Photonic-Crystal Exciton-Polaritons in Monolayer Semiconductors
PPhotonic-Crystal Exciton-Polaritons in MonolayerSemiconductors
Long Zhang , Rahul Gogna , Will Burg , Emanuel Tutuc and Hui Deng , ∗ Physics Department, University of Michigan,450 Church Street, Ann Arbor, MI 48109-2122, USA Applied Physics Program, University of Michigan,450 Church Street, Ann Arbor, MI 48109-1040, USA and Microelectronics Research Center, Department of Electrical and Computer Engineering,The University of Texas at Austin, Austin, Texas 78758, United States
Abstract
Semiconductor microcavity polaritons, formed via strong exciton-photon coupling, provide aquantum many-body system on a chip, featuring rich physics phenomena for better photonic tech-nology. However, conventional polariton cavities are bulky, difficult to integrate, and inflexible formode control, especially for room temperature materials. Here we demonstrate sub-wavelengththick one-dimensional photonic crystals (PCs) as a designable, compact and practical platform forstrong coupling with atomically thin van der Waals Crystals (vdWCs). Polariton dispersions andmode anti-crossings are measured up to room temperature. Non-radiative decay to dark excitonswas suppressed due to polariton enhancement of the radiative decay. Unusual features, includinghighly anisotropic dispersions and adjustable Fano resonances in reflectance, may facilitate hightemperature polariton condensation in variable dimensions. Combining slab PCs and vdWCs in thestrong coupling regime allows unprecedented engineering flexibility for exploring novel polaritonphenomena and device concepts. ∗ [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p . INTRODUCTION Control of light-matter interactions is elementary to the development of photonic devices. Ex-isting photonic technologies are based on weakly coupled matter-light systems, where the opticalstructure perturbatively modifies the electronic properties of the active media. As the matter-lightinteraction becomes stronger and no longer perturbative, light and matter couple to form hybridquasi-particles – polaritons. In particular, quantum-well (QW) microcavity exciton polaritons fea-ture simultaneously strong excitonic nonlinearity, robust photon-like coherence, and a meta-stableground state, providing a fertile ground for quantum many-body physics phenomena [1, 2] thatpromise new photonic technology [3]. Numerous novel types of many-body quantum states withpolaritons and polariton quantum technologies have been conceived, such as topological polaritons[4–6], polariton neurons [7], non-classical state generators [8–10], and quantum simulators [11–13].Their implementation require confined and coupled polariton systems with engineered properties,which, on one hand can be created by engineering the optical component of the strongly coupledmodes, on the other hand, is difficult experimentally using conventional polariton systems.Conventional polariton system are based on vertical FP cavities made of thick stacks of planar,distributed Bragg reflectors (DBRs), which have no free design parameter for mode-engineering andare relatively rigid and bulky against post-processing. Different cavity structures have been chal-lenging to implement for polariton systems as conventional materials are sensitive to free surfacesand lattice mismatch with embedding crystals. The recently emerged two-dimensional (2D) semi-conductor vdWCs [14, 15] are uniquely compatible with diverse substrate without lattice matching[16]. However, most studies of vdWC-polaritons so far continue to use FP cavities [17–23], whichare even more limiting for vdWCs than for conventional materials. This is because monolayer-thick vdWCs need to be sandwiched in between separately fabricated DBR stacks and positionedvery close to the cavity-field maximum. The process is complex, hard to control, and may changeor degrade the optical properties of vdWCs [22, 24]. Alternatively, metal mirrors and plasmonicstructures have been implemented [25–28]. They are more compact and flexible, but suffer fromintrinsically large absorption loss and poor dipole-overlap between the exciton and field [25, 29].Here we demonstrate sub-wavelength thick, one-dimensional dielectric PCs as a readily des-ignable platform for strong-coupling, which is also ultra-compact, practical, and especially wellsuited to the atomically-thin vdWCs. Pristine vdWCs can be directly laid on top of the PC with- ut further processing. Properties of the optical modes, and in turn the polariton modes, can bemodified with different designs of the PC. We confirm polariton modes up to room temperatureby measuring the polariton dispersions and mode anti-crossing in both reflectance and photolu-minescence (PL) spectra. Strongly suppressed non-radiative decay to dark excitons due to thepolaritonic enhancement was observed. We show that these polaritons have anisotropic polaritondispersions and adjustable reflectance, suggesting greater flexibility in controlling the excitationsin the system to reaching vdWC-polariton condensation at lower densities in variable dimensions.Extension to more elaborate PC designs and 2D PCs will facilitate research on polariton physicsand devices beyond 2D condensates. II. RESULTSA. The system
We use two kinds of transition metal dichalcogenides (TMDs) as the active media: a monolayerof tungsten diselenide (WSe ) or a monolayer of tungsten disulfide (WS ). The monolayers areplaced over a PC made of a silicon-nitride (SiN) grating, as illustrated in 1a. The total thicknessof the grating t is around 100 nm, much shorter than half a wavelength, making the structurean attractive candidate for compact, integrated polaritonics. In comparison, typical dielectricFP cavity structures are many tens of wavelengths in size. A schematic and scanning electronmicroscopy (SEM) images of the TMD-PC polariton device are shown in fig. 1. More details of thestructure and its fabrication are described in Methods. Since the grating is anisotropic in-plane, itsmodes are sensitive to both the propagation and polarization directions of the field. As illustratedin fig. 1a, we define the direction along the grating bars as the x -direction, across the grating barsas the y -direction, and perpendicular to the grating plane as the z -direction. For the polarization,along the grating corresponds to transverse-electric (TE), and across the bar, transverse-magnetic(TM). The TM-polarized modes are far off resonance with the exciton. Hence TM excitons remainin the weak coupling regime, which provides a direct reference for the energies of the un-coupledexciton mode. We focus on the TE-polarized PC modes in the main text and discuss the TMmeasurements in the Supplementary Figure S1. . WSe -PC polaritons We first characterize a monolayer WSe -PC device at 10 K.The energy-momentum mode struc-tures are measured via angle-resolved micro-reflectance (fig. 2a-b) and micro-PL (fig. 2c) spec-troscopy, in both the along-bar (top row) and across-bar (bottom row) directions. The data (leftpanels) are compared with numerical simulations (right panels), done with rigorous coupled waveanalysis (RCWA).Without the monolayer, a clear and sharp PC mode is measured with a highly anisotropicdispersion (fig. 2a, left panels) and is well reproduced by simulation (fig. 2a, right panels). Thebroad low-reflectance band in the background is an FP resonance formed by the SiO cappinglayer and the substrate. The PC mode half linewidth is γ cav = 6 . Q or finesse of about 270, much higher than most TMD-cavities [17, 19, 20, 25–27]and comparable to the best DBR-DBR ones [18, 21].With a WSe monolayer laid on top of the PC (fig. 1c), two modes that anti-cross are clearly seenin both the reflectance and PL spectra (fig. 2b-c) and match very well with simulations, suggestingstrong coupling between WSe exciton and PC modes. Strong anisotropy of the dispersion isevident comparing E LP,UP ( k x , k y = 0) (top row) and E LP,UP ( k x = 0 , k y ) (bottom row), resultingfrom the anisotropic dispersion of the PC modes. Correspondingly, the effective mass and groupvelocity of the polaritons are also highly anisotropic, which provide new degrees of freedom toverify polariton condensation and to control its dynamics and transport properties [30].To confirm strong-coupling, we fit the measured dispersion with that of coupled modes, and wecompare the coupling strength and Rabi-splitting obtained from the fitting with the exciton andphoton linewidth. In the strong coupling regime, the eigen-energies of the polariton modes E LP,UP at given in-plane wavenumber k (cid:107) and the corresponding vacuum Rabi splitting 2 (cid:126) Ω are given by: E LP,UP = 12 (cid:20) E exc + E cav + i ( γ cav + γ exc ) / (cid:21) ± (cid:115) g + 14 (cid:20) E exc − E cav + i ( γ cav − γ exc ) (cid:21) , (1)2 (cid:126) Ω = 2 (cid:112) g − ( γ cav − γ exc ( T )) / . (2)Here E exc is the exciton energy, γ exc and γ cav are the half-widths of the un-coupled exciton andPC resonances, respectively, and g is the exciton-photon coupling strength. A non-vanishing Rabisplitting 2 (cid:126) Ω requires g > | γ exc − γ cav | /
2; but this is insufficient for strong coupling. For the two esonances to be spectrally separable, the minimum mode-splitting needs to be greater than thesum of the half linewidths of the modes:2 (cid:126) Ω > γ cav + γ exc , or, g > (cid:112) ( γ exc + γ cav ) / . (3)In frequency domain, eq. (3) corresponds to requiring coherent, reversible energy transfer betweenthe exciton and photon mode. We first fit our measured PL spectra to obtain the mode disper-sion E LP,UP ( k x,y ), as shown by the symbols in fig. 2d. We then fit E LP,UP ( k x,y ) with (1), with g and E cav ( k x,y = 0) as the only fitting parameters. The exciton energy E exc and half-width γ exc are measured from the TM-polarized exciton PL from the same device, while the wavenumberdependence of E cav and γ cav are measured from the reflectance spectrum of the bare PC (Supple-mentary Figure S2b). We obtain g = 8 . ± .
23 meV and 7 . ± .
87 meV for dispersions along k x and k y , respectively, corresponding to a Rabi splitting of 2 (cid:126) Ω ∼ γ exc = 5 . γ cav = 3 .
25 meV. Therefore g is much greater than not only( γ exc − γ cav ) / . (cid:112) ( γ exc + γ cav ) / . C. Temperature dependence of WSe -PC polaritons At elevated temperatures, increased phonon scattering leads to faster exciton dephasing, whichdrives the system into the weak-coupling regime. We characterize this transition by the temperaturedependence of the WSe -PC system; we also show the effect of strong coupling on exciton quantumyield.We measure independently the temperature dependence of the uncoupled excitons via TMexciton PL, the uncoupled PC modes via reflectance from the bare PC, and the coupled modes viaPL from the WSe2-PC device. We show in fig. 3a the results obtained for k x = 3 . µm − , k y =0 µm − as an example. For the uncoupled excitons, with increasing T , the resonance energy E exc ( T )decreases due to bandgap reduction [31], as shown in fig. 3a, while the linewidth 2 γ exc broadensdue to phonon dephasing [32], as shown in fig. 3b. Both results are very well fitted by modelsfor conventional semiconductors (see more details in Methods). For the uncoupled PC modes,the energy E cav = 1 .
74 eV and half-linewidth γ cav = 6 . -PC device anti cross between 10-100 K and clearly split from he uncoupled modes, suggesting strong-coupling up to 100 K. Above 130 K, it becomes difficultto distinguish the modes from WSe -PC device and the uncoupled exciton and photon modes,suggesting the transition to the weak-coupling regime.We compare quantitatively in fig. 3b the coupling strength g with (cid:112) ( γ exc ( T ) + γ cav ) / γ exc − γ cav ) / g ( T ) drops to below (cid:112) ( γ exc ( T ) + γ cav ) / g > ( γ exc − γ cav ) / at low temperatures. Ithas been shown that the quantum yield of the bright excitonic states are strongly suppressed by10-100 fold in bare WSe monolayers due to relaxation to dark excitons lying at lower energiesthan the bright excitons [33, 34]. In contrast, the WSe -PC polariton intensity decreases by lessthan two-fold from 200 K to 10 K. This is because coupling with the PC greatly enhances theradiative decay of the WSe exciton-polariton states in comparison with scattering to the darkexciton states, effectively improving the quantum yield of the bright excitons. D. Room temperature WS -PC polaritons To form exciton-polaritons at room temperature, we use WS because of the large oscillator-strength to linewidth ratio at 300 K compared to WSe (Supplementary Figure S4). We use a1D PC that matches the resonance of the WS exciton at 300 K. The angle-resolved reflectancespectrum from the bare PC again shows a clear, sharp dispersion (fig. 4a). The broadband back-ground pattern is due to the FP resonance of the substrate. With a monolayer of WS placedon top, anti-crossing LP and UP branches form, as clearly seen in both the reflectance and PLspectra (fig. 4b-c). The data (left panels) are in excellent agreement with the simulated results(right panels). The dispersions measured from PL fit very well with the coupled oscillator modelin eq. (1), from which we obtain an exciton-photon interaction strength of g = 12 . ± .
36 meV, bove γ exc = 11 meV, γ cav = 4 . (cid:112) ( γ exc + γ cav ) / . (cid:126) Ω = 22 . E. Adjustable reflectance spectra with Fano resonances
Lastly, we look into two unconventional properties of the reflectance of the TMD-PC polaritonsystems: an adjustable reflectance background, and highly asymmetric Fano resonances. As shownin fig. 2 and fig. 4, a broadband background exists in the reflectance spectra for both WSe -PCand WS -PC polariton systems, arising from the FP resonances of the substrate. The heightand width of this broadband background is readily adjusted by the thickness of the SiO spacerlayer, un-correlated with the quality factor of the PC modes or the lifetime of the polaritons. Forexample, the WSe -PC polaritons are in the low-reflectance region of the FP bands (fig. 5a), whilethe WS -PC polaritons are in the high-reflectance region (fig. 5b). In contrast, in conventionalFP cavities, high cavity quality factor dictates that the polariton modes are inside a broad high-reflectance stop-band, making it difficult to excite or probe the polariton systems at wavelengthswithin the stop-band. The adjustability of the reflectance in PC-polariton systems will allow muchmore flexible access to the polariton modes and facilitate realization of polariton lasers, switchesand other polariton nonlinear devices.Another feature is the asymmetric Fano line shape of the PC and PC-polariton modes in thereflectance spectra (fig. 5). The Fano resonance arises from coupling between the sharp, discretePC or PC-polariton modes and the continuum of free-space radiation modes intrinsic to the 2D-slabstructure [35]. Such Fano line shapes are readily tuned by varying the phase difference betweenthe discrete mode and the continuum band. For example, the PC and WSe -PC polariton modeslocated at the valley of the FP band (fig. 5a) have a nearly-symmetric Lorentzian-like line shape, butthe PC and WS -PC polariton modes at the peak of the FP band feature a very sharp asymmetricFano line shape (fig. 5b). This is because of the π phase difference between the peak and valley ofthe FP bands.We compare the measured spectra with the standard Fano line shapes described by: R = R F (cid:18) ( (cid:15) + q ) (cid:15) + 1 − (cid:19) + R F P + I b , (4)The first term describes the Fano resonance, where R F is the amplitude coefficient, q is the asym-metry factor, (cid:15) = (cid:126) ( ω − ω ) γ is the reduced energy, (cid:126) ω and γ are the resonant energy and half inewidth of the discrete mode. R F P ( ω ) and I b are the FP background reflectance and a constantambient background, respectively. We use the transfer matrix method to calculate R F P , then fitour data to eq. (4) to determine the Fano parameters. For the WSe -PC spectrum, we obtain q cav = 5 . q LP = 3 . q UP = 4 .
1, for the PC, LP and UP modes, respectively. The large values of q suggest small degrees of asymmetry and line shapes close to Lorentzian, as seen in fig. 5a. Forthe WS device, we obtain q cav = 1 . q LP = 0 .
92, and q UP = 1 .
37, which are close to 1. Thiscorresponds to a much more asymmetric line shape with a sharp Fano-feature, as seen in fig. 5b.We note that, despite the striking Fano resonance in reflectance, strong coupling takes place onlybetween the exciton and the sharp, discrete, tightly confined PC modes. This is evident from thesymmetric line shape of the WS -polariton PL spectra fig. 4d. Fano resonances with polaritonstates as the discrete modes will enable control of the Fano line shape by angle and detuning. [36] III. DISCUSSION
In short, we demonstrate integration of two of the most compact and versatile systems – atom-ically thin vdWCs as the active media and PCs of deep sub-wavelength thicknesses as the opticalstructure – to form an untra-compact and designable polariton system. TMD-PC polaritons wereobserved in monolayer WS at room temperature and in WSe up to 110 K, which are the highesttemperatures reported for unambiguous determination of strong-coupling for each type of TMD,respectively. The TMD-PC polaritons feature highly anisotropic energy-momentum dispersions,adjustable reflectance with sharp Fano resonances, and strong suppression of non-radiative lossto dark excitons. These features will facilitate control and optimization of polariton dynamicsfor nonlinear polariton phenomena and applications, such as polariton amplifiers [37], lasers [38],switches [39] and sensors [35, 40].The demonstrated quasi-2D TMD-PC polariton system is readily extended to 0D, 1D andcoupled arrays of polaritons [41, 42]. The 1D PC already has many design parameter for mode-engineering; it can be extended to 2D PCs for even greater flexibility, such as different polarizationselectivity [43] for controlling the spin-valley degree of freedom [44]. The TMDs can be substitutedby and integrated with other types of atomically-thin crystals, including black phosphorous forwide band-gap tunability[45], graphene for electrical control[46], and hexagonal boron-nitride forfield enhancement. Cs feature unmatched flexibility in optical-mode engineering, while vdWCs allow unprece-dented flexibility in integration with other materials, structures, and electrical controls [47, 48].Combining the two in the strong coupling regime opens a door to novel polariton quantum many-body phenomenon and device applications [4–13].
METHODSSample fabrication.
The devices shown in fig. 1 were made from a SiN layer grown by low pres-sure chemical vapor deposition on a SiO -capped Si substrate. The SiN layer was partially etchedto form a 1D grating, which together with the remaining SiN slab support the desired PC modes.The grating was created via electron beam lithography followed by plasma dry etching. MonolayerTMDs are prepared by mechanical exfoliation from bulk crystals from 2D Semiconductors andtransferred to the grating using Polydimethylsiloxane (PDMS). For the WSe device, the gratingparameters are: Λ = 468 nm, η = 0 . t = 113 nm, h = 60 nm, d = 1475 nm. For the WS device,the grating parameters are: Λ = 413 nm, η = 0 . t = 78 nm, h = 40 nm, d = 2000 nm. Optical measurements.
Reflection and PL measurements were carried out by real-space andFourier-space imaging of the device. An objective lens with numerical aperture (N.A.) of 0.55 wasused for both focusing and collection. For reflection, white light from a tungsten halogen lampwas focused on the sample to a beam size of 15 µ m in diameter. For PL, a HeNe laser (633 nm)and a continuous-wave solid state laser (532 nm) were used to excite the monolayer WSe andWS , respectively, both with 1.5 mW and a 2 µ m focused beam size. The collected signals werepolarization resolved by a linear polarizer then detected by a Princeton Instruments spectrometerwith a cooled charge-coupled camera. RCWA simulation.
Simulations are carried out using an open-source implementation of RCWAdeveloped by Pavel Kwiecien to calculate the electric-field distribution of PC modes, as well asthe reflection and absorption spectra of the device as a function of momentum and energy. Theindices of refraction of the SiO and SiN are obtained from ellipsometry measurements to be n SiO2 = 1 .
45 + . λ and n SiN = 2 . . λ , where λ is the wavelength in the unit of µ m. TheWSe and WS monolayers were modelled with a thickness of .7 nm, and the in-plane permittivitieswere given by a Lorentz oscillator model: (cid:15) ( E ) = (cid:15) B + fE x − E − i Γ E . or WSe , we used oscillator strength f WS =0.7 eV to reproduce the Rabi splitting observed inexperiments, exciton resonance E WSe = 1 .
742 eV and full linewidth Γ
WSe = 11 . (cid:15) B, WSe = 25[49]. Likewise, for WS , we used f WS =1.85 eV , E WS = 2 .
013 eV and Γ WS = 22 meV measured from a bare monolayer, and (cid:15) B, WS = 16[50]. Modeling the temperature dependence of the WSe exciton energy and linewidth. The exciton resonance energies redshift with increasing temperature as shown in fig. 3a. It isdescribed by the standard temperature dependence of semiconductor bandgaps[31] as follows: E g ( T ) = E g (0) − S (cid:126) ω (cid:20) coth (cid:18) (cid:126) ω kT − (cid:19)(cid:21) . (5)Here E g (0) is the exciton resonance energy at T = 0 K, S is a dimensionless coupling constant,and (cid:126) ω is the average phonon energy, which is about 15 meV in monolayer TMDs[51, 52]. Thefitted parameters are: E g (0)=1.741 and S = 2 .
2, which agree with reported results[51, 52].The exciton linewidth γ exc as a function of temperature can be described by the followingmodel[52, 53]: γ exc = γ + c T + c e (cid:126) ω/kT − . (6)Here γ is the linewidth at 0 K, the term linear in T depicts the intravalley scattering by acousticphonons, and the third term describes the intervalley scattering and relaxation to the dark statethrough optical and acoustic phonons[53].The average phonon energy is (cid:126) ω =15 meV. The fittedparameters are: γ = 11 . c = 25 .
52 meV, and c is negligibly small in our case[52]. ACKNOWLEDGMENT
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Nature Communications , 13279 (2016). η , total thickness t and the grating thickness h . The SiO capping layer has athickness of d .(b) The TE-polarized electric field profile of the PC in the y − z plane. Thewhite lines mark the outline of the PC. The red line marks the position of the monolayerTMDs. (c) A side-view SEM image of the bare PC. (d) A top-view SEM image of theTMD laid on top of a PC.15IG. 2: Strong coupling between TE-polarized WSe exciton and PC modes measured byangle-resolved reflectance and PL at 10 K. The top/bottom row shows thealong-bar/cross-bar directions, respectively. The left/right panels of (a)-(c) show themeasured/simulated results, respectively. (a) Angle-resolved reflectance spectra of the barePC, showing a sharp, dispersive PC mode. (b) Angle-resolved reflectance spectra of theWSe -PC integrated device, showing split, anti-crossing upper and lower polariton modes.(c) Angle-resolved PL data (left) compared with the simulated absorption spectra of theWSe -PC integrated device, showing the same anti-crossing polariton modes as in (b). (d)The polariton energies E LP,UP vs. wavenumber k x , k y obtained from the spectra in (c).The lines are fits to the LP and UP dispersion with the coupled harmonic oscillator model,giving a vacuum Rabi splitting of 18.4 meV and 16.1 meV for the along-bar (top) andcross-bar (bottom) directions, respectively. The corresponding Hopfield coefficients | C | and | X | , representing the photon and exciton fractions in the LP modes, respectively, areshown in the top sub-plots.16IG. 3: Temperature dependence of the WSe -PC system. (a) The temperaturedependence of the exciton, cavity and polariton energies. The exciton energy E exc ( T ) ismeasured from the weakly coupled TM excitons (red circles) and fit by eq. (5) (dashed blueline). The cavity energy E cav ( T ) is found to be approximately constant with temperatureand indicated by the dashed blue line. The polariton energies E LP,UP ( T ) (black circles) areobtained from the PL spectra at k x = 3 . µm − , k y = 0 µm − . They anti-cross and splitfrom the exciton and PC mode energies at 100 K and below, showing strong coupling inthis range. (b) The strong coupling to weak coupling transition measured by thetemperature dependence of g (black stars), (cid:112) ( γ exc + γ cav ) / γ exc − γ cav ) / γ exc while γ cav is approximately constant with temperature. g drops to below (cid:112) ( γ exc + γ cav ) / g becomes smaller than ( γ exc − γ cav ) /
2. (c)Temperature dependence of the integrated PL intensity of the PC-WSe polariton,showing only mild change in intensity. The inset shows the spectrum at 10 K integratedover θ x = − ◦ to 30 ◦ and over the spectral range shown. The higher energy side shouldercorresponds to exciton-like LP emission at large θ x where the density of states is high.17IG. 4: Room-temperature strong coupling between TE-polarized WS exciton and PCmodes measured by angle-resolved reflectance and PL. The left/right panels are themeasured/simulated results, respectively. (a) Angle-resolved reflectance spectra of the barePC, showing a sharp, dispersive PC mode. (b) Angle-resolved reflectance spectra of theWS -PC integrated device, showing split, anti-crossing upper and lower polariton modes.(c) Angle-resolved PL data (left) compared with the simulated absorption spectra of theWS -PC integrated device, showing the same anti-crossing polariton modes as in (b). Thesolid lines are the fitted polariton dispersion, with a corresponding vacuum Rabi splittingof 22.2 meV. Dashed lines represent the exciton and cavity photon dispersion. (d)Normalized PL intensity spectra from (c) for k x = − . µm − (bottom) to k x = − . µm − (top). The red line marks the zero detuning at − . µm − . The dashedlines mark the fitted LP and UP positions, corresponding to the white solid lines markedin (c) .18IG. 5: Fano resonance in the reflectance of the (a) WSe and (b) WS systems, measuredat θ x = 24 ◦ and 6 ◦ , respectively. Blue dots are the reflectance spectra of the bare PCs, andred dots, the TMD-PC devices. The lines are comparison with Fano line shape given byeq. (4). The asymmetry parameters q for PC, LP, UP in WSe system are 5.0, 3.5, 4.1respectively, and in WS system, are 1 .
16, 0 .
92, and 1 ..