Download FYSP106 / K1 ENERGY GAP OF GERMANIUM 1 Introduction 2

FYSP106 / 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 10 2 109  cm . At
absolute zero the lattice structures of the most semiconductors are insulators (resistivity
 1014  cm ). Semiconductors are used in transistors, switches, diodes, photo cells,
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 ( 20  150 C ) the conductivity of a
semiconductor,  (the inverse number of resistivity) abides relation
   0e

 Eg 2 kT

,
(1)
where E g is the energy gap, k is the Boltzmann constant ( 8.625 *10 5 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