Hexagonal Warping Effects in the Surface States of the Topological InsulatorBi2Te3

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From λ and v, E* = 0.23 eV and EF = 1.2E* = 0.28 eV, which agrees fairly well with the measured Fermi energy 0.25 eV.

“From that we find E ∗ = 0 . 23 eV , and E F = 1 . 2 E ∗ = 0 . 28 eV which agrees fairly well with the measured Fermi energy 0 . 25 eV (shown in Fig.1.ii).”

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The out-of-plane spin polarization sz is momentum-dependent, given by sz = cos(3θ)/sqrt(cos^2(3θ)+1/(ka)^4), and can reach as high as 60% of the full polarization along ΓK for the Fermi surface in Fig.1.

“s z is then calculated from (3): s z = cos(3 θ ) / √ cos 2 (3 θ ) + 1 / ( ka ) 4 . The outof-plane spin polarization is momentum-dependent and can reach as high as 60% of the full polarization along Γ K for the Fermi surface in Fig.1.”

• AssumptionRests on an assumption: The pseudo-spin σ_i is proportional to the electron's spin: <sz> ∝ <σz> and <sx,y> ∝ <σx,y>.

Classified as probably not a substantial claim.

A single two-dimensional Dirac fermion state has been observed on the surface of the topological insulator Bi2Te3 by angle-resolved photoemission spectroscopy (ARPES).

“A single two-dimensinoal Dirac fermion state has been recently observed on the surface of topological insulator Bi 2 Te 3 by angle-resolved photoemission spectroscopy (ARPES).”

• cites [4] Y. L. Chen et al. , Science 325 , 178 (2009) sourceNot checked yet

  Not checked in the free run.

• cites [5] D. Hsieh et al. , Nature 460 , 1101 (2009) sourceNot checked yet

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Higher order terms in F break the U(3) symmetry and pick out a particular spatial ordering pattern.

“We now show that higher order terms in F break the U (3) symmetry and picks out a particular spatial ordering pattern.”

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For a non-convex constant energy contour above Ec, multiple pairs of stationary points exist, and inter-pair scattering is allowed, so the LDOS oscillation is restored at leading order.

“Now consider a non-convex constant energy contour above E c . As shown in Fig.3b, multiple pairs of stationary points exist. Since inter-pair scattering is still allowed, the LDOS oscillation will be restored at leading order.”

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The k·p Hamiltonian H(k) up to third order in k includes a particle-hole asymmetry term E0(k)=k^2/(2m*), a second-order correction to the Dirac velocity v_k=v(1+αk^2), and a hexagonal warping term H_w.

“We then find that H ( ⃗ k ) must take the following form up to third order in ⃗ k :”

• Also stated in: § Hexagonal Warping Effects in the Surface States of Topological Insulator Bi 2 Te 3 · p. 2 · plain

  “where E 0 ( k ) = k 2 / (2 m ∗ ) generates particle-hole asymmetry and the Dirac velocity v k = v (1 + αk 2 ) contains a second-order correction. The last term in (3), which we call H w , is most important.”

• AssumptionRests on an assumption: Bi2Te3 has a rhombohedral crystal structure with space group R-3m.

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The free energy to second order has an emerging U(3) symmetry, so single- and multiple-Q SDWs are degenerate.

“The free energy (8) to second order has an emerging U (3) symmetry φ iµ → U ij φ jµ . So single- and multipleQ SDWs are degenerate.”

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Alpichshev et al. imaged with STM the standing wave of surface states on Bi2Te3 near a line defect; the LDOS oscillation was found to exist in the energy range with snow-flake like constant energy contour, but strongly suppressed in the range with circular constant energy contour.

“During the final stage of this work, we learned that Alpichshev et al.[19] imaged with STM the standing wave of surface states on Bi 2 Te 3 near a line defect (instead of a point defect considered in this work). The LDOS oscillation was found to exist in the energy range with snow-flake like constant energy contour, but strongly suppressed in the range with circular constant energy contour.”

• cites [19]Not checked yet

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For a convex constant energy contour below Ec, only a single pair of stationary points exist at k and -k; because they carry opposite spins, scattering between them is forbidden by time reversal symmetry, so the LDOS oscillation vanishes at leading order.

“For a convex constant energy contour below E c as shown in Fig.3a, only a single pair of stationary points exist at ⃗ k and -⃗ k . However, since the two states at ⃗ k and -⃗ k here carry opposite spins, scattering between them is forbidden by time reversal symmetry-a fundamental property of surface states on a topological insulator. The LDOS oscillation then vanishes at leading order.”

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The leading order correction to the surface band dispersion near Γ due to H_w starts at fifth order in k and is angle-dependent.

“(4) predicts that the leading order correction due to H w starts at fi fth order in k and is angle-dependent:”

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Since the surface states at k and -k have opposite spins, a charge-density wave cannot connect them and is thus disfavored.

“Since the surface states at ⃗ k and -⃗ k have opposite spins, a charge-density wave cannot connect them and is thus disfavored.”

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Since spin polarization along ΓM has been found to be almost 100% polarized in ARPES, the doublet at Γ are almost pure spin eigenstates, i.e., sz ≈ <σz>.

“Since spin polarization along Γ M has been found to be almost 100% polarized in a very recent ARPES experiment[5], we conclude that the the doublet | ψ ↑ , ↓ 〉 at Γ are almost pure spin eigenstates, i.e., s z ≈ 〈 σ z 〉 , which agrees with a theoretical band structure calculation[16].”

• cites [5] D. Hsieh et al. , Nature 460 , 1101 (2009) sourceNot checked yet

  Not checked in the free run.

• cites [16] H. Zhang et al. , Nat. Phys. 5 , 438 (2009) sourceNot checked yet

  Not checked in the free run.

• AssumptionRests on an assumption: The pseudo-spin σ_i is proportional to the electron's spin: <sz> ∝ <σz> and <sx,y> ∝ <σx,y>.

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For u > 0, |ξ1| = |ξ2| = |ξ3| in the ordered phase, so the SDW forms a two-dimensional lattice.

“For u &gt; 0, | ξ 1 | = | ξ 2 | = | ξ 3 | in the ordered phase, so that SDW forms a twodimensional lattice.”

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Although the Hamiltonian H is threefold invariant, the band structure is six-fold symmetric under θ → θ + 2π/6 because of time reversal symmetry.

“Although the Hamiltonian H is threefold invariant, the band structure is six-fold symmetric under θ → θ +2 π/ 6 because of time reversal symmetry.”

• AssumptionRests on an assumption: Bi2Te3 has a rhombohedral crystal structure with space group R-3m.

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H_w vanishes in the mirror-symmetric direction ΓM because σz is odd under mirror, and reaches maximum along ΓK.

“We note that H w ( ⃗ k ) vanishes in mirror-symmetric direction Γ M , because σ z is odd under mirror, and H w ( ⃗ k ) reaches maximum along Γ K .”

• AssumptionRests on an assumption: Bi2Te3 has a rhombohedral crystal structure with space group R-3m.

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In an in-plane magnetic field B||, the Dirac point is shifted away from Γ to k* = g|| z×B|| /v, and a mass term Mσz = (g||B||)^3 sin(3φ)σz/E*^2 is generated at k*, opening an energy gap.

“From (3) we find the Dirac point is shifted away from Γ to ⃗ k ∗ ≡ g ‖ ˆ z × ⃗ B ‖ /v . In addition to that, a mass term is generated at ⃗ k ∗ : Mσ z = ( g ‖ B ‖ ) 3 sin(3 ϕ ) σ z /E ∗ 2 ( ϕ is the angle between ⃗ B ‖ and Γ K ), which opens up an energy gap.”

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The hexagonal warping term H_w is only invariant under three-fold rotation and is solely responsible for the hexagonal distortion of the otherwise circular Fermi surface.

“Unlike the other terms, H w is only invariant under three-fold rotation (as the Bi 2 Te 3 crystal structure does) and therefore is solely responsible for the hexagonal distortion of the otherwise circular Fermi surface.”

• AssumptionRests on an assumption: Bi2Te3 has a rhombohedral crystal structure with space group R-3m.

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At fourth order, the anisotropy term F4 = u sum |ξ_i|^4; for u < 0, only one φ_i is nonzero and the SDW forms a one-dimensional stripe that breaks C3 but is invariant under mirror symmetry.

“At fourth order, we find an anisotropy term F 4 = u ∑ 3 i =1 | ξ i | 4 . The sign of u determines the relative weight of φ i in the ordered phase. For u &lt; 0, only one of φ i , say ξ 1 , is nonzero. The resulting SDW forms a one-dimensional stripe, which breaks C 3 but is invariant under mirror symmetry.”

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Two surface bands are observed to touch at the origin of the surface Brillouin zone Γ, and the degeneracy is protected by time-reversal symmetry with the doublet forming a Kramers pair.

“Two surface bands are observed to touch at the origin of the surface Brillouin zone Γ. The degeneracy is protected by time-reversal symmetry and the doublet | ψ ↑ , ↓ 〉 form a Kramers pair.”

Classified as probably not a substantial claim.

The hexagonal warping term H_w describes cubic spin-orbit coupling at the surface of rhombohedral crystal systems.

“The hexagonal warping term H w describes cubic spinorbit coupling at the surface of rhombohedral crystal systems, and to the best of our knowledge, it has not been”

• AssumptionRests on an assumption: Bi2Te3 has a rhombohedral crystal structure with space group R-3m.