Phonon Scattering & Thermal Conductivity ME 381R Lecture 7: Phonon Scattering & Thermal Conductivity Dr. Li Shi Department of Mechanical Engineering The University of Texas at Austin Austin, TX 78712 www.me.utexas.edu/~lishi lishi@mail.utexas.edu Reading: 1-3-3, 1-6-2 in Tien et al References: Ch5 in Kittel

Phonon Thermal Conductivity Matthiessen Rule: Kinetic Theory: Phonon Scattering Mechanisms Decreasing Boundary Separation Boundary Scattering Defect & Dislocation Scattering Phonon-Phonon Scattering l Increasing Defect Concentration Boundaries change the spring stiffness (acoustic impedance) crystal waves scatter when encountering a change of acoustic impedance (similar to scattering of EM waves in the presence of a change of an optical refraction index) Phonon Scattering Defect Boundary 0.01 0.1 1.0 Temperature, T/qD

Specular Phonon-boundary Scattering Phonon Reflection/Transmission TEM of a thin film superlattice Acoustic Impedance Mismatch (AIM) = (rv)1/(rv)2

Phonon Bandgap Formation in Thin Film Superlattices min =50 100 50 n=2, n=1, =100 n=3, =66 =200 n=4, n = 2d cosq (i) (ii) wavevector, K frequency, w (A) (B) Courtesy of A. Majumdar

Diffuse Phonon-boundary Scattering Specular Diffuse Acoustic Mismatch Model (AMM) Khalatnikov (1952) Diffuse Mismatch Model (DMM) Swartz and Pohl (1989) E. Swartz and R. O. Pohl, “Thermal Boundary Resistance,” Reviews of Modern Physics 61, 605 (1989). D. Cahill et al., “Nanoscale thermal transport,” J. Appl. Phys. 93, 793 (2003). Courtesy of A. Majumdar

SixGe1-x/SiyGe1-y Superlattice Films Period AIM = 1.15 Alloy limit With a large AIM, k can be reduced below the alloy limit. Huxtable et al., “Thermal conductivity of Si/SiGe and SiGe/SiGe superlattices,” Appl. Phys. Lett. 80, 1737 (2002).

Effect of Impurity on Thermal Conductivity Why the effect of impurity is negligible at low T?

Phonon-Impurity Scattering Impurity change of M & C  change of spring stiffness (acoustic impedance) crystal wave scatter when encountering a change of acoustic impedance (similar to scattering of EM wave in the presence of a change of an optical refraction index) Scattering mean free time for phonon-impurity scattering: li ~ 1/(sr) where r is the impurity concentration, and the scattering cross section =  R2 [4/(4+1)] R: radius of lattice imperfaction l: phonon wavelength = 2R/l -> 0: s ~ 4 (Rayleigh scatttering that is responsible for the blue sky and red sunset)  -> : s ~  R2

Effect of Temperature u(w)= s  (R/l)4 for l >> R s  R2 for l << R l: phonon wavelength R: radius of lattice imperfection l Temperature, T/qD Boundary Phonon Scattering Defect Decreasing Boundary Separation Increasing Defect/impurity Concentration 0.01 0.1 1.0 u(w)= Increasing T wD w

Bulk Materials: Alloy Limit of Thermal Conductivity k [W/m-K] A B Alloy Limit Impurity and alloy atoms scatter only short- l phonons that are absent at low T!

Phonon Scattering with Imbedded Nanostructures Atoms/Alloys Nanostructures Frequency, w wmax eb v Spectral distribution of phonon energy (eb) & group velocity (v) @ 300 K Phonon Scattering Long-wavelength or low-frequency phonons are scattered by imbedded nanostructures!

Imbedded Nanostructures 5x1018 Si-doped InGaAs Si-Doped ErAs/InGaAs SL (0.4ML) Undoped ErAs/InGaAs SL (0.4ML) Nanodot Superlattice Data from A. Majumdar et al. AgPb18SbTe20 ZT = 2 @ 800K AgSb rich Hsu et al., Science 303, 818 (2004) Bulk materials with embedded nanodots Images from Elisabeth Müller Paul Scherrer Institut Wueren-lingen und Villigen, Switzerland

Phonon-Phonon Scattering The presence of one phonon causes a periodic elastic strain which modulates in space and time the elastic constant (C) of the crystal. A second phonon sees the modulation of C and is scattered to produce a third phonon. By scattering, two phonons can combine into one, or one phonon breaks into two. These are inelastic scattering processes (as in a non-linear spring), as opposed to the elastic process of a linear spring (harmonic oscillator).

Phonon-Phonon Scattering (Normal Process) Anharmonic Effects: Non-linear spring K1 K2 K3 = K1+K2 Non-linear Wave Interaction Because the vectorial addition is the same as momentum conservation for particles: Phonon Momentum = K Momentum Conservation: K3 = K1+ K2 Energy Conservation: w3= w1 + w2

Phonon-Phonon Scattering (Umklapp Process) that is outside the first Brillouin Zone K1 What happens if K3 = K1+K2 Then (Bragg Condition as shown in next page) K2 K1 K2 G U-Process K3 The propagating direction is changed.

Reciprocal Lattice Vector (G) G = 2p/a l: wavelength K = 2/l lmin = 2a Kmax = /a -/a<K< /a 2a

Normal Process vs. Umklapp Process Selection rules: K1 K2 K3 Normal Process: G =0 Umklapp process: G = reciprocal lattice vector = 2p/a 0 Ky Ky K1 K3 K3 Kx K1 K2 Kx K2 1st Brillouin Zone Cause zero thermal resistance directly Cause thermal resistance

Effect of Temperature phonon ~ exp(D/bT) l  phonon ~ exp(D/bT) Decreasing Boundary Separation l Increasing Defect Concentration  phonon ~ exp(D/bT) phonon ~ exp(D/bT) Phonon Scattering Defect Boundary 0.01 0.1 1.0 Temperature, T/qD

Phonon Thermal Conductivity Cl Kinetic Theory Decreasing Boundary Separation T l Increasing Defect Concentration Phonon Scattering Defect Boundary 0.01 0.1 1.0 Temperature, T/qD

Thermal Conductivity of Bulk Crystals 3 k