CHAPTER 1
Hartree-Fock Hamiltonians for the chiral 2DEGs in graphene and bilayer graphene
1.1 Basic notions
Lattice structure: honeycomb lattice of C atoms (triangular lattice with 2 C per unit cell: A and B) •
The crystal structure of graphene.
Primitive vectors: •
1 √3 a1 = (1.1) Ã2 − 2 !
a2 = (1 0) (1.2)
with = √3 where =142 Å is the spacing between carbons atoms. 2 1. Hartree-Fock Hamiltonians for the chiral 2DEGs in graphene and bilayer graphene
Nearest-neighbors: • 1 1 δ1 = (1.3) 2 2√3 µ ¶ 1 1 δ2 = (1.4) −2 2√3 µ ¶ 1 δ3 = 0 (1.5) −√3 µ ¶
sp2 hybridization, and bonds •
Reciprocal lattice: • y
b2 x
b1
Reciprocal lattice of graphene. §1.1 Basic notions 3
Valleys: •
2 2 K = 0 (1.6) 0 3 µ ¶ 2 2 K = 0 (1.7) −3 µ ¶
y K’ K
K K’ x
K’ K First Brillouin zone of graphene.
Band structure: eith 2 C’s per unit cell, we have 8 bandes: 6 bands and 2 bands: •
Fermi level: valence of C is 4: 3 and 1 bands are occupied in undoped graphene. • 4 1. Hartree-Fock Hamiltonians for the chiral 2DEGs in graphene and bilayer graphene
1.2 Tight-binding hamiltonian for the bands
Hamiltonian for the bands: • = † + (1.8) − hXi ³ ´ with 28 eV and 0 01 eV, 00 007 eV. ≈ ≈ ≈ Fourier transform to k and k operators: • 1 k R = · (1.9) √ k k X 1 k R = · (1.10) √ k k X with k† = k† = kk (1.11) k0 k0 0 Hamiltonian: n o n o • 0 Λ (k) k = k† k† (1.12) Λ∗ (k)0 k k µ ¶µ ¶ X ¡ ¢ where k Λ (k)= · (1.13) − X The TB Hamiltonian is diagonalized to give:
= (k) † kk + (k) † kk (1.14) k k X X Band structure: • √3 (k)=+Λ (k) Λ (k)=+ 1+4cos2 +4cos cos (1.15) ∗ v 2 2 2 u µ ¶ µ ¶ Ã ! p u t √ 2 3 (k)= Λ (k) Λ (k)= 1+4cos +4cos cos (1.16) − ∗ − v 2 2 2 u µ ¶ µ ¶ Ã ! p u t §1.3 Continuum approximation 5
1.3 Continuum approximation
We consider that the doping is small so that the Fermi level is near energy =0and we can use the band structure around the valley points Expandingtolinearorderin we get the linear dispersion: ±
(p)= } p (1.17) ± | | where the Fermi velocity is 3 6 = 1 10 m/s (1.18) 2} ' × For the Hamiltonians (basis )
0 − (p)= = σ p (1.19) − 0 − · µ ¶ 0 0 (p)=+ =+ σ∗ p (1.20) − 0 · µ ¶ where: tan = (1.21)
The eigenspinors give the wave function on site and For the and 0 valleys and the conduction (C) and valence (B) bands:
1 (p)2 1 (p)2 (p)= − (p)= − (1.22) √2 (p)2 √2 (p)2 µ − ¶ µ ¶ (p)2 (p)2 0 1 0 1 (p)= (p)2 (p)= (p)2 (1.23) √2 − √2 − µ ¶ µ − ¶
If we use the basis ( ) for and () for 0 then: ( =+1for and = 1 for 0) − 0 − (p)= = σ p (1.24) − 0 − · µ ¶ andwehaveforthesublattice spinors
1 (p)2 (p)= − (1.25) √2 (p)2 µ − ¶ 1 (p)2 (p)= − (1.26) √2 (p)2 µ ¶
1.4 Helicity
A general spin 12 spinor is given by
2 +2 + =cos − + +sin (1.27) | iu 2 | i 2 |−i ³ ´ 2 ³´ +2 =sin − + cos (1.28) |−iu 2 | i − 2 |−i ³ ´ ³ ´ For = 2:
1 2 +2 + = − + + (1.29) | iu √2 | i |−i h i 1 2 +2 = − + (1.30) |−iu √2 | i − |−i h i 6 1. Hartree-Fock Hamiltonians for the chiral 2DEGs in graphene and bilayer graphene
With the correspondence + (1.31) | i → | i (1.32) | i → |−i we have (p) = (p) = + (1.33) |−iu | iu ¯ E ¯ E ¯0 (p) = + ¯0 (p) = (1.34) ¯ | iu ¯ |−iu ¯ E ¯ E where ¯ ¯ ¯ u =p¯ (1.35)
The sublattice spinor is in (or opposite) the direction of the electronic momentum. For 0 : b
The helicity is defined as the projection of the pseudospin onto the direction of propagation p = σ (1.36) · p | | In graphene (p)= σ p = p (1.37) − · − | | and so (p) = = 1 (c.b.) and +1(v.b.) (1.38) p − − | | (p) = 0 =+1(c.b.) and 1 (v.b.) (1.39) 0 p − | | (in this picture, and 0 are inverted w.r.t. my definitions) §1.4 Helicity 7
The chirality is equivalent to the helicity only for masless particle. •
The eigenstates of the Dirac Hamiltonian are also eigenstates of the chirality operator and so chirality • is a good quantum number. It is conserved. Chirality is a quantum number that is conserved in elastic scattering processes induced by impurity potentials = (r) ( is the unit matrix) that vary smoothly on the lattice scale. This type of potential does not permit inter-valley scattering and so is fixed and = is conserved. This effect gives rise to the absence of backscattering in graphene and is at the origin of the Klein tunneling (perfect transmission through a high potential barrier at normal incidence).
Note that chirality is a good number only in the vicinity of the Dirac points. If we include higher-order • corrections, it is no longer conserved.
The rotation operator by an angle about an axis u for a spin 12 particle is given by •
u u ()=− 2 · =cos σ u sin (1.40) 2 − · 2 ³ ´ ³ ´
We see that a rotation in the spin space by =2 gives a phase factor of This means, in our case, that if the particule circles around the Dirac cone, the vector turns by 2 and so does the sublattice pseudospin. The eigenstate thus acquires a 1 sign. −
The chiral nature of low-energy electrons in graphene places an additional constraint on their scatter- • ing properties. If a given potential doesn’t break the A-B symmetry, then it is unable to influence the pseudospin degree of freedom which must, therefore, be conserved upon scattering. Considering only the pseudospin part of the chiral wave function , the probability to scatter in a direction =0, ± 2 where =0is the forwards direction, is proportional to ()= () (0) .Formonolayer graphene, ()=cos2 (2). This is anisotropic, and displays an absence± of| backscattering± ()=0]: ¯ ®¯ scattering into a state with opposite momentum is prohibited because¯ it requires¯ a reversal of the pseudospin. Such conservation of pseudospin is at the heart of anisotropic scattering at potential bar- riers in graphene monolayers, known as Klein tunneling. 8 1. Hartree-Fock Hamiltonians for the chiral 2DEGs in graphene and bilayer graphene §1.5 Family of chiral 2DEGs 9
(From P.E. Allain and J. N. Fuchs: Klein tunneling in graphene, Eur. Phys. J. B. 83, 201 (2011)). (Note : is the band index in this text.)
1.5 Family of chiral 2DEGs
In graphene (monolayer) (p) can be written as • () = } [cos () +sin() ] (1.41) − µ ¶ with =1where is the chirality index. () ∼
In Bernal-stacked bilayer graphene, a low-energy Hamiltonian (1) can be constructed where • (2) 2 21 are the low-energy sites. The minimal Hamiltonian is given by i.e. =2and () ∼
1 1 1 ⎛ ⎞ (1.42) 2 → 2 µ ¶ ⎜ 2 ⎟ ⎜ ⎟ ⎝ ⎠ 10 1. Hartree-Fock Hamiltonians for the chiral 2DEGs in graphene and bilayer graphene
In ABC-stacked trilayer graphene, the effective two-band model involves the low-energy sites 13 • and =3i.e. () 3 ∼ 1 1 ⎛ ⎞ 2 1 (1.43) ⎜ 2 ⎟ → 3 ⎜ ⎟ µ ¶ ⎜ 3 ⎟ ⎜ ⎟ ⎜ 3 ⎟ ⎜ ⎟ ⎝ ⎠ §1.6 Example of the importance of the spinor structure 11
1.6 Example of the importance of the spinor structure
In second-quantization, we define the field operators as
(p)2 1 p r − Ψ (r)= · (1.44) √ (p)2 p 2 p X X µ − ¶ The spinor structure modifies properties of the C2DEG wrt to the ordinary 2DEG. For example:
Non-chiral 2DEG: retarded density response function is given by •
1 (k q) (k) (q)=2 − − (1.45) + ( (k) (k q)) } k } X − − − Chiral 2DEG: • 1 (q)= (1 + 0 cos ((k k q))) (1.46) − − } k X0 X (k q) ( k) − − 0 × + ( (k) (k q)) } − 0 − − where = 1 for the conduction and valence bands. This affects the dielectric function ± 22 (q)=1 (q) (1.47) −
Similar form factors appear in the conductivity response functions (and so in the calculation of the • optical properties).
1.7 C2DEGs in a magnetic field
Peierls substitution p P = p + A} (1.48) → 12 1. Hartree-Fock Hamiltonians for the chiral 2DEGs in graphene and bilayer graphene
where A = B =z with 0.Weget ∇ ×
b 0 = − (1.49) − + 0 µ ¶ with the commutation relation }2 = (1.50) − 2 where 2 = } (1.51) is the magnetic length.
1.8 Landau levels and eigenfunctions
Ladder operators
= ( ) (1.52) √2} − † = ( + ) (1.53) √2} obeying the commutation relation
† =1 (1.54)
The Hamiltonian is now: £ ¤ √2} 0 = (1.55) − † 0 µ ¶ Landau gauge A = y Solutions are
√2 b () = () } (1.56) | | p with §1.8 Landau levels and eigenfunctions 13
() 0 0 (r)= (1.57) 0 (r) µ ¶ and () 1 () 1 (r) (r)= | |− (1.58) √2 (r) µ | | ¶ The functions 1 0 (r)= − 0 () (1.59) and p1 (r)= † | | 0 (r) (1.60) | | ! | | ¡ ¢ 2 are the usual Landau gauge wave functions for anp ordinary 2DEG with ()= the wave functions of the harmonic oscillator in 1D. − ¡ ¢ For the 0 valley (0) √2} = () (1.61) | | For =0 p (0) (r) (r)= 0 (1.62) 0 0 µ ¶ and (0) 1 (r) (r)= | | (1.63) √2 () 1 (r) µ − | |− ¶ For =0 valley and sublattice indices are equivalent. For graphene (monolayer) • √2} 0 = (1.64) − † 0 µ ¶ Note that there is a Landau level =0with energy =0: √2 0 0 0 } =0 (1.65) − † 0 0 (r) 0 (r) µ ¶µ ¶ µ ¶ (r) =0has degeneracy 2(spin) X 2 (valleys) X =4 .Thespinorfor is 0 valley and 0 0 sublattice indices are equivalent. µ ¶ 14 1. Hartree-Fock Hamiltonians for the chiral 2DEGs in graphene and bilayer graphene
For AB-bilayer graphene • 2 2 0 0 2 (1.66) ∼ 0 1 µ † ¶ ¡ ¢ For the =0Landau level:
0 2 0 0 2 0 2 =0; 2 =0 (1.67) 0 0 (r) 0 1 (r) µ † ¶µ ¶ µ † ¶µ ¶ ¡ ¢ ¡ ¢ The two "orbitals" 0 and 1 are degenerate with =0 The degeneracy is now 2 (spin) X 2 (valleys) (r) (r) X2(orbitals) =8 For the spinors are 1 2 and so valley and layer 0 0 0 indices are equivalent. µ ¶ µ ¶
For ABC-trilayer graphene • 3 3 0 0 = 3 (1.68) 2 0 1 µ † ¶ ¡ ¢ The =0spinors are for
0 0 0 ; ; (1.69) 0 (r) 1 (r) 2 (r) µ ¶ µ ¶ µ ¶
and the reversed for 0 so that valley and layer are equivalent. The three "orbitals" 0, 1 and 2 are degenerate with =0 The degeneracy is now 2 (spin) X 2 (valleys) X 3 (orbitals) =12
etc. for ABCA, ABCBA,.. • §1.9 Quantum Hall effects 15
1.9 Quantum Hall effects
1.10 Energies
In-plane hopping between nearest-neighbors: • = 0 =28 eV, (1.70)
Magnetic length: • 256 = (1.71) (Tesla) Gap between =1and =0: p • 2 1 =3 294 7 10− √ eV (1.72) × = 382 33√ K
Coulomb energy: • 2 2 5 624 9 10− √ = × eV (1.73) 652 74 = √ K Zeeman energy: • 4 =1 157 7 10− eV (1.74) × 13434 K 16 1. Hartree-Fock Hamiltonians for the chiral 2DEGs in graphene and bilayer graphene
1.11 Field operators
Here we use the same basis ( ) for the two valleys:
K r Ψ (r)= − · r ; (1.75) h | i X 1 K r 0 = − · − 2 0 0 µ − ¶ X 2 p 1 K r ¡ (¢) 1 + − · − | |− 2− 2 =0 µ | | − ¡ ¢ ¶ X6 p ¡ ¢ and
K0 r Ψ (r)= − · r 0; (1.76) 0 h | i 0 X 2 1 K0 r 0 = − · − − 0 0 0 µ ¡ ¢ ¶ X 2 p 1 K0 r + − · − | | − 2 0 2 () 1 =0 µ − ¡| |− ¢− ¶ X6 p ¡ ¢
1.12 Spinor structure and selection rules
The spinor structure plays again an important role in the calculation of the optical conductivity. For example
Non-chiral 2DEG: • 1 1 − 2 2 1 1 ←→ ()= ←→1 ⎡ µ − ¶ µ ¶ ⎤ (1.77) ( + ) − 2 ( + ) ( + )+1− ( + ) 1 ⎢ − ⎥ ⎢ ⎥ ⎣ ⎦ Absorption is at only.
Chiral 2DEG: • 222 (0) (0) ()= Λ() Λ() − (1.78) 2 ( + ) + ( ) } ® }® X − − with
() Λ = () 1 () 1 (1.79) | |− | | − | |− | | () Λ = £ () 1 + () 1 ¤ (1.80) | |− | | | |− | | Selection rule is given by: 1= (1.81) | |± | | For ex.: transitions 2 3 2 1 2 1 2 3 2 3 2 1 → → − →− − →− − → − → §1.13 Second-quantized Hamiltonian 17
1.13 Second-quantized Hamiltonian
If we put this expression in the first-quantized hamiltonian, we get the second quantized form
= rΨ† (r) Ψ† (r) (1.82) X Z √2} 0 † † = r r ; r ; 00 00 + 0 − h | i † 0 h | i X X0 0 ∙Z µ ¶ ¸ () = † X For the Coulomb interaction 1 = r r0Ψ† (r) Ψ† (r0) (r r0) Ψ (r0) Ψ (r) (1.83) 2 1 2 − 3 4 1X··· 4 X Z Z
(K K0) r Terms that do not conserve the valley index involves rapidly oscillating integrands − · These terms are very small and usually neglected. Thus:
1 = r r0Ψ† (r) Ψ† (r0) (r r0) Ψ (r0) Ψ (r) (1.84) 2 1 2 − 2 1 X1 2 X Z Z 18 1. Hartree-Fock Hamiltonians for the chiral 2DEGs in graphene and bilayer graphene
Now, write 1 22 q (r r0) (r r )= · − (1.85) 0 − q X so that with 22 (q)= (1.86) we have 1 q (r r0) = (q) r r0Ψ† (r) Ψ† (r0) · − Ψ (r0) Ψ (r) (1.87) 2 1 2 2 1 q X X1 2 Z Z which gives:
1 = (q) (1.88) 2 q X 1X 4 1X 4 X1 2 q r r ; 11 r · r ; 44 × h 1 | i h | 1 i Z q r0 r0 ; 22 r − · r ; 33 × h 2 | i h | 2 i Z † † × 111 222 2 3 3 1 4 4 The Coulomb interaction will depend on the matrix elements:
2 q r 2 (+0) r ; r · r ; 00 Ξ (q) (1.89) h | i h | i ≡ 0 0− Z
1.14 Hartree-Fock approximation (monolayer graphene)
We approximate the Coulomb interaction by the Hartree-Fock pairings:
Ψ† (r) Ψ† (r0) Ψ (r0) Ψ (r) 2 Ψ† (r) Ψ (r) Ψ† (r0) Ψ (r0) (1.90) 1 2 2 1 → 1 1 2 2 D E 2 Ψ† (r) Ψ (r0) Ψ† (r0) Ψ (r) − 1 2 2 1 D E After some algebra ... ( = 22 is the Landau level degeneracy) and if we ignore Landau level mixing, we get for the electrons in Landau level :
; = (q =0) (1.91) X 2 ; 0;0 + (q) (q) ( q) − q µ ¶ X X0 X 2 ® ;0 0; (q) (q) ( q) − − q µ ¶ X X0 X D E where ; 1 + 2 0 2 ( 0) † (q) − 0+ 00 (1.92) ≡ X0 The average values of these operators are the order parameters for the different broken-symmetry phases of the C2DEG. (q) = populations (1.93) h = i 6 (q) = coherences (1.94) ® §1.14 Hartree-Fock approximation (monolayer graphene) 19
Exemples of coherence •
0 = valley coherence, (1.95) D E ↑ ↓ = spin coherence, (1.96) =0=1 ® = orbital coherence (bilayer and trilayer) (1.97) The number of possible broken-symmetry® states is large (uniform states, CDW states, SDW, PSDW, crystals, etc.)
TheHartreandFockinteractionsaredefined by 1 (q)= Ξ (q) Ξ ( q) − µ ¶ and 1 2 (q)= (p) Ξ (p) Ξ ( p) p q (1.98) 2 × p − µ ¶ X with 2 2 2 2 1 22 22 − 4 0 0 − 4 Ξ (q)= Θ ( ) + 1 + 0 (1.99) 2 | | | | 2 | |− 2 ∙ µ ¶ µ ¶¸ The Hartree-Fock energy of the C2DEG in a magnetic field is: 1 = (1.100) X 2 1 ; 0;0 + (q) (q) ( q) 2 − µ ¶ 0 q X X X ® D E 1 2 2 (q) ;0 (q) 2 − q µ ¶ 0 ¯D E¯ X X X ¯ ¯ ¯ ¯ Interactions () () for =1 2 3 4 (blue: normal 2DEG, violet C2DEG) −
and in =0are identical to the normal 2DEG interactions • The Fock and Hartree interactions are indepenent of spin and valley indices • In this form, we can study uniform and well as non-uniform states of the C2DEG. • For a uniform state, • ; (q = 0) =0 (1.101) 6 and ® (0)=0 (1.102) (infinite terms are cancelled by the positive background). Also
2 ;0 (0) = (1.103) 0 ¯D E¯ X X ¯ ¯ Thus ¯ ¯ 1 1 2 = (0) (1.104) − 2 X µ ¶ The Fock term is constant. It follows that the ground states for =1 3 are spin polarized while the ground states for =2 4 are spin unpolarized. 20 1. Hartree-Fock Hamiltonians for the chiral 2DEGs in graphene and bilayer graphene
FIGURE 1.1.
For the spin polarized state, the energy is independent of how the electrons are distributed in the • two valleys. If we define a valley pseudospin, then at =1: = 1 2 + Υ (q) ( q) (q) (1.105) 4 q h − ih i µ ¶ X 1 2 + (q)[ S ( q) S (q) ] q h − i·h i µ ¶ X with (q)= (q) (1.106) − and Υ (q)=2 (q) (q) (1.107) − For the liquid state
2 2 1 2 = (0) + (0) S (0) (1.108) − 4 |h i| µ ¶ µ ¶
and so, because (0) 0, the minimal energy is obtained when S (0) is maximized in any direc- tion. The ground state is pseudo-ferromagnetic. WethenexpectaGoldstonemodewithaquadratic| | dispersion relation. (From R. Côté, J. -F. Jobidon, and H. A. Fertig, PRB 78, 085309 (2008).): • §1.14 Hartree-Fock approximation (monolayer graphene) 21
FIGURE 1.2. 22 1. Hartree-Fock Hamiltonians for the chiral 2DEGs in graphene and bilayer graphene
FIGURE 1.3.