Probing the annular electronic shell structure of a magnetic corral

We report on $B$-periodic oscillations in the magnetoresistance of a circular magnetic billiard. These arise from singly periodic orbits that return to their point of departure after $n$ bounces on the billiard. A semiclassical calculation of the magnetic field increment required to depin annulus $n$ and $n+1$ is in good agreement with the measured period of the resistance oscillations.

Figure 1

(Color online) (a) Magnetic field profile below a $2\mu \mathrm{m}$ dysprosium dot at $B_a=0.4\mathrm{T}$. One orbit belonging to the $n=18$ annular mode (lowest mode) encircles the magnetic corral. (b) Longitudinal resistance across the $2\mu \mathrm{m}$ disk for $V_g=0.12–0.24\mathrm{V}$. $B$-periodic oscillations ripple the second derivative at $B_a<0.6\mathrm{T}$ (lower curve, $V_g=0.24\mathrm{V}$). The resistance curves are offset by $-0.1\Omega$ per $\Delta V_g=0.01\mathrm{V}$ relative to the $V_g=0.12\mathrm{V}$ curve. Inset: device micrograph showing 2, 3, and $4\mu \mathrm{m}$ diameter dysprosium disks at the center of the $8\mu \mathrm{m}$ wide Hall bar. The dashed lines delimit the area modeled in (a). $V_g=0.12–0.24\mathrm{V}$, gives $n_s=9.07–10.50\times {10}^{15}{\mathrm{m}}^{-2}$ and $l=6.4–7.3\mu \mathrm{m}$.

Figure 2

Magnetization curve of a $75\text{nm}$ unpatterned dysprosium film measured in three different devices. ${\mu }_0{M}_{\text{sat}}=3.67\mathrm{T}\left(\text{Dy}\right)$.

Figure 3

“Top hat” approximation of the magnetic field profile that considers a constant magnetic field $\left(B_1+B_a\right)$ inside and $\left(B_2+B_a\right)$ outside the disk (grey area). In this example, the intercept angles $\phi$ have been calculated so as to plot the two orbits (full/dashed lines) belonging to the $n=4$ annular mode.

Figure 4

(a) The oscillation minima plotted as a function of $B_a$ for $V_g=0.24\mathrm{V}$ (filled squares), $0.16\mathrm{V}$ (open triangles), $0.14\mathrm{V}$ (filled diamonds), $0.12\mathrm{V}$ (open squares), and $0\mathrm{V}$ (hexagons). The full lines are the theoretical fit. Inset: (filled squares) the ratio of the orbit length to the mean free path plotted vs the gate voltage and compared to the relative amplitude of the oscillations. (b) The length of the orbit being expelled is plotted as a function of the magnetic field.