Metal-Insulator Transition in One-Dimensional Deformable Lattices

Abstract

Metal-insulator transition at 0 K in a one-dimensional classical deformable lattice (Peierls transition) is viewed through two complementary models which consider properly the discreteness of the lattice in the case of a Fermi wave-vector k_F incommensurate with the reciprocal lattice vector. The electron-phonon interaction is represented in both models by the same parameter λ while the electron-electron interaction and the spin effects are neglected. The first model assumes the knowledge at each atom of the potential λV(x) with amplitude λ. This potential is created by the charge density wave (CDW) of the electronic distribution and produces the periodic lattice distortion (PLD) at the same wave-vector 2k_F as the CDW. (The existence of the CDW and of the PLD was proved by Peierls). When 2k_F is incommensurate with the reciprocal lattice wave-vector and when V(x) is an analytic function, a phase transition versus λ, occurs at λ = λ_c in the PLD, which transforms from a quasi-sinusoidal modulation for λ < λ_c into an array of equidistant and locked phase defects for λ > λ_c. This transition is represented by the breaking of analyticity of the PLD hull function. The zero frequency mode corresponding to the phase rotation of the PLD-CDW (phason or Frölich mode), which carries a current, locks at λ = λ_c and transforms the system from a metal into an insulator. If V is not differentiable enough, this transition does not exist and the system always remains insulating. The second model assumes that the PLD is known and submits the electrons to a quasi-periodic potential with Fermi and lattice periods. This potential is represented in a tight-binding model by a potential defined at site i by the periodic hull function λV(i) with the Fermi period 2π/2k_F. For a particular analytic potential V(i) we show that a transition occurs versus λ: for λ < λ_c all the electrons are described by modulated plane waves with an analytic hull function,while for λ > λ_c all the electrons are exponentially localized. This transition induces also a metal-insulator transition.When the potential V is not differentiable enough, the electrons are expected to be always localized. On this basis we explain that the CDW and PLD hull functions must be both analytic or both discontinuous. A well-defined etal-insulator transition versus the electron-phonon coupling λ is expected. The conducting phase would then exhibit a Frölich mode with extended electronic wave functions and the insulating phase would consist in locked phase defects plus localized electrons. This last phase can be physically reinterpreted as a crystal of localized polarons superimposed onto the lattice.