Pointwise error estimation and high-precision resistivity
measurement with four-probe method
Xuefeng LIU, Taiga NAKANO (Niigata University)
Abstract
To measure the resistivity of semiconducor materials is one of the fundamental
problems in the semiconductor industry. In this research, we study the mathematicalmodels of resistivity measurement method and propose new computation algorithms
along quantitative error estimation to calculate several fundamental quantities required by the resistivity measurement. Particularly, as a general mathematical theory
on the pointwise error estimation for the finite element approximate solution to partial differential equation, we have developed a new theory to provide an optimal andexplicit error estimation.

1Introduction

The four-point probe method is widely used in the resistivity measurement. The principleof the four-point probe method is illustrated as in Figure 1, where the equidistant four
probes A, B, C and D are aligned on the surface of a semiconductor object and a constantcurrent IAD is imposed on (A, D) pair and the potential difference VBC between (B, C)
pair is measured.

IADVVBCA B C DFigure 1: Four-point probe method for semiconductor resistivity measurement
Suppose the semiconductor object has a uniform distribution of resistivity ρ. The
resistivity of the semiconductor object is expected to be calculated by using VBC and IAD ,ρ = Fc ·VBC,
IAD

where Fc is called by “correction factor”. The factor Fc dependends on the geometricshape of the object, the position of the probes and the layout of probes A, B, C and D.
The determination of Fc is reduced to solving the equation that describes the potentialdistribution.
1Assume that the probes contact with the surface of material at points with zero contact
area. Alghough such an assumption is not available in practical measurement, it works asa reasonable model to solve practical problems. The equation of potential Φ is given by∆Φ(x) = 2ρ IAD (δ(x − D) − δ(x − A)) in Ω;∂Φ= 0 on ∂Ω∂n(1)where δ is the Dirac delta function in R3 . By taking ρ IAD = 1, the concrete value of Fc
can be obtained as follows:1
Fc =.
Φ(B) − Φ(C)For object of regular shapes, the equation is solved with explicit formula and the results
are accepted by the JIS standard; see the discussion in , e.g., [1].

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Problems of current resistivity measurement method

The model in (1) is developed under ideal conditions. In practical resistivity measurement,
there are several factors that affect the precision of the measurement results.
• Irregular shapes of the wafers The wafers in general processing of semiconductor
industry usually have a cut edge or notch to mark the orientation; see Figure 2. Since
existing methods regard the wafer as a disk, the measurement error around the cutedge or notch is unignorable. To have a trustable measurement, one has to solve
the equation (1) over an iiregular shape by numerical scheme, for example, the finiteelement method.

Figure 2: Wafers with cut edge (left) and notch (right).

• The contact condtion of the probes Although the end of the proble is very sharp,
the contact area between the probe and the material is not ignorable (See Figure3) and the affection of the non-zero contact area should be properly estimated forthe purpose of high-precision mesaurement. To deal with the non-zero contact area
problem, one has to set up new mathemtatical model upon the contact condition.
For example, a reasonable assumption is that there is a equipotential on the contact
area of each proble, which leads to the Dirichlet boundary condition of the governing
equation.
• Non-uniforma distribution of resistivity Generally, the distribution of resistivity of semiconductor wafers is not uniform. Current method first measures the2Figure 3: Trace of probes

resistivity at one point under the assumption of a uniform distribition of resistivity,
then estimates the quality of wafers by analyzing the variation of measured valuesof resistivity. To give more reasonable model for the practical measurement, one
has to consider the non-uniform distribution of resistivity. For this purpose, theoptimiazation model will be needed to find the distribution of resistivity, along withthe usage of more measurements results on the whole surface of the material.

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Challenges to numerical analysis

By applying numerical schemes, one can easily obtain approximate solution to the model
problem (1) or updated models with equipotential on the contact area. In this case, theerror estimation for the numerical approximate solution is of great importance. Mostclassical numerical analysis only provides qualitical results such like convergence order,
while a concrete value of the error is usually not available.
In our research, we developed new quantitative error estimation to provide explicit
and sharp error estimation for the numerical results. In [3], Liu extended the idea ofKikuchi about the a posteriori error estimation [2] to develop a quantitative a priori error
estimation for the boundary value problem of Poisson’s equation. In [4], rather than aglobal error estimation of the numerical solutions, we focus on the local sub-domain aroundthe probes and have developed the local error estimation thoery to provide explicit local
error estimations under the energy norm.
Notice that the evaluaton of Fc requires the pointwise values of solution on the probe
B and C. However, to have an explicit error estimation of the pointwise vaule of solution
to partial differential equation is not an easy task. In our research, we consider the explicit
estimation of ponitwise value of solutions by using the hypercircle method. Although theL∞ norm estimation can produce uniform bound over the whole domain, the O(h2 |log(h)|)
convergence rate of L∞ is an overestimated result for solution value at interior point ofthe domain.
In 1950s, T. Kato considered a kind of boundary value problem associated with a selfadjoint operator H of Hilbert spaces defined in the form H = T ∗ T , where T ∗ denotes theself-adjoint operator of T [5]. Later, H. Fujita applied the theory of Kato to develop ahypercircle-like method for error estimation and further applied this method to provide
pointwise error estimation for the boundary value problems [6]. The method proposedin [6] only consider the model problem over a square domain along with the special basefunctions.

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In our research, we studied the two fundamental papers [5,6], and investigated thepossibility to apply Kato-Fujita’s theory to develop pointwise error estimation for FEM
solutions to boundary value problems. It is shown that, for linear conforming FEM, one
can obtain explicit lower and upper bound for the solution value at a specified point insidethe domain with O(h2 ) convergence rate.
In the presentation of this RIMS workshop, more details will be provided on the newly
developed local error estimation and the point-wise error estimation.
Acknowledgement:This reasearch is a joint work with NAPSON Corporation.

References[1] Masato Yamashita and Masahiro Agu, Geometrical Correction Factor for Semiconductor Resistivity Measurements by Four-Point Probe Method, Japanese Journal ofApplied Physics, 23(11) (1984), pp.1499–1504.
[2] Fumio Kikuchi and Hironobu Saito, Remarks on a posteriori error estimation for finiteelement solutions, J. Comput. Appl. Math., pp.329–336, vol. 199, 2007.
[3] Xuefeng Liu and Shin’ichi Oishi. Verified eigenvalue evaluation for the Laplacian overpolygonal domains of arbitrary shape, SIAM J. Numer. Anal. 51 (3):1634–1654, 2013.
[4] Taiga Nakano and Xuefeng Liu, Explicit a posteriori Local Error Estimation for Finite Element Solutions, Transactions of the Japan Society for Industrial and AppliedMathematics, Vol.29, No. 4, 2019, pp. 362–382.
[5] Tosio Kato. On some approximate methods concerning the operators T ∗ T , Mathematische Annalen, 126(1):253–262, 1953.
[6] Hiroshi Fujita. Contribution to the theory of upper and lower bounds in boundary
value problems. Journal of the Physical Society of Japan, 10(1):1–8, 1955.

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