Observation of ~100% valley-coherent excitons in monolayer MoS2 through giant enhancement of valley coherence time

In monolayer transition metal dichalcogenide semiconductors, valley coherence degrades rapidly due to a combination of fast scattering and inter-valley exchange interaction. This leads to a sub-picosecond valley coherence time, making coherent manipulation of exciton a highly challenging task. Using monolayer MoS2 sandwiched between top and bottom graphene, here we demonstrate fully valley-coherent excitons by observing ~100% degree of linear polarization in steady state photoluminescence. This is achieved in this unique design through a combined effect of (a) suppression in exchange interaction due to enhanced dielectric screening, (b) reduction in exciton lifetime due to a fast inter-layer transfer to graphene, and (c) operating in the motional narrowing regime. We disentangle the role of the key parameters affecting valley coherence by using a combination of calculation (solutions of Bethe-Salpeter and Maialle-Silva-Sham equations) and a careful choice of design of experiments using four different stacks with systematic variation of screening and exciton lifetime. To the best of our knowledge, this is the first report in which the excitons are found to be valley coherent in the entire lifetime in monolayer semiconductors, allowing optical readout of valley coherence possible.

The bound state of an electron and a hole, an exciton, is a superposition of the conduction and valence band states in the  and  ′ valleys in monolayer transition metal dichalcogenides (TMDs) [1][2][3] . and  ′ valley excitons are selectively generated by circularly polarized light excitation of opposite helicities [4][5][6][7] .On linearly polarized excitation, a hybrid  −  ′ exciton is generated in a state of valley coherence 8,9 .However, valley coherence degrades rapidly due to a combined effect of fast scattering and inter-valley exchange [10][11][12] .The reported values of valley coherence time lie in the range of 98 − 520 fs [12][13][14][15][16][17] , much shorter than the exciton radiative lifetime of ~1 ps [18][19][20] .This makes optical read out of strong exciton valley coherence a highly challenging task.To be able to use these coherent excitons as a qubit for quantum information processing, a longer valley coherence time is desirable to perform any manipulation on it.Any technique 21 that enhances this valley coherence time significantly is thus of high scientific importance.
Here we demonstrate a 100% degree of linear polarization (DOLP) in photoluminescence (PL) peak of  1 exciton in a monolayer of MoS2 encapsulated with few-layer-graphene (FLG) at the top and bottom.Such a complete retention of the generated valley coherence in steady-state PL implies the achievement of a large valley coherence time, the measured value of which is only limited by the lifetime of the exciton.This suggests that the valley coherence time has been significantly enhanced as compared to the reported values to date [12][13][14][15][16] .
Depending on the linear polarization direction of the excitation light, the excitons are generated at specific center-of-mass momentum () values [where  =   +   ,   (  ) denoting the electron (hole) crystal momentum] in the exciton band at time  = 0 (Figure 1).During its lifetime, the exciton undergoes scattering and exchange interaction, which, coupled together, degrades the valley coherence.The polarization state can be represented by the pseudospin vector  in the Bloch sphere.At  = 0, the direction of  is parallel to the exchange-induced magnetic field (denoted by the precession frequency ).Considering  − polarized excitation, the system is generated in a pure state represented by   = 1,   = 0,   = 0.When excitons scatter to different  values, the pseudospin precesses on experiencing a finite torque around () due to which it becomes a mixed state represented by a density matrix operator .The  -space and the Bloch sphere representations of this mechanism are shown in Figure 1a-b.
On decoupling the density matrix in terms of the number trace and traceless matrix . (where  denotes the Pauli matrices), the overall dynamics of the valley pseudospin as described by the Maialle-Silva-Sham (MSS) mechanism (see Supplementary Note 1 for proof) is given by 10 : Here W  ′ is the rate of any generic momentum scattering mechanism, e.g., exciton-impurity and exciton-phonon scattering. is the net exciton lifetime given by , where   ,   , and   are the radiative, non-radiative, and filtering timescale.
Filtering is a non-radiative process where the excitons are scattered out of the light cone, e.g., scattering to lower energy states, interlayer transfer to graphene, etc., in which case, the light collection is limited to  ≤   . represents the exciton generation rate.On recombination, the DOLP of this mixed state is given by 〈  〉, averaged over  values within the light cone (see Supplementary Note 2 for proof).
The possible ways to improve the valley coherence time are: (a) by minimizing scattering (W  ′ ) inside the light cone such that  does not accumulate random phase by precessing around (), or by enhancing W  ′ such that the whole operation is pushed towards the motional narrowing regime (simulation results in Figure 1c); (b) by screening the electron-hole interaction which results in reduced exchange interaction, and in turn a suppressed  (middle panel of Figure 1d).However, a side-effect of the enhanced screening is an increment in the exciton lifetime due to a reduction in the binding energy.This can be overcome by (c) reducing by introducing a fast-filtering mechanism [21][22][23] (bottom panel of Figure 1d).
Interestingly, there are several spots where we observe ~100% DOLP in the GMG stack (Figure 2d and Supplementary Figure 6).In Supplementary Figure 7a, we show similar results of ~100% DOLP obtained from few-layer graphene encapsulated monolayer WS2 (GWG) stack.We also perform polarization dependent time-resolved photoluminescence (TRPL) measurement and obtain a peak DOLP of 97.6% in the GWG stack (see Supplementary Figure 7b).
We would like to highlight some additional observations on the GMG stack before the main analysis begins: (1) As a result of FLG encapsulation, the PL spectra predominantly consist of the clean  1 exciton peak 22 .The spectra of FLG encapsulated monolayer MoS2, MoSe2, and WS2 on 532 nm excitation are shown in Figure 2f, clearly indicating suppression of spurious peaks from defect-bound excitons and other excitonic complexes.We also observe a clear  2 peak located at 44 (32) meV higher than the  1 peak in MoS2 (MoSe2 and WS2) due to enhanced screening 24 .
(2) We also get a very high degree of circular polarization (DOCP) of 81.6 (±2) % in the GMG stack, much larger compared to the 20.5 (±9) % DOCP in the HMH stack (Figure 2g-h and more spectra in Supplementary Figure 8-9).The in-plane nature of  explains this observation that DOCP is smaller than DOLP for 2D excitons, which is consistent with previous reports 10,25 (see Supplementary Note 3).This indicates that starting with a linear polarization (that is, on the equator of the Bloch sphere) is the most favourable scenario to maintain valley coherence compared with any other (elliptical) polarization.(3) The initial and the final state in the Raman scattering process coincides with the  2 and the  1 exciton level, respectively, on 633 nm excitation at 5 K in the GMG stack.This dual resonance enhances the intensity of the Raman peaks significantly (represented by the dashed lines in Figure 2d and 2h) and enables the observation of other less commonly observed modes distinctly (Supplementary Figure 10).The fact that the excitation laser is resonant with the 2s state in the GMG stack and we observe almost fully-coherent 1s excitonic emission from the stack, it is likely that the generated 2s excitons relax through polarization preserving processes, such as, dipole-coupled radiative transition (2s→2p→1s).
In order to establish the different degrees of screening in the stacks, we plot the PL spectra for the HMH, GHMHG, and the GMG stack obtained from 532 nm excitation in Figure 3a-c.The  2 −  1 energy separation obtained is 144.5 meV in the HMH stack, which reduces to 60 and 45 meV in the GHMHG and GMG stack, respectively.To get an estimate of the  1 exciton binding energy change (Figure 3d), we obtain the continuum of the exciton energy spectrum by numerically solving the Bethe-Salpeter equation 26 using a two-band Hamiltonian.
In the calculation, the parameters are fitted such that the experimentally obtained  2 −  1 energy separation matches with the calculated one (Supplementary Note 4 and Supplementary Figure 2).The calculated  1 exciton binding energy is 379 meV in the HMH system, which reduces to 162.5 and 122 meV in the GHMHG and the GMG stacks respectively due to graphene induced screening.
One immediate consequence of such a screening is the reduction in the inter-valley exchange interaction 25 .The exchange interaction is composed of two components -the short-range part and the long-range part.The short-range component is zero at  =  due to the three-fold rotational symmetry condition and is negligible at higher  values.The long-range part is given by 27 : (2) is the electron-hole wavefunction overlap at zero relative separation ( ℎ = 0),  is the reciprocal space wave-vector,   denotes the bandgap of MoS2, and () is the electron-hole coulomb interaction.The dielectric screening modulates the following factors: (a) |( ℎ = 0)| 2 − due to a reduction in the 2D exciton binding energy 24,28 ; (b)   − due to bandgap renormalization effect in monolayer TMDs 29,30 ; and (c) () − due to suppressed electron-hole interaction 31 .In Figure 3e, we show the variation in    with  within the light cone, and hence the screening induced suppression of the long-range exchange in our samples (see Supplementary Note 4).
Another consequence of screening is the enhancement of the exciton radiative lifetime due to a reduced electron-hole wavefunction overlap.Here, a longer lifetime is undesirable as it leads to a larger valley decoherence (Figure 1a).To estimate the exciton lifetime and its role in the valley decoherence, we carry out TRPL measurements on our samples (see Methods and Supplementary Figure 11).The TRPL values obtained in our stacks are as follows: < 5 ps in the HMH stack, 6 − 8 ps in the GHMHG stack, and < 5 ps in the GMG stack.The uncertainty in the lifetime in the HMH and GMG stacks arises as it is smaller than the 10% limit of our Instrument Response Function (IRF) width 32 .Nonetheless, several reports supporting these numbers are already available in literature 18,19,22,33 .Moreover, the qualitative trend in the exciton lifetime testifies the anticipated trend in Figure 1d, showing a clear enhancement in the exciton lifetime in the GHMHG stack, as compared to the other two stacks.
Obtaining an average DOLP of only 44.5 (±10) % implies an ultra-short valley coherence time in the HMH stack, in agreement with previous reports 12,15 .On the other hand, in the GHMHG stack, an increased exciton lifetime is an evidence of screening induced enhancement of exciton lifetime as a result of introducing top and bottom FLG.Due to the opposing roles of reduced exchange and increased lifetime, we do not observe any improvement in the exciton DOLP in this sample compared with the HMH sample.
This side-effect of screening driven enhanced exciton lifetime is eliminated in the GMG stack through filtering, where light collection from the long-lived excitons is prohibited due to a fast transfer of excitons to graphene.The extracted timescale corresponds to the graphene-transferlimited exciton lifetime in this system.This timescale is similar to that in the HMH stack, and in agreement with previous report 22,33 .Therefore, the significant exciton DOLP difference between the HMH and the GMG stacks is thus attributed to screening modified exchange interaction without any confounding effect due to a change in the exciton lifetime.
To obtain a quantitative understanding, we solve the steady-state form of the MSS equation: We obtain the DOLP (〈  〉) for the  1 exciton in the HMH and the GMG stack (Supplementary Note 4).For W  ′ , the exciton-impurity scattering rate expression is used (Supplementary Note 4.2). is the exciton lifetime,  is the angle between the initial () and the final ( ′ ) state and  is an overall scaling factor.We neglect the contribution of excitonphonon scattering in decoherence at 5 K. Plotted in Figure 4 is the variation in 〈  〉 as a function of the  for the two stacks.In both the cases, the V shaped variation is understood as follows: For small , an increase in the scattering degrades the valley coherence due to enhanced exciton precession around  (Figure 1a).However, this effect is non-monotonic, as on significantly enhancing the scattering rate, the DOLP starts increasing after reaching a minimum.This phenomenon is referred to as motional narrowing 10,34 , and arises due to a cancellation of the accumulated randomness in the phase information of .Mathematically, the system is in the motional narrowing regime when the exciton scattering frequency becomes larger than the precession frequency, leading to a longer pseudospin coherence time.
We take the extracted homogeneous (Lorentzian component of the Voigt fitting) linewidth (Γ ℎ ) of the co-polarized exciton PL peak as the experimental analogue of the scattering rate.
The experimental DOLP as a function of the Lorentzian linewidth, superimposed on the simulation results, is shown in Figure 4.Both in GMG and HMH stacks, the extracted value of Γ ℎ is much larger than the exciton lifetime limited linewidth, as obtained from TRPL.This suggests that the impurity scattering rate is similar in both the samples, and it dominates over other linewidth broadening mechanisms.This is also evident from the upward trend of the experimental DOLP with Γ ℎ , which is in excellent agreement with the rising side of 〈  〉 versus  in the simulation.This suggests that the whole operation lies in the motional narrowing regime in both the samples.
We conclude that the combined effect of enhanced screening, reduced lifetime due to interlayer transfer, and motional narrowing helps us to achieve ~100% exciton DOLP in our FLGcapped MoS2 sample.This is a direct evidence of valley coherence being maintained during the entire exciton lifetime by cutting down the decoherence channels before the spontaneous emission.Achieving ~100% steady-state DOLP also indicates that the true valley coherence timescale beyond few picoseconds in the GMG stack, which is in excellent agreement with simulation result obtained by solving time-dependent MSS equation (see Supplementary Figure 12).
The combination of such ~100% polarization, coupled with background-free, narrow linewidth emission, makes the GMG stack a promising substrate for spectral diffusion-free, indistinguishable single photon source.As the initialized coherence in the exciton is shown to be staying protected for a longer time, the results have intriguing prospects on performing experiments involving coherent manipulation of exciton and building quantum system operating at these timescales.

Sample preparation:
All the stacks in this paper are prepared first by mechanically exfoliating the layered material on a Polydimethylsiloxane (PDMS) film, and then its subsequent transfer in a controlled manner underneath a microscope on a Si substrate covered with 285 nm thick thermally grown SiO2.The thickness of the few-layer graphene is chosen to be 2-3 nm in all the stacks.The thickness of hBN layers is in the range of 20-30 nm apart from the GHMHG stack.To ensure strong screening in the GHMHG stack, the hBN thickness is kept at ~5 nm.
After the preparation of the entire stack, the samples are annealed at 200 0 C for 5 hours (pressure ~10 -6 torr) to ensure better adhesion between successive layers.

Sample characterization:
All the measurements are taken in a closed-cycle optical cryostat (Montana Instruments) at 5 K using a × 50 long-working-distance objective having a numerical aperture of 0.5.To measure the exciton DOLP �= ( / −  H/V ) � / +  H/V � ⁄ �, we place the analyzer in the parallel � / � and perpendicular � H/V � direction in the collection path relative to the excitation polarization direction.For the DOCP measurements, a quarterwave plate is inserted just before the objective lens, and aligned at 45 0 with respect to the incoming linearly polarized light.The time-resolved photoluminescence measurement is carried out using a 531 nm laser controlled by the PDL-800 D driver (laser pulse width is 48 ps).We use a single photon counting detector from Micro Photon Devices and the time correlated measurements are taken using the PicoHarp 300 TCSPC system (PicoQuant).We use a combination of two bandpass filters to get the time resolved counts of the  1 exciton at     The time evolution of the density operator  for the mixed state in the presence of exchange field and momentum scattering �W  ′ � s given by MSS as 1 : Here  represents the net exciton lifetime within the light cone, and the matrix  accounts for the exciton generation rate. is the total pseudospin Hamiltonian which can be decomposed into  =  0 +  1 (), where  0 is the diagonal matrix that accounts for the applied magnetic field in the out-of-plane direction: and  1 is the matrix containing the off-diagonal elements that is responsible for the exchange driven exciton pseudospin flip: where  is the angle between  and the  axis.
0 can be expanded as where Ω 0 = ( + −  − ) ћ, ⁄ and  1 () can be written as  where  is used as a fitting parameter.Finally, the DOLP (= 〈  〉) is calculated by averaging the calculated pseudospin at the -values within the light cone and is plotted in Figure 4 in the main text for the HMH and the GMG sample.

Figure 1 :
Figure 1: Mechanism of exciton valley decoherence and the factors affecting it.(a) Top view of a ring inside the light cone of the exciton band showing the exciton decoherence dynamics due to scattering W  ′ within the light cone (purple dashed arrows) and subsequent precession because of inter-valley exchange induced pseudo-magnetic field (black solid arrows).(b) The direction of  at generation ( = 0) and recombination ( = ) are shown in the Bloch sphere.(c) The calculated DOLP as a function of scattering rate with motional narrowing regime shown in shade.(d) The decoherence of  (green arrows) with time due to scattering (dashed purple arrow) and precession around  (black arrows) for an exciton for three different scenarios.The effect of screening versus screening + filtering on valley decoherence is compared in the three rows.

Figure 2 :
Figure 2: Excitonic emission from different stacks along with DOLP and DOCP.(a-d) PL spectra with near-resonant 633 nm linearly polarized excitation in co-(H/H) and cross-(H/V) polarized detection configuration in the (a) hBN-MoS2-hBN (HMH) stack, (b) FLG-hBN-MoS2-hBN-FLG (GHMHG) stack, (c) MoS2-FLG-hBN (MGH) stack, and the (d) FLG-MoS2-FLG (GMG) stack at  = 5 K.  1 and  1 − represent the exciton and the trion peak, respectively.(e) Bar graph comparing the experimental DOLP values along with the error bars in the four stacks.(f) PL spectra obtained from FLG-TMD-FLG stack (using 532 nm excitation) for monolayer MoS2, MoSe2, and WS2 showing the prominent  1 and  2 peaks.The  2 −  1 separation is around 44 (32) meV in MoS2 (MoSe2, WS2).The peaks marked as * and ⋅ are the 2D and the G Raman peaks of the FLG.(g-h) Representative PL spectra taken with circularly polarized excitation in co-( +/+) and cross-( +/−) polarized detection configuration.The corresponding DOCP value of the  1 exciton in (g) the HMH and (h) the GMG stack at  = 5 K is shown in the inset.The peaks indicated by the dashed lines in (d) and (h) represent the prominent Raman peaks in the GMG stack due to dual resonance.

Figure 3 :
Figure 3: Evidence of graphene induced screening of electron-hole exchange interaction.(a-c) Top panel: Schematic representation of the screening effect in the different stacks.Bottom panel: PL spectra taken with 532 nm excitation highlighting the different degree of dielectric screening in our samples.The  2 −  1 separation for the HMH stack (144.5 meV), GHMHG stack (63.5 meV), and the GMG stack (43.3 meV) is indicated by the dashed arrows.* in (c) is the 2D Raman peak of FLG.(d) Eigen energies (open circle) and the corresponding binding energies of the  1 exciton obtained from the Bethe-Salpeter equation.The solid symbols

Figure 4 : 2 �
Figure 4: Comparison of the experimental data with the steady-state solution of the MSS equation.Simulation results comparing the exciton DOLP (〈  〉) as a function of the scaling factor () of the scattering rate (bottom axis) for the HMH stack (solid black trace) and the GMG stack (dashed black trace).The downward trend in the left-hand side (unshaded region) is the low-scattering regime, and the upward trend in the right-hand side (shaded region) is the motional narrowing regime.Overlapped on the simulation results is the experimentally obtained DOLP variation with the Lorentzian linewidth (Γ ℎ ) of the corresponding copolarized PL spectrum (top axis) for the HMH stack (solid spheres) and the GMG stack (open circles).The upward trend of the experimental data suggests that both the stacks are operating in the motional narrowing regime.

2 4𝜋𝜋𝜖𝜖 0 2 4𝜋𝜋𝜖𝜖 0 1 + 2 ⁄ 2 2𝜖𝜖 0 2 ⁄ 2 ⁄
) || 2  −2  is the azimuthal angle in the -space.The screening modifies the exchange interaction through the change in the electron-hole wavefunction overlap |∑ ()  | 2  ⁄ , bandgap   and the Fourier potential ().In Figure 3e of the main text, the magnitude of the exchange interaction potential is plotted as a function of || within the light cone.During the calculation, we fit the parameters such that the  2 −  1 separation remains the same as that obtained experimentally. is the exciton lifetime.For W  ′ , we take the exciton-impurity scattering within the light cone as the only scattering mechanism at 5 K, neglecting exciton-phonon scattering.For simplicity, to get the exciton-impurity scattering matrix element, we use a similar form of the perturbing potential expression given for electron-ionized impurity scattering.We use the generic screened coulomb potential form as 7  0     −   ⁄  0 is the dielectric permittivity of vacuum and   is the Debye length.The matrix element for the above perturbing potential for an exciton scattering from  to  ′ state is given by   ′  =   � �  (− ′ ).  −   ⁄   Putting  ′ −  =  (note that, | ′ | = || since Coulomb scattering is elastic in nature), we obtain  ′  = �  0    � �  − cos   −   ⁄  The above expression after integration over  results in  ′  =  2   2 cos 2  �The result of the first integration is 2 �1 +  2   and the second one is zero.The matrix element thus becomes   ′  =  0    � 2 + 1   We assume the limiting case of   → ∞ and obtain the following expression for the scattering rate W  ′  as W  ′  ∝ �  ′  � 2 ,  is the angle between the initial () and the final ( ′ ) exciton state.The final form of the steady state MSS equation after plugging in the above form of scattering rate W  ′  as given in the main text is given as:  = 1  () − () × () − �   2 sin 2  2 [( ′ ) − ()]  ′ ∥ cos  , Ω ∥ sin , Ω 0 ].Similarly, decomposing the density matrix in terms of its trace  and the traceless part ., where  denotes the Pauli matrices, the commutator + � W  ′ ( ( ′ , ). − (, ).)  ′ − 1  ((, ).) + . where  = [      ] is the generation rate vector.By expanding the commutation bracket and using the vector identity (.)(.) = (.)I + ( × )., we get the final form of the MSS equation as: ()  = () × () + � W  ′ [( ′ ) − ()]