_{1}

^{*}

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 recent 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.

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 smart silicon Hall sensor circuits—we do not search for special material compositions, which provide large Hall mobility, such as III-V-heterojunctions, Graphene, or other 2-DEGs.

It is known that the main noise in macroscopic Hall plates with sizes in the order of millimeters is thermal noise [^{15} - 10^{17}/cm^{3}. This can give a relatively low number of one million charge carriers in the active region of the device. Therefore we also note a strong 1/f-noise contribution, as with all other micro-electronic circuit devices. 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 [

An optimization of the signal-to-thermal-noise-ratio (SNR) is known for conventional Hall plates with four contacts [

For Hall plates with three contacts the Hall signal and the thermal noise were

studied in [

Hall plates with eight contacts are reported in [

Hall plates with more than eight contacts were used in spinning current schemes studied by Munter [

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 (

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 (

Occasionally the question pops up if measuring current instead of voltage at the output contacts of a Hall plate might improve its performance [

[

The initial idea of this work is to supply multi-port Hall plates with electric current through only two opposite contacts like in [

We consider circular Hall plates with N = 2M contacts, where N is an even number and M is greater than 1 (see _{k}. Then the output voltage at the k-th port is V out , k = V k − V N − k . The overall output of the Hall plate is the linear combination of outputs of all ports V out = ∑ k = 1 M − 1 c k V out , k . An electronic circuit can readily sum up all the contributions of all ports as sketched in _{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 V out = V out , 1 = V 1 − V 3 . In this section we consider only symmetric devices where the voltages across the output ports vanish in the absence of impressed magnetic field. For conventional Hall plates with four contacts it holds

V out = R sheet G H tan ( θ H ) I supply (1)

with the Hall angle θ H = arctan ( μ H B ⊥ ) , the Hall mobility μ H , and the magnetic flux density B ⊥ perpendicular to the Hall plate. The Hall geometry factor G_{H} is a number between 0 and 1 and it accounts for the loss in output voltage caused by the finite size of the contacts. Large supply contacts short the Hall electric field and large output contacts shunt a considerable portion of the supply

current away from the Hall effect region, where it does not contribute to the Hall signal. If the contacts are point-sized and located on the boundary it holds G_{H} = 1 [_{H} tends to 1 [

α k = 360 ∘ × ( k − 1 ) / N and β k = α k + 180 ∘ / N ; k = 1 , 2 , ⋯ , N . (2)

Analogous to (1) we define the Hall geometry factor for the k-th port.

V out , k = R sheet G H , k tan ( θ H ) I supply (3)

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 between both supply contacts, because in the proximity of the supply contacts their short-circuiting action reduces the Hall signal. We write the sum of voltages of all ports analogous to (1)

V out = ∑ k = 1 M − 1 V out , k = R sheet ( M − 1 ) G H tan ( θ H ) I supply (4a)

G H = 1 M − 1 ∑ k = 1 M − 1 G H , k (4b)

In (4b) we normalize the total Hall geometry factor by the number of output ports in order to keep it less or equal to 1. In the sequel we refer to it as the average Hall geometry factor of the multi-port Hall plate in operating mode “single input current, M − 1 output voltages”. For Hall plates with four contacts (4a, b) become identical to (1).

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.

V = ( V 1 V 2 ⋮ V N − 1 ) = ( R 1 , 1 R 1 , 2 ⋯ R 1 , N − 1 R 2 , 1 R 2 , 2 ⋯ R 2 , N − 1 ⋮ ⋮ ⋱ ⋮ R N − 1 , 1 R N − 1 , 2 ⋯ R N − 1 , N − 1 ) ⋅ ( I 1 I 2 ⋮ I N − 1 ) = R ⋅ I (5)

In the operating mode of _{M} = I_{supply}. With (3) the Hall geometry factor of each port follows from the resistance matrix.

G H , k = R k , M − R N − k , M R sheet tan ( θ H ) (6)

The resistance matrix of a circular Hall plate with N contacts from

R = R sheet cos ( θ H ) B − 1 C (7)

with the elements of the matrices B and C given by

B k , m = ∫ α m β m h ( τ ) d τ sin ( ( τ − β N ) / 2 ) sin ( ( τ − β k ) / 2 ) (8)

C k , m = − ∑ q = m N − 1 ∫ β q α q + 1 h ( τ ) d τ sin ( ( τ − β N ) / 2 ) sin ( ( τ − β k ) / 2 ) (9)

h ( τ ) = ∏ j = 1 N | sin ( ( τ − β j ) / 2 ) sin ( ( τ − α j ) / 2 ) | 1 2 + θ H π (10)

for k , m = 1 , ⋯ , N − 1 . We use a different sign of θ H than [

lim B ⊥ → 0 G H = G H 0 and lim B ⊥ → 0 G H , k = G H 0 , k (11)

In _{H}_{0,k} versus the common mode cm, which we define as

N | G_{H}_{0,1} | G_{H}_{0,2} | G_{H}_{0,3} | G_{H}_{0,4} | G_{H}_{0,5} | G_{H}_{0,6} | G_{H}_{0,7} | G_{H}_{0,8} | G_{H}_{0,9} | G_{H}_{0,10} |
---|---|---|---|---|---|---|---|---|---|---|

4 | 0.666667 | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. |

6 | 0.736477 | G_{H}_{0,1} | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. |

8 | 0.757940 | 0.820246 | G_{H}_{0,1} | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. |

10 | 0.767379 | 0.847737 | G_{H}_{0,2} | G_{H}_{0,1} | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. |

12 | 0.772378 | 0.860488 | 0.878089 | G_{H}_{0,2} | G_{H}_{0,1} | n.a. | n.a. | n.a. | n.a. | n.a. |

14 | 0.775349 | 0.867537 | 0.892522 | G_{H}_{0,3} | G_{H}_{0,2} | G_{H}_{0,1} | n.a. | n.a. | n.a. | n.a. |

16 | 0.777259 | 0.871875 | 0.900677 | 0.907982 | G_{H}_{0,3} | G_{H}_{0,2} | G_{H}_{0,1} | n.a. | n.a. | n.a. |

18 | 0.778561 | 0.874748 | 0.905792 | 0.916832 | G_{H}_{0,4} | G_{H}_{0,3} | G_{H}_{0,2} | G_{H}_{0,1} | n.a. | n.a. |

20 | 0.779489 | 0.876753 | 0.909237 | 0.922450 | 0.926163 | G_{H}_{0,4} | G_{H}_{0,3} | G_{H}_{0,2} | G_{H}_{0,1} | n.a. |

22 | 0.780173 | 0.878211 | 0.911677 | 0.926273 | 0.932133 | G_{H}_{0,5} | G_{H}_{0,4} | G_{H}_{0,3} | G_{H}_{0,2} | G_{H}_{0,1} |

24 | 0.780692 | 0.879306 | 0.913475 | 0.929008 | 0.936227 | 0.938366 | G_{H}_{0,5} | G_{H}_{0,4} | G_{H}_{0,3} | G_{H}_{0,2} |

26 | 0.781095 | 0.880149 | 0.914840 | 0.931041 | 0.939175 | 0.942663 | G_{H}_{0}_{,6} | G_{H}_{0,5} | G_{H}_{0,4} | G_{H}_{0,3} |

28 | 0.781415 | 0.880813 | 0.915903 | 0.932598 | 0.941381 | 0.945773 | 0.947118 | G_{H}_{0,6} | G_{H}_{0,5} | G_{H}_{0,4} |

30 | 0.781672 | 0.881346 | 0.916748 | 0.933818 | 0.943079 | 0.948111 | 0.950356 | G_{H}_{0,7} | G_{H}_{0,6} | G_{H}_{0,5} |

32 | 0.781883 | 0.881779 | 0.917430 | 0.934795 | 0.944418 | 0.949919 | 0.952798 | 0.953697 | G_{H}_{0,7} | G_{H}_{0,6} |

34 | 0.782057 | 0.882137 | 0.917991 | 0.935589 | 0.945495 | 0.951351 | 0.954694 | 0.956225 | G_{H}_{0,8} | G_{H}_{0,7} |

36 | 0.782203 | 0.882436 | 0.918456 | 0.936245 | 0.946375 | 0.952507 | 0.956201 | 0.958193 | 0.958823 | G_{H}_{0,8} |

38 | 0.782327 | 0.882688 | 0.918848 | 0.936792 | 0.947105 | 0.953455 | 0.957420 | 0.959760 | 0.960851 | G_{H}_{0,9} |

40 | 0.782432 | 0.882903 | 0.919180 | 0.937255 | 0.947717 | 0.954244 | 0.958424 | 0.961033 | 0.962470 | 0.962929 |

c m = ( V k + V N − k ) / 2 − V N V M − V N (12)

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 _{in} of the Hall plate with N contacts in units of the sheet resistance.

R in = V M − V N I M = R M , M (13)

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

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 ( N − 1 ) + ( N − 2 ) + ⋯ + 2 + 1 = N ∗ ( N − 1 ) / 2 resistors. The symmetry of the Hall plate is reflected by the symmetry of the resistor network. For the Hall plates of

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

thermal noise source with a built-in noise voltage n(t) versus time t [

n i , j ¯ = lim t → ∞ 1 t ∫ τ = 0 t n i , j ( τ ) d τ = 0 (14a)

Thermal noise is characterized by its noise power, which is the mean of its squared value

n i , j 2 ¯ = lim t → ∞ 1 t ∫ τ = 0 t n i , j 2 ( τ ) d τ = 4 k b T r i , j Δ (14b)

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,

( n i , j + n k , l ) 2 ¯ = n i , j 2 ¯ + n k , l 2 ¯ = 4 k b T ( r i , j + r k , l ) Δ (15a)

because they are uncorrelated

n i , j n k , l ¯ = lim t → ∞ 1 t ∫ τ = 0 t n i , j ( τ ) n k , l ( τ ) d τ = 0 (15b)

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 [_{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 l by n_{k} and n l , respectively, it holds for the noise power of the sum of both ports

( n k + n l ) 2 ¯ = n k 2 ¯ + 2 n k n l ¯ + n l 2 ¯ ≠ n k 2 ¯ + n l 2 ¯ = 4 k b T ( R out , k + R out , l ) Δ (16)

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 N ∗ ( N − 1 ) / 2 resistors 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

port voltages with galvanic isolation. Note that the circuit in

V = ( R 1 , 1 ⋯ R 1 , N − 1 ⋮ ⋱ ⋮ R M − 1 , 1 ⋯ R M − 1 , N − 1 R M , 1 ⋯ R M , N − 1 R M + 1 , 1 ⋯ R M + 1 , N − 1 ⋮ ⋱ ⋮ R N − 1 , 1 ⋯ R N − 1 , N − 1 ) ⋅ ( I out ⋮ I out 0 − I out ⋮ − I out ) (17)

During the measurement of the output resistance the transformers force the current I_{out} into all contacts 1 , 2 , ⋯ , M − 1 and out of all contacts M + 1 , M + 2 , ⋯ , N − 1 . 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

R out = V out I out = ∑ j = 1 M − 1 ∑ k = 1 M − 1 R j , k + ∑ j = M + 1 N − 1 ∑ k = M + 1 N − 1 R j , k − ∑ j = 1 M − 1 ∑ k = M + 1 N − 1 R j , k − ∑ j = M + 1 N − 1 ∑ k = 1 M − 1 R j , k (18)

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 better impression on which elements of the definite resistance matrix are added and subtracted in (18).

R 1 , 1 ⋯ R 1 , M − 1 0 − R 1 , M + 1 ⋯ − R 1 , N − 1 ⋮ ⋱ ⋮ ⋮ ⋮ ⋱ ⋮ R M − 1 , 1 ⋯ R M − 1 , M − 1 0 − R M − 1 , M + 1 ⋯ − R M − 1 , N − 1 0 ⋯ 0 0 0 ⋯ 0 − R M + 1 , 1 ⋯ − R M + 1 , M − 1 0 R M + 1 , M + 1 ⋯ R M + 1 , N − 1 ⋮ ⋱ ⋮ ⋮ ⋮ ⋱ ⋮ − R N − 1 , 1 ⋯ − R N − 1 , M − 1 0 R N − 1 , M + 1 ⋯ R N − 1 , N − 1 (19)

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.

R = R even + R odd (20a)

( R even ) k , m = R k , m + R m , k 2 = ( R even ) m , k (20b)

( R odd ) k , m = R k , m − R m , k 2 = − ( R odd ) m , k (20c)

where all resistance elements are evaluated at the same magnetic field polarity B ⊥ . The principle of reverse magnetic field reciprocity states [

R k , m ( − B ⊥ ) = R m , k ( B ⊥ ) (21)

Inserting (21) in (20b, c) gives

( R even ) k , m = R k , m ( B ⊥ ) + R k , m ( − B ⊥ ) 2 (22a)

( R odd ) k , m = R k , m ( B ⊥ ) − R k , m ( − B ⊥ ) 2 (22b)

At arbitrary magnetic field the even matrix R_{even} has only terms B ⊥ 2 p and the odd matrix R_{odd} has only terms B ⊥ 2 p − 1 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 [_{odd} describes the Hall effect in the Hall plate. Interestingly, the summing scheme in (19) cancels out all contributions of R_{odd} due to the odd symmetry (22b). Therefore the thermal noise of a Hall plate is independent of the Hall-effect. It only depends on the magneto-resistance effect. For the calculation of the noise voltage of Hall plates we only need the resistor network, not the gyrator network.

First, we define what we mean by optimum SNR since the answer depends on some boundary conditions. The Hall signal increases with the electric power at which one operates the Hall plate, whereas the thermal noise does not depend on power. Thus, SNR rises with power. Large power leads to considerable self-heating and finally to the destruction of the Hall plate. We can push this limit further by more efficient means of cooling. Simple scaling of the size of the Hall plate reduces the power density and improves the heat delivery. In general, all these issues are irrelevant in industrial practice, because there, power and size relate to costs, which have to be kept small. For a Hall sensor circuit both supply voltage and supply current are already specified at the start of the design process. Therefore, we want to maximize SNR for a fixed value of input resistance of the Hall plate. With (4a), (14b), and (18) we write for multi-port Hall plates in

S N R = lim B ⊥ → 0 V out n out = μ H B ⊥ P Hall 4 k b T Δ η 1 , M − 1 (23a)

η 1 , M − 1 = ( M − 1 ) G H 0 R out / R sheet R in / R sheet (23b)

We call η 1 , M − 1 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) into (23a, b) gives the plot of SNR versus N in

N | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 20 | 36 | 40 |
---|---|---|---|---|---|---|---|---|---|---|

G_{H}_{0} | 0.666667 | 0.736477 | 0.778709 | 0.807558 | 0.828764 | 0.845136 | 0.858229 | 0.878002 | 0.919062 | 0.924960 |

R_{in} | 1.414215 | 1.666668 | 1.847760 | 1.988856 | 2.104399 | 2.202216 | 2.287017 | 2.428831 | 2.802727 | 2.869776 |

R_{out} | 1.414215 | 4.000003 | 7.708066 | 12.52199 | 18.43434 | 25.44113 | 33.53998 | 53.00815 | 174.3981 | 215.6139 |

η 1 , M − 1 | 0.471404 | 0.570472 | 0.619014 | 0.647285 | 0.665309 | 0.677454 | 0.685939 | 0.696415 | 0.706696 | 0.706503 |

S N R ( N ) S N R ( 4 ) | 1.000000 | 1.210155 | 1.313128 | 1.373100 | 1.411334 | 1.437097 | 1.455096 | 1.477320 | 1.499129 | 1.498721 |

S N R M / 2 ( N ) S N R ( 4 ) | 1.000000 | n.a. | 0.941683 | n.a. | 0.885150 | n.a. | 0.842198 | 0.808903 | 0.716911 | 0.711792 |

not be economical to use large N > 20 due to complexity and costs of the circuit. Yet, even small N = 6, 8, 10 has a notable improvement in SNR (21% to 37%) compared to the classical Hall plates with only four contacts.

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 c k ≠ 1 prior to summing them up.

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 [

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

V out , phase ( k ) = I supply ∑ j = k + 1 k + M − 1 R j , k − R j , k + M − R j + M , k + R j + M , k + M (24)

V out , spin = ∑ k = 1 M V out , phase ( k ) (25)

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

shows the final result when all indices are flipped back to the “primitive” interval [1, N − 1]. 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) [

R i , j ( B ⊥ ) − R j , i ( B ⊥ ) = R i , j ( B ⊥ ) − R i , j ( − B ⊥ ) (26)

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.

In a very general general case, a circular Hall plate may have N = 2M peripheral contacts. The N-th contact is grounded (see _{k} ( k = 1 , 2 , ⋯ , N − 1 ). M pairs of contacts are defined. A pair comprises contacts k and N + 1 − k for k = 1 , 2 , ⋯ , M − 1 . Voltmeters are connected to all contact pairs. Their voltage readings are multiplied by weighing factors c_{k}. These terms are summed up for all contact pairs to give the signal in a first operating phase. In total there are M operating phases, whereby the current sources and the voltmeters stay in place while the circular Hall plate rotates by one contact for each new operating phase. 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

R i , j = { R i , j ( B ⊥ ) if i ≤ j R j , i ( − B ⊥ ) if i > j (27)

which is the RMFR principle [

I N + 1 − k = I k for k ∈ [ 1 , M ] (28)

∑ k = 1 M I k = 0 (29)

(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.

V supply = max j = 1 N V j − min j = 1 N V j (30)

Note that from all N current sources only two have zero voltage across them—one 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.

P supply = V supply I supply (31)

I supply = ∑ j = 1 N If [ I j > 0 ; I j ; 0 ] = 1 2 ( | ∑ j = 1 N − 1 I j | + ∑ j = 1 N − 1 | I j | ) (32)

This gives us the SNR of a multi-port Hall plate with multiple input currents.

S N R = lim B ⊥ → 0 V out n out = μ H B ⊥ P supply 4 k b T Δ η N − 1 , M (33a)

η N − 1 , M = ∑ k = 1 M ∑ j = 1 M c k ( R j , k − R N + 1 − j , k ) I j P supply R out tan θ H (33b)

with the dimension-less noise efficiency η N − 1 , M for the operating mode with N − 1 input currents and M output voltages. The output resistance for the circuit in

R out = ∑ j = 1 M c j 2 ( ∑ k = 1 M R j , k − ∑ k = M + 1 N − 1 R j , k − ∑ k = 1 M R N + 1 − j , k + ∑ k = M + 1 N − 1 R N + 1 − j , k ) (34)

In (33b) and (34) we had to add an N-th row to the resistance matrix with R_{N}_{,k} = 0 for k = 1 , 2 , ⋯ , N − 1 . Analogous to (4a) we can define an average Hall geometry factor G_{H}.

G H = ∑ k = 1 M ∑ j = 1 M c k ( R j , k − R N + 1 − j , k ) I j M R sheet tan θ H I supply (35)

when searching for the optimum currents and weighing coefficients we can reduce their number due to the symmetry of the Hall plates at weak magnetic field. It holds

I M + 1 − k = − I k and c M + 1 − k = c k for k ∈ [ 1 , ⌊ M / 2 ⌋ ] . (36)

An algorithm is shown in Appendix F. The results of this optimization are shown in

N | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 |
---|---|---|---|---|---|---|---|---|---|---|

G_{H}_{0} | 0.33333 | 0.49098 | 0.58403 | 0.64605 | 0.69064 | 0.58209 | 0.62700 | 0.66259 | 0.69156 | 0.71563 |

V supply I supply | 0.70763 | 1.00058 | 1.19507 | 1.34224 | 1.46105 | 1.00088 | 1.09431 | 1.17497 | 1.24600 | 1.30955 |

R_{out} | 2.82843 | 5.66667 | 9.55583 | 14.5108 | 20.5387 | 27.6433 | 35.8270 | 45.0912 | 55.4370 | 66.8651 |

η N − 1 , M | 0.47123 | 0.61858 | 0.69130 | 0.73193 | 0.75645 | 0.77465 | 0.80109 | 0.81927 | 0.83209 | 0.84125 |

S N R ( N ) S N R ( 4 ) | 1 | 1.31270 | 1.46701 | 1.55324 | 1.60527 | 1.64388 | 1.69999 | 1.73859 | 1.76578 | 1.78522 |

I_{M}_{−}_{1} | −1 | 0 | 0 | 0 | 0 | 1.79641 | 1.76170 | 1.73860 | 1.72295 | 1.71180 |

N | 24 | 26 | 28 | 30 | 32 | 34 | 36 | 38 | 40 | |

G_{H}_{0} | 0.73599 | 0.75345 | 0.76861 | 0.78189 | 0.79365 | 0.73917 | 0.75226 | 0.76404 | 0.77469 | |

V supply I supply | 1.36709 | 1.41971 | 1.46819 | 1.51316 | 1.55510 | 1.29933 | 1.33729 | 1.37302 | 1.40679 | |

R_{out} | 79.3761 | 92.9704 | 107.648 | 123.410 | 140.256 | 158.186 | 177.201 | 197.300 | 218.484 | |

η N − 1 , M | 0.84784 | 0.85257 | 0.85593 | 0.85826 | 0.85982 | 0.87650 | 0.87963 | 0.88200 | 0.88376 | |

S N R ( N ) S N R ( 4 ) | 1.79920 | 1.80924 | 1.81637 | 1.82133 | 1.82462 | 1.86002 | 1.86666 | 1.87169 | 1.87543 | |

I_{M}_{−}_{1} | 1.70356 | 1.69727 | 1.69237 | 1.68846 | 1.68530 | 1.17899 | 1.17738 | 1.17604 | 1.17490 | |

I_{M}_{−}_{2} | 0 | 0 | 0 | 0 | 0 | 2.17131 | 2.16234 | 2.15487 | 2.14858 |

leads to potentials at all current input contacts, which are close to V_{supply}, and it leads to potentials at all current output contacts, which are close to ground. Conversely, the benefit of optimized weighing factors is tiny (see

For an experimental verification of our theory, we manufactured the two types of Hall plates in ^{15}/cm^{3} (three parts phosphorus and one part arsenic) and a metallurgical thickness of 1.5 µm. The Hall plates were pn-junction isolated with a roughly linear slope of the doping near the junction. At zero reverse bias voltage, the depletion layer reduced the effective thickness of the Hall plates to 0.9 … 1.0 µm. The contacts were made of standard CMOS n-wells and n^{+}-source/drain diffusions. Opposite contacts of the four-contacts device were spaced apart by 74.6 µm, and their size was 31 µm × 1.3 µm. Opposite contacts of the eight-contacts device were spaced apart by 72 µm and had a size of 11 µm × 9 µm. Unfortunately, we do not know the exact depth profile of the CMOS wells implanted through the small LOCOS openings of the n^{+} diffusions. This 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

Ohm, r_{3,N} = 96,680 Ohm, r_{4,N} = 89,165 Ohm at small supply voltage. Then we reconstructed the network with lumped resistors r_{1,N} = 6.2 kOhm, r_{2,N} = 47 kOhm, r_{3,N} = 100 kOhm, r_{4,N} = 92 kOhm. This gave a resistance of 6680 Ohm between opposite contacts. At 1 V supply voltage the Hall output signal of the four-contact device is 58 µV per milli-Tesla of magnetic flux density. The same supply voltage and magnetic flux density give 53.9 µV at ports 1 and 3 and 62.3 µV at port 2 (which is at cm = 0.5) of the eight-contacts Hall plate.

We built an electronic circuit according to the operating mode of _{1} = c_{2} = c_{3} = 11. The inputs of the three AD8429s were connected to the three ports of the eight-contacts Hall plate and the noise in the output of the circuit in

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

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

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 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

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

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.

We studied two circuits for multi-port Hall plates. One has a single input current and M − 1 output voltages (

currents and M output voltages (

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 [

One can readily apply our theory to SNR optimization and spinning schemes of Vertical Hall effect devices [

In this paper we have not dealt with important practical aspects of the circuits in

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 (

The author gratefully acknowledges the package assembly of the Hall plates, the build-up of the electronic circuit and resistor network, and the measurements reported in Section 7, which were all done by Michael Holliber.

The author declares no conflicts of interest regarding the publication of this paper.

Ausserlechner, U. (2020) 90% SNR Improvement with Multi-Port Hall Plates. Journal of Applied Mathematics and Physics, 8, 1568-1605. https://doi.org/10.4236/jamp.2020.88122

Here we compute the resistor network for a multi-port Hall plate of

Due to the symmetry of our multi-port Hall plates in Figure6 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 FigureA1, 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

( I 1 ⋮ I N − 1 ) = g ⋅ ( V supply ⋮ V supply ) with g = ( R even ) − 1 (A1)

it follows

r j , N = 1 ∑ k = 1 N − 1 g j , k (A2)

TableA1 gives numerical values for these resistances at vanishing magnetic

N | r_{1,}_{N} | r_{2,}_{N} | r_{3,}_{N} | r_{4,}_{N} | r_{5,}_{N} | r_{6,}_{N} | r_{7,}_{N} | r_{8,}_{N} | r_{9,}_{N} | r_{10,}_{N} |
---|---|---|---|---|---|---|---|---|---|---|

3 | 1.73205 | r_{1,}_{N} | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. |

4 | 2.00000 | 4.82843 | r_{1,}_{N} | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. |

5 | 2.12663 | 6.88191 | r_{2,}_{N} | r_{1,}_{N} | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. |

6 | 2.19615 | 8.19615 | 11.19615 | r_{2,}_{N} | r_{1,}_{N} | n.a. | n.a. | n.a. | n.a. | n.a. |

7 | 2.23833 | 9.06283 | 14.53565 | r_{3,}_{N} | r_{2,}_{N} | r_{1,}_{N} | n.a. | n.a. | n.a. | n.a. |

8 | 2.26582 | 9.65685 | 17.04789 | 20.10936 | r_{3,}_{N} | r_{2,}_{N} | r_{1,}_{N} | n.a. | n.a. | n.a. |

9 | 2.28471 | 10.07894 | 18.94221 | 24.72730 | r_{4,}_{N} | r_{3,}_{N} | r_{2,}_{N} | r_{1,}_{N} | n.a. | n.a. |

10 | 2.29825 | 10.38842 | 20.38842 | 28.47859 | 31.56876 | r_{4,}_{N} | r_{3,}_{N} | r_{2,}_{N} | r_{1,}_{N} | n.a. |

11 | 2.30828 | 10.62152 | 21.50956 | 31.51551 | 37.46256 | r_{5,}_{N} | r_{4,}_{N} | r_{3,}_{N} | r_{2,}_{N} | r_{1,}_{N} |

12 | 2.31591 | 10.80119 | 22.39230 | 33.98341 | 42.46870 | 45.57452 | r_{5,}_{N} | r_{4,}_{N} | r_{3,}_{N} | r_{2,}_{N} |

13 | 2.32186 | 10.94246 | 23.09767 | 36.00288 | 46.70167 | 52.74307 | r_{6,}_{N} | r_{5,}_{N} | r_{4,}_{N} | r_{3,}_{N} |

14 | 2.32658 | 11.05544 | 23.66900 | 37.66900 | 50.28257 | 59.01143 | 62.12672 | r_{6,}_{N} | r_{5,}_{N} | r_{4,}_{N} |

15 | 2.33039 | 11.14717 | 24.13755 | 39.05538 | 53.32123 | 64.46840 | 70.56945 | r_{7,}_{N} | r_{6,}_{N} | r_{5,}_{N} |

16 | 2.33351 | 11.22264 | 24.52615 | 40.21872 | 55.91128 | 69.21479 | 78.10392 | 81.22536 | r_{7,}_{N} | r_{6,}_{N} |

17 | 2.33610 | 11.28545 | 24.85174 | 41.20278 | 58.13026 | 73.34803 | 84.80086 | 90.94197 | r_{8,}_{N} | r_{7,}_{N} |

18 | 2.33827 | 11.33827 | 25.12707 | 42.04154 | 60.04154 | 76.95600 | 90.74480 | 99.74480 | 102.87047 | r_{8,}_{N} |

19 | 2.34010 | 11.38311 | 25.36186 | 42.76155 | 61.69666 | 80.11526 | 96.02143 | 107.69147 | 113.86076 | r_{9,}_{N} |

20 | 2.34167 | 11.42148 | 25.56362 | 43.38375 | 63.13752 | 82.89128 | 100.71141 | 114.85355 | 123.93336 | 127.06205 |

21 | 2.34302 | 11.45458 | 25.73821 | 43.92474 | 64.39823 | 85.33951 | 104.88786 | 121.30632 | 133.13604 | 139.32590 |

field. A comparison with published results shows perfect agreement: For regular Hall plates with three contacts (N = 3) Equation (9) in [

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 FigureA2 we connect all contacts to ground potential except contact 1, where we apply negative supply voltage. It holds

( I N I 1 ⋮ I N − 2 ) = V supply ( − ∑ j = 1 N − 1 r j , N − 1 r 1 , N − 1 ⋮ r N − 1 , N − 1 ) = g ⋅ ( − V supply 0 ⋮ 0 ) (A3)

where the currents I 1 , I 2 , ⋯ , I N are the currents flowing into contacts 1 , 2 , ⋯ , N in FigureA1. Comparison of (A1) and (A3) shows that g j , 1 = g j + 1 , 2 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

g = ( R even ) − 1 = ( g 11 − r 1 , N − 1 − r 2 , N − 1 − r 3 , N − 1 − r 4 , N − 1 ⋯ − r 1 , N − 1 g 11 − r 1 , N − 1 − r 2 , N − 1 − r 3 , N − 1 ⋯ − r 2 , N − 1 − r 1 , N − 1 g 11 − r 1 , N − 1 − r 2 , N − 1 ⋯ − r 3 , N − 1 − r 2 , N − 1 − r 1 , N − 1 g 11 − r 1 , N − 1 ⋯ − r 4 , N − 1 − r 3 , N − 1 − r 2 , N − 1 − r 1 , N − 1 g 11 ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ) (A4)

with g 11 = ∑ j = 1 N − 1 r j , N − 1 . Thus, g is a Toeplitz matrix with positive elements on the main diagonal and all other elements being negative. This holds also for the definite conductance matrix G = R^{−1} at arbitrary magnetic field. g has even symmetry and r j , N = r N − j , N .

These Hall plates can be readily treated with the theory of [

R in = R out = V supply I supply = 4 2 N R sheet (B1)

G H 0 = 2 3 R in R sheet = 8 3 N (B2)

Inserting (B2) into (1) gives the Hall output voltage. (B1) and (B2) also mean that the noise efficiency η is equal to the maximum one of Hall plates with four contacts.

η = G H 0 R in / R sheet R out / R sheet = 2 3 ≅ 0.471 (B3)

Hall plates from Figure5 can be mapped with conformal transformations to Hall plates from FigureC1(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 Figure5. 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 FigureC1(a) the MOS transistors act as current sources. The potential at one of the contacts M + 1 to N − 1 is maximal. If we neglect the saturation voltage of the PMOS transistors when their channels are pinched off, this maximum potential is also at the power supply. All other PMOS transistors on the k-th contact have non-vanishing drain-source voltages equal to max ( V j ) − V k , and therefore they dissipate the power ( max ( V j ) − V k ) ∗ I k for j and k from M + 1 to N − 1. The same applies to the NMOS transistors at the negative terminal of the power supply. The NMOS transistor on the k-th contact dissipates the power ( V k − min ( V j ) ) ∗ I k for j and k from 1 to M − 1. In other words, the power which we save in the Hall plate (a) is dissipated in the current sources! Thus, the system works suboptimally.

In FigureC1(b) we replace the Hall plate of FigureC1(a) by a circumscribed rectangle of length L and width W. All current contacts at positive supply are fused to a single large contact 3, and all current contacts at the negative supply are fused to a single large contact 1. The output contacts do not change. Due to the new shape we have shifted the power dissipation of the MOS transistors from (a) into the Hall plate (b). The entire circuit uses the same power, but only

in (b) the total power is available inside the Hall plate. Obviously, in FigureC1(b) big portions of contacts 1 and 3 are more distal to the output contacts than in FigureC1(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 [

Here we discuss the current mode operation of Hall plates from [_{a}, I_{b} to reference potential. The output signal I_{out} is the difference of the readings of both ampere-meters. Since the output contacts are tied to the very same reference potential via the low ohmic ampere-meters, we can skip half of the arrangement—this will double the output resistance and it halves the output signal and the supply voltage at identical supply current (see FigureD1(b)). The conductance matrix G = R^{−1} is given in (47) in [_{11}, G_{12}, and G_{21}. It holds

( I 1 I 2 I 3 ) = ( G 11 G 12 − G 11 − G 12 − G 21 G 21 G 11 G 12 − G 11 − G 12 − G 21 G 21 G 11 ) ⋅ ( 0 V supply / 2 V supply / 2 ) (D1)

with I supply = I 2 + I 3 and I out = − I 1 − ( I 1 + I 2 + I 3 ) it follows for FigureD1(b)

I out I supply = − G 12 + G 21 2 G 11 + G 12 + G 21 (D2)

If we repeat the same procedure for a conventional Hall plate operated the usual way (see

( 0 I supply 0 ) = ( G 11 G 12 − G 11 − G 12 − G 21 G 21 G 11 G 12 − G 11 − G 12 − G 21 G 21 G 11 ) ⋅ ( V 1 V supply V 3 ) (D3)

with V out = V 1 − V 3 it follows for

V out V supply = − G 12 + G 21 2 G 11 + G 12 + G 21 (D4)

The R.H.S.s of (D2) and (D4) are identical. Hence, the output signals given in per cent of supply quantities are identical for the current mode operation in FigureD1(b) and for the conventional Hall operation in Figure1. Of course the output signal doubles in FigureD1(a), because we send the same current through a second device and add their outputs. The supply voltage and the total power also double. Comparison of (D4) with (1) gives for weak magnetic field

V out V supply = I out I supply = G H 0 ( L / W ) e f f μ H B ⊥ (D5)

where we used the effective number of squares (L/W)_{eff} for the ratio of resistance between two opposite contacts over sheet resistance. G_{H}_{0} is missing in (3) in [_{H}_{0}/(L/W)_{eff}, which is 2 / 3 ≅ 0.471 . In silicon a phosphor doping of 2 × 10^{16}/cm^{3} gives a Hall mobility of 0.108/T. Therefore conventional silicon Hall plates with 90˚ symmetry have maximum magnetic field sensitivities of roughly 50 mV/V/T = 5%/T for both kinds of operation, voltage mode and current mode. Again, for FigureD1(a) this means 10%/T for the total circuit with two Hall plates. We have checked this also by 2D-FEM-simulations with COMSOL Multiphysics.

And how about SNR? At weak magnetic field the input resistance of a single device in current mode in FigureD1(b) is exactly half of the input resistance of the same device operated according to Figure1. This holds for contacts of arbitrary size. We can prove it with the resistor network in [_{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 FigureD1(a) and in Figure1 consume the same power at identical supply voltage V_{supply}. In FigureD1(a) the Hall output current I_{out} and the thermal noise output current I_{out,noise} are given by

I out = I a − I b = 2 G H 0 ( L / W ) e f f μ H B ⊥ V supply 2 R D + R H 2 R D R H (D6)

I out,noise = 2 4 k b T Δ 6 R D + R H 2 R D R H (D7)

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 FigureD1(a) is 2 R D R H / ( 6 R D + R H ) and it causes the thermal output noise current according to [^{(}^{current mode)} of the circuit in FigureD1(a). The SNR^{(}^{conventional)} of a conventional Hall plate with four contacts is given by (23). The ratio of both is

S N R ( currentmode ) S N R ( conventional ) = 2 R D + R H 6 R D + R H < 1 (D8)

which shows that at the same power consumption the SNR of the circuit in FigureD1(a) operated in current mode is smaller than the SNR of the same Hall plate operated in a conventional way like in Figure1. The conventional Hall plate circuit has maximum SNR for R H = r 1 , 4 = 2 R sheet and 2 R D = r 2 , 4 = 2 ( 1 + 2 ) R sheet ≅ 4.828 R sheet (see TableA1 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 [

FigureD2 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.

In Sections 2-5 all amplifiers in _{k} by which we multiply the output signals of the ports prior to summing them up? With (4a) we get the total output voltage

V out = ∑ k = 1 M − 1 c k V out , k = R sheet tan ( θ H ) I supply ∑ k = 1 M − 1 c k G H , k . (E1)

The output resistance also changes. We can recur to _{k} instead of the original 1:1. If output current I_{out} is injected into the secondary side of the transformers, it gives c_{k} × I_{out} flowing out of the primary sides. Multiplication of these currents with the resistance matrix gives the port voltages. They are present at the primary sides of the transformers, and they will be amplified again by the factor c_{k} to the secondary side. Thus, the output resistance of port k will appear c k 2 larger in the total output of the circuit, whereas the Hall signal is amplified only by c_{k}.

R out = V out I out = ∑ k = 1 M − 1 c k 2 ( ∑ j = 1 M − 1 R k , j + ∑ j = M + 1 N − 1 R N − k , j − ∑ j = 1 M − 1 R N − k , j − ∑ j = M + 1 N − 1 R k , j ) (E2)

with (23) the SNR is proportional to

∑ k = 1 M − 1 c k G H , k ∑ k = 1 M − 1 c k 2 ( ∑ j = 1 M − 1 R k , j + ∑ j = M + 1 N − 1 R N − k , j − ∑ j = 1 M − 1 R N − k , j − ∑ j = M + 1 N − 1 R k , j ) , (E3)

where we skipped all terms which are independent of the weighing factors c_{k}. We look for specific values of all c_{k} which maximize (E3). First we note from (E3) that once we have an optimum solution for all c_{k} we can multiply them with a constant factor without changing (E3). This means that for a unique solution we need to set c_{1} = 1. We determine the other M − 2 weighing factors by taking the square of (E3), differentiating it with respect to c l , and setting the result equal to zero. This gives a system of M − 2 second order algebraic equations for c l ( 2 ≤ l ≤ M − 1 ).

G H , l ∑ k = 1 M − 1 c k 2 ( ∑ j = 1 M − 1 R k , j + ∑ j = M + 1 N − 1 R N − k , j − ∑ j = 1 M − 1 R N − k , j − ∑ j = M + 1 N − 1 R k , j ) = c l ( ∑ j = 1 M − 1 R l , j + ∑ j = M + 1 N − 1 R N − l , j − ∑ j = 1 M − 1 R N − l , j − ∑ j = M + 1 N − 1 R l , j ) ∑ k = 1 M − 1 c k G H , k (E4)

The numerical solution is straightforward (e.g. with Mathematica), if we use different starting values smaller than 1 for all c_{k}. TableE1 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 TableE1 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.

N | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 26 | 32 | 40 |
---|---|---|---|---|---|---|---|---|---|---|

η 1 , M − 1 | 0.62012 | 0.65014 | 0.67025 | 0.68463 | 0.69539 | 0.70372 | 0.71035 | 0.72388 | 0.73203 | 0.73873 |

ratio | 1.00179 | 1.00441 | 1.00743 | 1.01059 | 1.01378 | 1.01693 | 1.02001 | 1.02867 | 1.03646 | 1.04561 |

c_{2} | 0.88232 | 0.82854 | 0.79709 | 0.77611 | 0.76094 | 0.74935 | 0.74013 | 0.72085 | 0.70843 | 0.69715 |

c_{3} | c_{1} | c_{2} | 0.75000 | 0.70411 | 0.67360 | 0.65161 | 0.63487 | 0.60176 | 0.58162 | 0.56405 |

c_{4} | n.a. | c_{1} | c_{2} | c_{3} | 0.64982 | 0.61377 | 0.58783 | 0.53988 | 0.51262 | 0.48983 |

c_{5} | n.a. | n.a. | c_{1} | c_{2} | c_{3} | c_{4} | 0.57406 | 0.50681 | 0.47157 | 0.44353 |

c_{6} | n.a. | n.a. | n.a. | c_{1} | c_{2} | c_{3} | c_{4} | 0.49214 | 0.44701 | 0.41309 |

c_{7} | n.a. | n.a. | n.a. | n.a. | c_{1} | c_{2} | c_{3} | c_{6} | 0.43378 | 0.39284 |

c_{8} | n.a. | n.a. | n.a. | n.a. | n.a. | c_{1} | c_{2} | c_{5} | 0.42960 | 0.37987 |

c_{9} | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | c_{1} | c_{4} | c_{7} | 0.37261 |

c_{10} | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | c_{3} | c_{6} | 0.37027 |

FigureF1 shows a Mathematica script to search for the optimum supply currents and weighing coefficients for Hall plates with N = 18 contacts operated in mode “multiple input currents—multiple output voltages”. The symmetry of the supply currents is needed for the spinning scheme. The symmetry of the weighing coefficients is a consequence. The results show that currents into contacts 3, 4, 5, 6, 7, 12, 13, 14, 15, 16 vanish. All currents are independent of the weighing coefficients. The noise efficiencies are 0.819275 and 0.829779, respectively.