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Supplementary Information for: Direct Measurement of Exciton Valley Coherence in Monolayer WSe2

Kai Hao1,*, Galan Moody1,†,*, Fengcheng Wu1, Chandriker Kavir Dass1, Lixiang Xu1, Chang-Hsiao Chen2, Ming-Yang Li3, Lain-Jong Li3, Allan H. MacDonald1 and Xiaoqin Li1

1 Department of Physics and Center for Complex Quantum Systems, University of Texas at Austin, Austin, TX 78712, USA.

2 Department of Automatic Control Engineering, Feng Chia University, Taichung 40724, Taiwan.

3 Physical Science and Engineering Division, King Abdullah University of Science & Technology (KAUST), Thuwal 23955, Saudi Arabia.

†Present address: National Institute of Standards & Technology, Boulder, CO 80305, USA.

*These authors contributed equally to this work.

Supplementary Note 1

Monolayer WSe2 Sample: CVD monolayer WSe2 triangular crystals were synthesized based on

previous work1. In brief, a double-side polished sapphire (0001) substrate (from Tera Xtal

Technology Corp.) was cleaned in a H2SO4/H2O2 (70:30) solution heated at 100 °C for one hour.

After cleaning, the sapphire substrate was placed on a quartz holder in the center of a 1” tubular

furnace. Precursor of 0.3 grams WO3 powder was placed in the heating zone center of the

furnace (99.5% from Sigma-Aldrich). Se powder (99.5% from Sigma-Aldrich) was placed in a

quartz tube at the upstream position of the furnace tube, which was maintained at 270 °C during

the reaction. The sapphire substrate was located at the downstream side, where the WO3 and Se

vapors were brought into contact with the sapphire substrate by an Ar/H2 flowing gas (H2 = 20

sccm, Ar = 80 sccm, chamber pressure = 3.5 Torr). The reaction heating zone was held at 925 °C

(ramping rate of 25 °C/min). The actual temperature of the sapphire substrate was ranged from

750 °C to 850 °C. The heating zone was held at 925 °C for 15 minutes after which the furnace

was then naturally cooled to room temperature. The reaction yielded monolayer WSe2 flakes

triangular in shape with a base width of ~20 μm. The thickness was determined using atomic

force microscopy, shown in Supplementary Fig. 1, which confirms the monolayer thickness of

the sample.

Direct measurement of exciton valley coherence in monolayerWSe2

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS3674

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http://dx.doi.org/10.1038/nphys3674

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Supplementary Figure 1: Atomic force microscope image of monolayer WSe2. (a) Individual triangular monolayer flakes are visible in the atomic force microscope image. A slice along the dashed line is shown in (b), illustrating the monolayer thickness of the flakes.

Linearly Polarized Photoluminescence: Linearly polarized photoluminescence (PL) from

monolayer WSe2 is measured using a 660 nm laser excitation source. Supplementary Fig. 2

shows polar plots of the exciton PL peak intensity (r) as a function of the detected angle (�) for two excitation polarization directions (indicated by the arrows). Linearly polarized PL following

the polarization of the excitation laser has been used as the key experimental evidence for valley

coherence in previous studies2. We repeat this linearly polarized PL experiment to demonstrate

the consistent properties for the monolayer WSe2 investigated in our nonlinear experiments. The

solid lines are fits using the equation � = � � (� � � � ��(� � �)), where � is the polarization angle corresponding to maximum exciton PL intensity. The results in Supplementary Fig. 2

demonstrate that � is determined entirely by the excitation laser polarization angle and is independent of crystal orientation. For both excitation angles, we measure a degree of linear

polarization�� = (��)(��) =�� , where IH (IV) is the PL intensity parallel (perpendicular) to the excitation laser polarization direction. PL spectra were collected in the range of 0 to 180

degrees in the detection angle, and spectra are duplicated in the range of 180 to 360 degrees in

the polar plot.

3

Supplementary Figure 2: Linearly polarized photoluminescence from monolayer WSe2.Normalized photoluminescence peak intensity (r) as a function of detection angle ( ) for a given excitation laser polarization (indicated by the arrows).

Two-Dimensional Coherent Spectroscopy: Optical two-dimensional coherent spectroscopy

(2DCS) is an enhanced version of three-pulse four-wave mixing in which the pulse delays are

varied with interferometric precision. A sequence of three pulses with variable delays is

generated using a set of folded and nested Michelson interferometers. The pulses are generated

by a mode-locked Ti:sapphire laser operating at an 80 MHz repetition rate with a pulse duration

of ~100 femtoseconds. This setup enables femtosecond control of the pulse delays with a

stabilization of λ/100, which allows for the four-wave mixing signal to be Fourier transformed.

Analysis of the signal in the Fourier spectral domain, combined with phase cycling of the pulse

delays to minimize scatter of the laser pulses into the spectrometer, suppresses optical frequency

noise overlapping with the exciton resonance and enables us to isolate the population

recombination and valley coherence signals.

In the experiments, three of the pulses with wavevectors k1, k2, and k3 are focused to a

single 35 μm spot on the sample that is held at a temperature of 10 K in an optical cryostat. The

interaction of the first pulse with the sample with wavevector k1 excites an electronic coherence

between the exciton ground and excited states. The second pulse with wavevector k2 converts the

optical coherence into a transient population grating (in the case of co-circular polarization) or a

valley coherence grating (in the case of alternating helicity of the pulses). After a time t2, the

third pulse with wavevector k3 generates an optical coherence that diffracts off the grating and is

2 NATURE PHYSICS | www.nature.com/naturephysics

SUPPLEMENTARY INFORMATION DOI: 10.1038/NPHYS3674



2

Supplementary Figure 1: Atomic force microscope image of monolayer WSe2. (a) Individual triangular monolayer flakes are visible in the atomic force microscope image. A slice along the dashed line is shown in (b), illustrating the monolayer thickness of the flakes.

Linearly Polarized Photoluminescence: Linearly polarized photoluminescence (PL) from

monolayer WSe2 is measured using a 660 nm laser excitation source. Supplementary Fig. 2

shows polar plots of the exciton PL peak intensity (r) as a function of the detected angle (�) for two excitation polarization directions (indicated by the arrows). Linearly polarized PL following

the polarization of the excitation laser has been used as the key experimental evidence for valley

coherence in previous studies2. We repeat this linearly polarized PL experiment to demonstrate

the consistent properties for the monolayer WSe2 investigated in our nonlinear experiments. The

solid lines are fits using the equation � = � � (� � � � ��(� � �)), where � is the polarization angle corresponding to maximum exciton PL intensity. The results in Supplementary Fig. 2

demonstrate that � is determined entirely by the excitation laser polarization angle and is independent of crystal orientation. For both excitation angles, we measure a degree of linear

polarization�� = (��)(��) =�� , where IH (IV) is the PL intensity parallel (perpendicular) to the excitation laser polarization direction. PL spectra were collected in the range of 0 to 180

degrees in the detection angle, and spectra are duplicated in the range of 180 to 360 degrees in

the polar plot.

3

Supplementary Figure 2: Linearly polarized photoluminescence from monolayer WSe2.Normalized photoluminescence peak intensity (r) as a function of detection angle ( ) for a given excitation laser polarization (indicated by the arrows).

Two-Dimensional Coherent Spectroscopy: Optical two-dimensional coherent spectroscopy

(2DCS) is an enhanced version of three-pulse four-wave mixing in which the pulse delays are

varied with interferometric precision. A sequence of three pulses with variable delays is

generated using a set of folded and nested Michelson interferometers. The pulses are generated

by a mode-locked Ti:sapphire laser operating at an 80 MHz repetition rate with a pulse duration

of ~100 femtoseconds. This setup enables femtosecond control of the pulse delays with a

stabilization of λ/100, which allows for the four-wave mixing signal to be Fourier transformed.

Analysis of the signal in the Fourier spectral domain, combined with phase cycling of the pulse

delays to minimize scatter of the laser pulses into the spectrometer, suppresses optical frequency

noise overlapping with the exciton resonance and enables us to isolate the population

recombination and valley coherence signals.

In the experiments, three of the pulses with wavevectors k1, k2, and k3 are focused to a

single 35 μm spot on the sample that is held at a temperature of 10 K in an optical cryostat. The

interaction of the first pulse with the sample with wavevector k1 excites an electronic coherence

between the exciton ground and excited states. The second pulse with wavevector k2 converts the

optical coherence into a transient population grating (in the case of co-circular polarization) or a

valley coherence grating (in the case of alternating helicity of the pulses). After a time t2, the

third pulse with wavevector k3 generates an optical coherence that diffracts off the grating and is





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detected in transmission in the wavevector-matching direction ks = -k1+k2+k3. The four-wave

mixing signal is interferometrically and spectrally resolved using a fourth phase-stabilized

reference pulse while the delay t2 is varied. Subsequent Fourier transformation of the signal

yields a rephasing zero-quantum spectrum with amplitude given by ES(t1, ħω2, ħω3). In the

present experiments, we set t1 = 0 fs to obtain maximum signal-to-noise; however using a value

of t1 = 100 fs, which is equal to the pulse duration, does not lead to qualitative difference in the

data other than an overall smaller amplitude.

Polarization control of the excitation pulses and detected signal allow for isolation of

valley coherence dynamics from population recombination. The coherent light-matter interaction

can be modeled by the density matrix,��. Here, the diagonal terms of � represents the exciton population in valley K and K′, while the off-diagonal terms describe exciton coherence. Using

perturbation theory in the applied field,3 the three-pulse excitation scheme of the K-valley

exciton for co-circular polarization can be described by the following sequence:

��)��

��)�� )��

��)�� , (1)

where Ei(t) corresponds to the ith excitation pulse and t is the pulse arrival time. It is clear from

supplementary equation (1) that the first two pulses excite an exciton population in the K valley;

therefore by recording the four-wave mixing signal while scanning the delay t2, we can extract

the exciton population recombination rate ΓK.

To resonantly excite and detect the exciton valley coherence, we use a pulse excitation

scheme in which the first and third pulses are co-circularly polarized (��) and the second pulse and detected signal are co-circularly polarized but with opposite helicity (��). This scheme can be described by the following density matrix sequence:

��)��

��)�� )��

��)�� . (2)

As shown in Fig. 2d of the main text, the first pulse generates a coherent superposition

between the ground and exciton state in the K valley. After a time t1, the second pulse transfers

this optical coherence to a non-radiative coherence between excitons in the K and K′ valleys, the

evolution of which we monitor during the time t2. The third pulse converts the valley coherence

to an optical coherence in the K′ valley, which is detected as the radiated four-wave mixing

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signal field. Thus by measuring the four-wave mixing signal while scanning the delay t2, we

directly probe the valley coherence dynamics and dephasing rate γv.


Discussion of impurity or phonon scattering: Supplementary Figs. 3 and 4 schematically

illustrate the effect of impurity or phonon scattering on valley dynamics, which include: (1)

change of the exciton center-of-mass-momentum, (2) suppression of exciton phase coherence,

(3) scattering between bright and dark exciton states, and (4) scattering between bright exciton

states in opposite valleys. Effect (1) is explicitly considered in our model. Effect (2) is the pure

dephasing rate which has been measured by our experiment. Effect (3) is part of the bright

exciton population decay process. Both effects (2) and (3) are phenomenologically described in

the theoretical model. Process (4) is likely to be slow, since it requires both electron and hole

spin flip as illustrated in Supplementary Fig. 4e.

An atomic-scale defect can localize excitons and lift their valley degeneracy. Recent

experiments4, 5, including ours (Fig. 2a in the main text), have found that impurity-bound exciton

states are energetically separated from the delocalized exciton states, and the PL of the former

(latter) have low (high) degree of correlation between excitation and emission circular

polarization. We conclude that the valley degree-of-freedom of delocalized excitons is not

destroyed by atomic-scale defects. A full understanding of the influence of atomic-scale defects

Supplementary Figure 3: Intra-valley scattering processes. (a) The center-of-massmomentum and the phase of a bright exciton can be changed by scattering. (b) A bright excitonis scattered to a dark one due to the spin flip of the constituent electron. The gray dot and emptycircle respectively represent electron and hole. The spin splitting of conduction bands isexaggerated for illustration purpose.





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on valley dynamics requires a detailed microscopic theory, which we leave for future

investigation.

Theory of exciton valley dynamics: We start from the exciton problem in monolayer TMDs. The

band extrema in valley K or K′ can be described by the massive Dirac model, given by

�� = �Δ �� + ��)

�� − ��) −Δ �, (3)

where the sign +/− corresponds to valley K/K′. To account for the finite thickness of monolayer TMDs, we use the following form of electron-hole interaction potential6:

�� = ��

��

��, (4)

Supplementary Figure 4: Inter-valley scattering processes. In (a), (b), and (c), either the constituent electron or hole in an exciton is scattered to the opposite valley. In (d) and (e), both the electron and hole in an exciton are scattered to the opposite valley. In (a), (b), (c), and (d), the final exciton state is optically dark. In (e), a bright exciton in valley Kis scattered to a bright one in valley K'.

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where � is the environment-dependent dielectric constant and �� is a parameter that is inversely proportional to �.

The characteristic length and energy scale are respectively the effective Bohr radius and

Rydberg energy, given by

��∗ = ��(ℏ��)�

�� , Ry∗ =��

��∗ . (5)

The effective fine structure can be defined as � = ��/(�ℏ��).

For monolayer WSe2, ℏ�� = 3.310Å × 1.19eV and Δ = 0.685eV, which are taken from Ref. (7). We note that this value of Δ obtained from DFT calculations underestimates the band gap, which, however, should not noticeablely affect the quantities we are interested in for this

study. We adjust the parameter �� so that the A-exciton binding energy for � = 2.5, which corresponds to WSe2 lying on an SiO2 substrate and exposed to air, is 0.370 eV as measured in

the recent experiments.8 We therefore find �� = 22.02Å/� by solving the Bethe-Salpeter equation for excitons in the massive Dirac model.9

In monolayer TMDs, the center-of-mass motion of an exciton is coupled to its valley

degree of freedom as described by the Hamiltonian

�� = �ℏ�� (2�)�� (2�)��], (6)

where �� and ��,� are identity matrix and Pauli matrices in the exciton valley space, respectively, ℏ�� is the excitation energy of the A exciton, �� = ℏ��/2� is the kinetic energy of the center-of-mass motion with � being the total mass of the exciton, � and � are the magnitude and orientation angle of the center-of-mass momentum �. The intra and inter-valley exchange interaction are described by the �� and ��,� terms, respectively. The coupling constant �� is linear in the magnitude of �:

�� = Ry∗ �� ∗ �(0)��(2��/Ry∗)�/�, (7)

where ��∗ �(0)�� is the probability that an electron and a hole spatially overlap.10





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For WSe2 lying on sapphire substrate and exposed to air, the dielectric constant � is approximately 5.5. The appropriate parameter values are thenRy∗ = 75.66 meV, � = 0.66, and |��∗ �(0)| = 0.88. Therefore, �� = 20�� × (2��/Ry∗)�/�.

The time evolution of excitons is governed by:

��(��)�� =

�ℏ ��(�� )� �� + ∑ �� [�(�� ) − �(�� )] −

�(��)� , (8)

where �(�� ) represents a 2 × 2 density matrix in the exciton valley space at momentum � and time �. The diagonal terms of � stand for the exciton population in valley K and K′, while the off-diagonal terms describe the coherence between K and K′ exciton states. The final term in

supplementary equation (8) phenomenologically captures the effects of exciton recombination

(Γ�) and pure dephasing (��∗ ) on valley coherence. We allow the decay rate ℏ/τ of diagonal and off-diagonal elements of the density matrix � to differ, as explained below.

We decompose the density matrix into�(�� ) = ��(�� )�� + �(�� ) � �. The equation of motion for the valley pseudospin vector � makes the physical picture more revealing:

��(��)�� = �(�) × �(�� ) + ∑ �� [�(�� ) − �(�� )] −

�(��)� , (9)

where �(�) = 2��(��(2�)� ��(2�)�0)/ℏ. We approximate the rate ℏ/�� for �� by the population decay rate Γ�, and the rate ℏ/�� for �� by Γ� + 2γ�∗ . The underlying assumption for the 2γ�∗ contribution is that the pure dephasing processes for K and K′ excitons are uncorrelated.

We make the assumption of elastic momentum scattering, i.e. ��is nonzero only if |�| = |��| . Thus we can write

��(�� )�� = �(�� ) × �(�� ) +

�� ∫ �� (� − ��)[�(�� ) − �(�� )] −

�(��)� . (10)

To take advantage of the rotational symmetry, we make the angular Fourier transformation:

�(�� ) = ∑ �(�)(�� )�� , (11)

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ℏ �� (�)(�, �)��(��)(�, �)��(��)(�, �)

� =

��−ℏ�� −�2�� 0−�� − ℏ�� −

ℏ��

��0 �2�� − ℏ�� −

ℏ��

��(�)(�, �)��(��)(�, �)��(��)(�, �)

�, (12)

where �±(�) = ��(�) ± ��(�), and 1�� being the momentum scattering rate:

��= �� ∫ ��

�� (�)�1 − ��(��)]. (13)

We assume δ-function impurity potentials. Then �(�) has no � dependence. Therefore,1�� =1�� 1��.

When � initially points to the x-direction, then Sx(Q, t) averaged over momenta has the form:

��(�)� = 12� ∫�2��+(�, �) = ∫��+(0)(�, �)]. (14)

The initial momentum dependence of �� is modeled by a Lorentzian distribution given by

��(�, 0) = ��(0,0)�(1 + ��), (15) where �� is the kinetic energy associated with the exciton center-of-mass motion. We approximate the parameter � by the homogeneous linewidth, assuming that finite-momentum states are able to radiate light only if their energy is within the homogeneous linewidth. This

approximation also assumes that excitons outside the light-cone can radiate due to interaction

with phonons and impurities. The time evolution of ��(�)� is shown in Fig. 4 of the main text.

Valley depolarization: When � initially points to the z-direction, the time dynamics are governed by:

ℏ �� (�)(�, �)��(��)(�, �)��(��)(�, �)

� =

��−ℏ�� −��2�� − ℏ�� −

ℏ��

0−�2�� 0 − ℏ�� −

ℏ��

��(�)(�, �)��(��)(�, �)��(��)(�, �)

�. (16)

We assume that the bright exciton population � also has a decay rate 1��:





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�� . (17)

The valley polarization can be quantified by the average of Sz(Q, t) over momenta:

��(�)� � 12� ��2��(�� ) � ��(0)(�� )]. (18)

If we denote the decay rate of ��(�)� by 1��∗, the degree of circularly polarization �� is ��∗��. Using the same parameter value for the dynamics of valley coherence ��(�)�, we calculated the time dynamics of valley polarization ��(�)�, which is shown in supplementary Fig. 5. �� is approximately 50% for ��= 10 meV and 57% for ��= 20 meV. These results demonstrate that ~ 100 fs exciton valley decoherence time is fully compatible with a higher degree of valley

polarization. This value of valley polarization in the resonant nonlinear experiments may

increase further by reducing the parameter � in the momentum distribution in supplementary equation (15), and adjusting other parameters accordingly. We emphasize that the degree of

valley polarization in resonant nonlinear experiments cannot be directly compared to that

observed in non-resonant PL experiments. It is known that the degree of valley polarization

observed in PL strongly depends on the excitation wavelength. In addition, the

photoluminescence signal includes contributions from incoherent exciton population dynamics,

which can be sensitive to scattering between optically bright and dark excitons. To achieve a

better agreement between theoretical and experimental values of �� , the time dynamics of dark

Supplementary Figure 5: Valley depolarization. Initially, excitons in valley K is generated bycircularly polarized light. The solid lines represent the time evolution of Sz for differentmomentum scattering rate ��, and the dashed line labels the time evolution of excitonpopulation N.

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excitons needs to be added to the theory.

Supplementary Information References

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device applications. ACS Nano 8, 923–930 (2013).

2. Jones, A. M. et al. Optical generation of excitonic valley coherence in monolayer WSe2.

Nat. Nanotech. 8, 634–638 (2013).

3. Scully, M.O. and Zubairy, M.S. Quantum Optics (Cambridge University Press, 1997).

4. He, Y.-M. et al. Single quantum emitters in monolayer semiconductors. Nat. Nanotech.

10, 497-502 (2015).

5. Srivastava, A. et al. Optically active quantum dots in monolayer WSe2. Nat. Nanotech.

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6. Berkelbach, T.C., Hybertsen, M.S., and Reichman, D.R. Theory of neutral and charged

excitons in monolayer transition metal dichalcogenides. Phys. Rev. B 88, 045318 (2013).

7. Xiao, D., Liu, G.-B., Feng, W., Xu, X., and Yao, W. Coupled spin and valley physics in

monolayers of MoS2 and other group-VI dichalcogenides. Phys. Rev. Lett. 108, 196802

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8. He, K., Kumar, N., Zhao, L., Wang, Z., Mak, K.F., Zhao, H., and Shan, J. Tightly bound

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