Introduction and Theory

A diode is a semiconducting material which in its simplest form converts ac to dc and allows current to flow only in one direction. A diode is created by joining a p-type semiconductor with an n-type semiconductor thus making a pn junction. (See Figure 1.1) The p-type semiconductor is a semiconducting material which is doped with some type of trivalent atom (usually indium or aluminum). The three valence electrons covalently bond with the semiconducting material and leave a "hole" in the fourth bond. The n-type semiconductor is a semiconducting material which is doped with a donor atom (usually arsenic). Four of the donor atom's electrons bind covalently with the semiconducting material while the fifth is free to move into the conduction band if given a small amount of thermal energy.

A pn junction may be considered as having a forward bias or a reverse bias. The forward bias condition represents a positive potential on the p side and a negative potential on the n side. For the case of a reverse biased system, the n side has a positive potential while the p side has a negative potential. (For this experiment we examined the forward bias condition.) Two types of current are present throughout the pn junction for both biases: the diffusion current and the drift current. The diffusion current (Idiff) refers to the movement of majority carriers. Majority carriers consist of electrons (in the conduction band) that travel from the n to the p side and holes (vacancies in the valence band) that travel from the p to the n side. The concentration of majority carriers that travel across the pn junction is voltage and temperature dependent. The drift current (Idrift) refers to the movement of minority carriers across the pn junction. The minority carriers also consist of electrons and holes yet the direction of their movement is opposite that of the majority carriers. Unlike the majority carriers, though, the concentration of minority carriers that cross the pn junction is independent of voltage and temperature. The total current (Itotal) through the pn junction for either bias may be written as:

Itotal = Idrift e + Idiff e exp(eV/hkT) + Idrift h + Idiff h exp(eV/hkT).

Since at equilibrium Idiff e = - Idrift e and Idiff h = -Idrift h we may write:

Itotal = Idrift e [ exp(eV/hkT) - 1 ] + Idrift h [ exp(eV/hkT) -1 ].

Simplifying we obtain:

Itotal = (Idrift e + Idrift h ) [ exp(eV/hkT) -1 ].

The quantity (Idrift e + Idrift h ) is referred to as the reverse saturation current of the system and is given the symbol Io. Thus, we obtain:

Itotal = Io[ exp(eV/hkT) -1 ]

where e is the charge on an electron (1.6 x 10-19 J), V is voltage (volts), h is the ideality factor which varies per diode, k is Boltzmann's constant (1.38 x 10-23 J/K), and T is temperature (degrees Kelvin).1 The ideality factor is due to such physical phenomenon as the surface effect, recombination, and tunneling.2 For forward bias voltages where V > (hkT/e) we can make the following approximation:

Itotal ~ Io exp(eV/hkT) or ln(Itotal ) = ln(Io) + (eV/hkT).3

The band gap energy (Eg) is related to the reverse saturation current by the following equation:

Io = BT(3/2) exp(-Eg /hkT) or ln(Io) = ln(BT(3/2)) + (-Eg /hkT).

Since the experiment, however, is being run with temperatures between 200-300 oK the differences in the term ln(BT(3/2)) are negligible thus giving us:

ln(Io) = -Eg /hkT.4

Therefore, by determining h and making calculations of Io at varying temperatures, a determination of the band gap energy for silicon and germanium diodes can be made.

Procedure

The apparatus consisted of a Radio Shack 1N4001 Silicon diode and 1N34A Germanium diode inside a Pyrex test tube filled with heat sink compound. The Pyrex test tube was placed inside a Dewar containing an acetone bath. The acetone bath (fp= 177 K)5 was cooled using a Cryocool apparatus thus providing a method of varying temperature. Each diode was connected in series outside of the Dewar to a 1000 ohm resistor. A Pasco Scientific Signal Interface was used to generate a sawtooth voltage and collect voltage readings across the resistor and diodes. The voltage across the resistor was used to calculated the current while the voltage across the diode gave the forward bias voltage. An Iron-Constantan thermocouple was used to measure the temperature of the diodes. (See Figure 1.2)

Results

Once the forward voltage and current had been collected at various temperatures for both the silicon and germanium diodes, natural log plots of the current versus the voltage were made. (See Figure 1.3 and Figure 1.4) After averaging the slopes of the best fit lines for the data, we were able to obtain the ideality factor (h) for the silicon and germanium diodes using the relation:

ln(I) = ln(Io) + (e/hkT)V

The values for the ideality factor of silicon and germanium were 1.90 and 2.2 respectively. Since these values are devices dependent no reference could be made to support their accuracy. In a similar experiment, h values were 1.44 for silicon and 1.76 for germanium.6

Using the y-intercepts obtained from the linear regression of the data, a natural log of the reverse saturation current, ln(Io), vs. the inverse temperature could be obtained. (See Figure 1.5 and Figure 1.6) A linear regression on the reverse saturation current vs. the inverse temperature for both the silicon and germanium diodes revealed the slope of both graphs. By using the relation:

ln(Io) = (-Eg /hk)(1/T)

we were able to set the slope equal to (-Eg /hk) and solve for the band gap energy. The band gap energies were found to be 1.14 eV and 0.77 eV. (See Figure 1.7 for complete results.)

Conclusion

By comparing the silicon diode data with the germanium diode data we can make several conclusions about the electronic properties of semiconducting devices. First, the energy (electrical or thermal) needed to transfer an electron from the valence band to the conducting band for the silicon diode is approximately twice that of the germanium diode. This would imply that the germanium diode is more of a conductor than the silicon diode. Second, the data shows that, unlike metals, the silicon and germanium diodes increase conductivity with an increase in temperature. From this, we may assert that a semiconducting device becomes more like an insulator as the temperature approaches absolute zero.

A suitable measurement for the band gap energy of the silicon and germanium diodes can be determined by following the preceding procedure and analysis. We were discouraged, however, by the performance of the iron-constantan thermocouple. While the thermocouple allowed us to make temperature readings without removing the test tube from the Dewar, we found that mere jostling of the reference wire in the ice bath would alter the temperature reported. Therefore, there may be no advantage of using a thermocouple over a regular thermometer.

References

1 Experimental Physics: Modern Methods. Dunlap, R. A., 16-35, (1988).

2 Physics of Semiconductor Devices. Sze S. M., 89-92, (1981).

3 "Elementary technique to measure the energy band gap and diffusion potential of pn junctions," Fischer, Charles W., American Journal of Physics. 50, 1103-1105, (1982).

4 "Simple Measurement of the band gap in silicon and germanium," Collings, Peter J., American Journal of Physics. 48, 197-199, (1980).

5 CRC Handbook of Chemistry and Physics. Weast, Robert C. ed. et al., C-51, (1989).

6 "Elementary technique to measure the energy band gap and diffusion potential of pn junctions," Fischer, Charles W., American Journal of Physics. 50, 1103-1105, (1982).

7 CRC Handbook of Chemistry and Physics. Weast, Robert C. ed. et al., E-110, (1989).
 

Literature Consulted

Horowitz, Paul and Winfield Hill. The Art of Electronics. Cambridge University Press: New York, pp. 43-50, (1995).

Serway, Raymond A., Physics For Scientists and Engineers. Saunders College Publishing: Philadelphia, pp. 1268-1276, (1990).

Abstracts