Faraday rotation spectroscopy of quantum-dot quantum wells

Time-resolved Faraday rotation studies of $\mathrm{Cd}\mathrm{S}∕\mathrm{Cd}\mathrm{Se}∕\mathrm{Cd}\mathrm{S}$ quantum-dot quantum wells have recently shown that the Faraday rotation angle exhibits several well-defined resonances as a function of probe energy close to the absorption edge. Here, we calculate the Faraday rotation angle from the eigenstates of the quantum-dot quantum well obtained with $\mathbf{k}\bullet \mathbf{p}$ theory. We show that the large number of narrow resonances with comparable spectral weights observed in experiment is not reproduced by the level scheme of a quantum-dot quantum well with perfect spherical symmetry. A simple model for broken spherical symmetry yields results in better qualitative agreement with experiment.

Figure 1

(Color online) (a) Schematic representation of the $\mathrm{Cd}\mathrm{S}∕\mathrm{Cd}\mathrm{Se}∕\mathrm{Cd}\mathrm{S}$ QDQWs of Ref. 13. (b) The radius of the central CdS core is denoted by $r_1$, the width of the CdSe QW by $r_2\text{−}{r}_1$, and the width of the CdS cap by $r_3\text{−}{r}_2$. The numerical values are given in the text. (c) The radial probability distribution of the conduction-band ground state $1S_e$. Because of the small conduction-band mass, the state is not well localized in the QW (indicated by dashed vertical lines). (d) The radial probability distribution of the $1S_{3∕2}$ valence-band state. The two components of the wave function $R_0$ and $R_2$ are shown in solid and dashed lines, respectively. Because the valence-band mass is large compared to the conduction-band mass, the valence-band states are more strongly localized in the QW. (e) The energies of the $1S_e$, $2S_e$, and $1P_e$ conduction-band levels as a function of QW width. (f) The energies of the lowest three valence-band levels with $S_{3∕2}$ symmetry (squares) as a function of QW width. The energies of the states $1P_{3∕2}$, $1D_{5∕2}$, $1P_{5∕2}$, and $1D_{7∕2}$ are also shown (from the bottom). The labels for $1D_{5∕2}$, $1P_{5∕2}$, and $1D_{7∕2}$ are omitted for clarity.

Figure 2

(Color online). (a) Schematic representation of the transitions which contribute to the FR angle ${\theta }_F\left(E\right)$ in the spectral representation [Eq. (13)]. The dipole transition matrix elements with $1S_e$ are finite only for $S_{3∕2}$ multiplets. (b) ${\theta }_F\left(E\right)$ calculated from Eq. (16) for $n_{\mathrm{Cd}\mathrm{Se}}=3$ and $\Gamma =15\mathrm{meV}$. $n{S}_{3∕2}$ multiplets with $n⩾4$ have been neglected. The transitions from (a) are indicated by arrows.

Figure 3

(Color online). (a) Cross section of a QDQW with broken spherical symmetry. A deformation of the central CdS QD implies a spatial variation of the QW width, which is modeled by Eq. (19). (b) A schematic representation of the level scheme for a spherical QDQW and a QDQW with broken spherical symmetry. The broken spherical symmetry mixes the $1S_{3∕2}$ and $1P_{3∕2}$ multiplets and renders transitions to $1S_e$ bright for all eight states of the multiplet.

Figure 4

(Color online) (a) Experimental data for the amplitude of the TRFR signal ${\theta }_F\left(E\right)$, measured at $T=294\mathrm{K}$, and the optical absorption as a function of probe energy. (b)–(d) The calculated amplitude of the TRFR signal ${\theta }_F\left(E\right)$ (solid line) and optical absorption (dashed line) for different strengths of an anisotropy potential, $v_0=0,40,70\mathrm{meV}$, respectively. The calculation is restricted to the $1P_{3∕2}$, $1S_{3∕2}$, $2S_{3∕2}$, and $3S_{3∕2}$ valence-band multiplets that dominate the optical response close to the absorption edge. For $\lambda <560\mathrm{nm}$, the transitions from higher valence-band multiplets become important for the absorption spectrum and the restriction to only four valence-band multiplets is no longer valid. Because $T=294\mathrm{K}$ is small compared to the conduction-band level splitting, all electron spins are assumed to occupy $1S_e$ in the calculation of ${\theta }_F\left(E\right)$.