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Optical control of room-temperature valley polaritons

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Supplementary Information:

OPTICAL CONTROL OF ROOM TEMPERATURE VALLEY POLARITONS

Zheng Sun1,2, Jie Gu1,2, Areg Ghazaryan1, Zav Shotan1, Christopher R. Considine1,

Michael Dollar1, Biswanath Chakraborty1, Xiaoze Liu1,2,*Pouyan Ghaemi1,2,

Stéphane Kéna-Cohen3, Vinod M. Menon1,2,†

1 Dept. of Physics, City College, City University of New York (CUNY), New York, USA

2 Dept. of Physics, Grad. Center, City University of New York (CUNY), New York, USA

3 Dept. of Engineering Physics, École Polytechnique de Montréal, Montréal, Quebec, Canada

† Corresponding author: vmenon@ccny.cuny.edu

Figure S1: Optical characterization of monolayer WS2.

Figure S2: Dependence of absorption and photoluminescence on SiO2 capping.

Figure S3: Tuning of cavity resonance by varying the top metal mirror thickness.

Figure S4: Helicity resolved photoluminescence with pump in resonance with bare exciton A energy.

Figure S5: Helicity resolved photoluminescence with pump in resonance with lower polariton branch.

Helicity Measurements

Figure S6: Laser spectrum through long pass filter.

Figure S7. Helicity resolved photoluminescence for quantum dot microcavity.

Rate Equation Model

* Present address: Department of Mechanical Engineering, University of California at Berkeley, Berkeley, CA.

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SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHOTON.2017.121

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Fig. S1. Optical characterization of monolayer WS2. (a) Raman spectrum of exfoliated WS2

showing the expected spectral signatures for monolayer samples. Photoluminescence spectrum (red

- solid) and absorption (blue - dashed) for the (b) uncapped WS2 and (c) SiO2 capped WS2. Upon

capping with SiO2 via plasma enhanced chemical vapor deposition, the main emission peak shifts due

to dielectric environment change from the capping layer. Inset in (c) shows the optical microscope

image of the exfoliated WS2.

Fig. S2. Dependence of absorption and photoluminescence on SiO2 capping. The main peak in

(a) absorption spectra and (b) emission spectra red shifts gradually with increasing the SiO2

thickness due to change in dielectric environment and strain.

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Fig. S3. Tuning of cavity resonance by varying the top metal mirror thickness. Simulated

reflectivity showing cavity resonance as a function of wavelength for the different top silver thickness

is shown.

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Fig. S4. Helicity resolved photoluminescence with pump in resonance with bare exciton A

energy. (a) and (b) correspond to the +16 meV detuned cavity while (c) and (d) correspond to the -105

meV detuned cavity. Black and red curves correspond to - and + emission. The pump was chosen

to be in resonance with the exciton energy (1.98 eV) and the helicity of the pump is shown in the

respective graphs. The peak helicity for the +16 meV is 20% and that for the -105 meV is 24%.

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Helicity Measurements:

Right and left circularly polarized excitation pump was obtained using the combination of an

achromatic quarter wave plate (QWP) and a linear polarizer (LP) in front of the tunable laser

(Toptica TVIS). On the collection side, a similar QWP and LP combination is used to resolve the

helicity of the lower polariton branch emission. A 633 nm (1.958 eV) long pass filter was used on

the emission side to filter out the excitation signal. Both helicities (right: + and left: -) were

resolved and the spectrum was recorded using a Princeton Instruments monochromator with Pixis

1024B EMCCD camera. To test our experimental set up and any residual helicity that the set up

might have, we carried out two tests. First one to establish that the laser tail does not leak through

the long pass filer at 633 nm. The result is shown in Fig. S6 where no residual laser tail is observed.

Following this we carried out helicity measurements on a microcavity containing colloidal

quantum dots which are not expected to show helicity. Results of helicity measurements on

quantum dot microcavity is shown in Fig. S7 where we do not see any helicity within our

experimental resolution.

Fig. S5. Helicity resolved photoluminescence with pump in resonance with lower polariton

branch. (a) +16 meV detuned cavity and (b) -105 meV detuned cavity. Black and red curves

correspond to - and + emission. The pump was chosen to be in resonance with the lower polariton

branch (1.95 eV) and the helicity of the pump is shown in the respective graphs. The peak helicity for

the +16 meV is 16% and that for the -105 meV is 12%.

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Fig. S7. Helicity resolved photoluminescence for quantum dot microcavity. (a) and (b)

correspond to - and + excitation, respectively. Black and red curves correspond to - and +

emission. The pump was chosen to be in resonance with the exciton energy (1.98 eV) and the

helicity of the pump is shown in the respective graphs. The peak helicity for the quantum dots is

below the noise level of our set up.

Fig. S6. Laser spectrum through long pass filter: Laser spectrum measured after a 633nm long

pass filter used in the experiments clearly showing no residual laser tail in the spectral region

where the helicity is observed from the polariton emission shown in Fig. 3 (main text).

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Rate Equation Model:

As discussed in the main text, the main valley relaxation mechanism of polaritons is comprised

from two components related to excitonic and photonic parts of the polariton. The excitonic part

arises due to the electron-hole exchange interaction, which results in coupling of exciton center of

mass motion to its valley degree of freedom. The resulting excitonic Hamiltonian in the valley

basis can be written in the form1,2

2 2

02 cos 2 sin 2x yH k M J J k k k k k , (S1)

where M is the total mass of the exciton, k is the orientation angle of center of mass momentum

k , i are Pauli matrices which act in valley basis and J kk

is the term resulting from

exchange interaction ( being material dependent constant). For non-zero center of mass

momentum the double degeneracy of excitonic bands is lifted. Lower band has quadratic and

higher band has both linear and quadratic dependence on the center of mass momentum. As was

noted in the main text excitons in each valley are directly linked to the right and left circular

polarization of the light. In optical microcavity, similar coupling is also present between light

momentum and circular polarization3 which leads to splitting between TE (transverse electric) and

TM (transverse magnetic) components of light. The splitting for photonic part is always quadratic

in momentum. For metallic cavities, the TE-TM splitting is considerably larger than the associated

splitting for excitons. Adding these two terms together the Hamiltonian for polaritons has the form

pol pol

eff2

kH E k

Ω k σ . (S2)

pol

kE is the bare dispersion of the polariton and eff ex

2 2

k k phX C Ω k Ω k Ω k . Here kX

and kC are the exciton and photon Hopfield coefficients. ex 2 cos 2 ,sin 2 ,0 /J k k k

Ω k

, phΩ k are related to the appropriate amplitudes of splitting for excitons and cavity photon. It is

clear from equation (S2) that effΩ k acts as a pseudomagnetic field for valley pseudospin and

combined with the scattering with impurities and structure disorder results in exciton valley

relaxation demonstrated below.

In order to calculate the effect of this valley relaxation mechanism for our system, we

define density matrices for polariton k t and for exciton reservoir R t and solve Liouville

– von Neumann equation for the system. In order to incorporate polariton-acoustic phonon

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scattering we adopt Born-Markov approximation and also make use of the cylindrical symmetry

of the problem to simplify the calculation. In the linear regime, when polariton density is small

and stimulated scattering can be ignored the equations for population and psudospin of the

polariton have the following form4,5

eff

eff

imp

12 2 ,

12 2 ,

12

1,

k k k k kk k Rk R kR k

kk

z z z z

k k k kk k k Rk R kR k

ksk

z

k k k kk k k

k

k

zzkk

kk

sk

N W N k W N k W kN W N

W S k W S k k S W kS W S

W S k W S k k S

dN

dt

dSS

dt

dSS

dt

(S3)

where k is the polariton lifetime defined as

2 2

1

k X

k k

ph

X C

, (S4)

X , ph being exciton and cavity photon lifetimes respectively. For the numerical simulations,

we use the values p5 sX and 1.5psph . sk is the pseudospin lifetime, which is defined as

1 1 1

sk k s and s is the lifetime of additional valley relaxation mechanisms not accounted by

the pseudomagnetic field. For s we use a definition similar to k

2 2

1s

k k

s

X hs p

X C

, (S5)

where s

X and s

ph are the exciton and cavity photon decoherence times respectively. We find the

best fit with the experimental results when we take 5pss

X and 0.5pss

ph . Combined with the

magnitude of pseudomagnetic field values for exciton and photon this shows that polariton

decoherence is mostly due to the photonic component, while excitonic part is immune to the

decoherence. This can be explained based on the fact that there are dark excitonic states observed

for WS2, which are lower in energy than the bright state and acts as a reservoir with higher

coherence lifetime6,7. We consider impurity and structure disorder scattering to be elastic and

circularly symmetric and treat it with a phenomenological scattering lifetime imp . In (S3) kkW and

kkW are defined as

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2 2

0 0

1 1cos 2

2 2,kk kkW W W Wd d

kk kk, (S6)

where W kk is the acoustic phonon-polariton scattering rate defined as

2

0,1 phon

p pW U n E k E k

kk k k k kk k , (S7)

where it is assumed that p pE k E k .phonnq is the Bose-Einstein distribution for phonons and

q is the phonon energy. In the (S7) 0 and – correspond to phonon absorption, whereas 1 and +

to emission and U k k is the matrix element of polariton-phonon interaction8.

The equations of dynamics of the population and pseudspin for reservoir have the form

12 ,

12 ,

2

R

X

zzR

R Rk R kR k

k k

z

Rk R kR k

k kX

Rs

dNP N W kN W N

W kS

dt

dS PS

dtW S

(S8)

where P defines the pump and we took exciton lifetime and decoherence time to be the same as

for the excitonic part of the polariton branch. Based on the experimental results we took the exciton

forward scattering time 0 ps0.5 , so there is fast population of the polariton branch after

excitation with the pump.

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