Dyadic Green's Functions and Guided Surface Waves for a Surface Conductivity Model of Graphene

0.34 MB·English

by  George W. Hanson

Dyadic Green’s Functions and Guided Surface Waves for a Surface Conductivity
Model of Graphene
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George W. Hanson
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J Department of Electrical Engineering, University of Wisconsin-Milwaukee

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3200 N. Cramer St., Milwaukee, Wisconsin 53211,USA.
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l 1 Abstract
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t. Anexactsolutionisobtainedfortheelectromagneticfieldduetoanelectriccurrentinthepresence
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m of a surface conductivity model of graphene. The graphene is represented by an infinitesimally-
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d thin, local and isotropic two-sided conductivity surface. The field is obtained in terms of dyadic
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o Green’s functions represented as Sommerfeld integrals. The solution of plane-wave reflection and
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[ transmission is presented, and surface wave propagation along graphene is studied via the poles

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of the Sommerfeld integrals. For isolated graphene characterized by complex surface conductivity
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σ =σ +jσ ,apropertransverse-electric(TE)surfacewaveexistsifandonly ifσ >0(associated
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2 ′′
withinterbandconductivity),andapropertransverse-magnetic(TM)surfacewaveexistsforσ <0
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(associatedwithintrabandconductivity). By tuning the chemicalpotentialatinfraredfrequencies,
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0 ′′
the sign of σ can be varied, allowing for some control over surface wave properties.
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2 Introduction
Fundamental properties and potential applications of carbon-based structures are of interest in
emergingnanoelectronicapplications. Particularlypromisingisgraphene,whichisa planaratomic
layer of carbon atoms bonded in a hexagonal structure. Graphene is the two-dimensional version
ofgraphite,anda single-wallcarbonnanotube canbe thoughtofasgraphenerolledinto atube [1].
Only recently has it become possible to fabricate ultrathin graphite, consisting of only a few
graphene layers [2]–[3], and actual graphene [4]–[8]. In graphene, the energy-momentum relation-
ship for electrons is linear over a wide range of energies, rather then quadratic, so that electrons
in graphene behave as massless relativistic particles (Dirac fermions) with an energy-independent
velocity. Graphene’s band structure, together with it’s extreme thinness, leads to a pronounced
electric field effect [5], [9], which is the variation of a material’s carrier concentration with electro-
static gating. This is the governing principle behind traditional semiconductor device operation,
and thereforethis effect in grapheneis particularlypromisingfor the developmentof ultrathincar-
bon nanoelectronic devices. Although the electric field effect also occurs in atomically thin metal
films, these tend to be thermodynamically unstable, and do not form continuous layers with good
transport properties. In contrast, graphene is stable, and, like its cylindrical carbon nanotube ver-
sions,canexhibitballistictransportoveratleastsubmicrondistances[5]. Furthermore,ithasbeen
shownthatgraphene’sconductancehasaminimum,non-zerovalueassociatedwiththeconductance
quantum, even when charge carrier concentrations vanish [6].
In this work, the interaction of an electromagnetic current source and graphene is considered.
TheelectromagneticfieldsaregovernedbyMaxwell’sequations,andthegrapheneisrepresentedby
a conductivity surface [10] that must arise from a microscopic quantum-dynamical model, or from
measurement. The method assumes laterally infinite graphene residing at the interface between
two dielectrics, in which case classicalMaxwell’s equations are solvedexactly for an arbitrary elec-
trical current. Related phenomena are discussed, such as plane-wave reflection and transmission
throughgraphene[11],andsurfacewaveexcitationandguidance,whichisrelevanttohigh-frequency
electronicapplications. Itisfoundthattherelativeimportanceoftheinterbandandintrabandcon-
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tributionstotheconductivitydictatesurfacewavebehavior[12],andthatsurfacewavepropagation
can be controlled by varying the chemical potential.
Although at this time only graphene samples with lateral dimensions on the order of tens of
microns have been fabricated, the infinite sheet model provides a first step in analyzing electro-
magneticpropertiesofgraphene. Itisalsorelevanttosufficientlylargefinite-sizedsheets,assuming
that electronic edge effects and electromagnetic edge diffraction can be ignored. In the following
all units are in the SI system, andthe time variation(suppressed)is ejωt, where j is the imaginary
unit.
3 Formulation of the Model
3.1 Electronic Model of Graphene
Figure 1 depicts laterally-infinite graphene lying in the x z plane at the interface between two
−
different mediums characterized by µ ,ε for y 0 and µ ,ε for y < 0, where all material
1 1 2 2
≥
parameters may be complex-valued.
The graphene is modeled as an infinitesimally-thin, local two-sided surface characterized by
a surface conductivity σ(ω,µ ,Γ,T), where ω is radian frequency, µ is chemical potential, Γ
c c
is a phenomenological scattering rate that is assumed to be independent of energy ε, and T is
temperature. The conductivity of graphene has been considered in several recent works [11]–[18],
and here we use the expression resulting from the Kubo formula [19],
je2(ω j2Γ) 1 ∞ ∂f (ε) ∂f ( ε)
d d
σ(ω,µ ,Γ,T)= − ε − dε (1)
c πℏ2 "(ω−j2Γ)2 Z0 (cid:18) ∂ε − ∂ε (cid:19)
∞
f ( ε) f (ε)
d d
− − dε
− Z0 (ω−j2Γ)2−4(ε/ℏ)2 #
whereeisthechargeofanelectron,ℏ=h/2πisthereducedPlanck’sconstant,fd(ε)= e(ε−µc)/kBT +1 −1
is the Fermi-Diracdistribution, andk is Boltzmann’sconstant. We assumethatnoex(cid:0)ternalmag- (cid:1)
B
netic field is present, and so the local conductivity is isotropic (i.e., there is no Hall conductivity).
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The firstterm in (1) is due to intrabandcontributions,and the secondtermto interbandcontribu-
tions.
For an isolated graphene sheet the chemical potential µ is determined by the carrier density
c
n ,
s
∞
2
n = ε(f (ε) f (ε+2µ ))dε, (2)
s πℏ2v2 d − d c
F Z0
wherev 9.5 105m/sistheFermivelocity. Thecarrierdensitycanbecontrolledbyapplication
F
≃ ×
ofagatevoltageand/orchemicaldoping. Forthe undoped, ungatedcaseatT =0K,n =µ =0.
s c
The intraband term in (1) can be evaluated as
e2kBT µc − µc
σintra(ω,µc,Γ,T)=−jπℏ2(ω j2Γ) k T +2ln e kBT +1 . (3)
− (cid:18) B (cid:16) (cid:17)(cid:19)
For the case µ = 0, (3) was first derived in [20] for graphite (with the addition of a factor to
c
account for the interlayer separation between graphene planes), and corresponds to the intraband
′ ′′
conductivityofasingle-wallcarbonnanotubeinthelimitofinfiniteradius[10]. Withσ =σ +jσ ,
′ ′′
it can be seen that σ 0 and σ < 0. As will be discussed later, the imaginary part of
intra ≥ intra
conductivity plays an important role in the propagation of surface waves guided by the graphene
sheet [12].
The interband conductivity can be approximated for k T µ ,ℏω as [21],
B c
≪| |
je2 2 µ (ω j2Γ)ℏ
c
σ (ω,µ ,Γ,0) − ln | |− − , (4)
inter c ≃ 4πℏ 2 µ +(ω j2Γ)ℏ
(cid:18) | c| − (cid:19)
such that for Γ = 0 and 2 µ > ℏω, σ = jσ′′ with σ′′ > 0. For Γ = 0 and 2 µ < ℏω,
| c| inter inter inter | c|
σ is complex-valued, with [22]
inter
πe2
σ′ = =σ =6.085 10−5 (S), (5)
inter 2h min ×
′′
and σ >0 for µ =0.
inter c 6
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3.2 Dyadic Green’s Function for a Surface Model of Graphene
For any planarly layered, piecewise-constant medium, the electric and magnetic fields in region n
due to an electric current can be obtained as [23]–[24]
E(n)(r)= k2 + π(n)(r), (6)
n ∇∇·
H(n)(r)=j(cid:0)ωε π(cid:1)(n)(r), (7)
n
∇×
where kn = ω√µnεn and π(n)(r) are the wavenumber and electric Hertzian potential in region n,
respectively. Assuming that the current source is in region 1, then
J(1)(r′)
π(1)(r)=πp(r)+πs(r)= gp(r,r′)+gs(r,r′) dΩ′, (8)
1 1 1 1 · jωε
ZΩnJ(1)(r′) o 1
π(2)(r)=πs(r)= gs(r,r′) dΩ′, (9)
2 2 · jωε
ZΩ 1
where the underscoreindicates a dyadic quantity, and whereΩ is the supportof the current. With
y parallel to the interface normal, the principle Greens dyadic can be written as [23]
gp(r,r′)=Ie−jk1R =I 1 ∞ e−p1 y−y′ H0(2)(kρρ)k dk (10)
1 4πR 2π Z−∞ | | 4p1 ρ ρ
where
p2 =k2 k2, ρ= (x x′)2+(z z′)2, (11)
n ρ− n − −
q
R= r r′ = (y y′)2+ρ2,
| − | −
q
and where k is a radial wavenumber and I is the unit dyadic.
ρ
ThescatteredGreen’sdyadicscanbeobtainedbyenforcingtheusualelectromagneticboundary
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conditions
y (H H )=Js,
× 1− 2 e
y (E E )= Js , (12)
× 1− 2 − m
where Js (A/m) and Js (V/m) are electric and magnetic surface currents on the boundary. For a
e m
local model of graphene in the absence of a magnetic field and associated Hall effect conductivity,
σ is a scalar. Therefore [10],
Js (x,y =0,z)=σE (x,y =0,z), (13)
e,x x
Js (x,y =0,z)=σE (x,y =0,z), (14)
e,z z
Js (x,y =0,z)=0, (15)
m
and (12) becomes
E y =0+ =E y =0− , α=x,z, (16)
1,α 2,α
H (cid:0)y =0−(cid:1) H (cid:0)y =0+(cid:1)=σE (y =0), (17)
2,x 1,x z
−
H (cid:0)y =0−(cid:1) H (cid:0)y =0+(cid:1)= σE (y =0). (18)
2,z 1,z x
− −
(cid:0) (cid:1) (cid:0) (cid:1)
Using (6), the boundary conditions on the Hertzian potential at (x,y =0,z) are
π =N2M2π , (19)
1,α 2,α
σ
ε π ε π = π (20)
1 1,y 2 2,y 1
− jω∇·
∂π ∂π σ
ε 2,α ε 1,α = k2π (21)
2 ∂y − 1 ∂y jω 1 1,α
∂π ∂π ∂π ∂π
1,y 2,y = 1 N2M2 2,x + 2,z , (22)
∂y − ∂y − ∂x ∂z
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:0) (cid:1)
α=x,z, where N2 =ε /ε and M2 =µ /µ . In the absence of magnetic contrast(e.g., if M =1)
2 1 2 1
and surface conductivity, boundary conditions (19)–(22) are identical to the Hertzian potential
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boundary conditions presented [25].
Enforcing (19)–(22) and following the method described in [25], the scattered Green’s dyadic is
found to be
∂ ∂
gs(r,r′)=yy gs(r,r′)+ yx +yz gs(r,r′)+(xx + zz)gs(r,r′), (23)
1 n ∂x ∂z c t
(cid:18) (cid:19)
bb bb bb bb bb
where the Sommerfeld integrals are
1 ∞ H(2)(k ρ)e−p1(y+y′)
gs(r,r′)= R 0 ρ k dk , (24)
β 2π Z−∞ β 4p1 ρ ρ
β =t,n,c, with
M2p p jσωµ NH(k ,ω)
1 2 2 ρ
R = − − = , (25)
t M2p +p +jσωµ ZH(k ,ω)
1 2 2 ρ
R = N2p1−p2+ σjpω1εp12 = NE(kρ,ω), (26)
n N2p +p + σp1p2 ZE(k ,ω)
1 2 jωε1 ρ
2p N2M2 1 + σp2M2
1 − jωε1
R = , (27)
c h(cid:0) ZHZE(cid:1) i
which reduce to the previously known results as σ 0.
→
The Green’s dyadic for region 2, gs(r,r′), has the same form as for region 1, although in (24)
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the replacement
R e−p1(y+y′) T ep2ye−p1y′ (28)
β β
→
must be made, where
(1+R ) 2p
t 1
T = = , (29)
t N2M2 N2ZH
p (1 R ) 2p
1 n 1
T = − = , (30)
n p ZE
2
2p N2M2 1 + σp1
1 − jωε1
T = . (31)
c h(cid:0) N2ZHZE(cid:1) i
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As in the case of a simple dielectric interface, the denominators ZH,E(k ,ω) = 0 implicate
ρ
pole singularities in the spectral plane associated with surface wave phenomena. Furthermore,
both waveparameters p = k2 k2, n = 1,2, lead to branch points at k = k , and thus
n ρ− n ρ ± n
q
the k -plane is a four-sheeted Riemann surface. The standard hyperbolic branch cuts [24] that
ρ
separate the one proper sheet (where Re(p ) > 0, such that the radiation condition as y
n
| | → ∞
is satisfied) and the three improper sheets (where Re(p ) < 0) are the same as in the absence of
n
surface conductivity σ.
In addition to representing the exact field from a given current, several interesting electromag-
netic aspects of graphene can be obtained from the above relations.
3.3 Plane-Wave Reflection and Transmission Coefficients
Normal incidence plane-wave reflection and transmission coefficients can be obtained from the
previous formulation by setting k =0 in (25) and (29). To see this, consider the current
ρ
j4πr
J(1)(r)=α 0δ(r r ), (32)
0
ωµ −
1
b
whereα=xory,r =yy ,andwherey 0. Thiscurrentleadstoaunit-amplitude,α-polarized,
0 0 0
≫
transverse electromagnetic plane wave, normally-incident on the interface. The far scattered field
b b b b b
in region 1, which is the reflected field, can be obtained by evaluating the spectral integral (24)
using the method of steepest descents, which leads to the reflected field Er =αΓe−jk1y, where the
reflection coefficient is Γ=R (k =0). In a similar manner, the far scattered field in region2, the
t ρ
b
transmitted field, is obtained as Et = αTejk2y, where the transmission coefficient is T = (1+Γ).
Therefore,
b
η η ση η
2 1 1 2
Γ= − − , (33)
η +η +ση η
2 1 1 2
2η
2
T =(1+R )= (34)
t
(η +η +ση η )
2 1 1 2
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whereη = µ /ε isthewaveimpedanceinregionn. Theplane-wavereflectionandtransmission
n n n
coefficients opbviously reduce to the correct results for σ =0, and in the limit σ , Γ 1 and
→∞ →−
T 0 as expected.
→
In the special case ε =ε =ε and µ =µ =µ ,
1 2 0 1 2 0
ση0 1
Γ= 2 , T = , (35)
−1+ ση0 1+ ση0
2 2
(cid:0) (cid:1)
where η = µ /ε 377 ohms. The reflection coefficient agrees with the result presented in [11]
0 0 0
≃
for normal ipncidence.
3.4 Surface Waves Guided by Graphene
PolesingularitiesintheSommerfeldintegralsrepresentdiscretesurfacewavesguidedbythemedium
[23]–[24]. From(25)–(27)and(29)–(31),thedispersionequationforsurfacewavesthataretransverse-
electric (TE) to the propagation direction ρ (also known as H-waves) is
ZH(k ,ω)=M2p +p +jσωµ =0, (36)
ρ 1 2 2
whereas for transverse-magnetic (TM) waves (E-waves),
σp p
ZE(k ,ω)=N2p +p + 1 2 =0. (37)
ρ 1 2
jωε
1
In the limit that ε =ε =ε , ZH,E agree with TE and TM dispersion equations in [12]
1 2 0
ThesurfacewavefieldcanbeobtainedfromtheresiduecontributionoftheSommerfeldintegrals.
For example, the electric field in region 1 associated with the surface wave excited by a Hertzian
dipole current J(r)=yA δ(x)δ(y)δ(z) is
0
E(1)(ρ )= A0kbρ2Rn′ e−p1y x x +z z H(2)′(k ρ ) y k12+p21 H0(2)(kρρ0) ,
0 4ωε1 (cid:18) ρ0 ρ0(cid:19) 0 ρ 0 − (cid:0) kρ (cid:1)k12−kρ2 
b b b q
 
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whereR′ =NE/ ∂ZE/∂k ,H(2)′(α)=∂H(2)(α)/∂α,andρ =√x2+z2. Theterme−p1y leads
n ρ 0 0 0
to exponential dec(cid:0)ay away f(cid:1)rom the graphene surface on the proper sheet (Re(p ) > 0,n = 1,2).
n
The surface wave mode may or may not lie on the proper Riemann sheet, depending on the value
of surface conductivity, as described below. In general, only modes on the proper sheet directly
result in physical wave phenomena, although leaky modes on the improper sheet can be used to
approximate parts of the spectrum in restricted spatial regions, and to explain certain radiation
phenomena [26].
3.4.1 Transverse-Electric Surface Waves
Noting that p2 p2 =k2(µrεr µrεr), where µr and εr are the relativematerialparameters (i.e.,
2− 1 0 1 1− 2 2 n n
µ = µrµ and ε = εrε ) and k2 = ω2µ ε is the free-space wavenumber, then if M = 1 (µr =
n n 0 n n 0 0 0 0 1
µr = µ ) the TE dispersion equation (36) can be solved for the radial surface wave propagation
2 r
constant, yielding
(εr εr)µ +σ2η2µ2 2
k =k µ εr 1− 2 r 0 r . (38)
ρ 0s r 1−(cid:18) 2ση0µr (cid:19)
If, furthermore, N =1 (εr =εr =ε ), then (38) reduces to
1 2 r
ση µ 2
0 r
k =k µ ε . (39)
ρ 0 r r
− 2
r
(cid:16) (cid:17)
For the case M =1, then (36) leads to
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2
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k =k µrεr M2ση µr (ση µr)2 (M4 1)(εrµr εrµr) . (40)
ρ 0s 1 1− (M4 1)2 0 2∓ 0 2 − − 1 1− 2 2
− (cid:18) q (cid:19)
Considering the special case of graphene in free-space, setting εr =εr =µr =µr =1,
1 2 1 2
ση 2
0
k =k 1 . (41)
ρ 0
− 2
r
(cid:16) (cid:17)
If σ is real-valued (σ = σ′) and (σ′η /2)2 < 1, then a fast propagating mode exists, and if
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(σ′η /2)2 > 1 the wave is either purely attenuating or growing in the radial direction. How-
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