Absorption in bulk semiconductors. Part 3: complex patterns of electric currents

The complex electric-field patterns of twisted light are imprinted in the semiconductor as complex electric currents that depend on the amplitude and orbital angular momentum of the light.

We have seen in Part 1 and Part 2 that in bulk semiconductors the excitation by paraxial beams of twisted light (TL) brings the electron to a superposition of Bloch-states with a clear absorption of orbital angular momentum. But most important to the present post is that separating the effects according to the intensity of the field (either proportional to  $A$ and $A^2$) helps to understand better the phenomena going on.

To recall what we found previously: coherences ($\rho_{v,c}$) and populations ($\rho_{v,v}$ and $\rho_{c,c}$) are first and second order in the vector potential amplitude $A$, respectively. And they evolve in time at a different rate, being the coherence much faster. We will do a similar separation, and describe the electric currents to order $A$ and $A^2$ separately.

The paramagnetic electric current [1] can be derived from the general expression
$$
\mathbf j(\mathbf r, t)
=
-i q \hbar/(2m_e) lim_{\mathbf r' \rightarrow \mathbf r}(\nabla-\nabla')\psi^\dagger(\mathbf r', t)\psi(\mathbf r, t)\,,  ~~~~~~~ (1)
$$
where $\psi^\dagger(\mathbf r, t)/\psi(\mathbf r, t)$ are field operators. Changing to creation/annihilation operators enables us to use our previous results and separate the current into a coherence and population contribution.

Coherence contribution

After some algebra on Eq. (1), which includes applying the $\nabla$ operator on the microscopic wave-function $u_b(\mathbf r)$ of electrons, one finds
$$
\mathbf j^{coh} (\mathbf r, t)
=
\frac{2q}{m_e} \sum_{\mathbf k'm',\mathbf km} {\cal R}[\mathbf p_{vc} F_{\mathbf k'm',\mathbf km}(\mathbf r) \rho_{v\mathbf k'm',c\mathbf km}(t)] ~~~~~~~ (2)
$$
where $\mathbf p_{vc}$ is the matrix element of $\mathbf p$ between microsopic functions $u_b(\mathbf r)$, and due to the selection rule $m-m'=\ell$ the function $F_{\mathbf k'm',\mathbf km}(\mathbf r) \propto J_{m'}(k'_r r) J_m(k_r r) e^{i (k_z-k'_z)z} e^{i \ell \phi}$. The current oscillates at the frequency of the field. Figure 1 plots the space-dependent part ${\cal R}[\mathbf p_{vc} F_{\mathbf k'm',\mathbf km}(\mathbf r)]$ for $\ell=1$ and polarization $\hat{x}-i\hat{y}$, where we see a clear circulation around the beam axis.

Figure 1: First-order paramagnetic current for light with $\ell=1$ and polarization $\hat{x}-i\hat{y}$. The center of the plot coincides with the beam axis. The modulation in r in dashed redline.

In contrast, for $\ell=1$ and polarization $\hat{x}+i\hat{y}$, the pattern does not show a circulation around the origin. For $|\ell|>1$ the electric current develops more complex patterns, which mimics the complex structure of the electric field of the TL beam, as illustrated in Fig. 2. The results are consistent with the net (global) transfer of orbital angular momentum described in the previous post, and explained in more detailed in [2]. 

Figure 2: First-order paramagnetic current for light with $\ell = 2$. The center of the plot coincides with the beam axis. The space pattern is more complex than in the $\ell = 1$ case and exhibits circulation around off-axis centers.

We can conclude that the "first-order" electric current is driven by the interband coherence, it oscillates in time with zero mean at the frequency of the EM field, and presents complex spatial patterns that are related to the peculiar electric field of the TL beam. For special cases, the spatial pattern displays a circulation of microscopic origin related to polarization effects (not to be confused with a macroscopic excursion of the electrons around the beam axis) with a concomitant instantaneous net transfer of orbital angular momentum from the beam to the material system.

Population contribution

In contrast to the coherence case, we here apply the $\nabla$ operators of Eq. (1) to the envelop wave-functions of electrons, and specifically look at the $\phi$ component of the electric current
$$
j_\phi(\mathbf r,t) =
{q \hbar \over m_e} \sum_{\mathbf km}
[{\cal N}J_m(k_r r)]^2 \rho_{c\mathbf km,c\mathbf km}(t)
+\{ c \rightarrow v \} ~~~~~~~ (3)
$$
where $\{ c \rightarrow v \}$ stands for a similar term exchanging $c$ and $v$. Note that $\ell$ doesn't appear explicitly in Eq. 3, but through the electron population $\rho_{c\mathbf km,c\mathbf km}(t)$ because an electron leaving a valence-band state with $m$ goes to a conduction-band state with $m + \ell$. There is no complex space dependence, as it happened in the coherence contribution with the extra $e^{i (k_z-k'_z)z} e^{i \ell \phi}\mathbf p_{vc}$. Therefore, the $\phi$ component of the electric current exists for any value of $\ell$,circulates either in the positive or negative sense all around the origin, depending on whether the electron population is unbalanced to positive or negative values of $m$, and oscillates with non-zero average.

In summary, we see that there are two qualitatively different contributions to the electric current, that can be termed microscopic and macroscopic contributions. The microscopic contribution relates to the interband coherence and mimics the complex patterns of the electric field; from this and other reasons, it parallels the induced polarization of a semiconductor in the presence of plane waves, as traditionally studied using the vertical-transition assumption. On the other hand, the macroscopic contribution signals a net transfer of OAM from the field to the electrons, and it parallels the photon-drag effect.

References

[1] This is the most common contribution to the current. The calculations for the other term, named diamagnetic, are explained in Ref. [2].

[2] G. F. Quinteiro and P. I. Tamborenea, Twisted-light-induced optical transitions in semiconductors: Free-carrier quantum kinetics, Phys. Rev. B 82, 125207 (2010).