A tunnel splitting calculation matching Wentzel, Kramers, Brillouin (WKB) theory to within a sub-percent level has been achieved using a new computational framework called TDSE-Z. Accurately modelling electron behaviour at material interfaces, particularly in systems with spatially varying effective mass like semiconductor heterostructures, previously posed a key challenge for existing software.

Zakaria Dahbi and Amelle Zaïr of King’s College London have unveiled TDSE-Z, software designed to accurately simulate how electrons behave within materials possessing complex structures such as semiconductors. The set of tools uses a specific mathematical approach, a ‘weak-form’ method, to correctly handle situations where an electron’s effective mass changes throughout a material; this avoids inaccuracies seen with previous methods at material boundaries.

The team overcame the challenge of non-uniform electron ‘weight, or effective mass, using a weak-form method and flexible computer modelling grids, similar to bending wire into precise shapes rather than relying on rigid squares, to correctly manage variations in effective mass. Validated against established theoretical benchmarks including Wentzel, Kramers, Brillouin (WKB) theory, an approximation technique for estimating quantum tunnelling, TDSE-Z achieves sub-percent accuracy in predicting tunnel splitting within semiconductor structures. This framework enables more accurate simulations vital to designing next generation attosecond optoelectronics.

B-Spline Meshes resolve interface discontinuity for accurate semiconductor modelling

Computed tunnel splittings within a gallium arsenide/aluminium indium arsenide double quantum well now align with Wentzel, Kramers, Brillouin (WKB) theory to within the sub-percent level, representing a strong improvement over previous modelling techniques. Existing software struggled with discontinuities at material interfaces, where an electron’s effective mass changes abruptly, causing inaccuracies in simulations of semiconductor heterostructures. The new framework, TDSE-Z, overcomes this limitation by employing flexible B-spline meshes that adapt to variations in effective mass and correctly handle probability flux across boundaries unlike rigid grid systems used before.

Agreement between computed tunnel splittings within a gallium arsenide/aluminium indium arsenide double quantum well and Wentzel, Kramers, Brillouin (WKB) theory now reaches sub-percent levels; this is a significant advance on prior modelling approaches. Validation against an analytical harmonic oscillator benchmark demonstrated machine-precision accuracy, confirming continuous probability conservation alongside essential algebraic scaling for position-dependent mass frameworks.

Further testing using the double quantum well structure revealed a ground state energy splitting of 2.8615 meV with a barrier width of 4nm, a result closely matching WKB predictions at only 1.2% deviation, and confirms accurate capture of exponential attenuation at heterojunctions.

Modelling variable electron behaviour in layered semiconductors with adaptive grid technology

Accurately simulating electrons within layered semiconductor materials is important when designing faster and more efficient electronic devices; however, modelling their behaviour at layer interfaces has long presented challenges. TDSE-Z excels at modelling materials where an electron’s ‘weight’, or effective mass, the property influencing its mobility, changes as it travels through different layers of a semiconductor device. Adaptable grids represent a strong step forward for specific applications. This acknowledges that simulating electrons remains computationally intensive and existing software may address some needs.

This new computational framework enables detailed simulations of electron behaviour in complex materials by addressing limitations encountered when modelling systems with varying electron effective masses across layers such as semiconductor heterostructures.

The researchers developed TDSE-Z, a high-performance computer program capable of accurately modelling how electrons behave within layered semiconductors where an electron’s ‘weight’, or effective mass, varies between material layers. Validating the program against established theoretical models and a gallium arsenide/aluminium indium arsenide double quantum well demonstrated its accuracy, tunnel splitting predictions agreed with theory to within sub-percent levels. This improved simulation capability means scientists can now better understand electron dynamics in these materials, potentially aiding the design of future electronic devices. The framework is designed for expansion towards more powerful computing architectures.

👉 More information
🗞 Unified Strong-Field Dynamics Simulations from Atoms to Heterostructures
✍️ Zakaria Dahbi and Amelle Zaïr
🧠 ArXiv: https://arxiv.org/abs/2608.18472

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