Solid State Communications, Vol. 46, No. 8, pp. 601-603, 1983. 0038-1098/83/200601-03503.00[0 Printed in Great Britain. Pergamon Press Ltd.

QU ',NTUM LOCALIZATION IN ONE-DIMENSIONAL QUASI-RANDOM SYSTEMS AND MAGNETIC BREAKDOWN

A.A. Slutskin and L.Yu. Gorelik

Physico-Technical Institute of Low Temperatures, Academy of Sciences of the Ukr.SSR, Kharkov 310164, U.S.S.R.

(Received 17 November 1982 by ~.A. Kaner)

The magnetic breakdown in metals is shown to cause the appearance of a new class of one-dimensional quasi-random "incommensurable" systems where the electrons are localized due to quantum interference effects. At this time both absolute localization and phase transition of "meta l - dielectric" type can be realized.

1. IT IS KNOWN [1] that in any disordered one-dimensional system, in which distant correlations between the values of the parameters characterizing dis- order are absent, Mott-Anderson localization takes place. All stationary states of a particle in such a system are localized. Recently much attention is paid to the study of quantum localization in the systems which can be naturally called quasi-random ones, because the parameters distribution in such systems is irregular, but with long-range correlations. For example, such a situ- ation takes place in a one-dimensional chain of atoms with some incommensurable periods. The presence of long-range correlations can essentially change the character of quantum localization. In accordance with [2, 3], in the above case of an "incommensurable" one- dimensional system the majority of stationary states of a particle with sufficiently large energies are generally delocalized.

In the model proposed by Aubry and Andre [4] a phase transition in the parameter g characterizing the mobility of a particle takes place. When g > ger, all the states are localized and when g < get delocalization takes place. In the present work we shall demonstrate that magnetic breakdown (MB) in metals causes the appear- ance of one-dimensional quasi-random systems of a new type in which a wide range of physical situations is real- ized. It is the range from the absolute quantum local- ization of conduction electrons to the localization admitting the phase transition described in [4].

2. The important class of one-dimensional quasi-random systems under consideration covers weakly aperiodic magnetic breakdown configurations (the systems of affiliated electron orbits in p-space) which arise in a typical experimental situation, when the magnetic field H = (0, 0, H ) forms a small angle 0 with the plane per- pendicular to one of the reciprocal lattice vectors b. For the brevity of description we shall consider the MB con-

Due to the smallness of semiclassical parameter = ehH/cb 2 (r ~ 10-4-10 -5 whenH ~ 10s-1060e)

the stationary wavefunction I~b ) of the electron of given magnetic breakdown configuration can be presented as:

I ~ = Y c~.IG.> (I) n,a

where subscript n enumerates the electron orbits, index a = I, 2 designates the parts of the nth orbit which are enclosed within two magnetic breakdown centers, ] ~ban) - semiclassical wavefunction describing the motion of the electron along the part (a, n). The main dynamic characteristic is carried by amplitudes can which determine the probability of finding the electron on the semiclassical parts of the magnetic breakdown configuration. The amplitudes Can satisfy the set of equations describing multiple scattering of the electron on the magnetic breakdown centers

_,[exp (iCn) r exp (--i~)n) ~ cn+, = Chin;in = P _~rexp(iCn ) e x p ( - i C n ) ];

r2 + p2 = 1 (2)

where c, = (cln, c2n); the quantity O 2 = If(H) deter- mines the probability of the transition between adjacent orbits (magnetic breakdown probability ~ [5 ]) and is the same for the whole magnetic breakdown configur- ation. The quantity ~b n is a quasi-classical phase determination by the relation

dpn = Sn(E, pz, O)/rb 2 = S(E,p z + nOb)/rb 2

where S n is the area of the nth orbit and E is the elec- tron energy. When 0 is small 0 ~ 1 the dependence Cn can be approximated by finite polynomial

figuration shown in Fig. 1.

601

[3ms l . S(E, pz) (2a) x \ a p ~ } ' 4,o - h Kb - - - r -

M ombm-2 ~,, = ~o + Y~ a , , , . " , a , . -

m=l m!K

602 QUANTUM LOCALIZATION IN ONE-DIMENSIONAL QUASI-RANDOM SYSTEMS Vol. 46, No. 8

H" x(n) =~=1 l n ( 11 + r exp ( ixk)12)W . (6)

Fig. 1. Thick lines in the figure designate small hole cavities of the Fermi surface which play the role of MB effective regions and connect large electron cavities into a single MB configuration. The areas of MB configur- ation electron orbits correspond to the given value of quantum number pz-pulse projections on the direction, I-I, are hatched. The direction of classical orbital motion is shown by arrows.

in which because of the smallness of K the order of some first coefficients (usually Bx, B2) and of the phase differ- ence £x0,(0) = 0n+l -- 0n becomes equal to unity when 0 ~< 1 °. Such dynamic system is essentially irregular, and its irregularity is of quasi-random character: the phase multipliers exp (iOn) in i n generally fill the circumfer- ence with a radius equal to unity with uniform density; but there exist long-range correlations in their distri- bution due to regular n-dependence of 0. We shall demonstrate below that in the system under consider- ation the electron stationary states are localized almost for all 0 as 0 and W 4: I, that is, they cover the finite number of orbits.

As shown by equation (2), the localization takes place, if the norm of transfer-matrix ~Pn = II~ =l t~ grow exponentially when n ~ oo for all values of energy E. Taking account of the indentity

[¢1n[ 2 - - i C 2 n [ 2 = const (3)

reflecting the unitarity of the scattering-matrices on the magnetic breakdown junctions, we can demonstrate that vectors have the form

e , = al exp [x(n, ),(l))]f~l) + a2 exp (x[n, X(2)])f~ 2)

(4)

where a~,2 are arbitrary coefficients, f ~ 'z) are vectors all components of which are modulo equal to unity, the phase difference of their components ),(n 1' 2) satisfying the following recursive expression that plays the main role in our consideration:

)tn+ 1 = )~n + 2 I m l n [ l + r e x p ( - - D t n ) ] + O n . (5)

The choice of initial conditions ),o = )t°'2)(;kO) 4= ),<2)) in equation (5) is unimportant. The quantity x(n) is directly expressed via )t n :

Therefore, the localization criterion is equivalent to the condition: x(n) --> oo when n --> oo.

As shown by equation (6), in the limit W ~ 0 the localization criterion wittingly fulfils. This fact has a simple physical interpretation. In the limit I¢ = 0 every magnetic breakdown orbit gives rise to a set of the electron stationary states according to its localization on the given orbit. The energy levels are determined by quasi-classical quantization rules Cn(Er) = 2;rr (r is an integer). When I¢ is small the electron can move along the magnetic breakdown configuration only at the expense of tunneling between the orbits n and n ' for which the relative shift of the systems of unperturbed levels A,_ n, = {0n(Er) -- 0,'(Er')/2zr} is smaller than the probability of transition n ~, n ' the order of which is equal to wIn-n'l({a} is a fractional part of a). It is easy to see that the condition An_ n' <~ 14] In-n'[ is not gener- ally fulfilled for the quasi-random system under consideration. This leads to the electron localization. The possibility of localization is not so obvious for arbi- trary W and, moreover, in the limit W ~ 1, when the electron motion along the magnetic breakdown configur- ation is almost free.

In the general case, owing to the "conservation law" (3), only two kinds of behaviour of the sum x(n) in equation (6) are possible when n ~ oo:x(n ) ~ co (localization), x(n) is a function with limited variation (delocalization). The limitation of ×(n) is, however, impossible because the arithmetic means

1 ore(n) = - L exp (im)~k) (7)

/'/ k = l

through which x(n) is expressed, fluctuate with the amplitude tim(n) decreasing more slowly than n -1. This property of ore(n) is probably fulfilled at r = 0 (W = 1) when

n

x . = ~ . - Y~ 0k (8) k = l

is a polynomial of degree M + 1 ~> 2. It can be shown that this property is preserved at 7 a s 0. Moreover, it is found that for almost all/9 the )t k phases are distributed uniformly: limn__,~ an(m) = 0". In this case lim n ~.0 x(n)/n = -- In W. It follows that the number of orbits in the magnetic breakdown configuration, on which the amplitudes of the stationary wavefunction in

* This fact can be established by the Fourier analysis of B m periodical function exp (ihk) (period is equal to 2zr)

21r N 2 and using the estimate fd d01~k=l exp [iOf(k)]l ~ N , valid for all the increasing functions f (k) .

Vol. 46, No. 8

equation (1) are not small, is of the order L ( W ) = 1 --

1/In W, that is, the localization radius increases as ( 1 - - W) -1 at W ~ 1. The given estimates are not valid if all coefficients B m in equation (2a) are small. In this case fluctuation addend (~m(n) in equation (7) increases and x(n) tends to linear asymptotic X = -- n in W only when

n ~ L(W)/Bma x > L(W)(Bmax = max {Bm/27r}l/m).

In the coordinate space the electron localization on the magnetic breakdown configuration corresponds to the localization across the magnetic field with the radius R ~ r~L(W)(1 + B~ax) (r~ is the characteristic Larmor radius). In the case R <~ l (l is the mean free path of the electron) the rise o f such "quantum traps" essentially tells on the majority o f the metal kinetic characteristics. It is interesting to note that in the general case W ~ 1, l aS/apz [ ~ bo(Bmax = BI -- O/K) the localization radius R ~ r~K/O becomes comparable to rn when 0 is rather small: 0 ~ K ~ 10 -4. By this reason for ~/K >> 0 >> K the main contribution in cross-conduction along the direc- tion [llb] and in analogous in structure kinetic coef- ficients is made by a narrow layer of Pz values, which lie in the vicinity of point P~o, where the square S(pz ) is extreme, Bma x "~ B~/~ = 0/~/~ and R increases ~/K/0 times: R ~ r i~/~/O. The width of this effective layer is VKb ~ lO-~b[l ~ r,~/VS~ ~ r~(VK/O)].

3. The above model is naturally generalized if we con- sider the phases ~b n in equation (2) as given parameters of the problem. The localization fluctuation criterion nfim(n) ~ oo at n ~ oo holds true; in particular, it is obviously fulfilled when the distribution of a n is true- random (Mot t -Anderson localization). Due to the small- ness of K the clearly expressed quasi-random situation

QUANTUM LOCALIZATION IN ONE-DIMENSIONAL QUASI-RANDOM SYSTEMS 603

can arise in the case of strictly periodic orbits of the magnetic breakdown configuation (0 = 0) when the elec- tron is affected by small regular external fields. Thus, conforming to the electric field E & It, b, phases q~k = A k (A = Ecb/KHeF)(eF is the Fermi energy) stimulate the same absolute electron localization as aperiodic mag- netic breakdown configurations. Upon propagation o f the acoustic waves with the wavevector q 1 El, b, phases Ck acquire the form q~k = ¢o + Ao cos (aqrnk), where the amplitude Ao ~ uo/eF (Uo is characteristic defor- mation potential in the acoustic wave a ~ 1 ). In this case n(rrn(n ) ~ 1, that is the fluctuation criterion breaks and the phase transition "localization-delocalization" of the type [4] takes place. The points of phase trans- ition on the plane (W, A 0) form the line which begins at point (0, 0). Detailed consideration o f this situation will be given in a separate paper. It follows from the pre- ceding discussion that the above localization phenom- enon creates large opportunities for the appearance of strong nonlinear effects in pure normal metals.

Acknowledgements - The authors are grateful to I.M. Lifshits and L.A. Pastur for numerous fruitful dis- cussions.

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