graphene and planar dirac equation - usalcampus.usal.es/~mpg/general/grafeno_dirac_2d2016.pdf ·...

Graphene and Planar Dirac Equation

Marina de la Torre Mayado

2016

Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 1 / 48

Outline

1 Introduction

2 The Dirac ModelTight-binding modelLow energy effective theory near the Dirac points

3 Modifications of the Dirac model

4 References


Graphene• Graphene is a one-atom thick layer of carbon atoms:

• It was isolated for the first time by A. K. Geim and K. S. Novoselov in 2004. Theywere awarded with the Nobel Prize in Physics in 2010:

“For groundbreaking experiments regarding the two-dimensional material graphene"


Graphene

• Graphene has many exceptional properties and it is one of the most interestingtopics in condensed matter physics.

• The principal features of graphene are:

1 The quasi-particle excitations satisfy the Dirac equation (Weyl).2 The speed of light c has to be replaced by the so-called Fermi velocity:

vF 'c

300' 106 m

s

• Therefore, the quantum field theory methods are very useful in the physics ofgraphene.


The Dirac Model

• The Dirac model for quasi-particles in graphene was elaborated around 1984:

D. P. DiVincenzo and E. J. Mele, Phys. Rev. B 29, 1685 (1984).G. W. Semenoff, Phys. Rev. Lett. 53, 2449 (1984).

• Its basic properties (linearity of the spectrum) were well known in 1947:

P. R. Wallace, Phys. Rev. 71, 622 (1947).

• For more details see the complete review:

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov and A. K. Geim, Rev.Mod. Phys. 81 , 109-162 (2009).


The Dirac Model• A Bravais lattice is an infinite arrangement of points (or atoms) in space that has thefollowing property:

The lattice looks exactly the same when viewed from any lattice point

A 1D Bravais lattice:



The Dirac Model


A Bravais lattice has the following property, the position vector of all points (oratoms) in the lattice can be written as follows:

1D: R = n a1, n ∈ Z.2D: R = n a1 + m a2, n,m ∈ Z.3D: R = n a1 + m a2 + p a3, n,m, p ∈ Z.

Where ai are called the primitive lattice vectors.


The Dirac Model

The honeycomb lattice is not a Bravais lattice:


The Dirac ModelBut the honeycomb lattice can be considered a Bravais lattice with a two-atom basis:

• A primitive cell of a Bravais lattice is the smallest region which when translated byall different lattice vectors can cover the entire lattice without overlapping. Theprimitive cell is not unique.

• The Wigner-Seitz (WS) primitive cell of a Bravais lattice is a special kind of aprimitive cell and consists of region in space around a lattice point that consists of allpoints in space that are closer to this lattice point than to any other lattice point. TheWigner-Seitz primitive cell is unique


The Dirac Model• I can take the blue atoms to be the points of the underlying Bravais lattice that has atwo-atom basis (blue and red) with basis vectors:

d1 = 0 , d2 = d i

However red and blue color coding is only for illustrative purposes: All atoms are the same.

Also, I can take the small black points to be the underlying Bravais lattice that has a two atombasis (blue and red) with basis vectors:

d1 =d2

i , d2 = −d2

i


Outline

1 Introduction



4 References


Tight-binding modelThen, in graphene the carbon atoms form a honeycomb lattice with two triangularsublattices A and B:

Where the lattice spacing is d = 1.42◦A.


Tight-binding modelIn this structure, each carbon atom has six electrons:Two electrons filling the inner shell 1s, three electrons engaged in the 3 in-planecovalent bons in the sp2 configuration, and a single electron occupying the pz orbitalperpendicular to the plane.

Hybridization sp2 in graphene


Tight-binding modelThe three quantum-mechanical states are given by

|sp21〉 =

1√3|2s〉 −

√23|2py〉

|sp22〉 =

1√3|2s〉+

√23

(√3

2|2px〉+

12|2py〉

)

|sp23〉 = − 1√

3|2s〉+

√23

(−√

32|2px〉+

12|2py〉

)


Tight-binding model• The nearest neighbors of an atom from the sublattice A belong to the sublattice B.The three vectors δ1, δ2 and δ3 relate A with the three nearest neighbors B:

δ1 = d(−1, 0)

δ2 = d

(12,

√3

2

), δ3 = d

(12,−√

32

)

• In the tight binding model only the interaction between electrons belonging to thenearest neighbors is taken into account. Thus, the Hamiltonian reads:

H = −t∑α∈A

3∑j=1

(a†(rα)b(rα + δj) + b†(rα + δj)a(rα)

)where t is the hopping parameter t ≈ 2.8 eV.


Tight-binding model• The operators a†(rα) and a(rα), b†(rβ) and b(rβ) are creation and annihilationoperators of the electrons in the sublattices A and B, respectively.

•We adopt natural units ~ = c = 1 and consider the usual anti-commutation relations:

{a(rα), a†(rβ)} = δα,β = {b(rα), b†(rβ)}{a(rα), a(rβ)} = {a†(rα), a†(rβ)} = 0{b(rα), b(rβ)} = {b†(rα), b†(rβ)} = 0{a(rα), b(rβ)} = {a†(rα), b†(rβ)} = 0 · · ·

• Example: we consider an atom of type A in the position r0 = (0, 0) and then

H0 = −t3∑

j=1

(a†(r0)b(r0 + δj) + b†(r0 + δj)a(r0)

)


Tight-binding model• In this model the possible states for the electron are:

{|1r0〉, |1r0+δ1〉, |1r0+δ2〉, |1r0+δ3〉}

So that,H|1r0〉 = −t|1r0+δ1〉 − t|1r0+δ2〉 − t|1r0+δ3〉

H|1r0+δj〉 = −t|1r0〉

and then

〈1r0+δ1 |H0|1r0〉 = −t

〈1r0+δ2 |H0|1r0〉 = −t

〈1r0+δ3 |H0|1r0〉 = −t


Tight-binding model• Let us represent the wave function through a Fourier transform:

|ψ〉 =

ψA(k)∑α∈A

eikrαa†(rα) + ψB(k)∑β∈B

eikrβ b†(rβ)

|0〉• The sublattice A is generated by shifts along the vectors δ2 − δ1 and δ3 − δ1:

where a1 and a2 are the basis vectors of the Bravais lattice

a1 = δ2 − δ1 = d

(32,

√3

2

), a2 = δ3 − δ1 = d

(32,−√

32

)Marina de la Torre Mayado Graphene and Planar Dirac Equation June 2016 18 / 48

Tight-binding model• Therefore two momenta k1 and k2 are equivalent if

[n1(δ2 − δ1) + n2(δ3 − δ1)] · (k1 − k2) ∈ 2πZ

for all integers n1 and n2.

• Representatives of the equivalence classes can be taken in a compact region in themomentum space, the Brillouin zone, which is an hexagon with the corners:

v1 =2π3d

(1,

1√3

), v2 =

2π3d

(1,− 1√

3

), v3 =

2π3d

(0,− 2√

3

)v4 =

2π3d

(−1,− 1√

3

), v5 =

2π3d

(−1,

1√3

), v6 =

2π3d

(0,

2√3

)

• The reciprocal lattice vectors are given by: bi · aj = 2πδij, i, j = 1, 2;

b1 =2π3d

(1,√

3)

, b2 =2π3d

(1,−√

3)


Tight-binding model• The reciprocal lattice:

The opposite sides of this hexagon has to be identified and the corners v1, v3 and v5are equivalent, as well as v2, v4 and v6.

•We calculate H|ψ〉:

H|ψ〉 = −t

ψB(k)

3∑j=1

eikδj∑α∈A

eikrαa†(rα) + ψA(k)

3∑j=1

e−ikδj∑β∈B

eikrβ b†(rβ)

|0〉


Tight-binding model• The stationary Schödinger equation H|ψ〉 = E|ψ〉 becomes the matrix equation:(

0 −tX−tX∗ 0

)(ψA

ψB

)= E

(ψA

ψB

)where

X =

3∑j=1

eikδj


Tight-binding model• If E 6= 0 the eigenvalues are:

E = ±t|X|

with the eigenfunctions: {−tXψB = EψA (1)−tX∗ψA = EψB (2)

1 Solution of Type 1 (ψB = 1):{ψ1+ =

(− X|X|1

), ψ2− =

( X|X|1

)}2 Solution of Type 2 (ψA = 1):{

ψ3+ =

(1− X∗|X|

), ψ4− =

(1

X∗|X|

)}


Tight-binding model• Let us write X = |X|eiφ, the Hamiltonian is then

H =

(0 −t|X|eiφ

−t|X|e−iφ 0

)The eigenfunctions can be taken,

Positive energies E = t|X| > 0:

ψ+ = − 1√2

e−i φ2 ψ1+ +1√2

ei φ2 ψ3+ ≡1√2

(ei φ2

−e−i φ2

)

Negative energies E = −t|X| < 0:

ψ− =1√2

e−i φ2 ψ2− +1√2

ei φ2 ψ4− ≡1√2

(ei φ2

e−i φ2

)


Tight-binding model

• These eigenfunctions satisfy:

S · n̂ ψ+ = −~2ψ+ , S · n̂ ψ− =

~2ψ−

where in the case of two dimensions

S · n̂ =~2

(σ1 cosφ+ σ2 sinφ)

=~2

(0 cosφ− i sinφ

cosφ+ i sinφ 0

)This is the pseudo spin index refers to the sublattice degree of freedom.


Tight-binding model• The spectrum is symmetric:

|X|2 = (

3∑j=1

eikδj)(

3∑j′=1

e−ikδj′ )

= 3 + 2 cos k(δ1 − δ2) + 2 cos k(δ1 − δ3) + 2 cos k(δ2 − δ3)

so that

E(k) = ±t√

3 + 2 cos k · a1 + 2 cos k · a2 + 2 cos k · (a1 − a2)

where

k · a1 =3d2

kx +

√3d2

ky , k · a2 =3d2

kx −√

3d2

ky , k · (a1 − a2) =√

3dky


Tight-binding model• The dispersion law is

E(k) = ±t

√√√√3 + 4 cos(

3d2

kx

)cos

(√3d2

ky

)+ 2 cos

(√3dky

)


Tight-binding model

Scanning probe microscopy image of graphene


Tight-binding model• The positive and negative parts of the symmetric spectrum coincide at the pointswhere X = 0 and E = 0.

Solutions of these conditions are the corners of the Brillouin zone:

,


Tight-binding model• Only two of these points are independent, let us take

K+ = v6 = −v3 =2π3d

(0,

2√3

)

K− = −v6 = v3 =2π3d

(0,− 2√

3

)

K+ and K− are two independent solutions called the Dirac points.


Tight-binding model• Section of the electronic energy dispersion of graphene for kx = 0, ky = 0 andkx = 2π

3d :


Outline

1 Introduction



4 References


Low energy effective theory near the Dirac points• The next step is to expand the wave function around the Dirac points:

ψ±A,B(q) ≡ ψ±A,B(K± + q)

where we suppose that |q| is small compared to 1d ∼ 1 KeV.

•We calculate X and X∗ for k = K± + q. Expanding to first order in momenta

X± =

3∑j=1

ei(K±+q)δj ≈ i

3∑j=1

eiK±δj(qδj)

=3d2

(−iq1 ∓ q2)

(X±)∗ =

3∑j=1

e−i(K±+q)δj ≈ −i

3∑j=1

e−iK±δj(qδj)

=3d2

(iq1 ∓ q2)


Low energy effective theory near the Dirac points

whereK±δ1 = 0 , K±δ2 = ±2π

3, K±δ3 = ∓2π

3and

qδ1 = −dq1 , qδ2 =d2

q1 +

√3d2

q2 , qδ3 =d2

q1 −√

3d2

q2

• The Hamiltonian describing the low energy excitations near the Dirac points isfound to be:

H± =3td2

0 iq1 ± q2

−iq1 ± q2 0


Low energy effective theory near the Dirac points• The single-electron Hamiltonian can be written in the compact form:

H± = vF(−σ2q1 ± σ1q2) ≡ vF~α± ·~q

where vF = 3td2 is the Fermi velocity, σi, i = 1, 2 are the standard Pauli matrices.

~α± = (α1, α±2 ) = (−σ2,±σ1).

With d = 1.42◦A and t = 2.8eV one obtains the value: vF ' 1

300 .

• Therefore near each of the Dirac points, one obtains a 2D Weyl Hamiltoniandescribing massless relativistic particles and the dispersion relation is:

E(q) = ±vF|q| = ±vF

√q2

1 + q22


Low energy effective theory near the Dirac points• The electronic lineal energy dispersion of graphene:


Low energy effective theory near the Dirac points• The eigenfunctions are:

H+:H+ = vF(−σ2q1 + σ1q2) , E+(q) = ±vF

√q2

1 + q22

1 Solution of Type 1 (ψB = 1):{ψ+

1+ =

(iq1+q2√

q21+q2

2

1

), ψ+

2− =

( −iq1−q2√q2

1+q22

1


ψ+3+ =

(1

−iq1+q2√q2

1+q22

), ψ+

4− =

(1

iq1−q2√q2

1+q22

)}


Low energy effective theory near the Dirac points• Let us write q1 + iq2 =

√q2

1 + q22 eiθ, with θ = arctan(q2/q1). Then,

H+ = vF

0√

q21 + q2

2 ei(π2 −θ)

√q2

1 + q22 e−i(π

2 −θ) 0

The eigenfunctions are:

Positive energies E =√

q21 + q2

2 > 0:

ψ++ =

1√2

(ei(π

4 −θ2 )

−e−i(π4 −

θ2 )

)

Negative energies E = −√

q21 + q2

2 < 0:

ψ+− =

1√2

(ei(π

4 −θ2 )

e−i(π4 −

θ2 )

)


Low energy effective theory near the Dirac points• The eigenfunctions are:

H−:H− = vF(−σ2q1 − σ1q2) , E−(q) = ±vF

√q2

1 + q22

1 Solution of Type 1 (ψB = 1):{ψ−

1+ =

(iq1−q2√

q21+q2

2

1

), ψ−

2− =

( −iq1+q2√q2

1+q22

1


ψ−3+ =

(1

−iq1−q2√q2

1+q22

), ψ−

4− =

(1

iq1+q2√q2

1+q22

)}


Low energy effective theory near the Dirac points• Again q1 + iq2 =

√q2

1 + q22 eiθ, the Hamiltonian is then

H− = vF

0√

q21 + q2

2 ei(π2 +θ)

√q2

1 + q22 e−i(π

2 +θ) 0

The eigenfunctions are


q21 + q2

2 > 0:

ψ−+ =1√2

(ei(π

4 +θ2 )

−e−i(π4 +

θ2 )

)


q21 + q2

2 < 0:

ψ−− =1√2

(ei(π

4 +θ2 )

e−i(π4 +

θ2 )

)


Low energy effective theory near the Dirac points• The pseudo-spin structure in this low energy limit represents that:

For H+: φ→ π2 − θ.

For H−: φ→ π2 + θ.

where

S · n̂ =~2

(σ1 cosφ+ σ2 sinφ)

=~2

(0 cosφ− i sinφ

cosφ+ i sinφ 0

)


Low energy effective theory near the Dirac points• By introducing a four-component spinor:

ψT =(ψ+

A ψ+B ψ−A ψ−B

)we can unify H± in a single Dirac Hamiltonian,

H =

(H+ 00 H−

)= vF~α ·~q ≡ −ivFγ

0γa∂a

where

γ0 =

(σ3 00 σ3

), γ1 =

(iσ1 00 iσ1

), γ2 =

(iσ2 00 −iσ2

)and qa = −i∂a , a = 1, 2 and αa = γ0γa.


Low energy effective theory near the Dirac pointsThe eigenfunctions are now


q21 + q2

2 > 0:

ψ1 =1√2

ei(π

4 −θ2 )

−e−i(π4 −

θ2 )

00

and

ψ2 =1√2

00

ei(π4 +

θ2 )

−e−i(π4 +

θ2 )

where θ = arctan(q2/q1).




q21 + q2

2 < 0:

ψ3 =1√2

ei(π

4 −θ2 )

e−i(π4 −

θ2 )

00

and

ψ4 =1√2

00

ei(π4 +

θ2 )

e−i(π4 +

θ2 )


Low energy effective theory near the Dirac points• By introducing a four-component spinor:

ψT =(ψ+

A ψ+B ψ−A ψ−B

)

S · n̂ =

( ~2 (σ1 cosφ+ σ2 sinφ) 0

0 (σ1 cosφ− σ2 sinφ)

)

=~2

0 cosφ− i sinφ 0 0

cosφ+ i sinφ 0 0 00 0 0 cosφ+ i sinφ0 0 cosφ− i sinφ 0

so that

S · n̂ ψ1 = −~2ψ1 , S · n̂ ψ2 =

~2ψ2

S · n̂ ψ3 =~2ψ3 , S · n̂ ψ4 = −~

2ψ4



• Here γµ, µ = 0, 1, 2 are 4× 4 gamma matrices in a reducible representation whichis a direct sum of two inequivalent 2× 2 representations:

{σ3, iσ1, iσ2} , {σ3, iσ1, −iσ2}

• Each of these two-component representations for graphene quasiparticleswave-function is somewhat similar to the spinor description of electron in QED.

• However in the case of graphene this pseudo spin index refers to the sublatticedegree of freedom rather than the real spin of the electrons.

• The whole effect of the real spin is just in doubling the number of spinorcomponents, so that we have 8-component spinors in graphene (N = 4 species oftwo-component fermions).


Modifications of the Dirac model

1 Interaction with the electromagnetic field: ∂ → ∂ + ieA.The electromagnetic field is not confined to the graphene surface.External magnetic field (Anomalous Hall Effect), electromagnetic radiation,quantized fluctuations.

2 Quasi-particles maybe have a mass. This mass is usually very small and could beconvenient to perform the Pauli-Villars regularization.

3 A more important mass-like parameter is the chemical potential µ, whichdescribes the quasi-particle density.

4 Impurities in graphene are describing by adding a phenomenological parameterΓ in the propagator of quasi-particles.

5 Most experiments with graphene are done at rather high temperatures: Thermalfield theory.


References

Quantum Field Theory in Graphene I. V. Fialkovsky, D.V. Vassilevich,November 21, 2011 1:18 WSPC/INSTRUCTION FILE qfext11’3.arXiv:1111.3017v2 (hep-th) 18 Nov 2011.

P. R. Wallace, Phys. Rev. 71, 622 (1947).

D. P. DiVincenzo and E. J. Mele, Phys. Rev. B 29, 1685 (1984).

G. W. Semenoff, Phys. Rev. Lett. 53, 2449 (1984).


References

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov and A. K. Geim,Rev. Mod. Phys. 81 , 109-162 (2009).

Introduction to Dirac materials and Topological Insulators Jérôme Cayssol.arXiv:1310.0792v1 [cond-mat.mtrl-sci] 2 Oct 2013.

Introduction to the Physical Properties of Graphene Jean-Noël FUCHS andMark Oliver GOERBIG. Lecture Notes 2008.http://web.physics.ucsb.edu/ phys123B/w2015/pdf_CoursGraphene2008.pdf.

Lecture notes. Farhan Rana 2009.https://courses.cit.cornell.edu/ece407/Lectures/Lectures.htm.


graphene and planar dirac equation - usalcampus.usal.es/~mpg/general/grafeno_dirac_2d2016.pdf ·...

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