www.elsevier.com/locate/ijengsci

International Journal of Engineering Science 43 (2005) 1059–1080

A continuum model for the Liquid Phase Diffusion growthof bulk SiGe single crystals

Mehmet Yildiz, Sadik Dost *

Crystal Growth Laboratory, University of Victoria, Victoria, BC, Canada V8W 3P6

Received 14 January 2005; received in revised form 1 June 2005; accepted 1 June 2005

Abstract

In this article, we present a computational model for the growth of SixGe1�x single crystals by LiquidPhase Diffusion (LPD) from the germanium-rich side of the binary SixGe1�x phase diagram. The modelaccounts for some important physical features of the LPD growth process of SixGe1�x such as agrowth-zone design on the thermal field, the buoyancy induced convective flow and its effect on the growthprocess, the evolution of growth interface, and the variation of growth velocity with time. Two- and three-dimensional numerical simulations were carried out. Simulation results show that the natural convection inthe solution is very strong during the initial stages of the growth process, and gets weaker as the growthprogresses. Diffusion becomes the dominant mass transport mechanism during the rest of the growth pro-cess. Computed growth interface shapes and growth velocities agree with experimental observations.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Crystal growth; Liquid phase; Diffusion; Convection; Modeling; Silicon; Germanium

1. Introduction

The SixGe1�x alloy is an emerging semiconductor material with many important potential elec-tronic and optoelectronic applications such as a base for the fabrication of Si/SiGe heterojunction

0020-7225/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijengsci.2005.06.001

* Corresponding author. Tel.: +1 250 721 8898.E-mail address: sdost@me.uvic.ca (S. Dost).

1060 M. Yildiz, S. Dost / International Journal of Engineering Science 43 (2005) 1059–1080

bipolar transistors (HBT) and high electron mobility field effect transistors [1,2], photodetectorand solar cell applications [3,4], and thermoelectric power generators, tuneable neutron andX-ray monochromators, high speed temperature sensors, and c-ray detectors [5–7]. In most ofthese applications, SixGe1�x single crystal substrates with a specific composition (x) are needed.In order to achieve high quality SixGe1�x crystals with uniform composition, a variety of crystalgrowth techniques, such as Czochralski, floating zone, Bridgman, multi component and liquidencapsulated zone melting, and others have been utilized with limited success in terms of growingcompositionally uniform and low defect density crystals [8]. The growth of high quality SiGe sin-gle crystals with uniform compositions, compatible with the requirements of device applications,is a challenging task. This is mainly due to the large miscibility gap in the Si–Ge phase diagram,and also due to the significant differences in physical properties of the constituent elements, suchas density, melting temperature, and lattice parameter. Because of the large miscibility gap, whichgives rise to segregation coefficients far from unity, any small changes in the solidification rate willlead to significant composition variations, and in turn various types of defects in the grown crys-tals. Thus, the crystal growth method to be chosen is of a crucial importance to obtain bulkSixGe1�x single crystals with desired qualities.

Liquid Phase Electroepitaxy (LPEE) was considered as a technique of choice for the growth ofSixGe1�x single crystals. LPEE is a solution growth technique and has been successfully utilizedfor the growth of bulk III–V alloy semiconductor materials [9,10]. It has a number of advantagesover melt techniques (see [9] for detail). However, the LPEE growth process needs a seed of singlecrystal of the same composition of the crystal to be grown (a small composition difference withinthe limit of acceptable lattice mismatch is tolerable). In order to address this issue, a crystalgrowth technique, called Liquid Phase Diffusion (LPD), was utilized in [11] with two objectivesin mind. One was the growth of bulk SixGe1�x single crystals with varying composition fromwhich the seed substrates with required compositions can be extracted. At the time this workwas initiated the growth of large size SixGe1�x crystals with a wide composition range was notreported, and such crystals were not available commercially. The second objective was the devel-opment of the first stage of a hybrid growth technique that combines LPD and LPEE in a singleprocess. In this hybrid technique in mind, a graded single crystal would be grown by LPD up tothe composition of interest, and then at this stage the LPEE process would be initiated by passingan electric current through the growth system at a uniform furnace temperature, leading to thegrowth of single crystals with desired uniform compositions. Such a single process would elimi-nate the adverse affects of growing crystals in two stages. Such a hybrid growth process will bedeveloped in the near future.

LPD is a crystal growth technique which can be considered within the family of ‘‘directionalsolidification’’, and developed by Nakajima et al. [12] under the name of ‘‘Multicomponent ZoneMelting’’. In this technique the solvent material (Ge) is sandwiched between a single crystal sub-strate (seed, Ge) and polycrystalline source material (feed, Si). In that sense the LPD techniqueutilized here can be considered as a solution growth technique, and called ‘‘Liquid PhaseDiffusion’’.

The vertically stacked charge materials held in a quartz crucible form the LPD growth cell, asillustrated in Fig. 1. Initially, all three layers are solid materials. Upon the application of a tem-perature profile on the growth zone, the solid germanium in the middle totally melts to form theliquid solvent for the growth. The germanium substrate partially melts. The silicon source, on the

Fig. 1. A schematic view of the LPD growth system.

M. Yildiz, S. Dost / International Journal of Engineering Science 43 (2005) 1059–1080 1061

other hand, remains in the solid state due to its higher melting point. The molten germanium dis-solves the silicon source according to thermodynamic equilibrium. Dissolved silicon species areincorporated into the germanium liquid, thereby forming a binary SixGe1�x mixture. Silicon spe-cies in the solution zone are transported to the growth interface by the combination of diffusionand convection. When the mixture at the growth interface is supersaturated, solidification takesplaces due to the presence of constitutional supercooling. For the detailed description of theLPD growth system and experiments the reader is referred to [11].

In this article, we present a computational model based on the thermo-mechanical balance lawsof a binary continuum mixture and the related irreversible thermodynamics of the transport phe-nomena (fluid flow, heat and mass transfer) occurring during the LPD growth process (see[13,14,16] for details). The model equations and the related interface and boundary conditionsare presented for the LPD system considered in [11]. The field equations were solved numericallyfor two and three-dimensional domains using a finite volume based commercial package (CFX)[15]. Simulations results are given for the dynamic evolution of the growth interface, time and spa-tial dependency of the crystal growth velocity, and solute distribution in the grown crystal. Sim-ulation results show that the natural convection in the solution is very strong during the initialstages of the growth process, and gets weaker as the growth progresses. Diffusion becomes thedominant mass transport mechanism during the rest of the growth process. The computed growthinterface shapes and growth velocities agree with experimental observations.

1062 M. Yildiz, S. Dost / International Journal of Engineering Science 43 (2005) 1059–1080

2. The model

In the model, the liquid phase is taken as the Ge-rich Si–Ge solution zone in the LPD growthcrucible, and the solid phase represents the Ge single crystal seed substrate, the Si polycrystallinefeed, and the crucible wall. The basic equations of each phase and the related assumptions arepresented below.

2.1. The liquid phase

The liquid phase is considered as a mixture of two viscous and heat conducting incompressibleNewtonian fluids, and is assumed to be a dilute binary solution (Ge-rich) of the solute (Si) and thesolvent (Ge). The mass density of the mixture q, and the mass fractions of the constituents, c(1) andc(2), are defined respectively as

qL ¼ qð1Þ þ qð2Þ and cð1Þ ¼ qð1Þ=q � C and cð2Þ ¼ qð2Þ=q ð2:1Þ

where c(1) + c(2) = 1, and C is the solute (silicon) mass concentration in the liquid phase.

The basic equations of the liquid phase are obtained from the thermomechanical balance laws ofa binary continuum mixture (see [8,11,13,14,16] for detail) under the following assumptions. Thecontribution of temperature gradient to mass flux is known as the Soret effect, and similarly thecontribution of concentration gradient to heat flux as the Dufour effect. Since there is no physicalevidence for the significance of these effects in the LPD growth process, their contributions wereneglected. As a result the Fick�s law for the mass flux, and the Fourier law for the heat flux wereadopted. Since the liquid phase has relatively low viscosity, the contribution of concentration andtemperature to the dissipative stress tensor were also neglected. In the momentum balance, the onlybody force is the gravitational force, and the well-known Boussinesq approximation is adopted.This is necessary in order to include the effect of density variations arising from the temperatureand concentration gradients across the liquid phase. The Boussinesq approximation assumes thatthe density of the mixture is constant everywhere in the field equations, except the one appearing inthe body force term. To account for density variations as a function of temperature and concen-tration, the density is expanded into a Taylor series about the reference concentration C0 and tem-perature T0. In the SixGe1�x liquid solution of the present setup, the density of the mixturedecreases as the temperature increases (positive thermal expansion). At the same time, silicon isthe lighter component in the solution; hence with the addition of silicon the density of the mixturealso decreases, (positive solutal expansion). The mixture density can then be expressed as

q ¼ qL þ oqoT

����T 0c0

�bTqL

ðT � T 0Þ þoqoC

����T 0c0

�bCqL

ðC � C0Þ ¼ qL½1 � bTðT � T 0Þ � bCðC � C0Þ� ð2:2Þ

where T stands for temperature, and bT and bC represent the thermal and solutal expansion coef-ficients evaluated at the reference state, respectively, and qL is the density of the reference (start-ing) liquid solution (which is the molten germanium).

Fig. 2 shows schematically the vertical cross section of the LPD growth system. Under theabove assumptions, the three-dimensional time-dependent governing equations describing thefluid flow, heat and mass transport for the liquid phase are written in cylindrical coordinates

Fig. 2. The schematics of the vertical cross section of the LPD growth system with the representative thermal profilewhere h(r,z) represents the location of the growth interface, t is the unit tangential vector to the growth interface, whilen is the unit normal to the growth interface pointing into the liquid domain.

M. Yildiz, S. Dost / International Journal of Engineering Science 43 (2005) 1059–1080 1063

(r,/,z). However, the field equations of the two-dimensional simulation model are not given sep-arately for the sake of space. It is straightforward to write two-dimensional equations from three-dimensional equations by simply dropping the dependency on the azimuthal coordinate (/) in thefield variables. In 2-D simulations, the growth cell is assumed axisymmetric with respect to thecenterline as shown in Fig. 2. When it is necessary, additional boundary conditions for the two-dimensional model are introduced. At the growing interface the contribution of latent heat (enthal-py of fusion) is also neglected due to very slow growth rate. The field equations of the model arethen written as

Continuity

1

ro

orðrV rÞ þ

1

roV /

o/þ oV z

oz¼ 0 ð2:3Þ

Momentum

oV r

otþ V r

oV r

orþ V /

roV r

o/þ V z

oV r

oz�V 2

/

r¼ � 1

qL

opor

þ m r2V r �V r

r2� 2

r2

oV /

o/

� �ð2:4Þ

oV /

otþ V r

oV /

orþ V /

roV /

o/þ V z

oV /

ozþ V rV /

r¼ � 1

rqL

opo/

þ m r2V / � V /

r2� 2

r2

oV r

o/

� �ð2:5Þ

oV z

otþ V r

oV z

orþ V /

roV z

o/þ V z

oV z

oz¼ � 1

qL

opoz

þ mr2V z þ bTðT � T 0Þg þ bCðC � C0Þg ð2:6Þ

1064 M. Yildiz, S. Dost / International Journal of Engineering Science 43 (2005) 1059–1080

Energy

oTot

þ V roTor

þ V /

roTo/

þ V zoToz

¼ kL

qLcLr2T ð2:7Þ

Mass transport

oCot

þ V roCor

þ V /

roCo/

þ V zoCoz

¼ DLSir2C ð2:8Þ

In Eqs. (2.3)–(2.8), vr, v/, and vz are, respectively, the velocity components in the radial (r), cir-cumferential direction (/), and vertical (z) directions, m is the kinematic viscosity of the solution,p is pressure, kL and cL represent the thermal conductivity and specific heat capacity of the solu-tion, respectively, DL

Si denotes the diffusion coefficient of silicon in the solution, and the gradientoperator is defined as

r2 � 1

ro

orro

or

� �þ 1

r2

o2

o/2þ o2

oz2ð2:9Þ

2.1.1. Boundary and interface conditionsThe required boundary and interface conditions corresponding to the governing equations in

Eqs. (2.3)–(2.8) are obtained from the associated jump conditions as follows (see [8,13] forderivations):

At the vertical wall: The no-slip boundary condition is employed at the vertical wall of thegrowth cell; namely, at the interface between the surrounding crucible and the liquid. As forthe boundary condition for the mass balance equation for solute (silicon), the vertical walls areassumed to be impermeable to the species in the solution. Thus,

vr ¼ 0; V z ¼ 0; V / ¼ 0;oCor

¼ 0 ð2:10Þ

At the axis of symmetry (for only two-dimensional model): At the axis of symmetry, we usesymmetry conditions, and also for physical (finite) results, it is required that the radial velocitycomponent be zero. Hence,

oV z

or¼ 0;

oTor

¼ 0;oCor

¼ 0; V r ¼ 0 ð2:11Þ

At the growth and dissolution interfaces: At both interfaces, the local thermodynamic equilib-rium is assumed. This implies that the concentration at the interface is determined from theSi–Ge binary phase diagram namely, ceq = f(Teq) which is written for the liquid ðcL

eqÞ and solidðcL

eqÞ equilibrium compositions for the Ge-rich side of the phase diagram as

cLeq ¼ aLT 3 þ bLT 2 þ cLT þ dL and cs

eq ¼ ðT � asÞ=bs ð2:12Þ

The numerical values of the phase diagram coefficients are given in Table 1.Growth interface: Since the growth rate in LPD is approximately five orders of magnitude smal-

ler than the fluid velocity, the radial and axial velocity components can be reasonably set to zeroat the growth interface. In addition, the diffusion of solute in the solid phase is far smaller than the

Table 1SiGe phase diagram coefficients

aL bL cL dL as bs

2.5984 · 10�9 �8.7189 · 10�6 0.0099041 �3.822 1211.45 639.63

M. Yildiz, S. Dost / International Journal of Engineering Science 43 (2005) 1059–1080 1065

one in the liquid phase, so, the solid phase diffusion at the interface can also be neglected. Regard-ing the energy balance at the interface, due to the slow growth rate, as mentioned earlier, thelatent heat dissipates without disturbing the continuity of the heat flux at the interface, so isneglected. In addition, since the curvature of the interface is fairly small, the effect of surface ten-sion in the energy balance is also excluded. The surface tension effect in the interface energy bal-ance, however, becomes important in the presence of an interface with large curvature such as inthe case of a dendritic growth or lateral overgrowth [17,18]. By solving the mass balance equationat the interface, the evolution of the growth interface is computed. Then, the conditions at thegrowth interface are written as

C ¼ CLgi ¼ CL

eq at T gi; V r ¼ 0; V / ¼ 0; V z ¼ 0;

qsugðCsgi � CL

giÞ ¼ qLDLSi

oCon

; ks oTs

on� kL oT

L

on¼ 0

ð2:13Þ

where CLgi is the equilibrium mass fraction of the solute (silicon) at the growth interface, Cs

gi is theequilibrium mass fraction of silicon in the solidified crystal, ug is the rate of displacement of thesubstrate–solution interface along the unit normal direction, also known as growth velocity. o C/onis the normal derivative of the solute mass fraction at the interface, Cs

gi is the equilibrium massfraction of silicon in the solid phase at the growth interface, and ks represents thermal conductivityof the solid phase, and TL and Ts are respectively the liquid and solid temperatures at the growthinterface.

Dissolution interface: Since the experimentally observed dissolution rate of the silicon source isvery small (approximately 0.3 mm/day), mass balance at the dissolution interface is excluded. Wethen have

C ¼ CLdi ¼ CL

eq at T di; V r ¼ 0; V / ¼ 0; V z ¼ 0; ks oTs

on� kL oT

L

on¼ 0 ð2:14Þ

where CLdi is the equilibrium mass fractions of the solute (silicon) at the dissolution interface.

2.2. The solid phase

As mentioned earlier, the solid phase represents the single crystal Ge seed substrate, the poly-crystalline Si source, the growing SixGe1�x single crystal, and the quartz crucible ampoule whichare modeled as a rigid, heat conducting solid. In this case the only remaining balance law is theenergy equation which takes the following form:

oTot

¼ ks

qscsr2T ð2:15Þ

where ks, qs and cs represent respectively the thermal conductivity, density and specific heat of thesolid phases.

1066 M. Yildiz, S. Dost / International Journal of Engineering Science 43 (2005) 1059–1080

2.2.1. Boundary conditionsThe thermal boundary conditions for the quartz crucible (vertical wall, top and bottom

surfaces) and the symmetry axis of the crucible are given as

TableDesig

TotalInitiaInitiaCrystTotal

�ksoTon

¼ h_

fT � T fðzÞg;oTon

¼ 0 ð2:16Þ

where Tf(z) is the ambient temperature inside the furnace along the quartz ampoule wall ortop and bottom surfaces, h

_

is the modified heat transfer coefficient, including the contributionof convective and radiation effect on heat loss, namely,

h_

¼ hþ �rfT þ T fðzÞgfT 2 þ T 2f ðzÞg ð2:17Þ

where e is the thermal emissivity of the quartz ampoule, and r is the Stefan Boltzmann constant.The modified heat transfer coefficient is approximated using preliminary experimental results(such as the measured thermal profile, solute distribution in a grown crystal, and the positionof the initial growth interface). In addition, perfect thermal contacts at the melt–ampoule, crys-tal–ampoule, and inner–outer crucible are assumed. Therefore at these boundaries, the heat fluxis continuous.

The initial conditions (at t = 0) are

C ¼ C0; V r ¼ 0; V / ¼ 0; V z ¼ 0: ð2:18Þ

2.3. Physical parameters of the LPD SixGe1�x system

Here, we list the physical properties of SixGe1�x and some input parameters used in the com-putations. Since the liquid mixture is dilute in silicon solute (maximum silicon molar fraction inliquid is xL

Si ¼ 0:04), the physical properties of germanium liquid are used whenever the requiredphysical properties of the SixGe1�x system are not available. The composition dependence of thespecific heat capacity of the solid and liquid phases is approximated by a linear expression interms of the values of end constituents at a given temperature as

cSixGe1�x¼ cSiX þ cGeð1 � X Þ ð2:19Þ

The composition dependence of the density of the growing crystal is calculated using the sameargument given in the above equation. The reported diffusion coefficient for silicon in a germa-nium melt is highly diverse varying between 1 · 10�8 and 5 · 10�8 m2/s. We adopted the value2.5 · 10�8 m2/s. Design parameters used in the simulated LPD growth system, and the physicalproperties of the charge materials and the quartz crucible are given in Tables 2 and 3, respectively.

2n parameters for LPD growth system

growth zone length 40 mml source height 3 mml substrate thickness 10 mmal diameter 25 mmwall thickness of quartz (quartz growth reactor plus crucible) 4 mm

Table 3Physical properties of growth charges and the quartz crucible

Physical parameter Source Si Substrate Ge Growingcrystal SixGe1�x

Liquid solutionSixGe1�x

Growth charges

Specific heat c (J/kg K) 967 at 1300 K,1037 (liquid)

396.1 at 1210 K,380 (liquid)

396.1–487 380–406

Thermal Conductivity k (W/m K) 23.7 at 1273 K 10.60 at 1210 K 10.60 42.8Mass diffusivity D (m2/s) N/A N/A 1 · 10�20 2.5 · 10�8

Mass density q (kg/m3) 2301.6 at 1300 K 5323 at 1210 K 4839–5323 5633Thermal expansion bT (1/K) N/A N/A N/A 1.2 · 10�4

Solutal expansion bC (1/mol% Si) N/A N/A N/A 0.0053Enthalpy of fusion DHF (kJ/kg) 1807.9 466.5 – –Viscosity lv = kg/m s N/A N/A N/A 7.35 · 10�4

Quartz crucible

Specific heat, c (J/kg K) 1200 at 1300 KThermal conductivity, k (W/m K) 2 at 1300 KMass density, q (kg/m3) 2200 at 1300 K

The physical parameters were compiled from Refs. [23–28].

M. Yildiz, S. Dost / International Journal of Engineering Science 43 (2005) 1059–1080 1067

3. Numerical analysis and results

In this section, we present the results of our numerical simulations. The field equations weresolved using a finite volume-based commercial package (CFX). Three-dimensional simulationswere conducted for only an initial period of growth due to the use of a fine mesh in the compu-tational domain, which requires significant computational time. In order to simulate the entiregrowth process we invoked two-dimensional simulations.

3.1. Order of magnitude analysis for the LPD growth system

Such a scaling analysis is important for shedding some light on how physical and transportparameters (i.e., density, viscosity, solutal and thermal expansion coefficients, etc.) contributesto the structure of field variables developed during a process before performing lengthy compu-tations. In this study, it will allow us to evaluate qualitatively the relative contributions of suchparameters to the transport phenomena occurring during the LPD growth of SiGe single crystals.For instance, two important material properties that relate the density of mixture to temperatureand solute concentration are the thermal and solutal expansion coefficients. The values of thesetwo coefficients (namely, being positive or negative) are dependent on whether the density ofthe solvent increase or decrease with temperature and solute concentration. Densities of the liquidsilicon and germanium decrease with increasing temperature. It is reasonable to expect that aliquid SiGe mixture shows the same behavior for the density variation as a function of tempera-ture, which means that the volume of the mixture expands on heating (positive thermal expan-sion). Solubility of silicon (solute) in germanium (solvent) increases with temperature, and sincethe density of silicon less than that of germanium, the mixture density also decreases with theincrease in the solute concentration (positive solutal expansion).

Table 4Non-dimensionless numbers and their characteristic values calculated for the growth germanium-rich SiGe singlecrystals by LPD

Dimensionless numbers Expression Characteristic values

Thermal Grashof GrT ¼ ðbTgl30Þv�2DT 5.09 · 107

Solutal Grashof GrC ¼ ðbCgl30Þv�2DT 10.7 · 107

Prandtl Pr = m/a 0.0075Schmidt Sc = m/D 7.6743

1068 M. Yildiz, S. Dost / International Journal of Engineering Science 43 (2005) 1059–1080

To account for the relative contributions of the solutal and thermal gradients to the existence ofnatural convection in the liquid solution, non-dimensionless thermal and solutal Grashof numberscan be compared. Definitions of the dimensionless numbers and their characteristic values (com-puted using the LPD experimental set-up and SiGe transport parameters evaluated about themelting temperature of germanium) are given in Table 4. In the table, l0 is the characteristic lengthtaken as the diameter of the liquid zone (25 mm), g is the gravitational constant, m is the kinematicviscosity, and DC = C1 � C0 and DT = T1 � T0 are the characteristic concentration and temper-ature. C1, C0 and T1, T0, are boundary values of concentration (in terms of silicon mass fraction)and temperature in Kelvin at the top and bottom of the liquid.

For an accurate scaling analysis, dimensionless numbers must be defined with respect to accu-rate characteristic parameters of the system. For instance, the thermal and solutal Grashof num-bers are usually defined with respect to the radial length and the thermal and solutal gradients inthis direction. However, in the LPD system, this would not be reliable since the radial temperatureand solutal gradients vary even over a small length scale. Due to this difficulty, in this work thesolutal and thermal Grashof numbers are defined in terms of the axial concentration and temper-ature gradients despite the fact that they do not scale the process very accurately. The computedvalues show that in the present system the solutal Grashof number is approximately two timesgreater than the thermal Grashof number, which implies that the contribution of the solutal gra-dient to natural convection, positively or negatively, is larger than that of thermal gradient.

Other important dimensionless parameters are the Schmidt and Prandtl numbers which aredefined as the ratios of momentum diffusivity to solutal diffusivity, and to thermal diffusivity,respectively. Small value of the Prandtl number suggests that the heat transport across the solutionis mainly due to conduction. This was verified by numerical simulations which led to the similartemperature distributions computed with and with no convection. The relatively large Schmidtnumber indicates that convection has a significant effect on solute transport during growth.

3.2. Computational procedure

Two- and three-dimensional time dependent partial differential equations and the associatedboundary and interface conditions, describing the fluid flow, heat and mass transport, are solvedusing CFX-4.4, a finite volume-based commercial software package developed by AE technology[15]. Simulation of the growth process of SiGe by LPD is a typical moving boundary problem, alsoknown as a Stefan problem where a particular portion of the computational domain changes as afunction of time. Therefore, several Fortran subroutines were developed and used to deal with themoving grid with time and complex temperature boundary conditions. For the discretisation of

M. Yildiz, S. Dost / International Journal of Engineering Science 43 (2005) 1059–1080 1069

time derivation, a fully implicit backward time difference stepping procedure has been imple-mented. The time iterations are initiated to calculate time dependence of concentration, flowvelocity in liquid and temperature in both liquid and solid. In so doing, it becomes possible toinvestigate the changes in these field variables, and the substrate thickness due to the motion ofgrowth interface.

The computational process starts with defining initial guesses and boundary conditions. Thenthe time dependent flow fields are computed. The computed flow fields are used to calculate ther-mal and concentration profiles in the liquid zone of the growth system since the energy and speciesmass balances and linear momentum equations are coupled. Since the liquid and solid phases havesignificantly different thermal conductivities, the thermal field has to be updated at each time stepto include the influence of increasing substrate thickness. The concentration at the interfaces isdetermined from the SiGe binary phase diagram. Hence, upon using the computed temperate fieldat the interface, the interfacial concentrations are computed, thereby the concentration field in theliquid domain at each time step as the growth thickness increases.

The selection of finite volume mesh has a direct effect on the accuracy of computed results. Theaccuracy of the numerical results is checked by refining the finite volume mesh until mesh-inde-pendent solutions are obtained, for both two- and three-dimensional simulations. The mesh sizeleading to mesh-independent results is used during the simulations. In the present work, we con-sidered an equidistant finite volume meshes. The mesh sizes used in the two-dimensional simula-tions for the substrate, liquid domain, source, and quartz ampoule are 1500, 4000, 500, and 1200,respectively. In the three-dimensional simulations, the mesh sizes are selected respectively as18,000, 54,000, 7200, and 42,240.

3.3. Results of two-dimensional simulations

3.3.1. Temperature fieldFig. 3 shows typical temperature distributions in the substrate, solution, source, and quartz cru-

cible for several hours of growth time. The initial temperature difference between the top and bot-tom of the liquid domain is about 45 K. As can be seen from Fig. 3a, the initial growth interfacedeformed according to a reference isothermal line is of a concave shape. The formation of a con-cave interface could be attributed to several reasons.

The large mismatch in thermal conductivities of the substrate, solution and surrounding quartzcrucible bends isothermal lines in the vicinity of interface. Recalling from the simulated experi-mental configuration in Fig. 1, there is an annular insulator material located inside the furnace,which encompasses the quartz crucible with a 3–5 mm clearance. The insulator material spansfrom approximately 5 mm above the substrate to the bottom of the source material. The convec-tive heat transfer in this semi-insulated region is handled by a proper estimation for the heat trans-fer coefficient, including both radiative and convective heat transfer. The presence of a ring-shaped insulator material distorts the isothermal lines around the boundary between the insulatedand uninsulated regions.

Furthermore, the convective heat transfer from the bottom of the crucible to surrounding fur-nace atmosphere is smaller than that from the periphery of the ampoule due to the presence of ahallow quartz pedestal underneath the crucible. Away from the substrate and source material,thermal lines are almost flat due to being inside the insulated region. In the source, thermal lines

Fig. 3. Computed thermal field within the entire computational domain (2-D) at various growth times (temperature inthe scales are given in Kelvin). (a) Growth time:0.5 h and (b) growth time: 7.5 h.

1070 M. Yildiz, S. Dost / International Journal of Engineering Science 43 (2005) 1059–1080

are convex since the top of the source material is in contact with the vacuum environment; there-fore, heat transfer from the surrounding to the source material through its surface is less than thatfrom its periphery through the quartz crucible.

As for the thermal profile nearby the dissolution interface, it has a convex shape and large ther-mal gradients. Convex thermal profile can also be reasoned out by envisaging the difference in thethermal boundary conditions between the regions with and without insulation. Besides, there is alarge mismatch in thermal conductivities of the solution, source and crucible as well.

Upon comparing the simulated thermal profiles of the growth zone for different growth times, itcan easily be concluded that the flow field does not have a strong influence on the thermal field.This is an expected result since the Prandtl number for the germanium rich SixGe1�x mixture is farsmaller than unity. Therefore, heat transport within the liquid domain is mainly due to conduc-tion with very little contribution of convective flow on the temperature profile.

It is interesting to note that as the growth proceeds, the growth interface is crossed by thermalisotherms, and hence a non-isothermal interface is developed. This situation gives rise to differentlevels of local saturations at the growth interface, and in turn varying growth velocity across theinterface as will be discussed in more detail in the next section.

Fig. 5 shows the decrease in the magnitude of the flow velocity near the growth interface as afunction of silicon mass fraction.

3.3.2. Flow and concentration fieldsFig. 4 presents the time evolution of the flow (left column) and solute concentration (right col-

umn) structures. The flow field is given in terms of the magnitude of the flow velocity, namely,

U ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV 2

r þ V 2z

q, and the concentration field in silicon mass fraction. As can be seen, the computed

flow field develops only one main eye-shaped convective cell which consists of radially-stacked

Fig. 4. Flow (left column) and concentration (right column) fields for different growth time. Flow field is given in terms

of magnitude of the flow velocity, U ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV 2

r þ V 2z

q, and the solute concentration is given in silicon mass fraction.

(a) Growth time: 0.5 h; (b) growth time: 5 h and (c) growth time: 6.5 h.

M. Yildiz, S. Dost / International Journal of Engineering Science 43 (2005) 1059–1080 1071

Fig. 5. Decrease in the magnitude of flow velocity as a function of silicon mass fraction.

1072 M. Yildiz, S. Dost / International Journal of Engineering Science 43 (2005) 1059–1080

several small cells. This circular flow cell is located near the growth interface. In the rest of theliquid domain, the convection is very weak, consequently is not notable numerically. The convec-tive flow cell circulates anti-clockwise, pushing the melt upward along the axis of symmetry, anddownwards along the vertical crucible wall. The fastest circulation of fluid particles is at the centreof the main cell.

The observation of weak convection in the rest of the liquid domain may be explained by keep-ing in mind the fact that silicon is constantly added into the germanium melt through dissolution.Silicon is the lighter constituent of the SiGe liquid mixture, and in turn, the addition of silicon intothe mixture decreases its density, thus causing an axial density gradient and making the mixtureheavier near the growth interface. This is an interesting feature of the LPD growth process ofSiGe. This axial density gradient due to the addition of solute acts as a stabilizing effect on theflow structure developed by the radial temperature gradient. At the early stages of the flow, con-vection is very strong since there is not enough silicon species in the vicinity of the growth inter-face, and this stabilizing effect of silicon cannot suppress the formation of convective cells. As thegrowth proceeds, more silicon species is transported towards the growth interface, thereby weak-ening the convective flow. Subsequently, the circular flow cell shrinks continuously and disappearscompletely after 5.2 h of growth. The suppression of this toroidal convective cell makes the LPDgrowth processes of SiGe diffusion-dominant. The corresponding concentration profiles are nat-urally are very stable giving rise to a very stable growth. The formation of an axial, stabilizingdensity gradient was also reported for the two-dimensional numerical simulation of the growthof SixGe1�x and GaInSb by the vertical Bridgman method [19,20]. As can also be seen fromFig. 4, due to the mixing effect of convection, the region with strong convection has a uniformconcentration distribution, while in that without convection, the concentration distributionresembles the one observed in a diffusion limited process.

M. Yildiz, S. Dost / International Journal of Engineering Science 43 (2005) 1059–1080 1073

In crystal growth, uncontrolled convection is not desired because it gives rise to growthinstability, inclusion trapping, local variation of growth velocity (which may cause poly-crystalline growth), even though convection enhances transport process, in turn, results in fastergrowth.

To elucidate clearly that the formation of solutal-based axial density gradient brings about astabilizing effect in the flow structure, two-dimensional simulations have also been performedfor the case where the contribution of solutal gradients on convection is neglected, i.e. webC = 0 take in the momentum equation (this assumption is made only in computations presentedin Fig. 6). Since the focus was only on the flow structure and concentration distribution in thiscase, the interface shape was assumed not to be deviating from its initial shape for early severalhours of the growth process. Therefore, the growth interface has been moved at a constant growth

Fig. 6. Flow and concentration fields for different hours of growth when bC = 0. (a) Growth time: 0.5 h and (b) growthtime: 3.5 h.

1074 M. Yildiz, S. Dost / International Journal of Engineering Science 43 (2005) 1059–1080

velocity, 0.9 mm/h. In Fig. 6, the flow field is presented in the left column, while the concentrationfield is in the right. The flow has two separate toroidal convective cells, one being close to the dis-solution interface, and the second adjacent to the growth interface. The top and bottom cellscirculate in opposite directions. The top cell rotates in clockwise, while the bottom cell in theanti-clockwise direction. The top flow cell moves the melt upward along the vertical quartzampoule wall and downwards along the axis of the ampoule. The bottom cell, on the other hand,pushes the melt downward along the quartz ampoule and upward along the axis of the ampoule.Existence of two convective cells can be attributed mainly to the radial thermal gradient. Magni-tude and direction of the velocity field are dependent on the steepness of the radial temperaturegradient and the geometrical shape of radial thermal isotherms, respectively. In the currentgrowth configuration, the radial thermal gradient near the dissolution interface is steeper thanthat near the growth interface so that the flow intensity of the top toroidal cell is greater than thatof bottom cell. There is no convective cell present in the middle region (between the top andbottom convective cells) because there is practically no radial temperature gradient in this region.This numerical result suggests that the existence of a convective cell is mainly due to the presenceof radial thermal gradients.

Because of the presence of two convective cells in the liquid domain, the concentration profiledeveloped exhibits a totally distinct behavior as compared to those introduced in Fig. 4. In thepresent case, each convection cell creates a concentration vortex. Since the flow field is strongerat the center of both convection cells, concentration gradients at the corresponding regions arethe smallest. Flow circulation pushes solute species towards the boundaries of the liquid domain,causing relatively larger solutal gradients. Concentration gradient is the largest in the areabetween the upper and bottom concentration vortexes for two reasons: the lack of large radialthermal gradients and the accumulation of solute caused by flow circulation.

3.3.3. Growth velocityThe interface displacement rate (growth velocity) is computed by solving the mass balance

equation coupled with the SiGe binary phase diagram at each time step. The initial computedgrowth interface is geometrically concave. Fig. 7 reveals both spatial and time variations of thenumerical and experimental growth velocity. Spatial variation of the growth velocity can easilybe observed by comparing the interface displacement values for the axis (r = 0) and periphery(r = r) of the crystal. At the onset of the growth, for several hours, the growth velocity at theperiphery of the grown crystal is faster than that at the center point. This situation makes theinterface become more rounded inward. As the growth progress, the center regions start growingmore rapidly than the periphery, leading to a flatter interface shape. As the growth advances fur-ther, the curve of the interface changes direction and becomes convex. The reason behind a fastergrowth along the vertical crucible wall at the early stages of growth can be attributed to the pres-ence of non-uniform mass flux associated with the convective flow cell which is of larger magni-tude nearby the crucible wall. In subsequent hours of the growth process, thermal field develops atthe interface such that it begins crossing the growth interface. As a result, the growth interface nolonger follows the shape of the thermal isotherm near the center region of the crystal. This createsa non-isothermal growth interface. Across the non-isothermal interface, concentration gradient(driving force for the growth) is different. In the regions with larger concentration gradients,the growth velocity is faster.

Fig. 7. Interface position as a function of growth time, at (a) r = 0 and (b) r = r.

M. Yildiz, S. Dost / International Journal of Engineering Science 43 (2005) 1059–1080 1075

The time variation of the growth velocity can be best visualized considering mass balance at thegrowth interface, i.e.,

ug ¼1

ðCsgi � CL

giÞqLDL

Si

qs

oCon

where Csgi and CL

gi are the equilibrium mass fractions of silicon in the solution and the solidifiedcrystal at the interface, computed from the phase diagram. According to the SiGe phase diagram,

1076 M. Yildiz, S. Dost / International Journal of Engineering Science 43 (2005) 1059–1080

the term ðCsgi � CL

giÞ increases with the increasing interface temperature as the growth interfacemoves upward. Considering that during the growth, the concentration gradient along the growthdirection remains nearly constant, one can conclude that the growth velocity will be inversely pro-portional to the term ðCs

gi � CLgiÞ.

In what follows is that the growth velocity must decrease as the growth interface moves to thehigher temperature regions. This purely mathematical definition refers to the fact that at the onsetof the growth process, less amount of silicon atoms are needed to saturate the interface for solid-ification; however, at the later stages of the growth, more silicon atoms are needed for super sat-uration, so that the growth velocity is expected to be not as fast as before. Fig. 7a shows theexperimentally [21] and numerically obtained interface displacements at different hours of thegrowth for r = 0 (at the centre). The computed averaged growth velocity over 29 h of growth is0.63 mm/h, while the experimental value is 0.67 mm/h. Fig. 7b presents the measured and com-puted growth velocities at the wall (for r = r). The numerically computed average growth velocityis 0.59 mm/h, while the measured value is 0.55 mm/h. The difference between experimental andnumerical growth velocities is approximately about 6%.

In Fig. 8, the computed time evolution of the growth interface is shown on the left, and thecross section of an LPD grown single crystal on the right (see [11,21] for crystal orientation, singlecrystallinity, and composition variation. All the grown crystals were single crystals, and weregrown on (111) and (100) oriented seeds. Comparison of growth striations (circular mark) onthe grown crystal with the computed growth interfaces shows that the numerical model developedsuccessfully simulates the tendency of the time evolution of growth interface. In experiments, the

Fig. 8. Time evolution of the growth interface for the half geometrical domain is shown on the left (the time intervalbetween each line is three hours, and total simulated growth time is 39 h), a cross section of an LPD grown crystal onthe right. Agreement between experimental and simulation results are quite good.

M. Yildiz, S. Dost / International Journal of Engineering Science 43 (2005) 1059–1080 1077

interface starts becoming flatter after 16 h of the growth [11], while numerically, the flat interfaceis observed around the 24th hour of the growth.

Fig. 9. Flow (left) and concentration (right) fields at (a) t = 0.5 h, (b) t = 3.5 h and (c) t = 5.5 h.

1078 M. Yildiz, S. Dost / International Journal of Engineering Science 43 (2005) 1059–1080

3.4. Results of three-dimensional simulation

In this section results of a three-dimensional simulation are presented. The computed temper-ature distributions in the vertical section (at / = p and at time t = 1 h) show the same patterns asthose obtained from two-dimensional simulations (Fig. 3). Therefore, they are not presented forthe sake of space.

The computed flow and concentration fields are presented in Fig. 9 in the vertical plane (where

/ = p) of the computational domain where the flow field is given in U ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV 2

r þ V 2z þ V 2

/

q(mag-

nitude of the flow velocity) and the concentration field in silicon mass fraction. As seen in Fig. 9a,similar to that of Fig. 4a, the flow field has two main cells bridged to each other at the vertical axisof the liquid domain (at 0.5 h). The cells circulate in the opposite directions. The cell on the rightmoves in clockwise pushing the fluid particles down along the crucible wall and up along the axisof symmetry, whereas the one on the left circulates in the anti-clockwise direction. Due to thepresence of convection, solute distribution in the mixture shows a perfect mixing. The regionsdeprived of convection cells show diffusion like solute distribution profiles. Two- and three-dimen-sional computational results show similar quantitative behavior regarding the flow structure andsolute distribution in the liquid domain. This is an expected result since there are no externaleffects such as applied magnetic field or crucible rotation.

Fig. 10. Flow fields at different growth times: 0.5, 3.5, 4.5 and 5.5 h. The horizontal plane is at a distance of 14 mmfrom the bottom of the substrate.

M. Yildiz, S. Dost / International Journal of Engineering Science 43 (2005) 1059–1080 1079

However, even in the present case, three-dimensional simulations clearly show the developmentof three-dimensional flow structures in the liquid zone in spite of the initially assumed axisymmetricboundary conditions. This is particularly obvious in the structures presented in the horizontal plane(Fig. 10) at a distance of 14 mm from the bottom of the substrate. The development of 3-D flowstructures in the present 3-D computations is of numerical nature, as known (see [22] for detail).

As the growth progresses, and in turn, more silicon species is transported towards the growthinterface, as was observed in two-dimensional simulations, the convective flow cells begin losingtheir intensity, and finally become almost suppressed when the solute concentration reaches acritical value in the vicinity of the growth interface.

4. Conclusions

In this article, we have introduced a computational model to investigate the transport phenom-ena occurring in the growth of SiGe single crystals by the growth technique Liquid Phase Diffusion.Results of the numerical simulations show that the natural convection in the liquid zone is verystrong at the beginning of the growth process, but it gets weaker as the growth progresses. Afterabout 5.2 h of growth time, convection becomes unobservable numerically, and diffusion becomesthe dominant growth mechanism. 2-D and 3-D simulation results are very similar quantitatively,however, 3-D simulations lead to the development of three-dimensional flow structures as expected.

Although LPD is mainly a diffusion dominated process, the presence of strong convection duringthe early stages of the growth, indeed, is not desired because it gives rise to non-uniform and uncon-trolled transport of species to the growth interface. The non-uniform mass flux to the growth inter-face will most likely result in local variations in the growth velocity. Such variations across thegrowth interface generally reveal itself as the driving force for the formation of low angle grainboundaries, or in extreme cases, of extensive polycrystalline structure. The suppression of the con-vection or reducing it to a minimum, in fact, can be helpful for increasing the likelihood of the singlecrystalline growth. The use of an applied magnetic field is a feasible option to achieve such a goal.

Acknowledgements

The financial support provided by BL Consulting Ltd. of Victoria, DL Crystals Inc. of Victoria,the Microgravity Science Program of the Canadian Space Agency, the Natural Sciences and Engi-neering Research Council of Canada (NSERC), and the Canada Foundation for Innovation(CFI) is gratefully acknowledged.

References

[1] J.D. Cressler, Re-engineering silicon: Si–Ge heterojunction bipolar technology, IEEE Spectrum (March) (1995)49–55.

[2] Y.J. Mii, Y.H. Xie, E.A. Fitzgerald, D. Monroe, F.A. Thiel, B.E. Weir, L.C. Feldman, Extremely high electronmobility in SixGe1�x structures grown by molecular beam epitaxy, Appl. Phys. Lett. 59 (1991) 1611–1613.

[3] M. Jutzi, M. Berroth, in: E. Kasper, K. Lyutovich, (Eds.), SiGe-based photodetectors for optical communications:properties of silicon germanium and SiGe:Carbon, INSPEC 2000, London, p. 342.

1080 M. Yildiz, S. Dost / International Journal of Engineering Science 43 (2005) 1059–1080

[4] K. Said, J. Poortmans, M. Caymax, et al., High quality relaxed SiGe epitaxial layers for solar cell application,Thin Solid Films 337 (1999) 85–89.

[5] C.M. Bhandari, D.M. Rowe, Silicon–germanium alloys as high-temperature thermoelectric materials, Contemp.Phys. 21 (1980) 219–242.

[6] M.L. Kozhukh, I.N. Belokurova, S.B. Vahrushev, A.N. Titkov, I.L. Shul�pina, Study of Ge–Si solid solutioncrystals by X-ray topography and neutron diffraction, Nucl. Instrum. Meth. 213 (1983) 483–487.

[7] J. Schilz, V.N. Romanenko, Review: bulk growth of silicon–germanium solid solutions, J. Mater. Sci.: Mater.Electron. 6 (1995) 265–279.

[8] M. Yildiz, A combined experimental and modeling study for the growth of SiGe single crystals by liquid phasediffusion (LPD), Ph.D. Thesis, 2005. Department of Mechanical Engineering, University of Victoria, Victoria, BC,Canada.

[9] H. Sheibani, S. Dost, S. Sakai, B. Lent, Growth of bulk single crystals under applied magnetic field by liquid phaseelectroepitaxy, J. Cryst. Growth 258 (3–4) (2003) 283–295.

[10] H. Sheibani, Y.C. Liu, S. Sakai, B. Lent, S. Dost, The effect of applied magnetic field on the growth mechanisms ofliquid phase electroepitaxy, Int. J. Eng. Sci. 41 (2003) 401–415.

[11] M. Yildiz, S. Dost, B. Lent, Growth of bulk SiGe single crystals by liquid phase diffusion, J. Cryst. Growth 280(1–2) (2005) 151–160.

[12] K. Nakajima, S. Kodama, S. Miyashita, G. Sazaki, S. Hiyamizu, Growth of Ge-rich SixGe1�x single crystal withuniform composition (x = 0.02) on a compositionally graded crystal for use as solar cells, J. Cryst. Growth 205(1999) 270–276.

[13] S. Dost, H.A. Erbay, A continuum model for electroepitaxy, Int. J. Eng. Sci. 33 (10) (1995) 1385–1402.[14] S. Dost, Z. Qin, A model for liquid phase electroepitaxial growth of ternary alloy semiconductors. Part I—Theory,

Int. J. Appl. Electromagnet. Mech. 7 (2) (1996) 109–128.[15] CFX-4.3 Solver manual. AEA Technology plc, Didcot. UK, 2000.[16] R.M. Bowen, Theory of mixtures in ‘‘Continuum Physics’’, in: A.C. Eringen (Ed.), Mixtures and Electromagnetic

Field Theories, vol. III, Academic Press, New York, 1976.[17] T. Nishinaga, Microchannel epitaxy: an overview, J. Cryst. Growth 237–239 (2002) 1410–1417.[18] Y.C. Liu, Z.R. Zytkiewicz, S. Dost, A model for epitaxial lateral overgrowth of GaAs by liquid phase

electroepitaxy, J. Cryst. Growth 265 (2004) 341–350.[19] P.M. Adornato, R.A. Brown, Convection and segregation in directional solidification of dilute and non-dilute

alloys: effects of ampoule and furnace design, J. Cryst. Growth 80 (1987) 155–190.[20] C. Steliana, T. Duffarb, Modeling of thermosolutal convection during Bridgman solidification of semiconductor

alloys in relation with experiments, J. Cryst. Growth 266 (2004) 190–199.[21] M. Yildiz, S. Dost, B. Lent, Evolution of the growth interface in liquid phase diffusion growth of bulk SiGe single

crystals, Cryst. Growth Technol., submitted for publication.[22] K. Nikfetrat, N. Djilali, S. Dost, Accuracy and non-uniqueness aspects of numerical solutions of natural

convection problems, Appl. Math. Modell. 20 (1996) 371–378.[23] Metals Handbook, 10th ed.Properties and Selection: Nonferrous Alloys and Special Purpose Materials, vol. 2,

ASM International, The Materials Information Society, 1990.[24] E. Yamasue, M. Susa, H. Fukuyama, K. Nagata, Thermal conductivities of silicon and germanium in solid and

liquid states measured by non-stationary hot wire method with silica coated probe, J. Cryst. Growth 234 (2002)121–131.

[25] V. Palankovski, R. Schultheis, S. Selberherr, Simulation of power heterojunction bipolar transistors on galliumarsenide, IEEE Trans. Electron. Dev. 48 (6) (2001).

[26] H. Nakanishi, K. Nakazato, S. Asaba, K. Abe, S. Maeda, Terashima, Temperature dependence of density ofmolten germanium measured by a newly developed Archimedean technique, J. Cryst. Growth 191 (1998) 711–717.

[27] S. Yesilyurt, L. Vujisic, S. Motakef, F.R. Szofran, M.P. Volz, A numerical investigation of the effect ofthermoelectromagnetic convection (TEMC) on the Bridgman growth of Ge1�xSix, J. Cryst. Growth 207 (1999)278–291.

[28] Supplier provided properties. Available from: <www.gequartz.com/en/fig11.htm>.