Variable-range hopping in quasi-one-dimensional electron crystals

We study the effect of impurities on the ground state and the low-temperature Ohmic dc transport in a one-dimensional chain and quasi-one-dimensional systems of many parallel chains. We assume that strong interactions impose a short-range periodicity of the electron positions. The long-range order of such an electron crystal (or equivalently, a 4k_F charge-density wave) is destroyed by impurities, which act as strong pinning centers. We show that a three-dimensional array of chains behaves differently at large and at small impurity concentrations N. At large N, impurities divide the chains into metallic rods. Additions or removal of electrons from such rods correspond to charge excitations whose density of states exhibits a quadratic Coulomb gap. At low temperatures the conductivity is due to the variable-range hopping of electrons between the rods. It obeys the Efros-Shklovskii (ES) law, -ln σ∼(T_ES/T)^1/2. T_ES decreases as N decreases, which leads to an exponential growth of σ. When N is small, the metallic-rod (also known as “interrupted-strand”) picture of the ground state survives only in the form of rare clusters of atypically short rods. They are the source of low-energy charge excitations. In the bulk of the crystal the charge excitations are gapped and the electron crystal is pinned collectively. A strongly anisotropic screening of the Coulomb potential produces an unconventional linear in energy Coulomb gap and an unusual law of the variable-range hopping conductivity -ln σ∼(T₁/T)^2/5. The parameter T₁ remains constant over a finite range of impurity concentrations. At smaller N the 2/5 law is replaced by the Mott law, -ln σ∼(T_M/T)^1/4. In the Mott regime the conductivity gets suppressed as N goes down. Thus, the overall dependence of σ on N is nonmonotonic. In the case of a single chain, the metallic-rod picture applies at all N. The low-temperature conductivity obeys the ES law, with log corrections, and decreases exponentially with N. Our theory provides a qualitative explanation for the transport properties of organic charge-density wave compounds of TCNQ family.