First-principles prediction of high oxygen-ion conductivity in trilanthanide gallates Ln₃GaO₆

Figures & data

Figure 1. Crystal structure of Ln₃GaO₆ with space group Cmc2₁. The supercell (1 × 1 × 2 conventional orthorhombic unit cell) with 24 Ln, 8 Ga, and 48 O atoms is shown. Different types of Ln and O sites are presented by different colors.

Figure 2. (a) Mean bond lengths of 15 Ln₃GaO₆ and (b) volume (eV/atom, left vertical axis) of 15 Ln₃GaO₆ and Ln₂O₃, and Ln³⁺ radius (right vertical axis). The experimental values of volume and the radius of Ln³⁺ are from Refs. [15] and [45], respectively.

Figure 3. Electronic PDOS of 15 Ln₃GaO₆ obtained by the GGA+U/w.f method. Only Ln 4f (red) and 5d (yellow), Ga 4s (blue), O 2p (black) levels are drawn for simplicity. Energy is shifted by the computational Fermi level. Occupied and unoccupied levels are drawn as solid and dashed lines, respectively. Positive and negative values of PDOS denote the up-spin and down-spin states of electrons, respectively. For easier viewing, the unoccupied La 5d, Ga 4s, and O 2p levels are 2, 30, and 2 times magnified, respectively, whereas the Ln 4f-levels are 0.15 times decreased.

Figure 4. E_g values of 15 Ln₃GaO₆ obtained by the GGA+U/w.f and GGA/w.o.f methods. The E_g values of eight Ln₃GaO₆ (Ln = La, Nd, Gd, Tb, Ho, Dy, Er, or Lu) are not significantly different from each other because occupied or unoccupied f-levels are not formed deeply inside E_g(O 2p–Ga 4s).

Figure 5. Schematic for the classification of Ln₃GaO₆ according to the location of occupied (red solid line) and unoccupied (red dashed line) f-level of Ln inside E_g(O 2p–Ga 4s). Black solid and blue dashed curves denote the HOMO (O 2p) and LUMO (Ga 4s), respectively. (a) The first group of Ln₃GaO₆ has E_g(O 2p–Ga 4s) less affected by the location of f-levels of Ln. (b) The second group of Ln₃GaO₆ has E_g(Ln 4f–Ga 4s). (c) The third group of Ln₃GaO₆ has E_g(O 2p–Ln 4f). (d) The last group of Ln₃GaO₆ has E_g(Ln 4f – Ln 4f), but they are not found.

Figure 6. Energy differences (eV/atom) of eight Ln₃GaO₆ (Ln = La, Nd, Gd, Tb, Ho, Dy, Er, or Lu) with respect to their competing reference states, namely, the coexisting states of ¾(Ln₂O₃) + ¼(Ga₂O₃) and ¾(Ln₄Ga₂O₉) + ¼(Ln₂O₃).

Figure 7. Phonon DOS of eight Ln₃GaO₆ (Ln = La, Nd, Gd, Tb, Ho, Dy, Er, or Lu). The vertical dashed line denotes zero phonon frequency. No imaginary phonon frequencies are found, which indicates that these eight oxides are dynamically stable.

Table 1. E_v of eight Ln₃GaO₆ (Ln = La, Nd, Gd, Tb, Ho, Dy, Er, or Lu). The computational results in this table are obtained using the GGA/w.o.f method. The Fermi level is located at the HOMO (VBM). The values are in eV.

O site		La3GaO6	Nd3GaO6	Gd3GaO6	Tb3GaO6	Dy3GaO6	Ho3GaO6	Er3GaO6	Lu3GaO6
	Eg	3.83	3.62	3.59	3.57	3.56	3.54	3.51	3.47
O(I)	Ev(VO^0)	2.89	2.91	2.92	2.91	2.91	2.91	2.91	2.91
	Ev(VO^+)	1.25	1.64	1.50	1.51	1.52	1.53	1.54	1.55
	Ev(VO^2+)	−0.49	−0.09	−0.01	0.00	0.01	0.02	0.04	0.04
	ε(2+/0)^a	1.69	1.50	1.46	1.46	1.45	1.45	1.44	1.43
O(II)	Ev(VO^0)	4.92	4.67	4.81	4.83	4.84	4.85	4.87	4.92
	Ev(VO^+)	1.65	2.00	1.86	1.89	1.90	1.92	1.94	1.99
	Ev(VO^2+)	−0.77	−0.44	−0.40	−0.39	−0.39	−0.39	−0.38	−0.40
	ε(1+/0)^a	3.27	2.67	2.95	2.94	2.94	2.93	2.93	2.93
	ε(2+/1+)^a	2.42	2.44	2.26	2.28	2.29	2.31	2.32	2.39
O(III)	Ev(VO^0)	2.68	2.82	2.85	2.86	2.86	2.86	2.86	2.86
	Ev(VO^+)	1.07	1.53	1.40	1.41	1.41	1.42	1.43	1.42
	Ev(VO^2+)	−0.71	−0.25	−0.18	−0.17	−0.17	−0.17	−0.14	−0.17
	ε(2+/0)^a	1.69	1.54	1.52	1.51	1.51	1.51	1.50	1.52
O(IV)	Ev(VO^0)	2.79	2.86	2.89	2.89	2.89	2.89	2.90	2.90
	Ev(VO^+)	1.50	1.90	1.81	1.83	1.84	1.85	1.88	1.90
	Ev(VO^2+)	0.55	1.00	1.16	1.20	1.23	1.26	1.31	1.36
	ε(1+/0)^a	1.29	0.96	1.08	1.06	1.05	1.04	1.02	1.00
	ε(2+/1+)^a	0.95	0.90	0.65	0.63	0.61	0.59	0.57	0.54

^a Locations are counted from the O 2p-HOMO.

Figure 8. E_v as a function of Fermi level of La₃GaO₆. μ_O is set to be ½E(O₂) − 1.75 eV to describe the pressure and temperature at 1 atm and 1500 K, respectively. The other Ln₃GaO₆ (Ln = Nd, Gd, Tb, Dy, Ho, Er, or Lu) series showed a similar trend (see Supplementary Figures S3). The gradient of the lines denotes the charge of the defect.

Table 2. E_m for various paths of eight La₃GaO₆. The computational results in this table are obtained using the GGA/w.o.f method. IS, TS, and FS denote the initial, transition, and final states for CI-NEB, respectively.

VO^2+ site of IS^a	Path	VO^2+ site of FS	E(FS) − E(IS) (eV)	O site distance (Å)	Em (eV)^b	E(TS) − E(IS) with VO^2+[O(II)] (eV)^c	Index
O(I)	I-1	O(I)	0.00	2.880	0.62	0.89
[+0.27 eV]	I-2	O(III)	−0.16	2.950	0.51	0.63
	I-3	O(II)	−0.27	2.987	1.03	1.03
	I-4	O(IV)	1.02	3.187	1.24	1.51
	I-5	O(IV)	1.02	3.298	1.89	2.16
	I-6	O(IV)	1.02	3.304	2.12	2.38
	I-7	O(I)	0.00	3.407	1.02	1.29
	I-8	O(II)	−0.27	3.462	0.94	0.94
O(II)	II-1	O(II)	0.00	2.867	0.59	0.59	The lowest Em
[+0.00 eV]	II-2	O(I)	0.27	2.987			Same as path I-3
	II-3	O(II)	0.00	3.004	1.02	1.02
	II-4	O(III)	0.11	3.269	0.76	0.76
	II-5	O(I)	0.27	3.462			Same as path I-8
O(III)	III-1	O(I)	0.16	2.950			Same as path I-2
[+0.11 eV]	III-2	O(III)	0.00	3.048	0.74	0.85
	III-3	O(IV)	1.18	3.086	1.67	1.78
	III-4	O(II)	−0.11	3.269			Same as path II-4
	III-5	O(IV)	1.18	3.271	1.66	1.77
O(IV)	IV-1	O(III)	−1.18	3.086			Same as III-3
[+1.18 eV]	IV-2	O(I)	−1.02	3.187			Same as I-4
	IV-3	O(III)	−1.18	3.271			Same as path III-5
	IV-4	O(I)	−1.02	3.298			Same as path I-5
	IV-5	O(I)	−1.02	3.304			Same as path IV-5

^a ΔE_ref, which can be defined as E(IS) − E(IS) with V_O²⁺[O(II)], with a unit of eV are in the brackets.

^b Larger value between E(TS) − E(IS) and E(TS) − E(FS).

^c E(IS) with V_O²⁺[O(II)] is the most stable among four types of O sites.

Figure 9. (a) E_m for various migration paths of eight Ln₃GaO₆ (Ln = La, Nd, Gd, Tb, Ho, Dy, Er, or Lu) and (b) the lowest E_m of the eight Ln₃GaO₆ and their counterpart Ln₂O₃ (left vertical axis) and the distances between O sites for the migration path in the crystal structure (right vertical axis).

Figure 10. (a) Relationship between D_O obtained by FPMD at 1873 K and the lowest E_m obtained by CI-NEB, and (b) predicted σ_O as a function of T based on D_O at 1873 K for eight Ln₃GaO₆ (Ln = La, Nd, Gd, Tb, Ho, Dy, Er, or Lu). Solid symbols denote computed or experimentally measured values from references. Dashed lines denote predicted values. The concentration of V_O²⁺ of our computation is ~2.1% from the computational cell of Ln₂₄Ga₈O₄₇. The concentration of V_O²⁺ of Nd_2.955Sr_0.045GaO_5.9775 [13] and Gd_2.9Sr_0.1GaO_5.95 [14] are ~0.4 and ~0.8%, respectively. σ_O of 10⁻² S/cm at 1000 K is described to show a reference level of YSZ [2]. The D_O values are averaged from three FPMD trials and error ranges are described between the minimum and maximum values.

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