Spontaneous polarization and pyroelectric coefficient of lithium niobate and lithium tantalate determined from crystal structure data

This work shows an alternative approach to extract the pyroelectric coefficient directly from the response of the crystal structure to temperature variations. The approach uses crystal structure parameters determined at different temperatures and Born effective charges calculated by density functional theory to evaluate the spontaneous polarization. On this basis, the pyroelectric coefficient is calculated using the first derivative of the polarization with respect to temperature. In this work, we utilize laboratory single-crystal x-ray diffraction in a temperature range from 80 to 400 $\mathrm{K}$ to demonstrate this approach on the standard pyroelectric materials lithium niobate and lithium tantalate. The accuracy of the presented method for the spontaneous polarization is on par with reported experimental data and relative errors are below $7\%$. Our determined pyroelectric coefficients have larger errors, but are within the range of reported literature values. The outlined semitheoretical method is recommended when electrical characterization under temperature change is not feasible, e.g., when the preparation of large area contacts is problematic or only small sample volumes are available ($<100$ $\mu {\mathrm{m}}^3$).

Figure 1

Temperature-dependent lattice parameters $a\left(T\right)$ and $c\left(T\right)$, unit-cell volume $V_{\text{UC}}\left(T\right),$ as well as coordinates $z_{\text{Li}}\left(T\right)$, $z_{\text{Nb/Ta}}\left(T\right)$, and $z_{\text{O}}\left(T\right)$, for experimentally determined ($•$) structures of C-LN (blue), S-LN (cyan), C-LT (orange), and DFT-calculated structures of LN (green) and LT (red). DFT-calculated structures were relaxed with experimental temperature dependencies and the extrapolated $0\text{−}\mathrm{K}$ unit cells ($\square$) as constraints. Additionally, fully relaxed theoretical structures in GS () are given as well. The SC-XRD structural parameters and DFT-computed data using the same structural parameters were smoothed with respect to the progression of thermal expansion by a quadratic fit (solid lines). The light colored areas indicated the $1\sigma$ confidence intervals of the fit.

Figure 2

Structure models for the SC-XRD refinement: (a) stoichiometric, (b) defect model for congruent ${\mathrm{LiNbO}}_3$ with Li (green) deficiency and Li as well as Nb (red) on the Li site, and (c) defect model for congruent ${\mathrm{LiTaO}}_3$ with Li deficiency and Ta (red) on the Ta site and within the vacant (gray) octahedron. For simplification, we do not show all O ions and oxygen bonds (black). Figures were produced using vesta3 [82].

Figure 3

Different models for the calculation of $P_S$ for C-LN (a) and C-LT (b): Model 1 (gray dots) combined BEC tensors calculated for every temperature step with fitted structural parameters from SC-XRD (according to Fig. 1). Model 2 (red solid line) combined BEC tensors calculated exclusively for each ion in the unit cell at the GS with fitted structural parameters. For C-LN we measured a second data set (red dashed-dotted line) to show the reproducibility of the presented method. All values are within the $1\sigma$ confidence interval (light red areas).

Figure 4

Temperature-dependent spontaneous polarization of ${\mathrm{LiNbO}}_3$ (a) as well as ${\mathrm{LiTaO}}_3$ (b). The values are determined by different methods: calculated from SC-XRD determined crystal structures and computed BEC of the DFT-GS for congruent (C-LN/LT SC-XRD) and stoichiometric (S-LN SC-XRD) materials as well as DFT-relaxed atomic positions (DFT) and directly calculated with DFT (Berry phase). The light colored areas indicate the $1\sigma$ error range. Additional experimental literature values for congruent (C-LN/LT Lit.) and stoichiometric (S-LN/LT Lit.) material, as well as theoretical (theo. Lit) and semitheoretical (semitheo. Lit.) literature values are plotted with the following markers: $\times$ [21], $★$ [24], $+$ [93], $\blacksquare$ [25], $▾$ [94], $⧫$ [95], $▴$ [26], $◂$ [96], $▸$ [97], $◯$ [98], $\square$ [99], $▵$ [100], $▿$ [101].

Figure 5

The pyroelectric coefficient for ${\mathrm{LiNbO}}_3$ (a) and ${\mathrm{LiTaO}}_3$ (b) derived from $P_S$. Data calculated for experimental atomic positions from SC-XRD and BEC tensors for congruent (C-LN/LT SC-XRD) and stoichiometric (S-LN SC-XRD) material are shown. The light colored areas indicate the $1\sigma$ error range. Additionally, $p$ was measured with the Sharp-Garn method for congruent (C-LN/LT SG) and stoichiometric (S-LN SG) ${\mathrm{LiNbO}}_3$ as well as congruent ${\mathrm{LiTaO}}_3$ crystals. Values from literature based on experiment for congruent (C-LN/LT Lit.) and stoichiometric (S-LN Lit.) as well as on theory (theo. Lit.) are shown for comparison: $★$ [102], $\blacksquare$ [103], $⧫$ [104], $▾$ [94], $+$ [105], $◂$ [106], $▴$ [107], $▸$ [108], $◯$ [109], $◊$ [110], $\square$ [26], $▿$ [26], $◃$ [111].

Figure 6

Fit of the thermal expansion coefficients ${\alpha }_a$ along the crystallographic axis $a$ and ${\alpha }_c$ along axis $c$ for congruent ${\mathrm{LiNbO}}_3$ (left) and ${\mathrm{LiTaO}}_3$ (right). The temperature-dependent lattice parameters are normalized to the lattice parameters at $T=300$ $\mathrm{K}$. The thermal expansion coefficients were determined with a linear fit, which resembles well the dependence in the range between 100 and $400\mathrm{K}$.

Figure 7

BEC data for both systems ${\mathrm{LiNbO}}_3$ (a) and ${\mathrm{LiTaO}}_3$ (b) using the experimentally (with SC-XRD) determined lattice parameters for the temperatures above $0\mathrm{K}$ and extrapolated lattice parameters at $0\mathrm{K}$.