Fermi level at high temperature

The Fermi level (also known as the Fermi energy) is a concept in solid-state physics that represents the highest energy state occupied by electrons at absolute zero temperature (0 Kelvin or -273.15 degrees Celsius). At high temperatures, several factors can influence the position of the Fermi level in a material:

1. In an n-type extrinsic semiconductor, the Fermi level (E_F) is a critical parameter that determines the distribution of electrons in the material. An n-type semiconductor is one that has been intentionally doped with donor impurities (such as phosphorus in silicon) to introduce excess electrons into the crystal lattice. At high temperatures, several factors influence the position of the Fermi level in an n-type extrinsic semiconductor:

   a. Intrinsic Carrier Concentration: The position of the Fermi level depends on the intrinsic carrier concentration (n_i), which represents the number of thermally generated electron-hole pairs in the absence of doping. As temperature increases, the number of thermally generated carriers also increases, affecting the position of E_F. In an n-type extrinsic semiconductor, the electron concentration (n) far exceeds the intrinsic carrier concentration (n_i).

   b. Donor Energy Levels: Donor impurities introduce additional energy levels called donor levels, which are located just below the conduction band. These donor levels supply extra electrons to the conduction band, making it easier for electrons to populate higher energy states at high temperatures.

   c. Temperature-Dependent Carrier Concentration: The concentration of electrons (n) in the conduction band increases with temperature due to thermal excitation. This leads to a shift in the Fermi level toward higher energies.

   d. Bandgap Shrinkage: At elevated temperatures, the semiconductor's bandgap may slightly decrease due to thermal effects. This can also influence the position of the Fermi level, albeit to a lesser extent.

In general, at high temperatures, the Fermi level tends to shift away from the bandgap towards the middle of the energy levels available for electrons. The precise position and behaviour of the Fermi level at high temperatures depend on the material's properties, such as its band structure, doping, and temperature-dependent carrier concentrations.

Expression for fermi level at high temperature

We have fermi level for incompletely ionized n type semiconductor is

At high temperature, Nc becomes large

sin h-1(x) becomes large

Using the condition

Equation 3 becomes

The specific behaviour of the Fermi level in an n-type extrinsic semiconductor at high temperatures will depend on factors such as the material, doping concentration, and temperature range. Researchers and engineers often consider these effects when designing and analyzing semiconductor devices like transistors, diodes, and integrated circuits.

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