How does fermi level vary with temperature
The Fermi level, also known as the chemical potential, is a fundamental concept in solid-state physics that describes the energy level at which electrons in a material have a 50% probability of being occupied. The position of the Fermi level with respect to the energy band structure of a material plays a crucial role in determining its electrical and thermal properties.
Generally, the position of the Fermi level varies with temperature due to the thermal excitation and generation of charge carriers (electrons and holes). The specific behavior of the Fermi level with temperature depends on the material and can be classified into three main cases:
1. Metals: In metals, the Fermi level lies within the conduction band, resulting in a high density of free electrons available for conduction. As the temperature increases, thermal energy promotes electrons from occupied states below the Fermi level to higher energy empty states above the Fermi level. Consequently, the rate of excitation exceeds the rate of recombination, leading to a small increase in electron density above the Fermi level. Therefore, the Fermi level in metals generally moves slightly towards higher energies with increasing temperature.
2. Intrinsic semiconductors: In intrinsic semiconductors, the Fermi level lies close to the middle of the energy bandgap, which separates the valence and conduction bands. As the temperature increases, thermal energy excites electrons from the valence band to the conduction band, generating free electrons and holes. This thermal excitation increases both the electron and hole densities, resulting in an approximate balance between them. Thus, the position of the Fermi level in intrinsic semiconductors remains relatively constant with temperature.
3. Doped semiconductors: The behavior of the Fermi level in doped semiconductors depends on the type of doping (n-type or p-type). In n-type semiconductors, the Fermi level lies closer to the conduction band due to the presence of excess electrons provided by the dopant. As the temperature increases, more thermal energy excites electrons from the dopant level to higher energy states in the conduction band, increasing the electron concentration above the Fermi level. Thus, the Fermi level in n-type semiconductors moves towards higher energies with increasing temperature.
In p-type semiconductors, the Fermi level lies closer to the valence band due to the presence of excess holes provided by the dopant. As the temperature increases, thermal energy promotes electrons from the valence band to the vacant dopant states, decreasing the hole concentration. Consequently, the Fermi level in p-type semiconductors moves towards lower energies with increasing temperature.
In summary, the Fermi level generally experiences a slight shift with temperature in metals, remains relatively constant in intrinsic semiconductors, and moves in a direction determined by the type of doping in doped semiconductors.