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 $$(\gamma_1, \gamma_2, \gamma_3) = (5.08, 1.60, 2.10)$$ for InP, and a deep valence band offset (VBO) of $0.9$ 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 $\overline{T}_d$) modifies the traditional angular momentum classification of states. Envelope functions with $L=0,1,2$ transform, respectively, as irreducible representations $\Gamma_1$, $\Gamma_5$, and $\Gamma_3 \oplus \Gamma_5$ under $\overline{T}_d$, 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 ($1S_e$-like) resides predominantly within the InP core. For representative tetrahedral QDs ($r_c = 3.0$ nm core, $r_s = 8$ nm shell), approximately $70\text{--}80\%$ of the $1S_e$ charge density is confined to the core, with moderate delocalization into the ZnSe shell. The $\Delta(1P_e - 1S_e)$ energy separation spans $0.3\text{--}0.5$ eV across core sizes, and redshifts with increasing $r_c$ while maintaining a minimum gap of $300$ meV between $1S_e$ and excited electron levels.

Hole States and Band Mixing

Hole ground states are dominated by heavy-hole/light-hole (HH/LH) character (~$92\%$) but exhibit an enhanced split-off-hole (SOH) admixture relative to, for example, CdSe QDs (SOH $\sim8\%$ for $r_c = 3$ nm, exceeding $20\%$ for excited states). In spherical cores, the ground state transitions from bright ($1S_{3/2}$) to dark ($1P_{3/2}$) as $r_c$ increases, but in tetrahedral cores, both $1S_{3/2}$ and $1P_{3/2}$ map onto the same $\Gamma_8$ 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—$2$-fold for $1S_e$, $4$-fold for $1S_{3/2}$, 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

• $E(1S_{3/2} \rightarrow 1S_e)$
• $$E(1S_{1/2} \rightarrow 1S_e) = E(1S_{3/2} \rightarrow 1S_e) + 60\text{--}70~\text{meV}$$
• $E(2S_{3/2} \rightarrow 1S_e)$ (approximately $0.3\text{--}0.5$ eV above the band edge).

Electron-hole Coulomb attraction, evaluated with $\epsilon_{in} = 10$ and $\epsilon_{out} = 2$, induces a quasi-rigid redshift of roughly $200$ meV, leaving the spectral spacings unchanged. This outcome reflects that $\Delta(1P_e - 1S_e) \gg |V_{eh}|$, 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 $1S_{3/2} \rightarrow 1S_e$ 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 $1P_{3/2}$ anticrossing. For red-emitting large QDs ($r_c \gtrsim 6$ nm), weak transitions such as $1P_{3/2} \rightarrow 1S_e$ (forbidden in the spherical limit, $\Delta L = \pm 1$) become faintly observable (oscillator strength $<5\%$ of the main band-edge transition), a direct manifestation of $\overline{T}_d$-driven envelope mixing.

The tetrahedral symmetry precludes the development of a dark $P_{3/2}$-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 $1S_e$ state ensures that the single-particle gap $\Delta(1P_e-1S_e) \approx 300\text{--}500$ meV is significantly larger than the mean electron–electron repulsion ($\langle V_{ee} \rangle \approx 250$ meV) and electron–hole attraction ($\langle V_{eh} \rangle \approx -270$ meV). This justifies the perturbative regime for exciton, trion, and biexciton binding energies.

Trion and Biexciton Binding

For a typical case ($r_c = 2$ nm, $r_s = 6$ nm):

• $\langle V_{eh} \rangle \approx -273$ meV
• $\langle V_{ee} \rangle \approx 249$ meV
• $\langle V_{hh} \rangle \approx 331$ meV

The first-order spectroscopic shifts are:

• $$\Delta E_{X^-} = \langle V_{eh} + V_{ee} \rangle \approx -24$$ meV (bound trion)
• $$\Delta E_{X^+} = \langle V_{eh} + V_{hh} \rangle \approx +58$$ meV (antibound trion)
• Configuration interaction refines these to $-20$ meV and $+44$ meV, respectively.

Biexciton ($XX$) binding $$\Delta E_{XX} = \langle 2V_{eh} + V_{ee} + V_{hh} \rangle$$ 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 $1S_e$ wave function remains nearly invariant between $X$ and $X^-$, 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 $S$–$S$ excitonic absorption, marked by a dominant $1S_{3/2} \rightarrow 1S_e$ transition, a $60\text{--}70$ meV fine structure split (arising from spin-orbit coupling), and a higher-energy $2S_{3/2} \rightarrow 1S_e$ resonance. Deviations manifest in the largest core QDs as weakly allowed $\Delta L = \pm 1$ transitions and the absence of a dark $P_{3/2}$ ground-state exciton, both stemming from $\overline{T}_d$ symmetry and cubic band warping.

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

Envelope $L$	$\overline{T}_d$ Irrep	Sample Allowed Transition(s)
0	$\Gamma_1$	$1S_{3/2} \rightarrow 1S_e$
1	$\Gamma_5$	($1P_{3/2} \rightarrow 1S_e$, symmetry-weak)
2	$\Gamma_3 \oplus \Gamma_5$	$1D_{3/2} \rightarrow 1S_e$

All irreducible representations for low $L$ mixing arise under $\overline{T}_d$, 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 $X$, $X^-$, $X^+$, and $XX$ 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).