(carolinesgorham.com) Energy Density and Thermal Diffusivity of Ioffe-Regel Confined Vibrations Caroline S. Gorham MRS Fall Meeting Symposium ii8: Phonons in Nano and Bulk Materials Session Chair(s): J. Khurgin and G. Sun December 03, 2015
Motivation [1] Energy Generation on Interplanetary Spaceflight *not to scale Thermo photovoltaic generators: HIGH T IR-Resonant Thermoelectric Highlight: Thermoelectric - Low, glass-like, thermal conductivity, κ, desired 2 IR-photovoltaic σs T ZT = κ [1] https://www.nasa.gov/content/nasas-orion-flight-test-and-the-journey-to-mars
Motivation: Glass-like thermal properties [2] Lower-temp (two-level) Higher-temp (random-walk) [2] Sidebottom, D. L. Fundamentals of Condensed Matter and Crystalline Physics: An Introduction for Students of Physics and Materials Science. Cambridge University Press, 2012.
Starting place: What do we know? [3] Perfect crystal Disordered material propagating phonons localized vibrations two-level system [2] Sidebottom, D. L. Fundamentals of Condensed Matter and Crystalline Physics: An Introduction for Students of Physics and Materials Science. Cambridge University Press, 2012.
Starting place: Whose shoulders do we stand on? (1) [4] A. Einstein (1911): *http://www.einstein.caltech.edu/ Original formulation of thermal conductivity, κ, using vibrations and a random walk [3]. In doing so - discovered (and used) first set of elementary excitations [4]. D. Cahill, S. Watson & R. Pohl (1992): Using a Debye spectrum, showed that Einstein s model applies to glasses [5]. [3] Einstein,. "Elementare Betrachtungen über die thermische Molekularbewegung in festen Körpern." Annalen der Physik 340.9 (1911): 679-694. [4] Anderson, P. W. Concepts in solids: lectures on the theory of solids. Vol. 58. World Scientific, 1997. [5] Cahill, David G., Susan K. Watson, and Robert O. Pohl. "Lower limit to the thermal conductivity of disordered crystals." Physical Review B 46.10 (1992): 6131.
Confinement by Strong Scattering: Mode Lifetime, τ [5] S(k) = 1 N Σ jk S(k) = ρ e ik(r j Rk ) V Γ dre ikr g(r) J. M. Larkin and A. J. H. McGaughey (2014): 20 Frequency [THz] 15 10 [6] Larkin, J. M., and A. JH McGaughey. "Thermal conductivity accumulation in amorphous silica and amorphous silicon." Physical Review B 89.14 (2014): 144303. 5 τ = I-R 2π Γ 0 0 0.5 1 k/ <Γ>
Confinement by Strong Scattering: Piecewise linear Γ : [6] Γ(k) = a k + b 5 Linewidth, Linewidth, <Γ> <Γ> [THz] [THz] 4 3 2 1 0 0 0.5 1 k/k k/ max [6] Larkin, J. M., and A. JH McGaughey. "Thermal conductivity accumulation in amorphous silica and amorphous silicon." Physical Review B 89.14 (2014): 144303.
Specific contribution(s): Case: Γ Vitreous silica [7] 1. Define characteristics of confined quasiparticle from. Γ 2. Describe a confining potential, µ. 3. Derive expressions for the density of states/heat capacity and thermal diffusivity resulting from confined quasiparticles.
1. Theory: Redundant forms of τ [8] Redundantly: τ = Λ v g = 2π Γ At the Ioffe-Regel Threshold ( Λ = λ = 2π ) [7]: k ω k = Γ(k) k [7] Ioffe, A. F., and A. R. Regel. "Non-crystalline, amorphous and liquid electronic semiconductors." Prog. Semicond 4 (1960): 237-291.
2. Theory: Effective energy, ω [9]! ω = a# " k $! &+b log# % " k $ &+R % 1 0.8 Where, R is $ R = Γ mid a k mid & % ' $ )+b log k mid & ( % log(ω/ω max ) ' ) ( 0.6 0.4 0.2 0 0.5 1 log(k/ )
3. Theory: 2 nd -order log fit to the effective energy [10] " log ω ω % " 0 $ ' = g log$ # & # ω max This simplifies to: " ω = H(k) ω max $ # Where, H(k) is k % ' & k m % ' & 2 +ω 0 " + m log$ # log(ω/ω max ) k 0 0.5 1 1.5 % ' & g = 0.265 m =1.25 ω 0 = 0.684 THz H(k) =! # " k $ & % 2g 2 3 2 1 0 log(k/ )
4. Theory: Dispersion character [11] Group Curvature: velocity: dω = (m + 2g)ω max dk 2 k v (km/s) 10 1 Note, Effective mass: 10 0 0 0 0.5 1 0 0.5 1 k/! # " v g v p $ & % (m+2 g) d 2 ω = (m + 2g)(m + 2g 1)ω k " max $ k 2 # m *! = 2 ω k 2 Effective mass, m* (MeV/c 2 ) 200 150 100 50 k % ' & k/ (m+2 g)
5. Theory: Implications on µ [12 ] Confining potential: µ(k) =!k(v p v p eff ) Confined Bose-Einstein Distribution: ( ) 1 " f BE = e #$ (E µ) k B T Implies [8]: * Conservation of particle number * Conservation of mass % &' 1 10 0 0.2 0.4 0.6 0.8 k/ [8] Chandler, David, and Jerome K. Percus. "Introduction to modern statistical mechanics." Physics Today 41.12 (2008): 114-118. (E - μ) [THz]
1. Results: Density of states: Boson peak (D/ω 2 ) [13] Energy density of states as a function of the effective frequency, D(ω) = ω 2 2π 2 v 3 dω D D / 3N / / 3N 3N [1/THz] / ω / ω 2 2 10 1 10 2 Boson peak: deviation from elastic crystal as linear in D/ω 2 10 5 Frequency, ω [THz] 10 3 0 20 40 [6] Larkin, J. M., and A. JH McGaughey. "Thermal conductivity accumulation in amorphous silica and amorphous silicon." Physical Review B 89.14 (2014): 144303.
2. Results: Heat capacity, C [14] Grand-canonical form: C = 3 1 (E(k) µ(k)) 2 k 2 2π 2 k B T 2 k IR " e $ # ( " * e $ ) # (E(k) µ(k)) k B T (E(k) µ(k)) k B T % ' & % + ' 1- &, 2 dk Heat Capacity, C [J g -1 K -1 ] 10 2 10 1 10 0 10 1 10 2 10 0 10 2 Temperature (K) [9] Zeller, R. C., and R. O. Pohl. "Thermal conductivity and specific heat of noncrystalline solids." Physical Review B 4.6 (1971): 2029.
3. Results: [15] Thermal diffusivity, Φ From basic kinetic theory: Φ = vλ At the Ioffe-Regel threshold: Φ = v 2π k Thermal Diffusivity, Φ [cm 2 s -1 ] 10 1 10 0 10 1 10 2 10 3 10 0 Frequency [THz] [6] Larkin, J. M., and A. JH McGaughey. "Thermal conductivity accumulation in amorphous silica and amorphous silicon." Physical Review B 89.14 (2014): 144303.
Summary and Conclusions [16] Confinement of vibration implies a finite and conserved mass. Boson peak in DOS is the result of confined vibrations. Self-consistent thermal diffusivity results from strong scattering of confined vibrations.
Acknowledgements: [A1] A sincere thank you for unbounded inspiration goes out to: - Prof. David E. Laughlin @ University I am grateful for funding from the NASA Office of Graduate Research through the Space Technology Research Fellowship.
References: [A2] [1] https://www.nasa.gov/content/nasas-orion-flight-test-and-the-journey-to-mars [2] Sidebottom, D. L. Fundamentals of Condensed Matter and Crystalline Physics: An Introduction for Students of Physics and Materials Science. Cambridge University Press, 2012. [3] Einstein,. "Elementare Betrachtungen über die thermische Molekularbewegung in festen Körpern." Annalen der Physik 340.9 (1911): 679-694. [4] Anderson, P. W. Concepts in solids: lectures on the theory of solids. Vol. 58. World Scientific, 1997. [5] Cahill, David G., Susan K. Watson, and Robert O. Pohl. "Lower limit to the thermal conductivity of disordered crystals." Physical Review B 46.10 (1992): 6131. [6] Larkin, J. M., and A. JH McGaughey. "Thermal conductivity accumulation in amorphous silica and amorphous silicon." Physical Review B 89.14 (2014): 144303. [7] Ioffe, A. F., and A. R. Regel. "Non-crystalline, amorphous and liquid electronic semiconductors." Prog. Semicond 4 (1960): 237-291. [8] Chandler, David, and Jerome K. Percus. "Introduction to modern statistical mechanics." Physics Today 41.12 (2008): 114-118. [9] Zeller, R. C., and R. O. Pohl. "Thermal conductivity and specific heat of noncrystalline solids." Physical Review B 4.6 (1971): 2029. *http://www.einstein.caltech.edu/
4. Results: Thermal conductivity, κ [xtra] κ = C v 2 τ = C Φ Thermal Thermal Conductivity, Conductivity, κ [W [W m -1-1 -1 K -1-1 -1 ] ] 10 0 10 1 10 2 2 10 0 10 1 10 2 Temperature [K] [8] Zeller, R. C., and R. O. Pohl. "Thermal conductivity and specific heat of noncrystalline solids." Physical Review B 4.6 (1971): 2029.