Modelling of the dynamics of HgCdTe growth by the vertical Bridgman method

Journal of Crystal Growth 114 (1991) 411 434 North Holland

j~

,o~

CRYSTAL GROWTH

Modelling of the dynamics of HgCdTe growth by the vertical Bridgman method Do Hyun Kim and Robert A. Brown Department of Chemical Engineering and Materials Processing Center, Massachusetts Institute of Technolo~’,Cambridge, Massachusetts 02139. USA Received 27 November l99O~manuscript received in final form 5 July 1991

The transients in vertical Bridgman growth of nondilute alloys of l-lgCdTe are studied by numerical integration of the time dependent equations for momentum, solute and energy transport and the conditions for the evolution of the melt crystal interface according to the pseudo binary phase diagram. The stabilizing axial density gradient caused by the rejection of heavier HgTc at the interface damps convection driven by the radial tempcrature gradients and by the density inversion at low CdTe concentrations. For typical conditions of crystal growth in small ampoules, the temperaturc and solute fields are controlled by conduction and diffusion, respectively. The major effects of the nondilute alloy are to increase the deflection of the solidification interface caused by the differences in thermal conductivities in the system and to couple the evolution of the crystal growth rate with the composition field to the long time scale for equilibration of the solutc field at the start of growth. The evolution in time of the flow field from the structure driven entirely by the temperature field to the weaker thcrmosolutal flow is demonstrated for terrestrial growth and lower gravity conditions. The importance of the ampoule translation rate and ampoule size on the predictions for solute segregation is emphasized.

1. Introduction Although the adjustable bandgap of mercury cadmium telluride (MCT; Hg1 ~Cd~Te) makes it an important semiconductor material for detector applications, the extremely high vapor pressure of the melt makes crystal growth by solidification extremely difficult. Directional solidification methods in sealed ampoules are the most used techniques and the vertical Bridgman system was among the first developed for this application [1,21. However, typical MCT crystals grown this way have limited utility because they have small diameter, high dislocation densities [11 and very nonuniform compositions [2,31.Extensive experimental studies [4~5]have focused on understanding solute redistribution during crystal growth and on lowering the defect densities in MCT crystals. Interestingly, the experiments of Lehoczky and coworkers [61demonstrated diffusioncontrolled axial segregation and large radial seg0022-0248/91/$03.5O © 1991

regation in crystals grown by the vertical Bridgman method. Our transient analysis described here and the pseudo-steady-state calculations described in ref. [71have been aimed at understanding these results and at providing a better quantitative picture of the interactions of heat transfer, convection in the melt, solute segregation and the shape of the melt/crystal interface during crystal growth. Analysis of directional solidification of MCT is particularly challenging because the phase equilibrium between melt and crystal is described as a nondilute, pseudo-binary mixture of CdTe HgTe and the details of the solidification and compositional uniformity of the material are coupled to this phase behavior. This coupling significantly complicates the description of heat and momentum transport in directional solidification of nondilute alloys. One of the most challenging problems is understanding the interactions of all the transport processes during the transients associated with

Elsevier Science Publishers By. All rights reserved

412

D H. Kuii. R.A. Brown

Modelling of dynamics of IIgC’dTe growth by i ertical Bridgman method

the beginning of crystal growth. Heat transfer in a small-scale Bridgman system for the growth of semiconductor materials is typically governed by conduction in the melt, crystal and ampoule and radiation between the components [81.The ternperature profile of the surrounding furnace provides the average axial temperature gradient necessary for solidication; however, the details of the temperature field near the melt/crystal interface are influenced by the different thermal conductivities of these three materials and by the release of latent heat during solidification. For a pure material or a dilute alloy, the interface shape is an isotherm in the temperature field and its shape is entirely decoupled from solute segregation in the melt. Hence, during transients in the growth process, the crystal growth rate and the interface shape equilibrate on time scales set by heat trans fer, which are typically on the order of minutes, at the longest. For a well-designed vertical Bridgman system, steady-state crystal growth with selfsimilar interface shapes and radial segregation of a dilute component are predicted by simulation [9] and achieved in experiment [10] The situation is much more complicated for a nondilute alloy, where the melting temperature depends on the composition at the interface. Here the interface shape and the crystal growth rate are coupled to the evolution of the solute field. Accordingly, the pattern and intensity of buoyancy-driven convection the melt couple to solute transport via convective transport. The effects of the solute field on buoyancy-driven convection are especially pronounced in MCT growth. Here, the lighter component of the pseudo-binary (CdTe) is preferentially incorporated into the crystal and the heavier component (HgTe) accumulates in the melt near the interface. The increase in the density caused by the HgTe concentration stabilizes the melt against thermally-driven buoyant motion. We have demonstrated the damping of melt convection caused by solutal segregation in steady-state analysis of vertical Bridgman growth of HgCdTe [7] and GeSi [8] alloys. Kim and Brown [7] have shown that the diffusion-controlled axial segragation and large radial segregation observed by Szofran et al. till can be

explained by the interactions of this thermosolutal convection with the large deflection of the melt/crystal interface caused by the thermal field and the MCT phase diagram. The curvature of the melt/crystal interface has been identified by others as a major factor in the radial nonuniform ity of composition observed in vertical Bridgman growth of MCT [3,12,131. Several efforts have been made to design crystal growth systems that give a flat interface by controlling the temperature field in the ampoule. For example, Jones et al. [12,14,15] analyzed the effect on interface shape of the ampoule shape and of increasing the heat transfer from the base of the ampoule. Szofran and Lehoczky [161 applied a vertical Bridgman furnace with heat pipes to control the interface shape. Unfortunately, the large differences in the thermal conductivities between melt and crystal [171 and the thick ampoule necessary to hold the high vapor pressure damp the advantages of tailoring the external furnace. Naumann and Lehoczky [171 demonstrated that the combination of the difference in thermal conductivities and thick anipoules leads to highly curved inferfaces. Understanding the effects of the variations of the thermophysical properties with temperature and composition is a major obstacle to the effec tive optimization of the growth of nondilute alJoys. Most important in the MCT system is the large separation between the liquidus and solidus curves that causes large changes in the partitioning of solute into the crystal and in the melting temperature during crystal growth. The effects of this phase diagram on the crystal growth rate and the mean interface location have been studied by analysis of one-dimensional transient models of heat conduction and solute diffusion [18 20]. These results clearly show the transition from thermal to solutal control of the transients in the crystal growth process as the coupling of the melting temperature to the solute composition is introduced. Dakhoul et al. [21] presented a twodimensional diffusion-dominated model for cornputing the melt/crystal interface shape for verti cal Bridgman growth of MCT and predicted significant increases in the interface curvature. The analysis described by Kim and Brown [7]

D.H. Kim, R.A. Brown

Modelling of dynamics of HgCdTe growth by sertical Bridgman method

was the first to include the effects of buoyancydriven convection in the melt in analysis of directional solidification of MCT. Here, a self-consistent model for steady-state directional solidification was developed that included species, heat and momentum transfer in the melt, heat transfer in the crystal and ampoule and the influence of the phase diagram on the location of the melt/crystal interface. Numerical analysis of the model for a vertical Bridgman system first demonstrated that the damping of thermallydriven convection by the HgTe solute layer can lead to the apparent diffusion-control of axial segregation that is obseived in the experiments of Szofran et al. [111. The analysis of Kim and Brown [7] is based on a pseudo-steady-state model that describes the long time state of the system, but cannot model the effects of transients at the beginning of solidification or during crystal growth, as the bulk composition changes. the form of the initial transient is particularly important because it is the evolution of the crystal composition in this region that has been used experimentally to infer the state of convection in the system [4,5]. The transport processes in this transient are not simple. At the onset of ampoule motion the solute field evolves from a uniform state to establish gradients that are consistent with the rejection of HgTe at the melt/crystal interface. These solute gradients damp the thermally-driven convection and, if sufficiently strong, lead to the diffusioncontrolled state at long times, as described by the analysis in ref. [7]. The details of this transition depend on the relative strength of the two convection mechanisms, which in turn depend on the crystal growth rate and heat transfer in the systern. This paper extends the analysis in ref. [7] to the study of the transients during crystal growth; in particular, to the transient at the beginning of growth. This is accomplished by detailed numerical simulations based on solution of the time-dependent, axisymmetric transport equations for the vertical Bridgman growth system that account for the coupling between the fields and the evolving shape of the melt/crystal interface. As such, the simulations reported here are the first complete

413

transient calculations of solidification of a nondilute alloy. The calculations are based on the application of a newly developed finite element method for simultaneous integration of the entire equation set describing transport in directional silidification [22]. The formulation of the mathematical model and the numerical method are described briefly in section 2. The calculations demonstrate the evolution from a well-mixed melt at the beginning of MCT growth to the diffusion-controlled solute segregation observed experimentally. Most importantly, the results show that, although the bulk of the flow field evolves ott the slow time-scale for solute diffusion, the changes near the melt-crystal interface occur quickly enough that the solute field there is almost entirely controlled by diffusion after only a small fraction of the melt has been solidified. The only measurable effect of convection is to increase the radial segregation and interface deflection predicted by calculations that include only solute diffusion. Another interesting issue is the effect on solute transport of the well established maximum in the dependence of the MCT melt density on CdTe mole fraction [23]. During crystal growth, this maximum in density leads to melt that is vertically unstable just above the interface. Convection in systems with non-monotonic density dependence on temperature is called penetrative and commonly occurs in geophysical and astrophysical applications [24]. Antar [25] performed the linear stability analysis for a layer of MCT in an idealized crystal growth system to compute the conditions for the onset of convective motion. We present simulations that include correlations for the dependence of the thermal expansion coefficient on temperature and concentration and which demonstrate that solute segregation is unaffected by penetrative convection, although the flow pattern is altered. The analysis presented here is for a prototype of the vertical Bridgman system developed by Lehoczky and Szofran [6,16]. We simplify the actual system by assuming that the melt fills the ampoule so that there is no free space above the melt and by modelling the heat barrier in the middle of the furnace as a perfect insulator. Heat

414

D.H. Kim, R.A. Brown

Modelling of dynamics of HgCdle growth hr i erocul Bridgman method

cause of the steady decrease of the melt volume

~

and the build-up of solute caused by the finite length of the ampoule. Both effects are accounted for in the model described below. In the model, melt and crystal are assumed to fill the ampoule completely, leaving no free surface above the melt. This assumption is consistent

z

MELT

U MELT CRYSTAL INTERFACE

with the approximation of equal densities of melt and crystal. The ampoule is modelled to be very thin at the top and bottom, so that the temperatures on these surfaces are uniform and equili-

A

~ TL

~0

L

CRYSTAL

brated with the temperatures at the corresponding locations of the furnace. The field variables are described in a cylindrical coordinate system (r, z) with origin located on the axis at the top of the ampoule as shown in fig 1 The translation of the ampoule is ac. counted for by time dependent changes tn the profiles of the furnace temperature and heat transfer coefficients between the furnace and thi ampoule as measured with respect to thts coordi nate system Variables are put in dimensionless form by scaling lengths with the radius of the crystal R .

..

AMPOULE

Vg

Fig. 1. Schematic diagram of prototype vertical Bridgman crystal growth system.

transfer coefficients appropriate for the hot and cold zones of the furnace are estimated by the methods developed for modelling the system of Wang [10]. The details of the furnace model are described in section 2.

2. Transient model of vertical Bridgman growth Of MCT Our transient model for the growth of MCT ~ developed for the prototypical vertical Bridgman growth system shown schematically in fig. 1. Here the melt, crystal and ampoule phases are characterized by different thermophysical properties. The transport of heat, solute and momentum in this growth system is time-dependent for any nonzero translation rate of the ampoule J/~be-

velocity ~(r, z) with v/Re, pressure ~(r, z) with 2/R~ and composition ë,(r, fl with the initial ~~

1v concentration

C

0, where v is the kinematic viscosity and ~m is the density of the melt. Time T ~ scaled with the momentum diffusion time, R~/v. The dimensionless concentration and temperature are defined as

S(r, z)

c(r, z)

1,

(1)

O(r, z) [T(r, z) TC~IdI/(ThUl T~01~1), (2) where ThOl and T~010are the dimensional temper-

atures of the hot and cold zones of the furnace, respectively. The equations governing time-dependent, axisymmetric convection in the melt, modelled using the Boussinesq approximation, are written in dimensionless form as V ~ di’

0,

+v.Vv

(3) 2

Vp+Vv+ x [Ra S

Pr RaT(O S) (0

1)Ie 4

D.H. Kim, R.A. Brown

Table

/ Modelling of dynamics of HgCdTe growth

415

by vertical Bridgman method

nary. The detailed form of Om(S) is given in

1

Definitions of dimensionless groups and characteristic values for analysis of thermosolutal convection of HgCdTe crystal growth Name

Symbol Definition

Thermal Rayleigh number Solutal Rayleigh number Thermal Peclet number Solutal Peclet number

3T LlTR~’/amv

RaT

gl

Ra~

g~3

Pc

5c55R~/a,,,v VgRc/a

section 3. The unit vectors normal N and tangential t to the surface are

Value

=

lOx i0~ 4 8.5x10 2.8x10

N

________ e~ HrCr

~

=

e,.

Hrez

(10)

,

~1 + H~

~/i + H~

where Hr dH/dr and (Cr, e~)are the unit vectors in the cylindrical coordinate system. In the coordinate system moving with the ampoule translation rate Vg~the energy and solute

Pc fr~R~/D

0.51

Pr Sc St

v/a,, v /D ~HS/C~,y~T

0.11 19.6 1.05

y

as/am

°~

P~/Pm

5 Prandtl number Schmidt number Stefan number Thermal diffusivify ratio Density ratio

balance at the interface are [N. VOIm

K~[N~V0]c

0H St Pr —(

N~er), (II)

0.21 1.0

[N. VS]

—

—Sc

8T

(Ne~)(1 —k)(S

+

1), (12)

[do

V20,

(5)

where (dH/8T)(N~ e) is the normal component of the interface velocity. The parameters appear-

V2S,

(6)

ing in the interfacial energy balance (11)therand solute balance eq. (12) are the ratio eq. of the

where V is the gradient operator in cylindrical coordinates, T is the dimensionless time and the dimensionless groups are defined in table 1. Heat transfer in the crystal and ampoule is by conduction and convection driven by the motion of the ampoule. The energy equations in the crystal and ampoule are written in the reference frame moving with the ampoule as

mal conductivities K~ ks/km and the equilibrium segregation coefficient k; other dimensionless groups are defined in table 1. The no-slip and no-penetration boundary condition on the velocity field at the interface and at the ampoule wall are

Pr~

+

v~VO)

+

v. vs)

(dS

Sc~

=

—

liT

Pr(dO/dT)~yeV26,

(7)

YaV’20,

(8)

Pr(dO/d’r)

—

where y~ ac/am and Ya aa/0~m are the ratios of the thermal diffusivities of crystal and ampoule to the value for the melt, respectively, and Y is defined in table 1. The shape of the melt/crystal interface z H(r, r), 0
—

v 0, (r, z)

(0
—

r < 1, H( r, r)) (13)

U(1,0
Application of eq. (13) at the melt/crystal interface assumes equal densities of the two phases. Because the temperatures along the ampoule and furnace at a specific axial location are not very different, the radiative interaction between these surfaces is linearized, as was done in ref. [261. The resulting thermal boundary condition between the ampoule and surrounding furnace is [dO/dr]a—Bi(z,

0(r, H(r, ‘r)) Om(S), (9) where Om(S) is the melting temperature of the alloy corresponding to the liquidus curve of the phase diagram for the CdTe—HgTe pseudo-bi-

1, 0) U (0

T)

[Om(Z, T)

—O(r, z,

(14) where a is the dimensionless outside radius of the ampoule, the Biot number Bi(z, r)

416

D.H. Kim, R.A. Brown

Modelling of dynamics of HgCdTe growth by i ertical Bridgman method

h~,,(z,T)R~/k~and the furnace temperature profile Om(z, T) are functions of time and axial loca-

tion and are determined by the furnace design and the ampoule translation rate. Solute is assumed not to penetrate the ampoule surfaces; this yields the boundary conditions [aS/dz]~

(15)

0.

~ — [dS/dr]r

The boundary conditions for heat transfer at the top and bottom of the ampoule are not easily defined. Most of the previous studies used Newton’s cooling law [27] with the magnitude of the heat transfer coefficient set to be same as in the adjacent part of the furnace [28 31]. Insulated ends also were used in other analyses [19,20]. In our analysis, the temperature on these boundaries is equilibrated with the furnace temperature at the same axial location. Because the temperature profile of the furnace is time-dependent, the top and bottom temperatures of the ampoule also vary with time. Finally, the axis of the cylinder is taken as a line of symmetry for all field variables, 1’ r

—

dr

liR

dr

dr —0

at

r

0,

(16)

gence of the calculations with decreasing mesh size is demonstrated in ref. [22]. The calculations presented here were performed with a mesh of 20 radial elements in the melt or crystal and four elements in the ampoule; 120 elements in the melt and 16 in the crystal were distributed axially in the system. This discretization leads to a set of 57,493 DAEs which are solved at each time step. The nonlinear algebraic equations which result from the implicit approximation at each step are solved by a modified Newton iteration.

3. Thermophysical properties The accuracy of calculations of the flow and solute fields and melt/crystal interface for a nondilute binary alloy hinges on the precision of the knowledge of the phase diagram and the thermophysical properties. In this section we review the data base for the properties of MCT alloys. 3.1. Phase diagram

which is consistent with the assumption of axisymmetric fields. The specification of the moving-boundary problem is completed by setting the initial (T 0) velocity, solute and temperature profiles and interface location. These initial conditions are ob-

recently, Brice et al. [33] reconstructed the CdTe—HgTe pseudo-binary phase diagram using previously published data. This phase diagram is the basis for calculations used in our analysis. Micklethwaite [34] and Brice et al. [33] reported formulae for the melting temperature T,,1 and the

tamed from steady-state calculations of velocity and temperature fields in the absence of ampoule translation. The initial state of the solute field is a uniform concentration Co throughout the melt. The moving-boundary problem defined by eqs. (1)—(16) was solved by the finite-element/Newton method described in ref. [22]. The field equations and boundary conditions are discretized U5 ing standard Lagrangian finite element approximations and Galerkin’s method, with the isotherm condition distinguished for locating the melt/solid [32] interface. The resulting set of differential-algebraic equations (DAEs) is solved by the fully implicit trapezoid rule integrator, which allows larger time steps because of its higher accuracy and numerical stability [20].The conver-

equilibrium segregation coefficient k as a function of the CdTe mole fraction x. The analysis by Brice et al. [33] disagreed significantly with the formula proposed by Micklethwaite [34]. We present a new formula for the liquidus curve based on the data suggested in ref. [33]. For mathematical convenience, the data are represented in terms of a cubic polynomial in the mole fraction of CdTe (x) as 2 + a 1, (17) 1(0 C) a0 + a1x + a2x 5x where the coefficients {a,) take the values {670.94, 671.46, 375.07, 110.54). The predicted phase diagram of the pseudo-binary CdTe HgTe alloy is shown in fig. 2. The circles in fig. 2 indicate the

D.H. Kim, R.A. Brown

417

Modelling of dynamics of HgCd Te growth by i ertical Bridgman method

median values of data within 90% confidence limits, as tabulated by Brice et al. [33]. The dimensionless melting temperature is represented as a cubic polynomial in the dimensionless interfacial concentration 5:

\~

eq. (20)

5,

g ~

2

Om~OmO+Omi(S+1)+Om2(S+1)

(18)

where S is defined by eq. (1) and the constants {Omi) are related to the (a,) by (a

OmO

TCOId) /( T~0, TCOId),

0

°m,1 — aicO/(ThQl °m,2 0m,3

—

(19a) (19b)

‘~oId)’

2 a2CO/(ThO~ — a,c~/(ThO~

~TCOId),

(19c) (19d)

The formula introduced by Brice et al. [33] for the equilibrium segregation coefficient agreed well with experimental data. We fit the same data to a fifth-order polynomial in x as ~ b x’,

k

0

2

(20)

0 I

0.0

05

i.0

fractiondistribution CdTe Fig. 3. Comparisonx,ofmole equilibrium coefficient values from the phase diagram and the formula eq. (20). Symbols 0 denote values of k from ref. [33].

correlation, eq. (20), is compared in fig. 3 with the data in ref. [33]. The representation of k in terms of the dimensionless concentration S is

‘

Where the coefficients (b } have the values (4.726 15.935, 39.335, 59.151, 46.752, 14.743). Our iioo

k

Ek~(S+ 1)’,

where the {k,) are defined as k, 0 5.

,,

(21) b,C~), i

eq.(17)

3.2. Thermophysical properties curve

i000

fit

2-.L

/

/7~

900

// ~ 5,

800

700

/ 00

/

/

/

/

,~

.~-

‘

‘

10

x, mole fraction CdTe Fig. 2. Pseudo binary phase diagram for CdTe HgTe alloy system. Symbols 0 denote median values within 90% confi dence limits, as reported in ref. [33].

Strong dependences are expected of the thermophysical properties of the melt and crystal on the composition of the CdTe HgTe pseudo-binary alloy; however, few measurements are available for accurate correlation of these properties. Only measurements of thermal diffusivity [35] and estimates of the heat capacity and thermal conductivity [36] of the melt are known as a function of temperature and alloy composition. For example, the thermal diffusivity of the HgTe melt (x 0) varies by 400 percent with temperature ranging 200 C above the liquidus temperature. The dependence of the thermal diffusivity of the CdTe—HgTe melt on temperature weakens considerably as the CdTe mole fraction increases. 0

418

D.H. Kim, R.A. Brown

Modelling of dynamics of HgCdTe growth by vertical Bridgman ,nethod

Unfortunately, the sparsity of data does not justify attempts at incorporating very detailed correlations of temperature and composition dependence of other thermophysical properties into analysis. Furthermore, because the simulations cover a fairly narrow composition range, the properties should remain relatively constant. We use a set of constant values for all properties, except the melt density. The set used here is listed in table 2 and was compiled by Antar [37] for an alloy with CdTe mole fraction x 0.2; these properties also were used in the steady-state calculations reported in ref. [7]. Because melt convection is driven entirely by density gradients the details of the dependence of density on temperature and composition may have a large effect on the flow pattern and the resulting compositional segregation. The Boussinesq form of the momentum equation (4) is based on writing the melt density as p(T, c)

p01[1 +131(T, c) (T

Tret)

~( T, c)

(c co)1, (22) where is the coefficient thermal expansion and PsPT is the coefficient of of solutal expansion. In general, these coefficients depend on both ternperature and composition. In fact, the density of MCT is known [23] to have a maximum with temperature in the composition range 0
x

~-.

-~

0.05

~

.~

~

o

x

02

1

78

LIQUIDUS

.i~

76

700

750

800

850

Temperature (°C) Fig. 4. Comparison of eq. (23) for the density of molten Hg, ~Cd,Te with experimental measurements (0) given in ref. [23].

The dependence of the coefficient of thermal expansion on temperature and composition was estimated from the density measurements of Chandra and Holland [23] and fit to the biquadratic polynomial. 2 + c 2, (23) ~ c0 + c1x + c2T+ c3xT+ c4x 5T where 15m is measured in units of g/cm3 and T is in C. The coefficients {c,) in eq. (23) have been estimated as (2.251, 4.478, 1.540 x 10 2 2.544 x 10 3.609 x 10 1.022 x 10 5}, The fit of 0

~,

the correlation to the experimental data in ref.

Table 2 Thermophysical property data uscd in analysis of HgCdTe growth Quantity

Symbol (units)

Value

Thermal conductivity of the melt Thermal conductivity of the solid Density of the melt Density of the solid Specific heat of the melt Specific heat of the solid Melting temperature Heat Kinematic of solidification viscosity

k,,, (W/ °Ccm) k5 (W Ccm) P,,, (g cm’) p~(g cmi) (J C~ g) C,, (J/ 0 C~ g) T5, (02C) s) ill v (cm 5 (J g) PT (0 C ‘) 2/s)fraction CdTe) ‘) Ps ((mole kD (cm

1.96x 10 2.91 x tO 7.55 7.63 0.257 0. t77 Eq. l.08x (17) 10

Thermal expansion coefficient Solutal expansion coefficient Diffusion coefficient Equilibrium distribution of CdTe coefficient in HgCdTe of CdTe

0

80

130 t.s ~“ 10 5.50.30 x 10 Eq. (20)

2

D.H. Kim, R.A. Brown

/ Modelling of dynamics of HgCdTe growth by o

x, mole fraction CdTe

ertical Bridgman method

—200000

,

Fig. 5. Contours of Hg

1 ,Cdje melt density generated from eq. (23) in the operating ranges of temperature and composition for the crystal growth system.

[23] is shown in fig. 4 and is extremely good. The maximum in the density with varying temperature and set values of composition with x <0.1 is clear in this figure and in the contours of the density surface shown in fig. 5. Curves of the coefficients of thermal and solutal expansion are obtained by differentiation of eq. (23) with respect to temperature and composition and are displayed in figs. 6a and 6b for selected compositions. The curves for PT intersect the line PT 0 for x <0.15. The dependence 0.0004

0

025 I

(b)

/

/

x x x

/ U

0 0.05 01

030/

___

800

Temperature (°C)

0.3

of PT on T and c leads to substantial variation in the driving force for convection from the bottom to the top of the ampoule. This variation is depicted by the shaded area in fig. 7 for the case of the one-dimensional, diffusion-controlled concentration field and temperature varying linearly from the liquidus temperature at this interface to 880 C at the top of the ampoule. The presence of the density inversion causes RaT to vary from negative (thermally destabilizing) to positive

0.0002

700

‘

Fig. 7. Bounds on the range of RaT caused by variations in temperature and composition in the melt.

/ xO x 0.05 ~ 0.1

0.2

‘

x, mole fraction CdTe

I

(a)

~

419

900

700

800

900

Temperature (°C)

Fig. 6. Coefficients of(a) thermal and (b) solutal expansion for Hg, ,CdrTe with compositions 0< x <0.2.

D. H. Kim, R.A. Brown

420

Modelling of dynamics of LIgCdTe growth, by vertical Bridgman method

(thermally stabilizing) within the ampoule. The coefficient of solutal expansion is almost mdcpendent of composition. The maximum variations of p~with temperature is about 10% in the temperature range considered; hence, f35 is as-

Table 3 Thermophysical data for ampoule and vertical Bridgman svs tem Parameter

Value

Ampoule length, L (cm) Crystal radius, R, (cm) Ampoule outer radius, R, (cm) Gradient zone length, 1., (cm) Temperature difference. T0, T, ( 0 C) Temperature in hot zone. T1, ( C)

So

PT

Temperature in cold zone, J~(0 C) Initial concentration, c~(mole fraction CdTe)

400 (1.21)

A sequence of simulations was performed to demonstrate the combined effects of convection, segregation and theofrole of the phase diagram on the crystal growth MCT alloys. These simulations were carried out with the constant value of the thermal expansion coefficient PT 1.5 X 10 which is appropriate for x 0.2 and T 830°C. Calculations with varying convection levels were performed by changing the gravitational acceleration g: (a) g 0 (RaT 0 and Ra 5 0), (b) g ~ (RaT— 1>< iO~ and Ra5 8.5 X 5 and Ra ]()5), and (c) g gearih ifiaT 1 x il) 5 8.5>< iO~),where gearth is the terrestrial level of the acceleration. The calculations simulate crystal growth of MCT in a prototypical vertical Bridgman system similar to the one used by Szofran and Lehoczky [161. Geometrical parameters and thermophysical properties of the quartz ampoule used in the simulations are listed in table 3. The effective heat transfer coefficients between the ampoule and the furnace are computed by the method described in ref. [22]. The values of the dimensionless groups appropriate for the growth of MCT with this furnace design are listed in table 1 for g gearth’ Transient simulations were carried out using the steady-state solution for a stationary ampoule as the initial condition (T 0). The stationary ampoule was positioned with the center plane of the gradient zone at z 13 (0.65 of the ampoule length) from the top of the ampoule. The initial composition is uniform throughout the melt and the flow field corresponds to the steady motion caused solely by the temperature gradients in the melt. The melt/crystal interface is highly concave to the crystal because of the factor of seven

Ampoule translation rate, V5 (~zm s) Ampoule material 5) K, (W Thermal conductivity of ampoule, Density of ampoule, p (g (‘i,cm., (J Specific heat of ampoule. Cg)

1.12

sumed to be constant in the calculations.

4. Flow structure and solute segregation: constant

0.25 0.50 1.1)1) 480 88))

0

2.2 C cm) Quartz ((.026 1.05

‘~,

—

~iii

±::: .

melt

..

______

—~

______

_____

—

~

______

_______

~,

_______

~

~,

~~“:‘

~.

—,

______

interface

______ _______ ______ ______

______

crystal

—

Fig.

0 ~

3043 s

(a)

(b)

6110

(c)

s

9003 s (d)

8. Evolution of finite element mesh during the transient

calculations.

D H. Kim, R.A. B, own

421

Modelling of dynamics of HgCdTe growth by ertic’al Bridgman method

increase in the thermal conductivity between melt and crystal and because conduction through the thick ampoule supplies a significant path for heat transport. The transient simulations reported here are continued until about one fifth of the melt is solidified at the midpoint of the radius of crystal (about 14 vol%) and so focus on the initial transients during growth. The deformation of the finite element mesh during a typical calculation is shown in fig. 8.

of the furnace; the temperature field in the remaining melt and crystal equilibrates with the furnace temperature in these regions. For the growth of a dilute alloy, this furnace geometry would lead to a self-similar temperature field near the melt/crystal interface, to a constant growth rate and a constant interface shape throughout the majority of the growth run, as demonstrated by the simulations in ref. [9]. However, for a nondilute alloy, the coupling between the solute and temperature fields through the melting temperature leads to changes in these variables during the initial transient. This transieiit is demunstiated by the tempetature and solute fields shown in fig. ~

4.1. temperature fields and crystal growth rate

Sample temperature dnd solute fields computed without buoyancy-driven convection (RaT Ra5 0) are shown in fig. 9. As shown in fig.

As crystal growth begins, a diffusion layer depleted in CdTe develops ahead of the interface, as shown by the isoconcentration curves in fig.

9A, the axial variation of the temperature is confined to the relatively short insulation region (A) ST

01

AT

01

AT

01

AT —01

melt

09

(B)

100 09

Ac

005 , 055 c,..~ 100 .

•

I

09

1

1

~ •

interface

0 9

A’

005

c

024 100

c,..~ • •

I ~

1,1

•

•

•

••

I

5 0 05 c,.,. 0 16 c~ 100 ••

—

N

•

I-.

N

o z

•

01 •

crystal •

01

Q95 •

095 •

095

01 01

•

a 0 ____

____

____

55

•

Os (a)

Os

3043 s

(a)

(b)

6110s (c)

9003s (d)

— Z

~ 3043s (b)

6110 ~ (c)

9003 (d)

Fig. 9. Sample (A) temperature and (B) solute fields for calculation without buoyancy driven convection: g

0.

422

D. H. Kim, R..4. Brown

Modelling of dynamics of HgCdTe growth by iet tica! Bridgman method I

9B. This solute layer evolves dynamically until it reaches an apparent steady-state form at long time. This concentration field is diffusion-controlled and essentially one-dimensional, except near the curved melt/crystal interface, where the condition normal incorporation, eq. (12), distorts the ofconcentration field. However, the nonuniformity of the composition field is not as with a similarly curved interface and a constant large as one would expect for a dilute alloy grown segregation coefficient k. There is less segregation of CdTe near the edges of the ampoule. Radial segregation in the crystal is reduced because k for MCT decreases with increasing cornposition. The decrease in the CdTe composition at the interface during the initial transient leads to a decrease in the melting temperature of the crystal, as dictated by the liquidus curve. Comparison of the interface location at r 0 between the plots for (a) t 0 s and (d) t 9000 s in fig. 9A shows a change of approximately 0.2 ~iT or 96°C in the melting temperature. The evolution of the melting temperature at r U.S is plotted in fig. 10 for Ra1 Ra5 0. Increasing the intensity of convection in the melt by increasing the gravitational acceleration 800 I

~

g g

0

g

ge~rrs

0.1

gearss

I

I

0 1

02

003

Ampoule Trans1~tion ~

‘t

5,~

002

/~_

i/go 001 .~

•

/

~

——

o

00 00

g g

0

g

1

g~rts

I

I

0.1

02

fraction solidified Fig. 11. Crystal growth rate as a function of fraction

03

solidified

for the three convcction levels listed ri fig. 10).

has only a slight effect on the temperature and concentration fields. Heat transfer continues to he dominated by conduction because of the low Prandtl number of the melt (Pr 0.11) and the relative weak convection in this small scale crystal growth system. As is shown below, the solute fields are altered by convection and these changes couple into the temperature field through the dependence of the melting temperature on concentration. The variation in the melting temperature with fraction solidified is shown in fig. 10 for acceleration levels of 0, O.lg51~11,and g51,,,11. The instantaneous growth rate of the crystal at the midpoint of the interface (r 0.5) is dis played in fig. 11 for the three sets of growth conditions; g 0,rate O.lg5.51~ gcario~ In each case, the growth evolves and through a transient weakly on the convection in the melt. As is disto the ampoule translation rate and depends only cussed in more detail below, the growth rate reaches a steady-state value because solute trans port in the bulk melt becomes controlled by diffusion, so that the composition away from the

j 700 O0

I

0 3

fraction solidified Fig. 10. Time evolution of the melting temperature ai the midpoint of interface (r 0.5). Calculations are shown for conditions corresponding to gravitation accelerations of 0, 0.lg.,,,,,,, and g.,,~s.

melt/crystal interface remains close to the initial value c11 during the first stages of growth. The deflection of the melt/crystal interface .

increases significantly once solidification begins. This increase is shown in fig. 12 and is a result of the coupling of the interface composition to the

D.H. Kim, R.A. Brown 15

/ Modelling of dynamics of HgCdTe growth

I

i

// /

0

the gravitational acceleration, increases the interface deflection. As is shown below, the concave interfaces lead to thermosolutal

/

convection that carries HgTe to the center of the ampou]e, which lowers the melting temperature and increases the interface deflection.

~‘

g g

‘/ .~

0

—

423

increase the curvature. Increasing convection, by I

increasing /7’

by’ vertical Bridgman method

0 1

~earth

4.2. Flow and solute fields

05 g

——

—

It is well documented that the radial temperature gradients in a vertical Bridgman system with

fraction solidified Fig. 12. Melt crystal interface deflection as a function of fraction solidified for the three convection levels listed in fig. 10.

melt above the crystal lead to two axisymmetric toroidal flow cells stacked axially in the melt [8,38]. The lower cell near the melt/crystal interface rotates so that melt moves down along the ampoule wall and up at the centerline and is driven by the mismatch in the thermal conductivi-

melting temperature. Radial nonuniformity of the composition along the interface adds to the effects of the disparate thermal conductivities to

ties of the melt, crystal and ampoule. The upper cell rotates in the opposite direction and is caused by the radial temperature gradients generated by the intersection of the hot and adiabatic zones of

0 0

00

I

I

0.1

02

54’— 005 4’,,,,, 0 40

C,,,,,,

4’,,,,,_041

0.3

1 00

c,,,,,, iOO ~

54’

~

‘1’,,,,,

~5(f3~

11111111

151151 ff4411

11511(1 F41

I ~

0.005

Ac

005 0.52 c,,,, i 00

0.033

c,,,,

‘i’,,,,, 0.0i6

I I

1t1881 I I

l~4 ~

I I

I

15551

I

1515(1 I

A4’— 0002 Ac 005 4’,,,,. 0.0i4 c,,,,, 0.Zi c,,,,,,-100

[f51SJ 4’,,,,,,—OOil

I

I I I M°’lW44dl I

144411 14144

54’ 4’,,,, 4’

0002 0.009 —0011

Ac 005 c,,,, 0.i4 c,,,,,—iOO

~H IH~I~H~IH 0.05

~

I IiI~

ds

Z 0

I III1~

°~

iii,,

I/Il

FLOW

SOLUTE 0

S

(a)

c12

FLOW

SOLUTE

3064 ~

FLOW

(b) Fig. 13. Sample flow and CdTe concentration fields for growth with g and

+

SOLUTE

6073 S (c) 0.1g~,~,

FLOW

SOLUTE

9024 s Cd)

5: (RaT, Ras) (lOx io~, 8.5x lOs). Symbols • denote the location of the minimum and maximum stream function value, respectively.

424

D I-I. Kim, R.A. Brown

Modelling of dynamics of J-i’gCdTe growth, by ertical Bridgman method

the furnace. This flow structure is displayed in fig. 13a for the stationary ampoule, where the composition is uniform. The two cells have almost equal intensity, as measured by the circulation rates. As solidification of MCT proceeds, the lighter CdTe is preferentially incorporated into the crystal, leaving the heavier HgTe accumulated near the interface. The resulting solute profile enhances the axial stability of the density gradient. Previous steady-state analysis for the effect of the stabilizing solute field [7,81showed that the two-

solute field; the isoconcentration curves near the interface are somewhat flatter and the contours in the upper portion of the ampoule are slightly distorted. The intensity of each flow cell decreases by more than an order-of-magnitude as the level of HgTe increases. The center of the upper cell moves toward the ampoule and its shape suggests the formation of a thin boundary layer along this wall. An analysis of the mechanism and scaling of this boundary layer was presented in ref. [7]. Calculations corresponding to g g5~1,(Ra 5, 8.5 x l0~),are shown in fig. 14 R5) (1 >< il) demonstrate the evolution of the and further solute gradient and its effect on convection. As the region of high HgTe composition diffuses into the melt, the lower flow cell is rapidly damped to ~,

cell structure flowcell was was maintained, but that flow intensityofoftheeach decreased. The transients in the stream function and the CdTe solute fields leading to this decrease in flow intensity are shown in fig. 13 for the conditions (RaT, Ra 5) (1 x i0~, 8.5 x i0~),which correspond to operation of the furnace at g 0.1 gcirlh’ The solute field evolves primarily by diffusion as the region of high HgTe concentration penetrates into the bulk melt. The presence of weak convection in the melt has only minor effects on the 54’ 4’,,,, 4’

02 0 98 178

c,,,, 1 00 c • 1.00

FLOW

SOLUTE 0

S

54’ 4’ ‘F

01 0.94 003

tic c c

,

the point that the details of its structure are not resolved on the plot for t 1723 s. At longer times the solute field has diffused to extend above the axial position of the junction of the hot and adiabatic zones of the furnace and the circulation in the top cell also is damped. The two-cell flow

002 073

i.oo

54’ 0002 4’ , 0027 ‘I’ =0018

c1~ FLOW1723 SOLUTE

Ac c,,,,

c

005 048 100

54’ 0002 ‘F,,,,,, 0.02i ‘F,,,,,=00i4

Ac 005 c,,,, 029 c,,,,,—100

+ T

FLOW

T

SOLUTE

FLOW

SOLUTE

3334 s

(a)

(b) (c)

499~ c (d)

Fig. 14. Samplc flow and CdTe concentration fields for growth with g g~,011:(RaT. Ra5) (lOx 10 . 8.5X lO(~).Symbols • and denote thc location of the minimum and maximum stream function value, respectively.

D.H. Kim, R.A. Brown

/ Modelling of dynamics of HgCdTe growth by vertical Bridgman

structure is reproduced and the top cell has the boundary-layer structure mentioned above. The details of the flow evolution caused by the diffus-

A+

02

_________ 4’,,,,— 4’,,,,, i.78 0.98

_________ c,,,,—iOO c,,,, 1 00

94 5* 0.iS 00 i87 Ac 0.005 4’,,.,— c,,.,—0. c,,,,, 100

,~,

method

425

ing solute field are shown in fig. 15 for short time intervals. The lower flow cell rapidly loses intensity to the point that it is not resolved by plotting.

~l’ 0 89 .~, Al’ 0063 Oi

c,,,,

0.87 c,,,,i00 Ac 0.01

54’ 00.92 1 4’,,,,— 4’,,,,, 0043

Ac 0 02 c,,,, c,,,,, 0.79 iOO

(ru

Z

o

III

-~

III

0-1 6-

o

1-2

11111

099

.995

098

‘t

Si 0

‘ctz

FLOW

-~0 in

54, 0 i 4’,,,, 094 4’,,,,, 0032

SOLUTE o s (a) tic 0 02 c,,, 073 C,,., 1.00

FLOW

SOLUTE S

(b) s+ ~

0 1

i.oo

‘9=,, 0026

Ac

0.02 c,,, 066 c,,,,, iOU

If)

FLOW SOLUTE 870 S (c) 54’ 0 i ‘F, 0.99 ‘I’,,,,, 0022

Ac

0 05 060 c,,,,, iOO

c,,,

C Z~

FLOW1307 SOLUTE S (d) 0 1 076 ‘i’,,., 0021 ~ 4~,,I,

e~

Ac 005 c,,,, 054 c,,., iOO

z o

-~

6-

0-i 0

0

1-2

o

098

095

095

0 0 II)

cj~

~

c~

C

FLOW SOLUTE 1723 s

FLOW SOLUTE 2128 S

FLOW SOLUTE 2529 s

FLOW SOLUTE 2915 s

(e)

(f)

(g)

(h)

Fig. 15. Sample flow and CdTe concentration showing the details of the initial transient: 4

0

z

g~,sand (RaT, Ra

5) (lOx lOs, 8.5 x lO~).Symbols s and + denote the location of the minimum and maximum stream function value, respectively.

426

D.H. Kim, R.A. Brown

Modelling of dynamics of HgCdTe growth by vertical Bridgman method

The upper cell migrates upward and weakens as the solute field diffuses into the bulk melt. The intensities of the flow fields for the two simulations described above are summarized in fig. 16 where the delay in the relaxation of the upper flow cell is obvious. Very small time steps are required in the numerical integration to resolve the rapid changes in the structure of the begins flow upper to interact cell that with occur it. when the solute fields

4

I

I

1 D diffusion

controlled

I

erowth (k

2.7) 1 D diffusion controlled

3

growth (k k(C,)) Well mixed melt (k Well mixed meit (k

-

0’ 2

2 7) k(C,,))

-

4.3. Solute segregation i

.

-

g

The axial segregation of CdTe, denoted as

C~,

in the crystal is computed by multiplying the average composition at the interface at each instant in time by the equilibrium segregation coefficient k(c). The results for the three simulations described above are shown in fig. 17 and are compared to modifications of the simple theories for solute segregation in diffusion-controlled [391 and well-mixed [40] melts. These theories originally were developed for analysis of idealized one-dimensional systems for dilute alloy systems with constant crystal growth rate and fixed equilibrium segregation coefficient. We have modified

I

100

0.1

g

lower flow cell upper flow cell

.

each analysis to account for the phase diagram of

f)

--

c~ kc0(1 00

03

dc/(c cs), (24) f is a fraction of the melt that has been solidified, c c(f) is the uniform composition of the melt and c~ c~(f)is the concentration in the crystal at the interface. For a constant segregation coefficient obeying the relationship c~ kc, eq. (24) is solved to give Scheil’s result:

I

102

02

fraction solidified Fig. 17. Axial profiles of CdTe concentration in thc crystal as a function of fraction solidified for g 0), 0.lg IrIS and 4~.IIil Results are shown for one dimensional thcories for the diffu sion controlled and well mixed limits with and without the effects of ihe phase diagram of llgCdTe.

df/(1

gearth

—

10

I

0 1

ge8rth

lower flow cell upper flow cell where ~

g,,,,.,5

I I

00

g

5

MCT. The solute balance in the melt and crystal, assuming complete mixing in the melt and no diffusion in the crystal, is I

g

~

g

0 0 1 g,,,.,

I

I

0.1

0.2

0

3

fraction solidified Fig. 16. Time evolution of the circulation rate in each flow cell 5), andwith g g~ 0.14c,rIh, Ra for growth (RaT. Ra5) (lOx io~, 8.5 x10 5) (lOx 1O~, 8.5x i0~).

f)k

i,

(25)

Co is the initial composition of the melt. No closed-form solution of eq. (24) exists if k is a function of c, We have solved this equation numerically using a fourth-order Runge Kutta

where

method.

D.H. Kim, R.A. Brown

/ Modelling of dynamics of HgCdTe growth

The one-dimensional model of transient, diffusion-controlled axial segregation is governed by the transport equation Sc(dS/dT) d2S/dz2, and the interface boundary condition dS

dH Sc

dz

dT

[1

k(S)] (1

+

s),

(26)

(27)

where dH/dr is the time-dependent growth rate and eq. (21) was used for k(s). Eqs. (26) and (27) were solved by a one-dimensional finite element analysis using the microscopic growth rate dH/dr determined in the convectionless simulations. The predictions of the one-dimensional analyses for diffusion-controlled and well-mixed growth are compared in fig. 17 to the results on the large-scale simulations, including convection in the melt. The diffusion-controlled nature of the solute field is obvious in all three simulations and appears qualitatively similar to the axial segregation profiles observed in the experiments of Szofran et ai. [11]. The results of the large-scale simulation agree surprisingly well with the onedimensional diffusion-controlled analysis that accounts for the variable segregation coefficient and growth rate. This agreement also was observed in the comparison of a similar one-dimensional analysis to the segregation measurements of Szofran et al. [11]. There is a discrepancy of ______________________________________ I I

ion

by vertical Bridgman method

427

approximately 10% in the steady-state value of the composition predicted by the one-dimensional analysis and the full simulations. This difference is caused by the highly curved melt/ crystal interface which leads to discrepancies between the radially-averaged interfacial composition computed for the simulations and the one-dimensional solution. The weak convection near the interface only slightly alters the concentration field there and the radial segregation is controlled most strongly by the interface curvature. The percent radial segregation iC is used to measure the lateral nonuniformity of thc concentration field as L1C( %)

max c (r, H(r))

mm

O
c(r, H(r))

(C)

x 100%,

(28)

where

(C)

J~~+H~rdr 1 °

v

+ H~r

dr

is the radially-averaged interfacial composition. The evolution of ~iC as a function of the fraction solidified is shown in fig. 18. The radial segregation rises from zero initially to over 100% at steady-state. Increasing the intensity of convection by increasing g has only a slight effect on ~C.

-

4.4. Effect of ampoule translation rate The importance of the damping of thermallydriven convection by the HgTe gradient emanatinggrowth of from the rateinterface on solute leads transport to a profound in the melt. effect It

Li

,

50

—

——

g

0 0.1

g

g,,,,,5

g,,,,,,5

is expected that a decrease in growth rate will result in a lower axial concentration gradient and to more intense convection in the melt. This

I 0

~

I

~

I

5

03

fraction solidified Fig. 18. Time evolution of the radial 0’1g,,r,h segregation andas~ a function of fraction solidified for g 0,

effect was identified in the steady-state analysis in ref. [7]. We investigated the effect of a reduction in the transient analysis ampoule by performing translation arate simulation in the with fully

428

D. H. Kim, R.A. Brown Al’ 005 4’,,,,— 0 40

Modelling of dynamics of HgCdTe growth by i ertical Bridgman method

c,,,, i 00 c,,,,, i 00 Al’ 000i Ac 005 4’,,,,— 0011 c,,,, 059 4’,,,,, 0014 c,,,,, 100 Al’ 0001 Ac 0.05 1’,,,, 0 007 c,,,, 0 35 1L,., OOii c,,,,,, iOO ‘

54’ O.OOi Ac 005 4’,,,,— 0.007 c,,,, 0 27 ‘i’,,,,, OOii c,,,, 100

I

085 Ni

0

1-’

095

—

FLOW

—

SOLUTE S

(a)

~—

FLOW

~

—

~‘

t

SOLUTE

5966 s

~—

7’

-

FLOW

(b)

SOLUTE

FLOW

12042 ~ (c)

SOLUTE

18003 s (d)

Fig. 19. Sample flow and CdTe concentration fields for growth with V 9 O.~6~.vms at g 0.14., uris Symbols S and location of the minimum and maximum stream function value, respectively.

J/~ 0.56 ,iLm/s0’1ge,,rti~ (compared to 1.12 ~sm/s used above) for flow g Sample and solute fields for this simulation are shown in fig. 19; the fractions solidified for the four samples correspond to the values used in fig. 13 for 1.12 Jsm/s. The decrease in the growth rate causes the solute diffusion layer to extend further into the melt, thereby decreasing the intensity of upper flow cell. The solute gradient is lower near the interface and an increase in convective intensity is expected. Contrary to the results for the steady-state analysis, the intensity of convection in the lower flow cell is unchanged by lowering the growth rate. Although the smaller axial concentration gradient at the lower growth rate weakens the stabilizing influence of the solute field, this effect is com-

‘V, (~vm s)

1 12 112 ~

—

+

denote thc

R, (cm) 05

0 25 0 2~

—

°

-

-

‘0 —

0000

~.

~

— —

—

.

0I2

pensated for in the transient analysis by lower deflection of the melt/crystal interface and the nearby radial temperature gradients due to less latent heat release. The evolution of the interface deflection ,iH as a function of fraction solidified

—

03

fraction solidified .

.

Fig. 20. Melt crystal interface deflection as a function of fraction solidified. Calculations are shown for conditions responding to larger diameter crystal and lower translation rate with respect to the base case with g 0.14r,Irth.

D.H. Kim, R.A. Brown iS

/ Modelling of dynamics of HgCdTe growth by

I

I

~A.mpoule Translation

1 00

———

00

I 0.1

R,(cm)

12

0.25 025

056 I 0.2

I I

I

Vg (pm/s)

R, (cm)

1 12

05

Rate (1.12 ~.vm s)

Cum/s)

429

Bridgman method

I I

s)

v2

i ertical

2 ~\~J~0~025

-

initital concentration in the melt

I 01 I

0.3

fraction solidified Fig. 21. Crystal growth rate as a function of fraction solidified. Calculations are shown for conditions corresponding to larger diameter crystal and lower translation rate with respect to the base case with g O.lg.,,~,11.

is shown in fig. 20 for the two ampoule translation rates. Halving the growth rate cuts the deflection by almost 30%. This effect is not seen in the steady-state analysis in ref. [7] because lower growth rates were used in those calculations. The transient in the crystal growth rate is relatively unchanged by lowering the ampoule translation rate. This is shown in fig. 21 by a plot of the growth rate at the midpoint of the interface (r 0.5) for both growth rates. The predictions for axial segregation of CdTe in the crystal are shown in fig. 22. The longer transient with the lower ampoule translation rate is expected from the one-dimensional transient analysis of Smith et al. [39]. The smaller interfacial area caused by the flatter interface at the lower translation rate amplifies this effect. The weak convection for the lower growth rate near the interface only slightly modifies the solute concentration field and the radial segregation is controlled by the interface curvature. The evolu tion of the radial segregation iiC is shown in fig. 23 as a function of the fraction solidified. The radial segregation with Vg = 0.56 ~tm/s is almost

00

I 02

03

fraction solidified Fig. 22. Axial profiles of CdTe concentration as a function of fraction solidified. Results are for conditions corresponding to larger diameter crystal and lower translation rate with respect to the base case with g 0.1g.,,,,5.

150

ioo

I I

I I

.

I

,..“

V~(tom/s)

.“ ~‘

.‘

~

.~

<

.~“

— — —

Rc (cm)

i 12

0.5

112 0 56

0 25 0 25

,“

— — —

50

——

—

,.“ ,.“

— —

—

— —

—

——

,, —

,“ //

0 I

I

01

I

02

fraction solidified .

.

03

Fig. 23. Time evolution of the radial segregation as a function . . .. .. of fraction solidified. Results are for conditions corresponding to larger diameter crystal and lower translation rate with respect to the base case with g OAg.,,,,5.

430

D H. Kim, R.A. Brown Al’ 02 4’,,,,, 174 4’,,,, i.54

/ Modelling of dynamics of HgCdTe growth

by i ertical Bridgman method

iii c,,,, 1 00 C,,,,

Al’ 0.2 Al’, 0.005 4’,,,,— 1.86 4’,,,, 0.034

Ac— 005 c,,,,—0 48 c,,,,,-i.OO

Al’ 0.2 Al’,~0.005 ‘I’,,,,, 1.74 4’,..,—O.047

Ac 0005 i8 c,,,,— c,,,,,,—i.99

0 -l

z 0

N

1-4 0

H

Z

I

P1 085

H

0950

z 0

in

N~C Cr’ z~, ,

i-in

FLOW

SOLUTE

If)

4:

4:

Os FLOW SOLUTE 3859 s

(a)

(b)

— 0.05 Al’, 0.005 $,,9,=~0.39 l’,,,,,—O.026

Z

6737 s (c)

Al’ 0.01 Al’, 0.005 l’,,,,——0.il 4’,,,,,—0023

Ac— 0.05 c,,,,—0.iZ c,,,,,—0.99

4: SOLUTE

FLOW

Ac= 0.05 c,,,,,=O.12 C —0.99

Al’— 0.0025 4’=——O.024 l’,,,,,—O.023

z 0

Ac— 0.05 c,,,,=0.li c,,,,, 0.99 0 -l

N)

Ni 0

I-’ 0

z p1

z 0

______ ______

______

Cr-i i/I

C)) Cr’ 0

4:

4:

CII

FLOW

SOLUTE

FLOW

z

4:

4: SOLUTE

CII

FLOW

SOLUTE

8334 s

8407 s

9078 s

(d)

(e)

(f)

Fig. 24. Sample flow and Cdte concentration fields for growth with R, 0.5 cm at g ~ Symbols • and location of the minimum and maximum Stream function value, respectively.

+

denote the

D.H. Kim, R.A. Brown

/ Modelling of dynamics of HgCdTe growth

by vertical Bridgman method

431

half the value for 1.12 ~tm/s because of the smaller interface deflection.

mixed region up in the melt for a period; see figs. 24c—24e, Further increasing g toward gearts is not expected to alter the conclusions of these

4.5. Effect of ampoule radius

calculations. Because both RaT and Ra5 are proportional to g, a similar balance of the two com-

Although the definitions of the thermal and solutal Rayleigh numbers given in table 1 both scale as R,~,it is clear that both driving forces do not depend on the ampoule radius. Buoyancydriven convection driven by the radial temperature gradient does scale with the ampoule radius because the magnitude of the radial temperature difference is proportional to R~ [8]. However, solutal convection depends on the axial solute gradient, which in the diffusion-controlled limit has a length scale of D/J/~ [7]. In ref. [7], we have shown that the intensity of the solutal damping is appropriately scaled by the modified solutal Rayleigh number Ra5

ponents of the density gradients is expected as long as the flows are weak. The segregation of CdTe in the crystal is mainly unaffected by the increase of the radius of the ampoule, because the strength of the lower flow cell near the interface is still rapidly damped. The radially-averaged concentration is plotted as a function of f in fig. 22. The radial segregation ~iC dues not increase proportionally to the interface deflection with increasing the ampoule radius; the history of i.~Ccomputed for the larger radius ampoule is shown in fig. 23.

5. Flow structure and solute segregation: BT Ra5

—

Ra51I~SI Ra5 Pe~1 k k

13T(T,C)

g/35c0R~V51

The effects of the dependence of the density of a MCT melt on temperature and composition

I

ar’D

I

k k

(29)

Thus, convection is expected to intensify with increasing ampoule size and fixed growth rate. Simulations were performed for an ampoule with R~ 0.5 cm with g 0.lg~~r~5 to test this effect; all other parameters were fixed at the values in table 2. Samples of the evolution of the flow and solute field for six points during the transient for growth in an ampoule with R~ 0.5 cmconditions are shownRaT in fig. 0’1gearth~ For these is 24 for g approximately equal to the value for g gearth with the smaller ampoule and the strength of the thermal driving force is expected to be similar; however, the solutal stabilization is stronger in the simulation with the small ampoule. The evolution of the flow field in the larger ampoule is considerably more complicated with a new seeondary flow cell forming near the junction of the adiabatic and hot zones of the furnace. This cell grows in intensity and for a time surpasses the strength of the lower cell during the transient. This flow becomes strong enough to create a well

were investigated through calculations using a variable thermal expansion coefficient that was derived by differentiation of eq. (23) as ~

=

1 / dp

—

Pm ~ dT)

1 =

—

(c2

+

c3x

+

/3~T,

+

2c5T)

Pm —

+

where /3~

C2/Pm,

/3i =

C3/Pm

(30) and f3~=

2C 5/Pm. The thermal Rayleigh number RaT that appears in the momentum equation (4) is replaced with the expression

RaT

3~ gf

L%TR3 C

Ra 0

+

Rai(S

+

1)

+

Ra20, (31)

where the constants {Ra,} are defined as Ra0 —g(1311 +f32T~)~iTR~/ctu,

(32a)

Ra1

(32b)

g~31c0~TR~/au,

432

D.H. Kim, R.A. Brown

Modelling of dynamico of HgCdTe growth by i ertical Bridgman method

(32c) Ra2 .._g/32(.iT)2R~/av, and er k/pc,,., is the thermal diffusivity. The definitions of other variables are found in tables 2 and 3. The flow and solute fields computed using the eq. (30) for 13T(T, c) and the parameters for g 0’1gcarih (RaT RaT(T(r, z), c(r, z)), Ra 5 8.5 X i0~)are shown in fig. 25. Because solute convection is controlled by diffusion, the only differences between these results and those shown in fig. 13 are observed in the flow field. For the initial composition x 0.2, the inversion of the melt cell density direction of themoving lower flow and reverses results the in two cells, both melt up along the ampoule wall and down at the interface. The maximum density in the melt is located slightly above the interface and melt convection in this gap is driven by both axial and radial temperature gradients. As solidification proceeds, the development of the diffusion layer depleted in CdTe leads to a Al’ 4’,,,,

ii 0.63

sequence of vertically stacked secondary flow cells that separate the interface from the upper cell in the ampoule. The upper flow cell develops the boundary layer structure that was described in ref. [7]. The flow structure of multiple cells near the interface differs distinctly from the single toroidal cell computed with constant PT. The time scale for the formation of the new secondary cells is similar to the scale for diffusion of the solute layer into the bulk, The flow near the interface is somewhat stronger in the simulation with constant PT The behavior of the upper flow cell is very similar for3T(T, bothc)calculations, with being approxithe flow yariable 1 mately twofor times more intense. The weaker convection level near the interface for ~ PT(T, c) leads to lower radial solute segregation and to lower interface deflection. These two effects are demonstrated by the plots of ~C and ,iH in fig. 26. These effects are small; at f 0.2 the radial segregation is reduced from ,iC 109% for the

c,, 1,, i.00 c,,,, 1 00

4’,,,, 0.00

Al’— 0.009 001 4’,,,,, 4’,,,, 0 i3

3 Ac— 005 c,,,,, 1 00 c,,,,,—0.S

Al’— 0.002 l’,,..,,——0.025 l’,,,,,=0.007

Ac— 0 05 c,.,,—0.22 c,,,,, 1 00 Al’ 0.002 Ac 0.05 4’.,,,— 0.016 c,,,,,—0 15 l’,,,,—0.006 c,,,,,—i.OO 0 ‘-I

!~

Ni 0 095

z p1

095

z 0

C ‘OZ Cr-i -~0 Cc

FLOW

SOLUTE 0

S

4:

z~

4: FLOW

(a)

SOLUTE

3110 s

C

FLOW

(b) Fig. 25. Sample flow and CdTe concentration fields for growth with g (30), is included in the simulation. Symbols s and

+

SOLUTE

6020 (c)

S

FLOW

SOLUTE

9052 s (dl

O.lg,,~,

5.The variable coefficient of thermal expansion, eq. denote the location of the minimum and maximum stream function value, respectively.

D.H. Kim, R.A. Brown 15

Modelling of dynamics of HgCdTe growth by vertical Bridgman method

I

I

I

(a)

(b)

433

I

-

--

IF

g

0.1 g,,)),,, fixed ~

g= 0

00

00

1

~

P~(T,c)

3T

1

/1

----

g= 0.1 g,,,..,,,,. fixed /3~

— —

g

0.1 g,,,th, P~ f3

1(T,c

I

I

0.1

0.2

0 03

,

00

I

I

01

02

0.3

fraction solidified fraction solidified Fig. 26. Evolution of (a) melt/crystal interface deflection LIH and (b) percent radial segregation z.IC as a function of the fraction solidified for g ~ The variable coefficient of thermal expansion is included in the simulation.

constant value of PT to 103% when the variable coefficient is included. The magnitude of this effect is well within the variation of the solution caused by the uncertainty of other thermophysical properties.

6. Conclusions

The time-dependent numerical simulations reported here show the development of this diffusion layer and the direct relationship between it and the intensity and pattern of convection in the melt. The simulations demonstrate that the spread of the solute layer occurs on a diffusive time-scale with the initial transient to the steady-state growth rate lasting several hours for the slow growth rates (fr~ t~(1j.~m/s))that are typical of MCT growth. The simulations demonstrate all the features seen in the experiments of Szofran et al. [11]; axial segregation appears to be diffusioncontrolled and large radial segregation is predicted. The radial segregation is a result of the curvature of the melt/crystal interface and not a direct result of convection. These conclusions are sensitive to other parameters that control the strengths of the thermal and solutal components of the density gradient. Most notably, changes of the ampoule translation rate or the ampoule radius alter the convective driving forces. The interactions between heat and solute transport can be complex and lead to Unexpected results. For example, little change in solute segregation is seen with decreasing transla tion rate because of the concomitant decrease of the radial temperature gradient. Only detailed simulation of heat and solute transport and the —

Thermosolutal convection during the crystal growth of a nondilute alloy, such as MCT, is governed by the interactions of buoyancy forces determined from the dependence of the melt density on temperature and concentration, the design of the ampoule and furnace, and the operating conditions for growth. Interactions of these variables can lead to complex behavior of the flow field, interface shape and solute segregation. In MCT growth, the preferential incorporation of the lighter component (CdTe) leads to an axially stabilizing solute contribution to the density and retards buoyancy-driven convection caused by radial temperature gradients and by the density inversion with temperature. For the small-scale crystal growth systems modelled here, almost diffusion-controlled solute segregation results after the transient needed to establish the solute diffusion layer is complete.

434

D.H. Kim, R.A. Brown

Modelling of dynamic,c of HgCdTe growth by i ertical Bridgman method

melt hydrodynamics can accurately unravel these cases. Including the precise dependence of the melt density on temperature and CdTe concentration is necessary to predict the exact flow pattern in the melt; however, the details of this flow have very little effect on measurable quantities such as the axial and radial composition profiles and the interface shape.

Acknowledgements This research was supported by the Microgravity Sciences atid Applications Program of the United States National Aeronautics and Space Administration and by a grant from the National Science Foundation for use of the Pittsburgh National Supercomputing Center, We are grateful to Dr. B. Antar for his interest in our calculations and for supplying the prototypical thermophysical properties for MCT.

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