Graphene: why πα? Louis Kang & Jihoon Kim Source: Science Vol. 320 no. 5881 p.1308 Physics 141A Spring 2013 Graphene: why πα? Louis Kang & Jihoon Kim

Graphene: why πα? Louis Kang & Jihoon Kim The Approach Let 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 Physics 141A Spring 2013 Graphene: why πα? Louis Kang & Jihoon Kim

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 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 u1 u2 l R , where t represents the hopping constant and atom of the unit cell Lattice position of the atom Physics 141A Spring 2013 Graphene: why πα? Louis Kang & Jihoon Kim

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 The diagonal entries are zero because there is no hopping from one sub lattice to itself Sample calculations are shown in Appendix A Physics 141A Spring 2013 Graphene: why πα? Louis Kang & Jihoon Kim

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. K q K’ Expanding around K’ would give similar dispersion relationship, which we will explore later on. Source: Munster University Physics 141A Spring 2013 Graphene: why πα? Louis Kang & Jihoon Kim

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 Sample calculations are shown in Appendix A , where σ represents Pauli matrices, Vf , Fermi velocity, is the slope of graphene’s linear dispersion relationship, and Physics 141A Spring 2013 Graphene: why πα? Louis Kang & Jihoon Kim

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 +E Density of States is the number of k vectors per unit k-area Explain direct transition vs indirect transition -E Physics 141A Spring 2013 Graphene: why πα? Louis Kang & Jihoon Kim

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, Density of States is the number of k vectors per unit k-area Physics 141A Spring 2013 Graphene: why πα? Louis Kang & Jihoon Kim

Graphene: why πα? Louis Kang & Jihoon Kim Finding Density of States (D) of Graphene Graphene 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 Density of States is the number of k vectors per unit k-area Physics 141A Spring 2013 Graphene: why πα? Louis Kang & Jihoon Kim

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. Density of States is the number of k vectors per unit k-area Physics 141A Spring 2013 Graphene: why πα? Louis Kang & Jihoon Kim

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 Density of States is the number of k vectors per unit k-area Physics 141A Spring 2013 Graphene: why πα? Louis Kang & Jihoon Kim

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. Density of States is the number of k vectors per unit k-area Physics 141A Spring 2013 Graphene: why πα? Louis Kang & Jihoon Kim

Graphene: why πα? Louis Kang & Jihoon Kim Matrix Element to πα Using 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 Density of States is the number of k vectors per unit k-area Physics 141A Spring 2013 Graphene: why πα? Louis Kang & Jihoon Kim

Graphene: why πα? Louis Kang & Jihoon Kim Matrix Element to πα Rearranging the terms in incident light energy by converting E to A, the vector potential according to the relationship: Then, now we have, (finally!) with Density of States is the number of k vectors per unit k-area Physics 141A Spring 2013 Graphene: why πα? Louis Kang & Jihoon Kim

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. Physics 141A Spring 2013 Graphene: why πα? Louis Kang & Jihoon Kim

Graphene: why πα? Louis Kang & Jihoon Kim Appendix A: Sample Calculations (2) Getting from Physics 141A Spring 2013 Graphene: why πα? Louis Kang & Jihoon Kim

Graphene: why πα? Louis Kang & Jihoon Kim Appendix A: Sample Calculations (3) Getting wave vectors of electrons in graphene from Physics 141A Spring 2013 Graphene: why πα? Louis Kang & Jihoon Kim

Graphene: why πα? Louis Kang & Jihoon Kim Appendix A: Sample Calculations (4) Calculating M2 PART 1 Physics 141A Spring 2013 Graphene: why πα? Louis Kang & Jihoon Kim

Graphene: why πα? Louis Kang & Jihoon Kim Appendix A: Sample Calculations (4) Calculating M2 PART 2 Physics 141A Spring 2013 Graphene: why πα? Louis Kang & Jihoon Kim

Graphene: why πα? Louis Kang & Jihoon Kim Appendix A: Sample Calculations (5) Finally! calculating πα Physics 141A Spring 2013 Graphene: why πα? Louis Kang & Jihoon Kim

Graphene: why πα? Louis Kang & Jihoon Kim Appendix B: References R.R.Nair et al.(2008). "Universal Dynamic Conductivity and Quantized Visible Opacity of Suspended Graphene".  Science 320, 1308 Wallace, P. R. (1947). "The Band Structure of Graphite". Physical Review 71: 622–634 Katsnelson, M.I. (2012). "Graphene: Carbon in Two dimension". Cambridge University Press.  Charles Kittel(2004) "Solid State Physics", Wiley Physics 141A Spring 2013 Graphene: why πα? Louis Kang & Jihoon Kim