Transcript QED in a Pencil Trace

QED
in a Pencil Trace
K.S. Novoselov
A.K. Geim
S.V. Morozov, D. Jiang, F. Schedin, T. Booth, L. Ponomarenko,
P. Blake, J. Shi, T. Ghulam, J. Meyer, M. I. Katsnelson
Two-Dimensional Form of Carbon
0d
“Buckyball”
Robert F. Curl
Harold W. Kroto
Richard E Smalley
1985
Nobel prize 1996
1d
Carbon
Nanotube
Multi-wall 1991
Single-wall 1993
2d
Graphene
3d
Graphite
1564
Two-Dimensional Form of Carbon
2d
Graphene
0d
“Buckyball”
1d
Carbon Nanotube
3d
Graphite
Outline
• 2-Dimensional Atomic Crystals: Do They Exist?
• Stability issues in 2D
• Massless Dirac Fermions in Graphene
• Relativistic Quantum Hall Effect
• Graphene Quantum Dots
• Applications
No Free Standing Monolayers Known
EPITAXIAL GROWTH
CHEMICAL DECOMPOSITION
up to Ø 2m; > 80 layers
Krishnan 1998
Nagashima 1992 (on TiC) ; Land 1992 (on Pt)
McConville 1986 (on Ni); Affoune 2001 (on HOPG)
Bommel 1975 (on SiC); Forbeaux 1998 (on SiC)
Charrier 2002 (on SiC); Berger 2004-2006 (on SiC)
Peierls-Landau-Mermin arguments not applicable
CHEMICAL EXFOLIATION
restacked and scrolled graphitic SOOT
individual folds: Horiuchi 2004
2-Dimensional Metallic Films
Why one shouldn’t try to obtain 2D Metallic Films
Clustering of Thin Metallic Films
Mermin-Wagner Theorem:
No Long-Range Order in 2D
Mermin N.D. & Wagner H.
Absence of Ferromagnetism or Antiferromagnetism
in One- or 2-Dimensional Isotropic Heisenberg Models
Phys Rev. Lett. 17 (22), 1133 (1966)
Mermin N. D.
Crystalline Order in 2 Dimensions
Phys. Rev. 176 (1), 250 (1968)
Buckling of disordered membranes
Radzihovsky L. & Nelson D.R.
Statistical Mechanics of Randomly Polymerised Membranes
Phys. Rev. A 44 (6), 3525 (1991)
Peierls instabilities
Layered Material
5 m
peeling off
to nm thickness?
how thin can you go
before they segregate,
decompose or scroll?
Extracting a Single Plane
Strongly layered material
Pull out one atomic plane
SEGREGATE?
DECOMPOSE?
SCROLL?
Cleavage to a Single Layer
Mechanical Cleavage by Drawing
PNAS 102, 10451 2005
Science 306, 666 (2004)
Graphene
1 μm
1 μm
1 μm
Optical
AFM
SEM
A monolayer of graphite
is stable
under ambient conditions
Mechanical Cleavage in Retrospect
Ohashi (Tanso 1997, 2000)
from 1000 down to 50 layers
Our work (Science 2004) down to 1 layer
Philip Kim’s & Paul McEuen’s groups
(PRL 2005 & Nanoletters 2005) down to 35 layers
for >10 layers,
electronic structure of bulk
graphite
no stability problems
Two Dimensional Crystallites
not just flakes, but
crystallites
crystal faces
armchair
zigzag
10 m
How can graphene exist?
 Is it stabilised by substrate?
 Is free-standing graphene stable?
Graphene: Membrane
1 m
Free-hanging graphene membrane.
Meyer et al, Nature 2007
Monolayer
Bilayer
Microscopic Crumpling of Graphene
local electron diffraction (beam Ø 250 nm)
normal incidence
26º tilt
tilt
axis
High crystallographic quality
diffraction peaks
away from tilt axis
become blurred
Microscopic Crumpling of Graphene
real space
reciprocal space
Intrinsic Microscopic Crumpling
atomic resolution TEM
ripple contrast appears for >1 layer
sample edge
height 5Å
size <5nm
strain 1%
Anharmonic approximation by Nelson (1987, 2004):
2D membranes can be stabilized by intrinsic crumpling in 3D
 buckling (dislocations would destroy mobility)
 bending (elastic strain)
Other 2D crystals
Other 2D Atomic Crystals
2D boron nitride in AFM
0Å
9Å
16Å 23Å
2D NbSe2 in AFM
0Å
8Å
23Å
1m
0.5m
1 m
2D Bi2Sr2CaCu2Ox in SEM
1m
1 m
2D MoS2 in optics
also,
can do
2,3,4
…
layers
Local Crystal Quality
STM image of 2D NbSe2
HRTEM image of 2D Bi2Sr2CaCu2Ox
NO
RECONSTRUCTION
(except for BISCCO)
electron diffraction Novoselov et al, PNAS 2005
from single-layer graphene
electron diffraction
from 2D Bi2Sr2CaCu2Ox
Macroscopic Quality & Homogeneity
Au contacts
Electric Field Effect
3
3
SiO2
Si
2
2
2D NbSe2
1
1
graphene
0
-80
-40
2D MoS2
0
Vgate (V)
40
80
0
2D crystal
 =neμ =0 μVg/d
 (1/M)
 (1/k)
1m
2D NbSe2 & MoS2:
0.5 to 3 cm2/Vs
as in bulk at 300K
graphene:
20,000 cm2/V·s thin films
50,000 cm2/V·s thick films
2D ATOMIC CRYSTALS
new class of crystalline materials
wide choice of materials properties
(electronic, mechanical, chemical, etc.)
www.GrapheneIndustries.com
Manchester production: <0.1 cm2/year
£0.50/m2
1 P£/g (1015 £/g)
GDP of US: $1013
About enough for a mg of graphene
2D ATOMIC CRYSTALS
new class of crystalline materials
wide choice of materials properties
(electronic, mechanical, chemical, etc.)
www.GrapheneIndustries.com
OR
book a free half-day session in Manchester
2D ATOMIC CRYSTALS
new class of crystalline materials
wide choice of materials properties
(electronic, mechanical, chemical, etc.)
www.GrapheneIndustries.com
Manchester production: <0.1 cm2/year
they exist, therefore
they can be made en masse
sublimation of Si from surface of SiC
C Berger et al, J. Phys. Chem. B 108 (2005)
Graphene FET
50 m
5-layers graphite
Contacts
formed by
silver epoxy
First graphitic FET
No clean room facilities required
Graphene Devices
Manchester Centre for Mesoscience and Nanotechnology
2 m
optical image
SEM image
design for contacts
and mesa
contacts and mesa formation
Au contacts
SiO2
Si
graphene
Graphene Field Effect Transistors
 (k)
10
6
electrons
T =10K
0
4
holes
1/xy (1/k)
B =2T
T =10K
-10
2
-100
-50
0
50
100
Vg (V)
0
-100
Au contacts
-50
0
Vg (V)
Novoselov et al, Science 2004
50
100
SiO2
Si
graphene
Graphene Field Effect Transistors
 (k)
First metallic field effect
transistor
6
T =10K
4
Mobility: 20.000 cm2/V·s
(order of magnitude better
than silicon)
2
0
-100
Au contacts
-50
0
Vg (V)
Novoselov et al, Science 2004
50
100
SiO2
Si
graphene
Cyclotron Mass
mc  ne
mc/m0
0.06
0.04
0.02
0
-6
-3
0
3
n (1012 cm-2)
6
Novoselov et al, Nature 2005
Zhang et al, Nature 2005
Cyclotron Mass
mc  ne  k
1 A kk
mc 

2 E E
E  k  mc
c*  10 m / s
6
EF  m c
2
c *
Half-Integer Quantum Hall Effect
quadruple degeneracy:
plateaus are expected at (h/4e2)N -1
2.0
1.5
Rxx, k
h/2e2
2
1.0
0.5
0.0
0
4
6
8
10
Magnetic Field, T
2
Rxy, h/4e
2
QHE occurs at
half-integer
filling factors
1
h/6e2
0
0
2
4
6
8
Magnetic Field, T
10
12
Related to the “odd”
Berry phase
Berry phase 
12
Half-Integer Quantum Hall Effect
Quantisation at  =N+1/2
xx (k)
xy (4e2/h)
3.5
12T
2.5
10
1.5
0.5
-0.5
5
-1.5
-2.5
-3.5
0
-4
-2
0
n (1012 cm-2)
2
4
Novoselov et al, Nature 2005
Zhang et al, Nature 2005
2-DEG: Parabolic and Linear Dispersions
Parabolic Dispersion
Linear Dispersion
E  c* p
2
p
E
*
2m
gv m
D( E ) 
2

c*  const
*
gv E
D( E )  2 2
 c*
2-DEG in Magnetic Field
Parabolic Dispersion
Linear Dispersion
ELL  c (n  12) ELL  c* 2eB n
Each level has a degeneracy
eB
eB
gv g s
4
h
h
Shared equally between
holes and electrons
Dirac Equation in Magnetic Field
eB( (NN 
) ) cNeB
 0,1,2
E E
 22cceB
2
E  cp
22
without
spin-splitting
11
22
1
2
2
with
spin-splitting
Chiral Dirac Fermions
E   2c eB( N  12  12 ) N  0,1,2
2 equivalent sublattices
pseudospin index
2


k
k

k

k
Chiral Dirac Fermions
Schrödinger
fermions
Spin
is a good
quantum number
Dirac fermions
Dirac fermions
in graphene
Helicity is a good quantum number
Chiral superposition
of spin projections
Chiral superposition
of sublattices
Quantum Hall Effect
von Klitzing constant:
RK  h / e 2  25812.807572 
Proposed to be used in metrology
However, requires
very low temperatures
Room Temperature Quantum Hall Effect
4
 xy  2e / h
2
xy , e2/h
2
Quantization at
room temperature
 Large inter-Landau level distance
E LL
0
-2
T=300K
-4
-40
-20
0
20
40
Vg, V
Rxx , k
30
ELL  c* 2eB
20
10
0
-60
-30
0
Vg, V
30
60
ELL  400[ K T ] B
Novoselov et al,
Science 2007
Room Temperature Quantum Hall Effect
4
 xy  2e / h
2
xy , e2/h
2
Quantization at
room temperature
 Large inter-Landau level distance
0
-2
T=300K
Requires only:
-4
-40
-20
0
20
40
Vg, V
Rxx , k
30
20
10
0
-60
-30
0
Vg, V
30
60
Novoselov et al,
Science 2007
Other
Relativistic-like Effects
Quantum-Limited Resistivity
12
max (k)
zero-gap
semiconductor
E =0
>50 devices
8
h/4e2
4
annealing
0
ρ (k)
6
ρmax
4,000
10K
no temperature
dependence
in the peak
(between
0.3 and 300K)
2
-40
0
Vg (V)
40
8,000
 (cm2/Vs)
4
0
-80
0
80
12,000
Quantum-Limited Resistivity
E =0
max (k)
zero-gap
semiconductor
12
>50 devices
8
h/4e2
4
annealing
0
0
4,000
8,000
 (cm2/Vs)
CONDUCTIVITY
WITHOUT
CHARGE CARRIERS
12,000
Quantum-Limited Resistivity
12
max (k)
quantized resistivity h/e2
(or conductivity)
NOT
the resistance
or conductance
>50 devices
8
h/4e2
4
2
e
  ne  k F l
h
Mott’s argument:
annealing
0
0
l  F
2
e

h
 Only in the absence of localization
4,000
8,000
 (cm2/Vs)
12,000
Absence of Localization (Klein paradox)
Massive particles in 2D:
Massless particles in 2D:
always localized
NEVER LOCALIZED
Absence of Localization (Klein paradox)
Consequence of pseudo-spin
conservation


k
k
Massless particles in 2D:
NEVER LOCALIZED
Klein paradox
(propagation of relativistic particles
through a barrier)
O. Klein, Z. Phys 53,157 (1929); 41, 407 (1927)
M.I.Katsnelson et al
Nature Physics 2006
Graphene
Quantum Dot
Graphene Quantum Dot
Graphene quantum dot
weakly coupled to the contacts
Conductance
Fermi
Energy
Gate Voltage
Graphene Quantum Dot
Single electron transistor
4
0.3 K
2
2
 (S)
Excitation Voltage, mV
3
0
1
-2
0
-4
16.88
16.90
16.92
Gate Voltage, V
16.94
0
0.05
Vg (V)
0.1
Graphene Quantum Dot
Room temperature
Single electron transistor
200nm
300 K
25.000 K
E 
L[nm]
 (nS)
60
40
20
0
-10
0
Vg (V)
10
CONCLUSIONS
 strictly-2D materials do exist
 NEW PARADIGM
“relativistic”
condensed matter
physics
realistic possibility of many applications
CONCLUSIONS
Yes Maybe No
Field Effect Transistor
√
Single Electron Transistor √
Spin-Valve
√
Gas Sensor
√
Metrology
√
TEM Membranes
√
Transparent Gate
√