Free Submitted: 09 July 2018 Accepted: 23 August 2018 Published Online: 06 September 2018
J. Chem. Phys. 149, 091101 (2018); https://doi.org/10.1063/1.5047802

Triple sum-frequency (TSF) spectroscopy is a recently developed methodology that enables collection of multidimensional spectra by resonantly exciting multiple quantum coherences of vibrational and electronic states. This work reports the first application of TSF to the electronic states of semiconductors. Two independently tunable ultrafast pulses excite the A, B, and C features of a MoS2 thin film. The measured TSF spectrum differs markedly from absorption and second harmonic generation spectra. The differences arise because of the relative importance of transition moments and the joint density of states (JDOS). We develop a simple model and globally fit the absorption and harmonic generation spectra to extract the JDOS and the transition moments from these spectra. Our results validate previous assignments of the C feature to a large JDOS created by band nesting.
Coherent multidimensional spectroscopy (CMDS) is a useful tool for exploring the rich many-body physics of semiconductors.1,21. S. T. Cundiff, Opt. Express 16, 4639 (2008). https://doi.org/10.1364/oe.16.0046392. G. Moody and S. T. Cundiff, Adv. Phys.: X 2, 641 (2017). https://doi.org/10.1080/23746149.2017.1346482 A new CMDS methodology is Triple Sum Frequency (TSF) spectroscopy. TSF is the non-degenerate analog of third harmonic generation (THG) and the four-wave mixing extension of three-wave mixing processes like sum-frequency generation and second harmonic generation (SHG).33. N. A. Neff-Mallon and J. C. Wright, Anal. Chem. 89, 13182 (2017). https://doi.org/10.1021/acs.analchem.7b02917 TSF uses independently tunable ultrafast pulses to create coherences at increasingly higher frequencies while discriminating against transient populations. Scanning the multiple input pulse frequencies enables collection of a multidimensional spectrum. Cross peaks in the spectrum identify the dipole coupling between states. TSF has studied vibrational and electronic coupling in molecules.4–74. E. S. Boyle, A. V. Pakoulev, and J. C. Wright, J. Phys. Chem. A 117, 5578 (2013). https://doi.org/10.1021/jp404713x5. E. S. Boyle, N. A. Neff-Mallon, and J. C. Wright, J. Phys. Chem. A 117, 12401 (2013). https://doi.org/10.1021/jp409377a6. E. S. Boyle, N. A. Neff-Mallon, J. D. Handali, and J. C. Wright, J. Phys. Chem. A 118, 3112 (2014). https://doi.org/10.1021/jp50185547. M. Grechko, T. Hasegawa, F. D’Angelo, H. Ito, D. Turchinovich, Y. Nagata, and M. Bonn, Nat. Commun. 9, 885 (2018). https://doi.org/10.1038/s41467-018-03303-y This work reports the first TSF spectroscopy of a semiconductor.
We investigate a polycrystalline MoS2 thin film. Transition metal dichalcogenides (TMDCs), such as MoS2, are layered semiconductors whose indirect bandgaps become direct in the monolayer limit.8,98. K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, Phys. Rev. Lett. 105, 136805 (2010). https://doi.org/10.1103/physrevlett.105.1368059. J. K. Ellis, M. J. Lucero, and G. E. Scuseria, Appl. Phys. Lett. 99, 261908 (2011). https://doi.org/10.1063/1.3672219 TMDCs exhibit strong spin-orbit coupling, and high charge mobility, and have novel photonic capabilities.10–1210. Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman, and M. S. Strano, Nat. Nanotechnol. 7, 699 (2012). https://doi.org/10.1038/nnano.2012.19311. F. Xia, H. Wang, D. Xiao, M. Dubey, and A. Ramasubramaniam, Nat. Photonics 8, 899 (2014). https://doi.org/10.1038/nphoton.2014.27112. K. F. Mak and J. Shan, Nat. Photonics 10, 216 (2016). https://doi.org/10.1038/nphoton.2015.282 The optical spectrum of MoS2 is dominated by three features: A (ℏωA ≈ 1.8 eV), B (ℏωB ≈ 1.95 eV), and C (ℏωC ≈ 2.7 eV).13,1413. A. Molina-Sánchez, D. Sangalli, K. Hummer, A. Marini, and L. Wirtz, Phys. Rev. B 88, 045412 (2013). https://doi.org/10.1103/physrevb.88.04541214. Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, Phys. Rev. B 90, 205422 (2014). https://doi.org/10.1103/physrevb.90.205422 A and B originate from high binding energy excitonic transitions between spin-orbit split bands.13,15–1813. A. Molina-Sánchez, D. Sangalli, K. Hummer, A. Marini, and L. Wirtz, Phys. Rev. B 88, 045412 (2013). https://doi.org/10.1103/physrevb.88.04541215. D. Y. Qiu, F. H. da Jornada, and S. G. Louie, Phys. Rev. Lett. 111, 216805 (2013). https://doi.org/10.1103/physrevlett.111.21680516. K. He, N. Kumar, L. Zhao, Z. Wang, K. F. Mak, H. Zhao, and J. Shan, Phys. Rev. Lett. 113, 026803 (2014). https://doi.org/10.1103/physrevlett.113.02680317. N. Saigal, V. Sugunakar, and S. Ghosh, Appl. Phys. Lett. 108, 132105 (2016). https://doi.org/10.1063/1.494504718. J. Kopaczek, M. P. Polak, P. Scharoch, K. Wu, B. Chen, S. Tongay, and R. Kudrawiec, J. Appl. Phys. 119, 235705 (2016). https://doi.org/10.1063/1.4954157 The stronger C feature is predicted to arise from a large joint density of states (JDOS) due to band nesting across a large section of the Brillouin zone (BZ).19–2219. L. Britnell, R. M. Ribeiro, A. Eckmann, R. Jalil, B. D. Belle, A. Mishchenko, Y.-J. Kim, R. V. Gorbachev, T. Georgiou, S. V. Morozov, A. N. Grigorenko, A. K. Geim, C. Casiraghi, A. H. C. Neto, and K. S. Novoselov, Science 340, 1311 (2013). https://doi.org/10.1126/science.123554720. A. Carvalho, R. M. Ribeiro, and A. H. C. Neto, Phys. Rev. B 88, 115205 (2013). https://doi.org/10.1103/physrevb.88.11520521. J. Jeong, Y.-H. Choi, K. Jeong, H. Park, D. Kim, and M.-H. Cho, Phys. Rev. B 97, 075433 (2018). https://doi.org/10.1103/physrevb.97.07543322. M. Bieniek, M. Korkusiński, L. Szulakowska, P. Potasz, I. Ozfidan, and P. Hawrylak, Phys. Rev. B 97, 085153 (2018). https://doi.org/10.1103/physrevb.97.085153 As of yet, no direct, experimental verification of the large JDOS defining the C feature has been accomplished. In this work, we demonstrate how first, second, and third order spectroscopies can be used together to determine whether the prominence of a feature is due to a large transition dipole or a large transition degeneracy.
The spectroscopies considered here can be understood in the electric dipole approximation using perturbation theory.23,2423. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2008).24. N. Bloembergen and Y. R. Shen, Phys. Rev. 133, A37 (1964). https://doi.org/10.1103/physrev.133.a37 In brief, an electric field (E) drives a polarization (P) in the material. The polarization is related to an oscillating coherence between two states. The polarization is expressed as an expansion in the electric field and susceptibility (χ) order. Absorption, SHG, and TSF (THG) depend on χ(1), χ(2), and χ(3), respectively. Absorption is proportional to Imχ(1). For very thin films (no interference or velocity-mismatch effects), SHG and TSF intensities are proportional to χ(2)2 and χ(3)2, respectively. All of the discussed spectroscopies detect state coupling through resonant enhancement. When the driving field is resonant with an interstate transition, χ is enhanced and the output intensity increases. In crystalline systems, interstate transitions are also subject to momentum conservation, which typically restricts interstate coupling to vertical transitions within the Brillouin zone (i.e., direct transitions).
Our work expands upon the extensive body of SHG and THG work on TMDCs25–3825. Y. Li, Y. Rao, K. F. Mak, Y. You, S. Wang, C. R. Dean, and T. F. Heinz, Nano Lett. 13, 3329 (2013). https://doi.org/10.1021/nl401561r26. L. M. Malard, T. V. Alencar, A. P. M. Barboza, K. F. Mak, and A. M. de Paula, Phys. Rev. B 87, 201401 (2013). https://doi.org/10.1103/physrevb.87.20140127. N. Kumar, S. Najmaei, Q. Cui, F. Ceballos, P. M. Ajayan, J. Lou, and H. Zhao, Phys. Rev. B 87, 161403 (2013). https://doi.org/10.1103/physrevb.87.16140328. R. Wang, H.-C. Chien, J. Kumar, N. Kumar, H.-Y. Chiu, and H. Zhao, ACS Appl. Mater. Interfaces 6, 314 (2013). https://doi.org/10.1021/am404254229. M. L. Trolle, G. Seifert, and T. G. Pedersen, Phys. Rev. B 89, 235410 (2014). https://doi.org/10.1103/physrevb.89.23541030. M. Grüning and C. Attaccalite, Phys. Rev. B 89, 081102 (2014). https://doi.org/10.1103/physrevb.89.08110231. D. J. Clark, V. Senthilkumar, C. T. Le, D. L. Weerawarne, B. Shim, J. I. Jang, J. H. Shim, J. Cho, Y. Sim, M.-J. Seong, S. H. Rhim, A. J. Freeman, K.-H. Chung, and Y. S. Kim, Phys. Rev. B 90, 121409 (2014). https://doi.org/10.1103/physrevb.90.12140932. M. L. Trolle, Y.-C. Tsao, K. Pedersen, and T. G. Pedersen, Phys. Rev. B 92, 161409 (2015). https://doi.org/10.1103/physrevb.92.16140933. G. Wang, X. Marie, I. Gerber, T. Amand, D. Lagarde, L. Bouet, M. Vidal, A. Balocchi, and B. Urbaszek, Phys. Rev. Lett. 114, 097403 (2015). https://doi.org/10.1103/physrevlett.114.09740334. J. Sun, Y.-J. Gu, D. Y. Lei, S. P. Lau, W.-T. Wong, K.-Y. Wong, and H. L.-W. Chan, ACS Photonics 3, 2434 (2016). https://doi.org/10.1021/acsphotonics.6b0068235. L. Karvonen, A. Säynätjoki, M. J. Huttunen, A. Autere, B. Amirsolaimani, S. Li, R. A. Norwood, N. Peyghambarian, H. Lipsanen, G. Eda, K. Kieu, and Z. Sun, Nat. Commun. 8, 15714 (2017). https://doi.org/10.1038/ncomms1571436. M. J. Shearer, L. Samad, Y. Zhang, Y. Zhao, A. Puretzky, K. W. Eliceiri, J. C. Wright, R. J. Hamers, and S. Jin, J. Am. Chem. Soc. 139, 3496 (2017). https://doi.org/10.1021/jacs.6b1255937. M. M. Glazov, L. E. Golub, G. Wang, X. Marie, T. Amand, and B. Urbaszek, Phys. Rev. B 95, 035311 (2017). https://doi.org/10.1103/physrevb.95.03531138. N. K. Balla, M. O’Brien, N. McEvoy, G. S. Duesberg, H. Rigneault, S. Brasselet, and D. McCloskey, ACS Photonics 5, 1235 (2018). https://doi.org/10.1021/acsphotonics.7b00912 by exploring the multidimensional frequency response. TSF and other spectroscopies that scan multiple driving field frequencies can identify multiresonant enhancement, whereby the driving fields resonate with more than one interstate transition.3939. J. C. Wright, Annu. Rev. Phys. Chem. 62, 209 (2011). https://doi.org/10.1146/annurev-physchem-032210-103551 Multiresonance selectively enhances coupled transitions and decongests dense spectra. In crystalline materials, multiresonant excitation is also subject to the momentum conservation mentioned earlier, so TSF can isolate multiple transitions from singular points in the Brillouin zone. Because of this selectivity, TSF is a potential method for mapping out band dispersion in crystals. With three independently tunable lasers, TSF can couple up to four states at a specific k-point together, so TSF could measure the dispersion of up to four bands. The present work is a step toward the goal of momentum-selective CMDS.
Nonlinear measurements of thin films are often complicated by non-resonant substrate contributions which are mitigated by measuring the coherent output in the reflected instead of transmitted direction.40–4440. A. Volkmer, J.-X. Cheng, and X. S. Xie, Phys. Rev. Lett. 87, 023901 (2001). https://doi.org/10.1103/physrevlett.87.02390141. K. J. Czech, B. J. Thompson, S. Kain, Q. Ding, M. J. Shearer, R. J. Hamers, S. Jin, and J. C. Wright, ACS Nano 9, 12146 (2015). https://doi.org/10.1021/acsnano.5b0519842. D. J. Morrow, D. D. Kohler, and J. C. Wright, Phys. Rev. A 96, 063835 (2017). https://doi.org/10.1103/physreva.96.06383543. J. D. Handali, K. F. Sunden, E. M. Kaufman, and J. C. Wright, Chem. Phys. 512, 13 (2018). https://doi.org/10.1016/j.chemphys.2018.05.02344. A. Honold, L. Schultheis, J. Kuhl, and C. W. Tu, Appl. Phys. Lett. 52, 2105 (1988). https://doi.org/10.1063/1.99549 For our experiment, TSF measured in the reflected direction has an effective penetration depth of ∼λfundamental/12 ∼ 100 nm; this small sampling length allows the resonant response from the thin film to be orders of magnitude more intense than the non-resonant response (see the supplementary material for more discussion). We prepared a 10 nm thick MoS2 thin film by first evaporating Mo onto the fused silica substrate followed by sulfidation of the Mo.41,4541. K. J. Czech, B. J. Thompson, S. Kain, Q. Ding, M. J. Shearer, R. J. Hamers, S. Jin, and J. C. Wright, ACS Nano 9, 12146 (2015). https://doi.org/10.1021/acsnano.5b0519845. M. R. Laskar, L. Ma, S. Kannappan, P. S. Park, S. Krishnamoorthy, D. N. Nath, W. Lu, Y. Wu, and S. Rajan, Appl. Phys. Lett. 102, 252108 (2013). https://doi.org/10.1063/1.4811410 Our sample substrate is a fused silica prism so that back-reflected, non-resonant TSF exits the substrate traveling parallel to the desired TSF signal but shifted spatially. Sample geometry, synthesis details, AFM measurements to determine film thickness, and a Raman spectrum are provided in the supplementary material.
For our TSF measurements, an ultrafast oscillator seeds a regenerative amplifier, creating pulses centered at 1.55 eV with a 1 kHz repetition rate. These pulses pump two optical parametric amplifiers (OPAs) which create tunable pulses of light from ∼0.5 to ∼1 eV with spectral width on the amplitude level of FWHM ≈ 46 meV. The two beams with frequencies ω1 and ω2 and wave vectors k1 and k2 are focused onto the sample. All beams are linearly polarized (S relative to sample) and coincident in time. The spatially coherent output with wave vector k1+2k2 is isolated with an aperture (the negative sign corresponds to the reflective direction), focused into a monochromator, and detected with a photomultiplier tube. The TSF intensity is linear in ω1 fluence and quadratic in ω2 fluence (see the supplementary material for details). The measured MoS2 TSF spectrum is normalized by the measured TSF spectrum of the fused silica substrate in order to account for spectrally dependent OPA output intensities and detector responsivity. The supplementary material contains additional experimental and calibration details. All raw data, workup scripts, and simulation scripts used in the creation of this work are permissively licensed and publicly available for reuse.4646. D. Morrow, D. Kohler, K. Czech, and J. Wright, “Multidimensional triple sum-frequency spectroscopy of MoS2 and comparisons with absorption and second harmonic generation spectroscopies” (published online). https://doi.org/10.17605/osf.io/2wf6g Our acquisition,4747. B. J. Thompson, K. F. Sunden, D. J. Morrow, and N. A. Neff-Mallon (2018). “PyCMDS,” Zenodo. https://doi.org/10.5281/zenodo.1198911. workup,4848. B. J. Thompson, K. F. Sunden, D. J. Morrow, N. A. Neff-Mallon, K. J. Czech, D. D. Kohler, and R. Swedin (2018). “WrightTools,” Zenodo. https://doi.org/10.5281/zenodo.1198905. and modeling software4646. D. Morrow, D. Kohler, K. Czech, and J. Wright, “Multidimensional triple sum-frequency spectroscopy of MoS2 and comparisons with absorption and second harmonic generation spectroscopies” (published online). https://doi.org/10.17605/osf.io/2wf6g are built on top of the open source, publicly available Scientific Python ecosystem.49–5149. E. Jones, T. Oliphant, and P. Peterson, SciPy: Open source scientific tools for Python, 2001, Online; accessed 28-09-2017.50. S. van der Walt, S. C. Colbert, and G. Varoquaux, Comput. Sci. Eng. 13, 22 (2011). https://doi.org/10.1109/mcse.2011.3751. J. D. Hunter, Comput. Sci. Eng. 9, 90 (2007). https://doi.org/10.1109/mcse.2007.55
Our main experimental result is a 2D TSF spectrum of a MoS2 thin film (Fig. 1). The TSF spectrum of MoS2 has a simple structure, with all features running parallel to a line with a slope of −1/2. This single variable dependence of our output intensity implies that there is negligible multiresonant enhancement within our spectral window. Our spectral window specifically rules out optical coupling between the A and C features. If the A and C features were coupled, we would see a peak that depends on two frequency conditions: ℏω2 = ℏωA/2 ≈ 0.9 eV and ω1=ωCωA0.9eV. Note that A and C features are believed to originate from different regions of the Brillouin zone, so resonant enhancement is not expected.
The slope of −1/2 implies that only the output frequency, ω1 + 2ω2, determines the resonant enhancement. The energy ladder diagram for such a resonant enhancement is shown in Fig. 1 inset (left). Peaks at output colors ∼1.85 and ∼2.0 eV are roughly consistent with three photon resonances with the A and B features, while a trough is present for output colors close to the C feature (∼2.7 eV). Curiously, we do not see contributions from two-photon resonances [energy level diagram, Fig. 1 inset (right)], even when 2ω2 or ω1 + ω2 traces over the A or B features. The two-photon resonances would manifest as horizontal or anti-diagonal (slope of −1) features in Fig. 1. The lack of two-photon resonances was surprising to us given the large two-photon absorption cross section of MoS2.5252. S. Zhang, N. Dong, N. McEvoy, M. O’Brien, S. Winters, N. C. Berner, C. Yim, Y. Li, X. Zhang, Z. Chen, L. Zhang, G. S. Duesberg, and J. Wang, ACS Nano 9, 7142 (2015). https://doi.org/10.1021/acsnano.5b03480 We address this observation later.
Since the dominant spectral features in Fig. 1 depend only on output color, we can generate a 1D THG spectrum by plotting the mean TSF amplitude for each output color. The THG spectrum is compared with other techniques in Fig. 2. Due to the unconventional prism substrate, we were unable to acquire an absorption spectrum of the sample, but we did acquire a reflection contrast spectrum shown in Fig. 2.53,5453. J. McIntyre and D. Aspnes, Surf. Sci. 24, 417 (1971). https://doi.org/10.1016/0039-6028(71)90272-x54. K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, Phys. Rev. Lett. 101, 196405 (2008). https://doi.org/10.1103/physrevlett.101.196405 The absorption spectrum presented in the work of Czech et al.4141. K. J. Czech, B. J. Thompson, S. Kain, Q. Ding, M. J. Shearer, R. J. Hamers, S. Jin, and J. C. Wright, ACS Nano 9, 12146 (2015). https://doi.org/10.1021/acsnano.5b05198 and Fig. 2 is of a sample prepared with similar conditions as our sample, but on a flat window substrate. The A and B feature peaks of the THG spectrum are blue-shifted compared to the absorption spectrum. Wang et al.2828. R. Wang, H.-C. Chien, J. Kumar, N. Kumar, H.-Y. Chiu, and H. Zhao, ACS Appl. Mater. Interfaces 6, 314 (2013). https://doi.org/10.1021/am4042542 observed a similar blue shift when they measured the THG spectrum of MoS2 around the A and B features. The C feature is dominant in the SHG3232. M. L. Trolle, Y.-C. Tsao, K. Pedersen, and T. G. Pedersen, Phys. Rev. B 92, 161409 (2015). https://doi.org/10.1103/physrevb.92.161409 and absorption spectrum,4141. K. J. Czech, B. J. Thompson, S. Kain, Q. Ding, M. J. Shearer, R. J. Hamers, S. Jin, and J. C. Wright, ACS Nano 9, 12146 (2015). https://doi.org/10.1021/acsnano.5b05198 while the A and B features are dominant in the THG spectrum. This observation is the main motivation for our analysis.
To explain why C dominates the absorption and SHG spectra but not the THG spectrum, we develop a simple model. To our knowledge, a unified model for comparing large dynamic-range absorption, SHG, and THG spectra of a semiconductor does not exist. Notable headway has been made to calculate SHG spectra5555. J. E. Sipe and A. I. Shkrebtii, Phys. Rev. B 61, 5337 (2000). https://doi.org/10.1103/physrevb.61.5337 and write closed equations of motion5656. V. M. Axt and S. Mukamel, Rev. Mod. Phys. 70, 145 (1998). https://doi.org/10.1103/revmodphys.70.145 for semiconductors, but simple formalisms are lacking. Most simple formalisms (cf. Refs. 5757. N. Peyghambarian, S. W. Koch, and A. Mysyrowicz, Introduction to Semiconductor Optics (Prentice Hall, 1993). and 5858. M. Dresselhaus, G. Dresselhaus, S. Cronin, and A. G. S. Filho, Solid State Properties (Springer Berlin Heidelberg, 2018).) approximate the dipole as a constant with respect to both transition energy and lattice momentum. This constant dipole approximation breaks down above the bandgap, where the lattice momentum of transitions is significantly different from that of the bandgap transitions. As the lattice momentum changes, Bloch waves alter their bonding symmetries and intralattice character (cf. Ref. 5959. J. E. Padilha, H. Peelaers, A. Janotti, and C. G. V. de Walle, Phys. Rev. B 90, 205420 (2014). https://doi.org/10.1103/physrevb.90.205420), which alters the transition dipole moments. Since the C feature is believed to originate from a different region of the Brillouin zone than the bandedge transitions, our theory requires both the dipole moment and JDOS to vary across our spectral range.
We develop our simple model by expanding the typical linear response formalism for direct transitions of semiconductors to include sum-frequency processes. In the case of χ(1), simple theories exist for expanding the single oscillator case to bulk conditions. For more than one oscillator, the total susceptibility is the sum of individual susceptibilities. For a semiconductor system, this is a summation over all wave-vectors, k, such that k ∈ BZ,
χ1ω1,ω1=a,gkμgak2Δgak1,(1)
where Δgak1=ωagkω1iΓ and Γ is a damping rate which accounts for the finite width of the optical transitions. It is common to replace the summation with a transition energy distribution function between conduction band x and valence band y, Jxy(E),6060. Equation (1), and all further theory developed, neglect indirect transitions. We find this a reasonable assumption since our multidimensional spectrum exhibited no cross-peaks between the K-point features (A and B) and the C band. such that
χ(1)(ω1,ω1)dEEω1iΓx,yJxy(E)μxy(E)2.(2)
In crystals, Jxy(E) is the JDOS. Note the dependence of the susceptibility on both Jxy(E) and μxy(E).22. G. Moody and S. T. Cundiff, Adv. Phys.: X 2, 641 (2017). https://doi.org/10.1080/23746149.2017.1346482 In the case that either the JDOS or the transition dipole is constant, spectroscopy techniques can be used to locate excitonic transitions or critical points in the JDOS. If the JDOS and transition dipole both vary, then traditional techniques fail.
The THG and SHG responses take the form of
χ2ω21,ω1,ω2=Pb,a,gkμgbkμbakμagkΔgbk12Δgak1,(3)
χ3ω321,ω1,ω2,ω3=Pc,b,a,gkμgckμcbkμbakμagkΔgck123Δgbk12Δgak1,(4)
where c, b, a, and g are the bands of the semiconductor. We have defined ω21ω2 + ω1 and ω321ω3 + ω2 + ω1. P is a permutation operator which accounts for all combinations of field-matter interactions. The additional detuning factors are defined by Δgck123ωcgkω321iΓ and Δgbk12ωbgkω21iΓ in which ωab is the frequency difference between bands a and b at point k in the BZ. The JDOS formalism employed in Eq. (2) can be abstracted to describe χ(2) and χ(3) with the introduction of multidimensional joint density functions. These joint densities depend not just on the energy difference between the initial and final states but also on the energy differences between the intermediate states reached during the sum-frequency process. See the supplementary material for further details.
To simulate the spectra, the sum over bands in Eqs. (2)–(4) is truncated at three total bands: the valence band, v, the conduction band, c, and a third higher-energy band, b, (note: bands b and c are not to be confused with the B and C absorption spectrum features). The SHG and THG spectra are measured close to the direct bandgap, so transitions between c and v are key to describing the response. The b band is taken to be a much higher energy (6 eV) than the valence band. We define the transition strength of low-lying states (c and v) to this nondescript high-lying band with the parameter μNR(ℏω). We note that μNR is not formally a dipole but instead contains all non-resonant transition factors involving band b (dipoles and degeneracies between c and b or v and b). While this is an improper parameterization of the actual band structure above the conduction band and below the valence band, its inclusion is crucial for reproducing details of our spectra, and the parameters offer insight into the role of virtual states in sum-frequency spectroscopies.
With this framework, we can now reason why THG, SHG, and absorption measurements are complementary for distinguishing degeneracy and dipole moments. The strength of absorption is proportional to μcv2 and Jcv [Eq. (2)]. SHG signals will be due to the sequence vbc, which informs on the non-resonant band b. THG has sequences such as vcvc, which scale as μcv4 but are still linear in Jcv. On the other hand, THG can depend on state b and consequently depends on the same non-resonant features of SHG. THG and absorption give different scalings for transitions between c and v, but SHG is also needed to constrain the non-resonant transitions of THG involving band b.
Figure 3 summarizes the fit of our model to experiment. Our simulation uses a discrete set of transition energies to approximate the integral of Eq. (2). We employ 140 discrete energies spaced 20 meV apart with Γ = 20 meV, ωbj=6eV, and Γb = 500 meV. Our strategy of a discretized set of transition energies is similar to the constrained variational analysis that is often employed to relate a material’s reflection spectrum to its absorption or dielectric spectra.14,61,6214. Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, Phys. Rev. B 90, 205422 (2014). https://doi.org/10.1103/physrevb.90.20542261. A. B. Kuzmenko, Rev. Sci. Instrum. 76, 083108 (2005). https://doi.org/10.1063/1.197947062. E. J. Sie, A. J. Frenzel, Y.-H. Lee, J. Kong, and N. Gedik, Phys. Rev. B 92, 125417 (2015). https://doi.org/10.1103/physrevb.92.125417 As a function of transition energy, the model extracts the dipole strength of the cv transitions, μcv; a weighting factor for transitions involving the non-resonant state, μNR; and the transition degeneracy, Jcv. See the supplementary material for further modeling details.
Figure 3(a) shows qualitative agreement between the model and experiment. Our model does not assume a functional form for the JDOS or dipole spectra, so we can compare the response from an excitonic transition to that of an interband transition. The fitted parameters are shown in Figs. 3(b) and 3(c); note that both μcv and μNR are peaked near the A and B features, while Jcv is minimized. As ℏω increases from the B feature, Jcv drastically increases, while μcv and μNR both decrease—the increase in Jcv is analogous to the large JDOS attributed to band nesting.19–2119. L. Britnell, R. M. Ribeiro, A. Eckmann, R. Jalil, B. D. Belle, A. Mishchenko, Y.-J. Kim, R. V. Gorbachev, T. Georgiou, S. V. Morozov, A. N. Grigorenko, A. K. Geim, C. Casiraghi, A. H. C. Neto, and K. S. Novoselov, Science 340, 1311 (2013). https://doi.org/10.1126/science.123554720. A. Carvalho, R. M. Ribeiro, and A. H. C. Neto, Phys. Rev. B 88, 115205 (2013). https://doi.org/10.1103/physrevb.88.11520521. J. Jeong, Y.-H. Choi, K. Jeong, H. Park, D. Kim, and M.-H. Cho, Phys. Rev. B 97, 075433 (2018). https://doi.org/10.1103/physrevb.97.075433 For comparison, in Fig. 3(c), we plot the JDOS as recently calculated by Bieniek et al.2222. M. Bieniek, M. Korkusiński, L. Szulakowska, P. Potasz, I. Ozfidan, and P. Hawrylak, Phys. Rev. B 97, 085153 (2018). https://doi.org/10.1103/physrevb.97.085153 for monolayer MoS2 within their tight-binding model. Both the extracted Jcv and the tight-binding, monolayer JDOS have a small value near the A and B features but form a peaked structure near the C feature. Because absorption, SHG, and THG spectra all scale linearly with Jcv but differently with transition dipole strength, the extracted structure of Jcv and μcv explains the glaring disparities between THG and the other two spectra. Our fitting procedure convincingly reproduces the large degeneracy of the C feature due to band nesting.
To the red of the A feature, the JDOS and μNR increase while μcv decays to zero. We attribute this behavior to an artifact of our finite spectral range. Variational approaches are known to have difficulty with the edges of spectra.6363. D. M. Roessler, Br. J. Appl. Phys. 16, 1119 (1965). https://doi.org/10.1088/0508-3443/16/8/310
Though our fitting procedure examined only harmonic generation, our fits also explain the notable lack of two-photon resonances in the 2D TSF spectrum. Figure 3(d) shows a 2D TSF spectrum simulated from our fit parameters. The TSF features produced by our model are primarily three photon resonances which lie parallel to the lines of constant output color. Some features from two-photon resonances (e.g., vbcb) such as the trough over the anti-diagonal line (ω1 + ω2) = 1.7 eV are visible but minor. The two-photon resonances that are prominent in the SHG spectra are suppressed because transitions involving b are severely detuned so sequences like vμcvcμcvvμcvc dominate the output over sequences like vμNRbμNRcμNRb. In the model, below the line (ω1 + ω2) = 1.7 eV, the dominant features come from coherence pathways like vcbc or vcvc, in which the third excitation is resonant.
Our model fails to capture some features of the three optical measurements. For instance, the model shifts the position of A and B absorption features and undershoots the THG spectrum at energies above 2.7 eV. The model falsely attributes the nature of the deep trough seen in Fig. 1 which runs along (ω1 + 2ω2) ≈ 2.56 eV. Specifically, Fig. 3(d) shows the dip of the simulated THG spectrum to be due to a two photon resonance running along (ω1 + ω2) ≈ 1.7 eV; this resonance is not seen in Fig. 1 which exclusively exhibits three photon resonances. A more complete fit would take into account our full 2D spectrum in order to distinguish between pathways with resonance enhancement from the second and third interaction. A more careful treatment of non-resonant transitions (μNR) may suppress these pathways. For instance, our model extracts a deep trough in μNR around 2.2-2.6 eV, yet we do expect the non-resonant transition strength to have a strong dependence on our output color. Despite these shortcomings, our approach provides a robust characterization that informs on the interplay of dipole strength and state density on the linear and non-linear spectra of our sample.
In summary, we performed TSF measurements to explore the electronic structure of MoS2 thin films. Our TSF measurements uncover a conspicuous difference between absorption, SHG, and TSF spectra: the C feature is prominent in absorption and SHG but not TSF. We address this conundrum by extracting the spectrally dependent dipole and JDOS using all three spectra. We find that the differences in the spectra arise because the C feature has a large JDOS and small dipole compared to the A and B features. We hope our measurements and analysis catalyze a renewed interest in elucidating the full spectral features of semiconductors by combining the results of many orders of complementary spectroscopies. Our measurements demonstrate the utility of non-linear, sum-frequency spectroscopies of semiconductor nanostructures over a wide range of excitation frequencies. In the future, our work can be extended to examine the coupling of multiple transitions which originate at the same point in the BZ and thus elucidate how different conduction or valence bands interact with each other.
See supplementary material for synthesis and characterization of our MoS2 thin film, more discussion of our ultrafast instrument and its calibration, discussion of data normalization scheme, discussion of our model, and additional simulation details.
This work was supported by the Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under Award No. DE-FG02-09ER46664. D.J.M. acknowledges support from the Link Foundation. The authors thank K. Lloyd for performing AFM measurements and T. Pedersen for sharing the SHG data used in this work.
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