Electronic standing waves on the surface of the...

Electronic standing waves on the surface

of the topological insulator Bi2Te3

Phys. Rev. B. 86, 085456 (2012)

P. Rakyta, A. Pályi, J. Cserti

Eötvös Loránd University, Hungary

Department of Physics of Complex Systems

1

Electronic standing waves on the surface of the topological insulator Bi2Te3 Peter Rakyta

Experimental background of electronic standing waves on a crystal surface

➠ a) Bi2Te3 cleavage step (line defect) on the surface of the crystal: to-

pography scans by current STM technique with constant

➠ b) studying electronic standing waves by measuring the differential ∂I∂U

∼

ρ(x) conductivity. (ρ(x) is the local density of states)

A. Varykhalov et al., Phys. Rev. Lett. 101, 157601 (2008).

Electronic standing waves on the surface of the topological insulator Bi2Te3 slide 2.


Three-dimensional topological insulator: Bi2Te3

➠ Effective Hamiltonian of the surface states: (L. Fu, Phys. Rev. Lett. 103, 266801 (2009).):

ĤTI(k) =k2

2m∗+ vk(kxσy − kyσx) +

λ

2(k3+ + k

3−)σz

ETI± (k) =k2

2m∗±√

v2kk2 + λ2k2x(k

2x − 3k

2y)

2

−0.1 0 0.1

−0.1

−0.05

0

0.05

0.1

0.15

kx [A−1]

k y [A

−1 ]

−0.5

0

0.5

ARPES data:




Asymptotic theory of electronic standing waves

ksc

ksc,1

ksc,2 y,k

y= k

x,kx= q

line

defe

ct

Ψ = Ψin + rΨr

Ψin = χin(q)ei(qx+kiny)

Ψr = χr(q)ei(qx+kry)

ksc = kin − kr

δρ ∼

∫

dqRe(

r(k) χin(q)† χr(q) e

ikscy)

|v(B)⊥ (q)|

; v⊥(q) =∂E(k, q)

∂q

∣

∣

∣

∣

E

1

|v(B)⊥ (q)|

≈δqα

|v(B)⊥,0 |

,

r(k, q) ≈ r0δqβ,

χin(q)† χr(q) ≈ Ω0δq

γ,

ksc ≈ ∆0 +∆1δqη

Stationary phase method around extremal points ksc(q).

(R. R. Biswas and A. V. Balatsky, Phys. Rev. B 83, 075439 (2011).)




δρ ∼extremal∑

points

abs(r0Ω0)

|v⊥,0|sin(∆0y + ϕ) y

−α+β+γ+1η

ksc

ksc,1

ksc,2 y,k

y= k

x,kx= q

line

defe

ct

1

|v⊥(q)|≈

δqα

|v(B)⊥,0 |

,



γ,


➠ Isotropic dispersion:, 2DEG electrons: α = 0, β = 0, γ = 0, η = 2.

δρ ∼ sin(kscy + ϕ) y−12




δρ ∼extremal∑

points

abs(r0Ω0)

|v⊥,0|sin(∆0y + ϕ) y

−α+β+γ+1η

ksc

ksc,1

ksc,2 y,k

y= k

x,kx= q

line

defe

ct

1

|v⊥(q)|≈

δqα

|v(B)⊥,0 |

,



γ,


➠ Isotropic dispersion, helical electrons: α = 0, β = 1, γ = 1, η = 2.

δρ ∼ sin(kscy + ϕ) y−32




δρ ∼extremal∑

points

abs(r0Ω0)

|v⊥,0|sin(∆0y + ϕ) y

−α+β+γ+1η

ksc

ksc,1

ksc,2 y,k

y= k

x,kx= q

line

defe

ct

1

|v⊥(q)|≈

δqα

|v(B)⊥,0 |

,

r(k, q) ≈ r0δqβ ,

χbe(q)† χr(q) ≈ Ω0δq

γ ,


➠ Hexagonally warped dispersion, helical electrons: α2 = 0, β2 = 0, γ2 = 0, η2 = 2.

δρ ∼ A1 sin(ksz,1y + ϕ1) y−32 + A2 sin(ksz,2y + ϕ2) y

−12 A1 ≪ A2



Electronic standing waves on the surface of Bi2Te3 crystal

δρ ∼ sin(2kfity + ϕ) y−1 →

Difference between experimental data and the

asymptotic theory


Energy contour for E = 330

meV.

−0.15 0.15

−0.1

0.1

2kΓ M

k [A−1]

knest

2kfit

q [A

−1 ]

➠ exact solution of the scattering problem, closer to the line defect.

➠ knest: the reflected and transmitted states from the other side cancels each other.



Exact solution of the scattering problem

➠ characteristics of the standing waves, that are independent of the exact shape of the

potential. (decay, oscillation wavenumber)

➠ We model the cleavage step with a step potential: V (y) = −V0θ(y) (which is trans-

lational invariant in x)

➠ Determine the 6 eigenstates of HTI with given energy E and parallel to line defect

momentum component q ‖ x. (ETI± (kr, q) = E, r = 1 . . . 6)

➠ Matching eigenstates at the potential step ⇒ ψ(L/R)E,q scattering wave function. (L/R

stand for the incident plane-wave coming from the left/right.)

➠ The electronic standing waves correspond to the oscillations in the local DOS:

ρ(E, y) =1

(2π)2~

∑

d=L,R

∫

Γ(d)E

dκ|ψ

(d)k,q(y)|

2

v(k, q)

Γ(L/R)E stand for the energy contour segments on the left/right side of energy E.



Pre-asymtotic contribution to the oscillations (E = 330 meV)

P. Rakyta et al., Phys. Rev. B. 86, 085456 (2012).

50 100 150 200 250

−0.5

0

0.5

y [A]

δρ [1

0−4 ρ

0] f1

f2

0 0.12 0.24 0.36

−15

−10

−5

0

2kfit

knest

k [A−1]

FF

T [A

−1 m

eV−

1 ]

−0.15 0.15

−0.1

0.1

2kΓ M

k [A−1]

knest

2kfit

q [A

−1 ]

150 200 250 300 3500

0.060.120.180.240.3 2k

fitknest

2kΓ M

E [meV]k

[A−

1 ]

(a)

(b)

(c)

(d)

f1(x, n) = A1 sin(2kfitx+ ϕ1)x−3/2 f2(x, L) = A2 sin(2kfitx+ ϕ2)e

−x/L



Pre-asymtotic contribution to the oscillations

0.09 0.1 0.11 0.12 0.13 0.14048

100

|r|

0.09 0.1 0.11 0.12 0.13 0.140

0.7

|χ+ χ

|

0.09 0.1 0.11 0.12 0.13 0.143

6

1/v |

|

0.09 0.1 0.11 0.12 0.13 0.140

0.2

0.4

kfit

k [A−1]

|Szo

rzat

|

(a)

(b)

(c)

(d)

−0.15 0.15

−0.1

0.1

2kΓ M

k [A−1]

knest

2kfit

q [A

−1 ]

The

characteristic kfit wavenumber is related to

the non-monotonic behaviour of the prallel

to line defect component of the group

velocity.

δρ(E, y) ∼

∫

dk

(

rk,q′χ†k,q′χ−k,q′e

−i2ky + c.c.)

|v‖(E, k)|; F2k ∼ Re

(

rk,q′χ†k,q′χ−k,q′

|v‖(E, k)|

)



Exponential decay of the oscilaltions

Direct data f2(x, L) = A2 sin(2kfitx+ ϕ)e−x/L

0 0.12 0.24 0.36

−15

−10

−5

0

2kfit

knest

k [A−1]

FT

[A−

1 meV

−1 ]

0 0.12 0.24 0.36

−3

−2

−1

0

12k

fit

k [A−1]

FT

[A−

1 meV

−1 ]

0 0.5 1

−0.2

−0.1

0

2kfit

k [A−1]

FT

[A−

1 meV

−1 ]

0 0.12 0.24 0.36−3

−2

−1

0

2kfit

k [A−1]

FT

[A−

1 meV

−1 ]

a) b)

c) d)

f1(x) = A2 sin(2kfitx+ ϕ)x−3/2 f1(x) = A2 sin(2kfitx+ ϕ)x

−1/2



Conclusions

Pre-asymptotic standing waves:

➠ The characteristic wavenumber in the oscillations (closer to the line defect) is related

to the non-monotonic behavior of the parallel to line defect component of the group

velocity.

➠ The decay of the pre-asymptotic contribution to the oscillations is exponential when a

dominant nesting vector is missing.

Asymptotic standing waves:

➠ The characteristic wavenumbers in the oscillations are given by nesting vectors on the

constant energy contour.

➠ The decay of the asymptotic oscillations is power-like (xn) for the dominant nesting

vectors.



Three-dimensional topological insulator: Bi2Te3

➠ The exact shape of the constant energy con-

tours is important to describe standing waves

around line defects. (vk = v0 + αk2)

ETI± (k) =k2

2m∗±√

v2kk2 + λ2k2x(k

2x − 3k

2y)

2

−0.1 0 0.1

−0.1

−0.05

0

0.05

0.1

0.15

kx [A−1]

k y [A

−1 ]

−0.5

0

0.5

ARPES data:




Bi2Te3 : band-structure parameters from ARPES data (+, ., DOS)

BCB: bulk conduction band, BVB: bulk valance band

0 0.05 0.1 0.150

100

200

300

k [A−1]

E [m

eV]

0 0.05 0.1 0.150

100

200

300

k [A−1]

E [m

eV]

0 100 3000

150

300

E [meV]

ρ 0 [m

eV−

1 µm

−2 ]

0 100 3000

200

400

600

E [meV]

ρ 0 [m

eV−

1 µm

−2 ]

a) b)

c) d)

BVB

BCB

BVB

BCB

BVB

BCB

BVB

BCB

theory ↔ ARPES

solid ↔ E(ΓM)

dashed ↔ E(ΓK)

Density of states

(only theory shown)

v0 = 2.55 eVÅ, λ = 250 eVÅ3,

12m∗ = 0, α = 0

v0 = 3.5 eVÅ, λ = 150 eVÅ3,

α = 21 Å2, γ = −19.5 eVÅ2



Fourier analysis of oscillating functions

−30 −20 −10 0 10 20 30 40 50−1

−0.5

0

0.5

1x

maxx

min2x

max

x

f 2(x

)

➠ The Fourier transform of the symmetrized function are real.



Electronic standing waves around line defect ⊥ ΓK (E = 330 meV)

50 100 150 200 250−1.5

−1

−0.5

0

0.5

1

y [A]

δρ [1

0−4 ρ

0]

knestΓ K

k

FT

−0.2 −0.1 0 0.1 0.2−0.2

−0.1

0

0.1

0.2

2kΓ K

knestΓ K

k [A−1]q

[A−

1 ]

a) b)

f1(x) = A2 sin(2kfitx+ ϕ)x−1/2

➠ Power-like decay recovered with nesting vector kΓKnest


Experimental background of electronic standing waves on a crystal surfaceThree-dimensional topological insulator: Bi2Te3 Asymptotic theory of electronic standing wavesAsymptotic theory of electronic standing wavesAsymptotic theory of electronic standing wavesAsymptotic theory of electronic standing wavesElectronic standing waves on the surface of Bi2Te3 crystalExact solution of the scattering problemPre-asymtotic contribution to the oscillations (E=330 meV)Pre-asymtotic contribution to the oscillationsExponential decay of the oscilaltionsConclusionsThree-dimensional topological insulator: Bi2Te3 Bi2Te3 : band-structure parameters from ARPES data (+, ., DOS)Fourier analysis of oscillating functionsElectronic standing waves around line defect K (E=330 meV)

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