Lead-free Zr-doped ceria ceramics with low permittivity displaying giant electrostriction

Electrostrictors, materials developing mechanical strain proportional to the square of the applied electric field, present many advantages for mechanical actuation as they convert electrical energy into mechanical, but not vice versa. Both high relative permittivity and reliance on Pb as the key component in commercial electrostrictors pose serious practical and health problems. Here we describe a low relative permittivity (<250) ceramic, ZrxCe1-xO2 (x < 0.2), that displays electromechanical properties rivaling those of the best performing electrostrictors: longitudinal electrostriction strain coefficient ~10−16 m2/V2; relaxation frequency ≈ a few kHz; and strain ≥0.02%. Combining X-ray absorption spectroscopy, atomic-level modeling and electromechanical measurements, here we show that electrostriction in ZrxCe1-xO2 is enabled by elastic dipoles produced by anharmonic motion of the smaller isovalent dopant (Zr). Unlike the elastic dipoles in aliovalent doped ceria, which are present even in the absence of an applied elastic or electric field, the elastic dipoles in ZrxCe1-xO2 are formed only under applied anisotropic field. The local descriptors of electrostrictive strain, namely, the cation size mismatch and dynamic anharmonicity, are sufficiently versatile to guide future searches in other polycrystalline solids.

1 Supplementary Note 1: Crystal structure of    −   ceramics Cubic lattice constants (a), determined under ambient conditions by X-ray diffraction (Rigaku, Ultima III), were observed to depend on dopant size relative to that of Ce.Local distortion of fluorite symmetry at low dopant concentration due, at least in part, to size mismatch of the cations, are not detected by XRD as they lack long range correlation.Shannon crystal radii (Å) for ligand coordination number 6-8: Zr +4 0.86 -0.98; Ce +4 1.01-1.11.Supplementary Figure 3. a) Grain size distribution; and b) mean grain size determined by the lineal intercept method, including the x1.56 correction factor which accounts for grains which may be only partially visible at the imaged surface 1 .Error bars are standard deviation for >100 grains.
3 Supplementary Note 3: Ultrasound pulse echo measurement of elastic moduli USTOF (ultrasound time of flight; ultrasound pulse echo) measurements.Shear (transverse, VS) and longitudinal, (VL) sound velocities were determined with accuracy better than 0.25% (pellet height measured with uncertainty ≤ 0.15%) with USTOF instrumentation and protocol as described in reference 2 and in previous reports. 3,4,5,6 USOF was measured using transducers coupled directly to the pellets with high viscosity commercial honey without external force.
Correction for porosity < 6 % was performed as described previously 3,4 .Measurements were performed on oxidized pellets.
To correct for porosity (p), the dynamic model developed by Ledbetter et al. 7,8 leads to: (Supplementary Eq. 3) The subscript "0" denotes the values of the elastic moduli before correction for porosity, calculated using (Supplementary Eq. 1 and (Supplementary Eq. 2, while the subscript "D" denotes the values corrected for porosity according to the dynamic model.To reliably apply the correction, porosity must be less than 6vol%, as was indeed the case for the ceramic pellets studied here, verified by Archimedes method.The magnetization of the re-oxidized samples,  = 2 K (Supplementary Figure 6), saturates at significantly lower values than the reduced samples.Magnetization drops sharply following codoping with La because the contribution of  3+ is suppressed.

Supplementary
Supplementary Figure 6.Saturation magnetization measurements as measured at T = 2K in the SQUID magnetometer for fragments of 10mol% Zr doped ceria pellets.Measurements were made both before (Red -red curve), and following, re-oxidation (Ox-black curve).
The data in Supplementary Figure 6 cannot be fit to Langevin-type curves with physically reasonable parameters, likely because of the common presence of ~100ppm of magnetic impurities 9 .However, point by point calculation of difference magnetization curves (reduced minus reoxidized Zr-doped ceramics) (Supplementary Figure 7) effectively cancels the contribution of the magnetic impurities; indeed, these can be fit to Langevin-type curves.The resulting fit parameters find approx.500ppm Ce 3+ ions for the reduced Zr samples while sample oxidation reduces this value to ~ 100 ppm.The latter value is comparable to known concentrations in undoped or aliovalent doped ceria under ambient conditions 9 .Supplementary Figure 7. Difference magnetization curves calculated with data presented in Supplementary Figure 6: magnetization of a fragment of a reduced (red curve) or oxidized (black curve) 10 mol% Zr doped ceria ceramic.Fitting parameters are listed in Supplementary Table 1.

𝜂
, where  is the ratio of the magnetic to thermal energy,  =  0 • ; μ0 is the vacuum magnetic permeability; kB is the Boltzmann constant; T is absolute temperature (K); and H is the magnetic field strength (A m -1 ),  0 is a temperature independent contribution which accounts for diamagnetic and Van Vleck susceptibility for curves in Supplementary Figure 7.The value of  0 was taken from previous studies 10 .where   = 41.50Å 3 is the volume per formula unit of CeO2 and ℂ (  11 = 343 GPa,  22 = 103 GPa,  44 = 54 GPa) is the elastic stiffness tensor of CeO2.The chemical strain,   , for Zr concentration   , can be obtained by (Supplementary Eq. 9)   =     .
here  −− is the closest O-Zr-O bond angle;  0 ≈ 70.528° is the bond angle in a perfect lattice.

Subsection 8.4 Supercell electric dipole moment
The overall supercell electric dipole moment can be calculated by taking into consideration all charges and positions of the 96 atoms: (Supplementary Eq. 12) , where   is the charge of atom i, and  ⃗  is the vector pointing to this atom from the origin, while the image atoms are considered due to the periodic boundary conditions.The electric dipole unit is selected to be debye (D), in CGS units, and

Supplementary Figure 1 . 2 Supplementary Note 2 :
(a) Lattice parameters of oxidized    1−  2 ceramics were calculated by linear regression based on the indexing of 10 diffraction peaks according to 3 ̅  symmetry from (b).Prior to XRD measurement under ambient conditions, samples were heated at 773K for 5 h in pure oxygen atmosphere to compensate for possible oxygen loss during sintering.For most cases, the error bars are smaller than the size of the data symbol.Grain size distribution of dense ceramic pellets Supplementary Figure2.SEM (Zeiss Sigma 500) micrographs of the circumferential surface of ceria ceramic pellets doped with: a) 5mol% Zr and b) 10mol% Zr.Prior to SEM measurements, samples were heated at 773K for 5 h in pure oxygen atmosphere to compensate for possible oxygen loss during sintering.Scale bars indicate 1µm.

Supplementary Figure 8 .
Room temperature impedance spectroscopy measurements were conducted with a Novocontrol Alfa dielectric analyzer in high voltage mode.Applied voltage was: 10 VAC, 1 MHz-1 mHz; 0 VDC.Room temperature Nyquist plots for 5 mol% (a) and 10 mol% (b) Zrdoped ceria pellets: 0 VDC bias, 10 VAC, frequency range 1 MHz-1mHz, measured with upper spring-loaded electrode.Measurements were made in the same device as the converse electrostriction measurements.

Supplementary Figure 9 .
Room temperature (a) conductivity and (b) real component of the relative dielectric permittivity for Zr-doped ceria pellets 0 VDC bias, 10 VAC, frequency range 1 MHz-100Hz, measured with stainless steel electrodes, the upper one being spring loaded.Measurements were made in the same device as the converse electrostriction measurements.

Table 6 .
The computed elastic dipole tensor  and the dopant-induced strain tensor per Zr ion,   , for different Zr structures.The diagonalization operation gives 3 eigenvectors for each diagonalized matrix, they are the principal directions that the strain tensor   is projected onto.