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Multiferroic Dislocations in Ferroelectric PbTiO
Takahiro Shimada, Tao Xu, Yasumitsu Araki, Jie Wang, and Takayuki Kitamura Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b00505 • Publication Date (Web): 14 Mar 2017 Downloaded from http://pubs.acs.org on March 16, 2017
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Multiferroic Dislocations in Ferroelectric PbTiO3 Takahiro Shimada,1,* Tao Xu,1,** Yasumitsu Araki1, Jie Wang,2 and Takayuki Kitamura1 1
Department of Mechanical Engineering and Science, Kyoto University, Nishikyo-ku, Kyoto 615-8540, Japan
2
Department of Engineering Mechanics, School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, China
Abstract Ultrathin multiferroics with coupled ferroelectric and ferromagnetic order parameters hold promise for novel technological paradigms, such as extremely thin magnetoelectric memories. However, these ferroic orders and their functions inevitably disappear below a fundamental size limit of several nanometers. Herein, we propose a novel design strategy for nanoscale multiferroics smaller than the critical size limit by engineering the dislocations in nonmagnetic ferroelectrics, even though these lattice defects are generally believed to be detrimental. First-principles calculations demonstrate that Ti-rich PbTiO3 dislocations exhibit magnetism due to the local nonstoichiometry intrinsic to the core structures. Highly localized spin moments in conjunction with the host ferroelectricity enable these dislocations to function as atomic-scale multiferroic channels with pronounced magnetoelectric effect that are associated with the antiferromagneticferromagnetic-nonmagnetic phase transitions in response to polarization switching. The present results thus suggest a new field of dislocation (or defect) engineering for the fabrication of ultrathin magnetoelectric multiferroics and ultrahigh density electronic devices.
T.Shimada and T.Xu contributed equally to this work. * E-mail:
[email protected] ** E-mail:
[email protected] 1 ACS Paragon Plus Environment
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Keywords multiferroics, dislocations, ferroelectrics, nonstoichiometry, magnetoelectric effect
Table-of-content Graphic:
Main text:
Multiferroics, in which different ferroic orders (such as ferroelectric and ferromagnetic) coexist, have been the subject of intensive research due to the fascinating basic physics arising from the cross-coupling between multiple ferroic degrees of freedom, and the significant potential for revolutionary new spintronic or energy-efficient memory devices in which electric fields control magnetism.[1-4] Both the surging demand for miniaturized, compact electronic devices and the discovery of a plethora of exotic physical phenomena in finite dimensional materials, such as unusual ferroic orders, have recently sparked an upsurge of interest in nanoscale multiferroics.[5-8] Unfortunately, the increased depolarization effects that accompany size reductions due to uncompensated surface charges tend to destabilize the ferroic orders in finite dimensions, such that 2 ACS Paragon Plus Environment
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there will be a critical size below which the ferroic orders completely disappear. The ferroelectric size limit, in particular, has been intensively investigated over the years. For example, Gregg et al.[9-11] systematically investigated the ferroelectric properties of nanoscale thin-film capacitors; Barthelemy and Bibes et al.[12-16] have made enormous attempts to probe the fundamental size effects of thin film systems experimentally and pushed lower limit of the size for ferroelectricity down to a few nanometers. The lowest experimental reported critical thickness up to now has been determined to be 1.2 nm for PbTiO3[17] thin films and 2 nm for BiFeO3[18] thin films, in good agreement with theoretical studies[19,20]. The ferroelectricty in nanostructures, such as nanodots, has also been found to disappear below size of 3-5 nm both theoretically and experimentally.[21,22] Therefore, additional miniaturization of certain devices while retaining desirable multiferroic properties is physically challenging. An alternative strategy that allows for atomic-scale multiferroics and ultrahigh density integration of multiferroic elements is thus highly desirable. Structural imperfections, or lattice defects (such as dislocations), are pervasive in perovskite oxides as the result of crystal deformations, and typically degrade material properties and device functionality due to atomic disorder.[23-26] However, in analogy to the topological defects that exhibit exotic transport properties[27-31], in some cases, dislocations can also present an opportunity to enhance the host material performance or can even produce exceptional phenomena that differ substantially from those exhibited by perfect crystals, such as unusual conductivity,[32,33] superconductivity[34] and enhanced mass transport[35]. These effects are attributed to the translational discontinuity of dislocation cores that locally alter the stoichiometry or electronic properties of the material.[36,37] Peculiar physical attributes such as these have provided novel opportunities to fabricate one-dimensional nanostructures with exotic properties inside crystalline 3 ACS Paragon Plus Environment
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solids through dislocation technology, as has been recently demonstrated through experimental trials.[33,38,39] The resulting novel device concepts based on dislocation engineering or control suggests the possibility of fabricating extremely thin multiferroic based on oxides in view of the atomic-scale geometry of the core structures. Here, we explored the magnetic properties of individual dislocations in the typical nonmagnetic ferroelectric material PbTiO3, and suggest a new pathway toward ultimately thin magnetoelectric multiferroics. Systematic first-principles calculations demonstrated that local nonstoichiometry intrinsic to naturally occurring dislocations generates (anti)ferromagnetism with highly localized spin moments at atomic-scale core centers, coexisting with the host ferroelectricity. The emerging magnetic properties strongly depend on the morphology of the constituent
dislocations
and
the
polarization
configurations,
accompanied
by
the
antiferromagnetic-ferromagnetic-nonmagnetic (AFM-FM-NM) phase transitions. These results suggest a new means of fabricating one-dimensional multiferroics thinner than the critical thickness of intrinsic multiferroics through the dislocation engineering. The PbTiO3 edge dislocations were simulated from first-principles calculations within the density functional theory (DFT) using the VASP code.[40,41] The exchange-correlation energy was treated by employing the generalized gradient approximation corrected with an on-site Hubbard correction U, applying values of U = 4.531 eV and J = 0.555 eV for Ti 3d orbitals, which were accurately derived from constrained random-phase approximation (cRPA) calculations.[42] It is necessary to employ the DFT+U scheme to predict the structural and electronic properties of wide band-gap oxides (such as PbTiO3) with defects or nonstoichiometry.[43,44] We also cross-checked the results using hybrid Hartree-Fock density-functional calculations proposed by Hyed, Scuseria, 4 ACS Paragon Plus Environment
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and Ernzerhof (HSE06). The dislocations were studied via the periodic supercell approach using the dipole model,[45] as shown in Figure S1, which results in stress-free conditions at the core structures. To avoid undesirable core-core interactions, the dislocation distances were set to greater than 20 Å, with the dimensions along the three cell vectors of the supercell being 11a, 5a and 2a, respectively. Both Ti-rich and Pb-rich dislocation core structures with O and Pb deficiencies, respectively,
were
considered
inspired
by
experimental
observations
and
theoretical
implications.[46-50] The stabilities of these core structures have further been validated by calculating the formation energy (Table S1). Spontaneous polarizations were aligned in the [100], [010], [001] and [0 1 0] directions, respectively, to study the effect of the polar directions. Full details of the computational methods are available in the Supplementary Information. The magnetic moments and spin configurations of dislocation systems with different polarization configurations were systematically explored, and the results are summarized in Table 1. No spin moment is observed in the case of the Pb-rich dislocations, regardless of the polarization configuration, and the system remains in a nonmagnetic state, just as in ideal defect-free PbTiO3. Surprisingly, the Ti-rich dislocation core structures are associated with rich magnetic behaviors depending on the polar direction. We initially focus on the Ti-rich dislocation with [001] polarization along the dislocation line. In this scenario, separate nonzero, nontrivial magnetic
moments
appear
around
the
dislocation
region
(Figure
1(a)),
conferring
antiferromagnetism on the intrinsically nonmagnetic PbTiO3. The emerging moments in the core centers form a checkerboard-like antiferromagnetic spin configuration in the (010) plane, in which the nearest-neighbor spins are aligned antiparallel. The remaining paired magnetic moments are also aligned in an antiferromagnetic configuration on the adjacent Ti sites at tensile regions below 5 ACS Paragon Plus Environment
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the core centers. As a result, there is no net overall spin polarization in the system. This localized emerging magnetism is embedded within the ambient ferroelectric host, enabling the dislocation structure to act as an atomic-scale multiferroic channel below the critical size of intrinsic multiferroics. In contrast, when the initial polarization is rotated to the perpendicular [100] polar axis, the antiferromagnetic state rapidly transforms into a ferromagnetic state with a total magnetic spin moment of 2.18 μB, exhibiting a remarkable magnetoelectric effect. The emerging magnetic moments are located solely on the Ti atoms in the left side of dislocation core (Figure 1(b)), while the moments on the other side are negligible, due to symmetry breaking along the [100] direction. On switching the initial polarization to the [010] direction, the emerging magnetic moments disappear and the system undergoes a magnetic phase transition to the same nonmagnetic state as the host ferroelectric PbTiO3 (Figure 1(c)). Interestingly, antiferromagnetism appears again in the Ti-rich dislocation core upon polarization reversal toward the [0 1 0] direction. The induced spin moments are strictly confined to two Ti-Ti pairs in the core center, forming the distinct striped antiferromagnetic structure presented in Figure 1(d). The magnetic orders can therefore be repeatedly switched on and off electrically, which is promising with regard to potential low-power electrical writing in spintronic devices. All these results suggest that both the dislocation core structure and the electric polarization are important for the emergence of magnetism. Therefore, the dislocations with Ti-rich morphology display magnetic features and undergoes the rich AFM-FM-NM phase transitions in response to polarization switching. To elucidate the underlying mechanism for the emergence of magnetism associated with the Ti-rich dislocations and the coupling between electric and magnetic polarizations, the electronic 6 ACS Paragon Plus Environment
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band structures of systems with different polarization configurations were analyzed comprehensively. In the case of an antiferromagnetic dislocation core structure with polarization along the [001] direction, four new, distinct states are observed within the bandgap ((i), (ii), (iii) and (iv)), among which (i), (ii) and (iii) represent spin-polarized defect states and (iv) is a degenerate state (occupied by identical spin-up and spin-down electrons at the same crystal site) with metallic characteristics (Figure 2 and Figure S2). These states are absent in a perfect PbTiO3 crystal and are therefore characteristic of defect states introduced by a dislocation structure. The (i) band is occupied by majority-spin and minority-spin electrons, whereas states (ii) and (iii) are partially filled by minority and majority spins, respectively. The net magnetic moment is thus zero and the dislocation appears antiferromagnetic state. These spin-polarized excess electrons originate from the nonstoichiometry of the dislocation core structures. The atomic composition of the Ti-rich dislocation system is composed of 110 Pb, 112 Ti and 326 O atoms, which deviates from the stoichiometric composition of bulk PbTiO3. This nonstoichiometric composition results in a net charge of +8 per dislocation based on the nominal ionic charges of Pb2+, Ti4+ and O2-, thus producing a charge imbalance and an n-type electronic character. As can be seen from the shape of the squared wave functions in Figures 2(b)-(d), the (i) band accommodating the majority and minority spin electrons is ascribed to d-dominant orbitals that are separately localized at the two Ti atoms in the tensile zone in opposite spin directions. The spin-down band (ii) and spin-up band (iii) are primarily distributed in the core center of the Ti atoms, forming a checkboard pattern in which the nearest-neighbor spins are aligned antiparallel. These localized and separated spin-polarized electrons result in spin polarizations arranged in antiferromagnetic ordering around the core center and the neighboring tensile region. 7 ACS Paragon Plus Environment
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In the case of a ferromagnetic dislocation with a [100] polarization configuration, eight new spin-polarized localized states appear within the band gap, due to the nonstoichiometry of the core structure (Figure 3). Five of these states ((i), (iii), (iv), (vi) and (vii)) are majority spin, and the other three ((ii), (v) and (viii)) are in minority spin states. These spin-polarized defect states contribute to a total magnetic moment of 2.18 μB in the dislocation system. Upon closer inspection of the squared wave function of these defect states, the excess electrons are also found to stem from d orbitals of dislocation core Ti atoms. Among these, the opposite sign paired states of (i) and (ii), (v) and (vi), (vii) and (viii) are localized on the same sites, and so do not contribute to the net magnetism, while the majority-spin charge density of the (iii) and (iv) states are situated to the left side of the core center, in line with the distribution of the emerging magnetism. Similar results are obtained in the case of antiferromagnetic dislocation systems with spontaneous polarization along the [0 1 0] direction, in which one defect state is spin-polarized with spin-up and spin-down electrons localized at different sites, and the remaining three defect states are degenerate and occupied by identical majority and minority electrons, as shown in Figures 4 and S3. It should also be noted that the defect states in the Ti-rich dislocations with polarization along the [010] direction are all degenerate and spin-unpolarized, leading to a nonmagnetic state for this polar configuration (Figure S4). Therefore, the unbounded Ti 3d spin-polarized electrons arising from the local nonstoichiometric and the crystal symmetry reduction associated with the intrinsic polar direction are responsible for the emergence of the unexpected magnetism. Similar reduction of the Ti oxidation state from +4 has also been observed at the dislocation cores of oxide materials experimentally.[33,48] Based on the above, ferromagnetic or antiferromagnetic spin moments are evidently intrinsic 8 ACS Paragon Plus Environment
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features of naturally-occurring Ti-rich PbTiO3 dislocations. To assess the ferroelectricity in the dislocation cores, the local vector field of the polarization around a dislocation was investigated, as presented in Figure S5. Although the local electric polarization is disturbed around the core structure with respect to the intrinsic ferroelectricity, due to the elastic field of the dislocations, ferroelectricity is retained in the core regions in each polarization configuration. The highly localized nature of the emerging magnetism, combined with the host ferroelectricity, thus turn the Ti-rich PbTiO3 dislocations into atomic-scale multiferroic channels based on their atomically-thin, one-dimensional structures. Note that the Pb-rich dislocations cannot be the candidate for atomic-scale multiferroics owing to the trivial nonmagnetic nature of such dislocations resulting from their charge balance electronic properties (Figure S6). Therefore, rather than degrading the host functionality, the purposeful engineering of dislocation morphologies can open up new possibilities for ultimately small multiferroics that are promising for novel technological paradigms. Such finest multiferroic filaments with magnetic properties being confined within one unit cell thickness (0.4 nm) cannot be achieved by the direct scale-down of intrinsic multiferroics due to the critical size limit of ferroic orders (1.2-5 nm). The remarkable magnetoelectric effect exhibited by multiferroic linear channels in the present work should enable atomic-scale spin manipulation by electric polarization, providing a means of electric manipulation for and ultimate multiferroic logic.[6] It is worth to mention that the PbTiO3 screw dislocations also exhibit similar magnetism (Figure S7), and thus the concept of dislocation engineering is promising in fabricating atomic-scale multiferroics. The results reported herein also suggest the possibility of tailoring multiferroic properties through the manipulation of both the density and distribution of dislocations. Dislocations with a 9 ACS Paragon Plus Environment
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sufficiently high density (up to 1012 cm-2) have been introduced experimentally.[39] This high concentration of dislocations could allow an ultrahigh density of spatially-arranged, atomic-scale multiferroic channels and therefore increase storage densities into the terabit range. Dislocations are known to be mobile and their arrangement can be controlled to some extent.[38,51] , which provides an opportunity to tailor the multiferroic properties dynamically. In summary, we have proposed a novel design strategy for atomic-scale magnetoelectric multiferroics through the engineering of otherwise detrimental dislocations in ferroelectric PbTiO3. Our theoretical predictions revealed that the magnetism is inherent to Ti-rich dislocations resulting from the local non-stoichiometry of core structures. The magnetism is highly localized around the dislocation cores and displays a strong sensitivity to the intrinsic polar direction, accompanied by rich AFM-FM-NM phase transitions. The localized character of the emerging magnetism, in conjunction with the atomic-scale geometry of the ferroelectric dislocation core structures, thus realizes atomically-fine multiferroic channels with a pronounced nonlinear magnetoelectric effect. The present results suggest new possibilities for the fabrication of extremely thin atomic-scale multiferroics and ultrahigh density magnetoelectric memories.
Acknowledgements The authors acknowledge financial support from JSPS KAKENHI (25000012, 26289006, and 15K13831) and the National Natural Science Foundation of China (11672264 and 11472242). Author Information Corresponding author. *E-mail:
[email protected] 10 ACS Paragon Plus Environment
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Author Contributions. T.S. and T.X. conceived the project, designed and directed computational experiments, and wrote the entire manuscript. Y.A. performed the theoretical calculations and interpreted the data. J.W. supported the calculations and discussed the results. T.K. supervised the work and provided critical feedback on the manuscript. All authors read and commented on the manuscript. T.S. and T.X. contributed equally to this work. Notes The authors declare no competing financial interest. Associated Content Supporting Information. This material is available free of charge via the Internet at htpp://pubs.acs.org.
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Table I. Total magnetic moments, M (μB), and magnetic phases of dislocations in Pb-rich and Ti-rich morphologies at different polarizations.
Morphology Polar direction
Pb-rich
Ti-rich
[001] [100]
[010] [0 1 0]
[001]
[100]
[010]
[0 1 0]
M (in μB)
0.00
0.00
0.00
0.00
0.00
2.18
0.00
0.00
Magnetic phase
NM
NM
NM
NM
AFM
FM
NM
AFM
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Figure 1. Magnetic spin-density distributions in Ti-rich dislocations with spontaneous polarization along the (a) [001], (b) [100], (c) [010] and (d) [0 1 0] directions. The cross-sectional views along and perpendicular to the dislocations are shown in the upper and lower panels in each picture. The red arrows indicate the spontaneous polarization, and the yellow and green regions denote isosurfaces with magnetic spin densities of 0.04 and -0.04 μB/Å3, respectively. 13 ACS Paragon Plus Environment
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Figure 2. (a) Electronic band structures of Ti-rich dislocations with polarization along the [001] direction. The green line denotes the Fermi level. The red and blue lines indicate the majority- and minority-spin orbitals, respectively, the pink line denotes majority- and minority-spin states, and the black line is the degenerate state. (b)-(d) The squared wave functions of the in-gap states. The orange and light-blue colors indicate isosurfaces of majority- and minority-spin densities of 0.04Å-3, respectively. The squared wave function of the degenerate state (iv) is shown in Figure S2.
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Figure 3. (a) Electronic band structures of Ti-rich dislocations with polarization along the [100] direction. The green line denotes the Fermi levels, the red and blue lines indicate the majority- and minority-spin orbitals, respectively. (b)-(i) The squared wave functions of the in-gap states. The orange and light-blue colors indicate isosurfaces of majority- and minority-spin densities of 0.04Å-3, respectively. 15 ACS Paragon Plus Environment
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Figure 4. (a) Electronic band structures of Ti-rich dislocations with polarization along the [0 1 0] direction. The green line denotes the Fermi level, the pink line denotes majority- and minority-spin states, and the black line is the degenerate state. (b) The squared wave function of the in-gap state. The orange and light-blue colors indicate isosurfaces of majority- and minority-spin densities of 0.04Å-3, respectively. The squared wave functions of the degenerate states are shown in Figure S3.
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[17] Fong, D. D.; Stephenson, G. B.; Streiffer, S. K.; Eastman, J. A.; Auciello, O.; Fuoss, P. H.; Thompson, C. Science 2004, 304, 1650. [18] Béa, H.; Fusil, S.; Bouzehouane, K.; Bibès, M.; Sirena, M.; Herranz, G.; Jacquet, E.; Contour, J.-P.; Barthélémy, A. Jpn. J. Appl. Phys. 2006, 45, L187. [19] Despont, L.; Koitzsch, C.; Clerc, F.; Garnier, M. G.; Aebi, P.; Lichtensteiger, C.; Triscone, J.-M.; Garcia de Abajo, F. J.; Bousquet, E.; Ghosez, Ph. Phys. Rev. B 2006, 73, 094110. [20] Junquera, J.; Ghosez, Ph Nature 2003, 422, 506-509. [21] Naumov, I. I.; Bellaiche, L.; Fu, H. Nature (London) 2004, 432, 737-740. [22] Polking, M. J.; Han, M.-G.; Yourdkhani, A.; Petkov, V.; Kisielowski, C. F.; Volkov, V. V.; Zhu, Y.; Caruntu, G.; Alivisatos, A. P.; Ramesh, R. Nat. Mater. 2012, 11, 700-709. [23] Hÿtch, M. J.; Putaux, J.-L.; Pénisson, J.-M. Nature 2003, 423, 270-273. [24] Neumark, G. F. Phys. Rev. Lett. 1968, 21, 1252. [25] Alpay, S. P.; Misirlioglu, I. B.; Nagarajan, V.; Ramesh, R. Appl. Phys. Lett. 2004, 85, 2044. [26] Lubk, A.; Rossell, M.; Seidel, J.; Chu, Y.-H.; Ramesh, R.; Hÿtch, M.; Snoeck, E. Nano Lett. 2013, 13, 1410-1415. [27] Seidel, J.; Martin, L. W.; He, Q.; Zhan, Q.; Chu, Y. H.; Rother, A.; Hawkridge, M. E.; Maksymovych, P.; Yu, P.; Gajek, M.; Balke, N.; Kalinin, S. V.; Gemming, S.; Wang, F.; Catalan, G.; Scott, J. F.; Spaldin, N. A.; Orenstein, J.; Ramesh, R. Nat. Mater. 2009, 8 (3), 229-234. [28] Maksymovych, P.; Morozovska, A. N.; Yu, P.; Eliseev, E. A.; Chu, Y.-H.; Ramesh, R.; Baddorf, A. P.; Kalinin, S. V. Nano Lett. 2011, 12 (1), 209-213. [29] Seidel, J.; Maksymovych, P.; Batra, Y.; Katan, A.; Yang, S.-Y.; He, Q.; Baddorf, A. P.; Kalinin, S. V.; Yang, C.-H.; Yang, J.-C. et al. Phys. Rev. Lett. 2010, 105, 197603. [30] Meier, D.; Seidel, J.; Cano, A.; Delaney, K.; Kumagai, Y.; Mostovoy, M.; Spaldin, N. A.; Ramesh, R.; Fiebig, M. Nat. Mater. 2012, 11, 284-288. [31] Salje, E. K. H. New J. Phys. 2016, 18, 051001. [32] Tokumoto, Y.; Amma, S.; Shibata, N.; Mizoguchi, T.; Edagawa, K.; Yamamoto, T.; Ikuhara, Y. J. Appl. Phys. 2009, 106, 124307. [33] Nakamura, A.; Matsunaga, K.; Tohma, J.; Yamamoto, T.; Ikuhara, Y. Nature Mater. 2003, 2, 453-456. 18 ACS Paragon Plus Environment
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[34] Fukuyama, H. J. Phys. Soc. Jpn 1982, 51, 1709-1710. [35] Hirth, J. P.; Lothe, J. Theory of Dislocations 2nd edition; Wiley, New York, 1982. [36] Jia, C. L.; Thust, A.; Urban, K. Phys. Rev. Lett. 2005, 95, 225506. [37] Hutson, A. R. Phys. Rev. Lett. 1981, 46, 1159. [38] Ikuhara, Y. Prog. Mater. Sci. 2009, 54, 770-791. [39] Sugiyama, I.; Shibata, N.; Wang, Z. C.; Kobayashi, S.; Yamamoto, T.; Ikuhara, Y. Nat. Nanotechnol. 2013, 8, 266-270. [40] Kresse, G.; Hafner, J. Phys. Rev. B: Condens. Matter Mater. Phys. 1993, 47, 558. [41] Kresse, G.; Furthmller, J. Phys. Rev. B: Condens. Matter Mater. Phys. 1996, 54, 11169. [42] Amadon, B.; Applencout, T. Phys. Rev. B, 2014, 89 125110. [43] Mitra, C.; Lin, C. Posadas, A. B.; Demkov, A. A. Phys. Rev. B 2014, 90, 125130. [44] Erhart, P.; Klein, A.; Åberg, D. Sadigh, B. Phys. Rev. B 2014, 90, 035204. [45] Yadav, S.; Ramprasad, R.; Misra, A.; Liu, X.-Y. Acta Mater. 2014, 74, 268-277. [46] Buban, J. P.; Chi, M. F.; Masiel, D. J.; Bradley, J. P.; Jiang, B.; Stahlberg, H.; Browning, N. D.; J. Mater. Res. 2009, 24, 2191-2199. [47] Choi, S. Y.; Kim, S. D.; Choi, M.; Lee, H. S.; Ryu, J.; Shibata, N.; Mizoguchi, T.; Tochigi, E.; Yamamoto, T.; Kang, S. J. L.; Ikuhara, Y. Nano Lett. 2015, 15, 4129-4134. [48] Du, H.; Jia, C.-L.; Houben, L.; Metlenko, V.; De Souza, R. A.; Waser, R.; Mayer, J. Acta Mater. 2015, 89, 344-351. [49] Szot, K.; Speier, W.; Bihlmayer, G.; Waser, R. Nat. Mater. 2006, 5, 312-320. [50] Waldow, S.P.; De Souza, R.A. ACS Appl. Mater. Interfaces 2016, 8, 12246-12256. [51] France, R. M.; McMahon, W. E.; Norman, A. G.; Geisz, J. F.; Romero, M. J. J. Appl. Phys. 2012, 112, 023520.
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Figure 1. Magnetic spin-density distributions in Ti-rich dislocations with spontaneous polarization along the (a) [001], (b) [100], (c) [010] and (d) [0-10] directions. The cross-sectional views along and perpendicular to the dislocations are shown in the upper and lower panels in each picture. The red arrows indicate the spontaneous polarization, and the yellow and green regions denote isosurfaces with magnetic spin densities of 0.04 and -0.04 µB/Å^3, respectively. 146x187mm (300 x 300 DPI)
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Figure 2. (a) Electronic band structures of Ti-rich dislocations with polarization along the [001] direction. The green line denotes the Fermi level. The red and blue lines indicate the majority- and minority-spin orbitals, respectively, the pink line denotes majority- and minority-spin states, and the black line is the degenerate state. (b)-(d) The squared wave functions of the in-gap states. The orange and light-blue colors indicate isosurfaces of majority- and minority-spin densities of 0.04Å-3, respectively. The squared wave function of the degenerate state (iv) is shown in Supporting Figure S2. 168x74mm (300 x 300 DPI)
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Figure 3. (a) Electronic band structures of Ti-rich dislocations with polarization along the [100] direction. The green line denotes the Fermi levels, the red and blue lines indicate the majority- and minority-spin orbitals, respectively. (b)-(i) The squared wave functions of the in-gap states. The orange and light-blue colors indicate isosurfaces of majority- and minority-spin densities of 0.04 Å^-3, respectively. 131x214mm (300 x 300 DPI)
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Figure 4. (a) Electronic band structures of Ti-rich dislocations with polarization along the [0-10] direction. The green line denotes the Fermi level, the pink line denotes majority- and minority-spin states, and the black line is the degenerate state. (b) The squared wave function of the in-gap state. The orange and lightblue colors indicate isosurfaces of majority- and minority-spin densities of 0.04 Å^-3, respectively. The squared wave functions of the degenerate states are shown in Supporting Figure S3. 138x116mm (300 x 300 DPI)
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Table-of-content Graphic 152x65mm (300 x 300 DPI)
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