Manifestation of axion electrodynamics throughmagnetic ordering on edges of a topological insulatorYea-Lee Leea,1, Hee Chul Parkb,1,2, Jisoon Ihma,3, and Young-Woo Sonb,3

aDepartment of Physics and Astronomy, Seoul National University, Seoul 08826, Korea; and bKorea Institute for Advanced Study, Seoul 02455, Korea

Contributed by Jisoon Ihm, August 10, 2015 (sent for review June 17, 2015)

Because topological surface states of a single-crystal topolog-ical insulator can exist on all surfaces with different crystalorientations enclosing the crystal, mutual interactions amongthose states contiguous to each other through edges can leadto unique phenomena inconceivable in normal insulators. Herewe show, based on a first-principles approach, that the dif-ference in the work function between adjacent surfaces withdifferent crystal-face orientations generates a built-in electricfield around facet edges of a prototypical topological insulatorsuch as Bi2Se3. Owing to the topological magnetoelectric cou-pling for a given broken time-reversal symmetry in the crystal,the electric field, in turn, forces effective magnetic dipoles toaccumulate along the edges, realizing the facet-edge magneticordering. We demonstrate that the predicted magnetic order-ing is in fact a manifestation of the axion electrodynamics inreal solids.

topological insulator | electronic structure | topological magnetoelectriceffect | axion electrodynamics | magnetic ordering

Atopological insulator (TI) hosts topologically protectedmetallic surface states on its boundaries between inner in-

sulating bulk and outer vacuum that can exist on all of the sur-faces with different crystal orientations enclosing the crystal (1, 2).Typically, the protected surface state has the relativistic masslessdispersion relation around the time-reversal invariant momenta inthe surface Brillouin zone, although its detailed features dependon surface characteristics (3–7). For example, the well-known TIswith the rhombohedral crystal structure such as Bi2Se3, Bi2Te3,and Sb2Te3 (8–10) have stacked quintuple layers along the ð111Þdirection and the low-energy surface state on the ð111Þ surfaceis isotropic in momentum space (9), whereas other surfaceshave quite anisotropic dispersions (3–7). Besides changes inits low-energy electronic dispersions, different facets in a singlecrystalline TI would have many different physical propertiesdepending on their orientations, and the facet-dependent workfunction (11, 12) is one interesting example among them. Inthe TIs mentioned above, such effects will be amplified becauseof their layered structure— surface atomic and electronic den-sities vary a lot depending on whether the surface is terminatedalong the layer or not.Although the physical properties of topological states on

a specific facet of 3D TIs have been studied intensively(1–10), mutual interactions among those contiguous to eachother through edges have not yet been examined well. A trivialexample is the coupling between two massless surface stateson the opposite surfaces resulting in an energy gap in thesurface energy band of the TI thin film (13). Even in a suf-ficiently large single 3D TI crystal where the interactionbetween opposite surfaces can be neglected, different mass-less surface states should meet and interact with each otherat edges between two adjacent facets. In this work, we dem-onstrate that the combined effects both from the usual surface-dependent properties such as facet-dependent work functiondifference and from the topological surface properties fora given broken time-reversal symmetry produce a topolog-ical magnetoelectric coupling (TME) (14–16) described by

the axion electrodynamics without external charge controlsas considered before (15). The resulting magnetic orderingalong the edges should be robust and strong enough tobe measured.Our study of TME couplings (14–16) on edges of TIs is based

on the ab initio pseudopotential density functional method (17).We examine Bi2Se3 as an example material for our investigation.For a rhombohedral crystal structure of Bi2Se3 (18), a surfacewith the (111) direction has a triangular lattice of Se atoms(typical cleavage surface) whereas one with the (110) or the(112) direction perpendicular to the (111) direction has a tetrag-onal surface unitcell (Fig. 1). In a single crystal of Bi2Se3 grownalong the (111) direction, the rectangular-shaped crystal has the(110) and (112) surfaces as side walls whereas the hexagonal (19)or triangular (20) column-shaped one has the (110) surfaces asside surfaces. We choose the (110) surface as a side wall in ourstudy (Fig. 1B). Then we solve the modified Maxwell’s equation ofthe axion electrodynamics (14, 15, 21) for a model geometry of theBi2Se3 single crystal with boundary conditions obtained fromthe first-principles calculations.

Results and DiscussionWe first examine the electronic structures of topological sur-face states on various facets. The calculated band structuresindicate that the protected massless metallic surface state onthe ð110Þ surface has a quite anisotropic dispersion re-lationship, unlike the well-known isotropic one on the ð111Þsurface (Fig. 2). The Fermi velocity of the surface states near

Significance

Interactions between two adjacent surfaces of differentsurface orientations in a single-crystal topological insulatorare investigated. We show that the edge between twosurfaces can host nontrivial axion electrodynamics withsizeable experimental signals owing to the unique inter-action between the two topological surface states. We findthat the large work function difference between facets in atopological insulator can generate strong electric fieldsaround the edges and that, in turn, the electric fields giverise to effective magnetic fields for a given broken time-reversal symmetry. Our theoretical work highlights a routeto reveal intriguing axion electrodynamics in a real solidand provides methods to exploit macroscopic topologicalstates.

The authors declare no conflict of interest.

Author contributions: Y.-L.L., H.C.P., J.I., and Y.-W.S. designed research; Y.-L.L., H.C.P., J.I.,and Y.-W.S. performed research; Y.-L.L. and H.C.P. contributed new reagents/analytictools; Y.-L.L., H.C.P., J.I., and Y.-W.S. analyzed data; and Y.-L.L., J.I., and Y.-W.S. wrotethe paper.1Y.-L.L. and H.C.P. contributed equally to this work.2Present address: Center for Theoretical Physics of Complex Systems, Institute for BasicScience, Daejeon 34056, Korea.

3To whom correspondence may be addressed. Email: jihm@snu.ac.kr or hand@kias.re.kr.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1515664112/-/DCSupplemental.

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the Γ point is isotropic with the magnitude of ∼ 5× 106 m/s onthe (111) surface, whereas on the ð110Þ surface it is ∼ 2× 105m/s for the Γ-to-X direction and ∼ 1× 105 m/s for the Γ-to-Ydirection. In addition to changes in the shape of the disper-sion, the energy level of the Dirac point of the surface statewith respect to the bulk chemical potential shifts dependingon the surface orientation owing to the intrinsic anisotropy ofthe system. Our estimation of the shift within slab geometrycalculations is 120 meV for Bi2Se3, which is comparable to aprevious estimation (3).Next, we study the work function difference between dif-

ferent facets of Bi2Se3. The work function is the energy re-quired to extract an electron from a specific surface of a solidand is determined by the difference between the vacuum andthe Fermi level (11) and obtained by the energy differencebetween the potential energy on the vacuum and the Fermilevel in our slab calculations. The calculated work function ofthe unrelaxed ð111Þ surface is Wð111Þ = 5.84 eV and that of theð110Þ surface is Wð110Þ = 5.04 eV. The relaxation of the atomicstructures turns out to change the calculated values slightly to5.81 eV for the (111) surface and 4.97 eV for the ð110Þ sur-face. Such small changes give negligible effects on our resultspresented below. We also find that the work function of the

ð112Þ surface is 5.11 eV, similar to that of the ð110Þ surface.Such a large work function difference mainly originates fromthe surface-dependent dipole moments when the inversionsymmetry is broken at the surface, and it has also been wellknown from previous first-principles calculations as well asexperiments on elemental metals (22).Whereas the electrostatic potential variation generated by

work function differences occurs only outside the bulk andnear edges due to metallic screening in elemental metals (12),the TI supports the variation inside as well as outside the bulkwith appropriate dielectric screening. The Hartree poten-tial distribution for the cross-section of the Bi2Se3 nanorodshown in Fig. 3A can be obtained from actual ab initio cal-culations. The resulting potential distributions exhibit spa-tially varying electric potential in the vacuum region owingto the large work function difference between ð110Þ and ð110Þsides of the nanorod and a constant electric potential suffi-ciently (greater than a few angstroms) inside the bulk (Fig.3B). Near the surface (both inside and outside the bulk within5 Å), the rapid potential variation is also confirmed by thiscalculation. Two boundary conditions, the calculated workfunction on the surfaces and a constant potential inside thebulk, will be considered in the next model below.Having established the electric potential distribution around

the edges of the 3D Bi2Se3, we now turn to its consequenceson TME couplings (14). As long as the TI has an energy gapopening at the surface by breaking the time-reversal sym-metry (14, 15) [achievable by depositing thin magnetic films(23–26), for example] and its Fermi level resides inside thegap, the low-energy physics can be described phenomeno-logically in terms of the axion electrodynamics (21). In ad-dition to the ordinary Maxwell Lagrangian (27), the axionelectrodynamics has an effective term Leff = κθE ·B (21), whereκ= e2=2πh, h is the Planck’s constant, e is the electric charge,E is the electric field, B is the magnetic induction, and θ is theaxion field. Although the constant θ plays no role in the equa-tions of motion, the gradient of θ gives rise to an extra chargedensity and a current density (16, 21) that are related to theTME couplings (i.e., magnetization induced by the electricfield and the electric polarization induced by the magneticfield). Consequently, the electric displacement D and the mag-netic field H can be rewritten assuming a linear dielectric forBi2Se3 (14, 15):

D= eE∓ κθB

H=1μB± κθE,

[1]

where the upper or lower sign corresponds to the outwardor inward surface magnetization, respectively. We introduce

(111)

(011) (110)

(111)

(112) (110)

x

yz

(111) (110)

Bi

SeBi

Se

A B

C D

Fig. 1. (A) Rectangular-shaped crystal of Bi2Se3 with the top (111), thefront (112), and the side (110) surfaces. (B) Hexagonal-shaped crystalwith the top (111) surface and the (110) sides. The dashed squares in Aand B are cross-sections of the crystal to be considered in the modelcalculations. (C ) Atomic structure of the (111) and (D) that of the (110)surface. The black parallelogram indicates the unit cell of each surface;the area for the ð111Þ surface is 14.83 Å2 and that for the ð110Þ surface is68.42 Å2.

E -

E F (eV

)

-1.0

-0.5

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1.0

M Γ K M

bulk(111) 6QL

-1.0

-0.5

0.0

0.5

1.0

X Γ Y X

(110) 15MLbulk

E -

E F (eV

)

A B

Fig. 2. Band structures of (A) the (111) surface of six quintuple layers (QL) and (B) the (110) surface of 15 monolayers (ML) on the energy-momentum space.Projected bulk states are shown by a gray region for both surfaces.

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electric (ϕ) and magnetic (ψ) scalar potentials such that theelectric field E is obtained from E=−∇ϕ and the magnetic fieldH=−∇ψ.* With these scalar potentials, we can convert theelectrodynamics into a variational problem for both potentialsδFðϕ,ψÞ≡ δFMðϕ,ψÞ+ δFAðϕ,ψÞ= 0. Here,

FM =12

ZZ

Ω

heð∇ϕÞ2 + μð∇ψÞ2

idΩ,

FA =12

ZZ

Ω

hμκ2θ2ð∇ϕÞ2 ∓ 2μκθð∇ϕ ·∇ψÞ

idΩ,

[2]

where Ω is the domain area. FM corresponds to the ordinaryMaxwell electrodynamics and FA to the axion term unique tothe TI (Supporting Information). Then we solve it numericallyusing the finite element method (28) with a given boundarycondition for this model system.†

We choose a rectangular cross-section of the 3D TI crystal suchas the dashed square in Fig. 1 to describe essential features of thelow-energy electrodynamics. In Fig. 4, each side of the dashed-linebox corresponds to the (111) or the (110) surface of Bi2Se3. We settwo additional boxes inside and outside the TI crystal with spacingof ds and dv chosen from ab initio calculations to be used inimposing boundary conditions for the model calculations.The electric potential on each side (outer box) is determined

by the negative work function value of each facet dividedby e, such as ϕð111Þ =−Wð111Þ=e=−5.84 V (Fig. 4, red line) andϕð110Þ =−Wð110Þ=e=−5.04 V (Fig. 4, blue line). Sufficiently insidethe TI crystal the dielectric screening leads to the constantelectric potential, so that the potential of the inner box isfixed at zero (ϕ0 = 0 V). Across the boundary between thedashed-line box to outer vacuum, the axion field changes fromπ to 0 and the permittivity changes from 100e0 (29) to e0,whereas the permeability (μ0) remains the same (Fig. 4). Alarge E originally exists around the edges of Bi2Se3, and the

numerical solution of δF = 0 with the abovementioned boundaryconditions gives rise to H as well as B (and consequently D)thanks to the presence of the axionic field θ. If the chemicalpotential of the TI crystal is located in the surface energy gapthat is opened with a certain time-reversal symmetry breakingoperation, we expect, no matter how weak the operation is,a strong TME effect around all of the edges connecting twofacets with different orientations. Hereafter, we choose the uppersign in Eqs. 1 and 2 for definiteness.The contour plot of the electric and the magnetic scalar po-

tentials are achieved through the aforementioned proceduresas shown in Fig. 5 A and B, respectively. The electric potentialrises continuously from the lower-potential (111) surface tothe higher-potential (110) surface. There is a discrete jump inthe electric potential at edges where the two surfaces meet onthe outer box boundary, and the field would diverge for an in-finitely sharp edge. A real solid does not have an infinitely sharpedge, and we have confirmed that the locally averaged fieldis practically independent of the sharpness of the edge, andsmoothing the potential variation near the edge does not change

x (Å)

y (Å

)

0 10 20 30 40 50 0

10

20

30

40

50

(111)

(110

)x (Å)

y (Å

)

0 5 10 15 20 25 25

30

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50

-0.2

0.40.20.0

0.6

5.7

5.55.4

5.3

5.2 5.1

5.6

5.0

A B

Fig. 3. (A) Supercell structure of the nanorod with ð110Þ and ð110Þ side surfaces. (B) Contour plot of the potential energy difference between the nanorodand the bulk shown for the upper left side. The contour interval is chosen as 0.2 eV for the contour values less than 5.0 eV, and 0.1 eV for those greater than5.0 eV. Dashed lines in A and B denote the outermost atomic positions and solid lines denote the boundary of the region of subtracting the bulk potentialfrom the nanorod potential.

TIθ=π

lTI

φ(111)

dv dsvacuumθ = 0

lvacφ(110)

φ0

φ0

φ0φ0

Fig. 4. Cross-section of the Bi2Se3 crystal as a model system. Inside a vacuumbox of lvac = 1 μm, the TI crystal (dashed-line box) of l TI = 100 nm is located atthe center. The two additional boxes for the boundary values lie on the TIcrystal with the spacing ds =dv = 5 Å, which are chosen to best fit ab initiocalculations. The boundary condition is given by the fixed potentials ofϕð111Þ =−5.84 V, ϕð110Þ =−5.04 V, and ϕ0 = 0 V on the red, blue, and blacklines, respectively.

*In ~∇ ×H= J+ ∂D∂t of the Maxwell equations, because we are considering a time-indepen-

dent situation without free electric charges and currents, the right-hand side vanishes.Then the magnetic scalar potential ψ may be defined and the effective magnetic chargemay be obtained from the relation −∇2ψ = ρM.

†For arbitrary variation δϕ and δψ at the boundary, δϕ+ = δϕ−, δψ+ = δψ−, D+ · n̂+ +D− · n̂− = 0,B+ · n̂+ +B− · n̂− = 0, and other standard boundary conditions are compatible with theabsence of electric and magnetic sources. Here + (−) denotes the outside (inside) closedloop and n̂ the normal to the boundary lines.

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our results (see Detailed Information for Numerical Results,Smoothing Boundary Conditions). Outside the material, themagnetic scalar potential profile exhibits a structure similar tothat of the electric potential, because the effective Lagrangianfrom the axion electrodynamics has a symmetric structure be-tween the electric and the magnetic parts. However, differentboundary conditions between electric and magnetic potentialscause different behaviors between two potentials on the surfaceand inside the material. Actually, the distributions of electricand magnetic charges (more properly, dipoles for the magneticcase) on the surfaces are different from each other and theyproduce different potentials or fields inside the materials (seeDetailed Information for Numerical Results, Electric and MagneticCharge Densities). When we closely examine Fig. 5, we find hugechanges in electric and magnetic potentials in the narrow regionnear the surfaces, and the contour lines are very dense. Thesehuge potential gradients near the surface lead to the large fieldnear the surfaces. The distributions of the electric field E andthe magnetic induction B are shown in Fig. 5 C and D, respec-tively. Both fields are very high at edges and decrease towardthe center of each surface. At 5 nm away from the edge, weobtain the electric field of 3.88× 107 V/m and the magneticinduction of 0.14 G. These values are orders of magnitudehigher than the previously predicted value for the topologicalmagnetic inductions induced by a point charge on a flat ge-ometry (15). Such an enormous enhancement of the field isattributed to the unusual geometric effect at the edges as wellas large intrinsic electrostatic potential differences betweentwo facets. Outside the TI crystal, the electric field heads fromthe (111) surface for the (110) surface, and the magnetic in-duction is parallel to the electric field. The electric field insidethe TI crystal near the surface is perpendicular to each surface(outward direction) because of the boundary structure. Themagnetic induction follows the right-hand rule with respect tothe quantized Hall current at edges, which is given by the electric

field and the gradient of the axion field such as jH =−κð∇θ ×EÞ,so that the asymmetric dipole-type magnetic induction arisesat the edges (Supporting Information). The magnetic inductionproposed here may possibly be measured experimentally usingsensitive techniques (30–35).

ConclusionsIn conclusion, we have examined electromagnetic propertiesof the single crystalline TI and showed that the large electricfield and the associated topological magnetic ordering arenaturally generated by the facet-dependent work functionnear the edges of the crystal without any external manipula-tion other than a tiny time-reversal breaking operation. Ourdemonstration can be a useful basis to realize the delicateaxion electrodynamics in real solids and to study various otheraxionic aspects of TIs. Furthermore, considering that thefacet-dependent work function difference is present in almostall crystals, we expect that an unusual magnetoelectric cou-pling could occur on edges of certain multiferroic materialswithout nontrivial Chern–Simons magnetoelectric couplingsuch as Cr2O3 (36).

MethodsFirst-Principles Calculations. Our study is based on the ab initio pseudopo-tential density functional method (17) within the local density approximation(37) including spin–orbit interactions as implemented in the VIENNA abinitio simulation package (38). The calculations are carried out with theexperimental lattice parameters of bulk Bi2Se3 for all directions (18). Weperform full atomic-relaxation calculations as well and confirm that the re-laxations do not change electronic structures of slab calculations apprecia-bly. The slab thickness is chosen as six quintuple layers and 15 monolayers for(111) and (110) surface, respectively, so that the interaction between top andbottom surfaces is ignorable. To reduce the spurious interaction betweenthe neighboring slabs in the z axis we introduce a large vacuum over 20 Å.The k-point grid is taken as 18 × 18 × 1 and 9 × 9 × 1 Monkhorst-Pack mesh(39) for (111) and (110) surfaces, respectively. We also note that the workfunction difference between different surface orientations of a material

-100

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-100 -50 0 50 100

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> 0.500.45

0.200.250.300.350.40

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m)

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> 1.441.42

0.320.34

0.380.400.42

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1.38

A B

C D

Fig. 5. Contour plots of (A) the electric (volts) and (B) the magnetic scalar potential (10−6 C/s) for the model in Fig. 4. The contour intervals are 0.05 V and10−8 C/s for A and B, respectively. The color scale jumps from −5.05 V (red) to zero (white) in A and from 0.42 ×10−6 C/s to 1.38 ×10−6 C/s in B because of thehuge potential change. (C ) The electric field (107 V/Å) and (D) the magnetic induction (G) with the strength denoted by colors and the direction by arrows.The length of the arrows is proportional to the strength of the fields, whereas the yellow arrows around four crystal edges in D are shortened to one halffor visual purposes.

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has been shown to be insensitive to the choice of exchange-correlationfunctionals (12, 40).

Finite Element Method. The variational problem of Eq. 2 is numerically solvedby applying the finite element method with triangular grids. All calculationsfollow Jin (28), and we expand to the electromagnetic problems withmagnetic scalar potentials instead of vector potentials. We obtain the tri-angular grids by the FEMM 4.2 program (41) based on the Triangle (42), andconstruct the 2N× 2N matrix, where N is the number of grid points. Thematrix elements consist of the electric and magnetic scalar potentials at each

node. The Dirichlet boundary condition is imposed and we obtain the so-lutions for electric and magnetic scalar potentials by inverting the matrixusing the Linear Algebra PACKage (43).

ACKNOWLEDGMENTS. We thank Y. Kim, Y. B. Kim, J. H. Han, B.-J. Yang,E.-G. Moon, and S. Coh for fruitful discussions. This work was supported bythe National Research Foundation of the Ministry of Science, ICT and FuturePlanning Grants 2006-0093853 (to Y.-L.L. and J.I.) and 2015001948 (to Y.-W.S.).Computations were supported by the Korea Institute of Science and Tech-nology Information and the Center for Advanced Computation of the KoreaInstitute for Advanced Study.

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0