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Axion Insulator State in a Ferromagnet/Topological Insulator/Antiferromagnet Heterostructure YUSHENG HOU, and Ruqian Wu Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.9b00047 • Publication Date (Web): 14 Mar 2019 Downloaded from http://pubs.acs.org on March 14, 2019

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Axion Insulator State in a Ferromagnet/Topological Insulator/Antiferromagnet Heterostructure Yusheng Hou† and Ruqian Wu†* †

Department of Physics and Astronomy, University of California, Irvine, CA 92697-

4575, USA

Phone number of corresponding author: +1-949-824-7640 Email address of corresponding author: [email protected]

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Abstract We propose to use ferromagnetic insulator MnBi2Se4/Bi2Se3/antiferromagnetic insulator Mn2Bi2Se5 heterostructures for the realization of the axion insulator state. Importantly, the axion insulator state in such heterostructures only depends on the magnetization of the ferromagnetic insulator and hence can be observed in a wide range of external magnetic field. Using density functional calculations and model Hamiltonian simulations, we find that the top and bottom surfaces have opposite half-quantum Hall conductances, b  xyt  e 2 2h and  xy  e 2 2h , with a sizable global spin gap of 5.1 meV opened for the

topological surface states of Bi2Se3. Our work provides a new strategy for the search of axion insulators by using van der Waals antiferromagnetic insulators along with threedimensional topological insulators. Keywords: Axion insulator state; ferromagnet; layered antiferromagnet; topological insulator; van der Waals heterostructure


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Three-dimensional (3D) topological insulators (TIs) with metallic topological surface states (TSSs) protected by the time-reversal symmetry 1, 2 have become prototypes for the realization of many emergent physical properties such as the quantum anomalous Hall effect

3-5,

quantized magneto-optical effect

5-9

and Majorana fermion state 10. One of the

most striking but underexplored phenomena is the topological magnetoelectric (TME) effect in the magnetized 3D TIs that is evoked by the topological term  in the axion electrodynamics

5, 11-20.

Simply put, this peculiar effect manifests in the so-called axion

insulators as that an external magnetic (electric) field induces an electric (magnetic) polarization 5. Such mutual controls between magnetic and electric polarizations make axion insulators promising candidates for spintronics and quantum information operations

21, 22.

Experimentally, the realization of axion insulators require a special

magnetic configuration that the magnetization over the surfaces of 3D TIs points either all inwards or all outwards so that their TSSs are globally gapped 5, 23. In practice, this is achieved by doping the two surfaces of 3D TIs with different magnetic elements, which produces different coercive fields in two sides

5, 11, 23-26.

As one sweeps the external

magnetic field, the axion insulator state (AIS) forms as the magnetizations of two surfaces align antiparallelly. Nevertheless, the coercive fields of doped TIs are not easy to control, and dopants may redistribute as the samples are aged. As a result, the AIS produced in this way is difficult to sustain. To put forward new strategies for the design of stable and controllable axion insulators, it is nature to use heterostructures that combine TIs and magnetic substrates 27-37. However, it has been revealed that the interfacial hybridization in most of such heterostructures is too strong to retain the characteristic TSSs, which are either damaged or shifted away from the Fermi level according to recent experimental and theoretical studies 30, 32, 38. It is intriguing that ferromagnetic (FM) hexagonal MnBi2Se4 (MBS) septuple layer (SL) can be fabricated on the topmost quintuple layer (QL) of Bi2Se3 (BS) film by either δ-doping or Mn and Se co-deposition, and a sizable spin gap (>50 meV) is opened for the TSSs of BS

39.

With essentially the same van der Waals multilayers for both magnetic and

topological parts, these systems are expected to be easier for growth and have better quality and thermal stability than other heterostructures. 3 ACS Paragon Plus Environment

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Here, we find that Mn2Bi2Se5 (M2BS) nonuple layer (NL) is also energetically stable, based on density functional theory (DFT) calculations. Furthermore, the M2BS NL prefers an antiferromagnetic (AFM) ordering between the two Mn layers. Nevertheless, it may introduce a spin gap of 6.2 meV for the TSSs of BS, as the two Mn layers have different distances from the BS surface. We may hence propose using MBS/BS/M2BS (or more generally FM/TI/AFM) heterostructures to make more stable AIS. In this case, external magnetic field is used only to “set” the AIS by flipping the magnetization of the FM part, rather than to “keep” the AIS as required in the conventional FM/TI/FM geometries. This phenomenon is indeed demonstrated through our DFT and model simulations. Our work introduces a new strategy for searching axion insulators to reach strong TME with very low energy consumption. We first demonstrate the advantage of the FM/TI/AFM heterostructure for the realization of AIS over the conventional FM/TI/FM heterostructure

5, 11.

In the latter case, the top

and bottom surfaces need to have antiparallelly aligned magnetizations with an external magnetic field 0 H in the range of 0 H 2c < 0 H < 0 H1c (Figure 1a). Obviously, it is rather challenging to observe the AIS in such heterostructure when the coercive field difference 0 H1c  H 2 c is small

23, 40.

Since layered AFM insulators play the same role

as FM insulators in magnetizing TSSs

30, 38,

we propose to realize the AIS in the

FM/TI/AFM heterostructure. When the FM insulator and the first-layer magnetic ions of the layered AFM insulator have opposite magnetizations (insets in Figs. 1b and 1c), the TSS of the TI film is globally gaped. Furthermore, because the layered AFM insulator has no net magnetization, its magnetic structure is not affected when sweeping the external magnetic field in a wide range. Consequently, the magnetic alignment between the FM and AFM parts can be easily reset by applying an external magnetic field 0 H >

0 H1c (H1c is the coercive field of the FM insulator) (Figure 1b) and the AIS in the FM/TI/AFM heterostructure may appear at zero external magnetic field and in a broad range 0 H < 0 H1c .


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Figure 1. (a) External field dependence of the Hall conductivity  xy of FM/TI/FM heterostructure. (b) and (c) show the external field dependence of the Hall conductivity  xy of FM/TI/AFM heterostructure. FM and AFM insulators are sketched by blue and yellow blocks, respectively. In (a), (b) and (c), the light cyan highlights the external magnetic field ranges that give rise to AIS and the insets show the magnetization configurations of AIS. Red arrows represent the magnetic moments of FM and AFM insulators.

There are two factors that are crucial for the realization of the AIS in the FM/TI/AFM heterostructure. (1) The AFM insulators should open a spin gap for the TSSs of 3D TIs but not damage their transport properties. It is nontrivial as previous studies showed that most layered AFM insulators such as MnSe 30, 38 hybridize too strongly with TIs and the Dirac points are shifted far away from the Fermi level. (2) The remaining hybridization between the top and bottom TSSs needs to be eliminated to obtain the quantized TME effect

11, 24.

This requires that the 3D TI films are thick enough. For BS, noticeable

hybridization between the two surfaces starts as the TI films is thinner than 6 QLs 41-44. In experiments, it is difficult to define the thickness of TI well, due to the existence of rotation twins

45, 46

but we recommend to use thick TI films for the observation of AIS.

From the band structures (see Figure S1), we find that the hybridization between the top and bottom TSSs is negligible across ten QLs of BS as we adopt in the present studies. For the best match with BS, we pick MBS SL and M2BS NL as the FM and layered AFM insulators, respectively. A recent experiment shown that FM MBS SL (Figure 2a) could be synthesized by depositing a MnSe bilayer onto the BS’s topmost QL

39.

DFT

calculations confirmed that MBS SL is an energetically stable FM insulator 39. Based on these findings, Our DFT calculations indicate that M2BS NL has a lower energy by 55.6 meV per atom than the MBS SL/MnSe bilayer heterostructure (see Figure S2) and we 5 ACS Paragon Plus Environment

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may hence assume that M2BS NL can be synthesized as well by depositing a MnSe bilayer onto MBS SL. After considering several different collinear spin orders and the typical noncollinear 120o-AFM order 47 (see Figure S3 and Table S1), we find that M2BS NL has a layered AFM order (Figure 3a). Our phonon and ab initio molecular dynamics simulations demonstrated that M2BS is thermodynamically stable (see Figure S4). Moreover, this layered AFM M2BS NL is an insulator (see Figure S3-h). Lastly, our DFT calculations show both MBS SL and M2BS NL have an out-of-plane magnetic anisotropy (see Figure S5).

Figure 2. (a) Side view of atom alignment at MBS/BS interface of MBS/BS/MBS. Magnetic moments of Mn in the FM MBS SL are represented by the red arrows. The vdW gap at the interface is shown by d1. (b) Band structure of MBS/BS/MBS. Left inset shows the spin-projected bands near the Fermi level and spin weights are indicated by the color bar. Right inset sketches the magnetization configuration. (c) Berry curvature of occupied bands is calculated based on the low-energy effective four-band model Eq. (1).

Figure 2b shows the band structure of MBS/BS/MBS. In consistent with previous studies 39,

the TSSs of BS are magnetized by the FM MBS SL and a large spin gap of 52.2 meV

is opened. Such a large gap is due to the large extension of the TSSs of BS into the FM MBS SL as they share common lattice structures and most chemical compositions (Figure 2a) 47. The vdW gap at the MBS/BS interface is d1=2.66 Å, which is markedly smaller than the vdW gap (about 3.20 Å) in CrI3/BS/CrI3

48.

The two-fold degenerate

bands near the Fermi level around the Γ point (left inset in Figure 2b) confirm that there is no noticeable hybridization between the top and bottom TSSs. By fitting the DFT band 6 ACS Paragon Plus Environment

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structures to the low-energy effective four-band model Eq. (1) (see Figure S6), we obtain a Chern number CN ,1  1 (Figure 2c). This nontrivial CN ,1  1 is also obtained by calculations with the Wannier90 package

49

(see Figure S7). This indicates the

applicability of the low-energy effective four-band model in the present work. Now we examine if the layered AFM M2BS NL may also magnetize the TSSs of BS. As shown by the band structure of M2BS/BS/M2BS in Figure 3b, a spin gap of 6.4 meV opens. Importantly, the Fermi level locates in the gap of the magnetized TSSs so there is no detrimental effect on the transport properties of BS in M2BS/BS/M2BS. This is completely different from the BS/MnSe heterostructure where the magnetized TSSs are shifted away from the Fermi level

30, 38.

Again, both the four-band model with fitting

parameters and Wannier90 package give a Chern number of

CN ,2  1

for

M2BS/BS/M2BS (see Figure S8 and S9). The vdW gap in M2BS/BS/M2BS is also small (d2=2.62 Å), but the spin gap of the TSSs is no as large as that in MBS/BS/MBS because of the mutual cancellation between opposite contributions from the two Mn layers. If we set M2BS NL in the FM state, the spin gap of TSSs of M2BS/BS/M2BS increases to 51.3 meV (see Figure S10).

Figure 3. (a) Side view of atom alignment at M2BS/BS interface of M2BS/BS/M2BS. Magnetic moments of Mn in the layered AFM M2BS NL are represented by the red arrows. The vdW gap at the interface is shown by d2. (b) Band structure of M2BS/BS/M2BS. Left inset shows the spinprojected bands near the Fermi level and spin weights are indicated by the color bar. Right inset sketches the magnetization configuration. (c) Berry curvature of occupied bands is calculated based on the low-energy effective four-band model Eq. (1).


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Finally, we assemble the FM-MBS/BS/AFM-M2BS heterostructure and explore the possibility of achieving the AIS. We set the magnetization of the FM MBS SL in the upward direction while the magnetization of the first Mn layer in the AFM M2BS NL in the downward direction, as shown in the right inset in Figure 4a. The band structure clearly suggests a global spin gap of 5.1 meV across the Fermi level, showing the appropriate magnetization of TSSs (Figure 4a). By projecting bands to the top and bottom QLs, we observe that the magnetized TSSs from the top and bottom surfaces are well separated (Figure 4b). Figure 4b and the left inset of Figure 4a suggest the spin components of TSSs from the top and bottom surfaces around the  point. Interestingly, the four branches of TSSs at the top and bottom surfaces sense opposite exchange fields, i.e.,  t  b  0 .

Figure 4. (a) Band structure of the AIS in MBS/TI/M2BS. Left inset shows the spin-projected four bands near the Fermi level and their Chern numbers. Right inset sketches the used magnetization configuration. Color bar indicates the spin-projected weights. (b) Top-QL-BS and bottom-QL-BS projected band structures are shown by the red dots in the left panel and the blue dots in the right panel, respectively. (c) Berry curvatures



of the occupied magnetized TSSs of top (up panel)

and bottom (bottom panel) surfaces.

By fitting the DFT-calculated band structure using the four-band model Hamiltonian Eq. (1), we obtain those: (I) the top and bottom exchange fields are  t  2.6 meV and

 b  26.8 meV, respectively; (II) the asymmetric interface potential 2V is 1.0 meV. Furthermore, the occupied TSSs from the top and bottom surfaces have opposite Berry 8 ACS Paragon Plus Environment

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curvatures, as shown in Figure 4c. Integrating the Berry curvatures of the occupied bands in the Brillouin zone gives that the Chern numbers of the top and bottom surfaces are CNt  1 2 and CNb   1 2 , respectively. In other words, the top and bottom surfaces have

opposite half-quantum Hall conductances  xyt  e 2 2h and  xyb  e 2 2h , respectively. This is the direct evidence of the AIS in MBS/TI/M2BS. This vanishing global Hall conductance  xy   xyt   xyb  e 2 2h  e 2 2h  0 is further verified by calculations using the Wannier90 package (see Figure S11). The small asymmetric interface potential 2V due to the difference between FM-MBS and AFM-M2BS appears not to affect the topological properties. Obviously, MBS/BS/M2BS can also be easily transformed to the Chern insulator state when the magnetization of MBS SL is reversed from upward to downward (see Figure S12). Mostly, their band structures of Chern and axion insulator states have very similar profiles, except the switch of spin component and Berry curvature of the occupied TSSs at the MBS/BS surface. As expected, integrating the Berry curvatures of the occupied bands in the Brillouin zone gives rise to the global Chern number CN  1 (see Figure S13). As the spin orientation of the FM layer can be easily reset, it is very convenient to switch MBS/BS/M2BS between the Chern insulator and AISs. To summarize, we demonstrated that MBS/BS/M2BS heterostructure may manifest stable AIS using DFT calculations and model simulations. As MBS and similar systems have been successfully grown in experiments

39, 50,

we believe that the fabrication of

MBS/BS/M2BS heterostructure is feasible. Our work significantly expands the range of searching axion insulators, from conventional FM/TI/FM heterostructures to FM/TI/AFM heterostructures. Methods. Density Functional Theory Calculations. Our DFT calculations are carried out by using the Vienna Ab Initio Simulation Package at level of the generalized gradient approximation

51-54.

We use the projector-augmented wave pseudopotentials to describe

the core-valence interaction

55, 56

and set the energy cutoff of 500 eV for the plane-wave 9

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expansion 54. Ten QLs of BS with lattice constant aBS=4.164 Å are used for building up the MBS/BS/MBS, M2BS/BS/M2BS and MBS/BS/M2BS slabs and the vacuum space between the adjacent slabs is 12 Å. The weak interaction across MBS/M2BS and BS layers is described by the nonlocal van der Waals (vdW) functional (optB86b-vdW) 57, 58. To consider the correlation effect among Mn 3d electrons, we employ the LSDA+U method with the on-site Coulomb interaction U=6.0 eV and the exchange interaction J=1.0 eV

30, 59.

Spin-orbit coupling is included in the calculations of all band structures.

To demonstrate the thermodynamic stability of the Mn2BS NL, we calculate its phonon dispersion, using a 4×4×1 supercell with the finite displacement method

60, 61.

Furthermore, we perform ab initio molecular dynamics simulations at 300 K for 5 ps with a time step of 2 fs, using the NVT canonical ensemble and the Nose-Hoover thermostat 62, 63.

Low-energy Effective Four-band Model Simulations. To study the topological properties, we utilize a low-energy effective four-band model

3, 64, 65.

This model consists of the

TSSs (Hsurf) of 3D TI BS, the exchange field (HZeeman) from magnetic MBS and M2BS and the interfacial potential (HInterface). With the basis set of { t ,  , t ,  , b,  , b,  } , the low-energy effective four-band model is H  k x , k y   H surf  k x , k y   H Zeeman  k x , k y   H Interface  k x , k y   0  iv k  A  kx2  k y 2    F   0   0

ivF k 0 0 0

0 0 0 ivF k

0  t 0   0  ivF k   0   0  0

0  t 0 0

0 0 b 0

0  V 0 0   0 V  0  0 0    b   0 0

0 0 V 0

0  0  (1) 0   V 

Here, t (b) denotes the top (bottom) surface state and ↑(↓) represent the spin up (down) states; vF is the Fermi velocity; kx, ky and k  k x  ik y are wave vectors;  t and  b are the exchange fields experienced by the top and bottom TSSs, respectively; 2V is the inversion asymmetric interface potential. In MBS/BS/MBS and M2BS/BS/M2BS, the inversion symmetry is kept so  t   b   and V=0. However, the inversion symmetry is broken in MBS/BS/M2BS. Berry curvatures and Chern numbers (CN) are calculated based on the formulas in Ref. 66, 67.


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ASSOCIATED CONTENT Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: xxxxxxxx

Band structure of ten-QL BS slab (Figure S1); structural comparison between M2BS NL and MBS/MnSe bilayer (Figure S2); various magnetic orders and band structure of M2BS NL (Figure S3); DFT calculated energies of the various magnetic orders of M2BS NL (Table S1); phonon dispersion and ab initio molecular dynamics simulations of M2BS NL (Figure S4); magnetocrystalline anisotropic energy of MBS SL and M2BS NL (Figure S5); band structure fitting of MBS/BS/MBS (Figure S6); Wannier90 fitted band structure and calculated Berry curvature of MBS/BS/MBS (Figure S7); band structure fitting of M2BS/BS/M2BS (Figure S8); Wannier90 fitted band structure and calculated Berry curvature of M2BS/BS/M2BS (Figure S9); band structure of M2BS/BS/M2BS in the case of M2BS with ferromagnetic ordering (Figure S10); Wannier90 fitted band structure and calculated Berry curvature of MBS/BS/M2BS (Figure S11); band structure and topological properties of Chern insulator state in MBS/BS/M2BS (Figure S12); Wannier90 fitted band structure and calculated Berry curvature of Chern insulator state in MBS/BS/M2BS (Figure S11) (PDF)

AUTHOR INFORMATION Corresponding Author *

Ruqian Wu, E-mail: [email protected]

ORCID Yusheng Hou: 0000-0001-7869-4741 Ruqian Wu: 0000-0002-6156-7874

Notes The authors declare no competing financial interest.


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Acknowledgement Work was supported by DOE-BES (Grant No. DE-FG02-05ER46237). Density functional theory calculations were performed on parallel computers at NERSC supercomputer centers.

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Topological Insulator

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