Effect of phonon confinement on the heat dissipation in ridges

Effect of phonon confinement on the heat dissipation in ridges P.-O. Chapuis 1*, A. Shchepetov 2*, M. Prunnila 2, L. Schneider 1, S. Lasko 2, J. Ahopelto 2, C.M. Sotomayor Torres 1,3 1 Institut Catala de Nanotecnologia (ICN), Centre d'investigacio en Nanociencia e Nanotecnologia (CIN2=ICN+CSIC), Bellaterra (Barcelona), Spain 2 VTT Microelectronics, Espoo, Finland 3 Institució Catalana de Recerca e Estudis Avaçats (ICREA), Barcelona, Spain *Contributed equally Eurotherm 91, Poitiers, August 2011

Outline Phonon transport confinement effects Measuring the temperature in ridge samples Thermal conductances of silicon ridges Conclusion/Perspectives

Phonon transport confinement effects Phonon lengthscales Acoustic phonons are the main heat carriers in nonmetals Thermal wavelengths Approximate mean free path by inversion of λ T 1 ( ) = ρc v Λ p s 3 [~0-20] nm @RT Mean free path (MFP) [10 500] nm =f(ω) Modifying the temperature is a way to change the average MFP Kn=Λ avg /D 2

Phonon transport confinement effects Si Example in suspended structures D 30nm Bulk~150 D= 500μm Li,, Majumdar, Appl. Phys. Lett. 83, 2934 (2003) etc. 3

Phonon transport confinement effects Supported nanostructures? e d w Heater L e Cross section ~ 100 nm w ~ 100 nm-1 μm Heater = phonon generator d ~ 100 nm-1 μm Ridge Confinement (phonon constriction) Heat flux propagation? 1 μ Substrate Si SAMPLE 4

Outline Phonon transport confinement effects Measuring the temperature in ridge samples Thermal conductances of silicon ridges Conclusion/Perspectives

Fabrication of the sample Different ways of sample fabrication Implanted heater Highly resistive silicon Ion implantation ρ e (Si doped, 100 nm) = 10 4 Ωm ρ e (Si doped, 200 nm) = 6 10 6 Ωm Doped silicon Highly resistive silicon 100 nm ICP etching Expitaxial heater Doped Si epitaxy Doped silicon Metal heater Highly resistive silicon Highly resistive silicon ρ e (Si, 100 nm) = 1.3 10 5 Ωm Al 90 nm TiW 10 nm 800 nm Silicon Silicon Mask Maximum roughness: Period~50nm, height~10% 5

Fabrication of the sample Different ways of sample fabrication Implanted heater Highly resistive silicon Ion implantation ρ e (Si doped, 100 nm) = 10 4 Ωm ρ e (Si doped, 200 nm) = 6 10 6 Ωm Doped silicon Highly resistive silicon 100 nm ICP etching Expitaxial heater Doped Si epitaxy Doped silicon Metal heater Highly resistive silicon Highly resistive silicon ρ e (Si, 100 nm) = 1.3 10 5 Ωm Al 90 nm TiW 10 nm 800 nm Silicon Silicon Mask Maximum roughness: Period~50nm, height~10% 5

Measuring the ridge temperature Wire temperature measurement 3ω method, Cahill, RSI (1990) R(T) = R 0 (1 + α ΔT) Electrical resistance depends on temperature I = I ω cos(ωt) P(t) = R I(t) 2 = ½R 0 (1 + cos(2ωt)) Joule heating of the electric T(t) = T 0 + T DC + T 2ω cos(2ωt+φ 2ω ) wire U = RI = R 0 [1 + α(t DC + T 2ω cos(2ωt+φ 2ω ))] I ω cos(ωt) = R 0 I ω [(1+ αt DC )cos(ωt) + ½ αt 2ω cos(ωt-φ 2ω ) + ½ αt 2ω cos(3ωtt+φ 2ω ) ] = U 1ω + ½ αr 0 I ω T 2ω cos(3ωt+φ 2 ω ) Lock-in detection Other possibility: Combining DC and AC current input I = I 0 + I ω cos(ωt) U ω = R 0 I ω (1+ αt DC )cos(ωt) Wire temperature = f(heat flux to the sample) 6

Measuring the ridge temperature Calculation of the conductance G th The conductance is determined from G th = P/ΔT = F geometry RI 2 / ΔT F geometry? Pads are heat baths. T wire is not always constant along its length. Temperature distribution T Temperature profile wire FEM simulation I 0 =1.3 10-3 A g =10 8 WK -1 m -2 (f =12 Hz) Analytical calculation 7

Outline Phonon transport confinement effects Measuring the temperature in ridge samples Thermal conductances of silicon ridges Conclusion/Perspectives

Thermal conductances of Si ridges Measured thermal conductances e d Heater Epitaxial heater, Superimposed AC-DC w Cross section 450 nm 200 nm 100 nm 400 nm 1600 nm G appears to follow a w/d trend (Fourier?) 8

Thermal conductances of Si ridges Deviation to ballistic theory Ratio S(cavity)/ S(contact) 400 nm 450 nm 3 2 1 4 2.8 5 9 33 200 nm 100 nm Thermal conductance normalized by theoretical Fourier s prediction Ballistic prediction w=d 1600 nm Temperature [K] Far from the Fourier prediction (<10%) Also different from a pure ballistic prediction (<50%) Impact of roughness on the violet curve 9

Thermal conductances of Si ridges Explanations of deviation to ballistic theory 1) Surface thermal resistance R th s - Until now not seen with 3ω - Si vs epitaxially-doped Si? 10

Thermal conductances of Si ridges Explanations of deviation to ballistic theory 2) Correction to theory of constrictions between macrobodies Standard ballistic theory simple particle approach Assumption : two heat baths All phonons are supposed to be transmitted through the constriction 11

Thermal conductances of Si ridges Explanations of deviation to ballistic theory 2) Correction to theory of constrictions between macrobodies Standard ballistic theory simple particle approach Assumption : two heat baths All phonons are supposed to be transmitted through the constriction Theory applied to a nanostructure ( Nano correction) Phonon backscattering? No thermalization in the nanostructure? R th = Ballistic Fourier deviation prediction(sharvin-like) R Fourier 1 ( 1+ βkn). C( W / BTE) 1 γ called Wexler s formula BTE correction to Wexler (heat source case) «Nano» correction to Wexler/BTE Related to the backscattering probability Volz, JAP (2008) 11

Conclusion/Perspectives The thermal conductance of silicon ridges has been measured for 100-450nm thick ridges Its temperature dependence (i.e. Knudsen number) was observed A two orders of magnitude decrease vs Fourier law was measured due to confinement As high as one order of magnitude difference with the ballistic prediction Relative contribution of the two effects : R th s and 1/(1-γ) Lower temperatures interplay with strong wave effect Higher temperatures larger range of Knudsen number Roughness consequences (violet curve)? 12

Conclusion/Perspectives Si D DIFFUSIVE D ~ MFP BALLISTIC D Particle Channels =modes Wave? Change of the relation dispersion Balandin, J. Super. Micro 26, 181 (1999) Refining the associated theory? Wave (freq.) dependence! 13

Thank you for your attention! Questions?