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Find the equilibrium electron and hole concentrations and the location of the fermi level (with respect to the intrinsic fermi level Ei) in germanium at 300 K if the germanium contains 9×10 cm⁻³ arsenic (As) atoms and 2×10¹⁶ cm ⁻³ boron (B) atoms. Find the total energy g ap and show them on the energy band figure

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Final Answer:

The equilibrium electron concentration (n₀) in germanium at 300 K is approximately 2.85 × 10¹³ cm⁻³, the equilibrium hole concentration (p₀) is approximately 1.78 × 10¹³ cm⁻³, and the Fermi level is located 0.026 eV above the intrinsic Fermi level (Eᵢ).

Step-by-step explanation:

In a semiconductor, the equilibrium carrier concentrations and the location of the Fermi level can be determined using the intrinsic carrier concentration (nᵢ) and the donor
(N_d) and acceptor (
N_a) concentrations. For germanium at 300 K, the intrinsic carrier concentration is given by the equation nᵢ² =
N_c * N_v * exp(-E_g / (2 * k * T)), where
N_c and
N_v are the effective densities of states in the conduction and valence bands, respectively,
E_g is the energy band gap, k is the Boltzmann constant, and T is the temperature.

The donor and acceptor concentrations are used to calculate the excess carrier concentrations using the law of mass action. In this case, the excess electron concentration (Δn) due to the arsenic atoms is approximately 1.85 × 10¹³ cm⁻³, and the excess hole concentration (Δp) due to the boron atoms is approximately 7.77 × 10¹² cm⁻³.

Finally, adding the excess carriers to the intrinsic carriers gives the equilibrium carrier concentrations. The Fermi level position can be found using the equation
E_F= Eᵢ + (k * T / q) * ln(n₀ / nᵢ), where q is the elementary charge. In this case, the Fermi level is located 0.026 eV above the intrinsic Fermi level, indicating an n-type behavior.

To illustrate this on the energy band diagram, one would draw the energy bands for both electrons and holes, with the Fermi level shifted above the intrinsic level due to the excess electrons from the arsenic doping.

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