Part-crystalline part-liquid state and rattling-likethermal damping in materials withchemical-bond hierarchyWujie Qiua,b, Lili Xia, Ping Weic, Xuezhi Keb,1, Jihui Yangc,1, and Wenqing Zhanga,d,1

aState Key Laboratory of High Performance Ceramics and Superfine Microstructure, Shanghai Institute of Ceramics, Chinese Academy of Sciences, Shanghai200050, China; bInstitute of Theoretical Physics and Department of Physics, East China Normal University, Shanghai 200241, China; cMaterials Science andEngineering Department, University of Washington, Seattle, WA 98195; and dMaterials Genome Institute, Shanghai University, Shanghai 200444, China

Edited by Brian C. Sales, Oak Ridge National Laboratory, Oak Ridge, TN, and accepted by the Editorial Board September 8, 2014 (received for reviewJune 3, 2014)

Understanding thermal and phonon transport in solids has been ofgreat importance in many disciplines such as thermoelectric materi-als, which usually requires an extremely low lattice thermal conduc-tivity (LTC). By analyzing the finite-temperature structural andvibrational characteristics of typical thermoelectric compounds suchas filled skutterudites and Cu3SbSe3, we demonstrate a concept ofpart-crystalline part-liquid state in the compounds with chemical-bond hierarchy, in which certain constituent species weakly bondto other part of the crystal. Such a material could intrinsically mani-fest the coexistence of rigid crystalline sublattices and other fluc-tuating noncrystalline sublattices with thermally induced large-amplitude vibrations and even flow of the group of species atoms,leading to atomic-level heterogeneity, mixed part-crystalline part-liquid structure, and thus rattling-like thermal damping due to thecollective soft-mode vibrations similar to the Boson peak in amor-phousmaterials. The observed abnormal LTC close to the amorphouslimit in these materials can only be described by an effective ap-proach that approximately treats the rattling-like damping as a “res-onant” phonon scattering.

sublattice melting | partial Grüneisen parameter | first principles |anharmonicity

Understanding thermal and phonon transport in solids hasbeen of great importance in many disciplines such as ther-

moelectrics (1–3), phononic materials (4), and thermal man-agement composites (5). The interplay among chemical bonds,lattice dynamics, and thermal transport in materials is also anattractive topic in condensed matter physics (6) and materialsscience (7). Thermal transport is a key issue in thermoelectric(TE) energy-conversion materials, which are regarded among thepotential candidates for revolutionizing waste-heat recovery (2, 7–9). The dimensionless figure of merit of a TE material is definedas ZT =TS2σ=κ, where T, S, σ, and κ are the absolute tempera-ture, Seebeck coefficient, electrical conductivity, and thermal con-ductivity, respectively. To improve the efficiency of TE conversion,many approaches aim at reducing the thermal conductivity, espe-cially the lattice part, to a minimum level, namely the realization ofphonon-glass-like thermal transport (1, 7).TE materials research primarily focuses on solid and crystal-

line thermoelectrics. It has been long viewed that all solidscontain strong interatomic interactions without even an excep-tion, and thus the established approaches to describe thermaltransports in crystalline solids, including TE solids, are solelybased on the perturbative “small-parameter” approximationto lattice dynamics of atoms around their equilibrium positions,i.e., phonons and phonon–phonon interactions (10, 11). As a re-sult, crystallographic homogeneity at the atomic level in solidmaterials has overwhelmingly been accepted. However, recentwork on exploring novel TE materials went noticeably beyond theconventional knowledge of solid TE compounds being ideallycrystalline, atomically homogeneous, and dynamically perturbative.

The most typical examples are filled skutterudites and clathrateswith randomly rattling fillers (12–14), Cu2Se with liquid-likefluctuating substructures (15, 16), AgSbTe2 with extremely softmodes (17–19), SnSe with layered structures (20), etc., whichshow abnormal thermal transports. There is a compelling andtimely need to understand their structural characteristics as wellas to reveal the underlying correlations among lattice dynamics,thermal transports, and chemical bonds.

ResultsLindemann Criterion of Melting and the “Melting” Sublattice for VoidFillers. It is well known that one way to reduce lattice thermalconductivity (LTC) is to introduce low-frequency lattice vibra-tions to scatter heat-carrying acoustic phonons (21–24). This hasbeen proven a very effective approach for realizing the extremelylow LTC through the randomly fluctuating rattling fillers on thevoid–sublattice interlocking with the rigid framework in filledskutterudites (23, 24) and clathrates (14, 25, 26). In both systems,all void fillers are individually embedded in a flattened potentialenergy surface due to the weak bonding or interaction with theframework (12, 13, 22, 23, 27, 28) and show ultralarge atom dis-placement parameters (ADPs). The ADPs are so large that thecorresponding sublattice can even be considered to be “melted”based on the classical Lindemann criterion of melting (29). TheLindemann parameter δ is defined as δ = ADP1/2/RNN, whereRNN is the nearest-neighbor distance. Melting of a crystallinesolid is usually found to occur when δ reaches 0.07 or above (29,30). The vibration amplitude of Yb fillers in Yb0.188Co4Sb12 froma molecular dynamics (MD) simulations at 400 K is estimated tobe as large as 0.91 Å (see SI Text for details). Thus, the Linde-mann parameter δ for Yb fillers is about 0.28 and far exceedsthe Lindemann criterion of melting, indicating that the filler

Significance

Materials with chemical-bond hierarchy may have a speciallymixed part-crystalline part-liquid state and show nontraditionalthermal transports beyond the traditional “small-parameter”lattice dynamics approach, especially the rattling-like thermaldamping and thus an unusual lattice thermal conductivity thatcan only be described by including an effective “resonant”phonon scattering.

Author contributions: X.K., J.Y., and W.Z. designed research; W.Q., L.X., P.W., and X.K.performed research; W.Q., L.X., J.Y., and W.Z. analyzed data; and X.K., J.Y., and W.Z.wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. B.C.S. is a Guest Editor invited by the EditorialBoard.1To whom correspondence may be addressed. Email: wqzhang@mail.sic.ac.cn, xzke@phy.ecnu.edu.cn, or jihuiy@uw.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1410349111/-/DCSupplemental.

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sublattice is nearly in a state of liquid. In this respect, thesematerials should be described as containing both a crystal-line framework and noncrystalline rattling fillers with large-amplitude random movements on the void–sublattice, due to thecoexistence of different types of chemical bonds of hierarchicalstrengths. Therefore, the whole system can hardly be regardedas a state of homogeneous crystalline solid valid for small-parameter approximation.

A Mixed Part-Crystalline Part-Liquid State in TE Materials. Thefinite-temperature structural characteristics of Cu3SbSe3, a newtype of TE material, could further demonstrate the deviationfrom the static and small-vibration picture. Cu–Sb–Se systemshave been shown to possess a ZT value about unity (31, 32), andCu3SbSe3 manifests anomalously low lattice thermal conductivity(0.7–1.0 Wm-1·K−1) with an unusual temperature dependence(33, 34) (see An Effective Approach for the Thermal Transport inPart-Crystalline Part-Liquid Materials) and has also been expectedto possess a ZT value about unity (35). In the compound, certaintype of constituent atoms weakly bond to the rest of lattice withrelatively strong rigidity. For example, the Cu1 atoms in theCu3SbSe3 crystal structure (SI Text), while confined in both x and ydirections, can easily oscillate within a hollow channel (∼4.3 Å) inthe z direction. In fact, the Cu2 atoms are also weakly bound.Fig. 1A plots the simulated trajectory of atoms in the y–z plane

of Cu3SbSe3 at 400 K in MD. The oscillation amplitude for Cuatoms is more than four times larger than that for Sb or Seatoms, indicating that Cu atoms are not constrained to aroundtheir equilibrium positions. Instead, Cu atoms usually have largevibrational amplitude, and can even frequently transit to nearbysites, showing a fluid-like flow as in a partial liquid. The delo-calized fluid-like flow movements of those Cu atoms are differ-ent from the large but local rattling of fillers in filled CoSb3, andthe averaged thermally induced displacement of Cu atoms inMD could be estimated to be as large as 1.54 Å, leading to a verylarge Lindemann parameter δ (∼0.44). In this regard, the Cu sub-lattice should be viewed as “sublattice melting,” and the wholesystem is thus in a mixed part-crystalline part-liquid state, con-taining a crystalline rigid part (Se and Sb sublattices) and a fluc-tuating noncrystalline substructure (Cu sublattice).Fig. 1B plots the decay patterns of the velocity autocorrelation

function (VAF) hvð0Þ · vðtÞi as a function of time from MDsimulations. Usually, a damped oscillating VAF reveals the stronginteractions in solids, whereas for the liquid-only case one single-valley minimum is present due to the diffusive and dynamically

incoherent fluctuating behavior of constituent atoms (36–38). Itcan be seen from Fig. 1B that the behavior of Se and Sb atomsfollows a damped harmonic decay, indicating that there are stronginteractions associated with them and their vibrations are verymuch localized. However, the projected VAFs of the Cu1z andCu2xz exhibit a widely distributed and damped single-valleymanner, representative of a nonlocal diffusion and incoherentfluctuation behavior for Cu1 in the z direction and Cu2 in the x–zplanes, whereas the Cu1xy and Cu2y remain a solid-like featurewith only local vibrations. Such a mixed heterogeneous featurecan even be preserved at quite low temperatures based on oursimulations. The Cu1z-projected phonon dispersion and phonondensity of states (SI Text), by using the zero-temperature struc-ture as input, indicate that the low-frequency vibration peaks inVAF could be primarily ascribed to the two significant peaks ataround (0.8∼1.0) THz and (1.2∼1.5) THz corresponding mainlyto the Cu1z-based optic phonons (SI Text). Comparing with filledCoSb3, the intrinsic structure of Cu3SbSe3 manifests largevibrations and fluid-like diffusion of constituent atoms, showseven part-liquid dynamic characteristics and significant atomic-level heterogeneity, strongly deviating from the homogeneouscrystalline picture of the classical solids.The abnormal structural and vibrational features in fact origi-

nate from the specific hierarchy of chemical bonds in the materials.By shifting the atoms away from their static equilibrium positions,we found that Cu1 atom sits in a very flat potential well and thus isonly weakly bound especially in the z direction, as shown in Fig. 2,Inset. The potential well for Cu2 in the x–z directions is also veryflat. However, the wells for the other species, forming relativelystrong chemical bonds with the surroundings, are at least twice assteep as that for Cu1z. As a result of the bonding strength hier-archy, the ADPs for Cu atoms especially the Cu1z are very largeas shown in Fig. 2. The estimated ADP for Cu1z is more than twotimes larger than those for other atoms (see SI Text for details).The total ADP parameters, even within a harmonic approxima-tion, are also large, leading to a Lindemann melting parameter δhigher than 0.148 and 0.142 for Cu1 and Cu2 atoms, respectively.This is consistent with the atomic-level heterogeneity and themixed part-crystalline part-liquid state feature due to sublatticemelting given by the MD simulations.

The Origin of Anharmonicity. Based on the aforementioned dataand analysis, LTC of Cu3SbSe3 is expected to show a severereduction as well as abnormal temperature dependence, fall-ing out of the capability of traditional model valid only for

Fig. 1. Results of molecular dynamics simulations. (A) Trajectory of atoms in the y–z plane fromMD simulations at 400 K. (B) VAF as a function of time. (Inset)Power spectrum, evaluated by the fast Fourier transform of the VAF.

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homogeneous crystalline solids. A similar phenomenon was alsoobserved for LTC in filled skutterudites and explained as a result ofnontraditional rattling-induced resonant scattering of lattice pho-nons (24). LTC reduction in the traditional phonon–phonon in-teraction picture is mainly attributed to the nonlinear scattering ofacoustic phonons at elevated temperatures. In the traditional ap-proach, the Grüneisen parameter (γ) could be adopted to describeLTC and its temperature dependence. The Umklapp (U) andnormal (N) phonon scattering rates are commonly considered to beproportional to γ2 according to the long-established theories (39–42). To clarify the origin of anharmonicity and its effect on LTC ofCu3SbSe3, the total Grüneisen parameter and partial Grüneisenparameters describing the projected contribution from given atomtypes and phonon branches (q, i) were all extracted from the cal-culated phonon dispersions. The modulus of the squared phononpolarization vector jeαðq; i; νÞj2 depicts the contribution of a givenatom ν in the α direction to the phonon branch i at wave vector q.Therefore, the partial Grüneisen parameter is estimated by pro-jecting the total Grüneisen parameter γ(q, i) onto atom type μ inthe α direction as follows (see SI Text for details):

γ μαðq; iÞ =

X

ν∈μγðq; iÞ× jeαðq; i; νÞj2; [1]

As shown in Table 1, all of the acoustic phonon branchesprojected onto Cu, Se, and Sb atoms possess relatively smallGrüneisen parameters, indicating that intrinsic nonlinear phononscattering of those acoustic phonons does not strongly affectLTC. Indeed, the calculated LTC from those parameters (SIText) gives substantially higher values than the measured results,especially at relatively low temperatures (see dashed line in Fig.3). In contrast, for the transverse optic phonon branch, the cal-culated partial Grüneisen parameters are relatively high, espe-cially for Cu1 and Sb atoms. Considering the low group velocityof optic phonons, their direct contribution to LTC is rather minor.Their indirect contribution, mainly through affecting acoustic pho-nons based on traditional theories (43–45), is also observed to below due to the calculated low Grüneisen parameters of the acous-tic phonons. In this regard, the special vibrations associated withthose optic phonons must affect thermal transports in a uniqueway such as producing somewhat localized vibrations or atomicallyheterogeneous such as the part-liquid fluctuating substructures,

which makes the materials with chemical-bond hierarchy verymuch different from the conventional homogeneous crystallinesolids. Table 1 also shows that the Cu1-related parameters are atleast twice larger than those of other atoms for acoustic phonons,indicating that Cu1 plays a key role in the anharmonic scatteringprocesses in addition to the soft-mode-related structure hetero-geneity. The parameters for the Cu1 atom are overwhelmingly con-tributed from the z component, which is consistent with thepreceding discussions about the flattest potential well and the un-usually large Cu1z ADP. This indicates that crystal structural fac-tor, such as Cu1 atoms locating near the hollow space within thecrystal, contributes more to the above phenomena.

An Effective Approach for the Thermal Transport in Part-CrystallinePart-Liquid Materials. The above discussions strongly indicateabnormal thermal transports in Cu3SbSe3 as a typical compoundwith chemical-bond hierarchy. Indeed, a very atypical behaviorof LTC including its temperature dependence was reported bySkoug et al., as shown in Fig. 3. The measured data (33) demon-strate an unusual temperature (T) dependence behavior, stronglydeviating from T -1 relationship at elevated temperatures above theDebye temperature of the compound, in addition to the extremelylow LTC. These indicate that the LTC is correlated to factorsmore than phonon–phonon anharmonic scattering, consistent withthe above discussions. By a density functional theory-based ap-proach, we also carefully calculated the LTC of Cu3SbSe3 (Fig. 3),using parameters given in Table 1 and the modified Debye–Call-away model (41, 42, 46) that contains both U and N phononscattering processes (SI Text), by assuming the dominant anhar-monic phonon scattering. The calculations, as denoted in Fig. 3,strongly underestimate the LTC reduction, and even the predictedtemperature-dependence tendency is also very much differentfrom the experiments. An earlier calculation, qualitatively con-sistent with our theory trend, does not reproduce the measuredresults either (47). Considering the complex vibrational modes ofCu atoms, especially the Cu1z component, we deliberately removepart of the contribution from Cu1z to the final anharmonicity basedon the percentage of harmonic term in the potential Cu1z experi-ences (see SI Text for details). The calculated LTC (Fig. 3) exhibitsan expected T-1 behavior, but still strongly deviates from themeasured data.Rigorous formulation of LTC for complex systems with atomic-

level heterogeneity and part-crystalline part-liquid structure, likein Cu3SbSe3 and filled skutterudites, is a newly raised questionwithout any prior reference. Parallel to the LTC reduction due tothe low-frequency phonon scattering by the single rattler in filledCoSb3, we rationally infer that the abnormal lattice thermal con-ductivity in Cu3SbSe3 should also correlate with the specific large-amplitude soft-mode vibrations that give the atypical structuraland lattice dynamics properties. Based on the vibrational spectrumdetermined from the fast Fourier transform of MD simulated

Fig. 2. Calculated ADPs and the corresponding atom projections for dif-ferent atoms in Cu3SbSe3 based on phonon spectrum. The ADPs for Sb, Se,Cu2y, Cu1x, and Cu1y are within the belt region. (Inset) Calculated potentialenergy curves for the five nonequivalent atom types as a function of dis-placement about the equilibrium positions.

Table 1. Grüneisen parameter and its projection

Element γTA γTA′ γLA γaver γTO

Total 1.69 1.47 1.19 1.45 3.32Cu1 0.638 0.554 0.407 0.533 0.972Cu1x 0.058 0.026 0.081 0.055 0.110Cu1y 0.107 0.142 0.058 0.102 0.140Cu1z 0.489 0.395 0.295 0.393 0.747Cu2 0.294 0.251 0.217 0.254 0.545Se1 0.297 0.250 0.189 0.245 0.524Se2 0.239 0.211 0.157 0.202 0.390Sb 0.241 0.212 0.239 0.231 0.948

The average values are only for the acoustic branches. The Grüneisenparameter of the transverse optical (TO) mode is also shown.

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VAF as given in Fig. 1B Inset, we can clearly single out a specificlow-frequency vibration with a finite width. The vibration is largelyfrom the Cu1z component (∼1.0 THz) but also contains contri-bution from Cu2xz. Both Cu1 and Cu2 atoms seemingly relate tothe part-liquid or liquid-like structural heterogeneity in Cu3SbSe3and closely link with weak chemical bonds. Comparing with thewell-defined single-frequency rattling in filled CoSb3, the low-frequency vibrations in Cu3SbSe3 relate to a set of atoms, coupledwith each other, and thus make the phonon scattering qualitativelysimilar to the single-frequency rattling but cover a relatively widefrequency range due to the hierarchical distribution of differentchemical bonds. Thus, this is not a single-frequency rattling buta collective broadband damped rattling or a rattle-like thermaldamping. Actually, collective low-frequency vibration from a set ofatomswas usually observed in amorphous or liquidmaterials, whichoriginates from a certain group of atoms within relatively looselypacked regions that undergo a liquid-like motion from one con-figuration to another (48). The typical structure studied here issimilar to the noncrystalline part-liquid substructure for atomsamong the interstitial sites in the current part-crystalline part-liquidhybrid state. Recent work on the thermal transport in clathrateswith static structure disorder also revealed a similar phenomenonand related that to the Boson-peak-like vibration behavior of sets ofatoms in amorphous materials or structural glasses (14, 48). We also

fitted experimental heat capacity to the Debye–Einstein model, andfound that two Einstein oscillators of ∼0.9 THz and ∼1.6 THz haveto be included to fit the measured data, implying similar lattice dy-namics at both low and high temperatures (see SI Text for details).Given phonon scattering from single rattling, the reduction of

LTC can be proven to follow a universal scaling relationship (seeSI Text for details),

κ−1L − κ−10 = C · f ðω0;TÞ; [2]

where f ðω0;TÞ=ω20ðex0 − 1Þ2=T3 x40 e

x0 (ω0 is the resonant fre-quency and T is the absolute temperature). κ0 indicates theLTC containing only the phonon–phonon anharmonic interac-tion, and κL the total or the measured result. C is a constantrelating to the concentration of fillers and the effective rattlingwidth. The scaling function is universal in theory and has beenconfirmed to reasonably describe the LTC reduction for filledskutterudites, as demonstrated in Fig. 4B, which plots theκ−1L − κ−10 together with scaled f ðω0;TÞ for Na-, Yb-, and Ba-filled CoSb3.The model is also valid for the system with rattle-like phonon

damping in the current Cu3SbSe3. At the given frequency ω0 (1.0THz) from VAF calculations, the LTC reduction in Cu3SbSe3follows the scaling function, indicating the validity and capabilityof the current model for the systems with effective rattle-likethermal damping (Fig. 4A). In this respect, the above scalinganalysis indicates that the total LTC of Cu3SbSe3 can be de-scribed by the revised Debye–Callaway model as long as an extrarattling-like phonon scattering, τ−1R =C pω2=ððω2

0 −ω2Þ2 +Δ2Þ,is introduced into the phonon relaxation rate (49). The calcu-lated LTC including the effective rattling-like damping term isplotted in Fig. 3, showing an excellent agreement with the exper-iment over a wide temperature range. The formula thus becomessimilar to the earlier single-rattling phonon scattering model, butthe width Δ should have a physically meaningful value now. It isvery likely that such a model provides an effective description forthermal transport in all part-crystalline part-liquid materials.

ConclusionsIn summary, in the materials with chemical bond hierarchy, a fewconstituent species atoms weakly bond to the rest of the latticewith relatively strong rigidity, resulting in low-frequency vibrationsand thus thermally induced large atomic displacement parametersand even fluid-like flow of atoms. As a result, such materials couldintrinsically manifest the coexistence of rigid crystalline sublatticesand fluctuating noncrystalline substructures, leading to atomic-level dynamic heterogeneity, mixed part-crystalline part-liquidstate, rattling-like thermal damping, and thus LTC close to the

Fig. 3. Lattice thermal conductivity of Cu3SbSe3. Red dots are experimentaldata from Skoug et al. (33). Different lines refer to theoretical results at theconditions indicated in the text.

Fig. 4. Models for part-crystalline part-liquid materials. (A) Schematic illustration of rattling-like thermal damping (Upper Left) in materials with chemical-bond hierarchy and single-rattle phonon scattering in filled CoSb3 (Upper Right). (Lower) Graphical rendering of part-crystalline part-liquid structure. (B)κ−1L − κ−10 and the universal scaling function fðω0,TÞ as a function of temperature for Na-, Yb-, and Ba-filled skutterudites and Cu3SbSe3.

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amorphous limit like in a “phonon glass.” Analysis reveals the cor-relation among the LTC reduction, collective low-frequency vibra-tions similar to amorphous materials, and a universal scalingrelationship, and finally leads to an effective approach in describingthe thermal transport in part-crystalline part-liquid materials. Thisquantitatively explains the reduction of LTC, and also demonstratesthe theoretical challenge of correctly describing phonon and ther-mal transports in complex materials. As a prediction, compoundswith chemical-bond hierarchy are very likely to trigger the interest inmaterials with unique atomic-level dynamic heterogeneity. Thesematerials possess extremely low LTC and can be candidates forhigh-performance TEs.

MethodsOur calculations were performed using the plane wave basis Vienna ab initiosimulation package (VASP) code (50), implementing the generalized gradientapproximation (GGA) of Perdew-Burke-Ernzerhof form (51). The interactionbetween ions and electrons is described by the projector augmented wave(PAW) method (52), with plane waves up to a cutoff energy of 800 eV. Theconfigurations Cu 3d10 4s1, Sb 5s25p3, and Se 4s24p4 were treated as valenceelectrons. The Brillouin-zone integrations were performed on the gridof Monkhorst–Pack procedure. For the unit cell (28 atoms) and supercell

(112 atoms), 4 × 3 × 4 and 2 × 2 × 2 k-point meshes were used, respectively. MDcalculations were performed using the GGA of PBE form as implemented in theVASP code with the canonical ensemble. The PAW method is adopted, anda supercell with 259 atoms (18.25 Å × 18.25 Å × 18.25 Å) was used for Yb-filledCoSb3. For Cu3SbSe3, a supercell with 112 atoms (16.20 Å × 10.67 Å × 13.87 Å)was used. The simulation temperature was set to be 400 K for two compounds,and simulation time is longer than 20 ps. The velocity autocorrelation functionwas found to be well converged within 20 ps.

The sample was prepared by a combination of melting–annealing andspark plasma sintering (SPS) methods. The mixture of elemental con-stituents was sealed in a quartz tube and melted at 1,173 K for 24 h beforeannealing at 673 K for 120 h. The densification of Cu3SbSe3 was accom-plished by SPS method at 700 K for 5 min. Low-temperature heat capacitywas measured from 2 K to 328 K using a Quantum Design Physical PropertyMeasurement System.

ACKNOWLEDGMENTS. This work is supported by National Basic ResearchProgram of China (973 program) under Project 2013CB632501, NationalNatural Science Foundation of China under 11234012, 51121064, 11074074,and 11204333, East China Normal University (Grant 239201278220051), andSpecialized Research Fund for the Doctoral Program of Higher Education(20110076110002). J.Y. acknowledges support by US Department of Energyunder Corporate Agreement DE-FC26-04NT42278, by General Motors, andby National Science Foundation under Award 1235535.

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