Relations between Hall Plates with Complementary Contact Geometries

Singly connected Hall plates with N peripheral contacts can be mapped onto the upper half of the z-plane by a conformal transformation. Recently, Homentcovschi and Bercia derived the General Formula for the electric field in this region. We present an alternative intuitive derivation based on conformal mapping arguments. Then we apply the General Formula to complementary Hall plates, where contacts and insulating boundaries are swapped. The resistance matrix of the complementary device at reverse magnetic field is expressed in terms of the conductance matrix of the original device at non-reverse magnetic field. These findings are used to prove several symmetry properties of Hall plates and their complementary counterparts at arbitrary magnetic field.


Introduction
The purpose of this work is to derive relations between Hall plates and their complementary counter-parts. A Hall plate is assumed to be a plane conductive region with thickness much smaller than its lateral dimensions. We discuss only singly-connected Hall plates without holes. An arbitrary number of at least two Several properties of complementary Hall plates at zero or weak applied magnetic field have already been studied in the past. We found the following ones.
Plain distributed resistive structures with complementary peripheral electrode geometries were studied at zero magnetic field in [1]. The number of extended electrodes was N > 1. The authors called the complementary device the dual device. They defined a first resistance between two non-neighboring contacts in the original device and a second resistance between two terminals in the dual device (see their Figure 2). The first terminal was connected to all contacts of the dual device, which are left of the two non-neighboring contacts in the original device. The second terminal was connected to all contacts of the dual device, which are right of the two non-neighboring contacts in the original device (see Figure 5 in this work). Finally, the product of these two resistances equals the square of the sheet resistance sheet , with the specific volume resistivity ρ and the thickness of the Hall plate H t . In [1] the authors also gave a general relation between indefinite impedance matrix ( ) N N × Z and indefinite admittance matrix ( ) N N × Y of original and dual devices at zero magnetic field. Thereby, the impedance matrix of the complementary Hall plate is fully given by the admittance matrix of the original Hall plate. The goal of the current paper is to find an analogue relation that also holds in the presence of applied magnetic field of arbitrary strength. The arguments in [1] were based on the idea that potential and stream function are swapped if contacts and insulation boundaries are swapped. This naturally raises the question of what happens in cases where no stream function exists [2]. Further insight into the symmetry of such devices is given in [3].
Van-der-Pauw measurement on Hall plates with 90˚ symmetry at zero magnetic field was discussed in [4]. There the authors focused on Hall plates with point-sized contacts and on their complementary counterparts of large contacts with no insulating boundaries in-between. Both cases can be readily computed in closed form. Then the authors found a smart power law, which interpolates the sheet resistance up to an astonishing accuracy of ±0.02% for all contact sizes ((7) in [4], also (24) in [5]).
In [6] Van-der-Pauw measurement on rectangular Hall plates was studied at zero magnetic field. The devices had four extended contacts with two orthogonal symmetry axes. There the original device was labeled with "even symmetry" and the complementary device with "odd symmetry". If opposite contacts are shorted each device has only two terminals with the so-called cross resistance between them. It was found that the sheet resistance at zero magnetic field is twice the square-root of the product of cross resistances of original device and comple-The Hall plate has N contacts with end points , n n a b with the sequential order . Thus, contacts 2 to N are lined up from left to right and contact 1 is at infinity. According to [12] the complex electric field ( ) E z in the point z x iy = + with 0 y ≥ (i being the imaginary unit) is given by This is opposite to the definition in [12] and therefore a different sign shows up in (1c). The applied magnetic flux density B ⊥ perpendicular to the Hall plate is counted positive, when it points out of the drawing plane.
In [12] Homentcovschi and Bercia derived their General Formula (1a) in a formal, "mathematical" way. Here we present a more intuitive derivation, which also sheds light on the physical meaning of the coefficients n c . Figure 2 shows a Hall plate in the w-plane with the shape of a skewed parallelogram according to the method of Wick [13]. It is obtained from the Hall plate in Figure 1 by conformal mapping (after Schwartz-Christoffel). The skew angle is identical to the Hall angle. If the top and bottom contacts are supplied with electric energy, the current streamlines will be homogeneous and parallel to the left and right edges of the parallelogram, and the equipotential lines will be horizontal and evenly spaced. Thus the electric field will be vertical and homogeneous. As for any Hall plate the angle between the current density ( ) S w and the electric field ( ) E w is identical to the Hall angle for all points in the Hall plate:  (2a) is the complex notation of the general Ohm's law in vector form In many materials ρ and H µ do not depend on B ⊥ as long as B ⊥ is sufficiently small (see (11.19) in [14]). However, at strong magnetic field and/or for certain doping concentrations ρ and H µ may well change versus B ⊥ (see Chapter 1.1c in [14], see also Chapter 3.3 in [15], also [16]).
In (2a)， the conjugate complex of ( ) ( ) where φ is the electric potential inside the Hall effect region (the potential on the contacts is denoted by 1 We are free to choose the ground potential and so we ground the N-th contact ( . The current flows out of the Hall plate through the lower contact N of the parallelogram, and it flows into the Hall plate through the upper contact j. The electric field in the upper half of the z-plane is given by whereby we used the fact that due to the conformal mapping between w-plane and z-plane it holds With the Schwartz-Christoffel formula [18] we immediately get the General Formula hereby K is a scaling constant of the Schwartz-Christoffel mapping, contact N is the current drain contact at ground potential, and contact j is the current input contact at supply voltage. All output contacts-i.e., floating contacts with zero current-are folded. Points n D are inner pointed tips of these folded contacts inside the parallelogram in the w-plane. in the numerator of (8a) is a real number and ( ) L z is a negative real number (see Table A1 in Appendix A), while the electric field must be perpendicular to contact N with 0 y E < (the electric field is directed towards the ground node) and therefore In the most general case currents are flowing through all contacts. Then the electric field is the following linear superposition: whereby the real-valued scaling constant j K and the stagnation points , n j d depend on the number j of the contact at which current j I was supplied in Figure 2. Equating the numerators of (1a) and (8b) gives the real constants n c as functions of It means that for z → ∞ the electric field declines with the dominant term The current density in the z-plane is given by (2a) if we replace w u iv = + by z x iy = + . Integration over the electric field and the current density gives the voltages between the contacts and the currents through the contacts. This was done in [12] with the following results: The real constants n c are linked to the currents n I into the Hall plate through the n-th contact and to the potentials n V at the contacts In the appendix we prove with the calculus of residues that the sums in front vanish, however, the sum in front of N c does not vanish. Therefore (12a) leads again to (9). Moreover, also the sum over all currents must vanish due to Kirchhoff's current law 1 2 0 Indeed, in the appendix we can show that the sums in front of all n c with n N < vanish.

The Resistance Matrix of a Hall Plate
From (12a) we see that we can skip the last equation in (11a) for n N = . With (9) it follows that Eliminating c in (13a) and (13b) gives (14) Inserting (B7a) into (14) gives the contact potentials ( ) (the index T denotes the transpose of a vector or a matrix).

U. Ausserlechner Journal of Applied Mathematics and Physics
In (15a) the resistance matrix R is given by It follows from inversion of (15b). Note that (15b) is computationally not very efficient, because we need to compute two matrices, invert one, and multiply two matrices. Another method is to determine the    Figure 1 and it is exposed to reverse magnetic field (denoted by ⊗

The Resistance Matrix of the Complementary Hall Plate
Comparison of (19a) and (1b) gives the important result Computing the voltages gives with (11a) and Table A1 in Appendix A In ↑ N all rows of N are shifted up once and the last row in ↑ N is equal to the negative sum of all rows in N . For the right equation in (22) we used with the definition This means cos cos (21) and (25) (21) and (25) the reverse magnetic field on complementary device theorem (RMFoCD).
With the RMFoCD theorem we can predict the behavior of complementary Hall plates from the original ones. Figure 4 shows a circuit where currents are injected into the contacts of a Hall plate with ideal current sources. This is the usual way how Hall plates are operated, because the supply current is forced through the device with a current source while zero current is guaranteed at the output contacts when one connects ideal voltmeters there. In general, it holds On the other hand, we can write for the complementary Hall plate at reverse magnetic field with (14) . Thus, the vectors on the right hand sides must be equal, if the vectors on the left hand sides are equal-and vice versa. This means whereby we introduced the square resistance at magnetic field ( ) The square resistance is the resistance of a square sample with contacts along two opposite edges (see [21], also (11.36) in [14], also (45) in [22]). Only at zero magnetic field the square resistance is equal to the sheet resistance. If we measure the sheet resistance on a sample with point-sized contacts according to van der Pauw [23], the result does not depend on the applied magnetic field provided ρ is a constant. This can be proven with (16c) in [17], which shows that the tapped voltage in a van der Pauw measurement is constant versus applied magnetic field. Note that (29) differs from the definition of sq R in [19]. We can express (28) also in terms of the resistance matrix of the complementary Hall plate at reverse magnetic field. It is defined as Now we express the left hand side in terms of parameters of the original Hall plate. With (B7c) and the left side of (28) we get We also express the current on the right hand side of (30a) in terms of parameters of the original Hall plate. With the right side of (28) In the last transformation we used the identity whereby all On the other hand we can multiply both sides of (32a) first with ∆ and then with ( ) Matrix inversion of both sides of (34) gives with (B10a) With (B10c) and (B7d) we can write this

Impedance Relation between Complementary Hall Plates
In the introduction we have already stated that there is a relation between impedances of original and complementary Hall plates at vanishing magnetic field [1]. With our findings above we can readily give a more general relation at arbitrary magnetic field. Both circuits are sketched in Figure 5.
From which we get It means: The product of the numbers of squares of both resistances is equal to 1. Thereby the number of squares of a resistance is defined as the ratio of this resistance over the square resistance-not over the sheet resistance (see (29)).
(40b) also means that the product of both measured resistances is equal to the square of the square resistance: This opens up a way to measure the sheet resistance and the Hall mobility of an unknown material. Note that we are free to choose contact m in Figure 5.

Complementary Hall Plates with Three Contacts
Let us consider general Hall plates with three peripheral contacts. Current is injected into contact 1, voltage is tapped between contact 2 and 3, whereby contact 3 is the current sink and ground node at zero volts (see Figure 6). In practice, these devices are not the first choice, but sometimes one is obliged to work with them, e.g., for Vertical Hall effect devices [25].

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If such Hall plates have a single mirror symmetry, and if we supply the original and the complementary Hall plate with the same supply voltage, their output voltages are identical. This was proven for weak magnetic field in [10] [11] with considerable mathematical effort, and with the above theory we will prove it also for strong magnetic field. For the original Hall plate in Figure 6 we write 11 12 supply supply which gives the output voltage At zero magnetic field the voltage 2 V does not vanish-this is called the offset voltage. Therefore 2 V is of little use in practical sensor applications. However, if one subtracts the voltage 2 V at positive and negative magnetic field, this procedure cancels out the offset voltage and renders the so-called Hall voltage In other words, the voltage 2 V may be decomposed into even and odd functions versus B ⊥ and the odd one is the Hall voltage (see also [2] [17]). With the principle of reverse magnetic field reciprocity (RMFR, see [26]) it holds where the quantities with overbar are at reverse magnetic field. It follows ( ) ( ) ( ) 21 where 2H V is the Hall voltage of the complementary device at reverse magnetic field, yet 11 12 21 , , G G G are the conductances of the original device at non-reverse magnetic field. Thus, for asymmetric Hall plates neither the voltages nor the Hall Note that the complementary device also has a single mirror symmetry, but this symmetry line does not go through the grounded reference contact 3 .

Complementary Hall Plates with 90˚ Symmetry
Let us consider Hall plates with four peripheral contacts and two perpendicular mirror symmetries. Moreover, the contacts should be such that the resistance between two non-neighboring contacts equals the resistance between the other two non-neighboring contacts. All devices with 90˚ symmetry belong to this group of Hall plates. However, also rectangular shapes with properly chosen contact sizes have such properties. Some popular shapes are shown in Figure 5 of [7]. The two axes of perpendicular mirror symmetry guarantee that the average of the potentials of the output contacts is half of the supply potential, i.e., the common mode output voltage is half the supply voltage. Even if these devices are not 90˚ symmetric, they can be mapped with a conformal transformation to devices with 90˚ symmetry. Therefore we only need to discuss devices with 90˚ symmetry. Then the complementary Hall plate exhibits the same kind of symmetry.
If we supply the original and the complementary Hall plate with the same supply voltage, their output voltages are identical. This was proven for weak magnetic field in [7] with considerable mathematical effort, and with the above theory we will prove it also for strong magnetic field. The circuit is shown in Figure 7. Note that a different amount of current flows through both devices.
Thus it is all the more astonishing that the short-circuiting action of the extended contacts in the device with higher resistance is reduced by exactly the same factor as the resistance goes up (the current goes down), so that the output voltage remains identical.
The conductance matrix of a Hall plate with 90˚ symmetry has the following symmetry.
which gives the output voltage where the quantities with overbar are at reverse magnetic field. With (32c) the resistance matrix of the complementary device at reverse magnetic field is ( ) ( ) which gives the output voltage

Conclusion
We gave a simple derivation of the electric field in the infinite upper half plane with N contacts on the real axis. The original formula has N real coefficients, but we could show that the N-th one vanishes. Based on these results the resistance and conductance matrices of an N-contact Hall plate can be expressed in terms

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.
For the integral in (A1) we use the calculus of residues [27]. We integrate along a closed path, which comprises the real axis and a large semi-circle with radius R → ∞ in the upper half of the z-plane. The ends of the contacts represent 2N isolated singular points, which we cut out of the contour integral as shown in Figure A1: we draw a small semi-circle of radius 0 ε → around each singular point. The integrals along the small semi-circles vanish as the following example on semi-circle n (denoted by s.c.n) around n a shows ( ) . .
where we used The integral along the closed contour is proportional to the sum of all enclosed residues. However, this sum is zero, because we cut out all singular points.

Appendix B
Here we summarize the mathematics to describe Hall plates with numerous contacts via impedance and conductance matrices. Let us consider a Hall plate with N contacts on the perimeter (see Figure B1) It is common practice to define the potential on one contact as reference potential and to set it equal to zero volts (=ground). We use the potential on the N-th contact as reference potential: The voltages between neighboring contacts are defined as According to Figure B1 the N-th voltage is . This is a consequence of our definitions (B2, B5a), which differ from (B1b, B3b). We could also redefine the reference potential N V analogous (B1b) but this would give more complicated matrices in (B7b) and (B7d). We can describe the shift-up procedure by a matrix multiplication: identity matrix, and ↑ 1 is obtained from 1 by shifting all rows up once and setting the elements of the bottom row equal to the negative sum of all elements per column in the original matrix 1 . In other words, we apply the shift-up procedure not only to vectors but also to matrices.
The shift-down manipulation is also equivalent to a matrix multiplication: Therefore, shift-up and shift-down annihilate each other However, we are not allowed to reverse the order: ( ) ( ) We can also write (B4a), (B4b) with (B5a) as a matrix relation Throughout this paper we use the following definitions of resistance matrix R , impedance matrix Z , and conductance matrix = G Y : For the complementary Hall plate at reverse magnetic field it follows Journal of Applied Mathematics and Physics where we used (32a), (B10c), and (B14a). This can be re-written as At zero magnetic field (B15b) differs slightly from (21) in [1] (in the second indices). There are two reasons for this discrepancy: 1) the contacts of the complementary device are shifted in the direction of lower indices in [1] (whereas they are shifted in the direction of larger indices in this paper), and 2) in Figure 2(a) and Figure 2 Here is another argument which proves that (21) [26]. With (B14a) this gives In other words, an element on the main diagonal of the Z -matrix of the complementary Hall plate cannot be proportional to an off-diagonal element of the Y -matrix of the original Hall plate, because this contradicts the RMFR principle (as long as the constant of proportionality is even in B ⊥ ).
In Appendix C we check our formulae against finite element simulations to guarantee their correctness. Moreover, we emphasize that all our equations work only if the following rules are applied: 1) The contacts are labeled in ascending order 1 to N.
2) If we walk along the perimeter of the Hall plate in the direction of rising contact labels, the conductive region lies at the left hand side.
3) The N-th contact is grounded and the N-th loop current vanishes.

4) The complementary
Hall plate is obtained from the original one, by exchanging all contacts with insulating boundaries and vice versa, whereby the contacts of the complementary device are shifted into the direction of rising contact labels against the respective contacts of the original device (compare Figure 1 with Figure 3).

Appendix C
Here Ω at zero magnetic field. We assume that ρ and H µ are constant with B ⊥ . We consider semi-infinite Hall plates according to Figure 1.
The first example has the following parameters: N = 4, 1 10 2 m a = µ , 2 10 m a = − π µ , 3 10 m a = − µ , 4  . . With (C3b) we may compute the electric field inside the Hall region (Table   C1). From The main reason for the difference between the FEM result and the analytical theory seems to come from the finite radius of the semi-disk ( Figure C1).
Next we check Figure 4. Therefore, we inject the currents (in units of amperes) ( ) ( )    Table C1). According to Figure 4 we expect the voltages