Abstract
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 twodimensional (2D) systems. Here, we demonstrate that Coulomb screening by 2D carriers plays a critical role in excitonic valley pseudospin relaxation processes in naturally carrierdoped WSe_{2} monolayers (1LWSe_{2}). The exciton valley relaxation times were examined using polarization and timeresolved photoluminescence spectroscopy at temperatures ranging from 10 to 160 K. We show that the temperaturedependent exciton valley relaxation times in 1LWSe_{2} under various exciton and carrier densities can be understood using a unified framework of intervalley exciton scattering via momentumdependent longrange 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 1LWSe_{2}. These findings may facilitate the development of TMDCbased optovalleytronic devices.
Introduction
A “valley” is an electronic degree of freedom in momentum space, and its potential applications as information carriers in future electronics or optoelectronics devices are called valleytronics^{1,2,3,4}. Monolayers of transition metal dichalcogenides (1LTMDCs) MX_{2} (M = Mo, W; X = S, Se, etc.) have recently emerged as promising twodimensional (2D) materials for developing valleytronics because they have hexagonal lattice structures similar to that of graphene but are semiconductors with finite direct band gaps in two inequivalent +K and −K valleys related by a timereversal operation in the 2D hexagonal Brillouin zone^{5,6}. Because of the reduced screening resulting from the atomically thin 2D structures of 1LTMDCs, their excitons, which are mutually attracting electron–hole pairs that interact through Coulomb interactions, have extremely large binding energies and dominate their optical responses even at room temperature^{7,8,9,10}. In addition, the strong spin–orbit interactions in these materials give rise to large spin splitting, reaching ~450 meV (~150 meV) for WX_{2} (MoX_{2}) in the valence band^{11}. This large valence spin splitting and a lack of inversion symmetry in these materials lead to spin–valley coupling that enables exclusive access to the excitonic valley pseudospins (+K〉 or −K〉) with right or leftcircularly polarized photons^{12,13,14,15,16}. These unique characteristics of 1LTMDCs have provided unprecedented platforms for the study of valleyexciton physics in 2D systems, as well as offering opportunities for developing future optoelectronic devices using the excitonic valley degrees of freedom^{4,12,13,14,15,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33}.
One of the most intriguing questions in valleyexciton physics is in regard to the relaxation mechanism of excitonic valley pseudospins in 1LTMDCs^{4,12,13,14,15,17,18,19,20,21,22,23,28,30,31,32,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48}. With regard to the theory on the subject, exciton (or hole) valley relaxations dominated by electron–hole (e–h) exchange interactions^{14,19,20,32,38,39}, as well as phononassisted intervalley scattering mechanisms^{13,17,37}, have been proposed and intensively studied. The importance of the Coulomb screening effect by naturally or intentionally doped carriers in 1LTMDCs has also been examined theoretically^{32}, although it has not yet been addressed experimentally. With regard to investigating the question experimentally, time and polarizationresolved transient absorption^{19,22}, reflection^{35}, photoluminescence (PL)^{28,41}, and Faraday^{43} and Kerr rotation (TRKR)^{38,44,46,47,49,50,51} techniques have been used to measure the depolarization (life) times of the spin or exciton (hole) valley states in 1LWSe_{2}^{38,46,47,50,52}, 1LWS_{2}^{22,49}, and 1LMoS_{2}^{19,35,43,44,51}, and spin or valley relaxation times ranging from several to 10 ps^{38,46,47} to tens of nanoseconds^{50} have been reported at cryogenic temperatures. The considerable variations in the reported spin or valley relaxation times have been discussed as being possible consequences of either the differences between the physical quantities observed by each measurement technique^{49}, the existence of dark exciton states^{49}, the spin/valley polarization of resident holes^{50,52}, ultrafast Coulombinduced intervalley coupling^{45}, the slow decay of localized states^{47}, or the high excitation density typically needed for pump–probetype experiments using femtosecond lasers (typically >~10^{12} cm^{−2} within the initial 100–200 fs)^{28,46,49}. Because of all or some of these complications, a comprehensive and unifying explanation for temperaturedependent exciton valley relaxation times that is applicable for various exciton and carrier density conditions has yet to be provided.
In the present paper, we provide experimental evidence that the excitonic valley relaxation times and their temperature dependence in naturally carrierdoped 1LTMDCs are dominated by momentumdependent e–h exchange interactions screened by a 2D electron gas. We measured the valley polarization, linewidth, and timedependent PL decay profile of excitons in 1LWSe_{2} to examine the valley relaxation times of an exciton at temperatures ranging from 10 to 160 K in the linear response regime. We show that the lowtemperature exciton valley relaxation times in 1LWSe_{2} can be excellently reproduced using a framework of an intervalley exciton scattering mechanism via momentumdependent longrange e–h exchange interactions screened by naturally doped 2D carriers. The temperature dependence of the Coulomb screening function deduced from the experimental data clearly demonstrates the 2D nature of the doped carriers; this is a unique manifestation of the true 2D structure of 1LTMDCs. Moreover, we demonstrate that the developed framework can be used to predict the valley relaxation times and polarizations under various experimental conditions with various exciton linewidths, exciton lifetimes, carrier densities, and exciton densities. We also demonstrate that the valley polarization in 1LWSe_{2} can be actually engineered via artificially modifying these parameters according to the theoretical prediction. Our findings therefore provide a unified framework through which the temperaturedependent exciton valley relaxation times and valley polarization in 1LTMDCs can be understood and predicted; this framework may facilitate the development of TMDCbased optovalleytronic devices.
Results
Measurements of exciton valley polarization
Figure 1a shows a schematic of the excitation of optically allowed (bright) excitons with the centerofmass wave vector k ≈ 0 in the excitonic Brillouin zone and the valley pseudospin +K〉 (for which both electron and hole are in the +K valley in the electronic Brillouin zone), initial ultrafast decays partially branching into bright excitons with the valley pseudospin +K〉 or −K〉 (for which both electron and hole are in the −K valley in the electronic Brillouin zone), the intervalley scattering between the +K〉 and −K〉 states causing exciton valley relaxation, circularly polarized emission processes from the +K〉 and −K〉 bright excitons, and the electron spinconserving scattering between the bright (k ≈ 0) and dark (k ≠ 0) exciton states in 1LWSe_{2}. The dark excitons with k ≠ 0 indicated by gray solid curves in Fig. 1 are optically forbidden because of the momentum mismatch with photons. Symbols σ_{+} and σ_{–} denote the circular polarizations, and \(I_{\sigma + }\) and \(I_{\sigma  }\) represent the corresponding PL intensities. Figure 1b shows the scattering processes and symbols for the relevant rates considered in this study. The valley relaxation time (τ_{v}) of the bright exciton is related to its intervalley scattering rate (γ_{s}) as τ_{v} = (2γ_{s})^{−1}. G (=G_{+} + G_{−}) is the generation rate of the bright excitons. Parameters G_{+} and G_{−} are the effective generation rate of the bright excitons with +K〉 and −K〉 pseudospins after the initial branching within the ultrafast timescale, respectively. We define ρ_{0} ≡(G_{+} − G_{−})/(G_{+} + G_{−}) to express the valley polarization loss in the initial ultrafast nonthermal timescale. In the ideal case under σ_{+} excitation, such as the resonant excitation condition to the bright exciton state, ρ_{0} should be equal to 1 because of the optical selection rule^{12,13,14,15}. However, ρ_{0} could be less than 1 in the event of an initial depolarization process due to defectinduced intervalley generation^{35}, ultrafast valley relaxation of hot excitons during their intraband cooling processes^{39}, and/or excitation to other spindegenerate valleys^{53}, potentially through phononassisted indirect absorption processes^{37}. In Fig. 1, the exciton scattering processes between the bright and intervalley dark states by phononassisted spinconserving electron scattering processes^{54} are tentatively indicated as a major bright–dark transition pathway. In this study, however, we do not attempt to distinguish the microscopic origin of the lowerlying dark states (intervalley spinallowed or intravalley spinforbidden) for the analysis under the condition that the hole’s valley is conserved. Recent experimental studies have revealed that the energy difference between the bright and dark states (Δ_{bd}) in 1LWSe_{2} is ~30–47 meV^{54,55,56,57}. For the valley depolarization of holes (excitons), we consider the scattering between the singletlike bright (k ≈ 0) excitons with +K〉 or −K〉 valley pseudospins via longrange e–h exchange interactions^{32,39} to be the dominant valley depolarization mechanism. Because this mechanism is effective only between the two bright states with opposite valley pseudospins, the hole valley relaxation of dark excitons is neglected as a first approximation in this study (see Supplementary Note 1 for additional details).
Figure 2a shows the polarizationdependent PL spectra of 1LWSe_{2} on a quartz substrate under circularly polarized σ_{+} excitation conditions at 10–160 K. At low temperatures, an exciton (Aexciton) peak (X, at 1.740 eV)^{34,58} and a charged exciton (trion) peak (T, at 1.705 eV)^{34} were observed. The emergence of the trion peaks and their peak positions indicate that 1LWSe_{2} was in a naturally electrondoped condition^{34,59}. The difference between the spectral integrated intensities of σ_{+} (I_{σ}_{+}) and σ_{–} (I_{σ}_{−}) exciton PL (valley polarization) was clearly observed at low temperatures. Under the excitation by σ_{+} photons, the valley polarization, ρ_{x}, is calculated as ρ_{x} = (I_{σ}_{+} − I_{σ}−)/(I_{σ}_{+} + I_{σ}_{−}). As shown in Fig. 3a, ρ_{x} values were nearly constant under the lowtemperature conditions (T < ~40 K) and gradually decreased with increasing temperature for T > ~60 K. Similar temperaturedependent behaviors of ρ_{x} have been reported for 1LMoS_{2}^{13} and 1LWSe_{2} (for T ≥ 70 K)^{31}.
The excitation photon energy dependences of ρ_{x} at various temperatures were also examined; the results are shown in Fig. 3b. Relatively high values of ρ_{x} were observed at the nearresonant excitation conditions with the 2s excited exciton state^{8,9} lying at an energy ~0.14 eV higher than the 1s excitons at 1.74 eV, whereas variations in ρ_{x} were small under the nonresonant conditions (corresponding to excitation photon energies of ~1.79–1.84 eV and 1.89–1.90 eV) within the measured range from 1.79 to 1.90 eV. Because we employed nonresonant excitation conditions (the excitation photon energy was 1.893 eV) for evaluating the temperature dependence of ρ_{x} shown in Fig. 3a, we ruled out the possibility that the observed temperature dependence in ρ_{x} was determined by the temperaturedependent exciton energy shift^{60}, which had previously been discussed as being the case in 1LMoS_{2}^{14,40}. The excitation photon energy dependence of ρ_{x} at low temperature is discussed in greater detail in Supplementary Note 2 and Supplementary Figure 1.
To deduce the valley polarization loss in the initial ultrafast nonthermal timescale (the value of ρ_{0}) for the following analysis, we also measured polarizationresolved PL decay profiles of the excitons in 1LWSe_{2} on a quartz substrate. Figure 2b shows the polarizationresolved PL decay profiles [I_{σ}_{+} (red curve) and I_{σ}− (black curve)] and the timeresolved valley polarization, ρ_{x}(t), (green circles) measured at 45 K under the σ_{+} excitation condition. By extrapolating ρ_{x}(t) to the onset time (t ≈ 0 ps) of the PL decay signals (dotted line), the lower limit of the maximum valley polarization of ρ_{0} ≡ ρ_{x}(0) ≈ 0.7 ± 0.1 is deduced. Since the similar value was also deduced at 120 K (see Supplementary Note 3 and Supplementary Figure 2 for additional details), temperaturedependent variation of the ρ_{0} parameter is considered to be small in the observed temperature range, and will be neglected in the following analyses (See also Supplementary Note 4 and Supplementary Figure 3 for the effects of the uncertainty in the ρ_{0} on the following analysis).
Temperature variations in exciton linewidth
Information about the temperaturedependent variations of the exciton homogeneous linewidths Γ_{h} were deduced from the PL spectral analysis. We evaluated Γ_{h} using a fitting procedure with Voigt functions (see Methods, Supplementary Note 5, and Supplementary Figure 4 for the detailed procedure). The inset in Fig. 2a shows an example of the peak deconvolution employed for the spectrum collected at 40 K. The reasonable uncertainty range of Γ_{h} estimated from the fit analysis (grayshaded region) is plotted as a function of temperature in Fig. 3c. The temperature dependence of Γ_{h} was phenomenologically modeled as Γ_{h} = Γ_{0} + Γ_{1}T + Γ_{2}[exp(E_{ph}/k_{B}T) − 1]^{−1}, where k_{B} is the Boltzmann constant, Γ_{0} is the temperatureindependent linewidth, Γ_{1} and Γ_{2} are the fitting parameters, and E_{ph} = 31 meV is assumed to be a typical optical phonon energy in 1LWSe_{2}^{61}. For convenience, we fit the mean values of Γ_{h} within a reasonable range at each temperature and obtained a best fit with Γ_{0} = 7.3 meV, Γ_{1} = 0.004 meV/K, and Γ_{2} = 39 meV. The fitted curve is plotted in Fig. 3c. The observed temperature dependence is consistent with trends previously reported for 1LWSe_{2}^{60,62,63,64}.
Valley relaxation time of the bright excitons
We now consider the relationship between the valley polarization ρ_{x} and the pure valley relaxation time τ_{v} of the bright excitons. The timeintegrated valley polarization of the excitons can be expressed by the following phenomenological equation^{14,15,40,65} (see Supplementary Note 1 for additional details):
We set ρ_{0} = 0.7 for 1LWSe_{2} in the following analyses according to the results discussed in the previous section. Parameter 〈τ_{x}〉 in Eq. (1) corresponds to an exciton population lifetime in the simplest case for which only the bright exciton levels are involved^{14}. For the multiexcitonlevel system we consider in this study (Fig. 1), 〈τ_{x}〉 is defined as an effective lifetime of the bright excitons that corresponds to the total (integrated) time in which a single exciton remains in its bright state before it recombines radiatively or nonradiatively (see also Supplementary Note 1). Equation (1) suggests that τ_{v}, which is defined as the pure valley relaxation time of the bright excitons, can be evaluated if ρ_{0} and 〈τ_{x}〉 are given. This procedure enables the evaluation of the pure valley relaxation time of an exciton in its bright state, which can be directly compared against the theoretical results. Notably, Eq. (1) is valid under the condition that the valley relaxation of holes composing excitons is dominated by the intervalley scattering of the bright excitons regardless of the microscopic origin of the scattering.
We evaluated 〈τ_{x}〉 from the timeresolved PL decay profile of the bright excitons, I(t), as \(\left\langle {\tau _{\mathrm{x}}} \right\rangle = {\int_{0}}^\infty {I\left( t \right)/I\left( 0 \right){\mathrm{d}}t}\), where I(t) is obtained by a fitting of the experimental data using a model PL decay function. Figure 2c shows representative PL decay profiles at 10, 40, 80, and 160 K in semilog plots (see also Supplementary Note 6 and Supplementary Figure 5 for the data in linear plots). The data at each temperature were fitted as a convolution of the instrumental response function (IRF) and a model double exponential function (lightgreen curves), I(t) = C_{1}exp(−t/τ_{1}) + C_{2}exp(−t/τ_{2}), and 〈τ_{x}〉 = (C_{1}τ_{1} + C_{2}τ_{2})/(C_{1} + C_{2}) was obtained by the fitting procedure. Figure 3d plots the obtained 〈τ_{x}〉 as a function of temperature (See Supplementary Note 7 and Supplementary Figure 6 for the consistency between the temperature dependences of the 〈τ_{x}〉 and the exciton PL intensity). The solid curves represent the fit obtained using a model for Eq. (5) described in the Methods section (and in Supplementary Note 1), where the exciton scattering pathways shown in Fig. 1b were considered. We found that the temperature dependence of 〈τ_{x}〉 can be well fitted using Eq. (5) with the bright–dark energy splitting^{54,55,56,57} Δ_{bd} in the range 30–47 meV, supporting the validity of the model of the multiexcitonlevel system shown in Fig. 1 (see Supplementary Note 1 for more detailed discussion on the results of the fitting and limitation of the model).
Valley relaxation mechanism of excitons
We then deduced the τ_{v} from the relation of Eq. (1), namely, τ_{v}^{−1} = 〈τ_{x}〉^{−1} (ρ_{0}/ρ_{x} − 1), using the experimentally deduced values of ρ_{0}, ρ_{x}, and 〈τ_{x}〉 at each temperature. Figure 3e shows the obtained valley relaxation times at various temperatures (green circles). The τ_{v} values under the lowtemperature conditions (<~40 K) were on the order of 10 ps and exhibited plateaulike behavior. Here, we discuss the origin of the exciton valley relaxation in greater detail on the basis of the temperature dependence of the bright exciton valley pseudospin relaxation times, τ_{v}. According to previous theoretical predictions^{32,39}, we consider longrange e–h exchange interactions (originally known as the Maialle–Silva–Sham (MSS) mechanism for exciton spindepolarization in semiconductors^{66}) screened by 2D electrons naturally doped in 1LWSe_{2} to be the dominant relaxation mechanism of the valley pseudospin of the bright excitons^{32}. In this theoretical framework, the inverse exciton valley relaxation time is given as, τ_{v}^{−1} = 〈 Ω_{k}^{2}τ_{h}〉, where Ω_{k} is the momentumdependent Larmor frequency under an effective magnetic field and τ_{h} is the momentum relaxation time, which is inversely proportional to Γ_{h} (See also Supplementary Note 8). Under lowtemperature conditions (Γ_{h} > ~k_{B}T) with screening by doped carriers, τ_{v} for bright excitons can be approximately expressed as^{32}
where \(A = 9a^2M^2/4\pi ^2\hbar ^4.\) is a materialdependent constant in which a is the lattice constant, M is the exciton mass, \(\hbar\) is Planck’s constant divided by 2π, J is the strength of the exchange interaction, and k_{TF} is the Thomas–Fermi wave vector; k_{TF} has a dimensionalitydependent function form and, for a 2D electron gas, k_{TF}(T, E_{F}) = k_{TF0}[1−exp(−E_{F}/k_{B}T)]^{32}. Parameter E_{F} is the Fermi energy measured from the bottom of the conduction band, and it is related to the carrier density, n_{c}, by n_{c} = g_{s}g_{v}m_{e}E_{F}/(2πħ^{2}), where g_{s} and g_{v} are the spin and valley degeneracies, respectively, and m_{e} is the electron mass. Parameter k_{TF0}, given by \(k_{{\mathrm{TF0}}} = g_{\mathrm{s}}g_{\mathrm{v}}m_{\mathrm{e}}e^2/\left( {4\pi \varepsilon \hbar ^2} \right)\), is a zerotemperature Thomas–Fermi wave vector. In the following analyses, a = 0.334 nm, m_{e} = 0.38m_{0}^{67}, and M = 2m_{e} were used for 1LWSe_{2}. Equation (2) is valid when k_{h} ≪ k_{TF}, where \(k_{\mathrm{h}} = \sqrt {2M\Gamma _{\mathrm{h}}/\hbar ^2}\) is a maximum wave vector of excitons determined by the collisional broadening; this condition is safely met for a carrier density of n_{c} > ~5 × 10^{10} cm^{−2} for Γ_{h} < ~35 meV, which is normally fulfilled in naturally carrierdoped 1LTMDCs.
The orangeshaded region in Fig. 3e is the prediction band for valley relaxation times reproduced using Eq. (2) and the experimentally deduced values of Γ_{h} within a reasonable uncertainty range (grayshaded region in Fig. 3c) with fitting parameters of E_{F} (∝n_{c}) and J for the screening function. The vertical width of the prediction band mainly originates from the uncertainty in Γ_{h}. The temperature dependence of τ_{v} deduced using Eq. (1) (solid circles) could be excellently reproduced using the theoretical model of Eq. (2) with the experimental Γ_{h}, given the fitting parameters E_{F} = 13 meV (n_{c} ≈ 2.1 × 10^{12} cm^{−2}) and J = 0.83 eV. The implied E_{F} and the carrier density on the order of 10^{12} cm^{−2} is consistent with the observed trion PL intensity (see also Supplementary Note 9) and previous results for asexfoliated 1LTMDCs^{68}. Within the k·p approximation, J can be roughly estimated as J ≈ 8π^{2}aE_{b}t^{2}/3a_{B}E_{g}^{2} ^{32,69}, where E_{b} is the excitonbinding energy, t is the hopping energy, E_{g} is the band gap, and a_{B} is the exciton Bohr radius. Using E_{b} = 0.37 eV^{8}, t = 1.19 eV^{12}, E_{g} = 2.11 eV, and 1 ≤ a_{B} ≤ 2 nm for 1LWSe_{2}, approximately 0.5 ≤ J ≤ 1 eV is deduced. Thus, the fitting result of J ≈ 0.83 eV is consistent with the theoretical prediction of the exchange interaction strength.
The lowerleft inset of Fig. 3e shows a plot of the E_{F} dependence of τ_{v} calculated using Eq. (2) and the empirical function for the mean values of Γ_{h}. The lowtemperature values of τ_{v} are clearly observed to be independent of E_{F}, and the lowtemperature plateau is extended for larger E_{F} values (higher carrier density). For the intermediate temperatures (~40 K < T < ~160 K), the valley relaxation times will be longer when the carrier density is greater. For further hightemperature conditions, Eq. (2) may no longer be applicable. At high temperatures, valley depolarization mechanisms other than the screened e–h exchange interactions, such as the intervalley scattering through the thermal activation of holes to the spindegenerate Γ valley^{22}, may also coexist, which would lead to Arrheniustype temperature dependence with an activation energy corresponding to the energy difference of the K_{e}K_{h}〉 and K_{e}Γ_{h}〉 excitons with electrons in the ±K valley and holes at the ±K or Γ valley. The activation energy has been estimated to be on the order of 0.14 eV for 1LWS_{2}^{22}; assuming a similar order for 1LWSe_{2}, this scattering pathway becomes gradually important for T > ~ 200 K.
To show the manifestation of the 2D screening in τ_{v}(T) more clearly, we plotted \(\sqrt {AJ^2\tau _{\mathrm{v}}\left( T \right)\Gamma _{\mathrm{h}}\left( T \right)/\hbar }\) in the righttop inset of Fig. 3e, which corresponds to k_{TF}(T, E_{F}) according to Eq. (2). The mean values of the estimated linewidth in the uncertainty range shown in Fig. 3c were used as Γ_{h}(T) at each temperature. The function form of the characteristic 2D Thomas–Fermi wave vector, k_{TF}(T, E_{F}), excellently reproduces the temperature dependence originating from the term \(\sqrt {\tau _{\mathrm{v}}\left( T \right)\Gamma _{\mathrm{h}}\left( T \right)}\). This plot and Eq. (2) also suggest that the dimensionless quantity \(k_{{\mathrm{TF}}0}^2/AJ^2 = \tau _{\mathrm{v}}\left( 0 \right)\Gamma _{\mathrm{h}}\left( 0 \right)/\hbar\) is a constant that is dependent only on the material and is directly accessible by experiment. From our current experimental results, we obtained τ_{v}(0)Γ_{h}(0)/ħ ≈ 118 for 1LWSe_{2}. Although the aforementioned considerations suggest that the presented model could capture important physics in the exciton valley relaxation phenomenon in doped 1LWSe_{2}, nevertheless, we note that our theoretical treatment within the static approximation neglects the dopingdependent changes of the excitonbinding energy and the quasiparticle band gap^{68} dominated by the dynamical screening effect on the direct Coulomb interactions^{70}, and any possible dynamical effect on the screening of the e–h exchange interactions; these potential shortcomings may change the formula of the valley relaxation time both in a qualitative and quantitative manner (See Supplementary Note 10 for more detailed discussion). Thus, it is still desired to develop a more rigorous theoretical approach for indepth understanding of the underlying physics on the valley relaxation phenomena in the carrierdoped 2D semiconductors.
Valley polarization engineering
On the basis of the aforementioned results, a semiempirical formula for the temperature, Fermi energy (carrier density), and linewidthdependent τ_{v} can be expressed as:
where C is a materialdependent dimensionless parameter C ≡ k_{TF0}^{2}/AJ^{2} = τ_{v}(0)Γ_{h}(0)/ћ (see Supplementary Note 10 for further discussion on the material dependence of C). From the experimental values of τ_{v} and Γ_{h} at the lowtemperature condition, C ≈ 118 for the 1LWSe_{2} was deduced in this study. Then, upon substitution of Eq. (3) into Eq. (1), the steadystate exciton valley polarization is predicted as:
Equation (4) suggests a condition in which the exciton valley polarization becomes high; the narrower linewidth Γ_{h}, the larger E_{F} (carrier density), and the shorter 〈τ_{x}〉 will be the keys to achieving a longer τ_{v} and a higher ρ_{x}. We first examined the expected correlation between the ρ_{x} and the Γ_{h} by changing the excitation power density at 10 K using Eq. (4) at the lowtemperature limit ρ_{x} ≈ ρ_{0}(1+〈τ_{x}〉Γ_{h}/Cћ)^{−1}. The expected inverse correlation of ρ_{x} and Γ_{h} was actually observed and could be well reproduced using this simple relation, as shown in Supplementary Note 11 and Supplementary Figure 7. Moreover, we could also successfully apply the above relation for reproducing the excitation photon energy dependence of the valley polarization observed at 15 K from the 〈τ_{x}〉 and Γ_{h} observed at each excitation photon energy (Supplementary Note 2 and Supplementary Figure 1). The wide applicability of the aforementioned relation under lowtemperature conditions thus strongly supports the validity of the presented mechanism.
To further verify Eq. (4) at the finitetemperature conditions and achieve realistic engineering on the excitonic valley polarization, we fabricated 1LWSe_{2} stacked on a multilayer graphene flake, as shown in Fig. 4a (see Methods). For this type of sample, we expect that the exciton linewidth could be narrow because of the reduced scattering on the atomically flat surface of multilayer graphene^{71} and because of the shorter 〈τ_{x}〉 resulting from charge or energy transfer to the graphene. Modulating the carrier density in 1LWSe_{2} may be possible because of the chemical potential difference between the graphene and 1LWSe_{2}.
Figure 4b shows the temperature dependence of the exciton valley polarization in 1LWSe_{2} stacked on the multilayer graphene substrate (ongraphene sample). For this sample, we observed a much higher exciton valley polarization ρ_{x} compared with those for the 1LWSe_{2} on the quartz substrate (onquartz sample), as compared in Fig. 4g. As expected, exciton linewidths much narrower than those on the quartz substrate were observed (Fig. 4c, d). In addition, the trion peak intensity was higher than the exciton’s peak intensity, which suggests that the carrier density in this sample was higher than that in the sample on the quartz substrate. The trion/exciton intensity ratio for the ongraphene sample shown in Fig. 4b is ~1.5 times greater than that for the onquartz sample shown in Fig. 2. We also observed the PL decay profiles (Fig. 4e). We could obtain the 〈τ_{x}〉 for temperatures only more than 80 K as shown in Fig. 4e, f because of the detection limit. The orangeshaded region in Fig. 4g is the prediction band for the ρ_{x} reproduced using the 〈τ_{x}〉 and Γ_{h} values shown in Fig. 4f, d with E_{F} = 19 meV (n_{c} ≈ 3.0 × 10^{12} cm^{−2}) as inputs to Eq. (4). For comparison, the prediction band for the onquartz sample (greenshaded region) calculated with the Γ_{h} shown in Fig. 3c and E_{F} = 13 meV (n_{c} ≈ 2.1 × 10^{12} cm^{−2}) is also shown.
The valley polarization of the onquartz sample and the increased valley polarization for the ongraphene sample could be consistently reproduced using Eq. (4). Importantly, the persistent valley polarization for T > ~100 K in the ongraphene 1LWSe_{2} cannot be explained by simply considering that the 〈τ_{x}〉 in the ongraphene sample is shorter than that in the onquartz sample. For instance, at 100 K, the ρ_{x} for the onquartz sample was only ~0.06 and 〈τ_{x}〉 was ~55 ps. The τ_{v} was then deduced to be ~5.5 ps, as shown in Fig. 3e. If 〈τ_{x}〉 becomes shorter, such as 〈τ_{x}〉 ≈ 30 ps, as observed in the ongraphene sample, and the other parameters in Eq. (1) are kept unchanged, then ρ_{x} is expected to increase only slightly to ~0.1. This prediction contradicts the much higher ρ_{x} ≈ 0.2 for the ongraphene 1LWSe_{2} at 100 K; thus, the narrower Γ_{h} and higher carrier density also contribute to the highvalley polarization, as predicted from Eq. (4). Figure 4h shows the E_{F} dependence of the ρ_{x} predicted by Eq. (4). For clarity, we used the mean values of the Γ_{h} in the uncertainty ranges shown in Fig. 3c (onquartz) and Fig. 4d (ongraphene). The high values of ρ_{x} for the ongraphene sample clearly cannot be reproduced with the same E_{F} (carrier density) as that for the onquartz sample. The inset of Fig. 4h shows the τ_{v} for various E_{F} calculated using Eq. (3) and the mean values of the Γ_{h} for the corresponding samples. The plateau region of τ_{v} is predicted to be extended as the E_{F} increased, consistent with the persistent ρ_{x} for the ongraphene sample at temperatures as high as ~60 K (Fig. 4g). These results confirm that the presented framework provides clear guidelines for engineering exciton valley polarization in 1LWSe_{2}. We note that there remains a possibility that the ρ_{0} and C also vary depending on the carrier density and/or the substrate’s dielectric constant. Thus, the prediction of Eq. (4) may be further modified when taking these effects into account (See Supplementary Note 10 for more detailed discussion).
Discussion
Finally, we discuss whether our results can be extended so as to predict the exciton valley relaxation times in 1LWSe_{2} under highexcitationdensity conditions employed in ultrafast spectroscopic studies using femtosecond lasers^{38,46}. The temperature dependence of τ_{v} deduced in this study, as shown in Fig. 3, is qualitatively similar to the temperature dependence of previous results obtained by TRKR measurements^{38,46}; however, the absolute values we obtained are a few times greater than those previously reported. We address this discrepancy through the exciton density dependence of the linewidth. A previous study on the intrinsic excitonic linewidth in 1LWSe_{2}^{62} reported that Γ_{h} strongly depends on the excitation density, N_{x}, as Γ_{h} = Γ_{s} + Γ_{col}N_{x}, under intense excitation conditions using femtosecond pulsed lasers, where Γ_{s} is a single exciton linewidth and Γ_{col} ≈ 5.4 × 10^{12} meV cm^{−2} is the constant for the densitydependent broadening due to exciton–exciton collisions^{62}. Thus, we assume that the density and temperaturedependent linewidths are expressed as Γ_{h}(T, N_{x}) = Γ_{s}(T) + Γ_{col}N_{x}, where Γ_{s}(T) is the linewidth observed under the weak excitation conditions and Γ_{col} is a temperatureindependent proportionality constant.
Figure 5 shows the τ_{v} for various exciton densities, N_{x}, in 1LWSe_{2} predicted using the excitondensitydependent Γ_{h}(T, N_{x}) as inputs for Eq. (3). For the temperaturedependent Γ_{s}(T), we used the empirical formula obtained by fitting the mean values of Γ_{h} in the uncertainty range shown in Fig. 3c for all cases. Along with the experimental data obtained in this study (circles), the data reported in previous ultrafast TRKR studies on 1LWSe_{2}^{38,46} are also plotted as rectangles and triangles. As shown in Fig. 5, all of the data sets of τ_{v} obtained in the experiments conducted under various excitation density conditions could be reproduced with appropriate N_{x} and E_{F} values as fitting parameters. The N_{x} values required to reproduce the data in ref. ^{38} (1.0 × 10^{12} cm^{−2}) and ref. ^{46} (2.5 × 10^{12} cm^{−2}) are in excellent agreement with the actual excitation densities used in these studies, which are on the orders of ~10^{12} cm^{−2}^{38} and ~2 × 10^{12} cm^{−2}^{46}. The ability to reproduce the experimental data for a wide range of excitation density conditions indicates the robustness and universality of the presented model even under highexcitondensity conditions. Our findings thus offer a unified framework to predict the exciton valley relaxation of 1LTMDCs observed under various experimental conditions. The results also provide clear guidelines for controlling the excitonic valley relaxation times via Fermi energy (carrier density) tuning and/or linewidth modulations via materials engineering and/or via optical means, which will facilitate the development of optovalleytronic technologies based on 2D semiconductors.
Methods
Sample preparation
We used the mechanical exfoliation method to prepare the 1LWSe_{2} samples on quartz substrates or on 300nmthick SiO_{2}/Si substrates^{72}. The 1LWSe_{2} samples stacked on multilayer graphene flakes were fabricated through a drytransfer method using dimethylpolysiloxane (PDMS) films^{73,74}. The 1LWSe_{2} and multilayer graphene flakes were mechanically exfoliated onto the PDMS films, and these flakes were stacked onto a quartz substrate under an optical microscope.
Optical measurements
For the circularly polarized excitation of 1LWSe_{2}, white light from a supercontinuum light source was sent through a monochromator, a linear polarizing prism, a quarterwave plate, and a 0.5 numerical aperture objective lens. The σ_{+} (σ_{−}) circularly polarized light was used for the initial excitation of the excitons with the valley pseudospin +K〉 (−K〉). The light emission from the sample was collected using the objective lens, and the σ_{+} and σ_{–} circularly polarized components were converted into two orthogonal linear polarized components using the quarterwave plate. These linear polarized components were separated by a calcite polarizing beam displacer and sent to a spectrograph and detected on separated regions of a liquidnitrogencooled chargecoupled device detector. A timecorrelated singlephoton counting method was used to perform the timeresolved PL spectroscopy under a pulsed excitation of duration of 20 ps and a frequency of 40 MHz. In the timeresolved measurements, the emitted light was sent though bandpass filters (10nm band width) that corresponded to the exciton resonance energies at each temperature and they were detected using a fibercoupled singlephoton avalanche photodiode device. Except for the results shown in Fig. 2b and Supplementary Fig. 2, light polarization was not resolved in the timeresolved PL measurements. For the temperature range below ~40 K, the fast components of the PL decays shown in Fig. 2c were approximately at the edge of the time resolution limit, whereas the slow components were more than a few hundreds of picoseconds. Thus, the values obtained for 〈τ_{x}〉 in the lowtemperature range may be an upper limit. The excitation power density was 10^{2} W cm^{−2} for both the spectral and timeresolved measurements, except for the excitationpowerdependent measurements shown in Supplementary Fig. 7. We confirmed the linear dependence of the exciton PL intensity on the excitation density. The exciton linewidth exhibited no detectable broadening as the excitation power density increased in the vicinity of the conditions employed in this study (~10^{2} W cm^{−2}) (except for the excitationpowerdependent observations using a power density >~4 × 10^{2} W cm^{−2} shown in Supplementary Fig. 7).
Linewidth analysis
We evaluated the homogeneous linewidths, Γ_{h}, of excitons using a fitting procedure with Voigt functions (convolutions of Lorentzian and Gaussian functions). The inset in Fig. 2a and Supplementary Figure 4 show the PL spectra that were decomposed by the Voigt fit at 40 K (Fig. 2a) or at four representative temperatures (Supplementary Fig. 4), respectively. The black circles are the experimental data and the green curves are the spectra reproduced by the peak fit. We considered the peak features for excitons (red curves) and the neighboring peaks of trions^{34} (gray curves) for the spectral decomposition; the major contributions of the neutral exciton (~1.740 eV) and negative trion (~1.706 eV), and one minor peak appearing at about 17 meV below the exciton peak (~1.723 eV) were assumed. The minor PL feature at this energy range was also observed in the carrierdensitydependent PL spectra of 1LWSe_{2}^{34,59}; the peak position implies a residual contribution from locally generated negative or positive trions with shifted energies at positions with relatively low carrier density, presumably because of spatial inhomogeneity of the carrier density on the 1LWSe_{2} on a quartz substrate. Disappearance of this minor component for the highcarrierdoped sample on the atomically flat surface of the multilayer graphene shown in Fig. 4 supports this interpretation. Further details of the fitting procedure are described in Supplementary Note 5.
Modeling for the temperature dependence of 〈τ _{x}〉
For convenience, in the discussion and validation of the model diagram shown in Fig. 1b, we considered a phenomenological formula to reproduce the experimental temperature dependence of 〈τ_{x}〉. Here, we model the temperature dependence of 〈τ_{x}〉 assuming a finite rate for phononmediated exciton scattering between the bright and dark states. We considered processes indicated by arrows in Fig. 1b as the major exciton scattering pathways to express a nonequilibrium bright exciton distribution in a steadystate condition; the expression for 〈τ_{x}〉 is obtained as:
where Γ_{b} and Γ_{d} are the decay rates of the bright exciton and the dark exciton, respectively (see Supplementary Note 1 for the detailed derivation of this expression), and γ is the rate constant for the intervalley phonon scattering. The solid curves in Fig. 3d are the fitted curves reproduced using Eq. (5). The fitted curve excellently reproduced the experimental temperature dependence of 〈τ_{x}〉 with an assumption of the previously reported bright–dark energy splitting Δ_{bd} in the range from 30 to 47 meV in 1LWSe_{2}^{54,55,56,57}.
Data availability
The data that support the findings of this study are available from the corresponding authors on reasonable request.
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Acknowledgements
This research was supported by JSPS KAKENHI Grant Numbers JP16H00911, JP15K13337, JP15H05408, JP15F15313, JP26390007, JP26107522, JP25400324, JP26390007, JP15K13500, JP16H00910, JP16H06331, and JP17K19055, by the Asahi Glass Foundation, JST Nanotech CUPAL, JST CREST (JPMJCR16F3), the Murata Science Foundation, the Research Foundation for OptoScience and Technology, and the Nakatani Foundation. G.E. acknowledges Singapore National Research Foundation, Prime Minister’s Office, Singapore for funding the research under NRF Research Fellowship (NRFNRFF201102) and its Medium Sized Centre Program. G.E. also acknowledges support from the Ministry of Education (MOE), Singapore, under AcRF Tier 2 (MOE2015T22123, MOE2017T21134). The authors thank Y. Miyata and K. Shinokita for fruitful discussions.
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Y.M. conceived the research and prepared the experimental setup. Y.M. and S.K. analyzed and interpreted the data and wrote the manuscript. F.W., W.Z., A.H., Y.H., and L.Z. prepared the samples, performed the optical measurements, and contributed to analyzing of the data. M.T. and G.E. provided the samples. Y.M., S.K., S.M., and K.M. contributed to the interpretations of the results and writing the manuscript.
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Miyauchi, Y., Konabe, S., Wang, F. et al. Evidence for line width and carrier screening effects on excitonic valley relaxation in 2D semiconductors. Nat Commun 9, 2598 (2018). https://doi.org/10.1038/s4146701804988x
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DOI: https://doi.org/10.1038/s4146701804988x
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