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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

1015 cm3to 1018 cm3the 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 D1018 cm3the 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 cm3and under

forward biased conditions, the I-Vcharacteristics demonstrate a vibration which

is more rigorous for N D1018 cm3. 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 5015cm3down to 51018cm3/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 D51015 cm3to

ND51016 cm3.InFig.1b the density of localized gap states has the value

ND51017 cm3and the corresponding I-V begins to vibrate; while for the

value N D51018 cm3the 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

Hr

E

XlE

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 D51015 cm3-N D51016 cm3(b)N D5

1017 cm3(c)ND51018 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.m1/ 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.

References

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(Si-MEMS). In: Advanced Structural Materials: Properties, Design Optimization, and Applica-

tions. Taylor & Francis Group, pp. 63–94 (2007)

2. Magafas, L.: Optical response study of the Al/a-SiC:H Schottky diode for different substrate

temperatures of the RF sputtered a-SiC:H thin ﬁlms. Active Passive Electron Components 26(2),

63 (2003)

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Summonte, C., Pinghin, R., Centurioni, E., Galloni, R.: Photocarrier collection in a-SiC:H/c-

Siheterojunction solar cells. J. Non-Crystalline Solids 227–230(part 2), 1291 (1998)

4. Magafas, L.: Study of optical sensors of the form Al/a-SiC:H/c-Si(n) with high sensitivity.

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6. Hanias, M.P., Tombras, J.S.: Chaos Solitons Fractals 40, 246–256 (2009)

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Cambridge (1997)