Extended states in one-dimensional random-segment models

PHYSICS LETTERS A

Physics LettersA 179 (1993) 217—220 North-Holland

Extended states in one-dimensional random-segment models Xiaoshuang Chen Department of Physics, Nanjing University, Nanjing 210008, China

and Shijie Xiong China Center ofAdvanced Scienceand Technology (WorldLaboratory), P.O. Box 8730, Beijing 100080, China and Department of Physics, Nanjing University, Nanjing 210008, China Received 12 March 1993; accepted for publication 2 June 1993 Communicated by L.J. Sham

We study here some one-dimensional (ID) disordered tight-binding models which can be constructed by randomly inserting a number ofidentical segments into an infinite purely periodic chain. We analytically show that under certain conditions there exist some completely unscattered states whose number is one less than the number of sites in each segment. The energies of these states are exactly determined. Some 1 D models with random periods can be considered as samples of the random-segment models. The energy spectrum and unscattered wave functions ofa sample model with different choices of parameters are illustrated.

The Anderson localization of states in disordered systems has proved to be of fundamental importance in our understanding of the role disorder plays in insulator—metal transitions in a wide range of materials [1,21. According to the scaling theory, the electronic states are always localized in one- and twodimensional (1 D and 2D) disordered systems [3], but maybe there exists an exceptionfor the states with zero measure. Azbel [4] and Azbel and Soven [5] have demonstrated that the extended states exist in some 1 D disordered systems, by which the absence of the localization in experiments [61 can be accounted for. Recently Dunlap, Wu and Phillips have also found that there do exist extended states in the lD random-dimer model [7,8], which is consistent with the theory of Dunlap et al. [9,10] and Flores [11]. This is a tight-binding model with site-diagonal disorder and is constructed by randomly inserting a number of identical dimers into a purely periodic chain. Basing on the random-dimer model, Wu and Phillips [12] and Lavarda et al. [13] can explain the insulator—metal transition in polyaniline and in highly doped tarns-polyacetylene. From nu-

merical calculation, the extended states have also been found in other 1 D random models, for example the 1 D tight-binding model with random periods [14]. However, we still cannot clearly answer the question posed by Azbel [4]: Is the existence of extended states in a 1 D disordered system a common feature or is it only an exception? In this paper, we focus on a larger class of 1 D random tight-binding models, the random-segment models, which can be constructed by randomly inserting a number ofidentical segments into an infinite purely periodic chain. Using the transfer matrix method, we show rigorously that there exist completely unscattered states under certain conditions. The number ofthese states is one less than the number of sites in each inserted segment. The random-dimer model of refs. [7,81 is in this class with two atoms in each inserted segment. The model in ref. [141 with some constraint can also be regarded as belonging to this class. From a numerical scheme, we illustrate the energy spectrum ofa sample model with different choices of parameters,where the locations of the unscattered states are marked. The wave functions of two unscattered

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217

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PHYSICS LETTERS A

are also shown to have the features of these We use a tight-binding Schrodinger equation

~)

(1)

to describe the lD electronic system, where yi,, denotes the amplitude at the nth site of the wave function, V~is the energy level of the Wannier orbital at the nth site, and t,,,,±1 is the nearest-neighbor hopping strength between site n and site n ±1. and ~

V,, are both random variables determined by the structure of the 1 D lattice. Here we only consider the diagonal disorder, and set t,~,,±1to be a constant ~ A random-segment lattice A is constructed by randomly inserting a number of identical segments ~ into a purely periodic lattice A0. If A0 is an infinite array of atoms of species B, and z~is a segment of rn atoms of species A, then A is

...AAA...BBB...AAA...BBB...AAA...BBB.... m

m

L,+i

m

9 August 1993

1) ...i~( N+

= ~1(N)~f(N—

—

1 )A~(— N)

(4)

x(w_N+I~,

~1’-N I with N—~oo.For the lattice A, the product of matrices in eq. (4) becomes

~

where MA or M8 is the matrix ~cf(n) with V,, equal to VA or VB. From the theory of matrices, the rnth power of the 2 x 2 unimodular matrix A~’Acan be written as [15] ‘~=Um_I(X)Ji~’A~Um_2(X)I, (5) where x= ~Tr(i1~A),I is a unit matrix, Um(X) is the mth Chebyshev polynomial ofthe second kind, which obeys the recurrence relation Um+i(X)=2XUm(X)Um_i(X),

rn?~0,

(6)

L~+2

Thus the array B is broken into segments with random lengths L, whose distribution may be expressed

with u_1=0 and u0=l. If lxi ~l, sin ( rn6)

by a stochastic function, P(L1)=>~p~o(L1—j)

um_i(x)= sin(8) where 8=cos’(x). If

Um(X)

has the form (7)

,

(2)

j

x1=cos(lx/m)

where ~5(1)=1, =0,

1=1,2

1=0,

we have

/~0,

rn—i

,

forrn~2, (8) and Um_2(X1)(_l)~’+1.

Um_i(Xi)O,

From eq. (5), one has and p~is the probability of finding a B segment having j atoms. V~takes the value VA or V8 depending on the species of the nth site. The tight-binding equation (1) may be rewritten in a transfer matrix form, = .Q( n) ‘p~

(3)

where ~ is a column vector (~“~) and A~’(n) is a transfer matrix ~Q(n)

((E—vn)/t

=

1

_l) 0

Repeated application of iQ( n) gives

218

= (~1)11. (9) Basing on these formulas, we can derive the condition of the existence of the extended states and their exact location in the energy spectrum. In fact, the enj1~

ergy corresponding to x1 in eq. (8) is VA + 21 cos( lx/m) 1= 1, 2, rn 1 (10) From eq. (9), for the state with eigenenergy E1, the E,

,

...,

—

.

matrix string of the 1D lattice is only composed of matrices ~j~r8 and (l)’I. Since matrix (—l)’Ihas no effect on the amplitudes of the wave function, this matrix string is equivalent to that ofthe periodic lattice A0, which only consists of matrix M~.For this periodic system, the exactly extended states exist in the energy range E— VB I ~ 2t. Therefore, if E, is lo-

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PHYSICS LETFERS A

9 August 1993

cated within this range, or the condition

____________________________

(11)

IVA+2tcos(lx/m)—VBi~<21

(a)

is satisfied, the Furstenberg theorem [16] implies that the electronic state at E, is extended. Thus, if rn ~ 2, and VA—VBI+2ItIcos(x/m).~2t (12) we can find m I extended states, which are cornpletely unscattered. The random-dirner model is just the case of rn = 2. In this case the energy of the unscattered state is VA, the same result as obtained in ref. [7]. From the estimation of ref. [7], we can also argue that in the vicinity of the energy of every unscattered state, there exist electronic states which remain extended over the total length of the sample. If the dimers of the random-dimer model are gath-

0

—

~

~

E2

E 1

(b)

~

J

5d00 SITE INDEX

I

5

0

ENERGY

_______________________________

Fig: 2. The wave functions in the random-segment model. The parameters in eq. (2) are the same as those of fig. I. (a) The unscattered state at E= —. 1.4, the parameters are the same as those in fig. Ia. (b) The unscattered state at E=0.0, the parameters are the same as those in fig. lb.

...AAABAAABBBAAABBBAAABAAABBB...

_____

m

—p-—— ,

10000

ered together to form identical segments of length 2M, then we have 2M— 1 unscattered states, among which one state has been considered in ref. [7]. From this one can guess that for a special model there may exist more unscattered states which have not been included in eq. (10). Another example51j+P3ô3j, can be crethe ated from lattice A by setting Pj=P1~ atom array becomes

(b)

—3.25

______

7iiiirnioiiiiii~iiii~

-3. 2

__________

00

INDEX

_______________________________

(a)

l[

~

SIrE

I

m

3

m

~n

3

1

m

3

5.0

ENERGY

Fig. 1. The density of states of the random-segment model with m=3 and t=l. The parameters in eq. (2) are p~=f,forj=4, for 3~j~5,j~4,p~=O,otherwise. (a) VB= — VA=O.4, there are two unscattered~states at E= —1.4 and E=0.6. (b) V 8= — VA =1.0, there is one unscattered state at E= 0.0.

where the segments B and BBB randomly appear. It is obvious that the states at energies of eq. (10) are .

unscattered. However, this lattice can also be constructed by inserting a number of segments BB in the following periodic array, 219

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PHYSICS LETTERS A

..AAABAAABAAABAAABAAABAAAB..., I

,n

I

m

1

m

I

,n

1

,,~

I

then the state at energy VB is also unscattered provided that Vb is in the energy band of this periodic lattice. In fact, the original lattice A0 can be any periodic array of any number of atomic species, and the inserted segments A can consist of any species of atoms. The unscattered states do exist provided that the transfer matrix of segment A can be expressed by a product of m (rn ~ 2) identical and unimodular 2 x 2 matrices, and the energies of eq. (10) are situated within the band of the original lattice A0. The random lattice A can also be constructed by randomly taking out (rather than inserting) a number of segments A from the original lattice A0, this is because if the condition of eq. (9) is satisfied, we have MA

_MA.

With some constraint, the random-period model considered in ref. [141 also belongs to this class. An example of the constraint is that the lengths of segments B should be even. In this case the random-pcnod model can be constructed by inserting a number ofdimers BB into the infinite string .AAAAAAA.... In order to display the energy spectrum and the extended wave functions, we use Dean’s method [17,18] to calculate the energy spectrum and the wave functions of a sample model with m = 3 for different parameters. The length of the chain considered is 10000, which is large enough to avoid the f~ nite-size effect. The densities of states for two values of V ( VA = V, V8 = V) are shown in fig. 1, in which the positions of the unscattered states are marked. It can be seen that the extended states appear in the smooth part of the spectrum. The wave functions of the unscattered states are shown in fig. 2. From fig. 2, it is obvious that the wave functions are delocalized in the disordered system. In summary, we have studied the 1D randomsegment lattice. It is found that the delocalized states still exist when the system is disordered. The condition for the existence of the extended states and ..

—

220

9 August 1993

the exact energy locations of them are analytically obtained. Usually, the extended states are in a smooth part of the spectrum. Thus, the rigorous result of ref. [7] for the random-dimer model is extended to a larger class of models. These extended states may play an important role in the transport process of lD or quasi-i D systems. It is interesting to study further the effect of interaction or an external field on these states. This work was supported by the National Fund of Natural Sciences of China.

References [1] P.W. Anderson, Phys. Rev. 109 (1958) 1492. [21T.V. Ramakrishnan and P.A. Lee, Rev. Mod. Phys. 57 (1985) 287. [31E. Abrahams, P.W. Anderson, D.C. Liccirdello and T.V. Ramakrishnan, Phys. Rev. Lett. 42 (1979) 673. [4] M.Ya. Azbel, Solid State Commun. 37 (1981) 789. [5] M.Ya. Azbel and P. Soven, Phys. Rev. Lett. 49 (1982) 751; M.Ya. Azbel, Phys. Rev. B 27 (1983) 3852. [6lD.R. Overcash, BA. Ratnam, M.J. SkoveandE.P. Stilwell, Phys. Rev. Lett. 44 (1980) 1348. [7] D.H. Dunlap, H.-L. Wu and P.W. Phillips, Phys. Rev. Lett. 65 (1990) 88.and H.-L. Wu, Science 252 (1991) 1802. [81 P.W. Phillips

[91D.H. Dunlap, K. Kundu and P.W. Phillips, Phys. Rev. B 40 (1989) 10999.

[101D.H. Dunlap and P.W. Phillips, J. Chem. Phys. 92 (1990) 1111 6093. J.C. Flores, J. Phys. Condens. Matter 1(1989) 8471. [121H.-L. Wu and P.W. Phillips, Phys. Rev. Lett. 66 (1991) 1366. [13] F.C. Lavarda, D.S. Galvao and B. Laks, Phys. Rev. B 45 (1992) 3107. [14] Xiaoshuang Chen and Shijie Xiong, Phys. Rev. B 46 (1992) 12004. [15] M. Born and E. Wolf, Principles of optics, 6th Ed. (Pergamon, Oxford, 1980) p.6’7; F. Abelès, Ann. Phys. (Paris) 5 (1950) 777. [161 H. Furstenberg, Trans. Am. Math. Soc. 108 (1963) 377. [17] P. Dean, Rev. Mod. Phys. 44(1972) 127. [18] K.S. Dy, S.Y. Wu and C. Wongtawatnugool, J. Phys. C 12 (1979) L141;

KS. Dy, S.Y. Wu and T. Spratlin, Phys. Rev. B 20 (1979) 4237.