Experiments on the Parallel Hall Effect in Three-Dimensional Metamaterials

The classical Hall effect in ordinary isotropic conducting materials describes the occurrence of a voltage perpendicular to the direction of the electric-current flow and perpendicular to the imposed magnetic-field vector. The Hall effect is routinely used in magnetic-field sensors. Here, we fabricate and characterize microstructured anisotropic metamaterials composed of a single semiconducting constituent ($n$-type ZnO) for which the direction and the sign of the Hall electric field can be tailored by microstructure. This class of metamaterials includes the possibility of a Hall voltage parallel—rather than perpendicular—to the magnetic-field vector. One possible future application arising from this far-reaching control of the effective electric-conductivity tensor is a sensor measuring the circulation of a magnetic field.

Figure 1

(a) Illustration of the usual (perpendicular) classical Hall effect. A constant current $I_x$ is injected along the $x$ direction and a static magnetic field is applied along the $z$ direction, i.e., $\overrightarrow{B}=(0,0,B_z{)}^T$. As the Hall voltage $U_H$ is proportional to $1/L_z$, thin platelets with $L_x>L_y\gg L_z$ are commonly used. (b) Unusual parallel Hall effect. Here, the Hall voltage is rather proportional to $1/L_y$, leading to the choice $L_x>L_z\gg L_y$. (c) Wrapped-up version of the geometry shown in (b). Here, the Hall voltage is proportional to the circulation of the magnetic field.

Figure 2

(a) One unit cell of a metamaterial enabling the observation of the parallel Hall effect. (b) Extended part of a crystal based on a simple-cubic translational lattice with the lattice constant $a$. For the configuration shown, we have $d_y<0$ and $d_z>0$.

Figure 3

Scanning electron micrograph of a fabricated structure after atomic-layer deposition and electron-beam evaporation. The shadow cast on the substrate by the top plate and the side contacts during electron-beam evaporation is clearly visible. The structure consists of $6\times 6\times 12$ unit cells. The design parameters are $a=100\mu \mathrm{m}$, $r=6.25\mu \mathrm{m}$, $h=40\mu \mathrm{m}$, $d_y=-10\mu \mathrm{m}$, $d_z=20\mu \mathrm{m}$.

Figure 4

(a) Measured parallel Hall resistance $R_H^z$ and (b) perpendicular Hall resistance $R_H^y$ versus the separation parameters $d_y$ and $d_z$. The red and blue bars correspond to two different nominally identical sample sets. $B_z=0.83\mathrm{T}$, $d_y=\pm 10\mu \mathrm{m}$, $d_z=\pm 20\mu \mathrm{m}$, and other parameters as in Fig. 3.

Figure 5

Cross section of an infinitely long current carrying hollow cylinder (compare to Fig. 2) oriented along the $x$ axis and subject to a static magnetic field $\overrightarrow{B}$ parallel to the $z$ direction. This geometry results in a Hall voltage $U_H^{\mathrm{cyl}}$ between the two black dots on the cylinder sides. Adding up this voltage (3) for many unit cells in series allows us to formulate the metamaterial Hall resistances (4) and (5), respectively.

Figure 6

Measured parallel Hall resistance $|R_H^z|$ versus $N_y$. For each $N_y$, two nominally identical samples (shown in red and blue) are fabricated. The two points for $N_y=8$ coincide within the dot size. The black solid straight line with a slope of $-1$ in this double-logarithmic representation demonstrates the scaling according to $R_H^z\propto 1/N_y$. The parameters are $a=100\mu \mathrm{m}$, $r=6.25\mu \mathrm{m}$, $h=40\mu \mathrm{m}$, $d_y=10\mu \mathrm{m}$, $d_z=20\mu \mathrm{m}$, $N_x=12$, and $N_z=6$.