Galerkin Method for Calculating Valence-Band Wavefunctions in Quantum-Well Structures Using Exact Envelope Theory
Bibliographic record
Abstract
The ability to calculate accurately the valence-band structure in semiconductors is important in the design of quantum-well (QW) semiconductor devices. The Galerkin method for calculating accurate analytic approximations for multicomponent valence-band wavefunctions is computationally fast and efficient. The Galerkin method is, in fact, an improved version of an earlier Raleigh-Ritz-type variational method. In this paper, we remark that both the variational method and the more recently proposed Galerkin method are formulated such that they imply symmetrized boundary constraints at material interfaces, with the symmetrized nature of the constraints arising from neglecting the ordering of the operators. Burt's exact envelope-function theory for semiconductor microstructures has been used to demonstrate, however, that the commonly used symmetrized boundary constraints for material interfaces are unphysical. We therefore present a modified version of the Galerkin method that implicitly assumes physically reasonable, exact envelope-function boundary constraints. Simulations show that the modified Galerkin method successfully produces physical, semi-analytic results that are consistent with exact envelope theory.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".