Impact of strain on the optical fingerprint of monolayer transition metal dichalcogenides
Maja Feierabend, Alexandre Morlet, Gunnar Berghäuser, Ermin Malic
IImpact of strain on the optical fingerprint of monolayer transition metal dichalcogenides
Maja Feierabend, Alexandre Morlet, Gunnar Bergh¨auser, and Ermin Malic Chalmers University of Technology, Department of Physics, 412 96 Gothenburg, Sweden ´Ecole Normale Suprieure de Cachan, D´epartement de Physique, 94230 Cachan, France Strain presents a straightforward tool to tune electronic properties of atomically thin nanomaterials that arehighly sensitive to lattice deformations. While the influence of strain on the electronic band structure has beenintensively studied, there are only few works on its impact on optical properties of monolayer transition metaldichalcogenides (TMDs). Combining microscopic theory based on Wannier and Bloch equations with nearest-neighbor tight-binding approximation, we present an analytical view on how uni- and biaxial strain influencesthe optical fingerprint of TMDs including their excitonic binding energy, oscillator strength, optical selectionrules, and the radiative broadening of excitonic resonances. We show that the impact of strain can be reducedto changes in the lattice structure (geometric effect) and in the orbital functions (overlap effect). In particular,we demonstrate that the valley-selective optical selection rule is softened in the case of uniaxial strain due to theintroduced asymmetry in the lattice structure. Furthermore, we reveal a considerable increase of the radiativedephasing due to strain-induced changes in the optical matrix element and the excitonic wave functions.
Atomically thin transition metal dichalcogenides (TMDs)have been in the focus of current research due to their efficientlight-matter interaction and the remarkably strong Coulombinteraction leading to tightly bound excitons [1–4]. Recently,the impact of strain on optical and electronic properties ofTMDs has gained importance, since these atomically thinmaterials are highly sensitive to deformations of their latticestructure suggesting strain-induced tailoring of TMD charac-teristics. Recent experimental [5–9] and theoretical [9–17]studies have revealed that strain can significantly change theelectronic band structure of TMDs. In particular, the directband gap decreases (increases) for tensile (compressive) strainresulting in a considerable red (blue) shift of optical reso-nance. So far, most theoretical studies on the impact of strainin TMDs are based on DFT calculations focusing on changesin the electronic band structure without taking into account thepredominant role of excitons in these materials. In this work,we present an analytic approach combining the Wannier andTMD Bloch equations for excitons with the nearest-neighbortight-binding wave functions. The goal is to provide a micro-scopic access to the impact of uni- and biaxial strain on theoptical fingerprint of TMDs including the excitonic bindingenergy, the oscillator strength, the optical selection rules, andthe radiative broadening of excitonic resonances.In a first step, we determine the electronic band struc-ture of a general TMD material MX with M=(Mo, W) andX=(S, Se), where Mo and W stand for molybdenum andtungsten transition metals, while S and Se denote the sul-fur and selenium chalcogen atoms. To this end, we solvethe stationary Schr¨odinger equation H Ψ λ k ( r ) = E λ k Ψ λ k ( r ) including the tight-binding (TB) wave functions Ψ λ k ( r ) = √ N (cid:80) j = M, X C λj k (cid:80) NR j e i k · R j φ λj ( r − R j ) . Here, the TB co-efficients C λj k express the contribution of each atomic sublat-tice j = (M,X), R j represents the position of the atoms withinthe sublattice j and φ λj ( r − R j ) stands for the atomic orbitalsthat are relevant for the considered bands λ . By inserting thenearest-neighbor TB approximation [18, 19], we obtain an ex- (b) biaxial strain BZ BZ (c) uniaxial strain BZ (a) unstrained geometric effectoverlapeffect FIG. 1. Strain affects electronic and optical properties of transi-tion metal dichalcogenides MX (with a transition metal M and twochalcogen atoms X) through (i) geometric changes in the real spacelattice (geometric effect) and (ii) changes in the orbital functions(overlap effect), upper and lower panel respectively. (a) The upperpanel shows the hexagonal lattice structure with M (orange) and X(yellow) atoms in real space and the corresponding Brillouin zone(BZ) in momentum space in the unstrained case. The lower panelrepresents the corresponding orbitals functions (orange for the M or-bitals, yellow for X orbitals) and their overlap (purple). (b) In pres-ence of tensile biaxial strain, atoms are uniformly moved apart inboth directions. Hence, the hexagonal lattice structure remains sym-metric. In momentum space, this leads to a decrease of the BZ size.Furthermore, due to the larger distance between M and X atoms, theorbital function overlap is reduced. (c) In the case of tensile uniax-ial strain, i.e. strain only along one direction (here x), the hexagonalstructure becomes antisymmetric both in real and momentum space.Beside the reduced orbital overlap, the atomic orbital functions alsobecome elliptic in the direction of strain. pression for the electronic band structure E λ k = ± (cid:113) E gap + 4 | t λ | | e ( k ) | . (1)The electronic band gap E gap = (cid:80) λ (cid:0) H λii − H λjj (cid:1) is determined by the on-site energies H λii . Further-more, the nearest-neighbor hopping integral reads H λij = (cid:104) φ λi | H | φ λj (cid:105) = t λ e ( k ) = t λ (cid:80) α e − i kb α with t λ = a r X i v : . [ c ond - m a t . m e s - h a ll ] J un (cid:104) φ λi ( r − R i ) | H | φ λj ( r − R j ) (cid:105) and the nearest-neighbor con-necting vectors b α . To obtain Eq. (1), we have exploited thesymmetry of the lattice resulting in H λij = H λ ∗ ji and neglect-ing the overlap of orbital functions of neighboring sites, i.e. S ij = (cid:104) φ λi | φ λj (cid:105) = δ ij . Finally, restricting our investigationsto the area around the high-symmetry K point in the Brillouinzone, we can further simplify the electronic band structure byperforming a Taylor expansion for small momenta k and byapplying the effective-mass approximation: E λ k ≈ σ λ (cid:18) E gap (cid:126) m λ k (cid:19) (2)with σ c = +1 , σ v = − and with the effective mass m λ = 2 (cid:126) E gap | t λ | , (3)that is given by the TB hopping parameter t λ . The latter de-termines the curvature of the electronic bands around the Kpoint. Solving the Schr¨odinger equation, we also obtain theTB coefficients: C λX k = (cid:0) | g λ k | (cid:1) − , C λM k = C λX k g λ k (4)with g λ k = t λ e ( k ) (cid:16) E gap − E λ k (cid:17) − .Now, we have access to the electronic band structure andthe electronic eigen function of unstrained TMDs. Puttingthese materials under strain leads to two effects having impacton electronic and optical properties of TMDs: (i) geometriceffect and (ii) orbital overlap effect, cf. Fig. 1. The geometriceffect describes the change in the geometry of the lattice com-pared to the unstrained case (Fig. 1(a)). For biaxial strain,i.e. strain applied both to x and y direction, this simply im-plies an increase in the lattice constant a (Fig. 1(b)), whilefor uniaxial strain, i.e. strain applied only in one direction, thelattice vectors change differently in both directions leading toa broken lattice symmetry (Fig. 1(c)). In momentum space,the Brillouin zone changes accordingly: biaxial strain leads toan uniform decrease of the zone, while uniaxial strain impliesa compression only in the direction of the applied strain. Be-side the pure geometric effect, strain has also an effect on theoverlap of the atomic orbitals, cf. the lower panel of Fig. 1.Here, the crucial property is the overlap of M and X orbitalfunctions. Due to the strain-induced increase in the distancebetween atoms, the overlap of the orbitals is reduced. The ef-fect is more pronounced for biaxial strain, while for uniaxialstrain the broken lattice symmetry is transferred to the orbitalshape (elliptic form). In this work, we focus on tensile strain,however the gained insights can be also applied to compres-sive strain.We implement the geometric effect of the strain by in-troducing the strain matrix S = (cid:18) s x s y (cid:19) , where s i =1 ± s [%]100% with s denoting the strain value and ± represent-ing tensile and compressive strain, respectively. The basis vectors in real (momentum) space transform to a i → S a i ( k i → S − k i ) which leads to an increase (decrease) of thehexagonal lattice, cf. the upper panel in Fig. 1. We neglectchanges in the z-direction, since the effect has been recentlyshown to be negligibly small [17].Within the nearest-neighbor TB approximation the atomic or-bitals φ λj appear in integrals of the form (cid:104) φ λi | H | φ λj (cid:105) . Since weare not interested in the exact shape of the orbitals, but only intheir strain-induced change, we assume effective 1s hydrogen-like atomic orbitals φ λj ( r ) = N j exp ( − r −S R j σ j,λ ) with the nor-malization constant N j , the atomic positions R j , and the or-bital width σ j,λ . We allow the width to change with strainand find a self-consistent solution by benchmarking the the-ory to experimentally observed strain-induced shifts in opti-cal spectra [5, 6]. Inserting this ansatz in Eq. (2) we can findanalytic expressions for the strain-dependent electronic bandgap E gap ( s ) and the TB hopping integral t λ ( s ) in the case ofsymmetric biaxial strain ( s x = s y = s ): E gap ( s ) = (cid:126) m s (cid:88) λ (cid:16) σ − i,λ − σ − j,λ (cid:17) , (5) t λ ( s ) = 2 (cid:126) sm ( σ i,λ + σ j,λ ) . (6)Both quantities show a clear dependence on strain predom-inantly via the orbital overlap effect (reflected by σ i,λ , σ j,λ ):the band gap decreases with s , while the hopping integral lin-early increases with s . The slope of the increase/decrease de-pends on the widths of the atomic orbitals and is hence TMDspecific - in agreement to experimental observations, wherewe find a band gap reduction of approximately 50 meV/ % strain in WSe and MoS [5, 9]. We exploit these experimen-tal findings to benchmark our theory by adjusting the widthsand the overlap of the atomic orbitals.The strain-induced change of the band gap (Eq. (5)) givesrise to a red-shift of electronic resonances in optical spectra,cf. the dashed gray line in Fig. 2 (a). The strain-inducedchange of the TB hopping integral (Eq. (6)) determines thevariation of the effective mass, i.e. the inverse band curva-ture. We observe a clear decrease in the effective mass of theconduction band (Fig. 2 (b)) and a corresponding increase inthe band curvature (Fig. 2 (c)). We predict a reduction ofthe effective mass by 3 % for WS (purple line) and 4 % forMoS (orange line) in the case of 1 % applied uniaxial strain.In case of biaxial strain, the effect is roughly twice as large.For the effective mass in the valence band, we obtain similarresults. Our results are in good agreement with DFT calcula-tions (dashed lines) [14, 15].By performing these studies, we have benchmarked our the-ory with available experimental and DFT studies regarding theimpact of strain on the electronic properties of TMDs (bandgap and band curvature). Now, we include excitonic effectsand investigate how they change in presence of bi- and uni-axial strain. Excitons are integrated by solving the Wannierequation providing access to eigenvalues and eigen functionsfor all available excitonic states [2, 20–23]. Furthermore, wederive the TMD Bloch equation for the microscopic polariza-tion p vc k , k ( t ) = (cid:104) a + c, k a v, k (cid:105) ( t ) giving access to excitonicoptical response of TMDs [18]. This microscopic quantity is ameasure for optically induced transitions from the state ( v, k ) to ( c, k ) that are characterized by the electronic momentum k i and the band index λ i = ( v, c ) denoting the valence andthe conduction band, respectively [18].Since excitonic effects are known to dominate optical prop-erties of TMDs [2, 24, 25], we project the microscopic po-larization into an excitonic basis [26] p vc k , k ( t ) → p vc qQ ( t ) = (cid:80) µ ϕ µ q p µ Q ( t ) with excitonic eigen functions ϕ q and the index µ representing the excitonic state. In this work, we focus onthe energetically lowest optically allowed A s state. Further-more, we introduce center-of-mass and relative momenta Q and q , where Q = k − k and q = m h M k + m e M k with theelectron (hole) mass m e ( h ) and the total mass M = m e + m h .The separation ansatz enables us to decouple the relative fromthe center-of-mass motion. For the relative coordinate includ-ing the reduced mass µ = m c m v m c + m v , we solve the Wannierequation [2, 20–22] E q ϕ q − (cid:88) k V exc ( k ) ϕ q − k = E bexc ϕ q (7)with the excitonic eigen function ϕ q , the excitonic bindingenergy E bexc , and the dispersion E q = (cid:126) q µ .To obtain the temporal evolution of the excitonic micro-scopic polarization p Q ( t ) , we solve the Heisenberg equa-tion of motion i (cid:126) ˙ p Q ( t ) = [ H, p Q ( t )] [18, 20]. This re-quires the knowledge of the many-particle Hamilton operator H = H + H c − l + H c − c including the free carrier contribution H , the carrier-light interaction H c − l and the carrier-carrierinteraction H c − c . To calculate the coupling elements, we ap-ply the nearest-neighbor tight-binding approach [18, 20, 21].Exploiting the fundamental commutator relations [20], we ob-tain the TMD Bloch equations for the excitonic microscopicpolarization [2]: ˙ p Q ( t ) = 1 i (cid:126) (cid:18) E exc + (cid:126) Q M − iγ (cid:19) p Q ( t ) + Ω( t ) δ Q , (8)The optical excitation is expressed by the Rabi frequency Ω( t ) = e m (cid:80) q ϕ ∗ q M vc ( q ) · A ( t ) including the optical ma-trix element M vc ( q ) and the external vector potential A ( t ) .Here, e denotes the electron charge and m the electron restmass, respectively. Taking into account only direct transitionswith Q = , we end up in an analytic expression for theabsorption coefficient corresponding to the well-known Elliotformula [20, 21]: α ( ω ) ∝ ω (cid:61) (cid:34) | (cid:80) q M vcσ ± ( q ) ϕ q | E exc − (cid:126) ω − iγ (cid:35) (9)Note that we have projected the optical matrix element to thepolarization direction of right- ( σ − ) and left-handed ( σ + ) cir-cularly polarized light, i.e. M vcσ ± ( q ) = M vcx ( q ) ± iM vcy ( q ) .The nominator determines the oscillator strength and crucially Energy [eV] (c) k E(k) m c [ m un s t r c ] Strain [%] WS MoS strained E gap E exc b Strain [%] E [ m e V ] A b s o r p ti on [ a . u . ] (a) E (b) s t r a i n FIG. 2. (a) Excitonic absorption spectra of unstrained (black) anduniaxially strained tungsten diselenide (WSe ) as exemplary TMDmaterial at 3 % strain. The observed red-shift stems from (i) a de-crease in the orbital overlap giving rise to a reduced band gap ( E gap )and hence a red-shift (dashed gray line) and (ii) the geometric effectleading to a decrease in the effective masses, which results in weakerbound excitons ( E bexc ) and hence a blue-shift of the unstrained peak.The inset shows the resulting energy shift ∆ E as a function of strainboth with (orange) and without (gray dashed) taking into account ex-citonic effects. (b) Strain-dependent decrease of the effective massin the conduction band for WS (purple) and MoS (orange). Ourresults (solid lines) are in good agreement with values obtained byDFT calculations (symbols and dashed lines) taken from Ref. [14](triangles) and Ref. [15] (squares). (c) Sketch of the effect of strainon the dispersion of the conduction band. depends on the TMD properties and the lattice symmetry. Thedenominator defines the energetic position E exc of the reso-nances in optical spectra E exc = E gap − E bexc with the elec-tronic band gap E gap and the excitonic binding energy E b exc .Furthermore, we have introduced a dephasing γ , which ac-counts for radiative decay of the excitonic polarization. Non-radiative channels have not been considered. They play a cru-cial role at higher temperatures [27].Evaluating Eq. (7), we have access to excitonic absorptionspectra of random TMDs. Figure 2 (a) shows the spectrum forthe exemplary WSe directly comparing the strained (orange)and the unstrained situation (black). Our approach enablesto extract the contribution of excitonic effects to the strain-induced shift of resonances. Interestingly, we find that thechange in the excitonic binding energy leads to a blue-shiftreducing the general strain-induced red-shift, cf. the inset ofFig. 2 (a). This can be ascribed to the smaller effective masses(Fig. 2 (b)) entering the Wannier equation through the reducedmass µ and resulting in smaller excitonic binding energies.Now, we discuss in detail the impact of strain on all quanti-ties appearing in the Elliot formula, i.e. the excitonic bindingenergies E b exc and excitonic wave functions ϕ q as well as theoptical matrix element M vcσ ± ( q ) and the radiative dephasing γ . The gained insights will allow us to understand the strain-induced change in the optical fingerprint of TMD materials. Strain-induced change of excitonic binding energies and wavefunctions
Strain enters in the Wannier equation (Eq. (7)) both throughthe geometric and the orbital overlap effect. From Eq. (6)follows directly for the reduced mass in case of biaxial strain µ ( s ) = µ s − , (10)where µ is the value for the unstrained case. For uniaxialstrain the geometric effect induces an anisotropy, i.e. E q = (cid:126) q µ → (cid:126) q x µ x + (cid:126) q y µ y . Exploiting the relation µ i = µ s − i andprojecting it to elliptic coordinates with the absolute value q and the angle φ q , the dispersion E q in the Wannier equationreads: E q ( s x , s y ) = (cid:126) q µ (cid:0) s x cos φ q + s y sin φ q (cid:1) (11)Here, we have to distinguish between uni- and biaxial strain,since the solution of the Wannier equation will be differentin an anistropic system. For biaxial strain with s x = s y ,the Wannier equation remains isotropic and only the reducedmass µ is smaller for tensile strain. As a result, biaxial strainaccounts for lighter and weaker bound excitons with radi-ally symmetric excitonic wave functions. The latter becomeslightly larger in amplitude and spectrally narrower (Fig.3(a)). In contrast, in the case of uniaxial strain the Wannierequation becomes anisotropic owing to the geometric effect.This results in anisotropic wave functions. To quantify thedegree of anisotropy, we calculate ∆ ϕ q = ( ϕ q x − ϕ q y ) /ϕ q y as a function of momentum (Fig. 3(b)). Note that we dividehere by ϕ yq (unstrained direction) to give percentage values ofthe strain-induced change. The anisotropy is zero for the un-strained case (black line) and increases with the applied uni-axial strain.Beside the change in the excitonic eigen function, strainalso induces a reduction of the excitonic binding energy E b exc ( s ) due to the smaller reduced mass that can be mainlyascribed to the orbital overlap effect, cf. Eq. (6). Our calcu-lations reveal a decrease of E b exc by 4 meV/ % applied uniaxialstrain in WSe . The effect is approximately twice as large inthe case of biaxial strain, cf. Fig. 3(c). The strain-dependentchange of the excitonic binding energy scales with s − ac-cording to Eq. (10), however, at the considered low strain val-ues s = 1 . − . the scaling is approximately linear. Theweaker bound excitons result in a blue-shift of excitonic res-onances, however, this strain-induced excitonic shift is muchsmaller than the general red-shift of the bandgap resulting inan overall red-shift of optical resonances, cf. Fig. 2(a). Thedashed lines in Fig. 3(c) show E b exc ( s ) in MoS , where we find δ ϕ [ % ] unstrained 1 %2 %3 %−20246 (c) q [nm ] -1 (a) E e x c [ e V ] strain [%]0.68 uniaxialbiaxial unstrained ϕ q [ a . u . ] b biaxial (b) uniaxial q FIG. 3. Strain induced changes in the (a) excitonic wave function φ q for biaxial strain, (b) ∆ ϕ q = ( ϕ q x − ϕ q y ) /ϕ q y expressing theanisotropy of the wave function for uniaxial strain, and (c) the exci-tonic binding energy for both bi- and uniaxial strain according to thesolutions of the Wannier equation (Eq. (7)) for WSe (solid lines)and MoS (dashed lines). The excitonic binding energy decreasesboth for uni- and biaxial strain due to the smaller reduced mass µ ( s ) .We find a decrease of 8 meV for 1% (25 meV for 3%) applied biaxialstrain for WSe and 11 meV (33 meV) for MoS . qualitatively the same behaviour, however the slope of the ap-proximately linear decrease is with 5 meV/ % uniaxial strainquantitatively larger. The difference to WSe can be tracedback to different atomic orbital functions, where the atomicmass of molybdenum is lighter than tungsten and thereforemolybendum-based TMDs tend to be generally more affectedby strain.Beside the change in the reduced mass µ ( s ) , the strain alsoaffects the Coulomb matrix element V exc appearing in Eq. (7).The latter is treated as a Keldysh potential including a consis-tent description of substrate-induced screening in quasi two-dimensional nanostructures [2, 29]. Hence, strain only entersvia the tight-binding coefficients. Our calculations reveal thatthis effect is negligible compared to the strain-induced changeof the reduced mass.Our work goes beyond the discussion of strain-inducedshifts, but rather focuses on the impact of strain on opticalproperties of TMDs that will be discussed in the followingsections. Strain-induced change of the optical matrix element
The optical matrix element M vc k = (cid:104) Ψ v k ( r ) |∇| Ψ c k ( r ) (cid:105) isgiven as the expectation value of the momentum operator p = − i (cid:126) ∇ [18, 30]. Exploiting the nearest-neighbor TB wavefunctions, we obtain an analytic expression for M vc ( k ) M vc k = c (cid:88) α =1 b α ( C v ∗ M k C cX k e i k · b α − C v ∗ X k C cM k e − i k · b α ) (12)where the constant c denotes the nearest-neighbour orbitaloverlap c = e √ a (cid:104) φ vj ( r − R j ) | p x | φ ci ( r − R i ) (cid:105) [2]. The opticalmatrix element exhibits, similar to the electronic band struc-ture, a strong trigonal warping effect, i.e. it shows a triangularshape around the K and K’ valleys. This reflects the threefoldsymmetry of the nearest neighbors in the real space lattice.Note that the optical matrix element is also strongly valleydependent, i.e. at the K (K’) point it is maximal, whereas itvanishes at the K’ (K) point for excitation with right (left)-handed circularly polarized light [2]. This is the microscopicorigin of the observed valley polarization in TMDs [31–33].Strain enters in Eq. (12) through the TB coefficients C λM ( X ) k and through the vectors b α connecting the nearestneighbors in the real space lattice. Figure 4 (a)-(d) shows thestrain-induced changes in the optical matrix element ∆ | M vcσ ± | projected to the direction of left- or right-circularly polarizedlight in the case of 1 % tensile strain. We find that strain has acomplex impact on the optical matrix element including areaswith positive (orange) and negative (purple) changes. In par-ticular, we observe an increase of the matrix element aroundthe K point both in uni- and biaxial case. This can be tracedback to a large extent to the orbital overlap effect: Insertingour ansatz for the atomic orbitals into the constant c appear-ing in Eq. (12) we find an analytic expression for the biaxialstrain: c ∝ s (cid:126) m ( σ λi + σ λj ) . (13)The larger the strain, the smaller are the orbital overlaps σ λi , σ λj resulting in an increase of the optical matrix element.Exploiting Eq. (4) and applying a Taylor expansion of Eq.(12) around the K point, we can further evaluate the opticalmatrix element yielding: M vc k = c C v ∗ X C cX i (cid:88) l,m k · ( b m − b l ) b m | b m | (14) × (cid:0) β v ∗ k e − i K · b l e i K · b m + β c k e i K · b l e − i K · b m (cid:1) , where we have introduced the abbreviation β λ k = t λ (cid:16) E gap − E λ k (cid:17) − . Neglecting the influence of strain on thetight-binding coefficients and focusing on the change of theconnecting vectors (geometric effect), which are responsiblefor the trigonal warping effect, i.e. b i → S b i , we find for thex (y) component of the optical matrix element: M x ( y ) ∝ i s x k x ( b xm − b xl ) + s y k y ( b ym − b yl ) (cid:112) ( s x b xm ) + ( s y b ym ) s x ( y ) b x ( y ) m . (15)For biaxial strain we find a simply relation M x ( y ) = sM x ( y ) where M x ( y ) is the unstrained optical matrix element. Here,strain has the same impact on x and y direction and so thetrigonal warping effect is fully conserved, cf. Fig. 4(a)-(b).For uniaxial strain, Eq. (15) shows that applying the strainin one direction affects the x and y components of the opticalmatrix element in different ways, which results in a distortedtrigonal warping effect, cf. Fig. 4(c)-(d). This feature can beclearly traced back to the geometric effect of the strain.Now, we evaluate the strain-induced changes of both | M vcσ − | and | M vcσ + | directly at the K point, cf Fig. 4(e)-(f). We find alinear increase of | M vcσ − | with strain, where the slope is largerin the case of biaxial strain: by applying 1% strain our calcu-lations reveal an increase of | M vcσ − | by 2 % (2.5%) for uniaxial | M - σ v c | M - σ v c un i a x i a l ( K ) | [ % ] unibipristine051015 ( K ) | [ % ] b i a x i a l K K' (f) strain [%] strain [%] (b) |M + | (d) |M - | (a) (c) (e) σ vc σ vc |M + | σ vc |M - | σ vc | M - σ v c | [ % ] FIG. 4. Influence of strain on the optical matrix element projected inthe direction of right ( σ − ) and left-handed ( σ + ) circularly polarizedlight. The change of | M vcσ − | and | M vcσ + | are shown for biaxial [(a)and (b), respectively] and uniaxial strain [(c) and (d), respectively,for 1% strain. Close to the K point, strain induces an increase (or-ange area) of the optical matrix element for both bi- and uniaxialstrain. Due to the valley selective excitation in TMDs, M vcσ + van-ishes at the K point in the unstrained case. In presence of uniaxialstrain, this valley-selective optical selection rule is softened resultingin | M σ + | (cid:54) = 0 . The change of | M vcσ − | and | M vcσ + | at the K point isplotted as a function of strain in (e) and (f), respectively. and 4 % (5%) for biaxial in WSe (MoS ). We find again thatthe effect is slightly more enhanced for MoS . Due to the op-tical valley-dependent selection rules, | M vcσ + | is zero at the Kpoint in unstrained TMDs. However, in the case of uniaxialstrain, we observe a softening of this selection rule due to thebroken symmetry stemming from Eq. (15). Even though theeffect is rather small (increase by 0.2- 0.3 % per 1 % appliedstrain in WSe and MoS ), it shows that strain can be princi-pally exploited to control the valley polarization.To further discuss this behavior, we find an analytic expres-sion for the optical matrix element directly at the K and K’point by evaluating the sums in Eq. (15) and directly pluggingin the coordinates of the these points: (cid:18) M x M y (cid:19) ∝ (cid:18) s x ± is y (cid:19) , (16)where +( − ) denotes the K (K’) point. For the projected opti-cal matrix element M vcσ ± = M x ± iM y follows:biaxial uniaxial M vcσ + ( K ) ∝ s x − s y (cid:54) = 0 (17) M vcσ − ( K ) ∝ s x + s y increase increase M vcσ + ( K (cid:48) ) ∝ s x + s y increase increase M vcσ − ( K (cid:48) ) ∝ s x − s y (cid:54) = 0 Here, we clearly see that the optical valley-dependent selec-tion rule does not apply anymore in the case of uniaxial strain.Figure 4(f) shows the strain-induced change of | M vcσ + | at theK point. For unstrained TMDs and for biaxial strain, | M vcσ + | is zero according to the optical selection rules. Applying uni-axial strain, | M vcσ + | increases with strain as predicted in Eq.(17). However, the effect is rather small and although it soft-ens the optical selection rules, it will be difficult to observethe decreased valley polarization at experimentally accessiblestrains. The effect also occurs in MoS (dashed line), where itis slightly more enhanced due to the larger overlaps of atomicorbital functions.In summary, the optical matrix element increases aroundthe K point as a function of applied strain. This can be tracedback to the orbital effect which leads to an increase of c (Eq.(13)) and the geometric effect changing the nearest-neighbourvectors b α and accounting for the distorted trigonal warpingeffect and the softening of valley-dependent optical selectionrules in the case of uniaxial strain. As the optical matrix ele-ment directly enters the Elliot formula (Eq. (9)), the observedchanges will have a direct impact on the the optical absorp-tion spectra. However, to fully understand the change in theoscillator strength of excitonic resonances, we investigate theinfluence of strain on the excitonic linewidth. Strain-induced change of the radiative dephasing
The excitonic linewidth is expressed by the dephasing con-stant γ in Eq. (9). We focus here on the radiative decay, whichis known to be the dominant dephasing channels at low tem-peratures, while at higher temperatures, phonon-induced non-radiative decay channels become important [27]. The impactof strain on these processes is beyond the scope of this work.The radiative decay rate is determined by spontaneousemission of light through recombination of carriers and hasbeen obtained by self-consistently solving the Bloch equationfor the excitonic polarization and the Maxwell equations in a2D geometry [27] γ = (cid:126) cµ E exc n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) q M vcσ ± ( q ) ϕ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (18)with cn describing the light velocity in the substrate materialand µ the vacuum permeability. As discussed in the previ-ous section, the optical matrix element increases with strainsuggesting an enhanced γ as a function of strain. However, total ϕ E γ r a d [ m e V ] A b s o r p ti on [ no r m a li ze d ] Energy [eV] 1 %2 %3 %00.20.40.60.811.644 1.646 1.648 1.65 1.652 1.654 1.656 (a)
WSe strain [%] (c) MoS (b) WSe M strain [%] pristine FIG. 5. Strain-induced broadening of the radiative linewidth. (a)Absorption spectra of strained WSe for 1-3 % of biaxial strain. Tofocus on the linewidths, the excitonic resonances are normalized andshifted to the peak in the unstrained case. We observe a clear strain-induced increase of the radiative linewidth. Quantitative evaluationof the strain-dependent change in (b) WSe and (c) MoS . We findan increase from 2 meV in the unstrained case to 2.3 meV (2.4 meV)for 1% biaxial strain in WSe (MoS ). The total broadening (solidpurple line) is due to the strain-induced change of the excitonic en-ergy (dashed orange), the optical matrix element (dashed red), andthe excitonic wave function (dashed blue), cf. Eq. (18). The ma-trix element turns out to have to play the predominant role for theobserved broadening. the appearing excitonic wave function and the excitonic res-onance also have an influence on the final broadening. Fig-ure 5(a) demonstrates that the radiative linewidth generallyincreases with biaxial strain in WSe . We find an enhance-ment from from 2 meV in the unstrained WSe (black line)to 3.2 meV for 3 % biaxial strain (purple line), cf. Fig. 5(b).We obtain a very similar behavior for MoSe (Fig. 5(c)) anduniaxial strain (not shown), where the increase of the broad-ening is smaller (2.6 meV and 2.7 meV for 3 % uniaxial strainin WSe and MoS , respectively).To get a deeper understanding of the underlying micro-scopic processes, we evaluate the increase of the radia-tive linewidth by considering separately the strain-inducedchanges in (i) the excitonic energy E exc , (ii) the optical matrixelement M vcσ ± ( q ) , and (iii) the excitonic wave function φ q , cf.the dashed lines in Figs. 5(b)-(c) for WSe and MoS , re-spectively. Our calculations demonstrate that the optical ma-trix element plays the crucial role for the observed generalincrease of the radiative linewidth. In contrast, the excitonicwave functions actually reduce the radiative decay due to thestrain-induced spectral narrowing of the wave functions (Fig.3(a)). Finally, the excitonic energy becomes smaller in pres-ence of strain resulting in a larger radiative broadening (Eq.(18)), however the effect is relatively small compared to theimpact of the optical matrix element.In conclusion, we have presented microscopic insights intothe impact of uni- and biaxial strain on the optical fingerprintof atomically thin transition metal dichalcogenides. Combin-ing Wannier and Bloch equations with the nearest-neighbortight-binding approximation, we derive analytic expressionsfor the strain-induced change in the (i) effective masses givingrise to a reduction in the excitonic binding energy, (ii) opticalmatrix element resulting in a softening of valley-dependentoptical selection rules in the case of uniaxial strain, and(iii) radiative broadening of the excitonic resonances. 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