FYSA2001 / K1 ENERGY GAP OF GERMANIUM
1 Introduction
This measurement is about conductivity of a semiconductor and its dependence on the temperature. As a quantitative result the energy gap of germanium is measured. Literature related: o Harris, Nonclassical Physics, chapters 9.4-9.9 o Young & Freedman, University Physics with Modern Physics, 11th ed., chapters 42.4, 42.6 and 42.7 (10th ed., chapters 44-5, 44-7 and 44-8) o Smith, Electronics, 3rd ed., chapter 5, Semiconductor Diodes o Kittel, Introduction to Solid State Physics, chapter 8, Semiconductor Crystals
2 Theoretical background
2.1 Semiconductors Semiconductors are materials, whose resistivity has a strong dependence on temperature. At room temperature the resistivity of a semiconductor is around 102 109 cm . At absolute zero temperature the lattice structures of the most semiconductors are insulators (resistivity 1014 cm ). Semiconductors are used in transistors, switches, diodes, photocells, sensors, processors, memory circuits etc. applications of electronics technology.
By doping a semiconductor with some other material, the conductivity of the semiconductor can be adjusted. E.g. silicon is often doped with phosphorus, boron, antimony or arsenic. On the valence band of silicon (and germanium), there are four electrons. When a substance with five valence electrons (phosphorus, arsenic, antimony) is added, in the end there is one extra electron which can carry a current. This is the case in the n-type semiconductor. If the added substance has three valence electrons (boron), there are positively charged holes formed in lattice structure, which can carry a current. This is the p-type semiconductor.
2.2 Temperature dependence of conductivity At absolute zero all electrons of insulators and semiconductors are on valence band. Higher conducting band is empty and therefore charges don’t move. Between valence band and conducting band there is an energy gap (Si: 1.12 eV, Ge: 0.67 eV). To cross this gap, electrons need energy. This is the reason why the resistance of semiconductors depends on temperature. Insulators have so broad energy gap (several eV:s) that electrons can’t get into conducting band at room temperature. Furthermore, the valence band of insulators is full, so electrons can’t move there either. Conductors have partially filled valence band and thus electrons can move regardless temperature.
At the temperature scale used in this measurement ( 20150 C ) the conductivity of a semiconductor, (the inverse number of resistivity) abides relation
Eg 2kT 0e , (1)
5 where E g is the energy gap, k is the Boltzmann constant ( 8.625*10 eV ) and T is temperature (in Kelvins).
3 Measurements
The measurement set-up consists of a voltage source, circuit board and module (all manufactured by Phywe), see figure 1. The bias is connected to the connectors on the back side of the module. Measured currents and temperatures are read from the display of the module. Selection between current and temperature is achieved by pushing the display button in the module. The current is set from the Ip knop. The voltage across the Ge crystal can be measured from the two lowest connectors in the module. The heating switch is on the back side of the module.
For determining the energy gap we need information of the resistance (R = U/I) of the Ge piece and the temperature. Set the control current to about 30 mA and heat the Ge crystal to the temperature of about 100 °C. While the temperature of the crystal decreases down to the room temperature record a number of data points. If needed, the crystal can be heated again. We can usually assume that the control current stays constant during the measurements. Check it! Prepare a (1/T, ln(1/R)) graph based on your measurements and determine the energy gap .
Figure 1: The circuit board and module used in this laboratory work. The Ge crystal is the grey piece on the circuit board