Critical Strength for Ideal Inconmensurate Structures

Volume 105A, number 8

PHYSICS LETTERS

5 November 1984

CRITICAL STRENGTH F O R IDEAL INCOMMENSURATE STRUCTURES L F . WEISZ and H2d. PASTAWSKI INTEC, Instituto de Desarrollo Tecnblogico para la lndustria Quimica, ( UNL-CONICET), Guemes 3450, 3000 Santa Fe, Argentina Received 23 February 1984

A concise method is developed to show the following in one dimension: (a) If there is a sharp metal-inmhtor transition in an ideal sinusoidal incommensurate structure then W/V = 2. Co) There is an infinite dc conductivity of electrons in an ideal incommensurate structure for T = 0 if W/V < 2. (c) Addition of impurities which scatter between all pairs of k values may lead to a finite conductivity for W/V < 2. (It tends to zero as L -* =). The concept of duality used by Aubry is then extended to the general problem of localization and the breakdown of extended states is illustrated.

Recently, several distinct problems such as Bloch electrons in a magnetic field [ 1 ] , electrons [2,3] and phonons [4,5] in incommensurate structures and the study o f superconductivity on a n e t w o r k * l [6] have been cast in the form o f the eigenvalue problem ( E - e n ) C n ÷ VCn+ 1 ÷ VCn_ 1 = 0 ,

On Fourier transforming the periodic potential W cos Qx one has Vkk, = ~ W S ( k - k '

(1)

(2)

E1-H

E - ek

A

A

E - ek+ Q

A

A

E - ek+2Q

_ =

I f Q = 21r/X and X/a is irrational the two periods are said to be incommensurate. Under these conditions it has been shown by Aubry [7] that Wc/V=2 is a critical value for a transition between extended and localized states. Many numerical calculations and other arguments support this conclusion [ 8 - 1 0 ] . In what follows we present a concise way o f understanding this result and its implications. In a tight binding representation one writes in coordinate space

(5) Here A = ~ W and e k = J cos ka where J = 2 V. Also as is well known a continued fraction expansion in coordinate space is 1

G0°=E-e "E - e_ 1 El-H-

V

V E-e V

(4)

Then i n k space one has

where e . = w cos Qna .

+Q)+~W6(k-k'-Q).

o-~00

'

where 0

V

(3)

E - eI

and I;~

#I In this problem it is inecisely the critical value of I¢ which is of interest. 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

-

V2 E - eI - V 2 E -

(6) e2 -

... 421

Volume 105A, number 8

PHYSICS LETTERS

Convergence of such an expression as (6) in the realm of real numbers means that Z~0 is real, and the states are local with an infinite lifetime. If it did not converge then Zoo may have an imaginary part Im Zoo = h/r L #: 0, where T£ is the lifetime of the localized states, so that diffusion could then take place. Similarly, for Gkk we have

Gkk = where A2 ~k =

(7)

E -- ek+ Q -- A 2 E -- ek+2Q -- ...

For k = 0 we notice a formal equivalence between the two continued fractions if Ae.V,

l¢'~J.

Convergence of (7) means that plane waves are an adequate basis now (as local functions were previously) and thus one has extended states. If it does n o t converge we may think of the existence of an imaginary part lm Zkk -- ~i/Te where r e is the lifetime of extended states. Let us suppose that we get convergence of ZOO for W/V > q and we want to Fred q. I4/which in coordinate space in coordinate is on the diagonal becomes A = 1W on the off-diagonal in k space. On the other h a n d , J = 2Vin the diagonal terms in k space goes to V = ~J on the off-diagonal in coordinate space. The same convergence criteria as before now applies in k space, because of the formal equivalence of the continued fractions, i.e. convergence in k space takes place if

IJ/Al>q

or

2V/~W>q

or

W/V