Hall-Effect Sign Inversion in a Realizable 3D Metamaterial

In 2009, Briane and Milton proved mathematically the existence of three-dimensional isotropic metamaterials with a classical Hall coefficient that is negative with respect to that of all of the metamaterial constituents. Here, we significantly simplify their blueprint towards an architecture composed of only a single-constituent material in vacuum or air, which can be seen as a special type of porosity. We show numerically that the sign of the Hall voltage is determined by a separation parameter between adjacent tori. This qualitative behavior is robust even for only a small number of metamaterial unit cells. The combination of simplification and robustness brings experimental verification of this striking sign inversion into reach. Furthermore, we provide a simple intuitive explanation of the underlying physical mechanism.

Popular Summary

Intuition tells us that increasing the porosity of a bulk material should lead to effective material properties in between those of the bulk and air. Indeed, such intuition is sufficient for the majority of porous materials. Here, we discuss a striking exception: By constructing a porous metal or a doped semiconductor consisting of interlocking micron-scale tori, the effective Hall voltage can assume any value and can even reverse its sign with respect to that of the bulk material. This Hall voltage has experimental relevance because measuring the Hall voltage makes it possible to calculate magnetic fields.

An electrical conductor placed in a magnetic field will experience a potential difference across its surfaces; this difference is the so-called Hall voltage. Textbook wisdom states that the sign of the Hall voltage allows researchers to determine whether negatively charged electrons or positively charged holes dominate the electrical conduction in a material. We theoretically investigate Hall voltage in a porous metamaterial composed of interlocking tori with parameters similar to those of phosphorus-doped silicon. Each torus has a diameter of $20\mu \mathrm{m}$, and we investigate a variety of separations between adjacent tori. The sign inversion that we observe between the porous metamaterial and the bulk material from which it was constructed demonstrates that this wisdom must be adopted with caution. Our findings are applicable to even only a small sampling of the material’s unit cells.

Such three-dimensional porous materials or “metamaterials” like these have yet to be realized experimentally, but we describe a feasible route to making them using existing technologies like three-dimensional direct laser writing. We expect that our results will motivate future experimental verification of this theory despite the challenges of microfabrication.

Figure 1

(a) Approximation of the crystalline three-dimensional simple-cubic metamaterial structure proposed by Briane and Milton [4]. (b) One cubic unit cell with lattice constant $a$. It is composed of three different isotropic constituent materials (the tori, the spheres, and the surrounding), all of which have the same sign of the Hall coefficient, yet different magnitudes. Briane and Milton showed analytically that the resulting effective metamaterial Hall coefficient is isotropic and can have the opposite sign.

Figure 2

(a) Scheme of a cuboid isotropic homogeneous material exhibiting the classical Hall effect with Hall voltage $U_H$, resulting from a constant electric current $I_x$ and a perpendicular static magnetic field $B_z$. The shown geometry and coordinate system is used throughout this paper. The small cubes with side length $a$ indicate the unit cells of the crystalline material with integer numbers of unit cells $N_x$, $N_y$, and $N_z$ in the $x$, $y$, and $z$ directions. This leads to the sample dimensions $L_x=N_{x}a$, $L_y=N_{y}a$, and $L_z=N_{z}a$. (b) Hall bar composed of $N_x=11$, $N_y=5$, and $N_z=1$ metamaterial unit cells like those shown in Fig. 1. The Hall bar contains four metal contacts (equipotential surfaces) to impose the current $I_x$ and to pick up the Hall voltage $U_H$, respectively.

Figure 3

(a) Calculated Hall voltage $U_H$ (dots) for the Briane-Milton metamaterial shown in Fig. 1 for the geometry in Fig. 2 versus applied magnetic field $B_z$ for $N_x=11$, $N_y=5$, and $N_z=1$, i.e., for a total of 55 metamaterial unit cells. The straight line is a guide to the eye. Each constituent alone would show a negative Hall voltage in bulk form. Material parameters for the spheres $R_H=-624\times {10}^{-6}{\mathrm{m}}^3{\mathrm{A}}^{-1}{\mathrm{s}}^{-1}$, ${\sigma }_0=200{\mathrm{AV}}^{-1}{\mathrm{m}}^{-1}$, for the tori $R_H=-3.5\times {10}^{-11}{\mathrm{m}}^3{\mathrm{A}}^{-1}{\mathrm{s}}^{-1}$, ${\sigma }_0=3.5\times {10}^7{\mathrm{AV}}^{-1}{\mathrm{m}}^{-1}$, and for the surrounding $R_H=0{\mathrm{m}}^3{\mathrm{A}}^{-1}{\mathrm{s}}^{-1}$ and ${\sigma }_0=2{\mathrm{AV}}^{-1}{\mathrm{m}}^{-1}$. The current is $I_x=0.1\mathrm{mA}$, the geometrical parameters are $R=10\mu \mathrm{m}$, $r=1.6\mu \mathrm{m}$, $\zeta =1.6\mu \mathrm{m}$ (for comparison, $\zeta =2r$ in Fig. 1); hence, $a=4R-2(\zeta +2r)=30.4\mu \mathrm{m}$. The cylindrical pickup contacts have radius $R+r$, height $0.2R$, and conductivity $3.5\times {10}^7{\mathrm{AV}}^{-1}{\mathrm{m}}^{-1}$ (aluminum). (b) Calculated Hall voltage (dots) for a bulk material with the same constituent-material parameters as the spheres in (a). Note the different vertical scales. The straight line is a guide to the eye.

Figure 4

(a) Simplified three-dimensional metamaterial (compare Fig. 1) composed of a single-constituent material only. We take $n$-doped silicon with conductivity ${\sigma }_0=200{\mathrm{AV}}^{-1}{\mathrm{m}}^{-1}$ and Hall coefficient $R_H=-624\times {10}^{-6}{\mathrm{m}}^3{\mathrm{A}}^{-1}{\mathrm{s}}^{-1}$. (b) One cubic unit cell with lattice constant $a$. The porous structure can lead to a Hall voltage with inverted sign with respect to that of the homogeneous bulk material (see Fig. 5). A crucial parameter is the separation $d$ between adjacent tori, which can be positive, zero, or negative (see Fig. 5). The depicted configuration corresponds to $d<0$. The torus diameter is $2R$ and the torus wire diameter $2r$. In this paper, we fix $R=10\mu \mathrm{m}$. The lattice constant $a$ results from $a=4R+2d$.

Figure 5

(a) Calculated Hall voltage $U_H$ (dots) for the simplified single-component metamaterial shown in Fig. 4 for the geometry in Fig. 2 and for fixed $I_x=0.1\mathrm{mA}$ versus applied magnetic field $B_z$ for various parameters. The straight lines are guides to the eye. In bulk form, the constituent material would lead to a negative Hall voltage. (b) Calculated Hall voltage (dots) versus separation parameter $d$ (see Fig. 4) divided by the torus radius $R$. The curves are guides to the eye. Obviously, the sign of the Hall voltage correlates with the ratio $d/R$, whereas the wire radius $r$ has a much smaller influence. The insets illustrate the geometry for three different values of $d/R$. Parameters are $R=10\mu \mathrm{m}$, $B_z=1\mathrm{T}$, $N_x=11$, $N_y=5$, $N_z=1$, contacts as shown in Fig. 2, and $n$-doped silicon with ${\sigma }_0=200{\mathrm{AV}}^{-1}{\mathrm{m}}^{-1}$ and $R_H=-624\times {10}^{-6}{\mathrm{m}}^3{\mathrm{A}}^{-1}{\mathrm{s}}^{-1}$.

Figure 6

Calculated Hall voltage $U_H$ (dots) for fixed geometry and parameters of the metamaterial unit cell versus $N_x$ for different values of $N_y$ as indicated in the legend. We fix $N_z=1$. The shape of the contacts is as shown in Fig. 2. The other fixed parameters are ${\sigma }_0=200{\mathrm{AV}}^{-1}{\mathrm{m}}^{-1}$, $R_H=-624\times {10}^{-6}{\mathrm{m}}^3{\mathrm{A}}^{-1}{\mathrm{s}}^{-1}$, $R=10\mu \mathrm{m}$, $r/R=0.1$, $d/R=-0.44$, $B_z=1\mathrm{T}$, and $I_x=0.1\mathrm{mA}$.

Figure 7

Hall potential map, i.e., difference of the calculated electrostatic potential $\varphi (\overrightarrow{r})$ with ($B_z=1\mathrm{T}$) and without ($B_z=0$) magnetic field, for the parameters as in Fig. 6 for the case of $N_x=11$, $N_y=5$, and $N_z=1$ plotted on a false-color scale.

Figure 8

Illustration of the sign inversion of the Hall voltage for different tori separations $d$. From top to bottom: $d>0$, $d\approx 0$, and $d<0$. The schemes in the left-hand side column can be compared with parts of the metamaterial structure shown in the right-hand side column.

Figure 9

(a) Hollow version of the single-component metamaterial in Fig. 4. The shown mesh has been used for the numerical calculations. (b) Calculated Hall voltage (dots) as in Fig. 5 but for the hollow structure shown in (a). Parameters are $N_x=11$, $N_y=5$, $N_z=1$, contacts as shown in Fig. 2, $I_x=0.1\mathrm{mA}$, $B_z=1\mathrm{T}$, $R=10\mu \mathrm{m}$, $r/R=0.14$, and $n$-doped silicon with ${\sigma }_0=200{\mathrm{AV}}^{-1}{\mathrm{m}}^{-1}$ and $R_H=-624\times {10}^{-6}{\mathrm{m}}^3{\mathrm{A}}^{-1}{\mathrm{s}}^{-1}$. The three different coating thicknesses $t/r$ are indicated in the legend.