Measurement of Resistivity and Determination of Band Gap Using Four Probe Method

The measurement of electrical resistivity and the determination of the band gap energy are fundamental to characterizing semiconducting and conducting materials. These parameters provide insight into the materials electrical behavior, quality, and suitability for various electronic applications. Among the several techniques developed for these measurements, the Four Probe Method stands out for its accuracy and effectiveness, especially in cases where contact resistance and surface effects might otherwise corrupt the results.

Introduction to Resistivity

Resistivity is an intrinsic electrical property of materials that quantifies how strongly a material opposes the flow of electric current. It is denoted by the symbol and expressed in ohm-meters (m). The lower the resistivity, the better the material conducts electricity.

Resistivity depends on the materials nature and temperature. For metals, resistivity generally increases with temperature, while for semiconductors, it decreases initially due to increased charge carriers and then rises at very high temperatures. Accurate resistivity measurement plays a crucial role in material science, semiconductor research, and device fabrication.

The Four Probe Method: Principle and Advantages

The Four Probe Method overcomes common limitations of two-probe methods by eliminating the effects of contact resistance and spreading resistance when measuring resistivity.

In a typical two-probe configuration, the same pair of contacts inject current and measure voltage across the sample. The measured voltage includes voltage drops across the contact resistance and the sample itself, leading to inaccurate resistivity estimates. The Four Probe Method uses four linearly arranged probes: the outer two probes supply a constant current, and the inner two probes measure the voltage drop. Since the voltmeter draws negligible current, the voltage measurement excludes the contact resistance voltage drops.

Setup Description

The probes are aligned collinearly on the sample surface, usually with equal spacing s. A current I is passed through the outer probes, and the potential difference V between the two inner probes is measured. The four probes lightly touch the materials surface to ensure good electrical contact without damage.

Calculation of Resistivity

For a sample thick enough relative to probe spacing and with a uniform surface, resistivity is given by:

= 2sV/I

where

= resistivity (m)
s = spacing between adjacent probes (m)
V = voltage measured between voltage probes (V)
I = current applied through current probes (A)

This equation assumes the sample is a semi-infinite bulk. For thin films or finite thickness samples, correction factors (geometry functions) are applied, depending on sample thickness t and probe spacing s. For instance, if the thickness is comparable to or less than the probe spacing, corrections involve multiplicative factors derived from theoretical models or numerical simulations.

Advantages of the Four Probe Method

Eliminates contact resistance: Since voltage measurement is separate from current injection, contact resistances do not affect the voltage reading.
Accurate for low-resistance samples: Particularly useful where contact resistance is significant compared to sample resistance.
Non-destructive and simple: Probes only touch the surface lightly, and the setup is relatively straightforward.
Sensitive to surface resistivity: Ideal for materials where surface conduction dominates.

Procedure for Measuring Resistivity Using Four Probe Method

Preparation

Clean the sample surface to remove dust, oxides, or contaminants for better contact.
Place the four probes carefully and evenly spaced, ensuring consistent and reproducible contact.
Connect the outer probes to a constant current source.
Connect the inner probes to a high-impedance voltmeter or nanovoltmeter.

Measurement Steps

Pass a known stable current I through the outer probes.
Measure the voltage V between the inner probes.
Calculate resistivity using the formula = 2sV/I or apply correction factors if necessary.
Multiple measurements with varying currents help verify linearity and eliminate Joule heating effects.
Repeat measurements at different points for sample uniformity.

Determination of Band Gap Using Four Probe Method

The band gap energy E_g is one of the key material parameters that defines the energy difference between the valence band and conduction band in semiconductors and insulators. It determines temperature-dependent electrical conductivity and optical properties. Although direct band gap measurements require optical techniques (like photoluminescence or absorption), the Four Probe resistivity measurement enables indirect estimation of the band gap by analyzing temperature-dependent resistivity.

Relationship Between Resistivity and Temperature

Semiconductor resistivity varies with temperature due to thermal excitation of charge carriers across the band gap. At high temperatures, intrinsic carrier concentration dominates and resistivity follows an Arrhenius-type behavior:

(T) = _0 exp(E_g / 2k_B T)

where:

(T) = resistivity at temperature T (K)
_0 = pre-exponential constant related to material
E_g = band gap energy (eV)
k_B = Boltzmann constant (8.617 10^-5 eV/K)
T = absolute temperature in Kelvin (K)

Taking natural logarithm on both sides, the relation can be linearized:

ln() = ln(_0) + (E_g / 2k_B) (1/T)

This suggests that plotting ln() versus 1/T yields a straight line whose slope is proportional to the band gap E_g.

Experimental Procedure

Perform Four Probe resistivity measurements over a range of temperatures, typically from room temperature upwards (e.g., 300 K to 500 K).
Ensure temperature uniformity by placing the sample in a temperature-controlled chamber or furnace.
Record resistivity values (T) at each temperature.
Calculate ln() and plot against reciprocal temperature 1/T.
Fit a linear line to the high-temperature regime where intrinsic conduction dominates.

Data Analysis and Band Gap Calculation

The slope m of the linear fit corresponds to:

m = E_g / 2k_B

Hence, the band gap energy can be estimated as:

E_g = 2k_B m

With k_B known, direct multiplication yields E_g in electronvolts (eV).

Considerations and Limitations

Temperature range: The intrinsic conduction regime must be identified correctly. At lower temperatures, extrinsic conduction due to impurities dominates, deviating from the Arrhenius behavior.
Sample preparation: Surface defects and contamination affect resistivity and may introduce errors.
Measurement accuracy: Four Probe reduces error from contact resistance but instrumentation accuracy and thermal stabilization are still critical.
Sample geometry: Uniform thickness and geometry enable reliable application of correction factors.

Summary

The Four Probe Method is a powerful technique for measuring the resistivity of semiconductors and conducting materials with high accuracy and minimal influence of contact resistance. By acquiring temperature-dependent resistivity data using this method, the band gap energy of semiconductor materials can be determined through analysis of the Arrhenius-type relationship between resistivity and temperature.

This approach facilitates non-destructive, electrical characterization crucial for research and development in semiconductor physics, materials science, and device engineering. Correct experimental setup, careful measurement, and appropriate data analysis enable accurate extraction of intrinsic material properties essential for optimizing electronic components.