Elastic interactions and control of the Mott transition

5.2 MB·

by  G. G. Guzmán-Verri

Elastic interactions and control of the Mott transition
G. G. Guzm´an-Verri1,2, R. T. Brierley3, P. B. Littlewood4,5
1Centro de Investigaci´on en Ciencia e Ingenier´ıa de Materiales and Escuela de F´ısica,
Universidad de Costa Rica, San Jos´e, Costa Rica 11501,
2Materials Science Division, Argonne National Laboratory, Argonne, Illinois, USA 60439,
3Department of Physics, Yale University, New Haven, Connecticut 06511, USA,
4Argonne National Laboratory, Argonne, Illinois, USA 60439, and
5James Franck Institute, University of Chicago, 929 E 57 St, Chicago, Illinois, USA 60637.
(Dated: January 11, 2017)
Metal to insulator transitions (MIT) driven a b
7 by strong electronic correlations are common in R/M
1
condensed matter systems, and are associated
0
with some of the most remarkable collective
2
phenomena in solids, including superconductivity
O
n
and magnetism. Tuning and control of the
a
J transition holds the promise of novel, low power, B O
ultrafast electronics1,2, but the relative roles of
9
doping,chemistry,elasticstrainandotherapplied

] fieldshasmadesystematicunderstandingdifficult
l to obtain. Here we point out that existing data
e
- on tuning of the MIT in perovskite transition FIG. 1. (a) Perovskite lattice showing the tilts of the BO6
tr metal oxides through ionic size effects provides (B=Mn,Ni) octahedra. R is a rare earch element such as La,
s evidence of systematic and large effects on the Pr,Nd,andSm;andMisanalkalineearthmetalsuchasCa,
.
t phase transition due to dynamical fluctuations Sr, Ba. (b) 2D representation of the tilts used in our model
a
whereφ isaninitialequilibriumantiferrodistortiverotation.
m of the elastic strain, which have been usually 0
neglected. This is illustrated by a simple yet
-
d quantitative statistical mechanical calculation in
n a model that incorporates cooperative lattice tune the phase transition.
o distortions coupled to the electronic degrees of The crystal structure of perovskite TMOs consists
c freedom. We reproduce the observed dependence of corner-sharing oxygen octahedra surrounding the B
[
of the transition temperature on cation radius transition metal ion, as shown in Fig. 1(a). In general,

1 in the well-studied manganite3–8 and nickelate9,10 the octahedra are tilted relative to their neighbors in
v materials. Since the elastic couplings are an alternating pattern, and the tilt angle increases
8 generically quite strong, these conclusions will with smaller A-site cation radius rA. The dramatic
1
broadly generalize to all MITs that couple to a changes in the functional behavior of perovskites when
3
2 change in lattice symmetry. varying the tilt angle have led to proposals to engineer
0 material properties by manipulating the tilt angle using
. Alargenumberoftransitionmetaloxides(TMO)with a combination of strain, doping and pressure16. In
1 theABO perovskitecrystalstructureallowtuningofthe
3 addition to variations of the atomic size, doping with
0 MITbynotonlybythechoiceandaveragevalenceofthe
A-site cations also introduces disorder in the cation size;
7
electronicallyactiveBion(usuallya3dtransitionmetal)
1 carefuldistinctionoftheeffectsofdopinganddisorderfor
but also by the size of the electronically inactive A ion
: the manganites demonstrated that disorder reduces the
v (usually a rare earth or alkaline earth)11–13. This size
transition temperature as effectively as does the average
Xi effect can shift the transition temperature by hundreds size change.17
of Kelvin, and the widely accepted explanation is that
r Although purely electronic mechanisms to describe
a it is due to a reduction in the electron bandwidth as
transition metal oxides are appealing in their theoretical
the bond bending induced by ionic size changes the
simplicity, it is known that the strong electron-phonon
orbital overlap1,14. However, the changes in bandwidth
coupling means that the effects of lattice distortions
are typically less than a few percent15, and it seems
cannot be neglected, and this is particularly well studied
remarkablethatacriticalvalueoftheratioofinteraction
in manganites and nickelates18–20. An electron that
strength to bandwidth can be crossed in every 3d
is localized by correlation effects in a unit cell will
transition metal oxide, solely by varying the counterion.
lower its energy further by the creation of a lattice
Instead, we propose here that the observed distortion, which may be of different symmetry in
dependence on the crystal structure arises from different materials. In the nickelates this is a simple
long-range, strain-mediated interactions, and that breathing distortion, and in the manganites a so-called
lattice fluctuations rather than electronic fluctuations Jahn-Teller distortion that lowers the cubic symmetry
2
of the octahedon. The competition between this a b
potential energy gain and the kinetic energy gained by
delocalization to form a metal gives rise to the complex
MIT phenomena in these materials. E
n
e
The corner-sharing constraint on the octahedra Jahn-Teller Breathing rg
introduces compatibility conditions between distortions y / m
at different lattice sites; when integrating out the
e
phonondegreesoffreedomtheseyieldhighlyanisotropic, V
long-range interactions21,22. Previous studies of phonon
cooperativity in the manganites have demonstrated that
Shear Rotation
they can explain the complex charge ordered phases and
mesoscopic structures that have been observed in the
FIG. 2. (a) Lattice distortions considered in our model. (b)
manganites23–26, and studied some effects of cooperative
Strain responses of a lattice to a local Jahn-Teller distortion
coupling on the transition27–29. However, these studies
as a result of rotations. The color of each square indicates
did not consider the effect of octahedral tilting on the
the strain energy associated with the local distortions of
long-range interaction of the distortions25. The purpose that square. The grey parallelogram at the centre has
of this work is to study such effects, and in doing so, to a Jahn-Teller distortion of fixed amplitude. Additional
constructacompletetheoryforcooperativeelasticeffects distortions on this site, such as shear, are allowed. Top:
at a phase transition. lattice with φ =0. Bottom: lattice with φ =15◦.
0 0
For illustration, we use a two-dimensional model
of a transition metal perovskite, where we replace
the octahedra by squares, as it is shown in a 𝜙0=0o b 𝜙0=15o
3 3 Vk /aS
Fig. 1(b). Although the physics of bulk perovskites
3.0
is three-dimensional, our approximation follows other 2 2
studies of strain interactions in the manganites25,30,31 1 1
2.0
and more generally21,32–34 and significantly simplifies a
the geometry while still capturing the relevant physical ky 0 0
-1 -1 1.0
behavior. Atalatticesiter,thesquarescanundergothe
distortions shown in Fig. 2(a): deviatoric/Jahn-Teller -2 -2
modes Tr, dilatation/breathing modes Dr, shear modes -3 -3 0.0
S , and small rotations R of the squares from an initial -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3
eqruilibrium antiferrodistorrtive rotation φ0, i.e., φr = kxa kxa
(−1)|r|φ +R . Assumingaharmonicenergypenaltyfor
0 r
FIG. 3. Dependence of the effective elastic potential energy
creating distortions from an equilibrium configuration,
for Jahn Teller distortions on rotations in momentum space.
(cid:88)
H = a T2+a D2+a S2, (1)
T r D r S r
r
reduced by changes in φ , the high-temperature phase
0
combined with the corner-sharing constraint, we can is favoured by a reduction in the polaron formation
find the effective interaction V (φ ) between different energy.37 To study this behaviour, we use V (φ ) to
rr(cid:48) 0 rr(cid:48) 0
types of distortion (see supplement). a , a , and a formastatisticalmechanicalmodelforthedistortionsin
D S T
are, respectively, the stiffness of breathing, shear, and this high-temperature phase, with a Hamiltonian,
Jahn-Teller distortions in a single, free octahedron and
(cid:20) (cid:21)
are independent of r. H =(cid:88) 1Π2− κQ2+ γQ4
2 r 2 r 4 r
Fig. 3 shows that the interaction strength is reduced
r
byanincreasetiltangleforJahn-Tellerdistortions. This (cid:88) (cid:88)
+ V (φ )Q Q − h Q , (2)
occursbecauseinthetiltedconfigurationitispossiblefor rr(cid:48) 0 r r(cid:48) r r
rr(cid:48) r
thedistortiontobeaccomodatedbyadditionalrotations
to the neighbouring sites, rather than changes in the where Q is a Jahn-Teller (breathing) distortion for the
r
shape. Characteristic strain responses of the lattice to manganites(nickelates)andΠ itsconjugatemomentum.
r
alocalJahn-Tellerdistortionwithandwithoutrotations To model the the compositional disorder that arises in
are shown in Fig. 2 (b). themanganitesfromchemicalsubstitutionofthealkaline
Bothmanganites35andnickelates36undergofirst-order earth element at the A site of the perovskite structure,
transitions from a characteristic low temperature phase weconsideralinearcouplingofthelatticedistortionsQ
r
to a high-temperature polaronic phase. This suggests toalocalquenchedrandomdistortionh . Wechoosethe
r
that the motion of conduction electrons through the h ’s to be normally distributed with mean h¯ = 0 and
r r
lattice is associated with the creation of local structural variance ∆2. The negative sign of the Q2 term describes
r
distortions. When the distortion interaction V (φ ) is thelocaltendencytowardsdistortionduetothepresence
rr(cid:48) 0
3
a b 400 c
600 EThxpeeorriyment ●●●● 600● EThxpeeorriyment ●
●
Kelvin500 RNiO3 ●● Kelvin500 ●R0 .3 M0 .7 MnO3 Kelvin300 ● R0 .3 M0 .7 MnO3
mperature/340000 PMM ●● PMI mperature/340000 ●●●●●● PMM mperature/200 ● PMM
Transitionte200 ○ ○ ○○○○ ○○○○ Transitionte200 FM ●●●●●●●●●●●●●●● Transitionte100 FM ● ● ●
100 AFI 100 ●● ●
●
0□ 0 0
8 10 12 14 16 18 4 6 8 10 12 0 5 10 15 20 25
Tiltangle/degrees Tiltangle/degrees Disorder/×10-3Å2
FIG. 4. Comparison between the results of our statistical mechanical calculation (solid lines) and experimental data for
transitionsinthenickelatesandmanganites. FittedmodelparametersaregiveninTableS1ofthesupplement. a: Comparison
forthenickelatesforthetransitiontemperatureasafunctionofoctahedraltiltangle. Filledandopencirclesareexperimental
transitiontemperatures10 fortheparamagneticinsulator(PMI)andantiferromagneticinsulator(AFI)phasesrespectively. The
extension of the green shading beyond the blue dashed line is an extrapolation. The open square denotes LaNiO , which is
3
always in the high temperature polaronic, paramagnetic metal (PMM) phase. For the manganites, the comparison is made
with results17 (red circles) that separate the effect of tilt angle (b) and compositional disorder (c) on the transition from the
paramagnetic, polaronic “bad metal” phase (PMM) to the ferromagnetic metal (FM) phase
of electrons. documented. However the couplings, including their
Using a variational pair-distribution function that rough order-of-magnitude, are generic, and the ideas
incorporates mean field behavior, Gaussian corrections presented here will surely be relevant to other classes
to the thermal and quantum fluctuations, and disorder of materials such as the titanates13, high temperature
at the level of the replica method, (see supplement) we superconductors38, ferroelectrics39, and molecular
calculate the free energy of the system as a function of fullerides40.
temperature, octahedral tilt angle φ and the strength
0 At low enough temperatures one should surely take
of disorder ∆2. This approximate value is compared
care of other low-energy degrees of freedom such as spin
to a constant free energy F associated with the
LowT fluctuations and electronic quantum fluctuations which
low temperature phase to identify the location of
our model does not take into account. Nonetheless,
the first-order phase transition, see Fig. 4. For the
the model we employ does generate a quantum critical
manganites, this phase corresponds to a ferromagnetic
point on account of elastic interactions alone. Moreover,
metal while for the nickelates it is a paramagnetic
the long range and anisotropy of these elastic couplings
insulator. Despite the over-simplicity of the model, the
will modify the critical dynamics away from that arising
relationship between tilt angle, disorder, and transition
from short-range models generated by purely electronic
temperature is well reproduced. We do not attempt
couplings.
to describe the effects of the strain interactions on the
MIT of the nickelates at low temperatures (see green Acknowledgements. Work at Argonne National
region in Fig. 4(a)), as its magnetic ordering is different Laboratory is supported by the U.S. Department of
from that of the insulating phase above it. Similarly for Energy, Office of Basic Energy Sciences under contract
the manganites, at low enough temperatures the PMM no. DE-AC02-06CH11357. RTB acknowledges the Yale
phase becomes either charge-ordered or glassy, beyond PrizePostdoctoralFellowship. WorkattheUniversityof
our approximations. CostaRicaissupportedbytheVice-rectoryforResearch
Conclusions. In this paper we have outlined a under the project no. 816-B5-222, GGGV and RTB
systematic theory for the incorporation of long-range acknowledge partial financial support from the Office of
elastic couplings into a simplified statistical mechanical International Affairs at the University of Costa Rica.
theory of Mott-like phase transitions, where the
Author contributions. PBL conceived the study;
electronic contributions to the free energy are
GGGV and RTB performed the calculations; all authors
incorporated at the level of Landau theory. That
constructed the model, wrote the manuscript, discussed
these elastic interactions are explicitly relevant for the
the results and implications at all stages.
manganitesandthenickelatesisconfirmedbytheability
of such a theory to systematically explain size effects Competing Interests. RTB is currently an editor at
or tolerance factor variations which have already been Nature Communications.
4
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1
Supplemental Information- Elastic interactions and control of the Mott transition
The purpose of this SI is to present details about the derivation of the elastic interactions, our model Hamiltonian,
its solution, and additional results.
I. DISTORTION INTERACTION
We consider a two-dimensional model of a perovskite, consisting of corner-coupled quadrilaterals that are the
analoguesofthethree-dimensionaloxygenoctahedra. Similartwo-dimensionalmodelshavebeenusedinearlierstudies
of strain in functional materials, including the manganitesS1,S2. The displacements of the corners of the quadrilateral
centred at a position r from their initial positions are the vectors t , u , v and w (see Fig. S1). To simplify the
r r r r
analysis, we assume that the allowed (i.e. low energy) configurations of the quadrilaterals are parallelograms, so that
t −w =u −v . (S1)
r r r r
Thiscorrespondstoassumingthereisinfiniteenergycostassociatedwith“shuffle”modeswithintheaquadrilateralS3.
With this assumption, deviations from an initial configuration can be described in terms of four degrees of freedom:
a rotation and three “strains” (see Fig. S1(a) of the main text), D (dilatation/breathing modes), S (shear
r r
modes) and T (deviatoric/Jahn-Teller modes). We assume that rotations of the squares from an initial equilibrium
r
antiferrodistortiverotationφ aresmall,writingφ =(−1)|r|φ +R andallequationsinvolvingR canbelinearized.
0 r 0 r r
We take the following simple form for the elastic energy,
FIG.S1. Aquadrilaterialalteredfromitsequilibriumstate(dashed)byabreathingdistortion. Thecorneratomsdisplacedby
displacement vectors t , u , v and w .
r r r r
(cid:88)
H = a T2+a D2+a S2, (S2)
T r D r S r
r
where a is the energy penalty for the corresponding distortion of a free octahedron. Note that we make the
i
approximation that the cost of rotations is small enough that it can be negelected. Such a term should exist in
order for there to be a state with a non-zero φ , as observed in experiment. In the perovskite model, the constraint
0
thatthesquaressharecornersmeansthatt =v andw =u . Theseconstraintsreducethenumberofdegrees
r r−y r r−x
of freedom per site to 2. Distortions of neighboring squares are coupled and as a result the energy given in Eq. (S2)
actually describes a system with long-range strain interactions.
We can find the constraint equations in terms of the rotation and strain degrees of freedom by expressing them in
terms of the atom positions t , u , v and w , then using the corner-sharing constraint to eliminate the positions.
r r r r
We assume that rotations of the squares from an initial equilibrium antiferrodistortive rotation φ are small, writing
0
2
φ =(−1)|r|φ +R . Then we assume that the equation for R is linear and our equations are,
r 0 r r
R =cosφ (uy−uy −vx+vx )−(−1)|r|sinφ (ux−ux +vy−vy ), (S3a)
r 0 r r−nx r r−ny 0 r r−nx r r−ny
S =cosφ (uy−uy +vx−vx )−(−1)|r|sinφ (ux−ux −vy+vy ), (S3b)
r 0 r r−nx r r−ny 0 r r−nx r r−ny
T =cosφ (ux−ux −vy+vy )+(−1)|r|sinφ (uy−uy +vx−vx ), (S3c)
r 0 r r−nx r r−ny 0 r r−nx r r−ny
D =cosφ (ux−ux +vy−vy )+(−1)|r|sinφ (uy−uy −vx+vx ), (S3d)
r 0 r r−nx r r−ny 0 r r−nx r r−ny
where we have used the convention r =(rx,ry) for the vector components.
r r r
After Fourier transformation, the parallelogram condition Eq. (S1), combined with the corner sharing constraint,
can be written as,
(1+eikxL)u =(1+eikyL)v ,
k k
where L is lattice spacing in the equilibrium configuration. Using this relationship, the Fourier transform of Eq. (S3)
is,
(cid:20) (cid:21)
R =2i(cid:0)1+eikxL(cid:1) cosφ (t uy −t ux)− sinφ0 (cid:0)t ux +t uy (cid:1) ,
k 0 x k y k t y M−k x M−k
y
(cid:20) (cid:21)
S =2i(cid:0)1+eikxL(cid:1) cosφ (t uy +t ux)− sinφ0 (cid:0)t ux −t uy (cid:1) ,
k 0 x k y k t y M−k x M−k
y
(cid:20) (cid:21)
T =2i(cid:0)1+eikxL(cid:1) cosφ (t ux−t uy)+ sinφ0 (cid:0)t uy +t ux (cid:1) ,
k 0 x k y k t y M−k x M−k
y
(cid:20) (cid:21)
D =2i(cid:0)1+eikxL(cid:1) cosφ (t ux+t uy)+ sinφ0 (cid:0)t uy −t ux (cid:1) ,
k 0 x k y k t y M−k x M−k
y
where M=(π,π) is the wavevector of the antiferrodistortive order and we have defined,
k L
t =tan i .
i 2
Finally we can eliminate u to obtain a set of (discrete) compatibility conditions,
k
R =f (cid:110)4t t (cid:2)−(t2 +t2)T +cos2φ (t2 −t2)D (cid:3)
k k x y x y k 0 x y k
(S4)
+2sin2φ (t2 −t2)(cid:2)(t2 +t2)T +cos2φ (t2 −t2)D (cid:3)(cid:111),
0 x y x y M−k 0 x y M−k
S =f (cid:110)4t t (cid:2)(t2 +t2)D −cos2φ (t2 −t2)T (cid:3)
k k x y x y k 0 x y k
(S5)
+2sin2φ (t2 −t2)(cid:2)(t2 +t2)D +cos2φ (t2 −t2)T (cid:3)(cid:111).
0 x y x y M−k 0 x y M−k
where,
1
f = .
k 8t2t2 +2(t2 −t2)2sin22φ
x y x y 0
Notethattheserelationshipshavesingularbehaviorwhenthek approach0orπ. Inparticular,thereisnorelationship
i
between the fields when k=0 or M, i.e. the cases of homogeneous and antiferrodistortive deformations; each strain
field is a separate degree of freedom in these casesS4.
In the long-wavelength limit, t →k L/2 and when φ =0 we can write Eq. (S5) as,
x x 0
(k2+k2)D −2k k S −(k2−k2)T =0,
x y k x y k x y k
which is the Fourier transform of the usual two-dimensional compatibility relationS5.
3
WecannowsubstituteEq.(S5)intotheFouriertransformofEq.(S2)tofindtheresultwithoutseparateconstraints:
(π,π)
F = (cid:88) (cid:2)a +2a f (t2 −t2)2cos22φ (cid:3)|T |2+(cid:2)a +2a f (t2 +t2)2(cid:3)|D |2
T S k x y 0 k D S k x y k (S6)
k=0
−2a f (t4 −t4)cos2φ (D∗T +h.c.).
S k x y 0 k k
In deriving this result, we have made use of the fact that the summation is over the entire Brillouin zone of the
undistorted, φ = 0, system and the property that t → 1/t when k → M−k. Since each power of t appears
0 i i i
equally often in the numerators and denominators of Eq. (S6), this relationship effectively means that (t2 −t2) →
x y
−(t2−t2)underthechangeofwavevector,producingcancellationsbetweentermsthatwouldotherwisecoupledifferent
x y
wavevectors.
To obtain the static interaction between the single strain fields T or D , we can minimize the energy with respect
k k
to the other field:
2a f (t4 −t4)cos2φ 2a f (t4 −t4)cos2φ
D = S k x y 0T , or T = S k x y 0 D .
k a +2a f (t2 +t2)2 k k a +2a f (t2 −t2)2cos22φ k
D S k x y T S k x y 0
Substituting these results into Eq. (S6), we obtain the energy for a single strain field, including the long-range
interaction term,
(cid:40) (cid:41)
H =(cid:88) a + aDaScos22φ0(t2x−t2y)2 |T |2 (S7a)
T T 4a t2t2 +a (t2 +t2)2+a (t2 −t2)2sin22φ k
k D x y S x y D x y 0
(cid:40) (cid:41)
H =(cid:88) a + aSaT(t2x+t2y)2 |D |2. (S7b)
D D 4a t2t2 +a (t2 −t2)2+(a −a )(t2 −t2)2sin22φ k
k T x y S x y T S x y 0
These strain interactions are highly anisotropic, as it is shown, respectively, in Figs. 3 and S2 for H and H
T D
and non-analytic; as |k| → 0, the value of the potential depends on the direction of k. The potential vanishes for
(anti-)ferrodistortive perturbations, corresponding to k = 0 (k = M = (π,π)), as the different distortion modes are
uncoupled at those wavevectors. This implies a discontinuity at k = M, which arises from our assumption that the
distortions in Fig. S1 are the only distortions, neglecting the shuffle modes for the octahedra.S6 However, as long as
such modes are significantly stiffer than the high-symmetry modes we do consider, this should not be an important
approximation. Inthelong-wavelengthlimitandnorotations,V (φ =0)forthemanganitesmatchesthatofprevious
k 0
work.S5,S7
a 𝜙0=13o b 𝜙0=20o
3 3 Vk /aS
1.00
2 2
1 1
a 0.75
ky 0 0
-1 -1
-2 -2
0.55
-3 -3
-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3
kxa kxa
FIG. S2. Dependence of the effective elastic potential energy on rotations in momentum space for the nickelates.
4
II. STATISTICAL MECHANICAL MODEL
When an electron is present on a lattice site, there is an energetic benefit to creating a lattice distortion. To model
this, we consider the following model Hamiltonian on a square lattice,
(cid:20) (cid:21)
(cid:88) 1 κ γ (cid:88) (cid:88)
H = Π2− Q2+ Q4 + V (φ )Q Q − h Q ,
2 r 2 r 4 r rr(cid:48) 0 r r(cid:48) r r
r rr(cid:48) r
where Q is a lattice distortion at site r associated with the strain modes and Π its conjugate momentum. V (φ )
r r rr(cid:48) 0
is a strain coupling with Fourier component V (φ ) given in Eq. (S7). The h ’s are local queched random distortions
k 0 r
thatarisefromchemicalsubstitutionofthealkalineearthelementattheAsiteoftheperovskitestructure. Wechoose
the h s to be independent random variables with a Gaussian distribution with zero mean and variance ∆2.
r
To study the statistical mechanics of the the model Hamiltonian (2) and handle compositional disorder, we use
the variational method developed in Ref. [S8]. This method includes leading correlations beyond the mean-field
approximation as well as disorder averaging at the level of a simple replica theory. In the absence of disorder, the
results are equivalent to those of the standard self-consistent phonon approximation. The method is described in
detail in Ref. [S8], however, we briefly described it here for the sake of completeness.
We consider a trial pair-probability distribution, ρtr = (Ztr)−1e−βHtr, where Htr is the Hamiltonian of coupled
harmonic oscillators in a random field, Htr = (cid:80) 1Π2 + 1(cid:80) Q D Q(cid:48) − (cid:80) h Q and Ztr = Tre−βHtr its
r 2 r 2 rr(cid:48) r r−r(cid:48) r r r r
normalization. TheFouriertransformofthefunctionD givesthefrequencysquaredofthemodeatwavevectork,
r−r(cid:48)
i.e., Ω2 =(cid:80) D eik·(r−r(cid:48)). Ω2 is a variational function and it is determined by minimization of the free energy of
k rr(cid:48) rr(cid:48) k
the lattice degrees of freedom, F =(cid:104)H(cid:105)+T(cid:104)k lnρtr(cid:105). Here, (cid:104)....(cid:105) denotes thermal and compositional averages.
lattice B
The free energy per site is therefore given by,
(cid:32) (cid:33) (cid:32) (cid:33)2
Flattice =−κ 1 (cid:88)η + 3γ 1 (cid:88)η + 1 (cid:88)V (φ )η (S8)
N 2 N k 4 N k 2N k 0 k
k k k
− 1 (cid:88)∆2 − 1 (cid:88)Ω coth(cid:18)βΩk(cid:19)+ kBT (cid:88)ln(cid:20)2sinh(cid:18)βΩk(cid:19)(cid:21),
N Ω2 4N k 2 N 2
k k k k
(cid:16) (cid:17)
whereη = 1 coth βΩk +∆2 aremeansquaredfluctuationsaveragedovercompositionaldisorderatwavevectork.
k 2Ωk 2 Ω4k
N isthenumberoflatticesitesandthesummationsrunoverthefirstBrillouinzoneofthesquarelattice. Minimization
of the free energy (S8) with respect to Ω gives the following result,
k
Ω2 =−κ+3γ 1 (cid:88)(cid:20) 1 coth(cid:18)βΩk(cid:48)(cid:19)+ ∆2(cid:21)+V . (S9)
k N 2Ω 2 Ω4 k
k(cid:48) k(cid:48) k(cid:48)
Equation (S9) determines the temperature and disorder dependence of the mode frequency Ω self-consistently.
k
III. MODEL PARAMETERS
Our model has six parameters (κ,γ,a ,a ,a , and F ), which are reduced to five as a (a ) is combined
D S T LowT T S
with κ for the manganites (nickelates). We begin by choosing a set of of physically reasonable parameters which give
phonon frequencies that are in order-of-magnitude agreement with the observed relevant modes.S9,S10 We then take
the resulting set of parameters and fine tune them to fit the observed dependence of T with the tolerance factor and
c
compositional disorder. The resulting values are given in Table S1.
TABLE S1. Model parameters obtained from fits to the experiments in manganites and nickelates.S11,S12 R is a rare earth
element such as La, Pr, Nd, and Sm; and M is an alkaline earth metal such as Ca, Sr, Ba.
κ [meV2] γ [meV3] a [meV2] a [meV2] a [meV2] F [meV]
D S T LowT
(R M )MnO 3.2×103 1.8×105 2.8×104 9.5×103 combined with κ 38
0.3 0.7 3
RNiO 7.1×103 6.02×105 4.3×104 combined with κ 2.1×104 78
3
5
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3
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3
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