SUPPLEMENTARY INFORMATION. Long-lived nanosecond spin relaxation and spin coherence of electrons in monolayer MoS 2 and WS 2

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1 Long-lived nanosecond spin relaxation and spin coherence of electrons in monolayer MoS and WS Luyi Yang, Nikolai A. Sinitsyn, Weibing Chen 3, Jiangtan Yuan 3, Jing Zhang 3, Jun Lou 3, Scott A. Crooker, National High Magnetic Field Laboratory, Los Alamos, NM 755 Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 755 and 3 Department of Materials Science and NanoEngineering, Rice University, Houston, TX 775 NATURE PHYSICS

2 WAVELENGTH DEPENDENCE OF TIME-RESOLVED KERR AND HANLE- KERR MEASUREMENTS Figure S shows time-resolved Kerr-rotation (KR) measurements in monolayer MoS, at B y = and 7 mt, using degenerate pump and probe pulses for different photon energies spanning the A exciton. At short wavelengths (high photon energies), the induced KR signals are small. At intermediate wavelengths λ just on the low-energy side of the exciton resonance (λ=5- nm), the KR signals at zero field are large and decay slowly, consistent with spin relaxation of polarized itinerant and resident electrons. At B y =7 mt, the large KR signals decay quickly due to dephasing caused by precession about the combined applied field B y and the large effective internal spin-orbit field ±ẑb so that is rapidly fluctuating due to fast intervalley scattering of itinerant electrons. At these wavelengths, the additional long-lived oscillatory (coherent) spin precession signal is barely visible. However at even longer wavelengths further on the low-energy side of the exciton resonance, the KR signals reduce until only this small oscillatory signal remains. These results are consistent with the laser pulses pumping (and probing) predominantly itinerant electrons in MoS at intermediate wavelengths (5- nm), but then pumping and probing more localized states at longer wavelengths (3-7 nm). Being localized, these electrons may not undergo rapid intervalley scattering and may not see the large spin orbit field ±ẑb so, and therefore they precess about the bare applied magnetic field B y. Localized states in monolayer TMDC materials have been observed in recent experimental studies []. Figure S shows Hanle-Kerr data from monolayer MoS at 5 K. A fixed cw pump laser was used (3. nm), and the different Hanle curves show data obtained with different cw probe laser wavelengths. The inversion of the induced KR signal is clearly observed (as in Fig. of the main text). These curves have somewhat greater width than those shown in the main text (e.g., Figure ), due to the use of higher pump and probe laser intensity. Minimum Hanle widths (implying long spin lifetimes of resident electrons) are obtained at long wavelengths on the low-energy side of the A exciton resonance. NATURE PHYSICS

3 a R/R.. "B" "A" Wavelength (nm) b nm B y (mt) 7 5 nm 5 nm Induced Kerr rotation (arb. units) nm nm nm... nm 7 nm Time (ns) FIG. S: Wavelength dependence of time-resolved Kerr rotation studies in MoS. a, Normalized reflectance spectrum R/R from a monolayer MoS crystal at 5 K. The arrows indicate the wavelengths λ at which the degenerate pump-probe KR measurements were performed. b, Time-resolved KR in monolayer MoS for various λ at B y = and 7 mt. Note the changing y-axis scales at the longest wavelengths. NATURE PHYSICS 3

4 Kerr rotation (arb. units) 5-5 λ probe (nm) λ pump =3. nm By (mt) FIG. S: Probe wavelength dependence of Hanle-Kerr measurements in MoS. Using cw pump and probe lasers, the plot shows the Hanle-Kerr curves measured in MoS at 5 K using different probe wavelengths spanning the A exciton resonance. SELECTIVELY PROBING POLARIZATION DYNAMICS IN K AND K VALLEYS Because Kerr signals scale as the difference between RCP and LCP optical constants, it is natural to ask which component contributes primarily. That is, does resonant pumping the K valley with RCP pump light affect only RCP probe light, or does it also affect LCP light (which probes the K valley)? Figure S3 shows time-resolved measurements of pump-induced reflectivity in MoS using co- and cross-circular pump and probe pulses. Interestingly, the reflectivity of both RCP and LCP probe light increases immediately following photoexcitation (albeit by different amounts), consistent with recent studies on very short (< ps) timescales [], suggesting that strong exciton Coulomb correlation effects while holes are present generate the initial non-equilibrium electron densities, which then, following recombination, relax on long timescales. NATURE PHYSICS

5 a B y = mt b B y =7 mt R pump/probe RCP/RCP RCP/LCP Time (ns) Time (ns) FIG. S3: Selective right- and left-circularly polarized (RCP/LCP) pump-probe studies of induced reflectivity. Using RCP pump pulses, the K valley of monolayer MoS is weakly photoexcited at the A exciton, while the induced change in reflectivity R of both RCP and LCP probe pulses is detected as a function of time. Both RCP and LCP probes (probing K and K valleys, respectively) show an induced longlived change. λ pump = λ probe = nm; T =5 K. a, Measurement at B y =. b, Measurement at B y =7 mt. INTER-VALLEY SCATTERING & SPIN RELAXATION OF DIRAC ELECTRONS The spin-dependent electronic part of the D Dirac Hamiltonian contains the spin coupling with an external magnetic field, which we assume to be applied in the plane of the sample (B =ŷb y ), and the valley-dependent spin-orbit coupling, Ω so, that acts as an effective out-of-plane magnetic field B so = ±ẑb so that is seen by electrons [3]: Ĥ =Ω L Ŝ y +Ω soˆτ(t)ŝz, (S) where Ω L = gµ B B y / and ˆτ = ± for K and K valleys respectively. The time dependence of the valley index ˆτ is due to stochastic inter-valley scattering. Analytical calculations based on the Dirac Hamiltonian provide an estimate Ω so = λϵ F /, where λ is the spin-orbit splitting of the hole bands, is the size of the semiconductor gap, and ϵ F is the Fermi energy of electrons [3]. As such, the effective field B so in this simple picture is due only to the slightly different curvatures of the spin-up and spin-down conduction bands within a given valley, and therefore depends on the density of resident electrons (that is, on ϵ F ). The different sign of the spin-orbit coupling in different valleys is guaranteed by time-reversal symmetry. More rigorous ab initio numerical calculations [ 7] predict slight deviations from this formula, and also predict an additional spin-orbit splitting of the conduction bands even at the minimum of the K and K valleys (called c in the main text). Though not yet experimentally observed, c has been calculated by density-functional theory for both MoS NATURE PHYSICS 5 5

6 and WS, and is predicted to be significantly larger in WS, and with opposite sign (of order + mev, compared with approximately -5 mev for MoS, although we note that substantial error bars exist in these predictions). c will necessarily also contribute to the net spin-orbit coupling Ω so felt by electrons. In the MoS and WS monolayers studied in this work, the resident electron density n e is estimated to be rather large, approximately 5 cm based on transport measurements of field-effect transistors fabricated from similarly-grown single-monolayer crystals. Given this value of n e and computing the Fermi energy, both the spin-up and spin-down conduction bands (in both the K and K valleys) are expected to be partially filled in our MoS and WS monolayers. As such, phase space exists at the chemical potential for fast spin-conserving inter-valley scattering (γ v ), motivating the model of spin relaxation that is outlined in the main text and described in full detail immediately below. At all events, the expected values of B so = Ω so /gµ B appear to be of order - Tesla for our D MoS and WS monolayers. This is much larger than the external magnetic fields B y applied in our experiments. An interesting and as-yet-unanswered question, however, is how electron spin dynamics depend in detail on n e, particularly at low densities when ϵ F < c. In that case, only one spin band in either valley is occupied, nominally suppressing the spinconserving intervalley scattering rate γ v at low temperatures. Future studies using gated samples will address this. Phenomenological Bloch equations for electron spin polarizations S K and S K in the two valleys are: ds K dt =Ω L ŷ S K +Ω so ẑ S K γ s S K γ v (S K S K ), (S) ds K dt =Ω L ŷ S K Ω so ẑ S K γ s S K + γ v (S K S K ), (S3) where, in addition to the spin precession described by the Hamiltonian (S), we introduce two phenomenological relaxation terms: one with a kinetic rate γ s that corresponds to spin relaxation processes within a given valley (unrelated to inter-valley scattering), and another with kinetic rate γ v that describes spin-conserving inter-valley scattering. These rates are depicted in Fig. a of the main text. Eqs. (S)-(S3) can be rewritten in terms of the total spin polarization, S = S K + S K, NATURE PHYSICS

7 and the difference, S = S K S K : ds dt ds dt =Ω L ŷ S +Ω so ẑ S γ s S, (S) =Ω L ŷ S +Ω so ẑ S (γ s +γ v )S. (S5) In the case of fast inter-valley scattering (γ v γ s ), it is safe to disregard the left-hand-side of (S5). Moreover, for Ω so Ω L we can also disregard the relative role of the external magnetic field term in (S5). Equation (S5) can then be solved as S = Ω soẑ S γ v. (S) Substituting (S) into (S), we find the dynamics of the total spin polarization: ds dt =Ω Lŷ S Γ v (ˆxS x +ŷs y ) γ s S, (S7) where Γ v = Ω so γ v. The solution of Eq. (S7) with initial condition S() = S ẑ shows either monotonic decay or oscillatory behavior: (S) ( S z (t) =S e (γs+γv)t cosh(ωt)+ Γ ) v sinh(ωt), Ω L < Γ v, (S9) Ω where Ω = Γ v Ω L, and ( S z (t) =S e (γs+γv)t cos(ωt)+ Γ ) v sin(ωt), Ω Ω L > Γ v (S) where Ω = Ω L Γ v. In particular, S z (t) =S e γ st at zero external field, and S z (t) = S e (γ s+γ v )t cos(ω L t) at large external fields (Ω L Γ v ). This demonstrates that inter-valley scatterings do not affect the observed spin relaxation at zero external field, but strongly enhance damping of spin precessions at large field values. In particular, the meaning of the parameter Γ v defined in (S) is the renormalization of the damping rate of spin precessions in a large external field. Figures Sa-c plot the spin polarization at K (Sz K, solid red curve) and K (Sz K, dashed blue curve) valleys by solving Eqs. (S) and (S3) with the initial condition S K () = S ẑ and S K () =, and the total spin polarization (S z = Sz K + Sz K, dotted black curve) assuming NATURE PHYSICS 7

8 a..5 S z K S z K' S z. b.5. c.5... Calculated spin polariztion (S ) d γ v = Ω SO = Ω L = GHz e.5 γ v = GHz Ω SO = Ω L = GHz f.5 γ v = THz Ω SO = Ω L = GHz 3 Calculated S z (S ) g γ v = Ω SO = THz Ω L = GHz γ v = THz Ω SO = THz Γ v = GHz Ω L (GHz) 3. h γ v = GHz Ω SO = THz Ω L = GHz Time (ns). i Ω SO = THz Ω L = GHz.5 γ v Γ v (THz) (GHz) x. -. γ v = THz Ω SO = THz Ω L = GHz 3 γ v = THz Ω L = GHz Ω SO Γ v (THz) (GHz)...x 5 Time (ns) FIG. S: Calculated spin polarization in units of S versus time for different parameters. a-f, Spin polarization at K (Sz K ) and K (Sz K ) valleys and the total spin polarization (S z ) for different γ v and Ω so. g-i, S z (t) for various parameters. γ s =.33 GHz for all the calculations. that Ω L = GHz and Ω so =. γ s =.33 GHz is set for all the calculations. As can be seen, S K z and S K z equilibrate on a time scale of γ v (i.e., ns and. ps), and S z oscillates about the external field with the decay rate γ s. Figures Sd-f demonstrate the corresponding cases when Ω SO = THz. S z changes from simply decaying (d and e) to oscillatory (f) dynamics. Figures Sg-i show S z by fixing two of the three parameters Ω L, γ v and Ω so and varying the third. In all cases, S z changes from simply decaying to oscillatory behavior as the value of Ω L exceeds Γ v. In Fig. Sg, S z decays rapidly with increasing Ω L, and shows the shallow dip and subsequent recovery at short timescales (the green curve), which captures NATURE PHYSICS

9 the main features of the experimental data. Interestingly, Fig. Sh illustrates that even for large Ω so, when the intervalley scattering rate is sufficiently large (the red curve), the fluctuating spin-orbit field is changing so rapidly that on average the electron spin does not see the spin-orbit field at all, and it just oscillates about the external field. Finally, we note that any non-zero valley polarization and Berry curvature can induce an orbital magnetic moment in these TMDC materials. However, unlike spin, this valleypolarization-induced orbital moment can only be strictly out-of-plane and has opposite sign for different valleys. It is not expected to couple to (i.e., cannot precess around) an external in-plane field, at least within the current understanding of the Dirac Hamiltonian of TMDCs. States in K and K valleys are separated by very large Bloch vector difference in the tightbinding Hamiltonian, and so an in-plane field does not have matrix elements between states of different valleys. Only an out-of-plane field can couple to this orbital angular momentum. The renormalization of the density of states is described by the scalar product (B*F), where F is the Berry curvature. The latter is a pseudovector that can point only out-of-plane in a D system. Hence, only out-of-plane component of an applied magnetic field can produce effects that depend on the valley index and Berry curvature, at least to linear order in B. [] He, Y.-M. et al., Single quantum emitters in monolayer semiconductors. arxiv:.9. [] Mai, C., Barrette, A., Yu, Y., Semenov, Y.G., Kim, K.W., Cao, L. & Gundogdu, K. Many-body effects in valleytronics: Direct measurement of valley lifetimes in single-layer MoS. Nano Lett., - (). [3] Tse, W.-K., Saxena, A., Smith, D.L. & Sinitsyn, N.A. Spin and valley noise in two-dimensional Dirac materials. Phys. Rev. Lett. 3, (). [] Kormányos, A., Zólyomi, V., Drummond, N.D., Rakyta, P., Burkard, G., & Fal ko, V.I. Monolayer MoS : Trigonal warping, the Γ valley, and spin-orbit coupling effects. Phys. Rev. B, 5 (3). [5] Liu, G-B, Shan, W-Y, Yao, Y., Yao, W. & Xiao, D. Three-band tight-binding model for monolayers of group-vib transition metal dichalcogenides. Phys. Rev. B, 533 (3). [] Kośmider, K., González, J. W. & Fernández-Rossier, J. Large spin splitting in the conduction band of transition metal dichalcogenide monolayers. Phys. Rev. B, 53 (3). NATURE PHYSICS 9

10 [7] Kormányos, A., Zólyomi, V., Drummond & Burkard, G. Spin-orbit coupling, quantum dots, and qubits in monolayer transition metal dichalcogenides. Phys. Rev. X, 3 (). NATURE PHYSICS

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