Experimental Evidence for Sign Reversal of the Hall Coefficient in Three-Dimensional Metamaterials

Effectively inverting the sign of material parameters is a striking possibility arising from the concept of metamaterials. Here, we show that the electrical properties of a $p$-type semiconductor can be mimicked by a metamaterial solely made of an $n$-type semiconductor. By fabricating and characterizing three-dimensional simple-cubic microlattices composed of interlocked hollow semiconducting tori, we demonstrate that sign and magnitude of the effective metamaterial Hall coefficient can be adjusted via a tori separation parameter—in agreement with previous theoretical and numerical predictions.

Figure 1

Blueprint for the three-dimensional chainmail-like metamaterial discussed in this Letter. It is composed of a simple-cubic lattice of hollow tori. $a$ is the lattice constant, $R$ the torus radius, $r$ the wire radius, $t$ the film thickness, and $d$ the separation parameter. The latter can be positive or negative; the depicted configuration corresponds to $d<0$.

Figure 2

Example images of fabricated three-dimensional metamaterial samples on glass substrates. (a)–(c) are oblique-view electron micrographs; (c) is colored for clarity, and (d) is an optical micrograph. (a) Sequence of polymer structures with different separation parameters $d$ (compare to Fig. 1). At fixed parameters $R$, $r$, and $t$, the lattice constant $a$ increases with increasing $d$ (from bottom right to top left). (b) Polymer structure coated with $n$-type ZnO with $4\times 4\times 5$ unit cells and $d<0$. (c) shows a Hall bar composed of $11\times 5\times 1$ unit cells. The roof on top casts a shadow during evaporation of the metal contacts onto the substrate and the two pick-up contacts. To ease the alignment of the tungsten contact needles, halfpipes have been fabricated on the left and right. (d) Structure as in (c) within the home-built electrical probe station. The four electrical contact probes are visible.

Figure 3

(a) Measured transverse voltage $U_y$ versus real time. The two examples shown correspond to two Hall bars [$11\times 5\times 1$ unit cells, cf., Figs. 2 and 2] with different separation parameters, namely $d=-22\mu \mathrm{m}$ (blue) and $d=4\mu \mathrm{m}$ (red). The magnet is moving in the shaded regions. Apart from the asymmetry offset voltage $U_0$, the voltage reverses sign upon reversing the sign of the magnetic induction $B_z$. The design parameters are $R=36\mu \mathrm{m}$, $r=6\mu \mathrm{m}$, $t=185\mathrm{nm}$, and $I_x=0.5\mathrm{mA}$. In the lower panel, the measured magnetic flux $B_z$ at the sample position vs time is shown. The measurement has been performed using a commercial Hall sensor. (b) Extracted Hall voltage (dots) vs injection current $I_x$ for the same samples. They exhibit different slopes of the Hall voltage $U_H$ vs $I_x$, i.e., effective Hall resistances $R_H$ of different sign. The effective Hall coefficient results as $A_H=R_H{L}_z{(0.83\mathrm{T})}^{-1}$, with the Hall bar thickness $L_z=a=4R+2d$. The straight lines are fits to the data. The underlying $n$-type ZnO constituent material has a negative Hall coefficient in bulk form. Thus, the case of $d=-22\mu \mathrm{m}$ corresponds to a sign reversal of the Hall coefficient.

Figure 4

(a) Hall resistance $R_H$ vs separation parameter $d$ as extracted from data as shown in Fig. 3. Each full blue dot corresponds to one sample and a measurement like in Fig. 3; the open red dots are calculated. The solid curve is a guide to the eye. Clearly, we find a sign reversal of the Hall resistance and hence of the Hall coefficient around $d=-12\mu \mathrm{m}$. The design parameters are as in Fig. 3. (b) Corresponding calculated Hall potential landscape in a part of the central region of the Hall bar for two different values of $d$.