Effects of magnetic field on the electronic structure of wurtzite quantum dots: Calculations using effective-mass envelope function theory

The Hamiltonian of the wurtzite quantum dots in the presence of an external homogeneous magnetic field is given. The electronic structure and optical properties are studied in the framework of effective-mass envelope function theory. The energy levels have new characteristics, such as parabolic property, antisymmtric splitting, and so on, different from the Zeeman splitting. With the crystal field splitting energy ${\Delta }_c=25\mathrm{meV}$, the dark excitons appear when the radius is smaller than $25.85\mathrm{\AA }$ in the absence of external magnetic field. This result is more consistent with the experimental results reported by Efros et al. [Phys. Rev. B 54, 4843 (1996)]. It is found that dark excitons become bright under appropriate magnetic field depending on the radius of dots. The circular polarization factors of the optical transitions of randomly oriented dots are zero in the absence of external magnetic field and increase with the increase of magnetic field, in agreement with the experimental results. The circular polarization factors of single dots change from nearly 0 to about 1 as the orientation of the magnetic field changes from the $x$ axis of the crystal structure to the $z$ axis, which can be used to determine the orientation of the $z$ axis of the crystal structure of individual dots. The antisymmetric Hamiltonian is very important to the effects of magnetic field on the circular polarization of the optical transition of quantum dots.

Figure 1

(a) Energies of electron states of quantum dots as functions of $b$ $(\cos \theta =1)$. (b) Energies of electron states of quantum dots as functions of $b$ $(\cos \theta =1∕2)$.

Figure 2

Energies of hole states of quantum dots $\left(21\mathrm{\AA }\right)$ as functions of $b$. (a) $M=-3∕2$. (b) $M=3∕2$. (c) $M=-1∕2$. (d) $M=1∕2$.

Figure 3

(a) The lowest two energy levels of hole states of quantum dots $\left(21\mathrm{\AA }\right)$ as functions of $b$. (b) Transition probabilities of dots $\left(21\mathrm{\AA }\right)$ as functions of $b$. (c) Transition probabilities of dots $\left(25\mathrm{\AA }\right)$ as functions of $b$.

Figure 4

(a) The normalized intensities of ${\sigma }^{-}$ and ${\sigma }^{+}$ transitions of dots $\left(28.5\mathrm{\AA }\right)$ as functions of $b$. (b) Circular polarization factors of dots $\left(28.5\mathrm{\AA }\right)$ as functions of $b$.

Figure 5

Circular polarization factors of single dots $\left(28.5\mathrm{\AA }\right)$ as functions of $\cos \theta$.