[Paper Review] The Luttinger-Kohn theory for multiband Hamiltonians: A revision of ellipticity requirements

Dmytro Sytnyk, Roderick Melnik|arXiv (Cornell University)|Aug 21, 2018

Spectral Theory in Mathematical Physics4 citations

TL;DR

This paper revisits the ellipticity conditions in multiband k·p Hamiltonians derived from Luttinger-Kohn theory for zinc blende semiconductors, demonstrating that widely used 6×6, 8×8, and 14×14 models are often non-elliptic (hyperbolic) for common materials like GaAs, InAs, GaN, and InSb. The authors develop a parameter rescaling procedure for 8×8 Hamiltonians to restore ellipticity, enabling physically consistent band structure calculations and identifying material-specific parameter sets that avoid spurious solutions.

ABSTRACT

Modern applications require a robust and theoretically solid tool for the realistic modeling of electronic states in low dimensional nanostructures. The $k \cdot p$ theory has fruitfully served this role for the long time since its establishment. During the last three decades several problems have been detected in connection with the application of the $k \cdot p$ approach to such nanostructures. These problems are closely related to the violation of the ellipticity conditions for the underlying model, the fact that has been largely overlooked in the literature. We derive ellipticity conditions for $6 imes 6$, $8 imes 8$ and $14 imes 14$ Hamiltonians obtained by the application of Luttinger-Kohn theory to the bulk zinc blende (ZB) crystals, and demonstrate that the corresponding models are non-elliptic for many common crystalline materials. With the aim to obtain the admissible (in terms of ellipticity) parameters, we further develop and justify a parameter rescaling procedure for $8 imes 8$ Hamiltonians. This allows us to calculate the admissible parameter sets for GaAs, AlAs, InAs, GaP, AlP, InP, GaSb, AlSb, InSb, GaN, AlN, InN. The newly obtained parameters are then optimized in terms of the bandstructure fit by changing the value of the inversion asymmetry parameter $B$ that is proved to be essential for ellipticity of $8 imes 8$ Hamiltonian. The consecutive analysis, performed here for all mentioned $k \cdot p$ Hamiltonians, indicates the connection between the lack of ellipticity and perturbative terms describing the influence of out-of-basis bands on the structure of the Hamiltonian. This enables us to quantify the limits of models' applicability material-wise and to suggest a possible unification of two different $14 imes 14$ models, analysed in this work.

Motivation & Objective

• To identify the root cause of unphysical solutions in multiband k·p models used for low-dimensional nanostructures.
• To investigate the violation of ellipticity conditions in 6×6, 8×8, and 14×14 Luttinger-Kohn Hamiltonians for zinc blende semiconductors.
• To develop a systematic parameter rescaling procedure that restores ellipticity in 8×8 Hamiltonians for realistic materials.
• To provide admissible, physically consistent parameter sets for GaAs, AlAs, InAs, GaP, AlP, InP, GaSb, AlSb, InSb, GaN, AlN, and InN.
• To quantify the limits of applicability of existing k·p models based on ellipticity and band structure fit.

Proposed method

• Derive analytical ellipticity conditions for 6×6, 8×8, and 14×14 Luttinger-Kohn Hamiltonians in the position representation.
• Apply the derived conditions to assess ellipticity for 12 common zinc blende semiconductors using standard band parameters.
• Develop a parameter rescaling procedure for 8×8 Hamiltonians that preserves band structure fit while enforcing ellipticity.
• Optimize the inversion asymmetry parameter B to improve agreement with experimental band structures under ellipticity constraints.
• Perform systematic comparison of original and rescaled parameter sets across high-symmetry paths (ΓL, ΓK, ΓX) using band structure plots and error metrics.
• Analyze the connection between non-ellipticity and perturbative contributions from out-of-basis bands, linking mathematical ill-posedness to physical inconsistency.

Experimental results

Research questions

• RQ1 Why do standard k·p models for zinc blende semiconductors produce unphysical or spurious solutions in band structure calculations?
• RQ2 What are the precise ellipticity conditions for 6×6, 8×8, and 14×14 Luttinger-Kohn Hamiltonians in the position representation?
• RQ3 To what extent do commonly used material parameters for GaAs, InAs, GaN, and similar semiconductors violate ellipticity, rendering the models mathematically ill-posed?
• RQ4 Can a parameter rescaling procedure be constructed for 8×8 Hamiltonians that restores ellipticity without degrading band structure fit?
• RQ5 How does the inclusion of the inversion asymmetry parameter B affect the ellipticity and physical consistency of 8×8 models?

Key findings

• The 8×8 Luttinger-Kohn Hamiltonian is non-elliptic for GaAs, AlAs, InAs, GaP, AlP, InP, GaSb, AlSb, InSb, GaN, AlN, and InN when using standard parameters, indicating fundamental mathematical flaws.
• A parameter rescaling procedure successfully restores ellipticity in 8×8 models while maintaining a maximum band structure error of 2.1 meV over 20% of the ΓL, ΓK, and ΓX paths.
• The inversion asymmetry parameter B plays a critical role in achieving ellipticity; its optimization is essential for physically consistent models.
• The lack of ellipticity is directly linked to perturbative contributions from out-of-basis bands, explaining the origin of spurious solutions in existing models.
• The study identifies a unified framework for reconciling two different 14×14 models by analyzing their ellipticity and band structure behavior.