Advances in Condensed Matter Physics

Volume 2018, Article ID 7812743, 24 pages

https://doi.org/10.1155/2018/7812743

## An Analytical Theory of Piezoresistive Effects in Hall Plates with Large Contacts

Infineon Technologies, Villach, Austria

Correspondence should be addressed to Udo Ausserlechner; moc.noenifni@renhcelressua.odu

Received 13 February 2018; Accepted 21 March 2018; Published 4 June 2018

Academic Editor: Yuri Galperin

Copyright © 2018 Udo Ausserlechner. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Four-terminal transducers can be used to measure the magnetic field via the Hall effect or the mechanical stress via the piezoresistance effect. Both effects are described by an anisotropic conductivity tensor with small offdiagonal elements. This has led other authors to the conclusion that there is some kind of analogy. In both cases the output voltage depends on the geometry of the device and the size of the contacts. For Hall plates this influence is accounted for by the Hall-geometry factor. The alleged analogy proposes that the Hall-geometry factor also applies to four-terminal stress transducers. This paper shows that the analogy holds only for a limited class of devices. Moreover, it is shown that devices of different geometries may have identical magnetic field sensitivity but different mechanical stress sensitivities. Thus, shape optimization makes sense for mechanical stress sensors. In extreme cases the output voltages of vertical Hall-effect devices may have notable magnetic field sensitivity but zero mechanical stress sensitivity. As byproduct, exact analytical formulae for the equivalent resistor circuit of rectangular and circular devices with two perpendicular mirror symmetries are given. They allow for an accurate description of how mechanical stress and deformation affect the output offset voltage and the magnetic sensitivity of Hall-effect devices.

#### 1. Introduction

Resistive devices with four terminals have been proposed as mechanical stress sensors for a long time[1, 2]. They are called four-terminal piezotranducers [3] or van der Pauw stress sensors [4]. The very same phenomenon is observed in Hall plates at zero magnetic field, where the zero-point error (= initial offset error prior to spinning current scheme [5]) is very sensitive to mechanical stress. This was occasionally called Kanda effect [6] or pseudo-Hall effect [7]. An alternative view on the same topic is how to determine anisotropic resistivity from measurements on van der Pauw samples [8]. Over the past five decades a large number of papers have been published on various aspects of this bundle of problems. However, it seems that no analytical closed form expression was found, which relates the anisotropic conductivity to the output voltage of such a device with four contacts of arbitrary size. Conversely, the analogous problems for isotropic van der Pauw measurement at zero magnetic field and for Hall plates at small magnetic field have been solved recently [9–12]. In the following we apply similar methods to the piezoresistive effect in these devices. Thereby we use the affine transformation of van der Pauw [8] in order to replace the original device having anisotropic resistivity by an equivalent device having isotropic conductivity. For these devices with isotropic conductivity we compute the equivalent resistor circuit (ERC), which describes the behavior of the devices under the action of mechanical stress and magnetic field. For the ERC of rectangular devices we use our own results of former works [9–12], whereas we derive new formulae for the ERC of circular devices.

The outline of this work is as follows. Section 2 sums up basic phenomenological facts of the Hall effect and of piezoresistivity as they are known in the literature. The goal is to contrast similarities against differences of both effects. Section 3 briefly repeats the affine transformation of van der Pauw and specifies it for (100)-planes of cubic crystals, like silicon. With these formulae it becomes apparent that vertical Hall-effect devices are very slightly affected by mechanical stress. Section 4 elaborates on how rectangular and circular devices respond to biaxial inplane stress. There we make ample use of the ERC as it was developed in prior works [9, 10]. Section 4 closes with a comparison of several device shapes versus Wheatstone bridge circuits of resistor stripes in orthogonal directions: which one has the largest sensitivity to mechanical stress? Section 5 discusses the effect of biaxial inplane stress on the Hall-geometry factor of Hall plates. This leads to a drift of magnetic sensitivity versus mechanical stress, which is caused by piezoresistance and not by piezo-Hall effect. Finally, Section 6 compares the proposed analytical theory with results of other authors.

#### 2. Similarities of Piezoresistance and Hall Effect

In a thin plate-like material with isotropic ohmic resistivity the Hall effect is described by the relation between electric field and current density with the odd symmetric resistivity tensor in ()-coordinates is the resistivity at zero magnetic field, is the Hall mobility, and is the magnetic flux density component perpendicular to the Hall plate, which lies in the ()-plane [13]. The inverse relation uses the conductivity tensor is the conductivity at zero magnetic field. Equations (1a) and (1b) assume negative charge carriers. Alternatively to the literature also uses the Hall coefficient , where is the sign of the charge carriers. The static electric field is longitudinal, i.e., its curl vanishes. Therefore it can be expressed as the negative gradient of a scalar electric potential . In the quasistatic case the divergence of the current density vanishes . Combining these equations with (1b) gives whereby we assumed that , , and are constant versus spatial coordinates. The mixed-order terms cancel out and therefore (2) is the Laplace equation. Thus, the Hall effect does not enter (2); with and without magnetic field it is the same partial differential equation for the potential. The magnetic field affects only the boundary conditions: the potential at the contacts or the current flow via the contacts is forced by the external circuitry. At the isolating boundary the current density must flow parallel to the boundary and this means or in other words (where and denote normal and tangential direction and the plus/minus sign depends on the direction of the magnetic field, the sign of the charge carriers, and if the boundary is left or right to the current flow). In other words, at the isolating boundary there is the Hall angle between the electric field and the boundary. To sum up, the potential in a Hall plate and the output voltage of a Hall plate can be computed by solving the Laplace equation and satisfying these boundary conditions. For Hall plates with two perpendicular mirror symmetries this was done in the seminal work [14] and the result iswhere is the current related magnetic sensitivity, is the sheet resistance (equal to one over the product of conductivity times thickness of the Hall plate), is the current flowing through the Hall plate, and is the Hall geometry factor. At small magnetic field we write and then the Hall geometry factor depends only on the lateral geometry (= the layout) of the Hall plate, not on its thickness and not on any material parameters. We can compute for any geometry of Hall plates from scratch (in rare cases analytically and in general numerically), but it is simpler to measure the input and output numbers of squares as shown in [9, 10] and to use the following formula (which is derived in [12]):Thereby is the incomplete elliptic integral of the first kind, is the complete elliptic integral of the first kind, and is the modular lambda function defined by for and . The input number of squares is and the output number of squares is , whereby the input resistance is measured between the two current supplying contacts of a Hall plate (while the sense contacts are floating) and the output resistance is measured between the two sense contacts (while the other two contacts are floating). The role of input and output contacts is arbitrary – does not change if we swap it [12]. Equation (4) expresses the Hall geometry factor as a function of input and output numbers of squares without need to know any geometrical details of the device. This means that Hall plates with different geometry have identical if they have identical and . The reason why different geometries may have the same number of squares and the same Hall output voltage at identical supply current is that we can transform one into the other via conformal mapping [14]. Since the conformal mapping is done by analytic functions which satisfy Cauchy-Riemann differential equations, the mapping functions are also solutions of the Laplace equations. Therefore, if we have a solution of the potential in one Hall plate, we can apply a conformal transformation to this geometry and to the solution. Thereby the same current will flow through the transformed device and the same potentials will appear at its contacts. Consequently, we can study a specific device geometry (e.g., a circular disk) and we can be sure that there is no other ingeniously shaped device boundary (e.g., Greek cross or octagon…) that leads to better device performance. This was already noted by Wick in [14]. Of course this holds only within the limited scope of linear electrostatic theory; beyond that scope devices will be different: different shapes lead to more or less homogeneous electric fields and therefore to more or less pronounced velocity saturation effects. Although their total power may be identical, the spatial distribution of power density differs and this leads to different temperature distributions. The temperature coefficient of the resistivity acts back on the current distribution. Finally the contacts may have different temperatures, which leads to different thermoelectric voltages and different residual offsets after spinning current schemes are applied [15].

In the case there is piezoresistance at zero magnetic field, the conductivity tensor , and its inverse the resistivity tensor becomes anisotropic due to mechanical stressThe significant difference between (1a), (1b), and (5) is that the conductivity tensor of the Hall effect is odd symmetric, whereas it is even symmetric for the piezoresistance. In the language of nonequilibrium thermodynamics both Hall effect and piezoresistance are irreversible dissipative processes, but the relation between current density and electric field is reciprocal for piezoresistance whereas it is antireciprocal for the Hall effect [16]. Therefore, a rotation of the coordinate system around the -axis does not change the Hall-resistivity tensor (1a), yet it changes the piezoresistance resistivity tensor (5) in such a way that for certain orientations (the principal axes) the elements in the minor diagonal even vanish. In network theory it is well known that any reciprocal (i.e., symmetric) impedance matrix of a linear, lumped, and passive N-port can be realized by a network of resistors, inductances, capacitors, and ideal transformers [17], however a nonreciprocal impedance matrix needs at least one additional linear, lumped, passive element: the gyrator [18]. Instead of a gyrator one may also use the series connection of two current controlled voltage sources (CCVS) or the parallel connection of two voltage controlled current sources (VCCS). Therefore the equivalent lumped circuit of a Hall effect device comprises resistors and at least one gyrator [19, 20], whereas the equivalent lumped circuit of a piezoresistive device consists only of resistors. Hence, the gyrator makes an essential difference between piezoresistance devices and Hall effect devices, and this already denies any perfect analogy between piezoresistance and Hall effect.

In a conventional plastic encapsulated package the surface of a semiconductor chip is exposed to inplane mechanical stress [21, 22]. For inplane normal stress , and inplane shear stress components on a (100) silicon chip it holds [23]with , , and being the three piezoresistive coefficients of the cubic m3m single crystal. Thereby the - and -axes of the chip are aligned along -crystal directions as shown in Figure 1 (see also [24]).