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O ti l i l ti f d t tOptical manipulation of edge state transport in topological insulatorsp p g
CARDEQ MeetingCARDEQ Meeting
Dresden, March 2009
Björn Trauzettel Koenig et al. Science 2007
Collaborators:Markus Kindermann (Georgia Tech)
Al N ik (Wü b )Alena Novik (Würzburg)Manuel Schmidt (Basel)
(my personal) motivation
Graphene Topological insulatorsvalleys (K and K’) Kramers partners (+ and )valleys (K and K ) Kramers partners (+ and -)sublattice (A and B) CB and VB states (E and H)spin (↑ and ↓) ---
Topological insulator in graphene
spin filtered edge states:topologically non-trivialtopologically non trivial
w/ respect to TRS
⇒no TRS-preserving (local)perturbation can open gap
( )x x x SOy z zy zH v p p sσ τ σ σ τΔ= + +
Kane & Mele PRL 2005
Outline• Effective Hamiltonian of HgTe Quantum Wells
• Experimental consequences → QSHI
• Optical manipulation of edge state transport
• Summary and Outlook
Band structure of HgTe QW I
Koenig et al. JPSJ 2008
a little bit of group theoryincluding SOI:
6Γ6
7 8Γ + Γ7 8
6 s-like: orbital part transforms Γ
7 8, : orbital part transfor p ikms -l eΓ Γ
k.p theory
( ) ( ) ( ) ( )2
, ,2 nn k n k
p V r r E k rm
ψ ψ⎡ ⎤
+ =⎢ ⎥⎣ ⎦
( )2m⎣ ⎦
( ) ( ), ,ikr
n k n kr e u rψ =Bloch functions
( ) ( ) ( ) ( )2 2 2
, ,2 2 nn k n k
p kV r u r E k um m
k p rm
⎡ ⎤+ + + =⎢ ⎥
⎣
⋅
⎦( )2 2m mm⎣ ⎦
• En(k) known at k=k0 (n: band index)
• band n has Γi symmetry ⇒ un,k(r) transforms as Γi
• in HgTe/CdTe: Γi = {Γ6, Γ7, Γ8}
Band structure of HgTe QW II
- 1.0 -0.5 0.0 0.5 1.0k( 0.01 )-1500
-1000
-500
0
500
1000
(eV)
H gTe
8
6
7 - 1.0 -0.5 0.0 0.5 1.0k( 0.01 )
H g0.3 2Cd0 .68T e
-1500
-1000
-500
0
500
1000
E(meV)
6
8
7
VB O
HgCdTeHgTe
Γ6 Γ6
HgTe
HgCdTe
H1E1
Γ8 Γ
HgCdTe
VBO = 570 meV
⇔Γ8 Γ8
inverted normalband structure
Effective model near Γ point( )
( ) ( ) ( ) ( )0
with aa
h kH h k d kk σε
⎛ ⎞⎜ ⎟= = +⎜ ⎟( ) ( ) ( ) ( )
*0a
h k⎜ ⎟⎝ ⎠
Bernevig, Hughes, Zhang Science 2006
( )( ) ( )
2
2, ,x y
C Dk
Ak Akd k
k
M Gk
ε = −
= − −( )
{ }
with basis states: comparisonto 8-band
Kane model{ }, , ,E H E H+ + − −
d K
Kane model
±: degenerate Kramers partnersNovik et al. PRB 2005
Schmidt, Novik, Kindermann & BT arXiv 2009
Experimental consequence:Experimental consequence:edge states at B=0g
Prediction: Observation:
Bernevig, Hughes & Zhang Science 2006 Koenig et al. Science 2007
see also: HL34 Spin controlled transport II, Wed 14:00-17:00, BEY 118
Effective model at finite B-field0
with 0 JH CO
hH hh
hh+
±−
±⎛ ⎞= = +⎜ ⎟⎝ ⎠
( )0 ,0,0A By= −⎝ ⎠
† † 31 12 2HO C DB a a M BG ah a σ⎡ ⎤⎛ ⎞ ⎛ ⎞= − + + − +⎜ ⎟ ⎜ ⎟⎢ ⎥2 22 2HO C DB a a M BG ah a σ+ + +⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
diagonal in E,H space and Kramers space
( )( )
†
†
2JCh B A a aσ σ+ + −= − +
comparisonto 8-band
Kane model( )†2JCh B A a aσ σ− + −= + +
couples HO levels, lifts ± degeneracycouples HO levels, lifts degeneracy
Schmidt, Novik, Kindermann & BT arXiv 2009
Edge states at finite B-field( )edgeM M V y→ +
i l l h l f h i l ⎛ ⎞varies slowly on the scale of the typical extentof Landau level wave function in y-direction
( )edge edgekV y VB
⎛ ⎞→ ⎜ ⎟⎝ ⎠
Optical transitions( ) ( )0 0 1 1 1 ˆ ˆ, with , 2 cosA A A r t A r t A n r t
cωε ω⎛ ⎞→ + = ⋅ −⎜ ⎟⎝ ⎠
ˆ ˆ ˆ ˆchoice: and y ze n eε = = −( )0 ,0,0A By= −
( ) ( )( ) ( )( ) ( ) ( ) ( )
† 31
† 3
21
2
2 cos
2
2 cos 2
2 2
h h i A t B a a D G
h h i A B
A A t
AD G A
ω σω σ+ +→ − − + −1
1( ) ( ) ( ) ( )† 311
22 c2 co s2 osh h i A t B a a A tD G A ωσ σω− −→ − − + +1
Relative importance of terms:
2 28 meVnm, 2 37 meV 364 menm, , for 1TVnmBD BG BA− − =2 28 meVnm, 2 37 meV 364 menm, , for 1TVnmBD BG BA
Selection rules
n = 1Δ ± similar to graphene:
Jiang et al. PRL 2007
Optical manipulation of edgeOptical manipulation of edge state transportp
f hfocus on two states that are resonantly connected
( ){ }b 0 k−Ψ ( ){ }1subspace: 0 , kΨ
( ) ( ) ( )( ) ( )3 20
1, ,2ti x t E k E k Q x tω σ σ⎡ ⎤∂ Ψ = + Δ − + Ψ⎢ ⎥⎣ ⎦
( ) ( ) ( ) ( ) ( ) ( ) 10 0 0 1 1
1 1with , , Q=AA cos2 2 2
E k E k k E k E k k φε ε − ⎛ ⎞+ Δ = − Δ = ⎜ ⎟⎝ ⎠2 2 2⎝ ⎠
depends on all system parameters
Transfer matrix approachlinearization of spectrum (at the edge) ⇒
⎡ ⎤⎛ ⎞ ( )1 2
2
00
0 x E
vE p Q x
vσ
⎡ ⎤⎛ ⎞− − Ψ =⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦
in transfer matrix notation:
( )20, 0x EE i Q T x xσ⎡ ⎤+ ∂ − =⎣ ⎦v
( ) ( ) ( )10 0
2
0with and ,
0 E E E
vx T x x x
v⎛ ⎞
= Ψ = Ψ⎜ ⎟⎝ ⎠
v
( ) ( )( )1 2
0
0, expL
E E xM T L T i dx E Q x σ−⎛ ⎞≡ = − −⎜ ⎟
⎝ ⎠∫ v
energy of scattering statesrelative to Fermi energy
x-dependent intensityof the FIR source
Transmissionin the off-resonant
case, current is
( )22
1E
E
tM
=
barely blocked
( )22
( )0
11 2 0
1
coshL
tA dxA xv v
=⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
∫
How to detect it? → Setup1
AssumptionsAssumptions• all chemical potentials tuned between 1h and 2h• μ1 > μ2= μ3= μ4 12 1 2I μ μ∝ −μ1 > μ2 μ3 μ4• L12=L13
• FIR ⇒ counterclockwise movers
12 1 2I μ μ
• FIR ⇒ counterclockwise movers scatter into clockwise movers• at resonance: I12 exponentially suppressed
121IA A L
∝⎛ ⎞2 1 12cosh
A A Lv
⎛ ⎞⎜ ⎟⎝ ⎠
How to detect it? → Setup2
AssumptionsAssumptions• all chemical potentials tuned between 1h and 2h• μ1 = μ2= μ3= μ4⇒ μ1 μ2 μ3 μ4• L12 à L13
• without radiation ⇒ no net current
⇒
• without radiation ⇒ no net current• FIR ⇒ parts of large counterclockwise background current is blocked• more efficient on long edge⇒ net current, e.g., from 3 to 1
Summary and Outlook
• Similarities and difference between TI’s d hand graphene
• Low-energy theory of TI’s• Optical manipulation of edge state
transportp
• Relaxation mechanism for excited edge
Schmidt, Novik, Kindermann & BT arXiv:0901.0621
• Relaxation mechanism for excited edge statesR l f R hb SOI• Role of Rashba SOI
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