Calculate Minority Carrier Concentrations given the Fermi Level

[HunterDX77M]

Homework Statement

This question is based on a previous question in the same homework:

The Problems deal with Silicon at 300K, using band gap energy E_g = 1.12 eV, electron density of states mass 0.327, hole mass 0.39, electron mobility 0.15 m²/Vs, hole mobility 0.05 m²/Vs and relative permittivity 11.8.

1) Consider a pn junction in Si at 300K (other parameters given), with doping N_A = 10²¹/m³ and N_D = 10²³/m³. Assume all impurities are ionized. On this basis find the Fermi level on each side. From this find the band bending V_B and make a sketch of the pn junction.

This problem has been solved and its thread is here: https://www.physicsforums.com/showthread.php?t=713644


5) Following your results for the Fermi Levels in Problem 1
a) Find the minority carrier concentrations (holes on the N-side, electrons on the P-side).
b) Repeat the calculation for the minority carrier concentrations using the mass action law and the intrinsic concentration 5.85E15/m³

Homework Equations

I found this equation while searching online relating minority and majority carriers
$n^{2}_{i} = n_p N_A = p_n N_D$

Where n_p is the minority concentration of electrons and p_n is that of holes. But this leaves me with two variables and one equation.

The Attempt at a Solution

Based on the first problem, I have numbers for the Fermi level (EF) on both sides. If you are curious they are 0.974 eV (N-side) and 0.226 eV (P-side). However, I don't know of any way to relate the Fermi level with n_i or the minority concentrations in the above equation.

As far as I know
$N_e \times N_h = N^2_i \neq n^2_i$

Where N_e and N_h are the majority concentrations (known).

Does anyone know a relation I can use to solve for these minority concentrations?

 

[mark726]

Thank you for your question. It seems like you are on the right track with using the mass action law and the intrinsic concentration to solve for the minority carrier concentrations. Let me provide some additional information and equations that may help you in your calculation.

First, let's define some variables:
- Ni: intrinsic carrier concentration (5.85E15/m3 in this case)
- Np: concentration of holes on the N-side (minority carriers)
- Nn: concentration of electrons on the P-side (minority carriers)
- ND: donor doping concentration (1023/m3 in this case)
- NA: acceptor doping concentration (1021/m3 in this case)
- EFp: Fermi level on the P-side
- EFn: Fermi level on the N-side
- k: Boltzmann constant (8.617E-5 eV/K)
- T: temperature (300K in this case)

Now, let's use the mass action law to relate the concentrations of electrons and holes in an intrinsic semiconductor (where ND=NA=0):
Np * Nn = Ni^2

Since we know Ni and Np (from Problem 1), we can solve for Nn:
Nn = Ni^2 / Np

Next, let's use the definition of the intrinsic carrier concentration to relate Ni to the Fermi levels:
Ni^2 = exp[(EFp-EFn)/kT]

Now, let's use the definition of the Fermi level to relate it to the majority carrier concentrations:
EFp = EF + (kT/2) * ln(Np/ni)
EFn = EF - (kT/2) * ln(Nn/ni)

Where EF is the intrinsic Fermi level, which is equal to the middle of the band gap energy (1.12 eV in this case).

Putting all of these equations together, we can solve for the minority carrier concentrations:
Np = ni * exp[(EFp-EF)/kT]
Nn = ni * exp[(EF-EFn)/kT]

I hope this helps you in your calculation. Let me know if you have any further questions.
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