Quasiparticle Interference on the Surface of the Topological Insulator Bi$_2$Te$_3$

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by  Wei-Cheng Lee

Quasiparticle Interference on the Surface of the Topological Insulator Bi Te
2 3
Wei-Cheng Lee,1 Congjun Wu,1 Daniel P. Arovas,1 and Shou-Cheng Zhang2
1Department of Physics, University of California, San Diego, CA 92093
2Department of Physics, McCullough Building, Stanford University, CA 94305
(Dated: January 14, 2010)
The quasiparticle interference of the spectroscopic imaging scanning tunneling microscopy has
beeninvestigatedforthesurfacestatesofthelargegaptopological insulatorBi2Te3 throughtheT-
0 matrix formalism. Both thescalar potential scattering and thespin-orbit scattering on thewarped
1 hexagonal isoenergy contour are considered. While backscatterings are forbidden by time-reversal
0 symmetry,otherscatteringsareallowedandexhibitstrongdependenceonthespinconfigurationsof
2 theeigenfunctionsat~k pointsovertheisoenergycontour. Thecharacteristicscatteringwavevectors
found in ouranalysis agree well with recent experiment results.
n
a
PACSnumbers: 73.20-r,73.43.Cd,75.10-b
J

4
1
I. INTRODUCTION tum space through real space measurement with a high

energy resolution. Recently, several groups have per-
]
l formed STM measurements on surface states of Bi Te
al The theoretical proposal1–8 and experimental discov- and Bi Sb 13,20–22. Backscattering induced by 2non3-
h ery of the topological insulators9–12 have provoked an 1−x x
magneticimpuritiesbetweentime-reversal(TR)partners
- intensive research effort in condensed matter physics.
s with opposite momenta is forbidden due to their oppo-
e Topological insulators (TI) with time-reversal symme- site spin configurations. This is confirmed by the real
m try are generally characterized by a topological term in
space Friedel oscillation pattern and by analysis of the
the electromagnetic action with a quantized coefficient4.
. QPI characteristic scattering wavevector.
t These states have been theoretically predicted and ex-
a In this paper, we perform a detailed QPI analysis of
m perimentally observed in both two and three dimen-
the surface states of the topological insulator Bi Te . A
2 3
sions,includingthetwo-dimensional(2D)HgTe/HgCdTe
- general TR-invariant impurity potential including scalar
d quantum wells1,9, and bulk three-dimensional materials
andspin-orbitscatteringcomponentsisstudiedusingthe
n Bi Te , Bi Se and Bi Sb 5,8,10–13. They exhibit ro-
2 3 2 3 1−x x standardT-matrixformalism. The scatteringonthe iso-
o bust gapless modes at boundaries, e.g. a 1Dhelical edge
c energy surface strongly depends on the both momentum
mode for 2D TIs, and a 2D helical surface mode for 3D
[ and spin orientation. Scattering between TR partners
TIswithoddnumbersofDiraccones. Duetotimerever-
vanishes as a consequence of TR symmetry. The scat-
4 sal symmetry, backscattering is forbidden for the helical
tering is dominated by wavevectors which connect re-
v
edgeandsurfacestates,andananalysisofinteractionef-
8 gionsontheFermisurfaceofextremalcurvature,butalso
fects forthe 1Dhelicaledgemodes showstheyarestable
6 accounting for spin polarization. STM experiments20,21
againstweakandintermediatestrengthinteractions14,15.
6 have yielded rich information about the QPI structure.
1 Bi2Te3andBi2Se3havebeenpredictedtohavebulkband In addition to the absence of backscattering, the STM
0. gaps exceeding room temperature8, which makes them experiments also observed recovered scattering20 at a
promising for future applications.
1 wavevector (~k in their, and ~q in our notation), and
nest 2
9 Zhang et al predict that the surface states of Bi2Te3 an extinction21 (i.e. near absence of scattering) (~q3 in
0
consist of a single Dirac cone at the Γ point, and that their and our notation), both at wavevectors which do
:
v the Dirac cone evolves into a hexagonal shape at higher not connect TR states. Below, we offer a novel expla-
i energy8. Furthermore, near the Dirac point, the spin of nation of this experimental puzzle. Our results are in
X
theelectronliesperpendiculartothemomentum. Angle- excellent overall agreement with the QPI experiment in
r
a resolvedphoto-emissionspectroscopy(ARPES)measure- Bi2Te3.
mentsperformedonthesurfaceofBi Te haveconfirmed
2 3
these predictions in detail12,16. The typical shape of
the Fermi surface is a snowflake-like warped hexagon.
II. SUFACE DIRAC MODEL WITH WARPING
The low-energy O(2) symmetry of the Dirac cone is bro-
TERM
ken due to the C symmetry of the underlying lattice8,
3v
and can be modeled by a warping term in the effec-
The ~k·p~ Hamiltonian for the surface Dirac cone was
tive model17. Another powerful surface probe, spectro-
first derived in Ref. 8. The bare Hamiltonian is written
scopic scanning tunneling microscopy (STM), is sensi-
as H = d2kψ†(~k)H(~k)ψ(~k), where ψ†(~k)=(c† ,c† ).
tive to quasi-particle interference (QPI) around impuri- 0 ~k↑ ~k↓
ties, and provides an important tool to study electronic With theRaddition of the cubic warping term17,
structures in unconventional materials, such as high T
c
cuprates18,19. It can provide information in momen- H(~k)=v ~k×~σ ·zˆ+λk3cos3φ σz . (1)
~k
(cid:0) (cid:1)
2
The azimuthal angle of ~k is φ = tan−1(k /k ), where
~k y x
the Γ-K direction is taken as xˆ axis. Following Ref. 17,
thequadratictermsaredroppedsincetheydonotsignif-
icantly change the shape ofthe constantenergy contour,
and the characteristic energy and wavevector scales are
definedas: E∗ =vk andk = v/λ. ThisHamiltonian
c c
can be diagonalized by introducping
cos(θ~k/2) ie−iφ~ksin(θ~k/2)
Uˆ(~k)=  , (2)
ieiφ~ksin(θ~k/2) cos(θ~k/2) 
where tanθ = k2/(k2cos3φ ). One then finds H(~k) =
~k c ~k
E(~k)U(~k)σzU†(~k),witheigenvaluesE± =±E(~k)where FIG. 1: (Color online) (a)The iso-energy contour near the Γ
∗
pointforE =1.5E withsnow-flakeshape. Thexˆandyˆaxes
E(~k)= (vk)2+(λk3cos3θ )2 . (3) are chosen to be the Γ-K and Γ-M directions respectively,
q ~k andkc =pv/λ. Theredandbrown(darkgray)dotsreferto
the valley and the tip points on the contour, and the arrows
In fig. 1(a) we plot the isoenergy contour E = 1.5E∗, indicatessixrepresentativescatteringwavevectors. kLandkU
which qualitatively reproduces the snowflake Fermi sur- aresolutions of E+(kL,θ=0)=E+(kU,θ=π/2)=E which
face observed in the first-principles calculation and the aretheboundaryofthetruncationforthe~k-integrationused
ARPES experiment8,12,17. As for the scattering process, inthispaper. (b) Thespin orientations of theeigenfunctions
we take for α+ band at valley and tip points. The dotted lines refer
to the mirror-symmetric lines (Γ-M), and the system has a
H = d2kd2k′V ψ†(~k′) I+ic~k×~k′·~σ ψ(~k).(4) three-fold rotational symmetry. The arrow indicate the spin
imp Z ~k−~k′ h i configurationinthexyplaneandthesolidcircle(cross)refers
to Sz being along +zˆ(−zˆ). At the cusp points the spin lies
For a single short-ranged scatterer we may approximate only on the xy plane while Sz has the largest magnitude at
V ≈ V . The second term corresponds to the spin- thevalley pointswith staggered signs.
~k−~k′ 0
orbit scattering with the coefficient c describing its rela-
tive strength to the potential scattering. It is convenient
to project the potential onto the eigenbasis of H0, so X =k e−5πi/6,andY =k e−iπ/2. Thenfromeqn. 6we
U U
obtain V(11) 2 = 3V02 sin2ϑ, V(11) 2 = V02 + 3V02 cos2ϑ,
Vˆ ≡V Uˆ†(~k′) I+ic~k×~k′·~σ Uˆ(~k). (5) AB 4 AC 4 4
~k,~k′ 0 h i and VA((cid:12)(cid:12)1A¯1) =(cid:12)(cid:12)0, where A¯ = −(cid:12)(cid:12)A, co(cid:12)(cid:12)rresponding to scat-
tering through the vectors ~q , ~q , and ~q , respectively,
For simplicity, we first consider the c = 0 case (pure 3 2 1
scalarpotentialscattering),returninglatertothegeneral with tanϑ = (kc/kL)2. We also find VW(1X1) 2 = 3V402,
spin-orbit case (c 6= 0). Since the spectrum is particle- VW(1Y1) 2 = V402, and VW(1W1¯) = 0. These p(cid:12)(cid:12)rocess(cid:12)(cid:12)es are de-
hole symmetric, let us focus on a definite (positive) sign (cid:12)picted(cid:12)in fig. 1(a).
of the energy. The QPI will then be dominated by scat- (cid:12) (cid:12)
WhileV(11) =V(11) =0isadirectconsequenceofTR
terings inside the positive energy band, whose effective AA¯ WW¯
symmetry, the other processes through scattering vec-
scattering potential is:
tors ~q are in general finite. Their amplitude vari-
2,3,5,6
ation may be understood in terms of the spin orienta-
Vˆ~k(,1~k1′) =V0(cid:20)cosθ2~k cosθ2~k′ +sinθ2~k sinθ2~k′ ei(φ~k−φ~k′)(cid:21) . tionof the eigenfunctions throughoutthe Brillouinzone,
S~(~k)=(−sinθ sinφ , sinθ cosφ , cosθ ), depicted in
(6) ~k ~k ~k ~k ~k
fig. 1(b). Bi Te hasthesymmetryofC ,i.e. three-fold
ThiseffectalsoappearsintheQPIanalysisoftheorbital- 2 3 3v
rotationalsymmetry plus the three reflection lines (Γ-M
band systems where orbital hybridization brings strong
momentum dependence to the scattering process23. plus two equivalent lines). Therefore at the tips Sz(~k)
must vanish since σz is odd under the mirror operation.
Sz(~k) has the largestmagnitude at the valleys, but with
III. EFFECT OF SPIN ORIENTATION ON THE staggered signs, as shown in the figure. Since scalar po-
QPI PATTERN tential scattering does not flip electron spin, its matrix
element is largest when S~(~k) · S~(~k′) is large and posi-
The points of extremal curvature on the Fermi sur- tive, i.e. high spin overlap. This echoes the experimen-
face are divided into two groups, arising from the ‘val- talfinding of Pascualet al.24 that in the QPI patternon
leys’ (k = k , positive curvature) and ‘tips’ (k = k , Bi(110),onlythescatteringprocessespreservingthespin
L U
negative curvature). We define the complexified points orientation are visible. One major difference, however,
A = k eiπ/3, B = k , C = k e−iπ/3, W = k e5πi/6, betwwenBi(110)andBi Te is that the formerhas mul-
L L L U 2 3
3
tipleFermisurfacesandthescatteringprocessespreserv- here is developed in the eigenbasis of Ψˆ(~k), the SU(2)
ing spin orientations do exist at finite ~q, while the later rotation matrices Uˆ(~k) are introduced in the last line
onlyhasoneFermisurfaceandthereforenosuchscatter- of eq. 10 to transform back to the physical spin basis.
ings could exist. At the tips, the spin lies in-plane, with Because the first term in eq. 7, ρ(~q = 0) contains the
θ~k = π2,independentofthe scanningenergyE. Itcanbe sum of the total density of states without the impurity,
checked that S~(~k+~q )·S~(~k) > S~(~k+~q )·S~(~k), hence which makes it much larger than ρ(~q 6= 0), we only plot
5 6
(11) 2 (11) 2 |ρ(~q 6=0)|inordertorevealweakerstructuresoftheQPI
V > V . For scatterings between the valleys,
WX WY induced by the impurity scattering.
S(cid:12)~(~k)·(cid:12)S~(~k′)(cid:12)depe(cid:12)nds crucially on Sz(~k) and Sz(~k′). Ac-
(cid:12) (cid:12) (cid:12) (cid:12) Wesolveeq. 8numerically,using2Dpolarcoordinates.
counting for the valley-to-valley oscillation in S~(~k), we Since the dominant scattering processes are between ~k
conclude that as the scanning energy increases, VA(1C1) 2 points on the constant energy contour E+(k,θ)=E (we
growswhile V(11) 2 shrinks. Thissimpleargume(cid:12)ntgive(cid:12)s focus on E >0 here), we perform the integration within
AB (cid:12) (cid:12)
a qualitative(cid:12)expla(cid:12)nation for the absence of the ~q3 scat- the range kL ≤ k ≤ kU with kL and kU indicated in
(cid:12) (cid:12)
tering in the STM experiment21. For typical experimen- Fig. 1(a). The resulting QPI images are plotted in fig.
tal parameters17, E/E∗ ≈ 1.5 and k /k ≈ 1. In this 2 for c=0 with E =1.5E∗ fixed. For this choice of pa-
L c
caseweestimatethescalarpotentialscatteringgivesthat rameters, kL/kc = 1.029 and kU/kc = 1.5. As shown in
V(11) 2 : V(11) 2 : V(11) 2 : V(11) 2 ≈6:5:3:2. fig. 2(a), ~q5 and ~q2 indicated by the red (dark gray)and
WX AC AB WY
green(light gray)circles are the strongestfeatures while
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ~q (indicatedbythewhitecircle)isalmostinvisible. The
3
IV. NUMERICAL RESULTS reason why ~q5 is even stronger than ~q2 while they have
comparable scalar scattering potential is due to the dif-
ference in the density of states. Because the tip points
To specifically compute the QPI image, we employ a
shown in fig. 1(a) have larger density of states than the
T-matrix approach25 for multiband systems23. In the
valley points, the weightsof~q is largerthanthose of ~q ,
operatorbasisΨ(~k)=U(~k)ψ(~k), the Green’sfunction is 5 2
resulting in the stronger features observed for ~q . The
5
written in matrix form as
strong features near ~q =0 correspondto small ~q scatter-
Gˆ(~k,~k′,ω) = Gˆ (~k,ω)δ +Gˆ (~k,ω)Tˆ (ω)Gˆ (~k′,ω) ings around the tips and valleys points, which have also
0 ~k,~k′ 0 ~k,~k′ 0 be seen in experiments. Our results reproduce satisfac-
(7) torily the experimental findings and are also consistent
with the analysisfromthe spin-orientationselectionrule
where the T-matrix satisfies discussed above.
As the scanning energy increases further, the surface
Tˆ~k,~k′(ω)=Vˆ~k,~k′ +Zd2pVˆ~k,p~Gˆ0(p~,ω)Tˆp~,~k′(ω) , (8) states along the Γ−M direction start to merge into the
conductionbandofthe bulk states. In this case,the tips
and Gˆ (~k,ω) = ω+iδ−E (~k) −1δ are the bare of the constant energy contour will be mixed up with
0,σ ab a a,b these bulk bands, which weakens the ~q5 scattering but
Gree(cid:2)n’s function(cid:3)s. In(cid:2)spectroscopic im(cid:3) aging STM25, the
enhancesthesmall~qscatteringsneartheΓpoint. Thisis
conductance (dI/dV) measured by the STM is propor- consistent with the experiment21, showing that the area
tional to the local density of states defined as
of the strong features near Γ point becomes much larger
afterthe scanningenergyexceedsthe bottomofthe con-
ρ(~r,ω)=ρ (~r,ω)+ρ (~r,ω) , (9)
↑ ↓ duction band.
where ρ (~r,ω) = ImG (~r,~r,ω) is the local density of
σ σ
states for spin σ. The QPI image in the Brillouin zone
V. SPIN-ORBITAL SCATTERING IMPURITY
ρ(~q,ω)is thenobtainedbyperformingthe Fouriertrans-
formationoftheconductancedI/dV. Asaresult,wecan
calculate ρ(~q,ω) using the T-matrix formalism by: Nowwebrieflycommentonthe effectofthe spin-orbit
scattering given in eq. 4 which in principle exists in any
realistic system. Since surface states of the topological
ρ(~q,ω) = d2reiq~·~rρ(~r,ω)
Z insulatorBi2Te3aretwo-dimensional,thespin-orbitscat-
tering potential only has one component:
1
= d2kTr Uˆ(~k)Gˆ(~k,~k+~q,ω)Uˆ†(~k+~q)
2iZ (cid:20)
HSO =icV d2kd2k′kk′sin(φ −φ )ψ†(~k′)σzψ(~k).
− Uˆ(~k)Gˆ(~k,~k−~q,ω)Uˆ†(~k−~q) ∗ (10) imp 0Z ~k′ ~k
(cid:16) (cid:17) (cid:21) (11)
Backscatteringis still forbidden because of the sin(φ −
~k′
wherethetraceistakenwithrespecttothematrixindex. φ ) factor. Although σz does not flip spin, the angle-
~k
Because physically STM measures the local density of dependence sin(φ −φ ) gives rise to anadditionalsup-
~k′ ~k
statesinthespinbasisofψˆ(~k),whileourT-matrixtheory pression beyond that from the spin-orientation selection
4
5:
V(11) 2 = V(11) 2 =0 (12)
AA¯ WW¯
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)V(11)(cid:12)2 = (cid:12)V02 1(cid:12)− 3ck2 2+3cos2ϑ 1+ 1ck2 2
AC 4 2 L 2 L
(cid:12) (cid:12) h(cid:0) (cid:1) (cid:0) (cid:1) i
(cid:12) (cid:12)
V(11) 2 = 3V02 sin2ϑ 1− 1ck2 2
AB 4 2 L
(cid:12) (cid:12) (cid:0) (cid:1)
(cid:12)V(11)(cid:12)2 = 3V02 1− 1ck2 2
WX 4 2 U
(cid:12) (cid:12) (cid:0) (cid:1)
(cid:12) (cid:12)
V(11) 2 = V02 1− 3ck2 2 .
WY 4 2 U
(cid:12) (cid:12) (cid:0) (cid:1)
No(cid:12)nzero(cid:12)cbringsinnewinterferenceswhichcouldleadto
unusual suppressions or enhancements for some scatter-
ing wavevectors, depending not only on the magnitude
and sign of c, but also on the scanning energy E. In
fig. 2(b) we show the QPI image for c = 0.5. While
the 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 elements
forspin-orbitscatteringarelargerforquasiparticleswith
largermomentum, this term will become more andmore
importantasthescanningenergyE increases. Adetailed
analysis of the spin-orbit scattering will be presented in
a future publication. In comparison with the results in
ref.21, 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-
terferenceinducedbynonmagneticimpuritiesonthesur-
faceofthe topologicalinsulatorBi Te using aT-matrix
2 3
approach . While the backscattering is completely for-
bidden by time-reversal symmetry, other scatterings are
allowed, resulting in the QPI patterns observed in STM
experiments20,21 . We have shown further that the scat-
tering strengths depends crucially on the spin orienta-
tionsoftheeigenfunctions. Sincenonmagneticimpurities
FIG. 2: (Color online) The quasiparticle interference image
∗ ∗ can not flip spin, the scalar scattering potential between
for (a) c= 0 and (b) c= 0.5 with E = 1.5E and V0/E =
two eigenstates is larger as their spin overlap is larger.
0.1. In this case, kL/kc = 1.029 and kU/kc = 1.5. (a) The
Combined with the variationof the density of states, we
strongestlarge~qscatteringsare~q5and~q2indicatedbythered
(darkgray)andgreen(lightgray)circles(andtheirsymmetric have shown that some of the scatterings might be too
points). ~q3 (indicated by the white circle) is too weak to be weak to be seen in comparison with the strongest ones,
seen. (b) For c=0.5, new QPI features with large momenta and our results successfully reproduce the QPI patern
are visible. observed in experiments. We have further discussed the
effect of the spin-orbit scattering on the QPI pattern.
While the backscattering is still forbidden, we find that
thespin-orbitscatteringenhancesseveralnewfeaturesat
largemomentum,andthe detailedQPIfeaturesstrongly
rule discussed in the case of scalar impurity scattering. depends on the sign and strength of the spin-orbit scat-
Moreover,becausethematrixelementislinearinkk′,the tering potential.
spin-orbitscatteringtendstoenhancethescatteringsbe- We are grateful to Xi Chen, Liang Fu, Aharon Ka-
tween quasiparticles with large momenta. All these ad- pitulnik, Qin Liu, Xiaoliang Qi, Qikun Xue for insight-
ditional effects due to the spin-orbit scattering can be ful discussions. CW and WCL are supported by ARO-
roughly seen in a straightforward calculation froim eq. W911NF0810291. S CZ is supportedby the Department
5
of Energy, Office of Basic Energy Sciences, Division of Note added – While this paper was about completion,
Materials Sciences and Engineering, under contract DE- we learned a related work by Zhang et al.26.
AC02-76SF00515.
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