Calculate Minority Carrier Concentrations given the Fermi Level

In summary, the conversation discusses solving a problem regarding a pn junction in silicon at 300K with given parameters such as band gap energy, electron density of states mass, hole mass, electron mobility, hole mobility, and relative permittivity. The Fermi level is calculated on each side and used to find the band bending and make a sketch of the pn junction. The conversation also explores using the mass action law and intrinsic concentration to solve for the minority carrier concentrations, with additional equations and variables provided.
  • #1
HunterDX77M
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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 Eg = 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 NA = 1021/m3 and ND = 1023/m3. Assume all impurities are ionized. On this basis find the Fermi level on each side. From this find the band bending VB 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
[itex]n^{2}_{i} = n_p N_A = p_n N_D[/itex]

Where np is the minority concentration of electrons and pn 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 ni or the minority concentrations in the above equation.

As far as I know
[itex]N_e \times N_h = N^2_i \neq n^2_i [/itex]

Where Ne and Nh are the majority concentrations (known).

Does anyone know a relation I can use to solve for these minority concentrations?
 
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  • #2


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.
Scientist at [Your Institution]
 

FAQ: Calculate Minority Carrier Concentrations given the Fermi Level

1. How do you calculate minority carrier concentrations given the Fermi Level?

To calculate minority carrier concentrations, you will need to use the Fermi-Dirac distribution function, which describes the distribution of electrons and holes in a semiconductor material. By plugging in the Fermi Level and other relevant parameters such as temperature and dopant concentrations, you can determine the minority carrier concentrations.

2. What is the Fermi Level and why is it important?

The Fermi Level is the energy level at which there is a 50% probability of finding an electron. It is an important parameter in understanding the electronic properties of a material, as it determines the conductivity and carrier concentrations. It is also a reference point for measuring the energy levels of electrons in a material.

3. What factors influence the Fermi Level in a semiconductor?

The Fermi Level in a semiconductor is influenced by factors such as temperature, doping concentration, and the presence of impurities or defects. Changes in these factors can cause the Fermi Level to shift, affecting the carrier concentrations and overall electronic properties of the material.

4. How does the Fermi Level affect minority carrier lifetimes?

The Fermi Level can affect minority carrier lifetimes by influencing the recombination rate of minority carriers. A higher Fermi Level can lead to a higher recombination rate, resulting in shorter minority carrier lifetimes. Conversely, a lower Fermi Level can lead to longer minority carrier lifetimes.

5. What are some applications of calculating minority carrier concentrations using the Fermi Level?

Calculating minority carrier concentrations using the Fermi Level is important in various applications, such as in the design and optimization of semiconductor devices, including solar cells and transistors. It can also be used in understanding the behavior of materials in different environments and in the development of new materials for electronic applications.

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