The signal-to-noise ratio and a hidden symmetry of Hall plates

- A theory of maximum SNR of Hall plates is given.
- It is proven that SNR gets maximal, if input and output resistances are equal.
- The SNR of Hall plates is given in closed analytical form.
- It is a sole function of input and output resistances over sheet resistance.
- No layout parameters are necessary.

In a Hall plate with finite size and contacts the Hall output voltage is given by the product of sheet resistance, input current, Hall mobility, magnetic flux density, and Hall geometry factor GH. GH∈[0,1] accounts for the loss in signal due to the contacts. At weak magnetic field GH→GH0 is a function of geometrical parameters only, which makes it the crucial point for layout optimization. We show how to express GH0 alternatively as a function of electrical parameters only, namely of input and output resistances over sheet resistance. This allows for an analytical optimization of signal-to-noise-ratio (SNR) without getting lost in the multitude of geometrical representations of equivalent Hall plates. In the course of this investigation we notice a hidden symmetry property of GH, which we prove rigorously in the limit of small magnetic fields. The physical meaning of this symmetry in the case of Hall plates with equal input and output resistances is also explained.