Stanford Synchrotron Radiation Lightsource

When atoms are displaced from the usual lattice site in a crystal, they form electric dipoles making the system paraelectric. If these electric dipoles become aligned parallel to each other at low temperatures, the system becomes ferroelectric – the electrical analogue of a ferromagnet.  Although off-center displacements sometimes do occur as the temperature is lowered, they generally do not form upon heating a material. However this unusual characteristic, which would be an important new result, was proposed by Bozin et al. [1] for the important thermoelectric material PbTe, following an analysis of neutron pair distribution function (NPDF) data. They suggested that Pb moves off-center in this simple rock-salt structure as the temperature increases above ~100 K and is displaced by approximately 0.18 Å at 300 K, resulting in the formation of significant Pb-Te dipoles.  For such a displacement there will be three Pb-Te bond lengths instead of one, split by ± 0.18 Å.  A difficulty for this system is that it is quite soft (weak springs between atoms) and the atomic vibrations have large amplitudes at room temperature. In addition, the vibrations become anharmonic [2,3] at higher temperatures, i.e. the vibration amplitude is not the same for expansion and compression of the Pb-Te bond.   

Most scattering experiments, including NPDF analysis, only have amplitude information. Consequently, it is difficult to differentiate between a broadened distribution and a split distribution of bond lengths for splittings on the order of 0.15 Å. In contrast, the extended x-ray absorption fine structure (EXAFS) technique employed in the present study also has phase information that provides a clear distinction between these two possibilities. The effects of a large splitting are clearly visible. In general, when there is significant destructive interference between components in the EXAFS r-space data, the amplitude is decreased and often there are kinks or phase shifts.

To investigate possible off-center displacements, Bridges and co-workers have recently performed EXAFS experiments at the Pb L_III- and Te K edges at SSRL’s Beam Line 4-1. The work, published in the journal Physical Review Letters, shows that Pb is in fact not off-center. 

Fig. 1 Plot of the r-space EXAFS data as a function of temperature for the Pb L_III edge, showing strong temperature dependence. The fast oscillating function is the real part of the Fourier transform (FT), R, and carries phase information.  The Pb-Te peak is near 3 Å. Right: Fits of the function R to various models at high temperature where Pb is proposed to be off-center. Best fit is for a broadened peak with some asymmetry (dashed line); poorest fit is for the split peak (black line) which has a significant kink near 2.6-2.8 Å.

In Fig. 1 we show the Pb L_III-edge r-space data as a function of temperature. In this plot the different peaks correspond to various shells of neighbors around the Pb atom, with the Pb-Te peak near 3 Å. As the temperature increases and the atoms begin to vibrate, the amplitude decreases, but there is no obvious change in the shape of the oscillatory function R other than the decrease in amplitude. To investigate off-center displacements, we fit R, the real part of the Fourier transform, to a broadened peak, a split peak, or a broadened peak with a small asymmetry, for temperatures from 250 to 310 K (Fig. 1, right).

The broadened peak fit (red line) models the data quite well but misses the data points in a few places.  The fit using the 100 off-center displacement model proposed by Bozin et al. [1] (one short bond, one long bond, and four bonds approximately unshifted) is shown as a black line and is much worse. It has a significant kink from 2.6-2.8 Å, and the disagreement between the fits using the split peaks (black line) and fits using a broadened peak (red line) increases as T increases.  The broadened peak fit can be improved by adding some asymmetry to the distribution (C3 parameter) on the high-r side of the distribution, consistent with the known anharmonic behavior.  This inproved fit passes through all the data points. Similar results were obtained for the Te K-edge data.  

In summary, we find no evidence, up to 310 K, of a significant off-center displacement of the Pb or Te atoms from the average crystal structure position. Displacements of the magnitude proposed by Bozin et al. [1] would produce significant changes in the phase of the real (and imaginary parts) of the Fourier transform; for the Pb edge data this displacement would introduce a large kink in R(r) near 2.7 Å. We agree that thermally induced vibrations grow rapidly with temperature and there is significant thermally-induced disorder at 300 K. Thus, this system does not have high-temperature-induced dipoles. However, the pair potential for Pb and Te atoms is asymmetric, leading to the anharmonic behavior observed by Delaire et al. [2] and Jensen et al. [3].