Strain-modulated ferromagnetism and band gap of Mn doped Bi2Se3

The quantized anomalous Hall effect (QAHE) have been theoretically predicted and experimentally confirmed in magnetic topological insulators (TI), but dissipative channels resulted by small-size band gap and weak ferromagnetism make QAHE be measured only at extremely low temperature (<0.1 K). Through density functional theory calculations, we systemically study of the magnetic properties and electronic structures of Mn doped Bi2Se3 with in-plane and out-of-plane strains. It is found that out-of-plane tensile strain not only improve ferromagnetism, but also enlarge Dirac-mass gap (up to 65.6 meV under 6% strain, which is higher than the thermal motion energy at room temperature ~26 meV) in the Mn doped Bi2Se3. Furthermore, the underlying mechanisms of these tunable properties are also discussed. This work provides a new route to realize high-temperature QAHE and paves the way towards novel quantum electronic device applications.

doped Bi 2 Se 3 with different strains are presented. The total energy of Mn doped Bi 2 Se 3 without strain is set as zero, namely the energies of Mn doped Bi 2 Se 3 with strains are given relative to the case without strain. The positive and negative strains respectively indicate the tensile and compressive lattice constants. When compressive strains are imposed, whether in-plane or out-of-plane strains, it usually decreases the stability of Mn doped Bi 2 Se 3 , however, the opposite situation happens for the tensile strain. From Fig. 1c, we can find that the Mn doped Bi 2 Se 3 with in-plane and out-of-plane tensile strains become more stable than that without strain. And the stability of studied system increases with strength of tensile strains (2% to 4%). With the tensile strain up to 6%, the stability of Mn doped Bi 2 Se 3 is the same as that of the case without strain. In the following discussion, we will find that this change in stability is related to the difference of electronic transport in Mn doped Bi 2 Se 3 with different tensile strains. Hence, our results show that modest in-plane or out-of-plane tensile strains will make Mn doped Bi 2 Se 3 becomes more stable. In the following discussion, we will find that this is because that strain has effect on the magnetic property of the Mn doped Bi 2 Se 3 , which furthermore change the relative stability of the Mn doped Bi 2 Se 3 with and without strain.
It is more interesting to observe that the magnetic property is modulated with strain, especially for tensile strain. In this study, we focus on the influence of strain on magnetic moment and ferromagnetic stability of Mn doped Bi 2 Se 3 . Firstly, in Fig. 1d, we provide the change in magnetic moment of Mn doped Bi 2 Se 3 with strain. For the most of Mn doped Bi 2 Se 3 with strain, the moment nearly does not alter except those with − 6% out-of-plane, − 2% in-plane, and 4% in-plane strains. For example, the moment of the case with 4% in-plane strain is about 4.8 μ B, which is almost twice larger than the case with 4% out-of-plane strain. The change in magnetic moment is usually related to the different d electrons occupation, which have been found that strains can change the d electrons occupation 31 . To show this is indeed the case. In Fig. 2, we compare the densities of states (DOSs) of Mn doped Bi 2 Se 3 under 4% out-of-plane strain with the case under 4% in-plane strain. The most obvious difference in Fig. 2a,b is the appearance of various degeneracy of Mn 3d orbitals under 4% out-of-plane and in-plane strains. For the case with 4% out-of-plane strain, the d xy , d yz and d xz orbitals is degenerated and form three fold-degeneracy t g 2 state, while d z 2 and − d x y 2 2 orbitals form twofold-degeneracy e g state. When 4% in-plane strain is applied, degeneracy of Mn 3d orbitals is fivefold, which results in larger moment.
To see whether ferromagnetism can also be tuned by strain in the Mn doped Bi 2 Se 3 , we next study the magnetic interaction between two Mn ions. As indicated in Fig. 1b, we have considered two kind of different Mn substitution sites (One Mn atom substituting a Bi at O is fixed and the other moves from A to B): in-plane (OA) and out-of-plane (OB) configurations. The magnetic interaction between the two Mn dopants at a given separation is evaluated by calculating the total energy difference between the ferromagnetic (FM) and the antiferromagnetic (AFM) configurations (∆ − E /2 FM AFM ) of the two Mn moments. The results are summarized in Fig. 3. Our results reveal that the magnetic coupling of OA and OB configurations always favors FM state under out-of-plane and in-plane strains. The FM stability of OA configuration is in the range of − 60 to − 20 meV under different strains. However, for the OB configuration with different strain, the FM stability (except − 4% and − 6% in-plane strains) is generally stronger (about − 60 to − 85 meV) than that in OA configuration. We find that the distance of two doped Mn atoms are respectively 3.97 and 3.89 Å in the OA configurations with − 4% and − 6% in-plane strains. While they are 4.40 and 4.37 Å in the OB configurations with − 4% and − 6% in-plane strains. Therefore, smaller Mn-Mn distance leads to stronger FM coupling in the OA configuration than that in the OB configuration. For the other cases with different in-plane strains from − 2% to 6%, the magnetic moments are larger than  those in the OA configuration. Thus, FM stability in OB configuration is stronger than that in the OA configuration. We have also calculated the Curie temperature under different strains within the mean-field theory (see Supplementary Fig. S1 and corresponding discussion). Usually, the T c from the mean-field theory is larger about 50% than that from Monte Carlo simulation 32 . Hence the T c of Mn doped Bi 2 Se 3 with different strains should be in the range of 30-150 K at 4.16% Mn doping. The magnetic anisotropy energy is also important to the QAHE. We have obtained the magnetic anisotropy energy (MAE) of A and B configurations. As listed in the Supplementary Table S1, the results show that out-of-plane ferromagnetism is more stable than in-plane one. This result agrees with experimental finding from Xu et al. 20 . Additionally, the FM strength can be strengthened with the change from compressive strain to tensile strain for this OB configuration. Hence, the ferromagnetic stability and tunable regularity by strain in OB configuration is better than that in OA configuration. Furthermore, we can see that tensile strains will make FM stronger than that without strain in OB configuration. Thus, we can say that modest in-plane or out-of-plane tensile strains will make the ferromagnetism of Mn doped Bi 2 Se 3 become stronger.
Above results have confirm that modest in-plane or out-of-plane tensile strains are propitious to the stability and FM strength of Mn doped Bi 2 Se 3 . We then turn to see the influence of in-plane or out-of-plane tensile strains on the band structures of the Mn doped Bi 2 Se 3 . Figure 4a shows the band structure calculated by using a 2 × 2 slab supercell of the ideal Bi 2 Se 3 surface with 6 QLs thick. As can be seen, the Fermi level (E f ) is just at the Dirac point, which is in agreement with previous results 33,34 and indicates that computational method used in this study is reasonable. After Mn is doped into Bi 2 Se 3 (Fig. 4b), a tiny Dirac gap about 3.9 meV is opened. If out-of-plane tensile strain is applied in Mn doped Bi 2 Se 3 , the Dirac-mass gap increases from 11.2 to 65.6 meV with strains as displayed in Fig. 4c-4e. Hence, the Dirac-mass gap 65.6 meV at 6% out-of-plane tensile strain is larger than that 27 meV i.e. band gap of room-temperature thermal exicition. But, in the case of in-plane tensile strain, for example 6%, our results show that the Dirac gap disappears, as shown in Fig. 4f. In fact, we calculate the band structures of Mn doped Bi 2 Se 3 with different (from 2% to 6%) in-plane tensile strains, no Dirac gap is opened. In addition, we find that E f is also shifted into the Dirac gap under 6% out-of-plane strain. This indicates that we can modulate the Dirac gap and E f by out-of-plane tensile strain.
We have calculated the Berry curvature of the OB configuration with 6% out-of-plane tensile strain. The corresponding results are shown in the Fig. 4e. The Berry curvature is obtained by integrating the occupied valence bands using the expression 35,36  where n is the band index, E n and ψ nk are the eigenvalue and eigenstate of the band n, y are the velocity operators along the x and y directions within the film plane, and f n = 1 for all n occupied bands. This integration gives the corresponding Hall conductance 36 . The Berry curvature distribution along the high symmetry directions shows a large negative peak near the Γ point and zero elsewhere else (See Fig. 4e). As a consequence, the total integration or the Hall conductance with the Fermi level lying inside the band gap must be nonzero. The system should be in a QAH state. In addition, in order to confirm the stability of Mn doped Bi 2 Se 3 under 6% out-of-plane tensile strain, we also perform the calculation of phonon dispersion curves Mn doped Bi 2 Se 3 under 6% out-of-plane tensile strain in a 6QLs 1 × 1 supercell(see Supplementary Fig. S3). It can be found that 6% out-of-plane tensile strain will not lead to any imaginary frequency dispersion for the Mn doped Bi 2 Se 3. Thus, this system should be stable in the 6% out-of-plane tensile strain.
Why can the out-of-plane tensile strain tune the Dirac gap, and why can not in-plane tensile strain Dirac gap in Mn doped Bi 2 Se 3 ? From previous study, FM exchange field can open surface gap in TIs 14 . In detail, lots of studies have proved that Bi 2 Se 3 is a TI system. Thus, Bi 2 Se 3 must be topologically nontrivial one. In such system, band structures are inverted and cross at the Dirac point (no gap is opened). And then if a sufficiently large exchange field is involved, the band inversion will be increased and leads to a topologically nontrivial band gap opened. Thus, for the Mn doped Bi 2 Se 3 without any strain, a small Dirac gap is opened due to ferromagnetic exchange field. Our above results have shown that in-plane and out-of-plane tensile strains can strengthen ferromagnetism of Mn doped Bi 2 Se 3 . It is reasonable that Dirac gap increases with out-of-plane tensile strains in Mn doped Bi 2 Se 3 . But, from Fig. 3, we can find that the ferromagnetism of OB configuration with in-plane tensile strain is stronger than that with out-of-plane tensile strain. The Dirac gap should become larger in the case of in-plane tensile strain. Then, why is Dirac gap in Mn doped Bi 2 Se 3 closed under in-plane tensile strain? In order to understand this difference, we provide band contribution analysis from different elements for the Mn doped Bi 2 Se 3 with 6% out-of-plane and in-plane tensile strains in Fig. 5. For the case with out-of-plane tensile strain, in Fig. 5a, it can be found that valence bands close to the Fermi level are mainly dominated by Bi and Se atoms, while Mn contributes mainly to the lower valence and higher conduction bands farther away from the Dirac gap. While the in-plane tensile strain is applied, from Fig. 5b, we can see that lots of hybrid states leaded by Mn, Se, and Bi will occupy some energy range around the E f . Hence, no Dirac gap is opened. It is noted that Mn doped Bi 2 Se 3 with in-plane tensile strain is metal transport, but Mn doped Bi 2 Se 3 with out-of-plane tensile strain is semiconductor. This difference can well explain why the ferromagnetism in the case with in-plane strain is stronger than that with out-of-plane strain.
In summary, by using first-principles calculations, the magnetic properties and electronic structures of Mn doped Bi 2 Se 3 with strain are explored. Our results firstly show that modest in-plane or out-of-plane tensile strains will make Mn doped Bi 2 Se 3 become more stable. Then it is found that modest in-plane or out-of-plane tensile strains will make the ferromagnetism of Mn doped Bi 2 Se 3 become stronger. Meanwhile, we find that out-of-plane tensile strain can modulates the Dirac gap and E f of Mn doped Bi 2 Se 3 . This study provides a new route to high temperature QAHE.

Methods
First-principles calculations were performed within the framework of DFT using the projector-augmented wave 37 method as implemented in the Vienna ab-initio simulation package (VASP) [38][39][40] . Generalized gradient approximation (PBE-GGA) was adopted 41 for treating the exchange correlation interactions. In our calculations we used a 2 × 2 × 1 slab supercell of Bi 2 Se 3 with 6 quintuple layers (QLs) (Fig. 1a). The atomic structures were fully optimized until the Hellmann-Feynman forces on each ion were smaller than 0.02 eV/Å. A plane-wave energy cut-off of 400 eV, a k-point grid of 5 × 5 × 1, and the Gaussian smearing method with a smearing width of 0.1 eV were used integrations over the Brillouin zone. Unless mentioned other wise, spin-orbit coupling (SOC) and GGA+ U 42 calculations with U = 4.5 eV, J = 0.5 eV are taken into account in all calculations. The viability of OA and OB configurations is also estimated by a larger 3 × 3 × 1 slab supercell of 6QLs Bi 2 Se 3 (See Supplementary  Fig. S2 and Table S2). Note that van der Waals (vdW) corrections 43 is not involved in our calculations. But our test calculations indicate that vdW corrections do not change the conclusions of this study and only have some effects on exact value of our results (See Supplementary Table S3). To simulate out-of-plane (in-plane) strains, the out-of-plane (in-plane) lattice constants c (a and b) are elongated or compressed synchronously, then a and b (c) lattice parameters is adjusted to keep the cell volume constant, as treating in some studies 44,45 . In our simulations, percent strain means the change of the lattice. For each strain case, the atom positions are fully optimized with the same force criterion as that in the case without strain.