Figure 1

Electron micrographs of (a) a typical device structure showing the leads and contact arrangement for a cross, and (b) a 550-nm-wide Hall bridge. Inset to (a): A $w$ = 171 nm cross. (c) A schematic of the Hall cross geometry. (d) A self-consistent Schrödinger-Poisson solution for the conduction-band profile of the heterostructure showing the confined energy levels.Reuse & Permissions

Figure 2

Longitudinal $R$${}_{\mathit{xx}}$ (left-hand axis) and transverse $R$${}_{\mathit{xy}}$ (right-hand axis) magnetoresistance of $w$ = 550 nm (solid lines) and 3 μm (dashed lines) Hall bridges at 2 K ($l$  $=$  8.4 μm). Inset: A schematic of the device structure and the relevant dimensions. The positions of $B$${}_{max}$ and $B$${}_{min}$ relate to features associated with boundary scattering (see the text).Reuse & Permissions

Figure 3

Subband depopulation diagram for the $w$ $=$ 550 nm Hall bridge at 2 K. The faint solid line shows the corresponding $R$${}_{\mathit{xx}}$ data (right-hand axis) from which the subband indices (left-hand axis) were assigned. The dashed and solid lines represent fits of Eq. (1b) to the high-field linear portion of the data and the low-field nonlinear portion of the data, respectively.Reuse & Permissions

Figure 4

Magnetoresistance, Δ$R$${}_{\mathit{xx}}$($B$)/$R$${}_{\mathit{xx}}$(0), of the $w$ = 550 nm Hall bridge plotted against the normalized field $B$/$B$${}_0$ at various temperatures after subtraction of a linear background displaying a peak at $B_{\max }$. Inset: The peak amplitude plotted against the mean free path (λ${}_0$) in the control sample at each temperature.Reuse & Permissions

Figure 5

The Hall resistance $R$${}_H$ in units of $h/e^2$ as a function of $B$ for a $w$ $=$ 171 nm (blue line, A), 400 nm (black line, B), 550 nm (red line, C), and 924 nm (green line, D) cross. The dashed lines represent the classical 2D result. Top inset: Low-field data illustrating the anomalies in $R_H$. Data for the $w$ $=$ 550 and 924 nm crosses are offset by 0.5 kΩ for clarity. Bottom inset: Dependence of $n$ on the inferred effective width ${\text{w}}_{\mathrm{eff}}=\mathit{w}-2{\text{w}}_{\mathrm{dep}}$ of the devices (crosses and bridges).Reuse & Permissions

Figure 6

(a) Bend resistance $R_B=V_{4,3}/I_{1,2}$ as a function of magnetic field for a $w$ $=$ 171 nm (blue line, A), 400 nm (black line, B), 550 nm (red line, C), and 924 nm (green line, D) cross. The $w$ = 171 nm data is plotted on a different scale for ease of comparison. The dotted curve (B′) is the reciprocal measurement $R_{B^{\prime }}=V_{1,2}/I_{4,3}$ for the $w$ = 400 nm cross, illustrating that asymmetries in $B$ originate from junction asymmetry. (b) Dependence of the experimental NBR amplitude Δ$R$${}_B$ ($◯$) on $1/N=\pi /k_F{\text{w}}_{\mathrm{eff}}$ and ${\text{w}}_{\mathrm{eff}}$. The results from billiard model simulations using the parameters given in Table  (x) and results for $p$ = 1 (+) are also shown. The dashed line is a guide to the eye, illustrating a 1/$N$ dependence. Inset: A schematic showing the definition of Δ$R$${}_B$.Reuse & Permissions

Figure 7

(a) Normalized NBR amplitude $\Delta R_B/R_0$ of the crosses plotted against $w$${}_{\mathrm{eff}}$. Horizontal lines represent the results from the billiard model for square and rounded junctions with $p$ $=$ 1 (solid lines) along with a square junction with $p$ = 0.7 and $l^{\prime }/\mathit{w}=3$ (dashed line). (b)–(d) Comparisons between experimental $R$${}_B$ (black lines) and billiard model simulations of $R$${}_B$ (solid red lines) for three crosses ($N\gg 1$) using parameters listed in Table . Simulations with $p$ = 1 are shown for comparison (dashed red lines). Inset: A schematic of the geometry used in the simulations. $R_0=(h/2e^2)(\pi /k_F{\text{w}}_{\mathrm{eff}})$.Reuse & Permissions