THE ZAK PHASE AND THE EDGE STATES IN GRAPHENEnanoctm-balaton.elte.hu/slides/slides_delplace.pdf · 2D Brillouin zone Relation between the edge and the Zak phase Γ⊥( , )n m r T

THE ZAK PHASE AND THE EDGE STATES IN GRAPHENE

Pierre DELPLACE

Collaborators: Gilles Montambaux & Denis UllmoUniversité Paris-Sud XI, CNRS, FRANCE

Nanoelectronics beyond the roadmap, June 13th, Lake Balaton, Hungary.

B

Quantum Hall Systems Quantum Spin Hall Systems

Spin-orbit

B. I. Halperin, Phys. Rev. B 25, 2189 (1982). M. König et al., J. Phys Soc. Jpn 77, 031007 (2008)

EF

EF

B

Quantum Hall Systems Quantum Spin Hall Systems

Spin-orbit

number of edge states

: Bulktopological number

= connectiondλ∫Ń

B. I. Halperin, Phys. Rev. B 25, 2189 (1982).

Bulk-Edge correspondence

M. König et al., J. Phys Soc. Jpn 77, 031007 (2008)

EF

EF

Graphene

K.S. Novoselov et al.

Nature 438, 197-200 (2005).

Mono-layer of graphite

Graphene

/ tε

-3

0

3

xk

Zigzag

0

2

3xka

π∆ =

Y. Fujita et al.

J. Phys. Soc. Jpn 65, 1920 (1996).

K. Nakada et al.

Phys. Rev. B 54, 17954 (1996).

0a

Graphene

/ tε

-3

0

3

xk

Zigzag

yk-3

0

3

/ tεArmchair

0

2

3xka

π∆ =

0yk∆ =

Y. Fujita et al.

J. Phys. Soc. Jpn 65, 1920 (1996).

K. Nakada et al.

Phys. Rev. B 54, 17954 (1996).

0a

What is ∆k for arbitrary edges ?

Bulk-edge correspondance with an edge dependance in graphene ?

S. Ryu and Y. Hatsugai, Phys. Rev. Lett. 89, 077002 (2002),

Topological origin of zero-energy edge states in particle-hole symmetric systems.

A. R. Akhmerov and C. W. J. Beenakker, Phys. Rev. B 77, 085423 (2008),

Boundary conditions for Dirac fermions on a terminated honeycomb lattice.

Topological number defined on a reduced (1D) space of parameter = « Zak » phase

What is the Zak phase? J. Zak, Phys. Rev. Lett. 62, 2747 (1988).

Zak phase = Berry phase on a closed path in one dimension

Γ

closed path in 1D

Berry connection

What is the Zak phase? J. Zak, Phys. Rev. Lett. 62, 2747 (1988).

What about the edge states?

What is the Zak phase?


Berry connection

Γ

closed path in 1D

t’ t

t’/t > 1 Z = 0Periodical system (Bulk)

What about the edge states?t’ t



Berry connection

Γ

closed path in 1D

t’/t > 1 Z = 0t’/t < 1 Z = π

Periodical system (Bulk)

What about the edge states?t’ t



Berry connection

Γ

closed path in 1D

t’ t

t’/t > 1 Z = 0t’/t < 1 Z = π

t’/t > 1-1/(M+1) Nstat= 2 M NO edge statet’/t < 1-1/(M+1) Nstat= 2 (M-1) 2 edge states





Berry connection

Γ

closed path in 1D

Open System

t’ t

t’/t > 1 Z = 0t’/t < 1 Z = π

t’/t > 1-1/(M+1) Nstat= 2 M NO edge statet’/t < 1-1/(M+1) Nstat= 2 (M-1) 2 edge states





Berry connection

Γ

closed path in 1D

Open System Z = 0 NO edge stateZ = π 2 edge states

Bulk-Edge correspondence




Berry connection

-3

0

3

xk

t’ t

t’/t/ tε / tε

Γ

closed path in 1D

How to define the edge in graphene?

Translations of the dimer A-Btimes along and times along

in an arbitrary order.

1 2( , )T m n ma na= +ur r r

m n1ar

2ar


1 22T a a= +ur r r

A. R. Akhmerov and C. W. J. Beenakker,

Phys. Rev. B 77, 085423 (2008),

« minimal » boundary conditions



1 2( , )T m n ma na= +ur r r

m n1ar

2ar


1 22T a a= +ur r r

1 22T a a= −ur r r

A. R. Akhmerov and C. W. J. Beenakker,

Phys. Rev. B 77, 085423 (2008),

« minimal » boundary conditions



1 2( , )T m n ma na= +ur r r

m n1ar

2ar


1 22T a a= +ur r r



1 2( , )T m n ma na= +ur r r

m n1ar

2ar

Period

Relation between the edge and the Zak phase

1 2( , )T m n ma na= +ur r r

Period 2( , ) 2T

n mT

πΓ =P

rr

r

Brillouin zone of the ribbon

Period

Period

k ∈ΓP P

Does an edge stateexist ?


1 2( , )T m n ma na= +ur r r

2( , ) 2T

n mT

πΓ =P

rr

r

( )Z kP

counts the number of missing bulk states


Period

k ∈ΓP P


Appropriate2D Brillouin zone


( , )n m⊥Γr

1 2( , )T m n ma na= +ur r r

( )Z kP

Period


2( , ) 2T

n mT

πΓ =P

rr

r


Period

k ∈ΓP P



1 2( , ) ( )n m nb m b⊥Γ = + −r rr

1 2( , )T m n ma na= +ur r r

( )Z kP

Period



2( , ) 2T

n mT

πΓ =P

rr

r


Period

k ∈ΓP P


kP

( ) kk kZ k i dk u u dπ

⊥⊥= ∂ = ±∫ r rP Ń


1 2( , ) ( )n m nb m b⊥Γ = + −r rr

1 2( , )T m n ma na= +ur r r

Period


2( , ) 2T

n mT

πΓ =P

rr

r


Period

k ∈ΓP P


kP


⊥⊥= ∂ = ±∫ r rP Ń


1 2( , ) ( )n m nb m b⊥Γ = + −r rr

1 2( , )T m n ma na= +ur r r

Period

(2,5)Tur

Same Zak phase


2( , ) 2T

n mT

πΓ =P

rr

r


Period

k ∈ΓP P


kP


⊥⊥= ∂ = ±∫ r rP Ń


1 2( , ) ( )n m nb m b⊥Γ = + −r rr

1 2( , )T m n ma na= +ur r r

Period

Bulk eignenvectors

of graphene(2,5)Tur

Same Zak phase


2( , ) 2T

n mT

πΓ =P

rr

r


BULK


Depends on the vectors basis, dimer A-B …

BULK

EDGE

SAME DIMER A-B


BULK

EDGE

SAME DIMER A-B

Winding

properties


How to evaluate the Zak phase in graphene?

yk

xk

Dirac Points


yk

xk

Dirac Points


Lines of discontinuities

yk

xk

Dirac Points=


Lines of discontinuities

degeneracy ofthe edge state

yk

xk

(1,0)⊥Γr (1,0)

(0,0)2(0,1)T a=

ur r

Example 1: zigzag ribbon

yk

xk

(1,0)⊥Γr (1,0)

(0,0)2(0,1)T a=

ur r


yk

xk

(1,0)⊥Γr

ΓP

r

(1,0)

(0,0)2(0,1)T a=

ur r


yk

xk

⊥Γr

ΓP

rkP

0

(1,0)⊥Γr

2(0,1)T a=ur r


yk

xk

⊥Γr

ΓP

rkP

Z = π

0

(1,0)⊥Γr


2(0,1)T a=ur r yk

xk

⊥Γr

ΓP

r

Z = π

0

(1,0)⊥Γr

kP

/ tε

-3

0

3

kP

0

1 Edge state


yk

xk

⊥Γr

ΓP

rkP

/ tε

-3

0

3

kP

Z = 0

0

0

NO Edge state

(1,0)⊥Γr


yk

xk

(1,5)T =ur

(5,1)⊥Γr

Example 2: (1,5)T =ur

K. Wakabayashi, et al.,Carbon 47, 124 (2009).

(1,5)T =ur

Z = 2π 2 Edge states

kP



(1,5)T =ur

1 Edge state

kP



Z = π

(1,5)T =ur

2 Edge states

kP



Z = 2π

Quantitative results

Total range of the existence of edge states

A. Akhmerov and C. Beenakker,

PRB 77, 085423 (2008)


OK with

Density of edge states per unit length:

Total range of the existence of edge states


Generalization to non-equal hopping parameters t1≠ t2≠ t3.

t3t2 t1


k∆ P

k∆ P

P. Delplace, PhD thesis, Université Paris Sud XI, (2010).H. Dahal et al., Phys. Rev. B 81, 155406 (2010)

t3t2 t1

Generalization to non-equal hopping parameters t1≠ t2≠ t3.

- simple graphical method.- non equal hopping parameters topological transitions.

CONCLUSION AND OUTLOOKS

• Bulk-edge correspondence in graphene in terms of Zak phase with arbitrary edges.

SameHBulk ( , )T m nur

Appropriate BZ Zak phaseEvery




• A large amount of different edges is considered (but not all of them!). What about edge disorder? …

• Other 2D systems: (p-wave superconductors, bi-layer graphene, square lattice with half a quantum flux per plaquette…)








Thank you for your attention!

-3

0

3/ tε

kP(1,1)⊥Γ

r

kP

0

Z = 0

(1,1)T =ur (0,0) (1,1)

NO Edge state

Example

NO Edge state⊥Γ

r

kP

0

Z = 0

(3,3)T =ur

(0,0) (1,1)

J. Cai et al.

Nature 466, 470 (2010)

D’

D’

D’

D

D

D

Dirac Points

First Brillouin zone of graphene

1br

2br

Documents

THE ZAK PHASE AND THE EDGE STATES IN GRAPHENEnanoctm-balaton.elte.hu/slides/slides_delplace.pdf · 2D Brillouin zone Relation between the edge and the Zak phase Γ⊥( , )n m r T