graphene slides

Christian Mendl
February 6, 2008
MPQ Theory Group Seminar
Mother of all Graphitic Forms
0.3 nm
Graphene is a 2D building material for carbon materials of all other dimensionalities. It can
be wrapped up into 0D buckyballs, rolled into 1D nanotubes or stacked into 3D graphite.
Landau and Peierls: strictly two-dimensional crystals are
thermodynamically unstable and cannot exist…
Landau, L. D. Zur Theorie der Phasenumwandlungen II. Phys. Z. Sowjetunion, 11, 26-35 (1937)
a) Novoselov, K. S. et al. Twodimensional atomic crystals. Proc.
Natl Acad. Sci. USA 102, 1045110453 (2005)
b) Meyer, J.C. et al. Nature 446, 6063 (March 2007)
c) T.J. Booth, K.S.N, P.
Blake & A.K.G.
One-atom-thick single crystals: the thinnest material you will ever see. a) Graphene visualized by atomic-force
microscopy. The folded region exhibiting a relative height of ≈4Å clearly indicates that it is a single layer. b) A graphene
sheet freely suspended on a micron-size metallic scaffold. c) scanning-electron micrograph of a relatively large graphene
crystal, which shows that most of the crystal’s faces are zigzag and armchair edges as indicated by blue and red lines and
illustrated in the inset. 1D transport along zigzag edges and edge-related magnetism are expected to attract significant
From: A.K. Geim and K.S. Novoselov: The Rise of Graphene
Micromechanical Cleavage
Graphene becomes visible in an optical microscope if placed on top of a Si wafer with a carefully
chosen thickness of SiO2, owing to a feeble interference-like contrast with respect to an empty wafer.
Tight-Binding Model
Tight-binding description
for π-orbitals of carbon:
V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte. AC conductivity
of graphene: from tight-binding model to 2+1-dimensional
quantum electrodynamics. International Journal of Modern Physics
B, 21, No.27:4611-4658, 2007
Effective Hamiltonian:
Dirac Equation
„Relativistic“ condensed matter physics: condensed matter analogue of
(2+1)-dimensional quantum electrodynamics
Conduction and valence
band touch each other at six
discrete points: the corner
points of the 1.BZ (K points)
Effective speed of light:
vF≈106 m/s
Ballistic Electron Transport
a) The rapid decrease in resistivity ρ with
adding charge carriers indicates high
electron mobility (in this case,
μ≈5000cm2/Vs and does not noticeably
change with temperature up to 300K).
Andre Geim et al. (University of Manchester),
Graphene Speed Record, Physics News Update
Number 854 #2, January 23, 2008:
μ≈200 000 cm2/Vs
b) Room-temperature quantum Hall
effect: quasiparticles in graphene are
massless and exhibit little scattering
even under ambient conditions
Chiral Quantum Hall Effects
a) The hallmark of massless Dirac fermions is QHE plateaux in σxy at half integers of 4e2/h b) Anomalous QHE for
massive Dirac fermions in bilayer graphene is more subtle (red curve): σxy exhibits the standard QHE sequence with
plateaux at all integer N of 4e2/h except for N=0. The zero-N plateau can be recovered after chemical doping, which
shifts the neutrality point to high Vg so that an asymmetry gap (≈0.1eV in this case) is opened by the electric field
effect (green curve) c-e) Different types of Landau quantization in graphene. The sequence of Landau levels in the
density of states D is described by EN ∝ √N for massless Dirac fermions in single-layer graphene (c) and by EN ∝
√N(N −1) N for massive Dirac fermions in bilayer graphene (d). The standard LL sequence EN ∝ (N+½) N is expected
to recover if an electronic gap is opened in the bilayer (e).
Spin Qubits in
Graphene Quantum Dots
• Spin-orbit coupling is weak in carbon (low atomic weight) → spin decoherence due to spinorbit coupling should be weak
• Natural carbon consists predominantly of the zero-spin isotope 12C → spin decoherence due
to hyperfine interaction of electron spin with surrounding nuclear spins should be weak
Björn Trauzettel, Denis V. Bulaev, Daniel Loss, and Guido Burkard:
Spin qubits in graphene quantum dots. Nature Phys., 3:192, 2007
Graphene double quantum dot: ribbon of
graphene (grey) with semiconducting armchair
edges (white). Confinement is achieved by
tuning the voltages applied to the “barrier” gates
(blue) to appropriate values such that bound
states exist. Additional gates (red) allow to shift
the energy levels of the dots. Virtual hopping of
electrons through barrier 2 (thickness d) gives
rise to a tunable exchange coupling J between
two electron spins localized in the left and the
right dot.
Idea: create ribbon of graphene with
semiconducting armchair boundary
 Valley degeneracy is lifted for all
modes (necessary to do two-qubit
operations using Heisenberg exchange
Generates energy gap → solves the
quantum dot confinement problem (Klein
Energy bands for single and double dot case
Björn Trauzettel, Denis V. Bulaev, Daniel Loss, and Guido Burkard:
Spin qubits in graphene quantum dots. Nature Phys., 3:192, 2007
Exchange Coupling
Exchange coupling based on Pauli principle:
with singlet-triplet splitting
(t is the tunneling matrix element and U
the on-site Coulomb energy)
Can estimate t for the ground state:
Room for tuning!
Long Distance Coupling
Tunnel coupling via Klein tunneling through the valence band!
Triple quantum dot setup. Dot 1 and dot 3 are strongly coupled via cotunnelling processes
through the valence bands of barrier 2, barrier 3, and dot 2. The center dot 2 is decoupled by
detuning. The energy levels are chosen such that ∆ε2 ≪ ∆ε1. The triple dot example illustrates
that in a line of quantum dots, it is possible to strongly couple any two of them and decouple
the others by detuning. This is a unique feature of graphene and cannot be achieved in
semiconductors such as GaAs that have a much larger gap.
• 2D → conceptually new material
• Extraordinary crystal and electronic properties
• Opens a door for testing QED phenomena
• Promising candidate for classical and quantum
computing (high mobility at room-temperature,
long distance coupling)

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