7.5 Shallow Defects
187
Fig. 7.7 Arsenic impurity
in silicon. Arsenic donates
one electron, and a fixed
positive charge remains
Si
Si
Si
As
Si
Si
Si
Si
7.5.1 Donors
Silicon doped with arsenic is denoted as Si:As. The situation is schematically shown in Fig. 7.7. The
arsenic atom has, after satisfying the tetrahedral bonds, an extra electron. This electron is bound to
the arsenic atom via the Coulomb interaction since the ion core is positively charged compared to the
silicon cores. If the electron is ionized, a fixed positive charge remains at the As site.
Without being in the silicon matrix, an arsenic atom has an ionization energy of 9.81 eV. However,
in the solid the Coulomb interaction is screened by the dielectric constant of the material, typically r is
of the order of 10 for typical semiconductors. Additionally, the mass is renormalized (effective mass)
by the periodic potential to a value that is smaller than the free electron mass. Within effective-mass
theory (Appendix I) the hydrogen problem is scaled with the (isotropic) effective mass m
∗
e and the
dielectric constant r , the binding energy (ionization energy) E
b
D of the electron to the shallow donor
is (relative to the continuum given by the conduction-band edge E C )
E
b
D =
m
∗
e
m 0
1
2
r
m 0 e
4
2 (4ππ 0 ) 2 .
(7.21)
The scaling with 1//
2 has been pointed out first in [576].
The absolute energy position of the level is E D = E C − E
b
D . The first factor in the right side of (7.21)
is the ratio of effective and free-electron mass, typically 1/10, the second factor is typically 1/100. The
third factor is the ionization energy of the hydrogen atom, i.e. the Rydberg energy of 13.6 eV. Thus,
the binding energy in the solid is drastically reduced by a factor of about 10
−3 to the 10 meV regime.
The excited states of the hydrogen-like spectrum can also be investigated experimentally (Sect. 9.8).
The extension of the wavefunction of the electron bound to the fixed ion is given by the Bohr radius
a D =
m 0
m ∗
e
r a B ,
(7.22)
where a B = 0.053 nm denotes the hydrogen Bohr radius. For GaAs a D = 10.3 nm. A similar value has
been determined for InP [577]. For semiconductors with a nonisotropic band minimum, such as Si, Ge
or GaP, an ‘elliptically deformed’ hydrogen problem with the masses m l and m t has to be treated [578].
An impurity that fulfills (7.21) is called an effective-mass impurity. For GaAs, the effective-mass
donor has a binding energy of 5.715 meV, which is closely fulfilled for several chemical species
(Table 7.3). In GaP, experimental values deviate considerably from the effective-mass donor (59 meV).
For silicon, considering the anisotropic tensor of the effective masses, the result for the effectivemass donor binding energy is 29 meV [578]. Some experimentally observed values are summarized
in Table 7.2. Deviations from the effective-mass theory are due to modification of the potential in the
immediate vicinity of the impurity atom and breakdown of the effective-mass formalism.
Different impurities can have quite similar binding energies. They can be distinguished, e.g., by
electron spin resonance (ESR). At low temperatures the electron is localized on the impurity and the
hyperfine interaction with the nucleus can be resolved in ESR. In Fig. 7.8 data are shown for As and P
187
Fig. 7.7 Arsenic impurity
in silicon. Arsenic donates
one electron, and a fixed
positive charge remains
Si
Si
Si
As
Si
Si
Si
Si
7.5.1 Donors
Silicon doped with arsenic is denoted as Si:As. The situation is schematically shown in Fig. 7.7. The
arsenic atom has, after satisfying the tetrahedral bonds, an extra electron. This electron is bound to
the arsenic atom via the Coulomb interaction since the ion core is positively charged compared to the
silicon cores. If the electron is ionized, a fixed positive charge remains at the As site.
Without being in the silicon matrix, an arsenic atom has an ionization energy of 9.81 eV. However,
in the solid the Coulomb interaction is screened by the dielectric constant of the material, typically r is
of the order of 10 for typical semiconductors. Additionally, the mass is renormalized (effective mass)
by the periodic potential to a value that is smaller than the free electron mass. Within effective-mass
theory (Appendix I) the hydrogen problem is scaled with the (isotropic) effective mass m
∗
e and the
dielectric constant r , the binding energy (ionization energy) E
b
D of the electron to the shallow donor
is (relative to the continuum given by the conduction-band edge E C )
E
b
D =
m
∗
e
m 0
1
2
r
m 0 e
4
2 (4ππ 0 ) 2 .
(7.21)
The scaling with 1//
2 has been pointed out first in [576].
The absolute energy position of the level is E D = E C − E
b
D . The first factor in the right side of (7.21)
is the ratio of effective and free-electron mass, typically 1/10, the second factor is typically 1/100. The
third factor is the ionization energy of the hydrogen atom, i.e. the Rydberg energy of 13.6 eV. Thus,
the binding energy in the solid is drastically reduced by a factor of about 10
−3 to the 10 meV regime.
The excited states of the hydrogen-like spectrum can also be investigated experimentally (Sect. 9.8).
The extension of the wavefunction of the electron bound to the fixed ion is given by the Bohr radius
a D =
m 0
m ∗
e
r a B ,
(7.22)
where a B = 0.053 nm denotes the hydrogen Bohr radius. For GaAs a D = 10.3 nm. A similar value has
been determined for InP [577]. For semiconductors with a nonisotropic band minimum, such as Si, Ge
or GaP, an ‘elliptically deformed’ hydrogen problem with the masses m l and m t has to be treated [578].
An impurity that fulfills (7.21) is called an effective-mass impurity. For GaAs, the effective-mass
donor has a binding energy of 5.715 meV, which is closely fulfilled for several chemical species
(Table 7.3). In GaP, experimental values deviate considerably from the effective-mass donor (59 meV).
For silicon, considering the anisotropic tensor of the effective masses, the result for the effectivemass donor binding energy is 29 meV [578]. Some experimentally observed values are summarized
in Table 7.2. Deviations from the effective-mass theory are due to modification of the potential in the
immediate vicinity of the impurity atom and breakdown of the effective-mass formalism.
Different impurities can have quite similar binding energies. They can be distinguished, e.g., by
electron spin resonance (ESR). At low temperatures the electron is localized on the impurity and the
hyperfine interaction with the nucleus can be resolved in ESR. In Fig. 7.8 data are shown for As and P