Supplementary materials to
Tunable oxygen vacancy configuration by strain engineering in
perovskite ferroelectrics from first-principles study
Q. Yang,1 J. X. Cao,1,a) Y. C. Zhou,1,a) Y. Zhang,1 Y. Ma,1 and X. J. Lou2
1
Key Laboratory of Low Dimensional Materials and Application Technology of Ministry of
Education, Xiangtan University, Xiangtan, Hunan 411105, China
2
Frontier Institute of Science and Technology, Xi’an Jiaotong University, Xi’an 710054, China
———————————————————————
a) E-mail addresses: jxcao@xtu.edu.cn (J. X. Cao) and zhouyc@xtu.edu.cn (Y. C. Zhou)
Section S1. Structure optimization details for PTO unit cell.
All the calculations in this work were performed on a Vienna Ab initio simulation
package code according to the projector augmented wave method.1 Local density
approximation (LDA) was performed. The cut-off energy for wave functions was set
at 450 eV; the tetrahedron method with Blöchl corrections, as well as a 15×15×15
Monkhorst-Pack k-point mesh,2 was used for the unit cell. A conjugate gradient
algorithm was used for structural optimization and the ions were fully relaxed until
the Hellmann-Feynman forces reached 0.005 eV/Å. Our calculations show that the
lattice constant a and the axial ratio c/a for the tetragonal PTO are 3.854 Å and 1.05,
respectively, which are in agreement with the results of a previous LDA calculation3
and experimental studies in the literature.4 The spontaneous polarization along the c
axis was 82.1 µC/cm2, which was calculated using the Berry phase method,5 and is in
1
good agreement with the results in previous studies: 85 µC/cm2 by experimental
measurement (at room temperature)6 and 82 µC/cm2 by theoretical calculation7 for the
same phase.
Section S2. Calculation details and results for strain effect on the defect-free
PTO.
Ab-biaxial strain (ε) ranging from –3% to +2% were applied on the defect-free PTO.
For each strained PTO, the lattice constant along the c-axis and all the atomic
positions are fully relaxed. The tetragonality (c/a ratio) and spontaneous polarization
for strained PTO are shown in Fig. S1. Results show that the tetragonality (c/a ratio)
and spontaneous polarization for a perfect tetragonal PTO increase almost linearly
with the increase of compressive strain. This spontaneous polarization-strain
relationship obtained in this work is in good agreement with the results calculated by
Bungaro and Rabe.8
Fig. S1
2
Fig. S1. The spontaneous polarization-strain relationship for a perfect tetragonal PTO. The blue
circles are the calculated values by Bungaro and Rabe.8 The insert gives the tetragonality
(c/a ratio) of the PTO unit cell under different strains.
Section S3. Charge density analyses for Vcud and Vcsw configurations.
a) Charge density
To shed light on the origin of energy stability of the Vcsw configuration, the
interactions between the Ti1 atom and its adjacent atoms were also investigated using
the charge density analyses. The total charge density and the charge difference are
illustrated in Fig. S2, which reveals the underlying mechanism for the stability of the
Vcsw configuration in the compressive-strained PTO. Here, the charge difference is
defined as diff   (Vc )   (Vc  Ti 1 )   (Ti1 ) , where  (Vc ) ,  (Vc  Ti 1 ) , and
 (Ti 1 ) are the total charge density of PTO with VO, PTO with VO and without Ti1,
and the charge density of a single Ti1 atom, respectively. The charge difference shows
the interaction between Ti1 and its neighboring atoms. Ti atoms interact strongly or
weakly with the upper and lower O atoms, thereby leading to the formation of short
and long bonds in the strain-free PTO. Under compressive strain, the short bond is
further enhanced and the long bond is weakened, as we can easily see from the
comparison between the total charge density in Figs. S2(a) and that in Fig. S2(c). The
strain-induced change in bond strength between Ti and O is further confirmed from
the partial density of states (pDOS; Fig. S3). The long bond and the short bond linked
to the defected oxygen atom are both broken when Vc is present in PTO, thereby
leaving a dangling bond, which makes Ti1 unstable. Since Ti1 downshifts towards the
3
lower oxygen ion (O1), an additional short bond (T1–O1), forms, which leads to the
formation of a Vcud configuration. Therefore, the atomic reconstruction induced by
Vc results in the formation of a tail-to-tail domain configuration, which has been
considered as the origin of polarization fatigue in ferroelectric materials 9,10. The short
bond reconstructed weakens the original strong Ti2–O1 bond because of the abundant
charge transfer that occurs from Ti1 to O1 [Fig. S2(e)]. In contrast to the Vcud
configuration, Ti1 remains at the position adjacent to VO in the Vcsw configuration.
Due to the formation of the dangling bond, the Ti2–O1 bond is recovered and intensive
charge transfer occurs from Ti1 to the Ti1–O2 linkage, which enhances the interaction
strength of the four Ti1–O2 bonds. As the compressive strain increases (therefore the
c/a-ratio increases, see the inset of Fig. S1), particularly for   3% , the
reconstructed T1–O1 short bond in the Vcud configuration is weakened and the Ti1–O2
interaction in the Vcsw configuration is further strengthened. The four neighboring
oxygen atoms (O2) tightly binding to Ti1 significantly enhance the diffusion barrier
between Vcsw and Vcud in compressively strained PTO [see Fig. 3(d)].
4
Fig. S2
S
Fig. S2. (a)–(d) show the charge densities of the ac-plane across oxygen vacancies (VO) for Vcud
5
and Vcsw at 1% and 3% compressive strains, respectively. (e)–(h) show the charge
differences of the planes that correspond to (a)–(d). Electron gain is given by yellow and
electron loss by blue. (a), (b), (e), and (f) at 1% compressive strain, as well as (c), (d), (g),
and (h) at 3% compressive strain. The black triangles and red circles represent Ti and O
atoms, respectively. The red dotted circles represent the VO.
b) Partial density of states (pDOS)
To further clarify the interaction between Ti1 and its neighboring oxygen atoms, we
illustrated the pDOS of Ti1 and the neighboring O1 and O2 atoms, as shown in Fig. S3.
Different from the pDOS of Ti in perfect PTO, the Fermi level shifts into the
conduction band in the oxygen-defective PTO, suggesting that VO acts as a donor in
PTO, which is consistent with the results in a previous study 9. In perfect PTO, the Ti1
3d orbitals are strongly coupled with both the O1 2p orbitals and the O2 2p orbitals.
Under compressive strain, however, the coupling between Ti1 and O2 is strengthened,
whereas the coupling between Ti1 and O1 is weakened (see Fig. S3). This finding is
attributed to the fact that the bond length of Ti1–O1 (along the c axis) is elongated,
whereas the bond length of Ti1–O2 (in the ab-plane) is shortened under compressive
strain. From the sharp resonation peak at –7 eV, we observed that the coupling
between Ti1 and O1 is stronger in the Vcud configuration than that in the Vcsw
configuration or in perfect PTO. However, the width of this coupling peak is
narrowed under a larger compressive strain. The coupling between Ti1–O2 in the Vcud
configuration is weaker than that in the Vcsw configuration, which eventually
stabilizes the Vcsw configuration.
6
Fig. S3
Fig. S3. Partial density of states (pDOS) of Ti1, O1, and O2 in 1%, 2%, and 3%
compressive-strained states. The first, second, and third rows show and Vcud (ud), Vcsw
(sw), and perfect (per) configurations. The Fermi level (Ef) is located at Energy = 0 eV.
1
G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
2
H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976).
3
P. Erhart, R. A. Eichel, P. Träskelin, and K. Albe, Phys. Rev. B 76, 174116 (2007).
4
S. A. Mabud and A. M. Glazer, J. Appl. Crystallogr. 12, 49 (1979).
5
R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993).
6
R. J. Nelmes and W. F. Kuhs, Solid State Commun. 54, 721 (1985).
7
Y. F. Duan, H. L. Shi, and L. X. Qin, J. Phys.: Condens. Matter 20, 175210 (2008).
8
C. Bungaro and K. M. Rabe, Phys. Rev. B 69, 184101 (2004).
9
C. H. Park and D. J. Chadi, Phys. Rev. B 57, R13961 (1998).
10
C. Brennan, Ferroelectrics 150, 199 (1993).
7