Nanoscale Syst.: Math. Model. Theory Appl. 2014; 3:44–54
Naijing Kang, Z. L. Mišković*, Ying-Ying Zhang, Yuan-Hong Song, and You-Nian Wang
Analyzing nonlocal effects in the plasmon
spectra of a metal slab by the Green’s
function technique for hydrodynamic model
Abstract: We study the dynamic response of a metal slab containing electron gas described by the hydrodynamic model with dispersion. The resulting wave equation for the perturbed electron density is solved by
means of the Green’s function that satisfies Neumann boundary conditions at the endpoints of the slab. This
solution is coupled with the electrostatic potential, which is expressed in terms of the Green’s function for
the Poisson equation for a layered structure consisting of three dielectric regions. As an illustration, a set of
dispersion relations for eigenfrequencies is deduced for the plasma oscillations in the electron gas, corresponding to both the surface and the bulk modes of even and odd symmetry with respect to the center of the
metal slab.
Keywords: hydrodynamic model, Green’s function, plasmon dispersion
PACS: 34.50.Bw, 05.60.Gg, 73.20.Mf
DOI 10.2478/nsmmt-2014-0005
Received November 14, 2014; accepted in revised form November 30, 2014
1 Introduction
The excitation of collective oscillations, or plasmons in the quasi-free electron gas (EG) of noble metal enables both the local enhancement and guiding of electromagnetic radiation across the surface of a metallic
nanoparticle at optical frequencies [1, 2], providing a great promise for the development of metallic nanostructures for photonic and bio-photonic applications [3]. At the level of mathematical modeling and computational schemes [4], recent achievements in the area of nanoplasmonics have brought the focus back to the
hydrodynamic model (HDM) of the EG in metallic nanostructures [5]. The ability of this model to describe the
spatially nonlocal effects in the plasma oscillations of an EG has been recently shown [6] to be particularly
useful in studying the enhancement of the electric field in nanostructures with sharp tips or narrow gaps
[7, 8].
The history of the HDM is quite long, dating back to the work of Bloch [9], which was taken up by Ritchie
in a seminal paper describing the plasma excitations by fast electrons in thin metallic films [10]. An important
next step was made by Eguiluz [11], who showed that using the HDM in the non-retarded limit provides an
adequate description of the nonlocal effects in the plasma oscillations in a metal film by making a direct
comparison with the more accurate random phase approximation (RPA) for the dynamic response function
of an EG, coupled with the specular reflection model (SRM) for the metal surface [12]. In the intervening years,
the HDM has become particularly popular owing to its physical transparency and analytical tractability in
*Corresponding Author: Z. L. Mišković: Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario,
Canada N2L 3G1
and Waterloo Institute for Nanotechnology, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1, E-mail:
[email protected]
Naijing Kang: Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Ying-Ying Zhang, Yuan-Hong Song, You-Nian Wang: School of Physics and Optoelectronic Technology, Dalian University of
Technology, Dalian, China 116024
© 2014 N. Kang et al., licensee De Gruyter Open.
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.
Analyzing nonlocal effects in the plasmon spectra of a metal slab |
45
studying metal surfaces [13] and metallic nanostructures of various shapes [14]. Partial reviews of the history
and applications of the HDM were given in, e.g., Refs. [13, 14]. More recently, the role of quantum effects in
the nonlocal response of an EG in a metal slab were studied by means of the quantum HDM [15].
In view of the revived interest for the HDM in nanoplasmonics, we provide a mathematical formulation
for dynamic polarization of a metallic slab sandwiched between two dielectrics by combining the Green’s
function (GF) technique for both the Poisson equation for the electric potential and the wave (Helmholtz)
equation for the induced charge density in the EG resulting from the HDM, where we combine the electrostatic matching conditions with the boundary condition for the EG arising from the SRM. In doing so, we
generalize the formulation given in Ref. [11] and outline a procedure for obtaining the full Green’s function
for an effective Poisson equation describing a metal slab with spatial nonlocality. As an illustration, we evaluate the dispersion curves for the plasmon modes in the slab and compare them with the results from a local
description of the EG, which is related to the Drude model of the dielectric response.
After outlining a linearization procedure for the HDM, we show how using the GF for the resulting equation for the induced charge density in the metal slab couples with the GF for the Poisson equation. This is
followed by a derivation of the latter GF for a three-layer structure of dielectrics, and by a detailed derivation
of the plasmon dispersion relations for the metal slab in free space, which are solved in order to illustrate the
nonlocal effects. Note that we use Gaussian electrostatic units throughout the paper.
2 Hydrodynamic model of electron gas
We study a metal slab of thickness h, occupied by a quasi-free electron gas (EG) with the electron charge
−e < 0 and effective mass m, having an equilibrium volume density n e0 = n0 , which moves over the positively
charged background of motionless ions with the volume density n i0 = n0 . We use a cartesian coordinate
system with coordinates R ≡ (r, z), where r ≡ (x, y), and assume that there are three regions, labeled 1, 2 and
3, which are defined by z lying in the intervals I1 = (−∞, 0), I2 = (0, h) and I3 = (h, ∞), respectively, so that
the metal slab occupies the region 2, as in Fig. 1. We further assume that the metal slab is surrounded by two
semi-infinite dielectrics in the regions 1 and 3, which are characterized by the relative dielectric constants
ϵ1 and ϵ3 , respectively, whereas the background dielectric constant for the metal slab is assumed to be ϵ2 ,
representing, e.g., a contribution of the tightly bound electrons in the metal.
We want to determine the electrostatic response of the metal due to the polarization of its EG by an external charged particle with the charge density ρext (R, t). As a result of the passage of the external particle,
the EG will be perturbed from its equilibrium state of a uniform density giving rise to a state with the velocity vector field u(R, t) and the volume density n(R, t), which may be described by the hydrodynamic model
(HDM) consisting of the continuity and the momentum balance equations for electrons [13, 14],
∂
n + ∇·(n v)
∂t
∂
m
+ v·∇ v
∂t
=
0,
(1)
=
−∇U − mγ v,
(2)
where γ is a phenomenological damping rate due to electron friction on the background ions, and
U(R, t) = −e Φ(R, t) +
2
2
~2
(3π 2 ) 3 n 3
2m
(3)
is the potential energy per electron with Φ(R, t) being the total electrostatic potential and the second term
representing a contribution of the electronic pressure within the Thomas-Fermi (TF) model of the EG. By defining the relative dielectric constant as a function of z in a piece-wise manner by ϵ(z) = ϵ1 Θ(−z) + ϵ2 Θ(z) Θ(h −
z) + ϵ3 Θ(z − h), where Θ(z) is the Heaviside unit step function, one may write the Poisson equation for the
electrostatic potential as
∇· ϵ(z)∇Φ(R, t) = −4πρext (R, t) + 4πe n(R, t) − n0 Θ(z) Θ(h − z).
(4)
46 | N. Kang et al.
Figure 1: Schematic diagram showing three regions with the metal slab occupying the interval 0 ≤ z ≤ h.
One notices that the above system is nonlinear due to the convective time derivative in the left-hand side of
Eq. (2) and due to the TF expression for the pressure in Eq. (3). Hence, assuming that the system is only weakly
perturbed by the external charge ρext , one may linearize the above system by letting n(R, t) = n0 +n1 (R, t)+. . .
and noticing that n1 , u and Φ are all quantities of the first order in ρext . Thus, we obtain the linearized HDM
as
∂
n1 + n0 ∇·u
∂t
∂
u
∂t
U1 (R, t)
=
0,
=
−
=
1
∇U1 (R, t) − γ u,
m
n
−e Φ(R, t) + mβ2 1 ,
n0
(5)
(6)
(7)
where we have defined the speed of propagation of the density disturbances in the EG due to the TF pressure
√
1
~
(3π 2 n0 ) 3 is the Fermi speed of the EG. Notice that this definition
as β = v F / 3, where v F = m
p of the parameter
β is often corrected in simulations of the EG dynamics at high frequencies so that β = v F 3/5 [13, 14]. Next,
we eliminate the velocity field u from the above system of equations and use the Poisson equation in Eq. (4)
in region 2 to obtain the following equation for the induced charge density, ρ(R, t) = −e n1 (R, t), in the EG as
∂ ∂ρ
γ+
= −ω p 2 ρ + β2 ∇2 ρ − ω p 2 ρext (R, t),
(8)
∂t ∂t
p
where we have defined the plasma frequency of the bulk of EG as ω p = 4πe2 n0 / (ϵ2 m). Notice that this definition of the plasma frequency involves screening of the electron-electron interactions by the background
dielectric constant ϵ2 in the metal slab. As a consequence of the TF pressure term in the linearized electron
potential energy in Eq. (7), one sees that Eq. (8) becomes a second-order partial differential equation describing a damped wave, or Helmholtz equation upon Fourier transforming its time dependence, where the second
order spatial derivative gives rise to nonlocal effects in the EG response. We note that removing the pressure
term in the HDM gives rise to the so-called local model for the EG, whose dynamic response may be described
by Eq. (8) with β = 0.
It appears that Eq. (8) for the induced charge density in the EG is decoupled from the Poisson equation (4)
for the electrostatic potential, but a coupling between those two quantities arises in the boundary conditions
(BCs) for Eq. (8), which needs to be solved in the region 2 occupied by the EG. Assuming that the boundary
Analyzing nonlocal effects in the plasmon spectra of a metal slab |
47
of the region 2 is impenetrable to electrons, it is physically plausible to assume that the normal component
of their velocity u vanishes at the interior surface of that boundary, B,
n̂·u(R, t)|R∈B = 0,
(9)
where n̂ is an outward pointing unit vector, which is perpendicular to the boundary at the point R ∈ B. Note
that this condition is equivalent to the well known specular reflection model (SRM) for the bounded EG [12].
Referring to Eq. (6), this condition becomes equivalent to
∂
U1 (R, t)
= 0,
(10)
∂n
R∈ B
∂
≡ n̂·∇. So, from Eq. (7) the BC for the induced charge
where we have defined the normal derivative by ∂n
density may be expressed as a nonhomogeneous Neumann BC
∂
k2
ρ(R, t)
= s D n (R, t)
,
(11)
∂n
4π
R∈ B
R∈B
where we have defined the TF inverse screening length in the EG by k s ≡ ω p /β and where D n (R, t) = n̂·D(R, t)
is the normal component of the electric displacement vector, which is defined throughout the region 2 by
D(R, t) = −ϵ2 ∇Φ(R, t).
3 Induced charge density in slab geometry
Taking advantage of the slab geometry, we define a Fourier transform (FT) of the induced charge density
ρ(R, t) = ρ(r, z, t) in the EG with respect to the 2D position vector r = (x, y) and time t as
¨
ˆ ∞
ρ̃(q, z, ω) =
d2 r e−iq·r
dt e iωt ρ(r, z, t),
(12)
−∞
where q = (q x , q y ) is a two-dimensional (2D) wavevector, with similar definitions for the FT of the external
e
charge density, ρ̃ext (q, z, ω), and the electric potential, Φ(q,
z, ω). In the following we shall drop the variables
q and ω in the notation for these functions and concentrate on solving the resulting equations for their dependence on the position z. Thus, we obtain from Eq. (8) a second-order differential equation for the induced
charge density,
2
∂
2
(13)
−
q
ρ̃(z) = k s 2 ρ̃ext (z),
s
∂z2
where we have defined the parameter
s
qs =
q2 −
ω(ω + iγ ) − ω p 2
β2
(14)
that generally takes complex values. Notice that in the static limit ω = 0 and in the limit of long waves (q → 0),
we obtain q s → k s . The equation (13) has to be solved for z ∈ I2 , subject to the BCs that follow from Eq. (11)
as
∂ρ̃(z) ∂ρ̃(z) k2s e
k2 e
=
= sD
D0 and
,
(15)
∂z z=0 4π
∂z z=h 4π h
where
e ∂ Φ(z)
e
D0 = −ϵ2
∂z and
z=0+
e ∂ Φ(z)
e
D h = −ϵ2
∂z (16)
z=h−
are the z components of the FT of the electric displacement vector at the interior end-points of the interval
I2 . Note that taking the one-sided derivatives in Eq. (16) is unnecessarily meticulous because the normal
48 | N. Kang et al.
component of the electric displacement vector is continuous at a boundary between two dielectric regions so
that one may simply use in Eq. (16) the proper derivatives of the potential at the end-points z = 0 and z = h.
We may express the solution of Eq. (13) that satisfies the BCs in Eq. (15) as
ˆ h
e
e
e z0 )ρ̃ext (z0 ) dz0 − H(z,
e h) D h + H(z,
e 0) D0 ,
e
ρ(z) =
H(z,
(17)
4π
4π
0
e z0 ) is the FT of the associated Green’s function (GF) with both z ∈ I2 and z0 ∈ I2 , which satisfies
where H(z,
the equation
2
∂
2
e z0 ) = k s 2 δ(z − z0 ),
−
q
H(z,
(18)
s
∂z2
along with the homogeneous Neumann BCs at the end points of the interval I2 ,
e
e
∂H
∂H
0 0 (z, z ) = 0 and
(z, z ) = 0.
∂z
∂z
z=0
(19)
z=h
e z0 ) in the form
Assuming H(z,
0
e z)=
H(z,

e > (z, z0 ) = A eq s z + B e−q s z ,
 H
0 ≤ z0 < z ≤ h,
 e
H < (z, z0 ) = C eq s z + D e−q s z ,
0 ≤ z < z0 ≤ h,
(20)
we may determine the coefficients A, B, C, D as functions of z0 by using the BCs in Eq. (19) and the continuity
e z0 ) at z = z0 , expressed as
and the jump conditions for H(z,

e > (z0 , z0 ) = H
e < (z0 , z0 ),

 H
(21)
e<
e>

H
H
 ∂∂z
− ∂∂z
= ks 2 .
(z, z0 )
(z, z0 )
0
0
z=z
Thus we find
z=z
e z0 ) = −k s 2 cosh (q s z < ) cosh [q s (h − z > )] ,
H(z,
q s sinh(q s h)
(22)
e z0 ) also depends on the wavenumber q and frewhere z > = max(z, z0 ) and z < = min(z, z0 ). Notice that H(z,
quency ω via the expression for q s in Eq. (14).
e 0 and D
e h at the endpoints of the interval I2 ,
In order to determine the FTs of the electric displacements D
we express the FT of the electrostatic potential in the integral form as
ˆ ∞
ˆ h
e z0 )ρ̃ext (z0 ) dz0 +
e z0 )ρ̃(z0 ) dz0 ,
e
Φ(z)
=
G(z,
G(z,
(23)
−∞
0
e z0 ) is a short-hand notation for the tensorial FT of the GF (FTGF), G(q;
e
where G(z,
z, z0 ; ω), for the Poisson
equation (4), which will be discussed in the following section.
By inserting in the second term of Eq. (23) the expression for e
ρ(z) given in Eq. (17), we obtain
ˆ ∞
ˆ h
ˆ h
e z0 )ρ̃ext (z0 ) dz0 +
e z0 )
e
e 0 , z00 )ρ̃ext (z00 ) dz00 dz0
Φ(z)
=
G(z,
G(z,
H(z
−∞
0
0
e ˆ h
e ˆ h
D
e z0 )H(z
e z0 )H(z
e 0 , h) dz0 + D0
e 0 , 0) dz0 .
G(z,
G(z,
− h
4π 0
4π 0
(24)
e
By differentiating Φ(z)
in Eq. (24) at the points z = 0+ and z = h− and using the definitions in Eq. (16), we
e 0 and D
e h , given by
obtain a system of equations for D
"
#
ˆ h
ˆ h
1
+
0 e 0
0
e
e (0+ , z0 )H(z
e
eh 1
e 0 , h) dz0
D0 1 −
G D (0 , z )H(z , 0) dz + D
G
4π 0
4π 0 D
ˆ ∞
ˆ h
ˆ h
e D (0+ , z0 )ρ̃ext (z0 ) dz0 +
e D (0+ , z0 )
e 0 , z00 )ρ̃ext (z00 ) dz00 dz0 ,
=
G
G
H(z
(25)
−∞
0
0
Analyzing nonlocal effects in the plasmon spectra of a metal slab | 49
#
"
ˆ h
ˆ h
1
−
0 e 0
0
e
e (h− , z0 )H(z
e 0 , 0) dz0
e
e0 1
G D (h , z )H(z , h) dz − D
G
Dh 1 +
4π 0
4π 0 D
ˆ ∞
ˆ h
ˆ h
e D (h− , z0 )ρ̃ext (z0 ) dz0 +
e D (h− , z0 )
e 0 , z00 )ρ̃ext (z00 ) dz00 dz0 ,
G
G
=
H(z
−∞
0
(26)
0
where we have defined an auxiliary function
e z0 ),
e D (z, z0 ) = −ϵ2 ∂ G(z,
G
∂z
(27)
which expresses the z component of the electric displacement vector for z ∈ I2 due to a unit point charge at
e 0 and D
e h in terms of
an arbitrary source point z0 . A solution of the system in Eqs. (25) and (26) expresses D
integrals involving the external charge density ρ̃ext (z). Feeding back those expressions into the last two terms
in the right hand side of Eq. (24) finally gives an integral expression for the FT of the electrostatic potential
e in terms of the external charge density ρ̃ext (z) and hence defines an effective, frequency dependent FTGF
Φ(z)
for the Poisson equation for a layered structure containing the metal slab described in a nonlocal manner by
the HDM.
4 Green’s function for Poisson equation
When applying the FT to the Poisson equation in Eq. (4) it may be worthwhile defining the electric potential
e
e j (z) when z ∈ I j for j = 1, 2, 3, as shown in Fig. 1. Thus, we find that
in a piece-wise manner, so that Φ(z)
=Φ
e
the functions Φ j (z) satisfy the equations
2
∂
4π 2 e
e
−
q
Φ j (z) = −
ρ(z)Θ(z)Θ(h − z) + ρ̃ext (z) ,
(28)
2
ϵj
∂z
which may be solved by using the method of Green’s functions, allowing us to express the FT of the potential
in the form given in Eq. (23). Noting that the GF in the space-time domain, G(R, R0 ; t, t0 ), is a solution of the
associated equation
∇ · ϵ(z)∇G(R, R0 ; t, t0 ) = −4πδ(R − R0 ),
(29)
and taking advantage of the time homogeneity and the translational invariance in the directions of the vector
r, we may write G(R, R0 ; t, t0 ) = G(r − r0 ; z, z0 ; t − t0 ), which may be expressed in terms of its counterpart in the
e
Fourier space, G(q;
z, z0 ; ω), as
ˆ
¨
d2 q iq·(r−r0 ) ∞ dω −iω(t−t0 ) e
G(r − r0 ; z, z0 ; t − t0 ) =
e
e
G(q; z, z0 ; ω).
(30)
2π
(2π)2
−∞
e z0 ) is actually a tensor
Dropping the variables q and ω to simplify the notation, we note that the FTGF G(z,
0
0
0
e z)=G
e jk (z, z ) when z ∈ I j and z ∈ I k for j, k = 1, 2, 3. Then, Eq. (23) may be rewritten
function given by G(z,
in a component form as
e j (z) =
Φ
3 ˆ
X
k=1
Ik
e jk (z, z0 )ρ̃(k) (z0 ) dz0 +
G
ext
ˆ
e j2 (z, z0 )ρ̃(z0 ) dz0 ,
G
(31)
I2
where ρ̃(k)
ext (z) defines the part of the external charge density function located in the interval z ∈ I k .
In general, the components of the FTGF for the Poisson equation are found as solutions of the set of
equations written in the compact form as
∂2 e
e jk (z, z0 ) = − 4π δ jk δ(z − z0 ),
G (z, z0 ) − q2 G
ϵj
∂z2 jk
(32)
e 1k (z, z0 ) → 0 when z → −∞ and
where δ jk is a Kronecker delta symbol. In addition to the BCs at infinity, G
e 3k (z, z0 ) → 0 when z → ∞, the components of the FTGF G
e jk (z, z0 ) also need to satisfy the usual electrostatic
G
50 | N. Kang et al.
matching conditions at the boundary points z = 0 and z = h that separate the intervals I1 , I2 and I3 , which
correspond to the continuity of the electrostatic potential,

e 1k (0, z0 ) = G
e 2k (0, z0 ),
 G
(33)
e 2k (h, z0 ) = G
e 3k (h, z0 ),
 G
and the continuity of the normal component of the electric displacement vector,

e 1k (z, z0 ) = ϵ2 ∂ G
e 2k (z, z0 ) ,
 ϵ1 ∂ G
∂z
∂z
z=0
z=0
0 e 2k (z, z0 ) = ϵ3 ∂ G
e
 ϵ2 ∂ G
,
∂z
∂z 3k (z, z )
z=h
(34)
z=h
0
when the source point z ∈ I k for k = 1, 2, 3. Note that in Eqs. (25) and (26) we take one-sided values of the
e D (z, z0 ) defined in Eq. (27), which is really not necessary because that function is continuous when
function G
its variable z crosses the boundaries between the dielectric regions, according to Eq. (34). On the other hand,
e over the metal slab region should
all the terms in Eqs. (25) and (26) that contain integrals of the function H
0
e
be evaluated with the function G D (z, z ), which is obtained by using the diagonal component of the FTGF
e 22 (z, z0 ) in the right hand side of Eq. (27).
G
Finally, when both z and z0 are in the interval I j for j = 1, 2, 3, one may define the diagonal components
of the FTGF as
(
e < (z, z0 ), z ≤ z0 ,
G
0
jj
e jj (z, z ) =
G
(35)
e > (z, z0 ), z0 ≤ z,
G
jj
which must satisfy the usual continuity and the jump conditions at z = z0 ,

e < (z0 , z0 ) = G
e > (z0 , z0 ),
 G
jj
jj
∂ e<
∂ e>
 ∂z
− ∂z
= − 4π
G jj (z, z0 )
G jj (z, z0 )
ϵj .
z=z0
(36)
z=z0
e jk (z, z0 ) of the FTGF for the Poisson equation that
We list here the final expressions for the components G
can be obtained from the above conditions after some tedious algebra. Noting that the Maxwell’s symmetry
e jk (z, z0 ) = G
e kj (z0 , z), we only give the results for j ≥ k. Upon defining the auxiliary
of the GF requires that G
parameters
ϵ1 − ϵ2
λb =
,
ϵ1 + ϵ2
ϵ3 − ϵ2
λt =
,
ϵ3 + ϵ2
ϵ1 + ϵ2
ϵ̄12 =
,
2
ϵ3 + ϵ2
ϵ̄23 =
,
2
∆ = e−2qh ,
the FTGF components may be expressed in a compact form as
0
0
e 11 = 2π e−q|z−z | + λb − λt ∆ eq(z+z ) ,
G
qϵ1
1 − λb λt ∆
−qz
2π e − λt ∆ eqz qz0
e 21 =
G
e ,
qϵ̄12 1 − λb λt ∆
0
e 31
G
=
e 22
G
=
e 32
G
=
e 33
G
=
2πϵ2 eq(z −z)
,
qϵ̄12 ϵ̄23 1 − λb λt ∆
(
)
0
0
2π
2λb λt ∆
λb e−q(z+z ) + λt ∆ eq(z+z )
−q|z−z0 |
0 e
+
cosh q(z − z ) −
,
qϵ2
1 − λb λt ∆
1 − λb λt ∆
0
0
0
0
2π e−q(h−z ) − λb ∆ eq(h−z ) −q(z−h)
2π eqz − λb e−qz −qz
e
≡
e ,
qϵ̄23
1 − λb λt ∆
qϵ̄23 1 − λb λt ∆
0
2π
λ − λb ∆ q(2h−z−z0 )
e−q|z−z | + t
e
.
qϵ3
1 − λb λt ∆
Analyzing nonlocal effects in the plasmon spectra of a metal slab |
51
5 Plasmon dispersion relation
When the external charge is absent, ρ̃ext (z) = 0, the first two terms on the right hand side of Eq. (24) vanish,
e 0 and
but the resulting electrostatic potential may remain finite if the components of the displacement vector D
e
e
e
D h are not both zero. Noticing that the system of equations (25) and (26) for D0 and D h becomes homogeneous
when ρ̃ext (z) = 0, one may seek a relation between the frequency ω and the wavenumber q = kqk that gives
e 0 and D
e h , and hence to finite Φ(z)
e
rise to a nontrivial solution for D
in Eq. (24). Such source-free, steady-state
response (with γ = 0) of the EG in a metal is characterized by the eigenmodes called plasma oscillations
(or plasmons), while the resulting expression for the eigenfrequency ω as a function of the wavenumber q
is called plasmon dispersion relation. This relation is obtained for an EG described by the nonlocal HDM by
setting the determinant of the matrix defining the left hand sides of the equations (25) and (26) to be zero.
To simplify the calculation, we only consider here the special case of a metal slab in vacuum by setting
the dielectric constants ϵ1 = ϵ2 = ϵ3 = 1, which gives
0
e z0 ) = 2π e−q|z−z |
G(z,
q
(37)
0
e D (z, z0 ) = 2πe−q|z−z | sgn(z − z0 ), where sgn is
for all values of z and z0 . Accordingly, one finds from Eq. (27) G
the signum function. Defining the auxiliary function
ˆ h
1
e (z, z00 )H(z
e 00 , z0 ) dz00 ,
Γ(z, z0 ) =
G
(38)
4π 0 D
one may express the coefficients in the left hand sides of the equations (25) and (26) as Γ µ,ν ≡ Γ(µ, ν) where
µ = 0, h and ν = 0, h. On using Eqs. (22) and (37) in Eq. (38), we find that Γ00 = −Γ hh , where
q cosh(q s h) − q s sinh(q s h) − q e−qh
,
2q s (q s 2 − q2 ) sinh(q s h)
(39)
e−qh q cosh(q s h) + e−qh q s sinh(q s h) − q
.
2q s (q s 2 − q2 ) sinh(q s h)
(40)
Γ hh = k s 2
as well as Γ h0 = −Γ0h , where
Γ0h = k s 2
Then, the condition that gives rise to a nontrivial solution of the homogeneous version of Eqs. (25) and (26)
2
= 0 giving two types of the relation between ω and q,
may be written as (1 + Γ hh )2 − Γ0h
1
q
Ω2 =
1 − e−qh 1 +
coth(q s h/2) ,
(41)
2
qs
q
1
Ω2 =
1 + e−qh 1 +
tanh(q s h/2) ,
(42)
2
qs
q
where Ω = ω/ω p is the reduced eigenfrequency, allowing to rewrite the parameter q s as q s = q2 + k s 2 (1 − Ω2 )
for γ = 0. The equations in Eqs. (41) and (42) are identical to those derived in Ref. [14] for the same system by
using a different method. In particular, the authors of the Ref. [14] showed that the solutions of Eq. (41) for Ω
give rise to the dispersion relation for the so-called even plasmon modes where the induced charge density
ρ̃(z) is symmetric function about the center of the metal slab, whereas the solutions of Eq. (42) give rise to
the dispersion relation for odd plasmon modes where the charge density is an antisymmetric function of the
position z relative to the slab center.
It is instructive to compare the plasmon dispersion relations in Eqs. (41) and (42) for an EG described by
the nonlocal HDM with those arising for a local model of EG. The latter model may be deduced by setting
β = 0 in Eq.
amounts to taking the limit k s → ∞ and hence q s → ∞ in Eqs. (41) and (42). This
q (8), which
−qh
gives Ω =
1∓e
/2 for the even and odd plasmon modes in the local model, respectively. Notice that
the parameter k s may be conveniently expressed in terms of the dimensionless electron-electron distance in
−1/3
4 1/3
3
the EG, r s = 4πa3B n0 /3
.
with a B = ~2 / me2 being the Bohr radius for electron, as k s = a B √
r s 9π
52 | N. Kang et al.
One may see then that the formal limit k s → ∞ corresponds to a very dense EG with small electron-electron
distance r s .
In Fig. 2, we show the results of solving Eqs. (41) and (42) for the reduced frequency Ω as a function of
the reduced wavenumber qh by choosing the screening parameter in the nonlocal HDM such that k s h = 20.
Those results are compared with the plasmon dispersion relations for the even and odd modes in the EG
described by the local model.
It is noteworthy that there are only one even and one odd plasmon dispersion relations in the local model,
which do not exceed the bulk plasma frequency ω p . The even and odd eigenfrequencies approach the values
ω = 0 and ω = ω p in the limit of long wavelengths, qh → 0, respectively, while they both converge to a
√
common value ω = ω p / 2 at short wavelengths, qh 1, which is characteristic of the so-called surface
plasmon in a thick metal slab within the local model.
On the other hand, one notices that the results for the nonlocal HDM give rise to multiple plasmon dispersion relations for both even and odd modes, labeled by n = 0, 1, 2, . . .. The modes labeled n = 0 seem to
be derived from the even and odd modes of the local model as they converge to their respective dispersion
relations in the limit of long wavelengths, or for a thin metal slab
q with qh 1. In particular, it appears that
the reduced frequency for the n = 0 modes is bounded by Ω < 1 + q2 /k s 2 , which makes the parameter q s in
Eq. (14) with γ = 0 real and positive, so that the corresponding solutions of Eq. (13) for the HDM are localized
near the surfaces of the metal slab. Hence the n = 0 modes are associated with the surface plasmons, which
are of interest in the nanoplasmonic applications.
On the other hand, the plasmon dispersion relations for the higher harmonics in q
the nonlocal model,
labeled n = 1, 2, 3 . . ., are characterized by the reduced frequencies larger than Ω = 1 + q2 /k s 2 , so that
the corresponding values of q s in Eq. (14) with γ = 0 are purely imaginary, giving rise to oscillatory solutions
of Eq. (13) throughout the metal slab, which are related to the bulk plasmon modes. One can see that in the
nonlocal model all the bulk plasmon modes and the even surface plasmon mode exhibit positive dispersion
characterized by an increase in frequency with increasing wavenumber q. This is not true, however, for the
odd surface mode, which has a negative dispersion for small q that passes through a minimum and joins
the even surface mode at higher q values. Finally, one notices that all the plasmon dispersion relations in the
nonlocal model become approximately linear functions of the wavenumber for sufficiently large values of qh.
6 Concluding remarks
We have presented a detailed derivation of the expression for the electrostatic potential as an integral over an
arbitrary external charge density for a structure consisting of three dielectric layers, with the middle region
occupied by a metal with the nonlocal dielectric response. The metal slab was described by a hydrodynamic
model of the electron gas that contains dispersive effects of the Thomas-Fermi pressure, giving rise to a wave
equation for the electron gas. By assuming the so-called specular reflection model, or the semiclassical infinite barrier model for the electron gas, we were able to express the induced charge density in the metal slab by
means of the Green’s function satisfying Neumann’s boundary conditions at the endpoints of the slab region.
This expression was further used in a piece-wise solution of the Poisson equation for the electrostatic potential in terms of a Green’s function for the three-layer domain, which satisfies the usual electrostatic boundary
and matching conditions. As an illustration, we studied the source-free formulation of the problem that gives
rise to nontrivial solutions for the electrostatic potential and the electron density when the eigenfrequency
and the wavenumber satisfy the dispersion relation(s), which describe both the surface and the bulk plasma
waves in the electron gas.
Our work illustrates the versatility of using the Green’s function for the Poisson equation, which may be
easily generalized to include effects of variation in the structure and composition of the dielectric materials
used with metallic nanostructures in the area of nanoplasmonics. In addition, using the hydrodynamic model
to describe the electron gas in metallic nanostructures requires a very careful consideration of the boundary
conditions, which is most systematically carried through within the framework of the Green’s function for the
Analyzing nonlocal effects in the plasmon spectra of a metal slab |
53
Local & Hydrodynamic Dispersion of Even Modes
3
n=8
2.5 n=7
n=6
ω/ωp
2
n=5
n=4
1.5
n=3
n=2
n=1
1
n=0
0.5
0
Local
Hydrodynamic
0
5
10
15
20
25
30
35
40
qh
Local & Hydrodynamic Dispersion of Odd Modes
3
n=8
n=7
2.5
n=6
2 n=5
ω/ωp
n=4
1.5 n=3
n=2
n=1
1
n=0
0.5
0
Local
Hydrodynamic
0
5
10
15
20
25
30
35
40
qh
Figure 2: Dispersion relations for both the (a) even and (b) odd plasmon modes in a metal slab of thickness h. The results are
shown for both the hydrodynamic model with nonlocal effects (blue solid lines) and the local model (red dashed lines) of the
electron gas with the density parameter fixed at r s = 3. The Thomas-Fermi screening length in the electron gas is chosen so
that k s h = 20.
resulting wave equation for the electron gas. This becomes particularly important when quantum effects are
included in the hydrodynamic model, which increase the order of the wave equation, as discussed in detail
elsewhere [15].
Acknowledgement: This work was supported by the National Natural Science Foundation of China (No.11275038),
National Basic Research Program of China (Grants No. 2010CB832901), and Program for New Century Excellent Talents in University (NCET-08-0073). This work was completed during the visit of Y.-Y. Zhang to the
University of Waterloo under the sponsorship of the China Scholarship Council. Z.L.M. acknowledges support by the Natural Sciences and Engineering Research Council of Canada.
54 | N. Kang et al.
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