90% SNR Improvement with Multi-Port Hall Plates

For Hall plates, the ratio of signal over thermal noise is determined by material properties, thickness, layout geometry, magnetic field, and the electric power at which the plate is operated. For traditional Hall plates with four contacts, the optimum choice is a symmetrical device with medium-sized contacts. This paper shows that the signal-to-noise-ratio (SNR) can be further increased by up to 90% for Hall plates with more than four contacts. Supply currents flow through several pairs of contacts, while a signal conditioning circuit taps output voltages at all pairs of contacts and sums them up. We compute the total thermal noise of the sum of correlated noise voltages and relate it to the total magnetic sensitivity. We also prove that for electrically linear devices a spinning current scheme cancels out zero point errors (offset errors) in a strict sense. All our investigations use the definite resistance matrix of multi-port Hall plates. We develop an analytical theory based on re-cent advances in the theory of Hall plates, and then we compute the integrals and matrices numerically for symmetrical Hall plates with six to 40 contacts. We also present measurements in accordance with our theory.


Introduction
In this work we look for ways about how to get less noisy signals from Hall plates. On the one hand we want to maximize the output signal per milli-Tesla of impressed magnetic field; on the other hand we want to minimize the noise in the signal while keeping the power consumption of the Hall plate constant. The focus of this paper is an optimum topology of the Hall plate that can be used in How to cite this paper: Ausserlechner, U.
(2020) 90% SNR Improvement with Multi-Port Hall Plates. Journal of Applied Mathematics and Physics, 8, 1568-1605. However, it is possible to operate a Hall plate at frequencies between 1 kHz and 1 MHz thereby cancelling out 1/f-noise. A simple way is to switch the Hall plate on and off at this frequency and process the output voltage with a sample and hold circuit or a simple low-pass filter. A more common method is the spinning current scheme, which greatly reduces the zero-point (offset) error of the Hall plate, and simultaneously it cancels out 1/f-noise, too [2] [3]. Note that all these operating modes allow for a detection of static and low frequency magnetic fields even though the Hall plate is electrically operated at elevated frequencies. Therefore we may ignore 1/f-noise of Hall plates and focus on thermal noise only.
An optimization of the signal-to-thermal-noise-ratio (SNR) is known for conventional Hall plates with four contacts [4]. For maximum SNR the Hall plate should be symmetric with medium-sized contacts. That means, it should have identical input and output resistance and the average potential of both output contacts should be half of the supply voltage; i.e. the common mode potential of the differential output signal should be exactly in the center between both supply potentials. The size of the contacts should be chosen such that the input and output resistances are 2 times as the sheet resistance sheet H R t ρ = ( ρ is the specific resistivity of the material in the Hall plate at zero impressed magnetic field and H t is the thickness of the Hall plate). There are many shapes, which fulfill these requirements. The most common ones are circles, crosses, octagons, and rectangles with specific sizes of contacts as shown in Figure 5 of [4]. Most of them have 90˚ symmetry. According to Wick, Hall plates whose shapes are linked via conformal transformations have identical impedances and magnetic field sensitivities [5]. Thus, they also have identical SNRs. Hence, we may focus on circular Hall plates with peripheral contacts. Once we find an optimum one, we may derive other equivalent shapes of Hall plates by conformal transformations. The optimum circular Hall plate with four contacts has contacts extending over 45˚ separated by insulating arcs of the same size (Figure 1(a)). These are surprisingly large contacts, which reduce the Hall signal per supply current by one third compared to the maximum possible one for point-sized contacts. On the other hand, point-sized contacts give infinitely large impedance, which leads to large thermal noise. The best trade-off between Hall signal and noise turns out to be medium-sized contacts that cover 50% of the perimeter of the disk. For Hall plates with three contacts the Hall signal and the thermal noise were Journal of Applied Mathematics and Physics studied in [6] [7]. The maximum SNR per Watt of dissipated power is obtained for symmetric devices operated like in Figure 2, but this optimum is still 1.51 times lower than for optimized Hall plates with four contacts. This is interesting because contacts are usually believed to deteriorate the Hall signal. Nevertheless the comparison of Hall plates with three and four contacts shows that the device with more contacts has better SNR.
Hall plates with eight contacts are reported in [8] [9]. They are supposed to have exceptionally low offset errors close to 1 µT when operated in an 8-phase spinning scheme [8]. In each phase current flows through two opposite contacts while voltage is tapped at two opposite contacts perpendicular to a line through the current carrying contacts ( Figure 3). The other four contacts are not used in this phase. In seven subsequent phases all contacts are moved by one instance clock-wise with regard to the preceding phase. Finally the output voltages of all eight phases are added, thereby very efficiently reducing offset errors.
Hall plates with more than eight contacts were used in spinning current schemes studied by Munter [10] [11]. But still he used only single pairs of output contacts in each phase (single input current, single output voltage).
Another idea uses an integer multiple of four contacts, e.g. 8, 12, 16 … contacts, whereby every fifth contact is connected to the same terminal ( Figure 4). An example of eight contacts is shown in [12] (see also  in [10]). In general, contacts 1, 5, 9, … are connected to a first terminal, contacts 2, 6, 10 ... are connected to a second terminal, contacts 3, 7, 11, ... to a third terminal, and contacts 4, 8, 12, … to a fourth terminal. Operation is analogous to a conventional Hall plate with four contacts: current is supplied through the even terminals and voltage is tapped across the odd terminals, and vice versa. We will show in Appendix B that regardless of the number of contacts this kind of device does not have better SNR than optimum Hall plates with four contacts.    Table   1 and Table 2).
Yet another idea uses Hall plates with a large numbers of contacts, where voltage is tapped only across a single pair of contacts while all other contacts carry supply current ( Figure 5) [13] (multiple input currents, single output voltage).
The goal of the authors was to avoid large current contacts because they reduce the Hall signal by their short-circuiting action. We will show in Appendix C that these topologies do not give better noise performance per Watt than optimized traditional Hall plates with four contacts from Figure 1.
Occasionally the question pops up if measuring current instead of voltage at the output contacts of a Hall plate might improve its performance [14]. As long as the Hall plate is a passive device with linear electric properties, its voltages and currents are linked via a linear resistance matrix. In contrast to the statements in U. Ausserlechner Journal of Applied Mathematics and Physics contacts. Input and output resistances are equal to sheet 2R n and the Hall geometry factor equals 2/(3n) (all at weak magnetic field, see Appendix B). For large numbers of contacts the current tends to flow near the perimeter. Then we may cut out the center portion of the plate. Then, we may also place contacts on the inner boundary-this gives Hall voltages with inverted polarity. Figure 5. Hall-effect devices with many supply contacts and only a single pair of output contacts after [13]. This concept has no better SNR then classical Hall plates with four contacts (see Appendix C).
[14] current mode operation does not change the boundary conditions of the

Hall Signals in a Multi-Port Hall Plate
We consider circular Hall plates with N = 2M contacts, where N is an even number and M is greater than 1 (see Figure 6). All contacts are labelled in sequential order along the periphery. The N-th contact is grounded and supply current is injected into the M-th contact. We define M − 1 output ports of the Hall plate: The k-th output port comprises the k-th and the (N-k)-th contact. The potential at each contact is labelled V k . Then the output voltage at the k-th port is out,k . The overall output of the Hall plate is the linear combination of outputs of all ports . An electronic circuit can readily sum up all the contributions of all ports as sketched in Figure 6. Let us set c k = 1 to start with.
For a conventional Hall plate with four contacts we have N = 4 and M = 2 and only one output port out out,1 1 3   [17]. Also at very strong impressed magnetic field G H tends to 1 [17]. In both cases the impedance of the Hall plate rises and this leads to excessive thermal noise. Therefore we have to use contacts which are neither too large nor too small-we have to trade off impedance and Hall signal. For circular Hall plates with four contacts it is known that optimum SNR at weak magnetic field is achieved for the device in Figure 1 [4]. Its contacts are equally large as the insulating arcs between the contacts. Therefore we keep this high degree of symmetry also for the multi-port Hall plates. Then the vertices of the contacts are at angles (see Figure 6) ( ) 360 1 and 180 ; 1, 2, , Analogous to (1) we define the Hall geometry factor for the k-th port.
( ) out, sheet , supply Although all contacts are equally large, they do not have the same Hall signal.
Ports closer to the supply contacts have less Hall signal than ports mid-way be- In (4b) we normalize the total Hall geometry factor by the number of output We may compute the potentials at all contacts in response to the supply current by use of the definite resistance matrix R of rank N − 1.
In the operating mode of Figure 6 all elements of the current vector I vanish except I M = I supply . With (3) the Hall geometry factor of each port follows from the resistance matrix.
The resistance matrix of a circular Hall plate with N contacts from Figure 6 can be computed in an elegant way with [16] ( ) with the elements of the matrices B and C given by  Table 1 and Figure 7. Thereby, we used a small Hall angle of 0.09˚, which corresponds to weak magnetic field where noise and SNR are most relevant in practice. We denote the weak field conditions by a subscript "0".
In Figure 7 we plotted G H0,k versus the common mode cm, which we define as U. Ausserlechner Table 1. Weak field Hall geometry factors G H0,k of all ports for symmetric Hall plates of Figure 6 (computed with (6)-(10) at 0.09˚ Hall angle). The operating mode is "single input current, M − 1 output voltages". Each line corresponds to a Hall plate with N = 2M contacts. Hall geometry factors of ports with larger index are labelled as non applicable "n.a". Example: A Hall plate with N = 12 contacts has 5 output ports. Due to symmetry it holds G H0,5 = G H0,1 , and G H0,4 = G H0,2 . Hence, only G H0,1 , G H0,2 , and G H0,3 are given explicitly.   Table 1. G H0,k is defined in (6) and (11), cm is defined in (12).

U. Ausserlechner Journal of Applied Mathematics and Physics
Thus, the common mode is the ratio of average potential of a port over the supply voltage drop. It is a value between 0 and 1. Obviously, the Hall signal drops for contacts closer to the current supply contacts. For N > 4 we have more output ports and larger Hall signal per output port. The average weak field Hall geometry factor versus N is shown in Figure 8. It starts at 2/3 for a Hall plate with four contacts and increases beyond 0.9 for N = 40. It is likely but not yet proven that it goes to 1 for N → ∞ . The same Figure 8 also plots the input resistance R in of the Hall plate with N contacts in units of the sheet resistance.
in , Although the sizes of the contacts shrink with N also their spacings shrink.
Thus, the impedance between the supply contacts goes up only moderately with N (see the logarithmic fit formula in Figure 8).

The Thermal Noise of a Multi-Port Hall Plate
At weak magnetic field the Hall plate is a resistive domain with N contacts. Then its deterministic electrical behavior is fully described by its resistance matrix. It links the voltages at the contacts with the currents into the contacts in a linear fashion. This corresponds to a resistor network with resistances r i,j between pairs (i, j) of contacts. Thus, a Hall plate with N contacts corresponds to a resistor network with ( ) ( ) ( ) The symmetry of the Hall plate is reflected by the symmetry of the resistor network. For the Hall plates of Figure 6 the resistor network has only 2 N     different resistance values (where x     means the integer part of x). For N = 3 to 21 they are given in Appendix A. There we also explain how to compute the resistance values of the network from the definite resistance matrix R.
In his seminal paper, Nyquist showed with general laws of thermodynamics that in thermal equilibrium each resistor r i,j between contacts i and j acts as a Figure 8. Average weak field Hall geometry factor G H0 of the Hall plates from Figure 6.
Also: normalized input resistance of these Hall plates at weak magnetic field. Both are plotted versus the number of contacts. Numerical values in Table 2. G H0 is computed with (4b), (6), and (7)- (10). R in is computed with (13) and (7)- (10). Journal of Applied Mathematics and Physics thermal noise source with a built-in noise voltage n(t) versus time t [18]. Its mean value vanishes ( ) Thermal noise is characterized by its noise power, which is the mean of its squared value ( ) where k b is Boltzmann's constant, T is the absolute temperature, and Δ is the observation bandwidth. The noise voltage is the square-root of (14b) which is also called the noise rms voltage (root mean squared). The noise voltages of different resistors in a network are uncorrelated, which means that their noise powers simply add up, Johnson said that whenever we look through a port into a complex network the thermal noise voltage at this port is the same as if we replace the network by the real part of its impedance at this port [19]. Thus the noise voltage at the k-th output port of our Hall plate at zero magnetic field is given by (14b) if we replace r i,j by the output resistance R out,k of this port.
However, a single resistor of the network will contribute to the noise on all ports. Therefore the noise voltages at the ports will exhibit some correlation. If we label the noise voltages at ports k and  by n k and n  , respectively, it holds for the noise power of the sum of both ports Hence, we would make an error in the noise calculation if we accounted only for the output resistances of the ports. A correct calculation would have to compute the noise in the sum of port voltages caused by each of the ( ) tors separately, and add up the squares of these noise contributions. To this end one has to compute the resistances of the network and the transfer functions of their noise voltages to all output ports-which is some computational effort without physical insight. We did it as a check for the results of the following paragraphs but it is not worthwhile to report it in detail.
A more rewarding method to compute the noise in the sum of all output ports is shown in Figure 9.  port voltages with galvanic isolation. Note that the circuit in Figure 9 is not supposed to be implemented in practice. It is just a theoretical tool to compute with least effort the thermal noise in the sum of voltages of all output ports. Next, we can apply the finding of Johnson: the noise is equivalent to the output resistance. We do not need the resistor circuit for the output resistance. It is simpler to use the definite resistance matrix R. With Figure 9 we write During the measurement of the output resistance the transformers force the current I out into all contacts 1, 2, , 1 M −  and out of all contacts No current flows into contacts M and N. We sum up all port voltages to get the total output voltage V out . This is the voltage, which an Ohm-meter would see during a measurement of R out , while it forces current I out into the output terminals of the circuit in Figure 9.
The thermal noise n out in the sum of voltages of all output ports is then again given by (14b) if we replace r i,j by R out of (18). The following scheme gives a bet-Journal of Applied Mathematics and Physics ter impression on which elements of the definite resistance matrix are added and subtracted in (18).
We can decompose the definite resistance matrix R into the sum of a matrix R even with even symmetry and a matrix R odd with odd symmetry.
where all resistance elements are evaluated at the same magnetic field polarity B ⊥ . The principle of reverse magnetic field reciprocity states [20] ( ) ( ) Inserting (21) in (20b, c) gives At arbitrary magnetic field the even matrix R even has only terms 2 p B ⊥ and the odd matrix R odd has only terms 2 1 p B − ⊥ with p being a non-negative integer. R even reflects a reciprocal network, which consists only of resistors. However, at large magnetic field these resistors depend on the magnetic field-they exhibit magneto-resistance. Conversely, R odd reflects an anti-reciprocal network, which can be modeled by gyrators or controlled sources [21]. R odd describes the Hall effect in the Hall plate. Interestingly, the summing scheme in (19) We call 1, 1 M η − the noise efficiency of the Hall plate with single input current and M − 1 output voltages. It is independent of material properties and thickness of the Hall plate. In the weak field approximation it is also does not depend on the magnetic field. It is a mere function of the lateral geometry, i.e. the layout of the Hall plate, and of the operating mode, i.e. the number of supply currents and output voltages. In (23a) the Hall mobility is the only material parameter, and P Hall is the power dissipated in the Hall plate.
Inserting (4b), (7) and (18) Table 2. Not visible in this scale: SNR(N) has a flat maximum in N = 36. The SNR is computed with (23b) and the output resistance is computed with (18). Table 2. Electrical parameters for Hall plates of Figure 6 (computed at 0.09˚ Hall angle): number of contacts N = 2M, average weak field Hall geometry factor G H0 , input and output resistances (for 1 Ω sheet resistance), noise efficiency 1, 1 M η − , and ratio of SNR(N) with N contacts over SNR(4) of conventional Hall plates with only four contacts-all for operating mode "single input current, M − 1 output voltages". The last line gives the SNR M/2 of Hall plates of Figure 6, if only the signal of the output port at 50% common mode is used for the output signal (this is port number M/2 compared to the classical Hall plates with only four contacts. Table 2 gives the numerical data for SNR, average weak field Hall geometry factor, and input and output resistances. Interestingly, the SNR has a flat maximum at N = 36. For more contacts it decreases, but this decrease is too small to be visibly in Figure   10. The last line in Table 2 gives the SNR for the same Hall plates of Figure 6 if only the single output port at 50% common mode provides the output signal (this is port number M/2). This is the case, which was studied in [10]. Then the SNR decreases monotonously with growing number of contacts, because the contacts become too small. For eight contacts (as in [8]) the loss in SNR is only 5.8% compared to optimum conventional Hall plates with four contacts.

U. Ausserlechner
In Appendix E we show that the SNR can be increased a bit further by multiplying the signals of the output ports with optimized weighing factors 1 k c ≠ prior to summing them up.

A Spinning Scheme for Offset Cancellation of Multi-Port Hall Plates with Single Input Current
Very low zero point error (offset error) is probably the strongest argument in favor of Hall plates when compared to magneto-resistive sensors. Due to small asymmetries of the Hall plate it has a relatively large initial offset in the order of several milli-Tesla. Yet, with the principle of spinning current Hall probes the residual offset can be reduced down to a few micro-Tesla [3] [8] [10]. The attractiveness of spinning schemes is very high for industrial manufacturing, because they implicitly reduce the offset error without need to measure it. During production a measurement of fields in the micro-Tesla range would be very costly, because standard equipment generates too strong magnetic disturbances.

U. Ausserlechner Journal of Applied Mathematics and Physics
Therefore it is of paramount importance to find spinning schemes for every new type of Hall-effect device. Luckily, we found the following one for multi-port Hall plates.
Let us define M operating phases of the spinning scheme, which are executed sequentially and their outputs V out,phase(k) are summed up. In the k-th operating phase current enters the Hall plate through contact k and leaves it through contact k + M, while all potentials at contacts k + 1 to k + M − 1 are added and all potentials at contacts k + M + 1 to k + 2M − 1 are subtracted. With (5) it holds In (25) no matrix element appears more than once, however, the indices go up to 3M − 1 > N. We have to delete all elements where the first index equals N, because V N = 0. We also have to delete all elements where the second index equals N, because the definite resistance matrix R already implicitly accounts for I N as being the negative sum of all other currents. Moreover, we subtract N from any index which is greater than N. This takes account for the fact that some output contacts go into "a second loop" beyond the N-th contact, where of course contact N + k is contact k. The resulting pattern of indices in (25) is shown for the case of N = 12 contacts in Figure 11(a), Figure 11(b). The first index in (24) corresponds to the horizontal axis and the second index to the vertical axis in Figure 11(a), Figure 11(b). A red "o" means that the matrix element is added and a blue "x" means that it is subtracted in (24). Figure 11(a) shows all matrix elements prior to subtracting N for indices greater than N. Figure 11(b) Figure 11. Occurence of matrix elements R i,j in the Hall spinning scheme (25) for M-1 output ports and N = 12. The horizontal axis in the plots gives the first index of R i,j , the vertical axis gives the second index. The red "o" means that R i,j is added in (25), the blue "x" means that R i,j is subtracted. No element appears twice or more often. . There we note the symmetry: for every element R i,j there is a corresponding element (−1) × R j,i . Applying the principle of reverse magnetic field reciprocity (RMFR) [20] to these pairs gives In other words, the output signal of the spinning scheme V out,spin is an odd function of the magnetic field. Thus it vanishes at zero magnetic field. Therefore the zero point error (offset error) vanishes. This holds also for asymmetric Hall plates, because we did not make use of any symmetry of R other than the RMFR. In (25) the sum goes only up to M, not up to N = 2M. This means that for each phase in (25) there is another one with inverted current flow polarity. In practice one will extend the sum over all N phases because it cancels out further errors caused by thermo-voltages, which were not accounted for in our simple linear theory.

Multi-Port Hall Plates with Multiple Input Currents and Multiple Output Voltages
In a very general general case, a circular Hall plate may have N = 2M peripheral contacts. The N-th contact is grounded (see Figure 12). All other contacts are connected to current sources which determine the currents I k ( 1, 2, , The overall output of this spinning scheme is the sum over all signals in all M operating phases. We have many parameters that can be optimized. How can we determine the currents and weighing coefficients to achieve 1) zero offset error, and 2) maximum SNR? Obviously the currents are unique only up to a common multiplicative factor, which would neither change offset nor SNR. Therefore we have to normalize one current, say I M = 1. The same applies to the weighing coefficients, thus we set c M = 1.
We start with the spinning scheme to figure out the restrictions on the currents and the weighing coefficients for zero offset error. First we apply the ideas of Section 5 explicitly to Hall plates with N = 6, 8, and 10. We compute the total signal, which is a large sum over all operating phases and contact pairs. The terms in the sum are currents multiplied by weighing factors multiplied by elements of the resistance matrix. We replace  (28) means that identical currents have to flow through contacts belonging to the same contact pair. Surprisingly, (28) and (29) are independent of the weighing coefficients. With (28) and (29) the spinning scheme is able to cancel out zero point errors regardless of the symmetry of the Hall plate. Moreover, (28) and (29) and the normalization I M = 1 leave M − 2 currents free to choose for maximum SNR. With the M − 1 free weighing coefficients we have in total N − 3 degrees of freedom (DoF) in the SNR optimization.
From the definition of the SNR in (23) we can start with SNR = V out /n out . Yet, we need to reconsider the input resistance for a device with many inputs. Figure   12 shows the circuit where the multi-terminal Hall plate is supplied by multiple current sources. Let us assume as a simplification that an MOS current mirror does not need any drain-source saturation voltage. Then the voltage of the power supply circuit is Note that from all N current sources only two have zero voltage across them-one Journal of Applied Mathematics and Physics at the positive and one at the negative pole of the battery. All others have non-vanishing voltage drop and therefore they all dissipate power. We want to get maximum SNR at minimum power dissipation in the system, not in the Hall plate alone. Therefore we need to account for the power P supply that is delivered by the power supply circuit.
This gives us the SNR of a multi-port Hall plate with multiple input currents.
with the dimension-less noise efficiency 1, N M η − for the operating mode with N − 1 input currents and M output voltages. The output resistance for the circuit in Figure 12 is obtained analogous to Section 3.
In (33b) An algorithm is shown in Appendix F. The results of this optimization are shown in Figure 13 and Table 3. We note that the maximum achievable SNR rises monotonously with the number of contacts. At N = 40 and for optimized weighing coefficients it is 89% larger than for conventional Hall plates with four contacts. But even for low N the SNR improvement is very good: with only eight contacts it is 47%. Another interesting aspect is that it does not bring any benefit   Figure 12, whereas the curve labelled "M − 1 outputs" assumes the circuit from Figure 6. The plot shows the ratio of SNR for Hall plates with N contacts over maximum SNR for a Hall plate with four contacts in conventional operating mode of Figure 1. Numerical values for "M outputs" in Table 3. There it is also noted how many supply currents are used and what are the optimum values of these currents. The SNR of the curves labelled "M outputs" is computed with (33b). Table 3. Parameters of multi-port Hall plates operated in the circuit of Figure 12   Conversely, the benefit of optimized weighing factors is tiny (see Figure 13) and not reported in Table 3.

Experimental Verification
For an experimental verification of our theory, we manufactured the two types of Hall plates in Figure 14. gives an uncertainty in the contact resistances of the Hall plates, which makes an exact quantitative comparison to our theory impossible. Yet, at least we can give a qualitative comparison in the following. According to measurements at room temperature, the resistance between opposite contacts of the four-contacts device is 5856 Ohm at small supply voltage. For the eight-contacts device we measured 6930 Ohm at small supply voltage. Both times the resistance increases by 5.5% if the positive supply contact is at 1 V due to the junction field effect at the pn-isolation. For the eight-contacts Hall plate we measured the equivalent resistor network r 1,N = 6562 Ohm, r 2,N = 46,591  Figure 15 were connected to the single output port of the four-contacts Hall plate, which gives an amplification factor 33. Both times, the Hall plates were supplied by a 1.5 V battery and placed inside a steel box to reduce line interference. We assembled the Hall plates in non-magnetic packages, and we conducted the measurements at ambient temperature. We divided the measured noise voltage spectra by the magnetic sensitivities of the Hall plates and the gain factors of the circuit to get the equivalent magnetic noise. These spectra are plotted in Figure 16.
First, we shorted all inputs of the three AD8429s, and measured a white noise voltage spectral density of 33 × 3 nV/sqrt(Hz) at the output of the circuit in Figure 15. Then, we connected three 10 kOhm resistors at the inputs of the three AD8429s, and measured 33 × 9.2 nV/sqrt(Hz). This is 24% larger than for ideal Figure 15. Electronic circuit that amplifies the port voltages of the eight-contacts Hall plate times eleven and sums them up. The operating mode of the Hall plate is the same as in Figure 6. Resistors of the same color are equal. Journal of Applied Mathematics and Physics Figure 16. Noise spectral densities measured on the Hall plates from Figure 14 with the circuit from Figure 15 at room temperature. The noise voltages were divided by the measured magnetic sensitivities to give the equivalent magnetic noise (for the resistor network the sensitivity of the eight-contacts Hall plate at 1.5 V supply voltage was used). "8C-Hall" denotes the eight-contacts Hall plate, "4C-Hall" the four-contacts Hall plate, and "Res. netw." denotes the equivalent resistor network for the eight-contacts Hall plate. "8C-Hall (port 2)" means that all three AD8429 amplifier inputs were connected to port 2 of the eight-contacts Hall plate. No battery was connected to the Hall plate and to the resistor network for the two flat curves. In all other cases the Hall plates were supplied with 1.5 V battery voltage. At high frequencies the measured noise of the eight-contacts Hall plate is 18.3% lower than of the four-contacts Hall plate in spite of the lower current through the eight-contacts Hall plate. noiseless amplifiers. Then we connected all inputs of the three AD8429s to a single 10 kOhm resistor, which increased the impedance level times three. In this mode there is the strongest correlation between all three channels. This led to 33 × 19.1 nV/sqrt(Hz) at the output, which is 48% larger than for a noiseless circuit.
To sum up, the noise of our circuit is not negligible. However, this is irrelevant if we compare noise measurements of Hall plates with identical impedances. Yet, if a Hall plate has larger output resistance, the noise input current of the circuit will add its own noise.
The measured noise spectra of Hall plates are plotted in Figure 16. Second, we note that the noise at 450 kHz of the eight-contacts Hall plate with 1.5 V battery supply is slightly larger than without any battery (10.2 versus 9.8 nV/sqrt(Hz)). This difference of 4.1% can be explained by the 8.2% larger impedance of the Hall plate at 1.5 V supply voltage.
The equivalent resistor network has slightly lower noise: 9.2 nV/sqrt(Hz) at U. Ausserlechner Journal of Applied Mathematics and Physics 340 kHz. This is 6.1% less than for the unpowered Hall plate. The resistor network has only 3.6% smaller input resistance than the unpowered Hall plate, which could explain 1.8% smaller noise voltage at port 2. The discrepancy between 1.8% and 6.1% may come from the fact that the lumped resistors deviate up to 6% from the exact values for the equivalent resistor network (see reported values above).
Anyhow, the discrepancy is much smaller than the differences in noise of powered Hall plates. At 1.5 V supply voltage and 450 kHz the equivalent magnetic noise of the circuit with the four-contacts Hall plate was 142 nT/sqrt(Hz), whereas it was only 120 nT/sqrt(Hz) for the circuit with the eight-contacts Hall plate.
Thus, the eight-contacts Hall plate has 18.3% better SNR despite its lower supply current. Of course, if we use only port 2 of the eight-contact Hall plate, its noise is worse than for the four-contact Hall plate.
The input resistance of the eight-contacts Hall plate is also 18.3% larger than for the four-contacts Hall plate. If we would make the eight-contacts Hall plate thicker by 18.3% it had identical supply current as the four-contacts Hall plate at 1.5 V supply voltage and its SNR would increase by another 8.8%. Then the eight-contacts Hall plate in operating mode according to Figure 6 would have 28.7% better SNR at identical power dissipation. This matches closely the predicted 31.3% from Table 2.
Moreover, we measured the residual offset of the eight-contacts Hall plate according to the spinning scheme of Section 5. Thereby we did not use the circuit of Figure 15. Instead, we supplied the current with a precision current source and measured voltages at the three ports with a single precision voltmeter. The contacts of the Hall plate were switched by a relais matrix with low thermo-voltages. The Hall plates were assembled in non-magnetic packages, and placed in double shielded zero-Gauss chambers at room temperature. The measured output voltages of the Hall plates were summed up according to (24) and (25). Yet, in (25) the sum extended to 2M instead of only M, to cancel out thermo-voltages. The result was divided by the measured magnetic sensitivity to obtain the residual offset equivalent magnetic field. We measured eight devices from a single wafer, and the standard deviation of the results is plotted versus supply voltage in Figure 17. The curve is similar to classic Hall plates with four contacts (as shown in Figure 11 in [6]): the residual offset increases with the supply voltage. The origin seems to be electrical non-linearity and self-heating of the Hall plates, which are not accounted for in our linear theory. Nevertheless, the residual offset is small: at 1 V supply voltage it is 3 µT, at 1.5 V it is 6 µT, and at 2 V it is 10 µT.
To sum up, this experimental section proves that multi-port Hall plates have less thermal noise at identical power dissipation and comparable residual offset to classic Hall plates with four contacts.

Discussion
We studied two circuits for multi-port Hall plates. One has a single input current and M − 1 output voltages (Figure 6), and the other one has multiple input U. Ausserlechner Figure 17. Residual offset equivalent magnetic field of the eight-contacts Hall plate from Figure 14 measured at room temperature. The spinning scheme of (24), (25) was applied, yet, with M replaced by 2M in (25). Note that we use supply voltage on the horizontal axis instead of supply current, because the main origin for residual offset error is the reverse biased pn-isolation junction at the bottom of the Hall plate. Nevertheless, the spinning scheme used constant supply current in all operating phases.
currents and M output voltages ( Figure 12). The first one shows an improvement in SNR of 50% the second one of 90%-both compared to the classical Hall plate with four contacts. Moreover, both circuits are compatible with spinning schemes, which cancel out offset errors as long as the resistances do no depend on the applied voltages. However, the full improvement in SNR holds only for large numbers of contacts (N = 36). Therefore, the spinning scheme needs to sum up the output signals of many operating phases, which reduces the signal bandwidth. This is the price we have to pay for low noise. However, the second circuit shows remarkable SNR improvement of 47% for Hall plates with only eight contacts, two current sources, and four voltmeters (cf. Table 3). Moreover, it is possible to move the spinning scheme from the simple classic forward path into a feedback path, where it has only a minor effect on signal bandwidth [24] [25]. For sensor systems with mega-Hertz bandwidth it is best to add pick-up coils for signals beyond 5 kHz and use the Hall plates only for dc to 5 kHz [26].
These systems have unprecedented SNR, zero point accuracy, and linearity at low costs [27]-better than modern tunnel-magneto-resistors (TMRs).
We have not touched upon Hall plates with odd numbers of contacts, but their treatment is straightforward and analogous to this paper.
So far, we have applied our theory only to Hall plates with contacts equally large as spacings between them, according to (2). For classical Hall plates with four contacts, this is known to give the best SNR per Watt [4]. However, it is not yet clear if this also gives optimum SNR per Watt for larger number of contacts.
One can readily apply our theory to SNR optimization and spinning schemes of Vertical Hall effect devices [28]. The only difference is that Vertical Hall effect devices have less symmetry in their contact arrangement. This will lead to smaller U. Ausserlechner SNR at the same power.
In this paper we have not dealt with important practical aspects of the circuits in Figure 6 and Figure

Conclusion
We have shown by calculation that under a given supply voltage and current drain a regular multi-port Hall plate can achieve up to 90% better signal-to-noise ratio (SNR) than classical Hall plates with four contacts. Alternatively, it can provide the same SNR at 3.6 times smaller power consumption. An experiment on a non-optimized silicon Hall plate with eight contacts showed 29% better SNR than for a classic Hall plate with four contacts, at identical power dissipation. The proposed optimum Hall plates have a regular shape with 2M identical contacts. Current flows through one, two, four, or more pairs of contacts, depending on the number of contacts and on which circuit is used ( Figure 6 or

Appendix A
Here we compute the resistor network for a multi-port Hall plate of Figure 6.
First we underline that a Hall plate exposed to magnetic field is non-reciprocal, which means that in general its impedance matrix has no symmetry. Yet, with (20a-c) we can decompose it into the sum of an even-and an odd-symmetric impedance matrix. The even one corresponds to a resistor network and the odd one to a gyrator network. Both networks are connected in series, because the sum of their impedance matrices gives the original one of the Hall plate [29].
Here we discuss only the resistor network, because the gyrator network does not contribute to the noise in the output signal.
Due to the symmetry of our multi-port Hall plates in Figure 6 the resistor network has only M different values of resistances. From all resistors r i,j between contacts i and j we only need to consider the ones r j,N connected to the N-th contact. All others are again given by the symmetry. For the calculation we assume an operation of the Hall plate according to Figure A1, where all contacts 1 to N − 1 are tied to the same potential V supply and the N-th contact is grounded. Since all contacts 1 to N − 1 are at identical potential, no current flows between them. Thus, the current into each of these contacts j is proportional to 1/r j,N .
From ( ) it follows  Table A1 gives numerical values for these resistances at vanishing magnetic Figure A1. A circuit to compute the resistor network of the Hall plates from Figure 6. All our multi-port Hall plates are symmetric, i.e. they do not change if we rotate them by integer multiples of 360˚/N. This also affects the symmetry of the g matrix. In Figure A2 we connect all contacts to ground potential except contact 1, where we apply negative supply voltage. It holds 1 1 , supply 1 1 1 1, supply  where the currents 1 2 , , , N I I I  are the currents flowing into contacts 1, 2, , N  in Figure A1. Comparison of (A1) and (A3) shows that ,1 1,2 j j g g + = for j > 1.
Continuation of this process means to apply negative supply voltage merely to contact 2, then to contact 3, and so on. Collecting all identities for g i,j gives ( )  These Hall plates can be readily treated with the theory of [16]. For symmetrical circular plates where contacts cover 50% of the perimeter we get from

Appendix C
Hall plates from Figure 5 can be mapped with conformal transformations to Hall plates from Figure C1(a). The idea behind this mapping is to get a Hall plate with homogeneous current density at zero impressed magnetic field. This is achieved by shifting contacts 1 to M − 1 down and contacts M + 1 to N − 1 up, both proportionally to the potential at these contacts. Moreover, the output contacts M and N are folded and their lengths are adjusted so that the same current passes through them as in the original Figure 5. Due to the folding, the output contacts do not disturb the homogeneous current density in the Hall region. At small magnetic field all current streamlines are vertical and the potential along horizontal lines is constant. The longest current streamline has length L, and the width of the Hall plate is W.
In Figure C1 In Figure C1(b) we replace the Hall plate of Figure C1 in (b) the total power is available inside the Hall plate. Obviously, in Figure   C1(b) big portions of contacts 1 and 3 are more distal to the output contacts than in Figure C1(a). Therefore, their short circuiting action on the Hall signal is lower, and this gives larger Hall signal per Hall input current. It means that the Hall plate from [13] has less Hall signal per Watt than a classical Hall plate with four contacts, for which we know the maximum noise efficiency to be 2 3 0.471 ≅ [4]. where we used the effective number of squares (L/W) eff for the ratio of resistance between two opposite contacts over sheet resistance. G H0 is missing in (3) in [14].
In [4] it is shown that Hall plates with four contacts have a maximum possible value for the noise efficiency G H0 /(L/W) eff , which is 2 3 0.471 ≅ . In silicon a phosphor doping of 2 × 10 16  And how about SNR? At weak magnetic field the input resistance of a single device in current mode in Figure D1(b) is exactly half of the input resistance of the same device operated according to Figure 1. This holds for contacts of arbitrary size. We can prove it with the resistor network in [31]: it has resistors r 1,4 = R H between all neighboring contacts and resistors r 2,4 = 2R D between non-neighboring contacts. Thus, for the same Hall plates the circuits in Figure D1(a) and in Figure 1 consume the same power at identical supply voltage V supply . In Figure D1(a) the Hall output current I out and the thermal noise output current I out,noise are giv- The noise current flows with opposite polarity through both ampere-meters, and this gives the factor 2 in (D7). The effective output resistance of the circuit in Figure D1 R R R R + and it causes the thermal output noise current according to [18]. The ratio of (D6) over (D7) gives the SNR (current mode) of the circuit in Figure D1 which shows that at the same power consumption the SNR of the circuit in Figure D1(a) operated in current mode is smaller than the SNR of the same Hall plate operated in a conventional way like in Figure 1. The conventional Hall plate circuit has maximum SNR for  Table A1 for N = 4). Inserting these values into (D8) shows that the maximum achievable SNR of current mode operation is roughly 1.55 times smaller. This finding is consistent with [6], where the maximum achievable SNR of symmetric Hall plates with three contacts was found to be 1.51 times lower than of conventional Hall plates with four contacts. After all the current mode operation in Figure D1 shorts two of the four contacts, thereby making a structure, which effectively has only three terminals. A four-contacts Hall plate with two contacts shorted is similar, but not identical, to a three-contacts Hall plate. This explains the reduction by 1.55 instead of only by 1.51. Figure D2 shows some other versions of current mode operation. For micro-electronic circuits it is difficult to make perfect shorts, whereas it is simpler to make perfect opens. Therefore, one prefers to measure open loop voltages instead of short circuit currents at the outputs of Hall plates.

U. Ausserlechner
The numerical solution is straightforward (e.g. with Mathematica), if we use different starting values smaller than 1 for all c k . Table E1 gives the optimum weighing factors and the relative increase in SNR for weighted over non-weighted output signals. Surprisingly, the improvement in SNR is only tiny, although the weighing factors deviate notably from 1. We found that the values reported in Table E1 are independent on the magnitude of the magnetic field. For practical use it seems needless to implement weighing coefficients, because the SNR improvement is too small. Table E1. Noise efficiency 1, 1 M η − of Hall plates operated with the circuit of Figure 6 with optimum weighing coefficients. "ratio" is the SNR with optimum c k over SNR with c k = 1. We normalized c 1 = 1. For c k -coefficients the symmetry of (36) holds. All other non-vanishing coefficients are given. contacts operated in mode "multiple input currents-multiple output voltages".