Thermal Energy at the Nanoscale Week 5: Carrier Scattering and Transmission Lecture 5.3: Phonon-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 energies • U processes do impede phonon momentum and do impede heat flow – Dominate thermal conductivity 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 phonons Type 2 Type 1 4) LA+ZATO, K =13.2 nm-1 1) LA ZO+LA, K =11.8 nm-1 1 1 5) LA+ZALO, 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+ZOLO, K1 =5.8 nm 7 Scattering Analysis and Models • Quantum mechanics + perturbation theory required for full scattering analysis, called “Fermi’s Golden Rule” (see Kaviany, Heat Transfer 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 crystal structure – High-temperature model (also from Klemens, 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 • b – boundary scattering = L/vg L = crystal size • Effective relaxation time, τ, by the Matthiesen rule – Assumes no interaction between scattering processes • Effective mean free path 14 Temperature Dependence of Thermal Conductivity Source: Kaviany, Heat Transfer Physics, 2008 15 .