Graphene bilayer with a twist and a Magnetic Field. Workshop on Quantum Correlations and Coherence in Quantum Matter

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1 Ultra-thin graphite: methods Graphene signature: Dirac Fermions Moiré patterns in the bilayer (H=0) Hexagonal Superlattice and Magnetic Field Results Conclusion Graphene bilayer with a twist and a Magnetic Field J.M.B. Lopes dos Santos, N.M.R. Peres and A.H. Castro Neto CFP e Departamento de Física, Faculdade de Ciências, Universidade do Porto Workshop on Quantum Correlations and Coherence in Quantum Matter Lopes dos Santos, Peres and Castro Neto Graphene bilayer with a twist - Évora 2008

2 Outline 1 Ultra-thin graphite: methods 2 Graphene signature: Dirac Fermions Single layer Bilayer 3 Moiré patterns in the bilayer (H=0) 4 Hexagonal Superlattice and Magnetic Field 5 Results Weak Potential Twisted bilayer 6 Conclusion

3 Outline Ultra-thin graphite: methods Graphene signature: Dirac Fermions Moiré patterns in the bilayer (H=0) Hexagonal Superlattice and Magnetic Field Results Conclusion 1 Ultra-thin graphite: methods 2 Graphene signature: Dirac Fermions Single layer Bilayer 3 Moiré patterns in the bilayer (H=0) 4 Hexagonal Superlattice and Magnetic Field 5 Results Weak Potential Twisted bilayer 6 Conclusion Lopes dos Santos, Peres and Castro Neto Graphene bilayer with a twist - Évora 2008

4 Graphene: mother of all Carbons. 2 Graphene: mother of them all Graphene (top left) consists of a 2D hexagonal lattice of carbon atoms. Each atom is covalently bonded to three others; but since carbon has four valence electrons, one is left free allowing graphene to conduct electricity. Other well-known forms of carbon all derive from graphene: graphite is a stack of graphene layers (top right); carbon nanotubes are rolled-up cylinders of graphene (bottom left); and a buckminsterfullerene (C 60) molecule consists of graphene balled into a sphere by introducing some pentagons as well as hexagons into the lattice (bottom right). Castro Neto, Guinea and Peres, Physics World, 2006

5 Scotch tape method (A K Geim's Manchester Group) ρ (kω) n 0 (T )/ n 0 (4K) 6 D T (K) σ (mω -1 ) 3 B V g (V) A Fig. 2. Field effect in FLG. (A) Typica dependences of FLG s resistivity D on gate voltage for different temperatures (T 0 5, 70, and 300 K for top to bottom curves, respectively). (B) Example of changes in the film s conductivity G 0 1/D(V g ) obtained by inverting the 70 K curve (dots). (C) Hall coefficient R H versus V g for the same film; T 0 5 K. (D) Temperature dependence of carrier concentration n 0 in the mixed state for the film in (A) (open circles), a thicker FLG film (squares), and multilayer graphene (d, 5 nm; solid circles) Red curves in (B) to (D) are the dependences calculated from our model of a 2D semimetal illustrated by insets in (C). ε F δε 0.5 Fig. 1. Graphene films. (A) Photograph (in normal white light) of a relatively large multilayer ε F graphene flake with thickness È3 nm on top of an oxidized Si wafer. (B) Atomic force microscope (AFM) image of 2 6m by 2 6m area of this flake near its edge. Colors: dark brown, SiO 2 surface 0 orange, 3 nm height above the SiO 2 surface. (C) AFM image of single-layer graphene. Colors: dark brown, SiO 2 surface; brown-red (central area), 0.8 nm height; yellow-brown (bottom left), 1.2 nm orange (top left), 2.5 nm. Notice the folded part of the film near the bottom, which exhibits a differential height of È0.4 nm. For details of AFM imaging of single-layer graphene, see (15). (D) C ε F Scanning electron microscope image of one of our experimental devices prepared from FLG. (E) V g (V) Schematic view of the device in (D). Novoselov, Geim et. al. Science vol. 306, 666 (2004) Novoselov, Geim et. al. Science vol. 306, 666 (2004) R H (kω / T )

6 Epitaxial Graphene SiC with scratches Flatten SiC by H 2 etching Graphitization Deposit contacts Resist spincoating B C Wire bonding Lift off HSQ mask O2 plasma etch Develop E-beam lithography IFM Materials Science Division Linkopings U (SE) Bare SiC HSQ Exposed HSQ Graphene Fig. 5. Patterning epitaxial graphene de Heer et. al. Sol. St. Comm., vol , (2007) Metal contact

7 Outline Ultra-thin graphite: methods Graphene signature: Dirac Fermions Moiré patterns in the bilayer (H=0) Hexagonal Superlattice and Magnetic Field Results Conclusion Single layer Bilayer 1 Ultra-thin graphite: methods 2 Graphene signature: Dirac Fermions Single layer Bilayer 3 Moiré patterns in the bilayer (H=0) 4 Hexagonal Superlattice and Magnetic Field 5 Results Weak Potential Twisted bilayer 6 Conclusion Lopes dos Santos, Peres and Castro Neto Graphene bilayer with a twist - Évora 2008

8 Dirac Cones a 2 a 1 2 E k k x k 2 y 4 sp 2 bonding: electrons hop between p z orbitals with amplitude t.

9 Cones de Dirac... a / 3 a i α e δ 2 δ 1 1 A B δ3 K e +i α ) εa i = t (b i+δ1 + b i+δ2 + b i+δ3 ) (a i δ1 + a i δ2 + a i δ3 εb i = t (( ) ( ) ( )) εa i = t bi+δ1 + bi+δ2 + bi+δ3 ) ε ( b i ) = t (a i δ1 + a i δ2 + a i δ3 K = ± 4π 3a (1,0) α = ± 2π 3 Wallace, PRB 71, 622, (1947) H ( K + q) = hv F σ q Two sublattices, particle-hole symmetry (E E ) and lack of inversion symmetry about C-atom.

10 Cones de Dirac... a / 3 a i α e δ 2 δ 1 1 A B δ3 K e +i α ) εa i = t (b i+δ1 + b i+δ2 + b i+δ3 ) (a i δ1 + a i δ2 + a i δ3 εb i = t (( ) ( ) ( )) εa i = t bi+δ1 + bi+δ2 + bi+δ3 ) ε ( b i ) = t (a i δ1 + a i δ2 + a i δ3 K = ± 4π 3a (1,0) α = ± 2π 3 Wallace, PRB 71, 622, (1947) H ( K + q) = hv F σ q Two sublattices, particle-hole symmetry (E E ) and lack of inversion symmetry about C-atom.

11 Cones de Dirac... a / 3 a i α e δ 2 δ 1 1 A B δ3 K e +i α ) εa i = t (b i+δ1 + b i+δ2 + b i+δ3 ) (a i δ1 + a i δ2 + a i δ3 εb i = t (( ) ( ) ( )) εa i = t bi+δ1 + bi+δ2 + bi+δ3 ) ε ( b i ) = t (a i δ1 + a i δ2 + a i δ3 K = ± 4π 3a (1,0) α = ± 2π 3 Wallace, PRB 71, 622, (1947) H ( K + q) = hv F σ q Two sublattices, particle-hole symmetry (E E ) and lack of inversion symmetry about C-atom.

12 Cones de Dirac... a / 3 a i α e δ 2 δ 1 1 A B δ3 K e +i α ) εa i = t (b i+δ1 + b i+δ2 + b i+δ3 ) (a i δ1 + a i δ2 + a i δ3 εb i = t (( ) ( ) ( )) εa i = t bi+δ1 + bi+δ2 + bi+δ3 ) ε ( b i ) = t (a i δ1 + a i δ2 + a i δ3 K = ± 4π 3a (1,0) α = ± 2π 3 Wallace, PRB 71, 622, (1947) H ( K + q) = hv F σ q Two sublattices, particle-hole symmetry (E E ) and lack of inversion symmetry about C-atom.

13 Landau levels q k k = eb h v k ω c = k k = eb h v k k q y q x B ( v k = hk m ω c = eb m ε n = hω c n + 1 ) 2 v k = v F ω c = ebv 2 F ε ε n = hv F ebn h

14 Landau levels q k k = eb h v k ω c = k k = eb h v k k q y q x B ( v k = hk m ω c = eb m ε n = hω c n + 1 ) 2 v k = v F ω c = ebv 2 F ε ε n = hv F ebn h

15 Landau levels q k k = eb h v k ω c = k k = eb h v k k q y q x B ( v k = hk m ω c = eb m ε n = hω c n + 1 ) 2 v k = v F ω c = ebv 2 F ε ε n = hv F ebn h

16 Landau levels Normal: Dirac: Bν max g s g v φ 0 Bν max g s g v ν = n φ 0 ( ν + 1 ) = n 2 Geim & Novoselov Nature Materials 2007

17 Shubnikov de Haas Novoselov et. al, Nature, vol 438, 197 (2005) mechanical exfoliation de Heer et. al. Sol. St. Comm. vol 143, 92, (2007) epitaxial Peres, Castro Neto & Guinea PRB (2006) Castro Neto, Guinea,Peres, Novoselov, and Geim (RMP to appear: arxiv: )

18 Bilayer A 1 t B 1 A 2 B2 + t = AB stacking: electrons hop with amplitude t entre A 1 e B 2.

19 Results (a) 400 (c) V 0.1 A2 B2 a A1 B1 t t (b) ev (mev) n (10 12 cm -2 ) E/t Γ ev g K M g (mev) V g (V) (d) n (10 12 cm -2 ) Otha et al. Science vol 313, 951 (2006) E.V. Castro et al. PRL (2007) E. McCann, PRB (2006)

20 Opening of gap 0.09 m c * /me screened unscreened n (10 12 cm -2 ) E.V. Castro et al. PRL (2007) Bilayer Quadratic dispersion; A gap opens in the presence of a perpendicular electric eld;

21 Outline Ultra-thin graphite: methods Graphene signature: Dirac Fermions Moiré patterns in the bilayer (H=0) Hexagonal Superlattice and Magnetic Field Results Conclusion 1 Ultra-thin graphite: methods 2 Graphene signature: Dirac Fermions Single layer Bilayer 3 Moiré patterns in the bilayer (H=0) 4 Hexagonal Superlattice and Magnetic Field 5 Results Weak Potential Twisted bilayer 6 Conclusion Lopes dos Santos, Peres and Castro Neto Graphene bilayer with a twist - Évora 2008

22 Moiré in graphite and FLG Rong and Kuiper, PRB 1993

23 Moiré in graphite and FLG Rong and Kuiper, PRB 1993 Varchon et. al, PRB (2008)

24 Moiré in graphite and FLG Rong and Kuiper, PRB 1993

25 Commensurability A B A B 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 = 15 θ = 2.13 o, L = 66 Å

26 Continuum limit (k p approximation) Layer 1 H 1 = t 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) +... [ hv F ψ (r) k 0 i x y i x + y 0 ] ψ (r).

27 Continuum limit (k p approximation) layer 2 ( rotated) H 2 = t j b 2 (r j ) [ a 2 (r j + s 0 ) + a 2(r j + s 0 a 1 ) + a 2(r j + s 0 a 2 )] + hc 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) +... [ hv F ψ (r) k 0 e iθ ( i x y ) e iθ ( i x + y ) 0 ] ψ (r).

28 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)

29 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, B(A, B ) H = α,β d 2 r t βα δ (r)eikθ αβ (r) i K r e ψ α(r)ψ β (r) + h.c. t βα (r)eikθ δ αβ (r) : period of Moiré pattern. ( K = K θ K); k k + K/2 layer 1 ; k k K/2 layer 2;

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

31 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 t BA (G) t t t t AB (G) t e i2π/3 t e i2π/3 t t AA (G) t e i2π/3 t e i2π/3 t t BB (G) t e i2π/3 t e i2π/3 t

32 Electronic structure ky kx New energy scale: hv F K 760 mev (θ = 3.9 0,L = 36Å) E(eV) k/ K E E(eV) A electric eld does not open gap. Renormalization of Fermi velocity. v ( ) 2 F t = 1 9 v F v F K

33 Conrmation θ = 32.2 ο L s = 13 J. Hass, et. al PRL (2008) Latil et al. PRB (2007)

34 Fermi velocity renormalization Zenhua Ni et al, PRB (2008) Zenhua Ni et al, PRB (2008)

35 Fermi velocity renormalization V F V F h ω h ω E L k v F = 0.95v F Zenhua Ni et al, PRB (2008)

36 Fermi velocity renormalization de Heer et. al. Sol. St. Comm. vol 143, 92, (2007) v F = ( )v F

37 Outline Ultra-thin graphite: methods Graphene signature: Dirac Fermions Moiré patterns in the bilayer (H=0) Hexagonal Superlattice and Magnetic Field Results Conclusion 1 Ultra-thin graphite: methods 2 Graphene signature: Dirac Fermions Single layer Bilayer 3 Moiré patterns in the bilayer (H=0) 4 Hexagonal Superlattice and Magnetic Field 5 Results Weak Potential Twisted bilayer 6 Conclusion Lopes dos Santos, Peres and Castro Neto Graphene bilayer with a twist - Évora 2008

38 Dirac Equation in funny coordinates t Low energy Hamiltonian, H (K) = ( i a 1 e i2π/3 + a i2π/3 2 ( 0 i ) ( i e hc A ) a 1 e i2π/3 + a +i2π/3 2 0 ) ( i e hc A ). a 2 a 1

39 funny coordinates=superlattice coordinates Superlattice of size L s a with hexagonal symmetry; t t 1 V (r + mt 1 + nt 2 ) = V (r) a 1 a 2 θ [ a 1 a 2 ] = 1Ls [ cosθ sinθ 2 3 sinθ sinθ cosθ 1 3 sinθ ] [ t 1 t 2 ] r = x 1 t 1 + x 2 t 2 H (K) = tl 1 s ( ie iθ t 1 e i 2π 2π i 3 + t 2 ( 0 ie iθ ) ( ) 3 e i hc A 2π 2π i +i t 1 e t 2 0 ) ( i e hc A ).

40 funny coordinates=superlattice coordinates Superlattice of size L s a with hexagonal symmetry; t t 1 V (r + mt 1 + nt 2 ) = V (r) a 1 a 2 θ [ a 1 a 2 ] = 1Ls [ cosθ sinθ 2 3 sinθ sinθ cosθ 1 3 sinθ ] [ t 1 t 2 ] r = x 1 t 1 + x 2 t 2 H (K) = tl 1 s ( ie iθ t 1 e i 2π 2π i 3 + t 2 ( 0 ie iθ ) ( ) 3 e i hc A 2π 2π i +i t 1 e t 2 0 ) ( i e hc A ).

41 Suitable gauge A = t 1 t 2 Bx 1 g 2 2π [ H (K) = tl 1 0 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 φ s φ 0 x 1 )] r = x 1 t 1 + x 2 t 2 Perturbing potential invariant under (x 1, x 2 ) (x 1 + m, x 2 + n) = Bloch waves on x 2.

42 Weak Potential Ψ(x 1, x 2 ) = e ik 2x 2 u(x 1, x 2 ) e ik 2x 2 e i2πmx2 Φ(x 1 ). 2 i(k 2 + 2πm) D, D a, a ; Ψ k2,m,n(x 1, x 2 ) = Ae ik 2x 2 e i2πmx2 [ 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. V (x 1, x 2 ) = V 0 [cos(2πx 1 ) + cos(2πx 2 ) + cos(2π(x 1 + x 2 ))] x 1 terms diagonal in m x 2 terms m m ± 1

43 Weak Potential Ψ(x 1, x 2 ) = e ik 2x 2 u(x 1, x 2 ) e ik 2x 2 e i2πmx2 Φ(x 1 ). 2 i(k 2 + 2πm) D, D a, a ; Ψ k2,m,n(x 1, x 2 ) = Ae ik 2x 2 e i2πmx2 [ 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. V (x 1, x 2 ) = V 0 [cos(2πx 1 ) + cos(2πx 2 ) + cos(2π(x 1 + x 2 ))] x 1 terms diagonal in m x 2 terms m m ± 1

44 Weak Potential Ψ(x 1, x 2 ) = e ik 2x 2 u(x 1, x 2 ) e ik 2x 2 e i2πmx2 Φ(x 1 ). 2 i(k 2 + 2πm) D, D a, a ; Ψ k2,m,n(x 1, x 2 ) = Ae ik 2x 2 e i2πmx2 [ 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. V (x 1, x 2 ) = V 0 [cos(2πx 1 ) + cos(2πx 2 ) + cos(2π(x 1 + x 2 ))] x 1 terms diagonal in m x 2 terms m m ± 1

45 Weak Potential Ψ(x 1, x 2 ) = e ik 2x 2 u(x 1, x 2 ) e ik 2x 2 e i2πmx2 Φ(x 1 ). 2 i(k 2 + 2πm) D, D a, a ; Ψ k2,m,n(x 1, x 2 ) = Ae ik 2x 2 e i2πmx2 [ 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. V (x 1, x 2 ) = V 0 [cos(2πx 1 ) + cos(2πx 2 ) + cos(2π(x 1 + x 2 ))] x 1 terms diagonal in m x 2 terms m m ± 1

46 Outline Ultra-thin graphite: methods Graphene signature: Dirac Fermions Moiré patterns in the bilayer (H=0) Hexagonal Superlattice and Magnetic Field Results Conclusion Weak Potential Twisted bilayer 1 Ultra-thin graphite: methods 2 Graphene signature: Dirac Fermions Single layer Bilayer 3 Moiré patterns in the bilayer (H=0) 4 Hexagonal Superlattice and Magnetic Field 5 Results Weak Potential Twisted bilayer 6 Conclusion Lopes dos Santos, Peres and Castro Neto Graphene bilayer with a twist - Évora 2008

47 Results In subspace of Landau Level (n) and k 2, V tight binding hamiltonian (TBH) in m. For commensurate ux, φ s /φ 0 = p/q, TBH has period p. Harper, Proc. Phys. Soc. A68, 874 (1955) Zak, Phys Rev. 134, A1602, A1607 (1964) Rauh et. al. phys. stat. sol.(b), 63, 215 (1974) Wannier, phys. stat. sol.(b), 88, 757 Thouless et. al Phys. Rev. Lett., 49, 405 (1982) cell i 1 cell i cell i+1 t *( 1) t *(m 1) t(m) t(p 1) m p Φ Φ Φ Φ

48 Results TBH single energy scale depends on φ s /φ 0. ( V 1 = V 0 exp π φ 0 ) 3φs ( 1F 1 n,1, 2πφ ) 0 3φs n=0 V 1 / V 0 n=1 0.4 n= φ s /φ 0

49 Results TBH single energy scale depends on φ s /φ 0. ( V 1 = V 0 exp π φ 0 ) 3φs ( 1F 1 n,1, 2πφ ) 0 3φs n=0 V 1 / V 0 n=1 0.4 n= φ s /φ 0

50 Results TBH single energy scale depends on φ s /φ 0. ( V 1 = V 0 exp π φ 0 ) 3φs ( 1F 1 n,1, 2πφ ) 0 3φs n=0 V 1 / V 0 n=1 0.4 n= v F = 10 6 m s 1 l B 250/ B(T)(Ȧ) φ s /φ 0 φ s φ 0 = BL 2 s (T Ȧ 2 ) Eects are strongly suppressed except for large elds and large superlattices.

51 The twisted bilayer Interlayer coupling has 3 dominant terms of identical amplitude: t αβ (r)ei K r = g tαβ (g)ei( K g) r e i(2πα 1x 1 +2πα 2 x 2 )/3 G 2 G 1 + G G 2 1 K K G 1 K G 1 G 2 Interlayer coupling has period 3t 1,3t 2! [ ] HI V H = (r) V (r) H II

52 The twisted bilayer cell i cell i t 1(m) t 2(m) m 2 m 1 m m+1 m+2 Layer Layer 1 p 1 cell i+1 TBH single energy scale depends on φ s /φ 0. ( ) ( ) φ 0 t 1 = t exp π 3 2πφ 0 1F 1 n,1, 3φ s 3 3φ s The unit lattice for the magnetic problem has 9 the area of the unit lattice of Moiré Pattern. High φ s /φ 0 achievable for moderate elds and small angles.

53 The twisted bilayer t 1 / t ~ L s =66 A, B=30 T n=0 n=1 n= φ s /φ 0 The unit lattice for the magnetic problem has 9 the area of the unit lattice of Moiré Pattern. High φ s /φ 0 achievable for moderate elds and small angles.

54 Outline Ultra-thin graphite: methods Graphene signature: Dirac Fermions Moiré patterns in the bilayer (H=0) Hexagonal Superlattice and Magnetic Field Results Conclusion 1 Ultra-thin graphite: methods 2 Graphene signature: Dirac Fermions Single layer Bilayer 3 Moiré patterns in the bilayer (H=0) 4 Hexagonal Superlattice and Magnetic Field 5 Results Weak Potential Twisted bilayer 6 Conclusion Lopes dos Santos, Peres and Castro Neto Graphene bilayer with a twist - Évora 2008

55 Summary A small twist dramatically changes electronic structure of bilayer. Formalism for Dirac equation in hexagonal symmetry superlattice in magnetic eld. For small elds, twisted layers are decoupled; for small angle twists interesting eects (soon to be calculated) will appear in moderate magnetic elds.

56 Summary A small twist dramatically changes electronic structure of bilayer. Formalism for Dirac equation in hexagonal symmetry superlattice in magnetic eld. For small elds, twisted layers are decoupled; for small angle twists interesting eects (soon to be calculated) will appear in moderate magnetic elds.

57 Summary A small twist dramatically changes electronic structure of bilayer. Formalism for Dirac equation in hexagonal symmetry superlattice in magnetic eld. For small elds, twisted layers are decoupled; for small angle twists interesting eects (soon to be calculated) will appear in moderate magnetic elds.

58 Summary A small twist dramatically changes electronic structure of bilayer. Formalism for Dirac equation in hexagonal symmetry superlattice in magnetic eld. For small elds, twisted layers are decoupled; for small angle twists interesting eects (soon to be calculated) will appear in moderate magnetic elds.

59 Acknowledgements Project PTDC/FIS/64404/2006 Condensed Matter Theory visitor's program Paco Guinea, Eduardo Castro, Andre Geim, Joanna Hass, Walt de Heer and Vitor Pereira, for discussions.