Anharmonic lattice dynamics of Ag$_2$O studied by inelastic neutron scattering and first principles molecular dynamics simulations

Abstract

Inelastic neutron scattering measurements on silver oxide (Ag$_2$O) with the
cuprite structure were performed at temperatures from 40 to 400\,K, and Fourier
transform far-infrared spectra were measured from 100 to 300\,K. The measured
phonon densities of states and the infrared spectra showed unusually large
energy shifts with temperature, and large linewidth broadenings. First
principles molecular dynamics (MD) calculations were performed at various
temperatures, successfully accounting for the negative thermal expansion (NTE)
and local dynamics. Using the Fourier-transformed velocity autocorrelation
method, the MD calculations reproduced the large anharmonic effects of Ag$_2$O,
and were in excellent agreement with the neutron scattering data. The
quasiharmonic approximation (QHA) was less successful in accounting for much of
the phonon behavior. The QHA could account for some of the NTE below 250 K,
although not at higher temperatures. Strong anharmonic effects were found for
both phonons and for the NTE. The lifetime broadenings of Ag$_2$O were
explained by anharmonic perturbation theory, which showed rich interactions
between the Ag-dominated modes and the O-dominated modes in both up- and
down-conversion processes.

Full text

arXiv:1402.3603v2 [cond-mat.mtrl-sci] 3 Mar 2014
Anharmonic lattice dynamics of Ag2O studied by inelastic neutron scattering and first
principles molecular dynamics simulations
Tian Lan,1, ∗Chen W. Li,2J. L. Niedziela,2Hillary Smith,1
Douglas L. Abernathy,2George R. Rossman,3and Brent Fultz1
1Department of Applied Physics and Materials Science, California Institute of Technology, Pasadena, California 91125, USA
2Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
3Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California 91125, USA
(Dated: March 4, 2014)
Inelastic neutron scattering measurements on silver oxide (Ag2O) with the cuprite structure were
performed at temperatures from 40 to 400 K, and Four ier transform far-infrared spectra were measured
from 100 to 300 K. The measured phonon densities of states and the infrared spectra showed unusually
large energy shifts with temperature, and large linewidth broadenings. First principles molecular
dynamics (MD) calculations were performed at various temperatures, successfully accounting for
the negative thermal expansion (NTE) and local dynamics. Using the Fourier-transformed velocity
autocorrelation method, the MD calculations reproduced the large anharmonic effects of Ag2O, and
were in excellent agreement with the neutron scattering data. The quasiharmonic approximation
(QHA) was less successful in accounting for much of the phonon behavior. The QHA could account
for some of the NTE below 250 K, although not at higher temperatures. Strong anharmonic effects
were found for both phonons and for the NTE. The lifetime broadenings of Ag2O were explained by
anharmonic perturbation theory, which showed rich interactions between the Ag-dominated modes
and the O-dominated modes in both up- and down-conversion processes.
I. INTRODUCTION
Silver oxide (Ag2O) with the cuprite structure has at-
tracted much interest after the discovery of its extraordi-
narily large negative thermal expansion (NTE),1,2 which
exceeds −1×10−5K−1and occurs over a wide range
of temperature from 40 K to its decomposition temper-
ature near 500 K. Besides its large NTE, Ag2O is com-
monly used as a modifier in fast-ion conducting glasses
and batteries,3,4 and its catalytic properties are also of
interest.5,6
In the cuprite structure of Ag2O shown in Fig. 1, the
fcc Ag lattice is expanded by the presence of the O atoms,
which form an interpenetrating bcc lattice. The O atoms
occupy two tetrahedral sites of the standard fcc unit cell
of Ag atoms. Each O atom is linked to four O atoms
through a bridging Ag atom, placing the Ag atoms in
linear O-Ag-O links with little transverse constraint. A
geometrical model of NTE considers tetrahedra of Ag4O
around each O atom that bend at the Ag atoms linking
the O atoms in adjacent tetrahedra. Rigid-unit modes
(RUMs) account for counteracting rotations of all such
tetrahedra.7,8 These RUMs tend to have low frequencies
owing to the large mass of the unit, and hence are excited
at low temperatures. Locally, the O-Ag bond length does
not contract, but bending of the O-Ag-O links pulls the
O atoms together, leading to NTE. This model correlates
the NTE and lattice dynamics. Similar models seem to
explain the large NTE of ZrW2O8and other systems.
The RUM model has value even if the Ag4O tetrahedra
do not move as rigid units, but the interpretation of NTE
becomes less direct. A related concern is that modes in-
volving bending of the O-Ag-O links may be strongly
anharmonic. In a recent study on ScF3, for example,
x
y
z
O
Ag
FIG. 1. Cuprite structure of Ag2O, showing standard cubic fcc
unit cell and O-Ag-O links that pass between cubes.
the NTE largely originated with the bending of linear
Sc-F-Sc links.9Frozen phonon calculations showed that
the displacement of F atoms in these modes followed a
nearly quartic potential, and the low mass of F made it
possible to approximate the problem as independent lo-
cal quartic oscillators. The heavy mass of the Ag atoms
in the O-Ag-O links implies that different Ag atoms will
move cooperatively, and delocalized anharmonic oscil-
lators are challenging to understand.
Recent measurements by high-resolution x-ray diffrac-
tometry and extended x-ray absorption fine structure
spectrometry (EXAFS) showed large deformations of the
Ag4O tetrahedra.10–13 Although these tetrahedra are not
distorted by a pure RUM, the simultaneous excitation
of other modes makes it unrealistic to view the dynam-
ics as motions of rigid framework units. Measurements

2
by EXAFS also showed that the average Ag-O nearest-
neighbor distance expands slightly upon heating, but
the Ag-Ag next-nearest neighbor distance contracts ap-
proximately as expected from the bulk NTE.
Anharmonic phonon behavior is known to be impor-
tant for the thermodynamics and the thermal conductiv-
ity of materials at elevated temperatures. It is also im-
portant for the thermodynamic stability of phases.14,15
Anharmonic phonon behavior is sometimes associated
with NTE, but such relationships are not well under-
stood, and helped motivate the present study.
Inelastic neutron scattering is a powerful method to
measure phonon dynamics, allowing accurate measure-
ments of vibrational entropy.16 Additionally, phonon en-
ergy broadening can be measured, allowing further as-
sessment of how anharmonic effects originate from the
non-quadratic parts of the interatomic potential. A re-
cent inelastic neutron scattering experiment on Ag2O
with the cuprite structure showed phonon softening (re-
duction in energy) with temperature.17 The authors in-
terpreted this result with a quasiharmonic model, where
they calculated harmonic phonons for reduced volumes
of the structure and obtained a negative Gr ¨uneissen pa-
rameter. These measurements were performed on the
neutron energy gain side of the elastic line, restrict-
ing measurements to temperatures above 150 K, and the
available energy range of 20 meV allowed about a quar-
ter of the Ag2O phonon spectrum to be measured.
Lattice dynamics calculations, based on either classical
force fields or density functional theory (DFT), have been
used to study materials with the cuprite structure.17–19
All these calculations were performed with the quasi-
harmonic approximation (QHA), where the interatomic
forces and phonon frequencies changed with volume,
but all phonons were assumed to be harmonic normal
modes with infinite lifetimes. This QHA ignores inter-
actions of phonons at finite temperatures through the
cubic or quartic parts of the interatomic potential, but
these interactions are essential to explicit phonon anhar-
monicity. Although the QHA calculation accounted for
the NTE behavior in ZrW2O8,20 for Ag2O with the cuprite
structure, the QHA was only partly successful. Molec-
ular dynamics (MD) simulations should be reliable for
calculating phonon spectra in strongly anharmonic sys-
tems ,21–23 even when the QHA fails. To our knowledge,
no MD investigation has yet been performed on Ag2O
with the cuprite structure.
To study phonon anharmonicity in Ag2O, and its pos-
sible relationship to NTE, we performed temperature-
dependent inelastic neutron scattering experiments at
temperatures from 40 to 400 K to obtain the phonon
density of states (DOS). (At temperatures below 40 K a
first-order phase transition occurs, giving a temperature-
dependent fraction of a second phase with different
phonon properties2,24.) Fourier transform infrared spec-
trometry at cryogenic temperatures was also used to
measure the frequencies and lineshapes of phonons at
the Γ-point of the Brillouin zone. First-principles ab-
initio MD simulations were performed, and by Fourier
transforming the velocity autocorrelation function, the
large temperature-dependent phonon anharmonicity
was reproduced accurately. An independent calcula-
tion of anharmonic phonon interaction channels was
performed with interacting phonon perturbation the-
ory, and semiquantitatively explained anharmonicites
of the different phonons. Most of the phonons have
many channels for decay and are highly anharmonic.
Although the QHA is capable of predicting about half of
the NTE at low temperatures, part of this NTE is asso-
ciated with anharmonicity, and most of the NTE above
250 K originates with anharmonic interactions between
Ag-dominated and O-dominated phonon modes.
II. EXPERIMENTS
A. Inelastic Neutron Scattering
Inelastic neutron scattering measurements were per-
formed with the wide angular-range chopper spectrom-
eter, ARCS,25 at the Spallation Neutron Source at Oak
Ridge National Laboratory. Powder samples of Ag2O
with the cuprite structure of 99.99% purity were loaded
into an annular volume between concentric aluminum
cylinders with an outer diameter of 29 mm and an in-
ner diameter of 27 mm, giving about 5% scattering of
the incident neutron beam. The sample assembly was
mounted in a bottom-loading closed cycle refrigerator
outfitted with a sapphire hot stage that can be controlled
independently of the second stage of the cryostat. Spec-
tra were acquired with two incident neutron energies of
approximately 30 and 100 meV. Measurements were per-
formed at temperatures of 40, 100, 200, 300 and 400 K,
each with approximately 1.6 ×106neutron counts. Back-
grounds with empty sample cans were measured at each
temperature. Ag has an absorption cross section of 63
barns, but this was not be a problem with low back-
ground and high neutron flux.
The raw data were rebinned into intensity Ias a func-
tion of momentum transfer Qand energy transfer E. Af-
ter deleting the elastic peak around zero energy, neutron-
weighted phonon densities of states curves were calcu-
lated from I(Q,E) by subtracting the measured back-
ground, and using an iterative procedure to remove
contributions from multiple scattering and higher-order
multiphonon processes.26
B. Fourier Transform Far-Infrared Spectrometer
The far-infrared spectrometry measurements were
performed with a Thermo-Nicolet Magna 860 FTIR
spectrometer using a room-temperature deuterated
triglycine sulfate detector and a solid substrate beam
splitter. The same Ag2O powder was mixed with
polyethylene fine powder with a mass ratio of 1:19 and

3
finely ground. The sample was compressed into a pellet
of 1 mm thickness, and mounted on a copper cold finger
of an evacuated cryostat filled with liquid nitrogen. The
cryostat had polyethylene windows that were transpar-
ent in the far-infrared. Spectra were acquired at temper-
atures from 100 to 300 K, and temperature was measured
with a thermocouple in direct contact with the sample
pellet. Backgrounds from a pure polyethylene pellet of
the same size were measured at each temperature.
C. Results
Figure 2 presents the “neutron-weighted” phonon
DOS of Ag2O with the cuprite structure from ARCS data
at two incident energies at temperatures from 40 to 400 K.
Neutron-weighting is an artifact of inelastic neutron scat-
tering by phonons. Phonon scattering scales with the
scattering cross section divided by atom mass, σ/m, so
the Ag-dominated modes around 8 meV are relatively
weaker than the O-dominated modes around 63 meV.
Since the instrument energy resolution is inversely re-
lated to both the incident energy and the energy transfer,
the spectra in Fig. 2(b) have generally higher resolution
than in Fig. 2(a). As shown in Fig. 2, the main features
(peaks 1, 2 and 3) of the DOS curve from inelastic neu-
tron scattering experiments undergo substantial broad-
ening with temperature, even below 200 K, indicating an
unusually large anharmonicity. Along with the broad-
ening, peak 1 stiffens slightly, but peaks 2 and 3 shift
to lower energy by more than 3.2meV. This is an enor-
mous shift over such a small temperature range. Over
the same temperature range, phonons of ScF3shifted by
about 1 meV, for example.9To quantify thermal shifts,
Gaussian functions were fitted to the three major peaks
in the phonon DOS, and Fig. 3 presents the peak shifts
relative to their centers at 40 K.
Figure 4 presents the infrared spectra of Ag2O between
50 and 650 cm−1. Two absorption bands at 86 cm−1and
540 cm−1are seen, consistent with previous measure-
ments at room temperature27. Analysis by group theory
showed they have F1usymmetry. The low frequency
mode at 86 cm−1is an Ag-O-Ag bending mode, while
the high frequency band corresponds to Ag-O stretching.
Consistent with the trend of phonon DOS measured by
neutron scattering, the high frequency band broadened
significantly with temperature. From 100 to 300K, the
two modes shifted to lower energy by about 4 cm−1and
13 cm−1, respectively.
III. FIRST-PRINCIPLES MOLECULAR DYNAMICS
SIMULATIONS
A. Methods
First-principles calculations were performed with the
generalized gradient approximation (GGA) of density
functional theory (DFT), implemented in the VASP
package.28–30 Projector augmented wave pseudopoten-
tials and a plane wave basis set with an energy cutoffof
500 eV were used in all calculations.
First-principles Born-Oppenheimer molecular dy-
namics simulations were performed for a 3 ×3×3 su-
percell with temperature control by a Nos´e thermostat.
The relatively small simulation cell could be a cause for
concern,31 but convergence testing showed that the su-
percell in our study is large enough to accurately capture
the phonon anharmonicity of Ag2O. The simulated tem-
peratures included 40, 100, 200, 300 and 400K. For each
temperature, the system was first equilibrated for 3 ps,
then simulated for 18 ps with a time step of 3 fs. The
system was fully relaxed at each temperature, with con-
vergence of the pressure within 1 kbar.
Phonon frequency spectra and their k-space structure
were obtained from the MD trajectories by the Fourier
transform velocity autocorrelation method.21–23,32 The
phonon DOS is
g(ω)=X
n,bZdt e−iωth~
vn,b(t)~
v0,0(0)i(1)
where h i is an ensemble average, and ~
vn,b(t) is the ve-
locity of the atom bin the unit cell nat time t. Indi-
vidual phonon modes could also be projected onto each
k-point in the Brillouin zone by computing the phonon
power spectrum.21,23 To better compare with data from
inelastic neutron scattering, the calculated DOS at each
temperature was convoluted with the ARCS instru-
mental broadening function, and was neutron-weighted
appropriately.9,26
Calculations in the quasiharmonic approximation
were performed two ways. In the lattice dynamics
method, the thermal expansion was evaluated by op-
timizing the vibrational free energy as a function of
volume.9Calculations were performed self-consistently
with a 6-atom unit cell with a 10 ×10 ×10 k-point grid.
Phonon frequencies were calculated using the small
displacement method implemented by the Phonopy
package.33 These phonon dispersions in the QHA were
also used for the anharmonic perturbation theory de-
scribed below. The second method used MD calculations
to implement the QHA. We removed the temperature-
dependent explicit anharmonicity by performing simu-
lations at 40 K for volumes characteristic of 400 K, which
produced a pressure of 0.45 GPa at 40 K. Further compu-
tational details are given in the Supplemental Material.34
At the lowest simulation temperature of 40 K, classi-
cal MD trajectories may require justification. In prin-
ciple, nuclear motions could be better treated by map-
ping each nucleus onto a classical system of several fic-
titious particles governed by an effective Hamiltonian,
derived from a Feynman path integral, for example.35,36
Such low temperature quantum effects are beyond the
scope of this work. Nevertheless, our results should
not be altered significantly by quantum effects for the

4
0 10 20 30 40 50 60 70 80 90
400K
300K
200K
100K
Phonon DOS (a.u.)
Energy (meV)
40K
0 3 6 9 12 15 18 21 24 27
400K
300K
200K
100K
Phonon DOS (a.u.)
Energy (meV)
40K
112 3 (a) (b)
FIG. 2. Neutron weighted phonon DOS of Ag2O with the cuprite structure from ARCS experimental data (black dots) and MD
simulations (red curves) at temperatures from 40 to 400 K. The dashed spectrum corresponds to the 40 K experimental result,
shifted vertically for comparison at each temperature. Vertical dashed lines are aligned to the major peak centers at 40 K from
experiments, and are numbered at top. The incident energy was 100meV for panel (a), and 30 meV for panel (b).
following reasons. Our particular interest is in anhar-
monic phonon-phonon interactions at higher tempera-
tures, and our new results concern the phonons and NTE
above 250 K. A classical MD simulation is usually appro-
priate at higher temperatures. The modes most subject
to quantum corrections are those involving the dynam-
ics of the lower mass O atoms, but these are at high
energies. They are not activated at 40 K, and show weak
anharmonic effects. Relatively larger anharmonic effects
at low temperatures are found in the modes below 10
meV. These are dominated by the Ag atoms, but with
their high mass only tiny quantum effects are expected.
There are several semi-quantitative methods to estimate
the magnitudes of quantum corrections. For example,
Berens, et al.37 and Lin, et al.38 suggest that quantum
effects could be evaluated from the difference between
the quantum and classical vibrational energy or free en-
ergy derived from the corresponding partition functions.
These methods do not account for all quantum effects on
nuclear trajectories, but for Ag2O at 40 K, by using both
classical and quantum paritition function for the same
phonon DOS, we found the vibrational energy difference
between classical and quantum statistics is only 1.2% of
the cohesive energy.
B. Results
Table I presents results from our MD simulations and
experimental data on lattice parameter, bulk modulus,
thermal expansivity, and phonon frequences at the Γ-
point. As shown in Fig. 5, the MD simulations predicted
the NTE very well. On the other hand, consistent with
a recent QHA calculation,17 the NTE calculated with the
QHA method was much smaller.
The phonon DOS curves calculated from first-
principles MD simulations are shown in Fig. 2 with the
experimental spectra for comparison. To facilitate visual
comparison, in Fig. 2(a) the energy axis of the calculated
spectra were scaled by 6.8% to correct for underestimates
of the force constants in the GGA method. Neverthe-
less, excellent agreement is found between the simu-
lated phonon DOS and the experimental data, and the
calculated thermal broadenings and shifts are in good

5
0 100 200 300 400
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
3
2
Temperature (K)
1
MD calc. QH calc.


DOS Peak Shihft (meV)

FIG. 3. Shifts of centers of peaks in the phonon DOS, relative
to data at 40 K. The filled symbols are experimental data, open
symbols (green) are from MD-based QHA calculations and
solid curves (red) are from MD calculations. Indices 1, 2, 3
correspond to the peak labels in Fig. 2, and are also represented
by the triangle, square and circle respectively for experimental
data and QHA calcuations.
agreement, too. Gaussian functions were also fit to the
calculated spectra, and Fig. 3 compares these thermal
shifts from experiment and calculation.
Because of the large mass difference between Ag and O
atoms, the O-dominated phonon modes are well sepa-
rated from the Ag-dominated modes. Partial phonon
DOS analysis showed that the Ag-dominated modes
had similar energies, forming the peak of the phonon
DOS below 20 meV (peak 1 in Fig. 2), whereas the O-
dominated modes had energies above 40 meV (peaks 2
and 3).
Figure 6 shows the normal modes from MD simula-
tions, projected to the Γ-point. Six vibrational modes are
evident, including the IR-active F(1)
1uand F(2)
1umodes. The
calculated frequencies of these modes at 40 K are listed
in Table I, showing good agreement with experiment.
Large thermal shifts and broadenings are apparent in
the simulated frequencies, consistent with experiment.
The calculated peaks were then fitted with Lorentzian
functions to extract the centroids and linewidths, which
compare well with the FT-IR data as shown in Fig. 7.
IV. ANHARMONIC PERTURBATION THEORY
A. Computational Methodology
Cubic anharmonicity gives rise to three-phonon pro-
cesses, which are an important mechanism of phonon-
phonon interactions. The strengths of the three-phonon
processes depend on two elements – the cubic anhar-
monicity tensor that gives the coupling strengths be-
Absorbance
Energy (meV)
0
0.8
0.4
Energy (cm-1)
100 200 300 400 500 600
10 20 30 40 50 60 70 80
Energy (meV)
400 450 500 550 600
52 56 60 64 68 72
170K
103K
Energy (meV)
300K
Energy (cm-1)
Energy (cm-1)
(a)
(c)
60 70 80 90 100 110
8 9 10 11 12 13
170K
103K
Absorbance
300K
(b)
FIG. 4. (a) FT-IR absorbtion spectra of Ag2O with the cuprite
structure at 300 K. (b), (c) Enlargement of two bands at selected
temperatures.
tween three phonons, and the kinematical processes de-
scribed by the two-phonon density of states (TDOS).40–42
From Ipatova, et al.43, an anharmonic tensor element
for a process with the initial phonon mode jat the Γ-point
and sphonons is
V(j;~
q1j1;...;~
qs−1js−1)=1
2s! ~
2N!s
2
N∆(~
q+~
q1+···+~
qs−1)
×[ω ω1···ωs−1]1
2C(j;~
q1j1;...;~
qs−1js−1) (2)
where the phonon modes {~
qiji}have quasiharmonic
frequencies {ωi}and occupancies {ni}. The C(.)’s
are expected to be slowly-varying functions of their
arguments.44 We assume the term C(j;~
q1j1;...;~
qs−1js−1)
is a constant for the initial phonon j, and use it as a pa-
rameter when fitting to trends from MD or experiment.
Although C(j;~
q1j1;~
q2j2) changes with j1and j2, an aver-
age over modes,
hC(.)i=P1,2C(j;~
q1j1;~
q2j2)
P1,21(3)

6
is found by fitting to experimental or simulational re-
sults, where 1, 2 under the summation symbol represent
~
qiji. We define the cubic fitting parameter as
C(3)
j=hC(j;~
q1j1;~
q2j2)i(4)
The second key element of perturbation theory is that
interacting phonons satisfy the kinematical conditions
of conservation of energy and momentum. This condi-
tion is averaged over all phonons with the two-phonon
density of states (TDOS), defined as
D(ω)=X
~
q1,j1X
~
q2j2
D(ω,ω1,ω2)
=1
NX
~
q1,j1X
~
q2,j2
∆(~
q1+~
q2)h(n1+n2+1) δ(ω−ω1−ω2)
+2(n1−n2)δ(ω+ω1−ω2)i(5)
The first and second terms in square brackets are from
down-conversion and up-conversion scattering pro-
cesses, respectively.
The strength of the cubic phonon anharmonicity can
be quantified by the quality factor Q, related to the
phonon lifetime as the number of the vibrational periods
for the energy to decay to a factor of 1/e, and Q=ω/2Γ,
where 2Γis the linewidth of the phonon peak. Consid-
ering Eqs. (2) to (5), the phonon linewidth is related to
the TDOS, D(ω), weighted by the coupling strength.23,44
To leading order, the inverse of the quality factor can be
expressed as a function of the TDOS
1
Qj
=π~
32 |C(3)
j|2X
~
q1,j1X
~
q2,j2
ω1ω2D(ω,ω1,ω2) (6)
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
Exp. data
QH calc.
MD calc.
(a(T)/a(T
40K)-1)X1000
0 150 300 450
Temperature (K)
FIG. 5. Temperature dependence of lattice parameter from
experimental data in Ref. [1], quasiharmonic calculations and
MD calculations, expressed as the relative changes with respect
to their 40 K values, i.e., a(T)/a(40 K)−1.
TABLE I. Properties of Ag2O with the cuprite structure from
present MD calculations, compared to experimental data.
Units: lattice parameters in Å, bulk modulus in GPa, ther-
mal expansion coefficients in 10−6K−1, vibrational frequencies
in meV.
Experiment aCalculation
Lattice Parameter
a 4.746 4.814
Bulk Modulus
K N/A 72
Bond LTE
βAg−O12.1–35.0 19.4
βAg−Ag -9.99 -14.7
Bond Variance
σAg−Ag (40 K) 0.0078 0.0073
σAg−Ag (400 K) 0.053 0.067
Mode Frequency
F2u5.60 5.58
Eu8.9 7.45
F(1)
1u11.2 11.5
A2uN/A 29.4
F2gN/A 48.4
F(2)
1u67 63.6
aLattice parameter at 40 K is from neutron scattering measurements
in the present work, which is in good agreement with Refs. [1 and
2]. Bond linear thermal expansion (LTE) and variance data are from
Refs. [11 and 12]. The IR active mode frequencies are from FT-IR
measurements in the present work, and the frequencies of the
lowest two modes are from the luminescence spectra in Ref. [39].
The TDOS at various temperatures was calculated from
the kinematics of all three-phonon processes, sampling
the phonon dispersions with a 16 ×16 ×16 q-point grid
for good convergence. The Qfrom MD simulations were
used to approximate the anharmonicity of the phonon
modes of different energies, and obtain the the coupling
strengths |C(3)
j|2for the different modes.
B. Results
Fig. 8(a) shows calculated phonon dispersion curves
of Ag2O with the cuprite structure along high-symmetry
directions. From these, the TDOS spectra, D(ω), were ob-
tained at different temperatures, presented in Fig. 8(b)
for 40 and 400 K. At low temperatures there are two
small peaks in the TDOS centered at 15 and 65 meV. Our
calculation showed that the peak at 15 meV is from the
decay processes of one Ag-dominated mode into two
with lower frequencies, i.e., Ag 7→ Ag +Ag. The peak at
65 meV originates from spontaneous decay of one O-
dominated mode into another O-dominated mode of
lower frequency and one Ag-dominated mode.
At high temperatures there are more down-conversion
processes, but an even greater change in up-conversion

7
0 10 20 30 40 50 60 70
F1u(IR)
F2g
A2u
F1u(IR)
Eu
F2u
40K+
0.45GPa
400K
200K
40K
Energy (meV)
P(ω) (a.u)
(1) (2)
FIG. 6. Phonon modes simulated by MD and projected on the
Γ-point, at temperatures and pressures as labeled. The normal-
mode frequencies calculated from harmonic lattice dynamics
are shown as vertical dashed lines in red. The group symmetry
for each mode is shown at the bottom.
processes. Figure 8(b) shows how the strong down-
conversion peaks at low temperatures grow approxi-
mately linearly with temperature, following the thermal
population of phonon modes involved in the interac-
tions. Near the peak at 65 meV, one up-conversion band
centered at 50 meV is also strong. This band comprises
scattering channels in which one O-dominated mode is
combined with a Ag-dominated mode to form a higher
frequency O-dominated mode, i.e., O 7→ O−Ag. At
the low energy side, there is another band below 15 meV
from two types of up-conversion processes. One is from
Ag-dominated modes alone, i.e., Ag 7→ Ag −Ag. The
other involves two O-dominated modes, i.e., Ag 7→ O−
O, owing to the increased number of higher energy O-
dominated modes that can participate in these processes
at higher temperatures.
Figure 9 shows the inverse of quality factors, 1/Q, of
the F(1)
1u,F2g,F(2)
1uand A2uphonon modes from FT-IR and
MD calculations, together with the best theoretical fits
with Eq. (6). These modes are near the centers of main
features of the phonon DOS shown in Fig. 2, i.e., the
peaks 1, 2, 3 and the gap in between, respectively, and
are useful for understanding the overall anharmonicity.
As shown in Fig. 9, at higher temperatures the quality
factors decrease substantially. At 400 K, the F(1)
1u,F2g,F(2)
1u
modes have low Qvalues from 12 to 15, but the A2u
mode has a much larger value of 26. With a single pa-
rameter |C(3)
j|2for each mode, good fittings to the quality
factors are obtained. The fitting curves and the corre-
0 100 200 300 400
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
F1u
F2g
Temperature (K)
Mode Shihft (meV)
F1u
(1)
(2)
(a)
F1u
F1u
(2)
(1)
(b)
0 100 200 300 400
1
2
4
8
16
F2g
Lifetime (ps)
Temperature (K)
FIG. 7. (a) Temperature dependent frequency shifts of the Ag-
dominated F(1)
1umode and the O-dominated F2gand F(2)
1umodes
from FT-IR (black), compared with the MD simulated peaks
(red) such as in Fig. 6. (b) The lifetimes of the corresponding
modes at temperatures from 40 to 400 K, from FT-IR (black)
and the MD simulated peaks (red).
sponding values of the parameters |C(3)
j|2are presented
in Fig. 9. The |C(3)
j|2values do not vary much among
different modes, so the typical assumption of a slowly
varying C(.) seems reasonable.
V. DISCUSSION
A. Quasiharmonic Approximation
In the quasiharmonic approximation (QHA), a mode
Gr¨uneisen parameter γjis defined as the ratio of the
fractional change of the mode frequency ωjto the frac-
tional change of volume V, at constant temperature,
γj=−∂(ln ωj)
∂(ln V). The usual trend is for phonons to soften
with lattice expansion, increasing the phonon entropy
and stabilizing the expanded lattice at elevated temper-
atures. A negative Gr¨uneisen parameter is therefore ex-
pected for the special phonon modes associated with
NTE, such as RUMs. If all the anharmonicity of Ag2O
with the cuprite structure is attributed to this volume

8
 (a.u.)
D
(ω)
0
10
20
30
40
50
60
70
R
X
Energy (meV)
M
(a)
0 20 40 60 80 100 120
0.0
0.5
1.0
1.5 Ag Ag Ag
Ag Ag+Ag
O O Ag
O O+Ag
Energy (meV)
Ag O O
ГГ
(b)
FIG. 8. (a) Calculated phonon di spersion along high-symmetry
directions of Ag2O with the cuprite structure. Γ(0,0,0),
M(0.5,0.5,0), X(0.5,0,0), R(0.5,0.5,0.5). (b) The TDOS spectra,
D(ω), at 40 K (dashed) and 400 K (solid). The down-conversion
and up-conversion contributions are presented separately as
black and green curves, respectively.
effect, however, the values of Gr ¨uneisen parameters are
approximately –9 for the high energy modes at peaks
2 and 3 of the phonon DOS, and –20 for the infrared-
active F1umode. These anomalous values may indicate
a problem with the QHA. The QHA method also signifi-
cantly underestimates the NTE, and misses the behavior
at temperatures above 250 K.
As seen in the figure in the Supplemental Material
and in Fig. 6, the giant negative Gr¨uneisen parame-
ters are inconsistent with the results of MD simula-
tions. The volume change alone doesnot affect much the
phonon lifetimes or frequency shifts. All features in the
phonon spectra generated with the MD-implemented
QHA showed little change except for small stiffening
about 0.6 meV at high energies, in agreement with the
recent lattice dynamics QHA calculations.17
0 150 300 450
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
1/Q
Temperature (K)
0 150 300 450
0.00
0.01
0.02
0.03
0.04
1/Q
0 150 300 450
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
1/Q
Temperature (K)
0 150 300 450
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
1/Q
F1u
(1)
F1u
(2)
F2g
A2u
|C | = 1.04
(3) 2|C | = 1.12
(3) 2
|C | = 0.67
(3) 2|C | = 0.51
(3) 2
FIG. 9. Temperature dependence of the inverse of the quality
factors 1/Qfor the F(1)
1u,A2u,F2gand F(2)
1uphonon modes from
FT-IR (black squares), MD calculations(red circles) and the the-
oretical fittings with a full calculation of the TDOS. The unit of
the fitting parameter |C(3)
j|2is 10−1eV−1.
B. Negative Thermal Expansion
Our eigenvector analysis of phonon modes showed
that the three low-energy Ag-dominated F2u,Euand F1u
modes correspond to two distinct types of vibrations.
The F2umode involves rigid rotations of Ag4O tetra-
hedra, and could be considered as RUM. The Eumode
involves the shearing of Ag4O units by changing the Ag-
O-Ag bond angles. The F1umode measured by infrared
spectrometry also shears the Ag4O units, and includes
some displacements of O atoms. Shearing the tetra-
hedra was shown to reduce the average vertex-vertex
distance27,45 and contribute to the NTE.
In the cuprite structure, the modes associated with the
rigid rotations and the distortions of the Ag4O tetrahedra
have similar energies below 10 meV. They are equally fa-
vorable thermodynamically, and both would be active
at very low temperature. As a consequence, there is si-
multaneously a large deformation of Ag4O units and a
strong contraction of the Ag-Ag shell, as observed exper-
imentally and computationally at low temperatures. The
large thermal distortions of the Ag4O tetrahedra involve
the Ag atoms at the vertices, and their large mass causes
these distortions to occur at low frequencies. Polyhe-
dral units in most other NTE materials are bridged by
the lightweight atoms, such as O and C-N, so the polyhe-
dra are distorted at significantly higher energies (usually
above 40 meV), and may not distort at lower tempera-
tures.
It is a thermodynamic requirement that the NTE of Fig.

9
5 must go to zero at T=0, but the intervening phase tran-
sition at 40 K impedes this measurement. Nevertheless,
the steep slope of the lattice parameter with tempera-
ture is consistent with the occupancy of phonon modes
of 10 meV energy, suggesting that the QHA model of
NTE involves the correct modes, such as the F2umode
(which is a RUM). These low-energy modes are dom-
inated by motions of the Ag atoms. This explanation
based on the QHA is qualitatively correct, but anhar-
monic interactions are large enough to cause the QHA
to underestimate the NTE by a factor of two.
C. Explicit Anharmonicity
At temperatures above 250 K, there is a second part
of the NTE behavior that is beyond the predictions of
quasiharmonic theory. This NTE above 250 K is pre-
dicted accurately by the ab-initio MD calculations, so
it is evidently a consequence of phonon anharmonic-
ity. The temperature-dependence of this NTE behavior
follows the Planck occupancy factor for phonon modes
above 50 meV, corresponding to the O-dominated band
of optical frequencies. In the QHA these modes above
50 meV do not contribute to the NTE. These modes are
highly anharmonic, however, as shown by their large
broadenings and shifts.
For cubic anharmonicity, the two-phonon DOS
(TDOS) is the spectral quantity parameterizing the num-
ber of phonon-phonon interaction channels available to
a phonon. For Ag2O with the cuprite structure, the peaks
in the TDOS overlap well with the peaks in the phonon
DOS. Most of the phonons therefore have many possible
interactions with other phonons, which contributes to
the large anharmonicity of Ag2O with the cuprite struc-
ture, and small Q(short lifetimes). Although the Qval-
ues of most phonon modes in Ag2O with the cuprite
structure are small and similar, the origins of these life-
time broadenings are intrinsically different. For peak 2 of
the phonon DOS, the anharmonicity is largely from the
up-conversion processes: O 7→ O−Ag, while for peak 3
it is from the down-conversion processes: O 7→ O+Ag.
The anharmonicity of peak 1 is more complicated. It
involves both up-conversion and down-conversion pro-
cesses of Ag-dominated modes. The TDOS also shows
why the A2umode has a larger Qthan other modes.
Figure 8(b) shows that the A2umode lies in the trough
of the TDOS where there are only a few phonon decay
channels.
Owing to explicit anharmonicity from phonon-
phonon interactions, the thermodynamic properties of
Ag2O with the cuprite structure cannot be understood as
a sum of contributions from independent normal modes.
The frequency of an anharmonic phonon depends on the
level of excitation of other modes. At high temperatures,
large vibrational amplitudes increase the anharmonic
coupling of modes, and this increases the correlations
between the motions of the Ag and O atoms, as shown by
perturbation theory. Couplings in perturbation theory
have phase coherence, so the coupling between Ag- and
O-dominated modes at higher energies, as seen in the
peak of the TDOS, causes correlations between the mo-
tions of Ag and O atoms. The ab-initio MD simulations
show that anharmonic interactions allow the structure to
become more compact with increasing vibrational am-
plitude. The mutual motions of the O and Ag atoms
cause higher density as the atoms fill space more effec-
tively. The large difference in atomic radii of Ag and O
may contribute to this effect. Perhaps it also facilitates
the irreversible changes in Ag2O at temperatures above
500 K, but this requires further investigation. For cuprite
Cu2O, which has less of a difference in atomic radii, the
thermal expansion is much less anomalous.
VI. CONCLUSIONS
Phonon densities of states of Ag2O with the cuprite
structure at temperatures from 40 to 400 K were mea-
sured by inelastic neutron scattering spectrometry. The
infrared spectra of phonon modes were also obtained
at temperatures from 100 to 300 K. Large anharmonic-
ity was found from both the shifts and broadenings of
peaks in the phonon spectra. A normal mode analysis
identified the rigid unit modes and the bending modes of
the Ag4O tetrahedra that play key roles in the negative
thermal expansion (NTE) at low temperatures. Some
of the NTE can be understood by quasiharmonic the-
ory, but this approach is semiquantitative, and limited
to temperatures below 250 K.
First principles MD calculations were performed at
several temperatures. These calculations accurately ac-
counted for the NTE and local dynamics of Ag2O with
the cuprite structure, such as the contraction of the Ag-
Ag shell and the large distortion of the Ag4O tetrahedra.
The phonon DOS obtained from a Fourier-transformed
velocity autocorrelation method showed large anhar-
monic effects in Ag2O, in excellent agreement with the
experimental data. A second part of the NTE at temper-
atures above 250 K is due largely to the anharmonicity of
phonon-phonon interactions and is not predicted with
volume dependent quasiharmonicity.
Phonon perturbation theory with the cubic anhar-
monicity helped explain the effects of phonon kinemat-
ics on phonon anharmonicity of Ag2O with the cuprite
structure. The phonon interaction channels for three-
phonon processes are given by the TDOS, weighted
approximately by the phonon coupling strength. The
phonons that are most broadened are those with ener-
gies that lie on peaks in the TDOS. The temperature-
dependence of the quality factors Qof individual
phonon modes measured by infrared spectrometry were
explained well by anharmonic perturbation theory. Per-
turbation theory also showed strong interactions be-
tween the Ag-dominated modes and the O-dominated
modes in both up-conversion and down-conversion pro-

10
cesses. In particular, the strong interactions of O-
dominated modes with Ag-dominated modes causes the
second stage of NTE at temperatures above 250K.
ACKNOWLEDGMENTS
This work was supported by DOE BES under con-
tract DE-FG02-03ER46055. The work benefited from
software developed in the DANSE project under NSF
award DMR-0520547. Research at Oak Ridge National
Laboratory’s SNS was sponsored by the Scientific User
Facilities Division, BES, DOE.
∗tianlan@caltech.edu
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arXiv:1402.3603v2 [cond-mat.mtrl-sci] 3 Mar 2014
Anharmonic lattice dynamics of Ag2O studied by inelastic neutron scattering and first
principles molecular dynamics simulations
Supplemental Material
Tian Lan,1, ∗Chen W. Li,2J. L. Niedziela,2Hillary Smith,1
Douglas L. Abernathy,2George R. Rossman,3and Brent Fultz1
1Department of Applied Physics and Materials Science, California Institute of Technology, Pasadena, California 91125, USA
2Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
3Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California 91125, USA
(Dated: March 4, 2014)
To separate the effects of quasiharmonicity and ex-
plicit anharmonicity, the mode frequency ωj=ωj(V,T)
is expressed as a function of volume and temperature
∂ln ωj
∂T!P
=−γjβ+ ∂lnωj
∂T!V
(1)
where jis the phonon mode index, βis the volume ther-
mal expansivity and γjis the mode Gr¨uneisen param-
eter. The left-hand side is the temperature-dependent
frequency shift at constant pressure, and includes con-
tributions from both quasiharmonicity and explicit an-
harmonicity. The first term on the right-hand side, the
isothermal frequency shift as a function of pressure, is
the quasiharmonic contribution to the frequency shift.
The second term on the right is the pure temperature
contribution to the frequency shift from the explicit an-
harmonicity. From the difference of the isobaric and
isothermal frequency shifts, the explicit anharmonicity
can be determined.
In a molecular dynamics simulation, the quasihar-
monic contribution can be evaluated explicitly by turn-
ing offthe temperature-dependent anharmonicity. In
principle, this method is equivalent to the QHA method
implemented with self-consistent lattice dynamics,1and
in practice we have found this to be true. For example,
we performed simulations at 40 K for volumes charac-
teristic of 400 K, which produced a pressure of 0.45 GPa.
This calculation therefore removed the temperature ef-
fect while preserving the quasiharmonic volume effect
at 400 K. The corresponding phonon DOS curves from
MD calculations are shown in Fig. 1. By comparing the
phonon spectrum of a simulation at 40 K and 0.45 GPa
with a simulation at 400 K, the pure temperature depen-
dence is identified. From the spectra of Fig. 1, it is found
that the explicit anharmonicity dominates the softening
and broadening of the phonon spectra. All features in
the phonon spectra with the QHA showed little change
with temperature, except for small stiffenings at high
energies.
∗tianlan@caltech.edu 1C. W. Li, X. Tang, J. A. Munoz, J. B. Keith, S. J. Tracy, D. L.
Abernathy, and B. Fultz, Phys. Rev. Lett. 107, 195504 (2011).

2
0 10 20 30 40 50 60 70 80 90
40K+0.45GPa
400K
Phonon DOS (a.u.)
40K
0 3 6 9 12 15 18 21 24 27
40K+0.45GPa
400K
Phonon DOS (a.u.)
Energy (meV)
40K
(a)
(b)
1
1
2 3
FIG. 1. Neutron weighted phonon DOS of Ag2O with the
cuprite structure from MD simulations. The green spectrum
is the MD simulated phonon DO S at 40 K and 0.45 GPa. Verti-
cal dashed lines are aligned to the major peak centers at 40 K
and labeled by numbers. The incident energy was 100 meV for
panel (a), and 30 meV for panel (b). The spectra were convo-
luted with the resolution function characteristic of ARCS for
the different energies of the incident neutron beam.