Ultrafast dynamics in monolayer TMDCs: the interplay of dark excitons, phonons and intervalley Coulomb exchange
Malte Selig, Florian Katsch, Robert Schmidt, Steffen Michaelis de Vasconcellos, Rudolf Bratschitsch, Ermin Malic, Andreas Knorr
UUltrafast dynamics in monolayer TMDCs: the interplay of dark excitons, phonons and intervalleyCoulomb exchange
Malte Selig , Florian Katsch , Robert Schmidt , Steffen Michaelis deVasconcellos , Rudolf Bratschitsch , Ermin Malic , and Andreas Knorr Nichtlineare Optik und Quantenelektronik, Institut f¨ur Theoretische Physik,Technische Universit¨at Berlin, 10623 Berlin, Germany Institute of Physics and Center for Nanotechnology, University of M¨unster, 48149 M¨unster, Germany and Chalmers University of Technology, Department of Physics, SE-412 96 Gothenburg, Sweden
Understanding the ultrafast coupling and relaxation mechanisms between valleys in transition metal dichalco-genide semiconductors is of crucial interest for future valleytronic devices. Recent ultrafast pump-probe exper-iments showed an unintuitive significant bleaching at the excitonic B transition after optical excitation of theenergetically lower excitonic A transition. Here, we present a possible microscopic explanation for this sur-prising effect. It is based on the joint action of exchange coupling and phonon-mediated thermalization intodark exciton states and does not involve a population of the B exciton. Our work demonstrates how intra- andintervalley coupling on a femtosecond timescale governs the optical valley response of 2D semiconductors. Introduction
Monolayer transition metal dichalcogenides(TMDCs) gained much attention due to their remarkable andrich intra- and intervalley exciton physics.[1–7] The TMDCelectronic bandstructure is illustrated in figure 1: The en-ergetically lowest optically addressable excitons are formedfrom electrons and holes located at the K and K (cid:48) points atthe corners of the hexagonal Brillouin zone [8, 9]. Thesetransitions exhibit a circular dichroism i.e. the K ( K (cid:48) ) pointcan be excited with left handed (right handed) polarized light σ + ( σ − ) allowing to selectively excite the opposite valleys atthe corners of the Brillouin zone [10]. On short time scales,polarization ( σ + , σ − ) resolved pump-probe signals revealedan ultrafast intervalley coupling finding a sub-picosecondtimescale for the intervalley transfer between the pumped andunpumped valley[11–17].In this Letter, we study the ultrafast intervalley transfer andrelaxation dynamics including simultaneous intervalley ex-change and phonon scattering and discuss their role for exper-imentally observed bleaching in transient pump-probe spec-tra: After optical excitation of the A transition, a significantbleaching at the B transition was observed.[18–21] Since theenergy of the probed B exciton is hundreds of meV larger[9]compared to the A exciton this type of response comes as asurprise. In particular the helicity resolved data in WS showthat the bleaching of the B transition occurs not only in the un-pumped valley but also within the pumped valley which, with-out considering further interactions, can only be attributed toan ultrafast spin flip of the electron[21] or to a mixture of A and B exciton states[22]. Interestingly, it was found that thebleaching of the B exciton in the unpumped valley rises fastercompared to the pumped valley after optical excitation of the A transition [19–21].There are several mechanisms which might explain theseresults: (a) A spin flip within the pumped valley[25, 26] wouldlead to a population of the opposite-spin conduction band,visible as a B transition bleaching in the same valley[21].(b) Dexter-like intervalley exciton coupling due to electronicwavefunction overlap in k-space where an A exciton in the FIG. 1.
Schematic illustration of the optical selection rules inMoSe A pronounced spin-orbit coupling lifts the degeneracy of theelectronic bands where the energetic ordering of the different spinbands (blue, red) is reversed between the K to the K (cid:48) point [9, 24].The spin splitting of the conduction band is on the order of few tensof meV and the splitting of the valence band is on the order of fewhundreds of meV. This results in two distinct optical transitions at the K ( K (cid:48) ) point where the energetically lower (higher) is called the A ( B ) transition [9]. pumped valley couples directly to the B exciton in the un-pumped valley, whereas the B exciton in the pumped valleyis driven through intervalley exchange coupling between B excitons[19, 20]. However, so far, only rough estimates forthe coupling strength are avilable[19, 20]. (c) Beyond inter-valley coupling also intravalley exchange coupling leads to anexciton-state mixing between A and B excitons but no polar-ization selective study of this effect is known yet[22]. Theprocesses (b) and (c) are strongly off-resonant and a carefulevaluation of the actual coupling strength via ab initio meth-ods is required. Another possible explanation for the observed B transition bleaching is, (d) that the B exciton populationis driven through stimulated exciton-exciton scattering[23].This process is strongly excitation dependent, and thereforecontributes only beyond a strict χ (3) limit.All proposed mechanisms (a)-(d) may explain specific fea-tures of the polarization resolved ultrafast experiments but sofar none of them has explained all related features. There-fore, in this Letter we introduce a combined mechanism (e),exhibiting several aspects of the mechanisms (a)-(d), at the a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug FIG. 2.
Schematic illustration of the intervalley exchange inMoSe (a-d) and the pump-probe experiment (e-f) (a) Excitationresonantly to the A transition with left handed polarized light σ + (b) Exciton phonon scattering mediates the thermalization of exci-tons: exciton densities in ( K ↑ , K ↑ ) states with non-vanishingcenter of mass momenta are created. (c) Excitons in ( K ↑ , K ↑ ) and ( K (cid:48) ↓ , K (cid:48) ↓ ) states are coupled through intervalley exchangecoupling. (d) Exciton phonon scattering between excitonic ↓ statesoccurs, which also leads to the population of ( K (cid:48) ↓ , K ↓ ) states. (e)Exemplary for the bleaching of the B transition with σ + probe pulse,we find contributions from electron of the ( K (cid:48) ↓ , K ↓ ) exciton. (f)For the bleaching of the A transition with σ + probe pulse, we findcontributions from electron and hole of the ( K ↑ , K ↑ ) exciton andfrom the hole of the ( K ↑ , K (cid:48) ↑ ) exciton. same time explaining three very specific signatures, seen inall related experiments:(i) the ultrafast intervalley transfer between the A excitons,(ii) the bleaching of the B transition after excitation of theA exciton as well as(iii) the temporal ordering of the rise times of the differentsignals and their temperature dependence.Here, we put these observations on a joint consistent the-oretical footing: As the dominating mechanism we identifythe combined action of the intervalley exchange coupling be-tween the A excitons in both valleys [13, 27–31] and phononmediated scattering to exciton states consisting of electron andhole at opposite K points in the 1. Brillouin zone[6, 32] sat-urating the B exciton transition. Figure 2 schematically illus-trates the proposed mechanism:First, figure 2 (a), an A exciton is excited in the K val- ley with σ + light. Second, figure 2 (b), exciton scatteringwith phonons creates excitons at non-vanishing center of massmomenta, including excitons with electron and hole being lo-cated at the same and different high symmetry point. Third,figure 2 (c), these excitons, having non-vanishing center ofmass momenta initialize the intervalley exchange coupling tothe unpumped valley. Finally, figure 2 (d), the excitons in theunpumped valley scatter to intervalley exciton states. We pro-pose, that this relaxation dynamics is experimentally visibleas a bleaching of the B transition on both valleys ( σ + , σ − ),cf. figure 2 (e), after optical excitation of the A transition in asingle valley ( σ + ).The bleaching results from Pauli blocking due to the co-bosonic nature of excitons[33]. In particular, for the probe ofthe B transition, we identify signatures from indirect excitons,where the electron of momentum space indirect excitons oc-cupies the conduction band referring to the B transition lead-ing to a bleaching of the latter in the pump-probe signal, cf.figure 2 (e). In contrast, for the probe of the A transition weidentify contributions from the hole of the momentum indi-rect excitons and from electron and hole of direct excitons, cf.figure 2 (f). Interestingly, we find that the response at the B transition in the pumped valley ( σ + ) rises slower compared tothe unpumped valley ( σ − ), which results from the step wiseprocess, cf. figure 2 (a-d), which is required for the formationof the B exciton σ + response. This result is in full agreementwith recent experiments[13, 18, 20, 21]. We present detailednumerical calculations for the exemplary material MoSe , butfind that our results are also applicable to other TMDC mate-rials. Theoretical Model
We introduce exciton operators P ξ h ξ e Q where ξ e/h = ( i e/h , s e/h ) denotes a compound indexconsisting of valley and spin of the electron/hole and theexcitonic center of mass momentum Q and an excitonicHamiltonian[33, 34] to derive the corresponding equations ofmotion in the Heisenberg picture for expectation values of theexcitonic transition (cid:104) P ξ h ξ e Q (cid:105) and the incoherent exciton den-sity N ξ h ξ e Q = δ (cid:104) P † ξ h ξ e Q P ξ h ξ e Q (cid:105) , cf. the supplementary materialI. The equation of motion for the pumped A exciton transition (cid:104) P ξ h ξ e Q =0 (cid:105) in the K valley (in the incoherent χ (3) limit [35])reads: i (cid:126) ∂ t P Aσ + = i (cid:126) ∂ t P K ↑ K ↑ = (cid:0) E K ↑ K ↑ − iγ K ↑ K ↑ (cid:1) P K ↑ K ↑ + d K ↑ · E pump ×× (cid:16) − (cid:88) K i e Ξ K ↑ , ( K ↑ i e ↑ ) K N K ↑ i e ↑ K − Ξ K ↑ , ( K ↑ K ↑ ) | P K ↑ K ↑ | (cid:17) + (cid:88) K ,ξ h ,ξ e W K ↑ , ( ξ h ,ξ e ) K (cid:0) | P ξ h ,ξ e K | + N ξ h ,ξ e K (cid:1) P K ↑ K ↑ . (1)For the B transition in the K valley, tested by the probepulse, the equation of motion reads: i (cid:126) ∂ t P Bσ + = i (cid:126) ∂ t P K ↓ K ↓ = (cid:0) E K ↓ K ↓ − iγ K ↓ K ↓ (cid:1) P K ↓ K ↓ + d K ↓ · E probe (cid:16) − (cid:88) K Ξ K ↓ , ( K (cid:48) ↓ K ↓ ) K N K (cid:48) ↓ K ↓ K (cid:17) + (cid:88) K ,ξ h ,ξ e W K ↓ , ( ξ h ,ξ e ) K (cid:0) | P ξ h ,ξ e K | + N ξ h ξ e K (cid:1) P K ↓ K ↓ . (2)The equations of motion for the A/B transition in the K (cid:48) val-ley can be obtained by exchanging the high symmetry points K ↔ K (cid:48) and spins ↑↔↓ . In both equations, the first line ac-counts for the excitonic dispersion with the excitonic energy E ξ h ξ e Q and the dephasing, where γ ξξ is determined by radiativedecay and exciton phonon coupling [6, 36, 37]. The secondline accounts for the optical excitation by the field E via thedipole moment d K ( , ) ↑ ( ↓ ) and for the excitonically modifiedPauli blocking of the constituent carriers forming the excitons.The appearing form factors Ξ ξ, ( ξ h ξ e ) K are a consequence of theco-bosonic commutation relation of excitons[33], equationS5. Evaluating the appearing sum over the excitonic valley inequation 1 we identify that the bleaching of the A exciton tran-sition results from intravalley ( K, K ) exciton and the holes ofintervalley ( K, K (cid:48) ) exciton, cf. figure 2 (f), whereas for thebleaching of the B transition we find contributions from theelectrons of intervalley ( K, K (cid:48) ) excitons only, cf. figure 2 (e).The third line in the equations 1 and 2 schematically describethe energy renormalization due to exciton-exciton interactionwith the coupling element W ξ, ( ξ h ξ e ) K , equation S6. Note, thatwe do not calculate the energy renormalization and furthermany body effects due to Coulomb interaction, which is alsorequired for the interpretation of pump-probe experiments, asshown in [18], but focus on the bleaching. The equations 1and 2 together with the equation of motion of the exciton den-sity, equation S13, could be solved iteratively in orders of theexciting optical field to obtain an analytic expression for thethird order susceptibility. Results
To clarify the temporal pathways of the ex-citons after optical exciation, we numerically evaluatethe coupled dynamics of the excitonic coherence P K ↑ K ↑ Q ,the exciton densities N ξ h ξ e Q and the intervalley coherence δ (cid:104) P † K ↑ K ↑ Q P † K (cid:48) ↓ K (cid:48) ↓ Q (cid:105) Eqs. S10, S13 and S14. As it is dis-cussed in the supplementary, all parameters are determined bymicroscopic coupling elements. For our numerical evaluationwe explicitly include excitons ( i h s, i e s ) for both spin bands s = ↑ , ↓ but identical spin for electron and hole forming theexcitons (optically spin allowed) with electron and hole in thesame and in different valleys i h (cid:54) = i e . For the hole valleys weinclude i h = K, K (cid:48) and for the electronic valleys we include i e = K, K (cid:48) , Λ (half way between K and Γ , also referred to as Q or Σ [7, 9]) , Λ (cid:48) . This way, exploiting equations 1 and 2 wehave access to the exciton bleaching B ξ = (cid:88) K ,ξ h ,ξ e Ξ ξ, ( ξ h ,ξ e ) K N ξ h ,ξ e K . (3)Figure 3 depicts the polarization ( σ + , σ − ) resolved bleaching of the B / A excitons seen by a probe pulse (pink wavy line) asa function of time after optical pump with a 20 fs σ + pulse res-onant to the A transition at an exemplary temperature of 77 Kin MoSe . On top figure 3 additionally illustrates the differentbleaching contributions of different intervalley ( K, K (cid:48) ) andintravalley ( K, K ) excitons. For all different contributions tothe bleaching of A and B excitons we find an ultrafast rise,faster than 1 ps:Figure 3 (a) depicts the temporal evolution of the bleachingof the B transition in the pumped valley. As the only con-tribution we find bleaching from ( K (cid:48) ↓ , K ↓ ) excitons. Thesignal rises relatively slow within approximately 700 fs, sincethe formation of these excitons requires a stepwise processas depicted in figure 2 (a-d): first excitons from the pumped ( K ↑ , K ↑ ) states have to couple to ( K (cid:48) ↓ , K (cid:48) ↓ ) exci-ton states through intravalley phonon scattering (or generate Q (cid:54) = 0 ) and intervalley exchange, cf. figure 2 (c), and a sub-sequent phonon scattering to intervalley ( K (cid:48) ↓ , K ↓ ) states,cf. figure 2 (d). Such a dynamics is observed as a block-ing of the probe pulse and contribute similar to the proposedexciton-upconversion or spin-flips[21, 23].Figure 3 (b) shows the temporal evolution of the bleachingof the B transition in the unpumped valley. The only con-tribution is the bleaching from ( K ↑ , K (cid:48) ↑ ) excitons. Thesignal rises faster compared to the signal in the pumped val-ley with a time constant of 120 fs, cf. figure 3 (a), sincehere the contributing ( K ↑ , K (cid:48) ↑ ) excitons can be directlyformed through intervalley phonon scattering from the opti-cally pumped ( K ↑ , K ↑ ) excitons, cf. figure 2 (b).Figure 3 (c) depicts the temporal evolution of the bleachingof the A transition in the pumped valley, where beyond theincoherent excitons N ξ h ξ e Q , also coherent excitons | P K ↑ K ↑ Q | contribute for the resonant probe. As the dominating contribu-tion we identify bleaching from ( K ↑ , K ↑ ) excitons and anadditional small contribution from ( K ↑ , K (cid:48) ↑ ) excitons. Thesignal rises with the pump pulse, since the ( K ↑ , K ↑ ) exci-tons are optically pumped. The additional small contributionfrom intervalley ( K ↑ , K (cid:48) ↑ ) excitons is due to phonon medi-ated thermalization of excitons. Since in MoSe these statesare located energetically above the bright state by some fewmeV, their contribution is small at cryogenic temperatures.Finally figure 3 (d) depicts the temporal evolution of thebleaching of the A transition in the unpumped valley. Wefind dominating contributions from ( K (cid:48) ↓ , K (cid:48) ↓ ) excitonsand a small contribution from ( K (cid:48) ↓ , K ↓ ) excitons. Thesignal rises relatively fast with a time constant of 200 fs,since the ( K (cid:48) ↓ , K (cid:48) ↓ ) excitons are formed from the pumped ( K ↑ , K ↑ ) excitons through intravalley phonon scatteringand intervalley exchange coupling. The small contributionfrom ( K (cid:48) ↓ , K ↓ ) excitons is due to the phonon mediatedthermalization, as for the pumped valley the ( K (cid:48) ↓ , K ↓ ) ex-citon states are located energetically above the ( K (cid:48) ↓ , K (cid:48) ↓ ) excitons, explaining their small contribution to the signal.All in all, the temporal sequence of the signatures at both B transitions was also observed experimentally recently inWS [20, 21]. FIG. 3.
Bleaching of the pump-probe signal at 77 K in MoSe . Contributions from exciton states with spin ↑ are denoted in blue, spin ↓ contributions are denoted in red, as in figure 2. In the upper panel, we illustrate the excitons leading to the bleaching of the excitonic transition,which is depicted in the lower panel. Contributions from intravalley excitons are illustrated with solid lines, contributions from intervalleyexcitons are denoted with dashed lines. (a,b) Pump-probe signal at the B transition, compare equation 2, in the pumped (a) and unpumped (b)valley after excitation of the A transition. (c,d) Pump-probe signal at the A transition, compare equation 1, in the pumped (c) and unpumped(d) valley after excitation of the A transition. Additionally we depict the additional bleaching contribution of the optically injected coherence | P | . The pump pulse with a HWHM of 20 fs centered at t = Extracted density rise times in MoSe . (a) shows the risetime of the direct exciton densities as a function of temperature and(b) shows the indirect exciton densities. Due to the large energetic separation of 150 meV from the A exciton in MoSe [9], we find that excitons involving theelectronic Λ valley have neglectible occupations and a van-ishing influence. For the same reason (180 meV detuning) weneglected the lower split off valence band.To get a more quantitative analysis with respect toexperiments[13, 18–21], we extract the rise times of thewavenumber integrated exciton occupations, c.f. figure 4, byfitting their initial temporal evolution exponentially. For the ( K ↑ , K ↑ ) exciton density, cf. figure 4 (a), we find decreas-ing rise times as a function of temperature (about 50 fs at 50 Kand about 10 fs at room temperature). This can be understoodfrom the fact, that the ( K ↑ , K ↑ ) exciton density is directlyformed and the optically pumped excitonic coherence fastlyconverted through intravalley phonon scattering to incoher-ent excitons[6, 32]. For the ( K (cid:48) ↓ , K (cid:48) ↓ ) density we findan increase from about 160 fs at 50 K to about 330 fs at roomtemperature. Here, several counteracting effects contribute:At elevated temperatures, since the intervalley exchange cou- pling increases with exciton momentum, a hotter exciton dis-tribution leads to a faster intervalley transfer. Counteractinghowever is the intervalley coherence damping, equation S8,due to phonon scattering at elevated temperatures. Furtherat elevated temperatures more excitons occupy the indirect ( K ↑ , K (cid:48) ↑ ) states, effectively slowing down the intervalleytransfer.In figure 4 (b) the risetimes of the intervalley exciton occu-pation is depicted. We find the rise time of the ( K ↑ , K (cid:48) ↑ ) states decreasing from 90 fs at 50 K to about 40 fs at room tem-perature, being consistent with the direct formation throughphonon scattering from the optically induced exciton coher-ence. The slowest risetime we find for the ( K (cid:48) ↓ , K ↓ ) ex-citons, ranging from about 680 fs at 50 K to about 460 fs atroom temperature. The reason for this comparably slow risetime is that for the formation of these excitons first IEC fromthe ( K ↑ , K ↑ ) states to ( K (cid:48) ↓ , K (cid:48) ↓ ) states and after thata phonon mediated scattering from the latter to ( K (cid:48) ↓ , K ↓ ) states is required. Our calculated time scale is in good agree-ment with the experimentally measured rise times of the B signal of about 200 fs at 77 K in WS [21]. We are awarethat in other TMDCs the excitonic landscape deviates fromthe situation in MoSe [32], and therefore a different tempera-ture dependence can occur. However, we expect, that at leastthe qualitative behavior including the temporal ordering of thedifferent signals as well as the order of magnitude of the risetimes are similar in other materials. Additional to the dis-cussed mechanism also spin-flip processes have been foundto contribute to the bleaching of the B transition[21]. In ouranalysis we do not observe any excitation power dependenceof the rise times, since we restricted our analysis to the lowdensity limit to keep the already high numerical complexityon a moderate level. Conclusion
The microscopic relaxation dynamics includ-ing simultaneous intervalley exchange and intervalley phononscattering significantly contributes to the theoretical under-standing of recent experimental findings, in particular to unin-tuitive experimental results in helicity-resolved ultrafast pump( A excitons) probe ( B excitons) experiments. All presentedcalculations are consistent with recent experimental observa-tion: (i) the fast transfer between the A excitons due to in-tervalley Coulomb exchange [13], (ii) The bleaching at the B transition after optically exciting the A transition [18, 20] and(iii) the faster response at the B transition in the unpumpedvalley compared to the pumped valley[21]. However, it doesnot explain the photoluminescence emission from the B exci-ton under A exciton excitation[23], since no B excitons (in-volving a hole in the lower valence band) are created. Acknowledgements
We acknowledge fruitful discussionswith Dominik Christiansen (TU Berlin). This work wasfunded by the Deutsche Forschungsgemeinschaft (DFG) -Projektnummer 182087777 - SFB 951 (project B12, M.S.,A.K.). This project has also received funding from the Eu-ropean Unions Horizon 2020 research and innovation pro-gram under Grant Agreement No. 734690 (SONAR, F.K.,A.K.). E.M. acknowledges financial support from the Euro-pean Unions Horizon 2020 research and innovative programunder grant agreement No. 696656 (Graphene Flagship) aswell as from the Swedish Research Council (VR). [1] Berkelbach, T. C., Hybertsen, M. S. & Reichman, D. R. Theoryof neutral and charged excitons in monolayer transition metaldichalcogenides.
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