Melt Motion Research Articles

Dopant transport during semiconductor crystal growth: Axial versus transverse magnetic fields

The present study investigates the effects of magnetic field orientation, magnetic field strength and growth rate on the dopant segregation in semiconductor crystals, and presents results of dopant composition in the crystal and in the melt at several different times during growth for several combinations of process parameters. The crystal's lateral segregation depends on the magnetic field's orientation and strength while the axial segregation depends on the magnetic field's strength and the growth rate. If either convective or diffusive transport truly dominates, then the crystal's dopant distribution is laterally uniform. The axial distribution in the crystal approaches the well-mixed limit if the melt motion is strong and the growth rate is slow, and the distribution approaches the diffusion-controlled limit if the melt motion is slow and the growth rate is fast. The deviations of the dopant distribution in the crystal from lateral uniformity and from the classical limits are quantified for several combinations of process parameters.

Inertial effects in the rotationally driven melt motion during the Czochralski growth of silicon crystals with a strong axial magnetic field

This article treats the melt motion driven by the rotations of the crystal and crucible about their common vertical axis during the Czochralski growth of silicon crystals with a strong, uniform, vertical magnetic field produced by a solenoid around the crystal growth furnace. Since molten silicon is an excellent electrical conductor, the interaction parameter N and the Hartmann number Ha are both large for typical magnetic field strengths, so that composite singular perturbation techniques for \(N\gg1\) and \(Ha\gg1\) are appropriate. An inertialess solution, which also assumed that \(N\gg Ha^{3/2}\), was presented in a previous paper. In the inertialess solution, the largest gradient of the azimuthal velocity \(v_\theta\) and the largest secondary flow with radial and axial velocities vr and vz both occur inside an interior layer with an \(O(Ha^{-1/2})\) radial thickness at the vertical cylinder directly beneath the periphery of the crystal, where the growth interface meets the free surface. For all current experimental studies, the assumption that \(N\gg Ha^{3/2}\) is not satisfied. The appropriate assumption is that \(N=O(Ha^{3/2})\), and inertial effects are not negligible inside the interior layer. An intersection region, which is formed by the intersection of the interior layer and a Hartmann layer with an \(O(Ha^{-1})\) axial thickness adjacent to the crystal-melt interface and free surface, is intrinsically coupled to the interior layer. This article treats inertial effects in the interior layer and intersection region for \(N=O(Ha^{3/2})\) and \(Ha\gg1\). Non-linear governing equations were derived and solved numerically. A fourth-order Adams-Bashforth-Moulton predictor-corrector method was used to solve the transport equations for the primary azimuthal velocity and for the secondary-flow vorticity. Poisson equations, which govern the stream functions for both the secondary flow and the electric current density, were solved using a matrix diagonalization technique. The effects of inertia on the melt motion are discussed. This type of study provides for a fuller understanding of the melt motion, without which defect-free crystals will be difficult to grow on a consistent basis.

High-temperature centrifuge simulation- new method to study multiphase flow of partially molten zones

Present-day physical theories suggest that the mantle was partially or completely molten about 4.5 billion years ago. As is expected, the phase separation could result in the stratification of the initially homogeneous mantle into reservoirs of different chemical composition. Because of the poor knowledge of the physical phenomena connected with the multiphase system differentiation, many aspects of the evolution of partially molten zones remain disputable. In this study the simulation of liquid and crystal separation in a hightemperature centrifuge was carried out. Experimental simulation with the pyroxene + olivine + plagioclase + magnetite + melt mixture demonstrates the gravitational instability of the polycrystalline framework of partially molten zones. The melt motion in partially molten rocks is though to be a multiphase flow involving both the melt and individual crystals. A high mobility of plagioclase crystals in the multiphase flow was ascertained. It is revealed as the formation of a monomineral plagioclase cumulate overlying the accumulated molten layer. The separation and motion of sulfides in the intercrystalline space is shown to be in an intimate relationship with the degree of melting of a silicate material.

The Effect of Gravitation Forced Convection on Structure of Al-Ni Eutectic

The effect of different kinds of experimental parameters such as gravitational acceleration (microgravity, highgravity), temperature gradient, solidification rate, primary phase, (aluminium dendrite, Al 3 Ni compound), mechanical instability of the liquid (micro gravity, high gravity, vertically stabilized and unstabilized, horizontal arrangement) on the melt motion and on the Al-Al 3 Ni rodlike eutectic structure was investigated.

Numerical study of the melt–thermal effect on a silicon crystal in Czochralski growth system

In this study, the melt–thermal effect on the silicon crystal was investigated in a cusp magnetic Czochralski process. The silicon melt motion and thermal field were numerically obtained for various system variables. The crucible rotational rate from 6 to 14 rpm, the silicon single-crystal rotational rates from −10 to −20 rpm, the cusp magnetic field strength with the range of 0.15–0.3 T and the radii and vertical distance from the melt–solid interface of the countercurrent loops were adopted as system variables. The melt–thermal effect on the solid was represented by total heat flow to the crystal at the melt–solid interface. It was found that the crucible rotational rate, the magnetic field strength and the radii of the loops had significant effects in the melt–thermal effect on the solid and the melt–thermal effect showed nonlinear behavior due to the nonlinearity of the melt motion for the magnetic field strength and the radii of the loops.

High temperature centrifuge simulation — New method to study multiphase flow of partially molten zones

Present-day physical theories suggest that the mantle was partially or completely molten about 4.5 billion years ago. As is expected, the phase separation could result in the stratification of the initially homogeneous mantle into reservoirs of different chemical composition. Because of the poor knowledge of the physical phenomena connected with the multiphase system differentiation, many aspects of the evolution of partially molten zones remain disputable. In this study the simulation of liquid and crystal separation in a high- temperature centrifuge was carried out. Experimental simulation with the pyroxene + olivine + plagioclase + magnetite + melt mixture demonstrates the gravitational instability of the polycrystalline framework of partially molten zones. The melt motion in partially molten rocks is thought to be a multiphase flow involving both the melt and individual crystals. A high mobility of plagioclase crystals in the multiphase flow was ascertained. It is revealed as the formation of a monomineral plagioclase cumulate overlying the accumulated molten layer. The separation and motion of sulfides in the intercrystalline space is shown to be in an intimate relationship with the degree of melting of a silicate material.

Melt Motion Due to Peltier Marking during Bridgman Crystal Growth with an Axial Magnetic Field

This paper treats a liquid-metal flow inside an electrically insulating cylinder with electrically conducting solids above and below the liquid region. There is a uniform axial magnetic field, and there is an electric current through the liquid and both solids. Since the lower liquid-solid interface is concave into the solid and since the liquid is a better electrical conductor than the adjacent solid, the electric current is locally concentrated near the centerline. The return to a uniform current distribution involves a radial electric current which interacts with the axial magnetic field to drive an azimuthal flow. The axial variation of the centrifugal force due to the azimuthal velocity drives a meridional circulation with radial and axial velocities. This problem models the effects of Peltier marking during the vertical Bridgman growth of semiconductor crystals with an externally applied magnetic field, where the meridional circulation due to the Peltier current may produce important mixing in the molten semiconductor.

A Model of Dopant Transport During Bridgman Crystal Growth With Magnetically Damped Buoyant Convection

Abstract This paper presents a model for the unsteady transport of a dopant during the vertical Bridgman crystal growth process with a planar crystal-melt interface and with an externally applied axial magnetic field. This dilute mass transport depends on the convective and diffusive mass transport of the dopant. The convective mass transport is driven by buoyant convection in the melt, which produces nonuniformities in the concentration in both the melt and the crystal. This convective transport is significant even for a strong magnetic field Bo=2 T. However, the electromagnetic damping of the melt motion produces a local region adjacent to the crystal-melt interface which is dominated by diffusion. Thus, this melt solidifies with a relatively radially uniform concentration, so that the radial distribution of dopants in the crystal is also relatively radially uniform. The transient model predicts the dopant distribution in the entire crystal. [S0022-1481(00)02301-X]

Growing range of correlated motion in a polymer melt on cooling towards the glass transition

Many liquids cooled to low temperatures form glasses (amorphous solids) instead of crystals. As the glass transition is approached, molecules become localized and relaxation times increase by many orders of magnitude1. Many features of this ‘slowing down’ are reasonably well described2 by the mode-coupling theory of supercooled liquids3. The ideal form of this theory predicts a dynamical critical temperature T c at which the molecules become permanently trapped in the ‘cage’ formed by their neighbours, and vitrification occurs. Although there is no sharp transition, because molecules do eventually escape their cage, its signature can still be observed in real and simulated liquids. Unlike conventional critical phenomena (such as the behaviour at the liquid–gas critical point), the mode-coupling transition is not accompanied by a diverging static correlation length. But simulation4,5,6,7,8,9,10 and experiment11,12 show that liquids are dynamically heterogeneous, suggesting the possibility of a relevant ‘dynamical’ length scale characterizing the glass transition. Here we use computer simulations to investigate a melt of short, unentangled polymer chains over a range of temperatures for which the mode-coupling theory remains valid. We find that although density fluctuations remain short-ranged, spatial correlations between monomer displacements become long-ranged as T c is approached on cooling. In this way, we identify a growing dynamical correlation length, and a corresponding order parameter, associated with the glass transition. This finding suggests a possible connection between well established concepts in critical phenomena and the dynamics of glass-forming liquids.

Comment on “Chain Motion in an Unentangled Polyethylene Melt: A Critical Test of the Rouse Model by Molecular Dynamics Simulations and Neutron Spin Echo Spectroscopy”

A Comment on the Letter by W. Paul, et al., Phys. Rev. Lett. 80, 2346 (1998).Received 18 May 1998DOI:https://doi.org/10.1103/PhysRevLett.82.2408©1999 American Physical Society

Models of melt motion, heat transfer and mass transport during crystal growth with strong magnetic fields

Models of melt motion, heat transfer and mass transport during crystal growth with strong magnetic fields

Thermoelectrically Driven Melt Motion During Floating Zone Crystal Growth With an Axial Magnetic Field

During semiconductor crystal growth with an externally applied magnetic field, thermoelectric currents may drive a melt circulation which affects the properties of the crystal. This paper treats a model problem for a floating zone process with a uniform axial magnetic field, with planar solid-liquid interfaces, with a cylindrical free surface, with a parabolic temperature variation along the crystal-melt interface, and with an isothermal feed rod-melt interface. The ratio of the electrical conductivities of the liquid and solid is a key parameter. The azimuthal velocity is much larger than the radial or axial velocity. There is radially outward flow near the crystal-melt interface which should be beneficial for the mass transport of dopants and species.

Effect of lateral migration of melt on chemical compositions: Implications for the local variations of mid‐ocean ridge basalts

The decompressional melting process involving dynamically induced migration of melt in the residual country rock is analyzed to formulate the effects of melt migration on the chemical compositions of mid‐ocean ridge basalt (MORB). It is rigorously proved that if the relative motion between melt and solid is unidirectional and steady, the composition of the melt is the same as that expected from the batch melting because it is determined by equations that are separable from those concerning the relative melt motion. The present proof is more general than the previous treatments and is applicable to any system consisting of arbitrary numbers of phases and components. The effect of lateral melt motion on the concentration of a minor component is formulated more specifically using two parameters. One of the parameters is the lateral gradient of melt flux that represents degree of convergence or divergence of melt in the flow. The other parameter that specifies the gradient of the chemical composition is introduced to evaluate the so‐called chromatographic effect that separates the minor component migration paths in the flow by their partition coefficients. The parameterization gives a good tool to predict the flow field from the known chemical variations. It is shown that some local variations of MORB compositions observed in individual mid‐ocean ridges can be modeled as a result of lateral melt migration convergent in some places and divergent in others.

Bridgman crystal growth with a strong, low-frequency, rotating magnetic field

This paper treats the melt motion driven by a low-frequency, spatially uniform, rotating magnetic field that is perpendicular to the axis of a cylindrical Bridgman ampoule. The magnetic field is sufficiently strong that inertial effects are negligible and that viscous effects are confined to thin boundary layers that are adjacent to the ampoule surfaces and to the crystal–melt interface. Outside the boundary layers, the melt nearly rotates as a rigid body with the angular velocity of the rotating magnetic field. Each boundary layer provides a transition from this rigid-body rotation to the zero azimuthal velocity at each solid surface. In the boundary layer adjacent to the crystal–melt interface, which is parallel to the magnetic field, there are nonaxisymmetric radial and axial velocities associated with the azimuthal variations of the azimuthal velocity. Because of its nonzero electrical conductivity, the static crystal acts as a generator in the rotating magnetic field and drives electric currents through the crystal–melt interface boundary layer. These electric currents from the slightly conducting crystal interact with the magnetic field to accelerate the azimuthal velocity. Some of the melt in this layer may have an angular velocity that is more than twice the angular velocity of the magnetic field. For the actual small electrical conductivities of solid semiconductors, the radial and axial velocities inside the crystal–melt interface layer are very different from those for an electrically insulating crystal. The results indicate that a strong, low-frequency, rotating magnetic field would produce poor crystals with severe rotational striations.

Chain Motion in an Unentangled Polyethylene Melt: A Critical Test of the Rouse Model by Molecular Dynamics Simulations and Neutron Spin Echo Spectroscopy

We have investigated the dynamic structure factor for single-chain relaxation in a polyethylene melt by means of molecular dynamics simulations and neutron spin echo spectroscopy. After accounting for a 20% difference in the chain self-diffusion coefficient between simulation and experiment we find a perfect quantitative agreement of the intermediate dynamic structure factor over the whole range of momentum transfer studied. Based on this quantitative agreement one can test the experimental results for deviations from standard Rouse behavior reported so far for only computer simulations of polymer melt dynamics.

Magnetic damping of buoyant convection during semiconductor crystal growth in microgravity. Steady transverse residual acceleration

In this paper we treat the steady, three-dimensional buoyant convection in a circular cylinder with an axisymmetric temperature, with a uniform axial magnetic field, and with a steady transverse acceleration or gravitational force. The magnetic field is sufficiently strong that inertial effects and convective heat transfer are negligible for the typical residual accelerations on an earth-orbiting vehicle. Since the governing equations are linear, the solution for any steady acceleration or gravitational vector is given by the superposition of the solutions for an axial acceleration and for a transverse acceleration. The solution for an axial acceleration is given by a rescaling of the well-known solution for a terrestrial vertical Bridgman crystal-growth process, and in this paper we present the solution for a transverse acceleration. With an axial magnetic field, the melt motion consists of two circulations in opposite directions around two axial lines on opposite sides of the diameter plane which is parallel to the transverse acceleration vector a: both circulations involve flows in the direction opposite to a near the diameter plane and flows in the same direction as a near the cylinder wall. The small electrical conductivity of the crystal is important because the crystal provides a path for a major part of the electric current responsible for the magnetic damping.

Thermomechanical Simulation of the Splashing of Ceramic Droplets on a Rigid Substrate

Finite element simulation techniques have been applied to the spreading process of single ceramic liquid droplets impacting on a flat cold surface under plasma-spraying conditions. The goal of the present investigation is to predict the geometrical form of the splat as a function of technological process parameters, such as initial temperature and velocity, and to follow the thermal field developing in the droplet up to solidification. A non-linear finite element programming system has been utilized in order to model the complex physical phenomena involved in the present impact process. The Lagrangean description of the motion of the viscous melt in the drops, as constrained by surface tension and the developing contact with the target, has been coupled to an analysis of transient thermal phenomena accounting also for the solidification of the material. The present study refers to a parameter spectrum as from experimental data of technological relevance. The significance of process parameters for the most pronounced physical phenomena is discussed as are also the consequences of modelling. We consider the issue of solidification as well and touch on the effect of partially unmelted material.

Melt‐Motion during the Czochralski Growth of Silicon Crystals with a Cusp Magnetic Field

The relative importance of the three melt‐motion components (thermocapillary convection, the buoyant convection, and centrifugal‐pumping flow) during the Czochralski growth of silicon crystals with a cusp magnetic field is investigated by combining the separate asymptotic solutions. The top boundary layer, which is adjacent to the crystal‐melt interface and to the free surface, is very important because the thermocapillary and centrifugal‐pumping flows are confined to this layer, and because the flow here determines the radial distributions of oxygen and dopants in the crystal, as well as the rate of oxygen evaporation from the free surface. The buoyant convection is not confined to the top layer, extends over the entire melt, and is the only component of the melt motion which affects the melt temperature for O(1) thermal Péclet numbers. The analysis of the thermal problem with convective heat transport is presented. The effects of varying the Hartmann number, the thermal Péclet number, and the angular velocity of the crystal on the balance between the three melt‐motion components in the top layer are investigated.

Inertial effects in magnetically stabilized thermocapillary convections during floating-zone semiconductor crystal growth in space

For the floating-zone growth of semiconductor crystals in space, striation-producing unsteady melt motions can be eliminated by applying a steady magnetic field which is parallel to the common centerline of the crystal and feed rod. A previous paper [T.E. Morthland and J.S. Walker, J. Crystal Growth 158 (1996) 471] presented numerical solutions for steady thermocapillary convections neglecting the inertial terms in the Navier-Stokes equation. With power limitations in space, achievable magnetic field strengths are large enough to eliminate unsteadiness in the melt motion, but are not large enough that inertial effects are negligible. The numerical solutions presented here include inertial effects, which reduce the magnitudes of the two circulations and shift their centers toward the solid-melt interfaces.

A Study of Convection, Striations and Interface Shape in InP Crystals Grown by the Double‐Crucible LEC Technique

AbstractA doublecrucible (DC) Czochralski setup has been used for the pulling of semi‐insulating Fe‐doped InP. The level of the Fe‐doped melt in the growth crucible is replanished with the undoped InP from the reservoir crucible. In this way, since the distribution coefficient of iron is very small (about 0.001), the Fe concentration in the growth crucible is virtually unchanged during the pulling so that the axial Fe concentration is much more uniform than in standard LEC crystals. The use of two concentric crucibles has great implications in terms of convective flows, stability of the melt temperature and interface shape. In this paper we report the results of a study on striations and structural defects in InP grown from a double‐crucible LEC arrangement. A dimensionless relationship for correlating striation features with melt motions is also proposed.