Module 3 C2

Density of States in Semiconductors

– The density of states (DOS) quantifies how many electron states are available per energy level for electrons and holes in a semiconductor. It is essential to know how many electrons can occupy these energy states and why they are crucial in determining electrical conductivity.

Electron and Hole Distribution

– Electrons and holes, being charged particles, cannot occupy the same quantum state due to the Pauli exclusion principle. Therefore, their distribution in the energy states is influenced by temperature. – At low temperatures, electrons are mostly bound to the nucleus, occupying the lowest available energy states, leading to a phenomenon known as freeze-out. – As temperature increases, electrons gain energy, allowing them to move among energy states, transitioning upwards as energy levels become unoccupied.

Importance of Temperature on Distribution

– The movement of electrons and holes is critical to the functioning of semiconductors and impacts the flow of electric current through them. – At absolute zero (0 Kelvin), all electrons reside below the Fermi level, indicating they have zero probability of being found in higher states. The Fermi energy (E_f) thus marks the maximum energy level filled with electrons at this temperature.

Fermi-Dirac Distribution

– The distribution of electrons in energy states follows the Fermi-Dirac statistics. This distribution is defined as:

[ f(E) = \frac{1}{1 + e^{(E - E_f)/(kT)}} ]

where:

  • (E) is the energy of the electron

  • (E_f) is the Fermi energy

  • (T) is the absolute temperature

  • (k) is the Boltzmann constant.

– At 0 Kelvin, (f(E)) equals 1 for energies less than (E_f) and 0 for energies above (E_f). This means that all lower energy states are filled and no higher states are occupied.

Effects of Increasing Temperature

– As the temperature rises, some electrons acquire enough energy to jump from below to above the Fermi level. This alters the probability of occupancy of various energy states:

  • At T > 0 K, the occupation probability for states at or above E_f increases, while it decreases for states below E_f.

  • Specifically, at E = E_f, the probability of finding an electron is 1/2, which serves as a reference point for occupation.

Summary of Electron Behavior with Temperature Changes

– When plotting the probability of finding electrons in energy states, the distribution curve shows that as temperature rises, more states above the Fermi level become occupied and the number of empty states below the Fermi level decreases. – This behavior underscores the symmetry of the occupancy and vacancy distributions across the energy band, which is essential for predicting semiconductor behavior in various conditions.