# Four-point turn into accurate measurement

Resistivity and Hall voltage measurements are key to determining the effectiveness of doping and thermal activation processes in semiconductor production. Keithley describes some of the factors that go in to making precise four-point measurements.

Semiconductor material research and device testing often involve determining the resistivity and Hall mobility of a sample. The resistivity of a semiconductor material is primarily dependent on the bulk doping. In a device, the resistivity can affect capacitance, series resistance and threshold voltages.

Four-point resistivity measurements

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

The four-point collinear probe is the most common way of measuring resistivity in semiconductor materials. This involves bringing four equally spaced probes in contact with the material. The probe array is placed in the centre of the material (Figure 1). The two outer probes are used for sourcing current and the two inner probes are used for measuring the resulting voltage drop across the surface of the sample. The volume resistivity is calculated as shown in Equation 1.

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 the geometric spacing of the contacts is unimportant. Effects due to a sample's size, which is the approximate probe spacing, are irrelevant. Using this method, the resistivity can be derived from a total of eight measurements that are made around the periphery of the sample with the configurations shown in Figure 4. Once all the voltage measurements are taken, two values of resistivity, rA and rB, are derived as in Equation 2. The symmetry factors, FA and FB, are related to the two resistance ratios QA and QB (Equation 3). For perfect symmetry, FA = FB = 1. A plot showing the relation between Q and F is shown in Figure 5. Once rA and rB are known, the average resistivity (rAVG) can be determined (Equation 4).

Test equipment

Keithley's Model 4200-SCS semiconductor characterisation system performs lab grade DC device characterisation, real-time plotting and analysis with high precision and sub-femtoamp resolution. The system includes a complete, embedded PC with Windows operating system and mass storage. A self-documenting, point-and-click interface simplifies the process of taking data, so users can begin analysing their results sooner. The Model 4200-SCS has a high input impedance (>1016½) and accurate low current sourcing, making it useful for both four-point methods on high resistance samples. It has four source-measure units (SMUs) and four preamps.

Collinear probe measurements can be made using either three or four SMUs. When using three SMUs, all three SMUs are set to current bias (voltmeter unit). However, one SMU will source current and the other two will be used to measure the voltage difference between the two inner probes. An example of how this can be set up with the Model 4200-SCS is shown in Figure 2. One SMU (SMU1) and the GNDU (ground unit) are used to source current between the outer two probes. Two other SMUs (SMU2 and SMU3) are used to measure the voltage drop between the two inner probes.

The electrical measurements for determining Van der Pauw resistivity require a current source and a voltmeter (Figure 4). To automate measurements, one might typically use a programmable switch to change the current source and the voltmeter to all sides of the sample. In principle, the Model 4200-SCS in a Van der Pauw set-up should be able to measure resistances greater than 1012½. Since each SMU can be configured as a current source or as a voltmeter, no external switching is required, thus eliminating leakage and offsets errors caused by mechanical switches. This removes the need for additional instruments and programming.

For high resistance materials, a current source that can output a very small current with high output impedance is necessary. This can be provided by a differential electrometer, minimising loading effects on the sample. On the lowest current source ranges (1pA and 10pA) of the Model 4200-SCS, the input resistance of the voltmeter is more than 1016½.

Hall voltage

Hall effect measurements are important to semiconductor material characterisation because it provides majority carrier type, carrier density and mobility. These measurements require an applied magnetic field (Figure 10).

With a positive magnetic field, B, a current is first applied between terminals 1 and 3 and the voltage drop (V2-4+) between terminals 2 and 4 measured. Then the current is reversed and the voltage drop (V4-2+) measured again. Then the current is applied between terminals 2 and 4 with the voltage drop (V1-3+) between terminals 1 and 3 measured. Finally, the current is reversed to getV3-1+. The procedure is then repeated with magnetic field reversed. From these eight measurements, the average Hall coefficient is calculated (Equations 5 and 6) giving the mobility (Equation 7).

Sources of error

For successful resistivity measurements, the potential sources of errors need to be considered from electrostatic interference, leakage current, light, temperature and carrier injection.

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 dissipates rapidly at low resistance levels. However, high resistance 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 minimise the effects of these fields, an electrostatic shield can be built to enclose the sensitive circuitry. The shield is made from a conductive material and is always connected to the low impedance terminal of the SMU. The cabling in the circuit must also be shielded.

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

To prevent currents generated by photoconductive, especially on high resistance samples, the sample should be placed in a dark chamber.

Gradients may result if the sample temperature is not uniform creating thermoelectric voltages. These 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, since a higher test current is needed to make voltage measurements easier. Temperature fluctuations in the laboratory environment may also affect measurements. Since semiconductors have a relatively large temperature coefficient, temperature variations in the laboratory may need to be compensated for by using correction factors.

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.

For high resistance samples, it is necessary to determine the settling time of the measurement. This can be accomplished 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. Settling time generally needs to be determined separately for different materials, but is not necessary for low resistance materials since they have a short decay.

Fig.1: Four-point collinear probe resistivity measurement. |

Fig.2: Collinear resistivity measurement on the Keithley Model 4200-SCS. |

Fig.4: Van der Pauw resistivity conventions. |

Fig.10: Hall voltage measurement configurations. |

Fig.5: Plot of F vs. Q from Equation 3. |

Eq.1: Resistivity formula for collinear method. The correction factors can be found in standard four-point probe resistivity test procedures such as SEMI MF84-02. |

Equ.2: Resistivity formula for the Van der Pauw method. FA and FB are geometrical factors based on sample symmetry. |

Equ.3: Calculation of Van der Pauw symmetry factors. |

Equ.4: Average resistivity. |

Equ.5: Hall measurement formulas. |

Equ.6: Hall coefficient average. |

Equ.7: Mobility. |

References:

ASTM, F76-86. Standard Method for Measuring Hall Mobility and Hall Coefficient in Extrinsic Semiconductor Single Crystals. Annual Bk. ASTM Stds., 1999: 10.05.

SEMI MF84-02: Test Method for Measuring Resistivity of Silicon Wafers With an In-Line Four-Point Probe. Last published by ASTM International as ASTM F 84-02.

Van der Pauw, L. J. A Method of Measuring Specific Resistivity and Hall Effects of Discs of Arbitrary Shape. Phips Rec. Repts., 1958: 13 1.

Schroder, Dieter K. Semiconductor Material and Device Characterisation. John Wiley & Sons, 1998.

Low Level Measurements, Keithley Instruments, Cleveland, Ohio, 1998.