A density wave in the topological Bi2Te3

Anjan Soumyanarayanan∗ Massachusetts Institute of Technology May 11, 2010

Abstract Topological insulators are materials with a bulk band gap, but with topologically protected, spin polarized surface states with Dirac dispersion. Bi2Te3 is such a with a single Dirac cone at the center of the Brillouin zone. ARPES studies have shown that the of Bi2Te3 changes from a circle to a hexagon, and then to a hexagram, when moving away from the Dirac point. This has been explained in literature by the addition of a warping term added to the Dirac Hamiltonian, which leads to more exotic spin texture. This manuscript elaborates on hexagonal warping in Bi2Te3, and examines a possible (SDW) state due to warping. Using Landau– Ginzburg formalism, we find that a correspondence between the SDW state and the cubic anisotropy problem. Using this, we construct a phase diagram of spin order in on the hexagonal Fermi surface of Bi2Te3.

CONTENTS critical phenomena and phase transitions with sponta- neous broken symmetries.

I. Topological Band Insulators 1 A. The Quantum Hall Insulator ...... 1 A. The Quantum Hall Insulator B. The Quantum Spin ...... 1 The quantum Hall state, discovered in 1980, was the C. 3D Topological Insulators ...... 2 first known macroscopic quantum state with no spon- taneously broken symmetry 4. The notion of order in II. A Warped Helical Fermi Surface 2 the quantum Hall state is topological in nature, and is

A. Fermi Surface Warping in Bi2Te3 ...... 2 insensitive to small perturbations (e.g. weak disorder) and smooth changes in tuning parameters (e.g. material III.SDW State in a Warped Dirac Cone 3 composition), unless the system passes through a quan- 4;5 A. Spin Density Waves ...... 3 tum .

B. Fermi Surface Nesting in Bi2Te3 ...... 4 Ordinary insulators are characterized by a Fermi en- C. Landau–Ginzburg Theory of SDW State ...... 4 ergy located in the bulk band gap. A two-dimensional D. Cubic Anisotropy in SDW State ...... 5 electron (2DEG) in a magnetic field has electrons undergoing cyclotron orbits, quantized to Landau lev- 6 IV. Conclusions 5 els . When a Landau level is filled, the Fermi energy is located between the Landau levels, separated by a gap References 6 h¯ ωc, where ωc is the cyclotron frequency. Despite being somewhat analogous to a bulk insulator, the Hall con- ductance of the quantum Hall system is perfectly quan- 2 I.T OPOLOGICAL BAND INSULATORS tized to σxy = n e /h. The robust quantization, even in the presence of disorder, was explained by the exis- The concept of broken symmetry has been central to tence of dissipationless states on the edge of the sample. . Systems of interest can be These chiral edge states are protected from backscatter- characterized by the symmetries that are spontaneously ing by a topological invariant known as Chern number, broken at a phase transition, e.g. translational symme- n 4. try for a crystalline and rotational symmetry for a magnet. Superfluids and superconductors that break a B. The Quantum more subtle gauge symmetry have been intensely stud- ied over the past century. All these systems can be de- The requirement of an external magnetic field can be scribed by the Landau–Ginzburg framework of phase circumvented by including a spin-orbit coupling term, transitions, where an order parameter emerges below L · S, which acts as an effective magnetic field, Beff. This 1–3 a critical temperature Tc . Landau–Ginzburg theory, was proposed in 1988 by Haldane for the honeycomb used in conjunction the renormalization group (RG) lattice 7, and later developed by Kane and Male as a technique 2, has been extremely successful in describing prediction of the quantum spin Hall (QSH) effect 8–10.

1 8.334 Term Paper SDW in the topological insulator Bi2Te3 May 11, 2010

Due to the chirality of spin-orbit coupling, the two edge persion, with the Dirac cones centered at the Kramers states with opposite spins counterpropagate. This leads points. Fu et. al. predicted BixSb1−x to be a 3D topo- 2 15 to a predicted quantized conductance σxx = 2 e /h. The logical insulator , and this was experimentally verified QSH effect was subsequently predicted and observed in by Hsieh et. al. using angle resolved photoemission HgTe quantum wells 11;12. spectroscopy (ARPES) with spin resolution 16;17. In 2009, Bi2Se3 and Bi2Te3 were discovered to be single Dirac cone topological insulators 13;18–20, with the cone located C. 3D Topological Insulators at the BZ Center, Γ, as shown in Fig. 1. Due to their much simpler bandstructure and large bandgap, Bi2Se3 and Bi2Te3 have been the subject of much experimental and theoretical investigation since then.

II.AW ARPED HELICAL FERMI SURFACE

(a) Schematic of TI bands (b) ARPES on Bi2Se3 FIG. 1: Bandstructure of a prototypical single Dirac cone topo- logical insulator. A schematic is shown in (a) with topologi- (a) Helical Dirac Cone in BZ (b) Fermi Surface of Bi Se cally protected surface states between bulk valence and con- 2 3 duction bands. The spin-polarized surface states can only in- FIG. 2: Helicity and spin momentum locking in a single Dirac tersect at Kramers points (k = 0, π/a) due to time reversal cone topological insulator, with schematic helical Dirac cone symmetry. An ARPES measurement of the bandstructure of in (a) and a spin-resolved ARPES measurement of the Fermi 13 19 Bi2Se3 by Xia et. al. is shown in (b). surface of Bi2−δCaδSe3 in (b) . The surface state momemtum, kx,y is locked to the spin σx,y due to the (k × σ)z term in the As Hall conductivity measured in an external mag- surface state Hamiltonian. netic field is odd under time reversal, it is topologi- Due to time reversal symmetry in a topological insu- cally distinct (topological class Z ) from the QSH state, 2 lator, the surface states at a given energy form Kramers where time reversal symmetry is unbroken 14. There is doublets (k , −k ). The Dirac cone is thus helical, a three-dimensional generalization of the QSH system ↑ ↓ and spin σ and momentum k are locked in the x − y with time reversal invariance (TRI). Physically, this cor- plane. For a mirror plane surface state, the Hamiltonian responds to a 3D bulk insulator with counterpropagat- reads 14 ing spin-polarized surface states. There are four Z2 in- Hsurface ∝ vF (k × σ)z (2) variants (ν0; ν1, ν2, ν3) associated with these 3D topolog- 15 ical insulators (TIs). ν0 = 1 defines a distinct phase The spin-momentum locking for Bi2Se3 seen by spin- called the strong topological insulator. resolved ARPES 19 is shown in Fig. 2. As expected, the Fermi surface is isotropic and spin-polarized, in agree- As the Z invariants require time reversal symme- 2 ment with Eqn. 3. Notably, the helical arrangement of try, the Hamiltonian for topological insulators is also spins around the Dirac cone leads to a geometric Berry’s required to be TRI. Kramers theorem requires that the phase of π. The spin polarization of the Fermi surface eigenstates of such TRI Hamiltonians have a two-fold has been found to persist up to room temperature, sug- degeneracy, and thus so do the surface states. How- gesting possible spintronics applications 19. ever for a crystal, there are certain TRI points in the Bril- louin Zone (BZ), where these states (k↑, −k↓) can cross, 15 viz. k = 0, ±π/a . Thus the lowest order TRI Hamil- A. Fermi Surface Warping in Bi2Te3 tonian for the surface states is 14 In contrast, ARPES studies of Bi2Te3 yielded slightly dif- ferent results. Chen et. al. 20;21 found that at energies ε Hsurface = −ihv¯ F σ · ∇ (1) significantly above the Dirac point energy εD, the Fermi Here, vF is the Fermi velocity and σ characterizes the surface of Bi2Te3 is no longer circular. It becomes hexag- fermion spin. Thus, the surface states have Dirac dis- onal at intermediate energies (ε − εD) and hexagram, or

2 Anjan Soumyanarayanan 8.334 Term Paper SDW in the topological insulator Bi2Te3 May 11, 2010

is TRI, and introduces the anisotropy that we require. Using Eqn. 4, the surface state dispersion becomes q 2 2 2 6 2 ε±(k) = ε0(k) ± v(k) k + λ k cos (3θ) (5)

Eqn. 5 contains the lowest order correction to the per- fect helicity of the Dirac cone in Eqn. 3. The cos2(3θ) FIG. 3: ARPES studies of the evolution Fermi surface of 21 term is has six-fold symmetry, due to the C3 symmetry Bi2Te3 . The Fermi surface evolves from an isotropic circle to a hexagon, and then to a hexagram (snow-flake shape) at of the Hamiltonian and time reversal symmetry. Since increasing energies away from the Dirac point (-335 mV). Hw couples to σz, we expect some out-of-plane spin 1 polarization at large enough (ε − εD) . A contour plot of Eqn. 5 generated using Mathematica c is shown in snow-flake like, at even higher energies, as shown in Fig. 4. We see that it reproduces the behavior seen in Fig. 3. The hexagonal deformation of the ARPES experiments by Chen et. al.rather well. We note density of states was confirmed by scanning tunneling that spin-orbit coupling is expected to be stronger in 21;22 spectroscopy (STS) experiments . Bi2Te3 than Bi2Se3, and thus the warping term Hw being The deformation of the Fermi surface was explained observed in the former, but not the latter, is not surpris- by Fu with addition of a warping term to the Hamil- ing. tonian 23 that introduces x − y anistropy. The isotropic surface state Hamiltonian in Eqn. 3 can be written as III.S PIN DENSITY WAVE STATE IN A  H0 ≡ vF kxσy − kyσx (3) WARPED DIRAC CONE An isotropic Dirac cone as in the linear dispersion in Noting that the Hamiltonian still has to obey the C 3 Eqn. 3 in uninteresting from the perspective of compet- crystal symmetry and TRI, the next order correction is ing order. However, the warping term H that intro- not quadratic in k, but is instead cubic 23 W duces a hexagonal deformation in the Fermi surface, can  lead to an instability in the Fermi surface. The helical H(k) = ε(k) + v(k) kxσy − kyσx + Hw nature of the Fermi surface leads us to consider the pos- λ   H = k3 + k3 σ (4) sibility of density waves with spin texture. w 2 + − z

A. Nesting Conditions and Spin Density Waves The one-dimensional electron gas has a Fermi surface consisting of two points separated by 2kF in momentum space. The response of the 1D gas to perturbations of the Fermi surface, known as the Lindhard function 6;24, is 2 q + 2kF χ(q) = −e $ (εF) log (6) q − 2kF

Here $ (εF) is the density of states at the Fermi energy εF. Thus the response function of a 1D gas diverges at wavevectors q = 2kF, and external perturbations lead to a divergent charge redistribution. Thus at zero temper- ature, the gas is unstable to the formation of a period- ically varying charge density or spin density 25;26, with the spatial periodicity λDW corresponding to 2kF π FIG. 4: A contour plot illustrating the evolution√ of the Fermi λDW = (7) k surface of Bi2Te3 using Eqn. 5, in units of v/λ using F Mathematica c . Note that it reproduces the behavior in Fig. 3. In higher dimensions, the number of states corre- The ε(k) term introduces particle-hole anisotropy and sponding to perfect Fermi surface nesting (q = 2kF) is 1 the second term corrects the Dirac velocity. The warp- The z-polarization hσzi has a periodicity of cos(3θ) around the ing term, Hw has the underlying C3 crystal symmetry, Fermi surface contour.

3 Anjan Soumyanarayanan 8.334 Term Paper SDW in the topological insulator Bi2Te3 May 11, 2010 significantly reduced, and thus the singularity in χ(q) C. Landau–Ginzburg Theory of SDW State (Eqn. ??) is removed. However, an anisotropic Fermi surface that does not disperse in certain directions re- sults in parallel regions of the Fermi surface (i.e. large 25 $ (εF)) separated by a nesting vector qDW . This leads to Fermi surface instabilities and a density wave in the qDW direction. Nesting across the Fermi surface with identical spins, i.e. ((k1, σ),(k2, σ)), leads to a (CDW) with no spin texture. In contrast, nesting with opposite spins, i.e. ((k1, σ),(k2, −σ)) leads to a spin den- sity wave (SDW) with no charge modulation. While a CDW state can be produced by electron- cou- FIG. 6: The three-component order parameter of spin-texture pling (Pierls distortion), a SDW requires significant for the nested hexagonal Fermi surface in Bi2Te3. The order electron-electron interactions (U). A textbook calcula- parameter components are illustrated for the nesting vector 25 tion finds the mean-field transition temperature to be K1 k e1.

MF We start by defining the nesting vector as kBTSDW ∼ exp(−1/U$ (εF)) (8)

Ki = 2kF ei, i = 1, 2, 3 (9) B. Fermi Surface Nesting in Bi Te 2 3 The order parameters of a spin-texture phase (e.g. SDW) would relate to the magnetization in the three direc- tions2. A density wave in momentum space corre- sponds to a particle-hole excitation cˆ† cˆ along the k+Ki k nesting vector Ki. We use this to define the three- component order parameter m~ i. We pick the three com- ponents to be parallel and perpendicular to Ki, and along z respectively, as illustrated in Fig. 6. Thus we write the components of m~ i as

m = hcˆ† e · σ cˆ i i,a ∑ k+Ki i k {k}BZ m = hcˆ† (e × e ) · σ cˆ i (10) i,b ∑ k+Ki z i k {k}BZ m = hcˆ† e · σ cˆ i i,c ∑ k+Ki z k FIG. 5: A near-hexagonal Fermi√ surface contour of Bi2Te3 gen- {k}BZ erated from Eqn. 5 for ε = 0.7 v/λ. The green arrows depict − the x y spin polarization around the surface. Parallel regions To define the Landau–Ginzburg free energy, we first of the Fermi surface support nesting in the directions labeled consider the symmetry operations that should leave the e . 1,2,3 free energy F invariant. From §II., we note that F should be invariant under three-fold rotation C , time- In Fig. 3, we see that the Fermi surface of Bi Te 3 2 3 reversal Θ, mirror reflection M and translation by an evolves from a circle to a hexagon, and then to a snow- √ arbitrary vector r, Tr. The components of the order pa- flake shape. In particular, for 0.55 < ε/ v/λ < 0.9, rameter, m transform under these symmetry opera- the constant energy contours are nearly hexagonal 23. i,α tions as follows As is clear in Fig. 5, there are three parallel regions of the Fermi surface, which support nesting, and thus the C3 : mi,α → mi+1,α formation of density waves in the directions (e , e , e ) 1 2 3 Θ : m → −m with magnitude 2k . We see in Fig. 5 that the surface i,α i,α F ∗ state flips spin across the nesting vector, it cannot sup- M : mi,α → mi+1,α (11) port a CDW. However this does allow us to consider the iKi·r Tr : mi,α → e mi,α possibility of a SDW state below a critical temperature 23 2 TSDW for finite electron-electron interactions . There is a finite spin-polarization in z due to HW (Eqn. 4).

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Due to TRI, F can only have even powers of mi,α, as can be seen from Eqn. 11. The lowest order term, apart from a constant, has to be quadratic in mi,α, i.e. of the ∗ form mi,α mj,β, which now also accounts for mirror sym- metry. The requirement of translational symmetry im- poses an additional factor of δij. Thus at quadratic order, the free energy can be written in terms of a susceptibility matrix, χα,β as

1 ∗ F0 = χα,β ∑ mi,αmi,β (12) 2 i FIG. 7: The phase diagram in (t, u) parameter space of the The susceptibility matrix changes at a critical tempera- anisotropic SDW instability of the Fermi surface. ture Tc from a positive definite form to having a neg- ative eigenvalue, and the ground state is in the SDW phase. The magnitude of the three order parameters {|m~ i|} being all equal in the ordered phase. A sixth or- (mi,a, mi,b, mi,c) are significantly different. Since the der term ensures stability in the phase diagram in the Fermi surface is helical with spin polarization in the b v < 0 region, given as direction, mi,b is likely to be the dominant term. The 2 ∗ ∗ ∗ ∗ 2 warping term HW makes the mi,c term non-zero. in F2 = u6(m1m2m3) + u6(m1m2m3) (15) comparison, mi,a is likely to be very small. This would correspondingly affect the elements of the susceptibility Considering the free energy up to sixth order F (6) = matrix χα,β. F0 + F1 + F2 and using the saddle point approxima- tion, we obtain the mean field phase diagram in the (t, v) plane shown in Fig. 7. The boundary between the D. Cubic Anisotropy in SDW State 2D SDW phase and the disordered phase corresponds to a second order transition, whereas the boundary be- To quadratic order, the free energy F is symmetric in 0 tween the stripe phase both the other phases is a first or- the order parameters {m~ }. We can have an SDW state i der transition. We note the presence of a tricritical point with the modulations in single or multiple directions be- in the phase diagram at t = v = 0. ing equally favorable in energy. This symmetry is bro- ken by the quartic term in the free energy. For the rest of this section, we ignore the spin-polarization index α IV.C ONCLUSIONSAND RELATED WORK to concentrate on the directional anistropy (in {ei}) of the SDW. Thus, we write the quadratic term of the free We have elaborated on the work of Fu 23 that explained energy as the evolution of the Fermi surface of Bi2Te3 as observed 2 by ARPES using a warping term, H in the Hamilto- F0 = t ∑ m~ i (13) rmW i nian. At intermediate energies (ε − εD), the Fermi sur- face is hexagonal, and following Fu, we have explored We would expect the quartic term to be of the form the possibility of spin density wave order below a mean 22 2 2 u ∑i m~ i . However, the cross terms m~ i m~ j are ex- field temperature Tc. We have studied the free energy of cluded due to translational symmetry and thus we get the SDW phase in the Landau-Ginzburg framework and the quartic correction find a close correspondence to cubic anisotropy. Using this, we have constructed a phase diagram of spin order 4 in on the hexagonal Fermi surface of Bi2Te3. F1 = v ∑ |m~ i| (14) i Looking to related topological insulators, the Fermi surface of BixSb1−x is much more complex. There are The quartic correction to the free energy (F1) breaks five surface states and a centrally deformed Dirac cone the SDW symmetry in the x − y plane, and corresponds that displays hexagonal warping 16;17;27. Due to the 2 to cubic anisotropy . For v < 0, the free energy is mini- complexity of its bandsstructure, BixSb1−x still lacks mized by stripe order, i.e. the alignment of the SDW as a tractable theoretical model, but it still displays a π a one-dimensional modulation along one of the nesting Berry’s phase and weak anti-localization as observed in 16;17 vectors ei. For v > 0, the energetically favorable order is the ‘simpler’ TIs – Bi2Se3 and Bi2Te3 . It is likely that diagonal, forming a two-dimensional lattice SDW with the nesting geometries would be more complex and also

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more exotic in BixSb1−x. [15] Liang Fu, C.L. Kane, and E.J. Mele. Topological Insulators in Three Dimensions. Physical Review Letters, 98(10):106803–4, There has also been some recent work in look- March 2007. ing at topological phases in the honeycomb lattice in 28;29 [16] David Hsieh, Dong Qian, L.A. Wray, Y. Xia, Y.S. Hor, R.J. Cava, the strongly correlated limit . Using the Hubbard and M. Zahid Hasan. A topological Dirac insulator in a quantum model, the authors find that in the large U limit, the spin Hall phase. Nature, 452(7190):970–4, April 2008. topological band insulator undergoes a phase transition [17] David Hsieh, Y. Xia, L.A. Wray, Dong Qian, A. Pal, J.H. Dil, J. Os- to a SDW state, with order in the x − y plane. There is terwalder, F. Meier, G. Bihlmayer, C.L. Kane, Y.S. Hor, R.J. Cava, an intermediate phase which makes this and M. Zahid Hasan. Observation of unconventional quantum spin textures in topological insulators. 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