Download Electronic properties of graphene, from `high` to `low` energies from

Electronic properties of graphene,
from ‘high’
high to ‘low’
low energies.
energies
Vladimir Falko, Lancaster University
Graphene for beginners: tight-binding model.
Berry phase π electrons in monolayers.
Trigonal warping. Stretched graphene.
PN junction in graphene.
Berry phase 2π electrons in bilayer graphene.
y asymmetry
y
yg
gap.
p
Landau levels & QHE. Interlayer
Lifshitz transition and magnetic breakdown in BLG. Stretched BLG.
Renormalisation group theory for interaction and spontaneous
symmetry breaking in BLG.
BLG
4 electrons in the outer s-p shell of carbon
sp 2 hybridisation forms strong directed bonds
which determine a honeycomb lattice structure.
C
 - bonds P x, y S   0 
P z ( ) orbitals determine conduction properties of graphite
A
B
Graphene: gapless semiconductor
Wallace, Phys. Rev. 71, 622 (1947)
Slonczewski, Weiss, Phys. Rev. 109, 272 (1958)
Wallace, Phys. Rev. 71, 622 (1947)
Slonczewski, Weiss, Phys. Rev. 109, 272 (1958)

 0

f (k ) 
0
ˆ


H  
*


f
(
k
)
0
 0

   |0 f |
Reciprocal lattice
 

 (k  GN1N 2 )   (k )



GN1N 2  N1G1  N 2G2

G2
1st Brilloun zone

G1
Hexagonal
reciprocal lattice
corresponding to
the hexagonal
Bravais lattice
M-point,
van Hove singularity
Fermi ‘point’
in undoped
Graphene:
p
K(K’) point

 (k )  EF
First Brillouin
zone

G
Valley
V
ll
  1

G'
Valley
  1
i2 / 3
e
e
i0
i2 / 3
e
H AB , K
  0 e

H BA, K 
3
2
i 23
e
 i ( a2 p x 
a
2 3
py )
e
 0 a ( p x  ip y )  v
i
a
3


 23  0 a ( p x  ip y )
py
i 23
e e
i ( a2 p x 
a
2 3
py )
Bloch function amplitudes (e.g., in the valley
K) on the AB sites (‘isospin’)
( isospin ) mimic spin
components of a massless relativistic particle.
A 
   
B 
0

Hˆ  v
 px  ipp y
McClure, PR 104, 666 (1956)
px  ip y 
 
  v  p
0 
v
3
2
 0 a ~ 10
8 cm
sec
(valleys)

p
valley index
‘pseudospin’
  1
0 

 0
ˆ
H  v


0




  

0 
  p x  ip y
   p x  ip y
  A,  


  B 
 
 B 
 
 A 

p

p
sublattice index
‘isospin’
Also, one may need to take into account an additional real spin degeneracy of all states
Electronic properties of graphene,
from ‘high’ to ‘low’ energies.
Graphene for beginners: tight-binding model.
Berry phase π electrons in monolayers.
Trigonal warping.
warping Stretched graphene
graphene.
PN junction in graphene.
Berry phase 2π electrons in bilayer graphene.
graphene
Landau levels & QHE. Interlayer asymmetry gap.
Lifshitz transition and magnetic breakdown in BLG. Stretched BLG.
R
Renormalisation
li ti group theory
th
for
f interaction
i t
ti and
d spontaneous
t
symmetry breaking in BLG.
   vp
0 
 
  v  p
H  v

0



px
py

p  ( p cos  , p sin  )
  p x  ip y  pei
   p x  ip y  pe i
   vp
conduction band


p
 
sublattice ‘isospin’ is
  n  1,   vp
linked to the direction
of the electron
 
  n  1,   vp 
momentum
p
valence band
 1 
 i 
 e 

p
Berry phase
2
 p 
1
2
d
  i  d 

d
0

 e
i
2  3
2
  ei   
3
Electronic states in graphene observed using ARPES
 1 
   i 
e 

p
1
2
I ARPES ~|A B |2
 



2 k  RBA
 
~ sin 
2
 2
  vp
   
k||  G  K  p
Mucha-Kruczynski, Tsyplyatyev, Grishin, McCann,
VF, Boswick, Rotenberg - PRB 77, 195403 (2008)
ARPES of heavilyy doped
p g
graphene
p
synthesized on silicon carbide
Bostwick et al - Nature Physics, 3, 36 (2007)
0  
 0
 
ˆ
    2

H  v



0
 0 
 ( )

2
i py
i 23 i ( 2 p x  2 3 p y )
i 23 i ( 2 p x  2 3 p y ) 
3

 0 e e
e
e e


2
 23  0 a ( p x  ip y )   08a ( p x  ip y ) 2
a
valley
  1
H AB, K
weak trigonal warping

 ( p)   F
a
a
a
a
  px  ipy  pei
  px  ipy  pei

p  ( p cos  , p sin  )
time-inversion
symmetry t  t


p   p; K   K 
A. Bostwick et al – Nature Physics 3, 36 (2007)
Slightly stretched monolayer graphene
0
 0''
0 0
 0e
 i 23
 0'  0'''
  0   0e
i 23
0
'  i 23
0
 e
''' i 23
0
  e
''
0
 u x  iu y  0








Hˆ  vp    u    v[ p  u ]  
v
shift of the Dirac point in the momentum space,
opposite in K/K’ valleys,
valleys like a vector potential
Beff
 
  [  u (r )]z
Ando - J. Phys. Soc. Jpn. 75, 124701 (2006)
Foster, Ludwig - PRB 73, 155104 (2006)
Morpurgo, Guinea - PRL 97, 196804 (2006)
Electronic properties of graphene,
from ‘high’ to ‘low’ energies.
Graphene for beginners: tight-binding model.
Berry phase π electrons in monolayers.
Trigonal warping.
warping Stretched graphene
graphene.
PN junction in graphene.
Berry phase 2π electrons in bilayer graphene.
graphene
Landau levels & QHE. Interlayer asymmetry gap.
Lifshitz transition and magnetic breakdown in BLG. Stretched BLG.
R
Renormalisation
li ti group theory
th
for
f interaction
i t
ti and
d spontaneous
t
symmetry breaking in BLG.
Monolayer graphene: two-dimensional gapless semiconductor
with Berry phase π electrons
  vp
py
 

H  v  p  1̂  U ( r )
px
 p 
 
 n 1
w( )

 1  ipr
 i e
e 
  p 1̂ U ( x) p  0
Due to the ‘isospin’ conservation,
A-B symmetric perturbation does
not backward scatter electrons,
Ando, Nakanishi, Saito
J. Phys. Soc. Jpn 67, 2857 (1998)
w( ) ~ cos 2 2 | U p  p ' |2
1
2
 
H  v  p  1̂  U ( x )
i
Potential which is smooth at the
scale of lattice constant (A-B
symmetric) cannot scatter Berry
phase
h
π electrons
l t
in
i exactly
tl
backward direction.
i
 p
22
wwpppp  
i i 
ii
 a  b  Ae
i 2  z
 b  a  Ae
i 2  z
 p
 p
2
 [
ab
  b a ] 
( a ,b )
2
0
0
( a ,b )
 a  b  e i  b  a   b  a
z
B
Berry
phase
h
π electrons
l t
PN junctions in the usual gap-full semiconductors are nontransparent for incident electrons
electrons, therefore
therefore, they are highly resistive
resistive.
PN junctions in in graphene are different.
Transmission of chiral electrons through the PN junction in graphene
 c  vp
eU '  vp
F
N h  p F2 / 
pF
pF
eU   vp
Ne  p
2
F
F
/
conduction band electrons

 
 n  1
p
 v   vp
 

H  v  p  1
1̂  U ( r )
Due to the isospin conservation,
conservation A-B
A B symmetric potential
cannot backward scatter electrons in monolayer graphene.
For graphene PN junctions: Cheianov
Cheianov, VF - PR B 74,
74 041403 (2006)
‘Klein paradox’: Katsnelson, Novoselov, Geim, Nature Physics 2, 620 (2006)
Transmission of chiral electrons through the PN junction in graphene
p x2  p F2 sin 2   U ( x )  
 

p
1
2
U 0
 1 
 i 
e 
d
Due to the ‘isospin’
p conservation,,
electrostatic potential U(x) which
smooth on atomic distances cannot
scatter electrons in the exactly
backward direction.
w( )  e
pF d sin 2 
1
cos 
2

1/ kF d
Transmission of chiral electrons through the PN junction in graphene
L
d
Due to transmission of electrons with a small
incidence angle, θ<1/pFd , a PN junction in graphene
should display a finite conductance (no pinch
pinch-off).
off).
A characteristic Fano factor in the shot noise:
g np
2
2e

L
h
I  I  (1 
pF
d
1
2
) eI
Cheianov, VF - PR B 74, 041403 (2006)
PN junctions should be
taken into consideration in
t t
two-terminal
i l devices,
d i
since metallic contacts
dope
p g
graphene,
p
due to the
work function difference.
Heersche et al - Nature Physics (2007)
w
PNP junction with a suspended gate: an almost ballistic regime: w~l.
A Young and P Kim - Nature Physics 5, 222 (2009)
Wishful thinking about graphene microstructures
Focusing and Veselago lens for electrons in ballistic graphene
Cheianov, VF, Altshuler - Science 315, 1252 (2007)

 c ( p )  vp

 
p
v   v
p
p
eU  vpc
N e  pc2 / 
Fermi
momentum
eU '  vpv
N h  pv2 / 
pc

 v ( p)  vp
Fermi
momentum

 
p
v    v
p
p
pv
The effect we’ll discuss would be the strongest in sharp PN junction,
with
ith d~λ
d λF .
p y  p  pc sin  c   pv sin  v
PN junction
'
y
sin  c
pv

n
sin  v
pc

Vc

p
Snell’s law with
negative
refraction index
c
v
n-type
 c  vpp;
p-type


p
Vc  v
p
 v  vp'


p'
Vv  v
p'

p'

Vv
pc  pv
pv  0.8 pc
Graphene bipolar transistor: Veselago lens for electrons
w
2w
Electronic properties of graphene,
from ‘high’ to ‘low’ energies.
Graphene for beginners: tight-binding model.
Berry phase π electrons in monolayers.
Trigonal warping.
warping Stretched graphene
graphene.
PN junction in graphene.
Berry phase 2π electrons in bilayer graphene.
graphene
Landau levels & QHE. Interlayer asymmetry gap.
Lifshitz transition and magnetic breakdown in BLG. Stretched BLG.
R
Renormalisation
li ti group theory
th
for
f interaction
i t
ti and
d spontaneous
t
symmetry breaking in BLG.