From Dot to Ring: Tunable Exciton Topology in Type-II InAs/GaAsSb Quantum Dots

Abstract

We present an experimental and theoretical study about the carrier confinement geometry and topology in InAs/GaAsSb quantum dots. The investigated sample consists of a field-effect device embedding a single layer of dot-in-a-well InAs/GaAsSb nanostructures. These nanostructures exhibit large electron-hole dipole moments and radiative lifetimes under externally applied electric fields. Both phenomena are related to the type-II band alignment existing between the two materials which, in principle, could also result in a change of the hole orbital confinement topology from simply to doubly connected. The latter aspect will be confirmed by ensemble magnetophotoluminescence experiments at 4.2 K. The oscillations observed in the photoluminescence intensity and degree of circular polarization will be described by an axially symmetric $\mathbf {k}\cdot \mathbf {p}$ model combining vertical electric and magnetic fields. Due to the large spin-orbit coupling of III-Sb nanostructures, the modulation of the orbital confinement geometry and topology reported here shall open a venue to control the spin dynamics by external voltages. This exciting idea will be theoretically discussed through band-effective models including spin-orbit coupling and anisotropic confinement effects.

Appendix: Axially-Symmetric Model Details

For the sake of simplicity, the SRL model presented in the Sect. 3.3 has been slightly modified to discuss magnetic field induced effects within the $\mathbf {k}\cdot \mathbf {p}$ formalism. Instead of a continuous layer conformal to the QD, we have consider an overlayer of finite extension. The number of geometrical parameters hence reduces to six as can be seen in Fig. 3.17. This approximation simplifies the strain calculation, as border effects on the overlayer are avoided. There is no impact on the results from a qualitative point of view, since, as we will show below, the hole wavefunction localizes either in the surroundings of the QD or in its interior. The material band parameters are extracted from [84]. The parameters used to describe the spin Zeeman effect are shown in Table 3.1. The values of the compounds GaInAs and GaAsSb are obtained through linear interpolation when no bowing parameters is reported.

Fig. 3.17

Depiction of the geometrical model to describe a QD of radius $R_\text {QD}$ and height $h_\text {QD}$, a SRL of thickness $t_\text {SRL}$ and a wetting-layer of thickness $d_\text {WL}$ embedded in hard-wall cylinder of radius $\mathcal {R}$ and height $\mathcal {Z}$

Table 3.1 Parameters related with the spin Zeeman effect. The values of GaAs and InAs are taken from [83] (p. 221) and those of GaSb from [85] (pp. 486 and 491)

In the implementation of the model we have adopted a further approximation, only the band edges are consider to the be discontinuous across the nanostructure interfaces. The motivation is two-fold. In first place we preserve the Hermiticity of the Hamiltonian avoiding the problem of the operator symmetrization. In second place, we simplify the calculation of the Hamiltonian matrix elements, which can be computed just once significantly speeding up the construction of the Hamiltonian. Such an approximation is justified for QDs exhibiting a type-I alignment, as the wavefunction is strongly localized in a homogeneous domain, inside of the nanostructure. However, our system could exhibit either type-I or type-II band alignment, meaning that in the latter case either the conduction and valence band states will be hosted by different materials. To solve that issue, we decided to decouple both bands which implies to nullify the Kane parameter. Hence the holes states will be described by a $6 \times 6$ Hamiltonian and the electron states by a $2 \times 2$ Hamiltonian. As most of the phenomenology observed in the experiments is associated to the properties of the hole states, we consider this approximation as safe in the context of the current study.

The strain field is calculated as if the semiconductor material were elastically isotropic. The procedure is based on the Eshelby’s inclusions method [86]. The lattice mismatch between the quantum dot and overlayer with respect to the GaAs matrix can be computed independently because of the linearity of elasticity equations. Analytical and compact expressions are obtained for the Fourier transform of the strain tensor [87]. In the past, we successfully used this method to explain Raman [88, 89] and middle energy ion scattering [90] experiments.

The basic equations of the $\mathbf {k}\cdot \mathbf {p}$ method are very compact. However, when many bands are considered to be in interaction a broad set of choices in terms of notation and basis expansion appears. To maintain the consistency in the development of the model, we follow the derivation of Trebin et al. [91, 92]. The final Hamiltonian results of adding the $\mathbf {k}\cdot \mathbf {p}$, the strain interactions (Bir-Pikus Hamiltonian) and the magnetic interactions:

$$\begin{aligned} H=H_{\mathbf {k}\cdot \mathbf {p}} + H_\varepsilon + H_B. \end{aligned}$$

(3.12)

We have neglected the linear $k_i$ terms in the valence-valence interaction, the quadratic $k_ik_j$ and $\varepsilon _{ij}$ terms in the conduction-valence interaction and the ${k_i}{\varepsilon _{jk}}$ in the strain-induced interactions. The solution of the Schrödinger-like equation is obtained by expanding the envelope: wavefunction of (3.2) in the eigenfunctions of a hard-wall cylinder:

$$\begin{aligned} \mathcal {F}^{(M)}_{m,k} = \sum _{\alpha ,\mu } \mathcal {N}^{(m)}_\alpha J_m\left( k_\alpha ^{(m)}\rho /\mathcal {R}\right) \sqrt{2/\mathcal {Z}} \sin \left[ \mu \pi \left( z/\mathcal {Z} - 1/2\right) \right] , \end{aligned}$$

(3.13)

where $J_m(x)$ is the Bessel function of order m, $k_\alpha ^{(m)}$ is its zero number $\alpha =1,2,\ldots$, $\mu =1,2,\ldots$, $\mathcal {R}$ and $\mathcal {Z}$ are the radius and height of the expansion cylinder and

$$\begin{aligned} N_\alpha ^{(m)}=\frac{\sqrt{2}}{\mathcal {R}\left| J_{m+1} \left( k_\alpha ^{(m)}\right) \right| } \end{aligned}$$

(3.14)

is the normalization of the radial part. This definitions ensure the orthonormality of the expansion basis. By writing the Hamiltonian (3.12) in the associated representation, the electron and hole states are obtained after solving the eigenvalue problem. A similar procedure was followed in  [93, 94]. Further details can be found in [95].