Thermal at the Nanoscale Week 5: Carrier Scattering and Transmission Lecture 5.3: -Phonon Scattering Fundamentals

By Tim Fisher Professor of Mechanical Engineering Purdue University

1 Anharmonic Scattering

• Spring stiffness, g, changes with strain

• A strain caused by one wave will be seen as a different stiffness

• Anharmonic scattering is also known as intrinsic scattering because it can happen in a pure crystal

2 3-Phonon Scattering

• 3-phonon interactions (phonon-  phonon)K  1    K3 K1 + K 2 = K 3 ω1 + ω2 = ω3  (A) Normal K 2  (N)  K3    K1 K1 = K 2 + K 3 ω = ω + ω  1 2 3 (B)  K K 2 K 2 1     Umklapp + = + ω + ω = ω (C) K1 K 2 K 3 G 1 2 3 (U)  Where G is the reciprocal lattice vector

3 Brillouin Zone

G

K2

K3 K2 K 1 Γ K1

K3

4 Consequences for Heat Conduction

• N processes do not impede phonon momentum, therefore, do not impede heat flow ‘directly’ – Affect heat flow by re-distributing phonon

• U processes do impede phonon momentum and do impede heat flow – Dominate of semiconductors and insulators

5 Finding the Scattering Rate

G

K2

K3 K2 K 1 Γ K1

K3

Surface of energy imbalance Δω for the process ZA(K1)+ZA(K2)→TA(K3)

6 Line Segment of Energy Balance: LA

Type 2 Type 1 4) LA+ZATO, K =13.2 nm-1 1) LA  ZO+LA, K =11.8 nm-1 1 1 5) LA+ZALO, K =11.8 nm-1  -1 1 2) LA ZO+TA, K1 =11.8 nm  -1 -1 6) LA+ZA LA, K1 =8.8 nm 3) LA  ZA+TA, K1 =11.8 nm -1 7) LA+ZOLO, K1 =5.8 nm

7 Scattering Analysis and Models • + perturbation theory required for full scattering analysis, called “Fermi’s Golden Rule” (see Kaviany, Physics, 2008 Appendix E). • Engineering models for scattering typically take a power law form

– Where B is a constant – n and m are integers

8 N-Process Scattering • Low temperatures – Longitudinal τ=ω−1B 23 T, B = const LN L L – Transverse −14 τ=ωTNB T T, B T = const • High temperatures – Longitudinal −12 τ=LNB' L ω T ,' B L = const – Transverse −1 τ=TNB' T ω T ,' B T = const

9 U-Process Scattering

• Many proposed models, but no universal one – Klemens (Solid State Physics, 7, 1–98, 1958) suggested

• A is a dimensionless constant that depends on atomic mass, lattice spacing, and Gruneisen constant • is a parameter that represents – High-temperature model (also from Klemens, 𝛾𝛾 1958)

10 Effective Relaxation Time • Clearly, relaxation times are complicated (and often not well established) • If scattering processes are independent, then scattering rates are additive (Matthiessen’s rule)

τ=−−11 τ ∑ j scat.. proc j • Often, scattering rates are assumed constant and made to be adjustable parameters to match experimental data

11 N Processes • Recall thermal conductivity

• If N processes were treated like U processes 1 11 = + τeff ττ N U • N processes would then erroneously product thermal resistance

12 Issues with N Process Modeing • U processes require large K vectors to occur – Rarely occur for small K – Need N processes to create long K’s

• Even though N processes do not directly impede heat flow, they are important because they ‘feed’ U processes

13 Effective Relaxation Time

• i – defect scattering • U – U (umklapp) scattering 𝜏𝜏 • b – boundary scattering = L/vg 𝜏𝜏 L = crystal size • Effective𝜏𝜏 relaxation time, τ, by the Matthiesen rule

– Assumes no interaction between scattering processes • Effective

14 Temperature Dependence of Thermal Conductivity

Source: Kaviany, Heat Transfer Physics, 2008

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