THE ZAK PHASE AND THE EDGE STATES IN GRAPHENE Collaborators: Gilles - - PowerPoint PPT Presentation

THE ZAK PHASE AND THE EDGE STATES IN GRAPHENE

Pierre DELPLACE

Collaborators: Gilles Montambaux & Denis Ullmo Université 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 : Bulk topological number =

connection dλ

∫ 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

/ t ε

• 3

3

x

k

Zigzag

2 3

x

k a π ∆ =

• Y. Fujita et al.
• J. Phys. Soc. Jpn 65, 1920 (1996).
• K. Nakada et al.
• Phys. Rev. B 54, 17954 (1996).

a

Graphene

/ t ε

• 3

3

x

k

Zigzag

y

k

• 3

3

/ t ε

Armchair

2 3

x

k a π ∆ =

y

k ∆ =

• Y. Fujita et al.
• J. Phys. Soc. Jpn 65, 1920 (1996).
• K. Nakada et al.
• Phys. Rev. B 54, 17954 (1996).

a

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?

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

Γ

closed path in 1D t’

t

t’/t > 1 Z = 0

Periodical system (Bulk)

What about the edge states?

t’

t

What is the Zak phase?

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

Γ

closed path in 1D

t’/t > 1 Z = 0 t’/t < 1 Z = π

Periodical system (Bulk)

What about the edge states?

t’

t

What is the Zak phase?

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

Γ

closed path in 1D

t’

t t’/t > 1 Z = 0 t’/t < 1 Z = π t’/t > 1-1/(M+1) Nstat = 2 M NO edge state t’/t < 1-1/(M+1) Nstat = 2 (M-1) 2 edge states

Periodical system (Bulk)

What about the edge states? What is the Zak phase?

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

Γ

closed path in 1D Open System

t’

t t’/t > 1 Z = 0 t’/t < 1 Z = π t’/t > 1-1/(M+1) Nstat = 2 M NO edge state t’/t < 1-1/(M+1) Nstat = 2 (M-1) 2 edge states

Periodical system (Bulk)

What about the edge states? What is the Zak phase?

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

Γ

closed path in 1D Open System

Z = 0 NO edge state Z = π 2 edge states Bulk-Edge correspondence

What about the edge states? What is the Zak phase?

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

• 3

3

x

k

t’

t

t’/t

/ t ε / t ε

Γ

closed path in 1D

How to define the edge in graphene?

Translations of the dimer A-B times along and times along in an arbitrary order.

1 2

( , ) T m n ma na = + u r r r m n

1

a r

2

a r

How to define the edge in graphene?

1 2

2 T a a = + u r r r

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

« minimal » boundary conditions

Translations of the dimer A-B times along and times along in an arbitrary order.

1 2

( , ) T m n ma na = + u r r r m n

1

a r

2

a r

How to define the edge in graphene?

1 2

2 T a a = + u r r r

1 2

2 T a a = − u r r r

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

« minimal » boundary conditions

Translations of the dimer A-B times along and times along in an arbitrary order.

1 2

( , ) T m n ma na = + u r r r m n

1

a r

2

a r

How to define the edge in graphene?

1 2

2 T a a = + u r r r

Translations of the dimer A-B times along and times along in an arbitrary order.

1 2

( , ) T m n ma na = + u r r r m n

1

a r

2

a r

Period Relation between the edge and the Zak phase

1 2

( , ) T m n ma na = + u r r r

Period

2

( , ) 2 T n m T π Γ =

P

r r r

Brillouin zone of the ribbon

Period Period

k ∈Γ

P P

Does an edge state exist ? Relation between the edge and the Zak phase

1 2

( , ) T m n ma na = + u r r r

2

( , ) 2 T n m T π Γ =

P

r r r

( ) Z kP

counts the number of missing bulk states

Brillouin zone of the ribbon

Period

k ∈Γ

P P

Does an edge state exist ?

Appropriate 2D Brillouin zone

Relation between the edge and the Zak phase

( , ) n m

⊥

Γ r

1 2

( , ) T m n ma na = + u r r r

( ) Z kP

Period counts the number of missing bulk states

2

( , ) 2 T n m T π Γ =

P

r r r

Brillouin zone of the ribbon

Period

k ∈Γ

P P

Does an edge state exist ? Relation between the edge and the Zak phase

1 2

( , ) ( ) n m nb m b

⊥

Γ = + − r r r

1 2

( , ) T m n ma na = + u r r r

( ) Z kP

Period

Appropriate 2D Brillouin zone

counts the number of missing bulk states

2

( , ) 2 T n m T π Γ =

P

r r r

Brillouin zone of the ribbon

Period

k ∈Γ

P P

Does an edge state exist ?

kP

( )

k k k

Z k i dk u u d π

⊥

⊥

= ∂ = ±

∫

r r P

Ń

Relation between the edge and the Zak phase

1 2

( , ) ( ) n m nb m b

⊥

Γ = + − r r r

1 2

( , ) T m n ma na = + u r r r

Period

Appropriate 2D Brillouin zone

2

( , ) 2 T n m T π Γ =

P

r r r

Brillouin zone of the ribbon

Period

k ∈Γ

P P

Does an edge state exist ?

kP

( )

k k k

Z k i dk u u d π

⊥

⊥

= ∂ = ±

∫

r r P

Ń

Relation between the edge and the Zak phase

1 2

( , ) ( ) n m nb m b

⊥

Γ = + − r r r

1 2

( , ) T m n ma na = + u r r r

Period

(2,5) T u r

Same Zak phase

Appropriate 2D Brillouin zone

2

( , ) 2 T n m T π Γ =

P

r r r

Brillouin zone of the ribbon

Period

k ∈Γ

P P

Does an edge state exist ?

kP

( )

k k k

Z k i dk u u d π

⊥

⊥

= ∂ = ±

∫

r r P

Ń

Relation between the edge and the Zak phase

1 2

( , ) ( ) n m nb m b

⊥

Γ = + − r r r

1 2

( , ) T m n ma na = + u r r r

Period

Bulk eignenvectors

• f graphene

(2,5) T u r

Same Zak phase

Appropriate 2D Brillouin zone

2

( , ) 2 T n m T π Γ =

P

r r r

Brillouin zone of the ribbon

BULK

Relation between the edge and the Zak phase

Depends on the vectors basis, dimer A-B …

BULK EDGE

SAME DIMER A-B

Relation between the edge and the Zak phase

BULK EDGE

SAME DIMER A-B Winding properties

Relation between the edge and the Zak phase

How to evaluate the Zak phase in graphene?

y

k

x

k

Dirac Points

How to evaluate the Zak phase in graphene?

y

k

x

k

Dirac Points

How to evaluate the Zak phase in graphene?

Lines of discontinuities

y

k

x

k

Dirac Points

=

How to evaluate the Zak phase in graphene?

Lines of discontinuities

degeneracy of the edge state

y

k

x

k

(1,0)

⊥

Γ r

(1,0) (0,0)

2

(0,1) T a = u r r

Example 1: zigzag ribbon

y

k

x

k

(1,0)

⊥

Γ r

(1,0) (0,0)

2

(0,1) T a = u r r

Example 1: zigzag ribbon

y

k

x

k

(1,0)

⊥

Γ r ΓP r

(1,0) (0,0)

2

(0,1) T a = u r r

Example 1: zigzag ribbon

y

k

x

k

⊥

Γ r ΓP r kP (1,0)

⊥

Γ r

2

(0,1) T a = u r r

Example 1: zigzag ribbon

y

k

x

k

⊥

Γ r ΓP r kP

Z = π

(1,0)

⊥

Γ r

Example 1: zigzag ribbon

2

(0,1) T a = u r r

y

k

x

k

⊥

Γ r ΓP r

Z = π

(1,0)

⊥

Γ r kP

/ t ε

• 3

3

kP

1 Edge state Example 1: zigzag ribbon

y

k

x

k

⊥

Γ r ΓP r kP

/ t ε

• 3

3

kP

Z = 0 NO Edge state

(1,0)

⊥

Γ r

Example 1: zigzag ribbon

y

k

x

k

(1,5) T = u r (5,1)

⊥

Γ r

Example 2:

(1,5) T = u r

• K. Wakabayashi, et al.,

Carbon 47, 124 (2009).

(1,5) T = u r

Z = 2π 2 Edge states

kP

Example 2:

(1,5) T = u r

• K. Wakabayashi, et al.,

Carbon 47, 124 (2009).

(1,5) T = u r

1 Edge state

kP

Example 2:

(1,5) T = u r

• K. Wakabayashi, et al.,

Carbon 47, 124 (2009).

Z = π

(1,5) T = u r

2 Edge states

kP

Example 2:

(1,5) T = u r

• K. Wakabayashi, et al.,

Carbon 47, 124 (2009).

Z = 2π

Quantitative results

Total range of the existence of edge states

• A. Akhmerov and C. Beenakker,

PRB 77, 085423 (2008)

Quantitative results

OK with Density of edge states per unit length: Total range of the existence of edge states

Quantitative results

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

Quantitative results

k ∆ P k ∆ P

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

t3 t2 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.

Same HBulk

( , ) T m n u r

Appropriate BZ Zak phase Every

• 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.
• 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…)

Same HBulk

( , ) T m n u r

Appropriate BZ Zak phase Every

• 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.

Same HBulk

( , ) T m n u r

Appropriate BZ Zak phase Every

Thank you for your attention!

• 3

3

/ t ε

kP (1,1)

⊥

Γ r kP

Z = 0

(1,1) T = u r

(0,0) (1,1)

NO Edge state Example

NO Edge state

⊥

Γ r kP

Z = 0

(3,3) T = u r

(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

1

b r

2

b r