{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T18:55:17Z","timestamp":1772823317620,"version":"3.50.1"},"reference-count":91,"publisher":"AIP Publishing","issue":"8","content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2010,4,15]]},"abstract":"<jats:p>We present a rigorous analysis of the thermal conductivity of bulk silicon (Si) and Si nanowires (Si NWs) which takes into account the exact physical nature of the various acoustic and optical phonon mechanisms. Following the Callaway solution for the Boltzmann equation, where resistive and nonresistive phonon mechanisms are discriminated, we derived formalism for the lattice thermal conductivity that takes into account the phonon incidence angles. The phonon scattering processes are represented by frequency-dependent relaxation time. In addition to the commonly considered acoustic three-phonon processes, a detailed analysis of the role of the optical phonon decay into acoustic phonons is performed. This optical phonon decay mechanism is considered to act as acoustic phonon generation rate partially counteracting the acoustic phonon scattering rates. We have derived the analytical expression describing this physical mechanism which should be included in the general formalism as a correction to the resistive phonon-point-defects and phonon-boundary scattering expressions. The phonon-boundary scattering mechanism is taken as a function of the phonon frequency, incidence angles, and surface roughness. The importance of all the mechanisms we have involved in the model is demonstrated clearly with reference to reported data regarding the isotopic composition effect in bulk Si and Si NW samples. Namely, our model accounts for previously unexplained experimental results regarding (i) the isotope composition effect on the thermal conductivity of bulk silicon reported by Ruf et al. [Solid State Commun.\u2008115, 243 (2000)], (ii) the size effect on \u03ba(T) of individual Si NWs reported by Li et al. [Appl. Phys. Lett.\u200883, 2934 (2003)], and (iii) the dramatic decrease in the thermal conductivity for rough Si NWs reported by Hochbaum et al. [Nature (London)\u2008451, 163 (2008)].<\/jats:p>","DOI":"10.1063\/1.3340973","type":"journal-article","created":{"date-parts":[[2010,4,21]],"date-time":"2010-04-21T22:30:51Z","timestamp":1271889051000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":101,"title":["Thermal conductivity of silicon bulk and nanowires: Effects of isotopic composition, phonon confinement, and surface roughness"],"prefix":"10.1063","volume":"107","author":[{"given":"M.","family":"Kazan","sequence":"first","affiliation":[{"name":"Universit\u00e9 de Technologie de Troyes 1 Laboratoire de Nanotechnologie et d\u2019Instrumentation Optique, ICD, CNRS (FRE2848), , 10010 Troyes, France"}]},{"given":"G.","family":"Guisbiers","sequence":"additional","affiliation":[{"name":"University of Aveiro 2 Department of Physics, CICECO, , Aveiro 3810-193, Portugal"}]},{"given":"S.","family":"Pereira","sequence":"additional","affiliation":[{"name":"University of Aveiro 2 Department of Physics, CICECO, , Aveiro 3810-193, Portugal"}]},{"given":"M. R.","family":"Correia","sequence":"additional","affiliation":[{"name":"University of Aveiro 3 Department of Physics, I3N, , Aveiro 3810-193, Portugal"}]},{"given":"P.","family":"Masri","sequence":"additional","affiliation":[{"name":"University of Montpellier II 4 Groupe d\u2019Etude des Semiconducteurs, CNRS-UMR 5650, , Montpellier 34095, France"}]},{"given":"A.","family":"Bruyant","sequence":"additional","affiliation":[{"name":"Universit\u00e9 de Technologie de Troyes 1 Laboratoire de Nanotechnologie et d\u2019Instrumentation Optique, ICD, CNRS (FRE2848), , 10010 Troyes, France"}]},{"given":"S.","family":"Volz","sequence":"additional","affiliation":[{"name":"Ecole Centrale Paris 5 Laboratoire d\u2019Energie Mol\u00e9culaire et Macroscopique, Combustion CNRS UPR 288, , Grande Voie des Vignes, F-92295 Ch\u00e2tenay-Malabry Cedex, France"}]},{"given":"P.","family":"Royer","sequence":"additional","affiliation":[{"name":"Universit\u00e9 de Technologie de Troyes 1 Laboratoire de Nanotechnologie et d\u2019Instrumentation Optique, ICD, CNRS (FRE2848), , 10010 Troyes, France"}]}],"member":"317","published-online":{"date-parts":[[2010,4,21]]},"reference":[{"key":"2023070515570707700_c1","first-page":"237","volume":"6","year":"1942","journal-title":"J. Phys. (Moscow)"},{"key":"2023070515570707700_c2","doi-asserted-by":"publisher","first-page":"773","DOI":"10.1103\/PhysRev.110.773","volume":"110","year":"1958","journal-title":"Phys. Rev."},{"key":"2023070515570707700_c3","doi-asserted-by":"publisher","first-page":"14850","DOI":"10.1103\/PhysRevB.47.14850","volume":"47","year":"1993","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c4","doi-asserted-by":"publisher","first-page":"3764","DOI":"10.1103\/PhysRevLett.70.3764","volume":"70","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"2023070515570707700_c5","first-page":"21","volume":"6","year":"1993","journal-title":"Synchrotron Radiat. News"},{"key":"2023070515570707700_c6","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1002\/pssa.2210830114","volume":"83","year":"1984","journal-title":"Phys. Status Solidi A"},{"key":"2023070515570707700_c7","doi-asserted-by":"publisher","first-page":"44","DOI":"10.1103\/PhysRev.125.44","volume":"125","year":"1962","journal-title":"Phys. Rev."},{"key":"2023070515570707700_c8","doi-asserted-by":"publisher","first-page":"2355","DOI":"10.1103\/PhysRevB.15.2355","volume":"15","year":"1977","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c9","doi-asserted-by":"publisher","first-page":"3702","DOI":"10.1103\/PhysRevB.50.3702","volume":"50","year":"1994","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c10","doi-asserted-by":"publisher","first-page":"2934","DOI":"10.1063\/1.1616981","volume":"83","year":"2003","journal-title":"Appl. Phys. Lett."},{"key":"2023070515570707700_c11","doi-asserted-by":"publisher","first-page":"168","DOI":"10.1038\/nature06458","volume":"451","year":"2008","journal-title":"Nature (London)"},{"key":"2023070515570707700_c12","doi-asserted-by":"publisher","first-page":"597","DOI":"10.1038\/35098012","volume":"413","year":"2001","journal-title":"Nature (London)"},{"key":"2023070515570707700_c13","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1038\/nature06381","volume":"451","year":"2008","journal-title":"Nature (London)"},{"key":"2023070515570707700_c14","doi-asserted-by":"publisher","first-page":"92","DOI":"10.1103\/RevModPhys.33.92","volume":"33","year":"1961","journal-title":"Rev. Mod. Phys."},{"key":"2023070515570707700_c15","doi-asserted-by":"publisher","first-page":"231922","DOI":"10.1063\/1.2822891","volume":"91","year":"2007","journal-title":"Appl. Phys. Lett."},{"key":"2023070515570707700_c16","doi-asserted-by":"publisher","first-page":"1046","DOI":"10.1103\/PhysRev.113.1046","volume":"113","year":"1959","journal-title":"Phys. Rev."},{"key":"2023070515570707700_c17","doi-asserted-by":"publisher","first-page":"2461","DOI":"10.1103\/PhysRev.132.2461","volume":"132","year":"1963","journal-title":"Phys. Rev."},{"key":"2023070515570707700_c18","doi-asserted-by":"publisher","first-page":"9431","DOI":"10.1103\/PhysRevB.56.9431","volume":"56","year":"1997","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c19","doi-asserted-by":"publisher","first-page":"195304","DOI":"10.1103\/PhysRevB.66.195304","volume":"66","year":"2002","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c20","doi-asserted-by":"publisher","first-page":"1713","DOI":"10.1021\/nl034721i","volume":"3","year":"2003","journal-title":"Nano Lett."},{"key":"2023070515570707700_c21","doi-asserted-by":"publisher","first-page":"113308","DOI":"10.1103\/PhysRevB.68.113308","volume":"68","year":"2003","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c22","doi-asserted-by":"publisher","first-page":"975","DOI":"10.1103\/PhysRev.156.975","volume":"156","year":"1967","journal-title":"Phys. Rev."},{"key":"2023070515570707700_c23","volume-title":"Thermal Conductivity in Solids","year":"1976"},{"key":"2023070515570707700_c24","doi-asserted-by":"publisher","first-page":"801","DOI":"10.1103\/PhysRev.186.801","volume":"186","year":"1969","journal-title":"Phys. Rev."},{"key":"2023070515570707700_c25","doi-asserted-by":"publisher","first-page":"2644","DOI":"10.1103\/PhysRevB.10.2644","volume":"10","year":"1974","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c26","doi-asserted-by":"publisher","first-page":"3133","DOI":"10.1103\/PhysRevB.19.3133","volume":"19","year":"1979","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c27","doi-asserted-by":"publisher","first-page":"1256","DOI":"10.1139\/p56-139","volume":"34","year":"1956","journal-title":"Can. J. Phys."},{"key":"2023070515570707700_c28","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1016\/S0031-8914(38)80162-2","volume":"5","year":"1938","journal-title":"Physica"},{"key":"2023070515570707700_c29","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1098\/rspa.1953.0180","volume":"220","year":"1953","journal-title":"Proc. R. Soc. London, Ser. A"},{"key":"2023070515570707700_c30","volume-title":"Electrons and Phonons","year":"1967"},{"key":"2023070515570707700_c31","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1007\/978-3-642-82912-3","volume-title":"Phonon Scattering in Condensed Matter V","author":"Anderson","year":"1986"},{"key":"2023070515570707700_c32","doi-asserted-by":"publisher","first-page":"180302","DOI":"10.1103\/PhysRevB.77.180302","volume":"77","year":"2008","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c33","doi-asserted-by":"publisher","first-page":"211903","DOI":"10.1063\/1.2937113","volume":"92","year":"2008","journal-title":"Appl. Phys. Lett."},{"key":"2023070515570707700_c34","first-page":"198","volume-title":"Encyclopedia of Physics","author":"Fl\u00fcgge","year":"1956"},{"key":"2023070515570707700_c35","doi-asserted-by":"crossref","first-page":"1113","DOI":"10.1088\/0370-1298\/68\/12\/303","volume":"68","year":"1955","journal-title":"Proc. R. Soc. London, Ser. A"},{"key":"2023070515570707700_c36","doi-asserted-by":"publisher","first-page":"1710","DOI":"10.1063\/1.1709746","volume":"38","year":"1967","journal-title":"J. Appl. Phys."},{"key":"2023070515570707700_c37","first-page":"27","volume":"14","year":"1992","journal-title":"Sverkhtverd. Mater."},{"key":"2023070515570707700_c38","doi-asserted-by":"publisher","first-page":"1451","DOI":"10.1103\/PhysRev.122.1451","volume":"122","year":"1961","journal-title":"Phys. Rev."},{"key":"2023070515570707700_c39","doi-asserted-by":"publisher","first-page":"A253","DOI":"10.1103\/PhysRev.133.A253","volume":"133","year":"1964","journal-title":"Phys. Rev."},{"key":"2023070515570707700_c40","first-page":"71","volume":"IIa","year":"1954","journal-title":"Nachr. Akad. Wiss. Goett. II, Math.-Phys. Kl."},{"key":"2023070515570707700_c41","doi-asserted-by":"publisher","first-page":"1499","DOI":"10.1103\/PhysRev.118.1499","volume":"118","year":"1960","journal-title":"Phys. Rev."},{"key":"2023070515570707700_c42","doi-asserted-by":"publisher","first-page":"603","DOI":"10.1103\/PhysRev.128.603","volume":"128","year":"1962","journal-title":"Phys. Rev."},{"key":"2023070515570707700_c43","doi-asserted-by":"publisher","first-page":"66","DOI":"10.1098\/rspa.1965.0249","volume":"289","year":"1965","journal-title":"Proc. R. Soc. London, Ser. A"},{"key":"2023070515570707700_c44","doi-asserted-by":"publisher","first-page":"2279","DOI":"10.1088\/0022-3719\/6\/14\/008","volume":"6","year":"1973","journal-title":"J. Phys. C"},{"key":"2023070515570707700_c45","doi-asserted-by":"publisher","first-page":"1906","DOI":"10.1103\/PhysRev.131.1906","volume":"131","year":"1963","journal-title":"Phys. Rev."},{"key":"2023070515570707700_c46","doi-asserted-by":"publisher","first-page":"1928","DOI":"10.1103\/PhysRevB.28.1928","volume":"28","year":"1983","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c47","doi-asserted-by":"publisher","first-page":"124302","DOI":"10.1063\/1.2749282","volume":"101","year":"2007","journal-title":"J. Appl. Phys."},{"key":"2023070515570707700_c48","doi-asserted-by":"publisher","first-page":"1169","DOI":"10.1016\/j.diamond.2005.11.014","volume":"15","year":"2006","journal-title":"Diamond Relat. Mater."},{"key":"2023070515570707700_c49","volume-title":"The Physics of Phonons","year":"1990"},{"key":"2023070515570707700_c50","doi-asserted-by":"publisher","first-page":"035319","DOI":"10.1103\/PhysRevB.65.035319","volume":"65","year":"2001","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c51","doi-asserted-by":"publisher","first-page":"845","DOI":"10.1103\/PhysRev.148.845","volume":"148","year":"1966","journal-title":"Phys. Rev."},{"key":"2023070515570707700_c52","doi-asserted-by":"publisher","first-page":"1348","DOI":"10.1103\/PhysRevB.57.1348","volume":"57","year":"1998","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c53","doi-asserted-by":"publisher","first-page":"413","DOI":"10.1016\/S0921-4526(02)00530-6","volume":"316\u2013317","year":"2002","journal-title":"Physica B"},{"key":"2023070515570707700_c54","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1016\/S0921-4526(98)01390-8","volume":"263\u2013264","year":"1999","journal-title":"Physica B"},{"key":"2023070515570707700_c55","doi-asserted-by":"publisher","first-page":"12977","DOI":"10.1103\/PhysRevB.59.12977","volume":"59","year":"1999","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c56","doi-asserted-by":"publisher","first-page":"1525","DOI":"10.1016\/j.diamond.2005.12.034","volume":"15","year":"2006","journal-title":"Diamond Relat. Mater."},{"key":"2023070515570707700_c57","doi-asserted-by":"publisher","first-page":"465","DOI":"10.1103\/RevModPhys.46.465","volume":"46","year":"1974","journal-title":"Rev. Mod. Phys."},{"key":"2023070515570707700_c58","doi-asserted-by":"publisher","first-page":"4789","DOI":"10.1103\/PhysRevB.15.4789","volume":"15","year":"1977","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c59","doi-asserted-by":"publisher","first-page":"13401","DOI":"10.1103\/PhysRevB.50.13401","volume":"50","year":"1994","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c60","doi-asserted-by":"publisher","first-page":"832","DOI":"10.1103\/PhysRev.166.832","volume":"166","year":"1968","journal-title":"Phys. Rev."},{"key":"2023070515570707700_c61","doi-asserted-by":"publisher","first-page":"871","DOI":"10.1103\/PhysRev.186.871","volume":"186","year":"1969","journal-title":"Phys. Rev."},{"key":"2023070515570707700_c62","doi-asserted-by":"publisher","first-page":"A1058","DOI":"10.1103\/PhysRev.134.A1058","volume":"134","year":"1964","journal-title":"Phys. Rev."},{"key":"2023070515570707700_c63","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1016\/S0038-1098(00)00172-1","volume":"115","year":"2000","journal-title":"Solid State Commun."},{"key":"2023070515570707700_c64","doi-asserted-by":"publisher","first-page":"092123","DOI":"10.1063\/1.2335972","volume":"89","year":"2006","journal-title":"Appl. Phys. Lett."},{"key":"2023070515570707700_c65","doi-asserted-by":"publisher","first-page":"1602","DOI":"10.1016\/S0925-9635(99)00040-0","volume":"8","year":"1999","journal-title":"Diamond Relat. Mater."},{"key":"2023070515570707700_c66","doi-asserted-by":"publisher","first-page":"84","DOI":"10.1023\/A:1022318402393","volume":"40","year":"2000","journal-title":"BIT"},{"key":"2023070515570707700_c67","doi-asserted-by":"publisher","first-page":"1413","DOI":"10.1016\/j.diamond.2006.12.013","volume":"16","year":"2007","journal-title":"Diamond Relat. Mater."},{"key":"2023070515570707700_c68","doi-asserted-by":"publisher","first-page":"195205","DOI":"10.1103\/PhysRevB.76.195205","volume":"76","year":"2007","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c69","doi-asserted-by":"publisher","first-page":"1155","DOI":"10.1021\/nl062823d","volume":"7","year":"2007","journal-title":"Nano Lett."},{"key":"2023070515570707700_c70","doi-asserted-by":"publisher","first-page":"276","DOI":"10.1021\/nl0725998","volume":"8","year":"2008","journal-title":"Nano Lett."},{"key":"2023070515570707700_c71","doi-asserted-by":"publisher","first-page":"205305","DOI":"10.1103\/PhysRevB.65.205305","volume":"65","year":"2002","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c72","doi-asserted-by":"publisher","first-page":"5279","DOI":"10.1103\/PhysRevB.52.5279","volume":"52","year":"1995","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c73","doi-asserted-by":"publisher","first-page":"9930","DOI":"10.1103\/PhysRevB.51.9930","volume":"51","year":"1995","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c74","doi-asserted-by":"publisher","first-page":"4695","DOI":"10.1103\/PhysRevB.51.4695","volume":"51","year":"1995","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c75","doi-asserted-by":"publisher","first-page":"2579","DOI":"10.1063\/1.369576","volume":"85","year":"1999","journal-title":"J. Appl. Phys."},{"key":"2023070515570707700_c76","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1006\/spmi.1999.0772","volume":"26","year":"1999","journal-title":"Superlattices Microstruct."},{"key":"2023070515570707700_c77","doi-asserted-by":"publisher","first-page":"4687","DOI":"10.1103\/PhysRevB.57.4687","volume":"57","year":"1998","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c78","doi-asserted-by":"publisher","first-page":"1733","DOI":"10.1103\/PhysRevB.50.1733","volume":"50","year":"1994","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c79","doi-asserted-by":"publisher","first-page":"1219","DOI":"10.1063\/1.1531810","volume":"93","year":"2003","journal-title":"J. Appl. Phys."},{"key":"2023070515570707700_c80","doi-asserted-by":"publisher","first-page":"2932","DOI":"10.1063\/1.1345515","volume":"89","year":"2001","journal-title":"J. Appl. Phys."},{"key":"2023070515570707700_c81","doi-asserted-by":"publisher","first-page":"1659","DOI":"10.1103\/PhysRevB.48.1659","volume":"48","year":"1993","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c82","doi-asserted-by":"publisher","first-page":"12456","DOI":"10.1103\/PhysRevB.39.12456","volume":"39","year":"1989","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c83","doi-asserted-by":"publisher","first-page":"1216","DOI":"10.1103\/PhysRevLett.58.1216","volume":"58","year":"1987","journal-title":"Phys. Rev. Lett."},{"key":"2023070515570707700_c84","doi-asserted-by":"publisher","first-page":"1237","DOI":"10.1103\/PhysRevB.47.1237","volume":"47","year":"1993","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c85","volume-title":"The Vibrational Principles of Mechanics","year":"1964"},{"key":"2023070515570707700_c86","doi-asserted-by":"publisher","first-page":"536","DOI":"10.1063\/1.1730391","volume":"31","year":"1959","journal-title":"J. Chem. Phys."},{"key":"2023070515570707700_c87","doi-asserted-by":"publisher","first-page":"2162","DOI":"10.1103\/PhysRevB.47.2162","volume":"47","year":"1993","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c88","doi-asserted-by":"publisher","first-page":"14513","DOI":"10.1103\/PhysRevB.43.14513","volume":"43","year":"1991","journal-title":"Phys. Rev. B"},{"key":"2023070515570707700_c89","first-page":"172","volume-title":"Thermal Conductivity","year":"1996"},{"key":"2023070515570707700_c90","doi-asserted-by":"publisher","first-page":"1542","DOI":"10.1063\/1.1427134","volume":"91","year":"2002","journal-title":"J. Appl. Phys."},{"key":"2023070515570707700_c91","doi-asserted-by":"publisher","first-page":"195901","DOI":"10.1103\/PhysRevLett.102.195901","volume":"102","year":"2009","journal-title":"Phys. Rev. Lett."}],"container-title":["Journal of Applied Physics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.aip.org\/aip\/jap\/article-pdf\/doi\/10.1063\/1.3340973\/13649074\/083503_1_online.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.aip.org\/aip\/jap\/article-pdf\/doi\/10.1063\/1.3340973\/13649074\/083503_1_online.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,7,5]],"date-time":"2023-07-05T20:18:58Z","timestamp":1688588338000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.aip.org\/jap\/article\/107\/8\/083503\/389506\/Thermal-conductivity-of-silicon-bulk-and-nanowires"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,4,15]]},"references-count":91,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2010,4,15]]}},"URL":"https:\/\/doi.org\/10.1063\/1.3340973","relation":{},"ISSN":["0021-8979","1089-7550"],"issn-type":[{"value":"0021-8979","type":"print"},{"value":"1089-7550","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2010,4,15]]},"published":{"date-parts":[[2010,4,15]]},"article-number":"083503"}}