# Dipolariton propagation in a van der Waals TMDC with $\Psi$-shaped channel
guides and buffered channel branches

Patrick Serafin  Physics Department, New York City College of Technology, The
City University of New York, Brooklyn, New York 11201, USA  German V. Kolmakov
Physics Department, New York City College of Technology, The City University
of New York, Brooklyn, New York 11201, USA

###### Abstract

Using a computational approach based on the driven diffusion equation for
dipolariton wave packets, we simulate the diffusive dynamics of dipolaritons
in an optical microcavity embedded with a transition-metal dichalcogenide
(TMDC) heterogeneous bilayer encompassing a $\Psi$-shaped channel. By
considering exciton-dipolaritons, which are a three way superposition of
direct excitons, indirect excitons and cavity photons, we are able to drive
the dipolaritons in our system by the use of an electric voltage and
investigate their diffusive properties. More precisely, we study the
propagation of dipolaritons present in a MoSe2-WS2 heterostructure, where the
dipolariton propagation is guided by a $\Psi$-shaped channel. We also consider
the propagation of dipolaritons in the presence of a buffer in the
$\Psi$-shaped channel and study resulting changes in efficiency. By our
consideration of a geometrically novel dipolariton channel guide, we introduce
novel designs for optical routers at room temperature as well as replicate the
dipolariton redistribution efficiencies of previously proposed polaritronic
applications.

##  I Introduction

Recent advances in the field of polaritronics have enabled us to realize many
proposed application of exciton and polariton physics at room temperature
scales tsintzos:5  Exciton-polariton physics has already found its application
in devices such as lasers Christopoulos:10 , optical transistors Serafin17 ;
Gao:85  and light emitting diodes jayaprakash:7 . The wide array of phenomena
found in exciton-polariton physics, such as superfluidity MacDonald:08 ;
Berman:08 ; Berman:10  and Bose-Einstein condensate byrnes:93 ; Richard7  has
drawn attention towards research into their implementation in various systems
as well as potential applications Sanvitto:4 .

Some obstacles that can arise with the use of conventional polaritons, such as
low exciton binding energies and electrical neutrality can be remedied by the
use of charged quasi-particles; that is, dipolaritons in van der Waals
transition metal dichalcogenide (TMDC) heterostructures. Dipolaritons present
us with the possibility of being driven by an external voltage, something that
is not possible in a conventional polariton as it is electrically neutral. The
emergence of Van der Waal TMDC heterostructures enables us to use properties
such as large binding energies and large Rabi splitting energies Vinod:2  to
transfer proposed applications of polaritons to room temperature scales. In
addition, these TMDC heterostructures provide for a type-II band alignment
Ceballos:7  which enables one to spatially separate electrons and their
respective holes upon laser pumping into the system.

In this paper, the diffusive dynamics of dipolaritons present in a TMDC inside
of an optical microcavity is computationally modeled. In particular, we
consider heterostructures with a $\Psi$-shaped channel guide in contrast to
heterostructures with Y-shaped channel guides previously investigated for
identical system parameters Serafin17 . This MoSe2-WS2 heteregenous bilayer
embedded inside of a optical microcavity enables one to redistribute
dipolaritons through the $\Psi$-shaped channel guide using an external voltage
as shown in Fig.1. The system parameters for this redistribution are
investigated in order to determine the conditions that optimize the efficiency
of redistribution of the dipolariton gas in our sytem. By considering a novel
$\Psi$-shaped channel guide, we are able to compare efficiency between
geometrically different channel guides. In particular, by considering various
driving forces, electric field angles, and channel buffers we can establish
optimal system parameters that promote efficient dipolariton routing in the
$\Psi$-shaped channel of the TMDC.

Figure 1: Schematic of a TMD heterogenous bilayer embedded inside an optical
microcavity with a $\Psi$-shaped channel guiding the dipolaritons. The opening
angle of the channel relative to the middle branch is $\theta_{0}=30$ and the
direction of the field $\bm{E}$ generated by an external voltage applied to
the bilayer is defined by an angle $\theta_{E}$ between the field vector and
the direction of the stem of the $\Psi$-shaped channel. The direction of the
electric driving force applied to the dipolaritons is opposite to the
direction of the electric field. Distributed Bragg Reflectors (DBR) are placed
between the TMD bilayer and laser pumping is applied to generate dipolaritons,
at the beginning of the stem of the channel at which point they propagate
along the $x$-axis towards the junction under the action of the electric field
$\bm{E}$.

##  II Simulation Method

We model our system of dipolariton propagation in a TMDC with a $\Psi$-shaped
channel using previously considered models for dipolariton based optical
transistors using TMDC’s Serafin17 . In particular, we consider the
propagation of dipolaritons controlled by an external electric field $E$
generated by an external voltage acting on the charges found in the MoSe2
layer of the TMDC heterostructure as seen in Fig.1. In our approach, we
consider the diffusive dynamics of the dipolariton gas at scales much larger
than the dipolariton mean path and describe the dipolariton gas using the
quasi-classical stochastic driven dissipative diffusion equation for a
polariton wave packet.Kolmakov:33 :

$d\textbf{r}=\eta_{\text{dip}}\textbf{F}(\textbf{r}(t),t))dt+\sqrt{2D}d\textbf{$\xi$}(t)$
(1)

The first term in Eq.(1) F(r,t) represents the external force acting on the
dipolaritons, the second term $\eta_{\rm dip}$ is the dipolariton mobility,
whilst $D$ is the diffusivity, and $d\textbf{$\xi$}(t)$ the differential of a
Weiner process Kolmakov:33 . The force acting on polaritons is set to
$\textbf{F}=-\nabla{U_{\rm eff}{(r)}}$. Polariton addition to the system is
modeled using $\delta t$ which represents a Gaussian probability distribution.
Eq.(1) has been shown to provide many advantages, among them being
computational effectiveness for a system of large particles and off lattice
properties Deng199 . The stochastic nature of Eq.(1) has been applied in other
models such as ones studying the diffusive properties of gases Carusotto2013
and nanoparticles in fluids Ladd10 . This computational model follows the same
physics and identical system parameters as described when considering this
system with a Y-shaped channel guide Serafin17 . The model parameters are
summarized in table 1 and their properties and derivations can be found in
investigations of similar systemsSerafin17 ; Kolmakov:33 .

Table 1: Simulation parameters for a cavity with embedded MoSe2-WS2 bilayer Quantity name | Value | Variable
---|---|---
Exciton mass | $m_{\rm ex}$ | $0.70m_{\rm 0}$
Photon mass | $m_{\rm ph}$ | $1.1234\times 10^{-5}m_{\rm 0}$
Dipolariton mass | $m$ | $2.4\times 10^{-5}m_{\rm 0}$
Dipolariton lifetime | $\tau_{\rm dip}$ |  $15.64\times 10^{-12}$ s
Indirect exciton lifetime | $\tau_{\rm IX}$ |  $80\times 10^{-12}$ s
Direct exciton lifetime | $\tau_{\rm DX}$ |  $4.0\times 10^{-12}$ s
Cavity Photon lifetime | $\tau_{\rm ph}$ |  $100\times 10^{-12}$ s
Exciton diffusion | $D_{\rm ex}$ |  $14$ cm2/s
coefficient |  |
Dimensionless dipolariton | $D\times dt/dx^{2}$ | $271.7$
diffusion coefficient |  |
Dimensionless dipolariton | $\eta_{\rm dip}\times eVdt/dx$ | $0.0015$
mobility |  |
Dielectric constant | $\epsilon$ | 4
Exciton Energy | $E_{\rm ex}$ | 1.58eV
Confining potential | $U$ | $25-500meV$
Numerical unit of | $dx$ | 0.15 $\mu$m
length |  |
Numerical time step | $dt$ | 9.63 fs

To characterize the propagation of the dipolaritons in the channel, we
numerically counted the total number of particles $n$, present in each branch
of the $\Psi$-shaped channel at locations $x_{\rm n}\geq 450\mu$m, where
$x_{\rm n}$ is chosen in such a manner as to only count particles that have
propagated at a sufficient distance through the junction of the channel.

Figure 2: Dipolariton propgation through the three branches of the $\Psi\
$channel for the case of (a) $F$=0.5eV/mm and $\theta_{E}$ = 0 (b)
$F$=2.0eV/mm and $\theta_{E}$ = 60

To characterize the redistribution of the dipolaritons in each branch of the
channel, we calculated the fraction of dipolaritons propagating through the
upper branch of the channel, or what we define as the efficiency in the
channel,

$\varepsilon={{n_{\rm up}\over{{n_{\rm up}+{n_{\rm mid}+n_{\rm low}}}}}}\times
100\%.$ (2)

where ${n_{\rm up}}$ is the number of dipolaritons in the upper branch of the
channel, ${n_{\rm mid}}$ is the number of dipolaritons in the middle branch of
the channel, ${n_{\rm low}}$ is the number of dipolaritons in the lower branch
of the channel. Fig.2 provides for an illustration of the density of
dipolariton particles in the $\Psi$-shaped channel for the specified cases.

We define $\epsilon$ in such a manner in order to find the percentage of
dipolaritons distributed through the upper branch of the channel relative to
the total number of dipolaritons in the $\Psi$-shaped channel. This enables us
to quantify to what extent we can re-route the total number dipolaritons in
the channel through the upper branch of the $\Psi$-shaped channel. Thus, we
can claim that a higher value for $\varepsilon\ $will indicate a greater
ability to drive the dipolaritons through the desired branch of the
$\Psi$-shaped channel.

Furthermore, we also investigate the efficiency of the channel when the middle
branch of the $\Psi$-shaped channel is taken to be a buffer, such that the
density of dipolaritons through the middle branch is not considered in the
calculation in what we define as the buffered efficiency, $\varepsilon_{2}$.
This consideration gives us an efficiency comparison of a $\Psi$-shaped
channel to that of Y-shaped channel for the case when the angle between the
upper and lower branches is sixty, where for certain cases dipolaritons were
able to be re-routed through the upper branch of the channel with greater than
90$\%$ efficiency Serafin17 . Thus, the buffered efficiency $\varepsilon_{2}$
for the channel in this case is calculated as,

$\varepsilon_{2}={{n_{\rm up}\over{{n_{\rm up}+{n_{\rm low}}}}}}\times 100\%.$
(3)

##  III Optimizing the efficiency of the $\Psi\ $ channel to effectively
reroute dipolaritons

In order to determine the optimal conditions for directing dipolaritons in the
$\Psi$-shaped channel, we varied the electric field angle $\theta_{E}$ in our
system and calculated the resulting efficiency in the channel. In order to
test the efficacy of changing the electric field angle $\theta_{E}$ on
$\varepsilon$, we varied the electric field angle from $\theta_{E}$ = $None$
to $\theta_{E}$ = $None$. The result of increasing the electric field angle on
efficiency at constant driving force can be shown in Fig.3(a) where we can
observe the dipolariton particle number and the channel efficiency as
functions of channel electric field angle for the case of a fixed electric
driving force of $F$=2.0eV/mm. We see in Fig.3(a) that an increase in the
electric field leads to a monotonic increase in efficiency. In particular we
can see that as the electric field angle $\theta_{E}$ is increased from
$\theta_{E}$ = $None$ to $\theta_{E}$ = $None$, the efficiency $\varepsilon\
$is improved by $\approx$70$\%$. Furthermore, we can observe that total number
of particles in the upper branch $n_{up}$ increases from
$1.5\text{\times}{10}^{4}$ particles to $7\text{\times}{10}^{4}$ particles as
the electric field angle $\theta_{E}$ is increased from $\theta_{E}$ = $None$
to $\theta_{E}$ = $None$. We can also observe that the number of particles
present in the lower branch of the channel $n_{low}$ decreases from
$3.3\text{\times}{10}^{3}$ particles to $1\text{\times}{10}^{2}$ particles as
$\theta_{E}$ is increased from $\theta_{E}$ = $None$ to $\theta_{E}$ = $None$.
In addition Fig.3(a) shows that the number of particles going through the
middle branch of the channel increase from $5.3\text{\times}{10}^{3}$
particles to $1.2\text{\times}{10}^{4}$ particles as the electric field angle
is increased from $\theta_{E}$ = $None$ to $\theta_{E}$ = $None$. Thus,
increasing the electric field angle in our $\Psi$-shaped channel from
$\theta_{E}$ = $None$ to $\theta_{E}$ = $None$ with the driving force $F$ =
2.0eV/mm provides for an efficiency of $\approx$85$\%$. This improvement of
efficiency as the electric field angle is increased can be attributed to the
fact that as the angle is placed more level to the direction of the upper
channel branch the dipolaritons have a greater statistical probability to be
driven through the upper and middle branch of the channel rather than the
lower branch of the channel. When the electric field angle is set to zero,
there is no driving direction and thus the dipolaritons will propagate
stochastically through the channel’s branches. Thus, we can claim that
increasing the electric field angle in the channel improves efficiency, while
increasing the upper branch population and decreasing the lower branch
population of dipolaritons.

Figure 3:  (a) Channel efficiency $\varepsilon\ $ versus the electric field
angle $\theta_{E}$ for the case of fixed driving force F = 2.0eV/mm and (b)
Buffered channel efficiency $\varepsilon_{2}\ $versus $\theta_{E}$ for the
case of a fixed driving force F = 2.0eV/mm

In order to determine the efficiency of the channel in the presence of a
buffer channel, we numerically calculated the buffered efficiency
$\varepsilon_{2}$ as a function of the electric field angle $\theta_{E}$. In
particular, we considered the middle branch of the $\Psi\ $-shaped channel to
not be considered in our calculation for $\varepsilon_{2}$ as seen in Eq.(3).
In Fig.3(b) we can observe the dipolariton particle number and the channel
efficiency as functions of channel electric field angle for the case of a
fixed electric driving force of $F$=2.0eV/mm. In particular we can observe
that as the electric field angle $\theta_{E}$ is increased from $\theta_{E}$ =
$None$ to $\theta_{E}$ = $None$ the efficiency with buffer $\varepsilon_{2}$
increases monotonically with an increase with the electric field angle
$\theta_{E}$. In Fig.3(b) we can see that $\varepsilon_{2}$ is maximized for
the case of $\theta_{E}$ = $None$ where we can obtain a buffered efficiency of
$\varepsilon_{2}$ $\approx$100$\%$ and total particle number of
$7\text{\times}{10}^{4}$particles. Thus, increasing the electric field angle
from $\theta_{E}$ = $None$ to $\theta_{E}$ = $None$ provides for a
$\approx$18$\%$ improvement of $\varepsilon_{2}$ with
$5.4\text{\times}{10}^{3}$ particles in the upper branch $n_{up}$. Compared to
the case of no buffer, we see that increasing the electric field angle with a
buffer in the $\Psi\ $-shaped channel has a lower impact on the improvement of
efficiency while clearly leaving particle number unchanged. Thus, we can state
that considering a buffer in the middle branch of the $\Psi$-shaped channel
makes the channel similar geometrically that of a Y-shaped channel with a
similar redistribution efficiency Serafin17

To determine the optimal conditions for directing dipolaritons in the
$\Psi$-shaped channel, we varied the driving force $F$ in our system and
studied the resulting distribution of dipolaritons in our system. In Fig.4(a)
we can see the efficiency $\varepsilon$ as a function of the driving force $F$
for the case of a fixed electric field angle $\theta_{E}$ = $None$. We can
observe that the efficiency $\varepsilon$ is a monotonically increasing
function with the driving force $F$ where $\varepsilon$ is maximized at a
value of $\approx$85$\%$ when the driving force is set to $F$ = 2000meV/mm. In
particular we can see that as the driving force $F$ is increased from
500meV/mm to 2000meV/mm, the efficiency $\varepsilon$ is improved by
$\approx$20$\%$. This improvement in efficiency is noted as less substantial
than the improvement we can gain by increasing the electric field angle as
previously discussed. In addition, we can see in Fig.4(a) that as the driving
force $F$ is increased from 500meV/mm to 2000meV/mm, the the number of
particles present in the upper branch of the channel increases from
$3.1\text{\times}{10}^{3}$ particles to $7\text{\times}{10}^{4}$particles,
whilst the number of particles present in the lower branch of the channel
decreases from $2\text{\times}{10}^{2}$ particles to $1\text{\times}{10}^{2}$
particles. This effect can be attributed to the fact that as the electric
driving force on the dipolaritons is increased, there are more particles
directed towards the direction of the field angle. Thus, we can claim that an
increase in the electric field angle in the channel increases the efficiency
$\varepsilon$ in the channel, while increasing the total upper branch particle
population and lowering the lower branch dipolariton population

In order to determine the efficiency of the channel in the presence of a
buffer channel, we numerically calculated the buffered efficiency
$\varepsilon_{2}$ as a function of the driving force $F$. In particular, we
considered the middle branch of the $\Psi$-shaped channel to not be considered
in our calculation for $\varepsilon_{2}$ as seen in Eq.(3). In Fig.4(b) we can
see the efficiency $\varepsilon_{2}$ of the $\Psi\ $-shaped channel with a
buffer as a function of the driving force, $F$ for the case of $\theta_{0}$ =
$None$. We can observe in Fig.4(b) that the efficiency $\varepsilon_{2}$ is
maximized when the driving force $F$ is to set to F = 2000meV/mm where we find
a buffered efficiency of $\approx$99$\%$. Thus we can claim an improvement in
buffered efficiency of $\approx$5$\%$ as the driving force $F$ is increased
from 500meV/mm to 2000meV/mm. This improvement in efficiency is noticeably
less significant when compared to the improvement we obtain with an increase
of driving force for the case with no buffer as the starting efficiency for
the case of the buffer is already quite high at $\varepsilon_{2}$
$\approx$95$\%$. Thus, we can report that an increase of the channel electric
driving force improves the buffered efficiency; however, this effect is not a
substantial as the effect of increasing driving force in the unbuffered
channel.

Figure 4: (a) Channel efficiency $\varepsilon\ $versus the driving force F for
the case of a fixed electric field angle $\theta_{E}$ = $None$ and (b)
Buffered channel efficiency $\varepsilon_{2}\ $ versus the driving force F for
the case of a fixed electric field angle $\theta_{E}$ = $None$

##  IV Conclusions

The optimum efficiency $\varepsilon$ for an unbuffered $\Psi$-shaped TMDC
channel has been shown to be maximized at particular values of $\theta_{E}$
and $F$. In particular, for values of $F$ = 2.0eV/mm and $\theta_{E}$ =
$None$, we can re-route $\approx$85$\%$ of dipolaritons through the upper
branch of the unbuffered channel. When considering a buffered channel, we are
able to re-route $\approx$100$\%$ of dipolaritons as demonstrated in other
geometrically similar channels when the channel parameters are set to $F$ =
2.0eV/mm and $\theta_{E}$ = $None$ Serafin17 . For all cases considered,
$\varepsilon$ and $\varepsilon_{2}$ were shown to be monotonically increasing
functions with an increase of $F$ as well as an increase of $\theta_{E}$. The
impact of increasing driving force and electric field angle was less
substantial on $\varepsilon_{2}$ when compared to $\varepsilon$. These results
closely resemble the results when considering Y-shaped channel guides
Serafin17 , with efficiencies over $95\%$ at optimal parameter ranges. Our
results demonstrate the possibility of replicating other proposed designs of
optical transistors in TMDC materials, whilst also opening the route toward
the design of novel optoelectronic applications using $\Psi$-shaped channel
guides and buffered channels.

##  V Acknowledgments

This work was supported in part by the Department of Defense under the grant
No. W911NF1810433. The authors are grateful to The Center for Theoretical
Physics of New York City College of Technology of The City University of New
York for providing computational resources. The authors are also grateful to
R. Ya. Kezerashvili, O. L. Berman, T. Byrnes for fruitful discussions.

##  VI References

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