Energy Density and Thermal Diffusivity of Ioffe-Regel Confined Vibrations

Transcription

1 (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

2 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]

3 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.

4 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.

5 Starting place: Whose shoulders do we stand on? (1) [4] A. Einstein (1911): * 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 (1911): [4] Anderson, P. W. Concepts in solids: lectures on the theory of solids. Vol. 58. World Scientific, [5] Cahill, David G., Susan K. Watson, and Robert O. Pohl. "Lower limit to the thermal conductivity of disordered crystals." Physical Review B (1992): 6131.

6 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] [6] Larkin, J. M., and A. JH McGaughey. "Thermal conductivity accumulation in amorphous silica and amorphous silicon." Physical Review B (2014): τ = I-R 2π Γ k/ <Γ>

7 Confinement by Strong Scattering: Piecewise linear Γ : [6] Γ(k) = a k + b 5 Linewidth, Linewidth, <Γ> <Γ> [THz] [THz] k/k k/ max [6] Larkin, J. M., and A. JH McGaughey. "Thermal conductivity accumulation in amorphous silica and amorphous silicon." Physical Review B (2014):

8 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.

9 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):

10 2. Theory: Effective energy, ω [9]! ω = a# " k $! &+b log# % " k $ &+R % Where, R is $ R = Γ mid a k mid & % ' $ )+b log k mid & ( % log(ω/ω max ) ' ) ( log(k/ )

11 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 % ' & g = m =1.25 ω 0 = THz H(k) =! # " k $ & % 2g log(k/ )

12 4. Theory: Dispersion character [11] Group Curvature: velocity: dω = (m + 2g)ω max dk 2 k v (km/s) 10 1 Note, Effective mass: 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 ) k % ' & k/ (m+2 g)

13 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 % &' k/ [8] Chandler, David, and Jerome K. Percus. "Introduction to modern statistical mechanics." Physics Today (2008): (E - μ) [THz]

14 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 Boson peak: deviation from elastic crystal as linear in D/ω Frequency, ω [THz] [6] Larkin, J. M., and A. JH McGaughey. "Thermal conductivity accumulation in amorphous silica and amorphous silicon." Physical Review B (2014):

15 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 ] 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.

16 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 ] Frequency [THz] [6] Larkin, J. M., and A. JH McGaughey. "Thermal conductivity accumulation in amorphous silica and amorphous silicon." Physical Review B (2014):

17 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.

18 Acknowledgements: [A1] A sincere thank you for unbounded inspiration goes out to: - Prof. David E. University I am grateful for funding from the NASA Office of Graduate Research through the Space Technology Research Fellowship.

19 References: [A2] [1] [2] Sidebottom, D. L. Fundamentals of Condensed Matter and Crystalline Physics: An Introduction for Students of Physics and Materials Science. Cambridge University Press, [3] Einstein,. "Elementare Betrachtungen über die thermische Molekularbewegung in festen Körpern." Annalen der Physik (1911): [4] Anderson, P. W. Concepts in solids: lectures on the theory of solids. Vol. 58. World Scientific, [5] Cahill, David G., Susan K. Watson, and Robert O. Pohl. "Lower limit to the thermal conductivity of disordered crystals." Physical Review B (1992): [6] Larkin, J. M., and A. JH McGaughey. "Thermal conductivity accumulation in amorphous silica and amorphous silicon." Physical Review B (2014): [7] Ioffe, A. F., and A. R. Regel. "Non-crystalline, amorphous and liquid electronic semiconductors." Prog. Semicond 4 (1960): [8] Chandler, David, and Jerome K. Percus. "Introduction to modern statistical mechanics." Physics Today (2008): [9] Zeller, R. C., and R. O. Pohl. "Thermal conductivity and specific heat of noncrystalline solids." Physical Review B 4.6 (1971): *

20 4. Results: Thermal conductivity, κ [xtra] κ = C v 2 τ = C Φ Thermal Thermal Conductivity, Conductivity, κ [W [W m K ] ] 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.