⇦Chinese Physics Letters, 2019, Vol. 36, No. 7, Article code 070501 A New Model of Ferroelectric Phase Transition with Neglectable Tunneling Effect * Hong-Mei Yin (尹红梅)1,2, Heng-Wei Zhou (周恒为)2, Yi-Neng Huang (黄以能)1,2** Affiliations 1National Laboratory of Solid State Microstructures, School of Physics, Nanjing University, Nanjing 210093 2Xinjiang Laboratory of Phase Transitions and Microstructures in Condensed Matters, College of Physical Science and Technology, Yili Normal University, Yining 835000 Received 12 February 2019, online 20 June 2019 *Supported by the National Natural Science Foundation of China under Grant No 11664042.
**Corresponding author. Email: ynhuang@nju.edu.cn Citation Text: Yin H M, Zhou H W and Huang Y N 2019 Chin. Phys. Lett. 36 070501    Abstract Due to the obvious deviations of the existing theoretical models from the experimental results of ferroelectric phase transition, a new model is proposed on the basis of the coupling between spontaneous polarization and spontaneous strain in ferroelectrics. The spontaneous polarization and specific heat of ferroelectric phase transition predicted by the model are in better agreement with the corresponding data of triglyceride sulfate, a typical ferroelectric. In addition, the model predicts a new type of ferroelectric in which a phase transition and a phase-like transition coexist. DOI:10.1088/0256-307X/36/7/070501 PACS:05.70.Fh, 77.80.B- © 2019 Chinese Physics Society Article Text Ferroelectrics have a wide range of applications, including random access memories,[1,2] pyroelectric detectors,[1] electrocaloric devices,[1] piezoelectric transducers,[1,3] manipulation of light devices,[4] tunable microwave devices,[5] bypass capacitors,[1,6] dynamic access memory capacitors,[1] electron emitters,[1] photovoltaic devices,[7] visible-light-absorbing devices,[7] filtering devices,[8] etc.,[1–18] and these applications are more or less related to ferroelectric phase transitions (FPTs). After long-term theoretical exploration, the theoretical models of FPTs have achieved great advances, such as (1) Landau phenomenological theory,[19,20] Weiss molecular field theory,[21] Ising model of pseudospin (equivalent of permanent dipole orientation),[22,23] transverse field Ising model of pseudospin,[24–26] and soft-mode theory of 2nd-order phase transition (PT),[27] (2) Devonshire phenomenological theory of 1st-order PT,[28] and (3) compressible Ising models,[29–32] and transverse field Ising model including four-pseudospin interaction (abbreviated as WQL model) of 1st or 2nd order PT.[33] However, for triglycine sulfate, a typical ferroelectric with 2nd-order PT,[34–36] the existing theoretical models still have obvious deviations from the experimental results of the spontaneous polarization ($P_{\rm s}$) and specific heat ($c_{\rm s}$) per dipole of FPT versus temperature ($T$) as shown in Fig. 1. Therefore, it is undoubtedly valuable to explore more precise theories or models of FPTs. In this work, based on the coupling between spontaneous polarization and spontaneous strain in ferroelectrics, a new model of FPTs with neglectable tunneling effect is proposed. Traditionally, there are two microscopic schemes, i.e., pseudospin and soft-mode, to describe FPTs. For example, the FPT of TGS and KH$_{2}$PO$_{4}$ families can be investigated by the pseudospin scheme[22–26] while that of ABO$_{3}$ ferroelectrics needs to be described by soft-mode theory.[27] However, Cohen found that, in BaTiO$_{3}$, a typical ABO$_{3} $ ferroelectric, the hybridization between the titanium $3d$ states and the oxygen $2p$ states is essential for ferroelectricity.[37] In other words, the phase transitions of ABO$_{3} $ ferroelectrics can also be investigated by covalent pseudospins, such as the Ising model of pseudospins.
cpl-36-7-070501-fig1.png
Fig. 1. Here circles and diamonds are the experimental results of $P_{\rm s}$,[34] the reciprocal of static susceptibility ($\chi_{\rm s}^{-1}$) of the paraelectric phase,[35] and specific heat ($c_{\rm s}+c_{\rm b}$)[36] for triglycine sulfate crystal versus temperature $T$, with $c_{\rm b}$ being the background irrelevant to FPTs. Solid lines ($J_{0}=322.3k_{_{\rm B}}$, $\alpha =0.289$, $\beta =0.135$, $\mu =1.32\times {10}^{-29}$ Cm, $N=2.80\times {10}^{27}$ m$^{3}$), dash-dotted line ($J_{0}=322.3k_{_{\rm B}}$, $\alpha =0.301$, $\mu =1.32\times {10}^{-29}$ Cm, and $N=2.80\times {10}^{27}$ m$^{3}$), and dotted line ($J_{0}=322.3k_{_{\rm B}}$, $\mu =1.32\times {10}^{-29}$ Cm, and $N=2.80\times {10}^{27}\,$m$^{3}$) correspond to the present model, WQL model, and Weiss theory, respectively.
In the pseudospin scheme, there are the transverse-field Ising model[26] (KH$_{2}$PO$_{4}$ families with obvious tunneling effect) and the Ising model (TGS and ABO$_{3} $ families with neglectable tunneling effect).[23] To simplify the mathematical representation, the Ising model of pseudospins is chosen as the starting point of the present study.[22,23] In fact, the results of this work can be easily extended to the transverse-field Ising model.[24–26] The Hamiltonian of the Ising model is $$\begin{align} H=-\frac{J}{2}\sum\limits_{i\ne j}^{\{nn\}} {s_{i}s_{j}},~~ \tag {1} \end{align} $$ where $J$ is the interaction energy constant between pseudospins, $s_{i}$ is the $i$th pseudospin in ferroelectrics, $s_{i}=\pm 1$, and $\{nn\}$ represents the sum over all the nearest neighbors. In ferroelectrics, $P_{\rm s}$ is always accompanied by spontaneous strain (changes in lattice parameters and symmetry versus $T$),[9,38] i.e., there is coupling between spontaneous polarization and spontaneous strain, while the strain inevitably leads to the variation of $J$, i.e., $J$ is related to $P_{\rm s}$. In view of the inversion symmetry of $P_{\rm s}$, the relationship between $J$ and $P_{\rm s}$ is expanded as $J=J_{0}(1+\alpha p_{\rm s}^{2}+\beta p_{\rm s}^{4})$ in this work, where $J_{0}$, $\alpha$, and $\beta$ are constants independent of $T$, and $p_{\rm s}\equiv \frac{P_{\rm s}}{N\mu}$ with $\mu$ being the permanent electric dipole moment of pseudospins and $N$ the dipole density per unit volume. In the following, we show that the $\alpha p_{\rm s}^{2}$ term is equivalent to the linear coupling of $J$ to the lattice spontaneous strain ($\delta_{\rm s}$) of the compressible Ising models,[29–32] which may originate from the electrostriction ($\delta_{\rm s}\propto p_{\rm s}^{2}$). Further, it could be imagined that the $\beta p_{\rm s}^{4}$ term is equivalent to the quadratic coupling of $J$ to $\delta_{\rm s}$, i.e., the elastic energy that is proportional to $\delta_{\rm s}^{2}\propto p_{\rm s} ^{4} $, and its influences on FPTs have not been reported as far as we know. Therefore, the FPT Hamiltonian including the coupling between spontaneous polarization and spontaneous strain up to $p_{\rm s}^{4}$ is $$\begin{align} H=-\frac{J_{0}(1+\alpha p_{\rm s}^{2}+\beta p_{\rm s}^{4})}{2}\sum\limits_{i\ne j}^{ \{nn\} } {s_{i}s_{j}},~~ \tag {2} \end{align} $$ which is an extended form of the three-dimensional Ising model, and it can be imagined that it is extremely difficult to solve it exactly. In this work, the single pseudospin mean field has been used,[39,40] i.e., $s_{i}s_{j}\approx s_{i}\langle s_{j}\rangle $, where $\langle s_{j}\rangle $ is the statistical average of $s_{j}$, and obviously, $p_{\rm s} \equiv \langle s_{j}\rangle$. In this way, the Hamiltonian can be expressed as $$\begin{align} H\approx -J_{0}p_{\rm s}(1+\alpha p_{\rm s}^{2}+\beta p_{\rm s}^{4})\sum\limits_{i=1}^\infty s_{i}.~~ \tag {3} \end{align} $$ Using Eq. (3) and based on the Boltzmann principle, $p_{\rm s}$, the average internal energy $u_{\rm s}$ per dipole, and $c_{\rm s}$, as well as the static polarizability $\chi_{\rm s}$ of ferroelectrics versus $T$ are $$\begin{align} &p_{\rm s}=\tanh [\frac{T_{0}}{T}p_{\rm s}(1+\alpha p_{\rm s}^{2}+\beta p_{\rm s}^{4})],~~ \tag {4} \end{align} $$ $$\begin{align} &u_{\rm s}=-\frac{J}{2}(1+\alpha p_{\rm s}^{2}+\beta p_{\rm s}^{4})p_{\rm s}^{2},~~ \tag {5} \end{align} $$ $$\begin{align} &c_{\rm s}\equiv \frac{\partial u_{\rm s}}{\partial T},~~ \tag {6} \end{align} $$ $$\begin{align} &\chi_{\rm s}=\frac{aC_{\rm w}}{T-T_{0}a(1+3\alpha p_{\rm s}^{2}+5\beta p_{\rm s}^{4})},~~ \tag {7} \end{align} $$ where $T_{0}\equiv \frac{J}{k_{_{\rm B}}}$, $C_{\rm w}\equiv \frac{N\mu^2}{k_{_{\rm B}}\varepsilon_{0}}$ is the Curie–Weiss constant, $k_{_{\rm B}}$ is the Boltzmann constant, $\varepsilon_{0}$ is the vacuum permittivity, and $a\equiv 1-\tanh^{2}[ \frac{T_{0}}{T}(1+\alpha p_{\rm s}^{2}+\beta p_{\rm s}^{4})]$. The detailed calculation of $\chi_{\rm s}$ can be found in Appendix A. Figure 1 shows the fits of the present model to the experimental results of $P_{\rm s}$,[34] $\chi_{\rm s}^{-1}$ of the paraelectric phase[35] (because the values of $\chi_{\rm s}$ in the ferroelectric phases are related to domain walls,[41,42] they are not fitted here), and $c_{\rm s}$[36] ($c_{\rm b} $ is the background irrelevant to FPT) of triglycine sulfate crystal. The fitting parameters used are shown in Fig. 1. It can be seen that, compared with the Weiss theory and the WQL model, the present model coincides better with the data.
cpl-36-7-070501-fig2.png
Fig. 2. The values of $p_{\rm s}$, $c_{\rm s}$ and $\chi_{\rm s}$ versus $T$ for $\beta =0$ and serial $\alpha$.
Obviously, the Weiss theory completely ignores the coupling of spontaneous polarization and spontaneous strain. Mathematically, the results (Eqs. (4)-(7)) of the present model when $\beta =0$ are equivalent to the compressible Ising models[29–32] and the WQL model[33] under the single pseudospin mean-field approximation. In other words, the $\alpha p_{\rm s}^{2} $ term in Eq. (2) is equivalent to the linear coupling of $J$ to the lattice strain of the compressible Ising models[29–32] and the four-pseudospin interaction of the WQL model.[33] To further study the effect of the $\beta p_{\rm s}^{4}$ term on FPTs, $p_{\rm s}$, $c_{\rm s}$, and $\chi_{\rm s}$ of the model system with $T$ are shown in Figs. 2–7, which indicate that: (1) For $\beta =0$ (Fig. 2), the results are consistent with those reported in Refs.  [22-26]. That is: (i) when $\alpha \leqslant \frac{1}{3}$, a 2nd-order PT occurs, and the PT temperature ($T_{\rm c}$) is equal to $T_{0}$ (see Appendix B for details), and (ii) when $\alpha >\frac{1}{3}$, a 1st-order PT appears, i.e., there is a temperature range of the transition (here the highest and the lowest temperatures of the range are expressed as $T_{\rm h}$ and $T_{\rm l})$, and $T_{\rm l}=T_{0}$. (2) For $\alpha =0$, in addition to the 2nd-order PT at $T_{0}$, (i) when $\beta$ is large (Fig. 6), a sharp peak of $c_{\rm s}$ and $\chi_{\rm s}$ appears at lower temperature, while $p_{\rm s}$ jumps from a nonzero value to another one, which is called a phase-like transition here. It is a 1st-order phase-like transition when the jump value is nonzero, or a 2nd-order phase-like transition when the jump value is zero but the slope tends to infinity, and (ii) when $\beta$ is small (Figs. 2–5), there is a diffuse peak of $c_{\rm s}$, while a diffuse or shoulder peak of $\chi_{\rm s}$ and a rapid change of $p_{\rm s}$ appears at lower temperature. This process and the temperature corresponding to the maximum value of $c_{\rm s}$ are called a diffuse phase-like transition and the transition temperature ($T_{\rm d}$) in this work, respectively.
cpl-36-7-070501-fig3.png
Fig. 3. The values of $p_{\rm s}$, $c_{\rm s}$ and $\chi_{\rm s}$ versus $T$ for $\beta =0.2$ and serial $\alpha$.
cpl-36-7-070501-fig4.png
Fig. 4. The values of $p_{\rm s}$, $c_{\rm s}$ and $\chi_{\rm s}$ versus $T$ for $\beta =0.4$ and serial $\alpha$.
Figures 8 and 9 show the phase diagrams predicted by the present model, i.e., the changes of $T_{\rm c}$, $T_{\rm h}$, $T_{\rm l}$, and $T_{\rm d}$ with $\alpha$ and $\beta$. It can be seen that the model system exhibits: (1) a 2nd-order PT and a diffuse phase-like transition when $\alpha$ and $\beta$ are smaller; (2) a 2nd-order PT, and a 1st- or 2nd-order phase-like transition when $\alpha$ is smaller but $\beta$ is relatively larger; and (3) only a 1st-order PT when both $\alpha$ and $\beta$ are larger.
cpl-36-7-070501-fig5.png
Fig. 5. The values of $p_{\rm s}$, $c_{\rm s}$ and $\chi_{\rm s}$ versus $T$ for $\beta =0.6$ and serial $\alpha$.
cpl-36-7-070501-fig6.png
Fig. 6. The values of $p_{\rm s}$, $c_{\rm s}$ and $\chi_{\rm s}$ versus $T$ for $\beta =0.8$ and serial $\alpha$.
It is worth mentioning that smaller $\alpha$ and larger $\beta$ correspond to the strong nonlinear spontaneous polarization-strain coupling, and whether this kind of ferroelectrics exists or not still needs further experimental exploration. As stated above, $P_{\rm s}$ is coupled with $\delta_{\rm s}$, thus the Hamiltonian describing ferroelectric phase transition can be expressed in the form $$\begin{alignat}{1} H_{\rm total}=-\frac{J(\delta_{\rm s})}{2}\sum\limits_{i\ne j}^{\{ nn \}} {s_{i}s_{j}} +H_{\rm strain}(\delta_{\rm s}),~~ \tag {8} \end{alignat} $$ where $J(\delta_{\rm s})$ is a function of $\delta_{\rm s}$, and $H_{\rm strain}(\delta_{\rm s})$ is the Hamiltonian related to $\delta_{\rm s}$.
cpl-36-7-070501-fig7.png
Fig. 7. The values of $p_{\rm s}$, $c_{\rm s}$ and $\chi_{\rm s}$ versus $T$ for $\beta =1.0$ and serial $\alpha$.
cpl-36-7-070501-fig8.png
Fig. 8. The phase diagrams predicted by the present model, i.e., $T_{\rm c}$, $T_{\rm h}$, $T_{\rm l}$, and $T_{\rm d}$ with versus $\alpha$ for serial $\beta$.
If we know the functions of $J(\delta_{\rm s})$ and $H_{\rm strain}(\delta _{\rm s})$ with $\delta_{\rm s}$ in a specific ferroelectric material, in principle we can calculate the change of $P_{\rm s}$ and $\delta_{\rm s}$ of the material with temperature by $H_{\rm total}$, that is, we can obtain the information of the ferroelectric phase transition related to the structure. However, it could be imagined that the mathematical description of the Hamiltonian is complex and difficult to solve. This study chooses a self-consistent approximation of $H_{\rm total}$, i.e., the influence of $H_{\rm strain}(\delta_{\rm s})$ and $\delta_{\rm s}$ on $J$ is equivalent to the contribution of $P_{\rm s}$ to $J$ by the approximation (Eq. (2)). This approximation is mainly for the conciseness of mathematical description and physical image.
cpl-36-7-070501-fig9.png
Fig. 9. The phase diagrams predicted by the present model, i.e., $T_{\rm c}$, $T_{\rm h}$, $T_{\rm l}$, and $T_{\rm d}$ versus $\beta$ for serial $\alpha$.
Appendix A When an external electric field ($E\rightarrow 0$) is applied to ferroelectrics along the direction of dipole moments, the FPT Hamiltonian becomes $$\begin{alignat}{1} H_E\approx -[J_0p(1+\alpha p^2+\beta p^4)+\mu E]\sum\limits_{i=1}^{\infty}s_i,~~ \tag {A1} \end{alignat} $$ where $p=\frac{P}{N\mu}$, with $p$ the polarization of ferroelectrics. Using Eq. (A1) and based on the Boltzmann principle, it is easy to obtain $$\begin{alignat}{1} p=\tanh\Big[\frac{J_0p(1+\alpha p^2+\beta p^4)+\mu E}{k_{_{\rm B}}T}\Big].~~ \tag {A2} \end{alignat} $$ From Eq. (A2) $$\begin{alignat}{1} \frac{\partial p}{\partial E}\Big|_{E\rightarrow 0}=\alpha\frac{J(1+3\alpha p_{\rm s}^2+5\beta p_{\rm s}^4)\frac{\partial p}{\partial E}+\mu}{k_{_{\rm B}}T},~~ \tag {A3} \end{alignat} $$ where $\alpha\equiv 1-\tanh^2[\frac{T_0}{T}p_{\rm s}(1+\alpha p_{\rm s}^2+\beta p_{\rm s}^4)]$. According to the definition, i.e., $\chi_{\rm s}\equiv \frac{N\mu}{\varepsilon_0}\frac{\partial p}{\partial E}|_{E\rightarrow 0}$, we obtain $$\begin{align} \chi_{\rm s}=\frac{aC_{\rm w}}{T-T_0a(1+3\alpha p_{\rm s}^2+5\beta p_{\rm s}^5)}.~~ \tag {A4} \end{align} $$ Appendix B When $\beta=0$ and $p_{\rm s}\rightarrow 0$, from Eq. (3) and $\tanh(x)\approx x-\frac{x^3}{3}$, we obtain $$\begin{align} p_{\rm s}=\frac{T_0}{T}p_{\rm s}+\alpha\frac{T_0}{T}p_{\rm s}^3-\frac{1}{3}\Big(\frac{T_0}{T}p_{\rm s}\Big)^3.~~ \tag {B1} \end{align} $$ That is, $$\begin{align} p_{\rm s}^{2}=\frac{1-\frac{T}{T_{0}}}{\frac{1}{3}(\frac{T_{0}}{T})^{2}-\alpha}.~~ \tag {B2} \end{align} $$ Thus $p_{\rm s}$ only has a nonzero real solution below $T_{0}$ when $\alpha \leqslant \frac{1}{3}$, i.e., a 2nd-order PT occurs in the system. References Applications of Modern FerroelectricsNucleation and growth mechanism of ferroelectric domain-wall motionOrigin of morphotropic phase boundaries in ferroelectricsAbove-room-temperature ferroelectricity in a single-component molecular crystalResonant domain-wall-enhanced tunable microwave ferroelectricsApplication of ferroelectric materials for improving output power of energy harvestersPerovskite oxides for visible-light-absorbing ferroelectric and photovoltaic materialsFerroelectricity from iron valence ordering in the charge-frustrated system LuFe2O4Recent Advances in Ferroelectric Nanosensors: Toward Sensitive Detection of Gas, Mechanothermal Signals, and RadiationFerroelectric polymers exhibiting behaviour reminiscent of a morphotropic phase boundaryComparisons of Criteria for Analyzing the Dynamical Association of Solutes in Aqueous SolutionsBerezinskii—Kosterlitz—Thouless Transition in a Two-Dimensional Random-Bond XY Model on a Square LatticeA Comparative Study on the Self Diffusion of N-Octadecane with Crystal and Amorphous Structure by Molecular Dynamics SimulationPhonon and Elastic Instabilities in Zincblende TlN under Hydrostatic Pressure from First Principles CalculationsTheoretical Analysis of a Modified Continuum ModelThe Ignition of Gases by Local Sources.The Ignition of Gases by Local Sources II. Ellipsoid SourcesBeitrag zur Theorie des FerromagnetismusDynamics of order-disorder-type ferroelectrics and anti-ferroelectricsCollective motions of hydrogen bondsCollective Excitations and Magnetic Ordering in Materials with Singlet Crystal-Field Ground StateThe Ising model with a transverse field. II. Ground state propertiesLattice vibrationsXCVI. Theory of barium titanateThermodynamics of Phase Transitions in Compressible Solid LatticesSpecific Heats of Compressible Lattices and the Theory of MeltingEffects of Lattice Compressibility on Critical BehaviorPhase diagrams for compressible Ising modelsFirst-order phase transition in order-disorder ferroelectricsThe Ferroelectric Phase Transition in (Glycine) 3 ·H 2 SO 4 and Critical X-Ray ScatteringDielectric and ferroelectric properties of phosphoric acid doped triglycine sulfate single crystalsMemory effect in triglycine sulfate induced by a transverse electric field: specific heat measurementOrigin of ferroelectricity in perovskite oxidesThe Effect of Thermal Agitation on Atomic Arrangement in AlloysInternal friction and dielectric loss related to domain wallsDomain freezing in potassium dihydrogen phosphate, triglycine sulfate, and CuAlZnNi
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