This article was downloaded by: [Indian Institute of Technology - Delhi] On: 08 March 2013, At: 05:10 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Ferroelectrics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gfer20
Multiferroic Properties of La-Rich BiFeO3-PbTiO3 Solid Solutions a
Anupinder Singh & Ratnamala Chatterjee
a
a
Magnetics and Advanced Ceramic Laboratory, Department of Physics, Indian Institute of Technology Delhi, New Delhi, 110016, India b
Department of Physics, Guru Nanak Dev University, Amritsar, 143005, India Version of record first published: 12 Sep 2012.
To cite this article: Anupinder Singh & Ratnamala Chatterjee (2012): Multiferroic Properties of LaRich BiFeO3-PbTiO3 Solid Solutions, Ferroelectrics, 433:1, 180-189 To link to this article: http://dx.doi.org/10.1080/00150193.2012.722462
PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.
Ferroelectrics, 433:180–189, 2012 Copyright © Taylor & Francis Group, LLC ISSN: 0015-0193 print / 1563-5112 online DOI: 10.1080/00150193.2012.722462
Multiferroic Properties of La-Rich BiFeO3 -PbTiO3 Solid Solutions
Downloaded by [Indian Institute of Technology - Delhi] at 05:10 08 March 2013
ANUPINDER SINGH∗∗ AND RATNAMALA CHATTERJEE∗ Magnetics and Advanced Ceramic Laboratory, Department of Physics, Indian Institute of Technology Delhi, New Delhi, India-110016 La rich (BF0.50 –LF0.50 )1-x –(PT)x (BLF-PT) (where x = 0.34, 0.40, 0.50 and 0.60) has been investigated. Solid solutions were prepared using conventional solid-state reaction route. Solid solutions x ≤ 0.50 exhibit rhombohedral structure and x = 0.60 has tetragonal structutre. It is established that the MPB of these solid solutions is shifted to 40:60 with La doping. Enhancement in dielectric, ferroelectric, magnetic properties have been observed. La doping also observed to reduce the dielectric losses and leakage current.
1. Introduction Multiferroics represent an appealing class of functional materials that exhibit different ferroic interactions simultaneously. The coexistence of several interactions, particularly existence of a cross-coupling between the magnetic and electric interactions, termed as magnetoelectric (ME) coupling, brings about novel physical phenomena and offer possibilities for new device functions [1–7]. BiFeO3 (BFO) is one of the most extensively studied multiferroic material in recent years [5–13] and it is the only material known to exhibit magnetic order (TN = 643 K) and ferroelectric order (ferroelectric transition temperature Tc FE = 1103 K) at room temperature. At 300 K, It has a rhombohedral structure with space group R3c, which permits coupling between magnetism and ferroelectricity. It shows G-Type antiferromagnetic spin configuration along [001]h direction with a long range (periodicity ∼620 Å) cycloidal spin structure incommensurate with the lattice along [110]h direction of the hexagonal unit cell of the rhombohedral structure below magnetic ordering temperature. It also exhibits a very weak magnetoelectric coupling [14,15] due to cycloidal spin modulation causing suppressed linear magnetoelectric coupling. Theoretical calculations [16] predict that a large difference between the transition temperatures Tc FE and TN causes weak ME coupling in BFO. In addition to this, BFO also has other shortcomings like its semiconducting behavior at T ≥ 300 K, which does not allow electric poling and causes high dielectric losses in the sample at room temperature. Due to this, it is difficult to measure the ferroelectric properties of BFO at and above room temperatures and limits its applications in devices. Thus the main aim of researchers in the field has been (i) to improve the dielectric properties of BFO/modified BFO and (ii) to bring ferroelectric
Received in final form July 31, 2012. ∗ Corresponding author E-mail:
[email protected] ∗∗ Presently with Department of Physics, Guru Nanak Dev University, Amritsar 143005, India
180
Downloaded by [Indian Institute of Technology - Delhi] at 05:10 08 March 2013
La-Rich BiFeO3 -PbTiO3 Solid Solutions
181
transition (Tc FE) and magnetic transition (TN ) temperatures in the modified BFO compositions closer to each other; thus achieving an enhancement of ME coupling. The incommensurate cycloidal spin structure can be suppressed with application of (a) magnetic field, (b) strain and probably (c) chemical substitution too [1–8]. In 1996, Sosnowska et al. [17] demonstrated that the La doping at A-site in BiFeO3 destroys the space modulated spin structure (SMSS) and allows the measurement of linear ME effect. In 2006, Zhang et al. [18] reported enhancement of magnetic and ferroelectric properties for La substituted BFO. In addition to this, various workers have shown that La substitution decreases the ferroelectric transition temperature in materials belonging to perovskite family [19]. In 2005, Wang et al. [20] reported an enhancement of magnetoelectric properties in the (1-x) BiFeO3 -(x) PbTiO3 (BF-PT) solid solutions and also claimed destruction of space modulated spin structure in them, due to addition of PbTiO3 . Using detailed powder x-ray diffraction studies, Zhu et al. [21] proposed phase diagram for the solid solution of (1-x) BiFeO3 -(x) PbTiO3 (BF-PT). Their results reveal the existence of a morphotropic phase boundary (MPB) in this system, in which rhombohedral (x≤0.20), orthorhombic (0.20 ≤ x ≤ 0.28) and tetragonal (x ≥ 0.31) phases exist with a large tetragonality in the tetragonal phase region. Ranjan et al. [22] have shown that for BiFeO3 concentration x = 0.40 and 0.50 have the tetragonal phase at room temperature. The compositions x = 0.40 and 0.50 have competing tendencies with regard to their high temperature phase transition behavior. Further, Cheng et al. [23] reported that tetragonality of BF-PT solid solution decreases as concentration of the La3+ ions increases (for x < 0.3). They suggest (but without any experimental evidence), that further increase of La content in BF-PT matrix may stabilize rhombohedral (R3c) phase. They also reported improved magnetic properties in (BF1−x –LFx )–(PT) [BLF-PT] system at MPB (45:55). Recently, detailed studies on the effect of further increasing La content in (BF0.50 –LF0.50 )1-x –(PT)x (BLF-PT) (where x = 0.34, 0.40, 0.50 and 0.60) has been investigated by the authors. Our studies on La rich BLF-PT solid solution revealed that by optimizing the processing parameters, and the sample properties were significantly improved. The samples showed improved spontaneous polarization and magnetization. The dopants help in bringing the ferroelectric and magnetic transition temperatures closer to each other; a condition favored for technological applications of the materials.
2. Experimental Procedure Solid solutions were prepared using conventional solid-state reaction route. The raw materials of 99.9% purity (Aldrich) (Bi2 O3 , La2 O3 , PbO, Fe2 O3 , and TiO2 ) were weighed in stoichiometric proportion and the parameters were optimized (calcination temperature was 950◦ C) to obtain single phase solid solution. 2% Poly vinyl alcohol was mixed into the calcined powder and this powder was hydraulically pressed into small discs of 12 mm diameter and 1 mm thickness. The discs were sintered at 1100◦ C in same composition environment to reduce the lead losses. The x-ray diffraction pattern (using Cu Kα radiation (Philips Xpert Pro PW3040)) of the sintered sample was obtained at a very slow scan rate with step size 0.008◦ and scan time per step 50. Scanning electron micrographs were taken using ZEISS EVO-50 scanning electron microscopes. Dielectric properties of the samples were measured with impedance analyzer (HP 4192A) and a interfaced furnace with temperature tolerance 1K. Ferroelectric tester (Precision premier II Radiant technologies) was used to measure the polarization vs. electric field hysteresis loops at room temperature. The magnetic hysteresis loops were taken using SQUID (Quantum Design MPMS-7).
Downloaded by [Indian Institute of Technology - Delhi] at 05:10 08 March 2013
182
A. Singh and R. Chatterjee
Figure 1. X-ray diffraction of (sintered samples) (BF0.50 -LF0.50 )1-x -(PT)x (BF-LF-PT) solid solutions for (a) x = 0.34, (b) x = 0.40, (c) x = 0.50 and (d) x = 0.60 at room temperature.
3. Result and Discussions XRD patterns for all sintered samples are shown in Fig. 1 (a–d). These patterns were taken at very slow scan rate (step size 0.008◦ , scan time per step 50 s). No secondary phase could be detected. For compositions x ≤ 0.50 all x-ray reflections seem to be singlet and there are no clear splitting in the peaks. This indicates that these compositions have cubic structure. However BiFeO3 is a well known rhombohedral perovskite with space group symmetry R3c and PbTiO3 exhibits tetragonal structure (space group symmetry P4mm). In Fig. 2
Figure 2. X-ray reflection corresponding to (111) crystal plane in BF-LF-PT solid solutions for (a) x = 0.34, (b) x = 0.40, (c) x = 0.50 and (d) x = 0.60
La-Rich BiFeO3 -PbTiO3 Solid Solutions
183
the evolution of x-ray reflection at 2θ ∼39.5◦ is shown for all samples. No superlattice reflection were found around this peak. If the structure is considered to be rhombohedral then these superlattice peaks should be present in the XRD patterns for compositions x ≤ 0.50. The Goldsmith tolerance factor for all samples is calculated using following relation.
Downloaded by [Indian Institute of Technology - Delhi] at 05:10 08 March 2013
RA + RO t=√ 2(RB + RO )
(1)
The tolerance factor is observed to increase with increasing x, because the average value of radius of A-site increases due to increase in Pb2+ (1.49 Å) substitution in place of Bi3+ (1.36 Å). Similarly the average size of B-site decreases due to the replacement of Fe3+ (0.645 Å) with Ti4+ (0.605 Å). Goldsmith tolerance factor is an important parameter to predict the structure and property relations of complex perovskites. It is predicted that the tetragonal and cubic symmetries are typically found for t ≥ 1.0, while lower symmetries such as rhombohedral, monoclinic and orthorhombic are common for t < 1.0. For composition x = 0.60 the tolerance factor is 0.977 and has tetragonal structure. So it is clear that the tetragonal structure in these solid solutions is stable even for t < 1.0. Suchomel et al. [24] suggested that due to the presence of lone pair configuration in Bi3+ and Pb2+ ions the higher symmetries are stable at lower tolerance factors. The tolerance factor decreases as the value of x decreases. For x = 0.50 composition the tolerance factor is ∼ 0.964. At such lower values of tolerance factor the solid solutions are not expected to be stable at higher symmetries (tetragonal and cubic). The tolerance factor also indicates towards the possibility that the cubic crystal structure is not stable for this solid solution. Cheng et al. [23] suggested (but without any experimental evidence), that further increase of La (more than 30%) content in BF-PT matrix may stabilize rhombohedral (R3c) phase. The detailed structural studies are required to estimate the correct crystal structure for these samples. So we preformed neutron diffraction studied for x = 0.50 sample and done Rietveld refinement on the sample. The composition is well fitted with rhombohedral structure (R3c). These studies have already been reported by the authors [25]. It was also noticed that the full width at half maximum (FWHM) of the peaks decrease with increase in PbTiO3 (PT) up to x = 0.50, indicating largest crystallite size for this composition. Figure 3 depicts the evolution of x-ray reflection corresponding to (210) crystal plane. There are clear superlattice reflections, confirm tetragonal structure (P4mm space group symmetry). Morphotropic phase boundary (MPB) of pure BF-PT solid solutions is reported at 70:30 [21]. The change in structure for composition x = 0.60, indicates the MPB of La rich BF-PT solid solutions in between x = 0.50 and x = 0.60. The substitution of La at A-site shifts the MPB towards PbTiO3 side in these solid solutions. Scanning electron micrographs for all sintered samples are obtained at 20 kX magnification (See Fig. 4 (a–d)). Average grain size increases with increase in x values up to x = 0.50. For composition x = 0.60 the average grain size decreases. The largest average grain size and better grain homogeneity is obtained for sample x = 0.50. Grain growth in these solid solutions is due to the substitution of La at A-site to BF-PT solid solutions. It is reported in literature that the partial substitution of La at A-site helps in grain growth and densification of the solid solutions [26, 27]. For x = 0.60 the decrease in grain size may be due to structural change. This also supported from the FWHM data of XRD reflection peaks. The decrease in FWHM with increasing x (up to 0.50) is an indicative of large grains. Figure 5 (a) shows dielectric constant (εr ) and vs. temperature (T) plots (300 K ≤ T ≤ 600 K) at different frequencies (1 kHz ≤ T ≤ 100 kHz) for representative composition
Downloaded by [Indian Institute of Technology - Delhi] at 05:10 08 March 2013
184
A. Singh and R. Chatterjee
Figure 3. X-ray reflection corresponding to (210) crystal plane in BF-LF-PT solid solutions for (a) x = 0.34, (b) x = 0..40, (c) x = 0.50 and (d) x = 0.60.
Figure 4. Scanning electron micrographs of BF-LF-PT (sintered samples) solid solutions for (a) x = 0.34, (b) x = 0.40, (c) x = 0.50 and (d) x = 0.60 at 20 KX magnification. The scale for all micrographs is same as shown in (a).
Downloaded by [Indian Institute of Technology - Delhi] at 05:10 08 March 2013
La-Rich BiFeO3 -PbTiO3 Solid Solutions
185
Figure 5. (a) Dielectric constant (εr ) and (b) dielectric loss (Tan δ) vs. temperature (T) plots (300 K ≤ T ≤ 600 K) at different frequencies (1 kHz ≤ T ≤ 100 kHz) for representative composition x = 0.50 (Color figure available online).
x = 0.50. The important features observed from the temperature dependence of dielectric constant are as follows. (i) In these plots the dielectric anomalies are clearly evident in the temperature range 500–550 K. (ii) It is noticed that the maxima of these dielectric anomalies are frequency dependent. The maxima of dielectric constant shift towards the higher temperatures with increase in the frequency. (iii) These dielectric anomalies occur in a broad range of temperature. Usually frequency dependent broad anomalies are reported as diffused phase transitions (relaxor type) with Tm as the temperature of dielectric maxima (εr max ). For this
186
A. Singh and R. Chatterjee
Downloaded by [Indian Institute of Technology - Delhi] at 05:10 08 March 2013
composition the Tm ranging from 500–550 K are observed with large (ε m ≤ 20000 at 1 kHz) ε m values. (iv) It is observed that the value of Tm decreases with increase in x values up to x = 0.50. Tm further increases for x = 0.60 sample. The sample x = 0.50 has lowest Tm (∼ 523 K, at 100 kHz) in this series of samples. Large increase in the dielectric loss (Tan δ) for T> 523 K, observed for all frequencies (see Fig. 5(b)) is attributed to an increase in conductivity at these temperatures. The increase in dielectric losses in the vicinity of frequency dependent broad anomalies may be an artefact of increased conductivity. Dielectric constant (εr ) and dielectric loss (Tan δ) values measured (100 kHz frequency) for all samples at room temperature are given in the Table 1. Sample x = 0.50 show largest dielectric constant with minimal dielectric losses. Ferroelectric hysteresis loops measured at room temperature for x = 0.50 and x = 0.60 are shown in Fig. 6 (a) and (b). The ferroelectric loops for x = 0.34 and x = 0.40 samples show lossy elliptical loops. The shape of the P-E loop for x = 0.50 sample is also not classical ferroelectric type and is indicative of high leakage current in the sample. The applied electric field could not be raised beyond 35 kV/cm due to leakage current. The sample shows high remnant polarization ∼ 61.5 µC/cm2 and relatively low coercive field ∼11.8 kV/cm. Sample x = 0.60 sample shows almost saturated ferroelectric loop with remnant polarization ∼19 µC/cm2 and coercive field ∼33 kV/cm. Room temperature M – H plots for all solid solutions are shown in the Fig. 7 (a–d). The symmetric magnetic hysteresis loops, although non-saturating (up to 10 kOe), indicate magnetic ordering in these samples. Both BiFeO3 and LaFeO3 are antiferromagnetic and PbTiO3 is a known diamagnetic. It has been shown in literature that single crystals of BF
Table 1 Summery of measured properties in BF-LF-PT solid solutions Property
x = 0.34
Lattice parameter ‘a’ Lattice parameter ‘b’ Lattice parameter ‘c’ α β γ Goldsmith tolerance factor εr at 100 kHz at room temperature Dielectric loss (Tan δ) at 100 kHz at room temperature Remnant Polarization ‘Pr ’ Coercive field Ec (Ferroelectric) TC FE at 100 kHz Remnant Magnetization (Mr ) Coercive field (Hc )
5.581 Å 5.581 Å 13.646 Å ◦ 90 ◦ 90 ◦ 120 0.944 278
5.583 Å 5.583 Å 13.648 Å ◦ 90 ◦ 90 ◦ 120 0.952 847
5.587 Å 5.587 Å 13.651 ◦ 90 ◦ 90 ◦ 120 0.964 1341
3.925 Å 3.925 Å 3.984 Å ◦ 90 ◦ 90 ◦ 90 0.977 537
0.0686
2.067
0.0401
0.0335
Lossy loop Lossy loop 633 K 0.32 emu/g 816 Oe
Lossy loop Lossy loop 570 K 0.27 emu/g 1485 Oe
61 µC/cm2 11.8 kV/cm 523 K 0.22 emu/g 2482 Oe
19 µC/cm2 33 kV/cm 533 K 0.21 emu/g 2124 Oe
x = 0.40
x = 0.50
x = 0.60
Downloaded by [Indian Institute of Technology - Delhi] at 05:10 08 March 2013
La-Rich BiFeO3 -PbTiO3 Solid Solutions
187
Figure 6. Polarization (P) vs. electric field (E) hysteresis loops for samples (a) x = 0.50 and (b) x = 0.60 at room temperature.
and solid solutions of BF-LF-PT show non saturating M-H loops up to ∼70 kOe. This has been attributed to cycloidal spin in case of single crystal BF [28] and is explained as basic antiferromagnetic nature of solid solutions in case of polycrystalline BF-LF-PT [29]. Compared to the best literature values of Mr for BF-LF-PT solid solutions, our samples
Figure 7. Magnetization (M) vs. magnetic field (H) hysteresis loops for (a) x = 0.34, (b) x = 0.40, (c) x = 0.50 and (d) x = 0.60 at room temperature.
188
A. Singh and R. Chatterjee
show larger remnant magnetization. The largest Mr (0.32 emu/g) and smallest coercive field (HC = 0.8 kOe) were observed for the samples with least PbTiO3 content (x = 0.34). However, the sample with best dielectric property (x = 0.50) also showed respectable values of Mr (0.22 emu/g) and HC (2 kOe). An induced net magnetization in this sample was observed at much lower fields of ∼ 80 Oe. The values of Mr and Hc for all the samples are listed in the Table 1. Magnetoelectric coupling of the x = 0.50 solid solutions is already reported by the authors [30].
Downloaded by [Indian Institute of Technology - Delhi] at 05:10 08 March 2013
4. Conclusions In summary, we conclude that the BF-LF-PT solid solutions are successfully synthesized using solid state reaction route. It is established that the MPB of these solid solutions is shifted to 40:60 (70:30 for BF-PT solid solutions). Both x = 0.50 and x = 0.60 samples exhibit the coexistence of ferroelectric and magnetic orders. However, microstructural, dielectric, ferroelectric and magnetic properties clearly evidenced that the sample x = 0.50 is the best composition in this series.
Acknowledgments I would like to acknowledge DST funded facility SQUID (Physics Department) at IIT Delhi for magnetic measurements.
References 1. 2. 3. 4.
5. 6. 7. 8.
9. 10. 11. 12. 13. 14. 15. 16. 17.
M. Bibes and A. Barthelemy, Nature Mater. 7, 4215 (2008). T. Kimura, Y. Sekio, H. Nakamura, T. Siegrist, and A. P. Ramirez, Nature Mater. 7, 291 (2008). M. Mostovoy, Nature Mater. 7, 269 (2008). Y. H. Chu, L.W. Martin, M. B. Holcomb, M. Gajek, S.-J. Han, Q. He, N. balke, C. H. Yang, D. Lee, W. Hu, Q. Zhan, Pei Ling Yang, A. F. Rodriguez, A. Scholl, S. X. Wang, and R. Ramesh, Nature Mater. 7, 478 (2008). R. Ramesh and N. A. Spaldin, Nature Mater. 6, 21 (2007). S. W. Cheong and M. Mostovory, Nature Mater. 6, 13 (2007). W. Eerenstein, N. D. Mathur, and J. F. Scott, Nature (London) 442, 759 (2006). J. Wang, J. B. Neaton, H. Zheng,V. Nagarajan, S. B. Ogale, B. Liu, D. Viehland, V. Vaithyanathan, D. G. Schlom, U. V. Waghmare, N. A. Spaldin, K. M. Rabe, M. Wuttig, and R. Ramesh, Science 299, 1719 (2003). I. A. Kornev and L. Bellaiche, Phys. Rev. B 79, 100105 (R) (2009). M. K. Singh, R. S. Katiyar, W. Prellier, and J. F. Scott, J. Phys.: Condens. Matter 21, 042202 (2009). R. Haumont, I. A. Kornev, S. Lisenkov, L. Bellaiche, J. Kreisel, and B. Dkhil, Phys. Rev. B 78, 134108 (2008). D. Lebeugle, D. Colson, A. Forget, M. Viret, A. M. Bataille, and A. Gukasov, Phys. Rev. Lett. 100, 227602 (2008). P. Ravindran, R. Vidhya, A. Kjekshus, and H. Fjellvag, Phys. Rev. B 74, 224412 (2006) and references therein. C. T. Munoz, J. P. Rivera, A. Monnier, and H. Schmid, Japanese J. of Appl. Phys. 24, 1051 (1985). S. V. Surynarayana, Bull. Mater. Sci. 17, 1259 (1994). S. J. Gong, Q. Jiang, Phys. Lett. A 333, 124–131 (2004). I. Sosnowska, R. Przeniosto, P. Fischer, V. A. Murashov J. of Magnetism and Magnetic Materials 160, 384–385 (1996).
Downloaded by [Indian Institute of Technology - Delhi] at 05:10 08 March 2013
La-Rich BiFeO3 -PbTiO3 Solid Solutions
189
18. Shan-Tao Zhang, Yi Zhang, Ming-Hui Lu, Chao-Ling Du, Yan-Feng Chen, Zhi-Guo Liu, YongYuan Zhu, Nai-Ben Ming, and X. Q. Pan, Appl. Phys. Lett. 88, 162901 (2006). 19. K. Keizer, G. J. Lansink, and A. J. Burggraaf, J. Phys. Chem. Solids 39, 59 (1978). 20. N. Wang, J. Cheng, A. Pyatakov, A. K. Zvezdin, J. F. Li, L. E. Cross, and D. Viehland, Phys. Rev. B 72, 104434 (2005). 21. W. M. Zhu, H. Y. Guo, and Z. Ye, Phys. Rev. B 78, 014401 (2008). 22. R. Ranjan and K. A Raju, Phys. Rev. B 82, 054119 (2010). 23. J. Cheng, Sh. Yu, J. Chen, Zh. Meng, and L. E. Cross, Applied Physics Letters 89, 122911 (2006). 24. M. R. Suchomel and P. K. Davias, J. Appl. Phys. 96, 4405 (2004). 25. Anupinder Singh, Ratnamala Chatterjee, S. K. Mishra, P. S. R. Krishna, and S. L. Chaplot, J. Appl. Phys. 111, 014113 (2012) 26. G. L. Yuan and Siu Wing Or, J. M. Liu, and Z. G. Liu, Appl. Phys. Lett. 89, 052905 (2006). 27. Yi-Hsien Lee, Jenn-Ming Wu, and Chih-Huang Lai, Appl. Phys. Lett. 88, 042903 (2006). 28. N. Wang, J. Cheng, A. Pyatakov, A. K. Zvezdin, J. F. Li, L. E. Cross, and D. Viehland, Phys. Rev. B 72, 104434 (2005). 29. J. Cheng, Sh. Yu, J. Chen, Zh. Meng, and L. E. Cross, Appl. Phys. Lett. 89, 122911 (2006). 30. Anupinder Singh, Arti Gupta, and Ratnamala Chatterjee, Appl. Phys. Lett. 93, 022902 (2008).