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Effect of the surface on the energy levels in PbS and PbSe nanocrystals

This study employs general boundary conditions within a four-band effective-mass model to demonstrate that surface properties significantly modify the size dependence and spin-orbit splitting of energy levels in PbS and PbSe nanocrystals, enabling a quantitative description of their optical transitions and the spectral differences between bare and Cl-passivated samples.

Original authors: O. Lundgren, A. V. Rodina, Al. L. Efros

Published 2026-10-30

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: O. Lundgren, A. V. Rodina, Al. L. Efros

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Tiny crystals, just a few billionths of a meter across, behave very differently from the bulk materials they are made of. When a piece of lead sulfide or lead selenide is shrunk down to the size of a few dozen atoms, the electrons and holes inside are squeezed into a tiny space, forcing them to occupy specific energy levels rather than a continuous range. This phenomenon, known as quantum confinement, is the reason these nanocrystals glow in specific colors and why they are so valuable for technologies like solar cells and medical imaging. For decades, scientists have tried to predict exactly what these energy levels would be, using mathematical models that treated the surface of the crystal as a hard, impenetrable wall where the electron waves simply stop. However, this standard approach has struggled to explain why the light these crystals absorb and emit sometimes shifts in ways that don't match the predictions, particularly when the crystals are very small or when their surfaces are treated with different chemicals.

A team of researchers has now revisited this problem by looking more closely at the boundary where the crystal meets the outside world. Instead of assuming the electron waves vanish instantly at the surface, they proposed that the surface acts more like a complex filter that modifies the waves as they reach the edge. By applying these more flexible rules, the researchers were able to calculate the energy levels of electrons and holes in spherical lead sulfide and lead selenide nanocrystals with much greater accuracy. Their work reveals that the surface of the crystal does more than just hold the particles in; it actively reshapes their energy landscape, creating a subtle splitting of energy levels that was previously overlooked. This new understanding allows them to match experimental data perfectly, explaining why bare crystals and those coated with chlorine atoms behave so differently.

The study began by acknowledging that previous models, which assumed the electron waves hit a dead end at the surface, failed to account for the full picture. In reality, the surface of a semiconductor is a unique environment that can influence the behavior of the particles inside. The researchers developed a new set of rules, which they call generalized boundary conditions, to describe how the electron and hole waves interact with this surface. These rules allow the waves to exist at the surface but require them to satisfy specific conditions regarding how they flow, rather than forcing them to disappear. This approach is crucial because the surface properties can change depending on how the crystal was made or what chemicals are attached to it.

When the team applied these new rules to their calculations, they discovered a significant effect: the surface induces a splitting of energy levels that depends on the spin of the particles. In simple terms, the energy levels that were once thought to be single, distinct states actually split into two slightly different levels. This splitting is caused by the interaction between the particle's spin and its motion near the surface, a phenomenon known as spin-orbit coupling. The researchers found that this effect is much stronger in these lead-based nanocrystals than in other materials like cadmium selenide, and it becomes more pronounced as the crystals get smaller. This splitting helps explain why certain optical transitions, which are the jumps electrons make between energy levels when they absorb or emit light, appear at slightly different energies than expected.

The power of this new model became clear when the researchers compared their calculations to real-world measurements. They looked at data for bare lead sulfide and lead selenide nanocrystals, as well as lead sulfide crystals that had been treated with chlorine atoms. The standard models, which assumed the waves vanished at the surface, consistently predicted energy levels that were too high compared to what was observed in experiments. In contrast, the new model, using the generalized boundary conditions, matched the experimental data with remarkable precision. For the bare crystals, the researchers found that the surface parameters they needed to fit the data were negative, suggesting that the effective size of the region where the electrons are confined is slightly larger than the physical size of the crystal measured by electron microscopy.

The situation changed dramatically when they looked at the chlorine-passivated crystals. These are crystals where the surface atoms have been bonded to chlorine, a treatment known to improve the efficiency of the light they emit. The researchers found that for these crystals, the surface parameter became positive. This shift indicates that the effective region where the electrons are confined has actually shrunk, becoming smaller than the physical crystal. This additional confinement pushes the energy levels higher, which explains the observed blue shift in the light these crystals absorb and emit. The model successfully described both the lowest energy transition and the second transition, which had been a subject of debate. The researchers determined that the second transition corresponds to specific allowed jumps between energy levels, rather than the forbidden jumps that some had suggested.

The implications of these findings extend beyond just getting the numbers right. The researchers noted that the surface-induced changes in energy levels also affect how the electrons and holes respond to magnetic fields, a property known as the g-factor. Because the surface parameters differ between bare and chlorine-treated crystals, the magnetic properties of these materials should also differ. Furthermore, the fact that the chlorine treatment effectively shrinks the confinement region suggests that the electrons and holes are held more tightly within the crystal. This tighter confinement could be the reason why chlorine-treated crystals are so efficient at emitting light, as it prevents the excited particles from escaping into the surrounding material where they would lose their energy without producing light.

By treating the surface not as a simple wall but as a dynamic interface that modifies the behavior of the particles inside, the researchers have provided a clearer picture of how these tiny crystals work. Their work demonstrates that the surface is not merely a container but an active participant in determining the electronic structure of nanocrystals. This insight is vital for designing better materials for future technologies, as it allows scientists to predict and tune the properties of these crystals by controlling their surface chemistry. The study confirms that to truly understand the behavior of nanocrystals, one must look beyond the bulk material and pay close attention to the subtle, yet powerful, influence of the surface.

Technical Summary: Effect of the Surface on the Energy Levels in PbS and PbSe Nanocrystals

Problem Statement
The electronic structure of spherical PbS and PbSe nanocrystals (NCs) has historically been modeled using a four-band effective-mass formalism (the Kang–Wise model) employing standard boundary conditions (SBCs). These SBCs assume that the carrier envelope wave functions vanish at the NC surface. While this approach reasonably describes the size dependence of the first exciton transitions, it fails to account for the specific influence of the NC surface on the level structure. Consequently, the model cannot fully explain the energy of the second exciton transition in PbSe or the differences observed between the absorption spectra of bare PbS NCs and those passivated by chlorine (Cl). Furthermore, the standard model does not predict the surface-induced spin–orbit splitting of excited states, which is expected to be significant in narrow-gap semiconductors like PbS and PbSe.

Methodology
The authors investigate the electron and hole quantum confinement levels within a four-band $k \cdot p$ effective-mass model, focusing on the Brillouin zone $L$ points. To address the limitations of the SBCs, the study derives and applies General Boundary Conditions (GBCs) at the abrupt spherical surface of the NC, modeled as an impenetrable barrier.

The methodology involves:

Derivation of GBCs: The authors derive GBCs by enforcing the continuity of the normal component of the envelope flux density matrix at the surface. This leads to a set of boundary conditions involving a Hermitian surface matrix characterized by three real, energy-independent parameters ( $t_c, t_v, t$ ).
Analytical Solutions: The authors solve the four-band Hamiltonian for the radial envelope functions under these GBCs. They analyze two limiting cases: an approximate model where inter-band coupling parameters $\alpha_c \alpha_v = 0$ , and the complete model where $\alpha_c \alpha_v \neq 0$ .
Phenomenological Fitting: The derived analytical equations for energy levels are used to fit experimental size-dependent absorption data. The authors extract phenomenological surface parameters ( $T_c$ $T_{c}$ and $T_v$ $T_{v}$ ) for:
- Bare PbS NCs.
- Bare PbSe NCs.
- Cl-passivated PbS NCs.
Comparison: The results obtained using GBCs are compared against the Kang–Wise model (SBCs) and various experimental datasets, including ground-state and excited-state optical transitions.

Key Contributions and Results

Surface-Induced Spin–Orbit Splitting: The application of GBCs reveals that the surface properties induce an additional spin–orbit splitting of the excited $1P_{e,h}$ levels. In the standard SBC model, the spin–orbit splitting of the first excited states is small and has different signs for electrons and holes. In contrast, the GBC model predicts a significant splitting where the surface parameter determines the magnitude and sign. This splitting arises from the admixture of valence band states near the surface, which is non-zero because the envelope wave function does not vanish at the boundary under GBCs.
Quantitative Description of Bare NCs: The GBC model successfully quantitatively describes the size dependence of the lowest optical transitions ( $1S_{e} - 1S_{h}$ ) in bare PbS and PbSe NCs. The authors find that negative values for the surface parameters ( $T_c$ and $T_v$ ) are required to match experimental data. These negative parameters imply that the effective electronic radius of the NC is larger than the ionic radius measured by transmission electron microscopy (TEM).
Explanation of Cl-Passivated NCs: The model explains the significant blue shift observed in the optical transitions of Cl-passivated PbS NCs. For these systems, the fitting yields a positive $T_c$ and a large negative $T_v$ . A positive $T_c$ indicates that the effective electronic radius is smaller than the ionic radius, leading to increased confinement and higher transition energies. The authors also incorporate a surface tension term to account for the lattice constant expansion observed in Cl-passivated NCs.
Assignment of Excited States: Using the GBC parameters derived from ground-state data, the authors quantitatively describe the size dependence of higher-order optical transitions in bare PbS. They identify the first six transitions as a combination of optically allowed ( $1P_{3/2}^h - 1P_{3/2}^e$ , etc.) and optically forbidden transitions, providing a better fit to experimental spectra than the SBC model.
Rejection of Alternative Assignments: The study demonstrates that assigning the second optical transition in Cl-passivated PbS to parity-forbidden transitions ( $1S_e - 1P_h$ or $1S_h - 1P_e$ ) results in a significantly worse fit (higher RMS error) compared to the assignment of allowed $1P$ -state transitions modified by surface effects.

Significance and Claims
The paper claims that the semiconductor surface plays a critical role in determining the energy structure of lead-halide NCs, an effect that cannot be captured by standard boundary conditions. By introducing GBCs, the authors provide a framework that:

Resolves discrepancies in the predicted energy of the second exciton transition.
Explains the differences in absorption spectra between bare and passivated NCs through the modification of effective electronic radii and surface tension.
Predicts surface-induced spin–orbit splitting and contributions to electron and hole g-factors.
Offers a potential explanation for the high photoluminescence quantum yield in Cl-passivated PbS NCs, suggesting that the reduced effective electronic radius enhances carrier confinement and suppresses nonradiative Auger recombination.

The authors conclude that the surface effect is particularly significant in small NCs and that the choice of boundary conditions is essential for the accurate theoretical description of narrow-gap semiconductor nanocrystals.

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