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Jiajie Pei, Xue Liu, Andrés Granados del Águila, Di Bao, Sheng Liu, Mohamed-Raouf Amara, Weijie Zhao, Feng Zhang, Congya You, Yongzhe Zhang, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Han Zhang, Qihua Xiong

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[Switching of K-Q intervalley trions fine structure and their dynamics in n-doped monolayer WS<sub>2</sub>](https://mdr.nims.go.jp/datasets/13dd64d9-9513-4cad-aa65-1d53d834a1ec)

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DOI: 10.29026/oea.2023.220034Switching of K-Q intervalley trions fine structureand their dynamics in n-doped monolayer WS2Jiajie Pei1,2, Xue Liu3, Andrés Granados del Águila3, Di Bao3, Sheng Liu3,Mohamed-Raouf AMARA3, Weijie Zhao3, Feng Zhang1, Congya You4,Yongzhe Zhang4, Kenji Watanabe5, Takashi Taniguchi5, Han Zhang1*and Qihua Xiong6*1Collaborative  Innovation  Center  for  Optoelectronic  Science  and  Technology,  International  Collaborative  Laboratory  of  2D  Materials  forOptoelectronic  Science  and  Technology  of  Ministry  of  Education  and  Guangdong  Province,  College  of  Optoelectronic  Engineering,  ShenzhenUniversity,  Shenzhen  518060,  China; 2College  of  Materials  Science  and  Engineering,  Fuzhou  University,  Fuzhou  350108,  China; 3Division  ofPhysics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore;4College  of  Materials  Science  and  Engineering,  Beijing  University  of  Technology,  Beijing  100124,  China; 5Research  Center  for  FunctionalMaterials,  International  Center  for  Materials  Nanoarchitectonics,  National  Institute  for  Materials  Science,  Tsukuba,  Ibaraki  305-0044,  Japan;6State Key Laboratory of Low Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China.*Correspondence: H Zhang, E-mail: hzhang@szu.edu.cn; QH Xiong, E-mail: qihua_xiong@tsinghua.edu.cnThis file includes:Section 1: Thermal dynamic of neutral and charged excitonsSection 2: Distribution of trions at different doping densitySupplementary information for this paper is available at https://doi.org/10.29026/oea.2023.220034Opto-Electronic Advances Supplementary informationApril 2023, Vol. 6, No. 4Open Access This article is licensed under a Creative Commons Attribution 4.0 International License.To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2023. Published by Institute of Optics and Electronics, Chinese Academy of Sciences.220034-S1 https://doi.org/10.29026/oea.2023.220034http://creativecommons.org/licenses/by/4.0/.  1.90 1.95 2.00 2.05 2.10 2.1502605200240480027054002905800270540032064003607200410820L3L2L1PL intensity (a.u.)Photon energy (eV)−60 V XTQXT X0XTQXTX0XTQXTX0−52 V−44 V−36 V−28 V−20 V−12 V−4 V1.90 1.95 2.00 2.05 2.10 2.150450900048096005101020057011400650130007201440084016800100020004 VPL intensity (a.u.)Photon energy (eV)12 V20 V28 V36 V44 V52 V60 V−60−40−20 0 20 40 602.012.022.032.042.052.062.072.08Peak energy (eV)Gate voltage (V)−60−40−20 0 20 40 600.0100.0150.0200.025FWHM (eV)Gate voltage (V)a b cdXQTXQTFig. S1 | Fitting result of the gate-dependent PL spectra of the monolayer WS2 taken at 10 K with a 25 μW excitation power. (a) Fittingresult of the PL spectra from –60 V to –4 V back gate voltages with Voigt function. (b) Fitting result of the PL spectra from 4 V to 60 V back gatevoltages  with  Voigt  function.  (c)  PL  peak  energy  of  X0,  XT,  and  emissions  as  a  function  of  gate  voltages.  (d)  Full  width  at  half  maximum(FWHM) of X0, XT and  emissions as a function of gate voltages. 0.1 1 10 100 10000.11101001000 X0 peak−60 V−12 VIntegrated PL intensity (a.u.)Laser power (μW)1 10 100 10001101001000−4 V−60 VIntegrated PL intensity (a.u.)Laser power (μW)XT peak−60 −40 −20 0 20 40 600.60.70.80.91.01.11.21.31.41.51.6Power law slopeGate voltage (V)XTQXTX0a b cXQTXQTFig. S2 | Power-law slope of X0,  XT and  at different back gate voltages. (a)  Log-log plot of the integrated PL intensity for X0 peak as afunction of excitation power from –60 V to –12 V. Note that the X0 peak could not be observed at gate from –4 V to 60 V. (b) Log-log plot of theintegrated PL intensity for XT peak as a function of excitation power from –60 V to –4 V. Note that the XT peak could not be observed at gate from4 V to 60 V. (c) Statistics of the power law slope for X0, XT and  at each gate voltage. Note that the X0 and XT peak was too weak to be re-solved at low powers and positive back-gate voltages.Pei JJ et al. Opto-Electron Adv  6, 220034 (2023) https://doi.org/10.29026/oea.2023.220034220034-S2   0 0.2 0.4 0.6 0.8 1.00.010.11 XT−60 V−50 V−40 V−30 V−20 V−10 V0 V10 V20 V30 V40 VNormalized PL intensityTime (ns)Fig. S3 | Gate-dependent time-resolved PL measurement for the XT. Measured time-resolved PL traces (dots) and corresponding double ex-ponential fitting (solid curves) for the XT at different back-gate voltages (from –60 V to 40 V). The signal was too weak to be detected when theback-gate voltage exceeded 40 V. 0 0.2 0.4 0.6 0.8 1.00.010.110.2 μW0.4 μW0.8 μW1.5 μW2.5 μW4.0 μWNormalized PL intensityTime (ns)XQT XQT0 0.2 0.4 0.6 0.8 1.00.010.11 XT0.2 μW0.4 μW0.8 μW1.5 μW2.5 μW4.0 μWNormalized PL intensityTime (ns)0.1 1 100204060200300400500600XTLifetime (ps)Laser power (μW)τ1τ2a b cXQTXQTXQTFig. S4 | Excitation power-dependent time-resolved PL measurement for the XT and . (a) Measured time-resolved PL traces (dots) andcorresponding double exponential fitting (solid curves) for the  at different excitation laser powers (from 0.2 μW to 4 μW). (b) Measured time-resolved PL traces (dots) and corresponding double exponential fitting (solid curves) for the XT at different excitation laser powers. (c) The statist-ical values of the fast decay lifetime τ1 and slow decay lifetime τ2 for the fitting results of XT and  at different excitation laser powers.Pei JJ et al. Opto-Electron Adv  6, 220034 (2023) https://doi.org/10.29026/oea.2023.220034220034-S3   Section 1: Thermal dynamic of neutral and charged excitonsnXnX− ne nP nBFirstly,  we  determine  the  relative  intensity  of  the  neutral  and  charged  excitonic  species  based  on  the  Mass  actionmodelS1.  From  charge  conservation  of  the  photoexcited  electrons  and  holes,  the  concentration  of  neutral  states  ( ),charged states ( ), free electrons ( ), laser intensity ( ), and doping level ( ) have the following relationship: nP = nX + nX− , nB = ne + nX− , ne + nX + 2nX− = nP + nB .Then, the equilibrium populations for the species are governed by the Saha equation: nXnenX−= AkBTexp(− ETkBT)= nA ,A =4MXmeπℏ2MX−≈ 6.18× 1011where kB is  Boltzmann constant, T is  the  temperature, ET is  the  trion binding energy, , nArepresents the temperature dependent equilibrium constant.Solving the above equations gives:  nX =12(nP − nB − nA +√(nP + nB + nA)2 − 4nPnB)nX− =12(nP + nB + nA −√(nP + nB + nA)2 − 4nPnB) .This fits well with a two-level system such as the monolayer MoSe2S1. However, due to the existence of dark states inthe monolayer WS2, both their populations will split into substructures: nX = nX0 + nXD , nX− = nXT + nXQT ,nX0 nXD nXT nXQTwhere  represents the bright exciton,  represents the dark exciton,  and  represent the two types of chargedexcitons,  respectively.  Resulting  from  the  difference  of  energy  levels  between  two  states,  their  concentrations  are 1.90 1.95 2.00 2.05 2.10 2.150991980110220096192092184010020001402800150300PL intensity (a.u.)Photon energy (eV)295 K260 K230 K200 K170 K140 K110 K1.90 1.95 2.00 2.05 2.10 2.150230460021042002605200350700047094005101020PL intensity (a.u.)Photon energy (eV)80 K60 K40 K30 K20 K10 KXTQ XTQXTXTX0X0a bFig. S5 | Fitting result of the temperature-dependent PL spectra of the monolayer WS2. (a) Fitting results of the PL spectra from 295 K to110 K temperatures with Voigt function. (b) Fitting results of the PL spectra from 80 K to 10 K temperatures with Voigt function.Pei JJ et al. Opto-Electron Adv  6, 220034 (2023) https://doi.org/10.29026/oea.2023.220034220034-S4 governed by the Boltzmann distributionS2:  nX0 = nXexp(− Δ1kBT)1+ exp(− Δ1kBT)constnXT = nX−exp(− Δ2kBT)1+ exp(− Δ2kBT)const,where Δ1 represents the energy difference between two exciton levels, Δ2 represents the energy difference between twotrion levels.XQTAt elevated temperatures, the value of Q-K valley energy difference (ΔEQK) changes mainly due to the thermalizationinduced  band renormalizationS3 that  switches  the  population  of  XT and .  Thus,  the  concentration  of  XT should  becorrected with a temperature-related function ΔEQK and becomes: nXT = nX−exp(−Δ2 − ΔEQKkBT)1+ exp(−Δ2 − ΔEQKkBT)const ,XQTand the population of  becomes: nXQT= nX−11+ exp(−Δ2 − ΔEQKkBT)const ,nX− = nXT + nXQTbased on the conservation of total population . The ΔEQK could be fit to a function ΔEQK = a+b(kBT)2,where a and b were fit to 15 and 0.24, respectively. The calculated results are shown in the figure below.  Section 2: Distribution of trions at different doping densityΔEF − ΔEQK ΔEQK ΔEFXQTThe  Fermi  energy  (EF)  increases  as  the  doping  density  is  increased 4, and  the  concentration  of  Q valley  electrons  in-creases accordingly. The formation of Q-valley trion relies on the relative position between the Fermi energy and the Qvalley energy level.  When the Q valley energy level  changes,  the proportion of  trions changes accordingly.  We set  theinitial Fermi level at the bottom of the K valley, then the relative position between the Fermi energy and the Q valley en-ergy level becomes: , where  is the Q-K valley energy difference,  is the change of Fermi energy as afunction of back-gate voltage. Then, the proportion of XT and  as a function of gate voltage increase could be calcu-lated based on the Boltzmann distribution:  0 50 100 150 200 250 30000.20.40.60.81.0 XX− X−XX0XTQXTPL intensity (a.u.)Temperature (K)×1012 ×1012 ×10120 50 100 150 200 250 30000.20.40.60.81.0PL intensity (a.u.)Temperature (K)0 50 100 150 200 250 30000.20.40.60.81.0PL intensity (a.u.)Temperature (K)a b cXQTFig. S6 | Calculated temperature-dependent PL intensity for the emission species. (a) The population of neutral and charged states withoutenergy splitting. (b) The population of bright exciton (blue circle line) and the original neutral exciton (blue dashed line). (c) The population of XTand  (green and red lines) splitted from the original negatively charged exciton (black dashed line). The amount of absorbed photons is set as1×1012 cm–2 for calculation, with an initial doping level 1×1012 cm–2.Pei JJ et al. Opto-Electron Adv  6, 220034 (2023) https://doi.org/10.29026/oea.2023.220034220034-S5 nXTnXQT= exp(−Δ2 + ΔEF − ΔEQKkBT),nXT + nXQT= nX−with the addition of hole conservation relationship: , we have:  nXT = nX−exp(−Δ2 + ΔEF − ΔEQKkBT)1+ exp(−Δ2 + ΔEF − ΔEQKkBT)constnXQT= nX−11+ exp(−Δ2 + ΔEF − ΔEQKkBT)const,Δ2Here  is energy difference of two types of trions at zero doping.ΔEX−TΔEX−X− = ΔE0X− + αΔEF ΔE0X−ΔEF = 2(ΔEX−X− − ΔE0X−)The Fermi energies at different doping density are estimated based on the energy separation ( ) between neutral(X) and charged state (X–). According to the energy and momentum conservation relationship of X and X– in refS4, wehave , where  is the energy separation of X and X− at zero doping, α is a constant. Here inour estimation, the calculated curves match well with the experimental data (main Fig. 2(c)) when the α=0.5. Thus, wehave . The Fermi energy as a function of gate voltage is shown below.According to the gate sweep, the back gate induced carrier density doping n can be estimated using the parallel-plate 2.00 2.04 2.08 2.1260 V−60 VReflectance contrast (a.u.)Photon energy (eV)XX−−60 −40 −20 0 20 40 60−60 −40 −20 0 20 40 60XX−Integrated absorbance (a.u.)Gate voltage (V)2.00 2.05 2.10 2.15Reflectance contrast_RVg-ROV (a.u.)Photon energy (eV)−60 V60 V25303540455055Energy separation (meV)Gate voltage (V)0204060Fermi energy (meV)a bc dFig. S7 | gate-dependent reflectance contrast spectra of monolayer WS2 at T=10 K. (a) Reflectance contrast spectra for the monolayer WS2at 10 K with back gate voltages from –60 V to 60 V. The spectra are vertically shifted for clarity. (b) Integrated absorbance for the neutral andcharged states. The dashed lines are fitting curves showing the trend. (c) The relative reflectance contrast spectra by subtracting the 0 V spec-trum (R0V) at each gate voltage (RVg) from (a). The peak intensities of the spectra are vertically expanded for clarity. (d) Energy separation of Xand X– at different gate voltages extracted from (c). The red curve is exponential fit with y=7.39e0.033x for the Fermi energy at each gate voltage.Pei JJ et al. Opto-Electron Adv  6, 220034 (2023) https://doi.org/10.29026/oea.2023.220034220034-S6 https://doi.org/10.29026/oea.2023.220034n = Cox (Vbg − Vbg,th) /eCox = ε0εr/doxΔEQKΔEQKcapacitor model , where Vbg is the back gate voltage, Vbg,th is the threshold voltage, e is the unitcharge, Cox is  the  dielectric  capacitance  per  unit  area,  which  could  be  calculated  from ,  where ε0 is  thedielectric constant of vacuum, εr is  the relative dielectric constant (3.9) of SiO2, dox is  the thickness (300 nm) of SiO2.The  carrier  density  is  estimated  to  be  ~0.7×1012 cm–2 per  10  V.  The  transition  trend  of  gate-dependent  neutral  andcharged excitons used for  the calculation is  based on the fitting curves  in Fig. S7(b) with the order  of  magnitude 1012cm–2. The amount of absorbed photons is set as 6×1012 cm–2 for calculation. The initial doping level is set as 1×1012 cm–2.For the calculation in Fig. S8, the energy range of  is set from –200 to 200 meV, and the change of Fermi energy isfrom 0 to 60 meV. The Fermi level at zero gate voltage is set to at the bottom of the conduction band edge. For the Qvalley energy levels of different TMDs, the  could be extracted from the density functional theory calculationS5, asindicated in Fig. S8(a, b) with dashed lines. The cross-sections of the image are shown in Fig. S8(c–f), which agrees wellwith the experimental observations. It should be noted that for a sample with high initial doping, the population of tri-ons would not be zero even applied with –60 V gate voltage, which is the case of our sample shown in main Fig. 2(c). 0 20 40 60 80 100 120−200−1000100200PL intensity (a.u.)Q-K valley energy difference (meV)Gate voltage increase (V)0E+008E+112E+123E+123E+124E+125E+12XQT1L WS22L WS21L MoSe21L WSe21L MoS21 3 7 20 55Fermi energy (meV)0 20 40 60 80 100 120−200−1000100200Q-K valley energy difference (meV)Gate voltage increase (V)0E+008E+112E+123E+123E+124E+125E+121L MoSe21L WS22L WS2XT1L WSe21L MoS21 3 7 20 55Fermi energy (meV)0 20 40 60 80 100 12001.02.03.04.05.0 XQTXTPL intensity (a.u.)WS20 20 40 60 80 100 12001.02.03.04.05.0 WSe2MoS2 MoSe20 20 40 60 80 100 12001.02.03.04.05.0PL intensity (a.u.)Gate voltage increase (V)0 20 40 60 80 100 12001.02.03.04.05.0×1012Gate voltage increase (V)Gate voltage increase (V) Gate voltage increase (V)a c db e fXQT XQTXQTFig. S8 | Calculated gate-dependent PL emissions of the trions XT and  for different TMDs. (a, b) Calculated population of  and XT asa function of gate voltage and the Q-K valley energy difference. The four types of TMDs with different Q-K valley splitting energies are indicatedwith dashed lines. (c–f) The transition curves of the  and XT as a function of gate voltage at the cross sections in (a) and (b). We found that thecalculated result for 1L WSe2 is in good agreement with the experimental result in Ref 6. For 1L MoS2 and MoSe2, the Q valleys have higher en-ergy level that are more difficult to access by back gate tuning.Pei JJ et al. Opto-Electron Adv  6, 220034 (2023) https://doi.org/10.29026/oea.2023.220034220034-S7 https://doi.org/10.29026/oea.2023.220034References Ross JS, Wu SF, Yu HY, Ghimire NJ, Jones AM et al. Electrical control of neutral and charged excitons in a monolayer semiconductor. NatCommun 4, 1474 (2013).S1. Zhang XX, You YM, Zhao SYF, Heinz TF. Experimental evidence for dark excitons in monolayer WSe2. Phys Rev Lett 115, 257403 (2015).S2. Peng GH, Lo PY, Li WH, Huang YC, Chen YH et al. Distinctive signatures of the spin- and momentum-forbidden dark exciton states in thephotoluminescence of strained WSe2 monolayers under thermalization. Nano Lett 19, 2299–2312 (2019).S3. Chernikov A, Van Der Zande AM, Hill HM, Rigosi AF, Velauthapillai A et al. Electrical tuning of exciton binding energies in monolayer WS2.Phys Rev Lett 115, 126802 (2015).S4. Roldán R, Silva-Guillén JA, López-Sancho MP, Guinea F, Cappelluti  E et al.  Electronic properties of single-layer and multilayer transitionmetal dichalcogenides MX2 (M = Mo, W and X = S, Se). Ann Phys 526, 347–357 (2014).S5. Jones AM, Yu HY, Ghimire NJ, Wu SF, Aivazian G et al. Optical generation of excitonic valley coherence in monolayer WSe2. Nat Nano-technol 8, 634–638 (2013).S6. 1.95 2.00 2.05 2.10−60−40−200204060Gate voltage (V)Photon energy (eV)95.00138.1200.6291.5423.6615.6894.6130025 μWXQT2L1.90 1.95 2.00 2.05 2.102004006008001000120014002LXQTXQT25 μWPL intensity (a.u.)Photon energy (eV)60 V−60 VXTX0KChQCeKVeEFa b cFig. S9 | Gate-dependent PL spectra of a bilayer WS2. (a) Color plot of the measured PL spectra for bilayer WS2 as a function of back gatevoltage at 25 μW excitation power at 10 K. The dashed line is guide to the eye showing the position of the emission peak. (b) PL spectra of bilay-er WS2 at –60 V and 60 V back gate voltage. (c) Schematic illustration of the Q-K valley energy difference for the bilayer WS2 and the corres-ponding carrier relaxation pathways.Pei JJ et al. Opto-Electron Adv  6, 220034 (2023) https://doi.org/10.29026/oea.2023.220034220034-S8 https://doi.org/10.1038/ncomms2498https://doi.org/10.1038/ncomms2498https://doi.org/10.1103/PhysRevLett.115.257403https://doi.org/10.1021/acs.nanolett.8b04786https://doi.org/10.1103/PhysRevLett.115.126802https://doi.org/10.1002/andp.201400128https://doi.org/10.1038/nnano.2013.151https://doi.org/10.1038/nnano.2013.151https://doi.org/10.1038/nnano.2013.151  Section 1:  Thermal dynamic of neutral and charged excitons  Section 2: Distribution of trions at different doping density