Model for organized current patterns in disordered conductors

We present a general theory of current deviations in straight current-carrying wires with random imperfections, which quantitatively explains the recent observations of organized patterns of magnetic field corrugations above micrometer-scale evaporated wires. These patterns originate from the most efficient electron scattering by Fourier components of the wire imperfections with wavefronts along the $\pm 45°$ direction. We show that long-range effects of surface or bulk corrugations are suppressed for narrow wires or wires having an electrically anisotropic resistivity.

Figure 1

(Color) Measured (Ref. 5) and simulated patterns of atomic density (normalized) $\sim 3.5\mathrm{\mu }\mathrm{m}$ above (a), (b) a thick wire $(H=2\mathrm{\mu }\mathrm{m})$, and (c), (d) a thin wire $(H=280\mathrm{nm})$, with grain size of $\sim 70$ and $\sim 40\mathrm{nm}$, respectively. Atomic density variations measure corrugations in the magnetic-field component $\delta B_x$ along the wire axis $\widehat{x}$. These corrugations are due to current directional deviations from the main current along $\widehat{x}$. The trapping potential ensures a cloud width of $\sim 1\mathrm{\mu }\mathrm{m}$ along $\widehat{y}$ and $\widehat{z}$, and hundreds of $\mathrm{\mu }\mathrm{m}$ in the $\widehat{x}$ direction. To create the above 2D maps, the different transverse locations are scanned across the wire at a constant height. The simulation assumes a combination of bulk resistivity inhomogeneity and geometrical perturbations of the wire with parameters chosen such that the power spectrum of the magnetic field along $\widehat{x}$, averaged over $\widehat{y}$ and over many realizations of the simulation, fits the measured power spectrum (right).

Figure 2

(Color) (a) A single Fourier component (plane wave) of the planar resistivity perturbations due to bulk inhomogeneity or wire thickness variations induces current flow directional changes (arrows). The current tilts along low-resistivity wavefronts and across high-resistivity wavefronts. (b) The amplitudes of transverse current component $\delta J_y\left(\mathit{\kappa }\right)$ $[\mathit{\kappa }\equiv (k_x,k_y\left)\right]$, generated by resistivity perturbations $\delta \rho \left(\mathit{\kappa }\right)\propto 1∕\kappa$ (see Ref. 5), are proportional to $\sin 2{\theta }_{\mathit{\kappa }}$ (color scheme). For a wire with a finite length $L$ and width $W$, these amplitudes are calculated at discrete values of $\mathit{\kappa }$, which are integer products of $2\pi ∕L$ and $2\pi ∕W$. (c) As a consequence of this discreteness, the corrugations at long wavelengths $(k_{x}W<1)$ are suppressed when the wire becomes narrower: the density of $k_y$ states becomes lower and no corresponding values of $k_y$ exist along the maximum scattering amplitude line $k_y\sim k_x$. Here, the power spectrum of the magnetic field corrugations at $3.5\mathrm{\mu }\mathrm{m}$ above the center of the wire is shown as a function of $k_x$ for resistivity perturbations $\delta {\rho }_{\mathit{\kappa }}∕{\rho }_0=3.4\times {10}^{-4}({\kappa }_0∕\kappa )$ with ${\kappa }_0=2\pi ∕680\mathrm{\mu }{\mathrm{m}}^{-1}$. (d) For comparison, the magnetic-field corrugations above the center of the wire are shown for wires of different widths for a model assuming edge fluctuations with $\mid \delta y_{\pm }\left(k_x\right)\mid =10\mathrm{nm}\times ({\kappa }_0∕k_x)$. Here, short-wavelength components are suppressed when the wire becomes wider.

Figure 3

(Color) For a current flowing through an electrically anisotropic wire, the perturbation wavefront angle ${\theta }_{\mathit{\kappa }}^{\max }$, giving rise to maximum transverse electron scattering, will depend on the ratio $r={\rho }_y∕{\rho }_x$ between the transverse and longitudinal resistivities (main plot). The two-dimensional maps show the predicted atomic density above a wire similar to that presented in Fig. 1 in the extreme cases ${\theta }_{\mathit{\kappa }}=0°$ $(r⪡1)$, $90°$ $(r⪢1)$, and the isotropic case ${\theta }_{\mathit{\kappa }}=45°$ $(r=1)$. Bottom inset: the magnetic corrugation amplitude as a function of ${\theta }_{\mathit{\kappa }}$, which is suppressed when $r>1$. For details see Ref. 14.