SUPPLEMENTARY INFORMATION SUPPLEMENTARY INFORMATION. Opto-valleytronic imaging of atomically thin semiconductors

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1 In the formt provided y the uthors nd unedited. DOI: 0.038/NNANO Opto-vlleytroni imging of tomilly thin semiondutors Andre Neumnn, Jessi Lindlu, Léo Colomier, Mnuel Nutz, Sin Njmei, Jun Lou, Adity D. Mohite, Histo Ymguhi, nd Alexnder Högele Smple frition nd si hrteriztion The hemil vpor deposition (CVD) of monolyer MoS 2 rystls ws dopted from Njmei nd oworkers. Briefly, SiO 2 /Si sustrtes with MoO 3 seeding prtiles were pled in qurtz furne in the presene of sulfur powder. The furne ws heted to 900 C for 5 min with flow of inert gses (N 2 nd Ar) under tmospheri pressure. After ooling down to room temperture, synthesized MoS 2 rystls were trnsferred onto p-doped SiO 2 /Si sustrtes with onventionl polymethyl methrylte (PMMA) trnsfer method 2. The PMMA ws removed y rinsing the resulting smple with MoS 2 on SiO 2 /Si in n etone th for three yles of 5 min. Optil nd tomi fore mirosopy (AFM) imges of typil MoS 2 monolyer flkes fter trnsfer onto SiO 2 /Si re shown in Supplementry Fig.. nm Supplementry Fig. :, Optil imge nd,, tomi fore mirogrphs of monolyer MoS 2 grown y hemil vpor deposition. The sle rs re 0 µm in, nd 2 µm in nd. Arrows indite seeds of ilyer formtion on top of monolyer flkes in nd. Note the 80 misorienttion of the lrge monolyer nd the smll ilyer tringles hrteristi of idel Bernl stking indited y the white dshed lines in. An exmple of n extended region of surfe ontmintion of single-rystl tringle is indited y the rrow t the t the lower edge in. NATURE NANOTECHNOLOGY

2 2 Experimentl setup nd settings Cryogeni onfol spetrosopy, rster-sn nd wide-field imging were performed in n ultr-low virtion losed-yle ryostt (ttoue systems, ttodry000) with se temperture of 3. K nd superonduting mgnet with fields of up to ±9 T. The smple ws positioned with nnopositioners (ttoue systems, ANP0 series) into the fol plne of lowtemperture pohromt with numeril perture of 0.65 (ttoue systems, LT-APO/VIS/0.65) nd onfol exittion nd detetion spots with 0.7 µm full-width t hlf-mximum (FWHM) dimeters. Liner polrizers (Thorls, LPVIS nd LPVISB), hlf- nd qurter-wvepltes (B. Hlle, RAC 3 series) mounted on piezo-rottors (ttoue systems, ANR240) were used to ontrol the photon polriztion in the exittion nd detetion pthwys. PL spetrosopy ws performed with ontinuous wve (w) exittion lsers t 532 nm (CNI, MLL-III ) or 637 nm (New Fous, Veloity TLB- 6704), nd stndrd spetrometer (PI, Aton SP-2558) with liquid nitrogen ooled silion CCD (PI, Spe-0:00BR/LN, gin setting of 4 e /ount). The spetrl resolution of the system ws 0.35 mev for onfol PL nd 0.6 m for Rmn mesurements. A strek mer (Hmmtsu, C S) nd femtoseond (fs) optil prmetri osilltor (OPO) operted t 630 nm or 637 nm (Coherent, Mir-OPO) were used for time-resolved PL studies. The fier-oupled system exhiited dispersion-limited temporl resolution down to 0 ps depending on the spetrl ndwidth nd the length of the single-mode fier. For wide-field imging, the OPO ws tuned to 637 nm nd defoussed to illuminte smple re of mm 2 t n verge exittion power density of 200 W/m 2. PL imges with 50 mgnifition were quired in the spetrl nds of tunle nd-pss filters (Semrok, VersChrome) with silion CCD (Point Grey, GS3- U3-4S5M-C, pixel size of ( ) µm 2 nd sturtion pity of 7 ke ). All wide-field imges were reorded with n integrtion time of 500 s(300 s in Supplementry Fig. 0) nd CCD gin setting of 0 db (exept for 5 db in Fig. 4 of the min text nd 3 db in Supplementry Fig. 0 - d). The imges digitized to 6 it formt with 4 it onversion were quired with liner gmm enoding (gmm enoding of 0.7 in Fig. 4 of the min text). The intensities were multiplied with sling ftors rnging from.0 to.7 to spn the full dynmi rnge of the grysle (identil ftors were used for o- nd ross-polrized imges of the sme exiton nds). 3 Spetrosopy nd polrimetry experiments 3. Bsi spetrl hrteristis The PL of monolyer MoS 2 rystls on SiO 2 /Si ws exited either non-resonntly t 532 nm (2.33 ev) or t 637 nm (.95 ev) in resonne with the lue shoulder of the A exiton. A non-resonnt spetrum reorded on regulr position of the tringulr flke is shown in Supplementry Fig. 2 (sme s in Fig. of the min text). The non-resonnt PL exhiits ontriutions of oth neutrl nd hrged exitons, A nd A due to unintentionl doping of MoS 2 rystls on p-doped SiO 2 /Si sustrtes 3. The symmetri lineshpe of the totl PL (spetrum in Supplementry Fig. 2) n e deomposed into two Lorentzins with equl FWHM linewidths of 62 mev seprted y the trion inding energy of 30 mev 4. The ner-resonnt spetrum reorded with long-pss filter t 652 nm (.90 ev) in Supplementry Fig. 2 (gry nd lue tres show for the sme NATURE NANOTECHNOLOGY 2

3 flke position non-resonnt nd ner-resonnt spetr, respetively) exhiited dditionl shrp fetures identified s Rmn sttered photons y the sustrte (Si) nd the MoS 2 monolyer, with ssignments given in Supplementry Fig. 2 dopted from Ref. 5. The Rmn spetr were useful to identify ilyer regions (s in the enter of the single-rystl tringle) where they exhiited hrteristi splitting of the A g (Γ) mode shown in Supplementry Fig. 2d under resonnt exittion with the fundmentl exiton 6. To void ontmintion of the opto-vlleytroni properties of A exitons y Rmn photons, we seleted spetrl nd of 3 mev width entered t.879 ev ner the PL mximum of the A exiton emission nd wy from Rmn resonnes (the red spetrl intervl in Supplementry Fig. 2 - is the sme s in Fig. of the min text ). Moreover, for the quntittive nlysis of the degree of irulr polriztion of A exitons presented in Fig. 3e of the min text, the ontriution of A ws removed y fitting the totl PL with two Lorentzins nd sutrting the ontriution of the trion PL from the spetrl nd hosen for the opto-vlleytroni nlysis s desried ove. PL intensity (r. units) 0 A A 2 A Energy (ev) Rmn intensity (r. units) 0 2E 2g (Г) Energy (ev) A g (M) + LA (M) E 2g (M) + LA (M) 2E g (Г) Rmn shift (m - ) A Si A 2u (Г) / 2LA (M) ꞌꞌ E 2g (Г) A g (Г) BL ML d A g (Г) E 2g (Г) Supplementry Fig. 2:, Photoluminesene spetrum reprodued from Fig. of the min text (gry) with Lorentzin fits (ornge) to the ontriutions of the neutrl nd hrged exitons, A nd A, entered round.882 ev nd.852 ev, respetively, with full-width t hlf-mximum linewidths of 62 mev., Comprison of the photoluminesene spetr for non-resonnt exittion t 532 nm (2.33 ev) nd exittion t 637 nm (.95 ev) in resonne with the lue shoulder of the A exiton (gry nd lue tres, respetively); the ner-resonnt spetrum ws reorded with long-pss filter t 652 nm (.90 ev)., Rmn spetrum for the ner-resonnt exittion t 637 nm is superimposed s shrp spetrl fetures on the photoluminesene pek in. The Rmn modes were ssigned ording to the MoS 2 ulk nottion of Ref. 5. The dt in nd were mesured on the sme regulr position of the monolyer tringle from the min text. A spetrl nd of 3 mev width (indited in red) ws seleted t the mximum emission of the A exiton nd wy from Rmn fetures to onstrut the polrimetri mps without ontmintion y Rmn photons. d, Vertilly offset Rmn spetr of the E2g(Γ) nd A g (Γ) modes for the ner-resonnt exittion t 637 nm on representtive monolyer (ML) nd ilyer (BL) positions of the tringulr flke (the dshed lines re guides to the eye). The splitting of the A g (Γ) mode is hrteristi of ilyer trnsition metl dihlogenides under resonnt exittion 6. All mesurements were otined t 3. K. NATURE NANOTECHNOLOGY 3

4 3.2 Time-resolved photoluminesene mesurements The PL dey dynmis of the monolyer MoS 2 tringle from the min text re presented in Supplementry Fig. 3. The A exiton PL, mesured in spetrl nd of 35 mev, exhiited rditive dey with time onstnt elow the resolution limit of 4 ps given y the dispersion-limited instrument response funtion (Supplementry Fig. 3). Lolized exitons, seleted y ndpss filter of 50 mev width, exhiited two dey timesles (Supplementry Fig. 3): the dominnt fst omponent ws resolution-limited (with prtil ontriution from A exitons), nd the slow omponent exhiited monoexponentil dey with lifetime of 74 ± 2 ps in greement with previous results 7. The slow dey hnnel ounted for 8% of the totl PL intensity on regulr position (L) nd for 2% in the puddle (L P ) of the tringulr rystl. 0 0 Normlized intensity 0 A irf 4 ps ± 2 ps Time (ps) L L P Supplementry Fig. 3:,, Time-resolved photoluminesene of A exitons (lue tre) nd lolized exitons in nd wy from the puddle, L P nd L (red nd lue irles). Bnd-pss filters with widths of 35 mev (50 mev) were used for spetrl seletion of A (L nd L P ) exitons. The dey dynmis in were limited y the instrument response funtion (irf) with mesured full-width t hlf-mximum of 4 ps (gry tre). The solid lines in re fits to the dt: the fst dey omponents of the lolized exitons were dispersion rodened, the slow omponents exhiited monoexponentil deys with dey onstnts of 74 ± 2 ps nd ontriutions reltive to the totl signl of 2%(L P, drk red tre) nd 8%(L, drk lue tre). All dt were mesured on the tringulr MoS 2 flke of the min text with lser exittion t 637 nm in nd 630 nm in ; the temperture ws 3. K. 3.3 Spetrl hrteristis of polriztion-resolved photoluminesene Supplementry Fig. 4 nd show o-polrized (I o ) nd ross-polrized (I r ) PL spetr reorded with n exittion lser t 637 nm (.95 ev) in irulr nd liner ses, respetively. The spetrl hrteristis of P nd P l, respetively shown in Supplementry Fig. 4 nd d, were lulted s the normlized differenes etween o- nd ross-polrized PL intensities ording to P =(I o I r )/(I o + I r ). Dt in Supplementry Fig. 4e nd f onfirm tht the degree of liner polriztion is independent of the hoie of the liner sis (the liner PL polriztion is prllel to the xis of the exittion lser set long 4 NATURE NANOTECHNOLOGY 4

5 π H nd π D in Supplementry Fig. 4e nd f, respetively). The red r in Supplementry Fig. 4 - d ws used for onfol opto-vlleytroni imging of A exitons (sme nd s in Supplementry Fig. 2 - nd Fig. of the min text). PL intensity (r. units) Degree of polriztion.0 A I o I r P.8.9 Energy (ev) d I o I r P l A.8.9 e f Supplementry Fig. 4:, Photoluminesene spetr reorded in o-polrized (solid tre, I o ) nd ross-polrized (dshed tre, I r ) onfigurtions with σ + exittion., Corresponding degree of irulr polriztion P., Co- nd ross-polrized photoluminesene spetr under liner (π H ) exittion nd d, degree of liner polriztion P l. The nds of 3 mev width indited in red (sme s in Supplementry Fig. 2 nd Fig. of the min text) were used to selet A exitons for onfol opto-vlleytroni imging. e, f, Polr plots of the normlized PL intensity within the spetrl nd s funtion of the rottion ngle θ of the liner nlyzer for π H, π D orienttions of the liner polrizer (indited y red rrows), respetively. The gry solid lines re fits to the dt with [ + P l os (2θ 2φ)] funtionl dependene, where φ is the polrizer ngle. All mesurements were reorded on the monolyer tringle wy from defets s disussed in the min text with n exittion lser t 637 nm (.95 ev) nd long-pss filter t 652 nm (.90 ev); the temperture ws 3. K. 3.4 Cirulr nd liner rster-sn polrimetry To onstrut polrimetri mps we rster-snned the smple with respet to fixed onfol exittion nd detetion spots, nd performed spetrl quisition of o- nd ross-polrized PL t eh rster pixel with n exittion power density of W/m 2. An verged kground spetrum ws sutrted from ll imge pixels, nd o- nd ross-polrized PL intensities of A nd L exitons were integrted within the spetrl nds shown in Fig. of the min text. Pixels with stndrd devition of P ove 0.05 (stemming from vnishingly smll PL intensities wy from the flke) were set to zero. In Supplementry Fig. 5 - d we reprodue the mps of Fig. 2 - d of the min text to indite speifi positions (numered from to 6) for quntittive omprison of site-to-site vritions of P (lue rs) nd P l (purple rs) summrized in Supplementry Fig. 5e for the of A exiton nd. Similr polrimetri mesurements were performed on other MoS 2 flkes. NATURE NANOTECHNOLOGY 5

6 Supplementry Fig. 6, nd Supplementry Fig. 6, d show profiles of P nd P l for poly-rystlline nd singlerystlline MoS 2 monolyer, respetively. The r hrt in Supplementry Fig. 6e summrizes the sttistis of P nd P l quired on five different monolyer flkes. e 2 d [] monolyer [2] tringle left edge [3] tringle right edge [4] puddle (point defets) [5] grin oundry [6] tringle ilyer P l P Degree of polriztion P P l Supplementry Fig. 5:,, Cirulr nd, d, liner polrimetri profiles for A exiton nds of single- nd poly-rystlline MoS 2, respetively. The dt re reprodued from Fig. 2 of the min text with speifi positions 6 dded for quntittive omprison (sle rs re 5 µm). e, P (lue rs) nd P l (purple rs) for A exitons t hrteristi positions of the flkes mrked with red irles in nd d. All dt were mesured t 3. K with lser exittion t 637 nm. e d P P l Degree of polriztion P P l Supplementry Fig. 6:,, Mps of P nd P l for segment of poly-rystlline MoS 2 flke (sle r is 0 µm)., d, Mps of P nd P l for single-rystl monolyer tringle (sle r is 3 µm). The r hrt in e summrizes vritions in P nd P l (lue nd purple rs, respetively) for five different MoS 2 monolyers t 3. K nd 637 nm exittion. 6 NATURE NANOTECHNOLOGY 6

7 3.5 Equivlene of polriztion ses t zero mgneti field In the sene of n externl mgneti field, time-reversl symmetry implies the equivlene of the degrees of irulr polriztion mesured in σ + nd σ onfigurtions. The set of dt in Supplementry Fig. 7 - demonstrtes tht the degree of the irulr polriztion is independent of the hoie of the irulr sis: P is identil (within the preision of our mesurement) for σ + nd σ polriztion ses (ompre Supplementry Fig. 7 nd, respetively). This ft is lso refleted y the vnishing differene profile P = P (σ + ) P (σ ) shown in Supplementry Fig. 7. The nlogous set of dt is respetively shown for the degrees of liner polriztion reorded under horizontl (π H ) nd digonl (π D ) linerly polrized exittions in Supplementry Fig. 7d nd e; their vnishing differene P l = P l (π H ) P l (π D ) is plotted in Supplementry Fig. 7f. The dt demonstrte the independene of P l of the hoie of the liner sis. σ + σ P P d π H e π D f P l P l Supplementry Fig. 7:,, P nd d, e, P l for irulrly polrized σ +, σ (right- nd left-hnded) nd linerly polrized π H, π D (horizontl nd digonl) lser exittion., Equivlene of the irulr polriztion ses, where hnges to P re prohiited y time-reversl symmetry, omputed s P = P (σ + ) P (σ ) 0. f, Sme for different liner polriztion ses: P l = P l (π H ) P l (π D ) 0. Note the hnged sle in nd f. All mesurements were spetrlly integrted nd reorded t zero mgneti field with lser exittion t 637 nm. The temperture ws 3. K. 3.6 Mgneti field dependene of the degree of irulr polriztion A finite externl mgneti field pplied in Frdy geometry (perpendiulr to the TMD monolyer surfe) lifts the vlley degenery nd gives rise to Zeemn shift of opposite sign for exitons in the K nd K vlleys. A positive mgneti field dereses (inreses) the energy of the K (K ) vlley exiton. Consequently, n pplied mgneti field hnges the degrees of irulr polriztion of the vlley exiton emission. The hnges re disussed in the min text for spetrlly seleted A nd L exitons under σ + nd σ exittion t +9 T (the orresponding dt of Fig. 3 re reprodued in Supplementry Fig. 8,, 7 NATURE NANOTECHNOLOGY 7

8 d, nd e). In Supplementry Fig. 8 nd f we plot for oth types of exitons the hnge in the degree of irulr polriztion, quntified s P = P (σ + ) P (σ ). For A exitons t B = +9 T, P is negtive (mgent-olored) throughout the MoS 2 flke (Supplementry Fig. 8). This orresponds to derese (inrese) of the σ + (σ ) degree of irulr polriztion for the lower (upper) K (K ) Zeemn rnh of A exitons. The L exitons show the sme trend wy from the puddle nd reversed fetures in the puddle (yn-olored region in Supplementry Fig. 8f) in ord with therml popultion distriution disussed in the min text. The right pnel of Supplementry Fig. 8 shows the sme set of dt ut for B = 9 T. The sign reversl of the mgneti field results in interhnged roles of σ + nd σ polriztions (Supplementry Fig. 8g, h, j, nd k) nd thus in sign reversl of P (Supplementry Fig. 8i nd l). A t +9T L t +9T A t -9T L t -9T σ + d σ + g σ + j σ +.0 σ e σ h σ k σ P P f i l -.0 Supplementry Fig. 8: Left pnel:,, nd d, e, Cirulr polrimetri profiles within the A nd L exiton nds under σ + nd σ exittions in mgneti field of B = +9 T reprodued from Fig. 3 of the min text., f, Corresponding hnges in the degrees of irulr polriztion of the photoluminesene within the A nd L exiton nds, respetively, omputed s P = P (σ + ) P (σ ). Right pnel: g - l, Sme s the left pnel - f, ut for B = 9 T. Sign reversl of mgneti field interhnges the roles of the Zeemn rnhes ssoited with σ + nd σ exittions, whih results in sign reversl of P s ompred to the left pnel. All dt were reorded t 3. K nd with 637 nm lser. 3.7 Wide-field imging nd polrimetry As desried previously the onfol setup ws modified for wide-field imging y defousing the exittion lser to illuminte smple re of mm 2 nd y repling the single-mode fier in the detetion pth with n imging lens nd silion CCD. A tunle nd-pss filter (with 5 nm FWHM ndwidth) ws inserted efore the imging lens to selet exiton-speifi nds. The orresponding setup shemtis re shown in Supplementry Fig. 9. A fs-opo operted t 637 nm ws used to exite the PL t n verge power density of 200 W/m 2. This verge power density ws 8 NATURE NANOTECHNOLOGY 8

9 ftor of 50 nd 500 lower thn in onfol mesurements with the exittion lser t 637 nm (w power density of W/m 2 ) nd 532 nm (w power density of W/m 2 ), respetively. In the following, we disuss the min implitions of the redued power density for the oservtions in wide-field imging nd polrimetry. To ompre the onfol nd wide-field imging modes we reorded dditionl PL spetr nd intensity mps in the onfol setup onfigurtion t redued power densities. In Supplementry Fig. 9 - g we present dt reorded in the regime of low power exittion for the single-rystl MoS 2 tringle disussed in the min text. A onfol PL spetrum reorded t n exittion power density omprle to tht of the wide-field imging mode is shown in Supplementry Fig. 9. Colored nds indite the spetrl intervls tht were used for the evlution of onfol PL intensity mps in Supplementry Fig. 9f nd g for A nd L exitons, respetively. The nds were seleted to mth the setting of the nd-pss filter in wide-field imging. At low exittion powers the PL intensity of L exitons is not sturted (Supplementry Fig. 9). This results in n inresed intensity rtio of L to A exiton PL in the onfol spetrum of Supplementry Fig. 9 s ompred to the dt in Fig. of the min text. Moreover, sine the sturtion responses of L exitons wy from the puddle nd in the puddle re different (the respetive dt re denoted s L nd L P in Supplementry Fig. 9), one expets rossover in the reltive PL intensities CCD rry nd-pss σ/π nlyzer σ/π polrizer lser ryostt (T=3.K) mgnet (B=±9T) B y x Norm mlized PL L A 500 W/m Energy (ev) d PL (r. units) Integrte L L P Power (W/m 2 ) d f A ndpss e g Normlized PL L ndpss 200 W/m W/m 2 Supplementry Fig. 9: Confol nd wide-field imging t low exittion power densities of 500 W/m 2 nd 200 W/m 2, respetively., Setup shemtis for wide-field imging: the exittion lser ws defoussed to illuminte spot of 00 µm dimeter. Exiton-speifi photoluminesene ws spetrlly seleted with nd-pss filter nd imged onto CCD mer with 50 effetive mgnifition., Confol low-power exittion spetrum with olored A nd L exiton nds used for exiton-seletive photoluminesene imging., Photoluminesene intensity of L exitons s funtion of exittion power wy from the puddle (L) nd in the puddle (L P ). d, f, Photoluminesene intensity profiles of A exitons for the tringulr flke of the min text under low-power exittion in wide-field nd onfol imging, respetively. e, g, Sme ut for the L exiton nd. The exittion wvelength ws 637 nm in d nd e, nd 532 nm in,, f, nd g. The imges in d nd e were quired with liner gmm enoding. All dt were reorded t 3. K. 9 NATURE NANOTECHNOLOGY 9

10 of defet-ound exitons in the puddle nd on regulr position of the flke. This effet is oserved in PL imging t low exittion power densities: oth wide-field (Supplementry Fig. 9e) nd onfol (Supplementry Fig. 9g) imges of the monolyer tringle show more intense L exiton PL wy from the puddle, wheres t high exittion power densities the L exiton PL in the puddle is more intense (Fig. e of the min text). The more pronouned pperne of hot-spots in the PL intensity profiles of A exitons under low-power illumintion (Supplementry Fig. 9d nd f) s ompred to the PL intensity mp under high-power illumintion (Fig. d of the min text) is lso ttriuted to sturtion effets. As in onfol polrimetry, polriztion-resolved wide-field imging n e used to onstrut polrimetri profiles of vlley polriztion nd vlley oherene. To demonstrte the quntittive hrter of the tehnique, we show in Supplementry Fig. 0 - f the P l nd P profiles of the single-rystl MoS 2 tringle of the min text otined with wide-field polrimetry. Note tht the opto-vlleytroni profiles re in exellent quntittive greement with onfol dt shown in Fig. 2 nd Fig. 3 of the min text if one tkes into ount the redution of P l nd P due to wider spetrl nds of the nd-pss filter used in wide-field polrimetry. The dvntge of the wide-field imging tehnique is tht it redues the integrtion time for opto-vlleytroni profiling whenever full spetrl informtion t eh pixel of the mp is not required. Ultimtely, for n imge size of (20 20) µm 2 with pixel size of ( ) µm 2 n quisition elertion of t lest n e otined with polrimetri profiling in wide-field imging s ompred to onfol rster-snning with the sme sptil resolution, spetrl width, noise level, nd trnsmission hrteristis. For this estimtion we negleted the onfol snning time nd ssumed equl exittion power densities of the imging tehniques s well s shot-noise limited signls. We note tht for our onfol rster-sn polrimetry the snning time ws ftor of two longer thn the integrtion time, nd the trnsmission properties were ftor of ten lower thn for wide-field imging. A t 0T A t +9T L t +9T σ + e σ + d σ f σ P l Supplementry Fig. 0:,, Liner polrimetri profiles P l of spetrlly filtered A exitons in single- nd poly-rystlline MoS 2, respetively, otined with wide-field imging., d, nd e, f, Cirulr polrimetri profiles P of A nd L exitons within the respetive nd-pss windows in n externl mgneti field of +9 T. The dt were reorded t 3. K with exittion t 637 nm nd σ + polriztion in nd e, nd σ polriztion in d nd f. P NATURE NANOTECHNOLOGY 0

11 4 Theoretil modeling In the following we model the degree of irulr polriztion of exitons in monolyer MoS 2 in the presene of n externl mgneti field pplied in Frdy geometry (the mgneti field is oriented perpendiulr to the TMD monolyer surfe). To this end we onsider four level system with the rystl ground stte 0, the K exiton stte, the K exiton stte 2, nd n exited stte 3 in one of the two vlleys. All levels re denoted in Supplementry Fig., where without loss of generlity the exited stte 3 is in the K vlley. In the following, exiton sttes in K (K ) tht ouple to σ + (σ ) polrized optil trnsitions re tgged with the vlley index κ = + (κ = ). 3 B = 0T B > 0T γ 3 γ 32 γ 2 K K 2 γ 2 0 γ 03 γ 0 γ 0 Supplementry Fig. : Shemtis of the model system: energy levels nd trnsition rtes of four level system with the rystl ground stte 0, the K exiton stte, the K exiton stte 2, nd the exited K exiton stte 3. The solid (dshed) red nd lue dispersions represent K nd K exitons without (with) n externl mgneti field. γ 03 denotes the sorption rte from the rystl ground stte into the exited stte of the K vlley exiton, γ 3 nd γ 32 re the vlley onserving nd vlley flipping relxtion rtes, γ 2 nd γ 2 re the vlley flipping rtes mong the K nd K sttes, nd γ 0 is the K nd K exiton dey rte (inluding oth rditive nd non-rditive hnnels). For σ + (σ ) polrized optil trnsition tht is ssumed to rete popultion exlusively in the exited K (K ) stte, the temporl evolution of the popultions in eh stte is given y the following set of rte equtions: ground stte 0 : ρ 0 = ρ γ 0 + ρ 2 γ 0 ρ 0 γ 03 K stte : ρ = ρ 3 γ 3 + ρ 2 γ 2 ρ γ 2 ρ γ 0 K stte 2 : ρ 2 = ρ 3 γ 32 ρ 2 γ 2 + ρ γ 2 ρ 2 γ 0 exited stte 3 : ρ 3 = ρ 0 γ 03 ρ 3 γ 3 ρ 3 γ 32, () where ρ i is the popultion of the ith stte, ρ i is the temporl derivtive of ρ i, nd γ ij is the trnsition rte from stte i to j. The popultion dynmis re governed y the following rtes: γ 03 denotes the sorption rte from the rystl ground stte into n exited stte in one of the two vlleys, γ 3 nd γ 32 re the vlley onserving nd vlley flipping relxtion rtes (or vie vers), γ 2 nd γ 2 re the vlley flipping rtes mong the K nd K sttes, nd γ 0 is the K nd K exiton dey rte (inluding NATURE NANOTECHNOLOGY

12 oth rditive nd non-rditive hnnels). The totl popultion of the system is normlized to ρ 0 + ρ + ρ 2 + ρ 3 =, nd stedy-stte solutions re otined for ρ i =0. The degree of irulr polriztion P =(I o I r )/(I o + I r ) is otined from stedy-stte popultions ρ nd ρ 2 of the K nd K vlleys: P = κ ρ ρ 2 ρ + ρ 2, (2) with the expliit expression equivlent to the one derived in the Supplementry Informtion of Ref. 8: ( ) γ 0 P = κ γ3 γ 32 γ 2 γ 2 +. (3) γ 0 + γ 2 + γ 2 γ 3 + γ 32 γ 0 + γ 2 + γ 2 We rewrite this expression s: ( γ 0 P = κ ) κ γ 2 γ 2 +, (4) γ 0 + γ 2 + γ 2 + κ γ 0 + γ 2 + γ 2 with rnhing prmeter κ given y the rtio of the vlley flipping to the vlley onserving relxtion rtes: ( ) κ γ32 κ =. (5) The definition of the rnhing prmeter implies κ = γ 32 /γ 3 ( κ = γ 3 /γ 32 ) for κ = + (κ = ). γ 3 First, we exmine the the degree of irulr polriztion given y Eqs. 3 nd 4 t zero mgneti field. In the presene of time-reversl symmetry, the vlley flipping proesses re symmetri nd thus γ 2 = γ 2. Moreover, in the limit of idel initil polriztion, i. e. for κ =0, Eqs. 3 nd 4 simplify to: P 0 = =, (6) +2γ l /γ 0 +2r 0 where r 0 = γ 2 /γ 0 = γ 2 /γ 0 is the rtio of the zero-field longitudinl vlley depolriztion rte γ 2 = γ 2 = γ l to the exiton dey rte γ 0. This expression is equivlent to the one derived in Ref. 9, nd it orresponds to the stedy-stte degree of irulr polriztion in optil spin orienttion with idel initil polriztion 0 : P 0 = +τ 0 /τ l, (7) if we identify τ 0 =/γ 0 s the exiton lifetime nd τ l =/(2γ l ) s the vlley depolriztion time. This limit where r 0 nd thus τ 0 /τ l re fixed y the degree of irulr polriztion P 0 ws used in the min text to disuss the dt t zero mgneti field. If the initil polriztion of the fundmentl exiton popultions in K nd K vlleys is non-idel euse of vlley flipping events upon relxtion from the optilly exited stte, we hve nlogous to imperfet spin orienttion 0 : P 0 = P 0 i = κ, (8) +2r 0 + κ +2r 0 with the initil polriztion Pi 0 = ( κ )/( + κ ) given y the yield of optil vlley polriztion of K nd K popultions t zero mgneti field in the presene of finite rnhing κ > 0. NATURE NANOTECHNOLOGY 2

13 In finite mgneti field B pplied perpendiulr to the smple, the vlley Zeemn splitting 8, 5 v introdues n imlne etween γ 2 nd γ 2. In the presene of non-zero rnhing, we return to Eqs. 3 nd 4 to repitulte the findings of Ref. 8 with respet to the mgneti field evolution of P. First we note qulittively tht due to the vlley Zeemn splitting, the imlne of γ 2 nd γ 2 inreses (dereses) the popultion trnsfer rte from the upper (lower) to the lower (upper) Zeemn rnh. Thus, provided suffiiently rpid vlley depolriztion on the timesle of the exiton lifetime, we expet the popultion to relx into the energetilly lower vlley. Consequently, the P of the upper (lower) Zeemn vlley should derese (inrese) with mgneti field. The opposite trend is oserved in our study of A nd L exitons in monolyer MoS 2 wy from defets in greement with the A exiton response in monolyer WSe 2 reported in Ref. 8. This ounter-intuitive popultion distriution ws ttriuted to (i) slow inter-vlley sttering nd (ii) polriztion proteting rnhing rtio κ tht dereses (inreses) for the upper (lower) vlley with mgneti field 8. Sine no expliit expression for the funtionl form of the evolution of κ with mgneti field ws given in Ref. 8, nd to ount for different mterils nd experimentl onditions, we pproximte κ with liner funtion in B: κ (B) = 0 + κδb, (9) where 0 is the zero-field rnhing rtio nd δ is proportionlity ftor. It is worth noting tht Eq. 9 should e interpreted s low-field pproximtion sine in it would yield unphysil negtive vlues for κ (B) in the limit of suffiiently high fields. Moreover, differene in δ for the K nd K vlleys ould ount for roken time-reversl symmetry in sene of mgneti field s pointed out in Ref. 8. To proeed with the nlysis of our dt using the model of Ref. 8 where γ l γ 0 (or orrespondingly r 0 0) ws ssumed, we fit the P evolutions of the A nd L exitons on regulr position of the tringulr flke with r 0, 0 nd δ s fitting prmeters. The results of est fits to the dt, otined from lest χ 2 red devition (defined y the unweighted sum of squred devitions etween fits nd dt nd divided y the numer of degrees of freedom), re presented in Supplementry Fig. 2. The non-therml popultion distriution of oth A nd L exitons is refleted y the ounter-intuitive X-pttern of higher P vlues for the Zeemn rnhes with higher energy. It is onsistent with slow longitudinl vlley depolriztion on the timesle of the exiton lifetime, nd est fits tully yield r 0 =0for oth A nd L exitons. The zero-field rnhing rtio for the seletively initilized popultion is smll for A exitons ( 0 =0.07) nd lrge for L exitons ( 0 =0.90) s expeted from the less resonnt exittion of the ltter. In this frmework, the evolution of the rnhing rtio with mgneti field ritilly determines the respetive P pttern: oth A nd L exitons exhiit n inresing protetion of the optil vlley polriztion in the upper Zeemn vlley s the rnhing rtio etween the vlley flipping nd onserving relxtion rtes dereses (δ >0 for oth A nd L) due to exhnge-modified exiton dispersions 8. The model of Ref. 8 within the frmework of Eqs. 4 nd 9 requires vnishingly smll vlues of r 0 without quntifying its smllness. To otin n estimte for the rtio of the longitudinl vlley depolriztion time to the exiton lifetime, we extend NATURE NANOTECHNOLOGY 3

14 P r σ + σ B (T) A χ 2 red r 0 P σ + σ L B (T) 0.5 d χ 2 red Supplementry Fig. 2:,, P s funtion of mgneti field for A nd L exitons; the dt re reprodued from Fig. 3 of the min text. Solid lines re lest χ 2 red fits ording to Eqs. 4 nd 9. Best fits were otined with 0 =0.07 nd 0.90, δ =0.006 T nd 0.08 T for A nd L exitons, respetively, nd r 0 =0for oth A nd L., d, Assoited χ 2 red s funtion of r 0 nd 0 for A nd L, nd δ vlues given ove. the model of Ref. 8 y ssigning therml imlnes to the longitudinl inter-vlley flipping rtes γ 2 nd γ 2 expliitly using the Boltzmnn ftor exp(β): γ 2 = r 0 γ 0 exp( β) γ 2 = r 0 γ 0 exp(+β), (0) where, t temperture T, the exponent β = ( v /2)/(k B T ) is determined y the Boltzmnn onstnt k B nd the vlley Zeemn splitting v = gµ B B liner in the mgneti field B (with g eing the effetive exiton g-ftor, nd µ B the Bohr mgneton) 8, 5. The zero-field rtio r 0, defined y the Eq. 6 ove, is likely to exhiit mgneti field dependene ut short of knowledge of its funtionl form is ssumed here s onstnt. The exponentil imlne of γ 2 nd γ 2 is responsile for n effetive unidiretionl thermliztion of popultion from the upper to the lower Zeemn rnh during the exiton lifetime, nd modifies Eq. 4 to: P = +2r 0 osh(β) κ 2r 0 sinh(β) + κ + κ +2r 0 osh(β). () The degree of irulr polriztion is now sensitive to the therml imlne of the inter-vlley sttering rtes through the Boltzmnn ftor, nd it ounts for the non-zero rnhing in the polriztion initiliztion with its hrteristi mgneti field dependene vi κ (B) given y Eq. 9. The seond term drives the therml popultion distriution mong the K nd K vlleys, while the first term ounterts this thermliztion vi the funtionl dependene of the rnhing rtio κ on the mgneti field in fvor of hot popultion distriution. With this expression for the field-evolution of P, we performed simultneous fit of our set of dt for A nd L exitons wy from the puddle nd L P exitons. To minimize the numer of free fitting prmeters, we ssumed no field-dependene 4 NATURE NANOTECHNOLOGY 4

15 P σ + σ A -0.2 σ + σ L -0.2 σ + σ L P B (T) -0 0 B (T) B (T) 0.5 d e f r χ 2 red Supplementry Fig. 3: -, Evolution of A, L, nd L P exiton P with mgneti field (the dt is reprodued from Fig. 3 of the min text). Solid lines re lest χ 2 red simultneous fit to the dt ording to the model given y Eqs. 9 nd. To redue the numer of free fitting prmeters, the g-ftor of the A exitons ws fixed to g =4, nd glol temperture T ws ssumed. Moreover, δ =0ws used for L P, nd g nd r 0 were set equl for L nd L P. Best fit ws otined for the following vlues of the remining free fitting prmeters: 0 =0.03, 0.90, nd 0.82 for A, L, nd L P ; r 0 =0.03 for A nd 0.0 for L nd L P ; δ =0.02 T for A nd T for L; g =4.4 for L nd L P ; T =9K. d - f, Corresponding χ 2 red vlues s funtion of r 0 nd 0 for A, L, nd L P with other prmeters s given ove. The gry region in d with unphysil vlues of P > ws not evluted. of κ for the se of L P exitons tht exhiit therml popultion distriution, nd set oth the g-ftors nd the rtios r 0 equl for oth L nd L P. Moreover, we fixed the A exiton g-ftor to 4.0 4,5, nd ssumed glol temperture T for the entire set of dt. With these prmeters fixed, est simultneous fit ws otined with 0 =0.03, 0.90, nd 0.82 for A, L, nd L P ; r 0 =0.03 for A nd 0.0 for L nd L P ; δ =0.02 T for A nd T for L; g =4.4 for L nd L P ; T =9K. The orresponding evolutions of P for A, L nd L P exitons re shown in Supplementry Fig. 3, nd, respetively. We note tht the est fit yielded n effetive temperture of T =9Konsistent with lol lser heting, nd g =4.4 for L nd L P exitons in greement with elevted vlues reported for exitons lolized in quntum dots 6 9. The min effet of the model of Eq. s ompred to the model of Eq. 4 is the inresed field dependene of the A exiton rnhing (δ inresed y ftor of two from T to 0.02 T ), wheres the vlues of the zero-field rnhing rtio of 0 for the vlley flipping to the vlley onserving rte re omprle for the two models. Most importntly, the model yields n estimte for r 0 for oth A nd lolized exitons: et fit vlues of r 0 =0.03 for A, nd 0.0 for L nd L P orrespond to τ l /τ 0 7 nd 50, respetively. Tking the PL dey time of 4.5 ps for the A exitons in MoS 7 2 tht we do not resolve in our time-orrelted PL experiments, we thus otin n estimte for the longitudinl vlley depolriztion time of τ l 80 ps for A exitons. With the sme sling we estimte τ l 230 ps for lolized exitons tht exhiit PL dey 5 NATURE NANOTECHNOLOGY 5

16 dynmis dominted y the fst omponent elow the resolution limit of our fier-sed setup with dispersion (s shown in Supplementry Fig. 3, only 0% of the totl PL intensity of oth L nd L P exitons ontriute to the slow dey omponent). The min differene for L nd L p popultions stems from different rnhing during relxtion: while the upper vlley polriztion of L exitons is wekly yet inresingly proteted with mgneti field (with slope of T ), the rnhing rtio of L P exitons is independent of mgneti field. Given similr PL dey dynmis in the L nd L P nds, nd similr rnhing rtios t zero mgneti field, we onlude tht hot nd therml vlley popultions of lolized exitons rise from field-dependent nd field-independent rnhing rtios, respetively. Referenes. Njmei, S. et l. Vpour phse growth nd grin oundry struture of molydenum disulphide tomi lyers. Nt. Mter. 2, (203). 2. Rein, A. et l. Trnsferring nd identifition of single- nd few-lyer grphene on ritrry sustrtes. J. Phys. Chem. C 2, (2008). 3. Sheushner, N. et l. Photoluminesene of freestnding single- nd few-lyer MoS 2. Phys. Rev. B 89, (204). 4. Berkelh, T. C., Hyertsen, M. S. & Reihmn, D. R. Theory of neutrl nd hrged exitons in monolyer trnsition metl dihlogenides. Phys. Rev. B 88, (203). 5. Gołs, K. et l. Multiphonon resonnt Rmn sttering in MoS 2. Appl. Phys. Lett. 04, (204). 6. Stiger, M. et l. Splitting of monolyer out-of-plne A Rmn mode in few-lyer WS 2. Phys. Rev. B 9, 9549 (205). 7. Lgrde, D. et l. Crrier nd polriztion dynmis in monolyer MoS 2. Phys. Rev. Lett. 2, (204). 8. Aivzin, G. et l. Mgneti ontrol of vlley pseudospin in monolyer WSe 2. Nt. Phys., (205). 9. Mk, K. F., He, K., Shn, J. & Heinz, T. F. Control of vlley polriztion in monolyer MoS 2 y optil heliity. Nt. Nnotehnol. 7, (202). 0. Meier, F. & Zkhrheny, B. P. (eds.) Optil Orienttion (Elsevier Siene Pulishers B.V., 984).. Li, Y. et l. Vlley splitting nd polriztion y the Zeemn effet in monolyer MoSe 2. Phys. Rev. Lett. 3, (204). 2. Srivstv, A. et l. Vlley Zeemn effet in elementry optil exittions of monolyer WSe 2. Nt. Phys., 4 47 (205). NATURE NANOTECHNOLOGY 6

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