Topic 12: Van der Pauw Method

• 学生：刘雪颖 周璐鹿 程楠
• 指导教师：陈元杰

范德堡方法.pptx

It is known that conductivity of a material can be measured independently of the sample shape, as long as the sample has one border(no holes). To what extent can such a method be applied? Investigate and explain such measurements if the sample has holes.

范德堡方法的证明要用到复变函数（复势平面可以通过保角映射从无穷大映射到任意形状，这个合理性可以通过保角变换下的拉普拉斯方程不变和静电唯一性来证明），目前没有学过，所以可以考虑用COMSOL进行模拟计算，绕过自己计算中遇到的问题。

1.本周要完成的工作： 搭建一套实验设备，验证范德堡法。 所需器材：

（1）、恒定电流源

（2）、电压表：灵敏度较高，能比较准确地测出小电压。

（3）、待测材料：本来考虑用硅片，但是硅片易碎且有氧化层，而且不能用焊锡。 看到有文献中用的是黄铜片，也可以考虑。 接触点的处理有两个方法，一个是用焊锡，一个是用夹子夹。焊锡的接触电阻比较小，但是不能变换位置。 还有一个在之后的探究中也比较重要，就是材料的可加工性。之后需要在材料上打孔。

2.打洞对测量的影响：

范德堡方法的本质是通过电流分布求得电导率。无孔薄片的电流分布较为简单，所以可以由范德堡方程直接算出电导率。打孔之后电流分布更为复杂，难以直接求得。 图为打孔对测量的影响：

由此图可以直观地看出有孔时测量值所满足的“范德堡不等式”，也是我们之后所希望能够得到的数据。

注意事项：
 1、实验前要画好表格，实验中做好记录
 2、由于时间问题，一组同学可能不能同时做实验。做完之后结果一定尽快上传至此页，有新的想法也一定要发到这里，有问题也在这里提，尽量少用微信群，让更多的人看到我们的进展方便讨论。

今天我去实验室试着做了一下范德堡法的实验，实验材料是一片4.8cm*2.2cm的铝箔，四角用焊锡连接了四根导线，用恒流源输出电流，电路接通。但是由于铝箔电阻太小，电压用万用表mV档示数为0.所以用这么小的铝箔做实验精度很难保证。

下一步准备换一块大一些的铝箔试试，同时做一个可以固定探针的装置。希望能够找到电阻率更大一些的金属材料。

用上次的铝箔铝箔做样品发现铝箔样品的电阻率过小，以至于电压表无法测出铝箔对应点之间的电压。听取乐老师的建议，用导电胶布自制样品，将导电胶布横向一层一层地粘到硬纸板上。

由于导电胶布样品缠绕的厚度难以测量，因此查出导电胶布的电阻率也仍然难以得到其面电阻率，然而，可以用另一种方法测量样品的面电阻率。

两次实验测量的实验数据如下

两次测量实验结果偏差很大，主要原因有以下两点，首先，样品制备得不够好，胶带粘贴在纸板上没有使各处保持相同厚度。其次，由于样品电阻太大，导致通过样品的电流很小，为0.0005A，而仪器的最小分度为0.0001A。

由于胶布电阻很大，通过样品的电流很小，使用恒压源比起使用恒流源而言能减小相对误差。

将旧仪器改装成有四个探针的装置。测量过程中发现探针与材料接触点的分压太大，导致材料两端电压无法测出。

具体数据呢？ — 乐永康 2016/05/15 10:46

It’s a technique commonly used to measure the resistivity and Hall coefficient of a sample.

vanderpauw.pdf

what_do_you_measure_when_you_measure_resistivity.pdf

1. The sample must have a flat shape of uniform thickness (approximately two-dimensional).
2. The sample must not have any isolated holes.
3. The sample must be homogenous and isotropic.
4. All four contacts must be located at the edge of the sample.
5. The area of contact of any individual contact should be at least an order of magnitude smaller than the area of the entire sample. (Any errors given by their non-zero size will be of the order D/L, where D is the average diameter of the contact and L is the distance between contacts.)

The proof is consisted of two parts.

1. Relation (1) is developed for a special case: a lamella in the form of an infinite half-plane, provided with four contacts at the periphery.
2. It is then shown that the relation must also apply to a lamella of any shape. This is done by means of conformal mapping of the arbitrarily shaped plate on the infinite half-plane with the aid of complex functions.

1. It obeys inequality similar in the form to the van der pauw equation.
2. It valid for any sample with an isolated hole and contacts located on the same edge.
3. experimental confirmation without rigorous proof

van_der_pauw_method_on_a_sample_with_an_isolated_hole.pdf

1. Wikipedia:Van der Pauw method, https://en.wikipedia.org/wiki/Van_der_Pauw_method
2. A method of measuring specific resistivity and Hall effect of discs of arbitrary shape. Philips Research Rep. 13, 1, 1-9 (1958), http://140.120.11.121/~chia/PDF/van%20der%20pauw.PDF,http://socrates.berkeley.edu/~phylabs/adv/ReprintsPDF/SHE%20Reprints/01-Measuring%20Discs.pdf
3. A method of measuring the resistivity and Hall coefficient on lamellae of arbitrary shape. Philips Techn. Rev. 20, 220–224(1958),http://electron.mit.edu/~gsteele/vanderpauw/vanderpauw.pdf
4. What do you measure when you measure resistivity? Rev.Sci. Instrum. 63, 1, 207-210 (1992)
5. Four-probe methods for measuring the resistivity of samples in the form of rectangular parallelepipeds. Instr. and Exp. Techniques 58, 1, 118-129 (2015), arXiv:1502.02600 [cond-mat.mtrl-sci] Sci. Instrum. 63, 1, 207-210 (1992)
6. On the Van der Pauw method of resistivity measurements. Thin Solid Films 239, 2, 272-275 (1994)

1. Application of the electrostatic Thompson–Lampard theorem to resistivity measurements. Meas. Sci. Technol. 23, 4, 045004 (2012)
2. A derivation of the van der Pauw formula from electrostatics. Solid-State Electronics 52,1,91–98 (2008)

1. Reduction of positional errors in a four-point probe resistance measurement. Appl. Phys. Lett. 84, 1695 (2004)
2. Accurate microfour-point probe sheet resistance measurements on small samples. Rev. Sci. Instrum. 80, 5, 053902 (2009)

1. Van der Pauw method on a sample with an isolated hole. Physics Letters A 377, 651-654 (2013), arXiv:1301.1625 [cond-mat.mes-hall]
2. Determination of the Riemann modulus and sheet resistance of a sample with a hole by the van der Pauw method. Meas. Sci. Technol. 26,5, 055003 (2015), arXiv:1412.0707 [physics.class-ph]

很赞。开工后效率很高啊！ — 乐永康 2016/01/07 13:43

程楠，可以通过尝试来学习网页编辑，但你做过的不需要的改变要回复回去。网页结构要保持。 — 乐永康 2016/05/20 14:51