7.5 Shallow Defects
195
Fig. 7.15 Carrier concentration as a function of temperature for p-type Ge. The net shallow level concentration is
2 × 10 10 cm −3 . Solid line is fit to the data, the dashed line indicates the intrinsic hole concentration p i . Adapted
from [604]
due to various impurities (cf. Table 7.4). The different impurities (B, Al, Ga) can be distinguished by
photothermal ionization spectroscopy [604] (cmp. Sect. 9.8).
In Fig. 7.12, the temperature dependence of the Fermi level is included for p-type Si. With increasing
temperature the Fermi level shifts from the valence-band edge (For T = 0, E F = E V + E
b
A /2) towards
the middle of the band gap (intrinsic Fermi level).
Also, the wavefunction at acceptors can be imaged using scanning tunneling microscopy [605].
In [606] images of ionized and neutral Mn in GaAs have been reported (Fig. 7.16b). The tunneling
I –V characteristics are shown in Fig. 7.16a. At negative bias, the acceptor is ionized and appears
spherically symmetric due to the effect of the A
− ion Coulomb potential on the valence-band states.
At intermediate positive voltages, tunneling is through the neutral state. The wavefunction of A
0 looks
like a bow-tie due to the admixture of d-wavefunctions [607]. The Mn atom is presumably in the third
subsurface atomic layer. At even higher positive bias the contrast due to the dopant is lost because the
image is dominated by a large tunneling current from the tip to the empty conduction band.
7.5.3 Compensation
When donors and acceptors are simultaneously present, some of the impurities will compensate each
other. Electrons from donors will recombine with holes on the acceptors. Depending on the quantitative
situation the semiconductor can be n- or p-type. This situation can be invoked by intentional doping
with donors or acceptors or by the unintentional background of donors (acceptors) in p-doped (n-doped)
material. Also the formation of pairs, exhibiting a new defect level different from the single donor or
single acceptor, has been described, e.g. for Se and B in silicon [308].
The charge-neutrality condition (now finally in its most general form) reads
− n + p − N
−
A + N
+
D = 0 .
(7.40)
We will now discuss the case of the presence of donors and acceptors, but limit ourselves to sufficiently
low temperatures (or wide band gaps) such that the intrinsic carrier density can be neglected. We assume
Boltzmann statistics and assume here N D > N A . Then it is a very good approximation to use N
−
A = N A
195
Fig. 7.15 Carrier concentration as a function of temperature for p-type Ge. The net shallow level concentration is
2 × 10 10 cm −3 . Solid line is fit to the data, the dashed line indicates the intrinsic hole concentration p i . Adapted
from [604]
due to various impurities (cf. Table 7.4). The different impurities (B, Al, Ga) can be distinguished by
photothermal ionization spectroscopy [604] (cmp. Sect. 9.8).
In Fig. 7.12, the temperature dependence of the Fermi level is included for p-type Si. With increasing
temperature the Fermi level shifts from the valence-band edge (For T = 0, E F = E V + E
b
A /2) towards
the middle of the band gap (intrinsic Fermi level).
Also, the wavefunction at acceptors can be imaged using scanning tunneling microscopy [605].
In [606] images of ionized and neutral Mn in GaAs have been reported (Fig. 7.16b). The tunneling
I –V characteristics are shown in Fig. 7.16a. At negative bias, the acceptor is ionized and appears
spherically symmetric due to the effect of the A
− ion Coulomb potential on the valence-band states.
At intermediate positive voltages, tunneling is through the neutral state. The wavefunction of A
0 looks
like a bow-tie due to the admixture of d-wavefunctions [607]. The Mn atom is presumably in the third
subsurface atomic layer. At even higher positive bias the contrast due to the dopant is lost because the
image is dominated by a large tunneling current from the tip to the empty conduction band.
7.5.3 Compensation
When donors and acceptors are simultaneously present, some of the impurities will compensate each
other. Electrons from donors will recombine with holes on the acceptors. Depending on the quantitative
situation the semiconductor can be n- or p-type. This situation can be invoked by intentional doping
with donors or acceptors or by the unintentional background of donors (acceptors) in p-doped (n-doped)
material. Also the formation of pairs, exhibiting a new defect level different from the single donor or
single acceptor, has been described, e.g. for Se and B in silicon [308].
The charge-neutrality condition (now finally in its most general form) reads
− n + p − N
−
A + N
+
D = 0 .
(7.40)
We will now discuss the case of the presence of donors and acceptors, but limit ourselves to sufficiently
low temperatures (or wide band gaps) such that the intrinsic carrier density can be neglected. We assume
Boltzmann statistics and assume here N D > N A . Then it is a very good approximation to use N
−
A = N A