3.3.1. The pyroelectric measurements at varying temperature
Pyroelectricity is strictly defined as the temperature dependence of spontaneous polarization in certain anisotropic solids. According to the definition, changes in the intensity of spontaneous polarization are related to changes in temperature as follows (Zhou et al. 2018; Chen et al. 2019; Jachalke et al. 2017)
$${P}_{s}=\frac{d\overrightarrow{D}}{dT}=\frac{d\overrightarrow{P}}{dT}$$
1
which \({P}_{s}\) is the pyroelectric coefficient, \(\overrightarrow{D}\) is the dielectric displacement, \(\overrightarrow{P}\) is the spontaneous polarization, T is Kelvin temperature.
As can be seen from Fig. 2, when the sample temperature rises from 300K to 370K, the polarization intensity increases linearly with the increase of temperature, and the average pyroelectric coefficient is 4.5µC/m2/K in the temperature range of 300-370K, and the pyroelectric coefficient changes little with temperature.
3.3.2. Comparison of calculated and experimental pyroelectric Coefficients
In pyroelectric crystals, electric dipoles usually form with the center of positive charge far away from the center of negative charge. In this study, the electric dipole of Mn3B7O13Cl mainly comes from the deformation of ClMn6 octahedron resulting in the non-coincidence of positive and negative electric centers(Wang;Qiao;Su;Hu;Yang;He and Long et al. 2018; Guo;Sun;Chen;Hao;Ma;Wang;Wu;Hou;Zhang;Liu;Li;Meng and Zhao et al. 2022). When the temperature changes, the electric dipole will stretch, resulting in an induced charge on the crystal surface, resulting in a pyroelectric effect, as shown in the Fig. 3(a). In addition, Mn3B7O13Cl has a unique axis of rotational symmetry along the C-axis, and polarity occurs at both ends of the c-axis. Therefore, only the contribution of the electric dipole moment in the c direction to the total electric dipole moment of the cell is considered. The calculation of electric dipole moment is by Formula (2).
where q is the charge capacity, l (=△z) is the distance between positive and negative charge center. It can be concluded that the calculation formula of the electric dipole moment in ClMn6 is Formula (3).
$$\sum p==[4\times 1\times (△{Z}_{X}\left)\right]\times 1.60\times {10}^{-19}$$
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∑P is the total electric dipole moment in ClMn6 octahedron, △z is the distance between positive and negative charge centers, which is caculated from the atomic positions at different temperatures by Table S1. The intrinsic electric dipole moment of ClMn6 polyhedron in unit cell along the c direction is obtained by using the Formula (3), as shown in Table S2.
According to the relation between pyroelectric coefficient and electric dipole moment, when the crystal volume is constant, the pyroelectric coefficient is proportional to the rate of change of electric dipole moment with temperature.
The intrinsic electric dipole moment fits the equation.
$${P}_{T}=\frac{d{p}_{z}}{VdT}$$
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Where PT is the pyroelectric coefficient, dpz is the variable of the intrinsic electric dipole moment at different temperatures, dT is the variable of the temperature during the test, and V is the unit volume.
According to the formula, the pyroelectric coefficients of Mn3B7O13Cl in different temperature ranges (300K-350K, 350K-400K) were calculated in this paper, as shown in Fig. 3(b). When the temperature increases, the pyroelectric coefficients in the two temperature ranges increase slightly.
In order to compare the experimentally measured pyroelectric coefficient with the calculated pyroelectric coefficient, we averaged the experimental data for each temperature range, as shown in Fig. 3(b). As can be seen from the figure, the experimental observed value is slightly greater than the theoretical calculated value, so it is inferred that the main structural unit of Mn3B7O13Cl pyroelectricity in ClMn6 octahedron.
3.3.3. Calculation of Mn3B7O13Cl polyhedron distortion parameters at different temperatures
The pyroelectric properties of materials can be derived from the generation of spontaneous polarization, which is fundamentally due to the symmetry breaking and polyhedral distortion in the crystal structure(Meirzadeh et al. 2019; Chang et al. 2009). Polyhedra in crystal structures are often obtained by distortion of ideal polyhedra(Ertl et al. 2002; Song and Liu et al. 2019; Zhang et al. 1993). A common way to calculate the distortion of a polyhedron is to count the difference between all bond lengths:
$$D=\frac{1}{n}\sum _{i=1}^{n}\left[\right({d}_{i}-{d}_{m})∕{d}_{m}{]}^{2}$$
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Which di is polyhedral bond length; dm average is bond length.
The distortion parameters of ClMn6 polyhedron at different temperatures are calculated. Figure 3(c) shows Mn3B7O13Cl boracite polyhedron linearly decrease with the increase of temperature, which indicates that the temperature change will directly cause the deformation of the inner polyhedron, and the degree of deformation of the polyhedron is linear.