Hybrid Numerical Simulation of Electrostatic Force Microscopes Considering Charge Distribution

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1 PIERS ONLINE, VOL. 3, NO. 3, Hybrid Numerical Simulation of Electrostatic Force Microscopes Considering Charge Distribution U. B. Bala, M. Greiff, and W. Mathis Institute of Electromagnetic Theory Leibniz University of Hannover, Germany Abstract The electrostatic force microscope (EFM) is an important tool for imaging and characterizing material surfaces. In this paper a hybrid numerical approach for the simulation of the EFM considering charge distribution inside the sample under investigtion is presented. In the simulation model electrical part is considered. In this paper first a basic knowledge on the EFM and then the numerical model of the EFM considering volume charge distribution are presented. At last several numerical simulation results of the EFM in 3D are pesented. DOI: /PIERS INTRODUCTION A significant progress in nanotechnology has been observed over the last few years. This progress has also been influenced by the development of new high resolution measurement instruments. Due to the rapid miniaturization of integrated devices into the mesoscopic regime and the increasing interest in very small structures, these instruments have become very important. An interesting example is the atomic force microscope (AFM). Based on the design of the scanning tunneling microscope (STM), the first AFM was developed in 1986 by G. Binnig and his coworkers in collaboration between IBM and Standford university. Since then a new era of topographical imaging, as well as for measuring force-separation interactions between a probe and substrate began. The AFM s ability to scan surfaces with nearly atomic resolution and its versatility make it one of the most important measurement devices in nanotechnics. If the sample under investigation holds a charge distribution and the distance between the AFM tip and the sample is kept large then all other interaction forces except the electrostatic force can be neglected. This special working mode of the AFM is known as electrostatic force microscope (EFM) which can be used for scanning electric field with nearly atomic resolution. The EFM has many materials-related applications including measuring the surface potential or contact potential, detecting charges on surfaces or nanocrystals etc. In this paper a 3D model of the EFM is presented considering charge distribution inside the sample. Several numerical methods are proposed to calculate the electric eld more efficiently. 2. WORKING PRINCIPLE AND MODEL OF THE EFM For the EFM the interaction force would be the electrostatic force between the biased atomically sharp tip and the sample. In addition the Van der Waals force between the tip and the sample are always present. The Van der Waals force and the electrostatic force have two different dominant regions. The Van der Waals force is proportional to 1/r 6 where as the elctrostatic force is proportional to 1/r 2. Thus when the tip is close to the sample the Van der Waals force is dominat and when the tip is moved away from the sample the electrostatic force is dominant. The scanning of the EFM is usually done in two steps. First the topography of the sample is done by tapping scanning mode which is also known as intermittent-contact (IC) mode. In this case the Van der Waals force plays a signifiant role. Second using this topgraphical information a constant tip-sample distance is maintained while scanning where the electrostatic force is dominat, a technique which is known as lift scanning [1]. In this technique it is assumed that the influence of all short-range forces can be neglected and only the electrostatic force plays the vital role for imaging. To detect the electrostatic force a voltage is applied between the cantilever tip and the sample. The cantilever oscillates near its resonance frequency which changes in response to any additional force gradient. Changes in cantilever resonant frequency can be detected using phase detection, frequency modulation, amplitude modulation etc. A diode laser is focused on the back of the reflective cantilever and the reflected light is collected by a position sensitive detector (PSD). This usually consists of two closely spaced photodiodes. Any angular displacement of the cantilever results in one photodiode

2 PIERS ONLINE, VOL. 3, NO. 3, collecting more light than the other and therefore in a high output voltage. This voltage then plots the topography of the sample. Some typical parameters of the EFM i.e., the length of the cantilever is some hundreds of 1 µm, the height of the tip is nearly 30 nm and the pick of the tip is usually less than 10 nm. So for modeling and simulating the EFM, multi physics aspects must be taken into consideration. From the numerical point of view additional problems arise since frequently we are confronted with multi-scale problems. Therefore the application of advanced numerical methods is necessary. As the cantilever frequently changes its position during scanning, the coupled mechanical and electrical behavior have to be taken into account. This can be achieved by dividing the model into an electrical part and a mechanical part [4]. The interaction between them can conveniently be realized by using a staggered simulation approach. In the present model the electrical part is considered, i.e., the cantilever deflection is kept fixed. For developing a model of the EFM different effects have to be considered. For example long distance interaction, charge distribution and possible non-linearity of the material properties, singularity etc. In order to take into consideration these effects the simulation region is divided into three regions as shown in Fig. 1. As high values of the electric field will occur at the pick of the tip, a special numerical method is needed to calculate this electric field more effectively. For this reason an augmented FEM will be applied to region Ω M. Since charge distribution and nonlinearities of the dielectric properties may have to be considerd, a versatile numerical method such as finite element method (FEM) should be applied to region Ω F. As boundary element method (BEM) works well when the boundary is infinite or semi-infinite, the large distance interaction between the tip and the cantilever can be conveniently treated by using BEM in region Ω B. Later all these three numerical methods will be coupled with each other. 3. NUMERICAL FORMULATION OF THE PROBLEM Consider a bounded domain Ω R be bounded with Lipschitz boundary Γ = Ω which is decomposed into three disjoint parts Ω = Ω M Ω F Ω B. At present Poisson s equation in Ω will be applied with mixed boundary conditions. Find u : Ω R such that u = f in Ω u = u 0 on Γ D u n = t 0 on Γ N (1) where f is the charge density, u 0 and t 0 are the Dirichlet and Neumann boundary conditions respectively. The detail Formulation of Laplace problem for FEM-BEM-MFS is shown in [2] by the same authors. Here only some important steps will be presented which is necessary to get the basic understanding of the formulation of Poisson problem. The energy-related functional in the electrostatic calculation domain Ω (Fig. 1) can be written as W = ( u) 2 dω u HD(Ω) 1 : { u H 1 } u ΓD = u 0 (2) Ω The electrostatic potential u in the spherical region Ω M of radius R can be approximated by [13] n l ( u(x, y, z) = u j ψ j (x, y, z) + M(r) A lm r l + B lm r (l+1)) Y lm (θ, φ), (3) where j=1 l=0 m= l 1, ( ( 0 r R 2 M(r) = cos 2 R r 1) π ) R, 2 < r < R 0, R r and Y lm (θ, φ) can be expressed by the Legendre functions P lm (cos θ) [9]. This leads to the stiffness matrix = 2 u u dω, = 2 u u dω, = 2 u u dω. (5) u i Ω u i A ik Ω A ik B ik Ω B ik (4)

3 PIERS ONLINE, VOL. 3, NO. 3, On the FEM-BEM transmission interface [4] Γ T = Γ B Γ F, u B = u F and ub n + uf n = 0. Using the Gauss theorem on Ω F M = Ω F Ω M one obtains [5] u F M n v dγ = div ( u F M v) dω = u F M v dω + u F M v dω (6) Ω F M Ω F M Ω F M Γ F i.e., for all v H 1 D,0 (Ω F M) := { v H 1 (Ω F M ) : v ΓD Γ F = 0 } u F M a (u F M, v) := u F M v dω = f v dω+ Ω F M Ω F M Γ F n v dγ=: (f, v) uf M Ω F M + n, v (7) Γ F where u F M includes u j, A lm and B lm. The representation formula of the Laplace s equation for the solution of u B inside Ω B { u B (x) = n(y) G(x, y)u B(y) G(x, y) u } B dγ, x Ω B (8) n(y) Γ B with the fundamental solution of the Laplacian in 3D given by G(x, y) = 1 4π x y 1 (9) For Poisson s problem the two boundary integral equations on the BEM region V u B n = (I + K)u B N 0 f (10) W u B = (I K ) u B n N 1f (11) where the single layer potential V and the hypersingular operator W are symmetric and the double layer potential K has the dual K [3]. The integral operators N 0 and N 1 are defined by N 0 f(x) := G(x, y)f(y)dy, N 1 f(x) := N 0 f (12) Ω B n x The saddle point formulation of the problem for all (w, v, ψ) H 1/2 H 1 D,0 (Ω F M) H 1/2 (Γ B ) 2a (u F M, v) + W u B, v ΓT + ( I + K ) ϕ, v Γ T = 2(f, v) ΩF M + 2 t 0, v Γn Γ F N 1 f, v ΓT (13) W u B, w ΓB Γ N + (I + K )ϕ, w Γ B Γ N = 2 t 0, w ΓB Γ N N 1 f, w ΓB Γ N (14) (I + K)u B, ψ ΓB V ϕ, ψ ΓB = N 0 f, ψ ΓB (15) If the bases are introduced as span{v , v F } = X F, span{w 1,......, w F } = X B and span{ψ 1,......, ψ F } = Y B, the basis functions of X F and X B are supposed to be ordered such that span{v 1,......, v F } = X F H 1 D,0(Ω F ) span{w 1,......, w B } = X B H 1/2 (Γ B ). If the coefficients of u F M and u B are denoted by u and the coefficients of ϕ are denoted by ϕ again then this system is equivalent to the original differential equation that can be used for descritization. This system corresponds to a matrix formulation which can be written as M B T B F NN F NC F CN F CC + W CC W CN ( K T + I ) C 0 0 W NC W NN ( K T + I ) N 0 0 (K + I) C (K + I) N V u m u F u T u B ϕ = where the subscript C means contribution from the coupling nodes and N means contribution from the non-coupling nodes. Finally the blocks W, V, K + I, and K T + I provide the coupling b m b F b Γ b B b ϕ (16)

4 PIERS ONLINE, VOL. 3, NO. 3, between the two ansatz spaces XF and XB. Here um includes the coecients Alm and Blm (3), uf and ub are the nodal potentials inside the FE domain and on the boundary of the BE domain respectively, ut are the nodal potentials on the FE-BE coupling interface and ϕ are the normal components of the electric field distribution on the boundary of the BE domain. The vector b includes the corresponding boundary conditions and the charge distribution. As the matrix in (16) is not positive definite, a specific algorithm such as the MINRES algorithm is required for the solution. A typical simulation of potential distribution and electric field distribution are shown in Figs. 2 and 4 respectively. In the present model some volume charges are injected on the top portion inside the sample. Equal amount of positive volume charge density are present on the left and right volumes where as the negative volume charge density are present in the center volume. As this charges will cause potential inside the volume, they will produce remarkable effects on the simulation result. The result of a typical simulation of potential distribution and electric field distribution considering a moving sample using FEMBEM coupling in the cut plane through the middle of the cantilever is shown in Figs. 3 and 5 respectively. Usually the highest value of electric field appears at the pick of the tip which plays the dominant role for cantilever deflection. But it may be the case due to the different values of volume charge density, the highest value of electric field may not appear on the pick of the tip but near the volume charges. Since the sample is moving there must be a deviation of electric field at the pick of the tip. By utilizing this deviation of electric field the EFM will plot the charge distribution on the moving sample. Since the scanning process of EFM is dynamic, one has to deal with a moving sample and moving boundaries. As a result the mesh has to be updated at each time step. The approach presented here for mesh updating is based on an arbitrary Lagrangian Eulerian (ALE) algorithm. 4. FIGURES Figure 1: 3D EFM model. Figure 2: Potential distribution on different places of an EFM. Figure 3: Potential distribution on the middle position of EFM with moving sample. Figure 4: Electric field distribution on different places of an EFM.

5 PIERS ONLINE, VOL. 3, NO. 3, Figure 5: Electric field distribution on the middle position of EFM with moving sample. 5. CONCLUSION A hybrid numerical approach for the simulation of the EFM considering charge distribution inside the moving sample is presented. In order to fulfill the special requirements different numerical methods are applied to different regions of the EFM. Here the simulation is implemented using FEMBEM coupling. As a high value of electric field is observed near the pick of the tip, implementing augmented FEM near the tip would obviously calculate this field more efficiently. Due to the moving sample and cantilever deflection the mesh needs to be updated at each time step. It can be obtained using ALE. ACKNOWLEDGMENT The authors would like to thank PD Dr. rer. nat. M. Maischak (Institute of Applied Mathematics, Hanover University) for his contribution and valuable discussions on numerical coupling. This work has been supported by the German Research Foundation (DFG), GRK 615. REFERENCES 1. Yan, M. and G. H. Bernstein, Apparent height in tapping mode of electrostatic force microscopy, Ultramicroscopy, Vol. 106, , Bala, U. B., M. Greiff, and W. Mathis, Hybrid numerical simulation of electrostatic force microscopes in 3D, Proceedings of Progress in Electromagnetic Research Symposium, Tokyo, Japan, August Stephan, E. P., Coupling of boundary element methods and finite element methods, Encyclopedia of Computational Mechanic, Vol. 01, Chapter 13, , Greiff, M., U. B. Bala, and W. Mathis, Hybrid numerical simulation of micro electro mechanical systems, Progress in Electromagnetic Research Symposium (PIERS), , Reddy, J. N., Finite Element Method, McGraw-Hill International Editions, Singapore, Bhushan, B., Handbook of Nano-technology, Springer, Berlin, Li, Z. C., Numerical Methods for Elliptic Problems with Singularities, World Scientific, Jackson, J. D., Classical Electrodynamics, Wiley, Jacobs, H. O., P. Leuchtmann, O. J. Homan, and A. Stemmer, Resolution and contrast in Kelvin probe force microscopy, Journal of Applied Physics, Vol. 84, No. 3, Morita, S., R. Wiesendanger, and E. Meyer, Noncontract Atomic Force Microscopy, Springer, Belytschko, T., K. W. Liu, and B. Moran, Nonlinear Finite Elements for Continua and Structures, Wiley, Sussex, Costabel, M., A symmetric method for the coupling of finite elements and boundary elements, The Mathematics of Finite Elements and Applications, Vol. VI, , Strang, G., G. and J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, 1973.