Power-law localization at the metal-insulator transition by a quasiperiodic potential in one dimension

Abstract

The localization properties of an electron in the tight-binding potential A, cos(man ), where 0 & v & 1, are investigated. Wave functions approaching the metal-insulator transition (MIT} A. , =2 -~E ( (or the mobility edges E, =+~2 -iL~) from the metallic region are found to localize with multiple well-separated power-law decaying peaks, contradicting previous conclusions obtained with the help of the Lyapunov exponent. The general expectation of exponential localization with divergent localization length at the MIT cannot be detected convincingly.The electron localization in a three-dimensional medium of a random potential is well established. ' The conductivity in such systems vanishes if the measure of disorder exceeds a certain critical value. This phenomenon is a manifestation of the metal-insulator transition (MIT). In the spectrum of three-dimensional (3D) random systems there exist mobility edges E, that separate extended states from localized ones. The MIT takes place when E, crosses the Fermi energy as the disorder is increased.On the metallic side of the MIT, the states are extended functions, while in an insulator they are assumed to have an exponential decay with a localization length which is divergent at the MIT. Therefore at the MIT one expects an abrupt change in the spatial characteristics of the states. Thus it is especially of great importance to find the properties of the wave function at the mobility edge suggesting the way how the conductivity tends to zero at the MIT.A random potential-induced MIT appears only in 3D systems, while in 1D an infinitesimal amount of disorder localizes all the states, and it is probably the same in two dimensions, as well. In 1D systems, ho~ever, one may obtain a MIT induced by deterministic quasiperiodic potentials. ' These potentials are of special experimental importance, since they appear naturally in crystals containing charge-density or spin-density waves, as well as in superlattices grown with molecular-beam epitaxy. In this paper we will study the localization properties of the wave functions of 1D quasiperiodic potentials at and in the vicinity of the mobility edge in the spectrum, i.e. , at the MIT.The discretized one-dimensional tight-binding Schrodinger equation within the nearest-neighbor approximation is written as t (u"~i+u",) -( V"E)u"=O, where u" is the amplitude of the eigenstate on the nth lattice site, V" is the on-site (diagonal}, quasiperiodic potential, and E is the eigenenergy. For sake of simplicity t = l is chosen as the unit of the energy scale without any loss of generality. V"=Xcos(~ctn'),(2) which produces a gapless spectrum containing mobility edges' ' in contrast to the more complicated spectrum of other quasiperiodic models. ' A perturbation theory calculation' yields localized states for v~2 and extended states for 0& v 1 near the band center. %KB theory, ' however, indicates a clear mobility edge at the critical energy E, =+ 2 -k~if 0& v & 1, which means that there is a MIT at A, , =2 -~E~f or a fixed energy.(assuming expo...

New Class of Level Statistics in Correlated Disordered Chains

“…Further, it has been shown that there is a localization-delocalization transition at a critical value of the degree of correlations imposed on the disorder in the system, and later works extended these results to other models [9][10][11][12]. Also, this type of transition can be found in quasiperiodic systems, as the Aubrey-Andre model and other models [13], indicating the importance of some type of ordering. These theoretical findings are supported also by experimental results showing delocalization and electronic transport driven by extended states in correlateddisordered GaAs=Ga 0:7 Al 0:3 superlattices [14].…”

Mathematical characterization and shape analysis of localized, fractal, and complex distributions in extended systems

“…As we can see, the generalized localization points (q, S , , ) of the ensemble are arranged along a reference curve that corresponds to the decay form f ( p ) = (1 + p)-'. Therefore, we expect a power-law decay of the wave function peaks, which was graphically demonstrated in [20]. We would like to mention here that the power-law localization in the extended range of the wave functions has an important effect on the conductance of the system.…”

“…In previous articles [18][19][20], we demonstrated that the characterization of one-particle states by the quantities ( q , S f , , ) gives meaningful results for the structure or shape of the wave function. As we will see in numerical applications, in some cases, most information about a given physical system is contained rather in the set { ( q p , S$)} of an appropriate ensemble of states {Ip)} .…”

Space–time, spectral, and energy characteristics of a gold-vapor ultraviolet laser

“…Lyapunov exponents of the form (4) as well as (5) are completely adequate choices for the study of localization phenomena only if exponential localization of the electrons is expected. When the electronic states exhibit a more complicated behaviour (like power-law localization) other forms of localization length may be opportunely used to characterize the shape of the electronic wavefunctions [13,15]. The primary aim of this work, however, was to ascertain whether the states beyond the second-order mobility edge were exponentially localized or not and the Lyapunov exponent (4) is a sensitive enough indicator for this purpose.…”

“…Analytical calculations are greatly simplified by truncating the expansion of the inverse localization length to the second-order term; the main drawback of this choice is that it leaves open the question of the true nature of the states of the 'extended' phase. In fact, a vanishing second-order inverse localization length can be related to various physical phenomena: it can indicate that the electronic states are completely delocalized, but it can also be a sign that the states of the extended phase undergo a power-law localization (evidence for a behaviour of this kind, for instance, has been found at the mobility edge in specific quasi-periodic systems [13]). Finally, the inverse localization length may be non-zero even if its second-order part vanishes, in which case the 'extended' states would still be exponentially localized, although on a much larger spatial scale than that characterizing the states of the 'localized' phase.…”