Diluted magnetic semiconductor quantum dots: An extreme sensitivity of the hole Zeeman splitting on the aspect ratio of the confining potential

The valence band states confined in infinitely deep quantum dots made of diluted magnetic semiconductors (DMS) are considered theoretically. A complex anisotropic structure of the valence bands in DMSs with cubic symmetry described by the full Luttinger Hamiltonian is taken into account. It is found that the Zeeman splitting is very sensitive to the shape of the confining potential and, in particular, to its orientation relative to the direction of an external magnetic field. This sensitivity has its origin in a mixing of different spin components of a hole wave function which takes place for finite hole wave vectors $\mathbf{k}$. Several consequences of the effect are discussed, including a possibility to control the interdot tunneling by an external magnetic field. It is shown also that the polarizations of optical transitions in a single DMS quantum dot depend on details of geometry of its confining potential as well as on the strength of the magnetic field.

Figure 1

Expectation value of the projection of the total angular momentum $⟨{\widehat{J}}_z⟩$ as a function of the QD size in the $z$ direction. The solid line corresponds to the structure with $L_x=50$ and $L_y=70\mathrm{\AA }$, while the dashed one corresponds to the structure with $L_x=40\mathrm{\AA }$ and $L_y=80\mathrm{\AA }$.

Figure 2

Zeeman splitting of the hole ground states in ${\mathrm{Cd}}_{0.97}{\mathrm{Mn}}_{0.03}\mathrm{Te}$ DMS quantum dots in the presence of a magnetic field $\mathbf{B}\Vert z$ at $T=2\mathrm{K}$. The solid lines correspond to the structure with $L_x=50\mathrm{\AA }$, $L_y=70\mathrm{\AA }$ and $L_z=65\mathrm{\AA }$, the dashed lines—to the structure with $L_x=65\mathrm{\AA }$, $L_y=70\mathrm{\AA }$ and $L_z=50\mathrm{\AA }$.

Figure 3

Schematic view and the evolution of the ground energy of the hole states in two ${\mathrm{Cd}}_{0.97}{\mathrm{Mn}}_{0.03}\mathrm{Te}$ quantum dots in a magnetic field $\mathbf{B}\Vert z$. Label 1 corresponds to $70\mathrm{\AA }\times 60\mathrm{\AA }\times 50\mathrm{\AA }$ quantum dot, label 2 to $70\mathrm{\AA }\times 50\mathrm{\AA }\times 65\mathrm{\AA }$ quantum dot. The resonant field, that couples the two quantum dots enabling the hole resonant tunneling, is about $1.2\mathrm{T}$ at $T=2\mathrm{K}$.

Figure 4

Probability of optical transitions associated with the fourfold degenerate exciton ground state as a function of $L_z$ for the QD with $L_x=70\mathrm{\AA }$ and $L_y=50\mathrm{\AA }$. The three lines correspond to three different linear polarizations of the light.

Figure 5

Probability of optical transitions in two circular, and two linear polarizations for the ground state of exciton in ${\mathrm{Cd}}_{0.97}{\mathrm{Mn}}_{0.03}\mathrm{Te}$ DMS QD with $L_x=70\mathrm{\AA }$, $L_y=60\mathrm{\AA }$, and $L_z=50\mathrm{\AA }$ in the magnetic field $\mathbf{B}\Vert z$ at $T=2\mathrm{K}$.