1

QUANTUM CHAOS IN GRAPHENESpiros Evangelou

is it the same as for any other 2D lattice?

2

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|

3

Anderson localization (averages over disorder W)

random matrix theory!

quantum chaos (averages over energy E)

energy level-statistics

4

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

5

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

6

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

6

fundamental physics & device applications

DOS

Exk

yk

7

armchair and zigzag edges

…edge states in graphene

chirality

nanoribbons

flakes:

8

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

…

9

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

11

energy level-statistics

participation ratios

)(14

EPRsitesall

i

Ei

)(1 SPEES iii

energyspacing

Amanatidis & Evangelou PRB 2009

L

W

12

participation ratio:

distribution of PR

the E=0 state

sitesall

i

Ei)E(PR

1400

13

From PR(E=0) vs L

fractal dimension

Kleftogiannis and Evangelou (to be published)

14

level-statistics

from semi-Poisson to Poisson

15

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)