Semiconductor Doping Ionization - Temperature Dependence [+ code]

1. Introduction and requirements

In this post the doping of a semiconductor material is analyzed as a function of temperature. You should be aware of semiconductor bands theory and the effects of doping on it.

The post summarize the original explanation that can be found in the source code. Indeed, in the latter more doping combinations are tested.

2. Fermi Energy vs. Temperature

The Fermi Energy is computed solving the neutrality equation.

Where:

• p is the concentration of holes in the valence band

• Nd+ is the ionized donor concentration

• n is the concentration of electrons in the conduction band

• Na- is the ionized acceptor concentration

Since both n and p depends on exponential and polynomial function of temperature the plot shown in the code is not linear. Moreover the Fermi Energy strongly depends on the concentration and type of doping.

3. Carrier Concentration vs. Temperature

In general both carriers concentration (p and n) grow as temperature increases. Whereas the ionized net doping has a behavior almost equivalent to the majority carrier curve. However, at a certain temperature it reaches a plateau because the intrinsic carrier concentration becomes dominant.

4. Doping Energy Levels and Concentrations

In this section it is discussed about carrier concentration and Fermi Energy (the two previous sections) varying doping concentrations.

4.1. Energy Levels

If the doping concentration is pretty high (n, p > 10^18) the temperature dependence is less significant. In other cases, the Fermi Energy almost reaches the Intrinsic Fermi Energy at temperature greater than 600K.

4.2. Concentrations (or Density)

If the doping concentration is pretty high (n, p > 10^18) the ionized net doping plateau is never reached. In other cases, at temperature higher than 600K the ionized net doping amount to a constant value, which is almost equal to the doping concentration (full-ionization). In the same temperature range it can be observed a surge in the intrinsic carrier density, which surpass the ionized net doping.

Moreover, since the law of mass action has to be satisfied, also the electron and holes follow the n_i exponential trend.

5. Fraction of Ionized Impurities

At ambient temperature (T=25°C), it is analyzed the amount of ionized impurities as a function of doping concentration (Na or Nd).

In both cases, the fraction of ionized impurities has a peak in the range 1-5*10^17 cm^-3. Then it decays exponentially.

6. Code

All the plots and python code to get them can be be found at the following link:

Google Colab