Theoretical and experimental study of Raman scattering in mixed (MoS₂)x(MoSe₂)₁₋x layered crystals

Theoretical and experimental study of Raman scattering in mixed (MoS₂)x(MoSe₂)₁₋x layered crystals

Raman scattering in mixed MoS₂/MoSe₂ layer type crystals was investigated in this work. The change of intensities and positions of bands for in-plane E¹₂g and outof-plane A₁g vibrations as functions of the “concentration” inherent to corresponding type layers has been studied. Estimation of interlay...

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Datum:2015
Hauptverfasser: Yaremko, A.M., Yukhymchuk, V.O., Romanyuk, Yu.A., Virko, S.V.
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Veröffentlicht: Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України 2015
Schriftenreihe:Semiconductor Physics Quantum Electronics & Optoelectronics
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spelling irk-123456789-1212462017-06-14T03:07:29Z Theoretical and experimental study of Raman scattering in mixed (MoS₂)x(MoSe₂)₁₋x layered crystals Yaremko, A.M. Yukhymchuk, V.O. Romanyuk, Yu.A. Virko, S.V. Raman scattering in mixed MoS₂/MoSe₂ layer type crystals was investigated in this work. The change of intensities and positions of bands for in-plane E¹₂g and outof-plane A₁g vibrations as functions of the “concentration” inherent to corresponding type layers has been studied. Estimation of interlayer interaction was obtained from comparison of experiment and theory, and effect of this interaction on the frequency of intralayer phonon was studied. 2015 Article Theoretical and experimental study of Raman scattering in mixed (MoS₂)x(MoSe₂)₁₋x layered crystals / A.M. Yaremko, V.O. Yukhymchuk, Yu.A. Romanyuk, S.V. Virko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2015. — Т. 18, № 3. — С. 354-361. — Бібліогр.: 20 назв. — англ. 1560-8034 DOI: 10.15407/spqeo18.03.354 PACS 71.36.+c, 78.30.Hv http://dspace.nbuv.gov.ua/handle/123456789/121246 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description Raman scattering in mixed MoS₂/MoSe₂ layer type crystals was investigated in this work. The change of intensities and positions of bands for in-plane E¹₂g and outof-plane A₁g vibrations as functions of the “concentration” inherent to corresponding type layers has been studied. Estimation of interlayer interaction was obtained from comparison of experiment and theory, and effect of this interaction on the frequency of intralayer phonon was studied.
format Article
author Yaremko, A.M.
Yukhymchuk, V.O.
Romanyuk, Yu.A.
Virko, S.V.
spellingShingle Yaremko, A.M.
Yukhymchuk, V.O.
Romanyuk, Yu.A.
Virko, S.V.
Theoretical and experimental study of Raman scattering in mixed (MoS₂)x(MoSe₂)₁₋x layered crystals
Semiconductor Physics Quantum Electronics & Optoelectronics
author_facet Yaremko, A.M.
Yukhymchuk, V.O.
Romanyuk, Yu.A.
Virko, S.V.
author_sort Yaremko, A.M.
title Theoretical and experimental study of Raman scattering in mixed (MoS₂)x(MoSe₂)₁₋x layered crystals
title_short Theoretical and experimental study of Raman scattering in mixed (MoS₂)x(MoSe₂)₁₋x layered crystals
title_full Theoretical and experimental study of Raman scattering in mixed (MoS₂)x(MoSe₂)₁₋x layered crystals
title_fullStr Theoretical and experimental study of Raman scattering in mixed (MoS₂)x(MoSe₂)₁₋x layered crystals
title_full_unstemmed Theoretical and experimental study of Raman scattering in mixed (MoS₂)x(MoSe₂)₁₋x layered crystals
title_sort theoretical and experimental study of raman scattering in mixed (mos₂)x(mose₂)₁₋x layered crystals
publisher Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
publishDate 2015
url http://dspace.nbuv.gov.ua/handle/123456789/121246
citation_txt Theoretical and experimental study of Raman scattering in mixed (MoS₂)x(MoSe₂)₁₋x layered crystals / A.M. Yaremko, V.O. Yukhymchuk, Yu.A. Romanyuk, S.V. Virko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2015. — Т. 18, № 3. — С. 354-361. — Бібліогр.: 20 назв. — англ.
series Semiconductor Physics Quantum Electronics & Optoelectronics
work_keys_str_mv AT yaremkoam theoreticalandexperimentalstudyoframanscatteringinmixedmos2xmose21xlayeredcrystals
AT yukhymchukvo theoreticalandexperimentalstudyoframanscatteringinmixedmos2xmose21xlayeredcrystals
AT romanyukyua theoreticalandexperimentalstudyoframanscatteringinmixedmos2xmose21xlayeredcrystals
AT virkosv theoreticalandexperimentalstudyoframanscatteringinmixedmos2xmose21xlayeredcrystals
first_indexed 2025-07-08T19:27:39Z
last_indexed 2025-07-08T19:27:39Z
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fulltext Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 3. P. 354-361. doi: 10.15407/spqeo18.03.354 © 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 354 PACS 71.36.+c, 78.30.Hv Theoretical and experimental study of Raman scattering in mixed (MoS2)x(MoSe2)1–x layered crystals A.M. Yaremko, V.O. Yukhymchuk, Yu.A. Romanyuk, S.V. Virko V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine, 45, prospect Nauky, 03028 Kyiv, Ukraine Abstract. Raman scattering in mixed MoS2/MoSe2 layer type crystals was investigated in this work. The change of intensities and positions of bands for in-plane 1 2gE and out- of-plane gA1 vibrations as functions of the “concentration” inherent to corresponding type layers has been studied. Estimation of interlayer interaction was obtained from comparison of experiment and theory, and effect of this interaction on the frequency of intralayer phonon was studied. Keywords: Raman scattering, interlayer interaction, phonon, vibration, layered crystals. Manuscript received 06.04.15; revised version received 10.07.15; accepted for publication 03.09.15; published online 30.09.15. 1. Introduction Spectroscopic studying the layer type crystals was performed for a long time. It began in the early 70-th of the last century. Investigations were related with both the electron and vibration properties of these crystals. Study of phonons was very intensively made for different types of layered crystals, GaSe [1], GaS [2], MoS2 [3-5], As2S3 [6], MoSe2, MoW2 [7], etc. and some models explaining the observed features, in particular Davydov’s splitting effect, were proposed. The new period of activity in studying these crystals arose when technological possibility appeared, using the method by Novoselov et al. for graphen, which enabled to prepare a very thin crystal structure having 1 to 10 atomic layers [8]. Especially perspective in this plan are layered crystals MoS2 and MoSe2 showing new spectroscopic features, if the crystal structure consists of only several n = 1…6 atomic layers [7, 9] (see also numerous references cited there). The electron band structure of these crystals differs from the bulk one, and they demonstrate very intensive luminescence. The number of works in which electron and phonon properties of such type crystal structures are studied using the spectroscopic methods, grows significantly for the last years. Recently, detailed study of vibration spectra of MoS2 layer crystals consisting of several layers, n = = 1…6 and analysis of its results were made in [9]. The authors observed strong signals of the in-plane ( 1 2gE ) and out-of-plane (A1g) Raman modes of all 6 layers. These modes exhibited a well-defined thickness dependence, with the two modes shifting away from each other in frequency with increasing the thickness. The behavior of frequency shifts with changing the layer thickness, as it was emphasized by the authors [9], cannot be explained solely in terms of weak van der Waals (vdW) interlayer interaction. The spectrum as a function of the film thickness has several features. It is noted that most strikingly that more low frequency 1 2gE vibration softens (red shifts), while the high frequency A1g vibration stiffens (blue shifts) with increasing the sample thickness. For the films consisting of four or more layers, the frequencies of both modes converge to the bulk values. Also, the rate Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 3. P. 354-361. doi: 10.15407/spqeo18.03.354 © 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 355 of frequency change is twice as large for the gA1 as for 1 2gE mode. The similar features were recently observed, too. The vibrations of bulk materials built up from vdW- bonded layers are often analyzed in terms of the work [10]. The vibrations of bulk materials built up from vdW-bonded layers are often analyzed in terms of the two-dimensional layers from which they are formed [1, 3, 5]. Within a classical (traditional) model for coupled harmonic oscillators [11], 1 2gE , and gA1 modes are expected to stiffen as additional layers are added to form the bulk material from individual layers, because the interlayer vdW interactions increase the effective restoring forces acting on the atoms. While the shift of gA1 mode was observed in experiments of the work [9] with increasing the layer number agrees with prediction, behavior of the 1 2gE mode does not. The failure of the model could indicate that the implicit assumption that stacking does not affect intra-layer bonding is incorrect. But in reality, even weak interlayer interaction in crystals can affect intra-layer bonding and lattice dynamics. It can be explained at least particularly on the base of results obtained in works [12-14] where Fermi- Davydov (FD) resonance in molecular type crystals was considered. Positions of bands and their intensities depend on the week vdW intermolecular interaction and intramolecular frequencies are renormalized. Indeed, as a result of intramolecular interaction between fun- damental vibrations and overtons (combination tones) of molecule, two strong bands (Fermi resonance doublet) arise with frequencies fω , and gω . In crystal, due to weak intermolecular interaction and exchange by excitations, even with one molecule per crystal unit cell two type of Davydov terms appear: diagonal, ffff MD , , gggg MD , and non-diagonal gfgf MD , . The diagonal terms result in shift like to all spectral bands, but the non-diagonal ones give rise to repulsion of new crystal states. Therefore, the shift of high- and low-frequency Fermi-doublet components should be in different sides. Similar facts were observed in experiments [9, 10]. A more complicated case in particular with taking into account of Fermi-Davydov resonance and strong interaction of H-bond vibrations with lattice phonons was later theoretically considered in [15]. In recent experiments, when studying mixed layer type crystals MoS2/MoSe2 new features related with complex variation of spectrum were observed in [10]. The problem related with influence of intra-interlayer interactions looks in this case especially complex for both the position of bands and their intensities, and it includes as a particular case the aspects noted in [9]. Therefore, we consider first a more general task for mixed crystals, and then the features related with thin layer type crystals will be discussed. 2. Intensity of Raman scattering RS intensity can be expressed by imaginary part of Fourier component Green function in the tensor of the crystal susceptibility βαχ , [16, 17], where Hamiltonian describing the interaction of electromagnetic (EM) field with crystal looks as follows ( ) ( ) ( ) =−′χλ′′λ−= βα βαλ ′′′λ + λ′′−λ− ∗ βα∑ kkPPkeke V H kk kk , ,,,,, ,,int ,,1 λ′′λ λ′′λ + λ′′−λ− χ−= ∑ ,,, ,,, ,, 1 kk kk kk PP V , (1) ( ) ( ) ( )kkkekekk −′χλ′′λ=χ βα βα ∗ βαλ′′λ ∑ , , ,., ,, ; λλ′′ ∗ λ′′λλ′′−λ− χ=χ=χ ,,,,,,,,, kkkkkk , (2) ( ) ( ) ( )∑ −χ=−χ=−′=χ βα ∗ βαβα n nQinQkkQ rrrrrr exp)(,,, , (3) ( )λ− + λλ −= ,,, 2 1 kkk aaP , ( )+ λ−λλ += ,, 2 1 kkk aaA . (4) Here, ),(,, λω αλ kek are the photon frequency and α- component of the electric field unit vector; λ + λ ,, , kk aa – Bose operators of creation and annihilation of electric field, respectively; λ,kP and λ,kA – operators of momen- tum and potential of EM field, correspondingly, satisfying the commutation relations [ ] λ′λ′λ′′λ δδ= ,,,, ; kkkk PA . In the layer type crystal where different layers can have various properties, two indexes (l, n) should be used to numerate unit cells of the crystal: the first index (l) points out the number of layer and the other one (n) – the number of the unit cell in layer. For conveniency, the wave vector is also presented by two components oriented in layer, nQ r , and normal to layer, lQ r , correspondingly, so that nl QQQ rrr += . Then, ( ) ( ) ( )( )[ ]=++−× ×+χ=−′=χ ∑ βαβα lnQQi lnkkQ nl ln rrr rrrrr exp , ,, ( ) ( )[ ]∑ +−+χ= βα ln ln QlQniln , , exp rrrrrr . (5) The value ( )ln rr +χ βα, can be presented as expansion in series on deviation of atoms in l layer, l knu α,, , from equilibrium position, which are then expressed by normal coordinates of phonon operators, l sQ ln ,ϕ . Here, sl are phonon quantum states of layer l. Using Eqs. (5) and (2), the following expression for susceptibility of crystal can be obtained: Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 3. P. 354-361. doi: 10.15407/spqeo18.03.354 © 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 356 ( ) ( ) ,exp, , , ,0,., l sQl sl ln l kkkk ln l ilQsQN ϕ−χ=χ ∑ λ′′λλ′′λ (6) N0 is the number of unit cell in layer. Because the lsN layers with quantum states ls are arbitrary distributed between full numbers of crystal layers, the Nl response of crystal on incident light will be averaged. Therefore, the expression (6) describing susceptibility should be averaged on their distribution. The probability that each layer at the same time will occupy each “layer cell” in crystal is equal, lss NNc ll /= . So, we obtain situation that the same response on the incident light is carried out by Nl crystal layers but with the probability lsc . Because all layers are now identical, the susceptibility is independent on l, i.e., ( ) ( )lnkkln l kk sQsQ ,, ,,,,,, λ′′λλ′′λ χ=χ . This allows making simplification of Eq. (6) as follows ( ) ( ) ( ) =ϕ× ×−χ=χ ∑ ∑ λ′′λλ′′λ ln l l sQp pl l sl lnkkskk ipl N ilQsQcN ,, , ,,,0,., exp1 exp, ( ) lnl l l sQQ s lnkksl sQcNN ,,,,,0 , ϕχ= ∑ λ′′λ , (7) For conveniency, we introduce ( ) ll slnkks sQc χ=χ λλ ~,',',, , where the index ls has a double meaning: l points out on the type of layers forming the crystal and { },..., 21 lll sss = characterize the different vibration states in this layer. As it was noted above, the intensity of RS can be expressed by Fourier component Green function from the tensor of susceptibility of crystal, and in our case the intensity of light scattering by one unit cell is described by the following expression λλ ,',', ppI ~ [ ]×ω+ π − )(11 n ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ χχ× ω + λλ′′λλ′′ )0();(Im ,,,,,, pppp t = = [ ] ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ϕϕχχω+ π − ω +∗ ′ ′ ′∑ )0();(~~Im)(11 ,, , ll ll ll sQsQ ss ss tn , (8) where values in brackets are Fourier components of retarded Green functions ω + ′′ ϕϕ=ω )0();(),( ,, llll sQsQss tQG . (9) It is known from the group theory that intermixing by anharmonicity of states in particular intralayer states are possible, if they have the same symmetry. Experiments show that, for MoS2 and MoSe2 layer crystals having hD6 space group, the most strong bands observed in RS spectra correspond to gA1 and gE2 symmetries. In this paper, we consider the case of mixing the states in mixed crystals with the noted symmetries. In other words, only two types of layers l = 0 and l = 1 corresponding to MoS2 and MoSe2, respectively, are taken into consideration, and their vibrations should have an identical symmetry, gA1 (or gE2 ). Then, for the given case, the intensity of RS is described by Eq. (8), which looks as follows λλ ,',', ppI ~ [ ]×ω+ π − )(11 n { },)(~~)(~~)(~~)(~~Im 1111010110100000 ωχχ+ωχχ+ωχχ+ωχχ× ∗∗∗∗ ssssssssssssssss GGGG (10) In Eq. (10), we took into account that ( )0→Q and used designation )(),0( ω=ω→ ′′ llll ssss GQG . 3. Hamiltonian and equations for Green functions of layer crystal vibrations Hamiltonian of layer crystal in the secondary quantum representation (SQR) is written as follows +ω=+= +∑ l sq l sq sql l sq ll l l bbHHH ,, ,, ,int0 ( ) l sq l sq llssq lll ssq ll ll ll V ′+ ′≠ ≠′ ′ ′ ′ ϕϕ′′+ ∑ ,, ,,, )(, ,, ~ 2 1 , (11) where l sq l,ω , l sq l b+ , , l sq l b , are phonon frequency and creation-annihilation operators of l layer phonons. The normal coordinate l sq l,ϕ and momentum l sq l,π look as follows ( )l sq l sq l sq lll bb + −+=ϕ ,,, 2 1 , ( ). 2 1 ,,, l sq l sq l sq lll bb − + −=π (12) The first sum in Eq. (11), H0, describes the system of non-interacting layer’s oscillators but the second term is responsible for interlayer interaction of phonons. Fourier components of Green functions (GF) describing the RS by averaged layer crystal are given in Eq. (15), but now we will study the GF of a more general form )0();()( ,,',, ' l sk l sksksk llll ttG ′ ′ ′′ ϕϕ= . (13) The equation for such GF looks as follows Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 3. P. 354-361. doi: 10.15407/spqeo18.03.354 © 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 357 ( ) = ∂ ∂ ′′ tG t i ll sksk ,, [ ] ( ) ( ) .0;)0();0()( ,,,, l sk l sk l sk l sk llll t t it ′+ ′ ′+ ′ ′′ ϕϕ ∂ ∂ +ϕϕδ= (14) The equation relating Fourier components of similar type GF, ω ′ ′ ′ ϕϕ )0();( ,, l sk l sk ll t , can be written as follows ( ) =ϕϕ⎥⎦ ⎤ ⎢⎣ ⎡ ω−ω ω ′+ ′ ′ )0();( ,, 2 , 2 l sk l sk l sk lll t { −δδδ−ω−= ′′′ lll ssllkk l sk ,,,, ( ) ⎭ ⎬ ⎫ϕϕ− ω ′+ ′ ′′ ≠′′ ′′ ′′′ ′′ ′∑ )0();(~ ,, )(, , ,, l sk l sk lls ll kss ll l ll tV . (15) It is seen that nonzero solution can be obtained, if intra- layer wave vectors are equal, kk ′= , but concerning interlayer interaction the situation is more complicate because layers in crystal are arbitrary distributed. Therefore, the response of crystal on the incident radiation will be averaged on all possible distributions of layers interacting with one another. Averaging Eq.(15) on all l ′′ and l layers, phonons of which characterizes ls , ls ′′ quantum states, gives rise to appearance of probabilities lss NNc ll /= and lss NNc ll / ′′′′ = . There- fore, Eq. (15) is transformed into the following one ( ) =ϕϕ⎥⎦ ⎤ ⎢⎣ ⎡ ω−ω ω + )0();( ' ,', 2 , 2 ' l sk l sk l sk lll t { −δδδ−ω−= ′′′ lll ssllkk l sk ,,,, ( ) ⎭ ⎬ ⎫ϕϕ− ω ′+ ′ ′′ ≠′′ ′′ ′′′ ′ ′′′′∑ )0();(~ ,, )(, , ,, l sk l sk lls ll kssss ll l llll tVcc . (16) In this approximation, we have the homogeneous crystal structure fully filled with identical layers of ),( lsl or ),( lsl ′′′′ types, but interlayer interaction become smaller and depend on the coefficients "ll ss cc .This situation is very similar to that studied in molecular crystal having several molecules in the unit cell and quasi-degenerated levels in molecules [12-15], because after averaging we obtain a new layer crystal, in which different type layers ),( lsl and ),( lsl ′′′′ can be considered as forming complex double layer “elementary cell”, but with more weak interlayer interaction. For a homogenous crystal structure, one can write, =′′ ′′ )(~ , ,, ll kss ll V )(~ ,, ll kss ll V ′′− ′′ . Besides, in this approximation in Eq. (16) all functions depend only on the difference between layers ll ′′− , therefore the following Fourier transformation can be made. =ϕϕ ω ′+ ′ )0();( ,, l sk l sk ll t ( )[ ]∑ ′−ω= ′ p ksks llippG N ll exp),(1 ,,, ' , (17a) ( ) ( )[ ]∑ ′′−ω= ′′′′ ′′− p ss l ll kss llipkpV N V llll exp),,(1~ ,,, , (17b) ( )[ ]∑ ′−=δ ′ p ll llip N exp1 , . (18) Then, Eq. (16) can be written in a more simple form (the conservation law for layer wave vectors, kk ′= , is taken into account) ( ) ( ){ }×ω+δω−ω∑ ′′ ′′′′′′′′ l llllllll s pk sssssksssk Vcc , ,,, 2 , 2 ~ lllll ssskss kpG ′′′ δω=ω× ,,, ),,( ' . (19) In Eq. (19), the index ls similar to Eqs. (7)-(13) has the double meaning: i) l points out the character of layer, ii) { },..., 21 ll sss = marks the type of vibrations in this layer, if several vibrations in the given layer are considered. We have noted in Section 2 that only two types of layers l = 0 and l = 1 corresponding to MoS2 and MoSe2, respectively, are taken into consideration and only one of layer vibrations that have an identical symmetry, gA1 (or gE2 ) are studied . The fundamental vibrations can be mixed by fourth order anharmonicity of layer but this value is very low and in current investigation is omitted. Taking into account that designations 10 , sssl = are, respectively, related with MoS2 and MoSe2 layer types, two pairs of equations are obtained. a) 0ssl =′ , the first system of equations looks as follows: 001100000 ,skssssssss GDG ω=+Δ , ( 0ssl = ), (20a) 0 01110001 =Δ+ ssssssss GGD , ( 1ssl = ), (20b) b) 1ssl =′ , the second system of equations looks as follows: 0 11101000 =+Δ ssssssss GDG , ( 0ssl = ) (21a) 111111001 ,skssssssss GGD ω=Δ+ , ( 1ssl = ) (21b) where designations in Eqs. (20a, b) and Eqs. (21a, b) have the following look: Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 3. P. 354-361. doi: 10.15407/spqeo18.03.354 © 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 358 ( ) ( )pk ssssskssskss llllllllll Vcc , ,, 2 , 2 ~ ′′′′′′′′′′ ω+δω−ω=Δ , (22a) ( )pk ssssskss lllllll VccD , ,, ~ ′′′′′′ ω= . (22b) In the following consideration, we will omit for conveniency the indexes k, p describing the components of the wave vector for crystal excitations. As it is seen from Eqs. (7), both wave vectors are related with components of exciting radiation 0→+= nl QQQ rrr where interlayer, nQk r = , and normal to layer, lQp r = , components respectively. Solutions of Eqs. (20a, b)-(21a, b) are as follows Δ Δω = 110 00 sss ssG , Δ ω− = 010 01 sss ss D G ; (23a) Δ Δω = 001 11 sss ssG , Δ ω− = 101 10 sss ss D G ; (23b) where written below designation was used 10011100 ssssssss DD−ΔΔ=Δ . (24) Insertion of Eqs. (23a, b) and (24) into Eq. (10) gives rise to the following expression for the spectral dependence of RS intensity λλ ,,',' ppI ~ [ ]×ω+ π − )(11 n ( ) ( ) ( ) ( )[ ] .~~~~~~~~1Im 00111010011011011000 ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ Δωχχ+ω−χχ+ω−χχ+Δωχχ Δ × ∗∗∗∗ ssssssssssssssssssss DD (25) For simplicity, we will suppose that the RS tensor components are real, ∗χ=χ ll ss ~~ . Now, to take into account the damping of phonon excitations, we will suppose the frequency to be a complex value, γ+ω→ω i , therefore, all the values depending on frequency in numerator and denominator become the complex ones, in particular ( ) ωγ+Δ=ωγ++ω−γ−ω=Δ 2~2 0000000 222 iiD sssssss , (26a) ( ) ωγ+Δ=ωγ++ω−γ−ω=Δ 2~2 1111111 222 iiD sssssss . (26b) Therefore, the final expression for intensity of RS can be written as follows It should be taken into consideration that in Eq. (27), ( ) llll sslnkkss csQc χ=→χ=χ λλ ,0~ ',',, , according to Eq. (7), therefore, the effective tensor of scattering lsχ ~ is changed with the “concentration” of the given sort of layers and real parameter is lsχ . Eq. (27) shows that RS intensity has an enough complicated dependence on the frequency ω , relation between concentrations of different type layers lsc and interaction between layers ', ~ ll ssV . The resonance frequencies are obtained from the first term in the denominator of Eq. (27) (if 0→γ ) and according to Eqs. (26a, b) are equal ( ) ( )[ ]{ ±−ω+−ω=γ−ω 111000 2222 2 1 ssssss DD ( ) ( )[ ] .416 0110111000 2222 ⎭ ⎬ ⎫ +γω+−ω−−ω± ssssssssss DDDD (28) In the case when crystals MoS2 or MoSe2 consist of only identical layers, some correction to resulting Eq. (28) should be made, because mixing the states by anharmonicity at corresponding their symmetries are possible. 4. Mixing of crystal states with identical layers and discussion of experiments The estimation of interlayer interaction parameters can be made on the base of results of theoretical calculations for phonons in one-layer and bulk MoS2 crystal obtained in work [20]. The actual phonon frequencies for one- layer MoS2, hD3 point group symmetry, are as follows 1cm3.410' 1 −≈ω A and 1cm7.391' 1 −=ω E , but the corresponding ones for bulk MoS2 with hD6 point group have very close values 1cm412 1 −=ω gA and 1cm8.387 2 −=ω gE to the former phonon pair. It means that change of frequencies due to interlayer interaction is close to 1cm5-3 − , if one takes into consideration some mistakes in numerical calculations. But experiments show that observed for one-layer frequencies are 1cm404' 1 −≈ω A and 1cm383' 1 −≈ω E , which are obviously lower than those predicted theoretically. One of the reasons can be anharmonic interaction that is are ( )[ ] ( ) ( ) [ ] ( )[ ]~~24~~ 4~~~~4~~~2 11~ 2222 22222222 ,,',' 110010011100 1100001101011100 ssssssssssss ssssssssssssssss pp DD DD nI Δ+Δωγ+−γω−ΔΔ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎥⎦ ⎤ ⎢⎣ ⎡ γωχ+χ−Δχω+⎥⎦ ⎤ ⎢⎣ ⎡ γωχ+χ−Δχωωγ ×ω+ πλλ (27) Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 3. P. 354-361. doi: 10.15407/spqeo18.03.354 © 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 359 important in one-layer crystal and is not taken into account in theoretical calculations. Indeed, from work [20] (Fig. 2, upper panel) follows that combination tones 1cm470)()()()( −≈−ω+ω≈−ω+ω MLAMLAKLAKLA qqqq are full symmetric and have frequency higher than ' 1A ω . Therefore, both vibrations can take part in FR interaction, and, as a result, the fundamental band should be shifted some down to the value 1cm404' 1 −≈ω A that is observed in experiment. The same shift down should occur for the one-layer fundamental vibration 1cm7.391' 1 −=ω E due to FR with combination tone ≈−ω+ω )()( KTAKLA qq 1cm420 − . The experimentally observed its one-layer final position is 1cm383' 1 −≈ω E . On the other hand, when increasing the layer numbers (to 6…8 layers), properties of this thin crystal become close to those of bulk MoS2 [9], and the bands discussed above are transformed into a new pair of strong gA1 and 1 2gE ones, and the final position of these bands is ruled by properties of bulk crystal. One can note that in the Raman spectra of crystals MoS2 and MoSe2, these two fundamental bands gA1 and gE2 (Fig. 1) cannot intermix with one another. The fundamental states, in principle, can be mixed by fourth order anharmonic terms in the potential energy, but both these states should have the same symmetry. It is obvious that direct interaction of gA1 and gE2 layer states are impossible, and they also cannot be intermixed by interlayer interaction, too. Experimental spectra and theoretical dependences describing their change with the “concentration” of layers in mixed crystals are presented in Figs 1 and 2. Fig. 1. Position and intensities of fundamental bands observed in Raman spectra of bulk crystals MoS2 and MoSe2 with real relation of their intensities at the same conditions of experiment. Fig. 2. The change of band intensities in spectra MoS2 and MoSe2 as a function of concentration 0scx = , 11 scx =− , 110 =+ ss cc . Spectra at x = 1 and x = 0 were fitted to experimental ones in Fig. 1. All parameters describing interlayer interaction are taken as high enough, ( ) 1, , cm8~ − ′ =pk lslsV . It should be noted that in MoS2 bulk crystal, with hD6 point group symmetry, close to considered states ( ) 1 1 cm412 −=ω gA and ( ) 11 2 cm389 −=ω gE , the combination tones ( ) ( ) ( ) 12 2 1 2 cm35389 −±=ω±ω gg EE are placed which can interact with both bands gA1 and gE2 due to Fermi resonance (FR). The latter is admitted by symmetry relations in hD6 point group, gggg EAEE 2122 +=× . The position and intensities of interacting bands depend on the anharmonic constant Γ responsible for this interaction [18, 19]. As a result of interaction, the fundamental vibration ( ) 1 1 cm412 −=ω gA with the combination tone ( ) ( ) ( ) 12 2 1 2 cm35389 −±=ω±ω gg EE the first is some shifted down to the value 1cm408 −≈ (Fig. 3, curve 2). The combination band 1cm424 − is also some shifted upper but its intensity is weak and band is broad (Fig. 2, curve 2). Space between fundamental ( )gA1ω and other combination tone ( ) ( ) 12 2 1 2 cm354 −=ω−ω gg EE is wide enough and influence of the last on ( )gA1ω can be neglected. The fundamental band ( ) 11 2 cm389 −=ω gE also can interact with these combination tones ( ) ( )=ω±ω 2 2 1 2 gg EE , ( ) 1cm35389 −±= but this interaction is more complica- ted because around this fundamental band two combina- tion tones are placed at the same distances: ( ) ( ) 12 2 1 2 cm424 −=ω+ω gg EE and ( ) ( )=ω−ω 2 2 1 2 gg EE Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 3. P. 354-361. doi: 10.15407/spqeo18.03.354 © 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 360 1cm354 −= . According to the theory of FR in crystals [18, 19], the intensity of scattering (absorption) is described by the renormalized constant Γ , and for each of the studied cases it is given by the following relations: for the first case ( ) ( )[ ]1 2 2 21 gg EnEn ++Γ=Γ→Γ + and for the second one ( ) ( )[ ]1 2 2 2 gg EnEn −Γ=Γ→Γ − , where ( )2 2gEn , ( )1 2gEn are filling numbers of corres- ponding phonons. It is obvious that −+ Γ>Γ , and so the fundamental band of MoS2, ( ) 11 2 cm389 −=ω gE , should be shifted some below, as a result of two noted FR inter- actions, to the value ( ) 11 2 cm383 −=ω gE (Fig. 3, curve 4). These new states that take part in FR with the fundamental ones gA1 and gE2 will be marked with the index B: Bω , BA ssD , BA ssΔ , BE ssD , BE ssΔ , etc. (see Eqs. (22a, b)). For example, the position of bands for crystal with one type of layer s0 for vibration of A-type, interacting with B-type vibrations are described by the following relation ( ) ( )[ ]{ ±⎢⎣ ⎡ −ω+−ω=ω± BBBAAA ssssss DD 22 2 1 ( ) ( )[ ] + ⎭ ⎬ ⎫ +γω+−ω−−ω± ABBABBBAAA ssssssssss DDDD 416 22222 ]212γ+ . (29) where 1,1 10 =→=→ BA ssss cccc , and instead of Eqs. (22a, b) we have ( ) ( )pk ssskssskss jjjjjjjj V , ,, 2 , 2 ''' ~ ω+δω−ω=Δ , (30a) ( )pk ssskss jjjjj VD , ,, '' ~ ω= . (30b) 360 380 400 420 440 0 10 20 30 In te ns ity , a rb . u ni ts Wavenumber, cm-1 1 23 4 412389 424 Fig. 3. Effect of FR interaction with combination tones at the position of fundamental 1 1 cm412 −=ω gA and 1 2 cm389 −=ω gE bands in bulk MoS2. 0 2 4 6 8 10 380 385 390 395 400 405 410 Fr eq ue nc y, a rb u ni ts number of layers 1 2 3 4 MoS2 Fig. 4. The change of intra-layer phonon frequencies 1 ' 1 cm404 −≈ω A and 1 ' 1 cm384 −≈ω E with increasing the layer numbers and conversion of them into the bulk ones 1 1 cm408)( −≈ω gA and 11 2 cm383)( −≈ω gE ; curves 1, 3 are ex- perimental, curves 2, 4 are theoretical; parameters: 2 – )( 1gAω , ( ) 1, 0,00 cm8~ −−=pk ssV , ( ) ( )== pk ss pk ss VV , ,0 , ,0 0110 ~~ 1cm6 −− , ( )pk ssV , 1,10 ~ = 1cm3 −−= ; 4 – ( )1 2gEω , ( )pk ssV , 0,00 ~ = 1cm5 −− , ( )=pk ssV , 1,00 ~ ( )pk ssV , ,0 01 ~ = = 1cm4 −− , ( )=pk ssV , 1,10 ~ 1cm2 −− . Interlayer interaction between different layers is described by the following expression (that will be considered in details elsewhere) ( ) ( ) 0,1(~~ 1 1 , ,0 , , '' >α≈ ∑ = α+ N n pk ss pk ssN n VV jjjj , (30c) ( )pk ss jj V , ,0 ' ~ describes the change of interaction energy between two layers when changing their excitation from js0 state to ' 0 js one; in our case 45.0=α . It is seen from Eqs. (29) and (30a-c), if interlayer interaction ( ) )0(,0~ , , ' →γ=pk ss jj V frequencies are correspondingly equal (if AsB ω<ω ): gA As 1 ω=ω=ω+ and Bω=ω− . With inclusion of interlayer interaction, ( ) 0~ , , ' ≠pk ss jj V , intermixing the layer fundamental vibrations occurs, and frequencies +ω and −ω are additionally shifted to some extent into different sides. Fitting dependence described by Eq. (29) to experiment [9] gave the possibility to obtain parameters of interlayer interactions ( )pk ss jj V , ,0 ' ~ , which are given as legend to Fig. 4 and parameter 45.0=α . Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 3. P. 354-361. doi: 10.15407/spqeo18.03.354 © 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 361 5. Conclusion The change of intensities and position of bands for in- plane 1 2gE and out-of-plane gA1 vibrations as a function of the “concentration” of corresponding type layers was studied. 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