Gauge fields at the surface of topological insulators
GGauge fields at the surface of topological insulators
M. I. Katsnelson
Radboud University Nijmegen, Institute for Molecules and Materials,Heyendaalseweg 135, NL-6525 AJ Nijmegen, The Netherlands
F. Guinea and M. A. H. Vozmediano
Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco, Madrid 28049, Spain (Dated: October 10, 2018)We study the emergence of gauge couplings in the surface states of topological insulators. Weshow that gauge fields arise when a three dimensional strong topological insulator is coated withan easy–plane ferromagnet with magnetization parallel to the surface. We analyze the modificationinduced by the gauge fields on the surface spectrum for some specific magnetic configurations.
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INTRODUCTION
The Dirac fermion description of novel condensed mat-ter systems have given rise to a great deal of new phe-nomena and ideas in the field that is acting as a bridgebetween different areas of physics including quantum fieldtheory, cosmology, elasticity, and statistical mechanics.One of the most successful examples can be found inthe physics of graphene [1] and, in particular, in the at-tractive relation between the structural - lattice - prop-erties and the electronics of this material. It is by nowwell known that elastic deformations and curvature ofthe graphene lattice give rise to effective gauge fields thatcouple to the fermionic degrees of freedom with the min-imal coupling typical of gauge theories [2].Another important example of Dirac fermions is that oftopological insulators. These materials are characterizedby a bulk insulating behavior with conducting bound-ary states protected by some topological property, theinteger quantum Hall effect [3, 4] being the prototype.In the recent developments (for a review, see Refs. 5–7) numerous systems have been found where the edgestates exist in the absence of a magnetic field and areprotected by time reversal symmetry. The simplest ex-ample is the spin Hall effect [8, 9] where the spin orbitcoupling plays a major role; most of the newly discoveredtopological insulators are based on materials with strongspin orbit coupling [10–12]. The topological protection ofedge states ensures their stability against non-magnetic– more generally, time-reversal invariant – perturbations,opening up the possibility for future spintronic devices.Novel exotic states appear at the surface of a threedimensional (3D) topological insulator when an energygap is induced by a perturbation breaking the time re-versal symmetry. They are obtained typically by apply-ing a magnetic field perpendicular to the surface whatgives rise to the half integer quantum Hall effect [13] orby proximity effects with a magnetic material what in-duces an anomalous quantum Hall effect [14]. A very re-cent experiment reports the observation of massive edge Dirac fermions in a three dimensional topological insula-tor doped with magnetic impurities [15]. In these casesthe exotic physics is due to a combination of the regularor space-dependent mass induced by the perpendicularmagnetic field to the Dirac fermions, and by the appear-ance of zero modes associated to the Dirac operator inmagnetic fields [16–18].A different situation arises when the surface of thetopological insulator is coated with an insulating ferro-magnet with an easy plane such that the magnetization isparallel to the surface. Although time reversal invarianceis also broken and edge states are no more protected, amass is not directly induced in the system. In this workwe will see that in such situation gauge fields appear cou-pling to the electronic degrees of freedom similar to theseproduced by elastic deformation on graphene. We willexplore the modification of the spectrum for the cases ofa vortex and a skyrmion configuration.
GAUGE FIELD INDUCED BY AN IN-PLANEMAGNETIZATION
When putting a thin ferromagnetic layer with magne-tization (cid:126)M on the surface of a strong topological insula-tor an interaction term of the form H (cid:48) = − J (cid:126)M .(cid:126)σ ariseswhere J is the coupling constant. There are two contri-butions to this magnetic term: the Zeeman energy, and s − d exchange interaction energy. The latter contributioncan be orders of magnitude higher than the former oneif the overlap between the electron wave function on thesurface of the topological insulator and that of the ferro-magnet is big enough. One can easily expect that JM ∼ σ z in the Dirac Hamiltonian resulting in the gap open-ing mentioned above. In the case of having an easy- plane ferromagnet, the vector (cid:126)M lies in the surface plane and a r X i v : . [ c ond - m a t . m t r l - s c i ] J un FIG. 1: (Color online) Sketch of the ferromagnetic vortexconfiguration. the s − d exchange interaction couples to the fermions asa vector potential.In the case of graphene where the “spin” degree offreedom is actually a pseudo spin related to the sublat-tice degree of freedom, gauge couplings are induced bystrains or other mechanical deformations of the lattice[2]. In topological insulators where spin is the real spinone needs in-plane magnetization to create such terms.Easy plane ferromagnetism is very natural for thinfilms of single layers as the shape anisotropy (magneticdipole–dipole interaction) always favor this situation [19].This will be the case for any soft magnetic material i. e.materials fulfilling the condition 4 πM s > K , K being theconstant of magnetocrystalline anisotropy. If the coatingis a circular symmetric disc, the magnetization vector willbe freely rotating in the plane. Magnetic textures at thesurface of a three dimensional topological insulator havebeen recently studied in Refs. 20, 21. In the latter pub-lication, the exchange field energy associated to parallelferromagnetic domains of 50 nm width was estimated tobe of the order of 6 meV. Vortex configuration
In easy–plane ferromagnets vortices can exist which arespin configurations with a net 2 π twist about a particularpoint or core similar to the Onsager-Feynman vortices insuperfluid helium or Abrikosov vortices in superconduc-tors. Examples are BaCoAsO and K CuF [22].In this case the magnetization is constant in magnitudeand rotates in direction as shown schematically in Fig. 1.The corresponding vector potential in polar coordinatesis A r = 0 , A θ = − JM .Let us assume, for simplicity, that without interactionwith magnetization the electron spectrum correspondsto that of isotropic (in plane) Dirac cone. Then, thedynamics of the low energy states of the system in thepresence of the vortex is described in polar coordinatesby the Hamiltonian H = i (cid:18) e − iθ (cid:0) ∂ r − i ∂ θ r + JM (cid:1) e iθ (cid:0) ∂ r + i ∂ θ r − JM (cid:1) (cid:19) . (1)where we have put (cid:126) = 1. From now on and since wewill not deal with interactions which can renormalize theFermi velocity [23] we will put it just to unity: v = 1.The usual ansatzΨ ( r, θ ) = e ilθ ϕ ( r ) , Ψ ( r, θ ) = e i ( l +1) θ ϕ ( r ) , (2)allows to write the Schrodinger equation H Ψ = E Ψ withΨ = (Ψ , Ψ ) † , as i (cid:18) ddr + l + 1 r + JM (cid:19) ϕ = Eϕ (3) i (cid:18) ddr − lr − JM (cid:19) ϕ = Eϕ , which can be reduced to (cid:20) d dr + lr ddr − l ( l + 1) r + JM (2 l + 1) r (cid:21) ϕ = (cid:2) ( JM ) − E (cid:3) ϕ. (4)for both components.Equation (4) is formally identical to the Schr¨odingerequation of a planar hydrogen atom with a nucleus ofcharge Ze [24, 25]: (cid:18) ∆ − m Ze r (cid:19) ϕ = − m(cid:15)ϕ, (5)with the substitution ε = E − ( JM ) , 2 mZe = JM (2 l + 1). From this formal analogy we can write theanswer for the eigenvalues of the energy: E ( n r , l ) = ( JM ) (cid:20) − ( l + 1 / ( n r + l + 1 / (cid:21) , (6)with n r = 0 , , .. .To ease the readout of the spectrum we show a plot ofthe numerical solution of eq. (1) in Fig. 2 (center). Theresults reproduce the analytical spectrum given in eq. (6)but only for one sign of the angular momentum. Thereare no bound states for the other sign. For comparison wealso plot the numerical results for the spectrum of a discas a function of the angular momentum in the presenceof a constant magnetic field (Fig. 2 (a)).We can see that for 2 l + 1 > JM > | E | < JM with accumulation points at the boundary values E = ± JM . The numerical solution shows also that the wavefunctions for the localized states decay exponentially atlarge distances. For any value of l there is a zero-energymode with n r = 0.For 2 l + 1 < FIG. 2: Top: Energy levels as function of angular momentumat the surface of a topological insulator in a constant in–planemagnetic field. Center: Energy levels in the presence of amagnetic vortex. Bottom: Energy levels in the presence of askyrmion configuration.
The natural spatial scale in the problem is ξ = (cid:126) v F | J | M which plays the same role as the correlation length forthe Abrikosov vortices. Skyrmion configuration
A novel situation arises when the (3D) magnetic coat-ing is in a skyrmion configuration [26]. 2D skyrmionpatterns can exist when the magnetic anisotropy is weak,and they have been recently observed with Lorentz trans-mission electron microscopy in a thin film of Fe . Co . Si[27]. The magnetization vector evolves from pointingalong a given direction at the center of the defect to theopposite direction at large distances. The absolute mag-netization does not change, and at intermediate distancesit has an in-plane component. A possible skyrmion tex- ture is M ( r, θ ) = (sin f ( r ) cos θ, sin f ( r ) sin θ, cos f ( r )) , (7)where f ( r ) can be chosen as a constant for simplicity.A numerical solution of the spectrum induced by askyrmion texture is shown in Fig. 2 (bottom). The cal-culation solves an effective one dimensional differentialequation for each value of the angular momentum[28].The figure also shows spectra for a constant magneticfield (top), and for the gauge field associated to a vor-tex (center). The skyrmion opens a gap in the spectrumand does not support a zero energy solution. Note that azero mode appears when the in-plane component of themagnetization is subtracted. CONCLUSIONS AND DISCUSSION
This work originated on the question of whether aneffective gauge field would be induced by strain on thesurface of a topological insulator. As it is known this isthe case in graphene which can be seen in many respectsof a precursor of the actual topological insulators. Itis known that elastic deformations and geometrical cor-rugations in graphene can be described by and inducedfictitious gauge field coupling to the electronic degrees offreedom [2]. The apparent paradox that the strains donot break time reversal symmetry while magnetic fieldsdo is solved in the graphene case by the fact that the in-duced fictitious field couples to the two Dirac cones withopposite signs and time reversal symmetry is preserved inthe complete system. This mechanism will probably oc-cur also in the case of weak topological insulators with aneven number of Dirac cones at the surface. The responseof a strong topological insulator to elastic deformationshas to be different and remains to be studied. In thiswork we have seen an alternative way to generate gaugefields in a strong topological insulator and we have stud-ied the modification of the energy spectrum caused bythe gauge fields.Magnetic coating of the surface three dimensional TIis often invoked to describe specific physical situationswhere a gap is needed in the 2D system [29–31]. Thediscussion presented in this work shows that in specificsituations the magnetic coating coating can give rise toa strong reorganization of the spectrum that has to betaken into account for each physical situation.
ACKNOWLEDGMENTS
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