Origin and evolution of ferroelectricity in the layered   rare-earth-titanate, $R_2$Ti$_2$O$_{7}$, Lichtenberg phases

Maribel N\'u\~nez Valdez; Nicola A. Spaldin

arXiv:1903.11847·cond-mat.mtrl-sci·March 9, 2021

Origin and evolution of ferroelectricity in the layered rare-earth-titanate, $R_2$Ti$_2$O$_{7}$, Lichtenberg phases

Maribel N\'u\~nez Valdez, Nicola A. Spaldin

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TL;DR

This study uses first-principles calculations to analyze the structural and ferroelectric properties of $R_2$Ti$_2$O$_{7}$ oxides, revealing the mechanisms behind their ferroelectricity and polarization trends across different rare-earth elements.

Contribution

It provides a detailed first-principles analysis of the origin and evolution of ferroelectricity in $R_2$Ti$_2$O$_{7}$, clarifying polarization values and the role of structural distortions.

Findings

01

Ferroelectric distortion driven by a single polar soft mode involving octahedral rotations and displacements.

02

Secondary centrosymmetric distortion significantly lowers the energy of the ground state.

03

The materials are confirmed as proper geometric ferroelectrics.

Abstract

We report a systematic first-principles study based on density functional theory (DFT) of the structural and ferroelectric properties of the $R_{2}$ Ti $_{2}$ O $_{7}$ perovskite-related oxides with $R =$ La, Ce, Pr, and Nd. We show that, in all cases, the ferroelectric distortion to the ground-state polar $P 2_{1}$ structure from its parent centrosymmetric $P 2_{1} / m$ phase is driven by a single polar soft mode consisting of rotations and tilts of the TiO $_{6}$ octahedra combined with displacements of the $R$ ions. A secondary centrosymmetric distortion, which is stable in the parent structure, contributes substantially to the energy lowering of the ground state. We evaluate the trends in structure and polarization across the series as a function of $R$ and reconcile discrepancies in reported values of polarization in the literature. Our results confirm that the family of $R_{2}$ Ti $_{2}$ O $_{7}$ materials…

Tables7

Table 1. Table 1: Lattice parameters ( a 𝑎 a , b 𝑏 b , and c 𝑐 c ), monoclinic angle ( β 𝛽 \beta ), and volume ( V 𝑉 V ) of polar ( P 2 1 𝑃 subscript 2 1 P2_{1} ) and centrosymmetric ( P 2 1 / m 𝑃 subscript 2 1 𝑚 P2_{1}/m ) phases of R 2 subscript 𝑅 2 R_{2} Ti 2 O 7 compounds for R = 𝑅 absent R= La, Ce, Pr, and Nd computed using the PBEsol functional compared to the standard PBE and experiment. Results from this work are labeled by [*] and experimental parameters by EXP. PW stands for the Perdew-Wang parametrization.

			$a$ (Å)		$b$ (Å)		$c$ (Å)		$β$ (deg)		$V$ (Å³)
		Ref.	$P 2_{1}$	$P 2_{1} / m$	$P 2_{1}$	$P 2_{1} / m$	$P 2_{1}$	$P 2_{1} / m$	$P 2_{1}$	$P 2_{1} / m$	$P 2_{1}$	$P 2_{1} / m$
La₂Ti₂O₇	PBEsol	[*]	7.77	7.82	5.51	5.50	13.05	13.14	98.58	98.55	552.23	558.50
	PBE	[*]	7.81	7.90	5.60	5.56	13.28	13.32	98.50	98.49	574.10	578.16
	PW	Bruyer and Sayede (2010)	7.82	7.89	5.59	5.56	13.26	13.31	98.51	98.51	573.16	577.55
	EXP	Nanamatsu et al. (1974)	7.81		5.55		13.02		98.70		558.00
Ce₂Ti₂O₇	PBEsol	[*]	7.75	7.83	5.52	5.50	13.10	13.16	98.56	98.55	554.66	561.14
	PBE	[*]	7.80	7.92	5.60	5.56	13.33	13.34	98.49	98.48	576.22	580.93
	EXP	Lichtenberg et al. (2008)	7.76		5.51		12.99		98.50		549.00
Pr₂Ti₂O₇	PBEsol	[*]	7.70	7.78	5.48	5.49	13.08	13.17	98.53	98.49	546.68	556.34
	PBE	[*]	7.74	7.86	5.56	5.54	13.30	13.33	98.47	98.56	565.96	573.70
	PBE	Patwe et al. (2015)	7.75	7.87	5.56	5.55	13.30	13.35
	EXP	Lichtenberg et al. (2008)	7.69		5.47		12.99		98.40		541.00
	EXP	Patwe et al. (2015)	7.72	7.71	5.49	5.49	13.00	13.00	98.55	98.53	544.47	544.40
Nd₂Ti₂O₇	PBEsol	[*]	7.66	7.74	5.45	5.47	13.06	13.18	98.51	98.45	540.01	552.34
	PBE	[*]	7.70	7.81	5.52	5.52	13.29	13.33	98.43	98.56	558.92	568.93
	PW	Bruyer and Sayede (2010)	7.68	7.78	5.47	5.54	13.06	13.08	98.54		542.65	557.70
	EXP	Lichtenberg et al. (2008)	7.67		5.46		12.99		98.50		538.00
	EXP	Kimura et al. (1974)	7.68		5.48		13.02		98.28		542.25

Table 2. Table 2: Symmetry modes linking the P 2 1 / m 𝑃 subscript 2 1 𝑚 P2_{1}/m and P 2 1 𝑃 subscript 2 1 P2_{1} phases of the titanate Lichtenberg phases. The distortion amplitudes shown are for the case of CeTi 2 O 7 .

Irrep	Isotropy	Amplitude %
	Subgroup
$Γ 1^{+}$	$P 2_{1} / m$	14.7
$Γ 1^{-}$	$P 2_{1}$	85.3

Table 3. Table 3: Polarization quanta ( P q subscript 𝑃 𝑞 P_{q} ), and spontaneous polarizations ( P S subscript 𝑃 𝑆 P_{S} ) obtained using the Berry-phase method and from formal charges (Eq.( 1 )), for R 2 subscript 𝑅 2 R_{2} Ti 2 O 7 with R = 𝑅 absent R= La, Pr, and Nd. Results from this work are labeled by [*]. All values are reported for the P 2 1 / m → P 2 1 → 𝑃 subscript 2 1 𝑚 𝑃 subscript 2 1 P2_{1}/m\rightarrow P2_{1} transformation, except for those from Ref. López-Pérez and J. Íñiguez ( 2011 ) , which are obtained for the C m c m → C m c 2 1 → 𝐶 𝑚 𝑐 𝑚 𝐶 𝑚 𝑐 subscript 2 1 Cmcm\rightarrow Cmc2_{1} transformation. Available literature experimental values, discussed in the Introduction, are also given for comparison.

$R_{2}$ Ti₂O₇	Method	Ref.	$P_{q}$ ( $μ$ C/cm²)	$P_{S}$ ( $μ$ C/cm²)	$P_{S}$ ( $μ$ C/cm²)	$P_{S}$ ( $μ$ C/cm²)	Ref.
				Berry Phase	Formal charges	Experimental
La₂Ti₂O₇	PBEsol	[*]	15.77	18.20	14.44	5.0, 4.4	Nanamatsu et al. (1974), Kim et al. (2002)
	PW	Bruyer and Sayede (2010)	15.60		7.72
	LDA	López-Pérez and J. Íñiguez (2011)	16.02		29.0
Ce₂Ti₂O₇	PBEsol	[*]	15.71	18.58	14.22	4.19	Kim et al. (2008)
Pr₂Ti₂O₇	PBEsol	[*]	15.80	18.17	14.18	7.0, 0.017	Sun et al. (2013), Patwe et al. (2015)
	PBE	Patwe et al. (2015)	15.42		8.3
Nd₂Ti₂O₇	PBEsol	[*]	15.88	17.77	14.16	9.0, 1.8	Kimura et al. (1974), Kim et al. (2002)
	PW	Bruyer and Sayede (2010)	16.10		7.42

Table 4. Table 4: La 2 Ti 2 O 7 Born effective charge tensors in units of e 𝑒 e .

Atom	$P 2_{1}$	Atom	$P 2_{1}$	Atom	$P 2_{1} / m$
La(1,2)	$(\begin{matrix} 4.28 & \pm 0.02 & 0.01 \\ \pm 0.04 & 4.56 & \pm 0.30 \\ - 0.03 & \pm 0.20 & 3.31 \end{matrix})$	O(1,2)	$(\begin{matrix} - 5.51 & 0.00 & 0.02 \\ 0.00 & - 2.38 & \pm 0.02 \\ 0.00 & \pm 0.05 & - 1.34 \end{matrix})$	La(1,…,4)	$(\begin{matrix} 4.39 & 0.00 & 0.00 \\ 0.00 & 4.21 & 0.00 \\ 0.00 & 0.00 & 3.66 \end{matrix})$
La(3,4)	$(\begin{matrix} 4.38 & \pm 0.02 & 0.01 \\ \pm 0.05 & 4.19 & \pm 0.59 \\ 0.02 & \pm 0.09 & 3.83 \end{matrix})$	O(3,4)	$(\begin{matrix} - 4.70 & \pm 0.02 & - 0.02 \\ 0.00 & - 2.54 & \pm 0.07 \\ 0.00 & \pm 0.15 & - 1.82 \end{matrix})$	La(5,…,8)	$(\begin{matrix} 4.61 & 0.00 & 0.00 \\ 0.00 & 4.67 & 0.00 \\ 0.00 & 0.00 & 4.20 \end{matrix})$
La(5,6)	$(\begin{matrix} 4.54 & \pm 0.07 & - 0.06 \\ \pm 0.19 & 4.61 & \pm 0.12 \\ - 0.09 & \pm 0.59 & 4.45 \end{matrix})$	O(5,6)	$(\begin{matrix} - 2.46 & 0.00 & - 0.09 \\ \pm 0.09 & - 3.40 & \pm 1.10 \\ 0.03 & \pm 0.99 & - 2.96 \end{matrix})$	Ti(1,…,4)	$(\begin{matrix} 6.67 & 0.00 & 0.00 \\ 0.00 & 7.91 & 0.00 \\ 0.00 & 0.00 & 5.44 \end{matrix})$
La(7,8)	$(\begin{matrix} 4.47 & \pm 0.11 & 0.04 \\ \pm 0.16 & 4.72 & \pm 0.34 \\ 0.11 & \pm 0.50 & 4.39 \end{matrix})$	O(7,8)	$(\begin{matrix} - 2.49 & \pm 0.01 & 0.09 \\ \pm 0.10 & - 3.34 & \pm 1.03 \\ - 0.02 & \pm 0.91 & - 2.88 \end{matrix})$	Ti(5,…,8)	$(\begin{matrix} 7.42 & 0.00 & 0.00 \\ 0.00 & 5.80 & 0.00 \\ 0.00 & 0.00 & 7.27 \end{matrix})$
Ti(1,2)	$(\begin{matrix} 6.44 & \pm 0.07 & 0.21 \\ \pm 0.33 & 6.63 & \pm 0.09 \\ 0.20 & \pm 0.34 & 5.00 \end{matrix})$	O(9,10)	$(\begin{matrix} - 1.83 & \pm 0.31 & - 0.31 \\ \pm 0.27 & - 3.20 & \pm 1.46 \\ - 0.32 & \pm 1.37 & - 3.22 \end{matrix})$	O(1,…,4)	$(\begin{matrix} - 5.42 & 0.00 & 0.00 \\ 0.00 & - 2.41 & 0.00 \\ 0.00 & 0.00 & - 1.68 \end{matrix})$
Ti(3,4)	$(\begin{matrix} 6.37 & \pm 0.07 & - 0.20 \\ \pm 0.26 & 6.42 & \pm 0.05 \\ - 0.20 & \pm 0.28 & 4.78 \end{matrix})$	O(11,12)	$(\begin{matrix} - 1.81 & \pm 0.28 & 0.36 \\ \pm 0.25 & - 3.10 & \pm 1.36 \\ 0.40 & \pm 1.22 & - 3.21 \end{matrix})$	O(5,…,8)	$(\begin{matrix} - 2.47 & 0.00 & 0.00 \\ 0.00 & - 3.83 & \pm 1.34 \\ 0.00 & \pm 1.08 & - 3.20 \end{matrix})$
Ti(5,6)	$(\begin{matrix} 6.44 & \pm 0.88 & 0.13 \\ \pm 0.06 & 5.50 & \pm 0.52 \\ - 0.17 & \pm 0.28 & 6.07 \end{matrix})$	O(13,14)	$(\begin{matrix} - 2.39 & \pm 0.23 & - 0.22 \\ \pm 0.17 & - 3.88 & \pm 1.45 \\ - 0.12 & \pm 1.37 & - 3.23 \end{matrix})$	O(9,…,16)	$(\begin{matrix} - 2.10 & 0.00 & 0.00 \\ 0.00 & - 4.13 & \pm 1.88 \\ 0.00 & \pm 1.68 & - 3.40 \end{matrix})$
Ti(7,8)	$(\begin{matrix} 6.28 & \pm 0.84 & - 0.03 \\ \pm 0.13 & 5.76 & \pm 0.51 \\ 0.24 & \pm 0.05 & 5.49 \end{matrix})$	O(15,16)	$(\begin{matrix} - 2.47 & \pm 0.26 & 0.22 \\ \pm 0.18 & - 3.82 & \pm 1.18 \\ 0.13 & \pm 1.23 & - 2.97 \end{matrix})$	O(17,…,20)	$(\begin{matrix} - 5.95 & 0.00 & 0.00 \\ 0.00 & - 1.59 & 0.00 \\ 0.00 & 0.00 & - 2.18 \end{matrix})$
		O(17,18)	$(\begin{matrix} - 2.16 & \pm 0.19 & 0.17 \\ \pm 0.20 & - 3.27 & \pm 0.88 \\ 0.22 & \pm 0.86 & - 2.91 \end{matrix})$	O(21,…,28)	$(\begin{matrix} - 2.53 & 0.00 & 0.00 \\ 0.00 & - 3.26 & \pm 0.76 \\ 0.00 & \pm 0.94 & - 3.36 \end{matrix})$
		O(19,20)	$(\begin{matrix} - 2.43 & \pm 0.26 & - 0.23 \\ \pm 0.30 & - 3.76 & \pm 1.12 \\ - 0.33 & \pm 1.17 & - 1.95 \end{matrix})$
		O(21,22)	$(\begin{matrix} - 2.38 & \pm 0.10 & 0.08 \\ \pm 0.21 & - 3.08 & \pm 0.68 \\ 0.10 & \pm 0.48 & - 3.28 \end{matrix})$
		O(23,24)	$(\begin{matrix} - 2.25 & \pm 0.02 & - 0.09 \\ \pm 0.10 & - 2.95 & \pm 0.81 \\ - 0.12 & \pm 0.50 & - 3.55 \end{matrix})$
		O(25,26)	$(\begin{matrix} - 5.14 & \pm 0.03 & - 0.08 \\ \pm 0.08 & - 1.88 & \pm 0.19 \\ - 0.05 & \pm 0.32 & - 1.99 \end{matrix})$
		O(27,28)	$(\begin{matrix} - 5.17 & \pm 0.05 & 0.02 \\ \pm 0.04 & - 1.80 & \pm 0.40 \\ 0.00 & \pm 0.36 & - 2.01 \end{matrix})$

Table 5. Table 5: Ce 2 Ti 2 O 7 Born effective charge tensors in units of e 𝑒 e .

Atom	$P 2_{1}$	Atom	$P 2_{1}$	Atom	$P 2_{1} / m$
Ce(1,2)	$(\begin{matrix} 4.05 & \pm 0.03 & 0.02 \\ \pm 0.05 & 4.50 & \pm 0.30 \\ - 0.04 & \pm 0.19 & 3.23 \end{matrix})$	O(1,2)	$(\begin{matrix} - 5.59 & 0.00 & 0.02 \\ 0.00 & - 2.31 & \pm 0.04 \\ 0.00 & \pm 0.08 & - 1.33 \end{matrix})$	Ce(1,…,4)	$(\begin{matrix} 4.38 & 0.00 & 0.00 \\ 0.00 & 4.25 & 0.00 \\ 0.00 & 0.00 & 3.63 \end{matrix})$
Ce(3,4)	$(\begin{matrix} 4.22 & \pm 0.03 & 0.02 \\ \pm 0.05 & 4.11 & \pm 0.58 \\ 0.03 & \pm 0.05 & 3.76 \end{matrix})$	O(3,4)	$(\begin{matrix} - 4.71 & \pm 0.02 & - 0.02 \\ 0.00 & - 2.45 & \pm 0.07 \\ 0.01 & \pm 0.15 & - 1.78 \end{matrix})$	Ce(5,…,8)	$(\begin{matrix} 4.47 & 0.00 & 0.00 \\ 0.00 & 4.66 & 0.00 \\ 0.00 & 0.00 & 4.23 \end{matrix})$
Ce(5,6)	$(\begin{matrix} 4.31 & \pm 0.06 & - 0.08 \\ \pm 0.19 & 4.53 & \pm 0.14 \\ - 0.08 & \pm 0.62 & 4.39 \end{matrix})$	O(5,6)	$(\begin{matrix} - 2.40 & \pm 0.02 & - 0.06 \\ \pm 0.08 & - 3.36 & \pm 1.08 \\ 0.05 & \pm 0.98 & - 2.91 \end{matrix})$	Ti(1,…,4)	$(\begin{matrix} 6.69 & 0.00 & 0.00 \\ 0.00 & 7.98 & 0.00 \\ 0.00 & 0.00 & 5.46 \end{matrix})$
Ce(7,8)	$(\begin{matrix} 4.26 & \pm 0.11 & 0.06 \\ \pm 0.16 & 4.63 & \pm 0.29 \\ 0.14 & \pm 0.50 & 4.40 \end{matrix})$	O(7,8)	$(\begin{matrix} - 2.44 & \pm 0.01 & 0.04 \\ \pm 0.08 & - 3.30 & \pm 1.01 \\ - 0.04 & \pm 0.88 & - 2.82 \end{matrix})$	Ti(5,…,8)	$(\begin{matrix} 7.46 & 0.00 & 0.00 \\ 0.00 & 5.78 & 0.00 \\ 0.00 & 0.00 & 7.29 \end{matrix})$
Ti(1,2)	$(\begin{matrix} 6.52 & \pm 0.11 & 0.23 \\ \pm 0.37 & 6.60 & \pm 0.06 \\ 0.20 & \pm 0.38 & 4.91 \end{matrix})$	O(9,10)	$(\begin{matrix} - 1.81 & \pm 0.30 & - 0.34 \\ \pm 0.29 & - 3.20 & \pm 1.41 \\ - 0.35 & \pm 1.31 & - 3.12 \end{matrix})$	O(1,…,4)	$(\begin{matrix} - 5.46 & 0.00 & 0.00 \\ 0.00 & - 2.42 & 0.00 \\ 0.00 & 0.00 & - 1.68 \end{matrix})$
Ti(3,4)	$(\begin{matrix} 6.44 & \pm 0.10 & - 0.22 \\ \pm 0.30 & 6.39 & \pm 0.01 \\ - 0.22 & \pm 0.30 & 4.65 \end{matrix})$	O(11,12)	$(\begin{matrix} - 1.80 & \pm 0.27 & 0.39 \\ \pm 0.27 & - 3.09 & \pm 1.30 \\ 0.43 & \pm 1.15 & - 3.11 \end{matrix})$	O(5,…,8)	$(\begin{matrix} - 2.48 & 0.00 & 0.00 \\ 0.00 & - 3.86 & \pm 1.36 \\ 0.00 & \pm 1.10 & - 3.22 \end{matrix})$
Ti(5,6)	$(\begin{matrix} 6.35 & \pm 0.96 & 0.14 \\ \pm 0.06 & 5.46 & \pm 0.58 \\ - 0.15 & \pm 0.21 & 5.95 \end{matrix})$	O(13,14)	$(\begin{matrix} - 2.29 & \pm 0.24 & - 0.21 \\ \pm 0.15 & - 3.86 & \pm 1.44 \\ - 0.12 & \pm 1.41 & - 3.25 \end{matrix})$	O(9,…,16)	$(\begin{matrix} - 2.08 & 0.00 & 0.00 \\ 0.00 & - 4.16 & \pm 1.91 \\ 0.00 & \pm 1.70 & - 3.41 \end{matrix})$
Ti(7,8)	$(\begin{matrix} 6.15 & \pm 0.88 & - 0.01 \\ \pm 0.18 & 5.76 & \pm 0.60 \\ 0.25 & \pm 0.10 & 5.30 \end{matrix})$	O(15,16)	$(\begin{matrix} - 2.38 & \pm 0.28 & 0.19 \\ \pm 0.15 & - 3.81 & \pm 1.13 \\ 0.13 & \pm 1.24 & - 2.93 \end{matrix})$	O(17,…,20)	$(\begin{matrix} - 5.99 & 0.00 & 0.00 \\ 0.00 & - 1.58 & 0.00 \\ 0.00 & 0.00 & - 2.16 \end{matrix})$
		O(17,18)	$(\begin{matrix} - 2.04 & \pm 0.19 & 0.16 \\ \pm 0.21 & - 3.21 & \pm 0.85 \\ 0.23 & \pm 0.84 & - 2.89 \end{matrix})$	O(21,…,28)	$(\begin{matrix} - 2.45 & 0.00 & 0.00 \\ 0.00 & - 3.24 & \pm 0.78 \\ 0.00 & \pm 0.95 & - 3.37 \end{matrix})$
		O(19,20)	$(\begin{matrix} - 2.30 & \pm 0.28 & - 0.27 \\ \pm 0.34 & - 3.78 & \pm 1.11 \\ - 0.38 & \pm 1.11 & - 1.83 \end{matrix})$
		O(21,22)	$(\begin{matrix} - 2.27 & \pm 0.13 & 0.07 \\ \pm 0.22 & - 3.06 & \pm 0.71 \\ 0.07 & \pm 0.51 & - 3.20 \end{matrix})$
		O(23,24)	$(\begin{matrix} - 2.15 & \pm 0.02 & - 0.07 \\ \pm 0.09 & - 2.90 & \pm 0.90 \\ - 0.10 & \pm 0.57 & - 3.49 \end{matrix})$
		O(25,26)	$(\begin{matrix} - 4.99 & \pm 0.04 & - 0.10 \\ \pm 0.10 & - 1.91 & \pm 0.23 \\ - 0.07 & \pm 0.32 & - 1.94 \end{matrix})$
		O(27,28)	$(\begin{matrix} - 5.13 & \pm 0.08 & 0.03 \\ \pm 0.04 & - 1.75 & \pm 0.36 \\ 0.00 & \pm 0.37 & - 1.99 \end{matrix})$

Table 6. Table 6: Pr 2 Ti 2 O 7 Born effective charge tensors in units of e 𝑒 e .

Atom	$P 2_{1}$	Atom	$P 2_{1}$	Atom	$P 2_{1} / m$
Pr(1,2)	$(\begin{matrix} 4.01 & \pm 0.03 & 0.02 \\ \pm 0.05 & 4.49 & \pm 0.31 \\ - 0.05 & \pm 0.21 & 3.17 \end{matrix})$	O(1,2)	$(\begin{matrix} - 5.62 & \pm 0.01 & 0.02 \\ 0.00 & - 2.31 & \pm 0.05 \\ 0.00 & \pm 0.11 & - 1.29 \end{matrix})$	Pr(1,…,4)	$(\begin{matrix} 4.38 & 0.00 & 0.00 \\ 0.00 & 4.22 & 0.00 \\ 0.00 & 0.00 & 3.61 \end{matrix})$
Pr(3,4)	$(\begin{matrix} 4.18 & \pm 0.03 & 0.01 \\ \pm 0.06 & 4.07 & \pm 0.59 \\ 0.03 & \pm 0.04 & 3.75 \end{matrix})$	O(3,4)	$(\begin{matrix} - 4.63 & \pm 0.03 & \pm 0.02 \\ 0.00 & - 2.45 & 0.09 \\ \pm 0.01 & \pm 0.18 & - 1.80 \end{matrix})$	Pr(5,…,8)	$(\begin{matrix} 4.40 & 0.00 & 0.00 \\ 0.00 & 4.61 & 0.00 \\ 0.00 & 0.00 & 4.18 \end{matrix})$
Pr(5,6)	$(\begin{matrix} 4.28 & \pm 0.06 & - 0.10 \\ \pm 0.20 & 4.48 & \pm 0.15 \\ - 0.08 & \pm 0.64 & 4.39 \end{matrix})$	O(5,6)	$(\begin{matrix} - 2.38 & \pm 0.02 & - 0.06 \\ \pm 0.08 & - 3.32 & \pm 1.06 \\ 0.05 & \pm 0.97 & - 2.90 \end{matrix})$	Ti(1,…,4)	$(\begin{matrix} 6.72 & 0.00 & 0.00 \\ 0.00 & 8.04 & 0.00 \\ 0.00 & 0.00 & 5.44 \end{matrix})$
Pr(7,8)	$(\begin{matrix} 4.18 & \pm 0.11 & 0.08 \\ \pm 0.16 & 4.60 & \pm 0.28 \\ 0.15 & \pm 0.48 & 4.37 \end{matrix})$	O(7,8)	$(\begin{matrix} - 2.41 & \pm 0.01 & 0.04 \\ \pm 0.08 & - 3.26 & \pm 0.98 \\ - 0.05 & \pm 0.86 & - 2.81 \end{matrix})$	Ti(5,…,8)	$(\begin{matrix} 7.64 & 0.00 & 0.00 \\ 0.00 & 5.69 & 0.00 \\ 0.00 & 0.00 & 7.44 \end{matrix})$
Ti(1,2)	$(\begin{matrix} 6.52 & \pm 0.15 & 0.23 \\ \pm 0.41 & 6.54 & \pm 0.10 \\ 0.22 & \pm 0.38 & 4.90 \end{matrix})$	O(9,10)	$(\begin{matrix} - 1.80 & \pm 0.31 & - 0.36 \\ \pm 0.29 & - 3.15 & \pm 1.38 \\ - 0.36 & \pm 1.31 & - 3.13 \end{matrix})$	O(1,…,4)	$(\begin{matrix} - 5.48 & 0.00 & 0.00 \\ 0.00 & - 2.42 & 0.00 \\ 0.00 & 0.00 & - 1.67 \end{matrix})$
Ti(3,4)	$(\begin{matrix} 6.44 & \pm 0.12 & - 0.22 \\ \pm 0.34 & 6.33 & \pm 0.04 \\ - 0.23 & \pm 0.31 & 4.63 \end{matrix})$	O(11,12)	$(\begin{matrix} - 1.80 & \pm 0.29 & 0.42 \\ \pm 0.28 & - 3.07 & \pm 1.26 \\ 0.45 & \pm 1.14 & - 3.11 \end{matrix})$	O(5,…,8)	$(\begin{matrix} - 2.47 & 0.00 & 0.00 \\ 0.00 & - 3.88 & \pm 1.36 \\ 0.00 & \pm 1.09 & - 3.21 \end{matrix})$
Ti(5,6)	$(\begin{matrix} 6.32 & \pm 1.04 & 0.15 \\ \pm 0.07 & 5.42 & \pm 0.59 \\ - 0.18 & \pm 0.25 & 5.94 \end{matrix})$	O(13,14)	$(\begin{matrix} - 2.29 & \pm 0.25 & - 0.22 \\ \pm 0.15 & - 3.82 & \pm 1.39 \\ - 0.13 & \pm 1.38 & - 3.23 \end{matrix})$	O(9,…,16)	$(\begin{matrix} - 2.08 & 0.00 & 0.00 \\ 0.00 & - 4.16 & \pm 1.92 \\ 0.00 & \pm 1.70 & - 3.42 \end{matrix})$
Ti(7,8)	$(\begin{matrix} 6.13 & \pm 0.96 & - 0.01 \\ 0.18 & 5.75 & \pm 0.62 \\ 0.26 & \pm 0.07 & 5.26 \end{matrix})$	O(15,16)	$(\begin{matrix} - 2.29 & \pm 0.25 & - 0.22 \\ \pm 0.15 & - 3.82 & \pm 1.39 \\ - 0.13 & \pm 1.38 & - 3.23 \end{matrix})$	O(17,…,20)	$(\begin{matrix} - 6.13 & 0.00 & 0.00 \\ 0.00 & - 1.53 & 0.00 \\ 0.00 & 0.00 & - 2.16 \end{matrix})$
		O(17,18)	$(\begin{matrix} - 2.37 & \pm 0.29 & 0.20 \\ \pm 0.15 & - 3.78 & \pm 1.09 \\ 0.15 & \pm 1.21 & - 2.91 \end{matrix})$	O(21,…,28)	$(\begin{matrix} - 2.44 & 0.00 & 0.00 \\ 0.00 & - 3.21 & \pm 0.77 \\ 0.00 & \pm 0.95 & - 3.39 \end{matrix})$
		O(19,20)	$(\begin{matrix} - 2.02 & \pm 0.20 & 0.17 \\ \pm 0.21 & - 3.19 & \pm 0.84 \\ 0.24 & \pm 0.82 & - 2.88 \end{matrix})$
		O(21,22)	$(\begin{matrix} - 2.25 & \pm 0.15 & 0.06 \\ \pm 0.23 & - 3.03 & \pm 0.71 \\ 0.08 & \pm 0.48 & - 3.18 \end{matrix})$
		O(23,24)	$(\begin{matrix} - 2.13 & \pm 0.03 & - 0.05 \\ \pm 0.09 & - 2.86 & \pm 0.90 \\ - 0.10 & \pm 0.55 & - 3.50 \end{matrix})$
		O(25,26)	$(\begin{matrix} - 4.92 & \pm 0.04 & - 0.10 \\ \pm 0.10 & - 1.94 & \pm 0.26 \\ - 0.07 & \pm 0.35 & - 1.95 \end{matrix})$
		O(27,28)	$(\begin{matrix} - 5.15 & \pm 0.07 & 0.04 \\ \pm 0.05 & - 1.75 & \pm 0.38 \\ 0.01 & \pm 0.37 & - 1.95 \end{matrix})$

Table 7. Table 7: Nd 2 Ti 2 O 7 Born effective charge tensors in units of e 𝑒 e .

Atom	$P 2_{1}$	Atom	$P 2_{1}$	Atom	$P 2_{1} / m$
Nd(1,2)	$(\begin{matrix} 3.96 & \pm 0.03 & 0.02 \\ \pm 0.05 & 4.47 & \pm 0.31 \\ - 0.05 & \pm 0.23 & 3.12 \end{matrix})$	O(1,2)	$(\begin{matrix} - 5.64 & \pm 0.01 & 0.03 \\ 0.00 & - 2.30 & \pm 0.07 \\ 0.00 & \pm 0.14 & - 1.25 \end{matrix})$	Nd(1,…,4)	$(\begin{matrix} 4.38 & 0.00 & 0.00 \\ 0.00 & 4.20 & 0.00 \\ 0.00 & 0.00 & 3.59 \end{matrix})$
Nd(3,4)	$(\begin{matrix} 4.14 & \pm 0.03 & 0.01 \\ \pm 0.06 & 4.03 & \pm 0.58 \\ 0.03 & \pm 0.03 & 3.74 \end{matrix})$	O(3,4)	$(\begin{matrix} - 4.56 & \pm 0.03 & - 0.02 \\ \pm 0.01 & - 2.45 & \pm 0.11 \\ 0.01 & \pm 0.20 & - 1.82 \end{matrix})$	Nd(5,…,8)	$(\begin{matrix} 4.34 & 0.00 & 0.00 \\ 0.00 & 4.56 & 0.00 \\ 0.00 & 0.00 & 4.13 \end{matrix})$
Nd(5,6)	$(\begin{matrix} 3.00 \\ 4.25 & \pm 0.06 & - 0.11 \\ \pm 0.20 & 4.44 & \pm 0.16 \\ - 0.07 & \pm 0.65 & 4.39 \end{matrix})$	O(5,6)	$(\begin{matrix} - 2.36 & \pm 0.01 & - 0.05 \\ \pm 0.08 & - 3.28 & \pm 1.03 \\ 0.05 & \pm 0.96 & - 2.89 \end{matrix})$	Ti(1,…,4)	$(\begin{matrix} 6.74 & 0.00 & 0.00 \\ 0.00 & 8.08 & 0.00 \\ 0.00 & 0.00 & 5.42 \end{matrix})$
Nd(7,8)	$(\begin{matrix} 4.11 & \pm 0.11 & 0.09 \\ \pm 0.15 & 4.57 & \pm 0.27 \\ 0.15 & \pm 0.47 & 4.34 \end{matrix})$	O(7,8)	$(\begin{matrix} - 2.39 & \pm 0.01 & 0.03 \\ 0.08 & - 3.22 & \pm 0.96 \\ \pm 0.05 & \pm 0.85 & - 2.79 \end{matrix})$	Ti(5,…,8)	$(\begin{matrix} 7.78 & 0.00 & 0.00 \\ 0.00 & 5.62 & 0.00 \\ 0.00 & 0.00 & 7.53 \end{matrix})$
Ti(1,2)	$(\begin{matrix} 6.52 & \pm 0.18 & 0.22 \\ \pm 0.44 & 6.48 & \pm 0.13 \\ 0.23 & \pm 0.39 & 4.89 \end{matrix})$	O(9,10)	$(\begin{matrix} - 1.79 & \pm 0.32 & - 0.38 \\ \pm 0.29 & - 3.12 & \pm 1.35 \\ - 0.37 & \pm 1.31 & - 3.15 \end{matrix})$	O(1,…,4)	$(\begin{matrix} - 5.51 & 0.00 & 0.00 \\ 0.00 & - 2.41 & 0.00 \\ 0.00 & 0.00 & - 1.66 \end{matrix})$
Ti(3,4)	$(\begin{matrix} 6.43 & \pm 0.15 & - 0.21 \\ \pm 0.38 & 6.28 & \pm 0.06 \\ - 0.25 & \pm 0.31 & 4.60 \end{matrix})$	O(11,12)	$(\begin{matrix} - 1.79 & \pm 0.31 & 0.44 \\ \pm 0.28 & - 3.05 & \pm 1.23 \\ 0.46 & \pm 1.13 & - 3.11 \end{matrix})$	O(5,…,8)	$(\begin{matrix} - 2.47 & 0.00 & 0.00 \\ 0.00 & - 3.89 & \pm 1.36 \\ 0.00 & \pm 1.08 & - 3.19 \end{matrix})$
Ti(5,6)	$(\begin{matrix} 6.31 & \pm 1.10 & 0.14 \\ \pm 0.08 & 5.39 & \pm 0.59 \\ - 0.20 & \pm 0.28 & 5.93 \end{matrix})$	O(13,14)	$(\begin{matrix} - 2.29 & \pm 0.26 & - 0.22 \\ \pm 0.15 & - 3.78 & \pm 1.36 \\ - 0.15 & \pm 1.35 & - 3.21 \end{matrix})$	O(9,…,16)	$(\begin{matrix} - 2.08 & 0.00 & 0.00 \\ 0.00 & - 4.16 & \pm 1.93 \\ 0.00 & \pm 1.70 & - 3.42 \end{matrix})$
Ti(7,8)	$(\begin{matrix} 6.11 & \pm 1.03 & - 0.01 \\ \pm 0.18 & 5.74 & \pm 0.64 \\ 0.27 & \pm 0.05 & 5.23 \end{matrix})$	O(15,16)	$(\begin{matrix} - 2.36 & \pm 0.29 & 0.20 \\ \pm 0.15 & - 3.74 & \pm 1.05 \\ 0.17 & \pm 1.18 & - 2.88 \end{matrix})$	O(17,…,20)	$(\begin{matrix} - 6.24 & 0.00 & 0.00 \\ 0.00 & - 1.49 & 0.00 \\ 0.00 & 0.00 & - 2.16 \end{matrix})$
		O(17,18)	$(\begin{matrix} - 2.00 & \pm 0.20 & 0.17 \\ \pm 0.22 & - 3.17 & \pm 0.84 \\ 0.24 & \pm 0.81 & - 2.86 \end{matrix})$	O(21,…,28)	$(\begin{matrix} - 2.42 & 0.00 & 0.00 \\ 0.00 & - 3.17 & \pm 0.75 \\ 0.00 & \pm 0.94 & - 3.40 \end{matrix})$
		O(19,20)	$(\begin{matrix} - 2.29 & \pm 0.28 & - 0.29 \\ \pm 0.35 & - 3.76 & \pm 1.10 \\ - 0.42 & \pm 1.10 & - 1.75 \end{matrix})$
		O(21,22)	$(\begin{matrix} - 2.23 & \pm 0.16 & 0.04 \\ \pm 0.25 & - 3.00 & \pm 0.70 \\ 0.08 & \pm 0.46 & - 3.17 \end{matrix})$
		O(23,24)	$(\begin{matrix} - 2.11 & \pm 0.04 & - 0.03 \\ \pm 0.09 & - 2.82 & \pm 0.90 \\ - 0.10 & \pm 0.54 & - 3.51 \end{matrix})$
		O(25,26)	$(\begin{matrix} - 4.85 & \pm 0.05 & - 0.10 \\ \pm 0.10 & - 1.96 & \pm 0.28 \\ - 0.06 & \pm 0.37 & - 1.95 \end{matrix})$
		O(27,28)	$(\begin{matrix} - 5.17 & \pm 0.07 & 0.05 \\ \pm 0.06 & - 1.75 & \pm 0.39 \\ 0.02 & \pm 0.37 & - 1.90 \end{matrix})$

Equations2

P_{S, j} = \frac{1}{V} \sum Z_{i} Δ X_{i, j} .

P_{S, j} = \frac{1}{V} \sum Z_{i} Δ X_{i, j} .

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Origin and evolution of ferroelectricity in the layered rare-earth-titanate, $R_{2}$ Ti2O7, Lichtenberg phases

Maribel Núñez Valdez

[email protected]

Materials Theory, ETH Zürich, Wolfgang-Pauli-Strasse 27, CH-8093 Zürich, Switzerland

Deutsches GeoForschungsZentrum GFZ, Telegrafenberg, 14473 Potsdam, Germany

Goethe-Universität Frankfurt, Altenhöferallee 1, 60438 Frankfurt am Main, Germany

Nicola A. Spaldin

[email protected]

Materials Theory, ETH Zürich, Wolfgang-Pauli-Strasse 27, CH-8093 Zürich, Switzerland

Abstract

We report a systematic first-principles study based on density functional theory (DFT) of the structural and ferroelectric properties of the $R_{2}$ Ti2O7 perovskite-related oxides with $R=$ La, Ce, Pr, and Nd. We show that, in all cases, the ferroelectric distortion to the ground-state polar $P2_{1}$ structure from its parent centrosymmetric $P2_{1}/m$ phase is driven by a single polar soft mode consisting of rotations and tilts of the TiO6 octahedra combined with displacements of the $R$ ions. A secondary centrosymmetric distortion, which is stable in the parent structure, contributes substantially to the energy lowering of the ground state. We evaluate the trends in structure and polarization across the series as a function of $R$ and reconcile discrepancies in reported values of polarization in the literature. Our results confirm that the family of $R_{2}$ Ti2O7 materials belong to the class of proper geometric ferroelectrics.

I Introduction and Background

Perovskite-structure titanates, with chemical formula $A$ TiO3, are of tremendous fundamental and technological interest. In those with divalent $A$ -site cations, in which the Ti ion is in the formally $d^{0}$ 4+ oxidation state, the properties range from quantum paraelectricity (in SrTiO3) to large and robust ferroelectricity (in BaTiO3 and PbTiO3) and even to multiferroicity (in strained EuTiO3). The $d^{1}$ configuration on the formally Ti3+ ions in the rare-earth titanates such as LaTiO3 leads to strong correlations and the associated Mott physics including magnetism and metal-insulator transitions.

The $A_{n}$ TinO3n+2 family of materials, or Lichtenberg phases Lichtenberg et al. (2008), of which the perovskite structure is the $n=\infty$ end member, offers additional structural flexibility and in turn additional functionality within this crystal chemistry class (for a thorough review see Ref. [Lichtenberg et al., 2008] and references therein). They are obtained by cutting the cubic perovskite $A$ TiO3 structure perpendicular to the [110] direction into slices that are $n$ perovskite unit cells thick, and then inserting additional compensating oxygen ions to complete the oxygen octahedra on either side of the cut (110) planes. The structures obtained for $n=2,4$ and $\infty$ are shown in Fig. 1(a). Many different properties have been reported within this family, depending on the identity of the $A$ -site cation and the value of $n$ , for example quasi-one-dimensional metallicity in the $n=5$ LaTiO3.4, and structural complexity in well-ordered multi-layered stacking sequences in $n=4\frac{1}{3}$ Ca-doped LaTiO3.46 Nanot et al. (1981).

Of particular interest are the $n=4$ members with trivalent rare-earth $A$ sites, $R_{4}$ Ti4O14, usually written as $R_{2}$ Ti2O7 or $R$ TiO3.5. For this chemistry, The $n=4$ layered perovskite phase is the stable phase at ambient pressure for ratios of $R^{3+}$ to Ti4+ cation radii of $r_{R^{3+}}/r_{\text{Ti}^{4+}}\geq 1.80$ Saha et al. (2011), and therefore occurs for $R=$ La, Ce, Pr, and Nd; smaller rare earths form the cubic pyrochlore structure Subramanian et al. (1983). In addition, $R_{2}$ Ti2O7 layered perovskites with $R$ = Sm, Eu, and Gd can be stabilized using high pressure synthesis Titov et al. (1987). (Under thin-film growth conditions, a metastable (001)-oriented layered structure has been reported for Sm2Ti2O7, Eu2Ti2O7 and Gd2Ti2O7 Shao et al. (2012a, b).) The $n=4$ materials are ferroelectric with Curie temperatures, $T_{C}$ , in the range 1600-1850 K, among the highest known. As a result, they are of interest for high-temperature applications such as piezoelectric transducers Damjanovic (1998), acoustic and vibration sensors Turner et al. (1994), high- $k$ dielectrics Atuchin et al. (2012), or as photocatalysts Hwang et al. (2003). The goal of this work is to use first-principles electronic structure calculations based on density functional theory to clarify the origin of the ferroelectricity and to extract and rationalize the trends in the structural and functional properties across the La2Ti2O7 - Ce2Ti2O7 - Pr2Ti2O7 - Nd2Ti2O7 series.

I.1 Crystal Symmetry of $A_{2}$ Ti2O7

The $R_{2}$ Ti2O7 Lichtenberg phases with $R=$ La, Ce, Pr, Nd are monoclinic with the polar $P2_{1}$ space group at ambient conditions. The earliest reports on growth and crystal symmetry are for La2Ti2O7 (LaTO) Nanamatsu et al. (1974) and Nd2Ti2O7 (NdTO) Kimura et al. (1974). LaTO and NdTO crystals of $\sim$ 5 mm in diameter and $\sim$ 50 mm in length were grown using the floating zone technique and analyzed using X-ray diffraction (XRD). These LaTO and NdTO crystals were shown to belong to the monoclinic system $P2_{1}$ with four formula units (f.u.) per unit cell (u.c.). More recently, CeTi2O7 (CeTO) (using powder XRD) and Pr2Ti2O7 (PrTO) (from XRD analysis of pellets) were identified to be isostructural with LaTO and NdTO Kim et al. (2008); Sun et al. (2013). The measured lattice parameters and monoclinic angles for all four materials are given in Table 1.

Additionally, it has been reported that LaTO undergoes a high-temperature transformation at $\sim$ 1053 K from its room-temperature monoclinic phase to a polar orthorhombic phase of symmetry $Cmc2_{1}$ also with four formular units per unit cell Ishizawa et al. (1982). The lattice parameters at 1173 K, measured using XRD were found to be $a=3.95$ Å, $b=5.61$ Å, and $c=25.95$ Å , that is approximately halved along the $a$ axis and doubled along $c$ compared with the $P2_{1}$ phase. Note that this $Cmc2_{1}$ phase is polar, and so can not be the paraelectric parent of the ferroelectric structure. It has not been observed for CeTO, PrTO or NdTO.

I.2 Ferroelectricity of $A_{2}$ Ti2O7

There have been some experimental and computational efforts towards measuring and understanding the ferroelectric properties of the titanate Lichtenberg phases, in particular for the cases of LaTO, NdTO, and to a lesser degree for PrTO and CeTO. From dielectric measurements, single crystals of monoclinic $P2_{1}$ LaTO have been reported to be ferroelectric with spontaneous polarizations, $P_{S}$ , of 5 $\mu$ C/cm2 and 9 $\mu$ C/cm2 Nanamatsu et al. (1974); Kimura et al. (1974). For PrTO nanocrystalline pellets, a $P_{S}$ of $\sim$ 7 $\mu$ C/cm2 was observed Sun et al. (2013), while a polycrystalline sample of PrTO yielded $P_{S}=0.017$ $\mu$ C/cm2 Patwe et al. (2015), however both works acknowledged a lack of saturation of the polarization-electric field loops due to leakage currents. Values of $P_{S}=4.4$ , 1.8 and 4.19 $\mu$ C/cm2 for thin films of LaTO, NdTO Kim et al. (2002) and CeTO Kim et al. (2008) respectively, have been reported. In other studies addressing the ferroelectric properties of LaTO, NdTO, PrTO and CeTO films Bayart et al. (2013, 2014, 2016), the authors declared the existence of ferroelectricity from observed piezoloops, but no values of $P_{S}$ were given.

Regarding the nature of the transition to the ferroelectric phase, in Fig. 2 we show a group-subgroup analysis Ivantchev et al. (2000) using the Bilbao Crystallographic Server Aroyo et al. (2006a, b, 2011) indicating possible pathways for reaching the reported $P2_{1}$ ground-state structure. The analysis indicates one direct single-step possibility for reaching the $P2_{1}$ ground state from a non-polar structure, which is via the $P2_{1}/m\rightarrow P2_{1}$ transition. While the $P2_{1}/m$ centrosymmetric phase has not been reported experimentally for the titanate Lichtenberg phases, it has been observed for the $n=4$ tantalate Sr2Ta2O7 Hushur et al. (2004). It is this transition that we explore in this work. We also see that a direct transition from the polar $Cmc2_{1}$ phase mentioned above to the $P2_{1}$ ground state is indeed allowed, and that this pathway would imply a $Cmcm$ paraelectric phase. An earlier first-principles theoretical study addressed the nature of the non-polar to polar $Cmcm\rightarrow Cmc2_{1}$ transition López-Pérez and J. Íñiguez (2011) and found that the leading instability causing the transition consists primarily of TiO6 octahedral rotations, which occur because the $A$ -site cation is too small to maintain the high-symmetry structure. The layered connectivity of the crystal structure means that, in contrast to the case of the three-dimensionally connected perovskites, such rotational modes can be polar. The mechanism (called topological ferroelectricity by the authors of Ref. [López-Pérez and J. Íñiguez, 2011]) is analogous to the geometric ferroelectric mechanism previously identified in the Ba $M$ F4 family ( $M$ =Mn, Fe, Co, or Ni) Ederer and Spaldin (2006), which have the same structure as the $n=2$ member of the $A_{n}B_{n}$ O3n+2 series. It is strikingly different from the conventional ferroelectricity in related perovskite-structure oxides such as BaTiO3, which occurs through off-centering of the Ti ion from the center of its oxygen octahedron, and is driven by the resulting enhanced covalent-bond formation with the closer oxygen ion(s) Ederer and Spaldin (2006); López-Pérez and J. Íñiguez (2011); Núñez Valdez et al. (2016).

Total energy calculations have confirmed the experimental observation that the $P2_{1}$ phase is the most stable structure for LaTO, NdTO Bruyer and Sayede (2010) and PrTO Patwe et al. (2015), and found the centrosymmetric $P2_{1}/m$ phase to be lower in energy than the centrosymmetric $Cmcm$ structure. In Fig. 3 we show our calculated total energies as a function of volume for the relevant phases of CeTO, where we again find that the lowest energy polar and non-polar phases are $P2_{1}$ and $P2_{1}/m$ respectively, further motivating our investigation of the details of this transition in this work.

A number of first-principles calculations of the magnitude of the spontaneous polarization for the observed $P2_{1}$ phase, taking the non-polar $P2_{1}/m$ phase as reference structure,have been performed, and $P_{S}$ magnitudes of 7.72, 7.42, and 8.3 $\mu$ C/cm2 for LaTO, NdTO Bruyer and Sayede (2010), and PrTO Patwe et al. (2015), respectively, were obtained. Interestingly, the spontaneous polarization calculated in Ref. [López-Pérez and J. Íñiguez, 2011] for the $Cmcm\rightarrow Cmc2_{1}$ in LaTO was $P_{S}=29~{}\mu$ C/cm2, almost four times larger than the previously reported calculated values for the $P2_{1}/m\rightarrow P2_{1}$ transition Bruyer and Sayede (2010). It was argued, however, that a direct comparison between the $P_{S}$ values for the $P2_{1}$ and $Cmc2_{1}$ phases might not be straightforward, and that although not typical of FE perovskites, the further transformation $Cmc2_{1}\rightarrow P2_{1}$ could involve a reduction in $P_{S}$ of LaTO López-Pérez and J. Íñiguez (2011).

In this work we provide a systematic investigation of the rare-earth titanates $R_{2}$ Ti2O7 for $R=$ La, Ce, Pr and Nd using density functional theory (DFT). Our goals are first to understand the ferroelectric mechanism with a particular focus on explaining the discrepancies in the reported values of polarization, and second to facilitate a systematic comparison between the members of the $R_{2}$ Ti2O7 family by revealing trends across the series. We single out Ce2Ti2O7 for detailed analysis, as its calculated ferroelectric properties have not been reported previously.

The remainder of this paper is organized as follows. First, we briefly describe the computational methods we used in our calculations. Then we present and discuss our results for the structural, electronic and ferroelectric properties of the $R_{2}$ Ti2O7 systems for $R=$ La, Ce, Pr, and Nd. In particular, we focus on the ferroelectric mechanism underlying the polar structural distortions. In the final section we summarize our conclusions.

II Computational Details

Our first-principles calculations are based on density functional theory (DFT) Hohenberg and Kohn (1964); Kohn and Sham (1964) using the Vienna Ab initio Simulation Package (VASP) Kresse and Hafner (1993, 1994); Kresse and Furthmüller (1996a, b). We use the generalized gradient approximation (GGA) Perdew and Wang (1992) in the Perdew, Burke and Ernzerhof (PBE) Perdew et al. (1996) format revised for solids (PBEsol) Perdew et al. (2008) for the exchange-correlation functional as it gives better agreement between the optimized and experimental lattice parameters for the systems under consideration compared with the standard PBE functional. We use the default PAW potentials Blöchl (1994); Kresse and Joubert (1999), including six valence electrons for the oxygen (2s22p4), four for Ti (3d34s1) and 11 for the $R$ atoms (5s25p65d16s2), with the $f$ electrons treated as core states. The cutoff energy for the plane-wave expansion of the wave-functions is 550 eV and 4 $\times$ 4 $\times$ 2 and 7 $\times$ 1 $\times$ 5 Monkhorst-Pack Monkhorst and Pack (1976) $\mathbf{k}$ -point meshes are used for the Brillouin zone sampling of the monoclinic and orthorhombic phases, respectively. In the structural optimizations, we fully relax the 44-atom unit cells, lattice parameters and ions, until all forces are converged to less than 0.1 meV/Å on each atom. For the calculation of the spontaneous polarization ( $P_{S}$ ) we use the Berry-phase formalism King-Smith and Vanderbilt (1993); Vanderbilt and King-Smith (1994); Resta (1994); Spaldin (2012) with integration along ten homogeneously distributed strings each of ten $\mathbf{k}$ -points parallel to the reciprocal $b$ axis in the Brillouin zone. Our reported $P_{S}$ values are the differences in polarization between the high-symmetry centrosymmetric reference structures belonging to the $P2_{1}/m$ space group and the corresponding low-symmetry ferroelectric (FE) $P2_{1}$ structures along the same branch in the polarization lattice.

III Results and Discussion

III.1 Energetics and Structural Properties

We begin by determining the lowest energy polar and non-polar structures for our series of rare-earth titanates. In Fig. 3 we present the calculated total energy per formula unit as a function of volume for CeTO for the symmetries of the outer branches in the group-subgroup chart of Fig. 2. We see clearly that the lowest energy polar phase has the monoclinic $P2_{1}$ space group, with the $Cmc2_{1}$ structure $\sim$ 193.0 meV/f.u. higher in energy, and that the lowest energy non-polar phase has $P2_{1}/m$ symmetry, $\sim$ 84.5 meV/f.u. lower in energy than the $Cmcm$ structure. The structures of these two lowest-energy phases, which consist of four formula units (44 atoms) per unit cell, are shown in Fig. 4. Ce, Ti and O occupy 2 $a$ Wyckoff positions in the $P2_{1}$ structure, while in the $P2_{1}/m$ centrosymmetric phase, Ce and Ti occupy 2 $e$ positions and O splits into 2 $a$ , 2 $e$ and 4 $f$ Wyckoff positions. We obtain the same energy ordering for the other members of the series, consistent with Refs. [Patwe et al., 2015] and [Bruyer and Sayede, 2010]. The $C222_{1}$ structure shown in Fig. 2, has never been observed/reported for the titanate Lichtenberg phases or other $n=4$ chemistries, therefore it is not investigated here.

Our calculated lattice parameters, with both PBE and PBEsol functionals and for the $P2_{1}$ and $P2_{1}/m$ symmetries are listed in Table 1 for all systems and compared with available experimental data. In general, the agreement between our PBEsol results and measurements is very good for $R=$ La, Ce, Pr, and Nd in $P2_{1}$ symmetry, with volume deviations of $\lesssim$ 1% (for PBE, the deviations are up to $\sim$ 4.6%). Fig. 5 shows the volume per formula unit trends as a function of $R$ . We notice that the experimental volume decreases in correspondence with the contraction of the $R^{3+}$ ionic radius from La $\rightarrow$ Ce $\rightarrow$ Pr $\rightarrow$ Nd. While this trend is captured by both the PBEsol and PBE functionals for the smaller rare earths, they both give anomalous results for the La compound, a behavior that has also been reported in calculations for the analogous $R_{2}$ O3 sesquioxides Hirosaki et al. (2003); Wu et al. (2007). We also observe that, independently of functional, the volume decreases in the transformation from the non-polar $P2_{1}/m$ phase to the polar $P2_{1}$ phase. This volume reduction for the PBEsol optimized structures ranges between $\sim$ 1% for La and $\sim$ 2% for Nd. Given the better structural results rendered by PBEsol, we use this functional for the rest of this work.

A comparison of the energy difference per formula unit, $\Delta E$ /f.u., between the centrosymmetric $P2_{1}/m$ and ferroelectric $P2_{1}$ phases of $R_{2}$ Ti2O7 for $R=$ La, Ce, Pr, and Nd is shown in Fig. 6. This energy difference increases as the ionic radius of the $R$ cation and volume decrease. Also one can observe that the overall change in energy per formula unit associated with the ferroelectric $P2_{1}/m\rightarrow P2_{1}$ transition in $R_{2}$ Ti2O7 compounds with $T_{C}$ between 1600 and 1850 K is considerably larger when compared, for example, to the corresponding energy change in two formula units of the conventional ferroelectric BaTiO3, ( $Pm\vec{3}m\rightarrow P4mm$ , $T_{C}\sim 400$ K) Zhang et al. (2017), in spite of BaTiO3’s larger ferroelectric polarization.

III.2 Ferroelectric Properties

In order to understand better the ferroelectric symmetry breaking from the high-symmetry $P2_{1}/m$ parent structure to the low-symmetry polar $P2_{1}$ structure in the rare-earth titanates, we employed symmetry-mode analysis using the Isotropy software from the Bilbao Crystallographic server Orobengoa et al. (2009); Perez-Mato et al. (2010). We found that the symmetry-lowering distortion consists of displacements from two zone-center irreducible representations (irreps) of the $P2_{1}/m$ parent group, the $\Gamma 1^{+}$ and $\Gamma 1^{-}$ modes. Table 2 gives the isotropy subgroup and the amplitude percentage of the total distortion amplitude for each mode, and their corresponding atomic displacements are shown in Fig. 7. Note that the symmetries and percentage contributions of these modes are very similar to those identified for the ferroelectric transition in the BaMF4 family Ederer and Spaldin (2006).

The $\Gamma 1^{-}$ mode, which consists of rotations and tiltings of the TiO6 octahedra combined with displacements of the $R$ ions along the $b$ direction, is polar and lowers the symmetry to $P2_{1}$ . It also lowers the energy from the parent $P2_{1}/m$ structure (Fig. 7(b) blue triangles), confirming that it is the primary order parameter for the transition, and indicating that Ce2Ti2O7 is a proper ferroelectric. In contrast, the $\Gamma_{1}^{+}$ distortion is a non-polar mode, consisting mainly of Ce-ion displacements in the $c$ direction (Fig. 7a) in such a way that it is symmetry conserving. In Fig. 7(b) we show the evolution of the energy per formula unit as this mode is frozen in from the high-symmetry structure, and we see that it is not energy lowering and therefore not the primary order parameter for the transition. We also show in Fig. 7(b) the change in energy per formula unit as one deforms the non-polar parent $P2_{1}/m$ structure to the ferroelectric $P2_{1}$ phase considering the total distortion amplitude, with 100% distortion amplitude corresponding to the distortion value required to reach the polar $P2_{1}$ ground state. We see that, while the $\Gamma 1^{+}$ mode contributes only $\sim$ 15 % of the total distortion amplitude, it approximately doubles the amount of energy lowering associated with the formation of the ground state compared with the $\Gamma 1^{-}$ distortion alone. The ground-state structure, with both $\Gamma 1^{+}$ and $\Gamma 1^{-}$ distortions, is shown in the right panel of Fig. 7(a). The behavior of the $P2_{1}/m\rightarrow P2_{1}$ transformation is analogous to that reported in BaMF4, in which the primary polar mode accounts for the majority of the total distortion, with a secondary non-polar mode also contributing to the ground-state structure Ederer and Spaldin (2006).

To determine the magnitude of the spontaneous polarization, $P_{S}$ , which occurs along the $b$ axis, we calculated the differences in polarization between the ferroelectric $P2_{1}$ , and the centrosymmetric $P2_{1}/m$ phases for the $R_{2}$ Ti2O7 series. Particular attention was paid in the mapping of the Berry-phase values onto the same branch of the polarization lattice, by calculating the polarization for a large number of intermediate structures along the deformation path between the $P2_{1}/m$ and $P2_{1}$ structures. Fig. 9 shows the calculated polarization branches as a function of the norm of the distortion vectors (distortion amplitude) from the high-symmetry structure (0%-distortion amplitude) to the ground state low-symmetry phase (100%-distortion amplitude) for Ce2Ti2O7 (for $R=$ La, Pr and Nd, the polarization branches, not shown here, are similar with the only difference being the polarization magnitude). The branches in Fig. 9 are separated by the polarization quantum $P_{q}=\frac{1}{V}eR_{b}$ , where $V$ is the volume of the unit cell, $e$ is the electron charge, and $R_{b}$ is the $b$ lattice parameter. (We use the expression for spin-polarized systems to facilitate copmarison with magnetic Lichtenberg phases.) Furthermore, we also computed $P_{S}$ by multiplying the atomic displacements $\Delta X_{i,j}$ (where $i$ labels the atom and $j$ the direction of displacement) in the $P2_{1}/m\rightarrow P2_{1}$ transformation, multiplied by the formal charges, $Z=3$ for $R^{3+}$ , $Z=4$ for Ti4+, and $Z=-2$ for O2-), that is,

[TABLE]

Table III gives our Berry-phase and formal charge (from Eq. (1)) results for $R_{2}$ Ti2O7 materials with $R=$ La, Ce, Pr, and Nd. For all materials, the Berry-phase polarizations are around 18 $\mu$ C cm*-2*, with the formal charge values slightly lower at around 14 $\mu$ C cm*-2*. We see that multiplying the displacements of the ions with their formal ionic charges yields spontaneous polarizations that are $\sim$ 80% of the Berry phase values. This close agreement is consistent with the geometric mechanism for ferroelectricity, in which the polarization arises from displacements of the ions without substantial electronic rehybridization Ederer and Spaldin (2006); Van Aken et al. (2004). Our computed Born effective charge tensors $\mathbf{Z}^{*}$ (see Appendix, Tables IV-VII) indicate that, while the $\mathbf{Z}^{*}$ s are close to their formal values for the majority of the ions, some Ti and O atoms have anomalously large effective charges of up to almost 8 for nominally Ti4+ and up to $-5.5$ for nominally O2-. These values are comparable to the anomalous Born effective charges in the prototypical conventional ferroelectrics BaTiO3 and PbTiO3 and indicate substantial rehybridization on displacement. These large effective charges correspond, however, to displacements along the $a$ direction in which the materials have infinite chains of TiO6 octahedra, Figs. 1(b) and 4, like those in the conventional perovskite structure. They do not contribute to the net polarization because the displacements along the $a$ direction occur in an antipolar arrangement.

We see that our calculated $P_{S}$ values for the $P2_{1}/m\rightarrow P2_{1}$ transformation are slightly more than double those reported in earlier DFT calcuations. This is not a result of our use of the PBEsol functional; our test calculations using PBE in fact gave larger $P_{S}$ values (not shown here) due to PBE’s overestimation of the lattice parameters of the $P2_{1}/m$ and $P2_{1}$ structures (Table I), and in consequence the distortion amplitude between the two phases. We notice however, that, since $b<a$ and $b<<c$ (Table I), the polarization quantum $P_{q}$ along the $b$ axis is smaller than the spontaneous polarization $P_{S}$ . This feature could be a source of problems in correctly connecting the polarization lattice points to obtain the polarization branches if insufficient intermediate distortions are taken, and might explain the lower-than-expected calculated values reported in the literature. Unfortunately no polarization lattices equivalent to those shown in Fig. 9 were presented in the earlier works Bruyer and Sayede (2010); Patwe et al. (2015).

Finally, we comment on the difference between our calculated polarization value for the $P2_{1}/m\rightarrow P2_{1}$ transition in LaTO (18.2 $\mu$ C/cm2) and the larger value obtained in the earlier study of the $Cmcm\rightarrow Cmc2_{1}$ transition in the same material (29.0 $\mu$ C/cm2) López-Pérez and J. Íñiguez (2011). The mode decomposition and energy profile results reported for the $Cmcm\rightarrow Cmc2_{1}$ transition also indicated the contribution of two modes, although in contrast to the $P2_{1}/m\rightarrow P2_{1}$ transition, in the former case both are polar and both are soft, with one being marginally unstable; we indicate the relative contributions of each mode to the final structure in Fig. 8. We begin by calculating the polarization for the $Cmcm\rightarrow Cmc2_{1}$ transition of LaTO using the computational parameters of Ref. [López-Pérez and J. Íñiguez, 2011], and find that our calculation (shown by the magenta line in Fig. 10) indeed closely reproduces the literature value (purple line). As a double check, we calculated the polarizations of the structures in all four space groups from summing the formal charges multiplied by the displacements of the atoms from their centrosymmetric positions taking the same lattice parameters for all structures: For $P2_{1}/m$ and $Cmcm$ we of course obtained values of zero by definition, for $P2_{1}$ we obtained 14.4 $\mu$ C/cm2, and for $Cmc2_{1}$ 19.5 $\mu$ C/cm2, both underestimating the full Berry phase values as discussed above; the polarization difference obtained by displacing the atoms along a pathway connecting the $P2_{1}\rightarrow Cmc2_{1}$ phases is consistent at 5.1 $\mu$ C/cm2. This leads us to the unusual conclusion that the higher-symmetry $Cmc2_{1}$ phase has a larger spontaneous polarization than the lower symmetry $P2_{1}$ phase, and that if the transition follows the pathway $Cmcm\rightarrow Cmc2_{1}\rightarrow P2_{1}$ on reducing temperature, the system will first show an increase in polarization from zero, then a decrease to a smaller non-zero value at the two successive phase transitions. The distortion modes that link the $Cmc2_{1}$ and $P2_{1}$ structures are shown in Fig. 11.

IV Summary and Conclusions

In summary, we investigated the structural and ferroelectric properties of the family of perovskite-related layered rare-earth titanate Lichtenberg phases, with chemical composition $R_{2}$ Ti2O7 for $R=$ La, Ce, Pr, and Nd using first-principles calculations. We studied the mechanism of the ferroelectric distortion between the lowest energy polar phase, belonging to the monoclinic space group $P2_{1}$ , and its parent high-symmetry non-polar $P2_{1}/m$ structure. We found that the energy lowerings associated with the ferroelectric transitions are consistent with their high Curie temperatures compared to those of conventional ferroelectrics such as BaTiO3. Interestingly, while the energy lowering between the non-polar and polar structures increases across the La - Ce - Pr - Nd series, consistent with the smaller rare-earth cation allowing larger structural distortions, the polarization is approximately constant across the series with a value of around 18 $\mu$ C/cm2. We found that the ferroelectric transition is driven by a polar distortion of $\Gamma 1^{-}$ symmetry consisting of tilts and rotations of the TiO6 octahedra combined with small $R^{3+}$ cation displacements; a secondary distortion of $\Gamma 1^{+}$ also contributes to the ground-state structure. This mode decomposition result is similar to that identified in the ferroelectric transformation of the Ba $M$ F4 family ( $M=$ Mn, Fe, Co, Ni) Ederer and Spaldin (2006), although the slightly anomalous Born effective charges indicate a higher degree of chemical rehybridization across the transition in this case.

We compared our results to an earlier literature study of the higher-energy $Cmcm\rightarrow Cmc2_{1}$ transition, and noted that the higher energy, higher symmetry $Cmc2_{1}$ has a higher polarization than the lower energy, lower symmetry $P2_{1}$ phase. This suggests a possible unusual decrease in ferroelectric polarization on cooling if this pathway is followed. We hope that our results motivate additional experimental studies of this fascinating material class, in particular to resolve the sequence of phases that occur during the ferroelectric phase transition, and to realize the substantial polarization values that have not yet been achieved experimentally.

Acknowledgements.

This work was supported by the ETH-Zürich and the Deutsches GeoForschungZentrum, Postdam. Calculations were performed on the ETH-Zürich Brutus cluster. The authors also gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for funding this project by providing computing time through the John von Neumann Institute for Computing (NIC) on the GCS Supercomputer JUWELS at J lich Supercomputing Centre (JSC) under project HPO24.

Appendix A Born effective charge tensors

Born effective charge tensors calculated in this work using the PBEsol functional (given in units of elementary charge $e$ ) for the polar $P2_{1}$ and the centrosymmetric reference $P2_{1}/m$ structures of $R_{2}$ Ti2O7 with $R=$ La, Ce, Pr, and Nd. The numbering of the atoms in Tables IV-VII is indicated in Fig. 12.

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TL;DR

Contribution

Findings

Abstract

Peer Reviews

Videos

Origin and evolution of ferroelectricity in the layered rare-earth-titanate, R2R_{2}R2​Ti2O7, Lichtenberg phases

Abstract

I Introduction and Background

I.1 Crystal Symmetry of A2A_{2}A2​Ti2O7

I.2 Ferroelectricity of A2A_{2}A2​Ti2O7

II Computational Details

III Results and Discussion

III.1 Energetics and Structural Properties

III.2 Ferroelectric Properties

IV Summary and Conclusions

Acknowledgements.

Appendix A Born effective charge tensors

Origin and evolution of ferroelectricity in the layered rare-earth-titanate, $R_{2}$ Ti2O7, Lichtenberg phases

I.1 Crystal Symmetry of $A_{2}$ Ti2O7

I.2 Ferroelectricity of $A_{2}$ Ti2O7