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Interpreting non-equilibrium thermoreflectance experiments to sort phonon contributions to thermal conductivity

Interpreting non-equilibrium thermoreflectance experiments to sort phonon contributions to thermal conductivity. Jonathan A. Malen with help from: Keith Regner , Justin Freedman, Ankit Jain, and Alan McGaughey PTES 2014, Shanghai. PhD Students In My Lab. Wee- Liat. Keith. Shub.

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Interpreting non-equilibrium thermoreflectance experiments to sort phonon contributions to thermal conductivity

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  1. Interpreting non-equilibrium thermoreflectance experiments to sort phonon contributions to thermal conductivity Jonathan A. Malen with help from: Keith Regner, Justin Freedman, Ankit Jain, and Alan McGaughey PTES 2014, Shanghai Jonathan A. Malen

  2. PhD Students In My Lab Wee-Liat Keith Shub Zonghui* Scott* Lili William Kevin Jillian Justin *alumni

  3. Thermal Conductivity of Gases and Solids Jonathan A. Malen

  4. Thermal Conductivity Accumulation Λ = ΛGray Λ* (μm) Dames and Chen, CRC TE Handbook 2006 Jonathan A. Malen

  5. Prior approaches to experimentally define Λ* [4] [3] [5] Minnich et al., PRL 107, 095901 (2011) Koh and Cahill, PRB 76, 075207 (2007) Johnson et al., PRL 110, 025901 (2013)

  6. Frequency Domain Thermoreflectance(up to 10-20 MHz)

  7. In our experiment, wecan readily control thermal penetration depth Lp = Λ* Larger range of f1 larger measurement range of kaccum Regneret al, Nature Communications (2013) Regner et al, Rev. Sci. Instr. (2013)

  8. Broad Band Frequency Domain Thermoreflectance (up to 200 MHz) Regneret al, Nature Communications (2013) Regner et al, Rev. Sci. Instr. (2013)

  9. Transforming raw BB-FDTR data (phase lag of T to q” vs.f) into kaccum Si is poorly fit over frange, and k=99 W/m-K SiO2 fits over all f, and k=1.4 W/m-K Analytical fit based on: Cahill, Rev. Sci. Inst. 2004 Regneret al, Nature Communications (2013) Regner et al, Rev. Sci. Instr. (2013)

  10. kaccum is flat for SiO2 and Pt, but varies for Si at 300K high f1 low f1 Regneret al, Nature Communications (2013)

  11. kaccum for Si at various temperatures Jonathan A. Malen

  12. How do we interpret these experiments? • Equating Lp=Λ* is not rigorously justified • What is the interplay between spot size and Lp? • What is the role of the transducer? Ideas from Keith Nelson’s Transient Grating Approach S(Λ,L)=keff/kbulk Maznev et al, PRB 2011, Collins et al, JAP 2013 Jonathan A. Malen

  13. The Suppression Function S provides a map between BTE and Diffusive Solutions S tells us how much the thermal conductivity of a phonon with MFP=Λ is suppressed in a non-diffusive experiment S=keff/kbulk Jonathan A. Malen

  14. Solving the Boltzmann Transport Equation is not new to our community The phonon BTE under the relaxation time approximation where f(t,r,v) is the distribution function The Equation of Phonon Radiative Transfer (EPRT) G. Chen, ASME J. Heat Trans., 1996 A. Majumdar, ASME J. Heat Trans., 1993 Jonathan A. Malen

  15. Suppressed k when Λ>d G. Chen, ASME J. Heat Trans., 1996 Jonathan A. Malen

  16. Solving the BTE for periodic heating of a semi-infinite solid Instead of using Intensity, write the BTE in terms of phonon energy density per unit frequency, per unit solid angle n(x,t,θ) Assumen(x,t,θ) =ñ(x,θ)eiΩt θ x Jonathan A. Malen

  17. Boltzmann Transport Eqn. in 1-D Planar Energy conservation in a gray medium n+ n- Assumeñis isotropic in θoverx+andx- θ x Jonathan A. Malen

  18. Boltzmann Transport Eqn. in 1-D Planar This two-flux method, known as the Milne-Eddington or Differential Approximation in Radiation, leads to two coupled differential equations in ñ0 and ñ1 n+ n- This is an eigenvalue problem θ x M.F. Modest, Radiative Heat Transfer, 2003 Jonathan A. Malen

  19. Eigenvalues give us the spatial decay Solve for the eigenvalues n+ n- Compare with the heat diffusion equation through . θ Recover diffusive decay in the limit that x Jonathan A. Malen

  20. Eigenvalues give us the spatial decay The real part of λ defines the spatial decay Lp-BTE. But what is the real part of this? n+ n- With some math tricks, we find θ x Jonathan A. Malen

  21. Apply Boundary Conditions to get solutions Boundary Conditions n+ n- Analytical Solution for ñ0 and ñ1 gives ñewhich gives T θ x Jonathan A. Malen

  22. Spatial temperature profiles Jonathan A. Malen

  23. What is the Suppression Function? The suppression function Smultiplieskbulk to create a keff=S×kbulkso that the diffusive thermal resistance equals the BTE thermal resistance at the heated surface. For a black surface ε=1 Jonathan A. Malen

  24. Temperature response for periodically heated black wall (ε=1) Jonathan A. Malen

  25. Temperature response for periodically heated gray wall (ε≠1) ε=1 ε=0.1 Jonathan A. Malen

  26. Boltzmann Transport Eqn.in 1-D Spherical Coordinates θ The BTE in spherical coordinates where n(r,μ=cosθ,t) r rspot n+ n- Analytical solution by the P1 approximation [1,2]; first consider with Ω=0, [1] Modest, Radiative Heat Transfer, 1993; [2] Marshak, Phys Rev. 1947 Jonathan A. Malen

  27. The effect of spot size at Ω=0 Jonathan A. Malen

  28. Spot size suppression function at Ω=0 Jonathan A. Malen

  29. Suppression function vs. MFP/Lp for various MFP/R Jonathan A. Malen

  30. How do we use the suppression function to predict keff for a given experiment? Jonathan A. Malen

  31. How do we use the suppression function to predict keff for a given experiment? Jonathan A. Malen

  32. Conclusions From Boltzmann Transport Theory • There is suppression of thermal conductivity due to periodic modulation • The smaller of spot size and Lp will generally dominate suppression • The emissivity of transducer plays a role in observed suppression Jonathan A. Malen

  33. Comparison with BB-FDTR Data vs. Lp Thermal conductivity per unit MFP from McGaughey’s First Principals Calculations Jonathan A. Malen

  34. Comparison with TDTR Data vs. rspot Thermal conductivity per unit MFP from McGaughey’s First Principals Calculations Jonathan A. Malen

  35. Thermal Conductivity Suppression in Nanograined Solids Lan, et al. Nano Letters 2009 Poudel, et al. Science 2008 Nanograined Si Samples from Garay & Dames, Nano Letters 2011 Jonathan A. Malen

  36. Thermal Conductivity Suppression in Nanograined Si bulk Lan, et al. Nano Letters 2009 Poudel, et al. Science 2008 550nm 100nm bulk Nanograined Si Samples from Garay & Dames, Nano Letters 2011 550nm 100nm Jonathan A. Malen

  37. Conclusions • It is possible to experimentally define a lengthscale that suppresses phonons contributions to heat transport • Rigorous suppression functions can be derived from the BTE. Analytical solutions were identified with several simplifications • Diffusion OK for MFP<(rspot,Lp,L…) • Both FDTR and TDTR data can be explained by these functions, yet some physics is unresolved • Collectively these techniques give us much more information about phonons for material/device development than we’ve ever had before Jonathan A. Malen

  38. Fitting Window Size Effect Jonathan A. Malen

  39. kaccum is not flat for amorphous-Si confirming “propagon” contributions Regneret al, Nature Communications 4, 1640 (2013)

  40. Universal high temperature shape of kaccum in small unit cell semiconductors Fit kBulk data to P & CU NondimensionalizeLp Freedman, J. P. et al. Scientific Reports (2013) Jonathan A. Malen

  41. Normalization by Lp,nondimensional collapses data to kuniversal Freedman, J. P. et al. Scientific Reports (2013) Jonathan A. Malen

  42. Other small unit cell semi-conductors Freedman, J. P. et al.,Nature Scientific Reports Jonathan A. Malen

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