Theory of second harmonic generation in few-layered MoS2
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Theory of second harmonic generation in few-layered MoS Mads L. Trolle, ∗ Gotthard Seifert, and Thomas G. Pedersen
1, 3 Department of Physics and Nanotechnology, Aalborg University, Skjernvej 4A, DK-9220 Aalborg East, Denmark Physikalische Chemie, Technische Universit¨at Dresden, D-01062 Dresden, Germany Center for Nanostructured Graphene (CNG), Aalborg University, DK-9220 Aalborg East, Denmark (Dated: November 5, 2018)Recent experimental results have demonstrated the ability of monolayer MoS to efficiently gener-ate second harmonic fields with susceptibilities between 0.1 and 100 nm/V. However, no theoreticalcalculations exist with which to interpret these findings. In particular, it is of interest to theoret-ically estimate the modulus of the second harmonic response, since experimental reports on thisdiffer by almost three orders of magnitude. Here, we present single-particle calculations of the sec-ond harmonic response based on a tight-binding band structure. We compare directly with recentexperimental findings and include in the discussion also spectral features and the effects of multiplelayers. The prediction and observation of a direct band gapin monolayer (ML) MoS has revitalized the interestin the optical properties of this material . Indeed,a substantial photoluminescence for 1H-MoS has beenobserved . Additionally, several papers have recentlydemonstrated how second harmonic generation (SHG)microscopy can be used to extract important informationregarding e.g. the number of layers and crystallographicorientation of few-layered MoS platelets. Furthermore,exfoliated MoS was shown experimentally to display aremarkably large second harmonic signal, with secondharmonic susceptibilities on the order of ∼
100 nm/Vreported in Ref. 4 while Refs. 5 and 6 report only ∼ . ∼ However, few-layered MoS grown by chemical vapourdeposition does not follow this trend , possibly due thestacking order of CVD grown films deviating from 2H.In Ref. 5, second harmonic spectra of both ML and tri-layer (TL) MoS were presented, demonstrating an in-tense peak in the second harmonic spectrum at pumpphoton energies near 1.45 eV, with a slight redshift forTLs compared to MLs.There exists no theoretical work, with which to inter-pret the experimental findings mentioned above. Hence,in this letter we consider the microscopic origins of thesecond harmonic response of few-layered MoS based on atight-binding sp d band structure recently published .At present, it is very computationally demanding to re-tain the k-space resolution needed to resolve delicatespectral features in a full exciton Bethe-Salpeter calcu-lation. Moreover, the single particle results remain animportant first step in the theoretical understanding ofthe nonlinear optical properties of MoS . We have there-fore chosen to omit excitonic effects in the present workand employ the single-particle second harmonic response E n e r gy [ e V ] −3−2−10123 Γ M K Γ B AD C
Figure 1. Band structure of ML MoS . Arrows indicate im-portant optical transitions. formalism developed by Moss and Sipe . We verifythat the experimental measurements in Refs. 5 and 6agree to within an order of magnitude with our model,and proceed to analyse the resonance structure of thesecond harmonic spectrum. We also analyse the depen-dence of χ (2) on the number of 2H stacked layers andfind little difference in the general magnitude of χ (2) forvarying odd-numbered layers, although slight changes tothe spectral features are observed.It is well known that the valence and conduction bandextrema of ML MoS are located at the K points of theBrillouin zone and are dominated by d -orbitals localizedon the Mo atoms. These bands are particularly im-portant for optical transitions in the visible range, andare well represented by the tight-binding band struc-ture of Ref. 10. However, a significant spin-orbit cou-pling, due to the heavy Mo atoms, causes a ∼
100 meVsplitting of the two highest valence bands near the K-points. To properly account for this we include spin-orbit coupling between d -orbitals localized on the same σ ′ ( ω ) [ e / ( h ) ] AB DC σ xx σ zz (a) χ ( ) [ n m / V ] Fundamental photon energy [eV] B / / / / B / / χ ( ) × Fund. photon energyRealImagAbs (b)
Figure 2. (a) Linear optical sheet conductivity tensor of a MoS ML, with the in-plane tensor elements denoted by σ xx = σ yy whereas the out-of-plane response is σ zz . (b) Second harmonic sheet response of MoS ML. The static response is χ (2) ( ω =0) = 0 . /V. Mo atom , instead of the p -orbitals used in the origi-nal parametrization . We fit the spin-orbit parameter λ d, Mo = 54 meV to the 112 meV splitting of the twohighest valence bands at K reported in Ref. 1. The bandgap of 1.8 eV generated using this method is comparableto the DFT band gap of 1 . − . of 1.9eV. For this reason, we choose to neglect quasi-particleeffects aiming instead to reproduce the position of exper-imental spectral features. However, it should be notedthat the agreement between the DFT band gap and theexperimental absorption peak position arises from theapproximate cancellation of the band gap increase dueto quasi-particle (GW) corrections and the red-shift ofoptical features due to the exciton binding energy .The band structure of ML MoS is displayed in Fig. 1together with some optical transitions to be discussedshortly. We calculate the real part of the diagonal, linearoptical sheet conductivity tensor using the well-knownexpression σ ′ aa ( ω ) = e πm ~ ω X c,v Z | p acv | δ ( ω − ω cv ) d k, (1)where ω ij and p aij denote, respectively, the transition fre-quency and the a -component of the momentum matrixelement between states in bands i and j (with an im-plicit k -dependency). Momentum matrix elements arecalculated as in Ref. 18. Furthermore, the band indices c and v indicate conduction and valence bands, respec-tively.The imaginary part of the interband sheet second har-monic susceptiblity tensor at fundamental pump fre- quency ω can be calculated using χ (2) ′′ abc ( ω ) = e πm ~ ǫ ω X c,v,l Z (cid:20) P vcl ω − ω lv δ (2 ω − ω cv )+ (cid:18) P vlc ω + ω cl + P clv ω + ω lv (cid:19) δ ( ω − ω cv ) (cid:21) d k. (2)Here, P ijl = Im n p aij (cid:16) p bjl p cli + p cjl p bli (cid:17)o / l runs over all bands with the restriction l = ( c, v ).The first term in Eq. 2 contributes when an electronic ex-citation frequency ω cv resonant with the second harmonicphoton frequency 2 ω can be found. This term is referredto as the 2 ω -term, and whenever the aforementioned cri-terion is satisfied while ω ≈ ω lv a particularly powerful,so-called double resonance is found. Similar commentscan be made regarding the ω -terms that contribute when-ever the condition ω = ω cv is fulfilled. Due to symmetry,the only non-vanishing second harmonic tensor elementsare χ (2) xxx = − χ (2) xyy = − χ (2) yyx = − χ (2) yxy ≡ χ (2) , with the x -axis aligned along an armchair direction. Malard et al. and Kumar et al. define this direction differently rela-tive to the indices of the contributing tensor elements.However, we follow the conventions of Ref. 5 and con-firm these to be correct by numerical testing (which isalso clear from symmetry, since the armchair directionspans a mirror plane together with the z -axis). The k -integrations are performed using the improved linear-analytic triangle method , where care is taken to analysedouble resonances of Eq. 2 by subsequent refinement ofthe integration mesh. Having calculated the imaginarypart of χ (2) , the real part is found by Kramers-Kronigtransformation. We generally apply a phenomenologicalbroadening of 5meV.The calculated linear optical response for a ML is plot-ted in Fig. 2(a). The absorption edge is dominatedby two step-like features in agreement with other single-particle results following from the relatively large split-ting of the highest valence bands in ML MoS due to spin-orbit interaction. Hence, transitions from the highest andsecond highest valence bands to the lowest conductionbands, as indicated in Fig. 1, give rise to the distinct Aand B steps in the single-particle spectrum of Fig. 2(a).It is noted that a full Bethe-Salpeter treatment enhancesthe A and B features into peaks red-shifted by 1.1 eVcompared to the quasi-particle spectrum , and in exper-iments these features are indeed observed as peaks .At larger energies corresponding to the band gap on theΓK-line, several transitions from nearly parallel bandscontribute to the absorption. The first of these are in-dicated as the C transition in Fig. 1, while a structureof subsequent peaks follow from spin-obit splitting simi-larly to the A and B transitions. Collectively, these giverise to a broad peak upon inclusion of additional phe-nomenological broadening denoted the C transition. Atphoton energies corresponding to the band gap at Γ, anabsorption peak arising from transitions indicated by Din Fig. 1 is seen, with a similar peak structure. | χ ( ) | [ n m / V ] C / ×
25 Experiment, Ref. 5Theory
Figure 3. Comparison between our theory and the experi-mental results of Ref. 5. A phenomenological broadening of25 meV has been applied to the theoretical results and thered line is a guide for the eye.
The calculated second harmonic susceptibility of aMoS ML is shown in Fig. 2(b). The low-energy re-sponse near half the band gap is due to the 2 ω -term ofEq. 2 only, and can be interpreted as optical transitions,for which the second harmonic photon energy matchestransitions from the highest valence band to the lowestconduction band near the K-points. Two peaks are ob-served at fundamental photon energies corresponding tohalf the photon energies of the A and B peaks in the linearspectrum, although these are much weaker relative to theremaining spectrum when compared to the correspond-ing step heights in Fig. 2(a). Hence, we have included a magnified view of this spectral region as insets in Fig.2(b). These peaks correspond to 2 ω -processes, and arehere termed the A / / ω -processes at twice the fundamental pho-ton energy. However, in this region 2 ω -processes closeto double resonance dominate making the ω -features dif-ficult to observe. Also, an intense peak is found in thesecond harmonic spectrum at fundamental photon ener-gies ∼ .
45 eV corresponding to the half the energy ofthe C transition. We denote this feature C /
2, and notethat its spectral position fits very well with the experi-mentally observed SHG peak at 1.45 eV of Ref. 5. Thiscan readily be seen in Fig. 3, where we compare our the-oretical calculations (now including a broadening of 25meV realistic for comparison with room-temperature ex-periments) with the spectrum recorded by Malard et al. .A good agreement in peak position and shape is gener-ally observed, however, a rather large intensity differenceis also found. We stress that Kumar et al. report sus-ceptibilities two order of magnitude larger than the thosecalculated by us, making it clear that some uncertaintiesin the experimentally determined susceptibilities must beexpected for an atomically thin material. Also, we herecompare theoretical results calculated for a free-standingMoS sheet with experiments performed on a substratewhere e.g., strain or substrate phonons may be impor-tant. We also stress that excitonic effects are neglectedin the current model, however, we believe that inclusionof these will have three main effects. Firstly, all spec-tral features arising from 2 ω processes will be red-shiftedby half the exciton binding energy, while features aris-ing from ω -processes will be red-shifted by the bindingenergy, compared to the quasi-particle spectrum (and,hence, remain at nearly the same spectral position com-pared to results derived directly from density-functionaltheory, as mentioned previously). Secondly, similarly towhat is observed in the linear case, low energy features(such as the A / / / χ (2) ( ω = 0) = 0 . /V. | χ ( ) | [ n m / V ] Fundamental photon energy [eV]0.8 0.9 1 1.10.20.40.6
N = N = N = N = N =
Figure 4. Sheet second harmonic response of few-layeredMoS with N layers. The scales on all boxes are identical. In Fig. 4, we present results for second harmonic gener-ation in few-layered MoS with varying number N of 2H-stacked layers. We observe that the general magnitude ofthe sheet second harmonic response remains unchangedwith increasing N , while the C / lay-ers; if these were completely decoupled, N layers wouldcontribute the same second harmonic polarization, butwith phases alternating by π due to adjacent layers be- ing mirror images of each other in the yz -plane. Hence,for an odd number of layers, the net second harmonic po-larization would equal the response of a single layer. Themost important effect of introducing a weak interlayercoupling is to increase the splitting of the top valencebands at K, introducing slight changes to the second har-monic response near half the band gap at K. This causesmodifications of the A / / / from a directto an indirect gap semiconductor upon including multi-ple layers , this has little consequence for the propertiesstudied here.In conclusion, we find clear signatures of the A andB excitations known from linear optics also in the sec-ond harmonic spectrum of ML MoS at photon ener-gies near half the band gap. With increasing numberof layers, the spectral separation between the A and Bpeaks predictably increases with increased splitting of thetwo highest valence bands. Furthermore, a very intensepeak corresponding to the C excitation in linear opticswas found at second harmonic photon energies near theband gap on the ΓK-line in k -space. This peak positionagrees very well with the experimentally observed peak at pump photon energies of 1.45 eV. The magnitude ofthe sheet second harmonic response was found to be ∼ /V off resonance and up to ∼ /V on resonance,placing it in between the values reported by Malard etal. and Li et al (agreeing to within an order of magni-tude) and those reported by Kumar et al. (agreeing towithin two orders of magnitude). ACKNOWLEDGEMENTS
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