To be submitted to Appl. Phys. Lett. Domain Dynamics in Piezoresponse Force Microscopy: Quantitative Deconvolution. and Hysteresis Loop Fine Structure

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1 To be submitted to Appl. Phys. Lett. Domain Dynamics in Piezoresponse Force Microscopy: Quantitative Deconvolution and Hysteresis Loop Fine tructure Igor Bdikin and Andrei Kholkin Department of Ceramics and Glass Engineering, CICECO, University of Aveiro, 8-9 Aveiro, Portugal, Anna N. Morozovska * and ergei V. vechnikov V. Lashkaryov Institute of emiconductor Physics, National Academy of cience of Ukraine, 4, Prospect, Nauki, 8 Kiev, Ukraine eung-hyun Kim INOTEK Inc., Gyeonggi Technopark, Ansan, Gyeonggi 46-9,. Korea ergei V. Kalinin The Center for Nanoscale Materials ciences and Materials cience and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 78, U..A. * Corresponding author, morozo@i.com.ua Corresponding author, sergei@ornl.gov

2 Domain dynamics in the Piezoresponse Force pectroscopy (PF) experiment is studied using the combination of local hysteresis loop acquisition with simultaneous domain imaging. The analytical theory for PF signal from domain of arbitrary cross-section is developed and used for the analysis of experimental data on Pb(Zr,Ti)O polycrystalline films. The results suggest formation of oblate domain at early stage of the domain nucleation and growth, consistent with icient screening of depolarization field within the material. The fine structure of the hysteresis loop is shown to be related to the observed jumps in the domain geometry during domain wall propagation (nanoscale Barkhausen jumps), indicative of strong domain-defect interactions.

3 Local polarization switching in ferroelectric materials by a biased Piezoresponse Force Microscopy (PFM) tip has emerged as a perspective approach for ultra high density data storage,, ferroelectric lithography, and nanostructure fabrication. 4,5 ubsequent imaging of the switched domain provides direct insight into domain growth mechanism 6 and relaxation kinetics 7, from which the ect of macroscopic defects, 8 microscopic disorder, 9 and surface state on domain wall dynamics and pinning can be established. The primary limitation of this approach is the (a) uncontrollable ect of surface charge migration, inevitable in ambient conditions on observed growth and relaxation kinetics, and (b) extremely large times (~s hours) required for studies even at a single location. ingle-point Piezoresponse Force pectroscopy (PF) allows probing the domain growth process by detecting the changes in local electromechanical response due to nucleation and growth of domain below the tip. Increase of acquisition speed to ~. s/loop enabled witching-pectroscopy PFM, in which PF loops are collected at each point in the image to yield D maps of parameters such as imprint, nucleation biases, or work of switching. The development of PF and PFM necessitated the quantitative interpretation of the spectroscopic data, establishing the relationship between the electromechanical response and the geometric parameters of the domain formed below the tip. Recently, the theoretical framework for the interpretation of the PF data for cylindrical domain geometry was established in Ref.[]. Here, we report the results of the combined spectroscopic-imaging experiments and correlate the changes in domain geometry and the loop shape. The PFM spectroscopy and domain imaging at each step of spectroscopic process were preformed as described elsewhere. 4 The films of Pb(Zr. Ti.7 )O (PZT) used in this study were produced by sol-gel method and commercialized by INOTEK 5. The thickness

4 of the films was about 4 µm and grain size varied in the range -5 nm. The films had columnar structure with random orientation of the grains. ufficiently big grains of the size about -4 nm with strong initial piezoelectric contrast were used for the local measurements. The representative local PF hysteresis loop, corresponding evolution of domain sizes, and in-situ domain images are shown in Fig.. The data were taken with poling pulse duration s and imaging of the resulting domain structure immediately after poling. Note that a sharp change in response in PF hysteresis loop at -5 V corresponds to domain nucleation. The subsequent lateral growth of the domain corresponds to the increase in PFM signal, which finally saturates at about V. (a) 6nm 6nm 6nm d (a.u.) Bias V (V) 6nm Domain diameter ( µm ),6,5,4,,,, Negative domain (b) Positive domain U dc ( Volts ) Fig.. (a) Effective piezoresponse d via applied bias V: filled squires is experimental data for examined Pb(Zr. Ti.7 )O 4µm film 6 ; solid and dashed curves are theoretical fitting based on Eqs.(-) and Eqs.(A.4), respectively. Domain radius r s for both curves was determined from PFM images and shown in part (b). olid curve in Fig. (a) corresponds to the size of nascent domain (dark and bright), while dashed curve is calculated for the difference between the fully switched area for bright domain and nascent dark domain. 4

5 To correlate local electromechanical response measured in a PF experiment to the size of ferroelectric domain formed below the tip, we utilized the decoupling approximation 7,8 combined with Pade approximation method. Here, we develop the generalized analytical relationship between the PFM signal and ective domain size R (V), as: d d πd 8R ( V ) d π d 8R ( V ) +. () 4 4 πd + 8R ( V ) 4 πd + 8R ( V ) ( R V )) d + ( + 4ν) 5 ( Equation () is valid for dielectrically isotropic materials with γ 9, are strain piezoelectric coicients, and ν.5 is the Poisson ratio. The bias-dependent ective domain size R(V) was derived as ( π 6) + l π 6 d nm lr R ( V ), () r ( ) where l(v) is the ective domain length, radius π r ( V ) dα r( α), α is the angle, as π defined in Fig. (a). Below we compare the hysteresis loop measurements with the in-situ imaging studies. For domains of irregular shape, ective radius, r s, was determined experimentally as a square root of domain cross-section determined from PFM images. Additionally, we introduce the domain center shift with respect to the probe apex axis, a(v), that can be significant at early stages of domain growth process due to the presence of local defects and inhomogeneities. Here, we regard a(v) as a fitting parameter. For high aspect ratio (r s /l<<) tip-induced domains R s (V) r s (V) and a(v), so the radius r s can be directly extracted from ective piezoresponse d ( ) deconvolution by means of Eqs.(-). For small shifts a<<rs, V 5

6 it is possible to estimate domain length using both ective radius r s (V) extracted from PFM images and R s (V) extracted from piezoresponse data d ( V ) as following: l( V ) 6 R r. () π r R Domain parameters determination for PZT films is shown in Fig.. The accuracy of the proposed deconvolution method and results presented in Fig. can be estimated from Fig.(a). Note that the solid and dashed curves in Fig.(a) do not coincide with experimental symbols at negative bias, indicating that the contribution of small dark domain into piezoresponse was underestimated (actually we have in-situ PFM data for r s of bright domain only). At the same time, dotted and solid curve coincide almost everywhere, indicative of relatively minor contribution of domain center displacement, a(v ), to measured signal. 6

7 x Length (µm) Defect (a) r(α). - - r α R x / l (c) a θ a x Probe Domain of irregular shape a= ize (µm) Eq.(A.4).4. Eq.(-) Bias V (V) (b) r Bias V (V). (d) R a r / /l r 9/ /l - 5 Bias V (V) Fig.. (a) Problem geometry. (b) Effective domain size via applied bias: filled squires are R values deconvoluted from d using Eq.(); empty squires are average radius r values obtained from in-situ PFM data shown in Fig. b; dotted curve is domain shift a deconvoluted from d using in-situ r values in Eqs.(A.4). (c) Effective domain length via applied bias: filled triangles were deconvoluted from d using in-situ r values in Eqs.(A.4) and a ; empty squares are deconvoluted from d using in-situ r values in Eqs.(-) and a=. (d) The ratios / r l (invariant for prolate domains in Molotskii model ) and 9 / r l (almost constant in our case) calculated from r values in Eqs.(A.4) (labels near the curves, arbitrary units). For Pb(Zr. Ti.7 )O constants d =-.4, d =6. pm/v 5 γ and d=4nm; vertical offset caused by electrostatic forces is -6. a.u. The deconvolution results are independent of d 5 within the range.5 d d 5.5d. 7

8 The fitting for strongly anisotropic domains, r s /l<<, is shown to lead to unphysically large tip sizes, d nm, or lateral domain shifts, a nm, inconsistent with d 5-4nm as determined from observed ferroelectric domain wall width. To account for this discrepancy, we treated domain penetration length as a fitting parameter, and deconvolution results are shown in Fig. (c-d). The fitting of piezoresponse loop and contrast of PFM images at small voltages V suggest that the domains are oblate at nucleation and initial growth stages. This domain geometry necessitates icient depolarization field screening in bulk of the sample, since in the rigid dielectric material depolarization field favors needle-like domain (depolarization energy is ~/l ). This behavior is possible when screening charges are captured by the moving domain wall. For d=4nm, maximal domain radius r s =nm~d at bias V can be explained by ective mechanism of depolarization field surface screening in examined PZT film. The careful inspection of domain evolution in Fig. illustrates that often domain growth proceeds through the rapid jumps of domain walls, resulting in irregular domain shapes. Interestingly, these jumps can be associated with the fine structures of the local hysteresis loop, as illustrated in Fig.. Almost all hysteretic curves taken at higher magnification demonstrate visible kinks under increasing voltages. These kinks are reproducible upon repeating voltage excursions and thus could be attributed to the defects and associated pinning of domain walls. These features can be naturally explained by the presence of local defects and long range electroelastic fields as it was theoretically predicted in the past (see, e.g., Ref. 4 ). Based on result shown in Fig., each reproducible feature on the hysteresis curve could be related to a separate metastable domain state pinned by the nearest 8

9 defect site. Future improvement of the PF technique is required to relate each kink in the hysteresis curve to an individual jump of the domain wall (nanoscale Barkhausen jump). d ( a.u. ) - - 6nm U dc ( V ) 6nm Fig.. Example of the proposed relationship between the fine structure of the piezoresponse hysteresis loops and jumps of domain walls observed in the investigated PZT films under increasing bias. To summarize, we have analyzed hysteresis loop formation in Piezoresponse Force Microscopy. Direct comparison of the induced domain size and the PF signal deconvoluted using self-consistent probe calibration has demonstrated that domain shape deviates significantly from simple needle-like geometry. In fact, the oblate domains are formed at nucleation and early growth stages. As a consequence, a new invariant 9 / r l was introduced for the description of the domain growth in polycrystalline PZT films. The fine structure features on piezoresponse spectra were demonstrated and tentatively related to the nanoscale jumps in domain geometry due to domain-defect interactions. Thus, high-resolution PFM 9

10 spectroscopy offers the pathway to study the nanoscale mechanism of defect-mediated domain nucleation and growth. Authors gratefully acknowledge multiple discussions and technical support of Dr. E. A. Eliseev. Research was sponsored in part (VK) by the Division of Materials ciences and Engineering, Office of Basic Energy ciences, U.. Department of Energy, under contract DE-AC5-OR75 with Oak Ridge National Laboratory, managed and operated by UT- Battelle, LLC. User proposal CNM7-65 (ORNL), FCT project PTDC/FI/844/6 and FAME Nework of Excellence (NMP-CT-4-559) are also acknowledged.

11 upplementary ) Appendix A Measured in a PF experiment is the electromechanical response related to the size of ferroelectric domain formed below the tip. Hence, to calculate the shape of the PFM hysteresis loop, the electromechanical response change induced by the domain is required. Within the framework of linearized theory by Felten et al, 7 the surface displacement vector ui ( x) at position x is u G ( x, ξ ) ij p i( x ) = dξ dξ dξ Ek ( ξ ) d kmn ( ξ ) c nmjl (A.) ξl where ξ is the coordinate system related to the material, ( x,ξ ) G is Green tensor in isotropic elastic approximation 5 (anisotropy of elastic properties is assumed to be much smaller then ij that of dielectric and piezoelectric properties), d nmp are strain piezoelectric coicients distribution, c nmjl are elastic stiffness and the Einstein summation convention is used. The electric field E ( ρ ) is created by the biased probe. Using an ective p k, x) = ϕ p ( ρ, x x k point charge approximation, the probe is represented by a single charge Q located at distance, d, from a sample surface. The potential ϕ p at x has the form: V d ϕ p( ρ, x). (A.) ρ + ( x γ + d ) Here x = ρ and is the radial and vertical coordinate respectively, V is the bias + x x applied to the probe, γ = ε ε is the dielectric anisotropy factor; d = R π for a flattened tip represented by a disk in contact.

12 Integration of Eq. (A.) for x = yields the expression for ective vertical piezoresponse, d u V, as = ( V ) d ( f w ( V )) + d ( f w ( V )) + d ( f w ( V )) d 5 =, (A.) The functions f = ( ( + γ) ν + ) ( + γ), f = γ ( + ), = ( + γ)( + γ) γ f define the electromechanical response in the initial and final states of switching process; 6 ν is the Poisson ratio. The bias dependence of PF signal, w i, is determined by the domain sizes and shape. Domain shape affected by defects is approximated by cross-section with radius vector r(α) and length l [see Fig.(a)]. For this case, wi components are 6 : w ( ) [ r( l] = dα dθ cos θ ( + ν) π π π cosθ R R G ( θ, r( l) ( θ, r( l), (A.4a) θ R ( θ, r( l) ( θ, r( l) π π γ d + cot w [ ] α θ r( l = d d cos θsin θ, (A.4b) π RG w [ r( l] = π π dα π dθcos R θ R G ( θ, r( l) ( θ, r( l). (A.4c) where the shape factors are introduced as ( θ, r, l) = rl tan θ r + l tan θ R ( ) + γ R ( θ, r l) ( θ, r, l) = γ d + cot θ R ( θ, r, l) G, Pade approximations theory, Eqs.(A.4) can be simplified as: w ( cos θ ( + ν) ) π cosθ [ r( l] θ α π d R ( γ d + cotθ R) + γ R R and. Using Lagrange mean point theorem, and [ r( )], (A.5a) π γ d + cot θ R [ ] [ r( α) ] w r( l dθcos θsin θ π, (A.5b) ( ) γ d + cot θ R + γ R

13 π dθcos θ R [ r( α) ] ( γ d + cotθ R ) + γ w [ r( l] π. (A.5c) R Where π * dα rl tan θi R i[ r( α) ] (A.5d) * r + l tan θ i and * θ i is mean point. Further interpolation by Pade at a=, γ, r(α) const and arbitrary l (or alternatively r<<l and arbitrary r(α)) leads to for i=,, where ( π 6) l r Ri (A.5e) r + ( π ) l 6 π r dα r( α). (A.5f) π This approximation is good (less than -5% discrepancy) for d contribution at arbitrary r/l ratio, moderate for d (about 5% discrepancy) and poor (more than 5% discrepancy) for d 5 for r>>l, while d 5 contribution is negligibly small at r>>l. Then direct integration on polar angle θ leads to expression: d d πd 8R ( V ) d π d 8R ( V ) + (A.6a) 4 4 πd + 8R ( V ) 4 πd + 8R ( V ) ( R V )) d + ( + 4ν) 5 ( Under the condition <a<<r and r <<l one obtains using concept of ective piezoresponse volume that and therefore: d ( ) ( r a) d ( r a) + d ( r + a) d ( r a),, (A.6b) 4r r a

14 ( πd + 4r ) a + r ( ) πd + 8r ( πd + 4r ) a πd ( π + ) r d 8r * 6r d + 8πd 4 ( ) πd + 8r d r, a (A.7) d 5 + 6r πd + 8r The case a~r should fitted numerically on the base of exact expressions (A.4). Appendix B Pt Pt Intensity ( a.u. ) PZT PZT PZT PZT + PZT PZT + PZT PZT + PZT PZT PZT + PZT Θ ( o ) Fig. X-ray diffraction pattern of Pb(Zr -x Ti x )O, x =.7, thickness 4 µm. This is no textured film. All orientations are present in the film. Data on this film Pb(Zr -x Ti x )O x =.7 (INOTEK): Thickness 4 µm Orientation Dielectric constant Random (XRD plots available) ~45 (poled) 4

15 ~6 (unpoled) Remanent polarization 5 µc/cm Coercive field d (clamped) e (wafer texture technique) 75 kv/cm 5 pm/v - C/m Assuming that values of piezoelectric modulus from the table are ective values, related to intrinsic ones as d film E s film c = d d, e = e e (see e.g. Ref. 7 ) and using E s + c E E E s elastic compliances from Ref. [ 8 ] s =8.5 and s =-.8 - Pa -, we obtained d =-.4, and d =6. pm/v. The deconvolution was performed for d 5 =.5d for bulk and d 5 =.5d for ceramics, and the results are shown to be weakly dependent on d 5. 5

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