Graphene: from simple to complex.
Graphene: from simple to complex. Bilayer graphene: material with flexible band structure. (with McCann, Mucha-Kruczynski and Aleiner)
Spontaneous symmetry breaking in pristine BLG. (with Lemonik and Aleiner) 2D crystals beyond graphene.
eV
meV
Wallace, Phys. Rev. 71, 622 (1947) Slonczewski, Weiss, Phys. Rev. 109, 272 (1958)
H AB ( K ) 0 e
i 23
A B
0 0e
i 23
0
cm v ~ 108 sec
(valleys)
p
0 Hˆ ( p ) v p v
McClure, PR 104, 666 (1956)
0
p x ip y pei
p
1 2
1 i e
Berry phase π winding number 1
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
a x ia y 0
Hˆ vp a v[ p a ] v shift of the Dirac point in the momentum space, opposite in K/K’ valleys (like vector potential).
Beff
[ a (r )]z
Ando - J. Phys. Soc. Jpn. 75, 124701 (2006) Foster, Ludwig - PRB 73, 155104 (2006) Morpurgo, Guinea - PRL 97, 196804 (2006)
BLG in ARPES T. Ohta et al Science 313, 951 (2006)
1eV
1 ˆ H z ( p y2 p x2 , 2 p x p y ) 2m 2 1 0 2 p 2 m 0
100meV
1 2
1 i 2 e
Berry phase 2π (winding number 2)
1 0.4eV m 0.035me McCann, VF - PRL 96, 086805 (2006)
eV Interlayer asymmetry gap in BLG
meV
Interlayer asymmetry gap in bilayer graphene
1 ˆ H z ( p y2 p x2 , 2 p x p y ) z z 2m
inter-layer asymmetry gap (can be controlled using electrostatic gates) McCann & VF - PRL 96, 086805 (2006)
T. Ohta et al – Science 313, 951 (2006)
Zhang, et al - Nature 459, 820 (2009)
Graphene under strain.
3
skew hopping and Lifshitz transition mv32 LiTr ~ 1meV 2
v 3 ~ 0 .1v 1 2 2 ˆ H z ( p y p x , 2 p x p y ) v 3 p 2m
2
McCann, VF - PRL 96, 086805 (2006)
2 mv3 10 nLiTr 2 cm2 ~ 10
Levitov: Is the value of skew hopping term γ3 ? Falko: The only known value is that for bulk graphite, γ3 =0.3eV. This is what we used for the estimation of Lifshitz transition energy quoted in the previous slide. In bilayer graphene, it may be different, but not by the order of magnitude, since the interlayer distance is very close to bulk graphite.
Strain effect on the BLG spectrum at low energies 3 3
3'
3
3''
3'''
2 w1 iw2 0 0 0 z ˆ v3 z H 2 0 2m ( ) 0 w1 iw2 0
u0
w1 0
w1 0
| w2 |
Mucha-Kruczynski, Aleiner, VF PRB 84, 041404 (2011)
Study the effect of homogeneous strain. What can be the magnitude of the strain-induced effects?
and DFT) Basko et al., PRB 80, 165413 (2009)
1% strain | w |~ 5meV Mucha-Kruczynski, Aleiner, VF - PRB 84, 041404 (2011)
Mele: What happens when the pair of Dirac minicones in the pervious slide merge at the critical value of strain? Is the pair of parabolas at the critical strain the same as in the approximate bilayer spectrum discussed in the simlified model (without skew hopping)? Falko: It is not the same, since the colliding points have opposite winding numbers (Berry phases, π and –π). Note that before strain was applied, the skew hopping term split the quadratic choral dispersion with Bery phase 2π into Dirac minicones, three in the corners with Berry phases, π and one in the middle with –π. When two collide at the critical strain, their Berry phases cancel, and the conduction-valence band degeneracy at zero energy is no more topologically protected. That is why at higher strains local in the momentum space gap appears, which persists over the red part of parametric space in the previous slide. At the same time, the remaining to Dirac cones carry Berry phase π each, making up the total equal to the conserving 2π. Mele: I shall describe similar behaviour in twisted bilayers, where the Dirac points can collide (merge and depart), preserving the total Beery phase.
eV
Spontaneous symmetry breaking in pristine BLG.
meV
Interaction-driven phases of electronic liquid in bilayer graphene
1 2 2 H ( p x p y ) x 2 p x p y y z v3 p 2m 2
e HC d 2rd 2r ' 2
2 H sr m
r r r' r'
g
l , s 0123
Irreps. R strain
| r r'| s l
d r 2
s l r
r
interlayer asymmetry (ferroelectric fluctuations)
2 1
2
A2
B2 charge-density wave
gO ~ 1
phase transition: spontaneous symmetry breaking,
E' ' E' 1
me 2 p2 h e2 Electron-electron e2 p p for V (r ) strong V ( ) ~ h aBohr m p h interaction in BLG r
Screening:
( p) Ry ~ 1eV [ 1 ~ 0.4eV ]
( q, )
2e q
2
2e 2 1 ~ q V ( q, ) 2e 2 N ( q , ) 1 N q
q2 m q2 8
0.22m
‘large’ N=4 (valley*spin)
1/N expansion
q
~ V (r ) 2e 2 2e 2 1 ~ 1 V (q ascr , 0) V ( q) 1 q 0 q NaBohr Nm q
2 e 2 ascr 2D screened r3
‘large’ N=4 1/N expansion justifying the use of perturbation theory
e2 r
aBohr ~ 30 A
d ln m ~ 10 2 d ln p
Lemonik, Aleiner, Toke, VF PRB 82, 201408 (2010)
r
Das Sarma: Charge density wave state driven by Coulomb repulsion – Wigner crystal – was discussed in usual semiconductors (GaAS) where the spectrum is also parabolic and Coulomb interaction takes over kinetic energy. What is the difference ? Falko: One difference is that in bilayer graphene there is no gap between conduction and valence band, and the density of states is constant. As a result screening always works, in contrast to GaAs or Si, where it fails at low enough densities. Halperin: In other words, one can never deplete graphene whereas one can deplete a finite band gap semiconductor. Falko: Another difference is that the phase transitions in bilayer graphene are driven by interaction between fluctuations in the electronic liquid which locally break the high symmetry of the graphene lattice. For example, when we talk about a charge density wave state, it has the wave number K of the Brillouin zone corner.
Renormalisation group approach. Vafek, Yang - PRB 81, 041401 (2010) Lemonik, Aleiner, Toke, VF - PRB 82, 201408 (2010)
at the shortest rang of applicability of two-band model
g R ( 12 1 ~ 2 / m21 ) 1 1 L ln 2 ln ( 1 )
g R ( ~ 2 / mL2 )
gO ( c ~ 2 nc / m ~ Tc ) c R g R ~ 1 signals phase transition into a broken symmetry state O.
L
Renormalisation of short-range interactions Lemonik, Aleiner, Toke, VF PRB 82, 201408 (2010)
2
2
2
c [0.02meV , 20meV ]
, mimics effect of strain: gaplesss with LiTr opposite spin polarisation on A and B sublattices in the opposite layers: gapped insulator, ′
′
Electrically polarised layers: gapped. almost never
Das Sarma: Does temperature, finite doping or disorder (such as inhomogeneity of carrier density) destroy the predicted phases? Falko: The RG flow in our calculations stops at the energy Ec where gO(Ec)~1. If temperature is low, kT< Ec, the phase persists, and Ec determines the critical temperature of the phase transition. For finite doping, one has to compare Ec with Fermi energy, if EF< Ec the phase persists, so that Ec determines critical density below which system falls into the new phase (for example, nematic or antiferromagnetic) and above which it is a metal. The same argument can be applied to a bilayer with inhomogeneous carrier density – electron/hole puddles. The new phases will persist if the root mean square of the inhomogeneous carrier density corresponds to the Fermi energy which is less than Ec. Levitov: There should be also critical strength of disorder locally breaking lattice symmetry. Note that the correlation length for the new phases depends on Ec and can be quite short.
Geim’s group
Yacobi’s group
Lau’s group
, mimics effect of strain: gapless with LiTr (or strain?)
? ? Maybe
opposite spin polarisation on A and B sublattices in the opposite layers: gapped.
Yacobi: Using local compressibility measurement on suspended bilayer graphene, we see a suppression of the density of states at low energy. This may be either a gap, or even a suppressed density of states in nematic phase. We also see some phase transition as a function magnetic field and transverse electric field. Although we do not do measure non-linear I(V) characteristics, like Lau, it is possible that Lau’s group sees the same thing as us.
Graphene: from simple to complex. Bilayer graphene: material with flexible band structure.
Spontaneous symmetry breaking in pristine BLG. 2D crystals beyond graphene.
eV
meV
Silicene: honeycomb 2D layer of silicon
Drummond, Zolyomi, VF arXiv:1112.4792
SO 1.5meV
boron nitride (‘white graphene’) 5eV band gap insulator MoS2, WS2, MoSe2, MoTe2, TaSe2 - semiconductors NbSe2 metal / superconductor
Complex systems: hetero- and superstructures of 2D crystals
Gr/(BN)4/Gr GrMoS2/Gr Manchester group 2011
Insulating state in graphene at n=0
Ponomarenko, Geim, Zhukov, Jalil, Morozov, Novoselov, Grigorieva, Hill, Cheianov, Fal’ko, Watanabe, Taniguchi, Gorbachev, Nature Physics 7,958 (2011) incl supplementary material
Heersche et al, Nature 446, 56-59 (2007)
Morozov et al, PRL 97, 016801 (2006)
Ki et al, PR B 78, 125409 (2008)
(B)
iv
cl
B
Inter-valley scattering -> weak localisation correction in G at high carrier density … McCann, Kechedzhi, VF, Suzuura, Ando, Altshuler, PRL 97, 146805 (2006)
Tikhonenko et al PRL 100, 056802 (2008)
… but ever insulating at n=0
Geim’s group
Kim’s group
Hwang, Adam, Das Sarma - PRL 98, 186806 (2007); Das Sarma’s group (2007 - …) Cheianov, VF, Altshuler, Aleiner – PRL 99, 176801 (2007)
Cheianov, VF, Altshuler, Aleiner PRL 99, 176801 (2007)
Random resistor network model of minimal conductivity in graphene (inhomogeneous doping) transparent NP boundary, due to electron chirality
e2 gnp h
Cheianov, VF PR B 74, 041403 (2006)
g pn g
e2 gnn ~ g pp g h Nn N p
ne p ~ 1 N n N p n
1
a
scaling of intrinsic conductance of a cluster x
a G( L) ~ g , x 0.97 L h L P( L) ~ a, h 74 a outer
g pn g
1
cluster perimeter
~ a /
1/( x h )
P( ) G( ) ~ g np a
min umin g , hx x 0.36
p 0 ne 0
a
min umin g hx x 0.36
( p) | p |
umin 1.3
a ( p) p 1 2 min p* 2
/2
, 1.3
p
Nn N p Nn N p
~
ne 1 n
g pn g
1
Weak localisation correction in the percolation regime
min G ( L) x ( / L ) min
min
L a L
D D ( L) x ( / L ) D
L a L
D 4e d q h 0 (2 ) 2 D(q 1 )q 2 1 1 2 a
2
Khmelnitski 1980
2e ln( L / ), L ~ x h ( L / ) , L 2
suppressed low-field magnetoresistance
2
e | min ( B) min (0) | h
Like in a point contact between two higherdimensional bulk electrodes
Ponomarenko, Geim, Zhukov, Jalil, Morozov, Novoselov, Grigorieva, Hill, Cheianov, Fal’ko, Watanabe, Taniguchi, Gorbachev, Nature Physics 7,958 (2011) - incl supplementary material
Graphene: from simple to complex. Bilayer graphene: material with flexible band structure.
Spontaneous symmetry breaking in pristine BLG. 2D crystals beyond graphene.
eV
meV
Spontaneous symmetry breaking in pristine BLG. (with Lemonik and Aleiner) 2D crystals beyond graphene.
eV
meV
Wallace, Phys. Rev. 71, 622 (1947) Slonczewski, Weiss, Phys. Rev. 109, 272 (1958)
H AB ( K ) 0 e
i 23
A B
0 0e
i 23
0
cm v ~ 108 sec
(valleys)
p
0 Hˆ ( p ) v p v
McClure, PR 104, 666 (1956)
0
p x ip y pei
p
1 2
1 i e
Berry phase π winding number 1
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
a x ia y 0
Hˆ vp a v[ p a ] v shift of the Dirac point in the momentum space, opposite in K/K’ valleys (like vector potential).
Beff
[ a (r )]z
Ando - J. Phys. Soc. Jpn. 75, 124701 (2006) Foster, Ludwig - PRB 73, 155104 (2006) Morpurgo, Guinea - PRL 97, 196804 (2006)
BLG in ARPES T. Ohta et al Science 313, 951 (2006)
1eV
1 ˆ H z ( p y2 p x2 , 2 p x p y ) 2m 2 1 0 2 p 2 m 0
100meV
1 2
1 i 2 e
Berry phase 2π (winding number 2)
1 0.4eV m 0.035me McCann, VF - PRL 96, 086805 (2006)
eV Interlayer asymmetry gap in BLG
meV
Interlayer asymmetry gap in bilayer graphene
1 ˆ H z ( p y2 p x2 , 2 p x p y ) z z 2m
inter-layer asymmetry gap (can be controlled using electrostatic gates) McCann & VF - PRL 96, 086805 (2006)
T. Ohta et al – Science 313, 951 (2006)
Zhang, et al - Nature 459, 820 (2009)
Graphene under strain.
3
skew hopping and Lifshitz transition mv32 LiTr ~ 1meV 2
v 3 ~ 0 .1v 1 2 2 ˆ H z ( p y p x , 2 p x p y ) v 3 p 2m
2
McCann, VF - PRL 96, 086805 (2006)
2 mv3 10 nLiTr 2 cm2 ~ 10
Levitov: Is the value of skew hopping term γ3 ? Falko: The only known value is that for bulk graphite, γ3 =0.3eV. This is what we used for the estimation of Lifshitz transition energy quoted in the previous slide. In bilayer graphene, it may be different, but not by the order of magnitude, since the interlayer distance is very close to bulk graphite.
Strain effect on the BLG spectrum at low energies 3 3
3'
3
3''
3'''
2 w1 iw2 0 0 0 z ˆ v3 z H 2 0 2m ( ) 0 w1 iw2 0
u0
w1 0
w1 0
| w2 |
Mucha-Kruczynski, Aleiner, VF PRB 84, 041404 (2011)
Study the effect of homogeneous strain. What can be the magnitude of the strain-induced effects?
and DFT) Basko et al., PRB 80, 165413 (2009)
1% strain | w |~ 5meV Mucha-Kruczynski, Aleiner, VF - PRB 84, 041404 (2011)
Mele: What happens when the pair of Dirac minicones in the pervious slide merge at the critical value of strain? Is the pair of parabolas at the critical strain the same as in the approximate bilayer spectrum discussed in the simlified model (without skew hopping)? Falko: It is not the same, since the colliding points have opposite winding numbers (Berry phases, π and –π). Note that before strain was applied, the skew hopping term split the quadratic choral dispersion with Bery phase 2π into Dirac minicones, three in the corners with Berry phases, π and one in the middle with –π. When two collide at the critical strain, their Berry phases cancel, and the conduction-valence band degeneracy at zero energy is no more topologically protected. That is why at higher strains local in the momentum space gap appears, which persists over the red part of parametric space in the previous slide. At the same time, the remaining to Dirac cones carry Berry phase π each, making up the total equal to the conserving 2π. Mele: I shall describe similar behaviour in twisted bilayers, where the Dirac points can collide (merge and depart), preserving the total Beery phase.
eV
Spontaneous symmetry breaking in pristine BLG.
meV
Interaction-driven phases of electronic liquid in bilayer graphene
1 2 2 H ( p x p y ) x 2 p x p y y z v3 p 2m 2
e HC d 2rd 2r ' 2
2 H sr m
r r r' r'
g
l , s 0123
Irreps. R strain
| r r'| s l
d r 2
s l r
r
interlayer asymmetry (ferroelectric fluctuations)
2 1
2
A2
B2 charge-density wave
gO ~ 1
phase transition: spontaneous symmetry breaking,
E' ' E' 1
me 2 p2 h e2 Electron-electron e2 p p for V (r ) strong V ( ) ~ h aBohr m p h interaction in BLG r
Screening:
( p) Ry ~ 1eV [ 1 ~ 0.4eV ]
( q, )
2e q
2
2e 2 1 ~ q V ( q, ) 2e 2 N ( q , ) 1 N q
q2 m q2 8
0.22m
‘large’ N=4 (valley*spin)
1/N expansion
q
~ V (r ) 2e 2 2e 2 1 ~ 1 V (q ascr , 0) V ( q) 1 q 0 q NaBohr Nm q
2 e 2 ascr 2D screened r3
‘large’ N=4 1/N expansion justifying the use of perturbation theory
e2 r
aBohr ~ 30 A
d ln m ~ 10 2 d ln p
Lemonik, Aleiner, Toke, VF PRB 82, 201408 (2010)
r
Das Sarma: Charge density wave state driven by Coulomb repulsion – Wigner crystal – was discussed in usual semiconductors (GaAS) where the spectrum is also parabolic and Coulomb interaction takes over kinetic energy. What is the difference ? Falko: One difference is that in bilayer graphene there is no gap between conduction and valence band, and the density of states is constant. As a result screening always works, in contrast to GaAs or Si, where it fails at low enough densities. Halperin: In other words, one can never deplete graphene whereas one can deplete a finite band gap semiconductor. Falko: Another difference is that the phase transitions in bilayer graphene are driven by interaction between fluctuations in the electronic liquid which locally break the high symmetry of the graphene lattice. For example, when we talk about a charge density wave state, it has the wave number K of the Brillouin zone corner.
Renormalisation group approach. Vafek, Yang - PRB 81, 041401 (2010) Lemonik, Aleiner, Toke, VF - PRB 82, 201408 (2010)
at the shortest rang of applicability of two-band model
g R ( 12 1 ~ 2 / m21 ) 1 1 L ln 2 ln ( 1 )
g R ( ~ 2 / mL2 )
gO ( c ~ 2 nc / m ~ Tc ) c R g R ~ 1 signals phase transition into a broken symmetry state O.
L
Renormalisation of short-range interactions Lemonik, Aleiner, Toke, VF PRB 82, 201408 (2010)
2
2
2
c [0.02meV , 20meV ]
, mimics effect of strain: gaplesss with LiTr opposite spin polarisation on A and B sublattices in the opposite layers: gapped insulator, ′
′
Electrically polarised layers: gapped. almost never
Das Sarma: Does temperature, finite doping or disorder (such as inhomogeneity of carrier density) destroy the predicted phases? Falko: The RG flow in our calculations stops at the energy Ec where gO(Ec)~1. If temperature is low, kT< Ec, the phase persists, and Ec determines the critical temperature of the phase transition. For finite doping, one has to compare Ec with Fermi energy, if EF< Ec the phase persists, so that Ec determines critical density below which system falls into the new phase (for example, nematic or antiferromagnetic) and above which it is a metal. The same argument can be applied to a bilayer with inhomogeneous carrier density – electron/hole puddles. The new phases will persist if the root mean square of the inhomogeneous carrier density corresponds to the Fermi energy which is less than Ec. Levitov: There should be also critical strength of disorder locally breaking lattice symmetry. Note that the correlation length for the new phases depends on Ec and can be quite short.
Geim’s group
Yacobi’s group
Lau’s group
, mimics effect of strain: gapless with LiTr (or strain?)
? ? Maybe
opposite spin polarisation on A and B sublattices in the opposite layers: gapped.
Yacobi: Using local compressibility measurement on suspended bilayer graphene, we see a suppression of the density of states at low energy. This may be either a gap, or even a suppressed density of states in nematic phase. We also see some phase transition as a function magnetic field and transverse electric field. Although we do not do measure non-linear I(V) characteristics, like Lau, it is possible that Lau’s group sees the same thing as us.
Graphene: from simple to complex. Bilayer graphene: material with flexible band structure.
Spontaneous symmetry breaking in pristine BLG. 2D crystals beyond graphene.
eV
meV
Silicene: honeycomb 2D layer of silicon
Drummond, Zolyomi, VF arXiv:1112.4792
SO 1.5meV
boron nitride (‘white graphene’) 5eV band gap insulator MoS2, WS2, MoSe2, MoTe2, TaSe2 - semiconductors NbSe2 metal / superconductor
Complex systems: hetero- and superstructures of 2D crystals
Gr/(BN)4/Gr GrMoS2/Gr Manchester group 2011
Insulating state in graphene at n=0
Ponomarenko, Geim, Zhukov, Jalil, Morozov, Novoselov, Grigorieva, Hill, Cheianov, Fal’ko, Watanabe, Taniguchi, Gorbachev, Nature Physics 7,958 (2011) incl supplementary material
Heersche et al, Nature 446, 56-59 (2007)
Morozov et al, PRL 97, 016801 (2006)
Ki et al, PR B 78, 125409 (2008)
(B)
iv
cl
B
Inter-valley scattering -> weak localisation correction in G at high carrier density … McCann, Kechedzhi, VF, Suzuura, Ando, Altshuler, PRL 97, 146805 (2006)
Tikhonenko et al PRL 100, 056802 (2008)
… but ever insulating at n=0
Geim’s group
Kim’s group
Hwang, Adam, Das Sarma - PRL 98, 186806 (2007); Das Sarma’s group (2007 - …) Cheianov, VF, Altshuler, Aleiner – PRL 99, 176801 (2007)
Cheianov, VF, Altshuler, Aleiner PRL 99, 176801 (2007)
Random resistor network model of minimal conductivity in graphene (inhomogeneous doping) transparent NP boundary, due to electron chirality
e2 gnp h
Cheianov, VF PR B 74, 041403 (2006)
g pn g
e2 gnn ~ g pp g h Nn N p
ne p ~ 1 N n N p n
1
a
scaling of intrinsic conductance of a cluster x
a G( L) ~ g , x 0.97 L h L P( L) ~ a, h 74 a outer
g pn g
1
cluster perimeter
~ a /
1/( x h )
P( ) G( ) ~ g np a
min umin g , hx x 0.36
p 0 ne 0
a
min umin g hx x 0.36
( p) | p |
umin 1.3
a ( p) p 1 2 min p* 2
/2
, 1.3
p
Nn N p Nn N p
~
ne 1 n
g pn g
1
Weak localisation correction in the percolation regime
min G ( L) x ( / L ) min
min
L a L
D D ( L) x ( / L ) D
L a L
D 4e d q h 0 (2 ) 2 D(q 1 )q 2 1 1 2 a
2
Khmelnitski 1980
2e ln( L / ), L ~ x h ( L / ) , L 2
suppressed low-field magnetoresistance
2
e | min ( B) min (0) | h
Like in a point contact between two higherdimensional bulk electrodes
Ponomarenko, Geim, Zhukov, Jalil, Morozov, Novoselov, Grigorieva, Hill, Cheianov, Fal’ko, Watanabe, Taniguchi, Gorbachev, Nature Physics 7,958 (2011) - incl supplementary material
Graphene: from simple to complex. Bilayer graphene: material with flexible band structure.
Spontaneous symmetry breaking in pristine BLG. 2D crystals beyond graphene.
eV
meV
