No Access Submitted: 07 June 2017 Accepted: 09 September 2017 Published Online: 27 September 2017
Appl. Phys. Lett. 111, 131902 (2017); https://doi.org/10.1063/1.4986884
more...View Affiliations
View Contributors
  • Long Chen
  • Jeffrey L. Braun
  • Brian F. Donovan
  • Patrick E. Hopkins
  • S. Joseph Poon
Computationally efficient modeling of the thermal conductivity of materials is crucial to thorough experimental planning and theoretical understanding of thermal properties. We present a modeling approach in this work that utilizes a frequency-dependent effective medium theory to calculate the lattice thermal conductivity of nanostructured solids. This method accurately predicts a significant reduction in the experimentally measured thermal conductivity of nanostructured Si80Ge20 systems reported in this work, along with previously reported thermal conductivities in nanowires and nanoparticles in matrix materials. We use our model to gain insights into the role of long wavelength phonons on the thermal conductivity of nanograined silicon-germanium alloys. Through thermal conductivity accumulation calculations with our modified effective medium model, we show that phonons with wavelengths much greater than the average grain size will not be impacted by grain boundary scattering, counter to the traditionally assumed notion that grain boundaries in solids will act as diffusive interfaces that will limit long wavelength phonon transport. This is further supported by using time-domain thermoreflectance at different pump modulation frequencies to measure the thermal conductivity of a series nanograined silicon-germanium alloys.
This material is based upon work supported by the Air Force Office of Scientific Research under Award No. FA9550-15-1-0079.
  1. 1. G. Joshi, H. Lee, Y. Lan, X. Wang, G. Zhu, D. Wang, R. W. Gould, D. C. Cuff, M. Y. Tang, M. S. Dresselhaus, G. Chen, and Z. Ren, “ Enhanced thermoelectric figure-of-merit in nanostructured p-type silicon germanium bulk alloys,” Nano Lett. 8, 4670–4674 (2008). https://doi.org/10.1021/nl8026795, Google ScholarCrossref, ISI
  2. 2. J. Callaway, “ Model for lattice thermal conductivity at low temperatures,” Phys. Rev. 113, 1046–1051 (1959). https://doi.org/10.1103/PhysRev.113.1046, Google ScholarCrossref, ISI
  3. 3. W. Kim, S. L. Singer, A. Majumdar, J. M. O. Zide, D. Klenov, A. C. Gossard, and S. Stemmer, “ Reducing thermal conductivity of crystalline solids at high temperature using embedded nanostructures,” Nano Lett. 8, 2097–2099 (2008). https://doi.org/10.1021/nl080189t, Google ScholarCrossref
  4. 4. R. Yang and G. Chen, “ Thermal conductivity modeling of periodic two-dimensional nanocomposites,” Phys. Rev. B 69, 195316 (2004). https://doi.org/10.1103/PhysRevB.69.195316, Google ScholarCrossref
  5. 5. C. Dames and G. Chen, “ Theoretical phonon thermal conductivity of Si/Ge superlattice nanowires,” J. Appl. Phys. 95, 682–693 (2004). https://doi.org/10.1063/1.1631734, Google ScholarScitation, ISI
  6. 6. R. Yang, G. Chen, M. Laroche, and Y. Taur, “ Simulation of nanoscale multidimensional transient heat conduction problems using ballistic-diffusive equations and phonon boltzmann equation,” J. Heat Transfer 127, 298–306 (2005). https://doi.org/10.1115/1.1857941, Google ScholarCrossref
  7. 7. R. Yang, G. Chen, and M. Dresselhaus, “ Thermal conductivity of simple and tubular nanowire composites in the longitudinal direction,” Phys. Rev. B 72, 125418 (2005). https://doi.org/10.1103/PhysRevB.72.125418, Google ScholarCrossref
  8. 8. Z. Wang, J. E. Alaniz, W. Jang, J. E. Garay, and C. Dames, “ Thermal conductivity of nanocrystalline silicon: Importance of grain size and frequency-dependent mean free paths,” Nano Lett. 11, 2206–2213 (2011). https://doi.org/10.1021/nl1045395, Google ScholarCrossref
  9. 9. Q. Hao, G. Zhu, G. Joshi, X. Wang, A. Minnich, Z. Ren, and G. Chen, “ Theoretical studies on the thermoelectric figure of merit of nanograined bulk silicon,” Appl. Phys. Lett. 97, 063109 (2010). https://doi.org/10.1063/1.3478459, Google ScholarScitation
  10. 10. J. D. Chung, A. J. H. McGaughey, and M. Kaviany, “ Role of phonon dispersion in lattice thermal conductivity modeling,” J. Heat Transfer 126, 376–380 (2004). https://doi.org/10.1115/1.1723469, Google ScholarCrossref
  11. 11. P. E. Hopkins, L. M. Phinney, P. T. Rakich, R. H. Olsson, and I. El-Kady, “ Phonon considerations in the reduction of thermal conductivity in phononic crystals,” Appl. Phys. A 103, 575–579 (2011). https://doi.org/10.1007/s00339-010-6189-8, Google ScholarCrossref
  12. 12. P. E. Hopkins, P. T. Rakich, R. H. Olsson, I. El-Kady, and L. M. Phinney, “ Origin of reduction in phonon thermal conductivity of microporous solids,” Appl. Phys. Lett. 95, 161902 (2009). https://doi.org/10.1063/1.3250166, Google ScholarScitation
  13. 13. S. Volz and G. Chen, “ Molecular dynamics simulation of thermal conductivity of silicon nanowires,” Appl. Phys. Lett. 75, 2056–2058 (1999). https://doi.org/10.1063/1.124914, Google ScholarScitation, ISI
  14. 14. A. Giri, J. L. Braun, J. A. Tomko, and P. E. Hopkins, “ Reducing the thermal conductivity of chemically ordered binary alloys below the alloy limit via the alteration of phonon dispersion relations,” Appl. Phys. Lett. 110, 233112 (2017). https://doi.org/10.1063/1.4985204, Google ScholarScitation
  15. 15. A. Giri and P. E. Hopkins, “ Spectral contributions to the thermal conductivity of C60 and the fullerene derivative PCBM,” J. Phys. Chem. Lett. 8, 2153–2157 (2017). https://doi.org/10.1021/acs.jpclett.7b00609, Google ScholarCrossref
  16. 16. J. M. Larkin and A. J. H. McGaughey, “ Thermal conductivity accumulation in amorphous silica and amorphous silicon,” Phys. Rev. B 89, 144303 (2014). https://doi.org/10.1103/PhysRevB.89.144303, Google ScholarCrossref
  17. 17. A. S. Henry and G. Chen, “ Spectral phonon transport properties of silicon based on molecular dynamics simulations and lattice dynamics,” J. Comput. Theor. Nanoscience 5, 141–152 (2008). https://doi.org/10.1166/jctn.2008.2454, Google ScholarCrossref
  18. 18. P. Ravindran, L. Fast, P. A. Korzhavyi, B. Johansson, J. Wills, and O. Eriksson, “ Density functional theory for calculation of elastic properties of orthorhombic crystals: Application to TiSi2,” J. Appl. Phys. 84, 4891–4904 (1998). https://doi.org/10.1063/1.368733, Google ScholarScitation, ISI
  19. 19. K. Esfarjani, G. Chen, and H. T. Stokes, “ Heat transport in silicon from first-principles calculations,” Phys. Rev. B 84, 085204 (2011). https://doi.org/10.1103/PhysRevB.84.085204, Google ScholarCrossref
  20. 20. L. Lindsay, D. A. Broido, and T. L. Reinecke, “ First-principles determination of ultrahigh thermal conductivity of boron arsenide: A competitor for diamond?,” Phys. Rev. Lett. 111, 025901 (2013). https://doi.org/10.1103/PhysRevLett.111.025901, Google ScholarCrossref, ISI
  21. 21. A. Ward and D. A. Broido, “ Intrinsic phonon relaxation times from first-principles studies of the thermal conductivities of Si and Ge,” Phys. Rev. B 81, 085205 (2010). https://doi.org/10.1103/PhysRevB.81.085205, Google ScholarCrossref, ISI
  22. 22. A. Ward, D. A. Broido, D. Stewart, and G. Deinzer, “ Ab initio theory of the lattice thermal conductivity in diamond,” Phys. Rev. B 80, 125203 (2009). https://doi.org/10.1103/PhysRevB.80.125203, Google ScholarCrossref
  23. 23. A. Seko, A. Togo, H. Hayashi, K. Tsuda, L. Chaput, and I. Tanaka, “ Prediction of Low-thermal-conductivity compounds with first-principles anharmonic lattice-dynamics calculations and Bayesian optimization,” Phys. Rev. Lett. 115, 205901 (2015). https://doi.org/10.1103/PhysRevLett.115.205901, Google ScholarCrossref
  24. 24. A. Minnich and G. Chen, “ Modified effective medium formulation for the thermal conductivity of nanocomposites,” Appl. Phys. Lett. 91, 073105 (2007). https://doi.org/10.1063/1.2771040, Google ScholarScitation, ISI
  25. 25. C.-W. Nan, R. Birringer, D. R. Clarke, and H. Gleiter, “ Effective thermal conductivity of particulate composites with interfacial thermal resistance,” J. Appl. Phys. 81, 6692–6699 (1997). https://doi.org/10.1063/1.365209, Google ScholarScitation, ISI
  26. 26. C.-W. Nan, “ Effective‐medium theory of piezoelectric composites,” J. Appl. Phys. 76, 1155–1163 (1994). https://doi.org/10.1063/1.357839, Google ScholarScitation
  27. 27. D. G. Cahill, “ Analysis of heat flow in layered structures for time-domain thermoreflectance,” Rev. Sci. Instrum. 75, 5119–5122 (2004). https://doi.org/10.1063/1.1819431, Google ScholarScitation, ISI
  28. 28. A. J. Schmidt, “ Pump-probe thermoreflectance,” Annu. Rev. Heat Transfer 16, 159–181 (2013). https://doi.org/10.1615/AnnualRevHeatTransfer.v16.60, Google ScholarCrossref
  29. 29. W. Kim and A. Majumdar, “ Phonon scattering cross section of polydispersed spherical nanoparticles,” J. Appl. Phys. 99, 084306 (2006). https://doi.org/10.1063/1.2188251, Google ScholarScitation, ISI
  30. 30. S. J. Poon and K. Limtragool, “ Nanostructure model of thermal conductivity for high thermoelectric performance,” J. Appl. Phys. 110, 114306 (2011). https://doi.org/10.1063/1.3662947, Google ScholarScitation
  31. 31. S. J. Poon, A. S. Petersen, and D. Wu, “ Thermal conductivity of core-shell nanocomposites for enhancing thermoelectric performance,” Appl. Phys. Lett. 102, 173110 (2013). https://doi.org/10.1063/1.4804150, Google ScholarScitation
  32. 32. D. Li, Y. Wu, P. Kim, L. Shi, P. Yang, and A. Majumdar, “ Thermal conductivity of individual silicon nanowires,” Appl. Phys. Lett. 83, 2934–2936 (2003). https://doi.org/10.1063/1.1616981, Google ScholarScitation, ISI
  33. 33. F. Yang and C. Dames, “ Mean free path spectra as a tool to understand thermal conductivity in bulk and nanostructures,” Phys. Rev. B 87, 035437 (2013). https://doi.org/10.1103/PhysRevB.87.035437, Google ScholarCrossref
  34. 34. M.-S. Jeng, R. G. Yang, D. Song, and G. Chen, “ Modeling the thermal conductivity and phonon transport in nanoparticle composites using Monte Carlo simulations,” J. Heat Transfer 130, 042410 (2008). https://doi.org/10.1115/1.2818765, Google ScholarCrossref
  35. 35. M. S. Dresselhaus, G. Chen, M. Y. Tang, R. G. Yang, H. Lee, D. Z. Wang, Z. F. Ren, J.-P. Fleurial, and P. Gogna, “ New directions for low-dimensional thermoelectric materials,” Adv. Mater. 19, 1043–1053 (2007). https://doi.org/10.1002/adma.200600527, Google ScholarCrossref, ISI
  36. 36. J. Ordonez-Miranda, M. Hermens, I. Nikitin, V. G. Kouznetsova, and S. Volz, “ Modeling of the effective thermal conductivity of sintered porous pastes,” in 20th International Workshop on Thermal Investigations of ICs Systems Terminic (2014), pp. 1–4. Google ScholarCrossref
  37. 37. R. B. Wilson and D. G. Cahill, “ Anisotropic failure of Fourier theory in time-domain thermoreflectance experiments,” Nat. Commun. 5, 5075 (2014). https://doi.org/10.1038/ncomms6075, Google ScholarCrossref, ISI
  38. 38. J. L. Braun and P. E. Hopkins, “ Upper limit to the thermal penetration depth during modulated heating of multilayer thin films with pulsed and continuous wave lasers: A numerical study,” J. Appl. Phys. 121, 175107 (2017). https://doi.org/10.1063/1.4982915, Google ScholarScitation, ISI
  39. 39. P. E. Hopkins, J. R. Serrano, L. M. Phinney, S. P. Kearney, T. W. Grasser, and C. T. Harris, “ Criteria for cross-plane dominated thermal transport in multilayer thin film systems during modulated laser heating,” J. Heat Transfer 132, 081302 (2010). https://doi.org/10.1115/1.4000993, Google ScholarCrossref, ISI
  40. 40. K. T. Regner, L. C. Wei, and J. A. Malen, “ Interpretation of thermoreflectance measurements with a two-temperature model including non-surface heat deposition,” J. Appl. Phys. 118, 235101 (2015). https://doi.org/10.1063/1.4937995, Google ScholarScitation, ISI
  41. 41. K. T. Regner, J. P. Freedman, and J. A. Malen, “ Advances in studying phonon mean free path dependent contributions to thermal conductivity,” Nanoscale Microscale Thermophys. Eng. 19, 183–205 (2015). https://doi.org/10.1080/15567265.2015.1045640, Google ScholarCrossref
  42. 42. K. T. Regner, A. J. H. McGaughey, and J. A. Malen, “ Analytical interpretation of nondiffusive phonon transport in thermoreflectance thermal conductivity measurements,” Phys. Rev. B 90, 064302 (2014). https://doi.org/10.1103/PhysRevB.90.064302, Google ScholarCrossref
  43. 43. K. T. Regner, D. P. Sellan, Z. Su, C. H. Amon, A. J. H. McGaughey, and J. A. Malen, “ Broadband phonon mean freee path contributions to thermal conductivity measured using frequency domain thermoreflectance,” Nat. Commun. 4, 1640 (2013). https://doi.org/10.1038/ncomms2630, Google ScholarCrossref
  44. 44. A. T. Ramu and J. E. Bowers, “ A compact heat transfer model based on an enhanced Fourier law for analysis of frequency-domain thermoreflectance experiments,” Appl. Phys. Lett. 106, 263102 (2015). https://doi.org/10.1063/1.4923310, Google ScholarScitation
  45. 45. Y. K. Koh and D. G. Cahill, “ Frequency dependence of the thermal conductivity of semiconductor alloys,” Phys. Rev. B 76, 075207 (2007). https://doi.org/10.1103/PhysRevB.76.075207, Google ScholarCrossref
  46. 46. Y. K. Koh, D. G. Cahill, and B. Sun, “ Nonlocal theory for heat transport at high frequencies,” Phys. Rev. B 90, 205412 (2014). https://doi.org/10.1103/PhysRevB.90.205412, Google ScholarCrossref
  47. 47. R. Cheaito, J. C. Duda, T. E. Beechem, K. Hattar, J. F. Ihlefeld, D. L. Medlin, M. A. Rodriguez, M. J. Campion, E. S. Piekos, and P. E. Hopkins, “ Experimental Investigation of size effects on the thermal conductivity of silicon-germanium alloy thin films,” Phys. Rev. Lett. 109, 195901 (2012). https://doi.org/10.1103/PhysRevLett.109.195901, Google ScholarCrossref
  48. 48. B. Abeles, “ Lattice thermal conductivity of disordered semiconductor alloys at high temperatures,” Phys. Rev. 131, 1906–1911 (1963). https://doi.org/10.1103/PhysRev.131.1906, Google ScholarCrossref, ISI
  49. 49. P. G. Klemens, “ The scattering of low-frequency lattice waves by static imperfections,” Proc. Phys. Soc., London, Sect. A 68, 1113–1128 (1955). https://doi.org/10.1088/0370-1298/68/12/303, Google ScholarCrossref
  50. 50. P. E. Hopkins, “ Dispersion considerations affecting phonon-mass impurity scattering rates,” AIP Adv. 1, 041705 (2011). https://doi.org/10.1063/1.3676171, Google ScholarScitation
  1. © 2017 Author(s). Published by AIP Publishing.