­­ First-principles study for surface tension and depolarizing effect on ferroelectric properties of BaTiO₃ nanowires

CAI Meng-qiu(蔡孟秋), DU Yong(杜 勇), HUANG Bai-yun(黄伯云)

State Key Laboratory of Powder Metallurgy, Central South University, Changsha 410083, China

Received 10 August 2009; accepted 15 September 2009

Abstract: The spontaneous polarization in ferroelectric (FE) nanowires (NWs) can be considerably enhanced due to the nanosize confinement by the first-principles calculations. The spontaneous polarization in a fully-relaxed BaTiO₃ NW with 2.0 nm in diameter is 1.18 times higher than that of bulk counterpart. The ferroelectric properties of the wire are found to generally depend on its dimensions through effects of the surface tension and near-surface depolarizing effect. The surface tension seems to be crucial for the giant enhancement of spontaneous polarization.

Key words: ferroelectric nanowires; surface stress; depolarization; first-principles

1 Introduction

Low-dimensional ferroelectrics (FEs) have attracted intense attention due to their potential applications in leading toward miniaturized devices for nanoscaled actuators, nanoscaled sensors, nonlinear optics, and especially non-volatile ferroelectric random access memories with large storage density[1-3]. Much experimental effort has been made recently in preparing and understanding FE nanorings[4], nanowires (NWs)[5-7], and nanoparticles[8-9]. Theoretically, NAUMOV and FU[10-12] have used a first-principles- derived effective-Hamiltonian approach to investigate the ferroelectricity in zero-dimensional (0D) BaTiO₃ nanodots, Pb(Zr, Ti)O₃ nanodisks, nanorods and one-dimensional (1D) NWs, and revealed the vortex structure dipoles in nanodots, nanodisks, and nanorods.

It is well known that the ferroelectricity is generally caused by atomic off-center displacements which result from a delicate balance between long-range (LR) Coulomb interaction and short-range (SR) covalent interaction[13]. In 1D FE nanostructures, the LR interaction is truncated due to the lack of periodicity and the SR one is significantly modified near the surface boundary. Thus, both interactions and the corresponding balance are changed with regard to the bulk. Therefore, the depolarizing effect would come into being because of the lack of periodicity and the modified surface boundary, which causes the spontaneous polarization decreasing  [2, 12]. On the other hand, the surface compressive stress caused by the surface restructuring of 1D nanostructures produces an effective tension in the length direction[14], which leads to a large off-center displacements. Importantly, the large off-center displacements would arise the spontaneous polarization. NAUMOV and FU[10] reported theoretically the scaling law of the spontaneous polarization in Pb(Zr, Ti)O₃ NWs based on the depolarizing effect. However, there have not been any theoretical and experimental studies of spontaneous polarization simultaneously involved in both the depolarizing effect and surface stress in 1D FE NWs.

In this work, the first-principles calculations were carried out to pursue the spontaneous polarization in the fully-relaxed 1D BaTiO₃ FE NWs, taking the near-surface depolarizing effect associated with finite thickness of wires in the radial direction and the large surface tension induced by the 1D confinement into account.

2 Model and calculating method

The BaTiO₃ NWs are extended to infinity along the longitudinal z axis through the periodic repetition of supercell. Wires are chosen to be of cylindrical shape with their x, y, and z axes along the pseudocubic [100], [010], and [001] directions of the perovskite structure, respectively. The schematic illustrations of NWs are shown in Fig.1. The polarity axis lies along the NWs length. The vacuum space with about 8 nm in thickness surrounds the nanowires, which ensures that the NWs in neighboring supercells do not interact with each other and cause the surface-induced atomic relaxation and the cell-shape changes. Thus, these treatments would affect both the local modes and the local inhomogeneous strains. We perform the simulations for a variety of diameters d, varying from 1.4 to 4.2 nm, while the z-axis periodical length h is chosen to be the bulk lattice constant. Other internal parameters used here are those of bulk BaTiO₃. First-principles density functional theory (DFT) with local density approximation is performed using the Vienna ab initio simulation package (VASP), within the projector augmented wave (PAW) pseudo potential method[15]. These calculations are performed with a 1×1×5 Monkhorst-Pack k-point mesh centered at Γ and a 460 eV plane-wave cutoff, both of which result in good convergence of the computed ground-state properties. The forces on the atoms are calculated using the Hellmann-Feynman theorem[15], which is used to perform a conjugate gradient relaxation, and shown to be feasible in our previous reports[16-18]. Structural optimizations are continued until the forces on the atoms converge to less than 1 meV/?. Moreover, the electronic polarization P via P= of the BaTiO₃ NWs is performed by the Berry phase method[19], where  is the Born effective charge of the local mode; and is the local model of the cell to describe the ferroelectric instability. The method is considered to be a good description for the electronic polarization of an insulating groundstate system[17]. It is noted that, in order to keep to be insulating and close to circular, the configurations of the nanowires are stoichiometric with the BaO and TiO₂ surface-terminations at the same time. It is not the only BaO- or TiO₂-surface-terminations. Of course, the whole structure would be kept symmetrical corresponding to the central z axis. In our simulated structures, the BaTiO₃ nanowires are insulating with stoichiometric configuration. As shown in Fig.2, we plot the orbital-resolved densities of states (DOSs) of BaTiO₃ nanowires(NWs) with diameter d=2.0 nm. There is a band-gap about 1.7 eV between the O 2p and Ti 3d  states. It is shown that the simulated wires are  insulators.

Fig.1 Schematic illustrations of 1D BaTiO₃ NWs viewed from [100] (a) and [001] (b) directions, and internal ferroelectric perovskite structure (c)

Fig.2 Orbital-resolved densities of states (DOSs) of BaTiO₃ nanowires(NWs) with diameter d=2.0 nm(Fermi energy is set to zero)

3 Results and discussion

Fig.3 shows the longitudinal z-axis FE spontaneous polarization as a function of the varying diameter of free-standing NWs under the open-boundary condition. In order to display the different ferroelectric order of the BaTiO₃ NWs, the ratio of P_nanowires/P_bulk is also presented. The spontaneous polarization is different along the radial direction, and thus the calculated spontaneous polarization is an average value. Remarkably, we can see the giant enhancement of spontaneous polarization in the BaTiO₃ NWs with 2.0 nm in diameter compared with that of the bulk counterpart. In detail, there is a critical size of about 1.4 nm in the spontaneous polarization of BaTiO₃ NWs, which further confirms the existence of the FE phase transition in 1D NWs when their diameter is less than a definite size[10]. Interestingly, a maximal value of spontaneous polarization that is about 1.18 times higher than that of bulk can be obtained for the BaTiO₃ NWs with 2.0 nm in diameter. Then, the spontaneous polarization gradually decreases with increasing the diameter of BaTiO₃ NWs and trends to the bulk value. However, the spontaneous polarization drops quickly when the diameter of NWs is less than 2.0 nm. It has been recognized that, in general, the near-surface depolarizing effects associated with finite thickness of wires are known to be strong in low-dimensional structures to cause the considerably smaller spontaneous polarization than that of bulk[2, 10, 12], which implies that FE nanostructures are not expected to be promising in the applications of nanodevices. Then, our studies show that the strong enhancement of spontaneous polarization can be achieved in 1D FE NWs, which actually opens a door toward the application of FE nanostructures.

Fig.3 Longitudinal z-axis spontaneous polarization as function of NWs diameter under open-boundary condition (Dot line: value of bulk polarization of BaTiO₃)

At the atomistic scale, the spontaneous polarization is mainly caused by the displacement of Ti atoms deviating from the centre positions (the paraelectric phase) in the displacive FE BaTiO₃[13]. Fig.4(a) shows the average displacement of Ti Ti> atoms along the x, y, and z directions of the BaTiO₃ NWs. Surprisingly, there are the same trends of Ti> as the spontaneous polarization along its length axis with varying the NWs diameter. However, the average displacement of Ti> is zero in the x and y directions. Thus, the results are responsible for the zero spontaneous polarization in the perpendicular direction of the BaTiO₃ NWs, which is similar to the previous studies[10, 12]. Fig.4(b) shows the schematic illustration of the displacements of the symmetric four Ti atoms along the x and y directions for the 2.6 nm BaTiO₃ NWs. Although there are considerable displacements for every Ti atom in the x and y directions, the whole displacement vector is zero in the parallel-to-surface direction to cause the zero polarization. Thus, our calculations indicate that the FE ordering in 1D infinite BaTiO₃ NWs does not form any toroid moments, which is different from the phase transition of 0D nanoparticles[10-12].

Fig.4 Average displacement of Ti atoms along x, y, and z directions of BaTiO₃ NWs (a) and schematic illustration (b) of displacements of symmetric four Ti atoms along x and y directions for 2.6 nm BaTiO₃ NWs ( ,)

NAUMOV and FU[10] reported the spontaneous polarization of the infinite length Pb(Zr,Ti)O₃ NWs and the scaling law of the spontaneous polarization in the (001) direction (P=P_bulk-A/d) based on the depolarizing effects, in which the spontaneous polarization in NWs is less than that of bulk and decreases with reducing NWs diameter. To understand the difference between our results and NAUMOV’s studies, and the origin of the enhancement of the polarization along z, we plot the in-plane lattice constant of the nanowires as a function  of NW diameter in Fig.5, where the d₁ represents the unoptimized NW diameter, and d₂ represents the optimized NW diameter. Evidently, the surface compressive stress is pointed to be at the origin of the enhancement of the polarization along z. The surface stress of NWs can greatly enhance the spontaneous polarization by counteracting the depolarizing effects. It is noted that there is very large surface stress below the critical size. Because of the large depolarized effects corresponding to the surface-stress effects, the polarization of the NWs is zero under the critical sizes. In fact, the enhanced polarization by the stress has been confirmed in the FE superlattices[20].

Fig.5 (d₂-d₁)/d₁ function of BaTiO₃ NW diameter

4 Conclusions

1) Taking the near-surface depolarizing effect and the surface tension induced by 1D confinement into account, there is the giant enhancement of spontaneous polarization in the fully-relaxed 1D BaTiO₃ NWs using first-principles calculation.

2) The calculated results show the spontaneous polarization in the BaTiO₃ NWs with 2.0 nm in diameter is 1.18 times higher than that of the bulk counterpart.

3) The physical mechanisms of the unusual spontaneous polarization in 1D FE NWs are suggested to be the competition between near-surface depolarizing effect and the surface tension on the basis of ab initio calculations.

Acknowledgments

CAI Meng-qiu expresses thanks to YANG Guo-wei, HU Wang-yu and WANG Ling-ling for helps and the Supercomputer Center of Hunan University and the Supercomputer Center of Shanghai for computations.

References

[1] SCOTT J F, de ARAUJO C A P. Ferroelectric memories [J]. Science, 1989, 246: 1400-1405.

[2] JUNQUERA J, GHOSEZ P. Critical thickness for ferroelectricity in perovskite ultrathin films [J]. Nature, 2003, 422: 506-509.

[3] SCHMISING C V K, BARGHEER M. Coupled ultrafast lattice and polarization dynamics in ferroelectric nanolayers [J]. Physical Review Letters, 2007, 98: 257601.

[4] ZHU X H, EVANS P B. Perovskite lead zirconium titanate nanorings: Towards nanoscale ferroelectric “solenoids”? [J]. Applied Physics Letters, 2006, 89: 122913-122915.

[5] SPANIER J E, KOLPAK A M. Ferroelectric phase transition in individual single-crystalline BaTiO₃ nanowires [J]. Nano Letters, 2006, 6: 735-739.

[6] NIKIFOROV M, LIU H Q, CRAIGHEAD H, BONNELL D. Polarization controlled transport in PANI-BaTiO₃ nanofibers [J]. Nano Letters, 2006, 6: 896-900.

[7] WANG Z Y, YU M F. Ferroelectric and piezoelectric behaviors of individual single crystalline BaTiO₃ nanowire under direct axial electric biasing [J]. Applied Physics Letters, 2006, 89: 082903.

[8] BANSAL V, PODDAR P. Room-temperature biosynthesis of ferroelectric barium titanate nanoparticles [J]. Journal of American Chemical Society, 2006, 128: 11958-11963.

[9] NURAJE N, SU K. Room temperature synthesis of ferroelectric barium titanate nanoparticles using peptide nanorings as templates [J]. Advanced Materials, 2006, 18: 807-811.

[10] NAUMOV I I, FU H X. Spontaneous polarization in one-dimensional Pb(ZrTi)O₃ nanowires [J]. Physical Review Letters, 2005, 95: 247602.

[11] PROSANDEEV S, BELLAICHE L. Properties of ferroelectric nanodots embedded in a polarizable medium: Atomistic simulations [J]. Physical Review Letters, 2006, 97: 167601.

[12] NAUMOV I I, BELLAICHE L, FU H X. Unusual phase transitions in ferroelectric nanodisks and nanorods [J]. Nature, 2004, 432: 737-740.

[13] COHEN R E. Origin of ferroelectricity in perovskite oxides [J]. Nature, 1992, 358: 136-138.

[14] MOROZOVSKA A N, GLINCHUK M D, ELISEEV E A. Phase transitions induced by confinement of ferroic nanoparticles [J]. Physics Review B, 2007, 76: 014102.

[15] KRESSE G, FURTHM?LLER J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set [J]. Physics Review B, 1994, 54: 11169-11186.

[16] CAI M Q, ZHANG Y J, YIN Z, ZHANG M S. First-principles study of structural and electronic properties of BaTiO₃(001) oxygen- vacancy surfaces [J]. Physics Review B, 2005, 72: 075406.

[17] CAI M Q, YANG G W. First-principles study of electronic and magnetic properties of BiNiO₃ [J]. Applied Physics Letters, 2007, 90: 242911.

[18] CAI M Q, YANG G W. First-principles study of pressure-induced metal-insulator transition in BiNiO₃ [J]. Applied Physics Letters, 2007, 91: 101901.

[19] VANDERBILT D, KING-SMITH R D. Electric polarization as a bulk quantity and its relation to surface charge [J]. Physics Review B, 1994, 48: 4442.

[20] LEE H N, CHRISTEN H M. Strong polarization enhancement in asymmetric three-component ferroelectric superlattices [J]. Nature, 2005, 433: 395-397.

Foundation item: Project(50802026) supported by the National Natural Science Foundation of China; Project(20090450187) supported by China Postdoctoral Science Foundation; Project(jj094001) supported by the Natural Science Foundation of Hunan Province, China; Project supported by the Postdoctoral Science Foundation of Central South University, China

Corresponding author: DU Yong; Tel: +86-731-88836213; E-mail: yong-du@mail.csu.edu.cn

DOI: 10.1016/S1003-6326(09)60081-9

(Edited by YANG Hua)