An Analytical Theory of Hall-Effect Devices with Three Contacts

Ausserlechner, Udo

doi:10.2174/1874843001804010014

Cited by 6 publications

(9 citation statements)

References 26 publications

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“…The theory is so general that it also covers former results on Hall plates with four contacts having two perpendicular mirror symmetries [1] and also former results on Hall plates with three contacts having only single mirror symmetry [31]. However, the mathematical expressions for the Hall plate parameters from the present theory show no apparent similarity to former work in [1] [31].…”

Section: Discussionsupporting

confidence: 49%

An Analytical Theory of the Signal-to-Noise Ratio of Hall Plates with Four Contacts and a Single Mirror Symmetry

Ausserlechner¹

2018

JAMP

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This work gives an analytical theory of the signal-to-thermal-noise ratio (SNR) of classical Hall plates with four contacts at small magnetic field. In contrast to previous works, the symmetry of the Hall plates is reduced to only a single mirror axis, whereby the average of potentials of the two output contacts off this mirror axis differs from the average of potentials at the two supply contacts on the mirror axis, i.e. the output common mode differs from 50%. Surprisingly, at fixed power dissipated in the Hall plate, the maximum achievable SNR is only 9% smaller for output common modes of 30% and 70% when compared to the overall optimum at output common modes of 50%. The theory is applied to Vertical Hall effect devices with three contacts on the top surface and one contact being the buried layer in a silicon BiC-MOS process. Geometries are found with large contacts and only a moderate loss in SNR.

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Section: Discussionsupporting

confidence: 49%

An Analytical Theory of the Signal-to-Noise Ratio of Hall Plates with Four Contacts and a Single Mirror Symmetry

Ausserlechner¹

2018

JAMP

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“…its layout) and the size of the contacts. The quantity G (4C) H0 can be computed as a function of geometrical parameters of the Hall plate, but it can also be expressed as a function of the resistances in the ERC [4]. The thermal noise of a Hall plate at small magnetic field is also described by the ERC.…”

Section: Physical Backgroundmentioning

confidence: 99%

“…In contrast to G (4C) H0 , the low field Hall geometry factor G (3C) H0 of devices with three contacts is a function of the resistances R e and R d of the ERC plus the sheet resistance R sh . For the case of devices having single mirror symmetry it is given explicitly in [4] that…”

Section: Physical Backgroundmentioning

confidence: 99%

Symmetries of certain double integrals related to Hall effect devices

2020

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One encounters iterated elliptic integrals in the study of Hall effect devices, as a result of conformal mappings of Schwarz-Christoffel type. Some of these double elliptic integrals possess amazing symmetries with regard to the physical parameters of the underlying Hall effect devices. We give a unified mathematical treatment of such symmetric double integrals, in the context of Hall effect devices with three and four contacts.

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“…In [8] it was also implicitly mentioned that the product of input resistances of original and complementary Hall plates of that particular symmetry (i.e., input resistance equals output resistance) at zero magnetic field equals twice the square of the sheet resistance (see the paragraph after (50) in [8]). Complementary Hall plates with three extended contacts on the perimeter were studied in [10]. If such a device has single mirror symmetry, also its complementary device has single mirror symmetry.…”

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confidence: 99%

“…Complementary Hall plates with three extended contacts on the perimeter were studied in [10]. If such a device has single mirror symmetry, also its complementary device has single mirror symmetry.…”

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confidence: 99%

Relations between Hall Plates with Complementary Contact Geometries

Ausserlechner¹

2019

JAMP

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Singly connected Hall plates with N peripheral contacts can be mapped onto the upper half of the z-plane by a conformal transformation. Recently, Homentcovschi and Bercia derived the General Formula for the electric field in this region. We present an alternative intuitive derivation based on conformal mapping arguments. Then we apply the General Formula to complementary Hall plates, where contacts and insulating boundaries are swapped. The resistance matrix of the complementary device at reverse magnetic field is expressed in terms of the conductance matrix of the original device at non-reverse magnetic field. These findings are used to prove several symmetry properties of Hall plates and their complementary counterparts at arbitrary magnetic field. Journal of Applied Mathematics and Physics mentary device (see (13a) in [6]).In [7] plane singly-connected Hall plates with four peripheral contacts and equal input and output resistances were considered. If magnetic field is impressed on such a device, it has the same output voltage as its complementary device, provided both are supplied by the same voltage source (see Figure 7 in this work). This was conjectured in [8] and proven in [7] and [9] for weak applied magnetic field (see (50) in [8], see Section 4, Appendix B, and Figure 8, all in [7]). Numerical inspection suggests that this also holds for strong magnetic field, but a rigorous proof has not been given so far. In [8] it was also implicitly mentioned that the product of input resistances of original and complementary Hall plates of that particular symmetry (i.e., input resistance equals output resistance) at zero magnetic field equals twice the square of the sheet resistance (see the paragraph after (50) in [8]).Complementary Hall plates with three extended contacts on the perimeter were studied in [10]. If such a device has single mirror symmetry, also its complementary device has single mirror symmetry. Then-analogous to above-the change of the potentials on the output contacts due to reversal of magnetic field polarity are identical in both original and complementary devices, if both devices are supplied with the same supply voltage on the other two contacts, and if the magnetic field is weak (see also Figure 6 in this work). This property also means that the ratio of Hall output signal over thermal noise under the constraint of fixed supply voltage and fixed input resistance is the same in the original Hall plate and in the complementary Hall plate [10] [11].In Section 2 we reconsider the General Formula of [12] for the electric field in the upper half of the z-plane with N contacts on the real axis. Thereby, we present a different derivation than the one given in [12]. This new approach shows how the stagnation points are linked to the electric field in the Hall plate. From this result we derive the resistance matrix of a general device at arbitrary magnetic field in Section 3. This is similar to [12]. In Section 4 we link the resistance matrices of original device and complementary device at reverse...

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An Analytical Theory of Hall-Effect Devices with Three Contacts

Cited by 6 publications

References 26 publications

An Analytical Theory of the Signal-to-Noise Ratio of Hall Plates with Four Contacts and a Single Mirror Symmetry

An Analytical Theory of the Signal-to-Noise Ratio of Hall Plates with Four Contacts and a Single Mirror Symmetry

Symmetries of certain double integrals related to Hall effect devices

Relations between Hall Plates with Complementary Contact Geometries

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