The Prime Function, the Fay Trisecant Identity, and the van der Pauw Method

The van der Pauw method is a well-known experimental technique in the applied sciences for measuring physical quantities such as the electrical conductivity or the Hall coefficient of a given sample. Its popularity is attributable to its flexibility: the same method works for planar samples of any shape provided they are simply connected. Mathematically, the method is based on the cross-ratio identity. Much recent work has been done by applied scientists attempting to extend the van der Pauw method to samples with holes (“holey samples”). In this article we show the relevance of two new function theoretic ingredients to this area of application: the prime function associated with the Schottky double of a multiply connected planar domain and the Fay trisecant identity involving that prime function. We focus here on the single-hole (doubly connected, or genus one) case. Using these new theoretical ingredients we are able to prove several mathematical conjectures put forward in the applied science literature.

Conductivity Transitions of La0.7Sr0.3MnO3±δ and La0.6Sr0.4Co0.2Fe0.8O3−δ in Ce0.9Gd0.1O2−δ Matrix for Dual-Phase Oxygen Transport Membranes

Using van der Pauw method, the conductivity of disk samples of La0.7Sr0.3MnO3±δ (LSM) and La0.6Sr0.4Co0.2Fe0.8O3−δ (LSCF) in a Ce0.9Gd0.1O2−δ (GDC) matrix was accurately quasi-continuously measured over 800 °C to −73 °C, and the transition points in Arrhenius behavior were systematically obtained from the extremum points of the second derivatives. While LSM-containing samples showed reproducible conductivity trajectories, the LSCF system exhibited unsystematic changes which may be related to the substantial oxidation/reduction reactions accompanying the ferroelastic–paraelastic transitions with a substantial thermal hysteresis at 650 °C to 750 °C, corresponding to conductivity maxima. A sudden decrease in activation energies on cooling corresponds to the para-to-ferromagnetic, weak insulator–metal transitions and the Curie temperature of LSM appears to gradually decrease in composites to 90 °C, while LSCF composites exhibit blurred transitions at approximately −40 °C. Relatively insulating paramagnetic phases are characterized by activation energy values ~0.2 eV, change to the high temperature phase exhibiting activation energy 0.1 eV for small polaron hopping mechanisms at 300 °C to 500 °C with increasing GDC content in the LSM composites and by two transitions at ∼60 °C and ∼245 °C for the LSCF composites. LSCF single phase shows distinctly lower transition points which appear to match with the singularly large c lattice parameter whereas the composites exhibit decreasing c with LSCF amount together with increasing lattice parameter of GDC. Van der Pauw conductivity is a feasible and sensitive in situ tool for monitoring the status of oxygen transport membranes.

Instrumentação para medidas de mobilidade eletrônica e concentração de portadores em amostras semicondutoras, pelo método de van der Pauw

Neste trabalho apresentamos os resultados da construção de um equipamento, de baixo custo, para medidas medida Hall, utilizando a técnica de van der Pauw. Para isso, foi desenvolvido um sistema composto por uma placa eletrônica controlada por uma placa DAQ que juntamente com um software em Labview realiza a automação de todas as medidas necessárias. O sistema é capas de realizar todas as operações necessárias entre os contatos para realizar a técnica de van der Pauw e obter parâmetros como mobilidade eletrônica, concentração de portadores e resistividades elétrica em função da temperatura. Tais medidas são essenciais para estudar o comportamento eletrônico de amostras, visando sua aplicação em dispositivos eletrônicos. Para verificar o comportamento do sistema, foi utilizado uma amostra de Indium tin oxide (ITO). Nesses testes verificou-se que o os tempos de sincronização utilizados para as medidas apresentam resultados com baixo ruído e que a montagem permite a realização de medidas didáticas para disciplinas de pós-graduação e também de pesquisas desenvolvidas pelos alunos de pós-graduação.

Fay meets van der Pauw: the trisecant identity and the resistivity of holey samples

The van der Pauw method is commonly used in the applied sciences to find the resistivity of a simply connected, two-dimensional conducting laminate. Given the usefulness of this ‘4-point probe’ method there has been much recent interest in trying to extend it to holey, that is, multiply connected, samples. This paper introduces two new mathematical tools to this area of investigation—the prime function on the Schottky double of a planar domain and the Fay trisecant identity—and uses them to show how the van der Pauw method can be extended to find the resistivity of a sample with a hole. We show that an integrated form of the Fay trisecant identity provides valuable information concerning the appearance of ‘envelopes’ observed in the case of holey samples by previous authors. We find explicit formulae for these envelopes, as well as an approximate formula relating two pairs of resistance measurements to the sample resistivity that is expected to be valid when the hole is sufficiently small and not too close to the outer boundary. We describe how these new mathematical tools have enabled us to prove certain conjectures recently made in the engineering literature.

Simple analytical method for determining electrical resistivity and sheet resistance using the van der Pauw procedure

This work reports an analytical method for determining electrical resistivity (ρ) and sheet resistance (RS) of isotropic conductors. The method is compared with previous numerical solutions and available experimental data showing a universal behavior for isotropic conductors. An approximated solution is also reported allowing one to easily determine ρ and RS for samples either with regular or arbitrary shapes.