The Classical Hall Effect in Multiply-Connected Plane Regions Part II: Spiral Current Streamlines

Multiply-connected Hall plates show different phenomena than singly connected Hall plates. In part I (published in Journal of Applied Physics and Mathematics), we discussed topologies where a stream function can be defined, with special reference to Hall/Anti-Hall bar configurations. In part II, we focus on topologies where no conventional stream function can be defined, like Corbino disks. If current is injected and extracted at different boundaries of a mul-tiply-connected conductive region, the current density shows spiral streamlines at strong magnetic field. Spiral streamlines also appear in simply-connected Hall plates when current contacts are located in their interior instead of their boundary, particularly if the contacts are very small. Spiral streamlines and circulating current are studied for two complementary planar device geometries: either all boundaries are conducting or all boundaries are insulating. The latter case means point current contacts and it can be treated similarly to singly connected Hall plates with peripheral contacts through the definition of a so-called loop stream function. This function also establishes a relation between Hall plates with complementary boundary conditions. The theory is explained by examples.


Introduction
Part II of this paper largely builds on part I, where we studied the stream func-tion of plane multiply-connected Hall plates [1]. In part I, we found that in the absence of spiral streamlines, the stream function obeys particularly simple rules when all boundaries are insulating except for the point-sized contacts. Then the stream function is independent of the applied magnetic field and it is linearly proportional to the Hall potential. Thus, there is no Hall voltage between points on the same current streamline. In the following, we will see that for a large group of Hall plates, no classical stream function exists. As a consequence, such devices show entirely different behavior with the most striking new feature being spiral current streamlines. Such spiral current streamlines were known in multiply-connected Hall plates where all boundaries were contacts [2] [3]. A well-known example of such a topology is the Corbino disk, which maximizes the magneto-resistance effect. However, there are numerous equivalent shapes-most of them are multiply-connected. This was already known at a very early time [2]. In [3], it was proven that there is no Hall voltage between any contacts of such a device. In [2], the author explicitly states that in such kinds of devices, the equipotential lines remain unaltered by the application of an external magnetic field. He also gave a general expression for the magnitude of the circulating currents that show up in these devices-we will pick up this thread in Section 3 after some basic definitions in Section 2. In Section 4, we discuss the converse case, i.e., multiply-connected Hall plates with insulating boundaries and point-sized contacts with internal current sources (the case without internal current sources was treated in part I). No theory exists on this kind of Hall plates so far. Finally, Section 5 summarizes all rules of parts I and II for convenient reference. Some symmetries and similarities will become apparent. Appendices A and B give analytical calculations of the current density in singly-and doubly-connected regions with insulating boundaries and spiral current streamlines. There, current is injected near the center and flows to a point on the unit circle. Appendix C deals with a specific detail of reverse magnetic field reciprocity for multiply-connected regions in two dimensions.

Assumptions and Basic Definitions
In this part II, the same assumptions and definitions apply as in part I [1]. Here we repeat the important ones. We assume only negative charge carriers. Then the Hall effect in a plane Hall plate in the (x, y)-plane with small thickness H t is described by ( ) The odd potential is also called Hall potential. The difference in Hall potential at two test points on two contacts is called Hall voltage. The odd electric field is also called the Hall electric field. All odd functions vanish at zero applied magnetic field due to their definition. We denote all even functions at zero applied magnetic fields with an index 0. Inserting (1) for both polarities of the applied magnetic field into (3b, c) and (4b, c) gives relations between even and odd vector fields. where unitary hole boundaries are split in two or more contacts at different potential. These cases are extremely challenging to study with FEM, due to insufficient meshing at the interface of contacts on the same boundary.
Furthermore, the electrical equivalent of such a Hall plate with n contacts is a pure resistor network with resistors between contacts i and j with 1 , ,0 ij R is the respective resistor at zero applied magnetic fields. This gives a network with ( ) 1 2 n n − resistors. Hence, the electric response of such a Hall plate with n contacts is fully described by ( ) = . (f) shows the same data as (e) in a different plot: here the lines and cones denote streamlines and orientation of the odd current density odd J , the height above/below the Hall plate and the color-coding denote the potential at , 10 Journal of Applied Mathematics and Physics controlled sources [8] [9]. If we supply such a Hall plate with a constant voltage source certain potentials will appear at the other (floating) contacts. If we change the applied magnetic field the ratios of resistances will remain the same and therefore the potentials at the floating contacts will also remain constant.
What happens if we supply the Hall plate with a constant current source instead of a constant voltage source? Due to the increase of resistances all potentials will rise by the same factor ( ) ( ) we used the fact that the total supply current flows out of the closed-loop L. Note that the loop L can be entirely within the conductive region, but it may also comprise portions of electrodes in a multiply-connected Hall plate. In the latter case, the loop enters and leaves the electrode in stagnation points of the current density at zero applied magnetic field (see the red 1.2348 V contour lines in Figure 1(a) & Figure 1(b)). Non-vanishing circulation means that the current streamlines are spirals around the current input electrode whenever there is a closed path around it within the conductive region. The same applies to the current output electrode (with opposite sign). If the entire outer perimeter is one supply contact, the current pattern has only one spiral pattern, otherwise, it has two spirals in opposite directions. There is no circulation and no spirals around electrodes, where no net current flows in or out. If a floating electrode encircles both current input and output contacts the circulation along it vanishes. Then this loop L can be split up in smaller loops with branching points being the stagnation points of the current density at zero magnetic field, and at least one of these smaller loops has non-vanishing circulation. At constant supply voltage, the equipotential lines in the Hall plate are also constant versus applied magnetic field. With (8b) the even current density is orthogonal to the equipotential lines. With (8a) the odd current density is parallel to the equipotential lines. Thus, odd J flows in closed loops along equipotential lines (see Figure 1(e)). Summing up all contributions between two fixed potentials 1 2 φ φ < at constant supply voltage gives with (8a) the respective circulating or loop current.
where the unit vector t is tangential to the path, z × = n n t , 0 (12c) is essentially identical to (7) in [2], however, Green expressed the loop current in terms of the capacitance matrix of the electrode configuration instead of its resistance matrix ,0 ij R .
The expression in (12c) is equal to the number of spiral loops of a current streamline within an annular region bordered by equipotential lines through points 1 and 2 (see also (B6a)).

Hall Plates with Point Current Contacts on Different Boundaries or in Their Interior
In Figure  3, which formerly were current contacts in Figure 9 of part I. This holds for arbitrary applied magnetic field irrespective of the strength of the spirals (a proof is given in Appendix C). It also holds, if we move the current contacts arbitrarily along the boundaries of their respective holes (because in part I the Hall potential was homogeneous on all hole boundaries).
If we shrink all holes with single current contacts on their boundaries we end up with a Hall plate where current is supplied at interior points. These arrangements act similar to multiply connected Hall plates even though they might be simply connected. As an example Figure 3 shows a circular disk where the current is injected in the center point and extracted at the rightmost point.    Figure 3 can be regarded as a single point current contact on a hole boundary in the limit of vanishing hole size. Thus the communality is that spiral current patterns are a product of internal hole boundaries being current sources or sinks.
In the case of non-zero net current through a hole boundary it is obvious that the stream function ψ has a problem, because in Section 5 of part I, we saw that ψ is constant on insulating boundaries between contacts, and it jumps across a contact by the amount of current through the contact (see (18) in part I).
If the net current through all contacts on this boundary differs from zero the stream function faces a dilemma: it is forced to change also somewhere in-between two contacts. So the entire concept of a stream function crumbles.
Let us recall from part I that the stream function is proportional to the vertical magnetic field z H caused by the currents , x y J J in an infinitely thick Hall plate with 0 z ∂ ∂ = . However, a stationary magnetic field due to a current makes sense only when this current flows in a closed loop. If the loop is opened, at its ends and this violates Maxwell's first law . This is explained in [10] and it seems to address our problem particularly well. In Figure 9(a) of part I, we were inexact, because we did not show the full current loop, i.e., how the current flows from a battery to the current input contact and from the current output contact back to the battery. Yet it is simple to add some curve through the big hole and insert a battery along this path (see Figure 9 The situation is different in Figure 2 and Figure 3. There we cannot close the current loop without cutting through the conductive region. We cannot solve this problem by pulling current straps out of the (x, y) drawing plane because the Hall plates are supposed to be infinitely thick for our z H versus ψ analogy in part I [1], we are not allowed to work with 3D tricks in a 2D world. Hence, the return current path goes right through the conductive region and this will affect z H and ψ there, too. Such a return current sheet will make z H discontinuous at infinitely many points in the conductive region.
On the other hand, ignoring the current return path altogether gives wrong results for z H and ψ , as the following example illustrates. Suppose an infinitely long hollow cylinder along z-direction. The cylinder consists of poorly conducting material like most high mobility semiconductors. No magnetic field is applied. The cylinder bore is clad with a perfectly conducting contact and also the outer surface is clad with the same material. If we tie one contact to ground and apply a voltage to the other contact a radial current density will result 1 r r − ∝ J n . This is an infinitely thick Corbino disk at zero applied magnetic fields. What is the magnetic field generated by the current density? Solving Maxwell's first is a solution. Thus, the solutions are discontinuous on x-and y-axes, respectively, and the solution is not even unique. Moreover, from physical intuition, we can consider a thin circular disk with radial current density. It will have only azimuthal magnetic field: CW above its top surface and CCW below its bottom surface.
If we pile up an identical second circular disk the azimuthal fields of both disks cancel at their interface. Piling up infinitely many of them will cancel out the field in all test points. In the end, there is no (!) magnetic field caused by this current distribution-which of course contradicts Maxwell's first equation H J (current must always be accompanied by a magnetic field). What went wrong? The current loop was not closed. In reality, currents need to flow in both contacts parallel to the cylinder axis. Both currents vary versus z. The current on the inner contact generates an azimuthal magnetic field on its own within the conductive region. This cannot be handled in a 2D model. A complete 3D model must account for the vertical current component (see [11] [12]).

U. Ausserlechner Journal of Applied Mathematics and Physics
To sum up, there exists no stream function, if the electrodes in a 2D conduction problem are placed in such a manner that the current return path via the battery cuts through the conductive region. If both supply contacts are on the same boundary we can add a current return path outside the conductive region or inside a hole. Then the current density is equal to the curl of a unique magnetic field perpendicular to the plane of conduction. This is the stream function.
It is continuous in nearly all points, i.e., in all points with exception of a finite number of points (namely a finite number of point supply contacts on the boundaries). Then we can interpret ψ as a current potential function. Conversely, problems like in Figure 2 and Figure 3 have internal current sources, i.e., single supply contacts on boundaries or within the conductive region. For them, we cannot close the current loop outside the conductive region or inside a hole in a 2D analysis. These current patterns are not identical to the curl of a unique vertical magnetic field.
In order to reconcile ( ) These are the Cauchy-Riemann differential equations. It means that the function ( ) In (17a), we were allowed to add odd J in the integrand because integration of odd ⋅ J n over a closed path gives zero according to (16). The circulation of a specific pattern of spiral current streamlines is computed in Appendix A. Note that the circulation in (17a) [13] is also a way to prove that the circulation around single point contacts is given by (17a) and that it vanishes when both supply contacts are encircled. In [13] this method was used for conductive regions with one boundary, but it can also be generalized for more boundaries (like ring domains) if one uses a superposition of infinitely many images [14] [15].
Inserting the right hand sides of (3b, c) into (6d) and eliminating the vector product by use of (15c) gives: In (18) (18), which is a von Neumann boundary condition that does not depend on the applied magnetic field. Therefore the term in the brackets of (18) does not depend on the applied magnetic field. Yet, at , 0 a z B = it is equal to 0 φ (up to an arbitrary additive constant) and therefore In ( = there. Therefore we have to ground a point on the perimeter of the multiply-connected Hall plate. (19) in (18) (20) which is identical to (10a) in Section 3 (both times the Hall plates are supplied by constant current sources). (19) can be seen as an alternative definition of the loop stream function, being the negative increase in even potential divided by φ φ − is also homogeneous there according to (19). With (19) we can eliminate the even potential in (15b).
Curves of constant Hall potential are orthogonal to curves of constant Next, we compute the Hall electric field perpendicular to an insulating boundary from (6a) with odd 0 ⋅ = J n and with (20).
Thus, the Hall potential is a harmonic function, which satisfies von Neumann boundary conditions with values that are linearly proportional to applied magnetic field. Therefore H φ is perfectly linear in With (20) and (25) where 0,12 0,12 , I I  are the currents flowing across a contour that connects points 1 and 2. Thereby the applied magnetic field vanishes and all boundaries of the Hall plate are electrodes (for 0,12 I  ) and insulating (for 0,12 I ), respectively. Thereby the same supply current flows into the same boundaries. Analogous to (7) and (19) in Part I, we can write for the Hall voltage between arbitrary points 1 and 2     where we define =0 Ψ in the ground node. These equations have greater similarity to (13) and (17a) in part I.

A Summary of Simple Rules
In parts I and II, we derived simple rules to understand the classical (non-quantum) Hall potential in thin, plane and homogeneous regions with linear material properties. Here we compile them.
It turns out that internal current sources are a key issue in multiply-connected

Discussion
We A major portion of this work was devoted to multiply-connected Hall plates with insulating boundaries and point-sized contacts, for which we developed a new theory. In part I, we showed that the Hall voltage vanishes if it is tapped between two points on a boundary without current contacts. This is the case in Hall/Anti-Hall bars. In part II, we introduced internal current sources and then a Hall voltage between the very same taps exists again! Our new theory can be applied to Hall plates with anisotropic conductivity as it is caused by mechanical stress in a cubic crystal. Then, in a preceding isotropization step, one can replace the original geometry by a distorted geometry with isotropic conductivity (see [16] and references therein). This procedure does not change the number of holes in a multiply-connected Hall plate. It also preserves point-sized contacts and boundaries, which are being entirely covered by electrodes. Therefore, we expect no new phenomena in multiply-connected Hall plates with anisotropic conductivity.
For engineering purposes, the magnitude of the Hall potential is less important than its ratio over noise at given impedance level and costs for chip area.
Under these boundary conditions, Hall plates with peripheral medium sized contacts turn out to be optimum [17].
with r n being the unit vector in radial direction. The current flowing out of any circle with 1 r < around the current input is given by: If we insert (A1) into (A2) into the integrand in (A3) all terms except 0 A vanish and we get: Next, we evaluate (A2) at the unit circle 1 r = and set it equal to the Fourier series of the outflowing current.
This means that the spacing between consecutive loops of the current streamlines gets smaller and smaller the more they approach the current input contact while the current density goes to infinity. In other words, every current streamline encircles the origin infinitely often, regardless if the magnetic field is weak or strong. The only difference is that at weak magnetic field the loops are much closer to the origin than at strong magnetic field. For a weak magnetic field with There is a remarkable difference between zero and non-zero This self-field phenomenon was studied in Corbino disks in [11] [18] [19].
There it was explained that the self-field adds constructively to the applied magnetic field if the charge carriers flow inward, and destructively if they flow outward. This holds for both polarities of charge carriers, current, and applied  [20], the authors discuss if this mechanism can be used in diodes, inductors, and energy storages.
If the point of the current input contact is not in the origin but somewhere else inside the unit disk, we can use a Möbius (bilinear) transformation to conformally map the unit disk on itself and the input contact onto the origin. If the Hall plate has some other simply-connected shape than a disk, there is always some conformal mapping onto the unit disk. If the current output is not on the outer perimeter, but also inside the unit disk, we can replace the problem by a superposition of two disks, where one disk has a current input contact on the unit circle and the other disk has a current output contact at the same location.
A general formula for all these cases can be found in [13].

Appendix B
If we cut out a circular hole around the origin and inject the current in any point on this insulating hole boundary, the current streamlines will not encircle the origin infinitely many times any more. The ring-shaped conductive region in  Both solutions do not give the correct radial current density on the inner circle U. Ausserlechner Journal of Applied Mathematics and Physics  180˚ towards 0˚ with decreasing speed (see also Figure 9).
The spiral current pattern in Figure 9 has two stagnation points, one on the outer perimeter and one on the inner hole boundary. Since With (B6b) we can estimate how small the hole must be that the streamlines encircle it at least once: for a magnetic field with this is a flux density of roughly 0.9 T in low n-doped silicon) the hole must have a radius 1 r smaller than 28 5.2 10 − × . It means that for a Hall plate as large as the universe the hole must be smaller than 6 cm. If this Hall plate had no hole the current spiral around its center point contact would have infinitely many turns within a circle with 6 cm diameter but only a single turn outside on its way to the end of the universe. If the applied magnetic field is only 100 times larger we need a disk of only 11.3 cm diameter with a 6 cm hole to observe one turn of the current streamlines. Hence, the effect is extremely sensitive to the strength of the applied magnetic field.