Simple Proofs of Upper and Lower Envelopes of Van Der Pauw’s Equation for Hall-Plates with an Insulated Hole and Four Peripheral Point-Contacts

For plane singly-connected domains with insulating boundary and four point-sized contacts,  , van der Pauw derived a famous equation re-lating the two R with the sheet resistance without any other parameters. If the domain has one hole van der Pauw’s equation becomes an inequality with upper and lower bounds, the envelopes. This was conjectured by Szymański et al. in 2013, and only recently it was proven by Miyoshi et al. with elaborate mathematical tools. The present article gives new proofs closer to physical intuition and partly with simpler mathematics. It relies heavily on conformal transformation and it expresses for the first time the trans-resistances and the lower envelope in terms of Jacobi functions, elliptic integrals, and the modular lambda elliptic function. New simple formulae for the asymptotic limit of a very large hole are also

It relies heavily on conformal transformation and it expresses for the first time the trans-resistances and the lower envelope in terms of Jacobi functions, elliptic integrals, and the modular lambda elliptic function. New simple formulae for the asymptotic limit of a very large hole are also given.

Introduction
The sheet resistance of a plane conductive layer is of prime importance in thin layer technologies. It is used pervasively in micro-electronic manufacturing to monitor the properties of thin conductive layers. It is given by tensors as they occur if magnetic fields are applied, but Section 4 extends the range of validity to include the Hall-effect.
The plane conductive region of a conventional Hall-plate has four peripheral point-sized contacts with consecutive labels 0, 1, 2, 3 in a positive mathematical direction (i.e. counter-clockwise). Thus, if we move along the boundary from contact 0 via 1 and 2 to 3 (in ascending order) the conductive region is on the left hand side. If current is forced to flow between two contacts and the voltage is tapped between two contacts, then van der Pauw used the ratio vdP : exp exp .
X and Y are abbreviations for the exponential terms. Then the basic result for Hall-plates without a hole in [1] is van der Pauw ' s equation, The peculiarity of (3) lies in the fact that it relates measurable electrical quantities 01, 23  However, occasionally one faces the problem that some of the above given requirements are not fullfilled. Then van der Pauw's method gives inaccurate or even wrong results for the sheet resistance. This is an inherent problem in materials science, where one needs to characterize novel materials. Often these samples have poor quality and poor homogeneity due to limitations in the manufacturing process, especially when the fabrication on a small laboratory scale is not yet mature [6] [7] [8]. From a practical standpoint, one would like to have a procedure that detects poor sample homogeneity and that gives error bounds for the derived sheet resistance. Inhomogeneous conductivity is supposed to have a similar effect to small voids. This is the motivation to study Hall-plates with holes.
The topic was pioneered over the last decade in a couple of papers by Szymański and coworkers [9] [10] [11] [12]. They introduced the concept of upper and lower envelopes (u.e., l.e.) for conductive samples with a hole, l.e. vdP u.e. 1. ≤ ≤ = (4) For constant hole size, the lower envelope depends only on X. Further contributions came from [13]. For nearly one decade (4) was just a conjecture, while a strict mathematical proof was missing, until only recently a thesis solved this problem (with its potential generalization to more than one hole) at a fairly elaborate mathematical level [14] [15]. The present article gives new and simpler proofs for samples with a single hole with less sophisticated mathematics and closer to the physical intuition of an electrical engineer. The employed mathematical tools are series expansions and conformal transformations which lead to Jacobi functions and elliptic integrals.
There is a certain similarity of the current topic with another topic called the Hall/Anti-Hall bar [16] [17]. This article starts with the easier case of a small hole, which leads us straight to the star-configuration and the minimum of the van der Pauw function. Then we compute the trans-resistances for arbitrary hole size with conformal transformations, and we prove the upper and lower envelopes. We discuss some properties of the trans-resistances and how they are affected by a magnetic field.

Series Expansion of the Potential
Let us start with a plane irregular ideal Hall-plate with a single irregular hole of arbitrary size. The entire inner and outer boundary is insulating except for four point-sized contacts 0 3 , , C C  . Current 01 I is injected by an ideal current source at 0 C and extracted at 1 C while the voltage from 3 C to 2 C is measured. We know from Riemann that a conformal map exists, which maps the irregular Hall-plate onto the unit disk with a central hole of radius r is the Riemann modulus of the singly-connected domain. Let us rotate the disk such that the current contacts 0 1 , C C are symmetrical to the real axis. Then, the azimuthal locations of the contacts are (see Figure 1) being the azimuthal angle of the location of C  . The electrostatic potential at zero magnetic field, 0 φ , is given in [17]. At the outer perimeter, it holds ( ) ( ) With (105) we get C . The potentials in 2 3 , 0  01 sheet  1   4  2  1  1  1  01 sheet  4  2  1  1  1  1   1 cos  1 ln  2  1 cos   1  2 cos  2  1  2  ln  .  4  1  2 cos The term 0 =  corresponds to the singly-connected Hall region with 1 0 r = . The measured van-der-Pauw voltage is If we inject the current at 1 C , extract it at 2 C and measure ( ) ( ) 1  1   2  1  2  1  2  3  2  3   2  1  1  2  3  1  3  3  2   2   2  2  3  2 2 U  ( ) ( ) In this equation, the coefficient of 2 1 r is positive, because 1 0 ϕ < < π and . This is the simple proof that a small insulating hole reduces the van der Pauw function below 1. Equation (17) is also derived in [9].
Which of all these sets causes the steepest drop of vdP for small holes? In other words, for fixed trans-resistances of a singly-connected Hall-plate, how do we have to place the contacts such that the van der Pauw function becomes most sensitive to a small nucleating hole? Keeping 0 X fixed implicitly defines the azimuthal position 1 ϕ as a funcion of the other two positions, ( ) The minimum of means the largest negative slope of vdP versus 2  3  1  2  3   cos 2  cos  cos  cos  cos  0,  ϕ  ϕ ϕ  ϕ  ϕ  ϕ  =  −  ∧  −  =  (20) with the only meaningful solution Let us call this specific pattern of contacts a star-configuration-the contacts are in the vertices of a rectangle inscribed into the perimeter of the Hall-plate.
This is the small-hole approximation of the lower envelope as it will be explained in Section 3.3.
For holes of arbitrary size a strict proof of vdP 1 ≤ appears to be difficult, because the trans-resistance 01,23 R may increase or decrease versus hole radius This asymmetric geometry can be mapped onto a symmetric one with a central circular hole (Any plane domain with a single hole can be mapped conformally to a circular ring with inner radius 1 r and outer radius 1 [18]). The radial slit may be placed in-between the two voltage taps. This increases the trans-resistance.
However, it may also be placed outside the two voltage taps. Then, a larger fraction of the total supply voltage drops outside the voltage taps, and therefore the trans-resistance becomes smaller. Thus, by the placement of the hole one can make the voltage 32 V smaller or larger.
Next we have a look at the trans-resistances of Hall-plates with contacts in a star-symmetry as defined in (21) and shown in Figure 3(a In contrast to the example given above, both trans-resistances increase with growing hole if the contacts are in a star-configuration. This can be readily seen in (27). The plot in Figure 3(b) visualizes this fact. The inequality (17) holds for small holes, and according to (27) both trans-resistances increase for larger holes.
Therefore the inequality vdP 1 ≤ holds also for large holes in the case of a symmetry.    , r ϕ   -plane. The surface in Figure 3  R surface at two different heights we get two curves, which have no common point (they do not intersect and they do not touch). If we mirror the second curve at 1 4 ϕ = π  this reflects the measurement of the second trans-resistance 12,30 R . If the two curves cross, due to their monotonicity they have to cross in a single uniquely defined point, which gives 1 r  and 1 ϕ  as shown in the example of . Equation (28)  , r ϕ   -plane clock-wise. Since This gives the unique solution of a hypothetical  -Hall-plate, which has the same trans-resistances as our original Hall-plate, albeit it has a different hole and a different sheet resistance.
This argument clearly shows that one cannot determine the sheet resistance of a doubly-connected Hall-plate with the measurement of both trans-resistances as in the singly-connected case, unless one has additional information about the hole or the contacts placements. In general it holds sheet sheet R R ≠  , either one can be larger than the other one. For the  -case the value of the van der Pauw function vdP is bounded: we insert (29) into (2) ( ) 12 which is fulfilled due to our assumption 01,23

Conformal Mapping of the Annular Hall-Plate
Next we apply conformal mapping to the general ring-shaped Hall-plate from Figure 1. Since the current contacts are symmetric to the real axis it is clear that all points the real axis are at the same potential, say 0 V. There we can insert a contact. We can further cut the ring apart at the positive real axis, apply contacts at both cut edges, and short them with a wire (see Figure 4(a)), without affecting the potential in the annular region. From the discussion in [17] we know that the fraction ( ) 1 1 ϕ − π of the current flows through the shorted wire, independent of the size of the hole. The conformal transformation ( ) maps the annulus in the z-plane to the rectangle in the w-plane shown in Figure   4(b). The width of this rectangle is ( ) 1 ln r and its height is 2π . The outer perimeter of the ring in the z-plane appears at the right edge of the rectangle in the w-plane, whereas the hole boundary appears at the left side of the rectangle. A Schwartz-Christoffel transformation maps this rectangle from the w-plane onto the upper half of the ζ -plane in Figure 4(c), In the w-plane the current contacts , C G are placed symmetrically to the large contacts , AB HJ . Thus, also in the ζ -plane they are symmetrically to the large contacts. From the sequential order of the points on the rectangular boundary in the w-plane it follows the same order in the ζ -plane, where , d . 1 1 Combining (34) with the modular lambda elliptic function ( ) , with the Jacobi-sine function In an analogous way we find the locations of the voltage taps , A final transformation maps the upper half of the ζ -plane onto the infinite stripe in the t-plane in Figure 4 The point of the input current is at G t → ∞ , the point of the output current The ultimate goal of all these transformations is to achieve homogeneous current density in the stripe in the t-plane. Then the distance between points D t and F t gives the voltage 32 V . There are still two unknowns 3 5 , c ζ to be determined. With 5 ζ we make the width of the slit zero, with the complete elliptic integral of the third kind whereby the integration path is an infinitely small semi-circle around 0 ζ , i.e., Figure 4(c)). We arbitrarily choose the width of the stripe equal to 1. With (the equality at the right side comes from (113) and (38) Both integrals (45) and (46) The hole size is reflected by 4 ζ (see (36)), and the three azimuthal positions of the point contacts are given by 0 2 3 , , ζ ζ ζ (see (37), (39)). Expressing the trans-resistance in terms of the physical parameters 1 In (48)

Proof of the Upper Envelope
The upper envelope was first conjectured in [9]. It reads vdP 1, ≤ for arbitrary placement of the point-contacts on the outer perimeter of a Hall-plate with one insulated hole of arbitrary size. The inequality (49) was proven recently in [14] by arguments using the prime function and Fay's trisecant identity. This Section presents an alternative proof based on the conformal mapping in Figure   4(d). It is short and elegant and it needs no numerical computations.
We start with a general contact arrangement in Figure 4(a), If accidentally 1 2 ϕ > π we shift all contacts by one instance to get 1 2 The current splits in two parts, one flowing left around the hole and the other one flowing right around the hole. Thus, there must be a point F' right of the hole, which has the same potential as point F (=contact 3 C ) left of the hole.
There must also be a point D' right of the hole, which has the same potential as point D (=contact 2 C ) left of the hole. Let us call the potential in point F 3 In Figure 4(d) we can easily localize points F' and D'. Point F' has the same horizontal position as point F, however, point F' is on the upper edge of the stripe, whereas point F is on the lower edge. The same applies to points D and D'.
When the second trans-resistance 12,30 R is measured, current flows between Journal of Applied Mathematics and Physics points C (=contact 1 C ) and D (=contact 2 C ) and the voltage is measured between points G (=contact 0 C ) and F (=contact 3 C ). Analogously, for 12 ,3 0 R ′ ′ current flows between points C and D′ and the voltage is tapped between points G and F'. However, Figures 4(a)-(d) do not apply in this case, because now the potential distribution is asymmetric. Hence the potential along the straight line HJ is not constant and therefore we are not allowed to insert an extended contact there. In fact we have to step back to (31) which maps the annulus of Figure 1 without a cut and without large contacts , AB HJ to an infinite stripe made up of rectangles like in Figure 4(b) lined up along the { } w ℑ direction yet without the extended contacts. Instead of the annulus we can think of a helical track that winds around the out-of-plane axis of Figure 1 infinitely often, whereby all four contacts repeat after every full revolution. This is shown in Figure 5(a), where we have infinitely many current and voltage contacts, each ones shorted with a pole, and the potential is periodic in each turn of the spiral.
The first turn of the spiral for azimuthal angles 0 2 ϕ ≤ < π is called the Riemann sheet #0. It is followed by Riemann sheet #1 for azimuthal angles 2 4 ϕ π ≤ < π and it is preceded by Riemann sheet ( ) # 1 − for azimuthal angles 2 0 ϕ − π ≤ < . This trick extablishes an equivalence between the doubly-connected domain in Figure 1 and the infinite singly-connected domain in Figure 5 The reciprocity principle [20] says that at zero magnetic field the voltage between F D ′ ′ for current flowing between GC is identical to the voltage between To sum up, we have two sets of contacts in Figure 1, the original points Let us repeat the contraction process infinitely often, until all four contacts are infinitely close together.
In this limit the contacts are so close together that the current arcs between the current contacts are tiny. Then the hole is comparatively distant and it does not affect the current distribution any more. Thus the potentials at the voltage contacts become identical to the potentials in a singly-connected Hall plate.
However, for singly-connected Hall plates the van der Pauw Equation (3) holds.
Since the second trans-resistance decreased during the contraction process, the inequality (49) must hold before contraction. This completes the proof.
The essential step in the proof was to show that 12 ,3 0

12,30
R R ′ ′ < holds. To this end we used the arguments of the multi-storey Hall-plate in Figure 5 to justify We have to prove that 03 03 V V ′ < . Per definition, the point 2 C ′ was obtained from 2 C by a contraction process, therefore it holds C ′ is in the lower half of the z-plane it holds . Consequently, in any case the first current is larger than the second current, because 2 C is more distant from 1 C than 2 C ′ is from

Derivation of the Lower Envelope
The upper envelope theorem implies that for any doubly-connected Hall-plate we can find a  -configuration which has the same trans-resistances 01,23 12,30 , R R and the same sheet resistance. This is not a specific property of  -configurations. Also other contact patterns have the same property: e.g. contacts with 1 2 2 ϕ ϕ = < π < π and 2 3 ϕ ϕ − = π − π , let us call them type 2 configurations, are also able to assume any physically meaningful pairs of values for the two trans-resistances (see Figure 7).
In the van der Pauw plane of Figure 7, a specific Hall-plate is represented by a dot. During the contraction process (c.p.) this point moves vertically up in the van der Pauw plane until it finally is on the straight line    9] conjectured that the lower envelope is given by  -contact arrangements and this was also proven in [14]. For small holes, our derivation of (21) leads straight to the same conjecture. A precise statement of the lower envelope reads: For a fixed trans-resistance 01,23 R the arrangement of the point contacts for largest 12,30 R is a  -arrangement.

A Proof of the Lower Envelope
In mathematical terms the lower envelope is defined like this: Equation (48) The replacements (13) were used to compute 12,30 R with the same formula as Analogous to (21) the  -configuration is specified by

U. Ausserlechner Journal of Applied Mathematics and Physics
We define the following function where we skipped the second argument in the Jacobi functions, e.g., The second part of the upper envelope theorem says that 12,30 R has an extremum, which means ( ) This gives in  -configuration. Adding both equations and using (59) and (60) gives

U. Ausserlechner Journal of Applied Mathematics and Physics
Re-inserting this into the first equation of (62) gives Combining (64) and (61) gives To sum up, we have to proof the validity of (63) and (65). From the contraction process we know that the extremum of 12,30 R cannot be a minimum, it must be a maximum. The nice feature is that both equations have an identical shape, they differ only in the test point 0 x . Thus we only have to prove for . 2 2 , , From the reciprocity principle in [20] Combining (67) and (59) gives For a Hall-plate with contacts in  -configuration it holds (55). Inserting this into (68) gives (66), which completes the proof. An alternative proof of (66) is given in Appendix C.

The Minimum of the Van Der Pauw Function
With (57)  and .
The lower envelope is identical to general  -configurations with 1 0 2 ϕ < ≤ π (insert the first line of (52) into (69)). For the specific  -configuration with (74) Figure 9 shows how the trans-resistances and the van der Pauw function change when the size of the hole grows from zero to full size while the contacts positions remain constant. In the van der Pauw plane of Figure 9(a) curves 1, 3 and 4 show that one of the two coordinates , X Y may increase initially before it decreases (the directions of growing 1 r are indicated by the arrows on the curves). Along the other curves, both coordinates , X Y decrease monotonically for all hole sizes 1 : 0 1 r → . In the limit of infinitely thin annular regions, all curves end in the origin ( ) ( ) In all cases, the van der Pauw function vdP decreases monotonically versus 1 r , Figure 9(b).
I have no rigorous proof of this conjecture. The plots in Figure 9 r < (see curves 1, 2, 4) or for 1 0.8 r > (see curve 6). In these cases, the van der Pauw function is not a very sensitive measure to detect holes.
A distinct feature of the curves in Figure 9 (16). The small-hole-angle 0 χ vanishes for ( ) ϕ ϕ ϕ ϕ χ ϕ ϕ which has two meaningful solutions ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ In words, if two non-neighbouring peripheral contacts lie on a straight line through the center of the annular Hall-plate, the curves in Figure 9(a) start perpendicularly from the upper envelope. This comprises all star-configurations, but it is more general than star-configurations. From (22) we know that for  -configurations in the asymptotic limit of a small hole the van der Pauw function vdP has the steepest decline versus hole size, as a quantity to detect small holes, vdP becomes most sensitive if the contacts are in a  -configuration. Therefore, for small 1 r the red curves 5  , 6  are below the blue curves 1, 2, 3, 4 in Figure 9 , X r Y r also start perpendicularly from the upper envelope, but for them the slope of the van der Pauw function differs from (22), ϕ ϕ ϕ ϕ χ ϕ ϕ This condition is fullfilled only if three or all four contacts approach infinitely closely. From Figure 9( For the blue curves 1, 2, 3, 4 in Figure 9(a) we get 0 72.29 χ = − , 18.08˚, 89.74˚, 77.33˚, respectively, whereby I define the sign of 0 χ equal to the sign of , this is identical to the sign of ( ) ( )

The Asymptotic Limit of a Very Large Hole
In the limit 1 1 r → the annular region of the Hall-plate degenerates to an infinitely thin ring. Then the trans-resistances grow unboundedly. We use 1 k → in (57) with (72) to compute the limit of ( ) , , f x y z . With (117) which is plotted as the green curve 3 in Figure 8. Let us define This gives 1 cos cos 2 A numerical inspection shows that we can find solutions 1 2 , ϕ ϕ of (90) for arbitrary 0 0 1 X < < . They define curves in the van der Pauw plane, which start from any point on the upper envelope and go towards the origin ( ) ( ) , 0,0 X Y = with 1 0 χ = . An example is curve 3 in Figure 9(a), which has 0 0.1 X = and 1 0 χ = . Interestingly, in Figure 9(b) curve 3 remains at vdP 1 ≈ for 1 0 0.95 r < < and only for very large holes 1 0.95 r > the van der Pauw function drops sharply. The numerical computation of curve 3 in Figure 9(a) and Figure 9(b) is tricky, it needs 5000 digits.

Checks for Correctness of the Derived Formulae
The formulae of the Section 2 are consistent with [9] for Journal of Applied Mathematics and Physics where the quantities on the left hand sides of (93) are from this article and the quantities on the right hand sides are from [9].
In Figure 1 which is identical to (A11b) in [17]. Thus, (57) holds in the limit of singlyconnected Hall-plates. Moreover, a series expansion of (57) for small k (small 1 r ) leads to (17). (It is lengthy and arduous and therefore I do not report it in detail here.) For hole sizes of 1 0, 0.1, 0.5 r = and 0.9 I computed the potential in 2 15 , 20 , ,175 ϕ =    analogous to the preceding paragraph (i.e., via 01,23 R with 3 ϕ = π ) and compared it with results of a finite element simulation with COMSOL Multiphysics. There I used a plane two-dimensional model in application mode "emdc" (static conductive media). Thickness and conductivity were set to 1 m and 1 S/m, respectively. Due to symmetry, only the upper half of the annular ring was modelled with a fine mesh of 917,504 elements. All boundaries were set insulating, except for the segments on the real axis, which were grounded to 0 V. A current of 1 A/m was extracted from contact 1 C at position 1 10 ϕ = . Figure 10 shows the potential along the perimeter for these four cases and the relative error between analytical and numerical results. The relative errors R -case the full geometry was modelled with a mesh of 1.1 million elements, see Figure 11(b). A current of 1 A was injected at point C and extracted at point D. Point D was also grounded to 0 V. According to the theory in [17], the current through the handle should be ( ) ( )     (27) and (57). It also relates to the prime function of the annular Hall region, see [15].

The Hall-Plate at Applied Magnetic Field
As it was shown in [17] the current density does not change when magnetic field is applied, whereas the potential indeed depends on the magnetic field. Thus we can compute the potential at zero magnetic field, 0 φ , derive its current density 0 J and its stream function ψ , and compute the potential [17] it is derived that the Hall potential is constant along a current streamline (the Hall potential is the difference in electric potential at positive and negative applied magnetic field, it comprises only terms of odd order of the magnetic field).
Because of the point-contacts the potential comprises only linear terms of the applied magnetic field, there are no even order terms of the magnetic field (no magneto-resistance terms). Since a current streamline flows from the input con-

Singly-Connected Hall-Plate with Extended Contacts in Star-Configuration
If a Hall-plate has no hole and if its contacts have finite size the van der Pauw function also deviates from 1. Thereby the extra degree of freedom from the hole is replaced by the additional parameters for the finite sizes of the contacts. For the simplified case of a star-arrangement of contacts (also called odd symmetry in [25]) at zero magnetic field closed analytical formulae are available in the literature (combine [25] with (C24), (C25) in [26]), tan tan whereby the angles 1 2 , α α are defined in Figure 12(a). It follows  Swapping contacts and isolating boundaries gives the complementary starconfiguration (also called even symmetry in [25]), see Figure 12(b with 1 2 , , x λ λ λ from (99). In summary, large contacts increase the van der Pauw function (at least for the symmetric cases of Figure 12), whereas a hole reduces it.

Conclusions and Suggestions
In this paper, I studied the case of an annular Hall-plate with insulating bounda-   JacobiZeta ArcSin , u k     . Frequently we are interested in aspect ratios of rectangles from conformal maps of Hall-plates.
Then the ratio ( ) ( ) y K k K k ′ = shows up. This function is monotonic. Thus, its inverse exists, this is the modular lambda elliptic function ( ) L y [27].
Inversion of (106) gives the Jacobi-sn function and the Jacobi amplitude

Appendix C
An alternative proof of (66) is by direct computation of the partial derivatives    ,   sc  sn  ,  sn  , ,  dn   sn  ,  sn  ,  ,   cn  sn  sn  ,  sn  , . dn x k which completes the proof of (66).