This paper presents an extensive collection of calculated correction factors that account for the combined effects of a wide range of non-ideal conditions often encountered in realistic four-point probe and van der Pauw experiments. In this context, “non-ideal conditions” refer to conditions that deviate from the assumptions on sample and probe characteristics made in the development of these two techniques. We examine the combined effects of contact size and sample thickness on van der Pauw measurements. In the four-point probe configuration, we examine the combined effects of varying the sample’s lateral dimensions, probe placement, and sample thickness. We derive an analytical expression to calculate correction factors that account, simultaneously, for finite sample size and asymmetric probe placement in four-point probe experiments. We provide experimental validation of the analytical solution via four-point probe measurements on a thin film rectangular sample with arbitrary probe placement. The finite sample size effect is very significant in four-point probe measurements (especially for a narrow sample) and asymmetric probe placement only worsens such effects. The contribution of conduction in multilayer samples is also studied and found to be substantial; hence, we provide a map of the necessary correction factors. This library of correction factors will enable the design of resistivity measurements with improved accuracy and reproducibility over a wide range of experimental conditions.

## I. INTRODUCTION

Electrical characterization of complex 3D samples is increasingly needed for a wide range of applications. In this regime, some of the approximations used in van der Pauw and four-point probe techniques fail, resulting in non-ideal measurement conditions. Moreover, as advances in the semiconductor industry continue to shrink the size of device features, there is a growing need for experimental thin-film characterization at scales and geometries where multiple non-ideal conditions may exist. However, the adaptation of bulk electrical measurement techniques to the study of nanoscale features is challenging and requires an understanding of the interplay between contact size and sample thickness, as well as micro- and nanoscale inhomogeneities in electrical properties. Additionally, the increasing complexity in device architectures can complicate interpretation of measurement results. Within this context, this work attempts to characterize the efficacy of two such measurement techniques under non-ideal conditions, namely, the van der Pauw and linear four-point probe techniques.

The van der Pauw (vdP) technique, published in 1958,^{1} is a non-destructive method used to perform direct thin-film sheet resistivity measurements. This method can be used to characterize any arbitrarily shaped samples and has been extended over the years to perform other types of experimental characterization such as thermal resistivity measurements.^{2–6} In this work, we focus only on electrical measurements using a van der Pauw setup as represented schematically in Fig. 1(a) on a square sample. For a complete description of the derivation of this method and how it applies to samples of arbitrary shapes, the reader is referred to the work of van der Pauw.^{1} Despite all its advantages, the vdP technique relies on very stringent experimental conditions such as infinitely thin (2D) samples and infinitely small contacts placed on the periphery of the sample, which are often impossible to achieve in practice. This creates a need for correction factors that account for deviations from these ideal conditions.

The four-point probe technique is similar to the van der Pauw method in the sense that they both use four probes and can measure electrical resistivity while minimizing the effect of parasitic resistance. However, the underlying assumptions and derivation of those two methods are different. The van der Pauw technique was developed based on the method of conformal mapping and allows for the Hall effect characterization as well as resistivity measurements to be performed in just one experiment with the application of a magnetic field.^{1} Another key difference between the two methods is based on the use of (usually linear and uniformly placed) probes ideally at the center of the sample in the case of four-point probes, as shown in Fig. 1(b), whereas the probes are placed in arbitrary locations on the periphery of the sample in the van der Pauw method. In addition, the ideal four-point probe geometry requires samples to have an infinite lateral dimension, whereas this is not a requirement for van der Pauw measurements. For more details on the derivation of the four-point probe method, the reader is referred to the original paper published in 1954 by Valdes where he describes the four-point probe technique as applied to resistivity measurements in semiconductors.^{7} Deviations from ideal conditions in the case of four-point probe measurements also require the use of correction factors.

Over the years, several authors have derived correction factors to address non-ideal conditions in both van der Pauw and four-point probe measurements.^{1,8–10} However, as characteristic device sizes shrink, or as samples become three-dimensional in nature, multiple non-idealities may be present, rendering most existing correction factors inaccurate. In this study, we present a comprehensive evaluation of, and provide practical considerations for, conducting van der Pauw and linear four-point probe measurements in situations where multiple non-ideal conditions are present.

## II. BACKGROUND

### A. The van der Pauw and four-point probe techniques

As shown in Fig. 1(a), a typical van der Pauw experiment places four-point contacts (*A*, *B*, *C*, and *D*) on the periphery of a thin-film sample. For illustration purposes, a square sample of side length *L* and thickness *t* is used, although samples of any arbitrary shape and size can be used. The voltage drop between two contacts is measured while a constant current is maintained through the other two contacts. Measuring *R*_{1} and *R*_{2} on different configurations allows the experimental sheet resistance $Rs,exp$ to be determined using Eq. (1). We remind the reader that the different assumptions made in the van der Pauw technique are as follows: (1) the sample is infinitely thin (2D shape), (2) contacts are placed on the circumference of the sample, (3) contacts are infinitely small (point contacts), (4) the sample is homogeneous in thickness, and (5) the sample does not contain holes. While applicable to a random geometry, symmetric forms are normally used with contacts placed on the corners or edges. In such cases, *R*_{1} and *R*_{2} are the same and Eq. (1) reduces to Eq. (2).

A schematic of the setup for a four-point probe experiment is presented in Fig. 1(b). In the most common configuration, linear probes that are uniformly spaced by a distance, *s*, are configured such that current passes through the outer probes, while the potential is measured between the inner probes. We define the probe placement normalized to the sample’s length *L* and width *W*, respectively, *f*_{l} and $fw$, to indicate how far from the center the probes are placed on the sample with $fl=fw=0.5$ corresponding to the symmetric case. The sheet resistance for an ideal four-point probe setup (with an infinitely thin 2D sample) is given by Eq. (2). The resistivity $\rho exp$ is simply $Rs,exp*t$, where *t* is the sample thickness. Complete derivations, as well as the underlying assumptions for those equations, are given by van der Pauw^{1} and Valdes,^{7}

### B. Correction factors

The experimental sheet resistance, $Rs,exp$, is equal to the true sample sheet resistance if all of the ideal conditions and assumptions are fully satisfied. In reality, one or more non-ideal conditions are usually present. In such cases, experimental sheet resistance, $Rs,exp$, differs from the true sample resistivity, $Rs$, by a factor which depends on the specific sample geometry. We define a correction factor, $fcorr$, as the value by which $Rs,exp$ is multiplied to obtain *R*_{s} [Eq. (3)]. Hence, a correction factor greater than 1.0 means that the true sample resistivity is underpredicted,

A non-exhaustive list of possible non-ideal conditions for van der Pauw measurements is presented in Table I. As mentioned before, some correction factors have previously been computed in the literature. For example, Chwang *et al.*^{8} have computed correction factors for finite contact size in van der Pauw measurements, whereas Kasl and Hoch^{9} calculated correction factors for the effect of finite thickness. Wu *et al.*^{11} looked at the effects of placing contacts away from the periphery of the sample, while Matsumura^{12} studied inhomogeneous samples. In the present work, we complement those previous studies by looking at the effects of both finite contact size and sample thickness on the accuracy of vdP measurements.

Non-ideal conditions . | References . |
---|---|

Finite contact size | Chwang et al.^{8} |

Contacts not on the periphery | Baojia et al., Koon et al.^{10,11} |

Finite thickness | Kasl and Hoch^{9} |

Inhomogeneous sample | Matsumura and Sato^{12} |

Finite thickness and finite contact size | This work |

Similarly, in Table II, we present a list of non-ideal sample geometries and conditions for the four-point probe technique. Previous work on the subject include Liu^{13} and Smits^{14} who have studied the effect of finite thickness (in the $<$40 nm regime) and finite sample size, respectively, on four-point probe measurements. Valdes^{7} and Uhlir^{15} have also published in-depth discussions on four-point probe correction factors for different non-ideal geometries and conditions including the effect of sample size, thickness, and conductive faces. For a comprehensive review of existing correction factors for the four-point probe method, the reader is referred to the work of Miccoli *et al.*^{16} Our work complements and completes those previous results in a few significant ways. First, in the treatment of both Uhlir and Valdes, the sample is always infinite in at least one dimension (except in the case of square filament with symmetric probe placement in Uhlir’s paper). Our studies focus on samples that are finite in all dimensions. Moreover, in Valdes and Uhlir’s discussion on the effects of conducting substrates, only the thickness of the measured layer is varied. The thickness of the conducting substrate is assumed to be very small (essentially corresponding to a 2D surface) and its resistivity is either infinitely large (insulator) or negligible. In our case, both the thickness and the conductivity of the substrate are varied to better understand their effects on measured resistivity.

Sample description (primary non-ideal conditions) . | References . |
---|---|

Infinite slice (thickness) | Valdes,^{7} Uhlir,^{15} and Smits^{14} |

Infinite slice with conducting bottom (thickness) | Valdes^{7} and Uhlir^{15} |

Semi-infinite slice with probes parallel to the edge (thickness) | Valdes^{7} and Uhlir^{15} |

Semi-infinite slice with probes perpendicular to the edge (thickness) | Valdes^{7} and Uhlir^{15} |

Quarter-infinite slice (finite thickness) | Uhlir^{15} |

Square filament with symmetric probe placement (sample size and thickness) | Uhlir^{15} |

2D rectangular sample with symmetric probe placement (sample size) | Smits^{14} |

2D circular sample with symmetric probe placement (sample size) | Smits^{14} |

2D rectangular sample with asymmetric probe placement (sample size and probe placement) | This work |

Rectangular slice with symmetric probe placement (sample size and thickness) | This work |

Square slice on infinite substrate with finite conductivity and thickness (substrate thickness and conductivity) | This work |

Sample description (primary non-ideal conditions) . | References . |
---|---|

Infinite slice (thickness) | Valdes,^{7} Uhlir,^{15} and Smits^{14} |

Infinite slice with conducting bottom (thickness) | Valdes^{7} and Uhlir^{15} |

Semi-infinite slice with probes parallel to the edge (thickness) | Valdes^{7} and Uhlir^{15} |

Semi-infinite slice with probes perpendicular to the edge (thickness) | Valdes^{7} and Uhlir^{15} |

Quarter-infinite slice (finite thickness) | Uhlir^{15} |

Square filament with symmetric probe placement (sample size and thickness) | Uhlir^{15} |

2D rectangular sample with symmetric probe placement (sample size) | Smits^{14} |

2D circular sample with symmetric probe placement (sample size) | Smits^{14} |

2D rectangular sample with asymmetric probe placement (sample size and probe placement) | This work |

Rectangular slice with symmetric probe placement (sample size and thickness) | This work |

Square slice on infinite substrate with finite conductivity and thickness (substrate thickness and conductivity) | This work |

In the present paper, correction factors are calculated for situations where multiple non-ideal conditions coexist and plotted as a function of two appropriately normalized non-ideal conditions. The proper normalization for the contact size in van der Pauw measurements is the square root of the sample’s top surface area. For a square sample, this is equivalent to using the length of the sides. The probe spacing, *s*, is a normalization constant in a four-point probe.

### C. Numerical methods

While analytical solutions exist for several of these non-idealities, numerical methods must be used to solve cases involving more complex geometries. In this paper, we use a Finite Element Method (FEM) (implemented in COMSOL$\u2009\u2009\u24c7$ Multiphysics)^{17} to solve for the three-dimensional distribution of the electrical potential within the sample. FEM has been successfully used in the past to solve for the electrical potential distribution during van der Pauw measurements with contacts placed away from the periphery.^{11} We assume isotropic conductivity and solve Eqs. (4)–(6), where *J* is the current density in A/m^{2}, *E* is the electric field in V/m, $\sigma $ is the electrical conductivity in S/m, and *V* is the potential V,

For both the van der Pauw and four-point probe method, cuboids are created in COMSOL according to Fig. 1(a). Electrical contacts are created using a 2D square surface at each corner. Correction factors are determined as a function of thickness of the sample as well as the size of the contacts. For the four-point probe method, rectangular monolayer or bilayer samples are created, but, unlike in the van der Pauw method, the contacts for the four-point probes are simulated using circular surfaces (to represent points) of radius at least 50 times smaller than the probe separation distance, *s*. Correction factors for the four-point probe method applied to monolayer samples are computed as a function of sample size and thickness as well as probe placement. In the case of bilayer samples, correction factors are computed as a function of conductivity and thickness of the substrate. For both the van der Pauw and four-point probe methods, one of the contacts is grounded while another one is used as the current source. The voltage is measured on the other two contacts while all other surfaces remain insulated.

To ensure the correctness of our methodology, sample data published by NIST^{18} were used for the electric current as well as the “true” sheet resistance of the samples. As per those data, a value of 4.103 × 10^{8} $\Omega /square$ was used for the sheet resistance *R*_{s} and a current of 1.005 nA was applied between the two outer probes in the case of four-point probes or any two consecutive probes in the case of vdP. The 3D FEM solution for the electric field distribution was then computed using COMSOL. After proper convergence of the solution, the ratio of the voltage (calculated between the remaining two probes) and the current was calculated as *R*_{A}. In the case of vdP, the probes used for current and voltage were then interchanged to calculate *R*_{B}. The experimental sheet resistance $Rs,exp$ was subsequently calculated using Eq. (1). For the four-point probe, Eq. (2) was used directly with *R*_{A} to calculate $Rs,exp$. We verified that $Rs,exp$ was equal to the true sheet resistance (*R*_{s} = 4.103 × 10^{8} $\Omega /square$) when the ideal conditions were met and that $Rs,exp$ differed, sometimes significantly, from *R*_{s}, otherwise. With both the true resistance *R*_{s} and the measured resistance $Rs,exp$ known, as per Eq. (3), the correction factor was computed as the ratio of those two values. This process was repeated for all the different cases of interest in this study leading to the correction factors reported here.

## III. RESULTS AND DISCUSSION

### A. Combined effect of sample thickness and contact size on van der Pauw

Figure 2 shows correction factors for van der Pauw resistivity measurements under non-ideal conditions. For a 2D sample, as previously reported in the literature,^{8,10} the measured sheet resistance decreases as the contact size increases. In such scenarios, van der Pauw measurements will underpredict the “true” resistivity and a correction factor greater than 1 will be required. For example, for a 2D sample, when the contact size is 25% of the side length of the sample, the measured resistivity is roughly 13% lower than the true resistivity. This is identical to previous calculations by Chwang *et al.*^{8} On the other hand, increasing sample thickness has the opposite effect, with the measured resistivity becoming larger than the true resistivity and hence requiring a correction factor less than 1.

Figure 2 presents our new results accounting concurrently for both sample thickness, *t*, and contact size. The reader is reminded that the normalization constant for the contact size in van der Pauw measurements is the square root of the sample’s top surface area (which corresponds to a side length, *L*, in the case of a square sample). The contours are lines of constant correction factor. Based on these results, sample thickness starts to have a significant effect on the accuracy of van der Pauw measurements only when the aspect ratio (*t*/*L*) is greater than 50%. Below an aspect ratio of 50%, errors are only on the order of 3% for small contact sizes. Moreover, for sufficiently thin samples, contact size can increase to 16% of the square sample size with virtually no effect on the accuracy of the technique. Above 16%, contact size starts to induce an increasingly large finite error. In some regimes, this error may be fully compensated by the opposite effect of increasing thickness. For example, the error made when the size is 20% is fully compensated by the opposite error due to a sample thickness of 40%. Such an error would be around 13% if the sample was infinitely thin. The use of a correction factor that only takes into account finite contact size in such a scenario would lead to the wrong resistivity value. This underscores the importance of employing the data in Fig. 2 to determine the appropriate value of the correction factor for van der Pauw measurements when multiple non-ideal conditions are present.

### B. Combined effect of finite sample size and asymmetric probe placement on the four-point probe

As mentioned earlier, the ideal four-point probe sample is a 2D sample with infinitely large lateral dimensions and infinitely small thickness. In this section, we explore the effects of finite sample size and compute relevant correction factors. The probe spacing, *s*, provides a natural scaling coordinate for all distances. Consequently, we define scaled coordinates, *x* and *y*, from physical coordinates (*X*, *Y*) as *x* = *X*/*s* and *y* = *Y*/*s* and scaled sample dimensions as $w=W/s$ and *l* = *L*/*s*. An analytical solution can be obtained using the method of images for samples of finite length and width, as given by Eq. (7). The parameters $fl$ and $fw$ account for asymmetric placement of the probes on the sample with $fw=fl$ = 0.5 corresponding to probes at the center of the sample. A detailed derivation of Eq. (7) is given in the Appendix. At the limit of infinite length and width, and for symmetric placement of the probes, the familiar expression given by Eq. (2) is recovered. Equation (7) and Smits’s analytical solution can be used interchangeably in the case of symmetric probe placement,^{14}

Figure 3 shows a graphical representation of Eq. (7) for the symmetric case. As expected, the correction factor is 1.0 when the dimensions of the sample approach infinity. The infinite sample size holds true as long as the normalized lateral dimensions remain above a value of approximately 25. Experimental measurements usually take place in the regime below a normalized size of 25 and should therefore be corrected by the appropriate factor. In the limit of small length values, more specifically below a normalized length of 5, further reducing the length of the sample has a dramatic effect on the accuracy of the four-point probe method. For example, a normalized length of 3 induces a 40% error, independent of the normalized width. However, in the limit of large sample size, more specifically, above a normalized length and width of 5, both width and length have relatively the same effect. The contour lines of constant error are parallel to one axis for any fixed value of the other.

To further validate the analytical result, for asymmetric probe placement, Figs. 4(a)–4(f) present plots of the correction factor given by Eq. (7) and corresponding numerical calculations for different sample sizes. Excellent agreement is observed between analytical and numerical results. Moreover, Figs. 5(a) and 5(b) show a comparison with experimental data for asymmetric probe placement. Good agreement is observed, with slight discrepancies attributed to size effects of the sample. The reader is reminded that these plots show validation data for only a few limited cases of Eq. (7) to confirm its validity. The equation obtained through analytical derivation remains applicable for combined effects of both asymmetric probe placement and finite sample size since we see very good agreement between the analytical, numerical, and experimental results in these validation tests.

As expected, and as shown in Figs. 4 and 5, in all cases, $fw$ has a larger effect than $fl$. In the limit when $fw$ approaches zero ($fw\u21920$), the current is forced to go through a smaller channel leading to a higher apparent resistivity. On the other hand, when the probes are placed sufficiently close to the center, any asymmetric effects become negligible and only finite lateral size effects matter. Based on these results, far enough from the center, additional correction factors have to be applied to any four-point probe resistivity measurement. The effect becomes even more significant if the probes are not placed at the center when measuring a sample of finite size. Predictably, the smaller the sample, the more pronounced the effect of asymmetric probe placement. For example, the accuracy of a measurement can become as low as 30% if the probes are placed at $fw$ = 0.25 (half-way to the center) on a sample with a normalized lateral dimension of 5.

### C. Combined effect of finite sample size and thickness on a four-point probe

Figure 6 gives correction factors for four-point probe measurements as a function of normalized sample size and thickness for square samples. As expected, and as discussed in Sec. III B, resistivity is overpredicted as the sample size approaches the probe spacing. However, deviations from the 2D limit enhance this effect and errors of as much as 40% can result when the thickness is just twice the probe spacing even with infinite lateral dimensions. Based on these results, we can summarize a few “rules of thumb” for best practices during four-point probe measurements:

It is preferable to have a sample size at least eight times greater than the probe spacing. Below a normalized lateral size of 8, the error increases significantly even if the thickness approaches zero.

The thickness should preferably be less than the probe spacing. Beyond a normalized thickness of 1, the accuracy of the four-point probe method also drops significantly. For example, once the normalized thickness is more than 3, the accuracy is less than 50% even for infinite lateral dimensions.

These rules of thumb only ensure that errors are on the order of 10%. A correction factor is still needed whenever the normalized sample size is less than 15 and/or the normalized sample thickness is more than 0.25.

### D. Non-ideal conductive substrates

Normally, thin conducting films are grown on insulating substrates where the conduction of the substrate can be ignored. However, for high resistivity films on substrates with finite conductivity, the current is shared between the film and the substrate. In order to study the influence of the presence of an additional layer, we use an otherwise ideal square sample with small thickness and small contact size. We vary both the resistivity and thickness of the substrate to probe the influence on the accuracy of the thin film resistance estimates. This is repeated for a substrate size corresponding to experimentally etched samples. An example of such samples is given in Figs. 7(a) and 7(b).

In Fig. 8, the correction factor for four-point probe measurements is plotted as a function of both substrate and film conductivity normalized by their respective thicknesses. Two cases are studied: in the first, substrate and film have the same lateral dimensions [Fig. 8(a)], and in the second, the substrate’s lateral dimension is five times that of the film [Fig. 8(b)]. The ideal bilayer sample would have a substrate of very low conductivity (very resistive) compared to the film. In this scenario, substrate thickness would be irrelevant as all the current would flow through the film. As the substrate becomes more conductive than the film, substrate thickness will start to increasingly influence the measured resistance. Larger substrate thickness will cause the measured resistivity to be lower than the true resistivity leading to correction factors greater than 1. The contour lines in Fig. 8 illustrate such behavior, showing that the error increases with both substrate conductivity and thickness. In the limit of small normalized substrate thicknesses (below 0.1), conductive substrates will have limited effect on the accuracy of four-point probe measurements. Increasing the substrate lateral size relative to the film will only enhance the behavior described above and lead to even higher error values, as shown in Fig. 8(b). Similar behavior is expected for van der Pauw measurements.

## IV. SUMMARY

We evaluate the influence of non-ideal conditions on the accuracy of van der Pauw and four-point probe techniques. We show that correction factors for van der Pauw measurements must include both the influence of finite contact size and sample thickness for accurate results. Similarly, in four-point probe measurements, infinitely large samples can still lead to erroneous results if the thickness is not taken into account. In addition, an analytical solution is derived and validated with numerical and experimental data for the effect of both finite lateral dimensions and asymmetric probe placements in four-point probe measurements of rectangular samples. Finally, we show that, in bilayer samples, proper tuning of the substrate resistivity and thickness can lead to electrical characterization within acceptable error. We hope these studies will provide practical guidelines for semiconductor thin-film resistivity measurements in increasingly common, non-ideal conditions.

## ACKNOWLEDGMENTS

Mardochee Reveil thanks Intel Corporation for generous funding through the GEM Fellowship and the Colman family for generous financial support through the Colman Fellowship at Cornell. Victoria C. Sorg acknowledges support, in part, through an Intel Foundation/Semiconductor Research Corporation Education Alliance (SRCEA) Graduate Research Fellowship and through a National Science Foundation Graduate Research Fellowship (NSF GRFP) under Grant No. DGE-1144153. E. R. Cheng acknowledges support through Cornell’s Engineering Learning Initiatives Undergraduate Research Program, funded through SRCEA. This work benefited from computing resources provided by the Cornell Institute of Computational Science and Engineering (ICSE).

### APPENDIX: DERIVATION OF THE ANALYTICAL SOLUTION FOR FINITE SIZE AND ASYMMETRIC PROBE PLACEMENT IN RECTANGULAR FOUR-POINT PROBE SAMPLES

The probe spacing, *s*, provides a natural scaling coordinate for all distances. Consequently, we define scaled coordinates *x* and *y* from physical coordinates (*X*, *Y*) as *x* = *X*/*s* and *y* = *Y*/*s* and scaled sample dimensions as $w=W/s$ and *l* = *L*/*s*. The origin is taken to be the center of the probe set with point-contact probes at coordinates (−3/2, 0), (−1/2, 0), (1/2), and (3/2, 0).

#### 1. Field from a single current source

Consider a current of magnitude, *I*, injected at the origin (of an arbitrary coordinate system) into an infinite sheet with resistivity, $\rho $ (isotropic), and thickness, *t*. The return current path is at $r=\u221e$ and the thickness, *t*, is assumed to be negligible. In a polar coordinate system, the current density *J* is then radially symmetric and the electric field *E* can be obtained via Ohm’s law where $\rho $ is the resistivity as

At a Cartesian coordinate (*x*, *y*), the polar transformation to $(r,\theta )$ requires

Re-expressing the electric field in Cartesian coordinates gives

For a current injection at any other coordinate (*x*_{0}, *y*_{0}), the electric field is

As the electric field is the gradient of the potential ($E\u2192=\u2212\u2207V$), the total potential difference between the two sense probes of the 4-point probe is given by the integral

Using linear superposition, the voltage for a collection of current source is just the sum of the integrals. Consider a positive current injected at (+3/2, 0) and returning from the contact at (−3/2, 0), the potential is then

which gives the sheet resistivity for an infinite sample as the familiar

#### 2. Superposition for a sample of finite width

For a sample of width, $w$, with the 4-point probes placed at the center, the finite sample can be extended to infinity by mirroring the sample in the *y* direction. This implies an infinite number of +*I* current sources at $\u2026,(+3/2,\u22122w),(+3/2,\u2212w),(+3/2,0),(+3/2,w),(+3/2,2w),\u2026$ and a similar set of −*I* sources at the *x* = −3/2 position. The potential is now just given by the infinite sum over all of these sources,

where we used the substitutions of *u* = (*x* − 3/2) and $v=u2$. The final result is remarkably simple and can be quickly verified to give the correct limit as $w\u2192\u221e$.

This can be written in terms of the correction factor, $fc$, as

The requirement that the probes be placed in the center of the sample width can easily be relaxed by recognizing the symmetry and can be achieved by pairing the mirror sources. Let the four-point probes be placed at a distance, *f*, from one edge of the sample and 1 − *f* from the other edge (*f* = 0.5 being the symmetric condition). In addition to current sources at *y* = 0, mirror current sources, in the positive direction, must be at $y=2fw$, $y=2w$, $y=2w+2fw$, $y=4w$, …with similar sources in the negative direction at $y=\u22122w+2fw$, $y=\u22122w$, $y=\u22124w+2fw$, $y=\u22124w$, …. In the limit of *f* = 0.5, this reproduces a symmetric set of sources and, in the limit $f\u21920$ or $f\u21921$, it reduces to a 2*I* current source with a sample width of 2$w$, as expected.

These source coordinates can be written more succinctly as pairs of sources at $y0=2nw$ and $y0=(2n+2f)w$ for all *n*. The integral over the +*I* and −*I* sources now just become four integrals,

and the equivalent correction factor can be written as

#### 3. Finite length samples

The extension to a finite length sample is straightforward although perhaps mathematically inelegant. We consider just a set of the sources mirrored in *x*. We have the primary negative source at *x* = −3/2 and positive source at *x* = +3/2. For symmetrically placed probes along the length, we need a matching positive mirror source at *l* − 3/2 in the positive direction and at *l* + 3/2 as the negative source. The next mirror source must reverse again with a negative source at 2*l* − 3/2 and a positive source at 2*l* + 3/2.

It is simpler to consider a repeat block consisting of four current sources that is mirrored with a 2*l* distance. Limiting ourselves initially to an infinitely wide case (the extension being obvious), we have an explicit set of sources at

which can be replicated for all $\u2212\u221e<m<\u221e$ as

It is easy at this point to take the asymmetric probe placement into account with the factor *f* ( *f* = 0.5 being symmetric). The current sources are then

Changing limits on the additional integrals gives

In the limit that $l\u2192\u221e$, we recover the expected infinite sample result,

In the symmetric case, *f* = 0.5, the result can be further reduced to

#### 4. Finite length and width

Finally, one can combine the two summations to express the result for the arbitrary placement of probes within a rectangular sample as

where $fl$ and $fw$ are the corresponding asymmetry placement factors along the length and width, respectively.

The corresponding correction factor, $fc$, is