ph h tt tiphase coherent transport in 2d toppgological ... · pdf fileph h tt tiphase coherent...

Ph h t t t iPhase coherent transport in 2D topological insulatorsp g

Mini-School on TopologicalInsulators and QSHE

Lyon, 9-11 December 2009

Björn Trauzettel Koenig et al. Science 2007

Collaborators:Ewelina Hankiewicz (Würzburg)

Markus Kindermann (Georgia Tech)Markus Kindermann (Georgia Tech)Alena Novik (Würzburg)Patrik Recher (Würzburg)Manuel Schmidt (Basel)

(my personal) motivationhelical

Luttingerliquidliquid

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 ↓) ---

Outline• Effective Hamiltonian of HgTe Quantum Wells

• Optical manipulation of edge state transport in QH regimeregime

• Bulk transport properties (conductance and noise)Bulk transport properties (conductance and noise) in QSHI regime

• Summary

Band structure of HgTe QW I

Koenig et al. JPSJ 2008

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

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}

Effective model near Γ point( )

( ) ( ) ( ) ( )0

with aa

h kH h k d kkε σ

⎛ ⎞⎜ ⎟= = +⎜ ⎟( ) ( ) ( ) ( )

*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 PRB 2009

Theory of edge states

( ) ( )2 2M G k A k Eψ ψ ψ⎡ ⎤∂ + ∂ =⎣ ⎦

strip geometry with open boundary in y direction ⇒

( ) ( )( ) ( )

1 2 1

2 21 2 2

x y x y

x y x y

M G k A k E

A k M G k E

ψ ψ ψ

ψ ψ ψ

+

−

⎡ ⎤− − ∂ + − ∂ =⎣ ⎦⎡ ⎤+ ∂ − − − ∂ =⎣ ⎦

G G D± = ±

ansatz: ⇒1,2yeλψ =

2 4 0 real solutionsA MG G G

λ+ −

> > ⇒

M: Einsteinian mass

Zh L Ch Sh & Ni PRL 2008

G: Newtonian mass

Zhou, Lu, Chu, Shen & Niu PRL 2008

Experimental evidence:Experimental evidence:edge states at B=0g

Prediction: Observation:Prediction: Observation:

B i H h & Zh S i 2006 K i t l S i 2007Bernevig, Hughes & Zhang Science 2006 Koenig et al. Science 2007

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


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 edgekB

V y V ⎛ ⎞→ ⎜ ⎟⎝ ⎠

LLs centeredaround y=k/B




• Summary

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 transport Ip

f hfocus on two states that are resonantly connected

( ){ }b 0 k−Ψ ( ){ }1subspace: 0 , kΨ

( ) ( ) ( ) ( )30

1, 2 cos2

H k t E k E k Q tσ ω= + Δ +

rewrite Hamiltonian in this subspace:

( ) ( ) ( ) ( ) ( ) ( ) 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 2⎝ ⎠

depends on all system parameters

Optical manipulation of edgeOptical manipulation of edge state transport IIp

i tω⎛ ⎞( )

2

2

0

0

t

i t

eU t

e

ω

ω−

⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

unitary transformation

( ) ( ) ( )( )2

†2 20 1

2i ti e

U Uω⎛ ⎞− +

⎜ ⎟

+ rotating wave approximation

( ) ( ) ( )( )

( )†2 2

22cos

1 0i tU t t U t

i e ωω σ σ

−⎜ ⎟=⎜ ⎟+⎝ ⎠

( ) ( ) ( )( ) ( )3 20

1, ,2ti x t E k E k Q x tω σ σ⎡ ⎤∂ Ψ = + Δ − + Ψ⎢ ⎥⎣ ⎦

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

01

1

cosht

AA Lv

=⎛ ⎞⎜ ⎟⎝ ⎠

For L=5μm, v=105m/s,and FIR power 4mW focused

1 2 A AL/ 1on 1mm2 ⇒ A1AL/v ≈ 1

How to detect it? → Setup I

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? → Setup II

AssumptionsAssumptions• all chemical potentials tuned between 1h and 2h• μ1 = μ2= μ3= μ4μ1 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

solar cell setup




• Summary

Ballistic transport theoryBallistic transport theory in TIs vs. graphenein TIs vs. graphene

( )H v V x= +k σ ( ).H v V x= +k σ

Tworzydlo, BT, Titov, Rycerz & Beenakker PRL 2006

• model the leads by highly doped graphene• difference to HgTe quantum wells: (i) no mass term

(ii) no quadratic terms

Predictions in graphene I

( )1

0

1N

n nn

T TS

F

−

=−∑

01

0

2n

N

nn

FeI

T

=−

=

= =

∑

124 NL e −

0

4n

n

L eT

W hσ

== ∑


Predictions in graphene II

( )1

0

1N

n nn

T TS

F

−

=−∑

01

0

2n

N

nn

FeI

T

=−

=

= =

∑

124 NL eTσ

−

= ∑0n

n

TW h

σ=

= ∑


Interesting questions

• What is the influence of the finite gap and the quadratic terms (in the effective HgTe QW Hamiltonian) on transport?Q ) p

d h b lk ff h d• How does the bulk transport affect the edge transport in the inverted regime?

Wh t i d d l f th l d ?• What is a good model for the leads?

How to model metallic leads?

( ) ( ) ( ) ( )0

ith ah k

H h k d kk⎛ ⎞⎜ ⎟( )

( ) ( ) ( ) ( )*

with 0

aaH h k

h kd kkε σ⎜ ⎟= = +

⎜ ⎟−⎝ ⎠

( )( )( ) ( )

2

2, ,x y

C Dk

Ak Akd k

k

M Gk

ε = −

= − −

{ }, , ,E H E H+ + − −basis:

Yokoyama, Tanaka & Nagaosa PRL 2009

N side: A=M=G=0⇒ artificial degeneracy⇒ artificial degeneracy

Alternative: highly doped HgTe

Assumptions:( )

( ) ( ) ( ) ( )*

0 with

0a

a

h kH h k

h kd kkε σ

⎛ ⎞⎜ ⎟= = +⎜ ⎟ p

• leads are modelled as highlydoped HgTe QWs

• periodic bc → no mode mixing

( ) ( ) ( ) ( )0 h k⎜ ⎟−⎝ ⎠

( ) 2C Dkkε = − periodic bc no mode mixing( ) ( )2, ,x yAk Akd k M Gk= − −

Conductance + noise at EF=0

M = 3.1meV: dashed linesM 0 lid liM = 0: solid lines

M = -3.1meV: dotted lines

Observations:

• bulk transport depends on thesign of M

• non monotonic function for M<0• non-monotonic function for M<0• bulk transport can substantially contribute to edge transport

Novik, Recher, Hankiewicz & BT in preparation2 21 m, 375meV, 1120meVnm , 3.75eV, 730meVnmW A G C Dμ= = = − = − = −

Can we understand theCan we understand the maximum?maximum?

Transmission of zero mode forGC A2 |M|G

( )2 2 M L

AA −

GC àA2 à |M|G:

( )0 e0 AAT MBC

> ≈

( )2

0 22

2

2

24 , for e 10

f

e

1

M LA

M L

M L

M LAA

AGCAT MA

− −

−

⎧⎪⎪< ≈ ⎨⎪ , fore e 1AA

BC⎪⎪⎩

21 m, 375meV, 1120meVnm , 3.75eV, 0W A G C Dμ= = = − = − =

Conductance + noise at EF¥0

MW/A = 8.3: dashed linesMW/A 0 lid liMW/A = 0: solid lines

MW/A = -8.3: dotted lines

Observations:

• minimum of conductance at EF≈|M|(in the inverted regime)• maximum of Fano factor at E ≈|M|• maximum of Fano factor at EF≈|M|(in the inverted regime)

2 21 m, 7, 375meV, 1120meVnm , 3.75eV, 730meVnmWW A G C DL

μ= = = = − = − = −

Summary( ) 0h k⎛ ⎞( )

( ) ( ) ( ) ( )*

0 with

0a

a

h kH h k

h kd kkε σ

⎛ ⎞⎜ ⎟= = +⎜ ⎟−⎝ ⎠


Novik, Recher, Hankiewicz & BT in preparation

ph h tt tiphase coherent transport in 2d toppgological ... · pdf fileph h tt tiphase coherent...

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