LOCALIZATION OF ELECTRONS IN TWO-DIMENSIONAL RANDON NETWORKS

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LOCALIZATION OF ELECTRONS IN

TWO-DIMENSIONAL RANDON NETWORKS

K. Tsujino, M. Yamamoto, A. Tokunaga, F. Yonezawa

To cite this version:

K. Tsujino, M. Yamamoto, A. Tokunaga, F. Yonezawa. LOCALIZATION OF ELECTRONS IN TWO-

DIMENSIONAL RANDON NETWORKS. Journal de Physique Colloques, 1981, 42 (C4), pp.C4-55-

C4-58. �10.1051/jphyscol:1981408�. �jpa-00220712�

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JOURNAL DE PHYSIQUE

CoZZoque C4, suppldment au nOIO, Tome 4 2 , octobre 1981 page c4-55

LOCALIZATION OF ELECTRONS IN TWO-DIMENSIONAL RANDOM NETWORKS

K. Tsujino, M. Yamamoto, A. Tokunaga and F. ~onezawa*

Osaka Industry University, Daito-shi 5 7 4 , Japan

'Research I n s t i t u t e for Fundamental Physics, Kyoto University, Kyoto 606, Japan

Abstract.- By regarding an infinite cluster of the bond percolation path as a possible model of a random network, we examine the effects of topological disorder on the shape of the density of states and on the behaviours of the wavefunctions.

Introduction.- The problem of electron localization has been studied rather exten- sively by analytic methods, by numerical approaches and recently by the scaling and renormalization group theory. In most cases, the random systems studied so far are substitutionally disordered where the positions of atoms form a regular lattice but the kinds of atoms on the lattice points are random. In three-dimensional random systems, the absence of quantum diffusion takes place when the degree of randomness exceeds some threshold. It has also been predicted that all states are localized in two-dimensional disordered systems, but this problem is still controversial. On the other hand, our knowledqe of local~zation in topologically disordered systems is next to nothing.

Wlth this situation in mind, it is a purpose of the present article to inves- tigate the general effects of topological disorder on localization by studying a model random network. In practice, we present;

(1) an examination of the characteristic features of the density of states (DOS) near the band edges and at the band centre; and

(2) the computer demonstration of the calculated wavefunctions of our model random network at E(energy)'O and EfO.

=.-

Our model system is described by the following one-electron single-band Hamiltonian;

where the orthogonal basis fuxctions

1

i> are associated with the sites of a regular lattice and ip denotes the nearest-neighbour site of i; we take Ei to be zero and assume that the distribution for v . . is given by; P(vij)=p&(vij-v)+

1 I

(1-p)6(v. . ) with V<O. This actually corresponds to the bond percolation problem.

1 I

Note that p is the probability of finding unbroken bonds. We regard an infinite percolation path as a model random network.

The present model is compared to a model of ideal amorphous semiconductors such as a-Si or a-Ge where the covalent bonds form a fully interconnected continu- ous random network (CRN) and the coordination member of an atom is the same throughout the sample. In the present model, on the other hand, the coordination number varies from atom to atom and the model may serve as a structural model of a-Si:H with broken bonds playing the part of dangling bonds. When the underlying lattice is a square or simple cubic lattice, there exist only even-membered rings in the present model while a fully interconnected CRN can hardly be free from odd- membered rings. In the discussion of localization, an introduction of odd membered rings yields a further source of scatterers. Accordingly, if all states are locallzed in a system with even-membered rings alone, then the states are even more localized in a corresponding system when odd-membered rings are introduced.

Therefore, the criterion for localization in the former system gives the upper

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981408

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C4-56 JOURNAL DE PHY SIQlJE

bound of the criterion for localization in the latter system.

It is also worth mentioning that the present model is advantageous as a model of a random network when compared to a site-percolation model. That is to say; the fact that the amplitude of a wavefunction on a site associated with a missing bond does not necessarily vanish indicates that no serious singular effects are brought about by an artificial removal of bonds and therefore we could expect that the es- sential aspects of the influences from topological disorder would be extracted.

This is-not the case with a site-percolation model where the amplitude of the wavefunction on a missing site is identically zero.

Density of States.- It has been shown that the DOS of an ideal CRN with odd-membered rings has the shape as sketched in Fig.l(a) with a normal band edge at the bottom (corresponding to the bonding state) and the Lifshitz limit at the top

(corresponding to the anti-bonding state).

''

The energy spectra of the present model of a (non-ideal) random network (RN) can be evaluated from the solutions of the SchrBdinger equation HY=EY where we write Y=Ciai

1

i>. Then, we have the energy in the form as

E = (Y,HY) =

v 1

^[ata.⁺^a.a?

^I ,

1 1 1 1 ( 2 )

, P P

where C(i,ip) indicates the summation over all unbroken bonds (i,ip). The wave function is normalized such that

1Y I '=zi 1

^ai

1

2=l. Using the Schwartz inequality and the Lifshitz theorem based upon the local density principle, we can draw a conclu- sion that the spectral bounds for our non-ideal RN is given by -z <E<z

~ v I

^where

z is the coordination number of the underlying lattice and that the SoTh band edges are the Lifshitz limits as illustrated in Fig.l(b). Naturally, the states in the tails are localized while there is a good reason to believe that the states near Eop are extended. The optical energy Eop is estimated to be p*zlVI.

In either case of an ideal CRN or of a non-ideal RN (the present model), topological disorder does not affect the width of the band but has the tendency of changing the characters of the band edges from the normal behaviour to the Lifshitz limit. This is rather marked when compared to the effects of substitutional disorder which mainly work to broaden the band.

The approximate DOS for our model was previously calculated3 using the homo- morphic cluster coherent potential approximation (HCPA) and given in Fig.2. A marked feature is that, for small values of p, there appears a peak at E=O and gaps take place on both sides of E=O. This feature of the DOS is the characteristic tendency found by a computer simulation for a bond-percolation model. lt It is also interesting to note that the same tendency appears in the DOS near E=O for a site- percolation model. 5'

I,

Fig.1 (a). The DOS for an

ideal random network (after Singh, Cohen and ~onezawa' )

.

(b) The DOS for the present model of a random network described in the text.

- ^flvl ^rlvl E

- ~ I V I Z J V ~

E ~ P EOP

(a) (b)

Fig.2. The average DOS for the present model of a random network in a cubic lattice.

The calculation was performed on the basis of the HCPA

(after Yonezawa and 0dagaki3).

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Wavefunctions.- We treat a square lattice composed of 50X50 (~2500) sites. We have checked the size effect and concluded that 50x50 is large enough to discuss the problem of localization. We impose a periodic boundary condition in one direction and a free-end boundary condition in the other direction. An eigenfunction is calculated for each energy by the Lanczos method. To display visually the spacial behaviour of the wavefunctions, we have made a graphical representation of the squared amplitudes

1

^ai

1

on a 50x50 square lattice. In Figs. 3 (a) and (b)

,

^{the two}

wavefunctions corresponding to the centre of the band (EzO) are shown for p=pc=

0.5, the critical percolation concentration for the bond problem on a square lattice. Note that r is the ratio of the scale enlargement. It is obvious from these figures that the states are still well localized even at the critical threshold for the classical percolation. In Figs.4(a) and (b) are shown the result for p=0.8.

The both figures represent the same eigenfunction but the ratio of enlargement for (b) is five times larger than that for (a). The ridge-like localization found in the site problem is also detected here. The results for a still larger value of p,

p-0.9, are given in ~igs.5 (a) to (c) where E=O. The ridge-like localization is still persistent even at this high concentration p of unbroken bonds. The wavefunction off the centre of the band (E'3) for p=0.9 is shown in Fig.6. The behaviour of the wavefunction is clearly different form that of the states near the centre; the wavefunction seems to be more extended in this case.

Summary.- To summarize, we can infer;

(1) that the critical concentration for the quanta1 percolation is definitely larger than that for the classical percolation;

,

(2) that even at a comparatively

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high concentration p of unbroken bonds the ridge-like localization and the line localization obviously take place; and

(3) that the wavefunctions off the centre of the band are apparently extended. However, as we discussed elsewhere, there are chances that the state is actually localized but the state looks as if it is extended simply because the localization length is larger than the linear dimension of the system size.

Fig.4(a). p=0.8, E'O, r=l. Fig.4 (b)

.

The same as Fig.4 (a) with r=5.

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JOURNAL DE PHYSIQUE

I n o r d e r t o say something more d e f i n i t e , it i s n e c e s s a r y t o

d e s c r i b e t h e s e r e s u l t s i n an a l g o r i - thmic language. This i s t h e r e a s o n why v a i o u s c r i t e r i a f o r l o c a l i z a t i o n have been proposed and s t u d i e d . One example of t h e c r i t e r i a i s t h e p a r t i c i p a t i o n numebr P= [Ci

1

a i

1

I - ' .

Besides, we have i n t r o d u c e d a s a new c r i t e r i o n 7 t h e e n t r o p y of mixing, S= -Ci

1

ai

1

'kn

1

ai

1 ' .

^This

c r i t e r i o n bv means of S h a s been

shown t o be advantageous s i n c e it F i g . 5 ( a ) . p=0.9, E=O, r=5.

p r o v i d e s u s w i t h t h e way t o p r e d i c t t h e i n f i n i t e l i m i t from t h e numeri- c a l r e s u l t s f o r f i n i t e - s i z e systems.

For want o f space, however, we g i v e a d e t a i l e d a n a l y s i s elsewhere.

References

I. Cohen M H, Singh J and Yonezawa F, S o l i d S t a t e Commun. 36 (1980) 923.

2. Yonezawa F and Cohen M H , "Fun- damental P h y s i c s o f Amorphous Semiconductors", ed F. Yonezawa, S p r i n g e r S e r i e s i n S o l i d S t a t e S c i e n c e s 25, S p r i n g e r , 1981, p119.

Yonezawa F and Odagaki T, S o l i d S t a t e Commun.

2

(1978) 1199.

Odagaki T, Ogita N and Matsuda H, J. Phys. C: S o l i d S t a t e Phys.

1 3 (1980) 189.

-

K i r k p a t r i c k S and E g g a r t e r T P, Phys. Rev. (1972) 3598.

Yoshino S and Okazaki M, S o l i d S t a t e Commun.

27

(1978) 557.

T s u j i n o K , Yamamoto M, Tokunaga A and Yonezawa F, S o l i d S t a t e Commun.

30

(1979) 531: Yonezawa F , J. Non-Cryst. S o l i d s

35

&

36

(1980) 29.