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# Origin of the boson peak in amorphous solids

## Abstract

It is widely known that the low-temperature physical properties, such as the heat capacity and thermal conductivity, of a disordered amorphous solid are markedly different from those of its ordered crystalline counterpart. However, the origin of this discrepancy is not known. One of the universal features of disordered solids is the excess vibrational density of states, known as the ‘boson peak’. Here we study the microscopic origin of the boson peak through numerical investigations of the dynamic structure factor of two-dimensional model glasses over a wide frequency–wavenumber range. We show that the boson peak originates from quasi-localized vibrations of string-like dynamical defects. Furthermore, we reveal that these dynamical defects provide a common structural origin for the three most fundamental dynamic modes of glassy systems: the boson peak, fast β relaxation and slow structural relaxation.

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## Data availability

The data supporting the findings of this study are available from the corresponding author upon reasonable request.

## Code availability

The simulation codes used in this study are available from the corresponding authors on reasonable request.

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## Acknowledgements

This work was partially supported by Specially Promoted Research (JP20H05619) and Scientific Research (A) (JP18H03675) from the Japan Society of the Promotion of Science (JSPS). Y.C.H. is grateful for financial support from a JSPS fellowship (JP19F19021).

## Author information

Authors

### Contributions

H.T. designed and supervised the project. Y.C.H. performed research. Y.C.H. and H.T. analysed data and wrote the paper together.

### Corresponding author

Correspondence to Hajime Tanaka.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Nature Physics thanks Lothar Wondraczek, Jack Douglas and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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## Extended data

### Extended Data Fig. 1 Dynamical and vibrational properties of the 2DPL model.

The detailed characterisation methods are provided in the Supplementary Information. a, Angell plot of the structural relaxation times τα, with a fit to the Vogel-Fulcher-Tammann (VFT) equation. b, Relationship of the Debye-Waller factor μDW with τα. The orange dashed line is a fit to $${\tau }_{\alpha }={\tau }_{{{{\rm{A}}}}}\exp [{({\mu }_{{{{\rm{A}}}}}^{2}/{\mu }^{{{{\rm{DW}}}}})}^{\alpha /2}-1]$$, with α = 3.51. c, The velocity auto-correlation function (VACF) of the glassy state at T = 0.1. d, The VDOS calculated from the fast Fourier transformation of VACF (FFT_VACF) compared to that from Hessian diagonalization (Hessian).

### Extended Data Fig. 2 The participation ratio of the vibrational modes of the 2DPL model.

a, The participation ratio of each vibrational mode in the full frequency range. Most of the vibrational modes are phononic at low frequencies with a PR ≈ 2/3, distributed discretely. PR decays to a plateau over a wide frequency range with increasing frequency. Phonons transform to diffusons gradually, and the quasi-localised modes emerge simultaneously. This suggests phonon scattering by quasi-localised modes. At high frequencies above ωDebye (the Mobility edge), all the modes become strongly localised, which may be referred to as ‘Anderson localization’. b, The participation ratio of the longitudinal vibrational modes. The longitudinal phononic modes last up to much higher frequencies than the transverse ones.

### Extended Data Fig. 3 Peak intensities of the dynamical structure factors at the characteristic frequencies of the 2DPL model.

a, The intensity of the phononic components of the transverse and longitudinal modes as a function of q. Both the transverse and longitudinal intensity decay in a q−2 manner at low q. b, The intensity of the transverse quasi-localised modes, which increases with increasing q. This indicates that the quasi-localised modes become more dominant at higher q, proving that these modes have a localised nature different from phonons and diffusons.

### Extended Data Fig. 4 The transverse dynamical structure factors at high q of the 2DPL model.

Equation (1) is used to describe ST(q, ω) at q = 2.962 and the other data below the high frequency peak (solid grey lines). A localised mode around the Debye frequency (ωHT) appears when q 3.0 which can be fitted well by a log-normal function (purple dashed lines). The strength of this highly localised mode is relatively weak compared to the quasi-localised and phononic modes.

### Extended Data Fig. 5 Properties of stringlets mainly contributing to the boson peak of the 2DPL model.

a, The spatial distribution of the contribution of each particle to the reduced transverse vibrational density of state, $${D}_{i}^{{{{\rm{T}}}}}(\omega )/\omega$$ around the boson peak frequency. This is the same plot as Fig. 6c in the main text, except that the atoms that mainly contribute to the boson peak vibrations are marked red. These atoms (stringlets) are quantified by $${D}_{i}^{{{{\rm{T}}}}}(\omega )/\omega > 0.013$$, whose fraction is 5.3% for this glass sample. b, The string-size distribution in our model system. The statistics is made from over 50 independent simulations.

### Extended Data Fig. 6 Atomic-scale features of string-like vibrational motion around ωBP of the 2DPL model.

These enlarged domains are chosen as typical examples from Fig. 6c in the main text. It is obvious that there are always additional vacancies (shown by blue circles) at the ends of a string (shown by red dashed lines). In each figure, the atom species are differentiated by the particle size.

### Extended Data Fig. 7 Atomic-scale properties of a glassy inherent structure of the 2DPL model.

a, Local Voronoi area. b, Local chemical composition. c, Local potential energy. d, Local simple shear modulus. e, Local pure shear modulus. f, Spatial correlation of the non-affine displacement fields excited by quasi-static athermal deformation in the form of uniaxial tension (γxx) and simple shear (γxy) at very small strain. g, Transverse component of the lowest non-zero frequency vibrational mode, which shows quasi-localised modes (QLMs) with four-leaf pattern. h, Non-affine displacement field induced by uniaxial tension along the horizontal axis (strain: 4 × 10−5). The major QLM in the right top corner is not excited. i, Non-affine displacement field induced by simple shear along the horizontal axis (strain: 3 × 10−5). The major QLM is excited in this case. The vector amplitudes have been adjusted for better visualisation. These results are obtained for the glassy state shown in Fig. 6 in the main text.

### Extended Data Fig. 8 Dynamical structure factors of the 2DKA model in the low-temperature glassy state.

a, The transverse dynamical structure factors at several low wavenumbers, with the fits (solid grey lines) to equation (1) in the main text. The blue dotted and purple dashed lines denote the log-normal and phonon contributions for q = 0.941, respectively. b, The transverse dynamical structure factors at higher wavenumbers, with the fits (solid grey lines) to equation (1) in the main text. The blue dotted and purple dashed lines denote the log-normal and phonon contributions for q = 1.566, respectively. c,d, The longitudinal dynamical structure factors at the studied wavenumbers corresponding to a and b.

### Extended Data Fig. 9 Vibrational properties of the 2DKA Model.

a, VDOS of the low-temperature glass. b, The corresponding VDOS with a fit to the log-normal function (red dashed line), from which we estimate the boson peak frequency as ~ 1.3. c, Two dimensional contour of the transverse dynamical structure factors. The orange shade indicates the boson peak frequency range. d, Dispersion relation in a wide frequency-wavenumber range, with the existence of quasi-localised modes around the boson peak frequency. e, Enlarged dispersion relation for transverse phonons, from which we estimate the transverse Ioffe-Regel limit as 1.6, close to ωBP.

### Extended Data Fig. 10 Properties of the 2DSL model in the low temperature glassy state.

a, Configuration of particles with directional bondings. Green particles form isolated pentagons. b, Transverse dynamical structure factors, with fits to equation (1) in the main text (solid grey lines). The blue dotted and purple dashed lines represent log-normal contribution and phonon contribution at q = 1.295, respectively. The red dashed line denotes the boson peak frequency. c, Spatial distribution of the atomic-level vibrability around the boson peak frequency.

## Supplementary information

### Supplementary information

Supplementary notes, Figs. 1–6 and references.

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Hu, YC., Tanaka, H. Origin of the boson peak in amorphous solids. Nat. Phys. 18, 669–677 (2022). https://doi.org/10.1038/s41567-022-01628-6

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