{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T07:05:16Z","timestamp":1769583916524,"version":"3.49.0"},"reference-count":39,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2019,12,6]],"date-time":"2019-12-06T00:00:00Z","timestamp":1575590400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We consider heat conduction in a superlattice with mobile defects, which reduce the thermal conductivity of the material. If the defects may be dragged by the heat flux, and if they are stopped at the interfaces of the superlattice, it is seen that the effective thermal resistance of the layers will depend on the heat flux. Thus, the concentration dependence of the transport coefficients plus the mobility of the defects lead to a strongly nonlinear behavior of heat transport, which may be used in some cases as a basis for thermal transistors.<\/jats:p>","DOI":"10.3390\/e21121200","type":"journal-article","created":{"date-parts":[[2019,12,6]],"date-time":"2019-12-06T10:41:44Z","timestamp":1575628904000},"page":"1200","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Nonlinear Heat Transport in Superlattices with Mobile Defects"],"prefix":"10.3390","volume":"21","author":[{"given":"David","family":"Jou","sequence":"first","affiliation":[{"name":"Grup de Fis\u00edca Estad\u00edstica, Universitat Aut\u00f2noma de Barcelona, 08193 Bellaterra, Catalonia, Spain"}]},{"given":"Liliana","family":"Restuccia","sequence":"additional","affiliation":[{"name":"Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, Viale F. Stagno d\u2019Alcontres, Salita Sperone 31, 98166 Messina, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2019,12,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1515\/jnet-2018-0002","article-title":"Non-equilibrium dislocation dynamics in semiconductor crystals and superlattices","volume":"43","author":"Jou","year":"2018","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_2","first-page":"90","article-title":"Non-equilibrium thermodynamics framework for dislocations in semiconductor crystals and superlattices","volume":"10","author":"Jou","year":"2018","journal-title":"Ann. Acad. Roman. Sci. Ser. Math. Its Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/j.mser.2010.06.007","article-title":"Trends in semiconductor defect engineering at the nanoscale","volume":"70","author":"Seebauer","year":"2010","journal-title":"Mater. Sci. Eng. Rep."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"9282","DOI":"10.1021\/jacs.8b05741","article-title":"Defect engineering for high-performance n-type PbSe thermoelectrics","volume":"140","author":"Zhou","year":"2018","journal-title":"J. Am. Chem. Soc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"9344","DOI":"10.1073\/pnas.1812554115","article-title":"Defect engineering using crystal symmetry","volume":"115","author":"Ramesh","year":"2018","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1038\/s41578-018-0026-7","article-title":"Point defect engineering in thin-film solar cells","volume":"3","author":"Park","year":"2018","journal-title":"Nat. Rev. Mater."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jeurceramsoc.2016.07.036","article-title":"Defect Engineering in Development of Low Thermal Conductivity Materials: A Review","volume":"37","author":"Zhao","year":"2018","journal-title":"J. Eur. Ceram. Soc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"11962","DOI":"10.1038\/srep11962","article-title":"Defect-engineered heat transport in Graphene: a route to high efficient thermal rectification","volume":"5","author":"Zhao","year":"2015","journal-title":"Sci. Rep."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1038\/s41528-018-0035-z","article-title":"Defects engineering for high-performance perovskite solar cells","volume":"2","author":"Wang","year":"2015","journal-title":"npj Flex. Electron."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"163101","DOI":"10.1063\/1.3703756","article-title":"Tunable thermal rectification in Graphene nanoribbons through defect engineering: a molecular dynamics study","volume":"100","author":"Wang","year":"2012","journal-title":"Appl. Phys. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.jechem.2018.09.010","article-title":"Defect engineering: a versatile tool for tuning the activation of key molecules in photocatalytic reactions?","volume":"37","author":"Zhang","year":"2019","journal-title":"J. Energy Chem."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"15919","DOI":"10.1038\/ncomms15919","article-title":"Engineering the thermal conductivity along an individual silicon nanowire by selective helium ion irradiation?","volume":"8","author":"Zhao","year":"2017","journal-title":"Nat. Commun."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2867","DOI":"10.1088\/0953-8984\/3\/17\/004","article-title":"Electron scattering by mobile defects","volume":"3","author":"Morosov","year":"1991","journal-title":"J. Phys. Condens. Matter"},{"key":"ref_14","first-page":"1539","article-title":"Dragging and deceleration of mobile defects in metals by conduction electrons. Role of electron dispersion law","volume":"80","author":"Fiks","year":"1981","journal-title":"Zh. Eksp. Teor. Fiz."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Granato, A. (1958). Thermal properties of mobile defcts. Phys. Rev., 740.","DOI":"10.1103\/PhysRev.111.740"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"134302","DOI":"10.1103\/PhysRevB.92.134302","article-title":"Phonon drag force acting on a mobile crystal defect: Full treatment of discreteness and nonlinearity","volume":"92","author":"Swinburne","year":"2015","journal-title":"Phys. Rev. B"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"6991","DOI":"10.1103\/PhysRevB.24.7007","article-title":"Simplified theory of dislocation damping including point-defect drag. I. Theory of drag by equidistant point defects","volume":"24","author":"Granato","year":"1981","journal-title":"Phys. Rev. B"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2041","DOI":"10.1142\/S0217984909020199","article-title":"Dynamic drag of dislocation by point defects in near-surface crystal layer","volume":"23","author":"Malashenko","year":"2009","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1002\/pssb.19680270145","article-title":"Effect of phonon drag on the kinetic properties of alloys","volume":"27","author":"Masharov","year":"1968","journal-title":"Physica Status Solidi B"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"195503","DOI":"10.1103\/PhysRevLett.115.195503","article-title":"Temperature-dependent transformation thermotics: from switchable thermal cloaks to macroscopic thermal diodes","volume":"115","author":"Li","year":"2015","journal-title":"J. Phys. Rev. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1045","DOI":"10.1103\/RevModPhys.84.1045","article-title":"Phononics: manipulating heat flow with electronic analogs and beyond","volume":"84","author":"Li","year":"2012","journal-title":"Rev. Mod. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1038\/nature12608","article-title":"Sound and heat revolutions in phononics","volume":"503","author":"Maldovan","year":"2013","journal-title":"Nature"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"214303","DOI":"10.1103\/PhysRevLett.108.214303","article-title":"Heat flux manipulation with engineered thermal materials","volume":"108","author":"Narayana","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_24","unstructured":"De Groot, S.R., and Mazur, P. (1962). Non-Equilibrium Thermodynamics, North-Holland Publishing Company."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Jou, D., Casas-Vazquez, J., and Lebon, G. (2010). Extended Irreversible Thermodynamics, Springer. [4th ed.].","DOI":"10.1007\/978-90-481-3074-0"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Lebon, G., Casas-Vazquez, J., and Jou, D. (2008). Understanding Non-Equilibrium Thermodynamics, Springer.","DOI":"10.1007\/978-3-540-74252-4"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1080\/00107514.2011.595596","article-title":"Mesoscopic transport equations and eontemporary thermodynamics: an introduction","volume":"52","author":"Jou","year":"2011","journal-title":"Contemp. Phys."},{"key":"ref_28","first-page":"1","article-title":"Fundamentals of Non-Equilibrium Thermodynamics","volume":"Volume 336","author":"Muschik","year":"1993","journal-title":"Non-Equilibrium Thermodynamics with Applications to Solids"},{"key":"ref_29","unstructured":"Kr\u00f6ner, E. (1988, January 11\u201315). Defects as Internal Variables, Lecture Notes for Internal Variables in Thermodynamics and Continuum Mechanics. Proceedings of the CISM, Udine, Italy."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"084303","DOI":"10.1063\/1.3386464","article-title":"Analytical expression for thermal conductivity of superlattices","volume":"107","author":"Alvarez","year":"2010","journal-title":"J. Appl. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"e1700027","DOI":"10.1126\/sciadv.1700027","article-title":"Heat conduction tuning by the wave nature of phonons","volume":"3","author":"Maire","year":"2017","journal-title":"Sci. Adv."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2200","DOI":"10.1063\/1.98939","article-title":"Thermal resistance at interfaces","volume":"51","author":"Swartz","year":"1987","journal-title":"Appl. Phys. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Wu, Y., Fang, L., and Xu, Y. (2019). Predicting interfacial thermal resistance by machine learning. npj Comput. Mater., 5.","DOI":"10.1038\/s41524-019-0193-0"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"024503","DOI":"10.1063\/1.4974011","article-title":"A simple model of thermoelastic heat switches and heat transistors","volume":"121","author":"Jou","year":"2017","journal-title":"J. Appl. Phys."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"4510","DOI":"10.1038\/s41467-018-06760-7","article-title":"An electrochemical thermal transistor","volume":"9","author":"Sood","year":"2018","journal-title":"Nat. Commun."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"200601","DOI":"10.1103\/PhysRevLett.116.200601","article-title":"Quantum thermal transistor","volume":"116","author":"Joulain","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"17002","DOI":"10.1209\/0295-5075\/122\/17002","article-title":"Coulomb-coupled quantum-dot thermal transistors","volume":"122","author":"Zhang","year":"2018","journal-title":"EPL (Europhys. Lett.)"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"14214","DOI":"10.1038\/ncomms14214","article-title":"Ionic thermoelectric gating organic transistors","volume":"8","author":"Zhao","year":"2017","journal-title":"Nat. Commun."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2017.10.004","article-title":"Non-equilibrium thermodynamics, heat transport and thermal waves in laminar and turbulent superfluid helium","volume":"726","author":"Jou","year":"2018","journal-title":"Phys. Rep."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/12\/1200\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:40:51Z","timestamp":1760190051000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/12\/1200"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12,6]]},"references-count":39,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2019,12]]}},"alternative-id":["e21121200"],"URL":"https:\/\/doi.org\/10.3390\/e21121200","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,12,6]]}}}