Lattice Dynamics of Bi2Те3 and Vibrational Modes in Raman Scattering of Topological Insulators MnBi2Te4·n(Bi2Te3)

This work is devoted to the experimental study and symmetry analysis of the Raman-active vibration modes in MnBi2Te4·n(Bi2Te3) van der Waals topological insulators, where n is the number of Te–Bi–Te–Bi–Te quintuple layers between two neighboring Te–Bi–Te–Mn–Te–Bi–Te septuple layers. Confocal Raman spectroscopy is applied to study Raman spectra of crystal structures with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n = 0,1,2,3,4,5,6$$\end{document}, and ∞. The experimental frequencies of vibration modes of the same symmetry in the structures with different n are compared. The lattice dynamics of free-standing one, three, and four quintuple layers, as well as of bulk Bi2Те3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n = \infty )$$\end{document} and MnBi2Te4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n = 0)$$\end{document}, is considered theoretically. Vibrational modes of the last two systems have the same symmetry, but different displacement fields. These fields in the case of a Raman-active mode do not contain displacements of manganese atoms for any finite \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}. It is shown that two vibrational modes in the low-frequency region of the spectrum (35–70 cm–1) of structures with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n = 1,\;2,\;3,\;4,\;5$$\end{document}, and 6 practically correspond to the lattice dynamics of n free quintuple Bi2Те3 layers. For this reason, the remaining two vibration modes, which are observed in the high-frequency region of the spectrum (100–140 cm–1) and are experimentally indistinguishable in the sense of belonging to quintuple or septuple layer or to both layers simultaneously, should also be assigned to vibrations in quintuple layers under immobile atoms of septuple layers.


INTRODUCTION
Prior to the discovery of the first intrinsic antiferromagnetic topological insulator MnBi 2 Te 4 with a Néel temperature of 25.4 K [1], various possibilities were considered to create a magnetic gap at the Dirac point of a topological insulator (TI) and to obtain the phase of the quantum anomalous Hall effect (QAHE). The simplest way to solve the problem was implemented both theoretically [2,3] and experimentally [4] by doping a nonmagnetic TI with magnetic atoms. However, since the distribution of magnetic atoms in real TIs is nonuniform, the magnetic gap varied irregularly along the surface [5], which caused too low a temperature of the QAHE. Another method based on the so-called magnetic proximity effect near the TI/ferromagnetic semiconductor interface generally shifts the topological surface state deeper into the TI and, correspondingly, weakens the magnetic effect [6,7]. The above drawbacks can be overcome to a large extent by depositing an ultrathin ferromagnetic insulator film on the surface of the nonmagnetic TI provided that the crystal structure and atomic composition of the former are close to those of the latter [6,[8][9][10][11]. The appearance of MnBi 2 Te 4 largely removed the problem of creating a magnetic gap in the spectrum of topological surface states and, at the same time, gave between the neighboring septuple Te-Bi-Te-Mn-Te-Bi-Te atomic layers (SLs) in a structure with a given n value [12]. The importance of studying this series as natural heterostructures was first emphasized in study [13] of topological surface states and magnetic and transport properties of MnBi 2 Te 4 ·n(Bi 2 Te 3 ) and, later, was underlined in [12,[14][15][16][17][18][19][20][21][22][23][24], including studies of Raman scattering [12,[19][20][21][22][23][24]. However, lattice dynamics, as well as other properties of structures with n > 3, still remains unexplored. A significant discrepancy between the number of Raman-active modes which should be observed according to the symmetry analysis of a bulk crystal and that of the modes found in Raman spectra of the structures with , 2, and 3 is also still unexplained. It should be noted that, although the MnBi 2 Te 4 ·n(Bi 2 Te 3 ) structures at room temperature are paramagnetic, their topology is also nontrivial [25,26].
In this work, we report Raman studies for MnBi 2 Te 4 ·n(Bi 2 Te 3 ) topological insulators with . In addition, the lattice dynamics of one, three, and four Bi 2 Те 3 QLs, as well as bulk Bi 2 Те 3 ( ) and MnBi 2 Te 4 ( ), is studied theoretically. The overall results obtained make it possible to establish a very unusual lattice dynamics of topologically nontrivial MnBi 2 Te 4 ·n(Bi 2 Te 3 ) systems and to explain the aforementioned discrepancy.

SAMPLE PREPARATION AND X-RAY ANALYSIS
Single-phase samples of each member of the series were cleaved under an optical microscope from MnBi 2 Te 4 ·n(Bi 2 Te 3 ) crystalline ingots grown by the = 1 n ∞ … = 0,1, ,6, n ∞ = n = 0 n Bridgman-Stockbarger method from preliminarily synthesized polycrystalline phases of each composition. The synthesis procedure and growth conditions differed from those for MnBi 2 Te 4 [12] only by the percentage of the initial components and a small change in the growth temperature, in accordance with the required values. Diffraction patterns from planes confirming the compliance of the structure of each of the studied samples with the required values were obtained using a Bruker XRD D2 Phaser X-ray diffractometer (Cu radiation; 5° ≤ 2θ ≤ 100°). The phase composition was determined and the structural analysis of the samples was carried out using the EVA and TOPAS-4.2 software packages. Structural data obtained for all members of the series are shown in Table 1.

METHODOLOGY OF EXPERIMENTAL AND THEORETICAL STUDIES OF RAMAN SCATTERING
Raman scattering was studied using a Nanofinder 30 confocal setup (Tokyo Instruments, Japan) with a diffraction grating of 1800 lines/mm and a spectral resolution of 0.5 cm -1 . Scattering was excited at a wavelength of 532 nm by the second harmonic of a Nd:YAG laser with a maximum power of 10 mW. The diameter of the laser spot on the sample did not exceed 4 μm. The signal was detected in the photon count mode by a CCD camera cooled to a temperature of -100°C. The signal collection time interval did not exceed 1 min, and the signal itself was filtered out using LP03-532RU-50 edge filters (Semrock Company). The filter location (https://www.semrock.com/filterspectra-at-non-normal-angles-of-incidence.aspx) made it possible to record 100% of the Raman signal in the spectral range of 40-5000 cm -1 and at least 50% of the signal in the spectral range of 35-40 cm -1 . The measurements were carried out in the backscattering n (000 ) l n α1 K  Theoretical calculations were carried out within the density functional perturbation theory. In the case of MnBi 2 Te 4 , the pseudopotential method was used within the ABINIT code [27] in the basis of plane waves. In the case of Bi 2 Te 3 , we used a pseudopotential approach and a mixed basis [28,29], in which the valence states are presented as a combination of plane waves and local s and p orbitals at Bi and Te nodes, which made it possible to reduce the cutoff energy for plane waves to 10 Ry without loss of accuracy. The PBE approximation [30] was used for the exchangecorrelation functional. All calculated values were obtained taking into account the spin-orbit coupling [31]. When integrating over the Brillouin zone, we used 96 × 96 and 12 × 12 inhomogeneous k-point grids near the center of the Brillouin zone (for small k values) and for the rest of the Brillouin zone, respectively.
The complete vibrational representation for all members of the MnBi 2 Te 4 ·n(Bi 2 Te 3 ) series is given in Table 2.

RESULTS AND DISCUSSION
Compounds with and are end members of the MnBi 2 Te 4 ·n(Bi 2 Te 3 ) series and their structures contain only QLs and only SLs, respectively ( Table 1, last column). The Raman spectra of these members are shown in Fig. 1.
The mode identification for MnBi 2 Te 4 ( Fig. 1, pink line) is based on the comparison of the experimental spectra with the calculated phonon frequencies and symmetry at the Γ point ( Fig. 2) and is in agreement with the data recently obtained in [24], where the polarization of the observed phonons was additionally measured. For Bi 2 Te 3 (Fig. 1, blue line), the observed modes are identified by comparison with the calculated frequencies and mode symmetries shown in Fig. 3. Table 2. Complete vibrational representation for each member of the MnBi 2 Te 4 ·n(Bi 2 Te 3 ) series; N is the number of atoms in a unit cell. The group-theory notation of each representation at the Γ point is given in parentheses after the spectroscopic notation (center of the Brillouin zone  As seen in Table 2, six and four modes are expected in the Raman spectra of MnBi 2 Te 4 ·n(Bi 2 Te 3 ) with (MnBi 2 Te 4 ) and (Bi 2 Te 3 ), respectively. In the spectra shown in Fig. 2, only five of six and three of four expected Raman modes are observed, respectively. As follows from the data in Fig. 1 and Table 2, the missing mode for each material is the mode with the symmetry. The observation of this low-frequency Raman mode in the bulk Bi 2 Te 3 sample is complicated by instrumental limitations (see the preceding section) and becomes impossible at a certain signal intensity. At least, this mode was not observed in [32][33][34][35][36][37]. At the same time, the observation of a mode with the symmetry and a frequency of 36.5 cm -1 in Bi 2 Te 3 was reported in [38][39][40][41]. In MnBi 2 Te 4 , the low-frequency mode with a calculated frequency of 28.2 cm -1 (Fig. 2, ) has not yet been observed in any published works. Therefore, in further discussions, we use the calculated value of the frequency of this mode. Figure 4 shows the Raman spectra of all members of the MnBi 2 Te 4 ·n(Bi 2 Te 3 ) series. As follows from this figure, the most significant difference is observed between the spectra of MnBi 2 Te 4 and MnBi 4 Te 7 . The spectra of MnBi 4 Te 7 and MnBi 2 Te 4 are presented in Fig. 5 (black circles) along with their decomposition into components (solid green lines).
Let us compare the lowest frequency modes of MnBi 2 Te 4 (Fig. 2) and Bi 2 Te 3 ( Fig. 3) with in-phase displacements of Bi and Te atoms with the lowest frequency modes in the spectra of MnBi 4 Te 7 and MnBi 2 Te 4 (Fig. 5, top and bottom spectra, respectively), which are also easily identified as the and modes.
In MnBi 4 Te 7 , these modes are blueshifted (by more than 20%) compared to MnBi 2 Te 4 . Such a significant increase in frequency when replacing one MnBi 2 Te 4 SL by one Bi 2 Te 3 QL is clearly associated with a decrease in the number of atoms involved in in-phase displacements, which takes place upon transition from MnBi 2 Te 4 to Bi 2 Te 3 (cf. the and modes in Figs. 2 and 3).   The decomposition of the spectra into components (Fig. 5, solid green lines) indicates that the number of modes observed for MnBi 4 Te 7 is clearly less than that for MnBi 2 Te 4 , contrary to expectations. At the same time, any additional broadening of the fundamental modes caused by the increased number of atoms in the unit cell of MnBi 4 Te 7 and the increased number of modes is absent (Table 2, second and fourth columns, respectively). The superposition of such modes with close frequencies should lead to the broadening of the main lines. According to the decomposition and the analysis of the broadening of the constituent components for all spectra in Fig. 4, the spectra of all MnBi 2 Te 4 ·n(Bi 2 Te 3 ) with n ≥ 1 contain four modes. It is noteworthy that the number of modes for MnBi 4 Te 7 and MnBi 14 Te 22 could reach 12 and 36, respectively (see data for n = 1 and 6 in the fourth column of  In fact, it can be seen in Table 3 that the calculated frequencies of these modes for one Bi 2 Te 3 QL almost coincide with the observed frequencies of the modes with a similar symmetry in MnBi 2 Te 4 ·n(Bi 2 Te 3 ) with . The frequencies of these modes tend to increase with an increase in the number of Bi 2 Te 3 QLs from 1 to 4, similar to that observed for low-frequency modes in MnBi 2 Te 4 ·n(Bi 2 Te 3 ) when varies from 1 to 4 (Fig. 4).
Thus, the overall data obtained for vibrational modes indicate that a part of the lattice dynamics of MnBi 2 Te 4 ·n(Bi 2 Te 3 ) with associated at least with low-frequency Raman modes is determined by the lattice dynamics of n Bi 2 Te 3 QLs. This is in good agreement with the experimental data for Raman scattering in ultrathin Bi 2 Te 3 films grown by the gas transport method on Si/SiO 2 or fluorphlogopite mica substrates [41]. Similar to the lowfrequency mode in MnBi 2 Te 4 ·n(Bi 2 Te 3 ) with , the low-frequency mode was observed for two Bi 2 Te 3 QLs (the minimum number of QLs studied in [41]) and its intensity dropped rapidly with a further increase in the number of QLs. The difference of 2 cm -1 between the frequencies of this mode in MnBi 2 Te 4 ·n(Bi 2 Te 3 ) with n = 2 ( Table 3, 38.5 cm -1 ) and two Bi 2 Te 3 QLs (36.5 cm -1 [37]) is apparently caused by a technological factor. In the former case, the substrate needed to obtain one (two, etc.) Bi 2 Te 3 QLs (the second case) is absent, and the resulting structure is free of its influence. The same as for MnBi 2 Te 4 ·n(Bi 2 Te 3 ) with , in the case of one or more Bi 2 Te 3 layers, the low-frequency mode is slightly blueshifted with an increase in the number of Bi 2 Te 3 QLs. Our data for MnBi 2 Te 4 ·n(Bi 2 Te 3 ) with nonzero n are given in Fig. 6 in comparison with the data from [41] for Bi 2 Te 3 with different numbers of QLs. It is clearly seen that the effect of the substrate, According to the calculations (Table 3, ninth column), the high-frequency mode in Bi 2 Te 3 does not experience frequency dispersion upon change in the number of Bi 2 Te 3 QLs in much the same way as similar modes in MnBi 2 Te 4 ·n(Bi 2 Te 3 ) in Fig. 4, the frequency of which (101 cm -1 ) does not change with increasing . Although the frequencies of the E g modes connected by vertical dashed line 5 in Fig. 4 are the same (101 cm -1 ), these modes cannot be considered as identical or be attributed to vibrations associated with MnBi 2 Te 4 SLs or Bi 2 Te 3 QLs in MnBi 4 Te 7 . These modes are indistinguishable in the experimental data obtained.
A similar situation is observed for the highest frequency mode in Bi 2 Te 3 ( Table 3, tenth column), and for similar modes in Fig. 4. High-frequency modes that have the same symmetry but different frequencies of 139 (MnBi 2 Te 4 ) and 136 cm -1 (MnBi 4 Te 7 ) and are connected by vertical dashed line 6 in Fig. 4 also cannot strictly speaking be attributed to MnBi 2 Te 4 SLs or Bi 2 Te 3 QLs or to both simultaneously. A slight (2%) decrease in the frequency of this high-frequency mode at the appearance of one Bi 2 Te 3 QL in MnBi 2 Te 4 and the formation of MnBi 4 Te 7 cannot be decisive for choosing any one of the above options for belonging to this mode. Nevertheless, the presence of low-frequency modes associated with vibrations in quintuple layers also requires the appearance of high-frequency modes of these layers. For this reason, high-frequency modes also have to be attributed to the vibrations of Bi and Te atoms in quintuple layers under stationary atoms in septuple layers. The same is required by the passage to the limit to the spectrum of Bi 2 Te 3 , which would otherwise be impossible.

CONCLUSIONS
Raman-active vibrational modes of the MnBi 2 Te 4 ·n(Bi 2 Te 3 ) system with in the studied region of the spectrum are almost completely frequency degenerate. As a consequence, the number of observed modes is limited and equals four as for Bi 2 Te 3 . The degeneracy covers both low-frequency modes with in-phase displacements of atoms and high-frequency modes with out-of-phase displacements of atoms. In both cases, the vibrational modes correspond to the displacements of the Bi and Te atoms in quintuple layers, while the atoms of septuple layers are completely or almost immobile. Note that the immobility of the Bi and Te atoms in septuple layers is in no way caused by the symmetry restrictions, as in the case of the central Mn, which is not involved in Raman-active vibrations in MnBi 2 Te 4 ·n(Bi 2 Te 3 ) with any n because of the symmetry of its position.

> 0 n
In general, an interesting situation is implemented in which septuple layers of MnBi 2 Te 4 are almost "silent," but this silence is not due to symmetry and is not related to the so-called silent modes that are inactive both in Raman scattering and in infrared reflection (absorption) and have nonzero displacements of atoms. We are unaware of examples of such behavior of vibrational modes in layered crystals.
The absence of the contribution of magnetic Mn atoms to the Raman spectra of lattice vibrations indicates the need for studies of infrared spectra of MnBi 2 Te 4 ·n(Bi 2 Te 3 ) According to our estimates, the upper limit of the frequencies of dipole vibrations involving Mn is approximately 200 cm -1 .
In the context of this work, which is based on data obtained at room temperature, there is no connection between vibrational modes and magnetic phase transitions. However, such a connection can be revealed in low-temperature studies of the entire MnBi 2 Te 4 ·n(Bi 2 Te 3 ) series, and the data of this work will serve as a starting point for the analysis of the results obtained. We have already started work in this direction. , and ∞ and (circles) data obtained in [41] for bulk Bi 2 Te 3 and two (2QL), three (3QL), four (4QL), five (5QL), six (6QL) quintuple layers of Bi 2 Te 3 . = 2,3,4,5,6 n