Proceedings of the 15th IEEE International Conference on Nanotechnology July 27-30, 2015, Rome, Italy
Fundamental Properties of Plasmonic Propagation in Graphene Nanoribbons Giampiero Lovat, Rodolfo Araneo, Paolo Burghignoli
George W. Hanson
Sapienza University of Rome Via Eudossiana 18, 00184 Rome, Italy Email:
[email protected]
University of Wisconsin-Milwaukee Milwaukee, Wisconsin 53211, USA E-mail:
[email protected]
graphene nanoribbon
Abstract—Fundamental properties of plasmon modes propagating along graphene nanoribbons are investigated by means of a Method-of-Moments approach together with a dyadic description of a nonlocal graphene conductivity. Modal propagation, field distributions, and characteristic impedances are studied in details showing the dramatic effects of spatial dispersion.
z y
I. I NTRODUCTION Planar electromagnetic structures incorporating graphene layers have received considerable attention in the last few years for waveguide, antenna, and shielding applications [1]–[5]. In particular, graphene nanoribbons (GNRs) have been proposed in various configurations, as possible interconnects in integrated circuits [6]–[8]. As is now well-known, GNRs support a variety of propagation modes, including surface-plasmon modes [9], [10]. Surface plasmon polariton (SPP) propagation in graphene nanostructures has recently attracted interest for the possibility of strong confinement of electromagnetic energy at subwavelength scales, tuned and controlled by a gate voltage or through chemical doping. Other applications of graphene plasmons include optical signal processing, light modulation, sensing, spectral photometry, quantum optics, and nonlinear photonics. Here we aim at studying the fundamental properties of the dominant SPP mode supported by a monolayer GNR placed on top of a dielectric substrate, employing a rigorous full-wave method-of-moments (MoM) approach in the spectral domain. The GNR is modeled through an appropriate tensor surface-impedance boundary condition which fully takes into account spatial-dispersion effects: this means that graphene is represented through a dyadic conductivity whose elements depend also on the spectral wavenumbers [11], [12]. Although many studies have been devoted to SPP propagation in GNRs [9], [10], to the best of our knowledge none has fully addressed the issue of nonlocality. In the present study, we investigate how spatial dispersion affects the propagation properties along a GNR giving particular attention to the dispersion curves, current profiles, field distributions, and characteristic impedances of the fundamental SPP mode. II. M ODAL A NALYSIS The electromagnetic problem under analysis is sketched in Fig. 1. It consists of a graphene sheet of width w along the x direction (i.e., a GNR) deposited on a a laterally-infinite
978-1-4673-8156-7/15/$31.00 © 2015 IEEE
w
εr dielectric substrate
d x
Fig. 1. Graphene nanoribbon (GNR) over a dielectric substrate. Parameters of the reference structure: w = 200 nm, d = 400 nm, and εr = 10.2.
dielectric substrate of thickness d along the z direction and relative permittivity εr . All the units are in the SI system and a time-harmonic variation ejωt is assumed and suppressed throughout. Propagation along the y direction, i.e., modes with a spatial dependence e−j qˆy y , is also assumed. In general, a graphene sheet can be modeled as a conductive sheet with a dyadic surface conductivity [11]. In the absence of magnetic bias and neglecting spatial dispersion, graphene can be simply characterized, using the Kubo formalism, by a scalar local conductivity σ, which depends on external and internal parameters, e.g., radian frequency ω, temperature T , a phenomenological scattering rate Γ = 1/τs (where τs is the relaxation time depending on a variety of factors and determined experimentally), and the chemical potential μc : π μc q 2 kB T σ (ω) = −j e 2 ln 2 1 + cosh π kB T ω − j/τs (1) thus showing a Drude-like behavior. In (1) −qe is the electron charge, vF 106 m/s is the Fermi velocity in graphene, while and kB are the reduced Planck and the the Boltzmann constants, respectively. When a more refined model is considered which takes into account spatial dispersion, the graphene conductivity has a dyadic form, whose elements have been derived in [12] for arbitrary q values and with the Bhatnagar-Gross-Krook (BGK) model (which allows for including charge diffusion and deriving the correct quantum capacitance). It can easily be shown that the Electric Field Integral Equation (EFIE) which solves the homogeneous problem can
176
be expressed in the form +∞
convergence of the spectral integral in (7) [13]. This is not trivial, since it can be shown that in a full-q formulation [12] ρxx = O qx2 , ρxy = ρyx = O (qx ) , ρyy = O (qx ) . (8)
˜ EJ (qx , qˆy ) − ρ (qx , qˆy ) · J ˜ S (qx ) e−jqx x dqx = 0 , G
−∞
(2) EJ ˜ for x ∈ , where G (qx , qy ) is the EJ-type spectral, domain Green’s function for planar layered media, ρ = σ −1 ˜ S (qx ) is the Fourier transform of the is the dyadic resistivity, J modal surface current JS (x), and qˆy = β − jα is the assumed propagation constant. Equation (2) is clearly a nonlinear eigenvalue equation, whose solution identifies a propagation mode of the nanoribbon through its propagation constant qˆy,i and its surface current distribution JS,i .
− w2
w 2
A. Numerical solution In principle, a standard MoM procedure may be used to solve the eigenvalue problem in (2). In particular, the x and y components of the modal current can be expanded in M and N entire-domain (e.g., sinusoidal) basis functions Jx,m and Jy,n , respectively, as JSx (x) = JSy (x) =
M
m=1 N −1
Am Jx,m (x)
(3a)
Bn Jy,n (x)
(3b)
n=0
where Am and Bn are unknown coefficients and 2πm Jx,m (x) = sin x m = 1, 2, . . . , M w 2πn Jy,n (x) = cos x n = 0, 1, . . . , N − 1 . w
Hence, if the classical Galerkin test procedure were used, the ln integrand functions of the matrix elements Z˜yy (ˆ qy ) would −1 show an asymptotic behavior as qx without alternating sign thus leading to non-convergent integrals. A suitable choice of test functions is instead 2πp Tx,p (x) = sin x p = 1, 2, . . . , M (9a) w 2π (l + 1) x + (−1)l l = 0, 1, . . . , N −1 . Ty,l (x) = cos w (9b) The Fourier transform of (9a) is the same as in (5a) and it has an asymptotic behavior as O qx−2 , while the Fourier transform of (9b) is w l 2 2 (−1) l sin qx 2 ˜ Jy,l (qx ) = 8π , (10) qx3 w2 − 4qx l2 π 2 with an asymptotic behavior as O qx−3 which ensures the convergence of all the integrals (7). The eigenvalues qˆy have to be found searching for the complex zeroes of the determinant of the square (M + N ) matrix in (6), i.e., det [Z (ˆ qy )] = 0 ,
(4a) (4b)
It should be observed that the Fourier transforms of the basis functions in (4) are m m (−1) sin qx w2 ˜ Jx,m (qx ) = 4πjw (5a) 2 2 (qx w) − (2mπ) n (−1) qx sin qx w2 J˜y,n (qx ) = 2w2 (5b) 2 2 . (qx w) − (2nπ) By applying a non-Galerkin test procedure with test functions Ti,p/l (x) (i = x, y), (2) can be recast in the following matrix form pm pn Z˜xx (ˆ qy ) Z˜xy (ˆ qy ) Am =0 (6) lm ln Bn Z˜yx (ˆ qy ) Z˜yy (ˆ qy )
(11)
and a dispersion analysis qˆy (f ) can easily be performed tracking the relevant complex pole as a function of the frequency f. B. Current, field, and characteristic impedance calculation Once the propagation constant qˆy has been found, the eigenfunction associated to the matrix Z (ˆ qy ) can easily be computed and the eigenfunction corresponding to the minimum eigenvalue (which is null within the machine accuracy) gives the expansion coefficients in (3); this way, both the longitudinal and transverse current components can easily be computed. Once the modal current JS (x, y) on the GNR surface is known, the modal field components can then be computed as e−j qˆy y Er (x, y, z) = 2π
where p = 1, ..., M , l = 0, ..., N − 1, and +∞
hq ˜ ˜ EJ Zrs (ˆ qy ) = T˜r,h (−qx ) G ˆy )−ρrs (qx , qˆy ) J˜s,q (qx )dqx rs (qx , q −∞
(7) with r, s = x, y, h = p, l, q = m, n, and ρrs are the components of the dyadic resistivity ρ. In particular, suitable test functions need to be chosen in order to ensure the
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e−j qˆy y Hr (x, y, z) = 2π
+∞
˜ EJ G ˆy ; z) J˜x (qx )+ rx (qx , q
(12a)
−∞
˜ EJ +G ˆy ; z) J˜y (qx ) e−jqx x dqx ry (qx , q +∞
˜ HJ G ˆy ; z) J˜x (qx )+ rx (qx , q
(12b)
−∞
˜ ˜ HJ (q , q ˆ ; z) J (q ) e−jqx x dqx +G x y y x ry (12c)
1.2
1000 q /k
Local model
|J /J
β /k
Full-q BGK model
| 1 0.8 Local model
0.6
100
Full-q BGK model
0.4 0.2
α/k 10 0.01
0.1
1
0 -100
10
-50
0
50
100
x [nm]
f [THz]
Fig. 2. Normalized phase (β/k0 ) and attenuation (α/k0 ) constants as functions of frequency f for a structure as in Fig. 2.
Fig. 3. Normalized y-component of the modal current Jy across the nanoribbon for a structure as in Fig. 1 at f = 10 THz. 10
˜ EJ and G ˜ HJ are the EJ- and HJwhere r = {x, y, z} while G type spectral-domain dyadic Green’s functions, respectively [14]. The characteristic impedance is then calculated according to the current-power definition [15] ∞ ∞ P Zc = 2 = | I| 2
−∞ −∞
Full-q BGK formulation
10
10
[E (r) × H∗ (r)] · uy dxdz 2 w 2 Jy (x) dx − w
Local model
|E|/|E|
10
,
(13)
GNR 10 300
2
where the integral in the numerator of (13) is recursively computed over an increasing surface centered on the GNR and truncated when the prescribed accuracy is reached. III. R ESULTS In order to show the effects of spatial dispersion on the electromagnetic properties of GNRs, we consider a GNR structure with τs = 0.5 ps, μ = 0 eV, T = 300 K, w = 200 nm , d = 400 nm, and εr = 10.2 (e.g., a common dielectric substrate such as alumina). The values of the scattering time τs corresponds to a mean free path of several hundred nanometers. In Fig. 2 the dispersion properties of the fundamental SPP mode supported by the considered GNR are reported in a logarithmic scale. In particular, the normalized phase (β/k0 = e{qy /k0 }) and attenuation (α/k0 = − m{qy /k0 }) constants are plotted as functions of frequency by adopting two different conductivity models for graphene, i.e., a local formulation and a full-q BGK formulation. The latter fully takes into account the nonlocal effects which are seen to be particularly pronounced in the lowest (microwave) frequency range. In particular, it can be seen that in the considered case ignoring spatial dispersion leads to relative errors in the calculation of the phase and attenuation constants which are both larger than 50% at 10 THz: at low frequencies such errors are still larger. In the considered frequency range, including spatial dispersion increases the attenuation constant of the plasmon mode, except in the higher part of the frequency spectrum:
350
400 z [nm]
450
500
Fig. 4. Amplitude of the normalized electric field as a function of z for a structure as in Fig. 1 at f = 10 THz.
in such a case, e.g., at f = 10 THz, spatial dispersion leads to an attenuation constant much lower than that of the local formulation. Instead, as concerns the phase constant, spatial dispersion makes the plasmon mode much slower at lower frequencies and much faster at higher frequencies. In Fig. 3, the amplitude of the longitudinal y-component Jy of the modal current along the nanoribbon (normalized to its maximum value) is reported at the operating frequency f = 10 THz for a structure as in Fig. 2. The transverse x-component Jx is not reported since its value is more than three-orders of magnitude smaller than the longitudinal component Jy . It can be seen that ignoring spatial dispersion leads to completely erroneous current profiles. In particular, the almost constant behavior typical of the fundamental mode in microwave microstrip lines is obtained using a purely local model, whereas the correct spatially-dispersive full-q conductivity formulation determines a variable current profile across the ribbon with a minimum at the edges and a maximum near the center of the graphene nanostrip. In Fig. 4 the amplitude of the normalized electric fields are reported as functions of z for x = 0 at f = 10 THz. It can be seen that the electric field is well concentrated near the ribbon, although the inclusion of spatial dispersion greatly reduces the amount of field confinement.
178
100
8 10 Z [Ω ]
Local model
Re[Z ]
4 10
q /k
Full-q BGK model
β /k
10
0
1 Local model
-4 10
α/k
Im[Z ]
Full-q BGK model 0.1
-8 10 0.01
0.1
1
0
10
0.5
μ [eV]
1
1.5
f [THz] Fig. 5. Real and imaginary parts of the characteristic impedance of a GNR as in Fig. 1.
Fig. 6. Normalized phase (β/k0 ) and attenuation (α/k0 ) constants as functions of the chemical potential μc for a structure as in Fig. 1 with εr = 1 at the operating frequency f = 1 THz.
R EFERENCES Finally, the real and imaginary parts of the characteristic impedance are reported as functions of frequency in Fig. 5 for a high-permittivity GNR as in Fig. 1. As it can be seen, at low freqeuncies both the real and imaginary parts are of the order of kΩ and there is a significant discrepancy between the local model and the full-q BGK formulation: such a discrepancy disappears beyond 1 THz for the real part and beyond 2 THz the imaginary part become one order of magnitude smaller than the real part. As is well known, one of the most attractive characteristic of graphene is the possibility of externally controlling its electrical conductivity by modifying the chemical potential μc : this can be obtained either by applying an electrostatic bias or by doping [11]. The analysis of inhomogeneous GNRs is out of the scope of the present investigation and it requires different numerical tools [16]; rather, it is here interesting to investigate the effects of a uniform variation of the chemical potential μc on the propagation features of the dominant plasmon mode in GNRs, i.e., we consider a GNR with a homogeneous conductivity whose value can be tuned at a fixed frequency by externally varying the chemical potential μc . In Fig. 6 the normalized phase and attenuation constants are reported as functions of the chemical potential μc for a structure as in Fig. 1, but with εr = 1 at the operating frequency f = 1 THz. As for the dispersion analysis, two different conductivity models are considered, i.e., a local model and a full-q BGK model. It can be seen that the influence of the chemical potential on the propagation properties of GNRs is the same for the local and the spatially-dispersive model. By increasing μc , the real part of the conductivity (which is responsible of losses) decreases and, accordingly, the attenuation constant of the propagating plasmon mode also decreases: such a decreasing is more pronounced for small values of μc . Although reported in Fig. 6 for a particular configuration of GNRs, such a behavior is general, as verified by numerous simulations.
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