Phononic filter effect of rattling phonons in the thermoelectric clathrate Ba 8 Ge 40+x Ni 6−x
Holger Euchner, Stephane Pailhès, Lien Nguyen, Wolf Assmus, Franz Ritter, Amir Haghighirad, Yuri Grin, Silke Paschen, Marc. de Boissieu
PPhononic filter effect of rattling phonons in the thermoelectric clathrateBa Ge x Ni − x H. Euchner, S. Pailh`es,
2, 3
L. T. K. Nguyen,
4, 5, 6
W. Assmus, F.Ritter, A. Haghighirad, Y. Grin, S. Paschen, and M. de Boissieu Institut f¨ur Theoret. und Angwandte Physik, Universit¨at Stuttgart, D-70550 Stuttgart, Germany LPMCN, UCBL, CNRS, UMR-5586, F-69622 Villeurbanne, Lyon, France LLB, CEA, CNRS, UMR-12, CE-Saclay, F-91191 Gif-sur-Yvette, France Physikalisches Institut,Johann Wolfgang Goethe-Universit¨at, D-60438 Frankfurt, Germany Max-Planck-Institut f¨ur chemische Physik fester Stoffe, Dresden, Germany Institute of Solid State Physics, Vienna University of Technology, 1040 Vienna, Austria SIMAP, UJF, CNRS, INP Grenoble, F-38402 St Martin d’H`eres, France (Dated: October 16, 2018)One of the key requirements for good thermoelectric materials is a low lattice thermal conductivity.Here we present a combined neutron scattering and theoretical investigation of the lattice dynamicsin the type I clathrate system Ba–Ge–Ni, which fulfills this requirement. We observe a stronghybridization between phonons of the Ba guest atoms and acoustic phonons of the Ge-Ni hoststructure over a wide region of the Brillouin zone which is in contrast with the frequently adoptedpicture of isolated Ba atoms in Ge-Ni host cages. It occurs without a strong decrease of theacoustic phonon lifetime which contradicts the usual assumption of strong anharmonic phonon–phonon scattering processes. Within the framework of ab–intio density functional theory calculationswe interpret these hybridizations as a series of anti-crossings which act as a low pass filter, preventingthe propagation of acoustic phonons. To highlight the effect of such a phononic low pass filter on thethermal transport, we compute the contribution of acoustic phonons to the thermal conductivity ofBa Ge Ni and compare it to those of pure Ge and a Ge empty-cage model system. A central issue in thermoelectrics research is to findmaterials that generate a high electromotive force un-der a small applied thermal gradient. The best thermo-electric materials (TE) should therefore combine a lowlattice thermal conductivity with high electrical conduc-tivity and thermopower. An efficient and economic wayto reduce the thermal conductivity of bulk TE materials,without degrading their electronic properties, is to takeadvantage of inelastic resonant scattering between heat-carrying acoustic and non-propagative phonons, arisingfrom isolated impurities with internal oscillator degreesof freedom. The librational vibration of molecules in-corporated in crystal structures is a well-known exampleof such scattering centers and was formerly observed inKCl ionic crystals, doped with anionic molecules [1], orin the filled water nanocages of clathrate hydrates [2, 3].In TE clathrates and skutterudites, in which the phonondispersions were recently investigated by means of in-elastic neutrons scattering (INS) [4–6, 10], the localizedresonators are so-called rattling phonon modes of looselybonded guest atoms in oversized atomic cages. The effectof inelastic resonant scattering [11, 12] was described the-oretically, using the relaxation-time approximation of theBoltzmann equation, in which the lattice thermal con-ductivity, κ L , for a cubic crystal can be expressed as κ L = 1 / (cid:90) ω max C v ( ω ) v ( ω ) τ ( ω ) ρ ( ω ) dω . (1)Here C v ( ω ) is the phonon specific heat per unit volumeand unit frequency, v ( ω ) is the mean group velocity for phonon frequency ω , τ ( ω ) is the mean time betweencollisions that destroy the heat current for all phonons offrequency ω and ρ ( ω ) is the normalized vibrational den-sity of states. The most common approach to evaluatethe scattering of acoustic phonons by a resonator withfrequency ω is to add a phenomenological relaxationrate, τ − R ( ω ) = C ω ( ω − ω ) , where C is a constant propor-tional to the concentration of the resonant defects andto the strength of the resonator-lattice coupling. Thishas been used to fit experimental data for clathratesand skutterudites [13, 14]. At a microscopic level, thisimplies the existence of a coupling and an anti-crossingbehavior between the non-dispersive phonons of theisolated resonators and the acoustic phonon branchesleading to the opening of a gap surrounding ω in thedispersion of the acoustic branch [17]. INS measurementson the clathrate Ba Ge Ga [5, 6] and the skutteruditeCeRu Sb [4] evidence such an anti-crossing. However,in these previous studies, the authors conclude thatthe remarkably low κ values of these materials cannotfully be ascribed to the effect of guest vibrations onlifetime τ ( ω ) or group velocity v ( ω ) of the acousticphonon modes (see also [17]). On the other hand, thepicture of isolated guest atoms in a host cage and themicroscopic mechanism responsible for the scatteringof heat carriers were revisited by INS studies of thephonon dynamics in polycrystals of the skutteruditesLa/CeFe Sb [8]. The authors demonstrated that theguest atoms are coherently coupled with the host-latticedynamics and associated with low energy optical phonon a r X i v : . [ c ond - m a t . m t r l - s c i ] N ov modes which are characterized by a small group velocity.It was then proposed that these low frequency phononbranches open new paths for the occurrence of Umklappscattering of heat-carrying acoustic phonons, that is,the backscattering of heat carriers by an elastic ora multiphonon process mediated by a lattice vector[4, 6, 8]. This mechanism obeys momentum and energypreservation; hence, it is in contrast to inelastic resonantscattering associated with energy dissipation at isolatedatoms. The reason to invoke such ph-ph processes incomplex unit cell materials is the presence of flat opticalphonon modes at low energies which are expected toenhance ph-ph scattering processes. However, to dateexperimental evidence for this claim is missing, - itwould require measuring the corresponding broadeningof the energy width of the phonons involved. Yet, theavailable INS measurements on single crystals [4, 6] aswell as our data demonstrate that the finite lifetime ofacoustic phonons is not small enough to explain such alow κ , meaning that no fingerprints of strong Umklappscattering could be evidenced. Thus, the microscopicorigin for the low κ remains elusive. In particular therespective roles played by guest atoms and unit cellcomplexity are ambiguous [18].In this paper, we present in section I the experimentalmapping of the transverse acoustic (TA) and the guestphonon dispersions in a single crystal of the clathrateBa Ge . Ni . (cid:3) . [24] by INS, covering the entire firstBrillouin zone (BZ) and a wide energy range. The highquality of the data allows us to analyse the relativechange in intensity of acoustic and optical phonons. Weevidence a gapless dispersion of the TA phonons in ab-sence of a strong broadening of their energy widths whenapproaching the energy of the optical phonons. Moreoverthe existence of a mechanism of spectral weight trans-fer between acoustic and optical phonons resulting fromstrong hybridizations is visible over a wide region in recip-rocal space. In section II, we interpret our experimentalresults in the framework of ab–intio DFT calculationscomparing the lattice dynamics of pure Ge, the emptyGe framework structure and the Ba Ge Ni clathrate.By a direct comparison with the experimental findings,we highlight the roles played by unit cell complexity andguest atoms for the lattice dynamics, respectively. Weconclude that the low lattice thermal conductivity is dueto a phononic filter effect. I. INELASTIC NEUTRON SCATTERING
Subsection a) of this paragraph, contains details onthe investigated sample, while in subsections b) and c) we provide the basic theoretical background of phononmeasurements by INS and explain how our study differsfrom previous measurements done on similar materials. In subsection d) we report our results together with ouranalysis and interpretation of the data. Finally, in sub-section e) we discuss technical details with respect to thevalidity and reliability of our data analysis. a) Sample preparation and physical properties A single crystal of the clathrate Ba Ni . Ge . (cid:3) . (space group Pm¯3n, a = 10.798(2) ˚A, ø= 8 mm andheight= 30 mm), as shown in Fig. 1 a), was grown fromthe melt, using the Bridgman technique. A detailedanalysis of its crystal structure and its thermoelectricproperties were reported in [24]. This study revealedBa Ni . Ge . (cid:3) . to be a thermoelectric (TE) n-typemetal with a relevant value for the dimensionless TE fig-ure of merit ZT at high temperatures, thanks to its verylow thermal conductivity of about κ ∼ . b) Neutron intensity, phonon dynamical structurefactor In case of coherent inelastic nuclear scattering by aphonon of branch j , with energy ω q,j and polarizationvector ξ jω q,j , the neutron diffusion function is written as: S ph ( (cid:126)Q, ω ) = n ( ω ) | F jD ( (cid:126)Q ) | ω q,j δ ( ω − ω q,j ) δ ( (cid:126)Q − (cid:126)q − (cid:126)G ) (2)where n ( ω ) = − exp ( − (cid:126) ω/k B T ) is the Bose factor, (cid:126)Q = (cid:126)q + (cid:126)G is the scattering vector given by the nearest reciprocallattice vector (cid:126)G and the phonon wave vector (cid:126)q . Finally F jD ( (cid:126)Q ) is the dynamical structure factor (DSF) definedas F jD ( (cid:126)Q ) = (cid:88) i e − W i ( (cid:126)Q ) b i √ M i e i (cid:126)Q.(cid:126)r i { (cid:126)Q.(cid:126)ξ ij ( (cid:126)Q ) } (3)where b i , (cid:126)r i , M i and W i ( (cid:126)Q ) are coherent scatteringlength, fractional coordinates, mass and Debye-Wallerfactor of the i’th element, respectively. The scalar prod-uct { (cid:126)Q.(cid:126)ξ ij ( (cid:126)Q ) } contains the phonon polarization and canthus be used to distinguish longitudinal and transversalphonon modes by choosing the the appropriate combina-tion of phonon wave vector (cid:126)q and reciprocal lattice vector (cid:126)G . The polarization vectors of longitudinal and transver-sal phonons are parallel and perpendicular to the phononwave vector (cid:126)q , respectively. In our case, as sketchedin Fig. 1 c), measurements were conducted around the (cid:126)G = (006) reflection, scanning along (cid:126)q [110] . In this set-ting longitudinal phonons, polarized parallel to (cid:126)q , areperpendicular to (cid:126)G and therefore only transverse compo-nents are selected.It is then worth to denote the diffusion function in FIG. 1: (Color online) a) : The measured single crystalof Ba Ni . Ge . (cid:3) . [24] b) : Elastic neutron scans aroundthe Bragg peak (600) along the transverse, [110], and lon-gitudinal, [100], directions. c) : Bragg peak intensities inthe { [100]; [011] } plane, calculated for a neutron wave vector k = 2 . − . The pointsize is proportional to the nuclearelastic structure factor. The intensity of the strongest Braggpeak (600) is then compared to those used in [5] and [6]. Assketched and explained in subsection I.b) , INS measurementsof TA phonons are obtained by scanning the reduced wavevector, q , along the [011] direction from the Bragg peak (600). the long wavelength limit, for | (cid:126)q | << | (cid:126)Q | . It can beshown that for acoustic phonon modes, close to the Bril-louin zone center, all atoms are vibrating in phase andwith equal displacements ξ ij ( (cid:126)Q ) / √ M i . Consequently, for T >> ω , the diffusion function, Eq. 2, becomes propor-tional to the nuclear structure factor (i.e. the Bragg in-tensity): S ph ( (cid:126)Q, ω ) ∝ G | F B ( (cid:126)Q ) | ω j ( (cid:126)q ) (4)Thus, the intensity is proportional to the nuclear struc-ture factor of the respective Bragg peak. Moreover, closeto the Brillouin zone center and for pure acoustic phonons S ph ( (cid:126)Q, ω ) × ω j ( (cid:126)q ) is constant. c) Measurement strategy and data analysis INS experiments were carried out on the thermal tripleaxis spectrometer 2T at the Orph´ee reactor of the Labo-
FIG. 2: (Color online) a) : Energy scans close to the Braggpeaks Q=(600) or Q=(222), at wave vector q = 0 . A − in transverse geometry. b) : Energy scans measured at theconstant wave vectors (cid:126)Q = (611), open circles, and (cid:126)Q =(5 , . , . E , E and E were usedto fit the total phonons line shape (black line). The instru-mental energy resolution, ∆ E inst =1.1 meV, is indicated bythe red line ratoire L´eon Brillouin (LLB, France). All measurementswere conducted at room temperature in fixed Q modewith a final wave vector of 2.662 ˚A − . To eliminatehigher-order harmonics graphite filters were used. Thesample was mounted in a vacuum chamber and aligned inthe plane { [100]; [011] } such that wave vectors of the formQ=2 π/a ( hkk ) were accessible as depicted in Fig. 1.c.Wide elastics scans around the Bragg peak (600) areshown in the Fig. 1. The bulk mosaicity is of 0.75 ◦ .In the long wavelenght limit the acoustic phonon inten-sity is directly proportional to the Bragg intensity asoutlined above, see Eq. 4. Therefore acoustic phononshave to be measured near strong nuclear Bragg reflec-tions. The main technical difference between the presentand previous INS experiments on clathrates is that wecould access a higher Q -range and thus measurementsaround the strongest Bragg peaks could be conducted.As sketched in Fig. 1 b, our measurement was conductedaround the strongest reflection (cid:126)G = (600), following thetransverse [0 kk ] direction, while previous INS studies oftransverse phonon branches were carried out around lessintense Bragg peaks (400), (222) and (330) [5, 6]. Asa consequence of the INS cross section detailed above,the TA phonons in our data appear much stronger thanthe guest phonon modes and with a significantly en-hanced resolution, when compared to the results in Refs.[5] and [6]. To emphasize this aspect, we compare inFig. 2 a) transverse phonon spectra measured at thesame wave vector | (cid:126)q | (0 . A − ) either around the Braggpeaks Q=(600) or Q=(222). Another consequence of themeasurement around the Bragg peak (cid:126)G = (600), whichfollows directly from energy and momentum conservationlaws, is that our measurements go up to 14 meV whiledata in Ref. [6] stop at 7 meV. This is an important ad-vantage which allows us to follow the spectral weight dis-tribution and transfer among the different acoustic andoptic phonon branches.Thanks to the high data quality, we are then also ableto perform an advanced and reliable analysis in order toextract properly the intrinsic energy width, the energyposition and the dynamical structure factor, or norm, ofthe contributing phonons. In fact, realistic INS measure-ments on a conventional (mono-detector) neutron tripleaxis spectrometer mean scanning the ( (cid:126)Q, ω ) space step bystep with a resolution function. The neutron intensity ata given point ( (cid:126)Q , ω ) is given by the convolution prod-uct of this instrumental resolution function, R ( (cid:126)Q, ω ), andthe diffusion function, S ( (cid:126)Q, ω ), which contains all the in-formation of the dynamics in the sample probed: I ( (cid:126)Q , ω ) = S ( (cid:126)Q, ω ) ⊗ R ( (cid:126)Q − (cid:126)Q , ω − ω ) (5)The resolution function is a 4-dimensional functionwhich couples the energy and the reciprocal space. Acut of this resolution function in the { (cid:126)q kk , ω } plane isshown in Fig. 5 a). As one can see, the size of thisfunction is not negligible with regard to the phononbranch dispersion. Therefore, in order to extractproperly the intrinsic position, width and DSF of aphonon from the measured scattering profile, we use afitting procedure which numerically convolutes a modelphonon cross section with the spectrometer resolution.It enables us to take into account the local curvature ofthe phonon branch over the range of the instrumentalresolution, i.e. the fact that the energy resolution for anacoustic phonon mode is not independent of its slope.Only this method allows for a reliable extraction of thecharacteristics of phonons from INS data. The DSFhas so far not been discussed in any neutron work onclathrates. (For a similar analysis on the skutteruditeCeRu4Sb13 we refer to [4]). (d) INS Results The energy scans, shown in Fig. 2 b), were performedfar from any acoustic phonon branch at the wave vectors(611) and (5 , . , .
1) to localize the low lying opticalphonon modes. The energy spectra at (611) were ana-lyzed by using three Gaussian profiles centered at ener-gies of E =5.5(0) meV, E =8.2(0) meV and E =10.8(5)meV with full width at half maximum (FWHM) val-ues of ∆ E ∼ E ∼ E ∼ E and E will beidentified as stemming from localized Ba motions in thelarge Ba(Ge,Ni) cages, while the peak E contains bothoptical phonon modes from the (Ge,Ni) host frameworkand guest phonon modes, in agreement with previous INS FIG. 3: (Color online)
Energy scans with constant Q =(600) + q (0 kk ) in the first BZ, k < E , E and E . For scans labeledby -E the negative energy region is shown (Stokes side). [5, 6, 9, 26, 27], Raman [28] and theoretical [9, 25] reports.In a next step the TA phonon branch was investigatedby performing energy scans around the Bragg peak (600)for a set of reduced phonon wave vectors, q = (0 kk ),covering an entire BZ (0 ≤ k ≤ k < k ≥ FIG. 4: (Color online)
Energy scans with constant Q =(600) + q (0 kk ) for k ≥ .
5. The solid black lines are fits asexplained in the text. Red lines and filled areas depict con-tributions of TA phonons while blue lines are representativesof the peaks at E , E and E . phonon cross section, applied for our fitting procedure,consists of a damped harmonic oscillator for TA phononsand three Gaussian functions (centered at energies E , E and E ) for the low-lying optical phonons as before-hand identified. In Fig. 2 red lines and shaded areasdepict the contributions of the TA phonons, while theblue lines represent the three optic-like excitations at E , E and E . The deconvoluted energy positions and dy-namical structure factors (DSF) of both the TA phononsand the three optic-like excitations are shown in Fig. 5 asfunctions of q .For k < FIG. 5: (Color online) a) Energy dispersions of the TA andthe three optic-like ( E , E and E ) phonons. The size ofthe error bars represent the phonon linewidth (FWHM). Theellipse is a cut of the instrumental resolution volume (axissizes: 0.03 ˚A − (0.034 rlu) along the (0 kk ) axis and of 0.24meV along the energy axis). b) DSF as function of q in thedirection (0 kk ). ing its dispersion. For wave vectors below about k =0.25,the energy width of the TA modes is resolution limitedand their DSF fluctuates around a constant value. In thelimit ω →
0, the slope of the TA phonon branch yieldsthe transverse sound velocity ν (011) t =2530 m/s, compa-rable to that one of Ba Ge Ga [6]. The dispersion ofthe TA mode deviates from linearity for energies aboveabout 4 meV ( ∼
46 K). For 0 . ≤ k ≤ .
4, althoughthere are still well defined TA phonons with resolutionlimited energy widths, one observes an abrupt decreaseof their DSF. The DSF of TA phonons drops by morethan 80% between k = 0 .
25 and k = 0 .
4, as seen in Fig. 5b). The analysis of the experimental data becomes moredelicate for wave vectors above k =0.4 where intensitiesand energy positions of the TA branch and the low lyingoptical excitations come closer. However, as can be ob-served in the raw data in Fig. 3 as well as in Fig. 6 whereonly the phonon spectrum measured at k=0.4062 is de-picted, there are always well defined TA phonon peakslocated at energies below the optical excitations. Theprofile of these peaks is continuous until a break at aboutthe energy E of the low lying optical excitation. Whilethe TA phonon branch fades away, the intensities of theguest phonon peaks at E and E change (see Fig. 5 b)).The DSF of the guest peak at E increases from a con-stant value for k ≤ q =0.43˚A − ( k =0.5) and then returns to roughly the same con-stant value.A similar q -dependence of the DSF is observed for theguest peak at E with the maximum slightly shifted toa higher q of about 0.54 ˚A − ( k =0.62). For the peakat E no sizable change is observed over the entire q -range. Note that the qualitative changes of the guestDSFs are clearly visible on the raw data in Figs. 3 and4 (compare e.g. data measured at k =0.34 to those ob-tained at k =0.46, k =0.62 and k =0.87). As evidenced inFig. 5 b), the q -dependence of the DSF seems to arisefrom a transfer of spectral weight between TA phononsand guest vibrations. As k increases, the spectral weightof the TA phonons is first transferred to the first guestphonon peak at E and then to the second one at E .Interestingly, similar effects have been observed in qua-sicrystals and large unit cell crystals [19, 20]. The energyof the guest phonons at E increases significantly duringthis process, shifting from 5.4 meV at the zone center to6 meV at k =0.5 without any sizable change in its energywidth (see Fig. 5 a)).To conclude, firstly, our data put in evidence a continu-ous dispersion of the TA phonon branch throughout thewhole BZ zone with a rapid bending together with a dras-tic decrease of the spectral weight as k increases beyond0 .
25 or for energies higher than 4 meV, as summarizedin Fig. 5 a) and b). The most surprising and relevant ef-fect observed is precisely that the spectral weight of TAphonons decreases abruptly without a strong broaden-ing of their energy profiles. In others words, within ourresolution, no finite energy damping or finite lifetime isneeded to fit the TA phonon profiles. Note that the lackof line broadening in the deconvoluted data is not meantto demonstrate the complete lack of anharmonic phononscattering processes, it only emphasizes that anharmoniceffects are by far not as big as assumed in previous stud-ies. Therefore it becomes evident that it is not correct toconsider a strong decrease of the relaxation time τ ( ω ) asfunction of energy in Eq. 1 to account for the effect ofguest phonon modes. Moreover, the TA phonon branchdisperses continously without any fingerprint of a gapinduced by a crossing with the low lying optical phononbranch in contrast to what is reported by [6]. This makesit obvious that adding resonant like relaxation times inEq. 1 to account for the effect of vibrations of the guestatoms is incorrect.Secondly, our data highlight a mechanism of spectralweight transfer between TA and guest phonon modeswhich occurs over a wide region of the BZ, from k =0.25to k =0.5. Such transfer of spectral weight was observedin the skutterudite CeRu Sb [4]. Note that along agiven direction across the BZ, neither an intensity changenor an energy dispersion would be expected for guestphonons, if they were assumed to be isolated oscillators.Since we observe changes of the spectral weight of both E and E guest phonons as well as a dispersion of thepeak E , the picture of a ”freely rattling” guest atom ina host cage is not applicable in the case of clathrates, aspreviously concluded from INS experiments on skutteru-dites [8]. FIG. 6: (Color online) a) Fit to the scan at k =0.4062 rlu,obtained by forcing a reduction of the intensity of the opticphonon mode at E (shaded in blue). The fits provide theupper limit of the intensity of the acoustic phonon as indicatedby error bars in Fig. 5 a). b) green area represents thesimulated profile of the phonon measured at k =0.3749 rluwith a finite energy width of Γ = 1 . (e) Validity and reliability of the data analysis Crucial for our data analysis is the ability to differen-tiate between the acoustic and optical parts of the mea-sured spectra even in the region where both contributionsare superimposed, i.e. for k ≥ k = 0 .
4, see Fig. 5b), with no evidence of any broadening in energy. Thusthe main effect of spectral weigth transfer between theTA and optical phonons occurs then well below k =0.4where the precision of the fits is excellent. We now dis-cuss the data analysis of the energy scan measured at k =0.4062, shown in Fig. 6, in which the distance in en-ergy between the TA phonon and the optical phonon at E is less than 1 meV. In our data analysis, the surpris-ing result is that, within our resolution, no finite energywidth is needed to fit the profile of the TA phonons. Theobserved energy width arises simply from the integrationof all the phonon modes at 0 . ± δk contained in theinstrumental resolution function. However, one can tryto force the occurrence of a finite energy width by forcingthe intensity of the E peak contribution to decrease ascompared to the fitting model proposed in Fig. 3. Theresult, shown in Fig. 6 a), clearly yields a poorer fit withan intrinsic energy width of 0.45 meV and an increaseof about 25% of the spectral weigth of the TA mode.The resulting limits for the TA phonon intensity are in-dicated by the red line on the y-axis in Fig. 5 b). We thusclearly evidence that, independtly of the fitting model,the acoustic phonon looses abruptly its intensity whenapproaching E .As demonstrated above, the upper limit for the TAphonon energy width is roughly 0.45 meV. In a sim-ple Debye approach of the thermal conductivity basedon Eq. 1 and on our measured TA phonons disper- FIG. 7: (Color online)
Calculated dispersion curves along thereciprocal space direction (6 kk ) for a) Ba Ge Ni , b) Ge and c) diamond Ge. Modes are color coded with respect totheir PR. While green means a high PR, red and black standfor intermediate and low PR, respectively (see text). TheEnergy axis of panel a) is rescaled by a factor 1.4 to matchthe experimental data. sion (see below), we estimate that the relaxation timethat would be required to reproduce the low experimen-tal value of the thermal conductivity in our material isof about τ ∼ . k =0.3749 with afinite energy width of 1.2 meV, dashed line, and we com-pare it with the result of our best fit. It evidences thatthe main microscopic mechanism responsible for the lowthermal conductivity cannot be a large relaxation time. II. DFT CALCULATIONS
To deepen our understanding of the INS measurementsand of the coupling mechanism between the guest andhost phonons, we have performed an ab initio densityfunctional theory (DFT) study of the lattice dynamicsand the INS cross section in the cubic high-symmetryconfiguration Ba Ge Ni and an empty Ge frameworkmodel system.Ab-initio density functional theory (DFT) calculationswere conducted using the DFT code VASP [29] apply-ing the projector augmented wave generalized gradientmethod (PAW-GGA). First the unit cell of Ba Ge Ni was relaxed to its electronic ground state, using high pre-cision settings with a 4 × × − eV/˚A. After reaching the ground state, symmetry non-equivalent displacements were introduced and the re-sulting Hellmann-Feynman forces were calculated andused to determine the dynamical matrix by use of thePHONON package [30]. Different displacements, rangingfrom 0.03 ˚A to 0.08 ˚A, were investigated, yet no signif-icant differences in the phonon properties could be ob-served. The presented calculations were then conductedfor displacements of 0.05 ˚A. By solving the dynamicalmatrix for selected q -values, phonon eigenfrequencies andcorresponding eigenvectors can be determined within theharmonic approximation. From these eigenvectors and FIG. 8: (Color online)
Calculated dispersion curves as in Fig.7, over the full energy range. eigenvalues it is then possible to calculate diffusion func-tion and DSF following Eqns. (2) and (3). a) Dispersion curves
Figures 7 and 8 depict a comparison of the phonon dis-persion curves of Ba Ge Ni and Ge along the recip-rocal space direction [0 kk ]. As for other complex systems[22], a rescaling of the energy axis is necessary to achievegood agreement between the absolute energy values ofcalculated and experimentally determined acoustic andlow-lying optical branches in Ba Ge Ni . The rescalingparameter of 1.4 is relatively large, pointing to an under-estimated Ba-Ge interaction within the DFT approach.Colors in Figs. 7 and 8 represent the phonon participationratio (PR), defined for an eigenmode j as [23] p j ( ω ) = (cid:32) N (cid:88) i =1 | ξ ij ( q ) | M i (cid:33) /N N (cid:88) i =1 | ξ ij ( q ) | M i . (6)The PR is calculated from the phonon eigenvector ξ ij ( (cid:126)q )of a given mode and the masses of the respective atoms.A PR close to 1 (green) means coherent displacementsfor all atoms of the same species and thus a collectiveexcitation. Localized excitations, such as rattling modes,have a PR close to 0 (black), meaning that only a fewatoms have large displacements, while the others areonly slightly or not at all displaced.The comparison of the two cage structures with thediamond Ge enables to disentengle different reasonsfor a reduction of κ l . If we first compare Ge anddiamond Ge we find that many phonon modes in thecomplex structure are less dispersive and evidence a lowPR. Interestingly this corresponds nicely to the resultsobtained in reference [16], where already the clathrateframework is found to reduce κ l . While neither pure Genore the empty Ge structure evidence modes with avery low PR, the flat dispersionless branches between6 and 8 meV in Ba Ge Ni clearly do and can thusbe identified as Ba rattling modes. These brancheshave strong impact on the acoustic phonon modes.First, they force the acoustic branch to bend over atlower energies, thus decreasing the group velocity ofthe acoustic phonons. Furthermore, they successivelyhybridize with the acoustic branch and lead to animportant reduction of the PR of the hybridized acousticwaves, which will contribute to the dissipation of theheat current through the material. More than a simpleanti-crossing, the rattling modes act as a low pass filterfor propagating phonons which prevents propagation ofheat carrying acoustic phonons with energies higher thanthe Ba rattling energies. Filtered phonons are not onlydispersionless modes with small group velocities but arealso characterized by a low or intermediate PR, meaningthat only few atoms are actually involved. Unfilteredphonons are unaffected by the rattling phonons andtheir efficiency to propagate heat depends on the usualanharmonic scattering processes.If we now moreover investigate the simulated INS scat-tering profiles, a sequence of spectral weight transfer fromacoustic phonons to the different rattling modes is visible,as was found for the experimental data of Ba Ge Ni ,as k is increased (see Fig. 9). However, these effects areclearly more pronounced in the experiment, in which hy-bridization starts at lower k values and leads to a fasterdecrease of the DSF of the TA phonon modes. This mightbe attributed to the fact that, in our simulation, we usea model which is free of disorder. Indeed it can be shownthat the Ni content strongly influences the cage size andthe related Ba rattling frequencies. b) Disorder Due to the size limit which is imposed to ab-initio cal-culation, an investigation of disorder is rather difficult.In the Ba-Ge-Ni system, it is yet possible to create ide-alized structures with different Ni content, which stillevidence high symmetry and can be described in a unitcell approach, thus enabling us to get some idea aboutthe influence of structural disorder. The structures ofBa Ge Ni and Ba Ge Ni differ on the 6c position,which is the only site that exhibits disorder in the Ba-Ge-Ni clathrate structure. While it is fully occupied byNi atoms in the case of Ba Ge Ni , it contains four Niand two Ge, which are symmetrically distributed in caseof our Ba Ge s Ni model system. Since the 6c posi-tion is located in the large 24-atom trapezohedron, thiscage indeed is influenced by the Ni content. An increas-ing Ge and vacancy content on this position thus intro-duces distortions and breaks the symmetry of this cage.This symmetry–breaking is expected to broaden the fre-quency distribution of the encaged Ba rattling modes.And indeed by comparing the DSF of Ba Ge Ni andBa Ge Ni this is evidenced (see Fig. 10). The low ly-ing Ba rattling mode which is seen on the top panel of FIG. 9: Constant q scans in the first BZ, determined forthe same q values as measured experimentally. The finitelinewidths is obtained by convolution with a Gaussian func-tion of 0.1 meV width. Fig. 10 is splitted in two branches on the bottom panel,thus confirming that disorder yields a broadening of therattling modes. c) Phononic filter effect
To highlight the effect of such a phononic low pass filteron the thermal transport, we compute the contribution ofLA phonons, propagating along the [011] direction, to thethermal conductivity of pure Ge, Ge and Ba Ge Ni .From our ab–initio calculations for these three systems,we extract ρ ( ω ) and v ( ω ) for the chosen phonon branch.We then evaluate Eq. (1) by assuming a constant phononlifetime, τ ( ω ) = 12 ps, estimated from the upper limit ofthe experimentally evidenced linewidth of TA phononsin the Ba-Ge-Ni clathrate. Moreover we replace ω max bya cutoff frequency ω cut , which is chosen to be the fre-quency at which the LA phonon branch looses its acous-tic character, i.e. the frequency at which the PR of theacoustic mode strongly decreases (change from green tored in Figs. 7 and 8). The introduction of this cutoff fre-quency takes the effect of the rattling modes into accout,which are responsible for a further flattening of the opti-cal branches, as was discussed in subsection . Figure 11shows the frequency dependence of κ [011] LA ( ω ) along a LAbranch at room temperature, as well as the resulting κ ( T )in the inset. The reduced thermal conductivity of Ge with respect to pure Ge shows that the cage structure FIG. 10: DSF of a) Ba Ge Ni and b) Ba Ge Ni alongthe direction (6 , ξ, ξ ). itself is significantly decreasing the lattice thermal con-ductivity. An additional reduction of κ in Ba Ge Ni arises from the low energy Ba rattling phonons whichlower the cutoff energy. Note that if for pure Ge we usethe same cutoff frequency as for Ge , this results, de-spite the differences in v ( ω ), in an similarly low thermalconductivity, as can be seen from the dashed curve inFig. 11. Thus, to lowest order, the thermal transport byacoustic phonons in the Ba-Ge-Ni-clathrates correspondsto that of a simple Debye system with the sound velocityof pure Ge and a cutoff frequency corresponding to theenergy of the first rattling mode of the guest atoms.It is of course true that to fully calculate the thermalproperties of a material optical phonon modes also haveto be taken into account, therefore our model calculationhas to be understood as a first order approximation. Thisis justified, since the main contribution to heat transportcan be attributed to acoustic phonons, while the opticalmodes above the introduced cutoff frequency can be neg-elected in a first order approach due to smaller lifetimesand group velocities. FIG. 11: (Color online)
Calculated frequency dependence ofthe lattice thermal conductivity per unit frequency for pureGe, Ge and Ba Ge Ni at room temperature along a LAbranch of the dispersion curve. The dashed blue line shows κ LA ( ω ) for pure Ge with the same cutoff that is applied forGe . The inset shows the resulting κ ( T ) for the three systemsobtained from the average over TA and LA branches of the(0 kk ) direction. The curves for Ge (black) are scaled by afactor 0.2 (main panel) and 0.1 (inset) for better readability. III. CONCLUSION
We have performed a high resolution INS study of thelattice dynamics of one of the prototypal thermoelectricclathrate systems, Ba-Ge-Ni. Our data exclude an in-terpretation in terms of an isolated oscillator, since co-herent modes of guest and host system are evidenced.Furthermore our results point out that Umklapp scat-tering is not able to account for the reduced κ L in theBa-Ge-Ni system. The evidenced phonon lifetimes are atleast an order of magnitude bigger than what would beexpected if such a low κ L was to be reached by Umk-lapp scattering processes. Therefore a new mechanismwas suggested to explain the low lattice thermal conduc-tivity in clathrates – the phononic low-pass filtering ofacoustic phonon modes, having its origin in both looselybound guest atoms and structural complexity of the hostframework. We are confident that these new insights willpromot further studies in clathrates and other guest-hostmaterials.We thank B. Hennion, retired from LLB, and S. Mer-abia, from the LPMCN, for discussions. This work wassupported by the EU Network of Excellence on Com-plex Metallic Alloys (Grant No. NMP3-CT-2005-500140)and within the European C-MAC network. L. N. and S.P. acknowledge support from the Austrian Science Fund(project P19458-N16). ∗ To whom correspondence should be addressed;E-mails: [email protected] or [email protected][1] D. Walton, H. A. Mook, R. M. Nicklow, Phys. Rev. Lett. , 412 (1974)[2] B. Chazallon, Phys. Chem. Chem. Phys. ( ), 4809 (2002)[3] C. Gutt, J. Baumert, W. Press, J. S. Tse, S. Janssen, J.Chem. Phys. , 3795 (2002)[4] C. H. Lee, I. Hase, H. Sugawara, H. Yoshizawa, H, Sato,J. Phys. Soc. Jpn. , 123602 (2006)[5] C. H. Lee, H. Yoshizawa, M. A. Avila, I. Hase, K. Kihou,T. Takabatake, J. Phys. Conf. Series , 012169 (2007)[6] M. Christensen, A. B. Abrahamsen, N. B. Christensen,F. Juranyi, N. H. Andersen, K. Lefmann, J. Andreasson,C. R. H. Bahl, B. B. Iversen, Nature , 811 (2008)[7] G. Nilsson, G. Nelin, Phys. Rev. B , 364 (1964)[8] M. M. Koza, M. R. Johnson, R. Viennois, Mutka, L.Girard, D. Ravot, Nature , 805 (2008)[9] M. M. Koza, M. R. Johnson, H. Mutka, M. Rotter, N.Nasir, A. Grytsiv, P. Rogl, Phys. Rev. B , 214301(2010)[10] C. P. Yang, H. Wang, K. Iwasa, M. Kohgi, H. Sugawara,H. Sato, J. Phys. Condens. Matter , 226214 (2007)[11] R. O. Pohl, Phys. Rev. Lett. , 481 (1962)[12] M. V. Klein, Phys. Rev. , 839 (1969)[13] J. Yang, D. T. Morelli, G. P. Meisner, W. Chen, J. S.Dyck, C. Uher, Phys. Rev. B , 165207 (2003)[14] J. L. Cohn, G. S. Nolas, V. Fessatidis, T. H. Metcalf, G.A. Slack, Phys. Rev. Lett. , 779 (1999)[15] N. J. English, J. S. Tse, Phys. Rev. Lett. , 015901(2009)[16] J. Dong, O. F. Sankey, C. W. Myles, Phys. Rev. Lett. , 2361 (2001)[17] E. S. Toberer, A. Zevalkink, G. J. Snyder, J. Mater.Chem , 15843 (2011)[18] E. Macia, M. de Boissieu, in Complex Metallic Alloys:Fundamentals and Applications’, Wiley-vch, p 41-115 (2011)[19] T. Takeuchi, N. Nagasako, R. Asahi, U. Mizutani, Phys.Rev. B , 054206 (2006)[20] M. de Boissieu, S. Francoual, M. Mihalkovic, K. Shibata,A. Q. R. Baron, Y. Sidis, T. Ishimasa, D. Wu, T. Lo-grasso, L. P. Regnault, F. Gahler, S. Tsutsu, B. Hennion,P. Bastie, T. J. Sato, H. Takakura, R. Currat, A. P. Tsai,Nature Mat. , 977 (2007)[21] L. T. K. Nguyen, U. Aydemir, M. Baitinger, E. Bauer,H. Borrmann, U. Burkhardt, J. Custers, A. Haghighirad,R. Hofler, K. D. Luther, F. Ritter, W. Assmus, Y. Grin,S. Paschen, Dalton Trans. , 1071 (2010)[22] H. Euchner, M. Mihalkovic, F. Gahler, M. R. Johnson,H. Schober, S. Rols, E. Suard, A. Bosak, A. P. Tsai, S.Ohhashi, C. Gomez, S. Lidin, J. Custers, S. Paschen M.de Boissieu, Phys. Rev. B , 144202 (2011)[23] J. Hafner, M. Krajci, Phys. Condens. Matter , 2489(1993)[24] L. T. K. Nguyen, U. Aydemir, M. Baitinger, E. Bauer,H. Borrmann, U. Burkhardt, J. Custers, A. Haghighirad,R. Hofler, K. D. Luther, F. Ritter, W. Assmus, Y. Grin.,Dalton Trans., ( ), 1071 (2010)[25] J. J. Dong, O. F. Sankey, G. K. Ramachandran, P. F.McMillan, J. Appl. Physics ( ), 7726 (2000)[26] M. Christensen, F. Juranyi, B. B. Iversen, Physica B,( ), 505 (2006)[27] S. Johnsen, M. Christensen, B. Thomsen, G. K. H. Mad-sen, B. B. Iversen, Phys. Rev. B ( ), 184303 (2010)[28] Y. Takasu, T. Hasegawa, N. Ogita, M. Udagawa, M. A.Avila, K. Suekuni, T. Takabatake, Phys. Rev. B ( ),134302 (2010)[29] G. Kresse, J. Furthmuller, Phys. Rev. B , 11169 (1996)[30] K. Parlinski, Software phonon, version 4.28 (2005). http://wolf.ifj.edu.pl/phonon/Public/phrefer.htmlhttp://wolf.ifj.edu.pl/phonon/Public/phrefer.html