Upload
others
View
4
Download
0
Embed Size (px)
Citation preview
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.
Electronic standing waves on the surface of the topological insulator Bi2Te3 Peter Rakyta
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:
A. Varykhalov et al., Phys. Rev. Lett. 101, 157601 (2008).
Electronic standing waves on the surface of the topological insulator Bi2Te3 slide 3.
Electronic standing waves on the surface of the topological insulator Bi2Te3 Peter Rakyta
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).)
Electronic standing waves on the surface of the topological insulator Bi2Te3 slide 4.
Electronic standing waves on the surface of the topological insulator Bi2Te3 Peter Rakyta
Asymptotic theory of electronic standing waves
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,
χin(q)† χr(q) ≈ Ω0δq
γ,
ksc ≈ ∆0 +∆1δqη
➠ Isotropic dispersion:, 2DEG electrons: α = 0, β = 0, γ = 0, η = 2.
δρ ∼ sin(kscy + ϕ) y−12
Electronic standing waves on the surface of the topological insulator Bi2Te3 slide 5.
Electronic standing waves on the surface of the topological insulator Bi2Te3 Peter Rakyta
Asymptotic theory of electronic standing waves
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,
χin(q)† χr(q) ≈ Ω0δq
γ,
ksc ≈ ∆0 +∆1δqη
➠ Isotropic dispersion, helical electrons: α = 0, β = 1, γ = 1, η = 2.
δρ ∼ sin(kscy + ϕ) y−32
Electronic standing waves on the surface of the topological insulator Bi2Te3 slide 6.
Electronic standing waves on the surface of the topological insulator Bi2Te3 Peter Rakyta
Asymptotic theory of electronic standing waves
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 ,
ksc ≈ ∆0 +∆1δqη
➠ 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 the topological insulator Bi2Te3 slide 7.
Electronic standing waves on the surface of the topological insulator Bi2Te3 Peter Rakyta
Electronic standing waves on the surface of Bi2Te3 crystal
δρ ∼ sin(2kfity + ϕ) y−1 →
Difference between experimental data and the
asymptotic theory
A. Varykhalov et al., Phys. Rev. Lett. 101, 157601 (2008).
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.
Electronic standing waves on the surface of the topological insulator Bi2Te3 slide 8.
Electronic standing waves on the surface of the topological insulator Bi2Te3 Peter Rakyta
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.
Electronic standing waves on the surface of the topological insulator Bi2Te3 slide 9.
Electronic standing waves on the surface of the topological insulator Bi2Te3 Peter Rakyta
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
Electronic standing waves on the surface of the topological insulator Bi2Te3 slide 10.
Electronic standing waves on the surface of the topological insulator Bi2Te3 Peter Rakyta
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)|
)
Electronic standing waves on the surface of the topological insulator Bi2Te3 slide 11.
Electronic standing waves on the surface of the topological insulator Bi2Te3 Peter Rakyta
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
Electronic standing waves on the surface of the topological insulator Bi2Te3 slide 12.
Electronic standing waves on the surface of the topological insulator Bi2Te3 Peter Rakyta
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.
Electronic standing waves on the surface of the topological insulator Bi2Te3 slide 13.
Electronic standing waves on the surface of the topological insulator Bi2Te3 Peter Rakyta
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:
A. Varykhalov et al., Phys. Rev. Lett. 101, 157601 (2008).
Electronic standing waves on the surface of the topological insulator Bi2Te3 slide 14.
Electronic standing waves on the surface of the topological insulator Bi2Te3 Peter Rakyta
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
Electronic standing waves on the surface of the topological insulator Bi2Te3 slide 15.
Electronic standing waves on the surface of the topological insulator Bi2Te3 Peter Rakyta
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 on the surface of the topological insulator Bi2Te3 slide 16.
Electronic standing waves on the surface of the topological insulator Bi2Te3 Peter Rakyta
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
Electronic standing waves on the surface of the topological insulator Bi2Te3 slide 17.
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)