PX262 - L2 - n- and p-type semiconductors

these include semiconductors, such as Si and Ge, doped with a very low concentration of impurities (1 part in $10^{5}$ ) such as P, As or B, Ga, which have either one more or one less valence electron per atom

the impurities can either donate an electron into the conduction band (P, As), or remove one (making a 'hole') from the valence band (B, Gn)
the substitution of a Si atom by an impurity (eg P) can be modelled by adding a fixed positive charge, $+ e$ , $(Z_{P} = Z_{S i} + 1)$ at a site along with an additional electron
such a centre of positive charge can bind one of the additional electrons
the donor impurities can introduce additional electronic states close to the bottom of the conduction bands
electrons are easily thermally excited into the conduction bands from those levels at room temperature and the majority of the carriers are electrons creating an n-type semiconductor

a similar model can be applied for the acceptor impurities, eg: B, Ga
the model is now a superposition of a fixed $- e$ $(Z_{a} = Z_{S i} - 1)$ on top of the host atom with the absence of 1e electron in the crystal
the affect of the 'missing' electrons is represented by additional unfilled energy levels, 'holes', at $E_{a}$ , just above the valence bands

$E_{a} - E_{v}$ is the energy needed to excite an electron from the top of the valence bands into an acceptor energy level filling a 'hole' near one of the acceptor impurities
it makes a free 'hole' / positively charge in the valence band

the difference between the n- and p-type semiconductors is the position of the chemical potential, $μ$ ( $E_{F}$ at $T = 0$ K)
in p-type semiconductors, the majority carriers are 'holes', where the number of holes $\propto \exp (- (E_{a} - E_{v}) / k_{B} T)$
the minority carriers are electrons, their number $\propto \exp (- (E_{c} - E_{v}) / k_{B} T)$
in n-type semiconductors, the majority carriers are electrons, their number $\propto \exp (- (E_{c} - E_{d}) / k_{B} T)$
the minority carriers are holes, their number $\propto \exp (- (E_{c} - E_{v}) / k_{B} T)$
dopants in Si: $Δ = 1.1$ eV
- $E_{c} = E_{d} = 0.044$ eV (P), $= 0.049$ eV (As)
- $E_{a} - E_{v} = 0.046$ eV (B), $= 0.065$ eV (Ga)