Vito Scarola wwwphysvtedu scarola Novoselov KS et al PNAS 05 Castro Neto et al RMP 09 Graphene Low energy electronic band structure of infinite graphene has Dirac cones ID: 526337
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Slide1
Flat Band Nanostructures
Vito Scarola
www.phys.vt.edu
/
scarolaSlide2
Novoselov, K.S. et al. PNAS ’05
Castro
Neto et al., RMP ‘09
Graphene Low energy electronic band structure of infinite graphene has Dirac cones Graphene:
honeycomb sheets of carbon atoms that allow electrons to hop from atom to atom
V. Scarola
Virginia Tech
P.R. Wallace, Phys. Rev. ‘47Slide3
Real graphene
nanostructure samples have disorder from substrates, defects, edge roughness, etc. Processing goal: reduce imperfections
J. H. Chen et al., Nature Nanotechnology ‘08
Graphene Nano-Flakes
V. Scarola
Virginia TechSlide4
Rough
Graphene
Nanoribbons
C. Stampfer et al. PRL ‘09Han et al. PRL ‘09 Transport in rough nanoribbons dominated by substrate disorder and edge roughness
V. Scarola
Virginia Tech
500 nmSlide5
C. Tao et al., Nature ‘09
Less rough graphene nanoribbons
from unzipped carbon nanotubes
L. Jiao et al., Nano Res ‘10 STM reveals magnetic propertiesGraphene Nanoribbons from Nanotubes
V. Scarola
Virginia TechSlide6
Graphene Nanostructures: Interesting Topics of Study
Transport properties
Device Applications
Disorder EffectsEdge RoughnessSurface and Edge ChemistryBand EffectsStrong InteractionMagnetism
V. Scarola
Virginia Tech
This TalkSlide7
Outline
Energy/t
Toy model of Coulomb interaction in armchair flat band
Propose and
t
est
Jastrow
-correlated
wavefunctions
: crystals and liquids
Energy/t
Model Coulomb interaction in
zig-zag
flat band
Prediction:
Ferromagnetic quantum crystals
Wang and Scarola, arxiv:1108.0088
Wang and Scarola, PRB ‘11
Overview of flat bands
V. Scarola
Virginia TechSlide8
Example Zero-Field Flat-Band Nanostructures
Pressed carbon nanotubes
Graphene
edges
Graphene quantum dots/antidots
Graphene
nanoribbons
Interactions are the de facto dominant energy scale
Absence of conventional screening
V. Scarola
Virginia TechSlide9
Classical vs. Quantum Flat Bands
Particles in well separated sites have small kinetic energy (flat bands) because they are classically localized
Interaction effects will then be classical because charge is localized
(commuting density operators)
Particles in
“
flat-band lattices” interfere quantum mechanically to give no
net
kinetic energy (
flat bands
)
Interactions can lead to quantum effects because single-particle charge is
delocalized
(non-commuting density operators)
site
(no hopping)
V. Scarola
Virginia Tech
Classical
QuantumSlide10
Fractional Quantum Hall Models: Quantum Flat Bands
Pan et
al.
PRL
‘06
Coulomb interaction leads to a superstructure in flat kinetic energy bands:
Landau
levels
Low energy physics captured by
composite fermion
wavefunctions
Jain PRL
‘
89
V. Scarola
Virginia TechSlide11
Armchair ribbons
K.
Nakada
et
al. PRB ‘96
Zig-Zag
ribbons
Nearest neighbor hopping on honeycomb lattices:
Energy/t
Energy/t
Zero-Field Flat Bands in Ribbons
V. Scarola
Virginia TechSlide12
Outline
Energy/t
Toy model of Coulomb interaction in armchair flat band
Propose and
t
est
Jastrow
-correlated
wavefunctions
: crystals and liquids
Energy/t
Model Coulomb interaction in
zig-zag
flat band
Prediction:
Ferromagnetic quantum crystals
Wang and Scarola, arxiv:1108.0088
Wang and Scarola, PRB ‘11
Overview of flat bands
V. Scarola
Virginia TechSlide13
Second quantized Hamiltonian
of polarized electrons in an
interacting
flat band:
Exact
Diagonalization
A Flat Band Model of
Armchair
Honeycomb Ribbons
Wang and Scarola, PRB ‘11
V. Scarola
Virginia Tech
(
eigenstate
)
Toy Gaussian model of
single-particle basis states:Slide14
N=12
particles
N
x=34 unit cells
Smooth transition from
crystal
to
a
uniform
liquid
as basis states delocalize.
Ground State in a Flat-Band of
an
Armchair Ribbon
Density
b=2
b=3.5
b=5
n=1/3
V. Scarola
Virginia Tech
Wang and Scarola, PRB ‘11Slide15
Jastrow
Correlated
Ansatz
States
A general scheme for attaching correlation holes with first quantized wavefunctions
m
c
correlation holes attached to particles in state
y
n
*
Wannier
Function
Variational
Parameter
V. Scarola
Virginia Tech
Lattice filling
Wang and Scarola, PRB ‘11Slide16
Δρ
c
Ansatz
wave
function accurately captures
exact ground state in all regimes
Verification of
Ansatz
Wavefunctions
(basis state width)
V. Scarola
Virginia Tech
Wang and Scarola, PRB ‘11Slide17
Outline
Energy/t
Toy model of Coulomb interaction in armchair flat band
Propose and
t
est
Jastrow
-correlated
wavefunctions
: crystals and liquids
Energy/t
Model Coulomb interaction in
zig-zag
flat band
Prediction:
Ferromagnetic quantum crystals
Wang and Scarola, arxiv:1108.0088
Wang and Scarola, PRB ‘11
Overview of flat bands
V. Scarola
Virginia TechSlide18
C. Tao et al. Nature Phys.
’
11
O.
Yazyev
et al
., PRB ‘11
Mean field theory with the Hubbard model: ferromagnetic coupling along edges but antiferromagnetic coupling perpendicular to edges.
Recent Work on
Graphene
Ribbons
V. Scarola
Virginia TechSlide19
Energy/t
Interactions Dominate at Low Filling
H=Kinetic + Corrections
H=Coulomb + Corrections
Focus on flat bands at low filling: Interactions dominate
u
d
u
d
V. Scarola
Virginia TechSlide20
(Coulomb units e
2
/
ea~ 27 eV)
Matrix Elements from Single-Particle
Wannier
Functions
V. Scarola
Virginia Tech
Wang and Scarola, arxiv:1108.0088Slide21
Flat Band Projection
Flat band Coulomb model of
zig-zag
ribbons
New projected operators are delocalized
V. Scarola
Virginia Tech
Wang and Scarola, arxiv:1108.0088
FBRSlide22
J~400K
Predictions from Model at 1/3 Filling of Upper Band
Ferromagnetic
Quantum
Crystals
Spin Waves
Ground State
Spin Excitations
V. Scarola
Virginia TechSlide23
A Broad Connection
Experiments?
V. Scarola
Virginia Tech
Quantum Hall
Zero-Field Flat-Band LatticesSlide24
Summary and Outlook
Jastrow
-correlated
wavefunctions
capture physics of quantum liquids and crystals in flat band latticesModels of
zig-zag ribbons at low fillings
Future Work: Explore new models and
wavefunctions for novel ground states
and excitations
V. Scarola
Virginia Tech
H. Wang
(Virginia Tech-> U. Hong Kong)
Hao
Wang and V. W. Scarola
Phys. Rev. B 83, 245109 (2011)
Hao
Wang and V. W. Scarola
arxiv:1108.0088