Introduction

Topological insulators (TIs) have attracted tremendous attention because of its robust surface states, which are protected by time-reversal symmetry (TRS) and induced by strong spin-orbit coupling (SOC)1,2,3,4,5,6,7,8. In particular, there is great interest in the effect of magnetic doping, which is considered to be an efficient way to open an energy gap in the Dirac cone by breaking TRS9,10,11, resulting in the emergence of massive Dirac fermions. This gap is expected to give rise to many interesting phenomena, such as Majorana fermion12, topological magnetoelectric effect13. Especially in a TI thin film, ferromagnetic order can also be established, which potentially could realize the quantum anomalous Hall state (QAHE)14,15.

Breaking the TRS of a suitable TI film by introducing ferromagnetism can naturally lead to the QAHE15. But, a crucial issue in the realizing of QAHE is how to obtain magnetic and insulated TIs. A conventional strategy to produce magnetism in TIs is by magnetic doping, similar to diluted magnetic semiconductors (DMS)16. Ferromagnetic TIs via magnetic doping have been reported in the past few years, including Fe doped Bi2Te39,17, V, Cr and Mn doped Sb2Te318 and Mn and Cr doped Bi2Se319,20. But, QAHE has been challenging to realize in transition metal doped TIs. Recently, the first experimental observation21 and further approve of QAHE in Cr or V doped (Bi, Sb)2Te3 have been reported22,23,24. A zero field Hall resistance exhibits a distinct plateau with the quantized value h/e2 in these experiments, which confirms the existence of the QAHE. However, as shown in their experiments, above 1 K, dissipative channels from surface states and bulk bands are thermally activated, which can smear out the transport properties of the QAHE edge states. In addition, low Curie temperature also influences the observation of QAHE. Hence, strong ferromagnetic TIs with large-size band gap need to be developed in order to realize high temperature QAHE, which will be the prerequisite to the development of low-power-consumption electronics based on QAHE in TIs.

It is well known that strain is a very important factor from the film growth and application aspect of nanodevices. This is because that strain is not avoidable during epitaxial growth if there is a lattice mismatch between grown nanostructure and substrate. Additionally, strain has also become a routine factor to engineer band gaps and influence some important properties such as magnetism and so on25,26. Recently, Strain modulated topological insulator properties of graphene/TI hybrid systems are also presented in the some works27,28. Here, we make a systematically study of the strain effects on magnetic property and electronic structures of Mn doped Bi2Se3 using first-principles calculations. We first show that the energetic stability and ferromagnetism of Mn doped Bi2Se3 can be tuned by out-of-plane tensile strain. Furthermore, out-of-plane tensile strain can enlarge the Dirac-mass gap of Mn doped Bi2Se3. The mechanisms of these tunable properties are also analyzed.

Results and Discussion

We start from studying the effect of strain on stability of Mn doped Bi2Se3. Experiments18,20 and theoretical calculations29,30 have suggested that the magnetic dopants, such as V, Cr, Mn and Fe, will mostly substitute the Bi or Sb ions in 3D TIs; we therefore concentrate on this situation. As shown in Fig. 1c, the total energies of Mn doped Bi2Se3 with different strains are presented. The total energy of Mn doped Bi2Se3 without strain is set as zero, namely the energies of Mn doped Bi2Se3 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 Bi2Se3, however, the opposite situation happens for the tensile strain. From Fig. 1c, we can find that the Mn doped Bi2Se3 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 Bi2Se3 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 Bi2Se3 with different tensile strains. Hence, our results show that modest in-plane or out-of-plane tensile strains will make Mn doped Bi2Se3 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 Bi2Se3, which furthermore change the relative stability of the Mn doped Bi2Se3 with and without strain.

Figure 1
figure 1

Energetics and magnetic behavior of Mn doped Bi2Se3 under strain.

(a) A 2 × 2 × 1 supercell of 6QLs Bi2Se3 used in the calculation. (b) Only 1 of 6QLs is used to clarity the doped configurations. One Mn is fixed at O and the others move from A to B. (c) Total energies relative to the case without strain versus in-plane and out-of-plane strains. (d) Moments versus in-plane and out-of-plane strains.

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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 Bi2Se3. Firstly, in Fig. 1d, we provide the change in magnetic moment of Mn doped Bi2Se3 with strain. For the most of Mn doped Bi2Se3 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 μB, 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 occupation31. To show this is indeed the case. In Fig. 2, we compare the densities of states (DOSs) of Mn doped Bi2Se3 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 , and orbitals is degenerated and form three fold-degeneracy state, while and orbitals form twofold-degeneracy state. When 4% in-plane strain is applied, degeneracy of Mn 3d orbitals is fivefold, which results in larger moment.

Figure 2
figure 2

Total and partial densities of states of Mn doped Bi2Se3.

(a) with 4% out-of-plane. (b) 4% in-plane strains.

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To see whether ferromagnetism can also be tuned by strain in the Mn doped Bi2Se3, 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 () 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 Tc from the mean-field theory is larger about 50% than that from Monte Carlo simulation32. Hence the Tc of Mn doped Bi2Se3 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 Bi2Se3 become stronger.

Figure 3
figure 3

Magnetic coupling between two Mn atoms versus in-plane and out-of-plane strains in the OA and OB configurations.

Configurations are shown in the Fig. 1b.

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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 Bi2Se3. We then turn to see the influence of in-plane or out-of-plane tensile strains on the band structures of the Mn doped Bi2Se3. Figure 4a shows the band structure calculated by using a 2 × 2 slab supercell of the ideal Bi2Se3 surface with 6 QLs thick. As can be seen, the Fermi level (Ef) is just at the Dirac point, which is in agreement with previous results33,34 and indicates that computational method used in this study is reasonable. After Mn is doped into Bi2Se3 (Fig. 4b), a tiny Dirac gap about 3.9 meV is opened. If out-of-plane tensile strain is applied in Mn doped Bi2Se3, 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 Bi2Se3 with different (from 2% to 6%) in-plane tensile strains, no Dirac gap is opened. In addition, we find that Ef is also shifted into the Dirac gap under 6% out-of-plane strain. This indicates that we can modulate the Dirac gap and Ef by out-of-plane tensile strain.

Figure 4
figure 4

Band structures of Mn doped Bi2Se3 under strain.

(a) pure Bi2Se3, (b) Mn doped Bi2Se3 without strain, (c–e) Mn doped Bi2Se3 with different out-of-plane tensile strains and (f) Mn doped Bi2Se3 with 6% in-plane tensile strain. Circle symbols: corresponding Berry curvatures along the high-symmetry directions.

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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 expression35,36

where n is the band index, En and are the eigenvalue and eigenstate of the band n, and are the velocity operators along the x and y directions within the film plane and fn = 1 for all n occupied bands. This integration gives the corresponding Hall conductance36. 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 Bi2Se3 under 6% out-of-plane tensile strain, we also perform the calculation of phonon dispersion curves Mn doped Bi2Se3 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 Bi2Se3. 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 Bi2Se3? From previous study, FM exchange field can open surface gap in TIs14. In detail, lots of studies have proved that Bi2Se3 is a TI system. Thus, Bi2Se3 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 Bi2Se3 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 Bi2Se3. It is reasonable that Dirac gap increases with out-of-plane tensile strains in Mn doped Bi2Se3. 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 Bi2Se3 closed under in-plane tensile strain? In order to understand this difference, we provide band contribution analysis from different elements for the Mn doped Bi2Se3 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 Ef. Hence, no Dirac gap is opened. It is noted that Mn doped Bi2Se3 with in-plane tensile strain is metal transport, but Mn doped Bi2Se3 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.

Figure 5
figure 5

The character of band structures of Mn doped Bi2Se3.

(a)with 6% out-of-plane strains. (b) Mn doped Bi2Se3 with 6% in-plane strain, obtained by projecting Kohn-Sham states to the local orbitals on a single atom of each type.

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In summary, by using first-principles calculations, the magnetic properties and electronic structures of Mn doped Bi2Se3 with strain are explored. Our results firstly show that modest in-plane or out-of-plane tensile strains will make Mn doped Bi2Se3 become more stable. Then it is found that modest in-plane or out-of-plane tensile strains will make the ferromagnetism of Mn doped Bi2Se3 become stronger. Meanwhile, we find that out-of-plane tensile strain can modulates the Dirac gap and Ef of Mn doped Bi2Se3. 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 wave37 method as implemented in the Vienna ab-initio simulation package (VASP)38,39,40. Generalized gradient approximation (PBE-GGA) was adopted41 for treating the exchange correlation interactions. In our calculations we used a 2 × 2 × 1 slab supercell of Bi2Se3 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+U42 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 Bi2Se3(See Supplementary Fig. S2 and Table S2). Note that van der Waals (vdW) corrections43 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 studies44,45. 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.

Additional Information

How to cite this article: Qi, S. et al. Strain-modulated ferromagnetism and band gap of Mn doped Bi2Se3. Sci. Rep. 6, 29161; doi: 10.1038/srep29161 (2016).