Chaotic Behavior of the Forward I-V Characteristic of the Al/a-SiC:H/c-Si(n) Heterojunction

Abstract

In this paper the electrical behavior of the Al/a-SiC:H/c-Si(n) heterojunction for different values of density of gap states (N) in a-SiC:H, is simulated and studied. It is observed that as the density of gap states in a-SiC:H increases from 10− 15 cm− 3 to 10− 18 cm− 3 the I-V characteristics, in the forward bias, present a deviation from the typical I-V of a diode, which is enhanced with the increase of N. For N = 10− 18 cm− 3 the forward I-V characteristic shows strong chaotic vibration that is attributed to the tunneling effect taking place in the junction a-SiC:H/c-Si(n) in the forward bias. With the method of delays correlation and minimum embedding dimension are calculated, while the influence of gap states in strengthening chaos is studied.

Full text

Chaotic Behavior of the Forward I-V
Characteristic of the Al/a-SiC:H/c-Si(n)
Heterojunction
M.P. Hanias, L. Magafas, S.G. Stavrinides, P. Papadopoulou, and M. Ozer
Abstract In this paper the electrical behavior of the Al/a-SiC:H/c-Si(n) heterojunc-
tion for different values of density of gap states (N) in a-SiC:H, is simulated and
studied. It is observed that as the density of gap states in a-SiC:H increases from
1015 cm3to 1018 cm3the I-V characteristics, in the forward bias, present a
deviation from the typical I-V of a diode, which is enhanced with the increase of N.
For N D1018 cm3the forward I-V characteristic shows strong chaotic vibration
that is attributed to the tunneling effect taking place in the junction a-SiC:H/c-Si(n)
in the forward bias. With the method of delays correlation and minimum embedding
dimension are calculated, while the influence of gap states in strengthening chaos is
studied.
1 Introduction
Amorphous thin films of hydrogenated silicon carbide (a-SiC:H) have been exten-
sively studied for about 40 years, due to very attractive properties that they
demonstrate, such as high values of hardness, thermal and chemical stability, high
resistance in radiations, wide optical band-gap and considerable absorption in the
blue region of the spectrum. Moreover, a series of applications in the field of
micro-electromechanical systems [1], such as high efficiency solar cells, thin film
transistors (TFTs), Schottky diodes [2] and optical sensors [3] have arisen. However,
even though this structure a-SiC:H/c-Si demonstrates very attractive applications
[3], only a few things have been done on the topic of isotype heterojunction [2].
M.P. Hanias L. Magafas S.G. Stavrinides ()P. Papadopoulou
Kavala Institute of Technology, Department of Electrical Engineering, Kavala, Greece
e-mail: mhanias@gmail.com;lmagafas@otenet.gr;stavros@physics.auth.gr
M. Ozer
Physics Department, Istanbul Kultur University (Atakoy Campus), Bakirkoy, Istanbul, Turkey
S.G. Stavrinides et al. (eds.), Chaos and Complex Systems,
DOI 10.1007/978-3-642-33914-1 66, © Springer-Verlag Berlin Heidelberg 2013
475

476 M.P. Hanias et al.
In the present research work, we studied for the first time the electrical behavior
of the a-SiC:H/c-Si(n) isotype heterojunction, for the case of a wide band gap of
amorphous semiconductor .EgD2:9eV/and for different values of the density
of localized gap states (N). The results show that for N >10
17 cm3and under
forward biased conditions, the I-Vcharacteristics demonstrate a vibration which
is more rigorous for N D1018 cm3. Simulation result analysis and the related
evaluation, indicated that these vibrations are chaotic, with a very interesting
behavior and a remarkable potential for applications.
2 I-V Characteristics
The I-V characteristics of the a-SiC:H/c-Si(n) heterojunction, for gap energy
Ega-SiCWHD2:9 eV and for different values of the density of localized gap states
.N/(from 5015cm3down to 51018cm3/are studied, by using an
advanced simulation program (S-PISCES - SILVACO), which work under the
ATLAS framework; ATLAS is a physically based device simulator. It should be
noted that the a-SiC:H/c-Si(n) heterojunction is assumed to be isotype, since the
sputtered a-SiC:H has been found to present an n-type behavior [4].
In Fig. 1a the corresponding I-V does not exhibits any vibration, as long
as, the density of localized gap states takes the values N D51015 cm3to
ND51016 cm3.InFig.1b the density of localized gap states has the value
ND51017 cm3and the corresponding I-V begins to vibrate; while for the
value N D51018 cm3the I-V characteristics demonstrate a robust vibration, as
showninFig.1c. It is clear that the density of localized gap states .N/is a parameter
that strongly affects the I-V characteristic pattern.
3 Evaluation-Nonlinear Analysis
We now proceed to the analysis of the obtained I-V [Fig. 1c], by applying the
Grassberger-Procaccia method [5]. According to Takens theory [6], a topologically
equivalent to the original, phase space is reconstructed by this I-V series. At first,
correlation integral C(r) is calculated, for the simulated characteristic, generally
defined by [6]:
C.r/ D1
Npairs
N
X
lD1;
jDlCW
Hr


E
XlE
Xj

(1)
where N is the number of the corresponding data points, W is the Theiler window
[4], H is the Heaviside function and is defined as follows:

Chaotic Behavior of the Forward I-V Characteristic of the Al/a-SiC:H/c-Si(n)... 477
ab
c
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.00E+000
1.00E-011
2.00E-011
3.00E-011
4.00E-011
I(A)
V(V)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
–2.00E-012
0.00E+000
2.00E-012
4.00E-012
6.00E-012
8.00E-012
1.00E-011
1.20E-011
I(A)
V(V)
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
–3.00E-011
–2.00E-011
–1.00E-011
0.00E+000
1.00E-011
2.00E-011
3.00E-011
4.00E-011
B
A
Fig. 1 I-V characteristic for (a)N D51015 cm3-N D51016 cm3(b)N D5
1017 cm3(c)ND51018 cm
Npairs D2
.N mC1/.N mCWC1/ (2)
with m being the embedding dimension.
It is clear that the summation in (1) counts the number of pairs for which the
distance, i.e. the Euclidean norm, is less than r in an m dimensional Euclidean
space. Here, the number of the experimental points is N D1; 000. Considering the
m dimensional space, each vector will be given by [5]
E
XDfI.i/; I.iC/;I.iC2/;:::;IŒiC.m1/  g(3)
and it will represent a point in the m dimensional phase space. In (3), £stands for
the delay time determined by the first minimum of Mutual Information function
I.£/. As shown in Fig. 2a, in this case Mutual Information exhibits a local minimum
at £D12 (steps). In Fig. 2b the phase portrait is presented which is reconstructed
with £D12. Thus, this value shall be considered as the optimum delay time. Since
there is no standard method for choosing Theiler window W, this was determined
by the first zero-crossing value of C2.£/, as suggested by Kantz and Schreiber [7].
As it is obvious by Fig.2c, this happens for W D14 steps. Hence, we can use these
values for phase space reconstruction. Then, if the attractor is a strange one, the

478 M.P. Hanias et al.
ab
cd
e
020406080100
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
I
av
t
–4.00E-011 –2.00E-011 0.00E+000 2.00E-011 4.00E-011
–4.00E-011
–3.00E-011
–2.00E-011
–1.00E-011
0.00E+000
1.00E-011
2.00E-011
3.00E-011
4.00E-011
I(i)
I(i-12)
0 20406080100
–0.2
0.0
0.2
0.4
0.6
0.8
1.0
C
R
t(i)
0246810
0
1
2
3
4
5
6
7
8
9
10
v
m
Fig. 2 (a) Average Mutual Information I vs. delay step £,(b) Phase space portrait, (c) Autocor-
relation function vs. delay step, (d) Relation between logC.r/ and logrfor different embedding
dimensions m,(e) Correlation dimension vvs. embedding dimension m
correlation integral will be proportional to r, where v is a measure of the attractor’s
dimension called correlation dimension. Figure 2d depicts the relation between the
logarithms of correlation integral C(r) and r for different embedding dimensions m.
Then, in Fig. 2e, the corresponding average slopes v are given as a function of the
embedding dimension m indicating that for high values of m, v tends to saturate
at the non integer value of v D3:33. For this value of v, the minimum embedding
dimension can be m D4[7], and thus, the minimum embedding dimension of the
attractor for one to one embedding will be equal to 4.

Chaotic Behavior of the Forward I-V Characteristic of the Al/a-SiC:H/c-Si(n)... 479
4 Conclusion
The I-V characteristics of the a-SiC:H/c-Si(n) heterojunction, for different values
of the density of localized gap states (N) exhibit a spatio chaotic behavior.
The scaling behavior of the correlation integral and the saturation of correlation
dimension , with increasing embedding dimensions m; reflect low dimensionality.
The strange attractor that governs the phenomenon has a correlation dimension
vD3:33 stretching and folding in a 4 dimension phase space. Thus, the number
of degrees of freedom of the whole domain structure is 4.
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