Impact of indirect transitions on valley polarization in WS2 and WSe2

Controlling the momentum of carriers in semiconductors, known as valley polarization, is a new resource for optoelectronics and information technologies. Materials exhibiting high polarization are needed for valley-based devices. Few-layer WS2 shows a remarkable spin-valley polarization above 90%, even at room temperature. In stark contrast, polarization is absent for few-layer WSe2 despite the expected material similarities. Here, we explain the origin of valley polarization in both materials based on the interplay between two indirect optical transitions. We show that the relative energy minima at the Λ- and K-valleys in the conduction band determine the spin-valley polarization of the direct K–K transition. Polarization appears as the energy of the K-valley rises above the Λ-valley as a function of temperature and number of layers. Our results advance the understanding of the high spin-valley polarization in WS2. This insight will impact the design of both passive and tunable valleytronic devices operating at room temperature.


. Resonant excitation of bilayer WSe2
Throughout our study, we report polarization values measured with excitation at a constant photon energy of 2.040 eV, which is close to resonance for bilayer WSe2 . Here, we check that bilayer WSe2 does not show spin-valley polarization either under resonant excitation conditions (Supplementary Figure S1). We used an excitation energy of 1.681 eV, which is close to the WSe2 bilayer K-K exciton emission around 1.62 eV.
Supplementary Figure S1. a, Polarization-resolved PL spectrum for bilayer WSe2 under nearresonant excitation (1.681 eV) at room temperature. b, Degree of circular polarization as a function of photon energy. No DOCP is measurable.

Thickness-dependent polarization
We determine the thickness of our WS2 and WSe2 samples by using a combination of reflection contrast microscopy, atomic force microscopy, and photoluminescence measurements. Similarly, for both WS2 and WSe2, the monolayers show bright emission due to their direct band gap. At room temperature, their emission spectrum shows a single peak. When increasing the thickness, a second peak emerges, which shifts to lower energy with an increasing layer thickness (Supplementary Figure S2). Only WS2 shows an increase in the DOCP with thickness, whereas in Supplementary Figure S4. Polarization-resolved PL spectra at room temperature for different thicknesses of WSe2 excited with 1.796 eV. a, Thickness dependent spectra. b, Thickness vs the DOCP and the direct-indirect energy difference for the spectra in a.

Bilayer photoluminescence spectra
When the temperature decreases, the two photoluminescence peaks shift with temperature (Supplementary Figure S4). In bilayer WS2, the polarization also increases. However, in bilayer WSe2, polarization only appears below T = 160 K (Supplementary Figure S5). Figure S5. Polarization-resolved PL spectra at different temperatures for bilayer samples. a, WS2. b, WSe2. Spectra are vertically shifted by a constant for clarity. Table S1. Fitting parameters obtained using Equation 2 in the main text in Figure 3. We contained the factor of 2 in the denominator of Equation 2 in the fitting parameter, c.

Fitting using the O'Donnel equation
We fit the peak position as a function of temperature using two equations. Fitting using the Varshni equation1 was presented in the main text. Here, we fit the peak position using the O'Donnell where T is the temperature, (0) is the excitonic band gap, S is the Huang-Rhys factor, 〈ћ 〉, is an average phonon energy, and is the Boltzmann constant. The obtained fitting parameters are listed in Supplementary Table S2  with the experimental data in Figure 3a-b.

Evidence of a dark ground state in bilayer WSe2
In W-based monolayers, the dark excitons lie lower in energy than the bright excitons and transitions between the lowest conduction band and the top valence band at K is spin-forbidden (dark K-K exciton) due to spin splitting 2 . As evidence for bright-dark excitons in bilayer WSe2, we observe a decrease of the K-K intensity with decreasing temperature consistent with reduced thermalization from dark to bright excitons 2-4 (Supplementary Figure S6). We fit the measured integrated PL intensity as a function of temperature to the expression ( )/ (0) − 1 = (− / ), where IPL(T) is the measured intensity as a function of temperature, IPL(0) is the intensity at T = 0 K, C is a constant, kB is the Boltzmann constant, and is the characteristic energy barrier that defines the slope of the emission. From the fit, we obtain = 37.9 meV, which is in good agreement with the bright-dark exciton splitting in monolayer WSe2 5 . We expect a similar value for bilayer WSe2 due to the limited effect of layer-layer interactions on the band structure near the K-point of the Brillouin zone. Supplementary Figure S6. Spectrally integrated PL of the A exciton emission as a function of temperature for both circular polarizations when excited with a 2.04 eV laser. The drop in emission intensity with temperature is consistent with a dark exciton ground state. The fitting is described in the text.

Temperature-dependent polarization with varying thickness in WSe2
For a fixed temperature, if we increase the WSe2 thickness to three or four layers, the K-Λ conduction band difference should become smaller. Similarly, the onset of an increase in DOCP should occur at a higher temperature compared to a bilayer. We confirm this trend by measuring the emission DOCP for three and four layers of WSe2 and by comparing it as a function of temperature to that of a bilayer (Supplementary Figure S7).
Supplementary Figure S7. Temperature-dependent DOCP measurements for 1, 2, 3, and 4 layers of WSe2 showing an increase in the onset temperature of DOCP with increasing layer thickness. The 2, 3, and 4 layer data was acquired using 2.04 eV excitation. The monolayer data was acquired using 1.796 eV excitation. The fits are made by assuming a Boltzmann distribution for the K-K' intervalley scattering, see details in the main text.