CONFINED ACOUSTIC PHONONS IN SILICON MEMBRANES Clivia M Sotomayor Torres
COLLABORATORS J Cuffe, E Chavez, P-O. Chapuis, F Alzina, N Kehagias, L Schneider, T Kehoe, C Ribéreau-Gayon, L Schneider A Shchepetov, M Prunnila, S Laakso, J Ahopelto MIT J Johnson, A A. Maznev J Eliason, A Minnich, K Collins, G Chen, K A Nelson, A Bruchhausen, M Hettich, O Ristow and T Dekorsy. El-Houssain,(U Oujda) Y Pennec, B Djafari-Rouhani A Mlayah, J Groenen, A Zwick and F Poinsotte, U P Sabatier, Toulouse
OUTLINE Motivation Hypothesis Methods Membranes Inelastic light scattering Dispersion relations Impact on heat transfer Perspectives and Conclusions
MOTIVATION Acoustic parameters at nm scale - Speed of Sound - Elastic constants/viscoelasticity of polymers Heat transport and nm scale group velocity of confined phonons Sharma, PRL 2007 4
MOTIVATION Double-gate SOI transistors top gate oxide, SiO2 Top gate n+ poly Si BOX (back gate ox) Al bonded interface n+ top gate (111) n+ Si subst. n+ contact Back gate n- or p- Si n+ back gate M Prunnila, J Ahopelto, K Henttinen and F Gamiz APL 85, 5442 (2004) Cross-sectional bright field TEM image of a DG-SOI FET with a 18 nm-thick channel
MOTIVATION Effect on charge carrier mobility L. Donetti et al J. Appl. Phys. 100(2006), 013701 6
MOTIVATION Effect of phonon confinement on ZT of quantum wells A Balandin and K L Wang 1998 Rather controversial but crucial for thermoelectric energy conversion in the nm scale. Suitable charge conduction in phonon glasses needed. See also, M.S. Dresselhaus et al, Adv Mat 19, 1043 (2007).
MOTIVATION Phononic crystals Acoustic and elastic analogues of photonic crystals stop bands in phonon spectrum (phonon mirrors); negative refraction of phonons (phonon caustics) Good theory available: Multiple scattering theory for elastic and acoustic waves. See, for example: Kafesaki & Economou PRB 60, 11993 (1999), Liu et al PRB 62, 2446 (2000) Psaroba et al PRB 62, 278 (2000). And for a database 2006-2008: http://www.phys.uoa.gr/phononics/phononicdatabase.ht ml cell phones have phononic crystal-like BAW filters
2D infinite phononic crystal: air holes in silicon matrix (B Djafari-Rouhani, Y Pennec, IEMN, U Lille) Square Hexagonal Honeycomb reduced frequency 1.0 square, f=0.3 0.8 0.6 0.4 0.2 0.0 M Γ X M wavenumber reduced frequency 1.0 0.8 0.6 0.4 0.2 0.0 X triangular, f=0.6 Γ J X wavenumber reduced frequency 1.0 0.8 0.6 0.4 0.2 0.0 X honeycomb, f=0.3 Γ J X wavenumber reduced frequency 1.0 square, f=0.6 0.8 0.6 0.4 0.2 0.0 M Γ X M wavenumber reduced frequency 1.0 0.8 0.6 0.4 0.2 0.0 X triangular, f=0.85 Γ J X wavenumber reduced frequency 1.0 0.8 0.6 0.4 0.2 0.0 X honeycomb, f=0.6 Γ J X wavenumber
MOTIVATION Coupled cavities: photon-photon cavities. Physics of weak to strong coupling regimes Trigo et al PRL 2002
MOTIVATION Trigo et al PRL 2002
MOTIVATION Optical forces control mechanical modes prospects for cooling, heating, M Eichenfield et al. Optomechanical Crystals, Nature 462, 78-82 (2009)
MOTIVATION Acoustic phonons have also an impact in: Noise and thermal limits in NEMS and nanoelectronics Coherence control in quantum information processing Phonon engineering: sources, detectors and other components Photon-phonon coupling: Phoxonic Crystals and Opto mechanical oscillators Energy harvesting and storage THz technologies for medical diagnostic and security
Previous work: 30 nm SOI membrane
OUTLINE Motivation Hypothesis Methods Membranes Inelastic light scattering Dispersion relations Impact on heat transfer Perspectives and Conclusions
HYPOTHESIS The confinement of phonons modifies their frequencies and density of states affecting group velocities of modes, scattering mechanisms, lifetimes and changes assumptions about boundary conditions and transport properties. Understanding of acoustic phonons confinement in nanostructures is crucial for phonon engineering.
OUTLINE Motivation Hypothesis Methods Membranes Inelastic light scattering Dispersion relations Impact on heat transfer Perspectives and Conclusions
MEMBRANES Freestanding Si membranes Corrugation due to residual compressive strain in SOI films Methods to avoid corrugation are being developed. 200nm 50nm 50nm with weak vacuum
MEMBRANES HRTEM image of freestanding Si membrane, thickness 6 nm A. Schcepetov, M. Prunnila, J. Ahopelto, VTT J. Hua, Aalto University
OUTLINE Motivation Hypothesis Methods Membranes Inelastic light scattering Dispersion relations Impact on heat transfer Perspectives and Conclusions
Scattering Mechanisms I s Photoelastic Scattering 2 + u( z) = α dzp( z) G( z, z') E( z) z r i q i des Corrugation (Ripple) Scattering 1 LDOS = Im π EH El Boudouti et al, Surf Sci Reports 64, 471 (2009) q = [ G ( z, z )] k i k s Related to power spectrum of normal displacement Benedek, G B & Fritsch, K Phys Rev, 149, 647 (1966) Rowell, N. L. & Stegeman, G. I. PRB 18 2598 (1978,)
Raman scattering of Silicon 300 K, 514 nm unanalysed A Balandin 2000
Raman scattering of Silicon Low frequency range Interpreted as confined acoustic phonons Balandin 2000
Thin film SOI sample cross-section 40 nm SOI Native oxide 3 nm 28 nm Buried (thermal) oxide (SiO 2 ) 400 nm Base Si wafer CZ p-type <100> 525 micrometer
Simulations Raman spectra SOI thin film Photoelastic model for scattering by LA phonos z I( ωqz) dz. EL. E z * φ( ) S. p( z). 2 Φ 1 (z) oxide Φ2(z) silicon E L (E S ) : laser (scattered) field p(z) : photoelastic constant Φ(z) : phonon displacement Φ3(z) oxide Silicon buffer F Poinsotte et al Proc Phonons 2004
Simulations Raman spectra SOI thin film Vibrational part - phonons displacement and stress boundary conditions - Assumptions Phonons stationary waves Free surface Dispersion relation Infinite silicon buffer { φ ω =q. v qz φ1( zox / φ1 C1 z Si ( z ) = φ ( z Ox / Si 2 ) Ox / Si = C 2 ) φ2 z iq1 z iq1 z 1 ( z) = Ae 1 + B1 e 1 C φ 1 ( z air / Ox) = z z ac 0 ( z sound velocity Vac(oxide) =5970 m.s -1 Vac(silicon) =8433 m.s-1 Ox / Si ) Electronic part F Poinsotte et al Proc Phonons 2004 { P P Ox Si ( z) = 0 ( z) = 1
Free standing silicon membranes SOI membranes and configuration 500 μm Back-scattering Laser spot Forward scattering Sotomayor Torres et al phys stat sol c 2004
Simulations of RS spectra of SOI membranes Treat SOI layer as a cavity for acoustic phonons, ie, confined since longitudinal v s in Si = 8433 m/s (cf. 332 m/s in air at 0 C). Displacement field of acoustic vibrations in a slab of thickness t is proportional to: n π cos( z) t n order of the confined frequencies can be derived from LA dispersion branch, considering discrete wave vectors q = n π/t
(z,t) E ), ( ), ( i z t z u p t z P z S = z t u z z ), ( * ), ( z t z u z 2 2 2 0 2 2 2 2 2 2 ), ( 1 ), ( ), ( t t z P c t t z Es c n z t z Es = ε Acoustic vibration periodic variation of strain polarisation field in presence of em wave p s photo-elastic coefficient of slab P(z,t) OK for anti-stokes part. Obtain Stoke part by changing by Thus, scattered field: Where n = slab index of refraction. Simulations of RS spectra of SOI membranes
RS spectra of 31.5 nm thick SOI membrane Forward scattering Back scattering J Groenen et al, PRB 2008 Wavenumber cm -1
RS data of SOI membranes Summary of five SOI membranes J Groenen et al, PRB 2008
Summary of early work Confined phonons observed in supported thin films of SOI of 30-40 nm thickness. - Frequencies of modes extremely sensitive to thickness of underlying layers, eg., BOX layer and Si substrate layer. (acoustic cavity) - Preliminary mode assignment Confined acoustic phonons observed in free-standing membranes by low frequency Raman scattering - Validation of photo-elastic model in thin membranes fro q=0. - Mode assignment still incomplete. Need dispersion relations of the confined phonons.
OUTLINE Motivation Hypothesis Methods Membranes Inelastic light scattering Dispersion relations (J Cuffe, E Chavez) Impact on heat transfer Perspectives and Conclusions
From bulk to membranes Elastic continuum approach Displacement Strain Relationship Hooke s Law Newton s Second Law Membrane (Lamb) z = +a/2 σ iz =0 Dispersion Relation z = a/2 σ iz =0 Flexural (Anti symmetric) Dilatational (Symmetric) 34
430 nm Si Membrane Spectra at 3mm Mirror Spacing Dispersion Relation LA Lens @ 35GHz (Reference Peak) Spectra observed with Brillouin Light Scattering spectroscopy Multiple modes observed (deviation from bulk behaviour) Good agreement with theoretical calculations (Lamb waves)
10 nm Si Membrane Spectra at 3 mm Mirror Spacing Shear Dilatational Spectra at 10 mm Mirror Spacing Flexural Dilatational Shear (SH) 36
Phase Velocity vs q.a Phase(Group) velocity decreases dramatically for thinner membranes
OUTLINE Motivation Hypothesis Methods Membranes Inelastic light scattering Dispersion relations Impact on heat transfer Perspectives and Conclusions
Impact on thermal conductivity Spatial confinement Modification of dispersion relation Modification of group velocity ω 10 12 rad/sec 6 4 2 0 0 1 2 3 DW FW SW aq // Group velocity Km/sec 8 6 4 2 0 0 20 40 60 80 100 aq // Increase of relaxation rate Decrease of thermal conductivity κ membrane / κ bulk 0,18 0,16 0,14 5 nm 4 nm 0,12 3 nm 0,10 300 400 500 600 700 800 Temperature K 12
Impact on thermal conductivity Change in dispersion relation and the emergence of more branches increases interaction between phonons increase in relaxation rates and a corresponding decrease in the thermal conductivity The thinner the membrane the lower the thermal conductivity K. Including all the confined modes and calculating Umklapp processes
Phonon anharmonic decay Optical phonons (10 s of mev) Acoustic phonons (few mev) τ opt ac ~ 5 ps in Si τ e-opt ph ~ 100s fs Optical phonon emission high-field Joule heating Acoustic phonons carry heat away from hot spots
Phonon anharmonic decay Decay can involve only acoustic phonons. Cubic case and frequency < Debye frequency Higher energy acoustic phonons (few mev) Lower energy acoustic phonons (few mev) v v τ ac ac ~ fs-us in Si But the smaller the acoustic phonons energy difference, the longer the lifetime & mfp. Caustics increasingly important. v Must understand and control anharmonic decay into and propagation of acoustic phonons.
OUTLINE Motivation Hypothesis Methods Membranes Inelastic light scattering Dispersion relations Impact on heat transfer Perspectives and Conclusions
Conclusions and trends Dispersion relations of confined phonons measured and simulated Group velocity dependence on q // : v g (q // ). 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. Phonon coherence studies in confined structures are expected to become important Need contribution from statistical and noise physics.
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