Calculating Electron and Hole Concentrations in a p-n Junction

Calculating hole and electron concentrations

Sometimes it can be complicated understanding and calculating hole and electron concentrations. My intent in this article is to briefly, but thoroughly describe what the variables used in these calculations mean and how to use them.

To begin I will introduce our variables

$n =$ concentration of free electrons (donors)
$p =$ concentration of holes (acceptors)
$n_i$ = number of free electrons and holes in a unit volume

In thermal equilibrium(or no doping)

$n=p=n_i$ and, therefore $n \cdot p=n_i^2$

However, doping is common in most examples. To increase the concentration of free electrons, an element with 5 valence electrons is used (i.e. Phosphorous). The resultant material is said to be n-type. To increase the number of holes, an element with 3 valence electrons is used (i.e. Boron). The resultant material is said to be p-type.

This introduces subscript n’s and p’s along with our concentration of free electron and hole variables.

n-type silicon:

$n_n =$ concentration of free electrons (in n-type silicon)
$p_n =$ concentration of holes (in n-type silicon)

p-type silicon:

$n_p =$ concentration of free electrons (in p-type silicon)
$p_p =$ concentration of holes (in p-type silicon)

Note: The subscript indicates whether the material is n-type or p-type.

Calculations

Typically you first want to identify whether the material you are working with is p-type or n-type. This introduces two new variables. $N_D$ which refers to the concentration of donor atoms and $N_A$ which refers to the concentration of acceptor atoms.

n-type silicon:

Here you will use the variables $n_n$ , $p_n$ , $n_i^2$ , and $N_D$ .

$n_n \approx N_D \quad \quad n_n \cdot p_n = n_i^2 \quad \quad p_n = \frac{n_{i}^{2}}{N_{D}}$

p-type silicon:

Here you will use the variables $n_p$ , $p_p$ , $n_i^2$ , and $N_A$ .

$p_p \approx N_A \quad \quad p_p \cdot n_p = n_i^2 \quad \quad n_p = \frac{n_{i}^{2}}{N_{A}}$

In most cases $n_i^2$ and $N_D$ or $N_A$ will be given and you will be able to find $n_n$ or $p_p$ . Then you will find $p_n$ or $n_p$ from the equations above.

Calculating hole and electron concentrations

In thermal equilibrium(or no doping)

n-type silicon:

p-type silicon:

n-type silicon:

p-type silicon:

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