Temperature Dependence of Carrier Concentrations

• The intrinsic carrier concentration has a temperature dependence, which can be given as

          (2.16)

• Thus, the explicitly, ni is proportional to T3/2 and to e 1/T, but, Eg also has a temperature dependence (decreasing with the increasing temperature, since interatomic spacing changes with the temperature).

Figure The intrinsic carrier concentration as a function of inverse temperature for Si, Ge, and GaAs.

• As n_i changes with temperature, so do n₀ and p₀.
• With and T given, the unknowns are the carrier concentrations and the Fermi level position with respect to  one of these quantities must be given in order to calculate the other.
• Example: Si doped with 10¹⁵/cm³ donors (N_d).

• At very low temperature, insignificant intrinsic EHPs exist, and all the donor electrons are bound to the donor atoms.
• As the temperature is raised, these electrons are gradually donated to conduction band, and  at  about  100  K  (1000/T  =  10),  almost  all  these  electrons  are  donated   this temperature range is known as the ionization region.
• Once all the donor atoms are ionized, the electron concentration n_o ≈ N_d, since for each donor atom, one electron is obtained.

Figure- Variation of the carrier concentration with the inverse temperature clearly showing three regions: extrinsic, ionization, and intrinsic.

• Thus, n₀ remains virtually constant with the temperature for a broad range of temperature (called extrinsic region), until intrinsic carrier concentration n_i starts to become comparable to N_d.
• For the high temperatures, n_i>> N_d, and material loses its extrinsic property (called as intrinsic region).
• Note that in the intrinsic region, the device loses the usefulness of it => determines the maximum operable temperature range.

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