a amorphous iv semiconductorsdownloads.hindawi.com/journals/apec/1997/065485.pdf · semiconductors...

Active & Passive Elec. Comp., 1997, Vol. 20, pp. 41-44

Reprints available directly from the publisherPhotocopying permitted by license only

(C) 1997 OPA (Overseas Publishers Association)Amsterdam B.V. Published in The Netherlands

under license by Gordon and Breach Science Publishers

Printed in Malaysia

A MOLECULAR-ORBITAL MODEL FORAMORPHOUS GROUP IVSEMICONDUCTORS

M.A. GRADO-CAFFARO* and M. GRADO-CAFFARO

Member of the Academy of Sciences of New York," C. Julio Palacios, 11, 9-B,28029-Madrid, Spain

(Received 19 Februa 1996; In final form 17 June 1996)

A theoretical model based on standard molecular-orbital theory and extended Htickel ap-proach is proposed. This model is valid for amorphous group IV semiconductors and repre-sents a substantial improvement of the state of the art.

1. INTRODUCTION

It is well-known that cluster approach, extended Htickel theory, and stan-

dard molecular-orbital theory may be employed to analyze the fundamen-tal aspects of electronic density of states and infrared spectra in amor-

phous group IV semiconductors [1][2]. On the other hand, to applystandard molecular-orbital theory, a conventional perturbation methodmust be used. The aim of this paper is to achieve a main formula as a

summary of a quantum-mechanical development that includes cluster ap-proach, extended Htickel theory, and conventional molecular-orbital

theory as basic ingredients.

*Corresponding author.

41

42 M.A. GRADO-CAFFARO and M. GRADO-CAFFARO

2. THEORY

First, let us examine the following expression [3]"

g(x, y, z, E) Z Iq/n(X, Y, z)12g(E En) (1)

where g denotes local density of states in a cluster, n are the eigenfunc-tions associated with EHT (Extended Htickel Theory), g is the Dirac func-tion, E is energy, E denotes energy eigenvalues, and x, y, z are the car-

tesian coordinates.

Hence, the total density of states of the cluster is given by:

g(E)- f f f g(x, y, z, E)dxdydz (2)D

where D is the volume of the cluster.The set of discrete eigenvalues E,1 arises from a consideration of clusters

isolated in vacuum. A cluster approach can be realized by using a numberof methods, but these methods must satisfy some conditions; one of theseconditions is that finite clusters of the same size can take up distinct

configurations, and it is important to have a configuration average of theseclusters [4].Now, we consider the basic equations associated with EHT-MO (mo-

lecular orbital), namely [3][5]:

N

(ij E,Sij)CI 0 (3)j=l

N

E (ij EaSij)C2j 0 (i 1, 2 N) (4)j=l

where tZlij denotes hamiltonian, and stand for the atomic orbitals, Sij arethe overlap integrals between the atomic orbitals, and C1j, C2.i are themolecular orbitals coefficients. Moreover, El and E2 represent the energiesassociated with molecular orbitals.

Let us now consider amorphous Si. By considering s orbitals and porbitals, we have the following molecular orbitals:

GROUP IV SEMICONDUCTORS 43

+ q- qp (5)

+2--qJp -Jr- qJs (6)

where q, is s orbital and *p is p orbital. Moreover, o and [3 are smalladmixture coefficients. Note that b and qb2 are the antibonding and bond-

ing orbitals, respectively. In this context, we shall use standard perturba-tion theory [6] by means of the following expression:

--t) -+- q/k) asp+1 Is no- q/p (7)

E Ep

with q,l} the kth perturbed term and asp <,slI2Isp *p>, I2Isp being the

corresponding hamiltonian of perturbation.From (5) and (7) we get:

Similarly, it is deduced:

aspo (8)

E Ep

asp[3 ot (9)

E,- Ep

Applying secondnorder perturbation theory, it is easy to compute the

energetic shift of the molecular orbital +1, namely:

2 (g gp)-I o/.2(gs gp) (10)AXE- asp

by considering eq. (8).Now we have: E EA, E2 EB, Clj

s, and 2 p for atomic orbitals with:CAj, Czj CB, N 2;

Sij <ilijlij> (i, 1, 2)

Now from (3), (4) and (10) we obtain:

44 M.A. GRADO-CAFFARO and M. GRADO-CAFFARO

2

E {I(c c) + s{[c + (, + )c]j=l

[( + >c + ]}}-o

since EA Es + AE and EB Ep AE.

(11)

3. CONCLUDING REMARKS

Formulae (10) and (1 l) are very important since these expressions allowus to perform numerical computations to obtain energy values for amor-

phous Si. Consider an amorphous Si molecule and examine Si-Si bond [7];our formulation is relevant for this examination.

References[1 M.A. Grado, M. Grado. Physics Letters A, 169, No. 5, 399-401 (1992).[2] R. Alben, E Steinhardt, D. Weaire. AIP Conf. Proc., No. 20, 213-217 (1974).[3] B.Y. Tong. AIP Conf. Proc., No. 20, 145-149 (1974).[4] W.H. Butler, W. Kohn. J. of Res. NBS 74 A, 443 (1970).[5] C.C.J. Roothaan. Reviews of Modern Phys. 23, 69 (1951).[6] J. Owen. J.M.H. Thornley. Rep. Prog. Phys. 29, 675 (1966).[7] Y. Hamakawa, ed., Amorphous semiconductor technologies and devices (North--Hol-

land, Amsterdam, 1986).

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttp://www.hindawi.com Volume 2010

RoboticsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation http://www.hindawi.com

Journal ofEngineeringVolume 2014

Submit your manuscripts athttp://www.hindawi.com

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

SensorsJournal of


Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


a amorphous iv semiconductorsdownloads.hindawi.com/journals/apec/1997/065485.pdf · semiconductors...

Documents