review article renormalized phonon microstructures at high...

Review ArticleRenormalized Phonon Microstructures at HighTemperatures from First-Principles CalculationsMethodologies and Applications in Studying StrongAnharmonic Vibrations of Solids

Tian Lan12 and Zhaoyan Zhu2

1Department of Applied Physics and Materials Science California Institute of Technology Pasadena CA 91125 USA2Ginkgo LLC Incline Village NV 89451 USA

Correspondence should be addressed to Tian Lan tianlancaltechedu

Received 14 August 2016 Accepted 28 September 2016

Academic Editor Mohindar S Seehra

Copyright copy 2016 T Lan and Z Zhu This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

While the vibrational thermodynamics of materials with small anharmonicity at low temperatures has been understood well basedon the harmonic phonons approximation at high temperatures this understandingmust accommodate how phonons interact withother phonons or with other excitations To date the anharmonic lattice dynamics is poorly understood despite its great importanceand most studies still rely on the quasiharmonic approximations We shall see that the phonon-phonon interactions give rise tointeresting coupling problems and essentially modify the equilibrium and nonequilibrium properties of materials for examplethermal expansion thermodynamic stability heat capacity optical properties thermal transport and other nonlinear propertiesof materials The review aims to introduce some recent developements of computational methodologies that are able to efficientlymodel the strong phonon anharmonicity based on quantum perturbation theory of many-body interactions and first-principlesmolecular dynamics simulations The effective potential energy surface of renormalized phonons and structures of the phonon-phonon interaction channels can be derived from these interdependentmethods which provide bothmacroscopic andmicroscopicperspectives in analyzing the strong anharmonic phenomenawhile the traditional harmonicmodels fail dramaticallyThesemodelshave been successfully performed in the studies on the temperature-dependent broadenings of Raman and neutron scatteringspectra high temperature phase stability and negative thermal expansion of rutile and cuprite structures for example

1 Introduction

Today our understanding of the vibrational thermodynamicsof materials at low temperatures is developing nicely basedon the harmonic model in which phonons are independentAt high temperatures however this understanding mustaccommodate how phonons interact with other phonons(so called anharmonic phonon-phonon interactions) or withelectron or magnon excitationsThese anharmonic processesand thermal excitations induce the frequency shifts andlifetime broadenings of the interacting quasiparticles andcontribute to most thermal properties at high temperatures

The anharmonic phonon couplings and excitations aremostly as a result of the nonlinear terms in the atomic

potentials The anharmonicity can be much more prominentin some materials than others owing to their physical andchemical properties For example an open lattice structuretends to give rise to large anharmonicity in general becauseof the flexibility adapting to atomic vibrations especiallyfor some rotational modes Some particular responses ofelectronic band structures to the lattice variation may causestrong phonon anharmonicity too Typical examples mayinclude rigid-unit modes rattling ions in structural cagesorbital driven ferroelectric-like lattice instability charge dis-proportionation valence fluctuations and vibrational drivenhybridization [1ndash6] At low temperatures the harmonic partis dominant in most materials but examples of extremelyanharmonic potentials are also seen However even in the

Hindawi Publishing CorporationAdvances in Condensed Matter PhysicsVolume 2016 Article ID 2714592 11 pageshttpdxdoiorg10115520162714592

2 Advances in Condensed Matter Physics

former cases with elevated temperatures the contributionof anharmonicity is growing fast and can become sub-stantial owing to its nonlinear nature The anharmonicityrenormalizes the vibrational quanta and serves to break theassumptions of harmonicity and independence in phonondynamics and substantially changes the characteristics ofphonons

These topics are rich and of great importance for therational design and engineering of next-generation materialsin energy based applications For example the anharmonicdynamics and other types of thermal excitations are the originof most thermal energy transport processes and thereforegreatly influence the performance of these materials in appli-cations of harvesting storing and transporting energy Areliable estimate of the anharmonic entropy is also crucial forsynthesizing materials For example for metals and oxidesit seems that pure anharmonic contributions become largeenough to affect phase stability at temperatures above halfthe melting temperature which is the temperature rangewhere materials are often processed or used [7] Anomaly inthermal expansion is another prominent example Recentlythe large negative thermal expansion (NTE) of ScF3 andAg2O was found to have strong dependence with these hightemperature vibrational dynamical properties [8 9]

Modern inelastic scattering techniques with neutrons orphotons are ideal for sorting these properties out Analysisof the experimental data can generate vibrational spectra ofthe materials that is their phonon densities of states (DOS)and phonon or spin wave dispersions We are developingthe data reduction software to obtain the high quality datafrom inelastic neutron spectrometers [10ndash13] With accuratephonon DOS and dispersion curves we can obtain the vibra-tional entropies of different materials The understanding ofthe underlying reasons for differences in DOS curves andentropies then relies on the development of the fundamentaltheories and the computational methods

To date most ab initio methods for calculating materialsstructures and properties have been based on density func-tional (DFT) methods and evaluating the internal energy 119864of materials at a temperature of zero Kelvin For example aharmonic or quasiharmonic model usually used to accountfor the vibrational thermodynamics at low temperatures andit is commonplace today to calculate harmonic phononsby methods based on DFT [7 14 15] The quasiharmonicapproximation (QHA) is based on how phonon frequencieschangewith volume and all shifts of phonon frequencies fromtheir low temperature values are considered as a result ofthermal expansion alone [7]

In theQHA the vibrational free energy can beminimizedas a function of volume

119865 (119881 119879)= 1198640+ int+infinminusinfin119892 (120596) (ℏ1205962 + 119896B119879 ln (1 minus 119890

minusℏ120596119896B119879)) d120596(1)

where 1198640 is the energy calculated from the relaxed structureat 119879 = 0K Thermodynamic properties are therefore derivedfrom here [7 16]

Although the QHA accounts for some frequency shiftsthe phonon modes are still assumed to be harmonic nonin-teracting and their energies depend only on the volume ofthe crystal This can be adequate when the temperatures ofservice of the materials are low or when differences of chem-ical potentials are much larger than kT Therefore QHA hasbeen found to be able to predict thermodynamic propertieswell consistent with the experiment results especially at highpressure [17 18] However for most applications of materialsin energy involving even modest temperature this 119864 aloneis insufficient because the anharmonic vibrational dynamicsand different types of thermal excitations play importantroles and have significant thermodynamic effects at elevatedtemperatures

Phonon-phonon interactions are responsible for pureanharmonicity that shortens phonon lifetimes and shiftsphonon frequencies especially at high temperatures Anhar-monicity competes with quasiharmonicity to alter the stabil-ity of phases at high temperatures as has been shown forexample with experiments and frozen phonon calculationson bcc Zr [19] and the possible stabilization of bcc Fe-Nialloys at conditions of the earthrsquos core [20] For PbTe ScF3and rutile TiO2 there are recent reports of anharmonicitybeing so large that both the QHA and anharmonic pertur-bation theory fail dramatically [6 8 21 22]

These cases are suitable for ab initio molecular dynamics(AIMD) simulations which should be reliable when theelectrons are near their ground states and the nuclearmotionsare classical The big advantage of ab initio MD is thatit can account for all effects of harmonic anharmonicand even some of the electron-phonon interactions How-ever advanced postprocessing methodologies are requiedto extract concrete information from these simulationsIn the few examples where comparisons have been madewith ab initio MD agreement has been surprisingly goodeven for highly anharmonic materials [6 22 23] Today byvalidating these calculated results from inelastic scatteringexperiments with facilities such as the Spallation NeutronSource for neutrons we can obtain sufficient details aboutphonon-phonon interactions electron-phonon interactionsand other excitations at elevated temperatures

To understand the microscopic picture of these inter-actions models of the effective vibrational potential energysurface and the fine structures of decay channels of phonon-phonon interactions have been proposed [23ndash26] Based onthe quantum perturbation theory of many-body interactionsand first-principles molecular dynamics simulations thesemethods are used to renormalize quasiharmonic phononsand to identify the three-phonon and four-phonon kinemat-ics We can assess the strengths of phonon-phonon interac-tions of different anharmonic orders or via different decaychannelsThesemethods with high computational efficiencyare promising directions to advance our understandingsof nonharmonic lattice dynamics and thermal transportproperties

Advances in Condensed Matter Physics 3

In this review article we discuss several first-principlescomputational techniques available recently which proved tobe useful for assessing the anharmonic vibrational thermody-namics of solids In particular the computational details arediscussed followed by concrete examples that demonstratehow the applications of these interdependent methods canunveil interesting anharmonic properties of materials andtheir relationships with NTE vibrational energy shift andphase stability

2 Renormalized Phonon Spectra fromMolecular Dynamics Simulations

21 Molecular Dynamics Simulation and Fourier TransformedVelocity Autocorrelation Method Pure anharmonicity con-tributes to phonon-phonon interactions that shorten phononlifetimes and shift phonon frequencies The vibrationalenergy spectra of these renormalized phonons can be pro-duced by velocity trajectories extracted from the MD sim-ulation at each temperature It is based on nonequilibriumstatistics initiated by Green and Kubo [27 28] In essencethe FTVACmethod transforms the vibrational representationfrom the time and spatial domain to the correspondingenergy andor momentum domain [29ndash31] Because theFTVACmethod does not assume a form for theHamiltonianit is a robust tool for obtaining vibrational spectra of renor-malized phonons from MD simulations even with stronganharmonicity In the FTVAC model the phonon DOS isgiven by

119892 (120596) = sum119899119887

int 119890minusi120596119905 ⟨V119899119887 (119905) V00 (0)⟩ d119905 (2)

where ⟨ ⟩ is an ensemble average and V119899119887(119905) is the velocity ofthe atom 119887 in the unit cell 119899 at time 119905 Further projection of thephonon modes onto each 119896 point in the Brillouin zone wasperformed by computing the phonon power spectrum withthe FTVACmethod with a resolution determined by the sizeof the supercell in the simulation

119892 ( 120596) = int119889119905 119890minusi120596119905sum119899119887

119890isdot119899 ⟨V119899119887 (119905) V00 (0)⟩ (3)

where 119899 is the equilibrium position of the cell 119899 and is thephonon wavevector Equation (3) is both a time and spaceFourier transform and gives the frequency and lifetime ofeach phonon mode

22 Temperature-Dependent Effective Potential Method Ingeneral the cubic phonon anharmonicity contributes to boththe phonon energy shift and the lifetime broadening whereasthe quartic anharmonicity contributes only to the phononenergy shift [26 32] To distinguish the roles of cubic andquartic anharmonicity the Temperature-Dependent Effec-tive Potential (TDEP) method [6 24] is used In the TDEPmethod an effective Hamiltonian model is used to samplethe potential energy surface not at the equilibrium positions

of atoms but at the most probable positions for a giventemperature in an MD simulation [24]

119867 = 1198800 + 12sum119894

119898p2119894 + 12 sum119894119895120572120573

120601120572120573119894119895 119906120572119894 119906120573119895

+ 13 sum119894119895119896120572120573120574

120595120572120573120574119894119895119896119906120572119894 119906120573119895 119906120574119896

(4)

where 120601119894119895 and 120595119894119895119896 are second- and third-order force con-stants p is momentum and 119906120572119894 is the Cartesian component120572 of the displacement of atom 119894 In the fitting the ldquoeffectiverdquoharmonic force constants 120601119894119895 are renormalized by the quarticanharmonicity The cubic anharmonicity however is largelyaccounted for by the third-order force constants 120595119894119895119896 and canbe understood in terms of the third-order phonon self-energythat causes linewidth broadening [32]

The above Hamiltonian was used to obtain the renormal-ized phonon dispersions (TDEP spectra) accounting for boththe anharmonic shifts Δ and broadenings Γ of the mode 119895These are derived from the real and imaginary parts of thecubic self-energies Σ(3) respectively [32]Δ (119895 Ω) = minus18ℏ2sdot sum11990211198951

sum11990221198952

1003816100381610038161003816119881 (119895 11198951 21198952)10038161003816100381610038162 Δ (1 + 2 minus )

sdot weierp [ 1198991 + 1198992 + 1Ω + 1205961 + 1205962 minus1198991 + 1198992 + 1Ω minus 1205961 minus 1205962 +

1198991 minus 1198992Ω minus 1205961 + 1205962

minus 1198991 minus 1198992Ω + 1205961 minus 1205962 ]

Γ (119895 Ω) = 18120587ℏ2sdot sum11990211198951

sum11990221198952

1003816100381610038161003816119881 (119895 11198951 21198952)10038161003816100381610038162 Δ (1 + 2 minus )

sdot [(1198991 + 1198992 + 1) 120575 (Ω minus 1205961 minus 1205962)+ 2 (1198991 minus 1198992) 120575 (Ω + 1205961 minus 1205962)]

(5)

where Ω is the renormalized phonon frequency and weierpdenotes the Cauchy principal part The 119881(sdot)rsquos are elementsof the Fourier transformed third-order force constants 120595119894119895119896obtained in the TDEP method Δ(1 + 2 minus ) ensuresconservation of momentum

23 Example The Quartic Phonons and Their Stabilizationof Rutile Phase of TiO2 at High Temperatures Althoughthe rutile structure of TiO2 is known to be stable athigh temperatures the QHA predicts that several acousticphonons decrease anomalously to zero frequency with ther-mal expansion incorrectly predicting a structural collapse attemperatures well below 1000K [33 34]

Inelastic neutron scattering was used to measure thetemperature dependence of the phonon density of states


2

Phon

on D

OS

(au

)

1

1373K

1073K

673K

300K

minus5 0 5 10 15 20 25minus10Energy (meV)

Figure 1 Neutron weighted phonon DOS of rutile TiO2 fromexperimental measurements at temperatures from 300 to 1373Kwith an incident energy of 30meV (black) The dashed spectrumcorresponds to the experimental result at 300K shifted verticallyfor comparison at each temperature Simulation results of peak 1 ofphonon DOS from the QHA at 300K and 1373K are shown in greenand compared with experimental spectra

(DOS) of rutile TiO2 from 300 to 1373K Surprisinglyas shown in Figure 1 these anomalous acoustic phononscentered at 14meV (peak 1 in the Figure) were found toincrease in frequencywith temperature instead [6]The greencurves in Figure 1 present the corresponding peak 1 of DOSspectra calculated with the quasiharmonic model at 300Kand 1373K It shows that QHA calculations predict that thetranslational acoustic (TA) branch softens dramatically tozero frequency with the thermal expansion of 1373K givingimaginary frequencies in the DOS that would destabilize therutile structure at high temperatures

Our QHA calculation is consistent with previous com-putational results [33 34] but in obvious contrary to theexperimetal observation The dramatic failure of QHA sug-gests the existence of strong anharmonicity that could intrin-sically alter the harmonic characteristics of these acousticphonons subject to the phonon renormalization It turnsout that the FTVAC and TDEP methods based on the first-principles MD simulations are reliable for this investigation

For MD first-principles calculations using the localdensity approximation (LDA) of density functional theory(DFT) were performedwith the VASP package [35 36] First-principles Born-OppenheimerAIMD simulations for a 2times2times4 supercell and a 2 times 2 times 1 119896 point sampling were performedto thermally excite phonons to the target temperatures of300 and 1373K For each temperature the system was firstequilibrated for 3 ps as an NVT ensemble with temperaturecontrol by a Nose thermostat and then simulated as an NVEensemble for 20 ps with time steps of 1 fs Fine relaxationswith residual pressures below 05GPa were achieved in each

calculation that accounted for thermal expansion With thesimulated atomic trajectories derived from MD simulationsthe FTVAC and TDEP methods were able to reproduce therenormalized phonon dispersions and effective vibrationalpotential surface as detailed in Sections 21 and 22

Figure 2 shows the vibrational energies of the TA branchcalculated by the FTVAC method with AIMD trajectoriesFrom 300 to 1373K the TA branch increases in energy byan average of about 21meV For this TA branch Figure 2(b)shows an enormous discrepancy of phonon energies betweenthe FTVAC calculation and the QHA (orange dashed line)at 1373K Apparently the unstable phonon modes predictedby the QHA are fully stable in the AIMD simulations at hightemperatures

Using the same MD trajectories as for the FTVACmethod the calculated TDEP dispersions agree well withthe FTVAC results as shown in Figures 2(a) and 2(b) At1373K the TAmodes below 20meVhave only small linewidthbroadenings suggesting the small cubic anharmonicity Fur-thermore they are close in energy to those calculated if all120595119894119895119896 are set to zero in (4) showing the dominance of quarticanharmonicity and the small cubic anharmonicity of the TAmodes

For more details about the anomalous anharmonicity ofthe TA modes the frozen phonon method was adopted tocalculate the potential energy surfaces for specific phononsas a typical example presented in Figure 2(c) It should beemphasized that the frozen phonon method is still basedupon the harmonic theory since this method isolates aparticular phononmode and tries to explore the anharmonicpotential landscape defined by its own harmonic eigenvec-tors The FTVAC and TDEP methods however are able tofully account for the couplings of all phonon modes andtherefore reproduce the potential energy surface renormal-ized by the interactions of all other phonons Neverthelessthe frozen phonon method is a convenient way to evaluatethe strength of anharmonicity although the potential energysurface which is derived from the frozen phonon method isnot the real anharmonic potential in general and cannot beused to investigate the phonon-phonon interactions

We shall see that at 300K the frozen phonon potentialenergy of the TAmode at the119877 point is nearly quadratic witha small quartic part With the lattice expansion characteristicof 1373K the potential energy curve transforms to beingnearly quartic In fact for all modes in the TA branchthat were evaluated by the frozen phonon method thepotential energy surface develops a quartic form with latticeexpansion For a quantum quartic oscillator the vibrationalfrequency stiffens with temperature owing to the increasingspread between the energy levels [8 37] We assessed a hightemperature behavior by assigning Boltzmann factors to thedifferent oscillator levels derived from frozen phonon poten-tials giving the energies of the quartic TAmodes at 1373K Asshown in Figure 2(b) they are reasonably close to the FTVACand TDEP resultsThe similarity of potential surfaces derivedfrom the TDEP method and the frozen phonon methodsuggests that the isolated frozen phonon potential does notreshape itself much in response to the renormalization as aresult of the interactions of other phonons This is unusual


AMZRX Z0

5

10

15

20

25

Ener

gy (m

eV)

(a)AMZRX Z

minus10

minus5

0

5

10

15

20

25

Ener

gy (m

eV)

(b)

minus02 minus01 00 01 02 03minus03Displacement (Aring)

0

20

40

60

80

100

Mod

e ene

rgy

(meV

)

(c)

Figure 2Diffuse curves areTDEPphonondispersions of rutile TiO2 below25meVat (a) 300Kand (b) 1373 K comparedwith the results fromthe FTVAC method (red circles) The white curves are phonon dispersions for the quasiharmonicity plus quartic anharmonicity calculatedwith all 120595119894119895119896 set to zero in (4) In (b) the dispersions are compared to the quasiharmonic dispersions (orange dashed curve) and the singlequartic oscillatormodel (orange triangles) (c) Frozen phonon potential (black) of TAmode at R point with 119902 = (05 0 05) at 1373K showingthe harmonic component (red) and quartic component (blue)The low temperature potential surface is also shown (dashed black) Reprintedfigure with permission from Lan et al [6] Copyright by the American Physical Society

and indicates a global phonon quarticity of acoustic branchin rutile

Based on FTVAC and TDEP further investigation revealsthat the positive and negative displacements of these acousticmodes show a significant accumulation of charge in the TindashObond of shorter distance and a depletion in the bond of longerdistance For rutile TiO2 the hybridization follows the atomdisplacements in thermal phonons instead of a displacivephase transition and this ldquophonon-tracked hybridizationrdquochanges with lattice parameter [6] With lattice expansion

the short-range repulsion is weaker and hybridization favorselectrons between the shorter TindashO pairs in the phonondisplacement pattern The hybridization in the TindashO bondis very sensitive to interatomic distance much as has beennoticed in the ferroelectric distortion of BaTiO3 [38] It notonly provides a source of extreme phonon anharmonicitybut also provides thermodynamic stability for rutile TiO2 Itmay occur in other transitionmetal oxides that show unusualchanges of properties with lattice parameter orwith structureBesides altering thermodynamic phase stability properties


such as ferroelectricity and thermal transport will be affecteddirectly

3 Fine Structures of Phonon-PhononInteraction Channels

31 Computational Methodology Anharmonicity tensorsdescribe the coupling strengths for phonon-phonon interac-tions but a prerequisite is that the phonons in these processessatisfy the kinematical conditions of conservation of energyand momentum as presented in (5) In the phonon-phononinteraction functional an anharmonicity tensor element foran 119904-phonon process can be expressed as [39]

119881 (119895 11198951 119904minus1119895119904minus1) = 12119904 (

ℏ2119873)1199042

sdot 119873Δ (1 + sdot sdot sdot + 119904minus1) [12059611989501205961 sdot sdot sdot 120596119904minus1]12

sdot 119862 (119895 11198951 119904minus1119895119904minus1)

(6)

whereΔ(1+sdot sdot sdot+119904minus1) enforcesmomentum conservation andthe 119862(sdot)rsquos elements of the 119904-phonon anharmonic tensor areexpected to be slowly varying functions of their arguments

If the anharmonicity tensor or its average does not varysignificantly for different phonon processes the couplingfactor and the kinematic factor are approximately separablein (6) The separation of the anharmonic coupling and thekinematics has been used with success in many studiesincluding recent reports on rutile TiO2 and SnO2 [25 26]Weconsider the term 119862(119895 11198951 119904minus1119895119904minus1) to be a constant ofthe Ramanmode 119895 and use it as a fitting parameter Although119862(119895 11198951 21198952) and 119862(119895 119895 11198951 minus11198951) change with 1198951 and 1198952an average over modes ⟨119862(sdot)⟩ = sum12 119862(119895 11198951 21198952)sum12 1is needed by the fitting where 1 and 2 under the summationsymbol represent 119894119895119894 We define the cubic and quartic fittingparameters as

119862(3)119895 = ⟨119862 (119895 11198951 21198952)⟩ (7a)

119862(4)119895 = ⟨119862 (119895 119895 11198951 minus11198951)⟩ (7b)

To the leading order of cubic and quartic anharmonicitythe broadening of the Raman peaks is 2Γ(3)(119895 Ω) Thefrequency shift of the Raman peaks is Δ119876 + Δ(3) + Δ(31015840) +Δ(4) where the quasiharmonic part is denoted by Δ119876These quantities can be written as functions of 119863(Ω 1205961 1205962)and 119875(Ω 1205961 1205962) weighted by average anharmonic couplingstrengths [25 26 40]

Γ(3) (119895 Ω) = 120587ℏ64 120596119895010038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 sum11990211198951

sum11990221198952

12059611205962119863(Ω 1205961 1205962)

= 1205961198950 10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162119863120596 (Ω)

(8a)

Δ(3) (119895 Ω)= minus ℏ641205961198950

10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 sum11990211198951

sum11990221198952

12059611205962119875 (Ω 1205961 1205962)

= 1205961198950 10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 119875120596 (Ω)

(8b)

Δ(31015840) (119895) = minus ℏ161198731205961198950

10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 sum11990221198952

1205961198952 (2) (11989921198952 + 12) (8c)

Δ(4) (119895) = ℏ81198731205961198950119862

(4)119895 sum11990211198951

1205961198951 (1) (11989911198951 + 12) (8d)

where 119863120596(Ω) and 119875120596(Ω) are functionals of 119863(Ω 1205961 1205962)and 119875(Ω 1205961 1205962) weighted by the kinematics of anharmonicphonon coupling119863120596(Ω) is the so called two-phonon densityof states (TDOS) spectra which characterize the size of thephonon-phonon interaction channels Δ(31015840) is an additionallow-order cubic term that corresponds to instantaneousthree-phonon processes [32] It is nonzero for crystals havingatoms without inversion symmetry as in the case for theoxygen atom motions in the 1198601119892 mode of rutile It ismuch smaller than other contributions however owing tosymmetry restrictions

32 Example 1 Anomalous Temperature-Dependent Broad-ening of B2119892 Mode of Rutile SnO2 Recently strong anhar-monicity of rutile SnO2was discovered by the high-resolutionRaman spectrometer and the anomaly of the temperature-dependent broadening of 1198612119892 mode is a prominent example[26] As shown in Figure 3(b) at high temperatures above500K the broadening of the 1198612119892 mode of rutile SnO2 showsan unusual concave downward curvature while the other twoRaman modes 119864119892 and 1198601119892 broaden linearly Moreoever atlow temperatures the 1198612119892 mode has a much larger linewidththan the other two modes The linewidth of the 1198612119892 modeextrapolated to 0K is approximately 8 cmminus1 whereas thelinewidths of the 119864119892 and 1198601119892 modes extrapolate to less than2 cmminus1

The TDOS function 119863120596(Ω) in Figure 4 shows largevariations with Ω which explains a trend in the thermalbroadening of Figure 3(b) Owing to the high frequency ofthe 1198612119892 mode at the temperatures of this study its phonon-phonon anharmonicity comes mostly from downconversionprocesses as shown in Figure 4 Ignoring the small upconver-sion contribution at high temperatures

119863120596 (Ω 119879) prop 119879 sum11990211198951

sum11990221198952

120575 (Ω minus 1205961 minus 1205962) equiv 1198791198630darr (Ω) (9)

where1198630darr(Ω) is the number of two-phonon downconversionchannelsTherefore the linewidth broadening resulting fromthe cubic anharmonicity is proportional to the temperaturemodulated by 119862(Ω) Because Ω undergoes a shift withtemperature 1198630darr(Ω) is an implicit function of 119879 Usuallythe line broadening is linear in 119879 because 1198630darr(Ω) does notvary much with temperature as in the cases of modes 119864119892and 1198601119892 However the 1198612119892 mode at 774 cmminus1 lies on a steep


150 900750600450300minus25

minus20

minus15

minus10

minus5

0

Temperature (K)

Δ120596

(cmminus1 )

Eg

A1g

B2g

(a)

150 900750600450300Temperature (K)

FWH

M (c

mminus1

)

Eg

A1g

B2g

0

5

10

15

20

25

30

35

(b)

Figure 3 Temperature dependence of (a) frequency shifts and (b) linewidths as FWHMs of the Raman modes 119864119892 1198601119892 and 1198612119892 of rutileSnO2 The solid and open symbols represent the experimental data from powder and single crystal samples respectively Solid curves are thetheoretical fittings with a full calculation of the kinematics of three-phonon and four-phonon processes The dashed curve was calculatedwithout considering the frequency dependence of1198630darr(Ω) and the number of decay channels at elevated temperatures Reprinted figure withpermission from Lan et al [26] Copyright by the American Physical Society

Eg

A1g

B2g

800K

0K

400 800 1200 16000Frequency (cmminus1)

minus1

0

1

2

(au

)

Figure 4 Two-phonon density of states 119863120596(Ω) of rutile SnO2 for0 K and 800K The arrowheads mark the positions of the threeRaman modes 119864119892 1198601119892 and 1198612119892 respectively The upconversionand downconversion contributions at 800K are shown in green andblack dashed curves respectively There is no upconversion processat 0 KThe phononDOS spectra are presented along with the TDOSbut take a negative sign and shaded blue to make the view of TDOSand DOS separated

gradient of 119863120596(Ω 119879) in Figure 4 On the other hand the1198612119892 mode undergoes a significant shift of frequency by morethan 25 cmminus1with temperature as shown in Figure 3(a) whichmoves the downconversion channels down the gradient of

119863120596(Ω 119879) These cooperative effects as a result narrow thecapacity of these decay channels Therefore its broadeningis significantly less than linear in 119879 exhibiting an unusualconcave downwards shape For comparison the dashedline in Figure 3(b) was calculated without considering thefrequency dependence of 1198630darr(Ω) at elevated temperaturesand it deviates substantially from the experimental trend

As discssed the unusual temperature dependence of thelinewidth of the 1198612119892 mode originates from the sharp peakin 119863120596(Ω 119879) centered at 800 cmminus1 We shall see that thisfeature in the TDOS originates with the phononDOSof SnO2which has a wide band gap between 360 cmminus1 and 450 cmminus1associated with the mass difference of Sn and O atoms asshown in Figure 4The shape of the TDOS can be understoodas the convolution of the phonon DOS with itself With twoapproximately equal regions above and below the gap theresult is a peak at 800 cmminus1 with steep slopes on both sidesFor comparison although the TDOS of rutile TiO2 is shapedas a broad peak [25] it does not have the sharp features ofFigure 4 because themass difference between Ti andO atomsdoes not cause a band gap in the phonon DOS of TiO2

The fine structures of phonon-phonon interaction chan-nels can also explain the physics of large linewidth of 1198612119892mode at low temperature limit following the same approachas detailed above In the low temperature limit upconversionprocesses are prohibited because 119899 rarr 0 The peak linewidthextrapolated to 0K is determined entirely by downconversionprocesses quantified by the downconversion TDOSThe 1198612119892mode has a significant broadening because its frequencyis near a peak in the downconversion TDOS as seen in


Figure 4 On the other hand the 119864119892 and 1198601119892 modes arenot broadened at low temperatures because their frequenciesare at low values of the TDOS The study on the structuresof phonon-phonon interaction channels successfully marksthe ldquohotrdquo downconversion decay activities even at low tem-perature limit and accounts for the significant difference oflinewidths between 1198612119892 and the other two Raman modes atlow temperatures

33 Example 2 NegativeThermal Expansion of Cuprite Ag2OSilver oxide (Ag2O) with the cuprite structure has attractedmuch interest after the discovery of its extraordinarily largenegative thermal expansion (NTE) [41 42] which exceedsminus1 times 10minus5 Kminus1 and occurs over a wide range of temperaturefrom 40K to its decomposition temperature near 500K

A rigid-unit modes (RUMs) model of NTE considerstetrahedra of Ag4O around each O atom that bend at theAg atoms linking the O atoms in adjacent tetrahedra RUMsaccount for counteracting rotations of all such tetrahedra[1 2] Locally the OndashAg bond length does not contract butbending of the OndashAgndashO links pulls the O atoms togetherleading to NTE These RUMs tend to have low frequenciesowing to the large mass of the unit and hence are excitedat low temperatures This model correlated the NTE withquasiharmonic approximation and should explain the mainphysics at low temperatures However as shown in Figure 5at temperatures above 250K there is a second part of theNTE behavior of cuprite Ag2O that is apparently beyond thepredictions of quasiharmonic theory [9]

First-principles Born-Oppenheimer molecular dynamicssimulations were performed for a 3 times 3 times 3 supercell withtemperature control by a Nose thermostat The simulatedtemperatures included 40 100 200 300 and 400K For eachtemperature the system was first equilibrated for 3 ps andthen simulated for 18 ps with a time step of 3 fs The systemwas fully relaxed at each temperature with convergence ofthe pressure within 1 kbar

As shown in Figure 5 the MD simulation predictsthe NTE very accurately The temperature dependence ofthis NTE behavior follows the Planck occupancy factorfor phonon modes above 50meV corresponding to the O-dominated band of optical frequencies In the QHA thesemodes above 50meV do not contribute to the NTE Thesemodes are highly anharmonic as shown by their largebroadenings and shifts

Simlar to rutile SnO2 because of the largemass differencebetween Ag and O atoms the O-dominated phonon modesare well separated from the Ag-dominated modes Partialphonon DOS analysis showed that the Ag-dominated modeshad similar energies forming the peak of the phonon DOSbelow 20meV (peak 1) whereas the O-dominatedmodes hadenergies above 40meV (peak 2 and 3) as shown in the DOSspectrum in Figure 6

For cubic anharmonicity as discussed the two-phononDOS (TDOS) is the spectral quantity parameterizing thenumber of phonon-phonon interaction channels available toa phonon For Ag2O with the cuprite structure the peaks in

Exp dataQH calcMD calc

a(T)a(T 0)minus

1

minus00030

minus00025

minus00020

minus00015

minus00010

minus00005

00000

150 300 4500Temperature (K)

Figure 5 Temperature dependence of lattice parameter of Ag2Ofrom experimental data in [41] quasiharmonic calculations andMD calculations expressed as the relative changes with respect totheir 40K values that is 119886(119879)119886(40K) minus 1

1 2 3

AgrarrAg minus Ag

AgrarrAg + Ag

OrarrO minus Ag

OrarrO + Ag

20 40 60 80 100 1200Frequency (meV)

minus05

00

05

10

15

(au

)

Figure 6 The TDOS spectra of Ag2O 119863(120596) at 40K (dashed) and400K (solid)The downconversion and upconversion contributionsare presented separately as black and green curves respectively Thephonon DOS spectra are presented along with the TDOS but take anegative sign and shaded blue to make the view of TDOS and DOSseparated

the TDOS overlap well with the peaks in the phonon DOSas illustrated in Figure 6 The phonon DOS is analogousto a reservior used as the supply of phonons while TDOSindicates the gates that can direct them We can thereforeunderstand that most of the phonons have many possibleinteractions with other phonons which contributes to the


large anharmonicity of Ag2O with the cuprite structure andshort lifetimes (or large linewidths) Although the lifetimes ofmost phonon modes in Ag2O with the cuprite structure aresmall and similar the origins of these lifetime broadeningsare intrinsically different For peak 2 of the phonon DOS theanharmonicity is largely from the upconversion processes O997891rarr O minus Ag while for peak 3 it is from the downconversionprocesses O 997891rarr O + Ag The anharmonicity of peak 1is more complicated It involves both upconversion anddownconversion processes of Ag-dominated modes

Owing to explicit anharmonicity from phonon-phononinteractions the thermodynamic properties of Ag2Owith thecuprite structure cannot be understood as a sum of contri-butions from independent normal modes The frequency ofan anharmonic phonon depends on the level of excitationof other modes At high temperatures large vibrationalamplitudes increase the anharmonic coupling of modesand this increases the correlations between the motions ofthe Ag and O atoms as shown by perturbation theoryCouplings in perturbation theory have phase coherenceso the coupling between Ag- and O-dominated modes athigher energies as seen in the peak of the TDOS causescorrelations between the motions of Ag and O atoms Theab initio MD simulations show that anharmonic interactionsallow the structure to become more compact with increasingvibrational amplitude The mutual motions of the O andAg atoms cause higher density as the atoms fill spacemore coherently as a result of anharmonic interactions Therelationship between macroscopic thermal expansion andmicroscopic anharmonic coupling of different modes requiesmore quantitative investigation on for example a particulardecay channel that contributes most to such effect It shouldbe noted that the large difference in atomic radii of Ag andO may substantially contribute to this effect in collaborationwith the coherent anharmonic interaction For cuprite Cu2Owhich has less of a difference in atomic radii the thermalexpansion is much less anomalous although the phononanharmonicity is found to be equally large

4 Conclusions

Today our understanding about the vibrational thermo-dynamics of materials at low temperatures is broad anddeep because it is based on the harmonic model in whichphonons are independent avoiding issues of anharmonic lat-tice dynamics and phonon-phonon interactions However inmost cases the failure of the harmonic theory also arises fromthe assumption of independent phonons which becomesincreasingly inaccurate at high temperatures We have seenthat the anharmonic renormalization of the vibrationalquanta could break the independent and harmonic phononassumptions and substantially change their characteristics ina coherent way

The complexity of phonon anharmonicity arises becausewe need to consider how phonons interact with otherphonons or with other excitations which is an exampleof notorious many-body interaction problem To this end

progress of anharmonic phonon theories and advanced com-putational methodologies discussed in this review provideboth the macroscopic and the microscopic perspectives ofphonon anharmonicty and its relationship with thermo-dynamic properties such as high temperature vibrationalenergy distribution thermal stability and anomalous thermalexpansion

Based upon quantum perturbation theory and first-principles molecular dynamics simulations microstructuresof renormalized phonons can be established In particular theeffective potential energy surface as well as the anharmonicenergy spectra of renormalized phonons can be built up byanalyzing the atomic vibration trajectories derived fromMDIncorporated with frozen phonon method we have seen thatthese methods give rise to the anharmonic potential land-scape fully accounting for the phonon-phonon interactionsThe evolution of the potential surface with temperature aswell as behaviors of different orders of anharmoncity can bederived in detail as well On the other hand the fine structureof the phonon-phonon interaction channels bymeans of two-phonon density of states provides unique insight into themicroscopic picture of pathways of phonon renormalizationprocessesWith phonon density of states these fine structuresgive crucial information about how phonons interact witheach other and the distribution of three-phonon and four-phonon kinematics in these interactions

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The study is in close collaboration with Dr B Fultz Dr OHellman and Dr CW Li The work benefited from softwaredeveloped in the DANSE project under NSF award Researchat the SNS at the Oak Ridge National Laboratory wassponsored by the Scientific User Facilities Division DOE

References

[1] A K A Pryde K D Hammonds M T Dove V Heine J DGale andM CWarren ldquoOrigin of the negative thermal expan-sion in ZrW2O8 and ZrV2O7rdquo Journal of Physics CondensedMatter vol 8 no 50 pp 10973ndash10982 1996

[2] V Heine P R L Welche and M T Dove ldquoGeometricalorigin and theory of negative thermal expansion in frameworkstructuresrdquo Journal of the American Ceramic Society vol 82 no7 pp 1793ndash1802 1999

[3] T Hotta and A Shudo ldquoChaos in Jahn-Teller rattlingrdquo Journalof the Physical Society of Japan vol 83 no 8 Article ID 0837052014

[4] CW Li J Hong A FMay et al ldquoOrbitally driven giant phononanharmonicity in SnSerdquoNature Physics vol 11 no 12 pp 1063ndash1069 2015

[5] R Yevych V Haborets M Medulych et al ldquoValence fluc-tuations in Sn(Pb)2P2S6 ferroelectricsrdquo httpsarxivorgabs160502367


[6] T Lan C W Li O Hellman et al ldquoPhonon quarticity inducedby changes in phonon-tracked hybridization during latticeexpansion and its stabilization of rutile TiO2rdquo Physical ReviewB vol 92 no 5 Article ID 054304 2015

[7] B Fultz ldquoVibrational thermodynamics of materialsrdquo Progress inMaterials Science vol 55 no 4 pp 247ndash352 2010

[8] C W Li X Tang J A Munoz et al ldquoStructural relationshipbetween negative thermal expansion and quartic anharmonic-ity of cubic ScF3rdquo Physical Review Letters vol 107 no 19 ArticleID 195504 2011

[9] T Lan C W Li J L Niedziela et al ldquoAnharmonic latticedynamics of Ag2O studied by inelastic neutron scatteringand first-principles molecular dynamics simulationsrdquo PhysicalReview B vol 89 no 5 Article ID 054306 2014

[10] D L Abernathy M B Stone M J Loguillo et al ldquoDesignand operation of the wide angular-range chopper spectrometerARCS at the Spallation Neutron Sourcerdquo Review of ScientificInstruments vol 83 no 1 Article ID 015114 2012

[11] B Fultz T Kelley J Lin et al Experimental Inelastic NeutronScattering Introduction to DANSE 2009 httpdocsdanseus

[12] M G Kresch Temperature dependence of phonons in elementalcubic metals studied by inelastic scattering of neutrons and xrays[PhD thesis] California Institute of Technology 2009

[13] M Kresch M Lucas O Delaire J Y Y Lin and B FultzldquoPhonons in aluminum at high temperatures studied by inelas-tic neutron scatteringrdquo Physical Review B vol 77 no 2 ArticleID 024301 2008

[14] A Togo F Oba and I Tanaka ldquoFirst-principles calculations ofthe ferroelastic transition between rutile-type and CaCl2-typeSiO2 at high pressuresrdquo Physical Review B vol 78 no 13 ArticleID 134106 2008

[15] S Baroni S de Gironcoli A Dal Corso and P Gian-nozzi ldquoPhonons and related crystal properties from density-functional perturbation theoryrdquo Reviews of Modern Physics vol73 no 2 pp 515ndash562 2001

[16] C W Li H L Smith T Lan et al ldquoPhonon anharmonicity ofmonoclinic zirconia and yttrium-stabilized zirconiardquo PhysicalReview B vol 91 no 14 Article ID 144302 2015

[17] T Tsuchiya J Tsuchiya K Umemoto and R M WentzcovitchldquoPhase transition in MgSiO3 perovskite in the earthrsquos lowermantlerdquo Earth and Planetary Science Letters vol 224 no 3-4pp 241ndash248 2004

[18] A Chopelas ldquoThermal expansivity of lower mantle phasesMgO and MgSiO3 perovskite at high pressure derived fromvibrational spectroscopyrdquo Physics of the Earth and PlanetaryInteriors vol 98 no 1-2 pp 3ndash15 1996

[19] Y Ye Y Chen K Ho B N Harmon and P Lindgrd ldquoPhonon-phonon coupling and the stability of the high-temperature bccphase of Zrrdquo Physical Review Letters vol 58 no 17 pp 1769ndash1772 1987

[20] L Dubrovinsky N Dubrovinskaia O Narygina et al ldquoBody-centered cubic iron-nickel alloy in earthrsquos corerdquo Science vol 316no 5833 pp 1880ndash1883 2007

[21] O Delaire J Ma K Marty et al ldquoGiant anharmonic phononscattering in PbTerdquo Nature Materials vol 10 no 8 pp 614ndash6192011

[22] C Li O Hellman J Ma et al ldquoPhonon self-energy and originof anomalous neutron scattering spectra in SnTe and PbTethermoelectricsrdquo Physical Review Letters vol 112 no 17 ArticleID 175501 2014

[23] O Hellman I A Abrikosov and S I Simak ldquoLattice dynamicsof anharmonic solids from first principlesrdquo Physical Review Bvol 84 no 18 Article ID 180301 2011

[24] O Hellman and I A Abrikosov ldquoTemperature-dependenteffective third-order interatomic force constants from firstprinciplesrdquo Physical Review B vol 88 no 14 Article ID 1443012013

[25] T Lan X Tang and B Fultz ldquoPhonon anharmonicity of rutileTiO2 studied by Raman spectrometry and molecular dynamicssimulationsrdquoPhysical Review B vol 85 no 9 Article ID 09430511 pages 2012

[26] T Lan C W Li and B Fultz ldquoPhonon anharmonicity ofrutile SnO2 studied by Raman spectrometry and first principlescalculations of the kinematics of phonon-phonon interactionsrdquoPhysical Review B vol 86 no 13 Article ID 134302 2012

[27] M S Green ldquoMarkoff random processes and the statisticalmechanics of timeminusdependent phenomena II Irreversible pro-cesses in fluidsrdquo The Journal of Chemical Physics vol 22 no 3p 398 1954

[28] R Kubo ldquoStatistical-mechanical theory of irreversible pro-cesses I General theory and simple applications to magneticand conduction problemsrdquo Journal of the Physical Society ofJapan vol 12 pp 570ndash586 1957

[29] N de Koker ldquoThermal conductivity of MgO Periclase fromequilibrium first principles molecular dynamicsrdquo PhysicalReview Letters vol 103 no 12 Article ID 125902 4 pages 2009

[30] J A Thomas J E Turney R M Iutzi C H Amon and A JH McGaughey ldquoPredicting phonon dispersion relations andlifetimes from the spectral energy densityrdquo Physical Review Bvol 81 no 8 Article ID 081411 2010

[31] T Lan Studies of phonon anharmonicity in solids [PhD thesis]California Institute of Technology 2014

[32] A A Maradudin and A E Fein ldquoScattering of neutrons by ananharmonic crystalrdquo Physical Review vol 128 no 6 pp 2589ndash2608 1962

[33] P D Mitev K Hermansson B Montanari and K Refson ldquoSoftmodes in strained and unstrained rutile TiO2rdquo Physical ReviewB vol 81 no 13 Article ID 134303 2010

[34] K Refson BMontanari P DMitev K Hermansson andNMHarrison ldquoComment on First-principles study of the influenceof (110)-oriented strain on the ferroelectric properties of rutileTiO2rdquo Physical Review B vol 88 no 13 Article ID 136101 2013

[35] G Kresse and D Joubert ldquoFrom ultrasoft pseudopotentials tothe projector augmented-wave methodrdquo Physical Review B vol59 no 3 pp 1758ndash1775 1999

[36] G Kresse and J Furthmuller ldquoEfficient iterative schemes forab initio total-energy calculations using a plane-wave basis setrdquoPhysical Review B vol 54 no 16 Article ID 11169 1996

[37] P Dorey and R Tateo ldquoAnharmonic oscillators the thermody-namic Bethe ansatz and nonlinear integral equationsrdquo Journal ofPhysics A Mathematical and General vol 32 no 38 pp L419ndashL425 1999

[38] R E Cohen ldquoOrigin of ferroelectricity in perovskite oxidesrdquoNature vol 358 no 6382 pp 136ndash138 1992

[39] I P Ipatova A A Maradudin and R F Wallis ldquoTemperaturedependence of the width of the fundamental lattice-vibrationabsorption peak in ionic crystals II Approximate numericalresultsrdquo Physical Review vol 155 no 3 pp 882ndash895 1967

[40] J Suda and T Sato ldquoTemperature dependence of the linewidthof the first-order raman spectra for CaWO4 crystalrdquo Journal ofthe Physical Society of Japan vol 66 no 6 pp 1707ndash1713 1997


[41] W Tiano M Dapiaggi and G Artioli ldquoThermal expansion incuprite-type structures from 10K to decomposition tempera-ture Cu2O and Ag2Ordquo Journal of Applied Crystallography vol36 no 6 pp 1461ndash1463 2003

[42] B J Kennedy Y Kubota and K Kato ldquoNegative thermalexpansion and phase transition behaviour in Ag2Ordquo Solid StateCommunications vol 136 no 3 pp 177ndash180 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


former cases with elevated temperatures the contributionof anharmonicity is growing fast and can become sub-stantial owing to its nonlinear nature The anharmonicityrenormalizes the vibrational quanta and serves to break theassumptions of harmonicity and independence in phonondynamics and substantially changes the characteristics ofphonons

These topics are rich and of great importance for therational design and engineering of next-generation materialsin energy based applications For example the anharmonicdynamics and other types of thermal excitations are the originof most thermal energy transport processes and thereforegreatly influence the performance of these materials in appli-cations of harvesting storing and transporting energy Areliable estimate of the anharmonic entropy is also crucial forsynthesizing materials For example for metals and oxidesit seems that pure anharmonic contributions become largeenough to affect phase stability at temperatures above halfthe melting temperature which is the temperature rangewhere materials are often processed or used [7] Anomaly inthermal expansion is another prominent example Recentlythe large negative thermal expansion (NTE) of ScF3 andAg2O was found to have strong dependence with these hightemperature vibrational dynamical properties [8 9]

Modern inelastic scattering techniques with neutrons orphotons are ideal for sorting these properties out Analysisof the experimental data can generate vibrational spectra ofthe materials that is their phonon densities of states (DOS)and phonon or spin wave dispersions We are developingthe data reduction software to obtain the high quality datafrom inelastic neutron spectrometers [10ndash13] With accuratephonon DOS and dispersion curves we can obtain the vibra-tional entropies of different materials The understanding ofthe underlying reasons for differences in DOS curves andentropies then relies on the development of the fundamentaltheories and the computational methods

To date most ab initio methods for calculating materialsstructures and properties have been based on density func-tional (DFT) methods and evaluating the internal energy 119864of materials at a temperature of zero Kelvin For example aharmonic or quasiharmonic model usually used to accountfor the vibrational thermodynamics at low temperatures andit is commonplace today to calculate harmonic phononsby methods based on DFT [7 14 15] The quasiharmonicapproximation (QHA) is based on how phonon frequencieschangewith volume and all shifts of phonon frequencies fromtheir low temperature values are considered as a result ofthermal expansion alone [7]

In theQHA the vibrational free energy can beminimizedas a function of volume

119865 (119881 119879)= 1198640+ int+infinminusinfin119892 (120596) (ℏ1205962 + 119896B119879 ln (1 minus 119890

minusℏ120596119896B119879)) d120596(1)

where 1198640 is the energy calculated from the relaxed structureat 119879 = 0K Thermodynamic properties are therefore derivedfrom here [7 16]

Although the QHA accounts for some frequency shiftsthe phonon modes are still assumed to be harmonic nonin-teracting and their energies depend only on the volume ofthe crystal This can be adequate when the temperatures ofservice of the materials are low or when differences of chem-ical potentials are much larger than kT Therefore QHA hasbeen found to be able to predict thermodynamic propertieswell consistent with the experiment results especially at highpressure [17 18] However for most applications of materialsin energy involving even modest temperature this 119864 aloneis insufficient because the anharmonic vibrational dynamicsand different types of thermal excitations play importantroles and have significant thermodynamic effects at elevatedtemperatures

Phonon-phonon interactions are responsible for pureanharmonicity that shortens phonon lifetimes and shiftsphonon frequencies especially at high temperatures Anhar-monicity competes with quasiharmonicity to alter the stabil-ity of phases at high temperatures as has been shown forexample with experiments and frozen phonon calculationson bcc Zr [19] and the possible stabilization of bcc Fe-Nialloys at conditions of the earthrsquos core [20] For PbTe ScF3and rutile TiO2 there are recent reports of anharmonicitybeing so large that both the QHA and anharmonic pertur-bation theory fail dramatically [6 8 21 22]

These cases are suitable for ab initio molecular dynamics(AIMD) simulations which should be reliable when theelectrons are near their ground states and the nuclearmotionsare classical The big advantage of ab initio MD is thatit can account for all effects of harmonic anharmonicand even some of the electron-phonon interactions How-ever advanced postprocessing methodologies are requiedto extract concrete information from these simulationsIn the few examples where comparisons have been madewith ab initio MD agreement has been surprisingly goodeven for highly anharmonic materials [6 22 23] Today byvalidating these calculated results from inelastic scatteringexperiments with facilities such as the Spallation NeutronSource for neutrons we can obtain sufficient details aboutphonon-phonon interactions electron-phonon interactionsand other excitations at elevated temperatures

To understand the microscopic picture of these inter-actions models of the effective vibrational potential energysurface and the fine structures of decay channels of phonon-phonon interactions have been proposed [23ndash26] Based onthe quantum perturbation theory of many-body interactionsand first-principles molecular dynamics simulations thesemethods are used to renormalize quasiharmonic phononsand to identify the three-phonon and four-phonon kinemat-ics We can assess the strengths of phonon-phonon interac-tions of different anharmonic orders or via different decaychannelsThesemethods with high computational efficiencyare promising directions to advance our understandingsof nonharmonic lattice dynamics and thermal transportproperties





119892 (120596) = sum119899119887

int 119890minusi120596119905 ⟨V119899119887 (119905) V00 (0)⟩ d119905 (2)


119892 ( 120596) = int119889119905 119890minusi120596119905sum119899119887

119890isdot119899 ⟨V119899119887 (119905) V00 (0)⟩ (3)




119867 = 1198800 + 12sum119894

119898p2119894 + 12 sum119894119895120572120573

120601120572120573119894119895 119906120572119894 119906120573119895

+ 13 sum119894119895119896120572120573120574

120595120572120573120574119894119895119896119906120572119894 119906120573119895 119906120574119896

(4)



sum11990221198952

1003816100381610038161003816119881 (119895 11198951 21198952)10038161003816100381610038162 Δ (1 + 2 minus )


1198991 minus 1198992Ω minus 1205961 + 1205962


Γ (119895 Ω) = 18120587ℏ2sdot sum11990211198951

sum11990221198952

1003816100381610038161003816119881 (119895 11198951 21198952)10038161003816100381610038162 Δ (1 + 2 minus )


(5)





2

Phon

on D

OS

(au

)

1

1373K

1073K

673K

300K












AMZRX Z0

5

10

15

20

25

Ener

gy (m

eV)

(a)AMZRX Z

minus10

minus5

0

5

10

15

20

25

Ener

gy (m

eV)

(b)


0

20

40

60

80

100

Mod

e ene

rgy

(meV

)

(c)









119881 (119895 11198951 119904minus1119895119904minus1) = 12119904 (

ℏ2119873)1199042


sdot 119862 (119895 11198951 119904minus1119895119904minus1)

(6)



119862(3)119895 = ⟨119862 (119895 11198951 21198952)⟩ (7a)

119862(4)119895 = ⟨119862 (119895 119895 11198951 minus11198951)⟩ (7b)


Γ(3) (119895 Ω) = 120587ℏ64 120596119895010038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 sum11990211198951

sum11990221198952

12059611205962119863(Ω 1205961 1205962)

= 1205961198950 10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162119863120596 (Ω)

(8a)

Δ(3) (119895 Ω)= minus ℏ641205961198950

10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 sum11990211198951

sum11990221198952

12059611205962119875 (Ω 1205961 1205962)

= 1205961198950 10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 119875120596 (Ω)

(8b)

Δ(31015840) (119895) = minus ℏ161198731205961198950

10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 sum11990221198952

1205961198952 (2) (11989921198952 + 12) (8c)

Δ(4) (119895) = ℏ81198731205961198950119862

(4)119895 sum11990211198951

1205961198951 (1) (11989911198951 + 12) (8d)




119863120596 (Ω 119879) prop 119879 sum11990211198951

sum11990221198952




150 900750600450300minus25

minus20

minus15

minus10

minus5

0

Temperature (K)

Δ120596

(cmminus1 )

Eg

A1g

B2g

(a)

150 900750600450300Temperature (K)

FWH

M (c

mminus1

)

Eg

A1g

B2g

0

5

10

15

20

25

30

35

(b)


Eg

A1g

B2g

800K

0K


minus1

0

1

2

(au

)















a(T)a(T 0)minus

1

minus00030

minus00025

minus00020

minus00015

minus00010

minus00005

00000



1 2 3

AgrarrAg minus Ag

AgrarrAg + Ag

OrarrO minus Ag

OrarrO + Ag

20 40 60 80 100 1200Frequency (meV)

minus05

00

05

10

15

(au

)






4 Conclusions





Competing Interests


Acknowledgments


References


















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119892 (120596) = sum119899119887

int 119890minusi120596119905 ⟨V119899119887 (119905) V00 (0)⟩ d119905 (2)


119892 ( 120596) = int119889119905 119890minusi120596119905sum119899119887

119890isdot119899 ⟨V119899119887 (119905) V00 (0)⟩ (3)




119867 = 1198800 + 12sum119894

119898p2119894 + 12 sum119894119895120572120573

120601120572120573119894119895 119906120572119894 119906120573119895

+ 13 sum119894119895119896120572120573120574

120595120572120573120574119894119895119896119906120572119894 119906120573119895 119906120574119896

(4)



sum11990221198952

1003816100381610038161003816119881 (119895 11198951 21198952)10038161003816100381610038162 Δ (1 + 2 minus )


1198991 minus 1198992Ω minus 1205961 + 1205962


Γ (119895 Ω) = 18120587ℏ2sdot sum11990211198951

sum11990221198952

1003816100381610038161003816119881 (119895 11198951 21198952)10038161003816100381610038162 Δ (1 + 2 minus )


(5)





2

Phon

on D

OS

(au

)

1

1373K

1073K

673K

300K












AMZRX Z0

5

10

15

20

25

Ener

gy (m

eV)

(a)AMZRX Z

minus10

minus5

0

5

10

15

20

25

Ener

gy (m

eV)

(b)


0

20

40

60

80

100

Mod

e ene

rgy

(meV

)

(c)









119881 (119895 11198951 119904minus1119895119904minus1) = 12119904 (

ℏ2119873)1199042


sdot 119862 (119895 11198951 119904minus1119895119904minus1)

(6)



119862(3)119895 = ⟨119862 (119895 11198951 21198952)⟩ (7a)

119862(4)119895 = ⟨119862 (119895 119895 11198951 minus11198951)⟩ (7b)


Γ(3) (119895 Ω) = 120587ℏ64 120596119895010038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 sum11990211198951

sum11990221198952

12059611205962119863(Ω 1205961 1205962)

= 1205961198950 10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162119863120596 (Ω)

(8a)

Δ(3) (119895 Ω)= minus ℏ641205961198950

10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 sum11990211198951

sum11990221198952

12059611205962119875 (Ω 1205961 1205962)

= 1205961198950 10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 119875120596 (Ω)

(8b)

Δ(31015840) (119895) = minus ℏ161198731205961198950

10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 sum11990221198952

1205961198952 (2) (11989921198952 + 12) (8c)

Δ(4) (119895) = ℏ81198731205961198950119862

(4)119895 sum11990211198951

1205961198951 (1) (11989911198951 + 12) (8d)




119863120596 (Ω 119879) prop 119879 sum11990211198951

sum11990221198952




150 900750600450300minus25

minus20

minus15

minus10

minus5

0

Temperature (K)

Δ120596

(cmminus1 )

Eg

A1g

B2g

(a)

150 900750600450300Temperature (K)

FWH

M (c

mminus1

)

Eg

A1g

B2g

0

5

10

15

20

25

30

35

(b)


Eg

A1g

B2g

800K

0K


minus1

0

1

2

(au

)















a(T)a(T 0)minus

1

minus00030

minus00025

minus00020

minus00015

minus00010

minus00005

00000



1 2 3

AgrarrAg minus Ag

AgrarrAg + Ag

OrarrO minus Ag

OrarrO + Ag

20 40 60 80 100 1200Frequency (meV)

minus05

00

05

10

15

(au

)






4 Conclusions





Competing Interests


Acknowledgments


References


















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




2

Phon

on D

OS

(au

)

1

1373K

1073K

673K

300K












AMZRX Z0

5

10

15

20

25

Ener

gy (m

eV)

(a)AMZRX Z

minus10

minus5

0

5

10

15

20

25

Ener

gy (m

eV)

(b)


0

20

40

60

80

100

Mod

e ene

rgy

(meV

)

(c)









119881 (119895 11198951 119904minus1119895119904minus1) = 12119904 (

ℏ2119873)1199042


sdot 119862 (119895 11198951 119904minus1119895119904minus1)

(6)



119862(3)119895 = ⟨119862 (119895 11198951 21198952)⟩ (7a)

119862(4)119895 = ⟨119862 (119895 119895 11198951 minus11198951)⟩ (7b)


Γ(3) (119895 Ω) = 120587ℏ64 120596119895010038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 sum11990211198951

sum11990221198952

12059611205962119863(Ω 1205961 1205962)

= 1205961198950 10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162119863120596 (Ω)

(8a)

Δ(3) (119895 Ω)= minus ℏ641205961198950

10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 sum11990211198951

sum11990221198952

12059611205962119875 (Ω 1205961 1205962)

= 1205961198950 10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 119875120596 (Ω)

(8b)

Δ(31015840) (119895) = minus ℏ161198731205961198950

10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 sum11990221198952

1205961198952 (2) (11989921198952 + 12) (8c)

Δ(4) (119895) = ℏ81198731205961198950119862

(4)119895 sum11990211198951

1205961198951 (1) (11989911198951 + 12) (8d)




119863120596 (Ω 119879) prop 119879 sum11990211198951

sum11990221198952




150 900750600450300minus25

minus20

minus15

minus10

minus5

0

Temperature (K)

Δ120596

(cmminus1 )

Eg

A1g

B2g

(a)

150 900750600450300Temperature (K)

FWH

M (c

mminus1

)

Eg

A1g

B2g

0

5

10

15

20

25

30

35

(b)


Eg

A1g

B2g

800K

0K


minus1

0

1

2

(au

)















a(T)a(T 0)minus

1

minus00030

minus00025

minus00020

minus00015

minus00010

minus00005

00000



1 2 3

AgrarrAg minus Ag

AgrarrAg + Ag

OrarrO minus Ag

OrarrO + Ag

20 40 60 80 100 1200Frequency (meV)

minus05

00

05

10

15

(au

)






4 Conclusions





Competing Interests


Acknowledgments


References


















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




AMZRX Z0

5

10

15

20

25

Ener

gy (m

eV)

(a)AMZRX Z

minus10

minus5

0

5

10

15

20

25

Ener

gy (m

eV)

(b)


0

20

40

60

80

100

Mod

e ene

rgy

(meV

)

(c)









119881 (119895 11198951 119904minus1119895119904minus1) = 12119904 (

ℏ2119873)1199042


sdot 119862 (119895 11198951 119904minus1119895119904minus1)

(6)



119862(3)119895 = ⟨119862 (119895 11198951 21198952)⟩ (7a)

119862(4)119895 = ⟨119862 (119895 119895 11198951 minus11198951)⟩ (7b)


Γ(3) (119895 Ω) = 120587ℏ64 120596119895010038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 sum11990211198951

sum11990221198952

12059611205962119863(Ω 1205961 1205962)

= 1205961198950 10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162119863120596 (Ω)

(8a)

Δ(3) (119895 Ω)= minus ℏ641205961198950

10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 sum11990211198951

sum11990221198952

12059611205962119875 (Ω 1205961 1205962)

= 1205961198950 10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 119875120596 (Ω)

(8b)

Δ(31015840) (119895) = minus ℏ161198731205961198950

10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 sum11990221198952

1205961198952 (2) (11989921198952 + 12) (8c)

Δ(4) (119895) = ℏ81198731205961198950119862

(4)119895 sum11990211198951

1205961198951 (1) (11989911198951 + 12) (8d)




119863120596 (Ω 119879) prop 119879 sum11990211198951

sum11990221198952




150 900750600450300minus25

minus20

minus15

minus10

minus5

0

Temperature (K)

Δ120596

(cmminus1 )

Eg

A1g

B2g

(a)

150 900750600450300Temperature (K)

FWH

M (c

mminus1

)

Eg

A1g

B2g

0

5

10

15

20

25

30

35

(b)


Eg

A1g

B2g

800K

0K


minus1

0

1

2

(au

)















a(T)a(T 0)minus

1

minus00030

minus00025

minus00020

minus00015

minus00010

minus00005

00000



1 2 3

AgrarrAg minus Ag

AgrarrAg + Ag

OrarrO minus Ag

OrarrO + Ag

20 40 60 80 100 1200Frequency (meV)

minus05

00

05

10

15

(au

)






4 Conclusions





Competing Interests


Acknowledgments


References


















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119881 (119895 11198951 119904minus1119895119904minus1) = 12119904 (

ℏ2119873)1199042


sdot 119862 (119895 11198951 119904minus1119895119904minus1)

(6)



119862(3)119895 = ⟨119862 (119895 11198951 21198952)⟩ (7a)

119862(4)119895 = ⟨119862 (119895 119895 11198951 minus11198951)⟩ (7b)


Γ(3) (119895 Ω) = 120587ℏ64 120596119895010038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 sum11990211198951

sum11990221198952

12059611205962119863(Ω 1205961 1205962)

= 1205961198950 10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162119863120596 (Ω)

(8a)

Δ(3) (119895 Ω)= minus ℏ641205961198950

10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 sum11990211198951

sum11990221198952

12059611205962119875 (Ω 1205961 1205962)

= 1205961198950 10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 119875120596 (Ω)

(8b)

Δ(31015840) (119895) = minus ℏ161198731205961198950

10038161003816100381610038161003816119862(3)119895 100381610038161003816100381610038162 sum11990221198952

1205961198952 (2) (11989921198952 + 12) (8c)

Δ(4) (119895) = ℏ81198731205961198950119862

(4)119895 sum11990211198951

1205961198951 (1) (11989911198951 + 12) (8d)




119863120596 (Ω 119879) prop 119879 sum11990211198951

sum11990221198952




150 900750600450300minus25

minus20

minus15

minus10

minus5

0

Temperature (K)

Δ120596

(cmminus1 )

Eg

A1g

B2g

(a)

150 900750600450300Temperature (K)

FWH

M (c

mminus1

)

Eg

A1g

B2g

0

5

10

15

20

25

30

35

(b)


Eg

A1g

B2g

800K

0K


minus1

0

1

2

(au

)















a(T)a(T 0)minus

1

minus00030

minus00025

minus00020

minus00015

minus00010

minus00005

00000



1 2 3

AgrarrAg minus Ag

AgrarrAg + Ag

OrarrO minus Ag

OrarrO + Ag

20 40 60 80 100 1200Frequency (meV)

minus05

00

05

10

15

(au

)






4 Conclusions





Competing Interests


Acknowledgments


References


















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




150 900750600450300minus25

minus20

minus15

minus10

minus5

0

Temperature (K)

Δ120596

(cmminus1 )

Eg

A1g

B2g

(a)

150 900750600450300Temperature (K)

FWH

M (c

mminus1

)

Eg

A1g

B2g

0

5

10

15

20

25

30

35

(b)


Eg

A1g

B2g

800K

0K


minus1

0

1

2

(au

)















a(T)a(T 0)minus

1

minus00030

minus00025

minus00020

minus00015

minus00010

minus00005

00000



1 2 3

AgrarrAg minus Ag

AgrarrAg + Ag

OrarrO minus Ag

OrarrO + Ag

20 40 60 80 100 1200Frequency (meV)

minus05

00

05

10

15

(au

)






4 Conclusions





Competing Interests


Acknowledgments


References


















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics












a(T)a(T 0)minus

1

minus00030

minus00025

minus00020

minus00015

minus00010

minus00005

00000



1 2 3

AgrarrAg minus Ag

AgrarrAg + Ag

OrarrO minus Ag

OrarrO + Ag

20 40 60 80 100 1200Frequency (meV)

minus05

00

05

10

15

(au

)






4 Conclusions





Competing Interests


Acknowledgments


References


















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






4 Conclusions





Competing Interests


Acknowledgments


References


















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



review article renormalized phonon microstructures at high...

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