Structure Change and Rattling Dynamics in Cu12Sb4S13 Tetrahedrite

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Energy, Environmental, and Catalysis Applications

Structure Change and Rattling Dynamics in Cu SbS Tetrahedrite; an NMR Study 12

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Nader Ghassemi, Xu Lu, Yefan Tian, Emily Conant, Yanci Yan, Xiaoyuan Zhou, and Joseph H. Ross ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.8b13646 • Publication Date (Web): 25 Sep 2018 Downloaded from http://pubs.acs.org on September 27, 2018

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ACS Applied Materials & Interfaces

Structure Change and Rattling Dynamics in Cu12Sb4S13 Tetrahedrite; an NMR Study Nader Ghassemi,∗,† Xu Lu,‡ Yefan Tian,† Emily Conant,† Yanci Yan,‡ Xiaoyuan Zhou,‡ and Joseph H. Ross, Jr.†,¶ †Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA ‡Department of Applied Physics, Chongqing University, Chongqing, 401331, China ¶Materials Science and Engineering Department, Texas A&M University, College Station, TX 77843, USA E-mail: [email protected]

Abstract We present a

63

Cu and

65

Cu NMR study of Cu12Sb4S13 the basis for tetrahedrite

thermoelectric materials. In addition to electronic changes observed at the Tc = 88 K metal-insulator transition, we find that locally there are significant structural changes occurring as the temperature extends above Tc , which we associate with Cu atom displacements away from symmetry positions. Spin-lattice relaxation rates (1/T1 ) are dominated by a quadrupolar process indicating anharmonic vibrational dynamics both above and below Tc . We used a quasi-harmonic approximation for localized anharmonic oscillators to analyze the impact of Cu rattling. The results demonstrate that Cu-atom rattling dynamics extend unimpeded in the distorted structural configuration below Tc , and provide a direct measure of the anharmonic potential well.

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ACS Applied Materials & Interfaces 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Introduction Massive utilization in the world’s energy market and tremendous greenhouse gas output, has intensified demand for more effective remedies of power generation. Thermoelectric technology has the potential to generate power in the form of electricity from heat. Large scale application of TE materials is limited by the high cost/efficiency ratio due to expensive/toxic components and complex synthesis processes. Tetrahedrites as a promising type of thermoelectric material have recently attracted great interest owing to the abundance of readily available copper, antimony and sulfur. 1–3 These materials allow a direct conversion between heat and electricity and thus could be good candidates to exploit wasted heat. 4 A favorable thermoelectric material has high electrical conductivity, high Seebeck coefficient and low thermal conductivity. Since these parameters are entangled inherently, it can be difficult to optimize one without affecting negatively the others. Recently, different methods, most notably the concept of electron-crystal and phonon-glass, have been applied to overcome this contraindicated nature. 5 We can quantify this collectively by a dimensionless figure of merit ZT =

S 2 σT , κ

where S is the Seebeck coefficient, σ the electric conductivity,

and κ the thermal conductivity, containing lattice (κl ) and electric (κe ) contributions. It is desired to maximize ZT and this can be done by decreasing thermal conductivity while increasing electric conductivity. Understanding phonon propagation is a key to achieving low κl . Materials with instabilities in their structure often have relatively low κl , and materials with large numbers of elements per unit cell and a complex crystal structure can create an enhanced phonon scattering probability leading to a low κl . 4 PbTe and SnSe are examples of thermoelectric materials where ferroelectric fluctuations and orbital interactions lead to lattice instability 6,7 which brings strong vibrational anharmonicity and thus leading to low thermal conductivity. Research on structure and atomic rattling led to discovery of anomalous vibrational properties and their connection to electronic behavior. In tetrahedrites very low thermal conductivity is associated with Cu anharmonicity, 8 in a process that is called "rattling" is 2


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a)

Cu I

Cu II

Sb

SI

S II

b) Cu II

Cu I

Figure 1: a) The room-temperature crystal structure of Cu12Sb4S13. b) Cu-I at the 12d crystallographic site shown tetrahedrally coordinated by S-I and Cu-II at 12e is trigonally coordinated by two S-I and one S-II. considered to play the vital role in this phenomenon. 3,9 As shown in figure 1, Cu12Sb4S13 is reported 10 to crystallize in a cubic structure (space group I-43m), wherein each Cu12d (Cu I) atom occupies a distorted tetrahedral site coordinated with four 24g S atoms and each Cu12e (Cu II) occupies a triangular planar site coordinated with one 2a S atom and two 24g S atoms. Note that a Cu II split-site variation 2,10 of the figure 1 structure is also proposed, while the structure becomes further modified 2,3 at the 88 K metal-insulator transition (MIT). In the current paper, we investigate NMR spectra and relaxation times relating to Cu atom motion in Cu12Sb4S13 tetrahedrite. We demonstrate gradual structure changes occurring above the MIT and show that a two-phonon Raman process involving strongly anharmonic local vibrational modes dominates NMR quadrupolar relaxation. A one dimensional anharmonic potential is presented to model the anharmonic local oscillators providing a measure of the dynamical atom displacements.

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Figure 2: X-ray powder diffraction data for tetrahedrite sample (blue crosses), with refined curve (green) and difference plot. In the inset, the most intense peak is Cu12Sb4S13 phase; small satellite at 29.8◦ due to Cu14Sb4S13. A third set of refined peaks is due to silicon sample holder.

Experiment Sample preparation and characterization Synthesis: A polycrystalline sample of tetrahedrite was synthesized by conventional solid state reaction. High purity elements (> 99.99%) weighed according to the stoichiometric ratio were sealed in quartz ampoules under a vacuum of ∼ 6.5×10−4 mbar. The sealed quartz tubes were firstly heated to 923 K at a rate of 0.3 K min−1 and kept at that temperature for 12 h before cooling to room temperature at 0.4 K min−1 . The ingots obtained were ground into fine powders and then cold pressed into pellets. The resulting pellets were further annealed at 723 K for a week in order to improve homogeneity. The annealed products were re-ground into fine powders for sintering. The final products were consolidated by the spark plasma sintering (SPS-625) system at 673 K for 5 minutes under a uniaxial pressure of 45 MPa.

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An electron microprobe (Cameca SXFive) was used to test composition. Measuring at 10 locations, we obtained a mean composition of Cu11.91Sb3.95S13, with statistical fluctuation corresponding to less than 1% relative uncertainty in these values. Systematic uncertainties for this instrument are also on order of 1%. Crystal Structural characterization was performed with powder x-ray diffraction (XRD) using a Bruker D8 diffractometer with CuKα radiation. GSAS II software was implemented for Rietveld refinement. 11 Tetrahedrite samples are known 12,13 to segregate into regions of Cu-rich Cu14Sb4S13 tetrahedrite (Cu14) in addition to Cu12Sb4S13, and the XRD results showed a small second Cu14 phase in addition to the main phase, with refinement indicating a Cu14 phase fraction of 8.7%, with the Cu14 presumably not visible in microprobe images due to segregated regions smaller than the measurement spot size. In agreement with reported results 12,13 the fitted Cu12 and Cu14 lattice constants are 10.3246 Å and 10.4458 Å respectively. Cu14Sb4S13 has the same space group and a set of atomic positions nearly equivalent to those of Cu12Sb4S13, with the addition of a small-occupation Cu interstitial (Cu III) which provides charge balance and thus insulating behavior and eliminating the metalinsulator transition in this phase. Figure 2 shows the result, with fitting details given in Table 1. For the minority Cu14 phase, the atomic positions are close to those reported in ref. 13, with small differences especially for Cu III. For the fit, we assumed full site occupation (except for the Cu III) due to relatively low sensitivity to these parameters.

Measurement methods A custom-made pulse spectrometer was used to conduct NMR experiments in a constant magnetic field of 8.9 T in a temperature range from 4.2 to 295 K using a multitemperature detecting probe. CuCl was utilized as the NMR shift standard for

63

Cu and

65

Cu spectra.

NMR measurements employed spin echo integration vs frequency and fast Fourier transformation (FFT), with the NMR spectra obtained by assembling and superposing FFT spectra at a sequence of frequencies. 5



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Table 1: Atomic parameters in the refinement of Cu12Sb4S13 and Cu14Sb4S13 at room temperature. Both structures cubic space group I-43 (# 217), with lattice constant a =10.3246 Å for Cu12Sb4S13 and 10.4458 Å for Cu14Sb4S13. SOF: site occupancy factor. RW = 7.88%.

Phase Wyckoff position Cu I 12d Cu II 12e Cu III 24g Sb 8c SI 24g S II 2a

x

Cu12Sb4S13 y z

0.5 0 – 0.2687 0.884 0

0 0 – 0.2687 0.884 0

0.25 0.2173 – 0.268 0.3623 0

SOF

Cu14Sb4S13 x y z

SOF

1 1 – 1 1 1

0.5 0 0.2885 0.2664 0.1149 0

1 1 0.082 1 1 1

0 0 0.2885 0.2664 0.1149 0

0.25 0.215 0.0054 0.2664 0.3603 0

NMR spin-lattice relaxation time (T1 ) measurements were obtained based on an inversion recovery sequence implementing a multi-exponential function for recovery of the central transition. 14 For magnetic shift contributions, we follow the convention K = (f − f0 )/f0 for Knight shifts, with f0 the standard reference frequency and positive shifts having paramagnetic sign. In the analysis, we used nuclear moment values (Q and γ) reported in ref. 15. The 63

63

Cu and

65

Cu gyromagnetic ratios γ and quadrupole moments Q used are as follows:

γ = 7.1118 rad/s G – 1 ,

65

γ = 7.6044 rad/s G – 1 , 63 Q = -22.0 fm2 , 65 Q = -20.4 fm2 . Electric

field gradients (EFGs) are given in terms of the standard parameters νQ =

3eQVzz 2I(2I+1)h

and

η = (Vxx − Vyy )/Vzz .

NMR RESULTS AND ANALYSIS Figure 3 shows the

63

Cu room temperature spectra. Seven clear peaks are observed. These

are edge singularities of quadrupole-broadened powder patterns, and the results were fitted as a superposition of two sites with equal site occupation, consistent with the reported structure (Figure 1). Fitting of the spectra was performed using the Quadfit package, 16 and with a custom NMR-line-shape code. Data were also taken using FFT methods for greater detail (shown 6


C u12S b4S13

Cu I

C u II

-4 0 0 0 0

Cu I

-2 0 0 0 0

0

Cu I

20000

C u II

40000

Cu I

C u II Sum D a ta

63

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


C u N M R S ig n a l ( A r b . U n its )

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C u II

-2 0 0 0

C u II

0

2000 S h if t (p p m )

Figure 3: 63Cu NMR line shape for Cu12Sb4S13 observed at room temperature. The inset shows the Iz = +1/2 to −1/2 central transitions while the satellite transitions can be observed in the wider spectrum. The solid curve corresponds to the two site fitting described in text. in figure 4 along with lower-temperature data), and by spin-echo integration with smaller H1 alternating field (Figure 3) for better resolution of the relative intensities. Fitting the latter measurements gave spectral areas differing by about 15% for the two fitted sites, due apparently to an enhanced broadening of the edge singularities of site II giving some deviation from the fit. Thus these are consistent with equal site occupation. The Cu I site (site assignment described below) exhibits the two outer satellite transitions (mz = ±3/2 ↔ ±1/2), in the upper plot of figure 3, as well as the central transition (mz = 1/2 ↔ −1/2) edge peaks near 700 and 1100 ppm. This site was fitted with quadrupole parameters νQ = 3.6 MHz and η = 0, magnetic shift Kiso = 970 ppm, and a small axial magnetic shift anisotropy δ = Kk − K⊥ = +141 ppm, where the principal values are parallel and perpendicular to the symmetry axis, respectively. Note that the satellite peaks at −18000 and 19000 ppm provide a very good match to the central transition in this fit and that there is no sign in this range of a second-order spectrum for νQ ≈ 18 MHz as observed at lower temperatures. 17 The Cu II site exhibits the outer singularities seen in both plots of figure 3. These were fitted with νQ = 7.3 MHz, Kiso = 970 ppm, and a small axial shift anisotropy of δ = −145 ppm, with

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Figure 4: Central region of 63Cu NMR line shapes at temperatures of 4.2 K, 77 K, and 291 K. Line shapes have been scaled for convenience to give comparable intensities. η = 0. Figure 4 displays the central part of the

63

Cu NMR lineshapes at three temperatures. As

seen in the figure the shifts become more positive as temperature decreases below the MIT, matching the previously reported 17 Knight shift changes for Cu12Sb4S13. Figure 5 shows the more detailed changes vs. temperature approaching the phase transition. There is a gradual lineshape change below about 200 K, then near T = 88 K the lineshape suffers an abrupt change at the position of the MIT 18 in synthetic Cu12Sb4S13. Over the same range the site II peaks appear to merge with a peak centered at about −700 ppm, with spectral area equivalent to the site-II fitted area at 291 K. However, this resonance disappears at 4 K and we could not further analyze these changes. The increase of the site I intensity below 200 K is due to decrease in EFG magnitude, causing the satellites to approach the central transition and increasing the central transition echo amplitude. 19 The smaller site I EFG below 200 K is consistent with computational results for the undistorted structure (Figure 1), as shown below. It is for these reasons that we assigned the observed spectra to site I and site II as indicated. The minority Cu14Sb4S13 phase would be expected to show Cu two resonances similar to the two Cu12Sb4S13 sites, but reduced by a factor of ∼ 10 and presumably not changing versus temperature due to the absence of the MIT. These peaks were not detected, perhaps hidden under the observed 8


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Figure 5: 63Cu NMR line shapes plotted at different temperatures. Signals are scaled by factor T to preserve the spectral intensity. peaks or broadened due to the partially-occupied interstitials. Below Tc the parameters νQ = 75 kHz, η = 1, along with δ = 225 ppm and η = 1, provide a good model for the 77 K data (inset, figure 4). With these parameters, the 77 K signal above 1300 ppm is a satellite resonance, and the measured 77 K T1 (0.20 s at 1700 ppm vs. 0.27 s at the peak position) is also consistent with this assignment, given that for a quadrupole mechanism, which dominates as discussed below, the central transition T1 exceeds that of the satellites 20 as observed here. Figure 6 shows the Cu-I most intense peak position versus temperature, along with its center of gravity (c.g.) position. At room temperature the c.g. is 120 ppm below the fitted isotropic K = 970 ppm due to the second-order quadrupole contribution, which adds a negative shift. 14 At intermediate temperatures the c.g. and peak position are very similar due to a small EFG, while at 4 K the spectrum develops considerable asymmetry, so the peak position and c.g. are again shown separately. As noted above, the Cu-I peak positions are the same as those of Kitagawa et al., 17 over the temperature range previously reported, and this can be seen in figure 6. The large jump at Tc is a Knight shift change, connected 17 to a large spin susceptibility change in the MIT. At higher temperatures the Cu-I peak merges with what becomes the lower edge of the Cu-I line, with its larger EFG signaling the development of a more asymmetric local environment.

9



Figure 6: Position of most intense spectral peak, and center of gravity position, for resonance of Cu-I, and comparison to data taken from Ref. 17

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63

Cu

In measurements between Tc and 120 K a second site was also observed 17 with νQ = 18.6 MHz. We did not make the large frequency sweep necessary to look for this site, however we assume this is the same as site II we observe with the smaller EFG at room temperature, which disappears close to Tc . As previously reported, the νQ = 18.6 MHz resonance also becomes very broad and difficult to observe at low temperatures. 17 Note also that Bastow et al. 21 observed site II with νQ ≈ 20 MHz at room temperature, but in a naturally-occurring tetrahedrite sample, in which due to heavy substitution the structural transformation and MIT would not be expected to occur. In unsubstituted Cu12Sb4S13, previously reported resistivity measurements have exhibited an unidentified feature at about 200 K, 22,23 and we assume that the line-shape changes we observe in this temperature range are also related to these observed features. The NMR results indicate that this is accompanied by a symmetry change which affects both Cu sites. A gradual softening of the elastic constants is also observed between room temperature and the MIT, 24 also pointing to local structure changes in this region. This would be consistent with the large changes in EFG which are indications that a significant change in site symmetry is involved. 10


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The transverse relaxation time (T2 ) for Cu-I remains long above Tc , increasing from 400 to 920 µs over the range 77 K to 291 K. This stands in contrast to the rapid reduction in T2 observed 25 in Cu2Se, as a low-temperature precursor to its super-ionic transition at 390 K. Note also tetrahedrite samples exhibit a transition near 400 K to a phase with mobile Cu ions. The large T2 for the Cu12 resonance up to room temperature indicates that this phase is stable against slow ion hopping, an important indicator of stability for device applications. At 4 K, the enhanced broadening (Figure 4) is consistent with the presence of dilute paramagnetic moments. To estimate the maximum defect density, we note that increase in line breadth relative to 77 K is approximately 1000 ppm ( full-width half maximum), and use established relations 26 for broadening due to dilute paramagnetic moments. Assuming 2 µB local moments, this gives a defect concentration 0.0014 per Cu ion. This matches previous results 22 indicating a nonmagnetic configuration for Cu12Sb4S13, with holes in the Cu d -based valence band remaining itinerant rather than forming localized Cu2+ moments, and does not appear to be compatible with a fully-magnetic Cu-II ions, 3 although a singlet configuration for site-II below Tc remains possible, and would still allow observation of the site I resonance. The most intense central transition peak was chosen to carry out NMR relaxation measurements. This was done for

65

Cu as well, with spectra for both nuclei showing similar

features. The T1 was fitted by a standard multiexponential function for magnetic recovery of the central transition. 27 For spin 3/2, this function is

M (t) M (0)

= A + 15 e(t/T1 ) + 95 e(6t/T1 ) . Using a

different function for quadrupole relaxation would lead to an overall scaling of the T1 which does not affect the fitting parameters described below. We also fitted the same function including stretched exponentials, but found that the ordinary exponential provided the better fit. The resulting rates can be seen in figure 7. There is a relatively broad jump in 1/T1 at around 88 K due to the MIT. 10 In addition, a peak can also be observed at a temperature around 12 K for both nuclei. The isotopic ratio 63 T1 /65 T1 , shown in figure 8, is uniform and stable over a large temper-

11



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ature range and approaching to the quadrupole moment ratio (Q65 /Q63 )2 = (20.4/22.0)2 = 0.86. Within the framework of relaxation 28,29 theory, this indicates a quadrupole mechanism. Spin or other magnetic fluctuations will lead instead to a ratio determined by gyromagnetic ratios, also shown in the figure. The quadrupole process is an indication of a lattice vibrations or atomic fluctuations as the dominant mechanism. As the temperature increases, T1 T becomes constant, a situation usually indicative of Korringa-like conduction electron relaxation, 28,30 while phonon processes normally lead to a T 2 or higher temperature dependence of 1/T1 . By contrast, the quadrupole-dominated T1 T ∼ constant behavior is a result that has been shown 31 to result from strongly anharmonic localoscillator behavior. The peak at the low temperatures is also characteristic of such behavior. This is interrupted by an additional maximum in the relaxation rate appearing near the MIT Tc , due to the slowing down of fluctuations often accompanying a phase transformation, for example as typically observed in incommensurate charge density wave systems. 32 Assuming additive quadrupole and magnetic contributions to the relaxation process, in order to further analyze we separated them by assuming 27,33

1 T1

=

1 T1Q

+

1 T1M

with

(T1M )−1 ∝ γ 2 and 28 (T1Q )−1 ∝ Q2 , where T1Q and T1M indicate the quadrupole and magnetic contributions. This separation is an approximation, since the two relaxation functions differ somewhat, however since the quadrupole process dominates for the entire temperature range, the correction is small, and as noted below, using the fitted T1 without separation gives results which are nearly identical to those obtained by this procedure.

Computational Results Ab initio density functional theory (DFT) simulations were conducted using WIEN2K, 34 applying the projector augmented wave (L)APW method and the Perdew-Becke-Ernzerhof (PBE) exchange-correlation functional. A cubic 58 atom unit cell of Cu12Sb4S13 in the ordered configuration of figure 1 with Cu at Wyckoff sites (12d) and (12e), S at (2a) and

12


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0 .0 6

63

-1

(

65

-1

T1T) T1T)

1 /T1T (K

-1

-1

s )

0 .0 5

(

0 .0 4

0 .0 3

0 .0 2

0 .0 1

0 .0 0 0

50

100

150

T(

200

K)

250

300

Figure 7: 63Cu and 65Cu NMR spin-lattice relaxation rates at the central transition frequency from 4.2 to 295 K 1 .2

1 .1 65

g /

63

)

g

2

T1 /

65

T1

( 1 .0

0 .9

63

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 .8 65

(

Q /

63

2

Q)

0 .7

0

50

100

150

200

250

300

T (K)

Figure 8: Isotopic ratio of overall relaxation rates, with limits for pure quadrupolar/magnetic relaxation indicated. (24g), and Sb at (8c) was investigated, using a 1000 k-point mesh and a plane wave cutoff of 100 eV. For a fixed lattice constant of 10.306 Å, corresponding to the 100 K experimental value, 10 the atomic positions were relaxed. The results show that the Fermi level is in the valence band as expected for Cu12Sb4S13, indicating p-type conduction for this composition, with band-structure agreeing with previous results. 8 The calculated EFG provides help in further understanding the experimental results, as well as measuring the charge anisotropy close to the nucleus. EFG results using the WIEN2K code, 35 for the two Cu sites are νQ = 0.3 and 17 MHz, with η = 0 and η = 0.15

13



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respectively. These are consistent with the values observed just above Tc in the pure tetrahedrite 17 and at ambient temperature in natural tetrahedrite. 21 We also see reduced EFG close to Tc and the computational results thus serve to reinforce the analysis showing that the spectrum near Tc corresponds to the undistorted structure.

Anharmonicity By employing real-space descriptors such as vibrational density of states for Cu12Sb4S13, it has been shown 10 that the Cu12e atom has a location probability concentrated close to the plane of its 3 S neighbors (Figure 1) with off plane dynamic displacement. To model the effect of such behavior on NMR relaxation Dahm and Ueda introduced a one dimensional double-well potential to study this type of problem and obtained qualitative agreement with results for β-pyrochlore oxide. 31 Following this result, 31 we define a Hamiltonian:

H=

1 1 P2 + ax2 + bx4 , 2M 2 4

(1)

where x, M and p are the coordinate, mass, and momentum of the Cu ion, respectively, and a and b > 0 are constants. This is treated in a self-consistent quasiharmonic approximation resulting in an effective harmonic Hamiltonian P2 1 H= + M ω02 x2 , 2M 2

(2)

where ω0 is phonon effective mode determined implicitly from M ω02 = a + bhx2 iT,ω0 . The dependency of the thermal average x2 on ω0 and T is given by

2

hx iT,ω0

h ¯ 1 = . M ω0 e¯hω0 /kB T − 1

14


(3)

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Eliminating b and a in favor of ω00 = ω0 (T = 0) in equation 1 gives:

ω 2 0

ω00

ω00 1 1 ω0 =1+β + − , ω0 e¯hω0 /kB T − 1 2 2ω00

(4)

where β = b M 2¯hω3 is a dimensionless factor characterizing the anharmonicity. NMR spin00

relaxation could be explained by two-phonon Raman process since quadrupole term is the dominant term, k T V 2 2 4Γ2 + ω 2 1 B 2 r 0 , = 2M Γ0 ωr6 T1R

(5)

where V2 is proportional to the second spatial derivative of the EFG, Γ0 is amount of phonon damping and ωr is phonon frequency renormalization which can be obtained by taking the real part of the phonon self-energy by ωr2 = ω02 + ω0 ReΠ(ω). In this model we obtain a good fit to our data, as displayed in figure 9 by the solid curve. This fit included the data shown in the figure, ignoring the points near the MIT for which 1/T1 is clearly dominated by fluctuations related to the transition. The resulting best-fit values are β = 55 , ω00 = 24 K, Γ0 = 5.0 K and ω0 Re(Π) = −(4.0 K)2 , corresponding to an anharmonic potential well given by,

V (x) = (−13 J/m2 ) x2 + (4.2 × 1022 J/m4 ) x4 .

(6)

This potential is shown in figure 10, also showing the energy levels obtained using a numerical package. 36 The separation between minima for this potential is 0.25 Å, and for comparison to crystallographic data, the rms displacement calculated for 291 K ( Eq. 3) p is hx2 iω,T = 0.16 Å. As further check on the effect of the separation at the quadrupolecontribution 1/T1Q , we also fitted data for the same temperatures using the unseparated

15



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1/63 T1 data alone, and obtained the very similar parameters β = 58 , ω00 = 23 K, Γ0 = 4.0 K and ω0 Re(Π) = −(4.7 K)2 , leading to a potential identical with equation (6) to within 2 significant figures. Since the MIT is believed to involve a lattice distortion, it is not clear whether the low temperature parameters should also apply above Tc . Thus, as alternative we used the method described above to fit the 1/T1Q data points only below the transition, with results shown in figure 9 as the dashed curve. This fitting gives β = 99 , ω00 = 29 K, Γ0 = 4.0 K and ω0 Re(Π) = −(8.0 K)2 , an anharmonic well with a distance 0.23 Å between minima, and room temperature rms displacement of 0.13 Å. The fitted potential is thus similar but slightly narrower than what is obtained by including the high-temperature data. For the data above the MIT, it is not possible to make a separate fitting of the rattling behavior based only on the (T1 T ) ≈ constant plateau at these temperatures, however since the low-temperature-only fit passes rather close to these data we infer that there is not a large change in the dynamics across the MIT boundary. Since the T1 is sensitive only to dynamical behavior, rather than static disorder, it is clear that the low-temperature structure retains the rattling processes, with the dynamics not frozen out by a distortion of the structure below the MIT or a large asymmetry in the potential well. Other measures of the rattling behavior are provided by crystallographic methods, which have been analyzed to yield 10,37 room-temperature thermal parameters for the 12e Cu site 2

in the range 0.14 Å , corresponding to a rms displacement on order of 0.37 Å. However, as described above 10 a split site analysis for this site, with probability maxima displaced by 0.30 Å bringing it closer to one of its neighbor S atoms, gives a better fitting. The latter 2

model has as its largest residual thermal parameter U33 = 0.0294 Å corresponding to a room temperature rms Cu 12d displacement of of 0.17 Å along one direction. This is quite close to the fitted result obtained here, making it appear likely that this indeed corresponds to a one-dimensional rattling behavior of the 12e Cu ions. As described in previous section, the split-site model also appears to provide a likely

16


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-1 -1 1 /T1 QT (K s )

0 .0 4

0 .0 3

0 .0 2

0 .0 1

0 .0 0

0

50

100

150

200

250

300

T (K )

Figure 9: Quadrupole NMR relaxation rate for 63Cu compared with the fitted anharmonic model (solid curve). Dashed curve obtained from fit of low-T data only.

300 250

K

)

200

E /kB (

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


150 100 50 0 -5 0 -1 0 0 -0 .0 2

0 .0 0

0 .0 2

x (n m )

Figure 10: Fitted 1D double-well potential of equation 6 with calculated energy levels. explanation for the much larger EFG observed for the 12e site at room temperature, with the large reduction in EFG approaching Tc implying that the distorted structure relaxes toward a more symmetric configuration as the MIT is approached. Although the orientation of these displacement may be random, the well defined site-I EFG at room temperature indicates that the displaced configurations are quite uniform.

17



Conclusions From investigations of

63

Cu and

65

Cu NMR in unsubstituted Cu12Sb4S13, we found that

above the metal-insulator Tc , which is accompanied by a large Knight shift change, there is a progressive change in site symmetry for both Cu sites. The Cu-II site exhibits a large reduction in electric field gradient (EFG), while Cu-I changes from a very small EFG/high symmetry configuration to a lower symmetry arrangement at room temperature. This shows that there is a change in the local atomic environments at these temperatures, probably corresponding to the development of a static split-site condition. DFT calculations confirmed that the low-EFG configuration corresponds to the undistorted configuration identified by crystallography. The T1 displays the presence of a strong quadrupole relaxation mechanism both above and below Tc . Analysis indicates that anharmonic rattling type motion is the main reason for this behavior. A double well anharmonic potential provides a good fit for the data, providing a measure of the rattling amplitude, and indicating that the dynamical behavior persists below Tc .

Acknowledgement This work was supported by the Robert A. Welch Foundation, Grant No. A-1526 and National Natural Science Foundation of China, Grant No.51772035. E. C. acknowledges support from the National Science Foundation under Award DGE-1545403. We are grateful to S. Chin for his numerical package.

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Graphical TOC Entry rattling potential

–0.02 nm

0.02 nm

T (K) NMR shift (ppm)

23


Structure Change and Rattling Dynamics in Cu12Sb4S13 Tetrahedrite

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