PHYSICAL REVIEW B VOLUME 51, NUMBER 10 1 MARCH 1995-II Explanation of delocalization in the continuous random-dimer model Angel Sanchez Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 Francisco Dominguez-Adame Departamento de Fisica de Materiales, Facultad de Fisicas, Universidad Complutense, E-28040 Madrid, Spain Gennady Berman Complex Systems Group T-13, Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 and Kirensky Institute of Physics, Krasnoyarsk 660036, Russia Felix Izrailev Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 and Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia (Received 18 November 1994; revised manuscript received 13 January 1995) We propose an explanation of the bands of extended states appearing in random one-dimensional models with correlated disorder, focusing on the continuous random-dimer model [A. Sanchez, E. Macia, and F. Dominguez-Adame, Phys. Rev. B 49, 147 (1994)].We show exactly that the transmission coefficient at the resonant energy is independent of the number of host sites between two consecutive dimers. This allows us to understand why there are bands of extended states for every realization of the model as well as the dependence of the bandwidths on the concentration. We carry out a perturbative calculation that sheds more light on the above results. In the conclusion we discuss generalizations of our results to other models and possible appli- cations which arise from insight into this problem. I. INTRODUCTION Since the beginning of this decade, a number of disor- dered one-dimensional (1D) models have been proposed' which exhibit nontrivial extended states. The key ingredient of those models is correlation: The defects or impurities are introduced in the host lattice at random, but always forming pairs, i.e., they never appear isolated. Further research in this and related models supported this unexpected result, includ- ing other grouping rules aside from pairing. All those results established on firm grounds the existence of bands of extended states in this class of models, at least in finite-size samples. However, the reasons for such bands to arise re- main unclear: To the best of our knowledge, there are only some perturbative results estimating the number of states, "' and some symmetry conditions for the existence of this kind of resonances. There is no need to stress the importance of achieving a good understanding of this delocalization phe- nomenon, both from theoretical and applied viewpoints: Such advance will certainly be helpful both to settle down theoretically its relevance and generality, as well as to design devices with specific transmission properties. On the other hand, we concern ourselves with a model which has been much less studied (indeed, we do not know of any perturba- tive calculation or related result regarding it) and which, on the other hand, has specific properties. We address these is- sues in this paper. To this end, in Sec. II we present the specific model we study, the continuous random-dimer model, ' and the properties we will be dealing with. In Sec. III we show how those features may be understood in terms of the structure of the transmission coefficients through lat- tice segments. Finally, in our conclusions we discuss corre- lated disordered models in general in view of our results, as well as possible applications of this work. II. THE CONTINUOUS RANDOM DIMER MODEL The continuous random-dimer model (CRDM) was intro- duced in Refs. 8 and 9 and is described by the following Schrodinger equation (we use units such that 6, = 2m = 1): 2++ )I „8(x—n) t/t(x) =Et/t(x), where )i.„)0 (the extension of the results to the X„(0case is straightforward; besides, the choice of the sign is irrelevant in most applications, e.g. , for superlattices ). To introduce paired correlated disorder k„ takes only two values at ran- dom, namely X and X', with the constraint that k' appears only in pairs of neighboring sites (dimer). This model is related to and inspired by the (tight-binding) random-dimer model (RDM) of Dunlap et al. , ' and we believe that our work on the CRDM will provide also relevant ideas for the RDM. However, there are a number of significant differences between both models. First, the CRDM exhibits an infinite number of resonances and their corresponding bands of ex- tended states, ' which makes it interesting from the view- point of applications as there are many options to match the Fermi level. Second, the fact that the CRDM is continuous and includes multiple scattering effects gives it a more real- 0163-1829/95/5 1(10)/6769(4)/$06. 00 6769 1995 The American Physical Society 6770 SANCHEZ, DOMINGUEZ-ADAME, BERMAN, AND IZRAILEV 51 gati t ii itive t t ~ i ~~ i iiiii iii 1.0 1.000 I -'0, 8 0.995 —0.8 —0.4 0.990 2.5 3.5 4.5 —0.2I ~ I I l L I I I i I I ~'0 0 5.5 I 3.0 3.5 4.0 4.5 0.985 0.980 5.0 FIG. 1. Transmission coefficient for the CRDM with a dimer concentration c=0.2. The 6 function strengths are X =1, X' = 1.5. Shown is an average over 100 realizations. Every realization con- sists of 15 000 scatterers. The first allowed band in the perfect lattice is [0.921,9.870]. FIG. 2. Transmission coefficient for two dimers with N host sites in between, placed in the middle of an otherwise periodic chain, for N=1, 2, 3, 4, 5, and 6 as indicated in the plot (even N, solid lines; odd N, dashed lines). The 8' function strengths are X= 1, A. '=1.5 as in Fig. 1. istic character, thus supporting the possibility of seeing these effects in a variety of actual physical systems. Finally, the RDM and the CRDM have different parameters: Whereas the RDM depends on the on-site energies e„and eb and on the hopping term V (see Ref. 2), the CRDM depends on the strengths of the 6 functions and the intersite distance. We now summarize the main features of the CRDM. In Ref. 9 we developed a generalized Poincare map formalism that allows us to map exactly general one-dimensional Schrodinger equations onto discrete equations, for any po- tential allowed in quantum mechanics. In particular, Eq. (1) is equivalent to the discrete map t/r +i+ t/r —t =20„t/r„, (2) where t/r„—= t/r(n) and A„—=cosq+(k„/2q)sinq, with q= +E That formalism allowed us to prove that there are an infinite number of resonant energies for which the reAection coeffi- cient of a single dimer vanishes. Resonant energies are given by the conditions ~ A ~ ~ 1 and II ' =0, where A=cosq+(X/2q)sinq and II' the same but replacing ) by X. '. The same result can be generalized in a different manner by using a technique valid for any equation cast in the form of Eq. (2) as explained in Ref. 13. We further showed that those resonances survive in the presence of a finite density of dirners, i.e., in the CRDM, and moreover, that they give rise to bands of finite width of truly extended states. This we established by analyzing several magnitudes, among which we take here the transmission coefficient as an example of the behavior of the model. We choose this magnitude be- cause it will subsequently be the main ingredient for our explanation of delocalization. An example of the mean behavior of the transmission coefficient around one of the resonant energies is shown in Fig. 1 for a dimer concentration c=0.2 (c is defined as the ratio between the number of X' and the total number of 8's in the lattice); typical realizations behave in the same way, the only effect of averaging being to smooth out particular features of realizations keeping only the main common char- acteristic, i.e., the wide transmission peak. This is the prop- erty we want to highlight: Close to single dimer resonances (in the case of Fig. 1, the first one, which occurs at E„=3.7626. . . for the chosen parameters X = 1.0, X ' = 1.5), there is an interval of energies that shows also very good transmission properties, similar to those of the resonant energy. Most important, such interval has always a finite width, for all values of dimer concentration, k and )i. ' (pro- vided they satisfy the above conditions), or number of sites in the lattice. The peak width depends on the order of the resonance (the higher the resonance, the wider the band of states with transmission coefficient close to unity) and the concentration of dimers (the larger the concentration, the nar- rower the peak, being always of finite width as already stated). Other magnitudes, such as Landauer resistance or Lyapunov coefficient behave accordingly. After collecting the main facts about the CRDM and its bands of extended states, we state what is it that we want to explain: First, why are there intervals of energies for which the transmission is very close to unity for every realization of the CRDM? Second, why does the bandwidth decrease with increasing dimer concentration? Third, why does the band- width not vanish when the dimer concentration goes to 1? It is clear that if we are able to answer those questions, we would have understood the physical reasons for the appear- ance of the extended bands we are concerned with. This we discuss in detail in the next section. III. TRANSMISSION COEFFICIENT THROUGH LATTICE SEGMENTS Pursuing answers to the above questions, we have com- puted the transmission coefficient of structures formed by N sites of type P sandwiched either between two dimers or two single impurities, with all that group embedded in a perfect infinite chain of ) sites. The calculation is once again a transfer matrix one, which yields an expression that can be evaluated with the help of a computer. The results, obtained by numerical evaluation of those exact analytical expres- sions, are plotted in Fig. 2 for the two dimer case and in Fig. 3 for the two single impurities case. Let us begin discussing the dimer results. It is apparent from Fig. 2 that in all cases considered, the transmission coefficient is very close to unity EXPLANATION OF DELOCALIZATION IN THE CONTINUOUS. . . 6771 i.00 eikn+ r& —ikn if n ~ 1 teikn if n~N+4, 0.9S where cosk=A, and we define the promotion matrices 0.96 (2n P= 1 (2n' 0 ) (4) Notice that P and P' are unimodular. The reflection ampli- tude can be found as follows: 3.0 I 3.5 -0 98 4.0 4.5 5.0 T22+ T]2e T2]er=e' ik —ik ~ T2i —T)2+ T22e —T)ie FIG. 3. Transmission coefficient for two single impurities with N host sites in between, placed in the middle of an otherwise peri- odic chain, for N=1, 2, 3, 4, 5, and 6 as indicated in the plot (even N, solid lines; odd N, dashed lines). The 8 function strengths are k=1, k'=1.5 as in Fig. 1. for energies in the neighborhood of the resonant one. This must be related to the fact that eigenfunctions corresponding to those energies acquire an extra phase which will be dif- ferent from the resonant energy condition (the change of phase has to be 7r), but very close to 7r anyway. The key point is that for any value of N considered, this interval is always located around the resonant energy. This must be compared to Fig. 3, where it can be seen that for different N values the position of the perfectly transmitted energies is also different. Therefore, we can conclude that the physical reason underlying the existence of bands of extended states is this overlap of good transmission properties that happens in the CRDM forced by the resonant energies of the dimers. The above results allow us also to understand why the width of the bands decreases with increasing concentration but being always finite: Note for the case with N=1 in Fig. 2 that even in this case, the band shows a nonzero extent. It is quite clear that in the high density limit most occurrences of the k sites will be of that type, i.e., one k between two dimers. This is the case that will then govern the total trans- mission coefficient of the chain (obviously, groups of dimers will be perfectly transparent around the resonant energy as this is placed in the X' band). We thus see that even in the case when the dimer concentration tends to unity, the struc- ture of the transmission coefficient for N=1 will preserve the band. Upon decreasing the concentration, those cases will be more and more rare, and the dominant ones wi11 have larger N. Figure 2 shows the dramatic increase of the band with increasing N, and this is in perfect agreement with the observations for the dilute chain. ' It is also possible to carry out a power expansion in E—E„ofthe transmission coefficient 7. close to the resonant energy, starting from the above-mentioned transfer-matrix re- sults. Importantly, the approach is general for any 1D model, because it can be first cast in the form of Eq. (2) (see Ref. 9) and then treated within the formalism we describe now. For the sake of brevity, we skip the general formulas and particu- larize Eq. (2) for the system ki=kz= kz+3= k~+4=k' and X„=) otherwise. We introduce reflection r and transmission t amplitudes through the relationships where T= (P ') P (P ') . Taking into account that P =Uz i(A)P —U& z(A)Iz, Iz and U„being the 2X2 unity matrix and the Chebyshev polynomial of second kind, respectively. Thus, the matrix elements T;, can be easily written down. So far, this result is exact for all energies; since we are interested in those values of E close to E„, a power expansion leads us after lengthy but straighforward calculations to the result r= 1 — ~ r ~ —1 f(N) (E—E„), — where f(N) is a positive, energy-independent function writ- ten as f(N) =A(E„)f(N) with A„[k (cosq„—sinq„/q„) —2q„sinq„] A(E„)= z z, (6a) f(N) = 1+(1—0z) I +rUN 1 UN 2 (6b) IV. CONCLUSIONS In summary, we have explained the existence of bands of extended states in the CRDM as arising from the property that transmission is almost perfect for all kinds of segments in the lattice around the resonant energy, which is not the case if the impurities are not paired. It is crucial to notice that this holds for every realization of the model. This explana- tion also accounts for the dependence of the bandwidth on the dimer concentration and its finiteness for any such con- centration. We have been able to estimate perturbatively the transmission coefficient and the divergence of the localiza- tion length in that energy interval. The relevance of this per- turbative calculation increases if one realizes that it is pos- sible to compute the mean transmission coefficient around the resonance by integrating f(N) with the probability of and the dependence of the Chebyshev polynomials on 0„ is understood. Here the subscript r means that all expressions must be evaluated at the resonant energy E, . Taking into account that f(N)~1 we obtain a very important result, namely that f(N) is bounded above. The upper bound is A(E,) =0.0247. . . with our chosen parameters [f(N) equals this value whenever N sat isfys U& i (0„)= 0]. Hence, the transmission in the vicinity of the resonant energy is very close to unity. In addition, the above perturbative treatment yields a divergence of the localization length of the form -(E E„), as foun—d in the RDM, as well as by different means. 6772 SANCHEZ, DOMINGUEZ-ADAME, HERMAN, AND IZRAILEV 51 finding a segment of N host sites in the infinite lattice. Fur- thermore, we believe that this explanation applies to all mod- els in the same kind of disordered systems with defect group- ing, because the calculations on those models will be formally very similar, as shown in Ref. 13. We note in closing that the structure of the transmission coefficient as depicted in Fig. 2 suggests that it is possible to build devices with tailored properties by designing an or- dered structure made up of unit cells formed by dimers with the appropriate number N of host monomers between them. In this context, quantum well superlattices can be a perfect example of such devices, as it has been shown ' that the CRDM can be realized in practice as a GaAs/A1GaAs sys- tem. On the other hand, there has been recently a significant increase of interest in nanotechnological ap lications of monomolecular assemblies on solid surfaces. Such self- assembled monolayers (SAM) can build up complicated quasi-1D and 2D structures. One of the important properties of SAM's is that they can show very different electron con- ductivity depending on their composition and structure. We hope that the above resonant mechanism of appearance of the extended states in correlated disordered models will be relevant to understand and design SAM's with the desired conduction properties. ACKNO%'LED GMENTS A.S. was partially supported by MEC (Spain)/Fulbright, by DGICyT (Spain) through project PB92-0248, and by the European Union Human Capital and Mobility Programme through Contract No. ERBCHRXCT930413. F.D.-A. was supported by Universidad Complutense through project PR161/93-4811. 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