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E.Y. Andrei
Electronic properties of Graphene and 2-D materials
Eva Y. Andrei
Rutgers University
2D materials – background
Carbon allotropes
Graphene Structure and Band structure
Electronic properties
Electrons in a magnetic field
Onsager relation
Landau levels
Quantum Hall effect
Engineering electronic properties
Kondo effect
Atomic collapse and artificial atom
Twisted graphene
Aug 27-29, 2018
E.Y. Andrei
Graphene: a theorists invention
2016
E.Y. Andrei
No long range order in 2-Dimensions
…continuous symmetries cannot be spontaneously broken at finite temperature
in systems with sufficiently short-range interactions in dimensions D ≤ 2.
David Mermin Herbert Wagner
… No Magnets
….No superfluids
…No superconductors
... No 2D crystals
No long range order in 2D
.. long-range fluctuations
can be created with little energy cost and
since they increase the entropy they are favored.
E.Y. Andrei
First 2D materials
Andre Geim Konstantin Novoselov
Graphene, hBN, MoS2,
NbSe2, Bi2Sr2CaCu2Ox,
ANY LAYERED MATERIAL
SLICE DOWN TO
ONE ATOMIC PLANE?
Can We Cheat Nature?
A. Geim
E.Y. Andrei
300nm oxide
White light300nm oxide560nm green light
200nm oxide
White light
Making graphene
Micromechanical cleavage by “drawing”
Novoselev et al (2005)
SiO2300 nm
Si
P. Blake et.al, 2007
D. S. L. Abergel 2007
S. Roddaro et. al Nano Letters 2007
Andre GeimK. Novoselov
Properties: Mechanical
S. Bunch et al,Nano Letters‘08
• Young’s modulus ~ 2 TPa
0.5 µm
T. Booth et al,Nano Letters‘08
Young’s modulus E ~ 2TPa
Stiffest materal
Unsupported graphene
with Cu particles
~5m ~1nm
1011 nano-particles
graphene
Properties: Optical
Transmittance at Dirac point:
T=1-ap =97.2%
P. Blake et al, Nano Letters, ‘08
,
R.R. Nair et al, Science (2008).
Properties: Chemical
F Schedin et al,Nature Materials ‘07
Single molecule detection
NO2, NH3, CO
E.Y. Andrei
2D Building Blocks
10
E.g.: Graphene, hBN,
Silicene, Germanene,
Stanene.
Graphene family Transition Metal Dichalcogenides
(TMDCs)
E.g.: MoS2 ,WSe2 , NbS2, TaS2
Phosphorene family
E.g.: Phosphorene, SnS, GeS, SnSe, GeSe
Group 13 Monochalcogenides
E.g.: GaS, InSe
Group 2 Monochalcogenides
E.g.: BaS, CaS, MgS
• About 40 2D
materials are
currently known.
• About half of them
have been isolated.
• Others shown to be
stable using
simulations.
E.Y. Andrei
Van der Waals heterostructures
Nature 499, 419 (2013)
E.Y. Andrei
Stacking 2D Layers
E.Y. Andrei
The most important element in nature
E.Y. Andrei
Carbon chemical bonds
2S + 2px + 2py + 2pz hybridize
4 sp3 orbitals: tetrahedron
Diamond
sp3
Carbon: Z=6
4 valence electrons 2s22p2
E.Y. Andrei
Carbon chemical bondsCarbon: Z=6
4 valence electrons 2s22p2
2S + 2px + 2py hybridize
3 planar s orbitals: tetragon
1 Out of plane 2pz [“p” orbital]
sp2
p electrons allow
conduction
s bonds: 2D and
exceptional rigidity
E.Y. Andrei
Carbon chemical bondsCarbon: Z=6
4 valence electrons 2s22p2
2S + 2px + 2py hybridize
3 planar s orbitals: tetragon
1 Out of plane 2pz [“p” orbital]
sp2
Graphite
E.Y. Andrei
Diamond
Carbon allotropessp2sp3
Graphite
p
s
Diamond is Metastable in ambient conditions!! break
E.Y. Andrei
3D
Graphite ~16 century
Carbon Allotropes sp2
2D
GrapheneSingle -layer 2005
“1D”
Carbon nanotube
Multi-wall1991
Single wall 1993
“0D”
Buckyball
1985
2010
1996
E.Y. Andrei
Graphene
2D materials – background
Carbon allotropes
Graphene Structure and Band structure Electronic properties
Electrons in a magnetic field
Onsager relation
Landau levels
Quantum Hall effect
Engineering electronic properties
Kondo effect
Atomic collapse and artificial atom
Twisted graphene
E.Y. Andrei
Tight binding model
Eigenstates of the isolated atom:
aaa EHat =
• Overlaps between orbitals a corrections to system Hamiltonian.
• Tight binding a small overlaps a small corrections DU
rUR-rHrHR
at ()()( D=
kkk EH =
Periodic potential
Atomic orbitals
E.Y. Andrei
Lattice vector
1. Normalization
2. Eigenstate of the Hamiltonian
3. Algebra aFind Solutions in terms of:
Tight binding model
Build Bloch waves out of atomic orbitals (Bravais lattice!)
kkk EH =
Solutions have to satisfy:
)()( j
R
Rrrj
=
aa jRik
k e
1=kk
• Nearest neighbor transfer (hopping) integral
• Nearest neighbour overlap integral
ijji tH =
ijji s=
E.Y. Andrei
Wallace, 1947
Band Structure
Bravais lattice
Parabolic dispersion
How is graphene different?
Simple metal
honeycomb lattice Not Bravais
strong diffraction by lattice
z unconventional dispersion
*m
pE
2
2
=
E
px
E.Y. Andrei
Graphene honeycomb lattice1. 2D2. Honeycomb structure (non-Bravais)3. 2 identical atoms/cell
A
B
2 interpenetrating Bravais (triangular) lattices
E.Y. Andrei
BB
N
R
Rki
B RreN
rkB
B
= .1,
Honeycomb lattice – two sets of Bloch functions1. 2D2. Honeycomb structure (non-Bravais)3. 2 identical atoms/cell
A
B
AA
N
R
Rki
A RreN
rkA
A
= .1,
sum over all type
B atomic sites
in N unit cells
atomic
wavefunction
= 0,
3
4
aK
p
E.Y. Andrei
1. 2D2. Honeycomb structure (non-Bravais)3. 2 identical atoms/cell
A
B
==
BBAAk
ccr )(
2 interpenetrating Bravais (triangular) lattices
New degree of freedom –like spin up and spin down -
but instead sublattice A or B a pseudo-spin
)0,1(3
4
aK
p=
Honeycomb lattice – two sets of Bloch functions
Tight binding model of monolayer graphene : Energy solutions
=
=
1
1;
** ksf
ksfS
ktf
ktfH
B
B
0det = ESH
Secular equation gives the eigenvalues:
0
0det
222
0
0
*
0
=
=
kfEstE
EkfEst
kfEstE
kfs
kftE
1
0 =
2
32/3/3
1
.cos2
akaikaikkixyy
j
j eeekf
=
==
BA
BA
BBBAAA
atomsidentical
Ht
HH
==
=
0:
;
Note:
If atoms on the two sublattices are different:
=
B
A
ktf
ktfH
*
Solutions more complicated
Edward McCann arXiv:1205.4849
E.Y. Andrei
Graphene tight binding band structure
129.0,033.3,00 === seVt
kfs
kftE
1
0 =
2
32/3/cos2
akaikaikxyy eekf
=
Typical parameter values :
Bonding
Anti-
bonding
E.Y. Andrei
==
BBAAk
ccr )(
New degree of Freedom
Pseudospin
Band Structure
KK’
Dirac cones
G
Dirac point
E
kx
ky
6 Dirac points, but only 2 independent
points K and K’, others can be derived with
translation by reciprocal lattice vectors.
Low energy excitations
The Dirac points are protected by 3
discrete symmetries: T, I, C3
E.Y. Andrei
Low energy band structure
KK’
Dirac cones
Fermi energy EF
Fermi “surface” =2 points
Conduction band a electrons
Valence band a holes
E.Y. Andrei
1;0,3
4', =
=
p
aKK
2
32/3/cos2
akaikaikxyy eekf
=
2/
2
3
paOipp
akf yx =
p
ak
= 0,
3
4p
Consider two non-equivalent K points:
and small momentum near them:
Linear expansion in small momentum:
p
Low energy band structure
1 Linear expansion near the K point
=
0
0
0
0*
yx
yx
F ipp
ippv
ktf
ktfH
smat
vF /102
3 6=
yip
xip yx
;
E.Y. Andrei
2. Dirac-like equation with Pseudospin
=
B
A
=
=
=
10
01;
0
0;
01
10; zyx
i
isss
pvppvipp
ippvH FyyxxF
yx
yx
F
.
0
0sss ==
=
For one K point (e.g. =+1) we have a 2 component wave function,
with the following effective Hamiltonian (Dirac Weyl Hamiltonian):
Bloch function amplitudes on
the AB sites (‘pseudospin’)
mimic spin components of
a relativistic Dirac fermion.
Low energy band structure
Pauli Matrices
Operate on sublattice degree of freedom
E.Y. Andrei
=
'
'
BK
AK
BK
AK
=
000
000
000
000
yx
yx
yx
yx
F
ipp
ipp
ipp
ipp
vH
To take into account both K points (=+1 and =-1) we can use a 4
component wave function,
with the following effective Hamiltonian:
Note: real spin not included.
Including spin gives 8x8 Hamiltonian
H(K)
H(K’)
pvH FK
.s= pvH FK
.*
' s=
2. Dirac-like equation with Pseudospin
Low energy band structure
HK and HK’ are related by time reversal symmetry
E.Y. Andrei
Hamiltonian near K point:
3. Massless Dirac Fermions
Massless Dirac fermions
yyxxF
yx
yx
F iivi
ivH =
= ss
0
0
==
i
i
Fse
epsvE1
rk
2
1
Py
Px
EDirac Cone
indexbands
anglepolarpp xy
=
= )/arctan( Pseudospin vector• Is parallel to the wave vector k in upper band
• Antiparallel to k in lower band (s=-1)
But 300 times slower
Massless particle : m=0 cpE =..Photons, neutrinos..
smvF /106
E.Y. Andrei
under pseudospin conservation,
backscattering within one valley
is suppressed
angular scattering probability:
= 0
2/cos0 22
==
4. Absence of backscattering within a Dirac cone
No backscattering within Dirac cone
=
i
i
see
1rk
2
1
E.Y. Andrei
npvpvKH FF
== ss)(
Helical electronspseudospin direction
is linked to an axis
determined by
electronic momentum.
for conduction band
electrons,
valence band (‘holes’)
1=n
s
1=n
s
Hamiltonian at K point:
5. Helicity
Helicity (Chirality)
nh
= sHelicity operator:
hpvKH Fˆ)( =
Helicity is conserved
E.Y. Andrei
pvE F=
xpyp
Helical electronspseudospin direction
is linked to an axis
determined by
electronic momentum.
for conduction band
electrons,
valence band (‘holes’)
1=n
s
1=n
s
p
p
Note: Helicity is reversed in the K’ cone
5. Helicity
Helicity (Chirality)
Helicity conserved a No backscattering between cones
E.Y. Andrei
Kkp
=
K’K
conserved Helicity hp
.s
X
X
No Backscattering
No backscattering
High carrier mobility
A
B
No backscattering between cones
No backscattering within cones
Dirac Weyl Hamiltonian
=
0.
.0* p
pvH F
s
s
E.Y. Andrei
Pristine graphene
Katsnelson, Novoselov, Geim (2006)
Klein Tunneling
Pseudospin + chirality a Klein tunneling
a no backscattering
a large mobility
Katsnelson, Novoselov, Geim (2006)
No electrostatic confinement
• No quantum dots
• No switching
• No guiding
E.Y. Andrei
Klein tunneling
Klein “paradox”
Transmission of relativistic particles
unimpeded even by highest barriers
Physical picture: particle/hole conversion
Katsnelson et al Nature physics (2006)
Angular dependence
of transmission
Carrier density outside the barrier is 0.5 x 1012 cm-2 .
Inside the barrier, hole concentrationn is 1 x 1012 and 3 x
1012 cm-2 for the red and blue curves.
E.Y. Andrei
The Berry phase
E.Y. Andrei
Berry phase
Examples
– Curved space
– Mobius strip
– Twisted optical fiber
– Bohm Aharonov phase
– Dirac belt
Berry Phase Tutorial:
https://www.physics.rutgers.edu/grad/682/textbook/ch-3.pdf
E.Y. Andrei
Berry phase
=
i
i
ee
1)( rk
2
1
1,02,02 pp
=
p
p
2
)2cos(
2
11i
ikr
ee
p
=
= d
ddphaseBerry Im Homework:
Prove this relationship
E.Y. Andrei
Graphene Relatives Many natural materials have Dirac cones in their bandstructure (e.g.
highTc compnds). But the Fermi surface of these materials contains
many states that are not on a Dirac cone. As a result the electronic
properties are controlled by the normal electrons and the Dirac electrons
are invisible.
Graphene is the ONLY naturally occurring material where the Fermi
surface consists of Dirac points alone. As a result all its electronic
properties are controlled by the Dirac electrons.
E.Y. Andrei
Graphene – bandstructureGraphene – offshoots
Graphene (2005)
Silicene (2010)
Germanine (2014)
Stanene (2013)
Phosphorene (2014)
Borophene B36 (2014)
1. 2D2. Honeycomb structure 3. 2 identical atoms/cell
Ingredients
E.Y. Andrei
Ingredients:1. 2D2. Honeycomb structure 3. Identical populations
Artificial Graphene
nma 2~
sma
tv
meVam
t
F /1032
3
120*
5
2
2
=
A
B
Cold atom “graphene” 87Rb
Also:
E.Y. Andrei
Graphene bandstructure
1. 2D2. Honeycomb structure 3. 2 identical atoms/cell4. t1=t2=t3= t Relax the condition: t1=t2=t3
Strain Impose external potential Hybridization
Electrostatic potential Defects
Can one control/manipulate the
bandstructure?
A
B
E.Y. Andrei
Density of states
Summary
1. 2D2. Honeycomb structure (non-Bravais)3. 2 identical atoms/cell
indexbands
seepsvE
i
i
F
1
1; rk
2
1
=
==
pvH F
.s=
Pseudospin ; Helicity
Chiral quasiparticles
No backscattering
Berry phase p
E
v
LEg
F
2
2
2)(p
=
Kkp
=
K’K X
![Electronic and transport properties of kinked graphene · outstanding electronic transport properties of graphene [1]. However, it is crucial to modify the semimetallic electronic](https://img.pdfslide.us/doc/110x75/5fd887529019ff628275f4c6/electronic-and-transport-properties-of-kinked-graphene-outstanding-electronic-transport.jpg)
