Bistable Photoconduction in Semiconductors
539
10-1
103
K
( ;r)
2
K 2
10-2
(1;0.25)
K
=1.5
1
K 1
(10;0.25) (100;0.25)
102
10-3
3
6
1x10-4
(0.1;0.031)
R 2
101
R 2
12
(1000;0.25)
1x10-5
25
100
50
10-6
100
(0.01;0.0016)
10-7
10-1
10-2
10-1
100
101
102
103
104
105
106
107
100
101
102
103
R
R
1
1
Fig. 7. (left) bistability by excited levels: lines which define the bistable region of the plane
R 1 R 2 for different values of K 2 and for the value of ρ which make maximum the area of the K 1
bistable region itself. (right) bistability by multiply charged centers: lines which define the
bistable region of the plane R 1 R 2 for different values of K 2 .
K 1
It is found that the inequality 32 is satisfied if, for given values of K 2 =
C 2 n and
K 1
C 1 n
ρ =
R 21
√
, the points ( R
N C
1, R 2), belong to the area below a given line. Since for a given
1 nC 2 n
recombination center the parameter ρ is inversely proportional to the doping density N, the
bistable zone below the line can be made maximum by tuning the parameter ρ. In figure 7
(left), several lines are represented with 0.01 ≤ K 2 ≤ 1000, with the values of ρ corresponding
K 1
to the maximum area. From their inspection, bistability results to be possible practically for
all the values of R 1, if R 2 is sufficiently small and/or K 2 is not too small or large respect to the K 1
unit. As a matter of fact, capture of an electron in the excited state by a positive center seems
much more likely than capture of a hole by the neutral excited state, especially if the latter is
closer to the conduction band than to the valence one, so that R 2 = C 2 p is likely very small. In C 2 n
conclusion, bistable photoconduction in semiconductors, if recombination is assured by deep
centers with metastable states, appears likely to be feasible by suitable choice of the doping
parameters M and N and of the irradiation conditions.
6. Bistability by multiply ionized recombination centers
Photoconductive instability is possible, in case of multiply charged recombination centers, if
the capture of a carrier by a neutral center is less likely than trapping by the center which
have the same charge of the carrier itself, in spite of columbic repulsion (see section 4). This
seemingly unlikely condition happens to be verified by several transition metal impurities in
germanium, silicon, silicon carbide (Grillenberger, 2004; Knack et al., 2002; Sachse et al., 2000;
Shiraishi et al., 1999; Wertheim, 1959; Yarykin et al., 1999), both in case of doubly and triply
ionized centers. A treatment of photoconduction in semiconductor in which recombination
is assured by multiply ionized centers can be found in ref.(Lagomarsino, 2008), where cases
of double or triple donors or acceptors, donor-acceptors, double donor-acceptors, ecc. are
studied in a general perspective. Here we restrict to the case of double acceptor impurities,
540
Optoelectronic Devices and Properties
which happens to comprehend the more interesting practical case. The rate equations of the
system, in this case, are the following (see also fig.5 right):
˙ n = g − C 2 nn ( n 1 − n 2) − C 1 nn ( M − n 1) ,
˙ n 2 = C 2 nn ( n 1 − n 2) − C 2 p pn 2,
˙ n 1 = C 1 nn ( M − n 1) − C 1 p p ( n 1 − n 2) ,
˙ p = g − C 2 p pn 2 − C 1 p p ( n 1 − n 2) ,
n + n 1 + n 2 − p − N = 0
(33)
We will consider all the parameters and the unknown quantities expressed in terms of the
following, non-dimensional parameters:
Cip
x = p , y = n , y
R
,
n
N
i = ni
M
i = Cin
Ki =
Cin
, G =
g
.
(34)
( C 1 nC 2 n)1/2
NM ( C 1 nC 2 n)1/2
As a consequence, the second and the third of Eqs. 33 give, at equilibrium ( ˙ ni = 0), the
following conditions:
y 1 =
1 + R 2 x
,
y
.
(35)
1 + R
2 =
1
2 x + R 1 R 2 x 2
1 + R 2 x + R 1 R 2 x 2
Their substitution in the first and the last of eqs.33 gives:
1 + R
y = G
2 x + R 1 R 2 x 2 ,
K 2 R 2 x + K 1 R 1 R 2 x 2
2 + R
y =
1
1 − M
2 x
.
(36)
1 − x
N 1 + R 2 x + R 1 R 2 x 2
Although the solving equation of the system, in the general case, is of the fifth degree in
the unknown x, if the concentration of deep centers is much lower than that of the dopants
( M << N), the second of the Eqs.36 reduces to: y = 1 , and the solving equation reduces to 1− x
one of the third degree of kind x 3 − αx 2 + βx − γ = 0 (eq.8), with
α =
K
1 − 1 − K 1 ,
β = 1
2 + 1 − 1 , γ = 1 .
(37)
R 1
G
R 1
G
R 2
R 1 R 2
In order to find conditions for bistability, one has to state which values of the parameters Ki
and Ri make the coefficients α, β, γ to satisfy the inequalities 10 for some positive value of the control parameter G. Since G is not contained in γ, we can consider, for a given value of γ, the 2-dimensional set ωγ of the couples ( α, β) which satisfies the inequalities 10. The shape of ωγ
is shown in fig.3b (right), with a cusp point C( γ) whose coordinates are
C ( γ) = ( αγ, βγ) = 3 γ 13 , 3 γ 23 .
(38)
Bistable Photoconduction in Semiconductors
541
By elimination of G in the first two equations of 37, we find that the coefficients α and β define a straight line r with a negative angular coefficient in the plane αβ. This line crosses the region ωγ if and only if the cusp point Cγ lies below it, that means
β
K 2
γ + K 2 αγ < 1
1 − 1
+ 1 − 1 .
(39)
K 1 R 1
R 1 K 1
R 1
R 2
From this expression, taking into account expressions 38 and 37, it turns out that
recombination bistability is possible if and only if the following inequality holds:
⎛
⎞
1
1
R
3
3
3 ⎝ K 1
1
+
1
⎠ + K 1 1 − 1 + 1 < 1
(40)
K 2
R 2
R
K
R
R
2
1 R 2
2
2
1
This inequality gives a necessary and sufficient condition for having photoconductive
multistability driven by the generation factor g. In the three-dimensional space-parameters
given by R 1, R 2 and K 2 , the region satisfying the inequality 40 is divided into two distinct K 1
branches. The first branch lies in the region with K 2 > 1 and R
K
1 > 1, the second one in the
1
"slice" 0 < R 2 < 1. Nevertheless, only the first branch has a physical meaning, because the values of the parameters corresponding to the second branch give bistability only if G < 0.
The projection of the first branch on the R 1 R 2 plane is shown in Fig.7 (right), for several values of K 2 .
K 1
From the inspection of fig.7 we can set the rule of thumb according to which bistability occur
if:
1. K 2 is large enough, and in any case greater than 1. That is, the charged center capture
K 1
another carrier of the same charge more easily than the neutral one in spite of columbic
repulsion.
2. R 1, R 2 are sufficiently large too, and R 1 is in any case greater than 1. That is, the columbic capture cross-section of the charged centers for the opposite-charged carriers is higher than
that for the carriers of the same polarity (which is generally true);
condition 1 has been stated in section 4 as a necessary condition for dynamical instability
referred to an arbitrary number of states of charge. Relation 40 also gives a sufficient condition
for doubly charged centers.
It will be useful for the following considerations to infer the order of magnitude of the
generation factor g and the time scale τ involved in the bistable transition. From Eqs. 38
and 37, we have, for the value Gγ at the cusp point C( γ):
−1
Gγ = K 1 1 − 1 −
3
,
(41)
R 1
( R 1 R 2) 13
which gives, in the limit R 1,2 >> 1 (condition 2):
Gγ ≈ K 1 thus gγ ≈ NMC 1 n.
(42)
The time scale τ can be obtained expressing Eq.33 in term of the adimensional parameters,
which gives:
2
τ dy = G − ∑ K
.
(43)
dt
iy ( yi−1 − yi) ,
with
τ =
1
i=1
M ( C 1 nC 2 n)1/2
542
Optoelectronic Devices and Properties
7. One example of possible photoconductive bistable behavior in germanium
In order to design a practical bistable photoconductive device, an accurate determination
of the recombination center’s trapping parameters is necessary in two respects: first, the
values of the capture coefficients of a center having either multiple states of charge or excited
metastable levels, define if the center is or not a candidate to exhibit multistability; second,
the same values determine which are the more suitable values of the doping densities in order
to obtain bistability for the desired radiation irradiances and with the appropriate response
time.
Even though several deep centers having metastable electronic states are known, especially in
large bandgap semiconductors (Ganichev et al., 1997; Ghosh et al., 1973; Jelezko & Wrachtrup,
2006; Pagonis et al., 2010), there is no plenty of detailed studies about the capture cross-section
of both the ground and the excited state. For instance, for native EL2 centers in gallium
arsenide, undoubtely one of the most studied (Delaye & Sugg, 1994; Lin et al., 2006; Sudzius
et al., 1999; Vincent et al., 1982), R 1 = C 1 p/ C 1 n (see section 5) is known to vary in the range 10−3 − 1 from the ambient to cryogenic temperatures , the dependence of its recovery rate R 21
(having an activation energy of about 0.3 eV) on temperature is also reported, but the capture
rates of the metastable level are only known to be much smaller than those of the ground level
(Delaye & Sugg, 1994), without any indication on the value of their ratio.
In comparison, the capture rates of muptiply charged recombination centers have been
studied in more detail, and several transition metal impurities in silicon, germanium and
silicon carbide have exhibited a favourable ratio of the higher level cross section to the lower
level one ( K 2/ K 1 > 1, see section 6) (Grillenberger, 2004; Knack et al., 2002; Sachse et al.,
2000; Shiraishi et al., 1999; Wertheim, 1959; Yarykin et al., 1999). Here we present a study on
a center which is likely the most deeply and the longest studied of these, namely Nickel in
Germanium. As a matter of fact, the conclusions reached about the capture capture rates of
this deep double acceptor center are not unanimous (see tab. 1). To my knowledge, there
are three works offering a quite complete picture of the recombination coefficients, based on
different experimental tecniques, along with a number of studies giving one or two among the
four capture rates. Klassen et al. (1961) obtains K 2 = 0.14 < 1 by generation-recombination
K 1
noise measurements, but Wertheim (1959), Tseng & Li (1972), and Eliseev & Kalashnikov
(1963) obtain K 2 = 6 > 1, with photoconductive methods. Wertheim and Tseng give three
K 1
of the four coefficients necessary for the calculation of rel.40 ( K 1, K 2, and R 2). Wertheim obtains 20 < R 2 < 80, which could give bistability, according to rel.40 if R 1 > 5, that is: σ 1 p > 5 × 10−16cm2. This inequality is largely satisfied by the values of σ 1 p independently obtained by all the other reserchers (see tab. 1) at sufficiently low temperatures. Tseng and Li,
on the other hand, give a value of R 2 ≈ 5, wich is slightly too small to permit bistability, also
with very large values of R 1. In conclusion, there is conclusive evidence, by fotoconductive
measurements, that the necessary condition K 2 > K 1 is satisfied for Nickel in Germanium;
moreover, there are suggestions that also the sufficient condition of rel. 40 is satisfied for this
center, at least at low temperatures (≈ 100 K).
A numerical simulation of the behavior of n-doped germanium, lightly compensated with
Nickel, has been carried out, solving eqs. 33. The parameters were those given by Wertheim
(1959), along with σ 1 p ≡ σp−/0 = 10−13cm2, which is a conservative estimate based on the
values of tab. 1. The doping centers and the recombination centers concentrations were,
respectively, N = 1016cm−3, M = 5 × 1014cm−3. Figure 8 (left) represent the hysteresis
cycle of the valence band population, as a function of the generation factor, and Fig.8 (right)
represents the response of the valence band population to pulses above and below the bistable
