Quasiparticle Interference on the Surface of the Topological Insulator Bi 2 Te 3
Wei-Cheng Lee, Congjun Wu, Daniel P. Arovas, Shou-Cheng Zhang
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Quasiparticle Interference on the Surface of the Topological Insulator Bi Te Wei-Cheng Lee, Congjun Wu, Daniel P. Arovas, and Shou-Cheng Zhang Department of Physics, University of California, San Diego, CA 92093 Department of Physics, McCullough Building, Stanford University, CA 94305 (Dated: October 24, 2018)The quasiparticle interference of the spectroscopic imaging scanning tunneling microscopy hasbeen investigated for the surface states of the large gap topological insulator Bi Te through the T -matrix formalism. Both the scalar potential scattering and the spin-orbit scattering on the warpedhexagonal isoenergy contour are considered. While backscatterings are forbidden by time-reversalsymmetry, other scatterings are allowed and exhibit strong dependence on the spin configurations ofthe eigenfunctions at ~k points over the isoenergy contour. The characteristic scattering wavevectorsfound in our analysis agree well with recent experiment results. PACS numbers: 73.20-r,73.43.Cd,75.10-b
I. INTRODUCTION
The theoretical proposal and experimental discov-ery of the topological insulators have provoked anintensive research effort in condensed matter physics.Topological insulators (TI) with time-reversal symme-try are generally characterized by a topological term inthe electromagnetic action with a quantized coefficient .These states have been theoretically predicted and ex-perimentally observed in both two and three dimen-sions, including the two-dimensional (2D) HgTe/HgCdTequantum wells , and bulk three-dimensional materialsBi Te , Bi Se and Bi − x Sb x . They exhibit ro-bust gapless modes at boundaries, e.g. a 1D helical edgemode for 2D TIs, and a 2D helical surface mode for 3DTIs with odd numbers of Dirac cones. Due to time rever-sal symmetry, backscattering is forbidden for the helicaledge and surface states, and an analysis of interaction ef-fects for the 1D helical edge modes shows they are stableagainst weak and intermediate strength interactions .Bi Te and Bi Se have been predicted to have bulk bandgaps exceeding room temperature , which makes thempromising for future applications.Zhang et al predict that the surface states of Bi Te consist of a single Dirac cone at the Γ point, and thatthe Dirac cone evolves into a hexagonal shape at higherenergy . Furthermore, near the Dirac point, the spin ofthe electron lies perpendicular to the momentum. Angle-resolved photo-emission spectroscopy (ARPES) measure-ments performed on the surface of Bi Te have confirmedthese predictions in detail . The typical shape ofthe Fermi surface is a snowflake-like warped hexagon.The low-energy O (2) symmetry of the Dirac cone is bro-ken due to the C v symmetry of the underlying lattice ,and can be modeled by a warping term in the effec-tive model . Another powerful surface probe, spectro-scopic scanning tunneling microscopy (STM), is sensi-tive to quasi-particle interference (QPI) around impuri-ties, and provides an important tool to study electronicstructures in unconventional materials, such as high T c cuprates . It can provide information in momen- tum space through real space measurement with a highenergy resolution. Recently, several groups have per-formed STM measurements on surface states of Bi Te and Bi − x Sb x . Backscattering induced by non-magnetic impurities between time-reversal (TR) partnerswith opposite momenta is forbidden due to their oppo-site spin configurations. This is confirmed by the realspace Friedel oscillation pattern and by analysis of theQPI characteristic scattering wavevector.In this paper, we perform a detailed QPI analysis ofthe surface states of the topological insulator Bi Te . Ageneral TR-invariant impurity potential including scalarand spin-orbit scattering components is studied using thestandard T -matrix formalism. The scattering on the iso-energy surface strongly depends on the both momentumand spin orientation. Scattering between TR partnersvanishes as a consequence of TR symmetry. The scat-tering is dominated by wavevectors which connect re-gions on the Fermi surface of extremal curvature, but alsoaccounting for spin polarization. STM experiments have yielded rich information about the QPI structure.In addition to the absence of backscattering, the STMexperiments also observed recovered scattering at awavevector ( ~k nest in their, and ~q in our notation), andan extinction (i.e. near absence of scattering) ( ~q intheir and our notation), both at wavevectors which donot connect TR states. Below, we offer a novel expla-nation of this experimental puzzle. Our results are inexcellent overall agreement with the QPI experiment inBi Te . II. SUFACE DIRAC MODEL WITH WARPINGTERM
The ~k · ~p Hamiltonian for the surface Dirac cone wasfirst derived in Ref. 8. The bare Hamiltonian is writtenas H = R d k ψ † ( ~k ) H ( ~k ) ψ ( ~k ), where ψ † ( ~k ) = ( c † ~k ↑ , c † ~k ↓ ).With the addition of the cubic warping term , H ( ~k ) = v (cid:0) ~k × ~σ (cid:1) · ˆ z + λk cos 3 φ ~k σ z . (1)The azimuthal angle of ~k is φ ~k = tan − ( k y /k x ), wherethe Γ- K direction is taken as ˆ x axis. Following Ref. 17,the quadratic terms are dropped since they do not signif-icantly change the shape of the constant energy contour,and the characteristic energy and wavevector scales aredefined as: E ∗ = v k c and k c = p v/λ . This Hamiltoniancan be diagonalized by introducingˆ U ( ~k ) = cos( θ ~k / ie − iφ ~k sin( θ ~k / ie iφ ~k sin( θ ~k /
2) cos( θ ~k / , (2)where tan θ ~k = k c / ( k cos 3 φ ~k ). One then finds H ( ~k ) = E ( ~k ) U ( ~k ) σ z U † ( ~k ), with eigenvalues E ± = ± E ( ~k ) where E ( ~k ) = q ( vk ) + ( λk cos 3 θ ~k ) . (3)In fig. 1(a) we plot the isoenergy contour E = 1 . E ∗ ,which qualitatively reproduces the snowflake Fermi sur-face observed in the first-principles calculation and theARPES experiment . As for the scattering process,we take H imp = Z d k d k ′ V ~k − ~k ′ ψ † ( ~k ′ ) h I + ic ~k × ~k ′ · ~σ i ψ ( ~k ) . (4)For a single short-ranged scatterer we may approximate V ~k − ~k ′ ≈ V . The second term corresponds to the spin-orbit scattering with the coefficient c describing its rela-tive strength to the potential scattering. It is convenientto project the potential onto the eigenbasis of H , soˆ V ~k,~k ′ ≡ V ˆ U † ( ~k ′ ) h I + ic ~k × ~k ′ · ~σ i ˆ U ( ~k ) . (5)For simplicity, we first consider the c = 0 case (purescalar potential scattering), returning later to the generalspin-orbit case ( c = 0). Since the spectrum is particle-hole symmetric, let us focus on a definite (positive) signof the energy. The QPI will then be dominated by scat-terings inside the positive energy band, whose effectivescattering potential is:ˆ V (11) ~k,~k ′ = V (cid:20) cos θ ~k cos θ ~k ′ + sin θ ~k sin θ ~k ′ e i ( φ ~k − φ ~k ′ ) (cid:21) . (6)This effect also appears in the QPI analysis of the orbital-band systems where orbital hybridization brings strongmomentum dependence to the scattering process . III. EFFECT OF SPIN ORIENTATION ON THEQPI PATTERN
The points of extremal curvature on the Fermi sur-face are divided into two groups, arising from the ‘val-leys’ ( k = k L , positive curvature) and ‘tips’ ( k = k U ,negative curvature). We define the complexified points A = k L e iπ/ , B = k L , C = k L e − iπ/ , W = k U e πi/ , FIG. 1: (Color online) (a)The iso-energy contour near the Γpoint for E = 1 . E ∗ with snow-flake shape. The ˆ x and ˆ y axesare chosen to be the Γ- K and Γ- M directions respectively,and k c = p v/λ . The red and brown (dark gray) dots refer tothe valley and the tip points on the contour, and the arrowsindicates six representative scattering wavevectors. k L and k U are solutions of E + ( k L , θ = 0) = E + ( k U , θ = π/
2) = E whichare the boundary of the truncation for the ~k -integration usedin this paper. (b) The spin orientations of the eigenfunctionsfor α + band at valley and tip points. The dotted lines referto the mirror-symmetric lines (Γ-M), and the system has athree-fold rotational symmetry. The arrow indicate the spinconfiguration in the xy plane and the solid circle (cross) refersto S z being along +ˆ z ( − ˆ z ). At the cusp points the spin liesonly on the xy plane while S z has the largest magnitude atthe valley points with staggered signs. X = k U e − πi/ , and Y = k U e − iπ/ . Then from eqn. 6 weobtain (cid:12)(cid:12) V (11) AB (cid:12)(cid:12) = V sin ϑ , (cid:12)(cid:12) V (11) AC (cid:12)(cid:12) = V + V cos ϑ ,and V (11) A ¯ A = 0, where ¯ A = − A , corresponding to scat-tering through the vectors ~q , ~q , and ~q , respectively,with tan ϑ = ( k c /k L ) . We also find (cid:12)(cid:12) V (11) W X (cid:12)(cid:12) = V , (cid:12)(cid:12) V (11) W Y (cid:12)(cid:12) = V , and V (11) W ¯ W = 0. These processes are de-picted in fig. 1(a).While V (11) A ¯ A = V (11) W ¯ W = 0 is a direct consequence of TRsymmetry, the other processes through scattering vec-tors ~q , , , are in general finite. Their amplitude vari-ation may be understood in terms of the spin orienta-tion of the eigenfunctions throughout the Brillouin zone, ~S ( ~k ) = ( − sin θ ~k sin φ ~k , sin θ ~k cos φ ~k , cos θ ~k ), depicted infig. 1(b). Bi Te has the symmetry of C v , i.e. three-foldrotational symmetry plus the three reflection lines (Γ- M plus two equivalent lines). Therefore at the tips S z ( ~k )must vanish since σ z is odd under the mirror operation. S z ( ~k ) has the largest magnitude at the valleys, but withstaggered signs, as shown in the figure. Since scalar po-tential scattering does not flip electron spin, its matrixelement is largest when ~S ( ~k ) · ~S ( ~k ′ ) is large and posi-tive, i.e. high spin overlap. This echoes the experimen-tal finding of Pascual et al. that in the QPI pattern onBi(110), only the scattering processes preserving the spinorientation are visible. One major difference, however,betwwen Bi(110) and Bi Te is that the former has mul-tiple Fermi surfaces and the scattering processes preserv-ing spin orientations do exist at finite ~q , while the lateronly has one Fermi surface and therefore no such scatter-ings could exist. At the tips, the spin lies in-plane, with θ ~k = π , independent of the scanning energy E . It can bechecked that ~S ( ~k + ~q ) · ~S ( ~k ) > ~S ( ~k + ~q ) · ~S ( ~k ), hence (cid:12)(cid:12) V (11) W X (cid:12)(cid:12) > (cid:12)(cid:12) V (11) W Y (cid:12)(cid:12) . For scatterings between the valleys, ~S ( ~k ) · ~S ( ~k ′ ) depends crucially on S z ( ~k ) and S z ( ~k ′ ). Ac-counting for the valley-to-valley oscillation in ~S ( ~k ), weconclude that as the scanning energy increases, (cid:12)(cid:12) V (11) AC (cid:12)(cid:12) grows while (cid:12)(cid:12) V (11) AB (cid:12)(cid:12) shrinks. This simple argument givesa qualitative explanation for the absence of the ~q scat-tering in the STM experiment . For typical experimen-tal parameters , E/E ∗ ≈ . k L /k c ≈
1. In thiscase we estimate the scalar potential scattering gives that (cid:12)(cid:12) V (11) W X (cid:12)(cid:12) : (cid:12)(cid:12) V (11) AC (cid:12)(cid:12) : (cid:12)(cid:12) V (11) AB (cid:12)(cid:12) : (cid:12)(cid:12) V (11) W Y (cid:12)(cid:12) ≈ IV. NUMERICAL RESULTS
To specifically compute the QPI image, we employ a T -matrix approach for multiband systems . In theoperator basis Ψ( ~k ) = U ( ~k ) ψ ( ~k ), the Green’s function iswritten in matrix form asˆ G ( ~k, ~k ′ , ω ) = ˆ G ( ~k, ω ) δ ~k,~k ′ + ˆ G ( ~k, ω ) ˆ T ~k,~k ′ ( ω ) ˆ G ( ~k ′ , ω )(7)where the T -matrix satisfiesˆ T ~k,~k ′ ( ω ) = ˆ V ~k,~k ′ + Z d p ˆ V ~k,~p ˆ G ( ~p, ω ) ˆ T ~p,~k ′ ( ω ) , (8)and (cid:2) ˆ G ,σ ( ~k, ω ) (cid:3) ab = (cid:2) ω + iδ − E a ( ~k ) (cid:3) − δ a,b are the bareGreen’s functions. In spectroscopic imaging STM , theconductance ( dI/dV ) measured by the STM is propor-tional to the local density of states defined as ρ ( ~r, ω ) = ρ ↑ ( ~r, ω ) + ρ ↓ ( ~r, ω ) , (9)where ρ σ ( ~r, ω ) = Im G σ ( ~r, ~r, ω ) is the local density ofstates for spin σ . The QPI image in the Brillouin zone ρ ( ~q, ω ) is then obtained by performing the Fourier trans-formation of the conductance dI/dV . As a result, we cancalculate ρ ( ~q, ω ) using the T -matrix formalism by: ρ ( ~q, ω ) = Z d r e i~q · ~r ρ ( ~r, ω )= 12 i Z d k Tr (cid:20) ˆ U ( ~k ) ˆ G ( ~k, ~k + ~q, ω ) ˆ U † ( ~k + ~q ) − (cid:16) ˆ U ( ~k ) ˆ G ( ~k, ~k − ~q, ω ) ˆ U † ( ~k − ~q ) (cid:17) ∗ (cid:21) (10)where the trace is taken with respect to the matrix index.Because physically STM measures the local density ofstates in the spin basis of ˆ ψ ( ~k ), while our T -matrix theory here is developed in the eigenbasis of ˆΨ( ~k ), the SU (2)rotation matrices ˆ U ( ~k ) are introduced in the last lineof eq. 10 to transform back to the physical spin basis.Because the first term in eq. 7, ρ ( ~q = 0) contains thesum of the total density of states without the impurity,which makes it much larger than ρ ( ~q = 0), we only plot | ρ ( ~q = 0) | in order to reveal weaker structures of the QPIinduced by the impurity scattering.We solve eq. 8 numerically, using 2D polar coordinates.Since the dominant scattering processes are between ~k points on the constant energy contour E + ( k, θ ) = E (wefocus on E > k L ≤ k ≤ k U with k L and k U indicated inFig. 1(a). The resulting QPI images are plotted in fig.2 for c = 0 with E = 1 . E ∗ fixed. For this choice of pa-rameters, k L /k c = 1 .
029 and k U /k c = 1 .
5. As shown infig. 2(a), ~q and ~q indicated by the red (dark gray) andgreen (light gray) circles are the strongest features while ~q (indicated by the white circle) is almost invisible. Thereason why ~q is even stronger than ~q while they havecomparable scalar scattering potential is due to the dif-ference in the density of states. Because the tip pointsshown in fig. 1(a) have larger density of states than thevalley points, the weights of ~q is larger than those of ~q ,resulting in the stronger features observed for ~q . Thestrong features near ~q = 0 correspond to small ~q scatter-ings around the tips and valleys points, which have alsobe seen in experiments. Our results reproduce satisfac-torily the experimental findings and are also consistentwith the analysis from the spin-orientation selection rulediscussed above.As the scanning energy increases further, the surfacestates along the Γ − M direction start to merge into theconduction band of the bulk states. In this case, the tipsof the constant energy contour will be mixed up withthese bulk bands, which weakens the ~q scattering butenhances the small ~q scatterings near the Γ point. This isconsistent with the experiment , showing that the areaof the strong features near Γ point becomes much largerafter the scanning energy exceeds the bottom of the con-duction band. V. SPIN-ORBITAL SCATTERING IMPURITY
Now we briefly comment on the effect of the spin-orbitscattering given in eq. 4 which in principle exists in anyrealistic system. Since surface states of the topologicalinsulator Bi Te are two-dimensional, the spin-orbit scat-tering potential only has one component: H SOimp = icV Z d k d k ′ kk ′ sin( φ ~k ′ − φ ~k ) ψ † ( ~k ′ ) σ z ψ ( ~k ) . (11)Backscattering is still forbidden because of the sin( φ ~k ′ − φ ~k ) factor. Although σ z does not flip spin, the angle-dependence sin( φ ~k ′ − φ ~k ) gives rise to an additional sup-pression beyond that from the spin-orientation selection FIG. 2: (Color online) The quasiparticle interference imagefor (a) c = 0 and (b) c = 0 . E = 1 . E ∗ and V /E ∗ =0 .
1. In this case, k L /k c = 1 .
029 and k U /k c = 1 .
5. (a) Thestrongest large ~q scatterings are ~q and ~q indicated by the red(dark gray) and green (light gray) circles (and their symmetricpoints). ~q (indicated by the white circle) is too weak to beseen. (b) For c = 0 .
5, new QPI features with large momentaare visible. rule discussed in the case of scalar impurity scattering.Moreover, because the matrix element is linear in kk ′ , thespin-orbit scattering tends to enhance the scatterings be-tween quasiparticles with large momenta. All these ad-ditional effects due to the spin-orbit scattering can beroughly seen in a straightforward calculation froim eq. 5: (cid:12)(cid:12) V (11) A ¯ A (cid:12)(cid:12) = (cid:12)(cid:12) V (11) W ¯ W (cid:12)(cid:12) = 0 (12) (cid:12)(cid:12) V (11) AC (cid:12)(cid:12) = V h(cid:0) − ck L (cid:1) + 3 cos ϑ (cid:0) ck L (cid:1) i(cid:12)(cid:12) V (11) AB (cid:12)(cid:12) = V sin ϑ (cid:0) − ck L (cid:1) (cid:12)(cid:12) V (11) W X (cid:12)(cid:12) = V (cid:0) − ck U (cid:1) (cid:12)(cid:12) V (11) W Y (cid:12)(cid:12) = V (cid:0) − ck U (cid:1) . Nonzero c brings in new interferences which could lead tounusual suppressions or enhancements for some scatter-ing wavevectors, depending not only on the magnitudeand sign of c , but also on the scanning energy E . Infig. 2(b) we show the QPI image for c = 0 .
5. Whilethe main features are still similiar to those of fig. 2(a),new prominent features associated with larger momen-tum scatterings are visible. Since the matrix elementsfor spin-orbit scattering are larger for quasiparticles withlarger momentum, this term will become more and moreimportant as the scanning energy E increases. A detailedanalysis of the spin-orbit scattering will be presented ina future publication. In comparison with the results inref. , we find that spin-orbit scattering from the impu-rity of the Ag atom is not very important in this partic-ular experiment. VI. CONCLUSION
In conclusion, we have analyzed the quasiparticle in-terference induced by nonmagnetic impurities on the sur-face of the topological insulator Bi Te using a T -matrixapproach . While the backscattering is completely for-bidden by time-reversal symmetry, other scatterings areallowed, resulting in the QPI patterns observed in STMexperiments . We have shown further that the scat-tering strengths depends crucially on the spin orienta-tions of the eigenfunctions. Since nonmagnetic impuritiescan not flip spin, the scalar scattering potential betweentwo eigenstates is larger as their spin overlap is larger.Combined with the variation of the density of states, wehave shown that some of the scatterings might be tooweak to be seen in comparison with the strongest ones,and our results successfully reproduce the QPI paternobserved in experiments. We have further discussed theeffect of the spin-orbit scattering on the QPI pattern.While the backscattering is still forbidden, we find thatthe spin-orbit scattering enhances several new features atlarge momentum, and the detailed QPI features stronglydepends on the sign and strength of the spin-orbit scat-tering potential.We are grateful to Xi Chen, Liang Fu, Aharon Ka-pitulnik, Qin Liu, Xiaoliang Qi, Qikun Xue for insight-ful discussions. CW and WCL are supported by ARO-W911NF0810291. S CZ is supported by the Departmentof Energy, Office of Basic Energy Sciences, Division ofMaterials Sciences and Engineering, under contract DE-AC02-76SF00515. Note added – While this paper was about completion,we learned a related work by Zhang et al. . B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science , 1757 (2006). C. L. Kane and E. J. Mele, Phys. Rev. Lett. , 146802(2005). B. A. Bernevig and S.-C. Zhang, Phys. Rev. Lett. ,106802 (2006). X. L. Qi, T. L. Hughes, and S. C. Zhang, Phys. Rev. B ,195424 (2008). L. Fu and C. L. Kane, Phys. Rev. B , 045302 (2007). J. E. Moore and L. Balents, Phys. Rev. B , 121306(2007). R. Roy , arXiv:0607531 (2006). H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C.Zhang, Nature Physics , 438 (2009). M. K¨onig, S. Wiedmann, C. Brune, A. Roth, H. Buhmann,L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Scinece ,766 (2007). D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava,and M. Z. Hasan, Nature , 970 (2008). Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Ban-sil, D. Grauer, Y. S. Hor, R. J. Cava, et al., Nat. Phys. ,398 (2009). Y. L. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K. Mo,X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, et al.,Science , 5937 (2009). P. Roushan, J. Seo, C. V. Parker, Y. S. Hor, D. Hsieh,D. Qian, A. Richardella, M. Z. Hasan, R. J. Cava, andA. Yazdani, Nature , 1106 (2009). C. Wu, B. A. Bernevig, and S.-C. Zhang, Phys. Rev. Lett. , 106401 (2006). C. Xu and J. E. Moore, Phys. Rev. B , 045322 (2006). D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil,J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, et al.,Science , 919 (2009). L. Fu, arXiv:0908.1418 (2009). T. Hanaguri, Y. Kohsaka, J. C. Davis, C. Lupien, I. Ya-mada, M. Azuma, M. Takano, K. Ohishi, M. Ono, andH. Takagi, Nature Physics , 865 (2007). Q.-H. Wang and D.-H. Lee, Phys. Rev. B , 020511(2003). Z. Alpichshev, J. G. Analytis, J. H. Chu, I. R. Fisher,Y. L. Chen, Z. X. Shen, A. Fang, and A. Kapitulnik ,arXiv:0908.0371 (2009). T. Zhang, P. Cheng, X. Chen, J.-F. Jia, X. Ma,K. He, L. Wang, H. Zhang, X. Dai, Z. Fang, et al.,arXiv:0908.4136 (2009). K. K. Gomes, W. Ko, W. Mar, Y. Chen, Z.-X. Shen, andH. C. Manoharan , arXiv:0909.0921 (2009). W.-C. Lee and C. Wu, Phys. Rev. Lett. , 176101(2009). J. I. Pascual, G. Bihlmayer, Yu. M. Koroteev, H.-P. Rust,G. Ceballos, M. Hansmann, K. Horn, E. V. Chulkov, S.Blugel, P. M. Echenique, and Ph. Hofmann, Phys. Rev.Lett. , 196802 (2004). A. V. Balatsky, I. Vekhter, and J.-X. Zhu, Rev. Mod. Phys. , 373 (2006).26