ACCEPTED MANUSCRIPT
Surface and bulk plasmon-polaritons in semiconductor photonic crystals
with embedded graphene sheets
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/ab198d
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.
Surface and bulk plasmon-polaritons in
semiconductor photonic crystals with embedded
graphene sheets
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. A theoretical study is made for the bulk and surface plasmon-polaritons in one-dimensional semi-infinite layered semiconductor photonic crystals with graphene interlayers. The retarded dispersion relations are calculated in a general way and applied to two cases, depending on whether a graphene sheet is present or absent at the first interface of the semi-infinite structure. We apply the result to doped GaAs as the constituent material, considering the layers to be arranged in a periodic fashion with alternating layer thicknesses. Both s- and p-polarized modes are evaluated with the effects of retardation on the polaritons being included in the general expressions. Simplifications, particularly for the surface modes, are studied for the case when retardation effects are small. Through numerical examples, we illustrate the important role of the graphene sheets in modifying the plasmon-polariton properties (focusing on the region near the bulk plasma frequency). A strong dependence on the mode polarization is found.
Keywords: Plasmon-polaritons, Graphene, Semiconductor, Surface modes, 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
The term metamaterials is often used for artificial materials that are modified such that they possess optical (or other) properties that do not exist in naturally-occurring materials [1]. These optical properties include the concentration of energy at communications frequencies [2], absorption [3], reflection [4] and filtering [5]. An important example of metamaterials are photonic crystals (PCs). These are structures that have periodicity and are artificially constructed using one or more dielectric materials as multilayer arrays (one-dimensional PCs), two-dimensional arrangements of rods or wires (two-dimensional PCs), or spheres in tridimensional arrangement (three dimensional PCs). In this paper, we focus on PCs in only one-dimensional arrays. Studies of PCs started in the late 1980s following work by Yablonovitch [6], who showed for the first time the possibility of controlling the propagation of electromagnetic waves in a semiconductor, including the ability to guide and confine the light, by forming bands and gaps regions in the spectrum analogous to the electrons within bulk semiconductors. Although PCs are different from the original metamaterials (limited to materials that exhibit negative refractive index), the term “metamaterial” nowadays is typically applied to any artificial structure with designed material properties (e.g., optical properties in our case) [1, 4, 7].
On the other hand, polaritons are excitations that have electric and magnetic fields extending beyond the boundaries of a PC. These fields can couple with the excitations of the other medium (or layer) in a PC, but through Bloch’s theorem to a set of collective excitations may arise that extends throughout the PC. These collective modes, called plasmon-polaritons in our case, are characterized in PCs by a Bloch wave number Q for the direction normal to the interfaces. The propagation of a polariton mode corresponds to Q lying within the new Brillouin zone associated with the PC periodicity such that we have the condition, 0 ≤ |Q| ≤ π/L, where L is the size of the unit cell of the PC. In a semi-infinite PC (which can be formed by truncating an infinite PC by an external surface), there is a reduction of symmetry that can lead to the appearance of so-called surface polaritons. These surface modes are characterized by the localization of their electromagnetic field amplitudes in the vicinity of the terminating interfaces of the PC. These amplitudes decay with distance away from the first truncated layer (usually in an exponential fashion). The modes can exist in the regions above, below, and between the bulk polariton bands. More specifically, surface plasmon polaritons (SPPs), are surface polaritons that can appear on metal-insulating interfaces. In this case, they can be understood as surface charge density waves that arise by coherent fluctuations of electrons that propagate along the interface between a conductor (or semiconductor) and a dielectric medium [8, 2]. They are of interest for the manipulation of light at small scales compared to the wavelength of light, such as sub-wavelength optics.
Graphene is a two-dimensional material that consists of a flat sheet of carbon atoms, forming a monoatomic layer [9, 10, 11]. Its electronic structure results in properties that translate to a higher mechanical strength than steel, higher electron mobility
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
than silicon, higher thermal conductivity than copper, a larger surface area than that observed for graphite, and still a lighter material than many others [9]. Also, it is well known in the literature that graphene has excellent transport properties. The term “zero bandgap” semiconductor is often used, due to graphene’s unique two-dimensional energy dispersion that incorporates the so-called Dirac cone in the Brillouin zone [10]. Also, in doped graphene or gated graphene, we should have collective excitations such as plasmons, and plasmon-polaritons [12] with interesting optical characteristics such as deep subwavelengths and high confinement of optical fields [13, 14, 15, 16], in similar way to surface plasmons on metal surfaces. With these properties graphene can serve as a one-atom-thick platform for the study of metamaterials in the infrared and terahertz regions [15, 16, 17]. All these interesting optical properties of graphene give us several applications in photonics [18, 19]. On the theoretical side, there are a few studies about photonic crystals that include graphene [20, 21]. Recently, one of us reported studies for light waves in photonic multilayers with graphene between the layers [22, 23]. On the experimental side, Shiramin et al. [24] demonstrated a compact optical switch realized by integrating a graphene layer with a silicon photonic crystal cavity, and Rybin et al. [25] reported the enhanced optical absorbance the graphene monolayer in combination with photonic crystal slab. Also Yan et al. [26] demonstrated that transparent photonic devices based on graphene/insulator stacks, which are formed by depositing alternating wafer-scale graphene sheets and thin insulating layers, may completely absorb infrared light at certain resonant wavelength structures. Chang et al. [27] reported the experimental realization of a multilayer structure of alternating graphene and Al2O3
layers, called hyperbolic metamaterials. These have a structure similar to the metal-dielectric multilayers designed to have an extremely anisotropic optical response in which the permittivities associated with different polarization directions exhibit opposite sign, in mid-infrared frequencies.
In this paper we consider the surface and bulk plasmon-polariton modes in semiconductor photonic crystal with embedded graphene sheets between the interfaces. In section 2 we present a modification of the plasmon-polariton theory for semiconductor superlattices with two-dimensional electron gas layers [28] to apply to our present case where there are graphene sheets at interfaces. The graphene sheets are characterized by a quite different optical conductivity with competing contributions. Then in section 3, we show numerical results when the semiconductor material is taken to be GaAs (either doped or intrinsic) in the various layers. These results illustrate the important role of the graphene sheets and also contrast with the behaviour for semiconductor-semiconductor interfaces with two-dimensional electron gas layers. Finally, in section 4, we present the conclusions of this work.
2. Theory
We consider a periodic photonic crystal (PC) consisting of alternating unit cells AB juxtaposed to form the periodic array ABABAB..., where the building blocks A and B
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
are layers of a semiconductor material (chosen as GaAs) with thicknesses dA and dB,
respectively, and corresponding dielectric function (ω), as in figure 1. At each interface between A and B it is assumed that there a monolayer of graphene (for which we assume the same Fermi energy) whose the thickness is negligible compared with dA and dB.
The graphene monolayer is modeled by the frequency-dependent optical conductivity, that is obtained from a Kubo formula [29], and includes contributions from both the intraband (σintra) and interband (σinter) transitions. We write σ(ω) = σintra + σinter,
where [30, 31, 32] σintra = e2 4¯h i 2π 16kBT ¯ hω ln[2 cosh( µc kBT )], (1) σinter = e2 4¯h ( 1 2 + 1 πtan −1 ¯hω − 2µc 2kBT ! − ln " (¯hω + 2µc)2 (¯hω − 2µc)2− (2kBT )2 #) . (2)
Here e is the electronic charge, kB is the Boltzmann constant, and T is the absolute
temperature. The chemical potential µc can be controlled by an gate voltage applied
between the edges of the graphene sheets. We stress that the interband contribution plays the leading role around the absorption threshold, ω ≈ 2µc , while the intraband
contribution is important at relatively low frequencies compared with µc, so there is a
competing effect between the two terms.
ە ۖ ۖ ۖ ۔ ۖ ۖ ۖ ۓ
…
B ە ۖ ۖ ۖ ۔ ۖ ۖ ۖ ۓ z=0 dA dB z=L z=2L z=(n-1)L z=nL A B A B 𝜎′ 𝜎 𝜎 𝜎 𝜎…
C x z ە ۖ ۖ ۖ ۔ 1 ۖ ۖ ۖ ۓ n=0 2 A 𝜎 𝜎 n 𝜎 m=0 m=1 m=2 m=3 m=4 m=2n-1 m= 2nFigure 1. (Color online) The geometry and choice of coordinate axes for the alternating 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.
The AB unit cells and the individual (A or B) semiconductor layers are labeled as indicated in figure 1. Initially, we consider an infinitely extended structure, for which Bloch’s theorem is applicable, and later the effects of an external surface are introduced. Both p- and s-polarizations are considered.
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
2.1. Bulk-like collective modes in p-polarization
In order to obtain the plasmon-polariton modes of the infinite PC described above, me may utilise Maxwell’s equations within each semiconductor medium, subject to the standard electromagnetic boundary conditions at the interfaces. For the transverse magnetic (TM) modes (or p-polarization) within medium A (or B) of the mth layer, the electric field component is
Exj(m) =hA(m)1j exp[ikzj(z − zm))] + A (m)
2j exp[−ikzj(z − zm)]
i
, (3)
where here, and henceforth, we omit for convenience an overall common factor of exp(ikxx − iωt) in expressions for the field components. The factors A
(m)
1j and A (m) 2j
(j = A or B; m = 1, 2, . . .) are the amplitudes for the forward- and backward-travelling waves, respectively, and L = dA+ dB is the periodicity length. We note that the index
m labels the individual layers, while the index n labels the unit cells (see figure 1), in such a way that zm−1 − zm is equal to dA if m is odd or dB if m is even. We have
employed the notation in equation (3) that kzj = (jω2/c2 − kx2)1/2, where kx is the
in-plane component of the wave vector (parallel to the x − y plane), ω is the angular frequency and c is the velocity of the light in vacuum. Using ~H = (0c2/iω) ~∇ × ~E, the
corresponding magnetic field component in medium A ( or B) can be expressed as
Hyj(m) = (0jω kzj )hAm)1j exp[ikzj(z − zm)] − A (m) 2j exp[−ikzj(z − zm)] i . (4)
We consider the magnetic permeability in each medium is equal to unity, i.e., µj = 1.
Also we have used ~∇ · ~D = 0 to relate the amplitudes of the electric and magnetic fields. By applying the electromagnetic boundary conditions, requiring the continuity of the tangential component of the electrical field Exj(m) and the discontinuity of the magnetic field to be equal to the current density ~J = σ(ω) ~E) at the interfaces z = (n − 1)L + dA and z = nL (see figure 1), we can relate the electromagnetic fields
for cell n to those for cell n + 1. After re-expressing the amplitudes in term of those for index n, we obtain the following equation
A(n)1A A(n)2A = MAMABMBMBA A(n+1)1A A(n+1)2A . (5)
Here the matrix Mαβ corresponds to propagation across the interface α|β (where α, β
denote A or B) and is given by
Mαβ = 1 2 " 1 + (βkzα/αkzβ) + σα 1 − (βkzα/αkzβ) + σα 1 − (βkzα/αkzβ) − σα 1 + (βkzα/αkzβ) − σα # , (6)
with σα = σkzα/α0ω. Also Mγ represents the propagation of the light wave across a
layer γ (= A or B) and is given by
Mγ = " exp(−ikzγdγ) 0 0 exp(ikzγdγ) # , (7)
where dγ is the corresponding thickness of the layer γ. 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
Finally, by taking into account the translational symmetry of the periodic system through the application of Bloch’s theorem, it is easy to obtain the dispersion relation in the case of the one-dimensional photonic crystal for the bulk plasmon-polariton modes as (see, e.g., [8, 33, 34])
cos(QL) = 1/2T r(M ), with M = MAMABMBMBA. (8)
Here M is a unimodular matrix and T r represent the trace of the matrix M [33]. As a check, on putting σ = 0 it is possible after some algebra to recover the results for the previous case without graphene (see [28, 33, 8]).
2.2. Surface modes in p-polarization
For the surface modes we consider a semi-infinite photonic crystal, having the same alternating structure as in figure 1, but with a terminating surface at z = 0 and the region z < 0 is assumed to be filled by a medium C (usually taken as vacuum). Considering p-polarization as before, the electric and magnetic fields in the region z < 0 can be written as ExC(0) =hA(0)1Cexp(kCz) i , (9) HyC(0) = −0ωs ikC h A(0)1Cexp(kCz) i . (10)
Here we omit the common overall factor exp(ikxx − iωt) as previously, and we define
kC = (kx2− Cω2/c2)1/2 inside medium C. For simplicity, we shall assume k2x > Cω2/c2
to ensure that kC takes only real and positive value. Then we apply the usual
electromagnetic boundary conditions, as stated beforehand, at the z = 0 surface to obtain A(0)1C = A(1)1A+ A(1)2A, (11) CA (0) 1C + A(A (1) 1A− A (1) 2A) = kxσ0 i0ω [A(1)1A+ A(1)2A]. (12)
We are assuming here that a layer of type A is the outermost layer in the PC. The localized surface plasmon-polaritons correspond to decaying solutions for the amplitudes of the electromagnetic fields with respect to distance from the surface plane z = 0. In other words, we must have
A(n+1)1A A(n+1)2A = exp(−βL) A(n)1A A(n)2A (13)
with Re(β) > 0 as a requirement for localization. Using equations (11) - (13), together with equation (5), we can find the implicit dispersion relation for the surface modes [33] as
M11+ M12/λ = exp(βL) = M22+ M21λ, (14)
where Mi,j (i, j = 1, 2) are the elements of the matrix M defined in equation (5), and λ
is a surface-dependent parameter given here by
λ = (r − σC+ 1)/(r + σC− 1). (15) 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
We have defined the ratio r = A/C, with σC = σ0kx/(iC0ω). Also σ0, which denotes
the optical conductivity of the graphene sheet at the surface z = 0 (see figure 1), will be chosen as equal either to σ (the same as in the interior of the PC) or zero (in the absence of a graphene sheet at the external surface). We will consider both cases numerically later. Note, for simplicity, we are quoting the surface mode results in the case where retardation effects are unimportant. Hence we chose kC ≈ kx in the surface case,
following [28], and this has already been used in deriving equations (11) and (12). It is possible to find an analytical form for the parameter exp(−βL), as in reference [28], but using the transfer matrix method adopted here this result is replaced by equation (14). For a special case where we have frequency-dependent optical conductivity σ(ω) ∝ ω−1, as in reference [35], the system will have a dispersion relation approximately like a two-dimentional electron gas at interfaces, and, therefore, we can define a cut-off for the surface mode in a similar way as reported by Giuliani and Quinn [36, 37].
2.3. Bulk-like collective modes in s-polarization
The theory in the case of s-polarization (TE modes) can be developed in an analogous fashion. First, for the bulk-like plasmon-polariton modes of an infinitely-extended PC, we assume now that the y-component of the electric field in the mth layer is given by Eyj(m) =hA(m)1j exp[ikzj(z − zm))] + A
(m)
2j exp[−ikzj(z − zm)]
i
. (16)
Also, using ~H = (0c2/iω) ~∇ × ~E, the relevant magnetic field component in the mth
layer is Hxj(m) = −kzj µ0ω h Am)1j exp[ikzj(z − zm)] − A (m) 2j exp[−ikzj(z − zm)] i , (17)
where µ0 is the vacuum permeability.
It is now a case of following similar steps as for the p-polarization in subsection 2.1. The procedure involves applying the standard electromagnetic boundary conditions at a pair of adjacent interfaces in a cell. The same form of matrix expression as in equation (5) is obtained, but with the matrix Mαβ redefined as
Mαβ = 1 2 " 1 + (kzβ/kzα) + σ0α 1 − (kzβ/kzα) + σ0α 1 − (kzβ/kzα) − σ0α 1 + (kzβ/kzα) − σ0α # , (18)
with σ0α = σωµo/kzα. Then equations (7) and (8) again apply for the dispersion relations
of bulk plasmon-polaritons in the PC.
2.4. Surface modes in s-polarization
The calculation again proceeds very similarly to that in subsection 2.2 for p-polarization. To summarize, we obtain the same implicit dispersion relation as in Eq. (14), provided we make the redefinition
λ = σ0C/(2 − σ0C), (19) 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
with σ0C = σ0/kxi. When equation (14) has been solved (usually numerically), the
derived value for β must again satisfy equation (13) with the requirement that Re(β) > 0 to ensure the localization. It should be noted that if we take the special case of σ0 = 0 in equation (19), we have λ = 0. In these circumstances it can readily be established (also see comments in [8, 33]) that equation (14) cannot be satisfied, implying no surface modes exist in s-polarization for this case.
3. Numerical results
In this section we present numerical examples to illustrate the results of the preceding theory for the semiconductor PC with embedded graphene sheets. We take the semiconductor material as gallium arsenide (GaAs) with a frequency-dependent dielectric function (ω) given by
(ω) = ∞ 1 − ω2 p ω(ω + iγ) ! . (20)
where γ is a damping constant. For the plasma frequency we take ωp = 9.48 THz
corresponding to doped GaAs with a charge density of 1.8 × 1017 cm−3 and ∞ = 12.9
(see, e.g., [38]). In the case of undoped (or intrinsic) GaAs we would simply have ωp ' 0.
For the region outside the semi-infinite PC we take C = 1 for vacuum. Also we choose
temperature T = 300 K and µc = 0.2 eV for the graphene sheets [30, 31, 32]. In
performing the numerical calculations we scale σ by dividing by the factor 0c. This is
done in order to rewrite expressions in terms of the fine structure constant e2/4π 0hc¯
(≈ 1/137), where c is the light speed in vacuum. This constant arises naturally from equations (1) and (2). Also we will ignore the role of damping by setting γ = 0, since it is typically small in GaAs compared to the frequencies of interest in this work.
In the following series of figures, we will present the numerical dispersion relations for plasmon polaritons in the one-dimensional PC with graphene, focusing on three cases: (i) all layers are considered as the intrinsic semiconductor GaAs with dielectric constant = ∞= 12.9 (figures 2 and 3); (ii) the A layers of the PC are intrinsic GaAs and the B
layers are doped GaAs (figures 4 and 5); and (iii) all layers of the PC are doped GaAs (figures 6, 7, and 8). In these figures, the bulk modes occur as bands separated by gaps, as expected, and are shown as the shaded areas (brown areas in color online version). These bulk bands are bounded by the curves for QL = 0 and QL = π. The surface modes, characterized by black lines, lie between and above the bulk bands, subject to the constraint k2
x > Cω2/c2. That means that any surface mode in these plots lies to
the right of the so-called vacuum light line (VLL), shown as a dashed straight line, given by ω = ckx/
√
0. For reference, we also show the semiconductor light line (SLL) for
the PC given by ω = ckx/
√
∞. Also it is convenient to introduce dimensionless units,
chosen as Ω = ω/ω0 for frequency and K = ckx/ω0 (with ω0 = ωp = 9.48 THz). Next,
we will detail the dispersion relationships for each case individually.
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 2. Plasmon-polariton bulk and surface dispersion relation for p-polarization in case (i). The straight lines VLL and SLL represent the vacuum light line and the semiconductor light line (see the text), respectively. Also, we have considered the optical conductivity at the z = 0 external surface to be (a) σ0 = σ and (b) σ0= 0.
Case (i)
Figure 2 shows the plasmon-polaritons dispersion relation in p-polarization for case (i), initially when we have σ0 = σ (see figure 2a) as the surface condition and then when σ0 = 0 (see figure 2b). As we see in these figures, the bulk bands have two regions of behaviour, for frequencies above and below Ω ≈ 0.38. For ω below about 0.38 and for K ≈ 0 there is a large gap, sometimes called the graphene induced gap [39, 23], which extends out to the SLL of GaAs, as defined earlier. This behaviour has not been
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
previously reported, to our knowledge. It is interesting that if there are no graphene sheets between the layers, this gap ceases to exist [35]. For K values to the right of the SLL, and Ω < 0.62, we have two bulk bands which start at the origin, and eventually merge into a narrow band, approximating a straight line with a constant group velocity given by
∂ω ∂kx
= υc, (21)
with υ ≈ 0.026 (estimated using the graphical approximation in Qtgrace software [40]) for large values of K. For frequencies with Ω > 0.38, we have a typical dispersion relation for a dielectric PC, with gaps and bulk modes [35], but limited by the SLL (for a review see [41]). Our results are consistent with the dispersion relation presented in the work of Bludov et al. [42], where they studied the reflection spectra of electromagnetic radiation from a stack of graphene layers at oblique incidence. Also, we can find in [43] similar results for dispersion relation presented here, but only for special case of bulk modes. For the surface modes, a direct comparation is not possible because of the different geometry and/or assumptions.
The surface modes in figure 2 provide a very rich and interesting behaviour. First, in figure 2a there is a graphene sheet at the exterior surface at z = 0 (see figure 1), whereas the graphene sheet at z = 0 is absent for figure 2b. For case (a) we have two surface modes in the region between the vacuum light line and the semiconductor light line, labeled as S2 and S5 in this figure. One of those surface modes, labeled by S2, is in the induced graphene gap, starting at the vacuum light line and finishing at the semiconductor light line. The other surface mode, labeled by S5, is between the bulk bands, around Ω = 1.0, starting near the vacuum light line, at the top of one the bulk bands, and finishing at the bottom of the other bulk band. For case (b) we have three surface modes. One of these is between the bulk bands as described above, S1 (see the labels in the figure 2a). It starts at the origin of the plot in figure 2b, and it comes together with the adjacent bulk bands, taking the same group velocity as given by equation (21) for large K. For the other two surface modes we have the following behavior: S3 starts close to the bulk band delimited by the semiconductor light line, and the other one, S4, starts close to the bulk band at Ω ≈ 0.38 and K ≈ 2.5. At larger K they combine into one straight-line branch, with a constant group velocity given by (21), but with υ ≈ 0.041. The differences between the case (a) and (b) can be explained by the fact when we have a graphene monolayer at z = 0 there are more possibilities for surface plasmon-polaritons to exist due to the graphene-air interface and its coupling with the other layers.
In figure 3 we present the bulk and surface dispersion relations for plasmon-polaritons in case (i), but considering now the results for s-polarization. The behaviour is much less interesting than for p-polarization, as can be seen by comparison with figure 2, and so we present only the results when σ0 = σ (the other case for σ0 = 0 is very similar). We note that there is only one surface mode that originates close to the third bulk band and near to the VLL, labeled as S1. It remains closely associated with this
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. The same as in Fig. 2, but for s-polarization taking σ0 = σ at the external
surface.
bulk band up to higher values of K. The bulk modes have the form of bands (and gaps) in a roughly parabolic fashion, similar to the case of a bulk semiconductor dispersion relation [33]. We note also that the lower boundary of the first (lowest) bulk band starts at Ω ≈ 0.38 and at larger K values it approaches asymptotically the semiconductor light line.
Case (ii)
The bulk and surface plasmon-polariton dispersion relations for p-polarization in case (ii) are presented in figure 4. We now have the A and B layers corresponding to intrinsic and doped GaAs, respectively. As in figure 2, we study the effects of including the graphene sheet at the first layer (capping layer) at z = 0 (see figure 4a), and in the absence of this graphene sheet (see figure 4b). In this case (ii) we again observe, as in figure 2, two bulk modes to the right side of the semiconductor light line, but now with different types of behavior. The lowest band in frequency, starting at the origin, has a constant group velocity as given by equation (21), but with υ ≈ 0.034 for large values of K, while the other one starts at around Ω ≈ 1.0 and asymptotically joins with the lowest band for large K.
In this same figure, we have plotted the surfaces modes, for which we can identify distinctive behaviours in different regions. In figure 4a we have two surface modes in the area between the vacuum light line and the semiconductor light line, identified by S2, S6, and S7. Also, we have three branches, labeled by S1, S3, and S5, to the right side of the semiconductor light line, that will eventually (for large K) join up (together also with two bulk-mode branches) with a group velocity corresponding to υ ≈ 0.034
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. Plasmon-polariton bulk and surface dispersion relations for p-polarization in case (ii), as explained in the text. Also, like in figure 2, we have considered the optical conductivity at the surface at z = 0 to be (a) σ0= σ and (b) σ0= 0
in equation (21). The other surface mode, S4, start close to VLL and finishing at the bottom of the other bulk band in Ω ≈ 0.9. The others bulk modes have the same behaviour as we have commented on for figure 2, but with more large gaps, except in the region around Ω ≈ 1.5 and K ≈ 3.75, where we have a narrowing in the gap region. In the case of figure 4b we have only one surface mode, S1, which starts at the SLL and joins with bulk bands for high values of K. We conclude, therefore, comparing cases (ii) and (i), that when we consider medium B as a doped semiconductor the group velocity of these plasmon-polariton modes for high K is modified. The induced graphene gap
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
still exists when K ≈ 0, and it is larger than in case (i).
Figure 5. The same as in figure 4, but for s-polarization taking σ0 = σ at the external surface.
In figure 5 we present both the bulk and surface plasmon-polariton dispersion relations for s-polarization in case (ii). We can see that there are no surface modes in frequency range considered in this paper. On the other hand, the bulk modes have gaps and bulk bands organized in a similar way to those in figure 3 with parabolic-like dispersion relations.
Case (iii)
Next, in figure 6 we present the surface and bulk modes for case (iii), described in the text, for p-polarization taking σ0 = σ, when we have a charge sheet at the external surface z = 0. For the bulk modes, we can see that the dispersion relation has three distinct regions. For high frequency (higher than Ω = 1.07) and low K, we have a “parabolic-type behavior” just as in the bulk dispersion relation (see figure 2 in [33]), but with an asymptotic limit that approaches the SLL as K increases. Also, we can see that have band gaps regions with this parabolic behavior, but it will disappear at high values of K. In this region we can see only one surface mode, labeled by S0, that is between the bands around Ω = 1.5. This mode is similar to the other ones in figures 2a, labeled by S5, and 4a, labeled by S7. For the low-frequency region, starting from Ω = 0 up to Ω = 1.0, we have a large band gap, where there are no surfaces modes. As we commented earlier, this behaviour was already reported in the literature in dielectric PCs and photonic quasicrystals as the induced graphene bandgap [39, 23]. The third region, is a narrow region that lies between ω/ωp = 1.0 and ω/ωp = 1.07, for low K 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
values. The dispersion relation in this range is rich in detail, and so for clarity, we present an expanded version of the rectangular region in figure 7).
Figure 6. Plasmon-polariton bulk and surface dispersion relations. Here we have considered the optical conductivity at the surface z = 0 as σ0 = σ . The lines represent the light line in vaccum (external medium) and the light line in semiconductor given by equation ω = ckx/
√
∞. The boxed area is expanded in figure 7.
First, in figure 7a for the expanded region, we note that for small K and Ω we have no bulk bands. At slightly larger values we have a surface mode that starts in the vacuum light line and finishes in a bulk band, labeled by S2. Second, for low K and frequencies Ω > 1, we have a gap extending from Ω = 1 up to about Ω = 1.07, which disappears for higher K. This behaviour has no counterpart for plasmon-polariton dispersion relation regarding the bulk modes in semiconductor superlattices (see [28, 33]). Third, for Ω > 1.07 we have a wide bulk band up to about Ω = 1.2. This behavior is common for the plasmon-polariton semiconductor dispersion relation, and also we have more gaps at higher frequencies (as shown in figure 6). In this gap region, we can observe a surface mode, labeled by S6, that starts at the vacuum light line and follows very close to the bulk band until disappearing at the end of this gap, at K ≈ 1.4 and Ω ≈ 1.07. For high frequency and high K, we have an asymptotic limiting of the bulk modes with the semiconductor light line, as commented earlier. For high K and in the frequency range 1.05 < Ω < 1.15 we have two surface modes, marked as S5 and S4, that eventually merge at large K and join up (not shown) having a small positive group velocity corresponding to υ ≈ 0.026. There are also two bulk bands that merge at large K with a group velocity corresponding to υ ≈ 0.016 (see also figure 6). Between these two bulk bands, for low K we have another surface mode, labeled by S3, that starts close to the semiconductor light line. Also, we have another surface mode that starts around to Ω = 1.0 at the VLL, labeled by S1, which follows very close to the bulk band and vanishes in a short
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 7. Plasmon-polariton bulk and surface dispersion relations. Here we have considered the optical conductivity at the surface z = 0 as (a) σ0 = σ and (b) σ0 = 0, with σ given by the sum of σintra and σinter, intra and interband transition
contributions, respectively, defined by equations (1) and (2).
range of K.
Figure 7b also shows the bulk and surface modes for Ω close to unity, but now for σ0 = 0. Here we have only two distinct surfaces modes, which is fewer than in figure 7a because the z = 0 boundary condition is more restrictive, as we have commented before. One of the surface modes is between the lowest two bulk bands, labeled by S1, starting at K = 1.0 and Ω = 1.01, and following the bulk bands for higher K. Eventually it has a constant group velocity corresponding to υ = 0.016. The other one, S2, has the same
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 8. Plasmon-polariton bulk and surface dispersion relation for s-polarization. Here we have considered the optical conductivity at the surface z = 0, as σ0= σ .
behaviour as in figure 7a, labeled by S6 in that figure.
For completeness, in figure 8 we present numerical results for plasmon-polariton bulk and surface modes for s-polarization, taking σ0 = σ. We observe that there is only one surface mode, S1, that originates close to the bulk band below ω/ωp = 1.0 and lies
to the right of the vacuum light line. The bulk modes have gaps and bands organized in a parabolic form, similar to our previous cases for s polarization.
4. Conclusions
We have extended previous theories for bulk and surface plasmon polaritons in superlattices (see [28]) to apply to one-dimensional PCs with embedded graphene sheets in place of a two-dimensional electron gas (2DEG). We have studied the dispersion relations for plasmon polaritons corresponding to three different arrangements of the semiconductor material: (i) where all layers are intrinsic (undoped) GaAs , (ii) where the layers A and B alternate intrinsic GaAs and doped GaAs, and (iii) where all the layers are doped GaAs. In our results, we compare all these cases and we find that the inclusion of a doped semiconductor medium in the PC will affect the dispersion relation mainly at low frequencies below ωp (equal to 9.48 THz in our examples). Also, we find differences
in the group velocities of various modes when the scaled wave vector K becomes large in all these cases. For frequencies above ωp, the doping in the semiconductor layers lead to
a similar parabolic behavior as in a bulk-doped semiconductor in the dispersion relation (for a review see [33]).
We particularly studied the effect of a graphene sheet being either present or absent
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
at the first layer on the surface modes when we have a truncation of the PC at the external surface z = 0. These results show, in general, that we can have more surface modes in the dispersion relations when we consider a graphene sheet present at this capping layer. This effect of the graphene-air interface can be understood as being analogous to the case of a metal-air interface, where it produces an extra coupling between the evanescent mode formed at this first interface with modes localized in the second and others layers inside the PC. Also, these surface properties can be significantly enhanced and extended by the engineering of this external surface, which can result in applications for nano-guiding, sensing, light-trapping and imaging based on near-field techniques [44].
Keeping in mind that is difficult to have perfect graphene sheets experimentally, we note that this work can be extended to include defects (sometimes called “disorder” [45]) in graphene layers or imperfections in semiconductor layers, as well as a surface defect layer at z = 0, in similar way as in the works of Hawrylak and Quinn [46]. We could consider this and other extensions (e.g., to include the plasmon-phonon interaction or phonon-polariton terms in the dielectric function) in further works.
We mention finally that inelastic light scattering, or Raman scattering, experiments (similar to those reported by Olego et al. [47] for other superlattices) could be used to test the predictions of the theory presented here. Furthermore, the structures proposed here can be experimentally fabricated, following [48] where the authors obtained a graphene-based one-dimensional photonic crystal, as confirmed by Raman spectra and optical image of CVD, and demonstrated the excitation of surface modes using a prism coupling technique. For a more complete comparison with experiment, the spectral intensities associated with the various plasmon-polariton excitations can be calculated using linear response theory, and this will be a topic for further studies.
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 Engineering Research Council (NSERC) of Canada (grant RGPIN-2017-04429).
References
[1] Engheta N and Ziolkowski R W 2006 Metamaterials: Physics and Engineering Explorations (Wiley)
[2] Han Z and Bozhevolnyi S I 2013 Reports on Progress in Physics 76 016402 [3] Kravets V G, Schedin F and Grigorenko A N 2008 Phys. Rev. B 78(20) 205405
[4] Joannopoulos J D, Johnson S G, Winn J N and Meade R D 2008 Photonic Crystals: Molding the Flow of Light 2nd ed (Princeton University Press)
[5] Xu T, Wu Y R, Luo X and Guo L J 2010 Nature communications 1 59 [6] Yablonovitch E 1987 Phys. Rev. Lett. 58(20) 2059–2062
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
[7] Wuestner S and Hess O 2014 Progress in Optics (Progress in Optics vol 59) ed Wolf E (Elsevier) pp 1 – 88
[8] Cottam M G and Tilley D 2004 Introduction to surface and superlattice excitations 2nd ed (CRC Press)
[9] Geim A K and Novoselov K S 2007 Nature Materials 6 183–191
[10] 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
[11] 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
[12] Xiao S, Zhu X, Li B H and Mortensen N A 2016 Frontiers of Physics 11 117801 [13] Hwang E H and Das Sarma S 2007 Phys. Rev. B 75(20) 205418
[14] Mikhailov S A and Ziegler K 2007 Phys. Rev. Lett. 99(1) 016803 [15] Jablan M, Buljan H and Soljaˇci´c M 2009 Phys. Rev. B 80 245435
[16] 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
[17] Low T and Avouris P 2014 ACS Nano 8 1086–1101
[18] Bonaccorso F, Sun Z, Hasan T and Ferrari A 2010 Nature Photonics 4 611–622 [19] Sule N, Willis K J, Hagness S C and Knezevic I 2014 Phys. Rev. B 90(4) 045431
[20] Zhan T, Shi X, Dai Y, Liu X and Zi J 2013 Journal of Physics: Condensed Matter 25 215301 [21] Namdar A, Onsoroudi R F, Khoshsima H and Sahrai M 2018 Superlattices and Microstructures
115 78 – 87
[22] Silva E F, Costa C H, Vasconcelos M S and Anselmo D H A L 2018 Optical Materials (in press) [23] Costa C H, Vasconcelos M S, Fulco U L and Albuquerque E L 2017 Optical Materials 72 756 –
764
[24] Shiramin L A, Xie W, Snyder B, Heyn P D, Verheyen P, Roelkens G and Thourhout D V 2018 IEEE Photonics Technology Letters 30 157–160 ISSN 1041-1135
[25] Rybin M G, Pozharov A S, Chevalier C, Garrigues M, Seassal C, Peretti R, Jamois C, Viktorovitch P and Obraztsova E D 2012 physica status solidi (b) 249 2530–2533
[26] Yan H, Li X, Chandra B, Tulevski G S, Wu Y, Freitag M, Zhu W, Avouris P and Xia F 2012 Nature nanotechnology 7 5 330–4
[27] Chang Y C, Liu C H, Liu C H, Zhang S, Marder S R, Narimanov E E, Zhong Z and Norris T B 2016 Nature Communications 7 10568
[28] Constantinou N C and Cottam M G 1986 Journal of Physics C: Solid State Physics 19 739 [29] Falkovsky L A and Pershoguba S S 2007 Phys. Rev. B 76 153410
[30] Madani A and Entezar S R 2013 Physica B: Condensed Matter 431 1 – 5 [31] Madani A and Entezar S R 2014 Superlattices and Microstructures 75 692 – 700
[32] 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
[33] Albuquerque E L and Cottam M G 1993 Physics Reports 233 67 – 135
[34] Albuquerque E L and Cottam M G 2004 Polaritons in Periodic and Quasiperiodic Structures (Elsevier Science)
[35] Fan Y, Wei Z, Li H, Chen H and Soukoulis C M 2013 Phys. Rev. B 88 241403 [36] Giuliani G F and Quinn J J 1983 Phys. Rev. Lett. 51 919–922
[37] Giuliani G F, Qin G and Quinn J 1984 Surface Science 142 433 – 437
[38] Kersting R, Heyman J N, Strasser G and Unterrainer K 1998 Phys. Rev. B 58 4553–4559 [39] Depine R A 2016 Graphene Optics: Electromagnetic Solution of Canonical Problems (Morgan &
Claypool Publishers)
[40] Winterr A 2017 QtGrace version 2.6 La Jolla, CA, United States of America URL https://sourceforge.net/projects/qtgrace/
[41] Yeh P, Yariv A and Hong C S 1977 J. Opt. Soc. Am. 67 423–438
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
[42] Bludov Y V, Peres N M R and Vasilevskiy M I 2013 Journal of Optics 15 114004
[43] Bludov Y V, Peres N M R, Smirnov G and Vasilevskiy M I 2016 Phys. Rev. B 93 245425 [44] Takayama O, Bogdanov A A and Lavrinenko A V 2017 Journal of Physics: Condensed Matter 29
463001
[45] Chaves A J, Peres N M R and Pinheiro F A 2015 Phys. Rev. B 92 195425 [46] Hawrylak P and Quinn J 1986 Solid State Communications 59 781–784
[47] Olego D, Pinczuk A, Gossard A C and Wiegmann W 1982 Phys. Rev. B 25(12) 7867–7870 [48] Sreekanth K V, Zeng S, Shang J, Yong K T and Yu T 2012 Scientific Reports 2 737
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
