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Graphene-photonic crystal switch
Optics Communications 321 (2014) 150–156
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Graphene-photonic crystal switch A. Wirth Lima Jr.a,b,n, A.S.B. Sombra a,b a b
Laboratory of Telecommunications and Materials Science and Engineering, Federal University of Ceará, Fortaleza, Ceará, Brazil Department of Teleinformatics Engineering, DETI, Center of Technology, Federal University of Ceará (U.F.C.), Fortaleza, Ceará, Brazil
art ic l e i nf o
a b s t r a c t
Article history: Received 13 July 2013 Received in revised form 17 December 2013 Accepted 21 January 2014 Available online 11 February 2014
We have designed and analyzed a graphene-based photonic crystal directional coupler working as a switch, which is embedded in a SOI photonic crystal slab with triangular lattice of dielectric rods. The directional coupler has two W1 (one missing row of dielectric rods) waveguides and the coupling region is coated with a graphene nanoribbon with width W ¼50 nm. We use an electric gate to modify the graphene chemical potential to obtain the desired change of the graphene equivalent permittivity. Thus, we can drive the directional coupler to get its transition from the bar state to the cross state and vice-versa. & 2014 Elsevier B.V. All rights reserved.
Keywords: Directional coupler Graphene Photonic crystal Switch
1. Introduction Photonic crystal structures are composed of dielectric constant basic block (unit cell), periodically repeated in space, so that the unit cell is a lattice with discrete translational symmetry. In a photonic crystal there is a dielectric constant lattice in one, two or three dimensions. That periodicity is defined by the lattice vectors, which have magnitudes on the order of a few hundred nanometers, working with modes usually (but not necessarily) in the infrared region. Indeed, even the physics of 3D photonic crystal slabs is based upon the 2D triangular lattice, where the Bloch modes are the eigenmodes, whose propagation direction is defined by the group velocity direction, so that the resulting electromagnetic energy distribution of the Bloch waves matches the periodicity of the lattice. In analogy to the atomic lattice, a bandgap in the dispersion diagram can be created, which does not allow light with a frequency in this bandgap to propagate. To allow the light confinement in the third direction in a 2D photonic crystal, we utilize the concept of total internal reflection at the interface of the photonic crystal slab and the cladding material. Photonic crystal slab platform is used in most photonic crystal (PhC) devices because its fabrication process is easier than that of the 3D PhC slab platform. Generally, the so-called slab structure consists of a thin 2D photonic crystal in a high-index membrane surrounded by air. However, this type of structure is not easily integrable into a chip. Hence,
n Corresponding author at: Laboratory of Telecommunications and Materials Science and Engineering, Federal University of Ceará, Fortaleza, Ceará, Brazil. E-mail address: [email protected] (A. Wirth Lima Jr.).
http://dx.doi.org/10.1016/j.optcom.2014.01.052 0030-4018 & 2014 Elsevier B.V. All rights reserved.
we used a SOI-based hybrid structure, where the lower cladding consists of a structured oxide layer, whose effective refractive index is close to 1 [1]. Thus, the device with this structure is easier to integrate into a chip than a membrane and is almost symmetrical. The PhC structure we choose is a silicon on insulator (SOI) slab with triangular lattice of dielectric rods. Given that this device will operate in the C-band of the ITU, which covers the wavelength range from 1528.77 nm to 1560.61 nm [2], we adopted the refractive index of the silicon dielectric rods, n ¼ 3.481 (at a wavelength equal to 1.55052 μm). The dielectric rods are embedded in a substrate (SiO2), whose refractive index is ns ¼1.528 (at a wavelength equal to 1.55052 μm). In a purely 2D PhC, the transverse magnetic (TM) modes (odd modes) have no electric field in the horizontal middle plane (x–y plane) of the PhC and have magnetic field in the x–y plane. Moreover, the TM modes have an electrical field in the z direction, and no magnetic field in the z direction. TE modes (even modes) present the reverse of what occurs in TM modes. On the other hand, the finite height of the photonic crystal slabs leads polarization mixing and the modes are not purely TM (or TE)-polarized anymore. However, if the (x, y)-plane in the middle of the slab is a mirror plane of the structure, the first-order modes are very similar to the corresponding modes existing in infinite 2D photonic crystals. Furthermore, in the (x, y)-mirror plane itself, these modes are a purely TM (or TE) polarized. In other words, in this case the polarization mixing is quite small and the approximation TE for even modes (TE-like) and TM for odd modes (TM-like) can be assumed. While the effective vertical wavelength of the TE-like modes is more dependent on the high dielectric material, this effective vertical wavelength of the TM-like modes is more dependent on
A. Wirth Lima Jr., A.S.B. Sombra / Optics Communications 321 (2014) 150–156
low dielectric material. For this reason, the gap in the hole slabs opens up at a smaller thickness than for the rod slabs. The light cone consists of optical power radiating vertically (region outside of the slab). The boundary between guided and radiation modes is called the light line. In a PhC slab consisting of a uniform material above and below the slab core, the light line is constituted by the modulus of the wave vectors divided by the cladding refractive index. The cladding refractive index must be lower than the average index of the slab core in order to confine light in the vertical direction. A defect in the PhC structure can enable modes within the bandgap. For example, by removing a line of rods, light can be guided in a PhC. Given the peculiar diffraction phenomena of the photonic crystals, the dimensions of the photonic crystal-based devices range from a few to tens of micrometers. Photonic crystal enabled the achievement of new optical devices, for example, waveguides [3], lasers [4], splitters [5], antennas [6], optical switches [7,8], etc. On the other hand, graphene is a two-dimensional (2D) allotropic form of carbon comprising of atoms periodically arranged in an infinite hexagonal structure. Taking into account that the lattice vectors of graphene are based on the distance between two neighboring carbon atoms (0.192 nm), graphene technology is embedded in the field of nanophotonics. The physics involving the two-dimensional graphene is based on the fact that electrons in this structure have a tiny rest mass and respond quickly to electric fields. On the other hand, plasmons in a graphene microribbon are collective oscillation of electrons, whose frequency depends on how rapidly waves in this electron sea travel back and forth between its edges. Light of the same frequency applied in the graphene microribbon gives rise to a “resonant excitation,” (a large increase in the strength of the oscillation). The strength of the light-plasmon coupling can be affected by the concentration of charge carriers (electrons and holes). We can modify the concentration of charge carriers in graphene, which can easily be increased or decreased simply by applying a strong electric field (electrostatic doping). It is noteworthy that when the chemical potential of graphene is greater than half the photon energy, intraband transitions at the conduction band dominate, so that graphene behaves like a metal. Hence, graphene can support transverse magnetic (TM) polarized surface plasmon polaritons (SPPs). Due to the graphene surface plasmons (GSPPs), graphene ribbons of micrometric widths can operate as a waveguide into photonic integrated circuits (PICs) [9–11]. However, the propagation length, where the field amplitude of the SPP falls to 1/e of the initial value is L¼VFτ, where VF ¼ 106 m/s is the Fermi velocity and τ is the momentum relaxation time. Graphene enables the propagation of modes in the terahertz frequency range between 300 GHz and 1.8714 Hz ¼ 187 THz (λ ¼ 1600 nm), which had never been explored for telecommunications, due to the lack of a technology that would provide devices and transmission media with the appropriate parameters for operation in this range of frequencies. We designed and analyzed a switch based on a directional coupler embedded in a photonic crystal structure with coupling length compatible with the propagation length allowed in a graphene waveguide. The coupling region of the directional coupler is coated with a graphene nanoribbon, thus yielding a graphene based waveguide. Even though we implanted a graphene based waveguide in the coupling region of the directional coupler described above, we use the graphene nanoribbon only to modify the coupling region dielectric constant. Thus, we can control the directional coupler to leave the cross state and to go to the bar state (and vice versa), via alteration of the graphene chemical potential.
151
We believe that this graphene-photonic crystal switch is one of the most compact switch achieved up to this present moment, so that it could be the basis for future communication networks operating in the range of terahertz and infrared. Moreover, the use of an electric gate in a graphene nanoribbon to control the medium permittivity may be useful in other types of compact devices, which can be used in photonic integrated circuits (PICs). This paper is organized as follows. In Section 2, we present the graphene_photonic crystal directional coupler working as a switch. The switch control is presented in Section 3. Finally, in Section 4 we show our conclusions.
2. Graphene_photonic crystal directional coupler The coupler is embedded in the silicon on insulator (SOI) slab as shown in Section 1. As the photonic band gap (PBG) and the gap–midgap ratio of a SOI slab is a function of the silicon rods radius we used the Plane Wave Expansion (PWE) to find the silicon rods radius that provides the highest gap–midgap ratio (rr ¼ 0.23, gap–midgap ratio ¼ 28.83%). On the other hand, the PBG width of a PhC slab depends on the height of the slab. However, by PWE we can observe that the slab height (h) between 1.5a and 2.2a, where “a” is the lattice constant, provides the highest PBG. Hence, we adopted h¼2a. The directional coupler has two W1 (one missing row of dielectric rods) waveguides and the coupling region is coated with a graphene nanoribbon with width W¼50 nm. The reason for the insertion of graphene nanoribbon is that we can easily control its refractive index by use of the electric field effect, since the work function of graphene can be adjusted as the gate voltage tunes the Fermi level across the charge neutrality point [12]. Hence, we can use the electrical gate as a command signal working in the coupling region. The increase of the refractive index graphene nanoribbon (due to the action of the electric gate) causes the decrease of the coupling coefficient value (and viceversa. See details below). Based on the above details if we consider that the coupler was designed to operate in the bar state, the decrease of the coupling coefficient should be sufficient to bring the coupler to work in the cross state. The substrate on which graphene is embedded must be carefully chosen to prevent deterioration of quality of the system. For example, graphene on SiO2 is highly disordered, exhibiting characteristics that are far inferior to the expected intrinsic properties of graphene [13–15]. Thus, we inserted a hexagonal boron nitride (h-BN) layer between SiO2 and the graphene nanoribbon, because graphene on h-BN substrates has mobilities and carrier inhomogeneities that are almost an order of magnitude better than devices on SiO2. h-BN is an insulating isomorph of graphite with boron and nitrogen atoms occupying the inequivalent A and B sublattices in the Bernal structure. Hexagonal boron nitride (h-BN) is also known as “white graphite” and has similar (hexagonal) crystal structure as of graphite. In this case, the density-independent mobility due to charged-impurity Coulomb (long-range) scattering is m¼ 60,000 cm2/V s [16]. According to the super mode method, we know that the minimal coupling length (LC) of a symmetrical directional coupler is given by
π π 2π ¼ ðbar stateÞ ðcross stateÞ; Lc ¼ βodd βeven Δβ Δβ
Lc ¼
ð1Þ
By means of the Equations (2) and (3), if it is considered a lowintensity continuous-wave (CW) beam, we can get the amplitude (A) and intensity (I) of the traveling signal as follows [17]: Aa ðzÞ ¼ A0 cos ðkxÞ and I a ðzÞ ¼ A20 ð cos ðkxÞÞ2
ð2Þ
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Ab ðzÞ ¼ iA0 sin ðkxÞ and I b ðzÞ ¼ A0 2 ð sin ðkxÞÞ2
ð3Þ
In Eqs. (2) and (3) we have neglected the self-phase modulation and the cross-phase modulation effects. We can notice that for kx¼ π/2, the input optical power travels to the exit port located in the other waveguide, so that the coupler is working in the cross state. However for kLc ¼ π the optical power goes to the exit port in the same waveguide, so that the coupler is working in the bar state. Therefore, the coupler may have a coupling length L ¼ mLc ¼ mðπ =2kÞ, where m is an integer and Lc is the minimal coupling length. According to what we have written above, the coupler is working in the cross state as long as “m” is odd. It is noteworthy that for even “m”, the coupler is working in the bar state. From Eq. (1) we have Δβ b Lc ¼ 2π , where Δβb is the difference between the values of the propagation constants referring to the odd and even modes in the bar state. Hence, we must decrease the value of Δβb to the half of its original value, so that our coupler can change from the bar state to the cross state (for a fixed Lc) i.e., we get Δβc L ¼ π , where Δβc ¼ Δβb/2 is the difference between the values of the propagation constants referring to the odd and even modes in the cross state. The photonic bandgap (PBG) of the PhC slab is shown in Fig. 1 (obtained by means of PWE). We chose u¼ 0.285, where u (normalized frequency) ¼a/λ so that a ¼356 nm. Note that this normalized frequency is located near the middle of the PBG and the coupler is working in the frequency range of the optical communications. The PhC coupler structure is shown in Fig. 2. Fig. 3 shows the dispersion relation related to odd and even modes of the coupler, obtained by means of PWE. We can apply Eq. (1) for each normalized frequency and thus obtain the required coupler length. For example, for u ¼0.285 we get Lc ¼1.78 μm (5a), which is the minimal coupler length (bar state). On the other hand, the following equation shows the dependences of the charge carrier density (ns) in graphene [18] ns ¼
μ2c π V 2F ℏ2
Fig. 2. Graphene_photonic crystal structure.
ð4Þ
The momentum relaxation time in graphene is given by
τ¼
μc μ
ð5Þ
eV 2F
being e the elementary charge, VF ¼106 m/s the Fermi velocity in graphene and mc the chemical potential of the graphene. Taking into account Eqs. (4) and (5), if we work, for example, with the chemical potential of the graphene mc ¼0.5 eV, we have that ns ¼1.837 1013 cm 2 and τ ¼3 ps. Hence, the propagation length, where the field amplitude of the SP falls to 1/e of the initial value is L¼ VFτ ¼3 mm. Therefore, even if the graphene nanoribbon was used as a waveguide, it would not be an obstacle to the
Fig. 3. Dispersion relation of the graphene_photonic crystal directional coupler.
propagation mode in the coupling region of the directional coupler. It is noteworthy that the relationship between the chemical potential and the carrier density at the room temperature T¼ 300 K and mc⪢KBT (KB is the Boltzmann constant) is [19]
μc ¼ ℏV F
pffiffiffiffiffiffiffiffi ns π
ð6Þ
We used the following equation in our simulations to find the value of the graphene equivalent permittivity [20]:
εeq ¼ 1 þ
Fig. 1. Dispersion relation of the PhC slab.
is w ε0 t
ð7Þ
In Eq. (7), s is the graphene conductivity, ε0 is the vacuum permittivity and t is the effective graphene thickness [20–22]. Undoped graphene has a simple optical absorption spectrum. Indeed, in the spectral range from infrared to visible light, the absorption exhibits a frequency independent magnitude given by απ ¼2.293%, where α ¼ e2 =ℏc is the fine-structure constant (in cgs
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units). The universal absorption is equivalent to an optical frequency independent sheet conductivity (s0) [23]. On the other hand, the graphene intraband conductivity as a function of the graphene chemical potential can be found by the Drude model, which provides its dependence on chemical potential and relaxation time as follows (mc⪢KBT) [24]:
sintra ðw; μc Þ ¼
iq2 μc ðw þ iτ 1 Þπ ℏ2
ð8Þ
On the other hand, the graphene interband conductivity is given by (mc⪢KBT) [9]:
sinter ðw; μc Þ ¼
q2 i ðℏw þ iτ 1 Þ 2μc 1 þ ln 4ℏ π ℏðw þ iτ 1 Þ þ 2μc
ð9Þ
As seen in Eq. (9) for undoped graphene with mc ¼0, s ¼ s0 ¼ ðq2 =4ℏÞ 6:08 10 5 Ω 1 . Otherwise, if μc o ℏω=2 ¼ 0:4 (at λ ¼1.55 mm) the interband transition occurs, which provides strong absorption of light. From the general equation, where the relationship between the absorption coefficient (for the propagation of light through an absorbing medium), the extinction coefficient k, real refractive index n and thickness d is shown, an equation for the absorption coefficient related to a single undoped graphene layer can be
Fig. 5. Chemical potential versus real and imaginary parts of the graphene equivalent permittivity in the interband domain as well as near the transition between the interband and intraband domains.
rewritten as k¼
λ
4π nd
λ
lnð1 παÞ C 1 ; n
ð10Þ
where real n ¼ 3 is the real part of the refractive index, d the graphene thickness (0.34 nm) and C1 E5.446 mm 1 [25]. However, we use Eq. (5) in our simulations when mc a 0. In Fig. 4 we show the real and imaginary parts of the graphene equivalent permittivity as a function of the chemical potential in the intraband domain. In Fig. 5 we show the real and imaginary parts of the graphene equivalent permittivity as a function of the chemical potential in the interband domain as well as near the transition between the interband and intraband domains.
3. Switch control
Fig. 4. Chemical potential versus real and imaginary parts of the graphene equivalent permittivity in the intraband domain.
A schematic of the switch is shown in Fig. 6. In Ref. [26] there is a detailed transfer process to fabricate graphene-on-BN devices. Gold contacts (not shown in Fig. 6) are attached to the graphene to apply the gate voltage. Exfoliated graphene onto SiO2 is the most used method for obtaining graphene nanoribbons (GNR), since the gate is relatively
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Fig. 6. Schematic of the graphene-photonic crystal switch.
Hence, the graphene conductivity increases with the increase of the concentrations of electrons (holes) induced by positive (negative) gate voltages. Several studies present a schematic view of the electric gate structure to get the electric field effect in graphene [29,30]. The changes in voltage change the conductivity and carrier density of the graphene, so that we can take the correct graphene chemical potential to obtain the desired change of the graphene equivalent permittivity. Thus, we can drive the directional coupler to get its transition from the bar state to the cross state and viceversa. From vf ðphase velocityÞ ¼ w=β and vf ¼ c=n we get w/β ¼ c/n ⇥ n¼ cβ/(2πc/λ)¼ βλ/2π, where n is the refractive index and β is the propagation constant. Therefore, for a given wavelength (λ) we have a relationship between the needed change in the material refractive index to obtain a given change in the propagation constant so that Δn ¼ Δβλ/2π. As already detailed in Section 2 we need to decrease the value of Δβb to the half of its original value in order to get the transition of the directional coupler from the bar state to the cross state. When the control signal is applied, the chemical potential of the graphene nanoribbon is changed. As the graphene conductivity is a function of the chemical potential (Eqs. (8) and (9)), it is
Fig. 7. Chemical potential versus real and imaginary parts of the graphene equivalent permittivity in the region near the transition between the interband and intraband domains.
Fig. 8. Electric field distribution inside the coupler (bar state; εeq ¼12.03 þ0.153i).
easy. Graphene manoribbons are mechanically extracted from bulk graphite crystals onto a SiO2/Si substrate [27]. The conductivity of the GNR does not depend on its crystallographic direction [28]. On the other hand, although the technology progresses rapidly towards obtaining monolayer GNR, thicker flakes are currently found in GNRs, so that the Raman spectra may vary along the GNR. Graphene provides the drive of the Fermi level from the valence to the conduction band simply by applying a gate voltage.
A. Wirth Lima Jr., A.S.B. Sombra / Optics Communications 321 (2014) 150–156
changed too. Hence, we can control the graphene equivalent permittivity according to Eq. (7). On the other hand, with the increase of the refractive index of the graphene nanoribbon the even mode shifts slightly to lower frequencies, whereas the odd mode remains almost unchanged [31]. Therefore, we need to increase the refractive index of the graphene nanoribbon from nb (refractive index in the bar state) to nc ¼ nb þ ðΔβb =2Þλ=2π
ð11Þ
to get the transition of the coupler from the bar state to the cross state. In Fig. 7 we show the real and imaginary parts of the graphene equivalent permittivity as a function of the chemical potential in the region near the transition between the interband and intraband domains. Since for μc 4 0.4 intraband absorption dominates, as we can observe from Figure 7, it is confirmed what has been already detailed above. Therefore, if μc is equal to 0.4027, the graphene equivalent permittivity (εeq) is equal to 12.03 þ 0.153i, so that the coupler is in the bar state, as we can see from Figure 8, which was obtained by means of the finite element method (COMSOL). Taking into account that this result was obtained for TM mode at λ ¼ 1.25 μm, the graphene conductivity is (9.9 7 7.12 5i)/(5 0.357 6) S/m
155
(Equations 8 and 9) and the graphene equivalent permittivity was shown above. We used Eqs. (7, 8, 11) to find the real part of the graphene equivalent permittivity, in order to pass the coupler from the bar state to the cross state (Re( ε eq ) ¼ 14.7). Thus, according to Fig. 7, mc ¼0.4011 if Re( ε eq ) ¼14.7 ( ε eq ¼14.7 þ0.371i; s ¼(2.396 6 8.852 5 i)/(5 0.356 6). In this case, the coupler works in the cross state, as we seen from Fig. 9, which was also obtained by means of the finite element method.
4. Conclusions We have designed and analyzed a graphene-based photonic crystal directional coupler working as a switch, which is embedded in a SOI photonic crystal slab with triangular lattice of dielectric rods, where the silicon dielectric rods are embedded in a SiO2 substrate. The radius of each silicon dielectric rods is 0.23a, where a is the lattice constant. The directional coupler has two W1 (one missing row of dielectric rods) waveguides and the coupling region is coated with a graphene nanoribbon with width W¼50 nm. The reason for the insertion of graphene nanoribbon is that we can easily control its refractive index by use of the electric field effect. We use an electric gate to modify the graphene chemical potential to obtain the desired change of the graphene equivalent permittivity. Thus, we can drive the directional coupler to get its transition from the bar state to the cross state and vice-versa.
Acknowledgments This work was partly sponsored by Coordination for the Improvement of Higher Level- or Education-Personnel (CAPES). The authors thank Dr. Victor Dmitriev Alexandrovic (UFPA) for calculations using the COMSOL. References
Fig. 9. Electric field distribution inside the coupler (cross state; εeq ¼14.7 þ 0.371i).
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Graphene-photonic crystal switch A. Wirth Lima Jr.a,b,n, A.S.B. Sombra a,b a b
Laboratory of Telecommunications and Materials Science and Engineering, Federal University of Ceará, Fortaleza, Ceará, Brazil Department of Teleinformatics Engineering, DETI, Center of Technology, Federal University of Ceará (U.F.C.), Fortaleza, Ceará, Brazil
art ic l e i nf o
a b s t r a c t
Article history: Received 13 July 2013 Received in revised form 17 December 2013 Accepted 21 January 2014 Available online 11 February 2014
We have designed and analyzed a graphene-based photonic crystal directional coupler working as a switch, which is embedded in a SOI photonic crystal slab with triangular lattice of dielectric rods. The directional coupler has two W1 (one missing row of dielectric rods) waveguides and the coupling region is coated with a graphene nanoribbon with width W ¼50 nm. We use an electric gate to modify the graphene chemical potential to obtain the desired change of the graphene equivalent permittivity. Thus, we can drive the directional coupler to get its transition from the bar state to the cross state and vice-versa. & 2014 Elsevier B.V. All rights reserved.
Keywords: Directional coupler Graphene Photonic crystal Switch
1. Introduction Photonic crystal structures are composed of dielectric constant basic block (unit cell), periodically repeated in space, so that the unit cell is a lattice with discrete translational symmetry. In a photonic crystal there is a dielectric constant lattice in one, two or three dimensions. That periodicity is defined by the lattice vectors, which have magnitudes on the order of a few hundred nanometers, working with modes usually (but not necessarily) in the infrared region. Indeed, even the physics of 3D photonic crystal slabs is based upon the 2D triangular lattice, where the Bloch modes are the eigenmodes, whose propagation direction is defined by the group velocity direction, so that the resulting electromagnetic energy distribution of the Bloch waves matches the periodicity of the lattice. In analogy to the atomic lattice, a bandgap in the dispersion diagram can be created, which does not allow light with a frequency in this bandgap to propagate. To allow the light confinement in the third direction in a 2D photonic crystal, we utilize the concept of total internal reflection at the interface of the photonic crystal slab and the cladding material. Photonic crystal slab platform is used in most photonic crystal (PhC) devices because its fabrication process is easier than that of the 3D PhC slab platform. Generally, the so-called slab structure consists of a thin 2D photonic crystal in a high-index membrane surrounded by air. However, this type of structure is not easily integrable into a chip. Hence,
n Corresponding author at: Laboratory of Telecommunications and Materials Science and Engineering, Federal University of Ceará, Fortaleza, Ceará, Brazil. E-mail address: [email protected] (A. Wirth Lima Jr.).
http://dx.doi.org/10.1016/j.optcom.2014.01.052 0030-4018 & 2014 Elsevier B.V. All rights reserved.
we used a SOI-based hybrid structure, where the lower cladding consists of a structured oxide layer, whose effective refractive index is close to 1 [1]. Thus, the device with this structure is easier to integrate into a chip than a membrane and is almost symmetrical. The PhC structure we choose is a silicon on insulator (SOI) slab with triangular lattice of dielectric rods. Given that this device will operate in the C-band of the ITU, which covers the wavelength range from 1528.77 nm to 1560.61 nm [2], we adopted the refractive index of the silicon dielectric rods, n ¼ 3.481 (at a wavelength equal to 1.55052 μm). The dielectric rods are embedded in a substrate (SiO2), whose refractive index is ns ¼1.528 (at a wavelength equal to 1.55052 μm). In a purely 2D PhC, the transverse magnetic (TM) modes (odd modes) have no electric field in the horizontal middle plane (x–y plane) of the PhC and have magnetic field in the x–y plane. Moreover, the TM modes have an electrical field in the z direction, and no magnetic field in the z direction. TE modes (even modes) present the reverse of what occurs in TM modes. On the other hand, the finite height of the photonic crystal slabs leads polarization mixing and the modes are not purely TM (or TE)-polarized anymore. However, if the (x, y)-plane in the middle of the slab is a mirror plane of the structure, the first-order modes are very similar to the corresponding modes existing in infinite 2D photonic crystals. Furthermore, in the (x, y)-mirror plane itself, these modes are a purely TM (or TE) polarized. In other words, in this case the polarization mixing is quite small and the approximation TE for even modes (TE-like) and TM for odd modes (TM-like) can be assumed. While the effective vertical wavelength of the TE-like modes is more dependent on the high dielectric material, this effective vertical wavelength of the TM-like modes is more dependent on
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low dielectric material. For this reason, the gap in the hole slabs opens up at a smaller thickness than for the rod slabs. The light cone consists of optical power radiating vertically (region outside of the slab). The boundary between guided and radiation modes is called the light line. In a PhC slab consisting of a uniform material above and below the slab core, the light line is constituted by the modulus of the wave vectors divided by the cladding refractive index. The cladding refractive index must be lower than the average index of the slab core in order to confine light in the vertical direction. A defect in the PhC structure can enable modes within the bandgap. For example, by removing a line of rods, light can be guided in a PhC. Given the peculiar diffraction phenomena of the photonic crystals, the dimensions of the photonic crystal-based devices range from a few to tens of micrometers. Photonic crystal enabled the achievement of new optical devices, for example, waveguides [3], lasers [4], splitters [5], antennas [6], optical switches [7,8], etc. On the other hand, graphene is a two-dimensional (2D) allotropic form of carbon comprising of atoms periodically arranged in an infinite hexagonal structure. Taking into account that the lattice vectors of graphene are based on the distance between two neighboring carbon atoms (0.192 nm), graphene technology is embedded in the field of nanophotonics. The physics involving the two-dimensional graphene is based on the fact that electrons in this structure have a tiny rest mass and respond quickly to electric fields. On the other hand, plasmons in a graphene microribbon are collective oscillation of electrons, whose frequency depends on how rapidly waves in this electron sea travel back and forth between its edges. Light of the same frequency applied in the graphene microribbon gives rise to a “resonant excitation,” (a large increase in the strength of the oscillation). The strength of the light-plasmon coupling can be affected by the concentration of charge carriers (electrons and holes). We can modify the concentration of charge carriers in graphene, which can easily be increased or decreased simply by applying a strong electric field (electrostatic doping). It is noteworthy that when the chemical potential of graphene is greater than half the photon energy, intraband transitions at the conduction band dominate, so that graphene behaves like a metal. Hence, graphene can support transverse magnetic (TM) polarized surface plasmon polaritons (SPPs). Due to the graphene surface plasmons (GSPPs), graphene ribbons of micrometric widths can operate as a waveguide into photonic integrated circuits (PICs) [9–11]. However, the propagation length, where the field amplitude of the SPP falls to 1/e of the initial value is L¼VFτ, where VF ¼ 106 m/s is the Fermi velocity and τ is the momentum relaxation time. Graphene enables the propagation of modes in the terahertz frequency range between 300 GHz and 1.8714 Hz ¼ 187 THz (λ ¼ 1600 nm), which had never been explored for telecommunications, due to the lack of a technology that would provide devices and transmission media with the appropriate parameters for operation in this range of frequencies. We designed and analyzed a switch based on a directional coupler embedded in a photonic crystal structure with coupling length compatible with the propagation length allowed in a graphene waveguide. The coupling region of the directional coupler is coated with a graphene nanoribbon, thus yielding a graphene based waveguide. Even though we implanted a graphene based waveguide in the coupling region of the directional coupler described above, we use the graphene nanoribbon only to modify the coupling region dielectric constant. Thus, we can control the directional coupler to leave the cross state and to go to the bar state (and vice versa), via alteration of the graphene chemical potential.
151
We believe that this graphene-photonic crystal switch is one of the most compact switch achieved up to this present moment, so that it could be the basis for future communication networks operating in the range of terahertz and infrared. Moreover, the use of an electric gate in a graphene nanoribbon to control the medium permittivity may be useful in other types of compact devices, which can be used in photonic integrated circuits (PICs). This paper is organized as follows. In Section 2, we present the graphene_photonic crystal directional coupler working as a switch. The switch control is presented in Section 3. Finally, in Section 4 we show our conclusions.
2. Graphene_photonic crystal directional coupler The coupler is embedded in the silicon on insulator (SOI) slab as shown in Section 1. As the photonic band gap (PBG) and the gap–midgap ratio of a SOI slab is a function of the silicon rods radius we used the Plane Wave Expansion (PWE) to find the silicon rods radius that provides the highest gap–midgap ratio (rr ¼ 0.23, gap–midgap ratio ¼ 28.83%). On the other hand, the PBG width of a PhC slab depends on the height of the slab. However, by PWE we can observe that the slab height (h) between 1.5a and 2.2a, where “a” is the lattice constant, provides the highest PBG. Hence, we adopted h¼2a. The directional coupler has two W1 (one missing row of dielectric rods) waveguides and the coupling region is coated with a graphene nanoribbon with width W¼50 nm. The reason for the insertion of graphene nanoribbon is that we can easily control its refractive index by use of the electric field effect, since the work function of graphene can be adjusted as the gate voltage tunes the Fermi level across the charge neutrality point [12]. Hence, we can use the electrical gate as a command signal working in the coupling region. The increase of the refractive index graphene nanoribbon (due to the action of the electric gate) causes the decrease of the coupling coefficient value (and viceversa. See details below). Based on the above details if we consider that the coupler was designed to operate in the bar state, the decrease of the coupling coefficient should be sufficient to bring the coupler to work in the cross state. The substrate on which graphene is embedded must be carefully chosen to prevent deterioration of quality of the system. For example, graphene on SiO2 is highly disordered, exhibiting characteristics that are far inferior to the expected intrinsic properties of graphene [13–15]. Thus, we inserted a hexagonal boron nitride (h-BN) layer between SiO2 and the graphene nanoribbon, because graphene on h-BN substrates has mobilities and carrier inhomogeneities that are almost an order of magnitude better than devices on SiO2. h-BN is an insulating isomorph of graphite with boron and nitrogen atoms occupying the inequivalent A and B sublattices in the Bernal structure. Hexagonal boron nitride (h-BN) is also known as “white graphite” and has similar (hexagonal) crystal structure as of graphite. In this case, the density-independent mobility due to charged-impurity Coulomb (long-range) scattering is m¼ 60,000 cm2/V s [16]. According to the super mode method, we know that the minimal coupling length (LC) of a symmetrical directional coupler is given by
π π 2π ¼ ðbar stateÞ ðcross stateÞ; Lc ¼ βodd βeven Δβ Δβ
Lc ¼
ð1Þ
By means of the Equations (2) and (3), if it is considered a lowintensity continuous-wave (CW) beam, we can get the amplitude (A) and intensity (I) of the traveling signal as follows [17]: Aa ðzÞ ¼ A0 cos ðkxÞ and I a ðzÞ ¼ A20 ð cos ðkxÞÞ2
ð2Þ
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Ab ðzÞ ¼ iA0 sin ðkxÞ and I b ðzÞ ¼ A0 2 ð sin ðkxÞÞ2
ð3Þ
In Eqs. (2) and (3) we have neglected the self-phase modulation and the cross-phase modulation effects. We can notice that for kx¼ π/2, the input optical power travels to the exit port located in the other waveguide, so that the coupler is working in the cross state. However for kLc ¼ π the optical power goes to the exit port in the same waveguide, so that the coupler is working in the bar state. Therefore, the coupler may have a coupling length L ¼ mLc ¼ mðπ =2kÞ, where m is an integer and Lc is the minimal coupling length. According to what we have written above, the coupler is working in the cross state as long as “m” is odd. It is noteworthy that for even “m”, the coupler is working in the bar state. From Eq. (1) we have Δβ b Lc ¼ 2π , where Δβb is the difference between the values of the propagation constants referring to the odd and even modes in the bar state. Hence, we must decrease the value of Δβb to the half of its original value, so that our coupler can change from the bar state to the cross state (for a fixed Lc) i.e., we get Δβc L ¼ π , where Δβc ¼ Δβb/2 is the difference between the values of the propagation constants referring to the odd and even modes in the cross state. The photonic bandgap (PBG) of the PhC slab is shown in Fig. 1 (obtained by means of PWE). We chose u¼ 0.285, where u (normalized frequency) ¼a/λ so that a ¼356 nm. Note that this normalized frequency is located near the middle of the PBG and the coupler is working in the frequency range of the optical communications. The PhC coupler structure is shown in Fig. 2. Fig. 3 shows the dispersion relation related to odd and even modes of the coupler, obtained by means of PWE. We can apply Eq. (1) for each normalized frequency and thus obtain the required coupler length. For example, for u ¼0.285 we get Lc ¼1.78 μm (5a), which is the minimal coupler length (bar state). On the other hand, the following equation shows the dependences of the charge carrier density (ns) in graphene [18] ns ¼
μ2c π V 2F ℏ2
Fig. 2. Graphene_photonic crystal structure.
ð4Þ
The momentum relaxation time in graphene is given by
τ¼
μc μ
ð5Þ
eV 2F
being e the elementary charge, VF ¼106 m/s the Fermi velocity in graphene and mc the chemical potential of the graphene. Taking into account Eqs. (4) and (5), if we work, for example, with the chemical potential of the graphene mc ¼0.5 eV, we have that ns ¼1.837 1013 cm 2 and τ ¼3 ps. Hence, the propagation length, where the field amplitude of the SP falls to 1/e of the initial value is L¼ VFτ ¼3 mm. Therefore, even if the graphene nanoribbon was used as a waveguide, it would not be an obstacle to the
Fig. 3. Dispersion relation of the graphene_photonic crystal directional coupler.
propagation mode in the coupling region of the directional coupler. It is noteworthy that the relationship between the chemical potential and the carrier density at the room temperature T¼ 300 K and mc⪢KBT (KB is the Boltzmann constant) is [19]
μc ¼ ℏV F
pffiffiffiffiffiffiffiffi ns π
ð6Þ
We used the following equation in our simulations to find the value of the graphene equivalent permittivity [20]:
εeq ¼ 1 þ
Fig. 1. Dispersion relation of the PhC slab.
is w ε0 t
ð7Þ
In Eq. (7), s is the graphene conductivity, ε0 is the vacuum permittivity and t is the effective graphene thickness [20–22]. Undoped graphene has a simple optical absorption spectrum. Indeed, in the spectral range from infrared to visible light, the absorption exhibits a frequency independent magnitude given by απ ¼2.293%, where α ¼ e2 =ℏc is the fine-structure constant (in cgs
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units). The universal absorption is equivalent to an optical frequency independent sheet conductivity (s0) [23]. On the other hand, the graphene intraband conductivity as a function of the graphene chemical potential can be found by the Drude model, which provides its dependence on chemical potential and relaxation time as follows (mc⪢KBT) [24]:
sintra ðw; μc Þ ¼
iq2 μc ðw þ iτ 1 Þπ ℏ2
ð8Þ
On the other hand, the graphene interband conductivity is given by (mc⪢KBT) [9]:
sinter ðw; μc Þ ¼
q2 i ðℏw þ iτ 1 Þ 2μc 1 þ ln 4ℏ π ℏðw þ iτ 1 Þ þ 2μc
ð9Þ
As seen in Eq. (9) for undoped graphene with mc ¼0, s ¼ s0 ¼ ðq2 =4ℏÞ 6:08 10 5 Ω 1 . Otherwise, if μc o ℏω=2 ¼ 0:4 (at λ ¼1.55 mm) the interband transition occurs, which provides strong absorption of light. From the general equation, where the relationship between the absorption coefficient (for the propagation of light through an absorbing medium), the extinction coefficient k, real refractive index n and thickness d is shown, an equation for the absorption coefficient related to a single undoped graphene layer can be
Fig. 5. Chemical potential versus real and imaginary parts of the graphene equivalent permittivity in the interband domain as well as near the transition between the interband and intraband domains.
rewritten as k¼
λ
4π nd
λ
lnð1 παÞ C 1 ; n
ð10Þ
where real n ¼ 3 is the real part of the refractive index, d the graphene thickness (0.34 nm) and C1 E5.446 mm 1 [25]. However, we use Eq. (5) in our simulations when mc a 0. In Fig. 4 we show the real and imaginary parts of the graphene equivalent permittivity as a function of the chemical potential in the intraband domain. In Fig. 5 we show the real and imaginary parts of the graphene equivalent permittivity as a function of the chemical potential in the interband domain as well as near the transition between the interband and intraband domains.
3. Switch control
Fig. 4. Chemical potential versus real and imaginary parts of the graphene equivalent permittivity in the intraband domain.
A schematic of the switch is shown in Fig. 6. In Ref. [26] there is a detailed transfer process to fabricate graphene-on-BN devices. Gold contacts (not shown in Fig. 6) are attached to the graphene to apply the gate voltage. Exfoliated graphene onto SiO2 is the most used method for obtaining graphene nanoribbons (GNR), since the gate is relatively
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Fig. 6. Schematic of the graphene-photonic crystal switch.
Hence, the graphene conductivity increases with the increase of the concentrations of electrons (holes) induced by positive (negative) gate voltages. Several studies present a schematic view of the electric gate structure to get the electric field effect in graphene [29,30]. The changes in voltage change the conductivity and carrier density of the graphene, so that we can take the correct graphene chemical potential to obtain the desired change of the graphene equivalent permittivity. Thus, we can drive the directional coupler to get its transition from the bar state to the cross state and viceversa. From vf ðphase velocityÞ ¼ w=β and vf ¼ c=n we get w/β ¼ c/n ⇥ n¼ cβ/(2πc/λ)¼ βλ/2π, where n is the refractive index and β is the propagation constant. Therefore, for a given wavelength (λ) we have a relationship between the needed change in the material refractive index to obtain a given change in the propagation constant so that Δn ¼ Δβλ/2π. As already detailed in Section 2 we need to decrease the value of Δβb to the half of its original value in order to get the transition of the directional coupler from the bar state to the cross state. When the control signal is applied, the chemical potential of the graphene nanoribbon is changed. As the graphene conductivity is a function of the chemical potential (Eqs. (8) and (9)), it is
Fig. 7. Chemical potential versus real and imaginary parts of the graphene equivalent permittivity in the region near the transition between the interband and intraband domains.
Fig. 8. Electric field distribution inside the coupler (bar state; εeq ¼12.03 þ0.153i).
easy. Graphene manoribbons are mechanically extracted from bulk graphite crystals onto a SiO2/Si substrate [27]. The conductivity of the GNR does not depend on its crystallographic direction [28]. On the other hand, although the technology progresses rapidly towards obtaining monolayer GNR, thicker flakes are currently found in GNRs, so that the Raman spectra may vary along the GNR. Graphene provides the drive of the Fermi level from the valence to the conduction band simply by applying a gate voltage.
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changed too. Hence, we can control the graphene equivalent permittivity according to Eq. (7). On the other hand, with the increase of the refractive index of the graphene nanoribbon the even mode shifts slightly to lower frequencies, whereas the odd mode remains almost unchanged [31]. Therefore, we need to increase the refractive index of the graphene nanoribbon from nb (refractive index in the bar state) to nc ¼ nb þ ðΔβb =2Þλ=2π
ð11Þ
to get the transition of the coupler from the bar state to the cross state. In Fig. 7 we show the real and imaginary parts of the graphene equivalent permittivity as a function of the chemical potential in the region near the transition between the interband and intraband domains. Since for μc 4 0.4 intraband absorption dominates, as we can observe from Figure 7, it is confirmed what has been already detailed above. Therefore, if μc is equal to 0.4027, the graphene equivalent permittivity (εeq) is equal to 12.03 þ 0.153i, so that the coupler is in the bar state, as we can see from Figure 8, which was obtained by means of the finite element method (COMSOL). Taking into account that this result was obtained for TM mode at λ ¼ 1.25 μm, the graphene conductivity is (9.9 7 7.12 5i)/(5 0.357 6) S/m
155
(Equations 8 and 9) and the graphene equivalent permittivity was shown above. We used Eqs. (7, 8, 11) to find the real part of the graphene equivalent permittivity, in order to pass the coupler from the bar state to the cross state (Re( ε eq ) ¼ 14.7). Thus, according to Fig. 7, mc ¼0.4011 if Re( ε eq ) ¼14.7 ( ε eq ¼14.7 þ0.371i; s ¼(2.396 6 8.852 5 i)/(5 0.356 6). In this case, the coupler works in the cross state, as we seen from Fig. 9, which was also obtained by means of the finite element method.
4. Conclusions We have designed and analyzed a graphene-based photonic crystal directional coupler working as a switch, which is embedded in a SOI photonic crystal slab with triangular lattice of dielectric rods, where the silicon dielectric rods are embedded in a SiO2 substrate. The radius of each silicon dielectric rods is 0.23a, where a is the lattice constant. The directional coupler has two W1 (one missing row of dielectric rods) waveguides and the coupling region is coated with a graphene nanoribbon with width W¼50 nm. The reason for the insertion of graphene nanoribbon is that we can easily control its refractive index by use of the electric field effect. We use an electric gate to modify the graphene chemical potential to obtain the desired change of the graphene equivalent permittivity. Thus, we can drive the directional coupler to get its transition from the bar state to the cross state and vice-versa.
Acknowledgments This work was partly sponsored by Coordination for the Improvement of Higher Level- or Education-Personnel (CAPES). The authors thank Dr. Victor Dmitriev Alexandrovic (UFPA) for calculations using the COMSOL. References
Fig. 9. Electric field distribution inside the coupler (cross state; εeq ¼14.7 þ 0.371i).
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