PHYSICAL REVIEW 8 VOLUME 45, NUMBER 6 1 FEBRUARY 1992-II Transmission through a Thue-Morse chain Y. Avishai Service de Physique Theorique, Centre d'Etudes Nucleaires de Saciay, F91191 Gif sur Yv-ette, CEDEX France; Labor atoire des Physique Theorique des Liquides, Universite Paris VI, Paris, France; Department of Physics, Ben Gu-rion University of the Negev, Beer Sh-eva, Israe! D. Berend Department of Mathematics and Computer Science, Ben Gu-rion University of the Negev, Beer Sheva-, Israel; Mathematical Sciences Research Institute, Berkeley, California 94720 {Received 27 June 1991) We study the reflection ~rn ~ of a plane wave (with wave number k ) 0) through a one-dimensional ar- ray of N 5-function potentials with equal strengths v located on a Thue-Morse chain with distances d& and dq. Our principal results are: (1) If k is an integer multiple of m /~d, — d2 ~, then there is a threshold value vo for v; if v ) vo, then ~ r„~ ~1 as N~ ~, whereas if v ( vo, then ~rN ~~1. In other words, the sys- tem exhibits a metal-insulator transition at that energy. {2j For any k, if v is su5ciently large, the se- quence of reflection coefficients ~r„~ has a subsequence ~r „~, which tends exponentially to unity. (3) Theoretical considerations are presented giving some evidence to the conjecture that if k is not a multi- ple of n l~d~ — dq ~, actually ~r „~ ~1 for any v ) 0 except for a "small" set (say, of measure 0). However, this exceptional set is in general nonempty. Numerical calculations we have carried out seem to hint that the behavior of the subsequence ~ r „~ is not special, but rather typical of that of the whole sequence 2 ~rn~ (4) A. n instructive example shows that it is possible to have ~r„~~l for some strength v while ~ rz ~ ~1 for a larger value of v. It is also possible to have a diverging sequence of transfer matrices with a bounded sequence of traces. The experimental advance in submicrometer physics that enables the fabrication of nearly ideal one- dimensional wires' naturally leads to increasing interest in their physical features, especially their Fourier spec- trum and their transport properties. The quantum- mechanical relation between electrical conductance at zero temperature and the transmission probability indi- cates that some measurable physical quantities can be ac- curately explained on the microscopic level once a one- dimensional wire is modeled as an infinite array of poten- tials. Systems consist of infinite one-dimensional array of potentials are of course extensively studied in the litera- ture in connection with Bloch theory (if they are period- ic), and Anderson localization (if they are completely disordered). Following the discovery of quasicrystals, interest has been focused on the mathematical and physi- cal nature of quasiperiodic sequences and com- mensurate-incommensurate systems which are the first class of structures on the way from periodic to random matter. In order to study transport properties of such systems one needs a theory of scattering from a semi-infinite one- dimensional array of potentials. One such a theory has been developed earlier for the study of transport in a ran- dom potential. The basic technique is to express the transmission and reflection amplitudes through %+1 scatterers in terms of the amplitudes for N scatterers (a combination of Mobius transformation and multiplica- tion by a phase equivalent to the transfer matrix) and to let N~ao (the so-called thermodynamic limit). If the system is not random and a trace map is available, this procedure is quite powerful. In two earlier publica- tions ' we concentrated on scattering from an infinite system of 5-function potentials located on the Fibonacci numbers x„=F„and on the Fibonacci chain x„=n+u [nlrb], n =1, 2, ... , N, as N~ao, where u is a real number, v=(1 +~5) /2, and [ . ] denotes integer value. To this end we have developed special mathemati- cal tools, basically analytical and number-theoretical techniques. In the present work we use the same mathematical framework and report the results of our study on scatter- ing of a plane wave with wave number k ) 0 from a one- dimensional sequence of 5-function potentials of strength U located on a Thue-Morse chain. " To be more specific, let g„be the Thue-Morse sequence i. e. , („=0 or 1 ac- cording to the number of 1's in the binary expansion of the integer number n being odd or even, respectively. Then the Thue-Morse chain [x„] is constructed such that y„=x„+, — x„=d, or d2 when („=0 or 1, respec- tively, where d„d2)0 and d, &d2. This is a prototype of a sequence generated by substitution, in this case d, ~d, d2 and dz ~dzd &, with highly nontrivial features. The basic difference between the Thue-Morse chain and the Fibonacci chain ' is expressed in terIns of their Fourier transforms. In the first case the Fourier trans- form is singular continuous and the sequence is termed as aperiodic. In the second case the Fourier transform is discrete (or atomic), and the system is said to be quasi- periodic and exhibits Bragg peaks. These aspects of 45 2717