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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
Schmidt, Novik, Kindermann & BT PRB 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 edgekB
V y V ⎛ ⎞→ ⎜ ⎟⎝ ⎠
LLs centeredaround y=k/B
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
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
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
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
== ∑
Tworzydlo, BT, Titov, Rycerz & Beenakker PRL 2006
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=
= ∑
Tworzydlo, BT, Titov, Rycerz & Beenakker PRL 2006
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
⎛ ⎞⎜ ⎟= = +⎜ ⎟−⎝ ⎠
Schmidt, Novik, Kindermann & BT PRB 2009
Novik, Recher, Hankiewicz & BT in preparation
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