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1 Ultrafast Dynamics of Defect-Assisted Electron-Hole Recombination in Monolayer MoS Haining Wang, Changjian Zhang, and Farhan Rana School of Electrical and Computer Engineering, Cornell University, Ithaca, NY, USA Supplementary Information Auger Carrier Capture Rates in an Electron-Hole Plasma in MoS Monolayer: Four basic Auger processes for the capture of electrons ((a) and (b)) and holes ((c) and (d)) at defects are depicted in Figure 1. In this Section, we obtain expressions for the capture rates of electrons and holes by defects via Auger scattering in two dimensional materials like MoS. Details are available in the preprint by Wang et al. 1 Strong Coulomb interactions in two dimensional materials can make Auger capture rates fairly large and consistent in magnitude with our experimental observations. We will look at process (c) in Figure 1 in detail in which a hole scatters off an electron and is captured by a deep defect and the electron is scattered to a higher energy. The rates of all other processes can be calculated in a similar manner. We assume a defect level at an energy E d. Since the Bloch states form a complete set, the wavefunction of the electron in the defect state can be written as, ψ d ( r) = 1 A c n,s ( k) ei( K s + k). r u n, k,s A n, ( r) (1) k,s Here, the sum over n runs over all the energy bands, s stands for valley and spin (σ) degrees of To whom correspondence should be addressed 1

2 E c Electron E v Hole (a) (b) (c) (d) Figure 1: Four basic Auger processes for the capture of electrons ((a) and (b)) and holes ((c) and (d)) at defects are depicted. 4 freedom, u n, k,s ( r) is the periodic part of the Bloch function, and the wavevector k is measured from the valley vector K s. Whereas shallow defect levels can usually be described well by limiting the summation above to a single band, deep mid-gap defect levels generally have contributions from multiple bands.,3 The above expression can usually be cast in much simpler forms for specific defect states. For example, for the E states associated with a sulfur vacancy in MoS the above expression can be written approximately as, 1 ψ d ( r) χ d ( r) n=c,v b n,s e i K s. r e iφ n, k,s / u n, ( r) () k,s s Here, the function χ d ( r) is localized near the defect site, the phase φ n, k,s cancels the phase associated with u n, k,s ( r),1 the line under k means that any wavevector near the band extrema can be chosen, and the sums over n and s run over only the valence and conduction bands, and the K and K valleys, respectively. The Hamiltonian describing the scattering process in Figure 1(c) can be written as, 1 Hhc = 1 A V ( q)m s,s ( k, k, q)c k+ q,s b k,s d σ c + h.c. (3) k,s k, k, q,s,s V (q) is the D Coulomb potential. 5 A is the sample area. c k,s, b k,s, and d σ are the destruction operators for the electron states in the conduction band, valence band, and the defect, respectively, and M s,s ( k, k, q) = b v,s χ d ( k + q)/ A. We assume that the initial many body state is the electron-hole plasma created after photoexcitation and consists of free as well as bound (excitons) carriers. An expression for the capture rate

3 can be derived that incorporates correlations between the electrons and holes. For this purpose, it is useful to switch to the exciton basis and express the products of electron and hole creation (and destruction) operators in terms of the exciton creation (and destruction) operators, respectively. 1,6 The desired hole capture rate R can then be obtained by a simple application of the Fermi s Golden rule. 1 Assuming a defect density n d and an average defect occupation F d the final result is, R π h n df d n p b v D c (q α ) V (q α ) χ d (q α ) α [ φ α ( r = 0) n ] α np + G α = Bn d npf d (4) We have assumed above, for simplicity, that b v,s is independent of the spin/valley index s. n and p are the electron and hole densities and include both free electrons and holes as well as bound electrons and holes (bound and ionized excitons). n α is the exciton density in the α-th exciton level. The wavevector q α is approximately given by the relation E c (q α ) E c (0) = E d E α, where E α is the exciton binding energy and E c ( k) is the conduction band energy dispersion. D c (q α ) is the density of states in the conduction band at wavevector q α measured from the K (K ) point. φ α ( r = 0) is the probability of finding and electron and a hole at the same location in the α-th exciton state and is related to the exciton wavefunction in the relative co-ordinates. 1 The factor G α describes the Coulomb enhancement in the probability of finding an electron and a hole, which may not belong to the same exciton, at the same location. 1 The terms in the square brackets therefore describe the enhacement in the probability of finding an electron and a hole at the same location due to Coulomb correlations. 1 If the electrons and holes are completely uncorrelated then the terms in the square bracket equal unity. On the other hand, if all the electrons and holes are bound and are in the lowest exciton level (1s) with wavefunction φ 1s ( r), then given that the exciton radii in monolayer MoS is 7-9 A (Zhang et al. 5 ), the terms in the square bracket can be larger than Electron-hole correlations induced by Coulomb interactions can therefore drastically increase the Auger capture rates. 3

4 In order to make an order of magnitude estimate of the value of the rate constant B, we assume a Gaussian defect wavefunction χ d ( r) of radius 3 A, which is appropriate for highly localized deep defects, 1, b v 1, and a defect energy 1 ev above the valence band edge (E d = 1.0 ev). Assuming m h = m e = 0.5m o, 5 and using the expression for the wavevector dependent dielectric constant for a MoS monolayer on a quartz substrate given by Zhang et al., 5 we obtain values of B equal to cm 4 /s and cm 4 /s for the assumed values of the Coulomb enhancement factor of 1 and 100, respectively. In comparison, the largest experimentally determined values in this work (see value of B f for the fast traps in Table 1 in the text) are in the range cm 4 /s (taking into account the uncertainty in the sample defect density listed in Table 1). Therefore, the estimates for the rate constants for Auger carrier capture processes in MoS determined from the theory agree fairly well with the experimental data. Interestingly, the smallest and the largest values of the Coulomb enhancement factor, used as a fitting parameter, needed to obtain an exact match between the theory and our experimental data are.1 and 6.8, respectively. Similar calculations shows that the rates of the other capture processes in Figure 1 can be written as, (a) R An d n (1 F d ) (5) (b) R An d np(1 F d ) (6) (c) R Bn d npf d (7) (d) R Bn d p F d (8) Intraband Absorption by Free Carriers and Excitons: In this Section, we derive an expression for the excitonic contribution to the intraband absorption and show that at optical frequencies much higher than the exciton binding energies and much lower than the optical bandgap, the intraband 4

5 conductivity of excitons looks similar to the intraband conductivity of free carriers. We assume a MoS monolayer with free electron density n f, free hole density p f, and bound exciton density n ex. The total electron density n is n f + n ex and the total hole density p is p f + n ex. The total intraband conductivity σ(ω) can be written as σ f (ω) + σ b (ω), where σ f (ω) is the contribution from the free carriers and σ b (ω) is the contribution from the bound excitons. We now show that the contribution to the real part of the intraband conductivity from all the carriers, free and bound, at the frequencies of interest can be written as, ( n σ r (ω) + p ) e τ m e m h 1 + ω τ (9) The real part of the intraband conductivity contribution from the free carriers must satisfy the partial sum rule, 7 0 σ r f (ω)dω = πe ( n f + p ) f m e m h (10) In addition, the high frequency limit of the imaginary part of the conductivity σ i f (ω) follows from the Lehmann representation of the current correlation function, 7 ( n lim σ f i f (ω) = i + p ) f e ω m e m h ω (11) An expression that satisfies both the above conditions is given by the Drude form, ( n f σ f (ω) = i + p ) f e m e m h ω + i/τ (1) The above well known expression for the Drude form of the free carrier conductivity can be derived in many different ways. 7 To find the intraband conductivity of the excitons we start from the relative coordinate operator for the exciton, r ˆ = r ˆ h r ˆ e (13) 5

6 where ˆ r h and ˆ r e are electron and hole position operators. We have, ] [ˆ p. r. ˆn,Ĥ = i h ˆ ˆn (14) m r where ˆ p is the relative momentum conjugate to ˆ r, ˆn is any unit vector, and m r is the reduced electron-hole mass, It follows that, 1 m r = 1 m e + 1 m h (15) ]] [ˆ r. ˆn, [ˆ r. ˆn,Ĥ = h (16) m r Suppose α represent all exciton states, tightly bound as well as ionized, with energies E α. Taking the matrix elements of the above commutator equation with the exciton states we get, (E β E α ) α ˆ r. ˆn β = h (17) m β r The above expression is a modification of the well known Thomas-Reich-Kuhn oscillator strength sum rule. Most of the oscillator strength on the left hand side comes from the terms in which both α and β are bound exciton states or low energy ionized states. Now consider an exciton gas with density n ex in which the state α is occupied with probability ρ α. We only consider bound exciton states to be occupied here since the contribution from the occupied exciton states that are ionized has been included in the intraband contribution from the free carriers considered above. The interaction between an exciton and a classical electromagnetic field of frequency ω is given by the dipole operator eˆ r. ˆnE(t), where ˆn is the field polarization unit vector. The optical intraband conductivity of the system can be found easily using standard linear response techniques and comes out to be, [ ] σ b (ω) = ie n ex β ˆ r. ˆn α ω(e β E α ) ρ α ( hω + i h/τ) α,β (E β E α ) (18) 6

7 We have introduced a phenomenological damping parameter τ. Using the sum rule given in 17, the bound exciton conductivity σ b (ω) is found to satisfy the conductivity sum rule, 0 σ rb (ω)dω = πe ( nex m r ) (19) Since most of the oscillator strength in the sum rule comes from the bound states or the low energy ionized states, one can take the large frequency limit of the expression in 18 and obtain, [ ] σ b (ω) i e h n ex β ˆ r. ˆn α (Eβ E α ) ρ α ω + i/τ α,β (0) The above expression is valid for frequencies ω much higher than the exciton binding energies but much lower than the material optical bandgap. Using the sum rule given in 17, we obtain, σ b (ω) i ( nex m r ) e ω + i/τ = i ( nex + n ) ex e m e m h ω + i/τ (1) Under the assumption that the damping parameters τ, appearing in the conductivity expressions for free carriers and bound excitons, are approximately the same, we can add σ b (ω) from 1 and σ f (ω) from 1, and obtain the simple expression for the intraband conductivity of the sample at the frequencies of interest, ( n σ(ω) = σ f (ω) + σ b (ω) i + p ) e m e m h ω + i/τ () Note that n and p above are now the total electron and hole densities including free carriers and bound carriers (excitons). The desired expression in 9 follows immediately by taking the real part of the above expression. Optically Induced Sample Damage and its Effect on the Optical Properties and on the Measured Dynamics: In our experiments we found that exfoliated MoS monolayers (SPI Supplies 7

8 and D Semiconductors) could easily get damaged permanently when pump fluences in excess of 50 µj/cm were used (45 nm wavelength). Once damaged in this way, the optical characteristics of the sample would change completely. So care needed to be exercised in ensuring that samples were not damaged during pump-probe, photoluminescence, or absorption measurements. An optical microscope image and a Raman spectroscopy image of a sample damaged by pump pulses is shown in Figure (a) and (b). The energy splitting between A 1g and E g mode is much larger at the damaged spots compared to the surrounding intact areas. Similar enhancement of the mode splitting in optically damaged samples has been reported previously. 8 In addition, damaged samples exhibited large optical absorption throughout the bandgap, as shown in Figure (c), indicating creating of midgap states. Finally, the transmission of the probe pulse at 905 nm wavelength in a damaged sample gets overwhelmed by the increased absorption inside the bandgap and, consequently, the dynamics observed by the probe pulse become completely different compared to the dynamics observed in an undamaged sample. A damaged sample shows a large slow component in the transient that has a time scale much longer than any of the time scales observed in an undamaged sample, as shown in Figure (d). The magnitude of the slow component in the measured transient is larger in samples damaged with a higher pump fluence. We strongly feel that it is very important that the absence or presence of optically induced sample damage is checked before/after optical measurements in order to ensure that reliable data has been obtained. 8

9 Width (µm) Microscope Monolayer (a) 5 Few Layers Length (μm) -1 A 1g - E g (cm ) Raman Image A 1g (b) - E g Damaged Spots Length (μm) Absorption Damaged by 18 μj/cm pump Probe: 1.37 ev Engergy (ev) (c) As Prepared Normalized ΔT/T Damaged by 64 μj/cm and 18 μj/cm pump As-prepared (d) Probe Delay (ps) Figure : (a) Optical microscope image of a sample damaged by a 18 µj/cm pump pulse is shown. No sign of damage is visible. (b) A scanned Raman image corresponding to the energy splitting between the A 1g and E g modes is shown for a sample damaged by a 18 µj/cm pump pulse. (c) A sample damaged by a 18 µj/cm pump pulse shows large absorption throughout the bandgap. (d) The transmission of the probe pulse at 905 nm wavelength in a damaged sample gets overwhelmed by the increased absorption inside the bandgap and, consequently, the dynamics observed by the probe pulse become completely different compared to the dynamics observed in an undamaged sample. A damaged sample shows a large slow component in the transient that has a time scale much longer than any of the time scales observed in an undamaged sample. The magnitude of the slow component in the measured transient is larger in samples damaged with a higher pump fluence. References (1) Wang, H.; Strait, J. H.; Zhang, C.; Chen, W.; Manolatou, C.; Tiwari, S.; Rana, F. arxiv: v 014. () Landsberg, P. T. Recombination in Semiconductors, 1st ed.; Cambridge University Press: cambridge, UK, 199. (3) Robbins, D. J.; Landsberg, P. T. Journal of Physics C: Solid State Physics 1980, 13, 45. (4) Riddoch, F. A.; Jaros, M. J. Phys. C: Solid St. Phys. 1980, 13, (5) Zhang, C.; Wang, H.; Chan, W.; Manolatou, C.; Rana, F. Phys. Rev. B 014, 89, (6) Kira, M.; Koch, S. Progress in Quantum Electronics 006, 30, (7) Giuliani, G. F.; Vignale, G. Quantum Theory of the Electron Liquid, 1st ed.; Cambridge University Press: New York, USA,

10 (8) Castellanos-Gomez, A.; Barkelid, M.; Goossens, A. M.; Calado, V. E.; van der Zant, H. S. J.; Steele, G. A. Nano Letters 01, 1,