1. Charge carriers

Find a great summary at Electronics Tour Book - 2.4. Charge carriers.

  • free electrons, n

  • (free) holes, p

    intrinsic

    Semiconductor with no doping, only has thermally-generated electron/hole pairs, which is a strong function of temperature.

    extrinsic

    Semiconductor with added impurities which add extra free electrons and/or holes to the solid.

2. Charge density with doping

The law of mass action says:[1]

\[n \cdot p = n_i^2\]

and a semiconductor material, possibly with doping, has no net charge:

\[p -n +N_D -N_A = 0\label{eq-sum-charges}\]

Combine these two equations and solve for electron density n:

\[n = \dfrac{(N_D - N_A) \pm \sqrt{(N_D - N_A)^2 + 4 n_i^2}}{2}\]

2.1. Example

For silicon near a temperature of 300 K, use \(n_i = 10^{10} \, \mathrm{cm^{-3}}\).

Add phosphorus doping at a density of \(10^{15} \, \mathrm{cm^{-3}}\).

  • Find the free electron and hole densities n and p.

Collect what we know:

  • ND =

  • NA =

Compute n

\[\begin{align} n &= \dfrac{(10^{15} - 0) + \sqrt{(10^{15} - 0)^2 + 4 (10^{10})^2}}{2} \\ \nonumber\\ &= \dfrac{10^{15} + \sqrt{10^{30} + 4 \!\times\! 10^{20}}}{2} \\ n &\approx 1.000\,000\,000\,100 \times 10^{15} \, \mathrm{cm^{-3}} \\ &\approx 10^{15} \; (= N_D) \end{align}\]

and p

\[\begin{align} p &= \frac{n_i^2}{n} \\ &\approx 9.999\,999\,999\,000 \times 10^{4} \\ &\downarrow \\ p &\approx \frac{\left(10^{10}\right)^2}{10^{15}} = 10^{5} \end{align}\]

What can you say about these numbers? Compare to the approximate equations.

2.2. When to approximate?

We as Engineers use approximations all the time. They make calculations easier or less complex, while still being accurate enough for the purpose.

But how do you know what is “close enough”?

The key skill is being aware of how much precision is needed for the final result or design. For example:

  • The clock’s frequency needs to be within ±X % so the timer is off by less than 1 second per day. (compute this X value!)

  • The amplifier’s gain needs to be 10±1 V/V.

  • The diode’s forward voltage at 25 °C and 1 mA of current needs to be 650 mV ±1%.

The Razavi book only uses the approximation and skips even showing the accurate formula for charge carrier density with doping. Assume that NA = 0 and the material is only doped with donors.

\[\begin{align} n &\approx N_D \\ p &\approx \frac{n_i^2}{N_D} \end{align}\]

  • So, when can you use the approximation for an error less than X%?

One nice way to figure this out is to come up with a relationship between the size of the variables that, when true, let you safely use the easy equation.

For less than 1% error, ND should be (less/greater) than compared to ni. Compute this condition.
1. See, for example, https://en.wikipedia.org/wiki/Mass_action_law_(electronics)