was introduced for convenience of analysis where are the

was introduced for convenience of analysis, where are the layer thickness values in dimensional units Two groups of phonon modes (I and II) can be identified in the obtained solutions. These modes have the following frequencies:
The first type of solutions deals with the phonons of A(LO) and E(ТО) symmetry. For the phonons of this symmetry, polar phonon frequencies are close to those of the phonons in the GaN, AlN and Al0.5Ga0.5N bulk materials (see Table 2). The difference with the case of the binary AlN/GaN SL is that phonons with the frequencies characteristic for the material of the cox 2 inhibitors layer appear in the spectra of the quaternary structures. Similar to the case of the binary SL, the frequencies of these modes do not change with a change in the relative thickness of the GaN and AlN layers. The modes of А(ТО) and E(LO) symmetry exhibit a qualitatively different character. These types of modes are sensitive to the ratio of layer thicknesses. The calculated frequencies of these modes in quaternary and binary SLs are shown in Fig.2. We have obtained three solutions for each of the А(ТО) and E(LO) modes in the quaternary periodic structure. One of these solutions falls within the ‘normal’ range: for the A(ТО) mode it lies in the TO frequency range, and for the E(LO) mode in the LO frequency range. The frequencies of the two other solutions are in the ‘abnormal’ range: one for the A(ТО) modes is in the LO frequency range, and the other, for the E(LO) modes, is in the TO frequency range. ‘Normal’ modes are characterized by in-phase amplitudes of atomic displacements, and therefore, as in the case of a binary SL [12], are the most intense in the RS spectra (from now on, these modes shall be referred to, respectively, as A(ТО)+ and E(LO)+). The interstitial layers present start to noticeably affect these modes only when the thickness of one of the layers becomes comparable to the thickness of the interstitial layer. However, in a wide range of thickness ratios of the GaN and AlN layers, the frequencies of these modes are almost insensitive to the presence of the interstitial layer. This result confirms it is possible to use these modes for describing the structure of the grown samples. Now let us analyze the ‘abnormal’ modes, denoting them, respectively, as А(ТО)1– and А(ТО)2–, and E(LO)1– and E(LO)2–. For these modes, the amplitudes of atomic displacements in the GaN and AlN layers are anti-phase. Introducing an interstitial layer between them leads to the splitting of these modes compared with the binary SL. This splitting is connected to the fact that for higher phonon frequencies (А(ТО)1–, E(LO)1–), the atoms in the interstitial layer are moving in phase with the atoms in the GaN layer, and in antiphase with those in AlN; for lower phonon frequencies (А(ТО)2–, E(LO)2–), the atoms in the interstitial layer move in phase with the atoms in the AlN layer and in antiphase with the GaN atoms (Fig. 3) It can be seen from Fig. 4 that the magnitude of the frequency splitting for ‘abnormal’ modes increases with an increase in interface thickness and reaches the value of 50 cm−1 at h=0.10. The simulation of quaternary systems with different thicknesses of the interstitial layers indicates that the magnitude of the frequency splitting for these modes provides information about the relative thicknesses of the interstitial layers (see Fig. 4).
Conclusions The simulation of the spectrum of polar optical phonons in quaternary SLs resulted in predicting the emergence of additional (compared to binary SLs) phonon modes which characterize thickness of the interstitial layer. It has been established that several modes corresponding to different phase relations of the atomic displacements both in the layers of the bulk constituents and in the interstitial layers are present in the А(TO) and Е(LО) phonon symmetry spectra of quaternary SLs. The relationship between the frequencies of the most intense A(ТО)+ and E(LO)+ lines and the ratio of the GaN/AlN layer thicknesses is almost the same as in the case of the binary SL. Because of this, these lines can be used for determining the ratios of the main GaN/AlN layer thicknesses even in the presence of diffuse interfaces. The frequency splitting values for low-intensity modes (А(ТО)1– and А(ТО)2–, and E(LO)1– and E(LO)2–) can be used for determining the relative thickness of the interstitial layer. Thus, the experimental determination of polar optical phonon frequencies provides us with opportunity for estimating the relative thicknesses of both the main layers of the grown structure and the interface smearing.