^{1}, Fei Wang

^{2}, Dirch H. Petersen

^{1,2}, Torben Mikael Hansen

^{2}, Daniel Kjær

^{1}, Rong Lin

^{1}, Jang-Yong Kim

^{2}, Peter F. Nielsen

^{1}and Ole Hansen

^{2,3,a)}

### Abstract

We show that accurate sheet resistancemeasurements on small samples may be performed using microfour-point probes without applying correction factors. Using dual configuration measurements, the sheet resistance may be extracted with high accuracy when the microfour-point probes are in proximity of a mirror plane on small samples with dimensions of a few times the probe pitch. We calculate theoretically the size of the “sweet spot,” where sufficiently accurate sheet resistances result and show that even for very small samples it is feasible to do correction free extraction of the sheet resistance with sufficient accuracy. As an example, the sheet resistance of a square sample may be characterized with an accuracy of 0.3% (0.1%) using a pitch microfour-point probe and assuming a probe alignment accuracy of .

The authors would like to thank Alessandra Satta and Antoine Brugere for preparation of the Ge sample. We are grateful for the financial support from Copenhagen Graduate School for Nanoscience and Nanotechnology (C:O:N:T), the Danish Research Agency (FTP), and Danish National Advanced Technology Foundation. Center for Individual Nanoparticle Functionality (CINF) is sponsored by The Danish National Research Foundation. We thank Peter Bøggild for continuous support, encouragement and fruitful discussions.

I. INTRODUCTION

II. THEORY

III. ANALYTIC, NUMERIC, AND EXPERIMENTAL METHODS

IV. RESULTS AND DISCUSSION

A. Single insulating boundary sample

B. Corner with top angle

C. Narrow stripe sample

D. Circular disk samples

E. Square samples

1. Square samples: Experiments

F. Rectangular samples

G. Practical measurement issues

V. CONCLUSION

### Key Topics

- Mirrors
- 26.0
- Electric measurements
- 5.0
- Electrodes
- 5.0
- Electrical resistivity
- 4.0
- Germanium
- 2.0

## Figures

Schematic of four-point probe measurement cases. In (A) the case discussed by van der Pauw is shown. In (B) a collinear four-point probe measurement on the trace of the sample mirror plane is shown. In (C) a van der Pauw measurement on half of the symmetric sample in (B) is shown.

Schematic of four-point probe measurement cases. In (A) the case discussed by van der Pauw is shown. In (B) a collinear four-point probe measurement on the trace of the sample mirror plane is shown. In (C) a van der Pauw measurement on half of the symmetric sample in (B) is shown.

Schematic of some simple sample geometries with mirror planes (dashed lines).

Schematic of some simple sample geometries with mirror planes (dashed lines).

The error as a function of angle between the line of the probe and a single insulating boundary with the distance between the boundary and the probe center as parameter. Calculations for are shown.

The error as a function of angle between the line of the probe and a single insulating boundary with the distance between the boundary and the probe center as parameter. Calculations for are shown.

Allowable misalignment angle as a function of the probe to boundary distance , with the error as parameter for measurements on a sample with a single straight insulating boundary. Calculations for are shown.

Allowable misalignment angle as a function of the probe to boundary distance , with the error as parameter for measurements on a sample with a single straight insulating boundary. Calculations for are shown.

Error parameter for a stripe of width with the probe parallel to the stripe as a function of probe displacement from the mirror plane. Calculations for normalized stripe widths are shown.

Error parameter for a stripe of width with the probe parallel to the stripe as a function of probe displacement from the mirror plane. Calculations for normalized stripe widths are shown.

Error parameter for a stripe of width with the probe center at as a function of the angle between the probe and the insulating boundaries. Calculations for are shown.

Error parameter for a stripe of width with the probe center at as a function of the angle between the probe and the insulating boundaries. Calculations for are shown.

Relation between allowable angular and lateral misalignments for a probe parallel to the boundaries of a stripe of width . The dotted curve shows the trajectory where . Calculations for are shown.

Relation between allowable angular and lateral misalignments for a probe parallel to the boundaries of a stripe of width . The dotted curve shows the trajectory where . Calculations for are shown.

Allowable angular misalignment as a function of stripe width . Probe initially parallel to the boundaries and in the middle of the stripe. Calculations for are shown.

Allowable angular misalignment as a function of stripe width . Probe initially parallel to the boundaries and in the middle of the stripe. Calculations for are shown.

Allowable lateral misalignment as a function of stripe width. Probe parallel to the boundary. Calculations for are shown.

Allowable lateral misalignment as a function of stripe width. Probe parallel to the boundary. Calculations for are shown.

Isoerror contours on a circular disk with radius . The full red curve indicates the sample perimeter, while the full blue curve is the trajectory of the center of the probe with one pin on the perimeter of the disk. Isoerror contours for are shown. To indicate the scale a probe positioned exactly at the center of the disk is shown.

Isoerror contours on a circular disk with radius . The full red curve indicates the sample perimeter, while the full blue curve is the trajectory of the center of the probe with one pin on the perimeter of the disk. Isoerror contours for are shown. To indicate the scale a probe positioned exactly at the center of the disk is shown.

Allowable misalignment of the four-point probe from the center of a circular disk sample of radius . Calculations are shown for .

Allowable misalignment of the four-point probe from the center of a circular disk sample of radius . Calculations are shown for .

Relation between relative radial probe position and relative off diagonal position for constant in four-point probe measurements on a circular disk of radius . Calculations for and are shown.

Relation between relative radial probe position and relative off diagonal position for constant in four-point probe measurements on a circular disk of radius . Calculations for and are shown.

Contour plot showing isoerror curves for the M4PP parallel to the diagonal of a square with the side-width . The dotted square indicates the position of the probe center in the extreme position with one probe pin on the boundary of the square, while the boundary of the square sample coincides with the boundary of the plot. Contour-lines corresponding to are shown. To assist visual interpretation of the graph the four-point probe is shown in its ideal position.

Contour plot showing isoerror curves for the M4PP parallel to the diagonal of a square with the side-width . The dotted square indicates the position of the probe center in the extreme position with one probe pin on the boundary of the square, while the boundary of the square sample coincides with the boundary of the plot. Contour-lines corresponding to are shown. To assist visual interpretation of the graph the four-point probe is shown in its ideal position.

Allowable misalignment from the ideal diagonal center position on a square sample for the errors . Calculations are shown for a sample width in the range from to .

Allowable misalignment from the ideal diagonal center position on a square sample for the errors . Calculations are shown for a sample width in the range from to .

Contour plot showing isoerror curves for the M4PP parallel to an edge of a square sample of width . Contour-lines corresponding to are shown as full and dashed lines. The dotted lines show the trace of the probe center with one probe pin on the edge of the sample, while the sample boundary coincides with boundary of the plot. To assist visual interpretation of the graph the M4PP is shown in its ideal position in the center.

Contour plot showing isoerror curves for the M4PP parallel to an edge of a square sample of width . Contour-lines corresponding to are shown as full and dashed lines. The dotted lines show the trace of the probe center with one probe pin on the edge of the sample, while the sample boundary coincides with boundary of the plot. To assist visual interpretation of the graph the M4PP is shown in its ideal position in the center.

Allowable misalignment from the ideal position at the center with the probe parallel to the edge of a square sample as a function of the sample size . Calculations for are shown.

Allowable misalignment from the ideal position at the center with the probe parallel to the edge of a square sample as a function of the sample size . Calculations for are shown.

Error parameter as a function of probe angle with the M4PP positioned at the center of a square or a rectangle. Calculations for squares with and rectangles with are shown.

Error parameter as a function of probe angle with the M4PP positioned at the center of a square or a rectangle. Calculations for squares with and rectangles with are shown.

A series of M4PP measurements on an approximately sample using a pitch probe arranged parallel to a sample edge; between each measurement the probe position is incremented by normal to the line of the probe. The full curve shows model calculations. Excellent agreement between measurement data and model is seen. Note, error bars on the experimental resistance data are drawn, but are not visible.

A series of M4PP measurements on an approximately sample using a pitch probe arranged parallel to a sample edge; between each measurement the probe position is incremented by normal to the line of the probe. The full curve shows model calculations. Excellent agreement between measurement data and model is seen. Note, error bars on the experimental resistance data are drawn, but are not visible.

Micrograph showing the pitch M4PP above a pad as seen on the screen of the measurement system.

Micrograph showing the pitch M4PP above a pad as seen on the screen of the measurement system.

Contour plots showing isoerror curves for a probe arranged parallel to the short (top panel) and long (bottom panel) edge of a rectangle of width and height , respectively. The boundary of the plots coincides with the sample boundary. The dotted lines show the trajectory of the probe center with one probe pin on the boundary of the sample. To ease visual interpretation of the plots the four-point probes are shown in their ideal positions in the center of the sample. Calculations for are shown.

Contour plots showing isoerror curves for a probe arranged parallel to the short (top panel) and long (bottom panel) edge of a rectangle of width and height , respectively. The boundary of the plots coincides with the sample boundary. The dotted lines show the trajectory of the probe center with one probe pin on the boundary of the sample. To ease visual interpretation of the plots the four-point probes are shown in their ideal positions in the center of the sample. Calculations for are shown.

Allowable misalignment for a M4PP arranged parallel to the edge of width for a rectangle of height . Calculations are shown for and sample width in the range from to .

Allowable misalignment for a M4PP arranged parallel to the edge of width for a rectangle of height . Calculations are shown for and sample width in the range from to .

Left, the upper half plane with an insulating boundary at . In the middle, the narrow stripe with a collinear four-point probe. Right, the circular disk with a collinear four-point probe.

Left, the upper half plane with an insulating boundary at . In the middle, the narrow stripe with a collinear four-point probe. Right, the circular disk with a collinear four-point probe.

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