Chinese Physics Letters, 2018, Vol. 35, No. 1, Article code 017502Express Letter Stepping Stone Mechanism: Carrier-Free Long-Range Magnetism Mediated by Magnetized Cation States in Quintuple Layer * Chunkai Chan(陈俊佳), Xiaodong Zhang(张小东), Yiou Zhang(张异欧), Kinfai Tse(谢建辉), Bei Deng(邓贝), Jingzhao Zhang(张璟昭), Junyi Zhu(朱骏宜) Affiliations 1Department of Physics, Chinese University of Hong Kong, Hong Kong Received 19 October 2017 *Supported by Chinese University of Hong Kong (CUHK) under Grant No 4053084, University Grants Committee of Hong Kong under Grant No 24300814, and the Start-up Funding of CUHK.
**Corresponding author. Email:
Citation Text: Chan C, Zhang X, Zhang Y, Tse K and Deng B et al 2018 Chin. Phys. Lett. 35 017502 Abstract The long-range magnetism observed in group-V tellurides quintuple layers is the only working example of carrier-free dilute magnetic semiconductors (DMS), whereas the physical mechanism is unclear, except the speculation on the band topology enhanced van Vleck paramagnetism. Based on DFT calculations, we find a stable long-range ferromagnetic order in a single quintuple layer of Cr-doped Bi$_2$Te$_3$ or Sb$_2$Te$_3$, with the dopant separation more than 9 Å. This configuration is the global energy minimum among all configurations. Different from the conventional super exchange theory, the magnetism is facilitated by the lone pair derived anti-bonding states near the cations. Such anti-bonding states work as stepping stones merged in the electron sea and conduct magnetism. Further, spin orbit coupling induced band inversion is found to be insignificant in the magnetism. Therefore, our findings directly dismiss the common misbelief that band topology is the only factor that enhances the magnetism. We further demonstrate that removal of the lone pair derived states destroys the long-range magnetism. This novel mechanism sheds light on the fundamental understanding of long-range magnetism and may lead to discoveries of new classes of DMS. DOI:10.1088/0256-307X/35/1/017502 PACS:75.50.Pp, 71.15.Mb, 71.70.Gm, 75.30.Hx © 2018 Chinese Physics Society Article Text The stable magnetism in 2D materials has been challenging and critical to realize new spintronic devices for decades. Recently, there are some developments in the realization of 2D magnetic transition metal compound.[1-3] Nevertheless, the carrier-independent ferromagnetism and anti-ferromagnetism in 2D or thin film dilute magnetic semiconductors remain to be extremely difficult to realize.[4] In such applications, an ideal purpose of free carrier manipulation is to realize the desired electronic functions with magnetic properties unchanged. However, in reality, it is almost unavoidable to tune both properties simultaneously. Very recently, it has been demonstrated that a long-range ferromagnetism, independent of both polarity and density of free carriers or even without carriers, can exist in Cr- or V-doped (Bi$_x$Sb$_{1-x}$$)_2$Te$_3$ thin film or multiple quintuple layers, leading to the discovery of the quantum anomalous Hall effect.[5-13] One of keys to the success is the long-range ferromagnetism, while the underlying physical mechanism is still largely unclear. (Bi$_x$Sb$_{1-x}$$)_2$Te$_3$ is a typical topological insulator with conducting surface states and insulating bulk states, protected by time reversal symmetry.[14] The introduction of magnetic dopants, such as Cr, may break the time reversal symmetry and affect the topological surface states.[15-19] In this DMS quasi 2D system, the long-range magnetic interaction network contains both cation sites and anion sites of host materials that connect magnetic dopants. The usual super exchange theory that focuses on response of mediated anions cannot explain the long-range ferromagnetism in this system.[20-22] As we shall show, the states near cations in the network may make significant contribution to the long-range magnetic interactions. On the other hand, the present theory proposed to explain the long-range ferromagnetism mechanism emphasizes on the topological non-trivial band structure.[15] In this attempt, however, only the enhanced paramagnetic susceptibility in undoped topological insulators was estimated, and no mutual interaction between magnetic dopants with large separations was calculated. In addition, no theoretical analysis was performed to study the difference between similar material family of Bi$_2$Te$_3$, Sb$_2$Te$_3$ and Bi$_2$Se$_3$. Later numerical works based on small supercells can hardly convince the existence of the long-range magnetism.[23,24] Also, one work estimated long-range magnetic coupling constants in the presence of carriers in a few configurations without analyzing the stability of the dopants.[25] Therefore, the systematic analysis of carrier free magnetism in such systems is missing. Also, in most of the previous works, the dopant concentration is higher than experimental value and may lead to some unphysical results.[23,24] However, in our calculations, the simulation cell is intentionally chosen to have the similar dopant concentration as experimental ones. In addition, the carrier free nature of our calculation guarantees the correct occupations of electrons and avoids the common mistakes in early literature on DMS.[26,27] In this Letter, we investigate the mechanism for the long-range ferromagnetism in Cr-doped topological insulators by performing ab initio calculations within the density functional theory (DFT) framework implemented in VASP code with a sufficiently large simulation cell[28] (refer to SI-A for computational details). To avoid further complexity of alloy effects in (Bi$_x$Sb$_{1-x}$$)_2$Te$_3$, we calculate the magnetic order of Cr doped Bi$_2$Te$_3$, Sb$_2$Te$_3$ and Bi$_2$Se$_3$, respectively. To our surprise, we discover a novel long-range ferromagnetic (FM) order in carrier free Bi$_2$Te$_3$ and Sb$_2$Te$_3$ single quintuple layer. In addition, these configurations are the most stable for a single quintuple layer only. This finding is independent of the band topology. The similar ferromagnetic order is relatively unstable for Bi$_2$Se$_3$, consistent with the experimental results.[29] Later, we use Bi$_2$Te$_3$ as a model system to illustrate the origin of the magnetism and qualitatively explain the intriguing interaction based on an extended Hubbard model.[30] To study the long-range magnetic order of the Cr-doped Bi$_2$Se$_3$, Bi$_2$Te$_3$, and Sb$_2$Te$_3$ thin films, we first replace two Bi or Sb atoms with two Cr atoms in the $4\times4$ host cell that contains one quintuple layer (details can be found in Fig. S1 of the Supplemental Material). The relative formation energy is defined as the energy difference between the formation energy of a configuration and that of the most stable configuration, as shown in Fig. 1(a). Significant formation energy drops on the seventh neighboring sites are found in all the systems. These configurations are the global minima in Bi$_2$Te$_3$ and in Sb$_2$Te$_3$, which are even more stable than the first or second nearest neighbor ones. On the other hand, the second nearest neighbor configuration is the most stable in Bi$_2$Se$_3$. As a function of the distance between the neighboring Cr atoms, the relative formation energy of Cr dopants in these three materials shows generally similar trends. Since the global minimum of the formation energy of Cr atoms in the Bi$_2$Se$_3$ thin film is the second nearest neighbor configuration, Cr atoms tend to form clusters in Bi$_2$Se$_3$, leading to nonmagnetic or paramagnetic orders. On the contrary, the global minimum of the formation energies of Cr atoms in the telluride suggests a long-range ferromagnetism order. These results qualitatively agree with the experimental observations.[5,29] We also calculate the ferromagnetic coupling constants vs the neighboring sites, as shown in Fig. 1(b). The coupling constants are defined as half of the energy difference between the two Cr atoms with the same spins (ferromagnetism) and the two atoms with opposite spins (anti-ferromagnetism). We find that the seventh nearest neighbor configurations favor ferromagnetism and yield coupling constants about 6 meV for all systems, despite the fact that the coupling strength is lower than the first or second nearest neighbor configurations. Also, the periodic images of the dopants may enhance the magnetism. This is physical because our simulated concentration is close to the experimental values. Also, calculations based on large cells without the enhancement of the periodic images demonstrate ferromagnetism (see SI-B and Fig. S2 for details).
Fig. 1. (a) The relative formation energy of two-Cr-atom doped ($4\times4$ supercell) single quintuple layer of Bi$_2$Se$_3$ (purple square), that of Bi$_2$Te$_3$ (green circle), and that of Sb$_2$Te$_3$ (red triangle) with respect to different neighboring sites of the Cr atoms. (b) The ferromagnetic coupling constants of the same systems
These findings are novel because the coupling strength in common DMS decays very fast in respect to the separation distances.[31] However, in our seventh nearest neighbor configuration, the two Cr atoms are separated by two anion atoms and one Bi atom. Also note that the difference between the formation energy on the seventh nearest neighboring sites and that on the first (or second) nearest neighboring sites in Sb$_2$Te$_3$ is the largest among these three systems. These results match with the experimental findings that quantum anomalous Hall effect can be realized in the Sb rich (Bi$_x$Sb$_{1-x}$$)_2$Te$_3$.[5] In order to check the accuracy of GGA results in the slab model, we also calculate the formation energy and the coupling constants using the GGA+U method with spin orbit coupling (SOC) included. We find that GGA+U and SOC do not change the conclusions of the GGA results (more details are shown in Fig. S3). Furthermore, calculations about Cr doped bulk Bi$_2$Te$_3$ also show the similar trend (refer to Fig. S4 and Fig.  S5). Based on the above discussions, both GGA and GGA+U plus SOC yield similar results. Therefore, we only include the GGA results on slab models in the following discussions. To understand the origin of such long-range ferromagnetism, we further study the intrinsic electronic properties of Bi$_2$Te$_3$. The projected density of states (pDOS) of Bi 6$s$ and Te 5$p$ are shown in Fig. 2(a). Despite the local symmetry difference, two Te atoms at different sites yield similar 5$p$ orbital components, so we choose one Te atom to show these orbitals. From the pDOS figure, it is clear that the dominant component of the valence band is Te 5$p$. Additionally, the majority of the Bi 6$s$ state, as a nonbonding lone pair state, is deep under the valence band maximum (VBM). However, a relatively small portion of Bi 6$s$ state is found to be slightly below the VBM. These results suggest that the small portion couples with the Te 5$p$ orbital and forms an occupied $sp$ anti-bonding (anti-$sp\sigma$) state.
Fig. 2. (a) The pDOS of the Bi $6s$ orbital (blue) and Te $5p$ orbital (green) in pure Bi$_2$Te$_3$. The total density of states is shown in the grey shading background. The dotted line shows the Fermi level. (b) The pDOS of the Cr $3d$ orbital (red) and the $6s$ orbital of Bi (blue) (Bi atom on the Cr-Te-Bi-Te-Cr path) in single Cr-doped Bi$_2$Te$_3$. The grey background is the total DOS of single Cr-doped Bi$_2$Te$_3$, and the positive and negative values represent the pDOS of spin-up and spin-down components, respectively.
Next, we study the single Cr-doped Bi$_2$Te$_3$. Cr dopant has 3 unpaired 3$d$ electrons with a magnetic moment of 3$\mu _{_{\rm B}}$. The pDOS of the Cr 3$d$ orbitals and the 6$s$ orbital of the Bi next to the Cr dopant are calculated, as shown in Fig. 2(b). The significant asymmetric nature of the spin-up and spin-down portion of the Bi 6$s$ orbital suggests that it is magnetized by the nearby Cr atom. Therefore, the anti-$sp\sigma$ state is also magnetized. It should be noted that the anti-$sp\sigma$ state is an intrinsic property of Bi$_2$Te$_3$. We calculate the partial charge density of the second band below VBM (Fig. S6). The charge density in vicinity to the Bi atom shows the shape of the anti-$sp\sigma$ state. The role of Cr dopant is to magnetize it, but not to create it. The pDOS of Cr 3$d$ also indicates that there is a small occupation of spin-down component, leading to the magnetization of the surrounding atoms. To understand the magnetization of surrounding atoms, we further calculate the spin density of single Cr-doped Bi$_2$Te$_3$, as shown in Fig. 3(a), where only the spin polarized atoms are displayed. We find that the neighboring $p$ orbitals of Te are magnetized along the bonding direction between Te and Cr atoms, which we label as $p_z$ orbital (note that $z$ axis is along the local bonding direction). The total magnetization of all the Te $p$ orbitals is anti-parallel to the Cr magnetic moment. These results are in good agreement with the experimental observations.[32] The spin density also suggests that the spin up electron cloud near the Bi atom is the anti-$sp\sigma$ state, which also couples to the Cr atom. Interestingly, the Cr-Te-Bi chain forms a right angle, as shown in Fig. 3(b). Note that such chains are fundamental building blocks to form the long-range magnetic order, as further studies suggest.
Fig. 3. Spin density of single and two Cr doped Bi$_2$Te$_3$. Large purple atom is Bi, cinnamon atom is Te, blue atom (central atom) is Cr. (a) The spin density of the single Cr-doped Bi$_2$Te$_3$ with Cr in the center. Center Cr atom is surrounded by six Te atoms. Bi atoms bonding to Te atoms. (b) The spin density of two adjacent magnetic blocks in ferromagnetism and anti-ferromagnetism states. The spin-up and spin-down density are yellow and blue, respectively, in both figures.
The spin polarized results can be qualitatively understood based on an electron hopping mechanism. Since the electronic environment around the Cr atom is approximately $O_h$ symmetry, the Cr 3$d$ orbitals are split into filled spin-up $t_{2g}$ states and empty $e_g$ states. Due to the local symmetry, the Te $p_z$ orbital is only coupled to the Cr $e_g$ states. Since the energy of the spin-up $e_g$ states is lower than that of the spin-down component, a portion of spin-up electrons of the Te $p_z$ orbital hops to the $e_g$ state. As a result, the $p_z$ orbital is magnetized towards spin down. Since the $t_{2g}$ states are already occupied by three spin up electrons, only spin-down electrons in the anti-$sp\sigma$ state can hop to the $t_{2g}$ states. As a result, the anti-$sp\sigma$ state is magnetized towards spin up. These results also indicate that the magnetic moments of the Cr atom are not fully localized at the Cr site and a small portion of them are distributed in the surrounding orbitals. Still, the magnetic moment in the Te $p$ states cannot be interpreted as spin polarized hole states,[32] because the Cr-doped Bi$_2$Te$_3$ thin films remain carrier-free. As shown in Fig. 3(b), when two of these building blocks are placed in adjacent to each other, long-range ferromagnetic interaction can occur, with the two Cr atoms at seventh nearest neighboring sites. In the ferromagnetic configuration, the spin of the anti-$sp\sigma$ orbitals in one block and the adjacent $p_z$ orbitals in the other block are anti-parallel. Therefore, electron hopping between these orbitals lowers their kinetic energy. However, in the anti-ferromagnetic configuration, these two spins are parallel and electron hopping cannot lower the energy. As a result, the ferromagnetic configuration is energetically favorable. Such building blocks further connect and form a strong ferromagnetic network in the whole crystal. In this novel scheme, the anti-bonding state near the middle cation serves as an essential stepping stone to mediate the magnetic interaction. Therefore, we name this mechanism as a stepping stone mechanism and these Bi atoms connecting the two building blocks are at stepping stone sites. Also, this effect is different from the traditional super exchange theory, which provides no discussions on the role of intermediate cation states.[20-22] To validate our new mechanism, we replace Bi atoms at the stepping stone sites with Ga atoms that do not have lone pair electrons, but share the same valence as Bi atoms. Due to the periodic boundary condition, the two adjacent building blocks in $4\times4$ Bi$_2$Te$_3$ supercell have four stepping stone sites. We replace Bi atoms at the sites one by one by Ga atoms and calculate the magnetic coupling strength, as shown in Fig. 4(a). The magnetic coupling constant decreases approximately linearly with the number of the replacements and finally vanishes when all four Bi atoms are replaced. In contrary to the Ga replaced case, the magnetic interaction remains approximately unchanged when Sb atoms substitute the stepping stone Bi atoms. These results are also qualitatively consistent with the ferromagnetism order observed in (Bi$_x$Sb$_{1-x}$$)_2$Te$_3$.[5] To further demonstrate the unique properties of the Bi atoms at the stepping stone sites, we also replace the Bi atoms that are not at the stepping stone sites, but still in the same quintuple layer, by Ga atoms and find that the ferromagnetic order preserves since the coupling constants almost do not change even when four non-stepping-stone Bi atoms are replaced. These results confirm our mechanism. Finally, to visualize the disappearance of the anti-$sp\sigma$ state when Bi at the stepping stone site is replaced, we calculate the spin density of two touching magnetic blocks with one Bi atom replaced by a Ga atom, as shown in Fig. 4(b) and Fig. 4(c). It clearly shows that the Ga atom does not have a spin polarized anti-$sp\sigma$ orbital, which still can be seen around the remaining Bi atom, as indicated by the arrow in Fig. 4(c). These results also suggest that the missing of such states can be the fundament reason to hinder the long-range magnetism in carrier-free DMS. Also note that, such a mechanism is a necessary condition for long-range magnetism and the formation energy of this configuration is required to be the global minimum to achieve DMS.
Fig. 4. (a) The ferromagnetic coupling constant with respect to the number of Ga and Sb atoms replacing the Bi atoms at the stepping stone sites (blue and brown, respectively), or at the non-stepping stone sites (red). (b) The $4\times4$ Bi$_2$Te$_3$ simulation cell with the seventh neighboring sites doped with Cr atoms and one Bi atom at the stepping stone site replaced by a Ga atom (green). (c) The spin density of two adjacent magnetic blocks with one Ga atom replacing one Bi atom at the stepping stone site. The yellow and blue colors represent the spin-up and spin-down densities, respectively. The $sp$ anti-bonding state is marked by the arrow
Our new discovery suggests that during the crystal growth stage, there exists a thermodynamic driving force that separates the magnetic dopants and forms long-range magnetic order. Of course, after the growth and during the transport measurement, other possible mechanisms, such as surface topological states, can also enhance the magnetism. Still, there are plenty of discussions[16,19] in this area and it is beyond the scope of this paper. In summary, we have found a stable carrier independent long-range ferromagnetic interaction in Bi$_2$Te$_3$ and Sb$_2$Te$_3$, which agrees well with the experimental results.[5,6,33,34] An electron hopping mechanism based on the intrinsic anti-$sp\sigma$ state is proposed to qualitatively explain this intriguing finding. Our further studies also suggest that this mechanism exists in other material systems (not published results). This discovery may greatly enhance the understanding of the hidden and carrier free long-range magnetism, lead to the discoveries of new classes of DMS materials, and widen the material choices of spintronic devices.
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