[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-127688-en":3,"doc-seo-127688-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},127688,962084925782,"Ava Thompson","https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc",8,"Research & Report","Machine learning the relationship between Debye temperature and superconducting transition temperature - Research report","Study investigates an asserted upper bound between Debye temperature ΘD and superconducting transition temperature Tc for conventional, phonon-mediated BCS superconductors. Data-driven machine learning models are trained on 10,330 Materials Project samples to predict ΘD, then applied to 9,860 known superconductors in the NIMS SuperCon database, where conventional systems follow the proposed Tc ≤ AΘD bound with A ~ 0.1. First-principles phonon calculations for H3S and LaH10 at 200 GPa show strong saturation of the predicted Tc–ΘD limit.","arXiv :2305 . 12977v2 [ cond-mat .supr-con] 27 Nov 2023  \nMachine learning the relationship between Debye temperature and superconducting  \ntransition temperature  \nAdam D. Smith, 1, ∗ Sumner B. Harris,2 Renato P. Camata, 1 Da Yan,3 and Cheng-Chien Chen 1,†  \n1 Department of Physics, University of Alabama at Birmingham, Birmingham, Alabama 35294, USA  \n2 Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA  \n3 Department of Computer Science, University of Alabama at Birmingham, Birmingham, Alabama 35294, USA (Dated: November 28, 2023)  \nRecently a relationship between the Debye temperature ΘD and the superconducting transition temperature Tc of conventional superconductors has been proposed [npj Quantum Materials 3, 59  \n(2018)] . The relationship indicates that Tc ≤ AΘD for phonon-mediated BCS superconductors, with A being a pre-factor of order ∼ 0.1. In order to verify this bound, we train machine learning  \n(ML) models with 10,330 samples in the Materials Project database to predict Θ D . By applying our ML models to 9,860 known superconductors in the NIMS SuperCon database, we find that the conventional superconductors in the database indeed follow the proposed bound. We also perform first-principles phonon calculations for H3 S and LaH10 at 200 GPa. The calculation results indicate  \nthat these high-pressure hydrides essentially saturate the bound of Tc versus ΘD .  \nI. INTRODUCTION  \nThe Eliashberg theory 1–3 describes frequencydependent phonon-mediated attractive interaction between electron Cooper pairs, by taking into account phonon dynamics and retardation effects. A key quantity in the theory is the Eliashberg spectral function α2 F (ω), which is related to the electronic density of states (DOS) near the Fermi level, the phonon spectra, and the electron-phonon (el-ph) coupling matrix elements. The Eliashberg function enables the determination of the total (isotropic) el-ph coupling parameter λ, which together with the effective Coulomb pseudopotential µ∗(typically chosen to be between ∼ 0.1 − 0. 15) can be utilized to estimate the Tc of conventional superconductors, via analytical expressions such as the McMillan4 , AllenDynes5 , or other related formula6 , obtained by fittings to numerical solutions of the Eliashberg equations.  \nThe Bardeen-Cooper-Schrieffer (BCS) theory7,8 dictates that Tc = 1 . 13ωce − ~~ 1~~λ , with ωc a characteristic phonon frequency. In the weak coupling limit of small λ, the Eliashberg theory Tc essentially reduces to the BCS Tc (up to a modified pre-factor9,10 ) . In the strong coupling limit of large λ, it was shown that Tc ∝ √λ in the Eliashberg theory5 . On the other hand, when λ becomes very large, the characteristic phonon frequency can be substantially re-normalized (softened) 11–15 , leading to a decrease in Tc. Moreover, in the presence of a significantly large λ, superconductivity can be suppressed by other instabilities such as charge density wave or by the formation of heavy polarons (localized charge carrier dressed by phonons) 16–20 , in which cases the Tc predicted by the Eliashberg theory is not applicable. More recently, determinant quantum Monte Carlo (DQMC) studies of two-dimensional Holstein models21–24 have shown that  \n∗ [smitha20@uab.edu](smitha20@uab.edu)[ ](smitha20@uab.edu)† [chencc@uab.edu](chencc@uab.edu)  \nthe Eliashberg theory might already fail when λ approaches a critical value λcr ∼ O(1), without the formation of any instability. In these regards, Tc would remain bounded even if λ can assume an arbitrarily large value.  \nBased on the DQMC results and the relationship between Tc and the Debye temperature ΘD for a few tens of selected compounds, Esterlis et al. have proposed that Tc ≤ AΘD for phonon-mediated superconductors25 . Here, A ∼ 0. 1 is a material-independent upper limit, while the actual material-specific ratio Tc /ΘD is determined by details of the band structure, el-ph coupling strength, Coulomb in","cbCaijAbU5cVebH9","https://ap.wps.com/l/cbCaijAbU5cVebH9","pdf",6216198,1,10,"English","en",105,"# Introduction\n# Computational Methods\n## Machine Learning Models\n## First-principles Phonon Calculations\n# Results and Discussion\n## Validation of the Tc–ΘD Bound\n## High-Pressure Hydrides (H3S, LaH10)","[{\"question\":\"What relationship between Debye temperature and Tc is examined in this study?\",\"answer\":\"The work tests the proposed bound Tc ≤ AΘD for phonon-mediated BCS superconductors, where A is a material-independent prefactor on the order of ~0.1.\"},{\"question\":\"How are machine learning models used to verify the Tc–ΘD bound?\",\"answer\":\"ML models are trained using 10,330 compounds from the Materials Project to predict ΘD, then applied to 9,860 superconductors in the SuperCon database with conventional superconductors shown to satisfy the bound.\"},{\"question\":\"What do first-principles phonon calculations for H3S and LaH10 show?\",\"answer\":\"Phonon calculations at 200 GPa indicate these high-pressure hydrides essentially saturate the Tc versus ΘD bound, giving Tc values roughly in the range 0.1 to 0.11 ΘD.\"}]","Machine learning the relationship between Debye temperature and superconducting transition temperature - Research report | PDF",1785940885,25,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"machine-learning-the-relationship-between-debye-temperature-and-superconducting-transition-temperature-research-report","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/machine-learning-the-relationship-between-debye-temperature-and-superconducting-transition-temperature-research-report/127688/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-22","2026-08-05",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What relationship between Debye temperature and Tc is examined in this study?","Question",{"text":76,"@type":77},"The work tests the proposed bound Tc ≤ AΘD for phonon-mediated BCS superconductors, where A is a material-independent prefactor on the order of ~0.1.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How are machine learning models used to verify the Tc–ΘD bound?",{"text":81,"@type":77},"ML models are trained using 10,330 compounds from the Materials Project to predict ΘD, then applied to 9,860 superconductors in the SuperCon database with conventional superconductors shown to satisfy the bound.",{"name":83,"@type":74,"acceptedAnswer":84},"What do first-principles phonon calculations for H3S and LaH10 show?",{"text":85,"@type":77},"Phonon calculations at 200 GPa indicate these high-pressure hydrides essentially saturate the Tc versus ΘD bound, giving Tc values roughly in the range 0.1 to 0.11 ΘD.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":93},[94,98,102,106,111,116,121,124,129,132,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":46,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":46,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":46,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":46,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":21,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":21,"slug":134},"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":107,"slug":138},19,"General","general"]