13 Jul 2026
Four-Probe Resistance of a Semiconductor and Determination of Band Gap
Aim
To measure the resistance of a semiconductor by the four-probe method at different temperatures and determine its energy gap.
Apparatus
Four-probe semiconductor unit, constant-current source, microvoltmeter, heater, thermometer, and regulated power supply.
Experimental arrangement
Theory
The four-probe method separates the current contacts from the voltage contacts. A known current enters through the outer probes and the inner probes draw negligible current, so contact resistance and lead resistance do not enter the measured voltage significantly. For a long thin sample, the resistivity is obtained from the measured voltage $V$, current $I$, and geometrical correction factor $G$:
\[\rho=G\frac{V}{I}.\]For an intrinsic semiconductor, thermal excitation creates electron-hole pairs across the gap. The conductivity varies approximately as $\sigma=\sigma_0e^{-E_g/(2kT)}$. Hence a plot of $\log_{10}\sigma$ against $1/T$ is linear and
\[E_g=-2.303k\frac{d(\log_{10}\sigma)}{d(1/T)}.\]Observations
| Temperature (K) | Current (mA) | Probe voltage (mV) | Resistivity (ohm m) |
|---|---|---|---|
| 303 | 2.0 | 18.2 | 0.91 |
| 313 | 2.0 | 12.8 | 0.64 |
| 323 | 2.0 | 8.8 | 0.44 |
| 333 | 2.0 | 6.0 | 0.30 |
| 343 | 2.0 | 4.0 | 0.20 |
For this trial sheet, the geometrical correction factor is $G=0.10\,\text{m}$.
Graph
Calculation
For the first reading,
\[\rho=G\frac{V}{I}=0.10\frac{18.2\times10^{-3}}{2.0\times10^{-3}}=0.910\,\Omega\,\text{m}.\]Therefore,
\[\sigma=\frac{1}{\rho}=\frac{1}{0.910}=1.10\,\text{S m}^{-1}.\]The graph is plotted against $1000/T$. Its slope is approximately $-3.36$ per unit of $1000/T$, which corresponds to $-3360$ K when the horizontal variable is $1/T$. Hence
\[E_g=-2.303(8.617\times10^{-5})(-3360)=0.67\,\text{eV}.\]Result
The semiconductor shows decreasing resistivity with increasing temperature, and the energy gap obtained from the graph is
\[\boxed{E_g\approx0.67\,\text{eV}}.\]Viva Questions
- Why are four probes used? The voltage contacts carry negligible current, so contact resistance has little effect.
- Why is the sample heated gradually? To maintain thermal equilibrium and avoid temperature gradients.
- What indicates semiconductor behaviour? Its resistance decreases as temperature increases.
Discussion