Interpretation of KPFM Data with the Weight Function for Charges

Abstract

The KPFM signal for systems containing local charges can be expressed as a weighted sum over all local charges. The weight function for charges quantifies the contribution of each charge, depending on its position. In this chapter, we evaluate the KPFM weight function for charges by analyzing several application-relevant model systems. The intention of this chapter is to provide insights into the KPFM contrast formation in order to facilitate the KPFM data interpretation. For this, we concentrate on three model systems: (A) a conductive sample in ultra-high vacuum, (B) a dielectric sample in ultra-high vacuum, and (C) a dielectric sample in water. We calculate the weight function for charges for each of these systems using a conductive sphere as a tip model. While the analysis substantiates a number of known experimental observations, it reveals surprising effects in some environments. For example, the sign of the FM-KPFM signal reflects the sign of the charges measured in the systems A and B, but in system C the sign of the KPFM signal is found to be tip-sample distance dependent. Additionally, we deduce the lateral KPFM resolution limits and finally discuss the lateral decay of the weight function to assess how charges contribute to the signal. Our discussion is accompanied by an interactive visualization available at www.pc1.uni-bielefeld.de/kpfm.

Appendix A: Electrostatic Model

In this appendix we consider different electrostatic models that can be used to calculate the weight function for charges. Moreover, we also discuss the model for the tip-sample system that we use in this chapter.

There is a large number of both approximative and exact solutions available in literature for the electrostatic problem of tip-sample systems of various compositions and geometries. As the weight function for charges depends only on the normalized electrostatic potential $\hat{\varPhi }_\text {void}$ and the capacitance $C_\text {void}$ of the void tip-sample system (see also the previous chapter), we only consider void, i.e., charge-free tip-sample systems. Thus, we here exclude solutions of the full electrostatic system including charges such as given by Pan et al. [24] for a hyperboloid tip and a point charge, by Uchida for a single charge in a series of parallel-plate capacitors [25], and by Terris et al. [26] for two point charges. We furthermore exclude the cases of a semiconducting substrate as well as non-metallic tips [27].

Models presented in literature can be separated into approximative and exact models, as we will discuss in the following.

1.1 A.1 Approximative Solutions

Several approximations have been put forward over the last years that allow for obtaining an approximative solution of the electrostatic potential and capacitance of the void tip-sample system. The equivalent charge model (ECM) [28,29,30,31], also known as the generalized image charge method (GICM) [32,33,34,35], has been employed by several authors, often to consider a more elaborate tip model by including a cone in addition to the tip sphere. In this model, a number of point charges are adjusted in their magnitudes and/or positions such that the boundary conditions for the electrostatic potential are matched on the tip and on the sample surface as good as possible. However, and in contrast to the infinite image charge series introduced later, the convergence of these models has not generally been tested.

Hudlet et al. [36] have presented a formula for the case of a metallic tip against a metallic surface using two assumptions: First, they consider the tip surface as a superposition of infinitesimal surface segments and second, they assume that the total electric field is given from the superposition of all electric field lines present between each surface segment and the sample in their respective geometry. This approach effectively models the field lines as circular segments. The authors could show a good agreement with the analytical model for a metallic sphere-metallic surface system [37].

Colchero et al. [38] and Gil et al. [39] solve the electrostatic problem of a system consisting of both cone and tip sphere under the same assumption that all field lines can be approximated by circular segments and that the potential varies linearly along these circular segments. Within this approximation, they calculate the electrostatic force for a rectangular cantilever tilted with respect to the surface, for a circular truncated cone representing the tip shank, and for a parabolic tip that smoothly joins the truncated cone.

1.2 A.2 Exact Solutions

For obtaining the electric potential and the capacitance of the tip-sample system, finite element and finite difference solvers [40] can be used. Most notably, a dedicated solver named CapSol has been published recently, which allows to specifically model scanning-probe relevant setups [41].

A computationally less expensive way to calculate the electric potential and the capacitance of a tip-sample system is, in the simplest case, to consider a parallel-plate capacitor [29, 42, 43]. Thereby, both a purely metallic tip and sample as well as additionally included dielectric materials can be modeled. The sizes of the plates are approximated by the dimensions of the tip. However, since the electric potential resulting from this model is laterally homogeneous, this model is not capable of explaining lateral contrast.

To include the dependence on the probe position, a straightforward approach is to represent the tip by a single point charge [29, 37, 44]. The potential distribution of a sphere of radius R at constant potential V in a medium with dielectric constant $\varepsilon _\text {m}$ can (outside of the sphere) be obtained by considering a single image charge with magnitude $q=4\pi \varepsilon _0\varepsilon _\text {m} VR$ located at the center of the sphere [37]. This model does, so far, not include the presence of the sample. However, when extending the finite number of charges to an infinite sum [37], the method of image charges is a powerful concept to solve the electrostatic problem of a metallic sphere in proximity to another metallic sphere [37, 45], a metallic plane [37], a dielectric slab [46], or a dielectric half-space [46]. The next section will revisit the solution for a metallic sphere against a dielectric or metallic sample following previous works [37, 46].

1.3 A.3 Infinite Charge Series Model

The method of infinite image charge series for both the sphere-conductor and the sphere-dielectric setup is based on two well-known textbook concepts for solving the electrostatic problem with boundary conditions using image charges: (a) the case of a point charge in front of a dielectric (or metallic) half-space and (b) the case of a point charge in front of a conducting sphere.

Point charge in front of conductive or dielectric half-space

For a point charge q in a medium with relative permittivity $\varepsilon _\text {m}$ at a distance b in front of a dielectric or conductive half-space, an image charge with magnitude $\xi ' = -\beta q$ placed at $b' = -b$ yields the correct boundary condition at the interface [44]. The factor $\beta$ is defined by the dielectric permittivities according to

$$\begin{aligned} \beta =\frac{\varepsilon _\text {s}-\varepsilon _\text {m}}{\varepsilon _\text {s}+\varepsilon _\text {m}}, \end{aligned}$$

(7.6)

where $\varepsilon _\text {s}$ is the permittivity of the lower half-space. For a metal, $\beta ={1}$ [46].

The potential for the upper half-space ($z\ge 0$) is then defined by the point charge q and image charge $\xi '$

$$\begin{aligned} \varPhi \left( \mathbf r\right) =&\, \frac{1}{4\pi \varepsilon _0\varepsilon _\text {m}}\left( \frac{q}{\sqrt{x^2 + y^2 + (b-z)^2}}+\frac{-\beta q}{\sqrt{x^2 + y^2 + (b+z)^2}}\right) . \end{aligned}$$

(7.7)

To calculate the electrostatic potential for the lower half-space ($z<0$) it is necessary to place a different image charge $\xi ''$ with magnitude $\xi '' = q-\xi '=q(1+\beta )$ in the upper half space at the same position as the charge q. The potential for $z<0$ then reads

$$\begin{aligned} \varPhi \left( \mathbf r\right) =&\, \frac{q(1+\beta )}{4\pi \varepsilon _0\varepsilon _\text {s}\sqrt{x^2+y^2+(b-z)^2}}. \end{aligned}$$

(7.8)

Point charge outside of a conductive sphere

For a point charge q located outside of a conducting sphere of radius R at a distance y from the center, an image charge $\xi '$ is placed on the line connecting the point charge q with the center of the sphere. This image charge of magnitude $\xi ' = -\frac{R}{y}\,q$ is placed at distance $d = \frac{R^2}{y}$ from the sphere center to match the boundary condition at the sphere surface [44]. Then, the total electrostatic potential for the conducting sphere of radius R at potential V in a medium with relative permittivity $\varepsilon _\text {m}$ and a point charge at distance y is given from the sum of potentials for three point charges:

$$\begin{aligned} \varPhi \left( \mathbf r\right)&= \frac{1}{4\pi \varepsilon _\text {0}\varepsilon _\text {m}}\left( \frac{q}{\left| \mathbf r-\mathbf y\right| }+\frac{-\frac{R}{y}q}{\left| \frac{R^2}{y^2}\mathbf y-\mathbf r\right| } + \frac{RV}{\left| \mathbf r\right| }\right) \end{aligned}$$

(7.9)

for $\left| \mathbf r\right| \ge R$. While the first two terms ensure the boundary condition on the sphere surface for a neutral sphere due to the external charge, the last term includes the potential distribution due to the charged surface.

Conductive sphere in front of dielectric or conductive half-space

Using these two concepts, the solution for the conducting sphere in front of the conductive or dielectric half-space can be found using series of image charges [37, 46].

Fig. 7.16

Geometry of the sphere-dielectric system including the positions of the image charge series

A single point charge $\xi _0=4\pi \varepsilon _0\varepsilon _\text {m}RV$ is placed at $z_0=R+z_\text {ts}$, representing a conducting sphere of radius R at constant potential V in a medium with $\varepsilon _\text {m}$ and with the center positioned at $R+z_\text {ts}$ from the lower half-space, see also Figs. 7.1 and 7.16. The dielectric material $\varepsilon _\text {s}$ is modeled with an infinite thickness where the metallic back contact resides at $z\rightarrow -\infty$. In practice, this approximation is usually fulfilled as the sample thickness is much larger compared to the sphere radius R and the tip-sample separation $z_\text {ts}$. Furthermore, and without loss of generality, the potential can be set to ground at this back electrode.

While the boundary condition at the sphere is fulfilled with the point charge $\xi _0$, the boundary condition at the dielectric boundary is not. The latter can be corrected by placing an image charge

$$\begin{aligned} \xi _0' =\, -\beta \xi _0\,\quad \text {at}\quad z_0' = -z_0 \end{aligned}$$

(7.10)

and $x=0, y=0$. The image charges series $\{\xi _i\}$ and $\{\xi _i'\}$ define the electrostatic potential in the upper half-space (above the sample, at $z\ge 0$). The electrostatic potential at $z<0$ is given from placing an image charge of magnitude

$$\begin{aligned} \xi _0'' =\, \xi _0 - \xi _0'=(1+\beta ) \xi _0\,\quad \text {at}\quad z_0'' = z_0 \end{aligned}$$

(7.11)

in the upper half-space. Now the image charge $\xi _0'$ violates the boundary condition on the sphere, which can again be fixed by placing according image charges. The concept of infinite charge series relies on an alternating correction of the two boundary conditions, whereby the infinite series fulfills all boundary conditions.

The infinite series of image charges continues with

$$\begin{aligned} \xi _1 = \frac{R}{2(R+z_\text {ts})}\beta \xi _0 \quad \quad \quad \text {at}\quad z_1=R+z_\text {ts}-\frac{R^2}{2(R+z_\text {ts})}, \end{aligned}$$

(7.12)

fulfilling the boundary condition at the sphere for $\xi _0'$. The series of image charges $\xi _i$, $\xi _i'$ and $\xi _i''$ is then continued to alternatively fulfill the boundary conditions at the sphere and at the dielectric boundary. The magnitudes $\xi _i$ and positions $z_i$ of these image charges placed inside the sphere are given by the following recursive equations (for $i>0$)

$$\begin{aligned} z_i&= z_0-\frac{R^2}{z_0+z_{i-1}} \quad \text {with} \quad z_0 = R+z_\text {ts}\end{aligned}$$

(7.13)

$$\begin{aligned} \xi _i&= \frac{R}{z_0+z_{i-1}}\beta \xi _{i-1} \quad \text {with} \quad \xi _0 = 4\pi \varepsilon _0\varepsilon _\text {m} RV. \end{aligned}$$

(7.14)

This charge series is accompanied by two further image charge series, namely $\xi _i'$ and $\xi _i''$ according to

$$\begin{aligned} z_i'&= -z_i,\end{aligned}$$

(7.15)

$$\begin{aligned} \xi _i'&= -\beta \xi _i,\end{aligned}$$

(7.16)

$$\begin{aligned} z_i''&= z_i,\end{aligned}$$

(7.17)

$$\begin{aligned} \xi _i''&= (1+\beta )\xi _i. \end{aligned}$$

(7.18)

Using the series of these charges, the potential $\varPhi _\text {void}$ for both half-spaces follows directly from the superposition of the point charge potentials, namely

$$\begin{aligned} \varPhi _\text {void}\left( \mathbf r\right)&= \frac{1}{4\pi \varepsilon _0\varepsilon _\text {m}}\sum _{i=0}^\infty \left[ \frac{\xi _i}{\sqrt{x^2+y^2+(z_i-z)^2}} +\frac{\xi _i'}{\sqrt{x^2+y^2+(z_i'-z)^2}}\right] \quad (\text {for}\,z\ge 0) \end{aligned}$$

(7.19)

for the upper half-space and

$$\begin{aligned} \varPhi _\text {void}\left( \mathbf r\right)&= \frac{1}{4\pi \varepsilon _0\varepsilon _\text {s}}\sum _{i=0}^\infty \frac{\xi _i''}{\sqrt{x^2+y^2+(z_i''-z)^2}}\quad (\text {for}\,z<0) \end{aligned}$$

(7.20)

for the lower half-space. The capacitance $C_\text {void}$ of the system is given by the sum over all image charges $\xi _i$ divided by the tip voltage V:

$$\begin{aligned} C_\text {void}=\frac{1}{V}\sum _{i=0}^{\infty }\xi _i. \end{aligned}$$

(7.21)

As the magnitudes of the image charges converge quickly to zero, it is practical to only consider a finite number of terms. Since the positions of the charges converge quickly, the high-index elements can furthermore be represented by a single charge holding the sum of the remaining infinite charge series [45, 46].

For all calculations shown within this chapter, we truncated the infinite image charge series after 100 image charges and consider one additional point charges holding the sum of the remaining charges in the series  [45, 46]. After calculating the normalized electric potential and the capacitance, we numerically determined the derivatives needed for computing the weight functions in (7.2) and (7.3) using a central finite difference scheme of second order.