Lattice expansion and local lattice distortion in Nb- and La-doped $\mathrm{SrTi}{\mathrm{O}}_3$ single crystals investigated by x-ray diffraction and first-principles calculations

Electron-doped $\mathrm{SrTi}{\mathrm{O}}_3$ (where the dopant can be Nb or La) has been widely investigated for both its fundamental interest in condensed matter physics and for industrial applications. Its electronic properties are closely related to the Ti-O bonding states in the $\mathrm{SrTi}{\mathrm{O}}_3$ crystal. To further develop and control these properties, it is crucial to understand the factors controlling the change in lattice parameters and local lattice distortion upon doping with various atoms. Herein, we report the changes in lattice parameters and local lattice distortion in Nb- and La-doped $\mathrm{SrTi}{\mathrm{O}}_3$ single crystals, investigated by in-plane x-ray diffraction and first-principles calculations. The lattice parameter of Nb- and La-doped $\mathrm{SrTi}{\mathrm{O}}_3$ single crystals increased with dopant concentration. The broad intensities around the Bragg peak observed in the in-plane x-ray-diffraction experiments indicated that the local lattice expansion and contraction, or local lattice distortions, in the crystal were caused by the dopant atoms. First-principles calculations similarly showed that the lattice expansion and local lattice distortions in the $\mathrm{SrTi}{\mathrm{O}}_3$ crystals were caused by doped Nb and La atoms. Atoms surrounding Schottky pairs of O and Sr vacancies were displaced both away from and towards the vacancies, resulting in a reduction in the lattice expansion of donor-doped $\mathrm{SrTi}{\mathrm{O}}_3$. Donor atoms and Schottky pairs thus play an important role in determining the lattice parameters of $\mathrm{SrTi}{\mathrm{O}}_3$ crystals. These fundamental structural analyses provide a useful basis to further investigate the electronic conductivity of electron-doped $\mathrm{SrTi}{\mathrm{O}}_3$.

Figure 1

(a) Linear and (b) log-scale intensities of out-of-plane and in-plane XRD patterns obtained for a pure $\mathrm{SrTi}{\mathrm{O}}_3$ single crystal. Reflections of the out-of-plane and in-plane XRD patterns are for 002 and 020 $\mathrm{SrTi}{\mathrm{O}}_3$, respectively. Rescaled data are shown in the inset at the upper right corner of (b).

Figure 2

In-plane XRD patterns obtained from 020 reflections of (a) Nb- and (b) La-doped $\mathrm{SrTi}{\mathrm{O}}_3$ single crystals. The red and black arrows in (a) and (b) indicate the intensities for the crystal lattice contraction and expansion, respectively. The rescaled data for Nb- and La-doped $\mathrm{SrTi}{\mathrm{O}}_3$ are shown in (c) and (d), respectively. The red-line XRD patterns in all the images are from the nondoped $\mathrm{SrTi}{\mathrm{O}}_3$ single crystal. The insets in the upper right corners of (c) and (d) show magnified images of the regions around the peaks of nondoped $\mathrm{SrTi}{\mathrm{O}}_3$.

Figure 3

Lattice parameters of Nb- and La-doped $\mathrm{SrTi}{\mathrm{O}}_3$ single crystals as a function of dopant concentration. The inset shows a magnified view of the low-concentration region.

Figure 4

(a) Fitting profile obtained using Gaussian functions for the experimental XRD pattern of 0.02-at.-% Nb-doped $\mathrm{SrTi}{\mathrm{O}}_3$. The dotted black, orange, and blue lines indicate the Gaussian functions used for the main, expansion, and contraction intensities, respectively. Gaussian curves were numbered as labeled. (b, c) The contraction (b) and expansion (c) intensity ratios calculated from the fitting results in Fig. S2 of Supplemental Material [34], as a function of dopant concentration. Horizontal axes in (b) and (c) are log-scale, except for the values for nondoped $\mathrm{SrTi}{\mathrm{O}}_3$ shown as a black diamond on the left side of each graph. (d) Ratios of the contraction intensities as a function of the FWHM of the rocking curves. Gray dashed lines in (b), (c), and (d) are included to aid comparison between Nb-doped, La-doped, and nondoped $\mathrm{SrTi}{\mathrm{O}}_3$.

Figure 5

Schematic diagram of electronic states in the valence band (VB) and conduction band (CB) of $\mathrm{SrTi}{\mathrm{O}}_3$, as used in the calculation models. There are no electrons in the CB for initial state i, i.e., perfect $\mathrm{SrTi}{\mathrm{O}}_3$; positively charged state 1+ (ii), i.e., $\mathrm{L}{{\mathrm{a}}_{\mathrm{Sr}}}^{•}$ and $\mathrm{N}{{\mathrm{b}}_{\mathrm{Ti}}}^{•}$; and positively charged state 2+ (iii), i.e., ${V_{\mathrm{O}}}^{••}$. The total charges of supercells ii and iii were neutralized using jellium background charges of 1− and 2−, respectively. (iv) Neutral state with one electron in the CB, i.e., $\mathrm{N}{{\mathrm{b}}_{\mathrm{Ti}}}^{•}+e^{\prime }$, and $\mathrm{L}{{\mathrm{a}}_{\mathrm{Sr}}}^{•}+e^{\prime }$. (v) Neutral state with two electrons in the CB, i.e., ${V_{\mathrm{O}}}^{••}$. (vi, vii) Neutral states with one electron and two electrons, respectively, in the CB, i.e., $e^{\prime }$ and $2e^{\prime }$. The total charge of the supercells of (vi) and (vii) was neutralized using jellium background charges of 1+ and 2+, respectively.

Figure 6

Calculated lattice parameter changes as a function of dopant concentration for positively charged defects $\mathrm{N}{{\mathrm{b}}_{\mathrm{Ti}}}^{•}$, $\mathrm{L}{{\mathrm{a}}_{\mathrm{Sr}}}^{•}$, and ${V_{\mathrm{O}}}^{••}$ when there are no electrons in the CB of $\mathrm{SrTi}{\mathrm{O}}_3$.

Figure 7

Calculated lattice parameter changes as a function of dopant concentration for neutral defects $\mathrm{N}{{\mathrm{b}}_{\mathrm{Ti}}}^{•}+e^{\prime },\mathrm{L}{{\mathrm{a}}_{\mathrm{Sr}}}^{•}+e^{\prime }$, and ${V_{\mathrm{O}}}^{••}+2e^{\prime }$ and free electrons $e^{\prime }$ and $2e^{\prime }$ when electrons are present in the CB of $\mathrm{SrTi}{\mathrm{O}}_3$.

Figure 8

Changes in (a) experimental (Exp.) and calculated (Cal.) lattice parameters of $\mathrm{SrTi}{\mathrm{O}}_3$ containing $\mathrm{N}{{\mathrm{b}}_{\mathrm{Ti}}}^{•}+e^{\prime }$ and $\mathrm{L}{{\mathrm{a}}_{\mathrm{Sr}}}^{•}+e^{\prime }$ defects, as a function of dopant concentration. (b) Changes in lattice parameters for ${V_{\mathrm{Ti}}}^{\prime \prime }+4h^{•},{V_{\mathrm{Sr}}}^{\prime \prime }+2h^{•},{V_{\mathrm{O}}}^{••}+2e^{\prime },{V_{\mathrm{Sr}}}^{\prime \prime }+{V_{\mathrm{O}}}^{••},\mathrm{N}{{\mathrm{b}}_{\mathrm{Ti}}}^{•}+e^{\prime },\mathrm{L}{{\mathrm{a}}_{\mathrm{Sr}}}^{•}+e^{\prime },2\mathrm{N}{{\mathrm{b}}_{\mathrm{Ti}}}^{•}+{V_{\mathrm{Sr}}}^{\prime \prime }$, and $2\mathrm{L}{{\mathrm{a}}_{\mathrm{Sr}}}^{•}+{V_{\mathrm{Sr}}}^{\prime \prime }$ defect models, as well as each dopant with a ${V_{\mathrm{Sr}}}^{\prime \prime }+{V_{\mathrm{O}}}^{••}$ partial Schottky defect. Dopant and vacancy concentrations in the theoretical models were 0.8 at. % in all cases. The experimental results for 1.0-at.-% Nb-doped (green circle) and La-doped (blue square) $\mathrm{SrTi}{\mathrm{O}}_3$ are included in (b) for comparison.

Figure 9

Three-dimensional vector plots of Sr displacements around defects/dopants: (a) $\mathrm{N}{{\mathrm{b}}_{\mathrm{Ti}}}^{•}+e^{\prime }$, (b) $\mathrm{L}{{\mathrm{a}}_{\mathrm{Sr}}}^{•}+e^{\prime }$, (c) $\mathrm{N}{{\mathrm{b}}_{\mathrm{Ti}}}^{•}+e^{\prime }+{V_{\mathrm{Sr}}}^{\prime \prime }+{V_{\mathrm{O}}}^{••}$, and (d) $\mathrm{L}{{\mathrm{a}}_{\mathrm{Sr}}}^{•}+e^{\prime }+{V_{\mathrm{Sr}}}^{\prime \prime }+{V_{\mathrm{O}}}^{••}$. Sr, Ti, and O ions are located at the corners, centers, and face centers of the lattice cells, respectively. Vectors indicate the magnitudes and directions of Sr displacements from their ideal cubic perovskite lattice positions and are color coded according to the scale bar on the right of each figure.

Figure 10

Lattice models of (a) Nb- and (b) La-doped $\mathrm{SrTi}{\mathrm{O}}_3$ crystals as viewed along the $\langle 100\rangle$ axis in the plane of the dopant atom. Background colors indicate the magnitude of $S_{\mathrm{Sr}-\mathrm{site}}/S_{\mathrm{ave}}$ corresponding to the color scale on the right of each figure. The vector on each atom shows the direction and magnitude of displacement from its ideal lattice position. Vectors of O overlapping with those of Ti along the projected direction have been removed for clarity. Vectors at the bottom of (a) and (b) serve as a scale for the magnitude of displacement. Crystal structures were drawn using the VESTA program [60].

Figure 11

(a) Histogram of cell lengths of Ti-centered and Sr-centered cells within relaxed supercells of Nb-doped $\mathrm{SrTi}{\mathrm{O}}_3$. For clarity, only a narrow range about the average lattice parameter (marked with a dotted line) is shown; results for the entire range are provided in Fig. S10(a) of Supplemental Material [34]. (b, c) Ratios of numbers of contracted and expanded cells, respectively, as a function of dopant concentration. The numbers of contracted and expanded cells were taken as those $\pm 0.25\mathrm{pm}$ to either side of the lattice parameter. As an aid to the eye, lengths of contracted cells and expanded cells are highlighted with purple and orange backgrounds, respectively.

Figure 12

Average distance between dopant atoms as a function of dopant concentration. Schematic models consisting of unaffected, expanded, and contracted regions around dopant atoms for each concentration are also shown.