Motivation. Confined acoustics phonons. Modification of phonon lifetimes Antisymmetric Bulk. Symmetric. 10 nm

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2 Motivation Confined acoustics phonons Modification of phonon lifetimes 0 0 Symmetric Antisymmetric Bulk 0 nm A. Balandin et al, PRB 58(998) 544 Effect of native oxide on dispersion relation Heat transport in nm scale Native oxide 30 nm M. Asheghi APL, 7(997), 798

3 Motivation Modification of dispersion relation (phonon engineering) Modification of group velocity Modification of relaxation rate Thermal conductivity Improve ZT 3

4 Outline Theory: Elastic continuum model Equation of motion Dispersion relation Relaxation rate & Thermal conductivity: Scattering mechanisms: mklapp & Normal Relaxation rate for -processes Thermal conductivity Numerical results: discussion Conclusions 4

5 Elastic continuum theory Theory Stress-strain relationship Macroscopic model i.e. continuum-based and isotropic Well-suited for nanoscale confinement studies Stress tensor / t T C ij Motion equation : S T Elastics Tensor ijkl S kl Strain tensor + ( S L S T ) T λs δ + µ S ij ( ) Vector amplitude of displacement ρ Density λ, µ Lamé constants S T, S L Transversal and longitudinal sound speed Isotropic media kk ij S ( λ + µ ) S T L µ ρ ij ρ 5

6 Wavevector q l, t q // l, t K l, t Single Layer Solutions / t S ( S // T + L l, t ω S K S q + q S T ) L, T l, t L, T // l, t ( ) z y T iz zsurface 0 i x, y, z Solutions y x z x z ( z) ( z) ( z) ( z) Asin( q ( z) - Aq iaq cos( q sin( q iaq sin( q Aq l l z cos( q z) or l z) + z) + z) + l l l z) B cos( q ibq z) cos( q Bqsin( q ibq t t z sin( q Bq cos( q t t z) t t z) Flexural Waves z) Shear Waves z) Dilatational Waves J.L. Rose, ltrasonic waves in solid media, Cambridge niversity press, K, 004 Coupled solutions Coupled solutions 6

7 Three layer Solutions,3 x,3 z x z (z) (z) (z ) (z) T iz DW DW DW DW x z z surface x z + + or or FW FW FW FW x z 3 x x 0 B. C + C. C + T iz i z interface z interface,, ω S T, L q t, l + q // T iz i i x, y, z z interface z interface equations System symmetry 6 linear equations, i.e., 6x6 matrix. DW x(z) : Dilatational Waves solutions en x (z) components. FW x(z) : Flexural Waves solutions en x (z) components. B. C. : Boundary Conditions. C. C. : Continuity Conditions. L. Donetti, F. Gámiz, J. B. Róldan and A. Godoy, J Appl. Phys. 00 (006),

8 Dispersion Relation & Group velocity Confinement More modes, hybridization of sagittal waves! Group velocity ( (v g ) dependant on q // // S L kms -, S T kms -, a0 nm DW: Dilatational Waves FW: Flexural Waves SW: Shear Waves 8

9 Experimental Results & Theory Flexural Waves 30 nm 0 nm Good match between theoretical model and experiments with Dispersion of Confined Acoustic Phonons in ltra-thin Silicon Membranes, J. Cuffe et al., in preparation 9

10 _3_layers Si v/s SiO Si-SiO SiO Oxide effect: decrease of frequency for large q // & high order modes // a8.5 nm b.6 nm a9.4 nm 0

11 Experimental Results & Theory 30 nm Si membrane Brillouin Scattering Spectroscopy ASOPS technique -L Prediction: 43.5 GHz 3-L Prediction: 33.8 GHz With:

12 Other systems

13 Outline Theory: Elastic continuum model Equation of motion Dispersion relation Relaxation Rate & Thermal Conductivity: Scattering mechanisms: mklapp & Normal Relaxation rate for -processes Thermal Conductivity Numerical results: discussion Conclusions 3

14 Scattering Mechanisms A: Lattice vacancy B: Point defect C: Crystal boundary D: mklapp E: Normal st Brillouin Zone τ eff κ Matthiessen's rule & Thermal Conductivity j 3 τ q, s j v q, s τ τ eff + τ C V b + τ i +... q q q q q q -process π/a π/a π/a π/a N-process -process Lattice Vector Consequence of anharmonicity Phonon-phonon interaction Finite phonon lifetime Possibilities:. Annihilation of two phonons and creation of a third phonon.. Annihilation of one phonon and creation of two phonons 4

15 Selection Rules: & N-Process ω q q G G q q R Q ω T L q q q V q ω P ω π/a r r r r -process q + q' G + q' ' P : Q : T T + T + T T R : T + L L If T is degenerate L If T is degenerate 0 0 ω + ω ω S π/a r q + : T + T r r q' q'' N-process L If T is degenerate V : T + L L 5

16 Relaxation Rate & Thermal conductivity τ ; q, s πγ h N Ωv ω ρ 0 q, s q' s', q'' s'', G ω q', s' ω q'', s'' ( n q' s' n q'' s'' ) δ ( ω qs + ω q' s' ω q'' s'' ) δ q+ q', q'' + G v s Phonon average group velocity q v qs n qs q n qs κ 3 Kinetic theory q, s v τ q C, s eff V κ hρv v ( + ) ( qsωqs nqs nqs nq ' s' 3πγ kbt qs q' s'; q'' s''; G n q'' s'' ) ω qs ω q' s' ω q'' s'' δ( ω) δ q+ q' q'', G More Branches More interactions Decrease thermal conductivity. Change of group velocity (v qs ) Change in thermal conductivity A. AlShaikhi and G.P. Srivastava, Phys. Rev. B, 76 (007), pp B. G. P. Srivastava, The Physics of Phonons, Taylor & Francis Group, NY,

17 Outline Theory: Elastic continuum model Equation of motion Dispersion relation Relaxation Rate & Thermal Conductivity: Scattering mechanisms: mklapp & Normal Relaxation rate for -processes Thermal Conductivity Numerical results: discussion Conclusions 7

18 Results: Relaxation Rates The modification of dispersion and group velocity lead to increase of relaxation rate compared with the bulk (left). The decrease in frequency of 3-layer membrane lead to decrease on relaxation rate compared with -layer membrane (right). 8

19 Results: Thermal Conductivity Intrinsic thermal conductivity The numerical calculations predict a strong decrease of the thermal conductivity of a -layer membranes. 9

20 Summary Spatial confinement Modification of dispersion relation Modification of group velocity ω 0 rad/sec DW FW SW aq // Group velocity Km/sec aq // Increase of relaxation rate Decrease of thermal conductivity κ membrane / κ bulk 0,8 0,6 0,4 5 nm 4 nm 0, 3 nm 0, Temperature K 0

21 Conclusions Hybridization of sagittal waves. Group velocity dependence on q // : v g (q // ). Oxide effect: decrease of frequency for large q // Decrease in relaxation rate. The change in dispersion relation and the emergence of more branches implies an increase of phonon-phonon interactions. This causes an increase in relaxation rates and a corresponding decrease in the thermal conductivity.

22 Flat Branches Future Work Phononic structures PWE solutions With: Band Gap Theoretical D dispersion relation for carbon cylinder in an epoxy matrix (sagittal waves) Y. Pennec et al, surface science reports 65(00), pp 9-9

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