A unified approach for the calculation of in-plane dielectric constant of films with interdigitated electrodes

The layout of a set of IDEs in the x-z plane is shown in figure 1(a). The IDEs consist of two comb electrodes with a total of N fingers. The fingers with positive potential overlaps with the fingers with negative potential in a region of length L in the z direction. Here we assume that the ratio between the periodicity of the electrodes and L is such that fringe effects at the end of the fingers are negligible and can be ignored. A cross section of the structure (x-y plane) is shown in figure 1(b). The spacing between the electrodes, a, and the finger width, b, are indicated. The figure shows a general design with one layer of finite thickness (medium 2, with thickness t₂), between two layers of infinite thickness (medium 1 and medium 3).

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For this work we describe the geometry by the cover fraction $\eta = b/(a+b)$ [19] and the normalized thickness of medium 2, $\tau_2 = t_2/(a+b)$. Normalizing the geometry also gives rise to the dimensionless coordinates $x' = x/(a+b)$ and $y' = y/(a+b)$.

The total capacitance of the structure can be estimated by considering the individual capacitance between each pair of fingers. Following the route of reference [19], we only consider edge effects at the outermost electrode pairs (capacitance per unit length C_E ), while periodic boundary conditions are used when considering the interior electrodes (capacitance per unit length C_I ). An equivalent circuit diagram is shown in figure 1(b), and the total capacitance of the structure can be estimated as follows:

$$\begin{equation} C_{total} = L((N-3)C_I+2C_E) \end{equation} \tag{ 1 }$$

The potential in the case where the thickness of medium 2 reaches infinity (i.e. $t_2\rightarrow\infty$) may be calculated as described by Engan [15], Wu et al [1], Wei [12], or Igreja and Dias [19]. We will use the subscript $_\infty$ to mark the capacitance per unit length, $C_{I\infty}$, and potential, $V_{I\infty}(x,y)$, for this case. The required expressions for calculating $C_{I\infty}$ and $V_{I\infty}(x,y)$ using both a conformal mapping approach [19] and a Fourier approach [15] are included in section 1 of the Supporting Information (stacks.iop.org/SMS/29/115039/mmedia).

Given a finite thickness of medium 2, as shown in figure 1(b), boundary conditions at the interface between medium 2 and medium 3 must be fulfilled. The same is true for the interface between medium 2 and 1 between the electrode fingers. Using the initial field $V_{I\infty}$ as a basis, we here use the method of recursive images to fulfil the boundary conditions between the layers [17]. This will distort the constant potential at the electrodes, and we will address this distortion later. In the method of recursive images, a forward and a reverse potential is generated whenever a potential interacts with the boundary conditions at an interface. These potentials are defined using the forward, $f_{ij} = \frac{2\varepsilon(i)}{\varepsilon(i)+\varepsilon(j)}$, and reverse, $r_{ij} = \frac{\varepsilon(i)-\varepsilon(j)}{\varepsilon(i)+\varepsilon(j)}$, coefficients for the respective interface defined by the dielectric constant ε(i) and ε(j) in medium i and j [16, 17]. A derivation of the coefficients from the boundary conditions is included in section 2 of the Supporting Information, where anisotropic materials are also considered. The three first generations of potentials in the method of recursive images are illustrated in figure 2(a)–(c). Equation (2) describes the sum of all partial potentials generated, V_I1(x, y), and this potential is illustrated in figure 2(d), while figure 2(e) shows the potential along the interface with the electrodes.

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$$\begin{equation} V_{I1}(x,y) = \left\{\begin{array}{l} V_{I\infty}(x,y) +f_{21}r_{23}\sum_{m = 0}^\infty \\[6pt] (r_{23}r_{21})^mV_{I\infty}(x, -y+2(m+1)t_2),\quad \mathrm{for}\ y\lt 0\\[6pt] V_{I\infty}(x,y) + \sum_{m = 0}^\infty (r_{23}r_{21})^{(m+1)}V_{I\infty}(x, 2(m+1)t_2+y) \\[6pt] + r_{23}^{m+1}r_{21}^{m}V_{I\infty}(x,2(m+1)t_2-y),\quad \mathrm{for}\ 0\lt y\lt t_2\\[6pt] f_{23}\sum_{m = 0}^\infty (r_{23}r_{21})^mV_{I\infty}(x, y+2mt_2),\quad \mathrm{for}\ y>t_2 \end{array}\right. \end{equation} \tag{ 2 }$$

The potential, V_I1(x, y), and the associated capacitance, C_I1, are denoted with the subscript 1, as we will soon place them in the context of a series of similar potentials. It can be seen from figure 2(d) and (e) that the potential V_I1(x, y) does not respect boundary conditions (± 0.5 V) at the electrodes. The total displacement field in the x direction remains constant when using the method of recursive images, so that C_I1 can be calculated using the expression for $C_{I\infty}$.

We now proceed to use linear algebra to address the non-constant potential at the electrodes. For this approach we generate a series of potentials V_Ii with ever decreasing cover fraction η_i with i = 1...M. Any linear combination of these potentials, will fulfil the boundary conditions at all interfaces with the exception of the electrodes, and we seek a linear combination that will approximate correct boundary conditions also here. The combined potential V_Ic (x, y) and capacitance C_Ic may be found if appropriate weights u_i are known:

$$\begin{equation} V_{Ic}(x,y) = \sum^{M}_{i = 1} u_i V_{Ii}(x,y) \end{equation} \tag{ 3 }$$

$$\begin{equation} C_{Ic} = \sum^{M}_{i = 1} u_i C_{Ii} \end{equation} \tag{ 4 }$$

The weights u_i may be found by using linear algebra to balance the potential at M locations along the electrode surface, denoted x_j with j = 1...M. A linear distribution of cover fractions η_i and locations x_j is chosen for simplicity, as detailed by equations (5) and 6:

$$\begin{equation} \eta_i = \eta_1\Big(1-\frac{i-1}{M}\Big) \end{equation} \tag{ 5 }$$

$$\begin{equation} x_j = -\frac{1}{2}\Big(a+b\frac{2j-1}{2{\rm M}}\Big) \end{equation} \tag{ 6 }$$

Solving the linear algebra problem in equation (7) will then yield the weights u_i . Here V₀ is the potential applied to the electrodes, and is set to 1 V.

$$\begin{equation} \begin{bmatrix} V_{I1}(x_1,0) & V_{I1}(x_2,0) & \dots & V_{I1}(x_{M},0) \\ V_{I2}(x_1,0) & V_{I2}(x_2,0) & \dots & V_{I2}(x_{M},0) \\ \dots & \dots & \dots & \dots & \dots \\ V_{IM}(x_1,0) & V_{IM}(x_2,0) & \dots & V_{IM}(x_{M},0) \\ \end{bmatrix} \begin{bmatrix} u_1 \\ u_2 \\ \dots \\ u_{M} \end{bmatrix} = \begin{bmatrix} V_0/2\\ V_0/2 \\ \dots \\ V_0/2 \end{bmatrix} \end{equation} \tag{ 7 }$$

An example of the potential described by a linear combination of M = 4 components is shown in figure 3(a) and (b), while each of the 4 components are shown separately in figures 3(c)–(f). It can be clearly seen that V_Ic (x, y) is in reasonable agreement with the boundary conditions that require +0.5 V at the left electrode and −0.5 V at the right electrode.

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We here propose a method for estimating the average in-plane polarization in the film at the plane midway between the electrode fingers (x = 0). At this plane, the displacement field is parallel to the x-axis, and distributed between the film, substrate, and air. By estimating the sum of the displacement fields in the substrate and air using the proposed model, these can be subtracted from the charge collected by the electrodes in order to calculate the polarization in the film as given in equation (8). Here $Q_{\textrm{exp}}$ is the experimentally collected charge at the electrodes, and $Q_\mathrm{sub}$ and $Q_\mathrm{air}$ are the integrated displacement fields in substrate and air, respectively. $Q_\mathrm{sub}$ and $Q_\mathrm{air}$ can be calculated based on the potential given by equation (3) by approximating the film as a homogeneous non-ferroelectric material. The details for calculating $Q_\mathrm{sub}$ and $Q_\mathrm{air}$ are provided in section 3 of the Supporting Information. The polarization, P, is may then be calculated according to equation (8):

$$\begin{align} P& = \frac{Q_{\textrm{exp}}-Q_\mathrm{air}-Q_{sub} }{t_2L(N-1)} \end{align} \tag{ 8 }$$

The capacitance of an exterior pair of electrodes was approximated by Igreja and Dias [19] by assuming a plane of constant potential at the center of the outermost electrode gap, and with no flux passing the plane at the center of the next-to-outermost electrode. Using these approximations, the edge capacitance can be calculated as follows:

$$\begin{equation} C_{Ec} = \frac{2C_{2c}C_{Ic}}{C_{Ic}+C_{2c}} \end{equation} \tag{ 9 }$$

where the subscript 2 denotes a set of IDEs with only two electrode fingers. Equations (2), 7 and 4 may easily be adapted to calculate C_2c using the initial potential $V_{2\infty}(x,y)$, as detailed in section 4 of the Supporting Information.

The linear algebra approach in equation (7) may be expanded to include additional electrodes at different interfaces in a multilayer stack, and the implementation of this model available on Github is to handle any number of layers with an arbitrary number of electrodes [25]. This may for example be interdigitated electrodes on top of one or more thin films grown on a conducting substrate, where the substrate acts as a separate electrode [13, 23], or alternatively systems with electrodes on both sides of a film [22]. The potential of such systems are illustrated in section 5 of the Supporting Information.

Anisotropic materials may be addressed by transforming the potential as $y\rightarrow \varepsilon_r y$, where $\varepsilon_r = \sqrt{\frac{\varepsilon_x}{\varepsilon_y}}$ [12, 15], and adapting the boundary conditions appropriately [16]. The code on Github is designed to support this while the constituent equations required are included in section 2 of the Supporting Information.

The FEM simulations were performed using the FEMM software [29]. For calculating the interior capacitance, C_I,FEM , a cell was made with dimensions (a + b)× 4(a + b) and with Neumann boundary conditions (zero flux) on all outside boundaries. For the vertical boundaries this is equivalent to a periodic boundary condition. The limited height of the cell reduces the accuracy, but with the height used here the reduction in accuracy was not found to be significant. The grid density was varied depending on τ₂, as thinner films require higher grid density to obtain sufficient accuracy. Up to 1 million nodes were used for the simulations. By assuming that the capacitance calculated by FEM is accurate, we can evaluate the accuracy of the model presented here as follows:

$$\begin{equation} \mathrm{Accuracy } = 100\% \frac{C_{Ic}}{C_{I,FEM}} \end{equation} \tag{ 11 }$$

The capacitances measured using geometrically different IDEs on the two films are shown in figure 4(a) and (b), where the capacitance has been normalized with respect to the number of gaps and the finger length. Based on the capacitance data, the dielectric constant was calculated using the model and the resulting dielectric constant-electric field, ε(2)–E, loops are shown in figure 4(c) and (d). The relative dielectric constant in the BaTiO₃ films at zero field is 1911± 118, and 3013 ± 148 for the thick and thin film, respectively. The dielectric constant as calculated using the PPC model is shown for comparison. The constituent equations of the PPC model are included in section 8 of the Supporting Information [19].

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A sensitivity analysis of the calculated dielectric constant with respect to the geometric parameters is included in section 9 of the Supporting information. Given the uncertainty of the geometric parameters, an uncertainty of ∼ 100 is expected for the dielectric constant of the thick film. The sensitivity analysis also examined the impact of anisotropy in the dielectric constant of the film, and it was found that the capacitance has much higher sensitivity to the in-plane component than the out-of-plane component.

Capacitance curves generated by the model using the field-dependent permittivity of figures 4(c) and (d) are presented in section 10 of the Supporting Information. The calculated capacitance curves represent a good fit to the experimental curves.

The charge collected at the electrodes when performing a voltage sweep is shown in figure 5(a) and (b) for the thick and thin films, respectively. A large capacitive contribution from the substrate can be seen in the slope of the curves. Figure 5(c) and (d) shows the polarization in the BaTiO₃ films as calculated by equation (8), where the contributions from the substrate and air have been subtracted using the model. The remnant polarization was found to be 10.54± 0.87 µC cm⁻² and 5.75± 0.67 µC cm⁻² for the thick and thin films, respectively, while the coercive field was 5.88± 0.14 kV cm⁻¹ and 5.48± 0.21 kV cm⁻¹.

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The accuracy of the model compared to FEM (equation (11)) for a structure with an infinite number of fingers (C_Ic ) is shown in figures 6(a)–(c) using a varying number of components M. The accuracy of the PPC model is shown in figure 6(d) for comparison. The accuracy of the proposed model is maximum when considering a thick film with a low cover fraction. When a single component is used, the accuracy is higher or comparable to that of the PPC model for the entire region probed. The accuracy increases with increasing number of components. When using M = 4 components, the model has more than 99 % accuracy for the geometries used to obtain the experimental data in figures 4 and 5. The PPC model has an accuracy greater than 92 % for the same geometries, a difference of 7 percentage points. The relative uncertainty of the FEM simulations used to compare the models is estimated to be 0.1 % based on the finite grid density and simulated cell height.

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