# Magnon-phonon hybridization in quasi-2D antiferromagnet MnPSe_3

Thuc T. Mai<EMAIL_ADDRESS>Nanoscale Device Characterization Division,
Physical Measurement Laboratory, NIST, Gaithersburg, MD  K.F. Garrity
<EMAIL_ADDRESS>Materials Measurement Science Division, Materials
Measurement Laboratory, NIST, Gaithersburg, MD  A. McCreary  Nanoscale Device
Characterization Division, Physical Measurement Laboratory, NIST,
Gaithersburg, MD  J. Argo  Department of Materials Science and Engineering,
Ohio State University, Columbus, OH  J.R. Simpson  Nanoscale Device
Characterization Division, Physical Measurement Laboratory, NIST,
Gaithersburg, MD  Physics, Astronomy, and Geosciences, Towson University,
Towson, MD  V. Doan-Nguyen  Department of Materials Science and Engineering,
Ohio State University, Columbus, OH  R. Valdés Aguilar  Center for Emergent
Materials, Department of Physics, The Ohio State University. Columbus, OH
A.R. Hight Walker<EMAIL_ADDRESS>Nanoscale Device
Characterization Division, Physical Measurement Laboratory, NIST,
Gaithersburg, MD

###### Abstract

Magnetic excitations in van der Waals (vdW) materials, especially in the two-
dimensional (2D) limit, are an exciting research topic from both the
fundamental and applied perspectives. Using temperature-dependent, magneto-
Raman spectroscopy, we identify the hybridization of two-magnon excitations
with two separate Eg phonons in MnPSe_3, a magnetic vdW material that could
potentially host 2D antiferromagnetism. Results from first principles
calculations of the phonon and magnon spectra further support our
identification. The Raman spectra’s rich temperature dependence through the
magnetic transition displays an avoided-crossing behavior in the phonons’
frequency and a concurrent decrease in their lifetimes. We construct a model
based on the interaction between a discrete level and a continuum that
reproduces these observations. The strong magnon-phonon hybridization reported
here highlights the need to understand its effects on spin transport
experiments in magnetic vdW materials.

Magnons, the quantized magnetic excitations in solids, have been the subject
of a plethora of exciting research in recent years. From the quantum computing
perspective, magnons have been shown to couple to a superconducting qubit in a
cavityTabuchi  _et al._ (2015); Lachance-Quirion  _et al._ (2017) to make up
the field of quantum magnonics Tabuchi  _et al._ (2016); Chumak and
Schultheiss (2017); Lachance-Quirion  _et al._ (2019); Clerk  _et al._ (2020).
Magnons have also received attention from the spintronics community due to
their ability to transport spin angular momentum efficientlyStamps  _et al._
(2014); Chumak and Schultheiss (2017); Bozhko  _et al._ (2020). Recently
discovered magnetic 2D materials can retain intrinsic magnetism down to the
monolayer limitMcGuire (2017); Susner  _et al._ (2017); Burch  _et al._
(2018); Gibertini  _et al._ (2019); Cortie  _et al._ (2020). Magnons have now
been observed in several materials of this class, including ferromagnetic (FM)
magnons in CrI_3Lee  _et al._ (2020); Cenker  _et al._ (2020); Li  _et al._
(2020); McCreary  _et al._ (2020a) and antiferromagnetic (AFM) magnons in both
$\alpha$-RuCl_3Banerjee  _et al._ (2017); Little  _et al._ (2017); Wang  _et
al._ (2017); Wu  _et al._ (2018); Wulferding  _et al._ (2020); Ponomaryov  _et
al._ (2020); Sahasrabudhe  _et al._ (2020) and the MPX3 family (where M is Fe,
Mn, or Ni and X is S or Se)Wildes  _et al._ (1998); Kim  _et al._ (2019a);
McCreary  _et al._ (2020b); Calder  _et al._ (2020). These observations have
resulted in an increased interest in using magnetic 2D materials for magnon
spintronics.

Along these lines, several experimental studies have explored the capability
of CrI_3 and Fe_3GeTe_2 as a tunable spin filterKlein  _et al._ (2018); Jiang
_et al._ (2018); Wang  _et al._ (2019); Albarakati  _et al._ (2019).
Additional experiment on MnPS_3 has provided evidence of magnon-mediated spin
transportXing  _et al._ (2019). Furthermore, magnon transport over macroscopic
scales has been linked to the hybridization between magnons and acoustic
phononsAn  _et al._ (2020); Rückriegel and Duine (2020). The same
hybridization could theoretically lead to topologically non-trivial
quasiparticle bandsTakahashi and Nagaosa (2016); Zhang  _et al._ (2020).
However, the interactions between magnons and phonons and their hybridization
are generally unfavorable as they can cause an increase in scattering rate and
a decrease in quasiparticle lifetime. Understanding magnon-phonon interactions
is essential for applications that rely on preserving the magnon coherence
over extended length and time scales.

In this letter, we use a novel combination of temperature and magnetic field-
dependent Raman spectroscopy and first principles calculations to investigate
the magnon-phonon interactions in the AFM vdW material MnPSe_3. We identify
the hybridization between a continuum of two-magnon excitations and two
separate Raman-active phonons as a function of temperature. A model with
magnon-phonon interaction is constructed to reproduce our experimental
results. We find that a broad range of two-magnon excitations are hybridized
with the two Raman-active phonons over a wide range of temperatures. This
interaction causes a dramatic reduction in quasiparticle lifetimes in the
system, impacting potential magnon spintronics devices.

Bulk crystals of MnPSe_3 are synthesized via the vapor transport methodSI .
The sample is mounted inside a closed-cycled magneto-optical cryostat and
immersed in He exchange gas. A static magnetic field can be applied either
normal or parallel to the vdW layers. Raman spectra are measured using a
triple grating spectrometer and in the backscattering geometry, where a 515 nm
Ar+ laser was used to excite an area of approximately 1 $\mu m^{2}$ on the
sample. A combination of linear polarizers and half-wave plates are used to
select parallel (VV) and crossed (VH) polarization configurations with respect
to the incoming and scattered light, as well as to correct for Faraday
rotation in the objective when the static field is normal to the plane of the
sample. Due to the relatively small momentum that the laser photons can impart
on the crystal, the excitations in a Raman scattering experiment are generally
limited to $q\approx 0$.

Figure 1: Temperature dependent Raman spectra of MnPSe_3. The Raman spectra
for crossed (VH) polarization collected through the Néel transition at 74 K.
The spectra are normalized to the peak intensity of E${}_{\mathrm{g}}^{2}$,
and offset for clarity. The two Raman phonon modes are labeled with the
irreducible representation of the point group $\overline{3}$. Below
$T_{\textrm{N\'{e}el}}$, a new scattering peak appears between 115 cm-1 \- 130
cm-1, while the E${}_{\mathrm{g}}^{1}$ and E${}_{\mathrm{g}}^{2}$ phonons
undergo changes in both intensity and frequency.

First principles density functional theory (DFT)Hohenberg and Kohn (1964);
Kohn and Sham (1965) calculations are carried out using the Quantum Espresso
codeet. al. (2020), with GBRV pseudopotentialsGarrity  _et al._ (2014). We
utilize the PBEsol exchange correlation function functionalPerdew  _et al._
(2008) and a Hubbard U correction of 4 eV on the Mn-d states (DFT+U)Cococcioni
(2012). A discussion of the effects of different U values is included in the
Supplementary Information (SI)SI . The phonon and magnon energies are
calculated using finite differences approachesGarrity (2019); Togo and Tanaka
(2015).

The MPSe_3 family of materials shows higher magnetic anisotropy in the bulk
Bhutani  _et al._ (2020); Jeevanandam and Vasudevan (1999) than its sulfide
counterpart, particularly when Mn or Fe occupies the transition metal site M.
Such anisotropy is a necessary ingredient to stabilize the long-range magnetic
ordering in the 2D limitBurch  _et al._ (2018); Gibertini  _et al._ (2019);
Cortie  _et al._ (2020). The crystal symmetry of MnPSe_3 is rhombohedral at
room temperature, with space group R$\overline{3}$, and remains so below
$T_{\textrm{N\'{e}el}}$. Previous elastic neutron scattering experiments have
determined that MnPSe_3 is a Néel type antiferromagnet within each vdW plane
below 74 KWiedenmann  _et al._ (1981); Calder  _et al._ (2020). The spin
$\frac{5}{2}$ Mn ions are arranged in a honeycomb lattice, with the magnetic
moments pointing nearly parallel to the honeycomb planeWiedenmann  _et al._
(1981); Calder  _et al._ (2020).

We measure the polarized Raman spectra of MnPSe_3 in its paramagnetic phase,
and characterize the symmetries of the Raman phonon modes with the assistance
from DFTSI . As the temperature is lowered through $T_{\textrm{N\'{e}el}}$
(Fig. 1), several striking features appear in the frequency region between 75
cm-1 and 140 cm-1: (1) the E${}_{\mathrm{g}}^{1}$ and E${}_{\mathrm{g}}^{2}$
phonons undergo significant changes in intensity and frequency and (2) a new
scattering intensity peak appears to split off from E${}_{\mathrm{g}}^{2}$ and
continues to increase in frequency to approximately 130 cm-1 at 10 K.

Figure 2: Two-magnon scattering (a) The scattering intensity around 130 cm-1
shows no discernible difference between 0 T and 9 T applied either parallel
and perpendicular to the honeycomb planeSI . (b) The magnon dispersion with
easy-plane anisotropy resulting from linear spin wave theory with highlighted
zone-edge magnon states. The gapless (blue) and gapped (green) bands
correspond to in-plane and out-of-plane spin excitations, respectively. (c)
The calculated two-magnon density of states, $DOS_{2M}$, with its highlighted
peak at twice the one-magnon energies at the Brillouin zone boundary (also
highlighted in (b)), corresponds to pairs of magnons with equal and opposite
momentum. (d) Top view of the normal mode vibrations for
E${}_{\mathrm{g}}^{2}$. The Mn atoms are shown here with different colors to
represent different spin directions. The arrows represent the directions of
the atomic vibrations. (e) The Mn2+ exchange coupling between nearest-neighbor
($J_{1}$), next-nearest-neighbor ($J_{2}$), third-nearest-neighbor ($J_{3}$).

The new mode, with Eg symmetrySI , cannot be attributed to a phonon according
to the DFT calculations. This mode is asymmetric, with a cutoff at high
frequency, a peak intensity around 130 cm-1, and a low-intensity tail that
merges with the E${}_{\mathrm{g}}^{2}$ phonon (Fig. 2a). A previous study
interpreted the origin of the 130 cm-1 peak as one-magnon scatteringMakimura
_et al._ (1993) based on its strong frequency shift as a function of
temperature. A one-magnon scattering mode in an AFM is expected to split in an
applied magnetic field due to its net magnetic moment, $\Delta S=\pm 1$Fleury
and Loudon (1968). Such behavior has been observed in three-dimensional AFMs
such as MnF_2 and FeF_2Fleury and Loudon (1968); Rezende  _et al._ (2019), and
more recently in vdW AFMsMcCreary  _et al._ (2020b, a); Cenker  _et al._
(2020); Li  _et al._ (2020). However, when we measure the low temperature
($T\,=2\,$K) Raman spectra as a function of static applied magnetic field,
both parallel and perpendicular to the honeycomb plane, we find no change in
the 130 cm-1 peak (Fig. 2(a) and SISI ). This result strongly suggests that
the peak at 130 cm-1 is not due to one-magnon scattering. Instead, we propose
that its origin is due to two-magnon scattering.

Qualitatively, pairs of magnon states with opposite momenta and opposite spins
are excited by the Raman scattering process, conserving both linear and
angular momentumFleury and Loudon (1968). The picture of opposite spin
magnons, with $\Delta S$ = 0, would explain the lack of dependency of the two-
magnon scattering on the applied magnetic field. We use a Heisenberg
Hamiltonian model to further support our interpretation of the 130 cm-1 peak
as two-magnon scattering. The Hamiltonian is comprised of spin exchange
coupling terms, $J_{ij}$, and on-site anisotropy, $D$:

$\mathcal{H}=\frac{1}{2}\sum_{i\neq
j}J_{ij}\mathbf{S}_{i}\cdot\mathbf{S}_{j}\;+\;D\sum{S^{z}_{i}}^{2}.$ (1)

We obtain the parameters ($J_{ij}$, $D$) by performing a least squares fit to
DFT energies with various magnetic configurationsGarrity (2019), with the
nearest-neighbor (NN) $J_{1}=0.758$ meV, next-nearest-neighbor (NNN)
$J_{2}=0.069$ meV, and third-nearest-neighbor (TNN) $J_{3}=0.474$ meV (see
Fig. 2e). The exchange coupling to adjacent the vdW planes is 0.002 meV, 0.033
meV, and 0.010 meV for the 1${}^{\textrm{st}}$, 2${}^{\textrm{nd}}$, and
3${}^{\textrm{rd}}$ nearest neighbor, respectively. The on-site anisotropy,
$D$, is 0.046 meV. The magnon bands are calculated using linear spin wave
theory (LSWT). Fig. 2b shows the magnon dispersion along the high symmetry
points in the reciprocal lattice space: $\mathbf{\Gamma}=(0,0,0)$, K
$=(1/3,1/3,0)$, M $=(1/2,0,0)$, and A $=(0,0,1/2)$.

As expected from the significantly weaker exchange interaction perpendicular
to the vdW plane, the magnon dispersion along A is smaller than along the in-
plane directions (Fig. 2b). The easy-plane anisotropy creates a gapless magnon
band (blue curve in Fig. 2b) for in-plane $\Delta S$, and a gapped magnon band
(green curve) for out-of-plane $\Delta S$. We did not observe the gapped
magnon mode at $\mathbf{\Gamma}$ in our experiment. In a recent inelastic
neutron scattering experiment on MnPSe_3Calder  _et al._ (2020), the measured
magnon dispersion agrees with our LSWT results quantitatively, on both the
zone-center and zone-edge magnon energies. The experimental $J_{ij}$ values
are different from ours (for a more detailed discussion, see the SISI ).

The two-magnon density of states, $DOS_{2M}(\omega)=\sum_{i,{\bf
k}}\delta(\omega-2\omega_{i,{\bf k}})$ with artificial linewidth broadening is
shown in Fig. 2c and qualitatively agrees with our observation of the broad
130 cm-1 peak (compare Fig. 2a,c). The numerical value of the first principles
two-magnon peak depends on the choice of Hubbard U as we discuss in the SISI .
Rather than a single sharp mode, the two-magnon scattering intensity is due to
a continuum of two-magnon excitations. In our calculation, the $DOS_{2M}$ has
its highest intensity around 130 cm-1. This energy corresponds to the single
magnon states near the Brillouin zone edge at approximately 65 cm-1(near
reciprocal points K, M), as highlighted in Fig. 2b, c. The intensity cutoff at
high frequency comes from the lack of magnon states above the zone-edge
energy, while the weaker but broad scattering intensity tail receives
contributions from pairs of lower energy magnons with opposite momenta.

Figure 3: Phonon and Two-Magnon Hybridization. (a) False color plot showing
Raman intensity between 75 cm-1 and 140 cm-1 as a function of temperature
through $T_{\textrm{N\'{e}el}}$. The phonons E${}_{\mathrm{g}}^{1}$ and
E${}_{\mathrm{g}}^{2}$ suddenly decrease in frequency as temperature is
lowered. The two-magnon scattering appears below $T_{\textrm{N\'{e}el}}$ and
increases gradually in frequency to 130 cm-1at 10 K. (b), (c) The results of
fitting E${}_{\mathrm{g}}^{1}$, E${}_{\mathrm{g}}^{2}$, and the two-magnon
peak intensity (labeled 2M), in addition to A${}_{\mathrm{g}}^{2}$. The jumps
in peak frequency of each phonon correlate to abrupt decreases in the
quasiparticle lifetime (inversely proportional to the scattering linewidth),
which occur around $T_{\textrm{N\'{e}el}}$ for E${}_{\mathrm{g}}^{1}$ and 55 K
for E${}_{\mathrm{g}}^{2}$ (highlighted by the black arrow).
A${}_{\mathrm{g}}^{2}$ shows no discernible change in this temperature range.
The error bars represent one standard deviation from the best fit values. (d)
Frequency versus temperature results using a Fano-like model that take into
account a broad scattering feature, with its center labeled by 2M, and its
interaction with discrete states (phonons) as a function of temperature. The
size of circles represents the line width.

A detailed look at the frequency range between 75 cm-1 and 140 cm-1 as the
temperature is varied from 100 K to 10 K is shown in Fig. 3, including the
E${}_{\mathrm{g}}^{1}$, E${}_{\mathrm{g}}^{2}$, and A${}_{\mathrm{g}}^{2}$
phonons and the two-magnon excitations. The false-color plot (Fig. 3a) shows
dramatic changes in the frequency and intensity of E${}_{\mathrm{g}}^{1}$ and
E${}_{\mathrm{g}}^{2}$ as a function of temperature, along with the appearance
of the two-magnon excitations below 60 K, which increases in frequency to
approximately 130 cm-1 at 10 K. The two-magnon excitations are of $\rm{E_{g}}$
symmetry, appearing in both parallel and crossed polarization configurationsSI
. Further supporting our assignment of the feature at 130 cm-1 is its
similarity to the two-magnon excitations measured in FeF_2 and MnF_2. In both
materials, the two-magnon excitations shows a strong decrease in frequency as
temperature increases to $T_{\textrm{N\'{e}el}}$Cottam  _et al._ (1983);
Lockwood and Cottam (1987).

Fitting results for the peak positions and lifetimes for
E${}_{\mathrm{g}}^{1}$, E${}_{\mathrm{g}}^{2}$, A${}_{\mathrm{g}}^{2}$, and
the two-magnon (2M) are shown in Fig. 3b, c (see SI for more fitting detailsSI
). The observed shifts in frequency for E${}_{\mathrm{g}}^{1}$ and
E${}_{\mathrm{g}}^{2}$ below $T_{\textrm{N\'{e}el}}$ are larger than any other
phonons in the spectra, as evidenced in the comparison with
A${}_{\mathrm{g}}^{2}$ (Fig. 3c). Immediately below $T_{\textrm{N\'{e}el}}$,
we see an abrupt decrease in the frequency of E${}_{\mathrm{g}}^{1}$ by more
than 1 cm-1(Fig. 3c). Concurrent with the change in frequency, the lifetime of
E${}_{\mathrm{g}}^{1}$ decreases sharply (Fig. 3b). Such a dramatic jump is
indicative that E${}_{\mathrm{g}}^{1}$ has coupled with another state or
states. Although E${}_{\mathrm{g}}^{2}$ exhibits the same behavior as
E${}_{\mathrm{g}}^{1}$, the onset occurs at a noticeably lower temperature of
approximately 55 K. These behaviors are reminiscent of an avoided-crossing
phenomenon, with a few key differences, as detailed below.

While the lifetimes of the E${}_{\mathrm{g}}^{1}$ and E${}_{\mathrm{g}}^{2}$
phonons recover to approximately the same value as other phonons at 10 K, the
peak frequencies remain below their high temperature values (Fig. 3b). We
propose that the temperature dependence of E${}_{\mathrm{g}}^{1}$ and
E${}_{\mathrm{g}}^{2}$ is caused by their hybridization with the strongly
temperature dependent two-magnon excitations. At $T_{\textrm{N\'{e}el}}$
$\approx$ 74 K, the two-magnon excitations are at a lower frequency than both
phonon modes, and thus there is no overlap. As the two-magnon peak increases
in frequency with decreasing temperature, it first crosses path with
E${}_{\mathrm{g}}^{1}$ slightly below $T_{\textrm{N\'{e}el}}$, and then
crosses with E${}_{\mathrm{g}}^{2}$ near 55 K. When the phonon modes overlap
with the two-magnon excitations, they become broad and asymmetric, and their
center frequencies shift away from the two-magnon peak intensity.

Since there is an overlap in energy, the two-magnon excitations and the
E${}_{\mathrm{g}}^{1}$, E${}_{\mathrm{g}}^{2}$ phonons can interact with each
other via the exchange striction effectBaltensperger and Helman (1968); Cheong
and Mostovoy (2007). The normal mode motion of E${}_{\mathrm{g}}^{2}$ phonon
shows the modulation in distance between NN and TNN Mn2+ atoms (Fig. 2d).
These modulations would affect the superexchange coupling due to the change in
various bond lengths and angles. This coupling results in strong hybridization
between the phonons and the two-magnon excitations when their energies are
degenerate.

Figure 4: Hybridization model. Calculated scattering intensity due to the two-
magnon (2M, blue dash) continuum and a discrete phonon state (solid red
filled). The relative energy between the 2M and phonon simulates different
temperatures. Select experimental spectra (black dash) from crossed
polarization (VH) are chosen to compare to the model. Significant
hybridization can be seen in the 55 K and 50 K spectra by the deviation of the
lineshape from a simple Lorentzian.

To analyze this interaction more quantitatively, we construct a model based on
the work by FanoFano (1961) on the interaction of a discrete resonance
(phonon) with a continuum (two-magnon). The model consists of a broadly
peaked, temperature-dependent two-magnon continuum that interacts with the
narrow, temperature-independent $\mathbf{\Gamma}$-point phonons. We assume the
squared of the interaction strength is proportional to the intensity of the
continuum peak (see SI for more detailsSI ). By assuming a simple, functional
form for the continuum, we solve the model analytically for weak coupling or
numerically for stronger coupling. The main results are presented in Fig. 3d,
which shows the temperature dependent phonon frequencies, with the linewidths
shown as circle sizes.

We find that this simple model reproduces several otherwise puzzling features
of the experiment, shown in Fig. 3a-c. (1) The frequencies of the
E${}_{\mathrm{g}}^{1}$ and E${}_{\mathrm{g}}^{2}$ modes are repelled by the
peak intensity in the two-magnon excitations, with the shift extending beyond
the temperature region where the two-magnon directly overlaps with the
phonons. The direction of this frequency shift abruptly reverses sign when the
peak of the two-magnon excitations passes through the phonon, but only slowly
returns to zero (as seen in E${}_{\mathrm{g}}^{1}$ in Fig. 3c,d). (2) The
phonon lifetimes drop significantly when the phonons overlap with two-magnon
excitations, but recover to their previous values outside of the overlap
region. (3) The phonon modes become asymmetric, with a Fano-lineshape, when
they overlap with the two-magnon excitations. (4) Finally, phonon modes that
never overlap with the two-magnon excitations, such as the
A${}_{\mathrm{g}}^{2}$ mode at 149 cm-1, are nearly temperature-independent.

The interaction with a two-magnon continuum explains the observation that the
frequencies of E${}_{\mathrm{g}}^{1}$ and E${}_{\mathrm{g}}^{2}$ remain
lowered after the two-magnon’s apparent peak crosses through. In a typical
avoided-crossing phenomenon, one would expect the interaction to lift the
degeneracy in the vicinity of the crossing, resulting in the hybridization of
the two levels and an avoided-crossing. However, away from the original
crossing point, the character of these discrete levels would be expected to
return to their unhybridized state. A comparison between experimental spectra
at 10 K, 45 K, 50 K, 55 K, and 70 K and our model results are shown in Fig. 4.
The strongest hybridization is shown in the phonon excitation channel that
corresponds to the 55 K experimental spectrum. The hybridization effect
becomes less prominent at lower temperatures but still presents as a shift in
the phonon energy. From our model and observations, we infer that the phonons
remain hybridized with the two-magnon excitations over a wide range of
temperatures below $T_{\textrm{N\'{e}el}}$. Since the two-magnon continuum
covers a broad range of frequencies, the bare phonon frequency is always
degenerate with some of the two-magnon excitations. Consequently, the phonons
E${}_{\mathrm{g}}^{1}$, E${}_{\mathrm{g}}^{2}$, and a broad range of states in
the two-magnon continuum are hybridized, leading to a reduction in the
lifetimes of the quasiparticles involved, phonons and magnons.

A similar magnon spectrum and exchange parameters have been measured in
MnPS_3Wildes  _et al._ (1998), a material with analogous crystallographic and
magnetic structure to MnPSe_3. In the sulfur compound, however, the weaker
spin-orbit coupling of the sulfur anions produces an on-site anisotropy that
favors the Mn2+ spins to point out of the vdW plane. Furthermore, an abrupt
decrease in phonon frequency as a function of temperature has been measured in
the Raman spectrum of MnPS_3 down to a single layer thicknessKim  _et al._
(2019b); Sun  _et al._ (2019). The same hybridization phenomenon between a
phonon and two-magnon excitations, as discussed herein, is likely to be
responsible for the frequency shift in MnPS_3 even though the two-magnon
excitations are not seen.

Interestingly, an experiment using MnPS_3 as the spin carrying channel has
provided evidence of magnon-mediated spin transportXing  _et al._ (2019). In
the study, spin angular momentum was carried by magnons over a length scale of
several microns, demonstrating the material capability in magnon spintronics
applications. It is still unknown as to how the magnon-phonon hybridization
studied herein would impact spintronic devices. Our results pave the way for
further examinations of the role of magnon-phonon hybridization on magnon
coherence in spin transport experiments within the MPX_3 family of materials.

In summary, using a novel temperature and magnetic field-dependent Raman
spectroscopy in the AFM phase of MnPSe_3, we accurately assign the origin of
the mode around 130 cm-1as scattering from a two-magnon continuum of
excitations. Using LSWT, we calculate the magnon spectrum and the two-magnon
DOS to support our assignment. The magnon calculation agrees well with the
latest published experimental resultsCalder  _et al._ (2020). The temperature-
dependent Raman scattering data reveal an avoided-crossing like behavior,
suggesting that the Raman-active phonons E${}_{\mathrm{g}}^{1,2}$ and the
magnetic excitations have hybridized in the AFM phase. By measuring the
phonons’ lifetime, we identify this hybridization as a source of magnon
lifetime reduction in the material. A Fano-like model is constructed, where we
discovered that the shift in frequency of two Raman phonons over a wide range
of temperatures is due to their hybridization with the two-magnon continuum.
This work highlights the identification and direct measurement of a source of
magnon lifetime reduction in 2D magnetic materials.

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# Quartic scaling of sound attenuation with frequency in vitreous silica

P.-J. Wang  Department of Electrical Engeneering and Graduate Institute of
Photonics and Optoelectronics, National Tawain University, Taipei 10617,
Taiwan  A. Huynh  Sorbonne Université, CNRS, Institut des Nanosciences de
Paris, F-75005 Paris, France  T.-C. Hung  Department of Electrical Engeneering
and Graduate Institute of Photonics and Optoelectronics, National Tawain
University, Taipei 10617, Taiwan  X. Lafosse  Université Paris-Saclay, CNRS,
Centre de Nanosciences et de Nanotechnologies (C2N), 10 Boulevard Thomas
Gobert, F-91120 Palaiseau, France  A. Lemaître  Université Paris-Saclay, CNRS,
Centre de Nanosciences et de Nanotechnologies (C2N), 10 Boulevard Thomas
Gobert, F-91120 Palaiseau, France  B. Perrin  Sorbonne Université, CNRS,
Institut des Nanosciences de Paris, F-75005 Paris, France  B. Rufflé
Laboratoire Charles Coulomb, Université de Montpellier, CNRS, F-34095
Montpellier, France  R. Vacher  Laboratoire Charles Coulomb, Université de
Montpellier, CNRS, F-34095 Montpellier, France  C.-K. Sun  Department of
Electrical Engeneering and Graduate Institute of Photonics and
Optoelectronics, National Tawain University, Taipei 10617, Taiwan  M. Foret
Laboratoire Charles Coulomb, Université de Montpellier, CNRS, F-34095
Montpellier, France

###### Abstract

Several theoretical approaches to disordered media predict that acoustic waves
should undergo a quartic increase in their attenuation coefficient with
increasing frequency in the sub-terahertz region. Such Rayleigh-type
scattering would be related to the anomalous low-temperature plateau in the
thermal conductivity and to the so-called boson peak, i.e. an excess of
vibrational modes above the Debye density of states at around
$1\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$. Brillouin scattering of light
allows the measurement of sound absorption and velocity dispersion up to about
$0.1\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$ while inelastic x-ray scattering
is limited to frequencies larger than about
$1\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$. We take advantage of the advent of
ultrafast optical techniques to explore the acoustical properties of amorphous
SiO2 layers in the difficult but crucial frequency region within this gap. A
quartic scaling law is clearly revealed between 0.2 and
$0.9\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$, which is further shown to be
independent of temperature. This strongly damped regime is accompanied by a
decrease in the sound velocity already starting from about
$0.5\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$, in line with theories.

###### pacs:

62.30.+d, 62.40.+i, 63.50.-x, 78.47.jg

##  I Introduction

It is well known that glasses have universal properties at low temperatures
that differ from those of crystals. Below 1 K, the specific heat $C_{p}$ of
dielectric glasses is approximately linear and the thermal conductivity
$\kappa$ almost quadratic with temperature $T$, in contrast to the cubic
dependence observed for both properties in crystals [1]. This ”anomalous”
behaviour of amorphous solids has been successfully explained by introducing
the well-known tunneling model [2, 3]. The model assumes the existence of
structural entities that can tunnel between two configurations of very similar
energy (two-level systems). Above 1 K, $C_{p}$ still deviates strongly from
the expected Debye model, $C_{p}/T^{3}$ being larger than expected from the
sound velocities. The excess is related to the so-called boson peak observed
in the THz frequency range by neutron spectroscopy [4]. On the other hand, a
plateau is found between 1 and $10\text{\,}\mathrm{K}$ in $\kappa(T)$ of all
glasses, indicating the existence of a strong attenuation mechanism for the
sound waves in the subterahertz range [1, 5, 6]. The acoustic phonon mean free
path $l(\omega)$ is presumed to dramatically decrease with increasing phonon
frequency $\omega$, at least as $l(\omega)\propto\omega^{-4}$, until the
Ioffe-Regel limit $\omega_{\textsc{ir}}$ is reached. Using inelastic x-ray
scattering (IXS), the expected $\omega^{4}$ dependence has been observed for
the longitudinal acoustic mode over a tiny frequency interval just below
$\omega_{\textsc{ir}}$ in a few glasses [7, 8, 9, 10]. It was also found that
$\omega_{\textsc{ir}}$ lies in the frequency region of the boson peak.
Different theories have been proposed to describe this peculiar attenuation
regime, strong scattering of sound waves by fluctuating elasticity [11, 12],
resonant coupling with soft modes [13, 14] (which can be regarded as an
extension of the tunneling model) or effect of connectedness and applied
stress on vibrational properties[15], to mention only the most important ones.

In this paper, we focus on vitreous silica, v-SiO2, in which sound waves
propagation has been extensively studied. For frequencies above a few tens of
GHz, Brillouin light scattering (BLS) is the most commonly used technique for
measuring the velocity and attenuation of acoustic waves. BLS using visible
[16, 17] and ultraviolet [18, 19, 20] incident radiations can cover the range
$10-130$ GHz, while frequencies above about $1$ THz can be studied by
inelastic X-ray scattering [21, 22] as shown in Fig. 1-(a). Pump-probe optical
methods, in which the propagation of acoustic phonon pulses generated by laser
light pulses is monitored, have so far made it possible to explore frequencies
up to $\sim 400$ GHz [23, 24, 25, 26]. However, the experimental study of the
onset of the $l(\omega)\propto\omega^{-4}$ regime up to the IR limit is still
missing and therefore there is a need for new advanced techniques to fill this
gap. It would be particularly beneficial to carry out measurements at low
temperature to reduce the acoustic absorption due to anharmonic phonon-phonon
interactions and thermal relaxation of two-level systems [17], thereby
increasing the visibility of the expected $\propto\omega^{4}$ attenuation
regime for which theories suggest it should be $T$-independent. Here we use
advanced ultrafast optical techniques to investigate the acoustic properties
of v-SiO2 in this crucial unexplored gap between a few hundred GHz and the IR
limit where a quartic scaling of sound attenuation with frequency and negative
velocity dispersion are expected from theory.

##  II Femtosecond acoustics results

Acoustic phonon spectroscopy in the time domain by using femtosecond lasers
for the generation and detection of acoustic waves in thin films is in
constant progress [27, 28, 29]. The principle consists in generating a
coherent hypersonic pulse by absorbing a femtosecond optical pulse (the pump)
and detecting its echoes produced at the film interfaces thanks to a second
time-delayed optical pulse (the probe). Thin metallic films and semiconductor
quantum-wells provide broadband photo-acoustic transducers with a high
frequency cut-off in the subterahertz range. To detect or generate acoustic
fields of definite frequencies, it is necessary to design specific periodic
structures such as super-lattices. We present here a combination of acoustic
results in thin amorphous SiO2 samples, acquired with two distinct setups: i)
broadband generation with a metallic film and enhanced detection of discrete
frequencies using a superlattice consisting of multiple quantum wells, and ii)
broadband generation and detection using an InGaN/GaN single quantum well.

Figure 1:  (color online) (a) Longitudinal acoustic (LA) phonon linewidth
$\Gamma/2\pi$ (full width at half maximum ) as a function of frequency in
v-SiO2 at $300\text{\,}\mathrm{K}$ deduced from different spectroscopies:
visible Brillouin light scattering (BLS) results ( $\bullet$) [30], ultra-
violet BLS ones ($\triangle$) [18, 19, 20]. The symbols ( $\bullet$) [31] and
( $\scriptstyle\square$) [25] are previous acoustic attenuation data from
picosecond acoustics experiments in amorphous SiO2 layers, converted in
linewidth, $\Gamma=\alpha v$ where $v$ is the LA velocity. The solid line
represents the Ioffe-Regel (IR) limit where the phonon mean free path $l$
reaches half the phonon wavelength, i.e. for
$\Gamma\cong\nicefrac{{qv}}{{\pi}}$ where $q$ is the phonon wave vector. (b)
Scheme of the pump-probe experiment using GaAs/AlAs superlattice for
detection. The acoustic pulse generated in the Al film by the pump light is
probed in the SL after propagation through the SiO2 layer of thicknesses $d$
and the GaAs substrate. (c) Typical power spectrum of the transient changes in
reflectivity of the SL (black solid line), in log scale, compared to the
folded acoustic branches of the SL (blue lines). A series of discrete
frequencies satisfying the selection rule $q=2k$ are detected as indicated by
the orange dotted line. These range up to
$630\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. (d) FT amplitude registered at
$15\text{\,}\mathrm{K}$ for two SiO2 layers with
$d=$100\text{\,}\mathrm{n}\mathrm{m}$$ and
$d=$300\text{\,}\mathrm{n}\mathrm{m}$$. The peaks at
$0.15\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$ and
$0.05\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$ has been divided by a factor 5
for clarity. (e) Attenuation coefficients. The dotted line is a least-square
fit to the data.

###  II.1 GaAs/AlAs superlattice as detection

In this first configuration, a GaAs/AlAs superlattice (SL) is used to detect
with high sensitivity some components of the broadband spectrum generated in
an aluminum film as sketched in Fig. 1-(b). The strain pulse generated in the
Al film propagates through the GaAs substrate and can be detected in the SL by
measuring the transient changes in its reflectivity using the probe light
pulse (see Methods for details). The coherent phonons generation and detection
processes in superlattices are now well understood [32, 28]. Due to the
periodicity of the acoustic parameters in the SL, the acoustic phonon
dispersion curve is folded into a mini-Brillouin zone whose boundary lies at
wavevector $\pi/d_{\textsc{SL}}$, $d_{\textsc{SL}}$ being the SL period. The
selection rule that determines the Brillouin frequency in transparent media,
$q=2k$ ($q$ and $k$ being the phonon and light wave vectors respectively),
still holds in the superlattice, allowing in this case the detection of every
frequency on the folded branches that satisfies this relation. Fig. 1-(c)
shows an experimental result when the SL detects an acoustic pulse generated
in the Al film. The Fourier transform of the signal is dominated by peaks at
frequencies verifying the selection rule. We choose a period
$d_{\textsc{SL}}=$25\text{\,}\mathrm{nm}$$ to sample the frequencies
approximately every $50\text{\,}\mathrm{GHz}$. Although the spectral content
generated in the Al film ranges mainly from about 25 to
$250\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$, components up to almost
$600\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ contained in its weak high-
frequency tail can be measured in the SL.

To access the acoustic attenuation coefficient of v-SiO2, the idea is to
insert a layer of SiO2 between the GaAs substrate and the Al film and to
compare the signal amplitude for different silica thicknesses (see Fig.
1-(b)). This design prevents the direct excitation of the SL by the laser pump
pulse. Low pump fluences are used to avoid non linear acoustic effects during
propagation in the substrate. Furthermore, measurements are limited to low
temperature since phonon absorption in the substrate would be too large at
room temperature. Power spectra registered at $15\text{\,}\mathrm{K}$ for two
v-SiO2 layers with a thickness of $100\text{\,}\mathrm{nm}$ and
$305\text{\,}\mathrm{nm}$ are shown in Fig. 1-(d). The amplitude of the peaks
is found to be lower the thicker the sample and the higher the frequencies.
Unfortunately, the frequency components above
$\sim$500\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ are no longer visible in
the thickest layer. By comparing the amplitudes for each detected frequency
component, the attenuation coefficients shown in Fig.1-(e) are obtained with
$\alpha=\frac{2}{d_{2}-d_{1}}\ln\frac{A_{1}(f)}{A_{2}(f)}$ where $A_{i}$ and
$d_{i}$ denote the Fourier amplitude and the silica layer thickness of two of
the three samples studied (100/305/467 nm). The overall results have been
fitted with an effective power law in frequency by using a least-square
method. We get $\alpha(f)=(99\pm 26)\times f^{2.9\pm 0.3}$ ($\alpha$ expressed
in ${\mathrm{\SIUnitSymbolMicro m}}^{-1}$ and $f$ in
$\mathrm{T}\mathrm{H}\mathrm{z}$), represented by the dotted line in
Fig.1-(e).

###  II.2 InGaN single-quantum-well as transducer

Figure 2:  InGaN single-quantum-well as transducer (a) Schematic showing the
cross-section of the GaN/v-SiO2 sample. The photoexcited acoustic pulse
generated in SQW travels through the GaN cap layer and reaches the GaN/v-SiO2
interface. Part of the energy is reflected back to SQW for detection (giving
the first echo $E1$), other transmitted part is reflected by the v-SiO2 free
surface and sent back to the SQW (first round-trip reflection gives $E2$, and
the second gives $E3$). (b) The echoes after removing background for room
temperature from samples with three different film thicknesses. Comparing to
$E1$, the following echoes have higher loss and result to a temporal
broadening. (c) frequency-wavenumber relation acquired from $E1/E2$ of 9.2 nm
thickness sample and the calculated sound velocity spectra showing a strong
negative dispersion above 600 GHz. (d) Power attenuation coefficients of
v-SiO2 found for 300K. The straight line is a least-squares fit to the data,
$\alpha=(429\pm 33)\times f^{4.0\pm 0.2}$ ($\alpha$ expressed in
${\mathrm{\SIUnitSymbolMicro m}}^{-1}$ and $f$ in
$\mathrm{T}\mathrm{H}\mathrm{z}$).

We have further conducted experiments on GaN-grown v-SiO2 layers with an
embedded $2.5\text{\,}\mathrm{n}\mathrm{m}$-thick InGaN single-quantum-well
(SQW) for opto-acoustic transduction as sketched in Fig. 2-(a). Thanks to
their piezoelectric properties, InGaN/GaN semiconductor heterostructures
enable the generation of large amplitude coherent LA phonons over the THz
range [33, 34, 35]. Here, the In0.1Ga0.9N/GaN SQW structure allows for the
generation of phonons with a broadband spectrum [26, 29]. By monitoring the
absorption changes in the well via the probe transmission, the SQW further
provides a localised detection probe [36, 35]. The band structure as well as
the photoacoustic coupling are dramatically modified as the acoustic echoes
travel back to the SQW, which benefits the acquisition of the THz acoustics
waveform. On the atomically flat surface of aforesaid GaN structure, we
performed THz acoustics experiments on GaN-grown v-SiO2 films with thicknesses
$6.4\text{\,}\mathrm{n}\mathrm{m}$, $9.2\text{\,}\mathrm{n}\mathrm{m}$ and
$13.8\text{\,}\mathrm{n}\mathrm{m}$. The interfaces morphology and thickness
of the v-SiO2 films were characterised by transmission electron microscopy and
atomic force microscopy to ensure a well-defined acoustic traveling distance
without additional interfacial layer.

To measure the THz hypersonic attenuation spectra of v-SiO2, the photoexcited
acoustic pulse with about $2\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$ bandwidth
is first partially reflected from the GaN/v-SiO2 interface, considered as $E1$
(see Fig. 2-(b)). Part of the acoustic pulse is further transmitted into the
v-SiO2 film, reflected from the v-SiO2 surface and detected as $E2$. Due to
the acoustic impedance mismatch between GaN and v-SiO2, multiple reflections
of the acoustic pulses with round trip time interval can also be observed
($E2$, $E3$, …). After round trip traveling inside the v-SiO2 film, the
acoustics not only go through a damping mechanism denoted as
$\exp[-\alpha(\omega)2d]$, but also lose energy from the interface scattering
loss. As the phonon frequency went higher, the scattering losses increase and
become more crucial. However, the scattering only affects the amplitude term
and induces zero phase change to the coherent phonon as for the processes of
reflection and transmittance. Thus, we could first utilise the $E1$ and $E2$
to obtain the sound velocity dispersion spectra plotted as Fig. 2-(c). Sound
velocity is related to the phase of the Fourier transform of each echo
$\phi_{i}$ via $v=\dfrac{\omega\times 2d}{|\phi_{2}-\phi_{1}|}$. A strong
negative dispersion at the frequency range above
$650\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ was measured. Furthermore, if the
roughness of the interfaces and surfaces of two v-SiO2 films of different
thickness were identical, then the only difference between the acoustic pulses
traveling through the v-SiO2 films was the propagation distance in the films.
The scattering effect can be removed by comparing the attenuation loss spectra
from different silica thicknesses for successive echoes (either $E1/E2$ or
$E2/E3$):

$\alpha=\dfrac{1}{2(d_{2}-d_{1})}\ln\left(\frac{({E_{i+1}}/{E_{i}})_{d_{1}}}{({E_{i+1}}/{E_{i}})_{d_{2}}}\right)$
(1)

Basically, thinner film provide a higher cut-off frequency when accessing the
high-frequency attenuation spectra and this property limits the suitable
thickness range of v-SiO2. On the other hand, thinner films do not provide
access to the attenuation of low-frequencies components because of the long
propagation lengths required to attenuate the acoustic energy to an observable
level. Therefore, the accessible frequency range is also limited at low
frequencies. We first used the $E1/E2$ combination of the 13.8 nm/9.2 nm and
6.4 nm/9.2 nm to obtain two sets of intrinsic attenuation data for v-SiO2 in
the range of $600\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ \-
$800\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ and around
$900\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$, respectively. Then, considering
$E2/E3$ of 13.8 nm/9.2 nm, we can extend the attenuation spectra down to
$330\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ \-
$440\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. After careful analysis of the
data, we finally obtained the three sets of attenuation results shown in Fig.
2-(d). A least-squares fit with an effective power law in frequency gives
$\alpha(f)=(429\pm 33)\times f^{4.0\pm 0.2}$ in which $\alpha$ is expressed in
${\mathrm{\SIUnitSymbolMicro m}}^{-1}$ and $f$ in
$\mathrm{T}\mathrm{H}\mathrm{z}$. Finally, by taking into account the velocity
dispersion noted above, we can express the attenuation law as a function of
the wave vector $q$ of the acoustic wave. A least-squares fit with an
effective power law in $q$ then gives $\alpha(q)\propto q^{3.7\pm 0.2}$.

##  III  Discussion

Figure 3:  (color online) (a) Energy attenuation coefficient as a function of
frequency combining the results at $15\text{\,}\mathrm{K}$ (
$\displaystyle\blacktriangle$) and at $300\text{\,}\mathrm{K}$ (
$\displaystyle\blacksquare$). Open symbols are the low frequency measurements
data as depicted in Fig. 1(a). The dotted lines show the estimated contibution
of $(\alpha_{\textsc{tar}}+\alpha_{anh})$ to the attenuation coefficient for
the two temperatures. The black dashed line is the fast growing attenuation
law due to interaction of sound waves on the modes forming the boson peak,
$\alpha_{bp}(\omega)=A\left(\frac{\omega}{2\pi}\right)^{4}$ as explained in
the text. We find the $\alpha_{bp}$ law independent of temperature. The solid
lines represent the resulting sum of the three contributions to attenuation
$(\alpha_{\textsc{tar}}+\alpha_{anh}+\alpha_{bp})$ for $15\text{\,}\mathrm{K}$
and $300\text{\,}\mathrm{K}$. Spectral linewidth (b) and velocity (c) of
quasilongitudinal acoustic phonons measured by inelastic x-ray scattering
spectroscopy at $300\text{\,}\mathrm{K}$ [22] ($\displaystyle\blacksquare$)
compared to these results ( $\displaystyle\blacksquare$).

The overall new results, shown as solid symbols in Fig. 3(a), extend the
existing experimental data (open symbols) on sound attenuation in silica glass
up to near THz frequencies. As a reminder, below a few hundred GHz,
$\alpha(\omega,T)$ is known to be driven by two main dissipation mechanisms :
thermally activated relaxations (TAR) of structural “entities” and anharmonic
interactions. It is well established that the TAR processes dominate the
damping in the sonic and ultrasonic regimes. They produce an attenuation
$\alpha_{\textsc{tar}}\propto\omega^{2}$ for $\omega\tau_{\textsc{tar}}\ll 1$
and $\alpha_{\textsc{tar}}\propto\omega^{0}$ for $\omega\tau_{\textsc{tar}}\gg
1$, where $\tau_{\textsc{tar}}$ is the characteristic time of the relaxing
entity. A large distribution of relaxing entities is in fact assumed thus
leading to an effective power law $\alpha_{\textsc{tar}}\propto\omega^{n}$
whose exponent $n$ varies with temperature $T$ and satisfies $n(T)\leq 2$. The
anharmonic losses result from the interaction of sound with the thermally
excited vibrations in the medium at $T$. Following the Akhiezer treatment, the
interaction can also be treated as a relaxation process characterised by the
mean thermal lifetime $\tau_{th}(T)$ of the dominant vibrations [17]. At room
temperature, $\tau_{th}$ is very short so that $\omega\tau_{th}\ll 1$ for
frequencies up to at least one hundred GHz, leading to
$\alpha_{anh}\propto\omega^{2}$. The power exponent is nearly independent of
$T$ as long as $\omega\tau_{th}\ll 1$. An estimate of the relative
contributions of the TAR and anharmonicity processes to the sound absorption
in v-SiO2 was given in Ref. 17 as part of the results of a Brillouin light
scattering experiment. The dotted lines in Fig. 3(a) are estimates of the
overall contribution $(\alpha_{\textsc{tar}}+\alpha_{anh})$ for
$300\text{\,}\mathrm{K}$ and $15\text{\,}\mathrm{K}$ using the models and the
parameters described in [17, 31, 37]. The new data therefore highlight the
presence of an additional dissipation process, which adds to the
$(\alpha_{\textsc{tar}}+\alpha_{anh})$ part, and which gives rise to an
attenuation that increases drastically at frequencies entering the boson peak
region. Let us note $\alpha_{bp}$ the associated attenuation constant. From an
adjustment of the whole data set, the
$(\alpha_{\textsc{tar}}+\alpha_{anh})$-part being fixed at its expected value
quoted above, we find the law
$\alpha_{bp}(\omega)=A(\frac{\omega}{2\pi})^{4}$, with the parameter
$A=0.33\pm 0.02$
$\mathrm{n}\mathrm{m}^{-1}\mathrm{T}\mathrm{H}\mathrm{z}^{-4}$ being
independent of temperature. The black dashed line in Fig 3(a) shows
$\alpha_{bp}(\omega)$ while the red and blue solid lines represent the overall
total attenuation
$\alpha_{tot}(\omega,T)=\alpha_{\textsc{tar}}+\alpha_{anh}+\alpha_{bp}$ for
$300\text{\,}\mathrm{K}$ and $15\text{\,}\mathrm{K}$, respectively.

The direct demonstration of this highly damped sound wave regime via phonon
spectroscopy techniques has been pursued for decades [38, 23, 39, 7, 8, 9,
10]. As suggested long ago, the rapid decay of the mean free path of sub-
terahertz phonons must be the cause of the existence of the universal
“plateau” observed in the thermal conductivity $\kappa(T)$ of almost all
glasses at few kelvin [1, 6]. The physical origin of such a strong phonon
attenuation regime, which should be closely linked to the glassy state
(disorder), has long been debated [12, 40, 41, 42, 15, 43]. The issue is
coming back into focus with the emergence of extensive work on the vibrational
properties of glasses observed in computer simulations. Recent investigations
on ensembles of computer-samples in different classes of amorphous materials
have conclusively demonstrated the existence of soft quasi-localised
vibrational modes (QLVs) exhibiting a universal quartic density of states,
$\mathcal{D}(\omega)\sim\omega^{4}$ [44, 45, 46, 47, 48], as early predicted
in the framework of a phenomenological theory, the so-called soft potential
model (SPM). The SPM is an extension of the tunneling model to include soft
localised vibrations in terms of soft anharmonic potentials with locally
varying parameters [49, 50, 13, 14]. The quasi-localised excitations can be
understood as soft localised vibrations hybridised with acoustic waves. The
QLVs explain the excess modes over the Debye model expectation that form to
the so-called “boson peak” [47]. The resonant scattering of sound waves on
these soft vibrations would therefore explain the strong acoustic damping
regime. The acoustic attenuation increases $\propto\omega^{4}$ which follows
from the corresponding quartic increase in the excess vibrational density of
state. On the basis of the SPM, the LA attenuation is expected to be
$\alpha_{\textsc{spm}}=A_{\textsc{spm}}\left(\frac{\omega}{2\pi}\right)^{4}$
with
$A_{\textsc{spm}}=0.23~{}$\mathrm{n}\mathrm{m}^{-1}\mathrm{T}\mathrm{H}\mathrm{z}^{-4}$$
for v-SiO2 [41, 51], independent of $T$. The latter value is derived from two
experimentally accessible quantities, the tunneling strength for LA modes
$C_{l}$ and the density of states of the soft vibrations which can be
extracted from low temperature heat capacity data [52]. Here we reveal this
strong damping regime along the whole low-frequency side of the boson peak.
The prefactor $A$ is found to be slightly larger than $A_{\textsc{spm}}$.
Another prominent theoretical framework, based on the concept of fluctuating
elasticity can also explain the observed phenomenology of sound damping and
dispersion in glasses [12]. It relies on the Rayleigh scattering of sound
waves, $\Gamma(\omega)\sim\omega^{d+1}$ for space dimension $d$, where the
prefactor quantifies the spatial fluctuations of the elastic moduli. The shear
elastic heterogeneity has been shown to be dominant [53, 54]. A unified view
was brought by Schirmacher, who argued that below resonance frequencies local
oscillators can be subsumed within the fluctuating-elasticity formalism [55].
Conversely, micro-mechanical frustration might play a central role in the
occurrence of QLVS [56, 57]. In the case of silica glass, the local
oscillatory motions that form QLVs are the librations of SiO4 tetrahedra [4].

Finally, the LA-linewidth and -velocity data in v-SiO2 at room $T$ using IXS
[22] are compared with our results in Fig. 3(b) and (c), respectively. Due to
resolution limitations, the IXS measurements are restricted to frequencies
above $\sim$1\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$$ and therefore cannot
access the $\omega\leq\omega_{\textsc{bp}}$ frequency region. The two data
sets are in good agreement with the Ioffe-Regel criterion being reached at
approximately $\omega_{\textsc{bp}}$. A decrease in the measured sound
velocity accompanies the increase in the damping, see Fig. 3(c). However, the
negative phase velocity dispersion from about
$500\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$, reported in this work seems to
be inconsistent with that extracted from the IXS data. The latter observation
can be rationalised by considering the fundamental differences between the two
techniques. In Brillouin x-ray scattering experiments, the momentum-transfer
$Q$ (fixed by the instrument setting) defines the wave-vector of the acoustic
mode under study, whose spectrum is treated in terms of the damped harmonic
oscillator model, leading to a constant-wavevector phase velocity
$v_{\textsc{q}}=\Omega(Q)/{Q}$ where $\Omega$ is the resonant frequency. It
should be emphasised that as the IR limit is approached from below, the
acoustic modes are expected to deviate increasingly from plane-waves so that
more and more modes contribute to the scattering intensity at the selected $Q$
value, leading thus to an apparent $v_{\textsc{q}}$. In this work, on the
other hand, the analyse of the phase of the acoustic echoes gives access to a
constant-frequency phase velocity $v_{\omega}$. In effective medium theories
based on either fluctuating elastic constants [12, 53, 15] or resonance on
QLVs [13, 14], the response of the disordered system is characterised by
$v_{\omega}$ and not by $v_{\textsc{q}}$ around the IR-crossover. As shown in
[15], $v_{\textsc{q}}$ and $v_{\omega}$ are expected to be somewhat different
for frequencies approaching this region, in agreement with our observations.
Similarly, the propagation of high frequency sound waves is better described
by an attenuation coefficient derived from a constant frequency measurement as
reported here, rather than by values extracted from IXS constant-$Q$ analysis,
$\alpha=\Gamma/v_{\textsc{Q}}$. It is therefore of great interest to have
direct access to the constant-frequency parameters, for both the phase
velocity and the scattering length (or attenuation). Above
$\omega_{\textsc{bp}}$, the spectral width observed in IXS is no longer
expected to be a true physical attenuation of a sound wave but rather to
reflect the superposition of many vibrational modes [39, 58, 59].

In short, this work has enabled us to disentangle the various contributions to
acoustic losses at frequencies entering the boson peak domain, in the
prototypical silica glass. As a result of the superposition of several loss
mechanisms, effective frequency power laws are observed for the attenuation
coefficient, the exponent of which varies with the ranges of $\omega$ and $T$.
This explains the non physical exponent of $\sim 2.9$ observed at
$15\text{\,}\mathrm{K}$ in the $150-400\text{ GHz}$ range. The expected
$\omega^{4}$-regime is finally uncovered at room-$T$ along the entire low-
frequency side of the boson peak, and is shown to be independent of $T$.
Quantitative agreement with the SPM prediction is found. Finally, this work
paves the way for phonon spectroscopy in disordered media in the challenging
terahertz frequency window by using ultrafast optical techniques. Broadband
detection configurations offer the unique advantage of direct access to
constant-frequency quantities for acoustic damping and velocity. This could be
highly relevant when vibrational modes cease to resemble plane waves as is
expected in glasses in the boson peak region and beyond.

###### Acknowledgements.

This work has been supported by the Agence Nationale de la Recherche (Grant
No. ANR-11-BS04-008-01, GlassPhon).

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### Methods

####  III.0.1 Superlattice-based samples

The AlAs/GaAs superlattice (SL) has been grown by molecular beam epitaxy on a
(001) oriented two-sided polished GaAs substrate. It consists of 40 periods of
a layer pair structure ($68\text{\,}\AA$ /$189\text{\,}\AA$ for AlAs/GaAs). On
the back of the GaAs substrate, thin films of amorphous silica with various
thicknesses were deposited using radio frequency cathodic sputtering. To
obtain films with properties close to those of bulk silica or thermal oxide
films [37], the deposition was done at
$300\text{\,}\mathrm{\SIUnitSymbolCelsius}$ using a cold O2/Ar plasma,
maintaining the two gas pressure at 0.1 Pa. At last, a thin Al film is
evaporated on the silica surface for use as a transducer. This last step was
made in the same time for all samples. The table 1 summarizes the studied
samples with various layers deposited on the back of the substrate.

Al film (nm) | SiO2 (nm) | substrate GaAs ($\text{\,}\mathrm{\SIUnitSymbolMicro m}$) | SL AlAs/GaAs (Å)
---|---|---|---
| 100 |  |
25 | 305 | 350 | 68/189($\times 40$)
| 467 |  |
Table 1: Samples structure.

A degenerated pump-probe set-up was used with a mode-locked Ti:Sapphire
femtosecond laser as the laser source. Two hundred femtosecond optical pump
pulses centered at $780\text{\,}\mathrm{n}\mathrm{m}$ are sent with a
$80\text{\,}\mathrm{MHz}$ repetition rate on the metallic film, causing it to
rapidly heat and expand. As a result, a longitudinal acoustic wave is launched
into the sample, propagates into the substrate and the silica layer before
reaching the SL. The incoming strain induces a displacement of the SL surface
and modifies the optical constants, resulting in changes in optical
reflectivity. These transient changes of the complex reflectivity $\Delta R/R$
are detected by the probe light pulse focused on the SL, time-delayed
relatively to the pump pulse. The detection setup, previously described in
[60], is based on a Sagnac interferometer to measure the real and imaginary
parts of the reflectivity. This interferometer presents the advantage of being
particularly insensitive to noise introduced by mechanical instabilities
because both arms describe the same path in counterpropagating senses. It is
particularly useful here as the sample is mounted in a cryostat. Indeed, the
sample is cooled down to low temperature (typically 7 to
$15\text{\,}\mathrm{K}$) in a 4 K helium-flow cryostat with sample in static
exchange gas to minimize sound attenuation during propagation in the
substrate. The transient changes of reflectivity being of the order of
$10^{-5}$ to $10^{-7}$, it is necessary to use a lock-in technique and signal
averaging, with a pump beam modulation of
$1\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$. We use typical energies per pulse
of
$0.31\text{\,}\mathrm{n}\mathrm{J}\mathrm{/}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{s}\mathrm{e}$
maximum for the pump and 0.1 to
$0.2\text{\,}\mathrm{n}\mathrm{J}\mathrm{/}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{s}\mathrm{e}$
for the probe. Light beams are focused to a diameter of
$60\text{\,}\mathrm{\SIUnitSymbolMicro m}$ on the sample surface. These low
pump intensities were required to avoid non-linearities during propagation
through the thick GaAs substrate. In addition, with these experimental
conditions, we have minimized the steady-state heating of the sample,
especially at low temperature. The laser spot sizes are much larger than the
layer thicknesses (Al or SiO2), so that the one-dimensional approximation is
appropriate for the description of acoustic wave propagation.

The signal of reflectivity was measured as a function of the time delay over
$1.5\text{\,}\mathrm{n}\mathrm{s}$ providing a Fourier transform with a
resolution of $0.7\text{\,}\mathrm{GHz}$. The attenuation was deduced from the
comparison of the spectra measured on two samples in the same experimental
conditions with the ratio of the peaks amplitude $A(f)$ for each detected
frequency by the SL :
$\alpha=\frac{2}{d_{2}-d_{1}}\ln\frac{A_{1}(f)}{A_{2}(f)}$, $d_{1}$ and
$d_{2}$ being the thickness of the silica layers.

####  III.0.2 InGaN single-quantum-well-based samples

Sample preparation
An In0.1Ga0.9N SQW was used as the photoacoustic transducer. The transducer
samples were grown by metal-organic chemical vapor deposition (MOCVD) on
430-$\mathrm{\SIUnitSymbolMicro m}$-thick (0001) double-sided polished
sapphire substrates. The InGaN SQW transducer was grown with a thickness of
$2.5\text{\,}\mathrm{n}\mathrm{m}$ by embedding it between a
2-$\mathrm{\SIUnitSymbolMicro m}$-thick gallium nitride (GaN) buffer layer and
a 63-nm-thick GaN cap layer. High Resolution Transmission electron microscopy
(HRTEM) was used to examine the structures of the samples, to confirm the
thicknesses of the InGaN SQW and the GaN cap layer, and to ensure the atomic
flatness of each interface. The samples were then cleaned with buffered HF,
acetone, isopropanol, and deionized water to remove the oxide layer and
organic residue on the surface of the GaN cap layer.

Next, v-SiO2 films with various thicknesses were grown on top of these
transducer samples using plasma-enhanced chemical vapor deposition (PECVD) at
$350\text{\,}\mathrm{\SIUnitSymbolCelsius}$, which has been widely used as a
reliable way to prepare v-SiO2 films in previous studies [25, 26]. The gas was
a mixture of N2O, SiH4, and Ar. First, the optical and acoustic properties of
the grown v-SiO2 films were compared with the known properties of bulk v-SiO2.
The refractive index of bulk v-SiO2 is 1.46 at
$633\text{\,}\mathrm{n}\mathrm{m}$ [61], and that of the v-SiO2 film could be
controlled to be 1.46 by adjusting the process pressure to 70 Pa in 8% SiH4
with Ar [61]. The sound velocity is another unique property of v-SiO2 glass.
By measuring the temporal difference between echoes reflected from the
GaN/v-SiO2 and v-SiO2/air interfaces and considering the film thicknesses
measured from the TEM images (Fig. S3), the calculated sound velocity of each
film was compared with the previously reported sound velocity of
$5970\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ in v-SiO2. Two samples
were examined by TEM, and the thicknesses of the v-SiO2 films were measured to
be $9.3(0.4)\text{\,}$ and $13.9(0.4)\text{\,}\mathrm{nm}$; the error is due
to the poor contrast between the v-SiO2 and the protection layer under TEM as
both v-SiO2 and the protection layer have disordered structures and the
interface between crystalline GaN and v-SiO2 is transparent. Under this
circumstance, here we took the reported v-SiO2 sound velocity value of
$5970\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ [25, 17] and the measured
echo time to calculate the actual thickness
($9.2\text{\,}\mathrm{n}\mathrm{m}$ and $13.8\text{\,}\mathrm{n}\mathrm{m}$).

Advanced femtosecond acoustics system – the optical part
A transmission-type femtosecond optical pump–probe system was constructed to
optically pump the photoacoustic transducer, thereby generating femtosecond
acoustic strain pulses. Split pulses were then used to optically probe the
same photoacoustic transducer for femtosecond-time-resolved detection of the
strain pulse echo. As depicted in Fig. 4, a mode-locked Ti:sapphire
femtosecond laser with a $76\text{\,}\mathrm{MHz}$ repetition rate was adopted
as the laser source. The output pulse had an energy of
$25\text{\,}\mathrm{n}\mathrm{J}$ and a central wavelength of
$790\text{\,}\mathrm{n}\mathrm{m}$. In order to reach the bandgap of the
indium-gallium-nitride single-quantum-well (InGaN SQW), the laser beam was
focused onto a beta-barium-borate (BBO) crystal, and its frequency was doubled
for the generation of $395\text{\,}\mathrm{n}\mathrm{m}$ ultraviolet (UV)
laser beam. A half-wave plate and a polarizing beam splitter were used to
adjust the relative energies of the p-polarized and s-polarized beams, which
served as the pump and probe beams. The pump beam was modulated at
$2\text{\,}\mathrm{MHz}$ using an acoustic-optic modulator (AOM) for the
purpose of acquiring better signal-to-noise ratio by the phase-sensitive
amplification technique. The pump beam was guided into a delay stage with a
retro-reflector, and the optical path length difference of the two beams can
thereby be translated to the time difference with femtosecond resolution. The
powers of the pump and probe beams were adjusted to be 10:1, and both the pump
and probe beams were focused onto the sample with the resulting spot sizes of
roughly $15\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Finally, the probe beams
passing through the samples, undergoing the effect resulting from the pump
beams, was collected by a silicon optical detector, and the data was recorded
for the further analysis.

Figure 4:  (color online) Optical pump–probe system for femtosecond acoustic
measurements and the autocorrelation of the UV pulse. The yellow area is the
prism-pair setup. The light is guided by the system to the prism pairs and
reflected by an end mirror, then propagates in the opposite direction at a
different height, and is guided into an autocorrelator, where its pulse width
is measured.

To broaden the bandwidth of the acoustic pulse, the widths of the pump and
probe pulses were shortened as much as possible. A pair of prisms was placed
after the BBO crystal to attain a transform-limited pulse width. Then, a UV
two-photon-absorption autocorrelator was used to record the autocorrelation
between the pulse and a signal having a Gaussian-like intensity and a full
width at half maximum (FWHM) of $240\text{\,}\mathrm{f}\mathrm{s}$, which is
equivalent to the $168\text{\,}\mathrm{f}\mathrm{s}$ pulse width of the UV
pump–probe pulses after being shortened as much as possible by the prism pair.