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QUANTUM CHAOS IN GRAPHENESpiros Evangelou
is it the same as for any other 2D lattice?
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DISORDER: diffusive to localized
quantum interference of electron waves in a random medium
TOPOLOGY: integrable to chaotic
quantum interference of classically chaotic systems
|ψ|
|ψ|
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Anderson localization (averages over disorder W)
random matrix theory!
quantum chaos (averages over energy E)
energy level-statistics
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localized to diffusive
P(S) level-spacing distribution
at the transition?
integrable to chaotic
𝑃 (𝑆 )=exp (−𝑆)
𝑃 (𝑆 )=𝐴𝑺 exp (−𝐵𝑆2)
Poisson
Wignerto
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Dirac fermions with 2 valleys & 2 sublattices etc.
graphene a sheet of carbon atoms
on a hexagonal lattice
𝐸± (𝑘𝑥 ,𝑘𝑦 )=∓𝛾 √1+4 cos𝑎𝑘𝑥
2cos
√3𝑎𝑘𝑦
2+4 cos
𝑎𝑘𝑥
2
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linear small-k dispersion near Dirac point
two bands touch at the Dirac point E=0
electrons with large velocity and zero mass
Dirac cones near E=0
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fundamental physics & device applications
DOS
Exk
yk
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armchair and zigzag edges
…edge states in graphene
chirality
nanoribbons
flakes:
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destructive interferencefor zigzag edges
𝑚=0
𝑚=1
𝑚=2
𝜓 1 𝜓 2
𝜓 3
…=0
𝜓𝑚≈−2 γ co𝑠2𝑚 𝑘2
𝑁𝑎𝑘𝑎𝑑𝑎𝑒𝑡𝑎𝑙 𝑃𝑅𝐵54 , 17954 ,1996h𝑊𝑎𝑘𝑎𝑏𝑎𝑦𝑎𝑠 𝑖 𝑒𝑡 𝑎𝑙𝑃𝑅𝐵 54 , 8271 ,1999
A atoms
B atoms
𝑘=𝜋 :𝜓𝑚≠0𝑜𝑛𝑙𝑦 𝑓𝑜𝑟𝑚=0
ψ ψ
edge states
…
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edge states move from to higher energies
in the presence of disorder(ripples, rings, defects,…)
what is the level-statistics of the edge states close to DP?
diagonal disorder (breaks chiral symmetry)
off-diagonal disorder (preserves chiral symmetry)
3D localization
Poisson 𝑃 (𝑆 )=𝐴𝑆exp (−𝐵𝑆2)𝑃 (𝑆 )=exp (−𝑆)
Wigner
intermediate statistics?
disordered nanotubes
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energy level-statistics
participation ratios
)(14
EPRsitesall
i
Ei
)(1 SPEES iii
energyspacing
Amanatidis & Evangelou PRB 2009
L
W
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participation ratio:
distribution of PR
the E=0 state
sitesall
i
Ei)E(PR
1400
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From PR(E=0) vs L
fractal dimension
Kleftogiannis and Evangelou (to be published)
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level-statistics
from semi-Poisson to Poisson
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zero disorder: ballistic motion (Poisson stat)
is graphene the same as any 2D lattice?
graphene lies between a metal and an insulator!
weak disorder: fractal states & weak chaos(semi-Poisson statistics)
strong disorder: localization & integrability(Poisson statistics)
Amanatidis et al (to be published)
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