Electronic properties of graphene stacks. A. Castro-Neto (Boston U.) - - PowerPoint PPT Presentation

Electronic properties of graphene stacks.

• A. Castro-Neto (Boston U.)
• N. M. R. Peres (U. Minho, Portugal)
• E. V. Castro, J. dos Santos (Porto), J. Nilsson (BU), A. Morpurgo (Delft), D.

Huertas-Hernando (Trondheim), J. González, F. G., M. P. López-Sancho, T. Stauber, M. A. H. Vozmediano, B Wunsch (CSIC, Madrid)

Outline Electronic structure of graphite.

• Electron-electron interaction in graphene.
• Graphene stacks. Interlayer coupling.

Electronic structure. Interaction effects.

• Disorder. Out of plane conductivity.
• Screening and surfaces.
• Transport in curved graphene sheets.

Weak antilocalization effects.

Some interesting references

Single layer graphene. Electrically doped.

And more interesting references

Integer Quantum Hall effect in graphene.

• The conduction band is built up

from the unpaired π orbitals at the C atoms.

• The crystal structure is stabilized

by the σ bonds within the plane.

• The hybridization between π
• rbitals in neighbouring planes

cannot be neglected.

Electronic band structure

• J. W. McClure, Phys. Rev. 108, 612 (1957)

Hybridization between in plane nearest neighbours: Hybridization between out of plane nearest neighbours: 0.3eV γ 2.4eV γ

1

≈ ≈

Electronic band structure

The Fermi surface has electron and hole like pockets at the edges of the Brillouin zone. The effective masses are small, The structure is consistent with Shubnikov-de Haas and photoemission experiments.

eff

0.06m m ≈

• D. E. Soule, J. W. McClure, and L. B. Smith, Phys. Rev. 134,

A454 (1964).

• D. Marchand, C. Frétigny, M. Lagües, F. Batallan, Ch. Simon,
• I. Roseman, and R. Pinchaux, Phys. Rev. B 30, 4788 (1984).

UPS SdH

Electronic band structure

• R. C. Tatar, and S. Rabii, Phys. Rev. B 25, 4126 (1982).

J.-C. Charlier, X. Gonze, and J.-P. Michenaud, Phys. Rev. B 43, 4579 (1991).

Graphite is a semimetal.

50nm λ cm 10 2.4 n atom) C states/(eV 10 1.2 ) N(ε

FT 3 18 4 F

≈ × ≈ × ≈

− −

A graphene plane. The Dirac equation.

a r b r

t

. c . h c c t H

s , j . n . n s , i

+ = ∑

+

( ) ( )

( ) ( ) ( ) [ ]

b a k cos b k cos a k cos t e e t e e t H

k b k i a k i b k i a k i k

r r r r r r r

r r r r r r r r r r

− + + + ± =         + + + + ≡

− −

2 2 2 3 1 1 ε

x

k

y

k

k r

( )

2 3 2 1 2 3 2 1 2 3 2 1 3 1 3 2 3 2 1 3 3 4 i e i e , a b , a a , a k k k k

b k i a k i

− − = + − =         = =         = + =

r r r r

r r r r r r π δ

        − + ≅ 2 3

y x y x

ik k ik k ta H Dirac equation:

Related systems. C

60 Threefold coordination The curvature is induced by five-fold rings

• There is a family of quasispherical

compounds

• The valence orbitals are derived from π

atomic orbitals.

The Dirac equation on a sphere?

Lattice frustration as a gauge potential.

A fivefold ring defines a disclination. The sublattices are interchanged.

• The Fermi points are also interchanged.
• These transformations can be achieved by

means of a gauge potential.

• J. González, F. G. and M. A. H. Vozmediano, Phys. Rev. Lett. 69, 172 (1992)

∫

= Φ         − ∇ → ∇ l d A A i i r r r r r 1 1

The flux Φ is determined by the total rotation induced by the defect.

Continuum model of the fullerenes.

Dirac equation on a spherical surface. Constant magnetic field (Dirac monopole).

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

a b F b a F

sin cos l i sin i R v sin cos l i sin i R v Ψ = Ψ       − + ∂ + ∂ Ψ = Ψ       + + ∂ − ∂ ε θ θ θ ε θ θ θ

φ θ φ θ

2 1 1 2 1 1 h h

( ) [ ] ( )

l J l l J J R vF

J

≥ + − + = 1 1 h ε

Coulomb interactions

Non Fermi liquid behavior of quasiparticle lifetimes.

Expts: S. Yu, J. Cao, C. C. Miller, D. A. Mantell, R. J. D. Miller, and Y. Gao, Phys. Rev. Lett. 76, 483 (1996). Theory: J. González, F. G., and M. A. H. Vozmediano Phys. Rev. Lett. 77, 3589 (1996)

N(E) E E F Absence of screening. Perturbation theory leads to logarithmic divergences. The expansion has similar properties to that for 1D metallic systems (Luttinger liquids). Large coupling constant: e /v =2-5 Deviations from Fermi liquid behavior. v v screened interaction

Single graphene planes: Limits of validity:

High energies > 0.3eV. Neglects electron-phonon interaction.

2 F

Renormalization of the Coulomb interaction.

( ) ( ) ( ) ( ) ( ) ( )

∫∫ ∫

Ψ Ψ − Ψ Ψ + + Ψ ∇ Ψ =

+ + + 2 2 2 1 1 1 2 2 1 2 2 2

1 8 r r r r r r r d r d e r r r d iv H

F

r r r r r r r r r r r r r π σ

Dimensional analysis:

[ ] [ ] [ ] [ ] [ ]

[ ]

2 2

1 1 l e l l H H t l

D int K

≡ ≡ Ψ ≡ ≡ ≡

The interaction is marginal in any dimension (as in QED). The interaction is mediated by photons in three dimensional space. The interaction breaks the Lorentz invariance of the Dirac equation.

Renormalization of the Coulomb interaction.

k r k′ r

Hartree Fock selfenergy:

( )

        Λ ≈ Σ k k e k r r r r log 8

2

σ π

The vortex corrections are finite (to all orders).

( )

2 2 2 2 2

8 ω ω − = Π k v k e i , k

F

r r r

Bare polarizability:

• J. González, F. G. and M. A. H. Vozmediano., Nucl. Phys. B 424, 595 (1994)

Renormalization of the Coulomb interaction.

( )

ω ω ω ∝ Σ ∂ Σ ∂ = Ψ = Ψ

− −

Im Z Z

R 1 2 1

One loop calculation: Renormalization of the particle residue.

F F

v e v e

2 2 2

8 π = Λ ∂ ∂ Λ

Lowest order RG flow:

The coupling constant goes to zero at low energies.

• C. L. Kane and E. J. Mele, Phys. Rev. Lett. 93,

197402 (2004) The electronic self energy due to the long range Coulomb interaction modifies the dependence

• f the gap on the radius in semiconducting

nanotubes.

Experimental consequences?

• A. Lanzara et al, unpublished.

The combined effects of disorder and electron- electron interactions lead to a non monotonous dependence of the quasiparticle lifetime on disorder.

Renormalization of the Coulomb interaction.

= + + + RPA summation:

• J. González, F. G. and M. A. H. Vozmediano, Phys. Rev. B 59, R2474 (1999)

π π 4 1 arccos 8

2 2

−         − + = Λ ∂ ∂ Λ g g g g

RG flow equation: ( which can be analytically extended to g > 1)

The coupling constant always flows to zero at low energies.

D.V. Khveshchenko, Phys. Rev. Lett. 87, 246802 (2001).

Compensation between low density of states and unscreened interaction

Stoner criterium: 1 v q q e N 1 ) N(E U

F 2 f F c

= ↔ = r r

For sufficiently large couplings, a charge density wave phase is induced

Non perturbative phase transitions.

In plane interactions reduce the interlayer coupling. Similar effect as in the cuprate superconductors.

t

Interchain hopping in Luttinger liquids

See also:

C.L. Kane and M.P.A. Fisher, Phys. Rev. Lett. 68, 1220 (1992) X.G. Wen, Phys. Rev. B 42, 6623 (1990).

• F. G. and G. Zimanyi, Phys. Rev. B 47, 501 (1993).
• S. Chakravarty and P.W. Anderson, Phys. Rev. Lett. 72, 3859 (1994).

J.M.P. Carmelo, P.D. Sacramento, and F. G., Phys. Rev. B 55, 7565 (1987) A.H. Castro-Neto and F. G., Phys. Rev. Lett. 80, 4040 (1998).

Extended hopping Local hopping

Interlayer hopping

• M. A. H. Vozmediano, M. P. López-Sancho, and F. G., Phys. Rev. Lett. 89, 166401 (2002); ibid,
• Phys. Rev. B 68, 195122 (2003).

Exchange instability in graphene.

• N. M. R. Peres, F. G. and A. H. Castro Neto, Phys. Rev. B 72, 174406 (2005)

E k E k

Kinetic energy Exchange energy

The exchange energy favors a ferromagnetic ground state. This instability is expected in a low density 2DEG.

Exchange instability in graphene.

The instability requires too high coupling values. The instability is enhanced in the presence of disorder (neglecting localized states). The interband exchange energy increases. The intraband exchange energy decreases. There is a competition between the two effects.

The graphene bilayer.

( ) ( ) ( ) ( )

              + − − + ≡

⊥ ⊥

ik k v ik k v t t ik k v ik k v H

y x F y x F y x F y x F

• Bilayer. Electronic structure.
• E. McCann and V. Fal’ko, Phys. Rev. Lett. 96, 086805 (2006)

⊥

≈ t k vF

k 2 2 r r

ε

The graphene bilayer.

• J. Nilsson, A. H. Castro Neto, N. M. R. Peres, and F. G. Phys. Rev. B 73, 214418 (2006)

3 3 4 ex kin

27π 2gQ 8ππ Q S E E S E − ≈ + =

⊥

Long range exchange interaction: Ferromagnetism ( )

        − ≈ + + − ≈

⊥ ⊥ ⊥ ⊥

∫

4t q ω t log t k t q k ω k d ω , q χ

2 2 2 2 AFM

r r r r r

Short range Hubbard repulsion: Antiferromagnetism

Similar effects expected in bulk graphite

The intraband exchange energy increases. The interband energy (proportional to the overlap between spinors) decreases. The problem resembles closely a dilute 2DEG.

Graphene multilayers.

Electronic structure and stacking order.

The bands of a trilayer with Bernal stacking are equivalent to those of a bilayer and a single layer superimposed. The bands of a bilayer with rhombohedral stacking show an incipient surface state.

Graphene multilayers.

The hamiltonian for each value of the parallel momentum defines a one dimensional tight binding model. A B A B

⊥

t

( )

y x F

ik k v +

( )

y x F

ik k v −

( )

F k

v t H H ,

||

⊥

∑

≡

1 2 1 1 1 2 2 2 2 1 1 3

Bernal stacking Rhombohedral stacking

Effective hamiltonian.

Graphene multilayers.

Bulk and surface density of states.

The low energy electronic states have vanishing amplitude on the atoms connected to the neighboring layers.

Graphene multilayers.

Surface states.

1 2 1 2 3

Surface states can be induced near stacking defects. Projected density of states: a) Bernal stacking (121212…) b) Rhombohedral stacking (123123…)

Graphene multilayers.

Some experimental results.

Conductance channels. Graphene multilayers.

• N. M. R. Peres, A. H. Castro Neto and F. G. ,Phys. Rev. B 73, 195411 (2006)
• N. M. R. Peres, A. H. Castro Neto, and F. G., Phys. Rev. B 73, 241403 (2006).
• F. G., A. H. Castro Neto and N. M. R. Peres, Phys. Rev. B 73, 245426 (2006), and to be published.

Landau levels associated to the Dirac equation can be observed in multilayers, or for special stackings, 123123123…

Graphene multilayers.

Fermi surface. Extremal orbits.

The Fermi surface, to lowest approximation, contains a regular orbit, and a Dirac orbit.

• I. A. Luk´vanchuk and Y. Kopelevich, cond-mat/0609037, and
• Phys. Rev. Lett. 93, 166402 (2004).

( ) ( ) ( )

        + − + + ≡

− ⊥ ⊥ ⊥

,

|| d ik y x F d ik y x F

z z

e t ik k v e t ik k v k k H

x

k

y

k

z

k

Graphene multilayers.

Rhombohedral graphite.

The hamiltonian of rhombohedral graphite is made up of a set of Dirac equations, one for each value of kz. There are surface states at the top and bottom layers of rhombohedral graphite.

Graphene multilayers.

Landau levels.

K 1 2 1 2 1 K’ There is a n=0 Landau level derived from valley K at one surface, while the corresponding n=0 Landau level from valley K’ is at the opposite surface (valley filtering).

2 1 3

B F

l n v 1 −

B F

l n v

B F

l n v 1 +

⊥

t

⊥

t

⊥

t

⊥

t

Rhombohedral stacking

.... 1

1 1 2 2 1 1 1 − − ⊥ − − ⊥ − + − ⊥

− + = + = + =

n B F n n m n B F n n m n B F n n m

a l n v a t a a l n v a t a a l n v a t a ε ε ε

B F m

l m v = ε

The Landau levels in a rhombohedral stack are quasi two dimensional.

Graphene multilayers.

Rhomobohedral stacking. Landau levels.

The Landau levels of rhombohedral graphite are quasi two dimensional.

Graphene multilayers.

• Disorder. Conductivity.
• J. Nilsson, A. H. Castro Neto, F. G., and N. M. R. Peres, cond-mat/0604106

The out of plane current requires the passage of the electrons through atoms with a semimetallic density of states. The in plane conductivity saturates at a value independent of the number of carriers. The out of plane conductivity is insulating.

( )(

)

∑

+ + ⊥ ⊥

+ − =

|| || 1 || 2 || 2 || 1

sin 2

k k A k A k A k A

c c c c k ed J

Graphene multilayers.

Induced charge at surfaces.

Gate

1 2 3 4 5 6

1

ε

2

ε

3

ε

4

ε

5

ε

6

ε

Electrostatic potential See also E. McCann, cond-mat/0608221

∑ ∑

= = + − + =

− + − = − j j ij i i i i i j j j i i

n dn e n d e ε χ ε ε ε ε ε

2 1 1 1 1 2 1

2

The potential has to be calculated self consistently. The induced charge is calculated within linear response theory (RPA). m i i j j n

χij

Graphene multilayers.

Induced charge at surfaces.

( ) ( ) ( ) ( ) ( ) ( ) ( )

' cos cos ' cos cos ' cos cosh sinh ' log 4 2

2 3 2 2 2 2 3 2 2

φ φ φ φ φ φ κ κ φ φ ε π χ ε ε

π π π π κ κ κ κ κ

− − −         = = = + − = =

∫ ∫ ∑

− ⊥ ⊥ ∞ = −∞ = − − − −

d d t v dt e e d e e e e n n e

F n n bulk n n n i n i

The decay of the charge into the bulk can be calculated analytically: Interlayer transitions

• nly lead to

anomalous, quasi insulating screening.

4 3 5 1 1

1 2

− ≈ ≈ ≈ ≈

− ⊥

N v d t v e

F F

κ

Intraband contribution at finite doping: ( )

09 . 4

2 2 2 2 2

≈ = ≈

⊥ F F

v dt e dD e π ε κ

The charge polarizability has inter- and intraband contributions.

        =

⊥ ⊥ 2

log ~ 4 ε χ π χ t v t

ij F ij

Graphene multilayers.

Induced charge in multilayers.

The charge distribution near a neutral surface shows a slow decay, and oscillations with period equal ro the interlayer distance. The charge oscillations persist in doped multilayers. Metallic screening localizes most of the charge within 3-4 layers.

Curvature and weak (anti)localization.

• A. Morpurgo and F. G., cond-mat/0603789.

See also: S. V. Mozorov, K. S. Novoselov, M. I. Katsnelson, F. Schedin, D. Jiang, and A. K. Geim, cond-mat/0603826

• E. McCann, .K. Kechedzhi, V. I. Fal’ko, H. Suzuura, T. Ando, and B. L. Altshuler, cond-mat/0604015

Smooth disorder should lead to antilocalization effects in graphene, H. Suzuura and T. Ando, Phys. Rev. Lett. 89, 266603 (2002). Neither localization nor antilocalization effects have been observed.

Antilocalization due to negative interference (Berry’s phase). On a curved surface, the accumulated rotation along a closed path is not π.

3 2 2 2 2

l Lh l h N δφ l h R 1 R l c π φ ≈       ≈ ≈       − ≈

l

nm 10 L 1nm h 10nm l

3

≈ ≈ ≈

Antilocalization effects disappear for:

Curvature and weak (anti)localization.

Effective gauge field:

Local rotations of the lattice axes. Topological defects: disclinations (pentagons, heptagons), dislocations.

Scattering at boundaries

• M. V. Berry, and R. J. Mondragon, Proc. R. Soc. Lond. A

412, 53 (1987).

Inequivalence between sublattices (mass term).

Other effects.

d k v τ

1 F F 1 gauge − −

≈

d: distance between dislocations.

• F. G., J. González, and M. A. H. Vozmediano, Phys. Rev. B 59, 134421 (2001)

Interaction effects can be important in multilayered systems. The electronic structure depends on the stacking order. Valley filtering occurs in a magnetic field.

• Quasi two dimensional behavior can be found when

stacking defects are present. Undoped graphite surfaces can show insulating behavior. Induced charge has oscillations from layer to layer. Screening in doped multilayers leads to a charge distribution localized in 3-4 layers.

• Diffusion in curved surfaces suppress weak

antilocalization effects.

Conclusions

• J. Nilsson, A. H. Castro Neto, N. M. R. Peres, and F. G. Phys. Rev. B 73, 214418 (2006)
• F. G., A. H. Castro Neto and N. M. R. Peres, Phys. Rev. B 73, 245426 (2006)
• J. Nilsson, A. H. Castro Neto, F. G., and N. M. R. Peres, cond-mat/0604106
• A. Morpurgo and F. G., cond-mat/0603789.