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Introduction
When measuring resistance-related parameters, engineers may have heard of the Kelvin 4-wire method for resistance measurement, the 4-probe method for resistivity measurement, the van der Pauw method, the Chen Cunli method, and so on. With so many methods to choose from, engineers often find it difficult to quickly select the correct test method.
These measurement methods may appear vastly different, yet their results seem "similar"—but this is actually a complete misunderstanding. In reality, the principles behind these methods are quite similar, while their measurement results and purposes each have different emphases.
In terms of scope of application, the Kelvin 4-wire method typically targets resistors and other discrete devices, while the 4-probe method, van der Pauw method, and Chen Cunli method target materials or sheet resistance.
In terms of the results obtained, the Kelvin 4-wire method yields a resistance value, the 4-probe method yields volume resistivity, the van der Pauw method yields average resistivity, and the Chen Cunli method yields specific contact resistivity. These parameters all seem related to resistance, but in fact they are used to measure and characterize capabilities in different directions.
Resistance value uses resistance to characterize the ability of a device, a segment of material, or another object to conduct current. Volume resistivity is typically used in semiconductor materials to measure the doping concentration and conductive properties of the material. Average resistivity, in the van der Pauw method, builds upon the four-probe method by taking into account variables such as the material's electron mobility and the material's own shape, thereby more accurately measuring indicators such as the doping concentration of a material or object. Specific contact resistance is an important means of studying the doping of semiconductor materials or thin films and judging ohmic contacts. It is also related to the effective carrier doping concentration of the material and can be well used to evaluate the ohmic contact between metal and semiconductor.
This article mainly analyzes the specific differences among these methods and the details engineers need to pay attention to during measurement.
The Kelvin 4-wire method is the method we use when we want to accurately measure the resistance value of an object while eliminating the error caused by the "loop body resistance" generated by the current flowing through the lines. A typical application scenario is when engineers measure very small resistances or perform high-current tests.
When measuring very small resistances, since the resistance of the device under test (DUT) itself is extremely small, a sufficiently high-precision source and meter or SMU is first required. Second, even when the SMU has sufficient precision, because the DUT resistance is small enough, the SMU output through cables and fixtures will produce a relatively large Rload, and the contact between the fixture and the DUT will also produce a relatively large Rcontact. The magnitudes of these two depend on their own size, shape, material, contact area, etc. During testing, the sum of the two may reach the level of hundreds of milliohms of interference, often larger than the DUT resistance—and sometimes far larger. Under such circumstances, it is naturally difficult to measure the resistance accurately.
In high-current testing, even if the DUT resistance itself is not small, it is still not large enough to ignore the error that may be caused by the large voltage drop of tens or hundreds of volts generated when a large current I of tens of amperes or even thousands of amperes flows through Rload and Rcontact. Naturally, measures are also needed to suppress the error caused by Rload and Rcontact.
The Kelvin 4-wire method for resistance measurement successfully minimizes the impact of Rload and Rcontact on the measurement. The so-called four-wire method separates the two wires originally shared by the source and the meter. The source output uses two wires (FORCE) placed at both ends of the DUT, and the meter measurement also uses two independent wires (SENSE) connected between the FORCE wires and the DUT terminals, closer to the DUT. With this method, FORCE is used to apply a constant current to the DUT, while SENSE is used only to measure the voltage across the DUT, thereby avoiding the influence of Rload and Rcontact on the FORCE lines that would affect correct results.
Since Rload and Rcontact are outside the SENSE path, and SENSE measures only the voltage across the DUT, the measurement is naturally more accurate.
Some engineers may wonder: doesn't SENSE also have Rload and Rcontact? Because SENSE has a very high input impedance, only a weak leakage current at the nA or pA level from the large current applied by FORCE flows through SENSE. Therefore, the impact of these two resistances in the SENSE path on the result is negligible.
The four-point collinear probe method, also known as the 4-probe method, is used when engineers need to evaluate the doping concentration and conductive properties of a known material; the four-point collinear method is then used to measure the volume resistivity of the material.
The resistivity of semiconductor materials mainly depends on bulk doping. In devices, resistivity affects capacitance, series resistance, and threshold voltage. Therefore, resistivity measurement is an important indicator for evaluating these.
When engineers use the four-point collinear probe method, they should note that the accuracy of the results is strongly correlated with the area and shape of the material under test as well as the arrangement of the probes.
During measurement, it is best that the sample material be flat, with a regular shape such as circular or rectangular, free of holes and scratches, and that the four probes be located on the centerline of the sample and arranged at equal spacing.
The SMU outputs a constant current to the two outer probes, and another SMU connected to the two inner probes measures the voltage drop across the sample surface. The resistivity is calculated according to the following formula:
Where:ρ = volume resistivity (Ω-cm)
V = measured voltage (V)
I = source current (A)
t = sample thickness (cm)
k* = correction factor based on the probe diameter ratio and the ratio of wafer thickness to probe separation.
This method is similar to using the Kelvin method to measure the voltage drop between two points on the sample surface to obtain the surface resistivity, and then multiplying the surface resistivity by the sample thickness to obtain the volume resistivity.
The van der Pauw method (VDP method) is suitable for flat samples of uniform thickness with four terminals. Probes only need to be placed at contact points on the four ends of the material to measure the volume average resistivity of the sample. This method is very useful for measuring very small samples because the geometric spacing of the contact points is not important, and the probes do not need to be arranged at strictly equal spacing.
The principle of the van der Pauw method is to apply a constant current to two contact points on the sample and measure the voltage drop between the other two contact points. After measuring and recording this voltage drop, the two points to which the constant current is applied are changed, and measurement and recording continue in this way until eight different voltages have been measured and recorded. The voltages are then substituted into the formulas to obtain rA and rB, that is, two different sets of volume resistivity.
Where: rA and rB are volume resistivities, in ohm-cm
t is the sample thickness, in cm
V1–V8 are the voltages measured by the voltmeter
I is the current flowing through the sample, in A
fA and fB are geometric factors for the symmetry of the sample, related to the two resistance ratios QA and QB, according to the following formulas (for perfect symmetry, fA = fB = 1).
QA and QB are calculated using the measured voltages, according to the following formulas:
After determining rA and rB, the average resistivity (rAVG) can be obtained according to the following formula:
Contact resistivity is often used to evaluate the quality of metal-semiconductor ohmic contacts. Contact performance directly affects device quality, material performance, and physics research. Therefore, its impact is far-reaching in both semiconductor manufacturing and research on material and physical properties. The Chen Cunli method is a simple model for measuring contact resistivity.
The Chen Cunli method uses a simple four-point circular transmission line model. It only requires fabricating the contact pattern and can measure contact resistivity without the traditional mesa insulation. Its sample preparation is simple, and its calculation is also more convenient and simpler than the multi-ring structure prototype transmission line model proposed by Reeves.
As shown in the figure above, a1, a2, a3, and a4 are the radii of the disks where four metal circular electrodes 1, 2, 3, and 4 of equal diameter and consistent spacing contact the sample. The spacings between them are S12, S13, S14, S23, S24, and S34, respectively. Current i enters the sample from 1 and flows out from 2, denoted as i12. When the semiconductor has a finite thickness, the resistances R12 and R34 are expressed as:
This method only requires s≥2a and a/w≤2.
In application, to ensure that the current flows in the direction specified by the method—from point a to point b—apply a test current at point a, and then set point b to the lowest potential or ground.
The procedures and principles of the four methods all lead to the same goal by different routes, but when classified by application scenario, engineers can quickly find the method they need.
For example, to measure a resistance value, whether of a material, a wafer, or a device, the Kelvin method can be used directly. To measure resistivity, the choice can be made according to the characteristics of the sample under test. If the sample is very small, flat, and regular, and it is very convenient to arrange probes at equal spacing along the centerline of the sample, use the four-probe method. If the sample is too small for the four-probe array to be placed at the center, the VDP method can be used. If the goal is to evaluate the ohmic contact performance of a material by measuring specific contact resistance, use the Chen Cunli method.
In actual measurement, when performing such highly precise measurements, we often use instruments such as the IWATSU CS-8000 semiconductor parameter curve tracer or the Keithley 4200 semiconductor parameter analyzer together with professional probe stations such as APW and Cascade, or, for devices or larger DUTs, use specially designed shielded safety test enclosures for measurement.
During testing, in addition to details such as selecting the appropriate range and the wiring and probing, attention should also be paid to the use of the SMU. For example, to pass a constant current between two terminals, flowing in from terminal a and out from terminal b, add an SMU at terminal a. The SMU's SOURCE needs to use CURRENT mode (CURRENT Bias), while terminal b is set to GND/Common, ensuring that the current flows from terminal a into terminal b.
To measure the potential difference between the other two points, two SMUs are needed, or one SMU can measure twice; but measuring twice means probing again, which also introduces many effects. Ideally, three SMUs and one GND are needed to measure all results relatively accurately in a single pass.
References:
van der Pauw method, A method of measuring the specific resistivity and Hall effect of disks of arbitrary shape using the L.J.A. method——Philips Rec.Repts.,1958:13 1.Schroder, Dieter K。
Semiconductor Material and Device Characterization——John Wiley & Sons, Inc., Third Edition, 2015。
Contact Resistivity of Metal-Semiconductor Ohmic Contacts——Chen Cunli. Chinese Journal of Semiconductors, 1983, Vol. 4, No. 2。
