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

Pierre DELPLACE THE ZAK PHASE AND THE EDGE STATES IN GRAPHENE Collaborators: Gilles Montambaux & Denis Ullmo Université Paris-Sud XI, CNRS, FRANCE Nanoelectronics beyond the roadmap, June 13 th , Lake Balaton, Hungary.

B Spin-orbit Quantum Hall Systems Quantum Spin Hall Systems E F E F B. I. Halperin, Phys. Rev. B 25 , 2189 (1982). M. König et al., J. Phys Soc. Jpn 77 , 031007 (2008)

B Spin-orbit Quantum Hall Systems Quantum Spin Hall Systems E F E F B. I. Halperin, Phys. Rev. B 25 , 2189 (1982). M. König et al., J. Phys Soc. Jpn 77 , 031007 (2008) Bulk number of d λ : = connection edge states Ń ∫ topological number Bulk-Edge correspondence

Graphene Mono-layer of graphite K.S. Novoselov et al . Nature 438 , 197-200 (2005).

Graphene Zigzag 3 ε π / t 2 ∆ = k 0 x 3 a 0 a 0 k x -3 Y. Fujita et al. J. Phys. Soc. Jpn 65 , 1920 (1996). K. Nakada et al. Phys. Rev. B 54 , 17954 (1996).

Graphene Zigzag 3 ε π / t 2 ∆ = k 0 x 3 a 0 a 0 k x -3 Y. Fujita et al. J. Phys. Soc. Jpn 65 , 1920 (1996). 3 K. Nakada et al. Phys. Rev. B 54 , 17954 (1996). Armchair ε / t ∆ = 0 k 0 y k y -3

What is ∆ k for arbitrary edges ? A. R. Akhmerov and C. W. J. Beenakker, Phys. Rev. B 77 , 085423 (2008), Boundary conditions for Dirac fermions on a terminated honeycomb lattice. 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. 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).

What is the Zak phase? J. Zak, Phys. Rev. Lett. 62 , 2747 (1988). Zak phase = Berry phase on a closed path in one dimension closed path in 1D Γ Berry connection

What is the Zak phase? Zak phase = Berry phase on a closed path in one dimension closed path in 1D Γ Berry connection What about the edge states? t’ t

What is the Zak phase? Zak phase = Berry phase on a closed path in one dimension closed path in 1D Γ Berry connection What about the edge states? t’ t Periodical system (Bulk) t’/t > 1 Z = 0

What is the Zak phase? Zak phase = Berry phase on a closed path in one dimension closed path in 1D Γ Berry connection What about the edge states? t’ t Periodical system (Bulk) t’/t > 1 Z = 0 t’/t < 1 Z = π

What is the Zak phase? Zak phase = Berry phase on a closed path in one dimension closed path in 1D Γ Berry connection What about the edge states? t’ t Periodical system (Bulk) t’/t > 1 Z = 0 t’/t < 1 Z = π Open System t’/t > 1-1/(M+1) N stat = 2 M NO edge state t’/t < 1-1/(M+1) N stat = 2 (M-1) 2 edge states

What is the Zak phase? Zak phase = Berry phase on a closed path in one dimension closed path in 1D Γ Berry connection What about the edge states? t’ t Periodical system (Bulk) t’/t > 1 Z = 0 Bulk-Edge t’/t < 1 Z = π correspondence Open System Z = 0 NO edge state t’/t > 1-1/(M+1) N stat = 2 M NO edge state Z = π 2 edge states t’/t < 1-1/(M+1) N stat = 2 (M-1) 2 edge states

What is the Zak phase? Zak phase = Berry phase on a closed path in one dimension closed path in 1D Γ Berry connection What about the edge states? t’ t 3 ε ε / t / t t’/t 0 -3 k x

How to define the edge in graphene? Translations of the dimer A-B r r m a a times along and times along n 1 2 in an arbitrary order. u r r r = + T m n ( , ) ma na 1 2

How to define the edge in graphene? A. R. Akhmerov and C. W. J. Beenakker, Phys. Rev. B 77 , 085423 (2008), « minimal » boundary conditions u r r r = + T a 2 a 1 2 Translations of the dimer A-B r r m a a times along and times along n 1 2 in an arbitrary order. u r r r = + T m n ( , ) ma na 1 2

How to define the edge in graphene? A. R. Akhmerov and C. W. J. Beenakker, Phys. Rev. B 77 , 085423 (2008), « minimal » boundary conditions u r r r = + T a 2 a 1 2 Translations of the dimer A-B r r m a a times along and times along n 1 2 u r r r in an arbitrary order. = − T 2 a a 1 2 u r r r = + T m n ( , ) ma na 1 2

How to define the edge in graphene? u r r r = + T a 2 a 1 2 Translations of the dimer A-B r r m a a times along and times along n 1 2 in an arbitrary order. u r r r = + T m n ( , ) ma na 1 2

Relation between the edge and the Zak phase u r r r = + Period T m n ( , ) ma na 1 2 r r T Γ = π ( , n m ) 2 Period r P 2 T Brillouin zone of the ribbon

Relation between the edge and the Zak phase u r r r = + Period T m n ( , ) ma na 1 2 r r T Γ = π ( , n m ) 2 Period r P 2 T Brillouin zone of the ribbon counts the number of Does an edge state Z k P ( ) k ∈Γ missing bulk states exist ? P P

Relation between the edge and the Zak phase u r r r = + Period T m n ( , ) ma na Appropriate 1 2 2D Brillouin zone r r T Γ = π ( , n m ) 2 Period r P 2 r T Γ ( , n m ) ⊥ Brillouin zone of the ribbon counts the number of Does an edge state Z k P ( ) k ∈Γ missing bulk states exist ? P P

Relation between the edge and the Zak phase u r r r = + Period T m n ( , ) ma na Appropriate 1 2 2D Brillouin zone r r T Γ = π ( , n m ) 2 Period r P 2 r r r T Γ = + − ( , n m ) nb m ( b ) ⊥ 1 2 Brillouin zone of the ribbon counts the number of Does an edge state Z k P ( ) k ∈Γ missing bulk states exist ? P P

Relation between the edge and the Zak phase u r r r = + Period T m n ( , ) ma na Appropriate 1 2 2D Brillouin zone r r T Γ = π ( , n m ) 2 Period k P r P 2 r r r T Γ = + − ( , n m ) nb m ( b ) ⊥ 1 2 Brillouin zone of the ribbon Does an edge state = ∂ = ± π Z k ( ) i dk u u d k ∈Γ r r ⊥ Ń ∫ exist ? P k k k ⊥ P P

Relation between the edge and the Zak phase u r r r = + Period T m n ( , ) ma na Appropriate 1 2 2D Brillouin zone r r T Γ = π ( , n m ) 2 Period k P r P 2 r r r T Γ = + − ( , n m ) nb m ( b ) ⊥ 1 2 Brillouin zone of the ribbon Does an edge state = ∂ = ± π Z k ( ) i dk u u d k ∈Γ r r ⊥ Ń ∫ exist ? P k k k ⊥ P P u r Same Zak phase T (2,5)

Relation between the edge and the Zak phase u r r r = + Period T m n ( , ) ma na Appropriate 1 2 2D Brillouin zone r r T Γ = π ( , n m ) 2 Period k P r P 2 r r r T Γ = + − ( , n m ) nb m ( b ) ⊥ 1 2 Brillouin zone of the ribbon Does an edge state = ∂ = ± π Z k ( ) i dk u u d k ∈Γ r r ⊥ Ń ∫ exist ? P k k k ⊥ P P Bulk eignenvectors u r Same Zak phase T (2,5) of graphene

Relation between the edge and the Zak phase BULK Depends on the vectors basis, dimer A-B …

Relation between the edge and the Zak phase BULK SAME DIMER A-B EDGE

Relation between the edge and the Zak phase BULK SAME DIMER A-B EDGE Winding properties

How to evaluate the Zak phase in graphene? k y k x

How to evaluate the Zak phase in graphene? Dirac Points k y k x

How to evaluate the Zak phase in graphene? Dirac Points Lines of discontinuities k y k x

How to evaluate the Zak phase in graphene? Dirac Points Lines of discontinuities k y = degeneracy of the edge state k x

Example 1: zigzag ribbon r Γ (1,0) (1,0) ⊥ u r k r = y (0,0) T (0,1) a 2 k x

Example 1: zigzag ribbon r Γ (1,0) (1,0) ⊥ u r k r = y (0,0) T (0,1) a 2 k x

Example 1: zigzag ribbon r Γ (1,0) (1,0) ⊥ u r k r = y (0,0) T (0,1) a 2 r Γ P k x

Example 1: zigzag ribbon r Γ r ⊥ Γ (1,0) ⊥ u r k r = y T (0,1) a 2 k P r Γ P 0 k x

Example 1: zigzag ribbon r Γ r ⊥ Γ (1,0) Z = π ⊥ u r k r = y T (0,1) a 2 k P r Γ P 0 k x

Example 1: zigzag ribbon 3 r Γ r ⊥ Γ ε (1,0) Z = π ⊥ / t k 0 y k P r Γ P k P 0 -3 0 1 Edge state k x

Example 1: zigzag ribbon 3 r Γ r ⊥ Γ ε (1,0) Z = 0 ⊥ / t k 0 y r k P Γ P k P 0 -3 0 NO Edge state k x

u r T = Example 2: (1,5) K. Wakabayashi, et al., Carbon 47, 124 (2009). r Γ (5,1) ⊥ u r T = (1,5)

u r T = Example 2: (1,5) K. Wakabayashi, et al., Carbon 47, 124 (2009). k P u r T = (1,5) Z = 2 π 2 Edge states

u r T = Example 2: (1,5) K. Wakabayashi, et al., Carbon 47, 124 (2009). k P u r T = (1,5) Z = π 1 Edge state

u r T = Example 2: (1,5) K. Wakabayashi, et al., Carbon 47, 124 (2009). k P u r T = (1,5) Z = 2 π 2 Edge states

Quantitative results Total range of the existence of edge states