Exchange interaction and its tuning in magnetic binary chalcogenides

Using a first-principles Green's function approach we study magnetic properties of the magnetic binary chalcogenides Bi2Te3, Bi2Se3, and Sb2Te3. The magnetic coupling between transition-metal impurities is long-range, extends beyond a quintuple layer, and decreases with increasing number of d electrons per 3d atom. We find two main mechanisms for the magnetic interaction in these materials: the indirect exchange interaction mediated by free carriers and the indirect interaction between magnetic moments via chalcogen atoms. The calculated Curie temperatures of these systems are in good agreement with available experimental data. Our results provide deep insight into magnetic interactions in magnetic binary chalcogenides and open a way to design new materials for promising applications.

Using a first-principles Green's function approach we study magnetic properties of the magnetic binary chalcogenides Bi2Te3, Bi2Se3, and Sb2Te3. The magnetic coupling between transition-metal impurities is long-range, extends beyond a quintuple layer, and decreases with increasing number of d electrons per 3d atom. We find two main mechanisms for the magnetic interaction in these materials: the indirect exchange interaction mediated by free carriers and the indirect interaction between magnetic moments via chalcogen atoms. The calculated Curie temperatures of these systems are in good agreement with available experimental data. Our results provide deep insight into magnetic interactions in magnetic binary chalcogenides and open a way to design new materials for promising applications. Tetradymite chalcogenides, in particular Bi 2 Te 3 , Bi 2 Se 3 , and Sb 2 Te 3 , are of great interest due to their outstanding structural and electronic properties. These compounds consist of repeated blocks of five atomic layers (quintuple layers) separated by a van der Waals gap. The electronic structure features a narrow band gap and strong spin-orbit coupling, which are responsible for the inverted band structure at the Brillouin zone center Γ. Tetradymite chalcogenides are attractive for thermoelectric applications [1] because of their high figure of merit at room temperature. Recently, topologically protected surface states have been observed in all of these chalcogenides, which makes them subject of intense research [2,3]. Especially, these bichalcogenides serve as a basis for new materials with desired properties [4]. This is feasible by stacking of different compounds or specific doping. In particular, doping with magnetic impurities can open new perspectives for spintronics and spin caloritronics applications [5][6][7][8].
Magnetic properties of magnetic chalcogenides can be efficiently described by first-principles methods. One of the first comprehensive studies was carried out by Larson and Lambrecht [27], who investigated the electronic and magnetic properties of bulk Bi 2 Te 3 , Bi 2 Se 3 , and Sb 2 Te 3 doped with 3d transition metal atoms; their results for magnetically doped Bi 2 Se 3 were confirmed by several groups [28,29]. Recently, it was shown that the Dirac surface state of the topological insulator Bi 2 Te 3 survives upon moderate Mn doping of the surface layer, but can lose its topological nontrivial character depending on the magnetization direction [30,31]. However, critical magnetic properties and the exchange interaction in magnetic chalcogenides were not studied in detail on a theoretical ab initio level and, thus, are still under debate.
In this work, doping bulk tetradymite chalcogenides with transition metals by means of a first-principles Green's function method, we show that the exchange interaction in these materials can be either long-range ferromagnetic, antiferromagnetic or paramagnetic, depending on the host and the impurity atoms. We identify in particular two main types of magnetic interactions and discuss ways to manipulate the magnetic properties of these systems.
The calculations were performed within the density arXiv:1306.6590v1 [cond-mat.mtrl-sci] 27 Jun 2013 functional theory using the local spin density approximation (LSDA) and the generalized gradient approximation (GGA) [32,33]. A self-consistent Green's function method in both relativistic and scalar-relativistic implementations was used to compute electronic and magnetic properties of the magnetic chalcogenides. Substitutional disorder was treated within the coherent potential approximation (CPA; e. g. [34]). Heisenberg exchange constants J ij were obtained using the magnetic force theorem as it is implemented within the multiple-scattering theory [35]. Inclusion of spin-orbit coupling leads to minor changes in the magnetic interaction (about 3-5% with respect to the scalar-relativistic case). Therefore, for the sake of clarity, here we present only exchange constants calculated within the scalar-relativistic approximation.
According to the available experimental data [9-20, 24, 25], 3d transition metal impurities in bulk tetradymite chalcogenides substitute typically cation atoms (Bi and Sb) and can supply 1-3 electrons for the bonding. The comparably smaller size of transition metal ions may lead to substantial relaxations of the underlying crystal structure [27]. We did not account for such structural deformations in our CPA calculations but investigated their impact on the magnetic interaction using a supercell approach, and found only minor changes of the exchange constant values. Therefore, the discussion below reports results from CPA calculations.
We performed extensive Green's function calculations of Bi 2−x TM x Se 3 , Bi 2−x TM x Te 3 , and Sb 2−x TM x Te 3 (TM = Ti, V, Cr, Mn, Fe, Co, Ni) for the range of concentrations 0 < x < 1.0. The electronic structures of these compounds calculated within the CPA agree for low and medium concentrations (x < 0.3) with those of previous supercell calculations by Larson and Lambrecht [27] (see the Supplementary Material). The self-consistently obtained Green's function was further used to calculate the magnetic exchange constants J ij . Their relevant directions are depicted in Fig. 1 on top of the lattice structure of Sb 2−x TM x Te 3 for clarity, where we distinguish among in-plane (within the Sb or Bi plane) and an out-of-plane coupling. Although experimental data are available for a large concentration range [15,16], we first focus discussion on a representative value of x = 0.2. The results presented in Fig. 2 can be summarized as follows.
(i) The effective exchange interaction is reduced with increasing number of d electrons per TM, going from positive to negative values. The strongest ferromagnetic interaction is found between Ti atoms, which is explained by the local density of states (DOS) of the impurities magnetization is 1.0 µ B /atom, indicating a valency of 3+ for each Ti atom. The strongly dispersed DOS at the Fermi energy leads to a large ferromagnetic coupling between the nearest magnetic moments within the Sb plane (Fig. 2).
(ii) With increasing number of d electrons per TM atom, the spectral weight at the Fermi level is reduced. This leads to a strong decrease of the exchange interactions in Sb 2−x Mn x Te 3 , in which the majority and minority spin d electrons are well separated in energy and show minor dispersion. In the case of Fe impurities, due to an occupied minority spin d state at the Fermi level, the magnitude of the exchange interaction increases but becomes negative because of the large exchange splitting and the isolated impurity-like character of the occupied d orbitals. For Co and Ni impurities, the valency changes from 3+ to 2+ and 1+, respectively, reducing, thereby, the magnitude of the exchange interaction, which remains negative. The exchange interaction of Sb 2−x Ni x Te 3 , Bi 2−x Ni x Te 3 , and Bi 2−x Ni x Se 3 is very weak and is not discussed here.
(iii) For almost all cases, the strongest exchange interaction is found between magnetic moments located in different Sb (Bi) planes but within the same quintuple layer (panel J 01 2 in Figs. 1 and 2). The coupling weakens systematically with the number of d electrons. This interaction occurs via a Te (Se) atom lying between two impurities and is of double exchange type. In addition, the magnitude of J 01 1 is as large as for the in-plane interaction between the nearest magnetic moments, which is an indirect exchange interaction mediated by free carrier sp states [36]. We thus conclude that two different exchange mechanisms, the double exchange interaction via an anion and the indirect exchange coupling via free carriers, determine the magnetic order in the TM doped chalcogenides.
(iv) The size of cation atoms is crucial for the exchange interaction. The large size of atoms and, thus, the more spatially extended wave functions, can lead to a strong hybridization with the electronic states of the neighboring atoms. On the one hand, this can increase the number of free carriers within the cation layer, favoring the indirect exchange of Zener type [36]. On the other hand, the strong binding between a cation (e. g. Bi) and an anion (e. g. Te) reduces the number of valence electrons of the anion and, thereby, reduces the strength of the double ex-change interaction. Therefore, in the case of Sb 2 Te 3 , the double exchange interaction via Te atoms is significantly larger than that in Bi 2 Te 3 and Bi 2 Se 3 .
(v) Surprisingly the exchange interaction between magnetic moments located in neighboring quintuple layers does not vanish (see J 02 1 and J 02 2 in Figs. 1 and 2).
(where ρ ↑ (z) and ρ ↓ (z) stand for the spin-up and -down charge densities, respectively, integrated over the lateral coordinates x and y) "bridges" the van der Waals gap and is responsible for the "interquintuple layer" magnetic interaction (Fig. 3). The spin density in anion layers is negative and has a magnitude comparable with that of the spin density in the van der Waals gap.
Considering a wider range of concentrations of the TM atom, we have estimated the critical temperatures T C using a Monte Carlo method [37][38][39]. To treat both ferromagnetic and antiferromagnetic materials, we investigate the spin-spin-correlation function where m i and Ω i are the magnetic moment and the interaction sphere around site i, respectively. We also account for percolation effects, using pair potentials, and compared estimated critical temperatures with the available experimental data. The results for ferromagnetic Sb 2−x TM x Te 3 (TM = Ti, V, Cr, Mn; Fig. 4) show a systematic increase of the T C with the concentration of dopants. Percolation effects do not affect strongly the behavior of T C at low concentrations; except in the case of Ti, for which percolation lowers T C . Calculations for Cr reproduce the experimentally measured trends for concentrations up to x = 0.6 [16]. For higher concentrations, we found a transition to antiferromagnetic order (area with a light red background in Fig. 4), which is understood as the results of an increasing antiferromagnetic interaction between magnetic moments from nearby quintuple layers. This explains why experimental data is unavailable for concentrations larger than x = 0.6.
Concerning the Sb 2−x V x Te 3 case, we reproduce the trend found in experiments for a broad concentration range [15]. However, the theoretical absolute values are underestimated by about a factor of 1/2. One could speculate that either structural imperfections due to sample preparations or limitations in the ab initio description could cause this mismatch. For the reported cases of Mn doping at low concentration regimes (x ≤ 0.1), our calculations are, again, in qualitative agreement with experiment [10,14,19,[40][41][42].
The systematic study of exchange interaction discussed for TM-doped magnetic chalcogenides elucidated in this Letter can open new possibilities for a specific design of their magnetic properties. We infer for instance that one way to control the magnetic interaction and the Curie temperature is to replace particular atoms or sheets of  Te Te  Te  Te Te Te  Te Te  Te Te  Te  Te Te Te  Te Te  Te  Te Te  Te  Te Te Te  Te Te  Te Te Te  Te Te Te  Te Te  Te  Te Te  Te  Te Te Te  Te Te  Te Te  Te  Te Te Te  Te Te  Te  Te Te  Te  Te Te Te  Te Te  Te Te  Te  Te Te Te  Te Te Te  Te Te  Te  Te Te Te  Te Te  Te Te  Te  Te Te Te  Te Te Te  Te Te Te  Te Te Te  Te Te  Te Te Te  Te Te Te  Te Te Te Sb  The critical temperature TC is determined from Monte Carlo simulations with randomly distributed impurities (black filled triangles) and with cluster percolation (red filled circles). TC is compared to experimental data (green and blue markers) [15,16]. In the case of Cr, there is a transition from a ferromagnetic state for x < 0.8 (light blue background) to an antiferromagnetic state at higher concentrations (light red background).
atoms, in order to tune the strength of the electronic hybridization; the latter is responsible for the exchange interaction mechanisms in this class of materials. Here, it has been shown that the overlap between electronic wave functions of anions and cations is crucial.
In an experimental realization, one could replace the anion layer between two cation sheets by atoms of the same group in the periodic table. As an example, we consider the layer between two Sb 1.80 Cr 0.20 sheets in the Sb 1.80 Cr 0.20 Te 3 alternated with various chalcogen atoms. We observe an increase of the associated Curie temperature with ionic size and spatial extension of the electronic wave function of the chalcogen ion: S, Se, Te, and Po yield T C = 69 K, 71 K, 76 K, and 79 K, respectively. A similar effect can be achieved with specific co-doping of the corresponding anion layer.
The in-plane exchange interaction can instead be tuned by co-doping of the cation layers in accordance with the exchange constant behavior presented in Fig. 2. To illustrate this aspect, we calculated the exchange interaction in Bi 2−x−y Cr x Sb y Te 3 for x = 0.2 and y = 0.0, 0.2, and 0.4 (detailed results are presented in the Supplementary Material). Here, the Curie temperature increases with Sb concentration from 35 K at y = 0.0 to 39 K at y = 0.2, and to 42 K at y = 0.4.
Another way to tune the magnetic interaction is to insert impurities into the van der Waals gap [43]. This can change the size of the van der Waals gap and, by a proper choice of the impurities, can supply free carriers, which are important for the indirect exchange of Zener type (see the Supplementary Material).
In summary, we have studied the exchange interaction in the Bi 2 Te 3 , Bi 2 Se 3 , and Sb 2 Te 3 tetradymite chalcogenides doped with transition metals. Our first principles calculations have shown that the magnetic interaction is long-range and is mainly mediated out-of-plane by the double exchange mechanism via an anion and in-plane by the indirect exchange coupling via free carriers. The calculated Curie temperatures as a function of the concentration are found in qualitative agreement with available experimental data. Finally, we presented several ways to tune the magnetic interaction in these systems: (i) replacing the anion layer between two cation sheets by atoms of the same group, (ii) co-doping of the cation sheet, and (iii) inserting impurities in the van der Waals gap. These results provide deep insight into the magnetic interactions in the magnetic binary chalcogenides, and open new ways to design new materials for promising applications.
We acknowledge support by the Ministry of Education and Science of Russian Federation (state task No.