The resolution function and effective response of piezoelectric thin films in Piezoresponse Force Microscopy

The resolution function and effective response of piezoelectric thin films in Piezoresponse Force Microscopy

The elastic Green function and resolution function in Piezoresponse Force Microscopy (PFM) of thin piezoelectric film capped on the rigid substrate are derived. The extrinsic size effect on the resolution function is demonstrated.

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Дата:2008
Автор: Morozovska, A.N.
Формат: Стаття
Мова:English
Опубліковано: Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України 2008
Назва видання:Semiconductor Physics Quantum Electronics & Optoelectronics
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Цитувати:The resolution function and effective response of piezoelectric thin films in Piezoresponse Force Microscopy / A.N. Morozovska // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2008. — Т. 11, № 2. — С. 171-177. — Бібліогр.: 11 назв. — англ.

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spelling irk-123456789-1188512017-06-01T03:03:06Z The resolution function and effective response of piezoelectric thin films in Piezoresponse Force Microscopy Morozovska, A.N. The elastic Green function and resolution function in Piezoresponse Force Microscopy (PFM) of thin piezoelectric film capped on the rigid substrate are derived. The extrinsic size effect on the resolution function is demonstrated. 2008 Article The resolution function and effective response of piezoelectric thin films in Piezoresponse Force Microscopy / A.N. Morozovska // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2008. — Т. 11, № 2. — С. 171-177. — Бібліогр.: 11 назв. — англ. 1560-8034 PACS 77.80.Fm, 77.65.-j, 68.37.-d http://dspace.nbuv.gov.ua/handle/123456789/118851 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description The elastic Green function and resolution function in Piezoresponse Force Microscopy (PFM) of thin piezoelectric film capped on the rigid substrate are derived. The extrinsic size effect on the resolution function is demonstrated.
format Article
author Morozovska, A.N.
spellingShingle Morozovska, A.N.
The resolution function and effective response of piezoelectric thin films in Piezoresponse Force Microscopy
Semiconductor Physics Quantum Electronics & Optoelectronics
author_facet Morozovska, A.N.
author_sort Morozovska, A.N.
title The resolution function and effective response of piezoelectric thin films in Piezoresponse Force Microscopy
title_short The resolution function and effective response of piezoelectric thin films in Piezoresponse Force Microscopy
title_full The resolution function and effective response of piezoelectric thin films in Piezoresponse Force Microscopy
title_fullStr The resolution function and effective response of piezoelectric thin films in Piezoresponse Force Microscopy
title_full_unstemmed The resolution function and effective response of piezoelectric thin films in Piezoresponse Force Microscopy
title_sort resolution function and effective response of piezoelectric thin films in piezoresponse force microscopy
publisher Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
publishDate 2008
url http://dspace.nbuv.gov.ua/handle/123456789/118851
citation_txt The resolution function and effective response of piezoelectric thin films in Piezoresponse Force Microscopy / A.N. Morozovska // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2008. — Т. 11, № 2. — С. 171-177. — Бібліогр.: 11 назв. — англ.
series Semiconductor Physics Quantum Electronics & Optoelectronics
work_keys_str_mv AT morozovskaan theresolutionfunctionandeffectiveresponseofpiezoelectricthinfilmsinpiezoresponseforcemicroscopy
AT morozovskaan resolutionfunctionandeffectiveresponseofpiezoelectricthinfilmsinpiezoresponseforcemicroscopy
first_indexed 2025-07-08T14:46:50Z
last_indexed 2025-07-08T14:46:50Z
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fulltext Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 2. P. 171-177. . © 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 171 PACS 77.80.Fm, 77.65.-j, 68.37.-d The resolution function and effective response of piezoelectric thin films in Piezoresponse Force Microscopy A.N. Morozovska V. Lashkaryov Institute of Semiconductor Physics, National Academy of Science of Ukraine, 45, prospect Nauky, 03028 Kyiv, Ukraine; e-mail: morozo@i.com.ua Abstract. The elastic Green function and resolution function in Piezoresponse Force Microscopy (PFM) of thin piezoelectric film capped on the rigid substrate are derived. The extrinsic size effect on the resolution function is demonstrated. Keywords: Piezoresponse Force Microscopy, thin piezoelectric films, resolution function. Manuscript received 28.09.07; accepted for publication 10.04.08; published online 30.07.08. 1. Introduction Verification of existing theoretical models, design of functional nanomaterials with predetermined properties, and application in various devices necessitate experimental and theoretical studies of piezoelectric coupling and ferroelectric properties in surface layers. These considerations of the local polarization switching behavior in thin films are possible in the context of Piezoresponse Force Microscopy (PFM) [1, 2]. Conventional framework for the PFM data analysis was based on the 1D models suggested originally by Ganpule, thus ignoring the 3D geometry of the PFM problem. Only recently, the decoupled theory [3, 4] was applied to derive analytical expressions for PFM response on semi-infinite materials of low symmetry, derive analytical expressions for resolution function and domain wall profiles, and interpret PFM spectroscopy data [5]. Further, the local piezoresponse dependence on film thickness was predicted in [6] for thin films capped on the nonpiezoelectric bulk with the same elastic and close dielectric properties. In the paper, the com- plementary case of thin piezoelectric films capped on rigid substrates is considered within the framework of decoupled approximation. For ferroelectric perovskite films like BaTiO3 or Pb(Ti, Zr)O3 rigid dielectric substrates are MgO oxide, sapphire Al2O3 or carbon with effective dielectric constant 105 −≈κb . Silicon ( 123 −≈κb ) and SiO2 ( 5≈κb ) have smaller elastic stiffness than typical perovskites. 2. Elastic Green function of thin film on rigid substrate Let us derive the elastic Green function for the layer on the rigid substrate. General equation for the field of elastic displacement vector is [7]: .)()1(2 )div(grad 21 1 ξ−δ⋅ ν+ −= = ν− +∆ x uu xxx F Y (1a) Here the vector x denotes the given location and ξ is the point at which the point force, F , is applied. The material is isotropic and ν is the Poisson coeffi- cient, Y is the Young modulus. Introducing the shear modulus ( )( )ν+=µ 12Y , Eq. (1a) can be rewritten as: )( 21 1 22 ξ−δ µ −= ∂∂ ∂ ν− + ∂∂ ∂ xi mi m kk i F xx u xx u . (1b) Introducing transversal Fourier transformation ( ) ,)(exp 2 1 ),,(~ 221121 321 ∫∫ ∞ ∞− ∞ ∞− ⋅+ π = = xi i uxikxikdxdx xkku (2a) ( ) ),,,(~exp 2 1)( 321221121 xkkuxikxikdkdku ii ∫∫ ∞ ∞− ∞ ∞− −− π =x (2b) Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 2. P. 171-177. . © 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 172 and using integral representation of the delta function ( ) ( ) ( )( ) .exp 2 1 )()( 222111212 2211 ξ−−ξ−− π = =ξ−δξ−δ ∫ ∫ ∞ ∞− ∞ ∞− xikxikdkdk xx (3) Eq. (1b) yields: ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ −δξ+ξ πµ − = ∂ ∂ α+++− ∂ ∂ α− ∂ ∂ α− −δξ+ξ πµ − = ∂ ∂ α− ∂ ∂ +α++−α− −δξ+ξ πµ − = ∂ ∂ α−α− ∂ ∂ ++α+− )(exp 2 ~ 1~ ~~ )(exp 2 ~~ ~1~ )(exp 2 ~ ~ ~ ~1 332211 3 2 3 3 2 3 2 2 2 1 3 2 2 3 1 1 332211 2 3 3 22 3 2 2 2 2 2 2 1121 332211 1 3 3 12212 3 1 2 1 2 2 2 1 ξxikik F x u ukk x u ki x u ki ξxikik F x u ki x u ukkukk ξxikik F x u kiukk x u ukk (4) where ( )ν−=α 211 is introduced. The solution of Eq. (4) is )exp(~~ 3xsui , where s values should be determined. Substitution into Eq. (4) with 0=iF (homogeneous system) yields the characteristic equation for s: ( ) ( ) 01322 2 2 1 =α+−+− skk . (5) Eq. (5) has thrice degenerated roots ks ±= , where 2 2 2 1 kkk += is the module of the vector k. After the simple, but cumbersome transformations one can find the general homogeneous solution of Eq. (4) as ( ) ( ) ,)exp( )exp(),,(~ 3333111 33131103211 xkCxkiC xkCxkiCxkku h ++ +−+= , (6) ( ) ( ) ,)exp( )exp(),,(~ 3333221 33132203212 xkCxkiC xkCxkiCxkku h ++ +−+= (7) ( ) ( ) ).exp(43 )exp( 43 ),,(~ 33333321 2 11 1 3 3133120 2 10 1 3213 xkCxkCC k k iC k k i xk CxkCC k k iC k k i xkku h ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −ν−+++ +−× ×⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ +ν−+−−= = (8) Note, that the equality ν−=α+ 4321 was used in Eqs. (6)-(8). To complement the general solution, we seek the particular solution )(xp iu of the inhomogeneous Eqs. (4). One of the simplest is the solution for the homogeneous space )(x∞ iu , since Eq. (1) is reduced to the system of algebraic equations when using the full 3D-Fourier transformation. Its solution has the form: ( ) ( ) ( ) ( ) ( ) ( ) . 12 1 2 1ˆ ,expˆˆ 422/3 ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎝ ⎛ ν− − δ µπ = = ∞ ∞∞ kk k kξkk jiij ij jiji kk G iFGu (9) The inhomogeneous solution (9) corresponds to the well-known Fourier image of Green’s tensor for infinite homogeneous isotropic media (see e.g. Ref. [8]). Looking for solution of the system, confined in 3x - direction, it is convenient to transform (9) to coordinate representation on 3x . Simple integration gives: ( ) ( ) ( ),exp,,~ ,,~ 22113321 321 ξ+ξξ−= = ∞ ∞ ikikFxkkG xkku jij i (10) where ( ) ( ) ( ) ( ) , 14 1 1 exp 4 1 ,,~ 2 33 2 133 332111 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ν− ξ−+ − ξ−− πµ = =ξ−∞ k xkk k xk xkkG (11a) ( ) ( ) ( ) ( ) , 14 1exp 4 1 ,,~ 2 332133 332112 k xkkk k xk xkkG ν− ξ−+ξ−− πµ −= =ξ−∞ (11b) ( ) ( ) ( ) ( ) , 14 1 1 exp 4 1 ,,~ 2 33 2 233 332122 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ν− ξ−+ − ξ−− πµ = =ξ−∞ k xkk k xk xkkG (11c) ( ) ( ) ( ) ( ) , 14 exp 4 1 ,,~ 33133 332113 ν− ξ−ξ−− πµ = =ξ−∞ xki k xk xkkG (11d) ( ) ( ) ( ) ( ) , 14 exp 4 1 ,,~ 33233 332123 ν− ξ−ξ−− πµ = =ξ−∞ xki k xk xkkG (11e) ( ) ( ) ( ) , 14 1 1 exp 4 1 ,,~ 3333 332133 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ν− ξ−− − ξ−− πµ = =ξ−∞ xk k xk xkkG (11f) and ( )( )ν+=µ 12Y . The general solution of the chosen elastic problem should satisfy the boundary conditions at the rigid substrate is 0)( 3 == hxui or )( 3 hxui = and )( 33 hxi =σ are continuous for the matched substrate; also 0)0( 33 ==σ xi at free upper surface. Keeping in mind the Hook law klijklij uc=σ , where the strain Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 2. P. 171-177. . © 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 173 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ = k l l k kl x u x u u 2 1 and elastic stiffness =ijklc ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ δδ+δδ+δδ ν− ν ν+ = jkiljlikklij Y 21 2 )1(2 , one obtains that: ( ) ( )( )22113333 )1( 21)1( uuuY +ν+ν− ν−ν+ =σ , 1331 )1( uY ν+ =σ , 2332 )1( uY ν+ =σ . For the rigid substrate case, the boundary conditions for ( ) ( )+= 321321 ,,~,,~ xkkuxkku h ii ( )321 ,,~ xkku p i+ have the form: ( ) ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ === =− ∂ ∂ =− ∂ ∂ =+ν− ∂ ∂ ν− === == = .0~,0~,0~ ,0~ ~ ,0~ ~ ,0~~ ~ )1( 333 33 3 321 0 32 3 2 0 31 3 1 0 2211 3 3 hxhxhx xx x uuu uik x u uik x u ukuki x u (12) Six constants Cij should be expressed via Fj from Eqs. (12). For the matched substrate only the first row of Eqs. (12) should be used. (I) First step. Let us find the Green function ( )3321 ,,~ ξ−xkkG s ij of the semi-space ( ∞→h ). The function is the solution for the matched substrate case. For the case 0332111 === CCC and then Eqs. (12) should be solved allowing for the partial solution )0,,(~)0,,(~ 2121 kkukku i p i ∞≡ . After cumbersome algebraic transformations one derives: ( ) ( ) ( ) .exp,,,~ ,,~ 22113321 321 ξ+ξξ= = ikikFxkkG xkku j s ij s i (13) Where ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ,14 116 exp 181222 43 116 exp ,,,~ 33 2 1 2 1 2 3 33 2 133 2 1 2 33 2 1 3 33 332111 ξ−−−ν− ν−πµ ξ−− + +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ νν−−+ξ+ν−+ +ν−ξ+− × × ν−πµ ξ+− =ξ xkkkk k xk kxkk xkk k xk xkkG s (14a) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ),1 116 exp 181 432 116 exp ,,,~ 33213 33 3333 2 21 3 33 332121 ξ−+ ν−πµ ξ−− + +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ νν−−+ +ν−ξ+−ξ × × ν−πµ ξ+− =ξ xkkk k xk xkxkkk k xk xkkG s (14b) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) , 116 exp 2114 432 116 exp ,,,~ 331 33 3333 2 12 33 332131 ξ− ν−πµ ξ−− + +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ν−ν−+ +ν−ξ−+ξ− × × ν−πµ ξ+− =ξ xik k xk xkxk ik k xk xkkG s (14c) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ,1 116 exp 181 432 116 exp ,,,~ 33213 33 3333 2 21 3 33 332112 ξ−+ ν−πµ ξ−− + +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ νν−−+ +ν−ξ+−ξ × ν−πµ ξ+− =ξ xkkk k xk xkxk kk k xk xkkG s (14d) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( ),14 116 exp 181222 43 116 exp ,,,~ 33 2 2 2 2 2 3 33 2 233 2 2 2 33 2 2 3 33 332122 ξ−−−ν− ν−πµ ξ−− + +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ νν−−+ξ+ν−+ +ν−ξ+− × × ν−πµ ξ+− =ξ xkkkk k xk kxkk xkk k xk xkkG s (14e) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , 116 exp 2114 432 116 exp ,,,~ 332 33 3333 2 2 2 33 332132 ξ− ν−πµ ξ−− + +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ν−ν−+ +ν−ξ−+ξ− × × ν−πµ ξ+− =ξ xik k xk xkxkik k xk xkkG s (14f) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , 116 exp 2114 432 116 exp ,,,~ 331 33 3333 2 1 2 33 332113 ξ− ν−πµ ξ−− + +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ν−ν−− +ν−ξ−+ξ × × ν−πµ ξ+− =ξ xik k xk xkxk ik k xk xkkG s (14g) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , 116 exp 2114 432 116 exp ,,,~ 332 33 3333 2 2 2 33 332123 ξ− ν−πµ ξ−− + +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ν−ν−− −ν−ξ−+ξ × × ν−πµ ξ+− =ξ xik k xk xkxk ik k xk xkkG s (14h) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ).43 116 exp 21141 432 116 exp ,,,~ 33 33 3333 2 33 332133 ξ−+ν− ν−πµ ξ−− + +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ν−ν−++ +ν−ξ++ξ × × ν−πµ ξ+− =ξ xk k xk xkxk k xk xkkG s (14i) Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 2. P. 171-177. . © 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 174 where 2 2 2 1 kkk += . Note that ( )=ξ332121 ,,,~ xkkG s ( )332112 ,,,~ ξxkkG s , ( ) =ξ332122 ,,,~ xkkG s ( )331211 ,,,~ ξxkkG s , ( ) =ξ332132 ,,,~ xkkG s ( )331231 ,,,~ ξxkkG s as expected. (II) Second step. Using Eq. (14) as the partial solution ( ) ( )321321 ,,~,,~ xkkuxkku s i p i = , let us find the surface vertical displacement =)0,,(~ 213 kku f )0,,(~)0,,(~ 213213 kkukku sh + for the film of the thickness h. Here )0,,(~ 213 kku h should be found from Eqs. (12), namely after cumbersome algebraic transformations we derived that ( ) ( ) ( )22113213213 exp,,~0,,~ ξ+ξξ= ikikFkkGkku j f j f . (15) Where the elastic Green function ( )3213 ,,~ ξkkG f j for the film on a rigid substrate has the form: ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) , ,,,,~,,,~ ,,,,~,0,,~ ,,~ 3212232111 332133213 3213 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ νφξ+ξ+ +νφξ−ξ = =ξ ⊥ hkhkkGkhkkGki hkhkkGkkG kkG s j s j s j s j f j (16) ( ) ( ) ( ) ( )( ) ( )( ) ( )( )( ) ,2211412)2exp()2exp(43 22)exp(22)exp(14 , 22 3 hkhkhk hkhkhkhk hk +ν−ν−+++−ν− +ν−+−ν−−ν− = =νφ (17a) ( ) ( ) ( ) ( )( ) ( )( ) ( )( )( )( ).2211412)2exp()2exp(43 21)exp(21)exp(14 , 22hkhkhkk hkhkhkhk hk +ν−ν−+++−ν− −ν−−+ν−−ν− = =νφ⊥ (17b) Here 2 2 2 1 kkk +≡ , ν is the Poisson ratio. Note that ( ) 00,,~ 3213 =<ξ< hkkG f j at 0=h as it should be expected. For the film on the matched substrate ( ) 0,3 =νφ hk and ( ) 0, =νφ⊥ hk . 3. Resolution function and electric field calculations for thin piezoelectric films The phenomenological resolution function theory for PFM based on linear imaging theory has been introduced in Ref. [9], where the resolution function and the effect of lock-in on resolution have been determined experimentally, and later on theoretically considered in Ref. [5] for the semi-infinite case. Let us consider the case when the film dielectric and piezoelectric properties differ from bulk or substrate ones. In this case, the strain piezoelectric coefficient ( )321 ,, xxxdklj is dependent on the depth 3x as follows: ( ) ( ) ⎪⎩ ⎪ ⎨ ⎧ ∞<< ≤≤ = 3 321 321 ,0 0,, ,, xh hxxxd xxxd S ijk klj (18) Here, ( )21, xxd S ijk are the film piezoelectric effect tensor components. The surface piezoresponse below the tip ( 03 =x ) is given by the convolution of piezoelectric coefficients kljd with the surface and bulk components of the resolution function [9]: ( ) ( ) ( ) .,, 0,, 22112121 21 ξ−ξ−ξξξξ= = ∫∫ ∞ ∞− ∞ ∞− xxdWdd xxu S lkj f ijkl S i (19) The film resolution function [6] components f ijklW are introduced as ( ) ( ) ( ) .,, ,, , 0 321 321 3 21 ∫ ξξξ ξ∂ ξξ−ξ−∂ ξ= =ξξ h l n f im kjmn f ijkl E G cd W (20) ( )xkE is the ac electric field distribution produced by the probe. Hereinafter the effective point charge model [10, 11] is used for electric fields description in the immediate vicinity of the tip-surface junction. Within the framework of the model, the charge value Q and its surface separation d are selected so that corresponding isopotential surface reproduces the tip radius of curvature in the contact point 0R (or contact radius for flattened tip) and potential U (see Fig. 1). For piezoresponse modeling, the electric field structure can be represented by the point charge model in which the effective charge value Q is equal to the product of tip capacitance )(hCt on applied voltage ( ) UhChQ t )(= . The electric field potentials ( )rbie ,,ϕ created by the point charge Q localized in ambient in the point ),0,0(0 dr −= outside the layer (film) hx ≤≤ 30 filled by transversely isotropic dielectric with 332211 ε≠ε=ε could be found from the boundary problem: x3 PFM probe x2 εij Q d x1 dijk (x1,x2) εb ij Fig. 1. Point charge model of the PFM probe and scheme of measurements. Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 2. P. 171-177. . © 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 175 ( ) ( ) ( ) .0),()( ,0),0()0( ,,0 ,0,0 ,0),,,( 3 3 3 33 3 3333 03 33 3 33 32 3 2 3311 32 3 2 3311 3321 0 =⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ϕ∂ ε− ∂ ϕ∂ ε=ϕ==ϕ =⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ϕ∂ ε− ∂ ϕ∂ ε=ϕ==ϕ ≥= ∂ ϕ∂ ε+ϕ∆ε ≤≤= ∂ ϕ∂ ε+ϕ∆ε ≤+δ εε −=ϕ∆ = = ⊥ ⊥ hx ibb bi x ie eie bb b b i i e e xx hxhx xx xx hx x hx x xdxxxQ r r r (21) Note, that 0)( 3 ==ϕ hxi for a conductive substrate. The solution of Eq. (21) can be found with the help of Hankel integral transformation. Inside the film hx ≤≤ 30 , the Fourier representation of the electric field ( )321 ,,~ xkkE j acquires the form ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )( ) . 2exp 2 expexp 2 ,,~ , 2exp 2 expexp 2 ,,~ 33 0 3213 33 0 2,13212,1 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ γ −κ−εκ−κ−κ+εκ+κγ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ γ − −−κ−κ+⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ γ −−κ+κ × × πε = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ γ −κ−εκ−κ−κ+εκ+κ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ γ − −−κ−κ−⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ γ −−κ+κ × × πε = kh xh kkd x kkd QxkkE khk xh kkd x kkd QikxkkE ebeb bb ebeb bb (22) Here eε is the dielectric constant of the ambient, 1133εε=κ is effective dielectric constant, 1133 εε=γ is the dielectric anisotropy factor of the film, bb b 1133εε=κ is effective dielectric constant of the substrate; d− is x3-coordinate of the effective point charge Q . Usually 1001~ −d nm. For the conductive disk of radius 0R representing flattened tip-surface contact area, the effective charge ( ) UhChQ t )(disk= . Calculations performed in Ref. [6] lead to the probe tip effective capacity ( ) ( )dhRhC et ,4)( 00 disk ψκ+εε≈ , where the function 1 0 )1(22 ),( −∞ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ++γ γ κ+κ κ−κ − +γ γ χ=ψ ∑ nhd d hnd ddh b b n n and the parameter ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ κ+ε κ−ε ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ κ+κ κ−κ =χ e e b b . The corres- ponding effective distance π= 02Rd , i.e. d is almost independent on the film thickness h. For the spherical tip with curvature 0R in point contact with film surface, the effective charge ( ) UhChQ t )(sph= . We obtained the probe tip capacity ( ) ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ε κ+ε κ−ε χ−κεγ +× ×⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ε κ+ε ε−κ ε+κ επε≈ e e e e e e e e et h R RhC 2 ln1ln21 2 ln4)( 2 0 00 sph and effective distance ×ε≈ 02 Rd e ( )( ) ( )eee ε−κεκ+ε× 2ln for the film thickness 01.0 Rh γ≥ in the actual range of high dielectric constants 1, >>κκb and 801 ≤ε≤ e [6]. For the case when the considered surface piezoelectric layer is inhomogeneous in the transverse directions { }21,xx (e.g. it is divided into polar regions or posses domain structure with different piezoelectric tensor values and signs ( )21,xxd S ijk ), the resolution function components f ijkW allow approximate calculation of the piezoresponse from those structures, which Fourier image ( )qS ijkd ~ exists in usual (e.g. domain stripes, rings etc.) or generalized (infinite plane domain wall) sense. Being more rigorous, one should use the Fourier image of tensorial object transfer function components )(~ qW f ijk , since the Fourier transform of film vertical piezoresponse ( )qeff 33 ~ d over transverse coor- dinates { }21, xx is ( ) ( ) ( ) ( )qqqq SfSfSf dWdWdWd 153513131333333 eff 33 ~~~~~~~ ++= , (23) where ( ) ( ) xxq qxdedd i∫= eff 33 eff 33 ~ is the Fourier transforms of effective vertical piezoresponse Uud )0(3 eff 33 == x ; the Voight notation is used. Object transfer function component ( )qW f ij3 ~ spectrum dependent on wavenumber absolute value 2 2 2 1 qqq += is shown in Fig. 2a-c for various film thicknesses h. In most cases, the component ( )qW f 333 ~ corresponding to the piezoelectric constant 33d provides the dominant (>50 %) contribution to the overall signal [5]. The two-point resolution rmin in PFM experiments is determined by the inverse halfwidth of ( )qW f 333 ~ . The dependence of corresponding two-point resolution rmin on h/d for different values of anisotropy γ is shown in Fig. 2d. It is clear that the information limit increases with the film thickness decrease because of rmin decrease. The dependence of 180º-periodic domain Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 2. P. 171-177. . © 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 176 10 -2 0.1 1 10 10 2 10 -2 0.1 1 4 3 2 1 Thickness h/d Tw o- po in t r es ol ut io n Anisotropy γ increase 0.1 1 10 0.1 1 1 4 3 2 Thickness h decrease W f 33 3( q) Wave number qd ~ (a) (d) W f 35 1( q) ~ 0.1 1 10 102 10 -2 0.1 1 W f 31 3( q) ~ 1 4 3 2 Thickness h decrease (b) (c) Wave number qd 0.1 1 10 10 2 10-4 10-2 1 1 4 3 2 Thickness h decrease Wave number qd Fig. 2. (a, b, c) Object transfer function components ( )qW f ij3 ~ spectrum for anisotropy γ = 1, permittivity 500=κ and relative thick- ness =dh 0.3, 1, 3, 10 (curves 1, 2, 3, 4); substrate permittivity 10=κb , ambient dielectric constant 1=εe . (d) The dependence of corresponding two-point resolution drmin on dh for different values of anisotropy γ = 0.25, 0.5, 1, 3 (curves 1, 2, 3, 4). 0 5 10 0 5 10 Film thickness h (nm) D om ai n st rip es p er io d a ( nm ) R0=9 nm R0=6 nm R0=3 nm 1 10 102 103 104 0.1 1 10 R es ol ut io n a m in (n m ) 180o-domains stripes Film thickness h (nm) (a) (b) amin +Pz -Pz -Pz +Pz +Pz y x Fig. 3. (a) The dependence of 180o-periodic domain structure resolution mina via PbTiO3 film thickness h on a rigid substrate for the effective distance d = 10 nm. (b) Information limit defined as a minimal domain stripes period a calculated from the condition ( ){ }had ,max eff 33 = noise level for the PbTiO3 film of the thickness h on a rigid substrate for the typical noise level 1 pm/V and different tip-surface contact radii R0 = 3, 6, 9 nm (effective distance π= 02Rd ). (b) PbTiO3 material parameters ν = 0.35, κ = 121, γ = 0.87, =Sd33 117, =Sd15 61, =Sd31 –25 pm/V, substrate permittivity 5=κb , ambient permittivity 1=εe . Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 2. P. 171-177. . © 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 177 structure resolution via the film thickness h and corresponding information limit defined as minimal domain stripes period mina calculated from the condition ( ){ }had ,max eff 33 = noise level is shown in Fig. 3 for the PbTiO3 film on a rigid substrate. 4. Conclusion The elastic Green function and resolution function in Piezoresponse Force Microscopy (PFM) of piezoelectric film capped on the rigid substrate with different dielectric properties are derived. The thickness dependence of resolution function of the thin piezoelectric films on rigid substrates is demonstrated: minimal lateral resolution (or higher information limit) is possible in thin films. However, the signal amplitude essentially decreases with film thickness decrease, eventually making the noise level relatively higher. Acknowledgements Author is grateful to Academician, Prof. S.V. Svechnikov (NAS of Ukraine) and Dr. S.V. Kalinin (Oak Ridge National Laboratory) for valuable remarks. References 1. Nanoscale Characterization of Ferroelectric Materials, ed. M. Alexe and A. Gruverman. Springer, 2004. 2. Nanoscale Phenomena in Ferroelectric Thin Films, ed. Seungbum Hong. Kluwer, 2004. 3. F. Felten, G.A. Schneider, J.M. Saldaña, and S.V. Kalinin // J. Appl. Phys. 96, p. 563 (2004). 4. D.A. Scrymgeour and V. Gopalan // Phys. Rev. B 72, 024103 (2005). 5. A.N. Morozovska, E.A. Eliseev, S.L. Bravina, and S.V. Kalinin // Phys. Rev. B 75, 174109-1-18 (2007). 6. 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), 054123-5 (2007). 7. L.D. Landau and E.M. Lifshitz, Theory of Elasticity. Theoretical Physics, Vol. 7. Butterworth- Heinemann, Oxford, U.K., 1998. 8. C. Teodosiu, Elastic Models of Crystal Defects. Springer-Verlag, Berlin, 1982. 9. S.V. Kalinin, S. Jesse, J. Shin, A.P. Baddorf, H.N. Lee, A. Borisevich, and S.J. Pennycook // Nanotechnology 17, p. 3400 (2006). 10. A.N. Morozovska, E.A. Eliseev, and S.V. Kalinin // Appl. Phys. Lett. 89, 192901 (2006). 11. A.N. Morozovska, S.V. Kalinin, E.A. Eliseev, and S.V. Svechnikov, Local polarization switching in Piezoresponse Force Microscopy // Ferroelectrics 354, p. 198-207 (2007).

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