Tight-Binding Model for Graphene

Franz Utermohlen

September 12, 2018

Contents

1 Introduction 2

2 Tight-binding Hamiltonian 2 2.1 Energy bands ...... 4 2.2 Hamiltonian in terms of Pauli matrices ...... 4 2.3 σz term ...... 5

3 Behavior near the Dirac points 6 3.1 Near K ...... 6 3.2 Near K0 ...... 6 3.3 Linear dispersion relation ...... 7 3.4 σz term gapping ...... 7

1 1 Introduction

The unit cell of graphene’s lattice consists of two diﬀerent types of sites, which we will call A sites and B sites (see Fig. 1).

Figure 1: Honeycomb lattice and its Brillouin zone. Left: lattice structure of graphene, made out of two interpenetrating triangular lattices a1 and a2 are the lattice unit vectors, and δi, i = 1, 2, 3 are the nearest-neighbor vectors. Right: corresponding Brillouin zone. The Dirac cones are located at the K and K’ points. Note: This ﬁgure and caption are from Castro et al. [2].

These vectors are given by a √ a = 3, 3 1 2 a √ a = 3, − 3 2 2 a √ δ = 1, 3 1 2 a √ δ = 1, − 3 2 2 δ3 = −a 1, 0 2π √ K = √ 3, 1 3 3a 2π √ K0 = √ 3, −1 , (1) 3 3a where K, K0 are the corners of graphene’s ﬁrst Brillouin zone, or Dirac points.

2 Tight-binding Hamiltonian

Considering only nearest-neighbor hopping, the tight-binding Hamiltonian for graphene is ˆ X †ˆ ˆ† H = −t (ˆai bj + bjaˆi) , (2) hiji

2 † where i (j) labels sites in sublattice A (B), the fermionic operatora ˆi (ˆai) creates (annihilates) ˆ† ˆ an electron at the A site whose position is ri, and similarly for bj, bj. We can rewrite the sum over nearest neighbors as

X †ˆ ˆ† X X †ˆ ˆ† (âi bj + bjaî) = (âi bi+δ + bi+δaî) , (3) hiji i∈A δ where the sum over δ is carried out over the nearest-neighbor vectors δ1, δ2, and δ3, and the ˆ operator bi+δ annihilates a fermion at the B site whose position is ri + δ. Using 1 † X ik·ri † aî = p e aˆk , (4) N/2 k

ˆ† where N/2 is the number of A sites, and similarly for bi+δ, we can write the tight-binding Hamiltonian for graphene (Eq. 2) as

t 0 0 X X i(k−k )·ri −ik ·δ † Hˆ = − [e e aˆ ˆb 0 + H.c.] N/2 k k i∈A δ,k,k0 X −ik·δ † ˆ = −t (e aˆkbk + H.c.) δ,k X −ik·δ † ˆ ik·δˆ† = −t (e aˆkbk + e bkaˆk) , (5) δ,k where in the second line we have used

0 N X i(k−k )·ri e = δ 0 . (6) 2 kk i∈A We can therefore express the Hamiltonian as X Hˆ = Ψ†h(k)Ψ , (7) k where aˆk † † ˆ† Ψ ≡ ˆ , Ψ = aˆk bk , (8) bk and 0 ∆k h(k) ≡ −t ∗ (9) ∆k 0 is the matrix representation of the Hamiltonian and

X ik·δ ∆k ≡ e . (10) δ

3 2.1 Energy bands

p ∗ The eigenvalues of this matrix are E± = ±t ∆k∆k. We can compute this by writing ∆k out more explicitly:

ik·δ1 ik·δ2 ik·δ3 ∆k = e + e + e = eik·δ3 [1 + eik·(δ1−δ3) + eik·(δ2−δ3)] h √ √ i = e−ikxa 1 + ei3kxa/2ei 3kya/2 + ei3kxa/2e−i 3kya/2 h √ √ i = e−ikxa 1 + ei3kxa/2(ei 3kya/2 + e−i 3kya/2) " √ # 3 = e−ikxa 1 + 2ei3kxa/2 cos k a . (11) 2 y The energy bands are therefore given by s √ √ 3 3 3 E (k) = ±t 1 + 4 cos k a cos k a + 4 cos2 k a , (12) ± 2 x 2 y 2 y or, as it is sometimes written, p E±(k) = ±t 3 + f(k) , (13) where √ √ 3 3 f(k) = 2 cos( 3k a) + 4 cos k a cos k a . (14) y 2 x 2 y These are two gapless bands that touch at the Dirac points K and K0 (see Fig. 2). In other words, the Dirac points are the points in k-space for which E±(k) = 0.

Figure 2: Energy bands for graphene from nearest-neighbor interactions. The bands meet at the Dirac points, at which the energy is zero.

2.2 Hamiltonian in terms of Pauli matrices

We can express the Hamiltonian 0 ∆k h(k) = −t ∗ (15) ∆k 0

4 in terms of Pauli matrices by expressing ∆k as

X ik·δ X ∆k = e = [cos(k · δ) + i sin(k · δ)] , (16) δ δ so the Hamiltonian reads X 0 cos(k · δ) + i sin(k · δ) h(k) = −t . (17) cos(k · δ) − i sin(k · δ) 0 δ In terms of Pauli matrices, the Hamiltonian is then X h(k) = −t [cos(k · δ)σx − sin(k · δ)σy] . (18) δ

2.3 σz term

Writing the Hamiltonian in the form X h(k) = −t [cos(k · δ)σx − sin(k · δ)σy] (19) δ it is natural to ask whether we can have also have a σz term. Let’s see what happens when we do: X hM (k,M) = −t [cos(k · δ)σx − sin(k · δ)σy + Mσz] δ X M cos(k · δ) + i sin(k · δ) = −t . (20) cos(k · δ) − i sin(k · δ) −M δ

From the matrix form of the Hamiltonian we can see that this additional σz term raises the energy of A sites and lowers the energy of B sites, thereby gapping the bands. We can think of this as being caused by on-site potential terms of the form ! ˆ X X Hpotential = M nˆa,α − nˆb,β , (21) α β {A sites} {B sites}

† ˆ† ˆ wheren ˆa,α =a ˆαaˆα andn ˆb,β = bβbβ. This could arise when there are two diﬀerent types of atoms on the A and B sites, such as in boron nitride, which has the same lattice structure as graphene, but whose A and B sites correspond to boron atoms and nitrogen atoms.

5 3 Behavior near the Dirac points

3.1 Near K

Let’s look at the behavior of ∆k about the Dirac point K. Deﬁning the relative momentum q ≡ k − K, we can write ∆k in terms of q as " √ # 3(K + q )a ∆ = e−iKxae−iqxa 1 + 2ei3(Kx+qx)a/2 cos y y K+q 2 " √ # π 3a = e−iKxae−iqxa 1 − 2e3iaqx/2 cos + q . (22) 3 2 y

Now, expanding this about q = 0 to ﬁrst order, we have 3a ∆ = −ie−iKxa (q + iq ) . (23) K+q 2 x y

The phase of ∆K+q carries no physical signiﬁcance (since, for example, the energy bands are p ∗ −iKxa given by E±(K + q) = ±t ∆K+q∆K+q ), so it is convenient to ignore the phase of ie . We thus have 3a ∆ = − (q + iq ) . (24) K+q 2 x y About the Dirac point K, the Hamiltonian is thus 0 qx + iqy h(K + q) = vF , (25) qx − iqy 0 where 3at v = (26) F 2 is the Fermi velocity. We can express this in terms of Pauli matrices as

h(K + q) = vF (qxσx − qyσy) , (27) or, deﬁning the vector q q¯ ≡ x , (28) −qy we can express h(K + q) more naturally as

h(K + q) = vF q¯ · σ . (29)

3.2 Near K0

0 Similarly, deﬁning the relative momentum q ≡ k − K and expanding ∆k about the Dirac point K0, we ﬁnd 3a ∆ 0 = − (q − iq ) , (30) K +q 2 x y 6 so the Hamiltonian about this Dirac point is 0 0 qx − iqy h(K + q) = vF = vF (qxσx + qyσy) = vF q · σ . (31) qx + iqy 0

3.3 Linear dispersion relation

From the matrix form of the Hamiltonian near the Dirac points (Eqs. 25, 31) we ﬁnd that the energy bands near the Dirac points are given by

E±(q) = vF |q| . (32)

Near a Dirac point, the dispersion relation is therefore linear in momentum, so the energy bands form cone (called a Dirac cone; see Fig. 3) with the vertex lying on the Dirac point.

Figure 3: Conical behavior of the bands near a Dirac point, known as a Dirac cone, where the energy is linear in momentum.

3.4 σz term gapping

If we add the σz term that we introduced in Section 2.3, the Hamiltonian near the Dirac points will be of the form M qx − iqy hM = vF (qxσx + qyσy + Mσz) = vF . (33) qx + iqy −M

As we saw, this additional term gaps the energies, so that near the Dirac points, the energies become q 2 2 2 EM,± = ± qx + qy + M , (34) See Fig. 4 for a plot of the band gap near a Dirac point.

7 Figure 4: Energy gap near a Dirac point produced by a σz term. Compare to Fig. 3.

References

[1] Anthony J. Leggett. Lecture 5: Graphene: Electronic band structure and Dirac fermions. http://web.physics.ucsb.edu/~phys123B/w2015/leggett-lecture.pdf, 2010. [2] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim. The electronic properties of graphene. https://journals.aps.org/rmp/abstract/10. 1103/RevModPhys.81.109, 2009.