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 $ν_F$ and the resistivity $ρ$, 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 $n_c\propto |λ|$, $λ\ll 1$ being the dimensionless amplitude of scattering. For $n\gg n_c$ the singularities are simply rounded at $\varepsilon\equiv E-E_N\sim τ^{-1}$ -- the Born scattering rate. For $n\ll n_c$ the non-Born effects in scattering become essential despite $λ\ll 1$. The peak of the resistivity is asymmetrically split in a Fano-resonance manner (however with a more complex structure). Namely, for $\varepsilon>0$ there is a broad maximum at $\varepsilon\propto λ^2$ while for $\varepsilon<0$ there is a deep minimum at $|\varepsilon|\propto n^2\ll λ^2$. The behaviour of $ρ$ below the minimum depends on the sign of $λ$. In case of repulsion $ρ$ monotonically grows with $|\varepsilon|$ and saturates for $|\varepsilon|\gg λ^2$. In case of attraction $ρ$ has sharp maximum at $|\varepsilon|\propto λ^2$. 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.