Tetrahedral InP/ZnSe Quantum Dots

• The study employs a multi-band k•p methodology to accurately model electron and hole states, revealing tunable excitonic features and energy separations in tetrahedral InP/ZnSe QDs.
• Tetrahedral symmetry relaxes traditional optical selection rules, enabling weak symmetry-violating transitions and distinctly sharp absorption resonances.
• The non-toxic design and robust charge confinement of these QDs make them ideal for high-brightness LEDs, bioimaging, and photovoltaic down-conversion applications.

Tetrahedral core/shell InP/ZnSe quantum dots (QDs) are nanoscale heterostructures characterized by an indium phosphide (InP) core and a zinc selenide (ZnSe) shell, both crystallizing in the zinc-blende lattice and exhibiting overall tetrahedral ($\overline{T}_d$) symmetry. This distinct geometry relaxes the optical selection rules compared to spherical QDs while retaining strong quantum confinement, leading to a robust and spectrally distinct excitonic structure. These QDs are of significant interest due to their non-toxic constituent materials and their tunable optoelectronic properties, relevant for applications in optoelectronics, photonics, and bioimaging (Planelles et al., 9 Dec 2025).

1. Electronic Structure Models and Theoretical Framework

The electronic states of tetrahedral InP/ZnSe QDs are accurately represented through a multi-band $k \cdot p$ methodology, combining a two-band position-dependent effective-mass description for electrons with a six-band Luttinger–Kohn Hamiltonian for holes. The electron Hamiltonian takes the form

$$H_e = \frac{\hbar^2}{2 m_0} \mathbf{p} \cdot \left[\frac{1}{m_e(\mathbf{r})}\right] \mathbf{p} + V_e(\mathbf{r})$$

where the effective masses and confining potentials are spatially dependent: $m_e(\text{InP}) = 0.08 m_0$, $m_e(\text{ZnSe}) = 0.16 m_0$, $E_g(\text{InP}) = 1.42~\text{eV}$, $E_g(\text{ZnSe}) = 2.82~\text{eV}$, and a conduction band offset (CBO) of $0.5$ eV.

For holes, the six-band Luttinger–Kohn Hamiltonian incorporates the coupled $\Gamma_8$ (HH/LH) and $\Gamma_7$ (split-off) bands, with parameterization $k \cdot p$0 for InP, and a deep valence band offset (VBO) of $k \cdot p$1 eV. Explicit strain effects are neglected, but finite-element numerical implementation (COMSOL) is applied with hard-wall potentials and uniform Luttinger parameters.

The tetrahedral symmetry (double group $k \cdot p$2) modifies the traditional angular momentum classification of states. Envelope functions with $k \cdot p$3 transform, respectively, as irreducible representations $k \cdot p$4, $k \cdot p$5, and $k \cdot p$6 under $k \cdot p$7, partially relaxing optical selection rules and allowing weakly observable symmetry-violating transitions (Planelles et al., 9 Dec 2025).

2. Single-Particle States: Energies and Spatial Distributions

Electron States

The ground electronic state ($k \cdot p$8-like) resides predominantly within the InP core. For representative tetrahedral QDs ($k \cdot p$9 nm core, $$H_e = \frac{\hbar^2}{2 m_0} \mathbf{p} \cdot \left[\frac{1}{m_e(\mathbf{r})}\right] \mathbf{p} + V_e(\mathbf{r})$$0 nm shell), approximately $$H_e = \frac{\hbar^2}{2 m_0} \mathbf{p} \cdot \left[\frac{1}{m_e(\mathbf{r})}\right] \mathbf{p} + V_e(\mathbf{r})$$1 of the $$H_e = \frac{\hbar^2}{2 m_0} \mathbf{p} \cdot \left[\frac{1}{m_e(\mathbf{r})}\right] \mathbf{p} + V_e(\mathbf{r})$$2 charge density is confined to the core, with moderate delocalization into the ZnSe shell. The $$H_e = \frac{\hbar^2}{2 m_0} \mathbf{p} \cdot \left[\frac{1}{m_e(\mathbf{r})}\right] \mathbf{p} + V_e(\mathbf{r})$$3 energy separation spans $$H_e = \frac{\hbar^2}{2 m_0} \mathbf{p} \cdot \left[\frac{1}{m_e(\mathbf{r})}\right] \mathbf{p} + V_e(\mathbf{r})$$4 eV across core sizes, and redshifts with increasing $$H_e = \frac{\hbar^2}{2 m_0} \mathbf{p} \cdot \left[\frac{1}{m_e(\mathbf{r})}\right] \mathbf{p} + V_e(\mathbf{r})$$5 while maintaining a minimum gap of $$H_e = \frac{\hbar^2}{2 m_0} \mathbf{p} \cdot \left[\frac{1}{m_e(\mathbf{r})}\right] \mathbf{p} + V_e(\mathbf{r})$$6 meV between $$H_e = \frac{\hbar^2}{2 m_0} \mathbf{p} \cdot \left[\frac{1}{m_e(\mathbf{r})}\right] \mathbf{p} + V_e(\mathbf{r})$$7 and excited electron levels.

Hole States and Band Mixing

Hole ground states are dominated by heavy-hole/light-hole (HH/LH) character (~$$H_e = \frac{\hbar^2}{2 m_0} \mathbf{p} \cdot \left[\frac{1}{m_e(\mathbf{r})}\right] \mathbf{p} + V_e(\mathbf{r})$$8) but exhibit an enhanced split-off-hole (SOH) admixture relative to, for example, CdSe QDs (SOH $$H_e = \frac{\hbar^2}{2 m_0} \mathbf{p} \cdot \left[\frac{1}{m_e(\mathbf{r})}\right] \mathbf{p} + V_e(\mathbf{r})$$9 for $m_e(\text{InP}) = 0.08 m_0$0 nm, exceeding $m_e(\text{InP}) = 0.08 m_0$1 for excited states). In spherical cores, the ground state transitions from bright ($m_e(\text{InP}) = 0.08 m_0$2) to dark ($m_e(\text{InP}) = 0.08 m_0$3) as $m_e(\text{InP}) = 0.08 m_0$4 increases, but in tetrahedral cores, both $m_e(\text{InP}) = 0.08 m_0$5 and $m_e(\text{InP}) = 0.08 m_0$6 map onto the same $m_e(\text{InP}) = 0.08 m_0$7 symmetry, producing anticrossing rather than inversion—the hole ground state remains optically active even at large core sizes (Planelles et al., 9 Dec 2025).

Degeneracies and Symmetry Effects

Degeneracies expected in the spherical limit—$m_e(\text{InP}) = 0.08 m_0$8-fold for $m_e(\text{InP}) = 0.08 m_0$9, $m_e(\text{ZnSe}) = 0.16 m_0$0-fold for $m_e(\text{ZnSe}) = 0.16 m_0$1, etc.—are only slightly split in the tetrahedral case, reflecting mild symmetry-induced envelope mixing.

3. Excitonic Spectrum and Optical Transitions

Near-Band-Edge Absorption

The QDs display prominent absorption resonances at

• $m_e(\text{ZnSe}) = 0.16 m_0$2
• $m_e(\text{ZnSe}) = 0.16 m_0$3
• $m_e(\text{ZnSe}) = 0.16 m_0$4 (approximately $m_e(\text{ZnSe}) = 0.16 m_0$5 eV above the band edge).

Electron-hole Coulomb attraction, evaluated with $m_e(\text{ZnSe}) = 0.16 m_0$6 and $m_e(\text{ZnSe}) = 0.16 m_0$7, induces a quasi-rigid redshift of roughly $m_e(\text{ZnSe}) = 0.16 m_0$8 meV, leaving the spectral spacings unchanged. This outcome reflects that $m_e(\text{ZnSe}) = 0.16 m_0$9, establishing first-order perturbative corrections as sufficient for excitonic and multiexcitonic energies (Planelles et al., 9 Dec 2025).

Oscillator Strength and Symmetry-Violating Transitions

The oscillator strength for the $E_g(\text{InP}) = 1.42~\text{eV}$0 band-edge transition rises with core size because of improved spatial e–h overlap and peaks, with no subsequent decay in tetrahedral geometry. In contrast, spherical QDs show a minor reduction due to $E_g(\text{InP}) = 1.42~\text{eV}$1 anticrossing. For red-emitting large QDs ($E_g(\text{InP}) = 1.42~\text{eV}$2 nm), weak transitions such as $E_g(\text{InP}) = 1.42~\text{eV}$3 (forbidden in the spherical limit, $E_g(\text{InP}) = 1.42~\text{eV}$4) become faintly observable (oscillator strength $E_g(\text{InP}) = 1.42~\text{eV}$5 of the main band-edge transition), a direct manifestation of $E_g(\text{InP}) = 1.42~\text{eV}$6-driven envelope mixing.

The tetrahedral symmetry precludes the development of a dark $E_g(\text{InP}) = 1.42~\text{eV}$7-like exciton ground state, diverging from the trends established in spherical geometries (Planelles et al., 9 Dec 2025).

4. Coulomb Terms and Multiparticle Excitonic Interactions

Perturbative Coulomb Interactions

The strong core/shell confinement of the $E_g(\text{InP}) = 1.42~\text{eV}$8 state ensures that the single-particle gap $E_g(\text{InP}) = 1.42~\text{eV}$9 meV is significantly larger than the mean electron–electron repulsion ($E_g(\text{ZnSe}) = 2.82~\text{eV}$0 meV) and electron–hole attraction ($E_g(\text{ZnSe}) = 2.82~\text{eV}$1 meV). This justifies the perturbative regime for exciton, trion, and biexciton binding energies.

Trion and Biexciton Binding

For a typical case ($E_g(\text{ZnSe}) = 2.82~\text{eV}$2 nm, $E_g(\text{ZnSe}) = 2.82~\text{eV}$3 nm):

• $E_g(\text{ZnSe}) = 2.82~\text{eV}$4 meV
• $E_g(\text{ZnSe}) = 2.82~\text{eV}$5 meV
• $E_g(\text{ZnSe}) = 2.82~\text{eV}$6 meV

The first-order spectroscopic shifts are:

• $E_g(\text{ZnSe}) = 2.82~\text{eV}$7 meV (bound trion)
• $E_g(\text{ZnSe}) = 2.82~\text{eV}$8 meV (antibound trion)
• Configuration interaction refines these to $E_g(\text{ZnSe}) = 2.82~\text{eV}$9 meV and $0.5$0 meV, respectively.

Biexciton ($0.5$1) binding $0.5$2 switches from positive (bound) in small QDs to negative (antibound) in larger QDs, tracking the relative strengths of Coulomb attractions and repulsions. Electron correlations are minimal: the $0.5$3 wave function remains nearly invariant between $0.5$4 and $0.5$5, with less than 5% change in shell delocalization (Planelles et al., 9 Dec 2025).

5. Optical Signatures and Symmetry-Driven Deviations

Tetrahedral InP/ZnSe QDs preserve the classic near-edge $0.5$6–$0.5$7 excitonic absorption, marked by a dominant $0.5$8 transition, a $0.5$9 meV fine structure split (arising from spin-orbit coupling), and a higher-energy $\Gamma_8$0 resonance. Deviations manifest in the largest core QDs as weakly allowed $\Gamma_8$1 transitions and the absence of a dark $\Gamma_8$2 ground-state exciton, both stemming from $\Gamma_8$3 symmetry and cubic band warping.

The table below summarizes the main symmetry properties and allowed transitions:

Envelope $\Gamma_8$4	$\Gamma_8$5 Irrep	Sample Allowed Transition(s)
0	$\Gamma_8$6	$\Gamma_8$7
1	$\Gamma_8$8	($\Gamma_8$9, symmetry-weak)
2	$\Gamma_7$0	$\Gamma_7$1

All irreducible representations for low $\Gamma_7$2 mixing arise under $\Gamma_7$3, diminishing strict angular momentum selection rules and enabling additional, though typically weak, optical transitions (Planelles et al., 9 Dec 2025).

6. Implications for Applications

The strong core/shell quantum confinement, absence of toxic cadmium, and robustly bound excitonic states make tetrahedral InP/ZnSe QDs promising for optoelectronic applications demanding narrow emission linewidths, spectrally stable exciton and multiexciton features, and tunable photoluminescence lifetimes. The structure allows for robust assignment and control of $\Gamma_7$4, $\Gamma_7$5, $\Gamma_7$6, and $\Gamma_7$7 spectral features. Key potential applications include high-brightness LEDs, biological labeling, and photovoltaic down-conversion layers.

The absence of “dark” excitonic ground states in large core QDs, tunable through symmetry-engineering rather than alloying or heavy element doping, offers a strategy for optimizing quantum dot emission efficiency while maintaining environmental safety (Planelles et al., 9 Dec 2025).