Microsecond dark-exciton valley polarization memory in two-dimensional heterostructures

Transition metal dichalcogenides have valley degree of freedom, which features optical selection rule and spin-valley locking, making them promising for valleytronics devices and quantum computation. For either application, a long valley polarization lifetime is crucial. Previous results showed that it is around picosecond in monolayer excitons, nanosecond for local excitons and tens of nanosecond for interlayer excitons. Here we show that the dark excitons in two-dimensional heterostructures provide a microsecond valley polarization memory thanks to the magnetic field induced suppression of valley mixing. The lifetime of the dark excitons shows magnetic field and temperature dependence. The long lifetime and valley polarization lifetime of the dark exciton in two-dimensional heterostructures make them promising for long-distance exciton transport and macroscopic quantum state generations.

for σ + excitation at out-of-plane magnetic field B z =+3 T, 0 T and -3 T. Here P j = indicates the excitation polarization and I j σ+ (I j σ− ) is the σ + (σ − ) polarized PL intensity when excitation with j polarization is used. b, Valley polarization P val = P σ + −P σ − 2 at out-of-plane magnetic field B z =+3 T, 0 T and -3 T. c, PL polarization P PL = P σ + +P σ − 2 at out-of-plane magnetic field B z =+3 T, 0 T and -3 T. d, The degree of polarization as a function of B z with pulsed laser excitation. The intensity is integrated from 21 ns to 7 µs. e, The PL polarization as a function of B z with pulsed laser excitation. The PL polarization shows linear dependence on magnetic field for small magnetic field before saturating at big magnetic field. N xx (N xx ) corresponds to the population of the additional exciton level at K (K') valley to account for the power decay with the decay rate equal to k 0 N xx for K valley and k 0 N xx for K' valley. The term k 3 is the intervalley scattering rate, k 1 (k 1 ) is the dark-to-interlayer scattering rate and k 2 (k 2 ) is the interlayer exciton decay rate. The term B and t denote the magnetic field and time dependence of the parameters. b, The ratio between the dark exciton population at K and K' valley at different excitation polarization. The population ratio shows exponential dependence on magnetic field.
Photon energy (eV)  Additionally, we can integrate the PL signal to get the total emission from each valley and plot the degree of polarization and PL polarization of this total emission as a function of out-ofplane magnetic field. This is shown in Supplementary Figure 2d and 2e respectively. As shown in Supplementary Figure 2d, the degree of polarization becomes larger when we apply a small magnetic field. This is similar with the CW excitation case. However, it becomes asymmetric for larger magnetic field. It increases to a saturation level for one magnetic field direction while it does not show saturation behavior for the opposite direction. This non-saturation behavior is mainly caused by the increased PL polarization, P P L , in the long timescale at a large magnetic field. The plot of the P P L against out-of-plane magnetic field is shown in Supplementary Figure 2e.
The energy degeneracy break down of the dark excitons in different valley is the primary reason for the magnetic field dependence of the observed PL polarization. At a finite magnetic field, the population ratio between these energy levels at thermal equilibrium is determined by the Boltzmann distribution. Because of this, the PL polarization will have a near-linear magnetic field dependence at small magnetic field and saturate to the value of 1 at big magnetic field. However, due to the valley polarization, the magnitude of the PL polarization will be less than 1. Following this line of reasoning, the PL polarization is fitted using equation with g v as the effective g-factor between K and K' valley, P s PL is the saturation level of the PL polarization, and P 0 PL is a small residual PL polarization at 0 T which is most probably caused by where k ± = , are the population of K (K') valley's dark exciton and interlayer exciton. The term N xx (N xx ) corresponds to the population of the additional exciton level at K (K') valley to account for the power decay with decay rate equal to k 0 N xx for K valley and k 0 N xx for K' valley. k 3 is the intervalley scattering rate, k 1 (k 1 ) is the dark-to-interlayer scattering rate and k 2 (k 2 ) is the interlayer exciton decay rate.
All c i terms are constants that depend on the initial condition.
It is possible to fit these equations straight away to the experimental data. However, due to the large number of the independent variables, there is a possibility that multiple local optimum solutions exist with some of the solution is not physically reliable. A different strategy can be employed by using the fact that for magnetic field with big enough magnitude the value of k 3 is negligible. In our case, we assume k 3 = 0 MHz for |B| = 0.8 T, which is reasonable given the saturation of valley polarization happens at |B| > 0.8 T. In this case the interlayer population N x (N x ) and the dark exciton population N d (N d ) can be approximated as As can be seen from Supplementary Equation 7-10, for big magnetic field, there is no coupling between the exciton population in different valley. Hence, in this case, the model is equivalent to a time-dependent function with two exponentials and one power decay as stated in the main text. Moreover, due to the absence of the coupling terms, the experimental PL data for different emission polarization can be fitted separately.
The fitting result at big magnetic field can be used to check the sanity of the model. The model predicts that the value of various decay rates (k 1 , k 1 , k 2 , k 2 , k 0 and k 0 ) and the population ratio between the interlayer exciton and dark exciton in one valley should be independent of the excitation polarization. We found that this is the case for all of these parameters other than the small difference for power decay rate (k 0 and k 0 ).
From the fitting result at big magnetic field, we extract the magnetic field dependence of . From these data, the values of these parameters at small magnetic field are interpolated by using linear interpolation. The value of k 0 , k 0 , N xx , and The ratio between the dark exciton population in K and K' valley obtained from this fitting (see Supplementary Figure 3b) can be used to get the dark exciton g-factor. Magnetic field will lift the energy degeneracy of dark excitons and cause the population difference between the two valleys. Based on this fitting, we find this g factor to be equal to 0.9 ± 0.04. This is quite close to the value of effective g-factor between K and K' (see Supplementary Note 1), which indicates that dark exciton dynamics dominates the total PL polarization.
We note that our rate equation model is similar to the rate equation model in [1] where it is used very well to explain the increase of valley polarization with increasing magnetic field in WSe 2 local exciton case. However, we found that this model has to be extended in order to explain our finding. Unlike in [1] where the scattering rate between the dark exciton and the local exciton is treated as a constant, in our case the scattering rate between the dark exciton and interlayer exciton (k 1 and k 1 ) depends on the magnetic field and the valley.

Supplementary Note 3. TEMPERATURE DEPENDENCE OF DARK-TO-INTERLAYER EXCITON SCATTERING RATE
In order to get the value of the energy level difference between bright and dark exciton at zero magnetic field (∆E 0 ), the pulse measurement were done in various temperatures at two different magnetic field settings: B = 0 T and B = 7 T. These data is then fitted with the theoretical model to obtain the value of dark-to-interlayer exciton scattering rate (k 1 ).
The measurement results for the case where σ − polarized pulsed excitation and σ − polarized PL detection are used can be seen in Supplementary Figure 4a (for B = 0 T) and 4c (for B = 7 T).
The corresponding temperature dependence of k 1 at two different magnetic field settings is shown in Supplementary Figure 4b (for B = 0 T) and 4d (for B = 7 T). Based on this fitting, we obtain ∆E 0 = 58.2 ± 20 meV. Considering the experimental uncertainty, this result is comparable to the value of this parameter reported in [2] and [3].

Supplementary Note 4. STACKING SEQUENCE OF THE SAMPLE
It is possible to determine the stacking sequence of the bilayer heterostructure using second harmonic generation (SHG) measurement [4]. If the two monolayers are 0-degree aligned (AA stacking), the SHG emission from the two monolayers will add together and therefore the SHG signal in the heterostructure area should be larger compared to the monolayer SHG signal. For 60-degree alignment (AB stacking), the SHG signal in the heterostructure is lower than that in monolayers because of destructive interference between the SHG signals from different layer. We have implemented this SHG measurement. The angle dependent SHG for WSe 2 and MoSe 2 and the SHG for WSe 2 /MoSe 2 heterostructure are shown in Supplementary Figure 5a, 5b, and 5c respectively. We found that the SHG signal from the heterostructure is larger than that from the individual MoSe 2 and WSe 2 monolayer. Based on this measurement result, we conclude that the stacking sequence of our heterostructure is AA stacking. In this section we presents the measurement result of one additional MoSe 2 /WSe 2 sample (see Supplementary Figure 6). In all of our experiments, we observed that applying an out-of-plane