Simulating and interpreting Kelvin probe force microscopy images on dielectrics with boundary integral equations

Simulating and interpreting Kelvin probe force microscopy images on dielectrics with boundary integral equations Yongxing Shen, David M. Barnett, and Peter M. Pinsky Citation: Rev. Sci. Instrum. 79, 023711 (2008); doi: 10.1063/1.2885679 View online: http://dx.doi.org/10.1063/1.2885679 View Table of Contents: http://rsi.aip.org/resource/1/RSINAK/v79/i2 Published by the American Institute of Physics. Related Articles Reduced leakage current, enhanced ferroelectric and dielectric properties in (Ce,Fe)-codoped Na0.5Bi0.5TiO3 film Appl. Phys. Lett. 100, 022909 (2012) Modified Johnson model for ferroelectric lead lanthanum zirconate titanate at very high fields and below Curie temperature Appl. Phys. Lett. 100, 022907 (2012) Engineering titanium and aluminum oxide composites using atomic layer deposition J. Appl. Phys. 110, 123514 (2011) High dielectric tunability of (100) oriented PbxSr1xTiO3 thin film coordinately controlled by dipole activation and phase anisotropy J. Appl. Phys. 110, 124107 (2011) Kelvin probe force gradient microscopy of charge dissipation in nano thin dielectric layers J. Appl. Phys. 110, 084304 (2011) Additional information on Rev. Sci. Instrum. Journal Homepage: http://rsi.aip.org Journal Information: http://rsi.aip.org/about/about_the_journal Top downloads: http://rsi.aip.org/features/most_downloaded Information for Authors: http://rsi.aip.org/authors Downloaded 17 Jan 2012 to 147.83.95.33. Redistribution subject to AIP license or copyright; see http://rsi.aip.org/about/rights_and_permissions Simulating and interpreting Kelvin probe force microscopy images on dielectrics with boundary integral equations Yongxing Shen,1,a兲David M. Barnett,1,b兲and Peter M. Pinsky2 1Department of Materials Science and Engineering, Stanford University, Stanford, California 94305, USA 2Department of Mechanical Engineering, Stanford University, Stanford, California 94305, USA 共Received 17 August 2007; accepted 4 February 2008; published online 28 February 2008兲 Kelvin probe force microscopy 共KPFM兲is designed for measuring the tip-sample contact potential differences by probing the sample surface, measuring the electrostatic interaction, and adjusting a feedback circuit. However, for the case of a dielectric 共insulating兲sample, the contact potential difference may be ill defined, and the KPFM probe may be sensing electrostatic interactions with a certain distribution of sample trapped charges or dipoles, leading to difficulty in interpreting the images. We have proposed a general framework based on boundary integral equations for simulating the KPFM image based on the knowledge about the sample charge distributions 共forward problem兲 and a deconvolution algorithm solving for the trapped charges on the surface from an image 共inverse problem兲. The forward problem is a classical potential problem, which can be efficiently solved using the boundary element method. Nevertheless, the inverse problem is ill posed due to data incompleteness. For some special cases, we have developed deconvolution algorithms based on the forward problem solution. As an example, this algorithm is applied to process the KPFM image of a gadolinia-doped ceria thin film to solve for its surface charge density, which is a more relevant quantity for samples of this kind than the contact potential difference 共normally only defined for conductive samples兲values contained in the raw image. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2885679兴 I. INTRODUCTION Kelvin probe force microscopy 共KPFM兲was invented by Nonnenmacher et al.1as a special design of electrostatic force microscopy for measuring the sample contact potential difference 共CPD兲. It has been applied to characterize various types of solids used in electronic devices such as ionic crystal thin films,2semiconductors,3organic solar cell structures,4and nanographenes5in order to study the sample lateral variations of surface potentials and work functions and infer the electronic structures. However, detailed analysis shows that an image obtained with KPFM is only an approximation to the actual sample surface potential due to the finite curvature of the probe tip and the long-range characteristic of the Coulombic interaction. In their model,6Jacobs et al. treated the sample as a multiconductor system, of which each of the constituent conductor has its own potential ⌽iand forms a capacitor with the probe tip, Cit. The KPFM signal of a particular configuration 共i.e., a particular probe tip position兲is the weighted average of all the ⌽i’s, the weights being the derivative of Cit with respect to the tip-sample separation. While the conductor potentials are assumed to be fixed, the weights are functions of the tip position. Under the assumption of a flat sample and a constant tip-sample separation and in the limit of continuous CPD variation, the KPFM image becomes a two-dimensional convolution of the “true” surface potential variation and a response function 共also called a transfer function兲. Such a convolution relation permits solving for the true sample surface potential by deconvolution. For example, Strassburg et al.7have developed a deconvolution algorithm for the special case of flat conductive samples. This development has employed the boundary integral equation approach with an image source in Green’s function to model the electrostatics, and the sample is assumed to have a fixed surface dipole layer that is solely responsible for the interaction with the KPFM probe. The response function is computed via the boundary element method in an efficient and robust way, and the deconvolution is performed in combination with certain extrapolation 共not stated兲and a Wiener filter to mitigate the edge effect and the instrument noise, respectively. While Strassburg et al.7have established a systematic methodology for reconstructing surface potentials from KPFM images of flat conductive samples, a counterpart treatment for another class of practically important materials—dielectrics 共insulators兲—still does not exist. An example of such materials is ionic crystals,2which are widely used in electronic devices and thus deserve a detailed model for studying their KPFM measurements. In this work, we will generalize the framework of Strassburg et al.7for solving both the forward 共predicting the image from the prescribed charge distributions兲and inverse 共solving the charge distributions from the image兲problems for dielectrics. In the case of a flat sample surface and a homogeneous isotropic dielectric constant, the counterpart a兲Electronic mail: [email protected]. b兲Also at Department of Mechanical Engineering, Stanford University. REVIEW OF SCIENTIFIC INSTRUMENTS 79, 023711 共2008兲 0034-6748/2008/79共2兲/023711/10/$23.00 © 2008 American Institute of Physics79, 023711-1 Downloaded 17 Jan 2012 to 147.83.95.33. Redistribution subject to AIP license or copyright; see http://rsi.aip.org/about/rights_and_permissions boundary integral equation can be derived using Green’s function with an appropriate image source term. On the other hand, the inverse problem for dielectrics will be shown to be ill posed because of insufficient data, which requires additional information about the material property and its treatment history, e.g., surface modulation that the sample has undergone, in order to obtain a meaningful solution. If the sample can be described by containing either only a surface dipole layer or only a surface trapped charge layer, analogous deconvolution algorithms can be employed to solve for the assumed layer of source. Sample application of this methodology to raw experimental data will be illustrated. II. IMAGING MECHANISM OF KPFM A. Imaging mechanism derived from the model of a simple capacitor KPFM was originally designed for detecting local CPD between a conductive sample and the tip. This apparatus consists of an atomic force microscope 共AFM兲and a control circuit. The KPFM scan is usually performed following a topographic scan by the AFM. After obtaining the sample surface topography, the AFM probe tip is lifted above the sample to some separation and is controlled to scan the sample by tracing the topography obtained from the previous scan. In the mean time, a voltage Vwith both direct current 共dc兲and alternating current 共ac兲components is applied to the probe, V=Vdc +Vac sin共 ␻ t兲,共1兲 where tdenotes time and ␻ is the angular frequency of the ac voltage. Let ⌽denote the tip-sample CPD; then if ⌽does not vary significantly at different locations, the system electrostatic energy can be approximated as W=1 2C共V−⌽兲2,共2兲 where Cis the probe-sample capacitance 共strictly speaking, the capacitance between the sample and the entire cantilever with the probe兲. The attractive capacitive force is then given by F=− ⳵ W ⳵ z,共3兲 where zis the probe-sample separation. Substituting Eqs. 共1兲 and 共2兲into Eq. 共3兲yields F=−1 2 ⳵ C ⳵ z关Vdc +Vac sin共 ␻ t兲−⌽兴2.共4兲 Expanding Eq. 共4兲and grouping terms in terms of their time dependence, Fcan be decomposed as F=Fdc +F ␻ sin共 ␻ t兲+F2 ␻ cos共2 ␻ t兲,共5兲 where Fdc =−1 2 ⳵ C ⳵ z 冋 共Vdc −⌽兲2+1 2Vac 2 册 ,共6兲 F ␻ =− ⳵ C ⳵ zVac共Vdc −⌽兲,共7兲 F2 ␻ =1 4 ⳵ C ⳵ zVac 2.共8兲 These components can be obtained individually using the control circuit, which is essentially obtaining a particular Fourier component by utilizing their orthogonality 共strictly speaking, it is the cantilever oscillation amplitudes that are being recorded兲. Note that in this model, F ␻ =0 if and only if Vdc=⌽.By adjusting Vdc using the feedback circuit to nullify F ␻ , one is able to measure ⌽by recording Vdc.共In reality, it is the cantilever’s oscillation amplitude at frequency ␻ that is being zeroed. Moreover, because of the unavoidable thermal fluctuation, the amplitude is never zero; thus, nullification is replaced with minimization.8In this work, however, we will neglect these errors for simplicity.兲Such an operation is performed for each sampling point on the sample surface before the tip is translated to the next position. All these Vdc values constitute the KPFM image, which is assumed to provide the spatial variation of the CPD ⌽. B. Limitations of the model of a simple capacitor The KPFM image obtained via the imaging mechanism described above is, however, only an approximation to the actual CPD between a tip and a conductive sample. This is because Eq. 共2兲is exact only when ⌽is constant over the sample surface; otherwise, the tip-sample system is no longer a simple capacitor. On the other hand, some materials, especially dielectrics, may possess a surface charge layer, a surface dipole layer, or interior charges, or any combination of these cases. Such samples may not have a well-defined CPD. When they interact with the KPFM, their surface charges, surface dipoles, and interior charges all participate in the overall Coulombic interaction. Thus, a model that takes into account these possible charge/dipole distributions is needed for the forward problem of predicting the KPFM image under certain experimental conditions and for the inverse problem of solving for such charge/dipole distributions from a KPFM image. III. BOUNDARY INTEGRAL EQUATION MODELING FOR THE FORWARD PROBLEM For the forward problem, we shall start with the boundary value problem that describes the probe-sample interaction and transform it to the equivalent boundary integral equation in order to solve for the electric field and predict the KPFM signal. A. Electrostatic boundary value problem Judging from the dimensions of the problem, we can apply the quasistatic approximation despite the timedependent voltage applied between the probe tip and the sample. This approximation allows us to neglect the magnetic field and to use a scalar potential ␾ such that −ⵜ ␾ 023711-2 Shen, Barnett, and Pinsky Rev. Sci. Instrum. 79, 023711 共2008兲 Downloaded 17 Jan 2012 to 147.83.95.33. Redistribution subject to AIP license or copyright; see http://rsi.aip.org/about/rights_and_permissions =E, where Eis the electric field. The KPFM cantilever with the probe tip is assumed to be a perfect conductor, whose potential is a constant given by Eq. 共1兲. The sample is assumed to be semi-infinite with a flat surface and an isotropic homogeneous dielectric constant ␧r. A conductive sample can be treated as a special case in which ␧r→⬁. The electrostatics involved in KPFM can be described by these governing partial differential equations: Poisson’s equation in the sample interior and Laplace’s equation in the open space 共vacuum兲outside the probe tip and the sample, i.e., ⵜ2 ␾ = 再 − ␳ /共␧r␧0兲in ⍀s 0in⍀v, 冎 共9兲 where ⵜ2is the Laplacian operator, ␳ is the trapped sample volumetric charge density, ␧0=8.8542⫻10−12 F/m is the vacuum permittivity, and ⍀vand ⍀sare the volumetric domains of the open space and the sample, respectively. The boundary value problem governed by Eq. 共9兲is a two-domain problem with a boundary condition on the surface of 共the cantilever plus the probe tip兲St, ␾ =Von St,共10兲 and two continuity conditions across the sample surface Ss, ␾ s− ␾ v=−⌽and ␧r␧0 ⳵ ␾ s ⳵ n−␧0 ⳵ ␾ v ⳵ n=⌺on Ss,共11兲 where subscripts vand sdenote whether the values are evaluated on the vacuum side or the sample side, nis the outward surface normal of the sample, and ⌽and ⌺are the sample surface dipole layer 共double layer兲density and charge 共single layer兲density, respectively. A schematic of the boundary value problem is shown in Fig. 1. B. Boundary integral equation If the sample surface Ssis flat, a boundary integral equation equivalent to this boundary value problem can be derived from Green’s second identity as 共see Appendix A兲 冕 St G ˜ 共r;r⬘兲 ␴ 共r兲dS共r兲=V−V⌽共r⬘兲−V⌺共r⬘兲 −V ␳ 共r⬘兲,∀r⬘苸St,共12兲 where the primary unknown function is ␴ 共r兲, the induced surface charge density on the probe tip, which is related to the electric field magnitude immediately outside the conductive probe as ␴ =␧0E. The integral kernel of Eq. 共12兲is G ˜ 共r;r⬘兲⬅G共r−r⬘兲−␧r−1 ␧r+1G共r−r⬘兲,共13兲 where G共r−r⬘兲⬅共4 ␲ ␧0兩r−r⬘兩兲−1 is the free space Green’s function for electrostatics with field point rand source point r⬘, and r⬘ ¯ is the mirror image position of r⬘with respect to Ss. V⌽,V⌺, and V ␳ denote the individual contributions from ⌽,⌺, and ␳ , respectively, V⌽共r⬘兲⬅2␧r␧0 ␧r+1 冕 Ss ⳵ G共r−r⬘兲 ⳵ nr ⌽共r兲dS共r兲,共14兲 V⌺共r⬘兲⬅ 2 ␧r+1 冕 Ss G共r−r⬘兲⌺共r兲dS共r兲,共15兲 V ␳ 共r⬘兲⬅ 2 ␧r+1 冕 ⍀s G共r−r⬘兲 ␳ 共r兲d⍀共r兲.共16兲 Equation 共12兲is a Fredholm integral equation of the first kind. For the case of a conductive sample, we can let ␧r→⬁so that Eq. 共12兲reduces to 冕 St G ˜ 共r;r⬘兲 ␴ 共r兲dS共r兲 =V−2␧0 冕 Ss ⳵ G共r−r⬘兲 ⳵ nr ⌽共r兲dS共r兲,∀r⬘苸St, 共17兲 where G ˜ 共r;r⬘兲=G共r−r⬘兲−G共r−r⬘兲.共18兲 Equation 共17兲agrees with the formulation of Ref. 7. The implementation of the boundary element method for solving Eq. 共12兲is given in Appendix B. For notational simplicity, we define G ˜ −1共r⬙;r⬘兲as the inverse of the kernel G ˜ 共r;r⬘兲such that 冕 St G ˜ −1共r⬙;r⬘兲G ˜ 共r;r⬘兲dS共r兲 = ␦ 共r−r⬙兲,∀r苸St,r⬙苸St,共19兲 where ␦ is the two-dimensional Dirac delta function with respect to St. From now on, G ˜ −1共r⬙;r⬘兲is assumed to be known, and we can formally invert Eq. 共12兲by multiplying G ˜ −1共r⬙;r⬘兲on both sides of Eq. 共12兲and integrate over St with respect to r⬘. Then, ␴ can be written as the sum of contributions from the individual sources V,⌽,⌺, and ␳ , ␴ =V ␴ 1− ␴ ⌽− ␴ ⌺− ␴ ␳ ,共20兲 where ␴ 1共r⬙兲⬅ 冕 St G ˜ −1共r⬙;r⬘兲dS共r⬘兲,共21兲 ␴ ⌽共r⬙兲⬅ 冕 St G ˜ −1共r⬙;r⬘兲V⌽共r⬘兲dS共r⬘兲,共22兲 0 2=∇ φ V= φ V= φ 0= φ t S s S s Ω v Ω )/( 0 2 εερφ r −=∇ FIG. 1. 共Color online兲Schematic of the electrostatic boundary value problem involved in KPFM. 023711-3 Interpreting KPFM images on dielectrics Rev. Sci. Instrum. 79, 023711 共2008兲 Downloaded 17 Jan 2012 to 147.83.95.33. Redistribution subject to AIP license or copyright; see http://rsi.aip.org/about/rights_and_permissions ␴ ⌺共r⬙兲⬅ 冕 St G ˜ −1共r⬙;r⬘兲V⌺共r⬘兲dS共r⬘兲,共23兲 ␴ ␳ 共r⬙兲⬅ 冕 St G ˜ −1共r⬙;r⬘兲V ␳ 共r⬘兲dS共r⬘兲.共24兲 To take into account the time dependence of ␴ due to that of V, we can substitute Eq. 共1兲into Eq. 共20兲to find ␴ 共r⬙,t兲= ␴ dc共r⬙兲+ ␴ ac共r⬙兲sin共 ␻ t兲,共25兲 where for all points on St, ␴ dc =Vdc ␴ 1− ␴ ⌽− ␴ ⌺− ␴ ␳ ,共26兲 ␴ ac =Vac ␴ 1.共27兲 C. Forces in terms of the Maxwell stress tensor The total force exerted on the probe tip can be obtained by integrating over Stthe traction associated with the Maxwell stress tensor.9The force component along the zdirection 共set to be parallel to the outward normal of the sample surface兲is given by F=␧0 2 冕 St E2nzdS =␧0 2T共E,E兲,共28兲 where Eis the electric field magnitude and nzis the zcomponent of the outward surface normal of St. The symmetric bilinear operator T共·,·兲is defined as T共u,v兲⬅ 冕 St uvnzdS,∀u,v苸L2共St兲.共29兲 Since the cantilever and the probe tip are both conductive, Eq. 共28兲can be written in terms of ␴ as F=1 2␧0 T共 ␴ , ␴ 兲.共30兲 Substituting Eq. 共25兲into Eq. 共30兲yields F=Fdc +F ␻ sin共 ␻ t兲+F2 ␻ cos共2 ␻ t兲,共31兲 where Fdc =1 2␧0 T共 ␴ dc, ␴ dc兲+1 4␧0 T共 ␴ ac, ␴ ac兲,共32兲 F ␻ =1 ␧0 T共 ␴ dc, ␴ ac兲,共33兲 F2 ␻ =− 1 4␧0 T共 ␴ ac, ␴ ac兲,共34兲 which will take the place of their counterparts in the model of a simple capacitor: Eqs. 共6兲–共8兲. D. Nullifying condition of KPFM From Eq. 共33兲, the nullifying condition F ␻ =0 according to the KPFM imaging mechanism is equivalent to T共 ␴ dc, ␴ ac兲=0. Solving Vdc from the above equation leads to the signal that the KPFM would register as the “surface potential” for the particular probe-sample configuration. Substituting Eqs. 共26兲and 共27兲into Eq. 共33兲yields F ␻ =Vac sin共 ␻ t兲 ␧0 T共 ␴ 1,Vdc ␴ 1− ␴ ⌽− ␴ ⌺− ␴ ␳ 兲.共35兲 Hence, F ␻ =0 implies Vdc =T共 ␴ 1, ␴ ⌽+ ␴ ⌺+ ␴ ␳ 兲 T共 ␴ 1, ␴ 1兲⬅VKPFM.共36兲 Equation 共36兲states that the KPFM signal VKPFM is a linear functional of the sources ⌽,⌺, and ␳ . E. Example predictions Judging from their spacings from the sample 共and thus the electrostatic forces they exert thereupon兲and for the sake of simplicity, only the tip apex and the conical portion of the probe will be included in the subsequent numerical treatment while the cantilever is neglected, as shown in Fig. 2共a兲共the tip apex radius of curvature, the cone height, and the cantilever length are on the orders of 40 nm, 15 ␮ m, and 100 ␮ m, respectively兲. A magnification for the region in proximity to the tip apex is shown in Fig. 2共b兲. For the first set of examples for the forward problem solution, we choose two dielectric samples, each of which contains a surface dipole layer of ⌽=1 V over a square and 0 otherwise, as shown in Figs. 3共a兲and 3共c兲. The side of the square in Fig. 3共a兲is 1.75 times the tip radius of curvature, whereas that of the square in Fig. 3共c兲is 17.5 times 共i.e., one order larger than兲the tip radius of curvature. In addition, these samples are assumed to have no other sources to interact with the probe. Hence, the surface potentials measured by KPFM, VKPFM, especially the second one, are expected to be faithful to the true potential ⌽. The predicted KPFM images are shown in Figs. 3共b兲and 3共d兲. The shapes of the “measured” potential inhomogeneity do resemble the true surface potential; however, the measured potential values differ from the true values by at least 0 2=∇ φ V= φ t S s S infinite)-(semi s Ω v Ω )/( 0 2 εερφ r −=∇ ( a ) 0 2=∇ φ V= φ t S s S infinite)-(semi s Ω v Ω )/( 0 2 εερφ r −=∇ ( b ) FIG. 2. 共Color online兲共a兲Schematic of the electrostatic boundary value problem simplified by 共i兲neglecting the cantilever and 共ii兲assuming the sample to be semi-infinite. 共b兲Magnification of a region of 共a兲in proximity to the tip apex 关schematically shown in a box in 共a兲兴. 023711-4 Shen, Barnett, and Pinsky Rev. Sci. Instrum. 79, 023711 共2008兲 Downloaded 17 Jan 2012 to 147.83.95.33. Redistribution subject to AIP license or copyright; see http://rsi.aip.org/about/rights_and_permissions 54% and 15% for these two cases. This can be attributed to the long-range characteristic of the Coulombic force, the consequence of which is that the portion of the sample surface outside the square also participates significantly in the interaction with the probe tip. As a result, the KPFM is registering a weighted average of the sample surface potential 共in this case, weighted averages of 1 and 0 V兲, leading to this instrumental error. In the second set of examples, we choose dielectric samples containing a surface charge layer, as shown in Figs. 4共a兲and 4共d兲, where the sizes of the charged areas are the same as those in Figs. 3共a兲and 3共c兲, respectively. The single layer potentials on the surfaces due to these charges in the absence of any instrument 关literally surface potential兴, Eq. 共15兲for r⬘苸Ss兲are plotted in Figs. 4共b兲and 4共e兲, respectively, which are assumed to be what an ideal instrument would register. The predicted KPFM images are shown in Figs. 4共c兲and 4共f兲, from which one can also see the broadening effect for small areas of charges; moreover, the errors of KPFM-measured potential values 共referenced to the surface potentials兲at the centers are also as large as 45% and 10%, respectively. Finally, the effect of the interior charge distribution ␳ is similar to that of the surface charge distribution ⌺since both of them enter the boundary integral equation 关Eq. 共12兲兴with the same kernel G共r−r⬘兲关see Eqs. 共15兲and 共16兲兴. Hence, the counterpart study for the effect of ␳ is omitted for conciseness. F. Convolution relation Equation 共36兲is the solution to the forward problem for a particular tip-sample configuration. To write the entire KPFM image in terms of the sample sources to facilitate further analysis, we will follow the treatment in Ref. 7to decompose a typical position vector r苸Stas r=rt+r*,共37兲 where rt=共xt,yt,0兲苸Ssis the nominal probe tip position, defined as the tip apex projected onto the sample surface Ss. The primary unknown of Eq. 共12兲, ␴ , can now be written as ␴ 共r*;rt,t兲, and all the integrations over Stwith respect to rcan now be written as integrations over r*, while the dependence of the probe tip position rtbecomes explicit. As a consequence, the KPFM signal VKPFM共xt,yt兲can be expressed by rewriting Eq. 共36兲as linear combinations of convolutions VKPFM =⌽ⴱR⌽+⌺ⴱR⌺+关 ␳ ⴱR ␳ 兴zt=0,共38兲 where the first two *’s are two-dimensional convolutions with respect to 共xt,yt兲, while the last *is a three-dimensional convolution with respect to 共xt,yt,zt兲. The response functions R⌽,R⌺, and R ␳ are defined as R⌽共rt兲⬅T关 ␴ 1,s⌽共·;rt兲兴/T共 ␴ 1, ␴ 1兲,共39兲 R⌺共rt兲⬅T关 ␴ 1,s⌺共·;rt兲兴/T共 ␴ 1, ␴ 1兲,共40兲 R ␳ 共rt,zt兲⬅T关 ␴ 1,s ␳ 共·;rt,zt兲兴/T共 ␴ 1, ␴ 1兲,共41兲 where s⌽共r*;rt兲⬅ ␧r 2 ␲ 共␧r+1兲 冕 St G ˜ −1共r*;r**兲z** ⫻关共xt+x**兲2 +共yt+y**兲2+共z**兲2兴−3/2dS共r**兲,共42兲 FIG. 3. 关共a兲and 共c兲兴 Prescribed surface dipole density 共true surface potential兲⌽of samples with dielectric constant ␧r=30.0 with no trapped charge distributions. ⌽=1 V over a square centered at 共0,0兲with a side of 共a兲70 nm, or 共c兲700 nm, and 0 otherwise. 关共b兲and 共d兲兴 Predicted KPFM image of these samples. The KPFM probe tip radius of curvature is taken to be 40 nm, the tip total length is 15 ␮ m, the tip half conical angle is 15°, and the tip-sample separation is 15 nm. The maximum predicted VKPFM 关the value at 共0,0兲兴 are 共b兲 0.46 V and 共d兲0.85 V, so that the minimum pointwise errors of the surface potential are 54% and 15%, respectively. If the pixel shown in 共a兲is used as the interpolation function for ⌽, then a typical component of the discrete response function, 关R⌽ d兴ij, equals the predicted KPFM intensity at point 共x,y兲=共i⌬,j⌬兲in 共b兲. 023711-5 Interpreting KPFM images on dielectrics Rev. Sci. Instrum. 79, 023711 共2008兲 Downloaded 17 Jan 2012 to 147.83.95.33. Redistribution subject to AIP license or copyright; see http://rsi.aip.org/about/rights_and_permissions s⌺共r*;rt兲⬅ 1 2 ␲ 共␧r+1兲 冕 St G ˜ −1共r*;r**兲⫻关共xt+x**兲2 +共yt+y**兲2+共z**兲2兴−1/2dS共r**兲,共43兲 s ␳ 共r*;rt,zt兲⬅ 1 2 ␲ 共␧r+1兲 冕 St G ˜ −1共r*;r**兲⫻关共xt+x**兲2 +共yt+y**兲2+共zt+z**兲2兴−1/2dS共r**兲. 共44兲 Here, we have extended the definition of ␳ such that ␳ 共x,y,z兲=0,∀z⬎0. One of the response functions, R⌽, has the property that 冕 Ss R⌽共rt兲dS共rt兲=␧r ␧r+1.共45兲 Hence, for a conductive sample for which ␧r→⬁, the above integral tends to unity. Furthermore, a conductive sample implies ⌺=0 and ␳ =0; thus, Eq. 共38兲becomes a weighted average of ⌽, as has also been reported by Ref. 6. IV. DATA INCOMPLETENESS OF THE INVERSE PROBLEM While the forward problem of predicting KPFM images from dielectric sample properties is a classical potential problem, solving the responsible sample properties from such images is an inverse problem with insufficient data. The primary insufficiency comes from the fact that for a particular sample, the surface dipole ⌽, the surface trapped charge ⌺, and the interior trapped charge ␳ contribute to the overall Coulombic interaction simultaneously, and their contributions are difficult to isolate from one another. As a result, without further information there will be no unique solution given a particular image. To overcome this difficulty, one needs a more detailed description of the sample properties in order to deduce more equations or less unknowns. For example, the interior charge contribution to the image can be absorbed in the surface charge contribution term by assuming that the detectable charges are distributed close to the surface. The justification is that KPFM, being a surface characterization technique, is usually not expected to be able to register the charge variations inside the sample due to the unavoidable thermal noises. Another incompleteness comes from the fact that only a FIG. 4. 关共a兲and 共d兲兴 Contour plots of prescribed surface charge densities ⌺with supports as pixels with dimensions ⌬⫻⌬, where ⌬=70 nm and 700 nm, respectively. For each case, ⌺equals some positive constant over the square and 0 otherwise. 关共b兲and 共e兲兴 The single layer potentials on the sample surface 共surface potential兲due to the charges in 共a兲and 共d兲关Eq. 共15兲for r⬘苸Ss兴, respectively. In both cases, the tip is absent. The potentials are normalized so that the maximum values are 1 V. 关共c兲and 共f兲兴 Predicted KPFM images due to charges described in 共a兲and 共d兲, respectively. The experimental setup and the sample property are the same as those in Fig. 3. The potential values at the centers of the squares of 共c兲and 共f兲are 0.55 and 0.90 V, respectively. If the pixel shown in 共a兲is used as the interpolation function for ⌺, then a typical component of the discrete response function, 关R⌺ d兴ij, equals the predicted KPFM intensity at point 共x,y兲=共i⌬,j⌬兲in 共c兲. 023711-6 Shen, Barnett, and Pinsky Rev. Sci. Instrum. 79, 023711 共2008兲 Downloaded 17 Jan 2012 to 147.83.95.33. Redistribution subject to AIP license or copyright; see http://rsi.aip.org/about/rights_and_permissions subset of the sample surface is imaged, but sources 共charges/ dipoles兲outside the imaging area also contribute to the overall Coulombic interaction. This incompleteness will be overcome in the next section when we implicitly introduce periodic assumption of the source in order to perform the discrete Fourier transform. V. DECONVOLUTION ALGORITHM FOR TWODIMENSIONAL DIPOLE OR CHARGE DISTRIBUTIONS If the sample of interest can be modeled as having either only a surface dipole layer distribution ⌽or only a surface charge layer distribution ⌺that is responsible for the Coulombic interaction with the probe tip, then there remains only one term on the right hand side of Eq. 共38兲共either the first term or second term, correspondingly兲, and Eq. 共38兲 becomes a one-to-one mapping from the source to the image, allowing one to solve for the source from the image via deconvolution. Such a deconvolution algorithm was first proposed by Strassburg et al.7to solve for the surface potential variations of flat conductive samples. They implemented it for the special case of one-dimensional surface potential variation, i.e., the case in which the surface potential is a function of only one Cartesian coordinate. We have generalized this algorithm for dielectric samples and performed, for the first time, a two-dimensional deconvolution study on a KPFM image to solve for the sample’s surface charge density. A. Discrete convolution relation The expression for the KPFM signal, Eq. 共36兲or 共38兲,is a continuous function of the nominal probe tip position 共xt,yt兲, whereas the pixels of the KPFM image are measurements at only a discrete set of sampling points. Thus, at the same level of interpolation accuracy, we can interpolate the true surface sources 共⌽and/or ⌺兲with piecewise constant functions defined over pixels centered at these sampling points. The following derivations will be specialized to the case of solving for ⌺while assuming ⌽=0 and ␳ =0. The case of solving for ⌽by assuming ⌺=0 and ␳ =0 is similar. We assume that there are N⫻N共Nis an even number兲 grid points 共xt i,yt j兲,i,j=0,1,...,共N−1兲, with spacings ⌬ along both the xand ydirections; then, the surface charge ⌺ can be written as ⌺共xt,yt兲=兺 i=0 N−1 兺 j=0 N−1 ⌺ij db共xt−xi,yt−yj兲,共46兲 where the pixel function bis defined as b共 ␰ , ␩ 兲⬅ 再 1兩 ␰ 兩艋⌬/2 and 兩 ␩ 兩艋⌬/2 0 otherwise. 冎 共47兲 The ⌺ij d’s are unknown coefficients to be determined via the deconvolution. In the case of ⌽=0 and ␳ =0, substituting Eq. 共46兲into Eq. 共38兲yields the expressions for the pixel intensities, Kmn ⬅VKPFM共xt m,yt n兲=兺 i=0 N−1 兺 j=0 N−1 ⌺ij d关R⌺兴m−i,n−j =关⌺d*R⌺ d兴mn,m,n=0,1, ...,共N−1兲,共48兲 where xt m⬅m⌬,yt n⬅n⌬, and the discrete response function R⌺ dis defined as the convolution of the continuous response function and the pixel function 关R⌺ d兴ij ⬅关R⌺ⴱb兴共xt i,yt j兲,i,j=−N/2, ...,N/2. 共49兲 The components of R⌺ din Eq. 共48兲with indices outside the range of 关−N/2,N/2兴2are defined by shifting either or both indices by an integer multiple of Nto render the indices in that range. For example, 关R⌺ d兴0,N−1⬅关R⌺ d兴0,−1. Using the convolution theorem, the discrete Fourier transform 共denoted F兲of Eq. 共48兲is FK=共F⌺d兲共FR⌺ d兲.共50兲 Equation 共50兲will be the basis of the deconvolution algorithm. B. Computing the discrete response function The discrete response function 关R⌺ d兴ij is defined in Eq. 共49兲via its continuous counterparts. However, if one has set up the computational facility for the forward problem, the response function can be directly obtained by assuming a pixel source located at 共i,j兲=共0,0兲, i.e., we input ⌺such that ⌺共xt,yt兲=b共xt,yt兲共51兲 or, equivalently, ⌺ij d= ␦ i0 ␦ j0,共52兲 where ␦ is the Kronecker delta; then, Eq. 共48兲becomes Kmn =关R⌺ d兴mn.共53兲 Thus, one can compute R⌺ mn by solving a corresponding forward problem. This forward problem is defined by substituting Eq. 共51兲and ⌽=0, ␳ =0 in Eq. 共12兲and placing the tip at 共xt m,yt n兲. An example calculation is given in Fig. 4共c兲for the pixel that we will use 关Fig. 4共a兲兴for the deconvolution in Sec. V D. It can be seen that a localized source in Fig. 4共a兲 becomes a broadened response, as shown in Fig. 4共c兲, due to the finite curvature of the probe tip. The counterpart for R⌽ d which is related to R⌽*bis shown in Fig. 3共b兲for the pixel source of surface dipole given in Fig. 3共a兲. C. Enforcing the sample electric neutrality To guarantee the electric neutrality while maintaining the consistency of the deconvolution algorithm, 关FK兴00 is forced to be zero, so that 关F⌺d兴00 is also zero. This preprocessing is consistent with the observation that the integral 冕 Ss R⌺共rt兲dS共rt兲共54兲 diverges, i.e., 关FR⌺兴共0,0兲=⬁, where Fdenotes the Fourier transform. As a result, from the convolution theorem, if 关FVKPFM兴共0,0兲⬍⬁, then 关F⌺兴共0,0兲=0, which automatically satisfies the electric neutrality. In the discrete counter023711-7 Interpreting KPFM images on dielectrics Rev. Sci. Instrum. 79, 023711 共2008兲 Downloaded 17 Jan 2012 to 147.83.95.33. Redistribution subject to AIP license or copyright; see http://rsi.aip.org/about/rights_and_permissions part, however, 关FR⌺ d兴00 is a finite number, so that the electric neutrality has to be explicitly enforced. D. Sample application to experimental data We will apply the deconvolution algorithm for a KPFM image obtained by Lee10 from experiments shown in Fig. 5共a兲. The sample is a gadolinia-doped ceria 共GDC兲thin film which is a popular material used as electrolyte in solid oxide fuel cells. This thin film has a thickness of 50–100 nm. It was fabricated by sputtering Gd0.2Ce0.8 alloy 共Kurt Lesker Co.兲on a 200 nm thick Pt layer 共serving as a reference electrode兲, which had been sputtered on a silicon nitride film. The Gd0.2Ce0.8 film was then oxidized in air at 650 °C for 5h. The surface modulation, topographic measurement, and the surface potential measurement on the GDC film were performed in ambient condition at room temperature using the same instrument, Molecular Imaging Inc. ’s PicoPlus II AFM with a probe coated with highly boron-doped diamond. In addition, commercial radio frequency lock-in amplifiers with dedicated circuitry were set up for imaging the surface potential using the KPFM mechanism described in Sec. II A. Before being imaged, the sample surface was modulated by placing the probe tip in contact with the sample and applying a positive dc voltage on the tip 共referenced to the Pt layer兲to facilitate some chemical reaction. After that, the topography and the surface potential were measured in the contact mode and the amplitude modulation mode, respectively. Such measurements were performed for each scan line to generate maps of topography and surface potential. As a result of the surface modulation, a peak centered at the former contact region is observed in the subsequent KPFM image, indicating charge concentration in this region compared to the rest of the imaging area. In order to obtain the surface charge density from the KPFM image, the deconvolution algorithm will be used. However, since this is a set of raw data, there would be instrument noise superposed to the measured VKPFM. This noise is sensitive to deconvolution operations; hence, we apply the Wiener filter11 prior to the subsequent treatment. Furthermore, to mitigate the edge effect due to the implied two-dimensional periodic assumption of both the source and the measurement 共implied by the discrete Fourier transform and the discrete convolution relation兲, we have extended zeros to the filtered data set. For the ideal case of a response function with compact support, the number of zeros needed in each direction is, according to Ref. 13, Chap. 13, at least half the maximum width of the response function in that direction. However, the response function R⌺not only lacks compact support but also decays to zero slowly. Thus, we cannot use the rule suggested by Ref. 13. Instead, we try to extend more zeros in each direction than there are number of data points in the same direction of the original data set. For concreteness, the raw data10 has 59⫻59 data points. We have examined the effects of extending zeros to render 128⫻128 or 256⫻256 data points 共N=128 or 256, respectively兲. The corresponding discrete response functions have nonzero values in all entries. According to Eq. 共49兲, their indices are in the range of 关−64,64兴2or 关−128,128兴2, respectively. After the deconvolution, only those entries in ⌺dcorresponding to the raw data 共rather than the zeros兲are retained. The surface charge density ⌺’s obtained using this algorithm with the two choices of N’s are plotted in Figs. 5共c兲 and 5共d兲, which do not appear to be significantly different to the naked eye. This means that zero extending to whether FIG. 5. 共a兲Raw 共Ref. 10兲and 共b兲 noise-filtered 共with the Wiener filter兲 KPFM images on a gadolinia-doped ceria thin film. The units of both are V. The probe is 15 ␮ m long and has a half conical angle of 15°. The probe tip radius is 40 nm. The tip-sample separation is 15 nm. The dielectric constant ␧ris 30.0. The spacing of the sampling points is ⌬=70 nm. The number of sampling points in either direction is 59. 关共c兲and 共d兲兴 Contour plots of the reconstructed surface charge density ⌺from 共b兲via deconvolution. The unit is e/nm2, where eis the elementary charge. Before deconvolution, the filtered data are extended with zeros to render 共c兲128⫻128 or 共d兲256⫻256 data points to mitigate the edge effect. The discrete response functions are arrays of the respective sizes. 023711-8 Shen, Barnett, and Pinsky Rev. Sci. Instrum. 79, 023711 共2008兲 Downloaded 17 Jan 2012 to 147.83.95.33. Redistribution subject to AIP license or copyright; see http://rsi.aip.org/about/rights_and_permissions