Valley phonons and exciton complexes in a monolayer semiconductor

The coupling between spin, charge, and lattice degrees of freedom plays an important role in a wide range of fundamental phenomena. Monolayer semiconducting transitional metal dichalcogenides have emerged as an outstanding platform for studying these coupling effects. Here, we report the observation of multiple valley phonons – phonons with momentum vectors pointing to the corners of the hexagonal Brillouin zone – and the resulting exciton complexes in the monolayer semiconductor WSe2. We find that these valley phonons lead to efficient intervalley scattering of quasi particles in both exciton formation and relaxation. This leads to a series of photoluminescence peaks as valley phonon replicas of dark trions. Using identified valley phonons, we also uncover an intervalley exciton near charge neutrality. Our work not only identifies a number of previously unknown 2D excitonic species, but also shows that monolayer WSe2 is a prime candidate for studying interactions between spin, pseudospin, and zone-edge phonons.


Supplementary Note1. Discussion of possible defect localization effect.
Here we discuss the experimental evidences which rule out the observed valley phonon replicas as states localized to defects. Supplementary Figure 3 shows the power dependence of the observed states. The power dependence of the phonon replicas is the same as their associated dark state, which supports their valley phonon origin. In specific, the power dependence of the phonon replicas of negative dark trion (D -) is linear (Supplementary Figure 3d). Localized states usually show sublinear behavior due to the long lifetime. Sublinear power dependence is indeed observed for the replicas in the neutral and positive charged regime. However, here the sublinear effects are from the long lifetime of the optically dark states (D + , D 0 , and I 0 ). Using replicas associated with positively dark trion (D + ) as example, D + photoluminescence intensity has sublinear power dependence I ~P α with α=0.9 ( Supplementary Figure 3f), which results from the long lifetime of D + . The PL intensity of the four lower energy peaks, all have same sublinear power dependence with α=0.9 as D + . This is consistent with all four peaks as valley phonon replica of D + .
More importantly, these phonon replica features (10 spectral peaks) are highly repeatable from sample to sample in terms of the spectral structure (Supplementary Figure 2), g factor, polarization, and their relative spectral energies. In fact, the spectral structure is the same for both positive and negative trions, as we demonstrated. Such repeatable and robust properties are distinct from the random spectral features localized to defects. We emphasize that all spectral features can be well explained by the valley phonon assisted intervalley scattering of electrons or holes. It is worth pointing out the sign of the g factor as well as the sign of PL polarization are different between different states, which are also consistent with different nature of the valley phonons involved in intervalley quasiparticle relaxation. The evidence of these states as valley phonon replicas, rather than defect bound states, are overwhelming.

Supplementary Note2. Landé effective g-factor analysis.
In monolayer WSe2, Zeeman splitting of excitonic states at the ±K-point valleys can be attributed to three main contributions [1][2][3][4] . The first part comes from spin of the composite quasiparticles in the excitonic state, which gives a Zeeman energy of 2SzµBB. The second part comes from the atomic orbital magnetic moment, giving a Zeeman energy of mµBB. In the case of monolayer WSe2, the conduction band edge is mainly composed of d-orbitals with m = 0 in both K-point valleys, whereas the valence band edges in the K(-K) point valley are mainly d-orbitals with m = 2(-2) in the upper valence band, and m = −2(2) in the lower valence band. Lastly, it is shown that the lattice structure also contributes a valley magnetic moment with a g-factor of αc(αv) for the conduction(valence) band, resulting in a Zeeman energy of αc(αv) µBB for an electron (hole), respectively.
Supplementary Figure 4a shows three main contributions to the Zeeman shift at the band edges in ±K point valleys. We identify three representative spin-valley configurations involved in the light emission process of all the excitonic states in Supplementary Figure 1a. Notice that for the trion states, the quasiparticle that remain unchanged after light emission does not affect the measured Zeeman splitting. Thus, we only consider spin-valley configuration of the electron-hole pair involved in the recombination. Intravalley bright exciton recombination, shown in Supplementary Figure 4b, refers to recombination of an electron in the conduction band and a missing electron (hole) in the valence band with the same spin and valley quantum numbers. The calculated effective g-factor is 2(αc-αv-2), and is experimentally measured to be about -4. This means the valley magnetic moments for the conduction and valence bands are almost the same, which agrees well with theory 3-6 .
Intravalley dark recombination refers to recombination of an electron and a missing electron (hole) in the same valley with opposite spin quantum number, as shown in Supplementary Figure  4c. Similarly, we can calculate that the effective g-factor is 2(αc-αv-4). The corresponding 0 , + and − states have an effective g-factor close to -9 7-9 . Finally, the intervalley recombination refers to recombination of an electron and a missing electron (hole) in the opposite valleys, but with the same spin quantum number. The effective g-factor can be written as 2(-αc-αv-4) for K3 and K2 valley phonon assisted intervalley recombination. It has the same amplitude but opposite sign for K1 valley phonon assisted intervalley recombination. The complication of the sign of g-factors comes from the circular polarization of the emitted photon, as a result of it being either an intervalley electron scattering process or intervalley hole scattering process. The magnitude of the effective g-factor for the intervalley recombination is measured be about 13.
In Supplementary Table 1, we summarized the measured effective g-factor of the excitonic states identified in Supplementary Figure 1a. We extract the + and − polarized PL peak energy, ( + ) and ( − ), respectively. The effective g-factor is then calculated from the extracted Zeeman splitting: ∆= ( + ) − ( − ).

Phonon-induced intravalley transitions between bright and dark excitons
Supplementary Table 2 shows the character table of D3h point double-group 10 . This table is used to derive selection rules for phonon-mediated transitions of intravalley excitons, corresponding to the case that both electron and hole reside in the same valley (e.g., both in the Kpoint valley) 11 . The transformation properties of optically-inactive dark excitons are captured by the irreducible representation (IR) Γ3, semi-dark excitons with out-of-plane optical transition dipole by Γ4, and bright excitons with in-plane optical transition dipole by Γ6 12 . From Supplementary Table 2, one can readily verify the selection rule, (Γ 6 * × Γ 3 ) * = (Γ 6 * × Γ 4 ) * = Γ 5 , (S1) implying that transitions between dark and bright excitons due to intravalley spin flip (of the electron component) can be induced by a zone-center phonon that transforms like the IR Γ5. This phonon is the homopolar optical mode, whose polarization vector is denoted by in-plane and outof-phase vibration of the chalcogen atoms, as shown in Supplementary Figure 5a 12 . Note that the spin-flip matrix element due to interaction with a long-wavelength flexural phonon, which transforms as Γ4, is nonzero only if the phonon wavevector is finite (i.e., when q ≠ 0) 11 . On the other hand, the phonon mode Γ5 is the only one for which the spin-flip matrix element is nonzero exactly at the zone center (q = 0).

Zone-edge phonons
Using Quantum Expresso 13 , Supplementary Figure 5b shows the calculated phonon spectrum in monolayer WSe2. Because the concept of acoustic and optical phonon modes near the Γ-point loses its meaning when qa is no longer much smaller than unity (a is the lattice constant), the notion of acoustic or optical phonons for longitudinal and transverse modes (i.e., LA, TA, LO and TO) is not valid when dealing with zone-edge phonons at the K-point whose wavenumber is q = K = 4π/3a. Instead, the use of K-point IRs is more appropriate. This nomenclature is presented in Supplementary Table 3, where the transformation properties of the zone-edge phonons are captured by the IRs 1 through K6 (these IRs also belong to the C3h point single group. The spin quantum number has no bearing on the polarization vectors of atoms in the unit cell) 11 .
The nine polarization vectors of the zone-edge phonons are presented in the last column of Supplementary Table 3. Mz/||(x,y,z) and X±,z/||(x,y,z) are polarization vectors for the transition-metal and chalcogen atoms, respectively. The subscripts z and || denote out-of plane (z) and in-plane (||) vibrations. The subscript '+(−)' of X denotes in(counter)-phase motion of the two chalcogen atoms in a unit cell. Supplementary Table 3 shows that each of the IRs K3−5 is associated with one polarization vector, and consequently, each of these IRs represents one zone-edge phonon. The IRs K1, K2 and K6 are different in that each contains two types of atomic displacements, and thus, each corresponds to two independent zone-edge phonon modes. From energy considerations that will be explained below and using Supplementary Figure 5b as a typical example for ML-TMDs, K3−5 represent zone-edge phonons from the mid-three bundle whereas the bottom (top) three zoneedge phonons are associated with the low-(high-) energy modes of K1, K2 and K6. The atomic displacement that corresponds to the mode Kl with energy ћωKl is proportional to Rα,j is a 2D vector denoting the equilibrium position of the α-atom in the j th unit cell, where α = (M,Xt,Xb) and Xt/b denotes the top/bottom chalcogen atoms. Vα(Kl) is the polarization vector (right column of Supplementary Table 3). Given that |K| = 4π/3a where a is the in-plane distance between nearby identical atoms, the phase difference obeys K · (RX(M),j − RX(M),j+n) = ±2πn/3, where n is the number of unit cells along a zig-zag chain. Consequently, the displacement of every third chalcogen (or transition-metal) atom along a zigzag chain is identical. In addition, when substituting the in-plane polarization vectors Vα = [1,±i,0] in Eq. (S2), the resulting atomic motion follows a clockwise or counterclockwise circular trajectory, δ α,j l ∝ cos( ⋅ α,j + ω l t) x ± sin( ⋅ α,j + ω l t) ŷ.
Supported by our experimental findings and the group theory analysis below, we will focus in this work on the modes K1−3. Supplementary Figure 6a shows the atomic displacements that correspond to low-and high-energy phonon modes with symmetry K1. The difference between the two is the phase difference in the circular motion of Se and W atoms, leading to mitigated deviation of the bond lengths from equilibrium in the low energy case. Supplementary Figure 6b shows the atomic displacement that corresponds to a phonon mode with symmetry K3. Here, only Se atoms go through in-plane circular motion around their equilibrium positions and the phonon energy is somewhere between that of the low-and high-energy phonon modes with symmetry K1.
Supplementary Figure 6c shows the atomic displacements that correspond to low-and highenergy phonon modes with symmetry K2. Here, the transition-metal atoms go through in-plane circular motion around their equilibrium positions, while chalcogen atoms vibrate in opposite directions along the out-of-plane axis. As shown in the figure, the difference between the low and high energy modes is the relative motion of the chalcogen and transition-metal atoms. Namely, the low energy mode corresponds to the case that when the transition-metal atoms move toward the chalcogen atoms, the chalcogen atoms move away from the mid-plane and vice versa. This combined motion keeps the bond length closer to its equilibrium value, and hence to lower phonon energy. Conversely, the high energy mode leads to stronger deviations from equilibrium because when the transition-metal atoms move toward the chalcogen atoms, the chalcogen atoms move closer to the mid-plane and vice versa.
Finally, Supplementary Figure 5b shows that the branch extensions of the ZA and TA modes anti-cross close to the K-point. This anti-crossing is not seen in the calculation of monolayer MoS2. The result is that the mode K6 has the lowest energy in monolayer MoS2, while K2 has the lowest energy in monolayer WeS2.

Band-edge electronic states and selection rules for intervalley transitions
Next, we derive selection rules for intervalley transitions of electrons and holes. We first note that K = 4π/3a is not only the wavenumber that connects each of the zone-edge ±K-points with the zone center, but it is also the wavenumber needed to connect the zone-edge +K and −K points. In other words, the conservation of crystal momentum due to intervalley transitions of electrons (or holes) states at the ±K points is mediated through the zone-edge K-point phonons. We use Supplementary Table 3  ( 12 * × 9 ) * = ( 10 * × 11 ) * = 3 , (S4) implying that the phonon mode K3 is the dominant mechanism for intervalley transitions in the conduction band.

Supplementary Note4. Qualitative comparison between theory and experiment
Analyzing the experimental findings of this work, we find very good agreement with the above theoretical analysis and previous predictions 11,12 . For example, Fig. 4 shows two phonon replicas of the indirect (intervalley) exciton, 0 , whose energies match the calculated values of the phonons modes K1 (17 meV) and K3 (26 meV) in Supplementary Figure 5b. These modes are suggested by the selection rules in Eqs. (S3) and (S4), and they are supported by the cross-polarized emitted light from the phonon replica 1 0 versus the co-polarized emitted light from the phonon replica 3 0 , as shown and analyzed in the main text. Similarly, we see that this behavior is consistent when dealing with dark trions states ( Fig. 2 and Fig. 3).
One aspect of the experiment that can be further supported by theory is the observation that the PL phonon-assisted peaks associated with the mode K3 are noticeably stronger in amplitude than those with K1. Invoking second-order perturbation theory 14 , the amplitude ratio of these peaks follows (S5) The reason for the stronger signature of K3 phonons compared with K1 is the small spinsplitting energy of the conduction band compared with that of the valence-band, ∆ ≪ Δ . As a result, the energies of the initial and intermediate virtual states are relatively similar when the intermediate hole states are kept in the top valleys of the valence bands (i.e., when the electron goes through intervalley transition whereas the hole is a spectator). Substituting empirical values for the energies in monolayer WSe2, | ± − ( ± + 3 )| ∼ 60 meV, and | ± − ( ± + 1 )| ∼ ∆v ∼ 400 meV, we get that Our experimental results show that the amplitude ratio of the peaks associated with K3 and K1 is about 4, implying that ℳ 1 should be about three times larger than ℳ 3 . This empirical analysis can be used to benchmark the results of future first-principles calculations of these matrix elements.

Supplementary Note5. The phonon replica K 2 .
In addition to the replicas predicted by the previous group-theory analysis, we have observed a weak phonon replica in both the positive and negative dark trions that emerges ∼13 meV below the no-phonon dark trion lines ( Fig.2 and Fig. 3). From the g-factor and polarization analysis, one can see that this replica, albeit weaker, shows the same characteristics of the replica K3. Namely, it involves intervalley scattering of the electron component. A question remains regarding the physical origin of this replica. Inspecting the low-energy modes at the K-point, as shown in Supplementary Figure 5b, we can associate these peaks with either the lowest-energy mode K2 or the second one K6 (their calculated energies are 11.7 and 15.3 meV).
The mode K6 involves out-of-plane vibration of the transition-metal atom, which breaks the mirror-inversion symmetry of the monolayer. In general, interactions of electrons (or holes) with such phonon modes can only couple to spin flips 11 . Unlike the zone-edge phonons with K6 symmetry in which only W atoms vibrate in the out-of-plane direction, zone-edge phonons with K2 symmetry involve counter motion of the Se atoms in the out-of-plane direction which retains the mirror inversion symmetry. Accordingly, the mode K2 can induce a spin-conserving intervalley transition, albeit its amplitude is measurable only when the initial and final states are relatively far from being time reversal partners. In more detail, the selection rule in Eq. (S4) does not only mean that the amplitude of the matrix element ℳ 3 is the dominant one for spin-conserving intervalley electron transitions, but that it is the only one for which the transition between the time-reversed +K and −K states does not vanish.
In conclusion, given that spin-conserving intervalley transitions consistently explain the recombination of electrons and holes from opposite valleys (see main text), we have attributed the low-energy phonon replica with K2 rather than K6.
Open discussion 1. Encapsulation by hBN breaks the mirror symmetry of the system. Thus, further experiments and theory analyses are needed to study if K6 can also contribute to the low-energy phonon replica.

2.
First-principle calculation of the matrix elements for spin-conserving intervalley transitions is needed for each of the nine zone-edge phonons. This calculation can be used to evaluate the ratio between the various phonon replicas, and explain the following question: why the phonon replicas K1 and K2 are observed only with the low-energy modes? For example, the selection rule in Eq. (S3) does not discern between the low-and high energy modes of K1 (whose calculated values are 17 and 30 meV). Calculating the matrix elements should shed light on this question.

3.
The localization of excitons (and trions) next to defects can enhance their coupling to phonons. Furthermore, defects alleviate the crystal symmetry and enable scattering of exciton complexes with phonon modes other than K3 and K1. As such, localization can explain the observation of the low-energy phonon replica that we have associated to the phonon mode K2. Further experiments and theory analyses are needed to study the effect of localization on the phonon replica.

Supplementary Note6. Fine structure of the .
Short-range exchange interaction between the electron and hole is predicted to lift the double degeneracy of 0 , the dark exciton 12 . Symmetry analysis further shows that the lower energy branch is strictly forbidden whereas the higher energy branch has an out of plane dipole and can give in-plane emission. As shown in Fig.4b, 0 exhibits fine structure, displaying a finite zero field energy splitting, consistent with the previous report. 7 Remarkably, we also observe the fine structure of the Γ 5 phonon replica of dark exciton,  Figure 9c) and reveals the hybridized nature of the two branches at low field. In contrast to the dark exciton 0 , which does not show circular polarization, Γ 5 0 becomes fully circularly polarized at high field. This shows that whereas the lower energy branch of 0 is strictly forbidden by symmetry, its Γ 5 phonon replica is allowed due to finite coupling to the bright exciton 0 .
This fine feature can also be well captured by the group theory analysis presented in SI-4. The dark (lower energy) and semi-dark (higher energy) branch of dark exciton 0 can be represented by irreducible representation Γ 3 and Γ 4 respectively, whereas the bright exciton 0 can be represented by Γ6. 11 The selection rule Eq.(S1) then naturally allows the two dark excitons to be coupled to 0 through a zone center Γ 5 phonon, leading to the observed fine structure of Γ 5 0 .

Supplementary Figure1|
Gate dependent photoluminescence. a, PL intensity plot as a function of gate voltage and photon energy, from the same device as shown in Fig. 1c. The excitation and detection are cross circularly polarized ( + excitation and − detection). We marked the excitonic states that have been identified in this paper as well as previously reported. b, The degree of circular of polarization as a function of gate voltage. − , 0 and + have negligible circular polarization due to their out of plane dipole moment. 1 0 and Γ 5 + stand out for their obvious cross polarization.    Table 3 and Supplementary Figure 6). c, Scheme of low-energy valleys in the conduction and valence bands. Spinflip intravalley transitions in the conduction band are mediated by the phonon-mode Γ 5 , while spinconserving intervalley transitions in the conduction (valence) bands are mediated by the phonon mode K 3 (K 1 ). Phonon-induced intervalley spin-flip scattering is relatively weak (the transition matrix element between time-reversed states vanishes).
a Energy (meV)    Supplementary Table1| Landé effective g-factor of identified excitonic states. Effective g-factor of identified excitonic states in Supplementary Figure 1a. For readers' convenience, g-factor of dark states and their phonon replicas in Table 1 are listed here as well.

Supplementary Table3|
Character table of C 3h point double-group. This table captures the symmetry properties of the K point in ML-TMDs where ω = exp(2πi/3). To prevent confusion with the notation in Supplementary Table 2, the Koster notation of the IRs is changed from Γ i to K i . The x-axis is along the zigzag edge direction and the y-axis is along the armchair direction.