van der Pauw measurements with a parameter analyzer are examined followed by a look at Hall effects measurements.

BY MARY ANNE TUPTA, Keithley Instruments Product Line at Tektronix, Cleveland, OH

Semiconductor material research and device testing often involves determining the resistivity and Hall mobility of a sample. The resistivity of a particular semiconductor material primarily depends on the bulk doping used. In a device, the resistivity can affect the capacitance, the series resistance, and the threshold voltage, so it’s important to perform this measurement carefully and accurately.

The resistivity of the semiconductor material is often determined using a four-point probe or Kelvin technique where two of the probes are used to source current and the other two probes are used to measure voltage. Using four probes eliminates measurement errors due to probe resistance, spreading resistance under each probe, and contact resis- tance between each metal probe and the semiconductor material. Because a high impedance voltmeter draws little current, the voltage drops are very small.

One useful Kelvin technique for determining the resistivity of a semiconductor material is the van der Pauw (vdp) method using a parameter analyzer with high input impedance and accurate low current sourcing. This article first looks at van der Pauw measurements with a parameter analyzer followed by a look at Hall effects measurements.

van der Pauw resistivity measurements

The van der Pauw method involves applying a current and measuring voltage using four small contacts on the circumference of a flat, arbitrarily shaped sample of uniform thickness. This method is particularly useful for measuring very small samples because geometric spacing of the contacts is unimportant, meaning that effects due to a sample’s size are irrelevant.

Using this method, the resistivity is derived from a total of eight measurements that are made around the periphery of the sample using the configurations shown in FIGURE 1.

FIGURE 1. van der Pauw resistivity conventions.

Once all the voltage measurements are taken, two values of resistivity, ρA and ρB, are derived as follows:

where: ρA and ρB are volume resistivities in ohm-cm

t_s is the sample thickness in cm

V₁–V₈ represents the voltages measured by the voltmeter

I is the current through the sample in amperes

f_A and f_B are geometrical factors based on sample symmetry. They are related to the two resistance ratios Q_A and Q_B as shown in the following equations (f_A = f_B = 1 for perfect symmetry).

Q_A and Q_B are calculated using the measured voltages as follows:

Also, Q and f are related as follows:

A plot of this function is shown in FIGURE 2. The value of f can be found from this plot once Q has been calculated.

FIGURE 2. Plot of f vs. Q.

Once ρ_A and ρ_B are known, the average resistivity (ρ_AVG) can be determined as follows:

The electrical measurements for determining van der Pauw resistivity require a current source and a voltmeter. To automate measurements, it’s possible to use a programmable switch to switch the current source and the voltmeter to all sides of the sample. However, a parameter analyzer offers greater efficiency.

A parameter analyzer with four source measure units (SMU) and four preamps (for high resistance measurements) is well-suited for performing van der Pauw resis- tivity measurements, and enables measurements of resistances greater than 1012Ω. A key advantage is that each SMU instrument can be configured as a current source or as a voltmeter with no external switching required. This eliminates leakage and offsets errors caused by mechanical switches as well as the need for additional instruments and programming.

For high resistance materials, a current source that can output very small current with a high output impedance is necessary. A differential electrometer with high input impedance is required to minimize loading effects on the sample.

Each terminal of the sample is connected to one SMU instrument, so a parameter analyzer with four SMU instruments is required. A diagram of how the four SMUs are configured for each of the tests is shown in FIGURE 3. For each test, three of the SMU instruments are configured as a current bias and a voltmeter. One of the SMUs applies the test current and the other two SMUs are used as high impedance voltmeters with a test current of zero amps on a low current range (typically 1nA range). The fourth SMU instrument is set to common. The voltage difference is calculated between the two SMU instruments set up as high impedance voltmeters. This measurement setup is duplicated around the sample, with each of the four SMU instruments changing functions in each of the four tests. The test current and voltage differences between the terminals from the four tests are used to calculate resistivity.

FIGURE 3. SMU Instrument Configurations for van der Pauw Measurements.

For high resistance samples, it’s necessary to determine the settling time of the measurement. This is done by sourcing current into two terminals of the sample and measuring the voltage difference between the other two terminals. The settling time can be determined by graphing the voltage difference versus the time of the measurement. A timing graph of a very high resistance material is shown in FIGURE 4. Note that settling time needs to be determined every time for different materials; however, it’s not necessary for low resistance materials since they have a short settling time.

FIGURE 4. Voltage vs. time graph of a very high resistance sample.

Hall voltage measurements

Hall effect measurements are important to semiconductor material characterization because from the Hall voltage, the conductivity type, carrier density, and mobility can be derived. With an applied magnetic field, the Hall voltage can be measured using the configurations shown in FIGURE 5.

FIGURE 5. Hall voltage measurement configurations.

With a positive magnetic field, B, current is applied between terminals 1 and 3, and the voltage drop (V_2–4+) is measured between terminals 2 and 4. When the current is reversed, the voltage drop (V_4–2+) is measured. Next, current is applied between terminals 2 and 4, and the voltage drop (V_1–3+) between terminals 1 and 3 is measured. Then the current is reversed and the voltage (V_3–1+) is measured again.

Then the magnetic field, B, is reversed and the procedure is repeated again, measuring the four voltages: (V_2–4–), (V_4–2–), (V_1–3–), and (V_3–1–).

From the eight Hall voltage measurements, the average Hall coefficient can be calculated as follows:

where: R_HC and R_HD are Hall coefficients in cm³/C

t_s is the sample thickness in cm

V represents the voltages measured by the voltmeter

I is the current through the sample in amperes

B is the magnetic flux in Vs/cm² (1 Vs/cm² = 10⁸ gauss)

Once R_HC and R_HD have been calculated, the average Hall coefficient (R_HAVG) can be determined as follows:

From the resistivity (ρAVG) and the Hall coefficient (R_HAVG), the mobility (μH) can be calculated:

For successful resistivity measurements, potential sources of errors need to be considered. Here are the errors sources you are most likely to encounter.

Electrostatic Interference — Electrostatic interference occurs when an electrically charged object is brought near an uncharged object. Usually, the effects of the interference are not noticeable because the charge dissi- pates rapidly at low resistance levels. However, high resis- tance materials do not allow the charge to decay quickly and unstable measurements may result. The erroneous readings may be due to either DC or AC electrostatic fields.

To minimize the effects of these fields, an electrostatic shield can be built to enclose the sensitive circuitry. The shield should be made from a conductive material and connected to the low impedance (FORCE LO) terminal of the test instrument. The cabling in the circuit must also be shielded.

Leakage Current — For high resistance samples, leakage current may degrade measurements. The leakage current is due to the insulation resistance of the cables, probes, and test fixturing.

Leakage current may be minimized by using good quality insulators, by reducing humidity, and by using guarding.

A guard is a conductor connected to a low impedance point in the circuit that is nearly at the same potential as the high impedance lead being guarded. Using triax cabling and fixturing will ensure that the high impedance terminal of the sample is guarded. The guard connection will also reduce measurement time since the cable capacitance will no longer affect the time constant of the measurement.

Light — Currents generated by photoconductive effects can degrade measurements, especially on high resistance samples. To prevent this, the sample should be placed in a dark chamber.

Temperature — Thermoelectric voltages may also affect measurement accuracy. Temperature gradients may result if the sample temperature is not uniform. Thermoelectric voltages may also be generated from sample heating caused by the source current. Heating from the source current will more likely affect low resistance samples, because a higher test current is needed to make the voltage measure- ments easier. Temperature fluctuations in the laboratory environment may also affect measurements. Because semiconductors have a relatively large temperature coeffi- cient, temperature variations in the laboratory may need to be compensated for by using correction factors.

Carrier Injection — To prevent minority/majority carrier injection from influencing resistivity measurements, the voltage difference between the two voltage sensing terminals should be kept at less than 100mV, ideally 25mV, since the thermal voltage, kt/q, is approximately 26mV. The test current should be kept as low as possible without affecting the measurement precision.

Conclusion

The van der Pauw technique in conjunction with a parameter analyzer is a proven method for determining the resistivity of very small samples because geometric spacing of the contacts is unimportant. Hall effect measurements are important to semiconductor material characterization for determining conductivity type, carrier density, and mobility. Some parameter analyzers may include built-in configurable tests that include the necessary calculations.

For successful measurements, it’s important to consider potential sources of error including electronics interference, leakage current and environmental factor such as light and temperature. Resistivity can impact the characteristics of a device, serving as reminder of the importance of making accurate and repeatable measurements.