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

TL;DR

This paper investigates how impurity concentration affects the smearing of Van Hove singularities in quasi-one-dimensional systems, revealing a crossover from simple rounding to complex non-Born effects with asymmetric Fano-like resonance features.

Contribution

It demonstrates the critical role of impurity concentration in the nature of scattering effects, highlighting the emergence of non-Born effects and Fano-like resonance structures in resistivity behavior.

Findings

At high impurity concentration, singularities are rounded at the Born scattering rate.

At low impurity concentration, non-Born effects dominate, causing asymmetric Fano-like resonance features.

Resonant scattering by quasistationary bound states significantly influences resistivity for attractive impurities.

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 $\varepsilon\equiv E-E_N\sim \tau^{-1}$ -- 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 $\varepsilon>0$ there is a broad maximum at $\varepsilon\propto \lambda^2$ while for $\varepsilon<0$ there is a deep minimum at $|\varepsilon|\propto n^2\ll \lambda^2$. The behaviour of $\rho$ below the minimum depends on the sign of $\lambda$. In case of repulsion $\rho$ monotonically grows with $|\varepsilon|$ and saturates for $|\varepsilon|\gg \lambda^2$. In case of attraction $\rho$ has sharp maximum at $|\varepsilon|\propto \lambda^2$. 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.