Absorption in bulk semiconductors. Part 1: electronic excitation

We predict new electronic phenomena in bulk semiconductors excited by twisted light. I will show, in part 1, how the excited state of electrons looks like; in part 2 the expected transfer of orbital angular momentum from light to electrons; and in part 3, the generation of non-trivial patterns of electric currents. [1]

We start with the excitation by a classical paraxial beam of twisted light. In this case, we can disregard the component of the field in the propagation direction. In the Coulomb gauge the vector potential is

\begin{eqnarray} \large {\bf{A}}({\bf{r}},t)= \varepsilon A_0 J_{\ell}(q_r r) e^{i(q_z z-\omega t)} e^{i\ell \phi} +c.c. \end{eqnarray}

where the radial profile is a Bessel function J(q_rr), and ε is the circular polarization vector. For the quantum state of electrons in the semiconductor of side L, we use a two-band model with electron wave-function φ_bk(r)=L^-3/2e^ikru_bk(r), where b is the band, either valence or conduction, and u_bk(r) is the Bloch-periodic part.

Figure 1: The coordinate r = r' + R, where R refers to
unit cells, and r' to points within the unit cell.

When the unexcited semiconductor is illuminated by the field Eq. (1) with energy hω>E_g, electrons in the valence band are promoted to the conduction band. The inter-band transition is modeled by the interaction Hamiltonian −(q/me) A(r)·p. To determine the matrix element between an initial valence-band state  φ_vk(r)=<r|vk> and a final conduction-band state φ_ck'(r) we: (i) disregard the term arising from the action of p on the envelope function (pe^ikr), (ii) separate the integral (matrix-element) into a sum over all unit cells of integrals within the unit cell, (iii) use the Jacobi-Anger identity to solve the in-plane part of the sum over all unit cells

\begin{eqnarray} \large N^{-2/3}\sum e^{-i\boldsymbol{\kappa}R} J_{\ell}(q_r R) e^{i\ell \Phi} = (-i)^\ell\frac{2\,\pi}{{L^2}}\,e^{i\,\theta \, l}\,\frac{1}{q_r}\delta(\kappa_r-q_r) \end{eqnarray}

see Fig. 1 for the definition of upper case coordinates; κ=k'-k. The final result for the matrix element is 

\begin{eqnarray} \large  \langle\,c{\bf\,k}^\prime|H_I|v{\bf\,k}\rangle=-(-i)\,^l\,\frac{q\,A_0}{m_e}\,\frac{1}{L}\,\frac{\delta_{\kappa_r\,q_r}}{q_r}\,\delta_{\kappa_z,\,q_z}e^{i\,\theta\,l}\,\left(\boldsymbol{\varepsilon}_\sigma\cdot{\bf\,p}_{c\,v}\right)\,e^{-i\,\omega\,t}\,, \end{eqnarray} for the absorption of twisted light. The transfer of linear momentum in the z direction causes a one-to-one correspondence between initial and final states, as signaled by the second delta function. In contrast, for the in-plane part we found the restriction κ_r=q_r. This tells us that an electron initially in a well-defined state of quasi-momentum k will be promoted to a superposition state in the conduction band, as seen in Fig. 2.

Figure 2: The transition induced by twisted light leads to a
superposition state. 

Even though it is somewhat weird, the cone-like transition makes sense! The light field Eq. (1) can be expanded in terms of plane waves using   
\begin{eqnarray}
\large  J_{\ell}(q_r r) e^{i\ell \varphi}=\frac{(-i)^\ell}{2\pi}\int_0^{2\pi}e^{-i\ell \eta} e^{iq_r r \cos(\eta+\varphi)}d\eta
\end{eqnarray}
and we get
 \begin{eqnarray} {\bf{A}}({\bf{r}},t)=\frac{(-i)^\ell}{2\pi} \varepsilon A_0 e^{-i \omega t}\int_{-2\pi}^0e^{i\ell \eta} e^{i{\bf{q}}(\eta) \cdot {\bf{r}}} d\eta + c.c.
\end{eqnarray}
 with $\mathbf q(\eta)=q_r \cos(\eta)\hat{x}+q_r \sin(\eta)\hat{y}+q_z\hat{z}$. So the EM-field is an infinite superposition of plane waves propagating in different directions $\mathbf q(\eta)$, and each excites an electron in a particular value of quasi-momentum k.

In the next post I will show, using the matrix element just calculated, that the orbital angular momentum is transferred to the electrons.

References

[1] G. F. Quinteiro and P. I. Tamborenea, Theory of the optical absorption of light carrying orbital angular momentum by semiconductors, EPL 85, 47001 (2009).