1. Graphene beyond the standard model: including trigonal warping, spin-orbit coupling and strain
Tobias Stauber
Nuno Peres (U. Minho), Paco Guinea (ICMM), Antonio Castro Neto (Boston U.), John Schliemann (U. Regensburg)
KITPC - PSGR – April 30th, 2010

2. Why is graphene interesting?

3. Truely two dimensional system
On top of non-crystaline substrate (e.g. SiO2)
K. S. Novoselov et al., PNAS 102, (2005).
Suspended graphene
R. R. Nair et al., Science 320, 1308 (2008).
Epitaxial graphene on SiC
C. Berger et al., J. Chem. 108, (2004).
K. I. Bolotin et al., Sol. Stat. Comm. 146, 351 (2008).
Xu Du et al., Nature Nanotechnol. 3, 491 (2008).

4. New physical phenomena
Chiral quantum Hall effect
K. S. Novoselov et al., Nature 438, 197 (2005).
Y. Zhang et al., Nature 438, 201 (2005).
Observable even at room temperature
K. S. Novoselov et al., Science 315, 1379 (2007).
Klein tunneling
M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Nature Physics 2, 620 (2006).
N. Stander, B. Huard, and D. Goldhaber-Gordon, Phys. Rev. Lett. 102, (2009).

5. Efficient nanoelectronics possible
Mobilities up to cm2/Vs
K. I. Bolotin et al., Sol. Stat. Comm. 146, 351 (2008).
Xu Du et al., Nature Nanotechnol. 3, 491 (2008).
High thermoconductivity
A. A. Balandin et al., Nano Letters 8, 902 (2008) .
Nanoribbons/dots show confinement gap
A. K. Geim and K. S. Novoselov, Nature Materials 6, 183 (2007).
X. Wang et. al, Phys. Rev. Lett. 100, (2008).

6. Optical transparency Gateable displays
P. Blake et al., Phys. Rev. Lett. 100, (2008).
Graphene films as electrodes for solar cells
Xuan Wang, Linjie Zhi, and Klaus Müllen, Nano Lett. 8, 323 (2008).
Junbo Wu et al., Appl. Phys. Lett. 92, (2008).

7. Easy to get started

8. High-school project
Three 16 year old students win special price in national competition

9. Tight-binding model Honeycomb-lattice: Brillouin-Zone: K K’
t=3 eV
Honeycomb-lattice:
P. C. Wallace, Phys. Rev. 71, 622 (1947).
a=1.42 A
Brillouin-Zone: K K’
Dispersion around K-point:

10. DC conductivity of graphene

11. Ambipolar field-effect
Novoselov et. al., Science 305, 666 (2004).
Novoselov et. al., Nature 438, 197 (2005).
Linear behavior of the conductivity:

12. Boltzmann equation Occupation number: Continuity equation:
Include collision term in relaxation-time approximation:

13. Relaxation time and conductivity
Short-range potential:
Long-range potential:
K. Nomura and A. H. MacDonald, Phys. Rev. Lett. 96, (2006).

14. Midgap states Midgap (zero-energy) states arise due to:
Vacancies
Zig-zag edges, boundaries, cracks, etc.
Impurities with a large potential difference with respect to the graphene layer
Corrugated graphene, wiggles
Covalently-bond hydrogen
Density of states
N. M. R. Peres, F. Guinea, and A. H. Castro Neto, Phys. Rev. B 73, (2006).

15. Phase shift Relaxation time: Short-range potential:
Vacancies (unitary scatterers): v r v r

16. Comparison Screened Coulomb scattering:
Scattering from impurities (midgap states):
T. S., N.M.R. Peres, and F. Guinea, Phys. Rev. B 76, (2007).

17. Conductivity Minimal conductivity:
T. S., N.M.R. Peres, and F. Guinea, Phys. Rev. B 76, (2007).

18. Current through nanoribbons

19. Model for strained graphene
V. M. Pereira et al., Phys. Rev. B 80, (2009).

20. Transmission for strained nanoribbons

21. Strain induced gap
Behavior independent of specific boundary condition. A certain amount of stress should make a ribbon conducting/non-conducting.

22. Optical conductivity

23. Experimental results
Photograph as directly seen in transmitted white light from a xenon lamp (λ= nm) using an optical microscope (Nikon Eclipse LV100). No contrast enhancement or any other image manipulation has been used.

24. Scattering problem Boundary conditions for p polarized light:
Continuity equation:
Ohm´s law:

25. Transmission amplitude:
Transmissivity
Transmission amplitude:
with
Transmissivity:

26. Universal optical conductivity
Within the Dirac-cone approximation:
V. P. Gusynin et al, Phys. Rev. Lett. 96, (2006).
A. B. Kuzmenko et al., Phys. Rev. Lett. 100, (2008).

27. Open Question The problem of the Manchester group:
The experimental measurements were being done in the visible region of the spectrum. Does the result for the Dirac cone hold?
There were two questions to be answered:
What is the effect of trigonal warping?
What is the effect of the second nearest neighbour hopping, since t´~0.13t with t=3eV ?

28. Beyond the Dirac cone approximation (1)
Energy dispersion around the K-point in two directions: Γ K M

29. Beyond the Dirac cone approximation (2)
Energy dispersion including second nearest neighbour hopping Γ K M

30. Photoemission experiments show trigonal warping:
ARPES
Photoemission experiments show trigonal warping:
M. Mucha-Kruczynski et al., Phys. Rev. B 77, (2008).

31. Kubo formula The Kubo formula for the conductivity is given by
with the current-current correlation function defined as:

32. Current operator The paramagnetic contribution reads:
t is the first nearest neighbour hopping.
t´ is the second nearest neighbour hopping.

33. Next-nearest neighbour hopping
The general structure of Λ is given by:
1) B1=-B2 due to gauge invariance.
2) C=0 due to triangular lattice.
t´ only enters through the band energy.

34. Effect of trigonal warping
There are two opposing effects:
1) (18-(ħω/t)2) descreases with frequency.
2) ρ(ħω/2) increases with frequency.
The asymptotic form is given by:
T. S., N. M. R. Peres, and A. K. Geim, Phys. Rev. B 78, (2008).

35. Experimental results

36. Infrared conductivity

37. Infrared absorption of gated graphene
σ(ω)
No absorption in the regime 0<ω<2μ due to Pauli-blocking.
Universal conductivity σ0=(π/2)e2/h for ω>2μ.
Temperature broading of the step function is negligible. σ0 2μ ω
Elastic and inelastic scattering broadens energy dispersion:
Scattering from short-range and long-range (Coulomb) impurities.
Scattering from optical and acoustic (in-plane) phonons.

38. Experimental results Z. Q. Li et al., Nat. Phys. 4,532 (2008).
T. S., N. M. R. Peres, and A. H. Castro Neto, Phys. Rev. B 78, (2008).
N. M. R. Peres, T. S., and A. H. Castro Neto, Europhys. Lett. 84, (2008).

39. Conductivity of strained graphene

40. Polarizability of graphene

41. Polarizibility of Dirac Fermions
The closed form can be expressed by the two analytic functions:
B. Wunsch, T. S., F. Sols, and F. Guinea, New J. Phys. 8, 318 (2006).

42. Dynamical polarizibility42

43. Prediction of novel plasmon mode

44. Experimental observation

45. Polarizability of honeycomb lattice
The band overlap reads with the dispersion

46. Interband contributions

47. Intraband contributions

48. Intraband contributionsΓ K M

49. Real part of the polarizability

50. Real part of the polarizability

51. Effective interaction

52. “Pseudo-Rashba” spin-orbit coupling

53. Graphene/Au/Ni(111)

54. Adatoms on graphene

55. The Hamiltonian is given by

56. Solution
Eigenenergies:
Eigenvectors:

57. Bulk Properties Bulk expectation values:
Sublattice and electron spin of freedom are entangled:

58. Spin dephasing at the boundary
Infinite mass boundaries:
Zigzag boundaries:
T. S. and J. Schliemann, New J. Phys. 11, (2009).

59. Energy dispersion

60. Refection on a hard wall
A general plane wave with fixed momentum kx and energy E>2λ reflected by a hard wall can be written as:
with

61. Refection on a hard wall
Consider a incident plane wave of type II (A1=0):
Appearance of evanescent modes for:

62. Spin dephasing at the boundary

63. Spin polarization at the boundary

64. Spin polarization at the boundary

65. Spin density of a nanoribbon

66. To be done Spin rotation without a magnetic field:
Transport in a ribbon for injected electrons.