Set of supplementary notes.

7.3. DOPED SEMICONDUCTORS 103
smaller than the free electron mass, so the kinetic energy of an electron in a given momentum
state is larger. The net effect is that the binding energy of the 1s impurity state is now
∆ d =
e 4 m ∗ c
2(4πɛɛ o h¯) 2 = m∗ c/m e
ɛ 2 × 13.6 eV (7.15)
which can be very small in comparison to the band gap, and often comparable or smaller than
thermal energies. Such donor impurities easily donate electrons to the conduction band.
The binding energy of the electronic states of the hydrogen atom express the energy difference
between the lowest vacuum state and the respective bound states. The hydrogen-like
bound impurity states we calculated now are referenced to the bottom of the conduction band,
because the electron unbinds from the impurity by occupying a conduction band state, just as
an electron unbinds from the Hydrogen atom by assuming a plane-wave vacuum state.
As donors in Si have an ionisation energy of 50 meV; donors in GaAs have an ionisation
energy of about 6 meV, approximately 50 Kelvin. When a donor atom is ionised, it releases its
formerly bound electron into the conduction band.
Acceptor levels. A trivalent impurity (e.g. B in Si) appears like a Si atom with an added
negative charge, and with a missing electron. It is the mirror image of the case of the donor
impurity, and corresponds to a positive charge (a ’hole’) circling a negatively charged nucleus.
As in the case of donor atoms, the binding energy of the hole is reduced by the combined
effects of high permittivity and low band mass in semiconductors. When the hole unbinds, the
impurity accepts an electron from the valence band. The accepted electron is used to complete
the covalent bond with the neighbouring atoms, and renders the site negatively charged. So,
while ionising a donor atom releases an electron into the conduction band, ionising an acceptor
atom absorbs an electron from the valence band, which leaves a hole in the valence band.
n- and p-type materials. Even for very low densities of impurities, since the donor or
acceptor energies are so much smaller than the gap, impurities in semiconductors are often
the principal source of electrically active carriers. If donor atoms predominate, the carriers are
predominantly electrons, and the material is said to be n-type. If holes are the dominant carrier
type, the material is called p-type. Experimentally, these regimes may be distinguished by a
measurement of the Hall effect, whose sign depends on the type of carrier.
Impurity ionisation. We just quote here the results for thermal ionisation of the carriers in
simple limits. If there are no acceptors, then the carrier concentration at low temperatures is
n = (n c N d ) 1/2 e − ∆ d
2k B T
(7.16)
where N d is the donor density and the factor n c = 2(m ∗ ek B T/2πh¯ 2) 3/2 is the effective density of
electron states within an energy k B T of the band edge. Notice again that the activation energy
is half the binding energy.
Since ∆ d is small and n c (T ) is usually large compared to N d , donor atoms are fully ionised
down to very low temperatures, and n ≈ N d . This is called the extrinsic regime. At a still
higher temperature, the intrinsic carrier generation by thermal activation from the valence
band into the conduction band takes over.
If there are only acceptors and no donors, then a similar formula can be obtained for holes by
inspection. When both donors and acceptors exist, the problem is in general more complicated.