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arXiv:cond-mat/0604261 · Physical Review B

Impurity scattering of wave packets on a lattice

Wonkee Kim, Lucian Covaci, F. Marsiglio

Published Nov 27, 2006• 20 citations
arXivHTMLPDF

Abstract

Quantum transport in a lattice is distinct from its counterpart in continuum media. Even a free wave packet travels differently in a lattice than in the continuum. We describe quantum scattering in a one-dimensional lattice and illustrate characteristics of quantum transport such as resonant transmission. In particular we examine the transport characteristics of a random trimer model. We demonstrate the real-time propagation of a wave packet and its phase shift due to impurity configurations. Spin-flip scattering is also taken into account in a spin-chain system. We show how individual spins in the chain evolve as a result of a spin-flip interaction between an incoming electron and a spin chain.

DOI
10.1103/physrevb.74.205120

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Quantum transport in a lattice is distinct from its counterpart in continuum media. Even a free wave packet travels differently in a lattice than in the continuum. We describe quantum scattering in a one-dimensional lattice and illustrate characteristics of quantum transport such as resonant transmission. In particular we examine the transport characteristics of a random trimer model. We demonstrate the real-time propagation of a wave packet and its phase shift due to impurity configurations. Spin-flip scattering is also taken into account in a spin-chain system. We show how individual spins in the chain evolve as a result of a spin-flip interaction between an incoming electron and a spin chain.
In this paper, we use a straightforward numerical method to solve scattering models in one-dimensional lattices based on a tight-binding band structure. We do this by using the wave packet approach to scattering, which presents a more intuitive physical picture than the traditional plane wave approach. Moreover, a general matrix diagonalization method that is easily accessible to undergraduate students taking a first course in quantum mechanics is used. Beginning with a brief review of wave packet transport in the continuum limit, comparisons are made with its counterpart in a lattice. The numerical results obtained through the diagonalization method are then benchmarked against analytic results. The case of a resonant dimer is investigated in the lattice, and several resonant values of the mean wave packet momentum are identified. The transmission coefficients obtained for a plane wave incident on a step potential and rectangular barrier are compared by investigating an equivalent scenario in a lattice. Finally, we present several short simulations of the scattering process, which emphasize how a simple methodology can be used to visualize some remarkable phenomena.
With the help of quantum-scattering-theory methods and the approximation of the stationary phase, we propose a one-dimensional quantum-random-walk (QRW) model, which describes for both tunneling and scattering above the potential, the coherent diffusion of independent particles described by wave packets in a periodic one-dimensional lattice. The QRW model describes for each lattice cell the time evolution of modulating amplitudes of two opposite-moving wave packets as they are scattered by periodic potential barriers. Since the QRW model is a coherent process, interference contributions in the probabilities bring about strong departures from classical results. For many identical free particles we obtain the theoretical and graphical Bose and Fermi two-body QRW probability distribution. The result is generalized to N identical free particles and we obtain the N-body Bose and Fermi QRW probability distribution.
We investigate the propagation of spin impurity atoms through a strongly interacting one-dimensional Bose gas. The initially well localized impurities are accelerated by a constant force, very much analogous to electrons subject to a bias voltage, and propagate as a one-dimensional impurity spin wave packet. We follow the motion of the impurities in situ and characterize the interaction induced dynamics. We observe a very complex nonequilibrium dynamics, including the emergence of large density fluctuations in the remaining Bose gas, and multiple scattering events leading to dissipation of the impurity's motion.
The wave function temporal evolution on the one-dimensional (1D) lattice is considered in the tight-binding approximation.The lattice consists of N equal sites and one impurity site (donor).The donor differs from other lattice sites by the on-site electron energy E and the intersite coupling C. The moving wave packet is formed from the wave function initially localized on the donor.The exact solution for the wave packet velocity and the shape is derived at different values E and C. The velocity has the maximal possible group velocity v = 2.The wave packet width grows with time ∼ t 1/3 and its amplitude decreases ∼ t -1/3 .The wave packet reflects multiply from the lattice ends.Analytical expressions for the wave packet front propagation and recurrence are in good agreement with numeric simulations.
The wave function temporal evolution on the one-dimensional(1D) lattice is considered in the tight-binding approximation. The lattice consists of N equal sites and one impurity site(donor). The donor differs from other lattice sites by the on-site electron energy E and the intersite coupling C. The moving wave packet is formed from the wave function initially localized on the donor. The exact solution for the wave packet velocity and the shape is derived at different values E and C. The velocity has the maximal possible group velocity v = 2. The wave packet width grows with time ～ t1/3and its amplitude decreases ～ t-1/3. The wave packet reflects multiply from the lattice ends. Analytical expressions for the wave packet front propagation and recurrence are in good agreement with numeric simulations.
We study, via a scattering transfer-matrix approach, the spin waves of a Heisenberg chain with randomly inserted impurity segments having spin and coupling strength different from the host ones. Numerical results show the existence of completely unscattered reflectionless modes, whose number is sensitive to the impurity parameters. If both the spin and the coupling strength of the impurities are different from those of the host we find a novel type of resonant propagating wave, for which the reflection by a single segment is minimum but not vanishing. The relevance of the present results to other systems and to possible applications are discussed.