Physics 141A Spring 2013

1Physics 141ASpring 2013

Graphene: why πα? Louis Kang & Jihoon Kim

Graphene: why πα?

Source: Science Vol. 320 no. 5881 p.1308



The ApproachLet a light wave with electric field (E) and frequency (ω) fall perpendicular to a sheet of graphene:

The incident energy

The absorbed energy , where

η indicates the absorbed events per unit time per unit area, which can be calculated using Fermi’s Golden Rule:

, where

M is the matrix element for graphene’s interaction between light and its Dirac fermions and D is the density of states of graphene.

Then, we do the absorption calculation to find πα!

We need to find η, D, and

the wave vectors of graphene.

Source: Science Institute of Physics



Finding η from Graphene’s Electronic Band Structure In a honeycomb lattice of graphene, its unit cell contains to atoms, a and b.

The unit cell’s lattice translational vectors are:

and Its reciprocal vectors

are canonically chosen as

and

Then, we use the tight-binding model1 on a and b to

find the Hamiltonian of graphene!

1 The tight binding model is an approach to the calculation of electronic band structure using a set of wave functions based upon superposition of wave functions for isolated atoms located at each atomic site

a

b

u1u2 l

, where t represents the hopping constant and atom of

the unit cell

Lattice position of the atom

R



Finding η from Graphene’s Electronic Band Structure Then, we can use the Bloch wave function2 to define the wavefunctions in reciprocal space. The Bloch theory says that:

, where

is the phase factor.

Applying this relationship to the the tight binding interaction we found earlier gives:

2 The Bloch wave function is the wave function of a particle placed in a periodic potential, which is written as the product of a plane wave envelope function and a periodic function

1) The diagonal entries are zero because there is no hopping from one sub lattice to itself

2) Sample calculations are shown in Appendix A



Finding η from Graphene’s Electronic Band Structure There are two high symmetry points, K and K’ in graphene’s Brillouin zone. We will taylor expand graphene’s Hamiltonian around one of such points, K, with respect to k.

Source: Munster University

Kq

K’

Expanding around K’ would give similar dispersion relationship, which we will explore later on.



Finding η from Graphene’s Electronic Band Structure Replace with K + q to make the equation applicable to any arbitrary position. K represents the K-point of graphene and q indicates how far the electron is from the K-point (as shown below): Then, H comes down to

which is equal to

, where σ represents Pauli matrices, Vf , Fermi velocity, is the slope of graphene’s linear dispersion relationship, and

Sample calculations are shown in Appendix A



Wave Vectors of Electrons in Graphene As mentioned earlier, the Hamiltonian of graphene around the K point is:

Then, the two entries on the Hamiltonian are complex conjugates of each other. When they are normalized:

Each k vector has two energy states. One corresponding to the higher energy state

The other corresponding to the lower energy state

2)(

2)(

21| ki

ki

e

ek

00

)(

)(

ki

ki

ee

-E

+E



Wave Vectors of Electrons in Graphene

Expansion around K’ point gives a different dispersion relationship,

which is the complex conjugate of Hamiltonian around K point.

So we now obtain the Hamiltonian around K’ point and hence the wavevectors around the point,

2)(

2)(

21| qi

qi

e

ek

00

)(

)(

qi

qi

ee



Finding Density of States (D) of GrapheneGraphene is a 2D material, so the only possible directions for q is qx and qy and

, where the numerator is the k-space area

with same value of q.

Therefore it suffices to calculate the density of states for only one of the two states with the same q.

Electrons around the K-points have the energy that is linearly proportional to q-vector, so

and the energy of the emission is twice the energy E, so

qvE f



Finding Density of States (D) of Graphene

Adding the unit of k-space (length over 2π) and taking into account the different spin orientations (factor of 2) and the K and K’ degeneracy,

The degeneracy is due to the two different points for each Brillouin zone , K’ and K. Due to the fact that these two sites have same density of states, we must multiply two to the overall number of states.

fvdqdE

ff vv

Eq

2

222 )2()

21(22

ff

q

vvq

dEdqq

dEd



The Matrix Element(M)From perturbation theory, H=H0 + H’ for which H0 is the original Hamiltonian and H’ is the first-order correction for some new interaction.

Earlier in the presentation, we had the Hamiltonian of electrons in graphene as:

When an electron interacts with light, it gains an extra momentum such that

qvH f

Acepp



The Matrix Element (M)

Since we are calculating the ‘absorbed’ energy, we calculate the change in Hamiltonian, H’, which is the matrix element in this first-order perturbation limit.Now we try to calculate the Matrix element, M, which is determined by

This describes the interaction between light and electrons in graphene.

iAcevfiHfM f |||'|

fk || ik ||



Matrix Element to παUsing the wavevectors that we have obtained earlier and averaging over all states(which is, over the ring of constant k, or phi from 0 to 2 pi), we obtain:

Then, using the formula for absorbed energy per unit area per unit time

And using

for D

222 )2()

21(22

ff

q

vvq

dEdqq

dEd



Matrix Element to παRearranging the terms in incident light energy by converting E to A, the vector potential according to the relationship:

Then, now we have, (finally!)

with

EtA

tieEE 0



Appendix A: Sample Calculations

(1) Getting from

and repeat the same process for the other components of the matrix.




(2) Getting from




(3) Getting wave vectors of electrons in graphene from



Appendix A: Sample Calculations(4) Calculating M2 PART 1



Appendix A: Sample Calculations(4) Calculating M2 PART 2



Appendix A: Sample Calculations(5) Finally! calculating πα



Appendix B: References1. R.R.Nair et al.(2008). "Universal Dynamic Conductivity and

Quantized Visible Opacity of Suspended Graphene". Science 320, 1308

2. Wallace, P. R. (1947). "The Band Structure of Graphite". Physical Review 71: 622–634

3. Katsnelson, M.I. (2012). "Graphene: Carbon in Two dimension". Cambridge University Press.

4. Charles Kittel(2004) "Solid State Physics", Wiley

Physics 141A Spring 2013

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