Topological Fermi-arc surface resonances in bcc iron
TTopological Fermi-arc surface resonances in bcc iron
Daniel Gos´albez-Mart´ınez,
1, 2
Gabriel Aut`es,
1, 2 and Oleg V. Yazyev
1, 2, ∗ Institute of Physics, Ecole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland National Centre for Computational Design and Discovery of Novel Materials MARVEL,Ecole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland (Dated: November 21, 2018)The topological classification of matter has been extended to include semimetallic phases char-acterized by the presence of topologically protected band degeneracies. In Weyl semimetals, thefoundational gapless topological phase, chiral degeneracies are isolated near the Fermi level andgive rise to the Fermi-arc surface states. However, it is now recognized that chiral degeneracies areubiquitous in the band structures of systems with broken spatial inversion ( P ) or time-reversal ( T )symmetry. This leads to a broadly defined notion of topological metals, which implies the presenceof disconnected Fermi surface sheets characterized by non-zero Chern numbers inherited from theenclosed chiral degeneracies. Here, we address the possibility of experimentally observing surface-related signatures of chiral degeneracies in metals. As a representative system we choose bcc iron,a well-studied archetypal ferromagnetic metal with two nontrivial electron pockets. We find thatthe (110) surface presents arc-like resonances attached to the topologically nontrivial electron pock-ets. These Fermi-arc resonances are due to two different chiral degeneracies, a type-I elementaryWeyl point and a type-II composite (Chern numbers ±
2) Weyl point, located at slightly differentenergies close to the Fermi level. We further show that these surface resonances can be controlledby changing the orientation of magnetization, eventually being eliminated following a topologicalphase transition. Our study thus shows that the intricate Fermi-arc features can be observed inmaterials as simple as ferromagnetic iron, and are possibly very common in polar and magnetic ma-terials broadly speaking. Our study also provides methodological guidelines to identifying Fermi-arcsurface states and resonances, establishing their topological origin and designing control protocols.
I. INTRODUCTION
Band degeneracies in condensed matter have attractedconsiderable interest due to their relation to the topolog-ical character of metallic systems , while quasiparticlesin proximity to these band degeneracies emulate differenttypes of relativistic particles and even extend beyondthe conceived high-energy schemes . The energy dis-persion around a band crossing, intimately related to thesymmetries of the system, determines the nature of thequasiparticles. Such fermion quasiparticles are topologi-cally protected and become especially relevant when theband degeneracies are close to the Fermi level giving riseto the notion of topological semimetals .The most generic case of a band degeneracy occurs insystems where PT symmetry, the combination of inver-sion ( P ) and time-reversal ( T ) symmetries, is broken. Inthis situation, the Kramers degeneracy is lifted, and bandcrossings between two bands may be present at isolatedpoints in the three dimensional (3D) Brillouin zone (BZ).The band dispersion around the crossing point is gener-ally linear and is described by the 2 × H ( q ) = (cid:80) i,j q i ν ij σ j , where q = k − k isthe position in k -space relative to the degeneracy point k , ν ij are real coefficients that determine the chirality ofthe Weyl fermion and σ j are the Pauli matrices includingthe identity matrix. Such Weyl node degeneracies cannotbe easily gapped by small perturbations. The Berry cur-vature Ω( k ) presents a singularity when two bands aredegenerate. Therefore, a Weyl node acts as a monopoleof the Berry curvature in momentum space, carrying a chiral charge χ that is determined by the total flux ofΩ( k ) in units of 2 π through a surface enclosing the de-generacy. The only possibility to open a gap at a Weylpoint is by annihilating it with another Weyl point ofopposite chiral charge.When PT symmetry is preserved the energy bands aredoubly degenerate, thus any band crossing involves a 4-fold band degeneracy. In this case, the fermion quasipar-ticles around the band touching point are described bythe 4 × .Both Weyl and Dirac fermions can be divided into twodifferent types according to the morphology of the en-ergy dispersion around the band touching point. Type-IWeyl fermions present the standard conical energy dis-persion. Such band degeneracies create closed Fermi sur-faces which collapse into a single point when the Fermilevel is at the energy of the Weyl point. Type-II Weylfermions have a tilted conical energy dispersion. In con-trast to the type I , these degeneracies produce open elec-tron and hole Fermi surfaces which undergo a Lifshitztransition at the energy of the Weyl point . Band degen-eracies carrying chiral charges of χ = ± ± C or C rotation symmetry axes are characterized byeither quadratic or cubic energy dispersions in the planeperpendicular to the symmetry axis. Such band cross-ings might be considered as double or triple Weyl pointsstabilized by the rotation symmetry . a r X i v : . [ c ond - m a t . m t r l - s c i ] N ov This taxonomic classification of quasiparticles can befurther extended to include non-conventional relativis-tic fermions. For example, nonchiral nodal lines canoccur in the presence of mirror or non-symmorphicsymmetries . In these cases, the symmetry reducesthe co-dimension of the system allowing the bands to bedegenerate along lines instead of points . Furthermore,3-, 6- and 8-band crossing with either linear or quadraticband dispersion can appear at high symmetry points un-der specific space groups. Some of these degeneraciescarry a chiral charge and can be considered as the spin-1and spin-3/2 generalizations of Weyl fermions .Band degeneracies, due to their topological nature, areassociated with novel physical properties. Point nodespresent a new type of surface states referred to as theFermi arcs whose energy dispersion is defined betweenthe projections of two nodes of opposite chirality on thesurface Brillouin zone (sBZ) . At the Fermi level, afinite Fermi arc contour that starts and ends at the pro-jections of the two chiral nodes is produced. Likewise,lines of degeneracies have been shown to exhibit surfacestates with a drumhead shape . Furthermore, newtransport properties such as the chiral anomaly andnovel quantum oscillations due to the presence of Fermiarcs , are present when charge carriers are described bychiral fermions. Ideally, to observe these properties, theband degeneracies have to be close to the Fermi level, andisolated from the rest of the bands. The search for ma-terials that present isolated band crossings at the Fermilevel has been exhaustive and many candidates have beenproposed. For further details, we would like to refer thereader to recent review articles on this topic .In general, however, it is fair to say that band degen-eracies may appear at any energy, and are not neces-sarily isolated from other bands. In the case of metal-lic systems, Fermi surface sheets provide closed surfaceson which Chern numbers can be defined (Fermi surfaceChern numbers), therefore providing an avenue to theirtopological classification. In this context, a metal thathas at least a pair of Fermi surface sheets with Chernnumbers different from zero is considered a topologicalmetal . This definition of topological metal comprisesthe topological semimetals, which are the limiting casewith Fermi surfaces collapsing to single points.Charge carriers enclosed by the nontrivial Fermi sur-face sheets retain their chiral nature. Similar to the chiralnodes, surface projections of nontrivial Fermi sheets withopposite Chern numbers are connected by the Fermi arcs.In this case, the topological surface states are well definedonly in the regions of the sBZ where no bulk states areprojected. Unfortunately, in many metals with extendedFermi surfaces, most of the sBZ is covered by projectedbulk states, and it is likely that the Fermi arc crossesregions of the sBZ where bulk states are present. At sur-face momenta where both a surface state and bulk statesexist, the former hybridizes with the bulk states becom-ing a surface resonance that has a bulk character with astrong weight on the surface. This resonance presents a finite lifetime on the surface prior to penetrating into thebulk, which depends on the coupling between the surfaceand bulk states and is manifested in the broadening ofthe surface state. When interaction between the surfaceand bulk states is strong, the broadening of the surfaceresonance is such that the resonance is indistinguishablefrom the bulk states. This is one of the main difficultiesin observing Weyl semimetals experimentally since theWeyl points often coexist with other bands and are notat the Fermi level. The presence of the bulk states alsoobscures the exact origin of the Fermi arc.Given the fact that chiral band degeneracies are ubiq-uitous in metals with broken PT symmetry, one cananticipate observing resonances originating from thesechiral degeneracies. In this work, we search for suchtopological resonances in a prototypical ferromagnet–body-centered cubic (bcc) iron–by studying its surfaceelectronic structure. As it was previously shown inRef. 30, bcc Fe is a topological metal whose band struc-ture presents a large number of band degeneracies includ-ing single and double Weyl nodes as well as non-chiralnodal lines. Furthermore, the Fermi surface of iron con-sists of several sheets, most of which are topologicallytrivial except two Fermi surface pockets with Chern num-bers C = ± . Severalsurface resonances have been observed experimentally inbcc iron for different surface orientations , althoughno direct link between these surface resonances and topo-logical band degeneracies has been established.If one is able to unambiguously associate a surface res-onance with a chiral degeneracy, it would be possible todetermine the chiral nature of band degeneracies evenaway from the Fermi level. It is of particular interest foridentifying time-reversal symmetry broken Weyl nodesthat remain elusive with rare exceptions . To asso-ciate the origin of a surface resonance with the Fermi arccreated by two chiral degeneracies, one needs to deter-mine the exact origin in momentum space of the surfaceresonance, at least at one end of the Fermi arc. Fur-thermore, ideally, one seeks to be able to manipulatethe topological state of the system by external param-eters, i.e. being able of annihilate the chiral degeneraciesin order to observe a change of the surface resonanceupon a topological phase transition. In the case of mag-netic metals, the orientation of the magnetization natu-rally provides such a control parameter. In fact, it hasbeen shown experimentally that the Fermi surface of bcciron changes appreciably upon varying the magnetizationorientation . We will follow this strategy in our work toidentify the topological origin of the surfaces resonancesin this simple magnetic system.Below, we investigate a surface resonance at the (110)surface of bcc iron that we shown to originate from topo-logically nontrivial Fermi surface sheets. The correspond-ing Fermi arc that hybridizes with the projected bulkstates connects the Weyl point enclosed by a nontrivialFermi surface sheet with a double Weyl point of oppositechiral charge enclosed by one of the the trivial surfaces.Interestingly, this composite Weyl point is tilted in thedirection parallel to the magnetization showing that notonly linear band degeneracies can be found in both fla-vors, either type-I and type-II, but also the double Weylnodes. When the direction of magnetization is tiltedaway from the easy axis, the composite Weyl node splitsinto two linear type-II Weyl points. For specific orienta-tions of the magnetization, one of the two Weyl pointsenters into the nontrivial pocket changing the net chiralcharge inside the pocket to zero, and therefore, resultingin a topological phase transition. The studied surfaceresonance changes its morphology following the transi-tion, i.e. it emerges and submerges into a small electronpocket indicating the trivial nature of this pocket.The rest of the article is organized as follows. In Sec-tion II we give some basic definitions of berryology, andpresent a brief reminder on the electronic structure bcciron and its topological properties. Exceeding details canbe found in Ref. 30 while in this work we focus on thecomposite Weyl node responsible for the abovementionedtopological phase transition. In Section III, we describethe surface density of states of the (110) surface of bcciron and discuss the nontrivial resonance. In order togain an insight into the origin of the surface resonance,in Section IV we address the topological phase transitiondriven in by the orientation of the magnetization andits effect on the Weyl points and surface states. Finally,main results of our work are summarized in Section V. II. BCC IRON AS A TOPOLOGICAL METAL
In order to relate a surface resonance to a chiral degen-eracy in a metal, one needs to know the exact distributionof band degeneracies in momentum space. For the case offerromagnetic iron, the survey of band degeneracies hasalready been performed in Ref. 30. Here, for the sakeof completeness, we present a summary of the electronicstructure of bcc iron and associated band degeneracies,preceded by the description of computational methodol-ogy and relevant definitions.
A. First-principles methodology
The electronic structure of bcc iron presented in thissection, and in the rest of the article, has been obtainedfrom first principles using the plane-wave pseudopo-tential method implemented in the
Quantum-Espresso package . Magnetization parallel to the easy axis [001]and the experimental lattice parameter a = 2 .
87 ˚Awere chosen. The exchange-correlation potential was de- scribed using the PBE functional within the generalized-gradient approximation . The spin-orbit coupling(SOC) was included through the fully relativistic norm-conserving pseudopotentials fused in Ref. 30. A plane-wave kinetic energy cutoff of 120 Ry, a Fermi smearingof 0.02 Ry, and a 16 × ×
16 Monkhorst-Pack grid havebeen employed.Maximally localized Wannier functions (MLWFs) havebeen used to interpolate the first-principles band struc-ture to a higher k -space resolution and to obtain theHamiltonian in a localized basis for computing the sur-face density of states as described below. To constructthe MLWFs we used the Wannier90 code . Startingfrom the overlaps of 28 Bloch functions computed on a8 × × k -point mesh, we obtained 18 Wannier func-tions using as initial spinor projection the sp d hybridorbitals and the d xy , d xz and d yz atomic orbitals of theiron atom as described in Ref. 43.In Sections III and IV we will use the Hamiltonianin the MLWF basis set to compute the surface densityof states for different surface orientations. The surfacedensity of states is obtained from the Green’s function ofa semi-infinite surface computed by the method describedin Ref. 44. The Hamiltonian used for computing theGreen’s functions was obtained from a principal layer ofa supercell containing 6 iron atoms repeated in the [110]direction. B. Basic definitions
The topological nature of a metal is given by the pres-ence of band degeneracies. When two bands n and n (cid:48) are degenerate in k -space, the Berry curvature Ω n ( k ) ofband n presents a singularity, therefore this band degen-eracy acts as a monopole of this field. This fact can beexplicitly seen defining the Berry curvature asΩ nαβ ( k ) = − (cid:88) n (cid:48) (cid:54) = n (cid:104) u n k | H α ( k ) | u n (cid:48) k (cid:105)(cid:104) u n (cid:48) k | H β ( k ) | u n k (cid:105) ( (cid:15) n − (cid:15) n (cid:48) ) , (1)where | u n k (cid:105) is the cell-periodic part of the Bloch func-tion | ψ n k (cid:105) of band n , and H α ( k ) = ∂H ( k ) ∂ kα with H ( k ) = e i k · r He − i k · r .An equivalent vector definition of the Berry curvatureis Ω n ( k ) = ∇ × A n ( k ) , (2)where A n is the Berry connection A n ( k ) = i (cid:104) u n k |∇ k | u n k (cid:105) . (3)The relation between these two definitions of Berry cur-vature is Ω n,γ ( k ) = (cid:15) αβγ Ω n,αβ ( k ), where (cid:15) αβγ is thethree-dimensional Levi-Civita symbol. According to theChern theorem, the total flux of the Berry curvaturethrough a closed and orientable surface is quantized in P (b) /2− /2 (a) E ne r g y ( e V ) −5−4−3−2−101 FIG. 1. (a) Band structure of bcc iron with magnetization along the [001] direction and including SOC. The color indicates the s z spin polarization, referring to the the spin down (majority) and spin up (minority) polarizations in red and blue, respectively.(b) Fermi surface sheets of the six bands that cross the Fermi level. The Brillouin zone is viewed along the [110] direction.The color indicates the side of the surface that is in contact with electrons (blue) and holes (red). Each Fermi surface sheetis labeled as S n,i , where n indicates the band, and i the individual sheet. Due to the orientation of the BZ some sheets areprojected on top of each other, and hence both labels are given. units of 2 π , and defines a topological invariant called theChern number of a given surface. In metals, the sheetsof the Fermi surface define closed surfaces on which it ispossible to compute this topological invariant. One candefine the Fermi surface Chern number C n,i of the Fermisurface sheet i of band n as flux integral C n,i = 12 π (cid:90) S n,i Ω n ( k ) · n dS. (4)This quantity can be defined as the total chiral chargeenclosed by the S n,i surface . In the same fashion,the chiral charge of a single chiral node can be obtainedby computing the flux through a small surface that en-closes just that single point alone. The methodology forevaluating numerically these quantities can be found inRefs. 30, 45, and 46. C. Electronic structure
The magnetic point group of ferromagnetic bcc iron is4 /mm (cid:48) m (cid:48) for the magnetization pointing along the easyaxis [001]. It contains a C rotation symmetry axis par-allel to the orientation of the magnetization, a mirrorsymmetry plane perpendicular to the magnetization, i.e. the (001) plane, and two antisymmetric mirror reflectionplanes, (010) and (110). However, these symmetries canbe eliminated by changing the direction of the magneti-zation to an arbitrary orientation. This will later help usin analyzing the topological protection of different typesof band degeneracies.The band structure of ferromagnetic iron, shown inFigure 1a, is an ideal playground for illustrating a broad range of band degeneracies that can be found in con-densed matter systems. At first glance, we can see thatband degeneracies are ubiquitous in this band structure.In the case of ferromagnetic bcc iron, only accidental de-generacies can be found when SOC is included in thedescription . A systematic study of all the band de-generacies in bcc Fe was reported in Ref. 30. Here, weprovide a brief overview of the electronic structure andband degeneracies for the following two purposes: first,to illustrate the different types degeneracies that mightappear in PT broken systems, and second, to point outprecisely the degeneracies that lead to the appearance ofsurface resonances.Bcc iron has six bands that cross the Fermi level giv-ing rise to a complex Fermi surface. In Figure 1b weillustrate the Fermi surface sheets associated with eachband. Labels S n,i refer to sheet i of band n following thenotation introduced in Ref. 30. Most sheets surroundthe inversion symmetric points Γ, H and N, and con-sequently, their Fermi surface Chern numbers are zero.The only Fermi sheets that can give rise to Chern num-bers different from zero are the 6 electron pockets S ,i ( i = 2 , ..., S ,i electron pockets can be related bysymmetry according to the orientation of the magnetiza-tion. In the case considered here, for the magnetizationalong the easy axis [001], the six S ,i pockets are di-vided into two inequivalent subgroups. The four S ,i ( i = 2 , ...,
5) pockets in the k z = 0 plane are related bythe C rotation symmetry. The S , and S , pocketslocated on the k z axis are related by the M z mirror sym-metry. In fact, as it was shown in Ref. 30, these twopockets along the Γ H direction parallel to the magneti-zation vector enclose Weyl points, and consequently haveFermi surface Chern numbers C , = − C , = +1. (b)(a) E ne r g y ( e V ) /2 WP1 WP2 NL − −0.05 H H H H /2 FIG. 2. Details of the band structure of bcc iron along theΓ H direction (a) parallel and (b) perpendicular to the mag-netization. The color denotes the s z component of the spinpolarization. Elementary (WP1) and double (WP2) Weylnodes are indicated in panel (a). In panel (b), a band cross-ing related to a nodal line in the k z = 0 mirror symmetryplane is shown. The four electron pockets on the k z = 0 mirror symme-try plane are connected to the Fermi surface sheet S through a nodal line. Due to the presence of such degen-eracy on the surface, strictly speaking, one cannot definethe corresponding Chern numbers, although a tiny shiftof the magnetization can break the symmetry detachingthe Fermi surface sheets. D. Band degeneracies
Below, we will focus on the topologically nontrivialFermi surface sheets S ,i ( i = 2 , ...,
7) and band de-generacies in their neighborhood. Band 10 crosses bothbands 9 and 11 at different points in the BZ. However,in a region close to the electron pockets, the only bandcrossings of band 10 are with band 9 along the two non-equivalent Γ H directions as shown in Figure 2. Theseband crossings illustrate the different types of band de-generacies that are present in bcc iron. Elementary Weyl nodes.
The band degeneracy en-closed by electron pockets S , and S , is shown inFigure 2 and will be referred to as WP1. This degen-eracy at k z ≈ . π/a ) and 63 meV below the Fermilevel is a typical example of Weyl point of chiral charge χ = ±
1. It is characterized by a linear band dispersionin all directions, as one can see in Figures 2(a) and 3(a).Since this point and its mirror symmetric counterpart arethe only degeneracies enclosed by the S , and S , elec-tron pockets, the resulting Fermi surface Chern numbersof these pockets are − Double Weyl nodes.
An example of composite Weylnode can be found along the same axis at k z ≈ . π/a ) and 71 meV above the Fermi level (Fig. 2(b)). Closeto this band touching point that we denoted WP2, theenergy dispersion is linear along the k z direction, thatis the C rotation symmetry axis. In the perpendicularplane, however, the dispersion is quadratic as one cansee in Figure 3(b). This type of band degeneracy canbe considered as two elementary Weyl points broughttogether by the C rotation symmetry, as explained inRefs. 13 and 14, and consequently, such composite Weylpoints carry a chiral charge χ = ± k z axis with both involved bandshaving positive Fermi velocities (Fig. 2(b)). Hence, weclassify this band degeneracy as the type-II compositeWeyl node. We note that such classification into twotypes according to the morphology of the Fermi surfacearound the band degeneracy has been developed exclu-sively for the elementary Weyl nodes with linear disper-sion of bands. Generalization of such classification tocomposite Weyl nodes is not straightforward, and to thebest of our knowledge has not been done. In the caseof the composite Weyl nodes, the Fermi surfaces are de-scribed by algebraic surfaces of 4 th or 6 th order whoseclassification is not complete.We can illustrate the complexity of the morphologyof composite degeneracies by considering the low-energyHamiltonian in the neighborhood of a double Weyl nodegenerated by the C rotation symmetry. Following thenotation of Ref. 14, we write the Hamiltonian as H ( q ) = d ( q ) I + f ( q ) σ + + f ∗ ( q ) σ − + g ( q ) σ z , (5)where I is the identity matrix, σ ± = σ x ± iσ y , with σ being the Pauli matrices, and q = k − k with k be-ing the position of the nodal point. As it was shown inRef. 14, due the C rotation symmetry, the momentumdependence of functions f ( q ) and g ( q ) is given by f ( q ) = aq + bq − , (6) g ( q ) = lq z + c ( q x + q y ) , (7)with q ± = q x + iq y . The quadratic energy dependence inthe q x q y plane stems from both the f ( q ) and g ( q ) func-tions, while the linear dispersion along k z appears onlyin g ( q ). Following analogous reasoning, it is straightfor-ward to show that the momentum dependence of d ( q ) isidentical to that of g ( q ), therefore d ( q ) = mq z + n ( q x + q y ) . (8)The energy dispersion around the band degeneracy isgiven by E ± ( q ) = d ( q ) ± (cid:112) f ( q ) + g ( q ) = T ( k ) ± U ( k ).For elementary, linear Weyl nodes, this energy disper-sion produces Fermi surfaces described by a quadric formwith a singular point when chemical potential is locatedat the degeneracy. In this scenario, the type-I casewith energy dispersion given by a conical surface pro-ducing closed Fermi surfaces appears when T ( k ) < U ( k ).The type-II case, with open Fermi surfaces, occurs when (a) (b) (c)WP1 WP2 NL FIG. 3. Band dispersion around (a) the elementary Weyl point, (b) the double Weyl point and (c) the nodal line that arepresent close to electron pockets S ,i ( i = 2 , ..., T ( k ) > U ( k ) . This last condition implies that the coneis tilted. In the case of composite Weyl nodes, the di-vision into type I and type II according to the relationbetween T ( k ) and U ( k ) terms still holds. However, themorphology of the Fermi surface is much more compli-cated due to the momentum dependence upto the 4 th order, and especially due to the presence of two differenttilting terms mq z and n ( q x + q y ), that can result in amuch richer variety of crossing types.In our case, the velocities of the two bands along the z direction are both positive, thus indicating the tiltedscenario with the mq z term of the Hamiltonian beingdominant. Thus, bcc iron presents a type-II compositeWeyl node between bands 9 and 10. Volovik pointed outthat a type-II Weyl node involves a Lifshitz transitionbetween two Fermi surfaces at the Weyl node energy andthe exchange of the Chern numbers of these surfaces .In bcc iron, Fermi surface sheets S , and S presentsuch behavior as shown in Ref. 30. Nodal line.
The third type of band degeneracy isshown in Figure 2(b). This degeneracy located at k y ≈ . π/a ) corresponds to a single point in a locus ofpoints forming a nodal line in the k z = 0 mirror symme-try plane. In this plane, since the mirror symmetry M z operator and the Hamiltonian commute, the eigenstatesof the Hamiltonian can be labeled by the ± i eigenval-ues of the mirror symmetry operator. Therefore, bandswith different labels may cross each other along lines inthe symmetry plane. Nodal lines do not present a chiralcharge and show a certain dispersion in energy. At theFermi energy, the nodal lines pierce the Fermi surfaceat points where two Fermi surface sheets intersect. Thisbehavior is illustrated in Figure 3(c), where we show asmall portion of the energy dispersion of bands 9 (red)and 10 (blue) in the k z = 0 plane around the S , elec-tron pocket. One can see that the bands are degeneratealong a line (black line) that connects two Fermi contours(red and blue lines).In summary, we have shown that a metal as simpleas ferromagnetic bcc iron hosts a variety of band degen-eracies, from elementary Weyl points to the new type-IIcomposite Weyl points. These two chiral degeneracies play an essential role assigning bcc iron its topologicalcharacter. We will now demonstrate the possibility ofobserving topological surface resonances in metals evenwhen numerous bands cross the Fermi level. III. SURFACE STATES AND RESONANCES
Fermi arc surface states that connect pairs of Weylpoints of opposite chirality appear in the surface Brillouinzone if these two points are not projected onto each other.When the Fermi level is not precisely at the Weyl pointenergy, the arc-like surface states instead connect pro-jected Fermi surface sheets characterized by the oppositeChern numbers . The Fermi arcs emerge tangentiallyfrom the Fermi surfaces since the velocity of the surfacestates has to match the one of the projected bulk states .In general, these surface states can overlap with the pro-jection of the bulk states present at the Fermi level. Inthis situation, the surface state becomes a surface res-onance hybridized with the bulk states. Indeed, this isan inherent difficulty encountered when identifying suchsurface states since the exact locations of the star and endpoints of the Fermi arc are obscured by the projectionsof bulk states.As discussed above, bcc Fe is a topological metal thathas two disconnected Fermi surface sheets with Chernnumbers C = ±
1. However, due to its metallic character,the sBZ is almost entirely filled by the projected bulkstates. One would naturally ask a question whether it isstill possible to observe a surface state, or most likely, asurface resonance whose origin can be traced back to thepresence of these two nontrivial electron pockets.To address this question, we investigated surfacesstates at the (100) and (110) surfaces, paying particu-lar attention to the regions where the nontrivial electronpockets are projected. While at the (110) surface wewere able to identify the small electron pocket amongthe other projected bulk states, on the (100) surface, thebulk states do not allow us to distinguish the S , and S , electron pockets. Therefore, below we will addressonly the (110) surface. −2.19−1.46−0.730.000.731.462.19−1.55 −0.77 0.00 0.77 1.55 (b)(a) (c) FIG. 4. (a) Surface density of states calculated for the (110) surface of bcc iron. (b) Atomic structure of the (110) surfaceand the definition of the magnetization vector. (c) Relation between the bulk and surface Brillouin zones.
A. Surface density of states
The surface density of states (DOS) of the (110) sur-face of bcc iron with the magnetization along the [001]direction is shown in Figure 4(a). In this plot, it is pos-sible to identify the projected bulk states from most ofpartially occupied bands according to the shapes of theirFermi surface sheets (see Fig. 1(b)). In addition, in Fig-ure 4(b) we illustrate the (110) surface and define theorientation of the magnetization. The relation betweenthe bulk BZ and sBZ is given in Figure 4(c).A higher density of projected bulk states is centeredaround the N point of the sBZ as shown in Figure 4(a).According to the Fermi surface sheets in Figure 1(b),bands 5, 6, 7 and 8 are present at point N , althoughthe contribution of each band is difficult to resolve. Inaddition, there are three distinct features that can beassociated with particular sheets of the Fermi surface.First, the tubular structure produced by band 8 ( S ) isprojected around the perimeter of the sBZ crossing thecentral part of the sBZ along the N − Γ − N path. Second,we can identify the projection of the bulk states of band 9enclosed by S centered around the Γ point. These statescover most of the sBZ nearly touching the tubular regionfrom band 8 at different points close to the middle of the P H lines. Third, we observe the contribution from thebulk states of band 10 that correspond to the diamond-shaped region centered at Γ, and four small nearly circu-lar features in the middle of the Γ H and Γ N lines. Thesefeatures correspond to the contributions originating frompockets S , and S ,i ( i = 2 , ..., S ,i ( i = 2 , ...,
5) trivial electronpockets are projected in pairs onto the same location thesBZ along the N − Γ − N line, while the two nontrivial S , and S , electron pockets are projected separatelyonto the H − Γ − H line.Next, we focus on the features that correspond to sur-face states or surface resonances observed in Figure 4(a).In the area around the N points, the contribution of theprojected bulk states to the surface density of states ishigh, and therefore, it is difficult to distinguish possi-ble surface resonances. However, in the rest of the sBZthere are three different regions where surface states orresonances can be identified. First, we notice the pres-ence of a diamond-shaped surface resonance that encir-cles the Γ point and the projection of the S , Fermi sur-face pocket. Since this surface resonance forms a closedloop, we conclude its topologically trivial origin. Fur-thermore, we found that this state is susceptible to thedetails of wannierization process, especially to the mini-mization of the spread function of the Wannier functions,which points to its possible relation to the surface dan-gling bonds. Second, in the regions indicated with blacksquares in Figure 4(a), where the projected bulk statesof bands 8 and 9 nearly touch each other, appear twodifferent surface states. These states behave differentlyin the left and the right parts of the sBZ (see insets inFigure 4(a)). The surface states on the left side of thesBZ emerge from the tubular surface S and immerseagain into S , while the surface states that appear onthe right side of the sBZ emerge from the S Fermi sur- E F − 70 meV E F + 50 meV E F E WP FIG. 5. Evolution of the surface resonance related to pockets S , and S , upon changing chemical potential µ . face sheet and connect back to S . Since these surfacestates emerge from the projected bulk states, it is diffi-cult to related their origin to the presence of any Weylpoint, thus not allowing to argue about their nature. In-terestingly, the existence of a surface resonance close tothe Fermi level located at about three quarters of the Γ S line has been observed experimentally . Third, the mostinteresting region is between the projections of pockets S , and S , , indicated with a red square in Figure 4(a)and shown in detail in Figure 5(c). One can observe, onone hand, a short surface state that emerges from theprojection of the bulk states of bands 8 and 9. Again,since this surface state emerges from two trivial S and S surfaces, it is difficult to argue about its nature andpossible relation to any band degeneracy. On the otherhand, we can also notice a surface resonance emergingfrom the projection of the nontrivial pockets S , and S , . Below, we will demonstrate that this surface reso-nance originates from the chiral degeneracies in bcc iron. B. Resonances at electron pockets S , and S , In Figure 5 we plot the surface DOS at different val-ues of chemical potential µ in the region of the sBZ wherethe states originating the S , pocket are projected. Theprojected bulk states correspond to three different bands.The lower part of each panel in Figure 5 is occupied bythe states of band 9 corresponding to the S sheet, whilein the upper part bulk states of band 8 are present. Inthe middle of the plots, it is possible to observe the pro- jected states from the small electron pocket S , . Theelectron pocket and the projected states of band 9 areconnected through a resonance that emerges from bothsurfaces tangentially. Below, we will prove that this res-onance originates from a band degeneracy enclosed bythe S , sheet. It is also possible to observe a surfacestate that connects the projections of band 9 and 8. Asexplained previously, this surface state cannot be unam-biguously associated to any chiral degeneracy, therefore,we will focus our attention only on the small resonancethat connects the projected states of bands 9 and 10.The first evidence of the topological origin of this sur-face resonance is revealed by the manner it is attached tothe projected bulk states of the S and S , sheets. Aspointed out by Haldane , the contact to the projectedsurfaces is tangential, so the Fermi velocities of the sur-face and bulk states are equal. However, the situationconsidered here is slightly different to the simple sce-nario with only two Weyl nodes and a Fermi arc connect-ing the two nontrivial Fermi surfaces. In the discussedcase, the surface resonance connects a nontrivial surfacewith a trivial one. One possible scenario to explain thisdiscrepancy could be that the resonance hybridizes withthe projected bulk states on the trivial surface, and re-emerges on the opposite side of the sBZ to connect withthe complementary nontrivial electron pocket S , .A surface state that gives rise to theFermi arcs can havea complex energy dispersion, but it must follow the evo-lution of the Fermi surface and intersect the Weyl points.Accordingly, the surface resonance that emerges from S , should follow this trend if it has topological origin.In order to address the evolution of the surface resonanceacross the degeneracy energy, we have computed the sur-face DOS for different values of chemical potential µ closeto the Fermi level E F . Figure 5 shows the surface DOSfor chemical potentials at E F −
70 meV, the exact energyof the
W P E W P = E F −
63 meV, E F and E F +50 meV (panels a–d, respectively). In Figure 5(a) weobserve that the surface resonance emerges from the leftside of a small hole pocket. Note that at E F −
70 meVthe chemical potential is below E W P , thus the Fermisurface pocket S , corresponds to hole pocket. Uponincreasing µ to E W P , the Fermi surface S , shrinksinto a single point in the sBZ from which the resonanceemerges (Fig. 5(b)). Finally, at higher energies the elec-tron pocket extends, and the resonance emerges tangen-tially from the right side of the projected S , pocket(Figs. 5(c) and 5(d)). In all these cases the resonanceterminates in surface S of the projected states of band9. This behavior indicates that the resonance follows theexpected behavior of a Fermi arc, especially at µ = E W P (Fig. 5(b)) where it emerges from the projection of the W P
W P k z direction in the region close to Weylpoints W P
W P
2. It is possible to observe a dif- E ne r g y ( e V ) −0.15 −0.10 −0.05 z (2 FIG. 6. Energy dispersion of the surface density of statescalculated for the (110) surface of bcc iron along the k z di-rection. ference in contrast in those regions where the numberof projected bands change. In particular, one can distin-guish the contributions of the bands that give rise to Weylpoints W P
W P
W P
1. Thissurface resonance disperses towards higher energies andmerges with the projected band structure close to Weylpoint
W P
2. Such energy dispersion suggests that thesurface resonance connects Weyl nodes
W P
W P
W P . IV. TOPOLOGICAL PHASE TRANSITION
The possibility of manipulating and switching betweentopological phases by means of external parameters is anappealing idea from the point of view of potential ap-plications. For example, the electronic structure of fer-romagnetic systems can be easily controlled by changingthe orientation of the magnetization by external magneticfield due to the spin-orbit coupling. This effect has beenmeasured in bcc iron by M(cid:32)ly´nczak et al. . Here, we willshow that a topological phase transition in bcc Fe can beinduced by manipulating adiabatically the orientation ofthe magnetization. A. Evolution of band degeneracies upon changingthe magnetization direction
When the magnetization of a bcc ferromagnet pointsalong an arbitrary direction, there are no symmetry oper-ations present . In this situation, the nodal lines and thecomposite Weyl nodes in bcc iron are no longer protectedby the mirror and C rotation symmetries, respectively.The nodal lines transform into individual Weyl pointswhen the magnetization is tilted with respect to the easyaxis [001] . Likewise, the two double Weyl nodes W P S , and S , split into pairs of elementary Weyl points ( W P I and W P II ). Due to the proximity in k -space of the twopoints W P
W P
2, we will have to follow the pathsof all three Weyl nodes (the
W P
W P
2) upon adiabaticchange of the magnetization vector.We define the magnetization vector by the polar andazimuthal angles θ and φ in spherical coordinates (seeFig. 4(b)). We have analyzed the positions of the Weylpoints at ten equidistant values of both the polar angle θ ∈ [0 ◦ , ◦ ] and the azimuthal angle φ ∈ [0 ◦ , ◦ ]. Inall the cases the double Weyl point W P
W P I and W P II upon tilt-ing the magnetization vector. As θ changes from 0 ◦ to90 ◦ , the W P k z values while the W P I and W P II points move away from each other.The W P II point displaces to higher values of k z and W P I tends to move to smaller values of k z approachingthe W P θ = 0 ◦ and an intermediate value θ = 50 ◦ . In Figure 7(b) we plot the energy dispersionin the k x k y plane around the ( k x , k y ) = (0 ,
0) point andat different values of k z . For the magnetization vectororiented along the easy axis [001], i.e. θ = 0 ◦ , bands 9(red) and 10 (blue) touch only at points k z = 0 . π/a and k z = 0 . π/a ), which correspond to Weyl nodes W P
W P
2, respectively. The linear dispersion ofthe elementary Weyl node
W P
FIG. 7. (a) Schematic illustration of the evolution of Weyl points
W P
W P k x k y plane at different values of k z for θ = 0 ◦ and θ = 50 ◦ , φ = 50 ◦ . persion of the composite Weyl node W P θ = 50 ◦ (in this case φ = 50 ◦ ), the W P k x , k y ) = (0 ,
0) while the k z valueincreases slightly to k z = 0 . π/a ). The W P I and W P II degeneracies are observed at k z = 0 . π/a )and k z = 0 . π/a ), respectively, both showing a lin-ear band dispersion. The W P I point displaced to theleft of the origin at ( k x , k y ) = (0 , W P II movedin the opposite direction and towards higher values of k z .As Weyl points W P
W P I with opposite chi-ral charges approach each other upon increasing θ , thesedegeneracies eventually meet and annihilate. This gen-eral behavior is similar for different azimuthal angles φ , FIG. 8. Illustration of the Lifshitz transition involving Fermisurface sheets of bands 9 and 10 driven by the change of themagnetization direction. although the exact value of polar angle θ at which theWeyl point annihilation takes place varies with φ . Be-fore the annihilation of the two Weyl points, W P I hasto enter the nontrivial electron pocket S , . This pro-cess is realized through a Lifshitz transition between thetwo Fermi surface sheets S , and S since W P I is atype-II Weyl node tilted toward the k z axis akin to theoriginal double node W P
2. At the Lifshitz transition thetwo Fermi surface sheets exchange Chern numbers withthe net value of the chiral charge of
W P I . Note thatthe effective change of the Fermi surface Chern numberof S is zero because it is compensated by the exchangewith the other nontrivial Fermi surface sheet S , . Af-ter the transition, the topological state of bcc iron haschanged since the net Chern number of S , is now zerobecause this Fermi surface sheet no longer contains anychiral degeneracy.Figure 8 illustrates the Lifshitz transition upon chang-ing θ for a fixed value φ = 50 ◦ . We plot the Fermi con-tours by slicing along the (001) and (110) directions at k x = k y = − . π/a ). The Fermi contours createdby bands 9 (red) and 10 (blue) are initially separatedat θ = 0 ◦ . As the value of θ increases, the Fermi con-tour of band 10 expands and eventually touches that ofband 9 at θ ≈ ◦ . The two Fermi contours detach uponfurther increase of θ . According to our calculations, thetwo Weyl points W P
W P I annihilate each othershortly after the Lifshitz transition. B. Evolution of the surface resonance uponchanging the magnetization direction
Finally, we address the evolution of the surface res-onances that emerge from the nontrivial electron pock-ets S , and S , across the topological phase transi-tion as a final confirmation of their topological origin.The change of the orientation of magnetization would1 FIG. 9. Evolution of the surface density of states around electron pocket S , upon changing θ at a fixed φ = 50 ◦ . not eliminate the surface resonances as long as the Fermisurface Chern numbers of pockets S , and S , do notchange. A qualitative change of the surface resonanceswould be observed only following the discussed topolog-ical phase transition.Figure 9 shows the surface DOS in the region of thesBZ around the projection of the S , pocket for differ-ent angles θ at a constant φ = 50 ◦ . At θ < ◦ the sur-face resonance connects the projections of the S , and S Fermi surface sheets. As θ increases, the projectionsof these two Fermi surface sheets approach each other,and the surface resonance shrinks accordingly. At theLifshitz transition point, the projections of both surfacesare linked at the projection of the W P I Weyl point, andthe surface resonance connects the S , pocket projec-tion with that of W P I . This is another indication thatthe W P
W P I points are linked by a Fermi arcthat gives rise to the surface resonance. At θ > ◦ theprojections of the two Fermi surface sheets disconnectfrom each other, while the surface resonance persists asa trivial state that starts and ends in the projection ofthe now trivial S , sheet.The evolution of the shape and connectivity of thesurface resonance can be understood in terms of thecriteria for detecting Fermi arcs given in Refs. 21 and51. These criteria consist in counting the number oftimes surface states intersect a closed path at constantenergy in the sBZ. A surface state counts as +1 or − θ < ◦ we can find a path that encloses the projected S , sheet and intersectsthe surface states only once. Past the topological phasetransition for θ > ◦ , the surface state is attached to S , from two sides, so we cannot find a path thatdoes not cross the projection of S , and intersects thesurface states an odd number of times. V. SUMMARY
We addressed the possibility of observing surface res-onances originating from the Weyl point band degenera-cies and nontrivial Fermi surfaces in a well-studied pro-totypical ferromagnet – bcc iron. We have shown thatbcc iron can host a surface resonance at the (110) sur-face located along the Γ H direction. This surface reso-nance is related to the Fermi arc that connects two Weylpoints of different types close to the Fermi level. Wefurther demonstrated that it is possible to manipulatethis Fermi-arc surface resonance inducing a topologicalphase transition driven by the change of the magnetiza-tion orientation. Our study thus shows that Fermi-arcsurface features can be observed in materials extendingfar beyond Weyl semimetals, and are possibly very com-mon in a broad range of polar and magnetic compounds.Our work also establishes the methodology for identify-ing Fermi-arc surface states and resonances, proving theirtopological origin and designing control protocols. ACKNOWLEDGMENTS
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