shows that these two conditions hold if the ionization and recombination coefficients satisfy
the following inequalities:
2
A 0 + B
I
T
+
2 A 0 I
− ND + NA
2 BT
+
σ (
1 < 0,
ND − NA)
σ ( ND − NA)
ND − NA σ ( ND − NA)
+
2
N
B
+ B
B
A
T
− A 0
1 −
A 0 I
T
T
>
ND− NA
I
σ (
8 +
A 0 I
27
ND − NA)
σ ( ND − NA)
σ ( ND − NA) .
(11)
In figure 4 the area in the A 0 B
I
T plane satisfying rel.11 is shown for several values of the ratio
ND/ NA. It is quite evident, from inspection of the figure, that once known the screening and
the recombination coefficients, the bandgap can be tailored, by suitable choice of ND and NA,
in order to manifest photoconductive bistability.
Let us consider, now, the Jacobian matrix T nn of the system 6:
t
∂
T =
nt ∂ ( A
n
I n) − BT ( ND − nt)
AIn + AR + BTn
−
;
(12)
n ∂
t ∂ ( A
n
I n) + BT ( ND − nt)
− AIn − AR − BTn
532
Optoelectronic Devices and Properties
c
g
W
a
b
(a)
b
g
r
wg
Bistable
Monostable
C
C
g
Bistable
a
r
w
Monostable
a
b
P
a
0
(b)
Fig. 3. (a) Diagram of the Ω region in the αβγ space, whose points correspond to three states
of equilibrium, two stable and one instable; χ is the geometric locus of the cusp points.
(b-left) Section of the Ω region in a plane with α = const. (b-right) Section of the Ω region in a
plane with γ = const.
according to the stability criterion of the non-negativity of the off-diagonal terms, stated in the
preceding section, the observation of the Jacobian matrix simply gives the indication that, in
order for the bistability to occur, the impact ionization term AI has to be nonnull. Compared
with eq. 11 this seems a quite poor and qualitative information, but we will verify how, in
more complex situations, with an arbitrary number of energy levels, for instance, this kind of
insight will be of unvaluable usefullness.
3.2 impact excitation and ionization of metastable states
A widely adopted model of impact ionization introduced by Schöll (1987), suggests that
the impact ionization processes of impurity multilevels can induce bistable conduction in
semiconductors. Here we extend the model to the photo-ionization of the impurity levels
and neglect thermal excitation and ionization of the deep donor centers. Given n, n 0 and n 1
the population of the conduction band, the ground and the excited level of the donor centers,
Bistable Photoconduction in Semiconductors
533
BT
9
A 0
s
I
B
+ =
T
N
( D-N
5
A)
3
2
1.5
ND
0
N =1.25
AI
A
Fig. 4. In grey, the areas in the AI BT plane which correspond to bistable situations, for
different values of the constant ND/ NA.
respectively, the rate equations of the system are:
˙ n = ( AI 1 n + AR 1) n 1 + ( AI 2 n + AR 2) n 2 − BTn ( ND − n 1 − n 2) ,
˙ n 2 = − ( AI 2 n + AR 2) n 2 + BTn ( ND − n 1 − n 2) − Rn 2,
˙ n 1 = − ( AI 1 n + AR 1) n 1 + Rn 2,
with
n + n 1 + n 2 = ND.
(13)
Given x = n/ ND, and considering that the ratio of the generation terms AR 2/ AR 1 ≡ ρ the solving equation at the equilibrium is again of the type of eq.8, with the coefficients α, β and
γ equalling:
α = ND AI 1 AI 2 − R ( AI 1 + BT) − AR 1 ( ρAI 1 + AI 2 + BT) ND AI 1 ( AI 2 + BT)
,
ρA 2 + A
β =
R 1
R 1 ( R − ρAI 1 ND − AI 2 ND) − AI 1 RND ,
N 2 A
D
I 1 ( AI 2 + BT )
ρA 2 + A
γ =
R 1
R 1 R
.
(14)
N 2 A
D
I 1 ( AI 2 + BT )
In this case, the curve describing the system in the αβγ space is of the second order in
the generation term, and the algebraic discussion of its intersection with the region Ω is
considerably more complex.
The stability criterion of the non-negativity of the off-diagonal entries of the Jacobian matrix
of the system can give useful insight in the physics of the processes. These have the following
values:
T nn = A
= A
1
I 1 n + AR 1 + BT n,
T nn 2
I 2 n + AR 2 + BT n,
∂
T n
( A
= R,
1 n = − n 1 ∂n
I 1 n) ,
T n 1 n 2
∂
T n
( A
= − B
2 n = BT ( ND − n 1 − n 2 ) − n 2 ∂n
I 2 n) ,
T n 2 n 1
T n,
(15)
534
Optoelectronic Devices and Properties
As expected, also for the Schöll model, the non-negativity of the off diagonal entries, along
with the stability of the system, are not guaranteed if the impact ionization coefficients AI 1,2
are nonnull. Moreover, one of the non-diagonal terms is negative independently on the
ionization mechanism, only due to the new recombination channel given by the metastable
state, determining the term − BTn in eqs.15. This seems to open the way to some kind of
different instability mechanism, independent on the electric field and determined only by
the competition of different recombination channels. The next section will be devoted to the
throughout investigation of all the possible mechanisms which can break the sign-simmetry
of the Jacobian matrix of a sistem subject to ionization-recombination processes. The analysis
will evidence the existence of two possible ways toward instability in photoconductors with
zero electric field.
4. Dynamical instability in ionization-recombination reactions
If rik represents the number of electrons that, in the unit time, change their state from k to i, the functions fi in eq.1 have the form
fi = ∑ ( rik − rki)
(16)
k= i
The transition rates rik, in their turn, depend on the occupation numbers Nl in one of the
following two ways:
excitation-relaxation
rik = Rik ({ Nl}) N A
k
(17)
ionization-recombination rik = Sik ({ Nl}) N A
k N F
i
(18)
where N A is the number of electrons in the state k which are available to make a transition
k
to another state, while N F is the number of free places in the state i. Transition probabilities
i
Sik and Rik may depend or not on the occupation numbers { Nl}, according to the nature
of the reactions involved. For example, in impact ionization and in Auger recombination, Sik
depends on the number of free electrons. N A and N F, on the other hand, depend on the nature
k
i
of the state labelled by the index. Figure 5 gives a schematics of the various cases involved in
neutral and singly ionized atoms (left side of the figure), multiply ionized atoms (right side)
and free states (top of the figure).
• Neutral or singly ionized atoms. Since all the valence electrons can be excited or extracted
from the atoms, the number N A of electrons available to make a transition is simply N
k
k.
On the other hand, the number of “free places" for an electron to recombine is given by the
number of ions, then
N A =
k
Nk
N F =
Δ
i
gi − ∑ ilNl,
(19)
l
where gi is the number of atoms of the specie labelled by i, while Δ il is equal to one only if i and l refer to states of a same atomic specie, otherwise is zero (notice that the index labels
both the atomic specie and the energy level).
Bistable Photoconduction in Semiconductors
535
N A l= Nl
N =
F
l
gl - Nl
N
l
l
configurations
r
r
r
r
r
il
il
ki
kl
lk
N
N
i
k+
+
k
i
levels
r
k
N
N
ii’
i’
k
i’
k -
Nk-
F
F
excited
A
N
F
A
F
Nk = N
i
N
=
k-
N A
i’
N A
N
k = N
i’
i
+
k
neutral-singly ionized
multiply ionized
Fig. 5. A schematics of the transitions described by the model represented by eq. 17 and 18.
On the left side, a set of atoms of a given specie is represented, each with its level diagram,
three of whom are neutral and two singly ionized. On the right side, a set of neutral, singly
or multiply ionized atoms are represented. In this case, the excited levels of each
configuration are disregarded, in the assumption that their population is ruled by lattice or
free charges temperature. The upper side represents the unbound state.
• multiply ionized atoms. Let the indexes k−, k and k+ indicate three adjacent energy levels in a given atomic configuration. An electron in the state k can make a transition in low energy
processes only if no other electrons occupy the state k+ in the same atom. Analogously, a
free state k can be reached by an unbound electron only if the state k− is occupied in its
turn. Consequently, we have
N A =
k
Nk − Nk+
N F =
i
Ni− − Ni
(20)
• Free states. If gi is the degeneration of an unbound state, we can obviously state
N A =
k
Nk
N F =
i
gi − Ni
(21)
It is worth noting that, if we formally state Nk+ = 0 and Ni− = gi, the present case reduces to the previous one, which will be useful in the following discussion.
Once the expressions for the transition probabilities rik are determined by eq.17-18, the entries
of the matrix T ij are calculated from their derivatives with respect to the occupation numbers,
as given in eq.2 , and can be then expressed as a sum of two contributions:
T ij = T nc + T c
ij
ij
with
∂N A N F
∂N A N F
∂N A
∂N A
T nc =
k
i −
i
k
+
k −
i
ij
∑ Sik ∂
S
R
R
and
N
ki
∂N
ik ∂N
ki ∂N
k= i
j
j
j
j
∂
T c =
Sik
− ∂Ski
+ ∂Rik
− ∂Rki
ij
∑ ∂ N A
N A
N A
N A
N
k N F
i
∂N i N F k
∂N
k
∂N
i
.
(22)
k= i
j
j
j
j
536
Optoelectronic Devices and Properties
The apices “nc" and “c" refer respectively to “non catalytic" and “catalytic" processes. In fact, T c is non-zero only if the transition probabilities S
ij
ik and Rik depend explicitly on { Nl }, that
is, if the transition of an electron from state k to i is catalyzed or somewhat influenced by
the presence of electrons in some specific quantum state. If this is not the case, only the T nc
ij
(non-catalytic) term is non zero. In this last case, the equations of the system 1 are of the
second degree in { Ni}. Indeed, non linearity of the second order has been known to be a cause
of instability in dynamical systems since the very beginning of chaos studies (Lorentz, 1963;
Roessler, 1976). Nevertheless, almost without exceptions, only catalytic processes have been
considered as possible mechanisms leading to instability in semiconductors. In the following
we will study non-catalytic processes, the only ones which can hold in absence of an externally
applied electric field.
• Neutral or singly ionized atoms. The rate equations assume the form (from eq.16,19,22)
˙
Ni = ∑
Sil gi − ∑ Δ ikNk + Ril Nl