﻿ 直排四探针法测量硅片电阻率及不确定度评定 Measuring Resistivity of Silicon Wafers with Four-Point Probe Method and Evaluation of Uncertainty in Measurement

Material Sciences
Vol. 09  No. 12 ( 2019 ), Article ID: 33382 , 7 pages
10.12677/MS.2019.912128

Measuring Resistivity of Silicon Wafers with Four-Point Probe Method and Evaluation of Uncertainty in Measurement

Ying Gao, Lanlan Li

National Institute of Metrology, Beijing

Received: Nov. 20th, 2019; accepted: Dec. 3rd, 2019; published: Dec. 10th, 2019

ABSTRACT

Resistivity measuring instruments with four-point probe is widely used in semiconductor industry, and it is a common device for measuring resistivity of semiconductor materials. In this paper, the measurement conditions and applicable calculation formulas for different samples are briefly described, and the uncertainty evaluation of the precise measurement results of wafer is discussed in detail.

Keywords:Four-Point Probe, Resistivity, Uncertainty

1. 经典直排四探针测试仪的基本原理

$\rho =2\pi S\frac{U}{I}$ (1)

Figure 1. Schematic diagram of classical four-point probe instrument

2. 四探针测试仪的实际应用

${\rho }_{23}={\rho }_{T}×{F}_{T}$ (2)

${\rho }_{T}$ 是实际温度T下测得的电阻率，Ω∙cm。

${F}_{T}$ 是温度修正因子。

${F}_{T}=1-{C}_{T}\left(T-23\right)$ (3)

T是被测样品的实际温度，℃。

${F}_{SP}=1+1.082\left(1-{\stackrel{¯}{S}}_{2}/\stackrel{¯}{S}\right)$ (4)

$\stackrel{¯}{S}=1/3\left({\stackrel{¯}{S}}_{1}+{\stackrel{¯}{S}}_{2}+{\stackrel{¯}{S}}_{3}\right)$

$\stackrel{¯}{S}$ 是探针间距的平均值。

${\stackrel{¯}{S}}_{1}$${\stackrel{¯}{S}}_{2}$${\stackrel{¯}{S}}_{3}$ 分别是探针的三个间距的十次测量平均值。

${\rho }_{23}=\frac{\stackrel{¯}{V}}{I}×W×F\left(W/\stackrel{¯}{S}\right)×F\left(\stackrel{¯}{S}/D\right)×{F}_{SP}×{F}_{T}$ (5)

$F\left(W/\stackrel{¯}{S}\right)$ 是厚度修正系数。

$F\left(\stackrel{¯}{S}/D\right)$ 是直径修正系数。

${F}_{SP}$ 是探针修正系数。

${F}_{T}$ 是温度修正系数。

3. 电阻率测量结果的不确定度评定 [4] [5]

1) 测量重复性引入的不确定度分量 ${u}_{A}$

Table 1. Measured resistivity values data

${u}_{A}=u\left(\stackrel{¯}{\rho }\right)=\frac{u\left(\rho \right)}{\sqrt{10}}=\frac{1}{\sqrt{\left(10-1\right)×10}}×\sqrt{{\sum }_{k=1}^{10}{\left({\rho }_{k}-\stackrel{¯}{\rho }\right)}^{2}}=0.000005\text{\hspace{0.17em}}\Omega \cdot \text{cm}$

${u}_{A}=\frac{0.000005}{0.009462}×100=0.053%$

2) B类不确定度分量 ${u}_{B}$

a) 电子设备引入的分量 ${u}_{B1},{u}_{B2}$

Table 2. Nano voltmeter calibration data

${u}_{B1}=\frac{0.00007}{2×5.00004}×100=0.0007%$

Table 3. Constant current source calibration data

${u}_{B2}=\frac{0.02}{2}=0.01%$

b) 温度变化带来的不确定度分量 ${u}_{B3}$

${u}_{B3}=0.0035%$

c) 测厚仪传递的不确定度分量 ${u}_{B4}$

$T=\stackrel{¯}{x}+\Delta {t}_{S}$

$\stackrel{¯}{x}$ 为测厚仪的读数，即测量结果。 $\Delta {t}_{S}$ 为测厚仪的误差对测量结果的影响，其数学期望值为零，但需考虑其不确定度。

629.1 μm 629.0 μm 628.9 μm

${u}_{\text{1}}\left(t\right)=u\left(\stackrel{¯}{x}\right)=s\left(\stackrel{¯}{x}\right)=0.04\text{\hspace{0.17em}}\mu \text{m}$

${u}_{2}\left(t\right)=\frac{0.6}{\sqrt{3}}=0.346\text{\hspace{0.17em}}\mu \text{m}$

d) 探针头引入的不确定度分量 ${u}_{B5}$

$\begin{array}{c}{u}_{C}=\sqrt{{u}_{A}^{2}+{u}_{B1}^{2}+{u}_{B2}^{2}+{u}_{B3}^{2}+{u}_{B4}^{2}+{u}_{B5}^{2}}\\ =\sqrt{{0.053}^{2}+{0.0007}^{2}+{0.01}^{2}+{0.0035}^{2}+{0.055}^{2}+{0.076}^{2}}\\ =0.11%\end{array}$

$U=k{u}_{C}=0.22%$

$\begin{array}{c}{u}_{B}=\sqrt{{u}_{B1}^{2}+{u}_{B2}^{2}+{u}_{B3}^{2}+{u}_{B4}^{2}+{u}_{B5}^{2}}\\ =\sqrt{{0.00065}^{2}+{0.01}^{2}+{0.166}^{2}+{0.055}^{2}+{0.076}^{2}}\\ =0.19%\end{array}$

$\begin{array}{c}{u}_{C\mathrm{max}}=\sqrt{{u}_{A}^{2}+{u}_{B1}^{2}+{u}_{B2}^{2}+{u}_{B3}^{2}+{u}_{B4}^{2}+{u}_{B5}^{2}}\\ =\sqrt{{0.2}^{2}+{0.00065}^{2}+{0.01}^{2}+{0.166}^{2}+{0.055}^{2}+{0.076}^{2}}\\ =0.28%\end{array}$

4. 结论

Measuring Resistivity of Silicon Wafers with Four-Point Probe Method and Evaluation of Uncertainty in Measurement[J]. 材料科学, 2019, 09(12): 1041-1047. https://doi.org/10.12677/MS.2019.912128

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