7.7 Deep Levels
207
(a)
(b)
Fig. 7.29 a Silicon cubic unit cell with an interstitial iron atom (red) at tetrahedral site. b EPR intensity (at T = 95 K
from interstitial iron in neutral state, Fe 0 with S = 1) versus Fermi level position for iron-doped silicon with varying
Fermi level due to different amounts of shallow impurity levels from to Al, B and P as labeled. The shaded areas indicate
the valence and conduction band. The dashed line at E t = E V + 0.375 eV indicates the trap level. The inset shows a
typical EPR spectrum of Fe 0 . Adapted from [649], inset adapted from [650]
where N 0 = N Z 0 and N + = N Z + are the number of available sites, given by the number N of atomic
sites and including possible internal degeneracies Z 0 and Z + , respectively. Degeneracy factors of deep
levels are not a simple subject [601] and , e.g., the degeneracy factors of Au donor and acceptor levels
in Si are under discussion [644–646]. G f denotes the free enthalpy of formation of the respective
defect, as in (4.3). We have written the free enthalpy of the separated pair V
+ and e
− as the sum
G(V
+
) + G(e
−
). μ e − = ∂G(e
−
)/∂n + is (by definition) the chemical potential of the electron, i.e. the
Fermi energy E F of Fermi–Dirac statistics.
6 From (7.62a,b) we find for the ratio of the concentrations
of defects c 0 = n 0 /N and c + = n + /N
c 0
c +
=
Z +
Z 0
exp
−
G
f
V + − G
f
V 0 + E F
kT
=
Z +
Z 0
exp
E t (V
0
) − E F
kT
,
(7.63)
where the trap level energy (for the particular charge transition), E t (V
0
) = G
f
V 0 − G
f
V + , is the free
enthalpy of ionization of V
0 . We note that c 0 can be obtained from (4.9) and E F is determined by the
charge-neutrality condition.
As example experimental data on the charge transition Fe
0
Fe
+
+ e
− of interstitial iron (in
tetrahedral position, Fig. 7.29a, cmp. Fig. 3.18) in silicon is shown. The concentration of Fe
0 is tracked
via the EPR signal from the neutral S = 1 state
7 with g-factor g = 2.07 [647]. For n-type samples
the iron is in neutral state and the maximum EPR signal is found. For strongly p-type samples, the
Fermi energy is below the trap level and all iron is in Fe
+ state, yielding no EPR signal at the given gfactor. From the investigation of various silicon samples with different doping levels and consequently
6 The chemical potential in a one-component system is μ = ∂G/∂n = G/n. In a multicomponent system it is, for the
ith component, μ i = ∂G/∂n i = G/n i .
7 The electron configuration is 3d 8 with two paramagnetic electrons. Under uniaxial stress along [100] the EPR line splits
into a doublet [647]. Further details can be found in [648].
207
(a)
(b)
Fig. 7.29 a Silicon cubic unit cell with an interstitial iron atom (red) at tetrahedral site. b EPR intensity (at T = 95 K
from interstitial iron in neutral state, Fe 0 with S = 1) versus Fermi level position for iron-doped silicon with varying
Fermi level due to different amounts of shallow impurity levels from to Al, B and P as labeled. The shaded areas indicate
the valence and conduction band. The dashed line at E t = E V + 0.375 eV indicates the trap level. The inset shows a
typical EPR spectrum of Fe 0 . Adapted from [649], inset adapted from [650]
where N 0 = N Z 0 and N + = N Z + are the number of available sites, given by the number N of atomic
sites and including possible internal degeneracies Z 0 and Z + , respectively. Degeneracy factors of deep
levels are not a simple subject [601] and , e.g., the degeneracy factors of Au donor and acceptor levels
in Si are under discussion [644–646]. G f denotes the free enthalpy of formation of the respective
defect, as in (4.3). We have written the free enthalpy of the separated pair V
+ and e
− as the sum
G(V
+
) + G(e
−
). μ e − = ∂G(e
−
)/∂n + is (by definition) the chemical potential of the electron, i.e. the
Fermi energy E F of Fermi–Dirac statistics.
6 From (7.62a,b) we find for the ratio of the concentrations
of defects c 0 = n 0 /N and c + = n + /N
c 0
c +
=
Z +
Z 0
exp
−
G
f
V + − G
f
V 0 + E F
kT
=
Z +
Z 0
exp
E t (V
0
) − E F
kT
,
(7.63)
where the trap level energy (for the particular charge transition), E t (V
0
) = G
f
V 0 − G
f
V + , is the free
enthalpy of ionization of V
0 . We note that c 0 can be obtained from (4.9) and E F is determined by the
charge-neutrality condition.
As example experimental data on the charge transition Fe
0
Fe
+
+ e
− of interstitial iron (in
tetrahedral position, Fig. 7.29a, cmp. Fig. 3.18) in silicon is shown. The concentration of Fe
0 is tracked
via the EPR signal from the neutral S = 1 state
7 with g-factor g = 2.07 [647]. For n-type samples
the iron is in neutral state and the maximum EPR signal is found. For strongly p-type samples, the
Fermi energy is below the trap level and all iron is in Fe
+ state, yielding no EPR signal at the given gfactor. From the investigation of various silicon samples with different doping levels and consequently
6 The chemical potential in a one-component system is μ = ∂G/∂n = G/n. In a multicomponent system it is, for the
ith component, μ i = ∂G/∂n i = G/n i .
7 The electron configuration is 3d 8 with two paramagnetic electrons. Under uniaxial stress along [100] the EPR line splits
into a doublet [647]. Further details can be found in [648].