Strain dependence of photoluminescence and circular dichroism in transition metal dichalcogenides: a k.p analysis

Within a two-band $k\cdot p$ method we analyze different types of strain for the $K$ valley optical characteristics of a freestanding monolayer MoS$_2$, MoSe$_2$, WS$_2$ and WSe$_2$. We predict that circular polarization selectivity for energies above the direct transition onset deteriorates/improves by tensile/compressive strain. Wide range of available strained-sample photoluminescence data can be reasonably reproduced by this simple bandstructure combined with accounting for excitons at a variational level. According to this model strain impacts optoelectronic properties through its hydrostatic component, whereas the shear strain only causes a rigid wavevector shift of the valley. Furthermore, under the stress loading of flexible substrates the presence of Poisson's effect or the lack of it are examined individually for the reported measurements.


Introduction
Transition metal dichalcogenides (TMDs) possess direct optical gap together with mechanical flexibility up to 10% range [1] which enables wide strain tunability of their optoelectronic properties [2,3]. Naturally, the associated body of literature is rapidly growing, among which some milestones can be mentioned. The tuning of the electronic structure by applying a uniaxial tensile bending to monolayer MoS 2 on flexible substrates has been demonstrated by several groups within a short time span [4][5][6][7][8][9][10]. For a suspended monolayer MoS 2 membrane, Lloyd et al. showed the continuous and reversible tuning of the optical bandgap over an ultralarge range of applied biaxial strain [11]. More recently, deterministic two-dimensional array of quantum emitters from thin TMDs due to a localized strain pattern is achieved that becomes instrumental to construct a scalable quantum architecture [12,13]. Additional experimental [14][15][16][17][18][19][20][21] as well as theoretical [22][23][24][25] studies substantiated strain as a viable control mechanism for these two-dimensional materials.
Our aim in this work is to consolidate accumulating experimental photoluminescence (PL) data on strained TMD samples with the aid of a simple K-valley-specific two-band k · p theory. Specifically, for the measurements performed by uniaxial bending of flexible substrates, this analysis can reveal the extend of Poisson's contraction over the TMD layer for each individual case. Moreover, it explains the circular dichroism which refers to the degree of optical polarization [26], and how it can be altered by various types of strain.

Theoretical details
2.1. Two-band strained k · p approach for TMDs Employing the two-band basis of d-orbitals, d z 2 and d x 2 −y 2 ± i d xy / √ 2 that largely governs the direct bandgap at the K ± valley, the strained k · p Hamiltonian matrix can be expressed as where a is the lattice constant, f i 's are the strained k · p parameters fitted to first-principles electronic bandstructure by Fang et al., which are for convenience listed in Table 1 [27]. In Eq.
(1) as well as in the remainder of this work, without loss of generality we refer to K + valley, which is assumed to be the origin for the wavevector ì k =xk x +ŷk y , with k x pointing along the Γ − K direction. The spin-splitting can be easily incorporated to this framework [27], but, for our purposes this is not necessary as we are interested in the so-called A-excitons only [5]. However, our treatment excludes other refinements such as trigonal warping, electron-hole asymmetry, and a cubic deviation in the band structure [28]. Table 1: k.p parameters f i (eV), lattice constant a (Å) [27], 2D polarizability χ 2D (Å) [29] for different TMDs.
Materials Neglecting any displacement perpendicular to TMD that lies on the two-dimensional (2D) xy-plane, the tensor strain components for most common types are: Biaxial strain : ε yy = ε xx , ε xy = ε yx = 0, Uniaxial strain : ε xx 0, ε yy = ε xy = ε yx = 0, Shear strain : ε yy = −ε xx , ε xy = ε yx 0, Uniaxial stress : where ν is the Poisson's ratio. We should caution that the term uniaxial strain is in widespread use in TMD literature [3,5,20,21], although with the assumed Poisson's effect, as explicitly mentioned in these works, it needs to be referred to as uniaxial stress; also note that we use tensor and not the engineering strain [30]. Some strained k · p expressions can be obtained analytically from Eq. (1): The direct bandgap becomes , then the energy dispersion for the conduction and valence bands are given by Hence, band extremum shifts from (k x = 0, k y = 0) to (k x0 , k y0 ) because of strain. So that for ε xx > ε yy , k x0 > 0, and band extremum at K shifts away from Γ (toward neighboring zone M ′ ) point, while for ε yy > ε xx , k x0 < 0 it shifts toward Γ point. The shear strain rigidly displaces it along k y direction without affecting the bandgap.

Degree of circular polarization
For a light polarized along a unit vectorû, the dipole matrix element connecting valence |U v and conduction |U c states is given by [31] where m 0 is the free electron mass. For the ± circularly polarized light defined by the unit vectorŝ can be expressed in terms of Pauli spin raising/lowering operatorsσ ± as This quantity has a use in k−resolved degree of optical polarization which is defined as [26], For isotropic and electron-hole symmetric bands, as in our case η( ì k) → η(E), so that we can simply consider excess energy ∆E from the band minima.

Exciton binding and PL energies
To compare with the experimental PL data under a given strain, we need to include excitonic effects as the associated binding energies significantly exceed the thermal energy at room temperature [5]. For that, we first extract numerically the band effective masses m c/v from the energy dispersion relation (Eq. 2) via, where the curvatures are evaluated at the strained band extremum, (k x0 , k y0 ). Within our twoband k · p model electron and hole effective masses are equal, i.e., m * e = m * c = m * h = −m * v , and furthermore they are spatially isotropic, but note that here these effective masses are straindependent. Thus, the corresponding exciton effective mass for its relative degrees of freedom follows from µ = m * e m * h / m * e + m * h . To retain the simplicity of our approach, the binding energies for neutral excitons in TMDs can be calculated following Ref. [29] by a variational method based on the exciton Hamiltonian (switching to Hartree atomic units in the remainder of this subsection), Here, the in-plane interaction between an electron and a hole separated by ρ = x 2 + y 2 is, where, κ a and κ b are dielectric constants for the media above and below the TMD (for a freestanding case, κ a = κ b = 1), H 0 and Y 0 are the Struve and the Bessel function of the second kind; the screening length is given by ρ 0 = 2π χ 2D , and χ 2D is the 2D polarizability of the TMD, which is listed in Table 1 [29].
The wave function for the neutral exciton with a single variational parameter λ is chosen as, In such a case the kinetic energy has the analytical form, T (λ) = 1/(2µλ 2 ), while potential energy requires the following integration to be evaluated numerically, The total exciton energy is found by minimizing E X (λ) = T (λ) + V(λ), where the optimum value of λ corresponds to the mean the exciton radius. For a bound exciton E X < 0, and the PL energy is obtained from E PL = E g + E X . Figure 1 shows the PL peak shift for the four TMDs under uniaxial strain, comparing our calculations with the data from numerous experimental references. For MoS 2 and WSe 2 we have a good agreement between our theory and the best fit to the experimental data, taking into account the spread in the latter. At variance to this, for WS 2 our results do not agree with two Refs. [18,19]. To resolve this case, we also plot the bandgap variation for WS 2 under uniaxial strain from a first-principles calculation [24] (yellow-dashed). If we add to this the strained excitonic correction we get a closer agreement with our calculations (purple-dotted vs. blue-solid). Therefore, we believe that some slipping may have occured on the TMD layer while applying strain to the substrate in Refs. [18,19], whereas other measurements in Fig. 1 such as Refs. [5,20,21] have taken measures to clamp the TMD to the substrate. For MoSe 2 , we see that the uniaxial stress condition using the Poisson's ratio of the substrate, ν = 0.37 (blue-dashed) matches perfectly with the data, in agreement with their assertion in Ref. [21]. In other words, unlike the other measurements in Fig. 1, for this experiment the TMD layer fully complies with the Poisson's contraction of the substrate. In the case of biaxial strain displayed in Fig. 2, for MoS 2 we have an excellent agreement with the widest-range strain data by Lloyd et al. [11] which goes up to 6%. Once again we plot the variation in the bandgap (green-dashed in upper-left panel); the notable offset from PL line indicates the extend of excitonic contribution on the strain variation of the PL energy. Here, our results as well as Ref. [11] are for freestanding monolayer TMDs, on the other hand the remaining biaxial strain data from Ref. [32] was originally reported with respect to polypropylene substrate strain. Therefore, to convert substrate strain results in Ref. [32], we multiplied all strain data from this reference by the 0.573 scale factor. This brings their data in agreement with the freestanding case of Ref. [11]. However, for MoSe 2 we still have a disagreement with Ref. [32]; noting the leveling off in their data beyond about 0.15% strain, we again suspect that a slipping might be responsible.

PL peak shift under strain
In Table 2 we compare our PL peak strain shift results with the quantities from various experimental references. For our results, both uniaxial strain and uniaxial stress cases are presented, where in the former no transverse contraction takes place in the direction perpendicular to axial deformation (i.e., ν = 0). For uniaxial stress we use ν = 0.37 value which is typical for the flexible substrates in use [20,21]. As mentioned above, the uniaxial stress condition applies only for the MoSe 2 experiment of Ref. [21]. We also quote in parantheses our results excluding the variation of exciton binding energy under strain. It can be observed that sulfur-TMDs are more responsive to strain for PL peak shift and the amount of change is larger for biaxial strain than uniaxial one for each material considered.  157 e 54 i 111 e a Ref. [5], b Ref. [4], c Ref. [6], d Ref. [11], e Ref. [32], f Ref. [21], g Ref. [18], h Ref. [19], i Ref. [20] 3.

Effect of strain on circular dichroism
To see how strain affects the bandedge light helicity selectivity in TMDs, we calculate η(∆E) using Eq. (5) at different excess energies ∆E, as measured from the conduction band minimum. In the unstrained case there is almost perfect selectivity (η → 1) which extends over the entire valley [26]. Figure 3 shows that this quantity deviates from unity with increasing ∆E. For both uniaxial and biaxial cases, the selectivity deteriorates/improves by tensile/compressive strain.
We also see that selenium-TMDs are more sensitive to strain in this respect, and the amount of change is larger for biaxial than uniaxial strain for all of these materials. It needs to be noted that any intervalley scattering [33][34][35] or other processes [36] not considered here will further

Conclusion
A simple two-band k · p approach shows that uniaxial and biaxial strain are both effective on the bandgap and effective masses, whereas the shear strain does not alter the optoelectronic properties, but merely shifts the wavevector of the K valley minima. Comparison of strain-dependent PL peak shifts with a wide range of experimental data for monolayer TMDs demonstrates a satisfactory agreement provided that excitonic effects are included, and it reveals whether Poisson's effect takes place in a certain experiment. As another finding, circular polarization selectivity beyond the K valley transition onset can be tuned in either direction by applying tensile or compressive strain which is more pronounced for the biaxial case. This analysis can easily be extended to other TMDs with the availability of their k · p parameters. It can also act as a benchmark for more refined theories.

Funding
We are grateful to Bilkent University, Department of Physics for its financial support.