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. ...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.展开更多
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 f...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.展开更多
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 de...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 multiply-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.展开更多
If Hall plates are used as magnetic field sensors they are usually powered up by a current source connected to a pair of non-neighboring contacts. The output voltage is tapped at another pair of non-neighboring contac...If Hall plates are used as magnetic field sensors they are usually powered up by a current source connected to a pair of non-neighboring contacts. The output voltage is tapped at another pair of non-neighboring contacts. In this paper we study more general operating conditions of Hall plates with an arbitrary number of contacts. In such hybrid operating modes current sources are connected to a first set of contacts and voltage sources to a second set of contacts. Output voltages are tapped at the first set of contacts and output currents are measured at the second set of contacts. All these output signals are multiplied by coefficients and added up. The purpose of this work is to figure out which operating mode and which Hall plate achieve maximum signal at minimum thermal noise and power dissipation. To this end we develop a theory, which gives the ratio of signal over noise and power as a function of the resistance matrix of Hall plates, of the supply voltages and currents, and of the coefficients. Optimization is done analytically in closed form and numerically for specific examples. The results are: 1) all operating modes have identical noise performance if their parameters are optimized;2) for any Hall plate one can measure its resistance matrix and insert its values into our formulae to obtain the optimum supply currents and coefficients for optimum noise performance.展开更多
For plane singly-connected domains with insulating boundary and four point-sized contacts, C<sub>0</sub> …C<sub>3</sub>, van der Pauw derived a famous equation relating the two trans-...For plane singly-connected domains with insulating boundary and four point-sized contacts, C<sub>0</sub> …C<sub>3</sub>, van der Pauw derived a famous equation relating the two trans-resistances R<sub>01,23</sub>, R<sub>12,30</sub> 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 given.展开更多
文摘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.
文摘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.
文摘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 multiply-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.
文摘If Hall plates are used as magnetic field sensors they are usually powered up by a current source connected to a pair of non-neighboring contacts. The output voltage is tapped at another pair of non-neighboring contacts. In this paper we study more general operating conditions of Hall plates with an arbitrary number of contacts. In such hybrid operating modes current sources are connected to a first set of contacts and voltage sources to a second set of contacts. Output voltages are tapped at the first set of contacts and output currents are measured at the second set of contacts. All these output signals are multiplied by coefficients and added up. The purpose of this work is to figure out which operating mode and which Hall plate achieve maximum signal at minimum thermal noise and power dissipation. To this end we develop a theory, which gives the ratio of signal over noise and power as a function of the resistance matrix of Hall plates, of the supply voltages and currents, and of the coefficients. Optimization is done analytically in closed form and numerically for specific examples. The results are: 1) all operating modes have identical noise performance if their parameters are optimized;2) for any Hall plate one can measure its resistance matrix and insert its values into our formulae to obtain the optimum supply currents and coefficients for optimum noise performance.
文摘For plane singly-connected domains with insulating boundary and four point-sized contacts, C<sub>0</sub> …C<sub>3</sub>, van der Pauw derived a famous equation relating the two trans-resistances R<sub>01,23</sub>, R<sub>12,30</sub> 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 given.
