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Transverse-resonance analysis of dominant-mode propagation in graphene
nano-waveguides
Conference Paper · September 2012
DOI: 10.1109/EMCEurope.2012.6396763
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Transverse-Resonance Analysis of Dominant-Mode
Propagation in Graphene Nano-Waveguides
Giampiero Lovat
Department of Astronautical, Electrical, and Energetic Engineering, “Sapienza” University of Rome
Via Eudossiana 18, 00184 Rome, Italy
[email protected]
Abstract—Dispersion properties of the dominant modes supported by two-dimensional graphene-based nano-waveguides are
studied by means of an exact approach based on the transverseresonance technique and using the equivalent circuit representation of graphene sheets. In particular, in the absence of magnetic
bias, the graphene is characterized by a scalar conductivity
obtained based on semi-classical and quantum-mechanical arguments. Such a description takes into account the possible presence
of static electric biasing fields and/or chemical doping. The modal
features of graphene nano-waveguides are investigated in detail
thus showing their potential in future nano-electromagnetic and
nano-photonic applications.
z
y
I. I NTRODUCTION
Graphene, an atomically thick layer of carbon atoms arranged in a honeycomb structure (see Fig. 1), is increasingly
gaining attention all over the world as potential material for
nanointerconnects from microwave to terahertz frequencies
and beyond [1]–[3]. This is due to its excellent electronic,
mechanical, optical, and thermal properties together with its
unique feature of being only one-atom thick.
From an electromagnetic point of view, graphene can be
described as an infinitesimally thin medium characterized by
a surface conductivity [4], [5]; in particular, a semiclassical
and a quantum-dynamical approach can be used to derive the
conductivity expression which also takes into account the presence of electrostatic and/or magnetostatic bias [5]. Recently,
applications of graphene structures have attracted the attention
of many researchers in the EMC community since graphenebased nanostructures (like graphite nanoplatelets (GNP), fewlayer graphene, and carbon nanowalls) could be investigated
for their potential as nanofillers in nanocomposites or as basic
building blocks in novel RF nanodevices [6].
In this contribution, dispersion properties of dominant modes supported by different two-dimensional (2-D)
graphene-based nano-waveguides are studied with an exact
approach based on the transverse-resonance technique and
using the transverse-equivalent-network (TEN) representation
of a graphene sheet [7]. In particular, the considered nanowaveguides are: i) a graphened dielectric slab, ii) a parallelplate waveguide with two graphene walls (graphene/graphene
PPW), and iii) a parallel-plate waveguide with one graphene
wall and one perfectly-conducting plate (graphene/PEC PPW).
Such an investigation expands the preliminary studies in [8],
[9] by means of a different and simpler approach.
978-1-4673-0717-8/12/$31.00 ©2012 IEEE
x
Fig. 1.
Graphene sheet.
The electromagnetic problem under analysis is thus
sketched in Fig. 2. In general, it consists of an infinite graphene
sheet deposited on a dielectric substrate (typically SiO2 ) of
thickness h and relative dielectric permittivity εr (graphened
slab). A bottom plane may also be present at z = 0, which can
be a perfectly-conducting (PEC) ground plane (graphene/PEC
PPW) or another graphene sheet (graphene/graphene PPW).
z
ε0
h
ε0 εr
0
ε0
graphene plate
Fig. 2.
PEC plate
Examples of two-dimensional (2-D) graphene nano-waveguides.
II. G RAPHENE C ONDUCTIVITY
In general, a graphene sheet can be modeled as a conductive
sheet with a dyadic conductivity. This dyadic form can be due
to the application of a magnetostatic bias together with a nonzero chemical potential μc or to spatial-dispersion effects [10]:
however, the latter can usually be neglected below the THz
regime and in the following analysis we will not consider the
presence of a possible magnetic bias. Therefore graphene will
be characterized by a scalar conductivity σ, which however
depends on different parameters, e.g., frequency f = ω/(2π),
temperature T , a phenomenological scattering rate Γ = 1/(2τ )
(where τ is the relaxation time depending on a variety of
factors and determined experimentally) [5], and the chemical
potential μc (which can be controlled either by doping or by
an applied bias electric field orthogonal to the graphene plate
[10]). In particular, it results
q 2 (ω − j2Γ)
σ=j e
π
+∞ 2
∂nF (−)
− Δ2 ∂nF ()
1
−
d
·
2
∂
∂
(ω − j2Γ) Δ
+∞ 2
+ Δ2
nF () − nF (−)
−
2
2 d
2
(ω − j2Γ) − 4 (/)
Δ
(1)
where −qe is the electron charge, vF 106 m/s is the Fermi
velocity in graphene, is the reduced Planck’s constant, and Δ
is an excitonic energy gap with no effects at room temperature
(so it will be assumed Δ = 0 in what follows). Finally, nF is
the Fermi-Dirac distribution given by
nF () =
1
,
(2)
1+
where is the energy and kB is the Boltzmann’s constant.
Moreover, when electronically controlled, the relation between
the chemical potential μc and the electrostatic bias field Ebias
is given by [10]
+∞
ε0 π2 vF2
Ebias =
[nF () − nF ( + 2μc )] d , (3)
qe
0
e(−μc )/(kB T )
which must be solved numerically for the chemical potential
μc (Ebias ). Also the value of the conductivity σ in (1) can
easily be computed through standard adaptive integration routines. Alternatively, as mentioned above, the chemical potential
is determined by the extra-carrier density ns as
+∞
2
ns =
[nF () − nF ( + 2μc )] d .
(4)
π2 vF2 0
Finally, for Δ = 0, (1) can be approximately evaluated in
a closed form as
qe2 kB T
μc
− kμcT
B
σ −j 2
+ 2 ln 1 + e
, (5)
π (ω − j2Γ) kB T
thus showing a Drude-like behavior. As it will be shown later,
(5) is very accurate at room temperatures and below the THz
regime.
978-1-4673-0717-8/12/$31.00 ©2012 IEEE
z
graphened slab
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
Y0 , kz 0
h
Yσ
Ys , kzs
0
BC at z
=0
Yσ
graphene/graphene PPW
graphene/PEC PPW
Y0 , kz 0
Fig. 3. Transverse-equivalent network (TEN) for the problem of modal
propagation along the waveguides in Fig. 2.
III. T RANSVERSE E QUIVALENT N ETWORK
As is well known, fields in a 2-D isotropic waveguide can
be decomposed into their TMz and TEz parts [11]. Both TMz
and TEz fields inside the slab as well as in free space can be
modeled using equivalent transmission lines (TLs) along the
vertical z axis, starting from the spectral form of Maxwell’s
equations. In general, this TEN model is valid for plane-wave
incidence as well as for the analysis of modal propagation.
As shown in [7], the TEN representation for a graphene
sheet operating in general conditions is represented by a fourport network (which in general couples TMz and TEz fields),
obtained by enforcing the correct boundary conditions in the
Maxwell equations; however, in the absence of both electric
and magnetic bias and neglecting spatial-dispersion effects, the
four-port network reduces to a simple shunt admittance Yσ (or
impedance Zσ = 1/Yσ ) with the same value for TMz and
TEz fields. The complete TEN for the considered waveguides
is then reported in Fig. 3, where the network parameters for
the two polarizations in the air and slab regions (subscripts 0
and s, respectively) are
kz0 = k0 k̂z0 = k0
kzs = k0 k̂zs = k0
1 − k̂ρ2 = k0
εr − k̂ρ2 = k0
1−
kρ
k0
kρ
k0
εr −
2
,
2
(6)
and
Y0TE =
Y0TM
=
k̂z0
,
η0
1
η0 k̂z0
YsTE =
,
YsTM
k̂zs
η0
=
εr
η0 k̂zs
(7)
,
√
where k0 = ω μ0 ε0 and η0 = μ0 /ε0 are the free-space
wavenumber and impedance, respectively. The TEN in Fig. 2
can thus be used to determine the (generally complex) radial
wavenumber kρ = β − jα of a mode traveling along the radial
direction ρ of the waveguide as e−jkρ ρ .
Y up + Y down = 0 or Z up + Z down = 0 .
(8)
Proper modes (which are of interest for this study) are
found by choosing the proper determination for the vertical
wavenumbers kz0 = k02 − kρ2 , i.e., the one with a negative
imaginary part, corresponding to fields that decay exponentially at infinity in the transverse z direction.
For all the structures considered below, the transverse
resonance condition is applied to determine the dispersion
equation of the modes supported by each nano-waveguide.
Such a dispersion equation will be expressed in terms of the
admittances of the individual TENs, which in turn depend on
the transverse wavenumbers kz0 and kzs via (7); finally, these
transverse wavenumbers are explicit functions of the radial
wavenumber kρ through (6), which is the actual unknown of
the dispersion equations.
A. Graphened slab
The TEN for the analysis of the graphened slab is reported
in Fig. 4(a). It clearly results
Y up = Y0 + Yσ
Y0 cos (kzs h) + jYs sin (kzs h)
Y down = Ys
Ys cos (kzs h) + jY0 sin (kzs h)
(9)
so that the transverse resonance condition reads
(Y0 + Yσ ) [Ys cos (kzs h) + jY0 sin (kzs h)]
+ Ys [Y0 cos (kzs h) + jYs sin (kzs h)] = 0 .
(10)
It can be noted that in the absence of a dielectric support
(h = 0 or Ys = Y0 ) (10) enormously simplifies to
2Y0 + Yσ = 0 .
(11)
Since Yσ = σ [7], from (7) it results
k0 η0
σ
2
2k0 1
=−
η0 σ
kz0 = −
for TE modes
kz0
for TM modes,
978-1-4673-0717-8/12/$31.00 ©2012 IEEE
h
Yσ
Y up
Y
Yσ
h
Y down
Ys , kzs
0
Y up
down
Ys , kzs
0
Y0
(b)
(a)
Fig. 4. TENs for dispersion analysis of the graphened-slab waveguide (a)
and of the graphene/PEC PPW (b).
B. Graphene/PEC PPW
The TEN for the analysis of the PPW with a graphene wall
and a PEC plate is reported in Fig. 4(b). It results
Y up = Y0 + Yσ
(13)
Y down = −jYs cot (kzs h)
so that in this case the transverse resonance condition reads
(Y0 + Yσ ) sin (kzs h) − jYs cos (kzs h) = 0 .
(14)
C. Graphene/graphene PPW
As it can easily be seen, the graphene/graphene PPW is
simmetric with respect to the z = h/2 plane. Because of
such a simmetry, the tangential component of either electric
or magnetic field must vanish: in the former case, the z = h/2
plane can be substituted by a PEC plane, while in the latter
by a perfectly magnetic conductor (PMC) plane. From a TEN
point of view, the TEN can be simplified and the transmission
line can be closed at z = h/2 either by a short ciruit or by
an open ciruit, as shown in Fig. 5. It can easily be seen that
the former gives rise to odd modes while the latter to even
modes. For odd modes we thus have
h
Y up = −jYs cot kzs
2
(15)
down
Y
= Y0 + Yσ
so that the transverse resonance condition reads
h
h
− jYs cos kzs
(Y0 + Yσ ) sin kzs
2
2
.
(16)
For even modes we instead have
Y up = jYs tan kzs
h
2
(17)
Y down = Y0 + Yσ
(12)
as also derived in [8].
Y0
Y0
IV. D ISPERSION A NALYSIS VIA T RANSVERSE R ESONANCE
The semi-infinite TLs in the air region of Fig. 3 can be
replaced by their characteristic admittances and the modal
dispersion equation can be found by enforcing the transverse
resonance condition (e.g., at z = h) of the resulting network
[12]: this means that the admittance Y up (or impedance Z up )
seen looking into the TL for z > h must be equal to the
admittance Y down (or impedance Z down ) seen looking into
the TL for z < h, i.e.,
z
z
and the transverse resonance condition reads
h
h
+ jYs sin kzs
(Y0 + Yσ ) cos kzs
2
2
.
(18)
z
Odd modes
h /2
z
Yσ
h
Yσ
0
Y up
Yσ
0
σR [Eq. (5)]
Y up
σJ [Eq. (5)]
0.01
Y down
0
Y0
Y down
Ys , kzs
σJ
0.02
Ys , kzs
Y0
σR
0.03
−1
[Ω ]
-0.01
z
-0.02
h /2
0
200
400
Ys , kzs
Y0
Yσ
0
Y up
800
1000
Fig. 6. Graphene conductivity σ = σR + jσJ as a function of frequency.
Parameters: T = 300 K, τ = 0.5 ps, and μc = 0.5 eV.
Y down
10
Y0
Even modes
Fig. 5.
600
f [GHz]
3
Graphened slab
Graphene/PEC PPW
Graphene PPW (odd mode)
Graphene PPW (even mode)
β/k0
10
2
10
1
10
0
TENs for dispersion analysis of the graphene/graphene PPW.
V. N UMERICAL R ESULTS
The variation of the isotropic complex graphene conductivity σ = σR + jσJ with frequency is reported in Fig. 6, where
it has been assumed T = 300 K, τ = 0.5 ps, and μc = 0.5
eV (i.e., from (4), a doping with ns 2 × 1013 cm−2 ). It
can be seen that significant changes of the conductivity arise
for frequencies larger than 10 GHz up to the THz regime.
The behavior of the complex conductivity as a function of
frequency simply follows a Drude model: such a behavior
reflects the conventional dispersive conductivity arising from
a Drude theory of metals [13], valid provided that spatialdispersion effects can be neglected, as for the frequency range
considered here. It can also be observed that the approximate
expression in (5) is very accurate in the whole considered
frequency range.
The following results for the dispersion curves of the
dominant modes of the structured described above have been
obtained with the graphene conductivity in Fig. 6 and with a
dielectric board having relative dielectric permittivity εr = 3.9
(SiO2 ) and thickness h = 100 nm. The normalized wavenumber k̂ρ = β̂ − j α̂ (with β̂ = β/k0 normalized radial phase
constant and α̂ = α/k0 normalized attenuation constant) have
been obtained by solving numerically (10), (14), (16), and (18)
using Müller’s method. The normalized phase and attenuation
constants are thus reported in Figs. 7 and 8, respectively, as
functions of frequency.
As already shown in [9] in connection with isolated
graphene sheets, the structures analyzed here do not support
proper TE modes below the THz regime (more specifically, as
978-1-4673-0717-8/12/$31.00 ©2012 IEEE
10
-1
0
200
400
600
800
1000
f [GHz]
Fig. 7. Normalized phase constant β/k0 as a function of frequency for the
considered graphene-based nano-waveguides with graphene conductivity as in
Fig. 6. Other parameters: εr = 3.9 and h = 100 nm.
long as the imaginary part of the graphene conductivity σJ is
negative); therefore all the modal solutions reported in Figs.
7-8 are proper TM modes.
The graphened slab (black solid line) supports a TM mode
which is quite fast (0.98 < β̂ < 1.14) and very slowly
attenuating (α̂ 1); this entails that the mode is not well
confined close to the waveguide; it should be noted that such
a solution is superimposed to the TM mode supported by the
isolated graphene sheet (see (12), although the relevant curve
is not reported in the figures), thus showing that the effects of
the dielectric nano-board are completely negligible.
On the other hand, a PPW with a PEC and a graphene wall
(gray solid line) supports a TM mode which is surprisingly
slow (23 < β̂ < 130) and highly attenuating (α̂ > 3 in the
whole frequency range), which is thus very well confined inside the nano-waveguide. Although the normalized attenuation
constant is very large, the mode is still capable of carrying
energy along a circuit of nanometric dimensions.
3
α/k0
10
Graphened slab
Graphene/PEC PPW
Graphene PPW (odd mode)
Graphene PPW (even mode)
2
10
1
10
0
10
-1
10
-2
10
-3
10
0
200
400
f [GHz]
600
800
1000
Fig. 8. Normalized attenuation constant α/k0 as a function of frequency for
the considered graphene-based nano-waveguides with graphene conductivity
as in Fig. 6. Other parameters: εr = 3.9 and h = 100 nm.
Finally, when considering a PPW with two graphene walls
with equal conductivity, two fundamental TM modes exist,
associated with an odd (black dashed line) and an even (gray
dashed line) field distribution, respectively. It turns out that the
odd mode is the perturbation of the TEM mode of the PPW
with two PEC walls (and therefore its dispersion properties
are also similar to those of the PEC-graphene PPW), whereas
the even mode is the perturbation of the TM mode supported
by the graphened slab (and therefore its propagation constant
is also very similar to that of the graphened slab).
VI. C ONCLUSION
The modal features of dominant surface waves for different two-dimensional graphene-based nano-waveguides (i.e.,
a graphened dielectric nano-slab, a nano-PPW with two
graphene walls, and a nano-PPW with one graphene wall and a
PEC plate) have been studied by means of an exact and simple
approach based on the transverse-resonance technique and a
transverse-equivalent-circuit representation of graphene sheets
which takes into account the possible presence of static electric
biasing fields and/or chemical doping. Dispersion properties
in terms of radial phase and attenuation constants are derived,
thus allowing for gaining an insight into the potentialities of
graphene nano-waveguides for future applications in nanoelectromagnetics.
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