Piezoelectric constant, elastic constant, and dielectric constant calculated by DFPT and experimentally measured values.
Abstract
Among the various computational methods in materials science, only first-principles calculation based on the density functional theory has predictability for unknown material. Especially, density functional perturbation theory (DFPT) can effectively calculate the second derivative of the total energy with respect to the atomic displacement. By using DFPT method, we can predict piezoelectric constants, dielectric constants, elastic constants, and phonon dispersion relationship of any given crystal structure. Recently, we established the computational technique to decompose piezoelectric constants into each atomic contribution, which enable us to gain deeper insights to understand the piezoelectricity of material. Therefore, in this chapter, we will introduce the computational framework to predict piezoelectric properties of polar material by means of DFPT and details of decomposition technique of piezoelectric constants. Then, we will show some case studies to predict and discover new piezoelectric material.
Keywords
- density functional perturbation theory
- ferroelectricity
- piezoelectricity
- first-principles calculation
1. Introduction
In this chapter, we will introduce how recent computational techniques can successfully predict response properties, represented as piezoelectricity, by means of perturbation method. Piezoelectricity is the polarization change in response to external mechanical force. Inversely, if electrical field is applied to piezoelectric material, mechanical strain is induced (inverse piezoelectric effect). Therefore, piezoelectric materials are widely used as vibrational censors, surface acoustic wave devices, and actuators. Only the material having no inversion symmetry shows piezoelectricity. For example, Figure 1 shows schematic illustration of the piezoelectric effect. Positions of positively charged ion (cation) and negatively charged ion (anion) are represented as plus and minus symbols. Figure 1a shows the paraelectric phase, where ions are orderly located with inversion symmetry. On the other hand, ions are slightly displaced by δ with respect to those in paraelectric phase, as shown in Figure 1b. Such small displacement induces microscopic polarization
Because ferroelectric phase is energetically more stable than paraelectric phase under low temperature,
First-principles calculation based on density functional theory (DFT [2, 3]) has been widely utilized as the computational method to predict the electronic properties of material under the ground state. Ideally, required information to conduct the first-principles calculation is only the crystal structure, including atomic species and position of periodic/nonperiodic structure unit. The most significant advantage of first-principles calculation is its predictability. Since King-Smith and Vanderbilt showed the theoretical methodology to calculate change in polarization per unit volume Δ
2. Formulation of piezoelectric constants
Formulation and calculation methodologies to obtain response properties of materials in the framework of DFPT have been developed in a step-by-step manner, because degrees of freedom by perturbations of atomic displacement, homogeneous electric fields, and strain are often strongly coupled. For example, piezoelectricity affects elastic and dielectric properties. Therefore, special care must be paid for the calculation of coupled properties. In 2005, Hamann et al. demonstrated that elastic and piezoelectric tensors can be efficiently calculated by treating homogeneous strain within the framework of DFPT [31]. At the same time, Wu et al. systematically formulated response properties with respect to displacement, strain, and electric fields [32]. In this section, we will briefly introduce how piezoelectric properties are formulated in the framework of DFPT. In each formulation, Einstein implied-sum notation is used. Cartesian directions {
Total energy of material under perturbation of atomic displacement
where
Clamped-ion term is a frozen quantity, which indicates that atomic coordinates are not allowed to relax as the homogeneous electric field or strain. Therefore, dynamical term should be added into the clamped-ion term in order to obtain proper response properties.
Simplest and physically well-understandable piezoelectric constant can be expressed as follows:
In this expression, it is easily understood that piezoelectric
Thus, proper piezoelectric constant can be obtained by Eqs. (7) and (9):
Here, the first and second terms on the right-hand side in Eq. (10) are the clamped-ion term and internal-strain term, respectively. The former shows the electronic contribution ignoring the atomic relaxation effect, and the latter shows the ionic contribution including the response of the atomic displacement to the strain. The Born effective charge
On the other hand, the internal-strain term of the piezoelectric stress constant
where
Because the subscript
Here, piezoelectric
where
Those formulations are implemented in specific first-principles simulation packages such as ABINIT [33] and Vienna ab initio simulation package (VASP) [34], and piezoelectric constants can be calculated on a daily basis. From the next section, we will show how DFPT calculation precisely gives piezoelectric properties of ferroelectric materials.
3. Introduction of target material
In this chapter, we selected LiNbO3 as a target material to show the predictability of DFPT calculation. LiNbO3 is one of ferroelectric materials and widely used as surface acoustic wave (SAW) and optical waveguide elements. Crystal structure of LiNbO3, which belongs to the space group of
Crystal structures shown in the present chapter was visualized by using VESTA software [35]. Curie temperature of LiNbO3 is quite high and ranges from 1140 [36] to 1210°C dependent on the quality of sample (variation of Li/Nb relation can shift Curie temperature [37]). Below Curie temperature, ferroelectric phase with
Due to the different bonding nature between Li-O and Nb-O, atomic positions of Li and Nb are off-centered within oxygen layers along
Cubic lattice is symmetric and usually high-temperature phase, same as LiNbO3. The “strained perovskite structure” expression for LiNbO3 means that LiO6 and NbO6 polyhedron are largely rotated with respect to the cubic perovskite structure. However, because of the simple atomic configuration of cubic structure, atoms can be displaced along various directions and change crystalline symmetry as shown in Figure 3a. Crystalline lattice is vibrated (referred as phonon) under finite temperature. Some lattice vibrations along specific directions are unstable. This specific phonon is called as soft mode with imaginary frequency. In such case, atoms are displaced along unstable phonon mode to lower the total energy. For example, cooperative atomic displacement along [001] direction shown in Figure 3b (referred as
The most convenient way to control and drastically change the crystal structure is imposing high pressure. Many compounds have found to be possible to form LiNbO3-type structure under the high-pressure synthesis [38], and some of them were quenchable phase. For example, LiNbO3-type structured ZnSbO3 was successfully synthesized [39] under high pressure, and improvement of the spontaneous polarization is suggested by enhancement of the covalency of Sn site from first-principles simulation [40]. Moreover, high-pressure synthesized research on LiNbO3-type structure is now extended to more complex compounds such as oxynitrides [41, 42].
The crystal structure of
In our previous theoretical study on high-pressure phase, analysis was mainly concerned with phase transition mechanism only from the viewpoint of subgroup symmetry and energy barrier [43]. It will be instructive to deal with this phase transition phenomenon from the viewpoint of lattice instability as discussed in the field of the ferroelectric instability analysis. In the following section, we will show investigation on the potential piezoelectric properties of LiNbO3 with various hypothetical crystal structures by the method of DFPT, and possible phase transition mechanism will be discussed from the viewpoint of soft mode.
4. Computational methodology
First-principles calculation was performed by using VASP code [34]. Interactions between ion and electron were treated by projector augmented wave (PAW) method [45]. PBEsol functional [46] was used to approximate exchanges and correlate interactions of electrons, which can be used to reproduce the lattice constants of various materials [45]. Precise calculation on the lattice constant is essential to predict piezoelectric properties because they depend on volume of unit cell Ω as shown in Eq. (1). The kinetic energy cutoff for plane waves was set at 500 eV, and the k-point mesh was set at ~0.03/Å intervals to obtain the converged total energy at less than 0.1 meV/atom. Before calculating the piezoelectric constants, atomic positions and cell parameters were optimized until the forces on each atom and cell converged at below 5 × 10−4 eV/Å.
Since VASP does not directly calculate Eq. (12), we added routine to calculate displacement-response internal-strain tensor
On the basis of cubic
5. Calculated piezoelectric properties of LiNbO3
Calculated piezoelectric properties of LiNbO3 in ferroelectric phase are summarized in Table 1. Some experimentally measured values are also shown in Table 1. All properties are confirmed to be well reproduced by calculation. In a technological importance, 33 components are the most important because
Calculated value | Experimental value | |
---|---|---|
Piezoelectric stress constant (C/m2) | ||
3.73 | 3.655 ± 0.022 [48], 3.7 [49] | |
2.51 | 2.407 ± 0.015 [48], 2.5 [49] | |
0.21 | 0.328 ± 0.032 [48], 0.2 [49] | |
1.69 | 1.894 ± 0.054 [48], 1.3 [49] | |
Elastic constant (GPa) | ||
190.7 | 198.86 ± 0.033 [48], 203 [49] | |
58.3 | 54.67 ± 0.04 [48], 53 [49] | |
62.4 | 67.99 ± 0.55 [48], 75 [49] | |
13.5 | 7.83 ± 0.02 [48], 9 [49] | |
220.0 | 234.18 ± 0.75 [48], 245 [49] | |
49.2 | 59.85 ± 0.01 [48], 60 [49] | |
Dielectric constant | ||
40.6 | 44.9 ± 0.4 [48], 44 [49] | |
24.1 | 26.7 ± 0.3 [48], 29 [49] |
Thus, Li vacancy is considered to have negligible influence on the piezoelectric properties. Decomposed ionic contribution of piezoelectric strain constant
Decomposed | Born effective charge | Displacement-response internal-strain constant | ||||||
---|---|---|---|---|---|---|---|---|
Li | Nb | O | Li | Nb | O | Li | Nb | O |
0.1 | 0.05 | 0.16 | 1.03 | 6.77 | −2.6 | 0.67 | −0.05 | −0.21 |
6. Piezoelectric properties of perovskite-LiNbO3
Next, we will show how piezoelectric properties are affected by crystal structure, while chemical composition is kept as LiNbO3. Various hypothetical crystal structures common for perovskite-type structure were constructed, and their energetic stabilities were examined by calculating enthalpy
Cubic,
LiNbO3-ferroelectric phase,
Orthorhombic,
where names of space groups are used to distinguish each structure. Crystal structure of each phase is shown in Figure 4a. Polyhedra shown in Figure 4a correspond to Nb-centered bonding structure of Nb-O bondings. Figure 4b shows the enthalpy difference of each phases as a function of external pressure. Here, external pressure is assumed to be isotropic. Standard of enthalpy was set to be the enthalpy of most stable
Imposing negative pressure can be achieved by solid solution with parent phase of larger lattice constant. At −6 GPa,
Within the eight phases shown in Figure 4a, only
are broken because of
Piezoelectric stress constant (C/m2) | ||||
3. 73 | 1.14 | 5.10 | 0.64 | |
2.51 | — | 1.19 | — | |
0.21 | 0.46 | 0.24 | 0.80 | |
1.69 | 3.28 | 1.92 | 2.99 | |
Elastic constant (GPa) | ||||
190.7 | 297.2 | 203.0 | 321.8 | |
58.3 | 48.9 | 169.0 | 91.6 | |
62.4 | 77.7 | 90.6 | 92.8 | |
13.5 | — | −42.1 | — | |
220.0 | 157.7 | 206.6 | 176.4 | |
49.2 | 39.5 | −29.6 | 32.5 | |
Dielectric constant | ||||
40.6 | 56.2 | 36.6 | 28.2 | |
24.1 | 12.3 | 16.2 | 13.3 |
Figure 5a and b show piezoelectric properties of
This giant piezoelectric constant is almost comparable to that of PZT material [51]. Giant piezoelectric constant is understood as a result of phase instability in morphotropic phase boundary [52]. The same as
Finally, we would like to show phase transition path between cubic perovskite structure and LiNbO3 structure. Figure 6a shows the energy change of
Frequency (THz) | Structure | Space group | Energy gain (eV/f.u.) | |
---|---|---|---|---|
Γ | −7.22 | Tetragonal | −0.422 | |
X | −5.38 | Orthorhombic | −0.220 | |
M | −6.56 | Orthorhombic | −0.125 | |
R | −5.51 | Rhombohedral | −0.682 |
Figure 7 shows schematic illustration of phase transition mechanism from cubic perovskite structure to LiNbO3 structure. On the contrary to the result of Figure 6a,
7. Summary and conclusion
In this chapter, we briefly introduced sophisticated method of density functional perturbation theory. DFPT can effectively calculate the second derivative of the total energy with respect to the atomic displacement within the framework of first-principles calculation. By using DFPT method, we can predict piezoelectric constants, dielectric constants, elastic constants, and phonon dispersion relationship of any given crystal structure. Moreover, we showed our established computational technique to decompose piezoelectric constants into each atomic contribution, which enable us to gain deeper insights to understand the piezoelectricity of material. By using LiNbO3 as a model material, we showed the predictability of DFPT for piezoelectric properties. In addition, we showed that superior piezoelectric properties are hidden in perovskite-structured LiNbO3. Structural relationship and possible phase transition path between LiNbO3 structure and perovskite structure were discussed and concluded that perovskite-structured LiNbO3 is thermodynamically unstable. Further studies are expected to control relative phase stability between perovskite and LiNbO3 structure by dopant substation and solid solution.
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