Exploration of the duality between generalized geometry and extraordinary magnetoresistance

We outline the duality between the extraordinary magnetoresistance (EMR), observed in semiconductor-metal hybrids, and nonsymmetric gravity coupled to a diffusive $U\left(1\right)$ gauge field. The corresponding gravity theory may be interpreted as the generalized complex geometry of the semidirect product of the symmetric metric and the antisymmetric Kalb-Ramond field: ($g_{\mu \nu }+{\beta }_{\mu \nu }$). We construct the four-dimensional covariant field theory and compute the resulting equations of motion. The equations encode the most general form of EMR within a well defined variational principle, for specific lower dimensional embedded geometric scenarios. Our formalism also reveals the emergence of additional diffusive pseudocurrents for a completely dynamic field theory of EMR. The proposed equations of motion now include terms that induce geometrical deformations in the device geometry in order to optimize the EMR. This bottom-up dual description between EMR and generalized geometry/gravity lends itself to a deeper insight into the EMR effect with the promise of potentially new physical phenomena and properties.

Figure 1

A semiconductor wafer with a concentric metallic disk at the center is shown in Fig. 1. In Fig. 1 The four-probe van der Pauw arrangement is displayed with current $I$ entering through port 1 and exiting through port 2. The contacts at ports 3 and 4 are used to measure the voltage drop across the device. (After Ref. [17].)

Figure 2

We plot the potential function solved for the device geometry in Fig. 1, for two cases in which the applied magnetic field [2] $H=0\mathrm{T}$ and [2] $H=1\mathrm{T}$. The color coding from blue to red represents the variation of the potential from negative to positive values.

Figure 3

We plot the current flow (arrows) for the device geometry in Fig. 1, for two cases in which the applied magnetic field [3] $H=0\mathrm{T}$ and [3] $H=1\mathrm{T}$. Note that the current avoids the metallic region at high fields and stays in the semiconductor region, where the Hall angle approaches $\pi /2$, leading to extraordinarily high magnetoresistance. The rescaling of the arrows everywhere manifests itself as the metallic region carrying a current even though it is reduced substantially there.