1Physics 141ASpring 2013
Graphene: why πα? Louis Kang & Jihoon Kim
Graphene: why πα?
Source: Science Vol. 320 no. 5881 p.1308
2Physics 141ASpring 2013
Graphene: why πα? Louis Kang & Jihoon Kim
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
3Physics 141ASpring 2013
Graphene: why πα? Louis Kang & Jihoon Kim
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
4Physics 141ASpring 2013
Graphene: why πα? Louis Kang & Jihoon Kim
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
5Physics 141ASpring 2013
Graphene: why πα? Louis Kang & Jihoon Kim
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.
6Physics 141ASpring 2013
Graphene: why πα? Louis Kang & Jihoon Kim
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
7Physics 141ASpring 2013
Graphene: why πα? Louis Kang & Jihoon Kim
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
8Physics 141ASpring 2013
Graphene: why πα? Louis Kang & Jihoon Kim
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
9Physics 141ASpring 2013
Graphene: why πα? Louis Kang & Jihoon Kim
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
10Physics 141ASpring 2013
Graphene: why πα? Louis Kang & Jihoon Kim
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
11Physics 141ASpring 2013
Graphene: why πα? Louis Kang & Jihoon Kim
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
12Physics 141ASpring 2013
Graphene: why πα? Louis Kang & Jihoon Kim
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 ||
13Physics 141ASpring 2013
Graphene: why πα? Louis Kang & Jihoon Kim
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
14Physics 141ASpring 2013
Graphene: why πα? Louis Kang & Jihoon Kim
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
15Physics 141ASpring 2013
Graphene: why πα? Louis Kang & Jihoon Kim
Appendix A: Sample Calculations
(1) Getting from
and repeat the same process for the other components of the matrix.
16Physics 141ASpring 2013
Graphene: why πα? Louis Kang & Jihoon Kim
Appendix A: Sample Calculations
(2) Getting from
17Physics 141ASpring 2013
Graphene: why πα? Louis Kang & Jihoon Kim
Appendix A: Sample Calculations
(3) Getting wave vectors of electrons in graphene from
18Physics 141ASpring 2013
Graphene: why πα? Louis Kang & Jihoon Kim
Appendix A: Sample Calculations(4) Calculating M2 PART 1
19Physics 141ASpring 2013
Graphene: why πα? Louis Kang & Jihoon Kim
Appendix A: Sample Calculations(4) Calculating M2 PART 2
20Physics 141ASpring 2013
Graphene: why πα? Louis Kang & Jihoon Kim
Appendix A: Sample Calculations(5) Finally! calculating πα
21Physics 141ASpring 2013
Graphene: why πα? Louis Kang & Jihoon Kim
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
Top Related