Vibrational properties and phonon transport of amorphous solids

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1 June 29th (Fri.), 2018 Yukawa Institute for Theoretical Physics, Kyoto University, Japan Rheology of disordered particles suspensions, glassy and granular materials 10:15-11:05, 40mins. talk and 10mins. discussion Vibrational properties and phonon transport of amorphous solids Hideyuki Mizuno, Hayato Shiba, Atsushi Ikeda The University of Tokyo, Tohoku University

Background Vibrational properties of amorphous solids 2/27 Crystals (lattice structure) Amorphous solids (amorphous structure) Molecules vibrate around lattice

2 Background Vibrational properties of amorphous solids 2/27 Crystals (lattice structure) Amorphous solids (amorphous structure) Molecules vibrate around lattice structure Molecules vibrate around amorphous structure Tanguy et al., EPL 2010 Vibrational modes are phonons Some modes are localized -> These modes are non-phonons

3 Background Vibrational properties of amorphous solids 3/27 Crystals (lattice structure) Amorphous solids (amorphous structure) Vibrational density of states (one component Lennard-Jones system) Transverse Longitudinal

4 Background Excess low-ω vibrational modes in amorphous solids Excess over Debye-theory prediction (Boson peak) is observed in many glasses (amorphous solids) 4/27 DOS Crystal Crystal Yamamuro et al., JCP 1996 Buchenau et al., PRL 1984

5 Zeller and Pohl, PRB, 1971 Background Low-T thermal properties of amorphous solids Specific heat Thermal conductivity 5/27 SiO2 Crystal SiO2 Glass SiO2 Crystal SiO2 Glass

6 Background Extension of Debye theory: Elastic heterogeneities 6/27 Crystals Debye theory Amorphous solids: Extended Debye theory Ø Ø Ø Homogeneous elastic media Elastic mechanics predict phonons or acoustic waves as vibrational modes This explains heat capacity of crystals T cubed behavior Ø Ø Ø Heterogeneous elastic media Vibrational modes are deformed to nonphonon modes by elastic heterogeneities This predicts heat capacity larger than Debye prediction Thermal properties of amorphous solids (Boson peak) Heterogeneous modulus Schirmacher et al., EPL 2006, PRL 2007

7 Background Multi-scale structure of amorphous solids 7/27 Molecules Local elastic modulus Homogeneous media? Micro-scale Meso-scale Macro-scale At meso-scale, local elastic modulus fluctuates in the space. -> Elastic heterogeneities control thermal properties of amorphous solids Mizuno, Mossa, Barrat, EPL 2013, PNAS 2014, PRB 2016 At macro-scale (continuum limit), amorphous solids behave as homogeneous elastic media? Mizuno, Shiba, Ikeda, PNAS 2017

8 Zeller and Pohl, PRB, 1971 Background Low-T thermal properties of amorphous solids Specific heat Thermal conductivity 8/27 SiO2 Crystal SiO2 Glass SiO2 Crystal SiO2 Glass

9 Contents 9/27 Questions l What is nature of low-frequency vibrations of amorphous solids? l What laws do they obey? We perform molecular-dynamics simulations on a simple model amorphous solid ü Molecules interact through harmonic pair potential ü System size : up to 4,096,000 ü Periodic boundary condition in all the directions

10 Zeller and Pohl, PRB, 1971 Background Low-T specific heat of amorphous solids Specific heat Within the harmonic approximation 10/27 vdos Low-temperature thermal properties are controlled by low-frequency vibrations: SiO2 Crystal SiO2 Glass

11 Background Two level system / Soft potential model Two level system 11/27 Debye theory -> Phonons Additional, non-phonon modes exist in amorphous solids -> Two level system, presumably considered as localized motions of particles -> C T is explained Anderson, et al., Phil. Mag., 1972 Density of two level system with very low energy barriers

12 Vibrational modes analysis 12/27 Spring-mass model u = r r 0 ( ) ü Linearized equation of motion ( 2 ) Φ ü = u r r ü Diagonalize Hessian matrix : (3N 3N matrix) 0 ( e k 2 ) Φ r r 0 ( )) e k = ω k2 ü Vibrational modes (normal modes, eigen modes) Eigen frequencies: k =1, 2,...,3N Eigen vectors:

13 Vibrational modes in Lennard-Jones crystals 13/27 Crystals (lattice structure) Eigen vectors: Transverse phonon Vibrational density of states Transverse Longitudinal Longitudinal phonon

14 Vibrational modes in simple amorphous solid 14/27 Ø A, Debye-like regime : Elastic-wave-like modes Silbert, Liu, Nagel, PRE 2009 Ø B, Plateau regime : Disordered extended modes Ø C, High frequency regime : Highly localized modes Elastic-wave-like mode Disordered extended mode Highly localized mode

15 Vibrational density of states 15/27 Debye level ü We observe the boson peak around ω = 0.1. ü Reduced vdos goes towards the Debye level, but does not converge in our frequency regime.

16 Characterize each vibrational mode: Phonon order parameter 16/27 Phonons in isotropic medium under periodic boundary: : wave vector Expansion by phonon modes (Fourier expansion) of vibrational mode k: Phonon order parameter: : Phonon : Non-phonon

17 Characterize each vibrational mode: Participation ratio 17/27 Participation ratio of vibrational mode k: Mazzacurati, Ruocco, Sampoli, EPL Only one particle vibrates: 2. All the particles vibrate equivalently: Fraction P k of total particles participate in the vibrational mode k

mode At ω < ωbp : Mixture of phonon mode and

18 Vibrational modes 18/27 At ω > ω* : Non-phonon, extended vibrational nature -> Disordered extended mode At ω ωbp < ω* : Hybrid character of phonon and disordered mode At ω < ωbp : Mixture of phonon mode and soft localized mode -> Consistent with the idea of two-level system

19 Close up to the low-frequency regime 19/27 N = 2,048,000, L = 114 Phonon mode Localized mode ü Non-phonon, localized modes sit between different phonon energy levels

20 Vibrational density of states 20/27

21 Vibrational density of states 21/27 Debye law Non-Debye law of localized modes

22 22/27 g(ω)/ω 2 ored the Vibrational low-frequencymodes vibrational properties in Lennard-Jones glass interacting via the LJ potential, We calculated d All modes figurations by the $!" # " # P > 10 σ 12 σ 6 nalize them to get < 10, φ(r) = 4ϵ P(1) r r ek = (ek1, ek2,, e A they are sorted in a ω vectors ω are normal lar dynamics simulations. When we use short0 eractions like the LJ potential, we usually 1trun ω otential at a fixed value rc [19]. Namely, we ω 0.6 ractions with particles at larger distances0.4than ω owever, when the potential is truncated, 0.2it has 0 nuity at r = rc and this discontinuity may lead 10 Given eigenfrequ 19]. Some methods are used to avoid such er10 λk, the vibration Firstly, we can truncate and shift the potential, 10 it vanishes at r = rc [19]. We call this 10type of 10 tential LJ0, Mixture of phonons and soft localized g(ω) = k 2 k BP 4 0 O k g(ω) Pk modes is observed in LJ glass % Shimada, Mizuno, Ikeda, PRE 2018 φ(r) φ(r c ) (r < rc ) ω 101

23 Conclusion 23/27 Questions l What is nature of low-frequency vibrations of amorphous solids? l What laws do they obey? ü Mixture of phonons and soft localized modes ü Phonons follow Debye law, while localized modes follow non-debye scaling law (ω4 law) ü This seems consistent with TLS/SPM picture ü In the continuum limit, glasses do NOT behave as homogeneous elastic media, but rather they behave as elastic media with ``defects Mizuno, Shiba, Ikeda, PNAS (2017)

24 Conclusion 24/27 Molecules Local elastic modulus Homogeneous media with defects Micro-scale Meso-scale Macro-scale At macro-scale (continuum limit), amorphous solids are essentially different from normal solids (elastic media). ü プレスリリース ( 東大東北大 ) ガラスと通常の固体の本質的な違いを発見

25 Low-T specific heat Can we explain linear-t term by localized modes? Specific heat Continuum limit Phonon modes Localized modes 25/27 + SiO2 Glass SiO2 Crystal

26 Low-T specific heat Can we explain linear-t term by localized modes? Localized modes Two level system 26/27 l Do two-level tunneling transitions happen in the localized modes? l What is role of non-debye law? Anderson, et al., Phil. Mag., 1972

27 27/27 Localized mode is key to understand amorphous solids Local elastic modulus Homogeneous media with defects Meso-scale Macro-scale Disordered extended modes Phonon modes Localized modes + ω BP ω ex0, Continuum limit

Inhomogeneous elastic response of amorphous solids

Inhomogeneous elastic response of amorphous solids Jean-Louis Barrat Université de Lyon Institut Universitaire de France Acknowledgements: Anne Tanguy, Fabien Chay Goldenberg, Léonforte, Michel Tsamados