Graphene is a planar (2D) material of carbon atoms arranged in a honeycomb lattice, see Fig.1. Because the honeycomb is not a Bravais lattice, one considers the structure as a triangular lattice with a basis of atoms A and B. As usually done, we describe optical transitions and electron kinetics near the K and K' Dirac points. Near these points the electron wavefunction can be written as a $2\times 1$ spinor, where each element corresponds to one sublattice A or B, for example $\psi=\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ represents an electron localized in atoms of type A.
Using the common substitution $\mathbf p \rightarrow \mathbf p - q \mathbf A$ into the free Dirac-like Hamiltonian, we find an expression for the light-matter interaction. This is a $2\times 2$ matrix with non-diagonal elements. For example, for the point $K$, and assuming the light is circularly polarized, it is
\begin{eqnarray}
H_I^{(K)} = \begin{bmatrix}
0 & A^{-} \\
A^{+} & 0
\end{bmatrix}
\end{eqnarray}
where $A^{+}$ ($A^{-}$) is the positive (negative) part of the vector potential. In the rotating wave approximation we retain for absorption only $A^{+}$; so if an unexcited electron in the valence band absorbs light, it will tend to become localized around atoms of type B in the conduction band
\begin{eqnarray}
H_I^{(K)} \begin{bmatrix} a \\ b \end{bmatrix}
\simeq
\begin{bmatrix}
0 & 0 \\
A^{+} & 0
\end{bmatrix}
\begin{bmatrix} a \\ b \end{bmatrix}
=
\begin{bmatrix} 0 \\ a \end{bmatrix}
\end{eqnarray}
Note the exchange of probabilities. However, the excitation also affects electrons that have momentum close to $K'$, and $H_I^{(K')}$ produces the opposite effect. Notice that we haven't said anything about the structure of light, and this discussion applies also to plane waves.
We can also set up equations of motion for the electron kinetics, as we did in the case of bulk and quantum well, and investigate from them the transfer of spin and orbital angular momenta from light to the electrons. We find that: 1) the situation is closer to the case of intraband transitions in semiconductors, since the two relevant energy bands of graphene originate from the same atomic orbitals, 2) a twisted light beam causes a transition between total angular momentum states, 3) because of the beam size (radial dependence) the final state is a superposition, and finally 4) light (all types, not just TL) affects the pseudo-spin of electrons.
I thank Pablo I. Tamborenea for useful comments.
References
Farías, M. B., G. F. Quinteiro, and P. I. Tamborenea. "Photoexcitation of graphene with twisted light" The European Physical Journal B 86, no. 10 (2013): 1-9.
 |
| Fig. 1. Graphene crystal structure: direct (left) and reciprocal (right) lattices, with primitive vectors T and G, respectively. |
\begin{eqnarray}
H_I^{(K)} = \begin{bmatrix}
0 & A^{-} \\
A^{+} & 0
\end{bmatrix}
\end{eqnarray}
where $A^{+}$ ($A^{-}$) is the positive (negative) part of the vector potential. In the rotating wave approximation we retain for absorption only $A^{+}$; so if an unexcited electron in the valence band absorbs light, it will tend to become localized around atoms of type B in the conduction band
\begin{eqnarray}
H_I^{(K)} \begin{bmatrix} a \\ b \end{bmatrix}
\simeq
\begin{bmatrix}
0 & 0 \\
A^{+} & 0
\end{bmatrix}
\begin{bmatrix} a \\ b \end{bmatrix}
=
\begin{bmatrix} 0 \\ a \end{bmatrix}
\end{eqnarray}
Note the exchange of probabilities. However, the excitation also affects electrons that have momentum close to $K'$, and $H_I^{(K')}$ produces the opposite effect. Notice that we haven't said anything about the structure of light, and this discussion applies also to plane waves.
We can also set up equations of motion for the electron kinetics, as we did in the case of bulk and quantum well, and investigate from them the transfer of spin and orbital angular momenta from light to the electrons. We find that: 1) the situation is closer to the case of intraband transitions in semiconductors, since the two relevant energy bands of graphene originate from the same atomic orbitals, 2) a twisted light beam causes a transition between total angular momentum states, 3) because of the beam size (radial dependence) the final state is a superposition, and finally 4) light (all types, not just TL) affects the pseudo-spin of electrons.
I thank Pablo I. Tamborenea for useful comments.
References
Farías, M. B., G. F. Quinteiro, and P. I. Tamborenea. "Photoexcitation of graphene with twisted light" The European Physical Journal B 86, no. 10 (2013): 1-9.
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