Introduction

Strain is a powerful platform for tailoring the emergent properties of condensed matter, with the capability of disclosing quantum phenomena unavailable in their unaltered states. Several remarkable instances of this paradigm have recently been demonstrated including, for example, strain-induced quantum paraelectric to ferroelectric phase transitions1, the stabilization of superconductivity2, reversible magnetic phase transitions3, and topological phase transitions4. Additionally, strain engineering has become a pivotal ingredient in the semiconductor manufacturing industry as a tool to enhance electron mobilities in transistors5,6.

Two dimensional materials (2D), and particularly transition metal dichalcogenides (TMDs) in their monolayer form, are particularly suitable for the exploration of novel strain-induced phenomena owing to the interplay of flexibility and structural stability, enabling to reach strain regimes inaccessible in the bulk. TMDs can withstand strain up to 10% before undergoing structural damage7,8. Additionally, high-strain regimes have been reached through a variety of techniques, including substrate lattice mismatching8,9, substrate thermal expansion10,11, van der Waals coupling12, and proton irradiation13.

Strain can be exploited to fine-tune specific features in the electron band structures14,15,16, providing a route to directly influence optical response, scattering channels, and transport properties of TMD monolayers. In particular, strain alters the energy level alignment between the K to the Q(Σ) high-symmetry points in the conduction band of monolayer WS2 17,18,19,20,21. This effect induces a transition from direct to indirect band gap under compressive strain13,22,23,24,25,26, with distinctive fingerprints in photoluminescence measurements27,28,29,30,31. Conversely, tensile strain increases the energy separation between the K and Q high-symmetry points in the conduction band, thereby, suppressing intervalley scattering and resulting in the increase of carrier mobility in transport measurements32,33,34,35,36.

More generally, the impact of strain on valley degrees of freedom in TMD monolayers emerges as a promising route to engineer physical phenomena that directly result from the multivalley character of the electron band structure19,35,37,38,39. Valley-dependent optical selection rules – arising from the coexistence of three-fold rotational invariance and a non-centrosymmetric crystal structure – allow to selectively inject carriers at either the K or \({{{{\rm{K}}}}}^{{\prime} }\) high-symmetry points via absorption of circularly-polarized photons, enabling the establishment of a finite valley polarization (namely, an anisotropic population of the K and \({{{{\rm{K}}}}}^{{\prime} }\) valleys) and the exploitation of valley degrees of freedom40,41. This effect serves as the foundation for a variety of physical phenomena that are special to TMD monolayers, including the emergence of chiral valley excitons and the valley Zeeman effect42,43, circular dichroism44, chiral phonon emission45,46,47,48, as well as different flavours of Hall effects40,49,50. In particular, chiral phonon emission can underpin a variety of novel physical phenomena in TMD monolayers. The finite angular momentum associated with chiral phonons imparts a finite magnetic moment to the lattice, if the ions carry nonzero Born effective charges51,52,53,54. This magnetic moment enables a direct coupling of the lattice to external magnetic fields, giving rise to phonon Zeeman effects51,55, which are closely analogous to their electronic counterparts42,43. Chiral phonons have further been proposed as a route towards nontrivial topological properties of the lattice, such as the phonon Hall effect47,56. Additionally, ultrafast phenomena in TMDs are governed by electron-phonon scattering processes involving inter- and intra-valley transitions accompanied by the emissions or absorption of a phonon57,58,59,60,61,62,63,64. These processes are extremely sensitive to the phase space available for phonon-assisted transitions, and are thus likely to be largely affected by the strain. Overall, these considerations suggest opportunities to directly influence intervalley scattering processes and the ensuing nonequilibrium dynamics of phonons and charge carrier using strain.

In this work, we explore strain-induced modifications of the ultrafast electron and phonon dynamics in monolayer WS2 via state-of-the-art ab-initio methods. Our investigations are based on the time-dependent Boltzmann equation (TDBE) and concentrate on realistic strain conditions that have already been achieved in experiments. We demonstrate that strain influences profoundly the phase space available for electron-phonon scattering, altering the accessible scattering channels for the relaxation of hot carriers. Specifically, for the case of valley-polarized electronic excitations strain can be exploited to switch among qualitatively different dynamical regimes, characterized by the emission of linearly polarized phonons at low strain and circularly polarized (chiral) phonons at high strain, as depicted in Fig. 1. Overall, these findings demonstrate a powerful route to enhance the chiral phonon emission in multivalley TMDs and open up opportunities to directly tailor the chirality of lattice vibrations in photo-driven solids.

Fig. 1: Influence of strain on the phonon emission in monolayer WS2 upon excitation with circularly polarized light.
figure 1

Top: in pristine (unstrained) monolayer WS2 carrier relaxation is accompanied by emission of linearly-polarized phonons. Bottom: a qualitatively different dynamical regime can be accessed via the application of uni- or bi-axial strain, whereby carrier relaxation results in the emission of chiral phonons.

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Results

Influence of strain on electrons and phonons at equilibrium

The band structure of monolayer WS2 as obtained from density functional theory (DFT) is reported in Fig. 2a. To illustrate the effects of tensile strain on the electronic properties and phonon dispersion of monolayer WS2, we perform DFT and density functional perturbation theory (DFPT) calculations for an increase of the in-plane lattice constant ranging from 0% to 2% with an increment of 0.5%. The phonon symmetry analysis is reported in Supplementary Note 1 and Supplementary Fig. 1. In Fig. 2b, c, we illustrate the influence of different strain conditions on the conduction and valence bands, respectively. Energies in the conduction (valence) band are relative to the CBM (VBM). The effect of strain manifests itself primarily in the conduction band. With increasing strain the spin-split Q valley – located at the midpoint between the Γ and K/\({{{{\rm{K}}}}}^{{\prime} }\) high-symmetry points in the conduction bands – is blue-shifted to higher energies, leading to an increase in the energy separation between the Q point and the CBM. Conversely, the spin-degenerate Γ valley in the valence bands follows the opposite trend. Tensile strain reduces the energy difference between the K and Γ high-symmetry points and for strain levels beyond 2% monolayer WS2 undergoes a transition from direct to indirect band-gap semiconductor, as demonstrated by recent photoluminescence experiments13. The origin of the band structure dependence on strain has been discussed extensively in earlier studies and it can be attributed to the reduction of monolayer thickness when strain is applied17,65,66. While we focus here on the case of bi-axial strain, uni-axial strain induces similar band structural renormalization as illustrated in Supplementary Fig. 5 and discussed in Supplementary Note 2.

Fig. 2: Influence of strain on the electron band structure and phonon dispersion of monolayer WS2.
figure 2

a Electronic band structures along the high symmetry path Γ-K-M-\({{{{\rm{K}}}}}^{{\prime} }\)-Γ, the red shading indicates the carrier population induced by a circularly polarized light pulse. Conduction b and valence bands c along the high symmetry path M-K-Γ, with energies relative to the CBM and VBM, respectively. The color coding reflects strain values ranging between 0 and 2 %. d Influence of strain on the phonon dispersion. Inset: Vibrations of motifs corresponding to the \({A}_{1}^{{\prime} }\) and \({E}^{{\prime} }\) modes.

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The phonon dispersion for different values of biaxial strain is reported in Fig. 2d. In general, strain induces a softening of the phonons which arises from the weakening of bonds as the interatomic distances are increased. The phonon frequencies, however, are not homogeneously softened for different modes. For the \({E}^{{\prime} }\) mode we observe a softening by 0.87 meV per % of strain. Conversely for the \({A}_{1}^{{\prime} }\) mode, the softening amounts to less than 0.1 meV per % of strain. This result agrees well with strain-dependent Raman spectroscopy experiments67,68 and the analysis of mode Grüneisen parameters69,70. The different dependence of \({E}^{{\prime} }\) and \({A}_{1}^{{\prime} }\) phonon energies on strain can be ascribed to the anharmonicity of atomic potentials and the increase of in-plane atomic distances. As a result, the \({E}^{{\prime} }\) mode with in-plane vibrations are strongly softened compared to the \({A}_{1}^{{\prime} }\) phonon which vibrates in the out-of-plane direction. The vibrations of these modes are schematically illustrated in the inset of Fig. 2d.

Having discussed the effects of strain on the electronic and vibrational properties under equilibrium conditions, we proceed next to inspect to which extent strain can influence the ultrafast dynamics of electrons and phonons out of equilibrium. Overall, while the minor changes of the phonon dispersion under strain are likely inconsequential for the ultrafast dynamics, the substantial strain-induced band-structure modifications are expected to profoundly influence the phase space available for phonon-assisted electronic transitions. In particular, in the following we show that the renormalization of band energy at the Q point can induce the selective activation/deactivation of additional scattering channels, leading to qualitatively different regimes for the non-equilibrium lattice dynamics.

Coupled electron-phonon dynamics in strained WS2

We focus below on the case of valley-selective circular dichroism and on the ensuing non-equilibrium dynamics of electrons and phonons. The interplay of threefold rotational invariance and the lack of inversion symmetry in hexagonal monolayer TMDs is responsible for distinctive valley-selective optical selection rules. In particular, photo-excitation with circularly-polarized photons and photon energies in vicinity of the fundamental gap can lead to the formation of electron-hole pairs located at either the K or \({{{{\rm{K}}}}}^{{\prime} }\) high-symmetry points depending on the light helicity. Pulses shorter than 100 fs have been shown to generate (valley-polarized) excited carrier density of the order of 1012 to 1013 cm−2 in TMDs58,71,72,73.

To investigate the non-equilibrium dynamics of electrons and phonons, we solve the TDBE in the time domain by accounting explicitly for electron-phonon and phonon-phonon scattering processes. Within the TDBE approach, the evolution of time-dependent electron distribution function fnk and phonon number nqν are described by the following set of coupled first-order integro-differential equations:

$${\partial }_{t}{f}_{n{{{\bf{k}}}}}(t)={\Gamma }^{{{{\rm{ep}}}}}[{f}_{n{{{\bf{k}}}}}(t),{n}_{{{{\bf{q}}}}\nu }(t)]$$
(1)
$${\partial }_{t}{n}_{{{{\bf{q}}}}\nu }(t)={\Gamma }^{{{{\rm{pe}}}}}[{f}_{n{{{\bf{k}}}}}(t),{n}_{{{{\bf{q}}}}\nu }(t)]+{\Gamma }^{{{{\rm{pp}}}}}[{n}_{{{{\bf{q}}}}\nu }(t)]$$
(2)

with ∂t = ∂/∂t. Here, Γep, Γpe, and Γpp denote the collision integrals due to electron-phonon, phonon-electron (ph-e), and phonon-phonon (ph-ph) interactions, respectively, for which explicit expressions have been reported elsewhere44,57,74,75,76,77. Radiative recombination is neglected hereafter as it takes place on nanosecond timescales which are much longer the characteristic time of electron-phonon scattering. Additionally, intervalley electron-electron scattering processes (K → \({{{{\rm{K}}}}}^{{\prime} }\) or vice versa) do not influence the valley depolarization dynamics owing to momentum conservation. These effects are thus neglected in our simulations. We further neglect electron-hole interaction. This choice is compatible with the large carrier densities considered in our work and in earlier pump-probe investigations (>1013 cm−2). Recent photoluminescence studies indicate that at these carrier densities electron-hole interactions are screened, suppressing the formation of excitons, leading to formation of an electron-hole plasma78. We solved Eqs. (1) and (2) fully from first principles based on our recent implementation of the TDBE approach within the EPW code79. The electron-phonon coupling matrix elements are obtained from DFT and DFPT with norm-conserving Vanderbilt (ONCV) pseudopotentials80 and Perdew-Burke-Ernzerhof generalized gradient approximation (GGA-PBE) to the exchange-correlation81, as implemented in Quantum ESPRESSO82,83 and interpolated using maximally-localized Wannier functions within EPW using the Wannier90 code as a library84,85. Equations (1) and (2) are solved as an initial value problem by discretizing the time derivative using Heun’s method and recalculating the collision integrals at each time step on homogeneous Monkhorst-Pack grids consisting of 120 × 120 × 1 k and q points. We used a time step of 1 fs and considered a total simulation time of 6 ps. All computational details are reported in the Method section.

As initial state, we consider electron and phonon distribution functions that mimic the conditions established following absorption of a circularly-polarized light pulse. The lattice is initially at thermal equilibrium for t = 0, with phonon occupations defined according to Bose-Einstein statistic at room temperature (300 K). For momenta within the hexagonal BZ, the initial electron distributions fnk(t = 0) in the conduction and valence bands are illustrated in Fig. 3a, b, respectively. fnk(t = 0) is defined by considering a photoexcited electron (hole) density of 2.0 × 1013 cm−2 at the K valley in the conduction (valence) band, whereas electron and hole occupations at \({{{{\rm{K}}}}}^{{\prime} }\) are unchanged with respect to thermal equilibrium.

Fig. 3: Influence of strain on the valley-depolarization dynamics of electrons and holes in monolayer WS2.
figure 3

a, b Momentum-resolved electron distribution in the conduction and valence bands, generated by ultrashort circularly-polarized light pulses resonant with the band gap at the K point, for a carrier density of 2.0 × 1013 cm−2. The electron populations fnk are summed over both spin channels for the lowest conduction and highest valence bands. c–e Momentum-resolved electron distribution in the conduction bands under 0%, 1% and 2% biaxial strain for a time delay of 1 ps after photo excitation. f–h Momentum-resolved electron distribution in the valence bands for a time delay of 1 ps for different values of strain.

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Figure 3c–e illustrate the electron distribution function in the conduction band after 1 ps following photoexcitation for 0, 1, and 2% strain, respectively. The valley polarization of carriers in the conduction bands, reflected by the anisotropy in the population of the K and \({{{{\rm{K}}}}}^{{\prime} }\) valley, is found to decay within 1 ps for all values of strain. Remarkably, strain directly influences the decay path of excited carriers. In unstrained WS2, electrons at K can directly scatter to either the \({{{{\rm{K}}}}}^{{\prime} }\) or Q valley by emitting phonons. As strain increases, however, the scattering of electrons from K to Q is gradually suppressed, as reflected by the decrease of electron occupation at the Q high-symmetry point in Fig. 3d, e. This behaviour can be attributed to the increase in the energy separation ΔEKQ ≡ − (EK − EQ) between the K and Q valleys in the conduction band. ΔEKQ increases linearly from 60 meV for the unstrained monolayer up to 390 meV for 2% strain, as illustrated in Fig. 4(e). For phonon-assisted transitions, energy conservation requires εfin − εin ≃ ℏωqν, where ε denote the initial and final electron energy and ℏωqν < 50 meV is the phonon energy. This condition cannot be satisfied for strain values larger than 1% leading to the deactivation of the K → Q scattering channel. This result is also consistent with the increase of electron mobility in strained monolayer WS235.

Fig. 4: Nonequilibrium phonon dynamics in strained monolayer WS2 due to electron-phonon and phonon-phonon interactions.
figure 4

a–c Momentum-resolved effective vibrational temperature Tq,TO for the strongly-coupled TO mode for crystal momenta in the Brillouin zone for a time delay of 1 ps and different levels of strain. d Schematic representation of the vibrational motif arising from linearly-polarized phonons at the M high-symmetry point. e Energy difference between the K and Q (Γ) valleys in the conduction (valence) bands as a function of strain. The y-axis marks the effective vibrational temperature Tq,TO obtained at the \({{{{\rm{K}}}}}^{{\prime} }\) valley under different levels of strain for 1 ps time delay. f Schematic representation of the strongly-coupled chiral TO phonons at the \({{{{\rm{K}}}}}^{{\prime} }\) high-symmetry point.

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The electron distribution in the valence bands 1 ps after excitation is shown in Fig. 3f–h for 0, 1, and 2% strain, respectively. Also in the valence bands strain influences profoundly the carrier dynamics by altering the accessible scattering channels. In unstrained monolayer, K → \({{{{\rm{K}}}}}^{{\prime} }\) transitions are spin forbidden, whereas K → Γ transitions are energy forbidden. Correspondingly, the valley polarization is preserved beyond 1 ps, as shown in Fig. 3f. In presence of strain, the energy difference ΔEKΓ between K and Γ valley decreases from 240 meV to about 1 meV at 2%, as shown in Fig. 4e, activating the K → Γ scattering channel and substantially accelerating the electron dynamics in valence bands (Fig. 3g–h).

Recent theoretical and experimental studies revealed that the non-equilibrium phonon population established following photoexcitation are extremely sensitive to the phase space available for phonon-assisted electronic transitions57,86. These considerations, alongside with the findings discussed above, indicate that strain could serve as a means to selectively influence the mechanisms of phonon emissions and the ensuing phonon populations in monolayer WS2 and related compounds. In the following, we proceed to corroborate this hypothesis by quantifying the non-equilibrium phonon population established upon thermalization of valley-polarized electrons and holes.

As an indicator for the nonequilibrium phonon populations established as a result of the coupled electron-phonon dynamics, we consider below the mode resolved effective phonon temperature57\({T}_{{{{\bf{q}}}}\nu }=\hslash {\omega }_{{{{\bf{q}}}}\nu }{[{k}_{{{{\rm{B}}}}}{{{\rm{\ln }}}}(1+{n}_{{{{\bf{q}}}}\nu }^{-1})]}^{-1}\), where kB is the Boltzmann constant, ωqν the phonon frequency, and nqν is the phonon population obtained from Eq. (2). In short, Tqν is directly related to the phonon population, it is constant at thermal equilibrium, whereas it may differ for distinct vibrational modes out of equilibrium. In particular, the quantity Tqν enables to directly monitor transient changes of phonon population with full momentum resolution. The nonequilibrium phonon temperature for the strongly coupled TO branch established after 1 ps – reported in Fig. 4a–c for 0, 1, and 2% strain, respectively – indicates that strain profoundly influences the phonon populations excited throughout the valley depolarization dynamics. The strain effects induce similar changes to other phonon branches, as demonstrated in Supplementary Fig. 2 and Supplementary Fig. 3. In the absence of strain, the effective phonon temperature [Fig. 4a] reflects an enhancement of the phonon population for crystal momenta q in the vicinity of the M, Q, and Γ high-symmetry points. While Γ phonons are emitted throughout intravalley relaxation (and thus inconsequential for the valley depolarization), the emission of M and Q phonons can only result from electronic transitions mediated by the Q valley owing to momentum conservation. These results indicate that the most likely scattering pathway for valley-polarized carriers in pristine WS2 involves a Q-mediated two-step process (\({{{\rm{K}}}}\to {{{\rm{Q}}}}\to {{{{\rm{K}}}}}^{{\prime} }\)), whereas the direct transition (\({{{\rm{K}}}}\to {{{{\rm{K}}}}}^{{\prime} }\)) occurs at a significantly lower rate.

As lattice strain is introduced [Fig. 4b, c], the emission of M and Q phonons is gradually suppressed, and for 2% strain the nonequilibrium phonon population is almost exclusively characterized by hot phonons at the \({{{{\rm{K}}}}}^{{\prime} }\) and Γ points. Figure 4e further indicates that with the strain-induced increase of ΔEKQ and the ensuing suppression of K → Q scattering processes, the temperature of \({{{{\rm{K}}}}}^{{\prime} }\) phonons (right axis) increases linearly with strain, saturating at about 2% strain. In other words, strain effectively deactivate the \({{{\rm{K}}}}\to {{{\rm{Q}}}}\to {{{{\rm{K}}}}}^{{\prime} }\) scattering channel, boosting the emission of \({{{{\rm{K}}}}}^{{\prime} }\) phonons. Overall, these findings reveal the possibility to exploit strain to selectively switch between two distinct dynamical regimes: at low strain, the electron and phonon dynamics are governed by two-step scattering processes \({{{\rm{K}}}}\to {{{\rm{Q}}}}\to {{{{\rm{K}}}}}^{{\prime} }\), resulting in the emission of M and Q phonons; beyond a critical strain threshold of about 1%, conversely, the scattering pathways involve \({{{\rm{K}}}}\to {{{{\rm{K}}}}}^{{\prime} }\) transitions accompanied by the emission of \({{{{\rm{K}}}}}^{{\prime} }\) phonons. This latter regime is characterized by valley-polarized phonon populations (with \({T}_{{{{\rm{K}}}},{{{\rm{TO}}}}}\ne {T}_{{{{{\rm{K}}}}}^{{\prime} },{{{\rm{TO}}}}}\)) which can persist for several picoseconds (Supplementary Fig. 6), and that constitute the vibrational counterpart of the valleytronic paradigm in 2D materials40. The full time dependence of the phonon population at the high-symmetry points in the BZ is illustrated in Supplementary Fig. 4. Additionally, valley-polarized phonons have been predicted to give rise to vibrational circular dichroism in ultrafast diffuse scattering experiments44, which could provide a direct route for the experimental validation of these predictions.

The crossover from M-phonon to \({{{{\rm{K}}}}}^{{\prime} }\)-phonon regimes also indicates a transition from linearly-polarized phonon emission to chiral phonon emission. The phonon chirality can be characterized by introducing phonon angular momentum \({L}_{{{{\bf{q}}}}\nu }=\langle {\epsilon }_{{{{\bf{q}}}}\nu }\vert {\hat{{{{\bf{r}}}}}}\times {\hat{{{{\bf{p}}}}}}\vert {\epsilon }_{{{{\bf{q}}}}\nu }\rangle\), obtained by projecting the phonon eigenvector on a circularly polarized basis. In short, Lqν > 0 (Lqν < 0) corresponds to chiral phonon with right-handed (left-handed) circular polarization, whereas non-chiral phonons are characterized by Lqν = 0. For strained WS2, the phonon emission is governed by the strongly-coupled TO(\({{{{\rm{K}}}}}^{{\prime} }\)) mode, which is characterized by \({L}_{{{{\bf{q}}}}\nu }^{z}\, > \,0\) and by a distinctive circular polarization, as illustrated in Fig. 4f. The chiral character of the emitted phonons by electronic transitions among the K, K’, and Γ high-symmetry points can further be corroborated by a group-theoretical analysis, as discussed in details in previous works62,87,88. At the M point, all phonons display linearly-polarized vibrations, as required by time-reversal symmetry (TRS) which enforces the identity L−qν = − Lqν = 0 for all TRS-invariant points, including M. The vibration motif of the strongly-coupled M phonons is illustrated in Fig. 4d. Consequently, the angular momentum lost by valley polarized electrons is primarily carried by phonon at \({{{{\rm{K}}}}}^{{\prime} }\) in strained WS2, whereas in unstrained materials the angular momentum of phonon is distributed throughout all modes in the BZ.

Discussion

In conclusion, we presented an ab-initio investigation of the ultrafast electron and phonon dynamics in strained WS2 monolayers. Our study demonstrates the possibility to exploit strain as a route to selectively control the relaxation pathways of photoexcited carriers. Remarkably, even strain values as low as 1% have the capacity to significantly alter available scattering channels and relaxation pathways in the valence and conduction bands. This has profound implications for the lattice dynamics, and it can lead to qualitatively different nonequilibrium phonon populations. Specifically, in the context of valley-polarized electronic excitations – as induced, e.g., by the absorption of circularly-polarized photons –, strain can trigger a transition between distinct dynamical regimes of the lattice characterized by the emission of chiral phonons under high strain and linearly-polarized phonons under low strain.

More generally, multivalley materials19 emerge as a powerful platform to exert control over nonequilibrium phenomena through the manipulation of the electron-phonon scattering phase space via strain. It is worth noting that the strain conditions considered in our study have already been achieved in experiments, thus the experimental validation of these findings can readily be attempted. Specifically, the excitation of chiral phonon is expected to give rise to distinctive fingerprints in time-resolved diffuse scattering experiments, leading to vibrational circular dichroism in the scattering intensities upon excitation with the circularly polarized light. Additionally, we estimate that the lattice remains in a non-thermal state characterized by chiral phonon populations persisting for 10 ps or beyond, namely, over timescales that are within reach of time-resolved scattering experiments44.

Methods

Computational details

The ground-state properties of the monolayer WS2 are calculated from density-functional theory (DFT) within the plane-wave pseudopotential method as implemented in the Quantum Espresso code82. We use fully-relativistic norm-conserving Vanderbilt (ONCV) pseudopotentials80 and Perdew-Burke-Ernzerhof generalized gradient approximation (GGA-PBE) to the exchange-correlation functional81. We use the DFT-relaxed crystal structure, a plane-wave kinetic energy cutoff of 120 Ry and a Brillouin zone sampling of 20 × 20 × 1 points. The phonon dispersion is calculated based on density-functional perturbation theory (DFPT) on a 6 × 6 × 1 Monkhorst-Pack grid83. Electron band structure and phonon dispersion are interpolated onto dense grids consisting of 120 × 120 × 1k and q points via maximally-localized Wannier functions, as implemented in Wannier9085. Electron-phonon matrix elements are computed with the EPW code79,84. The inclusion of phonon-phonon interactions requires the third-order force constants, which we obtain with finite displacement methods by generating displaced patterns on the 4 × 4 × 1 super-cell with thirdorder.py from the ShengBTE module89. Spin-orbit coupling is included at all levels of calculations. The ultrafast electron-phonon dynamics is investigated via solution of the time-dependent Boltzmann Equation (TDBE) – which we implemented in the EPW code – for both electrons and phonons in presence of electron-phonon and phonon-phonon scattering. A detailed overview of this approach and our implementation is provided in earlier works44,57,74,75,76.

Initial conditions of the ultrafast dynamics simulations

We consider light pulses resonant with the band gap of WS2, populating the highest valence band (VB) and second lowest conduction band (CB+1). This choice is assuming spin-conserving photoexcitation conditions. To simulate the initial carrier distributions excited by circular polarized light, we use a Gaussian function centered at the K point to describe electron-hole excitations: \(f({{{\bf{k}}}})=\exp \{-\frac{| | {{{\bf{k}}}}-{{{\rm{K}}}}| {| }^{2}}{{\eta }^{2}}\}\). Correspondingly, the initial distribution is defined by fV B(k) = 1 − f(k) and fCB+1(k) = f(k), while leaving other bands unexcited. The parameter η is related to the excited carrier density via the condition \(n=\frac{1}{{N}_{{{{\bf{k}}}}}\Omega }{\sum }_{c{{{\bf{k}}}}}{f}_{c{{{\bf{k}}}}}\), where Nk and Ω are respectively the number of k points sampled (120 × 120 × 1) and the area of the two-dimensional (2D) unit cell. The carrier density of 2 × 1013 cm−2 considered in this work is equivalent to around 0.017 electrons (holes) per formula unit, and the nonadiabatic phonon renormalization can be neglected according to previous studies77,90. Hence the assumption of fixed electronic and lattice band structure used in TDBE is justified. The initial phonon distributions are considered to be thermally distributed with Bose-Einstein function at 300 K.