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测量PV材料电阻率的另外一种技术是范德堡方法。这种方法利用平板四周四个小触点加载电流并测量产生的电压，待测平板可以是厚度均匀任意形状的PV材料样本。

Another technique for determining a PV material’s resistivity is the van der Pauw method. It involves applying a current and measuring the resulting voltage using four small contacts on the circumference of a flat, arbitrarily shaped PV material sample of uniform thickness. The spacing of the four probes is not critical, making this approach useful for small samples where the probes can’t be evenly placed.

范德堡电阻率测量方法需要测量8个电压。测量V₁ 到 V₈是围绕材料样本的四周进行的，如图7所示。

Eight voltage measurements are required in the van der Pauw resistivity measurement method. The measurements, V₁ through V₈, are made around the periphery of the material sample as shown in Fig 7.

图7. 范德堡电阻率常用测量方法

Fig 7. Van der Pauw resistivity conventions.

按照下列公式可以利用上述8个测量结果计算出两个电阻率的值：

Two values of resistivity are then found from the eight measurements, according to

ρ_A = (π/ln2)(f_At_s)[(V₁ – V₂ +V₃ – V₄)/4I]

ρ_B = (π/ln2)(f_Bt_s)[(V₅ – V₆ +V₇ – V₈)/4I]

其中，ρ_A 和 ρ_B分别是两个体积电阻率的值，t_s =样本厚度，单位是cm，V₁ – V₈是测得的电压，单位是V，I=流过光伏材料样品的电流，单位是A，f_A 和 f_B是基于样本对称性的几何系数，它们与两个电阻比值Q_A 和 Q_B相关，如下所示：

where ρ_A and ρ_B = the two values of volume resistivity, t_s = the sample thickness in cm, V₁ – V₈ are the measured voltages in V, I = the current through the photovoltaic material sample in A, and f_A and f_B = the geometric factors based on sample symmetry and are related to the two resistance ratios, Q_A and Q_B, by

Q_A = (V₁ – V₂)/(V₃ – V₄)

Q_B = (V₅ – V₆)/(V₇ – V₈)

当已知ρ_A 和 ρ_B的值时，可以根据下列公式计算出平均电阻率（ρ_AVG）：

Once the values of ρ_A and ρ_B are known, the average resistivity (ρ_AVG) can be found from

ρ_AVG = (ρ_A + ρ_B)/2

高电阻率测量中的误差可能来源于多个方面，包括静电干扰、漏电流、温度和载流子注入。当把某个带电的物理拿到样本附近时就会产生静电干扰。要想最大限度减少这些影响，应该对样本进行适当的屏蔽以避免外部电荷。这种屏蔽可以采用导电材料制作，应该通过将屏蔽层连接到测量仪器的低电势端进行正确的接地。电压测量中还应该使用低噪声屏蔽线缆。漏电流会影响高电阻样本的测量精度。漏电流来源于线缆、探针和测试夹具，通过使用高质量绝缘体，最大限度降低湿度，启用防护式测量，包括使用三轴线缆等方式可以尽量减少漏电流。

Errors in high resistivity measurements can arise from a number of sources, including electrostatic interference, leakage current, temperature, and carrier injection. Electrostatic interference occurs when an electrically charged object is brought near the sample. To minimize these effects, the sample should be properly shielded to avoid external charges. The shield can be made from a conductive material and should be properly grounded by connecting the shield to the low terminal of the measuring instrument. Low-noise shielded cables should also be used in the voltage measurements. Leakage current can degrade the measurement accuracy on high resistance samples. The leakage current is due to the cables, probes, and test fixtures and can be minimized by using high-quality insulators, minimizing humidity, and by enabling guarded measurements, including the use of triax cables.