{"paper":{"title":"Electrostatics in semiconducting devices II: Solving the Helmholtz equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Mapping the self-consistent electrostatic problem to a non-linear Helmholtz equation permits provably convergent iterations that reach the exact solution in one or two steps.","cross_cats":["physics.comp-ph"],"primary_cat":"cond-mat.mes-hall","authors_text":"Antonio Lacerda-Santos, Xavier Waintal","submitted_at":"2025-07-03T19:14:37Z","abstract_excerpt":"The convergence of iterative schemes to achieve self-consistency in mean field problems such as the Schr\\\"odinger-Poisson equation is notoriously capricious. It is particularly difficult in regimes where the non-linearities are strong such as when an electron gas in partially depleted or in presence of a large magnetic field. Here, we address this problem by mapping the self-consistent quantum-electrostatic problem onto a Non-Linear Helmoltz (NLH) equation at the cost of a small error. The NLH equation is a generalization of the Thomas-Fermi approximation. We show that one can build iterative "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show that one can build iterative schemes that are provably convergent by constructing a convex functional whose minimum is the seeked solution of the NLH problem. In a second step, the approximation is lifted and the exact solution of the initial problem found by iteratively updating the NLH problem until convergence. We show empirically that convergence is achieved in a handfull, typically one or two, iterations.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The self-consistent quantum-electrostatic problem can be mapped onto the Non-Linear Helmholtz equation at the cost of only a small error that is later removed by iterative lifting to the exact solution.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Maps the quantum-electrostatic self-consistency problem to a non-linear Helmholtz equation, enabling construction of a convex functional for provable convergence that lifts to the exact solution in typically one or two iterations.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Mapping the self-consistent electrostatic problem to a non-linear Helmholtz equation permits provably convergent iterations that reach the exact solution in one or two steps.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"8205d6d856d531d60c536a333bf3d5175efedb05e9b04eaa5132241b2ca99a79"},"source":{"id":"2507.03131","kind":"arxiv","version":5},"verdict":{"id":"13d0b867-787e-4276-9f4c-3bd708bb190c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T05:44:30.937350Z","strongest_claim":"We show that one can build iterative schemes that are provably convergent by constructing a convex functional whose minimum is the seeked solution of the NLH problem. In a second step, the approximation is lifted and the exact solution of the initial problem found by iteratively updating the NLH problem until convergence. We show empirically that convergence is achieved in a handfull, typically one or two, iterations.","one_line_summary":"Maps the quantum-electrostatic self-consistency problem to a non-linear Helmholtz equation, enabling construction of a convex functional for provable convergence that lifts to the exact solution in typically one or two iterations.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The self-consistent quantum-electrostatic problem can be mapped onto the Non-Linear Helmholtz equation at the cost of only a small error that is later removed by iterative lifting to the exact solution.","pith_extraction_headline":"Mapping the self-consistent electrostatic problem to a non-linear Helmholtz equation permits provably convergent iterations that reach the exact solution in one or two steps."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.03131/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":37,"sample":[{"doi":"","year":2025,"title":"Electrostatics in semiconducting devices I : The Pure Electrostatics Self Consistent Approximation","work_id":"53a01970-5bc2-4ad1-8801-615e859866b3","ref_index":1,"cited_arxiv_id":"2502.15897","is_internal_anchor":true},{"doi":"","year":2025,"title":"A. Lacerda-Santos, C. Groth and X. Waintal, Electrostatics in semiconducting devices iii : The pescado open source library, In preparation (2025)","work_id":"a7d7715d-9ce9-4917-9cbb-67f0f149bea3","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.48550/arxiv.2407.16257","year":2024,"title":"Computational quantum transport: a scattering approach perspective","work_id":"d740f6b3-1244-42d4-b6c9-b7e91d571add","ref_index":3,"cited_arxiv_id":"2407.16257","is_internal_anchor":true},{"doi":"","year":2014,"title":"C. W . Groth, M. Wimmer, A. R. Akhmerov and X. Waintal,Kwant: a software package for quantum transport, New Journal of Physics 16(6), 063065 (2014), doi:10.1088 /1367- 2630/16/6/063065","work_id":"b3f54de7-b502-485c-8233-236560fc8816","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"E. Chatzikyriakou, J. Wang, L. Mazzella, A. Lacerda-Santos, M. C. d. S. Figueira, A. Trel- lakis, S. Birner, T . Grange, C. Bäuerle and X. Waintal,Unveiling the charge distribution of a gaas-based nan","work_id":"650a885e-f94c-4a74-aaf1-7d9b24143e90","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":37,"snapshot_sha256":"4ed22fc1c198422a1e2aaf9cbac7a83d5453a7f1b996546558c3c96fafb9c815","internal_anchors":2},"formal_canon":{"evidence_count":2,"snapshot_sha256":"653a733a99e40ef04ca8077932d9d22800276e0d4a9d5102639cd73b9bc5fbae"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}