An unexpected turn for twisted graphenes
An unexpected turn for twisted graphenes Gene Mele University of Pennsylvania
Twisted Graphenes: I. Introduction: what, where, why II. Family behavior of low energy physics III. New approach to an old Hamiltonian IV. Compensated class near a topological transition V. Uncompensated new topological state in twisted FLGs
The interlayer coherence scale is large for a Bernal bilayer
Single layer
but it is very small in faulted multilayers e.g. M. Sprinkle et al. PRL 103, 226803 (2009) (ARPES)
Bernal Bilayer
The interlayer coherence scale is large for a Bernal bilayer
Single layer
Bernal Bilayer
Some faulted graphene bilayers Epitaxial graphene on SiC
(000 1)
C. Berger et al. Science 312, 1191 (2006) (synthesis) D.L. Miller et al. Science 324, 924 (2009) (Landau level spectroscopy) M. Sprinkle et al. PRL 103, 226803 (2009) (ARPES)
more faulted graphene bilayers Graphene on Graphite G. Li, A. Luican and E. Andrei, Phys. Rev. Lett. 102, 176804 (2009) Landau level (Scanning Tunneling Spectroscopy)
Folded Exfoliated Graphene H. Schmidt et al., Appl. Phys. Lett. 93, 172108 (2009)
Orthodoxy: Layer decoupling comes from the momentum mismatch across a rotational fault
! ! ! H eff u( r ) = "i"vF (! ## ) u( r ) mismatch pseudo-spinor wf
J.M.B. Lopes dos Santos, N.M.R. Peres and A.H. Castro Neto Physical Review Letters 99, 256802 (2007)) R. Bistritzer and A. H. MacDonald, PNAS 108, 12233 (2011)
Highest Symmetry Bilayers
AB (Bernal) Stacking
AA Stacking
Rotationally Faulted Bilayers
32.204o
27.796o
Faults are indexed by a 2D graphene translation vector (m,n) and occur in complementary partners with the same commensuration area but opposite sublattice exchange symmetry
What is the smoothest density wave that matches the commensuration cell ? Periodic on commensuration supercell Maxima for overlapping sites Minima for misregistered zones
! nµ (r ) =
$ $e
! Gm = G1 !
! ! ! iGµ , m "( r #" µ ,! )
(real, lattice-symmetric)
! ! ! nsc (r ) = n 1 (r ) + n 2 (r ) " " " T! (r ) = C0 exp !#C1 ( n 1 (r ) + n 2 (r ) )"$
Exponential suppression is misregistered regions.
Nonlinear mixing of overlapping lattices: all G_s present.
Interlayer Potentials AA stacking
Bernal
SE odd
SE even
30o ± 8.213o
More Interlayer Potentials SE-even near 30o
Interlayer potential for a small angle fault (SE-even) BA
AB AA
Interlayer potential @ 30o: quasiperiodic
Bilayer Hamiltonian in the pseudospin basis SE even
SE odd
+ Hˆ int
Hˆ even
Hˆ odd
ˆ+ " $i"v! # # ! H % F 1 int ' =& † + & Hˆ int ' $i"v! F ! 2 #% ( )
( )
ˆ" # "i"v! $ # ! H % F 1 int ' =& † " * & Hˆ int ' i"v! ! F 2 #% ( )
i! / 2 $ e = V! ei" & ( 0
( )
0 % # i! / 2 ' e )
0$ " i! # 1 ˆ H int = V! e % & 0 0 ' (
Interlayer & Intra-valley (two copies) Interlayer & Inter-valley (two copies)
Transport on two sublattices via x-y pseudospin rotation Transport on the dominant (eclipsed) sublattice
Two types of low energy physics
no pseudospin rotation in AA stacking
Topics for today: I. Introduction: what, where, why II. Family behavior of low energy physics III. New approach to an old Hamiltonian IV. Compensated class near a topological transition V. Uncompensated new topological state in twisted FLGs
Small angle partners
Spatially modulated coupling
6 ! ! ! 2 ! 2 Matrix Coupling: Tˆ (r ) = tˆ0 + " tˆn eiGn "r n=1
with expansion coefficients ! ! $ iGn "r!
!z tG # %z
! ! $ iGn "r"
%e e tˆn = tG ' ! ! ! ! ' e $ iGn "r# e $ iGn "r! ) 1" !z (even n) & tG # $ z& %z
! caa t0 = # % cba
& ( ( * 1" (odd n) $ z&
cab " ; caa = cbb , cab = cba $ cbb &
Slonczewski-Weiss-McClure model
SWMcC model contains a (strong) threefold lattice anisotropy
Slonczewski-Weiss-McClure model reflects a threefold anisotropy of its lattice Wannier states
Slonczewski-Weiss-McClure model
isotropic two center tight binding model
Model I: q=0 term in K-point basis
Tij # Tij e
! ! ! ! iG ' "! i ' "iG "! j
!1 1" T1 = w # $ 1 1 % &
!z T2 = w # %z
w=
!1 3
1" z $&
!z T3 = w # %z
1" z $&
= 130 meV (TB est. ! 110 meV)
Scaling Relations 8! #" $ %K = sin & ' 3a (2)
q Q= !K
8" !vF #! $ E! = !vF %K = sin & ' 3a (2)
E "= E!
c!=
3ac #! $ 8" "vF sin % & '2(
only !-dependent parameter: strong coupling at small angles
Canonical (isotropic) picture 8! !vF #" $ E/ sin % & 3a '2(
8! #" $ q/ sin % & 3a '2(
van Hove sing.
coupling scale
Interlayer coupling models Three geometrically distinct states conventional, Isotropic
!1 1" w# $ 1 1 % &
!0 1" cab # $ 1 0 % &
anisotropic (-)
! 1 0 " anisotropic (+) caa # $ 0 1 % &
Model II q=0 term in K-point basis #1 1$ #0 1$ % cab & = cab! x" x T1 = w & ' ' (1 1) (1 0)
! 0 1" T2 = cab # $ 0 z % &
!0 1" T3 = cab # $ 0 z % &
Topological Transition c! !x % ! "(#i$) & H =' ( ! ˆ " c ! ! ( # i $ # (1) e ) "K * x ) "I 0 # ! H =$ H % &0 !x '
"I 0 # $0 ! % x' &
" !I ( ) H q c $ % K " H!= & " ' ! x H K (q " #K )! x ) ) ( c!I Compensated (opposite helicities)
Evolution of Dirac points
Decrease angle: increase coupling strength
Spectral reconstruction starts here annihilates here
c!=
1 2
re-emerge here
Pair annihilation for compensated DPs Converge/collapse/diverge with increasing c
Comments: The SWMcC model contains a (large) threefold lattice anisotropy Breakdown of first generation models : Relativistic (linear) bands ! Massive (curved) bands Dirac point annihilation and regeneration. Gauge potential from twisted neighbor
Topics for today: I. Introduction: what, where, why II. Family behavior of low energy physics III. New approach to an old Hamiltonian IV. Compensated class near a topological transition V. Uncompensated new topological state in twisted FLGs
Reverse sign of anisotropy
isotropic
!1 1" w# $ 1 1 % &
!0 1" cab # $ 1 0 % &
anisotropic (-)
! 1 0 " anisotropic (+) caa # $ 0 1 % &
Topological Transition c!I % ! "(#i$) & H =' ( ! ˆ ! " ( # $ + ) c I i e "K * )
! c"I # H K (q ) $ ! & H =% ! H K (q ! "K ) ( ' c"I Uncompensated (same helicity)
Symmetry-protected crossing
Bilayer version of absence of backscattering
Second generation Dirac points uncompensated
new DP
compensated
Velocity renormalization (C) #Hˆ vˆ+ = = vF !ˆ + $ vF (1 " c!2 )!ˆ + #q" #Hˆ vˆ" = = vF !ˆ " $ vF (1 " c!2 )!ˆ " #q+
vF* = vF (1 ! 9c!2 )
perturbative, isotropic reduction
Velocity renormalization (U) #Hˆ vˆ+ = = vF !ˆ + $ vF (!ˆ + " c!2!ˆ " ) #q" #Hˆ vˆ" = = vF !ˆ " $ vF (!ˆ " " c!2!ˆ + ) #q+
vF* = vF (1 " c!4 cos(2! ))
perturbative with twofold anisotropy (vanishes after sum over "K triad)
Signatures of the uncompensated state SYMMETRY-PROTECTED BAND CROSSING SECOND GENERATION DIRAC POINTS: DEGENERATE AT DISCRETE POINTS (NOT LINES)1. NO PERTURBATIVE VELOCITY RENORMALIZATION EXTENDED (STRONGER) VAN HOVE SINGULARITY
These are all properties of the SiC (000 \bar 1) epitaxial graphenes! 1
ARPES pending
Simons Symposium Questions During Talk by Gene Mele 3:55 Das Sarma: Are you going to be addressing this problem just at the single particle level? Mele: Yes. The band structure of these systems is rich and not completely understood. Our interest in studying it is to understand the role of interactions on top of this. 7:30 Altshuler/Halperin/Freedman What are the rotation angles? Are the layers also buckled? Sheared? Mele: In an epitaxial multilayer there are typically a distribution of rotation angles, where the distribution is peaked around magic angles that correspond to low order commensurate rotations. The layers are reasonably flat over a large coherence width. The scattering measurements identify the rotation angle, the layers can also be displaced laterally. 11:48 Altshuler: Could the faulted structure be quasiperiodic Mele: Yes. That occurs for example at a 30 degree rotation which is a crucial angle in the classification of all faulted bilayer structures. For the most part our theory focuses on commensurate rotations where we can say something about the physics. 14:50 Freedman: Do you allow for slip of one layer relative to another. Mele: The construction is to make a rotation away from the Bernal structure holing one pair of eclipsed sites fixed. There is no lateral slip with respect to this fixed rotation axis. 16:19 Aleiner: Do the partners have the same energy, and is it clear which one is preferred. Mele: Undoubtedly the two have different energies. We are going to freeze in these two structures and compare their electronic spectra. 25:18 Altshuler: Is there a Hofstadter-like band structure to bands for incommensurate structures. Mele: Yes. I’m steering your attention away from such a singular point to understand the situation where the structure is commensurate, though possibly with a long period. 27:40 Aleiner: Aren't the interlayer coupling matrices a function of position.
Mele: No, the position dependence has been integrated out. These are the matrix elements for tunneling between Dirac points of the two layers. It is an interlayer mass operator. 29:00 Aleiner: Do these structures have different point group symmetries. Mele: Yes. That’s the essential difference that allows us to classify their low energy Hamiltonians. 38:10 Das Sarma: Gamma_4 isn't very well known. Mele: It is known to be small. The important idea in this model is that gamma_3 and gamma_4 are inequivalent. Gamma_4 is not known nearly as well as gamma_3, but it is known to be a lot smaller than gamma_3. 50:11 Altshuler: Doesn't a Peierls type instability break the degeneracy, gapping the crossing point. Mele: It’s not like the one dimensional problem. There’s very little spectral weight at the crossing point, so one does not generically expect and instability. 51:50 Falko: Couldn't this be gapped by short range variations of the type you discussed in the first part of the talk. Mele: At small angle the physics is controlled by the cell average of the interlayer potential and one should be insensitive to the spatially modulated terms. 53:50 Aleiner: Compare the interactions from the first part of the talk with the discussion in the section.
Mele: For small angles we can focus only on the cell average of the interlayer potential. For large angle rotations the key quantity is the Fourier component of the crystal potential and the momentum that spans the offset Dirac points. 55:00 Halperin: How do these depend on the rotation angle. Mele: The angle itself is the wrong metric since nearby angles can have vastly different periods and different physical properties. There are a hierarchy of commensurations which is the right language for trying to understand these effects. 56:00 Freedman: Could one look for topological transitions in the band structure. Mele: We haven’t done that, but I think it could be useful. 57:00 Levitov: Do the gauge fields produce the analog to magnetic translations. Mele: I don’t think so since there is no net flux through the cell. 59:14 Das Sarma: The literature on states and waves in quasiperiodic lattices should be relevant here. Mele: Yes, I think applications along those lines could be quite useful.
Twisted Graphenes: I. Introduction: what, where, why II. Family behavior of low energy physics III. New approach to an old Hamiltonian IV. Compensated class near a topological transition V. Uncompensated new topological state in twisted FLGs
The interlayer coherence scale is large for a Bernal bilayer
Single layer
but it is very small in faulted multilayers e.g. M. Sprinkle et al. PRL 103, 226803 (2009) (ARPES)
Bernal Bilayer
The interlayer coherence scale is large for a Bernal bilayer
Single layer
Bernal Bilayer
Some faulted graphene bilayers Epitaxial graphene on SiC
(000 1)
C. Berger et al. Science 312, 1191 (2006) (synthesis) D.L. Miller et al. Science 324, 924 (2009) (Landau level spectroscopy) M. Sprinkle et al. PRL 103, 226803 (2009) (ARPES)
more faulted graphene bilayers Graphene on Graphite G. Li, A. Luican and E. Andrei, Phys. Rev. Lett. 102, 176804 (2009) Landau level (Scanning Tunneling Spectroscopy)
Folded Exfoliated Graphene H. Schmidt et al., Appl. Phys. Lett. 93, 172108 (2009)
Orthodoxy: Layer decoupling comes from the momentum mismatch across a rotational fault
! ! ! H eff u( r ) = "i"vF (! ## ) u( r ) mismatch pseudo-spinor wf
J.M.B. Lopes dos Santos, N.M.R. Peres and A.H. Castro Neto Physical Review Letters 99, 256802 (2007)) R. Bistritzer and A. H. MacDonald, PNAS 108, 12233 (2011)
Highest Symmetry Bilayers
AB (Bernal) Stacking
AA Stacking
Rotationally Faulted Bilayers
32.204o
27.796o
Faults are indexed by a 2D graphene translation vector (m,n) and occur in complementary partners with the same commensuration area but opposite sublattice exchange symmetry
What is the smoothest density wave that matches the commensuration cell ? Periodic on commensuration supercell Maxima for overlapping sites Minima for misregistered zones
! nµ (r ) =
$ $e
! Gm = G1 !
! ! ! iGµ , m "( r #" µ ,! )
(real, lattice-symmetric)
! ! ! nsc (r ) = n 1 (r ) + n 2 (r ) " " " T! (r ) = C0 exp !#C1 ( n 1 (r ) + n 2 (r ) )"$
Exponential suppression is misregistered regions.
Nonlinear mixing of overlapping lattices: all G_s present.
Interlayer Potentials AA stacking
Bernal
SE odd
SE even
30o ± 8.213o
More Interlayer Potentials SE-even near 30o
Interlayer potential for a small angle fault (SE-even) BA
AB AA
Interlayer potential @ 30o: quasiperiodic
Bilayer Hamiltonian in the pseudospin basis SE even
SE odd
+ Hˆ int
Hˆ even
Hˆ odd
ˆ+ " $i"v! # # ! H % F 1 int ' =& † + & Hˆ int ' $i"v! F ! 2 #% ( )
( )
ˆ" # "i"v! $ # ! H % F 1 int ' =& † " * & Hˆ int ' i"v! ! F 2 #% ( )
i! / 2 $ e = V! ei" & ( 0
( )
0 % # i! / 2 ' e )
0$ " i! # 1 ˆ H int = V! e % & 0 0 ' (
Interlayer & Intra-valley (two copies) Interlayer & Inter-valley (two copies)
Transport on two sublattices via x-y pseudospin rotation Transport on the dominant (eclipsed) sublattice
Two types of low energy physics
no pseudospin rotation in AA stacking
Topics for today: I. Introduction: what, where, why II. Family behavior of low energy physics III. New approach to an old Hamiltonian IV. Compensated class near a topological transition V. Uncompensated new topological state in twisted FLGs
Small angle partners
Spatially modulated coupling
6 ! ! ! 2 ! 2 Matrix Coupling: Tˆ (r ) = tˆ0 + " tˆn eiGn "r n=1
with expansion coefficients ! ! $ iGn "r!
!z tG # %z
! ! $ iGn "r"
%e e tˆn = tG ' ! ! ! ! ' e $ iGn "r# e $ iGn "r! ) 1" !z (even n) & tG # $ z& %z
! caa t0 = # % cba
& ( ( * 1" (odd n) $ z&
cab " ; caa = cbb , cab = cba $ cbb &
Slonczewski-Weiss-McClure model
SWMcC model contains a (strong) threefold lattice anisotropy
Slonczewski-Weiss-McClure model reflects a threefold anisotropy of its lattice Wannier states
Slonczewski-Weiss-McClure model
isotropic two center tight binding model
Model I: q=0 term in K-point basis
Tij # Tij e
! ! ! ! iG ' "! i ' "iG "! j
!1 1" T1 = w # $ 1 1 % &
!z T2 = w # %z
w=
!1 3
1" z $&
!z T3 = w # %z
1" z $&
= 130 meV (TB est. ! 110 meV)
Scaling Relations 8! #" $ %K = sin & ' 3a (2)
q Q= !K
8" !vF #! $ E! = !vF %K = sin & ' 3a (2)
E "= E!
c!=
3ac #! $ 8" "vF sin % & '2(
only !-dependent parameter: strong coupling at small angles
Canonical (isotropic) picture 8! !vF #" $ E/ sin % & 3a '2(
8! #" $ q/ sin % & 3a '2(
van Hove sing.
coupling scale
Interlayer coupling models Three geometrically distinct states conventional, Isotropic
!1 1" w# $ 1 1 % &
!0 1" cab # $ 1 0 % &
anisotropic (-)
! 1 0 " anisotropic (+) caa # $ 0 1 % &
Model II q=0 term in K-point basis #1 1$ #0 1$ % cab & = cab! x" x T1 = w & ' ' (1 1) (1 0)
! 0 1" T2 = cab # $ 0 z % &
!0 1" T3 = cab # $ 0 z % &
Topological Transition c! !x % ! "(#i$) & H =' ( ! ˆ " c ! ! ( # i $ # (1) e ) "K * x ) "I 0 # ! H =$ H % &0 !x '
"I 0 # $0 ! % x' &
" !I ( ) H q c $ % K " H!= & " ' ! x H K (q " #K )! x ) ) ( c!I Compensated (opposite helicities)
Evolution of Dirac points
Decrease angle: increase coupling strength
Spectral reconstruction starts here annihilates here
c!=
1 2
re-emerge here
Pair annihilation for compensated DPs Converge/collapse/diverge with increasing c
Comments: The SWMcC model contains a (large) threefold lattice anisotropy Breakdown of first generation models : Relativistic (linear) bands ! Massive (curved) bands Dirac point annihilation and regeneration. Gauge potential from twisted neighbor
Topics for today: I. Introduction: what, where, why II. Family behavior of low energy physics III. New approach to an old Hamiltonian IV. Compensated class near a topological transition V. Uncompensated new topological state in twisted FLGs
Reverse sign of anisotropy
isotropic
!1 1" w# $ 1 1 % &
!0 1" cab # $ 1 0 % &
anisotropic (-)
! 1 0 " anisotropic (+) caa # $ 0 1 % &
Topological Transition c!I % ! "(#i$) & H =' ( ! ˆ ! " ( # $ + ) c I i e "K * )
! c"I # H K (q ) $ ! & H =% ! H K (q ! "K ) ( ' c"I Uncompensated (same helicity)
Symmetry-protected crossing
Bilayer version of absence of backscattering
Second generation Dirac points uncompensated
new DP
compensated
Velocity renormalization (C) #Hˆ vˆ+ = = vF !ˆ + $ vF (1 " c!2 )!ˆ + #q" #Hˆ vˆ" = = vF !ˆ " $ vF (1 " c!2 )!ˆ " #q+
vF* = vF (1 ! 9c!2 )
perturbative, isotropic reduction
Velocity renormalization (U) #Hˆ vˆ+ = = vF !ˆ + $ vF (!ˆ + " c!2!ˆ " ) #q" #Hˆ vˆ" = = vF !ˆ " $ vF (!ˆ " " c!2!ˆ + ) #q+
vF* = vF (1 " c!4 cos(2! ))
perturbative with twofold anisotropy (vanishes after sum over "K triad)
Signatures of the uncompensated state SYMMETRY-PROTECTED BAND CROSSING SECOND GENERATION DIRAC POINTS: DEGENERATE AT DISCRETE POINTS (NOT LINES)1. NO PERTURBATIVE VELOCITY RENORMALIZATION EXTENDED (STRONGER) VAN HOVE SINGULARITY
These are all properties of the SiC (000 \bar 1) epitaxial graphenes! 1
ARPES pending
Simons Symposium Questions During Talk by Gene Mele 3:55 Das Sarma: Are you going to be addressing this problem just at the single particle level? Mele: Yes. The band structure of these systems is rich and not completely understood. Our interest in studying it is to understand the role of interactions on top of this. 7:30 Altshuler/Halperin/Freedman What are the rotation angles? Are the layers also buckled? Sheared? Mele: In an epitaxial multilayer there are typically a distribution of rotation angles, where the distribution is peaked around magic angles that correspond to low order commensurate rotations. The layers are reasonably flat over a large coherence width. The scattering measurements identify the rotation angle, the layers can also be displaced laterally. 11:48 Altshuler: Could the faulted structure be quasiperiodic Mele: Yes. That occurs for example at a 30 degree rotation which is a crucial angle in the classification of all faulted bilayer structures. For the most part our theory focuses on commensurate rotations where we can say something about the physics. 14:50 Freedman: Do you allow for slip of one layer relative to another. Mele: The construction is to make a rotation away from the Bernal structure holing one pair of eclipsed sites fixed. There is no lateral slip with respect to this fixed rotation axis. 16:19 Aleiner: Do the partners have the same energy, and is it clear which one is preferred. Mele: Undoubtedly the two have different energies. We are going to freeze in these two structures and compare their electronic spectra. 25:18 Altshuler: Is there a Hofstadter-like band structure to bands for incommensurate structures. Mele: Yes. I’m steering your attention away from such a singular point to understand the situation where the structure is commensurate, though possibly with a long period. 27:40 Aleiner: Aren't the interlayer coupling matrices a function of position.
Mele: No, the position dependence has been integrated out. These are the matrix elements for tunneling between Dirac points of the two layers. It is an interlayer mass operator. 29:00 Aleiner: Do these structures have different point group symmetries. Mele: Yes. That’s the essential difference that allows us to classify their low energy Hamiltonians. 38:10 Das Sarma: Gamma_4 isn't very well known. Mele: It is known to be small. The important idea in this model is that gamma_3 and gamma_4 are inequivalent. Gamma_4 is not known nearly as well as gamma_3, but it is known to be a lot smaller than gamma_3. 50:11 Altshuler: Doesn't a Peierls type instability break the degeneracy, gapping the crossing point. Mele: It’s not like the one dimensional problem. There’s very little spectral weight at the crossing point, so one does not generically expect and instability. 51:50 Falko: Couldn't this be gapped by short range variations of the type you discussed in the first part of the talk. Mele: At small angle the physics is controlled by the cell average of the interlayer potential and one should be insensitive to the spatially modulated terms. 53:50 Aleiner: Compare the interactions from the first part of the talk with the discussion in the section.
Mele: For small angles we can focus only on the cell average of the interlayer potential. For large angle rotations the key quantity is the Fourier component of the crystal potential and the momentum that spans the offset Dirac points. 55:00 Halperin: How do these depend on the rotation angle. Mele: The angle itself is the wrong metric since nearby angles can have vastly different periods and different physical properties. There are a hierarchy of commensurations which is the right language for trying to understand these effects. 56:00 Freedman: Could one look for topological transitions in the band structure. Mele: We haven’t done that, but I think it could be useful. 57:00 Levitov: Do the gauge fields produce the analog to magnetic translations. Mele: I don’t think so since there is no net flux through the cell. 59:14 Das Sarma: The literature on states and waves in quasiperiodic lattices should be relevant here. Mele: Yes, I think applications along those lines could be quite useful.
