# Plasmon damping rates in Coulomb coupled 2D layers in a heterostructure

Dipendra Dahal1, Godfrey Gumbs2, Andrii Iurov3 and Chin-Sen Ting1 1Texas
Center for Superconductivity and Department of Physics, University of Houston,
Houston, TX 77204, USA
2 Department of Physics and Astronomy, Hunter College of the City University
of New York, 695 Park Avenue, New York, New York 10065, USA
3Department of Physics and Computer Science, Medgar Evers College of City
University of New York, Brooklyn, NY 11225, USA

###### Abstract

The Coulomb excitations of charge density oscillation are calculated for a
double layer heterostructure. Specifically, we consider two-dimensional (2D)
layers of silicene and graphene on a substrate. From the obtained surface
response function, we calculated the plasmon dispersion relations which
demonstrate the way in which the Coulomb coupling renormalizes the plasmon
frequencies. Additionally, we present a novel result for the damping rates of
the plasmons in this Coulomb coupled heterostructure and compare these results
as the separation between layers is varied.

###### pacs:

73.21.-b, 71.70.Ej, 73.20.Mf, 71.45.Gm, 71.10.Ca, 81.05.ue

##  I Introduction

Figure 1: (Color online) Schematic illustration of a heterostructure
consisting of a pair of 2D layers separated by a a dielectric medium
$\epsilon_{1}(\omega)$. This structure lies on a substrate with dielectric
function, $\epsilon_{2}(\omega)$

.

A huge number of researchers from various disciplines have been showing their
interest in the new material silicene especially, after the development of its
fabrication process in 2012. Vogt  Because of its exceptional potential
applications in electronic and optoelectronic devices, many industries are
making substantial investments to harness its properties. Additionally, before
making investments for commercial gain, both theoreticians and
experimentalists have been exploring this material for many years. A credit of
foremost importance goes to Takeda and Shiraishi,Takeda  who, in 1994, dealt
with the atomic and electronic structure of the material for the first time.
These authors calculated the band structure of silicon in the corrugated stage
having optimized atomic geometry. This work, though very novel, did not
receive the attention it deserves until in 2004 when single-layer carbon atoms
named graphene were fabricated in the laboratory from graphite by Novoselov et
al.novoselov  Their research not only validated the stability of two-
dimensional (2D) material but also opened the door for new research on thin
film materials, silicene being one of them.

Both silicene and graphene were studied in parallel. The former has a buckled
crystal geometry whereas the latter has the honeycomb planar geometry. Due to
this, differences arise between them. Ab initio calculations showed that the
bandgap of silicene is electrically tunableNi ; Drummond ; Liu  which is an
advantageous property for designing a field effect transistor which works at
room temperature. Another distinct difference between these two materials is
the strength of the spin-orbital coupling (SOC), which is very weak in
graphene. Consequently, the quantum spin Hall effect occurs at extremely low
temperaturekane ; Yao . In contrast to this, silicene displays quantum spin
Hall effect at temperature $18$K, far higher than that for graphene.

Several investigations have been carried out on both graphene and silicene
with respect to transport phenomenaBao ; Dora ; Tian ; Rosenstein ; Dahal1 ;
Wakamura11 ; Kim11 ; Yan11 ; Bao11 , as well as their magnetic and electric
field effects Gumbs1 ; Gumbs11 ; Checkelsky ; Nakamura ; Yan ; Yu ; Novoselov
; Liu2 , the fabrication processEletskii ; Jia ; Cai ; Aliofkhazraei , and on
plasmonic behavior Garcia ; Farmer ; Gumbs3 ; Andrii1 ; Andrii2 ; Andrii3 ;
Gumbs4 ; Dahal ; Andrii11 . An intensive literature search on the plasmonic
studies suggests that no study on the plasmon dispersion and its rate of
damping was carried out for composite silicene and graphene materials. This
hybrid material could have significant benefits for use in the advancement of:
quantum information technology, Calafell ; Alonso ; Hanson ; Christensen
sensing devices, Esfandiari ; Tong ; Hu  and protein analytic clinical
devices, Hu ; Andoy ; Viswanathan ; Huang  etc. With this motivation, in this
research work, we choose a system composed of silicene and graphene
accompanied by a conducting substrates.

The plasmon mode is tunable by the thickness of the substrate and the
variation of material behavior. We first determine the surface response
function of the structure, the same technique used recently by Gumbs et.
al.Gumbs2 ; Hwang  which gives us the condition for the existence of the
plasmon dispersion. The analytical result for the surface response function is
further used for different limiting cases and a comprehensive comparison is
made with a variety of structures composed of different graphene-silicene
compositions. Furthermore, the same function is used to obtain the Landau
damping rate of the plasmon modes whose numerical calculation demonstrates
that the it’s variation depends on the layer separation, types of dielectric
used and the type of 2D layer employed.

We have organized the rest of our paper as follows. In Sec. II, we present the
core idea about our work where we show the analytical result for the surface
response function for the chosen structure. Under limiting conditions, the
result is used to derive the results for a variety of conditions. The
graphical results and their interpretation are presented in Sec. III. We
conclude our paper with a summary of our main results and conclusions in Sec.
IV.

##  II theory

In this article, we have reported the plasmonic behavior of heterostructure
consisting of graphene and silicene together for which the Hamiltonian in the
low energy regime near the K point is considered. One significant difference
between the Hamiltonian of graphene and silicene is: a small band gap,
$\Delta$ is present in the Silicene energy band structure which is due to
spin-orbit coupling and applied external electric field. This band gap is not
seen in intrinsic graphene.

###  II.1 Silicene

We now briefly describe the case pertaining to silicene whose Hamiltonian in
the continuum limit is given by

$H_{\xi}=\hbar v_{F}(\xi
k_{x}\hat{\tau}_{x}+k_{y}\hat{\tau}_{y})-\xi\Delta_{so}\sigma_{z}\tau_{z}+\Delta_{z}\hat{\tau}_{z}\
,$ (1)

where $\hat{\tau}_{x,y,z}$ and $\sigma_{x,y,z}$ are Pauli matrices
corresponding to two spin and coordinate sub-spaces, $\xi=\pm 1$ for the $K$
and $K^{\prime}$ valleys, $v_{F}(\approx 5\times 10^{5})$ m/sec is the Fermi
velocity for silicene Ezawa ; Drummond , $k_{x}$ and $k_{y}$ are the wave
vector components measured relative to the K points. The first term represents
the low-energy Hamiltonian whereas the second term denotes the Kane-Mele
system Kane  for intrinsic spin-orbit coupling with an associated spin-orbit
band gap of $2\Delta_{so}$. The last term in the expression describes the
sublattice potential difference that arises from the application of a
perpendicular electric field. Equation (1) for the Hamiltonian is a block
diagonal in $2\times 2$ matrices labeled by valley $(\xi)$ and spin
$\sigma=\pm 1$ for up and down spin, respectively. These matrices are given by

$\hat{H}_{\sigma\xi}=\bigg{(}\begin{array}[]{cc}-\sigma\xi\Delta_{so}+\Delta_{z}&\hbar
v_{F}(\xi k_{x}-ik_{y})\\\ \hbar v_{F}(\xi
k_{x}+ik_{y})&\sigma\xi\Delta_{so}-\Delta_{z}\end{array}\bigg{)}\ .$ (2)

This gives the low-energy eigenvalues as

$E_{k}=\pm\sqrt{\hbar^{2}v_{f}^{2}|k|^{2}+\Delta_{\xi\sigma}^{2}}$ (3)

where $\Delta_{\sigma\xi}=|\sigma\xi\Delta_{so}-\Delta_{z}|$.

###  II.2 Graphene

The low-energy model Hamiltonian for monolayer graphene is similar to that in
Eq. (2) with the diagonal terms replaced by zero and $\xi$ labeling the
valley. In this regime, the Hamiltonian for intrinsic graphene is given by

$\hat{H}=\hbar v_{F}\bigg{(}\begin{array}[]{cc}0&(\xi k_{x}-ik_{y})\\\ (\xi
k_{x}+ik_{y})&0\end{array}\bigg{)}$ (4)

with the linear energy dispersion, $E_{k}=\pm\hbar v_{f}|k|$ in either valley.

###  II.3 Polarization function: $\Pi(q,\omega)$

Considerable work has been done on the dynamical properties involving the use
of the dielectric function $\epsilon(q,\omega)$ of various types of free-
standing 2D systemsWunsch ; Balassis ; Roldan  under different conditions.
These include temperature effects,Patel ; Ramezanali  the role of an ambient
magnetic field for the 2D electron gas (2DEG), graphene, silicene,Tabert  and
the dice latticeBalassis . For a single 2D layer, one can extract the plasmon
dispersion relation and damping rate by employing the dielectric function.
However, the situation is more complicated for a multi-layer heterostructure
which relies on a knowledge of the surface response function that we have
presented in detail below. However, in either case, we need to calculate the
polarization function obtained in the random phase approximation (RPA). For a
2D layer surrounded by a medium with dielectric constant $\epsilon_{b}$, the
dynamic dielectric function is given by

$\epsilon(q,\omega)=1-V(q)\Pi(q,\omega),$ (5)

where $V(q)=\frac{2\pi e^{2}}{4\pi\epsilon_{0}\epsilon_{b}q}$ is the Coulomb
interaction potential and $\epsilon_{0}$ is the permittivity of free space,
$q$ is the wave vector and $e$ is the electron charge. The polarization
function is an important quantity in calculations of the transport, collective
charge motion and, charge screening properties of the material. In the one-
loop approximation, the polarization function for gapped graphene is given by
Pyatkovskiy

$\displaystyle\Pi^{0}(q,\omega)$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}{\bf k}}{2\pi^{2}}\sum\limits_{s,s^{\prime}=\pm
1}\left\\{\frac{\hbar^{2}v_{f}^{2}(\bf k+\bf q)\cdot\bf
k+\Delta^{2}_{\sigma,\xi}}{E_{k}\cdot
E_{|k+q|}}\right\\}\frac{f_{0}(sE_{k}-E_{F},T)-f_{0}(s^{\prime}E_{|k+q|}-E_{F},T)}{sE_{k}-s^{\prime}E_{|k+q|}-\hbar(\omega+i\delta)}\
,$

where $\theta_{k,k+q}$ is the angle between ${\bf k}$ and ${\bf k}$+${\bf q}$.
At zero temperature, the Fermi function $f_{0}(z)$ is just a step function.
The analytical expression for the polarization function for silicene and
graphene monolayer is given by Tabert et. al. Tabert  and Wunsch et. al.Wunsch
respectively.

We now turn our attention to a crucial consideration in this paper regarding
the structure consisting of a silicene layer, a graphene layer and substrates
as depicted in Fig. 1. By employing the boundary condition of continuity of
the potential and the discontinuity of the electric field at the interface, we
solved for the various coefficients appearing in the potential. The result for
the surface response function $g(q,\omega)$ gives the required conditions for
the plasmon dispersion for our case, namely

$\displaystyle\phi_{<}(z)=e^{-qz}-g(q,\omega)e^{-qz}\ ,z\lesssim 0\,,$ (7)
$\displaystyle\phi_{>}(z)=a_{1}e^{-qz}+b_{1}e^{qz}\ ,0\leq z\leq d\,,$
$\displaystyle\phi_{1>}(z)=k_{1}e^{-qz}\ ,z\geq d\,.$

Here, $\phi_{<}(z)$, $\phi_{>}(z)$ and $\phi_{1>}(z)$ correspond to the
electrostatic potential of region (I),(II), and (III) respectively as shown in
figure 1. In order to conduct numerical computation, we make use of linear
response theory, for which we have $\sigma_{1}=\chi_{1}\phi_{<}(0)$,
$\sigma_{2}=\chi_{2}\phi_{1>}(2)$, with $\chi_{1}$, $\chi_{2}$ are 2D
susceptibilities. Generalising, $\chi_{i}=e^{2}\Pi^{0}_{i}$ for convenience,
we obtain the solution of these equations for different coefficients, leading
to

$g(q,\omega)=\frac{1}{D(q,\omega)}\left\\{[q\epsilon_{0}(\epsilon_{1}-1)-\chi_{1}][q\epsilon_{0}(\epsilon_{1}+\epsilon_{2})-\chi_{2}]-[q\epsilon_{0}(\epsilon_{1}+1)+\chi_{1}][q\epsilon_{0}(\epsilon_{1}-\epsilon_{2})+\chi_{2}]e^{-2dq}\right\\}$
(8)

$D(q,\omega)\equiv[q\epsilon_{0}(\epsilon_{1}-1)-\chi_{1}][q\epsilon_{0}(\epsilon_{1}+\epsilon_{2})-\chi_{2}]-[q\epsilon_{0}(\epsilon_{1}-1)+\chi_{1}][q\epsilon_{0}(\epsilon_{1}-\epsilon_{2})+\chi_{2}]e^{-2dq}\
,$ (9)

where $\epsilon_{1}(\omega)$ is the dielectric function of the substrate
between layers “1” and “2”, $\chi_{1}$ and $\chi_{2}$ correspond to the
susceptibilities of these two layers and $d$ is the thickness of the
substrate. The plasmon dispersion equation is obtained from the solutions of
$D(q,\omega)=0$ which we solve below. We note that when we set $\chi_{2}=0$
and take the limit $d\to\infty$, Eq. (10) yields the well established form
BoPersson

$g_{2D}(q,\omega)=1-\frac{1}{\frac{\epsilon_{1}+1}{2}-\frac{\chi_{1}}{2q\epsilon_{0}}}$
(10)

which is the surface response function for a 2D layer embedded in a medium
whose average background dielectric constant is
$\epsilon_{b}=(\epsilon_{1}+1)/2$. The plasma resonances, which Eq. (10) gives
from its poles, are in agreement with the zeros of the dielectric function in
Eq. (5).

###  II.4 Damping rate

We now turn to a critical issue in this paper which concerns the rate of
damping of the plasmon modes by the single-particle excitations. If this rate
of damping for a plasmon mode with frequency $\Omega_{p}$ is denoted as
$\gamma$, then $D(\Omega_{p}+i\gamma,q)=0$ in the complex frequency space.
Carrying out a Taylor series expansion of both the real and imaginary parts,
we have

$\displaystyle D(\Omega_{p}+i\gamma,q)$ $\displaystyle=$
$\displaystyle\text{Re}\ D(\Omega_{p}+i\gamma,q)+i\text{Im}\
D(\Omega_{p}+i\gamma,q)$ (11) $\displaystyle=$
$\displaystyle\texttt{Re}D(\Omega_{p})+i\gamma\left.\frac{\partial}{\partial\Omega}\texttt{Re}D(\Omega)\right|_{\omega=\Omega_{p}}+i\texttt{Im}D(\Omega_{p})-\gamma\left.\frac{\partial}{\partial\Omega}\texttt{Im}D(\Omega)\right|_{\omega=\Omega_{p}}+\cdots$

Therefore, setting the function in Eq. (11) equal to zero, we obtain $\gamma$
to lowest order as

$\gamma=-\frac{\texttt{Im}D(\Omega_{p})}{\partial\texttt{Re}D(\omega)/\partial\omega|_{\Omega_{p}}}\
.$ (12)

With these formal results, we now evaluate the plasma spectra for double layer
heterostructure. The expression shows the dependence of $\gamma$ on the
imaginary part of $D(\Omega_{p})$ and the Real part of $D(\omega)$ which in
turn are dependent on the type of layer and the substrate considered.
Eventually, we can infer that the viability of plasmon modes can be tuned by
the dielectric substrate thickness and by the choice of 2D layer. In addition,
the rate of decay also helps us in maintaining the intensity and the frequency
of the obtained plasmon mode. This could have great impact in the development
of the quantum information sharing technology and the data storing devices.

##  III Numerical results and discussion

Figure 2: (Color online) Plasmon frequency $\omega_{p}(q)$ and damping rates
$\gamma(\omega_{p}(q),q)$ for an isolated graphene layer(SLG) with
$E_{F}=1.0E_{F}^{(0)}$(left panels (a)and (c)) and
$E_{F}=1.5E_{F}^{(0)}$(right panels (b) and(d)). Two top panels(a) and
(b)demonstrate the plasmon dispersion (either damped or undamped obtained as
$Re(\epsilon(q,\omega))=0$, the lower plots (c) and (d)describe the
corresponding damping rate along the plasmon branches, also calculated and
shown as insets(i1) and (i2).

In our numerical calculations, energy is scaled in units of $E_{F}^{(0)}$ and
wave vector is scaled with $k_{F}^{(0)}=\sqrt{\pi n}$ which is in the
experimental range for electron hole doping densities $n=10^{10}$ per cm2.
This gives $k_{F}^{(0)}=10^{6}$ per cm and $E_{F}^{(0)}$ is equivalent to
$\sim$60meV. From the preceding discussion, in Sec. II, it is clear that the
plasmon mode for any system is given by the pole of the dielectric function
obtained from Eq. 9. Thus, making use of it, we computed the plasmon mode
dispersion for a heterostructure based on graphene and silicene with/without a
substrate. For this, we have first obtained a graphical result for graphene as
shown in Fig. 2. One can clearly see that a single branch plasmon mode
originate from the origin in $q-\omega$ space which increases monotonically
and decays out when the plasmon mode reaches the interband particle-hole
excitation region. In Fig. 2, the plasmon branches for two values of the Fermi
energy are shown in panels (a) and (b). The damping rates of these plasmon
modes are demonstrated in panel (c) and (d), correspondingly, where it is
distinctly shown by an arrow pointing at the boundary of the region where
Landau damping takes place. The rate of decay for both types of graphene are
monotonically increasing signifying that the deeper into the single particle
excitation region where the plasmon mode enters, the rate of plasmon decay
becomes larger. That is the lifetime of the plasmon mode is decreased in the
same manner.

Going next to the case when we have a structure with two graphene layers
together, separated by various distances, a set of plots as shown in Fig. 3
are obtained with two branches of plasmon modes originating from the origin in
the $q-\omega$ space. One can clearly see that the closer the graphene sheets
are, the further apart are the plasmon modes. In Figs. 3(a), (b), (c) and (d),
plasmon modes for a structure with two graphene layers separated by a distance
of $0.5(k_{F}^{(0)})^{-1}$, $1.0(k_{F}^{(0)})^{-1}$, $2.0(k_{F}^{(0)})^{-1}$
and $5.0(k_{F}^{(0)})^{-1}$ are shown, respectively, and all the plots
portrays that the further apart the graphene layers are, the closer the two
plasmon branches become. For this same set of figures with the other
parameters remaining the same, the plasmon decay rate is shown in Fig. 4 which
shows that the plasmon decay for the lower plasmon branch always starts at
larger wave vector value in comparison to upper plasmon branch. As the
distance of separation is increased, the two plasmon branches come closer and
their decay also starts from the same value of wave vector and the rate of
decay is almost same in value.

Figure 3: (Color online) Acoustic (lower) and optical (upper) plasmon modes
for a pair of identical graphene layers with $E_{F}=1.0E_{F}^{(0)}$. Each
panel corresponds to a different values of the separation between the layers
corresponding to $d=0.5(k_{F}^{(0)})^{-1}$, $1.0(k_{F}^{(0)})^{-1}$,
$2.0(k_{F}^{(0)})^{-1}$ and $5.0(k_{F}^{(0)})^{-1}$ as labeled. The plasmon
dispersion (either damped or undamped is obtained by solving
$Re(D(q,\omega|d))=0)$. Figure 4: (Color online) The damping rates
corresponding to the acoustic and optical plasmon branches shown in Fig. 4 for
two identical graphene layers with $E_{F}=1.0E_{F}^{(0)}$. Each panel
corresponds to a different values of the separation between the layers with
$d=0.5(k_{F}^{(0)})^{-1}$, $1.0(k_{F}^{(0)})^{-1}$, $2.0(k_{F}^{(0)})^{-1}$
and $5.0(k_{F}^{(0)})^{-1}$ as labeled.

Now, in addition, we carried out an investigation of the plasmon modes and
their decay rate for the structure with two silicene layer for various
separations. The graphical results for these calculations are shown in Fig. 5
where we again have two plasmon modes originating from the origin of
$q-\omega$ plane. As in the case for a two-graphene-layer structure, we again
notice a similar effect on two plasmon branches coming closer to each other
when their separation increases. This clearly is demonstrated in Figs. 5(a),
(b), (c) and (d) for their distance apart of $0.5(k_{F}^{(0)})^{-1}$,
$1.0(k_{F}^{(0)})^{-1}$, $2.0(k_{F}^{(0)})^{-1}$, and $5.0(k_{F}^{(0)})^{-1}$
respectively.

As a representative calculation, we investigated the decay rate of plasmon
modes for silicene-silicene structure when their separation is
$d=0.5(k_{F}^{(0)})^{-1}$ and $d=5.0(k_{F}^{(0)})^{-1}$. Figure 6 shows that
the upper plasmon mode does not decay at all and the lower plasmon branch
decays after reaching a critical wave vector. This behavior is due to the
presence of a band gap for silicene, resulting in an opening in the single
particle excitation region, which provides a larger area in $q-\omega$ space
for the plasmon mode to survive. The upper plasmon branch in this case has a
larger space and is more likely to self-sustain for a longer period of time
without damping. On the other hand, the lower plasmon branch enters the
intraband single-particle excitation region where it decays. The rate of decay
starts from a critical value of the wave vector and the magnitude of this
decay rate monotonically increases.

Figure 5: (Color online) Acoustic (lower) and optical (upper) plasmon
dispersions for two silicene layers with $E_{F}=1.0E_{F}^{(0)}$ and the band
gaps $\Delta_{SO,1}=\Delta_{SO,2}=0.7E_{F}^{(0)}$,
$\Delta_{z,1}=0.2E_{F}^{(0)}$ and $\Delta_{z,2}=0.4E_{F}^{(0)}$. Each panel
corresponds to a different values of the separation between the layers
$d=0.5(k_{F}^{(0)})^{-1}$, $1.0(k_{F}^{(0)})^{-1}$, $2.0(k_{F}^{(0)})^{-1}$
and $5.0(k_{F}^{(0)})^{-1}$ as labeled.  Figure 6: (Color online) The damping
rates corresponding to the acoustic and optical plasmon branches calculated in
Fig. 6 for two silicene layers with $E_{F}=1.0E_{F}^{(0)}$, and the band gaps
$\Delta_{SO,1}=\Delta_{SO,2}=0.7E_{F}^{(0)}$, $\Delta_{z,1}=0.2E_{F}^{(0)}$
and $\Delta_{z,2}=0.4E_{F}^{(0)}$. Each panel corresponds to a different
values of the separation between layers with $d=0.5(k_{F}^{(0)})^{-1}$, and
$5.0(k_{F}^{(0)})^{-1}$ as labeled.

A comparison of plasmon modes and their decay for graphene-graphene and
silicene-silicene structures is shown along with the single-particle
excitation regions in Fig. 7. The figure in panel (a) of Fig.7 shows that two
plasmon modes which originate from the origin of the frequency-momentum space
is steadily increased but decays after it reaches the boundary of the single-
particle excitation region. Corresponding red lines are drawn to further
clarify the point where the actual decay begins. The dark triangular region is
the area where the plasmon mode survives without Landau damping and
mathematically, in this region, the imaginary part of the polarization
function of graphene is zero. This means that the plasmon mode has self-
sustaining oscillations. The green region where the imaginary part of the
polarization function is nonzero is the single particle excitation region
where the plasmon mode decays into particle hole mode. The corresponding decay
rate figure below this panel shows that the rate of decay for the upper
plasmon branch is greater and its critical wave vector is smaller compared
with the lower plasmon branch.

Figure 7: (Color online) Plasmon modes for (a) graphene-graphene structure and
(b) silicene-silicene structure with corresponding plasmon damping rate in Fig
(c) and (d), respectively.

Similar plots for silicene-silicene structures were demonstrate figure 7 (b)
where one can clearly see the opening of a gap in the single-particle
excitation region yielding two parts which is a significant effect arising
from band gap. The imaginary part of the polarization function in this gap
region is zero where the plasmon mode can sustain its oscillation for long
time. The upper break away region is single-particle excitation region due to
interband transitions of electrons from the valence to the conduction band and
the lower break away region is the intraband single particle excitations
region which is due to transistions within the same band from below to above
the Fermi level. In Fig. . 7, two plasmon modes originate from the origin as
demonstrated in the figure. The upper plasmon mode survives without damping
over a wider range of wave vector and the plasmon branch enters the gap
created by the opening within the single-particle excitation region. The
corresponding decay rate appearing below the plasmon dispersion shows that the
upper plasmon branch does not decay at all. Whereas for the case of the lower
plasmon branch, the plasmon mode near the origin lies closer to the intraband
single-particle region. Consequently, there is small plasmon decay rate as
illustrated in the corresponding figure in the panel fig 7(d) below. As the
plasmon mode rises, it gets separated from the single-particle excitation
region where the decay rate is zero and as it moves further away from the
origin the plasmon branch comes in contact with the single particle excitation
region where we notice the Landau damping again. Correspondingly, the decay
rate is increased monotonically, reaches a maximum before drops down,
indicating the reappearance of an undamped plasmon branch at a larger value of
wave vector. Another noticeable effect observed here is the closeness of the
plasmon branches and the plasmon decay rate which can be altered by changing
the layer separation, this effect may be used as another plasmon mode tuning
parameter.

In order to extract more information about the plasmonic behavior, in Fig. 8
(a), we have presented the figure to show the result highlighting the changes
in the plasmonic nature for a structure with different types of layer and a
substrate. In Fig. 8(a), we demonstrate the plasmon mode for a structure with
silicene and graphene separated by a distance of $1.0(k_{F}^{(0)})^{-1}$ with
vacuum in between. One can clearly observe two plasmon modes originate from
the origin in $q-\omega$ space. A special effect of overcoming the single
particle excitation region of silicene by the single-particle excitation
region of graphene is observed which causes the shortening of the lower
plasmon branch which used to be there for silicene-silicene structure. AS soon
as the plasmon branch reaches this region, the plasmon mode decays into
particle hole mode just because of the replacine one silicene layer by a
graphene layer in silicene-silicene structure. The effect due to the band gap
in silicene is just nullified. Furthermore, the result of adding a substrate
between the silicene and graphene layer is illustrated in panel (b) of Fig. 8.
In this case, we could see a new plasmon branch originating from the bulk
plasma frequency and one plasmon branch originating from the origin. Here, due
to the presence of a substrate, the lower plasmon branch bend sharply towards
the intraband single-particle excitation region where it decays causing
complete disappearance. The upper plasmon branch and the plasmon from the bulk
plasmon frequency become closer and move towards the interband single-particle
excitation region where they get damped.

These new effects on the plasmon branches in this type of structure was not
reported previously. Results of this type are helpful in developing electronic
and quantum computing devices where knowledge of plasmonic behavior of
material is very essential.

##  IV Concluding Remarks

In summary, we have shown that the effect due to the addition of a substrate
and the difference from introducing different types of 2D materials in the
heterostructure resulting a novel tunning technique of the plasmon excitations
associated with these 2D systems. A completely new effect of a plasmon branch
emerging from the bulk plasmon frequency is seen which would be very helpful
in engineering computing devices. Another discovery is the disappearance of
the lower plasmon branch and the suppression of the silicene band gap effect.
These are other interesting new effects seen from our calculations. We have
also developed an approach for calculating the decay rates for the plasmons
due to Landau damping by the particle-hole modes.

Additionally, our results infer that the number of plasmon branches emerging
from the origin can be varied by choosing the number of 2D layer. In brief, we
can say that our study gives an important idea about the plasmonic behavior of
a graphene-silicene based heterostructure which would be very helpful in
carrying out further study of other type of heterostructure including various
low dimensional material.

Figure 8: (Color online) Plasmon mode dispersion for two layer of of 2D
materials separated by a distance of $2k_{F}^{-1}$. In (a) Both layers are
silicene, $\epsilon_{1}(\omega)=1-\Omega_{p}^{2}/\omega^{2}$ and
$\epsilon_{2}=1$ for the substrate. (b) The same as (a) except that the
silicene layers are replaced by graphene.

## Acknowledgement(s)

G.G. would like to acknowledge the support from the Air Force Research
Laboratory (AFRL) through Grant No. FA9453-21-1-0046

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