Evidence for anisotropic polar nanoregions in relaxor Pb(Mg${}_{1/3}$Nb${}_{2/3}$)O${}_3$: A neutron study of the elastic constants and anomalous TA phonon damping in PMN

We use neutron inelastic scattering to characterize the acoustic phonons in the relaxor Pb(Mg${}_{1/3}$Nb${}_{2/3}$)O${}_3$ (PMN) and demonstrate the presence of a highly anisotropic damping mechanism that is directly related to short-range polar correlations. For a large range of temperatures above $T_c\sim 210$  K, where dynamic, short-range polar correlations are present, acoustic phonons propagating along [1$\overline{1}$0] and polarized along [110] (TA${}_2$ phonons) are overdamped and softened across most of the Brillouin zone. By contrast, acoustic phonons propagating along [100] and polarized along [001] (TA${}_1$ phonons) are overdamped and softened for a more limited range of wave vectors $q$. The anisotropy and temperature dependence of the acoustic phonon energy linewidth $\Gamma$ are directly correlated with neutron diffuse scattering cross section, indicating that polar nanoregions are the cause of the anomalous behavior. The damping and softening vanish for $q\to 0$, i.e., for long-wavelength acoustic phonons near the zone center, which supports the notion that the anomalous damping is a result of the coupling between the relaxational component of the diffuse scattering and the harmonic TA phonons. Therefore, these effects are not due to large changes in the elastic constants with temperature because the elastic constants correspond to the long-wavelength limit. We compare the elastic constants we measure to those from Brillouin scattering experiments and to values reported for pure PbTiO${}_3$. We show that while the values of C${}_{44}$ are quite similar, those for C${}_{11}$ and C${}_{12}$ are significantly less in PMN and result in a softening of (C${}_{11}-C_{12}$) over PbTiO${}_3$. The elastic constants also show an increased elastic anisotropy [2C${}_{44}$/(C${}_{11}-C_{12}$)] in PMN versus that in PbTiO${}_3$. These results are suggestive of an instability to TA${}_2$ acoustic fluctuations in PMN and other relaxor ferroelectrics. We discuss our results in the context of the current debate over the “waterfall” effect and show that they are inconsistent with acoustic-optic phonon coupling or other models that invoke the presence of a second, low-energy optic mode.

Figure 1

Neutron elastic diffuse scattering intensity contours measured near (a) $\vec{Q}=(0,0,1)$ and (b) $\vec{Q}=(1,1,0)$ at $T=300$ K on SPINS ($E_f=4.5$ meV). The sample (S-II) was aligned in the (HHL) scattering plane.

Figure 2

(a) Neutron elastic diffuse scattering intensity measured in zero field at $\vec{Q}=(0.025,0.025,1.05)$ plotted versus temperature. The data were measured on SPINS using a final energy $E_f=4.5$ meV. The sample (S-II) was aligned in the (HHL) scattering plane. The dashed line is a schematic representation of the field-cooled data presented in Ref. 31. (b) The thermal expansion coefficient is plotted as a function of temperature, illustrating no observable structural distortion to a sensitivity of $\alpha \sim {10}^{-4}$. The data were taken on C5 with $E_f=14.6$ meV using a perfect germanium analyzer for improved $Q$ resolution. The sample (S-II) was aligned in the (HK0) scattering plane.

Figure 3

DCS neutron scattering data measured on PMN sample S-II in the (HK0) scattering plane near (200) and along [010] at $T=300$ K. The constant inelastic scattering intensity contours reveal well-defined TA${}_1$ and TO modes near the zone boundary at $(2,\pm 0.5,0)$, but overdamped TO modes near the zone center. The vertical dashed lines indicate where constant-$\vec{Q}$ scans were made; these are discussed in the next figure.

Figure 4

Temperature dependence of TA${}_1$ and TO phonons measured at two different wave vectors on PMN sample S-II in the (HK0) scattering plane using the DCS spectrometer. The constant-$\vec{Q}$ scans integrate over the regions $(2\pm 0.025,0.10\pm 0.02,0)$ and $(2\pm 0.025,0.20\pm 0.025,0)$. The vertical dashed lines indicate the average position in energy of the TA${}_1$ phonon.

Figure 5

Neutron elastic diffuse scattering intensity contours measured near (110) on SPINS ($E_f=4.5$ meV) in the (HK0) scattering plane. The white dashed line represents the direction in reciprocal space where the TA${}_2$ and TO phonons were measured. The open black circles represent typical reciprocal lattice points where constant-$Q$ scans were measured.

Figure 6

Constant inelastic scattering intensity contours measured on DCS in the (110) Brillouin zone are shown at (a) 600 K and (b) 300 K after integrating over $(1\pm 0.1,1\pm 0.1,0)$. Underdamped, dispersive TA${}_2$ phonons at 600 K become heavily damped at all $q$ at 300 K except for $q\sim 0$. The TO mode is not visible in this zone due to a weak structure factor. The wide, black streak at $E=0$ meV is where the intensity is dominated by the incoherent, elastic scattering cross section from the sample and the aluminium sample mount. These data were obtained on the S-II PMN crystal aligned in the (HK0) scattering plane.

Figure 7

Constant-$\vec{Q}$ scans measured at $\vec{Q}=(1.15,0.85,0)$ and (1.4,0.6,0) at temperatures ranging from 300–600 K. The solid lines are fits to a damped harmonic oscillator as described in the text. The data were taken with the C5 thermal-neutron, triple-axis spectrometer located at Chalk River using $E_f=14.6$ meV. The PMN S-I sample was aligned in the (HK0) scattering plane.

Figure 8

Constant-$\vec{Q}$ scans measured at $\vec{Q}=(1.1,0.9,0)$ and $(1.25,0.75,0)$ at 600, 300, and 100 K. The solid lines represent fits to a damped harmonic oscillator as described in the text. The data were taken with the C5 thermal-neutron, triple-axis spectrometer (Chalk River) using $E_f=14.6$ meV. The PMN S-I sample was aligned in the (HK0) scattering plane.

Figure 9

Elastic diffuse scattering intensity contours measured near (100) on SPINS ($E_f=4.5$ meV) in the (HK0) scattering plane. The gray dashed line represents the direction in reciprocal space where the TA${}_1$ and TO phonons were measured. The open black circles represent typical reciprocal lattice points where constant-$Q$ scans were measured.

Figure 10

Constant inelastic scattering intensity contours measured on DCS in the (100) Brillouin zone are shown at (a) 600 K and (b) 300 K after integrating over $(0,1\pm 0.15,0)$. Underdamped, dispersive TA${}_1$ phonons at 600 K remain well defined near the zone boundary at 300 K, but become heavily damped for intermediate wave vectors. The wide black streak at $E=0$ is where the intensity is dominated by the incoherent, elastic scattering cross section from the sample and the aluminium sample mount. These data were obtained on the S-II PMN crystal aligned in the (HK0) scattering plane.

Figure 11

Constant-$\vec{Q}$ scans measured at $\vec{Q}=(1.0,0.14,0)$ and $(1.0,0.3,0)$ at 600, 400, 200, and 100 K. The solid lines represent fits to a damped harmonic oscillator as described in the text. The data were taken with the C5 triple-axis spectrometer located at Chalk River Laboratory using $E_f=14.6$ meV. The PMN S-I sample was aligned in the (HK0) scattering plane.

Figure 12

Constant-$\vec{Q}$ scans measured at $\vec{Q}=(1.0,1.0,-0.05)$ and $(1.0,1.0,-0.15)$ at 600, 300, and 100 K. The solid lines represent fits to a damped harmonic oscillator plus a Gaussian function centered at $E=0$. The data were taken with the SPINS cold-neutron, triple-axis spectrometer located at NIST using $E_f=4.5$ meV. The PMN S-II sample was aligned in the (HHL) scattering plane.

Figure 13

The temperature dependence of the linewidth $\Gamma$ (upper panels) and energy ${\Omega }_0$ (lower panels) of the TA${}_1$ and TA${}_2$ phonons was determined by fitting a damped harmonic-oscillator line shape to constant-$\vec{Q}$ scans measured at $\vec{Q}=(1.0,0.21,0)$ and $(1.15,0.85,0)$, respectively. The horizontal dashed lines in the two upper panels represent the average TA phonon energy and thus indicate when the phonon becomes overdamped ($\Gamma \ge {\Omega }_0$). The error bars are derived from the least-squares fit to the data.

Figure 14

The temperature dependencies of the linewidth $\Gamma$ (upper panels) and energy ${\Omega }_0$ (lower panels) of the TA${}_1$ and TA${}_2$ phonons measured in the (200) and (220) Brillouin zones, respectively. The values are obtained from constant-$\vec{Q}$ scans measured at the exactly same reduced wave vector $\vec{q}$ as those shown in Fig. 13. The error bars are derived from the least-squares fit to the data; for the energy positions, the error bars are equal to the size of the data points.

Figure 15

The TA${}_2$ phonon in PMN-60$\%$PT measured in the (110) Brillouin zone. The data were taken using the N5 thermal-neutron, triple-axis spectrometer with the sample aligned in the (HK0) scattering plane.

Figure 16

A schematic illustration of the phonon dynamics for the TA${}_1$ and TA${}_2$ phonons measured in Brillouin zones where the diffuse scattering is strong [(100), (110), and (300)] and weak [(200)]. The TO mode in the (110) and (100) zones is too weak to provide any information on the damping. The diagram above assumes the intensity of the phonons scales as $Q^{2}F{\left(Q\right)}^2$, with the line shape the same in the (100) and (300) zones. Further discussion of the curves is provided in the text.