Crossover from Regular to Chaotic Behavior in the Conductance of Periodic Quantum Chains

The conductance of a waveguide containing finite number of periodically placed identical point-like impurities is investigated. It has been calculated as a function of both the impurity strength and the number of impurities using the Landauer-Büttiker formula. In the case of few impurities the conductance is proportional to the number of the open channels N of the empty waveguide and shows a regular staircase like behavior with step heights ≈ 2e2/h. For large number of impurities the influence of the band structure of the infinite periodic chain can be observed and the conductance is approximately the number of energy bands (smaller than N) times the universal constant 2e2/h. This lower value is reached exponentially with increasing number of impurities. As the strength of the impurity is increased the system passes from integrable to quantum-chaotic. The conductance, in units of 2e2/h, changes from N corresponding to the empty waveguide to ∼ N/2 corresponding to chaotic or disordered system. It turnes out, that the conductance can be expressed as (1− c/2)N where the parameter 0 < c < 1 measures the chaoticity 1 of the system. In recent years transport in a wide variety of mesoscopic systems has been investigated. One of the interesting questions is the behavior of the conductance of disordered identical blocks coupled in chain. This problem may arise whenever identical chaotic or disordered nanostructures are organized in a sequentially built structure. So far mostly chains with completely disordered/chaotic blocks have been studied [1,2]. Our aim in this letter is to study a system in which chaoticity of the blocks can be tuned by varying a single parameter c ranging from zero (completely regular) to one (completely chaotic). Well known examples of such systems are provided by point-like impurities with variable strength placed in regular waveguides [4,5]. We study here the low temperature conductance of a waveguide containing finite number of periodically placed identical point-like impurities. In this model the conductance depends on the strength and on the number of impurities. It will be shown that when passing from the regular toward the quantum-chaotic case by increasing the impurity strength the average conductance (in units of 2e/h) behaves like ∼ αN , where α = 1− c/2, interpolating between the conductance of the regular (empty) waveguide N and the universal conductance of chaotic systems ∼ N/2 determined by Random Matrix Theory [3] (RMT). The direct universal link between c, characterizing the chaoticity, and the average conductance is our main result. We consider an ideal 2D waveguide of width W which is divided into L blocks of length a. We place an impurity with Dirac-delta potential U(r) = λδ(r − r0) in each block, where r0 is its position within a block (see Fig. 1) and λ is the strength of the potential. Inside the block the potential is assumed to be zero and the wave functions should fulfill Dirichlet boundary condition (ψ = 0) on the walls of the waveguide. The system is adiabatically matched to 2D electron reservoirs on both ends. The conductance is given by the Landauer-Büttiker formula [6] G(EF ) = 2e h T, (1) where T is the total transmission coefficient of the system at Fermi energy EF . The total 2 transmission can be expressed in terms of the partial transmission amplitudes tnm of open modes between the entrance and the exit of the waveguide