Spin-textures, Berry's phase and Quasiparticle Interference in Bi2Te3: A Topological Insulator with Warped Surface States
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec PRL-Viewpoints: Physics , 108 (2009). http://physics.aps.org/articles/v2/108 Spin-textures, Berry’s phase and Quasiparticle Interference in Bi Te : A TopologicalInsulator with Warped Surface States M.Z. Hasan,
1, 2
H. Lin, and A. Bansil Joseph Henry Laboratories : Department of Physics, Princeton University, Princeton, NJ 08544 Princeton Center for Complex Materials, Princeton University, Princeton, NJ 08544 ∗ Department of Physics, Northeastern University, Boston, MA 02115 (Dated: November 6, 2018)The energy-momentum relationship of electrons on the surface of an ideal topological insulatorforms a cone - a Dirac cone, which, when warped (no longer described by the Dirac equation), canlead to unusual phenomena such as enhanced electronic interference around defects and a magnet-ically ordered broken symmetry surface. A detailed spin-texture and hexagonal warping maps onBi Te are presented here. Unlike common ordered states of matter such as thecrystalline solids and magnets characterized by some bro-ken symmetry, the topological states of quantum mat-ter are characterized by boundary properties that arehighly robust or stable in the presence of disorder, fluc-tuations, and perturbations [1, 2]. The first exam-ple of a two-dimensional topological insulator was pro-vided by the quantum Hall effect in an electron gas [1],which has no distinct three-dimensional topological gen-eralization [3, 4, 5, 6]. Recent theoretical and exper-imental work indicates that strong spin-orbit couplingin insulators can produce a three-dimensional topologi-cal insulator, which is a completely new state of matter[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. This state is char-acterized by a surface Berry’s phase, realized via an oddnumber of Dirac cones protected against disorder such asalloying [3, 4, 5, 6]. This is in sharp contrast to the caseof graphene, which possesses an even number of Diraccones.The thermoelectric Bi-Sb alloys discovered to be topo-logical last year provided the first example of a trulythree-dimensional topological insulator [7, 8, 9]. Theexperimental discovery of another thermoelectric fam-ily followed the Bi X (X=Se,Te) series - harboring thesame topological insulator states with a simpler surfacespectrum [10, 11, 12, 13]. The large band gap (0.3 eV)topological insulator Bi Se features a nearly ideal Diraccone [10, 12], but in the smaller band gap (0.15 eV) ma-terial Bi Te , with trigonal crystal potential, the coneis warped [11, 12]. Even though the cone is deformedwith hexagonal symmetry, the Berry’s phase quantifyingthe topological invariant remains unchanged, consistentwith its topological order [12, 13]. Now, in an article ap-pearing in the current issue of Physical Review Letters[14], Liang Fu of Harvard University in the US presents aperturbation calculation, which yields a hexagonal mod-ulation of the surface states analogous to the existenceof a nonlinear term in the spin-orbit interaction in the ∗ Electronic address: [email protected] bulk crystal. (See Fig. 1 for an illustration using ourresults of a full-potential, self-consistent computation inBi Te .) This leads to a number of remarkable predic-tions concerning the unusual physical properties inducedby the warping of the surface states, which are distinctfrom the simpler hydrogen-atom version of the topologi-cal insulator discovered previously [10].The spin and linear momentum of an electron on thesurface of a topological insulator are locked-in one-to-one,as revealed in recent spin-resolved photoemission mea-surements [8, 9, 12, 13]: the electron is forbidden frommoving backwards or taking a U-turn an effect that hin-ders its localization [15]. This can be realized in a realmaterial if the spins of electrons lying on the Dirac cone,but away from the apex (the Dirac node), rotate aroundin a circle at the Fermi level. Spins arranged in this waylead to a geometrical quantum phase known as Berrysphase with a value of π [8, 12]. The work of Fu effectivelyshows that in the presence of a hexagonally deformedcone, the spins must acquire finite out-of-the-plane com-ponents to conserve the net value of Berry’s phase, thuspreserving the bulk topological invariant. The resultingfinite value of the out-of-the-plane component opens upnew possibilities for observing spectacular quantum ef-fects.Fascinating classes of new particles such as a Majo-rana fermion, or a magnetic monopole image, or a mas-sive spin-textured Dirac particle can live on the surfaceof a topological insulator if a finite gap can be inducedin the surface band with the chemical potential near theDirac node [3, 4, 5, 6, 16]. One approach is to create aninterface between the topological insulator and a moreconventional material such as a magnet or a supercon-ductor, which can induce an energy gap in the Diracbands. Alternatively, a magnetic field can be applied per-pendicular to the sample surface. Until now it had beenthought that a parallel magnetic field would neither opena gap nor lead to a surface Hall effect. However, whenthe surface states are warped significantly in a truly bulkinsulating sample, as the work of Fu shows, a parallelmagnetic field would open up a gap in the surface spec-trum, enabling quantum Hall experiments to be carried FIG. 1:
Spin-textures, Berry’s phase and Quasiparticle Interference in Bi Te : (Left) In the simplest (hydrogen-atom) case, the energy-momentum relationship of the surface states in a topological insulator takes the form of a Dirac cone.The constant energy surfaces are then circles of different radii. (Right) Results of a first-principles computation of the spintexture of the surface states in Bi Te that we have carried out, showing in quantitative detail how the Dirac cone is warpeddue to the effect of the crystal potential. The actual shape of the Fermi surface is determined by the natural chemical potentialand can be a hexagon or a snowflake. As a result of deviations from the ideal Dirac cone dispersion, a nonzero out-of-the planespin polarization develops and the conventional nesting channels (red arrows) open up possibilities for magnetic order on thesurface. out using relatively modest magnetic fields (less than 7tesla) accessible in a typical laboratory setting. This isan exciting prospect since such experiments have so farnot been possible because very high magnetic fields areneeded for opening a gap.In a truly bulk insulating sample, where the chemicalpotential is such that the hexagonal warping of the Fermisurface survives, one would expect to see a rare realiza-tion of the quantum Hall effect on the surface withoutthe presence of Landau levels. In this way, one couldobserve a half-integer quantum of conductance per sur-face, which has not been possible so far in any knowntwo-dimensional electron gas, including graphene, andthus allow experimental tests of currently debated sce-narios for unusual antilocalization-to-localization tran-sitions on a topological insulator [5]. Presently real-ized Bi Te is quite bulk-metallic with small resistivity[11, 12], whereas spin-orbit insulators with high bulk re-sistivity values coupled with surface carrier control areneeded to realize a functional topological insulator withwhich many more interesting experiments could be pos-sible. Recent progress in making highly resistive Bi Se [12, 17] coupled with surface carrier control [18] heraldsthe potential dawn of a topological revolution. Similarimprovements in sample resistivity are also called for inBi Te . The hexagonal warping of the surface cone and theresulting deviations from a circular Fermi surface pro-vide a natural explanation for the enhanced magnitudeof surface electronic interference effects recently reportedaround crystalline defects in Bi Te [19, 20]. For a cir-cular Fermi surface (Fig. 1, left), the electronic states atthe Fermi surface cannot backscatter since in a fully spin-polarized band [12, 13] a consequence of time-reversalsymmetry is that there are no available states into whichelectrons can scatter elastically. As a result, interfer-ence effects around the defects are small. With hexago-nal warping, however, new electronic scattering channelsopen up, connected by various simple nesting vectors onthe Fermi surface (see Fig. 1), and consistent with recentscanning tunneling measurements, the scattering due todefects becomes enhanced [19, 20].The ideal topological insulators displaying a perfectDirac cone dispersion cannot support the formation ofcharge- or spin-density waves. However, in the presenceof hexagonal or other forms of deformations of the surfaceband, such ubiquitous ordering instabilities are no longerforbidden, but rather present a competing ground state,which can circumvent the topological protection. Sinceelectrons with opposite momenta possess opposite spins,a spin-density-wave state is favored if the Fermi surfacehas parallel segments that can be nested, as is seen to bethe case for hexagonal deformation (see Fig 1, right). Asone goes around the hexagonal Fermi surface, electronswill experience rapid changes in the out-of-the-plane spincomponent ( σ z ) to maintain their overall Berry’s phasevalue at π [8, 12], making the resulting spin-density waveappear quite exotic. A stripelike one-dimensional orderon the topological surface may also be energetically pos-sible.Even more complex spin-density-wave orderings couldbe supported by other topological insulators thatpresently lack a tractable theoretical model. For exam-ple, Bi − x Sb x alloys display a central hexagonally de-formed Dirac cone and additional surface states of com-plex character [8]. Despite the complexity of its five sur-face bands and hexagonal warping, the Bi − x Sb x alloytopological insulator precisely exhibits π Berry’s phaseand antilocalization just like the hydrogen-atom single-surface-Dirac-cone systems [8, 10, 12]. This is a remark-able manifestation and demonstration of a deep and pro-found principle of physics - topological invariance. Thusthe power and beauty of the topological order and invari-ance in nature are rather firmly established by studyingsystems that have complex geometries, i.e., complex andwarped surface states as in the Bi − x Sb x alloy series orBi Te . The most remarkable property of topological surfacestates is that they are expected to be perfect metals in thesense that they are robust and stable against most per-turbations and instabilities towards conventional orderedstates of matter. The warping of the surface states, eventhough it opens up interesting possibilities for study-ing this unusual state of matter in detail, ironically alsoopens up competing channels which provide a route fordeparture from the much sought after paradigm of per-fect topological stability of the spin-textured metal [3, 4].Most topological insulators lying hidden in nature arelikely to harbor complex and warped surface states as inBi − x Sb x and Bi Te .Interestingly, even though the first three-dimensionaltopological insulators and their spin-textures carryingnontrivial Berry’s phases were discovered only recently,we already have new materials emerging with physicalproperties lying in a regime where quantum effects as-sociated with competing orders such as the spin-densitywaves can be foreseen. 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Zhang, Physics FIG. 2:
Spin-textures, Berry’s phase and Topological Invariance in Bi Te :(Left) In the simplest (hydrogen-atom)case, the energy-momentum relationship of the surface states in a topological insulator takes the form of a Dirac cone. Theconstant energy surfaces are then circles of different radii. (Right) Results of a first-principles computation of the spin textureof the surface states in Bi Te that we have carried out, showing in quantitative detail how the Dirac cone is warped due tothe effect of the crystal potential. The actual shape of the Fermi surface is determined by the natural chemical potential andcan be a hexagon or a snowflake. As a result of deviations from the ideal Dirac cone dispersion, a nonzero out-of-the planespin polarization develops and the conventional nesting channels (red arrows) open up possibilities for magnetic order on thesurface. [M.Z. Hasan, H. Lin, A. Bansil, Physics2