1D) graphene photonic crystals (PCs) are multilayered systems where one of the constituent materials is graphene, the well-known one-atom-thick metallic film discovered in 2004 [1]. These graphene PCs form an emergent research area of intense interest [2, 3, 4, 5, 6, 7, 8]. Systems such as PCs [9] or quasicrystals [10] that incorporate the atomically-thin graphene material have been studied because they exhibit novel optical properties and may be valuable for modern applications in the field of tunable plasmonic and graphene electronics at far-infrared frequencies, for example. Also, these structures can support highly confined plasmon polaritons with very low loss [11] due to graphene’s unique electronic dispersion with gapless features near the so-called Dirac points in the Brillouin zone [12].

Polaritons are hybrid (or mixed) modes that occur when a photon strongly couples to excitation in the material [13, 14]. Examples include exciton-polaritons, when a photon couples to an electron-hole pair in a semiconductor, or plasmon-polaritons, when the coupling is with the electron plasma in a metal or doped semiconductor. On the other hand, graphene constitutes a very special kind of 2D electron gas system. For example, the capability to change its carrier concentration using an applied voltage-gate opens the way to precise control of the frequency range for excitations like surface plasmon-polaritons (SPP) [15]. Also long propagation lengths can be achieved [16] compared with conventional SPPs .

On the other hand, it is known that when an external magnetic field is applied perpendicular to a 2D electron gas [16] a mix between cyclotron excitations and the electronic oscillations (or plasmons) occurs in that system, leading to magnetoplasmon-polaritons [17]. An effect of the magnetic field is to cause an anisotropy in the optical conductivity tensor and strong resonant absorptions in the dielectric functions. Consequently the dispersion properties and frequencies of the polaritons become very sensitive to the applied magnetic field, especially in systems that incorporate graphene sheets [18, 19, 20]. For example, Ferreira et al. [19] reported an exotic polariton mode with weak damping in a single-layer graphene film with an applied static magnetic field; they termed it a quasi-transverse-electric mode because of the resemblance to conventional transverse electric modes. Also Melo [21] studied the excitation of surface polaritons in magnetically biased graphene at the interface separating two dielectric media. The focus was on the influence of the magnetostatic field for the surface polaritons in graphene in the 500 GHz to 5 THz frequency range, where the author calculated the reflectance in an Attenuated Total Reflection (ATR) set up [14] and associated the reflectance dip with the excitation of the surface polaritons. Michalski [22] used the modal transmission line theory to calculate the reflectivity and transmissivity rate in a multilayered system with multiple anisotropic conductive graphene sheets at the interfaces. He noted that the analysis of such structures is rendered non-trivial, because of the coupling between the transverse magnetic (TM) and transverse electric (TE) polarization fields caused by the graphene sheets. It was concluded that a TE 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Accepted Manuscript

plane wave should also excite surface plasmon-polaritons modes, due to this coupling of the TE and TM waves, and this coupling becomes enhanced with increased magnetic field bias.

In this work we emphasize the role (and its interesting consequences) of an applied magnetic field on the surface and bulk plasmon-polariton modes in a photonic crystal with embedded graphene sheets at the interfaces, henceforth referred to as a graphene photonic crystal (GPC). The plasmon-polaritons are studied here by using Maxwell’s equations in a macroscopic formalism, where graphene is characterized by a frequency-dependent optical conductivity tensor. In section 2 we present a modification of the plasmon-polariton theory for dielectric photonic crystal with graphene sheets [9] to relate to our present case where we have an external magnetic field applied. Here the graphene sheets are characterized by a modified optical conductivity, which includes the Landau levels’ quantization due to the applied magnetic field, and we concentrate initially on the bulk polaritons. Next the localized surface polaritons of a semi-infinite GPC are considered in section 3. Numerical results are then presented in section 4 when the dielectric material is taken to be SiO2 in the various layers of the GPC. These results illustrate the important role of the applied magnetic field in GPCs in modifying the polariton dispersion curves, especially when we have a magnetic field of order 1 T. Also we report a sensitive dependence on the polariton dispersion relation curves on the electronic scattering rate parameter and on the intensity of the applied magnetic field in the graphene sheets. In the last section we present the conclusions of this work. 2. General theory

We consider a photonic crystal (PC) consisting of alternating unit cells AB juxtaposed to form the periodic ABABAB... array, with graphene sheets at each of the interfaces and a static magnetic field ~B0 applied in the direction perpendicular to the layers (see figure 1). The building blocks A and B are layers of a dielectric material with thicknesses a and b, respectively.

2.1. Field-dependent conductivity of graphene

When a static magnetic field B0 is applied perpendicular to the graphene sheets, it is known that the conductivity becomes a tensor with the components in the xy-plane satisfying [23] e σ = σxx σxy σyx σyy ! . (1)

Here, the general expressions for the diagonal and off-diagonal terms (sometimes called Hall conductivities) can be obtained using the Kubo formula [18, 24] as

σxx = σyy = e2_v2 F|eB0|(¯hω + 2iΓ) πi 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Accepted Manuscript

ە ۖ ۖ ۖ ۔ ۖ ۖ ۖ ۓ

…

…

Figure 1. The geometry and coordinate axes for the 1D PC array (ABABAB...) with graphene sheets at each of the interfaces. In the case of a semi-infinite array with a surface at z = 0 the external medium is denoted as C.

× ∞ X n=0 n[n_F(M_ν) − n_F(M_ν+1)] − [n_F(−M_ν) − n_F(−M_ν+1)] (Mν+1− Mν)3− (¯hω + 2iΓ)2(Mν+1− Mν) (2) +[nF(−Mν) − nF(Mν+1)] − [nF(Mν) − nF(−Mν+1)] (Mν+1+ Mν)3− (¯hω − 2iΓ)2(Mν+1+ Mν) o , σxy = − σyx= −ξ e2_v2 F|eB0| π × ∞ X n=0 n [nF(Mν) − nF(Mν+1)] + [nF(−Mν) − nF(−Mν+1)] o (3) × 2[M 2 ν + Mν+12 − (¯hω + 2iΓ)2] [M2 ν + Mν+12 − (¯hω + 2iΓ)2]2− 4Mν2Mν+12 .

The summations over n are for the energies Mν = (2¯hν|eB0|vF2)1/2 of the Landau level, nF is the Fermi-Dirac distribution function at temperature T , vF is the Fermi velocity of the electrons in graphene, Γ is the electron scattering rate, ξ = sgn(B0) is the sign function of ~B0, and e is the electron charge. Here we have considered the excitonic energy to be equal to zero, i.e., ∆ = 0 in reference [24].

When B0 → 0, the sums over n for σxy go to zero and σxx converges very slowly. In this case (not studied here) we could alternatively use the Drude form given in references [25, 21]. To illustrate the behaviour of the optical conductivity for graphene for the field values of interest, we consider T = 300 K, chemical potential µc = 0.2 eV for the graphene sheets [26, 27, 28], and electron scattering rate Γ = 0.11 meV [21]. Figure 2 shows σxx and σxy plotted versus frequency for the values of the applied magnetic field, B0 = 0.5, 1.0, and 1.5 T, where the solid (dashed) lines correspond to the real (imaginary) part of σxx (in figure 2 (a)) and σxy (in figure 2 (b)). The vertical lines (located at the frequencies 0.390 and 0.558 THz) characterize the maximum and minimum value for the imaginary part of σxx (real part of σxy), respectively, when B0 = 1.5 T. These values will turn out to be important later for interpreting the bulk 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Accepted Manuscript

polariton dispersion relations.

Figure 2. The diagonal, σxx (a), and off-diagonal, σxy (b), components of the graphene conductivity tensor as functions of the frequency for several values of the applied magnetic field B0. The solid (dashed) lines are the real (imaginary) part of σxxand σxy. The vertical lines are explained in the text.

2.2. Bulk polariton modes

In order to study the bulk polariton modes for an infinite GPC, we need to consider solutions of Maxwell’s equations within each dielectric medium, subject to the standard electromagnetic boundary conditions at the interfaces with graphene. As a consequence of the Landau levels leading to a non-diagonal conductivity tensor for the graphene sheets, we need to consider the electric and magnetic field solutions within the 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Accepted Manuscript

constituent media A and B of the lth layer in a general way. Specifically, we do not make a separation into particular polarizations like transverse magnetic (TM) or transverse electric (TE). Instead, in both media we write ~Ej = (Exj, Eyj, Ezj) for the electric fields, where Exj = h A(l)_xjeikzj(z−zl)_{+ B}(l) xje −ikzj(z−zl)i_ei(kxx−ωt)_{, (4)} Eyj = h A(l)_yjeikzj(z−zl)_{+ B}(l) yje −ikzj(z−zl)i_ei(kxx−ωt)_{, (5)} Ezj = −(kx/kj) h A(l)_xjeikzj(z−zl)_{− B}(l) xje −ikzj(z−zl)_]i_ei(kxx−ωt)_{. (6)}

The factors A(l)_xj, B(l)_xj, A(l)_yj, and B_yj(l) (with j = A or B; l = 1, 2, 3, . . .) are the amplitudes for the forward- and backward-travelling waves, respectively, and L = a + b is the periodicity length. Without loss of generality, we have taken the in-plane wave vector to be along the x direction, and so kzj = (jω2/c2− k2x)1/2 for the (complex) wave vector in the perpendicular direction.

Correspondingly, for the magnetic fields components ~Hj = (Hxj, Hyj, Hzj) in medium A ( or B), we deduce from ~H = (0c2/iω) ~∇ × ~E that

Hxj = − kzj0c2 ω h A(l)_yjeikzj(z−zl)_{− B}(l) yje −ikzj(z−zl)i_ei(kxx−ωt)_{, (7)} Hyj = ( 0jω kzj )hA(l)_xjeikzj(z−zl)_{− B}(l) xje −ikzj(z−zl)i_ei(kxx−ωt)_{, (8)} Hzj = kx0c2 ω h A(l)_yjeikzj(z−zl)_{− B}(l) yje −ikzj(z−zl)i_ei(kxx−ωt)_{, (9)}

From the electromagnetic boundary conditions at any A − B interface we require ~

EyA = ~EyB, E~xA= ~ExB, (10)

−(HyB − HyA) = σxxExB + σxyEyB, (11)

HxB− HxA= σyxExB+ σyyEyB. (12)

Then, by considering these condition at the interfaces z = nL + a within a cell and at z = (n + 1)L between cells (see figure 1), we can relate the electromagnetic fields for cell n to those for cell n + 1. From equation (10) we deduce

A(n)_xAeikzAa_{+ B}(n) xAe −ikzAa _{= A}(n) xB+ B (n) xB, (13) A(n)_yAeikzAa_{+ B}(n) yAe −ikzAa _{= A}(n) yB + B (n) yB. (14)

Next from equations (11) and (12) we can write A(n)_xAeikzAa_{− B}(n) xAe −ikzAa_{= (σ} xx+ BkzA AkzB )A(n)_xB + (σxx− BkzA AkzB )B_xB(n) (15) + σxy(A (n) yB + B (n) yB), A(n)_yAeikzAa_{− B}(n) yAe −ikzAa_{= (σ} yy+ kzB kzA )A(n)_yB + (σyy − kzB kzA )B_yB(n) (16) + σyx(A (n) xB+ B (n) xB), 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Accepted Manuscript

where we define σxx = σxxkzA/A0ω, σxy = σxykzA/A0ω, σyy = σyyω/0c2kzA, and σyx= σyxω/0c2kzA. The above equations can be conveniently re-expressed in a matrix form as MA        A(n)_xA B_xA(n) A(n)_yA B_yA(n)        = MAB        A(n)_xB B_xB(n) A(n)_yB B_yB(n)        . (17)

Here the 4 × 4 propagation matrix MA is simply given by

MA=       eikzAa _{0 0 0} 0 e−ikzAa _{0 0} 0 0 eikzAa ₀ 0 0 0 e−ikzAa       , (18)

while the more complicated 4 × 4 interface matrix due to the graphene sheet is defined in block form as MAB = M11 AB MAB12 M21 AB MAB22 ! , (19)

where the forms of the 2 × 2 matrices M_ABi0j0 (with i0 and j0 taking the values 1 or 2) are specified in Appendix A.

An analogous matrix equation arises when we apply the same boundary conditions at the interfaces where z = (n + 1)L. In this case it relates the amplitudes A(n)_pB and B_pB(n) (p = x, y) to A(n+1)_pA and B_pA(n+1), giving MB        A(n)_xB B_xB(n) A(n)_yB B_yB(n)        = MBA        A(n+1)_xA B_xA(n+1) A(n+1)_yA B_yA(n+1)        . (20)

Here MBand MBAare the same as MAand MAB, respectively, provided the substitutions A → B, B → A, and a → b in the matrices (19) and (18) are made. On defining the kets formed by the undetermined set of coefficients as

|A(n)_j >=        A(n)_xj B_xj(n) A(n)_yj B_yj(n)        , (21)

we find that equations (17) and (20) can be re-expressed as the matrices equations MA|A (n) A >= MAB|A (n) B >, (22) MB|A (n) B >= MBA|A (n+1) A > . (23)

It is easy to show that, using these equations, we obtain

|A(n+1)_A > = T |A(n)_A >= exp (iQL)|A(n)_A >, (24) 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Accepted Manuscript

where in the last step Bloch’s theorem (see [14]) was employed, with Q being the modulus of the Bloch wave vector. Here the so-called transfer matrix T is given by

T = M_BA−1MBMAB−1MA, (25)

and therefore

[T − exp (iQL)I]|A(n)_A > = 0, (26)

where I is the 4 × 4 unit matrix. The equation above is the secular equation that defines the bulk polariton dispersion relation. From the property that T is an unimodular matrix (det T = 1) it can be shown [29] that its eigenvalues {s1, s2, s3, s4} occur in reciprocal pairs, allowing us to write s3 = s−11 and s4 = s−12 . Therefore, as we conclude below, it is necessary only to find two eigenvalues, which are related to the Bloch wave vectors through [29]

exp(iQjL) = sj, j = 1, 2. (27)

By solving equation (26) we find that the eigenvalues satisfy the polynomial equation s4 _{+ αs}3_{+ βs}2_{+ γs + 1 = 0, where the coefficients are related to the elements of the} transfer matrix by

α = γ = −T r(T ), (28)

β = T11T22+ T33T44+ (T11+ T22)(T33+ T44) − T24T42

−T34T43− T23T32− T12T21− T13T31− T14T41. (29) After some straightforward algebra the above polynomial can be rearranged as

H2+ αH + (β − 2) = 0 (30)

with Hi = si + s−1i . This roots of this equation, together equation (24), gives us the dispersion relations for the bulk bands in the form

cos(Q1L) = H1/2 and cos(Q2L) = H2/2, (31)

where

H1,2 =

−α ±qα2_{− 4(β − 2)}

2 . (32)

In this case we have two independent solutions for the Bloch wave vector Qi, as illustrated later by numerical examples.

It is worth noting that, as a special case of the above theory, we may use equation (17) to consider the polaritons propagating at a single interface (at z = a) between two different semi-infinite media (A and B). The general result reduces to

       0 B_xA(1) 0 B_yA(1)        = M_A−1MAB        A(1)_xB 0 A(1)_yB 0        , (33) 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Accepted Manuscript

which leads to a 2 × 2 matrix secular equation for the polariton modes. After some algebra, we can recover the result given in [19] for the single graphene sheet at the interface between two media, i.e.,

(A qA + B qB + iσxx A0ω )(qA+ qB− iσyyω 0c2 ) = σyxσxy (0c)2 , (34)

where we define qA= −ikzA and qB = −ikzB. 3. Surface polariton modes

For a study of the surface modes we consider a semi-infinite GPC with the geometry as in figure 1, where the region z < 0 is taken to be a dielectric medium C. In practice, a surface mode in a photonic crystal is excited by an incident electromagnetic wave either in p-polarization (a TM mode) or s-polarization (a TE mode), e.g., as in an Attenuated Total Reflection (ATR) experiment. Here we will consider both the cases of p-polarization and s-polarization in the external medium C.

3.1. Case of p-polarization in the external medium

We consider now a p-polarized wave incident on the PC sample from the external medium C. The electric and magnetic fields are given by ~EC = (ExC, 0, EzC) and

~ HC = (0, HyC, 0), where ExC = A (0) xCe kzCz_ei(kxx−ωt)_{, (35)} EzC = − kx kzC A(0)_xCekzCz_ei(kxx−ωt)_{, (36)} HyC = − 0ωC ikzC A(0)_xCekzCz_ei(kxx−ωt)_{. (37)}

Here kzC = (k2x− Cω2/c2)1/2 and we assume k2x > Cω2/c2 so that kzC takes only real and positive values. To obtain equations (36), and (37), we used ~H = (0c2/iω) ~∇ × ~E together ~∇ · ~D = 0. By applying the previous electromagnetic boundary conditions as in equations (10), (11) and (12) to the interface at z = 0, we have

A(0)_xC = A(1)_xA+ B_xA(1), (38) 0 = A(1)_yA+ B_yA(1), (39) 0ωA kzA (A(1)_xA− B_xA(1)) + 0ωC ikzC A(0)_xC = −σxx(A (1) xA+ B (1) xA) − σxy(A (1) yA+ B (1) yA), (40) −kzA0c 2 ω (A (1) yA− B (1) yA) = σyx(A (1) xA+ B (1) xA) + σyy(A (1) yA+ B (1) yA). (41)

From these coupled equation we can deduce a relationship between the A(1)_xA and B_xA(1) amplitudes. The result is

A(1)_xA= λB_xA(1), (42) where λ =1 − kzAC ikzCA − σxxkzA 0ωA  /1 + kzAC ikzCA +σxxkzA 0ωA  . (43) 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Accepted Manuscript

Then from equations (40), (41) and (42) we can write B_yA(1) = ∆B_xA(1) and A(1)_yA = −∆B_xA(1), (44) where ∆ = σyxω 2kzA0c2 (λ + 1). (45)

For the semi-infinite GPC in figure 1, instead of using Bloch’s theorem as in the bulk-mode case, we now seek solutions for localized surface plasmon-polaritons that have decaying amplitudes for the electromagnetic fields with respect to distance from one unit cell to the next interior cell, starting from the surface plane z = 0. Therefore, taking equation (24) with Q = iβ0, we must have

T |A(n)_A >= exp (−β0L)|A(n)_A >, (z > 0) (46)

with Re(β0) > 0 (or equivalently | exp(−β0L)| < 1) as a requirement for localization. On applying this equation to the first layer, we have (following [13])

T        A(1)_xA B_xA(1) A(1)_yA B_yA(1)        = exp (−β0L)        A(1)_xA B_xA(1) A(1)_yA B_yA(1)        . (47)

From this equation, we can find two conditions for localized surface modes. One

condition comes from writing out the first and second lines of equation (47) in full, and leads to

T11+ T12/λ − T13∆/λ + T14∆/λ = T21λ + T22− T23∆ + T24∆, (48) together the condition

|T11+ T12/λ − T13∆/λ + T14∆/λ| < 1. (49)

Also, from the third and fourth lines of (47), we obtain a second implicit dispersion relation for surface modes as

T41λ + T42− T43∆ + T44∆ = −T31λ − T32+ T33∆ − T34∆, (50) together with

|T31λ/∆ + T32/∆ − T33+ T34| < 1. (51)

Two brief comments on the above results are useful. First, if we put σxy = σyx = 0, we just recover the polariton results in our previous paper [9] where there was no applied magnetic field. Secondly, our present results for the surface modes simplify in the regime where retardation effects are small (as at larger in-plane wavenumbers). In this case, choosing kzC ≈ kx and kzA = ikx, following [30], the parameters in equations (43) and (45) reduce to λ = (1 − C A − σxxikx 0ωA )/(1 +C A +σxxikx 0ωA ), (52) ∆ = σyxω 2ikx0c2 (λ + 1). (53) 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Accepted Manuscript

3.2. Case of s-polarization in the external medium

To obtain the surface modes in a semi-infinite GPC in the case of an incident wave with s-polarization (TE) we write ~EC = (0, EyC, 0) and ~HC = (HxC, 0, HyC), where

EyC = A (0) yCe kzCz_ei(kxx−ωt)_{, (54)} HxC = − kzC0c2 iω A (0) yCe kzCz_ei(kxx−ωt)_{, (55)} HzC = kx0c2 ω A (0) yCe kzCz_ei(kxx−ωt)_{. (56)}

To obtain equations (55) and (56) we used ~H = (0c2/iω) ~∇ × ~E. Now, by applying the same boundary conditions, (10), (11), and (12), we have

0 = A(1)_xA+ B(1)_xA, (57)

A(0)_yC = A(1)_yA+ B_yA(1), (58)

for the electrical fields amplitudes, and 0ωA kzA (A(1)_xA− B_xA(1)) = −σxx(A (1) xA+ B (1) xA) − σxy(A (1) yA+ B (1) yA), (59) −kzA0c 2 ω (A (1) yA− B (1) yA) + kzC0c2 iω A (0) yC = σyx(A (1) xA+ B (1) xA) + σyy(A (1) yA+ B (1) yA), (60)

for the magnetic fields amplitudes. In similar way as for p-polarization, we have from (57), B_xA(1) = −A(1)_xA. Therefore, applying it in (59) and (60) we have

20ωA kzA (A(1)_xA) = −σxy(A (1) yA+ B (1) yA), (61) −kzA0c 2 ω (A (1) yA− B (1) yA) + kzC0c2 iω A (0) yC = σyy(A (1) yA+ B (1) yA). (62)

Now, using (58), we can rewrite (3.2) as −kzA0c 2 ω (A (1) yA− B (1) yA) + kzC0c2 iω (A (1) yA+ B (1) yA) = σyy(A (1) yA+ B (1) yA). (63) After some algebra we can find

A(1)_yA(1 − kzC ikzA + σyyω kzA0c2 ) = B_yA(1)(1 + kzC ikzA − σyyω kzA0c2 ), (64) or A(1)_yA = λ0B_yA(1), (65) with λ0 = (1 + kzC ikzA − σyyω kzA0c2) (1 − kzC ikzA + σyyω kzA0c2) . (66)

By using (57) and (58) we can find

A(1)_xA= −∆0B_yA(1), (67) 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Accepted Manuscript

and B_xA(1) = ∆0B_yA(1), (68) with ∆0 = σxykzA 20ωA (λ0+ 1). (69)

The conditions for having localized s-polarized surfaces modes can be obtained using the transfer matrix in a similar fashion to the analysis in the previous subsection for the case of p-polarization. The analogous equations to (48)-(51) are found to be

(T11− T12)∆0− T13λ0 − T14= (T22− T21)∆0+ T23λ0 + T24, (70) (T32− T31)∆0/λ0+ T33+ T34/λ0 = (T42− T41)∆0+ T43λ0+ T44, (71) with the localization conditions

|(T11− T12) − T13λ0/∆0− T14/∆0| > 1, (72)

|(T32− T31)∆0/λ0 + T33+ T34/λ0| > 1. (73)

Again there are simplifications in the case of weak retardation. Specifically, equations (66) and (69) reduce to give

λ0 = 2ikx0c 2 σyyω − 1 (74) ∆0 = −σxy σyy k2 x A(ω/c)2 ! . (75) 4. Numerical results

In this section we apply the previous theory to a semi-infinite GPC to illustrate the results for the bulk and surface polaritons. We take both of the constituent materials, A and B, to be silica (SiO2), with a dielectric constant A= B ≡  = 2.2, as appropriate for THz frequencies [31]. We mention here, as a first approximation, our results will not include the effects of absorption near 4 THz and 8 THz due to SiO2, since our goal here is to simulate a lossless material and focus on the role of the graphene layers in a magnetic field. Indeed, absorption processes will modify the properties of polaritons most sensitively when the frequencies lie close to a resonance. Generally, absorption leads to attenuation of the polaritons (surface and bulk modes) as it propagates through the crystal (see, e.g., [32]). The absorption effects could be studied in future work.

We choose vacuum (C = 1) for the external medium. We ignore the role of

absorption on the silica since it is typically small for other distinct crystalline forms and composition [33] in the temperature and frequencies of interest in this work. Also we choose the temperature T = 300 K and a chemical potential µc = 0.2 eV for the graphene sheets [26, 27, 28]. The other physical parameters used here are a = b = 10 µm and applied magnetic field B0 = 1.5 T. We have put our expressions for the optical conductivity in terms of the fine structure constant e2/4π0¯hc (≈ 1/137), where c is 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Accepted Manuscript

the light speed in a vacuum. This constant arises naturally from our equations. We consider the electron scattering rate (the inverse of the intrinsic relaxation time) to be Γ = 0.11 meV [21].

In the following figures, we present numerical dispersion relations for plasmon polaritons in the 1D GPC, focusing on the effects of the applied magnetic field. The bulk modes occur here as bands separated by gaps and they are shown as the shaded areas (brown areas in the online version). These bulk bands are bounded by the curves for QjL = 0 and QjL = π, where QjL is given in equation (31) with L = a + b. The surface modes, characterized by black lines, lie between and above the bulk bands, subject to the constraint k2_x > Cω2/c2. Consequently any surface mode in these plots lies to the right of the vacuum light line, shown as a dashed straight line, given by ω = ckx/

√ 0. For reference, we also show the dielectric light line for the PC given by ω = ckx/

√ . We have chosen to plot the frequency in THz units and kxa is the scaled 2D in-plane wave vector.

The bulk and surface plasmon-polariton dispersion relations for the frequency (f ) region up to 10 THz are presented in figure 3 for the in-plane wave vector ranging from kxa = −5.0 to kxa = 5.0. A reciprocal propagation effect is found for the bulk bands, as expected, with respect to the sign of the wave-vector component, but this is not the case for the surface modes where we have a non-reciprocity effect. In describing the behaviour of these modes we conveniently separate this spectrum into three distinct regions: (a) to the right (or left) side of the dielectric light line for kx > 0 (or kx < 0) and frequencies greater them 1.0 THz; (b) inside the region bounded by the the dielectric light lines for kx > 0 and kx < 0 with f > 1.0 THz; and (c) the region where the frequencies are less than 1 THz.

In region (a) in figure 3 we have two bulk bands that start at the dielectric light line and merge into a narrow band approximating a straight line with a constant group velocity for high values of kxa. Those bulk bands show a behaviour that is reciprocal with respect to kx. However, we also have five surface modes, labeled by S1, S2, S3, S4 and S5, that are non-reciprocal with respect to the substitution kx → −kx. The explanation for this behaviour is related to the symmetry properties of the optical conductivity tensor. The origin is similar to the well-known non-reciprocity in surface modes [34] and for waves in electromagnetic systems in general [35]. The modes S3, S4 and S5 are p-polarized modes in the external medium (vacuum), while S2 and S3 are s-polarized modes in this external medium. Note that these latter two modes have a particular limit when kx (or −kx) goes to infinity (minus infinity): specifically they tend to f3 = 8.32 THz (long-short-dashed in figure 3) line. We draw attention here to the fact that we have s-polarized (TE) plane surface polariton modes, for which there is no counterpart in non-graphene PCs (or dielectric superlattices) [13, 14]. We also point out the existence of two other very interesting surface modes, labeled by S6 and S7, which can more easily be studied by expanding the frequency region from zero to 1 THz as done in figure 4.

In region (b) we have a bulk region terminating at the dielectric light line, for kx > 0 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Accepted Manuscript

Figure 3. Polariton dispersion relation for the 1D graphene photonic crystal with applied magnetic field B0 = 1.5 T. Here the dashed and dotted sloping lines are the vacuum light line (VLL) and dielectric light line (DLL), specified by ω = ckx/

√ 0and ω = ckx/

√

, respectively. The horizontal thin chain lines correspond to the frequencies f1= 0.39 THz, f2= 0.55 THz and f3= 8.32 THz. The first of these is for the minimum in Imσxx(or the minimum in Reσxy) and the second is for the maximum in Imσxx(or the maximum in Reσxy), both being highlighted in figure 2. The surface modes are labelled S1 and S2, S3, S4, S5, S6, and S7 (see the text).

or kx < 0. In figures 3 and 4 we can also see the so-called graphene induced gap, by analogy with results in [8, 9], but in the present case the gap starts at about 0.32 THz and finishes at about 2.48 THz for kxa = 0. Naturally, this gap depends on the GPC parameters (such as layer thicknesses, dielectric constants, etc.), but here we have the added feature that it depends on the applied magnetic field as well. For the case where there is zero applied magnetic field, this gap starts at the origin [9], i.e., where kxa = 0 and f = 0.

The low-frequency region (c) is shown in figure 4. Here we can see the specific behaviour for the bulks modes close to the frequency values f1 = 0.39 THz and f2 = 0.55 THz. The bulk bands, labeled by B1 and B2 are reciprocal and tend to f2 when kx goes to infinity. This value represents a resonance frequency corresponding to the maximum 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Accepted Manuscript

Figure 4. The same as in figure 3, but focusing on a different frequency region. Here, B1, B2, B3, B4, and B6 are labels for the bulk bands, while S6, S7, and S8 are the labels for the surface modes discussed in the text.

in Im(σxx), or the maximum in Re(σxy). The other bulk bands, labeled by B3 and B4 are also reciprocal and they tend to f1 at larger magnitude of kxa. Here f1 is another resonance frequency that corresponds to the minimum in Imσxx (or the minimum in Reσxy). Clearly, these bulk bands depend on the resonant frequencies f1 and f2, and consequently on the intensity of the applied magnetic field. These bulk modes are very different from the bulk modes in dielectric (semiconductor) PCs, as well as those reported in previous works [13, 14]. We infer that these bulk modes have this different behaviour due to the absorption, as characterized by the scattering rate Γ in equation (2), in the graphene sheets. Also, we see that the applied magnetic field is a very important parameter for tuning the bulk (and surface) polariton modes in the GPC. We note the small gap region that starts at 0.32 THz and finishes at 0.51 THz, for kxa = 0, and it is limited by the lowest bulk band and the dielectric light lines. It depends sensitively on the applied magnetic field (see figure 5). On the other hand, for low frequencies, below to 0.05 THz (or 50 GHz), we have more two bulk bands, labeled by B5 and B6 (which are reciprocal), and tend to 0.045 THz when |kxa| becomes large. The surface modes in the region (c) are also quite different. We have three surface modes, labeled by S8, S7, and S6. The mode S8 starts at f = 0.38 THz for kxa = 0.11 and finishes at f = 0.49 THz for kxa = 0.133. The surface mode S7 starts at f = 0.51 THz for 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Accepted Manuscript

kxa = 0.139 and finishes at the vacuum light line. These modes, S8 and S7, have a behaviour that is typical for polariton surface modes when we consider the damping (or absorption ) in the dielectric function [36]. Indeed, here we have taken account of an equivalent damping effect through the scattering rate Γ in the optical conductivity tensor (see equations (2) and (3)). We should mention here that these surface modes are very sensitive to this parameter. For example, we will not have surface modes if Γ = 1 meV.

The non-reciprocity aspects of the surface modes, as described above, do not change with the increasing of the magnetic field. However, these modes will be shifted (except to the surface modes S7 and S8 that will disappear). We can interpret this using figure 2, because, essentially, we have a shift in the optical conductivity (including the peaks and dips in the imaginary and real parts of σ). Therefore, we present now a study to find the behaviour of these modes for bulk bands with the applied magnetic field. This behaviour will be similar for all the spectra.

Finally, in figure 5 we show the bulk polariton modes (and the band gap) for the region below 1 THz, for kxa = 0 plotted as a function of the applied magnetic field. Here, we can see that it is possible to tune the bulk bands (and gap) by changing the applied magnetic field intensity. Note that the frequencies of these bulk bands will increase approximately linearly as a function of the applied magnetic field when kxa = 0. In general terms, we comment that the absence of an applied magnetic field in this system would make this spectrum more simple, mainly in the region around f < 0.6 THz, in figure 4. Further, we would not have surface modes for s-polarization [9] in that case, as discussed also by Albuquerque and Cottam [13, 14]. On the other hand, the bulk spectrum with applied magnetic is rather similar to the case in the absence of an applied magnetic field for p-polarization (see ref. [9] for more detail).

5. Conclusions

In summary, we have presented a general theory for the plasmon-polaritons in graphene photonic crystals subjected to an external applied magnetic field. Specifically we have considered a periodic photonic crystal consisting of alternating unit cells arranged in a periodic array with embedded graphene sheets at the interfaces and a static applied magnetic field in the direction perpendicular to the layers. On comparing our results with previous work we find that the inclusion of an external magnetic field in the GPC will affect strongly the dispersion relations of the polaritons, mainly at low frequencies (below about 1 THz, for B = 1.5 T). Also, we find a novel nonreciprocity effect for the propagation of the surface modes. Further, there is the possibility of having s-polarized (TE) surface polariton modes that no have counterpart in the dispersion relations in semiconductor (or dielectric) photonic crystals [13].

In the context of applications, we mention that surface magnetoplasmons (or magnetoplasmon polaritons) in multilayer structures are typically difficult to realize because they require very high magnetic fields. The explanation is that in order to 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Accepted Manuscript

Figure 5. The polariton frequencies below 1 THz for kxa = 0.0 plotted versus the applied magnetic field B (in T).

observe the effects due to an external magnetic field for metals, for example, the frequencies would need to be of the order of 1016 _{Hz, which is the order of the} characteristic cyclotron frequency in metals [37]. The characteristic cyclotron frequency is a function of the applied magnetic field, and, therefore, in the case of metals, it needs a magnetic field of order 103 _{T, which is exceptionally difficult to realize in laboratories.} One solution to avoid this generally unreachable magnetic field strength is to use doped semiconductors (instead of metals) where the characteristic cyclotron frequency is in THz regime (for a review see [38]). In this case, a more modest external magnetic field of less than 2 T will suffice. From our results, by using GPCs instead semiconductor PCs, we can see that there is the possibility for excitation of surface and bulk magnetoplasmon polaritons in this same frequency range and the same order for the specified external magnetic field.

Acknowledgments

M. S. Vasconcelos thanks the Department of Physics and Astronomy at the University of Western Ontario for hospitality during his sabbatical as visiting professor. This study was financed in part by the Coordena¸c˜ao de Aperfeioamento de Pessoal de N´ıvel Superior (CAPES) of Brasil (Finance Code 88881.172293/2018-01) and the Natural Sciences and 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Accepted Manuscript

Engineering Research Council (NSERC) of Canada (grant RGPIN-2017-04429). Appendix A.

Here we provide the definitions for the matrix blocks constituting the interface matrix in equation (19). First M11

AB and MAB22 , which involve the diagonal elements of the graphene conductivity tensor, are given by

M_AB11 = 1 2   1 + BkzA AkzB + σxxkzA A0ω 1 − BkzA AkzB + σxxkzA A0ω 1 −BkzA AkzB − σxxkzA A0ω 1 + BkzA AkzB − σxxkzA A0ω  , (A.1) M_AB22 = 1 2   1 + kzB kzA + σyyω 0c2kzA 1 − kzB kzA + σyyω 0c2kzA 1 −kzB kzA − σyyω 0c2kzA 1 + kzB kzA − σyyω 0c2kzA  . (A.2)

We note that these matrices are similar to those in [9] for p-polarization and s-polarization, respectively. The other matrices M12

AB and MAB21, which involve the off-diagonal elements of the graphene conductivity tensor, are

M_AB12 = 1 2   σxykzA A0ω σxykzA A0ω −σxykzA A0ω − σxykzA A0ω  , (A.3) M_AB21 = 1 2   σyxω 0c2kzA σyxω 0c2kzA − σyxω 0c2kzA − σyxω 0c2kzA  . (A.4) References

[1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V and Firsov A A 2004 Science 306 666–669

[2] Hanson G W 2008 Journal of Applied Physics 104 084314

[3] Wang B, Zhang X, Garc´ıa-Vidal F J, Yuan X and Teng J 2012 Phys. Rev. Lett. 109(7) 073901 [4] Wachsmuth P, Hambach R, Benner G and Kaiser U 2014 Phys. Rev. B 90(23) 235434

[5] Fuentecilla-Carcamo I, Palomino-Ovando M and Ramos-Mendieta F 2017 Superlattices and Microstructures 112 46 – 56

[6] Bludov Y V, Peres N M R, Smirnov G and Vasilevskiy M I 2016 Phys. Rev. B 93 245425 [7] Bludov Y V, Peres N M R and Vasilevskiy M I 2013 Journal of Optics 15 114004

[8] Fan Y, Wei Z, Li H, Chen H and Soukoulis C M 2013 Phys. Rev. B 88 241403

[9] Vasconcelos M S and Cottam M G 2019 Journal of Physics D: Applied Physics 52 285104 [10] Costa C H, Vasconcelos M S, Fulco U L and Albuquerque E L 2017 Optical Materials 72 756 –

764

[11] Sunku S S, Ni G X, Jiang B Y, Yoo H, Sternbach A, McLeod A S, Stauber T, Xiong L, Taniguchi T, Watanabe K, Kim P, Fogler M M and Basov D N 2018 Science 362 1153–1156

[12] Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V and Firsov A A 2005 Nature -London- 438 197–200

[13] Albuquerque E and Cottam M 1993 Physics Reports 233 67–135

[14] Cottam M G and Tilley D 2004 Introduction to surface and superlattice excitations 2nd ed (CRC Press)

[15] Fei Z, Andreev G O, Bao W, Zhang L M, McLeod A S, Wang C, Stewart M K, Zhao Z, Dominguez G, Thiemens M, Fogler M M, Tauber M J, Castro-Neto A H, Lau C N, Keilmann F and Basov D N 2011 Nano Letters 11 4701–4705 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Accepted Manuscript

[16] Koppens F H L, Chang D E and Garc´ıa de Abajo F J 2011 Nano Letters 11 3370–3377 [17] Chiu K W and Quinn J J 1974 Phys. Rev. B 9(11) 4724–4732

[18] Stauber T, Peres N M R and Geim A K 2008 Phys. Rev. B 78 085432

[19] Ferreira A, Peres N M R and Castro Neto A H 2012 Phys. Rev. B 85(20) 205426 [20] Berman O L, Gumbs G and Lozovik Y E 2008 Phys. Rev. B 78(8) 085401 [21] Melo L G C 2015 J. Opt. Soc. Am. B 32 2467–2477

[22] Michalski K A 2019 Journal of Quantitative Spectroscopy and Radiative Transfer 226 19 – 28 ISSN 0022-4073

[23] Mikhailov S A and Ziegler K 2007 Phys. Rev. Lett. 99 016803

[24] Gusynin V P, Sharapov S G and Carbotte J P 2006 Journal of Physics: Condensed Matter 19 026222

[25] Gusynin V P, Sharapov S G and Carbotte J P 2006 Phys. Rev. Lett. 96(25) 256802 [26] Madani A and Entezar S R 2013 Physica B: Condensed Matter 431 1 – 5

[27] Madani A and Entezar S R 2014 Superlattices and Microstructures 75 692 – 700

[28] Mak K F, Sfeir M Y, Wu Y, Lui C H, Misewich J A and Heinz T F 2008 Phys. Rev. Lett. 101(19) 196405

[29] Sesion P D, Albuquerque E L, Vasconcelos M S, Mauriz P W and Freire V N 2006 The European Physical Journal B - Condensed Matter and Complex Systems 51 583–591

[30] Constantinou N C and Cottam M G 1986 Journal of Physics C: Solid State Physics 19 739–747 [31] Li Y, Qi L, Yu J, Chen Z, Yao Y and Liu X 2017 Opt. Mater. Express 1228–1239

[32] Mills D L and Burstein E 1974 Reports on Progress in Physics 37 817–926

[33] Davies C L, Patel J B, Xia C Q, Herz L M and Johnston M B 2018 Journal of Infrared, Millimeter, and Terahertz Waves 39 1236–1248

[34] Camley R 1987 Surface Science Reports 7 103 – 187 ISSN 0167-5729

[35] Caloz C, Al`u A, Tretyakov S, Sounas D, Achouri K and Deck-L´eger Z L 2018 Phys. Rev. Applied 10(4) 047001

[36] Arakawa E T, Williams M W, Hamm R N and Ritchie R H 1973 Phys. Rev. Lett. 31(18) 1127–1129 [37] Hu B, Zhang Y and Wang Q 2015 Nanophotonics 4(1) 383–396

[38] Armelles G, Cebollada A, Garc´ıa-Mart´ın A and Gonz´alez M U 2013 Advanced Optical Materials 1 10–35 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Accepted Manuscript