Evidence for line width and carrier screening effects on excitonic valley relaxation in 2D semiconductors

Monolayers of transition metal dichalcogenides (TMDC) have recently emerged as excellent platforms for exploiting new physics and applications relying on electronic valley degrees of freedom in two-dimensional (2D) systems. Here, we demonstrate that Coulomb screening by 2D carriers plays a critical role in excitonic valley pseudospin relaxation processes in naturally carrier-doped WSe2 monolayers (1L-WSe2). The exciton valley relaxation times were examined using polarization- and time-resolved photoluminescence spectroscopy at temperatures ranging from 10 to 160 K. We show that the temperature-dependent exciton valley relaxation times in 1L-WSe2 under various exciton and carrier densities can be understood using a unified framework of intervalley exciton scattering via momentum-dependent long-range electron–hole exchange interactions screened by 2D carriers that depend on the carrier density and the exciton linewidth. Moreover, the developed framework was successfully applied to engineer the valley polarization of excitons in 1L-WSe2. These findings may facilitate the development of TMDC-based opto-valleytronic devices.


Supplementary Note 1: Time-integrated valley polarization of excitons
Steady-state exciton valley polarization r x has been deduced in previous literatures 1-3 from standard rate equation models (in which only bright excitons are considered) as, where ρ 0 , t v are the initial valley polarization and the valley relaxation time for the bright excitons, respectively. t x is the population lifetime of the bright excitons; g r times t x corresponds to a PL quantum yield, where g r is the radiative decay rate of the bright exciton.
Here, we consider the steady-state exciton valley polarization for the system with multiple exciton levels including bright and dark states as shown in Fig. 1 in the main text. G b and G d are the relaxation rates of the bright and dark excitons to the ground state, respectively.
For simplicity, G b and G d are assumed to be independent of temperature. g is a phonon scattering rate constant, n is a phonon number n º [exp(E ph /k B T) -1] -1 , where E ph is the phonon energy that corresponds to the energy splitting of the bright and dark states, D bd . G + and Gare effective generation rates of bright excitons in +K and -K valleys, respectively, and g s is the intervalley scattering rate of bright excitons (holes), considered to be dominated by the electron-hole exchange interactions.
In regards to the possible origin of the lower energy dark states, both intervalley dark excitons (k ≠ 0) with electron-spin conservation or intravalley dark excitons (k ≈ 0) with spin flip can be considered [4][5][6][7][8][9] . Here we do not attempt to provide a rigorous discussion as to where the lower energy dark state originates from, but the generation of intervalley dark excitons with spin conservation is expected to be dominant, because of the availability of relevant phonons 5 .
Intravalley phonon-assisted scattering with electron spin-flip could also be considered, but this process has been predicted to be mitigated for the monolayer sample supported on a substrate 10 .
In Fig. 1, we thus tentatively assumed that the electron-spin-conserving processes are dominant such intervalley spin-flip process is forbidden by the selection rules at the lowest-order 10 . Thus, under low temperature conditions focused in this study, contribution of the intervalley scattering of excitons with hole spin-flip due to zone-edge phonon scattering is expected to be smaller than that of the electron-hole exchange interactions that should be effective at the lowest-order, and we neglected this process in our theoretical treatment as a first approximation. Under relatively high temperature conditions (T > ~ 200 K), however, multiple intervalley scattering processes through thermal activation 4 or carrier-carrier scattering 2 could also be efficient, and the electron-hole exchange interaction may not be the unique mechanism for the exciton (hole) valley relaxation.
The effects of the exciton scattering between the bright and dark states are taken into account through the consideration of phonon-assisted scattering between the bright and dark states by the use of a finite scattering rate constant, g. The rate equations corresponding to the scattering processes shown in Fig. 1(b) are below: where N b± and N d± are the numbers of bright and dark excitons in which holes composing the excitons are at the +K and −K valleys in the electronic Brillouin zone, respectively. The above formulation eventually does not distinguish the spin configuration of the lower lying dark excitons as far as the holes are in the same valley. Therefore, the scattering rate constant g is defined as an effective quantity, and the analytical result is not affected by whether the scattering processes between the bright exciton and the intravalley dark excitons (not indicated in Fig. 1) are included or not.
Then, under the steady-state condition, we obtain the following two equations: where we define a quantity, át x ñ, by the following equation: Using Supplementary Eqs. (6)(7)(8), the valley polarization of the excitons r x can be derived as Further defining r 0 º (G + − G − ) / (G + + G − ) and t v º (2 g s ) −1 , the expression of the valley polarization in the main text (Eq. (1)) can be obtained as the following equation: Supplementary Eq. (6) suggests the physical meaning of the quantity, át x ñ.
Modification of Supplementary Eq. (6) gives the following relationship: This expression clearly indicates that át x ñ is the total number of bright excitons divided by the total exciton generation rate, in other words, the total (integrated) time in which an exciton can be in its bright state before it recombines radiatively or nonradiatively. Thus, g r times át x ñ corresponds to the PL quantum yield of a bright exciton (k ≈ 0) generated in the ±K valley in the electronic Brillouin zone, where g r is the radiative decay rate of the bright excitons. In this study, we evaluated this quantity using a photoluminescence (PL) decay profile I(t), because I(t) yields a quantity, as described below: where I(0) is the PL intensity at t = 0. The numerator in Supplementary Eq.

Supplementary Note 5: Line width analysis
We evaluate the homogeneous line widths, G h , of excitons using a fitting procedure with Voigt functions (which are a convolution of Lorentzian and Gaussian functions). Supplementary Figure 4 shows the PL spectra that were decomposed by the Voigt fit at four representative temperatures; 10 K, 40 K, 80 K, and 160 K. The black circles are the experimental data and the green curves are the data reproduced by the peak fit. We considered the peak features for  Fig. 2 and Fig. 4 in the main text and in Supplementary Fig. 1, and 2.0 meV for the spectra in Supplementary Fig. 7). The temperature-independent Gaussian width in the range of ~11-14 meV was found to yield reasonable fits for the data shown in Fig.   2 at all of the temperatures between 10-160 K, as shown in Supplementary Fig. 4. The Lorentzian widths determined for the lower (upper) limit of the constant Gaussian width of 10.8 meV (14.3 meV) were considered to be the upper (lower) limit of the Lorentzian widths at each temperature in the analyses. Similar procedure was also used for the line width analysis on the data shown in Fig. 4 (Supplementary Fig. 7), for which the constant Gaussian width of ~10-12 meV (~9 meV) was found to yield reasonable fit results. Here we compare the temperature dependences of the át x ñ and the integrated intensity of bright exciton emission I x at low temperature conditions (Supplementary Figure 6) to confirm that these quantities show the expected relation of I x µ g r át x ñ, where g r is the radiative decay rate.

Supplementary
Under the conditions in which G h > k B T is fulfilled, g r in two-dimensional systems has been predicted to follow g r µ [1 -exp(-G h /k B T)] / G h º r(T) 23,24 . Thus, I x µ r(T)át x ñ is expected. This expression was derived by considering the effects of intraband thermalization of excitons, exciton coherence length limited by collisions (that also yield G h ), and the corresponding uncertainty in the exciton momentum that determines the range of exciton states that can contribute to the optical transitions 23,24 . We note that it becomes r(T) µ 1/T in the coherent limit, G h << k B T, which is commonly fulfilled in the case of conventional semiconductor quantum wells (in the limit of G h ® 0, exciton states only within the light cone can contribute to the optical transitions). As expected, the temperature dependent variation of the I x at low temperatures is in good agreement with that of r(T)át x ñ; this confirms the consistency between the quantities át x ñ and I x obtained in the experiments. Figure 6 | át x ñ, I x , and r(T)át x ñ plotted as functions of temperature.

Supplementary
Inset shows the r(T) as a function of temperature. The mean values of G h in the reasonable range shown in Fig. 3(c) in the main text were used to calculate r(T).

Supplementary Note 8: t v under the condition without carrier screening effect
Here we evaluate the t v for 1L-WSe 2 under the condition with no carrier screening 25 ; t v determined by the unscreened e-h exchange interaction strength in the range of J ~ 0.5 -1 eV is predicted to be t v = M/AћJ 2 ~ 2.5 -10 fs and independent of the temperature 25 . Thus, t v determined by the unscreened e-h exchange interaction is much shorter than the value calculated with the carrier screening effect, and unrealistic considering the observable valley polarization of the excitons for át x ñ more than 10 ps (t v ~ 10 fs and át x ñ ~ 10 ps result in negligible valley polarization r x ~ 10 -3 according to Eq. (1)), which clearly contradict the experimental observations.

Supplementary Note 9: Evaluation of the naturally doped carrier density
The emergence of the negative trion peaks in the PL spectra shown in Fig. 1 indicates that 1L-WSe 2 was in a naturally electron-doped condition as discussed in the main text. Here we provide an estimation of the naturally doped electron density from the ratio of the observed trion and exciton PL intensities at 160 K at which thermal equilibrium condition is approximately fulfilled. According to the mass action law [26][27][28] , electron density, n c , is related to the exciton (N x ) and trion (N x-) populations in the equilibrium conditions as where m e , M x (~2m e ), M x-(~3m e ) are electron, exciton, and trion masses, respectively, and E bx-is the trion binding energy. The relation between the exciton (I x ) and trion (I x-) intensities and their populations N x and N x-are expressed as where g x and g x-are the effective radiate decay rate of excitons and trions, respectively. Using Supplementary Eqs. (13) and (14), n c is evaluated as Since C º k TF0 2 /AJ 2 » g s 2 g v 2 e 4 /144a 2 e 2 J 2 , the constant C in Eq. (3) and Eq. (4) does not explicitly depend on the exciton (or carrier) effective mass, and should scale as C µ a -2 e -2 J -2 . Considering J » 8p 2 aE b t 2 /3a B E g 2 , the approximate scaling of C is predicted as For the change of the carrier density from ~2 ´ 10 12 to ~3 ´ 10 12 cm -2 , reduction of E b and E g on the order of 1 % and 0.1 %, were observed 34 , respectively. It is roughly estimated that the 1% reduction of the E b leads to the reduction of the width of the exciton wave function in the k space on the order of 0.5% for the parabolic dispersion relation (because ∆k/k ≈ (1/2)∆E/E for small ∆E/E). This may cause about 0.5% increase of the a B through the uncertainty relation.
Thus, the change in the C parameter by the carrier density increase from ~2 ´ 10 12 to ~3 ´ 10 12 cm -2 is roughly estimated to be by a factor of (1.005) 4  thus the difference between the graphene and the quartz substrates should be smaller than that between the graphene and vacuum.

Supplementary Note 11: Linewidth dependence of the exciton valley polarization
In the discussions of the main text, we demonstrated that our framework based on the and high (2.5 kW cm -2 ) excitation power densities at 10 K. This sample showed a narrower line width than that of 1L-WSe 2 shown in Fig. 2 and the exciton peak was well separated from other PL features; this allowed us to evaluate the small change in the exciton line width with better accuracy. As shown in Supplementary Fig. 7(a), the exciton peak position was almost unchanged within the power density range examined in this study; this suggests that the sample temperature was not increased even under the highest excitation power density employed in the measurements. Unfortunately, the effective lifetime át x ñ for this sample could not be obtained because it was faster than the detection limit. Instead, we observed excitation power dependence of the integrated PL intensity as shown in Supplementary Fig. 7(b). The integrated intensity showed linear increase as the excitation power increased. This suggests that át x ñ was almost constant regardless of the excitation power.
Supplementary Fig. 7(c) compares the PL line shapes at low (0.4 kW cm -2 ) and high (2.5 kW cm -2 ) power excitation conditions. The spectrum under the high density condition clearly exhibited line width broadening. Supplementary Fig. 7(d) plots the r x and G h as functions of the excitation power density. Clear anti-correlation between these quantities was observed. G h was increased as the power density increased, while r x decreased. Using Eq. (4) at the low temperature limit with the coefficients r 0 = 0.7 and C = 118 (≈t v (0)G h (0)/ħ), the observed reduction in r x could be excellently reproduced from the power density dependent variation in G h with át x ñ ≈ 10 ps as a fit parameter. Thus, within the excitation densities tested in this study, the assumption of the constant r 0 regardless of the excitation density seems to be consistent. However, we note that there still remains a possibility that the r 0 depends on the exciton density when it is much higher than those examined in this study.