Impact of Size Distribution on Optical Properties of Quantum Dots: A Comparative Study of Statistical Models

Impact of Size Distribution on Optical Properties of Quantum Dots: A Comparative Study of Statistical Models | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Impact of Size Distribution on Optical Properties of Quantum Dots: A Comparative Study of Statistical Models Omar Qasaimeh This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6319636/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 28 Jun, 2025 Read the published version in Optical and Quantum Electronics → Version 1 posted 8 You are reading this latest preprint version Abstract This study investigates the effects of size fluctuations on the optical properties of PbS/ZnS quantum dots (QDs) by employing Gaussian, log-normal, Gamma, and Weibull probability density functions to model size distributions. Unlike most published studies, which apply averaging only to the gain coefficient, we perform statistical averaging across all terms in the rate equations—including stimulated emission, escape, and relaxation rates—to fully account for size fluctuations. This averaging is carried out using solutions of the Schrödinger wave equation as a function of quantum dot size. The resulting averaged rate equations are then consistently solved for various device parameters. The results reveal that the Weibull distribution exhibits the highest unsaturated optical gain, while the Gaussian distribution demonstrates lower gain with a peak shifted to higher energy. Log-normal and Gamma distributions produce similar optical spectra, with broader bandwidths but lower gain peaks. Larger size fluctuations result in reduced gain peaks and shifts toward lower energies. The Weibull distribution's pronounced contribution from larger particles explains its higher gain and shifted energy peak. Further analysis indicates that the log-normal distribution achieves the highest 3dB gain-saturation density product and the largest gain bandwidth, making it particularly advantageous for broadband applications. The Weibull distribution, while offering higher gain, exhibits narrower bandwidth and higher four-wave mixing efficiency. These findings underscore the critical influence of size distribution on the optical performance of QDs and provide valuable insights for designing QD-based devices, particularly at 1550 nm. Colloidal quantum dot model size fluctuation optical gain probability density distribution Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 28 Jun, 2025 Read the published version in Optical and Quantum Electronics → Version 1 posted Editorial decision: Revision requested 19 May, 2025 Reviews received at journal 01 May, 2025 Reviewers agreed at journal 12 Apr, 2025 Reviewers agreed at journal 10 Apr, 2025 Reviewers invited by journal 10 Apr, 2025 Editor assigned by journal 28 Mar, 2025 Submission checks completed at journal 28 Mar, 2025 First submitted to journal 27 Mar, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-6319636","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":442072544,"identity":"f2d74e06-4f93-4846-bc17-d498d2513adf","order_by":0,"name":"Omar Qasaimeh","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA9ElEQVRIiWNgGAWjYBACfhCRAET8DAfAAowNYD4eINkA1SLZANRygBgtBhCzGRLADKgW/MDg+PGHHx7m2OQZHzxj/PkDg43shgMMDx/g1XImx1gicVtasdmBM2YSBxjSjIFakg3wOyyHAajlcOI2oBagww4nArWkSeDTYnb++eMfidv+J25uOGP84QDDf8JajG8kmAFtOZC4geGMAdBhBwhrkZzxxswicVty4owDx8okzhgkG888TMAv/P7pj2/+3GaX2D/j8OYPFRV2sn3HexIf4NOCAEBXAUMDiJl5EojTwcDfAGOxHyBSyygYBaNgFIwQAAAfCFrd+vdyGwAAAABJRU5ErkJggg==","orcid":"","institution":"Jordan University of Science and Technology","correspondingAuthor":true,"prefix":"","firstName":"Omar","middleName":"","lastName":"Qasaimeh","suffix":""}],"badges":[],"createdAt":"2025-03-27 10:38:09","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6319636/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6319636/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11082-025-08331-9","type":"published","date":"2025-06-28T15:57:02+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":85686104,"identity":"a1df4fa6-17b8-45d7-9944-9099db0ac5b5","added_by":"auto","created_at":"2025-06-30 16:03:04","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":521687,"visible":true,"origin":"","legend":"","description":"","filename":"OQEColloidal.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6319636/v1_covered_2ef3657a-7ce5-4a2d-bdab-f743bec99b68.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Impact of Size Distribution on Optical Properties of Quantum Dots: A Comparative Study of Statistical Models","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"optical-and-quantum-electronics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"oqel","sideBox":"Learn more about [Optical and Quantum Electronics](https://www.springer.com/journal/11082)","snPcode":"11082","submissionUrl":"https://submission.nature.com/new-submission/11082/3","title":"Optical and Quantum Electronics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Colloidal quantum dot, model, size fluctuation, optical gain, probability density distribution","lastPublishedDoi":"10.21203/rs.3.rs-6319636/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6319636/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis study investigates the effects of size fluctuations on the optical properties of PbS/ZnS quantum dots (QDs) by employing Gaussian, log-normal, Gamma, and Weibull probability density functions to model size distributions. Unlike most published studies, which apply averaging only to the gain coefficient, we perform statistical averaging across all terms in the rate equations\u0026mdash;including stimulated emission, escape, and relaxation rates\u0026mdash;to fully account for size fluctuations. This averaging is carried out using solutions of the Schr\u0026ouml;dinger wave equation as a function of quantum dot size. The resulting averaged rate equations are then consistently solved for various device parameters. The results reveal that the Weibull distribution exhibits the highest unsaturated optical gain, while the Gaussian distribution demonstrates lower gain with a peak shifted to higher energy. Log-normal and Gamma distributions produce similar optical spectra, with broader bandwidths but lower gain peaks. Larger size fluctuations result in reduced gain peaks and shifts toward lower energies. The Weibull distribution's pronounced contribution from larger particles explains its higher gain and shifted energy peak. Further analysis indicates that the log-normal distribution achieves the highest 3dB gain-saturation density product and the largest gain bandwidth, making it particularly advantageous for broadband applications. The Weibull distribution, while offering higher gain, exhibits narrower bandwidth and higher four-wave mixing efficiency. 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