Band Gap of Germanium and the Hall Effect
Friday, Section 006
TA: Yilikal Ayino
with Daniel Erickson & Kevin Haar
February 14, 2014 PreLab
With the new understanding of quantum mechanics and solid-state physics in the 1930’s came the development of semiconductors. Semiconductors are materials such as silicon and germanium in which the current density is non-zero, unlike an insulator, yet still several magnitudes smaller than that of a conductor. There exists a significant energy gap between the filled valence – highest range of electron energies normally present at absolute zero – and empty conduction bands – range of energies which are enough to unbind the electron from the atom and allow it to move freely within the atomic lattice – in a semiconductor, unlike a conductor which has very small or no band gaps and/or an insulator which has very large band gaps.
Semiconductors are of great interest as they allow a material’s conductivity/resistivity to be manipulated. Due to the Pauli exclusion principle, a filled valence band cannot carry a current. Thus, in a semiconductor, electrons must be either added to the conduction band or removed from the valence band in order for current to flow. This can be done by increasing the temperature or by “doping”.
The former will be achieved in this experiment by connecting circuit board-mounted germanium to electrical leads, sending a current through it. A heater and a thermistor (a resistor used as a thermometer) are in adequate thermal contact with the crystal, allowing energy to be exchanged as heat. After correctly connecting an ammeter and voltmeter, the current and voltage can be measured as a function of temperature (controlled by adjusting the power of the heater).
Applying statistical mechanics, the number of moving charge carriers is given by:
where is a function of the temperature which scales as , is Boltzmann’s constant, and is the band gap (the energy...
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