Excitonic luminescence upconversion in a two-dimensional semiconductor

Transcription

1 Excitonic luminescence upconversion in a two-dimensional semiconductor Authors: Aaron M. Jones 1, Hongyi Yu 2, John R. Schaibley 1, Jiaqiang Yan 3,4, David G. Mandrus 3-5, Takashi Taniguchi 6, Kenji Watanabe 6, Hanan Dery 7, Wang Yao 2*, Xiaodong Xu 1,8* The supplementary materials include: Supplementary Figures 1-7 Supplementary Text I- V Supplementary Figures: Supplementary Figure 1 Figure S1 Polarization-Resolved Luminescence Up-Conversion. a-d Data from four additional samples. Upper and bottom rows show spectra under σ + and vertically polarized excitation, respectively. The conservation of circular, but not linear, polarization implies the up-converted exciton inherits the valley properties of the negatively charged exciton. NATURE PHYSICS 1

2 Supplementary Figure 2 Figure S2 Fine Structure of Negatively Charged Exciton in WSe 2 on Boron Nitride. Photoluminescence spectrum of boron nitride supported monolayer WSe 2, clearly showing a split negatively charged exciton (X - ) peak. Blue lines denote a bi-lorentzian fit to the doublet. Inset: microscope image of the device, with boron nitride appearing light blue. Supplementary Figure 3 Figure S3 Overlap of Reverse PLE Spectrum with in Additional Samples. Blue lines show bi-gaussian peaks fit to the doublet structure of the negatively charged exciton (X - ) in standard PL spectra (black data). Red data are the integrated X o luminescence intensity for varying laser excitation energy. In all instances, the luminescence upconversion follows the high-energy peak, with maximal efficiency occurring near its peak energy. The ratios of standard X o PL to peak reverse PL intensity shown here are 20, 25, 10, 61, 77, and 41 for samples a-f, respectively. The variation in efficiency likely comes from different doping levels and sample quality. 2 NATURE PHYSICS

3 SUPPLEMENTARY INFORMATION Supplementary Figure 4 Figure S4 Negatively Charged Exciton Doublet at Varying Excitation Energies. a, Polarization resolved photoluminescence spectra as a function of excitation energy with σ + polarized excitation. Black and red curves represent co-circular and cross circular detection. These data show that the X - doublet has a strong excitation wavelength and polarization dependence. When the excitation energy moves from far above X o to lower energy, the intensity of relative to increases concomitantly with X o. In the limit of resonant X o excitation, the X - emission is almost entirely composed of the higher energy state, as shown in b, Overlay of charged exciton spectrum for excitation resonant with the neutral exciton (blue) with PL spectrum for excitation above the neutral exciton (black). (also see Figs. 2c and d). Supplementary Figure S5 Figure S5 Peak Splitting in Additional Samples. a, The doublet splitting from 14 monolayer WSe 2 samples. The distribution yields a mean splitting of 6.9 ± 0.5 mev. b, Extracted doublet splitting of a single sample from 20 K to 80 K, showing constant splitting. Due to linewidth broadening, we cannot obtain reliable peak fitting for temperatures above 80 K. NATURE PHYSICS 3

4 Supplementary Text I. The coupling of the and phonon to the -valley exciton From ab initio results, the phonon mode has a rather strong deformation potential coupling with the -valley electron 1. From the viewpoint of symmetry considerations 2,3, the out-of-plane phonon mode preserves the lattice symmetry while the in-plane mode breaks the lattice symmetry. Thus, an electron at the point which has the symmetry of the lattice can emit/absorb an Γ-phonon, while emission/absorption of the Γ-phonon is forbidden exactly at the point. Consequently, we expect the coupling strength of to be much weaker than for low energy electrons/holes and excitons in valleys. Experiments in Ref. 4 have shown that the Raman mode is more intense than when the exciting laser energy lies near the resonance of the A or B exciton (X ). Meanwhile, in our experiment, Stokes Raman emission at X preserves both circular and linear polarization of the excitation laser, signaling that the involved phonon scattering preserves both valley polarization and coherence. Combined with observations that the phonon preserves both circular and linear polarization while the phonon is cross-polarized (un-polarized) for circular (linear) polarization 5,6, we conclude that the phonon should dominate the resonant anti-stokes emission observed here. Our experimental observations are in agreement with the above theoretical analysis. II. Phonon induced coupling between a negatively charged exciton and a neutral exciton Here, we use the convention that the electron center-of-mass (COM) wave vector is measured from the band edge point, while that of the neutral exciton X is measured from its energy minimum. We denote as the electron annihilation operator with wave vector, ( ) as the X (negatively charged exciton X ) annihilation operator with COM wave vector. We also write as the annihilation operator of an unbound electron-x pair with COM wave vector and relative wave vector. The system Hamiltonian can be written as +, with the non-interacting Hamiltonian + ħ + ħ + + ħ + ħ. Here the 1 st term describes the energy of an unbound electron-x pair with COM wave vector and relative wave vector. The 2 nd term describes the energy of X (bound electron-x pair) with COM wave vector, and the 3 rd term denotes the energy of optical phonons. The X (electron) effective mass is estimated as. (. ), while + is the X effective mass. We write the electron-phonon interaction Hamiltonian +. Here is the phonon annihilation operator, and area. is the coupling parameter with quantized 4 NATURE PHYSICS

5 SUPPLEMENTARY INFORMATION We assume a bath of optical phonons in the monolayer described by:. Here is the number state of the phonon mode with wavevector where the occupation number is. The phonon has a nearly flat dispersion which can be neglected here for simplicity, i.e. we take = for all values. The average phonon occupation number is given by the Bose distribution = eħ, where is the lattice temperature (often distinct from the carrier temperature). First we analyze the X /X phonon interplay under resonant excitation at the X resonance. In this case, we consider the initial state of the X /X phonon system as: with energy = + ħ, i.e. X with COM wave vector ( 0 ) plus the phonon bath. After absorbing a -wave vector phonon from the bath, X can be discharged into an unbound X -electron pair, 0 with COM wave vector + and relative wave vector (Fig. S6 (a)). This final state can be written as,. which has the energy, = + ħ. Here is the phonon state + ħ where a phonon with wavevector is removed from the initial bath. Momentum conservation puts no constraints on the relative momentum q, so there is a continuum of final states starting from. Alternatively, when the applied laser is resonant with the X state, we consider the initial state eb with eb denoting an electron bath. For the final state, we have, e eb, where e eb denotes extraction of a wave vector electron from the electron bath, indicates the addition of a phonon of wavevector Q to the phonon bath, and creates X with COM wave vector. The state, has an energy, = ħ ħ + +, Again, as momentum conservation puts no constraint on the value of q, the possible, states form a continuum near the energy +. In either of the two resonant excitation scenarios discussed above, we can model the X X coupling as a discrete state coupled to a continuum (Fig. S6 (b)). Such a model has already been thoroughly discussed in Ref. 7. In order to analyze the dynamics of the initial state, below we give an estimate of the coupling strength and how it varies with the final state energy in the continuum. NATURE PHYSICS 5

6 Fig. S6 (a) with wave vector can absorb a -wave vector phonon and become an unbound electron- pair, and vise versa. (b) This is equivalent to a model in which a single state couples with an energy continuum, with the coupling strength dependent on the energy. III. Up-conversion ( ) We consider the initial state of X with wave vector 0, dropping its -subscript. As shown in the previous section, the final states of an unbound X -electron pair are characterized by the state with an energy = + ħ + ħ which forms a continuum. The X X coupling strength is then. In order to calculate, the X state is expanded using the basis of unbound X 0 -electron pairs: =. In the expansion, X excited states (2p orbitals etc.) are dropped as they are hundreds of mev above the ground state and are not substantially mixed into X by the Coulomb binding 8. The electronphonon interaction Hamiltonian is taken to have the deformation potential coupling form ħ ~300 mev A, where we have used ħ = 30 mev for WSe 2, while the mass density = 3. 0 g cm and electron-phonon coupling matrix element = 4.8 ev A are taken from the ab initio values of MoS 1 2 (we expect these values to have the same order of magnitude for WSe 2 ). We then have =. The spectral density for the coupling of with the continuum is then Γ = =. where we have changed the summation to an integral: =. 6 NATURE PHYSICS

7 SUPPLEMENTARY INFORMATION Below we analyze the behavior of Γ as a function of. For slightly above the threshold energy (that is, small ħ ), both and are close to zero so + ħ and are nearly constant, leaving Γ 0 ħ which increases linearly with. On the other hand, when ħ + ħ is large, either or becomes large, so we need to analyze the behavior of and. From the ab initio calculation in Ref. 1, is almost independent of the -direction, and attains a maximal value 300 mev A for 0, and then decays with increasing. The half-width half-maximum is ~0.2 A. and also decays with. Numerical estimations have shown average electronhole separations in X of ~ and ~2. 9, giving the half-width of to be ~. ~0.03 A. Thus decays much faster than, and for a rough estimation we can approximate by and replace the phonon occupation with its thermal average e ħ 1, yielding Γ 2, 2ħ which means Γ 0 when is significantly higher than the threshold energy. So Γ shows a single peak (Fig. S6 (b)). The expression above leads to an estimation of a half width half maximum ~100 mev. The height (maximum value) of the spectral density Γ can be estimated as maxγ~ 1 Γ 1 2, mev ~ ~ 1 mev. Above the threshold energy, the spectral density of the final states continuum has an approximately linear rising Γ~0.1, reaching the same order (few mev) as the peak value for. Since maxγ is much smaller than the width (~100 mev) of the spectral density of the continuum, in this parameter regime 7, the population of the initial X state will irreversibly decay to the X -electron continuum as long as the trion energy lies above the threshold energy, where the effective decay rate is well described by Fermi s Golden rule: Γ ħ ~ 1 mev. One shall also keep in mind that the above threshold energy will be ħ blurred by the finite linewidth of X and X. For Γ ~maxγ and assuming a lattice temperature ~100 K, then ~0.03 and we estimate an effective X X up-conversion rate Γ ħ 0.03 ps. On the other hand, if the X energy is well below the threshold energy (by a detuning much larger than the X and X linewidths), it will experience a small red shift instead of decaying to the continuum. This is consistent with the observation that upconversion is efficient only for the higher energy peak and not the lower energy one in the X- doublet. We note that the above analysis focuses only on the process of X to X conversion, while the luminescence up-conversion or spontaneous anti-stokes relies also on the interconversion of photons with X and X. The latter processes happens only within the appropriate light cones. X can be efficiently created only when the laser frequency is within ±Γ, where Γ is the ħ, NATURE PHYSICS 7

8 linewidth of the X absorption peak. If the X energy is much higher than +Γ (Γ being the linewidth of the X luminescence peak), after phonon induced discharging, the resultant X has a large kinetic energy and is dark, i.e. well outside the light cone. This also leads to the double resonant condition for luminescence up-conversion (spontaneous anti-stokes), i.e. the two broadened resonances ±Γ and ±Γ will have significant overlap. The enhancement at doubly resonant conditions comes from energy conservation of all three consecutive steps: photon to X, X to X, X to photon. Finally, we emphasis that the discussions and estimation of the up-conversion rate here are for a 2D system where the in-plane translational invariance requires momentum conservation of both the X to X conversion and their interconversion with photons. In reality, the trapping of X and X by local potential fluctuations will relax the momentum conservation constraint, which will help to enhance the efficiency of the spontaneous anti-stokes process. IV. Down-conversion under resonant excitation of ( ) When the X state is resonantly excited, the generated bright X with zero COM wavevector can combine with a free electron to become X, accompanied by the emission of a phonon (Fig. S6 (a)). We write the initial state as eb with eb an electron bath and the phonon bath, and the final state as, e eb, where e eb denotes annihilation of a -wave vector electron from the electron bath and corresponds to X with COM wave vector. This state, has an energy, = ħ ħ + +, All, states form a continuum near the energy +. We expand with the unbound electron-x pair as = e,. The matrix element, =, with ebe e eb = 0 or 1 the electron occupation number in the initial electron bath eb. Following the treatment in the previous section, we write the coupling spectral density as Γ =,,, =,. We approximate the electron occupation by its thermal average ħ + 1 with the chemical potential. For small or moderate electron density, is only significant when ħ. Also decays to its half-maximum when 0.0, so we expect Γ to be significant when ħ + ħ + 10 mev. So Γ bears the shape of a peak with half width 10 mev (Fig. S6 (b)). 8 NATURE PHYSICS

9 SUPPLEMENTARY INFORMATION We note that is significant only when + ~0.03 A and ħ 0.0 A. For such a value, is almost a constant, so the maximum spectral density maxγ~ 1 Γ 2 = 2 = 2 ~ 600 mev A. Here = is the electron density. For 10 cm = 10 A or lower density, maxγ ~10 mev. In this parameter regime 7, the population of the initial X state will irreversibly decay to the X continuum (Fermi s golden rule regime), with an effective decay rate Γ ħ ~ ħ 600 mev A. Assuming a doping with electron density = 10 cm = 10 A, then the decay rate Γ ħ ~. V. Temperature-dependence First, we focus on the up-conversion case (reverse PL). When resonantly exciting X, the up-converted exciton PL is determined by three factors: (1) The conversion rate from X to an electron- pair, which is proportional to the phonon occupation e ħ 1. This conversion rate increases when raising the temperature. (2) The absorption rate. An electron at wavevector is optically coupled to X with the same COM wave vector, which introduces finite absorption at the energy of (where ħ ) with the absorption proportional to the electron occupation at kinetic energy. For low doping levels and relatively high, the electron has a thermal distribution e. Assuming the CW excitation laser energy is at, the X absorption then scales as ~1. (3) The radiative decay rate. The kinetic energy of X will follow a thermal distribution, but only carriers inside the light cone, whose kinetic energy lies close to 0, can radiatively recombine. So the X effective radiative decay rate Γ is proportional to the X thermal population in the light cone which is 1. On the other hand, the nonradiative decay rate Γ is usually much larger than Γ, giving an exciton PL intensity proportional to Γ Γ +Γ Γ Γ 1. Combining the above three factors, the temperature dependence of the reverse PL is e ħ 1, as shown in Fig. S7a. Below, we discuss the temperature dependence of the standard PL, where an excitation laser with energy ~0.2 ev higher than the X resonance is applied to generate X and X. The large excess laser energy can facilitate the dynamical equilibrium between X and X, so we assume NATURE PHYSICS 9

10 that X and X populations satisfy the mass action law. Then the equilibrium exciton density, X density, and electron density within the system satisfy: = e. Here, = 30 mev is the X binding energy. We also write + as the total background electron density and + as the total laser excitation. Then = , 2 = Obviously and depend on several parameters: background electron density, laser excitation density, and the temperature. In Fig. S7b we show the exciton PL ( ) as a function of. Note that in Fig. 3d of the main text, the standard PL intensity remains nearly constant below 50 K. We consider this behavior to be a potential signature of laser heating from the ~0.2 ev excess photon energy. Fig. S7 (a) Estimated reverse PL intensity as a function of temperature. (b) Standard exciton PL intensity as a function of temperature, for background electron density = & total excitation density = ). 1 Kaasbjerg, K., Thygesen, K. S. & Jacobsen, K. W. Phonon-limited mobility in n-type single-layer MoS2 from first principles. Phys. Rev. B 85, , doi: /physrevb (2012). 2 Song, Y. & Dery, H. Transport Theory of Monolayer Transition-Metal Dichalcogenides through Symmetry. Phys. Rev. Lett. 111, , doi: /physrevlett (2013). 3 Chakraborty, B. et al. Symmetry-dependent phonon renormalization in monolayer MoS2 transistor. Phys. Rev. B 85, , doi: /physrevb (2012). 4 Carvalho, B. R., Malard, L. M., Alves, J. M., Fantini, C. & Pimenta, M. A. Symmetry- Dependent Exciton-Phonon Coupling in 2D and Bulk MoS2 Observed by Resonance Raman Scattering. Phys. Rev. Lett. 114, , doi: /physrevlett (2015). 10 NATURE PHYSICS

11 SUPPLEMENTARY INFORMATION 5 Zhao, W. et al. Lattice dynamics in mono- and few-layer sheets of WS2 and WSe2. Nanoscale 5, , doi: /c3nr03052k (2013). 6 Chen, S.-Y., Zheng, C., Fuhrer, M. S. & Yan, J. Helicity-Resolved Raman Scattering of MoS2, MoSe2, WS2, and WSe2 Atomic Layers. Nano Lett. 15, , doi: /acs.nanolett.5b00092 (2015). 7 Cohen-Tannoudji, C., Dupont-Roc, J. & Grynberg, G. Atom-photon interactions: basic processes and applications. (New York: Wiley, 1992). 8 Shiau, S.-Y., Combescot, M. & Chang, Y.-C. Trion ground state, excited states, and absorption spectrum using electron-exciton basis. Phys. Rev. B 86, , doi: /physrevb (2012). 9 Berkelbach, T. C., Hybertsen, M. S. & Reichman, D. R. Theory of neutral and charged excitons in monolayer transition metal dichalcogenides. Phys. Rev. B 88, , doi: /physrevb (2013). NATURE PHYSICS 11