Equation for finding electrical resistance

Question

I want to know how the flow of electrons will change when I change the tempertature from 100 F to 250 F in a silicon semi-conductor (ex: computer mouse)

How can I find this out?

Answer 1

If you consider a homogeneous piece of silicon the total flow of electrons through it is: $$ I = \frac{U}{R} = n \mu \frac{S}{d} U $$ where
$R$ - the resistance of the piece, $U$ - external voltage applied to it.

The resistance depends on:
$n$ - the concentration of electrons (number of electrons per m$^3$),
$\mu$ - the mobility of electrons (ratio of velocity of the electron and electric field that makes it move),
$S$ and $d$ - cross-section and length of the sample.

The changes of geometric size with temperature are negligible. So the values that affect the resistance are concentration $n$ and mobility $\mu$.

The mobility depends on the temperature and also on the concentration of various defects in the sample. At 100 F the temperature dependence is dominating.

The concentration is the most complicated point. There are the following cases:

1. Pure silicon. All the electrons (and the same amount of holes) are thermally generated. Their concentration depends on the temperature exponentially. If you need total current don't forget about the holes.
2. Silicon doped with donors. The amount of thermally generated electrons is negligible. The concentration does not depend on the temperature.
3. Semiconductor device with p-n junction or/and heterojunction (connection of different materials). The laser/LED of optical computer mouse is this case. The sample is not homogeneous and the concentration is determined mainly not by temperature but by more interesting things like voltage polarity. This case requires more formulas and exact data concerning the sample structure.

   ! The laser is made of GaAs and similar materials not silicon. The attempts to make silicon laser never stop though.

Edited (2011/12/15):

For the temperature dependence of electron mobility Wikipedia gives $$ \mu(T) \approx \mu_0 T^{-2.4} $$ where
$\mu_0 = 9.46 \cdot 10^{6} \text{m}/\text{(V s)}$, hope I've calculated it correctly from first point set (black circles) here
This formula takes into account only electron scattering on the oscillations of the ions of the crystal. This effect is dominating at room temperature and higher. The temperature must be in Kelvin degrees here.

For the electron $n$ and hole $n_h$ concentration in pure silicon (case 1.) at room temperature and higher one can use the following formula: $$ n = n_h = N_\text{eff} \; T^{\;3/2} \exp \left( -\frac{E_g}{2k_B T} \right) $$ where
$N_\text{eff}$ - some constant describing the shape of conduction and valence bands of silicon (I have not found explicit value yet),
$T$ - temperature in Kelvin degrees,
$E_g = 1.12\;\text{eV} = 1.79 \cdot 10^{-19} J$ - energy gap of silicon,
$k_B$ - Boltzmann constant.