Three-dimensional metamaterial Hall-bar devices

We realize classical Hall bars, made of chain-mail-inspired cubic-symmetry 3D metamaterials composed of interlinked hollow tori, contacted to printed circuit boards. This step is a prerequisite for refined experiments compared to our previous probe-station based measurements as well as for potential applications of 3D Hall-effect metamaterials in general. On this basis, we systematically study the dependence of the Hall voltage on the relative orientation of the 3D metamaterial crystallographic axes with respect to the applied static magnetic field vector and the imposed current flow direction. Our findings are consistent with an isotropic sign-reversed Hall coefficient, which has been predicted by homogenization theory.

Figure 1

Illustration of the 3D metamaterial crystal unit cell. The cubic lattice constant $a$, the major torus radius $R$, the minor torus radius $r$, the distance parameter $d$, and the thickness $t$ of the semiconductor layer are indicated. Adapted with permission from Ref. [20].

Figure 2

Metamaterial Hall-bar samples in different stages of fabrication. (a) Scanning electron micrograph of a Hall-bar on a silicon substrate after 3D laser lithography and ALD, before transfer of the structure. The structure includes a handling arm (on top) and four pedestals. The unit cell orientation corresponds to case D (compare Fig. 3). (b) Optical image of the printed circuit board with four contact pads featuring an electroless nickel immersion gold (ENIG) surface plating. (c) Scanning electron micrograph of a sample on a printed circuit board after transfer and curing of the conductive epoxy. The unit cell orientation corresponds to case C (compare Fig. 3). The geometrical target parameters (compare Fig. 1) for all samples are $a=104\mu \mathrm{m}, R=36\mu \mathrm{m}, r=6\mu \mathrm{m}, d=-20\mu \mathrm{m}$, and $t=0.17\mu \mathrm{m}$.

Figure 3

(Left) Schematic representation of the four different unit-cell orientations A-D with respect to the Hall bar that were studied experimentally (compare Fig. 1). (Right) Scanning electron micrographs of the corresponding metamaterial Hall-bar samples. Results for the four orientations A-D are shown in Fig. 4 and are summarized in Table 1.

Figure 4

Measured Hall voltage $U_{\text{H}}$ vs electric current $I$ for four different Hall-bar samples corresponding to the four different unit-cell orientations A-D illustrated in Fig. 3. The dots are measured, the straight lines are guides the eye. The slope of the straight lines is the Hall resistance $R_{\text{H}}$. In the ideal case of an isotropic Hall coefficient and identical sample thicknesses $L_z$, the slopes should be identical. Due to the different sample thicknesses $L_z$ (compare Table 1), the slopes of the pairs (A,C) and (B,D) are slightly different. The upper inset shows raw data for case A, current $I=0.5\mathrm{mA}$, and a magnetic flux density $B_z=\pm 0.83\mathrm{T}$. The lower inset is an example measurement of $R_{\mathrm{H}}$ versus $B_z$ for $I=0.5\mathrm{mA}$ and sample type A, showing the expected proportional behavior. Here, $B_z$ has been varied by changing the position of the magnet with respect to the sample. The relation between position and $B_z$ has been determined by a calibration measurement using a standard Hall sensor. All of the derived effective Hall coefficients $A_{\text{H}}^{*}=R_{\text{H}}{L}_z/B_z$ are summarized in the fourth row of Table 1. The fifth row there corresponds to a second generation of nominally identical samples A-D.