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First-principles lattice dynamics and thermodynamic properties of pre-perovskite PbTiO3
Accepted Manuscript First-principles lattice dynamics and thermodynamic properties of pre-perovskite PbTiO3 Meng-Jun Zhou, Yi Wang, Yanzhou Ji, Zi-Kui Liu, Long-Qing Chen, Ce-Wen Nan PII:
S1359-6454(19)30200-9
DOI:
https://doi.org/10.1016/j.actamat.2019.04.008
Reference:
AM 15218
To appear in:
Acta Materialia
Received Date: 2 November 2018 Revised Date:
2 April 2019
Accepted Date: 5 April 2019
Please cite this article as: M.-J. Zhou, Y. Wang, Y. Ji, Z.-K. Liu, L.-Q. Chen, C.-W. Nan, First-principles lattice dynamics and thermodynamic properties of pre-perovskite PbTiO3, Acta Materialia, https:// doi.org/10.1016/j.actamat.2019.04.008. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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First-principles lattice dynamics and thermodynamic properties of pre-perovskite PbTiO3
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Meng-Jun Zhou1,2, Yi Wang2,*, Yanzhou Ji2, Zi-Kui Liu2, Long-Qing Chen2,*, Ce-Wen Nan1,* 1
School of Materials Science and Engineering, State Key Lab of New Ceramics and Fine
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Processing, Tsinghua University, Beijing 100084, China 2
Department of Materials Science and Engineering, The Pennsylvania State University,
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University Park, Pennsylvania 16802, United States Abstract
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It was recently found that nanowires of PbTiO3 synthesized through an intermediate pre-
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perovskite phase exhibit enhanced spontaneous polarization. Here we investigated the pre-
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perovskite PbTiO3 (PP-PTO) nanowire phase at finite temperatures employing first-principles
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quasiharmonic calculations. We calculated its band gap, phonon dispersions, phonon density of
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states, Debye temperature, and thermodynamic properties. The corresponding calculations for
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cubic and tetragonal PbTiO3 were also carried out for comparison. In the current calculations, the
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amount of imaginary frequencies associated with the ideal cubic PTO structure, i.e., a cubic cell
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shape with ion positions at the ideal cubic perovskite lattice sites, was decreased to a negligible
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level by employing a constrained cubic structure, a structure with the same cubic cell shape as
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the ideal cubic PTO structure but allowing the ion positions to relax to thermodynamically more
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stable tetragonal positions at 0 K. In contrast to the general observation that a higher volume
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phase would have relatively higher entropy, it is found that the PP-PTO phase possesses the
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lowest entropy while having the largest volume compared to cubic and tetragonal PbTiO3 phases.
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Furthermore, the temperature-pressure phase diagram for the three PbTiO3 phases was obtained,
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which demonstrates that PP-PTO could be stabilized under a large volume or a negative pressure.
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This study provides insights to experimentally synthesizing the PP-PTO phase and to better
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understanding its phase transition into the converted tetragonal PbTiO3 nanowires with enhanced
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piezoelectric and ferroelectric properties.
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Key words: Pre-perovskite PbTiO3 (PP-PTO); First-principles calculations; lattice dynamics;
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phase diagram
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*
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[email protected] (Ce-Wen Nan)
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Corresponding authors: [email protected] (Yi Wang), [email protected] (Long-Qing Chen) and
7 1. Introduction
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Lead titanate (PbTiO3, PTO) is a benchmark ferroelectric insulator. The phase transition
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between its two known perovskite structures, i.e., the ferroelectric tetragonal phase PbTiO3 (TP-
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PTO) and the para-electric cubic phase PbTiO3 (CP-PTO) [1], has been considered as a classical
12
example to understand structural and ferroelectric phase transitions in perovskites [2–7].
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Recently, a metastable PTO phase, referred to the pre-perovskite phase for PbTiO3 (PP-PTO),
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was synthesized by a hydrothermal method in the form of nanowires [8,9]. In addition, the PP
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nanowires can be synthesized by doping Zr (0%-15%) [10,11] and Ba (0%-15%) [12]. It was
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noted that this new phase has the same structure as the PX phase discovered by Cheng et al [13]
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in the form of powders, and was reported to transform into the TP-PTO phase after being
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annealed at temperatures higher than 350 °C [8,14]. Interestingly, this transformation would lead
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to the TP-PTO nanowires with enhanced ferroelectric and piezoelectric properties, e.g., an
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increased tetragonality (c/a - 1) as high as 0.13, a doubling of spontaneous polarization to ~ 160
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µC/cm2, an increase of Curie temperature by 40 °C and piezoelectric constant d33 by 20 pm/V
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[11,15]. Zero thermal expansion was also observed in the TP-PTO nanowires converted from
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PP-PTO [16].
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The three-dimensional CP-PTO and TP-PTO perovskite structures, as shown in Fig. 1(a) and
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Fig. 1(b), are characterized by the corner-shared TiO6 octahedra: macroscopically, the Ti atoms
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of CP-PTO are located at the center of the TiO6 octahedra with displacements along the c-axis in
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TP-PTO [17]. In comparison, the octahedra in PP-PTO are connected by sharing their edges and
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packed along the c-axis in an interlaced manner to form a one-dimensional column [8] as shown
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in Fig. 1(c) and Fig. 1(d).
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Existing first-principles calculations [18–21] of PP-PTO were mostly carried out to
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investigate its vibrational stability and other mechanical properties at 0 K. In this work, we focus
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on the phase stability of PP-PTO with respect to the stable PTO phases at finite temperatures and
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pressures. Systematic first-principles quasiharmonic calculations were carried out to investigate
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the lattice dynamics and thermodynamic properties of the PP-PTO phase together with the cubic
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and tetragonal PbTiO3. The predicted thermodynamic properties and pressure-temperature phase
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diagram for the three types of PbTiO3 confirmed the preferable formation of PP-PTO under a
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negative pressure (or a large volume), which could provide feasible strategies for the
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experimental fabrication of the PP-PTO phase and obtaining the converted TP-PTO nanowires
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from PP-PTO.
2. Method
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To determine the phase equilibria of PP-PTO, we first compute the Helmholtz free energy F
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using first-principles calculations which can be decomposed into the sum of the static energy
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Estatic, the lattice vibrational free energy Fvib, and the thermal electron free energy Felec induced
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by the thermal electronic excitation [22–27]
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F (T ,V ) = Estatic (V ) + Fvib (T , V ) + Felec (T ,V )
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(1)
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where T is temperature and V is volume. Based on phonon theory, Fvib can be written as [28]
ℏω Fvib (T ,V ) = kBT ∫ ln 2sinh g(ω,V )dω , 2 k T B
2
(2)
where ℏ is the reduced Planck’s constant, kB is the Boltzmann constant, and g ( ω , V ) represents
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the phonon density-of-states (PDOS). Since the electronic contribution for semiconductors and
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insulators is not important, Felec is ignored in the present calculations. The vibrabtional entropy
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Svib and heat capacity Cvib can be calculated by:
Cvib (T ,V ) = kB ∫
∞
( ℏω / kBT )
0
(e
) × g(ω,V)dω ,
(3)
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ℏω / k BT
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eℏω / kBT
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−1
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g (ω ,V )dω .
(4)
In the present work, the Helmholtz free energy is calculated as a function of volume. Then the results are fitted using the modified Birch-Murnaghan equation of states [29,30],
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∞ ( ℏω / k T ) Svib (T,V ) = kB ∫ ℏω/kBT B − ln 1− e−ℏω/kBT 0 −1 e
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F (T ,V ) = a + bV −2/3 + cV −4/3 + dV −2 + eV −8/3 .
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(5)
Thus, the equilibrium volume Veq at zero pressure and specific temperature can be obtained
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by solving
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∂F (T,V ) − = 0. ∂V T
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(6)
Thermodynamic properties including bulk modulus BT, Debye temperature
ΘD
and thermal
expansion coefficient α can be calculated following the Eqs. (7) - (9):
∂2 F (V , T ) BT (T ,V ) = V , 2 ∂ V T
(7)
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T Cvib,Debye (T ) = 9kB ΘD (T )
a(T) =
2
∫
ΘD (T )/T
0
x4ex dx , (ex −1)2
1 ∂Veq (T) , 3Veq (T) ∂T P
(8)
(9)
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where the Debye temperature ΘD in Eq. (8), can be calculated by fitting the calculated Cvib in Eq.
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(4).
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All calculations were carried out by the projector-augmented wave (PAW) method [31,32]
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implemented in the Vienna ab initio simulation package (VASP, version 5.4.1). Following
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previous experiences, the local density approximation (LDA) for exchange-correlation function
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[5,33,34] is usually more suitable for oxides while the generalized gradient approximation (GGA)
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is more appropriate for metals [35]. We employed the quasiharmonic approximation, and all
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VASP calculations were performed at 11 different volumes with an increment rate of 6%. To
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ensure the efficiency of calculation, the primitive unit cell of PP-PTO containing 20 atoms was
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considered [18]. For the crystal structure relaxation [36], the Γ-centered k-point mesh of 3×3×6
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was adopted with the default cutoff energy of 400 eV. A conjugate gradient method was used to
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optimize the lattice parameters and atomic coordinates and minimize the Hellmann-Feynman
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Forces and stress tensor components. The change of total free energy was converged to 10-8 eV
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per cell, and all the forces were relaxed to at least 10-4 eVÅ-1. After crystal relaxation, a k-point
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mesh of 6×6×12 and a cut off energy of 520 eV were set for high accuracy static calculations.
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The lattice dynamics and properties of CP-PTO and TP-PTO were also calculated for
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comparison. For TP-PTO and CP-PTO, the unit cell containing five atoms was considered.
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YPHON package [37–39] which adopted the small displacement approach (supercell method)
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was used for calculating phonon dispersion and thermal properties, considering both plus and
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minus displacements by default. A 3×3×3 k-point mesh and a 2×2×2 supercell of the primitive
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cell were considered in our phonon calculations for the three phases. Since the primitive unit cell
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of PP-PTO already contains 20 atoms, supercells larger than the 2×2×2 supercell for PP-PTO are
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computationally prohibitive. Therefore, we employed a 2×2×2 supercell for all the three
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compounds in our calculations. We tested the effect of supercell size on the amount of imaginary
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phonon frequencies for CP-PTO since the unit cell of CP-PTO only contains 5 atoms. As we
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increased the supercell size from 2×2×2 to 4×4×4, no significant changes were observed for the
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phonon dispersions and the amount of the imaginary phonon frequencies.
9 Results and Discussion
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3.1 Static energy and electronic density of states
The calculated lattice constants and atomic coordinates of PP-PTO at the 0 K static
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equilibrium volume are collected in Table 1, which are in good agreement with the results from
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Neutron diffraction [8]. The static energy-volume curves for the three perovskites at 0 K are
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plotted in Fig. 2(a). It shows that PP-PTO has the highest static energy at its equilibrium volume
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among the three phases, which is 0.20 eV (∆EPP-TP) per formula unit and 0.14 eV (∆EPP-CP) per
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formula unit higher than those of TP-PTO and CP-PTO, respectively. The equilibrium volume
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for PP-PTO is much larger than those of TP- and CP-PTO. ∆EPP-TP and ∆EPP-CP are a little higher
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than those reported by Wang et al. [19] and Liu et al. [18], perhaps due to the usage of different
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pseudo-potentials and software packages for first principles calculations. The total and atomic
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electronic density-of-states (EDOS) for PP-PTO, TP-PTO and CP-PTO are given in Fig. 2(b). It
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shows that the band gaps for PP-, TP- and CP-PTO are 1.96 eV, 1.30 eV and 1.62 eV,
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respectively. These calculated band gaps are comparable to those from previous calculations
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[21,40].
3 3.2 Lattice dynamics
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The vibrational contribution to free energy plays the major role in dictating the relative
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thermodynamic stability of a compound at finite temperatures [41]. Fig. 3(a) and Fig. 3(c) show
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the partial PDOS due to the individual atoms for PP- and TP-PTO, respectively. It is
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demonstrated that in both PP- and TP-PTO, the Pb atoms only contribute to the low frequency
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region of <5 THz due to the large atomic mass. Compared with TP-PTO, the contributions of the
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Ti atoms to the frequency regions ranging from 18 THz to 22 THz and from 23 THz to 25 THz
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are more prominent in PP-PTO. In addition, it seems that the vibrational modes of Ti and O in
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PP-PTO are stronger than those in TP-PTO, indicating that the bonding between the Ti atoms
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and the O atoms in PP-PTO is more complicated.
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In addition, Fig. 3(b) and Fig. 3(d) show the calculated total PDOS and the generalized
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phonon density-of-states (GPDOS) [42] of PP-PTO and TP-PTO. The total PDOS is the simple
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summation of contributions from Pb, Ti and O atoms. Note that usually for compounds, PDOS
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cannot be measured directly through experiments, in which case the measured data mostly refer
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to the neutron scattering cross section weighted PDOS, i.e., GPDOS. The GPDOS for PP-PTO
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and TP-PTO can be calculated by
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GPDOS = ∑i
σi mi
pDOSi ,
(10)
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where σi and mi refer to the atomic scattering cross section and the atomic mass for each
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element, respectively; pDOSi represents the projected partial PDOS for individual atoms. Since
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the Pb atoms possess the largest atomic mass, their contributions to the GPDOS are much weaker
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than the Ti and O atoms. The calculation of the PDOS is for the purpose of evaluating the free energy which normally
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does not allow imaginary phonon modes. However, for the high temperature CP-PTO phase,
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when the ideal cubic perovskite structure is employed, the calculated PDOS for the CP-PTO
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phase shows a significant amount of imaginary phonon modes [43]. It is noted that the more the
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amount of the imaginary phonon modes is, the higher the uncertainty of the calculated free
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energy becomes. To circumvent this difficulty, we performed a calculation using a structure
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defined as the constrained cubic PTO structure to calculate the phonon density-of-states for CP-
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PTO. The constrained cubic PTO structure is obtained by keeping the cell shape of the ideal
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cubic PTO structure while the internal atomic coordinates are allowed to relax as those of TP-
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PTO, i.e., allowing all atoms to relax from their ideal cubic positions under the constraint of the
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cubic cell shape. The idea of constrained calculation for CP-PTO is inspired from the X-ray
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absorption fine-structure (XAFS) measurements [44,45] for PbTiO3 and BaTiO3 that showed
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disorder of local distortions. In particular for PbTiO3, XAFS measurement of Sicron et al.[44]
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pointed out that the Ti atoms can be displaced relative to the oxygen octahedra cage center both
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below and above the transition temperature. The calculated partial PDOS, total PDOS and
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GPDOS based on the constrained cubic PTO structure are illustrated in Fig. 4(a) and Fig. 4(b).
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The amount of imaginary modes is significantly reduced when compared with those calculated
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using the ideal CP-PTO structure as shown in Fig. 4(c) and Fig. 4(d). Specifically, the amount of
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imaginary phonon modes in the total phonon density-of-states has been decreased from 6.46% to
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0.46% when the constrained structure is employed instead of the ideal cubic structure.
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To our best knowledge, no theoretical and experimental results have been reported on the
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complete phonon spectrum for PP-PTO. Fig. 5 shows the calculated phonon dispersions for the
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PP-PTO, TP-PTO, the ideal and the constrained cubic PTO structures. Phonon dispersion
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represents the one-dimensional evolution of phonon frequency along the high symmetry
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directions in the wave vector space. In Fig. 5(a), the phonon modes at Γ point for PP-PTO are
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classified into 43 phonon modes, i.e., 25 Raman-active modes including 10 Ag, 10 Bg and 5 Eg;
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13 IR-active modes including 9 Eu and 4 Au; and 5 Bu optically silent modes. Table 2 lists the
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calculated phonon frequencies for the total 43 phonon modes at the Γ point of PP-PTO. The
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experimental data by Raman scattering for PP-PTO in references [19,20] are represented by the
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hollow dots in Fig. 5(a), which are in good agreement with the calculated results. According to
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the phonon dispersion for TP-PTO in Fig. 5(b), the calculated results for TP-PTO are consistent
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with the experimental data (marked by the hollow dots) [46,47]. No imaginary phonon modes
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were found for either PP-PTO or TP-PTO, which indicates that PP-PTO and TP-PTO are
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vibrationally stable at 0 K. Knowing the fact that the static energy (discussed in Section 3.1) at 0
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K for PP-PTO is much higher than that for TP-PTO, it can be concluded that PP-PTO is a
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metastable phase at 0 K. The findings explain the experimentally verified phase transition
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process during heating treatment in which the PP-PTO transforms to the CP-PTO as temperature
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increases from 25 °C to 700 °C, and when the CP-PTO is cooled back down to 25 °C, it becomes
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the TP-PTO rather than the original PP-PTO [8,11,14,15].
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Figure 5(c) and Fig. 5(d) show the phonon dispersions calculated using the ideal and
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constrained cubic PTO structures, respectively. The ideal cubic PTO structure exhibits imaginary
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phonon frequencies at Γ, M and P points, and the largest imaginary phonon frequency is about -4
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THz while the constrained cubic PTO structure exhibits imaginary phonon frequency only at Γ
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point, with a value of about -1 THz. In contrast to the ideal cubic PTO structure, there are very
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few imaginary phonon modes for the calculated result using the constrained cubic PTO structure.
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Therefore, only the constrained cubic PTO structure is used in calculating the vibrational energy
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for CP-PTO.
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3.3 Free energies and thermodynamic properties
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Since the electron excitation free energy is ignored due to the insulating nature of all PTO
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considered in this work, the calculated entropy S is the vibrational entropy as defined in Eq. (3).
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Fig. 6(a) shows the variation of entropy with temperature ranging from 0 K to 1200 K for PP-
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PTO, TP-PTO and CP-PTO. Among these three phases, PP-PTO possesses the lowest entropy
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while CP-PTO has the highest. This result is beyond conventional thought by which it was
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anticipated that a large volume phase (the PP-PTO phase has the largest volume in the present
12
case) would have relatively high entropy. This can be attributed to the change of the low
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frequency vibrational mode distribution due to the change of the Pb-O bonds from TP-PTO to
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PP-PTO. As shown in Fig. 3, the PDOS values near 2 THz (due to Pb) and 15 THz (due to O) are
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quite different between PP-PTO and TP-PTO; this difference further leads to less contribution of
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these phonon vibration frequencies to the entropy of PP-PTO by the frequency-dependent
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entropy contribution
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f (ω, T ) =
( ℏω / kBT ) − ln 1− e−ℏω/k T
(
eℏω/ kBT −1
B
) g(ω,V ) ,
(11)
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As illustrated in the inset plot in Fig. 6(a), the lower f(ω,T) values of PP-PTO near 2 THz and 15
20
THz are the reasons for the lower entropy of PP-PTO at 500 K.
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In literature, the comparison of the 0 K total energies for these three perovskites was already
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reported. In this work, we want to consider the effect of temperature. The Gibbs free energy G at
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finite temperatures can be obtained by 10
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G (T , P ) = F (T ,V ) + PV ,
1 2
(12)
where P is pressure. Figure 6(b) plots the Gibbs free energies as a function of temperature at P = 0 for the three
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phases. The Gibbs free energies all show a continuous decrease with the increase of temperature
5
from 0 K to 1200 K since the contribution of the negative term −TS becomes more dominant as
6
temperature increases. The Gibbs free energy of PP-PTO ranks the highest among these three
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within the considered temperature range. Based on the calculations, there will be an intersection
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located at 960 K between the curves of TP-PTO and CP-PTO. This calculated phase transition
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temperature between TP-PTO and CP-PTO is higher than the reported experimental values of
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763 K [48,49], probably due to the underestimation of the entropy of the constrained CP-PTO
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phase.
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Table 3 lists the values of Gibbs free energies, lattice constants and bulk moduli for PP-PTO,
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TP-PTO and CP-PTO at 0 K and room temperature 300 K. Compared with the calculated values
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at 0 K, the energy differences ∆GPP-CP and ∆GPP-TP nearly remain the same at 300 K. Specifically,
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the energy difference ∆GPP-TP changes from 0.21 eV/f.u. to 0.22 eV/f.u. while ∆GPP-CP changes
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from 0.16 eV/f.u. to 0.18 eV/f.u.. Bulk moduli for these three perovskites are calculated by Eq.
17
(7). Bulk modulus indicates the resistance of a material to the uniform compression. According
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to the calculated results, PP-PTO is the easiest to be compressed while CP-PTO is the hardest
19
one. The absence of decrease in the bulk modulus for TP-PTO from 0 to 300 K is due to the
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calculated nearly zero thermal expansion coefficient within this temperature range, which agrees
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well with the experimentally reported values [6].
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The curves of Debye temperatures and thermal expansion coefficients at finite temperatures
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for these three structures are plotted in Fig. 7, respectively. When the temperature is close to 0 K, 11
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the differences in the Debye temperatures for these three perovskites are very small as seen in
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Fig. 7(a). The values of thermal expansion coefficients plotted in Fig. 7(b) indicate that the
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thermal expansions of PP-PTO and CP-PTO are both higher than those of TP-PTO in the
4
calculated region from 0 K to 1200 K. The change of thermal expansion for these three phases is
5
not so significant when the temperature is higher than room temperature, since the slopes of
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theses curves are quite flat when T>300 K. It should be noted that the calculation predicts a
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negative expansion for TP-PTO structures in the temperature range from 30 K to 150 K. The
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experimental results on TP-PTO nanoplates from Ren et. al also showed a typical negative
9
thermal expansion with a thermal expansion coefficient of -1.92×10-5 K-1 from 293 K to 573 K
10
[16]. Previous studies [5,16,50,51] have also demonstrated that the origin of the negative thermal
11
expansion in TP-PTO is closely related to the ferroelectric transition between TP-PTO and CP-
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PTO, which corresponds to the two-phase equilibrium line with a negative slope in the
13
temperature-pressure phase diagram.
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3.4 Pressure-temperature phase diagram
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Besides the effect of temperature, the influence of pressure on the Gibbs free energies for
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PP-PTO, TP-PTO and CP-PTO are also studied in this work. Following Eq. (12), Gibbs free
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energies at different temperatures and pressures can be obtained. The most stable phase is
18
defined as the phase with the minimum Gibbs free energy at given temperature and pressure. Fig.
19
8 shows the calculated pressure-temperature phase diagram. The calculated ranges of
20
temperature and pressure cover from 0 K to 1500 K and from -2.5 GPa to 2.5 GPa, respectively.
21
At the bottom of the phase diagram, where the pressure is from -1.6 GPa to -2.5 GPa, PP-PTO is
22
the most stable phase. On the left top of the PP-PTO phase, it is TP-PTO. Next to the TP-PTO
23
phase is the CP-PTO. The predicted transition pressure between PP-PTO and TP-PTO at 0 K is
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1.6 GPa, which is comparable to the calculated value of 1.4 GPa by Liu et al. [18]. This
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experimental data point is marked with the red-and-blue circular point in Fig. 8. The
3
experimental data in Ref. [48] are also marked in Fig. 8. Most of the data are in good agreement
4
with the present calculation except for a slight difference for the transition temperature from CP-
5
PTO to TP-PTO at room temperature.
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The established pressure-temperature (P-T) diagram indicates that a comparison among the
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temperature-pressure space is important in studying the phase stability of PP-PTO. The presence
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of PP-PTO under a negative pressure or a high volume proposes the possible strategies for
9
effectively synthesizing the PP-PTO phase. Interphase strain recently proposed by Zhang et.al
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[52] could be a feasible way to realize a negative pressure although achieving a negative pressure
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remains a challenge in experiments. [15,34,52]
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12 4. Summary
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A systematic investigation of lattice dynamics for the PP-PTO structure is performed based
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on first-principles quasiharmonic approach. A novel methodology is presented to reduce the
16
amount of soft modes in the calculation of phonon density-of-states. The influences of
17
temperature and pressure on phase stability and thermodynamic properties of PP-PTO were
18
studied. It is found that:
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1. At 0 K and zero pressure, the PP-PTO is a vibrationally metastable phase while the TP-PTO
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is vibrationally stable, and the CP-PTO is vibrationally unstable. 2. The PP-PTO phase possesses lower entropy and larger volume compared to the CP-PTO and TP-PTO phases.
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3. A pressure-temperature phase diagram for PTO is constructed, indicating that the PP-PTO
2
phase can be stable under a negative pressure in the temperature range of 0 K through 1500
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K.
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Acknowledgements The author M. J. Zhou gratefully acknowledges the financial support from China
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Scholarship Council (No 201706210108). This work was partially supported by the NSF of
4
China (Grant Nos. 51332001 and 51472140) and in part by the Hamer Professorship (Wang and
5
Chen) and the National Science Foundation (NSF) through Grant Nos. DMR-1310289 and CHE-
6
1230924 (Wang and Liu). First-principles calculations were carried out partially on the LION
7
clusters at the Pennsylvania State University. The authors also gratefully acknowledge Prof.
8
Zhaohui Ren at Zhejiang University for useful discussions.
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Figure Captions
2
Fig. 1. Projections of crystal structures of (a) CP-PTO (Space group Pm3m ഥ ), (b) TP-PTO (Space
3
group P4mm), and (c) PP-PTO (Space group I4/m) along the a-axis; (d) PP-PTO along the c-axis.
4
Fig. 2. (a) Calculated static energy for PP-PTO, TP-PTO and CP-PTO; and (b) corresponding
5
electronic density-of-states (the solid, long dashed, short dashed, and dot-dashed lines represent
6
the total, Pb, Ti, and O contribution to the EDOS, respectively).
7
Fig. 3. (a), (b) Calculated partial, total and generalized phonon density-of-states (PDOS) for PP-
8
PTO; and (c), (d) Calculated partial, total and generalized phonon density-of-states for TP-PTO.
9
The label “total” refers to PDOS due to the simple summation of contributions from Pb, Ti and O
10
atoms and the label “General” refers to the neutron scattering cross section weighted PDOS (the
11
generalized PDOS, see equation (10) in the main text)
12
Fig. 4. (a), (b) Calculated partial, total and generalized phonon density-of-states for the
13
constrained cubic PTO structure; and (c), (d) Calculated partial, total and generalized phonon
14
density-of-states for the ideal cubic PTO structure. (The constrained cubic PTO structure adopts
15
the lattice constants of the ideal cubic PTO structure with the internal atomic coordinates being
16
allowed to relax as those of TP-PTO.)
17
Fig. 5. Calculated phonon dispersion of (a) PP-PTO, (b) TP-PTO, (c) the ideal cubic PTO
18
structure, and (d) the constrained cubic PTO structure along the selected principal symmetry
19
directions. (Blue hollow points: ref. [19]; green hollow points: ref. [20]; magenta hollow points:
20
ref. [46]; red hollow points: ref. [47].)
21
Fig. 6. (a) Calculated entropy, and (b) Gibbs free energy as a function of temperature for PP-
22
PTO, TP-PTO and CP-PTO at ambient pressure.
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Fig. 7. (a) The calculated Debye temperature, and (b) thermal expansion coefficient as a function
2
of temperature for PP-PTO, TP-PTO and CP-PTO at ambient pressure.
3
Fig. 8. Calculated pressure-temperature phase diagram by first-principles calculations for PP-,
4
TP- and CP-PTO.
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Table Captions
6
Table 1: The calculated and experimental fractional atomic coordinates and lattice parameter of
7
PP-PTO
8
Table 2: Calculated phonon frequencies at the Γ point for PP-PTO
9
Table 3: Calculated Gibbs free energies, lattice constants and bulk moduli at 0 K and 300 K for PP-, CP-, and TP-PTO, respectively.
11 12
16 17 18 19 20
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Table 1: The calculated and experimental fractional atomic coordinates and lattice parameter of PP-PTO Neutron diffraction [8]
x
y
z
x
y
z
Pb
0.161
0.151
0.5
0.165
0.151
0.5
Ti
0.470
0.143
0.5
0.471
0.142
0.5
O1
0.605
0.027
0.5
0.605
0.028
0.5
O2
0.166
0.288
0.0
0.169
0.289
0
O3
0.551
0.265
0.5
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Theoretical calculation
0.543
0.260
0.5
a = b = 12.367 (Å), c = 3.808 (Å)
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Table 2: Calculated phonon frequencies at the Γ point for PP-PTO Sym
THz
(cm-1)
EP
No.
10 Ag Modes (Raman active) Ag
1
Sym
THz
(cm-1)
9 Eu Modes (IR active)
25.9081
(864.20)
0
Eu
23.7522
(792.29)
Ag
19.0225
(634.52)
1
Eu
19.4970
(650.35)
2
Ag
14.5517
(485.39)
2
Eu
12.5981
(420.23)
3
Ag
11.9404
(398.29)
3
Eu
11.3287
(377.88)
4
Ag
8.3986
(280.15)
4
Eu
10.8704
(362.60)
5
Ag
7.8379
(261.45)
5
Eu
7.6321
(254.58)
6
Ag
5.7355
(191.31)
6
Eu
5.9574
(198.72)
7
Ag
5.2023
(173.53)
7
Eu
4.024
(134.23)
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No.
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Ag
3.2213
(107.45)
9
Ag
1.8314
(61.09)
10 Bg Modes (Raman active)
8
Eu
1.9827
(66.14)
4 Au Modes (IR active）
Bg
23.8081
(794.15)
0
Au
15.0737
(502.81)
1
Bg
20.9799
(699.81)
1
Au
9.2764
(309.43)
2
Bg
13.3511
(445.35)
2
Au
6.6266
(221.04)
3 4 5
Bg Bg Bg
12.3043 11.3077 7.7745
(410.43) (377.19) (259.33)
3
Au
2.2444
(74.86)
6
Bg
6.8289
(227.79)
0
Bu
20.1405
(671.82)
7
Bg
5.2077
(173.71)
1
Bu
9.5517
(318.61)
8
Bg
3.1799
(106.07)
2
Bu
7.1807
(239.52)
9
Bg
1.788
(59.64)
3 4
Bu Bu
5.8358 1.0022
(194.66) (33.43)
Eg
17.6166
1
Eg
8.9472
2
Eg
8.2887
3
Eg
5.7436
4
Eg
1.366
(298.45) (276.48)
(191.59) (45.56)
EP AC C
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(-587.63)
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5 Bu Modes (silent)
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Γopt =10Ag + 4Au +10Bg + 5Bu + 5Eg + 9Eu Raman Active = 10Ag + 10Bg + 5Eg IR Active = 4Au + 9Eu Silence Modes = 5Bu
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Table 3: Calculated Gibbs free energies, lattice constants and bulk moduli at 0 K and 300 K for PP-, CP-, and TP-PTO, respectively.
（eV/f.u.）
Lattice constant* (Å)
Bulk modulus (GPa)
RI PT
Gibbs free energy*
300 K
0K
300 K
0K
300 K
PP-PTO
-41.28
-41.46
a=12.208 c=3.768
a=12.240 c=3.770
64
63
TP-PTO
-41.49
-41.68
a=3.859 c=4.049
a = 3.860 c = 4.049
119
119
CP-PTO
-41.44
-41.64
a=3.894
a=3.897
213
204
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0K
AC C
EP
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* The results of Gibbs free energies and lattice constants at 0 K in Table II have included the contribution of vibration energy at 0 K.
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S1359-6454(19)30200-9
DOI:
https://doi.org/10.1016/j.actamat.2019.04.008
Reference:
AM 15218
To appear in:
Acta Materialia
Received Date: 2 November 2018 Revised Date:
2 April 2019
Accepted Date: 5 April 2019
Please cite this article as: M.-J. Zhou, Y. Wang, Y. Ji, Z.-K. Liu, L.-Q. Chen, C.-W. Nan, First-principles lattice dynamics and thermodynamic properties of pre-perovskite PbTiO3, Acta Materialia, https:// doi.org/10.1016/j.actamat.2019.04.008. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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First-principles lattice dynamics and thermodynamic properties of pre-perovskite PbTiO3
2
Meng-Jun Zhou1,2, Yi Wang2,*, Yanzhou Ji2, Zi-Kui Liu2, Long-Qing Chen2,*, Ce-Wen Nan1,* 1
School of Materials Science and Engineering, State Key Lab of New Ceramics and Fine
5
Processing, Tsinghua University, Beijing 100084, China 2
Department of Materials Science and Engineering, The Pennsylvania State University,
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University Park, Pennsylvania 16802, United States Abstract
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It was recently found that nanowires of PbTiO3 synthesized through an intermediate pre-
9
perovskite phase exhibit enhanced spontaneous polarization. Here we investigated the pre-
10
perovskite PbTiO3 (PP-PTO) nanowire phase at finite temperatures employing first-principles
11
quasiharmonic calculations. We calculated its band gap, phonon dispersions, phonon density of
12
states, Debye temperature, and thermodynamic properties. The corresponding calculations for
13
cubic and tetragonal PbTiO3 were also carried out for comparison. In the current calculations, the
14
amount of imaginary frequencies associated with the ideal cubic PTO structure, i.e., a cubic cell
15
shape with ion positions at the ideal cubic perovskite lattice sites, was decreased to a negligible
16
level by employing a constrained cubic structure, a structure with the same cubic cell shape as
17
the ideal cubic PTO structure but allowing the ion positions to relax to thermodynamically more
18
stable tetragonal positions at 0 K. In contrast to the general observation that a higher volume
19
phase would have relatively higher entropy, it is found that the PP-PTO phase possesses the
20
lowest entropy while having the largest volume compared to cubic and tetragonal PbTiO3 phases.
21
Furthermore, the temperature-pressure phase diagram for the three PbTiO3 phases was obtained,
22
which demonstrates that PP-PTO could be stabilized under a large volume or a negative pressure.
23
This study provides insights to experimentally synthesizing the PP-PTO phase and to better
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understanding its phase transition into the converted tetragonal PbTiO3 nanowires with enhanced
2
piezoelectric and ferroelectric properties.
3
Key words: Pre-perovskite PbTiO3 (PP-PTO); First-principles calculations; lattice dynamics;
4
phase diagram
5
*
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[email protected] (Ce-Wen Nan)
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Corresponding authors: [email protected] (Yi Wang), [email protected] (Long-Qing Chen) and
7 1. Introduction
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Lead titanate (PbTiO3, PTO) is a benchmark ferroelectric insulator. The phase transition
10
between its two known perovskite structures, i.e., the ferroelectric tetragonal phase PbTiO3 (TP-
11
PTO) and the para-electric cubic phase PbTiO3 (CP-PTO) [1], has been considered as a classical
12
example to understand structural and ferroelectric phase transitions in perovskites [2–7].
13
Recently, a metastable PTO phase, referred to the pre-perovskite phase for PbTiO3 (PP-PTO),
14
was synthesized by a hydrothermal method in the form of nanowires [8,9]. In addition, the PP
15
nanowires can be synthesized by doping Zr (0%-15%) [10,11] and Ba (0%-15%) [12]. It was
16
noted that this new phase has the same structure as the PX phase discovered by Cheng et al [13]
17
in the form of powders, and was reported to transform into the TP-PTO phase after being
18
annealed at temperatures higher than 350 °C [8,14]. Interestingly, this transformation would lead
19
to the TP-PTO nanowires with enhanced ferroelectric and piezoelectric properties, e.g., an
20
increased tetragonality (c/a - 1) as high as 0.13, a doubling of spontaneous polarization to ~ 160
21
µC/cm2, an increase of Curie temperature by 40 °C and piezoelectric constant d33 by 20 pm/V
22
[11,15]. Zero thermal expansion was also observed in the TP-PTO nanowires converted from
23
PP-PTO [16].
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The three-dimensional CP-PTO and TP-PTO perovskite structures, as shown in Fig. 1(a) and
2
Fig. 1(b), are characterized by the corner-shared TiO6 octahedra: macroscopically, the Ti atoms
3
of CP-PTO are located at the center of the TiO6 octahedra with displacements along the c-axis in
4
TP-PTO [17]. In comparison, the octahedra in PP-PTO are connected by sharing their edges and
5
packed along the c-axis in an interlaced manner to form a one-dimensional column [8] as shown
6
in Fig. 1(c) and Fig. 1(d).
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Existing first-principles calculations [18–21] of PP-PTO were mostly carried out to
8
investigate its vibrational stability and other mechanical properties at 0 K. In this work, we focus
9
on the phase stability of PP-PTO with respect to the stable PTO phases at finite temperatures and
10
pressures. Systematic first-principles quasiharmonic calculations were carried out to investigate
11
the lattice dynamics and thermodynamic properties of the PP-PTO phase together with the cubic
12
and tetragonal PbTiO3. The predicted thermodynamic properties and pressure-temperature phase
13
diagram for the three types of PbTiO3 confirmed the preferable formation of PP-PTO under a
14
negative pressure (or a large volume), which could provide feasible strategies for the
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experimental fabrication of the PP-PTO phase and obtaining the converted TP-PTO nanowires
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from PP-PTO.
2. Method
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To determine the phase equilibria of PP-PTO, we first compute the Helmholtz free energy F
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using first-principles calculations which can be decomposed into the sum of the static energy
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Estatic, the lattice vibrational free energy Fvib, and the thermal electron free energy Felec induced
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by the thermal electronic excitation [22–27]
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F (T ,V ) = Estatic (V ) + Fvib (T , V ) + Felec (T ,V )
3
(1)
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where T is temperature and V is volume. Based on phonon theory, Fvib can be written as [28]
ℏω Fvib (T ,V ) = kBT ∫ ln 2sinh g(ω,V )dω , 2 k T B
2
(2)
where ℏ is the reduced Planck’s constant, kB is the Boltzmann constant, and g ( ω , V ) represents
4
the phonon density-of-states (PDOS). Since the electronic contribution for semiconductors and
5
insulators is not important, Felec is ignored in the present calculations. The vibrabtional entropy
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Svib and heat capacity Cvib can be calculated by:
Cvib (T ,V ) = kB ∫
∞
( ℏω / kBT )
0
(e
) × g(ω,V)dω ,
(3)
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ℏω / k BT
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eℏω / kBT
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−1
2
g (ω ,V )dω .
(4)
In the present work, the Helmholtz free energy is calculated as a function of volume. Then the results are fitted using the modified Birch-Murnaghan equation of states [29,30],
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∞ ( ℏω / k T ) Svib (T,V ) = kB ∫ ℏω/kBT B − ln 1− e−ℏω/kBT 0 −1 e
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F (T ,V ) = a + bV −2/3 + cV −4/3 + dV −2 + eV −8/3 .
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(5)
Thus, the equilibrium volume Veq at zero pressure and specific temperature can be obtained
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by solving
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∂F (T,V ) − = 0. ∂V T
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(6)
Thermodynamic properties including bulk modulus BT, Debye temperature
ΘD
and thermal
expansion coefficient α can be calculated following the Eqs. (7) - (9):
∂2 F (V , T ) BT (T ,V ) = V , 2 ∂ V T
(7)
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T Cvib,Debye (T ) = 9kB ΘD (T )
a(T) =
2
∫
ΘD (T )/T
0
x4ex dx , (ex −1)2
1 ∂Veq (T) , 3Veq (T) ∂T P
(8)
(9)
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where the Debye temperature ΘD in Eq. (8), can be calculated by fitting the calculated Cvib in Eq.
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(4).
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All calculations were carried out by the projector-augmented wave (PAW) method [31,32]
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implemented in the Vienna ab initio simulation package (VASP, version 5.4.1). Following
7
previous experiences, the local density approximation (LDA) for exchange-correlation function
8
[5,33,34] is usually more suitable for oxides while the generalized gradient approximation (GGA)
9
is more appropriate for metals [35]. We employed the quasiharmonic approximation, and all
10
VASP calculations were performed at 11 different volumes with an increment rate of 6%. To
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ensure the efficiency of calculation, the primitive unit cell of PP-PTO containing 20 atoms was
12
considered [18]. For the crystal structure relaxation [36], the Γ-centered k-point mesh of 3×3×6
13
was adopted with the default cutoff energy of 400 eV. A conjugate gradient method was used to
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optimize the lattice parameters and atomic coordinates and minimize the Hellmann-Feynman
15
Forces and stress tensor components. The change of total free energy was converged to 10-8 eV
16
per cell, and all the forces were relaxed to at least 10-4 eVÅ-1. After crystal relaxation, a k-point
17
mesh of 6×6×12 and a cut off energy of 520 eV were set for high accuracy static calculations.
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The lattice dynamics and properties of CP-PTO and TP-PTO were also calculated for
19
comparison. For TP-PTO and CP-PTO, the unit cell containing five atoms was considered.
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YPHON package [37–39] which adopted the small displacement approach (supercell method)
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was used for calculating phonon dispersion and thermal properties, considering both plus and
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minus displacements by default. A 3×3×3 k-point mesh and a 2×2×2 supercell of the primitive
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cell were considered in our phonon calculations for the three phases. Since the primitive unit cell
3
of PP-PTO already contains 20 atoms, supercells larger than the 2×2×2 supercell for PP-PTO are
4
computationally prohibitive. Therefore, we employed a 2×2×2 supercell for all the three
5
compounds in our calculations. We tested the effect of supercell size on the amount of imaginary
6
phonon frequencies for CP-PTO since the unit cell of CP-PTO only contains 5 atoms. As we
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increased the supercell size from 2×2×2 to 4×4×4, no significant changes were observed for the
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phonon dispersions and the amount of the imaginary phonon frequencies.
9 Results and Discussion
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3.1 Static energy and electronic density of states
The calculated lattice constants and atomic coordinates of PP-PTO at the 0 K static
13
equilibrium volume are collected in Table 1, which are in good agreement with the results from
14
Neutron diffraction [8]. The static energy-volume curves for the three perovskites at 0 K are
15
plotted in Fig. 2(a). It shows that PP-PTO has the highest static energy at its equilibrium volume
16
among the three phases, which is 0.20 eV (∆EPP-TP) per formula unit and 0.14 eV (∆EPP-CP) per
17
formula unit higher than those of TP-PTO and CP-PTO, respectively. The equilibrium volume
18
for PP-PTO is much larger than those of TP- and CP-PTO. ∆EPP-TP and ∆EPP-CP are a little higher
19
than those reported by Wang et al. [19] and Liu et al. [18], perhaps due to the usage of different
20
pseudo-potentials and software packages for first principles calculations. The total and atomic
21
electronic density-of-states (EDOS) for PP-PTO, TP-PTO and CP-PTO are given in Fig. 2(b). It
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shows that the band gaps for PP-, TP- and CP-PTO are 1.96 eV, 1.30 eV and 1.62 eV,
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respectively. These calculated band gaps are comparable to those from previous calculations
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[21,40].
3 3.2 Lattice dynamics
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The vibrational contribution to free energy plays the major role in dictating the relative
6
thermodynamic stability of a compound at finite temperatures [41]. Fig. 3(a) and Fig. 3(c) show
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the partial PDOS due to the individual atoms for PP- and TP-PTO, respectively. It is
8
demonstrated that in both PP- and TP-PTO, the Pb atoms only contribute to the low frequency
9
region of <5 THz due to the large atomic mass. Compared with TP-PTO, the contributions of the
10
Ti atoms to the frequency regions ranging from 18 THz to 22 THz and from 23 THz to 25 THz
11
are more prominent in PP-PTO. In addition, it seems that the vibrational modes of Ti and O in
12
PP-PTO are stronger than those in TP-PTO, indicating that the bonding between the Ti atoms
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and the O atoms in PP-PTO is more complicated.
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In addition, Fig. 3(b) and Fig. 3(d) show the calculated total PDOS and the generalized
15
phonon density-of-states (GPDOS) [42] of PP-PTO and TP-PTO. The total PDOS is the simple
16
summation of contributions from Pb, Ti and O atoms. Note that usually for compounds, PDOS
17
cannot be measured directly through experiments, in which case the measured data mostly refer
18
to the neutron scattering cross section weighted PDOS, i.e., GPDOS. The GPDOS for PP-PTO
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and TP-PTO can be calculated by
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GPDOS = ∑i
σi mi
pDOSi ,
(10)
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where σi and mi refer to the atomic scattering cross section and the atomic mass for each
22
element, respectively; pDOSi represents the projected partial PDOS for individual atoms. Since
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the Pb atoms possess the largest atomic mass, their contributions to the GPDOS are much weaker
2
than the Ti and O atoms. The calculation of the PDOS is for the purpose of evaluating the free energy which normally
4
does not allow imaginary phonon modes. However, for the high temperature CP-PTO phase,
5
when the ideal cubic perovskite structure is employed, the calculated PDOS for the CP-PTO
6
phase shows a significant amount of imaginary phonon modes [43]. It is noted that the more the
7
amount of the imaginary phonon modes is, the higher the uncertainty of the calculated free
8
energy becomes. To circumvent this difficulty, we performed a calculation using a structure
9
defined as the constrained cubic PTO structure to calculate the phonon density-of-states for CP-
10
PTO. The constrained cubic PTO structure is obtained by keeping the cell shape of the ideal
11
cubic PTO structure while the internal atomic coordinates are allowed to relax as those of TP-
12
PTO, i.e., allowing all atoms to relax from their ideal cubic positions under the constraint of the
13
cubic cell shape. The idea of constrained calculation for CP-PTO is inspired from the X-ray
14
absorption fine-structure (XAFS) measurements [44,45] for PbTiO3 and BaTiO3 that showed
15
disorder of local distortions. In particular for PbTiO3, XAFS measurement of Sicron et al.[44]
16
pointed out that the Ti atoms can be displaced relative to the oxygen octahedra cage center both
17
below and above the transition temperature. The calculated partial PDOS, total PDOS and
18
GPDOS based on the constrained cubic PTO structure are illustrated in Fig. 4(a) and Fig. 4(b).
19
The amount of imaginary modes is significantly reduced when compared with those calculated
20
using the ideal CP-PTO structure as shown in Fig. 4(c) and Fig. 4(d). Specifically, the amount of
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imaginary phonon modes in the total phonon density-of-states has been decreased from 6.46% to
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0.46% when the constrained structure is employed instead of the ideal cubic structure.
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To our best knowledge, no theoretical and experimental results have been reported on the
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complete phonon spectrum for PP-PTO. Fig. 5 shows the calculated phonon dispersions for the
3
PP-PTO, TP-PTO, the ideal and the constrained cubic PTO structures. Phonon dispersion
4
represents the one-dimensional evolution of phonon frequency along the high symmetry
5
directions in the wave vector space. In Fig. 5(a), the phonon modes at Γ point for PP-PTO are
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classified into 43 phonon modes, i.e., 25 Raman-active modes including 10 Ag, 10 Bg and 5 Eg;
7
13 IR-active modes including 9 Eu and 4 Au; and 5 Bu optically silent modes. Table 2 lists the
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calculated phonon frequencies for the total 43 phonon modes at the Γ point of PP-PTO. The
9
experimental data by Raman scattering for PP-PTO in references [19,20] are represented by the
10
hollow dots in Fig. 5(a), which are in good agreement with the calculated results. According to
11
the phonon dispersion for TP-PTO in Fig. 5(b), the calculated results for TP-PTO are consistent
12
with the experimental data (marked by the hollow dots) [46,47]. No imaginary phonon modes
13
were found for either PP-PTO or TP-PTO, which indicates that PP-PTO and TP-PTO are
14
vibrationally stable at 0 K. Knowing the fact that the static energy (discussed in Section 3.1) at 0
15
K for PP-PTO is much higher than that for TP-PTO, it can be concluded that PP-PTO is a
16
metastable phase at 0 K. The findings explain the experimentally verified phase transition
17
process during heating treatment in which the PP-PTO transforms to the CP-PTO as temperature
18
increases from 25 °C to 700 °C, and when the CP-PTO is cooled back down to 25 °C, it becomes
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the TP-PTO rather than the original PP-PTO [8,11,14,15].
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Figure 5(c) and Fig. 5(d) show the phonon dispersions calculated using the ideal and
21
constrained cubic PTO structures, respectively. The ideal cubic PTO structure exhibits imaginary
22
phonon frequencies at Γ, M and P points, and the largest imaginary phonon frequency is about -4
23
THz while the constrained cubic PTO structure exhibits imaginary phonon frequency only at Γ
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point, with a value of about -1 THz. In contrast to the ideal cubic PTO structure, there are very
2
few imaginary phonon modes for the calculated result using the constrained cubic PTO structure.
3
Therefore, only the constrained cubic PTO structure is used in calculating the vibrational energy
4
for CP-PTO.
5
3.3 Free energies and thermodynamic properties
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Since the electron excitation free energy is ignored due to the insulating nature of all PTO
7
considered in this work, the calculated entropy S is the vibrational entropy as defined in Eq. (3).
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Fig. 6(a) shows the variation of entropy with temperature ranging from 0 K to 1200 K for PP-
9
PTO, TP-PTO and CP-PTO. Among these three phases, PP-PTO possesses the lowest entropy
10
while CP-PTO has the highest. This result is beyond conventional thought by which it was
11
anticipated that a large volume phase (the PP-PTO phase has the largest volume in the present
12
case) would have relatively high entropy. This can be attributed to the change of the low
13
frequency vibrational mode distribution due to the change of the Pb-O bonds from TP-PTO to
14
PP-PTO. As shown in Fig. 3, the PDOS values near 2 THz (due to Pb) and 15 THz (due to O) are
15
quite different between PP-PTO and TP-PTO; this difference further leads to less contribution of
16
these phonon vibration frequencies to the entropy of PP-PTO by the frequency-dependent
17
entropy contribution
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f (ω, T ) =
( ℏω / kBT ) − ln 1− e−ℏω/k T
(
eℏω/ kBT −1
B
) g(ω,V ) ,
(11)
19
As illustrated in the inset plot in Fig. 6(a), the lower f(ω,T) values of PP-PTO near 2 THz and 15
20
THz are the reasons for the lower entropy of PP-PTO at 500 K.
21
In literature, the comparison of the 0 K total energies for these three perovskites was already
22
reported. In this work, we want to consider the effect of temperature. The Gibbs free energy G at
23
finite temperatures can be obtained by 10
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G (T , P ) = F (T ,V ) + PV ,
1 2
(12)
where P is pressure. Figure 6(b) plots the Gibbs free energies as a function of temperature at P = 0 for the three
4
phases. The Gibbs free energies all show a continuous decrease with the increase of temperature
5
from 0 K to 1200 K since the contribution of the negative term −TS becomes more dominant as
6
temperature increases. The Gibbs free energy of PP-PTO ranks the highest among these three
7
within the considered temperature range. Based on the calculations, there will be an intersection
8
located at 960 K between the curves of TP-PTO and CP-PTO. This calculated phase transition
9
temperature between TP-PTO and CP-PTO is higher than the reported experimental values of
10
763 K [48,49], probably due to the underestimation of the entropy of the constrained CP-PTO
11
phase.
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Table 3 lists the values of Gibbs free energies, lattice constants and bulk moduli for PP-PTO,
13
TP-PTO and CP-PTO at 0 K and room temperature 300 K. Compared with the calculated values
14
at 0 K, the energy differences ∆GPP-CP and ∆GPP-TP nearly remain the same at 300 K. Specifically,
15
the energy difference ∆GPP-TP changes from 0.21 eV/f.u. to 0.22 eV/f.u. while ∆GPP-CP changes
16
from 0.16 eV/f.u. to 0.18 eV/f.u.. Bulk moduli for these three perovskites are calculated by Eq.
17
(7). Bulk modulus indicates the resistance of a material to the uniform compression. According
18
to the calculated results, PP-PTO is the easiest to be compressed while CP-PTO is the hardest
19
one. The absence of decrease in the bulk modulus for TP-PTO from 0 to 300 K is due to the
20
calculated nearly zero thermal expansion coefficient within this temperature range, which agrees
21
well with the experimentally reported values [6].
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The curves of Debye temperatures and thermal expansion coefficients at finite temperatures
23
for these three structures are plotted in Fig. 7, respectively. When the temperature is close to 0 K, 11
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the differences in the Debye temperatures for these three perovskites are very small as seen in
2
Fig. 7(a). The values of thermal expansion coefficients plotted in Fig. 7(b) indicate that the
3
thermal expansions of PP-PTO and CP-PTO are both higher than those of TP-PTO in the
4
calculated region from 0 K to 1200 K. The change of thermal expansion for these three phases is
5
not so significant when the temperature is higher than room temperature, since the slopes of
6
theses curves are quite flat when T>300 K. It should be noted that the calculation predicts a
7
negative expansion for TP-PTO structures in the temperature range from 30 K to 150 K. The
8
experimental results on TP-PTO nanoplates from Ren et. al also showed a typical negative
9
thermal expansion with a thermal expansion coefficient of -1.92×10-5 K-1 from 293 K to 573 K
10
[16]. Previous studies [5,16,50,51] have also demonstrated that the origin of the negative thermal
11
expansion in TP-PTO is closely related to the ferroelectric transition between TP-PTO and CP-
12
PTO, which corresponds to the two-phase equilibrium line with a negative slope in the
13
temperature-pressure phase diagram.
14
3.4 Pressure-temperature phase diagram
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Besides the effect of temperature, the influence of pressure on the Gibbs free energies for
16
PP-PTO, TP-PTO and CP-PTO are also studied in this work. Following Eq. (12), Gibbs free
17
energies at different temperatures and pressures can be obtained. The most stable phase is
18
defined as the phase with the minimum Gibbs free energy at given temperature and pressure. Fig.
19
8 shows the calculated pressure-temperature phase diagram. The calculated ranges of
20
temperature and pressure cover from 0 K to 1500 K and from -2.5 GPa to 2.5 GPa, respectively.
21
At the bottom of the phase diagram, where the pressure is from -1.6 GPa to -2.5 GPa, PP-PTO is
22
the most stable phase. On the left top of the PP-PTO phase, it is TP-PTO. Next to the TP-PTO
23
phase is the CP-PTO. The predicted transition pressure between PP-PTO and TP-PTO at 0 K is
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1.6 GPa, which is comparable to the calculated value of 1.4 GPa by Liu et al. [18]. This
2
experimental data point is marked with the red-and-blue circular point in Fig. 8. The
3
experimental data in Ref. [48] are also marked in Fig. 8. Most of the data are in good agreement
4
with the present calculation except for a slight difference for the transition temperature from CP-
5
PTO to TP-PTO at room temperature.
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The established pressure-temperature (P-T) diagram indicates that a comparison among the
7
temperature-pressure space is important in studying the phase stability of PP-PTO. The presence
8
of PP-PTO under a negative pressure or a high volume proposes the possible strategies for
9
effectively synthesizing the PP-PTO phase. Interphase strain recently proposed by Zhang et.al
10
[52] could be a feasible way to realize a negative pressure although achieving a negative pressure
11
remains a challenge in experiments. [15,34,52]
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12 4. Summary
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A systematic investigation of lattice dynamics for the PP-PTO structure is performed based
15
on first-principles quasiharmonic approach. A novel methodology is presented to reduce the
16
amount of soft modes in the calculation of phonon density-of-states. The influences of
17
temperature and pressure on phase stability and thermodynamic properties of PP-PTO were
18
studied. It is found that:
19
1. At 0 K and zero pressure, the PP-PTO is a vibrationally metastable phase while the TP-PTO
21 22
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is vibrationally stable, and the CP-PTO is vibrationally unstable. 2. The PP-PTO phase possesses lower entropy and larger volume compared to the CP-PTO and TP-PTO phases.
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3. A pressure-temperature phase diagram for PTO is constructed, indicating that the PP-PTO
2
phase can be stable under a negative pressure in the temperature range of 0 K through 1500
3
K.
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Acknowledgements The author M. J. Zhou gratefully acknowledges the financial support from China
3
Scholarship Council (No 201706210108). This work was partially supported by the NSF of
4
China (Grant Nos. 51332001 and 51472140) and in part by the Hamer Professorship (Wang and
5
Chen) and the National Science Foundation (NSF) through Grant Nos. DMR-1310289 and CHE-
6
1230924 (Wang and Liu). First-principles calculations were carried out partially on the LION
7
clusters at the Pennsylvania State University. The authors also gratefully acknowledge Prof.
8
Zhaohui Ren at Zhejiang University for useful discussions.
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Figure Captions
2
Fig. 1. Projections of crystal structures of (a) CP-PTO (Space group Pm3m ഥ ), (b) TP-PTO (Space
3
group P4mm), and (c) PP-PTO (Space group I4/m) along the a-axis; (d) PP-PTO along the c-axis.
4
Fig. 2. (a) Calculated static energy for PP-PTO, TP-PTO and CP-PTO; and (b) corresponding
5
electronic density-of-states (the solid, long dashed, short dashed, and dot-dashed lines represent
6
the total, Pb, Ti, and O contribution to the EDOS, respectively).
7
Fig. 3. (a), (b) Calculated partial, total and generalized phonon density-of-states (PDOS) for PP-
8
PTO; and (c), (d) Calculated partial, total and generalized phonon density-of-states for TP-PTO.
9
The label “total” refers to PDOS due to the simple summation of contributions from Pb, Ti and O
10
atoms and the label “General” refers to the neutron scattering cross section weighted PDOS (the
11
generalized PDOS, see equation (10) in the main text)
12
Fig. 4. (a), (b) Calculated partial, total and generalized phonon density-of-states for the
13
constrained cubic PTO structure; and (c), (d) Calculated partial, total and generalized phonon
14
density-of-states for the ideal cubic PTO structure. (The constrained cubic PTO structure adopts
15
the lattice constants of the ideal cubic PTO structure with the internal atomic coordinates being
16
allowed to relax as those of TP-PTO.)
17
Fig. 5. Calculated phonon dispersion of (a) PP-PTO, (b) TP-PTO, (c) the ideal cubic PTO
18
structure, and (d) the constrained cubic PTO structure along the selected principal symmetry
19
directions. (Blue hollow points: ref. [19]; green hollow points: ref. [20]; magenta hollow points:
20
ref. [46]; red hollow points: ref. [47].)
21
Fig. 6. (a) Calculated entropy, and (b) Gibbs free energy as a function of temperature for PP-
22
PTO, TP-PTO and CP-PTO at ambient pressure.
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Fig. 7. (a) The calculated Debye temperature, and (b) thermal expansion coefficient as a function
2
of temperature for PP-PTO, TP-PTO and CP-PTO at ambient pressure.
3
Fig. 8. Calculated pressure-temperature phase diagram by first-principles calculations for PP-,
4
TP- and CP-PTO.
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Table Captions
6
Table 1: The calculated and experimental fractional atomic coordinates and lattice parameter of
7
PP-PTO
8
Table 2: Calculated phonon frequencies at the Γ point for PP-PTO
9
Table 3: Calculated Gibbs free energies, lattice constants and bulk moduli at 0 K and 300 K for PP-, CP-, and TP-PTO, respectively.
11 12
16 17 18 19 20
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Table 1: The calculated and experimental fractional atomic coordinates and lattice parameter of PP-PTO Neutron diffraction [8]
x
y
z
x
y
z
Pb
0.161
0.151
0.5
0.165
0.151
0.5
Ti
0.470
0.143
0.5
0.471
0.142
0.5
O1
0.605
0.027
0.5
0.605
0.028
0.5
O2
0.166
0.288
0.0
0.169
0.289
0
O3
0.551
0.265
0.5
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Theoretical calculation
0.543
0.260
0.5
a = b = 12.367 (Å), c = 3.808 (Å)
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Table 2: Calculated phonon frequencies at the Γ point for PP-PTO Sym
THz
(cm-1)
EP
No.
10 Ag Modes (Raman active) Ag
1
Sym
THz
(cm-1)
9 Eu Modes (IR active)
25.9081
(864.20)
0
Eu
23.7522
(792.29)
Ag
19.0225
(634.52)
1
Eu
19.4970
(650.35)
2
Ag
14.5517
(485.39)
2
Eu
12.5981
(420.23)
3
Ag
11.9404
(398.29)
3
Eu
11.3287
(377.88)
4
Ag
8.3986
(280.15)
4
Eu
10.8704
(362.60)
5
Ag
7.8379
(261.45)
5
Eu
7.6321
(254.58)
6
Ag
5.7355
(191.31)
6
Eu
5.9574
(198.72)
7
Ag
5.2023
(173.53)
7
Eu
4.024
(134.23)
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0
No.
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8
Ag
3.2213
(107.45)
9
Ag
1.8314
(61.09)
10 Bg Modes (Raman active)
8
Eu
1.9827
(66.14)
4 Au Modes (IR active）
Bg
23.8081
(794.15)
0
Au
15.0737
(502.81)
1
Bg
20.9799
(699.81)
1
Au
9.2764
(309.43)
2
Bg
13.3511
(445.35)
2
Au
6.6266
(221.04)
3 4 5
Bg Bg Bg
12.3043 11.3077 7.7745
(410.43) (377.19) (259.33)
3
Au
2.2444
(74.86)
6
Bg
6.8289
(227.79)
0
Bu
20.1405
(671.82)
7
Bg
5.2077
(173.71)
1
Bu
9.5517
(318.61)
8
Bg
3.1799
(106.07)
2
Bu
7.1807
(239.52)
9
Bg
1.788
(59.64)
3 4
Bu Bu
5.8358 1.0022
(194.66) (33.43)
Eg
17.6166
1
Eg
8.9472
2
Eg
8.2887
3
Eg
5.7436
4
Eg
1.366
(298.45) (276.48)
(191.59) (45.56)
EP AC C
SC
(-587.63)
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0
5 Bu Modes (silent)
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Γopt =10Ag + 4Au +10Bg + 5Bu + 5Eg + 9Eu Raman Active = 10Ag + 10Bg + 5Eg IR Active = 4Au + 9Eu Silence Modes = 5Bu
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Table 3: Calculated Gibbs free energies, lattice constants and bulk moduli at 0 K and 300 K for PP-, CP-, and TP-PTO, respectively.
（eV/f.u.）
Lattice constant* (Å)
Bulk modulus (GPa)
RI PT
Gibbs free energy*
300 K
0K
300 K
0K
300 K
PP-PTO
-41.28
-41.46
a=12.208 c=3.768
a=12.240 c=3.770
64
63
TP-PTO
-41.49
-41.68
a=3.859 c=4.049
a = 3.860 c = 4.049
119
119
CP-PTO
-41.44
-41.64
a=3.894
a=3.897
213
204
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* The results of Gibbs free energies and lattice constants at 0 K in Table II have included the contribution of vibration energy at 0 K.
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