Avoided Crossing of Rattler Modes in Thermoelectric Materials

Transcription

1 1 Supplementary Material for Avoided Crossing of Rattler Modes in Thermoelectric Materials M. Christensen, 1 A. B. Abrahamsen, N. Christensen,,3,4 F. Juranyi, 3 N. H. Andersen, K. Lefmann, J. Andreasson, 5 C. R. H. Bahl, B. B. Iversen. 1* 1 Center for Energy Materials, Department of Chemistry and Interdisciplinary Nanoscience Center inano, University of Aarhus, DK-8000 Aarhus C, Denmark Materials Research Department, Risø DTU, DK-4000 Roskilde, Denmark, 3 Laboratory for Neutron Scattering, Paul Scherrer Institute, CH-53, Villigen, Switzerland 4 Nano-Science Center, Niels Bohr Institute, University of Copenhagen, DK-100 Copenhagen, Denmark 5 Departement of Applied Physics, Chalmers University of Technology, Göteborg, Sweden * bo@chem.au.dk Synthesis The n-type Ba 8 Ga 16 Ge 30 samples were obtained from stoichiometric amounts of pure metals Ba (99.5%), Ga (99.99%) and Ge ( %). The metals were handled in a glove box under inert atmosphere and placed in an Al O 3 crucible, which in turn was placed in a stainless steel bomb. The bomb was heated in a vertical tube furnace to 1100 C followed by cooling to room temperature at 50 C/hour. The resulting product was bulk clathrate material. The large single crystal of n-type Ba 8 Ga 16 Ge 30 was grown by the Czochralski method. The starting material was 30 g of sample synthesized by the method described above. The presynthesized compound was placed in a glassy-carbon crucible and heated by induction to the melting point. A seeding crystal was mounted on a pulling rod and brought to the surface of the melt. The crystal was pulled with an approximate speed of 5 mm/hour. Figure 1a shows a photograph of the Czochralski pulled single crystal used for neutron triple axis spectroscopy. The single crystal has a mass of 13 g. Crystal structure The clathrate type I structure belongs to the cubic spacegroup Pm-3n and the lattice spacing is a = Å. The Ga/Ge framework is defined by the 6c (¼,0,½), 16i (x,x,x) = 0.184, and 4k (0,0.309,0.118) sites. The Ba guest atoms at a (0,0,0) resides inside the small 0 atom dodecahedral cages and the rattling Ba guest atom is located around the 6d (¼,½,0) inside the larger 4 atom tetradecahedral cages. Inelastic neutron measurements of phonons A phonon is a wave of displacements of atoms in a solid and it can be detected by scattering neutrons on the displaced atomic planes. The neutron-phonon interaction will cause changes in energy and propagation of the neutrons through creation or annihilation of phonons. Generally the energy of a phonon is given by a dispersion relation ω s (q), which gives the relation between the energy E = ħω s and the propagation vector q of the phonon. The number of phonon branches s = 1,.., 3n is determined by n, the number of atoms in the unit cell. For coherent elastic nuclear scattering the corresponding nuclear structure factor for neutrons takes the form 1 : i j Wj( Q) F ( ) b e Q = Qd e (1) N j j

2 where b j, d j and W j (Q) are the scattering length, fractional coordinates, and the Debye-Waller factor of the j th element, respectively. The sum is taken over all elements in the unit cell and the scattering vector is given by Q. When the scattering vector (Q) and reciprocal lattice vector (G) coincide (Q = G) Bragg scattering occurs. The coherent inelastic nuclear scattering for such a lattice can be written using a similar expression known as the dynamical structure factor: bj iqd j Wj( Q) FD ( Q) = e e ( Q ε js ) () j m j Now the j th element s mass m j and the phonon polarization vector of mode s enter the equations. The scattering vector Q = G + q is given by the nearest reciprocal lattice vector G and q the directional wave vector. The ε js is known as the phonon polarization term and it causes the largest problems in evaluating the dynamical structure factor as it can only be evaluated if the force constants between the individual atoms are known. The integrated intensity for a constant-q scan can be written as 1 : A nb + 1 neutronenergy loss I = FD ( Q ) (3) ωqs nb neutronenergy gain where A is a scale factor depending on sample size, source flux, counting time and other features of the experimental setup. The ω qs is the energy of the phonon mode and n B = (exp(ħω qs /k B T)-1) -1 is the Bose-factor. In the high temperature case the situation can be simplified and the intensity can be approximated by: kt B I ~ A F ( ) D Q (4) ωqs Thus, the intensity is inversely proportional to the energy of the mode squared. The dynamic structure factor has a Q-dependence both from exponential term exp(iq d j ) and the phonon polarization term (Q ε js ). The phonon polarization vector may change considerably as function of Q and therefore cause abrupt changes in the integrated intensity. For acoustic modes close to the Brillouin zone center all atoms are vibrating in phase. Consequently the phonon polarization ε js is described by a unit vector and the equation can be rewritten as: lim kt B G ( ) I = A F 0 N Q (5) q ωqs M The question is to which q the intensity can be evaluated in this way. The assumption that the atoms are vibrating in phase is certainly not valid as q approaches the zone boundary. In the low q-region, the phonon polarization vector term can be used to distinguish longitudinal and transversal modes by choosing the wave vector direction q at specific reciprocal lattice points G with care. The long wavelength longitudinal and transverse modes are parallel and perpendicular to the wave vector q, respectively. Therefore by scanning along q = [110] at the G = (004), the ε T being perpendicular to q becomes parallel to G and since Q = G + q the scan gives only the transverse mode. If the polarization vector ε js is assumed to be constant the transverse intensity will change due to rotation of the Q-vector; I Q =I G (G ε js ) /(Q ε js ). For I ½½4 this corresponds to a reduction of the transverse mode at the zone boundary of about 1%, compared to I 004. For a scan at (330) along [110] the phonon polarization vector is parallel to the longitudinal direction hence the transverse mode is suppressed. In case of the () reflection both the transversal and longitudinal modes are expected to contribute from these arguments. Figure 1b) schematically shows reciprocal space with the relevant vectors. However the assumption of a constant polarization vector is in most

3 3 cases not valid at higher Q values. The changes in ε js are difficult to predict as they in principle depend on the force constant between all atoms. a) b) Figure 1a) Photograph of the 13 g single crystal used in neutron triple axis spectroscopy experiment. b) Reciprocal space spanned by [110] and [001] for Ba 8 Ga 16 Ge 30. With the chosen configuration the green area gives the reachable area of the spectrometer. The red boundary shows an area within which the energy range is limited due to equipment collisions. Reciprocal scattering vectors (G) have been drawn to the reciprocal lattice points (004), (), and (330). The chosen wave vector q = [110] is shown along with the resulting scattering vector Q = G + q. The phonon polarization vectors in the long phonon wavelength limit are drawn for the transversal (ε T ) and longitudinal (ε L ) acoustic mode perpendicular and parallel to the wave vector q, respectively. The polarization term is evaluated as the dot product between the polarization vector and the scattering vector (Q ε js ), therefore at the (330) along [110] only the longitudinal mode is observed. Likewise (004) is dominated by the transverse mode and for () both transverse and longitudinal modes will be observed. The triple axis neutron spectrometer RITA-II The cold neutron triple axis spectrometer RITA-II at the Paul Scherrer Institute (PSI), Switzerland was used for the experiment. The energy of the incoming neutrons was selected by a 5 blade vertically focusing pyrolytic graphite (PG) monochromator with a mosaic spread of 40. The beam was conditioned by an 80 collimator before passing through the monitor, primary slit, and hitting the sample. A cooled Be filter with a coarse radial collimation was placed between the sample slits and the detector tank to suppress neutrons with energies higher than 5. mev. The analyzer consists of 7 PG blades (Mosaic spread 40 ) scattering neutrons with incoming energy of 5 mev onto different parts of the dimensional detector with 18x18 channels. A coarse radial collimator was mounted in front of the detector. The setup is referred to as imaging mode, because each of the analyzer blades collects a different momentum transfer vector q i = (hkl), i = 1-7 at the same neutron energy transfer ω. The data were collected with fixed analyzer energy set to 5 mev for all blades and neutron energy loss was changed by varying the incoming neutron energy from mev. The efficiency of the 7 analyzer blades were normalized by the incoherent scattering from the sample. The analyzer blades cannot be positioned individually, but they are aligned fairly well along the [110] direction. The Ba 8 Ga 16 Ge 30 sample of mass 13 g was glued onto an aluminum mount with epoxy and shielded by Cadmium foil to avoid scattering from the glue and the aluminum mount. The sample was

4 4 mounted in an evacuated closed-cycle cryo-refrigerator to control the temperature and reduce background scattering from air. The sample was oriented with the [110] and [001] directions spanning the scattering plane. Energy scans were performed with 0.5 mev steps and monitor counts of (corresponding to 15-5 min per data point.). Different scattering vectors were used in order to cover the Brillouin zone and repetitions of scans were performed to improve statistics. The individual data scans were merged and shown on a grid. The data treatment was carried out using Matlab and an example for () is shown in figure. Figure : Example of the data treatment (left) the projection of the original scans at () along [110], the circles correspond to individual measured data points in q-energy space and the color gives the intensity. (middle) The merged data and (right) the linear interpolation between the merged data. Gaussian fits, were made to constant-q scans extracted from the data. Examples of the constant-q scans are shown in figure 3 of the main paper. Instrument resolution and phonon lifetimes: The instrumental broadening of the energy of the flat modes depends of details in the instrumental set-up and varies with energy transfer. The broadening can be measured experimentally only at the elastic line (E = 0 mev), using the incoherent scattering. The incoherent scattering from the Ba 8 Ga 16 Ge 30 sample gave an energy resolution of 0.18() mev. To estimate the broadening at nonzero energy transfers, we have performed Monte-Carlo ray-tracing simulations with the package McStas. 3 We have used a detailed model of RITA-, validated against several sets of standard instrument alignment data. 4 Further, we have used a model of the sample with realistic values for beam attenuation from absorption and incoherent scattering. The resolution widths were obtained through virtual experiments, i.e. simulated scans through a flat inelastic Einstein-like mode. 5 The scans were taken as 1 point scans though the peak, using 10 7 rays (1 minute) per point. For simplicity, only the central analyzer blade was used in the simulations. E (mev) width (mev) 0.19(4) 0.341(7) 0.395(7) 0.406(9) The very good agreement between the simulated and measured resolution of the elastic line is a good indicator that the used model is correct. The McStas simulation is only strictly valid for flat modes.

5 5 Possible rattler modes in Ba 8 Ga 16 Ge 30 The unit cell of Ba 8 Ga 16 Ge 30 as shown on figure 1 in the article contains 0 atom cages holding two of the 8 Ba rattler atoms and larger 4 atoms cages holding the remaining 6 of the Ba rattler atoms. The larger cages have three different orientations with respect to the main unit cell a-axis. One way to illustrate these orientations is to project the cages onto the ab-plane and indicate the plane of the cages by unit vectors denoted e x and e y, whereas the vertical movement in a cage is denoted e z. This is shown on figure 3, where an acoustic phonon is assumed to propagate along the a-axis. The displacement of the atoms is assumed to be either along the propagation direction causing a longitudinal polarization vector e L or perpendicular to the propagation direction causing a transverse polarization vector e T. Thus it is readily seen that the acoustic phonon will interact differently with the rattlers in the three types of cages. The 3 types of cage orientations have two rattler atoms each and with 3 degrees of freedom (x,y,z) there are 18 possible rattler modes. The energy of the rattlers movement has been predicted using molecular dynamics simulations by Dong et.al. 6 and it was found that a rattler movement in the xyplane spanned by e x and e y of the cage would correspond to an energy of ε xy = mev, whereas a movement along the e z direction have ε z ~ 8. 0 mev. The triple axis neutron scattering technique will probe the atomic displacements component along the scattering vector Q and one can thereby separate the longitudinal and transverse motion by measuring along the principal axis of a cubic crystal. The different rattler modes are listed in table 1 for both a longitudinal and transverse propagation along the principal [100] axis. Figure 3. Projection of cage chains in the unit cell oriented along the [100] (blue), [010] (red) and [001] (green) directions. By assuming an acoustic phonon propagating along the [100] axis then the longitudinal polarization vector e L indicate displacements of the cage atoms primarily along the propagation direction, whereas the transverse polarization vector e T indicate displacement perpendicular to the propagation direction (there is an additional e T pointing out of the paper). The interaction of the acoustic phonon with the local rattler motion will now depend on the cage orientation. Thus an e z motion inside the blue cage and an e xy motion in the red and green cage will correspond to the longitudinal polarization of the rattler motion. The yellow atoms indicate the projected position of the Ba in the small cages.

6 6 Polarization Cage type Rattler motion Longitudinal e L Blue e z, a/ ~ 8 mev Red e xy, a ~ mev Green e xy, a Transverse e T Blue e xy,a/, e xy,a/ Red e z, a, e xy, a Green e xy, a, e z, a Table. Rattler motions along the longitudinal and transverse polarization of an acoustic phonons propagating along the [100] direction as shown on figure 3. The color refers to the three different orientation of the 4 atom cages. The direction of the rattler motion is indicated relative to the cage plane e xy spanned by the e x and e y vectors and the cage symmetry axis denoted e Z. The distance between the rattlers along the propagation direction is indicated by either a/ or by a. The energy of motions along e z has been predicted to be ε z ~ 8. 0 mev and the motion in the cage plane e xy is expected 6 in the range ε = mev. xy From table 1 it is seen that longitudinal modes with motion along the e z direction are expected at ε z ~ 8. 0 mev, whereas 4 longitudinal modes with rattler motion in the e xy plane and a separation of a are expected at lower energies ε = mev. xy There will be 4 transverse modes with rattler motion along the e z direction and a separation of a and they are again expected to have an energy of ε z. The remaining 8 transverse modes with motion in the e xy plane are divided into 4 modes where the separation of the rattlers is a/ and 4 modes where the separation is a. Thus there is a possibility to have an energy split between the two rattler motions in the e xy plane whereby εxy, a/ εxy, a. A minimal model for a rattler dispersion relation Using classical physics the longitudinal phonons can be simplified as a linear chain consisting of the cage wall and the guest atom. Each cage wall has mass M and the cages interact with one another through the spring constant K 1. The guest atom has mass m and interacts only with the neighboring cage walls through the weaker spring K. The situation is sketched in figure 4. Figure 4: Left: Illustration of a longitudinal rattler model with the hexagons representing the cage walls (M) and the yellow atom the guest atom (m). Interactions between cages are denoted K 1 and guest cage spring constants are denoted K. The displacement from the mean positions is denoted u j and v j for the cage and guest, respectively. A unit cell with lattice spacing a is shown in grey. Right: Illustration of a transverse rattler model where the displacement is perpendicular to the propagation direction along x. If the displacement along x is much smaller than d then one can write the force at u 1 as F = K1 d + ( u u1). The force along the y axis then becomes F y = F sin( α ) = K1( u u1 ). Thus the mathematical description is identical for the two models, but the spring constants represent longitudinal and transverse interactions respectively.

7 7 Based on the above assumption the equations of motion can be written as: d M u j() t = K1[ uj 1() t + uj+ 1() t uj()] t + K[ vj 1() t + vj() t uj()] t dt (6) d m v () j t = K[ uj+ 1() t + uj() t vj()] t dt In the case where K << K 1 the situation is equal to the classical one-atomic chain and if K 1 << K the situation is equivalent of the two-atomic linear chain. Furthermore, if m << M, with a finite K, the smaller mass will behaves as an independent oscillator with a constant q-independent frequency. For a general solution, we define the vibration frequencies as ω 1 =K 1 /M and ω =K /m solutions to the equation of motion will take the form: uj( t) = αaexp( iqjd iωqt) (7) vj( t) = Aexp( iq( j+ ½) d iωqt) Inserting this in the above equation and simplifying using β = m/m, we obtain the equations for the two vibration frequencies: 1 ωq = ω1 [1 cos( qd)] + βω [1 cos( qd /) α ] (8) ωq = ω [1 cos( qd / ) α] We now introduce γ = ω 1 /ω and solve the above equations with respect to α: 1 β γ[1 cos( qd)] ± (1 β γ[1 cos( qd)]) + 4βcos ( qd / ) α = (9) cos( qd / ) The final results are then found by inserting the two values for α into either of the dispersion relations above. Figure 5 shows the dispersions for different model parameters. The agreement between the simple model and the measured data is very convincing. Figure 5: Dispersion relation for different values of β and γ. (left β = 1/3 and γ =), (center β =1/3 and γ =1.5), and (right β = 1/1 and γ =1.5). [ABA can we get q/d* on x-axis?] The mass of ratio of β = 1/3 corresponds to β = 8m Ba /(16m Ga + 30m Ge ), this is not taking into account that the guest atom in the small cages might have a different interaction, than the atoms in the large cages. The β = 1/1 is obtained by assuming that each barium guest atom is surrounded by 4 host structure atoms β = m Ba /(16m Ge + 8 m Ga ) ~ 1/1. The small β value neglects that most of the host structure atoms are shared between different guest atoms. However it takes into accounts the interaction with the closest neighboring atoms. The spring constant difference between K 1 and K found from fits to the data points to the host structure spring being 15 times stronger than the spring attaching the guest atom to the host structure.

8 8 Powder time-of-flight inelastic neutron scattering measurements The cold neutron time-of-flight spectrometer FOCUS 7 at PSI, Switzerland, was used to measure the inelastic neutron scattering (INS) spectrum of n-type Ba 8 Ga 16 Ge 30. Approximately 10g of sample was loaded into a cylindrical 10 mm diameter aluminum sample holder. For the 300 K data the sample was placed in a closed-cycle refrigerator. The 00 and 100 K data were measured at a later beamtime and the experiment was performed using an Orange type ILL cryostat. The white beam from the quasi-continuous spallation source was first conditioned by a disc chopper running at 9568 Hz. This ensures sufficient spacing of the pulses in order to avoid frame overlap. The instrument was used in time focusing mode with the PG(00) monochromator selecting neutrons with wavelength equal to 4 Å. The Fermi-chopper had a frequency of Hz, resulting in the best energy resolution of about 0.5 mev at an energy transfer of 4.5 mev. The slits were set to match the sample size. The signal from the sample is detected by 375 detectors arranged in an upper, middle and lower bank in the Debye-Scherrer geometry. The efficiency of the detectors was normalized using a vanadium standard. The spectrum from the empty aluminum sample holder was measured and subtracted to remove the background. Each spectrum was measured for approximately 16 hours. The program NATHAN was used for data treatment. Raman scattering spectroscopy: The Raman scattering measurements were performed using a DILOR-XY800 spectrometer in a double-subtractive mode with a foremonochromator step employed to rid the spectra of the elastically scattered signal. For the excitation the λ = 515 nm (photon energy hν =.41 ev) laserline from a Ar + /Kr + laser was used and a liquid nitrogen cooled multi channel CCD (charge coupled device) detector was used for data collection. The energy resolution for this particular setting is about 0. mev (1.7 cm -1 ). For all measurements the samples were placed in an evacuated CryoVac microcryostat to avoid scattering from air-vibrations. Initial measurements at room temperature, using parallel and perpendicular relative polarizations of the incident and scattered signals, showed similar phonon activity with the employed scattering configuration. However, due to the highly reflective nature of the sample surfaces, the low energy part of the spectra obtained in the parallel scattering configuration have considerable background intensity from elastic scattering and direct reflectance. In order to limit this impact the temperature dependent measurements were done in a pseudo backscattering configuration with perpendicular relative polarization directions between the incident and scattered signal. 1. Shirane, G., Shapiro & S. M., Tranquada, J. M. in Neutron scattering with a triple-axis spectrometer : basic techniques, 73 (Cambridge University Press, Cambridge, 00).. Bahl, C. R. H. et al. Inelastic neutron scattering experiments with the monochromatic imaging mode of the RITA-II spectrometer. Nucl. Instrum. Methods Phys. Res, Sect. B 46, (006). 3. K Lefmann & K. Nielsen, Neutron News 10/3, 0 (1999); 4. L. Udby et al, in preparation (008) 5. K. Lefmann et al, accepted for J. Netr. Res. (008) 6. J. Dong, et. al., Journal of Applied physics 87, 776 (000) 7. Juranyi, F. et al. The D small angle detector project at the FOCUS time-of-flight spectrometer at SINQ. J. Neutron Research 14, (006).