The Classical Hall Effect in Multiply-Connected Plane Regions Part I: Topologies with Stream Function

Typical Hall plates for practical magnetic field sensing purposes are plane, simply-connected regions with peripheral contacts. Their output voltage is the sum of even and odd functions of the applied magnetic field. They are commonly called offset and Hall voltage. Contemporary smart Hall sensor circuits extract the Hall voltage via spinning current Hall probe schemes, thereby cancelling out the offset very efficiently. The magnetic field response of such Hall plates can be computed via the electric potential or via the stream function. Conversely, Hall plates with holes show new phenomena: 1) the stream function exists only for a limited class of multiply-connected domains, and 2) a sub-class of 1) behaves like a Hall/Anti-Hall bar configuration, i.e., no Hall voltage appears between any two points on the hole boundary if current contacts are on their outer boundary. The paper studies the requirements under which these effects occur. Canonical cases of simply and doubly connected domains are computed analytically. The focus is on 2D multiply-connected Hall plates where all boundaries are insulating and where all current contacts are point sized.

general relation between resistivity and voltages measured between peripheral contacts in a plane plate with a hole is difficult. Attempts and a conjecture are given in [18] [19].
In fact the authors in [17] did not give an ab initio explanation for the Hall effect in doubly connected regions. They interpreted the regions near the four contacts as four separate Hall crosses. The two Hall crosses at the current supply contacts are operated in an unusual way: current enters in one branch of the cross, then it is split up and exits through the neighboring branches, while the fourth branch is not connected. In [17] the authors found experimentally that these two Hall crosses add certain voltages to the left and right branches of the ring, which are identical to half of the Hall voltages on the left and right Hall crosses, respectively. Adding up all voltages gives the correct Hall voltage as observed in [17] and also in [10]. However, the main question is still open: why do the two Hall crosses at the supply contacts add exactly the required voltages to match the experiments in [10]? Also the exact way how to add up the voltage contributions is obscure. These questions are not tackled in [17]-the following paper answers them in a rigorous mathematical framework.
In this paper we do not discuss quantum Hall effect phenomena, which are frequently mentioned in the context of multiply connected Hall effect regions [20]. We stay within the realm of classical stationary flow of electric current on a macroscopic scale in 2D. Section 2 explains the "anti-Hall bar within a Hall bar" whose unexpected behavior bewildered several engineers with decades of experience in Hall sensor technology (including the author). Section 3 states the assumptions of the theory. Section 4 presents numerical simulations on a very general doubly connected device to give us an idea of the roles of symmetry, contacts size, and magnetic field strength. Section 5 develops a theory for Hall plates, where all boundaries are insulating. Sections 6-8 apply this theory to simply-connected, doubly-connected, and multiply-connected plane Hall plates without interior current sources. Section 9 explains the role of extended contacts in the limit of large applied magnetic field. Section 10 sums up the main achievements of the paper. In Appendix A we compute the current pattern in the unit disk with and without hole, if point current contacts are on the unit circle.

The "Anti-Hall Bar within a Hall Bar": A Surprising Example Kindles Our Interest
In [10] a rectangular Hall plate with a symmetric rectangular hole was discussed.
We repeat the calculation with the commercial finite element (FEM) code COMSOL

U. Ausserlechner Journal of Applied Mathematics and Physics
MULTIPHYSICS. The geometry of such an "anti-Hall bar within a Hall bar" is shown in Figure 1(a). There the conductive region is a symmetric ring with the outer boundary being a 14 × 2 rectangle and the inner rectangular hole boundary is 13.6 × 1 large (the units are arbitrary). We intentionally exaggerate the aspect ratio of the device to emphasize the astonishing result. The model is 2D with the thickness normalized to 1. Let us also normalize the resistivity to parts of the device, and there the contours of the potential are also straight lines tilted against the current streamlines by 90˚ minus the Hall angle. This is identical to simply connected Hall plates. In Figure 2(a) we plot the potential at the four points C, D, E, F. At zero magnetic field the potential is identical at all four points due to symmetry. With rising magnetic field the potential at the lower point C on the outer boundary decreases while the potential at the upper point F on the outer boundary increases. This gives an increasing output voltage between C and F in exactly the same way as it occurs in simply connected Hall plates.
However, the potentials in the points D and E at the inner boundary remain zero versus magnetic field. Thus, the voltage between D and E does not respond to an applied magnetic field of arbitrary strength.
In Figure 1(b) we inject the current at point G and extract it at point H. Both points are on the inner boundary. Then the potentials in D and E on the inner boundary respond to applied magnetic field, whereas the voltage between points C and F on the outer boundary does not respond to applied magnetic field (see Figure 2(b)). Hence, the stunning result is that the voltage between C and F depends on whether the current is supplied via points on the inner or outer boundary, although the current streamlines near C and F are perfectly horizontal in both cases. This holds even if the aspect ratio of the device becomes infinite and the points of current injection move to left and right infinity-nevertheless at applied magnetic field the voltage between C and F changes markedly if we swap the current contacts between inner and outer boundaries. These numerical findings are in perfect agreement to experimental observations [21]. In this paper we try to clarify this phenomenon. In particular we ask if this behavior is valid only for certain symmetries, for single holes (single ring topologies), or for

Assumptions and Basic Definitions
In this paper we deal only with plane Hall plates whose size in the (x, y)-plane is much larger than their small and homogeneous thickness H t in z-direction. All quantities are constant versus z-position: is applied with external means to the Hall plate. It has a component only in n ( z n is the unit vector in z-direction) and it is homogeneous in the entire (x, y)-plane. It may be weak but it may also be very large.
Then the classical Hall effect is described by We consider Hall plates with four contacts, the contacts having infinite conductivity. Via two supply contacts we force a current supply I through the Hall plate and at the other two sense contacts we tap an output voltage out V . Perfectly symmetric Hall plates have zero output voltage at zero magnetic field. However, in practice Hall plates always suffer from unavoidable asymmetries so that in general one has to account for a non-vanishing output voltage in the absence of applied magnetic field-this is called offset. If magnetic field is applied the output voltage will change. We decompose the output voltage into a sum of even and odd functions of the applied magnetic field. The even part is commonly called the offset and the odd part is the Hall voltage H V . do not need to reverse the polarity of the applied magnetic field to obtain it. This principle is used in most smart Hall sensor ASICs because it improves the zero-point error by a factor of ~500 [29].
We can also measure the potential φ with a single point-sized probe at any location r in the Hall plate. Analogous to (3a-c) we define the even potential even φ and the odd potential odd If the location 1 r is on the first sense contact and 2 r on the second sense contact then Other authors define the Hall potential as the difference of potential with and without magnetic field, which differs from our definition by the magnetic field change of offset This difference is small of order for small magnetic field.
All odd functions vanish at zero applied magnetic field due to their definition.
We denote all even functions at zero applied magnetic field with an index 0 The

Doubly-Connected Asymmetric Hall Plates with Large Contacts
In Figure 3(a) we have an entirely asymmetric Hall plate with asymmetric hole.
It has two large contacts for current supply at the outer boundary and two large sense contacts on the inner boundary. Figure 3 . This becomes more apparent if we compute the Hall geometry factor H G according to its definition in (7), again with the rule that the sense contact at the RHS of global current flow between current contacts is grounded. Figure   3(  This could be used in an auto-calibration scheme of a smart sensor system. To sum up, the behavior on inner and outer sense contacts for this asymmetric Hall plate with large contacts differs, but the difference is milder than in Figure 1: the Hall response is smaller on the inner contacts but it does not vanish altogether unless the magnetic field grows unboundedly. If we make one current contact point sized, skip the large sense contacts, and sample the Hall potential only with point probes, the behavior is similar. Figure   4(a) shows the potential and the current streamlines at large magnetic field and  Therefore, the particular behavior of the "anti-Hall bar within a Hall bar" seems to have two origins: in the case of irregular geometry it is the infinitely small contacts, and in the case of large contacts it is the symmetry according to Haeusler [9].

Plane Hall Plates Where All Boundaries Are Insulating
If all boundaries are insulating the contacts must be point sized. Then it is better to use the stream function instead of the potential, because the boundary conditions specify the stream function and not the potential (see also Appendix A).
with n being the unit vector perpendicular to the curve. Thereby the closed curve may also encircle holes. If a multiply-connected Hall plate has one current input contact and one current output contact and both are on the boundary of the same hole, a stream function exists. The same holds if both contacts are on the outer perimeter. If the current input contact is on the boundary of a different hole than the current output contact, no stream function exists. If the entire boundary of a hole is a single current contact, no stream function exists. This holds also in the limit of vanishing hole size. Topologies with no stream function will be focused on in the follow-up part II of this publication. A proof of (8) will become clear later (see (18) In a thin plane Hall plate it holds: 1) 0  (12) A′ is the area in the (x, y)-plane. For which is a simple multiplication by H t .
Since J is obtained from z H by spatial differentiation in (9), and z H is proportional to ψ in (11), the stream function ψ takes over the role of a 2D "potential" that can be used in a way similar to the potential φ . It holds n n (13) , , x y z n n n are the Cartesian unit vectors. Note the different viewpoint: in (9) H is generated by J , yet in (13) J is generated by ψ . In fact both equations Defining J as the curl of ( ) z ψ ρ − n implies that J is orthogonal to ψ (14) This explains the name stream function, because it is constant along each current streamline. With Maxwell's second equation and (2a) we get for 2D geometries and because of (10). With  Figure 5(a) and Figure 6(c) in [30]). On the other hand the Lorentz force acts on the charge carriers in the whole volume.
Thus the Hall effect appears to be a peculiar interplay between volume and boundary.
If a Hall plate has only insulating boundaries and point-sized contacts where current is impressed, both the Laplace equation and the boundary conditions for the stream function do not contain the applied magnetic field any more. In such a Hall plate the stream function, the current density, and the current streamlines are constant versus applied magnetic field. For the case of a circular disk with peripheral point current contacts this was already stated in [30]. Now we see that it is also valid for multiply-connected regions with point sized contacts whenever a stream function exists.
If we express the electric field in (2a) by the potential and the current density by the stream function this gives At vanishing applied magnetic field (16a) reads For the particular case of all-insulating boundaries we do not need an index 0 Journal of Applied Mathematics and Physics for ψ , because it is identical with and without applied magnetic field. From (16a, b) we get In the ground node it holds Another property of the stream function is its relation to the total current 12 I flowing across any contour (extruded into thickness direction) that starts at point 1 and ends at point 2. Using (13) we get whereby we used n J = ⋅ J n , z = × n t n , 1 ⋅ = ⋅ = n n t t , and t is tangential to the path and points from point 1 to point 2. The current 12 I flowing across the contour connecting the two points 1, 2 is the difference in stream function at these two points divided by the sheet resistance. Inserting (18) into (17b) and comparing with (7) Thus, the Hall geometry factor between the points 1 and 2 is equal to the ratio of the current between the two points and the total current through the Hall plate. This holds for multiply connected conductive regions without extended contacts and without internal current sources (i.e., whenever a stream function exists). Obviously, 1) this ratio cannot exceed 1 and it is smaller if both points 1, 2 are in the interior of the Hall plate with peripheral contacts, 2) it is zero along any insulating boundary without current contacts between points 1 and 2, and 3) it is ±1 if points 1 and 2 are on the same boundary and a single point-sized supply contact is between them on this boundary.
In Hall plates with all insulating boundaries and point sized contacts the current pattern does not change with applied magnetic field (see above). Therefore (19) implies that the Hall geometry factor is constant versus applied magnetic field. Consequently the Hall voltage is directly proportional to the applied magnetic field whenever the current through the Hall plate is kept constant. These U. Ausserlechner devices are linear versus applied magnetic field-within the scope of our assumption that the material parameters , H ρ µ are constant. In engineering practice one may partly compensate for the magnetic field dependence of the material parameters with the magnetic field dependence of the Hall geometry factor in Hall plates with extended contacts [31].
If the integration path in (18) is a closed loop, points 1 and 2 are identical and this leads to (8). Hence, we see that in multiply-connected domains a stream function makes sense only in the absence of internal current sinks and sources.
The stream function jumps in point-sized current contacts according to (18), and so we must not use it there, because it would lead to self-contradictory pre- Hence, the Hall voltage across supply contacts vanishes. The argument holds for point-sized supply contacts, but the Hall voltage also vanishes across extended supply contacts (this can be shown with the methods developed in [32]).
In the absence of an applied magnetic field ( ) These are the Cauchy-Riemann differential equations and therefore the functions In the presence of an applied magnetic field E and J are not parallel, and therefore φ and ψ do not fulfill the Cauchy-Riemann differential equations.
Nonetheless we can construct a complex potential whenever the stream function ψ exists. There are two possibilities: either the real or the imaginary part of the complex potential is equal to the electric potential φ , and the remainder is defined such as to satisfy the Cauchy-Riemann differential equations (enter (13), (16c), and J = 0 J into (2a)).
Hence, the complex potential has no quadratic dependence on applied magnetic field. This gives a simple relation between complex potentials with and without applied magnetic field.
(24) is compatible with (C9) in [15]. If we define a complex electric field by with z x iy = + . If we define a complex current density by 0 E and 0 J are the complex electric field and the complex current density, respectively, at zero applied magnetic field. The leftmost equality in (26) is identical to (2a). If we take the conjugate of (26) we see that the current density is equal to the electric field rotated CCW by the Hall angle (and scaled in length and dimension).
In the review process of this paper the author became aware of [33]. In section 5 of [33] some of the above given arguments are used to explain the "anti-Hall bar" of [10] for irregular shapes and point-sized contacts. In [33] the authors based their arguments on the assumption that for point contacts the current density does not change with applied magnetic field-which holds only for cases, where a stream function exists (they did not explicitly refer to the concept of a stream function).

Hall Plates without Holes and with Point Current Contacts on the Boundary
Riemann's mapping theorem says that all simply connected bounded plane domains can be mapped onto the unit disk with a conformal transformation. Such a mapping is described by an analytical function and it preserves angles. The potentials at corresponding points are identical and also the currents into corresponding contacts are identical. The Hall angle between E and J is also identical, but E and J themselves are generally not identical. If we are only interested in the potential, we can limit the discussion to the circular unit disk of n J s n n n (27) The Hall geometry factor between any two points inside the disk is smaller

Hall Plates with One Hole and with Point Current Contacts on the Same Boundary
These are plane domains with the shape of a ring. After Riemann's mapping theorem, a ring of general shape can always be mapped onto the annular region between two concentric circles [34] [35]. One radius is arbitrary but the ratio of both radii is fixed by the so-called modulus of the doubly connected region. The modulus is a measure of the size of the hole, and it relates to the resistance between inner and outer boundary at zero magnetic field, if both boundaries are thought to be contacts at different potentials (like in a Corbino disk). We set the outer radius equal to 1 and the inner radius is First we assume that all current contacts are on the outer circle as shown in   A5a).
Since the current streamlines do not change with applied magnetic field, we can cut the Hall plate of Figure 7 along the specific current streamline, which encircles the hole, in two parts without changing the pattern of current flow (see Figure 8). The supply currents of both disjunct and simply connected regions are equal to the current partitioning in the original device. The current density fields exactly match before and after cutting. We know that the Hall potential is constant along current streamlines. Therefore it is constant along the cut line and along the hole before and after cutting, because the current streamlines were not changed by the cut. Before the cut we had a single ground node-after the Figure 8. A circular Hall unit disk with insulating hole and with point-sized current contacts at azimuthal angles ± 30˚ is split up in its left and right current path around the hole. The current partitioning is 1 to 5 (see also (A6)). The figures show the current streamlines in grey color and the potential at

Hall Plates with Several Holes and with Point Current Contacts on the Same Boundary
As long as all current contacts are on boundaries and the net current into each boundary is zero, the problem can be described by hole boundary until the current through the straight black lines L1, L2 became equal. Thus there is a single streamline (in red color) being split in two parts by the first hole, then joining up again to a single streamline and being split up again in two parts by the second hole. Therefore, the Hall potential is identical on the boundaries of the square and triangular holes. On the outer boundary the Hall potential is also constant but not equal to the one on the hole boundaries.

Hall Plates with Extended Contacts at Large Hall Angle
In Figure 3 and Figure 4 we saw that extended contacts tend to behave similar to point sized contacts for large Hall angles. This is explained in Figure 10 This behavior can be studied analytically by conformal mapping of the z-plane in Figure 11(a) onto the w-plane in Figure 11 [10]. For symmetrical circular ring domains with extended electrodes on perpendicular axes on both boundaries a rigorous solution via conformal mapping has been known since long ago [9]. For the much larger group of asymmetrical, multiply-connected Hall plates the present paper works out a rigorous theory based solely on the classical laws of macroscopic flow of electric current.
For the first time the three necessary requirements are identified: 1) a stream function must exist, 2) all boundaries must be insulating, and 3) the current across any contour starting at one sense contact and ending at the other sense contact must vanish. The first requirement means that the net current through all closed boundaries must vanish (e.g., current input and output contacts must be at the boundary of the same hole or they must be both on the outer perimeter). The second requirement means that all contacts are point-sized. The third requirement means that no Hall voltage drops along any current streamline. If the Hall plate has boundaries without current contacts, there is always a current streamline flowing along these boundaries, and therefore the Hall potential on these boundaries has no spatial change.
In principle one can do the calculation with the potential φ or with the stream function ψ . However, with the potential one encounters the following problem: If we make a Fourier separation ansatz for φ we need to fullfill the conditions for the radial current density r J r φ = ∂ ∂ at the insulating boundaries. Thereby, we have to differentiate the series term wise, which reduces the convergence of the series drastically-it even diverges at 1 r = . Whenever we do not manage to find a closed formula for the sum of the φ -series we should avoid boundary conditions that use derivatives of φ . Conversely, for the stream function we do not need derivatives at the insulating boundary, which facilitates the calculation.
We use polar coordinates ( ) , r ϕ . The current input contact is located in 1, r ϕ α = = − , the current output is in 1, r ϕ α = = with 0 π α < < . The annular conductive region is in 1 1 r r ≤ ≤ with 1 0 1 r < < . For the stream function ψ we make the ansatz (see [36]) . With (A1) this gives 0 0 A = . Thereby it is allowed to differentiate the ansatz (A1) term-wise because with k A obtained below in (A4b) the term-wise differentiated series converges uniformly in 1 1 r r < < [37].  We draw a small circle of radius 0 r δ → around the negative current contact in ( ) ( ) cos ,sin α α − = r . At points within this circle we compute the current density with (A5b) and (13). A series of the current density in powers of r δ has the dominant term