Charged skyrmions on the surface of a topological insulator

Hilary M. Hurst,1, 2 Dmitry K. Efimkin,1, 2 Jiadong Zang,3 and Victor Galitski1, 2

1Joint Quantum Institute and Condensed Matter Theory Center, Department of Physics,University of Maryland, College Park, Maryland 20742-4111, USA

2School of Physics, Monash University, Melbourne, Victoria 3800, Australia3Institute for Quantum Matter, Department of Physics and Astronomy,

Johns Hopkins University, Baltimore, Maryland 21218, USA

We consider the interplay between magnetic skyrmions in an insulating thin film and the Diracsurface states of a 3D topological insulator (TI), coupled by proximity effect. The magnetic textureof skyrmions can lead to confinement of Dirac states at the skyrmion radius, where out of planemagnetization vanishes. This confinement can result in charging of the skyrmion texture. The pres-ence of bound states is robust in an external magnetic field, which is needed to stabilize skyrmions.It is expected that for relevant experimental parameters skyrmions will have a few bound statesthat can be tuned using an external magnetic field. We argue that these charged skyrmions canbe manipulated directly by an electric field, with skyrmion mobility proportional to the number ofbound states at the skyrmion radius. Coupling skyrmionic thin films to a TI surface can provide amore direct and efficient way of controlling skyrmion motion in insulating materials. This providesa new dimension in the study of skyrmion manipulation.

PACS numbers: 75.70.Kw, 75.70.Cn

Introduction. Topological insulators (TIs) are a newclass of matter with a band structure that can be char-acterized by the topological invariant (See Refs. [1, 2] andreferences therein). As with usual insulators, TIs have aforbidden band gap in the bulk separating the filled va-lence and empty conduction band. Contrary to the for-mer, the topology dictates them to posses very unusualedge (2D) or surface (3D) states protected from nonmag-netic disorder. The surface spectrum of a 3D TI has anodd number of Dirac points, in the vicinity of which elec-trons are described by the Dirac equation for relativisticmassless particles. Strong spin-momentum locking makesthe electronic spectrum sensitive to magnetic perturba-tions. Particularly, exchange coupling between the Diracelectron and a uniform magnetization opens a gap in thesurface spectrum and leads to the Anomalous quantumHall effect [3]. This has a number of interesting mani-festations [4, 5], including quantized Faraday and Kerreffects [6–8], and has recently been directly observed ex-perimentally [9, 10].

Profound physics emerges when the Dirac electronscouple to spin textures with spatially non-uniform mag-netization distributions [11, 12]. A magnetic domain wallin proximity to the Dirac electron generates a chiral state,in analogy to the Jackiw-Rebbi zero mode [13], whichin turn alters its dynamics and stability [14–17]. Onecould thus expect more novel properties when the Diracstates of a TI couple to nontrivial spin textures, such asskyrmions [18, 19], which is the main issue of this work.

A magnetic skyrmion is a topological spin texturewith whirlpool-like structure, illustrated in Fig. 1-a. Itstopology is characterized by the topological invariant

NS =

∫dr

4πn

[∂n

∂x× ∂n

∂y

], (1)

where n(r) is the unit vector describing direction of themagnetization [20]. The skyrmion has been experimen-tally observed in various chiral magnets [21, 22] as a re-sult of the the competition between the Dzyaloshinskii-Moriya (DM) interactions, Heisenberg exchange, andZeeman interaction [23–25]. It was first discovered inMnSi in a narrow window at finite temperatures [21].However, once on a thin film, the skyrmion phase canbe greatly extended even down to zero temperature, aslong as the fine tuning of the external magnetic fieldis achieved [22]. Although it has been demonstratedthat metallic skyrmions can be driven by the spin trans-fer torque (STT) from an electric current [26, 27], re-cent appearance of the insulating helimagnet materialCu2OSeO3 raises a more challenging question of how tomanipulate insulating skyrmions [28–31]. It is impor-tant not only from a fundamental point of view, but alsofor applications in spintronics and memory devices (seeRef. [32] for a recent review).

Here we show that the skyrmion texture in an insu-lating helimagnet, proximity coupled to TI surface, canbecome charged due to Dirac surface states localized atthe skyrmion radius. The number of localized statescan be controlled by an external magnetic field, whichis needed for skyrmion phase stabilization, and in therealistic conditions there are a few localized states. Ex-perimentally accessible electric fields can drive individ-ual charged skyrmions in insulating materials at speedscomparable to those seen in metallic systems. Our workshows the first mechanism of manipulating skyrmions inthin films directly via electric field, which opens the doorto further investigation of skyrmion manipulation with-out relying on the STT mechanism.

The electronic spectrum of TI surface states. In thepresence of the skyrmion texture n(r) and external per-

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FIG. 1: (color online) (a) Sketch of the wave function |Ψ(r, φ)|2 (gray) of a TI surface state localized at the skyrmion radius.For clarity we plot |Ψ|2 for 0 < φ < π only. The vector field represents the direction of local magnetization n(r) of theskyrmion. (b) Electronic spectrum of TI surface proximity coupled to the skyrmion texture as a function of skyrmion radiusand magnetic field. Without magnetic field states with orbital quantum number |m| ≤ 4 are presented, with magnetic fieldB = 0.2 T states with |m| ≤ 10 are presented, and with magnetic field B = 0.8 T states with |m| ≤ 25 are presented. Withoutmagnetic field localized states split from continuous bands |E| ≥ ∆S in pairs. In the presence of an external magnetic fieldlocalized states split from the zeroth Landau level, which has the energy E0m = −∆S −∆Z and is highly degenerate. Higherenergy LLs are also shown. (c) The energy of bound states as a function of m at large skyrmion radius RS without magneticfield. Numerical solution (blue circles) from equation (3) matches almost exactly with the semiclassical quantization (redsquares) as in equation (5).

pendicular magnetic field B, corresponding to the vectorpotential A = B(xey− yex)/2, the surface states of a TIcan be described by the Hamiltonian

H = vF

[(p− e

cA)× σ

]z−∆Sn(r) · σ −∆Zσ

z. (2)

Here vF is the Fermi velocity, σ is the vector of Paulimatrices, corresponding to electron’s spin, ∆Z = gµbB isthe Zeeman shift due to the external field, and ∆S > 0parametrizes ferromagnetic exchange coupling betweenthe thin film and TI surface [33]. The out-of planecomponent of the magnetic texture nz(r) plays the roleof position dependent Dirac mass, while the in-planecomponent n|| can lead to an emergent magnetic fieldBS(r) = c∆Sdivn||(r)/e~vF. A skyrmion stabilized byDM interaction, as it is for insulating helical magnets, hasa solinoidal in-plane magnetization, shown in Fig. 1-a.Therefore, its magnetic texture can be parametrized byn(r) = (− sinφ

√1− n2z(r), cosφ

√1− n2z(r), nz(r)). In

this case the emergent magnetic field BS is zero and thein-plane component of magnetic texture can be gaugedaway. For the out-of-plane component we will use thehard-wall approximation nz(r) = 2Θ(r −RS)− 1, whereRS is the skyrmion radius. The applicability of thisapproximation is justified below. In materials whereskyrmions are stabilized via DM interaction the radius ofthe skyrmion is not dependent on the external magneticfield. Rather, skyrmion radius depends on the relative

strength of the Heisenberg and DM exchange interactionswhich in turn depends on the material parameters [32].In the following we therefore regard the skyrmion radiusas an independent parameter.

The Hamiltonian conserves total momentum, and thewave function spinor can be presented as Ψ(r, φ) =(F (r)eimφ/

√2π,G(r)ei(m+1)φ/

√2π), where m is the or-

bital momentum quantum number connected with thetotal momentum of Dirac particles as j = m + 1

2 . Theradial wave functions Fm(r) and Gm(r) satisfy the equa-tion (

−∆Snνz −∆Z − E vFeB

2c r − ~vF ∂∂r

vFeB2c r + ~vF ∂

∂r ∆Snνz + ∆Z − E

)(F νmGνm

)

−(

0 ~vFm+1r

~vFmr 0

)(F νmGνm

)= 0. (3)

Here ν = ± for wave function of electrons outside andinside the skyrmion, where n±z = ±1. Continuity ofthe Dirac wave function at the skyrmion radius RS re-sults in the boundary condition F+(RS) = F−(RS) andG+(RS) = G−(RS). The energy at each skyrmion radiusis then numerically calculated by solving the transcen-dental equation G−(RS)/F−(RS) = G+(RS)/F+(RS),generating the spectra shown in Fig. 1-b. We considerthe skyrmion radius and external magnetic field to beseparate controlling parameters of the surface state en-ergy. Although skyrmions are usually stabilized in the

3

presence of an external magnetic field, it is instructive toconsider at first the electronic spectrum without it.

For the following calculations we use vF ≈ 0.5×106 m/sand gTI ≈ 5, corresponding to the TI material Bi2Se3,and exchange coupling ∆S = 10 meV. Insulatingskyrmions with radius RS ≈ 25 nm have been stabi-lized in Cu2OSeO3 films [28, 34–36] at low temperaturesT < 40 K and for magnetic fields 0.05 T ≤ B ≤ 0.2 T.

Electronic structure in the absence of the magneticfield. At B = 0 the skyrmion texture does not mod-ify the continuous gapped spectrum |E| ≥ ∆S [37] ; ifthe skyrmion radius exceeds the critical radius R∗S =~vF/2∆S it can lead to the appearance of localized statesas presented in Fig. 1-b for B = 0 T. The localized statesappear in pairs, since the Dirac Hamiltonian possesseselectron-hole symmetry {HD, K} = 0 with K = σxCwhere C is complex conjugation. The states are localizedin the vicinity of the skyrmion radius RS, as presentedin Fig. 1-a with length ~vF/∆S. These states are wellsplit from the continuous bands and are bound to theskyrmion in case of its motion, with wave functions

Ψ−m ∝[(ε+ n−z )Im(kρ)Im+1(kρ)

]; Ψ+

m ∝[(ε+ n+z )Km(kρ)Km+1(kρ)

],

(4)where k2 = n2z − ε2, ε = E/∆S, ρ = ∆Sr/~vF, and Im(Km) are modified Bessel functions of the the first (sec-ond) kind.

The second term in equation (3) originates from the an-gular motion and can be neglected if RS � 2mR∗S, 2(m+1)R∗S. In that case the system of equations (3) can be re-duced to the system of supersymmetric Schrodinger equa-tions (SUSY) by squaring H to solve H2Ψm = E2

mΨm,with the SUSY superpotential W (x) = ∆Snz(r), as itis for a domain wall [13, 38]. The SUSY nature of theequations guarantees the presence of a localized state forevery angular momentum. Their presence is not sensi-tive to the shape of the skyrmionic texture nz(R), sothe physics is well captured by the hard-wall anzatz inthe RS � R∗S regime. When the the angular motion be-comes important, at RS ≈ R∗S, the reduction to SUSYis lost and localized states are pushed to the continuum.The energy gain due to the localization becomes smallerthan additional kinetic energy due to orbital motion.

For large skyrmions RS � R∗S, the energy of boundstates can be directly calculated from the semiclassicalpicture. Along the skyrmion boundary there are chiralstates inheriting the dispersion law of massless Dirac par-ticles Eφ = ~vFkφ, and the Bohr-Sommerfeld quantiza-tion of their motion is respected;

2πRSkφ + φB = 2πm, (5)

where φB = −π is the Berry phase for massless Dirac par-ticles, which appears due to the rotation of the electron’sspin when it makes a closed loop in momentum space.This gives Eφ = ~vF(m + 1/2)/RS which is in excellent

FIG. 2: (color online) (a) Electronic spectrum of TI surfacestates a function of magnetic field B, for skyrmion radiusRS = 25 nm. Landau levels with n ≤ 6 are presented. Withincreasing magnetic field the number of states split from thezeroth Landau level with the energy E0m = −∆S − ∆Z in-creases. The black lines correspond to the spectrum with-out the texture. (b) Number of split states, bound to theskyrmion, as a function of magnetic field.

agreement with exact numerical calculation, as presentedin Fig. 1-c. It should be noted that the semiclassical ar-guments are also weakly dependent on the actual shapeof the magnetic texture.

Electronic structure in the presence of the externalmagnetic field. Without the skyrmion texture, but in thepresence of an external magnetic field, all surface statescondensed into Landau Levels (LL) indexed by main andorbital quantum numbers n and m, respectively. Landaulevels have macroscopic degeneracy m = [0,Φ/Φ0 − 1],where Φ is the total flux through the TI surface andΦ0 = hc/e is the flux quantum. In the radial gauge, LLare localized at a cyclotron radius Rm ≈ lB

√2(m+ 1)

with magnetic length lB =√

~c/eB. Their energeticspectrum is given by

E±nm = ±vF

√2~2l2B|n|+

(∆Z

vF

)2

; n 6= 0,

E0m = −sgn(B)∆Z ; n = 0.

(6)

4

The zeroth LL is spin polarized due to the orbital effectof magnetic field and its shift is sensitive to Zeeman cou-pling ∆Z, while other LLs are not sensitive to its sign.

In the presence of the magnetic field and the skyrmiontexture, discrete states appear which are well split fromthe macroscopically degenerate n = 0 LL. For thesestates, the cyclotron radius Rm approximately coincideswith the skyrmion radius RS, therefore the states arelocalized at the skyrmion boundary and move with theskyrmion. The zeroth LL states with Rm � RS approachE0m = −sgn(B)(∆Z + ∆S). These states are weakly af-fected by the skyrmion texture and are not correlatedwith its motion, in analogy to the delocalized states inthe absence of a magnetic field. The energetic spectrumof TI surface states as function of skyrmion radius is pre-sented in Fig. 1-b for magnetic field values B = 0.2 T and0.8 T. States bound to the skyrmion no longer appearin pairs since electron-hole symmetry is broken by themagnetic field. States with |E| ≥ ∆S condense into LLswith n 6= 0 which are weakly affected by the skyrmiontexture. States localized at the skyrmion boundary havethe wave functions

Ψ−m ∝ ρme−ρ2b2

[F1(α−, 1 +m; ρ2b)

ρF1(α− + 1, 2 +m; ρ2b)

];

Ψ+m ∝ ρme−

ρ2b2

[U(α+, 1 +m; ρ2b)

ρU(α+ + 1, 2 +m; ρ2b)

],

(7)

where F1, U are confluent hypergeometric functions ofthe first and second kind, parameterized by α± = ((n±z +δ)2 − ε2)/4b. b = B/B0 where B0 = 2∆2

Sc/~v2Fe,δ = ∆Z/∆S and ρ = ∆Sr/~vF.

The dependence of the TI surface spectrum as a func-tion of magnetic field B for RS = 25 nm is presented inFig. 2-a. The presence of states bound to the skyrmionis robust to the external magnetic field. Moreover due tothe interplay between LL and skyrmion confinement thenumber of bound states increases with magnetic field aspresented in Fig. 2-b. The appearance of bound statescan lead to charging of the skyrmion texture. The elec-tric charge is eNB, where NB is the number of additionaloccupied electronic states in comparison to the TI sur-face without the skyrmion texture. The value NB de-pends on local chemical potential and can be controlledby magnetic field. Our mechanism of the texture charg-ing differs from the one in Ref. [12] based on quantizedresponse of electronic density to the emergent magneticfield BS(r), which originates from a magnetic texture butis zero BS(r) = 0 in our case.

The radius of skyrmions in Cu2OSeO3, RS = 25 nm,exceeds the critical radius R∗S = ~vF/2∆S ≈ 16 nm andthere is one bound state for magnetic field 0.05 T ≤B ≤ 0.2 T. The skyrmionic charge can be equal to thesingle charge of an electron. If the radius of individualskyrmions is larger, the skyrmion phase is stabilized in awider magnetic field interval, or for greater values of ex-change coupling ∆S, the skyrmion will host a few bound

states and the skyrmion charge becomes more tunable bymagnetic field or doping.

Dynamics of charged skyrmions. Being chargedparticle-like objects, skyrmions with bound surface statescan be described by the following Lagrangian

L = QS

[R×R

]z

+ eNB

(φ(R)− R

cA(R)

)+mSR

2

2.

(8)Here the first term represents the Magnus force originat-ing from spin dynamics [39–41], and QS = 4πρSS~NS

where NS is the skyrmion topological invariant, withρS and S the density and amplitude of spins in theskyrmionic magnet. The second term represents the in-teraction of skyrmion with external electromagnetic field,while the last term corresponds to the possible addi-tional kinetic energy with phenomenologically inducedskyrmion mass mS. In the presence of an electric fieldE = (Ex, 0) the skyrmion acquires the velocity

Ry = − eNBc (2QSc+ eNBB)

(2QSc+ eNBB)2

+ c2Γ2Ex; (9)

Rx = − eNBc2Γ

(2QSc+ eNBB)2

+ c2Γ2Ex. (10)

Here Γ is a phenomenological friction parameter. Notethat the mass term in equation (8) only providesthe initial acceleration of the skyrmion. However theskyrmion’s final steady velocity is determined from thebalance between electric field and damping, thus it is in-dependent of the massmS. In the limit Γ→ 0, the systemexhibits Hall motion where Rx = 0 and Ry = −µSEx,defining skyrmion mobility µS = eNBc/ (2QSc+ eNBB).Taking NB = 1, B = 0.2 T, ρ ∼ d/a3, where d ≈ 100 nmis the film thickness and a ∼ 8.9 A is the crystallographiclattice constant, we have µS ∼ 1 × 10−6 m2/Vs. Elec-tric fields as low as 102 V/m can induce drift veloci-ties vH ∼ 0.1 mm/s, comparable to skyrmions drivenby conduction electrons in metallic systems [42], accord-ing to the STT mechanism (Ref. [26, 27, 32, 43, 44] andreferences therein). In the system under considerationthe skyrmionic material is insulating and the the surfacespectrum of TI is gapped away from the skyrmion, whichmakes the STT mechanism unimportant.

It has recently been proposed that insulatingskyrmions can be driven by a temperature gradient [30].For electric fields E ∼ 105 V/m or greater the velocityof a skyrmion driven by electric field is higher than theexpected velocity due to thermal gradient (0.1 m/s inref. [30]) and, more importantly, an electric field is easierto manipulate than a temperature gradient. It has alsobeen shown that in multiferroic materials, which includeCu2OSeO3, skyrmion textures have an intrinsic dipolemoment which enables them to couple to to the gradientof electric field [29, 35, 36, 45]. Regardless of the difficultyof applying a gradient in electric field, the estimated Hall

5

velocity vH in our approach is two orders of magnitudegreater than vH due to magnetoelectric coupling [29],where vH ≈ (λRS/2πS~)∆E ∼ 10−3 mm/s with RS =25 nm, dipolar coupling λ ∼ 10−33 ≈ C ·m [34], andfield strength difference ∆E ∼ 102 V/m. We thereforeconclude that our mechanism is very effective due to thedirect coupling of skyrmions with electric field.

In an experiment one can use a TI such as Bi2Se3as a substrate and epitaxially grow the Cu2OSeO3 thinfilm on top of it. An ultra-thin buffer layer inserted be-tween these two can help to avoid lattice mismatch whilekeeping the coupling between Dirac electrons and themagnetization. Concerning the experimental difficulty ingrowing Cu2OSeO3 films, one can also use monolayers offerromagnet instead, which is generally insulating. Theinterfacial DM interaction can also generate skyrmionstherein [46]. The whole sample is sandwiched in a pairof electrodes, which generates an electric field.

It should be noted that the skyrmion phase can be in-trinsic to surface states of TI in the presence of strongCoulomb repulsion and hexagonal warping, which takesplace at high doping [47]. In that regime our mechanismis not of importance, since the skyrmion texture changesthe surface spectrum in the vicinity of the Dirac pointwhich is deeply buried under the Fermi level.

We have focused on the behavior of a single chargedskyrmion. Skyrmions most frequently appear in a lat-tice which can be pinned by disorder and the underly-ing atomic lattice. Therefore in an experiment this ef-fect would manifest as a rotation of the skyrmion latticecaused by Hall motion of individual charged skyrmions.We predict that the angle of skyrmion lattice rotationwill be proportional to the electric field and the numberof bound states at the skyrmion radius. The angle ofrotation should change within the skyrmion phase in re-sponse to increased DC electric field. To rule out magne-toelectric effects, an experiment at constant electric fieldbut varying magnetic field could be conducted, providedthat the stabilizing magnetic field range of skyrmions isbig enough to tune the number of bound states. In thiscase we predict a change in rotation angle in response toan increase in magnetic field, indicating additional boundsurface states.

This research was supported by the U.S. Departmentof Energy DOE-BES (Grant No. DE- 317 SC0001911)and the Simons Foundation (D.E. and V.G). H.H. ac-knowledges support from the National Science Founda-tion CAREER grant DMR-0847224 and additional fel-lowship support from the National Physical Science Con-sortium and NSA. J.Z. was supported by the Theoret-ical Interdisciplinary Physics and Astrophysics Center,the U.S. Department of Energy under Award DEFG02-08ER46544, and the National Science Foundation underGrant No. ECCS-1408168.

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