Mixed Type of the Magnetic Order in Intrinsic Magnetic Topological Insulators Mn(Bi,Sb)2Te4

It is known that Mn(Bi1 –xSbx)2Te4 is an intrinsic magnetic topological insulator, where the Dirac point can be localized at the Fermi level by substituting Bi atoms for Sb atoms to implement the quantum anomalous Hall effect and other unique quantum effects. It has already been shown that Mn(Bi1 – xSbx)2Te4 samples has antiferromagnetic order in a wide range of Sb concentration x = [0, 0.5]. In this paper a mixed type of magnetic ordering is experimentally demonstrated. This implies that there is both a ferromagnetic and an antiferromagnetic order in the system. We revealed the main reproducible magnetic characteristics of this magnetic structure, as well as their dependence on the concentration of Sb atoms.


INTRODUCTION
The intrinsic magnetic topological insulator MnBi 2 Te 4 [1,2] is currently the most promising platform to implement a number of quantum effects caused by the combination of the magnetic and topological properties in this material [3][4][5][6][7][8][9][10]. Different modifications of the stoichiometry of MnBi 2 Te 4 are actively studied to improve the properties of the material [11][12][13][14][15] or to follow changes in the electronic structure of the topological insulator depending on the imperfections of the structure such as defects and deviation from stoichiometry [16].
One of the directions of such studies is the investigation of the modification of the stoichiometry of MnBi 2 Te 4 by substituting Bi atoms for Sb atoms [12,[17][18][19]. This substitution is a reliable and relatively simple method to compensate electron doping of a MnBi 2 Te 4 single crystal synthesized by the Bridgman method. Mn(Bi 1 -x Sb x ) 2 Te 4 compound at certain concentration of Sb can be in charge neutral state, which is a necessary condition for the quantum anomalous Hall effect.
It was shown in the paper [12] that the antiferromagnetic (AFM) order remains in the Mn(Bi 1 -x Sb x ) 2 Te 4 compound up to x = 0.5 with a slight decrease in the Néel temperature T N (from 24.7 K for MnBi 2 Te 4 to 22.1 K for MnBiSbTe 4 ). It is known that the MnSb 2 Te 4 compound can exist in the AFM, ferromagnetic (FM), and ferrimagnetic phases with the magnetic phase transition temperature T C from 19 to 50 K [20][21][22][23]. It was assumed that this variety of results in the different papers was caused by fact that Mn with Sb atoms are much more likely to form intermixing defects than Mn and Bi atoms [18,20,24]. Thus, the CONDENSED MATTER magnetic properties of MnSb 2 Te 4 samples depend on the concentration and type of defects.
The high variation in magnetic properties observed in several works for MnSb 2 Te 4 can be reasonably expected for materials with mixed stoichiometry Mn(Bi 1 -x Sb x ) 2 Te 4 . Indeed, in the paper [25] it was shown that a transition of the Mn(Bi 1 -x Sb x ) 4 Te 7 compound to the FM phase is possible at a low (x = 0.15) concentration of Sb atoms.
In this work, the magnetic properties of Mn(Bi 1 -x Sb x ) 2 Te 4 samples with x = 0.1-0.5 were studied. Hereinafter, the concentration of Sb atoms is the concentration stated at growth. Particular attention was paid to the Mn(Bi 0.8 Sb 0.2 ) 2 Te 4 sample since at that Sb concentration (x = 0.2) the Dirac point is located at the Fermi level. The magnetic properties were studied using SQUID magnetometry and the ferromagnetic resonance technique by analogy with [26,27].

EXPERIMENTAL RESULTS AND DISCUSSION
The magnetic properties of Mn(Bi 1 -x Sb x ) 2 Te 4 samples with growth-stated values x = 0.1, 0.2, 0.3, 0.4, and 0.5 were measured using a SQUID magnetometer and the ferromagnetic resonance technique. By repeating the experiment on different samples of the same stoichiometry, it was shown that there is reproducibility for some parameters, and strong fluctuations for others.
The growth of the MnBi 2 Te 4 sample consisting of Te-Bi-Te-Mn-Te-Bi-Te septuple layers can be accompanied by the additional growth of Te-Bi-Te-Bi-Te quintuple layers [28]. The introduction of quintuple layers between septuple layers results in the formation of MnBi 2 Te 4 (Bi 2 Te 3 ) m , m = 1, 2, …, phases with magnetic properties different from those of MnBi 2 Te 4 [29]. As the number of quintuple layers increases, the exchange coupling constant decreases and the AFM phase becomes less stable. If there are regions with different m in the crystal structure, it will be natural to detect a mixed type of magnetic ordering in the measurements. Thus it is essential to determine the ratio of crystalline phases in the studied materials.
The stoichiometry of all samples was preliminarily tested for consistency with the stated growth concentrations. The crystal structure was studied using X-ray single-crystal diffraction (XRSCD). This method is able to certainly separate different MnBi 2 Te 4 (Bi 2 Te 3 ) m , m = 0, 1, 2, …, phases [29]. The samples were examined using angle-resolved photoelectron spectroscopy (ARPES) in order to reveal features of the electron structures, in particular, topological surface states inherent in these compounds. The sample composition on the surface was probed by X-ray photoelectron spectroscopy (XPS). Figure 1 shows (a) the XRSCD spectrum, the (b) ARPES and (c) XPS spectra of the Mn(Bi 0.8 Sb 0.2 ) 2 Te 4 sample. According to the XRSCD spectrum, the MnBi 2 Te 4 -type phase is the main phase in the sample, while the MnBi 2 Te 4 (Bi 2 Te 3 ) m (m > 1)type gives a minor inclusions (peaks marked by asterisks in Fig. 1a). The ARPES spectrum Fig. 1b demonstrates the lower part of the Dirac cone of topological surface states that merges with the upper part of the valence band. The band gap separating the upper and lower parts of the cone is located at the Fermi level, which is specific to this compound. The concentrations of atoms on the measured region of the surface were estimated from the intensities of the peaks of the core levels from the XPS spectrum in Fig. 1c. The calculation was performed using Trzhaskovskaya's database of photoionization cross sections [31]. Obtained concentrations are 15, 23, 7, and 55% for Mn, Bi, Sb, and Te atoms, respectively. They differ from growthstated ones by 1-2%, which is within the error of the used method; consequently, the relation between the concentrations of atoms on the surface of the sample can be treated as corresponding to the value declared during growth. The real concentration of Sb atoms is higher than the growth-stated value, but this is the known and expected tendency at the growth of these compounds [32,33].  Fig. 2b). For comparison with the published data, the magnitude of the magnetic moment obtained in the measurement was recalculated from emu units to units of Bohr magneton μ B per MnBi 2 Te 4 formula unit (one Mn atom per layer). It is worth to note that in the presence of quintuple layers, which do not contain the Mn layer, the total magnetic moment after recalculation is underestimated. However, according to XRSCD data (Fig. 1a), samples do not include a significant number of quintuple layers. The recalculation was performed by the expression (1) where is the mass of the sample, k is the recalculation coefficient, and is the mass of one formula unit calculated as the sum of the mass of all atoms in the stoichiometric formula.   Furthermore, the magnetic field of the spin-flop transition H SF was estimated for each curve. H SF and the saturation magnetization of the AFM phase M S provide an experimental estimate of the main characteristics of the antiferromagnet, namely, anisotropy H A and exchange interaction H E effective fields [34]. However, the M S cannot be determined because the applied magnetic field is insufficient to saturate the magnetic moment [16]. Thus, only the H SF can be analyzed. Its position is marked by the dashed vertical lines in Fig. 2b. The H SF value is the same for different samples of the identical stoichiometry. Figure 2f shows the dependence of the H SF on the concentration of Sb atoms. It is monotonically decreasing at low concentrations of Sb atoms.
Thus, in spite of obvious fluctuations of the FM and AFM contributions for the samples with the same stoichiometry, the main parameters of these contributions demonstrate a stable dependence on the concentration of Sb atoms.
To analyze the behavior of the AFM transition temperature, we measured the temperature dependencies of the magnetic susceptibility χ for Mn(Bi 1 -x Sb x ) 2 Te 4 samples with different x values in a magnetic field of 50 Oe applied along the c crystallographic axis. The results are presented in Figs. 3a-3e in the form of the temperature dependence of the inverse susceptibility 1/χ. It is seen that the AFM transition temperature decreases with an increase in the concentration of Sb atoms, which was also observed for the samples with the pure AFM phase in the paper [12]. However this decrease is less pronounced in the case of the mixed phase. Figure 3b shows data for different samples of the same stoichiometry. It is clearly seen that the AFM transition temperature is the same for all  Figure 3f presents the temperature dependence of the magnetic susceptibility for samples with different compositions, which demonstrates the behavior of the FM phase. The FM transition temperature for all samples is near T C = 17 K. It is worth to note that the FM transition temperature is below the AFM transition temperature.
The ferromagnetic resonance study of the magnetic properties of the Mn(Bi 1 -x Sb x ) 2 Te 4 samples with x = 0.2 and 0.4 was carried out in a manner similar to [26,27]. The samples for microwave measurements were flakes about 100 μm in thickness with lateral dimensions of 2-3 mm whose c crystallographic axis was perpendicular to the plane of the sample. The thinfilm geometry was thus implemented. Figure 4a present the microwave spectrum of the Mn(Bi 0.8 Sb 0.2 ) 2 Te 4 sample at a temperature of 13 K in the field parallel to the waveguide. Figure 4b shows the cross section of the spectrum at a frequency of 11.52 GHz. To determine the resonance field, we simulated the dependence by the complex resonance susceptibility [35] including the background.
The microwave spectrum was analyzed for fields below the saturation field of the FM phase and below the field of the spin-flop transition (see Fig. 2) in order to exclude the effect of possible exchange interaction between the FM and AFM phases and the domain state on the position of the resonance line. Figure 4с presents the experimental field dependence of the resonance frequency f r (H). Under these conditions, the FM phase resonance is described by the simple Kittel formulas for the thin-film geometry [36]: for the field parallel to the waveguide (perpendicular to the c axis) and (3) For the field perpendicular to the waveguide (parallel to the c axis). Here, γ is the gyromagnetic ratio, and M eff = M S -2K u /(μ 0 M S ) is the effective saturation magnetization, where M S is the saturation magnetization of the FM phase, μ 0 is the permeability of free space, and the second term is the field of the perpendicular uniaxial anisotropy with the coefficient K u . The red line in Fig. 3с is the approximation of the experimental dependence f r (H) by Eq. (2).
The analysis of the spectra shows that M eff < 0. This certainly indicates that the c axis is the easy axis of the FM phase. The resulting temperature dependences of the absolute value of M eff for Mn(Bi 1 -x Sb x ) 2 Te 4 samples with x = 0.2 and 0.4 are shown in Fig. 4d. According to the experimental data, the Curie temperature of the FM phase T C is about 20 K for both compounds in the field perpendicular to the c axis. At this temperature, the transition of the FM resonance to the paramagnetic one occurs.
The approximation of the M eff (T) dependence by the empirical expression (4) allows one to estimate the effective saturation magnetization of the FM phases at zero temperature (μ 0 M 0 = -0.49 and -0.36 T for x = 0.2 and 0.4, respectively) and the Curie temperature (T C = 20.4 and 20.1 for x = 0.2 and 0.4, respectively). We note that the effective magnetization is much larger for the sample with the lower concentration of Sb atoms. The difference between T C values obtained for the FM phase from the measurements of the magnetization and ferromagnetic resonance can be due to the anisotropy of the properties of the FM phase.
The behavior of the magnetic structure revealed in this work can be described within the model of electron phase separation [37][38][39]. However, the observed change in the magnetic structure allows other explanations such as the effect of additional inclusions of MnBi 2 Te 4 (Bi 2 Te 3 ) m phases [29] or the effect of point defects [40]. Future studies of the microstructure of these compounds can allow one to separate the contributions of possible effects.

CONCLUSIONS
To summarize, it has been shown that a mixed magnetic structure can arise in Mn(Bi 1 -x Sb x ) 2 Te 4 samples. The characteristics of this structure have been studied using SQUID magnetometry and the ferromagnetic resonance technique. The relation Magnetic characteristics were measured at the Center of Diagnostic of Functional Materials for Medicine, Pharmacology, and Nanoelectronics, Science Park, St. Petersburg State University, using a SQUID magnetometer with a helium cryostat (Quantum Design) at a temperature of 2 K in the magnetic field with the strength up to 7 T applied along the c crystallographic axis.
X-ray diffraction studies were performed at the National University of Science and Technology MISiS with a multifunctional Rigaku Ultima IV diffractometer (Cu K radiation) using Bragg-Brentano focusing and a graphite monochromator on the diffracted beam. Spectra were recorded in the cw regime in the 2 angle range of 10°-100° and processed using the PDXL program.
Ferromagnetic resonance measurements were carried out at the National University of Science and Technology MISiS. The sample was glued on the 0.5-mm transmission line of the coplanar waveguide with an impedance of 50 Ω, which was made of Arlon AD1000 foil dielectric and was equipped with SMP high-frequency connectors. The holder was placed in a superconducting solenoid inside a closed-cycle cryostat (Oxford Instruments Triton, a base temperature of 1.2 K). The response of the samples was studied by analyzing the transfer coefficient of the microwave signal using a VNA Rohde & Schwarz ZVB20 analyzer.
Angle-resolved and X-ray photoelectron spectra were measured at the Rzhanov Institute of Semiconductor Physics, Siberian Branch, Russian Academy of Sciences using a SPECS Proven X-ARPES instrument at hν = 21.22 and 1486.7 eV for angle-resolved and X-ray photoelectron spectra, respectively. Pure surfaces of the samples were obtained by cleavage in ultrahigh vacuum. The basic pressure in the experiment was (3-5) × 10 -11 mbar.