_{1}

^{*}

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

This paper focuses on impedances, magnetic sensitivity, and thermal noise of Hall plates with four contacts at small magnetic field. The specific question we want to answer is, how much does the signal to noise ratio (SNR) deteriorate for devices with reduced symmetry. This is a matter of layout and shape―not of technology. Conventional Hall plates are 90˚ symmetric (

have at least two perpendicular mirror symmetries (^{1}, and they offer still lower zero point errors when operated in spinning current schemes [

Hall plates with four extended contacts and four-fold symmetry with equal input and output resistance R i n = R o u t are discussed in [

the Hall plate, which is an out-of-plane parameter. Conversely, λ i n describes the lateral geometry of the Hall plate in the plane of the current flow. λ i n is an in-plane parameter. Although Vertical Hall effect devices of

The magnetic sensitivity of a Hall plate at weak magnetic field is defined as change in output voltage V o u t per change in flux density B ⊥ orthogonal to the Hall plate [

S 0 = lim B ⊥ → 0 d V o u t d B ⊥ = μ H G H 0 λ i n V i n = μ H R sh G H 0 I i n (1)

with the Hall mobility μ H , the Hall input or supply voltage V i n , and the Hall input or supply current I i n . The low field Hall geometry factor G H 0 depends only on the in-plane geometry of the Hall plate, just like λ i n does. It holds 0 ≤ G H 0 ≤ 1 . G H 0 can be expressed as a function of geometrical parameters of the Hall plate, but it can also be expressed more generally as a function of the dimensionless DoFs. Thus, for a symmetric Hall plate with R i n = R o u t the Hall geometry factor G H 0 is a function of the in-plane parameter λ i n , not of the out-of-plane parameter R sh [

In [_{1}-C_{3} and C_{2}-C_{4}, respectively) are at half of the input voltage: ( V 1 + V 3 ) / 2 = ( V 2 + V 4 ) / 2 for current flowing between contacts C_{2} and C_{4}, or between contacts C_{1} and C_{3}. Thereby, the potential at contact C_{n} is denoted by V_{n} for n = 1, 2, 3, 4. The ERC of these Hall plates has three resistances R H , R D 1 , R D 2 as in

The topic of this paper is Hall plates with single mirror symmetry whereby the symmetry axis goes through the centers of two opposite contacts C_{1} and C_{3} like in

c m = ( V 1 + V 3 ) / 2 − V 2 V 4 − V 2 (2)

for current flowing from C_{4} to C_{2}. Generally, it holds 0 ≤ c m ≤ 1 . Devices with two perpendicular mirror symmetries have c m = 1 / 2 . The ERC of devices with only a single mirror symmetry has six resistors with four resistances: R H , γ R H , R D 1 , R D 2 with c m = γ / ( 1 + γ ) , λ f R sh = R H ( 1 + γ ) / 2 | | ( 2 R D 2 ) , λ p R sh = ( 2 R H γ ) | | ( 2 R H ) | | ( 2 R D 1 ) . The low field Hall-geometry factor is a function of three in-plane parameters G H 0 = G H 0 ( λ f , λ p , c m ) .

Amongst all useful Hall plates with zero output signal at zero magnetic field there are some with no mirror symmetry at all (see

In the most general case a resistive device with four terminals has an ERC composed of 3 + 2 + 1 = 6 resistors: between each couple of terminals there is one resistor, as was shown in [

R sh = R sh ( R 1 , R 2 , ⋯ , R 6 ) (3)

Moreover, if we normalize R 2 / R sh , ⋯ , R 6 / R sh by R 1 / R sh we do not lose any DoF. Therefore, the set

{ R 2 / R sh R 1 / R sh , R 3 / R sh R 1 / R sh , ⋯ , R 6 / R sh R 1 / R sh } = { R 2 R 1 , R 3 R 1 , ⋯ , R 6 R 1 } (4)

also covers the five in-plane DoF and finally it holds

G H 0 = G H 0 ( R 2 / R 1 , R 3 / R 1 , ⋯ , R 6 / R 1 ) . (5)

The low field Hall geometry factor is a function of ratios of resistances of the ERC. Inserting (3) and (5) into (1) leads to the remarkable conclusion that at small Hall angles the output voltage is fully determined by the ERC, the Hall angle, and the Hall supply current or voltage. We do not need any information on the geometry of the Hall plate. The electrical parameters of the ERC fully determine the sensitivity of the Hall plate output signal with respect to changes in the Hall angle. A similar conclusion was drawn in [

The rest of the paper mainly elaborates on the relation G H 0 = G H 0 ( λ f , λ p , c m ) for Hall plates with four contacts of finite size and only one mirror symmetry having ERCs like in

First we compute the resistance between the flush contact pair and the common mode potential of the partial contacts―both at zero magnetic field. To this end we map the rectangular plate in the z-plane of

transformation onto the upper half of the zeta plane and from this into another rectangle in the w-plane. The partial contacts in the w-plane are parallel to lines of constant potential, which greatly facilitates the calculation of their potential and the calculation of the resistance between the other contacts.

A mapping from the ζ-plane onto the z-plane is given by Schwartz-Christoffel’s formula

z = C 1 ∫ 0 ζ d ζ ζ 2 − 1 ζ 2 − ζ 6 2 . (6)

Applying (6) to the width of the rectangle in the z-plane gives

W 2 = C 1 ∫ 0 1 d ζ ζ 2 − 1 ζ 2 − ζ 6 2 = C 1 ζ 6 K ( 1 ζ 6 ) . (7)

The complete elliptic integral K is defined in Appendix A. Applying (6) to the length of the rectangle in the z-plane gives

l = C 1 ∫ 1 ζ 6 d ζ ζ 2 − 1 ζ 6 2 − ζ 2 = C 1 ζ 6 K ′ ( 1 ζ 6 ) (8)

K ′ is the complementary complete elliptic integral defined in Appendix A. Dividing (8) by (7) gives

2 l W = K ′ ( 1 / ζ 6 ) K ( 1 / ζ 6 ) . (9)

Equation (9) defines ζ 6 . Applying (6) to Z 5 Z 6 ¯ and using (61) in [

b = C 1 ( ∫ 1 ζ 6 d ζ ζ 2 − 1 ζ 6 2 − ζ 2 − ∫ 1 ζ 5 d ζ ζ 2 − 1 ζ 6 2 − ζ 2 ) = C 1 ζ 6 F ( ζ 6 2 − ζ 5 2 ζ 6 2 − 1 , 1 − 1 ζ 6 2 ) .

(10)

with the incomplete elliptic integral F defined in Appendix A. (10) can be solved for ζ 5 by use of the Jacobi-sn function (see Appendix A)

ζ 5 = ζ 6 1 − ( 1 − 1 ζ 6 2 ) sn 2 ( b l K ′ ( 1 ζ 6 ) , 1 − 1 ζ 6 2 ) . (11)

In an analogous way one can apply (6) to Z 3 Z 6 ¯ . This gives

ζ 3 = ζ 6 1 − ( 1 − 1 ζ 6 2 ) sn 2 ( b + s l K ′ ( 1 ζ 6 ) , 1 − 1 ζ 6 2 ) . (12)

From (12) one can compute b + s from which one can subtract (10). With [

s l = 1 K ′ ( 1 ζ 6 ) F ( ζ 6 ζ 5 ζ 5 2 − 1 ζ 6 2 − ζ 3 2 − ζ 3 ζ 3 2 − 1 ζ 6 2 − ζ 5 2 ζ 6 2 ( ζ 3 2 + ζ 5 2 − 1 ) − ζ 3 2 ζ 5 2 , 1 − 1 ζ 6 2 ) . (13)

From (11), (12), and [

2 b + s l = 1 K ′ ( 1 ζ 6 ) F ( ζ 6 ζ 5 ζ 5 2 − 1 ζ 6 2 − ζ 3 2 + ζ 3 ζ 3 2 − 1 ζ 6 2 − ζ 5 2 ζ 6 2 ( ζ 3 2 + ζ 5 2 − 1 ) − ζ 3 2 ζ 5 2 , 1 − 1 ζ 6 2 ) . (14)

For the symmetric case c m = 1 / 2 it holds 2 b + s = l . Then the first argument in the incomplete elliptic integral in (14) is equal to unity. This gives

ζ 3 ζ 5 = ζ 6 for c m = 1 / 2 . (15)

By now we have expressed all parameters defining the contacts in the ζ-plane by parameters of the z-plane. Turning to the mapping from the ζ-plane onto the w-plane in

w = C 2 ∫ 0 ζ ζ 2 − ζ 4 2 ζ − 1 ζ − ζ 3 ζ − ζ 5 ζ − ζ 6 ζ + ζ 6 ζ + ζ 5 ζ + ζ 3 ζ + 1 d ζ . (16)

Note that ζ − 1 ζ + 1 ≠ ζ 2 − 1 , because for ζ = − 2 we get ζ − 1 ζ + 1 → − 3 − 1 = i 2 3 = − 3 which is different from ζ 2 − 1 → 3 . The integral (16) is more difficult than (6). It contains additional factors which are caused by the 90˚ corners at points W 3 , W 5 and by the 180˚ turn at point W 4 (plus their symmetric counterparts at W 8 , W 9 , W 10 ).

The parameter ζ 4 = − ζ 9 is defined by the requirement W 3 W 4 ¯ = W 4 W 5 ¯ or W 3 W 5 ¯ = 0 which means that the contacts are folded in such a way that points W 3 and W 5 become identical. This gives

0 = ∫ ζ 3 ζ 5 ζ 2 − ζ 4 2 ζ 2 − ζ 3 2 ζ 5 2 − ζ 2 d ζ ζ 2 − 1 ζ 6 2 − ζ 2 . (17)

Equation (17) can be solved to give the parameter ζ 4

ζ 4 2 = ∫ ζ 3 ζ 5 ζ 2 d ζ ζ 2 − 1 ζ 2 − ζ 3 2 ζ 5 2 − ζ 2 ζ 6 2 − ζ 2 ∫ ζ 3 ζ 5 d ζ ζ 2 − 1 ζ 2 − ζ 3 2 ζ 5 2 − ζ 2 ζ 6 2 − ζ 2 . (18)

With the abbreviations (B4) defined in Appendix B Equation (18) becomes

ζ 4 2 = G 2 / G 0 . (19)

The number of squares between the flush contacts is defined as the resistance between these contacts divided by the sheet resistance. With

λ f = W 2 W 6 ¯ W 1 W 2 ¯ = W 2 W 3 ¯ + W 5 W 6 ¯ W 1 W 2 ¯ = ∫ 1 ζ 3 ( ζ 4 2 − ζ 2 ) g ( ζ ) d ζ + ∫ ζ 5 ζ 6 ( ζ 2 − ζ 4 2 ) g ( ζ ) d ζ 2 ∫ 0 1 ( ζ 4 2 − ζ 2 ) g ( ζ ) d ζ . (20)

For the common-mode potential of the partial contacts W 3 W 4 W 5 ¯ at zero magnetic field we assume that contact W 6 W 7 ¯ is at potential V i n and contact W 1 W 2 ¯ is at zero potential, i.e. ground potential. From

1 c m = W 2 W 6 ¯ W 2 W 3 ¯ = 1 + W 5 W 6 ¯ W 2 W 3 ¯ = 1 + ∫ ζ 5 ζ 6 ( ζ 2 − ζ 4 2 ) g ( ζ ) d ζ ∫ 1 ζ 3 ( ζ 4 2 − ζ 2 ) g ( ζ ) d ζ . (21)

With the functions defined in Appendix B we may write (20), (21) like this

c m 1 − c m = − G ¯ 2 G 0 + G ¯ 0 G 2 G ¯ ¯ 2 G 0 − G ¯ ¯ 0 G 2 , (22)

2 λ f c m = G ¯ 2 G 0 − G ¯ 0 G 2 G ′ 2 G 0 − G ′ 0 G 2 . (23)

For large impedance between the two contacts on the axis of mirror symmetry it follows from (9)

l / W → ∞ ⇔ ζ 6 → ∞ ⇔ λ f → ∞ ( for ζ 3 > 1 ∨ ζ 5 < ζ 6 ) . (24)

For ζ 3 → 1 ∧ ζ 5 → ζ 6 all contacts touch and inputs and outputs are shorted.

We compute the resistance λ p between the partial contacts at zero magnetic field. For this purpose we map the interior of the rectangle in

w = C 5 ∫ 0 ζ ζ d ζ ζ − 1 ζ − ζ 3 ζ − ζ 5 ζ − ζ 6 ζ + ζ 6 ζ + ζ 5 ζ + ζ 3 ζ + 1 (25)

The center of the folded bottom contact W 1 W 2 ¯ is mapped to the origin of the ζ-plane, whereas the center of the folded top contact W 6 W 7 ¯ is mapped to infinity of the ζ-plane. Therefore it does not show up in the integrand of the Schwartz-Christoffel integral (see also [

λ p = 2 W 2 W 3 ¯ W 3 W 5 ¯ = 2 ∫ 1 ζ 3 ζ g ( ζ ) d ζ ∫ ζ 3 ζ 5 ζ g ( ζ ) d ζ = 2 G ¯ 1 G 1 . (26)

In (26) both integrals can be solved explicitly as functions of ζ 3 , ζ 5 , ζ 6 (see Appendix B). Thus, in general the three parameters ζ 3 , ζ 5 , ζ 6 are linked via λ p

L ( 2 λ p ) = ζ 6 2 − ζ 5 2 ζ 6 2 − ζ 3 2 ζ 3 2 − 1 ζ 5 2 − 1 . (27)

The modular lambda function L is defined in Appendix A. The resistance between the partial contacts gets large for (see (13) or

s / l → 0 ⇔ ζ 3 → ζ 5 ⇔ λ p → ∞ ( for ζ 6 > 1 ) . (28)

c m < 1 / 2 ⇔ G ′ 1 > G ″ 1 ⇔ ζ 3 ζ 5 < ζ 6 (29)

c m > 1 / 2 ⇔ G ′ 1 < G ″ 1 ⇔ ζ 3 ζ 5 > ζ 6 (30)

Obviously the transformation c m → 1 − c m keeps λ f , λ p , G H 0 constant, while it swaps G ′ 1 with G ″ 1 . This corresponds to the transformation ( ζ 3 , ζ 5 , ζ 6 ) → ( ζ 6 / ζ 5 , ζ 6 / ζ 3 , ζ 6 ) , which has the fix point ζ 6 = ζ 3 ζ 5 for symmetric devices with c m = 1 / 2 .

For c m = 1 / 2 it holds Z 4 = W / 2 + i l / 2 according to

ζ 4 = ζ 6 for c m = 1 / 2 . (31)

Such a device can be mapped onto a disk with two perpendicular mirror symmetries like in

λ f = K ′ ( ζ 3 + ζ 5 1 + ζ 3 ζ 5 ) K ( ζ 3 + ζ 5 1 + ζ 3 ζ 5 ) (32)

λ p = K ′ ( ζ 3 − ζ 5 ζ 3 + ζ 5 1 + ζ 3 ζ 5 1 − ζ 3 ζ 5 ) K ( ζ 3 − ζ 5 ζ 3 + ζ 5 1 + ζ 3 ζ 5 1 − ζ 3 ζ 5 ) (33)

Inverting (32), (33) gives ζ 3 , ζ 5 as functions of λ f , λ p . With L f = L ( λ f ) , L p = L ( λ p ) we get for c m = 1 / 2 :

ζ 3 = 1 + L f L p + ( 1 − L f ) ( 1 − L f L p ) L f ( 1 + L p ) (34)

ζ 5 = 1 − L f L p + ( 1 − L f ) ( 1 − L f L p ) L f ( 1 − L p ) (35)

For c m = 1 / 2 ∧ λ f = λ p it holds α 1 = α 2 in

ζ 6 = ( ζ 5 + ζ 3 ) / ( ζ 5 − ζ 3 ) . (36)

Inserting (15) into (36) gives

ζ 6 = ζ 5 ζ 3 = 1 4 ( 1 + ζ 3 + ζ 3 2 + 6 ζ 3 + 1 ) 2 (37)

which holds for devices with 90˚ symmetry.

Here we consider only the case when current flows between the original partial contacts off the symmetry axis and the output voltage is tapped between the original flush contacts on the axis of symmetry. At magnetic field the rectangle in

A Schwartz-Christoffel mapping from the upper half-plane of

w = C ″ 3 ∫ 0 ζ ( ζ − ζ 12 ) ( ζ − ζ 67 ) d ζ ( ζ − 1 ) 1 / 2 − m ( ζ − ζ 3 ) 1 / 2 + m ( ζ − ζ 5 ) 1 / 2 − m ( ζ − ζ 6 ) 1 / 2 + m ( ζ + ζ 6 ) 1 / 2 − m ( ζ + ζ 5 ) 1 / 2 + m ( ζ + ζ 3 ) 1 / 2 − m ( ζ + 1 ) 1 / 2 + m + C ″ 4 . (38)

For small magnetic field it holds | m | ≪ 1 and we may approximate x m = exp ( m ln x ) ≅ 1 + m ln x . ζ 12 and ζ 67 are determined by the requirements ∫ − 1 1 ⋯ = 0 and ∫ ζ 6 ∞ ⋯ + ∫ − ∞ − ζ 6 ⋯ = 0 with integrands like in (38). With the abbreviations in Appendix B this leads to the equations

G ′ 2 + ζ 12 ζ 67 G ′ 0 − m H ′ 1 ( ζ 12 + ζ 67 ) = 0 (39)

G ″ 2 + ζ 12 ζ 67 G ″ 0 − m H ″ 1 ( ζ 12 + ζ 67 ) = 0 (40)

with the solutions

ζ 12 ζ 67 = G ″ 2 H ′ 1 − G ′ 2 H ″ 1 G ′ 0 H ″ 1 − G ″ 0 H ′ 1 (41)

ζ 12 + ζ 67 = 1 m G ′ 0 G ″ 2 − G ″ 0 G ′ 2 G ′ 0 H ″ 1 − G ″ 0 H ′ 1 . (42)

From (38) one gets

w 3 − w 5 = C ″ 3 ∫ ζ = ζ 5 ζ 3 ( ζ − ζ 12 ) ( ζ − ζ 67 ) ζ 2 − 1 ζ 2 − ζ 3 2 ζ 2 − ζ 5 2 ζ 2 − ζ 6 2 { 1 + m [ ln ( ζ − 1 ) − ln ( ζ + 1 ) + ln ( ζ + ζ 3 ) − ln ( ζ − ζ 3 ) + ln ( ζ − ζ 5 ) − ln ( ζ + ζ 5 ) + ln ( ζ + ζ 6 ) − ln ( ζ − ζ 6 ) ] } d ζ (43)

which is identical to

C ″ 3 ∫ ζ = ζ 5 ζ 3 ζ 2 − ( ζ 12 + ζ 67 ) ζ + ζ 12 ζ 67 i 2 ζ 2 − 1 ζ 2 − ζ 3 2 ζ 5 2 − ζ 2 ζ 6 2 − ζ 2 { 1 + m [ ln ( ζ − 1 ζ + 1 ζ + ζ 3 ζ − ζ 3 ) + i π + ln ( ζ 5 − ζ ) − ln ( ζ + ζ 5 ) + ln ( ζ + ζ 6 ) − i π − ln ( ζ − ζ 6 ) ] } d ζ = − C ″ 3 ∫ ζ = ζ 3 ζ 5 ζ 2 − ( ζ 12 + ζ 67 ) ζ + ζ 12 ζ 67 i 2 ( g ( ζ ) + m h ( ζ ) ) d ζ (44)

The arbitrary scaling w 3 − w 5 = 1 gives the real valued constant C ″ 3 .

C ″ 3 = 1 G 2 − ( ζ 12 + ζ 67 ) G 1 + ζ 12 ζ 67 G 0 + m ( H 2 − ( ζ 12 + ζ 67 ) H 1 + ζ 12 ζ 67 H 0 ) (45)

C ″ 3 → 0 for m → 0 due to (42). Next we compute the potential at the sense contacts in

w 1 − w 10 = C ″ 3 ∫ ζ = − ζ 3 − 1 ( ζ − ζ 12 ) ( ζ − ζ 67 ) ζ + ζ 6 ζ + ζ 5 ζ + ζ 3 ζ + 1 ζ − 1 ζ − ζ 3 ζ − ζ 5 ζ − ζ 6 × { 1 + m [ ln ( ζ − 1 ) − ln ( ζ + 1 ) + ln ( ζ + ζ 3 ) − ln ( ζ − ζ 3 ) + ln ( ζ − ζ 5 ) − ln ( ζ + ζ 5 ) + ln ( ζ + ζ 6 ) − ln ( ζ − ζ 6 ) ] } d ζ

= C ″ 3 ∫ ζ = − ζ 3 − 1 ζ 2 − ( ζ 12 + ζ 67 ) ζ + ζ 12 ζ 67 i 5 ζ + ζ 6 ζ + ζ 5 ζ + ζ 3 − ζ − 1 − ζ + 1 − ζ + ζ 3 − ζ + ζ 5 − ζ + ζ 6 × { 1 + i m π ( 1 − 1 − 1 + 1 − 1 ) + m [ ln ( − ζ + 1 ) − ln ( − ζ − 1 ) + ln ( ζ + ζ 3 ) − ln ( − ζ + ζ 3 ) + ln ( − ζ + ζ 5 ) − ln ( ζ + ζ 5 ) + ln ( ζ + ζ 6 ) − ln ( − ζ + ζ 6 ) ] } d ζ

= C ″ 3 ∫ ζ = 1 ζ 3 ζ 2 + ( ζ 12 + ζ 67 ) ζ + ζ 12 ζ 67 i − ζ + ζ 6 − ζ + ζ 5 − ζ + ζ 3 ζ − 1 ζ + 1 ζ + ζ 3 ζ + ζ 5 ζ + ζ 6 × { 1 − i m π + m [ ln ( ζ + 1 ) − ln ( ζ − 1 ) + ln ( − ζ + ζ 3 ) − ln ( ζ + ζ 3 ) + ln ( ζ + ζ 5 ) − ln ( − ζ + ζ 5 ) + ln ( − ζ + ζ 6 ) − ln ( ζ + ζ 6 ) ] } d ζ = C ″ 3 ∫ ζ = 1 ζ 3 ζ 2 + ( ζ 12 + ζ 67 ) ζ + ζ 12 ζ 67 i [ ( 1 − i m π ) g ( ζ ) − m h ( ζ ) ] d ζ = ( − i − m π ) ( G ¯ 2 + ( ζ 12 + ζ 67 ) G ¯ 1 + ζ 12 ζ 67 G ¯ 0 ) + i m ( H ¯ 2 + ( ζ 12 + ζ 67 ) H ¯ 1 + ζ 12 ζ 67 H ¯ 0 ) G 2 − ( ζ 12 + ζ 67 ) G 1 + ζ 12 ζ 67 G 0 + m ( H 2 − ( ζ 12 + ζ 67 ) H 1 + ζ 12 ζ 67 H 0 )

(46)

Thus, the potential at this sense contact 12 ¯ is

Im { w 1 − w 10 } = − ( G ¯ 2 + ( ζ 12 + ζ 67 ) G ¯ 1 + ζ 12 ζ 67 G ¯ 0 ) + m ( H ¯ 2 + ( ζ 12 + ζ 67 ) H ¯ 1 + ζ 12 ζ 67 H ¯ 0 ) G 2 − ( ζ 12 + ζ 67 ) G 1 + ζ 12 ζ 67 G 0 + m ( H 2 − ( ζ 12 + ζ 67 ) H 1 + ζ 12 ζ 67 H 0 ) = λ p 2 + 2 ( G ¯ 2 − ( ζ 12 + ζ 67 ) m H ¯ 1 + ζ 12 ζ 67 G ¯ 0 ) + λ p ( G 2 − ( ζ 12 + ζ 67 ) m H 1 + ζ 12 ζ 67 G 0 ) 2 G 1 ( ζ 12 + ζ 67 ) m m + O ( m ) 2 (47)

where we used (26) and neglected higher powers in m. At zero magnetic field the potential is λ p / 2 , which is half the supply voltage. Hence, the change in potential at sense contact 12 ¯ caused by small magnetic field is d V o u t = Im { w 1 − w 10 } − λ p / 2 . With μ H d B ⊥ = tan β and (1) we define the low field Hall geometry factor for this single sense contact

G H 0 , 12 = − Im { w 1 − w 10 } − λ p / 2 R sh I i n tan β . (48)

In (48) the negative sign in front of the fraction is needed to make the Hall geometry factor positive, even though the potential at this contact 12 ¯ decreases with rising field according to

The Hall supply current is equal to the integral of the current density component perpendicular to the contact along the contact: I i n = t H ∫ W 5 W 3 d J v = t H J ( w 3 − w 5 ) cos β with J being the homogeneous current density. Computing the scalar product of both sides of general Ohm’s law J = σ 0 E − μ H J × B with J we get J = σ 0 E cos β ( J , E , B are the vector fields of current density, electric field and magnetic flux density with J = | J | and E = | E | ). Finally, E = 1 in

I i n = ( w 3 − w 5 ) σ 0 t H cos 2 β (49)

with w 3 − w 5 = 1 from (45) [

G H 0 , 12 = 1 π G 1 { − G ′ 0 G ¯ 2 − G ′ 2 G ¯ 0 G ′ 0 G 2 − G ′ 2 G 0 H 1 + H ¯ 1 − 2 λ f c m H ′ 1 } . (50)

For the other sense contact 67 ¯ we get

w 7 − w 8 = C ″ 3 ∫ ζ = − ζ 5 − ζ 6 ( ζ − ζ 12 ) ( ζ − ζ 67 ) ζ + ζ 6 ζ + ζ 5 ζ + ζ 3 ζ + 1 ζ − 1 ζ − ζ 3 ζ − ζ 5 ζ − ζ 6 × { 1 + m [ ln ( ζ − 1 ) − ln ( ζ + 1 ) + ln ( ζ + ζ 3 ) − ln ( ζ − ζ 3 ) + ln ( ζ − ζ 5 ) − ln ( ζ + ζ 5 ) + ln ( ζ + ζ 6 ) − ln ( ζ − ζ 6 ) ] } d ζ

= − C ″ 3 ∫ ζ = ζ 5 ζ 6 ζ 2 + ( ζ 12 + ζ 67 ) ζ + ζ 12 ζ 67 i 7 { g ( ζ ) [ 1 + i m π ( 1 − 1 + 1 − 1 + 1 − 1 − 1 ) ] − m h ( ζ ) } d ζ = ( G ¯ ¯ 2 + ( ζ 12 + ζ 67 ) G ¯ ¯ 1 + ζ 12 ζ 67 G ¯ ¯ 0 ) ( − i − m π ) + i m ( H ¯ ¯ 2 + ( ζ 12 + ζ 67 ) H ¯ ¯ 1 + ζ 12 ζ 67 H ¯ ¯ 0 ) G 2 − ( ζ 12 + ζ 67 ) G 1 + ζ 12 ζ 67 G 0 + m ( H 2 − ( ζ 12 + ζ 67 ) H 1 + ζ 12 ζ 67 H 0 ) (51)

Im { w 7 − w 8 } = λ p 2 + 2 ( G ¯ ¯ 2 − ( ζ 12 + ζ 67 ) m H ¯ ¯ 1 + ζ 12 ζ 67 G ¯ ¯ 0 ) + λ p ( G 2 − ( ζ 12 + ζ 67 ) m H 1 + ζ 12 ζ 67 G 0 ) 2 G 1 ( ζ 12 + ζ 67 ) m m + O ( m ) 2 (52)

We define

G H 0 , 67 = Im { w 7 − w 8 } − λ p / 2 R sh I i n tan β , (53)

which gives

G H 0 , 67 = 1 π G 1 { G ′ 0 G ¯ ¯ 2 − G ′ 2 G ¯ ¯ 0 G ′ 0 G 2 − G ′ 2 G 0 H 1 − H ¯ ¯ 1 − 2 λ f ( 1 − c m ) H ′ 1 } . (54)

Obviously, (50) and (54) are different: The magnetic sensitivity of the Hall signal at the two contacts on the axis of mirror symmetry is different. For c m > 1 / 2 more current is shunted via contact 67 ¯ than via 12 ¯ , and therefore G H 0 , 12 > G H 0 , 67 . Thus, the output contact that is closer to a supply contact has smaller Hall signal. It also follows that contacts on the axis of mirror symmetry have a common mode of ½ at zero magnetic field, but their common mode changes when magnetic field is applied. It depends on the sign of the magnetic field if the common mode increases or decreases.

The total Hall geometry factor is the sum of Hall geometry factors of both output contacts

G H 0 = G H 0 , 12 + G H 0 , 67 = 1 π G 1 { G ′ 0 ( G ¯ ¯ 2 − G ¯ 2 ) − G ′ 2 ( G ¯ ¯ 0 − G ¯ 0 ) G ′ 0 G 2 − G ′ 2 G 0 H 1 + H ¯ 1 − H ¯ ¯ 1 − 2 λ f H ′ 1 } (55)

With (B23), (B24), and (B41) in Appendix B we finally get

G H 0 = 1 π G 1 { H ¯ 1 − H ¯ ¯ 1 + ( G ¯ ¯ 2 − G ¯ 2 ) ( G ″ 0 H ′ 1 − G ′ 0 H ″ 1 ) − ( G ¯ ¯ 0 − G ¯ 0 ) ( G ″ 2 H ′ 1 − G ′ 2 H ″ 1 ) G ″ 0 G ′ 2 − G ′ 0 G ″ 2 } . (56)

For c m = 1 / 2 it holds H ′ 1 = − H ″ 1 . This is readily proved by a transformation of the integration variable ζ = ζ 3 ζ 5 / ζ ′ in (B35) and by (15). With (B41) it follows H 1 = 0 . Moreover, (50) and (54) must be equal due to the symmetry. Thus, H ¯ 1 = − H ¯ ¯ 1 . Finally

G H 0 = 2 π G 1 { H ¯ 1 − λ f H ′ 1 } for c m = 1 / 2 (57)

(57), (34), and (35) give the weak field Hall geometry factor of devices with c m = 1 / 2 , i.e. with two perpendicular mirror symmetries, as function of the numbers of squares. An alternative formula for this Hall geometry factor was given in (5a-c) in [

The Hall geometry factor G H 0 , the common mode c m , and the numbers of squares λ f , λ p at low magnetic field are functions of three parameters ζ 3 , ζ 5 , ζ 6 which are subject to the relations 1 ≤ ζ 3 ≤ ζ 5 ≤ ζ 6 . On the other hand G H 0 is an implicit function of the three parameters λ f , λ p , c m , which we study in the following. Thereby the quantity G H 0 / λ f λ p plays an important role. This ratio of Hall geometry factor over the square-root of the product of numbers of squares of inputs and outputs sums up the effect of the shape of the device on the signal-to-noise ratio at a fixed power dissipation (SNRP ? P denotes fixed power). This is explained in [

Here we are looking for the maximum of G H 0 / λ f λ p for all allowed values of the three parameters ζ 3 , ζ 5 , ζ 6 under the additional constraint that c m has a fixed value between zero and one. To this end we use (22), (23), (26) and (56). However, the numerical evaluation of the involved integrals is tricky and therefore we need to use some identities and transformations of Appendix B. For the optimization algorithm it is also important to deal with the allowed range of parameters 1 ≤ ζ 3 ≤ ζ 5 ≤ ζ 6 . If the algorithm leaves this allowed range during its search, G H 0 / λ f λ p will get complex valued and the algorithm will have troubles to find the maximum. We avoid this problem by introducing additional parameters x , y , z via ζ 3 = 1 + x 2 , ζ 5 = 1 + x 2 + y 2 , ζ 6 = 1 + x 2 + y 2 + z 2 . The new parameters x , y , z can attain any real values without leaving the allowed region of 1 < ζ 3 < ζ 5 < ζ 6 . In spite of all these provisions the search for maximum G H 0 / λ f λ p lasted several hours for c m → 0 or c m → 1 (for details of the computer used see section 6).

In

For c m = 1 / 2 ∧ λ f = λ p = 2 the SNRP is maximal. The exact location of this global maximum follows from (27) and (37).

ζ 3 , max = − 1 + 2 2 + 2 2 − 2 ≅ 3.35916 (58)

ζ 5 , max = 1 + 2 2 + 2 2 + 2 ≅ 7.52395 (59)

ζ 6 , max = 7 + 4 2 + 2 20 + 14 2 ≅ 25.27410 (60)

On the other hand, it is known from [

G H 0 ( ζ 3 , max , ζ 5 , max , ζ 6 , max ) = 2 / 3 . (61)

A strict mathematical proof of (61) based on (57) seems to be challenging and has not yet been accomplished.

Let us consider a first rectangular Hall plate of _{H} = W = 1 m, ℓ = 1.770130 m, b = 0.162917 m, s = 0.239699 m. This gives a common mode cm = 0.85 with maximum G H 0 / λ f λ p and λ f ≅ 1.726 , λ p ≅ 1.270 (see italic line in _{H} = W = 1 m, ℓ = 1.435180 m, b = 0.099209 m, s = 0.113905 m also has a common mode cm = 0.85, yet with λ f = λ p ≅ 1.414 and with slightly smaller G H 0 / λ f λ p (see italic line in ^{2}) in the main diagonal and ±0.01/(1 + 0.01^{2}) in the minor diagonal. This corresponds to a small Hall angle of 0.573˚. The triangular meshing was fine: it had more than 3.6 million elements. Lagrange multiplier was used to get accurate values of the current through the supply contacts. The values of the analytical theory were obtained with MATHEMATICA 8.0 (see also details in Appendix B.3). All computations were carried out on a desktop computer with Intel Core i7-4930K CPU, ASRock X79 Extreme 6 mainboard, PNY Quadro K2000D graphics board with 64 GB RAM and 3.4 GHz clock cycle.

The differences between the analytical theory and the FEM simulation are smaller than 0.03% for all values of

From a practical view point this example shows that a Hall plate with a fairly pronounced common mode of 0.85 may still achieve a good noise performance. It has the largest SNRP if input and output resistances differ by 36%. Then its SNRP is only 30% less than for an optimum Hall plate with 90˚ symmetry and 1.4 squares impedance. If the electronic circuit requires equal input and output resistances at the same common mode of 0.85, the SNRP loss is only slightly larger (31% instead of 30% for λ f = λ p = 2 ).

This work gave an analytical theory on the signal-to-thermal-noise ratio (SNR) of Hall plates with reduced symmetry, where the common mode potential of the two output contacts was not midway between the two supply potentials. The method

c m | SNRP loss | max G H 0 λ f λ p | λ f , o p t | λ p , o p t | G H 0 , o p t | ζ 3 , o p t | ζ 5 , o p t | ζ 6 , o p t |
---|---|---|---|---|---|---|---|---|

0.01 | −81.3% | 0.08820403 | 13.5684191 | 0.97714897 | 0.3211689 | 1.0126975 | 1.4154623 | 9.316E+17 |

0.02 | −73.5% | 0.12473934 | 6.78427221 | 0.97715402 | 0.3211712 | 1.0126981 | 1.4154656 | 51666217 |

0.03 | −67.6% | 0.15277365 | 4.52444172 | 0.97734957 | 0.3212595 | 1.0127186 | 1.4155907 | 426524.66 |

0.05 | −58.2% | 0.19716117 | 2.83246755 | 1.00077903 | 0.33195 | 1.0153938 | 1.4313833 | 2094.4638 |

0.10 | −41.4% | 0.27614336 | 1.96616646 | 1.16343939 | 0.4176540 | 1.0549779 | 1.6190136 | 137.64742 |

0.15 | −29.7% | 0.33163219 | 1.72558712 | 1.27010636 | 0.4909590 | 1.1338450 | 1.9121322 | 65.026289 |

0.20 | −20.8% | 0.37346478 | 1.60523057 | 1.33229454 | 0.5461582 | 1.2528145 | 2.2935627 | 44.890437 |

0.25 | −13.9% | 0.40569653 | 1.53222002 | 1.36884971 | 0.5875431 | 1.4165467 | 2.771975 | 35.952151 |

0.30 | −8.7% | 0.43040045 | 1.48401527 | 1.3903129 | 0.6182273 | 1.6326833 | 3.3638489 | 31.098303 |

0.35 | −4.8% | 0.44875400 | 1.45144223 | 1.4027452 | 0.6403208 | 1.9129024 | 4.0933519 | 28.218344 |

0.40 | −2.1% | 0.46146003 | 1.43019915 | 1.4096867 | 0.6552298 | 2.2746276 | 4.9945976 | 26.49560 |

0.45 | −0.5% | 0.46893594 | 1.41813302 | 1.41316212 | 0.6638470 | 2.7437445 | 6.1154838 | 25.567890 |

0.50 | 0.0% | 0.47140451 | 1.41384147 | 1.41435319 | 0.6666119 | 3.3591589 | 7.5239390 | 25.274102 |

0.55 | −0.5% | 0.46893594 | 1.41813133 | 1.41316432 | 0.6638471 | 4.1808368 | 9.3185630 | 25.567722 |

0.60 | −2.1% | 0.46146003 | 1.4301982 | 1.40968893 | 0.6552301 | 5.3048460 | 11.648275 | 26.495482 |

0.65 | −4.8% | 0.44875400 | 1.45144239 | 1.40274751 | 0.6403214 | 6.8937092 | 14.751561 | 28.218323 |

0.70 | −8.7% | 0.43040045 | 1.48401705 | 1.39031541 | 0.6182282 | 9.2449019 | 19.047404 | 31.098440 |

0.75 | −13.9% | 0.40569653 | 1.53222002 | 1.36884971 | 0.5875431 | 12.969869 | 25.380138 | 35.952151 |

0.80 | −20.8% | 0.37346478 | 1.60523879 | 1.33229834 | 0.546160 | 19.572788 | 35.832430 | 44.891549 |

0.85 | −29.7% | 0.33163219 | 1.72558712 | 1.27010636 | 0.4909589 | 34.007214 | 57.350248 | 65.026289 |

0.90 | −41.4% | 0.27614336 | 1.96616645 | 1.16343939 | 0.4176540 | 85.019310 | 130.47422 | 137.64742 |

0.95 | −58.2% | 0.19716117 | 2.83241167 | 1.00076829 | 0.3319452 | 1462.9986 | 2062.3557 | 2094.1005 |

0.97 | −67.6% | 0.15277365 | 4.52444219 | 0.97734963 | 0.3212595 | 301305.52 | 421168.63 | 426525.29 |

0.98 | −73.5% | 0.12473934 | 6.78430801 | 0.97715692 | 0.3211725 | 36505282 | 51024117 | 51672041 |

0.99 | −81.3% | 0.08820403 | 13.5684191 | 0.97714897 | 0.3211689 | 6.583E+17 | 9.202E+17 | 9.318E+17 |

c m | SNRP loss | G H 0 λ f λ p | G H 0 | ζ 3 | ζ 5 | ζ 6 | l / W | b / l | s / l |
---|---|---|---|---|---|---|---|---|---|

0.025 | −76.7% | 0.109905743 | 0.155430192 | 1.002335535 | 1.013536098 | 21.31686947 | 1.4150 | 0.9630 | 0.0216 |

0.05 | −62.7% | 0.175994946 | 0.248894440 | 1.009521765 | 1.054276928 | 21.46355938 | 1.4171 | 0.9263 | 0.0427 |

0.1 | −43.6% | 0.266102531 | 0.376325808 | 1.040865803 | 1.218890953 | 21.99558033 | 1.4250 | 0.8547 | 0.0816 |

0.15 | −30.6% | 0.327057404 | 0.462529016 | 1.101607638 | 1.497368164 | 22.71303885 | 1.4352 | 0.7869 | 0.1139 |

0.2 | −21.2% | 0.371430098 | 0.525281482 | 1.202125012 | 1.893182063 | 23.44626216 | 1.4453 | 0.7237 | 0.1384 |

0.25 | −14.1% | 0.404827226 | 0.572512154 | 1.353665966 | 2.412091843 | 24.07721415 | 1.4538 | 0.6650 | 0.1558 |

0.3 | −8.8% | 0.430064674 | 0.608203294 | 1.567884442 | 3.06612838 | 24.56018366 | 1.4601 | 0.6099 | 0.1674 |

0.35 | −4.8% | 0.448649712 | 0.634486508 | 1.857742246 | 3.875385898 | 24.89883052 | 1.4645 | 0.5573 | 0.1749 |

0.4 | −2.1% | 0.461439423 | 0.652573890 | 2.238880866 | 4.867146861 | 25.1162362 | 1.4672 | 0.5066 | 0.1795 |

0.45 | −0.5% | 0.468934647 | 0.663173737 | 2.730987815 | 6.073054588 | 25.2360367 | 1.4687 | 0.4572 | 0.1820 |

0.5 | 0.0% | 0.471404521 | 0.666666667 | 3.359160854 | 7.523945255 | 25.27414237 | 1.4692 | 0.4086 | 0.1827 |

0.55 | −0.5% | 0.468934647 | 0.663173737 | 4.155410812 | 9.240625886 | 25.2360367 | 1.4687 | 0.3609 | 0.1820 |

0.6 | −2.1% | 0.461439423 | 0.652573890 | 5.160361279 | 11.21821021 | 25.1162362 | 1.4672 | 0.3139 | 0.1795 |

0.65 | −4.8% | 0.448649712 | 0.634486508 | 6.42486482 | 13.40273689 | 24.89883052 | 1.4645 | 0.2677 | 0.1749 |

0.7 | −8.8% | 0.430064674 | 0.608203294 | 8.01016155 | 15.66453688 | 24.56018366 | 1.4601 | 0.2227 | 0.1674 |

0.75 | −14.1% | 0.404827226 | 0.572512154 | 9.981881174 | 17.78667319 | 24.07721415 | 1.4538 | 0.1792 | 0.1558 |

0.8 | −21.2% | 0.371430098 | 0.525281482 | 12.38457865 | 19.50401324 | 23.44626216 | 1.4453 | 0.1379 | 0.1384 |

0.85 | −30.6% | 0.327057404 | 0.462529016 | 15.16864015 | 20.61808402 | 22.71303885 | 1.4352 | 0.0992 | 0.1139 |

0.9 | −43.6% | 0.266102531 | 0.376325808 | 18.04556863 | 21.13200402 | 21.99558033 | 1.4250 | 0.0637 | 0.0816 |

0.95 | −62.7% | 0.175994946 | 0.248894440 | 20.35855931 | 21.26111604 | 21.46355938 | 1.4171 | 0.0310 | 0.0427 |

0.975 | −76.7% | 0.109905723 | 0.155430165 | 21.03217264 | 21.26719592 | 21.31686619 | 1.4150 | 0.0154 | 0.0216 |

relies on conformal mapping theory introduced by [

Quantity | First Hall plate | Second Hall plate | ||||
---|---|---|---|---|---|---|

Theory | FEM | Difference | Theory | FEM | Difference | |

l / W | 1.77012983 | 1.43517967 | ||||

b / l | 0.09203691 | 0.09920905 | ||||

s / l | 0.13541332 | 0.11390467 | ||||

c m | 0.85000 | 0.85000 | 0.0003% | 0.85000 | 0.84999 | 0.001% |

λ f | 1.72559 | 1.72560 | −0.001% | 1.41421 | 1.41419 | 0.001% |

λ p | 1.27011 | 1.26983 | 0.022% | 1.41421 | 1.41399 | 0.016% |

G H 0 | 0.49096 | 0.49083 | 0.026% | 0.46253 | 0.46251 | 0.005% |

G H 0 , 67 | 0.13312 | 0.13325 | −0.097% | 0.11091 | 0.11097 | −0.058% |

G H 0 , 12 | 0.35784 | 0.35771 | 0.038% | 0.35162 | 0.35154 | 0.024% |

G H 0 / λ f λ p | 0.33163 | 0.33158 | 0.015% | 0.32706 | 0.32708 | −0.006% |

3 G H 0 / 2 λ f λ p | 0.70350 | 0.70339 | 0.015% | 0.69379 | 0.69384 | −0.006% |

λ f / λ p | 1.35862 | 1.35892 | −0.022% | 1.00000 | 1.00014 | −0.014% |

effect devices.

With this theory, it was shown that the SNR is only slightly impaired if the common mode output potential deviates moderately from half of the supply potential. Despite the lack in symmetry for c m ≠ 1 / 2 it is possible to keep input and output resistances equal without significant further loss in SNR. In Appendix C, the theory was applied to optimize of Vertical Hall effect devices with three top contacts and one buried layer contact. There it was possible to specify geometries with sufficiently large contacts for practical use and with only moderate loss in SNR. Moreover, it was shown that for c m ≠ 1 / 2 the magnetic sensitivity of both contacts on the axis of mirror symmetry is not the same.

The theory is so general that it also covers former results on Hall plates with four contacts having two perpendicular mirror symmetries [

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

Ausserlechner, U. (2018) An Analytical Theory of the Signal-to-Noise Ratio of Hall Plates with Four Contacts and a Single Mirror Symmetry. Journal of Applied Mathematics and Physics, 6, 2032-2066. https://doi.org/10.4236/jamp.2018.610174

Here are the definitions of some functions used in the text. The imaginary unit is denoted by i = − 1 . The incomplete elliptic integral of the first order is defined as

F ( w , k ) = ∫ 0 w ( 1 − α 2 ) − 1 / 2 ( 1 − k 2 α 2 ) − 1 / 2 d α . (A1)

The complete integral of the first order is given by

K ( k ) = F ( 1 , k ) . (A2)

It holds K ( 0 ) = π / 2 . Asymptotic limit K ( k → 1 ) = ln ( 4 / 1 − k 2 ) . We also use the common notation for the complementary elliptic integral

K ′ ( k ) = K ( 1 − k 2 ) . (A3)

Inversion of (A1) gives the Jacobi-sn function

w = sn ( F ( w , k ) , k ) . (A4)

From (A4) it follows 0 = sn ( 0 , k ) and ± 1 = sn ( ± K ( k ) , k ) .

The modular lambda function is defined as the inverse of the monotonic function K ′ ( k ) / K ( k ) in the interval 0 ≤ k ≤ 1

L ( K ′ ( k ) / K ( k ) ) = k 2 . (A5)

Further useful properties of this function are given in [

We introduce the following functions which appear in the calculation of the resistances at zero magnetic field.

g ( ζ ) = 1 | ( ζ 2 − 1 ) ( ζ 2 − ζ 3 2 ) ( ζ 2 − ζ 5 2 ) ( ζ 2 − ζ 6 2 ) | > 0 (B1)

with 1 ≤ ζ 3 ≤ ζ 5 ≤ ζ 6 . For n = 0, 1, 2

G ′ n = ∫ ζ = 0 1 ζ n g ( ζ ) d ζ (B2)

G ¯ n = ∫ ζ = 1 ζ 3 ζ n g ( ζ ) d ζ (B3)

G n = ∫ ζ = ζ 3 ζ 5 ζ n g ( ζ ) d ζ > 0 (B4)

G ¯ ¯ n = ∫ ζ = ζ 5 ζ 6 ζ n g ( ζ ) d ζ (B5)

G ″ n = ∫ ζ = ζ 6 ∞ ζ n g ( ζ ) d ζ (B6)

The integrands of G ′ n and G ″ n have only a single pole, whereas the integrands of G ¯ n , G n , G ¯ ¯ n have two poles. All G-functions are positive. For n = 1 these integrals can be solved explicitly by [

G ¯ 1 = 1 ( ζ 6 2 − ζ 3 2 ) ( ζ 5 2 − 1 ) K ′ ( ζ 5 2 − ζ 3 2 ζ 6 2 − ζ 3 2 ζ 6 2 − 1 ζ 5 2 − 1 ) (B7)

G 1 = 1 ( ζ 6 2 − ζ 3 2 ) ( ζ 5 2 − 1 ) K ( ζ 5 2 − ζ 3 2 ζ 6 2 − ζ 3 2 ζ 6 2 − 1 ζ 5 2 − 1 ) (B8)

From

W 2 W 3 ¯ = W 5 W 6 ¯ ⇔ G ¯ 1 = G ¯ ¯ 1 . (B9)

One can get (B9) by direct calculation of the integrals according to [

G ′ 1 = 1 ( ζ 6 2 − ζ 3 2 ) ( ζ 5 2 − 1 ) F ( 1 ζ 3 ζ 6 2 − ζ 3 2 ζ 6 2 − 1 , ζ 5 2 − ζ 3 2 ζ 6 2 − ζ 3 2 ζ 6 2 − 1 ζ 5 2 − 1 ) , (B10)

G ″ 1 = 1 ( ζ 6 2 − ζ 3 2 ) ( ζ 5 2 − 1 ) F ( ζ 5 2 − 1 ζ 6 2 − 1 , ζ 5 2 − ζ 3 2 ζ 6 2 − ζ 3 2 ζ 6 2 − 1 ζ 5 2 − 1 ) . (B11)

and with the addition theorems for elliptic integrals in [

G ′ 1 ± G ″ 1 = 1 ( ζ 6 2 − ζ 3 2 ) ( ζ 5 2 − 1 ) F ( ζ 3 ζ 5 ± ζ 6 ζ 5 ± ζ 3 ζ 6 , ζ 5 2 − ζ 3 2 ζ 6 2 − ζ 3 2 ζ 6 2 − 1 ζ 5 2 − 1 ) . (B12)

The physical meaning of G ′ 1 / G 1 and G ″ 1 / G 1 is the size of the folded contacts W 0 W 2 ¯ and W 6 W ∞ ¯ on the axis of symmetry in

Another identity between G-functions can be proven with

w ′ = C ′ ∫ 0 ζ ζ 2 − ζ 3 2 ζ 2 − 1 ζ 2 − ζ 5 2 ζ 2 − ζ 6 2 d ζ . (B13)

With (B13) we compute the following segments

W ′ 0 W ′ 2 ¯ = | C ′ | ∫ 0 1 | ζ 2 − ζ 3 2 | | ( ζ 2 − 1 ) ( ζ 2 − ζ 3 2 ) ( ζ 2 − ζ 5 2 ) ( ζ 2 − ζ 6 2 ) | d ζ = | C ′ | ( ζ 3 2 G ′ 0 − G ′ 2 ) (B14)

W ′ 3 W ′ 5 ¯ = | C ′ | ∫ ζ 3 ζ 5 | ζ 2 − ζ 3 2 | | ( ζ 2 − 1 ) ( ζ 2 − ζ 3 2 ) ( ζ 2 − ζ 5 2 ) ( ζ 2 − ζ 6 2 ) | d ζ = | C ′ | ( G 2 − ζ 3 2 G 0 ) (B15)

W ′ 6 W ′ ∞ ¯ = | C ′ | ∫ ζ 6 ∞ | ζ 2 − ζ 3 2 | | ( ζ 2 − 1 ) ( ζ 2 − ζ 3 2 ) ( ζ 2 − ζ 5 2 ) ( ζ 2 − ζ 6 2 ) | d ζ = | C ′ | ( G ″ 2 − ζ 3 2 G ″ 0 ) (B16)

According to

ζ 3 2 ( G ′ 0 − G 0 + G ″ 0 ) = G ′ 2 − G 2 + G ″ 2 . (B17)

A similar mapping is shown in

w ″ = C ″ ∫ 0 ζ ζ 2 − 1 ζ 2 − ζ 3 2 ζ 2 − ζ 5 2 ζ 2 − ζ 6 2 d ζ (B18)

with the lengths

W ″ 1 W ″ 2 ¯ = 2 | C ″ | ∫ 0 1 1 − ζ 2 | ( ζ 2 − 1 ) ( ζ 2 − ζ 3 2 ) ( ζ 2 − ζ 5 2 ) ( ζ 2 − ζ 6 2 ) | d ζ = 2 | C ″ | ( G ′ 0 − G ′ 2 ) (B19)

W ″ 3 W ″ 5 ¯ = | C ″ | ∫ ζ 3 ζ 5 ζ 2 − 1 | ( ζ 2 − 1 ) ( ζ 2 − ζ 3 2 ) ( ζ 2 − ζ 5 2 ) ( ζ 2 − ζ 6 2 ) | d ζ = | C ″ | ( G 2 − G 0 ) (B20)

W ″ 6 W ″ 7 ¯ = 2 | C ″ | ∫ ζ 6 ∞ ζ 2 − 1 | ( ζ 2 − 1 ) ( ζ 2 − ζ 3 2 ) ( ζ 2 − ζ 5 2 ) ( ζ 2 − ζ 6 2 ) | d ζ = 2 | C ″ | ( G ″ 2 − G ″ 0 ) (B21)

With

G ′ 0 − G 0 + G ″ 0 = G ′ 2 − G 2 + G ″ 2 . (B22)

The combination of (B17) and (B22) gives two remarkable identities.

G 0 = G ′ 0 + G ″ 0 (B23)

G 2 = G ′ 2 + G ″ 2 (B24)

B2. The H-FunctionsIn the calculation of the Hall signal the following function appears.

h ( ζ ) = g ( ζ ) ln | 1 − ζ 1 + ζ ζ 3 + ζ ζ 3 − ζ ζ 5 − ζ ζ 5 + ζ ζ 6 + ζ ζ 6 − ζ | (B25)

with 1 < ζ 3 < ζ 5 < ζ 6 . h ( ζ ) has four zeros.

h ( 0 ) = h ( ζ 13 , 0 ) = h ( ζ 35 , 0 ) = h ( ζ 56 , 0 ) = 0 (B26)

ζ 1 < ζ 13 , 0 = ζ 3 − ζ 5 + ζ 3 ζ 5 + ζ 6 − ζ 3 ζ 6 + ζ 5 ζ 6 − sqrt / 2 < ζ 3 (B27)

ζ 3 < ζ 35 , 0 = ζ 3 ζ 5 ζ 6 + ζ 3 ζ 6 − ζ 3 ζ 5 − ζ 5 ζ 6 − 1 + ζ 3 − ζ 5 + ζ 6 < ζ 5 (B28)

ζ 5 < ζ 56 , 0 = ζ 3 − ζ 5 + ζ 3 ζ 5 + ζ 6 − ζ 3 ζ 6 + ζ 5 ζ 6 + sqrt / 2 < ζ 6 (B29)

with

sqrt = − 4 ζ 3 ζ 5 ζ 6 + ( ζ 3 − ζ 5 + ζ 6 + ζ 3 ζ 5 − ζ 3 ζ 6 + ζ 5 ζ 6 ) 2 . (B30)

We define for n = 0, 1, 2

H ′ n = ∫ ζ = 0 1 ζ n h ( ζ ) d ζ (B31)

H ¯ n = ∫ ζ = 1 ζ 3 ζ n h ( ζ ) d ζ (B32)

H n = ∫ ζ 3 ζ 5 ζ n h ( ζ ) d ζ (B33)

H ¯ ¯ n = ∫ ζ = ζ 5 ζ 6 ζ n h ( ζ ) d ζ (B34)

H ″ n = ∫ ζ = ζ 6 ∞ ζ n h ( ζ ) d ζ (B35)

The integrands of H ′ n and H ″ n are singular only at one end of the integration interval, whereas the integrands of H ¯ n , H n , H ¯ ¯ n are singular at both ends. The integrands of H ′ n and H ″ n have no zeros, yet the integrands of H ¯ n , H n , H ¯ ¯ n have one zero. Therefore, the signs of H ¯ n , H n , H ¯ ¯ n are not obvious, whereas H ′ n < 0 and H ″ n > 0 .

There are several relations between these integrals such as H ¯ 0 = H ¯ ¯ 0 and H ¯ 2 = H ¯ ¯ 2 (given without proof). For the purpose of this work we need relations of H-functions with index 1. To this end we consider the conformal transformation of

w ‴ = ∫ 0 ζ C ‴ 1 ( ζ − 1 ) 1 / 2 + m ( ζ − ζ 6 ) 1 / 2 − m d ζ ( ζ + 1 ) 1 / 2 + m ( ζ − ζ 3 ) 1 / 2 + m ( ζ − ζ 5 ) 1 / 2 − m ( ζ + ζ 6 ) 1 / 2 − m ( ζ + ζ 5 ) 1 / 2 + m ( ζ + ζ 3 ) 1 / 2 − m + C ‴ 0 (B36)

with m = β / π . In the limit of small m it holds

W ‴ 7 W ‴ 6 ¯ = | C ‴ 1 | | ∫ − ∞ − ζ 6 + ∫ ζ 6 ∞ ( ζ − 1 ) ( ζ − ζ 6 ) [ 1 + m ln ( ζ − 1 ζ + 1 ζ + ζ 3 ζ − ζ 3 ζ − ζ 5 ζ + ζ 5 ζ + ζ 6 ζ − ζ 6 ) ] d ζ ( ζ 2 − 1 ) 1 / 2 ( ζ 2 − ζ 3 2 ) 1 / 2 ( ζ 2 − ζ 5 2 ) 1 / 2 ( ζ 2 − ζ 6 2 ) 1 / 2 | = | C ‴ 1 | [ G ″ 2 + ( 1 + ζ 6 ) G ″ 1 + ζ 6 G ″ 0 − m ( H ″ 2 + ( 1 + ζ 6 ) H ″ 1 + ζ 6 H ″ 0 ) + G ″ 2 − ( 1 + ζ 6 ) G ″ 1 + ζ 6 G ″ 0 + m ( H ″ 2 − ( 1 + ζ 6 ) H ″ 1 + ζ 6 H ″ 0 ) ] (B37)

W ‴ 5 W ‴ 3 ¯ = | C ‴ 1 | | ∫ ζ 3 ζ 5 ( ζ − 1 ) ( ζ − ζ 6 ) ( g ( ζ ) + m h ( ζ ) ) d ζ | = | C ‴ 1 | [ − G 2 + ( 1 + ζ 6 ) G 1 − ζ 6 G 0 + m ( − H 2 + ( 1 + ζ 6 ) H 1 − ζ 6 H 0 ) ] (B38)

W ‴ 2 W ‴ 1 ¯ = | C ‴ 1 | | ∫ − 1 0 + ∫ 0 1 ( ζ − 1 ) ( ζ − ζ 6 ) [ 1 + m ln ( ζ − 1 ζ + 1 ζ + ζ 3 ζ − ζ 3 ζ − ζ 5 ζ + ζ 5 ζ + ζ 6 ζ − ζ 6 ) ] d ζ ( ζ 2 − 1 ) 1 / 2 ( ζ 2 − ζ 3 2 ) 1 / 2 ( ζ 2 − ζ 5 2 ) 1 / 2 ( ζ 2 − ζ 6 2 ) 1 / 2 | = | C ‴ 1 | [ G ′ 2 + ( 1 + ζ 6 ) G ′ 1 + ζ 6 G ′ 0 − m ( H ′ 2 + ( 1 + ζ 6 ) H ′ 1 + ζ 6 H ′ 0 ) + G ′ 2 − ( 1 + ζ 6 ) G ′ 1 + ζ 6 G ′ 0 + m ( H ′ 2 − ( 1 + ζ 6 ) H ′ 1 + ζ 6 H ′ 0 ) ] (B39)

W ‴ 8 W ‴ 10 ¯ = | C ‴ 1 | | ∫ − ζ 5 − ζ 3 ( ζ − 1 ) ( ζ − ζ 6 ) ( g ( ζ ) + m h ( ζ ) ) d ζ | = | C ‴ 1 | [ G 2 + ( 1 + ζ 6 ) G 1 + ζ 6 G 0 − m ( H 2 + ( 1 + ζ 6 ) H 1 + ζ 6 H 0 ) ] (B40)

With

H 1 = H ′ 1 + H ″ 1 . (B41)

B3. Numerical Evaluation of the IntegralsThe numerical evaluation of the integrals (B2-B6) and (B31-B35) may get tricky for ζ 3 → 1 (which means c m → 0 ), ζ 5 → ζ 6 (which means c m → 1 ), and ζ 3 → ζ 5 (which means small partial contacts). The situation improves if we transform the integrals in the following way, which is explained exemplarily for H ¯ ¯ n (n = 0, 1, 2). This integral extends from ζ 5 to ζ 6 with poles of the integrand at both ends. We split up the integration interval into two parts ( ζ 5 , ζ 56 , 0 ) and ( ζ 56 , 0 , ζ 6 ) , because then we have only a single pole of the integrand in each of these intervals. Next we substitute the integration variables: for the lower interval we set ζ = ζ 5 cosh α , and for the upper interval we set ζ = ζ 6 cos β . (Alternatively one could also substitute ζ = ζ 5 / cos α and/or ζ = ζ 6 / cosh β .) This gives

H ¯ ¯ n = ζ 5 n ∫ α = 0 arccoshyp ζ 56 , 0 ζ 5 cosh n α ln ( cosh α − 1 cosh α + 1 ζ 5 cosh α − 1 ζ 5 cosh α + 1 ζ 5 cosh α + ζ 3 ζ 5 cosh α − ζ 3 ζ 6 + ζ 5 cosh α ζ 6 − ζ 5 cosh α ) ζ 5 2 cosh 2 α − 1 ζ 5 2 cosh 2 α − ζ 3 2 ζ 6 2 − ζ 5 2 cosh 2 α d α + ζ 6 n ∫ β = 0 arccos ζ 56 , 0 ζ 6 cos n β ln ( cos β + 1 cos β − 1 ζ 6 cos β − 1 ζ 6 cos β + 1 ζ 6 cos β + ζ 3 ζ 6 cos β − ζ 3 ζ 6 cos β − ζ 5 ζ 6 cos β + ζ 5 ) ζ 6 2 cos 2 β − 1 ζ 6 2 cos 2 β − ζ 3 2 ζ 6 2 cos 2 β − ζ 5 2 d β (B42)

Thus, we avoided the poles in the denominators. Only the arguments of the logarithms have poles, which are easier to deal with by numerical integration routines. For the G ¯ ¯ n functions we simply skip the logarithms in (B42), and then we have no singularities at all. For the functions H ′ n , H ″ n , G ′ n , G ″ n we do not need to split up the original integration interval. Then the integration intervals become ( 0 , π / 2 ) or ( 0 , ∞ ) . The numerical integration routines of MATHEMATICA can handle the singularities in the H-integrals, when we choose the integration method Double Exponential, however, the maximum recursion limit and the working precision for numerical computations also have to be increased sufficiently.

Here we look for Vertical Hall effect devices of

q = z ˜ / l , (C1)

t = sn ( 2 q K ( κ ) , κ ) , (C2)

w ˜ = − 1 / ( κ t ) , (C3)

with the parameter κ given by the aspect ratio of the Hall tub

d / l = K ′ ( κ ) / ( 2 K ( κ ) ) ⇔ κ = L ( 2 d / l ) . (C4)

For the contacts we get

T a = sn ( ( l a / l ) K ( κ ) , κ ) , (C5)

T b = sn ( ( l a / l + 2 l b / l ) K ( κ ) , κ ) , (C6)

T c = sn ( ( l a / l + 2 l b / l + 2 l c / l ) K ( κ ) , κ ) . (C7)

A comparison of

ζ 3 = − W ˜ c = 1 / ( κ T c ) , (C8)

ζ 5 = − W ˜ b = 1 / ( κ T b ) , (C9)

ζ 6 = − W ˜ a = 1 / ( κ T a ) . (C10)

Inserting (C4-C7) into (C8-C10) gives three relations between the parameters ζ 3 , ζ 5 , ζ 6 and the geometrical parameters of the device in

l a = l F ( κ − 1 ζ 6 − 1 , κ ) / K ( κ ) (C11)

l b = l [ F ( κ − 1 ζ 5 − 1 , κ ) − F ( κ − 1 ζ 6 − 1 , κ ) ] / ( 2 K ( κ ) ) (C12)

l c = l [ F ( κ − 1 ζ 3 − 1 , κ ) − F ( κ − 1 ζ 5 − 1 , κ ) ] / ( 2 K ( κ ) ) (C13)

Before we turn to the case of general common mode, we study the special case c m = 1 / 2 . Such a device may have a circular shape as in

w ′ = − i tan ( α 2 / 2 ) z ′ + i z ′ − i . (C14)

The mappings of Z ′ a → W ′ a , Z ′ b → W ′ b , and Z ′ c → W ′ c give

W ′ a = cos α 2 + 1 cos α 2 − 1 , (C15)

W ′ b = sin α 2 cos α 2 − 1 1 + cos α 1 + sin α 1 1 + cos α 1 − sin α 1 , (C16)

W ′ c = sin α 2 cos α 2 − 1 1 + cos α 1 − sin α 1 1 + cos α 1 + sin α 1 . (C17)

For c m = 1 / 2 the w ˜ -plane in

w = ( 1 − z 2 1 + z 2 ) 2 (C18)

t = A − w C + κ 2 w (C19)

A = 1 − ( cos α 1 sin α 2 ) − 1 (C20)

C = κ 2 ( A − 2 ) (C21)

κ 2 = cos α 1 − sin α 2 cos α 1 + sin α 2 (C22)

q = F ( t , κ 2 ) (C23)

Equations (C19-C22) define a Möbius transformation. The number of squares between the vertical contacts (the ones with half aperture angle α 2 ) is

λ 2 = 2 K ( κ 2 ) K ′ ( κ 2 ) = K ′ ( ( 1 − κ 2 ) / ( 1 + κ 2 ) ) K ( ( 1 − κ 2 ) / ( 1 + κ 2 ) ) = K ′ ( sin α 2 / cos α 1 ) K ( sin α 2 / cos α 1 ) , (C24)

where we used [

we only have to swap indices 1 and 2.

λ 1 = K ′ ( sin α 1 / cos α 2 ) K ( sin α 1 / cos α 2 ) (C25)

We can solve (C24, C25) for the half aperture angles with the help of the modular lambda function and insert this into (C15-C17), insert this again into (C8-C10) and finally into (C11-C13). With the abbreviations L 1 = L ( λ 1 ) , L 2 = L ( λ 2 ) we get

l a = l K ( κ ) F ( 1 κ ( 1 − L 1 L 2 − 1 − L 2 ) 2 L 2 ( 1 − L 1 ) , κ ) (C26)

l b = l 2 K ( κ ) F ( 1 κ 1 − L 1 L 2 − 1 − L 2 L 2 1 − L 1 L 1 1 − L 2 − 1 − L 1 L 2 + 1 − L 1 L 1 1 − L 2 + 1 − L 1 L 2 − 1 − L 1 , κ ) − l a 2 (C27)

l c = l 2 K ( κ ) F ( 1 κ 1 − L 1 L 2 − 1 − L 2 L 2 1 − L 1 L 1 1 − L 2 + 1 − L 1 L 2 + 1 − L 1 − L 1 1 − L 2 + 1 − L 1 L 2 + 1 − L 1 , κ ) − l b − l a 2 (C28)

This gives the geometrical parameters of the device in

l a → l F ( κ − 1 ζ 6 , max − 1 , κ ) / K ( κ ) , (C29)

l b → l F ( κ − 1 ζ 5 , max − 1 , κ ) / ( 2 K ( κ ) ) − l a / 2 , (C30)

l c → l F ( κ − 1 ζ 3 , max − 1 , κ ) / ( 2 K ( κ ) ) − l b − l a / 2 , (C31)

with the numbers ζ 3 , max , ζ 5 , max , ζ 6 , max from (58-60). For l = 20 µm and d = 5 µm one gets l a = 0.254 µm, l b = 0.302 µm, l c = 0.556 µm. In practice these values are smaller than the feature size of many CMOS technologies, and they give rise to large electric fields when operated at typical supply voltages of around 2 V. Large electric fields lead to velocity saturation and electrical non-linearity and local self-heating, which reduce the efficiency of the spinning current Hall probe scheme [

l ≥ 2 K ( − 1 + 2 2 − 2 2 − 2 ) K ′ ( − 1 + 2 2 − 2 2 − 2 ) d ≅ 1.22004 d (C32)

For maximum SNRP the length of the Hall tub must be at least 22% larger than its depth. Moreover, the outer contacts should reach towards the end of the tub l a + 2 l b + 2 l c = l . This follows from (C31) when the first argument of the incomplete elliptic integral equates 1. For d = 5 µm one gets l a = 0.506 µm, l b = 0.628 µm, l c = 2.170 µm and l = 6.100 µm. In the following we will see that it is possible to increase the contacts even further if we depart from c m = 1 / 2 .

In the general case of Hall effect devices with different input and output resistances it is advantageous to operate them in a stacked way according to _{1}-C_{3} of one device and through C_{2}-C_{4} of the other device. Then a spinning Hall probe scheme can be implemented, where input and output

terminals of both devices are swapped synchronously while a constant supply current is forced through them, and the output voltages at the remaining terminals are added for both devices and both operating phases. This cancels out offset errors and low frequency 1/f-noise [

of both devices in each operating phase is 2 μ H R s h G H 0 I i n stacked B ⊥ and the thermal noise voltage in this sum of output voltages is 4 k b T ( λ f + λ p ) R s h E N B W , with Boltzmann’s constant k b , the absolute temperature T, and the effective noise bandwidth ENBW of the signal path. This gives the signal-to-noise ratio

S N R stacked = μ H B ⊥ 4 k b T E N B W G H 0 ( λ f + λ p ) / 2 V i n stacked R i n stacked (C33)

The last factor in (C33) is equal to the square-root of the power dissipated in both devices. Equation (C33) gives the SNR of stacked Hall effect devices related to the dissipated power. This SNRP of the stack uses the arithmetic mean ( λ f + λ p ) / 2 whereas the SNRP of single devices discussed above used the geometric mean λ f λ p (see also (8) in [

# | SNRP loss | c m | λ f | λ p | G H 0 | l [µm] | l a [µm] | l b [µm] | l c [µm] | ( l − l a ) / 2 − l b − l c [µm] | min ( l a , l b , l c ) [µm] |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | −4.1% | 0.483 | 1.173 | 1.716 | 0.6532 | 7 | 1 | 1 | 2 | 0 | 1 |

2 | −4.5% | 0.486 | 1.175 | 1.751 | 0.6589 | 7 | 1 | 1 | 1.5 | 0.5 | 1 |

3 | −5.9% | 0.535 | 1.253 | 1.918 | 0.7035 | 6 | 1 | 1 | 1.5 | 0 | 1 |

4 | −6.0% | 0.497 | 1.181 | 1.872 | 0.6763 | 7 | 1 | 1 | 1 | 1 | 1 |

5 | −6.9% | 0.540 | 1.255 | 1.983 | 0.7104 | 6 | 1 | 1 | 1 | 0.5 | 1 |

6 | −9.0% | 0.529 | 1.056 | 1.868 | 0.6274 | 7 | 1.5 | 1 | 1.5 | 0.25 | 1 |

7 | −10.6% | 0.537 | 1.059 | 1.968 | 0.6380 | 7 | 1.5 | 1 | 1 | 0.75 | 1 |

8 | −10.6% | 0.451 | 1.187 | 2.133 | 0.6996 | 7 | 1 | 1.5 | 1.5 | 0 | 1 |

9 | −11.3% | 0.600 | 1.370 | 2.225 | 0.7511 | 5 | 1 | 1 | 1 | 0 | 1 |

10 | −11.7% | 0.455 | 1.188 | 2.203 | 0.7056 | 7 | 1 | 1.5 | 1 | 0.5 | 1 |

11 | −12.7% | 0.586 | 1.134 | 2.130 | 0.6718 | 6 | 1.5 | 1 | 1 | 0.25 | 1 |

12 | −14.6% | 0.567 | 0.974 | 2.005 | 0.5999 | 7 | 2 | 1 | 1.5 | 0 | 1 |

13 | −15.1% | 0.508 | 1.262 | 2.486 | 0.7502 | 6 | 1 | 1.5 | 1 | 0 | 1 |

14 | −15.7% | 0.571 | 0.975 | 2.071 | 0.6048 | 7 | 2 | 1 | 1 | 0.5 | 1 |

15 | −16.9% | 0.499 | 1.065 | 2.360 | 0.6705 | 7 | 1.5 | 1.5 | 1 | 0.25 | 1 |

16 | −20.0% | 0.628 | 1.052 | 2.324 | 0.6365 | 6 | 2 | 1 | 1 | 0 | 1 |

17 | −20.4% | 0.432 | 1.192 | 2.679 | 0.7259 | 7 | 1 | 2 | 1 | 0 | 1 |

18 | −21.5% | 0.604 | 0.912 | 2.195 | 0.5750 | 7 | 2.5 | 1 | 1 | 0.25 | 1 |

19 | −23.6% | 0.539 | 0.979 | 2.573 | 0.6392 | 7 | 2 | 1.5 | 1 | 0 | 1 |

20 | −28.3% | 0.639 | 0.863 | 2.373 | 0.5469 | 7 | 3 | 1 | 1 | 0 | 1 |

21 | −0.2% | 0.519 | 1.363 | 1.453 | 0.6627 | 7 | 0.5 | 0.5 | 1 | 1.75 | 0.5 |

22 | −0.2% | 0.530 | 1.418 | 1.369 | 0.6554 | 6 | 0.5 | 0.5 | 1.5 | 0.75 | 0.5 |

23 | −0.3% | 0.497 | 1.340 | 1.324 | 0.6259 | 7 | 0.5 | 0.5 | 1.5 | 1.25 | 0.5 |

24 | −0.3% | 0.523 | 1.408 | 1.323 | 0.6416 | 6 | 0.5 | 0.5 | 2 | 0.25 | 0.5 |

25 | −0.6% | 0.547 | 1.437 | 1.484 | 0.6846 | 6 | 0.5 | 0.5 | 1 | 1.25 | 0.5 |
---|---|---|---|---|---|---|---|---|---|---|---|

26 | −0.8% | 0.484 | 1.323 | 1.257 | 0.6033 | 7 | 0.5 | 0.5 | 2 | 0.75 | 0.5 |

27 | −1.1% | 0.478 | 1.315 | 1.228 | 0.5925 | 7 | 0.5 | 0.5 | 2.5 | 0.25 | 0.5 |

28 | −1.5% | 0.576 | 1.541 | 1.449 | 0.6939 | 5 | 0.5 | 0.5 | 1.5 | 0.25 | 0.5 |

29 | −2.0% | 0.422 | 1.380 | 1.582 | 0.6842 | 7 | 0.5 | 1 | 2 | 0.25 | 0.5 |

30 | −2.1% | 0.587 | 1.553 | 1.534 | 0.7122 | 5 | 0.5 | 0.5 | 1 | 0.75 | 0.5 |

31 | −2.2% | 0.428 | 1.385 | 1.637 | 0.6965 | 7 | 0.5 | 1 | 1.5 | 0.75 | 0.5 |

32 | −2.3% | 0.469 | 1.462 | 1.749 | 0.7398 | 6 | 0.5 | 1 | 1.5 | 0.25 | 0.5 |

33 | −2.6% | 0.542 | 1.134 | 1.355 | 0.5715 | 7 | 1 | 0.5 | 2 | 0.5 | 0.5 |

34 | −2.6% | 0.539 | 1.131 | 1.337 | 0.5665 | 7 | 1 | 0.5 | 2.5 | 0 | 0.5 |

35 | −2.7% | 0.551 | 1.392 | 1.740 | 0.7186 | 7 | 0.5 | 0.5 | 0.5 | 2.25 | 0.5 |

36 | −2.7% | 0.553 | 1.144 | 1.415 | 0.5868 | 7 | 1 | 0.5 | 1.5 | 1 | 0.5 |

37 | −2.8% | 0.587 | 1.220 | 1.451 | 0.6120 | 6 | 1 | 0.5 | 2 | 0 | 0.5 |

38 | −3.0% | 0.591 | 1.224 | 1.481 | 0.6181 | 6 | 1 | 0.5 | 1.5 | 0.5 | 0.5 |

39 | −3.2% | 0.441 | 1.394 | 1.771 | 0.7221 | 7 | 0.5 | 1 | 1 | 1.25 | 0.5 |

40 | −3.3% | 0.477 | 1.467 | 1.848 | 0.7552 | 6 | 0.5 | 1 | 1 | 0.75 | 0.5 |

41 | −3.3% | 0.576 | 1.463 | 1.759 | 0.7341 | 6 | 0.5 | 0.5 | 0.5 | 1.75 | 0.5 |

42 | −3.7% | 0.573 | 1.160 | 1.541 | 0.6135 | 7 | 1 | 0.5 | 1 | 1.5 | 0.5 |

43 | −4.2% | 0.605 | 1.235 | 1.585 | 0.6370 | 6 | 1 | 0.5 | 1 | 1 | 0.5 |

44 | −5.0% | 0.609 | 1.574 | 1.790 | 0.7527 | 5 | 0.5 | 0.5 | 0.5 | 1.25 | 0.5 |

45 | −5.5% | 0.528 | 1.581 | 1.994 | 0.7966 | 5 | 0.5 | 1 | 1 | 0.25 | 0.5 |

46 | −5.5% | 0.644 | 1.347 | 1.606 | 0.6578 | 5 | 1 | 0.5 | 1.5 | 0 | 0.5 |

47 | −6.3% | 0.582 | 1.028 | 1.443 | 0.5457 | 7 | 1.5 | 0.5 | 2 | 0.25 | 0.5 |

48 | −6.3% | 0.642 | 1.743 | 1.638 | 0.7467 | 4 | 0.5 | 0.5 | 1 | 0.25 | 0.5 |

49 | −6.3% | 0.649 | 1.352 | 1.663 | 0.6655 | 5 | 1 | 0.5 | 1 | 0.5 | 0.5 |

50 | −6.7% | 0.590 | 1.033 | 1.491 | 0.5550 | 7 | 1.5 | 0.5 | 1.5 | 0.75 | 0.5 |