Solute trapping and diffusionless solidification in a binary system

Mathematical model of solidification of melt with high-speed cooling

A new mathematical model of supercooled melt crystallization based on the variational principles of thermodynamics has been developed. The model takes into account the crystal formation and diffusion growth regularities, as well as the diffusionless crystal growth with the deviation from the local equilibrium at the surface. The model also takes into account the growing crystals mutual influence on the components concentration in the melt. The calculations for the supercooled eutectic melt Fe₈₃B₁₇showed that the nucleation and growth of the phases Fe and Fe₂B with a metastable phase Fe₃B occur in the melt. The local equilibrium on the surface of the growing Fe₃B crystals with the melt probably does not maintained. The regularities of the nucleation and mutual influence of the growing crystals of the phases are studied. The nucleation and growth rate of the Fe₃B nuclei differs from the growth of Fe and Fe₂B nuclei due to the diffusionless capture of boron atoms by the growing Fe₃B crystals surface. The model will help to calculate the melt cooling technology mode for producing amorphous ribbons on a copper rotating drum. The calculation made it possible to analyze changes in the temperature and the crystallization degree in the various ribbon layers. The calculation results have been verified experimentally by x-ray diffraction and calorimetric studies of the obtained ribbons. The correspondence of the calculation and the experimental results confirms the effectiveness developed methodology for studying the regularities of crystal growth in supercooled melts.

Enhancing the incorporation of Sn in vapor-liquid-solid GeSn nanowires by modulation of the droplet composition

We report on the influence of the liquid droplet composition on the Sn incorporation in GeSn nanowires (NWs) grown by the vapor-liquid-solid (VLS) mechanism with different catalysts. The variation of the NW growth rate and morphology with the growth temperature is investigated and 400 °C is identified as the best temperature to grow the longest untapered NWs with a growth rate of 520 nm min^-1. When GeSn NWs are grown with pure Au droplets, we observe a core-shell like structure with a low Sn concentration of less than 2% in the NW core regardless of the growth temperature. We then investigate the impact of adding different fractions of Ag, Al, Ga and Si to Au catalyst on the incorporation of Sn. A significant improvement of Sn incorporation up to 9% is obtained using 75:25 Au-Al catalyst, with a high degree of spatial homogeneity across the NW volume. Thermodynamic model based on the energy minimization at the solid-liquid interface is developed, showing a good correlation with the data. These results can be useful for obtaining technologically important GeSn material with a high Sn content and, more generally, for tuning the composition of VLS NWs in other material systems.

Thermodynamic description of metastable fcc/liquid phase equilibria and solidification kinetics in Al-Cu alloys

The thermodynamic description of the fcc phase in the Al-Cu system has been revised, allowing for the prediction of metastable fcc/liquid phase equilibria to undercoolings of ΔT = 421 K below the eutectic temperature. Hypoeutectic Al-Cu alloys that are prone to pronounced microsegregation were solidified containerlessly in electromagnetic levitation. Solidus and liquidus concentrations were experimentally determined from highly undercooled samples employing energy-dispersive X-ray analysis. Solid concentrations at a rapidly propagating solid/liquid interface were additionally calculated using a sharp interface model that considers all undercoolings and is based on solvability theory. Modelling results (front velocity versus undercooling) were also corroborated by in situ observation with a high-speed camera. A newly established thermodynamic description of the fcc phase in Al-Cu is compatible with existing CALPHAD-type databases. Inconsistencies of previous descriptions such as a miscibility gap between Al-fcc and Cu-fcc on the Al-rich side, an unrealistic curvature of the solidus line in the same composition range or an azeotropic point near the melting point of Cu, are amended in the new description. The procedure to establish the description of phase equilibria at high undercoolings can be transferred to other alloy systems and is of a general nature. This article is part of the theme issue 'Transport phenomena in complex systems (part 2)'.

Dendrite tips as elliptical paraboloids

This review article summarizes current theories of the steady-state growth mode of dendrites in the form of elliptical paraboloids. The shape of dendrite tips is analyzed, temperature and solute concentration distributions are described in its vicinity, and a solution of the hydrodynamic problem of a viscous incompressible fluid flowing against a dendrite tip is developed. A significant difference in analytical solutions describing a dendrite tip as an elliptic paraboloid as compared to an axisymmetric morphology is shown. The system of nonlinear equations for determining the stationary velocity of dendrite growth and the radii of curvature of the dendrite tip along the major and minor axis of the ellipse, respectively, is derived. The developed theory is compared with experimental data on the growth of ice crystals consisting of D₂O or H₂O.

Stretching the Equilibrium Limit of Sn in Ge1-x Sn x Nanowires: Implications for Field Effect Transistors

Ge1-x Sn x nanowires incorporating a large amount of Sn would be useful for mobility enhancement in nanoelectronic devices, a definitive transition to a direct bandgap for application in optoelectronic devices and to increase the efficiency of the GeSn-based photonic devices. Here we report the catalytic bottom-up fabrication of Ge1-x Sn x nanowires with very high Sn incorporation (x > 0.3). These nanowires are grown in supercritical toluene under high pressure (21 MPa). The introduction of high pressure in the vapor-liquid-solid (VLS) like growth regime resulted in a substantial increase of Sn incorporation in the nanowires, with a Sn content ranging between 10 and 35 atom %. The incorporation of Sn in the nanowires was found to be inversely related to nanowire diameter; a high Sn content of 35 atom % was achieved in very thin Ge1-x Sn x nanowires with diameters close to 20 nm. Sn was found to be homogeneously distributed throughout the body of the nanowires, without apparent clustering or segregation. The large inclusion of Sn in the nanowires could be attributed to the nanowire growth kinetics and small nanowire diameters, resulting in increased solubility of Sn in Ge at the metastable liquid-solid interface under high pressure. Electrical investigation of the Ge1-x Sn x (x = 0.10) nanowires synthesized by the supercritical fluid approach revealed their potential in nanoelectronics and sensor-based applications.

Diffusionless (chemically partitionless) crystallization and subsequent decomposition of supersaturated solid solutions in Sn-Bi eutectic alloy

Results of a study on microstructural evolution of eutectic Sn-57 wt.% Bi processed with cooling rates of 10^-2, 1 K s^-1 and approximately 10⁵ K s^-1 are presented. In order to distinguish different mechanisms of microstructure formation, a comparison with microstructures of different hypoeutectic alloys with compositions down to below the maximum solubility of Bi in Sn-Bi is undertaken. It is found that at the cooling rates of 10^-2 and 1 K s^-1, coupled eutectic growth occurs, leading to lamellar structures with different length scales. At the rapid quenching rates of approximately 10⁵ K s^-1, structure formation in the eutectic alloy is qualitatively different. Partitionless solidification resulting in a supersaturated solid solution with the initial composition is observed in both eutectic and hypoeutectic alloys. It is shown that the observed microstructure of the rapidly solidified alloys forms by the decomposition of the supersaturated solid solution. This article is part of the theme issue 'Heterogeneous materials: metastable and non-ergodic internal structures'.

Experimental and modelling studies for solidification of undercooled Ni-Fe-Si alloys

We present experimental results, analytical calculations and phase-field simulations for undercooled Ni-Fe-Si alloy system. Undercooling experiments are performed using flux encapsulation along with in situ measurement of recalescence speed using a high-speed camera followed by microstructural characterization. Dendrite growth calculations are performed using a modified Boettinger, Coriell and Trivedi theory to incorporate constitutional undercooling due to multiple segregating elements and a modified kinetic undercooling term. Phase-field simulations are performed using a multi-component phase-field model to generate dendrites in this alloy. High growth velocities are observed and the analytical calculations are in good agreement with experiments. The microstructure evolution from the phase-field simulations indicates that there is a difference in solute segregation during growth of dendrites. This article is part of the theme issue 'Heterogeneous materials: metastable and non-ergodic internal structures'.

Kinetic transition in the order-disorder transformation at a solid/liquid interface

Phase-field analysis for the kinetic transition in an ordered crystal structure growing from an undercooled liquid is carried out. The results are interpreted on the basis of analytical and numerical solutions of equations describing the dynamics of the phase field, the long-range order parameter as well as the atomic diffusion within the crystal/liquid interface and in the bulk crystal. As an example, the growth of a binary A₅₀B₅₀ crystal is described, and critical undercoolings at characteristic changes of growth velocity and the long-range order parameter are defined. For rapidly growing crystals, analogies and qualitative differences are found in comparison with known non-equilibrium effects, particularly solute trapping and disorder trapping. The results and model predictions are compared qualitatively with results of the theory of kinetic phase transitions (Chernov 1968 Sov. Phys. JETP26, 1182-1190) and with experimental data obtained for rapid dendritic solidification of congruently melting alloy with order-disorder transition (Hartmann et al. 2009 Europhys. Lett.87, 40007 (doi:10.1209/0295-5075/87/40007)).This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.

Crystal growth kinetics in undercooled melts of pure Ge, Si and Ge-Si alloys

We report on measurements of crystal growth dynamics in semiconducting pure Ge and pure Si melts and in Ge_100-x Si _x (x = 25, 50, 75) alloy melts as a function of undercooling. Electromagnetic levitation techniques are applied to undercool the samples in a containerless way. The growth velocity is measured by the utilization of a high-speed camera technique over an extended range of undercooling. Solidified samples are examined with respect to their microstructure by scanning electron microscopic investigations. We analyse the experimental results of crystal growth kinetics as a function of undercooling within the sharp interface theory developed by Peter Galenko. Transitions of the atomic attachment kinetics are found at large undercoolings, from faceted growth to dendrite growth.This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.

The boundary integral theory for slow and rapid curved solid/liquid interfaces propagating into binary systems

The boundary integral method for propagating solid/liquid interfaces is detailed with allowance for the thermo-solutal Stefan-type models. Two types of mass transfer mechanisms corresponding to the local equilibrium (parabolic-type equation) and local non-equilibrium (hyperbolic-type equation) solidification conditions are considered. A unified integro-differential equation for the curved interface is derived. This equation contains the steady-state conditions of solidification as a special case. The boundary integral analysis demonstrates how to derive the quasi-stationary Ivantsov and Horvay-Cahn solutions that, respectively, define the paraboloidal and elliptical crystal shapes. In the limit of highest Péclet numbers, these quasi-stationary solutions describe the shape of the area around the dendritic tip in the form of a smooth sphere in the isotropic case and a deformed sphere along the directions of anisotropy strength in the anisotropic case. A thermo-solutal selection criterion of the quasi-stationary growth mode of dendrites which includes arbitrary Péclet numbers is obtained. To demonstrate the selection of patterns, computational modelling of the quasi-stationary growth of crystals in a binary mixture is carried out. The modelling makes it possible to obtain selected structures in the form of dendritic, fractal or planar crystals.This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.

Boron Partitioning Coefficient above Unity in Laser Crystallized Silicon

Boron pile-up at the maximum melt depth for laser melt annealing of implanted silicon has been reported in numerous papers. The present contribution examines the boron accumulation in a laser doping setting, without dopants initially incorporated in the silicon wafer. Our numerical simulation models laser-induced melting as well as dopant diffusion, and excellently reproduces the secondary ion mass spectroscopy-measured boron profiles. We determine a partitioning coefficient k p above unity with k p = 1 . 25 ± 0 . 05 and thermally-activated diffusivity D B , with a value D B ( 1687 K ) = ( 3 . 53 ± 0 . 44 ) × 10 - 4 cm 2 ·s - 1 of boron in liquid silicon. For similar laser parameters and process conditions, our model predicts the anticipated boron profile of a laser doping experiment.

Microstructural and Mechanical-Property Manipulation through Rapid Dendrite Growth and Undercooling in an Fe-based Multinary Alloy

Rapid dendrite growth in single- or dual-phase multicomponent alloys can be manipulated to improve the mechanical properties of such metallic materials. Rapid growth of (αFe) dendrites was realized in an undercooled Fe-5Ni-5Mo-5Ge-5Co (wt.%) multinary alloy using the glass fluxing method. The relationship between rapid dendrite growth and the micro-/nano-mechanical properties of the alloy was investigated by analyzing the grain refinement and microstructural evolution resulting from the rapid dendrite growth. It was found that (αFe) dendrites grow sluggishly within a low but wide undercooling range. Once the undercooling exceeds 250 K, the dendritic growth velocity increases steeply until reaching a plateau of 31.8 ms(-1). The increase in the alloy Vickers microhardness with increasing dendritic growth velocity results from the hardening effects of increased grain/phase boundaries due to the grain refinement, the more homogeneous distribution of the second phase along the boundaries, and the more uniform distribution of solutes with increased contents inside the grain, as verified also by nanohardness maps. Once the dendritic growth velocity exceeds ~8 ms(-1), the rate of Vickers microhardness increase slows down significantly with a further increase in dendritic growth velocity, owing to the microstructural transition of the (αFe) phase from a trunk-dendrite to an equiaxed-grain microstructure.

Non-equilibrium induction of tin in germanium: towards direct bandgap Ge(1-x)Sn(x) nanowires

The development of non-equilibrium group IV nanoscale alloys is critical to achieving new functionalities, such as the formation of a direct bandgap in a conventional indirect bandgap elemental semiconductor. Here, we describe the fabrication of uniform diameter, direct bandgap Ge_1−xSn_x alloy nanowires, with a Sn incorporation up to 9.2 at.%, far in excess of the equilibrium solubility of Sn in bulk Ge, through a conventional catalytic bottom-up growth paradigm using noble metal and metal alloy catalysts. Metal alloy catalysts permitted a greater inclusion of Sn in Ge nanowires compared with conventional Au catalysts, when used during vapour–liquid–solid growth. The addition of an annealing step close to the Ge-Sn eutectic temperature (230 °C) during cool-down, further facilitated the excessive dissolution of Sn in the nanowires. Sn was distributed throughout the Ge nanowire lattice with no metallic Sn segregation or precipitation at the surface or within the bulk of the nanowires. The non-equilibrium incorporation of Sn into the Ge nanowires can be understood in terms of a kinetic trapping model for impurity incorporation at the triple-phase boundary during growth.

Direct band gap nanostructures compatible with Si-based electronics are actively investigated. Here, Biswas et al. incorporate unusually large amounts of tin in germanium nanowires by non-equilibrium kinetic trapping, and optical characterizations suggest that the nanowires exhibit a direct band gap.

Metastable crystalline AuGe catalysts formed during isothermal germanium nanowire growth

We observe the formation of metastable AuGe phases without quenching, during strictly isothermal nucleation and growth of Ge nanowires, using video-rate lattice-resolved environmental transmission electron microscopy. We explain the unexpected formation of these phases through a novel pathway involving changes in composition rather than temperature. The metastable catalyst has important implications for nanowire growth, and more broadly, the isothermal process provides both a new approach to growing and studying metastable phases, and a new perspective on their formation.

Solute trapping in rapid solidification of a binary dilute system: a phase-field study

The phase-field model of Echebarria, Folch, Karma, and Plapp [Phys. Rev. E 70, 061604 (2004)] is extended to the case of rapid solidification in which local nonequilibrium phenomena occur in the bulk phases and within the diffuse solid-liquid interface. Such an extension leads to the fully hyperbolic system of equations given by the atomic diffusion equation and the phase-field equation of motion. This model is applied to the problem of solute trapping, which is accompanied by the entrapment of solute atoms beyond chemical equilibrium by a rapidly moving interface. The model predicts the beginning of complete solute trapping and diffusionless solidification at a finite solidification velocity equal to the diffusion speed in bulk liquid.

Phase-field-crystal and Swift-Hohenberg equations with fast dynamics

A phenomenological description of transition from an unstable to a (meta)stable phase state, including microscopic and mesoscopic scales, is presented. It is based on the introduction of specific memory functions which take into account contributions to the driving force of transformation from the past. A region of applicability for phase-field crystals and Swift-Hohenberg-type models is extended by inclusion of inertia effects into the equations of motion through a memory function of an exponential form. The inertia allows us to predict fast degrees of freedom in the form of damping perturbations with finite relaxation time in the instability of homogeneous and periodic model solutions.