Boundary scattering effects on magnetotransport of narrow metallic wires and films

Electron transport in thin metallic wires and films is strongly influenced by the quality of their surface. Weak localization and magnetoconductivity are also sensitive to the electron scattering at the edges of the sample. We study weak localization effects in a two-dimensional electron gas patterned in the form of a narrow quasi-one-dimensional channel in transverse magnetic field. The most general boundary conditions interpolating between the limits of mirror and diffuse edge scattering are assumed. We calculate magnetoconductivity for an arbitrary width of the sample including the cases of diffusive and ballistic lateral transport as well as the crossover between them. We find that in a broad range of parameters, the electron mobility is limited by the boundary roughness, while the magnetotransport is only weakly influenced by the quality of the edges. In addition, we calculate magnetoconductivity for a metallic cylinder in the transverse field and a quasi-two-dimensional metallic film in the parallel field.

Figure 1

Diffusion coefficient in units of $v_{0}l$ as a function of $W/l$ for different values of $\lambda$ computed using Eq. (23).

Figure 2

The function $g\left(x\right)$ that determines the magnetic scattering rate in Eq. (28) for a cylinder sample, Eq. (30), with the circumference $W$. Dashed lines show the asymptotic behavior in the diffusive $W\gg l$ and ballistic $W\ll l$ limits.

Figure 3

The function $g\left(x\right)$ for a flat sample, Eq. (33), at different values of $\lambda$. In the diffusive limit $x=W/l\gg 1, g\left(x\right)$ attains the value $1/6$ shown by the dashed line. At $x\ll 1$, the function vanishes linearly according to Eq. (34).

Figure 4

The function $c\left(\lambda \right)$ [Eq. (34)] that determines the magnetic scattering rate in a flat ballistic sample in the weak field limit, Eq. (36). The two limiting values for diffuse ($\lambda =0$) and mirror ($\lambda =1$) boundaries are given by Eq. (35). We add the straight dashed line connecting the two endpoints to emphasize the small deviation of $c\left(\lambda \right)$ from the linear function.

Figure 5

Schematic illustration of the trajectories providing the major contribution to the magnetic scattering rate in the diffusive and ballistic regimes.

Figure 6

Crossover function $F\left(x\right)$ defined in Eq. (48) for the magnetic scattering rate in the flat ballistic sample. Dashed lines show the limiting values for mirror ($x=0$) and diffuse ($x=\infty$) boundaries.

Figure 7

Magnetic scattering rate in units of $1/\tau$ as a function of the mean free path $l$ (in units of $W/\pi$) for a cylinder sample with $\pi l_B/W=10$. The crossover from weak field diffusive to weak field ballistic and further to strong field ballistic limits occurs as $l$ is increased.

Figure 8

Magnetic scattering rate (in units of $W/\pi l\tau$) as a function of the magnetic field $B$ (in units of $B_0=h/eWl$) for a cylinder sample with fixed $W/2\pi l={10}^{-3}$. The crossover from weak (quadratic) to strong (linear) field behavior occurs at $B\sim B_0$.

Figure 9

Magnetic scattering rate (in units of $\pi W/l\tau$) as a function of the magnetic field $B$ (in units of $B_0=h/eWl$) for a flat sample with fixed $W/l={10}^{-3}$ at different values of boundary roughness parameter $\lambda$.

Figure 10

(a) Diagram for the Drude conductivity involving two current operators $j$, Eq. (B8), and a diffuson propagator (d), Eq. (B3). Fully (b) and partially (c) dressed current operators, Eqs. (B8) and (B11).

Figure 11

(a) Diagram for the quantum correction to the conductivity. The Hikami box (b) contains one empty (c) and two equal crossed (d) diagrams. Momentum labels in (b) correspond to Eqs. (B10) and (B14). Real space labels in (c) and (d) are used in Eqs. (B12) and (B15).