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
Motivation Modification of dispersion relation (phonon engineering) Modification of group velocity Modification of relaxation rate Thermal conductivity Improve ZT 3
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
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
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
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), 0370 7
Dispersion Relation & Group velocity Confinement More modes, hybridization of sagittal waves! Group velocity ( (v g ) dependant on q // // S L 8.440 kms -, S T 5.845 kms -, a0 nm DW: Dilatational Waves FW: Flexural Waves SW: Shear Waves 8
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
_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
Experimental Results & Theory 30 nm Si membrane Brillouin Scattering Spectroscopy ASOPS technique -L Prediction: 43.5 GHz 3-L Prediction: 33.8 GHz With:
Other systems
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
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
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
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. 9505-. B. G. P. Srivastava, The Physics of Phonons, Taylor & Francis Group, NY, 990. 6
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
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
Results: Thermal Conductivity Intrinsic thermal conductivity The numerical calculations predict a strong decrease of the thermal conductivity of a -layer membranes. 9
Summary Spatial confinement Modification of dispersion relation Modification of group velocity ω 0 rad/sec 6 4 0 0 3 DW FW SW aq // Group velocity Km/sec 8 6 4 0 0 0 40 60 80 00 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 0,0 300 400 500 600 700 800 Temperature K 0
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.
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