Non-Born effects in scattering of electrons in a conducting tube with a low concentration of impurities

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 $n_c\propto \left|\lambda \right|,\lambda \ll 1$ 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 single-impurity 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 $\left|\varepsilon \right|\propto n^2\ll {\lambda }^2$. The behavior of $\rho$ below the minimum depends on the sign of $\lambda$. In case of repulsion, $\rho$ monotonically grows with $\left|\varepsilon \right|$ and saturates for $\left|\varepsilon \right|\gg {\lambda }^2$. In case of attraction, $\rho$ has a sharp maximum at $\left|\varepsilon \right|\propto {\lambda }^2$. The latter feature is due to resonant scattering at quasistationary bound states that inevitably arise just below the bottom of each subband for any attracting impurity.

Figure 1

$\rho \left(\mathrm{\Phi }\right)$ dependence for clean and dirty cases. (Top) Clean case, $\rho \left(\mathrm{\Phi }\right)$ is periodic with a period $2{\mathrm{\Phi }}_0$ and Van Hove square root singularities are present for $\mathrm{\Phi }=2n{\mathrm{\Phi }}_0$. (Bottom) Dirty case, $\rho \left(\mathrm{\Phi }\right)$ is periodic with a period ${\mathrm{\Phi }}_0$—odd harmonics are suppressed.

Figure 2

Dependence of the resistivity on the position of the Fermi level for repulsing impurities in the case of low concentration of impurities $n\ll n_c$ (strongly non-Born regime). Note that $\rho \left(\epsilon \right)$ vanishes as $\epsilon \to 0$: it is an artefact of the single-impurity approximation that is not applicable in the narrow vicinity of $\epsilon =0$: for $\epsilon \lesssim {\epsilon }_{min}\ll 1$. The horizontal asymptote (dashed line) corresponds to $\rho ={\rho }_0$.

Figure 3

The same as in Fig. 2 but for attracting impurities. The sharp maximum at $\epsilon <0$ arises due to resonant scattering at quasistationary states.

Figure 4

Thin conducting tube, threaded by a magnetic field $H$. Impurities (shown as stars) are embedded in the tube. Electrons live on the surface of the cylinder.

Figure 5

Spectrum of an electron on a surface of an ideal cylinder. Subbands of the transverse quantization are shown. The Fermi level $E$ crosses all the subbands with $m\le N$.

Figure 6

The “quasifold”: in the vicinity of the bifurcation point, $\varepsilon ={\varepsilon }_{\mathrm{bi}}$, the slope of the curve $\rho \left(\varepsilon \right)$ is anomalously steep.

Figure 7

The shape of a smeared Van Hove singularity within the Born approximation.

Figure 8

The energy dependence of resistivity near the minimum. Dashed line—for $n\to 0$, solid line—for finite $n$.