Theoretical and Experimental Study of Phonon Spectra of Bulk and Nano-Sized MoS2 Layer Crystals

Theoretical analysis of Raman scattering spectra (RS) for single-crystal MoS2 sample and atomically thin MoS2 sample consisting from one to few layers was performed in order to explain the change of MoS2 vibrations at transition from a monoatomic layer to a bulk crystal. Experiments have shown that changes of frequencies of the most intensive bands arising from the in-plane, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {E}_{2 g}^1 $$\end{document}E2g1, and out-of-plane, A 1g, vibrations, as a function of number n of layers looks differently. Thus, the frequency of ω(A 1g) is increasing with growth of n, whereas the frequency of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \omega \left({E}_{2 g}^1\right) $$\end{document}ωE2g1 is decreasing. Such a change of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \omega \left({E}_{2 g}^1\right) $$\end{document}ωE2g1 frequency was explained as the effect of “strong increase of the dielectric tensor when going from single layer to the bulk” sample. In the present work, we show that the reason of different dependences of frequencies can be related to both the van der Waals (vdW) interlayer interaction and the anharmonic interaction of noted fundamental vibrations with the corresponding combination tones (CT) of layer that manifests itself due to Fermi resonance in the layer. Overjumping of these phonon pairs (s, s ') owing to interlayer interaction, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tilde{V}\left({}_{s, s\hbox{'}, q}^p\right) $$\end{document}V˜s,s',qp, to other layers at growth of number n, results in the change of frequencies for each interacting pair of A 1g or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {E}_{2 g}^1 $$\end{document}E2g1 symmetry. The alteration of pair frequencies depends on the ratio of constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tilde{V}\left({}_{s, s\hbox{'}, q}^p\right) $$\end{document}V˜s,s',qp describing the interaction of studied states s and s '. Electronic supplementary material The online version of this article (doi:10.1186/s11671-016-1808-8) contains supplementary material, which is available to authorized users.

The new period of activity in the study of such crystals arose again when Novoselov et al [1] published their results related to graphene. The usage of their method allows to prepare a very thin crystal structure that has 1-10 atomic layers [1]. The layer type crystals MoS 2 and MoSe 2 are especially perspective in this sphere, as they show very interesting spectroscopy features, if the sample consists of only few atomic layers. In particular, electronic band structure of such crystals differs from the bulk ones, and they demonstrate very intensive luminescence [2,3]. The number of works in which electronic and phonon properties of such type of crystal structures that are studied by spectroscopy methods significantly increased during the last years.
Recently, a detailed study of vibrational spectra of MoS 2 layer crystals consisting of several layers, n = 1-6 (atomically thin crystals), and analysis of former results were made in [4,5]. Strong signals of the in-plane and out-of-plane (A 1g ) Raman modes were observed for all 1-6 layer samples. These modes exhibited well-defined thickness dependence, thus the frequencies of these two modes are shifting away from each other with increasing thickness. The behaviour of frequency shifts with the layer thickness as emphasized in [4] cannot be explained solely in the terms of a weak van der Waals (vdW) interlayer interaction. The most striking is that frequency of the low frequency E 1 2g vibration decreases (red shifts), while that of the high frequency A 1g vibration increases (blue shifts) with the increasing of the sample thickness. Besides, the frequencies of both these modes converge to bulk values, when films have four or more layers. Within a classical model for coupled harmonic oscillators [3], the frequencies of E 1 2g and A 1g modes are expected to increase as additional layers are added to form the bulk material from individual layers. That means, that the interlayer vdW interactions increase the effective restoring forces acting on the atoms. Therefore, the shift of A 1g mode observed in experiments of works [4,5] with increasing of layer number seems to be in agreement with the theoretical prediction; however, the behaviour of the E 1 2g mode does not agree. The reason of such a behaviour of E 1 2g may be attributed [4] to the long-range Coulombic interlayer interaction [6,7] what manifests as anomalous Davydov splitting (DS) [8,9]. Similar features have also been observed in crystals of GaSe [6] and GaS [10]. Anomalous DS means that there are additional factors, besides of "pure DS effect", which are displayed in experiment.
Indeed, not only vdW interlayer interaction, but also its combination with anharmonic interactions that occur in complex molecules or layers, can give rise to new features in dynamics of crystal lattice and observed in infrared (IR) and Raman spectroscopy (RS). Such approach was developed in a number of works [11][12][13] in 70-es of the last century, where not only DS effect but also its combination with intramolecular Fermi resonance (FR) was taken into account. Such effect is known as Fermi-Davydov (FD) resonance in molecular type crystals. It was shown in the cited works that frequencies of crystal vibrations and their intensities depend on treatment of the vdW inter-molecular interaction and intra-molecular anharmonicity. More complicated situation of displaying FD resonance with participation of lattice phonons was studied in crystals with hydrogen bond in which anharmonic constants can be greater than in the case of lattice phonons [14]. Therefore, we suppose that features observed in recently published Raman spectra [4,5] can be explained in the framework of approach developed in following works [11][12][13]. However, the problem connected with the influence of weak interlayer interactions on strong intralayer interaction looks more complex in this case, because each layer is a two-dimensional crystal. Therefore, in the present work, we at first consider the response of the bulk layer crystal on incident radiation, writing the Hamiltonian for the layer type crystal in a form which separates strong intralayer interactions and weak interlayer ones. Then, by variation of the vdW interaction value between layers (the effect of dielectric function [7] is considered to be included in both the intralayer interactions and in the vdW parameters), we can analyse the change of layer frequencies in the framework of the developed approach below.

Intensity of Raman Scattering
RS intensity is expressed by the imaginary part of Fourier component of Green function (GF) on tensor of susceptibility of crystal χ k,λ,k ',λ ' [15,16].
Here, ē α (k, λ) is α-component of electric field E α (m) unit vector; χ α,β (m) are the components of susceptibility tensor of the m-th unit cell of a crystal.
For a layer type crystal, the tensor (2) has some specific features; in particular, two indexes must be used (m → l, n) to number the unit cells of a crystal: the first index (l) points out the number of layer and other one (n) numbers the unit cell in given layer. The wave vector is also convenient to present by two components oriented in layer, Q ! n , and normal to layer, Q ! l , respectively, so that Q Then Taking into account an identical crystal layers, making the Fourier transformation of Eq. (3) and inserting the results into Eq. (1), the expression for susceptibility of layer crystal can be written as follows: Where N 0 and N l are numbers of unit cells in layer and number of layers in crystal, respectively; besides, in the following, we will write the operator of normal coordinate and tensor of scattering in more simple forms, ϕ Q l ;Q n ;s ¼ ϕ Q;s and χ k;λ;k 0 ;λ 0 Q n ; s ð Þ¼χ s . It was noted in [15,16] that the intensity of RS is expressed by Fourier component of Green function from tensor of susceptibility of a crystal. In our case, the intensity of light scattering by one unit cell is described by the following expression: It is seen from the last line of Eq. (5) that intensity is expressed by the Fourier component of the retarded Green functions from operators of the normal coordinate (for convenience all indexes besides of ones describing the phonon states of layer were omitted and wave is supposed to be small). In following consideration, we will write:

Hamiltonian and Equations for Green Functions
Potential energy of crystal vibrations, V(l, n), can be written as series on atom deviations from an equilibrium position, u k l;n;α ; therefore, the crystal energy in harmonic approximation is written as follows: k 0 ;n 0 ;β l;n;k;α u k l;n;α u k 0 l;n 0 ;β Þ þ 1 2 X l; l 0 ≠l; n; k; α; n 0 ; k 0 ; β V } l 0 ;k 0 ;n 0 ;β l;n;k;α u k l;n;α u k 0 l 0 ;n 0 ;β Þ ð 7Þ In a crystal with identical layers, the phonon frequencies are independent on index layer l but depend only on quantum states: ω l q;s j →ω q;s , s l → s. Besides, interlayer interaction is a function of space between layers,Ṽ s l ;s l} ;k l;l} À Á →Ṽ l−l} s;s 0 ;k . Therefore, after Fourier transformation, the Eq. (7) can be written as follows (more details see in Additional file 1): In valueṼ } p s;s 0 ;q , the down indexes (s, s′, q) characterize the quantum states of layer, but upper one (p) points out on the transmission of excitation between layers due to their interaction.
Operators of normal coordinate ϕ q,p,s and momentum π q,p,s are expressed by creation-annihilation phonons operators of layer, b þ q;p;s ; b q;p;s , by relations: Fourier components of GF in layer type crystal were defined by Eq.(6), but it is more convenient to consider the GF of more general form: The equation for such GF looks as follows: Because commutators [ϕ k,p,s (0); ϕ k ',p ',s ' (0)] = 0 and [ϕ k,p,s ; H int ] = 0 equation for GF is described by simple relation: The equation for new GF arising in Eq. (12) is likely obtained and looks as follows: After Fourier transformation of Eqs. (12,13), we obtain the system of coupled equations for Fourier component of GF which can be reduced to one simple equation In Eq. (14), we took into account that nonzero solution is possible if conservation law for wave vectors k = k ', p = p ' takes place.

Spectral Dependence of Raman Scattering
If only two states (s 0 , s 1 ) of layers interact one with another, Eq. (14) results in the following solution of equation (in the following consideration we will mean that indexes describing the components of wave vector of crystal excitations k, p → 0; because, Eq. (5), both wave vectors are connected with corresponding components of incident ra- Where we used the designations Insertion of Eqs. (15,16) and Eq. (17) into Eq. (5) results in the following relation: For simplicity, we will assume that the RS tensor components are real, χ s ¼ χ Ã s . Now, in order to take into account the damping of phonon excitations, we will consider the frequency as a complex value, ω → ω + iγ; therefore, all values depending on frequency in numerator and denominator became complex ones, in particular: After separating the imaginary part in Eq. (20), the intensity of RS in final form can be written as follows: Eq. (23) and Eqs. (21,22), (18,19) show that RS intensity has an enough complicated dependence on the frequency ω, and on interaction between layers described by valueṼ p s;s 0 ;k , Eqs. (18,19).

Results and Discussion
The estimation of interlayer interaction parameters can be made on the base of results of phonon theoretical calculations for one layer and bulk MoS 2 crystal obtained in work [17]. The actual phonon frequencies for onelayer  Table 2). Experiments however showed [4] that observed frequencies ω A 0 1 ≈403 cm −1 and ω E 0 1 ≈384 cm −1 for one layer differ significantly from ones predicted by theory ( ω A 0 1 ;calc − ω A 0 1 ; exp ¼ 7:3 cm −1 ). One of the reasons of such difference can be connected with anharmonic interactions, which are in monolayer and which were not taken into account in theoretical calculations. Indeed, from work [17], (Fig. 2, upper panel) follows that combination tones ω LA (q M ) + ω LA (−q M ) ≈ ω LA (q K ) + ω LA (−q K ) ≈470 cm −1 are fully symmetric and have frequency greater than ω A 0 1 (for points K, M symmetry group can be subgroup of D 3h , for example, C 3h ). Therefore, both vibrations can take part in Fermi resonance (FR) interaction, and as a result, the calculated fundamental band, ω A 0 In the case of bulk MoS 2 discussed above, the A 1g and E 1 2g bands are observed as two strong fundamental ones, Fig. 1. Insertion in Fig. 1, taken from our work [5], shows the transformation of these band intensities as function of layer numbers (thickness). It was shown in [5] that for atomically thin crystal MoS 2 consisting of the few layers that there are some different forbidden rules for even and odd layer numbers. That is clearly seen for ratio of intensities for cases 1L, 3L and 2L in insertion. In the last case, the intensity of E 1 2g band increases significantly. For other cases beginning from thickness 4L and up to bulk, the ratio of band intensities E 1 2g and A 1g is practically unchanged. The transformation of other bands, E 1g and B 1 2g , allowed for atomically thin crystals but disappearance for bulk case is also seen in insertion.
One can also note in Fig. 1   These CT can interact with fundamental vibrations A 1g and E 2g symmetry due to FR, admitted for D 6h point group symmetry of bulk MoS 2 by the relation E 2g × E 2g = A 1g + E 2g . As a result, both fundamental ones should be shifted. Position and intensities of A 1g and E 2g bands depend on anharmonic constant Γ responsible for FR interaction [18,19]. Effect of FR on fundamentals is shown in Fig. 2. As a result of anharmonic interaction of the fundamental ω(A 1g ) = 412 cm −1 with + ω E 2 2g = 423 cm −1 , the first is shifted down to meaning ω(A 1g ) ≈408 cm −1 , Fig. 2 = (388 ± 35) cm −1 , but such an interaction is more complicated, because two CT are placed around this fundamental band at the same distances: According to the theory of FR in crystals [18,19], intensity of Raman scattering (absorption) and shift of interacting bands are described by renormalized constant Γ. For each of the studied cases, this constant is given by the following relations: for the first case of FR, ; n E 1 2g are occupation numbers of the corresponding phonons. Because is shifted slightly below to experimentally observed Fig. 2, curves 3 and 4. Thus, taking into consideration the anharmonic interactions, the better correlation of calculation and experiment can be obtained.
One can note that the frequencies ω E 1 2g and ω(A 1g ) are caused by presence of two layers in the crystal unit cell; therefore, anharmonic interactions in bulk MoS 2 are also related with these two layers. However, according to Fig. 1 of present work and Figs. 4-6 work of [5], the number of layers has influence on the spectrum. For example, the odd or even number of layers in nanothin crystals is very important for intensities of bands, but it is not so important for their frequencies which are changing smoothly enough. Therefore, for description of frequencies dependence of the discussed ω E 1 2g and ω(A 1g ) bands on the layer numbers, one can use the monolayer properties of MoS 2 . According to [4], the experimental values of monolayer frequencies are equal ω A 0 1 ≈ 403 cm −1 and ω E 0 1 ≈ 384 cm −1 as what is shown in Fig. 3 by thick points for L = 1. Then, on going from monolayer to bulk MoS 2 , two noted bands are transformed into other doublet ≈ 382 cm −1 .
(These limit frequencies can be slightly changed on 1-2 cm −1 at different experiments [5]). It is also seen from with growth of layer numbers was explained in [17] as "anomalous Davydov splitting", and it was related with effect of dielectric tensor [7]. In the present work, we show that different behaviour of bands can be connected with two effects: with increasing of interlayer interaction growing up of layer numbers and also with appearing of additional with concrete quantum states s 0 and s 1 . Because one of these states, s 0 , in our description is connected with fundamental vibration and other, s 1 , characterizing the CT, we obtain some special form of FD resonance for layer type crystal. It should be noted that at comparison of theoretical calculations with experiment, [17] Fig. 6, the greatest difference between both results was observed at L = 1 for both ω A 0 , the marked difference between theory and experiments is observed also for L = 2 and 3. It is clear that with growth of the layer numbers the value of interlayer interaction increases and the intralayer anharmonic interaction effect, which was not taken into account in theory [17], became not so important. Therefore, the agreement of theoretical calculations with experiment for big L is better. The effects of influence of the intramolecular anharmonism on Davydov splitting were studied before in works [11,12,14] for molecular type crystals.
Increasing of parametersṼ p s;s 0 ;k →V s;s 0 ; k; p→0 ð Þ describing the interlayer interaction with growth of layer numbers can be seen from the following expression (more details are presented in Additional file 2) Fitting of the dependence described by Eq. (24) with using Eq. (25) to experiment [4] gives the possibility to obtain the parameters of interlayer interactions V s,s ' . For MoS 2 crystal, the best fit is observed at α = 0.45.
One can note that change of layer frequencies at transition from monolayer to bulk was recently studied in layer type crystal V 2 O 5 . The situation with this crystal is more complicated [20] because it has enough low crystal symmetry, point group D 2h , 39 optical vibrations and one dimensional representation. But for this crystal, there are some features related with product of irreduced representations, for example, B 1g ⊗ B 1g =, B 1u ⊗ B 1u = and B 3u ⊗ B 3u = A 1g . Therefore, many combination tones appear which can interact by anharmonicity with fundamental vibrations of corresponding symmetry and which result in shifting them to different sides. That depends on their initial frequencies: ω fund > ω comb. tone or contrary. In particular, it is seen in Fig. 9 work [20]. The highest band of monolayer, B 3u symmetry, is located at 1088 cm −1 , but at some upper of this fundamental, the combination tone ω(B 3u ) + ω(A 1g ) = (747 + 471) cm −1 of the same symmetry B 3u is placed. As a result of going from monolayer to bulk, the interlayer and anharmonic interactions increase and the fundamental band is shifted down. But if two CT interacting with fundamental are located upper and lower of it, the last can be immobile at transition from layer to bulk. Such type of FR was recently discussed in work [21].

Conclusions
In the present article on the example of MoS 2 layer type crystal, we show that change of phonon bands position, in-plane E 1 2g and out-of-plane A 1g , as a function of number of layers for thin layer crystals can be understood if one uses the Hamiltonian written in the form separating the strong intralayer and weak interlayer interactions. It is also shown that taking into account anharmonic effects of layer together with wdW interlayer interaction gives for ω(A 1g ), V s0;s0 = 8, V s0;s1 = V s1;s0 = 6, V s1;s1 = 3; for ω E 1 2g , V s0;s0 = 5, V s0;s1 = V s1;s0 = 4, V s1;s1 = 2; (experimental data are taken from [4]) possibility to describe correctly enough the observed changes of phonon frequencies with increasing of layer numbers. Estimation of parameters describing the interlayer interaction was made by comparison of theory with experiment, and it was obtained that these values for MoS 2 layer crystal are significantly smaller than studied intralayer frequencies.
Additional files