Chemistry Reference
In-Depth Information
removes a negatively-charged electron from the filled valence band. This
is equivalent to introducing a positively-charged 'hole'. Boron is referred
to as a p-type dopant ('p' for positive). Atoms which contribute an extra
electron are often referred to as 'donors', while those which remove an
electron from the valence band are called 'acceptors'.
In addition, there are other impurity atoms such as nitrogen, which intro-
duce defect states in the Si energy gap, well away from the band edges.
These are referred to as deep levels. They can often act as non-radiative
recombination centres, trapping free carriers moving through the crys-
tal, and hence nullifying the effect of any shallow impurities present. The
control of impurity dopant atoms is the key to almost all semiconductor
technology. Many of the key characteristics of shallow and deep impu-
rity levels can be understood based on the models developed in earlier
chapters.
4.5.1 Shallow impurities
Consider adding an extra electron to a semiconductor. The electron is free to
move, with an effective mass,
m
c
, at the bottom of the conduction band. In
practice the extra electron is introduced via an impurity atom, for example,
by replacing a Si atomwith an arsenic atom in a Si crystal. Aneutral arsenic
atom has five valence electrons which can contribute to bonding. Four of
the electrons can form bonds with the neighbouring Si atoms, leaving one
electron free to move through the crystal. This electron will then see a
net positive charge associated with the arsenic impurity atom, and will
experience a potential
e
2
r
is
the dielectric constant of the Si crystal. It can be shown that we can write
the Hamiltonian
H
describing the motion of the electron as
−
/
4
πε
ε
r
r
due to that positive charge, where
ε
0
e
2
2
m
c
∇
2
H
ψ(
r
)
=
−
−
ψ(
r
)
=
E
ψ(
r
)
(4.19)
4
πε
ε
r
r
0
that is, treating the electron as if it were a particle with mass
m
c
moving in
the impurity potential, with the zero of potential in eq. (4.19) taken to be
at the bottom of the conduction band.
Equation (4.19) is identical to the Hamiltonian for an electron in an
isolated hydrogen atom discussed in Appendix A, if we replace the free
electron mass
m
by the effective mass,
m
c
, and the free space permittivity
ε
r
in eq. (A.2). The energy of the
n
th
bound state of an isolated hydrogen atom,
E
n
, is given in Appendix B by
0
by the semiconductor permittivity
ε
ε
0
me
4
E
n
=−
(4.20)
2
8
(ε
0
hn
)
