ME 381R Lecture 7: Phonon Scattering & Thermal Conductivity

1. 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

2. 1.0 0.01 0.1 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) PhononScattering Defect Boundary Temperature, T/qD

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

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

5. Diffuse Phonon-boundary Scattering Specular Diffuse Diffuse Mismatch Model (DMM) Swartz and Pohl (1989) Acoustic Mismatch Model (AMM) Khalatnikov (1952) E. Swartz and R. O. Pohl, “Thermal Boundary Resistance,” Reviews of Modern Physics61, 605 (1989). D. Cahill et al., “Nanoscale thermal transport,” J. Appl. Phys. 93, 793 (2003). Courtesy of A. Majumdar

6. SixGe1-x/SiyGe1-y Superlattice Films Superlattice 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).

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

8. 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

9. Decreasing Boundary Separation l Increasing Defect/impurity Concentration PhononScattering Defect Boundary 1.0 0.01 0.1 Temperature, T/qD Effect of Temperature s (R/l)4forl >> R s R2forl << R l: phonon wavelength R: radius of lattice imperfection u(w)= Increasing T wD w

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

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

12. Imbedded Nanostructures 5x1018 Si-doped InGaAs Si-Doped ErAs/InGaAs SL (0.4ML) Undoped ErAs/InGaAs SL (0.4ML) Hsu et al., Science303, 818 (2004) AgPb18SbTe20 ZT = 2 @ 800K AgSb rich • Nanodot Superlattice Data from A. Majumdar et al. • Bulk materials with embedded nanodots Images from Elisabeth Müller Paul Scherrer Institut Wueren-lingen und Villigen, Switzerland

13. 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).

14. K1 K3 = K1+K2 K2 Phonon-Phonon Scattering (Normal Process) Anharmonic Effects: Non-linear spring 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

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

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

17. 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

18. 1.0 0.01 0.1 Effect of Temperature Decreasing Boundary Separation l Increasing Defect Concentration  phonon~ exp(D/bT) phonon~ exp(D/bT) PhononScattering Defect Boundary Temperature, T/qD

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

20. Thermal Conductivity of Bulk Crystals 3 k