# Stratified Graphene-Noble Metal Systems for Low-Loss Plasmonics Applications

L. Rast<EMAIL_ADDRESS>Applied Chemicals and Materials Division,
National Institute of Standards and Technology, Boulder, CO 80305  T. J.
Sullivan  Mathematics Institute, University of Warwick, Coventry, CV4 7AL,
United Kingdom  V. K. Tewary  Applied Chemicals and Materials Division,
National Institute of Standards and Technology, Boulder, CO 80305

###### Abstract

We propose a composite layered structure for tunable, low-loss plasmon
resonances, which consists of a noble-metal thin film coated in graphene and
supported on a hexagonal boron nitride (hBN) substrate. We calculate electron
energy loss spectra (EELS) for these structures, and numerically demonstrate
that bulk plasmon losses in noble-metal films can be significantly reduced,
and surface coupling enhanced, through the addition of a graphene coating and
the wide-bandgap hBN substrate. Silver films with a trilayer graphene coating
and hBN substrate demonstrated surface plasmon-dominant spectral profiles for
metallic layers as thick as $34\,$nm. A continued-fraction expression for the
effective dielectric function, based on a specular reflection model which
includes boundary interactions, is used to systematically demonstrate plasmon
peak tunability for a variety of configurations. Variations include substrate,
plasmonic metal, and individual layer thickness for each material. Mesoscale
calculation of EELS is performed with individual layer dielectric functions as
input to the effective dielectric function calculation, from which the loss
spectra are directly determined.

##  I Introduction

Plasmonic devices hold promise for a host of metamaterials applications, due
to their ability to control light propagation on a subwavelength scale. Noble
metal nanomaterials are in some ways ideal components in such devices – they
possess tunable, large amplitude plasmon resonances, which can be excited at
optical wavelengths. Johnson and Christy (1972) Nevertheless, metallic
plasmonic devices at infrared and visible wavelengths present significant
challenges due to bulk plasmon losses. These large losses severely limit the
practicality of these materials for a wide variety of applications,
particularly in telecommunications and photovoltaics. West  _et al._ (2010)

Bulk losses may be mitigated through the use of very thin noble-metal films;
however, fabrication of uniform thin metal films is experimentally
challenging. Thin films tend to form islands and often require adhesion layers
that alter the electronic structure of the device. Boragno  _et al._ (2009);
Rha and Park (1997) Additionally, thin noble-metal films still suffer
significant resistive losses in the visible regime. Atwater and Polman (2010)
Composite materials designed to shift plasmon resonances to a low-loss regime,
while increasing coupling to surface plasmons and diminishing bulk resonances,
provide clear advantages from a fabrication standpoint. Systems that combine
the strong and tunable optical-wavelength plasmon resonances of noble metals
with materials possessing improved transport properties could mitigate
surface-plasmon losses in the visible regime.

Graphene is an ideal candidate for such a composite structure due to its
unparalleled carrier mobility. This allows for extremely enhanced and tunable
electromagnetic response spectra when doped with other plasmonic materials, or
fabricated as a component of a multilayer structure. Monolayer graphene has a
response spectrum dominated by absorption peaks at $\approx 4.5\,$eV and
$\approx 15\,$eV, the $\pi$ and $\pi$ \+ $\sigma$ surface plasmons. Graphene
plasmon resonance provides low losses in the frequency regime below the
optical phonon frequency of $0.2\,$eV, where large losses are typically
present for metallic plasmonic materials. Geim and Novoselov (2007)

Hexagonal boron nitride (hBN) provides several advantages as a substrate for
graphene-based plasmonic materials, as it is an ultra-flat wide bandgap
insulator with excellent thermal transport properties. Xue  _et al._ (2011)
The flat morphology and uniformity provided by hBN are are desirable as key
factors in preserving the transport properties of graphene. Xue  _et al._
(2011) Results of our numerical study clearly elucidate the advantages of the
hBN spectral response for low-loss plasmonics applications. We demonstrate in
this work that through the combined advantages of graphene and noble-metal
films, along with careful choice of substrate material and layer thickness,
multilayer systems can be tuned for low-loss surface plasmon resonance (SPR).

Metamaterials derive their exotic properties in part from advantageous
behaviors of their constituent materials, as well as collective behaviors that
emerge due to interactions within the system. The complex task of designing
metamaterials tailored for applications as diverse as photovoltaics,
biosensing, and microscopy gives rise to a need to develop efficient methods
for predicting metamaterial properties. These methods must realistically treat
both the individual material properties and the interactions among the
constituent materials. We detail and employ such a method for the calculation
of electron energy loss spectra (EELS) of multilayer structures consisting of
graphene layers on noble-metal (silver, gold, and copper) films with silicon
and hBN substrates.

The effective dielectric function is based on a specular reflection model,
first derived by Lambin et al. Lambin  _et al._ (1985), and takes into account
the boundary conditions across each layer in the stratified structure. The use
of the efficient continued-fraction expression along with pre-prepared
libraries of dielectric functions for the individual materials opens up the
possibility of multilayer graphene composites by design.

##  II EELS Calculation Details

###  II.1 General Procedure

Figure 1: Multilayer structure: EELS are calculated for a structure consisting
of graphene top layer(s), noble-metal middle layer, and semiconductor
substrate.

EELS are calculated for a variety of multilayer sandwich structures as
depicted in Fig. 1. In general, the sandwich structure consists of a noble-
metal middle layer, a graphene coating of an integer number of layers, and a
semiconducting substrate. Graphene layer numbers vary from 1 to 20. Noble
metals employed include silver, gold, and copper, and are varied in thickness.
The effect of two different substrates is considered: (1) a more traditional
substrate, Si with a $4\,$nm top layer of SiO2 (Fig. 1 (a)), and (2) hBN (Fig.
1 (b)). $4\,$nm is a typical thickness for the SiO2 layer, which arises from
the thermal processing of the Si substrate. Seah and White (2002) The Si layer
is assumed to be semi-infinite. In the case of the hBN substrate, an integer
number of hBN layers is considered, with a semi-infinite vacuum layer below
the hBN layer. The single-layer thickness for both graphene and hBN are taken
to be $3.4\,$Å, the natural inter-layer spacing for graphite. Berashevich and
Chakraborty (2011) In all cases, a semi-infinite vacuum layer exists above the
top ($z=0$) layer.

Individual complex dielectric functions are obtained for each layer. We
calculate the values through ab-initio methods for graphene and hBN. Empirical
values from the literature are used for the metal and silicon substrate
layers. These values are then stored for use as input to a continued-fraction
algorithm, which yields the effective dielectric function. This algorithm is
outlined in Section II.2.

###  II.2 The Effective Dielectric Function

The effective dielectric function, $\xi(\omega,k,z)$, of the stratified
structure in Fig. 1 is that of Lambin et al. Lambin  _et al._ (1985) The
expression for $\xi$ was derived from EELS theory in a reflection geometry.
The expression has been shown to be applicable to both phonons Lambin  _et
al._ (1985) and polaritons Dereux  _et al._ (1988) in stratified structures
with histogram-like dielectric functions (continuous within each layer) and
interacting interfaces. Though Lambin’s initial paper containing the
derivation focused on semiconducting materials, the expression and the
formalism from which it is derived also apply to surface plasmon structure of
alternating of metal-insulator layers. Dereux  _et al._ (1988); Economou
(1969) It is worth noting that the specular reflection model agrees well with
the spectroscopic predictions of the Bloch hydrodynamic model in the small
wave vector regime considered in our work. Ritchie and Marusak (1966)

The $z$ coordinate is in the direction perpendicular to the free surface of
the sample, extending from the $z=0$ surface to $-\infty$. $\mathbf{k}$
denotes the surface excitation (plasmon or phonon) wave vector and $\omega$ is
the frequency of excitation.

$\xi(\mathbf{k},\omega,z)=\frac{i\mathbf{D}(\mathbf{k},\omega,z)\cdot\mathbf{n}}{\mathbf{E}(\mathbf{k},\omega,z)\cdot\mathbf{k}/k},$
(1)

where
$\mathbf{D}(\mathbf{k},\omega,z)=\epsilon(\omega,z)\mathbf{E}(\mathbf{k},\omega,z)$,
and $\epsilon(\omega,z$) is the long wavelength dielectric function (tensor)
of the material at $z$. $\xi$ remains continuous even in the case of sharp
interfaces parallel to the $x$-$y$ directions below the surface (as is the
case in our multilayer system). This is due to the interface boundary
conditions: continuity of $D_{\perp}$ and $E_{\parallel}$.

The effective dielectric function $\xi_{0}(k,\omega)$ (Eq. 3) is a solution to
the Riccati equation (Eq. 2), in the long-wavelength approximation $k\approx
0$, at the $z=0$ surface. Lambin  _et al._ (1985) We fix $k$ as $k=0.05\,$Å-1
for both the ab-initio calculations and the composite calculation. Eq. 2 was
derived for heterogeneous materials made of a succession of layers (with
homogeneous dielectric functions within each layer), the layers having
parallel interfaces. $\epsilon(z)$ are complex functions, with positive
imaginary parts at $z=0$. Lambin  _et al._ (1985)

$\displaystyle\frac{1}{k}\frac{\mathrm{d}\xi(z)}{\mathrm{d}z}+\frac{\xi^{2}(z)}{\epsilon(z)}=\epsilon(z)$
(2)
$\xi_{0}=a_{1}-\frac{b_{1}^{2}}{a_{1}+a_{2}-\frac{b_{2}^{2}}{a_{2}+a_{3}-\frac{b_{3}^{2}}{a_{3}+a_{4}-\cdots}}}$
(3)

where

$\displaystyle a_{i}=\epsilon_{i}\coth(kd_{i})$ (4)

and

$\displaystyle b_{i}=\epsilon_{i}/\sinh(kd_{i}).$ (5)

Once individual dielectric functions are obtained, this procedure allows for
the performance of mesoscale EELS calculations of a wide variety of
structures. Layer thickness and materials can easily be substituted in the
calculation, with each EELS calculation running in a fraction of a second
(nearly independent of the spectral range). EELS are calculated directly from
the effective dielectric function as

$\mathrm{EELS}=\mathrm{Im}\left[\frac{-1}{\xi(\omega,k)+1}\right].$ (6)

Inspection of Eq. 3 reveals that for $\mathrm{Im}[\epsilon_{i}]>0$,
$\mathrm{Im}[\xi_{0}]>0$. The EELS spectra given by Eq. 6 are then positive.

###  II.3 Noble-Metal Dielectric Functions

The copper, silver, and gold complex dielectric functions are empirical values
by Johnson and Christy Johnson and Christy (1972) obtained by reflection and
transmission spectroscopy on vacuum-evaporated films at room temperature.
Film-thickness in the Johnson and Christy study ranged from $185\,$Å–
$500\,$Å. Dielectric functions in the film-thickness range $250\,$Å – $500\,$Å
did not vary significantly. In our work, the intermediate value $340\,$Å was
chosen to represent bulk mode dominant (yet still nanoscale) metallic thin
films.

###  II.4 SiO2 and Si Dielectric Constants

Relative static permittivities of 3.9 and 11.68 are used for the SiO2 and Si
dielectric constants, respectively. These are reasonable and widely-used
values from the literature. Murarka (2003); Yi (2012)

###  II.5 Graphene and hBN Individual Layer Dielectric Functions

Complex dielectric functions for graphene and hBN are displayed in Fig. 2 (a)
and Fig. 2 (b), respectively. These ab-initio calculations use the time-
dependent density functional theory in the local density approximation (LDA),
and are implemented in the Python code GPAW, a real-space electronic structure
code using the projector augmented wave method. 111Certain commercial
equipment, instruments, or materials are identified in this paper in order to
specify the experimental procedure adequately. Such identification is not
intended to imply recommendation or endorsement by the National Institute of
Standards and Technology, nor is it intended to imply that the materials or
equipment identified are necessarily the best available for the purpose.,
Mortensen  _et al._ (2005); Enkovaara  _et al._ (2010); Walter  _et al._
(2008); Yan  _et al._ (2011) Both graphene and hBN dielectric functions are
calculated in the armchair configuration for a momentum transfer value of
$0.05\,$Å-1, along the $\bar{\Gamma}$-$\bar{M}$ direction of the surface
Brillouin zone. The armchair configuration and value of $k$ are selected for
comparison of EELS with existing data in the literature obtained via density
functional theory (DFT) methods. The $k$-point sampling with $30\times
30\times 1$ Monkhorst–Pack grid was chosen for the band-structure and EELS
calculations for both graphene and hBN.

Our model utilizes dielectric functions due to surface parallel excitations
only, as the effective dielectric function is derived in a specular reflection
geometry. This is a reasonable approximation as out-of-plane excitations are
minimal in graphene at energies less than $\approx 10\,$eV, and extreme UV
radiation is outside of the regime of interest for this study. Gass  _et al._
(2008) A lattice constant of $2.46\,$Å is used for graphene, and $2.50\,$Å for
hBN, as hBN is nearly isomorphic to the graphene, except for the slightly
larger lattice constant. Song  _et al._ (2010)

The dielectric function we have obtained for graphene (see Fig. 2 (a)) is
nearly identical to those of Yan et al. Yan  _et al._ (2011), where the
authors used the projector augmented wave methodology implemented in GPAW in
terms of linear combinations of atomic orbitals, with a momentum transfer of
$0.046\,$Å-1, along the $\bar{\Gamma}$-$\bar{M}$ direction of the surface
Brillouin zone. Both our results and that of Yan et al. display collective
peaks for free-standing single-layer graphene at $\approx 5\,$ eV and $\approx
15\,$ eV. These values also agree with experimental EELS results for single-
layer graphene plasmons (with in-plane excitation), for example Eberlein et
al. Eberlein  _et al._ (2008) find the $\pi$ plasmon at 4.7 eV and $\pi$ \+
$\sigma$ at 14.6 eV.

A low-energy peak (below 1 eV) is also apparent in Fig. 2 (a). This feature,
which has been observed in ab-intio calculations by others Trevisanutto  _et
al._ (2008); Yan  _et al._ (2011), corresponds to a broad shoulder in the
EELS, and is due to the low-energy $\pi$ $\rightarrow$ $\pi^{\star}$ single-
particle excitation. Low energy graphene features, including intraband
transitions are generally quite dependent upon the value of momentum transfer.
Hwang and Das Sarma (2007) These features are outside the energy regime of
interest for the surface and bulk plasmon peaks in this paper, work in which
very low-energy excitations are a focus should not neglect momentum transfer
dependence. In the context of these calculations, increasing momentum transfer
values lead to a shift towards higher energies of the $\pi$ plasmons for
intrinsic graphene. Yan  _et al._ (2011); Gao (2011) Others, for example Gao
et al. Gao (2011) have found (via the time-dependent local density
approximation) linear dispersion for the $\pi$ plasmon (the key feature in our
energy regime of interest) in single-layer graphene for both
$\bar{\Gamma}$-$\bar{M}$ and $\bar{\Gamma}$-$\bar{K}$ directions. The
calculations of Gao et al. were found to agree well with experiment.

Our hBN dielectric function (see Fig. 2 (b)) compares well with that of Yan et
al. Yan  _et al._ (2012), who obtained their results in the long-wavelength
limit, using the LDA-adiabatic local density approximation method. Both our
results for hBN spectra and that of Yan et al. display a broad absorbtion peak
with onset at $\approx 4.5\,$ eV and maximum at $\approx 5.75\,$ eV, which is
in good agreement with previous literature. Wirtz  _et al._ (2006) The
dielectric function obtained for hBN is also similar to the experimental
results of Tarrio and Schnatterly Tarrio and Schnatterly (1989), where the
authors measured a peak in the imaginary part of the dielectric function at
$\approx 6\,$ eV. Differences between the complex dielectric function used in
this study and that of Tarrio and Schnatterly (1989) are attributable to the
larger momentum transfer value of $0.13\,$Å-1 in the Tarrio and Schnatterly
study.

Figure 2:  Complex relative dielectric function $\epsilon(\omega)$ for
graphene (a) and hBN (b). Real and imaginary parts
($\epsilon^{\prime}(\omega)$ and $\epsilon^{\prime\prime}(\omega)$) are
represented by solid and dotted lines, respectively.

##  III RESULTS

###  III.1 Decreasing metallic film thickness and the begrenzung effect

Figure 3: EELS: (a), (b), and (c) demonstrate the effect of differing
thickness for the Ag, Au, and noble-metal layer, respectively, with a SiO2/Si
substrate, and without a graphene coating. Film thicknesses are $34\,$nm
(solid line), $20\,$nm (long dashes), $10\,$nm (short dashes), and $4\,$nm
(dotted line).

Figures 3 (a)–(c) (respectively, Ag, Au, and Cu on SiO2/Si, without graphene
top coating) demonstrate the effect of decreasing noble metal layer thickness.
These data should serve as a basis for comparison with Fig. 4 (a)–(c), which
are discussed in detail in the next section.

As the noble metal film thickness is reduced in Fig. 3 (a), the so called
_begrenzung effect_ is apparent. An increase in the surface-to-volume ratio in
the metal causes enhanced coupling to the surface resonance and diminished
coupling to the bulk modes. Ritchie (1957); Osma and Garcia de Abajo (1997)

In the case of a thin metallic slab, empirical models have been quite
thoroughly explored. Upon the introduction of a boundary to an infinite
metallic slab, a negative (begrenzung) peak is introduced at the same energy
as the bulk peak, and a trailing surface peak appears. Ritchie (1957) The
surface peak becomes more pronounced with decreasing thickness, as does the
negative begrenzung peak, decreasing the net bulk-plasmon amplitude. A sharp
transition can be observed, between $34\,$nm and $10\,$nm Ag film thickness,
in which surface modes become dominant. This transition can also be observed
for Au and Cu films, between $34\,$nm and $20\,$nm. This is consistent with
observations in the well-validated and widely-used empirical data by Johnson
and Christy. Johnson and Christy (1972)

###  III.2 SPR Enhancement due to the graphene coating

####  III.2.1 SiO2/Si Substrate

Figure 4: EELS for varying numbers of layers for the graphene film: (a)
$34\,$nm Ag layer and SiO2/Si substrate, (b) the surface peak in (a), and (c)
$34\,$nm Ag layer and monolayer hBN substrate. In both cases the graphene
layer numbers are 0 (solid line), 1 (long dashes), 3 (intermediate-length
dashes), 10 (short dashes), and 20 (dotted line).

Figures 4 (a) and (b) demonstrate the effect of the addition of graphene
coatings of varyious thicknesses to the Ag surface, in the case of a SiO2/Si
substrate. For up to three layers of graphene, the surface peak remains
virtually unchanged. However, at three layers, the bulk peak amplitude is
reduced by approximately $50\%$. This effect is attributed to the introduction
of a thin boundary layer, decreasing the bulk resonance amplitude. Imposition
of a boundary leads to simultaneous diminished coupling to bulk modes and
enhanced coupling to surface modes, physically similar to the aforementioned
begrenzung effect. Ritchie (1957); Osma and Garcia de Abajo (1997); Egerton
(2011) At 20 layers, as we would expect, the bulk peak broadens significantly
(indicating increased losses) and the system has spectral properties
(broadening and peak position) resembling those of graphite on silver. Peak
positions calculated for the 20-layer graphene coating ($\approx 2.6\,$eV and
$\approx 3.5\,$eV) are very close to those measured in EELS of thin films of
Ag nanoparticles evaporated on graphitic surfaces, for example $2.2\,$eV and
$3.4\,$eV. Palmer  _et al._ (2002) Differences in peak position (particularly
for the lower-energy peak) are attributable to differences in morphology
between the Ag slab in our calculations and the film of Ag nanoparticles in
the experimental samples.

####  III.2.2 hBN Substrate

Figure 4 (c) demonstrates the effect of the addition of graphene coatings of
varying thickness to the silver surface, in the case of a hBN substrate. The
use of the hBN substrate dramatically enhances the SPR peak, as well as
shifting the surface peak to an energy very close to that of the bulk peak,
even the case where no graphene coating is used. This blue-shift of the SPR is
due to the much smaller real part of the hBN dielectric function in comparison
to Si. Decreasing substrate dielectric function is known to dramatically blue-
shift SPR. G. Xu and Yoshimura (2003)

Addition of a single-layer graphene coating serves to further diminish the
bulk peak without significant degradation of the surface peak. For up to 3
layers of graphene, the surface peak does not broaden significantly. As in the
case of the SiO2/Si substrate, broadening of the surface peak for more than 3
layers of graphene corresponds to increased losses. Indeed, at 10 graphene
layers the response is what one would expect from a graphite coating. A
similar transition in plasmonic behavior at 10 layers has been observed
experimentally for multilayer graphene EELS. Eberlein  _et al._ (2008)

####  III.2.3 Ideal configurations for Ag: enhancement of surface modes

Figure 5: EELS:(a) Comparison of the use of a SiO2/Si substrate (solid line),
a mono-, tri-, 10-, and 20-layer hBN substrate (long dashes, intermediate-
length dashes, short dashes, and a dotted line, respectively). No graphene
coating is used and the Ag film is $34\,$nm thick. (b) Close-up of EELS peaks
for the hBN substrates. The suspended sample (“vacuum substrate”) is
represented by the solid line in this case. Mono-, tri-, 10-, and 20-layer hBN
substrates are represented by long dashes, intermediate-length dashes, short
dashes, and a dotted line, respectively.

Figure 5 depicts EELS for a $34\,$nm Ag film (with no graphene coating) on
various substrates: SiO2/Si, mono-layer, tri-layer, and 10-layer hBN, and
semi-infinite vacuum (as would occur for a suspended sample). Introduction of
the high band-gap hBN substrate dramatically enhances surface modes as
compared to SiO2/Si. The result is striking — the sample with a monolayer hBN
substrate has a spectral profile nearly identical to that of a suspended
sample. The EELS remain very nearly identical to the suspended case for up to
10 layers. This is in accordance with experimental results that have found the
electron mobility of graphene on hBN to be nearly that of suspended graphene.
Xue  _et al._ (2011) The similarity of the hBN substrate to the vacuum is
expected due to its ultra-wide bandgap, and in the case of our model partly
due to the atomic layer thickness. Upon inspection of Fig. 2 (b), it is clear
that both real and imaginary parts of hBN’s dielectric function are small;
therefore at a thickness of $3.4\,$Å, the properties of this substrate
approach those of the vacuum. The hBN substrate appears to dramatically
enhance coupling to the surface plasmon resonance when compared to the SiO2/Si
substrate, as well as reducing the peak strength of the lossy bulk plasmon.

Figure 6: (a)EELS: trilayer graphene on Ag layers (of varying thickness) with
a mono-layer hBN substrate. (b) Free-standing Ag films of various thicknesses.
In both (a) and (b) Ag layer thicknesses are $50\,$nm (solid line), $34\,$nm
(long dashes), $20\,$nm (intermediate-length dashes), and $10\,$nm (short
dashes), and $4\,$nm (dotted line).

Fig. 6 (a) demonstrates the combined effects of graphene coating and hBN
substrate for various Ag film thicknesses. Comparison of Fig. 6 (a) with Fig.
3 (a) reveals that the introduction of a tri-layer graphene coating and hBN
substrate to the Ag film results in a surface plasmon intensity and relative
surface to bulk intensity similar to a significantly thinner Ag film on the
SiO2/Si substrate. For example, the surface plasmon intensity and relative
surface to bulk intensity of the $34\,$nm Ag case in Fig. 6 (a) is comparable
to the $20\,$nm case in Fig. 3 (a). Additionally, the surface plasmon
intensity and relative surface to bulk intensity of the $20\,$nm Ag case in
Fig. 6 (a) is comparable to the $10\,$nm case in Fig. 3(a).

Comparison of Fig. 6 (a) with Fig. 6 (b) (which shows spectra for free-
standing Ag of various thicknesses) further elucidates the effect of a few
graphene layers. For thicker Ag films, a few-layer graphene coating diminishes
the bulk peak without significant broadening of the SPR. However, for very
thin Ag films, of 4nm, for example, the effect of diminishing the bulk peak is
negligible and the addition of the graphene coating of three layers only
slightly broadens the SPR. This is in keeping with the idea that for bulk Ag,
the addition of a graphene layer enhances surface coupling, through the
begrenzung effect. This is further demonstrated by the lack of significant
broadening of the SPR peak for a single-layer graphene coating (see Fig. 4
(c)).

###  III.3 Au and Cu Noble-Metal Film Composites

####  III.3.1 Spectral changes of Au and Cu films due to the graphene coating

One may wish to employ a noble metal other than Ag in multilayer structures
such as depicted in this paper. Cu has the advantage of lower cost and is
therefore attractive for industrial applications. Both Cu and Au may also be
of interest due to inherent surface plasmons that occur at longer wavelengths
than those of Ag. In this section we demonstrate the effect of graphene
coatings and the hBN substrate on Au and Cu films. For reference, Fig. 3 (b)
and (c) demonstrate the effect of reduction of noble-metal film thickness for
Au and Cu films (respectively) on a SiO2/Si substrate. The begrenzung effect
is again increasingly apparent in both Au and Cu film plasmonic response as
film thickness decreases. At $20\,$nm, both Au and Cu display strong SPR and
broad absorption for higher energies, rather than a bulk plasmon peak.

Inspection of Fig. 7 (a) and (b) reveals that the introduction of a graphene
coating to an Au or Cu surface does measurably strengthen surface modes while
reducing bulk plasmon intensity, although the effect is not as dramatic as in
the Ag case (see Fig. 4). Coatings as thick as 20 layers further enhance and
do not significantly broaden the surface peak. A jump in broad absorption
(corresponding to increased losses) at higher energies is apparent in the case
of both Au and Cu for more than three graphene layers.

Figure 7: EELS: (a) The effect of various numbers of layers for the graphene
film, in the case of a $34\,$nm Au layer and SiO2/Si substrate, and (b) The
effect of varyious numbers of layers for the graphene film, in the case of a
$34\,$nm Cu layer and SiO2/Si substrate. Graphene layer numbers are 0 (solid
line), 1 (long dashes), 3 (intermediate-length dashes), 10 (short dashes), and
20 (dotted line).

####  III.3.2 Metallic films with hBN substrate: noble-metal comparison

Figure 8: EELS: $34\,$nm Au (a) and Cu (b) films, with monolayer graphene
coatings and various substrates including SiO2/Si (solid line), hBN monolayer
(dashes), and vacuum (dotted line) substrates.

Figure 8 (a) and (b) are the EELS for Au and Cu films of thickness $34\,$nm
(respectively), with monolayer graphene coating, comparing the SiO2/Si
substrate, hBN substrate, and suspended samples. Surface peaks at $\approx
2\,$eV for both Au and Cu are surface peaks due to change in dielectric
function across the metal/SiO2/Si substrate interface. The addition of the hBN
layer in these cases appears to sharpen and enhance Au and Cu inherent
plasmons. Notably, in the Au and Cu cases, the hBN substrate appears to
produce a similar spectral profile to the suspended “vacuum substrate”, as in
the case of the Ag film. Rigorous optimization of both the surface coating and
substrate for both Au and Cu films is the subject of further investigation.

##  IV DISCUSSION

In this study we systematically investigated the effect of the use of
different plasmonic materials, different semiconducting substrates, and
different layer thicknesses in a multilayer graphene-based plasmonic composite
structure. For mono-, bi-, and tri-layer graphene, in the case of $34\,$nm Ag
layers (where bulk modes would dominate in the absence of a graphene coating),
bulk plasmon modes are significantly diminished, while maintaining the
strength of surface modes.

Reduced plasmon losses for graphene coated plasmonic metals on hBN substrates
likely originates within the context of our mesoscopic model from two primary
physical effects: (1) The addition of a graphene boundary layer on the
metallic surface reduces coupling of excitations to bulk plasmons through the
begrenzung effect. The origin of the begrenzung effect is a reduction of the
degrees of freedom for excitations, and thus further surface confinement comes
at the expense of bulk oscillations, leading to reduced losses. (2) The
strongly insulating hBN substrate diminishes bulk losses and enhances surface
confinement particularly through a reduction in scattering.

Traditional substrates such as Si are known to degrade the electron transport
properties of graphene as compared to suspended samples. Xue  _et al._ (2011)
Reduced transport properties are due to various scattering mechanisms. There
is strong evidence that scattering in these systems is due in large part to
various substrate interactions including interfacial phonons (which are taken
into account in our work), surface charge traps, and substrate stabilized
ripples. Bolotin  _et al._ (2008)

It is therefore reasonable to mitigate scattering in graphene-based systems
for plasmonics (and devices for plasmonics in general) by suspending samples,
or through the use of alternative substrates. Bolotin  _et al._ (2008) In this
work, for Ag films, the use of a hBN substrate (rather than the more
traditional silicon) is found to shift plasmonic coupling towards surface
modes, both diminishing bulk losses and enhancing the SPR peak amplitude. This
effect is particularly dramatic in the case of the $34\,$nm Ag film (including
those with no graphene top-coating), as an Ag film at this thickness on an
SiO2/Si substrate produces EELS that are extremely dominated by the bulk
resonance. Johnson and Christy (1972) This is interesting in light of recent
experimental results, showing electron mobility of graphene on hBN to be
similar to that of suspended graphene. Xue  _et al._ (2011) We expect SPR
enhancement for other wide-bandgap substrates due to enhanced surface plasmon
field confinement. Further comparison of alternate wide-bandgap substrate
materials, such as SiC, are the subject of future work.

As expected, bulk modes were quenched for $4\,$nm Au and Cu films. For Ag,
bulk modes very nearly vanished at this thickness, and are nearly undetectable
when coupled with the hBN substrate. However, the morphology of very thin
metallic films deposited on graphene is difficult to control, often forming
island-like structures of various sizes. Boragno  _et al._ (2009) This is also
the case for graphene deposited on a metallic substrate, as agglomeration
below a critical thickness is a general property of thin films. Rha and Park
(1997) Results of this study indicate that Ag films as thick as $34\,$nm, when
coated with a graphene film and placed on a wide-bandgap substrate such as
hBN, may also be employed for low-loss plasmon resonance applications. An hBN
substrate, due to its ultra-flat morphology, would also lend additional
uniformity to the structure.

The mesoscopic model used in these calculations has several limitations that
are worth discussing. Results of this study are valid in the long-wavelength
limit for which the continued fraction expression by Lambin et al. was
derived. Additionally, coupling between layers is classical (via boundary
conditions), and as a result inter-layer hopping is neglected. This tunnelling
has been determined by angle-resolved photoemission spectroscopy to be
important for graphene band-structure, leading to $\pi$ band splitting, which
increases with layer number ($\approx 0.7\,$eV at 4 layers). Ohta  _et al._
(2007) Multilayer graphene excitation spectra have been calculated by Ohta et
al. with the inclusion of inter-layer hopping in the kinetic Hamiltonian.
Tunneling is found to be primarily important in accurately producing the low
energy region of the excitation spectrum ($\omega\sim v_{F}k$), where $v_{F}$
is the Fermi velocity). Yuan  _et al._ (2011) Though tunnelling is not as
critical in the regime of interest for this work, calculations for graphene-
based multilayer plasmonic systems geared towards lower-energy applications
should take care to include inter-layer hopping.

More general EELS, accurate for a wider range of $k$ values and angles of
incidence will require first principles calculations for the entire structure,
and incorporate higher level quasiparticle interactions (including the effect
of excitons). Using guidance from the results of this study, future work
consisting of 1-3 layers of graphene on silver, with the hBN substrate are the
subject of future work. The band-gap for hBN is underestimated by about 33% in
the LDA Giovannetti  _et al._ (2007) — the effect of hBN on surface
confinement of plasmons may be even more dramatic than is demonstrated in this
work. A quasiparticle GW correction to the LDA calculation will be employed,
as it has been demonstrated to bring the hBN bandgap into close agreement with
experimental results. Arnaud  _et al._ (2006)

As the individual layers in these structures are nearly isomorphic, rather
than exactly isomorphic, future work will incorporate the effect of lattice
strain on the optical properties of the composite. Strain engineering is
expected to provide a further means of plasmon resonance tunability.
Sciammarella  _et al._ (2010) Additionally the effect of configurations other
than the flat-armchair morphology for graphene is the subject of ongoing work.
The effect of rippling and defects in graphene-based multilayer structures is
of particular interest, as is a rigorous quantification of the uncertainties
involved in the modelling and manufacturing stages, and optimal design against
those uncertainties.

###### Acknowledgements.

The authors would like to thank Katie Rice, Ann Chiaramonti Debay, and Alex
Smolyanitsky for helpful discussions. This research was performed while the
first author held a National Research Council Research Associateship Award at
the National Institute of Standards and Technology. This work represents an
official contribution of the National Institute of Standards and Technology
and is not subject to copyright in the USA.

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