The twisted bilayer: an experimental and theoretical review. Graphene, 2009 Benasque. J.M.B. Lopes dos Santos

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1 Moiré in Graphite and FLG Continuum theory Results Experimental and theoretical conrmation Magnetic Field The twisted bilayer: an experimental and theoretical review J.M.B. Lopes dos Santos CFP e Departamento de Física, Faculdade de Ciências, Universidade do Porto Graphene, 2009 Benasque Lopes dos Santos, Peres and Castro Neto The twisted bilayer -Benasque 2009

2 Outline 1 Moiré in Graphite and FLG 2 Continuum theory 3 Results 4 Experimental and theoretical conrmation 5 Magnetic Field

3 Outline Moiré in Graphite and FLG Continuum theory Results Experimental and theoretical conrmation Magnetic Field 1 Moiré in Graphite and FLG 2 Continuum theory 3 Results 4 Experimental and theoretical conrmation 5 Magnetic Field Lopes dos Santos, Peres and Castro Neto The twisted bilayer -Benasque 2009

4 Moiré in graphite and FLG L = a 0 2 sin(θ/2) a 0 θ Rong and Kuiper, PRB 1993

5 Moiré in graphite and FLG L = a 0 2 sin(θ/2) a 0 θ J. Hass, et. al PRL (2008) Latil et al. PRB (2007) Rong and Kuiper, PRB 1993

6 Moiré in graphite and FLG L = a 0 2 sin(θ/2) a 0 θ θ = 32.2 ο L s = 13 Rong and Kuiper, PRB 1993

7 Moiré in graphite and FLG L = a 0 2 sin(θ/2) a 0 θ Varchon et. al, PRB (2008) Rong and Kuiper, PRB 1993

8 Moiré in graphite and FLG L = a 0 2 sin(θ/2) a 0 θ AA AB AB BA AA AA AB Rong and Kuiper, PRB 1993

9 Controversy A 2 B 2 B 1 A 1 Bilayer Graphite Xhie et. al (PRB, 93) Rong Kuiper (PRB,93) BA(Ah) BAB(AhA) Bright (M β) Gray AA (BB) AAB (BBh) Dark (M h) Bright AB(Bh) ABh(BhA) Gray (M α) Dark

10 Moire in Exfoliated Graphene Zenhua Ni et al, PRB (2008) Poncharal et al, PRB (2008)

11 Moire in Exfoliated Graphene Zenhua Ni et al, PRB (2008) Raman 2D band looks like SLG, not like the bilayer, but blue shifted. Poncharal et al, PRB (2008)

12 Outline Moiré in Graphite and FLG Continuum theory Results Experimental and theoretical conrmation Magnetic Field 1 Moiré in Graphite and FLG 2 Continuum theory 3 Results 4 Experimental and theoretical conrmation 5 Magnetic Field Lopes dos Santos, Peres and Castro Neto The twisted bilayer -Benasque 2009

13 Commensurability LdS, Peres, Castro Neto, PRL, 2007 A 2 B 2 B 1 A 1 θ cos(θ i ) = 3i 2 + 3i + 1/2 3i 2 + 3i + 1 t 1 = ia 1 + (i + 1)a 2 t 2 = (i + 1)a 1 + (2i + 1)a 2. i = 1 θ = 21.8º, L = 8 Å

14 Commensurability LdS, Peres, Castro Neto, PRL, 2007 A 2 B 2 B 1 A 1 θ cos(θ i ) = 3i 2 + 3i + 1/2 3i 2 + 3i + 1 t 1 = ia 1 + (i + 1)a 2 t 2 = (i + 1)a 1 + (2i + 1)a 2. i = 1 θ = 21.8º, L = 8 Å Other angles are possible Shallcross, PRL 2008.

15 Continuum limit (k p approximation) Layer 1 H 1 = t P i a 1 (r i ) [b 1(r i + δ 1) + b 1(r i + δ 2) + b 1(r i + δ 3)] + hc a 1(r) v 1/2 c ψ a(r) exp(i K r) +... b 1(r) v 1/2 c ψ b (r) exp(i K r) +... v F X k» ψ (r) 0 i x y i x + y 0 ψ (r).

16 Continuum limit (k p approximation) layer 2 ( rotated) X H 2 = t b 2 (r j) ˆa 2(r j + s 0) + a 2(r j + s 0 a 1) + a 2(r j + s 0 a 2) + hc j a 2(r) v 1/2 c ψ a (r) exp(i K θ r) +... b 2(r) v 1/2 c ψ b (r) exp(i K θ r) +... v F X k» ψ (r) 0 e iθ ( i x y ) e iθ ( i x + y ) 0 ψ (r).

17 Parametrization of t (r) z + 1 c 0 R θ + y t (δ) / t (0) δ δ/ a t (δ) = V ppσ(d) cos 2 θ + V ppπ(d) sin 2 θ t Tang, et al Phys. Rev B 53,979 (1996)

18 Inter-layer hopping Electrons hop from atom in layer 1 to closest atomof either sub-lattice in layer 2; r (r) = r + δ(r) (plane coordinates) t t [δ βα (r)] t βα (r) α(β) = A 1, B 1 (A 2, B 2 ) H = α,β ˆ d 2 r t βα (r)e ikθ δ αβ (r) e i K r ψ 1,α (r)ψ 2,β (r) + h.c. t βα (r)e ikθ δ αβ (r) : period of Moiré pattern. ( K = K θ K); k k + K/2 layer 1 ; k k K/2 layer 2;

19 Inter-layer hopping H = α,β k,g tαβ (G) = 1 ˆ V c uc tβα (G)φ α,k+g φ β,k + h.c. d 2 r t αβ (r)e ikθ δ AB (r) e ig r Dirac electrons with periodic interlayer coupling.

20 Results for tαβ (G) Exact symmetries δ AB δ BA e δ AA δ BB ; limit θ 1, δ AA δ BA ; G 1 + G G 2 1 G 2 K K /2 K /2 G 0 G 1 G 1 G 2 tba (G) t t t tab (G) t e i2π/3 t e i2π/3 t taa (G) t e i2π/3 t e i2π/3 t tbb (G) t e i2π/3 t e i2π/3 t

21 Real Space Coupling t BA AA t, BB t t AB tβα (r) = βα t G (G)e ig r e i K r

22 Outline Moiré in Graphite and FLG Continuum theory Results Experimental and theoretical conrmation Magnetic Field 1 Moiré in Graphite and FLG 2 Continuum theory 3 Results 4 Experimental and theoretical conrmation 5 Magnetic Field Lopes dos Santos, Peres and Castro Neto The twisted bilayer -Benasque 2009

23 Moiré in Graphite and FLG Continuum theory Results Experimental and theoretical conrmation Magnetic Field Electronic Structure Linear Dispersion relation at low energies. New energy scale, v F K 0.19 ev θº. Lopes dos Santos, Peres and Castro Neto The twisted bilayer -Benasque 2009

24 Moiré in Graphite and FLG Continuum theory Results Experimental and theoretical conrmation Magnetic Field Electronic Structure Linear Dispersion relation at low energies. New energy scale, v F K 0.19 ev θº. Electric eld bias does not open a gap. Lopes dos Santos, Peres and Castro Neto The twisted bilayer -Benasque 2009

25 Moiré in Graphite and FLG Continuum theory Results Experimental and theoretical conrmation Magnetic Field Electronic Structure Linear Dispersion relation at low energies. New energy scale, v F K 0.19 ev θº. Electric eld bias does not open a gap. v F v F 1 9 ( t v F K ( t v F K) ) 2 v F is reduced relative to the single layer. Lopes dos Santos, Peres and Castro Neto The twisted bilayer -Benasque 2009

26 Outline Moiré in Graphite and FLG Continuum theory Results Experimental and theoretical conrmation Magnetic Field 1 Moiré in Graphite and FLG 2 Continuum theory 3 Results 4 Experimental and theoretical conrmation 5 Magnetic Field Lopes dos Santos, Peres and Castro Neto The twisted bilayer -Benasque 2009

27 Epitaxial graphene Epitaxial graphene often displays SLG behaviour. de Heer et. al. Sol. St. Comm. vol 143, 92, (2007)

28 Raman Raman in Moire bilayers ( ) EL ω R = 2E ph v f v F E L δω R δv F v F For a 5% reduction, θ 7º, v F K 1.4 ev, E L = 2.33 ev. q q h ω R Dependence of δω R seems wrong. with E L k k q = 2k E L= 2v k F

29 LDA Calculations Laissardière et al. (2009) Latil et al (PRB 2007) ShallCross et al (PRL, 2008), LDA calculations conrm linear dispersion. Laissardière, Mayou & Magaud (2009) Reduction of v F ; localization for very small angles?

30 DOS by STS

31 DOS by STS Low energy VHP Low energy Van Hove peaks (saddle point) predicted at meeting of two cones.

32 DOS by STS DOS (AU) E (ev) E. Andrei et. al. (unpublished)

33 DOS by STS DOS (AU) E (ev) E. Andrei et. al. (unpublished)

34 DOS by STS E. Andrei et. al. (unpublished) Bias shifts cones DOS does not vanish, and is dierent in the two layers.

35 Three data points!

36 Outline Moiré in Graphite and FLG Continuum theory Results Experimental and theoretical conrmation Magnetic Field 1 Moiré in Graphite and FLG 2 Continuum theory 3 Results 4 Experimental and theoretical conrmation 5 Magnetic Field Lopes dos Santos, Peres and Castro Neto The twisted bilayer -Benasque 2009

37 Modulated interlayer hopping Two planes of Dirac massless fermions coupled by modulated hopping ˆ H = d 2 r t βα (r)e ikθ δ βα (r) i K r e ψ (r)ψ α1 β2 (r) + h.c. t βα (r)e ikθ δ BA (r) period of superlattice: t 1, t 2 ; e i K r period 3 that of superlattice, 3t 1, 3t 2, because K = (g 1 + 2g 2 ) /3. G 2 G 1 + G G 2 1 K K /2 K /2

38 Funny gauge H (K) I 2 4 = tl 1 s ie iθ t 1e i 2π 3 + t i 2π ie t iθ 1e i 2π +i 3 + 2π t 3 2 ` A e 0 i c ` A e i c 3 5 A = ( t 1 t 2 Bx 1 /2π) g 2 [ H (K) 0 = tl 1 D s D 0 ] g 2 t 2 A(r) t 1 g 1 D = ie iθ [e i 2π 3 1 i + e i 2π 3 ( 2 i 2π φs φ 0 x 1 )] r = x 1 t 1 + x 2 t 2 Perturbation invariant under (x 1, x 2 ) (x 1 + m, x 2 + n) = Bloch waves on x 2.

39 Symmetry adapted basis Ψ k 2,m(x 1, x 2) = e ik 2x2 u(x 1, x 2) = e ik 2x2 e i2πmx 2 Φ(x 1).

40 Symmetry adapted basis Ψ k 2,m(x 1, x 2) = e ik 2x2 u(x 1, x 2) = e ik 2x2 e i2πmx 2 Φ(x 1). 2 i(k 2 + 2πm) D, D a, a ;

41 Symmetry adapted basis Ψ k 2,m(x 1, x 2) = e ik 2x2 u(x 1, x 2) = e ik 2x2 e i2πmx 2 Φ(x 1). 2 i(k 2 + 2πm) D, D a, a ; Ψ I k2,m,n(x 1, x 2) = Ae ik 2x2 e i2πmx 2 " φ n(x 1 k 2 φ 0 2π φ n 1 (x 1 k 2 2π φ s m φ 0 φ s ) φ 0 φ s m φ 0 φ s ) #

42 Symmetry adapted basis Ψ k 2,m(x 1, x 2) = e ik 2x2 u(x 1, x 2) = e ik 2x2 e i2πmx 2 Φ(x 1). 2 i(k 2 + 2πm) D, D a, a ; Ψ I k2,m,n(x 1, x 2) = Ae ik 2x2 e i2πmx 2 " φ n(x 1 k 2 φ 0 2π φ n 1 (x 1 k 2 2π φ s m φ 0 φ s ) φ 0 φ s m φ 0 φ s ) # k 2 is a good (Bloch) quantum number in presence of perturbation. tαβ (0)e i K r + tαβ ( G 1)e i( K G1) r + tαβ ( G 1 G 2 )e i( K G1 G2) r

43 Symmetry adapted basis Ψ k 2,m(x 1, x 2) = e ik 2x2 u(x 1, x 2) = e ik 2x2 e i2πmx 2 Φ(x 1). 2 i(k 2 + 2πm) D, D a, a ; Ψ I k2,m,n(x 1, x 2) = Ae ik 2x2 e i2πmx 2 " φ n(x 1 k 2 φ 0 2π φ n 1 (x 1 k 2 2π φ s m φ 0 φ s ) φ 0 φ s m φ 0 φ s ) # k 2 is a good (Bloch) quantum number in presence of perturbation. tαβ (0)e i K r + tαβ ( G 1)e i( K G1) r + tαβ ( G 1 G 2 )e i( K G1 G2) r r-dependence = exp (α 12πx 1 + α 22πx 2) x 1 terms diagonal in m x 2 terms m m α 2 α 1 = 2, 1; α 2 = 1, 2

44 Harper Problem cell i 1 cell i cell i+1 m Layer II Layer I φ /φ 0= p/q period p Each landau Level becomes a set of 2p (tight-binding) bands with ɛ n,r(k 1, k 2 ) for commensurate ux,φ/φ s = p/q.

45 Results n = 0; p/q = 1/10; L = 77 A; B 8 T ; ω c = 100 mev; o

46 Results n = 2; p/q = 1/10; L = 77 A; B 8 T ; ω c = 100 mev; o

47 Results n = 1; p/q = 1/4; L = 77 A; B 19 T ; ω c = 150 mev; o

48 Results n = 1; p/q = 2/9; L = 77 A; B 19 T ; ω c = 150 mev; o

49 Results n = 0; p/q = 3/11; L = 77 A; B 19 T ; ω c = 150 mev; o

50 Moiré in Graphite and FLG Continuum theory Results Experimental and theoretical conrmation Magnetic Field Acknowledgements Collaborators Nuno Peres, Antonio Castro Neto. Rutgers Group Eva Andrei, Guohong Li, A. Luican. Discussions Paco Guinea, Vitor Pereira, Eduardo Castro, Joanna Hass, Ed Conrad, Claire Berger. Sponsors Condensed Matter Theory visitor's program Lopes dos Santos, Peres and Castro Neto The twisted bilayer -Benasque 2009