The Montgomery method is extensively employed to determine the electrical resistance tensor of anisotropic samples. This technique relies on two essential parameters describing an isotropic system: the geometric factor (*H*_{1}) and the effective thickness (*E*). The numerical values of these parameters are intricately linked to the dimensions of an isotropic block equivalent to the studied anisotropic specimen. While these parameters hold importance, the physical interpretation of these terms still lacks clarity. In this study, we utilized the finite element method to simulate electrical transport experiments across samples of various shapes. Utilizing the Electric Currents physics interface in the COMSOL program, we were able to provide a comprehensive analysis of the physical meaning of these parameters to accurately determine the electrical properties of thin films and wafers. The presented findings related to the physical interpretation of *H*_{1} and *E* terms make substantial contributions to the field of electrical transport experimental techniques, which are fundamental to design advanced materials for technological applications and understand their physical properties.

## I. INTRODUCTION

The Montgomery and Van der Pauw methods are widely recognized for their effectiveness in investigating the electrical properties of both isotropic and anisotropic conductors.^{1–3} The Montgomery method, specifically tailored to identify tensor electrical resistivity components based on the conformal mapping of parallelepipeds, yields valuable insights into the electrical behavior of anisotropic systems.^{4,5} Moreover, this approach has demonstrated its effectiveness in diverse practical applications within both fundamental physics research and engineering contexts.^{6–14} Conversely, the primary benefit of the Van der Pauw method lies in its capability to analyze either anisotropic or isotropic thin discs of complex geometries,^{15,16} which has attracted great theoretical and experimental attention.^{17–26}

The shared significant benefit of the Montgomery and Van der Pauw methods in the study of electrical properties stems from their electrode probe arrangements. These methods use electrode point probes positioned along the edge of the specimen,^{2} offering a marked advantage over the conventional four-probe method, which uses aligned line electrodes and is typically better appropriated for samples in bar or wire shapes.^{27} The strategic edge probe placement reduces the risk of contact short-circuiting, an essential feature that enhances the suitability of these methods for the study of small and irregularly shaped samples such as monocrystalline samples produced by floating zone, flux, or vapor-transport crystal growth techniques.^{28–30}

^{1,27,31}Typically, this setup involves directing an electrical current between two neighboring corners, such as

*A*and

*B*, while simultaneously measuring the voltage across another adjacent pair of corners, such as

*C*and

*D*[see, for instance, Fig. 1(a)]. The Van der Pauw method, while similar in using edge-mounted probes, allows for more flexibility in

*A*,

*B*,

*C*, and

*D*probe placement. The resistance, denoted as

*R*

_{1}, is calculated based on the ratio of the measured voltage to the applied current. A secondary resistance value,

*R*

_{2}, is obtained by rerouting the current through a different set of points, say

*A*and

*D*, and then measuring the voltage between a separate pair of points

*B*and

*C*. These two resistance values,

*R*

_{1}and

*R*

_{2}, are key in determining the aspect ratio

*L*

_{2}/

*L*

_{1}of an isotropic rectangle, which equivalently represents the electrical properties of a sample of arbitrary shape and anisotropy using the following equation:

^{1}

^{,}

^{2}

^{31}which is based on the foundational calculations made by Logan and co-workers.

^{32}This equation is expressed as

^{1,31}

*E*represents the effective thickness of the rectangle, and

*H*

_{1}is the geometric parameter. Although

*H*

_{1}can be described through an infinite series of sums or by closed-form expressions,

^{25}for simplicity in practical applications, it is approximated as

^{1}

*H*

_{2}is

Despite the longevity of the Montgomery method and numerous mathematical studies and developments conducted since its publication, there remains a lack of clear physical interpretation for these two geometric parameters, *H*_{1} and *E*.

Unfortunately, conducting experimental investigations of *H*_{1} and *E* using the Montgomery method can be challenging. This is due to the necessity of preparing and measuring a multitude of samples varying both in size and type (conducting isotropic materials), a task that is far from straightforward in a practical laboratory setting. In contrast, physical simulations, particularly those utilizing the finite element method, offer several advantages. They can efficiently replicate experiments on rectangular blocks of various sizes, significantly reducing time and effort. Moreover, this method enables a detailed and systematic examination of equipotential and electric field patterns in a conducting material under external electrical current flow, a level of analysis often unattainable in standard transport measurements. The efficacy and depth of the finite element analysis, especially when conducted using COMSOL software, are clearly demonstrated in our recent publication,^{3} highlighting the robust capabilities of this approach in the development of electrical transport experimental characterization techniques.

In this study, we executed a comprehensive numerical modeling analysis to deepen our understanding on the geometric factor and effective thickness in the Montgomery method. We employed the Electric Currents physics module of the COMSOL software for this purpose. Our simulations, focusing on electrical transport measurements in rectangular sheets and parallelepipeds of varying dimensions, showed that voltage measurements at corner points on the side opposite to where external current application probes are located are consistent with results from the conventional four-probe method in a corresponding bar configuration. Furthermore, our exploration of the electrical resistivity of three-dimensional samples with different thicknesses has elucidated the concept of effective thickness in parallelepiped conductive blocks. We observed that the voltage measured at the corners decreases as the thickness increases. This behavior is a consequence of the electrical field profile along the third axis. These findings are essential for a comprehensive understanding of the electrical transport properties of materials in two-dimensional (2D) and three-dimensional (3D) structures.

## II. METHODOLOGY

We employed the COMSOL Multiphysics software for the analysis of two-dimensional conducting isotropic samples, as illustrated in Fig. 1(a). Our focus was on two main dimensions: *L*_{1} and *L*_{2}. The length *L*_{1}, extending between corners *A* and *B*, as well as *C* and *D*, was consistently set at 1 m. In contrast, the dimension *L*_{2}, linking corners *A* to *D* and *B* to *C*, was varied within a spectrum ranging from 0.01 to 8 m. By adjusting *L*_{2} using a fixed *L*_{1}, we were able to investigate the impact of spatial dimensions on the conductive properties of these samples, offering insights into their behavior under varied geometric configurations.

Numerical simulations were conducted utilizing the Electric Currents module, focusing on a stationary state study in 2D space dimension. This module applies the electrical current continuity equation in a stationary regime.^{3} We established an external current (*I*_{1}) of 1 A as the boundary condition. This was implemented by designating point *A* with a contact area of 0.01 × 0.01 m^{2} as the electric potential reference. Here, the Terminal function was adjusted to set the zero-voltage terminal type. An identical contact area at point B was set to 1 A electrical current terminal type. For all other boundaries, the Electric Insulation option was selected. The computational 2D mesh was built using quadrilateral elements, each with four nodes, with 0.01 m maximum element size, culminating in a total of 10 000 domain elements and 408 boundary elements in a square sheet.

The observed voltage drop, designated as *V*_{1}, between contact points *C* and *D* of identical dimensions is a consequence of the consistent electrical current traversing between points *A* and *B*, as illustrated in Fig. 1(a). The voltages at 0.01 × 0.01 m^{2} electrodes in points *C* and *D*, labeled as *V*_{C} and *V*_{D}, respectively, were ascertained using the average values obtained from voltage domain probes. This experimental arrangement employed isotropic samples with a defined electrical resistivity of 1 Ω-m. The chosen color scheme shown in Fig. 1(a) was meticulously selected to highlight the boundaries and to emphasize the equipotential lines at points *C* and *D* within the sample.

The addition of the third dimension, denoted as *L*_{3}, into the analysis was essential to explore the role of effective thickness in the Montgomery method application for thicker samples. To conduct this investigation, finite element modeling simulations were performed on a variety of isotropic three-dimensional systems of 1 Ω-m, each characterized by distinct dimensions. *L*_{1} was consistently maintained at 1 m while *L*_{2} varied between 1 and 4 m. The range for *L*_{3} was more extensive, spanning from 0.01 to 8 m, allowing for a comprehensive assessment of the impact of thickness on the measurement technique. Here, we have employed both tetrahedral and triangular elements with 0.02 m as the maximum element size to build up mesh grids with more than 2 × 10^{6} tetrahedral elements, ∼36 thousand triangular elements, and 702 edge elements in a cubic sample.

In alignment with the methodology of the 2D simulations, the contact at point *A*, measuring 0.01 × 0.01 × 0.01 m^{3}, was designated as the ground connection via the terminal function. The contact at point *B*, possessing the same dimensions, was configured as the current terminal. This configuration enables a current flow of 1 A through the initial insulated 1 Ω-m 3D conducting block, as illustrated in Fig. 1(b). Analogously, the potential drop, *V*_{1}, was determined by measuring the difference between the average values obtained from voltage domain probes situated at contacts of identical dimensions at *C* and *D*.

## III. RESULTS AND DISCUSSION

### A. Interpretation for the geometric factor (*H*_{1})

Figure 2 illustrates the schematic diagrams of the samples used in both the traditional four-probe technique and the Montgomery methods.

^{33}

*w*is the sample width,

*ℓ*is the distance between the voltage contacts, and

*t*=

*L*

_{3}is the sample thickness.

*E*=

*L*

_{3}=

*t*), as depicted in Fig. 2. Employing the

*H*

_{1}term expressed by Eq. (3a) and comparing the electrical resistivity values obtained via the Montgomery method [Eq. (1)] with those from the standard four-probe method [Eq. (4)], it becomes straightforward to demonstrate that

This finding is notable as it assigns a tangible physical interpretation of the geometrical parameter *H*_{1} in the Montgomery method. Specifically, Eq. (5) is related to the ratio of the sample's width to the distance between voltage contacts in the four-probe method, as detailed in Fig. 2. A parallel interpretation can also be applied to the complementary geometrical parameter *H*_{2}.

We conducted a series of simulations analogous to those presented in Fig. 1(a) across a variety of isotropic rectangular conducting sheets, each differing in its aspect ratios (*L*_{2}/*L*_{1}). This approach was undertaken to gain a more comprehensive understanding of the implications of the geometrical parameter *H*_{1}, as previously discussed. The outcomes of these simulations, pertaining to various sample dimensions, are showcased in Fig. 3.

By analyzing the simulation data for the electric voltage along segments AB and CD, we can accurately determine the distance *ℓ*. Figure 4(a) presents these findings, illustrating the method used to ascertain *ℓ* for a 2D sample where *L*_{1} and *L*_{2} are both 1 m. It is notable that the voltage variation is linear with the distance between the C and D projections along the AB segment [see Fig. 4(a) lower inset], which is consistent with the four-probe method geometry (see Fig. 2). Furthermore, Fig. 4(b) displays the results concerning *V*_{1} (equivalent to *V*_{D} − *V*_{C}), *ℓ*, and *w* for samples that vary in their aspect ratios (*L*_{2}/*L*_{1}). This comparative analysis offers insights into how these parameters interact and vary with changes in the aspect ratios of the samples.

*V*

_{1}aligns with the anticipated trends observed in electrical transport measurements of isotropic samples, particularly those with extended dimensions along

*L*

_{2}. In these scenarios, the electric field intensity decreases exponentially as the distance from the current source increases. Consequently, there is a significant reduction in voltage at points farther from the source, as evidenced in Fig. 4(b). Furthermore, the variation in

*V*

_{1}as a function of

*L*

_{2}/

*L*

_{1}(black symbols), displayed in Fig. 4(b), can be effectively fitted using Eq. (7) (black line), also detailed in recent studies,

^{3}using accurate values for 1/

*H*

_{1}obtained from sums of infinite series or Jacobi elliptical functions.

^{25}This implies that the parameter

*ℓ*is directly proportional to

*V*

_{1}, particularly when the ratio

*L*

_{2}/

*L*

_{1}exceeds 1. This relationship is clearly illustrated in Fig. 4(b), as indicated by the red symbols and the red line. We discovered that this proportional relationship is expressed by the following equation:

The comparison of this result with Eq. (5) leads to a notable conclusion: for all cases where *L*_{2}/*L*_{1} > 1, regardless of their dimensions, *w* = *L*_{1}/2 holds true. The inset of Fig. 4(b) reinforces this conclusion, emphasizing the universality of this relationship across various sample sizes. Conversely, for conditions where *L*_{2}/*L*_{1} < 1, it is observed that *w* ≈ *L*_{2} consistently applies. In addition, in these cases, *ℓ* tends toward *L*_{1}, especially for *L*_{2}/*L*_{1} ≤ 0.2, as prominently displayed in the main panel of Fig. 4(b).

### B. Interpretation of the effective thickness (*E*)

Figure 5 presents three-dimensional simulations of rectangular blocks with varying thicknesses, denoted as *L*_{3}. The illustration clearly demonstrates the electric field lines, indicated by arrows, and highlights the electrical field intensity along the *x*-direction across a plane at the center of these blocks, where the palette of colors maps the *x*-component of the electrical field intensity. This visualization effectively captures the spatial variations in the electric field intensity in different sample thicknesses.

We observe significant variations in the electric field profiles when comparing thin and thick samples. For thin samples, the electric field intensity is consistent across their thickness, as indicated by the uniform color gradient visible along the *z*-axis. However, a noticeable decrease in intensity is observed across the width (*y*-axis), where the color transitions from one side to the other, illustrating a gradient along the width. This effect is clearly depicted in Fig. 5(a).

In contrast, Fig. 5 also shows that thicker samples demonstrate a pronounced reduction in electric field intensity with both increasing depth and width. The fading color intensity from the top (near the source) to the bottom and from one side to the other signifies a gradual decrease in the electric field as one moves further away from the source and deeper into the center of the block. The progression of this electric field pattern with increasing thickness is systematically demonstrated in Figs. 5(b)–5(f).

Values of *V*_{1} from our simulations facilitate an investigation into the behavior of electric voltage and field across various samples with different thicknesses. Figure 6 depicts how *V*_{1} varies in response to the thickness *L*_{3} of samples possessing different *L*_{2}/*L*_{1} ratios.

This expression serves as a robust approximation for cases where the ratio *L*_{2}/*L*_{1} is greater than or equal to 1.^{25} The linear relationship observed in Fig. 6 emerges when we set *E* = *L*_{3}, which corresponds to the actual thickness of the sample. Practically, this allows for the determination of electrical resistivity using the sample's true thickness, even when the sample's cross section may have an arbitrary shape or exhibit in-plane anisotropic properties, which is consistent with previous experiments using thickness-controlled discs.^{19}

*V*

_{1}occurs at the maximum effective thickness, illustrated by full yellow circles in the inset. This specific threshold thickness, denoted as

*E**, corresponds to

This observation is consistent with findings from previous studies.^{1}

These two phenomena distinctly represent distinct physical states of the sample. First, the thin sample limit, as elucidated in Eq. (7), is characterized by a decrease in electrical resistance correlating with a reduction in sample thickness. Second, the maximum effective thickness, defined earlier, marks a threshold beyond which the measured electrical resistance of the sample remains invariant, irrespective of any further increase in thickness.

*V*

_{1}presented in the inset of Fig. 6 is consistent across all samples, we can establish a universal scaling. This is achieved by considering $V1/V1*$ as a function of

*L*

_{1},

*L*

_{2}, and

*L*

_{3}, as shown in the main panel of Fig. 6. Here, $V1*$ represents the saturation value for each sample, defined as

*L*

_{3}, it becomes clear that $V1/V1*$ should scale with

*L*

_{3}/$L1L21/2$. The main panel of Fig. 6 illustrates this scaling, exhibiting a remarkable overlap of data across all samples.

Further analysis of this data collapse suggests that the function defining this universal behavior is straightforward and is related to the normalized effective thickness *E*/*L*_{3} previously discussed.^{1} With the integration of the Van der Pauw and Montgomery methods, it becomes clear that the normalized effective thickness is well-suited for accurately determining the electrical resistivity in thicker specimens that exhibit both random shapes and in-plane anisotropies. Ongoing research is exploring this aspect deeper, with results to be shared in future publications.

## IV. CONCLUSIONS

In conclusion, our numerical simulations conducted using COMSOL software have provided significant insights into the geometric parameters essential to the Montgomery method, particularly in describing electrical transport in isotropic parallelepiped blocks. These parameters are vital for the accurate determination of electrical resistivity in specimens, regardless of their shape or anisotropy level, as long as they can be equated to equivalent isotropic regular shapes. This discovery not only emphasizes the adaptability of the Montgomery method when combined with the van der Pauw technique, but it also highlights its broad potential for application in diverse electrical properties characterization scenarios.

These simulations unequivocally demonstrate that parameter *H*_{1} correlates with both the width and the distance between voltage probes corresponding to the conventional four-probe method. In addition, by adjusting the thickness of the rectangular samples in our 3D simulations, we gained valuable understanding about the effective thickness *E* in the Montgomery method. We observed two distinct behaviors: the first corresponds to the limit of thin samples where *E* = *L*_{3}, and the second relates to a saturation point of voltage *V*_{1} at $L3=2/2L1L21/2$, beyond which the resistance of the sample becomes independent of its thickness. These findings are in remarkable accordance with previous studies, underscoring the robustness and reliability of our simulation approach.

## ACKNOWLEDGMENTS

F. S. Oliveira is a post-doc at UNICAMP (FAPESP Grant No. 2021/03298-7). M. S. da Luz is a CNPq fellow (Proc. 311394/2021-3).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**F. S. Oliveira**: Conceptualization (equal); Data curation (equal); Methodology (equal); Software (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). **L. M. S. Alves**: Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). **M. S. da Luz**: Methodology (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). **E. C. Romão**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). **C. A. M. dos Santos**: Conceptualization (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding authors upon reasonable request.