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Showing papers on "Van der Pauw method published in 2015"


Journal ArticleDOI
09 Feb 2015-ACS Nano
TL;DR: Time-resolved terahertz spectroscopy measurements were used to study the mobility and lifetime of PbSe nanocrystal films prepared with five common ligand-exchange reagents and found that films treated with different displacing ligands show more than an order of magnitude difference in the peak conductivities and a bifurcation of time dynamics.
Abstract: Colloidal semiconductor nanocrystals have been used as building blocks for electronic and optoelectronic devices ranging from field-effect transistors to solar cells. Properties of the nanocrystal films depend sensitively on the choice of capping ligand to replace the insulating synthesis ligands. Thus far, ligands leading to the best performance in transistors result in poor solar cell performance, and vice versa. To gain insight into the nature of this dichotomy, we used time-resolved terahertz spectroscopy measurements to study the mobility and lifetime of PbSe nanocrystal films prepared with five common ligand-exchange reagents. Noncontact terahertz spectroscopy measurements of conductivity were corroborated by contacted van der Pauw measurements of the same samples. The films treated with different displacing ligands show more than an order of magnitude difference in the peak conductivities and a bifurcation of time dynamics. Inorganic chalcogenide ligand exchanges with sodium sulfide (Na2S) or ammon...

69 citations


Journal ArticleDOI
TL;DR: A spray pyrolysis technique was employed to deposit cerium (Ce) doped cadmium oxide (CdO) thin films with low level doping concentrations.
Abstract: In the present work, a spray pyrolysis technique was employed to deposit cerium (Ce) doped cadmium oxide (CdO) thin films with low level doping concentrations (0.25, 0.50, 0.75 and 1.0 wt%). The crystallite size and lattice parameter values were estimated from X-ray diffraction analysis. X-ray diffraction patterns reveal the shift of preferential growth orientation from (111) to (200) planes on incorporating Ce in the CdO matrix. The oxidation state of Ce, Cd and O in the deposited films was determined by X-ray photoelectron spectroscopic (XPS) studies. Surface microstructures of the films were analyzed by atomic force microscopy and their surface nature was studied by field emission scanning electron microscopy (FE-SEM). The electrical properties of the deposited films were determined by Hall measurements in the van der Pauw configuration. The charge carrier concentration of the CdO thin film increased from 1.0 × 1020 cm−3 to 3.85 × 1020 cm−3 on doping 0.50 wt% Ce; whereas, resistivity decreased from 9.32 × 10−4 Ω cm to 3.81 × 10−4 Ω cm. The deposited Ce-doped CdO thin films for various concentrations showed an increase in the average optical transmittance to 85% from that of 72% for the CdO film in the visible and NIR region. The band gap was gradually increased from 2.38 eV to 2.63 eV due to an increase in the Ce doping at various levels. The emission properties of CdO and Ce doped CdO thin films were studied by a photoluminescence spectrum recorded at room temperature. The figure of merit estimated for 0.50 wt% of Ce doped CdO film is 9.18 × 10−3 Ω−1.

59 citations


Journal ArticleDOI
TL;DR: The mixing of dopant-containing alkenes withAlkenes that lack this functionality at various ratios to control the dopant concentration in the resulting monolayer and concomitantly the dopants dose in the silicon substrate is reported.
Abstract: Molecular monolayer doping (MLD) presents an alternative to achieve doping of silicon in a nondestructive way and holds potential for realizing ultrashallow junctions and doping of nonplanar surfaces. Here, we report the mixing of dopant-containing alkenes with alkenes that lack this functionality at various ratios to control the dopant concentration in the resulting monolayer and concomitantly the dopant dose in the silicon substrate. The mixed monolayers were grafted onto hydrogen-terminated silicon using well-established hydrosilylation chemistry. Contact angle measurements, X-ray photon spectroscopy (XPS) on the boron-containing monolayers, and Auger electron spectroscopy on the phosphorus-containing monolayers show clear trends as a function of the dopant-containing alkene concentration. Dynamic secondary-ion mass spectroscopy (D-SIMS) and Van der Pauw resistance measurements on the in-diffused samples show an effective tuning of the doping concentration in silicon.

57 citations


Journal ArticleDOI
TL;DR: Wafer-scale, non-contact mapping of essential carrier transport parameters, carrier mobility (µdrift), carrier mobility, carrier density, DC sheet conductance, and carrier scattering time in CVD graphene is demonstrated, using spatially resolved terahertz time-domain conductance spectroscopy.
Abstract: We demonstrate wafer-scale, non-contact mapping of essential carrier transport parameters, carrier mobility (µdrift), carrier density (Ns), DC sheet conductance (σdc), and carrier scattering time (τsc) in CVD graphene, using spatially resolved terahertz time-domain conductance spectroscopy. σdc and τsc are directly extracted from Drude model fits to terahertz conductance spectra obtained in each pixel of 10 × 10 cm2 maps with a 400 µm step size. σdc- and τsc-maps are translated into µdrift and Ns maps through Boltzmann transport theory for graphene charge carriers and these parameters are directly compared to van der Pauw device measurements on the same wafer. The technique is compatible with all substrate materials that exhibit a reasonably low absorption coefficient for terahertz radiation. This includes many materials used for transferring CVD graphene in production facilities as well as in envisioned products, such as polymer films, glass substrates, cloth, or paper substrates.

52 citations


Journal ArticleDOI
TL;DR: The analysis of Hall effect data at low temperatures indicates that the donor energy level for nitrogen dopants is located at 189 meV below the conduction band, consistent with the literature value.
Abstract: This Report presents a nitrogen-doping method by chemically forming self-assembled monolayers on silicon. Van der Pauw technique, secondary-ion mass spectroscopy and low temperature Hall effect measurements are employed to characterize the nitrogen dopants. The experimental data show that the diffusion coefficient of nitrogen dopants is 3.66 × 10−15 cm2 s−1, 2 orders magnitude lower than that of phosphorus dopants in silicon. It is found that less than 1% of nitrogen dopants exhibit electrical activity. The analysis of Hall effect data at low temperatures indicates that the donor energy level for nitrogen dopants is located at 189 meV below the conduction band, consistent with the literature value.

33 citations


Journal ArticleDOI
TL;DR: In this article, a method for contact-free determination of the sheet resistance of large area and arbitrary shaped wafers or sheets coated with graphene and other (semi) conducting ultrathin layers is described, which is based on an open dielectric loaded microwave cavity.
Abstract: A method for contact-free determination of the sheet resistance of large-area and arbitrary shaped wafers or sheets coated with graphene and other (semi) conducting ultrathin layers is described, which is based on an open dielectric loaded microwave cavity. The sample under test is exposed to the evanescent resonant field outside the cavity. A comparison with a closed cavity configuration revealed that radiation losses have no significant influence of the experimental results. Moreover, the microwave sheet resistance results show good agreement with the dc conductivity determined by four-probe van der Pauw measurements on a set of CVD samples transferred on quartz. As an example of a practical application, correlations between the sheet resistance and deposition conditions for CVD graphene transferred on quartz wafers are described. Our method has a high potential as measurement standard for contact-free sheet resistance measurement and mapping of large area graphene samples.

32 citations


Journal ArticleDOI
TL;DR: In this paper, the electrical conductivity of the gas diffusion layer (GDL) and bipolar plates in stainless steel foam (SSF), as the heart of a Proton Exchange Membrane Fuel Cell (PEMFC), depends in fact strongly on mechanical compression arising in an operating system.

27 citations


Journal ArticleDOI
TL;DR: In this paper, the influence of deposition parameters, especially the amount of nitrogen during film synthesis, to mechanical and electrical properties is investigated, and the crystallographic phase and lattice constants are determined by x-ray diffraction analyses.
Abstract: Molybdenum and molybdenum nitride thin films are presented, which are deposited by reactive dc magnetron sputtering. The influence of deposition parameters, especially the amount of nitrogen during film synthesization, to mechanical and electrical properties is investigated. The crystallographic phase and lattice constants are determined by x-ray diffraction analyses. Further information on the microstructure as well as on the biaxial film stress are gained from techniques such as transmission electron microscopy, scanning electron microscopy and the wafer bow. Furthermore, the film resistivity and the temperature coefficient of resistance are measured by the van der Pauw technique starting from room temperature up to 300 °C. Independent of the investigated physical quantity, a dominant dependence on the sputtering gas nitrogen content is observed compared to other deposition parameters such as the plasma power or the sputtering gas pressure in the deposition chamber.

26 citations


Journal ArticleDOI
TL;DR: In this paper, the influence of applied tensile and compressive strain on the offset voltage of the van der Pauw square device was investigated using the bending beam method, and it was shown that the effect of applied compressive and tensile strain is significant.
Abstract: This letter reports for the first time the strain dependence of the offset voltage in p-type 3C-SiC van der Pauw square device. The p-type 3C-SiC thin film was epitaxially grown on a p-type Si(100) wafer using low pressure chemical vapor deposition followed by a conventional photolithography and dry etch processes, forming four-terminal van der Pauw device. The influence of applied tensile and compressive strain on the offset voltage of the van der Pauw device was investigated using the bending beam method. Experimental results showed that the offset voltage of the device is significantly changed by applied compressive and tensile strain, indicating the feasibility of using this effect for mechanical sensing applications. The sensitivity of the device to the applied strain has been found to be 70 (mV/A)/ppm.

25 citations


Journal ArticleDOI
TL;DR: A series of CoFe/Cu multilayer was electrodeposited as a function of ferromagnetic and non-ferromagnetic layer thicknesses on Ti substrates from a single electrolyte containing Co (from CoSO4), Fe (from FeSO4) metal ions under potentiostatic control as discussed by the authors.
Abstract: A series of CoFe/Cu multilayer was electrodeposited as a function of ferromagnetic and non- ferromagnetic layer thicknesses on Ti substrates from a single electrolyte containing Co (from CoSO4), Fe (from FeSO4), Cu (from CuSO4) metal ions under potentiostatic control. The deposition was carried out in a three electrode cell at room temperature. The Cu and CoFe layers were deposited at a cathode potential of −0.3 and −1.5 V with respect to saturated calomel electrode respectively. The structural studies by X-ray diffraction revealed that the multilayers have face-centered-cubic structure. The magnetic characteristics of the films were investigated using a vibrating sample magnetometer and their easy-axis was found to be in the film plane. Magnetoresistance measurements were carried out at room temperature with magnetic fields up to ±955 kA/m using the Van der Pauw method. All multilayers exhibited giant magnetoresistance (GMR). The GMR values up to 22 % and a GMR sensitivity of 52 % per 1 kA/m were obtained.

22 citations


Journal ArticleDOI
TL;DR: Hetero-epitaxial TiC thin films were deposited at 100°C on MgO(001) by DC reactive magnetron sputtering in a mixture of Ar and CH 4 as mentioned in this paper.

Journal ArticleDOI
01 Feb 2015-Carbon
TL;DR: In this article, quasi-free-standing monolayer and bilayer graphene is grown on homoepitaxial layers of 4H-SiC and the epilayers themselves are grown on the Si-face of nominally on-axis semi-insulating substrates using a conventional SiC hot-wall chemical vapor deposition reactor.

Journal ArticleDOI
TL;DR: In this article, a new method of contactless electric conductivity measurement is developed, essentially based on the association of the coupled electric field forward model, with a simple and efficient research algorithm.
Abstract: In this paper, a new method of contactless electric conductivity measurement is developed. This method is essentially based on the association of the coupled electric field forward model, which we have recently developed, with a simple and efficient research algorithm. The proposed method is very fast because 1.3 s are sufficient to calculate electric conductivity, in a CPU of 2 GHz and RAM of 3 GB, for a starting research interval of 1.72–17.2 %IACS and tolerance of 1.72 × 10− 5 %IACS. The study of the calculation time according to mesh density and starting interval width has showed that an optimal choice has to be made in order to improve the rapidity while preserving its precision. Considering its rapidity and its simplicity of implementation, this method is more adapted, in comparison to direct current techniques using Van der Pauw geometry, for automated applications.

Journal ArticleDOI
TL;DR: In this paper, Sn doping In 2 O 3 (ITO) thin films with a thickness of 200nm were deposited on glass substrates by electron beam evaporation (EBE) method at different substrate temperatures.
Abstract: In this work, Sn doping In 2 O 3 (ITO) thin films with a thickness of 200 nm were deposited on glass substrates by electron beam evaporation (EBE) method at different substrate temperatures. The crystal structure of these films was studied by X-ray diffraction technique. The sheet resistance was measured by a four-point probe. Van der Pauw method was used to measure carrier density and mobility of ITO films. The optical transmittance spectra were recorded in the wavelength region of 300–800 nm. Scanning electron microscope (SEM) has been used for the surface morphology analysis. The prepared ITO films exhibited body-centered cubic (BCC) structure with preferred orientation of growth along the (2 2 2) crystalline plane. The grain size of the films increases by rising the substrate temperature. Transparency of the films, over the visible light region, is increased with increasing the substrate temperature. It is found that the electrical properties of ITO films are significantly affected by substrate temperature. The electrical resistivity decreases with increasing substrate temperature, whereas the carrier density and mobility are enhanced with an increase in substrate temperature. The evaluated values of energy band gap E g for ITO films were increase from 3.84 eV to 3.91 eV with increasing the substrate temperatures from 200 °C to 500 °C. The SEM micrographs of the films revealed a homogeneous growth without perceptible cracks with particles which are well covered on the substrate.

Journal ArticleDOI
TL;DR: In this article, an approach to extracting the resistivity tensor using van der Pauw measurements is presented, which reduces the number of required measurements and can be applied to single crystals.
Abstract: Anisotropy in properties of materials is important in materials science and solid-state physics. Measurement of the full resistivity tensor of crystals using the standard four-point method with bar shaped samples requires many measurements and may be inaccurate due to misalignment of the bars along crystallographic directions. Here an approach to extracting the resistivity tensor using van der Pauw measurements is presented. This reduces the number of required measurements. The theory of the van der Pauw method is extended to extract the tensor from parallelogram shaped samples with known geometry. Methods to extract the tensor for both known and unknown principal axis orientation are presented for broad applicability to single crystals. Numerical simulations of errors are presented to quantify error sources. Several benchmark experiments are performed on isotropic graphite samples to verify the internal consistency of the developed theory, test experimental precision, and characterize error sources. The presented methods are applied to a RuSb_2 single crystal at room temperature and the results are discussed based on the error source analysis. Temperature resolved resistivities along the a and b directions are finally reported and briefly discussed.

Journal ArticleDOI
01 May 2015
TL;DR: In this paper, the van der Pauw Hall effect measurement was used to measure the electrical properties of the grown germanium crystals and photo-thermal ionization spectroscopy (PTIS) were used to identify what the impurity atoms are in the crystal.
Abstract: In the crystal growth lab of South Dakota University, we are growing high purity germanium (HPGe) crystals and using the grown crystals to make radiation detectors. As the detector grade HPGe crystals, they have to meet two critical requirements: an impurity level of ~109 to 10 atoms /cm3 and a dislocation density in the range of ~102 to 104 / cm3. In the present work, we have used the following four characterization techniques to investigate the properties of the grown crystals. First of all, an x-ray diffraction method was used to determine crystal orientation. Secondly, the van der Pauw Hall effect measurement was used to measure the electrical properties. Thirdly, a photo-thermal ionization spectroscopy (PTIS) was used to identify what the impurity atoms are in the crystal. Lastly, an optical microscope observation was used to measure dislocation density in the crystal. All of these characterization techniques have provided great helps to our crystal activities.

Journal ArticleDOI
TL;DR: In this article, the exfoliation of oxygenated functional groups from 60-W hydrogen plasma treated graphene oxide films was investigated using X-ray diffractometric (XRD), Raman spectroscopic and atomic force microscopic techniques.
Abstract: The exfoliation of oxygenated functional groups from 60 W hydrogen plasma treated graphene oxide films was investigated using X-ray diffractometric (XRD), Raman spectroscopic and atomic force microscopic techniques. The interlayer spacing of the graphene oxide sheets was found from the XRD pattern to decrease from 0.88 to 0.35 nm after plasma treatment. The reduced intensity ratio of the D and G peaks of the Raman spectra indicates a decrease in the crystallite size of the sp 2 domains due to plasma treatment. Atomic force microscope showed the continuous morphology of the plasma treated film. The electrical properties of plasma treated samples spin-coated on silicon were studied using Van Der Pauw and non contacting microwave techniques. The sheet resistivity determined from Van der Pauw measurements was $$1.62\,{\text{M}}\Omega /{\text{sq}}$$ , yielding the value of $$3.1\,{\text{S}}\,{\text{m}}^{ - 1}$$ for the bulk conductivity. The charge mobility of $$3.8\,{\text{m}}^{2} \,{\text{V}}^{ - 1} \,{\text{s}}^{ - 1}$$ has been determined from Hall measurement technique.

Journal ArticleDOI
TL;DR: In this paper, the van der Pauw method for 2D samples of arbitrary shape with an isolated hole is considered, and the relationship between extreme values of the resistances allow one to determine the specific resistivity of the sample and the dimensionless parameter related to the geometry of the isolated hole, known as the Riemann modulus.
Abstract: The van der Pauw method for 2D samples of arbitrary shape with an isolated hole is considered. Correlations between extreme values of the resistances allow one to determine the specific resistivity of the sample and the dimensionless parameter related to the geometry of the isolated hole, known as the Riemann modulus. The parameter is invariant under conformal mappings. Experimental verification of the method is presented.

Journal ArticleDOI
TL;DR: In this article, the microstructural and electrical stability of a functionally gradient platinum-zirconium (Zr) composite thin film electrode designed for resistive-type chemical sensors was investigated.
Abstract: Stable metal interconnect thin films are critical in the development of various micro-machined devices that may operate continuously at elevated temperatures. The main objective of this work was to investigate the microstructural and electrical stability of a functionally gradient platinum (Pt)–zirconium (Zr) composite thin film electrode designed for resistive-type chemical sensors. Thin film electrodes were fabricated using a DC magnetron sputtering process. Zirconium was used as both the conventional adhesion promoter and the Pt grain modifier within the bulk electrode microstructure. The thin film deposition was completed on highly polished alumina substrates at 200 °C. The various composite Pt thin films were further annealed at 1200 °C after deposition for 1–24 h for rapid evaluation of the microstructure stability. This temperature was chosen since the electrodes are expected to operate beyond 1000 °C for high-temperature MEMS applications. Scanning electron microscopy (SEM), energy-dispersive X-ray spectroscopy (EDS) and X-ray photoelectron spectroscopy (XPS) were conducted to characterize the alterations in chemistry, microstructure and distribution of the constituent elements through the film thickness. The electrical resistivity of the as-deposited and thermally processed Pt thin films was measured by utilizing a van der Pauw's four-point probe technique. The work identified a Zr/Zr + Pt/Pt composite thin film with the 525 nm total film thickness that demonstrated resistivity

Journal ArticleDOI
TL;DR: In this paper, the authors studied the feasibility of low temperature doping of boron (B) atoms into crystalline silicon (c-Si) by using decomposed species generated similarly from diborane (B 2 H 6 ) molecules.

Journal ArticleDOI
TL;DR: In this paper, the impact of heat treatment on the crystal silicon (c-Si) with developed surface by Aluminum /Porous silicon (Al/PS) and evaluating their influence on the solar cell performance was described.

Journal ArticleDOI
TL;DR: In this article, a degenerated wide bandgap semiconductor (3C-SiC) was found to saturate in 3CSiC at ∼7.5×1019 cm−3.

Journal ArticleDOI
TL;DR: In this paper, the van der Pauw method was used to measure the anisotropic transport of epitaxial graphene on silicon carbide, and the calculated maximum mobility is 2876cm2·V-1·s-1, which is 1.43 times higher than that obtained by the standard van der-Pauw method.
Abstract: The electronic transport of epitaxial graphene on silicon carbide is anisotropic because of the anisotropy of the surface structure of the substrate. In this Letter, we present a new method for measuring anisotropic transport based on the van der Pauw method. This method can measure anisotropic transport on the macroscopic scale without special equipment or device fabrication. We observe an anisotropic resistivity with a ratio of maximum to minimum of 1.62. The calculated maximum mobility is 2876cm2·V-1·s-1, which is 1.43 times higher than that obtained by the standard van der Pauw method.

Journal ArticleDOI
TL;DR: In this paper, a six-point generalization of the van der Pauw method is presented for 2D homogeneous systems with an isolated hole, which is applicable for the experimental determination of the sample inhomogeneity.
Abstract: A six-point generalization of the van der Pauw method is presented. The method is applicable for 2D homogeneous systems with an isolated hole. A single measurement performed on the contacts located arbitrarily on the sample edge allows us to determine the specific resistivity and a dimensionless parameter related to the hole, known as the Riemann modulus. The parameter is invariant under conformal mappings of the sample shape. The hole can be regarded as a high resistivity defect. Therefore, the method can be applied for the experimental determination of the sample inhomogeneity.

Journal ArticleDOI
TL;DR: In this article, the authors studied the sensitivity of charge transport measurement to local physical defects for van der Pauw circular and square cloverleafs with rounded internal corners and unclovered geometries, using copper-foil specimens.
Abstract: We have undertaken the first systematic computational and experimental study of the sensitivity of charge transport measurement to local physical defects for van der Pauw circular and square cloverleafs with rounded internal corners and unclovered geometries, using copper-foil specimens. Cloverleafs with rounded internal corners are in common use and reduce sampling of the material near their boundaries, an advantage over sharp corners. We have defined two parameters for these cloverleafs, one of which, the 'admittance', is the best predictor of the sensitivity at the center of these specimens, with this sensitivity depending only weakly on the central 'core' size when its diameter is less than about 60% of the specimen's lateral size. Resistive measurement errors in all four geometries are linear in areas for errors up to about 50% in sheet resistance, and superlinear above. An ASTM-based 'standard' cloverleaf geometry, in which the central core diameter of the specimen is 1/5 the overall length and the slit widths are 1/10 the overall length, narrows the effective area sampled by the resistive measurement by a factor of about 16 × in the small-hole limit and over 40 × for larger holes, relative to unclovered goemetries, whether square or circular, with a smooth transition in these numbers for geometries intermediate between the standard cloverleaf and unclovered specimens. We believe that this work will allow materials scientists to better estimate the impact of factors such as the uniformity of film thickness and of material purity on their measurements, and allow sensor designers to better choose an optimal specimen geometry.

Journal ArticleDOI
TL;DR: Using gated van der Pauw structures, this article showed that the transient behavior of AlGaN/GaN heterostructure field effect transistors is determined by transport, rather than emission capture.
Abstract: Transient phenomena in AlGaN/GaN heterostructure field-effect transistors are attributed to trapping and detrapping of electrons from deep localized states, and the measured activation energies are conventionally associated with the electron capture and emission processes. This standard interpretation ignores, however, transport between the two-dimensional electron gas and the trap. Using gated van der Pauw structures, we demonstrate that the transient behavior is determined by transport (at least for the trapping process). The measured activation energy is, therefore, a characteristic of the transport process rather than the emission-capture processes.

Journal ArticleDOI
TL;DR: In this paper, a Corbino geometry sample was fabricated and measured in an ultra-high mobility GaAs heterostructure, with a focus on transport in the second and higher Landau levels.

Journal ArticleDOI
TL;DR: Six types of commercially available multiwall carbon nanotube soot were obtained and prepared into buckypapers by pellet pressing and by filtration into a paper and results by two-point and four-point measurement methods as a function of preparation parameters are presented.
Abstract: Six types of commercially available multiwall carbon nanotube soot were obtained and prepared into buckypapers by pellet pressing and by filtration into a paper. These samples were evaluated with respect to thickness, compressibility and electrical conductivity. DC conductivity results by two-point and four-point (van der Pauw) measurement methods as a function of preparation parameters are presented. Topology was investigated qualitatively by way of scanning electron microscopy and helium ion microscopy and from this, some generalizations about the nanotube structural properties and manufacturing technique with respect to conductivity are given.

Journal ArticleDOI
TL;DR: In this paper, a six point generalization of the van der Pauw method is presented for two-dimensional homogeneous systems with an isolated hole and a single measurement performed on the contacts located arbitrarily on the sample edge allows to determine the specific resistivity and a dimensionless parameter related to the hole, known as the Riemann modulus.
Abstract: Six point generalization of the van der Pauw method is presented. The method is applicable for two dimensional homogeneous systems with an isolated hole. A single measurement performed on the contacts located arbitrarily on the sample edge allows to determine the specific resistivity and a dimensionless parameter related to the hole, known as the Riemann modulus. The parameter is invariant under conformal mappings of the sample shape. The hole can be regarded as a high resistivity defect. Therefore the method can be applied for experimental determination of the sample inhomogeneity.

Journal ArticleDOI
TL;DR: In this article, the surface exchange reaction and diffusion of oxygen through oxide ceramic samples as well as the relaxation of the dc conductivity upon an instantaneous change of the oxygen activity in the surrounding atmosphere have been modelled simultaneously by means of the finite element method.