Stability, Electronic and Magnetic properties of magnetically doped topological insulators Bi{}_{2}Se{}_{3}, Bi{}_{2}Te{}_{3} and Sb{}_{2}Te{}_{3}

Stability, Electronic and Magnetic properties of magnetically doped topological insulators BiSe, BiTe and SbTe

Jian-Min Zhang Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China    Wenmei Ming Department of Materials Science and Engineering, University of Utah, Salt Lake City 84112, USA    Zhigao Huang Department of Physics, Fujian Normal University, Fuzhou 350007, China    Gui-Bin Liu School of Physics, Beijing Institute of Technology, Beijing 100081, China    Xufeng Kou Department of Electrical Engineering, University of California, Los Angeles, California 90095, USA    Yabin Fan Department of Electrical Engineering, University of California, Los Angeles, California 90095, USA    Kang L. Wang Department of Electrical Engineering, University of California, Los Angeles, California 90095, USA    Yugui Yao ygyao@bit.edu.cn School of Physics, Beijing Institute of Technology, Beijing 100081, China Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
Abstract

Magnetic interaction with the gapless surface states in topological insulator (TI) has been predicted to give rise to a few exotic quantum phenomena. However, the effective magnetic doping of TI is still challenging in experiment. Using first-principles calculations, the magnetic doping properties (V, Cr, Mn and Fe) in three strong TIs (BiSe, BiTe and SbTe) are investigated. We find that for all three TIs the cation-site substitutional doping is most energetically favorable with anion-rich environment as the optimal growth condition. Further our results show that under the nominal doping concentration of 4%, Cr and Fe doped BiSe, BiTe, and Cr doped SbTe remain as insulator, while all TIs doped with V, Mn and Fe doped SbTe become metal. We also show that the magnetic interaction of Cr doped BiSe tends to be ferromagnetic, while Fe doped BiSe is likely to be antiferromagnetic. Finally, we estimate the magnetic coupling and the Curie temperature for the promising ferromagnetic insulator (Cr doped BiSe) by Monte Carlo simulation. These findings may provide important guidance for the magnetism incorporation in TIs experimentally.

pacs:
71.20.Nr, 61.72.U-, 75.50.Pp, 73.43.-f

I Introduction

In recent years, topological insulators characterized by insulating bulk states and gapless conducting surface states have been studied intensively both theoretically and experimentally. Bernevig2006 (); Moore2007 (); Fu2007a (); Zhang2009 (); Xia2009 (); Hsieh2009 (); Hasan2010 (); Qi2011_RevModPhys () Specifically tetradymite compounds BiSe, BiTe and SbTe are found to be three-dimensional strong topological insulators with realistically large (a few hundred meV) bulk gap and simple surface electronic structure. Zhang2009 (); Xia2009 (); Chen2009 () On the other hand, even before the concept of topological insulator, great efforts were made to incorporate magnetism into these systems for potential diluted magnetic semiconductors (DMS). For example, ferromagnetism was reported in Cr doped BiSe,Haazen2012 () Mn and Fe doped BiTe Choi2005 (); Kulbachinskii2002 (); Choi2004 (); Hor2010d (); Niu2011 () and V, Cr and Mn doped SbTe. Dyck2002 (); Dyck2005 (); Chien2007 (); Choi2005 (); Choi2004 () The ferromagnetism in topological insulator will break the time-reversal symmetry, this intricate interplay between topological order and ferromagnetism aroused a few proposals to realize exotic quantum phenomena, Qi2008 (); Yu2010b (); Liu2009 (); Qi2009 (); Garate2010 (); Wray2011b (); Jin2011 (); Zhang2013 (); Kim2013 (); Lang2013 () such as, magnetoelectric effect Qi2008 () and quantum anomalous Hall effect (QAHE). Yu2010b () Experimentally the massive Dirac fermion spectrum was reported in both Mn and Fe doped BiSe surface, Chen2010i () complex spin texture was revealed in Mn doped BiTe and QAHE was recently observed in Cr(BiSb)Te film under 30 mK. Chang2013 ()

However, in experiment it is still challenging to incorporate stable ferromagnetism in the TIs aforementioned. For example, ferromagnetism in Fe doped BiTe and SbTe is hardly detected even in low temperature. Zhou2006 (); Chien2007 (); He2011 () For BiSe with Mn doping a spin glass state rather than ferromagnetic state is observed. Choi2005 () Also both antiferromagnetism Choi2011 () and ferromagnetism Haazen2012 (); Kou2012 () were observed in Cr doped BiSe. The similar controversy also exists from different groups for Fe doped BiSe. Kulbachinskii2002 (); Sugama2001 (); Salman2012 (); Choi2011 () This may be related to different magnetic atoms distribution within the host material caused by the sample preparation, such as, temperature, flux ratio, and chemical potentials of constituent atoms.

In order to clarify this issue, we systematically investigate the stability, electronic and magnetic properties of 3 transition metal (TM) elements V, Cr, Mn and Fe doped BiSe, BiTe and SbTe using DFT calculations and Monte Carlo simulations. We first assess the feasibility of magnetic doping in BiSe, BiTe and SbTe under different growth environment according to formation energy calculations. Zhang1991 (); VandeWalle2004 () The preferred site for the doping magnetic atoms and the optimal growth conditions are identified. Further the electronic band structure results show that Cr and Fe doped BiSe, BiTe, and Cr doped SbTe remain as magnetic insulator with substantially reduced band gap , while all TIs doped with V and Mn as well as Fe doped SbTe become magnetic metal. Additionally the magnetic coupling strength between magnetic atoms is studied and Curie temperature for typical concentration is estimated using Monte Carlo simulations.

This paper is organized as follows: In Sec. II we describe the method for all the calculations proceeding. In Sec. III, we first identify the native defects of BiSe, BiTe and SbTe, which may be responsible for the intrinsic non-insulating bulk states observed in experiment. Then we calculate the formation energies, electronic and magnetic properties for magnetic atom doped TIs. We additionally show the Monte Carlo simulations for the estimation of magnetic coupling strength and Curie temperature. Finally we conclude our paper with a brief summary of those findings.

Ii Method

All the first-principles calculations are performed using projected augmented wave (PAW) Blochl1994 () potentials with Perdew-Burke-Ernzerhof type generalized gradient approximation (GGA) Perdew1996 () as implemented in the Vienna ab initio simulation package (VASP). Kresse1993a (); Kresse1996 () In particular, spin orbit coupling (SOC) is explicitly included due to the strong relativistic effect in Bi and Sb elements, and the significant impact on electronic structure and formation energy, SOCnote () as also revealed by West et alWest2012a () We choose hexagonal cell with the experimental lattice constants a=4.138 Å, c=28.64 Å for BiSe; a=4.383 Å, c=30.487 Å for BiTe and a=4.250 Å, c=30.35 Å for SbTe. The cutoff energy for the plane wave expansion of electron wavefunction was set at 300 eV. A gamma-centered k-mesh was adopted to sample the Brillouin zone for BiSe, BiTe and SbTe supercells as illustrated in Fig. 1. As calculating the energies of charged defects/dopants, a jellium background charge is added. All atoms in each doped supercell are fully relaxed through the conjugate-gradient algorithm until the residual force on each atom is less than 0.02 eV/Å. The numerical errors of calculated formation energy are controlled to be less than 20 meV.

Figure 1: (color online). (a) Crystal structure Illustration of a supercell for modeling a single dopant in bulk BiSe, with (b) for side view and (c) for top view. (d) Brillouin zone and high symmetry points of the supercell. There are two types of Se atoms, including Se1 involving the van der Waals bonding and the other Se2.

The formation energy of defect or impurity in charge state as a function of the Fermi energy and the chemical potential of atom is defined as VandeWalle2004 ()

(1)

where (defect+host) is the total energy of a supercell of host material with one defect or impurity in charge state , and (host only) is the total energy of the equivalent supercell containing only pure host. denotes the chemical potential for species (host atoms or dopants), and indicates the corresponding number that have been added to () or removed from () the supercell. Here, it is noted that is given with respect to the value of solid phase , i.e., the absolute value of the chemical potential . is the Fermi energy, referenced to the valence band maximum (VBM) of the pure host crystal . is a potential alignment due to different energy references in defect containing supercell and pure supercell in DFT calculations.

The chemical potentials depend on the experimental growth condition. The values of are determined as follows, as taking BiSe for example, first, and to avoid precipitation of solid elements. To maintain equilibrium growth of BiSe, it requires . Here, (BiSe) is the formation energy of BiSe. Furthermore, to ensure that the competing phases can not precipitate, where is the dopant atom, i.e., V, Cr, Mn and Fe in the paper.

is a function of charge q and Fermi energy, then we can determine the transition energy as the Fermi energy at which , i.e., where the charge state of defect spontaneously transforms from to . The concentration of defects or dopants at growth temperature under thermodynamic equilibrium can be estimated from VandeWalle2004 ()

(2)

where N is the number of sites that can be occupied in the lattice (per unit volume), is defined in Eq.(1) and is Boltzmann constant.

Iii Results and Discussions

iii.1 Native defects

Figure 2: (color online). The formation energy as a function of anion chemical potential for all the possible intrinsic defects in (a) BiSe, (b) BiTe and (c) SbTe. , , and stand for bismuth vacancy, antimony vacancy, selenium vacancy and tellurium vacancy, respectively, while Bi, Bi, Sb, Se and Te are antisites defects. and are labeled to distinguish Se (Te) in different layers. -rich ( for Bi, Sb, Se or Te) indicate the extreme growth condition with taking the maximum value in Eq.(1). Vertically dotted lines highlight the boundary of carrier types.
-type tendency -type tendency
Theory Experiment Theory Experiment
BiSe Our111most growth conditions,, Ref. West2012a,; Wang2012-arXiv, Ref. Xia2009,; Hor2009,; Hsieh2009,; Wang2010,; Urazhdin2004, Our222extremely Bi-rich condition,, Ref. Wang2012-arXiv,
BiTe Our333Te-rich condition,, Ref. West2012a,; Hashibon2011, Ref. Chen2009,; Giani1999,; Yoo2005,; Wang2011b,; Urazhdin2004, Our444Bi-rich condition., Ref. West2012a,; Hashibon2011, Ref. Chien2007,; Lee2008,; Wang2011b,
SbTe Our, Ref. West2012a, Ref. Giani1999,; Gasenkova2001,; Jiang2012,
Table 1: A list of theoretical and experimental reports of carrier type tendency in pure BiSe, BiTe and SbTe.

Experimentally, BiSe, BiTe and SbTe are always dominated by conducting bulk carriers rather than being insulating even though they are all intrinsically narrow-band semiconductors. This is related to the unintentional doping induced by native defects. BiSe often shows -type conductivity and is difficult to be tuned into -type via compensation doping, Xia2009 (); Hor2009 (); Hsieh2009 (); Wang2010 (); Urazhdin2004 () while SbTe shows strong -type tendency. Giani1999 (); Gasenkova2001 (); Jiang2012 () For BiTe, it is reported to be either -type Giani1999 (); Yoo2005 (); Chen2009 (); Urazhdin2004 () or -type Chien2007 (); Lee2008 () depending on the growth method and environment. We then identify how the carrier type varies with the chemical potentials. The most possible native point defects including atom vacancies and antisites defects are considered. The formation energy versus chemical potential is plotted in Fig. 2.

As shown in Fig. 2(a), donor-like defects and Se dominate in BiSe in the most range of growth conditions according to their lowest formation energies among all the native point defects, as also revealed by Ref. Wang2012-arXiv, ; West2012a, . This will lead to an intrinsic -type doping as experimentally observed. In the extreme Bi-rich condition, acceptor-like defect Bi will be preferred and the resulting doping will be -type.

For BiTe in Fig. 2(b), antisite defects Bi and Te are more preferred than other native point defects. We identify that acceptor-like Bi is likely to appear in Bi-rich condition and donor-like Te in Te-rich condition, leading BiTe to be intrinsic -type and intrinsic -type, respectively. Our result explains the experimentally reported native - amphoteric type conductivity of BiTeGiani1999 (); Yoo2005 (); Chen2009 (); Urazhdin2004 (); Chien2007 (); Lee2008 () Our result agrees with Ref. West2012a, , while calculation without the inclusion of SOC gives rather different values of formation energy. Hashibon2011 () Experimentally Ref. Wang2011b, reported the co-existence of Te antisite and Bi antisite, rendering BiTe to be either -type tendency or -type tendency. This result further confirms our predication.

For SbTe in Fig. 2(c), we find that antisite defect Sb is dominant with the lowest formation energy in most range of the growth conditions especially in Sb-rich condition. This can be explained qualitatively by the similar atomic sizes of Sb atom and Te atom. As the growth atmosphere evolves to be extremely Te-rich, antimony vacancy becomes to be the most energetically stable. Note that both Sb and are acceptor-like defects, SbTe is thus always intrinsic -type. West2012a (); Gasenkova2001 (); Jiang2012 () Our results provide an important guidance to carrier tuning in BiSe, BiTe and SbTe as well as intrinsic carrier environments for magnetic doping. Meanwhile, these findings also provide a clear explanation to experimental reports, as listed in Table 1.

iii.2 Formation energies of magnetic doping in BiSe, BiTe and SbTe

Figure 3: (color online). (a) Possible interstitial sites for Fe in BiSe after relaxation. (b) The formation energy as a function of anion chemical potential () for interstitial Fe and Fe substitutional for Bi in BiSe. Fe, Fe, and Fe stand for different sites for interstitial Fe in (a), while Fe denotes that Bi atom is doped by Fe atom.
Figure 4: (color online). Calculated formation energies of the most stable configurations of single V, Cr, Mn, and Fe impurities doped (a) BiSe, (b) BiTe and (c) SbTe as a function of the host element chemical potentials.

In this section, we will calculate the formation energies for the incorporation of TM atoms into the three TIs. First, the site preference of TM atom (substitutional or interstitial site) is studied. Notice that intercalated and interstitial sites were reported to be preferred for Cu in BiSe. Wang2011a () We have considered all the possible interstitial sites in bulk BiSe, including interstitial sites between different layers (intercalated sites) and interstitial sites on the same layer. Relaxed structures indicate that all the interstitial atoms are relaxed to the three main sites, as shown in Fig. 3(a). Formation energies for both substitution and interstitial cases are shown in Fig. 3(b). We find that Bi substitutional site is strongly preferred West2012 () regardless of the changes of growing condition, as compared to all the possible interstitial sites. Our results are well in line with recently experimental findings. West2012 (); Song2012 () Then in the following we will mainly focus on cation substitutional doping.

Formation energy of TM doping as a function of chemical potential is shown in Fig. 4. Similar to TM doping in BiSe, Zhang2012 () the formation energies exhibit the same size effect in BiTe and SbTe, that is, the formation energy is lowest for V atom doping while highest for Fe atom doping. It is attributed to the closest atom radius of V atom to the substituted Bi or Sb atom than other dopants. Also we find that V and Cr have negative formation energies in BiSe and SbTe for the whole range of chemical potential, indicating the doping of them can occur spontaneously. Recently, heavy Cr doping of BiSe with the concentration up to 23 was reported Liu2012 () and AFM measurement indicated Cr atoms of 20 doping concentration were uniformly distributed. Kou2012 () For SbTe, even in SbCrTe, Cr atoms can homogeneously distribute without clustering. Chien2007 () However, in BiTe, only V can be spontaneously doped. Also it’s rather different that the formation energies of Mn and Fe doping are positive values in all three TIs, suggesting the doping of them is not spontaneous except Mn in BiSe at extremely Se-rich atmosphere. Indeed, Hor et al. showed that 9% Mn can substitute for Bi atoms with randomly distributing in BiTeHor2010d () Fe is confirmed even more difficult to be doped in BiSe with the effective doping concentration less than 2%. Cha2010 () Notice that BiTe has entire higher formation energies for all these dopants than that in BiSe and SbTe, suggesting it is relatively more difficult to dope those atoms in BiTe.

Figure 5: (color online). Calculated defect formation energy as a function of Fermi energy for magnetic doped BiSe under (a) Bi-rich condition and (b) Se-rich condition; BiTe under (c) Bi-rich condition and (d) Te-rich condition and SbTe under (e) Sb-rich condition and (f) Te-rich condition. Here, is referenced to the valence-band maximum () in the bulk. The shaded areas highlight band gaps, where can range, as we mainly focus, from VBM () to CBM (). Calculated values of band gaps with SOC for BiSe, BiTe and SbTe are 0.32, 0.15 and 0.12 eV, respectively. Solid dots denote the defect transition energies between different charge states.

From section III A, we have known that the native defects are responsible for the various intrinsic carriers doping. Such carrier environment is expected to affect the formation energy of magnetic dopant with nonzero charge state according to Eq.(1). Then we study the possible charge states by calculating the formation energy as a function of Fermi energy, as shown in Fig. 5.

(i) For BiSe, BiTe and SbTe, anion-rich growth conditions (Se-rich or Te-rich) with lower formation energies are revealed better than cation-rich conditions (Bi-rich or Sb-rich) for magnetic atoms doping, which is consistent with experimental reports. Chien2007 (); Song2010 ()

(ii) For BiSe, we find that as the Fermi energy ranges from VBM at 0.0 (left edge of the shaded area) to CBM at (right edge of the shaded area), V, Cr, Mn and Fe atoms are almost stable with charge state of , i.e., neutral substitute for Bi atoms, which indicate that dopants do not introduce free carriers to the host materials. This result agrees with theoretical study from Larson et al. a-Larson2008 () and has been experimentally confirmed in Cr doped BiSeKou2012 () Although, as shifts very close to CBM, i.e., under extremely -type condition, dopants tend to act as acceptors with valency, especially Mn and Fe. Experimentally, Mn was indeed found to show hole doping effect in BiSeChen2010i ()

(iii) For BiTe, the formation energies of TMs are larger than that in BiSe or SbTe both at Bi-rich and Te-rich conditions. Mn and Fe can be neutral doped in very -type conditions. Mostly, Mn tends to act as an acceptor (Mn) with valence state Mnin BiTe. This agrees well with the experimental result from Hor et alHor2010d ()

(iv) For SbTe, a similar size effect among V, Cr, Mn and Fe dopants is observed. V and Cr with negative formation energies can be spontaneously incorporation under Te-rich atmosphere. Cr can be neutrally doped (CrDyck2005 (); Chien2007 () in the most range of Fermi level.

Figure 6: Schematic representation of the thermodynamic transition levels for magnetic doped (a) BiSe, (b) BiTe, and (c) SbTe, which corresponding to the solid dots in Fig. 5. The transition levels shown are referenced to valence-band maximum (VBM) in the bulk. The distances of to VBM, for example, indicate the thermal ionization energy of simple acceptors, respectively. The shaded area highlight band gaps of BiSe, BiTe and SbTe, as shown in Fig. 5.
Figure 7: (color online). SOC band structures for relaxed V, Cr, Mn and Fe doped BiSe, BiTe and SbTe with the nominal doping concentration of 4 (x=0.083). The size of red dots denotes the contribution of TM-d states.
System Cr Fe
gap gap (+U) (+U) gap gap (+U) (+U)
BiSe 0.010 eV 0.025 eV 2.94 2.99 0.028 eV 0.028 eV 4.73 4.99
BiTe 0.017 eV 0.019 eV 2.93 2.98 0.0024 eV 0.041 eV 4.17 4.09
SbTe 0.077 eV 0.100 eV 3.06 3.16 0 0 4.17 4.03
Table 2: Relaxed band gaps and magnetic moments () of Cr and Fe doped BiSe, BiTe and SbTe with SOC. The results of GGA+U with U=3 eV and J=0.87 eV are also listed.

From Fig. 5 we can easily determine the thermodynamic transition level from one charge state to another for different dopants, which can be observed in deep-level transient spectroscopy (DLTS) experiments or temperature-dependent Hall measurements. VandeWalle2004 () We schematically illustrate them in Fig. 6. BiSe is revealed above to be mostly intrinsic -type, nevertheless, from Fig. 6(a) we can see that V, Cr, Mn and Fe are neutral stable in most carrier environment for BiSe, even in -type condition ( near to CBM). From Fig. 6(b) for BiTe, it is indicated that under -type condition, only V and Cr can neutrally substitute for Bi atom, while Mn and Fe are energetically stable with charge state of . Conversely, neutral substituting is more likely to appear by V, Cr and Fe doping than Mn in -type BiTe. From Fig. 6(c) for SbTe, as compared to V, Mn and Fe, Cr is especially deep and more difficult to be ionized from charge state Cr to Cr, suggesting Cr is the best candidate atom for the realization of QAHE in SbTe. Actually, Cr has already been experimentally confirmed can substitute for Sb with Crin SbTe. Dyck2005 (); Chien2007 ()

iii.3 Electronic structure of magnetically doped BiSe, BiTe and SbTe

To elaborate the electronic properties with magnetic atoms introduced in BiSe, BiTe and SbTe, we further calculate the band structures for all those magnetically doped TIs and they are shown in Fig. 7. In Fig. 7, additional states appear in the band gaps of TM doped BiSe, BiTe and SbTe and give rise to semiconducting or metallic ground states, comparing to the pure host materials. Zhang2009 (); Zhang2012 (); Zhang2010 () The plots of TM-d orbital projected band structures show the states near the band gaps are from sizable hybridization between TM states and p states of host materials, most obvious at the Gamma point of the Brillouin zone. The results indicate that V and Mn doped BiSe, BiTe and SbTe are metals, as shown in Figs. 7(a)-7(c) and Figs. 7(g)-7(i), respectively. However, Cr doped BiSe exhibits an insulating magnetic state with the energy gap 0.01 eV. Compared to pure BiSeZhang2012 () inverted bands are remain observed in the doped system, indicating the topological nontrivial property. Accounting to our calculations, we get similar results in Cr doped BiTe with the band gap of 0.017 eV and Cr doped SbTe with a larger band gap of 0.077 eV. From Figs. 7(j)-(l), we find that Fe doped BiSe manifests insulating behavior with the band gap of 0.028 eV, whereas magnetic moments of Fe doped BiTe and SbTe are less than 5 , rendering Fe doped BiTe to be semi-metal with a narrow gap 0.0024 eV and Fe doped SbTe to be gapless. The phenomenon of gap closing may lead to a topological phase transition. Jin2011 (); Zhang2013 (); Kim2013 () The resulting values of band gaps and magnetic moments upon doping are listed in Table 2. In order to investigate the effect of electron-electron correlation on band gap and magnetic moment, we further perform GGA+U calculations with U ranging from 3 to 6 eV and J=0.87 eV. We find only slight modification of the band gaps and magnetic moments.

Notice that the band gaps are 0.32 eV, 0.15 eV and 0.12 eV for BiSe, BiTe and SbTe respectively. However we find the band gaps are substantially reduced to several meV upon doping. This result hints that QAHE should be observed under low temperature in magnetically doped BiSe family, which is consistent with recent experimental reports. Chang2013 () In order to uncover the reason which causes this band gap reduction, we study the effect of structural relaxation on the band gap. In Fig. 8, we show the schematic structures of doped BiSe before and after relaxation. The structural relaxation shows Se atoms neighboring to dopants move inward to the dopants by sizable distances (See Table 3), as consistent with Ref. a-Larson2008, . This suggests that hybridization between TM dopant and the neighboring Se will be strengthened and thus the impurity bands may be broadened. Zhang2012 () Specifically as reported in our previous paper, Zhang2012 () calculated band gaps of Cr and Fe doped BiSe without relaxation are 0.28 and 0.18 eV, respectively. While after structural relaxation, the band gaps are reduced to 0.01 and 0.028 eV, respectively. GGA+U calculation for the relaxation gives essentially the same results shown in Table 3 as only GGA calculation. Additional SOC relaxations for Cr and Fe doped BiSe suggest the relaxed distances change within only the order of 0.001 Å comparing to non-SOC cases. We thus conclude that the band gap reduction is induced by Se-dopant hybridization.

Figure 8: (color online). Dynamics illustration of the magnetically doped BiSe. Note that those Se atoms neighboring to the dopants tend to close to the dopants, while Bi atoms hardly move.
Bond length Cr Fe
(+U) (+U)
  0.009 0.009 0.000 0.002
  -0.342 -0.305 -0.404 -0.360
  -0.390 -0.362 -0.239 -0.263
Table 3: Amplitude of the bond length variation (in Å) from GGA and GGA+U calculations for Cr and Fe doped BiSe with all the atoms fully relaxed. Here, , where negative value means a decrease of the bond length.

iii.4 Magnetic properties of magnetically doped BiSe

Configuration (, ) (Å) (meV)
(0, 1) 4.138 22.0
(0, 2) 4.456 13.8
(0, 3) 5.785 8.7
(0, 4) 6.081 -3.8
(0, 5) 7.113 5.5
(0, 6) 7.167 2.1
(0, 7) 7.355 -1.8
Table 4: Calculated magnetic coupling strength [] between two substitutional Cr atoms with different distances for BiCrSe (). Two Bi atoms are replaced by dopant atoms at site =0 and site (= 1, 2, 3…) in a supercell of BiSe, which gives distinct configurations. The first seven nearest neighbor configurations of Cr atoms are considered as listed. Farther neighbor configurations are ignored for their large distances of Cr atoms [ of configuration (0, 8) is already larger than 9 Å] and little contributions to the magnetic coupling strengths (less than 2 meV).

As proposed in Ref. Yu2010b, , both insulator and ferromagnetism are required to realize QAHE. After possible candidates have been achieved, we then further investigate the feasibility of establishing ferromagnetism for Cr and Fe in the most promising and concerned TI BiSe, which has the largest band gap among all the discovered TIs. First, the magnetic ground state of single magnetic dopants in BiSe is identified. We have tried different initial directions of the magnetic moments. The results indicate that both Cr and Fe prefer the direction perpendicular to the BiSe quintuple layers (axis 001). The magnetic moments are about 3 for Cr and 5 for Fe, respectively. The magnetization anisotropy energies for Cr and Fe are about 7 meV and 16 meV, respectively. Then, we investigate the magnetic coupling between the two TM dopants. The favored magnetic state [either ferromagnetic (FM) or anti-ferromagnetic (AFM)] is studied by calculating the total energy difference of the two configurations at the same TM-dopant separation. Our calculations indicate that weak AFM is favorable in Fe doped BiSe, while the magnetism is experimentally difficult to be observed Kulbachinskii2002 (); Sugama2001 () due to the scarce effective doping concentration. Zhang2012 () Our previous work Zhang2012 () found that Cr doped BiSe prefers to be FM state, which has been also predicted by Lu et alLu2011 () and confirmed by some recent experiments. Liu2012 (); Haazen2012 (); Kou2012 () The magnetic coupling strengths [] for two Cr atoms within the same QL at the first three nearest neighboring distances are on the order of 10 meV. In our calculations, an appropriate supercell of BiSe was employed. Additional calculations from a larger supercell of BiSe indicate that magnetic coupling strengths only change within 2 meV. From the QAHE point of view, a spontaneous FM ground state is required. We therefore carried out Monte Carlo simulations MonteCarloSimulationinStatisticalPhysics-AnIntroduction (); Wu2007 () to determine the Curie temperature () in Cr doped BiSe. (=20) BiSe cells with periodic boundary conditions are used. Then the magnetic Cr atoms are randomly distributed on the Bi lattice sites of BiSe with the ratio of Cr:Bi to be , where in the simulation. The Heisenberg Hamiltonian of the system is described as MonteCarloSimulationinStatisticalPhysics-AnIntroduction ()

(3)

where is the exchange coupling constant between the th and th dopant atoms, taken from the first-principles calculations as shown in Table IV. The thermodynamic magnetization per atom can be calculated by , where N is the number of the magnetic dopant atoms, and is the statistical average over different states which are generated during the Markov process. MonteCarloSimulationinStatisticalPhysics-AnIntroduction () To define the Curie temperature, an accumulation of magnetization of the fourth order (Binder-cumulant) are calculated by . Binder1981and1987 (); Schliemann2001 (); Fukushima2004 () Fig. 9 shows the simulated magnetization and as a function of temperature for BiCrSe with . We get an estimated Curie temperature at about 76 K.

Figure 9: Monte Carlo simulation of BiCrSe with . is the simulated magnetization and , Binder cumulant, is the normalized fourth-order cumulant of the magnetization. The Curie temperature is estimated at about 76 K.

Iv Summary

In summary, we systematically studied the stability, electronic and magnetic properties of magnetically doped topological insulators BiSe, BiTe and SbTe using first-principles calculations in combination with Monte Carlo simulation. Our calculations showed that cation site substitutional doping was energetically most favorable. Further we suggested a recipe of effective magnetic doping for experimental study with the optimal growth conditions. In addition, our results indicated that under the nominal doping concentration of 4%, Cr and Fe doped BiSe, BiTe, and Cr doped SbTe were remain insulators, although the band gaps were substantially reduced due to Se-dopant hybridization. Instead, all TIs doped with V and Mn as well as Fe doped SbTe became metals. Finally, we explored the magnetic coupling between dopants, suggesting FM was favorable in Cr doped BiSe while AFM in Fe doped material. Using Monte Carlo simulation, we estimated that the Curie temperature of 7.4% Cr doped BiSe was about 76 K. Our results provide important guidelines towards further experimental efforts of incorporating magnetism in TI, in particular for the realization of QAHE based on magnetic topological insulators.

Acknowledgements.
We would like to thank W.G. Zhu, D. Xiao and F. Yang for the helpful discussions. This work was supported by the MOST Project of China (Grants Nos. 2014CB920903, 2011CBA00100) , the NSFC (Grant Nos. 11174337, and 11225418), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grants No. 20121101110046). Z.H. was supported by NSF of China (Grants No. 11004039), National Key Project for Basic Research of China under Grant No. 2011CBA00200. We acknowledge support from Supercomputing Center of Chinese Academy of Sciences.

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