Determination of Energy Band Gap of Germanium by Four Probe Method

Introduction

The energy band gap is a fundamental property of semiconductors, defining the minimum energy required to excite an electron from the valence band to the conduction band. For Germanium (Ge), a semiconductor material, this gap determines its electrical conductivity as a function of temperature. The Four-Probe method is a standard experimental technique used to measure the resistivity of a semiconductor sample accurately, eliminating errors associated with contact resistance.

Principle of the Four Probe Method

In a standard two-probe measurement, the contact resistance between the metal probes and the semiconductor can significantly influence the result. The Four-Probe method overcomes this by separating the current-carrying probes from the voltage-sensing probes. Four collinear probes are placed on the surface of the Germanium crystal. A constant current (I) is passed through the two outer probes, while the potential difference (V) is measured across the two inner probes.

For a thin semiconductor wafer, the resistivity () is calculated using the formula:

Where 's' is the spacing between the probes, 'w' is the thickness of the sample, and G(w/s) is a correction factor based on the geometry of the sample.

Temperature Dependence and Band Gap

The resistivity of a semiconductor is inversely proportional to the concentration of charge carriers, which increases exponentially with temperature according to the relation:

In this equation:

• Eg is the energy band gap in electron volts (eV).
• k is the Boltzmann constant (8.617 10 eV/K).
• T is the absolute temperature in Kelvin.
• is a constant related to the material properties.

By taking the natural logarithm of both sides, we get: ln() = ln() + (Eg / 2k) * (1/T). This linear relationship allows us to determine Eg by plotting ln() versus (1/T). The slope of the resulting straight line is equal to Eg / 2k.

Experimental Procedure

1. Setup: The Germanium crystal is placed on the base of the four-probe apparatus. The four probes are lowered onto the center of the crystal surface.
2. Heating: The crystal is gradually heated using a controlled oven. It is essential to allow the temperature to stabilize at each interval to ensure accurate readings.
3. Measurement: At regular temperature intervals (e.g., every 5 degrees), the current (I) is kept constant, and the voltage (V) is recorded.
4. Data Collection: The temperature is measured using a thermocouple placed in close proximity to the Germanium crystal to ensure the recorded temperature reflects the crystal's actual state.

Data Analysis and Calculation

Once the data (V, I, T) is collected:

1. Calculate the resistivity () for each temperature point.
2. Convert the temperature (T) to inverse temperature (1/T) in units of K.
3. Calculate ln() for each resistivity value.
4. Plot a graph with ln() on the Y-axis and (1/T) on the X-axis.
5. Calculate the slope of the linear fit line: Slope = (ln ) / (1/T).
6. Determine the band gap using the derived slope: Eg = 2k * Slope.

For Germanium, the expected value for the energy band gap at room temperature is approximately 0.67 eV. Deviations from this value are typically attributed to contact alignment, temperature fluctuations, or impurities within the crystal lattice.

Conclusion

The Four-Probe method provides an accurate and reliable way to study the temperature dependence of semiconductor resistivity. By utilizing the linear relationship between the log of resistivity and inverse temperature, we can effectively determine the energy band gap of Germanium, confirming its intrinsic semiconductor nature.