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