Strain-Induced Exciton Hybridization in WS 2 Monolayers Unveiled by Zeeman-Splitting Measurements

Mechanical deformations and ensuing strain are routinely exploited to tune the band gap energy and to enhance the functionalities of two-dimensional crystals. In this Letter

In this Letter, we present unprecedented ZS measurements on WS 2 MLs under high tensile strain (biaxial strain ε biax up to ∼2%) and intense magnetic fields (up to 28.5 T). The mechanical deformation is attained by creating hydrogen-filled, one-layer-thick WS 2 microbubbles [35,36], which are studied by microphotoluminescence (μ-PL) measurements. The strain-induced crossover in the VB maximum from K to Γ permits us to select the light emission from direct (A, involving the K point in both CB and VB) or indirect (I, involving K in CB and Γ in VB) excitons [37]. For the A exciton in a strain-free WS 2 ML, we find g A ¼ −3.913, while values between −2 and −3 are measured for A and I excitons for ε biax ∼ 2%. Based on first-principles calculations and phenomenological modeling, we ascribe these sizable strain-dependent variations of g exc to a hybridization of A and I excitons occurring when strain brings them into near-resonance conditions. Our evidences of exciton hybridization in a 2D system open the doors to novel research in the field, where, e.g., exciton hybridization could influence the pair lifetime and efficiency or could represent an important ingredient in understanding the coupling between different materials in 2D heterostructures.
Experimental details.-We investigated strained WS 2 MLs in the shape of microbubbles [35,36]. The bubbles were created by hydrogen-ion irradiation of bulk WS 2 flakes, as reported in Ref. [35]. One-layer-thick bubbles with diameters up to several microns and a height-to-radius ratio of ∼0.16 swell from the flake due to the trapping of hydrogen at tens of atm [38], resulting in a nonuniform tensile strain [39]. Samples with lower strain (described later) and an unstrained WS 2 ML were also measured for comparison. Magneto-μ-PL was performed at room temperature either in a Bitter magnet [40] up to 28.5 T or in a superconducting magnet up to 12 T; see Supplemental Material [41], Methods. The field was perpendicular to the sample surface; see Supplemental Material [41], Note I.
Strain dependence of energy bands and excitons.-We performed density functional theory calculations [7,[42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61] to evaluate the band structure of WS 2 ML under different biaxial strainvalues; see Fig. 1(a) and details in Supplemental Material [41], Notes I and II. The calculations were performed on planar geometries, the curvature being negligible at the atom scale due to the small height-to-radius ratio and relatively large spatial extent of the deformation (few μm) of the bubbles. The CB minimum sits at K, where the spin-split upper (CB þ ) and lower (CB − ) energy states are separated by 33 meV. The VB maximum is initially represented by the upper spin-split state at K (VB þ ). However, when a strain ε biax ¼ 2.0% is reached, the VB maximum shifts to Γ (VB Γ ), and a crossover from a direct to an indirect band gap occurs. The exciton crossover occurs at lower strains of ∼1.1% due to the larger binding energy of indirect excitons [33]; see Supplemental Material [41], Note I. This transition was recently experimentally observed in WS 2 bubbles [37]. Figure 1(b) shows the microscope image of a typical flake with one-layer-thick bubbles. The bubbles host a similar strain distribution regardless of their size. The strain is predictable via numerical [35,39] or analytical [39] calculations and changes from uniaxial at the bubble edge to biaxial at its summit, where it reaches its maximum value [36,37,39]; see Supplemental Material [41], Note I. Since the electronic response to strain is independent of the strain character (uniaxial or biaxial), as shown in Supplemental Material [41], Note I, and in agreement with Ref. [62], the relevant information can be embedded in an "equivalent biaxial strain" ε biax defined as the average of the two inplane strain components. The strain increases from the bubble edge toward the center, influencing the optoelectronic properties of the ML. The PL spectrum of a strainfree ML is dominated by the A exciton at 2.02 eV. The same transition is observed at the edge of the bubbles, but it is redshifted due to strain [ Fig. 1(c)]. Toward the summit, a new band appears and totally dominates the spectrum at the center. This band is attributed to the indirect I exciton, features longer decay times and is less intense [37]. In small bubbles, where the spatial resolution is not sufficiently high, the A exciton dominates due to its higher recombination efficiency. Therefore, A and I excitons can be separately observed in bubbles of different dimensions.
g factors from first-principles.-While the direct-toindirect exciton transition has been predicted in many theoretical works [51,[62][63][64][65][66], a compelling analysis of the I exciton in terms of bands and symmetries and of the g factor of A and I excitons under strain is lacking. To characterize the g factor, one has to quantify the spin (S z ) and orbital (L z ) angular momentum of the energy bands involved in the exciton transitions. Figure 1(d) shows the schematic representation of the relevant CBs and VBs, indicating the corresponding orientation of S z and L z . The A exciton stems from CB þ and VB þ (with same spin), whereas the I exciton stems from CB − (the lowest energy CB) and VB Γ ; see also Supplemental Material [41], Notes III and IV. The band g factors are calculated using state-ofthe-art ab initio techniques [53][54][55][56], which allows us to compute L z not only at the K valleys but also at the Γ point. Particularly, we found L z ≈ 0 for VB Γ over the whole range of analyzed strain; see Supplemental Material [41], Note IV. This result agrees with previous reports on unstrained MoSe 2 [54] and WSe 2 [55] MLs. For the A exciton, the spin-conserving selection rules that determine the g-factor sign are readily determined by the direct transitions involving CB þ and VB þ at the AEK valleys (with σ AE polarization) [67,68]. Conversely, to understand the magneto-PL emission of the I exciton, it is also necessary to account for the influence of phonons [69]. From a systematic analysis of the phonon-mediated selection rules using group theory [70][71][72][73], we found that different phonons with wave vector K (namely, the K 4 or a pair of K 5 , K 6 ) could mediate the radiative recombination between CB − and VB Γ and lead to a ZS of two oppositely circularly polarized components. Combining the symmetry analysis of the optical transitions and first-principles calculations of the orbital angular momentum, we summarize the dependence of the effective g factors of A and I excitons in Fig. 1(e), for both biaxial and uniaxial strain. Specifically, L z being calculated at different strain values. Our results reveal that strain slightly alters the g factors of both A and I excitons. Since the radiative emission of the indirect exciton can be mediated by different phonons, we could not determine its g factor sign and thus considered both the case of g I > 0 and g I < 0. For details on the selection rules, see Supplemental Material [41], Note IV. Experimental g factors.-To probe the g factors of strained WS 2 , we measured different bubbles, where the A or I transitions could be separately observed. Figure 2(a) shows the room-temperature μ-PL spectra at 0 T and 28.5 T recorded on an unstrained planar ML (gray-shaded) and on two different bubbles, showing A (light-blue-shaded) and I (pink-shaded) transitions. The planar ML was deposited on a SiO 2 =Si substrate. While the substrate might induce some strain transfer, such strain is negligible with respect to the bubble strains. The planar ML is thus referred to as "unstrained." In all cases, a clear splitting between the σ þ and σ − components is observed at 28.5 T, but the strained bubbles feature a smaller ZS than the unstrained ML. At 0 T, no splitting is observed for the unstrained ML, whereas a small Zeeman-like splitting can be noticed for the bubbles; see also Supplemental Material [41], Notes V and VI. We speculate that strain-induced pseudogauge fields [74,75] are at the origin of this phenomenon, which will be the object of future studies. Figure 2(b) depicts the field dependence of the ZS for the different excitons considered. Via Eq. (1), we find g A;ML ¼ −3.913 AE 0.004 for the unstrained ML; see Supplemental Material [41], Note V, where a full characterization (g factor, diamagnetic shift, degree of circular polarization) is reported. The experimental g factor agrees well with the theoretical calculations (−3.84) for the A exciton at zero strain [ Fig. 1(e)]. Our g factor is also close to those estimated in previous works [6,8,9,12,13], where, however, the measurements were performed at ∼4.2 K. This implies that temperature does not influence much the physics of g exc . This agrees with recent predictions [53], showing how g exc remains nearly unchanged when the band gap is modified, at variance with band g factors. Figure 2(b) also shows the ZS obtained for the bubbles. Interestingly, we find g A;bubble ¼ −2.47 AE 0.10, which amounts to a 40% decrease with respect to the unstrained ML. Equally interestingly, g I;bubble ¼ −2.49 AE 0.12, remarkably close to g A;bubble . We studied several other bubbles and estimated g exc for A-type or I-type excitons subject to slightly different strain values, finding similar g factors, as detailed in Supplemental Material [41], Note VI. These results clearly contrast with the pure density functional theory picture given in Fig. 1(e). To understand the role of strain in determining the g-factor variation, we studied samples characterized by lower strains. In particular, in a WS 2 bubble with elongated geometry we observed a redshift of 90 meV (ε biax ≈ 0.7%) and in a WS 2 ML deposited on a hBN bubble [76] a small strain transfer was achieved, resulting in a redshift of 7 meV (ε biax ≈ 0.06%). In both cases, A exciton g factors close to −4 were measured; see Supplemental Material [41], Note VI. This proves that high strains are necessary to affect g exc .
It should be noticed that Ref. [19] reported on g factors between −2 and −4 measured at 4.2 K in some WSe 2 MLs subject to small uniaxial strains (redshifts < 6 meV) (g A;ML ¼ −4.02 and −4.6 were measured in unstrained MLs), accompanied by an exciton splitting, which was attributed to a strain-induced intervalley electron-hole exchange interaction [19]. In our case, however, neither an exciton splitting nor a g-factor variation was observed in the WS 2 ML deposited on a hBN bubble [76], characterized by similar strains. We also repeated the same experiment by depositing a WSe 2 ML on hBN bubbles [76], achieving a redshift of 10 meV. Again, we found no splitting and a g factor comparable to that of an unstrained ML; see Supplemental Material [41], Note VII. The discrepancy between our results and those of Ref. [19] requires further investigation, which is beyond the scope of this work.
Exciton hybridization and g-factor renormalization.-To interpret the experimental g factors of ∼ − 2.5 in highly strained WS 2 bubbles [ Fig. 2(b)], we recall that our firstprinciples calculations [ Fig. 1(e)] clearly establish that neither g A nor g I is strongly altered by strain at the orbital level. This suggests that additional mechanisms, beyond purely orbital effects, are responsible for the g-factor renormalization. Since the two exciton species become energy-resonant for increasing strain, we consider exciton hybridization. For a generic coupling between excitons, either even or odd with respect to time-reversal symmetry, the strain and magnetic field dependence of the coupled A and I excitons can be described by the upper (U) and lower (L) branches following these relations: Here, the subindex AE refers to the σ AE component, Δ is the coupling parameter encoding the exciton hybridization, and E AE and G AE are given by  where E AðIÞ ðεÞ and g AðIÞ ðεÞ depend linearly on strain (E AðIÞ ðεÞ ¼ E AðIÞ ð0Þ þ ΔE AðIÞ ε and g AðIÞ ðεÞ ¼ g AðIÞ ð0Þþ Δg AðIÞ ε) [see, e.g., Fig. 1(e)]. See Note VIII of the Supplemental Material [41] for the explicit form of the hybridization Hamiltonian and details. Figure 3(a) shows the effect of exciton hybridization on the energy levels E U and E L (at zero magnetic field). At zero strain, the A exciton is below the I exciton. With increasing strain, the two excitons hybridize, marked by a distinct anticrossing in the energy levels. Eventually, for sufficiently large strain, the two exciton species are decoupled again and the I exciton is below the A exciton. Interestingly, this anticrossing has never been observed experimentally in a 2D system, which is different from, e.g., bulk semiconductor alloys [77]. Indeed, strain-induced exciton transitions (direct-to-indirect for MLs, indirect-todirect for bilayers) were predicted [51,62,65,66] and observed [37,[78][79][80] in several TMDs. Clear-cut evidence was, however, reported only at room temperature and in microscopic structures with variable strain [37,78]. In these circumstances, exciton funneling makes it difficult to establish experimentally a relationship between energy and strain [33,37,81], and the width of PL bands is comparable to the L=U separation in the hybridization region. These factors possibly prevented the observation of exciton anticrossing.
In the presence of magnetic fields, the g factors associated with the U (L) exciton branches are given by g UðLÞ ðεÞ ¼ ½E UðLÞþ − E UðLÞ− =μ B B. Figure 3(b) compares the theoretical trends with the experimentally measured g factors. The strain corresponding to each experimental datum was deduced from the theoretical relationship between exciton energy and strain displayed in panel (a), where the experimental exciton recombination energy E X is determined from the PL measurements; see further details in Supplemental Material, Note IX. For small strains (≤ 1%), g L remains nearly unchanged. As strain increases and we enter the hybridization region, g L increases (toward g I ) and g U decreases (toward g A ). The experimental g factors nicely fall onto the calculated curves of g L , which corresponds to the E L branch favored in PL. Interestingly, the majority of the measured g factors lies in the strain range 1.5% to 2%, where the largest hybridization is achieved. In our analysis, we still consider the two possible signs of g I but we emphasize that in the strong hybridization regime the g I sign uncertainty is less pronounced. In Note VIII-B in the Supplemental Material [41] we study the behavior of g U and g L with different parameter sets.
The crucial ingredient for exciton hybridization is the coupling parameter Δ. Although we introduced such coupling phenomenologically, we further elaborate on its possible microscopic origin. Recent studies showed that pure exciton-exciton interactions are of the order of few meV [82], whereas phonon-mediated exciton-exciton interactions may reach tens of meV [83]. In Fig. 3, we found the best agreement for Δ ¼ 35 meV, thus suggesting that phonons are mediating the exciton hybridization. Interestingly, for such phonon-mediated coupling, decreasing the temperature would decrease Δ [83], resulting in a steeper slope of g UðLÞ vs strain. Phenomenological approaches akin to ours have recently been used to explain hybridization signatures between dark and B (CB − , VB − ) excitons [84] and between A and B excitons [85] in bilayer MoS 2 . These observations call for further theoretical efforts to provide a deeper microscopic understanding of exciton hybridization mechanisms in van der Waals materials.
Conclusions.-We showed that strain induces a hybridization of k-direct and k-indirect excitons in mechanically deformed WS 2 MLs. The hybridization-otherwise hard to unveil by the exciton energy behavior-was apparent from ZS measurements on WS 2 microbubbles. Indeed, a drastic reduction of the exciton g-factor modulus was found and well reproduced by a combination of first-principles calculations and phenomenological modeling of the exciton interaction. The strain dependence of g exc highlighted the pivotal role that mechanical deformations play in the encoding and reading of the valley quantum number of excitons. Our study also demonstrates that ZS measurements can be a fine tool to highlight exciton hybridization phenomena, thus providing a reference for future research in the field of 2D materials and heterostructures, where hybridization phenomena between resonant exciton species could play a fundamental role in tuning the system properties.