Finite element simulations of the resolution in electrostatic force microscopy

Abstract

In this paper, we present simulation results for the electrostatic force between two conducting parts placed at different voltages: an atomic force microscope (AFM) sensor and a metallic sample. The sensor is composed of a cantilever supporting a conical tip terminated by a spherical apex. The simulations are based on the finite element method. For tip-sample distances (5-50 nm) and for an electrically homogeneous plane, the electrostatic force can be compared to the results obtained with the equivalent charge model and experiment. By scanning a plane with a potential step, the variation of the electrostatic force near the discontinuity gives the spatial resolution in electrostatic force microscopy (EFM). We establish then the relationships between the resolution, tipsample distance, and tip apex radius.The electrostatic force microscope results from one of many specializations of tip sensor in near-field microscopy [1, 2]. More precisely, this type of microscope is realized by applying a voltage on a conducting AFM tip. It is a good tool for imaging samples that present a gradient of electrical properties [3][4][5]. Variations of flexion of the cantilever holding the tip during a scan allow us to construct an electrical image [6] on inhomogeneous materials as well as on nanostructures (superlattices, nanoelectronics, etc.) [7][8][9]. In the simple case where the tip is in front of a conductive plane sample, we can deduce the force applied on the sensor by means of analytical expressions [10][11][12] or an equivalent charge model [13]. As soon as the geometry of the sample becomes complex (integrated circuits, dielectrics), the theoretical behavior of the system can be obtained by numerical methods such as the surface charge method [14], finite difference method [15], or finite element method [16].To determine the properties of the electrostatic force microscope in front of a sample with areas at different potentials, we propose to use the finite element method. In Sect. 1 we verify the results obtained by this numerical method in the simple case of a tip in front of a plane sample at constant potential [13]. In Sect. 2 we consider the response of the microscope near a potential step [17]. For this, we study the 3-dimensional tip-object system and determine the force applied on the tip by the finite element method and then we deduce the resolution for a potential step. Mathematical model The electrostatic problemThe problem consists in determining the interaction between an AFM sensor (tip + cantilever) and an infinite plane (both conducting). If the tip is long enough or the distance d between the tip and the sample is small, we can neglect the effect of the cantilever [18]. Then, the study is reduced to the calculation of the force exerted on a conical tip in front of a metallic plane. We treat the problem in 3-dimensional space because heterogeneities, as introduced in Sect. 2, cause the revolution symmetry to disappear. First, we must obtain the potential distribution in the space between the tip and the plan...

Multiscale approach for simulations of Kelvin probe force microscopy with atomic resolution

“…13,27,31,32 Several researchers developed models and computational schemes based on classical electrostatics which treated the tip and the sample (sometimes also the cantilever) as macroscopic bodies in order to interpret the resolution of KPFM images of inhomogeneous surfaces on lateral scales of several nanometers and above. [33][34][35][36][37][38][39][40] On the other hand, only few authors considered atomistic nano-scale tip-sample systems, either neglecting 16,41 or including the macroscopic contributions via simple approximations. In the first theoretical study of combined NCAFM-KPFM on an ionic crystal sample, 5,29,42 a formally correct partitioning was proposed between capacitive and short-range electrostatic forces induced by the effective macroscopic bias V .…”

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Evaluation of Kelvin probe force microscopy for imaging grain boundaries in chalcopyrite thin films

“…The contact potential is obtained by minimizing the electrostatic forces; thus, a weighted average of the contact potentials in a certain region below the tip is obtained. [16][17][18] We have applied three dimensional finite element methods to simulate the electrostatic field of the tip-sample geometry, assuming a flat sample topography. The tip potential to minimize the electrostatic forces is deduced for each position of the tip on the sample.…”

Exact solutions of AFM scanning probes subjected to tip-sample forces

“…These forces are usually modeled by a phenomenological Lennard-Jones force which is composed of a repulsive force and a van der Waals force. The Lennard-Jones force F L is [Belaidi et al 1998]…”