Matano Method Research Articles

Defect-mediated diffusion of implanted Mg in GaN: Suppressing dopant redistribution by sequential thermal and microwave annealing

The diffusion behavior of Mg in Mg/N co-implanted GaN is investigated in response to a set of annealing conditions and methodologies, namely, 1000 °C/30 min thermal anneal, by high-temperature pulsed gyrotron microwave annealing at 1420 or 1500 °C, or by thermal and microwave annealing, sequentially. After 1000 °C annealing, the diffusion of Mg in GaN is found to be negligible, as measured by secondary ion mass spectrometry. Annealing by gyrotron microwave annealing alone induces the diffusion of Mg at a rate on the order of 10−12 cm2/s. However, the use of a thermal anneal before microwave gyrotron annealing reduces this rate by an order of magnitude to 10−13 cm2/s. We find that a model that considers Mg diffusion from an inhomogeneous medium that contains a defect-rich implanted region near-surface to a relatively pristine region below the implant range better explains the observed diffusion behavior than a conventional model that assumes a homogeneous medium. By analyzing the diffusion behavior using the Boltzmann–Matano method, we present a discussion of reduction in [VGa] by thermal annealing at 1000 °C, leading to a suppressed diffusion coefficient during subsequent high-temperature annealing relative to diffusion after 1420/1500 °C annealing alone. This effect holds potential for improvement in the precision of selectively doped regions for future applications based on the (Al)GaN material system. An improved field profile control in real devices can increase the breakdown and current-handling capabilities in power electronic applications.

Physically driven enhancement of the stability of Bi2O3-based ionic conductors via grain boundary engineering

Fast oxygen-ion conductors for use as electrolyte materials have been sought for energy conversion and storage. Bi2O3-based ionic conductors that exhibit the highest known oxygen-ion conductivities have received attention for use in next-generation solid electrolytes. However, at intermediate temperatures below ~600 °C, their conductivities degrade rapidly owing to a cubic-to-rhombohedral phase transformation. Here, we demonstrate that physical manipulation of the grain structure can be used to preserve the superior ionic conductivity of Bi2O3. To investigate the effects of microstructural control on stability, epitaxial and nanopolycrystalline model films of Er0.25Bi0.75O1.5 were fabricated by pulsed laser deposition. Interestingly, in situ impedance and ex situ XRD analyses showed that the grain boundary-free epitaxial film significantly improved the stability of the cubic phase, while severe degradation was observed in the conductivity of its polycrystalline counterpart. Consistently, the cation interdiffusion coefficient measured by the Boltzmann–Matano method was much lower for the epitaxial thin film compared to the polycrystalline thin film. Furthermore, first-principles calculations revealed that the presence of grain boundaries triggered the structural resemblance between cubic and rhombohedral phases, as evidenced by radial distribution functions. Additionally, phase transition energetics predicted that the thermodynamic stability of the cubic phase with respect to the rhombohedral counterpart is reduced near grain boundaries. Thus, these findings provide novel insights into the development of highly durable superionic conductors via microstructural engineering.

Intermetallic Phases Identification and Diffusion Simulation in Twin-Roll Cast Al-Fe Clad Sheet.

Aluminium steel clad materials have high potential for industrial applications. Their mechanical properties are governed by an intermetallic layer, which forms upon heat treatment at the Al-Fe interface. Transmission electron microscopy was employed to identify the phases present at the interface by selective area electron diffraction and energy dispersive spectroscopy. Three phases were identified: orthorhombic Al5Fe2, monoclinic Al13Fe4 and cubic Al19Fe4MnSi2. An effective interdiffusion coefficient dependent on concentration was determined according to the Boltzmann–Matano method. The highest value of the interdiffusion coefficient was reached at the composition of the intermetallic phases. Afterwards, the process of diffusion considering the evaluated interdiffusion coefficient was simulated using the finite element method. Results of the simulations revealed that growth of the intermetallic phases proceeds preferentially in the direction of aluminium.

InfPolyn, a Nonparametric Bayesian Characterization for Composition-Dependent Interdiffusion Coefficients.

Composition-dependent interdiffusion coefficients are key parameters in many physical processes. However, finding such coefficients for a system with few components is challenging due to the underdetermination of the governing diffusion equations, the lack of data in practice, and the unknown parametric form of the interdiffusion coefficients. In this work, we propose InfPolyn, Infinite Polynomial, a novel statistical framework to characterize the component-dependent interdiffusion coefficients. Our model is a generalization of the commonly used polynomial fitting method with extended model capacity and flexibility and it is combined with the numerical inversion-based Boltzmann–Matano method for the interdiffusion coefficient estimations. We assess InfPolyn on ternary and quaternary systems with predefined polynomial, exponential, and sinusoidal interdiffusion coefficients. The experiments show that InfPolyn outperforms the competitors, the SOTA numerical inversion-based Boltzmann–Matano methods, with a large margin in terms of relative error (10× more accurate). Its performance is also consistent and stable, whereas the number of samples required remains small.

Growth stress and interdiffusion analysis of NiCoCrAlYTa coating during oxidation

ABSTRACT The NiCoCrAlYTa coating deposited on Ni-based superalloy using arc ion plating (AIP) technology was subjected to isothermal oxidation in air at 1050°C for up to 200 h. The microstructure, oxides scale thickness, growth stress and interface diffusion behaviour were characterized by using scanning electron microscopy, X-ray diffraction, electron probe micro-analyser and photo-stimulated luminescence piezo-spectroscopy. The results revealed that the microstructure of oxides scale was predominantly composed of α-Al2O3 with (Y, Ta)-rich oxides and Cr2O3 irregularly distributed during the oxidation process. The oxides scale thickness of NiCoCrAlYTa coating conformed to parabolic law, which indicated the coating had good oxidation resistance. Also, the relationship between the thickness and growth stress, as well as the oxidation time, was studied based on the Clark oxidation theory and Wagner oxidation model, developing a methodology to measure the high-temperature stress in NiCoCrAlYTa coating. The theoretical calculation value of the stress was basically consistent with the value of the experiment, which could effectively predict the growth stress in the coating and help understand the failure mechanism of NiCoCrAlYTa coating during the service life. In the oxidation process, the interdiffusion behaviour occurred at the interface between the oxides scale and the NiCoCrAlYTa coating was also discussed. The diffusion coefficient of typical elements was calculated by using Boltzmann–Matano method, which was helpful to analyse the inner oxidation of the NiCoCrAlYTa coating, providing in-depth understanding of the oxidation behaviour.

Determination of accurate interdiffusion coefficients in fcc Ag-In and Ag-Cu-In alloys: A comparative study on the Matano method with distribution function and the numerical inverse method with HitDIC

In this paper, 3 fcc binary Ag-In and 3 fcc ternary Ag-Cu-In diffusion couples were firstly prepared. Based on the measured composition-distance profiles, both the traditional Matano method with distribution functions and the numerical inverse method with HitDIC were then utilized to determine the accurate interdiffusion coefficients in fcc Ag-In alloys at 873 K, 973 K, and 1073 K and Ag-Cu-In alloys at 1073 K. The determined interdiffusivities in fcc Ag-In alloys increases as the composition of In increases. The impurity diffusivities of In in fcc Ag extrapolated from the evaluated interdiffusivities based on the Matano method agree well with the tracer experimental data from the literature. As for fcc Ag-Cu-In alloys, the three-dimensional main interdiffusivities planes at 1073 K were constructed over the investigated composition range. The diffusion rate of In is generally faster than that of Cu. Moreover, the main interdiffusivity of In increases with the increment of both In and Cu concentrations, while the main interdiffusivity of Cu increases with In concentration but decreases with Cu concentration. After that, a comprehensively comparative study on the two methods were performed. The interdiffusivities in both the fcc Ag-In and Ag-Cu-In alloys evaluated from the two methods are consistent with each other, indicating that both methods can result in accurate interdiffusivities in binary and ternary alloys. However, as the number of elements increases, the Matano method becomes inconvenient. While for the numerical inverse method with HitDIC, no such limitation exists, and it can be conveniently applied in alloys with any number of components.

Effects of Sc Addition on the Diffusion Kinetics and Mechanical Properties of bcc Ti Alloys

In this study, binary interdiffusivities of bcc titanium-scandium (Ti-Sc) alloys in the temperature range of 1273 K to 1473 K were determined by combining the diffusion couple technique and Boltzmann–Matano method. In addition, the composition-dependent Young’s modulus and hardness of bcc Ti-Sc alloys were obtained using the nanoindentation technique. The results reveal that a small amount of Sc additive can be very beneficial to bio-Ti alloys with good properties.

Simultaneous measurement of interdiffusion and intrinsic diffusion coefficients in liquid metals on the ground

Simultaneous measurements of interdiffusion and intrinsic diffusion coefficients in liquid Sn-Pb were performed using the shear cell method and a stable density layering. The binary diffusion couples of Sn-Pb and Sn0.6Pb0.4-Sn0.4Pb0.6 were used in two experiments with the average molar fraction of diffusion couples NSn = 0.5. The concentration dependency of the interdiffusion coefficient was confirmed to exhibit a downward convexity through the comparison of the interdiffusion coefficients. However, the Boltzmann–Matano method reveals a diminished influence. The sample measurements of the intrinsic diffusion coefficients of a diffusion couple of pure metals demonstrate a dependency on concentration. On the other hand, the difference of diffusion coefficients was small in a diffusion couple of alloys. The difference between the values calculated by fitting the error function based on the superposition of fluxes and by the conventional analysis methods was small. We propose that a reasonable intrinsic diffusion coefficient can be calculated using the superposed equation of error function. The theoretical interdiffusion coefficients were calculated by substituting the measured intrinsic diffusion coefficients into the Darken’s equation and comparing the results with the measured values. As a result, the difference between the theoretical and measured interdiffusion coefficients is not very large.

Simultaneous Measurement of Isotope-Free Tracer Diffusion Coefficients and Interdiffusion Coefficients in the Cu-Ni System

A new formalism recently developed by Belova et al., based on linear response theory combined with the Boltzmann–Matano method, allows determination of tracer and interdiffusion coefficients simultaneously from a single, isotope-free, traditional diffusion couple experiment. An experimental methodology with an analytical approach based on the new formalism has been carried out using the model Cu-Ni system to effectively determine tracer diffusion coefficients from an isotope-free diffusion couple experiment. Cu thin films were deposited in between several binary diffusion couples with varying terminal alloy compositions (Cu-25Ni, Cu-50Ni, Cu-75Ni, Ni). Diffusion couples were annealed at 800, 900 and 1000 °C, and the superimposed concentration profiles of thin film and interdiffusion were analyzed for the simultaneous determination of tracer and interdiffusion coefficients. Processed concentration profiles obtained from the diffusion experiments were also fitted with simple Gaussian distribution function. Results were compared to existing literature data obtained independently by radiotracer experiments, and an excellent agreement has been observed.

Coordinate dependent diffusion analysis of phosphorus diffusion profiles in gallium doped germanium

We have analyzed phosphorus diffusion profiles in an In0.01Ga0.99As/In0.56Ga0.44P/Ge germanium structure during phosphorus co-diffusion with gallium for synthesis of the germanium subcell in multi-junction solar cells.. Phosphorus diffused from the In0.56Ga0.44P layer simultaneously with gallium diffusion into the heavily gallium doped germanium substrate thus determining the specific diffusion conditions. Most importantly, gallium and phosphorus co-diffusion produces two p–n junctions instead of one. The phosphorus diffusion profiles do not obey Fick’s laws. The phosphorus diffusion coefficient DP depth distribution in the specimen has been studied using two methods, i.e., the Sauer–Freise modification of the Boltzmann–Matano method and the coordinate dependent diffusion method. We show that allowance for the drift component in the coordinate dependent diffusion method provides a better DP agreement with literary data. Both methods suggest the DP tendency to grow at the heterostructure boundary and to decline closer to the main p–n junction. The DP growth near the surface p–n junction the field of which is directed toward the heterostructure boundary and its decline near the main p–n junction with an oppositely directed field, as well as the observed DP growth with the electron concentration, suggest that the negatively charged VGeP complexes diffuse in the heterostructure by analogy with one-component diffusion.

Binary diffusion behaviour in Ti–X (X = Al, Mo, V, Cr, Fe) alloys

Abstract Interdiffusion coefficients of the β single-phase in the Ti-rich portion of the binary Ti–X (X = Al, Mo, V, Cr, Fe) systems at 1 523 K were determined by annealing infinite diffusion couples between appropriate pairs of alloys. The values were derived from the concentration profiles by electron probe microanalysis coupled with the Boltzmann–Matano method. The comparison of the interdiffusion coefficients in these binary Ti–X (X = Al, Mo, V, Cr, Fe) systems show that the Mo diffuses the least while the Fe diffuses fastest in titanium alloys. Furthermore, the values of the interdiffusion coefficients of the Ti–Al, Ti–V and Ti–Cr binary systems were similar.

Reactive diffusion in Mg–Gd binary system at 773 K

The reactive diffusion in Mg–Gd binary system was studied at 773 K by optical microscopy (OM), scanning electron microscopy (SEM) and electron probe micro-analysis (EPMA). After annealing at 773 K for 12–48 h, four different intermetallic layers, Mg5Gd, Mg3Gd, Mg2Gd and MgGd, form at the Mg/Gd interfaces in the diffusion couples. The thicknesses of intermetallic layers Δi (i stands for the phases of Mg5Gd, Mg3Gd, Mg2Gd and MgGd, respectively) are proportional to the square root of annealing time t1/2, which indicates that the growth behavior of the intermetallics is controlled by the diffusion rate. The ratio of thickness of each intermetallic layer to the total thickness is constant with increasing the annealing time, which means that the growth behavior is constant at a certain annealing temperature. The diffusion coefficient of Gd in different intermetallics was calculated by Matano method.

The intrinsic diffusivities in multi component systems

The complete understanding of the phenomenological process related to the position of the Matano and Kirkendall plane in multicomponent system is still lacking. In this paper some aspects of Matano and lattice plane migration are studied by means of numerical simulations. New approach to calculate intrinsic diffusion coefficients in binary Ni–Au system and ternary Cu–Fe–Ni and Co–Fe–Ni is presented. Diffusivities were calculated using modified Boltzmann–Matano method. Self diffusion coefficients of the elements in diffusion couple must be known as well as position of the Kirkendall plane after the experiment to use this method. Diffusion profile was calculated using presented equations. Calculations compared with experimental data prove validity of the presented approach. This work proves that intrinsic diffusion coefficients should be given by the concentration function, not the exact value.

Discussion of “On the Boltzmann–Matano Analysis of Diffusion in a Semi-infinite Medium”

Ernst et al. analyzed the concentration-dependent diffusion of carbon in austenitic steel in the semi-infinite geometry by the Boltzmann–Matano method. They assumed that the origin of the position coordinate be at the surface, and tried to justify it from the particular nature of interstitial diffusion. We point out that their ‘assumption’ is prerequisite to applying the method to the problem, irrespective of the mechanism.

Generalization of Matano's method: interdiffusion in solutions with volume change

Generalization of Matano's method: interdiffusion in solutions with volume change

Interdiffusion and impurity diffusion in polycrystalline Mg solid solution with Al or Zn

Interdiffusion and impurity diffusion in Mg binary solid solutions, Mg(Al) and Mg(Zn) were investigated at temperatures ranging from 623 to 723K. Interdiffusion coefficients were determined via the Boltzmann–Matano Method using solid-to-solid diffusion couples assembled with polycrystalline Mg and Mg(Al) or Mg(Zn) solid solutions. In addition, the Hall method was employed to extrapolate the impurity diffusion coefficients of Al and Zn in pure polycrystalline Mg. For all diffusion couples, electron microprobe analysis was utilized for the measurement of concentration profiles. The interdiffusion coefficient in Mg(Zn) was higher than that of Mg(Al) by an order of magnitude. Additionally, the interdiffusion coefficient increased significantly as a function of Al content in Mg(Al) solid solution, but very little with Zn content in Mg(Zn) solid solution. The activation energy and pre-exponential factor for the average effective interdiffusion coefficient in Mg(Al) solid solution were determined to be 186.8 (±0.9)kJ/mol and 7.69×10−1 (±1.80×10−1)m2/s, respectively, while those determined for Mg(Zn) solid solution were 139.5 (±4.0)kJ/mol and 1.48×10−3 (±1.13×10−3)m2/s. In Mg, the Zn impurity diffusion coefficient was an order of magnitude higher than the Al impurity diffusion coefficient. The activation energy and pre-exponential factor for diffusion of Al impurity in Mg were determined to be 139.3 (±14.8)kJ/mol and 6.25×10−5 (±5.37×10−4)m2/s, respectively, while those for diffusion of Zn impurity in Mg were determined to be 118.6 (±6.3)kJ/mol and 2.90×10−5 (±4.41×10−5)m2/s.

The generalization of the Boltzmann–Matano method

The phenomenological Boltzmann–Matano (B–M) analysis allows approximating the concentration dependent interdiffusion coefficients related to infinite binary diffusion couple experiments. However, the complete understanding of the phenomenological process related to the Kirkendall plane position coupled with the B–M method is still lacking. A purpose of the present study is to generalize the Boltzmann–Matano analysis and propose a method of estimating the unique intrinsic diffusion coefficients of components from the experimental concentration profile and the positions of Matano and Kirkendall planes in multicomponent systems. The proposed physico-chemical approach is used to approximate the intrinsic diffusivities in a single phase of Ni–Pd binary and Cu–Fe–Ni ternary diffusion couples.

Solid state reaction studies in Fe3O4–TiO2 system by diffusion couple method

Abstract The solid state reactions in Fe 3 O 4 –TiO 2 system has been studied by diffusion couple experiments at 1323–1473 K, in which the oxygen partial pressure was controlled by the CO–CO 2 gas mixture. The XRD analysis was used to confirm the phases of the inter-compound, and the concentration profiles were determined by electron probe microanalysis (EPMA). Based on the concentration profile of Ti, the inter-diffusion coefficients in Fe 3 O 4 phase, which were both temperature and concentration of Ti ions dependent, were calculated by the modified Boltzmann–Matano method. According to the relation between the thickness of diffusion layer and temperature, the diffusion coefficient of the Fe 3 O 4 –TiO 2 system was obtained. According to the Arrhenius equation, the estimated diffusion activation energy was about 282.1 ± 18.8 kJ mol −1 .

Diffusivities and atomic mobilities in fcc_A1 Ni–X (X=Ge, Ti and V) alloys

Based on three groups of Ni/Ni–9at% Ge diffusion couples, interdiffusion coefficients were determined in the face-centered (fcc_A1) Ni–Ge alloys at 1173, 1273 and 1373K by means of the Boltzmann–Matano method coupled with electronic probe microanalysis (EPMA) technique. On the basis of various critically-reviewed experimental diffusivities as well as the available thermodynamic descriptions, atomic mobilities in fcc_A1 Ni–X (X=Ge, Ti and V) alloys were assessed by means of DIffusion Controlled TRAnsformation (DICTRA) software package. For the self-diffusivities in dynamically unstable fcc_A1 Ge, Ti and V, a semi-empirical correlation was employed. Moreover, the first-principles calculations were also used to calculate the activation energy for self-diffusion in fcc_A1 Ge. Comprehensive comparison between the calculated and the measured diffusivities shows that most of the experimental data can be reproduced by the atomic mobilities obtained in the present work. In addition, the reliability of the currently obtained atomic mobilities in fcc_A1 Ni–X (X=Ge, Ti and V) alloys were further validated by comparing the model-predicted concentration–distance profiles and the experimental ones.

Extracting interdiffusion coefficients from binary diffusion couples using traditional methods and a forward-simulation method

A MatLab program is developed to allow effective extraction of both interdiffusion coefficients and impurity diffusivities from concentration profiles of binary diffusion couples. In addition to implementing the Boltzmann–Matano method, the Sauer–Freise method, the Hall method, and the Wagner method, a forward-simulation method was developed and implemented in the program. Eight binary systems (Fe–Ni, Co–Ni, Nb–W, Ni–W, Ni–Mo, Co–Mo, Fe–Nb and Ni–Nb) were used to test the program and to evaluate the advantages and disadvantages of the various methods. The forward-simulation method shows advantages including the ability to extract the impurity diffusion coefficients. Valuable impurity diffusivities and interdiffusion coefficients data at 1100 °C for the eight binary systems are obtained and they are in good agreements with available literature data. The successful diffusivity extraction from several complicated composition profiles with several intermetallic compounds demonstrates the reliability and robustness of the program.