Phonons mean free paths and sound absorption Theory and Experiments B. Perrin Institut des NanoSciences de Paris

Transcription

1 Phonons mean free paths and sound absorption Theory and Experiments B. Perrin Institut des NanoSciences de Paris Phonon School 2017 : Wave Phenomena and Phonon Thermal Transport Oléron - September 3-8, 2017

2 Outline Macroscopic description of sound absorption Non-equilibrium thermodynamics Boltzmann equation approach coupling between an acoustic wave and a phonon gas the Akhiezer regime Microscopic description of sound absorption a perturbation approach the Landau Rumer regime Some diagrams Experimental determination of sound absorption in GaAs in the subterahertz range Herring processes breakdown in the terahertz range

3 Sound absorption α Units Phonon meanfree path «l» and phonon lifetime «τ» Quality factor of an acoustic resonator FWHM Quality factor

4 Sound absortion measurements in the ultrasonic range using piezoelectric transducers Piezoelectric transducers detected acoustic pulses sample Electrical pulse 1 cm Piezoelectric films Time delay (s) a few GHz Pioneering work : H. E. Bömmel and Dransfeld (1959) Highest frequency studied so far with this technique : 114 GHz J. Ilukor and E. H. Jacobsen (1966) in quartz at low temperatures. Acousto-optic Bragg interaction Possible artifacts : Diffraction effects Non linear up conversion Surfaces losses Reflection and conversion on the transducer Idiffr exp 0 z z

5 Equation of state Energy balance equation Fourier law Equation of motion Expansion

6 Plane waves solutions Cubic system D : thermal diffusivity

7 phonon viscosity loss thermoelastic loss (= 0 for shear waves) NaCl CdS KCl Ge ZnO GaAs LiF MgO Si SrTiO 3 TiO 2 Al 2 O 3 Sound velocity dispersion Diamond

8 Quality factor due to mechanical coupling 0 Q Quality factor due to intrinsic losses Q v Q 2 1 Q factor at 1GHz ( 10 6 ) Q factor at 20 GHz Diamond Sapphire TiO SrTiO MgO ZnO silicium LiF Germanium NaCl CdS KCl Silica

9 Boltzmann equation approach Grüneisen parameters

10 Boltzmann equation Local equilibrium Collision operator Phonon collisions conserves the energy Symmetrized collision operator Phonon group velocity

11 Collision time approximation Akhiezer description B C n n C C x x kt C C 1 T T, C T, T T D v Comparison with the macroscopic approach

12 1 T T, C T T D v 3 4 The transition between the Landau-Rumer and Akhiezer regimes Sound absorption of transverse waves in quartz H.E. Bömmel and K. Dransfeld, Phys. Rev. Lett. 2, 298 (1959) I. S. Ciccarello, K. Dransfeld, Phys. Rev. 134, A1517(1964)

13 Microscopic approach Landau-Rumer Inelastic scattering Annihilation N N 1 Fermi Golden Rule Creation N N 1 Balance

14 Phonon collision operator N n N N t Coll. M ' N ' Fission process Landau-Rumer expression Scattering process For a low frequency acoustic phonon : Fission processes can be neglected Assuming phonons λ and " λ belong to the same branch

15 There are two contributions to the lifetime : 1 2 The two-phonon density of states : number of processes which satisfy : The coupling parameters long wavelength approximation (LWA) V q q q 2 ' " ' " Deviation from the LWA of coupling parameters in Silicon A. Ward, D.A. Broido, Phys. Rev. B 81, (2010) 1 3 eff ' "

16 Diagrammatic approach to phonon-phonon interactions Harmonic propagator Dyson equation Neglecting modes hybridation The self-energy Phonon energy renormalisation 1 2

17 Diagrammatic approach to phonon-phonon interactions 3 phonons interactions 0.2 1, f THz T K, q 1 Only this lowest order diagram has to be considered This diagram contributes only to the real part of the self energy Bubble diagram, q, q q q : : : :, q q Acoustic frequency G Acoustic wave vector Thermal phonon frequency Thermal phonon wave vector Contribution of this diagram to the imaginary part 36 2 V 2 ' " n ' n " ' " ', " 1 2

18 4-phonons interactions If : 1 thermal phonons have to be dressed Multiphonon processes T<100K, θ = 360K this diagram can be neglected Relaxation of the energy conservation Modification of the Landau-Rumer expression 1 ' ' " 2 2 ( ' " ' " ) ' " CT C CT C 1 v ' 2 v ' 2 ' 3 ' ' 3 ' 2 2 2v0 C 2v0 C 1' 1'

19 Example : non dispersive isotropic medium The 3 LA phonons should be collinear Introducing a finite lifetime τ A LLL L L L ac. th th C111 3C 11 kt LLL 6 240v l C Carctan 2 C ln 1 C 240v 4 kt l F C 3C C C 3C C , B, C11 C B 2 C1 A B 4 B, C2 2 BB A, C3 6A4B 3B 3 Presence of B term comes from the angular dependence of the coupling parameter lim F 1 4 lim 0 F 15A 20AB 8B

20 Higher order phonons interactions If : 1 An infinite series of ladder diagrams has to be taken into account Boltzmann equation approach Akhieser mechanism Jp B. Perrin, in Studies in Physical and Theoretical Chemistry 46, 105 (1987) J p 1 Jp This recurrence relation leads to a Bethe-Salpeter equation for corresponding to the Boltzmann equation J At the end : H.J. Maris, Physical Acoustics VIII, Edited by W. P. Mason and R.N. Thurston, Academic Press, pages J.W. Tucker, V.W. Rampton, «Microwave Ultrasonics in Solid State Physics», North Holland Publishing Company

21 Direct measurement of phonon mean path in GaAs in the subterahertz range

22 What do we know about phonon mean free paths? Broadband phonon mean free path contributions to thermal conductivity measured using frequency domain reflectance K. T. Regner et al., Nature Communications 4, 1640 (2013) MFP of 40 and 20 nm in Si and GaAs

23 assuming p x T TD / T x e 3kB v 2 0 x g x dx T D e 1 x x T 0 kt D GaAs : T 370K f 7.5THz D D p WK.. m à 300K kt 10 nmand (1 THz) 0.38m WK.. m à 100K kt 56nmand 1 THz 2.2m And we can extrapolate the sound wave attenuation 1 (1 THz,50 K) 0.11m 2 1

24 Thermal conductivity accumulation distribution A. J. Minnich et al.. PRL 107, (2011) (Measurements in Silcon wafers)

25 Justin P. Freedman et al. Scientific Reports 3, 2963 (2013) 2

26 Pump pulse Measurements of phonon mean freepaths using coherent terahertz acoustics probe pulse 5 «Forward scattering q = 0» Backscattering modes q = 2k Frequency (THz) Brillouin superlattice backward scattering k : electromagnetic wave vector 0 q=2k 0 /d Acoustic wave vector forward scattering

27 Experimental configurations PUMP Probe PUMP emitting superlattice or 980 µm detecting Superlattice (thickness gradient) emitting and detecting superlattice PUMP optical microcavity

28 Pump and probe GaAs substrate Superlattice Substrate Fourier Transform (arb. units) q vector q = 2k q = 2k q = 2k q = 2k q = 2k q = 0 (mod 2π/d) Frequency (GHz)

29 Measurement of the inverse meanfree path dependence in terms of temperature Pump pulse probe pulse 1.5e-4 r/r 1.0e-4 5.0e e-5-1.0e ps (Filtered signal 1 THz x10) Fourier transform FWHM=3.45 GHz -1.5e-4-2.0e Time (ps) We measure the amplitude change A(T) in terms of temperature lt AT ln AT / 1 1 lt 0 this is not a determination of the absolute value of the meanfree path d 10K 2 T Frequency (THz)

30 Experimental results (below 0.4 THz) Temperature dependence Frequency (THz) µm Frequency dependence Round trip with a large period superlattice 0.3 Optical cavity 0.35 Round trip , 0.7, 0,785, Transmission 0.82, 1.0 Attenuation at 50 K Relative change of attenuation (m -1 ) GHz GHz 292 GHz GHz GHz Temperature (K) Relative change of attenuation (m -1 ) Best fit : A Frequency (GHz)

31 Experimental results (up to 1 THz) Temperature dependence Frequency dependence Attenuation at 50K Relative change of attenuation (m -1 ) THz THz THz THz THz THz THz THz THz Temperature (K) Relative change of attenuation (m -1 ) Frequency (THz)

32 Experiments at 0.4 THz performed on different propagation distances PUMP 360 or 983 µm probe Pump and probe GaAs substrate Real(r/r) 2e-4 1e-4 0-1e-4 Superlattices Sample 360 µm 983 µm 346*2µm Frequency (THz) e Temps (ps) 0.393THz Attenuation (m -1 ) Superlattice Substrate Attenuation at 0.4 THz Transmission through 360 mm Transmission through 983 mm Round trip through 346 mm Temperature (K) Errors bars 15%

33 Fission process 18 Scattering process V 2 ' " n' n" ' " n' n" ' " vl ', " Two-phonons density of states : How it can be explained? We are interested only in the temperature dependence. Thus elastic scattering by isotopes and impurities can be neglected and contribution of 3-phonons interactions to sound absorption can be only considered. Contribution of 3-phonons interactions to sound absorption ac ac q', ' q", " dq' ', " Phonons populations V i S qˆ qˆ qˆ e e e Anharmonic coupling parameter : ' ", ', ",, ', ", ijklmn ijklmn j l n i k m S ijklmn : Linear combination of second and third order elastic constants which can be determined experimentally

34 Spontaneous fission processes in an isotropic system Example : LA LA+ TA zaxis 18 2 V 2 ' " ac q l qac q v l ', " ', ' q ac q " q ' q'cos sin 2 q ' q'sin sin dq ' 2 q' dudq' with u cos BZ BZ cos vl vt q" q 1 ac q' 2 qacq' u x 2 vl vt ' x 1 1 x xu c ' " ac vt v l ac 0 u u x ac 0 ac ac 1 BZ vlvt v v ' " ' du u u x q q q dq f x dx Strong dependence in frequency, small contribution l t

35 The Herring processes Conyers Herring, Phys. Rev. 95, 954 (1954) LA velocity [110] The dominant process for low energy longitudal acoustic phonons is the «Herring Process» LA+STA FTA STA and FTA surfaces touch along A 4 axes If we neglect dispersion, the thermal phonon frequencies can be scaled by the frequency ɷ ac. of the acoustic phonon under consideration q' xac. A 4 axis STA velocity [100] q q' Close to a A 4 axis : x 2 FTA velocity q θ q ' T q 5 q Herring q depends on the crystal symmetry

36 Asymptotic expression for the contribution of Herring processes to α For low acoustic frequencies ω ac., processes with q, q close to A 4 axes are dominant. in cubic crystals : Consequently Herring Prefactor calculation by Simons (in GaAs) 2 3 f T 1 C C C (S. Simons, Proc. Cambridge Philos. Soc. 53, 702 (1957)) Herring 3.96 f ( THz) T 2 3

37 Frequency and temperatures ranges where Simons formula is valid Exact calculation of ρ 2 (x) (neglecting dispersion) x = 20 First condition : f(ghz) <T (K)

38 GaAs dispersion curves ɷ Tc = 3 THz Dispersion cannot be neglected for transverse phonons above 1THz There is a frequency cutoff for transverse phonons at 3THz Second condition : T < 10 K Drastic conditions should hold for this asymptotic formula to be valid in GaAs: T < 10 K f(ghz) <T (K)

39 STA and FTA surfaces intersect on A 3 axes LA velocity [110] A 3 axis [111] TA velocity q q' Close to a A 3 axis : TA velocity [001] q θ q ' x x > x if x x 2, A 2, A 3 4 9

40 Attenuation at 50K Simons formula (S. Simons, Proc. Cambridge Philos. Soc. 53, 702 (1957)) Herring 3.96 f ( THz) T 2 3 T < 10 K f(ghz) <T (K) Relative change of attenuation (m -1 ) Frequency (THz) The asymptotic expression is not valid for T >10K and f(ghz) >T(K) but the Herring contribution of Herring processes is in fact larger than what this expression predicts. What happens at higher acoustic frequencies? Could the plateau we observed experimentally be related to a Herring processes breakdown?

41 Calculation over the whole Brillouin zone neglecting phonon dispersion 1 THz 0.7 THz THz THz 70 THz GHz 40 GHz 20 GHz Breakdown of Herring processes ɷ Tc = 3 THz

42 Calculation with realistic dispersion curves from ab initio calculations Maximum around 0.5 THz 0 above 1.2 THz Herring density of states ɷ thermique f thermique (THz) f acoustique (THz) The maximum position depends critically on dispersion curves

43 Coupling parameters Approximations : Long wavelength expansion Cut off for the largest wavevectors if if 0 0 0

44 All processes Herring LA+FTA LA LA+STA LA Fission

45 1 (1 THz,50 K) 0.11m 2 1 Attenuation at 50K Relative change of attenuation (m -1 ) Experimental data Theoretical calculations for q c =0.38q D Frequency (THz)

46 Longitudinal mean free path extrapolated at room temperature Mean free path (m) Frequency (THz) T. Luo, J. Garg, J. Shiomi, K. Esfarjani and G. Chen Euro. Phys. Lett., 101 (2013) Average value at 1THz about 5 μm

47 The Simons formula paradox C C C C ac th m C C 2C m

48 Conclusions : Measurements of sound absorption can be difficult. Different regimes exist depending on the temperature and frequency ranges. Coherent terahertz acoustics experiments allow to zoom on phonon mean free paths in the subterahertz range. The mean free path for longitudinal modes exhibits a plateau in GaAs between 0.7 and 1 THz and is unexpectedly long for this frequency range. The macroscopic penetration depth of subaterahertz acoustic radiation opens the way for imaging embedded nano objects at room temperature. This plateau results from the breakdown of Herring processes and is likely to occur at different frequencies in other systems.