Electric current and OAM in quantum rings

As in bulk semiconductors, twisted light can transfer Orbital Angular Momentum to quantum rings and generate electric currents.

Our previous post on tilted transitions in quantum dots made clear the important difference between excitation in bulk/quantum dots and quantum rings. This is illustrated in Figure 1: In bulk the transition brings the electron in a single valence-band state to a superposition of  states in the conduction band. Due to their cylindrical symmetry, an electron in a QD is excited to a superposition of radial modes, if a broadband laser is used. Finally, the simplest transition occurs in 1D quantum rings, that is a 1-to-1 transition.

Figure 1: Three different systems and their response to twisted light. In bulk the transition brings the electron in a single valence-band state to a superposition of  states in the conduction band. In a QD the electron is excited (with a broadband laser) into a superposition of radial modes (number between parenthesis). Finally, 1D-QRs should exhibit a clear 1-to-1 transition.

The transfer of OAM and the production of electric currents is easily understood. Suppose that we have a laser that emits light near the band-gap energy $E_g$ and its spectral width is not that narrow. Furthermore, consider a QR with diameter in the hundreds of nanometer; then, the energy levels will be close together. And the light pulse will excite a multitude of electrons; we show this Fig. 2 where the vertical length of blue arrow is not fixed to a single value but to a range.

Figure 2: The optical excitation produces an unbalanced population of electrons and holes.

An unbalanced population of electrons and holes is created. Electrons with positive and negative envelope orbital angular momentum $m$ will compensate each other and will not contribute to the total angular momentum of the QR; the same is true for holes. However, there are extra electrons at positive $m$'s and extra holes at negative $m$'s. Remember that holes have the opposite value of momentum. Thus, both electrons and holes contribute with positive (in this example) OAM.

The OAM gained by the system depends on the amount $A$ of excited electron, and the value of the topological constant $\ell$. If $A$ is large, the extra (not-compensated) particles exist at a higher $m$, and if $\ell$ is large we get more extra particles.

The analysis in terms of unbalance population is in agreement with the idea that each $e-h$ pair gains $\ell$ from the light. Then, the number of pairs $A$ times $\ell$ yields the total OAM.

Figure 3: OAM of the QR electronic 

state: In red coherence OAM $L_z$, in 

black and blue population OAM $L_z$.

A quantum mechanical calculation shows that the OAM is of two types: 1) coherence and 2) population. The intuitive picture in Fig. 2 represents the last one (2), where the population in each band changes with the Rabi frequency. The coherence population follows the polarization, and it has zero mean value. Fig. 3 shows a numerical calculation of the OAM. To get the electric current one may use the simple model $I=q L_z/(2 \pi r_0 m)$, that shows the proportionality between current and OAM.

In the next posts we will talk about current and future applications of QRs, and the possible coupling of OAM to spin -via for example spin-orbit coupling.

Acknowledgment

I thank Mark Siemens for interesting comments.