III. RESULTS AND DISCUSSION

The XRD pattern of the SrZrO₃ ceramic (10–100°), as shown in Fig. 1, was indexed according to the Joint Committee on Powder Diffraction Standard (JCPDS) card No. 70-0283, which proves the main phase of the SZ sample has an orthorhombic structure (space group Pbnm, no. 62).

FIG. 1.

XRD pattern for the SZ ceramic sintered at 1500 °C for 3 h and the unit cell of SZ.

The XRD pattern of the SZ ceramic and the Rietveld refinement results are illustrated in Fig. 2. The orthorhombic structure model (JCPDS# 70-0283) with space group Pbnm (62) was chosen for refinement, and the cell parameters of a = 5.7910 Å, b = 5.8108 Å, c = 8.1964 Å and α = β = γ = 90° were obtained. A unit cell figure of SZ is included in Fig. 1. ^{Reference Kamishima, Hattori, Ohta, Chiba and Ishigame9} The Wyckoff positions for each atom of the sample are listed as below: Sr (4b) (0.004 0.524 0.250), Zr (4a) (0 0 0), O1 (4e) (−0.089 −0.012 0.250), O2 (8i) (0.220 0.292 0.025). In this model, both Sr and Zr atoms are at special sites, but only O is at a general site, in which the sites were not refined. ^{Reference Diao, Wang, Lu, Shi and Jing2} According to the literature, the quality of the data from structural refinement is generally checked by R-values (R _p, R _exp, R _wp) and the goodness of fit (GoF = R _wp/R _exp). ^{Reference Ferrari and Lutterotti13} The refinement results has a high quality with R _p = 6.04, R _wp = 9.25, R _exp = 6.43, and GoF = 1.44, which indicates that the measured pellet of SZ has an orthorhombic perovskite structure. The refinement results are listed in Table I.

FIG. 2.

Observed (cycles) and calculated (line) XRD patterns as well as the difference profile (line below the pattern) for the Rietveld structure analysis of the SZ sample.

TABLE I.

Crystallographic data of SrZrO₃ derived from the Rietveld refinement of XRD data.

Each unit cell of SrZrO₃ contains four molecules at room temperature. The irreducible representation Г of the phonon modes at the zone center can be decomposed as follows according to a factor group analysis: Г = 7A_g + 5B_1g + 7B_2g + 5B_3g + 8A_u + 10B_1u + 8B_2u + 10B_3u, where 24 are Raman active (7A_g + 5B_1g + 7B_2g + 5B_3g) with 3 acoustic modes (B_1u + B_2u + B_3u), and 25 are infrared active (9B_1u + 7B_2u + 9B_3u) with 8 silent modes (8A_u). ^{Reference Amisi, Bousquet, Katcho and Ghosez8,Reference Kamishima, Hattori, Ohta, Chiba and Ishigame9}

The Raman spectroscopy is widely used to determine the relationship between the lattice vibrational modes and the resonant patterns. ^{Reference Diao and Shi3} The Raman spectrum of SZ ceramic is shown in Fig. 3 from 100 cm⁻¹ to 600 cm⁻¹, and the peaks are assigned according to the related compounds of SZ. ^{Reference Zhao and Weidner14–Reference Slodczyk, Colomban, Lamago, Limage, Romain, Willemin and Sala16} The Raman shifts and full width at half-maximum (FWHM) values are two important parameters to distinguish Raman normal modes. ^{Reference Diao and Shi3} The parameters of the phonon modes were fitted by Lorentzian function. All phonon modes and their descriptions are shown in Table II.

FIG. 3.

Raman spectrum of the SZ samples in the range of 100–600 cm⁻¹.

TABLE II.

Phonon mode parameters from Raman data of SrZrO₃.

In our study, 10 main bands out of 24 possible Raman vibrators were observed from 100 cm⁻¹ to 600 cm⁻¹, another three vibrators between Q1 and Q2 were neglected due to their weak intensities. Based on recent studies, ^{Reference Parida, Rout, Subramanian, Barhai, Gupta and Gupta7–Reference Kamishima, Hattori, Ohta, Chiba and Ishigame9} in the range of 100 cm⁻¹ to 200 cm⁻¹, the lowest frequency vibrator Q1 at 114 cm⁻¹ is attributed to the lattice vibration related to Sr²⁺ ion in the A-site with ZrO₆ octahedron. As shown in Table II, the vibrators Q2 and Q3 are also generated by the vibrations of Sr²⁺ with the ZrO₆ octahedron. From 200 cm⁻¹ to 400 cm⁻¹, the vibrators Q4, Q6, and Q7 are essentially associated with O²⁻ and the mode Q5 at 275 cm⁻¹ relates to Sr²⁺ ion. In the range of 400 cm⁻¹ to 600 cm⁻¹, the three phonon vibrators are all related to Zr–O. The vibrators Q8 and Q9 are caused by the bending vibrations of Zr–O, while the vibrator Q10 is the stretching vibrations of Zr–O. In a word, the phonon vibrators were assigned as follows: B_2g (114.28 cm⁻¹), B_2g (141.97 cm⁻¹), A_g (165.71 cm⁻¹), A_g (239.75 cm⁻¹), A_g (275.39 cm⁻¹), B_3g (312.04 cm⁻¹), B_2g (391.38 cm⁻¹), A_g (409.71 cm⁻¹), B_2g (440.07 cm⁻¹), and B_1g (549.22 cm⁻¹).

The infrared reflection spectra are obtained by the absorption of the polar lattice vibration. ^{Reference Kamba, Hughes, Noujni, Surendran, Pullar, Samoukhina, Petzelt, Freer, Alford and Iddles17} The wave number of infrared reflection spectra has a close relationship with the dielectric constant. Thus, the research of infrared reflection spectra is also very important. ^{Reference Prasanth, Kumar, Pazhani, Solomon and Tomas18} As shown in Fig. 4, the FTIR reflection spectrum is from 30 to 700 cm⁻¹. The solid line shows the four-parameter fitting curve, and the small circle represents the original data curve. According to the relevant research, ^{Reference Parida, Rout, Subramanian, Barhai, Gupta and Gupta7,Reference Amisi, Bousquet, Katcho and Ghosez8} the modes, below 160 cm⁻¹, are related to the ion movement of Sr²⁺ ions with ZrO₆ octahedron. The modes between 180 and 300 cm⁻¹ are associated with O–Zr–O– bending vibrations. The modes above 500 cm⁻¹ are assigned as the Zr–O stretching modes.

FIG. 4.

Fourier transform far-infrared reflectivity between 30 and 700 cm⁻¹ for the SZ ceramic and fitting results for the FTIR reflection spectrum of SZ by using the four-parameter model.

Each vibration mode is used as a damped harmonic oscillator. According to the curve fittings with the four-parameter semiquantum model, the vibrational frequencies of the IR modes were obtained, ^{Reference Li, Gao, Li, Pan, Yan and Zhu19} which describes the connection between the complex permittivity ε(ω) and the vibrational modes of the materials as:

ε_∞ is the optical permittivity, n is the number of vibrational modes, Ω _jTO and γ _jTO are the frequencies and damping factors of the jth transverse modes of vibration, respectively. Ω _jLO and γ _jLO are the frequencies and damping factors of the jth longitude modes, respectively. ^{Reference Diao, Wang, Lu, Shi and Jing2} Reflectivity (R) is an important parameter in the infrared reflectance spectrum. The Fresnel equation [Eq. (2)] represents the correlation between the IR reflection spectrum and the complex permittivity. This fitting procedure is carried out by FOCUS program and the fitting results from the four-parameter model are shown in Fig. 4 and Table III.

TABLE III.

Parameters of the IR-active modes based on the four-parameter model.

There are 25 IR possible modes (9B_1u, 7B_2u, and 9B_3u) which are polarized respectively along the x, y, and z directions. ^{Reference Amisi, Bousquet, Katcho and Ghosez8} From 30 cm⁻¹ to 700 cm⁻¹, six main bands of these modes were observed. The modes near F_1u ⁽¹⁾ and F_1u ⁽⁵⁾ were ignored due to their weak relative strength. Based on the fitting results, the IR-active modes were assigned as follows: F_1u ⁽¹⁾ (102 cm⁻¹), F_2u ⁽²⁾ (120 cm⁻¹), F_1u ⁽³⁾ (140 cm⁻¹), F_3u ^(4)′ (228 cm⁻¹), F_3u ^(4)″ (287 cm⁻¹), F_1u ⁽⁵⁾ (326 cm⁻¹), and F_1u ⁽⁶⁾ (527 cm⁻¹). The first-principle calculations indicated that almost all the symmetric coordinates are involved in each vibrational mode. ^{Reference Diao, Wang, Lu, Shi and Jing2} The mode F_1u ⁽¹⁾, F_2u ⁽²⁾, and F_1u ⁽³⁾ are primarily represented as the Sr–ZrO₆ inverted translational vibration. The F_3u ⁽⁴⁾ mode, that breaks into F_3u ^(4)′ and F_3u ^(4)″ modes, is regarded as the O–Zr–O bending mode. The mode F_1u ⁽⁵⁾ is described as the inverted ion movement of Zr⁴⁺ atoms in the ZrO₆ octahedron. The mode F_1u ⁽⁶⁾ is due to the Zr–O stretching vibration, which is caused by the antisymmetrical breathing vibrations of the ZrO₆ octahedron. ^{Reference Lei, Fan and Chen20}

The complex permittivity is determined from the attenuation of electromagnetic waves in the ceramic. The equation of the dielectric constant frequency is ε(ω) = ε′(ω) + iε′′(ω). Kramers–Krönig analysis (K–K) is a widely adopted standard method to obtain the vibrator parameters from the reflection spectra. ^{Reference Diao and Shi3} In this study, we performed K–K analysis by the formula deduction of Mathematica Software to obtain both the imaginary and real parts of the dielectric constant, Eqs. (1) and (2) were used.

The imaginary part characterizes the absorption characteristics of the vibrator to electromagnetic wave, as shown in Fig. 5. The peaks are numbered from the lowest to the highest frequency, i.e., from 1 to 6. Each peak corresponds to an IR-active mode. The real part is shown in Fig. 6 with three modes. Based on $\varepsilon '' = {{\left( {{\varepsilon _{\rm{s}}} - {\varepsilon _\infty }} \right)\varepsilon _{\rm{T}}^2\gamma } \over {\varepsilon \left[ {4{{\left( {{\varepsilon _{\rm{T}}} - \varepsilon } \right)}^2} + {\gamma ^2}} \right]}}$ , when ε″(ω) reaches a maximum value, the resonant frequency ω_TO of the vibrator can be determined from the imaginary part of the dielectric constant, ^{Reference Diao and Shi3} so peak frequency is concerned with resonant frequency ω_TO. In these six modes, mode 5 has the maximum value of the imaginary part.

FIG. 5.

Imaginary part of dielectric constants ε″(ω) calculated from K–K analysis for the SZ ceramic in the far-IR region (30–700 cm⁻¹).

FIG. 6.

Real part of dielectric constants ε′(ω) calculated from K–K analysis for the SZ ceramic in the far-IR region (30–700 cm⁻¹).

According to the fitting results of four-parameter models, the contribution of each vibrational mode to the permittivity (ε_r) and dielectric loss tan(δ _j ) is discussed. The results can be deduced from:

The calculated values of ε_r and ${{{\rm{tan}}\left( {{\delta _j}} \right)} \mathord{\left/ {\vphantom {{{\rm{tan}}\left( {{\delta _j}} \right)} \omega }} \right. \kern-\nulldelimiterspace} \omega }$ are also listed in Table III. The first five modes are all below 400 cm⁻¹. They are mainly affected by Sr²⁺ and Zr⁴⁺ and contribute most of the permittivity and dielectric loss. However, the last-mode is caused by the breathing vibrations of the ZrO₆ octahedron and has little effect on the dielectric properties. ^{Reference Fontana, Metrat, Servoin and Gervais21–Reference Silva, Moreira, Almeida, Paniago and Paschoal23} It can be seen that the vibration modes with large contributions to the permittivity also have large contribution to the dielectric loss. The vibrational modes at low frequencies have larger contributions to ε_r and ${\rm{tan}}\left( {{\delta _j}} \right)$ . The dielectric properties are mainly affected by the modes involving more movements of heavier metal atoms. The mode F_1u ⁽¹⁾, that can be represented as the Sr–ZrO₆ inverted translational vibration, has the highest contribution to the permittivity and dielectric loss.

The value of IR fitting by summing the ε _j values of all the modes is 23.7 (as shown in Table III). The permittivity value (ε_r) obtained in this study is 22.4. The calculated value of dielectric loss is $\sum {{{{\rm{tan}}\left( {{\delta _j}} \right)} \mathord{\left/ {\vphantom {{{\rm{tan}}\left( {{\delta _j}} \right)} \omega }} \right. \kern-\nulldelimiterspace} \omega }} = 1.9 \times {10^{ - 3}}$ and the experimental data of dielectric loss is 3.1 × 10⁻³. Parida et al. ^{Reference Parida, Rout, Subramanian, Barhai, Gupta and Gupta7} reported that the measured permittivity is 25 and the dielectric loss is 2.4 × 10⁻³. The calculated values agree well with the experimental data and previous results reported by Ref. Reference Parida, Rout, Subramanian, Barhai, Gupta and Gupta7.