The influence of size effects on thin films local piezoelectric response of thin films

We discuss the influence of size effects on the local piezoelectric response of thin films. In calculations of the electrostatic potential in the triple system “PFM probe tip – film – substrate,” the effective point charge model is used. The obtained expressions for the local piezoelectric respon...

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Опубліковано в: :	Semiconductor Physics Quantum Electronics & Optoelectronics
Дата:	2007
Автори:	Morozovska, A.N., Svechnikov, S.V.
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Мова:	Англійська
Опубліковано:	Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України 2007
Онлайн доступ:	https://nasplib.isofts.kiev.ua/handle/123456789/118331
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Цитувати:	The influence of size effects on thin films local piezoelectric response of thin films / A.N. Morozovska, S.V. Svechnikov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2007. — Т. 10, № 4. — С.36-41. — Бібліогр.: 17 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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author	Morozovska, A.N. Svechnikov, S.V.
author_facet	Morozovska, A.N. Svechnikov, S.V.
citation_txt	The influence of size effects on thin films local piezoelectric response of thin films / A.N. Morozovska, S.V. Svechnikov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2007. — Т. 10, № 4. — С.36-41. — Бібліогр.: 17 назв. — англ.
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container_title	Semiconductor Physics Quantum Electronics & Optoelectronics
description	We discuss the influence of size effects on the local piezoelectric response of thin films. In calculations of the electrostatic potential in the triple system “PFM probe tip – film – substrate,” the effective point charge model is used. The obtained expressions for the local piezoelectric response of a surface layer (film) capped are intended for the calculations of Piezoresponse Force Microscopy signals of thin polar films epitaxially grown on thick substrates. Theoretical predictions are in qualitative agreement with typical experimental results obtained for perovskite Pb(Zr, Ti)O₃ and multiferroic BiFeO₃ films.
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fulltext	Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 36-41. © 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 36 PACS 77.80.Fm, 77.65.-j, 68.37.-d The influence of size effects on local piezoelectric response of thin films A.N. Morozovska, S.V. Svechnikov V. Lashkaryov Institute of Semiconductor Physics, National Academy of Science of Ukraine 41, prospect Nauky, 03028 Kyiv, Ukraine, e-mail: morozo@i.com.ua Abstract. We discuss the influence of size effects on the local piezoelectric response of thin films. In calculations of the electrostatic potential in the triple system “PFM probe tip – film – substrate,” the effective point charge model is used. The obtained expressions for the local piezoelectric response of a surface layer (film) capped are intended for the calculations of Piezoresponse Force Microscopy signals of thin polar films epitaxially grown on thick substrates. Theoretical predictions are in qualitative agreement with typical experimental results obtained for perovskite Pb(Zr,Ti)O3 and multiferroic BiFeO3 films. Keywords: local piezoelectric response, size effects. Manuscript received 14.06.07; accepted for publication 19.12.07; published online 31.01.08. 1. Introduction It is well recognized that film piezoelectric and dielectric properties can be strongly modified as compared with those in bulk. For instance, the surfaces of non- piezoelectric materials can reveal a built-in dipole moment due to the inversion symmetry breaking and thus possess surface piezo- and flexoelectricity. In centrosymmetric materials, symmetry breaking at surfaces and interfaces can give rise to surface piezo- electric coupling even in non-polar materials [1]. Several theoretical groups analyzed the effect of properties of the films of polar materials and hysteresis loop features within the framework of the Landau-Ginzburg- Devonshire phenomenology. (see, e.g., Refs [2–9]). For the verification of theoretical models, elaboration of functional nanomaterials with predetermined properties, and application in various devices (such as ferroelectric micro- and nanocapacitors, sensors, actuators, etc.), the recognition and diagnostics of piezoelectric films and their multilayer structures are rather important. In thin films, the vertical shift of piezoelectric and ferroelectric hysteresis loops was interpreted in terms of a non-switchable layer by Saya et al. [2]. Alexe et al. [3] analyzed the hysteresis loop shape in ferroelectric nanocapacitors with top electrode and obtained the estimation for a switchable volume. A similar analysis was applied to ferroelectric nano- particles developed by the self-patterning method [4] by Ma [5]. In all cases, the results were interpreted in terms of 2~20 nm non-switchable layers, presumably at the ferroelectric-electrode interface. Recently, the development of Piezoresponse Force Microscopy (PFM) has allowed the 2D mapping of the switching behavior in piezoelectric thin films. However, the existing framework for the data analysis is invariably based on the 1D models suggested originally by Ganpule, thus ignoring the 3D geometry of the PFM problem. Recently, we have applied [6] the decoupled theory [7, 8] to derive analytical expressions for the PFM response on semi-infinite low-symmetry materials. However, to the best of our knowledge, there are no analytical results of calculations of the effective piezoelectric response of thin piezoelectric films epitaxially grown on thick substrates. Here, we analyze the local effective piezoresponse (PFM signal) of a piezoelectric film using the decoupling approximation. The obtained results can be successfully used for calculations of the effective piezoelectric response in the case of a ferroelectric or piezoelectric film capped on the nonpiezoelectric substrate with close elastic properties (like PbTiO3 or BaTiO3 on SrTiO3 or SrRuO3). 2. Problem 2.1. Decoupling approximation The PFM signal, e.g., the surface vertical displacement ( )yx,3u at a point x induced by the tip at the position ( )21, yy=y , is given by Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 36-41. © 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 37 ( ) ( ) ( ) ( ).,, ,, , 32211 322113 0 3213 ξξ+ξ+ξ× × ξ∂ ξξ−ξ−∂ ξξξ= ∫∫∫ ∞∞ ∞− ∞ ∞− yydcE xxG dddu lnmkjmnl k f jyx (1) Here, the coordinate ( )321 ,, xxx=x is linked to the tip apex, and the coordinates ( )21, yy=y denote the tip position in the sample coordinate system y (see Fig. 1a); the coefficients dmnk and cjlmn are components of the piezoelectric strain constant and elastic stiffness tensors, respectively; ( ) ( ) kik xE ∂ϕ∂−= xx is the ac electric field distribution produced by the probe ( iϕ is its potential); and the Green’s function for a film, ( )ξx,3 f jG , is derived in Ref. [9] and depends on mechanical boundary conditions on the film-substrate interface. For the surface piezoelectric layer appeared on the non-polar bulk, as well as for the films on substrates with matched elastic properties, one can use the Green function S jG3 of a semiinfinite medium given in Ref. [6]. 2.2. Electrostatic potential In the case of a dielectrically transversely isotropic piezoelectric film, the inner electrostatic potential iϕ created by a point charge Q located at distance d outside the layer has the following form: Here, ρ=+ 2 2 2 1 xx and 3x are the radial and vertical coordinates, respectively, ( )xJ 0 is the zero- order Bessel function, eε is the dielectric constant of the ambient, 1133εε=κ is the effective dielectric constant, 1133 εε=γ is the dielectric anisotropy factor of the film, and bb b 1133εε=κ is the effective dielectric constant of the bulk. Since the electrostatic potential is a linear function of the applied electric field, the point charge potential (2) provides the basic model, and the results for realistic tip geometries can be obtained using an appropriate image charge model with the help of the additional summation or integration over the set of real or image charges representing the tip. Unfortunately, in the considered case of two boundaries (the triple system “ambient – dielectric 1 – dielectric 2”), the image charge method is not suitable. An alternative approach to the description of electric fields in the immediate vicinity of the tip-surface junction is the effective point charge model [10, 11], in which the charge value Q and its surface separation d are selected so that the corresponding isopotential surface reproduces the tip radius of curvature R0 and potential U. We succeed to evolve the effective point charge approach for the spherical tip potential corresponding to the considered triple system “ambient – film – bulk” and obtained the exact series in image charges for the determination of Q and d. For the tip that touches the surface, their Pade approximations have the form: ( )22 0 2 2 0 22 hR Rh dd b b +γκ κγ+κ ≈ ∞ , (3a) ( ) .1ln 1, 11 1- −− ∞ ⎟⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠ ⎞ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ κ+ε κ−ε ⋅ κ+κ κ−κ −× ×⎜ ⎜ ⎝ ⎛ ⎜⎜ ⎝ ⎛ κ κ−ε γ + κ−κ ε+κ +≈ e e b b e b eb d h QdhQ (3b) Saturation values κε=∞ 0Rd e and ( ) κε+κεπε=∞ ee URQ 002 correspond to the semiinfinite system “ambient-dielectric” [10]. The dependences of the effective charge surface separation d and its value Q on the layer thickness h are shown in Fig. 2 for both the relatively low dielectric permittivity κ = 30 (a, c) and high dielectric permittivity κ = 3000 (b, d) of the film. ( ) ( ) ( ) ( ) ( )( ) ( )( ) ∫ ∞ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ γ −κ−εκ−κ−κ+εκ+κ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ γ − −−κ−κ−⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ γ −−κ+κ ρ πε =ρϕ 0 33 0 0 3 2 exp 2 expexp 2 , kh xh kkd x kkd kdkJ Q x ebeb bb i . (2) z, ξ3 Z Y2 Y1 x1, ξ2 x2, ξ2 Coordinate systems Surface piezo- electric layer y1 y2 h Applied bias U (a) (b) dij S εij b εij Nonpolar bulk or substrate Polar bulk (c) PSPS dij S εij b εij dij B Fig. 1. (a) Coordinate systems in the PFM experiment. (b, c) structure of the considered systems: (b) the piezoelectric layer (film) capped on a nonpolar bulk (substrate) with the same elastic properties; (c) unswitchable piezoelectric layer capped on a polar bulk. Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 36-41. © 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 38 Q /Q ∞ 10-3 10-2 0.1 1 1 1.5 2. (a) 2 1 3 4 thickness h/(γR0) 10-3 10-2 0.1 1 0.6 0.8 1. (b) 2 3 1,4 thickness h/(γR0) 10-3 10-2 0.1 1 1.003 1.006 2 1 3 4 thickness h/(γR0) (d) 10-3 10-2 0.1 1 0.4 0.6 0.8 1. (c) 2 1 3 4 d/ d ∞ thickness h/(γR0) Fig. 2. Dependence of the effective charge ∞QQ (a, b) and ∞dd (c, d) on the surface layer thickness with dielectric permittivity 3000,30=κ [(a, c) and (b, d), respectively]. Other parameters: dielectric anisotropy 1=γ , ambient dielectric constant 1=εe , and bulk dielectric constant 260=κb . Curves 1, 2, 3 were calculated for the systems of 50, 4, and 2 image charges. Dotted curves 4 represent approximations (3). Under the condition 0Rh γ≥ , the values ∞→ dd and ∞→ QQ . Since the dielectric anisotropy γ is less that unity ( 10 ≤γ< ) for the majority of perovskites and the dielectric constant 10>κ for the majority of polar materials, our calculations essentially improve the heuristic condition imposed on the layer thickness, h >> γR0, for the semiinfinite approximation to be valid in films. Namely, when the piezoelectric film thickness h is close or more than the tip curvature R0 (h ≥ R0), the substrate (or bulk) does not affect the surface electrostatic potential ( )03 =ϕ xi almost independently on the ratio of film/substrate dielectric permittivities. 2.3. Piezoelectric properties Here, we consider the case where the dielectric and piezoelectric properties of a layer (or film) differ from those of the bulk or substrate. In this case, the strain piezoelectric coefficient ( )321 ,, xyydklj is dependent on the depth 3x as follows: ( ) ( ) ( )⎪⎩ ⎪ ⎨ ⎧ ∞<< ≤≤ = .,0or,, 0,, ,, 3321 321 321 xhxyyd hxyyd xyyd B ijk S ijk klj (4) Here, ( )21, yyd S ijk and ( )321 ,, xyyd B ijk are the piezoelectric effect tensors of the film and substrate, respectively. For a non-piezoelectric substrate, ( ) 0,, 321 ≡xyyd B ijk (see Fig. 1b, c). 3. Effective local piezoelectric response 3.1. Homogeneous surface layers and the piezoelectric response of thin films Assuming that the piezoelectric coupling is uniform inside the film regions (domains) with transverse sizes much greater than the tip curvature, i.e., ( ) const,, 321 ≈≤ hxyyd S lkj at 0 2 2 2 1 Ryy >>+ , the film vertical piezoresponse ( )0)0(3 eff 33 =ϕ== rr iud has the form: S i f S i f S i f d W d W d W dhd 15 351 31 313 33 333eff 33 ),( ψ + ψ + ψ = , (5) where the components f ijW3 are ( ) ( )( ) ( )( ) , 21 1 21 122 ),( 0 333 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ γ+++γ γ + γ+++γ γ × ×⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ++γκ+κ κ−κ + +γ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ κ+ε κ−ε ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ κ+κ κ−κ −= ∑ ∞ = hmd h hmd h mhd d hmd d dhW b b m m e e m b bf (6) ( ) ( )( ) ( ) ( )( ) , 21 21 21 122 ),( 0 313 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ γ+++γ γ −ν+ γ+++γ γ × ×⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ++γκ+κ κ−κ + +γ × ×⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ κ+ε κ−ε ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ κ+κ κ−κ −= ∑ ∞ = hmd h hmd h mhd d hmd d dhW b b m m e e m b bf (7) ( ) ( )( ) . 21 122 ),( 2 23 0 351 hmd h mhd d hmd d dhW b b m m e e m b bf γ+++γ γ × ×⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ++γκ+κ κ−κ − +γ × ×⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ κ+ε κ−ε ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ κ+κ κ−κ −= ∑ ∞ = (8) At the same time, iψ has the form: . )1(22 ),( 0 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ++γ γ κ+κ κ−κ − +γ γ × ×⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ κ+ε κ−ε ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ κ+κ κ−κ =ψ ∑ ∞ = mhd d hmd d dh b b m m e e m b b i (9) Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 36-41. © 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 39 We note that the response theorem ( )333 ,0,0~),0,0( xxu iϕ formulated in Ref. [6] for semi- infinite piezoelectric materials is not valid for a layer of finite thickness h, because the ratio i f jkW ψ3 is dependent on h and d in accordance with Eqs. (6)–(9). This becomes clear since some image charges are located below the layer (i.e., in the region x3 > h). When the dielectric permittivity κ lies in the interval be κ<κ<ε , the first two terms for m = 0 (point charge + its first image) provide accuracy not less than 5 % even if 95.0≅ κ+κ κ−κ b b . For bκ>κ , series (6)–(9) converge more slowly, and the accuracy not less than 10% corresponds to the case where 5.0≤ κ+κ κ−κ b b . Under the condition 1>>γdh , i.e., for a thick film, Eq. (10) coincides with the expression for eff 33d obtained earlier in Ref. [6] for a semiinfinite system ( ) ( ) ( ) S SS d dd dhd 312 33 2 15 2 eff 33 1 21 1 11 ),( ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ γ+ ν+ −− γ+ − γ+ γ −≈ , as it should be expected. For ultra-thin layers (h << d), Eq. (10) yields ( ) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ γ +ν++ γ −≈ SSS d d h dd d h d 153133 eff 33 21 . 3.2. Size effects of the piezoelectric properties of thin films Analyzing the basic results of Section 3.1, we would like to underline once more that they have independent application to calculations of the effective piezoelectric response )(eff 33 hd in the case of a ferroelectric or piezoelectric film capped on a non-piezoelectric bulk. Such a situation for the nanostructure BaTiO3/SrTiO3 is shown in Fig. 3. As follows from Fig. 3, the condition 210≥dh is sufficient for the effective piezoresponse of a layer to become thickness-independent. As usual, 1001~ −d nm, thus the piezoelectric response appeared thickness- dependent for the thickness of films to be less than 100 nm (extrinsic size effect). The main reasons of the considered extrinsic size effect are the thickness- dependent structure of elastic strains and the electrostatic potential, as well as the finiteness of the signal generation volume [12]. The extrinsic size effect should be clearly distinguished from the intrinsic size effects related to the inhomogeneous polarization distribution in nanostructures. Moreover, the extrinsic size effect can interfere with several intrinsic ones caused by film- substrate misfit strains, a decrease of the correlation volume, and the depolarization field [13, 14]. For thin 10-2 0.1 1 10 102 0 25 50 d 3 3ef f , p m /V 3 2 1 4 h/d Fig. 3. Effective piezoelectric response )(eff 33 hd of BaTiO3 (κ = 700, γ = 0.24) film capped on a non-piezoelectric bulk with close elastic properties (ν = 3), but different dielectric constants κb = 3; 30; 300; 3×103 (curves 1, 2, 3, 4, respectively); εe = 1. Note that the scaling κε→ 0Rd e for the effective point charge model of a tip is valid under the conditions 101.0 <κκ< b and 02.0 Rh γ≥ . films, the dielectric permittivity εii (h) and the spontaneous polarization PS (h) are thickness-dependent. Since the piezoelectric constants S ij S ij Pd 3~ ε , the dependence )(hd S ij should be included in Eq. (5). The size-driven phase transition into the paraelectric (non-polar) phase appeared in thin films with thicknesses crhh ≤ , where the critical thickness hcr depends on temperature, stress, etc. For instance, 21cr −=h nm for PbTiO3 on SrTiO3 substrate and 5cr ≤h nm for BaTiO3 on SrRuO3 substrate at room temperature (see Refs. [15, 16] for more details). If the values of hcr and d are of the same order, the extrinsic and intrinsic size effects would interfere. Their contributions can be separated for a homogeneous (single-domain) film by fitting the experimental data provided by Eq. (5) accompanied with appropriate dependence )(hd S ij . Such a situation is demonstrated in Fig. 4 for PbZr0.52Ti0.48O3 film on SrTiO3 substrate. 3.2. Vertical shift of the piezoresponse for “surface piezoelectric layer + switchable bulk” In the case of an irreversible surface polar layer, the vertical shift Vd33 caused by the surface piezoeffect should be distinguished from the downward loop shift Dd33 originated from the onset of a nested domain inside the existed one and its domain wall pinning (bulk polarization switching in the kinetic limit) considered in details in Refs. [10, 17]. However, the vertical asymmetry ( )max33 Ud D considered by the onset of a Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 36-41. © 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 40 nested domain inside the existed one depends on the maximal dc bias maxU applied to the probe (i.e., on the domain size) and decreases with increase in maxU , whereas the shift eff 3333 dd V = described by Eq. (5) is voltage-independent, but depends on the layer thickness (e.g., ( )( )SS dd d h d 3133 eff 33 21 ν++ γ −≈ for ultrathin films) and the tip position when ( )21, yydd S ijk S ijk = . It is clear that the value of Vd33 can be found as the difference between positive and negative piezoresponse saturating values. For the more complex case of a hardly switchable surface polar layer, the shift Vd33 described by Eq. (5) is voltage-independent only at SUU ≤ and should be distinguished from the aforementioned downward loop shift Dd33 . At SUU > , the surface layer can switch and the piezoresponse loop can become almost symmetric. It is clear that the value of Vd33 can be found as the difference between negative piezoresponse saturating values: ( ) ( )SS V UUdUUdd <−>>= eff 33 eff 3333 . Such an example is schematically shown in Fig. 5a. Relevant experimental loops are shown in Fig. 5b, c. The size dependence of the relative hysteresis loop shift (imprint) is shown in Fig. 5d. 4. Conclusion Piezoresponse size effects can be extrinsic (as considered in the present paper) or intrinsic ones (e.g., related to the inhomogeneous polarization distribution in nanostructures). If the values of critical thickness crh for the intrinsic size-induced paraelectric phase transition and the effective charge-surface separation d are of the same order, extrinsic and intrinsic size effects should interfere for realistic experiments in thin films. Their contributions can be separated by fitting the experimental data provided by the proposed analytical expressions accompanied with the appropriate “intrinsic” dependence of the piezoelectric coefficient d 3 3ef f , p m /V thickness h, Å (a) (b) thickness h, Å Fig. 4. (а) Effective piezoelectric response )(eff 33 hd versus the thickness of a PbZr0.52Ti0.48O3 film on SrTiO3 substrate. Symbols are the experimental data from Ref. [14], solid curve is the theoretical fitting with the help of Eq. (5) at d = 2 nm allowing for the dependence (b) for polarization PS(h). The dependence for was fitted by the formula hhPPhP b SiS cr1)( −+≈ , where the bulk polarization 135=b SP µC/cm2 and the surface polarization Pi ≈ 5 µC/cm2 (appeared from symmetry breaking) and the critical thickness hcr = 5 nm. -2 0 -1 0 0 1 0 2 0 -3 0 -2 0 -1 0 0 1 0 2 0 3 0 - 2 0 - 1 0 0 1 0 2 0 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 d 3 3ef f , p m /V Udc, V d33 V=5pm/V d33 V=7pm/V Udc, V -20 -10 0 10 20 -30 -20 -10 0 10 20 30 d 3 3ef f , p m /V Udc, V 1 4 5 3 2 Umax<Us Umax≥Us d33 eff d33 D d33 V (b) (c) (a) PZT-sell size, nm R el at iv e lo op sh ift Proposed theory Exp.fit Exp.data (d) Fig. 5. Piezoresponse loops at different maximal biases maxU (a) – results of theoretical calculations in the kinetic limit; (b,c) – experiment for a multiferroic BiFeO3 film from Ref. [17]. (d) –size dependence of the relative hysteresis loop shift (imprint) in mesoscopic PZT cells; symbols and solid curve – experimental data and their empirical fit from Ref. [3], dashed curve – results of our theoretical calculations of the relative response )()( 0 eff 3333 hhdhdV − ( h is the nonswitchable surface layer thickness, and 0h is the cell thickness). Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 36-41. © 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 41 )(hd S ij . Appropriate theoretical calculations are in quantitative agreement with the piezoresponse of a PbZr0.52Ti0.48O3 film on SrTiO3 substrate. Thus, the obtained simple Pade approximations and the exact series can be applied to calculations of the local piezoelectric response of polar layers capped on the non- piezoelectric bulk. We have proposed an analytical expression for the local vertical shift Vd33 of effective piezoelectric response hysteresis loops caused by the surface piezoeffect S ijkd . The theoretical result is in qualitative agreement with the asymmetry of typical piezoresponse loops obtained for multiferroic BiFeO3 and mesoscopic PZT cells. References 1. A.K. Tagantsev, Piezoelectricity and flexo- electricity in crystalline dielectrics // Phys. Rev. B 34, p. 5883 (1986). 2. Y. Saya, S. Watanabe, M. Kawai, H. Yamada, K. Matsushige, Investigation of nonswitching regions in ferroelectric thin films using scanning force microscopy // Jpn J. Appl. Phys. 39, p. 3799 (2000). 3. M. Alexe, C. Harnagea, D. Hesse, U. Gosele, Polarization imprint and size effects in mesoscopic ferroelectric structures // Appl. Phys. Lett. 79, p. 242 (2001). 4. I. Szafraniak, C. Harnagea, R. Scholz, et al., Ferroelectric epitaxial nanocrystals obtained by a self-patterning method // App. Phys. Lett. 83, p. 2211 (2003). 5. W. Ma, and D. Hesse, Polarization imprint in ordered arrays of epitaxial ferroelectric nano- structures // App. Phys. Lett. 84, p. 2871 (2004). 6. A.N. Morozovska, E.A. Eliseev, S.L. Bravina, and S.V. Kalinin, Resolution function theory in piezoresponse force microscopy: domain wall profile, spatial resolution, and tip calibration // Phys. Rev. B. 75(17), 174109-1-18 (2007). 7. F. Felten, G.A. Schneider, J.M. Saldaña, and S.V. Kalinin, Modeling and measurement of surface displacements in BaTiO3 bulk material in piezoresponse force microscopy // J. Appl. Phys. 96, p. 563 (2004). 8. D.A. Scrymgeour and V. Gopalan, Nanoscale piezoelectric response across a single antiparallel ferroelectric domain wall // Phys. Rev. B 72, 024103 (2005). 9. A. N. Morozovska, E.A. Eliseev, and S.V. Kalinin, The piezoelectric surface layers recognition by piezoresponse force microscopy // J. Appl. Phys. 102(7), 074105-1-12 (2007). 10. A. N. Morozovska, E.A. Eliseev, and S.V. Kalinin, Domain nucleation and hysteresis loop shape in piezoresponse force spectroscopy // Appl. Phys. Lett. 89, 192901 (2006). 11. A.N. Morozovska, S.V. Kalinin, E.A. Eliseev, and S.V. Svechnikov, Polarization screening effect on local polarization switching mechanism and hyste- resis loop measurements in piezoresponse force microscopy // Ferroelectrics 354, p.198-207(2007). 12. A.N. Morozovska, S.V. Svechnikov, E.A. Eliseev, and S.V. Kalinin, Extrinsic size effect in piezoresponse force microscopy of thin films // Phys. Rev. B 76(5), 054123-1-5 (2007). 13. N.A. Pertsev, A.G. Zembilgotov, and A.K. Ta- gantsev, Effect of mechanical boundary conditions on phase diagrams of epitaxial ferroelectric thin films // Phys. Rev. Lett. 80, p.1988 (1998). 14. V. Nagarajan, J. Junquera, J.Q. He, C.L. Jia, R. Waser, K. Lee, Y.K. Kim, S. Baik, T. Zhao, R. Ramesh, Ph. Ghosez, and K.M. Rabe, Scaling of structure and electrical properties in ultrathin epitaxial ferroelectric heterostructures // J. Appl. Phys. 100, 051609 (2006). 15. C. Lichtensteiger, J.-M. Triscone, Javier Junquera and Ph. Ghosez, Ferroelectricity and tetragonality in ultrathin PbTiO3 films // Phys. Rev. Lett. 94, 047603 (2005). 16. D.D. Fong, G.B. Stephenson, S.K. Streiffer, J.A. Eastman, O. Auciello, P.H. Fuoss, and C. Thompson, Ferroelectricity in ultrathin perovskite films // Science 304, p. 1650 (2004). 17. A.N. Morozovska, E.A. Eliseev, S.V. Svechnikov, V. Gopalan, S.V. Kalinin // E-print arxiv: 08014086.
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institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1560-8034
language	English
last_indexed	2025-12-07T15:15:33Z
publishDate	2007
publisher	Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
record_format	dspace
spelling	Morozovska, A.N. Svechnikov, S.V. 2017-05-29T19:17:08Z 2017-05-29T19:17:08Z 2007 The influence of size effects on thin films local piezoelectric response of thin films / A.N. Morozovska, S.V. Svechnikov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2007. — Т. 10, № 4. — С.36-41. — Бібліогр.: 17 назв. — англ. 1560-8034 PACS 77.80.Fm, 77.65.-j, 68.37.-d https://nasplib.isofts.kiev.ua/handle/123456789/118331 We discuss the influence of size effects on the local piezoelectric response of thin films. In calculations of the electrostatic potential in the triple system “PFM probe tip – film – substrate,” the effective point charge model is used. The obtained expressions for the local piezoelectric response of a surface layer (film) capped are intended for the calculations of Piezoresponse Force Microscopy signals of thin polar films epitaxially grown on thick substrates. Theoretical predictions are in qualitative agreement with typical experimental results obtained for perovskite Pb(Zr, Ti)O₃ and multiferroic BiFeO₃ films. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics The influence of size effects on thin films local piezoelectric response of thin films Article published earlier
spellingShingle	The influence of size effects on thin films local piezoelectric response of thin films Morozovska, A.N. Svechnikov, S.V.
title	The influence of size effects on thin films local piezoelectric response of thin films
title_full	The influence of size effects on thin films local piezoelectric response of thin films
title_fullStr	The influence of size effects on thin films local piezoelectric response of thin films
title_full_unstemmed	The influence of size effects on thin films local piezoelectric response of thin films
title_short	The influence of size effects on thin films local piezoelectric response of thin films
title_sort	influence of size effects on thin films local piezoelectric response of thin films
url	https://nasplib.isofts.kiev.ua/handle/123456789/118331
work_keys_str_mv	AT morozovskaan theinfluenceofsizeeffectsonthinfilmslocalpiezoelectricresponseofthinfilms AT svechnikovsv theinfluenceofsizeeffectsonthinfilmslocalpiezoelectricresponseofthinfilms AT morozovskaan influenceofsizeeffectsonthinfilmslocalpiezoelectricresponseofthinfilms AT svechnikovsv influenceofsizeeffectsonthinfilmslocalpiezoelectricresponseofthinfilms

The influence of size effects on thin films local piezoelectric response of thin films

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