Chirality Sum Rule in Graphene Multilayers

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1 Title 28 APS March Meeting ongki Min and A.. MacDonald Deartment of Physics, The University of Texas at Austin Phys. Rev. B 77, (28)

2 Outline The low energy electronic structure of arbitrarily stacked grahene multilayers are described by a set of chiral 2D electron systems. 1. Monolayer Grahene 2. Multilayer Grahene 3. Chirality Sum Rule 4. Alications: Pseudosin Magnetism

3 1. Monolayer Grahene 1) Grahene K Grahene is a two-dimensional honeycomb lattice of carbon a 2.46A Γ M K atoms. σ, σ* At low energies near K/K', energy bands are described by a 2D Dirac-like equation with linear disersion. Energy (ev) π, π* K Γ M K

4 1. Monolayer Grahene 2) Effective theory At low energies near K, grahene is described by v F e i e 1 σ x, σ y 1 i n( ) (cos, sin ) i v F i σ n v E in-lane velocity F tan 1 ( y / x ) a 2.46A v F K Γ K M K σ refers to sublattice and β instead of sin and. seudosin

5 1. Monolayer Grahene 3) Chirality Chirality is defined by a rojection of sin (σ) on the direction of motion (n). n i e σ n i e ( ) (cos, sin ) Energy sectrum is given by ( ) tan 1 ( y / x ) E ( ) Monolayer grahene is described by =1 chiral 2D electron gas.

6 2. Multilayer Grahene 1) Stacking diagrams Layers are linked by vertical interlayer hoing 3 C β 3 1 β 1 3 A β 3 A Ordered stacks: Guinea et al., PRB 73, (26); Koshino et al., PRB 76, (27); Mañes et al., PRB 75, (27); akamura et al., PRB 77, (28). 4 A β 4 C 3 β 3 4 B β 4 C 3 β 3 4 B β 4 A 3 β 3 4 C β 4 A 3 β 3 β β β B A C

7 2. Multilayer Grahene 2) Effective theory Identify zero-energy states from the diagram. Obtain the effective theory using degenerate state erturbation theory. Examle: ABC stacked -layer Isolated 1 and β are zero-energy states. ABC 1 i e i ABC stacked -layer grahene is described by -chiral 2D electron gas. e Min et al. PRB 77, (28) β -1 β -1 3 β 3 2 β 2 1 β 1 zero-energy states

8 2. Multilayer Grahene 3) Examle: =3 and =4 layers Min et al. PRB 77, (28) The system is decomosed to different chiral systems. 3 C β 3 1 β 1 =3 =2+1 4 A β 4 3 A β 3 A 4 B β 4 4 B β 4 4 C β 4 β β β B A C C 3 β 3 C 3 β 3 A 3 β 3 A 3 β 3 =4 =3+1 =2+2 =1+3 B A C

9 2. Multilayer Grahene 4) Energy band structure Energy sectrum for -chiral system E () 4 B β 4 3 C β 3 Examle: ABCB tetralayer E ABCB ( ), 3 =3+1 E/t Min et al. PRB 77, (28) E ka ABCB tetralayer is described by a combination of =3 and =1 chiral 2D systems 3 E

10 3. Chirality Sum Rule 1) Chirality sum rule -chiral system is described by Min et al. PRB 77, (28) σ n ( ) n ( ) (cos, sin ) Arbitrarily stacked grahene multilayers are described by a set of chiral seudosin doublets. eff 1 2 D D umber of doublets Although the number of doublets in an D i -layer system deends on the stacking i1 sequence, the seudosin chirality sum is always.

11 3. Chirality Sum Rule 2) Quantum all effect Chirality sum rule imlies a new quantum all conductivity. D i1 xy i n=,1,2,3, e h n σ xy (4e 2 /h) B=1T Min et al. PRB 77, (28) ABCA ABCB, ABAC ABAB Examle: Tetralayer =4 quantum all conductivity E F /t EF t Fermi energy intralayer hoing

12 4. Alications: Pseudosin Magnetism 1) Examle: Bilayer grahene (=2) Min et al. PRB 77, 4147(R) (28) k y k c Pseudosin olarization k x k c σ n ( ) n Gate voltage ( ) ( n cos, n sin, n z The charge density sontaneously shifts to one of the two layers. )

13 4. Alications: Pseudosin Magnetism 2) Phase diagram Min et al. PRB 77, 4147(R) (28) ormal f : chirality : interaction strength : doing Ferro Pseudosin magnetism is stable for stronger interaction strength, for smaller doing, for larger chirality. ABC stacked multilayers are excellent candidates.

14 Summary 1) Chirality sum rule Arbitrarily stacked grahene multilayers are described by a set of chiral systems. eff 2) Quantum all effect D 1 2 i D i1 Chirality sum rule imlies a new quantum all conductivity. xy e h n n=,1,2,3, Min et al. PRB 77, (28) 3 C β 3 3 A β 3 σ xy (4e 2 /h) 1 β 1 A =3 =2+1 B=1T ABCA ABCB, ABAC ABAB E F /t

Pseudospin Magnetism in Graphene

Title Phys. Rev. B 77, 041407 (R) (008) Pseudospin Magnetism in Graphene Hongi Min 1, Giovanni Borghi, Marco Polini, A.H. MacDonald 1 1 Department of Physics, The University of Texas at Austin, Austin