Simple analytical method for determining electrical resistivity and sheet resistance using the van der Pauw procedure

This work reports an analytical method for determining electrical resistivity (ρ) and sheet resistance (RS) of isotropic conductors. The method is compared with previous numerical solutions and available experimental data showing a universal behavior for isotropic conductors. An approximated solution is also reported allowing one to easily determine ρ and RS for samples either with regular or arbitrary shapes.

Simple analytical method for determining electrical resistivity and sheet resistance using the van der pauw procedure f. S. oliveira, R. B. cipriano, f. t. da Silva, e. c. Romão & c. A. M. dos Santos * this work reports an analytical method for determining electrical resistivity (ρ) and sheet resistance (R S ) of isotropic conductors. the method is compared with previous numerical solutions and available experimental data showing a universal behavior for isotropic conductors. An approximated solution is also reported allowing one to easily determine ρ and R S for samples either with regular or arbitrary shapes.
The study of electrical transport properties is of great importance since is routinely used in many theoretical and experimental researches as well as in many applications. They are especially important for characterizing many materials such as isotropic and anisotropic polycrystalline samples, metal plates, single crystals in bulk and platelike forms [1][2][3][4] , homogeneous and heterogeneous thin films [5][6][7] , 2D electrical conductivity in exfoliated materials such as graphene and MoS 2 8-10 , conventional and non-conventional superconductors [11][12][13][14][15] , organic conductors and superconductors 5,11 , Si-based compounds for electronic applications 16,17 , new materials for energy storage such as modern batteries, fuel cells, and solar panels [18][19][20] , and even for biological specimens 21,22 . Furthermore, the study of phase transitions 23,24 , structural properties 25,26 , and anisotropic properties 1,27 have been important for materials science and engineering applications. With regard to the physical properties, measuring electrical conductivity is of great relevance for understanding several physical phenomena such as superconductivity 14,15,28,29 , topological materials 9,13,30,31 , 1D and 2D conductivity 3,10,11,28 , quantum Hall effect 10,32 , electronic and quantum phase transitions 3,12,23,33 , and many other physical effects. In many of these applications one of the broadest interests is dealing with samples of irregular shapes and small sizes.
In this particular work, we report a simple analytical procedure to determine electrical resistivity and sheet resistance of isotropic conductors either with regular or arbitrary shapes.
One of the procedures used to determine of electrical resistivity and sheet resistance of conducting materials with regular or arbitrary shape was proposed more than 60 years ago by van der Pauw (vdP) 34,35 . In this procedure four electrical contacts are placed on the sample, as shown in Fig. 1. Current and voltage contacts are cycled through switches 1 and 2.
Based upon the measured R 1 and R 2 values, electrical resistivity (ρ) and sheet resistance (R S = ρ/d) can be determined by 34,35 where d is the thickness of the sample and f(R 2 /R 1 ) is a geometric factor that is, up to now, determined numerically or graphically. Since the vdP method was introduced 34 others have suggested various ways to determine f(R 2 / R 1 ). Ramadan et al. 37 provided values of f(R 2 /R 1 ) for 1 ≤ R 2 /R 1 ≤ 200. de Vries and Vieck 38 reported some results for f(R 2 /R 1 ) using polyethylene samples with parallelepiped shapes with aspect ratio (length/width) between approximately 0.6 and 2.6. At present, f(R 2 /R 1 ) is the best determined using the method of Chan 39 who solved numerically the transcendental equation  Montgomery 45 , using a similar procedure as illustrated in Fig. 1, but for parallelepiped samples, developed a graphical method to determine the electrical resistivity of both isotropic and anisotropic samples, which was supported by the calculation of the electrostatic potential for rectangular blocks reported by Logan et al. 46 and the Wasscher transformation from isotropic to anisotropic sample 47 . A modification of the Montgomery method 45 has allowed us to determine the electrical resistivity and sheet resistance of parallelepiped samples based upon simple equations 48 . Using the Montgomery method 45,47 one can find electrical resistivity of isotropic parallelepiped samples by where E is the effective thickness, which measures the penetration depth of the electrical current into the sample; it is equal to d in Fig. 1 for thin samples H 1 and H 2 are functions of L 2 /L 1 given by 46,48 and where L 1 and L 2 are the length and width of a particular parallelepiped sample 46,48 (see Fig. 2).

Results and discussion
(2)  which shows that f(H 1 ,H 2 ) given by Eq. (7) is exact solution of the transcendental Eq. (2). A confirmation can be observed in the lower inset of Fig. 3, which shows that adding the two terms on the left side of the Eq. (8) yields unity, no matter the R 2 /R 1 ratio.
Additionaly, from Eq. (8), one can write H 2 as a function of H 1 , as which can be used to eliminate H 2 in Eq. (7), yielding the geometric factor since H 1 is only a function of L 2 /L 1 , given by Eq. (4). Eliminating H 2 in Eq. (7) is of special interest because the convergence of 1/H 1 series is much faster than 1/H 2 (see Supplementary material). Thus, electrical resistivity and sheet resistance can be exactly determined by Figure 3. Geometric factor f (R 2 /R 1 ) and f (L 2 /L 1 ) in the van der Pauw method versus R 2 /R 1 . Red line represents the behavior predicted by Eq. (11)  www.nature.com/scientificreports/ An important aspect associated with above results is the fact that the equations in the Montgomery method are related to parallelepiped samples, but solve the transcendental equations in the vdP method, which work not only for parallelepiped samples but also for any regular and arbitrary shapes. This is valid because there is a conformal transformation, in a similar way as described by others 34,38,49 of the isotropic sample, with arbitrary shape and measurements R 1 and R 2 in the vdP method, into an equivalent parallelepiped sample with the same electrical resistances and correspondent sides L 1 and L 2 in the Montgomery method. This idea is displayed in Fig. 2. Once the equivalent parallelepiped sample is defined, equations of the Montgomery method can be used to solve Eqs. (1) and (2) by vdP. Such as H 1 and H 2 are functions of the L 2 /L 1 (see Eqs. (4) and (5) again), f(H 2 ,H 1 ) and f(L 2 /L 1 ) given by Eq. (7) or (11) shows that this parameter is indeed a geometric factor, in agreement with previous reports 34,45,46 .
Two cases of this conformal transformation are of particular interest: (i) a sample with parallelepiped shape is its own equivalent sample, and (ii) a non-parallelepiped sample with R 1 = R 2 (f(1) = 1) is represented by an equivalent square sample. Furthermore, a sample with square shape (L 1 = L 2 ) has also special interest, since R 1 and R 2 must have same values, no matter the size of the sample (see, for instance, results for Al foils in Table II of the reference 48 ). Since R 1 = R 2 for any square sample, R 1 + R 2 must be constant regardless the length of the squares, which is expected because sheet resistance must be the same for a given conducting material with homogeneous thickness (see Eq. (1) again).
In order to understand this transformation better, we compare the f(L 2 /L 1 ) from the exact solution given by Eq. (11) with f(R 2 /R 1 ) from different numerical calculations 34,35,[37][38][39] . Figure 3 displays the high precision numerical solution reported by Chan 39 and the expected behavior for Eq. (11) (see also Supplementary material). Available experimental data reported previously 38,48 and measured in this work are also shown in Fig. 3.
In Fig. 3, one can see a clear the overlap between the expected behavior by Eq. (11), indicated by red line, and the numerical calculation by Chan 39 shown by the blue dashed line. Overlap (not shown) with other results reported previously were also observed 34,35,37,38 . Additionally, symbols shown in Fig. 3 are related to some experimental data available in literature for polyethylene (PE) samples (green triangles) 38 , Cu metal sheet plates (blue squares), and Al foil (red diamonds) 48 . Yellow circles and pink stars are due to measurements with samples made out of Al foil with a circular and arbitrary shape, respectively, as displayed in Fig. 4. In order to test the transformation proposed in this work, we measured a sample with arbitrary shape, as used by van der Pauw (see Fig. 3 in Ref. 34 ).
The electrical resistivity used to determine f(R 2 /R 1 ) for the experimental data plotted in Fig. 3 was measured using standard four-probe method, measuring square samples with Montgomery method 45 , or searching for R 1 = R 2 during the application of the vdP method, in which f(1) = 1 (equivalent square sample). As shown in Fig. 3, all the data for the several isotropic samples collapse into a universal behavior, despite the shapes and materials.
The upper inset of the Fig. 3 displays the results of the sheet resistance calculated by Eq. (12) for the five experimental sets of data. The results show that R S is almost constant for each material over the measured range: 1 ≤ R 2 /R 1 ≤ 10 3 , as expected.
Regarding practical purposes, as reported previously 48 H 1 and H 2 can be truncated in the first term of each series, as H 1 ≈ π 8 sinh(πG) and H 2 ≈ π 8 sinh(π/G) , providing an approximated solution for the geometric factor from Eq. (7) as where G ≈ L 2 /L 1 is a dimensionless geometric parameter for the equivalent parallelepiped sample which can be easily estimated by . www.nature.com/scientificreports/ that is a good approximation to find L 2 /L 1 from 1 up to 4, corresponding to R 2 /R 1 from 1 up to ~ 10 5 . Figure 5 displays the behavior of f(G) given by Eq. (13) and compares with the exact solution of f(L 2 /L 1 ) predicted by Eq. (11).
The behavior of the black line in Fig. 5 demonstrates that the Eq. (13) for f(G), along with (14), is a good approximation to determine f(L 2 /L 1 ), which is very close to the red line. The error of f(R 2 /R 1 ) shown in inset of the Fig. 4 is less than 3.3% over the 10 -5 ≤ R 2 /R 1 ≤ 10 5 range. For the practical range of R 1 and R 2 measurements (typically R 2 /R 1 ≤ 10 3 ) the error is less than 1%. Thus, the benefits of using the approximated solution can be considered excellent since the uncertainties due to the size of the electrical contacts during samples preparation are typically much higher. A numerical comparison between f(L 2 /L 1 ) and f(G), for selected values in the R 2 /R 1 range shown in Fig. 5 is displayed in Table 1.
Based upon above discussion, one can rewrite Eq. (1) as which is a simple approximated function that allows the determination of R S or ρ for thin samples (E = d) directly from the R 1 and R 2 measurements. If R 1 and R 2 are measured with high precision, the error of ρ in Eq. (15) is the same as the f(G), which is displayed in upper inset of the Fig. 5 and Table 1. Finally, the conformal transformation described above is verified analyzing the measurements of the samples displayed in Fig. 4, with the method reported here. Table 2 and 3 provided the results for both samples, whose different electrical contacts were used to measure R 1 and R 2 , as indicated in the corresponding tables.

Figure 5.
Comparison between exact solution f(L 2 /L 1 ) and approximated solution f(G) given by Eqs. (11) and (13), respectively. In lower inset is displayed a zooming from 10 4 to 10 5 . Upper inset shows the percentage difference between f(R 2 /R 1 ) and f(L 2 /L 1 ). Error is less than 3.3% over the 10 -5 ≤ R 2 /R 1 ≤ 10 5 range. License: GF3S5-3078-7903112, Origin 2018 (9.5), https ://www.origi nlab.com. Table 1. Some selected values for the parameters discussed in this work. L 2 /L 1 , H 1 , H 2 , and f(L 2 /L 1 ) are numerically calculated using Eqs. (4), (10), and (11) with n = 0. G and f(R 2 /R 1 ) are calculated using the approximations given by Eqs. (13) and (14). Error of f(G) was obtained after comparison with f(L 2 /L 1 ) values. For details of the calculations refer to Section II of the Supplementary material.  Table 2 and 3 show that the Eqs. (13) to (15) allow one to determine easily the sheet resistance of the aluminum foil in both samples. All values of the sheet resistances are close to each other and the averages, < R S > = 3.25(3) and < R S (G) > = 3.26(3) mΩ for the circular sample, and < R S (G) > = 3.34(3) mΩ for the sample with arbitrary shape, agree very well, within of uncertainty, with regard to the value determined from standard four-probe method, R S (4P) = 3.2(3) mΩ. Especially interesting is the result in the first line of the Table 3, related to the shape and the electrical contacts of the sample that is similar to the one reported by vdP 34 (yellow contacts in Fig. 4b). In such a case, R 1 and R 2 measurements are similar and yield f(R 2 /R 1 ) ~ 1, suggesting that the electrical contacts in the historical sample by vdP 34 has to do with an equivalent square sample.

conclusion
In summary, a simple method for determining electrical resistivity and sheet resistance of isotropic samples has been obtained by comparing electrical resistivity equations given by vdP and Montgomery methods. The transcendental equation reported in the vdP method has been solved both exactly and approximately by Eqs. (12) and (15), respectively. The geometric factor, regardless of the shape of the sample, can be determined based upon H 1 and H 2 series, which depend on L 2 /L 1 , the ratio between the length and width of the equivalent parallelepiped sample. A comparison of the geometric factor with previous numerical calculations and experimental data shows a universal behavior for any isotropic conducting materials. The method reported can be used for measuring electrical properties of many materials, as noted in the introduction. Compared with the best numerical calculation reported by Chan 39 , the method described here is much simpler to use. As far as we know, this is the first time since the report for the vdP method 34 more than 60 years ago, that an analytical solution for this method has been reported.
Received: 22 April 2020; Accepted: 24 August 2020 Table 2. Experimental data set measured using the sample with the circular shape shown in Fig. 4a. α defines the position of the electrical contacts A to D. f(R 2 /R 1 ) is calculated using Eq. (1) with R S (4P) determined through conventional four probe measurement. R S is obtained using Eq. (12), after determining H 1 in a similar manner as described in Table 1. Values of G, f(G), and R S (G) are calculated using Eqs. (13) to (15). α (degree) R 2 (mΩ) R 1 (mΩ) R 2 /R 1 f(R 2 /R 1 ) R S (mΩ) G f(L 2 /L 1 ) R S (G) (mΩ)  Table 3. Experimental data for a sample with arbitrary shape shown in the Fig. 4b. Letters and the colors in next to R 1 and R 2 values represent the positions of the electrical contacts. f(R 2 /R 1 ), G, and R S were calculated using Eqs. (1), (14), and (15), respectively. Uncertainties in R 1 and R 2 are determined by the linear regression of IV curves and uncertainties in f(R 2 /R 1 ) are determined by propagating the error due to R S (4P) determined by standard four probe method.