Quantum rings: the paradigmatic system

Because of its geometry, the one-dimensional semiconductor quantum ring is perhaps the quintessential system to investigate the interaction of twisted light with solids. We explore briefly the system providing relevant literature.

The original work and theory in optical properties of solids had to do almost entirely with homogeneous fields and extended systems, as we repeatedly have seen in class and books [1-2]. To many of us, fundamental semiconductor optics equates to plane-waves interacting with bulk. If we are forced to consider instead twisted light, we still may have an urge to work on bulk, in order to keep doing "basic" physics.

But it turns out that the fundamental theory of twisted light - solid interaction is better done in one-dimensional quantum rings.

Quantum rings (QRs) are among the semiconductor nanostructures more extensively studied in the last few decades. QRs are relevant for they help clarify basic principles –e.g. the Aharanov-Bohm effect and persistent currents [3]–, and because they promise various uses in nanotechnology, for instance the control of spin states near the ring [4], or the possibility to build lasers out of a stack of rings [5].

Semiconductor QR structures have been experimentally realized by many groups [6-8]. A two-dimensional mathematical representation makes a very precise model of most of the systems [9, 10], but in several situations the ring can be considered one-dimensional [11], where the radius $r_0$ is much larger than the height $h$ and width $w$. For example Ref. 12 shows an AFM picture of a highly symmetrical ring of $r_0>150$nm, $w=10$nm and $h<10$nm. In such a system the single-particle wavefunction represents an electron moving either to the right or left according to
\begin{equation}
\large
\psi(\mathbf r) = \frac{1}{\sqrt{2\pi}} e^{i m \varphi} R(r) Z(z) u_b(\mathbf r) \,,   ~~~~~~~ (1)
\end{equation}
and given in cylindrical coordinates $\{r,\phi,z\}$. The only relevant quantum number is the orbital quantum number $m= 0,  \pm 1, \pm 2, \ldots$  for the envelope (macroscopic) wavefunction. The motion in $r$ and $z$ is frozen. And we add the Bloch (microscopic) wavefunction $u_b(\mathbf r)$ for the study of interband processes, where under light irradiation the electron changes state between bands $b=v,c$. The energy levels are
\begin{equation}
\large
\epsilon_{vm} = \frac{\hbar^2 m^2}{2 m^*_v r_0^2 } \, ,   ~~~~~~~
\epsilon_{vc} = E_g + \frac{\hbar^2 m^2}{2 m^*_c r_0^2 } \,.   ~~~~~~~ (2)
\end{equation}
We have left aside various effects such as Coulomb interaction [13, 14], impurity scattering [13], electron-phonon interaction [15], and spin-orbit coupling [16].

With this simple one-dimensional model we will show in future posts tilted one-to-one optical transitions (compare to the complex situation in bulk, figure 2), the exact solution to twisted-light excitation, electric currents, generated magnetic fields, and by adding spin-orbit coupling the interplay with the spin degree of freedom. These results will make clear why the QR is the preferred structure to study twisted light.


References

[1] Ashcroft, Neil W., and N. David Mermin. "Solid State Physics" (Holt, Rinehart and Winston, New York, 1976).

[2] Haug, Hartmut, and Stephan W. Koch. "Quantum theory of the optical and electronic properties of semiconductors". Vol. 5. Singapore: World scientific, 1990.

[3] N. A. J. M. Kleemans, I. M. A. Bominaar-Silkens, V. M. Fomin, V. N. Gladilin, D. Granados, A. G. Taboada, J. M. García, P. Offermans, U. Zeitler, P. C. M. Christianen, J. C. Maan, J. T. Devreese, and P. M. Koenraad, PRL 99, 146808 (2007). // Hendrik Bluhm, Nicholas C. Koshnick, Julie A. Bert, Martin E. Huber, and Kathryn A. Moler PRL 102, 136802 (2009). // Georg Schwiete and Yuval Oreg, PRL 103, 037001 (2009). // Ding, F., N. Akopian, B. Li, U. Perinetti, A. Govorov, F. M. Peeters, CC Bof Bufon et al. Physical Review B 82, no. 7 (2010): 075309.

[4] E. Rasanen, A. Castro, J. Werschnik, A. Rubio, and E. K. U. Gross, PRL 98, 157404 (2007).

[5] Ferran Suarez, Daniel Granados, Marıa Luisa Dotor and Jorge M Garcıa, Nanotechnology 15 S126 (2004).

[6] W. Rabaud, L. Saminadayar, D. Mailly, K. Hasselbach, A. Benoˆıt, and B. Etienne, “Persistent currents in mesoscopic connected rings”, Phys. Rev. Lett. 86, 3124 (2001).

[7] A. Fuhrer, S. Luscher, T. Ihn, T. Heinzel, K. Ensslin, W. Wegscheider, and M. Bichler, “Energy spectra of quantum rings”, Nature (London) 413, 822 (2001).

[8] Mano, T., Kuroda, T., Sanguinetti, S., Ochiai, T., Tateno, T., Kim, J., Noda, T., Kawabe, M., Sakoda, K., Kido, G. and Koguchi, N., 2005. "Self-assembly of concentric quantum double rings. Nano letters", 5(3), pp. 425-428.

[9] W-C Tan and J C Inkson, "Electron states in a two-dimensional ring—an exactly soluble model", Semicond. Sci. Technol. 11 (1996) 1635–1641.

[10] Carlos M. Duque, Miguel E. Mora-Ramos, and Carlos A. Duque, "Quantum disc plus inverse square potential. An analytical model for two-dimensional quantum rings: Study of nonlinear optical properties", Ann. Phys. (Berlin) 524, No. 6–7, 327–337 (2012).

[11] Splettstoesser, J., Governale, M. and Zülicke, U., 2003. "Persistent current in ballistic mesoscopic rings with Rashba spin-orbit coupling". Physical Review B, 68(16), p.165341.

[12] Yu, L. W., K. J. Chen, J. Song, J. Xu, W. Li, X. F. Li, J. M. Wang, and X. F. Huang. "New self-limiting assembly model for Si quantum rings on Si (100)."Physical review letters 98, no. 16 (2007): 166102.

[13] Müller-Groeling, A., Weidenmüller, H.A. and Lewenkopf, C.H., 1993. "Interacting electrons in mesoscopic rings". EPL (Europhysics Letters), 22(3), p.193.

[14] Chakraborty, Tapash, and Pekka Pietiläinen. "Electron-electron interaction and the persistent current in a quantum ring." Physical Review B 50.12 (1994): 8460.

[15] Piacente, G., and G. Q. Hai. "Phonon-induced electron relaxation in quantum rings." arXiv preprint cond-mat/0703133 (2007).

[16] Zhen-Gang Zhu and J. Berakdar, “Photoinduced nonequilibrium spin and charge polarization in quantum rings,” Phys. Rev. B 77, 235438 (2008).