Non-Born effects in scattering of electrons in a clean conducting tube

Abstract

Quasi-one-dimensional systems demonstrate Van Hove singularities in the density of states ${\nu }_F$ and the resistivity $\rho$, occurring when the Fermi level $E$ crosses a bottom $E_N$ of some subband of transverse quantization. We demonstrate that the character of smearing of the singularities crucially depends on the concentration of impurities. There is a crossover concentration $n_c\propto |\lambda |$, $\lambda \ll 1$ being the dimensionless amplitude of scattering. For $n\gg n_c$ the singularities are simply rounded at $\epsilon \equiv E-E_N\sim {\tau }^{-1}$ -- the Born scattering rate. For $n\ll n_c$ the non-Born effects in scattering become essential despite $\lambda \ll 1$. The peak of the resistivity is asymmetrically split in a Fano-resonance manner (however with a more complex structure). Namely, for $\epsilon >0$ there is a broad maximum at $\epsilon \propto {\lambda }^2$ while for $\epsilon <0$ there is a deep minimum at $|\epsilon |\propto n^2\ll {\lambda }^2$. The behaviour of $\rho$ below the minimum depends on the sign of $\lambda$. In case of repulsion $\rho$ monotonically grows with $|\epsilon |$ and saturates for $|\epsilon |\gg {\lambda }^2$. In case of attraction $\rho$ has sharp maximum at $|\epsilon |\propto {\lambda }^2$. The latter feature is due to resonant scattering by quasistationary bound states that inevitably arise just below the bottom of each subband for any attracting impurity.