accepted for publication in jpcm

The 100th anniversary of the four-point probe technique

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TOPICAL REVIEW The 100th anniversary of the four-point probe technique: The role of probe geometries in isotropic and anisotropic systems† I. Miccoli*,1,2, F. Edler1, H. Pfnür1 and C. Tegenkamp 1 1 2

Institut für Festkörperphysik, Leibniz Universität Hannover, Appelstrasse 2, D-30167 Hannover, Germany Dipartimento di Ingegneria dell’Innovazione, Università del Salento, Via Monteroni, I-73100 Lecce, Italy

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Received zzz, revised zzz, accepted zzz Published online zzz

Key words bulk and surface resistivity, 4 point probe techniques, correction factor, nanostructures

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The electrical conductivity of solid-state matter is a fundamental physical property and can be precisely derived from the resistance measured via the four-point probe technique excluding contributions from parasitic contact resistances. Through the ages, this method has become an interdisciplinary characterization tool in material science, semiconductor industry, geology, physics, etc. and is employed for both fundamental and application-driven research. However, the correct derivation of the conductivity is a demanding task, which faces several difficulties, e.g. the homogeneity of the sample or isotropy of the phases. Besides, these sample-specific characteristics are intimately related to technical constraints like the probe geometry and size of the sample. Notably, the latter is of importance for nanostructures that nowadays can be probed technically on very small length scales. On the occasion of the 100th anniversary of the four-point probe technique introduced by Frank Wenner, we revisit and discuss various correction factors in this review, which are mandatory for an accurate derivation of the resistivity from the measured resistance. Among others, sample thickness, dimensionality, anisotropy as well as the relative size and geometry of the sample with respect to the contact assembly are considered. Along this line, we also managed to derive the correction factors for 2D-anisotropic systems on circularly shaped finite areas with variable probe spacings. All these aspects are illustrated by state-of-the-art experiments carried out via a 4-tip STM/SEM system. We are aware that this review article can only cover some of the most important topics. Regarding further aspects, e.g. technical realizations, the influence of inhomogeneities or different transport regimes, etc., we refer to other review articles in this field.

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Copyright line will be provided by the publisher

† *

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This paper is dedicated to Prof. em. Dr. Martin Henzler on occasion of his 80th birthday Corresponding author: [email protected]

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I. Miccoli et al.: The role of probe geometries in isotropic and anisotropic systems

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Contents

1. 2. 3.

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6.

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4. 5.

Introduction 4-probes methods for isotropic semi-infinite 3D bulks and infinite 2D sheets Correction factors for finite isotropic samples 3.1. Samples of finite thickness: the correction factor 𝐹𝐹! 3.2. Probes in the proximity of a single sample-edge: the correction factor 𝐹𝐹! 3.3. Samples of finite lateral dimension: the correction factor 𝐹𝐹! 3.3.1. In-line and square 4P probe geometry inside a finite circular slice 3.3.2. Square 4P probe array inside a finite square slice Van der Pauw theorem for isotropic thin films of arbitrary shape Four point probe technique on anisotropic crystals and surfaces 5.1. Formulas for anisotropic semi-infinite 3D bulks and infinite 2D sheets 5.2. Classical approaches for finite anisotropic samples 5.2.1. The Montgomery method 5.2.2. The Wasscher method 5.3. Experimental comparison among finite and infinite regime for anisotropic 2D systems 5.4. Correction factors for a square 4P array inside an anisotropic circularly shaped area Conclusion and outlook

Introduction

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The specific electrical resistance or the resistivity ρ of a solid represents one of the most fundamental physical properties whose values, ranging from 10-8 to 1016 Ω ⋅ 𝑐𝑐𝑐𝑐 [1], are used to classify metals, semiconductors and insulators. This quantity is extremely important and manifoldly used for the characterization of materials as well as sophisticated device structures, since it influences the series resistance, capacitance, threshold voltage and other essential parameters of several devices, e.g. diodes, LEDs and transistors [2].

1.1

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𝜌𝜌 = 𝐸𝐸 𝐽𝐽

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From a fundamental point of view, a precise measurement of the resistance is closely related to other metrological units. In general, when an electric field 𝐸𝐸 is applied to a material it causes an electric current. In the diffusive transport regime, the resistivity ρ of the (isotropic) material is defined by the ratio of the electric field and the current density 𝐽𝐽:

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Thereby, the resistivity of the material is measured in Ω ⋅ 𝑐𝑐𝑐𝑐, the electric field in 𝑉𝑉 ⋅ 𝑐𝑐𝑐𝑐 !! and the current density in A ⋅ 𝑐𝑐𝑐𝑐 !! . Experimentally, a resistance 𝑅𝑅 is deduced from the ratio of an applied voltage 𝑉𝑉 and the current 𝐼𝐼. Only in case that the geometry of the setup is well-known, the resistivity can be accurately calculated as we will show below.

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As sketched in Fig.1a, the resistance 𝑅𝑅 is determined by measuring the voltage drop 𝑉𝑉 in between two electrodes, which impinge a defined current 𝐼𝐼 into the sample. However, the identification of this value with the resistance of the sample is usually incorrect as it includes intrinsically the contact resistances 𝑅𝑅! at the positions of the probes, which are in series with the resistance of the sample. This problem was encountered and solved for the first time in 1915 by Frank Wenner [3], while he was trying to measure the resistivity of the planet earth. He first proposed an in-line four-point (4P) geometry (Fig.1b) for minimizing contributions caused by the wiring and/or contacts, which is to date referred in the geophysical community as the Wenner method [4, 5]. In 1954, almost 40 years later, Leopoldo Valdes uses this idea of a 4P geometry to measure the Page 2 of 44


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resistivity ρ of a semiconductor wafer [6] and from 1975 this method was established throughout the microelectronics industry as a reference procedure of the American Society for Testing and for Materials Standards [7]. For the sake of completeness, also the C. Schlumberger method shall be mentioned here. He proposed already in 1912 an innovative approach to map the equipotential lines of the soil, however, his approach relied only on 2 probes. At first 8 years later he also measured the earth resistivity using a four-point probe configuration. In contrast to Wenner, the Schlumberger method uses non-equidistant probe spacings. The interested reader is referred to Ref. [8].

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Fig. 1 Schematic of (a) a two-point probe and (b) a collinear four-point probe array with equidistant contact spacing.

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Technically, if the voltage drop 𝑉𝑉 between the two inner contacts is measured while a current 𝐼𝐼 is injected through the two outer contacts of the proposed in-line 4P geometry, the ratio of 𝑉𝑉/𝐼𝐼 is a measure of the sample resistance 𝑅𝑅 only (providing that the impedance of the voltage probes can be considered infinite). Having this in mind, still the question remains how the resistivity 𝜌𝜌 of the material can be determined from the resistance 𝑅𝑅. This review summarizes the different mutual relations between these two quantities for isotropic and anisotropic materials in various dimensions. Thereby, the description covers various geometric configurations of the voltage and current probes, e.g collinear and squared arrangements. As we will show the four point (4P) probe resistivity measurements are intrinsically geometry-dependent and sensitive to the probe positions and boundary conditions. The relationship between 𝑅𝑅 and 𝜌𝜌 is defined by details of the current paths inside the sample.

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We will start with the recapitulation of homogeneous 3-dimensional semi-infinite (3D) bulk and infinite 2dimensional (2D) systems which can be exactly solved. Thereafter, the effect of limited geometries is taken into account for technically relevant cases (e.g. finite square- and circularly-shaped samples) followed by basics regarding the Van der Pauw method which can be applied for thin films of completely arbitrary shaped. Finally, we will revisit the regime of anisotropic phases based on the theoretical approaches of Wasscher and Montgomery. The careful re-analyses and application of their methods allowed us to derive for the first time the correction factors for a contact assembly inside a circular lamella hosting an anisotropic 2D-metallic phase. Our theoretical conclusions will be corroborated and illustrated by latest experiments done with a 4-tip STM/SEM performed either in our group or by our colleagues.

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We want to emphasize that this review rather highlights the progress made in the field of geometrical correction factors during the last century and their latest application on low dimensional, anisotropic and spatially confined electron gases. The inclusion of further aspects would definitely go beyond the constraints of this journal. As mentioned, this technique is used in related disciplines and readers with a geophysical background might be interested in Refs. [9, 10]. For technical aspects please see, e.g. Refs. [11, 12, 13]. Readers working in the field of surface science are referred to Refs. [14, 15], which address further aspects of semiconductor surface conductivity. At this point we like to acknowledge the contributions from our colleagues working also in the field of low dimensional systems, [16, 15, 17]. In comparison to the diffusive transport regime, further attention needs to be paid for the probes interacting with ballistic systems, where the probes may have either invasive or non-invasive character [18]. In this review we restrict ourselves to Page 3 of 44

I. Miccoli et al.: The role of probe geometries in isotropic and anisotropic systems homogeneous phases. The conclusions, of course, change drastically if inhomogeneities are present, as mentioned in Ref [19].

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2 4-probes methods for isotropic semi-infinite 3D bulks and infinite 2D sheets

For the ideal case of a 3D semi-infinite material with the four electrodes equally spaced and aligned along a straight line (4P in-line array – see Fig.1b), the material resistivity is given by [6]: 𝑉𝑉 𝐼𝐼

2.1

𝜌𝜌𝜌𝜌 𝑑𝑑𝑑𝑑 =− 𝑑𝑑𝑑𝑑 2𝜋𝜋𝑟𝑟!!

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𝑑𝑑𝑑𝑑 𝐼𝐼𝐼𝐼 ⟹ 𝑉𝑉 𝑃𝑃 = ! 𝑟𝑟 2𝜋𝜋𝑟𝑟!

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!"#$ = 2𝜋𝜋𝜋𝜋 𝜌𝜌!!

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where 𝑉𝑉 is the measured voltage drop between the two inner probes, 𝐼𝐼 is the current flowing through the outer pair of probes and 𝑠𝑠 is the probe spacing in between the two probes. Equation (2.1) can be easily derived considering that the current +𝐼𝐼, injected by first electrode in Fig.1a, spreads spherically into a homogeneous and isotropic material. Therefore, in a distance 𝑟𝑟! from this electrode, the current density 𝐽𝐽 = 𝐼𝐼 2𝜋𝜋𝑟𝑟!! and the associated electric field, i.e the negative gradient of the potential, can be expressed as: 𝐸𝐸 𝑟𝑟! = 𝜌𝜌𝜌𝜌 =

By integrating both sides of the equation (2.2), the potential at a point P reads:

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𝜌𝜌𝜌𝜌 2𝜋𝜋

!!

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𝑑𝑑𝑑𝑑 = −

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For the scenario shown in Fig. 1a, the voltage drop is then given by the potential difference measured between the two probes, i.e.: 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 1 1 − = − 2𝜋𝜋𝑟𝑟! 2𝜋𝜋𝑟𝑟! 2𝜋𝜋 𝑟𝑟! 𝑟𝑟!

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𝑉𝑉 𝑃𝑃 =

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1 1 1 1 − − − 𝑠𝑠! 𝑠𝑠! 𝑠𝑠! 𝑠𝑠!

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𝐼𝐼𝐼𝐼 2𝜋𝜋

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𝑉𝑉 = 𝑉𝑉! − 𝑉𝑉! =

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This concept can be easily extended to 4P geometries where usually the problem of contact resistances (see above) is avoided. According to Fig.1b, the concept presented above can be generalized and the voltage drop between the two inner probes of 4P in-line array is:

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which, for the special case of an equally spaced 4P probe geometry (with 𝑠𝑠! = 𝑠𝑠! = 𝑠𝑠 and 𝑠𝑠! = 𝑠𝑠! = 2𝑠𝑠), is equivalent to eq. (2.1).

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In order to calculate correctly the resistivities from the resistances other aspects are of importance. For instance, when the thickness 𝑡𝑡 of the sample is small compared to the probe spacings, i.e. for simplicity when Page 4 of 44


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𝑡𝑡 ≪ 𝑠𝑠 (see Sec. 3.1 below for a more accurate definition), the semi-infinite 3D material appears as an infinite 2D sheet and the current can be assumed to spread cylindrically instead of spherically from the metal electrode as depicted in Fig. 2. The current density in this case is given by 𝐽𝐽 = 𝐼𝐼 2𝜋𝜋𝜋𝜋𝜋𝜋, which yields an electric field of: 𝐸𝐸 𝑟𝑟 = 𝜌𝜌𝜌𝜌 =

𝜌𝜌𝜌𝜌 𝑑𝑑𝑑𝑑 =− 2𝜋𝜋𝜋𝜋𝜋𝜋 𝑑𝑑𝑑𝑑

2.6

Repeating the same steps as for eqs. (2.3) to (2.5) a logarithmic dependency is obtained for the voltage drop in between the two inner probes: 𝑉𝑉 = 𝑉𝑉! − 𝑉𝑉! =

𝑠𝑠! 𝑠𝑠! 𝐼𝐼𝐼𝐼 𝑙𝑙𝑙𝑙 𝑠𝑠! 𝑠𝑠! 2𝜋𝜋𝜋𝜋

(2.7)

In case of an equally spaced in-line 4P geometry the bulk resistivity is given by:

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!"#$ 𝜌𝜌!! =

𝜋𝜋𝜋𝜋 𝑉𝑉 , 𝑙𝑙𝑙𝑙2 𝐼𝐼

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i.e. the resistance is not dependent on the probe distance which directly underlines the 2D character of the specimen. In case of a homogenous and finitely thick sample the resistivity can be assumed to be constant, thus the bulk resistivity is often replaced by the so-called sheet resistance R !! defined as: ρ (2.9) R !! = 𝑡𝑡 Ω

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This quantity is also used to describe the spatial variation of the dopant concentration in non-homogeneously doped thick semiconductors (e.g. realized by ion-implantation or -diffusion). Please note that the dimension of the sheet resistance is also Ohm, but often denoted by Ω/𝑠𝑠𝑠𝑠 (Ohm per square) to make it distinguishable from the resistance itself. The origin of this peculiar unit name - Ohm per square - relies on the fact that a squareshaped sheet with a sheet resistance of 1 Ω/𝑠𝑠𝑠𝑠 would have an equivalent resistance, regardless of its dimensions. Indeed, the resistance of a rectangular rod of length 𝑙𝑙 and cross section 𝐴𝐴 = 𝑤𝑤 𝑡𝑡 can be written as R = ρ𝑙𝑙/A, which immediately simplifies to R = R !! for the special case of square-shaped lamella of side 𝑙𝑙 = 𝑤𝑤 (see Fig.3). The four electrodes are often arranged in a square configuration rather than along a straight line. Indeed, the square arrangement has the advantage to require a smaller area (the maximum probe spacing is only √2s against 3𝑠𝑠 of the collinear arrangement) and reveals a slightly higher sensitivity (up to a factor 2, see below). The corresponding expression of the bulk resistivity 𝜌𝜌 (sheet resistance R !! ) for the 4P square configuration on a semi-infinite 3D bulk (infinite 2D sheet) is easily derived from eq. 2.5 (2.7) with 𝑠𝑠! = 𝑠𝑠! = 𝑠𝑠 and 𝑠𝑠! = 𝑠𝑠! = 2𝑠𝑠 (see Fig. 3).

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1D wire*

𝑉𝑉 I

𝜋𝜋 𝑉𝑉 𝑙𝑙𝑙𝑙2 I Σ 𝑉𝑉 𝑠𝑠 I

2𝜋𝜋𝜋𝜋 𝑉𝑉 2− 2 I 2𝜋𝜋 𝑉𝑉 𝑙𝑙𝑙𝑙2 I −

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Fig. 3 Schematic of a square four-point probe configuration with 𝑠𝑠! = 𝑠𝑠! = 𝑠𝑠 and 𝑠𝑠! = 𝑠𝑠! = 2𝑠𝑠.

2𝜋𝜋𝜋𝜋

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2D sheet**

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3D bulk*

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Table 1 Bulk resistivity 𝜌𝜌 or sheet resistance R !! for the case of a linear and square arrangement of 4 probes on a semi-infinite 3D material, infinite 2D sheet and 1D wire.

* Bulk resistivity 𝜌𝜌, ** Sheet Resistance R !! , Σ= π𝑎𝑎 ! wire section

All relations derived so far for the infinite 3D and 2D systems are summarized in Table 1. From these equations it is evident that the measured resistance 𝑅𝑅 does not depend on the probe spacing for the 2D case Page 5 of 44


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(𝑅𝑅!! ∝ 𝜌𝜌 ∙ 𝑙𝑙𝑙𝑙2 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐), while it decreases as 𝑠𝑠 !! by increasing the probe spacing for the 3D case (𝑅𝑅!! ∝ 𝜌𝜌 𝑠𝑠). Naively, one would expect that the resistance should increase as the paths for the electric charges are increased, irrespective of the dimension. This counter-intuitive scenario can be rationalized by inspection of the sketches shown in Fig. 4 (for a linear arrangement of the probes): for an infinite 2D sheet (Fig. 4a) the expected increase of the resistance (like in the 1D case - see below), is exactly compensated by the current spreading in the direction perpendicular to the probes. In case of 3D this effect is overcompensated by the spread into the sample, which causes the 𝑠𝑠 !! probe dependence. By contrast, a linear increase of the resistance with increasing probe distance is found only for the onedimensional (1D) case, where the current density is constant and non-dependent on the distance 𝑠𝑠 from the electrodes that impinge the electric current. Hence, for a circular wire with radius 𝑎𝑎, much smaller than the probe spacing (i.e. for 𝑎𝑎 ≪ 𝑠𝑠), the wire appears quasi-1D and the current density simply reads 𝐽𝐽 = 𝐼𝐼 π𝑎𝑎 ! . From eq. 1.1, it’s easy to see that the resistance is now proportional to the probe spacing and equals to 𝑅𝑅!! = 𝜌𝜌𝜌𝜌/π𝑎𝑎 ! (cf. with Tab. 1). Please note, the conclusions drawn so far are valid both on macroscopic as well as on microscopic scales.

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Fig. 4 Sketch of the current flow pattern in (a) an infinite 2D sheet and (b) a semi-infinite 3D material.

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As an example for an 1D system, Fig. 5(a) shows the corrected two-probe resistance R×Σ!"# versus the probe spacing 𝑠𝑠 of a semiconductor GaAs nanowire (NW) [20, 21]. The transport measurements have been realized via a multi-probe scanning tunneling microscope (STM) system (Fig. 5b) by placing with nanometric precision two tungsten tips on a freestanding NW (i.e. vertically-oriented with respect to the GaAs substrate). The NW is 4 𝜇𝜇m long, while its radius 𝑎𝑎 decreases from 60nm down to 30nm by moving from the NW pedestal to the top and is at least 10 times smaller than probe spacing (i.e. 𝑎𝑎 ≪ 𝑠𝑠). We point out that the resistance of the GaAs NW is orders of magnitude larger than the contact resistances in the present case and that a two-probe configuration is in our case sufficient to infer the inherent resistivity of the NW. Examples for 4-probe measurements on 1D structures can be found in Refs [22, 23, 24, 25].

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Furthermore, in order to illustrate the 2D/3D transition due to the finite thickness of the sample, Fig. 6 shows the resistance measured on a n-type Si 111 wafer (nominal resistivity of 5-15 Ω ∙ 𝑐𝑐𝑐𝑐, 4×15×0.4 𝑚𝑚𝑚𝑚 ! in size) as function of the probe spacing 𝑠𝑠 [26]. The experimental data points were recorded again via a similar nano-4P STM microscope and follow a 𝑠𝑠 !! dependence, expected for a semi-infinite 3D semiconductor, as long as the probe spacing 𝑠𝑠 is within 10-60 𝜇𝜇𝜇𝜇, i.e. small compared to the sample thickness. The resistivity is around 7 Ω ∙ 𝑐𝑐𝑐𝑐 in accordance with eq. 2.1 [26], Contrary, for larger probe spacings, the current penetrates deeper into crystal reaching the bottom and edges of the wafer. The current pattern gets compressed and the resistance increases.

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Fig. 5 (a) Two-probe resistance - corrected for the average wire section Σ!"# - versus probe spacing 𝑠𝑠 of a free-standing GaAs nanowire. The solid red line is the linear best fit of experimental data and shows the expected 𝑠𝑠 dependence for a quasi-1D system. The inset in (a) is a false-color SEM image (60.000 × magnification, 45°-tilt view) of a freestanding GaAs NW with two STM tips positioned on its lateral facet. [20]. (b) Photograph of a multi-probe STM system mounted in the focus of a scanning electron microscope (SEM) for the navigation and placement of tungsten tips.

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Conventional macroscopic 4P setups for wafer testing reveal typically probe distances in the mm range, which are comparable to the overall specimen dimensions [27]. The effect of confinement for the current paths is not covered by the equations derived so far. The following sections will stepwise introduce the so-called correction factors for thin/thick films necessary to reveal precisely the resistivities of both isotropic and anisotropic materials on various length scales.

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JP Fig. 6 Electrical resistance of a Si (111) wafer crystal measured via a nano 4P STM microscope as a function of the probe spacing. The two sketches display the current flow pattern inside the Si (111) wafer for different probe spacings. The solid line shows the expected s-1 dependence for a semi-infinite 3D material, while the dashed curve is only a guide for the eyes. Only experimental data associated with the bulk-states (i.e. for probe-spacing larger than 10 𝜇𝜇𝜇𝜇) are taken from Ref. [26]. Electrical transport measurements using a smaller probe-spacing are dominated by semiconductor surface-states and intentionally not reported here.

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3

Correction factors for finite isotropic samples

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Real specimens are not infinite in either lateral or vertical directions and the equations reported in Tab. 1 need to be corrected for finite geometries. Equivalently, correction factors become necessary also if the probes are placed close to boundary of a sample like in the case of truly nano-scaled objects and/or the probe spacing itself is comparable to the size of samples. In such cases of finite and arbitrarily shaped samples the bulk resistivity is generically expressed as: 𝑉𝑉 𝜌𝜌 = 𝐹𝐹 3.1 𝐼𝐼

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where 𝐹𝐹 = 𝐹𝐹! ∙ 𝐹𝐹! ∙ 𝐹𝐹! is a geometric correction factor, which usually is split up into three different factors taking account of the finite thickness of the sample 𝐹𝐹! , the alignment of the probes in the proximity of a sample-edge 𝐹𝐹! and the finite lateral width of the sample 𝐹𝐹! . Formally, 𝐹𝐹 is dimensionally equivalent to a length, however, the correction factors 𝐹𝐹! , 𝐹𝐹! , 𝐹𝐹! are defined dimensionless (see below). Further correction factors related to the case of anisotropic and finite materials will be introduced and discussed in Sec. 5. The evaluation of the correction factors 𝐹𝐹! , 𝐹𝐹! and 𝐹𝐹! has triggered many studies. Several mathematical approaches over a time span of almost 40 years, such as the method of images [6,28,29,30], conformal mapping theory [31,32,33], the solving of Laplace’s equations [34,35], the expansion of Euler-Maclaurin series [36] or the method of finite elements [37] have been used to determine accurately the values of 𝐹𝐹!!!,!,! for different geometric configurations and probe arrangements.

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3.1 Samples of finite thickness: the correction factor F1 The resistivity of an infinite sheet of finite thickness 𝑡𝑡 can be formally expressed as: !"#$ ∙ 𝑡𝑡 ∙ 𝐹𝐹! 𝜌𝜌 = 𝑅𝑅!!!!!

𝑡𝑡 𝜋𝜋 𝑉𝑉 𝑡𝑡 = ∙ 𝑡𝑡 ∙ 𝐹𝐹! 3.2 𝑠𝑠 𝑙𝑙𝑙𝑙 2 𝐼𝐼 𝑠𝑠

𝑙𝑙𝑙𝑙2 𝑠𝑠𝑠𝑠𝑠𝑠ℎ 𝑡𝑡 𝑠𝑠 𝑙𝑙𝑙𝑙 𝑠𝑠𝑠𝑠𝑠𝑠ℎ 𝑡𝑡 2𝑠𝑠

(3.3)

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!"#$ where 𝑅𝑅!!!!! is the sheet resistance of an infinite 2D sheet (measured by the in-line geometry). 𝐹𝐹! is now a dimensionless function of the normalized sample thickness 𝑡𝑡 𝑠𝑠 which reduces to 1 as 𝑡𝑡 approaches zero (at moment we assume 𝐹𝐹! = 𝐹𝐹! = 1). A detailed derivation of the thickness correction factor 𝐹𝐹! 𝑡𝑡 𝑠𝑠 was given for the first time by Valdes in 1958 [6] using the methods of images. This method is historically the first and to date still the most frequently used one for the calculation of the correction factors 𝐹𝐹. Also the factor 𝐹𝐹! will be explicitly evaluated through this method as we will show below. However, this method results in a power series expression for 𝐹𝐹! so that it is not really suitable for numerical computation. Instead, the expression found by Albers et al. in 1985 [35] through an approximated solution of the Laplace’s equation shall be reported here. For the case of a 4P in-line array on an infinite sheet of thickness 𝑡𝑡 (electrically decoupled from a substrate), the correction factor 𝐹𝐹! 𝑡𝑡 𝑠𝑠 can be written as [35]:

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A quite similar dependence is obtained for the case of a 4P square configuration [34]. The first experimental verification of the latter formula has been obtained so far only by Kopanski et al. in 1990 [38]. In 2001, Weller [36] has re-calculated 𝐹𝐹! through an expansion of the Euler–Maclaurin series confirming the validity of the eq. 3.3.

Page 8 of 44


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Fig. 7 The solid curve in the figure is the correction factor 𝐹𝐹! versus normalized sample thickness 𝑡𝑡 𝑠𝑠 ; where 𝑡𝑡 is the wafer thickness, 𝑠𝑠 is the probe spacing. The dashed lines represent the two limit cases, i.e. 𝐹𝐹! = 1 for 𝑡𝑡 𝑠𝑠 < 1 5 and 𝐹𝐹! = 2𝑙𝑙𝑙𝑙2 𝑠𝑠/𝑡𝑡 for 𝑡𝑡 𝑠𝑠 > 4.

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Fig. 7 shows a plot of the correction factor F1 and nicely demonstrates that for 𝑡𝑡 𝑠𝑠 ≫ 1 the curves follows 𝐹𝐹! 𝑡𝑡/𝑠𝑠 ≈ 2𝑙𝑙𝑙𝑙2 𝑠𝑠/𝑡𝑡 thus eq. 3.2 reduces to the expression for a semi-infinite 3D specimen. On the other hand, for thin samples, i.e. for 𝑡𝑡 𝑠𝑠 ≪ 1, the term 𝑠𝑠𝑠𝑠𝑠𝑠ℎ 𝑡𝑡 𝑠𝑠 of eq. 3.3 can be approximated by 𝑡𝑡 𝑠𝑠. 𝐹𝐹! becomes unity and eq. 3.2 reduces to the expression of an infinite 2D sheet (see Sec. 2, Tab 1). This approximation still holds until t/s < 1 5 (with an approximation error around 𝜖𝜖 ≈ 1%), which means that real semiconductors with a finite thickness 𝑡𝑡 can be considered thin and approximated by a quasi-2D sheet until this condition is satisfied. Similarly, the sample can be considered of infinite thickness if 𝑡𝑡 𝑠𝑠 > 4 (𝜖𝜖 ≈ 1%).

3.2 Probes in the proximity of a single sample-edge: the correction factor F2

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The correction factor 𝐹𝐹! accounts for the positioning of the probes in the proximity of an edge on a semiinfinite sample. Albeit this idealized configuration can be realized only approximately, the equally spaced 4P in-line configuration with a distance 𝑑𝑑 from a non-conducting boundary, as sketched in Fig. 8a, serves nicely as a reference model to illustrate the concept of image probes, which is intensely used in the forthcoming section (see Sec. 3.3.). The non-conducting (reflecting) boundary is mathematically modeled by inserting two current image sources of the same sign at a distance – 𝑑𝑑 for the current probe 4 and – (𝑑𝑑 + 3𝑠𝑠) for probe 1, respectively [6]. Because of this mathematical trick, eq. 2.3 still holds for a semi-infinite 3D specimen and the potential at probe 2 is given by: 𝐼𝐼𝐼𝐼 1 1 1 1 − − + 2𝜋𝜋 𝑠𝑠 2𝑠𝑠 2𝑑𝑑 + 𝑠𝑠 2𝑑𝑑 + 5𝑠𝑠

IN

𝑉𝑉! =

3.4

𝐼𝐼𝐼𝐼 s s 1 1 1+ − − + 2𝜋𝜋𝜋𝜋 2d + 𝑠𝑠 2𝑑𝑑 + 2𝑠𝑠 2𝑑𝑑 + 4𝑠𝑠 2𝑑𝑑 + 5𝑠𝑠

!"#$ ∙ 𝐹𝐹! 𝑑𝑑 𝑠𝑠 with: and the bulk resistivity can be written as ρ = 2𝜋𝜋𝜋𝜋 ∙ 𝑉𝑉 𝐼𝐼 ∙ 𝐹𝐹! 𝑑𝑑 𝑠𝑠 = 𝜌𝜌!!

𝐹𝐹!

𝑑𝑑 s s 1 1 = 1+ − − + 𝑠𝑠 2d + 𝑠𝑠 2𝑑𝑑 + 2𝑠𝑠 2𝑑𝑑 + 4𝑠𝑠 2𝑑𝑑 + 5𝑠𝑠

3.5

CM

𝑉𝑉 = 𝑉𝑉! − 𝑉𝑉! =

JP

A similar equation is obtained for the potential at probe 3, so the total voltage drop 𝑉𝑉 = 𝑉𝑉! − 𝑉𝑉! between the two inner probes reads:

3.6

Page 9 of 44

I. Miccoli et al.: The role of probe geometries in isotropic and anisotropic systems The case of a 4P in-line geometry oriented parallel to a non-conducting boundary is solved in the same way. More details can be found in the original paper of Valdes [6].

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The dimensionless correction factor 𝐹𝐹! 𝑑𝑑 𝑠𝑠 for both geometric configurations (i.e. perpendicular and parallel to a non-conducting boundary) are plotted in Fig. 8b. It is evident that as long as the probe distance from the wafer boundary is at least four times the probe spacing, the correction factor 𝐹𝐹! reduces to unity (with an error of around 𝜖𝜖 ≈ 1%). This also explains why the data points in Fig. 6 follow the tendency for a semi-infinite 3D semiconductor when the probe spacing 𝑠𝑠 is in the 10-100 𝜇𝜇𝜇𝜇 range.

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Fig. 8 (a) Sketch of a 4P in-line array perpendicular to a distance 𝑑𝑑 from a non-conducting boundary of a semi-infinite 3D specimen. Probes from (1) to (4) are real, while tips (5) and (6) are imaginary and they are introduced to mimic mathematically the presence of the non-conducting edge. (b) Correction factor 𝐹𝐹! versus normalized distance 𝑑𝑑 𝑠𝑠 from the boundary (𝑑𝑑= edge-distance). The solid (dashed) curve refers to the case of 4 probes perpendicular (parallel) to the sampleedge.

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For instance, if the 4P array is centered on the Si wafer, which is 4×15×0.4 𝑚𝑚𝑚𝑚 ! in size, the probe distance from the closest sample-edge is about thirty times the probe spacing and 𝐹𝐹! ≈ 1 for each of four edges, while the thickness 𝑡𝑡 remains four times or more the probe spacing, then 𝐹𝐹! ≈ 2𝑠𝑠 𝑙𝑙𝑙𝑙2/𝑡𝑡. The resistivity equation 3.1 thus clearly reduces to that for a semi-infinite 3D sample. Noteworthy, the correction factor 𝐹𝐹! reaches its minimum 𝐹𝐹! !"# = 1 2 when the 4P array is aligned parallel along the sample-edge. This means that the measured resistance R can increase up to a factor 2 compared to the case of a semi-infinite 3D sample by moving the 4P array from a faraway location towards the sample-edge. Qualitatively, this behavior can be easily rationalized since the current paths are restricted to one half of the semi-infinite 3D sample.

3.3 Samples of finite lateral dimension: the correction factor F3

IN

The condition for 𝐹𝐹! to be unity (𝑡𝑡 𝑠𝑠 < 1 5) is easily fulfilled in a macroscopic 4P setup with probe spacings in the mm-cm range [27] on wafers with typical thicknesses of 200-300 𝜇𝜇𝜇𝜇. Furthermore, the 4P probes positioned closely to a single edge of the sample is also an idealized approximation and the correction factor 𝐹𝐹! is not sufficient for a realistic description. Therefore, a further correction factor 𝐹𝐹! is needed, which takes into account the entire effect caused by all lateral boundaries of the sample. In this section the correction factor 𝐹𝐹! will be discussed for two special geometric configurations which are, however, representative for a variety of practical situations, i.e. in-line or square 4P probe geometry inside a finite circular slice (3.3.1) and square 4P probe array inside a finite square (3.3.2). These configurations are usually used for semiconductor wafer or integrated circuit characterizations where the test windows are usually squares or rectangles.

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Page 10 of 44


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3.3.1 In-line and square 4P probe geometry inside a finite circular slice

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In 1958, F. Smith [29] first calculated the correction factor 𝐹𝐹! for an in-line 4P probe array placed in the center of a circularly shaped sample using the concept of the current image sources. M. Albert et al. [39], and independently L. J. Swartzendruber [40], obtained in 1964 the same result applying the conformal mapping theory [41] and transforming the circular sheet into an infinite half-plane (see Sec. 2). Here, we report on the more general solution proposed by D. Vaughan [42], which is valid also for a squared 4P configuration and displacement of the 4P probes away from the sample center. The model is based on the following assumptions: (i) The resistivity of the material is constant and uniform (isotropic material), (ii) the diameter of the contacts should be small compared to the probe distance (point contacts), (iii) the 4P probes are arranged in a linear (equally spaced) or square configuration and (iv) the sample thickness is much smaller than the probe spacing (𝑡𝑡 𝑠𝑠 < 1 5: 𝐹𝐹! = 1) and thus equivalent to a quasi-2D scenario.

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Likewise, the mathematical approach used by Vaughan is based on the method of the images: the resistivity formula for an infinite 2D sheet is thus extended to the case of a finite circular quasi-2D sample by introducing an appropriately located current image dipole for describing the effect of a finite boundary. This concept finally adds an additional term to eq. 2.7 (with 𝑠𝑠! = 𝑠𝑠! = 𝑠𝑠 and 𝑠𝑠! = 𝑠𝑠! = 2𝑠𝑠 for an in-line array) yielding the following voltage drop between the inner probes (𝑉𝑉! = 𝐻𝐻, 𝑉𝑉! = 𝐺𝐺) for the situation shown in Fig. 9a:

𝐼𝐼𝐼𝐼 𝐹𝐹 ! 𝐻𝐻 ∙ 𝐸𝐸 ! 𝐺𝐺 𝑙𝑙𝑙𝑙4 + ln ! (3.7) 2𝜋𝜋𝜋𝜋 𝐹𝐹 𝐺𝐺 ∙ 𝐸𝐸 ! 𝐻𝐻

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𝑉𝑉 = 𝑉𝑉! − 𝑉𝑉! =

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Fig. 9 Schematics of a (a) 4P probe in-line and (b) square array onto a finite circular slice. The current sources outside the circle, namely 𝐸𝐸 ! 𝐹𝐹 ! in (a) and 𝐻𝐻 ! 𝐵𝐵 ! in (b), represent two additional image dipoles introduced for describing the effect of finite boundary.

IN

𝑉𝑉 2𝜋𝜋 ∙ (3.8) 𝐼𝐼 𝑙𝑙𝑙𝑙 4𝐿𝐿!,!

Where the term 𝐿𝐿!,! is a function of the position of the 4P probes: 𝐿𝐿!,! =

CM

!"!!"#$ !! 𝑅𝑅!! =

JP

Now, for a 4P probe in-line geometry with an inter-probe spacing 2𝑠𝑠 ! (= 𝑠𝑠) on a circle of diameter 𝑑𝑑, where the the mid-point of the 4P geometry (𝐸𝐸, 𝐻𝐻, 𝐺𝐺, 𝐹𝐹) is displaced at a distance 𝛼𝛼𝑠𝑠 ! 𝛽𝛽𝑠𝑠 ! in the 𝑥𝑥- (𝑦𝑦-) direction with respect to circle center (see Fig. 9a for reference), equation 3.7 can be written as [42]:

𝐸𝐸𝐸𝐸𝐸𝐸 ! − 𝐸𝐸 + 𝐻𝐻 − 16 𝑅𝑅 ! + 1 𝐹𝐹𝐹𝐹𝐹𝐹 ! − 𝐹𝐹 + 𝐺𝐺 − 16 𝑅𝑅 ! + 1 3.9 𝐸𝐸𝐸𝐸𝐸𝐸 ! − 𝐸𝐸 + 𝐺𝐺 − 4 𝑅𝑅 ! + 1 𝐹𝐹𝐹𝐹𝐹𝐹 ! − 𝐹𝐹 + 𝐻𝐻 − 4 𝑅𝑅 ! + 1

Page 11 of 44


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with 𝐸𝐸 (𝐹𝐹) equal to 3 + − 𝛼𝛼 ! + 𝛽𝛽 ! , 𝐺𝐺 𝐻𝐻 equal to 1 + − 𝛼𝛼 ! + 𝛽𝛽 ! and 𝑅𝑅 = 𝑠𝑠 𝑑𝑑. Furthermore, when the linear probe array is centered with respect to the circular sample 𝑖𝑖. 𝑒𝑒. 𝛼𝛼 = 𝛽𝛽 = 0 , eq. !"!!"#$ !! 3.9 greatly simplifies yielding the same result for the correction factor 𝐹𝐹!!!"#!$% done by Smith [29]: !"!!"#$ !! 𝑅𝑅!!!!"#!$% =

𝑉𝑉 𝑉𝑉 𝑑𝑑 2𝜋𝜋 𝜋𝜋 𝑙𝑙𝑙𝑙2 !"!!"#! !! !"!!"#$ !! ∙ = 𝐹𝐹!!!"#!$% ∙ ∙ ⟹ 𝐹𝐹!!!"#!$% = 𝑑𝑑 𝑠𝑠 𝐼𝐼 𝑙𝑙𝑙𝑙 4𝐿𝐿!,! 𝑠𝑠 𝑙𝑙𝑙𝑙2 𝐼𝐼 𝑙𝑙𝑙𝑙2 + 𝑙𝑙𝑙𝑙 𝑑𝑑 𝑠𝑠

!

+3 !−3

(3.10)

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!"!!"#$ !! Fig. 10a shows a plot of the latter equation and clearly reveals that for 𝑑𝑑 𝑠𝑠 > 25: 𝐹𝐹!!!"#!$% ≈1 !"#$ (approximation error 𝜖𝜖 ≈ 1%), thus, as expected, the sheet resistance 𝑅𝑅!! reduces to the expression of eq. 2.8 for an in-line array of four probes inside an infinite 2D sheet. As a rule of thumb, a finite sample can be considered as infinite when the overall width is at least one-order of magnitude larger than the half probespacing. For instance, for a 4-inch wafer, the maximum probe spacing should not exceed 5 mm. Noteworthy, !"!!"#$ !! !"!!"#$ !! reaches a minimum value of 𝐹𝐹!!!"#!$% = 1 2 (like 𝐹𝐹! ) when the external current probes lie 𝐹𝐹!!!"#!$% !"# on the sample circumference (𝑑𝑑 = 3𝑠𝑠). In other words, the measured resistance increases by a factor of two by increasing the probe distance and moving the 4P array from the center (𝑑𝑑 ≫ 𝑠𝑠) to the sample periphery !"!!"#$ !! 𝑑𝑑 𝑠𝑠 has not a physical meaning. (𝑑𝑑 = 3𝑠𝑠). For 𝑑𝑑 < 3𝑠𝑠, the correction factor 𝐹𝐹!!!"#!$% The case of a 4P square geometry, as sketched in Fig. 9b, can be solved in an analogue way [42]. Again, a current image dipole is introduced to maintain the necessary boundary conditions and an additional term appears into eq. 2.7 (where 𝑠𝑠! = 𝑠𝑠! = 𝑠𝑠 and 𝑠𝑠! = 𝑠𝑠! = 2𝑠𝑠):

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𝐼𝐼𝐼𝐼 𝐵𝐵 ! 𝐷𝐷 ∙ 𝐻𝐻 ! 𝐶𝐶 𝑙𝑙𝑙𝑙2 + ln ! (3.11) 2𝜋𝜋𝜋𝜋 𝐵𝐵 𝐶𝐶 ∙ 𝐻𝐻 ! 𝐷𝐷

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𝑉𝑉 = 𝑉𝑉! − 𝑉𝑉! =

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Fig. 10 Correction factor 𝐹𝐹!!!"#!$% versus normalized wafer diameter 𝑑𝑑 𝑠𝑠 for an (a) in –line and (b) square 4P probe array on a finite circular slice (𝑠𝑠 is the probe spacing for the in-line configuration and the square edge for the square configuration, respectively).

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Vaughan [42] has shown that the latter formula can be still written in the following form:

𝑉𝑉 2𝜋𝜋 ∙ (3.12) 𝐼𝐼 𝑙𝑙𝑙𝑙 2𝑆𝑆!,!

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!"#$%& !!

𝑅𝑅!!!!"#!$% =

CM

where the parameter 𝑆𝑆!,! is again a non-trivial function of the square 4P array displacement 𝛼𝛼𝑠𝑠 ! , 𝛽𝛽𝑠𝑠 ! with respect to the circle center. Further details can be found in the original paper of Vaughan [42]. Here, we restrict ourselves to the case of a 4P square array placed in the center of the circle (i.e. 𝛼𝛼 = 𝛽𝛽 = 0), so that the !"#$%& !! correction factor 𝐹𝐹!! !"#!$% reduces to: Page 12 of 44

The 100th anniversary of the four-point probe technique !"#$%& !!

𝑉𝑉 𝑉𝑉 2𝜋𝜋 𝑙𝑙𝑙𝑙2 !"#$%& !! 2𝜋𝜋 !"#$%& !! 𝑑𝑑 ∙ = 𝐹𝐹!!!"#!$% ∙ ∙ ⟹ 𝐹𝐹!!!"#!$% = (3.13) 𝑑𝑑 𝑠𝑠 ! + 2 ! 𝐼𝐼 𝑙𝑙𝑙𝑙 2𝑆𝑆!,! 𝑠𝑠 𝑙𝑙𝑙𝑙2 𝐼𝐼 𝑙𝑙𝑙𝑙2 + 𝑙𝑙𝑙𝑙 𝑑𝑑 𝑠𝑠 ! + 4

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𝑅𝑅!!!!"#!!" =

!"#$%& !!

The correction factor is plotted in Fig. 10b as a function of 𝑑𝑑/𝑠𝑠. As obvious, 𝐹𝐹!!!"#!$% 𝑑𝑑 𝑠𝑠 ≈ 1 for 𝑑𝑑 𝑠𝑠 > 25 (approximation error 𝜖𝜖 ≈ 1%) and the sheet resistance converges, as expected, to the expression for an infinite 2D sheet (see Tab. 1). On the other hand, when the 4P probes are located on the edge of the !"#$%& !! circularly shaped sample for 𝑑𝑑 𝑠𝑠 = 2, 𝐹𝐹!!!"#!$% = 1 2 and the sheet resistance is: !"#$%& !!

𝑅𝑅!!!!! !" !"#!$% =

𝜋𝜋 𝑉𝑉 𝑙𝑙𝑙𝑙2 𝐼𝐼

(3.14)

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This equation refers to the case of 4P probes lying on the circumference of a circular sample and remains valid for an arbitrarily shaped sample provided with a symmetry plane. We will show this explicitly by introducing the Van der Pauw theorem in Sec. 4. Moreover, since the sheet resistance represents an intrinsic material !"#$%& !! !"!!"#$ !! and 𝑅𝑅!!!!"#!$% are revealing that the current property, both expressions (3.10) and (3.13) for 𝑅𝑅!!!!"#!$% densities are increased when the 4P probe array is placed inside a finite sample (where 𝐹𝐹! ≤ 1), yielding to a larger voltage drop V and thus to a larger resistance. Naturally, this would results to an apparently increased sheet resistance (up to a factor two), if we would simply apply the formula of Tab.1. Finally, although formally equal to the eq. 2.8, eq. 3.14 should not be mixed up with that for an in-line arrangement of 4P probes on an infinite sheet.

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The method of images can be also applied to the case of a rectangular 4P array inside a circle. Interested !"#$%&'(" !! readers are referred to Appendix D, where the correction factor 𝐹𝐹!!!!"#!$% for a rectangular 4P array placed in the center of a circular lamella is explicitly derived further generalizing the results of Vaughan theory [42].

3.3.2 Square 4P probe array inside a finite square slice

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The case of a square 4P probe array inside a finite square-shaped sample is mathematically a non-trivial scenario. In 1960, Keyweel et al. [28] first determined the correction factor 𝐹𝐹! by using the method of images. The authors correctly introduce an infinite series of current image sources to model the boundaries of the square, however, the result suffers from convergence problems, which finally were overcome by Buehler et al. [30] in 1977 solving the problem in the complex plane.

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Here, we concentrate on an alternative approach for calculating 𝐹𝐹! on a square sample, which was proposed by Mircea in 1964 [31] and relies on the so-called conformal mapping theory. Interested readers are referred to Refs. [41, 43] for a detailed description of this theory. In brief, the method is based on a conformal transformation that merely maps a square specimen onto a circular geometry for which the problem was already solved [31].

IN

According to the conformal mapping theory, each point 𝐵𝐵 𝑥𝑥, 𝑦𝑦 of a square can be mapped uniquely to a point 𝐵𝐵! 𝑟𝑟, 𝜃𝜃 of a circle as illustrated in Fig.11. Consequently, if 𝐻𝐻, 𝐵𝐵, 𝐶𝐶, 𝐷𝐷 are the 4P probes placed on a square lamella, we can determine four corresponding points H! , B! , C! , D! on a circular lamella. For this scenario of 4 probes on a circle, a formula equivalent to eq. 3.11 can be written and the voltage drop between 𝑉𝑉! (= 𝐷𝐷! ) and 𝑉𝑉! (= 𝐶𝐶! ) reads:

(3.15)

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𝐼𝐼𝐼𝐼 H! C! ∙ B! D! 𝐵𝐵! ! 𝐷𝐷! ∙ 𝐻𝐻! ! 𝐶𝐶! 𝑙𝑙𝑙𝑙 + ln ! 2𝜋𝜋𝜋𝜋 H! 𝐷𝐷! ∙ B! C! 𝐵𝐵! 𝐶𝐶! ∙ 𝐻𝐻! ! 𝐷𝐷!

JP

𝑉𝑉 = 𝑉𝑉! − 𝑉𝑉! =

where H! C! , B! D! , H! 𝐷𝐷! , B! C! correspond to s! , s! , s! , s! respectively and 𝐻𝐻!! , 𝐵𝐵!! is the current image dipole. At this stage it should be evident that the last equation remains valid also for the original square sample, since points H! , B! , C! , D! correspond by definition to 𝐻𝐻, 𝐵𝐵, 𝐶𝐶, 𝐷𝐷 and the problem reduces to find a transformation between the 𝐵𝐵(𝑥𝑥, 𝑦𝑦) and 𝐵𝐵! 𝑟𝑟, 𝜃𝜃 planes. Page 13 of 44


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Fig. 11 Schematic of a 4P square array centered with respect to a square lamella (a) and conformally mapped onto a circular lamella (b). The points H! , B! , C! , D! in the circle correspond to the contact points 𝐻𝐻, 𝐵𝐵, 𝐷𝐷, 𝐶𝐶 of the 4P square array inside the square lamella, although their position is only illustrative of the mapping procedure. 𝐻𝐻!! , 𝐵𝐵!! is the current image !"#$%& !! dipole that needs to be introduced for describing the circular boundary. (c) Correction factor 𝐹𝐹!!!"#$%& for a 4P square array on a finite square lamella as function of 𝑠𝑠 𝑑𝑑 ratio and 𝜙𝜙 tilt angle. Here, 𝑠𝑠 and 𝑑𝑑 are the side lengths of square 4P array and lamella respectively.

𝑟𝑟 ! =

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Mircea [31] has shown that the transformation between the coordinates can be approximated by the following expressions: cosh 𝑥𝑥 − cos 𝑦𝑦 cos 𝜋𝜋 − 𝑥𝑥 + cos 𝑦𝑦 cos 𝜋𝜋 + 𝑥𝑥 +cos 𝑦𝑦 𝑒𝑒 !!! 1−8 cosh 𝑥𝑥 cos 𝑦𝑦 1 + 𝑒𝑒 !! cosh 𝑥𝑥 + cos y cos 𝜋𝜋 − 𝑥𝑥 − cos 𝑦𝑦 cos 𝜋𝜋 + 𝑥𝑥 −cos 𝑦𝑦 sin 𝑦𝑦 sin 𝑦𝑦 sin 𝑦𝑦 𝑒𝑒 !!! + tan!! − tan!! −4 sinh 𝑥𝑥 sin 𝑦𝑦 sinh 𝑥𝑥 sinh 𝜋𝜋 − 𝑥𝑥 sinh 𝜋𝜋 + 𝑥𝑥 1 + 𝑒𝑒 !!

where 𝑥𝑥 = 𝜋𝜋𝜋𝜋 cos 𝜙𝜙

C LI

θ = tan!!

2𝑑𝑑 and 𝑑𝑑 is now the side length of the squared lamella.

2𝑑𝑑, 𝑦𝑦 = 𝜋𝜋𝜋𝜋 sin 𝜙𝜙

3.16𝑎𝑎

(3.16𝑏𝑏)

!

!

𝐵𝐵! 𝐷𝐷! ∙ 𝐻𝐻! 𝐶𝐶! 𝐵𝐵! ! 𝐶𝐶! ∙ 𝐻𝐻! ! 𝐷𝐷!

!"#$%& !!

= 𝐹𝐹!!!"#$%& ∙

𝑉𝑉 !"#$%& !! ⟹ 𝐹𝐹!!!"#$%& = 𝐼𝐼

2𝜋𝜋 (3.17) 𝑟𝑟 ! + 1 ! 𝑙𝑙𝑙𝑙2 + 𝑙𝑙𝑙𝑙 ! 𝑟𝑟 + 1

IN

𝑙𝑙𝑙𝑙2 + ln

2𝜋𝜋

N

𝑉𝑉 ∙ 𝐼𝐼

IO

!"#!"# !!

𝑅𝑅!!!!"#$%& =

AT

Now, if the 4 point probes form a square array, which is centered with respect to the square sample (as depicted in Fig.11a), these points are mapped for symmetry reasons again on a square array which is still centered on !"#$%& !! the circular specimen and eq. 3.16 greatly simplifies yielding a correction factor 𝐹𝐹!!!"#$%& [31]:

CM

JP

where r is given by eq. 3.16a. Except for the constant factor 2𝜋𝜋 𝑙𝑙𝑙𝑙2, which is now included in the definition of !"#$%& !! !"#$%& !! the term 𝐹𝐹!!!"#$%& , the last expression looks very similar to that obtained for a circle 𝐹𝐹!!!"#!$% . When inserting eq. 3.16a into eq. 3.17, the correction factor finally is a function of both the normalized side 𝑠𝑠 𝑑𝑑 and the tilt angle 𝜙𝜙 of the 4P square array. The factor has been calculated and plotted in Fig. 11c. As obvious, 𝐹𝐹! changes only by around 5% when rotating the square array 𝜙𝜙 of 45°. Likewise to the case of the 4P square array inside a circle, moving the four probes from the sample center to the square periphery or equivalently decreasing the sample sizes from infinite (for d≫ 𝑠𝑠) to fit the dimensions of 4P square array (for d= 𝑠𝑠), the measured resistance increases again by factor of two. Page 14 of 44

The 100th anniversary of the four-point probe technique !"#$%& !!

Indeed, this effect is compensated by 𝐹𝐹!!!"#$%& when changing from 2𝜋𝜋 𝑙𝑙𝑙𝑙2 for d≫ 𝑠𝑠 to 𝜋𝜋 𝑙𝑙𝑙𝑙2 for d= 𝑠𝑠. For !"#$%& !!

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the latter case we get for the sheet resistance 𝑅𝑅!!!!"#$%& the same expression which was obtained above for the circle (see eq. 3.14) and which is expected for a thin sample of arbitrary shape provided with a symmetry plane [44, 45] (see Sec. 4): !"#$%& !!

𝑅𝑅!!!!! !" !"#$%& =

𝜋𝜋 𝑉𝑉 (3.18) 𝑙𝑙𝑙𝑙2 𝐼𝐼

ED

In 1992, Sun et al. [46] obtained independently similar correction factors by mapping a squared sample onto a semi-infinite half plane. They carried out also first experimental measurements (on a 25𝑥𝑥25 mm2 silicon substrate) to check their theoretical calculations. Moreover, the correction factor 𝐹𝐹! for a square sample with a square 4P probe was evaluated numerically [37, 47, 48] using the flexible finite element method (FEM). Fig. 12 summarizes the theoretical results obtained so far by these authors using different methods (i.e. method of the images, conformal mapping theory and FEM) and compares them with the experimental result of Sun et al. [46]. The extremely good agreement between theory and experiment is evident. The conformal mapping theory and FEM method can be also extended to the case of rectangular samples. However, the calculations become even more complicated. For details the reader is referred to Refs. [47, 49, 50].

R FO

IO

AT

C LI

B PU !"#$%& !!

N

Fig. 12 Correction factor 𝐹𝐹!!!"#$%& for a square 4P probe array on a thin square sample as function of 𝑠𝑠 𝑑𝑑 ratio (with 𝜙𝜙 fixed to 45° degree). The dashed and solid curves represent the theoretical curves obtained by Mircea [31] and Sun [46] using the conformal mapping theory, while the dotted and dashed-dotted curves are the theoretical results obtained by Green [47] and Shi [37] through the finite element method. The open squares are the experimental values measured by Sun [46] on a 25𝑥𝑥25 mm2 silicon substrate.

IN

Van der Pauw theorem for isotropic thin films of arbitrary shape

JP

4

CM

Of great importance for resistivity measurements is the Van der Pauw theorem [44, 45], which virtually extends the formulas for evaluating the correction factor 𝐹𝐹! for the special case of square/circular samples to a specimen of arbitrary shape, as long as the four probes are located on the sample’s periphery and are small compared to the sample size. Moreover, the Van der Pauw theorem requires samples, which are homogeneous, thin (i.e. 𝑡𝑡 𝑠𝑠 < 1 5: 𝐹𝐹! = 1), isotropic and singly connected, i.e. the sample is not allowed to have isolated holes. Page 15 of 44


ED

PT CE AC Fig. 13 (a) Typical Van der Pauw arrangement of the four-point probes placed along the periphery of a thin and arbitrarily shaped sample. (b) Schematics of a thin sample provided with a line of symmetry. (c) Alternative Van der Pauw arrangement with the four-point probes placed along a symmetry line of the sample. For details see text.

R FO

If 𝐼𝐼!" is the current flowing between contacts 𝐴𝐴 and 𝐵𝐵, while 𝑉𝑉!" is the voltage drop between contacts 𝐶𝐶 and 𝐷𝐷, the resistance is given by 𝑅𝑅!",!" = 𝑉𝑉!" 𝐼𝐼!" (cf. with Fig. 13a). Analogously, we define 𝑅𝑅!",!" = 𝑉𝑉!" 𝐼𝐼!" . Van der Pauw has shown that these resistances satisfy the following condition (ρ is the resistivity): 𝑒𝑒

! !! !!!,!" !

+ 𝑒𝑒

! !! !!",!" !

=1

(4.1)

B PU

For samples provided with a plane of symmetry (where 𝐴𝐴, 𝐶𝐶 are on the line of symmetry while 𝐵𝐵, 𝐷𝐷 are placed symmetrically with respect to this line - see Fig.13b), we immediately obtain by using the so-called reciprocity theorem 𝑅𝑅!",!" = 𝑅𝑅!",!" = 𝑅𝑅 and eq. 4.1 reads: 𝜋𝜋𝜋𝜋 𝑉𝑉 𝑙𝑙𝑙𝑙2 𝐼𝐼

(4.2)

C LI

ρ=

𝜋𝜋𝜋𝜋 𝑅𝑅!",!" + 𝑅𝑅!",!" 𝑓𝑓 2 𝑙𝑙𝑙𝑙2

(4.3)

IO

ρ=

AT

This equation coincides exactly with eqs. 3.14 and 3.18 obtained in the previous part for the special cases of circular and square samples, respectively (with the four probe located on its periphery). In the case of symmetrized samples, i.e. 𝑅𝑅!",!" = 𝑅𝑅!",!" , a single resistance measurement is sufficient for evaluating the sample resistivity. For non-symmetric samples, the resistivity is generally expressed as [44, 45]:

𝑙𝑙𝑙𝑙2 𝑅𝑅!",!" /𝑅𝑅!",!" − 1 1 !"! = 𝑒𝑒 ! 𝑓𝑓 𝑅𝑅!",!" /𝑅𝑅!",!" + 1 2

IN

cosh

N

where 𝑓𝑓 is now a function of the 𝑅𝑅!",!" /𝑅𝑅!",!" ratio and satisfies the relation:

(4.4)

CM

JP

Summarizing, eq. 4.3 allows the determination of ρ for an arbitrarily shaped thin sample from two simple resistance measurements. Van der Pauw has explicitly calculated the result of eq. 4.3 in two famous articles [44, 45] and interested readers are referred to those for further details. In brief, the proof of the theorem consists of two parts. First, eq. 4.3 is derived for the special case of a semi-infinite half sheet, with four probes located at the edge. Its demonstration is given explicitly in Appendix A. ρ is easily obtained using the same mathematical approach as described in Sec. 2 assuming that the current spreads cylindrically. Finally, it needs to be shown that equation 4.3 remains valid for a lamella of any shape. This is done by means of a conformal mapping in the complex field of the arbitrarily shaped specimen into an infinite half-sheet.

Page 16 of 44


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Noteworthy to mention is a recent revision of the Van der Pauw method for samples with one or more planes of symmetry as elaborated by Thorsteinsson et al. [51]. In this case, eq. 4.1 still holds (with the exception of a factor of 2, see below) if the four probes are placed along one of the planes of symmetry. The current density component normal to the mirror plane is zero, i.e. 𝐉𝐉 ∙ 𝐧𝐧 = 𝟎𝟎, for a linear four-point arrangement as shown in Fig. 13c and the potential remains unchanged by replacing the mirror plane by an insulating boundary. Consequently, the resistances are lowered exactly by a factor of two compared to the situation where the probes are positioned on the boundary. For this special scenario depicted in Fig. 13c, eq. 4.1 can be rewritten as: 𝑒𝑒

! !!! !!!,!" !

+ 𝑒𝑒

! !!! !!",!" !

=1

(4.5)

ED

As an example, if we consider the case of an in-line 4P probe array aligned along the diameter of a finite !"!!"#$ !! circular slice (see Fig. 10a), the evaluation of correction factor 𝐹𝐹!!!"#!$% (eq. 3.10) is no longer required and, according eq. 4.5, the resistivity can be precisely extracted from two independent 4-point configurations. Moreover, this geometry is robust to probe positioning errors. Note, this aspect is of importance but has not been addressed so far in the derivation of correction factor 𝐹𝐹! (see Sec. 3). For details see the original work of Thorsteinsson et al. [51], where the error due to small probe misalignments in circular- and square-shaped specimens is evaluated.

R FO

5

Four point probe technique on anisotropic crystals and surfaces

B PU

AT

C LI

Till the end of 80’s only little attention was paid to anisotropic materials, whose transport properties were seldom studied and measured. However, the growing interest into these classes of solids, revealing pronounced electronic correlation effects (such as high-temperature superconductors [52] and low dimensional organic and metallic conductors [53, 54, 55] renewed the interest into transport measurements. Moreover, also applicationdriven research has sustainably triggered the technique of anisotropic conductivity measurements, e.g. the industrial application of anisotropic textiles inside high-tech woven [56, 57] or highly oriented paper-like carbon nanotubes (so-called buckypapers) or carbon fiber papers inside fuel cells [58], supercapacitor electrodes [59], or even artificial muscles [60].

(5.1)

IN

𝐽𝐽! 𝐽𝐽! 𝐽𝐽!

N

ρ!! ρ!" ρ!" 𝐸𝐸! 𝐸𝐸! = ρ!" ρ!! ρ!" ρ!" ρ!" ρ!! 𝐸𝐸!

IO

The evaluation of the electrical resistivity in case of an anisotropic solid is in general more complex and demanding. For instance, the resistivity ρ is no longer a scalar, but instead needs to be substituted by a symmetric second-rank tensor, whose components ρ!" represent the resistivities along different directions of the solid; thus, Ohm’s law (1.1) can be rewritten as:

CM

JP

where 𝐸𝐸! and 𝐽𝐽! are the electric field and the density current along the 𝑖𝑖 −th direction, respectively. Crystallographic symmetries fortunately further reduce the number of the resistivity components ρ!" . For example, two quantities 𝜌𝜌! = 𝜌𝜌! and 𝜌𝜌! are sufficient for the complete description of trigonal, tetragonal and hexagonal systems while three, four and six are necessary for orthorhombic, monoclinic and triclinic crystals, respectively [61, 62]. As seen, for isotropic materials the 𝐼𝐼/𝑉𝑉 ratio measured with 4P probes along one axis is directly proportional to the material resistivity if appropriate correction factors are included (cf. with Sec. 3). This linear relationship fails for anisotropic materials where the 𝐼𝐼/𝑉𝑉 ratio measured along one arbitrary axis simultaneously depends now on other resistivity components (e.g. 𝜌𝜌! , 𝜌𝜌! , 𝜌𝜌! for orthorhombic crystals). Page 17 of 44


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The main question here is how to disentangle the different components in order to fully determine the resistivity tensor. So far, this problem has been solved for crystals with a maximum of three components [63]. In this section we follow the same scheme presented in the context of isotropic crystals, i.e we first consider the case of a 3D semi-infinite half-plane and, thereafter, infinite 2D sheet. In the end, we will finally extend our focus to finite and anisotropic samples with dimensions that are comparable to typical probe distances. We first recapitulate two of the most important and relevant methods, proposed by Wasscher [32] and Montgomery [63], respectively. In the course of this section we finally derive also the correction factor for a square 4P array inside an anisotropic 2D system with variable probe spacings. The theoretical results are underlined by latest experiments on finite and 2D anisotropic systems carried out with a 4-tip STM/SEM system.

Formulas for anisotropic semi-infinite 3D bulk and infinite 2D sheet

ED

5.1

R FO

In 1961 Wasscher [64] first solved the problem regarding decoupling and measuring the components of the resistivity tensor and extended the formulas reported in Tab. 1 to the case of anisotropic materials. The original solution is based on an idea of Van der Pauw [65], who suggested a transformation of the coordinates (cf. with Fig. 14) of the anisotropic cube with dimension 𝑙𝑙 and resistivities 𝜌𝜌! , 𝜌𝜌! , 𝜌𝜌! (along the 𝑥𝑥, 𝑦𝑦 and 𝑧𝑧 -axes) onto an isotropic parallelepiped of resistivity 𝜌𝜌 and dimensions 𝑙𝑙!! via: 𝜌𝜌! 𝑙𝑙 𝑖𝑖 = 𝑥𝑥, 𝑦𝑦, 𝑧𝑧 𝜌𝜌

5.2

B PU

𝑙𝑙!! =

N

IO

AT

C LI

where 𝜌𝜌 = ! 𝜌𝜌! ∙ 𝜌𝜌! ∙ 𝜌𝜌! . It is important to underline that these transformations preserve voltage and current, i.e. do not affect the resistance 𝑅𝑅 [64, 65].

IN

Fig. 14 Schematic of the mapping procedure of an anisotropic cubic sample into an equivalent isotropic parallelepiped.

CM

JP

We first will start with an in-line geometry of four probes on an anisotropic semi-infinite 3D half plane. For the sake of simplicity we further assume that the resistivities 𝜌𝜌! , 𝜌𝜌! , 𝜌𝜌! are directed along the 𝑥𝑥, 𝑦𝑦, 𝑧𝑧-high symmetry axes of the solid. According to eqn. 5.2, the 4P probes, which shall be aligned along the 𝑥𝑥-axis of the anisotropic solid with a probe distance 𝑠𝑠! , are still aligned along the 𝑥𝑥′-axis after transformation with a distance 𝑠𝑠!! = 𝜌𝜌! 𝜌𝜌 𝑠𝑠! . As 𝑉𝑉! and 𝐼𝐼! are preserved, the resistivity according to eq. 2.1 is for isotropic samples given by: 𝑉𝑉! 𝜌𝜌 = 2𝜋𝜋 𝜌𝜌! 𝜌𝜌 𝑠𝑠! , (5.3) 𝐼𝐼! Page 18 of 44


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which can be immediately rearranged giving now for the resistance 𝑅𝑅! = 𝑉𝑉! 𝐼𝐼! along the 𝑥𝑥-axis of the anisotropic sample: 1 𝑅𝑅! = 𝜌𝜌! 𝜌𝜌! (5.4) 2𝜋𝜋𝑠𝑠! The resistance measured with an in-line arrangement of 4P probes along the 𝑥𝑥-axis of an anisotropic sample is thus the geometric mean of the resistivity components along the other two principal axes. The remaining cases (4P probe in-line array on an infinite 2D sheet and 4P probe square array on a semi-infinite 3D plane and infinite 2D sheet) can be solved using a similar approach. Table 2 summarizes all formulas for the four geometric configurations considered here (in-line and square geometry in 2D and 3D). The equations are derived by assuming the 4P in-line (square) array aligned along the 𝑥𝑥-axis (the 𝑥𝑥- and 𝑦𝑦-axis) of the anisotropic solid (further details are reported in the Appendix B).

ED

Table 2 Electrical resistances 𝑅𝑅! = 𝑉𝑉! 𝐼𝐼! for the cases of a linear and square arrangement of 4 probes on an anisotropic semi-infinite 3D material and infinite 2D sheet. Sample shape

4P in-line*

R FO

3D bulk

2D sheet

1 𝜌𝜌 𝜌𝜌 2𝜋𝜋𝜋𝜋 ! !

1 𝜌𝜌! 1− 1+ 𝜋𝜋𝜋𝜋 𝜌𝜌!

!

! !

𝜌𝜌! 𝜌𝜌!

1 𝜌𝜌! 𝜌𝜌 𝜌𝜌 𝑙𝑙𝑙𝑙 1 + 2𝜋𝜋𝜋𝜋 ! ! 𝜌𝜌!

B PU

𝑙𝑙𝑙𝑙2 𝜌𝜌𝑥𝑥 𝜌𝜌𝑦𝑦 𝜋𝜋𝜋𝜋

4P square**

C LI

* The 4P probes are aligned along the 𝑥𝑥-axis of the anisotropic solid with a probe distance 𝑠𝑠. ** The 4P probes are arranged in a square-shaped configuration whose sides are aligned along the 𝑥𝑥- , 𝑦𝑦-axis respectively. Current is applied through two probes aligned along the 𝑥𝑥-axis, while the remaining probe couple measures the voltage drop. Here 𝑠𝑠 is the side length of the square.

N

IO

AT

From the comparison of the formulas reported in Tab.2 with those for an isotropic sample (Tab.1), it is evident that the measured resistances still decrease by increasing the probe distance on a semi-infinite half plane, while it remains constant for the case of an infinite 2D sheet. The reason of this behavior is still owed to the current spreading in the direction normal to the probe array and into the sample while the probe distance is increased (see Fig. 4). In order to reveal information about the anisotropy either the current/voltage probes need to be exchanged or the 4P probe geometry needs to be rotated. For instance, rotation of the 4P probe array by 90° reveals a resistance which is now defined by 𝑅𝑅! = 𝑉𝑉! 𝐼𝐼! . The corresponding expressions similar to those of Tab. 2 are obtained by exchanging 𝜌𝜌! and 𝜌𝜌! .

IN

Finally, the anisotropy ratio 𝑅𝑅! 𝑅𝑅! , which directly refers to the anisotropy of the resistivities, is easily obtained. The equations are summarized in Table 3 and plotted in Fig. 15 as function of the resistivity anisotropy 𝜌𝜌! 𝜌𝜌! . The dependence on 𝜌𝜌! (for 3D materials) cancels out by evaluating the resistance ratio 𝑅𝑅! 𝑅𝑅! . It is evident that the square arrangement reveals a higher sensitivity compared to the linear arrangement. In case of an infinite 2D sheet, the anisotropy cannot be determined at all with the in-line 4P geometry. In the section 3.3, we showed that the impact of finite boundaries can be neglected if the sample size is larger by one-order of magnitude compared to the probe-spacing. This argument still holds in the case of an anisotropic sample (see Sec. 5.4 below).

CM

JP

In general, the equations reported in Tab. 2 can be used to fully determine the resistivity tensor of large and thick (𝑡𝑡/𝑠𝑠 > 4, see Sec. 3.1) 3D samples, albeit three distinct measures are necessary at least for the most general case of an anisotropic material with three resistivity components. In order to determine all principal resistivity directions 𝜌𝜌!!!,!,! , which are for the sake of simplicity assumed to be parallel to each of the three Page 19 of 44

I. Miccoli et al.: The role of probe geometries in isotropic and anisotropic systems principal 𝑥𝑥, 𝑦𝑦 and 𝑧𝑧 –axes, first the geometric mean

𝜌𝜌! 𝜌𝜌! is determined by eq. 5.4 using an in-line

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arrangement of 4P probes aligned along the principal 𝑥𝑥-axis, thereafter 𝜌𝜌! 𝜌𝜌! is determined by rotating the inline 4P array of 90° degree, finally the last term 𝜌𝜌! 𝜌𝜌! is obtained by cutting a thin lamella (t/s < 1 5 – see Sec. 3.1) from the thick sample and repeating the first measurement. Table 3 Electrical resistance ratio 𝑅𝑅! 𝑅𝑅! of an anisotropic semi-infinite half-plane and infinite 2D sheet measured through an in-line and square arrangement of 4 probes. Sample shape

𝜌𝜌! 𝜌𝜌! 1∗

1 + 𝜌𝜌! 𝜌𝜌! − 1 1 + 𝜌𝜌! 𝜌𝜌! − 1

R FO

2D sheet

4P square**

ED

3D bulk

4P inline*

ln 1 + 𝜌𝜌! 𝜌𝜌! ln 1 + 𝜌𝜌! 𝜌𝜌!

* this configuration is not sensitive at all to the material anisotropy.

B PU

Fig. 15 Electrical resistance ratio 𝑅𝑅! 𝑅𝑅! versus the resistivity anisotropy degree 𝜌𝜌! 𝜌𝜌! for the infinite 3D half-plane and 2D sheet depending on the adopted 4P probe geometric configuration (squarevs. in-line geometry).

𝜌𝜌! 𝜌𝜌!

2𝜋𝜋𝜋𝜋

𝑙𝑙𝑙𝑙

𝜌𝜌! 𝜌𝜌!

!

− 4 cos ! 𝜃𝜃 sin! 𝜃𝜃 1 −

sin! 𝜃𝜃 +

𝜌𝜌! cos ! 𝜃𝜃 𝜌𝜌!

!

𝜌𝜌! 𝜌𝜌!

!

5.5

IN

𝑅𝑅 𝜌𝜌! , 𝜌𝜌! , 𝜃𝜃 =

1+

N

IO

AT

C LI

In this context, the characterization of anisotropic 2D-materials with only two components 𝜌𝜌! , 𝜌𝜌! is easier. If the square 4P probe geometry is aligned with respect to the principal axes of the anisotropic surface, it is sufficient to perform the measurement twice by rotating the square array of 90° degree or exchanging the combination of selected current and voltage probes. In case that the contact geometry is not aligned accurately, the equations reported in Tab. 2 cannot be applied any longer and the evaluation of the data becomes tedious and extremely arduous. As an example, we consider the latter case of a 4P probe square array on an anisotropic 2D sheet, and we assume that the 4P array is rotated by an arbitrary angle 𝜃𝜃 with respect to the two orthogonal resistivity components. In this case the expression relating the measured resistance and the material resistivity becomes a function of the angle 𝜃𝜃 and reads [66] (the readers are referred to appendix C for the derivation of eq. 5.5.):

CM

JP

The expected resistance for arbitrary orientations of the square 4P geometry and for various resistivity anisotropy parameters is plotted in Fig. 16a. As mentioned, the anisotropy is best seen for two orthogonal contact configurations. Furthermore, it is evident that a negative resistance appears at some 𝜃𝜃 for extremely anisotropic materials, i.e. 𝜌𝜌! 𝜌𝜌! > 20. This artifact is explained by a deformation of the electrostatic potential in case of very large anisotropies. This unexpected behavior was observed for the first time by Kanagawa et al. while they were studying the transport properties of atomic indium chains on Si 111 [66]. Page 20 of 44


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Fig. 16 (a) Theoretical angle dependence of the electrical resistance 𝑅𝑅 𝜃𝜃 for an anisotropic infinite 2D sheet. The different dashed curves are plotted by using eq. 5.5 with various (𝜌𝜌! 𝜌𝜌! ) values. (b) Measured electrical resistance 𝑅𝑅 on a single domain Si 111 4×1-In surface as function of 4P square array angular position 𝜃𝜃 respect to indium atomic chains. Full symbols are the experimental values obtained by Kanagawa et al. [66], while open symbols are obtained by our group [67]. Solid lines in the figure are the best-fitting curves of experimental data, obtained by using eq. 5.5.

B PU

N

IO

AT

C LI

As a general remark, highly anisotropic 2D atomic chain ensembles have recently attracted a great interest because of their exotic electronic properties, such as charge-density waves [68], spin-density waves or also signatures of a Luttinger-liquid [69]. In this review we restrict ourselves to the In/Si(111) system taking up the part of a benchmark system as it has been comprehensively studied over last decade. A single domain Si 111 4×1-In surface is obtained by depositing one monolayer of indium onto 𝑆𝑆𝑆𝑆(111) (miscut 0.5° ÷ 2°) at 350-400°C [70]. This 2D-system is highly anisotropic because the In-chains are preferentially oriented along the Si atomic steps and also electrically decoupled from the Si-bulk bands by a Schottky barrier [66].

IN

JP

Fig. 17 SEM micrograph (× 2.000 magnification, plan-view) of four STM tungsten tips placed on a Si 111 4×1-In surface. Insets (a), (b), (c) and (d) show how the 4P square array of nano multi-probe STM system is rotated of almost 180° degree. 𝜃𝜃 is the angle between the square side and the indium chains, which are aligned along the Si atomic steps.

CM

Fig. 16b shows the resistances measured via a nano-4P STM system on such a single domain Si 111 4×1-In surface as function of the angular position 𝜃𝜃 of the assembly with respect to indium chain orientation. Some of the probe configurations are imaged by a SEM and shown in Fig. 17. The probe spacing is around 40 𝜇𝜇𝜇𝜇 and therefore much smaller than the dimension of the sample itself (1.5×0.8 𝑐𝑐𝑐𝑐! in size) mimicking an infinite Inlayer. Full symbols of Fig. 16b represent values measured by Kanagawa et al. [66], while open symbols are obtained in our group under similar experimental conditions [67]. At a low degree of anisotropy (open symbols in Fig. 16b), the resistance only lightly changes with 𝜃𝜃 and remains positive, while at a high level of anisotropy Page 21 of 44


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(full symbols in Fig. 16b), the resistance varies strongly and becomes negative at some 𝜃𝜃 in accordance with theory. The absolute values of the resistances and the degree of anisotropy for the Si 111 4×1-In surface depends significantly on the substrate cleaning procedure [71], miscut angle (single domain), the amount of deposited indium as well as the deposition temperature [67]. The lower anisotropy measured for our samples is most likely ascribable to the smaller miscut angle of Si(111) substrates (namely 1° degree versus the 1.8° degrees of Ref. [66]). Irrespective of further details for the different anisotropies, it is important to note that the experimental behavior is in excellent agreement with theory shown in context of Fig. 16a.

5.2

Classical approaches for finite anisotropic samples

ED

R FO

As outlined in Sec.3, the common way of extending the concepts elaborated for a semi-infinite 3D bulk and/or an infinite 2D sheet to the case of finite samples is done by the introduction of correction factors depending on the 4P geometry/placement and sample shape. The correction factors introduced for isotropic finite samples can be related to the anisotropic case, i.e. by mapping these anisotropic samples on equivalent isotropic ones according to the Wasscher transformations [64]. However, attentive readers may have noticed that this transformation typically maps an anisotropic square-shaped sample on an isotropic rectangle or an anisotropic circularly shaped specimen on an equivalent elliptic one, respectively. Therefore, first a revision of the !"#$%& !! !"#$%& !! correction factors 𝐹𝐹!!!"#$%& (𝑠𝑠/𝑑𝑑) (eq. 3.17) or 𝐹𝐹!!!"#!$% (𝑠𝑠/𝑑𝑑) (eq. 3.13) is mandatory before we extend this concept towards anisotropic square- or circular-shaped samples (see also Sec. 5.4 below).

B PU

IO

AT

C LI

As matter of fact, to date the resistivity of finite and anisotropic materials is exclusively calculated via the methods introduced by Montgomery [32] and by Wasscher [63], which will be explained hereafter. In order to allow an easy analytical treatment of the problem both methods rely on some simplifications, e.g. (i) the sample has the shape of a parallelepiped [63] (or of a thin circular lamella [32]), (ii) the components of resistivity are aligned w.r.t. the edges of the parallelepiped (or along two orthogonal diameters of circular lamella), and (iii) the 4P probes must be placed at the corners of one rectangular face (or at the boarders of two perpendicular diameters of the circular lamella). Hence, for the sake of simplicity, both these methods do not evaluate the correction factors for any arbitrary value of probe spacing 𝑠𝑠 over sample dimension 𝑑𝑑 ratio (i.e. 𝑠𝑠 𝑑𝑑), but only with probes located on the sample periphery, i.e. for 𝑠𝑠/𝑑𝑑 = 1. In section 3, we showed that the resistance 𝑅𝑅 of isotropic 2D materials becomes larger by increasing the probe distance, i.e. by moving the 4P probe geometry from the sample center (for 𝑑𝑑 ≫ 𝑠𝑠) towards the sample periphery (for 𝑑𝑑 = 𝑠𝑠). A comparable effect takes place for finite anisotropic materials, which similarly offers a much higher sensitivity with respect to the infinite case and will be elucidated in detail in Sec. 5.4.

N

In the present section, we first will briefly introduce both theoretical concepts, i.e. Montgomery and Wasscher methods, and we will subsequently compare these with recent experimental results obtained for anisotropic ensembles of In wires grown on 𝑆𝑆𝑆𝑆(111) Mesa structures of finite widths (Sec 5.3). In particular, the gradual rotation of the squared 4P probe geometry allows us to determine the conductivity components for this finite anisotropic system. In Sec. 5.4 these concepts are further generalized for arbitrary probe spacings 𝑠𝑠 for a 4P probe geometry inside an anisotropic circular lamella with diameter 𝑑𝑑. By the latter method we introduce a complementary method to measure independently the conductivity components for an anisotropic system.

IN

CM

JP

5.2.1 The Montgomery method

In 1970 Montgomery proposed a graphical method [63] for specifying the resistivities of anisotropic materials cut in the form of a parallelepiped with the three orthogonal edges 𝑙𝑙′! , 𝑙𝑙′! , 𝑙𝑙′! collinear to the three resistivity directions 𝜌𝜌!!!,!,! . The Montgomery approach is the most used method for determining the electrical resistivity Page 22 of 44


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of anisotropic materials (more than 500 citations [72]). Here we report on the revised version done by Dos Santos et al. in 2011 [73] that allows to solve the problem analytically. Although this method can be applied to a rectangular prism of finite thickness, here we will derive the formulas for the case of a thin rectangular film with two distinct resistivity components 𝜌𝜌! , 𝜌𝜌! (= 𝜌𝜌! ). For the more general case the readers are referred to Refs. [63, 73]. The revised Montgomery method is based on the Wasscher transformation (cf. with eqn. 5.2) and the theoretical work of Logan, Rice and Wick [74], who showed that the resistance 𝑅𝑅! = 𝑉𝑉! /𝐼𝐼! of an isotropic rectangular prism (with dimensions 𝑙𝑙! , 𝑙𝑙! , 𝑙𝑙! - cf. with Fig. 18a,b) is related to the resistivity 𝜌𝜌 by means of two correction factors, 𝐸𝐸 and 𝐻𝐻! , via:

𝜌𝜌 = 𝐸𝐸𝐻𝐻! 𝑅𝑅! (5.6)

ED

Thereby, the current 𝐼𝐼! is applied via two contacts placed on one edge of the facet 𝑙𝑙! 𝑙𝑙! , while the voltage drop 𝑉𝑉! is probed by the other two contacts on the opposite edge of the same facet (as depicted in Fig. 18a,b). As we will see below the correction factor 𝐸𝐸 is comparable to the correction factor 𝐹𝐹! (cf. with Sec. 3) and accounts for the finite thickness of the isotropic sample. Furthermore, 𝐻𝐻! is the analogue to the correction factor 𝐹𝐹! and corrects the finite lateral dimensions. An equivalent relation can be written by exchanging the current and voltage probes with each other (i.e. 𝜌𝜌 = 𝐸𝐸𝐻𝐻! 𝑅𝑅! with 𝑅𝑅! = 𝑉𝑉! /𝐼𝐼! ).

R FO

C LI

B PU AT

Fig. 18 (a) Schematics of the contact geometry for the Montgomery method and (b) Wasscher mapping procedure of an isotropic parallelepiped on an anisotropic one and vice versa.

N

IO

Since the contacts are placed on the corners of the parallelepiped (i.e. 𝑠𝑠! = 𝑙𝑙!!!,!,! and fixed), both 𝐸𝐸 and 𝐻𝐻! (or 𝐻𝐻! ) do not depend on the 𝑠𝑠 𝑑𝑑 ratio, but they are a function of the ratios between sample dimensions 𝑙𝑙! , 𝑙𝑙! , 𝑙𝑙! . Logan, Rice and Wick [74] applied the method of images (see Sec. 3.2) for the evaluation of the correction factors 𝐻𝐻! (or 𝐻𝐻! ), which reads: ! 1 2 4 = (5.7a) 𝑙𝑙! 𝐻𝐻! 𝜋𝜋 !!! 2𝑛𝑛 + 1 sinh 𝜋𝜋 2𝑛𝑛 + 1 𝑙𝑙!

IN

𝐸𝐸 𝑙𝑙! = 𝑙𝑙!

! !!!

2𝑙𝑙 + 1 sinh 𝜋𝜋 2𝑛𝑛 + 1

𝑙𝑙! 𝑙𝑙!

!!

JP

1/𝐻𝐻! is obtained by substituting 𝑙𝑙! 𝑙𝑙! with 𝑙𝑙! 𝑙𝑙! . Similarly, the 𝐸𝐸 factor can be expressed as: ! !!!

CM

(5.7b) !! 𝜖𝜖! 𝑠𝑠 sinh 𝜋𝜋𝜋𝜋𝑙𝑙! where 𝑠𝑠 = 2𝑙𝑙 + 1 /𝑙𝑙! ! + 𝑛𝑛/𝑙𝑙! ! !/! , 𝜖𝜖! = 1, and 𝜖𝜖! = 2 in case of 𝑛𝑛 >0. As mentioned, the values of 𝐸𝐸 and 𝐻𝐻 were determined by graphical interpolation in the original paper of Montgomery. The revision by Dos Santos et al. [73] has revealed that both mathematical series can be greatly simplified and expressed through analytic equations. An in-depth analysis of eq. 5.7a has finally revealed that 𝐻𝐻! can be approximated by: ! !!!

Page 23 of 44

I. Miccoli et al.: The role of probe geometries in isotropic and anisotropic systems 𝜋𝜋 𝑙𝑙! sinh 𝜋𝜋 (5.8) 𝑙𝑙! 8 A similar expression is obtained for 𝐻𝐻! when substituting 𝑙𝑙! 𝑙𝑙! with 𝑙𝑙! 𝑙𝑙! . Both equations 5.7a (full squares) and 5.8 (solid curve) are plotted in Fig. 19a, which demonstrates the extremely good agreement between the exact and the approximated expressions over several orders of magnitude.

ED

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𝐻𝐻! ≈

R FO B PU

Fig. 19 (a) Correction factor 𝐻𝐻 versus 𝑙𝑙! 𝑙𝑙! ratio according to series 5.7a (full squares) and the approximated expression 5.8. (solid red curve). (b) Correction factor 𝐸𝐸 𝑙𝑙! as function of normalized thickness 𝑙𝑙! 𝑙𝑙! 𝑙𝑙! . The dashed black lines represent the two limit cases, i.e. 𝐸𝐸 𝑙𝑙! = 1 for 𝑙𝑙! 𝑙𝑙! 𝑙𝑙! < 0.2 and 𝐸𝐸 𝑙𝑙! ≈ 2/2 𝑙𝑙! 𝑙𝑙! !/! /𝑙𝑙! for 𝑙𝑙! / 𝑙𝑙! 𝑙𝑙! !/! > 2, while the solid red curve describes the transition regime for 0.2 < 𝐸𝐸 𝑙𝑙! < 2 and it is approximated by eq. 5.9.

1 + 𝑒𝑒 !![ !!!!

! ! /! ! !/!] !

(5.9)

AT

𝐸𝐸 𝑙𝑙! ≈ 1

C LI

Similarly, eqn. 5.7b reduces to unity (i.e. 𝐸𝐸 𝑙𝑙! ≈ 1) for 𝑙𝑙! / 𝑙𝑙! 𝑙𝑙! !/! < 0.2, while it can be approximated by 𝐸𝐸 𝑙𝑙! ≈ 2/2 𝑙𝑙! 𝑙𝑙! !/! /𝑙𝑙! for 𝑙𝑙! / 𝑙𝑙! 𝑙𝑙! !/! > 2. These two cases correspond to that of a thin and a thick film, respectively. For 0.2 < 𝑙𝑙! / 𝑙𝑙! 𝑙𝑙! !/! < 2, eq. 5.7b is well described by the following expression:

IO

Fig. 19b shows the correction factor 𝐸𝐸 𝑙𝑙! as function of the 𝑙𝑙! / 𝑙𝑙! 𝑙𝑙! !/! ratio and strictly reproduces the lineshape of correction factor 𝐹𝐹! introduced above for isotropic samples (see Fig. 7 for comparison) but also confirming the formal equivalence between the two theories.

N

Based on these approximations, finally the resistivity can be related with the resistance: by means of Wasscher eq. 5.2, the thin anisotropic rectangle with dimensions 𝑙𝑙′! , 𝑙𝑙′! and 𝑙𝑙 ! ! (≪ 𝑙𝑙 !! 𝑙𝑙 ! ! ) along the three resistivity directions 𝜌𝜌! , 𝜌𝜌! , 𝜌𝜌! can be always mapped onto an isotropic parallelepiped with a resistivity 𝜌𝜌 = ! 𝜌𝜌! 𝜌𝜌! 𝜌𝜌!

IN

!

and dimensions 𝑙𝑙!!!,!,! = 𝑙𝑙′! 𝜌𝜌! 𝜌𝜌 (cf. with Fig. 18b). As 𝐸𝐸 ≈ 𝑙𝑙! for the present case, it follows that: 𝜌𝜌!

𝑙𝑙! 𝑙𝑙′! 𝑙𝑙! 𝑙𝑙′!

JP

𝜌𝜌! =

𝜌𝜌 = 𝑙𝑙 ! !

Page 24 of 44

!

𝜌𝜌! 𝐻𝐻 𝑅𝑅 ⟹ 𝜌𝜌 ! !

⟹ 𝜌𝜌! ≈

𝜌𝜌! 𝜌𝜌! = 𝑙𝑙 ! ! 𝐻𝐻! 𝑅𝑅!

𝜋𝜋 𝑙𝑙 ! ! 𝑙𝑙′! 𝑙𝑙! 𝑙𝑙! 𝑅𝑅! sinh 𝜋𝜋 𝑙𝑙! 8 𝑙𝑙′! 𝑙𝑙!

CM

so that finally the Logan relation 5.6 can be expressed in terms of resistivity component 𝜌𝜌! :

(5.10)

5.11


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A similar equation is obtained for the second component 𝜌𝜌! by exchanging 𝑙𝑙! (𝑙𝑙′! ) with 𝑙𝑙! (𝑙𝑙′! ). The unknown 𝑙𝑙! /𝑙𝑙! term in eq. 5.11, which represents the length ratio of the equivalent isotropic rectangular prism, can be still determined via the same resistances 𝑅𝑅! and 𝑅𝑅! measured on the face 𝑙𝑙′! 𝑙𝑙′! of the anisotropic thin rectangle. According the analytical expressions derived by Dos Santos et. al., the 𝑅𝑅! /𝑅𝑅! resistance ratio can be written as:

𝜋𝜋𝑙𝑙! 𝑅𝑅! sinh 𝑙𝑙! ≈ (5.12) 𝑅𝑅! sinh 𝜋𝜋𝑙𝑙! 𝑙𝑙!

ED

which is easily solved by using the hyperbolic relation sinh 𝑥𝑥 = 𝑒𝑒 ! − 𝑒𝑒 !! expression for 𝑙𝑙! /𝑙𝑙! length ratio [73]: 𝑙𝑙! 1 1 𝑅𝑅! ≅ 𝑙𝑙𝑙𝑙 + 𝑙𝑙! 2 𝜋𝜋 𝑅𝑅!

1 𝑅𝑅! 𝑙𝑙𝑙𝑙 𝜋𝜋 𝑅𝑅!

!

2 and yields the following

+ 4 (5.13)

R FO

In summary, the revised Montgomery method through eqs. 5.11, 5.13 and two simple resistance measurements permit to fully and easily determine the resistivity components of a finite thin anisotropic rectangular lamella assuming that their directions are well defined and known. However, if the directions of the resistivity tensor are unknown but still orthogonally-oriented, the problem can only be solved using the approach proposed by Wasscher which is described in the following.

B PU

5.2.2 The Wasscher method

AT

C LI

J.D. Wasscher described in his Ph.D thesis an alternative method for determining the electrical resistivity components of an anisotropic thin film [32]. Although his solution was proposed one year before that of Montgomery, very few works make use of his technique [72] probably because of non-trivial mathematics required for its effective application. However, his studies in the field of resistivity measurements were of uttermost importance and have allowed first quantitative comparisons between the infinite and finite regimes of anisotropic thin films. This is particularly of importance for nanostructures (see below).

N

IO

In a more general way, the Wasscher method can be considered as a special case of the Van der Pauw method for anisotropic samples introduced in Sec. 4. Wasscher uses the reversed mathematical approach and showed that an anisotropic rectangular or circular thin lamella can be always mapped on an isotropic semi-infinite sheet, where the Van der Pauw eqs. are valid. The demonstration is not trivial and uses both the coordinate transformation of eq. 5.2 and the conformal mapping theory in the complex field: if 𝑃𝑃 !! , 𝑄𝑄 !! , 𝑅𝑅 !! , 𝑆𝑆′′ denote the locations of four probes on the edge of a semi-infinite sheet the resistances 𝑅𝑅! = 𝑅𝑅!",!" = 𝑉𝑉! − 𝑉𝑉! 𝐼𝐼!" and 𝑅𝑅! = 𝑅𝑅!",!" = 𝑉𝑉! − 𝑉𝑉! 𝐼𝐼!" respectively, can be expressed as (see Appendix A): 𝜌𝜌 𝑅𝑅′′𝑃𝑃′′ 𝑆𝑆′′𝑄𝑄′′ 𝑙𝑙𝑙𝑙 5.16𝑏𝑏 𝜋𝜋𝜋𝜋 𝑅𝑅′′𝑄𝑄′′ 𝑆𝑆′′𝑃𝑃′′

JP

𝑅𝑅! = 𝑅𝑅!",!" =

𝜌𝜌 𝑆𝑆′′𝑄𝑄′′ 𝑃𝑃′′𝑅𝑅′′ 𝑙𝑙𝑙𝑙 5.16𝑎𝑎 𝜋𝜋𝜋𝜋 𝑆𝑆′′𝑅𝑅′′ 𝑃𝑃′′𝑄𝑄′′

IN

𝑅𝑅! = 𝑅𝑅!",!" =

CM

Let us consider again a thin (i.e. 𝑙𝑙! / 𝑙𝑙! 𝑙𝑙! !/! < 0.2) anisotropic rectangular lamella of dimensions 𝑙𝑙! , 𝑙𝑙! with its edges parallel to the resistivity directions 𝜌𝜌! , 𝜌𝜌! and provided with probes 𝑃𝑃, 𝑄𝑄, 𝑅𝑅, 𝑆𝑆 on its four corners (see Fig. 20a). First, the anisotropic rectangular lamella will be mapped onto an equivalent isotropic rectangle by using the transformation of coordinates given by eq. 5.2 (Fig. 20b). Secondly, a transformation of the coordinates, which makes use of the properties of Jacobian sine-amplitude elliptic function 𝑠𝑠𝑠𝑠(𝐾𝐾(𝑘𝑘), 𝑘𝑘) in the complex field, maps the four probes 𝑃𝑃′, 𝑄𝑄′, 𝑅𝑅′, 𝑆𝑆′ onto the upper half-plane of the complex plane 𝑃𝑃′′, 𝑄𝑄′′, 𝑅𝑅′′, 𝑆𝑆′′. At this Page 25 of 44


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point, it should be evident that the resistances 𝑅𝑅! = 𝑉𝑉! 𝐼𝐼! and 𝑅𝑅! = 𝑉𝑉! 𝐼𝐼! of the anisotropic rectangular lamella (where 𝐼𝐼!!!,! is the current injected current via two probes along one edge and 𝑉𝑉!!!,! the corresponding voltage drop on the opposite edge) can be measured via eq. 5.16a,b, since all the transformations preserve both currents and voltages. Thus, the problem reduces to find the general correspondence formula between the original four probes 𝑃𝑃, 𝑄𝑄, 𝑅𝑅, 𝑆𝑆 on the anisotropic rectangle and the corresponding 𝑃𝑃′′, 𝑄𝑄′′, 𝑅𝑅′′, 𝑆𝑆′′ probes on the semi-infinite sheet. A quite similar case was already described in section. 3.3.2. We will not report on the details here, but instead call the attention of interested readers to the original thesis [32]. In brief, the coordinates of the original probes mapped onto upper half-sheet of complex plane are expressed by: 𝑃𝑃 !! 1 𝑘𝑘 𝑠𝑠𝑠𝑠 𝑲𝑲 𝑘𝑘 , 𝑘𝑘 , 0 (5.17𝑎𝑎)

ED

𝑄𝑄!! −1 𝑘𝑘 𝑠𝑠𝑠𝑠 𝑲𝑲 𝑘𝑘 , 𝑘𝑘 , 0 (5.17𝑏𝑏)

𝑅𝑅′′(−𝑠𝑠𝑠𝑠 𝑲𝑲 𝑘𝑘 , 𝑘𝑘 , 0) (5.17𝑐𝑐)

𝑆𝑆′′(𝑠𝑠𝑠𝑠 𝑲𝑲 𝑘𝑘 , 𝑘𝑘 , 0) (5.17𝑑𝑑)

where 𝑲𝑲(𝑘𝑘) is the complete elliptic integral defined as:

𝑲𝑲 𝑘𝑘 ∶= 𝑭𝑭 𝜋𝜋 2 |𝑘𝑘 ∶=

! !

𝑑𝑑𝑑𝑑

R FO

(5.18𝑎𝑎) 1 − 𝑘𝑘 ! 𝑠𝑠𝑠𝑠𝑠𝑠! 𝜃𝜃 and where 𝑠𝑠𝑠𝑠 𝑧𝑧, 𝑘𝑘 is the so-called sine-amplitude function defined as the inverse of the incomplete elliptic integral of first kind: !

B PU

𝑠𝑠𝑠𝑠 𝑧𝑧, 𝑘𝑘 ∶= sin 𝑥𝑥 , where 𝑧𝑧 ∶= 𝑭𝑭 𝑥𝑥|𝑘𝑘 =

! !

!"

!!! ! !"#! !

(5.18𝑏𝑏)

∶= 𝑒𝑒 !"#$%&'("

!!

𝑲𝑲

!!! ! 𝑲𝑲 !

= 𝑒𝑒

! ! !!! ! ! !! ! !

5.18𝑐𝑐

N

IO

AT

𝑞𝑞 𝑘𝑘

C LI

We point out that for the incomplete elliptic integral, the upper limit 𝑥𝑥 becomes a function of 𝑧𝑧, called amplitude of z. So the Jacobi sine-amplitude is the sine of the upper bound of the incomplete elliptic integral obtained by inverting the F function (i.e. 𝑥𝑥 = 𝐹𝐹 !! 𝑧𝑧, 𝑘𝑘 ) [75, 76]. Both equations depend on the modulus 𝑘𝑘, which is only a function of the resistivities and lengths of the anisotropic rectangular lamella and it is given by the inverse of so-called Jacobi’s nome 𝑞𝑞 𝑘𝑘 :

IN

JP

Fig. 20 (a) Schematics of an anisotropic rectangular sample with edges parallel to resistivity directions, (b) of the equivalent isotropic rectangular sample after Wasscher transformation and (c) final mapping onto upper half-plane of the complex plane.

CM

When the Wasscher method was published, the values of the nome 𝑞𝑞 𝑘𝑘 as well as the numerical values of elliptic integrals 5.18a and b as function of 𝑘𝑘 were given in mathematical tables [77], while nowadays, computer software, such as Matlab [78] or Wolfram Mathematica [79], allows their rapid and easy evaluation with high numerical precision.

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Evaluating the distances 𝑃𝑃′′𝑄𝑄′′ ∶= 𝑃𝑃 !! 1 𝑘𝑘 𝑠𝑠𝑠𝑠 𝐾𝐾, 𝑘𝑘 , 0 − 𝑄𝑄 !! −1 𝑘𝑘 𝑠𝑠𝑠𝑠 𝐾𝐾, 𝑘𝑘 , 0 , etc. and replacing into eqs. 5.16, we finally get: 𝑅𝑅!

𝑅𝑅!

!"#$%&'("

=

!"#$%&'("

=

𝜌𝜌! 𝜌𝜌! 1 + 𝑘𝑘𝑠𝑠𝑠𝑠! 𝑲𝑲 𝑘𝑘 , 𝑘𝑘 2𝑙𝑙𝑙𝑙 𝜋𝜋𝑙𝑙! 1 − 𝑘𝑘𝑠𝑠𝑠𝑠! 𝑲𝑲 𝑘𝑘 , 𝑘𝑘

5.19a

𝜌𝜌! 𝜌𝜌! 1 + 𝑘𝑘𝑠𝑠𝑠𝑠! 𝑲𝑲 𝑘𝑘 , 𝑘𝑘 2𝑙𝑙𝑙𝑙 𝜋𝜋𝑙𝑙! 1𝑘𝑘 ! ! 𝑠𝑠𝑠𝑠 𝑲𝑲 𝑘𝑘 , 𝑘𝑘

5.19b

ED

Similarly to the Montgomery method, the numerical solution of both eqs. 5.19a and b allows the evaluation of both resistivity components 𝜌𝜌! , 𝜌𝜌! from two single resistance measurements (i.e. 𝑅𝑅! and 𝑅𝑅! ).

R FO B PU

C LI

Fig. 21 (a) Schematic of an anisotropic circular lamella with the 4P probes placed on two vertical diameters and rotated by 𝜙𝜙 degree w.r.t. resistivity directions. For sake of clarity, it is also shown the angle 𝜃𝜃 = 5𝜋𝜋/4 − 𝜙𝜙 used in Fig. 17 to define the angular position of square 4P array w.r.t. resistivity directions. (b) Calculated angle dependence of electrical resistance 𝑅𝑅 𝜙𝜙 for an anisotropic finite circular thin film. The several solid curves are plotted using eq. 5.20a with different (𝜌𝜌! 𝜌𝜌! ) resistivity ratios.

IO

AT

The main advantage of this mathematical approach relies on its simple generalization to the case of a circle. Indeed, if we consider now a thin anisotropic circular lamella, of radius 𝑟𝑟 and thickness 𝑡𝑡(< (𝑑𝑑/5)), with all four point probes placed on its circumference 𝑑𝑑 along two orthogonal diameters, and we call 𝜙𝜙 the angle between the two orthogonal resistivity components 𝜌𝜌! , 𝜌𝜌! and the lines intersecting the two opposite contacts as sketched in Fig.21a, the corresponding resistances measured on a circle reads [32]:

𝑅𝑅!

!"#!

%$=

𝜌𝜌! 𝜌𝜌! 2 𝑙𝑙𝑙𝑙 𝜋𝜋𝜋𝜋 1 − 𝑘𝑘𝑘𝑘𝑘𝑘 4𝑲𝑲 𝑘𝑘 𝜙𝜙 𝜋𝜋 , 𝑘𝑘 𝜌𝜌! 𝜌𝜌! 2 𝑙𝑙𝑙𝑙 𝜋𝜋𝜋𝜋 1 + 𝑘𝑘𝑘𝑘𝑘𝑘 4𝑲𝑲 𝑘𝑘 𝜙𝜙 𝜋𝜋 , 𝑘𝑘

𝑞𝑞 𝑘𝑘

!"#!

%$∶= 𝑒𝑒

!!

𝑲𝑲

!!!!

𝑲𝑲 !

= 𝑒𝑒

!"#!

%$!! ! ! !!! ! ! !! !!! !

!

:

5.20a

CM

where now the modulus 𝑘𝑘 is given by inverse of Jacobi’s nome 𝑞𝑞 𝑘𝑘

5.20a

JP

=

IN

!"#!

%$N

𝑅𝑅!

(5.21) Page 27 of 44


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Fig. 21b reveals the equation 5.20a as function of 𝜙𝜙 with different (𝜌𝜌! 𝜌𝜌! ) resistivity ratios. The graph is only shifted of 𝜋𝜋/4 with respect to the infinite case of Fig.16a which is plotted as function of 𝜃𝜃 = 5𝜋𝜋/4 − 𝜙𝜙 (see Fig.21a for reference). Interestingly, contrary to the infinite case, the resistance remains always positive, also for high anisotropies. As in the case of an infinite sample the resistivity directions can be easily found and their values consequently determined by rotating the entire 4P array. As obvious the 𝑅𝑅! !"#!$% / 𝑅𝑅! !"#!$% ratio takes its maximum for 𝜙𝜙 = 𝜋𝜋/4 (or 𝜃𝜃 = 𝜋𝜋/2), because 𝑅𝑅! is maximum (see Fig. 21a), while 𝑅𝑅! , which has a similar dependence on 𝜙𝜙 but shifted by 90 degree respect 𝑅𝑅! , has its minimum. Since 𝑠𝑠𝑠𝑠 4𝑲𝑲 𝑘𝑘 𝜙𝜙 𝜋𝜋 , 𝑘𝑘 = 1 for 𝜙𝜙 = 𝜋𝜋/4, one immediately gets from eqs. 5.20:

ED

(𝑅𝑅! 𝑅𝑅! )!"# =

𝑙𝑙𝑙𝑙 1 − 𝑘𝑘 𝑙𝑙𝑙𝑙 1 + 𝑘𝑘

2 (5.22) 2

R FO N

IO

AT

C LI

B PU IN

Fig. 22 Electrical resistance ratio 𝑅𝑅! 𝑅𝑅! (= 𝑅𝑅! 𝑅𝑅! ) as function of resistivity anisotropy degree 𝜌𝜌! 𝜌𝜌! (= 𝜌𝜌! 𝜌𝜌! ) for an infinite 2D sheet and a finite circular- and square-shaped lamella. Solid curves referring to finite geometries are plotted using eqs. 5.19 (with 𝑙𝑙! = 𝑙𝑙! = 𝑙𝑙) and 5.22 of Wasscher method. Full symbols are obtained from eqs. 5.10 and 5.12 of Montgomery’s method (here assuming 𝑙𝑙′! = 𝑙𝑙′! ).

JP

CM

Similarly, the resistance anisotropy as a function of resistivity anisotropy degree 𝜌𝜌! 𝜌𝜌! for a rectangular or square-shaped (with 𝑙𝑙1 = 𝑙𝑙2 = 𝑙𝑙) lamella with the probes located on its corners is easily derived from eqs. 5.19. Both dependencies are plotted in Fig. 22 together with those obtained on anisotropic infinite 2D sheets (cf. with Tab. 3). As obvious, the electrical resistance ratios 𝑅𝑅! 𝑅𝑅! (= 𝑅𝑅! 𝑅𝑅! ) of a finite anisotropic film can be increased by several orders of magnitude compared to infinite film although its resistivity ratio 𝜌𝜌! 𝜌𝜌! (= 𝜌𝜌! 𝜌𝜌! ) is the same. The dependency on the geometry of the sample is much weaker. Therefore, a much higher sensitivity is expected when the 4P probes are placed on the sample periphery (i.e. 𝑠𝑠 = 𝑑𝑑 or 𝑠𝑠 = 𝑙𝑙) compared to the infinite case (i.e. 𝑠𝑠 ≪ 𝑑𝑑 or 𝑠𝑠 ≪ 𝑙𝑙). However, care should be taken while measuring Page 28 of 44


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resistivities of finite anisotropic materials. While for isotropic samples the error can be only a factor two, it can hugely increase for the case of finite anisotropic lamella structures. Finally, the full squares in Fig. 22 display the values determined by using Montgomery method (with 𝑙𝑙′! = 𝑙𝑙′! , eqs 5.10 and 5.12). The perfect agreement with the solid curve, obtained with Wasscher method relying on a completely different approach, underlines once more the robustness of both mathematical solutions.

5.3 Experimental comparison among finite and infinite regime for anisotropic 2D systems

ED

We have introduced in section 5.1 the In/Si(111) system as an anisotropic 2D-benchmark system to determine the resistivity tensor and will use the same system now also on spatially restricted areas to proof the increase of the resistance ratio discussed in context of Fig. 22. Please refer to Sec. 3 for any detail on the infinite (i.e. unconfined) In/Si(111) system. The spatially restricted 4×1-In wire system grown on circular 𝑆𝑆𝑆𝑆(111) mesa structures, i.e. elevated Si(111) islands, is illustrated by Fig. 23c. The resistivity components have been again determined by means of a 4-tip STM/SEM system. The mesa structures in this case have a diameter of 30 µm and a height of 500 nm (Fig. 23 a,b) and were fabricated by optical lithography and reactive ion etching (RIE, using SF6) onto a 𝑆𝑆𝑆𝑆(111) semiconductor wafer (miscut of 1° degree toward the [-1-12] direction and P-doped, 500-800 Ω ∙cm). A clean Si(111)7x7 surface [80, 81], is thus obtained through a vigorous chemical treatment of the sample including standard RCA-1 cleaning [82] in air and subsequent flash annealing cycles up to 900°C under UHV conditions. The 4𝑥𝑥1-In phase is finally obtained by adsorption of an Indium monolayer at 400 °C.

R FO

N

IO

AT

C LI

B PU

IN

Fig. 23 (a) SEM micrograph (1.000× magnification, plan-view) of circular mesa-structures fabricated by optical lithography and reactive ion etching onto a 𝑆𝑆𝑆𝑆(111) substrate. (b, c) SEM image and schematic of the four tungsten tips placed with nanometric precision on the periphery of mesa structure. (d) Electrical resistance of 4𝑥𝑥1-In system as function of 4P array angular position 𝑅𝑅 𝜃𝜃 on (open symbols) and faraway from (full symbol) the Si mesa structure. The solid red and blue lines are the best-fitting curves of experimental data, obtained by using eqs. 5.20 and 5.5 for the finite and infinite regimes, respectively.

JP

CM

By means of a high-resolution scanning electron microscope (SEM), the four tungsten tips of the multi-probe STM microscope are independently navigated with nm-precision to the mesa periphery as shown in Fig. 23b and individually approached to the 4𝑥𝑥1-In surface via feedback control approach mechanisms. The electrical resistance of 4𝑥𝑥1-In atomic chains is thus measured as a function of 4P square array angular position on the circular mesa periphery. In order to compare these measurements on finite areas we performed similar measurements with the same contact geometry on the lower quasi 2D infinite areas at least 300 µm away from the circular-mesa structure. In this way we can realize resistivity measurements on infinite as well as finite Page 29 of 44


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areas on the same sample and ensure that the average defect densities are approximately the same. Both measurements are plotted in Fig. 23d showing clearly the effect of confinement. The resistance anisotropy = degree for the 4×1-In system grown onto the circular mesa surface (open symbols) is around 𝑅𝑅! /𝑅𝑅! !"#$

ED

= 3.5 measured on the large and flat area of the same sample. The solid curves 17.4 compared to 𝑅𝑅! /𝑅𝑅! !! reveal best-fits of experimental data by using eqs. 5.20 and 5.5 for the finite and infinite case, respectively, for the resistivity components 𝜌𝜌! = 6.2 ±1.2 ×10! Ω/𝑠𝑠𝑠𝑠 and 𝜌𝜌! = 16.7 ±1.0 ×10! Ω/𝑠𝑠𝑠𝑠 yielding a resistivity anisotropy degree 𝜌𝜌! /𝜌𝜌! ~2.7. The present result is of importance, because it directly confirms that a much higher sensitivity (of one order of magnitude or more) can be easily achieved when the sample size becomes comparable to probe distance and validates both theoretical models introduced for the description of infinite and finite regimes. Furthermore, we notice that the minimum resistance (i.e. parallel to the Indium chains) for the present 4×1-In system is of ~5×10! Ω, one order of magnitude larger than the value reported in the Fig. 16b for an Indium layer deposited under similar conditions. We point out that the difference is not related to the probe spacing and sample geometry, but most likely due to the lower annealing temperature (900 °C vs 1200°C) in order to preserve the mesa structures. This leaves a higher concentration surface defects behind which is responsible for the larger measured resistance [71], although relative values (i.e. the resistivity anisotropy degree 𝜌𝜌! /𝜌𝜌! ) remain comparable. Nonetheless, the spatial constriction reveals the unique possibility to control and correlate the impact of defects in surface transport.

R FO

B PU

5.4 Correction factors for a square 4P array inside an anisotropic circularly shaped area

C LI

In order to derive the resistivity components of a finite and anisotropic material by means of the Montgomery [63] or Wasscher [32] method, the 4P probes must be precisely positioned on the sample periphery (see Sec. 5.2). To the best of our knowledge, the correction factor for arbitrary probe spacings 𝑠𝑠 on samples with size 𝑑𝑑, i.e. 𝐹𝐹(𝑠𝑠/𝑑𝑑), again for the case of a finite anisotropic medium, has so far not been derived.

N

IO

AT

In the following we will explicitly calculate the correction factor for the case of a square 4P probe array placed in the center of an anisotropic circular slice with two distinct and mutually orthogonal resistivity components 𝜌𝜌! , 𝜌𝜌! (= 𝜌𝜌! ). As we will show further, our theoretical result is fully in line with latest experiments done on the 4×1-In wire system selectively grown on circular 𝑆𝑆𝑆𝑆(111) mesa structures.

IN CM

JP Fig. 24 (a) Schematics of 4P probe square array placed in the center of an anisotropic circular sample. The square edges are assumed parallel to resistivity directions 𝜌𝜌! and 𝜌𝜌! . (b) Sketch of electrically equivalent elliptical isotropic sample and rectangular 4P array and (c) related subsequent mapping onto a unit circle.

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Fig. 24a shows the schematics of a 4 “point” probe square array placed in the center of “thin” anisotropic circular medium. Like in the context of the Wasscher method, we still assume that both the probe size and sample thickness are small w.r.t. the probe spacing, i.e. we restrict our analysis to a quasi 2D-scenario. In agreement with the nomenclature adopted throughout the review, we call 𝑠𝑠 the distance in between next neighbor probes, while 𝑑𝑑 is the diameter of circle. Furthermore, for sake of simplicity, we suppose that the square edges of 4P probe array are parallel to the two resistivity components, 𝜌𝜌! and 𝜌𝜌! . At the end we will generalize this to arbitrary angles of rotation (cf. Sec. 5.3). By applying eq. 5.2 (Wasscher method), the anisotropic circle (Fig.24a) is first transformed into an electrically equivalent isotropic ellipse of resistivity 𝜌𝜌 = ! 𝜌𝜌! 𝜌𝜌! 𝜌𝜌! with the semi-axes 𝑎𝑎′ = 𝑟𝑟 𝜌𝜌! /𝜌𝜌, 𝑏𝑏′ = 𝑟𝑟 𝜌𝜌! /𝜌𝜌 (and

ED

thickness 𝑡𝑡′ = 𝑟𝑟 𝜌𝜌! /𝜌𝜌), while the square 4P probe assembly is simultaneously stretched into an electrically equivalent rectangular 4P probe assembly. In Fig. 24b, the special case 𝜌𝜌! /𝜌𝜌! = 2 is depicted. In general, the aspect ratio of the rectangle depends on the anisotropy degree of original circle.

R FO

According to the conformal mapping theory (cf. also with Sec.5.2.2), we know that the interior of an ellipse (i.e. every point 𝑧𝑧 of an ellipse) can be mapped onto a unit circle (Chap. VI p. 296 of Ref. [41]) by the function: 𝑤𝑤 = 𝑓𝑓 𝑧𝑧 = 𝑘𝑘𝑠𝑠𝑠𝑠

2𝐾𝐾(𝑘𝑘) 𝑧𝑧 𝑠𝑠𝑠𝑠𝑠𝑠!! , 𝑘𝑘 with 𝑤𝑤 < 1 ! 𝜋𝜋 𝑎𝑎′ − 𝑏𝑏′!

(5.23)

B PU

where 𝑧𝑧 is the position of an arbitrary point inside the ellipse (expressed in complex coordinates), 𝐾𝐾(𝑘𝑘) denotes the complete elliptic integral of modulus 𝑘𝑘 (eq.5.18a), whose value is given by the inverse of Jacobi’s nome 𝑞𝑞 𝑘𝑘 !"#!$% (see eq. 5.21) and depends on the resistivity values of the original anisotropic circle. It is easy to demonstrate that the four initial points 𝐴𝐴, 𝐵𝐵, 𝐶𝐶, 𝐷𝐷 sketched in Fig. 24a are mapped onto the points 𝐴𝐴’’, 𝐵𝐵’’, 𝐶𝐶’’, 𝐷𝐷’’ in the unit circle with the following coordinates:

5.24𝑏𝑏

𝜌𝜌! + 𝕚𝕚 𝜌𝜌! 𝑠𝑠 2𝐾𝐾 𝑘𝑘 𝑠𝑠𝑠𝑠𝑠𝑠!! , 𝑘𝑘 𝜋𝜋 𝜌𝜌! − 𝜌𝜌! 𝐷𝐷

5.24𝑐𝑐

− 𝜌𝜌! + 𝕚𝕚 𝜌𝜌! 𝑠𝑠 2𝐾𝐾 𝑘𝑘 𝑠𝑠𝑠𝑠𝑠𝑠!! , 𝑘𝑘 𝜋𝜋 𝐷𝐷 𝜌𝜌! − 𝜌𝜌!

5.24𝑑𝑑

N IN

𝐷𝐷 !! = 𝑓𝑓 𝐷𝐷′ = 𝑘𝑘𝑠𝑠𝑠𝑠

𝜌𝜌! − 𝕚𝕚 𝜌𝜌! 𝑠𝑠 2𝐾𝐾 𝑘𝑘 𝑠𝑠𝑠𝑠𝑠𝑠!! , 𝑘𝑘 𝜋𝜋 𝜌𝜌! − 𝜌𝜌! 𝐷𝐷

IO

𝐶𝐶 !! = 𝑓𝑓 𝐶𝐶′ = 𝑘𝑘𝑠𝑠𝑠𝑠

5.24𝑎𝑎

AT

𝐵𝐵 !! = 𝑓𝑓 𝐵𝐵′ = 𝑘𝑘𝑠𝑠𝑠𝑠

− 𝜌𝜌! − 𝕚𝕚 𝜌𝜌! 𝑠𝑠 2𝐾𝐾 𝑘𝑘 𝑠𝑠𝑠𝑠𝑠𝑠!! , 𝑘𝑘 𝜋𝜋 𝐷𝐷 𝜌𝜌! − 𝜌𝜌!

C LI

𝐴𝐴!! = 𝑓𝑓 𝐴𝐴′ = 𝑘𝑘𝑠𝑠𝑠𝑠

CM

JP

These equations have been plotted in Fig. 24c (by means of Wolfram Mathematica). The dashed black curves show how the coordinates of four points change as the D/𝑠𝑠 ratio is varied. Noteworthy, the 4P array has still a rectangular appearance on the unit circle, however, the paths of the corners when going from zero up to the circle diameter 𝑑𝑑 (i.e. for 𝑠𝑠/𝑑𝑑 = 2) are curved as indicated by the arrows. Furthermore, the angle between the two curved paths depends on the anisotropy degree of original circle (see the gray dashed curves in Fig 24c for reference). Fortunately, the case of a rectangular 4P array placed in the center of an equivalent isotropic circular slice can be easily solved by applying the method of images, further generalizing the Vaughan solution [42] for a 4P square array (see Sec. 3.3.1 above). If the current 𝐼𝐼! is injected, according Fig. 24c, via the probes 𝐵𝐵’’ and 𝐶𝐶′′, Page 31 of 44


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while the voltage 𝑉𝑉! is probed between 𝐴𝐴′′ and 𝐷𝐷′′, the resistance 𝑅𝑅! for the present rectangular probe configuration inside a circle reads: !! !"#$%&'(" !! 𝜌𝜌! 𝜌𝜌! 𝜌𝜌! 𝜌𝜌! 𝐹𝐹!!!"#!$% d′′/b !! 𝐹𝐹! (𝑠𝑠/𝑑𝑑) = (5.25) 𝑅𝑅! = 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋 𝑙𝑙𝑙𝑙 1 + 𝑟𝑟 ! where 𝑟𝑟 = 𝑎𝑎/𝑏𝑏 (with 𝑎𝑎 = 𝐴𝐴′′𝐷𝐷′′ and 𝑏𝑏 = 𝐴𝐴′′𝐵𝐵′′, see Fig. 24c) is the aspect ratio of the rectangle mapped on the !"!"#$%&' !! is the general correction factor for a rectangular 4P array placed in the center of unite circle and 𝐹𝐹!!!"#!$% an isotropic circular slice, which reads: !"#$%&'(" !!

𝐹𝐹!!!"#!

%$𝑑𝑑′′ = 𝑏𝑏

𝑙𝑙𝑙𝑙 1 + 𝑟𝑟 !

𝑙𝑙𝑙𝑙 1 + 𝑟𝑟 ! (5.26) 1 + (𝑑𝑑′′/𝑏𝑏)! + 𝑟𝑟 ! ! + 𝑙𝑙𝑙𝑙 1 + (𝑑𝑑′′/𝑏𝑏)! ! + 2 1 − (𝑑𝑑′′/𝑏𝑏)! 𝑟𝑟 ! + 𝑟𝑟 !

ED

For the present case 𝑑𝑑 !! = 2 which is the diameter of unit circle. For further details regarding the derivation of these equations the reader is referred to the appendix D (eqs. 5.25 and 5.26 correspond to eqs. 𝐷𝐷8, with 𝑅𝑅!! replaced by 𝜌𝜌/𝑡𝑡′ = 𝜌𝜌! 𝜌𝜌! /𝑡𝑡, and 𝐷𝐷7, respectively).

R FO

As obvious from eq. 5.25, the correction factor 𝐹𝐹! (𝑠𝑠/𝑑𝑑), albeit formally linked to the correction factor !"#$%&'(" !! 𝐹𝐹!!!"#!$% for a rectangular 4P array in an isotropic circle, represents (via the eqs. 5.24) the correction factor for a 4P square array placed in the center of an anisotropic circle and aligned as depicted in Fig. 24c (i.e. with the current probes parallel to the 𝜌𝜌! component). Based on eqs. 5.24 and 5.26, this correction factor can be numerically calculated with high precision as we will show below.

B PU

N

IO

AT

C LI

In order to determine the entire resistance anisotropy ratio the orthogonal 𝑅𝑅! component is needed. By exchanging the current and voltage probes an equation similar to eq. 5.25 can be written for the resistance 𝑅𝑅! parallel to the 𝜌𝜌! component: !! !"#$%&'(" !! 𝜌𝜌! 𝜌𝜌! 𝜌𝜌! 𝜌𝜌! 𝐹𝐹!!!"#!$% (𝑑𝑑′′/𝑎𝑎) !! 𝐹𝐹! (𝑠𝑠/𝑑𝑑) = (5.27) 𝑅𝑅! = 𝑙𝑙𝑙𝑙 1 + 𝑟𝑟 !! 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋

IN JP

CM

Fig. 25 (a) Theoretical electrical resistance ratio 𝑅𝑅! 𝑅𝑅! versus normalized wafer diameter 𝑑𝑑 𝑠𝑠 for a square 4P probe array placed in the center of an anisotropic circle (𝑠𝑠 is the square edge of 4P array). The several solid curves are plotted using eq. 5.29 with different (𝜌𝜌! 𝜌𝜌! ) resistivity ratios. (b) Experimental electrical resistance ratio 𝑅𝑅! 𝑅𝑅! versus normalized wafer diameter 𝑑𝑑 𝑠𝑠 measured on a 4×1-In wire system selectively grown on a circular 𝑆𝑆𝑆𝑆(111) mesa structure (𝑑𝑑 = 20𝜇𝜇𝜇𝜇). Open and full symbols refer to two different mesas of the same sample. Solid line in the figure is the best-fitting curve of experimental data, obtained by using eq. 5.29.

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Similarly to the former case, now a correction factor 𝐹𝐹! (𝑠𝑠/𝑑𝑑) is introduced which depends on !"#$%&'(" !! (𝑑𝑑′′/𝑎𝑎): 𝐹𝐹!!!"#!$% !"#$%&'(" !!

𝐹𝐹!!!"#!

%$𝑑𝑑′′ = 𝑎𝑎

𝑙𝑙𝑙𝑙 1 + 𝑟𝑟 !!

𝑙𝑙𝑙𝑙 1 + 𝑟𝑟 !! (5.28) 1 + (𝑑𝑑′′/𝑎𝑎)! + 𝑟𝑟 !! ! + 𝑙𝑙𝑙𝑙 ! ! !! ! ! 1 + 𝑟𝑟 + 2 1 − 𝑟𝑟 (𝑑𝑑′′/𝑎𝑎) + (𝑑𝑑′′/𝑎𝑎)

From the ratio of eqs. 5.27 and 5.25, we get a compact expression for the anisotropy ratio 𝑅𝑅! 𝑅𝑅! , which directly refers to the resistivity components 𝜌𝜌! and 𝜌𝜌! inside the anisotropic circle and normalized circle diameter 𝑑𝑑/𝑠𝑠 and simply reads:

ED

𝑅𝑅! 𝐹𝐹! 𝑙𝑙𝑙𝑙 1 + 𝑟𝑟(𝑑𝑑/𝑠𝑠, 𝜌𝜌! , 𝜌𝜌! )!! = = (5.29) 𝑅𝑅! 𝐹𝐹! 𝑙𝑙𝑙𝑙 1 + 𝑟𝑟(𝑑𝑑/𝑠𝑠, 𝜌𝜌! , 𝜌𝜌! )!

R FO

This resistance anisotropy ratio can be numerically derived using eqs. 5.24. Fig.25a shows the ratio as a function of the normalized circle diameter 𝑑𝑑/𝑠𝑠 for various resisitivity anisotropy values. It is apparent that the measured electrical resistance ratio 𝑅𝑅! 𝑅𝑅! rises exponentially as the squared probe assembly is moved from the center (𝑑𝑑 ≫ 𝑠𝑠) to the circle periphery (𝑑𝑑 = 2𝑠𝑠). As expected, the values we get from the eq. 5.29 for the limits 𝑑𝑑 ≫ 𝑠𝑠 (infinite 2D sheet) and for 𝑑𝑑 = 2𝑠𝑠 (i.e. probes on the circle circumference) coincide with those values plotted in Fig. 22.

N

IO

AT

C LI

B PU

In order to confirm our considerations from above, we used again the 4×1-In reconstruction grown on a circular 𝑆𝑆𝑆𝑆(111) mesa structure (diameter 𝑑𝑑 = 20𝜇𝜇𝜇𝜇) and measured precisely the resistance components parallel (𝑅𝑅! ) and perpendicular (𝑅𝑅! ) now for various probe spacings 𝑠𝑠. The correct alignment of the probes was done before using the rotational square 4P method discussed in context of Fig. 23d. We thereafter measured sequentially both resistance values by approaching the tips from the periphery toward the mesa center. As shown in Fig. 25b, the ratio decreases exponentially from a value of 𝑅𝑅! 𝑅𝑅! ~11 with the tips at about 1 𝜇𝜇𝜇𝜇 from the circular border down to a minimum value of 𝑅𝑅! 𝑅𝑅! ~3.5 when the probes are distant only 3 𝜇𝜇𝜇𝜇. The solid curve is the best-fit by using eq. 5.29 and yields a resistivity-anisotropy of 𝜌𝜌! /𝜌𝜌! = 2.5 ± 0.1 which perfectly agrees with those values determined (independently) with the rotational square 4P method. We notice that some data points appear slightly scattered w.r.t. the fitting curve and is caused most likely due to uncertainties regarding the positioning of the tips or possibly inhomogeneities within the In-reconstruction on the Si(111)-mesa itself.

IN CM

JP Page 33 of 44


ED

PT CE AC R FO

B PU

Fig. 26 Theoretical calculation of the resistance for a 4P square array placed in the centre of a finite anisotropic circular lamella as function of 𝑑𝑑 𝑠𝑠 ratio and angle of rotation 𝜃𝜃 w.r.t. the directions of anisotropy. Here, 𝑠𝑠 and 𝑑𝑑 denote the side length of the square 4P array and the diameter of circular lamella, respectively. The plot is obtained by setting the resistivity values to: 𝜌𝜌! = 6.2 ±1.2 ×10! Ω/𝑠𝑠𝑠𝑠 and 𝜌𝜌! = 16.7 ±1.0 ×10! Ω/𝑠𝑠. These values coincide with those obtained for the In-4x1 phase shown in Fig. 23d.

6

Conclusions and outlook

N

IO

AT

C LI

As just seen, the probe-distance dependent measurements of the resistance in a finite area can be used to determine correctly the resistivities. Finally, we like to demonstrate that this technique can be combined with the rotational square method mentioned in Sec. 5.2.2. and Sec. 5.3. In this case, function 5.23 maps the 4P square array on an electrically equivalent parallelogram inside the unit circle and the new coordinates of eqs. 5.24 depend explicitly on the azimuthal orientation of the 4P assembly. Technically, the approach is similar to what we outlined in the appendix for the rectangular geometry used above. Thus we will not repeat all steps in detail but show the final result for the resistance 𝑅𝑅(𝜌𝜌! , 𝜌𝜌! , 𝜙𝜙, 𝑠𝑠/𝑑𝑑) in Fig.26 exemplarily for one set of resistivity components 𝜌𝜌! = 6.2 ±1.2 ×10! Ω/𝑠𝑠𝑠𝑠 and 𝜌𝜌! = 16.7 ±1.0 ×10! Ω/𝑠𝑠.

IN

In this review paper we have revisited the four-point probe technique with the special emphasize on geometrical correction factors for both isotropic and anisotropic systems in order to determine the electrical resistivity from apparently simple resistance measurements. Despite its long history of almost one-hundred years, the four-point probe transport technique is to date still a leading method in both fundamental and application-driven research. Along with the development of sophisticated multi-probe scanning tunneling microscopes, this technique can nowadays be applied successfully even on a truly nanometer scale. Particularly for these structures and, in a more general sense, for spatially restricted areas, the aspects of anisotropy and the current paths need to be considered to reveal reliable values for the resistivity.

CM

JP

As outlined in detail in Sec.3, the resistance 𝑅𝑅 and the resistivity 𝜌𝜌 are generally linked by dimensionless correction factors 𝐹𝐹, which in turn depend on the probe geometry and assembly as well as the structure of the sample. Even more important, the dimensional crossover which occurs along the transition from the infinite- to finite- regime, depending on ratio between the probe spacing s and sample size d, can be accurately taken into Page 34 of 44


PT CE AC

account. As seen for infinite (i.e. 𝑠𝑠/𝑑𝑑 ≪ 1) and finite scenarios (i.e. 𝑠𝑠/𝑑𝑑 = 1), the apparent resistance of isotropic materials may increase up by a factor two which, on the other hand, can be caused also by different probe geometries or sample structures. An exception is the well-known Van der Pauw method, where the shape of the sample is not important as long as the four probes are located on the sample periphery or along one of its planes of symmetry. Details have been mentioned in Sec.4.

ED

On the other hand, anisotropic materials as discussed in Sec.5 still remain challenging since two or three resistivity components are needed for an entire characterization. As recently as 40 years ago full analytical treatment was possible only for the special case of infinite samples (i.e. 𝑠𝑠/𝑑𝑑 ≪ 1) or finite rectangular/circular- shaped specimens with the 4P probes located on the sample periphery (i.e. 𝑠𝑠/𝑑𝑑 = 1). Although Van der Pauw theorem has been shown to be applicable to anisotropic materials of arbitrary shape as well, it allows to measure only the geometric mean 𝜌𝜌! 𝜌𝜌! without being capable to disentangle the individual resistivity components [83, 84]. The methods introduced by Wasscher and Montgomery are routinely applied for the measurement of resistivity components in anisotropic materials, but the validation of some effects such as the negative resistance at a high degree of anisotropy [66] as well as the increase of the sensitivity of the 4P probe technique in a finite geometry setup were only lately demonstrated and also subject of this review article. Especially, the latter aspect is important as high resistance anisotropies are induced also by spatial restrictions (e.g. steps, inhomogeneities, etc.) which in turn could lead to wrong resistivity values. Only few attempts are made to date for the calculation of correction factors 𝐹𝐹 for anisotropic materials as a function of the probe distance compared with the sample dimension 𝑑𝑑 [32, 85, 86]. In the course of this review we have calculated for the first time the general case of a square 4P geometry inside an circularly shaped anisotropic system.

R FO

B PU

AT

C LI

In this review we restrict ourselves to homogeneous phases. However, in particular large samples may exhibit various phases causing a spatial variations regarding transport properties, e.g. carrier mobilities and densities. In this case, the interpretation is not straightforward. First studies, which date back over fifty years, have shown that 4-point probe measurements are only qualitatively sensitive to non-uniformities in case that their diameter is larger than probe spacing and yielding less information (i.e. averaging their effect) when their diameter is much smaller [40]. Therefore, two-dimensional or three dimensional contour map techniques have been developed in course of time in order to precisely characterize variation of the wafer resistivity [87]. Only recently, this problem was encountered theoretically providing analytical functions in order to be capable to simulate the sensitivity of 4-point probe measurements towards local inhomogeneities [19].

N

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Furthermore, the effect of the finite widths of the contacts themselves needs to be taken into account in order to improve further the accuracy in resistivity measurements. In this work we assumed ohmic and sufficiently small contacts, e.g. point contacts, compared to sample size. If this condition is not fulfilled, the resistivity expressions for the infinite 3D half-plane and 2D sheet, further generalizing those reported in Tab. 1, can be derived [88, 89, 90]. In case of finite isotropic samples, Van der Pauw has given a first rough estimate of the error associated with large contacts anchored to the sample periphery, being of the order Δ𝜌𝜌 𝜌𝜌 ~ 𝑙𝑙 ! 𝑑𝑑 ! (where 𝑙𝑙 and 𝑑𝑑 are the contact width and sample size, respectively) [44, 45]. A more detailed analysis by conformal mapping theory [91, 92, 93, 94] and numerical simulations [95, 96] was carried out for rectangularly or circularly shaped samples. A method that is valid for samples of arbitrary shape and that generalizes the Van der Pauw approach was proposed at first recently by Cornils et al. [97, 98]. The impact of finite contacts for anisotropic materials is analyzed only in part and actually solved for the special case of circular-shaped thin films [99].

IN

JP

CM

Further efforts and development are certainly required and expected in the near future in this field considering the increasing interest in anisotropic materials and striking applications of four point probe techniques. Acknowledgements: We gratefully acknowledge the financial support of this project by the DFG through the projects FOR1700 and Te 386/9-1. Furthermore, I.M. wishes to acknowledge the Institut für Festkörperphysik at Leibniz Universität of Hannover for their hospitality and the “Angelo della Riccia” Italian foundation for supporting his stay there during the period in which this review was written.

Page 35 of 44


Appendix A

PT CE AC

ED

The Van der Pauw formula can be easily derived for the special case of a semi-infinite thin film. This trivial calculation reveals the origin of the exponential factor in the eq. 4.1 and helps to understand the method proposed by Wasscher for the evaluation of resistivity components for the case of a finite thin anisotropic rectangular or circular lamella (see Sec. 5.2.2). Taking as reference Fig. A1, we assume that the 4P probes are placed along the edge of a semi-infinite 2D sheet. The current is injected and collected through the Q and R probes respectively, while the voltage drop is measured between the probes S and P. The sample thickness is assumed much smaller than the probe distance, thus the current density J can be immediately expressed as 𝐽𝐽 = 𝐼𝐼 𝜋𝜋𝑟𝑟𝑟𝑟 yielding for the voltage drop 𝑉𝑉!" : ! 𝜌𝜌𝐼𝐼!" 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝐸𝐸𝐸𝐸𝐸𝐸 = 𝑙𝑙𝑙𝑙 − 𝑙𝑙𝑙𝑙 (𝐴𝐴. 1) 𝑉𝑉!" = 𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃 ! where the first term is the voltage drop due to the current injected by probe Q, while the second term is the voltage drop due to the current leaving the sample via probe R. Rearranging eq. A1, we thus obtain: 𝜌𝜌 𝑆𝑆𝑆𝑆 𝑃𝑃𝑃𝑃 𝑙𝑙𝑙𝑙 𝐴𝐴. 2 𝑅𝑅!",!" = 𝜋𝜋𝜋𝜋 𝑆𝑆𝑆𝑆 𝑃𝑃𝑃𝑃 If we change now current and voltage probes, we can similarly write: 𝜌𝜌 𝑅𝑅𝑅𝑅 𝑆𝑆𝑆𝑆 𝑅𝑅!",!" = 𝑙𝑙𝑙𝑙 𝐴𝐴. 3 𝜋𝜋𝜋𝜋 𝑅𝑅𝑅𝑅 𝑆𝑆𝑆𝑆 At this point we notice that:

R FO

! !! !!",!" !

+ 𝑒𝑒

! !! !!",!" !

B PU

since:

𝑒𝑒

= 1 (A. 4)

𝑆𝑆𝑆𝑆 𝑃𝑃𝑃𝑃 𝑅𝑅𝑅𝑅 𝑆𝑆𝑆𝑆 + = 1 (𝐴𝐴. 5) 𝑆𝑆𝑆𝑆 𝑃𝑃𝑃𝑃 𝑅𝑅𝑅𝑅 𝑆𝑆𝑆𝑆

N

IO

AT

C LI

The last equation is derived by substitution of 𝑄𝑄𝑄𝑄, 𝑅𝑅𝑅𝑅, 𝑆𝑆𝑆𝑆 with the probe spacing 𝑠𝑠, and it demonstrates the Van der Pauw formula for the special case here considered.

IN

Fig. A1 Schematic of the semi-infinite 2D sheet here considered for the special proof of Van der Pauw equation

CM

JP

Page 36 of 44


Appendix B

PT CE AC

ED

In this appendix, the formulas reported in Tab. 2 for infinite anisotropic solids will be calculated. The case of an in-line arrangement of four probes on an anisotropic semi-infinite 3D half plane is described in Sec. 5.1. Here, we first consider the case of an in-line arrangement of four probes on an anisotropic infinite 2D sheet. We further assume that the resistivity components 𝜌𝜌! , 𝜌𝜌! , 𝜌𝜌! are directed along the 𝑥𝑥, 𝑦𝑦, 𝑧𝑧-axes of the sample. If the 4P probes are aligned along the 𝑥𝑥-axis with a probe distance 𝑠𝑠! , the same probes will have a distance 𝑠𝑠!! = 𝜌𝜌! 𝜌𝜌 𝑠𝑠! on an equivalent isotropic lamella of thickness 𝑡𝑡!! = 𝜌𝜌! 𝜌𝜌 𝑡𝑡! after applying equation 5.2. On this isotropic sheet, 𝑉𝑉! and 𝐼𝐼! are preserved while the eq. 2.8, which is still valid, yields for the singular resistivity 𝜌𝜌: 𝜌𝜌 =

𝜋𝜋𝑡𝑡!! 𝑉𝑉! 𝜋𝜋 𝜌𝜌! 𝜌𝜌 𝑡𝑡! 𝑉𝑉! = 𝐵𝐵. 1 𝑙𝑙𝑙𝑙2 𝐼𝐼! 𝑙𝑙𝑙𝑙2 𝐼𝐼!

R FO

which can be immediately rearranged yielding the following equation for the measured resistance along the 𝑥𝑥axis of original anisotropic sheet: 𝑅𝑅! =

𝑙𝑙𝑙𝑙2 𝜌𝜌 ! 𝑙𝑙𝑛𝑛2 = 𝜌𝜌! 𝜌𝜌! 𝐵𝐵. 2 𝜋𝜋𝑡𝑡! 𝜌𝜌! 𝜋𝜋𝑡𝑡!

C LI

B PU

i.e., the resistance in an anisotropic 2D sheet is still the geometric mean of resistivity components, but now lying on the same plane of the 2D lamella. Although similar, the demonstration of the two remaining equations summarized in Tab. 2 for a square arrangement of the probes is slightly more complex. Taking as reference the schematics of Fig. B1, the 4P square array placed on an anisotropic sample and aligned along the 𝑥𝑥, 𝑦𝑦-axes is transformed into a 4P rectangular array, still aligned along the 𝑥𝑥, 𝑦𝑦-axes of the equivalent isotropic sample of resistivity 𝜌𝜌 and dimensions: 𝑠𝑠!! = 𝑠𝑠!! = 𝑠𝑠 𝜌𝜌! 𝜌𝜌 𝑡𝑡!! =

𝜌𝜌! 𝜌𝜌 + 𝜌𝜌! 𝜌𝜌

𝐵𝐵. 3

AT

𝑠𝑠!! = 𝑠𝑠!! = 𝑠𝑠

𝜌𝜌! 𝜌𝜌 𝑡𝑡!

N

IO

where 𝑠𝑠! = 𝑠𝑠! = 𝑠𝑠 is the side while 𝑠𝑠! = 𝑠𝑠! = 𝑠𝑠 2 is diagonal of the original square array.

IN CM

JP Fig. B1 Mapping of (a) 4P square array placed on an anisotropic sample into (b) a 4P rectangular array placed on an equivalent isotropic specimen.

If we now insert eqs. B.3 into expressions 2.5 for a 3D semi-infinite half plane, which remains valid on the transformed isotropic sample, we obtain: Page 37 of 44

I. Miccoli et al.: The role of probe geometries in isotropic and anisotropic systems 𝐼𝐼𝐼𝐼 2𝜋𝜋

1 1 1 1 ! − ! − ! − ! 𝑠𝑠! 𝑠𝑠! 𝑠𝑠! 𝑠𝑠!

PT CE AC

𝑅𝑅! =

=

𝜌𝜌 2 2𝜋𝜋 s

1

𝜌𝜌! 𝜌𝜌

−

1

𝜌𝜌! 𝜌𝜌 + 𝜌𝜌! 𝜌𝜌

𝜌𝜌! 𝜌𝜌! 1 1− 𝜋𝜋𝜋𝜋 1 + 𝜌𝜌! 𝜌𝜌!

=

𝐵𝐵. 4

Eq. B.4 represents the relation between the measured resistance and the resistivity components of the anisotropic sample for the case of a very large and thick 3D specimen. While inserting eqs. B.3 into expressions 2.7, we finally obtain the resistance expression for the case of square arrange of 4 probes on an anisotropic infinite 2D sheet: 𝑅𝑅! =

ED

𝑠𝑠!! 𝑠𝑠!! 𝐼𝐼𝐼𝐼 𝜌𝜌 𝑙𝑙𝑙𝑙 ! ! = 𝑙𝑙𝑙𝑙 𝑠𝑠! 𝑠𝑠! 2𝜋𝜋𝜋𝜋 2𝜋𝜋 𝜌𝜌! 𝜌𝜌 𝑡𝑡!

𝜌𝜌! 𝜌𝜌 + 𝜌𝜌! 𝜌𝜌 𝜌𝜌! 𝜌𝜌

𝜌𝜌! 𝜌𝜌!

=

2𝜋𝜋𝑡𝑡!

𝑙𝑙𝑙𝑙 1 +

𝜌𝜌! 𝐵𝐵. 5 𝜌𝜌!

Appendix C

R FO

The formulas for infinite anisotropic samples reported in Tab.2 are deduced assuming the principal resistivities 𝜌𝜌! and the 4P probe array aligned along the 𝑥𝑥! -axes, but this condition is not always satisfied. Here we derive the more general equation (5.5) for the special case of a square probe array placed on an anisotropic infinite 2D sheet and rotated of an arbitrary angle 𝜃𝜃 respect to one of two orthogonal resistivity components 𝜌𝜌! , 𝜌𝜌! . If 𝐴𝐴, 𝐵𝐵, 𝐶𝐶, 𝐷𝐷 are the starting probe positions of 4P square array on the anisotropic sample, their position 𝐴𝐴’, 𝐵𝐵’, 𝐶𝐶’, 𝐷𝐷’ after an arbitrary rotation of 𝜃𝜃 degree can be easily evaluated applying the 2D rotation matrix:

B PU

𝑥𝑥 ! cos 𝜃𝜃 − sin 𝜃𝜃 𝑥𝑥 = 𝐶𝐶. 1 𝑦𝑦′ sin 𝜃𝜃 + cos 𝜃𝜃 𝑦𝑦

which gives:

𝐵𝐵 𝑠𝑠, 0 𝐷𝐷 0, 𝑠𝑠

! !"#

! !"#

𝐴𝐴′ 0,0

𝐵𝐵′ 𝑠𝑠 cos 𝜃𝜃 , 𝑠𝑠 sin 𝜃𝜃 𝐶𝐶′ 𝑠𝑠 cos 𝜃𝜃 − 𝑠𝑠 sin 𝜃𝜃 , 𝑠𝑠 sin 𝜃𝜃 + 𝑠𝑠 cos 𝜃𝜃

! !"#

𝐷𝐷′ −𝑠𝑠 sin 𝜃𝜃 , 𝑠𝑠 cos 𝜃𝜃

(𝐶𝐶. 2)

N

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AT

𝐶𝐶 𝑠𝑠, 𝑠𝑠

! !"#

C LI

𝐴𝐴 0,0

IN JP

CM

Fig. C1 (a) Rotation of the 4P square array of an arbitrary angle 𝜃𝜃 respect to the 𝑥𝑥, 𝑦𝑦-axes and (b) successive mapping onto an equivalent isotropic sample.

Next, we map the contacts points of rotated square array on an equivalent isotropic sample via the eq. 5.2: Page 38 of 44


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𝐴𝐴′ 0,0 → 𝐴𝐴′′ 0,0

𝐵𝐵′ 𝑠𝑠 cos 𝜃𝜃 , 𝑠𝑠 sin 𝜃𝜃 → 𝐵𝐵′′ 𝑠𝑠 𝜌𝜌! 𝜌𝜌 cos 𝜃𝜃 , 𝑠𝑠 𝜌𝜌! 𝜌𝜌 sin 𝜃𝜃

𝐶𝐶′ 𝑠𝑠 cos 𝜃𝜃 − 𝑠𝑠 sin 𝜃𝜃 , 𝑠𝑠 sin 𝜃𝜃 + 𝑠𝑠 cos 𝜃𝜃 → 𝐶𝐶′′ 𝑠𝑠 𝜌𝜌! 𝜌𝜌 cos 𝜃𝜃 − sin 𝜃𝜃 , 𝑠𝑠 𝜌𝜌! 𝜌𝜌 sin 𝜃𝜃 + cos 𝜃𝜃

(𝐶𝐶. 3)

𝐷𝐷′ −𝑠𝑠 sin 𝜃𝜃 , 𝑠𝑠 cos 𝜃𝜃 → 𝐷𝐷′ −𝑠𝑠 𝜌𝜌! 𝜌𝜌 sin 𝜃𝜃 , 𝑠𝑠 𝜌𝜌! 𝜌𝜌 cos 𝜃𝜃

As sketched in Fig. C1b, we now calculate the probe spacing 𝑠𝑠′′! , 𝑠𝑠′′! , 𝑠𝑠′′! , 𝑠𝑠′′! of the rotated and transformed 4P array: 𝜌𝜌! sin! 𝜃𝜃 + 𝜌𝜌! cos ! 𝜃𝜃 𝜌𝜌

ED

𝑠𝑠 !!! = 𝐴𝐴!! 𝐷𝐷 !! = 𝑠𝑠

𝑠𝑠 !! !

𝜌𝜌! cos 𝜃𝜃 − sin 𝜃𝜃 𝜌𝜌

R FO

𝑠𝑠 !! ! = 𝐴𝐴!! 𝐶𝐶 !! = 𝑠𝑠

𝜌𝜌! = 𝐵𝐵 !! 𝐷𝐷 !! = 𝑠𝑠 cos 𝜃𝜃 + sin 𝜃𝜃 𝜌𝜌

!

+

𝜌𝜌! cos 𝜃𝜃 + sin 𝜃𝜃 𝜌𝜌

𝜌𝜌 ! + ! cos 𝜃𝜃 − sin 𝜃𝜃 𝜌𝜌

! !

(𝐶𝐶. 4)

𝜌𝜌! sin! 𝜃𝜃 + 𝜌𝜌! cos ! 𝜃𝜃 𝜌𝜌

B PU

𝑠𝑠 !! ! = 𝐵𝐵 !! 𝐶𝐶 !! = 𝑠𝑠

Finally, we insert eqs. C.4 into eq. 2.7, which remains valid on the equivalent isotropic sample where the 4P square array is mapped. After some rearrangements, we get:

𝜌𝜌

2𝜋𝜋𝜋𝜋 𝜌𝜌! 𝜌𝜌

𝜌𝜌! 𝜌𝜌!

𝑙𝑙𝑙𝑙

! !

𝜌𝜌!! + ! cos ! 𝜃𝜃 − sin! 𝜃𝜃 𝜌𝜌

!

! !

⟹

1+

𝜌𝜌!! 𝜌𝜌!!

!

!

⟹

𝜌𝜌!! 4 cos ! 𝜃𝜃 sin! 𝜃𝜃 𝜌𝜌!! 𝐶𝐶. 5 ! 𝜌𝜌! sin! 𝜃𝜃 + cos ! 𝜃𝜃 𝜌𝜌! − 1−

JP

2𝜋𝜋𝜋𝜋

𝜌𝜌! cos ! 𝜃𝜃 − sin! 𝜃𝜃 ! + 2 1 + 4 cos ! 𝜃𝜃 sin! 𝜃𝜃 𝜌𝜌! 𝜌𝜌! sin! 𝜃𝜃 + cos ! 𝜃𝜃 𝜌𝜌!

!

IN

⟹ 𝑅𝑅(𝜃𝜃, 𝜌𝜌! , 𝜌𝜌! ) =

𝑙𝑙𝑙𝑙

𝜌𝜌!! 𝜌𝜌!! 𝜌𝜌!! 𝜌𝜌!! + ! cos 𝜃𝜃 − sin 𝜃𝜃 ! + ! cos 𝜃𝜃 + sin 𝜃𝜃 𝜌𝜌 𝜌𝜌 𝜌𝜌 𝜌𝜌! ! sin! 𝜃𝜃 + cos ! 𝜃𝜃 𝜌𝜌 𝜌𝜌

N

𝜌𝜌! 𝜌𝜌!

2𝜋𝜋𝜋𝜋

𝜌𝜌!! +1 𝜌𝜌!!

!

IO

⟹ 𝑅𝑅(𝜃𝜃) =

𝑙𝑙𝑙𝑙

𝜌𝜌!! cos ! 𝜃𝜃 − sin! 𝜃𝜃 𝜌𝜌 !

AT

=

𝜌𝜌 𝑠𝑠 !! ! 𝑠𝑠 !! ! 𝑙𝑙𝑙𝑙 !! !! = ! 2𝜋𝜋𝑡𝑡 𝑠𝑠 ! 𝑠𝑠 !

C LI

𝑅𝑅(𝜃𝜃, 𝜌𝜌! , 𝜌𝜌! ) =

CM

which coincides with the equation 5.5 and describes how the measured resistance changes as function of 4P square array angular position 𝜃𝜃 and resistivity anisotropy degree of 2D infinite sheet.

Page 39 of 44


Appendix D

PT CE AC

!"#$%&'(" !!

The derivation of the correction factor 𝐹𝐹!!!"#!$% for a rectangular 4P array placed in the center of an isotropic circular lamella is based on the following assumptions: (i) uniform resistivity, (ii) point contacts and (iii) sample thickness much smaller than the probe spacing (thus equivalent to a quasi-2D scenario). Hence, Fig. D1a shows a rectangular 4P array placed in the center of an isotropic circular lamella, where 𝐵𝐵 ad 𝐶𝐶 are the current probes while 𝐴𝐴 and 𝐷𝐷 the voltage probes of the 4P array, both are positioned at a distance 𝑝𝑝 from the center (denoted by 𝑂𝑂). If we neglect for a moment the circular finite boundary, we can easily get by means of eq. 2.7 (with 𝑠𝑠! = 𝑠𝑠! = 𝑏𝑏 and 𝑠𝑠! = 𝑠𝑠! = 𝑎𝑎 ! + 𝑏𝑏 ! ) the voltage drop 𝑉𝑉! for a given current 𝐼𝐼! , which is:

ED

𝑉𝑉! = 𝑉𝑉! − 𝑉𝑉! =

𝐼𝐼! 𝜌𝜌 𝐼𝐼! 𝜌𝜌 𝑠𝑠! 𝑠𝑠! 𝑎𝑎 ! + 𝑏𝑏 ! 𝑙𝑙𝑙𝑙 = 𝑙𝑙𝑙𝑙 (𝐷𝐷. 1) 𝑠𝑠! 𝑠𝑠! 𝑏𝑏 ! 2𝜋𝜋𝜋𝜋 2𝜋𝜋𝜋𝜋

R FO

where 𝑎𝑎 and 𝑏𝑏 are the sides of the rectangle. It is evident that eq. D.1 immediately converges to the expression for a 4P square array when 𝑎𝑎 = 𝑏𝑏 (which is strictly valid only for an infinite or unlimited 2D sheet). According to the method of images, the effect of an isolated circular finite boundary can be mathematically described by adding two current images 𝑃𝑃 and 𝑄𝑄, both placed at a distance 𝑑𝑑 ! 4𝑝𝑝 from the circle center 𝑂𝑂 (where 𝑑𝑑 is the circle radius) and along two straight lines connecting the circle center with the two current probes 𝐶𝐶 and 𝐵𝐵 (see Fig. D1a). It is easy to verify that the adopted configuration of real and image currents compensates the electrostatic potential along the circumference of circular lamella [100] and allows to correctly evaluate the potential inside the circle. For this scenario of 4 current probes, the voltage drop between 𝑉𝑉! (= 𝐷𝐷) and 𝑉𝑉! (= 𝐴𝐴) now reads:

B PU

𝑉𝑉! = 𝑉𝑉! − 𝑉𝑉! =

𝐼𝐼! 𝜌𝜌 𝑎𝑎 ! + 𝑏𝑏 ! 𝑃𝑃𝑃𝑃 ∙ 𝑄𝑄𝑄𝑄 𝑙𝑙𝑙𝑙 + 𝑙𝑙𝑙𝑙 (𝐷𝐷. 2) 𝑏𝑏 ! 2𝜋𝜋𝜋𝜋 𝑃𝑃𝑃𝑃 ∙ 𝑄𝑄𝑄𝑄

N

IO

AT

C LI IN

JP

Fig. D1 (a) Schematics of a 4P rectangular array onto a finite circular slice. The current sources 𝑃𝑃 and Q outside the circle, !"#$%&'(" represent the additional image dipole introduced to mimic the boundary of the lamella. (b) Correction factor 𝐹𝐹!!!"#!$% versus the normalized wafer diameter 𝑑𝑑 𝑎𝑎 (𝑑𝑑 𝑏𝑏), where 𝑎𝑎 (𝑏𝑏) is shorter (longer) edge of the rectangle 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴.

while, Page 40 of 44

𝑃𝑃𝑃𝑃 = 𝑄𝑄𝑄𝑄 = 𝑝𝑝 +

𝑑𝑑 ! = 4𝑝𝑝

𝑎𝑎 ! + 𝑏𝑏 ! 1 +

4

CM

The various line segments in eq. D.2 are obtained by simple geometric considerations (see Fig. D1a):

𝑑𝑑 ! (𝐷𝐷. 3) + 𝑏𝑏 !

𝑎𝑎 !

The 100th anniversary of the four-point probe technique 𝑃𝑃𝑃𝑃 ! + 𝑂𝑂𝑂𝑂 ! − 2𝑃𝑃𝑃𝑃 ∙ 𝑂𝑂𝑂𝑂 cos 𝜋𝜋 − 𝛼𝛼 (𝐷𝐷. 4)

PT CE AC

𝑃𝑃𝑃𝑃 = 𝑄𝑄𝑄𝑄 =

Eq. D.4 can be also expressed as function of side lengths, 𝑎𝑎 and 𝑏𝑏, of the rectangular 4P array by noticing that 𝑐𝑐𝑐𝑐𝑐𝑐 𝜋𝜋 − 𝛼𝛼 = −𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼 = − 2𝑐𝑐𝑐𝑐𝑐𝑐 ! 𝛼𝛼 2 − 1 and 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼 2 = 𝑏𝑏 𝑎𝑎 ! + 𝑏𝑏 ! and we immediately get: 𝑃𝑃𝑃𝑃 = 𝑄𝑄𝑄𝑄 =

𝑎𝑎 ! + 2𝑎𝑎 ! 𝑏𝑏 ! − 𝑑𝑑 ! + 𝑏𝑏 ! + 𝑑𝑑 ! 4 𝑎𝑎 ! + 𝑏𝑏 !

!

(𝐷𝐷. 5)

ED

If we insert eqs. D.3 and D.5 into D.2 and we define the aspect ratio of the rectangle 𝑟𝑟 = 𝑎𝑎/𝑏𝑏, we finally get a compact expression for the voltage drop 𝑉𝑉! between A and D as function of geometric parameters of the rectangular 4P probe array, which writes: 𝑉𝑉! =

𝐼𝐼! 𝑅𝑅!! 1 + 𝑥𝑥 ! + 𝑟𝑟 ! ! 𝑙𝑙𝑙𝑙 1 + 𝑟𝑟 ! + 𝑙𝑙𝑙𝑙 (𝐷𝐷. 6) 2𝜋𝜋 1 + 𝑥𝑥 ! ! + 2 1 − 𝑥𝑥 ! 𝑟𝑟 ! + 𝑟𝑟 ! !"#$%&'(" !!

R FO

with 𝑥𝑥 = 𝑑𝑑/𝑏𝑏. We can thus define the correction factor 𝐹𝐹!!!"#!$% !"#$%&'(" !!

𝐹𝐹!!!"#!

%$𝑑𝑑 ≝ 𝑏𝑏

𝑙𝑙𝑙𝑙 1 + 𝑟𝑟 ! (𝐷𝐷. 7) 1 + 𝑥𝑥 ! + 𝑟𝑟 ! ! + 𝑙𝑙𝑙𝑙 1 + 𝑥𝑥 ! ! + 2 1 − 𝑥𝑥 ! 𝑟𝑟 ! + 𝑟𝑟 !

B PU

Hence, the measured resistance 𝑅𝑅! reads:

𝑙𝑙𝑙𝑙 1 + 𝑟𝑟 !

in the following form:

!"#$%&'(" !!

𝑑𝑑/𝑏𝑏 𝑅𝑅!! 𝐹𝐹!!!"#!$% 𝑅𝑅! = ! 2𝜋𝜋 𝑙𝑙𝑙𝑙 1 + 𝑟𝑟

!!

(𝐷𝐷. 8)

AT

C LI

It is evident that eq. D.8 again reduces to that of a square (see eq. 3.13 above) for 𝑟𝑟 = 1. Eq. 𝐷𝐷. 7 is plotted as a function of the normalized wafer diameter (𝑥𝑥 = 𝑑𝑑 𝑏𝑏) in Fig.D1b, revealing also the effect of the aspect ratio 𝑟𝑟 !"#$%&'(" !! of the rectangle, which becomes obvious. The interval of existence of the function 𝐹𝐹!!!"#!$% shrinks as the rectangle aspect ratio rises up. If we rotate the rectangle of 90° degree (equivalently exchange current and voltage probes), an equation similar to eq. D.8 results: !"#$%&'(" !!

𝑑𝑑 ≝ 𝑎𝑎

!"#$%&'(" !!

𝑙𝑙𝑙𝑙 1 + 𝑟𝑟 !!

𝑙𝑙𝑙𝑙 1 + 𝑟𝑟 !! (𝐷𝐷. 10) 1 + 𝑥𝑥 !" + 𝑟𝑟 !! ! + 𝑙𝑙𝑙𝑙 ! ! ! !" !" 1 + 𝑟𝑟 + 2 1 − 𝑟𝑟 𝑥𝑥 + 𝑥𝑥

!"#$%&'(" !!

IN

!"#$%&'(" !!

𝐹𝐹!!!"#!

%$ (𝐷𝐷. 9)

N

is now a function of 𝑥𝑥′ = d/a:

!!

IO

!"#$%&'(" !!

where 𝐹𝐹!!!"#!"#

𝑑𝑑/𝑎𝑎 𝑅𝑅!! 𝐹𝐹!!!"#!$% 𝑅𝑅! = 2𝜋𝜋 𝑙𝑙𝑙𝑙 1 + 𝑟𝑟 !!

CM

JP

It is easy to proof that 𝐹𝐹!!!"#!$% 𝑑𝑑/𝑎𝑎 = 𝐹𝐹!!!"#!$% 𝐷𝐷/𝑏𝑏 and eqs. D.8 and D.9 differ only for the logarithmic factor (besides the resistance term 𝑅𝑅!,! ). Fig. D1b thus shows the trend of correction factor !"#$%&'(" !! irrespective of the adopted configuration for the current and voltage probes. 𝐹𝐹!!!"#!

%$Page 41 of 44


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