Plane multiply-connected Hall plates with constant supply current

All boundaries are insulating

All boundaries are conducting

Reverse Magnetic Field Reciprocity holds

Reciprocity holds

No internal current sources

Internal current sources

w/o internal current sources

$J_{\text{even}}=J_0$

$J_{\text{even}}=J_0$

${\tilde{J}}_{\text{even}}={\tilde{J}}_0$

$J_{\text{odd}}=n_z\times \nabla {\psi }_{\text{loop}}/\rho \Rightarrow J_{\text{odd}}=0$

$J_{\text{odd}}=\tan {\theta }_H{n}_z\times {\tilde{J}}_0={\tilde{J}}_{\text{odd}}\propto \tan {\theta }_H$

${\psi }_{loop}=0$

$J_{\text{odd}}=n_z\times \nabla {\psi }_{\text{loop}}/\rho \Rightarrow {\psi }_{\text{loop}}=-\tan {\theta }_H{\tilde{\varphi }}_0$

$J=J_0=n_z\times \nabla \psi /\rho$ constant versus ${\theta }_H$

$J=J_0+\tan {\theta }_H{n}_z\times {\tilde{J}}_0$ changes with ${\theta }_H$

$\tilde{J}={\tilde{J}}_0+\tan {\theta }_H{n}_z\times {\tilde{J}}_0$ changes with ${\theta }_H$

$E_{\text{even}}=\rho J_0$

$E_{\text{even}}=\rho J_0+{\tan }^2{\theta }_H\rho {\tilde{J}}_0$

$\tilde{E}={\tilde{E}}_{\text{even}}=\rho {\tilde{J}}_0/{\cos }^2{\theta }_H$

${\varphi }_{\text{even}}-{\varphi }_0=0$

${\varphi }_{\text{even}}-{\varphi }_0={\tilde{\varphi }}_0{\tan }^2{\theta }_H={\tilde{\varphi }}_{\text{even}}-{\tilde{\varphi }}_0$

$E_H=\rho \tan {\theta }_H{J}_0\times n_z\propto \tan {\theta }_H$

$E_H=\rho \tan {\theta }_H\left(J_0-{\tilde{J}}_0\right)\times n_z\propto \tan {\theta }_H$

${\tilde{E}}_H=0$

${\varphi }_H=-\psi \tan {\theta }_H\propto \tan {\theta }_H$

${\varphi }_H\propto \tan {\theta }_H$

${\tilde{\varphi }}_H=0$

$G_{H,12}=\frac{-I_{12}}{I_{\text{supply}}}$ with $I_{12}=I_{12,0}$

$G_{H,12}=\frac{{\tilde{I}}_{0,12}-I_{0,12}}{I_{\text{supply}}}$

$G_{H,12}=0$

$\psi$ does not depend on ${\theta }_H$

${\psi }_{\text{loop}}\propto \tan {\theta }_H$

$\tilde{\varphi }{\cos }^2{\theta }_H$ does not depend on ${\theta }_H$

${\varphi }_H$ and $\psi$ are homogeneous along boundaries without current contacts

${\varphi }_{\text{even}}-{\varphi }_0$ and ${\psi }_{\text{loop}}$ are homogeneous along all boundaries

$\tilde{\varphi }$ is homogeneous along all contacts

${\varphi }_H$ is homogeneous on streamlines of $J_0$

${\varphi }_H$ is homogeneous on streamlines of $J_0-{\tilde{J}}_0$

${\tilde{\varphi }}_H=0$ everywhere

$F_R=\left(1+i\tan {\theta }_H\right)\left({\varphi }_0+i\psi \right)$

$F_{R,H}={\varphi }_H+i\tan {\theta }_H\left({\varphi }_0-{\tilde{\varphi }}_0\right)$

-

$\Gamma =0$ no current spirals

$\Gamma =\tan {\theta }_H{I}_{\text{supply}}/{\text{t}}_{\text{H}}$ current spirals

w/o current spirals