Single Crystal Growth of Semiconductors from Metallic Solutions

(4.2.34)

which relates the total material derivative to the one written with respect to the component. Balance of Momentum The balance of linear momentum states that the time rate of change of the linear momentum of a body is equal to the resultant force acting upon the body. Here we assume that the external forces consist of surface tractions acting on the surface of the body, and the gravitational and magnetic body forces acting at the points of the body. It is also possible to include surface effects such as surface tension in the formulation. Under these assumptions, the balance of linear momentum for a single component can be written as Da Dt

J pav?dV = J {patf +

ff*^+{p°+rav°}dV

v -a

-a i

-+-

f

/ +(%

I (/

i

jQYn(OL)\

-I- #

7

.

f/\~7

iw nn -I- I i V

^,OCG

1/

. o/^«^C^<7

-h/vJn/

\

7

w 1/7/7

//i

/•> 1

C\

(4 7 S i i

where ft" and f*m^ are respectively the gravitational and magnetic body force components acting on the a th constituent. /?" is the rate of momentum production, j a o is the thermodynamic surface tension, Q is the mean curvature, V^ is the surface gradient, and t% and t^a>} represent respectively the stress and Maxwell stress tensor components. The use of the transport equation, Eq. (4.2.20), after some manipulations, yields

J G

.th

from which the local momentum balance for the a component is obtained as -a

k

Using the following definitions

al

l

on a

(4.2.38)

104

Sadik Dost and Brian Lent IN

N

b,=-lp

N

..

N

bt , tu = I (/H - p vk v, ), /H = I ta , tk/ =^tkl

I (p? + v?ra ) = 0, pv, = £ (pav;a - ( p " ^ ^ ) + v «r"a), F™ = I //m(a) (4.2.39) in Eqs. (4.2.37) and (4.2.38), we obtain the overall momentum balance for the mixture as

where Z?z, ftem , ^ , t°, ^zm, and y° represent, respectively, the gravitational body force, the magnetic body force, the stress tensor, the interface stress tensor, the Maxwell stress tensor, and the surface tension for the mixture. For the metallic liquids, the magnetic body force and the Maxwell stress can be defined, respectively, as and

t% = EkEk + BkBk-^(E2

+ B2)8kl

(4.2.42)

where £lmn and SM represent respectively the permutation symbol and Kronecker delta. The balance of the moment of linear momentum becomes e

ta.. + ma =0 Imn ml

inF

n

—o o

(4.2.43)

v

y

where m® is the moment of momentum supply vector. Summing Eqs. (4.2.43) over a gives that the stress tensor is symmetric, i.e. tu =tlk since N

^ma ^

n

Balance of Energy The law of balance of energy states that the time rate of change of the sum of the internal and kinetic energies of a material body is equal to the sum of the rate of power of all forces and couples and the energies that enter or leave the bodyper unit time. We then write Da J -{pa(vfvf Dt VJ 2 -a

+ea) + (Ea2 +Ba2)}dV = J V -a

Single Crystal Growth of Semiconductors from Metallic Solutions

J {tau +tekfa)}v?nkdV + jvfiY?

105

+2aya(7nl)da

+ J {paha + JfEf +fa(- vfvf + ea) +ea}dV + J qfada V

(4.2.44)

S

-a

-a

Using the Green-Gauss theorem in Eq. (4.2.19), the transport equation in Eq. (4.2.20), and satisfying the conservation of mass and the balance of momentum, after some manipulations we obtain

J {pa£'a - (t^k

+ qaKk + Pahaha

V

-o

J a

1 2

a

+J 2jaav™Q.nlda = 0

nkda (4.2.45)

The localization of the above equation yields the balance of energy for the a * component as

+ ea+J?E?

inF_

(4.2.46)

on o

(4.2.47)

1 2 = 7at7vff72Q«/

where e", q" and /z" represent the internal energy density function per unit mass, the heat influx vector and heat source for the ath constituent, respectively. £a is the rate of energy supply to the ath constituent due to the local interaction of the ath constituent with other species within the body. The term on the right hand side of Eq. (4.2.47) gives the rate at which work is done upon the body due to the surface tension. The effect of surface tension in the energy balance is included under the following assumptions: the discontinuity surface contains surface particles of different species, and is a material surface in which surface particles are confined. Therefore, the normal component of the velocity of surface particles is equal to the normal component of the velocity of the discontinuity surface. The velocity of surface particles for the ath species is referred to as the surface velocity \ao. It is also assumed that tangential components of the surface velocity are continuous across the discontinuity surface, which means that the discontinuity surface does not have a discontinuity line. In the derivation of Eq. (4.2.47) we assumed that the surface

106

Sadik Dost and Brian Lent

tension is independent of position, and that the surface mixture is incompressible. Using the following definitions

+ - Y (oavvavav vav) o ^ ^" ^ a=l

i

i

a = Y (aa + paec avva

k h

Hi

Z. \Hk a=l

^ H

k

-tava)

l

klvl

/'

J a=\

^ P cc=l

CT ff V/

= £ y^vf"7, cc=l

cc=l

£ {(£a +^-)fa cc=l

+ p?v? + ea} = 0,

2

N

p£j-

L{p

£

-(p

£ vk )k+£

r }

(4.2.48)

and the summing of Eqs. (4.2.46) and (4.2.47) over a gives the overall balance of energy for the mixture as in V_a

(4.2.49)

)}(vt - « , ) - < ? , on a

(4.2.50)

Entropy Inequality The law of entropy states that the time rate of the total entropy is never less than the sum of the entropy supply due to body sources and the entropy flux through the surface of the body. This statement yields the local form of the entropy inequality for the ath species, in the absence of electromagnetic effects, as

p«rfa-(qakiea)

-pa(ha/6a)>0

paria(vak-uk)-qak/ealnk>0

in V_

(4.2.51)

on a

(4.2.52)

where we have not included the electromagnetic effects for simplicity. H]a and 6a denote the specific entropy and temperature (absolute) of the ath constituent, respectively. Neglecting the higher order terms in diffusion velocities and assuming the mixture is at the same temperature (i.e., 0a =0), and using the following definitions

Single Crystal Growth of Semiconductors from Metallic Solutions _

N

107

N

—a—a\ • a N

(4-2.53) a=\

a=\

we obtain the entropy inequality for the mixture by summing Eqs. (4.2.51) and (4.2.52) over a as y d

a

6a=l

1

n=0

k

in V-O

(4.2.54)

on

(4.2.55)

a

k

a=\

where q, and \i are, respectively, the influx vector, and the mass-based chemical potential for the a constituent Defining the specific Helmholtz free energy function for the mixture as y/ = £-6ri and eliminating ph using the energy equation, Eq. (4.2.49), the entropy inequality takes the following form (including the electromagnetic effects) (4.2.56) C7

a=\

a=\

The above presented balance laws can easily be written explicitly for a binary or ternary mixture by simply writing them for a = 1, or a = 1,2 respectively. We will not present them in this section. However, when we introduce the constitutive equations we will write them explicitly for binary and ternary metallic liquid mixtures. 4.2.4. Summary of Balance Laws For the benefit of the reader we here present a summary of the local balance laws. Conservation of Mass for the a th Constituent dc a dt

pc

a

(vk-uk)+ika

)CC

inV -O

(4.2.57)

on a

(4.2.58)

Sadik Dost and Brian Lent

108

Conservation of Mass for the Mixture (Continuity)

dP.

inF

(4.2.59)

on a

(4.2.60)

-a

=° Balance of Momentum

in V-O

(4.2.61)

on a

(4.2.62)

Balance of Energy N

inF-CT

a=\

+ e) + (E2 + B2) + I (

(4.2.63)

)}(vt - M , ) - ^ - (tkl + f™)V/ on cr

(4.2.64)

Entropy Inequality

1

> 0 (4.2.65)

a=\

4.3. Constitutive Equations In this section we introduce the development of constitutive equations without going into details of the constitutive theory of mixtures. We first present the development of the constitutive equations for a binary metallic solution. Binary systems are much simpler than ternary systems and give a better understanding for the physics of the development. We begin with the introduction of a Newtonian heat conducting mixture, and determine the restrictions imposed by the entropy inequality. The contributions of the electromagnetic fields will be included within the framework of the magnetohydrodamic approximation. These equations will be linearized about a reference state and the physical significance of various constitutive coefficients will be discussed. The equations will be simplified for specific growth techniques step by step. The final constitutive equations for a ternary system will be given without the detailed derivations. In a solution crystal growth technique, there are two phases in the growth crucible to be considered in modeling. The first is the liquid phase that

Single Crystal Growth of Semiconductors from Metallic Solutions

109

represents the liquid metallic solution from which crystals are grown. The second is the solid phase that represents the seed single crystal, the grown single crystal, the source polycrystalline material, and the crucible walls. We begin with the liquid phase. 4.3.1. Binary Liquid Mixture- The Liquid Phase A complete set of nonlinear constitutive equations of an electromagnetic fluid is given by Eringen and Maugin [1989] for a single conductive continuum, taking into account both polarization and magnetization. The linear constitutive equations of a binary metallic liquid mixture were obtained by Dost and Erbay [1995] after lengthy manipulations, where the mixture was assumed to be a nonpolarizable and nonmagnetizable metallic liquid. Following the same procedure, we now present general constitutive equations for a nonpolarizable and nonmagnetizable binary metallic liquid mixture. First, from the equations developed in Section 4.2, we write the balance equations for a binary system in the liquid domain. The mass balance for the solute, from Eq. (4.2.29), yields the mass transport equation as f*— m

\_

+ vkCk) = ikt if

^"v

if

if

/

(4.3.1)

if

\

/

where C is the mass fraction for the solute (C — cx — pxl p where p is the density of the mixture). The entropy inequality in Eq. (4.2.56) becomes / . V

-\

1

;

0

k

,k

k

k ,k

M M

k k -

where Jl = \ix— ji2 is the effective chemical potential, and 2du = vkl + vlk is the symmetric deformation rate tensor. We now assume that all the dependent variables, i.e., y/, £, 7], tM, qk, ik, and Jk are functions of the following dependent variables p" 1 , 0 , 0 k, C ,

Ck,

and dM . For instance, if( n~l O O C C E B d ) K

K

K

K

Kl '

(4 3 3) ^

s

Similar expressions are written for £, 7], tu , qk, ik, and Jk . Constitutive equations are subject to a number of invariance requirements (see Eringen and Maugin [1989] for details). Among these, the restrictions arising from the second law of thermodynamics, i.e., Eq. (4.3.2), are important. In other words the form assumed in Eq. (4.3.3) for the dependent variables must

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Sadik Dost and Brian Lent

satisfy the entropy inequality. Substituting y/ from Eq. (4.3.3) into Eq.(4.3.2) we obtain

dy/ • dy/ \ B£ p ^ ^^- 7d 7« + r\xlk

^ —^r> ^r>

r~ k'

±

ijUj n k^± +(t v^ kl k~ ,k kl

-, -\—)d kl 7'

-\

+JkEk>0

—1

i

(4.3.4)

This inequality is linear in 6, C, (0k), (Ck), Ek, Bk, and dM. The necessary and sufficient conditions for Eq. (4.3.4) to be satisfied for all possible thermodynamic processes are that their respective coefficients must vanish, i.e., d

77

¥

de

A

dc ^

'

d

¥

= f t

dek ,k

dck ,k

(4.3.5) j k

dBj

ddjj kl

k

and 1

DtMdu

0

+ JkEk > 0

(4.3.6)

where Dtu is the dissipative stress tensor and n is thethermodynamic pressure, defined respectively by iJu=tu D kl

kl

^-^/+^<577, -x - l

dp

kl

kl '

and

n-

^— -\ -1

(4.3.7) v

y

dp

From the last five equations of Eqs. (4.3.5), it is clear that the free energy function y/ is a function of density, temperature andconcentration only; y/ = y/(p-\6,C)

(4.3.8)

and

However, the remaining dependent variables, i.e., the mass flux i, the heat flux q, the electric current density J, and the dissipative stress tensor Dtwill still be functions of all the independent variables until proven physically otherwise.

Single Crystal Growth of Semiconductors from Metallic Solutions

111

We now present these equations without derivation (see Chp. 5 of Eringen and Maugin [1989] for derivation): Mass Flux l

+ D2VT + D3E + £>4dVC + D5dVT + D6dE + D7 VC xB J

8

^

^ ^ .

T

X^-/

I

* . ^ ^ -* -|

+Z)l6{d(VC x B) - d(B x VC)} + Z)l7 {d(V7 x B) - d(B x VT)} + Dlg{d(ExB)-d(BxE)}

(4.3.10)

Flux q = JtjVT1 + k2 VC + yt3£" + £ 4 dVr + yt5dVC + yt6d£ + k7 VT x B

+ksVC xB +k9E xB +kwd2VT + knd2VC + kud2E +ku(B • VC)B + Jt15(B • E)B +kl6{d(VT x B) - d(Bx +£ 17 {d(VCxB)-d(BxVC)} + £ 1 8 {d(£xB)-d(Bx£)}

(4.3.11)

Electric Current Density a4dE + a 5 dVC + o6dVT + onE x B +C78VC x B + a 9 V r x B + C7iod2£" + O"nd2Vr + o"12d2Vr + O"13(B +C714(B • VC)B + a 15 (B • V7)B + C716{d(£ x B) - d(B x E)} +c7 17 {d(VCxB)-d(BxVC)} + a 1 8 {d(VTxB)-d(BxVT)}

(4.3.12)

Dissipative Stress Tensor Dt

= aol + axd +a2d2 +a3E ® E +a4B®B + a5VT ® : 10 (Vr®d 2 Vr) s +a n (VC®(dVC)) s +a 12 (VC®(d 2 VC)) s

®VT)s+alg(E

-VT ®E}d\ + a

®VC)s+al9(WE

®EW)S

iW(E®VT-

(4.3.13)

112

Sadik Dost and Brian Lent

where WJJ=£JJ B kl

and the subscript S indicates symmetrization. In these

Mm m

r

J

equations the coefficients Dj, ..., D18; k\, ..., ki$\ (71,...,cr18; and a 1? ...,a 25 are functions of temperature T, concentration C, and also the joint invariants of d, E, B , V r , and VC. These invariants can be read from Table El in Eringen and Maugin [1989]. A full list of these invariants is not necessary in this book. We will however later present some of these invariants that play significant roles in the modeling of LPEE and THM. We first simplify these equations based on physical grounds for the solution growth process under consideration, as much as possible. When we examine each constitutive equation, we also discuss, to the best of our knowledge, the physical significance of these coefficients in view of crystal growth. We then simplify them further for the LPE, LPEE, THM, and LPD growth processes. The above constitutive equations in Eqs. (4.3.10)-(4.3.13) must not violate the entropy inequality in Eq. (4.3.6), thus the restrictions imposed by the inequality on the constitutive coefficients must be taken into account. In crystal growth, the use of linear constitutive coefficients (in their simplest forms) may be sufficient in most modeling purposes. However, in some cases nonlinear coefficients (cross interaction terms) must be taken into account for accurate predictions. For instance, such nonlinear coefficients may play a significant role in crystal growth under strong (magnetic) and weak (reduced gravity) fields. Previous numerical simulations have shown that the concentration field (mass transport) is more sensitive to such cross interactions (non linear) than the thermal and flow fields (Minakuchi et al [2004, 2005]). We therefore begin with the mass flux. In this constitutive equation, the first three terms, D^C , D^S/T, and D3E are linear in VC, VT, and E, but the coefficients Dx, D2, and D3 are still arbitrary functions of T, C, and the joint invariants of d , E , B , VT , and VC. We first expand these coefficients into a Taylor series about a reference temperature To, and concentration Co. This process is straightforward but very lengthy. We will only present the process for the mass flux, and then write the resulting equations for the others. Let us start with

= D3(T,C,Il,I2,...,IK)

(4.3.14)

where some of the invariants are Il=tr(d) = I, 7 2 =/r(d 2 ) = / 2 -2(/7), 73 = /r(d3) = 73 -2(/)(//) + /// IA=EE,

/ 5 =B-B, / ^ ( f - B ) 2 , o?/zer5.

(4.3.15)

Single Crystal Growth of Semiconductors from Metallic Solutions

113

The remaining invariants can be read from Table El of Eringen and Maugin [1989], of course, by adding the concentration gradient to the list of independent variables. We now expand L\, D2, and D 3 into a Taylor series:

(4.3.16) D ={DT+DTT7E+ v

z

1

ltL

DTRB+ ... ID

....

(4.3.17)

+{DET + DETE B + ...}T +.... ETEEE + DD ETB ET ETB

(4.3.18)

where E = I4V2, B = / 5 1/2 , and the material constants appearing in the above equations are functions of the reference temperature and the reference concentration only. Now using Eqs. (4.3.16)-(4.3.18) in Eq. (4.3.10), and also dropping some higher-order terms we obtain the mass flux as l

\ = {(Dc + DCEE + DCBB) + (Dcc + DCCEE + DCCBB)C +(DCT 2 +DTCE E + DTCBB)C + DTCC C + ....}Vr + {(DE + DEEE + DDER ERB) TCEE

+(DEC + DECEE + DECBB)C

+ (DET+

D

ETEE + D ETBB)T +

+DrAdVC + ^ . d V r + DrAE + DriVC x B + Dr.VT x B C4

CJ

CO

C/

Co

x B + £) cl3 (B-VC)B + £) c l 4 (B-Vr)B + £) c l 5 (B-£)B

(4.3.19)

It would be beneficial to mention the convention used for the subscripts in the material constants. In the expanded parts, we used only letters, and the first letter indicates the direct contribution of the field to the related flux, while the second and third letter describe higher-order contributions of the other fields. For instance, Dc is the coefficient of the direct contribution of VC to the mass flux, and DRC represents the interactive contribution of E with C to the mass flux. In addition, the number of letters describes the rank of the order of contribution. For instance Dn is a first-order contribution while Z> and £> R C

EC

ECB

are second- and third-order contributions. For the coefficients in the cross terms, the first index (letter) refers to the dependent variable (fluxes), and for the second indices we kept the numbering indexing to make the identification tractable.

114

Sadik Dost and Brian Lent

Eq. (4.3.19) is still complicated. It can be further simplified, of course, based on physical grounds and experimental observations. At this point, considering the applications only in solution growth techniques, we will leave only the terms up to second-order with the exception of two third-order coefficients in the coefficient of E, and one in the coefficient of VT. The significance of higher order coefficients will be discussed later. It is important to mention that the decision of leaving coefficients in a model in or out depends on how the model is being developed. This can either be the result of experimental observations that may force us to reexamine the significance of such coefficients in a model to make more accurate predictions, or can be brought about in the development of a general theory which can be tried to be proven by experiments. The former reason is the justification used in this work. Based on the purpose in mind, Eq. (4.3.19) is simplified further to p"4 = {Dc + DCCC + DCTT}VC + {DT + DTCC + DTTT + DTCCC2}VT HDE + (DEC + DECBB)C + (DET + DEmB)T}E

+ DC4dVC

+DC5AVT + DC6dE + DC7 VC x B + £>C8V7 x B + DC9E x B

(4.3.20)

We will leave the mass flux in its form at the moment. We will later make further simplifications specific to each crystal growth technique. Also when we use Eq. (4.3.20) in the mass balance equation, further simplifications can be made by dropping higher-order terms depending on their significance to the process under consideration, and also due to the restrictions imposed by the entropy inequality on material coefficients. Following the same procedure and arguments, the heat flux and the electric current are reduced to the following: = {kT+ kTTT + kTCC}VT + {kc + kCTT + +{kE + (kET + kEWB)T + (kEC + kECBB)C}E + kT4dVT +kT5dVC + kT6dE + kT7VTxB

+ kTSVCxB + kT9ExB

(4.3.21)

and J = {aE + (aEC + oECBB)C + (aET + oETBB)T}E + {aT + aTCC +OTTT]S/T + {oc + GCCC + <7C7T}VC + <JJ4dE + o^dVC +ci / 6 dVr + cJ7E x B + CJ J 8 VC x B + cJ9VT x B

(4.3.22)

The stress tensor can be written in its simplest form since the other terms do not have any significance in solution growth, i.e., as (4.3.23)

Single Crystal Growth of Semiconductors from Metallic Solutions

115

Eqs. (4.3.21)-(4.3.23) must satisfy the entropy inequality in Eq. (4.3.6). The material constants are functions of the reference temperature and concentration only. 4.3.2. Significance of Material Coefficients We now discuss the significance of some constitutive constants that represent certain physical effects or observations. This would be beneficial for model development and may also provide insight towards a better understanding for the phenomenological constitutive models developed for crystal growth. We mainly discuss the coefficients that are significant to crystal growth, but we also mention a few for their historical importance, and some that have significance in processes in the presence of a free surface or in microgravity. The intention here is not to make a comprehensive list of such coefficients encountered in many areas of physics and engineering, just to show the reader the importance of nonlinear modeling. Mass Flux, Eq. (4.3.20): • When all the nonlinear interacting terms in the coefficient of VC are neglected, the well-known Fick's law is obtained. This law implies that concentration gradient produces mass flux. It is a direct (first-order) contribution of concentration gradient to mass flux. D is the effective Fick's diffusion coefficient. In many mass transport applications, this is the only term kept in models. • Similarly, if we neglect all the interacting terms in the coefficient of V r except the constant term DT, we obtain the well-known Soret effect, which represents the contribution of temperature gradient to the mass flux. DT is known as the Soret coefficient. It may be important in some crystal growth and solidification processes depending on the materials considered. It is a first-order contribution of VT to mass transport. • The nonlinear second- and third-order Soret terms, i.e., DTCC and DTCCC may be significant in solidification under weak force fields. Indeed, these effects were considered in Timchenko et al. [2000, 2002] in a solidification process in microgravity by not only taking DC into account but also the third-order term DTCCC in the form of DTCC(\ + DTCCC) with DTCC = - 1 . It was shown that the contribution of these terms was necessary to predict their experimental results in microgravity. It is interesting to note that the inclusion of the first-order term DT was not necessary, and also taking the second-order term alone into account could not predict the experimental results observed in Timchenko et al. [2002]. The inclusion of the third-order

116

•

•

•

•

Sadik Dost and Brian Lent

term was necessary. This clearly shows the importance of nonlinear terms in modeling of Bridgman crystal growth. It is also possible that these terms may have significant contributions in other growth techniques under the effects of either strong fields such as magnetic field, or weak fields such as micro gravity. As will be discussed later in detail, the iQrm(DEC + DECBB)C in the coefficient of E is also very significant in LPEE growth. DEC represents the interactive (second-order) contribution of electric current with C to the mass flux, and is known as electromigration. It is an important mass transport mechanism in LPEE crystal growth. The third-order coefficient DECB is also a very important coefficient representing an indirect (higher-order) interactive contribution of an applied magnetic field to the mass transport in the presence of an electric current. This term is zero in the absence of electric current, so it is not a direct contribution of magnetic field. As we will see later the contribution of this term to mass transport in LPEE is very significant, and is twice that of DRC in Sheibani et al. [2003a,b], Liu et al. [2004], Dost et al. [2004], and Dost [2005]. DE represents the first-order contribution of the electric field to the mass flux. Its contribution may be important for some materials, but for metallic liquids it may be insignificant if V E is negligible. In such a case this term drops out when the mass flux is used in the mass transport equation DC4, DC5 and DC6 denote respectively the contributions of the interaction of concentration gradient, temperature gradient and electric field with the fluid flow. To the best of our knowledge, these are not important in either LPEE or THM. However, in passing we would like to mention that they may be significant in the presence of a free surface, or possibly in preferential growth processes such as lateral overgrowth or dendritic growth, or directional growth in micro gravity. For instance, the contribution of DC4 (also that of A^4) was included in solidification in microgravity (see Prud'homme and El Ganaoui [2005], and references therein). DC7 represents the contribution of concentration gradient that crosses the magnetic field lines. It is the result of interaction between the concentration gradient and the magnetic induction. It implies that mass transports perpendicular to VC and B . It may have a contribution to the mass transport (growth rate) but this needs to be proven by experiments. For instance the growth rate in THM may be enhanced by the application of a strong external magnetic field.

Single Crystal Growth of Semiconductors from Metallic Solutions

117

•

Dcs is the contribution of temperature gradient that crosses the magnetic field lines, giving rise to the mass flux perpendicular to VT and B. • DCQ implies that mass fluxes are perpendicular to E and B. It may be significant if the magnetic field is not aligned with the growth direction. In the LPEE set up (in Sheibani et al. [2003a,b], for instance) the electric and magnetic fields are aligned with the axis of the growth cell (axis of symmetry), then the contribution of this coefficient become negligible. Heat Flux, Eq. (4.3.21): • If we neglect all the nonlinear interacting terms in the coefficient of VT , we arrive at the well-known Fourier's law, which implies that the temperature gradient produces heat flux. It is a direct contribution of the temperature gradient to the heat flux. kT is the thermal conductivity coefficient. • Similarly, if we also neglect all the interacting terms in the coefficient of VC, except the constant term kc, we obtain the well-known Dufour effect, which represents the contribution of concentration gradient to the heat flux. It may be important in crystal growth. However, the significance of higherorder Dufour coefficients is not known in LPEE or THM. • Constant kE in the coefficient of E is very significant in LPEE growth; it implies that the electric current contributes to the heat flux. This term gives rise to the well-known Joule heating in the bulk, and also to the well-known Peltier effect (cooling or heating) at the interfaces of two media with different kE values (for details, see Dost and Erbay [1995]). As will be seen in Chapter 6 in detail, Peltier cooling is one of the main growth mechanisms of LPEE. In addition, the combined effect of Peltier heating and Joule heating affects the growth process of LPEE significantly. In the growth of thick crystals, the Joule heating in the grown crystal may become a limiting factor for the achievable growth thickness (see Zytkiewicz [1999]). Of course, the Peltier effect is also very significant in solid/solid interfaces used in cooling or heating applications. • kT4 , kT5 and kT6 represent respectively the contributions of the interaction of temperature gradient, concentration gradient and electric field with the fluid flow. There is no physical evidence for any significant contributions of these interactions in metallic liquids. However, as mentioned in the mass flux section above, kT4 was considered in solidification in microgravity (Prud'homme and El Ganaoui [2005]). kT6 implies that fluid flow produces

118

Sadik Dost and Brian Lent

an anisotropic Peltier effect. Again this may not be significant in metallic liquids. • kT7 represents the contribution of temperature gradient that crosses the magnetic field lines. It is the result of the interaction between the temperature gradient and the magnetic induction. It implies that heat flows perpendicular to V r and B. This is known as the Righ-Leduc effect. There are experimental evidences that the presence of an applied magnetic field also enhances heat transfer in metallic liquids (Terashima et al. [1987], Tagawa and Ozoe [1998]). • kTS is the contribution of concentration gradient that crosses the magnetic field lines, giving rise to heat flow perpendicular to VC and B. • kTQ implies that heat flows perpendicular to E and B fields. This is known as the Ettings-hausen effect. It may be significant if the magnetic field is not aligned with the electric field. Electric Current, Eq. (4.3.22): • aE is the direct contribution of electric field to the current. It implies that the electric field gives rise to current, and is very significant in LPEE. It is known as the electric conductivity of the medium, and its value determines the levels of contributions of electromigration, magnetic body forces, and the Joule heating in LPEE. • oT in the coefficient of VT suggests that the temperature gradient produces a current. This is the well-known Seebeck effect. Its contribution in metallic liquids may not be as significant as it is in solids. • aJ7 implies that current flows perpendicular to E and B fields. This is the Hall effect, which is used to determine mobilities in semiconductors. There is no known significance of this effect in solution growth. • aJg and oJ9 are the interacting terms that give rise to current flowing perpendicular to VC and B, and VT and B. oJ9 is known as the Nernst effect. Some of these above mentioned effects may be significant in metallic liquids and some may not. We will discuss them specifically when we examine each solution growth process in detail. It must be mentioned that all of the above effects represent bulk properties of the metallic liquids under consideration. There are also a number of surface effects that must be taken into account depending on the growth process. For instance in the growth of thin layers (epitaxial) and also in preferential growth processes such as lateral epitaxial overgrowth or dendritic growth, surface kinetics and curvature effects may be

Single Crystal Growth of Semiconductors from Metallic Solutions

119

significant; they must be included in the prediction of growth rates. We will discuss some of them when we discuss the growth of thin layers by LPE and LPEE using selective area growth. 4.3.3. Simple Constitutive Equations for a Binary System Based on our experience, the constitutive equations given in Eqs. (4.3.20)(4.3.23) can be written in their simplest forms that can be used for modeling in various crystal growth processes. For instance the mass flux can be simplified to p - l \ = DCVC + {DT + DTCC + DTCCC2}VT

+ (DEC + DECBB)CE

(4.3.24)

This equation can be used for the modeling of LPE, LPD and THM growth by simply selecting only the first term. For LPEE growth, the first and the last terms should be included. The middle term, the coefficient of VT, will only be included when the contribution of the Soret effect is significant. The remaining heat flux, electric current, and stress tensor can be written as and

Dt

= 2^ v d

(4.3.25)

Again these equations will, in most cases, be sufficient for modeling of LPE, LPEE, THM and LPD. 4.3.4. Simple Constitutive Equations for a Ternary System Following the procedure given in this section, the constitutive equations of a ternary system can be written after lengthy but straightforward manipulations. We will give again the simplest form that will be useful for the growth methods considered here (see Dost and Qin [1996] for the detailed derivations). For a ternary system, there are two independent components (solutes), and therefore there will be two mass fluxes. The equations related to mass transport can easily be written from Section 2. The mass transport equations in Eqs. (4.2.30) take the following explicit forms

and

Kk

Dt

DC

p p^— = Pl Kk Dt Dt

(4.3.26)

and the associated interface conditions are

n"k = 0 , and

pC ( 2 ) (v,-^)-/< 2 ) k=0

(4.3.27)

The mass fluxes, in the absence of the Soret effects, can be written as

= 4 n ) VC ( 1 ) + ECB

ECB

(4.3.28)

120

P -Y

Sadik Dost and Brian Lent

2)

= (4.3.29)

where D^\ D^\ and ^ ^ are the components of, respectively, the diffusion tensor, the electric mobility tensor, and the electromagnetic mobility tensor. The heat flux, electric current and the stress will take the same forms given in Eqs. (4.3.25). The free energy function, the entropy density function, and the chemical potentials become

4.3.5. Constitutive Equations for the Solid Phase In crystal growth the solid phase is usually assumed as a heat and electric conducting rigid solid. In a binary system, the solid phase involves only the heat conduction and electric balance equations. However, in a ternary system, in order to allow mass transport from/to the liquid phase, the mass transport of the solid composition must be considered in the solid phase. This gives rise to an additional diffusion equation in the solid phase. Due to its importance in ternary solution growth, we present the solid phase equations in detail (Dost and Qin [1996], Dost and Erbay [1995]). The solid phase represents the solid single crystal substrate and the grown layer in the form of a ternary system of AxBj_xC, and also the solid source as either a ternary or a binary system, as either AB or AC. In this section we will refer to the ternary system as the solid phase. The solid phase is assumed to be a rigid, heat and electric conducting material with the property of diffusive mass transport. The model will allow diffusion in the solid phase as well as the diffusion between the solid and liquid phases. Since the solid phase is assumed to be rigid, the relative motions of the material points will be neglected. However, the diffusion velocities will not be neglected in the mass transport equation. Only one compositional variable is needed in the solid phase to define the solid composition since = 1/2,

and

C^3) = l / 2

(4.3.31)

where C ^ , C^, and C^ are the solid compositions of the components A, B, and C. If we select Cy =C as the independent solid composition, there will be only one unknown in the solid phase, C . For instance, for an InxGa\.xAs system, the indium composition in the solid phase can be selected as the

Single Crystal Growth of Semiconductors from Metallic Solutions

111

independent composition, while in the liquid phase, for a Ga-rich solution, the As and In compositions can be selected as the independent liquid compositions. Following the derivation given for the liquid phase, the mass conservation for component A yields the following mass transport equation: d

P

s dt

(4.3.32)

°k.k

where rk } is the flux density in the solid phase. Since the solid phase is assumed to be rigid and the overall density remains unchanged, the overall mass conservation and the momentum equations will not enter the list of basic equations. The entropy inequality is then obtained as

dy/ dO

1

(4.3.33)

where ji{S) is the effective chemical potential in the solid phase. The above entropy inequality requires that (4.3.34)

and then the inequality reduces to (4.3.35)

Similarly the energy balance in the solid phase is obtained as (4.3.36) The linearization of the constitutive equations about a reference state yields the following free energy function

dr\ dT

2 7

o

djX o

o

where we have neglected the terms higher than second-order, and we have

r =T

o

djl

dT

0

0

dT

C^ o ; , 0

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Sadik Dost and Brian Lent IS)

dT

T+

(S)

0

C(S)

(4338)

o

where a zero subscript indicates the values are evaluated at the reference temperature and concentration while y/Q, r]0 and Ji0 represent the free energy, the entropy density and the chemical potential of the reference state, y denotes the specific heat for the solid phase, and the constitutive equations for the heat flux, the electric field and the solid diffusion flux are obtained as s

a.

SE

1

E

(4339)

where nonlinear terms were neglected (Dost and Qin [1996]). 4.4. Interface Conditions The jump conditions presented in Section 4.2 will be used to write the required boundary and interface conditions for a given setup. In order to write boundary conditions, the growth setup (crucible) for a specific growth technique must be specified. Thus, the development of the specific boundary conditions will be given when we discuss specific growth setups. However, it would be beneficial at this point to write general interface conditions (growth and dissolution) for solution growth. These conditions can then be specialized for a specific growth technique. In all the growth techniques considered here the growth interface moves with a velocity of u = V g . Let us consider the growth interface. The overall mass balance at the interface can be written as n rg

Solution (liquid phase)

Grown crystal (solid phase)

Fig. 4.4.1. Schematics of the growth interface.

Single Crystal Growth of Semiconductors from Metallic Solutions ^

L

123 (4.4.1)

The momentum balance takes the following form g

e m e m

n

(4.4.2)

where I denotes the unit tensor, and the Maxwell stress tensors will be removed from Eq. (4.4.2) in the absence of electric and magnetic fields. The energy balance at the interface yields (q(s) -q(L)yn

= psLv(Xg*n)-v(tesm*n)

(4.4.3)

where L =£,-£, - - v 2 + — ^ +^ v

L

S

2

2pL

(4.4.4)

2ps

is the Latent heat. The latent heat in most solution growth techniques can be neglected on the basis of a very small growth velocity. However, for high growth rates it may be included in the model. In the absence of electric field, the last two terms in Eq. (4.4.4) will be deleted. The conditions for the mass transport at the interface are obtained as .n = p s (C< s >(4.4.5) It must be mentioned that in the above interface conditions, some surface effects such as surface tension, surface curvature, and surface kinetics are not included. Such effects will be considered in detail in Chapters 5 and 6 in epitaxial lateral overgrowth (ELO) of single crystal layers by LPE and LPEE. 4.5. Application of Magnetic Fields The use of an applied magnetic field in crystal growth is of great interest mainly for two reasons; to suppress convection, and to provide better mixing in the liquid solution (or melt). A static but strong field is used for suppressing convection, and a rotating but weak field is employed for better mixing and species distribution. The literature in this field is very rich, comprising a large number of experimental and modeling studies. In this book however, we will confine ourselves only to the use of a magnetic field in metallic solutions associated with the solution growth techniques of LPEE, THM, and LPD. In this section we present the magnetic force components in general without

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Sadik Dost and Brian Lent

introducing specific models. This will be done later in Chapters 6, 7 and 8 when we discuss the specific models of LPEE, THM and LPD. The application of a magnetic field gives rise to a body force that acts on the points of the liquid solution. This force is known as the Lorenz force, and is usually written for a metallic solution under the assumptions mentioned earlier, such as being non-magnetizable and non-polarizable, as (see Eq. (4.2.42)i) F e m = J x B = <7 (E + v x B ) x B

(4.5.1)

In this equation the applied magnetic field B can be either a static field Bsta or a rotating field Bro\ or both. In general we can write \Sta , T%rot

= B4IU+B™

(4.5.2)

The electric field may also have two sources; a field induced by the applied magnetic field Em , and an applied electric field Eapp as in the case of LPEE growth: E = Emd+Eappl

(4.5.3)

The applied electric field can be very strong compared with the induced field as is the case in LPEE. In such a case, as we will see later in Chapter 6, the contribution of the induced electric field can be neglected. In the absence of an applied electric field, however, the contribution of the induced field should normally be taken into account in the model; it may or may not be significant depending on the growth technique and material in concern. For instance, in the growth of semiconductors from metallic solutions, such as THM and LPD growth of semiconductors the effect of the induced field may not be significant as shown by Liu et al. [2002a,b], and Kumar et al. [2006]. However, in the growth of crystals (particularly oxides) from the melt this may not be the case, as the induced field may have a significant effect on species transport in the system as pointed out by Lan et al. [2003] (see Chapters 7 and 8). 4.5.1. Static Magnetic Field Let us consider first the case of a static magnetic field. The magnetic body force components can be written as F;m = GEejqr{E^1

+E™d+eqmnvmBn)Br

(4.5.4)

where B = Bsta. When the applied electric field (electric current) is very strong as in LPEE, the applied field will be dominant and the induced field can be neglected. The other option is to make the applied electric field perfectly aligned with the applied magnetic field (Sheibani et al. [2003a,b]). In this case the term EapplxB will vanish. In such a case, Eq. (4.5.4) will become

Single Crystal Growth of Semiconductors from Metallic Solutions em =

F,sta,-T? +F

125 rot

x3 (p

sta.T? rot -*• (p ^ (p

Fig. 4.5.1. Schematic view of body forces acting at a point.

BiviBj -

(4.5.5)

where (|) is the electric potential, and a} is defined by a

(4.5.6)

B

=-£ J

j

Eq. (4.5.5) can be written explicitly as 2

B2v2 F =

B3v3)B2

F3 = aJa

2

B

-

>2

(4.5.7)

- oE{B] exp //

implicit

The above form of the magnetic body force components is very appropriate for computational reasons (see Kumar et al. [2006]). The second group terms are called "implicit" since they can be treated implicitly (taking them to the lefthand side of the discretized momentum equations). These terms will always be negative and therefore if added to source terms, they are likely to create a tremendous convergence problem or the solution may not converge at all in some cases. In order to avoid possible convergence problems in numerical simulations, the implicit part should be moved to the left-hand side of the algebraic system of field equations. The magnetic force components in Eqs.

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Sadik Dost and Brian Lent

(4.5.7) will be appropriate for use in the THM growth of semiconductors (Kumar et al. [2006]). If the magnetic field can be considered uniform in the solution region, and is in the vertical direction, Eqs. (4.5.7) will simplify to

Fx =

~OEBlvx ~OEB23v2

F2= exp licit

(4.5.8)

implicit

where we have neglected the contribution of induced field. In the case of a uniform applied electric current in the solution zone (LPEE), aligned perfectly with the vertical applied magnetic field, the magnetic body force components will take the simple forms

= -oEB23v2

(4.5.9)

implicit

Other specific forms of magnetic body force components in specific growth systems will be given later when we discus specific models in Chapters 6, 7 and 8. 4.5.2. Rotating Magnetic Field In the presence of a small, rotating magnetic field, the magnetic body force in Eq. (4.5.1) will have two parts, one from the applied strong field, and the other from the rotating field so that sta

¥em

(4.5.10)

rot vnt

V

/

where F™ is the same given in Eq. (4.5.4), and F™ is given by (4.5.11)

rot

where we have neglected the effects of temperature and concentration gradients in J. Since the field is not stationary the Maxwell equations must be added to the system equations; dt

= 0,

V-E m r f =0,

V-B ro ' =

Single Crystal Growth of Semiconductors from Metallic Solutions V x B r o / — - J = 0,

V J =0

111 (4.5.12)

Now assuming that electric and magnetic fields can be obtained from a scalar potential 0, and a vector potential A as follows B r o ' = V x A , and

E ^ = -(V0 + — ) , dt

(4.5.13)

Eqs. (4.5.12) reduce to dt

=0

(4.5.14)

Let us now consider some special cases. For instance, if the rotating field is applied in the horizontal plane (perpendicular to the growth direction), the magnetic field can be expressed by ot B rrot =

(4.5.15)

where Brot and co are respectively the magnitude and frequency of the applied rotating field. The field defined in Eq. (4.5.15) satisfies Eq. (4.5.13)i identically, and the solution of Eq. (4.5.14) and the electric field in Eq. (4.5.13)2 yield, respectively = -Bw'rcos((p-cot)ez

(4.5.16)

and s

(4.5.17)

Substitution of the electric field Emd and the magnetic field Bwt into the magnetic body force given in Eq. (4.5.11) yields a complex expression for this force. Under certain assumptions, this expression will take simplified forms (specific applications in THM and LPD growth of crystals will be given in Chapters 7 and 8). For instance, when the skin depth 8 = (u a Ym » R

(4.5.18)

is much larger than the radius of the crucible well, R (for instance, in the THM growth system modeled by Dost et al. [2003], 8 = 23 cm), the magnetic field distribution can be assumed not affected by fluid flow, that is, the field penetrates the liquid solution unchanged. In this case, assuming that

128

Sadik Dost and Brian Lent l

(4.5.19)

2

the mean magnetic body force components are obtained as rrrot

r

±

~2

T?rot V

E

2

~Rrot~\

Trrot

e

dz Y?J

2r

rr

~2

T>rot r

'1

j

nmt

^

(4.5.20)

Upon imposing further modeling assumptions, such as taking the amplitude of the oscillating component of the magnetic body force as negligible, the fluid rotation being considerably slower than the magnetic field frequency co, the solid phases being electric insulators, and the magnetic field continuous at all boundaries (see Dost et al. [2003] for justifications), the mean magnetic body force components become

F ; ° ' = O,

F;ot = l<7E(Bro>)2rco,

F ; ° '= O

(4.5.21)

4.6. Numerical Techniques In crystal growth from solution mainly two numerical techniques have been utilized: finite volume and finite elements. Each technique has its own advantages and disadvantages. Due to fast matrix conversion, the finite volumebased techniques became more popular for simulation in crystal growth from solution. Finite element-based codes were also used, particularly when an adaptive mesh was beneficial. Many authors have used either in-house or commercial packages. Details of the numerical procedures and codes used will be given in a particular section.

PART II

NUMERICAL SIMULATIONS

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131

Chapter 5

LIQUID PHASE EPITAXY

In this chapter we present the simulation models developed for the liquid phase epitaxial (LPE) growth of semiconductor single crystals. The focus is on the growth of silicon, binary systems such as GaAs and GaSb, and also ternaries such as GaInAs and GaInSb. We also introduce some models developed for the epitaxial lateral growth (ELO) and the conversion of semiconductor layers by LPE. The effect of gravity on convection is studied in an LPE growth system, known as the yo-yo technique. The effect of surface kinetics is also considered, and included in a model developed for an LPE sandwich system, and also in ELO growth of crystals. Various simulation results are presented, and discussed. 5.1. Introduction In LPE, the growth cell is normally a rectangular cavity where the solution is placed. Therefore, modeling the LPE growth process is actually a three dimensional problem. However, most models in the literature developed for LPE are two dimensional. This is mainly for two reasons; the first is for simplicity, and the second is due to the physical nature of the growth system. Crystals grown by LPE show 3-D and edge effects in the regions very close to the crucible wall. Except the very edges, the LPE process in general produces uniform and flat crystalline layers. Thus, two-dimensional models provide reasonably accurate predictions for most purposes. Since the focus here is on the growth of crystals from metallic solutions, and also mainly from dilute solutions, although the density of the solution changes during growth, the well-known Bousinessq approximation is adopted. That is, the density of the mixture (solution) is assumed constant everywhere in the field

Sadik Dost and Brian Lent

132

equations except for in the body force term in the momentum equations. This term allows density changes, and couples the momentum equations with the mass transport and energy equations (see below and Chapter 8 for further discussion on this approximation). Metallic solutions in the models are also assumed to be Newtonian, viscous, incompressible liquids. The solid phases, normally the grown crystal and the source (feed) crystal, are assumed to be rigid and heat conducting solids. Mass transport (diffusion) is also considered in the solid phase in the growth of ternary materials, since the composition of the grown crystal changes during growth. In LPE, growth is achieved by the gradual reduction of the overall furnace temperature. Therefore, many models developed for LPE have considered isothermal modeling in which the energy equation has been neglected. A schematic view of an LPE growth cell was shown in Fig. 3.2.1. 5.1.1. Field Equations of the LPE Growth Process In this section we derive the two dimensional field equations governing LPE growth process. In most LPE models found in the literature, constitutive equations used for the liquid phase are usually Fourier’s law heat flux, the Newtonian stress for incompressible flow, and Fick’s law(s) mass flux(es), e.g.,

q = kT ,

t =  pI + 2 μ d ,

1 (1) 1 (2) i = DC(11)C (1) + DC(12)C (2) , i = DC(21)C (1) + DC(22)C (2) L L

the the for for

(5.1.1) (5.1.2)

These equations can be written from Eqs. (4.3.25), (4.3.28) and (4.3.29) by simply neglecting the contributions of the applied electric field and magnetic field. The mass fluxes can also be expressed in terms of mole fractions. As we will see later, this is particularly convenient for the LPE growth of ternary crystals since the phase diagrams are included in the numerical solution procedure. For a binary system, Eqs. (5.1.2) reduce to a single mass flux equation as (from Eq. (4.3.28) and (4.3.29) in the absence of electric field)

i =  L DC C

(5.1.3)

In the solid phase, the heat flux q, and if necessary, the mass flux i, will normally be represented by Fourier’s and Fick’s laws, respectively, as given in Eqs. (5.1.1)1 and (5.1.3), of course, with their proper corresponding materials coefficients. Some other thermoelectric effects such as the Soret and Dufour effects can also be included when justified. We now derive the two dimensional model equations of the liquid phase.

Single Crystal Growth of Semiconductors from Metallic Solutions

133

The continuity equation in Eq. (4.2.31), under the assumption of incompressibility reduces to the well-know incompressibility condition, i.e.,

vk ,k = 0

(5.1.4)

or, using the Cartesian velocity components u and v in 2-D ( v = ui + vj , i and j being the unit vectors in the x- and y-directions),

u v + =0 x y

(5.1.5)

In the absence of electromagnetic effects, and using Eq. (5.1.1)2 the momentum equations in Eqs. (4.2.40) become

L (

vl t

+ vl,k vk ) =  p,l + 2 μ d kl,k  g  L  c (C  C0 ) l3

(5.1.6)

where we have assumed the Bousinessq approximation holds (in other words we allowed density changes in the gravitational body force term, bl in the momentum equations), and C is the solutal expansion coefficient defined by

C = 

1 d  L dC

(5.1.7)

In two dimensions, Eqs. (5.1.6) yield the following momentum equations in the x- and y-directions, respectively

u u 1 p  2u  2u u +u +v = + ( 2 + 2 ) x y t  L x x y

(5.1.8)

v v 1 p 2 v 2 v v +u +v =  +  ( 2 + 2 )  g C (C  C0 ) x y t  L y x y

(5.1.9)

Using the mass flux constitutive equation for a binary system given in Eq. (5.1.3) the mass transport equation in Eq.(4.3.1) becomes

L (

C + vk C,k ) =  L DCLC,kk t

(5.1.10)

or in two-dimensions,

C C  2C  2C C +u +v = DCL ( 2 + 2 ) x y t x y

(5.1.11)

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Sadik Dost and Brian Lent

We now first present some early theoretical models developed for binary and ternary systems using mainly a one-dimensional approach. 5.1.2. Early Models A number of pioneering modeling studies have been carried out for the LPE growth process (see, for instance, Small and Barnes [1969], Minden [1970], Crossley and Small [1971], Ghez and Lew [1973], Ghez [1973], Petrosyan et al. [1973], and Small and Crossley [1974]). These studies shed light on various aspects of the LPE process, and laid a solid foundation for future sophisticated computer simulation models when the required computational means became available. Their significant contributions are discussed in the coming sections when the two-dimensional simulation models are presented in detail. We would like, however, to discuss first their key contributions very briefly. Small and Barnes [1969] proposed a one-dimensional diffusion model for the LPE growth of GaAs (from a Ga-rich solution). An analytical solution, under certain assumptions, was given for the one-dimensional diffusion equation for an infinitely long domain. Growth rate and the thickness of grown GaAs layers were obtained for various cooling rates. Results shed some light on the necessary conditions for controlling the growth conditions used in LPE, such as the degree of supercooling that can be tolerated in the solution and its relationship with the cooling rate. Through a one-dimensional diffusion analysis, the issue of constitutional supercooling was studied by Minden [1970] for the LPE growth of GaAs (from a Ga-solution). Both, finite and semi-infinite domains were considered. The diffusion equation was solved analytically using the Laplace transform under the assumptions of equilibrium at the interface and a constant cooling rate. Results showed that for a semi-infinite domain, the concentration is obtained as an exponential function of time, and the maximum concentration gradient is given by

Cmax  0.6C0 / DC where DC is the diffusion coefficient of arsenic in liquid gallium, and C0 is the initial solute concentration. For a finite solution zone with width w, the concentration gradient is always less than this value, and depends on the parameter w/D. However, the minimum temperature gradient required to avoid constitutional supercooling continually increases as the liquid is cooled. As expected, it is without bound for the semi-infinite domain, while for the finite domain it approaches a limiting value. Following these pioneering works, Crossley and Small [1971] considered the same problem, i.e., the LPE growth of GaAs, and the one-dimensional diffusion equation was solved numerically by finite difference for a finite domain. It was perhaps the first numerical solution given for LPE growth. For the boundary

Single Crystal Growth of Semiconductors from Metallic Solutions

135

conditions, it was assumed that the solution concentration is the equilibrium concentration, given by

Ci = 4.82  1022 exp[8.42  (1.32  104 / T )] below 1170 °C, and there is no mass flux at the other boundaries of the domain. The model assumes a diffusion coefficient of 510-5 cm2/s, a 5-mm solution thickness, and a linear cooling with various rates. Numerical results obtained for arsenic concentration and growth thickness were also compared with experiments. It was shown that the best fit between simulation and experimental results (for the growth thickness) was obtained using a value of 4.510-5 cm2/s for the diffusion coefficient. As we will see in the next section in detail, an equilibrium condition at the interface can only be assumed when interface kinetics are very fast. Ghez and Lew [1973] considered, for the first time, the inclusion of interface kinetics in a model of LPE growth of GaAs, based on Minden’s analysis (Minden [1970]). The one-dimensional diffusion equation was solved for the same semi-infinite domain with an initial concentration C0, and under the following boundary conditions: at the boundary far from the interface C = C0, and at the interface (x is the growth direction)

DC (C / x)

x =0

= k[C(0,t)  Ce ]

(5.1.12)

where k and Ce are respectively the reaction constant and the equilibrium concentration at a given temperature. Modeling results obtained by numerical integration have shown that the concentration at the interface can deviate from the equilibrium value of the concentration, given by the phase diagram. It was found that the growth rate first increases linearly with time, its slope being proportional to the initial reaction rate. As time proceeds, the growth rate decays exponentially and the decay rate is inversely proportional to the activation energy of the kinetic process. It was also shown that their results reduce to those given by Minden [1970] in the case of infinitely fast kinetics. A full, analytical analysis (solution) was given for Minden’s problem by Ghez [1973]. Petrosyan et al. [1974] considered a one-dimensional diffusion model for an LPE growth system under a temperature gradient. Results showed that in such a system the time required to establish thermal equilibrium is much shorter than the diffusion time. The composition of the grown layer was also determined as a function of layer thickness. The LPE yo-yo cycle shown in Fig. 3.2.1 has been applied many times, and successive layers have been grown on the upper substrate. The lower substrate dissolves and supplies the required material to the solution. The time evolution of the grown silicon layer from an indium solution was shown in Fig. 3.2.2. As seen, between 20 to 30 cycles of growth have been achieved and silicon layers of up to a 400-μm thickness have been grown. This is due to the effect of

136

Sadik Dost and Brian Lent

gravity which raises the convective cells developed in the solution towards the upper substrate. This enhances the growth on the upper substrate, and consequently the growth thickness obtained on the upper substrates is larger than that on the lower substrates. At the same time during the dissolution period (ramping up the temperature) more material dissolves from the lower substrate. This way at the end of each cycle a net growth is achieved at the upper substrate while a net dissolution occurs at the lower substrate. However, the difference in the thicknesses of the upper and lower substrates depends on the spacing between the substrates. For instance, the thicknesses of the grown layers are almost the same for a 2-mm spacing, and increases with the increasing spacing. A typical experimental data was shown in Fig. 3.2.3 for the growth of silicon from a tin-solution. 5.2. A Convection Model for the Growth of Silicon in a Sandwich System In this section we present a convection model for the LPE sandwich system used for the growth of Si from a Sn solution shown in Fig. 5.2.1 (Sukegawa et al. [1988, 1991a], Kimura et al. [1990, 1994]). The model also includes surface kinetics based on the model given by Small and Ghez [1979] for the LPE growth of a GaAlAs system. The sandwich system consists of two horizontal Si substrates set 4 mm apart (h = 4 mm) in a rectangular graphite boat. The solution length is L = 20 mm. After introducing a saturated solution between the substrates, growth proceeds by gradually lowering the furnace temperature to maintain supersaturation. Growth is achieved after one growth cycle. The growth cycle is shown in Fig. 5.1.3 for a 4-mm spacing. 5.2.1. Governing Equations The governing equations of the LPE model are respectively the twodimensional equations of continuity, momentum, and mass transport given in the previous section, i.e,

u v + =0 x y

(5.2.1)

u u 1 p  2u  2u u +u +v = + ( 2 + 2 ) x y t  L x x y

(5.2.2)

v v 1 p 2 v 2 v v +u +v =  +  ( 2 + 2 )  g C (C  C0 ) x y t  L y x y

(5.2.3)

2 C C  2C C L  C +u +v = DC ( 2 + 2 ) x y t x y

(5.2.4)

Single Crystal Growth of Semiconductors from Metallic Solutions y

137

Surface reaction limited

CL

single crystal substrate

Transport Growth interface liquid solution

h

Ci g

Surface reaction

Ce Dissolution interface

Transport limited

single crystal substrate

x y

0 L

(b)

(a) Fig. 5.2.1. (a) Schematic view of the LPE sandwich growth cell, (b) solute concentration near the crystal interface with the driving gradients for diffusion and kinetic processes. C e, Ci, and CL correspond respectively to the equilibrium, interfacial, and bulk concentrations (redrawn from Kimura et al. [1994]).

where u(x,y,t) and v(x,y,t) are the velocity components in the x- and ydirections, p(x,y,t) is the pressure, and C(x,y,t) is the Si concentration in the Sn solution. 5.2.2. Boundary and Interface Conditions The field equations, Eqs.(5.2.1)-(5.2.4), are subjected to initial and boundary conditions. For the system shown the initial conditions are

u = v = 0,

p = p0 ,

C(x, y,0) = C0

at

t=0

(5.2.5)

where C0 is equal to the initial solute concentration corresponding to the initial temperature of the system. The boundary conditions on the cell wall are

u = v = 0,

C =0 x

(5.2.6)

where we assume no slip conditions for the flow velocity field, and no mass flux through the wall. At the boundaries of the liquid zone along the substrates, we also assume zero flow velocity components since the growth and dissolution interfaces move very slowly, that is u = v = 0 . For the concentration field at these boundaries, a simple condition is the assumption of equilibrium, In other words, the

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138

concentration along these boundaries (usually referred to as interface concentration) is assumed equal to the equilibrium concentration, Ci=Ce. However, such a simple condition would not be appropriate when surface effects are to be included. In such a case, the following general inequilibrium condition can be used

(i  n)

interface

= S g =  L DCL (C  n)

interface

(5.2.7)

where S g represents a surface term at the substrate-solution interface through which the effect of surface kinetics can be incorporated into the model, with of course, an appropriate specification of S g as described in the next section. For the LPE system considered here, since the interfaces are horizontal, Eq. (5.2.7) becomes

 L DCL

C y

= Sg

(5.2.8)

y =0,h

5.2.3. Surface Reaction The LPE growth process is not limited by bulk mass transport only. As is the case of vapor phase epitaxy, LPE growth proceeds by combined mass transport and subsequent incorporation reaction at the crystal-solution interface (surface). The purely transport-limited (diffusion) growth is a rather unusual case, made possible only when the interface kinetics is extremely fast. The reaction rate at the interface depends on several processes: adsorption and desorption, dissociation and chemical reaction, nucleation, surface migration, and capture at the growth site. As discussed in the previous section, a simple model that accounts for the essential effects through a lumped parameter, the surface reaction constant k, was proposed by (Ghez and Lew [1973], and Ghez and Giess [1974]). This model can be obtained by selecting a power representation for the term S g in Eq. (5.2.7), i.e.

S g =  L k(Ci  Ce ) n

(5.2.9)

where n is the order of the reaction, k is the surface reaction constant (rate), and Ce and Ci represent the equilibrium and interfacial concentrations respectively. The physical concept underlying the kinetics model is illustrated in Fig.5.2.1b. The growth process depletes the solute near the interface and Ci lies below C L . C L and Ce are determined from the phase diagram using initial conditions. However, Ci depends on both C L and Ce . From the mass balance point of view, the supply of growth units by diffusion and convection to the interface must balance their possible rate of incorporation into the crystal. In the fast kinetics case, the interface concentration of Ci will remain equal to the equilibrium concentration Ci , but in general departure from Ce would be

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139

expected. Using Eq. (5.2.9) to substitute for S g in Eq. (5.2.7)2 leads in analogy to the kinetic theory of chemical reactions, to the following representation of two-step (mass transport and surface reaction) process:

C y

= y =0,h

1 DCL

k(Ci  Ce ) n , or

Ci = Ce + (

DCL C k y

)1/ n

(5.2.10)

y =0,h

In the fast kinetics case ( k   and Ci = Ce ) the process is transport limited. The slow kinetics case, on the other hand ( k  0 ) implies Ci = C L ; the process is then surface-reaction limited. Following Ghez and Lew [1973] and Baliga [1978], first order kinetics (n = 1) can be assumed, i.e.

C y

= y =0,h

1 DCL

k(Ci  Ce ) ,

or

Ci = Ce +

DCL C k y

(5.2.11) y =0,h

Eq. (5.2.11) replaces the condition given in Eq. (5.2.7), and is used as an interface condition (boundary condition) at the upper and lower substratesolution interfaces. Since temperature changes are small during LPE growth, the surface reaction rate is assumed to be constant throughout the process. However, during the computations, the value of Ci is updated at each time step, based on the computed concentration distribution in the solution and on the equilibrium value Ce determined from the phase diagram. As we will see later, a similar interface kinetics condition is also used in the modeling of ELO growth of GaAs by LPE and LPEE (Liu et al. [2004b, 2005]). The growth rate G(t) and growth thickness H (t) are computed at the end of each time step in the computation of the field equations, Eqs.(5.2.1)-(5.2.4), using the following interface relations obtained from the jump conditions, given in Chapter 4, representing the solute conservation at the interface (see Wilcox [1983]):

 L DCL C , and G(t) = S (1 C) y

 L DCL t 1 C H (t) = dt  S 0 1 C y

(5.2.12)

5.2.4. Computational Procedure The nonlinear, time-dependent equations, Eqs. (5.2.1)-(5.2.4), governing the flow and concentration fields are solved numerically using a finite volume method. A hybrid central/upwind differencing scheme is used to discretize the convective and diffusive terms and a fully implicit scheme is employed for time stepping. A non-uniform mesh arrangement is used to allow for efficient meshing in high gradient regions, and a staggered grid arrangement, with velocity nodes offset from scalar nodes, is used to avoid spurious oscillations in the pressure field. The discretized set of algebraic equations are solved using a

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line-by-line Gauss-Seidel procedure which allows the use of an efficient tridiagonal matrix inversion procedure (Thomas algorithm). At each time step, a solution is obtained using the iterative SIMPLE algorithm (Kimura et al. [1994]), with the appropriate amount of underrelaxation to ensure numerical stability. The iteration cycle is terminated and the time step advanced when the normalized sum of the absolute difference between two successive iterations is smaller than a set tolerance value, taken as 5  105 . The above solution procedure is computationally intensive due to the nonlinearity of the problem and it is, therefore, important to minimize the number of computational nodes, while maintaining a sufficiently small grid spacing to ensure the accuracy of the solution. Since the computational study given by Erbay et al. [1993a,b] showed that the computations covering the entire domain of Fig.5.1.1 yield symmetrical solutions at all time steps, the computations given by Kimura et al. [1994] were performed for a domain covering half of the system only with the use of appropriate symmetry conditions. In both studies, a 71 29 grid was found to be sufficient. 5.2.5. Numerical Results for an LPE Cycle The effect of a finite surface reaction rate was investigated for two values of the surface reaction rate: k = 3.6  104 and k = 3.6  103 cm/s corresponding to estimates by Baliga [1978] for the growth of silicon from a silicon-tin solution. Two different diffusion coefficients were also used: DCL = 1.627  104 , and 3.0  105 cm2/s. The first is the value given by Smithells [1976] for the self-diffusion coefficient of tin at the growth temperature, while the second is the estimated value determined using Baliga’s method (Baliga [1978]). The various parameters used for the Sn-Si system are taken as (Kimura et al. [1994]): L = 6.486 gr/cm3 , C = -1.859, and μ = 8.01710-3 gr/cm.s. The initial temperature of the system was 994 °C, and the growth process was simulated for 60 min with a constant cooling rate of 0.33 °C/min. In order to see the significance of surface reactions the time evolution of the averaged growth thickness was computed for the above-mentioned surface reaction rates and diffusion coefficients. Results show that LPE growth is triggered and sustained by supercooling of the solution, and takes place initially by diffusion. As growth proceeds, the lower solute concentrations in the vicinity of the interface result in an unstable density stratification near the upper interface which, eventually, induces buoyancy-driven natural convection. Convective transport helps maintain higher concentration gradients near the upper substrate; this in turn produces higher growth rates. The thickness of the layer grown on the upper substrate is much larger than that on the lower substrate, about twice for the system given in Fig. 3.2.1. This agrees with the experimental results of Sukagewa et al. [1988, 1991a,b], and is due to gravity which will be discussed later in detail.

Single Crystal Growth of Semiconductors from Metallic Solutions

141 -4

3.0

k = 3.6x10 cm/s

2.5

k = 3.6x10 cm/s

-3

2.0

H(t) = growth rate in 10 μm/s -4

upper substrate

H(t) 1.5 1.0 lower substrate

0.5 0.0 0

1000

2000

3000

4000

time (s)

Fig. 5.2.2. Time history of averaged growth rates for the upper and lower substrates using DC = 3.010-5 cm2/s (redrawn from Kimura et al. [1994]).

The numerical simulation results show that both substrates are affected by the reaction rate, but the sensitivity of the upper substrate is much more pronounced and a notable reduction of the substrate thickness is observed for the lower reaction rate. A schematic diagram for the computed growth rates, for DCL = 3.0  105 cm2/s, is given in Fig. 5.2.2 (it is not presented here, but the difference between the numerically computed growth thicknesses at the upper and lower substrates is much smaller in the case of a larger diffusion coefficient, i.e., DCL = 1.627  104 cm2/s, implying that diffusion is dominant, for more information see Kimura et al. [1994]). At the beginning of the process, growth rates increase rapidly and remain identical for both substrates by diffusion and surface reaction. After the onset of convection, at about 200 s, the growth rate for the upper substrate first increases and then decays gradually. The overall trend is similar to that obtained in the diffusion-reaction case by Ghez and Lew [1973]; the important distinction is the difference between the upper and lower substrates when convective transport is accounted for. Table 5.2.1. Computed values of the parameter r (Kimura et al. [1994]). L

r

k (cm/s)

DC (cm2/s)

11.10

0.00036

0.00003

111.00

0.0036

0.00003

4.76

0.00036

0.000627

47.60

0.0036

0.000627

Sadik Dost and Brian Lent

142 -1

10

r = 11.10

r = 4.76

-2

10

i

r = 47.60

r = 111.00

-3

10

upper substrate: lower substrate: 0.0 0

1000

2000

3000

4000

time (s)

Fig. 5.2.3. Variation of the averaged interfacial supersaturation during the LPE growth process (redrawn from Kimura et al. [1994]).

Further insight into the relative importance of diffusion, convection, and surface reaction can be gained by introducing the following parameters (following Ghez and Lew [1973]):

r = k m / DCL where m = (DCL m )1/ 2 and  m = RT02 / bH

(5.2.13)

where r is a dimensionless parameter that provides a measure of the relative importance of rate limitation due to kinetics to those due to diffusion. In these expressions m is a diffusion length,  m is a characteristic relaxation time, R is the gas constant, T0 is the initial temperature, b is the cooling rate, and H is the heat of solution. Large values of r ( r  1 ) imply fast kinetics and interfacial concentrations close to the equilibrium value (see Fig.5.2.1b). It is generally considered that surface kinetics is very important in the early stage even for r  1 (Ghez and Lew [1973]), but once the concentration field is established (either by diffusion, or by combined diffusion/convection), the process will be controlled by mass transport for r  1 . The values of r presented in Table 5.2.1 imply that the process should be essentially limited by mass transport and that the role of interface kinetics is minimal except in the case of r = 4.76 . However, the computational results indicate otherwise, and clearly show that surface reaction is an important ratelimiting process in all cases investigated. Even when r = 111.00 the growth thickness at a given time is notably different from that obtained with k   .

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Further insight can be obtained by examining the behavior of the averaged interfacial and bulk supersaturations defined respectively as:

 i = (Ci  C0 ) / C0

and

 b = (C L  C0 ) / C0

(5.2.14)

The value of C L required to evaluate  b corresponds to the region of the solution where the concentration is uniform. The time history of the averaged interfacial supersaturation  i (computed for the cases shown in Table 5.2.1) is presented in Fig. 5.2.3. Growth starts in all cases with equilibrium conditions at the interface and  i is therefore equal to zero initially. However, due to the constraints imposed by surface kinetics,  i increases, first abruptly and then more gradually, and tends asymptotically towards a constant value determined by both surface and bulk (convective and diffusive) mass transport. This asymptotic value is small (about 10-3) for the relatively fast kinetics case (r = 111.00). But even in this case, the drastic change of  i in the early stage affects growth rate strongly. The relative deviation of the interfacial concentration from its equilibrium value is given by the ratio of  =  i /  b  (Ci  Ce ) / (C L  Ce ) . This ratio is initially close to unity and decreases very rapidly during the pre-convection phase. Clearly the growth process during this phase is limited by surface reaction for all cases. At the onset of convection, the curves corresponding to the lower and upper substrates bifurcate, with the upper substrate attaining higher values of  . For r = 4.76 and 11.10,  is larger than 0.5, and surface kinetics are evidently the dominant rate-limiting process throughout growth. Though reduced for large values of r, the effects of surface reaction cannot be neglected even for r = 111.00. Table 5.2.2. Comparison of r and  m with their effective values r* and  m (Kimura et al. [1994]). *

Substrate position

 m (cm)

r

 m (cm)

r*

 = i / b

Upper Lower Upper Lower Upper Lower Upper Lower

2.150 2.150 2.150 2.150 0.923 0.923 0.923 0.923

4.760 4.760 11.100 11.100 47.600 47.600 111.000 111.000

0.07 0.11 0.065 0.080 0.030 0.085 0.030 0.070

0.160 0.240 0.360 1.020 1.440 2.210 3.610 8.420

0.870 0.800 0.740 0.500 0.410 0.310 0.220 0.110

*

It must be mentioned that the observed inconsistency between the numerical simulations and theoretical expectations stems from an inadequate definition of the diffusion length. The value of m obtained following Ghez and Lew [1973] corresponds to one-dimensional pure diffusion in a semi-infinite domain. This

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provides a good description of the process before the onset of natural convection. Once convection is established, however, the process becomes twodimensional and the diffusion fields near both substrates interact with each other via non-linear convective effects. The effective diffusion lengths ( m* ) are consequently much smaller than in the pure diffusion case. The diffusion lengths were re-evaluated using the averaged concentration profiles corresponding to t = 3000 s. These values are presented in Table 5.2.2 together with the new estimates of the reaction rate parameter (r*). For laminar convection, linear concentration gradients can be assumed near the substrates and  can be estimated from  = 1 / (r * +1) ; this relation is derived from Eq. (5.2.11). These values of  are in good agreement with the numerical simulation results of Kimura et al. [1994], and provide further support of the above interpretation. Convective transport appears, therefore, not only to enhance growth rates and alter the shapes of the interfaces, but it also changes the process at large times from what would be, in the case of pure diffusion, a bulk transport limited process to one where interface kinetics becomes an important rate-limiting mechanism. 5.3. A Parametric Study The computational model developed by Erbay et al. [1993a,b] to account for both diffusive and convective transport for In-Si systems provided valuable information on the interaction between mass transport at the interfaces and natural convection in the solution, and predicted the formation of wavy substrates as a result of transient, non-uniform growth rates. In a subsequent study of LPE in a Sn-Si system by Kimura et al. [1994], as discussed in Section 5.2, the computational model was extended to account for finite reaction rate limitations, via a first-order interface kinetics model. The simulations showed that during the initial phase, LPE proceeds according to diffusion reaction rate model predictions, with both substrates growing at identical rates. With the onset of natural convection, complex, transient concentration patterns develop predominantly in the upper region of the cell, and the evolution of the two substrates bifurcate. The time at which the bifurcation takes place, subsequent growth rates and substrate shapes depend on both the strength of convective transport and the surface reaction rate. Whereas earlier diffusion based models indicate that LPE growth is a bulk transport limited process at large times, as presented in Section 5.2, Kimura et al. [1994] found that in fact when convection is accounted for, interface kinetics becomes an important rate-limiting mechanism. In this section, we discuss the results of a combined experimental and computational parametric study on the effect of solutal convection on mass transport during the growth of silicon from a Sn-Si solution for various spacing between the upper and lower substrates.

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The experimental setup modeled was presented in Chapter 3. The graphite boat used in this experiment is a split barrel with a slider as shown in Fig. 3.2.4. The barrel has a solution chamber, and the solution is contacted with the source silicon crystal during the heating phase to establish an exact saturation at the growth temperature. The solution can be brought into contact or removed from the substrates by moving the slider through the barrel (for further details on the experimental apparatus and procedure, see Kanai et al. [1993]). Ramp cooling growth experiments were performed using (111) oriented 22cm2, n-type silicon substrates, and tin as a solvent in a flow of Pd-diffused H2. Two substrates were set face to face horizontally in a sandwich configuration in the graphite boat; the gap between substrates (solution height) was varied from 2 to 8 mm. The temperature cycle shown in Fig. 3.2.3 was used. Both substrates and the solution were heated to 994°C and the tin solution, saturated with silicon, was inserted between the two substrates. After a three hours waiting period to ensure good wetting, the temperature was lowered to the final temperature (974°C) at a constant cooling rate of 0.33°C/min. Due to the low cooling rate and the good thermal conductivity of the apparatus, no vertical temperature gradient could be detected across the solution during the cooling process. After the growth period, the solution was removed from the substrates by sliding the boat. To assess the validity of the numerical predictions, the variation in surface contour of the grown layers near the edge was measured

Fig. 5.3.1. Dependence of the average growth thickness on the spacing between the upper and lower substrates (after Kimura et al. [1996a]).

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using a profilometer. The mathematical model and the associated boundary and interface conditions for the LPE domain shown in Fig. 3.2.1 and Fig. 5.2.1 are the same as those given in the previous section. The above-described growth process is simulated for different cell solution heights. The computational procedure is the same as presented in the previous section, and the diffusion coefficient of DCL = 3.0  105 cm2 /s and surface reaction rate of k = 3.610-4 are used. 5.3.1. Growth Rates and Substrate Shapes The experimental and simulated averaged growth thicknesses are compared in Fig. 5.3.1 for various solution heights. For h = 2 mm, the upper substrate is only marginally thicker than the lower one. As expected, as h increases the difference becomes more pronounced due to enhanced convective mass transport in the upper region. At h = 8 mm, the upper substrate is about 70% larger. The overall agreement between measured and computed growth thicknesses is good and

Fig. 5.3.2. Time evolution of grown layers for four different solution heights (spacing): (a) 2 mm, (b) 4 mm, (c) 6 mm, and (d) 8 mm. It must be noted the scale along the vertical axis is in micrometer while it is in centimeter along the horizontal axis. Due to this large difference in scaling, the computed shapes of the grown layers are dramatically exaggerated. These wavy shapes cannot be seen if the actual scaling is used. The shapes provide information about the relative accuracy of the simulation results (after Kimura et al. [1996a]).

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147

indicates that the model captures the essential physics of the process. Examination of the computed evolution of the substrates in Fig. 5.3.2 reveals several interesting features. In all cases, a wavier upper substrate develops due to convective transport (due to the effect of convective cells near the upper substrate in the solution). However the roughness of the substrate does not appear to simply increase with convection (i.e. with spacing). The roughness in the upper substrate depends on spacing in the following order: (1) h = 8 mm, (2) h = 6 mm, (3) h = 2 mm, and (4) h = 4 mm. As will be seen in the next section, this is intrinsically linked to the complex mass transfer patterns associated with solutal convection. Along the bottom substrates, the shape remains essentially flat except in the case for h = 2 mm. Again this is due to the fact that convection cells in the solution are closer to the upper substrates in all cases, of course except the case of h = 2 mm since there is not enough room in the solution for the cells to move up due to gravity. The surface contours near the edge of the experimentally grown layers are shown in Fig. 5.3.3. These confirm the overall shapes shown in Fig. 5.3.2, except very close to the edge where the ideal boundary conditions assumed in the model are difficult to reproduce experimentally and where non-negligible three-dimensional effects are expected to occur. This can only be predicted by inclusion of the edge effects into the model. However, this is not an easy task. 5.3.2. Convective Transport Many of the interesting features of the grown substrates can be explained by

Fig. 5.3.3. Surface contours of grown layers for different solution heights (after Kimura et al. [1996a]).

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Sadik Dost and Brian Lent

Fig. 5.3.4. Streamline (left) and concentration (right) patterns at various stages of the growth process for the spacing of 2 mm. Patterns are symmetrical about the centre of the growth cell. In this case, the Gr and Sc numbers are estimated as 1.42104 and 41.2, respectively. The difference between each stream function contour is 1.510-5 and the maximum velocity in (d) is 5.3210-3 cm/s (after Kimura et al. [1996a]).

examining the detailed evolution of the convection patterns in the solution. The computed flow and concentration patterns are presented for two spacing values in Fig. 5.3.4 and Fig. 5.3.5. In the case of the small spacing (2 mm), convective transport is essentially negligible during the first 270 seconds, and the concentration distribution (typical of a diffusive processes) is established. By time t = 900 s, a multi-cellular convection pattern is established, and the concentration contours are significantly altered along both substrates. The

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149

Fig. 5.3.5. Streamline (left) and concentration (right) patterns at various stages of the growth process for the spacing of 8 mm. Patterns are symmetrical about the centre of the growth cell. In this case, the Gr and Sc numbers are estimated as 9.11105 and 41.2, respectively. The difference between each stream function contour is 1.6510-4 and the maximum velocity in (d) is 9.6610-2 cm/s (after Kimura et al. [1996a]).

convection pattern evolves in a complex fashion with a continuous formation and amalgamation of convection cells, but the number of cells remains almost constant (18 for the entire domain). Although the convection cells are offset towards the top substrate where the destabilizing density stratification occurs, convective transport affects the lower region as well due to the relatively small spacing. Both top and bottom concentration patterns show regular peaks (lower concentration gradients C / y ) and troughs (higher concentration gradients) which induce lower and higher growth rates respectively. Because these patterns persist in roughly the same location, sustained high or low growth rates

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Sadik Dost and Brian Lent

Fig. 5.3.6. Time variation of maximum flow strength (the maximum of the magnitude of the velocity vector) in the calculation domain. S and F imply the initial and final points, respectively. The maximum point was calculated at 22 time points during the 1 h growth period (after Kimura et al. [1996a]).

occur locally and result in the wavy upper and lower substrates shown in Fig. 5.3.2a. In the case of the large spacing (8 mm), Fig. 5.3.5, the convection and concentration patterns are radically altered (note the smaller scale used in the yaxis). First, due to the higher Rayleigh number, the onset of convection occurs earlier and its effect is already evident by t = 270 s. Convection is also more chaotic and the number of cells varies from 4 to 12. Despite the enhanced strength of convection, there is no incursion of the convection patterns into the lower part of the solution as in the previous case because the stronger stabilizing density stratification which is established along the bottom substrate suppresses convection more effectively. Consequently, growth is controlled by diffusion along the lower substrate resulting in the flat substrate shown earlier in Fig. 5.3.2d. The enhanced convection along the upper substrate increases growth rate and yet results overall in a flatter substrate than in the 2 mm spacing case. This is due to the higher mixing associated with the more chaotic convective transport which results in continuous changes in the location of maximum and minimum growth rates. A measure of the degree of unsteadiness is given by tracking the location of the maximum velocity point in the solution throughout the growth process. This is plotted in Fig. 5.3.6. In the case of the smallest spacing (2 mm), the location of this point changes slowly and only in a localized area of the growth cell. As spacing increases, the locus of this point becomes more mobile. In the wider

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151

Fig. 5.3.7. Normalized standard deviation of growth thickness (after Kimura et al. [1996a]).

spacing case (8 mm), the locus alternates continuously from the sidewall to the center while gradually migrating downwards. The roughness of the substrates can be monitored from Fig. 5.3.7 by examining the variation of the normalized standard deviation of the growth thickness with time. In the early part of the process, the upper substrate roughness is highest for the 8-mm spacing due to earlier onset of convection. As growth proceeds however, the highest roughness occurs in the 4-mm case; this is consistent with the reduced degree of unsteadiness illustrated in Fig. 5.3.7. For the lower substrate, the 2-mm spacing results in the highest roughness throughout the process due to the deeper incursion of convection cells into the lower region of the growth cell. 5.4. A Diffusion Model for the Growth of Ternary Crystals In the LPE growth of ternary semiconductor alloys, the solid and liquid phases are close to equilibrium at the interfaces. Thus the underlying material (either the substrate or the last formed layer) can interact with the liquid solution and hence can actively participate in the formation of the current layer. The process of growing ternary alloys is much more complex than that of binary crystals. The active interaction between the solid and liquid phases leads to complicated interface conditions which make numerical simulations extremely difficult. Small and Ghez carried out a series of studies on the growth and dissolution kinetics of III-V heterostructures formed by LPE (see Small and Ghez [1979,

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Sadik Dost and Brian Lent

1980, 1984], and Ghez and Small [1981, 1982]). They developed a 1-D diffusion model which combines analytic solutions with numerical simulations. The analytic solutions are only valid around t = 0, but provide a good ”starting” point for numerical simulations, i.e., using the analytic solutions to initiate the numerical simulations. In this way, numerical problems that occur initially can be avoided. These pioneering works have shed light on the growth and dissolution kinetics of ternary alloys from liquid solution, and showed the importance of numerical simulations for better understanding of ternary alloy growth processes. The numerical simulations that rely on a “starting solution” are not general enough to simulate the complex processes of growing ternary alloys. In LPE crystal growth and also in other solution growth techniques, natural convection in the liquid phase, along with other effects, plays an important role (Sukegawa et al. [1988], Erbay et al. [1993a,b], Kimura et al. [1994, 1996c]). An accurate simulation must include convective transport in the model which will be presented in the next section. As seen in Chapter 4, depending on the technique considered, the governing equations involve nonlinear coupling of heat and mass transfer, fluid flow, and also electric and magnetic fields. A comprehensive numerical model is the only viable approach to solve such equations. However, a diffusion model would be very beneficial for a few good reasons. First, in most LPE growth systems only thin layers are grown in a relatively short period of time. Particularly, during the initial stage of growth the effect of convection is minimum, and a diffusion model can provide the needed information on the transport process accurately. This was the main reason for obtaining reasonably accurate results, within the limits of the simplifying assumptions of course, from the 1-D diffusion models of the above mentioned works of Ghez and Small. In addition, such a model can also provide information for the process under microgravity. Finally, it prepares a base for the needed experience for further modeling of ternary systems where convection is included. Motivated by the above-mentioned reasons, a numerical model for the LPE growth process that includes only solid–liquid diffusion was presented by Kimura et al. [1996c]. The model is tested by repeating all calculations of Small and Ghez [1979], and then is used to simulate LPE growth of Al-Ga-As alloy. The computed solid composition and growth rate are in good agreement with available experimental data of Ijuin and Gonda [1976]. The results showed that the numerical procedure developed by Kimura et al [1996c,d] is suitable for simulations of solution growth of ternary alloys. In this section we present this diffusion model and the associated simulations results. 5.4.1. Governing Equations We consider here a III-III-V ternary alloy. As mentioned earlier, for convenience and also for the benefit of the reader, the system equations will be

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153

given here in terms of mole fractions. In an AxB1-xC system, three compositional variables in the liquid phase must satisfy

x A + x B + xC = 1

(5.4.1)

where x A , x B , and xC are respectively the mole fractions of components A, B, and C in the liquid phase. In the solid phase, only one compositional variable (x) is needed to define the solid composition distribution since

x sA = 0.5x,

x Bs = 0.5(1 x),

xCs = 0.5

(5.4.2)

where x sA , x Bs , and xCs represent respectively the mole fractions of components A, B, and C in the solid phase. Here we choose x A and xC as the independent variables in the liquid phase, and x sA , instead of x, in the solid phase. Note that in this section in order to prevent the inflation of indices, we use indices a little bit different than what was adopted in Chapter 4, without causing any confusion. The mass transport equations are obtained in the absence of convection in the liquid phase as

x A t

= D AL (

x A x 2

+

x A

xC

y

t

), 2

= DCL (

xC x 2

+

xC y 2

)

(5.4.3)

and, in the solid phase as

x sA t

=

s s x A DA ( 2

x

+

x sA y 2

(5.4.4)

)

where D AL , DCL , and D As represent respectively the diffusion coefficients of components A and C in the liquid solution of A-B-C, and of C in the solid grown crystal. 5.4.2. Interface and Initial Conditions The interface conditions can be written from the jump conditions given for the mass balance in Chapter 4. This simply says that mass fluxes in the liquid and solids phases at the interface must be equal. Considering the fact that the interface moves with a velocity of Vi, and also assuming that the interface remains flat during growth, we can write

w

Vi (x so A

so



x oA )

=w

o

D AL

x A y

w 0

so

D As

x sA y

(5.4.5) 0

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154

w

Vi (xCso

so



xCo )

so

=w

o

DCL

xC y

w

so

DCs

0

xCs y

(5.4.6) 0

o

where w and w are the initial solid and liquid mole densities respectively. The growth rate can be written as

G(t) = (Vi / D AL )L

(5.4.7)

At the interface, the compositions of the two phases must also satisfy the phase diagram (two equations). Therefore, keeping in mind that xCso = 0.5 , we can solve for the interface concentrations x oA , xCo , and x so , and the growth rate G(t) A once the composition gradients are computed by solving the governing equations in Eqs. (5.4.3). The specification of the problem is complete when the initial condition of the problem is determined. We assume that the liquid phase is in equilibrium with the desired composition of the epitaxial layer to be grown. It is important to note that, in general, the initial substrate has a different composition from the new layer. The initial condition is determined by the phase diagram based on the growth temperature and the solid composition with which the liquid should be in equilibrium. The phase diagram for a ternary III–V alloy system AxB1-xC can be described by the following equations (Casey and Panish [1978], Panish and Ilegems [1972]):



s x AC

=

4 AL CL x A xC AC ) sL( AC )  sL( C A

s  BC (1 x) =

exp[

4 BL CL x B xC

F F (TAC T) S AC

exp[ sL( BC )

 BsL( BC ) C

RT

(5.4.8)

]

F F (TBC T) S BC

RT

]

(5.4.9)

F F where  AL and  CL are the activity coefficients, S AC and S BC are the entropies F F of fusion, TAC and TBC are the melting points, R is the gas constant, and T is the growth temperature. The activity coefficients are defined as follows L L L L L RT ln( AL ) =  AB x B +  AC xC + ( AB +  AC   BC )x B xC

(5.4.10)

L L L L L RT ln( BL ) =  BC xC +  AB x A + ( BC +  AB   AC )xC x A

(5.4.11)

L L L L L RT ln( CL ) =  AC x A +  BC x B + ( AC +  BC   AB )x A x B

(5.4.12)

AC ) L RT ln( sL( ) = RT ln( CsL( AC ) ) = (1 / 4) AC A

(5.4.13)

L RT ln( BsL( BC ) ) = RT ln( CsL( BC ) ) = (1 / 4) BC

(5.4.14)

Single Crystal Growth of Semiconductors from Metallic Solutions s RT ln( sAC ) =  sAC  BC (1 x)2 , and RT ln( BC ) =  sAC  BC x 2

155

(5.4.15)

L L L ,  AC ,  BC ,  sAC  BC are the interaction parameters in the liquid and where  AB solid phases, respectively. Since the activity coefficients are related to the compositions and temperature, the phase diagram equations are nonlinear. Newton–Raphson method is used to solve for compositions of A and C in the liquid phase. In the solid phase, the initial condition is the substrate composition.

5.4.3. Numerical Solution Method The governing equations given in the preceding section define the process completely. However, it is difficult to solve these equations because of the complex interface conditions and the nonlinear coupling of the solid and liquid phases. Since the diffusion coefficient of component A in the liquid is much larger than that in the solid, the initial-value problem of this type is called stiff. This will result in poorly conditioned matrix equations in numerical analysis. To overcome this problem, the diffusion equations are solved separately for the solid and liquid phases. By doing so, different mesh sizes and different time scales are used in the two phases. The two phases are however coupled by the growth rate and the interface condition. Before computing composition distributions in the two phases, the interface conditions must be determined. The simulations are initiated by a guess interface condition. An iteration procedure is then applied to obtain convergent solutions in the two phases and at the interface for each time step. The initial guess interface condition is crucial for a successful simulation. A poor guess would lead to non-convergence in the interface condition iteration. In order to overcome this difficulty, the thermodynamic model developed by Small and Ghez [1980] to determine the guess interface concentrations is extended. The concept of the model is as follows. A substrate of composition x is placed in contact with a saturated liquid of compositions x A and xC which would be in equilibrium with the desired solid of composition xe . The thermodynamic model describes a virtual process which is the transfer of a relative molar quantity μ of the substrate into the liquid to form a supersaturated mixture of compositions x mA and x Bm . This mixture is then relaxed to equilibrium forming the solid and a saturated liquid of compositions x oA and x Bo . In this process, transport processes are generally very much more rapid in the liquid than in the solid. So μ must be very small. When the supersaturated solution relaxes, the relative molar quantity forming the solid is μ . If μ > μ then the solid would tend to grow as the system approaches equilibrium; while if μ < μ , it would tend to dissolve. Based on the virtual process, the interface concentrations may be determined as follows.

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In AxB1-xC system, based on interface mass conservation, the compositions of a mixture of the substrate and the liquid are given by

x mA = x A (1 μ ) + μ x sA = x A (1 μ ) + (1 / 2) μ x

(5.4.16)

xCm = xC (1 μ ) + μ xCs = xC (1 μ ) + (1 / 2) μ x

(5.4.17)

A calculation is started by taking a given substrate composition x and the liquid compositions x A and xC which are in equilibrium with the solid composition xe , determined by the phase diagram. The compositions of the mixture are then calculated using Eqs. (5.4.16)-(5.4.17).

x mA = x A (1 μ ) + (1 / 2) μ x , and xCm = xC (1 μ ) + (1 / 2) μ

(5.4.18)

When this mixture is allowed to relax to equilibrium, conservation of mass necessitates that Eqs. (5.4.16)-(5.4.17) still be satisfied, thus we have

x oA (1 μ ) + μ x so = x mA , A

and

xCo (1 μ ) + (1 / 2) μ = xCm

(5.4.19)

In this case the compositions of the two phases produced must also satisfy the phase diagram. So the problem is reduced to solving Eqs. (5.4.19) together with the phase diagram equations for x oA , xCo , x so and μ . As indicated above, the A value of μ  μ shows the direction of interface motion (growth or dissolution) when the solid is just brought into contact with the liquid. The computed compositions provide a good guess for the interface condition in the first time step. As one can see from Eqs. (5.4.18), μ is the only uncertain parameter which affects the solution of Eqs. (5.4.19). It was found that the magnitude of

Fig. 5.4.1. Mole fraction of As in the liquid for temperature rate of 0.1 °C/min (after Kimura et al. [1996c]).

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157

Fig. 5.4.2. Mole fraction of Al in the liquid for temperature rate of 0.1 °C/min (after Kimura et al. [1996c]).

μ is related to the ratio of diffusion coefficients of the solid to the liquid,  s . Using a relation μ = 10 n/ 2 , for  s = 10 n , very good initial guess solutions were obtained. The use of this choice always led to quick convergence of interface iteration in all test cases with values ranged from 10-6 to 10-10. As indicated by Small and Ghez [1979], the combination of finite supersaturation together with local surface equilibrium implies infinite initial compositional gradients at the surface and the initial growth rate is singular. Very fine meshes have to be used for these kinds of problems. This requires small time steps and causes accumulation of round-off errors when the explicit finite difference methods are used. Therefore, the use of the implicit finite element method, to which no time increment restriction is applied for stable numerical procedure, has advantages. In addition, the separated solution procedure for the solid and liquid phases makes it possible to use a very fine mesh to catch the great composition gradients in the solid phase, since there is only one unknown diffusion variable. In summary, the numerical simulations are started by determining the initial conditions based on the phase diagram. Then the guess solution for the interface concentrations is calculated by using the thermodynamic model. The time integration is then proceeded by the iteration procedure of the finite element solution for the liquid and solid phases, and Newton-Raphson method for the interface concentrations. 5.4.4. Simulation Results The solid–liquid diffusion model is applicable to most III-V ternary alloys for which their phase diagrams may be calculated (Casey and Panish [1978]).

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Fig. 5.4.3. Mole fraction of Al in the solid for temperature rate of 0.1 °C/min (after Kimura et al. [1996c]).

However, AlxGa1-xAs was selected as the first simulation case because of its technological importance and simple thermodynamic properties due to a nearly constant lattice parameter. Further, some analytic results for this alloy are available for comparison. The simulations are carried out for two cases: constant temperature, and temperature decreasing with time. In the case of constant temperature, the growth and dissolution processes depend only on the concentration state of the system. Although the liquid composition is about the same as the interface concentration of the liquid, the substrate composition is different from the interface concentration of the solid. This gives rise to a driving force (constitutional non-equilibrium, or concentration differences) for growth or dissolution. In the case of varying temperature, on the other hand, the temperature–liquidus concentration relation varies with time, and so do the interface concentrations of the liquid and the solid. The decreasing temperature with time acts as an external driving force for growth in addition to the one due to concentration differences. Case I. Constant Temperature Small and Ghez [1979] have engaged in a detailed study for this relatively simple case. They investigated growth and dissolution kinetics using both analytic and numerical methods in conjunction. In this case, an analytic solution valid for just the beginning of the process is readily obtained. The analytic solution is then used to start a numerical solution in order to avoid numerical difficulties. There are basically two possible results. If the liquid is prepared at a temperature the same as or lower than the growth temperature, the substrate tends to dissolve. If the liquid is prepared at a temperature which is higher than

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159

Fig. 5.4.4. Growth rate with time for temperature rate of 0.1 °C/min (after Kimura et al. [1996c]).

the growth temperature, then the substrate may be grown or dissolved depending on the degree of supersaturation and the ratio of diffusion coefficient of the solid to that of the liquid. As a test case, Kimura et al. [1996c] repeated all calculations carried out by Small and Ghez [1979]. The solutions are readily obtained when care is paid to the guess interface concentrations which are important for the solution convergence. The iteration for interface concentrations at each time step converges in a few steps when the guess interface concentrations are calculated

Fig. 5.4.5. Mole fraction of Al in the solid for temperature rate of 1°C/min (after Kimura et al. [1996c]).

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Fig. 5.4.6. Comparison of growth rates for different temperature rates (after Kimura et al. [1996c]).

as described in Section 3. The numerical results of Kimura et al. [1996c] are almost identical with those of Small and Ghez [1979]. This gives the confidence that a purely numerical procedure can be effectively used in simulations of the growth process of ternary alloys. Case II. Decreasing Temperature In this case, the temperature decreases with time to supersaturate the liquid solution. The change of temperature with time can be expressed as T = T0  rt where T0 is the initial growth temperature, and r is the temperature rate of change which may be a function of time. This case is much more complicated because of the interaction of the constitutional nonequilibrium and the changes in the thermal field (temperature change). The growth process may actually start with dissolution followed by re-growth when the temperature change is sufficient to overcome the natural tendency to dissolve. An analytic solution of the governing equations for this case is very difficult to obtain, since the growth temperature is a function of time which leads to an interface condition that changes with time. An approximate solution, through a complicated mathematical derivation, is given by Ghez and Small [1982] which may be valid for a constant temperature rate of change and a very short growth time. The reliability of the approximate solution is case–dependent and solutions beyond a varied time limitation are physically incorrect, as indicated by Ghez and Small [1982]. However, the numerical method developed by Kimura et al. [1996c] exhibits no additional difficulties in simulations for this case than for the case with constant temperature. The only difference is the need for calculating from the phase diagram based on changing temperature. In the following, we present a

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161

Fig. 5.4.7. The computed solid composition at different temperature rates (after Kimura et al. [1996c]).

complete LPE process of growing Al0.35Ga0.65As on a substrate of composition x = 0.1 for different growth conditions. The computed composition profiles in the solid and liquid phases, and the effect of initial temperature and temperature rate on the growth rate and composition distributions are presented. Figs. 5.4.1-5.4.3 show the computed composition profiles in the solid and liquid phases for the growth conditions given below:

T0 = 850°C, D AL

=

DCL

r = 0.1°C/ min

= 5  105 cm 2 / s,

D As = 1 1011 cm 2 / s

It can be seen from Fig. 5.4.1 that the gradients of As composition change direction during the process. The negative gradients at the initial stage mean that As comes from the solid whereas the positive gradients appeared later, and imply that As goes to the solid. This indicates that the process reverses direction from dissolution to growth. In Fig. 5.4.2, the gradients of Al composition in the liquid are always positive. Al in the liquid diffuses towards the interface (regardless of growth or dissolution) because of diffusion of Al in the solid. From Fig. 5.4.3, it is evident that Al always diffuses into the substrate since the interface concentration is much larger than the substrate concentration. Fig. 5.4.4 shows the growth rate as a function of time. The negative growth rate indicates that dissolution takes place at the initial stage. The growth rate becomes positive after about 180 second, and then increases with time until reaching a quasi–steady growth. In Fig. 5.4.3, the interface concentration hardly varies and the composition profiles in the solid are smooth due to the relatively small temperature rate, and large diffusion coefficient of Al in the solid. For a larger temperature rate,

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Fig. 5.4.8. The computed growth rate for three initial temperatures at a temperature rate of 0.5°C/min (after Kimura et al. [1996c]).

however, the interface concentration decreases significantly due to changes in the temperature–liquidus relation. Fig. 5.4.5 shows the composition profiles in the solid for r =1 °C/min and s = 108 . The drop of interface concentration and the slow diffusion in the solid make the highest composition occur inside the newly grown layer. The growth rate is obviously a function of the temperature rate of change. Fig. 5.4.6 shows a comparison of growth rates due to different temperature rates. The faster temperature rate results in a faster growth rate which is desirable. However, it also affects the composition distribution in the solid, as shown in Fig. 5.4.7. The computed growth rates for three different initial growth temperatures are shown in Fig. 5.4.8 (at a temperature rate of 0.5 °C/min). The growth rate increases with increasing initial temperature and the transient process lasts longer at a higher initial temperature because of the increasing solubility. To validate the numerical model, the computed solid composition and growth thickness are compared with the available experimental data of Ijuin and Gonda [1976]. The experiment was performed under a diffusion–limited growth condition (using a very small volume of solution to minimize the effect of convection). The growth temperature and the temperature rate used in the experiment were T0 = 850°C and r = 0.5 °C/min, respectively. The desired layer composition was x = 0.4 grown on a GaAs substrate. The initial layer composition measured in the experiment is in fact slightly higher then 0.4. This is because the initial solution was not exactly in equilibrium with the desired layer composition. This was possibly due to the error in measuring weights of the solute materials during the preparation of the solution. In the simulation, the initial layer composition determined from the experiment is used as the initial equilibrium composition xe. Two sets of diffusion coefficients are used in the liquid phase, namely,

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163

Fig. 5.4.9. Variation of the computed solid composition compared with the experimental data of Ijuin and Gonda 1976 (after Kimura et al. [1996c]).

D AL = DCL = 5  105 cm 2 /s and D AL = DCL = 7  105 cm 2 /s . The mass diffusion is neglected in the solid phase. The variation of the solid composition in the grown layer is shown in Fig. 5.4.9. The computed solid composition decreases with increasing thickness, which is in good agreement with the experimental data. Fig. 5.4.10 shows the growth thickness as a function of the growth time. The error bars are from the experimental measurement. The predicted growth rate is within the range of experimental data. The solid–liquid diffusion model presented provides valuable information in spite the fact that it did not include the effect of convection and also the coupling with the energy balance. For instance, it clearly shows how to handle the problems associated with initial guess and convergence of iterations. Secondly the numerical treatment of the interface mass balance (when there is a mass transfer between the liquid and solid phases) is a crucial issue in the modeling of crystal growth of ternary alloys. It also shows that under certain conditions, such as short growth times and in the growth of thin layers, a diffusion model can provide sufficient information. An extension of this model to include the convection effect is presented next.

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Fig. 5.4.10. The computed thickness of grown layer as a function of growth time (after Kimura et al. [1996c]). The error bars represent the experimental data of Ijuin and Gonda [1976].

5.5. Convection Models for the Growth of Ternary Alloys 5.5.1. Growth of GaxIn1-xAs The computational model developed by Qin et al. [1996b] investigated the effect of natural convection on the dissolution and growth processes of liquid phase epitaxy of ternary semiconductor alloys. This model, for the first time, accounted for diffusive and convective mass transport in the liquid phase, mass diffusion in the solid phase, and phase equilibrium and mass conservation at the moving liquid–solid interfaces. Numerical simulations were carried out for a horizontal sandwich LPE system used for the growth of GaxIn1xAs by a temperature modulation technique. The simulation results show that solutal convection plays an important role in LPE growth. It enhances the growth rate of the upper substrate during the cooling of the solution and gives rise to a faster dissolution rate of the lower substrate during the heating of the solution. Natural convection was also found to influence the compositional uniformity of the grown crystals. In the growth of III-V binary compounds, the composition of the epitaxial layer is not significantly altered by the change in the solution composition as growth proceeds due to the near-stoichiometry of the deposit. This is not the case in the growth of ternary alloys, since the distribution coefficients relating the compositions of the various elements in the solid to their concentrations in the solution may differ from each other. As a consequence, the alloy composition may vary significantly as growth proceeds, with a rate of change of

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165

alloy composition depending on the initial composition of the solution and the growth temperature. The continuous cooling and the solute depletion of the solution during LPE growth usually limit the thickness of the layer and sharply grade its composition profile. It is difficult to grow thick epitaxial layers with uniform composition by the conventional LPE method because of the depletion of limited solute elements during the growth. Sukegawa et al. [1998, 1991] and Kimura et al. [1990] have developed a novel LPE temperature modulation technique, in which both growth and dissolution phases are combined in a cyclic manner. The temperature of the solution is raised and lowered periodically resulting in successive growth and dissolution cycles. In a horizontal sandwich system consisting of a substrate– solution–substrate arrangement, dissolution occurs mainly on the lower substrate, while growth on the upper substrate is larger than that on the lower substrate. Thus in a cycle, the lower substrate is dissolved to feed the solution which makes it possible to produce thicker epitaxial layers on the upper substrate. This yo-yo solute feeding method eliminates the difficulty of solute depletion during LPE growth and allows a sustained growth of thick layers with a uniform composition (Sukegawa et al. [1991]). The key feature in the LPE temperature modulation technique is the different transport rates at the upper and lower substrates. These phenomena were attributed to natural convection driven by solutal concentration gradients (see Erbay et al. [1993a,b], and Kimura et al. [1994]). In LPE, the mass transfer between the crystal substrate and the surrounding liquid causes a change in the density of the solution which gives rise to natural convection. For a sufficiently large solution height, convection enhances the mass transport rate in the solution near the upper substrate during growth, whereas it increases the mass transport rate in the vicinity of the lower substrate during dissolution. The temperature modulation technique relies on the beneficial use of solutal convection. Effective control of convection in LPE growth of bulk crystals is crucial for its success. Most of the theoretical studies of LPE growth of ternary alloys of the time were based on the assumption that epitaxial growth is controlled by diffusive mass transport (e.g., Small and Ghez [1979], Ghez and Small [1982], Nakajima [1990], and Kimura et al. [1995]). In order to simulate the “yo-yo” growth process accurately the incorporation of convection in the model is necessary. Qin et al. [1996b] has extended the earlier diffusion model of Kimura et al. [1995] to include natural convection. This study was the first comprehensive numerical study of solutal convection in the LPE growth and dissolution processes of semiconductor alloys. It has served as a basis for numerous numerical simulations performed on LPEE and THM growth of ternary alloys (see Chapters 6 and 7). Next we present a model (Qin et al. [1996b]) which accounts for diffusive and convective mass transport and fluid flow in the liquid, diffusive mass transport in the solid, and phase equilibrium and mass conservation at the

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interfaces. The finite element method is used to solve the governing equations numerically. Simulations are carried out for the LPE growth of GaInAs by the temperature modulation technique. 5.5.2. Field Equations In addition to the assumptions made in Section 5.4, since LPE growth or dissolution is often realized through programmed gradual temperature changes, and these processes are very slow and also the solution is a good heat conductor, we assume that the temperature distribution is approximately uniform throughout the system, but its value changes with time. These temperature changes do not affect the concentration distribution and fluid flow significantly; they mainly alter the phase relations at the crystal–liquid interface. Because they displace equilibria, they can be viewed as external driving forces that act simultaneously with internal forces due to constitutional nonequilibria, i.e., the system initial compositions. In general, macroscopic dissolution and growth of an epitaxial layer are affected by mass and heat transfer as well as fluid flow. Since the temperature of the solution is increased or decreased very slowly during dissolution and growth, in simulations these processes are therefore assumed to be isothermal. Since the velocity of the growth interface is very small in LPE, the contribution of latent heat to the energy balance at the interface is not significant, and therefore is also neglected. Furthermore, the solution is assumed to be an incompressible, Newtonian fluid, and the wellknown Boussinesq approximation is adopted. Under these conditions, the system of equations governing the processes consist of only the unsteady, two–dimensional, incompressible fluid flow and mass transport equations. The energy balance is satisfied identically. The fluid flow equations take the following form in Cartesian coordinates:

u v + =0 x y

(5.5.1)

u u 1 p  2u  2u  2 v u +u +v = +  (2 2 + 2 + ) x y t xy  L x x y

(5.5.2)

v v 1 p 2 v  2 v  2u v +u +v =  + ( 2 + 2 2 + ) x y t xy  L y x y +g[ A (x A  x 0A ) + C (xC  xC0 )]

(5.5.3)

Note that the continuity equation is embedded into the momentum equations, Eqs. (5.5.2) and (5.5.3), for numerical convenience. The mass transport equations take the following forms: In the liquid phase

Single Crystal Growth of Semiconductors from Metallic Solutions

x A t xC t

+u +u

x A x xC x

+v +v

x A y xC y

=

 D AL (

2

xA

)

(5.5.4)

)

(5.5.5)

D As ws x sA ws x sA ( + ) y y ws x x

(5.5.6)

x 2

= DCL (

+

2 x A

167

 2 xC x 2

+

y 2  2 xC y 2

In the solid phase

x sA t

= D As (

 2 x sA x 2

+

 2 x sA y 2

)+

where ws is the mole density of the solid phase, and for most III-V alloys, it can be associated with the solid composition, x, in the following form:

8 = ws N Av [xd AC + (1 x)d BC ]

(5.5.7)

where N Av is the Avogadro number, and d AC and d BC are the lattice parameters of components AC and BC, respectively. It was assumed in Eq. (5.5.7) that the lattice parameter of an alloy d AC  BC changes linearly from d AC to d BC . The above equations will be complete with the addition of the phase diagram equations given in Eqs. (5.4.8)-(5.4.9), and the interface/boundary conditions given in Section 5.2.4. They are solved by the Finite Element Method (FEM). 5.5.3. Finite Element Solution The modeled LPE process involves mass transport in the liquid and solid phases, fluid flow in the liquid phase, and phase equilibrium and mass balance at the moving interfaces. The governing equations are highly nonlinear and field variables such as velocity, pressure and concentrations, are coupled. The solution of these types of problems requires great computational effort. An efficient and reliable numerical method is necessary for successful transient simulations. The finite element method based on the penalty function formulation is used to solve the field equations. The advantage of the penalty function formulation is that the additional flow variable pressure is eliminated and so is the need for solving the continuity equation, which avoids the wellknown difficulties of the mixed velocity/pressure formulation in incompressible flows and makes it easy to implement the complicated problem (Qin et al. [1995a,b, 1996]). The Galerkin method and 4-node quadrilateral elements are used to discretize the governing equations. The resulting set of first-order simultaneous ordinary differential equations with time derivatives are further discretized by the fully implicit time-marching algorithm based on the finite difference method. The non-linear algebraic equations resulting from the

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implicit approximation at each time step are solved by the Newton-Raphson iteration. Table 5.5.1. Physical parameters used in the growth of GaInAs (Qin et al. [1996b]). Parameter

Value

Solution viscosity; 

1.0  10-3 cm2/s L

L

Solution Diffusion Coefficients: D A , DC s

8.2  10-5 cm2/s

Solid Diffusion Coefficient: D A

1.0  10-12 cm2/s

Solutal Expansion Coefficient: A

-0.12

Solutal Expansion Coefficient: C

-0.19

Mole density of the solution: w

0.0596 mol/cm2

Lattice parameter of GaAs: d AC

5.6533 

Lattice parameter of InAs: dBC

6.0584 

Since the physical parameters are very different in the solid and the liquid phases, a separated solution procedure is applied to avoid numerical difficulties due to large difference in the matrix elements, i.e. the governing equations are solved separately for the solid and liquid phases. These two bulk phases are however coupled at the interface, requiring the satisfaction of phase equilibrium and mass conservation at the interfaces. This necessitates an iterative procedure y Temperature (°C) single crystal InAs substrate

t4

t1

t2

t3

t4

Tg= 700

In-rich Ga-In-As solution

h g

Td= 690

single crystal GaAs substrate

One Cycle = 120 min t1=40min, t2=20min, t3=20min, t4=40min

x Time

L

(b) (a)

Fig. 5.5.1. Schematic view of the LPE growth configuration (a) and the time history of the temperature modulation (b) for the growth of Ga0.1In0.9As crystals (redrawn from Qin et al. [1996b]).

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Fig. 5.5.2. Evolution of velocity field during growth: (a) t = 10 min, (b) t = 20 min, (c) t = 30 min (after Qin et al. [1996b]).

to solve Eqs. (5.4.8) and (5.4.9), and the equations of phase diagram for interface concentrations and the growth rate, which are determined by the transport rates in the liquid and solid phases and phase equilibrium. Therefore, the computations in the bulk phases and at the interfaces must be performed iteratively to obtain a convergent solution for each time step. Upon the convergence of the solution, the growth thickness is readily computed by integrating the growth rate. Then the finite element mesh moves to track the moving interfaces. The transient solution of the governing equations for LPE growth of ternary alloys is very time consuming. A combined full and modified Newton-Raphson iteration scheme is used, namely, the full Newton-Raphson method is used in the first iteration for each time step and the modified Newton- Raphson method is used during the subsequent iterations until the solutions for the bulk phases and the interfaces have converged. This solution scheme significantly reduces computation time, since the modified Newton-Raphson method requires fewer

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reformations and factorizations of the tangent stiffness matrix and still provides a reasonable convergence speed. The overall computational procedure consists of the following steps: 1. Set initial growth configuration and finite element mesh. 2. Calculate initial composition of the solution using phase diagram. 3. Guess initial interface condition based on the thermodynamic model. 4. Start time integration. 5. Perform Newton-Raphson iterations for finite element solutions of the bulk phases. 6. Calculate interface concentrations and growth rate. 7. Check convergence of the solutions for the bulk phases and the interfaces. 8. Modify interface conditions and return to step 5 if convergence is not achieved. 9. Update finite element mesh and forward to next time step if convergence is achieved. 5.5.4. Simulation Results The simulations are carried out for a horizontal sandwich system consisting of a substrate- solution-substrate arrangement for growth of Ga0.1In0.9As from an In-rich solution. The upper and lower substrates are single crystals of InAs and GaAs, respectively. In a cycle of the temperature modulation technique, the temperature decreases in the first stage to allow deposition of solutes on the substrates, then it is held constant to stabilize the solution, followed by an increase of temperature to feed the depleting solution. The growth cell configuration and the temperature history are shown in Fig. 5.5.1. We first discuss the effect of natural convection on growth and dissolution and present velocity and composition fields. Then we consider growth rate and solid composition variations in a temperature modulation cycle. The initial temperature is 700°C and the initial solution composition is determined by the L L phase diagram as xGa = 0.0194 , x InL = 0.8732 and x As = 0.1074 . The cooling rate during growth is 0.25°C/min and the heating rate during dissolution is 0.5°C/min. The physical parameters used in the simulations are listed in Table 5.5.1. As mentioned earlier, the temperature modulation technique includes a number of growth and dissolution processes in a cyclic manner. In a cycle, as shown in Fig. 5.5.1, crystal growth takes place during the first period when the temperature drops with a constant rate and dissolution occurs during the third period when temperature increases linearly with time. In the growth process, the deposition of the lower density solutes on the substrates increases the density of the solution in the vicinity of the substrates. Therefore, unstable stratification develops in the upper region, leading to buoyancy-induced convective flow, which in turn affects mass transport in the solution. In the dissolution process, the density of the solution in the vicinity of the substrates is reduced by the

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Fig. 5.5.3. Evolution of concentration patterns during growth: (a) t = 10 min, (b) t = 20 min, (c) t = 30 min (after Qin et al. [1996b]).

dissolved solutes. This results in unstable stratification in the lower region, causing convective flow. The evolution of the velocity field during the growth process is shown in Fig. 5.5.2. The flow cells are mainly located in the vicinity of the upper substrate and move along the interface. The convective flow in the solution results in higher concentration gradients near the upper substrate. Fig. 5.5.3 shows the isoconcentrations of Ga in the solution. The concentration patterns for As are similar since the same diffusion coefficient is used for the solutes in the simulation. The higher concentration gradients at the upper substrate interface result in a faster growth rate for the upper substrate than for the lower substrate. The evolution of the velocity field for the dissolution process is shown in Fig. 5.5.4 (concentration patterns are not given for the sake of space, see Qin et al. [1996]). Contrary to the growth process, the flow cells are mainly located in the vicinity of the lower substrate which result in higher concentration gradients near the lower substrate. Thus the average dissolution rate for the lower substrate is much larger than that for the upper substrate.

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Fig. 5.5.4. Evolution of velocity field during dissolution: (a) t = 65 min, (b) t = 70 min, (c) t = 75 min (after Qin et al. [1996b]).

In addition to enhancing the mass transport rate, convection also influences the composition uniformity of the grown crystals. Fig. 5.5.5 shows a comparison between the composition profiles computed by the diffusion and convection models of LPE (with a constant cooling rate). As expected, the composition profiles of the grown crystals are significantly graded due to the continuous cooling and the solute depletion as the growth proceeds. Since convection brings additional solutes to the vicinity of the growing substrate and gives rise to effective mixing of the solution, the convection model predicts that convection leads to a better composition uniformity in the grown crystals. The computed growth rates, averaged along the substrate surfaces, are shown in Fig. 5.5.6 for one yo-yo cycle. At the beginning of the growth process, growth rates increase rapidly and remain identical for both substrates, indicating that growth is controlled by diffusion. After the onset of natural convection, bifurcation of the growth rate curves for the upper and lower substrates takes place and the growth rate of the upper substrate increases continuously. After the development of convective flow, the growth rate varies about an almost

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Fig. 5.5.5. Comparison of computed solid composition profiles (after Qin et al. [1996b]).

constant average value with small amplitudes, and nearly steady state growth takes place. The oscillatory patterns of the average growth rate curves are due to the nonuniform growth rates along the substrates, which are attributed to the unsteady convective flow. The temperature is held constant before solute elements are depleted in the solution. Growth rates decrease rapidly in the initial stage and stop when the solution is stabilized. During the heating of the solution, dissolution rates vary in a similar path as the growth process. After an initial

Fig. 5.5.6. Time history of average growth rates for upper and lower substrates (after Qin et al. [1996b]).

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Fig. 5.5.7. Time history of average layer thickness for upper and lower substrates (after Qin et al. [1996b]).

diffusion-controlled period, the dissolution rate curves of the two substrates are separated by convection. The more oscillatory average dissolution rate of the lower substrate indicates the strong effect of the unsteady convective flow. Since the heating rate is twice that of the cooling rate, the overall dissolution rate during the heating of the solution is faster than the growth rate during the cooling of the solution. The variation of layer thickness with time is shown in Fig. 5.5.7. It is clear that the upper substrate grows more when the temperature decreases and dissolves less when the temperature increases, whereas the lower substrate shows the opposite tendency. In an entire cycle, the upper substrate grows about 16 μm while the lower substrate dissolves about the same amount. This is very close to an experimental observation which is about 14 μm (Kimura [1995]). The lower substrate plays in fact the role of source material during LPE growth by the temperature modulation technique. The composition variation in the grown crystal after 3 cycles is shown in Fig. 5.5.8. It can be seen that the composition varies slightly in a cycle corresponding to the temperature drop and keeps similar profiles in the following cycles. The composition in the new layer grown by the LPE temperature modulation technique is almost uniform, which is consistent with experimental observations (Kimura et al. [1996a]). One can state that the numerical model presented in this section is applicable to all III-V ternary systems. The simulation results demonstrate that natural convection plays a crucial role in liquid phase epitaxial growth of ternary alloys. Natural convection influences growth and dissolution rates, and compositional uniformity of the grown crystals. The temperature modulation technique takes

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Fig. 5.5.8. Composition variation of grown crystal by temperature modulation (after Qin et al. [1996b]).

advantage of natural convection, and makes it possible to grow thick layers with uniform compositions. 5.5.5. Structure of the Liquid Solution in LPE As discussed in the previous section, experimental and numerical studies have clearly shown that gravity plays a significant role in LPE growth of Si, GaAs, and GaInAs crystals by the Yo-Yo technique (Sukegawa et al. [1988, 1991a,b], Kimura et al. [1990, 1994, 1996], and Qin et al. [1996]). As seen from Figs.5.5.2-5.5.4, convective cells developed in the liquid solution enhance growth on the upper substrate during the growth period (cycle) while dissolving more material from the lower substrate during the dissolution period. This was attributed, correctly, to the effect of gravity, which causes the lighter components to rise in the solution cell. Of course, this was based on assuming that the structure of the liquid solution remains elemental, in other words, components move in the elemental forms and there is no association between them (solute(s) and solvent). If we assume that the components of the solution remain elemental, this explanation is true in the case of Si grown from either an In or Sn solution since Si is lighter than both In and Sn. GaAs and GaInAs crystals were grown from a Ga-rich solution. The explanation is also valid in this case since Ga is heavier, though only slightly, than both In and As. Based on this understanding, using the LPE sandwich system, Kanai et al. [1997] grew GaSb crystals from a Ga-rich solution, with the expectation that this time contrary to what was observed in the growth of Si or GaAs, growth would be more on the lower substrates that on the upper substrate since in this

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Fig. 5.5.9. Flow field, and concentration patterns for Cd at t = 2 min. Initial solution composition is in equilibrium with solid phase CdxHg1-xTe; x = 0.187 at T = 782 K. CdTe substrate is brought into contact with the solution at T0 = 788 K, i.e. the amount of superheating is T = 6 K. Contact of the substrate with liquid phase is followed immediately by cooling at the cooling rate of 1.0 K/min (after Denison et al. [2004]).

case Sb (solute) was heavier that the solution (Ga-rich). However, this did not happen, and the thickness of the grown layer on the upper substrate was still larger than on the lower substrate, similar and comparable to the growth of GaAs. In order to observe such a phenomenon, the solute must have a smaller density than that of the solvent. The density of liquid Sb at 500°C is estimated approximately as 6.55 g/cm3 which is larger than that of liquid Ga, 5.80 g/cm3 (Smithells [1976]). If the components are in the elemental form in the solution, the following logic should work out. Assuming that the density of the Ga-Sb solution increases linearly with the increasing concentration of Sb, and varying from the value of the liquid density of Ga to the liquid density of Sb, the density of the Ga-Sb solution would be larger than that of the liquid Ga. Since the density of molten GaSb at the melting point is 6.60 g/cm3 (Meskimin et al. [1968], there might be attraction between Ga and Sb atoms. In this case the density of the liquid Ga-Sb is still larger. Then the dissolution of GaSb would occur mainly on the upper substrate, while the growth on the lower substrate would be much larger than that on the upper substrate. However, the experiments of Kanai et al. [1997] did not agree with this argument. So the question is then: what was really happening? The only logical explanation is that the components in the liquid solution must not be in their elemental forms, instead, Ga and Sb may form an “associated” structure

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Fig. 5.5.10. Flow field and concentration patterns at t = 4 min, and the dissolution and growth interfaces at respectively t = 2 min and t = 4 min (after Denison et al. [2004]).

somewhere between liquid and crystal. Indeed, the density of solid GaSb at 500°C is 5.57 g/cm3 (Smithells [1968] and Glazov [1968]) which is smaller than that of liquid Ga. If the solute forms Ga-Sb molecules with a “crystal-like”, associated structure in the solution, the density of the Ga-Sb solution will be smaller than that of the liquid gallium at the same temperature. This theory would then explain the experimental results of Kanai et al. [1997]. It appears that in the absence any other explanations, in this LPE process the Sb species are transported in the solution in the form of an associated structure of Ga-Sb even in the under-saturation condition. 5.5.6. Growth of CdxHg1-xTe The role of convection was also investigated numerically by Denisov et al. [2000, 2004] during the LPE growth process of CdxHg1-xTe crystals from a Terich solution (Denisov et al. [2002]). In order to minimize the adverse effect of convection, the solution height was selected very small at 2 mm. The twodimensional model equations, and the associated boundary and initial conditions and model assumptions used are almost the same given in the previous sections. However, in this model diffusion in the solid is neglected. The field equations were given using a stream function formulation, and were solved using a finite-volume-based technique. The growth rate and the thickness of the grown layers were calculated from the mass balance at the interface (see Denisov et al. [2000, 2004] for details). The physical parameters used in simulation are presented in Table 5.5.2. Here we present some of the results.

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Table 5.5.2. Physical parameters used in the CdHgTe system (after Denisov et al. [2004]). Parameter

Value

Solution viscosity; 

2.8410-3 cm2/s L

L

Solution Diffusion Coefficients: DHg , DCd

5.010-5 cm2/s

Density of liquid:  L

6.567 gr/cm2

Density of solid:  s

(8.10-2.25x) gr/cm2

Solutal Expansion Coefficient for Cd:  Cd

0.15 (mol.frac)-1

Solutal Expansion Coefficient for Hg:  Hg

0.08 (mol.frac)-1

In order to illustrate the contribution of natural convection to the dissolution process, the CdTe substrate is brought in contact with a solution heated above the liquidus temperature. Dissolution of the substrate begins in an undersaturated liquid phase, and lasts for 2.5 min. During this period, density stratifications established in the solution gives rise to natural convection (Fig. 5.5.9). Gradually, as the solution reaches equilibrium, the convection gets weaker and the growth becomes almost diffusion-limited (Fig. 5.5.10). From t = 2 min to t = 4 min, convection weakens by an order of magnitude. However, an interface of a wavy shape of the grown layer appears to be a result of the influence of convection during the dissolution period (Fig. 5.5.10). Simulation results show that during dissolution a superheating of 1–2 K leads to an interface non-uniformity less than 1 μm for a solution depth between 0.15 and 0.3 cm. Nonuniformity increases with the increasing undersaturation and solution height values, as shown in Fig. 5.5.11, since they make convection stronger.

Fig. 5.5.11. Variation of the nonuniformity of the dissolution interface with initial superheating (from Desinov et al. [2004]).

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Fig. 5.5.12. Variation of the nonuniformity of the dissolution interface with cooling rate (after Desinov et al. [2004]).

However, it is shown that higher cooling rates reduce the time the substrate and the undersaturated solution remain in contact (Fig. 5.5.12). This leads to smoother growth interfaces as expected. Consequently, the cooling rate also affects the thickness of the dissolved substrate, and hence the composition and thickness of the growing layer. The time history of dissolution and growth thicknesses is shown in Fig.

Fig. 5.5.13. Time history of dissolution and growth thicknesses in the middle of the substrate (for H = 0.15 cm, L = 2.5 cm, and initial superheating T = 6 K). The system is cooled at a given rate from T0 = 788K to T1 = 780K. and then kept at the constant temperature T 1. The initial solution composition is in equilibrium with solid phase CdxHg1-xTe: x = 0.187 at T = 782 K (after Desinov et al 2004).

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Fig. 5.5.14. Composition distribution in the grown layer (middle of the substrate) (after Denisov et al. [2004]).

5.5.13. In this figure the vertical axis represents the dissolution or growth depths. Thus the decreasing depth corresponds to dissolution, while the increasing value corresponds to growth. As shown in Fig. 5.5.14, the amount of substrate material supplied to the solution during the dissolution process affects the composition of the growing film. The composition of the growing layer changes only when the system is cooled. As stated by Desinov et al. [2000, 2004], an LPE growth at a constant temperature, just by leaving the system to reach equilibrium, produces a uniform solid composition. Lower cooling rates result in higher CdTe concentrations in the grown layer, and in addition, slow cooling increases the amount of Cd dissolved in the solution. 5.6. The Conversion Phenomenon in LPE III-V ternary and/or quaternary semiconductor alloy crystals are of vital importance in fabrication of optoelectronic and high-speed devices since the desired band-gap energy and lattice constants can be achieved by controlling their composition. However, the misfit between the alloy layer and the available binary substrate still remains the main problem. In the design and fabrication of new alloy devices, the availability of alloy substrates with controlled lattice parameters can remove the limitation of the lattice-matching condition with simple compound substrates. As covered in detail in Chapter 3, numerous experimental studies on bulk alloy crystals have been carried out to address this issue using various new concepts (for instance, see Nakajima et al. [1986], Sukegawa et al. [1991, 1993], Kusunoki et al. [1991], Udono et al. [1993], Tanaka et al. [1994], and Kodama et al. [1998]).

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Fig. 5.6.1. Schematic view of the computational domain (after Kimura et al. [1996, 1997]).

It was observed by Sukegawa et al. [1993] and Udono et al. [1993] that a GaAs layer grown on a GaP substrate changes its composition to GaAsP when it is put in contact with a Ga-As-P solution. This conversion stops at the surface of the GaP substrate, leaving a GaAsP layer on GaP. The desired final composition of GaAsP is achieved by adjusting the liquid composition of the solution. Using this conversion technique, the GaAsxP1-x alloy layers with the middle composition range of 0.25< x <0.75 were easily obtained on GaP substrates. These experimental results suggested the possibility of preparing an alloy layer lattice-mismatched to a substrate. The growth following the conversion process has confirmed the possibility of obtaining a conversion layer as an alloy substrate for optoelectronic devices. Further, as mentioned earlier it has been demonstrated by applying the “yo-yo” solute feeding method of Sukegawa et al. [1988, 1991], that thick layers of GaAsP alloy with a uniform composition profile can be grown on GaP substrates, making this technique very promising in the preparation of alloy substrates. The above-mentioned studies were followed by a number of experimental and numerical studies on the conversion of various alloys (Kimura et al. [1996, 1997], Motogaito et al. [1997]). In this section we present the numerical simulations conducted in order to have a better understanding for the Conversion Method. 5.6.1. Conversion of GaAs to GaAsP on a GaP Substrate The conversion process is relatively simple compared with other epitaxial growth techniques. However, the phenomena occurring during the conversion process seem to be quite complex. As frequently observed in the LPE growth of an alloy on a binary substrate with a lower melting point than that of the growing alloy, such as GaInAs on InAs and GaInSb on InSb, Ga atoms in the growth solution diffuse rapidly into the unstable weak-bonded lattice of the substrate to relax the nonequilibrium state to form the alloy in the substrate (Bolkhovityanov [1981, 1983]). Especially in the case of diffusion of Ga into InSb substrates, the diffusion coefficient of 108 - 107 cm2/s was measured at 506°C (Hayakawa et al. [1994, 1996]). This value is extremely high compared with self-diffusion coefficients of In and Sb in InSb, which range from the order

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Fig. 5.6.2. Variation of dissolution rate with time (after Kimura et al. [1996, 1997]).

of 1016 to 1014 cm2/s in the temperature range of 475–517°C (Kendall and Huggins [1969]). Based on these facts, P and As will have larger diffusion coefficients in the GaAs layer than those in the GaP substrate when the GaAs layer is placed in contact with a Ga-As-P solution, since the alloy, which would be in equilibrium with the Ga-As-P solution, has a higher melting point than GaAs. Furthermore, in the conversion process, dissolution of GaAs must occur because of the nonequilibrium condition between the GaAs layer and the GaAs-P solution, unlike the situation observed in the AlGaAs/GaAs system of Kimura et al. [1996c]. All these show the complexity of the conversion process. The simulation is based on the solid-liquid diffusion model of Kimura et al. [1996] which was presented in Section 5.4, however, the governing equations of the conversion model also include changes in the solid mole density (as in the convection model of Qin et al. [1996] presented in Section 5.5), making the governing equations more complex, and in turn, the numerical procedure more challenging. The finite element method is used to solve the governing equations numerically. A schematic view of the computational domain is shown in Fig. 5.6.1. The coordinate system is selected on the GaAs layer, and the solid phase represents both the conversion layer and GaP substrate. The liquid phase is the Ga-As-P liquid solution. The model considers a III-V-V ternary system AByC1-y where A, B, and C represent respectively Ga, As and P, and correspondingly the liquid compositions x A , x B , and xC , and the solid compositions x sA = 0.5 , x Bs = 0.5y , and xCs = 0.5(1 y) . We select x B (As) and xC (P) as the independent variables in the liquid phase, and xCs as the solid composition. In Section 5.4 we assumed that the total mole densities in the liquid and solid phases were constant. This assumption is still reasonable for the liquid phase

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Fig. 5.6.3. Variation of dissolution depth with time (after Kimura et al. [1996, 1997]).

since the solute concentrations are very small, and the density of solution is almost equal to that of the solvent. In the solid phase, this assumption may be valid if the lattice parameters change slowly with compositional variation, such as AlGaAs. The lattice parameter, however, changes significantly from AB (GaAs) to AC (GaP) for the GaAsP system. The mole density is a function of solid composition, and it changes considerably in different layers of a heterostructure. Therefore the change in the solid mole density is taken into account in this model. Inclusion of these effects makes the governing equation of the solid phase more complicated since the lattice parameters are changing during the process. In light of the above remarks, the mass transport equations given in Eqs. (5.5.4)-(5.5.6) for the conversion model simplify to

x B t

= DBL

2 x B y 2

,

and

xC t

= DCL

 2 xC y 2

,

(5.6.1)

in the liquid phase, and to

xCs t

= DCs

 2 xCs y 2

+

DCs ws xCs ws y y

(5.6.2)

in the solid phase. The molar density will be related to the solid composition by Eq. (5.5.7), and the interface mass balance conditions are the same as those given in Eqs. (5.4.5) and (5.4.6). Eqs. (5.6.1) and (5.6.2) together with two phase diagram equations given in Eqs. (5.4.8) and (5.4.9) in terms of interface concentrations, are solved by the Finite Element Method. The numerical procedure is the same discussed in Section 5.4. We assume that a 3μm thick GaAs layer on a GaP substrate is

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put in contact with a Ga-As-P liquid solution which would be in equilibrium with a ternary alloy of GaAsyP1-y (0.3< y <0.7) at 800°C. Physical parameters used in this study are given in Table 5.6.1. Table 5.6.1. Parameter used in the conversion model (Kimura et al. [1996, 1997]). Parameters

Values

Conversion temperature

800 °C

Thickness of GaAs layer

3 μm

Solution height: L

4 mm L

Diffusion coefficient of As in solution: DB L

Diffusion coefficient of P in solution: DC

s

Diffusion coefficient of GaAs in solid: DC s

Diffusion coefficient of GaP in solid: DC

410-5 cm2/s 410-5 cm2/s 110-10 cm2/s 110-15 cm2/s

The time-dependence of the computed dissolution rate is presented in Fig. 5.6.2 where the initial liquid composition, represented by the solid composition ym in equilibrium with the liquid composition, is used as a parameter. Once the Ga-As-P solution is put in contact with the GaAs layer, the dissolution of the GaAs layer is initiated, since the Ga-As-P solution is not in equilibrium with GaAs. Compared with the case of the AlGaAs/GaAs system presented in Section 5.4, this non-equilibrium condition between the solution and the GaAs layer is quite large due to lattice mismatch, and therefore promotes faster dissolution.

Fig. 5.6.4. Variation of As concentration in the solution with time (ym = 0.5) (after Kimura et al. [1996, 1997]).

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Fig. 5.6.5. Variation of P concentration in the solution with time (ym = 0.5) (after Kimura et al. [1996, 1997]).

Initially the dissolution rate is quite large, however, it decreases abruptly with time, and almost decays after 400s. Features of the dissolution process occurring during conversion can be better understood by examining the variation of dissolution depth with respect to time (Fig. 5.6.3). The dissolution depth is increasing initially, however, becomes gradually saturated after 400s. This implies that, in the early stage, the sudden change of compositional profile occurs in the GaAs (GaAsP conversion) layer, and the conversion process is almost completed in 400s. The total dissolution depth is around 1.5μm and the thickness of the GaAsP conversion layer is approximately half of the original GaAs thickness. It is clear that the dissolution depth becomes larger for higher ym. This is expected since the initial liquid solution with smaller ym is farther from the equilibrium with GaAs layer. Fig. 5.6.4 and Fig. 5.6.5 show respectively the time variation of As and P concentration distributions in the solution for the case of ym = 0.5. In these figures, lines for 0s represent initial uniform concentration distributions. As clearly seen from these figures, upon the solution being put in contact with the substrate, the initial nonequilibrium between the solution and substrate forces the As atoms from the substrate into the solution, and the P atoms into the solid from the solution. In spite of the drastic change of dissolution depth in the early stage, the interface atomic fractions of As and P are pinned to certain points until 180s. As the process proceeds, these concentration distributions of As and P become uniform after 600s, and are almost constant at 1800s. This implies that the condition of the system changes towards the final state where the GaAsP converted layer is in quasi-equilibrium with the Ga-As-P solution. Fig. 5.6.6 shows the time variation of the composition distribution in the solid phase for the case of ym = 0.5. As expected, the composition profile changes

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rapidly and becomes uniform in the GaAsP conversion layer after 600s. In other words the conversion is almost complete after 600s. This result is consistent with the experimental observations of Sukegawa et al 1993 and Udono et al 1993. In addition, the surface composition changes rapidly in the early stage of the process and hardly varies afterwards. These results suggest that the surface composition is constrained mainly by the phase equilibrium condition and is almost fixed around ym. Then the composition gradient in the vicinity of the surface is acting as a driving force for diffusion of As and P. During this process, the conversion layer is being dissolved, and the GaAsP conversion layer with half thickness of the GaAs is finally formed. Eventually the conversion process is initiated by the non-equilibrium condition between the solution and the solid, and promoted by the rapid diffusion of V elements in the GaAs layer. It should be noted that this process takes place only in the conversion layer, and the GaP substrate is not affected during the process. Furthermore P atoms in the GaAsP conversion layer are not supplied from the GaP substrate, but from the Ga-As-P solution. Fig. 5.6.7 shows the computed solid compositions of GaAsyP1-y compared with the experimental results of Udono et al. [1993]. In these experiments, the solid composition was determined by X-ray diffraction and EPMA measurements. The GaAs layer is dissolved due to the nonequilibrium condition at the interface. Further constraints of phase equilibrium enhance the outdiffusion of As. As concentration in the solution then becomes larger than that corresponding to ym. This causes a higher As composition in the GaAsP layer in the final quasi-equilibrium condition. Differences in the interface concentrations of As and P between 1800s and 0s correspond to the difference in the solid

Fig. 5.6.6. Time variation of solid composition in the GaAsyP1-y conversion layer (ym = 0.5) (after Kimura et al. [1996, 1997]).

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Fig. 5.6.7. Comparison of the computed solid composition with experimental data of Udono et al. [1993] (after Kimura et al. [1996, 1997]).

composition, y-ym. As seen from Fig. 5.6.7, the numerical simulation results are in good agreement with the experimental data, predicting relatively well the experimental results obtained from the conversion process of GaAs to GaAsP. However, one cannot help but notice that the predicted value of the solid composition y is slightly lower than the experimental value of y for each case. This small but consistent difference between the experimental and simulation results is probably due to the fact that some experimental conditions have not been taken into account in the simulation. For instance, the influence of vapor pressure of P, which was not considered in the simulation, may well be the main factor for observing such a small discrepancy. 5.6.2. Growth of GaInP Crystals on GaP by Conversion of InP GaInP is one of the important materials that is used for fabricating visible light emitting devices or ultra-fast devices. The heterostructure of Ga0.5In0.5P alloy on a GaAs substrate can be used for systems of HBT and HEMT. Because of the difficulties involved in the LPE growth of alloys of different compositions on commercially available substrates, as mentioned earlier the conversion technique offers a feasible solution (Sukegawa et al. [1993], Udono et al. [1993], Motogaito et al. [1997]). The conversion technique has been used by Motogaito et al. [1997] to grow GaInP alloys with desired compositions. In this study, an InP layer was first grown on a GaP substrate, with a 7.5% lattice mismatch. Then the InP layer

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was put in contact with a Ga-In-P solution in isothermal conditions. After a period of time, the InP layer changed its composition to GaInP. The experimental setup and the temperature program used are shown in Fig. 5.6.8. The LPE setup was vertical and consisted of two main sections. The upper section was filled with the Ga-In-P solution. In the bin, 5.0 gr of In, InP and Ga were loaded into the Ga-In-P solution. The solution composition was adjusted according to the Ga-In-P ternary phase diagram so as to keep the solution in equilibrium with the GaInP alloy to be grown. The substrate was placed in the lower section. Prior to the conversion process, an InP layer was grown on a 1213 mm2 GaP (111)B substrate. The grown layer had about 2.5at% Ga (5 mol% GaP). After raising the temperature to Tc, the Ga-In-P solution was kept

Fig. 5.6.8. Schematic view of the growth system (a) and the temperature program (b) (after Kimura et al. [1996, 1997]).

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at a constant temperature for 1.5 hours (represented by Region A in Fig.5.6.8). Then the InP epitaxial layer was brought into contact with the Ga-In-P solution by rotating the substrate, and kept at Tc (Region B). After a period of time, the solution was removed by rotating the sliding boat. Table 5.6.1. Parameters used in simulation (Motogaito et al. [1997]). Parameter

Value

InP layer thickness

24 μm

Solution height (L)

6 mm

Diffusion coefficient of Ga in the solution

2.010-4 cm2/s (Pan et al. [1986])

Diffusion coefficient of P in the solution

2.510-4 cm2/s (Pan et al. [1986])

Diffusion coefficient of Ga in the solid InP

1.010-7 cm2/s

Diffusion coefficient of Ga in the solid GaP

1.010-15 cm2/s

Ga-In-P liquid solution zone

y x

InP (Ga0.05In0.95P) layer GaP substrate

Fig. 5.6.9. Computational domain (redrawn from Motogaito et al. [1997]).

Fig. 5.6.10. A cross-section view of the conversion layer (after Motogaito et al. [1997]).

The above described process has been numerically simulated. The governing mass transport equation and the associated boundary conditions are the same given in the previous section. The schematic diagram of the computational domain is shown in Fig. 5.6.9. The simulation model assumes that the total mole densities in the liquid and solid phases are constant, and the liquid phase is initially in equilibrium with an alloy of specified composition, and the initial condition is therefore determined by the phase diagram based on the conversion temperature and alloy composition. The solid phase is the 24-μm thick InP layer and the GaP substrate as shown in Fig. 5.6.9. The governing equations are solved by the finite element method.

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Fig. 5.6.11. X-ray 333 diffraction curve of the GaInP converted layer on the GaP substrate (after Motogaito et al. [1997]).

The physical parameters used in the simulation are given in Table 5.6.1. The diffusion coefficient of Ga into InSb was measured in the order of 10-8-10-7 cm2/s at 506 °C (Hayakawa et al. [1994, 1996]). This value is very high compared with the self-diffusion coefficient of In or Sb in InSb, which is in the order of 10-16-10-14 cm2/s in the range of 475-517 °C (Kendal and Huggins 1969). The diffusion coefficient of Ga in the InP layer was estimated by the Ga profiles obtained from an experiment using InP(100) substrates which were put in contact with a Ga-In-P solution at 750 °C for 1 min. From EPMA profiles of Ga, the diffusion coefficient of Ga was estimated based on the assumption of a Gaussian distribution. Its value is much larger than the self-diffusion coefficients of In and P in InP, which vary in the order of 10-15-10-13 cm2/s at

Fig. 5.6.12. Time-variation of Ga concentration in the solution (after Motogaito et al. [1997]).

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750 °C (Goldstein [1961]). It is possible that the diffusion of Ga is causing distortion of the In-P lattice. Diffusion under this condition becomes much faster. The InP layer contained 2.5at% Ga (5 mol% GaP), because of the meltback of the GaP substrate at the earlier stages of growth. For the sample shown in Fig. 5.6.10, the conversion temperature (Tc) was 850 °C, and the conversion time was 3 hours. The thickness of the InP layer was 25 μm. The Ga-In-P solution was in equilibrium with the alloy composition of 0.7 GaP mole fraction at 850 °C in the phase diagram. A rapidly cooled layer can be seen on the top of the conversion layer. This is due to the fact that the solution could not be removed completely at the end of the conversion process. The interface between the substrate and the conversion layer is almost flat. The EPMA measurement made in the direction of cross-section shows that the composition is almost constant, and that the InP layer was converted to a Ga0.74In0.26P layer on the GaP substrate. Fig. 5.6.11 shows the x-ray diffraction curve measured using 333 diffraction of the CuK line ( = 1.54056 ). As seen, two peaks were detected: one was 333 diffraction from the GaP substrate and other one from the GaInP conversion layer, with a shoulder corresponding to the rapidly cooled layer not removed completely. No other peaks were detected by the wide-angle diffraction, which concluded that the converted GaInP layer is a single crystal. The alloy composition calculated from these peaks by Vegard’s law agreed with that of the EPMA measurement. A peak of an InP layer was not detected. This means that the InP layer was completely converted to GaInP at the end of the 3h conversion period. 1.0

Solution composition (x)

GaP substrate

GaInP conversion layer

0.8 1800

dissolution 600 s

0.6

300 s 60 s

0.4

10 s

0.2

0s

0.0 -30

-20

-10 Distance (μm)

0

Fig. 5.6.13. Time variation of the computed solid composition in the GaxIn1-xP conversion layer (xm = 0.7) (redrawn from Motogaito et al. [1997]).

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The computed Ga concentration distribution in the solution is presented in Fig. 5.6.12 (for xm = 0.7). The line 0s represents the initial uniform concentration distribution. Upon the initiation of the conversion process, Ga atoms diffuse into the solid phase and the diffusion continues until the Ga concentration becomes uniform (about 1800 s). The computed solid composition in the conversion layer (for xm = 0.7) is shown in Fig. 5.6.13. The composition profile changes rapidly and becomes uniform after 30 min, indicating that the conversion process was completed at this period. The conversion time of 3 h was more than sufficient to have a complete conversion. The process takes place only in the conversion layer and the GaP layer remained unaffected during conversion. This is different in the conversion of GaAsP, Fig. 5.6.6, as discussed in the previous section. The computed solid composition values are compared with experiments in Fig. 5.6.14. In the experiments, the solid composition was determined by X-ray diffraction and EPMA measurements. The horizontal axis represents the prepared liquid composition (xm), while the vertical axis describes the obtained solid composition of the alloy (x). The relationship between the solid and liquid compositions is almost linear (solid line) in the equilibrium state. Results show that in the conversion process the alloy composition can be controlled with the liquid composition. In the middle range, the solid composition is slightly lower than the liquid composition. In comparison with that of GaAsP, the difference between the liquid and solid compositions is large. In the Ga-In-P ternary phase diagram, the slope of the solid line in the middle range is sharper. Thus, the decrease of Ga concentration by the

Solid composition of GaxIn1-xP conversion layer (x)

1.0 0.8

0.6

0.4

0.2

Experiment Simulation

0.0 0 0.2 0.4 0.6 0.8 1.0 Liquid composition represented by equilibrium solid composition, xm

Fig. 5.6.14. Comparison of the computed solid composition with experiments (redrawn after Motogaito et al. [1997]).

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193

dissolution of the InP layer results in a shift in the solid composition to the InP side. 5.7. Epitaxial Lateral Overgrowth (ELO) of Semiconductors Today’s semiconductor devices require high quality, thin multilayers grown epitaxially on substrates. Defects in the device structure affect the device performance adversely, and lead to faster degradation. Such undesirable defects are very often generated at the substrate/layer interface due to the lattice mismatch between the available substrate and the epitaxial layer. Such defects propagate to the next-grown layer during epitaxy. In order to prevent the propagation of defects, the Epitaxial Lateral Overgrowth (ELO) technique has been developed (see for instance Nishinaga et al. [1988], Ujiie et al. [1989], Nishinaga [1991], Alam et al. [1999], Zytkiewicz [1999], Yan et al. [1998, 1999a,b, 2000], Khenner et al. [2002], Greenspan et al. [2003], and references therein). In ELO, an amorphous mask is deposited on a substrate, and then a narrow line-window structure is created by opening up windows of desired spacing in the mask (Fig.5.7.1). The epitaxial growth begins in these line windows, and then proceeds in the lateral direction over the mask. The lateral growth leads to a new epitaxial layer on the masked substrate, and may fully cover the masked substrate if a sufficient growth time is given for coalescence of the adjacent ELO strips. Since the mask effectively blocks the propagation of substrate dislocations, laterally overgrown sections of the ELO layers exhibit a much lower dislocation density than that observed in standard planar epilayers grown on the substrate. Therefore, if combined with the well-developed methods of buffer layers engineering, the ELO technique offers the possibility of producing high quality substrates with an adjustable value of lattice constant required by modern electronics (see Yan et al. [2000] and Liu et al. [2004]). This is the main reason for a widespread interest in a deeper understanding of the ELO mechanism and in the development of efficient ELO techniques. In this section we present the models developed to examine the transport y liquid mask

boundaries of simulation domain

solution

zone

ELO layer

window in the mask

substrate

x

Fig. 5.7.1. Schematic view of the cross section of a typical ELO set up (after Liu et al. [2004]).

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Sadik Dost and Brian Lent

Fig. 5.7.2. Computed concentration patterns after 120 s of growth time (after Yan et al. [2000]).

mechanisms of the epitaxial lateral overgrowth of InP and GaAs by LPE (Yan et al. [2000], Zytkiewicz et al. [2005], and Liu et al. [2005]). The models are based on the experimental set up of Yan et al. [1998, 1999a,b], and Zytkiewicz [1999]. The numerical simulation results are compared with those of experiments. A typical computational domain of an ELO is shown in Fig. 5.7.1. Due to symmetry, only the half sections of the liquid solution and the ELO crystal are considered. It is assumed that the ELO crystal starts growing from a window seed, and that there is no nucleation on the masking film. Growth is initiated and sustained by a gradual cooling of the system. The system is therefore considered quasi-isothermal, and the contribution of fluid flow to mass transport in the liquid zone is neglected. In this case the energy and momentum equations are not considered, and the only governing equation is the mass transport for the solute in the liquid phase:

 2C  2C C = DCL ( 2 + 2 ) t x y

(5.7.1)

5.7.1. Growth of InP To shed light on the growth mechanism of microchannel epitaxy (ECM), a two-dimensional model for the ELO of InP was developed by Yan et al. [2000]. The governing mass transport equation, Eq. (5.7.1), is solved under the following growth and boundary conditions. At the growth interface (crystal/solution) two different boundary conditions were adopted. At the lateral

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195

rough surface the concentration is assumed to be C = Ce, and at the faceted smooth surface it was assumed that the mass flux is balanced as

DCL

C = iy y

(5.7.2)

where the values of the equilibrium concentration, Ce, and the flux of the solute towards the growing interface, iy, are obtained respectively from the phase diagram of P in In solution (Hall [1963]) and the experimentally measured growth rate. The lateral growth rate is calculated by

Vg =

iy Cs



DCL C Cs y

(5.7.3) x =0

The distance between mesh lines and the time increment were selected respectively as 1 μm and 1.010-5 s, and the diffusion coefficient as DCL = 5.0  105 cm2s-1. The growth temperature and the cooling rate were selected as 550 °C and 0.1 °C/min, respectively. Simulation results, based on the above growth conditions, show that during the vertical growth (initial) stage, concentration patterns in the liquid solution are centro-symmetric with respect to the seed crystal, as expected (se Fig. 5.7.2). However, as can be seen from the concentration contours near the crystal (microchannel), we note a flux from the edge to the centre of the microchannel, as pointed out by arrows. It would be beneficial to discuss this flux near the growing interface.

near surface diffusion top facet MCE island rough surface (b)

Fig. 5.7.3. (a) Concentration patterns after 500 s of growth time, and (b) a schematic illustration of near surface diffusion at the growing interface (after Yan et al. [2000]).

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196

computation

9

W/T ration

experiment

6

3

0 0.0

0.1

0.2

0.3

Cooling rate (°C/min) Fig. 5.7.5. Cooling-rate dependence of the calculated W/T ratio (redrawn after Yan et al. [2000]).

During the formation of a facet inside the microchannel, the growth velocity will be the same everywhere. However, when a location on the facet has excess supply, instead of further growth at this point, the excess mass supply will diffuse away. In ELO, this occurs near the edges of the growing area, which is known as the edge effect (Nishinaga and Pak [1979]). This excess mass supply near the edges will be transported to the center where the concentration is lower. The mass flow seen in Fig. 5.7.2 represents such mass transport, which occurs from the points of higher supersaturation to those of lower supersaturation in the solution near the growing surface. This diffusion is called near-surface diffusion by Yan et al. [2000], and as pointed out by them, the value of the diffusion coefficient in the bulk solution is of the order of 10-4-10-5 cm2/s, approximately 103-104 times higher than those of surface diffusion coefficients reported in vapor growth. This shows the significance of near-surface diffusion as an essential mechanism in providing a uniform supply to the growing interface for the growth of atomically flat surfaces in the epitaxial lateral overgrowth by LPE. Fig. 5.7.3a shows the computed concentration distribution after 500 s of growth time. The intensity of the contour lines, which is larger near the lateral interface, indicates a faster growth in the lateral direction. The near-surface diffusion is visible again, however, but in the opposite direction to that seen in Fig. 5.7.2 where the growth occurs only in the vertical (radial) direction. In Fig. 5.7.3a however, growth is in the direction from facet to rough side surfaces, indicating a faster growth in the lateral direction. In addition, since the interface supersaturation is almost zero near the atomically rough side wall, the lateral growth is almost diffusion limited, and therefore, depending on the level of contribution of the near-surface diffusion, lateral growth will be enhanced

Single Crystal Growth of Semiconductors from Metallic Solutions

197

45 W/T ration

computation experiment

30

15

0 430

480

530

580

Growth temperature (°C) Fig. 5.7.6. Growth rate dependence of the W/T ratio (redrawn after Yan et al. [2000]).

while the growth in the vertical direction will be reduced. The near surfacediffusion will determine the ratio of lateral and vertical growth rates:

W / T = VgL / VgV Using Eq.(5.7.3), the growth rate in the lateral direction is determined from the computed concentration profiles, and plotted in Fig. 5.7.4. As seen, the growth is not uniform on the lateral (side) surface, being larger near the top surface. This is due to the high solute supply due to the near-surface diffusion and the

20

Growth rate (nm/s)

experimental growth rate in the vertical direction

15

computed growth rate in the lateral direction experimental growth rate in the lateral direction

10

5 0 420

460 500 Growth temperature (°C)

540

Fig. 5.7.7. Dependence of growth rate on growth temperature (redrawn from Yan et al. [2000]).

Sadik Dost and Brian Lent

198

bulk diffusion. It was shown numerically that the growth rate ratio, W/T, is a function of cooling rate (Fig. 5.7.5). The simulation results agree with experiments. The dependence of growth rate on growth temperature was also examined numerically. For a cooling rate of 0.13 °C/min, the ratio of W/T was computed for various growth temperatures. It was found that W/T increases significantly with the decreasing growth temperature, as shown in Fig. 5.7.6. As seen, the simulation results agree with experiments at high growth temperatures, however, there is a discrepancy at low growth temperatures (lower than 500 °C). In order to investigate this disagreement, the lateral and the vertical growth rates are plotted separately in Fig. 5.7.7. The main reason is due to the fact that the model assumes that the lateral growth is bulk diffusion limited, which is correct only for high growth temperatures. At low growth temperatures, the growth kinetics begin to play an important role, and the lateral growth rate becomes small. As we will see in the next section, when the contribution of growth kinetics is taken into account, this disagreement will disappear. 5.7.2. Growth of GaAs, and the Role of Surface Kinetics In this model, it is assumed that the ELO crystal starts growing from a 5-μm wide seed, and the growth is initiated and sustained by cooling the system gradually by T = T0   t , where T0,  and t represent respectively the initial growth temperature, cooling rate and growth time (Fig. 5.7.8). In the mass transport equation in Eq. (5.7.1) C represents the mass fraction of the solute (As), and the value of the diffusion coefficient of As in Ga, DCL is taken after Rode [1973] as

(

DCL = 5000exp 2  104 T

)

(5.7.4)

where T is temperature in Kelvin. y Wy Ga-As solution

Cin h

L seed

Ceq ELO layer

x

mask Wx

Fig. 5.7.8. Computational domain of the GaAs ELO model. Near surface diffusion is marked by an arrow (after Liu et al. [2004]).

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199

For the boundary conditions associated with the field variable at the vertical symmetry plane (x = 0) and on the mask surface, it is assumed that there is no material deposition on the mask and no solute transfer through the symmetry plane, i.e.,

(Cin) 

C C C = nx + n =0 n x y y

(5.7.5)

where  / n represents differentiation in the normal direction. At the upper (y = Wy) and the side (x = Wx) boundaries we assume C = C0 which is the equilibrium solute concentration at the initial temperature T0 as dictated by phase diagram. By this approach, for the present growth conditions, the liquid zone is assumed to be very large compared with the characteristic diffusion length Ldiff = DCl t . Indeed, for the GaAs system considered here (with the selected growth time and growth temperature), the characteristic diffusion length Ldiff will always be less than 2 mm. Therefore, the size of solution zone Wx Wy (1010 mm2) is large enough to satisfy this assumption. The phase diagram relation is given by Crossley and Small [1971]:

(

Ceq = 1.0657 exp 8.42  1.32  104 T

)

(5.7.6)

where again T is in Kelvin. Furthermore, a boundary condition at the solid/liquid interface is needed. For an optimal ELO, the orientation of the seeding lines in the mask is selected such that the atomically rough fast-growing planes cover the sidewalls of the growing ELO layer, while the slowly grown facet is formed on the top of the layer (as described in the previous section, Nishinaga [1991] and Yan et al. [2000]). This requires the use of different boundary conditions on these two faces. For numerical simplicity, we use a single condition of the same form for both ELO walls, but take into account their different properties by a proper choice of input parameters. Thus, at the ELO faces we use a condition of the linear surface kinetics similar to one defined in Eq. (5.2.11) (Kimura et al. [1994]):

(Cin) r Ceq

interface



C n

= interface

k

r r + (C  Ceq ) , or, C = Ceq L

DC

DCL C k n

(5.7.7) interface

where is the local equilibrium solute concentration at the growth interface and k the interfacial kinetics coefficient. Since an ELO layer is strongly curved in its upper corner, the effect of phase equilibrium on local crystal curvature (Gibbs-Thomson effect) is also included using the following expression for r Ceq :

Sadik Dost and Brian Lent

200

Fig. 5.7.9. Solute concentration distribution near the ELO layer (T0 = 6500C, kf = 310-4,  = 0.50 C/min, t = 150 min). Contour spacing is 1.510-4 (after Liu et al. [2004]).

(

r Ceq = Ceq 1+  r

)

(5.7.8)

where Ceq is the equilibrium concentration for a flat surface and its temperature dependence is given by Eq. (5.7.6), and  is the capillarity coefficient which is taken equal to 2.010-7 cm (Liu et al. [2004]). In order to reflect the difference in boundary conditions on rough and smooth surfaces, the kinetic coefficient is defined as

k = k f + (kr  k f )(1 cos  )

(5.7.9)

where kf and kr are the kinetic coefficients at the upper (faceted) and lateral (rough) ELO faces respectively. is the angle by which a local surface segment is misoriented relative to the low index facet (e.g., the (100) plane). In other words, =0 represents a perfectly faceted surface with an interface kinetics coefficient of k = k f , while = /2 describes a perfectly rough lateral growth interface with a coefficient of k = kr . It must be mentioned that for the case of

= /2, the value of kr is very large, thus it is much simpler for computation efficiency to assume that the boundary condition in Eq. (5.7.7) at the ELO sidewall reduces to C=Ceq, the equilibrium condition commonly used at the atomically rough interface. To satisfy the condition of k f >> kr a value of kr=1000 cm/s is selected in the computations, while kf is varied between 110-3110 –5 cm/s. The mass balance at the growth interface, Eq.(5.2.12), determines the growth rate

Single Crystal Growth of Semiconductors from Metallic Solutions

 L DCL C Vg = s (1 Ceq ) n

201

(5.7.10) interface

The initial concentration at the liquid phase is given by C = C0 , and the initial crystal shape is chosen to be a 50.7 μm rectangular. The model equations presented above are solved by the Galerkin finite element method. The width and thickness of the grown crystal are updated at each time step according to the growth rate Vg given by Eq. (5.7.10). Nine-node Lagrangian biquadratic basis functions are used to approximate the unknown fields. A backward Euler time stepping algorithm is employed in the calculation of transient term. Computed As concentration patterns in the liquid solution are presented in Fig. 5.7.9 after 2.5 hours of an epitaxial lateral overgrowth. In the simulation, we used an initial growth temperature of 650oC, a cooling rate of 0.5oC/min, and a surface kinetic coefficient kf of 310-4 cm/s at the upper ELO surface. As seen from the figure, iso-concentration contours first follow a pattern close to a rectangular crystal shape, but then at the points away from the crystal they become almost circular since at this length scale the ELO system feels the crystal like a point sink located at the lower left corner of the computational domain. Close to the crystal, however, the concentration distribution, C(x,y), shows

Fig. 5.7.10. Time evolution of the growth interface of the ELO crystal computed with (a) and without (b) the contribution of Gibbs-Thomson effect: time interval between two lines is 4 minutes (after Liu et al. [2004]).

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Sadik Dost and Brian Lent

additional features. At the centre, the concentration gradient is normal to the ELO surface as a result of the enforced boundary condition in Eq. (5.7.5). Away from the centre, we see a significant gradient of solute concentration along the upper ELO surface. This behavior can be explained as follows. Due to a finite rate of surface kinetics, the surface solute concentration Cin on the upper ELO face is larger than the equilibrium value of Ceq. On the other hand, the sidewall of ELO is atomically rough, so the solute concentration there is equal to Ceq. This gives rise to solute diffusion along the upper ELO face to the sidewall (called near-surface diffusion, Yan et al. [2000]) as described by an arrow in Fig. 5.7.8. Of course, the magnitude of near-surface diffusive solute flux increases with the decreasing value of kf. It also depends on the solute supply rate, which varies with the cooling rate  and temperature via DCl and Ceq. As will be shown later, the presence of near-surface diffusion significantly enhances the lateral growth rate of ELO layers. Fig. 5.7.10a shows the computed time evolution of the growth interface during the ELO of a GaAs crystal. As seen, the shape of the upper ELO surface is very flat, which closely agrees with experiments. Moreover, the lateral growth rate is much larger than that in the vertical direction, so a thin and wide layer is obtained numerically. This shape is similar to that of the shape obtained by Yan et al. [2000] through a two-dimensional simulation of the ELO growth of InP. However, in this case the numerically predicted flat top face is due to the inclusion of surface kinetics and diffusion processes in the model. In Fig. 5.7.10b, we show, for comparison, the shape of ELO layer calculated for the same growth conditions but without the contribution of the GibbsThomson effect. A spiky growth interface, starting from the ELO corner, is then obtained, which does not agree with experiments. This comparison shows the

Fig. 5.7.11. Aspect ratio L/h of GaAs ELO layer versus surface kinetics coefficient kf for various LPE growth temperatures (after Liu et al. [2004]).

Single Crystal Growth of Semiconductors from Metallic Solutions

203

significance of the Gibbs-Thomson effect on the ELO layer shape, and how the crystal surface curvature influences solid/liquid equilibrium. Note that for a shorter growth time, the spiky growth at the corner is quite small. When the growth proceeds, however, concentration gradients in the liquid zone will increase. This may lead to faster corner growth and give rise to growth instability if the Gibbs-Thomson effect is not taken into account. Fig. 5.7.11 shows the aspect ratio (width/thickness RA=W/T = L/h) of the GaAs ELO layer, computed in terms of the surface kinetic coefficient kf for various growth temperatures. As seen, the aspect ratio is high at smaller values of kf. Such a result is expected since a smaller kf means a higher barrier for solute incorporation into the solid, which leads to lower vertical growth rates and larger surface solute concentrations Cin at the upper face. Then, the nearsurface diffusion towards the ELO sidewall becomes stronger and, consequently, the lateral growth rate increases. It is also worth mentioning that, for a fixed value of the surface kinetics coefficient, the aspect ratio increases with growth temperature. The published experimental data show a disagreement on the dependence of W/T on temperature (Zytkiewicz [1999], and Yan et al. [2000]). This discrepancy arises since in the ELO growth experiments the surface kinetic coefficient kf increases exponentially with temperature, which reflects the thermally activated nature of the surface processes (Ghez and Lew [1973]). Due to the lack of precise experimental data, this phenomenon is not included in the present model. It is possible that, by comparison of the experimental values of RA=W/T given at a growth temperature with the predictions from numerical simulations, one can evaluate with a reasonable accuracy the value of kf for a particular growth condition.

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Chapter 6

LIQUID PHASE ELECTROEPITAXY

In this chapter we present the numerical simulation models developed for the Liquid Phase Electroepitaxial growth of semiconductors. The numerical simulations carried out to date for binary and ternary systems are presented in a chronological order. The role of a static vertical magnetic field is also examined, and its effect on the growth process is discussed. The models of lateral overgrowth of semiconductor layers are also presented. 6.1. Early Modeling and Theoretical Studies We first present a brief summary of the conceptual analyses and simple modeling of LPEE developed prior to the initiation of the continuum based numerical simulation models seen in the literature. A number of conceptual/modeling studies have been carried out to have a better understanding for the relative contributions of electromigration, Peltier cooling, diffusion, and convection (see for instance, Bryskiewicz [1978], Jastrzerbski et al. [1978b], Lagowski et al. [1979], Bryskiewicz et al. [1979, 1987a], Zytkiewicz [1983], and Nakajima [1989, 1991]). These early studies have contributed significantly to the understanding of the LPEE growth process. In the presention of these subjects here, we tried to be loyal, as much as we could, to the original terminology used for the various concepts and definitions of LPEE. 6.1.1. Peltier-Induced Growth Kinetics: Electromigration Mechanism Bryskiewicz [1978] presented an analytical treatment for the Peltier-induced growth kinetics in LPEE growth of GaAs under the conditions that the natural convection in the solution can be neglected. In this model, the growth rate

Sadik Dost and Brian Lent

206

(which is the relationship between the layer thickness, time, and applied electric current density) was examined considering diffusion, electromigration, and electrotransport of As in a Ga-rich solution (Fig. 6.1.1). The mass balance for the solution and the grown layer is written as L

 LC L (T0 )L = sCs R(t) +  L  C(x,t)dx

(6.1.1)

R

The distribution of charged-species (solute) C(x,t) in the solution is expressed writing the conservation of mass for the ionized species as (after Levich [1962], and Wever [1973]):

C  2C  = DC 2  Z eff μ E (EC) t x x

(6.1.2)

where

μE =

FDC RT

, and Z eff = Z  Z0

 1 (C)   1 (0) C 1 (0)

(6.1.3)

and μ E , F, and  are respectively the electric mobility, the Faraday number, and conductivity of the solution, Z, and Z0 represent the valances of solute and solvent ions, and Zeff is the effective charge of solute species. Differentiating Eq. (6.1.1) with respect to time and using Eq. (6.1.2), we obtain the growth rate as

V (t) =  L

DC (C / x)

x= R

 μ E Z eff E(R)C L (T1 )

S CS   LC L (T1 )

(6.1.4)

As can be seen, the calculation of the growth rate requires the electric field intensity E to be known, defined by charge distribution in the melt, and the solute concentration C(x,t). Since the Ga (or Ga-rich) solution exhibits metallic properties (also any other metallic mixtures alike), the participation of free electrons in the electric conductivity is much greater than that of the ionized solute (arsenic in this case) atoms. One may then assume that the electric field intensity E is constant across the solution and its value can be estimated from the following expression J =  E E which is simply the constitutive equation used for the electric current in Chapter 4, with the conductivity E of the solution. In that case, the growth expression reduces to that given in Chapter 4 (also Eq. (6.2.10)) with the assumption of E = Z eff E . Then, Eq. (6.1.2) reduces to the one-dimensional mass transport equation, which can be deduced from the one that was given in Chapter 4, as

Single Crystal Growth of Semiconductors from Metallic Solutions

207

C(x,t) solid

CS

source

solution CL(T0)

CL(T1) 0 R(t)

E(x)

x

L

Fig. 6.1.1. Distribution of the solute concentration C(x,t) in the solution (liquid) and solid: CL = equilibrium composition of the solution, R = thickness of the grown layer, L = solution height, t = time, E = electric field intensity (redrawn from Bryskiewicz [1978]).

C  2C C = DC 2  μ E E t x x

(6.1.5)

Now, Eq. (6.1.2) can be solved numerically under the following conditions:

C(x,0) = C L (T0 ) , C(R,0) = C L (T1 ) , C(L,t) = C L (T0 ) ,

C x

=0

(6.1.6)

x= L

One can see that in order to calculate the growth rate in Eq. (6.1.4) one must know the values (or functions) of DC (T ) , C L (T ) , T  T1  T0 ,  E (C) , and also Z eff . Some of these values can be determined experimentally (such as DC (T ) ,  E (C) , and Z eff ) and some of them (such as C L (T ) and T  T1  T0 ) can be estimated through calculations (or numerical simulations). Bryskiewicz [1978] estimated these values as follows. The temperature difference T  T1  T0 was estimated directly by taking into account the fact that the growth of an epitaxial layer without the source on the solution surface occurs until the following condition is fulfilled (Fig. 6.1.2).

DC

C = Z eff μ E EC x

(6.1.7)

which means the diffusion, electromigration and electrotransport streams balance each other. Solving Eq. (6.1.7) and substituting into Eq. (6.1.2) yields (see Bryskiewicz [1978] for details)

Sadik Dost and Brian Lent

208 C(x,t) solid CS solution CL(T0) CL(T1)

DC

C

Z μ EC eff

x

E 0

Rmax

E

x L

Fig. 6.1.2. Distribution of arsenic concentration in the solution after a very long time (redrawn from Bryskiewicz [1978]).

C L (T1 ) =

C L (T0 )  (CS S Rmax ) / (  L L) 1 ( μ E LZ eff E) / (2DC )

(6.1.8)

where Rmax is the maximum thickness of the layer. Eq. (6.1.8) and the exact knowledge of the liquidus curve C L (T ) will determine T  T1  T0 . The electric conductivity of the Ga-solution and the parameter Z eff were not known, and Bryskiewicz [1978] estimated them as follows. The product EZ eff , which is linearly dependent on current density, was treated as the parameter to fit the experimental data of the growth rates obtained (see Bryskiewicz [1978] for the experimental data) as

J Z eff E

= const. ,

at

T0 = const.

(6.1.9)

It follows that the fitting parameter Z eff E has to be proportional to the current density J. Bryskiewicz [1978] suggests that an agreement within ±20% between the calculated and measured values can be achieved. Actually in the growth of very thick crystals (during prolonged growth periods), as will be seen later (Sheibani et al. [2003a]), this agreement is very close, within a very small margin. Once the values of Z eff E are known from the fitting procedure, the electrical conductivity  E , the effective charge Z eff , and the electric field intensity E can be estimated. Using J =  E E , and Eq. (6.1.3)2 we obtain

 1 (C) =  1 (0)(1+ ZC / Z0 )[1+ (1  )1/ 2 ] / 2 and

(6.1.10)

209

Single Crystal Growth of Semiconductors from Metallic Solutions

Z eff = Z  Z0

(1+ ZC / Z0 )[1+ (1  )1/ 2 ]  2

(6.1.11)

2C

where



4Z eff EC

(6.1.12)

Z0 1 (0)J (1+ ZC / Z0 )2

Using Eqs. (6.1.10)-(6.1.12), the calculated values of Z eff are tabulated in Table 6.1.1. Table 6.1.1. Estimated values of electrical conductivity and effective charge (Bryskiewicz [1978]). 3

1

3-

T0 (°C)

C L (T0 ) (at %)

(Z eff E / J )  10 (cm)

 (C )  10 (cm)

Z eff (As )

750

1.38

-1.2

5

-24

800

2.23

-1.2

5.2

-23

850

2.75

-1.2

5.5

-22

5

These results showed that mass transport in the LPEE growth of GaAs is due to both the diffusion and electromigration of solute (arsenic) in the liquid towards the seed substrate. The high value of the effective charge, estimated by fitting of the theoretical values to those measured ones, justifies the conclusion that the migration of arsenic species in the liquid solution is realized mainly due to collisions with electrons flowing across the solution. Indeed, this conclusion of Bryskiewicz [1978] on the mechanism of

Layer thickness

30

L = 55 mm J = 10A/cm2 T = 18°C T0 = 850°C ZeffE = 1 V/m

20

Total transport

electromigration

10

diffusion

0

30

60 Time (min)

90

Fig. 6.1.3. Layer thickness versus time, and the relative contributions of electromigration and diffusion (redrawn from Bryskiewicz [1978]).

210

Sadik Dost and Brian Lent

electromigration established the understanding of the electromigration process. As we have discussed in Chapter 3, the high growth rates achieved under an applied magnetic field in LPEE growth of GaAs (Sheibani et al. [2003a]) can only be explained by such a mechanism as suggested in Dost et al. [2005a,b] and Dost and Sheibani [2006]. The resistance during the collision of electrons with the species of the solution determines the mobility of the species and in turn the growth rate. This resistance is very much reduced under an applied static magnetic field due to the possibility that the charged species are aligned along the magnetic field lines (which are almost uniform along the growth direction in the growth system used in Sheibani et al. [2003a]), and leads to a very high mobility. Thus, the mass transport due to electromigration increases tremendously and the total mobility of species depends not only on the electric mobility coefficient which is measured in the absence of an applied magnetic field, but also on the magnetic field intensity and a new material coefficient which is called the magnetic mobility by Dost et al. [2005a,b] and Dost and Sheibani [2006]. This understanding supports the earlier definition of electromigration in LPEE. 6.1.2. LPEE Growth Using a Very Thin Solution In order to minimize the effect of convection, an LPEE growth system for GaAs was designed by Zytkiewicz [1983] using a very small solution height, 0.5 mm. For such a system, a theoretical treatment, based on Bryskiewicz [1978], was given and the relative contributions of electromigration, diffusion and the Peltier effect were studied. It was shown that the LPEE growth of GaAs from a very thin solution has the following advantages: i) it is quite easy to minimize the effect of convection which may lead to interface instabilities in a regular LPEE set up, ii) reduction of stabilization time of the growth rate which should result in a better crystal uniformity and control of layer thickness, iii) reduction of As diffusive streams towards the substrate during the early stages of growth, and iv) retention of the same, steady state growth rate. 6.1.3. A One-Dimensional Model of LPEE Based on the conservation of species mass, a one-dimensional model for the LPEE growth process was given by Jastrzebski et al. [1978] where the contributions of the Peltier effect at the solid-solution interfaces and that of solute electromigration in the solution to the overall growth process were defined. According to this one-dimensional model, the contribution of electromigration to growth is dominant in the absence of convection and the contribution of the Peltier effect can be dominant in the presence of convection (the relative contributions of electromigration and the Peltier effect studied through two-dimensional numerical simulation models are presented in the next section).

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211

The one-dimensional mass transport equation was written as

C C C  2C +V g = DC 2  μ E E x t x x

(6.1.13)

The following boundary conditions were considered

 DC

C + μ E EC L = (CS  C L )V g , x 0

(6.1.14)

C = C0 at t = 0 for all x, and at t > 0 for x =  (absence of convection), or C = C0 at t > 0 for x >  (presence of convection) C = C L at t > 0 and x = 0 (growth follows the liquidus line) Eq. (6.1.13) was solved analytically with the assumption of a small growth velocity so that the second term on the left-hand-side of term in Eq. (6.1.13) was neglected. The growth velocity was obtained for two cases: i) for an infinitely long boundary layer (  =  , no convection), and ii) for a finite boundary layer (with convection), which are given respectively as

VTg =

Tp

CL dC DC 1/ 2 ( ) + μE E CS  C L dT L  t CS  C L

(6.1.15)

and

VTg =

Tp

dC CS  C L dT

DC L



+ μE E

CL

(6.1.16)

CS  C L

Growth velocity (μm)

0.8

0.6

0.4

0.2

Electromigration plus Peltier cooling

Electromigration Peltier cooling

0.0 0

15

30 45 Time (minutes)

60

75

Fig. 6.1.4. Growth velocity of GaAs calculated from Eq. (6.1.15) (redrawn from Jastrzebski et al. [1978]).

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In Eqs. (6.1.15) and (6.1.16) the first terms represent the contribution of the Peltier effect, and the second terms represent the contribution of electromigration to the growth velocity. Both terms depend linearly on the electric current through Tp and E, respectively. Eq. (6.1.15) was solved for the LPEE growth of GaAs from a Ga-As solution at the 800°C growth temperature and using J = 10 A/cm2, Tp = 3°C , DC = 6.0  105 cm 2 / s , and μ E E = 105 cm/ s . Results were plotted in Fig. 6.1.4. Eq. (6.1.16) was also solved for various boundary layer thicknesses. It was shown that as the boundary layer gets thinner, the contribution of Peltier cooling to the growth rate increases. The contribution of Peltier cooling to the growth rate also increases with increasing Peltier cooling at the growth interface (see Jastrzebski et al. [1978] for details). According to the model presented above the relative contributions of the Peltier cooling at the growth interface and electromigration of solute species in the liquid towards to the interface depends on the experimental conditions. On the basis of this model, Jastrzebski et al. [1978] gave a quantitative criterion to determine the relative contributions of the Peltier effect and electromigration. A critical substrate thickness, dc, was defined, for which the contributions of the Peltier effect and electromigration are equal. When the substrate thickness is greater than the critical thickness, the Peltier effect dominates, and when the thickness is smaller than the critical thickness, electromigration dominates. Their conclusion then was that in the growth of thick crystals and in the presence of significant convection, electromigration is dominated by Peltier cooling. On the other hand, with thin substrates and in the absence of significant 1.0

Growth rate (μm/min)

0.8

n-type substrate p-type substrate

0.6

0.4

0.2

10

20 current density (A/cm2)

30

Fig. 6.1.5. Growth rate versus electric current density: 800°C with a substrate thickness of 300 μm (redrawn from Jastrzebski et al. [1978]).

Single Crystal Growth of Semiconductors from Metallic Solutions

213

950 °C

Growth rate (μm/min)

5

900 °C

4

3

850 °C

2 800 °C 1

10

20 30 Current density (A/cm2)

40

Fig. 6.1.6. Growth rate versus current density at various growth temperatures (redrawn from Jastrzebski et al. [1978]).

convection, electromigration dominates the electroepitaxial growth. In order to verify the predictions of the one-dimensional model, specific LPEE experiments were devised by Jastrzebski et al. [1978]. Under specific growth conditions, using n- and p-type seed substrates, it was shown that the growth in the case of the p-type substrate was smaller than in the case of the ntype substrate. From the difference in growth rates, it was concluded that the

Normalized max. layer thickness

0.006

0.004

0.002

0

4.0 8.0 Solution height (mm)

12.0

Fig. 6.1.7. Normalized maximum GaAs layer thickness as a function of solution height under the experimental conditions of 800°C growth temperature, 0.3 mm thick n-type substrate (redrawn from Jastrzebski et al. [1978]).

Sadik Dost and Brian Lent

214 Boundary layer thickness (mm) 1.1 0.5

Growth rate (μm/min)

4.0

3.5 Contribution of Peltier effect 3.0

2.5 0

5

Contribution of electromigration 10 15 Solution height (mm)

20

Fig. 6.1.8. Growth rate of GaAs from a Ga-As solution (at 900 °C and 25 A/cm2) as a function of solution height. Substrates were Cr-doped and 0.3 mm thick. The estimated boundary layer thicknesses are also indicted at the top (redrawn from Jastrzebski et al. [1978]).

contribution of Peltier cooling to the growth rate is less than 15%. The almost linear relationship between the growth rate and electric current density observed experimentally is consistent with the theoretical predictions (Fig. 6.1.5). It was shown experimentally that the growth rate is also proportional (almost linear) to current density for all growth temperatures used, and that, for a given current density, the growth rate increases with temperature. Results are summarized in Fig. 6.1.6 for an LPEE growth of GaAs on a 0.3-mm thick ntype substrate. Another important issue is the selection of a proper solution height when an LPEE crucible is designed since it influences the growth rate, among of course also other growth parameters. The LPEE experiments designed for the growth of GaAs at 800°C using an n-type substrate showed that the growth rate is linearly proportional to the solution height up to a certain height (about 10 mm) above which the growth rate remains constant. The results of experimental measurements of Jastrzebski et al. [1978] are presented in Fig. 6.1.7. Naturally, the relative contribution of the Peltier effect to the growth rate varies with solution height due to the contribution of convection in the solution. Experiments carried out by Jastrzebski et al. [1978] (at 25 A/cm2 electric current density, and at 900 °C) show that the contribution of Peltier cooling increases with solution height (Fig. 6.1.8). 6.1.4. Dopant Segregation in LPEE The control of dopant distribution in semiconductors by passing an electric current during crystal growth has been known for many years (for instance, see

Single Crystal Growth of Semiconductors from Metallic Solutions

215

Joffe [1956], Pfann et al. [1957], and Pfann and Wagner [1962]). It has also been shown that LPEE can provide a control of dopant segregation (Blom et al [1973], Lawrence and Eastman [1976], and Jastrzebski and Gatos [1977]). Following these studies, Lagowski et al. [1980] presented a theoretical model for dopant segregation in LPEE which is an extension of the model of Jastrzebski et al. [1978]. In this model the incorporation of impurities into the grown crystal is described by introducing an interface segregation coefficient

k0 =

CS C L0

where CS and C L0  C L

(6.1.17)

x =0

are the impurity concentrations in the solid and in the solution at the interface, respectively. The transport equation for the impurity concentration is written as given in Eq. (6.1.13), and the growth rate and the associated interface and boundary conditions as in Eqs. (6.1.14), where the sign in front of the electric mobility is taken positive when the impurities move toward the interface. The initial temperature distribution (T0) in the system changes as soon as the electric current is applied, due to Peltier cooling/heating at the interfaces. Accordingly, the interface segregation coefficient changes. An analytical expression for the effective segregation coefficient was given as

1   keff = k01{1 erfc[ (DC t)1/ 2 + erfc[ (DC t)1/ 2 ] 2 2( +  )

+

 exp[DC t( 2   2 )]erfc[ (DC t)1/ 2 ]}  +

(6.1.18)

where

k V g  μE E V g + μE E 1 1 1 g , k0  (k0 + 0 = {(k0  1)V + },  = T DC 2 2DC

Tp ) T0

For high or low applied current densities, the above expression can be simplified significantly (see Lagowski et al. [1980]). For instance, in the use of high electric current densities, under the assumption of

μ E Et  2(DC t)1/ 2 Eq. (6.1.18) simplifies to

keff  k01{1+

μE E k01V

[1 exp( g

k01V g μ E E DC

t)]}

(6.1.19)

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216

Thus, the effective segregation coefficient increases exponentially with time to a value of keff = k01 + μ E E / V g , which can be observed in non-metallic solutions. In the LPEE growth process of semiconductors, which is of interest in this book, we use metallic liquids for which the conductivity of the solutions are very high, and thus low electric current densities are employed. Particularly, in the growth of thick (bulk) crystals by LPEE the electric current density is less than 10 A/cm2, and usually is either 3 or 5 A/cm2 (see Sheibani et al. [2003a]). For low electric current densities, Eq. (6.1.18) can be simplified to

keff = k0 (1+  E + V + T )

(6.1.20)

where

 E  2(

1 k0 t 1/ 2 t 1/ 2 g ) μ E E , V  2( ) V (1 k0 ) ,  T  k0 T  DC  DC

Tp (6.1.21) T0

which represent respectively the contributions of electromigration, growth velocity (rate), and the Peltier effect. From Eq. (6.1.20) one can see that if k0  1 , the contribution of growth velocity can be more than that of electromigration since the ratio of  E / V becomes very small, in other words, the depletion of impurities at the interface becomes large (due to segregation) compared to the amount of impurities transported to the interface by electromigration. If k0 is smaller that unity, keff may not necessarily increase with growth velocity, as is the case in the absence of electric current, since the sign of the electromigration term,  E , is not a function of the value of k0 but depends on the direction of the applied electric current. It means that this term will be positive if impurities migrate toward the growth interface and negative if they migrate away from the interface. The sign of the Peltier term,  T , on the other hand is determined by the direction of the electric current (for a given substrate type) and by the temperature dependence of the segregation coefficient. In the presence of convection, the effective segregation coefficient was given as (Lagowski et al. [1980])

keff = k01 (1+

μE E Vg

)[k01 + (1+

μE E Vg

 k01 )exp(

 (V g + μ E E) 1 )] DC

(6.1.22)

For low current densities the above expression can be simplified to

keff = k0{1+

k  [ μ E E  V g (k0  1)] + 0 DC T

Tp T0

(6.1.23)

Single Crystal Growth of Semiconductors from Metallic Solutions 1.4

217

μEE > 0, k0 = 0.1

1.2

k eff

1.0

k0

0.8

T{

μEE > 0, k0 = 10 μEE < 0, k0 = 0.1

0.6

μEE < 0, k0 = 10

0.4 1

2

3

4

 (mm)

Fig. 6.1.9. Dependence of the effective segregation coefficient on the boundary layer thickness, calculated from Eq. (6.1.23) for LPEE growth of GaAs at 900°C, and using Tp = -3 °C, and the 2 parameter (1 / k 0 )(k 0 / T ) = 2  10 (redrawn from Lagowski et al. [1980]).

Eq. (6.1.23) indicates that in the case of weak convection or no convection the direction of change in the effective segregation coefficient is primarily determined by the sign of the electromigration velocity μ E E . However, in the presence of strong convection (for a large boundary layer thickness) the Peltier term,  T , which is independent of convection, becomes dominant. The ratio of keff / k0 is presented in Fig. 6.1.9 for an LPEE growth of GaAs from a Ga-rich solution. 6.1.5. Multi-Component Systems: Growth of GaxAl1-xAs The relative contributions of electromigration and the Peltier effect to the growth rate in LPEE growth of a ternary system, GaxAl1-xAs, have been examined by Bryskiewicz et al. [1980]. Following a similar approach presented in the previous subsection an expression for the growth velocity was derived. For instance, the growth velocity for a multicomponent system at low electric current densities was given, in the absence of convection, as

V g  [

Tp ( t)1/ 2

n1 μ

i  C 0 n1  i (Cis0  Ci0 ) 1 E i i ][

] 1/ 2 1/ 2 i=1 (D ) i=1 ) (D C C

+E

(6.1.24)

and, in the presence of convection, as

V  [ g

where

Tp



n1 μ

+E

i=1

i  C0 E i i

DC

n1 

][

i=1

i

(Cis0  Ci0 ) DC

]1

(6.1.25)

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218

 i  (T / Ci ) C =C 0 ,T i

i

(6.1.26)

0

For n = 2, Eqs. (6.1.24) and (6.1.25) reduce to those given for a binary system. Theoretical and experimental results given for the LPEE growth of the Ga0.7Al0.3 As system show that the composition of the grown crystal is controlled by the Peltier cooling at the growth interface, the diffusion and electric mobility constants of the component solutes, and the applied current density (Bryskiewicz et al. [1980]). For a selected composition, the crystal composition can be varied by varying the current density. For a given current density, the composition remains constant even in the case very thick crystals (see also Bryskiewicz [1994] and Sheibani et al. [2003a] in the case of Ga0.96In0.04As). 6.1.6. Source-Current-Controlled (SCC) Growth Nakajima et al. [1984, 1987], and Nakajima and Yamazaki [1985, 1986] have developed a technique called the source-current-controlled (SCC) method which is a version of LPEE. A schematic view of the model growth system is shown in Fig. 6.1.10. In this set up the applied electric current passes through the source, the solution and then through the graphite on top, but bypasses the substrate. Due to the passage of electric current through the source the temperature T1 just above the source is higher than that of T2 just below the substrate. This temperature difference generates a temperature gradient between the source and the substrate. For this system, the thickness of the grown layer is calculated by solving the one-dimensional mass transport equation given in Eq. (6.1.5) (where we assumed the mobility is positive when the current flow from source to substrate, in other words, from negative to positive polarity, but Nakajima Substrate

Epitaxial layer

+

+

G

G

T2 Diffusion and electromigration

BN

L

BN: Boron nitride G: Graphite

x

Solution height

BN

Solution T1 Source

Dissolution

0

BN

a) Growth crucible

T2

T1

b) Temperature profile

Fig. 6.1.10. Schematic view of the SCC (LPEE) growth crucible domain used in the model (redrawn from Nakajima [1980]).

Single Crystal Growth of Semiconductors from Metallic Solutions

219

[1980] used an opposite sign convention to this, in other words, the mobility was taken positive from positive to negative polarity: μ E   μ ) under the following boundary conditions:

C(0,t) = C0  (R0 / m0 )t

(6.1.27)

C(L,t) = C L  (RL / mL )t

(6.1.28)

C(x,0) = C0 + (C L  C0 )(x / L)

(6.1.29)

where C0 and CL are the initial solution concentration in the vicinity of the growth interface and the source, respectively, and R0 and RL represent respectively the cooling rates at the growth and dissolution interfaces. The solution height is denoted by L, and m0 and mL are the slopes of the liquidus curves near the growth and dissolution interfaces, and are given by

m0 

1 dT wL dX L

, and mL  x =0

1 dT wL dX L

(6.1.30) x= L

where XL represents the atomic fraction of the solute, and wL is the solution density. Through an approximate analytical solution to Eq. (6.1.5), an expression for the layer thickness was given as

L = M t / Cs

(6.1.31)

where

M t = DC (

Rt D Rt μE 1 μE  )(C0  0 )t + C exp( )(C L  L )t 2DC L 2DC L 2m0 2mL

  nTn (0) 2  BC L  C0 +  }[exp( DC At)  1] ( L n=1 L n=1 A A R 1 1   2n2 R0 +[ 3  exp( DC At)  ] ][  B L ][t + 2 DC A DC A m0 mL L n=1 A {

(6.1.32)

and Cs is the concentration of C atoms per unit volume of the grown layer. For an AxB1-xC system, Cs is given by Cs = 4/d3 where d is the lattice constant of the ternary compound. The layer thickness for the diffusion and electromigration limited growth in a temperature-graded solution can be calculated using Eqs. (6.1.31) and (6.1.32) (for details, see Nakajima [1980]). The thickness of the grown layers of an In0.53Ga0.47As system was calculated by Nakajima [1980] and various numerical data were given for the dependence of concentration on growth time, layer thickness as a function of temperature and cooling rate, the relative contributions of electromigration and diffusion to growth rate, and the composition of the grown crystal along the growth

Sadik Dost and Brian Lent

220

direction. These calculations were made using the values of μE = 0.1 - 0.001 cm/min for the mobility of As in the Ga-As solution. Nakajima [1980] stated that “the real value of μ can be determined through a comparison of experimental layer thicknesses with computed ones”. Indeed, as shown by Sheibani et al. [2003a,b], Dost et al. [2005] and Dost and Sheibani [2006] the mobility was calculated using experimental results of LPEE of a GaInAs system with and without the application of an external magnetic field. The above analysis of Nakajima [1980] was extended in Nakajima [1991] to a GaxIn1-xAs system to determine the composition variation in the grown crystals. In this model, for the first time, phase equilibrium between the crystal and the solution is maintained while having a consistency between the transported and incorporated mass or solute atoms at the growth interface. The comparison of experiments with computed results revealed that in the Ga-In-As solution the diffusion coefficient of Ga is about twice that of As, and the electric mobility of Ga is larger than that of As. The above results were very instrumental in the development of future 2-D and 3-D numerical models and simulations for such systems. 6.1.7. A Thermal Analysis for LPEE The work of Bryskiewicz and Laferriere [1993] aimed at improving our understanding of the LPEE growth process for growing ingots of ternary semiconductors with uniform composition and low defect density. In addition to a review of LPEE to date (1999), theoretical considerations were given for the role of convection, and also for controlling convection by means of an applied static magnetic field. A 2-D thermal analysis was carried out for an LPEE growth crucible. The computational domain is shown in Fig. 6.1.11. The following assumptions were made in the analysis of Bryskiewicz and Laferriere [1993]: the electrical resistivity of the solution and graphite are negligible, heat 6.0 mm

36.0 mm Graphite: 19.0 mm Ga-rich solution: Boron nitride

Single crystal substrate: 0.5, 2.0, 4.0 mm Ga+Al contact liquid: 2.0 mm Graphite: 19.0 mm

Fig. 6.1.11. The computational domain for the thermal analysis (half of the growth cell, redrawn from Bryskiewicz and Laferriere [1993]).

Single Crystal Growth of Semiconductors from Metallic Solutions

221

flow in the solution is mainly due to conduction, and the difference in thermal conductivities of the Ga-rich and In-rich solutions is negligible. Numerical results for the radial temperature distribution were given for two crystal diameters (12 and 48 mm), and three substrate thicknesses 0.5, 2.0, 4.0 mm. The current density was selected as 5.0 A/cm2. Results show that the maximum radial temperature change in the solution along the growth interface is about 0.5 °C. A simple analysis for the effect of an applied magnetic field was also given for a Ga-rich solution in LPEE assuming an applied magnetic field of intensity 30 kG. Based on the assumed data, Bryskiewicz and Laferriere [1993] have concluded that a 30-kG field intensity will be sufficient to suppress convection significantly. The application of a magnetic field in LPEE will be discussed in Sections 6.4 and 6.5 in detail. 6.2. Simulations Based on Continuum Models As described earlier, the growth process of LPEE is quite complex, and involves the interactions of various thermomechanical and electromagnetic fields. These include fluid flow, heat and mass transfer, electric and magnetic fields, various thermoelectric effects and their interactions in the liquid phase, and the heat and electric conduction with various thermoelectric effects in the solid phase. In addition, the moving growth and dissolution interfaces with possible finite mass transport rates complicate the process further. This nature of the LPEE growth process has brought about a number challenges for researchers. To gain a better understanding of the growth process of LPEE, over the past 10 years a number of numerical simulation modeling studies have been carried out. In this section we present some important features of these models following historical developments and understanding for the LPEE process. For the first time, a two-dimensional computer simulation model for the LPEE growth process of GaAs has been introduced by Dost et al. [1994]. This simulation model was based on the rational mathematical model presented in Chapter 4 for a binary system (see also by Dost and Erbay [1995]). The objective of this diffusion model was to examine the relative contributions of electromigration and Peltier cooling without introducing the additional complexity of natural convection. The model includes heat transfer, diffusive mass transport, electromigration, and Peltier and Joule effects. The governing equations are solved numerically using a finite volume method. Simulations are presented for three different growth cell configurations to investigate: (i) temperature and concentration distribution in the growth cell, (ii) the effect of applied electric current density and substrate thickness, and (iii) the contribution of electromigration and Peltier cooling to the overall growth rate. Based on the simplifying assumptions used in this work, the simulation results showed that the magnitude of the relative temperature at the growing

Sadik Dost and Brian Lent

222

interface is controlled mainly by Peltier cooling for thin substrates (less than 2 mm) and small electric current densities (less than 20 A/cm2). As expected, Joule heating becomes significant only for thick substrates and high electric current densities. For all configurations investigated, electromigration is found to be the dominant growth mechanism. In critical regions of the growth cell, relatively small changes in the configuration are found to have a significant impact on the process, and on the degree of non-uniformity of the grown crystal. This diffusion model has been followed by a number of numerical simulation models specific to particular growth cell configurations, as will be discussed later. As an extension of the diffusion model of Dost et al. [1994], the effect of natural convection in LPEE was introduced for the first time by Djilali et al. [1995]. In this work, the effect of thermosolutal convection in LPEE growth of GaAs was investigated through a two-dimensional numerical simulation model that accounts for heat transfer and electric current distribution with Peltier and Joule effects, diffusive and convective mass transport including the effect of electromigration, and fluid flow coupled with temperature and concentration fields. Simulations were performed for two growth cell configurations and the results are analyzed to determine growth rates, substrate shape evolution, and relative contributions of Peltier cooling and electromigration. The simulations

-

-

Graphite GaAs source Boron Nitride

Ga-As solution

Graphite Dissolution Interface

Boron Nitride

Boron Nitride

GaAs seed

GaAs source Ga-As solution

Boron Nitride

GaAs seed Growth Interface

Graphite

Graphite

+

+

(a) SETUP1

(b) SETUP2

Fig. 6.2.1. Schematic views of the computational domain of two setups: (a) the electric current bypasses the source, and (b) the electric current passes through the source (Dost et al. [1994]).

Single Crystal Growth of Semiconductors from Metallic Solutions

223

predicted and helped explain a number of experimentally observed features, which previous diffusion-based models failed to reproduce. As we will see later in detail, in general, electromigration is found to be the dominant growth mechanism, but the contribution of Peltier cooling to the overall growth rate is found to be significantly enhanced by thermosolutal convection in the solution, and Peltier cooling can in fact become the dominant growth mechanism for certain growth conditions and growth cell configurations. The overall growth rate is found to increase with increasing furnace temperature and applied electric current density. This thermosolutal convection model predicts an increased non-uniformity of the grown layers compared with the pure diffusion model of Dost et al [1994]. The shape of the grown layers is also shown to be very sensitive to changes in growth cell configurations. 6.2.1. Computational Domain The two vertical growth cell configurations simulated are shown schematically in Fig. 6.2.1. The dilute Ga-As solution is bounded at the top by a polycrystalline GaAs source and at the bottom by an n-type GaAs single crystal seed substrate. The graphite sections, placed at the top and bottom of the growth cell, play the role of electrodes. The liquid contact zone between the seed and the lower graphite provides a uniform, low resistance electrical contact. In SETUP1, the electric current passes through the lower graphite, the contact zone, the substrate and the solution, but bypasses the source material. The contact between upper graphite and the solution is achieved through a relatively small area along the upper part of the vertical wall. The bypassing of the source by the current reduces Joule heating in the source, and eliminates Peltier heating at the solution-source interface. In SETUP2, the electric current passes directly through the source and the source solution interface. This configuration is considered to illustrate the effect of growth cell configuration on the growth process, and also because of its historical value as an early LPEE growth cell developed. In an equilibrated isothermal LPEE system, the Peltier effect results in reduced temperatures at the growth interface, and increased temperatures at the dissolution interface. The flow of electric current through the substrate and the source induces Joule heating proportional to the square of electric current density and inversely proportional to electrical conductivity. Thus in SETUP1, where Joule heating in the source and Peltier heating at the dissolution interface are avoided, an axial temperature gradient is induced due solely to the Peltier cooling at the growth interface. In SETUP2, however, the temperature is increased by the Joule heating in the source and Peltier heating at the dissolution interface, and is reduced by Peltier cooling at the growth interface, thereby resulting in higher axial temperature gradients than in SETUP1. The axial temperature gradients induce supersaturation of the solution at the

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substrate solution interface, which in turn results in solute diffusion towards the substrate, leading to epitaxial growth. The second dominant growth mechanism of LPEE is the electromigration of solute species in the solution towards the substrate. Under the influence of the electric field E, species of the GaAs (Ga and As in these cases) solution migrate towards the anode. Hence, when the substrate has a positive polarity, the solution tends to become supersaturated in the vicinity of the substrate-solution interface; this results in a further contribution to epitaxial growth. 6.2.2. Preliminary Analysis LPEE experiments show that growth of bulk crystals is distinctly different from the growth of thin crystalline layers. As pointed out by Bryskiewicz et al. [1987] initially, and later by Sheibani et al. [2003a], certain additional problems are encountered in growth of bulk crystals which are not necessarily of significance in the growth of thin layers. For example, Joule heating in the substrate and source, and natural convection, driven by unstable density stratification due to temperature and concentration gradients in the solution, may become significant. Cases in which both temperature and concentration gradients have relatively equal contributions to convection are more complex than those involving either one as the dominant driving force. Two distinct cases of thermosolutal convection can be delineated. First, when the destabilizing density gradients are increased by the combined effect of temperature and concentration gradients, natural convection is enhanced. This is known as the augmenting effect. On the other hand, when the density gradients are reduced by the interaction of temperature and concentration gradients, this is referred to as the case of opposing effect (Ostrach, [1983]). For a Ga-As solution, solubility increases with temperature and the solute specific density is lower than that of the solvent. Therefore, the density of the solution decreases with higher solute concentrations (positive solutal expansion coefficient c = - (1/o)(d/dC)) or with higher temperature (positive thermal expansion coefficient t = - (1/o)(d/dT)). One can therefore expect that buoyancy-induced convective flow during LPEE growth of GaAs would be enhanced by the augmenting effect of temperature and concentration gradients. Before introducing the results of the two-dimensional thermosolutal convection numerical model, it would be instructive to assess the relative contributions to natural convection of the thermal and concentration gradients. The Grashof number representing the contribution of natural convection is given by

Grt = ( t gL3 /  2 ) T ,

and

Grc = (  c gL3 /  2 ) C

(6.2.1)

for temperature and concentration, respectively. In Eqs. (6.2.1), L is the characteristic dimension of the cell, g is the gravitational constant, v is the

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225

kinematic viscosity, and  T and  C are characteristic temperature and concentration differences. The Prandtl and the Schmidt numbers are defined in terms of thermal and solutal diffusivities,  and D, as Pr = / and Sc = /D. Table 6.2.1. Physical properties of the GaAs system (Dost et al. [1994]). Parameter

Symbol

Value

Growth temperature Current density Peltier coefficient Solution electric conductivity Solid (crystal and source) electric conductivity Thermal diffusivity Solutal diffusion coefficient Solution kinematic viscosity Solution density Solute electric mobility Thermal expansion coefficient Solutal expansion coefficient Thermal conductivities: Graphite Ga-As solution and contact zone GaAs substrate Boron nitride: the r-direction Boron nitride: the z-direction Crystal radius Furnace radius Graphite height Solution height Substrate thickness Source thickness Contact zone height

Tg J

E S  DC L μE t c

800°C 10 A/cm2 0.3 V 25 000 -1 cm -1 40 -1 cm-1 0.30 cm2/s 4.010-5 cm2/s 1.2110-3 cm2/s 5.63 g/cm3 0.027 cm2/V.s 9.8510-5 K-1 - 8.410-2 0.225 W/cm.°C 0.526 W/cm.°C 0.082 W/cm.°C 0.282 W/cm.°C 0.440 W/cm.°C 6.0 mm 36.0 mm 19.0 mm 6.0 mm 0.3 mm 5.0 mm 2.0 mm

In the LPEE growth of GaAs, the Peltier effect results in a typical temperature change of about 1°C across the cell. The corresponding concentration difference evaluated from the phase diagram (at 800°C) is 2.4510-4. Using the physical parameters for the GaAs solution given in Table 6.2.1, and taking the characteristic length of 0.6 cm as the height of the solution in the growth cell, we obtain Grt = 14200 and Grc = 3000. These values indicate the dominance of thermally driven natural convection over solutal convection, and a slower establishment of solutal convection is therefore to be expected. The Prandtl number for a GaAs solution at 800°C is Pr = 0.004. Heat transport is therefore expected to take place mainly via conduction, with very little influence of convective flow on the temperature distribution. This was corroborated by a

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numerical analysis which resulted in identical temperature distributions being computed with both the diffusion equation and the convective heat transfer equation. The value of the Schmidt number on the other hand, Sc = 30, indicates that convective transport has a significant influence on the solute concentration profiles, and thus on the growth rates. The Rayleigh numbers corresponding to the temperature and concentration fields are Rat= PrGr t= 57 and Rac = ScGrc = 9104. These values are a few orders of magnitude lower than the critical value of 108, and the flow may, therefore, be assumed to be laminar. The above dimensional analysis shows that convection, which is driven mainly by temperature gradients in the solution, is an important mass transport process but has a negligible effect on heat transfer. This allows the use of a numerically efficient computational model in which heat transfer is considered to take place via conduction only in both growth cell and solution. Mass diffusion and laminar convective transport are considered in the solution. By eliminating the direct dependence of the temperature field on convection, the computing requirements for the numerical simulations are significantly reduced. 6.2.3. Governing Equations The governing equations of the liquid phase of the two-dimensional numerical simulation models are obtained from the laws of mass conservation for solute species, and conservation of mass, balance of linear momentum, and balance of energy for the solution given in Chapter 4. Here for the sake of efficiency we only present the explicit forms of the governing equations of the convection model (Djilali et al. [1995]) from which the field equations of the diffusion model (Dost et al. [1994]) can easily be deduced. The energy and electric charge balance equations (steady-state and including only the Joule and Peltier effects) are used to determine temperature and electric current distributions in the cell; these two equations are identical to those presented in the diffusion model of Dost et al. [1994]. The magnetic field induced by the electric current is neglected. As discussed previously, due to the low Prandtl number of the Ga-As solution, the convective transport term in the energy equation is neglected, and the only coupling of the temperature and convective fields is through the thermal buoyancy source term in the momentum equation. The Ga-As solution is assumed to be a Newtonian fluid, and the flow is assumed to be incompressible and laminar. The assumption of incompressibility is relaxed for the source term in the momentum equations in order to include the effect of natural convection arising from density gradients (Boussinesq approximation). Surface kinetics effects are not incorporated into this model. Under the above conditions, the conservation of solute species (mass transport) gives the equation of mass transport as

Single Crystal Growth of Semiconductors from Metallic Solutions

C C C + (u + μ E Er ) + (w + μ E Ez ) = DC  2C r z t

227

(6.2.2)

where

2 

2 r 2

+

1  2 + r r z 2

(6.2.3)

The associated boundary and interface condition are taken as C = C1,

at the solution-substrate (growth) interface,

C = C2,

at the solution-source (dissolution) interface,

C/ r =0,

at the axis of symmetry and the vertical wall.

The initial condition is C = C0

at

t = 0,

where C is the solution mass concentration, μE is the solute mobility, Er and Ez are the electric field intensities in the r and z directions respectively, u and v are the r- and z-direction velocity components, C0 is the saturation concentration of GaAs solution at the growth temperature, and C1 and C2 are the concentrations at the growth and dissolution interfaces which are assumed to be in equilibrium with the solution at their respective temperatures. C1 and C2 are determined from the liquidus equilibrium condition based on the interface temperatures; these temperatures are obtained from the thermal analysis. The conservation of mass and the balance of momentum yield respectively the equations of continuity

u u w + + =0 r r z

(6.2.4)

and momentum

u u 1 p u u +u + w =  +  ( 2u  2 ) r z t  L r r

(6.2.5)

w w 1 p w +u +v = +  2 w  g{t (T  T0 ) +  c (C  C0 )} r z t  L z

(6.2.6)

with the boundary conditions

u = 0 and w / r = 0 u = 0 and w = 0 and the initial conditions

at the axis of symmetry at the interfaces and vertical wall

Sadik Dost and Brian Lent

228

u = 0 and w = 0

at t = 0

The above equations are nonlinear in terms of the velocity, temperature and concentration fields. These fields are coupled through the convective and source terms in Eqs. (6.2.2) and (6.2.6). The incorporation of the convective terms significantly increases the complexity of the problem from the numerical solution standpoint. The steady heat conduction (energy) and electric charge balance equations are identical to those used in Dost et al. [1994]. The energy equation is given by

kr (

 2T

+

r 2

1 T  2T J 2 ) + kz 2 + =0 r r E z

(6.2.7)

with the following boundary conditions

T / r = 0 T = Tg

T  k zL z T k zs  k zL z k zs

at the axis of symmetry along the outside of the furnace

T =  J z T =J z

at the substrate-solution (growth) interface at the substrate-contact zone, and the sourcesolution interfaces

kr and kz are the general conductivities in the radial and axial directions, k zs and k zL are the values of thermal conductivities in the z-direction corresponding to the substrate and solution; Tg is the growth temperature, J is the applied electric current density, and is the Peltier coefficient. The last term in Eq. (6.2.7) represents the Joule heating in the bulk, and the term J is the Peltier heating/cooling at the interfaces. With the definition of electric potential  as

Er = 

  , and Ez =  r z

(6.2.8)

the equation of electric charge balance takes the following form

 2 = 0

(6.2.9)

The associated boundary conditions are

 =J z  / z = 0

E

at the solution-substrate interface at the solution-source interface

Single Crystal Growth of Semiconductors from Metallic Solutions

 / z = 0 =0

229

at the boron nitride wall at the graphite-solution contact zone

Once the electric field equation, Eq. (6.2.9), is solved, the electric field components are computed from Eq. (6.2.8). Upon solving Eq. (6.2.2) for the concentration C, the growth rate is computed via  C 1 (6.2.10) V g = L {DC ( )0 + μ E Ez C L } n (Cs  C L ) s 6.2.4. Physical Parameters The accuracy of the computed velocity, temperature and concentration fields hinges on the accuracy of the phase diagram and the thermophysical properties. Because few measurements are available to correlate accurately the physical parameters of Ga-As solutions, we have to make some assumptions and rely on approximate expressions to evaluate those parameters for which experimental data are not available. Since the Ga-As solution is dilute, the density, viscosity and thermal expansion coefficient of the solution are taken as those of Ga. Following Smithells [1976], the variation of the density with temperature and concentration is represented by a linear equation

 L = 0 + (T  T0 )(d  / dT ) + (C  C0 )(d  / dC)

(6.2.11)

where 0 is the density of the liquid metal at its melting point T0. For liquid gallium, 0 = 6.09 g/cm3, T0 = 29.8°C, and d/dT = -0.610-3 g/cm3.K. C0 is the corresponding reference concentration. The variation of viscosity μ with temperature is given by

μ = μ0 e A/ RT

(6.2.12)

where μ0 and A are constants, and R = 8.3144 J/K. mol is the gas constant. For -3 liquid gallium, μ0 = 4.35910 g/cm.s, and A = 4 kJ/mol. The temperature dependence of the solute diffusion coefficient, DC, is calculated using the relation given by Rode [1973]

DC = 5000e210

4

/T

(6.2.13)

Following Crossley and Small [1971], the liquidus equilibrium mass concentration of As species, X(T), can be written as

X (T ) = 6e8.421.3210

4

/T

(6.2.14)

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230

The mass fraction C(T) is then obtained by dividing the mass concentration X(T) by the density of the solution, L. The phase diagram obtained using the above relations is in good agreement with that given in Hall [1963]. In the Boussinesq approximation adopted in the model, the solution density L is expressed in the last term in Eq. (6.2.6) where the thermal and solutal expansion coefficients are evaluated using the following definitions

t = 

1 d 1 d , and  c =  0 dT 0 dC

(6.2.15)

The value of t is calculated using the first term of Eq. (6.2.11) and Eq. (6.2.15) 1. The value  c is estimated through the procedure given by Long et al 1974:

c =

M As  (V As  VGa ) M Ga

(6.2.16)

CGa M Ga + C As M As

where M As and M Ga , V As and VGa , and CGa and C As are the molecular weights, the molecular volumes, and the concentrations of Ga and As, respectively. The values of the various physical parameters corresponding to a growth temperature of 800°C are listed in Table 6.2.1. 6.2.5. Numerical Solution Method The governing partial differential equations, Eqs. (6.2.2), (6.2.4)-(6.2.7), and 0.0 at z = 0.03 cm

T (°C)

-0.5

at z = 0.12 cm -1.0 at z = 0.30 cm -1.5 0.0

0.1

0.2

0.3

0.4

0.5

0.6

Radial distance (cm) Fig. 6.2.2. Temperature profiles at the growing interface for SETUP 1. T = T-T0 is the relative temperature with respect to the furnace temperature T0 . Details can be found in Dost et al. [1994].

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231

(6.2.9), together with the appropriate boundary, interface and initial conditions, and the incorporation of relations in Eqs. (6.2.11)-(6.2.16), given previously, are solved numerically by a finite volume method (Patankar [1980]) in which the calculation domain is divided into a finite number of rectangular cells, i.e., of finite volumes. The governing differential equations are integrated over the finite volumes, and a central differencing scheme is used to discretize the equations. It should be noted that no upwinding is required for the convective terms since the cell Peclet numbers have maximum values one order of magnitude lower than the critical value of 2, beyond which numerical instabilities can occur. The discretization scheme is therefore second-order accurate in space. A fully implicit time marching procedure is employed in conjunction with the iterative SIMPLE (Semi-implicit method for pressure-linked equations) algorithm for the pressure-velocity coupling (Patankar [1980]). A non-uniform mesh arrangement is used to allow efficient meshing in high gradient regions, and a staggered grid arrangement, with velocity nodes offset from scalar nodes used to avoid spurious oscillations in the pressure field. The discretized set of algebraic equations are solved using a block-iterative solution procedure which allows the use of an efficient tri-diagonal matrix inversion procedure (Thomas algorithm). At each time step, the velocity and pressure field are solved iteratively using the SIMPLE algorithm. Once converged solutions are obtained, the solute transport equation is solved, and the time step is then advanced. The thermal field is computed using the numerical strategy 0.0

Growth interface Dissolution interface J = 10 A/cm2

T (°C)

-0.2

J = 20 A/cm2

-0.4

J = 10 A/cm2

-0.6

J = 20 A/cm2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Radial distance (cm) Fig. 6.2.3. Comparison of interface temperature profiles for J = 10 and 20 A/cm2 (redrawn from Dost et al. [1994]).

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Sadik Dost and Brian Lent

outlined in Dost et al. [1994]. This strategy takes advantage of the fact that (i) the effect of convection on the temperature field is negligible (low Prandtl number), and (ii) the thermal field is quasi-steady for any given growth thickness. This allows the decoupling of the thermal field from the flow and solute fields, while accounting for the effect of varying source and substrate thicknesses. 6.2.6. Simulation Results Numerical simulations are carried out for the two cell configurations shown in Fig.6.2.1. The growth temperature and electric current density are taken as 800°C and 10 A/cm2. The results of the thermal analysis are identical to those presented in the diffusion model (Dost et al. [1994]). The relative temperature profiles at various locations at the growing interface are given in Fig. 6.2.2. In SETUP1, no Joule heating takes place in the source material. The growth interface temperature drop due to Peltier cooling is of the order of 1°C. In

Fig. 6.2.5. Evolution of velocity field and isoconcentrations with growth time for SETUP1. Convection increases with time and maximum velocities of about 0.03 cm/s are attained after 60 hours (after Djilali et al. [1995]).

Single Crystal Growth of Semiconductors from Metallic Solutions

233

SETUP2, Peltier heating takes place at the dissolution interface and Joule heating in the source. The dissolution interface temperature increases typically by about 2°C; this results in higher temperatures at the growth interface due to heat transfer into the solution. Although the temperature fields are different for each of the two configurations, the temperature differences between the dissolution and the growth interfaces are negative in both cases, i.e., the growing interface is cooler than the dissolving interface, resulting in an axial temperature gradient favourable for growth. The thermal analysis also shows significant radial temperature gradients along the growth interface in both configurations. In SETUP1, the temperature in the solution increases gradually from the cell center to the cell wall, whereas the reverse is observed in SETUP2. The magnitude of this radial temperature gradient is of the order of l °C/cm, which is comparable to the axial temperature gradients. It is the radial temperature gradient that causes convective flow in the solution and nonuniform thicknesses of grown crystals. As mentioned in the previous section, in LPEE the magnitude of Peltier cooling and the electric mobility of species in the solution are directly proportional to the current density, leading to a well established linear variation of growth rate with electric current density (Jastrzebski et al [1978]). Increasing current density appears therefore to be a simple and expedient way of obtaining faster growth rates. However, the increasing contribution of Joule heating, which varies with the square of current density, may affect both growth rate and temperature distribution, leading to growth instabilities. Indeed the LPEE experiments of Sheibani et al. [2003a,b] have shown that in the growth of thick crystals, the use of higher electric current densities leads to growth instabilities. For instance, as we will see in later sections, in the growth of GaAs, it was not possible to exceed the electric current density of 7 A/cm2. 0.0

T (°C)

Dissolution interface

-1.0

Growth interface

-2.0

0.0

0.1

0.2

0.3

0.4

Radial distance (cm) Fig. 6.2.4. Interface temperature as a function of substrate thickness at J = 20 A/cm2 (redrawn from Dost et al. [1994]).

Sadik Dost and Brian Lent

234

The overall effect of increasing electric current density was investigated by increasing the current density from 10 A/cm2 to 20 A/cm2 in Dost et al. [1994]. Results are shown in Fig. 6.2.3. As can be seen, the temperatures decrease significantly at both dissolution and growth interfaces with increasing electric current density. The increased electric current density also results in a larger effective temperature difference in the solution. The variation of the relative temperature at the interfaces as a function of substrate thickness, is also obtained numerically from this diffusion model, and is shown in Fig. 6.2.4. It was found that the temperature decreases with increasing substrate thickness (thickness of the grown crystal) up to a certain value, and then increases thereafter. The critical value is attained earlier for the dissolution interface. This behaviour illustrates the important role played by the Joule heating in the thermal analysis. The growth rates obtained by applying an electric current density of 20 A/cm2 are almost twice those obtained at J = 10 A/cm2. The corresponding contributions of Peltier effect and electromigration remain the dominant driving mechanism for growth. The proportional increase of the two contributing mechanisms was confirmed with further calculations at J = 30 A/cm2. SETUP 1 Both temperature and concentration are higher at the wall, resulting in a higher density solution in the upper central region of the cell. The unstable density stratification causes the onset of a toroidal convective flow pattern with an upward moving boundary layer at the wall of the growth cell as depicted in Figs. 6.2.5a and 6.2.5b. 0.

Growth rate (μm/min)

r = 0 (centre) 0. r = 0.3 average

0.

diffusion model 0. r = 0.6 (wall) 0. 0

1000

2000 3000 Growth time (min)

4000

5000

Fig. 6.2.6. Variation of local and average growth rates with time (SETUP1) (redrawn from Djilali et al. [1995] and Dost et al. [1994]).

Single Crystal Growth of Semiconductors from Metallic Solutions

235

Fluid motion gradually gathers strength with growth time since the temperature gradient increases with increasing substrate thickness. The overall flow pattern remains unchanged during growth. The corresponding isoconcentration patterns are shown in Figs. 6.2.5c and 6.2.5d. Compared to the diffusion model predictions of Dost et al. [1994], higher concentration gradients at the substrate-solution and the source-solution interfaces are achieved due to convective transport. These higher concentration gradients lead to faster deposition and dissolution at the two interfaces. The growth rate on a substrate with an initial thickness of 300 μm is plotted against growth time in Fig.6.2.6. Consistent with the enhanced solute transport due to convection, higher average growth rates are achieved compared to the diffusion case. Although the average growth rate is almost constant, resulting in an essentially linear variation of average growth thickness with time, the local growth rates at various locations across the interface vary significantly due to the non-uniform mass flux associated with convective flow. The maximum growth rate occurs at the cell center in this configuration. While the growth rate at the center increases with time, that at the wall decreases. This results in growth of an increasingly non-uniform substrate as shown in Fig. 6.2.7 which is markedly different from the essentially flat substrate predicted using the diffusion model of Dost et al. [1994]. The contributions of electromigration and Peltier effect to the overall growth rate are shown in Fig. 6.2.8. The electromigration contribution is constant and identical to that obtained in Dost et al. [1994], i.e. it is independent of convection. It also appears to be the dominant growth mechanism in this setup. The contribution of Peltier cooling, on the other hand, does depend on

grown crystal thickness (cm)

0.30

0.20

0.10

0.00 -0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

radial coordinate (cm) Fig. 6.3.7. Evolution of the shape of the substrate during growth (SETUP1; the time increment for each curve is 10 h, redrawn from Djilali et al. [1995]).

Sadik Dost and Brian Lent

Contribution to growth rate (μm/min)

236

electromigration 0.3

0.2 Peltier effect 0.1

0

1000

2000 3000 Growth time (min)

4000

5000

Fig. 6.2.8. Contributions of electromigration and Peltier cooling to the overall growth rate in SETUP1 (redrawn from Djilali et al. [1995]).

convection since the concentration gradient at the growth interface is a function of the convective flow in the solution. Though the contribution of Peltier cooling to the growth rate increases gradually with time (or substrate thickness) due to increasing temperature gradients and, thus, enhanced convective solute transport, it remains small compared to the electromigration contribution. Simulations were also performed for different growth temperatures and electric current densities. The results are compared with the experimental data of Bryskiewicz et al. [1987a] and Jastrzebski et al. [1978b] in Fig. 6.2.9. The average growth rate varies linearly with electric current density for all growth temperatures, and for a given current density, the growth velocity increases with increasing temperature due to higher interface concentration C1, diffusion coefficient DC and electromigration velocity μE. The thermosolutal model simulations are in good agreement with experimental data. SETUP 2 In this configuration the electric field is uniform in the solution. However, due to Peltier heating at the source-solution interface and to Joule heating in the source, the temperature distribution in the horizontal direction is opposite to that of SETUP1, i.e. the temperature in the solution decreases gradually from the cell center to the cell wall. The consequences are, first, a convection pattern in the opposite direction to that in SETUP1, and second, much higher intensity of convection during the initial phase (maximum velocities of about 0.08 cm/s compared to 0.015 cm/s for SETUP1) followed by a noticeable reduction of convection with time. This reduction is due to the reduced influence of Joule heating in the dissolved source material.

Single Crystal Growth of Semiconductors from Metallic Solutions 1.5

237

Convection model

Average growth rate (μm/min)

Diffusion model Experiments

1.0

0.5

0.0 0

2

4 6 Current density (A/cm2)

8

10

Fig. 6.2.9. Effect of current density and furnace temperature on growth rate in SETUP1 (redrawn from Djilali et al. [1995], experimental data from Bryskiewicz et al. [1987a] and Jastrzebski et al. [1978b]). Growth rates given for GaAs at 900 °C by Sheibani et al. [2003b] are in good agreement with these results.

Evolution of the velocity field and the associated isoconcentrations are shown in Fig. 6.2.10 at two different growth times. Unlike SETUP1, higher concentration gradients occur near the wall. This results in higher growth rates near the wall and the double-hump shape observed experimentally and reproduced by the present simulations as illustrated in Fig. 6.2.11. We also note that the strong convection observed right at the onset of the growth process causes thickness non-uniformity earlier than in SETUP1, but with the gradual reduction in convection, the shape stabilizes eventually. The contributions of electromigration and the Peltier effect to the overall growth rate are shown in Fig. 6.2.12 for two applied current densities. Comparing the case J = 10 A/cm2 to the results obtained for SETUP1, a much more significant contribution of Peltier cooling is observed. This, naturally, is a consequence of the higher temperature gradients and the enhanced convective transport in the solution. However, as expected, the contribution of Peltier cooling to the growth rate decreases rapidly with time since the temperature gradients become smaller as the substrate grows and the source dissolves. Higher growth rates can be achieved by applying higher electric current densities, as shown in Fig. 6.2.12 for J = 20 A/cm2. The higher Joule heating in the source material enhances the strength of convection considerably, to the point where the contribution of Peltier cooling becomes comparable with that of

238

Sadik Dost and Brian Lent

Fig. 6.2.10. Evolution of velocity field and isoconcentrations with growth time for SETUP2. Convection decreases with time and maximum velocities of about 0.08 cm/s occur during the initial phase (after Djilali et al. [1995]).

electromigration during the initial stages of growth. A further increase in the applied electric current density would, of course, make Peltier cooling the dominant growth mechanism in this configuration, but would also result in stronger convection and therefore higher substrate nonuniformity. We also note that the relationship between growth rate and the applied electric current density is non-linear for this cell configuration, since Joule heating is proportional to the square of current density. The two-dimensional (axisymmetric) simulation model presented for the two LPEE growth cell configurations demonstrates the importance of natural convection resulting from density variations in the solution due to both thermal and solutal gradients. The results show that the overall growth rate increases significantly due to thermosolutal convection which also results in the formation of non-uniform substrates. It is important to mention that, as also shown numerically in this section, the shape of the interface of the grown crystal is strongly influenced by the growth

Single Crystal Growth of Semiconductors from Metallic Solutions

239

grown crystal thickness (cm)

0.30

0.20

0.10

0.00 -0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

radial coordinate (cm) Fig. 6.2.11. Evolution of the shape of the substrate during growth (SETUP2; the time increment for each curve is 10 h, after Djilali et al. [1995]).

cell configuration. For instance, as we will see later in this chapter, the LPEE growth cell used in Sheibani et al. [2003a,b] has produced many large crystals of very flat interfaces. Considering the two cell configurations simulated here, one can conclude that in SETUP2 convection enhances the contribution of Peltier cooling to the growth rate, whereas in SETUP1, where the current bypasses the source, the process is essentially controlled by electromigration. Although the induced electric field is non-uniform in SETUP1, this configuration is superior to SETUP2 for growing better quality thicker crystals with a higher growth rate because of the weaker convection in the solution for a given applied electric current density. The simulations predict that the growth rate is almost proportional to the applied electric current density, and increases with increasing growth temperature. These predictions, as well as the calculated growth rates, are consistent with experimental observations (Bryskiewicz et al. [1987a], Jastrzebski et al. [1978b], and Sheibani et al. [2003a,b]). The above thermosolutal simulation model did not account for the moving boundaries associated with the growing crystal and dissolving source. Therefore it is expected that the interface shape could be better predicted by a numerical simulation model that includes interface movement. This was studied numerically by Qin et al. [1995] and the simulation results are presented in the next section.

Sadik Dost and Brian Lent

240 electromigration

Contribution to growth rate (μm/min)

Peltier effect J = 10 A/cm2

0.7

J = 10 A/cm2

0.5 J = 20 A/cm2 0.3 J = 20 A/cm2 0.1

0

500

1000 1500 Growth time (min)

2000

2500

3000

Fig. 6.2.12. Contributions of electromigration and Peltier cooling to the overall growth rate in SETUP2 for two applied current densities (after Djilali et al. [1995]).

6.2.7. Effect of Moving Interface In early simulation models developed by Dost et al. [1994] and Djilali et al. [1995], the shape of the moving interfaces was not taken into consideration. In general, the assumption of flat interfaces in numerical simulations is appropriate when the changes in the substrate and source thicknesses during growth are very small compared to the height of the solution zone. This is the case in growing thin epitaxial layers. The variation of interface shape in LPEE depends on the design of the growth crucible, and may be very significant in the growth of bulk (mm-thick) crystals. For instance, this is the case in the growth crucibles presented in Fig. 6.2.1. Therefore, simulation models of these crucibles, for accurate predictions, must include the effect of the evolving shape of the solidliquid interfaces, on the growth process. However, as presented in Chapter 3 and also later in this chapter, an LPEE crucible can be designed to obtain flat growth interfaces as shown experimentally by Sheibani et al. [2003a,b]. The finite element method provides a good vehicle for simulations involving moving and curved interfaces due to its flexibility in dealing with moving boundaries and irregular domains. Below we present such a model for the growth crucible SETUP1 (Fig. 6.2.1) in which the applied electric current bypasses the source at the top (Qin et al. [1995]). In this model the governing equations of the liquid and solid phases and most of the interface/boundary conditions are the same given in Section 6.3.2.

Single Crystal Growth of Semiconductors from Metallic Solutions

241

However, due to the inclusion of interface curvature the energy balance and electric field interface conditions are different, and are given below. The interface condition corresponding to the balance of energy is given as follows (see Eq.(4.4.3)):

(q s  q L )  n = s L (n  V g ) + v  (t em  n) s

(6.2.17)

where q s and q L are the heat fluxes in the solid and liquid phases, n is the exterior normal to the interface, V g is the velocity of the interface, L is the latent heat, and t em is the Maxwell stress in the solid phase. In LPEE s experiments the velocity of the growth interface (also that of dissolution interface) is very small. For instance, in the growth of GaAs bulk crystals (4.0 to 6.0 mm) and in the absence of an applied magnetic field, the experimentally achieved growth velocity is 0.5 mm/day at J = 3 A/cm2, 0.7 mm/day at J = 5 A/cm2, and 1.2 mm/day at 7 A/cm2 (in other words, on average it is on the order of 10-8 m/s) . On the other hand, the flow velocity in the liquid solution is about (numerical) 0.7  10-3 m/s. Therefore the effect of moving interface to the energy equation, as well as to the other field variables, may be neglected. Under this assumption the interface condition in Eq. (6.2.17) becomes At the growth interface

(ksT  k LT )  n +  Ls (J  n) = 0

(6.2.18)

At the substrate-contact zone interface

(ksT  k LT )  n +  sL (J  n) = 0

(6.2.19)

Since in this model the Ga-As solution and Ga-Al contact zone are dilute (Garich solutions), we take  Ls =  sL =  . In SETUP1 there is no Peltier heating (current bypasses the source), the interface condition at the dissolution interface thus becomes At the dissolution interface

(ksT  k LT )  n = 0

(6.2.20)

The conditions associated with the electric potential are given by

 =J n  / n = 0  / n = 0 =0

E

at the solution-substrate interface, at the solution-source interface, at the boron nitride wall, and at the graphite-solution contact zone interface.

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6.2.8. Numerical Solution and Simulation Results The moving boundary problem defined above is solved by the finite element method. The governing partial differential equations are discretized using the Galerkin method and developed using a nine-node isoparameter finite element model. The simulation employs piecewise continuous biquadratic approximations for velocity, temperature, and concentration, and a bilinear approximation for pressure to avoid the well-known difficulties of mixed velocity/pressure formulation in compressible flows (Zienkiewicz and Taylor [1989], Olson and Tuann [1978]).

(a)

(b)

Fig. 6.2.13. Nine-node quadrilateral element mesh (a), and Six-node triangular element mesh (b) for the complete growth cell (Qin et al. [1995]).

The resulting set of first-order simultaneous ordinary differential equations are further discretized by the fully implicit time-marching algorithms based on the finite difference method, which allows larger time steps because of its higher accuracy and numerical stability. The non-linear algebraic equations resulting from the implicit approximation at each time step are solved by a modified Newton-Raphson iteration in which the stiffness matrix is updated only at the start of every time step. This is more efficient for the problem under consideration due to fewer reformations and factorizations of the tangent stiffness matrix, although the convergence speed is

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243

slower as compared to the quadratic convergence of the full Newton-Raphson iteration. The resulting linear algebraic equations in each Newton-Raphson iteration are solved by applying the linear algebra package LAPACK (Anderson [1992]). The convergence is declared and the time step is advanced when

f i n+1  f i n / f i n  103

(6.2.21)

where f i denotes variable to be solved for, and n is the iteration number. The computational procedure employed here takes advantage of the fact that the effect of convection on the temperature field is negligible so that the heat transfer equation is decoupled from the momentum equations. Thus two different meshes can be used in the simulations. For thermal analysis, the mesh includes the complete cell configuration while the mesh for the mass transport

Fig. 6.2.14. Evolution of mesh covering the Ga-As solution: (a) 5 min, (b) 20 h, (c) 40 h, (d) 60 h (after Qin et al. [1995]).

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analysis covers only the solution zone. For computational convenience, identical mesh structures are used in the overlapping regions for thermal and mass transport calculations. Since growth rates are very small, the growing and dissolving interfaces move very slowly, and thus the motion of the solution, substrate and source regions are very small from one time step to the next in the numerical calculation. The finite element meshes are not regenerated completely, but the nodes are moved following the shapes of the interfaces in each time step so that the meshes adapt to the deformed substrate, solution and source. The change in the location of nodes between two time steps is very small, so the values of all variables computed in the previous step are used directly in the present step without any interpolation to reduce computing time. The overall computational procedure consists of the following steps:

Fig. 6.2.15. Temperature distribution at various steps of the growth process: (a) 5 min, (b) 20 h, (c) 40 h, (d) 60 h (after Qin et al. [1995]).

Single Crystal Growth of Semiconductors from Metallic Solutions

1) 2) 3) 4) 5) 6) 7) 8) 9)

245

Set initial growth cell configuration and initial concentration distribution, Generate initial meshes for the physical domain, Calculate electric field, Calculate temperature distribution, Calculate velocity field and concentration distribution, Calculate growth rate and growth thickness, Determine interface shape, Update growth cell configuration and generate new meshes, and Return to Step 3 and begin calculations for the next step.

Fig. 6.2.16. Temperature profiles at (a) the growing and (b) dissolution interfaces, top curve is at t = 0, and others are at the increment of 10 h (after Qin et al. [1995]).

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The physical and growth parameters used in SETUP1 were given in Table 6.2.1. A sequence of mesh design experiments showed that the mesh of nine-node quadrilateral elements shown in Fig. 6.2.13a is quite adequate (results on finer meshes differed little). Although the mesh contains elements which have high slenderness in the boron nitride regions, numerical analysis has shown that these elements do not cause significant numerical error since temperature gradients are very small in LPEE (this is not the case for simulations of a THM system, Meric et al. [1999]). In a test case with the node numbers doubled in the r-direction, the maximum temperature difference was found to be in the order of 10-5 and maximum concentration difference of the order of 10-10. The mesh of six-node triangular elements shown in Fig. 6.2.13b was also used in the

Fig. 6.2.17. Velocity field at various steps of the growth process: (a) 5 min, (b) 20 h, (c) 40 h, (d) 60 h (after Qin et al. [1995]).

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247

computations, and the numerical results based on the two types elements were identical. Evolution of the finite element meshes in the solution zone during a typical computation is shown in Fig. 6.2.14. The substrate thickness increases and the source thickness decreases with growth time, and the interface becomes curved. The finite element meshes deform significantly during the growth process to adapt to the evolving interfaces.

Fig. 6.2.18. Concentration contours at various steps of the growth process: (a) 5 min, (b) 20 h, (c) 40 h, (d) 60 h (after Qin et al. [1995]).

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Sample temperature contours in the solution are shown in Fig. 6.2.15. The relative temperatures at the interfaces decrease with growth time (or substrate thickness), indicating the temperature distribution in the solution is mainly controlled by Peltier cooling (Fig. 6.2.16). The velocity and isoconcentration contours are shown respectively in Figs. 6.2.17 and 6.2.18. The strength of fluid motion increases with growth time due to the increasing radial temperature gradients. The convective flow causes higher concentration gradients, leading to faster growth rate near the center of the growing interface, while electric flow through the solution-graphite contact region (at the upper part) results in higher concentration gradients and dissolution rate near the upper right corner of the solution region. Despite the relatively high concentration gradients in these regions, the problem is not convection-dominated, and numerical solutions are stable without upwinding. The evolution of the growing substrate is shown in Fig. 6.2.19. Thickness non-uniformity is due to the effect of convection and the surface profiles reflect the flow patterns in the solution. In comparison with the earlier computations of Dost et al. [1994] (Fig. 6.2.7) which were based on a fixed, flat interface assumption, the present computations (Qin et al. [1995]) which track the motion and shape of the interfaces predict enhanced convective transport. The predicted variations in crystal thickness increase because of the enhanced convective flow. This could cause interfacial instability in the growth of bulk crystals. The contribution of electromigration and Peltier cooling to the overall growth rate is identical to that given in Fig. 6.2.8, where the contribution of electromigration, as expected, is almost constant; however, the contribution of the Peltier effect increases with time due to enhanced convection caused by increasing thickness and interface uniformity. Hence, the ratio of electromigration induced growth rate to the total growth rate decreases with time.

Fig. 6.2.19. Evolution of the grown crystal surface with time: the bottom curve is at the 10th hour of growth, and other curves are at the increments of 10 h (after Qin et al. [1995]).

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249

Fig. 6.2.20. A GaAs crystal grown at 10 A/cm2 (after Sheibani et al. [2003a,b]).

The interface stability analysis of Okamoto et al. [1982] suggests that this ratio is an important measure for interface stability, and its decreasing value indicates a tendency for unstable growth. Bryskiewicz et al. [1987] have observed experimentally the presence of a critical crystal thickness above which the growth becomes unstable. For higher electric current densities, the contribution of Peltier cooling to the growth rate increases due to the enhanced convection in the solution (a detailed section on growth stability in LPEE will be presented later in this Chapter). It must be mentioned here that, although the above conclusions can generally be applied to an LPEE growth system, they are drawn from the simulations carried out for SETUP1 in which the electric current bypasses the source as shown in Fig. 6.2.1. Thus, the relative contribution of Peltier cooling will depend on the setup used, since the intensity of convective flow in the solution will be different. Nevertheless, as the contribution of Peltier cooling increases with the increasing electric current density, the application of a higher electric current density may lead to interface instabilities, particularly in the growth of thick crystals. Indeed, as shown in Sheibani et al. [2003a,b], the growth interface loses its stability at electric current densities higher than 7 A/cm2. A sample GaAs crystal grown at 10 A/cm2 electric current density is shown in Fig. 6.2.20. As seen, the interface lost its stability after about 3 to 4 mm growth. 6.3. LPEE Growth of Ternary Alloys The LPEE growth of ternary alloys was numerically simulated by Qin and Dost [1996] using a diffusion model for the growth of AlGaAs, and by Dost and Qin [1998] including the effect of convection in growth of GaInAs. Below, we present the important features of these models. First, the general governing equations derived in Chapter 4 are written explicitly for a ternary system including the effect of convection.

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6.3.1. Model Equations of an AxB1-xC Ternary System In the growth of AxB1-xC ternary systems of III-V, for computational convenience we choose mole fractions since the phase diagrams are given in mole fractions. xA, xB and xC represent respectively the mole fractions of elements A, B, and C in the liquid phase, and xCs is the mole fraction of element C in the solid phase. For instance, A, B, and C will represent respectively Al, Ga, and As in the case of AlxGa1-xAs, and Ga, In, and As in the GaxIn1-xAs system. There will be only two independent variables in the liquid phase since

x A + x B + xC = 1

(6.3.1)

In the solid phase, the solid element C is always 50 at.% if the solid-state defects are neglected, so that

x sA = x / 2 , x Bs = (1 x) / 2 , xCs = 1 / 2

(6.3.2)

where x is the solid composition of the grown crystal. The importance of mass diffusion in the solid phase was demonstrated by considering the components at the moving interface (Small and Ghez [1978], Ghez and Small [1981]). It plays an important role when one considers constitutional conditions of a heterostructure. Mass diffusion takes place between elements A and B in a III-III-V ternary crystal. Therefore, if one neglects the defects in the solid, then the element C sublattice must be filled, and thus the mole fraction of C, xC, will be constant. Diffusion in the solid phase will then be governed by a single mass transport equation

s

Dx sA Dt

= i( s D AsC)

(6.3.3)

where  s is the mole density, D As is the diffusion coefficient of element A in the solid, and the  is the gradient operator given in Eq. (6.2.3). Since there is no particle motion in the solid phase, Eq. (6.3.3) takes the following form in an axisymmetric system (in cylindrical coordinates with the assumption of axisymmetry)

x sA t

= D As 2 x sA +

D As x sA  s x sA  s ( + ) z z  s r r

(6.3.4)

where we allowed changes in the mole density. The relationship between the mole density and the solid composition x was given in Eq. (5.5.7). The energy equation becomes

Single Crystal Growth of Semiconductors from Metallic Solutions

 s s

J2 T = k s  2T + s t E

251

(6.3.5)

where  s is the specific heat of the solid phase. The equation of continuity in the liquid phase is the same given in Eq. (6.2.4). However, the momentum equations will be slightly different since the body force term was written for two components, and also expanded in mole fractions, we write them again here for convenience, u u 1 p u u (6.3.6) +u + w =  +  ( 2u  2 ) r z t  L r r

w w 1 p w +u +v = +  2 w r z t  L z  g{t (T  T0 ) +  A (x A  x 0A ) + C (xC  xC0 )}

(6.3.7)

where x 0A and xC0 are the initial mole fractions of A and C. The electric field equation is the same given in Eq. (6.2.9) with the definitions of Eq. (6.2.8). The conservation of mass for species yields two mass transport equations

x A x x x x + u A + w A = D A 2 x A + μ EA Er A + μ EA Ez A t r z r z

(6.3.8)

xC x x x x + u C + w C = DC  2 xC + μ CE Er C + μ CE Ez C t r z r z

(6.3.9)

where μ EA and μ CE , and D A and DC are the electric mobilities, and the diffusion coefficient of species A and C in the liquid solution (mixture of B-rich A-B-C), respectively. The energy equation is

k T T T +u +w = T  2T r z  L L t

(6.3.10)

where  L is the specific heat of the liquid phase. Interface Conditions The interface conditions given in Eqs. (4.4.5), which were written for conservation of mass at the interfaces, yield explicitly the following conditions for the LPEE setup considered:

μ A Ez x A +  D A

x A z

=

s

D As

x sA z

+  sV g (x sA  x A )

(6.3.11)

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252

μC Ez xC +  DC

xC

1 =  sV g (  xC ) 2 z

(6.3.12)

where  is the mole density of the liquid phase. The left-hand sides of Eqs. (6.3.11) and (6.3.12) represent, respectively, the contributions of electromigration and diffusion from the liquid phase. The first term on the righthand side of Eq. (6.3.11) represents solid diffusion, which does not appear in Eq. (6.3.12) because the composition of element C is assumed to be constant in the crystal. The last terms in both equations represent solidification with growth rate V g and relative concentrations at the interface. The interface concentration in the liquid side is coupled, through the phase diagram, with the solid concentration which changes with time. The interface conditions consist of phase equilibrium between the substrate and solution and a mass balance between the transported and incorporated solute species. One of the most difficult problems in numerical simulations of growth of ternary crystals is to approximately satisfy the interface conditions. The interface conditions for energy balance at the interfaces were given in Eqs. (6.2.18)-(6.2.20). The remaining boundary and initial conditions used in the model are the same those given in Section 6.2. Table 6.3.1. Phase diagram parameters for Al-Ga-As (Qin and Dost [1996]). Parameter

Symbol

Value

Entropy of fusion of AlAs

S AC

15.6 (cal/mole K)

Entropy of fusion of GaAs

S BC

16.64 (cal/mole K

Melting point of AlAs

F F

F

TAC

2043 (K)

F

Melting point of GaAs

TBC

1511 (K)

Interaction coefficient of Al-Ga

 AB

104 (cal/mole)

Interaction coefficient of Al-As

 AC

600-12T (cal/mole)

Interaction coefficient of Ga-As

 BC

5160-9.16T (cal/mole)

Interaction coefficient of AlAs-GaAs

 AC  BC

0

L

L

L

s

6.3.2. Phase Diagram, and Physical and Growth Parameters The concentrations of the solid and liquid phases at the interface must satisfy the phase diagram. For binary systems, the assumption of equilibrium deposition is valid and the interface concentration can be determined simply by using the phase diagram based on the growth temperature. In the growth of ternary alloys, however, the solid composition varies in the crystal and is therefore determined by both the mass transport ratio of element A to element C and the phase diagram. The growth rate cannot be calculated by the mass

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253

transport rate alone, and it must be determined by the ratio of solute elements transported through the solution to the interface and the diffusion in the solid. Mass transport equations at the interface and phase diagram must be solved simultaneously for concentrations x A , xC , and x sA and growth rate V g . The phase diagram for a ternary III-V alloy system AxB1-xC is described by the expressions given in Eqs. (5.4.8) and (5.4.9) (Panish and Ilegems [1972]). Table 6.3.2. Growth and thermophysical parameters of the Al-Ga-As and Ga-In-As systems (Qin and Dost [1996], and Dost and Qin [1998]). Parameter

Symbol

Value

Thermal conductivity of solution

kL

0.526 (W/cm °C)

Thermal conductivity of crystal

ks

0.082 (W/cm °C)

Thermal diffusivity of Ga-In-As



0.3 (cm2/s)

Kinematic viscosity of Ga-In-As



1.0  10-3 (cm2/s)

Electrical conductivity of Al-Ga-As solution

E

25,000 (1/ cm)

Electrical conductivity of AlGaAs crystal

E

40 (1/ cm)

Electrical conductivity of GaInAs crystal

E

12.8 (1/ cm)

L

s s

Growth temperature in LPEE growth of AlGaAs

850 °C

Growth temperature in LPEE growth of GaInAs

780 °C

Peltier coefficient

 sL

0.3 (V)

Solutal expansion coefficient of Ga in Ga-In-As

A

0.12

Solutal expansion coefficient of As in Ga-In-As

C

0.19

Thermal expansion coefficient

T

9.85  10-5

Mobility of Al in Al-Ga-As

μE

A

0.018 (cm2/V s)

Mobility of As in both Al-Ga-As, and Ga-In-As

μE

0.027 (cm2/V s)

Mobility of Ga in Ga-In-As

μE

A

0.0135 (cm2/V s)

Diffusion coefficient of Al in Al-Ga-As

DA

5.0  10-5 (cm2/s)

Diffusion coefficient of As in Al-Ga-As and Ga-In-As

DC

5.0  10-5 (cm2/s)

Diffusion coefficient of Ga in Ga-In-As

DA

5.0  10-5 (cm2/s)

Lattice parameter of GaAs

dAC

5.6533 Å

Lattice parameter of InAs

dBC

6.0584 Å

C

Initial GaAs substrate thickness

0.6 mm

GaInAs source thickness

10.0 mm

Ga-In-As solution height

12.7 mm

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254

Here we present the required physical and growth parameters for Al-Ga-As and Ga-In-As systems. Some of these parameters have already been given in Chapter 5, and also in Table 6.2.1, but for the sake of completeness and continuity, we will present a complete set in this section. Thermodynamic parameters for the phase diagram calculations of the Al-Ga-As system are presented in Table 6.3.1. The thermophysical and growth properties of Al-GaAs and Ga-In-As systems are given in Table 6.3.2. 6.3.3. Dimensional Analysis and Computational Procedure For computational convenience, we express the governing equations in dimensionless forms so that the relative importance of various phenomena observed in the LPEE growth process can be better understood. We define the following nondimensional variables:

u =

P =

T  T0 x  x 0A x  xC0 wL uL , w= , = , A = A , C = C   T x A xC

pL

 L

, = 2

t

r z   ,  = ,  = , Sc = , Sc = A C L L DA DC L2

μ AE Er L μ AE Ez L μCE Er L μCE Ez L A C C = , ez = , er = , ez = ,     g  A L3x g  A L3x g  L3T  GrA = C 2 A , GrC = C 2 C , GrT = T 2 , Pr =     erA

(6.3.13) (6.3.14)

(6.3.15) (6.3.16)

where L, and T , x A , and xC denote respectively the characteristic length, temperature, and concentration differences. Using the above dimensionless variables, the governing equations of the simulation model take the following forms in the liquid phase

u w + =0  

(6.3.17)

u u p u + u + w = +  2u     w w p w + u + w = +  2 w  GrA A  GrCC  GrT     

(6.3.18)

  1 2  + u + w =      Pr

(6.3.19)

Single Crystal Growth of Semiconductors from Metallic Solutions

 A  C 

+ u + u

 A  C 

+ w + w

 A  C 

= erA = erC

 A  C 

+ ezA + ezC

 A  C 

+

1  2 A Sc A

+

1  2C ScC

255

(6.3.20)

where

 2 

2  2

+

1  2 + 2   

(6.3.21)

Using the thermophysical properties of the GaxIn1-xAs system, given in Table 6.3.2 (at a growth temperature 800 °C, and a solid composition of x = 0.9), we calculated the dimensionless parameters to estimate the relative significance of buoyancy and viscous effects in the momentum equations and evaluate the relative importance of convection and diffusion in the mass and heat transfer equations, as follows:

GrA  GrC  7  104 , GrT  1 105 , Sc A  ScC  20 , Pr  3  103

(6.3.22)

The close values of the Grashof numbers based on concentration and temperature gradients indicates the equal importance of thermally driven natural convection and solutal convection. The small Prandtl number shows the insignificant influence of convection on the temperature distribution. The values of the Schmidt numbers, on the other hand, are rather large, indicating that convective transport has a significant influence on the solute concentration profiles, and thus on the growth rate and compositional uniformity. The dimensional analysis shows that natural convection, which is driven by both temperature and concentration gradients, is an important mass transport process, but has a negligible effect on heat transfer. This allows the use of a numerically efficient computational model in which heat transfer is considered to take place via conduction only. Such a computational model significantly reduces the computing requirements for numerical simulations (Dost et al. [1994], Djilali et al. [1995]) Simulations for the LPEE growth of the AlxGa1-xAs system were carried out in the absence of convection in the liquid phase. The growth cell was considered axisymmetric, and therefore cylindrical coordinates (r and z, with no dpendence on the circumferential coordinate) are adopted in writing the governing equations (as presented in Eqs. (6.3.17), (6.3.19), and (6.3.20)). However, in the simulation of the GaInAs system, the problem is assumed to be two-dimensional for the sake of simplicity for numerical convenience, and therefore the governing equations are expressed in the Cartesian system (x, z) with the x-axis along the horizontal direction corresponding to the largest dimension of the growing crystal while the z-axis is in the vertical direction.

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This two-dimensional computational domain is the plane in which the strongest convective flow is expected to take place because of the large dimension in the x-direction, since the Grashof number is proportional to the cube of the characteristic length. For both simulations, the finite element technique was used to handle the moving boundary problem in which the conditions at the moving interfaces are changing with time. The discretization of the governing equations is based on the Galerkin approximation. Four node-quadrilateral elements are used. The resulting set of first-order simultaneous ordinary differential equations are further discretized by the fully-implicit-time-marching algorithm based on the finite difference method which results in stable and accurate solution. The moving boundaries are tracked by moving the nodes following the shapes of the interfaces in each time step (Qin et al. [1995]). The numerical simulations are started by determining the initial conditions. The initial solution is assumed to be saturated at the growth temperature and in equilibrium with the desired composition of the crystal to be grown. Therefore, the initial concentrations of the solution are calculated using the phase diagram based on the growth temperature and the solid composition. Since the phase diagram equations are non-linear, the Newton-Raphson iteration is used in calculations. The complicated interface conditions and the different mass transport rates in the liquid and solid phases give rise to numerical difficulties. Since the values of physical parameters are very different in the solid and liquid phases, the initial value problem of this type is called stiff, and will result in poorly conditioned matrix equations in numerical analysis. To overcome this problem the governing equations for the solid and liquid phases are solved separately. This separated solution procedure makes it possible to use different mesh sizes and different time scales in the two phases. For instance, in both systems (AlGaAs or GaInAs) growth is initiated from a binary commercially available substrate (GaAs), and the new grown layer (either AlGaAs or GaInAs) has a different composition than the seed GaAs crystal. Therefore, there are large interfacial concentration differences in the solid phase. This requires the use of a fine mesh in the solid phase. The variables of the solid and liquid phases are coupled through the liquid/solid interface conditions and growth rate. Therefore, the governing partial differential equations in the liquid and solid phases, and the set of differential equations from the interface conditions, must be solved sequentially and iteratively. Specifically, the interface conditions of liquid compositions xA, xC and solid composition x are determined by solving the interface equations, using the solute distributions obtained from the solution for the bulk phase equations at each time step. As mentioned earlier in Section 6.2 the heat transfer in the solution is mainly due to diffusion. Numerical tests have confirmed that convection

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257

effects are negligible in heat transfer analysis (Djilali et al. [1995], Qin et al. [1995], Qin and Dost [1996]). Therefore, the heat transfer equation is decoupled from the momentum equations and the temperature field calculations are carried out before the mass transport and fluid flow field calculations. Thus, as done in Section 6.2, two different meshes are used in the simulations. For thermal analysis, the mesh includes the complete cell configuration while the mesh for the mass transport analysis covers only the solution and the substrate (Fig. 6.2.14). This consideration has the advantage of reducing computational cost without a significant loss in accuracy. Since the temperature field is influenced by the Joule heating which is proportional to the volume of grown crystal, the change in growth thickness must be taken into account in the thermal analysis. This is done by modifying the substrate thickness and following the moving interfaces which are determined by the computed growth rate at each time step. Thus, the temperature fields in both the liquid and the solid are calculated, at each time step, as the growth thickness increases. In Fig. 6.3.1 we present the computational flow chart for the simulation of the LPEE growth of GaxIn1-xAs. The computational flow chart is the same for LPEE growth of AlGaAs with the exception that the flow field is not included in computations (diffusion limited; no convection, see Qin et al. [1996] for details). The computational procedure for GaxIn1-xAs is follows. We start with the condition of the initial In-rich solution with gallium 0 composition xGa and arsenic composition x 0As at a given growth temperature. The initial condition is determined from the Ga-In-As phase diagram, which is in equilibrium with the desired crystal composition x. Because the substrate (GaAs) composition is different from the desired crystal (GaxIn1-xAs, x<1) composition, the initial solution is not in equilibrium with the substrate. When the solution and the substrate are brought into contact, the interface concentrations of the liquid and the solid will change. The initial interface condition is a function of the difference between the compositions of the substrate and the desired crystal and the ratio of the diffusion coefficients of the liquid to the solid. To initiate the simulations, the interface conditions are initially assumed. An iterative procedure is then applied to obtain a convergent solution in the two bulk phases and at the interface for each time step. The virtual process for determining the assumed interface compositions, described in Kimura et al. [1996b], is used in the simulations. The steady-state electric potential,  and electric field, E, are first computed by solving the Laplace Eq. (6.2.9). They are then used in the mass transport equations considering the electromigration of the solutes. Then time iterations are started to calculate time-dependent temperature, concentration, and flow velocities in both the liquid and solid phases. In this way, the changes in these field variables, the growth thickness and the crystal composition due to changes in growth conditions, are investigated. For example, the cooling condition at the interface continuously changes as the growth thickness increases.

Sadik Dost and Brian Lent

258 start compute initial condition

compute guess interface concentrations

compute electric field

time integration

compute temperature field

Interface condition iteration

compute composition and flow field in liquid

compute composition in solid

compute interface concentration and growth rate

no converged?

yes t = t +t

t >tmax

stop Fig. 6.3.1. Computational flow chart for the GaInAs system (after Dost and Qin [1998]).

With the initially assumed interface conditions, the calculated electric and temperature fields, the governing equations for coupled solute and convective flow fields in the solution and mass diffusion in the substrate are solved. The convective mass transfer is significant as indicated by the large Schmidt numbers given earlier. Using the concentration gradients at the interface obtained from the solution, the interface compositions and growth rate are calculated by solving the phase equilibrium and mass conservation equations at the interface. The new interface conditions are then used to compute

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concentrations and fluid flow in the liquid and solid again. The iteration procedure continues until the interface conditions are converged to within a small tolerance of 10-3. The computation step is therefore determined and growth thickness can be calculated using the converged value of the growth rate. The finite element meshes are then modified by moving the nodes following the shapes of the interfaces so that the meshes adapt to the deformed substrate, solution and source. Computations continue for the sequential growth steps until the prescribed growth time is reached. The calculation data for each time step are recorded and plotted. The evolution of interface temperature profile, interface shape, growth rate, growth thickness, convective flow pattern, and most importantly the solid composition variation with time are obtained from the simulation. As mentioned earlier, the governing equations of the LPEE growth process are coupled and highly nonlinear. In the finite element method, these partial differential equations (PDEs) are discretized into a set of nonlinear algebraic equations which represent an approximation to the original PDEs within the domain of the growth cell. The Galerkin weighted residual method is used to reduce these partial differential equations into a set of first-order ordinary differential equations. The time derivative is then discretized by the implicit time marching algorithm based on the finite difference method. Further, the penalty function formulation is used to eliminate the pressure as an unknown, and consequently to reduce computation cost. The selection of finite element mesh is an important issue for the accuracy of results obtained. For the Penalty function formulation, the domain of the growth cell is discretized into a mesh of four-node quadrilateral elements. The interpolation (shape) functions for the element are used for the implementation of the finite element and variables approximation. Details of these applications of the finite element techniques can be found in Qin et al. [1995] and Hughes [1979]. The accuracy of numerical solutions is checked through refining the element mesh. When the mesh is fine enough to give mesh-independent solutions, it is used for growth simulations. The meshes are nonuniformly arranged with the denser mesh near the interface regions where higher gradients and property variations arise. The set of nonlinear algebraic equations obtained from the finite element approximation are solved by the Newton-Raphson iteration technique for temperature, solutes, and fluid flow velocities. The transient solution of the governing equations is very time consuming. A combined full and modified Newton-Raphson iteration scheme is used, that is, the full Newton-Raphson method is used in the first iteration for each time step and the modified NewtonRaphson is used during the subsequent iterations until the solutions for the bulk phases and interfaces converge. This solution scheme significantly reduces computation time since the modified Newton-Raphson method requires fewer reformations and factorizations of the tangent stiffness matrix, and still provides a reasonable convergence speed.

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Fig. 6.3.2. Growth rate versus time for different solid compositions (after Qin and Dost [1996]).

6.3.4. LPEE Growth of AlxGa1-xAs: A Diffusion Model The growth of AlxGa1-xAs is considered from a Ga-rich solution. The solution is dilute with 0.345% Al and 2.445% As. The growth temperature is 850 °C, the electric current density is 10 A/cm2, and the solid composition is up to x = 0.4. Simulations are carried out in the absence of convection (Qin and Dost [1996]). The variation of growth rates with time is presented in Fig. 6.3.2. As can be seen, the growth is affected by the solid composition and is lower than that for growing GaAs (Dost et al. [1994]). This is consistent with the experimental growth rates observed for ternaries in Bischopink and Benz [1993b]. The growth rate increases with the increasing solid composition. This is because in the solution, the concentration of Al is much lower that that of As. The mobility of Al is slower, therefore the growth rate is determined by the transport rate of Al. This leads to slower growth rates in the growth of AlGaAs, since more Al species are needed. This result is supported by the computed concentration profiles in the solution. Fig. 6.3.3 shows the evolution of concentration profiles of Al and As in the solution with time. The concentration of Al is higher near the dissolution interface all the time, i.e., always positive gradients, as shown in Fig. 6.3.3a, meaning that Al always diffuses toward the growing interface. The direction of the solute diffusion is the same as that of electromigration. On the other hand, As species diffuse towards the growing interface during the initial stage corresponding to a much faster growth rate due to the Peltier effect, and then diffusion changes direction and starts to counteract electromigration slowing down the transport of the solute towards the substrate. The higher concentration in the vicinity of the growth interface is because electromigration brings excess

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As species which are not able to incorporate into the substrate due to fewer Al species available for crystallization. This is quite different from the LPEE growth of a binary system such as GaAs, in that As always diffuses towards the growing interface and the transport rate of diffusion and electromigration determines the growth rate.

(a)

(b)

Fig. 6.3.3. Evolution of concentration profiles in the solution: (a) Al, and (b) As (after Qin and Dost [1996]).

The relative contributions of electromigration and Peltier cooling to the overall growth rate are shown in Fig. 6.3.4a. The relative growth rates due to electromigration and Peltier cooling, computed through this diffusion-limited simulation model, are essentially time-independent. At the beginning of the growth process, the lower contribution of electromigration is the result of overcoming the natural tendency to dissolve because of constitutional nonequilibrium of the system (Small and Ghez [1979], Kimura et al. [1996b]) whereas a higher contribution of Peltier cooling is attributed to supersaturation of the solution due to a temperature drop at the growth interface. It is evident that electromigration is the dominant growth mechanism under the diffusion and electromigration growth condition. The effect of Peltier cooling is significant only in the initial stage of the growth process. This is the same feature observed in LPEE growth of binary systems (Dost et al. [1994]). The main advantage of LPEE, its ability to produce bulk crystals with uniform compositions, can clearly be seen in Fig. 6.3.4b. The solid line shows the computed composition variation with layer thickness for crystals grown by LPEE. It is quite uniform throughout the growth period, except that there is a

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slight drop at the beginning of the growth period. This reduction in composition is due to the initial temperature drop caused by Peltier cooling and constitutional non-equilibrium between the substrate and the solution. Because of this, as will be discussed in the next subsection, in LPEE experiments of ternary crystals the composition of the source material is kept slightly higher than the composition of the crystal to be grown. Thus, the target composition can be realized. The long dashed line represents the composition variation in crystal grown by conventional LPE (Kimura et al. [1996b], Ijuin and Gonda [1976]). It can be seen in Fig. 6.3.4b that LPEE results in much better compositional uniformity over LPE. This is mainly due to its constant growth temperature and small temperature gradients in the system. Indeed, the GaInAs crystals grown by Sheibani et al. [2003a] show remarkable compositional uniformity.

(a)

(b)

Fig. 6.3.4. (a) Relative contributions of electromigration and Peltier cooling to overall growth rate, and (b) comparison of variations of the solid composition (data taken from Qin and Dost [1996], Kimura et al. [1996b], Ijuin and Gonda [1976]).

The use of source material in LPEE is necessary. However, in order see to the effect of the absence of source material, computations were also carried out with no source, and the result is plotted in Fig. 6.3.4b (dashed line). As expected, the solid composition changes and gets smaller as growth proceeds. In order to compare simulation results with experiments, the growth of AlxGa1-xAs with a crystal composition of x = 0.12 is considered at a growth temperature of 780 °C and three electric current densities 10, 20, and 30 A/cm2. The variation of growth rate with electric current density is shown in Fig,

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6.3.5a. The growth rate increases linearly with the increasing current density due to enhanced mass transport by electromigration. The computed growth rates are in good agreement with the experimental data of Daniele [1975] for an essentially diffusion and electromigration limited growth.

(a)

(b)

Fig. 6.3.5. (a) Growth rate versus electric current density, and (b) the effect of current density on the solid composition (after Qin and Dost [1996]).

Fig. 6.3.5b shows the computed solid composition at three electric current densities. As seen, the solid composition is uniform at all electric current densities. However, the initial drop in the composition value gets higher with the increasing electric current density. This implies that at higher electric current densities, the difference between the composition of the source material and that of the grown crystal must be increased to achieve the desired crystal composition. The growth temperature also has a similar effect as the applied electric current density on the growth rate and the composition uniformity. The higher growth temperature results in higher solubility of the solution, leading to faster growth rates. However, the concentration difference between two solutes also increases with growth temperature, which causes a greater ratio of mass transport rates, leading to a slightly increased uniformity of solid composition. 6.3.5. LPEE Growth of GaxIn1-xAs: A Convection Model Simulations for the LPEE growth of GaxIn1-xAs (x = 0.94) were carried out for the growth configuration SETUP1 shown in Fig. 6.2.1. The computational domain covers one-half of the crucible geometry, assuming symmetry of the

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Fig. 6.3.6. Relative temperature distribution in the solution (after Dost and Qin [1998]).

geometric and boundary conditions with respect to the y-axis. The geometric dimensions and physical properties of the GaInAs system used in the simulation are given in Table 6.3.2. In the simulations, the initial temperature is chosen as 780 °C corresponding to the experimental growth conditions. The solution compositions, determined from the Ga-In-As phase diagram for the growth at 780 °C, are assumed to be uniformly distributed in the liquid zone. In this experimental set up the Ga-InAs solution is non-dilute with 29.85% Ga and 7.07% As. The LPEE growth starts with the application of a direct electric current to the growth system, which gives rise to Peltier cooling at the growth interface, Peltier heating at the substrate/contact zone interface, Joule heating in the

Fig. 6.3.7. Flow field in the solution at J = 3 A/cm2 (after Dost and Qin [1998]).

Crystal thickness (cm)

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experiment simulation

0.30 0.20 0.10 0.00 -2

-1

0

1

2

Horizontal distance, x (cm) Fig. 6.3.8. Shape of the grown crystal: evolution of grown crystal surfaces from simulation (dashed lines), and the profile of a GaInAs crystal grown under the simulation conditions (circles) (after Dost and Qin [1998]). The interface profile was obtained by Mr. N. Audet from a crystal provided by Dr. T. Bryskiewicz).

substrate, and electromigration in the solution. The growth rate at the electric current density of 3 A/cm2 is on average 0.5 mm/day, which is in good agreement with experimental observations. Similar to the growth of GaAs, electromigration is still the dominant mechanism in LPEE of GaInAs. Fig. 6.3.6 shows the temperature field with temperature drops from the initial temperature of 780 °C in the liquid zone for an average growth thickness of 3 mm. The temperature difference in the growth direction (vertical) at the liquid zone centre is higher than that at the crucible wall. The temperature gradient along the interface is also higher near the wall. As a result, the convective flow pattern shown in Fig. 6.3.7 is observed in the region near the wall, giving rise to stronger convective mass transfer there. In the central region of the liquid zone, the effect of fluid flow is not significant. Thus the mass transport in this region is almost driven by diffusion due to the temperature difference between the two interfaces. Relatively different contributions from the diffusion and convection transports in different regions bring about a nonuniform growth across the growing interface. The evolution of growth thickness with time, obtained from the simulation, is shown in Fig. 6.3.8 (dashed lines). With the faster growth rate due to the contribution of convective mass transfer, thicker layers occur in the region near the wall, and thinner layers in the central region, forming a surface shape qualitatively similar to the experimentally observed double-humped pattern (the measured profile is shown by circles). The nonuniform distribution of convection is the reason for the nonuniform growth. Nonuniform growth may bring about interface instability that leads to compositional inhomogeneities and solvent inclusions (Bryskiewicz and Laferriere [1993]). Note that the patterns of the grown crystal surface obtained from the simulation are in good agreement with experiments, although the simulation over estimates the effect of natural convection on the growth thickness. It is possible that this difference is due to

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Fig. 6.3.9. Temperature profiles at different stages of growth (after Dost and Qin [1998]).

the fact that some growth parameters for the GaInAs solution used in the simulation had to be estimated since they are not available. Fig. 6.3.9 shows the temperature profiles in the solution at different stages of growth. Since the heat transfer is mainly due to diffusion, the temperature distributions are linear from the lower interface to the upper interface. The temperature change between the interfaces is very small initially because of the small applied current density, but increases gradually with the increasing substrate thickness.

(a)

(b)

Fig. 6.3.10. Concentration profiles at different stages of growth (after Dost and Qin [1998]).

The concentration profiles of Ga and As in the solution are shown in Fig. 6.3.10. Unlike the temperature profiles, the solutes are not linearly distributed through the solution height due to the significant effect of natural convection. The concentration of As is higher near the upper interface, and gives rise to positive gradients, meaning that As species always diffuse towards the growing

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Fig. 6.3.11. Variation of the indium composition in the grown crystal (after Dost and Qin [1998]).

interface. The direction of solute diffusion is the same as that of electromigration (Nakajima [1991]). On the other hand, Ga species diffuse towards the growing interface only at the initial stage and then diffusion changes direction and starts to counteract electromigration slowing down the transport of the solute towards the substrate. The higher concentration in the vicinity of the growing interface is due to the electromigration effects which bring excess Ga species to the interface, and these cannot be incorporated into the substrate due to fewer As species being present. For the growth conditions under consideration, the growth rate is mainly determined by the transport rate of As species (Nakajima [1991]). Fig. 6.3.11 describes the indium composition variation with changing growth thickness. The composition is very uniform, as expected, due to the insignificant effect of temperature variations, and to the dominant effect of electromigration. Contrary to what we observed in the case of the diffusionlimited model of AlGaAs growth (Fig. 6.2.2), the indium composition increases slightly with the growth thickness; about 1% change over a growth thickness of 2 mm. This may be attributed to the effect of convection leading to a small change in the interface temperature resulting from the change in substrate thickness during growth. The applied electric current density is the most important control parameter for the LPEE growth process; the growth rate increases with increasing current density. In the mean time, Joule heating (proportional to J2) may play a more important role as well as the natural convection due to greater temperature and concentration gradients. To investigate the effect of current density on the growth process, in the simulations a current density of 5 A/cm2 is applied while other growth conditions are unchanged. The relative temperature in the solution becomes positive, indicating that the Joule heating has a dominant effect over Peltier cooling.

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Fig. 6.3.12. Flow field at J = 5 A/cm2 (after Dost and Qin [1998]).

The computed flow field is shown in Fig. 6.3.12. There are two flow cells with about the same intensity in the solution. The predicted growth rate increases by about 70% as compared to the rate for J = 3 A/cm2 . Indeed, the experiments carried out by Sheibani et al. [2003a] for the growth of GaInAs (from a Ga-rich Ga-In-As solution) at a 5 A/cm2 current density yielded a growth rate, on average, about 0.72 mm/day, which is roughly 50% greater than the 0.5 mm/day growth rate at the 3 A/cm2 current density. 6.4. Two-Dimensional Simulations Under Magnetic Fields In this section we examine the two-dimensional simulation models developed for the LPEE growth of crystals under magnetic fields. The feasibility of using a magnetic field in suppressing convection in LPEE was first studied through a model for the growth of GaAs in Dost and Qin [1995]. The model developed in Dost and Erbay [1995] was extended to include the effect of a magnetic field in the fundamental equations. The LPEE growth process of GaAs in SETUP1 (Fig. 6.2.1) was numerically simulated under an applied static magnetic field using the finite element method based on the penalty function formulation (Qin et al. [1995]). It was found that this formulation is more robust and efficient than a mixed velocity-pressure formulation. The results of these early simulations have shown that the effect of natural convection can be reduced significantly by the application of an external static magnetic field. For the crucible selected, the uniformity of grown layers improves with an increasing magnitude of the applied magnetic field. Results also showed that the complete elimination of natural convection in the liquid phase requires the application of a very large magnetic field (about and more than 20 kG). However, this prediction is numerical, based on the model assumptions and simplifications made, and as we will see later in growth of

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bulk crystals by LPEE, the strong interaction between the applied magnetic and electric fields does not allow the use of a magnetic field intensity above a critical (maximum) value. For instance, this value was about 4.5 kG in LPEE growth of GaAs in experiments conducted by Sheibani et al. [2003a,b], and growth at higher field intensities led to unstable growth. However, this early simulation model of Qin et al. [1995] has shed light on a number of issues in LPEE growth under magnetic field and laid the foundation for further experimental and numerical simulation studies in LPEE. It has also provided a valuable insight on the use of the finite element technique for future studies. Therefore it would be beneficial to review the results of this early study. 6.4.1. Computational Model The applied magnetic field is constant in time, but its magnitude may vary along the vertical axis. At the time of the work of Qin et al. [1995], the magnetic field was assumed uniform in the solution zone using the following arguments. Since the length of the solution zone is small, compared with the size of the system, the magnetic field is taken as uniform in the solution. Furthermore, for this particular setup, the electric balance equation governing the current distribution is decoupled from the other field equations, and gives a uniform electric field distribution in the solution. Hence the problem was considered as completely axisymmetric. Indeed, as we will see later, the superconducting magnets developed (see Sheibani et al. [2003a], and Dost [2005a,b]), provide almost a uniform field distribution in the magnet opening (in the absence of growth crucible) and also the LPEE crucible developed in Sheibani et al. [2003a] allows the passage of electric current very uniformly through the growth crucible. Therefore, these early assumptions used in simulations were accurate. However, the application of the vertical fixed magnetic field gives rise to magnetic body force components in both the radial and circumferential directions. This makes the problem actually three dimensional. As we will see later this was addressed in the works of Dost et al. [2005a,b], and Liu et al. [2002a,b, 2003, 2004]. Under the axisymmetry conditions assumed, the field equations are almost the same as those given in Sections 6.2 and 6.3, except for the momentum equations due to the effect of an applied magnetic field. The continuity, electric charge balance, and energy equations are, respectively, the same as those given in Eqs. (6.2.4), (6.2.9), and (6.3.10). Here we only present the momentum equations.

u u 1 p  2u  2 w 1 u 2u u +u + w =  +  ( 2u + 2 + +  ) r z t rz r r r 2  L r r   E B 2u

(6.4.1)

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w w 1 p  2 w  2u 1 u w +u +w = +  ( 2 w + 2 + + ) r z t rz r z  L z z + g[ t (T  T0 ) +  c (C  C0 )]

(6.4.2)

where the gradient operator is the same defined in Eq. (6.2.3), and B is the applied magnetic field intensity. Here note that the continuity equation is embedded into the above momentum equations for convenience in the finite element procedure. Eqs. (6.4.1) and (6.4.2) become, respectively, the same equations given in Eqs. (6.2.5) and (6.2.6) when the continuity is satisfied on the right hand sides of these equations. The last term in Eq. (6.4.1) represents the magnetic body force in the radial direction, and due to the imposed axisymmetry and field uniformity, it reduces to a single term in the radial direction (for derivation of magnetic body force terms, see Section 4.5.1). The energy equation in the solid phase is given by

krs (

 2T r 2

+

1 T  2T J 2 T ) + k zs 2 + s = s s r r t E z

(6.4.3)

The associated boundary and interface conditions are the same as those presented in Section 6.3.2. The growth rate is computed using Eq. (6.2.10). 6.4.2. Penalty Function Formulation and Numerical Solution The transient solution of the governing equations in general requires considerable computational effort in long processes. For instance, a typical LPEE growth process of bulk crystals lasts between 3 to 8 days (see Sheibani et al. [2003a]). An effective numerical method is therefore crucial to the successful performance of a numerical simulation. The finite element method based on the penalty function formulation is especially attractive because it eliminates the pressure as an unknown field variable through the use of a "penalty" parameter and solves the modified momentum equations for the velocity components. This method does not require an independent solution of the continuity equation. The principal advantages of the penalty function formulation are the ease in programming, lower storage requirements, and fewer equations to solve. This makes the approach very cost effective. We briefly present the formulation. Following Hughes et al. [1979], we write

p = (

u u w + + ) r r z

(6.4.4)

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where  is a large penalty parameter. To formulate the finite element equations in the penalty form, we start with the momentum equations, and proceed by using Galerkin's method

 [(



u u w 1 p +u + w )+ r t z  L r

 2 w 1 u 2u  ( u + 2 + +  2 ) +  E B 2u]N i d  = 0 rz r r r r 2

 2u

(6.4.5)

w w w  2 w  2u 1 u 1 p 2 + w )+   ( w + 2 + + )  [( + u t r z rz r z  L z z  +g[ t (T  T0 ) +  c (C  C0 )]N i d  = 0

(6.4.6)

where N i is the shape function. Integrating the pressure and viscous terms by parts and assuming zero boundary tractions, we write

 [(



+

w N i 2u + N ) +  E B 2uN i ]d  = 0 r z r i

 [(



+

u u u p N i N i u N i u N i + u + w )N i  ( + ) +  (2 + r t z r r z z  L r r (6.4.7)

w w w p N i w N i w N i +u + w )N i  + ( +2 r t z r r z z  L z

u N i ) + g t (T  T0 )N i  g  c (C  C0 )N i ]d  = 0 z r

(6.4.8)

Eliminating the pressure by introducing Eq. (6.4.4) into Eqs. (6.4.7) and (6.4.8), we obtain

 [(



u u u  u u w N i N i + u + w )N i + ( + + )( + ) r t z  L r r z r r

+ (2  [(



u N i u N i w N i 2u + + + 2 N i ) +  E B 2uN i ]d  = 0 r r z z r z r

w w w  u u w N i +u + w )N i + ( + + ) r t z  L r r z z

(6.4.9)

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w N i w N i u N i +2 + ) + g t (T  T0 )N i r r z z z r g  c (C  C0 )N i ]d  = 0

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+ (

(6.4.10)

The weak formulation for the energy and mass transport equations can be readily obtained by applying Galerkin's method and integrating the diffusion terms by parts. The field variables u, w, T, and C are approximated by piecewise bilinear shape functions. All integrals are evaluated using a 22 Gaussian quadrature over each element, except for the penalized terms which have to be integrated in an alternative form to insure that the element possesses the mean incompressibility property (Hughes et al. [1979]). The fully implicit time-marching algorithm based on the finite difference method is used to discretize the first-order differential equations after finite element approximation. The resulting nonlinear algebraic equations are solved at each time step by Newton- Raphson iteration. The computational procedure employed in this study takes advantage of the fact that the effect of convection on the temperature field is negligible (the Pr number is of the order of 10-3), so that the heat transfer equation is decoupled from the momentum equations. This reduces computational requirements significantly. Upon solving the governing equations for concentration, the growth rate is calculated based on the mass conservation at the solution/substrate interface, Eq. (6.2.10). The interface shapes are then determined by integrating the growth rate. Since growth rates are very small, the growing and dissolved interfaces move very slowly, and thus the motion of the solution, substrate and source regions are very small from one

Fig. 6.4.1. Oscillating flow field from the mixed velocity/pressure formulation. The induced magnetic force brings about a poor satisfaction of the incompressibility constraint and leads to numerically oscillatory solution (after Qin et al. [1995]).

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Fig. 6.4.2. Comparison of computed velocity profiles along the axis of symmetry. The mixed formulation gives rise to numerical oscillation while the penalty formulation results in oscillation-free solution even for large magnetic fields (after Qin et al. [1995]).

time step to the next in the numerical calculation. The finite element meshes are not re-generated completely, but the nodes are moved following the shapes of the interfaces in each time step so that the meshes adapt to the deformed substrate, solution and source. The change in the location of nodes between two time steps is very small, so the values of all variables computed in the previous step are used directly in the current step as initial values without any interpolation to reduce computing time. 6.4.3. Simulation Results The numerical simulations are performed for the growth cell configuration shown (SETUP1) in Fig. 6.2.1 for different values of the magnetic field. The geometries and thermophysical properties of this growth cell were given in Tables 6.2.1, and 6.3.2. The growth temperature and electric current density are taken as 800°C and 10 A/cm2. To investigate the validity of the numerical procedures, a comparison is made between the results computed using the mixed velocity/pressure formulation and the penalty function formulation. In the absence of the applied magnetic field, the results from both formulations are identical for all practical purposes, and for a relatively small magnetic field (B < 1.0 kG), the results from both formulations differ only slightly. The solutions from the mixed velocity/pressure formulation, however, become oscillatory in space with increasing magnetic field intensity, although the numerical convergence is still achieved (this will be discussed further in 3-D simulations of the LPEE and also THM growth processes). This is illustrated by a typical velocity field shown in

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Fig. 6.4.3. Penalty solution for B = 0 after growth of 20 hours, (a) flow field, (b) streamlines, (c) concentration contours and (d) temperature contours (after Qin et al. [1995]).

Fig. 6.4.1, from which it is evident that the velocity field does not have zero divergence (i.e. the solution is not compatible with the condition of incompressibility). The maximum value of the discrete divergence is of the order of 10-4 for the mixed formulation, while it is of the order of 10-8 for the penalty formulation. The solutions from the penalty function formulation are always oscillation-free, even for large applied magnetic fields. A comparison of the computed profiles of the axial velocity component is shown in Fig. 6.4.2. The numerical results show that the magnetic body force acting on fluid points, which is proportional to the radial velocity component in this special case, brings about the poor satisfaction of the incompressibility constraint in the mixed formulation. This is because the magnetic force causes an imbalance of the various terms in the momentum equation, Eq. (6.4.9). The widely different magnitudes of the pressure and magnetic force terms leads to significant roundoff error accumulation. The global matrix terms corresponding to the pressure terms are much smaller than those of the magnetic force terms. Thus the pressure terms are "drowned" because of floating point round-off errors

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Fig. 6.4.4. Penalty solution for B = 2.0 kG after growth of 20 hours, (a) flow field, (b) streamlines, (c) concentration contours and (d) temperature contours (after Qin et al. [1995]).

(Pelletier et al. [1989]). For the penalty function formulation, the large penalty parameter (109L ) enforces the incompressibility constraint, so that satisfactory velocity fields are obtained. A study of solution sensitivity to mesh density established that the present mesh with 5925 nodes in the solution zone gives an adequate resolution for the problem under consideration. The computed velocity fields, streamlines, concentration and temperature contours are shown in Figs. 6.4.3 and 6.4.4 for B = 0 and B = 2.0 kG. The plotting scale for the velocity field in Fig. 6.4.4a is 10 times that of Fig. 6.4.3a. The applied magnetic field reduces the intensity of the convective flow, and results in a more uniform concentration field, which in turn reduces the effect of convection on the growth process. Fig. 6.4.5 shows the contribution of Peltier cooling to growth rate through diffusion and convection. As mentioned earlier in Sections 6.2 and 6.3, the contribution of Peltier cooling to the growth rate increases with increasing convection. Under the applied magnetic field, the growth rate due to Peltier cooling decreases because of a suppressed convective mass transport, and

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therefore, the reduced contribution of Peltier cooling to the growth rate enhances the interface stability (Djilali et al. [1995], Qin and Dost [1995], Dost and Qin [1998]). With decreasing interference of convection, mass transfer becomes increasingly diffusion controlled, and the surfaces of the grown crystal become flatter. The computed interface shapes (the evolution of interfaces with time) are plotted in Fig. 6.4.6. As can be seen, the well-known double-humped crystal shape observed in SETUP1 gets flatter as the magnetic field intensity increases, leading to a more stable crystal growth. Similarly, the single-humped crystal interface shape observed in SETUP2 will also get flatter under the effect of a magnetic field. These non-flat (curved) interface shapes observed in LPEE (in SETUP1 and SETUP2) are due to the non-uniformity of electric current passing in the solution. These shapes can be improved if the electric field becomes more uniform in the solution, leading to flatter interfaces, even in the absence of an applied magnetic field. The influence of an applied magnetic field on the growth process can be evaluated based on the mass transfer Peclet number

Pem = U max r / DC

(6.4.11)

where U max is the maximum value of the axial velocity component. Fig. 6.4.7 shows the variation of Peclet number Pem with magnetic field B. The Peclet number decreases with increasing magnetic field intensity. The contribution of convective flow to the crystal growth rate can be disregarded when the mass transfer Peclet number is smaller than 1 (Motakef [1990]). For the growth condition under consideration, a diffusion controlled growth can be achieved

Fig. 6.4.5. Contribution of Peltier cooling to growth rate for B = 0 and B = 2 kG. Under the applied magnetic field, the growth rate decreases due to the suppressed convective mass transport (after Qin et al. [1995]).

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Fig. 6.4.6. Evolution of grown crystal surface, (a) B = 0, (b) B = 2 kG and (c) B = 4 kG (the time increment for each curve is 10 hours). The shape of the grown layer in (a) is a typical one observed experimentally (Bryskiewicz et al. [1987]). The double-humped shape becomes essentially flat when large magnetic fields are applied (after Qin et al. [1995]).

when B > 15 kG. The intensity of convection increases when larger electric current densities are applied to achieve faster growth rates, due to the increasing temperature and concentration gradients. Simulations show that, as expected, in the absence of a magnetic field the maximum magnitude of the flow velocity field (convection

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intensity) in the solution increases with increasing electric current density (Qin et al. [1995]). Numerical simulations indicate that the mass transfer Peclet number is also a function of the electric current density in the presence of an applied magnetic field. Fig. 6.4.8 shows the computed variation of Pem with J under magnetic field intensities of 2 and 5 kG. The magnetic field must be larger than 25 kG in order to achieve a diffusion-controlled growth for J = 20 A/cm2. 6.4.4. Effect of Magnetic Field Nonuniformity As mentioned in the previous section, a static vertical magnetic field has been considered to suppress convection in the solution of an LPEE system. In the modeling studies of Qin et al. [1995], Dost [1999], and Dost and Sheibani [2000], it was assumed that the applied magnetic field is perfectly aligned with the vertical axis of the growth system, and is uniform in the solution zone of the growth cell. However, due to either a misalignment of the externally applied fixed magnetic field with the symmetry axis of the growth system (an axisymmetric “tilt” in the field with respect to the symmetry axis of the system), or the nonuniformity of the magnetic field distribution which is of the order of 1 to 2 % in the system at CGL of the University of Victoria, a field non-uniformity may give rise to small but additional magnetic body force components in the vertical and circumferential directions. It is then important for the development of

Fig. 6.4.8. Mass transfer Peclet number versus electric current density for B = 2 and B = 5 kG. The mass transfer Peclet number is a function of current density. Larger magnetic fields are required to suppress convection in growth processes applying higher current densities (after Qin et al. [1995]).

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Fig. 6.4.7. Mass transfer Peclet number versus magnetic field intensity. As can be seen, magnetic fields exceeding 5 kG reduce the influence of convection significantly (after Qin et al. [1995]).

experimental setups to determine the effect of such small magnetic body forces on the overall growth systems, and also on the flow patterns and concentration distribution in the solution zone. Such information is very beneficial in determining the level of the convective flow in the vertical plane, which may affect the shape of the growth interface, and the level of convective flow in the horizontal plane that gives rise to mixing in the solution that is beneficial for stabilizing the growth interface. The present section examines numerically the effects of the applied magnetic field and the applied electric current density in LPEE growth of single crystal semiconductors (Dost et al. [2002]). The SETUP1 is considered for the growth of GaAs. All the required physical and growth parameters are given in Table 6.3.2. The crucible used has the following dimensions: solution height is 6 mm, cell diameter is 12 mm, contact zone is 2 mm, and seed crystal thickness is 0.6 mm. In order to examine the effect of magnetic field nonunformity or misalignment, small magnetic field components in the r- and  -directions are introduced. The applied fixed magnetic field vector is then expressed as

(

B = B  r e r +  e + e z

)

(6.4.12)

where B is the magnitude of the applied magnetic field, and  r and  are the introduced small numbers that represent small magnetic field components in the r- and  -directions. Since the electric current passing through the system is uniform in this set up, the electric field is vertical: E = Ez e z . It must be noted that the system is still axisymmetric. Thus, all the field variables are independent of the  -coordinate. In the growth system considered here, the

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magnetic body force due to the external applied magnetic field B and the electric field E is given by (Eq. 4.5.1)

(

)

F em =  E E + v  B  B

(6.4.13)

Using Eq.(6.4.12), E = Ez e z , and v = (ue r + ve + we z ) in Eq.(6.4.13) we obtain

F em = Frem e r + Fem e + Fzem e z

(6.4.14)

where

(

) B [  u  (1+  ) v +  w] + 

Frem =  E B 2 [ 1+ 2 u +  r  v +  r w]   E BEz 

(6.4.15)

Fem =  E

(6.4.16)

2

r 

2 r



E

BEz  r

Fzem =  E B 2 [ r u +  v  ( r2 + 2 )w]

(6.4.17)

The system is axisymmetric, thus all field variables are functions of the space coordinates, r and z, and time only. The solution is assumed to be an incompressible, Newtonian fluid. Under these assumptions, the axisymmetric field equations are the same as those given in Sections 6.2 and 6.3, namely the continuity in Eq. (6.2.4), mass transport in Eq. (6.2.2), electric field in Eq. (6.2.9), and energy in Eq. (6.3.10). The gradient operator is also the same as that given in Eq. (6.2.3). However, the momentum equations are different in form due to the addition of magnetic body force components: in the r-direction em  2 u u v 2 u 1 p Fr u , +u + w  =   u  2  + r z r t  r   L r  L

(6.4.18)

in the z-direction em w w 1 p Fz w +u +w =  2 w  + r z t  L z  L

(

)

(

 g t T  T0 + g  c C  C0

)

(6.4.19)

The velocity component v is computed from the momentum balance in the  -direction em  2 v v uv v  F v +u + w  =   v  2 + r z r t r L

(6.4.20)

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The temperature distribution on the vertical wall of the growth cell is given as

T = Tg  (

z  z0 H

)T

(6.4.21)

where H is the solution height and Tg is the growth temperature, and the remaining boundary and interface conditions are the same as those given in Section 6.3 for the same growth setup.

(a) B = 0.0 kG

(b) B = 10 kG

(c) B = 20 kG

(d) B = 40 kG

Fig. 6.4.9. Concentration contours under various magnetic field values (t =20 h, J =10 6 A/cm2). Contour spacing is 5.0  10 mass fraction, and the minimum and maximum concentrations are 0.0218858 and 0.0219603, respectively (after Dost et al. [2002]).

The field equations are solved using the finite element technique based on the penalty function formulation. The growth rate is calculated using Eq. (6.2.10). It must be mentioned that the field equations are solved by a two-dimensional solver due to axisymmetry. The momentum equation in the circumferential direction, Eq. (6.4.20), is then solved for the circumferential velocity

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component, v, using the computed values from the previous step. computed values of v are then used in the next step.

The

Simulation Results As mentioned earlier, various electric current density values have been used in the literature, ranging from 3 A/cm2 to 20 A/cm2. The electric current density values of 3, 5 and 7 A/cm2 used in the growth of thick crystals led to the growth rates of about 0.5, 0.75, and 1.2 mm/day, respectively, with stable growths up to 5 mm crystal thickness (Bryskiewicz [1994], Dost and Sheibani [2000], and Sheibani et al. [2003a]). Current densities higher than 7 A/cm2 led to growth instabilities (Sheibani et al. [2003b]).

(a) B = 0.0 kG,  min = 0.02663

(b) B = 10 kG,  min = 0.002228

(c) B = 20 kG,  min = 0.0007495

(d) B = 40 kG,  min = 0.00002591

Fig. 6.4.10. Stream function contours under various magnetic field values (t = 2 h, J =10 4 A/cm2): Contour spacing is 2.0  10 cm2/s (after Dost et al. [2002]).

However, when convection in the solution is suppressed, by means of an applied external magnetic field for instance, higher electric current densities can be used for higher growth rates (Dost and Sheibani [2000]). This was the goal

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of the experimental work of the LPEE growth of GaInAs under an external magnetic field in Sheibani et al [2003a]. In order to determine the optimum level of the applied magnetic field, the growth of a GaAs system is simulated for various levels of magnetic field and electric current densities. The results for concentration and flow patterns are presented in Figs. 6.4.9 and 6.4.10 for J =10 A/cm2. Note that the simulation results are presented only for the half of the liquid zone due to axisymmetry. As expected, for higher magnetic fields the convection is weakening in the solution.

(a) B = 10 kG, J = 10 A/cm2

(b) B = 40 kG, J = 10 A/cm2

(c) B = 10 kG, J = 3 A/cm2

(d) B = 40 kG, J = 3 A/cm2

Fig. 6.4.11. Evolution of growth interface with time: (a), (b), (c), (d). The time increment between two curves for each case is 10 h (after Dost et al. [2002]).

The evolution of the growth interface is also computed for four different combinations: (a) B = 10 kG, J = 10 A/cm2, (b) B = 40 kG, J = 10 A/cm2, (c) B = 10 kG, J = 3 A/cm2, (d) B = 40 kG, J = 3 A/cm2. Results are presented in Fig. 6.4.11. As can be seen, for a higher magnetic field, the shape of the growth

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interface becomes flatter. In addition, higher electric current densities lead to higher growth rates, as can be seen from Figs. 6.4.11b and 6.4.11d. The growth rate for J = 10 A/cm2, for instance, is about three times higher than that of J = 3 A/cm2, as expected. This has been indeed verified by the experiments of Sheibani et al. [2003a], as mentioned earlier. The results shown in Figs. 6.4.9-6.4.11 were presented for the cases with no tilt in the applied magnetic field. Now, in order to determine the effect of an axial tilt, four levels of magnetic field values were selected: (a)  r =  = 0 , (b)  r =  = 0.02 , (c)  r =  = 0.05 , (d)  r =  = 0.08 . The concentration distributions corresponding to these cases are computed and depicted in Fig. 6.4.12. The concentration distribution for the case with no tilt is given in Fig.6.4.12a for comparison.

(a)  r =   = 0

(b)  r =   = 0.02

(c)  r =   = 0.05

(d)  r =   = 0.08

Fig. 6.4.12. Concentration contours for various levels of tilt values (t = 20 h, J =10 A/cm2, B =10 6 kG): Contour spacing is 5.0  10 mass fraction, and the minimum and maximum concentrations are 0.0218863 and 0.0219603, respectively (after Dost et al. [2002]).

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As can be seen from Fig. 6.4.12a,b,c, and d, the concentration patterns are changing with the increasing level of tilt values. For instance, for  r =  = 0.02 , the concentration patterns are still similar to those of the case with no tilt. However, when the level of tilt is high, for instance,  r =  = 0.08 , the concentration patterns are different, and become smoother in the vicinity of the growth interface. This implies that the growth interface may become flatter. This is obvious from the flow patterns presented in Fig. 6.4.13. Stream function contours become smoother with the increasing levels of tilt values.

(a)  r =   = 0

(b)  r =   = 0.02

(c)  r =   = 0.05

(d)  r =   = 0.08

Fig.6.4.13. Stream function contours for various levels of tilt values (t = 2 h, J =10 A/cm2, B =10 4 kG): Contour spacing 2.0  10 cm2/s after Dost et al. [2002]).

As will be seen later, this is due to the strong contribution of the applied electric current density to the magnetic body force. The last terms in the magnetic body force components Frem and Fem in Eqs. (6.4.17) are large compared with other terms, and are independent of the flow velocity components. When the applied magnetic field is uniform and aligned with the vertical axis, these components vanish. However, when the field is tilted, these

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components are present, and become significant. These two terms in the magnetic body force F em act like additional body forces and suppress convection further. However, in the absence of the contribution of the electric field, such stabilizing body force components will not be present, and large tilt values in the applied magnetic field may have adverse effects on the growth interface shapes. This may imply that in growth techniques without an applied electric current, such as liquid phase epitaxy (LPE) or the travelling heater method (THM), any tilt in the magnetic field may have more prominent effects on the growth interface shapes. However, one should keep in mind that, in THM for instance, the temperature gradients are very large compared with LPEE, and consequently the contributions of the magnetic body force components due to the higher convective flow velocities in the liquid phase may become significant (Lent et al. [2002]). In order to see the effect on the growth interface, these three levels of tilt values were considered for various levels of electric current densities and applied magnetic fields. Simulation results are depicted in Fig. 6.4.14. In this figure dashed lines represent the growth with no tilt. As can be seen from Fig. 6.4.14a, the effect of tilt less than 2% is not significant on the shape of the growth interface. This effect becomes obvious with increasing values of tilt values and the interface becomes flatter for values higher than 8%, see Figs. 6.4.14b and c. For lower electric current densities, these tilt values have similar effects on the growth interface, see Figs. 6.4.147d, e, and f. From these results, one can conclude that the additional magnetic body force components due to such tilts in the applied magnetic field will not have adverse effects in LPEE growth due to the strong contribution of the applied electric current. In fact, no matter how perfectly we try to align the system in the experimental set up, there will be some small tilt. It appears that such a small possible tilt is not affecting our growth experiments since the system is presently allowing us the growth of very good crystals at various electric current levels. One should however keep in mind that the numerically introduced tilt values in the applied magnetic field do not lead to a deviation from the axisymmetry. This was the main assumption in simulations in this section. The inclusion of any deviation from the axisymmetry (vertical misalignment) of the growth system requires a three-dimensional numerical simulation. This in turn requires huge computational power that was not available at time of the work of Dost et al. [2002]. As mentioned earlier, in LPEE a mechanism for forced mixing in the solution is not essential. This is mainly due to electromigration that moves species in a controlled and steady manner towards the seed substrate, and achieves growth (Jastrzebski et al. [1976]). As shown by Djilali et al. [1995] and Dost and Qin [1998], the convective flow in the solution (in the plane of axisymmetry) enhances the growth rate. When the level of convection is however reduced in

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this plane due to the application of an external magnetic field, it is important to know its effect on the growth process, and also the level of convection in the horizontal plane that provides mixing for the solution.

(a)

(d)

(b)

(c)

(e)

(f)

Fig. 6.4.14. Effects of tilt values on the growth interface shape: (a) J = 10 A/cm2, B = 10 kG,  r =   = 0.02 ; (b) J =10 A/cm2, B =10 kG,  r =   = 0.04 ; (c) J = 10 A/cm2, B = 10 kG,  r =   = 0.08 ; (d) J = 10 A/cm2, B = 40 kG,  r =   = 0.02 ; (e) J = 3 A/cm2, B = 10 kG,  r =   = 0.02 ; (f) J = 3 A/cm2, B = 10 kG,  r =   = 0.08 . The dashed lines in each figure represent the interfaces corresponding to the case with  r =   = 0 . The time increment between two curves for each case is 10 h (after Dost et al. [2002]).

To this end, numerical simulations were carried out for two tilt values:  r =  = 0.02 and  r =  = 0.08 , and for two current values: J = 3 A/cm2, and J = 10 A/cm2. Results are depicted in Fig. 6.4.15. As can be seen, while the velocity profiles are very similar in the two cases, the velocity intensities are different, being higher for larger tilt values, as expected.

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Fig. 6.4.16 presents the simulation results for various combinations of tilt values. As can be seen, it appears that when both  r and  are present, the effect of these field components is more prominent in suppressing the convection in the liquid phase.

Fig. 6.4.15. Variation of the velocity component in the r-direction (B = 10 kG, t = 2 h): (a)  r =   = 0.02 and at the middle of the solution zone; (b)  r =   = 0.02 , and near the growth interface; (c)  r =   = 0.08 , and at the middle of the solution zone; (d)  r =   = 0.08 , and near the growth interface. Dashed lines represent the case of J = 3 A/cm2, and the solid lines that of J=10 A/cm2 (after Dost et al. [2002]).

6.5. Three-Dimensional Simulations Under Magnetic Field The simulation results presented in the previous section show that the level of applied magnetic field has a significant effect on natural convection in the solution; both in its intensity and structure. A stronger applied magnetic field leads to weaker convection in the solution, and more uniform interfaces. These two-dimensional numerical simulations have shed light on various aspects of the applied magnetic field in LPEE. However, these models, being two dimensional, have naturally neglected the contributions of the circumferential velocity and magnetic body force components. Therefore, the results presented earlier were qualitative and did not include information about three-dimensional effects such as deviations from axisymmetry, variations of the magnetic field components, and mixing in the solution.

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In order to shed light on these three-dimensional effects, three-dimensional models were developed by Liu et al. [2003, 2004] for the LPEE growth of a binary system. To the best of our knowledge, these were the first threedimensional simulation models developed for LPEE. Here, we present the key features of these models.

(a)

(c)

(b)

(d)

(e)

Fig. 6.4.16. Effect of combinations of tilt values (J =10 A/cm2, B =10 kG): (a)  r =   = 0.08 ; (b)  r = 0.08 ,   = 0 ; (c)  r = 0 ,   = 0.08 ; (d)  r = 0 ,   = 0.20 ; (e)  r = 0.20 ,   = 0 . Dashed lines represent the interfaces corresponding to each case with  r =   = 0 . The time increment between two curves for each case is 10 h (after Dost et al. [2002]).

6.5.1. Simulation Model The interest in our experimental study (Sheibani et al. [2003a,b]) was focused on the LPEE growth of a ternary In0.04Ga0.96As system under a static vertical magnetic field. However, since the emphasis is on the variations of flow

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patterns and intensities in the solution, and also for computational simplicity, the three-dimensional simulation study was carried out for the growth of a binary system (GaAs). The effect of magnetic field non-uniformity is also investigated. The SETUP2 in Fig. 6.2.1 is selected for simulation. The governing equations are written explicitly in cylindrical coordinates for a binary system as (see Chapter 4): Continuity

1 v w 1  ru + + =0 r  z r r

( )

(6.5.1)

Momentum 2  u v u u v 2 u 2 v 1 p  E B u u (6.5.2)  +u + +w  =   2u  2  2  r r  z r t

L  r r   L r 2  2 1 p  E B v v v v v uv v 2 u

v (6.5.3) +u + +w + =   v  2 + 2   r r  z r t

L  r r   r L 

w v w w 1 p w +u + +w =  2 w   g t T - T0 + g  c C - C0 r r  z t  L z

(

)

(

)

(6.5.4)

Mass transport

C v C C C + (u + μ E Er ) + + (w + μ E Ez ) = DC  2C r r  z t

(6.5.5)

Energy

T v T T T +u + +w =  2T r r  z t

(6.5.6)

Electric charge balance

 2 r 2

+

1   2 + =0 r r z

(6.5.7)

where the gradient operator is defined as

2 =

1   1 2 2 (r ) + 2 + r r r r  2 z 2

(6.5.8)

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The associated boundary and interface conditions are the same as those given earlier for two-dimensional cases in the previous sections. However, we write the list for the sake of completeness. Along the vertical wall

u = 0, v = 0, w = 0, T = Tg 

z  z0 H

T ,

 C = 0, =0 r r

(6.5.9)

Along the growth interface

u = 0, v = 0, w = 0, ks

T T   kL =  J ,    = J , C = C1 z z r

(6.5.10)

Along the dissolution interface

u = 0, v = 0, w = 0, ks

T T  kL = + J ,  = 0, C = C2 z z

(6.5.11)

Initial conditions at t = 0

C = C0 , u = 0, v = 0, w = 0, T = Tg

(6.5.12)

6.5.2. Deviation from Axisymmetry Measurements show that the magnetic field distribution in the magnet opening where the growth crucible is located is almost uniform (see Sheibani et

P R’ P0 R



0

O(0,0,z)

Fig.6.5.1. The coordinate system in the horizontal plane .

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al. [2003a]). However, when the growth system is lowered into the magnet opening, it is not possible to measure the field distribution, and it is likely that the field uniformity would be altered. Furthermore, the center of the growth system may not be aligned perfectly with the vertical axis due to possible unintentional errors. This will result in further deviations from axisymmetry. This in turn may affect the growth adversely and lead to unsatisfactory growth. It is therefore desirable to investigate the influence of such asymmetries. In order to accommodate such deviations in the model, it is assumed that the center of the magnetic field is offset with respect to the symmetry axis, and is located at P0(R0, 0, z) in the horizontal plane (see Fig. 6.5.1). The magnitude of the magnetic field is assumed to be in the form of

(

)

B = B0 1+ AR , R =

( R sin   R sin  ) + ( R cos  R cos ) 2

0

0

0

0

2

(6.5.13)

where A is the coefficient measuring the nonuniformity, R is the distance in the r   plane between point P(R, , z) and the center of the magnetic field at P0 , R is the distance between point P and the center of the circular plane O(0,0, z) , and R0 is the distance between point P and O. 6.5.3. Numerical Method The commercial CFX software is used to solve the field equations. The computation mesh in the liquid is 1204080 in the r-, -, and z-directions, respectively, which is demonstrated to be sufficient for an accurate and stable solution. Since the focus is on the flow field only, the evolution of the growth and dissolution interfaces is not included in the computations. The mass transport equation in the solution is solved simultaneously in order to take into account the influence of concentration field on the flow field (through the solutal Grashof number); however, the concentration field is not presented here for the sake of space. Transient terms are considered in the energy, mass transport, and momentum equations in order to account for possible unsteady flows and their influence. The simulation results are presented at t = 1 h since the flow field has fully developed by that time. Most of the physical and growth parameters of the GaAs system were given in Table 6.2.1. The parameters that are different and were not given earlier are presented in Table 6.5.1. Dimensionless numbers are the same defined earlier in Eqs. (6.3.16), except that they are defined in this section with respect to solution height H. The Hartmann number is defined by

Ha = BH  E  L

(6.5.14)

The simulations are carried out for the half of the cylindrical cell domain for computational efficiency. However, to ensure that the half domain solution represents fully the flow structure of the full domain, a full domain solution was

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carried out for a case that asymmetry was assured (under a large magnetic field, B = 4 kG). Results demonstrated that the half plane treatment is reliable. Computed temperature distributions in the horizontal plane near the growth interface and also in the vertical plane at  = 0 are given in Fig. 6.5.2. Temperature distributions agree with earlier 2D-solutions (Djilali et al. [1995], Qin et al. [1995]). In addition, since the electric current is passing through the source in this setup (SETUP2), the computed isotherm patterns indicate that the shape of the growth interface will be single-humped (concave towards the crystal), as expected.

(b)

(a)

Fig. 6.5.2. Temperature distribution (K): (a) near the growth interface in the horizontal plane, (b) in the vertical plane at  = 0 (after Liu et al. [2003, 2004a]). Table 6.5.1. Parameters of the GaAs system (Liu et al. [2002]). Parameter

Symbol

Value

Crystal radius

Rc

12.0 mm

Solution height

H

10.3 mm

Magnetic field intensity

B

0 to 12 kG

Nonuniformity coefficient

A

0 to 4

Thermal Grashof number

GrT

6.84104

Solutal Grashof number

GrC

7.10103

Hartmann number

Ha

0 to 871.5

Prandtl number

Pr

4.0010-3

6.5.4. Effect of Magnetic Field Strength To have a better sense for the flow field in the growth cell, the simulation results are presented in three distinct planes, the vertical (r-z) plane at  = 0 that represents typical flow structures along the growth direction, the horizontal (r-) plane at the middle of the growth cell (z = 5.15 mm) where the changes in the flow field will be more prominent for radial and circumferential velocity components representing mixing, and finally the horizontal plane (r-) near the

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growth interface (z = 0.4 mm) where the flow field is often closely related to the quality of the grown crystals. Fig. 6.5.3 presents the simulation results for the flow field that are presented in the horizontal plane at the middle of the growth cell (left column) and in the vertical plane at  = 0 (right column) for four levels of magnetic field strengths (B = 0.0, 0.5, 1.0, and 2.0 kG), while Fig. 6.5.4 shows the results for the flow field in the horizontal plane near the growth interface (left column) and in the vertical plane at  = 0 (right column) for two levels of magnetic field intensities (B = 2.5 and 3.0 kG), and in the horizontal plane at the middle of the growth cell (left column) and in the vertical plane at  = 0 (right column) for B = 4.0 and 8.0 kG. Local magnitudes of the flow velocity (which will be referred to as “flow strength”) are computed in m/s, and the scales of flow strengths are shown in each figure.

(a) B = 0 kG

(b) B = 0.5 kG

(c) B = 1.0 kG

(d) B = 1.0 kG Fig. 6.5.3. Flow field in the horizontal plane in the middle of the solution zone (left column) and in the vertical plane at  = 0 (right column) (after Liu et al. [2002, 2004a]).

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The computed flow patterns in the vertical plane appear more complex than those of the 2D simulations presented in Section 6.2 for the same LPEE growth system (Fig. 6.5.3a). This complexity can be attributed mainly to the inclusion of the contribution of the circumferential velocity component. In addition, the size of the growth crucible used in this study is twice the one used in Section 6.2; this might have also contributed towards these differences. Such differences in the flow field of 2D- and 3D-models have also been observed in Dost et al. [2002], Ben Hadid and Henry [1996], and Ben Hadid et al. [2001]. The strongest flow is seen in the lower part of the crucible cell, near the centre of the half domain along the r-direction. This is similar to what was observed in Dost et al. [2002]. A weak flow cell forms just above the strongest flow cell. The flow in the horizontal planes is nearly homocentric. The flow in the horizontal plane near the growth interface is stronger in the middle region along the radial direction and becomes weaker and weaker near the growth cell wall or the axial center (Fig. 6.5.3a—left column). On the other hand, in the horizontal plane at the middle of the growth cell (Fig. 6.5.3a—left column) there are two maximum points for the flow intensity along the radial direction, with a relatively strong flow in the central region. Note that the flow field in Fig. 6.5.3a is not strictly axisymmetric, especially in the middle of the growth cell, although the flow is stable. This result suggests that the present system is near the Grashof number that is a little bit lower than the critical Grashof number (2.5105) under the same conditions (Prandtl number and geometry aspect ratio) given in Gelfgat et al. [2000] in the analysis of axisymmetry breaking of natural convection in a vertical Bridgman growth configuration. Figs. 6.5.3b–d and Figs. 6.5.4a and b show the influence of the applied magnetic field on the flow field. Results are also summarized in Figs. 6.5.5 and 6.5.6 for various aspects of the flow field. Simulation results show three distinct characteristics depending on the level of applied magnetic field: (a) the ‘‘weak’’ magnetic field, with intensities from 0.0 to 2.0 kG, (b) the ‘‘intermediate’’ magnetic field, with levels from 2.0 to 3.0 kG, and (c) the ‘‘high’’ magnetic field, with intensities above 3.0 kG. Flow characteristics are quite different at each of these field levels. Let us first focus on Figs. 6.5.3b–d which represent results for magnetic field levels from 0.5 to 2.0 kG. In this category, the weak magnetic field level, an increase in the applied magnetic field strength results in significant reduction in the flow strength, which is, in general, desirable for a stable and controlled crystal growth. Flow cells are the strongest near the vertical wall and forms the so-called Hartmann layer (Kim et al. [1988], Ben Hadid and Henry [1996], Davoust et al. [1999], and Ben Hadid et al. [2001]). At higher magnetic field strengths, the relative strength of these flow cells becomes increasingly stronger compared with the flow in the rest of the growth cell domain, and the Hartmann layer becomes thinner which is in accordance with the scaling analysis in Ben

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Hadid and Henry [1996]. Furthermore, the strongest flow cells appearing in the lower part of the growth cell (Fig. 6.5.3a) move further down towards the growth interface with increasing strength, and new flow cells form near the dissolution interface. These strong flow cells form very visible boundary layers near the growth and dissolution interfaces (the so-called end layers Ben Hadid and Henry [1996], Davoust et al. [1999]), and hence give rise to strong vertical velocity gradients in the vicinity of the growth interface, which may have an adverse effect on the crystal growth process.

(a) B = 2.5 kG

(b) B = 3.0 kG

(c) B = 4.0 kG

(d) B = 8.0 kG Fig. 6.5.4. Flow field in the horizontal plane near the growth interface at z = 0.4 mm (left column) and in the vertical plane at  = 0 (right column) for (a) B = 2.5 kG, and (b) B = 3.0 kG, and in the horizontal plane at the middle of the growth cell (left column) and in the vertical plane (right column) for (c) B = 4.0 kG, and (d) B = 8.0 kG (after Liu et al. [2002, 2004a]).

One may state that the application of a magnetic field may not always be beneficial for the growth process (Kim et al. [1988], and Dost et al. [2003]). It was shown by Kim et al. [1988] that the radial nonuniformity in vertical

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Bridgman is the most significant at the intermediate levels of magnetic field strength. Flow fields become perfectly homocentric (and hence axisymmetric) with the increase of magnetic field strength, and hence the flow strength decreases. The flow velocities in the central region of the cell (both in the r- and z-directions) become more uniform at higher magnetic field levels, and form a core region of the uniform flow field (Kim et al. [1988], and Ben Hadid and Henry [1996]), leading to a domain in which the flow is suppressed with the application of an applied magnetic field. Indeed, an applied magnetic field at the weak intensity levels can suppress fluid flows in a growth system, as shown in Fig. 6.5.3.

(a) B = 0.0 kG

(b) B = 1.0 kG

(c) B = 2.0 kG

(d) B 3.0 kG

Fig. 6.5.5. Variation of the flow strength U = (u + v + w ) along the radial direction at  =  / 2 under various magnetic field strengths: (a) B = 0.0 kG, (b) B = 1.0 kG, (c) B = 2.0 kG, and (d) B = 3.0 kG. Solid lines at z = 5.05 mm (in the middle of the growth cell), and dashed lines at z = 0.4 mm (near the growth interface) (after Liu et al. [2002, 2004a]). 2

2

2 1/ 2

To the best of knowledge, a magnetic field level higher than 8.0 kG is very strong for an LPEE growth of thick crystals, due to the strong interaction between the applied magnetic and electric fields. Thus far it was not possible to exceed the level of 4.5 kG in experiments (Sheibani et al. [2003a]). Higher magnetic field levels led to unsatisfactory growth (Sheibani et al. [2003b]). The

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issue of LPEE growth under higher magnetic fields and also the higher growth rates observed under magnetic fields will be further discussed later in this section. Fig. 6.5.4 represents the simulation results for the intermediate magnetic intensity levels selected as B = 2.5, 3.0, 4.0, and 8.0 kG. In this case, flow strength increases with magnetic field intensity. This result has not been widely reported in the relevant literature. It is possible that it is numerical in nature, and can be fixed by some innovative numerical treatments (see Vivek et al. [2005]), but it is also possible physically due to the following reasoning. The magnetic body force and the body force due to buoyancy (maybe others such as surface tension) acting on the points of the liquid, act in different directions to balance each other in a closed container (growth cell for instance). Up to a certain level of magnetic body force, the magnetic body force suppresses the fluid flow by counterbalancing the buoyancy force. However, when the level of magnetic field exceeds a certain value, it surpasses the buoyancy force and becomes an excessive force, feeding the convection in the liquid. Thus, the flow strength increases further. Some experimental and numerical studies indeed demonstrated the enhancement of heat transfer (and hence flow strength) in a melt under a stationary magnetic field (Tagawa and Ozoe [1997, 1998], and Terashima et al. [1987]). As can be seen from Fig. 6.5.4, the flow patterns show dramatic changes, as two flow cells were formed in each half of the vertical plane, with the upper cells getting larger and lower cells getting smaller with the increasing magnetic field intensity. In the vertical plane, some strong unidirectional flows

Fig. 6.5.6. Variation of the maximum velocity with magnetic field intensity: three distinct regions of stability of the flow field are obvious. The flow field is stable up to Ha = 150; in the region between Ha = 150 and 220 the flow is transitional, and above Ha = 220 the flow is unstable (after Liu et al. [2002, 2004a]).

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appear in the middle region along the r-direction, and some with very weak intensity in the middle forming a small cylindrical region where the flow is nearly stationary. The flow fields are no longer axisymmetric and homocentric. Such an axisymmetry breaking may be caused by the unsteadiness of the flow field according to Gelfgat et al. [2000]. The flow stability was also examined at B = 2.5 and 3.0 kG levels. Comparing the results at different times, it was found that the flow field is essentially stable in spite of some small changes with time. One can then speculate that the reason for the axisymmetry breaking of the flow field at these magnetic field levels may be physical and due to a very delicate balance between the buoyancy and magnetic forces, even though the flow still remains stable.The variation of the flow strength along the radial direction for B = 0.0; 1.0, 2.0, and 3.0 kG levels are shown in Fig. 6.5.5. Solid and dashed lines represent, respectively, the values at the middle and the lower regions of the solution zone. As can be seen, the flow strength fluctuates and shows differences in these regions. However, at higher magnetic field levels, the difference becomes less obvious. The variation of the flow strength in the radial direction near the growth interface is almost symmetric in the absence of a magnetic field. However, this symmetry is broken at higher magnetic field intensity levels. Although higher magnetic field intensity levels, higher than 4.5 kG, appear not to be practical for the LPEE growth of thick crystals (Sheibani et al. [2003b]), for the sake of completeness and also for the purpose of comparison with other studies, higher magnetic field intensity levels of B = 4.0 kG and B = 8.0 kG were also considered. The flow patterns become dramatically different and show large fluctuations with time, and temperature distributions show asymmetric behavior. The maximum flow strength (the maximum magnitude of the local velocity vector) values, Umax, are presented in Fig. 6.5.6 for all magnetic field levels. As seen, the variation of the maximum flow strength shows three distinct regions of stability. The flow strength decreases with increasing magnetic field in the stable region (up to Ha = 150), but in the intermediate and unstable regions (between Ha = 150 and 220, and above Ha = 220; respectively) the flow strength increases. If one examines the logarithmic plot of Umax as a function of the Hartmann number, Ha, given in Fig. 6.5.6, we can see that within the stable region, the relationship between Umax and Ha obeys a power law of

U max  Ha 5/ 4

(6.5.15)

which has been demonstrated by many authors (Kim et al. [1988], Ben Hadid and Henry [1996], Ben Hadid et al. [2001], Davoust et al. [1999]), although the index of the power law is slightly different due to different system parameters and conditions. In the unstable region, this relationship becomes

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U max  Ha5/ 2

(6.5.16)

In the transitional (intermediate) region, on the other hand, as one expects, the change of the maximum strength of the flow field with the Hartmann number is so dramatic that the power law is not suitable. The results given here for the intermediate and unstable regions were obtained for the first time by Liu et al [2002], and, to the best of our knowledge, are not corroborated by anyone else in the literature.

(a) B = 0.0 kG, t = 20 hours

(b) B = 1.0 kG, t = 20 hours

(c) B = 3.0 kG, t = 20 hours Fig. 6.5.7. Flow field in the horizontal plane in the middle of the growth cell (left column) and in the vertical plane at  = 0 (right column) (after Liu et al. [2002, 2004a]).

6.5.5. Evolution of Interfaces As mentioned earlier, the above simulation results were given at the end of a one-hour growth period (t = 1.0 h) and also for a stationary interface, for computational efficiency. This was sufficient for examining the flow field and the effect of a magnetic field on the flow structures. However, in order to draw meaningful conclusions for future experiments, and also examining the concentration fields in the solution and the evolution of interfaces, simulations were carried for a longer period of growth (t = 20.0 h) for three levels of

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magnetic fields, namely B = 0.0 kG (no applied magnetic field), B = 1.0 kG (the mid-field level in the stable region), and B = 3.0 kG (a field at the beginning of the unstable region), and also the evolution of growth and dissolution interfaces are included. Fig. 6.5.7 summarizes the computed results for the flow field in the horizontal plane at the middle of the growth cell (left column) and in the vertical plane at  = 0 (right column) for three levels of magnetic field intensities (B = 0.0, 1.0, and 3.0 kG). Flow strengths are computed in m/s, and the scales are shown in each figure. The inclusion of the evolution of growth and dissolution interfaces affected the flow patterns to a certain extent, although it is not significant in terms of magnitudes. The effect of the magnetic field in the stable region is obvious in suppressing the natural convection in the solution. At the B = 3.0 kG level, however, the flow patterns show the signs of unstable flows.

(a) B = 0.0 kG, t = 20 hours

(b) B = 1.0 kG, t = 20 hours

(c) B = 3.0 kG, t = 20 hours Fig. 6.5.8. Concentration distributions in the horizontal plane near the growth interface (left column) and in the vertical plane (right column) (after Liu et al. [2002, 2004a]).

The concentration distributions at the B = 0.0, 1.0, and 3.0 kG magnetic field levels are presented in Fig. 6.5.8 at t = 20 h. Comparing Fig. 6.5.8a and b, one can see that the concentration gradients near the center of the growth interface and near the growth cell wall in the vicinity of the dissolution interface decrease

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with the increasing magnetic field levels. This is due to the reduction in flow strength, resulting in slower mass transfer towards the growth interface, and consequently slowing down both the growth and dissolution rates. The concentration vortices in Fig. 6.6.8a nearly disappear in Fig. 6.5.8b. At the B = 3.0 kG level (Fig. 6.5.8c), concentration distributions exhibit significant changes compared with those of Fig. 6.5.8a and b. The concentration vortices near the center of the symmetric axis, in the vicinity of the growth interface form strong concentration gradients, and lead to fast growth rates in that region. Concentration layers are formed in the vicinity of both the growth and dissolution interfaces. Concentration distributions determine the shape of the growth interface. Fig. 6.5.9 presents the evolution of growing interface of the crystal under the selected three different magnetic field levels. Comparing the growth interface shapes in Fig. 6.5.9a and b, one can see that in the stable region, at the B = 1.0 kG magnetic field level, the growth interface is flatter than that with no field, as expected. At the B = 3.0 kG field level, however, the growth interface becomes very wavy in the central region (Fig. 6.5.9b), implying that growth might become unstable with time. The growth rates obtained from simulations (Fig. 6.5.9) under magnetic field do not predict the experimental growth rates (Dost et al. [2004]). We will discuss this issue later in detail.

(a)

(b)

Fig.6.5.9. The evolution of the growth interface under (a) B = 0.0 kG (- - -) and B = 1.0 kG (––), and (b) B = 3.0 kG (after Liu et al. [2002, 2004a]).

6.5.6. Effect of Magnetic Field Nonuniformity As mentioned earlier, in order to see the effect of field non-uniformities, certain non-uniformity levels are incorporated into the simulation model. Selecting  0 = 0 , and R0 = 0.008 m, which, together with the crystal radius of R = 0.012 m, gives the maximum difference of the magnetic field as 2A% (see Eq. (6.5.13), for instance A = 1.0 and 3.0, correspond to the nonuniformity levels of

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2.0% and 6.0%, respectively. These non-uniformity levels were examined for the magnetic field level of B = 1.0 kG. The results of the numerical simulations are presented in Fig. 6.5.10. Results show that even for a 6% non-uniformity, the changes in the flow field are small, indicating that the natural convection becomes weaker under the application of an external magnetic field, and the influence of such nonuniformities at this level of external magnetic field is not significant. However, the flow field becomes asymmetric with increasing non-uniformity (see Fig. 6.5.10c). As one expects, at higher levels of applied magnetic field intensity, such non-uniformities may have significant effects on the flow field; however, due to the practical insignificance, the simulation results conducted for B = 8.0 kG are not presented here. Comparison with 2D simulations for the same growth system shows that the flow patterns of the 3D model appear to be much complex. The flow structures at high magnetic field strengths could not be observed through the 2D models, partially due to the omission of the velocity and magnetic force components along the circumferential direction.

(a) B = 1.0 kG, A = 0.0

(b) B = 1.0 kG, A = 1.0

(c) B = 1.0 kG, A = 3.0 Fig.6.5.10. Flow field under B = 1.0 kG and three different non-uniformity values: (a) A = 0.0; (b) A = 1.0; and (c) A = 3.0: The left column describes the flow patterns in the horizontal plane near the growth interface at z = 0.4 mm, and the right column shows the flow patterns in the vertical plane at  = 0 (after Liu et al. [2002, 2004a]).

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Fig.6.5.11. Computed flow strength values at three levels of electric current densities: (a) J = 3 A/cm2, (b) J = 5 A/cm2, and (c) J = 7 A/cm2 (after Liu et al. [2004a]).

6.5.7. Effect of High Electric and Magnetic Field Levels The flow field was also numerically simulated under various electric current densities in Liu et al. [2004]. The maximum flow strength shows the same trend for at J = 3, 5, and 7 A/cm2 electric field levels, and are shown in Fig. 6.5.11. The time evolution of the flow field is computed under various magnetic field intensity levels, and the one corresponding to B = 4.0 kG and J = 7 A/cm2 is shown in Fig. 6.5.12. As can be seen, the flow patterns change with time and begin to become localized after 120 s of growth. This localization point is near the growth interface and is approximately at about a distance of  of the diameter of the crystal from the edge. This point is almost at the locations where the holes (or damages) were observed in crystals grown under high fields by Sheibani et al. [2003a,b]. Simulations are shown in Fig. 6.5.12 for the liquid solution zone, but it is also possible that similar flow patterns can also be computed in the liquid contact zone below the seed crystal. The flow patterns computed for B = 3.0 kG and J = 5 A/cm2 show similar patterns (Liu et al. [2004]). The numerical simulations are in qualitative agreement with experiments, but do not agree quantitatively on the critical value of the magnetic field. Numerical simulations predict a lower value (just over 2 kG) than the maximum experimental value of 4.5 kG. Crystals were grown by Sheibani et al. [2003a] at field values up to B = 4.5 kG, and J = 7 A/cm2. However, experiments at higher fields failed (two samples of such crystals are shown in Fig. 6.5.12f).

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Three-dimensional numerical simulation results have shown that magnetic field intensities up to 2.0 kG suppress the flow structures in the solution, and the flow structures are stable and get weaker with the increasing magnetic field level. These levels of magnetic field are beneficial in suppressing the natural convection. However, field intensities higher than 2.0 kG change the flow patterns significantly, and at intensities higher than 3.0 kG the flow structures become unstable (Liu et al. [2002, 2004a], Sheibani et al. [2003b]).

(a) t = 70 s

(b) t = 120 s

holes

(c) t = 600 s

(d) t = 1800 s

(e) t = 3600 s

(f) Samples of crystals with holes (top view). Holes are approximately at the distance half the radius from the edge of the crystal. Crystal diameter is 25 mm, and thickness is about 4.5 mm.

Fig.6.5.12. Time evolution of the flow field (iso-strength contours). The flow field is localized near the growth interface at about a distance of half of the crystal radius approximately where holes are observed (after Liu et al. [2004a], samples from Sheibani et al. [2003b]).

As discussed in detail in Chapter 3 (Section 5), the LPEE growth system developed by Sheibani et al. [2003a] allowed the growth of a large number of GaAs and Ga0.96In0.04As single crystals of thicknesses up to 9 mm. It was possible to apply electric current densities of 3, 5, and 7A/cm2, and the corresponding growth rates in these experiments with no magnetic field were respectively about 0.57, 0.75 and 1.25 mm/day. Growth interfaces were very flat, and the growth experiments were reproducible in terms of crystal thickness and growth rate. Experiments at higher electric current intensities were not

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successful. Experiments at 3, 5 and 7 A/cm2 electric current densities were repeated under various applied static magnetic field levels. Results showed that LPEE experiments at the 4.5 kG and lower magnetic field levels were successful, but those under higher magnetic field levels were not. In fact the experimental study of the LPEE growth of GaAs and Ga0.96In0.04As single crystals (Sheibani et al. [2003a]) supported qualitatively the results of the three-dimensional numerical simulations. It seems that the 4.5 kG field intensity is a maximum (critical) level above which the growth is not stable. This experimental ‘critical’ magnetic field level is higher than that predicted from the numerical simulations performed under the same growth conditions, which was somewhere between 2.0 and 3.0 kG (Liu et al. [2002]). Considering the complexity of the LPEE growth process, this is a good qualitative agreement. As mentioned above, the LPEE experiments of Sheibani et al. 2003a,b have shed light on the issue of the presence of an optimum applied magnetic field in suppressing convection for prolonging the LPEE growth for larger crystals, and supported the observations of three-dimensional simulations.

(a)

(b) Fig.6.5.13. Samples of LPEE grown GaAs crystals under high fields: grown under (a) B = 20 kG and J = 3 A/cm2, and (b) B = 0, and J = 10 A/cm2 (after Sheibani [2002]).

However, as discussed in Chapter 3, the LPEE experiments of Sheibani et al. [2003a] yielded a very significant result that was not predicted from the modeling studies conducted so far: the experimental LPEE growth rates under magnetic field were much higher than the expected values. For instance, the

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growth rate at the 4.5-kG magnetic field level (at J = 3A/cm2) was about 6.1 mm/day, which is about 12 times higher than that with no magnetic field. Experiments performed at B = 1.0 and 2.0 kG field levels (at J = 3 A/cm2) were also successful, and the growth rates were also higher: 1.62 and 2.35 mm/day, respectively. Such growth rates have not been predicted from any models so far. One more interesting observation of the LPEE experiments was that the direction of the applied magnetic field, either up or down, was not relevant. The growth rate was almost the same, being about 5–6% less when the magnetic field was in the direction of the applied electric field (Liu et al. [2004]). As predicted from the three-dimensional models, at higher magnetic field levels (even with the J = 3A/cm2 electric current density level), and higher electric current density levels (J = 10 A/cm2 or higher), experiments did not lead to successful growth, but showed very interesting outcomes (Sheibani et al. [2003b], Liu et al. [2004]). Although very thick crystals were grown, even up to a 9 mm thickness, the growth processes were not stable, and led to uneven growth (Figs. 6.5.12 and 6.5.13). From the visual inspection of the grown crystals, the adverse effect of natural convection was obvious, causing either one-sided growth or leading to holes in the grown crystals. It was considered that such growth (one-sided and with holes) is because of the strong and unstable convection in the liquid zones (solution and contact zones) due to the strong interaction between the applied magnetic field and the applied electric current. Such predictions were also confirmed qualitatively by the numerical simulations carried out by considering field non-uniformities in (Liu et al. [2002]), and also by using a newly defined electromagnetic mobility in (Liu et al. [2004], Dost et al. [2005a,b]) and a new model in Dost [2005]. The simulated flow structures show the possibility of causing such non-uniform growth of crystals. 6.6. High Growth Rates in LPEE: Electromagnetic Mobility In the numerical simulation models discussed so far, the contribution of an applied static magnetic field was incorporated into the momentum equations through the magnetic body force components. These magnetic body force components together with the gravitational body force, act on the fluid particles in the liquid solution, and as a result, suppress the fluid flow in the solution zone. Numerical simulations have provided quite accurate predictions for the fluid flow and temperature distributions in the solution zone. However, these models failed to predict the experimental mass transport rates (growth rates) in LPEE under an applied magnetic field. This was mainly for the following reason. Although the contribution of the applied electric current to mass transport was included very accurately through the electric mobility, the contribution of applied magnetic field did not appear in the mass transport equation. Therefore, the high growth rates could not have been predicted through numerical simulation models.

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This issue in LPEE was addressed by introducing a new model that includes the contribution of applied magnetic field in mass transport in Dost et al. [2004], and numerical simulations based on this model were carried in Liu et al. [2004a]. Later, a complete nonlinear theory of LPEE growth under magnetic field was given by Dost [2005] and Dost and Sheibani [2006] as presented in Chapter 4. Table 6.6.1. Numerical values of mobilities (Sheibani et al. [2003b], Dost et al. [2005], and Dost and Sheibani [2006]). Experimental values Magnetic field (kG)

0.0

1.0

2.0

4.5

Electric current density (A/cm )

3.0

3.0

3.0

3.0

Growth rate (mm/day)

0.50

1.62

2.35

6.10

Electric mobility constant, μ E ( m2/V.s)

0.710-5

0.710-5

0.710-5

0.710-5

Total mobility, μT = μ E + μ B B ( m2/V.s)

0.710-5

2.310-5

3.410-5

7.110-5

Electromagnetic mobility, μ B B (m2/V.s)

0.0

1.610-5

2.710-5

6.410-5

1.410-5

1.410-5

1.410-5

3

5

10

2

Computed values

Electromagnetic mobility constant, μ B (m2/V.s.kG) Dimensionless mobility, μ = μT / μ E  1 + 2B

1

As discussed in Chapter 4, the contribution of electromigration under magnetic field was obtained through a nonlinear model (for the binary GaAs system, from Eq. 4.3.24 in the absence of the Soret effect) as

i =  L DC C +  L (DEC + DECB B)CE

(6.6.1)

The second term in Eq. (6.6.1) represents the contribution of applied electric current density to mass transport under the effect of a static external magnetic field. This term represents electromigration in the mass transport equation. Its coefficient, which is called the total mobility (Dost et al. [2004, 2005]), is written in the following form for convenience:

μT  DEC + DECB B  μ E + μ B B

(6.6.2)

where the material constant μ E (a second order material coefficient) is the classical electric mobility of the solute (As) in the liquid solution (Ga-As solution) due to the applied electric current in the absence of an applied magnetic field. The constant μ B is a third-order material coefficient that represents the contribution of the applied magnetic field intensity to the

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electromigration of species. It is zero (or insignificant) in the absence of an applied electric current. This term is called the electromagnetic mobility of solute (Dost [2006]). Its value is determined using the experiments of Shiebani et al. [2003a]). The mass transport equation then becomes

( μ E + μ B B)(E  C) + DC  2C =

C + v  C t

(6.6.3)

6.6.1. Estimation of the Electromagnetic Mobility Value Experiments show that the growth rate is proportional to the applied electric current density, and we have evaluated the value of μ E in the Ga-As solution in the absence of an applied magnetic field. The numerical simulations based on this value verify the experimental growth rates at all three electric current levels (J = 3, 5, and 7 A/cm2). Of course, diffusion (the second term in Eq. (6.6.3)) and also natural convection (the last term on the right-hand side of Eq. (6.6.3)) contribute to the growth rate. However, as shown many times numerically, in LPEE the contribution of the first term (electromigration) is dominant, and the growth rate can be assumed approximately proportional to this term. Experiments also show that the growth rate increases significantly in the presence of a static magnetic field, and is also proportional to the field intensity level as long as the field level is below a critical value, above which the growth is not stable (Liu et al. [2002], Sheibani et al. [2003a,b]). Numerical values of the mobilities are calculated using the results of a large number of experiments of Sheibani et al. [2003a] in which the magnetic field vector B was applied both upward and downward. The growth rates in these experiments were almost the same whether B was up or down. In other words the mass transport due to electromigration was only dependent on the magnetic field intensity but not on its direction. This is also in compliance with the defined constitutive equations in Chapter 4. Using the measured growth rates, the mobility values were computed (see Table 6.6.1). The dimensionless mobility is defined as

μ=

μT μ = 1+ B  1+ 2B μE μE

(6.6.4)

and is plotted in Fig. 6.6.1. As seen the total mobility is almost linearly dependent on the magnetic field intensity, of course, within the limit of experimental measurements. The first term in the mass transport equation reads explicitly as

( μ E + μ B B)(E + v  B)  C

(6.6.5)

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Dimensionless Mobility

1.1 10.1

9.0

7.0

5.0 4.9 3.0 3.3 1.0 1.0

2.0

3.0

4.0

Magnetic Field Intensity (kGauss)

5.0 B

Fig. 6.6.1. Dependence of the total mobility on magnetic field intensity (Dost et al. [2005]).

where the term (v  B)  C is the contribution of the applied magnetic field due to the motion of the fluid particles (coupling term). Its contribution was found to be very small compared to that of E  C (in the order of 3% based on a maximum velocity of 0.01 m/s and a 10 kG field level) in Dost et al. [2002] and Liu et al. [2002]. Therefore, its contribution can be neglected in the model for computer simulations. Then, the electromigration term in the mass transport equation is written as

( μ E + μ B )E  C = μT E  C

(6.6.6)

and the growth rate is computed by

Vng =

L 1 C (DC + μT CEn ) Cs  C n s

(6.6.7)

or simply by, for the purpose of evaluating the mobility constants,

Vg =

L 1 C (DC + μT CEz ) Cs  C n s

(6.6.8)

6.6.2. Simulations of High Growth Rates in the GaAs System Simulations presented in the previous sections are repeated using the total mobility values in the mass transport equation, Eq. (6.6.3). In earlier simulations only the electric mobility μ E was used. A summary of the growth

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rates from these numerical simulations is presented in Fig. 6.6.2a. The values under no magnetic field are the experimental growth rates and are used to compute the value of μ E . Naturally, they are coincident with the computed values. As seen, the growth rate decreases first with the magnetic field level and then increases with the magnetic field above the critical value. This pattern is similar to the pattern of experimental growth rates under various magnetic field levels, and also agrees with the numerical simulation results. In order to predict the high experimental growth rates under an applied magnetic field, the mass transport equation in Eq. (6.5.5) was replaced with Eq. (6.6.3), and then the 3-D simulations were repeated using the total electromagnetic mobility μT = μE + μB values given in Table 6.6.1. The growth rates computed using the total mobility μT are presented in Fig. 6.6.2b (the full circles), and agree with those of experiments. The growth rates using only the electric mobility μE are also given in Fig. 6.6.2b for comparison (denoted by empty squares). For the sake of completeness and for comparison, the experimental growth rates under a magnetic field are also presented in Fig. 6.6.2b (note that the full and empty circles are coincident).

(a)

(b)

Fig. 6.6.2. (a) Computed growth rates are presented versus applied magnetic field with the use of constant electric mobility μ E. Squares represent the values at J = 3 A/cm2 and circles denote for the values at J = 7A/cm2. (b) Growth rates computed using the total mobility, μT = μE + μB, are presented by full circles. These values are in agreement with the experimental growth rates (hallow circles; but are coincident with the full circles, see Table 6.7.1 for their values). The growth rates under no magnetic field are also shown (squares) for comparison (Liu et al. [2004a]).

Transient behavior of the maximum flow strength is shown in Fig. 6.6.3. It is interesting to see that under the effect of an applied magnetic field, the fluid flow in the solution reaches an almost-steady state much faster than that in the absence of a magnetic field. This behavior also shows that there is a critical magnetic field value below which the flow is suppressed. The flow reaches steady-state much faster under an applied magnetic field.

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Fig.6.6.3. Transient behavior of the flow field is presented under three magnetic field levels, B = 0.0, 2.0, and 4.0 kG. It is obvious that flow is suppressed under low magnetic field levels (less than the critical value). It is also interesting that the flow under magnetic field reaches an almost-steady-state much faster than no magnetic field (after Liu et al. [2004a]).

The above results show that the inclusion of the new total mobility that includes the contributions of both the applied electric current and also its interaction with the applied magnetic field, provide much more accurate predictions for growth rates in LPEE under a magnetic field. As mentioned in Section 6.6., when only the electric mobility is used in simulations, not only the growth rates but also the shapes of the involving growth interface were not predicted accurately. As can be seen from Fig. 6.5.9a, the interface shapes are not as flat as those of the grown crystals. However, when the total mobility was introduced, in addition to better predictions for growth rates, the growth interface shapes were also much closer to those of experiments. In Fig. 6.6.4a the evolution of the computed growth interface is presented. As seen, the computed interfaces are flatter than those obtained through earlier simulations (Fig. 6.5.9a), and they agree with the shapes and interfaces of the crystals grown in Sheibani et al. [2003a], which are almost perfectly flat. These results show that the introduction of a bulk constitutive coefficient representing the total electromagnetic mobility due to electromigration under magnetic fields in LPEE (the nonlinear model for LPEE under magnetic field by Dost [2005], and Dost and Sheibani [2006]) is a step in the right direction. It is by no means complete, since the interaction of electric and magnetic fields must also affect mass transport at the growth interface. Therefore, in addition to this model in the bulk (solution), a closer look may also be needed at various surface phenomena under the combined effect of applied electric and magnetic fields, in order to obtain better predictions from modeling.

Single Crystal Growth of Semiconductors from Metallic Solutions

313

(b) 25 mm in diameter, 4.5 mm thickness.

(a) Computed interface shapes.

(c) 25 mm in diameter, about 4 mm thickness. The flat interface between the single crystal (bottom) and the secondary grown section (top) is clearly seen.

Fig. 6.6.4. (a) Evolution of the computed growth interface using the total mobility μT = μE + μB (solid lines) and using only the electric mobility μE (dashed lines). Time increment between lines is 40 hours. The shapes of the computed interfaces are in excellent agreement with those of experiments with and without magnetic field. Two sample GaAs crystals are shown in (b) and (c), Sheibani et al. 2003a. This shows that the model introduced in Liu et al. [2004a], Dost et al. [2005], Dost and Sheibani [2006] also predicts the experimental interface shapes well.

6.7. Morphological Instability in LPEE Morphological instability of the growing interface is an important phenomenon in LPEE growth of semiconductors. The theoretical and numerical models mentioned in this section so far have not considered the morphological instability analysis of the growing interface in LPEE. The complexity and nonlinearity involved in the governing equations makes it difficult to develop comprehensive, analytical stability models. Furthermore, the solutions for such models can only be obtained numerically. To the best of our knowledge, there is no such numerical stability study in the literature conducted for LPEE. However, the literature on the analytical morphological stability analysis of growing interfaces in crystal growth is relatively rich. There are a number of excellent studies reported. It is difficult to present a comprehensive coverage for these studies within the limit of this book. We will only attempt to cover very briefly some essential features of the morphological instability in LPEE. The first of such stability analyses is the linear theory given by Mullins and Sekerka [1964]. This benchmark work was followed by a number of articles. For instance, Wollkind and Segel [1970] conducted a weakly nonlinear analysis, which is valid near the onset of instability. Utilizing the idea of a stagnant

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boundary layer during mixing, Coriell et al. [1976] studied the Mullins-Sekerka linear stability for finite length cells in liquid phase epitaxy. Okamoto et al. [1982] extended the linear stability analysis of Mullins and Sekerka [1964] to include the effect of electrical current, and Coriell and Sekerka [1983] considered various nonequilibrium effects. Wheeler et al. [1988] presented a stability analysis for a planar growth interface for an infinitely long growth cell to examine the effect of electric field on stability. The effects of diffusion and electromigration on the interface stability were also studied. A more comprehensive linear stability analysis, which also includes the effects of Joule heating and other thermoelectric effects, such as the Thomson, Peltier and Seebeck effects, was presented by Coriell et al. [1989], where the formulation of the thermoelectric effects, electromigration, and Joule heating was based on the works of de Groot and Mazur [1962], and Hurle et al. [1967]. The relative contributions of these thermal effects to the linear stability of growth interface were clearly identified. Wollkind and Wang [1988] has also studied the linear and weakly nonlinear stability analyses for LPEE. The stability analysis of Mullins and Sekerka [1964] and Wheleer et al. [1988] was extended for a growth cell of finite dimensions by Dost and Su [1996]. The growth cell of a GaAs system modeled in Dost et al. [1994] was selected for the analysis. Dost and Su [1996] used the same assumptions of Wheleer et al. [1988] for compative purposes. The stability studies in LPEE to date could not consider a number of important effects of LPEE such as crucible cell configuration, natural convection, magnetic field, etc., due to the complexity of the governing equations. They could also not be handled analytically. Such stability analyses for LPEE growth can be best carried out numerically. 6.7.1. Model Equations The governing equations and the associated assumptions for the present stability analysis for LPEE are based on the works of Wheleer et al. [1988] and Dost and Su [1996]. The assumptions are as follows: (1) no fluid flow in the solution, which eliminates the continuity and momentum equations from the list, (2) no diffusion in the solid phase, (3) except electromigration in the liquid phase, all other thermoelectric effects such as Joule heating, Peltier heating/cooling, Thomson effect, etc., are neglected, (4) temperature field does not change with time, and is not coupled with the mass transport, (5) the thermal properties of the solid and liquid phases are the same, (6) the contribution of the latent heat is negligible since the growth velocity is small, (7) no magnetic effects and quasi-steady electric field due to small growth velocity, and (8) planar growth interface.

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315

Under the above assumptions, the remaining balance equations are the mass transport of solute in the liquid phase, and the electric charge balance in both phases, namely

C = DC  2C + (Vk  μ E  )  C t

(6.7.1)

and

 2 L = 0 , and  2 S = 0

(6.7.2)

where k is the unit vector in the z-direction. Note that Eq. (6.7.1) is written for a frame of reference moving with the velocity of interface V. The interface and boundary conditions considered are as follows:

DC (C  n) = C(k  1)(V g  n) + μ E C( L  n)

at z = w(x,t)

(6.7.3)

GL w = m(C  C1 ) + T0

at z = w(x,t)

(6.7.4)

 ES ( S  n) =  EL ( L  n)

at z = w(x,t)

(6.7.5)

at z = l

(6.7.6)

as z   

(6.7.7)

C = C0 ,

 L / z =  E0

 S / z   ( EL /  ES )E0

where k is the segregation coefficient, w(x,t) the position of material points on the deformed interface, n the exterior unit normal to the interface, GL the temperature gradient at the interface, m the slope of the liquidus curve, T0 the melting temperature,  the capillarity constant, C1 the concentration at the growth interface C0 is the concentration at z = l , and  and Vg the curvature of the interface and the growth velocity, defined respectively by

=

 2 w / x 2 [1+ (w / x) ]

2 3/ 2

,

V g = (V +

w )k t

(6.7.8)

Dividing the quantities of length dimensions by DC/V, the time by DC/V2, and by defining

C=

μ E C V ,  = , = E 0, C0 E0 DC V

 ES GL DC V U= , G= , = L DC VT0 E

M=

mC0 T0

, (6.7.9)

the dimensionless forms of the field equations are obtained as follows. In the domain

Sadik Dost and Brian Lent

316

C C = DC  2C + +   C, z t

 2 L = 0,

 2 S = 0

(t  0)

(6.7.10)

at the interface, z = w(x, t)

d w dC = C(k  1)(V + )(k  n) +  C L ,  L =  S t dn dn Gw = M (C  C1 ) + U ,  (d S / dn) = d L / dn

(6.7.11)

at the boundaries

C = 1,  L / z = 1 (at z = l),

 S / z   1 (as z  )

(6.7.12)

where for convenience we have dropped the overbars. The steady-state solutions of the governing equations in Eqs. (6.7.10)-(6.7.12) are (Dost and Su [1996])

C = 0

1 C1e( +1)l 1 e

( +1)l

+

C1  1 1 e

( +1)l

e( +1)z ,  L0 = z,  S0 = z /  , w0 = 0

(6.7.13)

where

C1 = C 0

z =0

= (1+  )[(k  1  )(1 e(1+  )l ) + (1+  )]1

(6.7.14)

In the case of infinitely long solution zone, the length goes to infinity, and then Eqs. (6.7.13) and (6.7.14) reduce to those given by Wheeler et al. [1988]).

z solid (0, 2l)

liquid

E n

k (0, 0)

growth interface

solid

z = w(x, t) x

Fig. 6.7.1. Model domain for the LPEE growth cell of GaAs (redrawn from Dost and Su [1996]).

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317

6.7.2. The Linear Analysis For the stability analysis we assume that the dependent variables C, L, S, and w(x, t) can be expressed in terms of small perturbations superimposed on the steady solutions in the form of A = A0 +  A1 + O( 2 ) where A represents any of these solutions, and  is a small parameter. We also assume that the firstorder perturbations accept solutions in the form

(C 1 ,  1L ,  1S , w1 ) = [C 1 (z),  L1 (z),  S1 (z), w1 (z)]eiax +  t

(6.7.15)

Using the above solutions, the field equations and the interface and boundary conditions yield the following first-order solutions: in the solution

( (

d2 dz 2 d2 dz

2

+ 

d dC 0 d L  a 2   )C +  = 0, dz dz dz

 a 2 ) L = 0,

(

d2 dz

2

 a 2 ) S = 0

(6.7.16)

at the interface (z = 0)

d d d C + (  k)C +  C1 L  [kGc +  )k  1)C1 ]w = 0,  S = L z dz dz dz  1  MC + (G  MGc + Ua 2 )w = 0,  L   S  w=0 

(6.7.17)

at the boundaries

C = 0, d L / z = 0 (at z = l),

d S / dz  0 (as z  )

(6.7.18)

where

 = 1+  and Gc = (dC 0 / dz)

z =0

.

Eqs. (6.7.16) form an eigenvalue problem for  and yield the following solutions and dispersion relation, respectively

C = A1e

 1z

+ A2 e

 2 z

 L = b1e az + b2 eaz , and





+ A3[B1e( a +  )z + B2 e( a   )z ],

 S = c1eaz ,

w = A3

(6.7.19)

Sadik Dost and Brian Lent

318

(a11a23  a21a13 )  [

a31 a32

(a12 a23  a22 a13 ) +

a33 a32

(a11a22  a21a12 )] = 0

(6.7.20)

where

Q1 Q2  aGc b1 1  1,2 =  ± R, B1 = , B2 = , Q1 = 2 a   a   A3

 aGc b2 A (1  ) 1 R = [ 2 + 4(a 2 +  )]1/ 2 , Q2 =  , b1 = 3 , 2 A3 ˆ b2 =

A3 (1  )e2al A (1  )(e2al  1) ˆ , c1 = 3 ,  = (1  )e2l  (1+  ) (6.7.21) ˆ ˆ  

and

a11 =   k   1 , a12 =   k   2 , a21 = a22 =  M , a31 = e

  1l

,

a13 = (a + k)B1 + (a  k)B2 +  aC1 (b2  b1 )  kGc   (k  1)C1 



a23 = G  MGc + Ua 2  M (B1 + B2 ), a33 = B1e( a +  )l + B2 e( a   )l

b1 = (1  ) / ˆ ,b2 = [(1  ) / ˆ ]e2al , a32 = e

 2 l

(6.7.22)

and A1, A2, and A3 are constant perturbation amplitudes, but may depend on l . As a result of being an eigenvalue problem, once one of them, say A3, is given, others will be determined. From Eqs. (6.7.19) and (6.7.21) we see that the perturbed electric field arises solely from the deformation of the interface (Wheleer et al. [1988]). However, it includes the effect of finite growth cell length (Dost and Su [1996]). When l   , the above equations reduce to those given in Wheleer et al. [1988]). The solute transport is affected by the perturbation of electric field through the last term of the first equation of Eq. (6.7.19). The dispersion relation in Eq. (6.7.20) is quite complex. This is because the cell dimension in the z-direction was considered finite, z = l . However, it is not difficult to show that as l   with the assumption of R  0 , the dispersion relation in Eq. (6.7.20) reduces simply to (Wheleer et al. [1988])

a11a23  a21a13 = 0

(6.7.23)

The resulting equation is a cubic polynominal in R. A strategy to solve the cubic equation was given by Wheleer et al. [1988] for a particular wavenumber a and inverse Sekerka number G/MGc (Hurle [1985]) with other parameters being fixed. All the roots were determined for Re(R)>0, the corresponding

Single Crystal Growth of Semiconductors from Metallic Solutions

319

value of . Solutions for various special cases can be found in Wheleer et al. [1988]. However, for the case of finite solution length, the dispersion relation in Eq. (6.7.20) is very complex to solve analytically. A numerical iterative solution method was employed to obtain the roots of this dispersion equation (Dost and Su [1996]). The dispersion relation is written as

 = F1 (a, R)  e2 Rl F1 (a, R)  F2 (a)e Rl R = 0

(6.7.24)

where

2 MQ1 1 F1 (a, R) = (   k  R)(G  MGc + Ua 2 )  2  + 2a + 2R 2 MQ2  +  MaC1 (b2  b1 )  M[kGc +  C1 (k  1)]   2a + 2R 



F2 (a) = 2 M[B1e( a +  / 2)l + B2 e( a   / 2)l ]

(6.7.25)

(6.7.26)

Eq. (6.7.24) implies that if R is one of the roots, -R will also be a root of this equation. For marginal dispersion curves, this equation must be solved for the temperature gradient G for  R = Re( ) = 0 . When  I = Im( ) = 0 , which corresponds to a stationary instability mode, G may be expressed in terms of wavenumber and other physical parameters, and then be calculated. For the case  I = Im( )  0 , corresponding to an overstable instability mode, G cannot be calculated analytically. We obtain G as follows. We first set

1 F(G, a,  ,l) = (  + a + R) (G, a,  ,l) 2

(6.7.27)

where

1 1 F(G, a,  ,) = (  + a + R)[S0 (   k  R) 2 2 2Q1  M( + S  + S2 )]  + 2a + 2R 1

(6.7.28)

Next we write

2R = ( 2 + 4a 2 + 4 )1/ 2 = X + iY ,

 =  R + i I

Then setting  R = 0 , we write

Re[F(G, a,  ,)] = Re[F(G, a,  ,)  F(G, a,  ,l)]

(6.7.29)

Sadik Dost and Brian Lent

320 G

Eq.6.8.36 using Eq.6.8.37 G0

Eq.6.8.36 using Eq.6.8.38



0

Fig.6.7.2. Temperature gradient curves from dispersion relation (redrawn from Dost and Su [1996]).

Im[F(G, a,  ,)] = Im[F(G, a,  ,)  F(G, a,  ,l)] 4( X 2  Y 2 ) =  2 + 4a 2 ,

2 XY =  1

(6.7.30)

where

1 1 Re[F(G, a,  ,)] = S0 [   k  R)(  + a + X ) + Y 2 ] 2 2 1  M[Q1 + S2 (  + a + X )  S1Y  1 ] 2 1 Im[F(G, a,  ,)] = S0 [k  a  2 X )Y  MYS2  MS1 1 (  + a + X ) (6.7.31) 2 The iterative procedure for solving the system in Eq. (6.7.30) for a given a and l is as follows: (1) we compute from the second equation

X n+1 = X (Gn , X n ,Yn ,  1,n )

(6.7.32)

(2) from Eq. (6.7.30)3

Yn+1 =

1 2 X n+1  (  2 + a 2 ) 4

(6.7.33)

(3) from Eq. (6.7.30)4

 I ,n+1 = 2 X n+1Yn+1 (4) and from Eq. (6.7.30)1

(6.7.34)

Single Crystal Growth of Semiconductors from Metallic Solutions

321

Fig. 6.7.3. Marginal stability curves for Alloy 1 ( = 0.9, k = 0.1,  = 1.0) from dispersion relation in Eq. (6.7.36). Results of Dost and Su [1996] are plotted as lines while those of Okamoto et al. [1988] as discrete data points. Note in this case that for large wavenumbers, a decrease in the cell size stabilizes the growth since the stable region above the marginal curve is enlarged (after Dost and Su [1996]).

Gn+1 = G(Gn , X n+1 ,Yn+1 ,  I ,n ,  I ,n+1 )

(6.7.35)

where the subscript n indicates the iterative step. Iteration continues until the solution converges to a required accuracy. The initial temperature gradient G0 is set to the stationary mode value. X0, Y0 and  I ,0 are set to equal to the values of the case for l   . (i) Consider the case of l   , E0 = 0 , and  EL =  ES (Mullins and Sekerka [1964]). In this case Eq. (6.7.24) reduces to

 MS1 = (G  MGc + U 2 )(1 k   )  MkGc

(6.7.36)

where

2 = 1+ 1+ 4(a 2 +  ),

S1 = Gc =

k 1 k

(6.7.37)

as given in Dost and Su [1996], and

2 = 1+ 1+ 4a 2 ,

S1 = Gc =

k 1 k

(6.7.38)

Sadik Dost and Brian Lent

322

Fig. 6.7.4. The effectiveness of the CS criterion as a function of  for the case k = 0.1 and  = 1.0 (after Dost and Su [1996]).

as given in Mullins and Sekerka [1964]. Eq. (6.7.37) is nonlinear due to the presence of  while Eq. (6.7.38) is linear. This significant difference is due to the fact that Eq. (6.7.36) was obtained from a transient problem while Eq. (6.7.38) is based on an á priori steady-state assumption for concentration by neglecting the partial time derivative of concentration in the mass transport equation. However, in order to predict the overstable instability mode (where  I  0 ,  R = 0 which may occur under certain conditions, see Wheleer et al. [1988]), one has to use Eq. (6.7.38). By taking  = R , Eq. (6.7.36) becomes quadratic in R, and is plotted in Fig.6.7.2 using both Eqs. (6.7.37) and (6.7.38). Two dispersion curves become identical only when  = 0 (stationary stability mode, G = G0). As can be seen, when  >0 the priori steady-state assumption underestimates the rate of increase of perturbation, while, when  < 0 it overestimates the rate of decrease of perturbation. (ii). Assuming identical electric conductivities in the solid and liquid phases, and setting  = 0 at the limit (Okamoto et al. [1982]), the dispersion relation in Eq. (6.7.27) yields

= where

(G  MGc + Ua 2 )[(  k   1 )  E(  k   2 )] M (k  1)(1 E)C1



kGc (k  1)C1

(6.7.39)

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323

Fig. 6.7.5. Variation of dimensionless temperature gradient with  for k = 0.1, and  = 1.0 (after Dost and Su [1996]).

E = el

 2 + 4a 2

,

2 1,2 =  ±  2 + 4a 2

(6.7.40)

Eq. (6.7.39) in the limit when l   reduces to (Dost and Su [1996])

=

(G  MGc + Ua 2 )(  k   1 ) M (k  1)C1



kGc (k  1)C1

(6.7.41)

The same dispersion relation under the same conditions was given by Okamoto et al. [1988] as

=

(G  MGc + Ua 2 )[(   1 ) M (k  1)C1

(6.7.42)

The above dispersion relations depend on l through C1. Note the significant difference between the dispersion relations in Eqs. (6.7.39) and (6.7.42) is that Eq. (6.7.42) cannot give a solution for concentration satisfying the far field condition at z = l when a disturbance is introduced. Even for an infinitely long cell ( l   ) the difference between Eqs. (6.7.41) and (6.7.42) is still significant. This is due to the fact that the solid concentration (CS = kCL) is allowed to vary in Dost and Su [1996] while it was kept constant in Okamoto et al. [1988].

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Sadik Dost and Brian Lent

Fig. 6.7.6. Marginal stability curves for the case of  = - 0.9, k = 0.2,  = 1.0 (after Dost and Su [1996]).

6.7.3. Simulation Results Numerical simulation results are presented for different values of the dimensionless growth cell length. The influence of the cell length on the stability region and the existence of an overstable mode are discussed. For all cases, the material and physical parameters given in Wheleer et al. [1988] are adopted: U = 10-8, and M = 2.0  104 for k > 1 or M = - 2.0  104 for k < 1. Note that the dimensionless length does not only depend on the actual cell length but also on the diffusion coefficient and the growth rate. The marginal stability curves for an alloy, Alloy 1 ( = 0.9, k = 0.1,  = 1.0) are plotted in Fig. 6.7.3 versus wavenumber for four different values of dimensionless length (Dost and Su [1996], Okamoto et al. [1988]). In this figure, B = G/(MGc) describes the stability of the system by measuring the effectiveness of the constitutional supercooling (CS) criterion with B = 1 corresponding to the CS criterion. For large values of the dimensionless length, l , we observe an overstable branch (AB) at small wavenumbers. As l decreases, the overstable branch diminishes in the same region. The effect of the length on the stability region depends strongly on wavenumber. For wavenumbers between 1 and 100, the effect of the dimensionless length is insignificant. This effect decreases with decreasing length. However, for smaller or larger wavenumbers, its effect is obvious. Comparison of the results of Dost and Su [1996] and Okamoto et al. [1988] shows that the results agree for large wavenumbers. This is due to the fact that for large wavenumbers the term Ua 2 is

Single Crystal Growth of Semiconductors from Metallic Solutions

325

Fig. 6.7.7. Marginal stability curves for the case of  = 0.9, k = 0.2,  = 1.0 (after Dost and Su [1996]).

dominant in the dispersion relations. However, for small wavenumbers, the results show significant disagreement. Furthermore, for a smaller dimensionless length, the results agree for the whole spectrum of wavenumber. This can easily be verified by considering the limiting case in Eq. (6.7.39) as the length approaches zero. At the limit, both dispersion relations, i.e., Eqs. (6.7.39) and (6.7.42) yield the same result; G = MGc  Ua 2 . As mentioned earlier, the dimensionless length does not only depend on the actual cell height but also the diffusion coefficient and the growth rate. Therefore, depending on the values of these parameters, each of the cases mentioned above may represent an actual physical problem. For instance, using the physical parameters given in Wheleer et al. [1988] (taking the growth velocity about 10-3 cm/s , diffusion coefficient about 2  10-5 cm2/s ), one can see that the dimensionless length is about 50 for the 1-cm high growth cell. On the other hand, for a typical growth cell configuration used for LPEE growth of GaAs crystals (Dost and Sheibani [2000], Sheibani et al. [2003a], for instance, for a sytem with a growth rate about 10-6 cm/s, a diffusion coefficient of 4  10-5 cm2/s, and a cell of 1.0 cm height), the dimensionless length is about 0.025, for which case there is almost no overstability region and both dispersion curves agree. The numerical results for a different alloy ( = - 0.85 , k = 0.1, and,  = 1.0) also show the same pattern of Fig. 6.7.3. In determining the marginal curves of B versus wavenumber , the value of B representing the most unstable mode of the system is needed. This value of B at a certain wavenumber is called the critical value. It is denoted by B* and plotted

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Sadik Dost and Brian Lent

in Fig. 6.7.4 as a function of  for the case of k = 0.1 and  = 1.0. As can be seen from the figure, the critical values for different lengths vary about the unity. This is an indication of the fact that the CS criterion provides a good approximation for this case. A second important parameter, defined as G = (Gk)/[M(k- 1)] for a fixed bulk concentration in the solution and interface velocity, describes the temperature gradient required for stability. Similarly, the critical value of G, denoted by G*, is also plotted in Fig. 6.7.5 versus  for the same alloy composition. The values of G* increase with , indicating a destabilization of the system since a larger temperature gradient is required for stability. The critical value of temperature gradient is also increasing with dimensionless length (which depends on the cell height, growth velocity and diffusion coefficient). This means that larger values of the dimensionless length increase the destabilization of the system. For comparison, the results of Okamoto et al. [1988] and Dost and Su [1996] are presented in Fig. 6.7.5., as seen they agree remarkably. Numerical results from the dispersion relation in Eq. (6.7.24) are presented in Figs. 6.7.6 and 6.7.7 respectively for Alloy 2 ( = -0.9, k = 2.0,  = 1.0) and Alloy 3 ( = 0.9, k = 2.0,  = 1.0), along with the corresponding results from Eq. (6.7.42) that of Okamoto et al. [1988]. Note from Fig.6.7.6 that there exist overstable branches (AB for l =  and CD for l = 5) for larger dimensionless length, however the branch vanishes when l gets smaller. In the case of Alloy 3 (Fig. 6.7.7), there is no overstable branch. The marginal curves for large wave numbers remain unchanged as l varies.

Fig. 6.7.8. Effectiveness of the CS criterion as a function of  for the case of k = 0.2,  = 1.0 (after Dost and Su [1996]).

Single Crystal Growth of Semiconductors from Metallic Solutions

327

Fig. 6.7.9. Variation of dimensionless temperature gradient with  for k = 0.2, and  = 1.0 (after Dost and Su [1996]).

The values of B* and G* for the above composition (k = 2.0,  = 1.0) are also presented respectively in Fig. 6.7.8 and Fig. 6.7.9 as functions of . From Fig.6.7.8, we observe that B* approaches unity as l goes to zero. However, for larger values of l , i.e. l > 1, Fig. 6.7.8 shows that the SC criterion does not provide a good approximation. This result contradicts that of the model presented in Okamoto et al. [1988], in which B* is always equal to unity, implying that the CS criterion will always hold. The reason for this contradiction is as follows. As noted earlier, in Okamoto et al. [1988] the solid concentration CS was fixed rather than fixing the concentration C0 at z = l as done in Dost and Su [1996]. Although the analysis of Dost and Su [1996] is two-dimensional, due to the axisymmetric nature of the actual problem, fixing the concentration C0 at z = l is more appropriate. However, this is not the point in mentioning the difference between the results Dost and Su [1996] and Okamoto et al. [1988]. There are two reasons for this discrepancy: (i) the solution given in Dost and Su [1996] satisfies the boundary condition at z = l (note that this not the dissolution interface, see Fig. 6.7.2) exactly while the solution in Okamoto et al. [1988] does not, and (ii) Dost and Su [1996] fixes the segregation coefficient k as in Mullins and Sekerka [1964] instead of fixing CS as in Okamoto et al. [1988]. The second point suggested in Mullins and Sekerka [1964] is more important physically, and adopted by a number of researchers. In contrast to the results presented in Fig. 6.7.5, in Fig. 6.7.9 we observe that G* decreases as  increases for l < 1.0, indicating a stabilization of the system. For large l , an extremum is apparent, and G* has a maximum at  = 0 for

328

Sadik Dost and Brian Lent

Fig. 6.7.10. Variation of dimensionless temperature gradient with  for k = 0.2, and  = 1.0 (after Dost and Su [1996]).

l   , taking smaller values for both  < 0 and  > 0 . It is also noted that in general the value of G* increases with decreasing l , indicating that the system stabilizes with larger l . In Fig. 6.7.10, as seen the results of Okamoto et al. [1988] do not agree with those of Dost and Su [1996], and overestimate the values of G*. It must be mentioned that the above analysis considered only the case where the electrical conductivities are equal, i.e.  = 1.0 (see Wollkind and Wang [1988]). When Y ~ 1.0, there exist four singularities at R = ±[(1 / 2) + a] and R = ±[(1 / 2)  a] for a finite l, two of which are positive and physically meaningful. As a summary, the results show that an a priori steady-state approximation cannot predict the overstable regions which may occur under certain conditions. This has to be obtained as a limiting case. Furthermore, such an a priori steadystate assumption overestimates the rate of decrease of the perturbation while underestimating the rate of increase. In the analysis, the far field conditions are strictly satisfied by the perturbations. The effect of a finite dimensionless cell length on the CS criterion has also been discussed. The results support the view that the CS criterion is approximate, and cannot be used under certain conditions. It has also been shown that the finite length of the growth cell influences the stability region, the existence of the overstable mode, and the effectiveness of the CS criterion, and it has significant influence on marginal stability curves for small wave numbers.

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6.8. Epitaxial Lateral Overgrowth by LPEE As discussed in Chapter 5, epitaxial lateral overgrowth (ELO) is a technologically promising technique to grow high-quality epilayers with very low defect densities on substrates of high defect densities. In ELO, the epitaxial growth is achieved on a selected substrate which is partially masked by covering by a thin masking film. The substrate is patterned by photolithography and selective etching to form on the substrate the mask-free seeds. Then an epitaxial layer is deposited on this substrate. Growth of the layer is restricted in the windows in the beginning, and then the ELO layer is free to spread along the lateral direction over the masking film. In this section we would like to give to the reader a brief overview for the modelling aspect of epitaxial lateral overgrowth by LPEE. To the best of our knowledge, ELO was initiated in 1980 when McClelland et al. [1980] and Bozler et al. [1981] used metalorganic vapor-phase epitaxy to grow epitaxial GaAs layers laterally over oxide film on a GaAs substrate. Since then, the ELO technique has been successfully used to grow high-quality epilayers of Si, GaN, GaAs, GaP, InP, etc. on various substrates (for a review see Bauser et al. [1987], Jastrzerbski et al. [1988], Nishinaga [1991, 2002], Zytkiewicz [1999, 2002], Beaumont et al. [2001], Hiramatsu [2001], and references therein). Computational analyses, in association with quite a large body of experimental research, have been conducted to better understand the mechanism of the ELO process. For the selective area growth (SAG) or for ELO from the vapor phase, the contributions of surface migration and vapor phase diffusion have been recognized as two main growth mechanisms (Kayser et al. [1991], Greenspan et al. [2003]), although there exist experimental data showing that for large distances (say, larger than 100 μm) diffusion in the vapor dominates growth kinetics (Mitchel et al. [2001]. Some mathematical models (Thrush et al. [1993], Zybura and Jones [1994], Alam et al. [1999]) considering only diffusion in the vapor phase have also been proposed, which successfully predicted and interpreted some physical phenomena occurring during selective epitaxy. Other computational research work has taken into account surface migration effects (Greenspan et al. [2003], Coronell and Jensen [1991], Fujii et al. [1995], Khnener et al. [2002a,b]). As for ELO from the liquid phase, on the other hand, to the best of our knowledge, the number of numerical articles reported in the literature is very small despite quite a large number of reports on the growth of semiconductor ELO structures by liquid-phase epitaxy (LPE). In Yan et al. [2000], a twodimensional numerical calculation was carried out for the ELO growth of InP epilayers by LPE. Within a certain temperature range the calculated values of width-to-thickness ratio of the layers were in good agreement with experimental ones. Liu et al. [2005] carried out a two-dimensional numerical simulation for the growth of GaAs.

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Recently, there is also increasing interest in lateral overgrowth of semiconductor layers by LPEE (Sakai et al. [1991], Mauk and Curran [2001], Dobosz et al. [2003], Liu et al. [2004], Khenner and Braun [2005]). Although the LPEE process is quite complex and involves the interactions of various thermomechanical, electromagnetic, and chemical phenomena (Bryskiewicz [1986], Djilali et al. [1995], Dost [2005a.b]), it offers a unique possibility to control the growth kinetics and the properties of crystals produced. In particular, since electric current flow affects solute distribution in the liquid solution, it can be efficiently used to produce areas with local supersaturation in the liquid zone, which is crucial for selective epitaxy techniques such as SAG and ELO. Sakai et al. [1991] discussed the SAG growth mechanism of GaAs on SiO2-masked GaAs/Si substrates by LPEE. They have introduced a simplified analytical model to show the influence of mask geometry and electrical current density on LPEE lateral overgrowth of GaAs. For a long period of time, to the best our knowledge, we did not see any modeling studies published in the literature describing LPEE growth of ELO layers on substrates coated by electrically conductive masks, while this is the best way to control current flow in the system and, as experiments show (Mauk and Curran [2001], Dobosz et al. [2003]), to increase the lateral growth rate. Recently, Liu et al. [2004] developed a mathematical model for lateral overgrowth of semiconductor layers by LPEE. The robustness and validity of their model are demonstrated by simulating LPEE growth of GaAs ELO layers on GaAs substrates. In particular, it was shown that by controlling the mask conductivity, the ELO layer shape and its growth rate can be efficiently tailored. This is precisely the phenomenon observed experimentally during the LPEE growth of GaAs ELO layers on GaAs substrates coated by SiO2 or tungsten y

liquid

boundaries of simulation domain solution ELO layer

mask

zone window in the mask

substrate

x

Fig. 6.8.1. Schematic view of a LPEE ELO growth cell (not to scale). Due to symmetry of the system, the computational domain is limited to the area marked by dashed lines. Thickness of the liquid, the mask and the substrate are 1.5 mm, 0.1 mm and 400 mm, respectively. The windows are 6 mm wide and are spaced at 500 mm (after Liu et al. [2004b]).

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masking films (Dobosz et al. [2003]). In this section we will present the essential features and the results of the work of Liu et al. [2004]. The study of Khenner and Braun [2005] presents a diffusion model for ELO of GaAs by LPEE. In the opinion of the authors, this work may present an excellent model for dopant or/and impurity segregation control by passing an electric current through a two-phase system consisting of a solid GaAs substrate and a liquid Ga-As solution in contact, and also includes an innovative treatment at the liquid/solid interface. However, the model presented in Khenner and Braun [2005] does not represent the LPEE growth of GaAs at all. It considers an unphysical electromigration in the GaAs solid, and the physical parameters used are very unrealistic and also unphysical. The reader is referred to Khenner and Braun [2005] for details. 6.8.1. The Model A schematic view of the configuration for growing ELO GaAs epilayers is shown in Fig. 6.8.1. LPEE crystal growth is a complex process which involves fluid flow, heat and mass transfer, interface kinetics, electromigration, and thermoelectric effects. However, in order to be able to focus on the most important and intrinsic factors involved in the ELO growth by LPEE (e.g. electric and concentration fields), several simplifications are introduced in the simulation model. Convection in the solution is neglected on the basis of the nature of the growth cell and process (small solution height, and shorter growth period) and also for computational efficiency. In addition, the interaction between concentration and electric fields around ELO crystals grown from adjacent windows is not taken into account. This assumption allows us to consider the growth from one single window only. Furthermore, due to the symmetry of the problem the computations are restricted to the region marked by the dashed rectangle in Fig. 6.8.1 (Liu et al. [2004]). Although the effect of Peltier cooling at the growth interface is not taken into account, a very small temperature difference along the solution height is considered for computational convenience. The solution height is chosen to be 1.5 mm, above which all the variable fields are assumed to be uniform. The computational region is divided into four domains, namely the substrate (S), the masking film (M), the ELO crystal (E), and the liquid solution (L). This way, different values of physical parameters can be used in each domain to study their influence on the phenomena under discussion. In particular, the electrical conductivities of the substrate, the ELO crystal, and the mask can be varied independently. Based on the above assumptions, the field equations of the model reduce to the steady-state electric field equation written for the whole computational domain, and the mass transport equation written only in the liquid solution. These equations were given earlier in this chapter, however, for convenience, we present them here.

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The electric field equation takes the form of

 E  2  = 0 , where   l, e, s, m

(6.8.1)

where  represents respectively the domains of l (liquid), e (ELO), s (substrate), and m (mask). Mass conservation of the solute (As) yields the mass transport equation in the liquid phase (Ga–As liquid solution):

C  C  C  μE  μE = DC  2C t x x z z

(6.8.2)

The boundary conditions associated with the field variables are as follows: At the two vertical symmetry planes

C / x = 0,

 / x = 0

(6.8.3)

At the upper (horizontal) liquid surface

C = C2  C0 (T0 + T ),

 EL ( L / y) = J

(6.8.4)

where C2 is the equilibrium solute concentration C0 at the temperature ( T0 + T ), with T0 being the growth temperature and T a small temperature difference applied across the solution. The phase diagram relation is given by (Kimura et al. [1994])

C0 = 1.0657 exp[8.42  (1.32  104 ) / T ]

(6.8.5)

with T in Kelvin degrees. At the bottom (horizontal) substrate surface

 ES ( S / y) = J

(6.8.6)

Eqs. (6.8.4) and (6.8.6) indicate that the electric current density is kept constant in the growth system. At the inner boundaries of the simulation domain, namely the growth interface, liquid-mask interface, mask-substrate interface, and crystal-substrate interface, we assume that the electric current is continuous:

( E  )+ = ( E  )

(6.8.7)

where  and  represent respectively the two neighboring domains of the interface under consideration. As discussed by Nishinaga [1991], for an optimal ELO growth, the orientation of the windows in the mask is chosen such that the side walls of the appearing ELO layer are covered by atomically rough fast-growing planes,

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while the slowly grown facet is formed on the upper surface of the layer. Therefore, on the growth interfaces, the effect of surface kinetics is taken into account by considering the following condition:

C = C1 +

DC k

(C) n  C0 (T0 ) +

DC C ( ) k n int

(6.8.8)

where k is the interfacial kinetic coefficient. At the faceted upper surface k takes a small but finite value. However, at the rough vertical growth interface the kinetics is very fast ( k   ) so the ratio of DC/k in Eq. (6.8.8) is very small, being almost zero. Therefore, the surface concentration in Eq. (6.8.8) becomes

C = C1 = C0 (T0 )

(6.8.9)

which is equal to the equilibrium concentration obtained from the phase diagram at the growth temperature. Eqs. (6.8.8) and (6.8.9) are, respectively, used at the horizontal and vertical interfaces. At the horizontal growth interface the use of Eq. (6.8.8) allows us to predict the experimental slower growth rate observed there. To the best of our knowledge there is no independent data in the literature for k in a GaAs system. Thus, the value of k is selected equal to 510-3 cm/s (see Table 6.8.1). This value corresponds to the best prediction of the growth rate of GaAs ELO layer in the LPEE experiments of Dobosz et al. [2003]. Note that the numerical value of k used in this work is quite close to that obtained for LPE growth of planar Si layers from tin-silicon solution on Si(100) substrates (Kimura et al. [1994]). Effect of Interface Curvature on Growth Rate The condition at the growth interface is obtained from the mass balance, which determines the growth rate, for instance by Eq. (6.6.7), using only the electric mobility constant. As experiments show (Zytkiewicz [1999], Silier [1996]) the Gibbs–Thompson effect, i.e. the dependence of equilibrium solute concentration on surface curvature, should be considered for the ELO technique because the thickness of the grown epilayers is usually thin and their walls are sharply rounded. This can be done by replacing C1 in Eqs. (6.8.8) and (6.8.9), being a function of growth temperature T0 only, by a generalized equilibrium condition

C1 (T0 ,r) = C0 (T0 )(1+  / r)

(6.8.10)

where r is the radius of local curvature along the growth interface, and  is the capillary coefficient, given by (Randolph and Larson [1971])

 =  / k BT

(6.8.11)

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where  is the surface energy of the crystal,  the molecular volume of solid, and kB the Boltzman constant. Eq. (6.8.10) shows that, due to the Gibbs– Thompson effect, the equilibrium solute concentration is larger than C0 for positive surface curvature (i.e. for convexparts of the layer) and smaller than C0 for negative curvature (i.e. for concave parts). Consequently, a decrease (increase) of growth rate is expected at convex (concave) parts of the epilayer, as compared to the growth rate of its planar parts. As shown in Liu et al. [2004b], this approach leads to computational problems such as a significant increase in computation time and a particularly insurmountable numerical instability. Therefore, to simplify the numerical procedure, the Gibbs–Thompson effect is incorporated into the growth rate equation in Liu et al. [2004b] following the treatment of Khenner et al. [2002 and 2005]. Thus, Eq. (6.6.7) is modified as

Vng =

L C 1  / r [DC ( ) Int + μ E EnC] n CS  C S

(6.8.12)

The growth rate is computed with this equation in which the effect of surface curvature is explicit through the numerator of the last term. Eq. (6.8.12) reduces to Eq. (6.6.7) when the interface surface is flat. It gives higher growth rates for the concave (r < 0) sections of the ELO layer, and lower growth rates for the convex (r > 0) sections. At the interface between the masking film and liquid phase, the mass balance across the interface yields the following boundary condition:

DC (

C ) + μ E En C = 0 n mask

(6.8.13)

This condition can also be obtained from Eq. (6.8.12) by making the growth velocity zero ( Vng = 0 ) at the mask since the ELO growth starts selectively from the seed, and there is no growth occurring at the mask. The initial values of As concentration and electric potential are given by C = C1 and  = 0 . The initial crystal shape is chosen as a 4.00.3 μm rectangle, unless specified otherwise. Some of the physical properties and operating parameters for the growth of GaAs ELO layers by LPEE can be found in Table 6.2.1, and those specific to this system are given in Table 6.8.1. Numerical simulations were performed for GaAs ELO layers grown by LPEE on GaAs substrates covered by insulating or electrically conductive masks.

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Table 6.8.1. Parameters of the GaAs ELO/LPEE system. Parameters

Symbol

Values

Growth temperature

T0

973 (K)

Applied electric current density

J

3.73 (A/cm2)

Capillarity coefficient (computed from Eq. (6.8.11) using  = 500 erg/cm2, and  = 410-23 cm3.



210-7 (cm)

Surface kinetics coefficient

k

510-3 (cm/s)

Electric conductivity of conductive mask(a)

0.1 (1/cm)

Electric conductivity of insulating mask

510-6 (1/cm)

Crystal mass density

S

5.32 (g/cm3)

Solutal electric mobility

μE

0.036 (cm2/Vs)

Temperature difference in the solution

T

0.05 (K)

Substrate thickness

Ls

400 (μm)

Mask thickness

Lm

0.1 (μm)

Solution height

Ll

1.5 (mm)

Window spacing

S

500 (μm)

Window width

W

6.0 (μm)

(a) this value corresponds to mask/GaAs contact resistivity of 10 cm achievable by modern technologies (Okamoto et al. 1982). -4

The field equations in Eqs. (6.8.1) and (6.8.2) are solved numerically for concentration C and electric potential  under the boundary conditions given in Eqs. (6.8.3)-(6.8.9), and (6.8.13), and the initial conditions mentioned above. Using the computed values of C and , the growth rate is then computed from Eq. (6.8.12). The Galerkin finite element method is employed for the computations. The thickness of the grown crystal is updated at each time step according to the growth rate given by Eq. (6.8.12). There are significant differences in the dimensions of the computation domains. Thus, different finite element meshes are generated. In the region of the growing epilayer and in its vicinity, a mesh of 3228 (in the x- and y-directions, respectively) is used. Simulations showed that this is sufficient for a stable solution. One mesh is given along the y-direction in the mask region (thickness of the mask is 0.1 μm). Nine-node Lagrangian biquadratic basis functions are used to approximate the unknown fields. A backward Euler time stepping algorithm is employed in the calculation of transient terms. The Newton–Raphson method is used to iteratively solve the discretized algebraic equations until a selected convergence criterion is reached.

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(a)

(b)

(c)

Fig. 6.8.2. Electric potential and solute concentration field distributions after 15 min of LPEE growth of GaAs ELO layer on a GaAs substrate coated with an insulating mask: (a) electric potential distribution in the substrate and lower part of the solution, (b) electric potential distribution in the liquid solution domain, (c) solute concentration distribution in the liquid zone (after Liu et al. [2004b]).

6.8.2. Simulation Results The simulation results for the LPEE growth of ELO GaAs layers on a GaAs substrate coated by an insulating mask are presented in Fig. 6.8.2. The distribution of electric potential in the substrate is given in Fig. 6.8.2a, and that in the liquid zone in Fig. 6.8.2b. The normal to the iso-potential lines indicates the direction of electric current. As seen from the figure, the simulation model predicts high electric potential gradients in the lower left corner of the liquid zone, so the electric current flows through the growth window only. This behavior is expected, since the mask electrically insulates the liquid from the substrate. Consequently, the local electric current intensity in the vicinity of the window is larger than that at the upper solution boundary by a factor of (S+W)/W  80 (S and W are the spacing and width of growth windows, as described in Table 6.8.1). It is worth noticing that the electric potential drop in the substrate is much larger than that in the liquid zone, because the electric conductivity of the Ga–As solution is much larger than that of solid GaAs. Fig. 6.8.2c shows the computed As concentration distribution in the liquid zone. As seen, while in the upper part of the solution the As distribution is uniform along the x-direction; at the region near the growing ELO crystal the concentration contours are curved. This is due to the imposed boundary conditions at the growth interface (electric current flows only though the crystal). Note that the electric potential distribution near the grown crystal is also curved (Fig. 6.8.2b). The highest As concentration in the solution appears in its upper part due to a slightly higher temperature, and the presence of source material there.

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(a)

(b)

337

(c)

Fig. 6.8.3. Electric potential and solute concentration field distributions after 15 min of LPEE growth of GaAs ELO layer on a GaAs substrate coated with a conductive mask: (a) electric potential distribution in the substrate and lower part of the solution, (b) electric potential distribution in the liquid solution domain, (c) solute concentration distribution in the liquid zone (after Liu et al. [2004b]).

The simulation results for the LPEE growth of ELO GaAs layers on a GaAs substrate coated by an electrically conductive mask are presented in Fig. 6.8.3. The computed electric potential in the substrate and that in the liquid zone are respectively shown in Figs. 6.8.3a and 6.8.3.b. Since the electric current passes through the whole substrate, a nearly-uniform electric potential distribution is obtained both in the liquid and solid phases. However, a slight nonuniformity in the potential distribution is seen in the substrate near the growth window. This is because the electrical conductivities of the GaAs crystal and the mask are different. Fig. 6.8.3c presents the As concentration distribution in the liquid zone. As observed earlier, the solute concentration in the upper part of solution is again uniform in the x-direction. This time, however, the highest As concentration appears near the mask in the left corner of the L domain (i.e. in the middle of the masked substrate area between the adjacent windows). This is a direct consequence of the boundary condition at the mask (Eq. (6.8.13)), which physically means that electromigration moves the solute species towards the surface of the mask. Since the electrical current can pass through the conductive mask while the solute cannot, a high solute concentration is observed in this region. This solute flux is, in turn, counterbalanced by the diffusion, so a kind of quasi-steady-state As distribution appears along the ydirection close to the right vertical wall of L domain. It is important to mention that according to the available data, the highest As concentration due to the mass accumulation near the mask corresponds to a local solution supercooling of 0.42K only, which is not sufficient for GaAs nucleation. Therefore, the accumulated solute diffuses towards the nearest As sink which is the growing ELO layer. As will be shown later, this process leads to a significant enhancement in the lateral growth rate of the grown layer.

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(b)

(a) Fig. 6.8.4. Time evolution of shape of ELO layer grown on a GaAs substrate with an insulating mask (a) and conductive one (b). The time interval between two lines is 2 min (after Liu et al. [2004b]).

Fig. 6.8.4 shows the evolution of the GaAs ELO layer grown by LPEE on a GaAs substrate coated by insulated (a) and electrically conductive (b) masks. As can be seen, ELO growth on the substrate masked with an insulating film (e.g. SiO2) is nearly isotropic, while under the same conditions a wide and thin ELO layer is grown on the substrate with an electrically conductive (e.g. metallic) mask. These simulation results can be explained as follows. If the substrate is masked by an insulating film, the electric current flows through the growth window only (Fig. 6.8.2a). As the lateral growth proceeds more electric current passes to the seed through the upper surface of the layer than through its laterally overgrown parts. Thus, a faster growth rate, mainly due to a larger solution supersaturation induced by electromigration, is found at the upper face of the ELO layer than at its sidewalls. If, on the other hand, the layer thickness becomes larger than the width, then more electric current flows to the seed through the laterally overgrown parts, enhancing their growth. This process is self-regulating, favors growth isotropy and explains the shape of the layer shown in Fig.6.8.4a. On the other hand, after some initial vertical growth on the metal-masked substrate, more electric current flows to the substrate directly through the mask rather than through the ELO layer. Thus, a higher melt supersaturation is observed at the mask surface between the adjacent ELO stripes than at the upper surface of the ELO layer (Fig. 6.8.3c), which enhances the lateral growth rate. This, together with the effect of interfacial kinetics along the upper growth interface, leads to thin and wide ELO layers as shown in Fig. 6.8.4b.

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(b)

(a)

Fig. 6.8.5. Experimental (a) and calculated shape (b) of GaAs ELO layer grown on a GaAs substrate coated by an insulating (Si02) mask. The scale in both figures is selected the same for convenience. The time interval between lines in (b) is 8 min (after Liu et al. [2004b]).

(a)

(b)

Fig. 6.8.6. Experimental (a) and calculated shape (b) of the GaAs ELO layer grown on a GaAs substrate coated by an electrically conductive (tungsten) mask. The scale in both figures is the same for easier comparison. The time interval between lines in (b) is 8 min.

Fig. 6.8.5a shows a photograph of the cross section of a GaAs ELO layer grown by LPEE on a SiO2-masked GaAs substrate. The layer was grown at 700 °C with a current density of 3.73 A/cm2. Details of the growth procedure can be found in the experimental work of Dobosz et al. [2003]. To reproduce the experimental condition, the evolution of the ELO layer is simulated for a growth time of 4h. Fig. 6.8.5b shows the computed shape of the GaAs ELO layer grown on the GaAs substrate coated by an insulating mask. For convenience, both surface profiles are plotted in the same scale. As can be seen, the agreement between the results of the simulation and the experiment is

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excellent; the simulation model not only predicts accurately the shape of the grown layers but also the actual (experimental) dimensions of the ELO layer (cross section). (b)

(c)

(a)

Fig. 6.8.7. Influence of the initial crystal size (rectangular) on the computed shape of the ELO layer at t = 14 min: (a) insulating mask; (b) conductive mask. Bold, dashed and dash-dotted lines represent respectively the crystals with initial sizes of 40.3, 30.3, and 30.6 μmμm. (c) Influence of initial crystal shape on calculated shape of ELO layer. An insulating mask is used for the first 4 min of growth, which is then numerically replaced by a conductive mask. The time interval between lines is 2 min.

Similarly, Fig. 6.8.6 presents a comparison for the experimental and simulation results of the GaAs ELO layer grown by LPEE on a GaAs substrate with an electrically conductive mask. In this case a tungsten film was used to mask the GaAs substrate in the experiment of Dobosz et al. [2003]. Again, the same scale was used in the graphs to allow a direct comparison. As can be seen, the model predicts the shape of ELO layer very accurately. It must be mentioned that in this case the model slightly overestimates the lateral growth rate. Such a difference can however be attributed to the many simplifying assumptions made in the model and also to the values of various parameters used in simulations that are not known accurately. For instance, the hightemperature resistivity of GaAs is known to vary by a factor of 10 depending on temperature and doping (Okamoto et al. [1982]). Even more importantly, there are no techniques to measure the metal/GaAs contact resistivity at high

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temperatures, so a reasonable estimate had to be made for this parameter. In fact, the prediction from the model can be fitted to the experimental data to estimate the value of the mask conductivity for the present growth conditions if all the other parameters are known. However, the conductivity values of the substrate, the ELO layer and the mask are of no importance if the ELO layer is grown on a substrate with an insulating mask. In that case, simulations lead to results that are in closer agreement with experiments. The question whether the shapes of ELO crystals predicted by simulations would depend on the dimensions and/or the shape of the initial crystal is also examined. Fig. 6.8.7 presents the computed shapes of the ELO layer obtained by choosing rectangular initial crystals of 40.3, 30.3, and 40.6 μmμm for insulating (Fig. 6.8.7a) and conductive (Fig. 6.8.7b) masks. As seen, the dimensions of the initial crystal do not have a significant affect on the shapes of the ELO layer. The shapes almost remain the same although some very small changes are observed in their sizes. The GaAs ELO growth is first simulated for a GaAs substrate with an insulating mask for 4 min, at which time the insulating mask is replaced by a conductive mask, and the growth is run for an additional 20 min. As can be seen from Fig. 6.8.7c the layer initially takes a rounded shape which is typical for an isotropic ELO layer overgrowth with an insulating mask. However, after the mask was changed to a conductive one, the layer quickly starts to grow laterally much faster than in the vertical direction. The simulation results show that the model presented by Liu et al. [2004] is insensitive to the choice of initial crystal size. Furthermore, the simulation results verify that the distributions of both the electric current in the system, and the solute concentration in the liquid zone play a significant role and are the major factors for determining the growth behavior of ELO layers by LPEE.

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Chapter 7

TRAVELING HEATER METHOD

In this chapter we present the recent developments in modeling of the traveling heater method (THM), particularly during the last two decades. The basic theoretical considerations regarding the modeling issues are presented first. Then, two and three dimensional simulation models and numerical simulation results are presented for binary and ternary systems. The challenges in modeling of the THM growth of ternary systems are emphasized. The use of static and rotating magnetic fields has also found a great interest in THM. First, we cover the models under static and strong magnetic fields, and then those under weak and rotating magnetic fields. 7.1. Introduction As discussed in Chapter 3, the Traveling Heater Method falls into the category of solution growth, and is a relatively new, promising technique for commercial production of high quality, bulk compound and alloy semiconductors. Due to its importance, a number of experimental and theoretical studies have been carried out for the THM growth process (see for instance Wald and Bell [1975], Cherepanova [1982], Bischopink and Benz [1989, 1991, 1993], Chang et al. [1989], Sugiyama et al. [1989], and Danilewski et al. [1992]). In one of the earliest studies, Wald and Bell [1975] have investigated the steady-state temperature profiles within the growth ampoule of CdTe for various heater positions. They also performed a series of experiments under an introduced forced-convection using the accelerated rotation technique. Cherepanova [1982] has modeled the THM growth of PbTe and examined the effect of gravity on heat and mass transfer in the liquid zone. The field

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equations were presented in two dimensions, but a solution was given for a onedimensional case. Boeck and Rudolph [1986] have also considered the THM growth of PbTe. Through a simple mathematical model, they examined the influence of the thermal diffusion effect (Soret effect) in this system. Sell and Muller [1989] carried out a numerical modeling study and a one-dimensional numerical simulation for the growth of GaxIn1-xAs. Chang et al. [1989] have presented a finite element, quasi steady-state thermal model to simulate the growth of HgCdTe, where transients in the temperature field caused by the displacement of the ampoule in the furnace were not taken into account, and the solvent/crystal interfaces were assumed to be set at the equilibrium liquidus temperature. Apanovich and Ljumkis [1991] have performed a finite difference simulation of heat and mass transfer during the growth process of a binary system. They have presented results for a crystal (and a solution) with thermophysical properties similar to CdxHg1-xTe, with the assumption that the pseudobinary CdTe-HgTe phase diagram is valid. Ye et al. [1996] have studied the influence of thermosolutal convection on CdTe growth by adopting a quasisteady-state model of Chang et al. [1989]. In their finite element simulation model, a fixed length of liquid zone was assumed. In a finite difference simulation model, Matsumoto et al. [1997] have analyzed the transient response in the THM growth of InP for a stationary heater. Barz et al. [1997] have conducted a numerical study of convection using the control volume method during the THM growth of CdTe with the accelerated crucible rotation technique. The above mentioned theoretical and numerical studies have naturally assumed various simplifying assumptions, such as one-dimensional modelling (Sell and Muller [1989]), a quasi-steady-state assumption (Chang et al. [1989] and Ye et al. [1996]), and a stationary heater profile (Matsumoto et al. [1997]). In addition, the interface mass transport equations, the thermal and mass convection effects, and/or the proper phase diagram equations have not been incorporated into some of the above referred studies. In order to have a better understanding of the complex transport phenomena involving diffusion and convection of heat and mass transfer during the growth of ternary alloys, Meric et al. [1999] have presented a mathematical model for the growth of GaxIn1-xSb by THM. For numerical simulations of the THM growth process, an adaptive finite element technique was employed. Comparisons with experiments are also provided to assess the validity of the model and computations. MartinezThomas et al. [2002] studied the THM growth of HgTe numerically using a three-step computational scheme in order to minimize the computational demand. The field equations were solved using a quasi steady-state approximation and the growth conditions leading to solvent inclusions in the grown crystal were examined. Okano et al. [2002] have carried out a numerical simulation study for the THM growth of GaSb from a Ga-solution where the effects of crucible temperature, crucible rotation, and crucible material on the crystal/solution interface shape were investigated.

Single Crystal Growth of Semiconductors from Metallic Solutions

345

Due to the high temperature gradients used in THM, strong convective flows develop in the liquid zone. In order to minimize the adverse effect of convection, which may adversely affect the quality of the grown crystals, and also for better mixing in the liquid solution, applied magnetic fields have also been used in THM (see for instance Salk et al. [1993, 1994], Lan and Yang [1995], Fiederle et al. [1996], Ghaddar et al. [1999], Senchenkov et al. [1999], Dost et al. [2003]. Liu et al. [2003], Abidi et al. [2005], and Kumar et al. [2006]). A small rotating magnetic field (at 400 Hz) of 20G in magnitude was used in THM under microgravity conditions to enhance mixing (Salk et al. [1993, 1994], Fiederle et al. [1996]). The effect of a rotating magnetic field on the radial compositional uniformity in CdHgTe crystals grown by THM was investigated by Senchenkov et al. [1999]. Lan and Yang [1995] used a pseudosteady-state model to simulate fluid flow, heat and mass transfer, and interface shapes in the Travelling Solvent Method (TSM) growth of a CdTe/Te system under various gravity intensity levels. The effects of some growth parameters such as growth rate, initial solvent volume, and heater temperature were studied for a crucible of 1.5 cm diameter. In the simulation of larger crystals, computational difficulties were encountered due to strong convection in the solution. The two-dimensional numerical simulation of the THM growth of CdTe by Ghaddar et al. [1999] focused on the influence of rotating magnetic fields on flow patterns and compositional uniformity in the solution. It was found that under microgravity conditions, the application of rotating magnetic fields can suppress the residual buoyancy convection in the solution, and may result in complex flow structures and enhanced compositional nonuniformity at high gravity levels. Dost et al. [2003] have carried out two-dimensional numerical simulations for the growth of CdTe by THM to examine the effect of applied stationary and rotating magnetic fields, as well as that of small nonuniformities in the strong stationary magnetic field. Three dimensional simulations for the THM growth process have been carried out by Liu et al. [2003], Abidi et al. [2005], and Kumar et al. [2006] under static but strong applied magnetic fields. It was shown that the applied stationary magnetic field aligned perfectly with the axis of the growth cell gives rise to two magnetic body force components in the horizontal plane: one in the radial direction inward, and the other one in the circumferential direction. These two horizontal force components together with the vertical gravitational body force affect the structure of the fluid flow in the solution, and consequently suppress it. The flow field can be suppressed further using higher magnetic field levels. The computational domain for an axisymmetric THM ampoule including the source (feed), the substrate, the liquid solution and the quartz ampoule wall (half domain), and the applied temperature profile was shown in Fig. 3.4.1 in Chapter 3. The applied temperature profile inside the heater liner tube was also shown in the figure. As described in Chapter 3, after reaching a thermal and

346

Sadik Dost and Brian Lent

chemical quasi steady-state equilibrium within the ampoule, the ampoule is lowered (downward with respect to the heater) with a very small velocity, usually in the order of 1-3 mm per day. During growth, a temperature difference between the upper (dissolution) and the lower (growth) liquid-solid interfaces is created due to the asymmetrical thermal profile with the higher temperature at the dissolution interface. The source material is thereby dissolved at the upper interface, where the solubility of the liquid solution increases due to the heater movement. The material is transported through the liquid zone by both thermosolutal convection and diffusion. Recrystallization then occurs at the lower interface, which is at a lower temperature than the dissolution interface. In essence, the heater thermal gradient (temperature profile) and its movement are two important factors controlling the growth process. The temperature gradient in the vicinity of the growth interface must be properly controlled so as to avoid constitutional supercooling and thermal stresses. This can be achieved by an optimum thermal design for the THM growth crucible. For instance, thermal signatures taken in the THM system of Liu et al. [2003] show that the shape of the growth interface can be controlled to provide a favorable growth interface shape (see Fig. 3.4.3). 7.2. One-Dimensional Models In this section we present some fundamental aspects of the THM growth process considering only diffusion and steady-state transport. 7.2.1. Effect of Thermal Diffusion The transport of species in the liquid zone of a THM crucible was modeled by Boeck and Rudolph [1986] by considering a diffusion controlled mass transport. The thermal diffusion, known as the Soret effect, was also included in the model. The growth of a binary system of PbTe from a Te-rich solution was modeled under a simplified triangular thermal profile (Fig. 7.2.1). In the model the mass flux is expressed as (see Chapter 4 Section 3)

i =  DC C  DCCT C(1 C)T

(7.2.1)

where C = C A represents the solute (Pb) concentration with the solvent concentration of C B = 1 C . The ratio of DCCT / DC = ST is called the Soret coefficient. The use of Eqs.(7.2.1) in the mass balance in the absence of convection leads to

C  2C 2T = DC + D 0 t z  2 z  2

(7.2.2)

Single Crystal Growth of Semiconductors from Metallic Solutions T

V

T0

g

- GL

+ GL

T2

347

T

T1

a

z z0

z1 seed

z2

solution

l0

feed

l

L Fig. 7.2.1. Schematic representation of the THM growth system (redrawn from Boeck and Rudolph [1986]).

where we defined D0  C(1 C)DCCT . Eq. (7.2.2) can be expressed with respect to a coordinate system at the interface moving with the velocity of V g ;

d 2C B dz 2

V g dC B + =0 DC dz

(7.2.3)

where z is the distance from the interface. In addition to the unknown interface concentrations and temperatures, the positions of the phase boundaries are to be determined. To this end, the mass balances at the interfaces are written using Eq. (7.2.1) as

dC B dz

= z = z1

dC B D Vg C B1 + 0 G L , and DC DC dz

= z = z2

D Vg C B2  0 G L DC DC

(7.2.4)

where GL is the applied temperature gradient, and CB1 and CB2 are the interface concentrations. At the thermal centre of the liquid zone, the continuity requirement yields

dC B dz

 z 0

D0 DC

GL =

dC B dz

+ z +0

D0 DC

GL

(7.2.5)

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348

and the mass balance gives z2

 C B (z)dz = l0

(7.2.6)

z1

where we have assumed C BS = 0 and C Bi = 1 (pure Te in the initial zone). The temperature distribution within the zone, and the phase diagram are represented by (with a linear liquidus curve)

T1 = T0 + G L (z1  z0 ) = T  mC B1 ,

T2 = T0 + G L (z2  z0 ) = T  mC B2 (7.2.7)

X (%) 10

200

T (K)

55

8 6 a (mm) 4

100 50

2 VcrOD

z1

z (mm)

z2

Fig. 7.2.2. The concentration distribution in the liquid zone at Vg = 0 (dashed-dotted lines) and at Vg = 4 mm/day (continuous lines). Heavy lines are with the Soret effect, and the thinner lines are without it (redrawn from Boeck and Rudolph [1986]).

10

20

Vg (mm/day)

VcrTD 30

Fig..7.2.3. The difference in temperature T (continuous lines) and the asymmetry a (dashed-dotted lines) as functions of the growth velocity. Heavy lines are with the Soret effect, and the thin lines are without it (redrawn from Boeck and Rudolph [1986]).

The solution of Eqs. (7.2.3)-(7.2.7) determines the zone length as

l=

Ts  T0 GL +

T T 1 1 +{( s 0 )2 (D0 m / DC ) + 1 G L (D0 m / DC ) + 1 V g ml0 1 m D0 m [2l0 + ( )2 ]}1/ 2 (D0 m / DC ) + 1 G L 2DC G L (D0 m + DC )

and the asymmetry (shown in Figs. 7.2.1 and 7.2.3) is given by

(7.2.8)

Single Crystal Growth of Semiconductors from Metallic Solutions

V g ml0 1 a= 2 G L (D0 m + DC )

349

(7.2.9)

The unknown concentration and temperature distributions are calculated using the following physical parameters (Boeck and Rudolph [1986]): Ts = 1198 K, T0 = 900 K, GL =200 K/cm, m =770 K, DC =510-5 cm2/s, ST =510-3 1/K, and l0 = 0.5 cm. The concentration distributions at two growth interfaces (with and without the influence of the Soret effect) are given in Fig. 7.2.2. Similarly, the temperature distributions are presented in Fig. 7.2.3. The growth velocity corresponding to the maximum temperature difference Tmax ( T2 = T0 ) is called the “critical growth rate”, and at this growth rate value, the asymmetry becomes half of the zone length l . As can be seen from Fig. 7.2.3, both the temperature difference and the asymmetry value increase with the increasing growth velocity, and their values are lower when the Soret effect (thermal diffusion) is included. This implies that the Soret effect may contribute to the stability of the liquid zone positively. 7.2.2. A Steady-State Model for the THM Growth of GaxIn1-xAs Sell and Muller [1989] presented a steady-state model describing the THM growth of ternary alloy crystals of GaxIn1-xAs. The model is one-dimensional and does not take convection into account. The liquid solution is modeled with two moving interfaces which are determined through the consideration of the phase diagram. The model is aimed at the determination of growth conditions to grow ternary crystals with uniform crystal compositions (at x = 0.2 in a system of AxB1-xC). Initially, at static equilibrium, the temperatures of the growth and dissolution interfaces are the same. When the heater moves, however, the temperature at the growth interface, Tg, decreases while the temperature of the dissolution interfaces, Td, increases. This temperature difference is the driving force for growth. In this model, it is assumed that the system reaches a dynamic equilibrium (i.e., a steady-state). The thermal boundary conditions of the liquid zone at the interfaces are given by

dTd dt

= Gd [VH  Vd (t)] , and

dTg dt

= Gg [VH  Vg (t)]

(7.2.10)

where VH is the heater velocity, Vg and Vd are the velocities of the growth and dissolution interfaces, respectively, and Tg and Td are the growth and dissolution interface temperatures. Gd and Gg denote the temperature gradients at the interfaces.

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The mass balance in the liquid zone leads to two mass transport equations in terms of two independent mole fractions of species Ga and As which are represented by x A and x B as

x A t

= DA

2 x A z 2

+ vw

x A z

,

and

xC t

= DC

2 xC z 2

+ vw

xC z

(7.2.11)

which are written with respect to a coordinate system at the growth interface moving with a velocity of v w , where z is the axial coordinate, and D A  DGa and DC  D As are the diffusion coefficients of Ga and As in the In-rich solution (In is the solvent, the component B). In THM, the growth rate is also an unknown in addition to two mole fractions, and the differential equations in Eqs. (7.2.11) are coupled with the equations describing the phase diagram, as discussed in Chapters 5 and 6 in detail. In the growth of ternary crystals, the variation of the crystal composition ( x s ) in the grown crystal must also be taken into account, which normally requires inclusion of the mass transport equation in the solid phase, coupled with the equations of the liquid phase through interface and phase diagram equations, as we have seen in Chapters 5 and 6). However, in this onedimensional model, x s is used by considering only stoichiometric III-V alloy s semiconductors with x III : xVs = 1 . The phase diagram is included in the calculations by

x s = 2{[(D A

x A z

) / (DC z =0

xC z

z =0

1 )](  xCg ) + x Ag } 2

(7.2.12)

where xCg and x Ag denote mole fractions at the growth interface. The velocity of the dissolution interface is estimated in correlation with the phase diagram as

x 1 Vd = [(D A A 2 z

z= L

x 1 ) / ( x s  x dA ) + (DC C 2 z

z= L

1 ) / ( x s  xCd )] 2

(7.2.13)

with mole fractions xCd and x dA at the dissolution interface. The differential equations in Eqs. (7.2.11) are solved by the finite difference method with the use of Eqs. (7.2.10), (7.2.12) and (7.2.13), and the physical parameters given in Table 7.2.1. Solutions for the interface temperatures and velocities and the solid composition in the grown crystal are obtained. The axial crystal composition x s (z) in ternary crystals is very important since it must be uniform for efficient device applications. The computed values of the solid composition x s (z) of GaAs in the grown GaxIn1-xAs crystal are presented in Fig. 7.2.4. Results show that during the initial period of growth the composition decreases (between 0  z  0.4 cm ) and then increases with time approaching the feed composition of 0.8 ( z  0.4 cm).

Single Crystal Growth of Semiconductors from Metallic Solutions

351

During the initial period of growth, the composition is determined by the different diffusivities and concentrations of Ga and As in the vicinity of the growth interface. Table 7.2.1. Parameters of the GaxIn1-xAs system (Sell and Muller [1989])*. Parameter

Value

Diffusion coefficients

DGa = 5  10 , and D As = 1  10

Feed material

x = 0.8

Heater velocity

VH = 2 mm/day

Average temperature in the liquid solution

Tm = 1125 K

Temperature gradients at the interfaces

G g = Gd = 20 K/cm

Initial zone length

L = 1 cm

5

4

(cm2/s)

s

*Details of the phase diagram calculations can be found in Sell and Muller [1989].

After a 0.4-cm growth, the thermal conditions of the interfaces approach their final steady-state values at which the temperature difference between the dissolution and growth interfaces reaches a value corresponding to a growth rate which is identical to the heater velocity. At this point, the composition approaches the initial composition of 0.8. This initial deviation from the initial composition is determined by the phase diagram and the ratio of diffusion coefficients of the A and B components of the ternary alloy, Therefore this initial deviation can also be in the other direction (i.e., it can first increase and then approaches the initial composition). This one-dimensional model also shows that the initial deviation from the 0.800 L = 1 cm 0.798

xs 0.796

0.794 0.0

0.4

0.8

1.2 1.6 Axial position (cm)

2.0

2.4

Fig. 7.2.4. Variation of the computed crystal composition of GaAs in the grown crystal (redrawn from Sell and Muller [1989]).

Sadik Dost and Brian Lent

352

initial composition increases with the increasing solution height, and naturally the position of the maximum variation moves along the growth direction with the increasing solution height (see Sell and Muller [1989]). This maximum deviation value is almost linearly proportional to the solution height, i.e., 0.0055 at L = 1 cm, 0.011 at L = 2 cm, and 0.030 at L = 5 cm. 7.3. Two-Dimensional Numerical Simulation Models In this section, based on the well-known thermomechanical balance equations of the continuum given in Chapter 4, we first present the explicit field equations for the THM growth of III-V ternary alloys under general boundary conditions and assumptions. The field equations of a binary system can easily be deduced from these equations. In the development of the model, the solution (liquid phase) is considered as a ternary fluid mixture, and the solid phase as a rigid, heat conducting material allowing solid diffusion. The simulation domain in this section is considered two-dimensional. All field variables are then assumed to be functions of the spatial coordinates r and z and time only (in a cylindrical coordinate system). For computational convenience, the two-dimensional governing equations are written in terms of mole fractions which are selected as compositional variables since they describe naturally the phase diagram. For a ternary alloy crystal of AxB1-xC, x A , x B , and xC are the mole fractions of elements of A, B and C in the liquid phase, respectively, and xCs is the mole fraction of element C in the solid phase. Since

x A + x B + xC = 1 ,

(7.3.1)

either x A and xC or x B and xC can be selected as the two independent compositional variables in the liquid phase, and in the solid, since the atoms of C are always 50% if the solid-state defects are neglected, we have only one variable (either x sA or x Bx ) since

x sA = x / 2 ,

x Bs = (1 x) / 2 ,

xCs = 1 / 2

(7.3.2)

where x is the solid composition. 7.3.1. Field Equations We now present the two-dimensional field equations in general form, from which the field equations of the numerical simulation models will be deduced. Mass diffusion takes place between elements A and B in a ternary crystal. Therefore, if one neglects the defects of the solid, then the element C sublattice must be filled, and thus

Single Crystal Growth of Semiconductors from Metallic Solutions

xCs r

=

xCs z

=0

353

(7.3.3)

This leads to a uniform distribution of the mole fraction xCs in the solid phase. Diffusion in the solid phase is governed by only one mass transport equation which can be written in terms of either x sA or x Bs . If we select x sA , we write (in cylindrical coordinates)

x sA t

= D As (

2 x sA r 2

+

2 s s D As  s x sA  s x sA 1 x A  x A + ) + ( + ) r r z z z 2  s r r

(7.3.4)

where D As is used to denote the diffusion coefficient of element A in the solid,  s denotes the mole density of the solid, r is the horizontal radial direction and z is the axis of symmetry in the vertical direction. The last term in Eq. (7.3.4) represents the change in mole density. For instance for most III-V alloys, the solid mole density  s can be related to the solid composition, x, in the following form

s =

8 {xd AC + (1 x)d BC }3 N Av

(7.3.5)

where NAv is the Avogadro number, and dAC and dBC are the lattice parameters of components AC and BC, respectively. In Eq. (7.3.5), we have assumed that the lattice parameter of an alloy, dAC-BC changes linearly from dAC to dBC. The energy balance yields the following equation

2T 1 T T 2T =  rs ( 2 + ) +  zs 2 r r t r z

(7.3.6)

where the thermal diffusivity is defined by

s 

ks

 s s

(7.3.7)

where ksr and ksz are the thermal conductivities in the r- and z-directions, respectively, and  s is the specific heat of the solid. The liquid phase is assumed to be an incompressible, Newtonian fluid. In this axisymmetric case, the equation of continuity becomes

u u w + + =0 r r z

(7.3.8)

where u and w are the flow velocity components in the r- and z-directions, respectively.

Sadik Dost and Brian Lent

354

In the momentum balance, the well-known Boussinesq approximationassuming a constant fluid density in all equations except in the body force term due to buoyancyis adopted. This allows us to take into account the buoyancy forces due to density changes induced by temperature and concentration gradients in the solution. The momentum equations then become

u u 1 p 2u 2 u 2u 2 w u u +u + w =  +  (2 2 + + 2+ 2 2) r z r r z t rz  L r r r

(7.3.9)

w w 1 p 2 w 1 w 1 u 2 w 2u w +u +w = +  (2 2 + + + + ) r z r r r z r 2 rz t  L z z  g{ A (x A  x 0A ) + C (xC  xC0 ) + T (T  T0 )}

(7.3.10)

where the equation of continuity is embedded, and we use the same notations for the physical parameters as defined in earlier chapters. The conservation of mass for species A and C yield two mass transport equations as

x A t xC t

+u +u

x A r xC r

+w +w

x A z xC z

= DA ( = DC (

2 x A r 2 2 xC r 2

+

2 1 x A  x A + ) r r z 2

(7.3.11)

+

2 1 xC  xC + ) r r z 2

(7.3.12)

The associated interface and boundary conditions will be given later explicitly when we discuss specific simulation models. The energy equation takes the following form

T T 2T 1 T 2T T +u +w = L( 2 + + ) r z r r z 2 t r

(7.3.13)

where  L is the thermal diffusivity of the liquid phase, and we have neglected the effects of concentration and density changes on the temperature field. 7.3.2. A Thermal Analysis for the THM Growth Process A two-dimensional thermal analysis of a THM crucible of HgCdTe has been carried out by Chang et al. [1989]. The energy equation of the liquid phase was solved based on a quasi-steady-state model using the finite element technique. The model assumes a quasi-steady state where the transients in the temperature field due to the movement of the ampoule and also the contributions of the flow and concentration fields are neglected, and the energy equation (from Eqs.

Single Crystal Growth of Semiconductors from Metallic Solutions

355

(7.3.6) and (7.3.12)) is written in a general form (for the liquid and solid domains) in an axisymmetric configuration as

V g(

T T 2T 2T 2T + ) =  (i) ( 2 + 2 + ) r z rz z 2 r

(7.3.14)

where  (i) represent the thermal diffusivity of each domain (crystals, wall, and solution). In the solution of the above equation, it is assumed that the solvent is at a uniform concentration and the composition of the solid phase is in equilibrium with the liquidus temperature calculated from the phase diagram. It is also assumed that at the interfaces the solid and liquid temperatures are equal to the liquidus temperature, and the energy balances are then written as

n  Ts  n  TL = V g H s

(7.3.15)

where H S represents the latent heat of solidification. Along the crucible wall we take convective and radiative heat transfer into account so that

kT = hC (T f  T ) + hR (T f4  T 4 )

(7.3.16)

where Tf is the surrounding measured temperature profile and hC and hR are the convective and radiative heat transfer coefficients. The calculated shape of the growth interface (the thermal evolution of the growth interface) through this quasi-steady state thermal analysis is given as a function of the grown crystal length in Fig. 7.3.1. Results imply that at the initial periods of growth the interface shape changes rapidly, and then as time 0.2 14 mm

0.1 18 mm

0.0 23 mm

h (cm)

- 0.1 - 0.2

33 mm

- 0.3

53 mm 73 mm

- 0.4 - 0.5

0.0

0.5 1.0 1.5 Radial coordinate (cm)

2.0

Fig. 7.3.1. Evolution of the growth interface from a thermal analysis; only half of the crystal is shown (redrawn from Change et al. [1989]).

Sadik Dost and Brian Lent

356

progresses this change gets smaller and the interface takes on a concave shape. It must be noted that the evolution of the interface shape is determined from a thermal analysis only, and the effects of transients and the flow and concentration fields were not taken into account. Based on the thermal analysis, the interface shapes seen in Fig. 7.3.1 are explained by Chang et al. [1989] as follows. HgCdTe is not a good thermal conductor. At the beginning, heat conduction through a CdTe seed and quartz stem is rather effective. As growth progresses, the heat conduction becomes ineffective as the grown crystal length increases. Consequently, in order to distribute the heat input into the solution, the solution zone expands and the interface shape becomes concave. The computed interface shape from the thermal analysis is compared with experiments in which the interface shapes were obtained by quenching the ingot at the end of the selected growth period. Chang et al. [1989] report that the model predictions agree with experiments. 7.3.3. A Steady-State Convection Model The effect of natural convection on temperature and solute distributions, growth rate and interface shape was studied by Ye et al. [1996] through a steady-state simulation model for the THM growth of CdTe single crystals. Since the model considers a binary system of CdTe, in the solid phase (the CdTe feed and seed crystals, and the quartz wall) the only field equation is the steadystate energy equation which will be in the same form as Eq. (7.3.6). The liquid phase is modeled as a binary mixture of Te-rich solution of liquid CdTe (solute) and liquid Te (solvent). A solution thermodynamic analysis, using the associated liquid model, shows that the main solute is the CdTe liquid compound (about 18%) in the Te solution at around 800°C with a small amount of liquid Cd (less than 1%) (Ye et al. [1995]). Thus, the small amount of Cd present in the solution is neglected. The continuity equation will be the same as given in Eq. (7.3.8). However, the momentum equations are written for a steady-state case, and also for one species. Thus, Eqs. (7.3.9) and (7.3.10) reduce to

u

u u 1 p 2u 1 u 2u u +w = + ( 2 + + 2 2 2) r z r r z  L r r r

u

w w 1 p 2 w 1 w 2 w +w = + ( 2 + + ) r z r r r 2  L z z

 g{C (C  C0 ) + T (T  T0 )}

(7.3.17)

(7.3.18)

where C represents the concentration of the solute (CdTe) in the Te-solution (mass fraction was used here for convenience), and C is the solutal expansion of CdTe in the solution.

Single Crystal Growth of Semiconductors from Metallic Solutions

357

The mass transport equations in Eqs. (7.3.11) reduce to the single equation

u

C C 2C 1 C 2C +w = DC ( 2 + + ) r z r r z 2 r

(7.3.19)

For the thermal boundary conditions, it is assumed that the furnace temperature varies linearly along the axial direction with respect to the thermal centre of the liquid zone ( z0 ), i.e.,

T f = Tmax ± G L (z  z0 )

(7.3.20)

and along the crucible wall both the convective and radiative heat transfer are included as given in Eq. (7.3.16). Table 7.3.1. Physical parameters used in simulations (Ye at al. [1996]). Parameter

Value

Liquid conductivity

6.310-2 W/cm °C

Liquid mass density

5.64 g/cm3

Liquid thermal diffusivity

0.0475 cm2/s

Liquid kinematic viscosity

0.15610-2 cm2/s

Thermal expansion coefficient

1.44810-4 1/K

Solutal expansion coefficient

0.056

Diffusion coefficient of CdTe in Te

510-5 cm2/s

Crystal thermal conductivity

0.013 W/cm °C

Crystal thermal diffusivity

0.0284 cm2/s

Thermal diffusivity of the feed

0.02 cm2/s

Thermal diffusivity of quartz

0.0108 cm2/s

The model assumes a fixed liquid zone, so that the interface temperatures are therefore determined from a heat transfer analysis. The liquid concentrations at the growing and dissolving interfaces are determined from the phase diagram by

C = (1 xTe ) / xTe

(7.3.21)

where the mole fraction of Te given by (Bell [1974])

xTe = 5.125  1012 T 4  1.55  108 T 3 + 1.603  105 T 2  0.0074T + 2.273 At the axis of symmetry we write

(7.3.22)

Sadik Dost and Brian Lent

358

C / r = 0,

u = 0,

w / r = 0

(7.3.23)

and at the crucible wall u = w = 0 . The governing equations are solved by the finite element method using the penalty function formulation. In the model domain, the solution zone was taken as 7.0 mm (radius) by 12.0 mm (length), and is located in the growth ampoule between the 40-mm-long crystal at the bottom and the 48-mm source rod at the top. The heater temperature profile, outside the ampoule, has a peak temperature of 800°C and is assumed to decrease linearly in the axial direction at the constant gradient of 67 °C/mm. The heater temperature profile is shifted towards the source (feed) material by the asymmetric distance of 2.0 mm, relative to the solution zone centre. The shift is related to the temperature difference between the two interfaces which drives the THM growth process. The convective heat transfer coefficients at the ampoule outside wall are estimated through comparison of the calculated and measured temperature distributions at the wall (Schwenkenbecher and Rudolph [1985]). The physical parameters used in the simulation are given in Table 7.3.1. The simulated temperature field in the solution zone is shown in Fig. 7.3.2a. As seen, the temperature gradient is almost uniform in the radial direction near the growth interface, indicating favorable growth conditions since it leads to uniform solute distribution across the growth interface. The temperature gradient is relatively large in the region near the dissolving interface. The calculated temperature difference between the two interfaces is about 14°C

Fig. 7.3.2. Simulation results (half domain): (a) temperature contours; (b) flow field; (c) concentration contours at the heater temperature gradient of 67 °C/cm and the thermal asymmetry of 0.2 cm (after Ye et al. [1996]).

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Fig. 7.3.3. The flow fields with (a) the combined convections, (b) thermal convection only and (c) solutal convection only (after Ye et al. [1996]).

which is close to the experimentally measured value of 15°C of Schwenkenbecher and Rudolph [1985]. As seen in Fig. 7.3.2b, in the simulated fluid flow field one flow cell forms in the half liquid zone. The flow moves upward along the ampoule wall where the temperature is higher, and moves downward at the centerline of the ampoule where temperature is lower. The center of the cell is located slightly closer to the dissolving interface, as a result of the shift of the heater temperature profile. The maximum velocity of the fluid flow inside the zone is about 0.4 cm/s. As seen, both temperature and concentration gradients are important driving forces for the fluid flow, as indicated by the Grashof numbers as high as 106. The simulated concentration field is plotted in Fig. 7.3.2c. A uniform concentration region in the interior, and steep concentration gradients observed near the growing interface reflect the effect of convection. Since the diffusivity of CdTe in the Te solution is low, the fluid flow makes an important contribution to the mass transport, as described by a relatively large Schmidt number of 30. The solute field near the interface tends to move the heavier liquid over the lighter liquid since the liquid density increases with decreasing solute concentration, as shown by the positive sign of the solute expansion coefficient (see Table 7.3.1). Due to the weak flow near the growth interface, diffusive mass transfer is dominant in this high concentration gradient region, which constitutes the important feature of THM for growing high quality crystals, despite the use of relatively large temperature gradients. The computed concentration difference between the two interfaces is about 0.013, corresponding to the temperature difference of 14°C.

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ampoule wall

Growth rate (cm/s)

9.0x10-6

G = 67 °C, a=0.2 cm

7.0x10-6 5.0x10-6 3.0x10-6 1.0x10-6

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Radial distance from centre (cm)

Fig. 7.3.4. The growth rate distribution along the growing interface (after Ye et al. [1996]).

The relative contributions of thermal and solutal convection were also examined numerically in this system. As can be seen in Fig. 7.3.3., while the flow due to thermal convection is strong everywhere in the solution zone, the flow induced by solutal convection is weak in the solution zone except the regions near the dissolution and growth interfaces. In these regions, the computed solutal convection is very strong, and leads to a nonuniform growth interface as shown in Fig. 7.3.4. This is probably due to the assumed simple temperature profile in this model. As we will see later, the interface shape is related to the thermal characteristics of the furnace, including its cooling system and temperature profile. If they are designed properly, the interface shape can be controlled, and be made close to ideal (slightly concave to the liquid) (see Fig. 3.4.3). 7.3.4. A Model for the THM Growth of Ternary GaxIn1-xSb An axisymmetric (2-D) numerical simulation model for the THM growth of ternary GaxIn1-xSb single crystals was developed by Meric et al. [1999]. In this thermodynamically rational mathematical model, the momentum equations including the solutal and thermal buoyancy terms, the mass transport equations in terms of the mole fractions of components, the interface mass conservation equations and the phase equilibrium equations are presented for an axisymmetric THM growth cell configuration. The cell configuration and the furnace thermal profile were adopted from the experimental setup of Amistar Research Inc., Victoria, BC, Canada (See Fig. 3.4.2). The field equations are solved numerically by an adaptive finite element procedure as the interfaces between the solid and liquid phases change in time. A fully-implicit time integration technique is adopted to solve the transient

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361

equations, and the resulting non-linear algebraic equations are solved by the Newton-Raphson method. Numerical results show that the furnace thermal profile, and the thermal and solutal convection in the liquid solution have significant effects on the growth process. Results are only presented for small growth times. Modeling Domain The computational domain for the axisymmetric THM ampoule, including the source (feed), the substrate, the Ga-rich liquid solution and the quartz ampoule wall are shown in Fig. 7.3.5. The growth temperature profile inside of the heater liner tube is also shown in the figure. After reaching a thermal and chemical steady-state equilibrium, the ampoule moves relative to the heater with a small velocity (in the order of 1-3 mm per day), and the growth process begins. During growth, the asymmetrical growth temperature profile gives rise to a temperature difference between the upper (dissolution) and the lower (growth) liquid-solid interfaces. The source material at the top dissolves into the solution, and then the solutes move in the liquid zone towards the seed substrate, by both thermo-solutal convection and diffusion. Crystallization then occurs at the lower interface which is at a lower temperature than the dissolution interface. In THM, the temperature profile and its translation rate are two important factors controlling the growth process. The temperature gradient near the growth

Fig. 7.3.5. Computational region and the furnace thermal profile (after Meric et al. [1999]).

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362

interface must be controlled to avoid constitutional supercooling in the solution and thermal stresses in the grown crystal. The governing equations of the liquid and solid phases were given in Eqs. (7.3.3)-(7.3.13). The specific boundary and interface conditions of the model are as follows. Interface Conditions Conservation of mass for the solutes at the dissolution and growth interfaces yields two conditions (see Chapter 4),

 DA  DC

x A

x sA

+  sV g (x sA  x A ) n 1 =  sV g (  xC ) 2 n

n xC

=  s D As

(7.3.24) (7.3.25)

where  and  s are respectively the mole densities of the liquid and solid phases which are approximately constant. The left-hand sides of Eqs. (7.3.25) represent the contributions of diffusion from the liquid phase. The first term on the right-hand side of Eq. (7.3.25)1 represents solid diffusion. But such a term does not appear in Eq. (7.3.25)2 because the composition of element C is assumed to be constant in the crystal. The last terms in both equations represent solidification with a growth rate V g and relative concentrations at the interface. The interface concentration in the liquid side is coupled, through the phase diagram, with the solid concentration, which changes with time. The interface conditions consist of phase equilibrium between substrate and solution, and mass balance between the transported and incorporated solute species. The balance of energy at the interfaces is given by

(qs  q L )  n = 0

(7.3.26)

where q denotes the heat flux and n is the unit normal to the interface. The contribution of latent heat is neglected due to the very small growth velocity Vg. Phase Diagram As also mentioned in Chapters 5 and 6, in the growth of ternary alloys the concentrations of the solid and liquid phases at the interfaces must satisfy the phase diagram. The solid composition varies within the crystal and is therefore determined by both the mass transport ratio of element A to element C and the phase diagram. The growth parameters cannot be calculated by the mass transport rate alone. They must be determined by the ratio of solute elements transported through the solution to the interface, and the diffusion in the solid. The mass transport equations at the interfaces and the phase diagram must be

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363

solved simultaneously for the concentrations x A , x B , xC and the growth rate V g . The expressions describing the phase diagram of a ternary III-V alloy system Ax B1 x C were given in Section 5.4 (see Eqs. (5.4.3)-(5.4.15)). Boundary Conditions For the solution of the mass transport equation, Eq. (7.3.4), in the source and substrate, the homogeneous Neumann boundary condition with the zero normal derivative of x sA is taken along the boundary of the subregions, except at the dissolution and growth interfaces, where the computed values of x sA from the mass conservation equations, Eqs. (7.3.25), and the phase equilibrium equations, Eqs.(5.4.3) and (5.4.4), are prescribed. The temperature field is solved for the entire ampoule region, including the quartz wall. The growth ampoule is heated by the traveling heater with a furnace temperature profile T f (z,t) depicted in Fig. 7.3.5. The boundary condition at the lateral periphery of the ampoule is taken in two different forms as indicated below: Case a:

T = T f (z,t)

(7.3.27)

Case b:

ks

T = q(z,t) + h[T  T f (z,t)] n

(7.3.28)

where q is an effective boundary heat flux to compensate for radiation effects, and h is the heat transfer coefficient. The boundary heat flux q is only considered for the lateral boundary segment opposite to the solution region and is taken in the following form:

q(z,t) = 

T f (z,t) T f max

q0

(7.3.29)

where T f max is the maximum value of the furnace temperature profile; and q0 is a given value. At the top of the ampoule, the temperature is set at the furnace temperature level T f (zt ,t) , while boundary convection with the ambient temperature of T f (zb ,t) is considered at the bottom, where zt and zb refer to the top and bottom z positions of the ampoule. At r = 0 , the symmetry condition with the zero normal derivative of temperature is adopted. For the momentum equations, Eqs. (7.3.9) and (7.3.10), no-slip boundary conditions with zero velocity components are adopted on the boundary of the liquid solution zone, except at the symmetry axis, r = 0 , where only u = 0 is

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prescribed. The effect of the slowly moving interfaces on the velocity components is neglected. For the solution of the mass transport equations, Eqs. (7.3.11) and (7.3.12), the mole fractions x A and xC at the interfaces are determined from the mass conservation equations, Eqs. (7.3.25), and the phase equilibrium equations, Eqs. (5.4.3) and (5.4.4). At the remaining boundary segments of the liquid solution zone, the normal derivatives of the mole fractions are taken as zero. Initial Guess Interface Conditions The initial conditions for the mole fractions x A and xC in the solution may be determined from the initial guess interface conditions. Thermodynamically valid initial conditions are crucial for a successful simulation of the growth process. As described in Section 5.5, the thermodynamic model of Kimura et al. [1996] describes a virtual process which is the transfer of a relative molar quantity μ of the substrate into the liquid to form a supersaturated mixture of compositions x mA and xCm . This mixture is then relaxed to equilibrium, forming the solid and a saturated liquid of compositions x 0A and xC0 . When the supersaturated solution relaxes, the relative molar quantity forming the solid is μ . When μ > μ , then the solid would tend to grow as the system approaches equilibrium; if μ < μ , it would tend to dissolve. Based on this virtual process, the interface concentrations may be determined from the following equations: μ x x mA = x 0A (1 μ ) + μ x s0 = x A (1 μ ) +   A 2 μ μ (7.3.30) xCm = xC0 (1 μ ) + = xC (1 μ ) +  2 2 The compositions of the two phases must also satisfy the phase diagram. By taking a substrate composition x and liquid compositions x A and xC which are in equilibrium with the solid composition x determined by the phase diagram, the problem is reduced to solving Eqs. (7.3.30) together with the phase equilibrium equations, Eqs. (5.4.3) and (5.4.4), for x 0A , xC0 , x s0 and μ . As the A equations are nonlinear, the Newton-Raphson (N-R) method may be used for the solution. The relative molar quantity μ may be related to the ratio of the diffusion coefficients of the solid to the liquid, and is a very small number. Initial Conditions To find the initial temperature condition within the entire ampoule, the steady-state conduction equation given by

Single Crystal Growth of Semiconductors from Metallic Solutions

 2T r 2

+

1 T  2T + =0 r r z 2

365

(7.3.31)

is solved for the two different cases of peripheral boundary conditions given by Eqs. (7.3.27) and (7.3.28). The computed temperature distributions are then adopted as the initial temperature conditions for the two cases. The initial velocity components in the liquid are taken as zero. The initial conditions for the molar compositions x A and xC in the liquid and x sA in the solid are obtained as described in the previous subsection. Dimensional Analysis For computational convenience, the governing equations of the model developed for the THM growth of the GaxIn1-xSb system are made dimensionless. For this purpose, the following non-dimensional quantities are adopted:

x A  x 0A xC  xC0 uL vL T T0 U= , V= , = , A = , C =   T x A xC P=

pL2

 L

Pr =

2

, =

t 2

L

, =

r z   ,  = , Sc A = , ScC = , L L DA DC

g  A L3x A g C L3xC g T L3T  , GrA = , Gr = , Gr = C T  2 2 2

(7.3.32)

where L represents the characteristic dimension of the liquid solution,  is the thermal diffusivity of the solution, T denotes the characteristic temperature difference in the solution, and x A and xC are used to denote the characteristic concentration differences in the solution. GrA and GrC are the Grashof numbers related to the solutal concentration changes while GrT is the Grashof number related to the temperature change. Finally Sc A and ScC represent the Schmidt numbers related to solutes A and C in the solution, respectively. Using the above dimensionless variables, the governing equations in the liquid phase become: Continuity

U V U + + =0   

(7.3.33)

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Momentum

U U P  2U 2 U  2U  2V 2U U +U +V = +2 2 + + +         2   2 

(7.3.34)

V V P  2V 1 V U  2V  2U V +U +V = +2 2 + ( + )+ 2 +            GrA A  GrC C  GrT 

(7.3.35)

Energy

  1  2 1   2  +U +V = ( + + )   Pr  2    2 

(7.3.36)

Mass Transport 2 2 1   A 1  A   A ( + + ) Sc A  2    2

(7.3.37)

2 2 1  C 1 C  C +U +V = ( + + ) ScC  2       2

(7.3.38)

 A  C

+U

 A  C

+V

 A  C

=

In THM, the magnitude of temperature differences in the solution is in the order of 30K which causes solutal concentration differences in the order of 10-4, depending on the furnace temperature profile and the desired solid composition. Using the thermophysical properties listed in Table 7.3.2, the dimensionless parameters may be calculated rendering information on the relative significance of buoyancy and viscous effects in the momentum equations and the convection and diffusion in the mass and heat transfer equations. For a solid composition of x = 0.94, the following values for the dimensionless parameters may be found:

GrA  GrC  3  104 ,

GrT  9  105 , Sc A  ScC  80, Pr  3  102 (7.3.39)

The high values of the Grashof numbers, GrA and GrC , related to concentration and temperature gradients indicate the importance of both thermally driven natural convection and solutal convection in the solution zone. Furthermore, the comparatively high value of GrT with respect to GrA and GrC shows the relative importance of natural convection in the THM growth of ternary alloys. The values of the Schmidt numbers are rather high, indicating that convective transport has a significant influence on the solute concentration profiles, and thus on the growth rate and compositional uniformity. The small Prandtl number, on the other hand, shows that heat transfer takes place mainly via diffusion rather than convective flow.

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Table 7.3.2. Dimensions and Parameters of the THM Growth of GaInSb (Meric et al. [1999]). Parameter

Symbol

Values

Liquid density

L

6.06 g/cm3

Liquid kinematic viscosity



1.3210-3 cm2/s

Liquid thermal diffusivity

s

3.7810-2 cm2/s

Liquid thermal diffusivity

L

5.1210-2 cm2/s

Thermal diffusivity of Si02

q

1.3210-2 cm2/s

Diffusion coefficient of Ga in the liquid

DA

1.7210-5 cm2/s

Diffusion coefficient of Sb in the liquid

DC

1.6010-5 cm2/s

Diffusion coefficient of Ga in the solid

DA

s

1.010-11 cm2/s

Solutal expansion coefficient of Ga

A

-0.12

Solutal expansion coefficient of Sb

C

-0.19

Thermal expansion coefficient of the liquid

T

1.810-5 1/K

Lattice constant of GaSb

dAC

6.905

Lattice constant of InSb

dBC

6.497

Relative molar density

μ

1.010-5

Solid composition

x

0.94

Ratio of mole densities of solid to liquid Solution height

1.0 L

1.5 cm

Initial substrate thickness

1.0 cm

Initial source thickness

2.0 cm

Inner diameter of crucible

2.6 cm

Outer diameter of crucible

3.0 cm

Numerical Solution Method The model presented here for the THM growth of ternary alloys is a moving boundary problem. The moving boundaries of the two liquid and solid phases must be taken into account. The Galerkin finite element method (FEM) (see Zienkiewicz and Taylor [1989]) is adopted for the solution of the governing partial differential equations. Four-node, quadrilateral elements are used for the interpolation of all variables within elements. The penalty function method is employed to eliminate the pressure as an unknown, and consequently to reduce computational cost. The resulting first-order simultaneous ordinary equations

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Sadik Dost and Brian Lent

Fig. 7.2.7. Computational flowchart (after Meric et al. [1999]).

are further discretized by the fully-implicit time-marching algorithm based on the finite difference method. This allows larger time steps, because of its higher accuracy and numerical stability. The Newton-Raphson (N-R) method is used to solve the non-linear algebraic equations. The resulting linear matrix equations in each N-R iteration are solved by applying the linear package LAPACK (Anderson [1992]). The relative convergence of variables in iterations is taken as 10-3. Different finite element meshes are used for the substrate, the source, and the liquid solution regions. For thermal analysis, the mesh includes the complete cell configuration, overlapping all the submeshes. For computational convenience, identical mesh structures are used in the overlapping regions for thermal and mass transport equations.

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The dissolution and growth interfaces move very slowly as the growth progresses due to small growth rates. Thus, the motion of the substrate, source, and solution regions are very small from one time step to the next in the numerical calculations. The finite element meshes are not regenerated completely, but the nodes are moved following the shapes of the interfaces in each time step so that the meshes adapt to the deformed substrate, source, and solution regions. The values of all variables computed in the previous step are used directly in the present step without any interpolation to reduce computational time. A cubic spline method is used to retrieve the nodal boundary or ambient temperatures from the moving furnace temperature profile given in Fig. 7.3.6. The overall computational procedure may be summarized as follows (Fig. 7.3.7): 1. Compute the initial guess concentrations x A and xC by solving the phase equilibrium equations using the N-R method. 2. Compute the initial guess interface conditions for x 0A , xC0 , x s0 , and μ by A solving the phase equilibrium and interface mass conservation equations using the N-R method. 3. Evaluate the initial temperature distribution by solving the steady-state conduction equation in the entire cell configuration using FEM. 4. Start the time integration and move the traveling heater with a constant velocity. 5. Evaluate the temperature distribution in the ampoule, including thermal convection effects in the liquid solution region, using FEM. 6. Start the iteration for the dissolution and growth interface conditions. 7. Evaluate the mole fraction distribution within the substrate using FEM. 8. Evaluate the mole fraction distribution within the source using FEM. 9. Evaluate the velocity components u and v, and the mole fractions x A and xC within the liquid solution using FEM. 10. Compute the interface conditions for the mole fractions x A , xC , and x sA , and the growth velocity V g by solving the phase equilibrium and mass transport equations using the N-R method. 11. Determine the dissolution and growth interface shapes. 12. Update the growth cell configuration and generate new meshes. 13. Check for the convergence of the interface iteration condition. If the solution has converged, return to Step (6). 14. Increase the time step and return to Step (4) if t < tmax. Otherwise, stop. The computational procedure is outlined in the flow chart given in Fig. 7.3.7. Simulation Results The numerical simulations for the THM growth of the GaxIn1-xSb ternary system have been carried out using the computational procedure described in the previous section. The material properties and the geometric dimensions of the THM system are tabulated in Table 7.3.2.

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Fig. 7.3.8. Initial finite element mesh, (a) the entire cell and (b) the zoomed liquid zone (after Meric et al. [1999]).

Quadrilateral finite element grids typically have about as many elements as nodes, while triangular elements typically have twice as many elements as nodes. Since the solution accuracy is primarily a function of the number of nodes, quadrilateral elements are chosen over triangles for the finite element discretization. The finite element mesh for the growth ampoule with 4-noded quadrilateral elements is shown in Fig. 7.3.8. The entire mesh configuration is used only for the solution of the temperature field, while the submesh configurations are used for the substrate, source and liquid solution regions. The number of quadrilateral elements for the entire mesh is 3953, while for each of the substrate, source and liquid zones the number of elements is 1475. The initial furnace temperature profile shown in Fig. 7.3.6 has temperature gradients of approximately 25 and -20 °C/cm in the z-direction at the position of the growth and dissolution interfaces, i.e., at z = 0 and 1.5 cm. The temperatures at these levels are respectively 556 and 611 °C, with a difference of 55 °C. A higher temperature level is thus rendered at the dissolution interface. A steep temperature gradient at the growth interface also results in high growing rates. The furnace temperature profile is moved with a velocity of 1 mm/day upwards along the z-direction. The initial temperature distributions corresponding to the peripheral boundary conditions of Cases (a) and (b) are depicted in Fig. 7.3.9 (a) and (b), respectively. As stated earlier, these distributions are obtained by solving the steady-state heat conduction equation. The heat transfer coefficient has been taken as h =10 W/cm2K, while the value of qo is 50 W/cm2 for Case (b). From Fig. 7.3.9, it may be seen that the isotherms differ only slightly for the two cases.

Single Crystal Growth of Semiconductors from Metallic Solutions

Fig. 7.3..9 Initial temperature distribution, (a) Case (a) and (b) case (b) (after Meric et al. [1999]).

371

Fig. 7.3.10. Temperature distribution in the cell at (a) t = 3 s, and (b) t = 6 s (after Meric et al. [1999]).

The incremental time step has been taken as t = 0.12 s in the solution procedure. With this rather high value of t, an underrelaxation of the NewtonRaphson solution was required with a parameter of 0.5. The convergence of the Newton-Raphson solutions of the various subproblems within the procedure has been found highly dependent on the relevant initial guesses. The numerical solutions of the transient problem are found to be almost the same for the two cases of the thermal boundary conditions given by Eqs. (7.3.27) and (7.3.28). Only the results pertaining to the heat

Fig. 7.3.11. Temperature distribution in the liquid zone at (a) t = 3 s, and (b) t = 6 s (after Meric et al. [1999]).

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Fig. 7.3.12. Velocity field in the liquid zone at (a) t = 3 s, and (b) t = 6 s (after Meric et al. [1999]).

convection/boundary flux boundary condition, i.e., Case (b), will henceforth be presented. The temperature contours within the entire ampoule are shown in Fig. 7.3.10 for t = 3 s and t = 6 s. Comparing the isotherms for the initial and transient cases given in Fig. 7.3.9 and Fig. 7.3.10, the effects of thermal convection on the heat transfer in the cell can be seen. Apart from the difference in the general behaviour, the contours next to the bottom of the ampoule have much lower temperatures in the transient case. In addition, the isotherms at the position of the growth interface (at approximately z = 0) become rather horizontal. This is very promising in that the growth interface will eventually represent a flat isotherm as time progresses. The temperature contours in the solution region are depicted for t = 3 s and t = 6 s in Fig. 7.3.11. The isotherms are affected by the varying fluid velocity field through the thermal convection terms in the energy equation as time progresses.

Fig. 7.3.13. Steamtraces in the liquid zone at (a) t = 3 s, and (b) t = 6 s (after Meric et al. [1999]).

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373 (c)

Fig. 7.3.14. Composition variation in the liquid zone at t = 6 s of (a) Ga and (b) Sb, and (c) the solid composition variation in the source and substrate at t = 6 s (after Meric et al. [1999]).

The fluid velocity vectors and the corresponding streamtraces in the liquid solution region are plotted at different times in Fig. 7.3.12 and Fig. 7.3.13, respectively. As can be seen from Fig. 7.3.12, there is a downward movement of the fluid next to the centerline position at r = 0 due to convection. This is inconsistent with the counterclockwise circulation of the fluid going up next to the outer boundary of the solution region at r = 1.3 cm as depicted in Fig. 7.3.13. The fluid velocities next to the growth and dissolution interfaces, at approximately z = 0 and 1.5 cm, respectively, are very small due to negligible convection.

(b) (a) Fig. 7.3.15. Time evolution of (a) the growth interface and (b) the dissolution interface (after Meric et al. [1999]).

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Fig. 7.3.16. Temperature distribution along the growth interface at different times (after Meric et al. [1999]).

The compositional distributions of Ga and Sb, i.e., x A and xC , in the solution zone are shown for t = 6 s in Figs. 7.3.14a and 7.3.14b. At this small value of time, the mole fraction contours are mostly positioned next to the interfaces at the top and bottom, and in the center of the growth cell. In Fig. 7.3.14c, the corresponding solid composition x contours are plotted for the source and substrate regions. The time evolution of the grown crystal for small times is depicted in Fig. 7.3.15a. As can be seen from the figure, the crystal is steadily growing in time. It is noted that the thickness of the grown crystal is shown in a rather small scale. The interface shapes are essentially horizontal if viewed on a larger scale comparable to the size of most bulk crystals, i.e., 2 or 3 cm. The time evolution of the dissolution interface is also plotted for different times in Fig. 7.3.15b.

(a)

(b)

Fig. 7.3.17. (a) Temperature distribution along the dissolution interface at different times, and (b) variation of the temperature difference between the dissolution and growth interfaces in the radial direction (after Meric et al. [1999]).

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In Fig. 7.3.16, the temperature of the growth interface as a function of the radial coordinate is plotted for different times. The interface temperature decreases as time progresses. It is also seen that the interface temperature is not constant, hence the interface does not represent an isotherm at these small times. However, it is expected that the interface becomes an isotherm as a thermal and solutal equilibrium is reached as time progresses. The temperature of the dissolution interface is plotted as a function of r for different times in Fig. 7.3.17a. In this case, the interface temperature is mostly increasing with time, except at the outer position of the solution region at r = 1.3 cm. In Fig. 7.3.17b, the temperature difference between the dissolution and growth interfaces is shown as a function of the radial coordinate. The temperature difference is increasing with time, while becoming more uniform with the radial position for larger times. In this section we have presented the results of the two-dimensional numerical simulations carried out for the THM growth of ternary single crystals of GaxIn1-xSb. To the best of our knowledge, this is the only numerical simulation study that has appeared in the literature, which has been carried out for the THM growth of a ternary system. An in-house 2-D finite element code was used. Such a simulation, even in 2-D, represents formidable challenges due to the required iterations at the growth and dissolution interfaces, since the field equations along with the phase diagram must be solved simultaneously in both the liquid and solid phases. Due to this difficulty, numerical results were presented for only small times by Meric et al. [1999]. However, in spite of this, their results showed the feasibility of using the present numerical simulation model for the solution growth of bulk ternary crystals. Thus far, to the best of our knowledge there are no 3-D commercial packages capable of solving simultaneous mass transport in the solid and liquid phases. 7.3.5. A Quasi Steady-State Model for the THM Growth of HgTe A numerical simulation study was carried out by Martinez-Tomas et al. [2002] for the THM growth of HgTe using a commercial package (FLUENT). The whole growth system including the furnace, the ampoule and the charge, was considered through a quasi-steady-state approximation. In the model, mass conservation for the solute in the liquid zone determines the evolution of the crystallisation isotherms (growth rate) as a function of the heater position. The objective was to understand the relationship between the inclusions and the translation rate of the heater. The simulation is carried out in three steps in order to reduce the computational demand. At the first step, heat transport is assumed to be by conduction, convection and radiation between the furnace and the ampoule, and by conduction through the ampoule wall, coating, solid and liquid zones. The temperature field calculated at this level in the air/ampoule boundary is used as a boundary condition for the second and third levels. At these two levels the ampoule and its content are studied in detail. Convection in the liquid

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Sadik Dost and Brian Lent

Fig. 7.3.18. The computational domain and the applied temperature profile (after MartinezTomas [2002]).

zone is considered at the second level and thermosolutal convection is included at the third level. Fig. 7.3.18a illustrates a schematic view of the modeled THM system. The domain consists of the furnace with a cylindrical enclosure, and an ampoule containing the liquid–solid HgTe. The growth ampoule is made of silica, with a top plug of the same material, and is coated with graphite. Parameters of the system are given in Table 7.3.3. The travel of the ampoule is simulated by moving the applied temperature profile upward in time at a fixed rate (see Fig. 7.3.18b). Initially, the amount of the solvent material (Te) is situated at the bottom of the ampoule, but as growth progresses, the liquid solution zone travels upward and becomes larger in volume by the amount of source material dissolved into it. The two-dimensional governing equations, namely the equations of continuity, momentum, energy and mass transport in the liquid phase, and the energy equation in the solid phase are solved using the commercial code FLUENT. The model takes into account conduction, convection and radiation heat transfer, as well as mass transport and phase change. The Boussinesq approximation is used. At the first level computations, a grid of 200x75 cells was used, and at the second level the grid was 167x57. In the third level, the total number of cells is the same as the second level, but in the liquid zone the number of cells is typically 40x30. At each level, the grid was refined near the wall to increase the accuracy.

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Fig. 7.3.19. A view of the adapted computational grid for the liquid zone when the heater is positioned at 12.5 cm (after Martinez-Thomas et al. [2002]).

For simplicity, radiation is considered through a discrete transfer radiation model, assuming that the radiation leaving a surface element in a defined range of solid angles can be approximated by a single ray. The radiant intensity of each ray is estimated along a path to predict radiation heat transfer between surfaces without using the explicit view-factor calculations. The accuracy of the model is limited mainly by the number of rays traced and the chosen computational grid. It was assumed that the transient effects on heat and mass transport and fluid flow are negligible due to the small growth rates, and a steady-state approximation can offer snapshots of the process (for details of the numerical procedure used, see Martinez-Tomas [2002]). The liquid zone is defined as the space between isotherms at which the crystallisation and solvent processes take place (Tc and Ts, respectively). The length of the zone is defined as the axial distance between isotherms Tc and Ts. The initial length of the solvent zone without any solute incorporated is l0 = 6.8 mm, calculated from the initial mass of Te and the internal section of the ampoule. HgTe was considered as the solute and Te as the solvent, thus the concentration C refers to the mass fraction of HgTe in Te. In order to overcome the required computational demand, Martinez-Thomas et al. [2002] carried out the computations in three stages. In the first and second stages, the heater is assumed to be stationary. In this case, the crystal and solvent temperatures have the same value, Tsta =Tc =Ts. The solute concentration is constant along the zone, Csta. Assuming the solubility of solvent in the solid phase is negligible, the volume of the zone is equal to the initial volume of the solvent plus the volume of solute incorporated (Lan and Yang [1995]), i.e.,

V = V0 +  Csta dV

(7.3.40)

Once the temperature field is calculated, the values of Tsta and Csta are determined from Eq. (7.3.40) and the phase diagram. The effect of convection was considered at the second stage, and the isotherms were computed using an

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378

adaptive grid (Fig. 7.3.19). At the final stage, species transport was included. Since the crystallisation and solvent temperatures are different (Tc < Tsta and Ts > Tsta), the corresponding concentrations are also different; Cc < Csta and Cs > Csta. Values of Tc, Ts, Cc, and Cs are obtained from the quasi-steady growth and solvent rates, and the liquidus line of the phase diagram. Table 7.3.3. System parameters (Martinez-Thomas et al. [2002]). Component

Value

Unit

Furnace length Furnace inner radius Ampoule length Ampoule inner radius Ampoule silica wall thickness Graphite thickness Furnace translation rate Prandtl number Schmidt number Thermal Grashof number Solutal Grashof number

30 2 15 7.5 1.5 50 2 0.039 26 1.25106 6.20105

cm cm cm mm mm μm mm/day

Simulation Results for a Stationary System The operating parameters and physical properties of the system are given in Tables 7.3.3 and 7.3.4. The measured temperatures along the wall of the empty furnace was approximated by a piecewise profile. The position of the heater referenced with the respect to the centre of the ring, was modeled by shifting the temperature profiles. The coordinate system is located at the bottom of the furnace wall (z = 0). The computations at the first stage show that the highest temperature at the centre can be 51–77 K lower than the heater temperature (830 K). This difference depends on the heater position, and also affects the interface temperature. Fig. 7.3.20 shows the temperature distribution along the interface when the heater is stationary (Tsta). As seen, the temperature increases at the beginning similar to one from a transient process; it increases first and then decreases sharply (a hump). This behavior happens when the heater is located between the positions of 10.5 and 11.5 cm. It corresponds to the growth of an ingot about 2.2–3.2 cm in length. This thermal behavior is related to the heating characteristics of the system. The heat flow going out is presented in Fig. 7.3.21. This heat flux is a function of the heater position, and exhibits a dip at the heater positions where the interface temperature has the hump. Since the heat fluxes going out and coming in must be balanced, one may argue that the

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379

incoming heat flux from the furnace to the ampoule is also a function of the heater position. Table 7.3.4. Physical properties (see reference sources in Martinez-Thomas et al. [2002]). Magnitude (unit) 3

Density (kg/m )

Thermal expansion coefficient (1/K)

Solutal expansion coefficient Viscosity (kg/m.s)

Thermal conductivity (W/m K)

Specific heat (J/kg K)

Heat of fusion (J/kg)

Prefixed temperature (K) Emissivity

Diffusion coefficient (m2/s)

Material

Value

Te HgTe CdTe Air for HgTe and CdTe Te HgTe CdTe Air for HgTe and CdTe HgTe (that of CdTe) CdTe HgTe+Te (l) (Te only) CdTe+Te (l) (Te only) Air for HgTe and CdTe Te HgTe (s) HgTe+Te (l) (50%, 50%) CdTe (s) CdTe+Te (l) (20%, 80%) Air for HgTe and CdTe Silica Graphite Te HgTe (s) HgTe+Te (l) (50%, 50%) CdTe (s) CdTe+Te (l) (20%, 80%) Air for HgTe and CdTe Silica Graphite HgTe CdTe Te Hg0.275Te0.725 Cd0.175Te0.825 HgTe CdTe Mullite Ni Silica Graphite HgTe in Te CdTe in Te

640 8 087 5 680 0.4446. and 0.415 0.00015 0.00015 0.0005 0.0025 and 0.0011 0.056/XHgTe 0.056/XCdTe 0.00088 0.00088 0.000033, and 0.000039 6.3 2 4 1.5 5.4 0.0506 and 0.055 2.8 120 372 176 274 160 330 1074 and 1122 770 712 108 360 209 200 137 061 793 1 365 0.6 0.6 0.9 0.15 0.7 0.6 5x10-9 5x10-9

Sadik Dost and Brian Lent

380 770

760

computed

Tsta (K)

expected

750

740 8

10

12

14

16

Heater position (cm)

Fig. 7.3.20. Computed temperature at the interface when the heater is stationary (Tsta) and expected behavior, as a function of the heater position (redrawn from Martinez-Thomas et al. [2002]).

The dip in the incoming–outgoing heat fluxes is considered to be due to the removal of heat at the lower part of the ampoule. Fig. 7.3.21 also shows the outgoing heat flux for the lower part of the ampoule, which was numerically separated into two parts; one from radiation and the other one from conduction plus convection. Both fluxes show a dip at the same location as that of the total flux. The outgoing heat flux at the upper part of the ampoule, not shown here, exhibits a uniform behavior as also shown by Chang et al. [1989] in the growth of HgxCd1-xTe. This is due to the poor conductivity of this system. They observed this behavior through the expansion of the solvent zone and through the shape of interfaces which become concave. To investigate the origin of the dip in the outgoing heat flux, MartinezThomas et al. [2002] also modeled the THM growth of CdTe. The grown CdTe crystals in their experiments were generally free of solvent inclusions. It was found that, contrary to what was observed in the HgTe system, for the CdTe system, the contributions of conduction and convection, and also that of radiation to the heat flux have shown a smoothly increasing behavior with respect to the heater position. This also leads to the same behavior in the total heat flux, which exhibits no dip. The typical growth temperature for the CdTe system is about 1000 K, while it is about 775 K for the HgTe system. It was concluded that the growth temperature used in the HgTe system was responsible for the numerically observed anomaly in the thermal behavior of this system. Computed isotherms in the liquid zone are curved due to the effects of fluid flow and radial temperature gradients (Martinez-Thomas et al. [2002]). The growth interface was concave, except towards the end of the growth period, where it becomes slightly convex. Such a change in the shape of the growth

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381

2

Heat Power (A.U.)

1

0 total heat flux conduction plus convection outcoming heat flux radiative outcoming heat flux

-1 8

10

12

14

16

18

Heater position (cm)

Fig. 7.3.21. Total heat flux per second, and outgoing heat flux per second by radiation, and that by conduction plus convection (redrawn from Martinez-Thomas et al. [2002]).

interface may be due to the thermal characteristics of the growth system. Similar behavior of the growth interface was also observed in the LPD growth of SiGe by Yildiz et al. [2005]. The maximum flow velocity in the solution was about 1 cm/s. The Prandtl number Pa = 0.039 indicates the importance of thermal diffusion and convection. Fig. 7.3.22 presents the computed zone length as a function of the

Zone length (cm)

1.8

1.4

1.0

zone length (cm) Tsta or Csta (A.U.)

0.6 8

10

12

14

16

18

Axial position (cm)

Fig. 7.3.22. Liquid zone length. The dashed line indicates the Tsta or Csta behavior, shown in Fig. 7.3.20 (redrawn from Martinez-Thomas et al. [2002]).

Sadik Dost and Brian Lent

382

heater position. As seen the zone length first increases from 1.25 to 1.4 cm and then gets smaller down to 1.25 cm. The hump in the zone length begins when the hump in the temperature Tsta ends. From the location of the growth and dissolution isotherms (Tsta), the ‘‘quasisteady’’ growth and dissolution rates are computed, and the ratio of these rates and the heater rate is shown in Fig. 7.3.23. Examination of the zone length and of the growth and dissolution rates indicates that the increase in the liquid zone is the result of increasing dissolution rate as growth proceeds. Similarly, the reduction in the zone length is due to an increase in the growth rate. However, at the beginning of the ingot (at the heater position of 8.25 cm) both rates are increasing until the system reaches a steady-state, similar to that observed in Martinez-Thomas et al. [2001]. Simulation Results for a Non-Stationary System Martinez-Thomson et al. [2002] stated that they also computed the concentration field in the liquid solution assuming that the heater is not stationary, and the results indicated that the concentration distribution is uniform in the center but exhibits step-like gradients near the interfaces (a boundary layer). The thickness of the boundary layers near the interfaces was in the order of 400-500 μm. It was concluded that the presence of such layers shows the significance of convection in THM. The significance of convection in the liquid solution was confirmed by Liu et al. [2003] through a 3-D numerical model in the THM growth of CdTe. 1.8

Zone length (cm)

Tsta or Csta (A.U.) relative “quasi-state” growth rate relative “quasi-state” solvent rate

1.4

1.0

0.6 8

10

12

14

16

18

Axial position (cm)

Fig. 7.3.23. Quasi-steady growth and solvent rates with respect to the heater translation rate. The dashed line indicates the Tsta or Csta behavior, shown in Fig. 7.3.20 (redrawn from MartinezThomas et al. [2002]).

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383

7.4. Three-Dimensional Axisymmetric Models In modeling the THM growth of crystals, instead of using a full scale 3-D model, it would be computationally efficient to use an axisymmetric domain whenever it can be justified on physical grounds. In such an axisymmetric system, although all the system variables will be independent of the circumferential coordinate, the circumferential flow velocity component can be obtained from the momentum balance in this direction. In this section, we first present the general governing equations for both the liquid and solid phases. The associated boundary and interface conditions will be presented later when a specific case is considered. 7.4.1. Axisymmetric 3-D Field Equations As usual, we assume that the liquid phase is a binary dilute mixture. The Boussinesq approximation holds, and the flow is incompressible, laminar, and Newtonian. The solid phase is a heat conducting rigid material with no mass transport. Under these assumptions we present the general forms of the governing equations. In the solid phase, the energy equation will be the same given in Eq. (7.3.6). In the liquid phase, the continuity equation is the same given in Eq. (7.3.8). The momentum balance yields

u u v 2 1 p  2u 1 u  2u u u +u + w  = + ( 2 + +  ) r z r r r z 2 r 2 t  L r r

+

1 (F ms + Frmrot ) L r

(7.4.1)

w w 1 p  2 w 1 w  2 w w +u +w = + ( 2 + + ) r z r r z 2 t  L z r

+

1 (F ms + Fzmrot )  g{T (T  T0 ) + C (C  C0 )} L z

v v uv  2 v 1 v  2 v v 1 v +u + w + = ( 2 + + 2  2)+ (F ms + Fmrot ) r z r r r z t L  r r

(7.4.2) (7.4.3)

where we also included the components of the static and rotating magnetic body forces, since these equations will be needed in the next section. The magnetic body force components in the r-, z- and  - directions, namely Frms , Fzms , and Fms , and Frmrot , Fzmrot , and Fmrot will be defined in Section 7.5. The energy equation is in the same form of that given in Eq. (7.3.13). We write the mass transport equation as

384

C C 2C 1 C 2C C +u +w = DC ( 2 + + ) r z r r z 2 t r

Sadik Dost and Brian Lent

(7.4.4)

where C represents the solute concentration. The associated boundary and interface conditions will be written in the next sections for specific models. 7.4.2. Growth of InP Matsumoto et al. [1997] carried out a 3-D numerical simulation for the THM growth of InP. The governing equations of the liquid phase were solved for the velocity, temperature and concentration fields by the finite difference method and the boundary fitting method under the assumption of stationary heater. The effect of thermal convection on the crystal growth rate and the constitutional supercooling occurring in the indium solvent was investigated. The heat flux on the ampoule wall is given by (based on the experimental system using a mirror furnace)

q = q0 f (Z  Z0 ) = q0 (aZ 8 + bZ 6 + cZ 4 + dZ 2 + e)

(7.4.5)

where Z is the dimensionless coordinate in the vertical direction (see Fig. 7.4.1) and f(Z) is a dimensionless function. The coefficients in this function are taken as

a = 2.48  1014 , b = 2.17  1011 , c = 7.29  107 , d = 1.20  104 , e = 1.0 where q is the total heat flux including radiative and convective heat transfer at the surface and the temperature and heat flux are nondimensionalised using q0. In the model heat is removed from the top and the bottom of the ampoule. The heat input from the heater is taken equal to the amount of heat output removed from the top and the bottom. The lateral wall is thermally insulated, except in the heated regions. The ampoule is rotated at an angular frequency  . The temperature at the solution-solid interface and the concentration of phosphorus at the solution side of the interface are determined by the liquidus curve of the InP phase diagram (see Matsumoto et al. [1997]). The associated field equations of the liquid phase, i.e., Eqs. (7.3.8), (7.3.13), (7.4.1), (7.4.2), (7.4.3), and (7.4.4), are expressed in dimensionless form using the stream function approach. Non-dimensionalization is made using

R = r / r0 ,  = t, u = Ur0 , v = Vr0 , w = Wr0

(7.4.6)

The dimensionless velocity components and the vorticity are expressed in terms of dimensionless stream function  , respectively, as

Single Crystal Growth of Semiconductors from Metallic Solutions

U=

385

1  1  U W 1 2 1  2 , V = , =   (  + ) (7.4.7) R Z R R Z R R R 2 R R Z 2

The use of Eqs. (7.4.6) and (7.4.7) in Eqs. (7.3.8), (7.3.13), (7.4.1), (7.4.2), (7.4.3), and (7.4.4), in the absence of magnetic body force components, yields

  U  1 V  +U +W   V  R Z R 2R Z RaT  L RaC C L 1  2 1   2  = { 2+ + 2  2 }  2 Ta R R R Z R PrTa R ScTa 2 R

V V UV 1  2V 1 V  2V V V +U +W + = { 2+ +  } R Z R Ta R R R Z 2 R 2   L  C L 

+U +U

 L R C L R

+W +W

 L Z C L Z

2 2 1   L 1  L   L { 2 + + } Pr Ta R R R Z 2

= =

2 2 1  C L 1 C L  C L { 2 + + } ScTa R R R Z 2

(7.4.8)

(7.4.9)

(7.4.10)

(7.4.11)

and the energy equation in the solid phase (feed and seed) is written as

 S 

=

2 2    S 1  S   S { 2 + + } R R Pr Ta R Z 2

(7.4.12)

where  L ,  S and C L are the dimensionless temperatures and concentration, defined as

 L  kL

TL  T f q0 r0

,  S  kS

TS  T f q0 r0

, and C L  1 2cLph

(7.4.13)

where cLph is the P concentration in the In solvent, and T f is the melting temperature of InP. The P composition in the solid phase is cSph = 0.5 , and therefore according this definition (i.e. the dimensionless solid composition is zero)

CS  1 2cSph = 1 2(0.5) = 0 The dimensionless numbers are defined as

(7.4.14)

Sadik Dost and Brian Lent

386 z Heat flux Q2=q2/q0

AS2

Dissolution interface

Solid II

F2* Heat flux Q =q/q0 =f(z-z0)

Initial interface

A AL

Solution Growth interface

q0

4.8

F1*

Initial interface AS1

z0 Solid I r

r0

0

Heat flux Q1=q1/q0 Fig. 7.4.1. Computational domain (redrawn from Matsumoto et al. [1997]).

RaT =

q0 r0 c p T gq0 r04  gcr03  r 2 , RaC = C , Ta = 0 ,  = S , Sf L = k L L L DL L L L kL H

(7.4.15)

where RaT and RaC are the thermal and solutal Rayleigh numbers, Ta is the Taylor number, Sf is the Stefan number, H represents the latent heat of fusion, and cp is the specific heat. The following boundary conditions are used. At R = 0 :

U = 0, V = 0,  = 0,  = 0,

 L R

= 0,

C L R

= 0,

 S R

=0

At R = 1 :

U = 0,

 L C L V = 0,  = 0,  = 0, = f (Z  Z0 ), =0 R R R

(7.4.16)

Single Crystal Growth of Semiconductors from Metallic Solutions

 S R

= 0 (nonheated), and

 S R

= s(Z  Z0 ) (heated)

387

(7.4.17)

At the interfaces at Z = F1* and Z = F2* :

U = 0, V = 0, W = 0,  = 0,  =

1  2 R Z 2

(7.4.18)

At Z = 0 and Z = A , respectively:

 S R

= Q1 , and

 S R

= Q2

(7.4.19)

where the change in interface positions in time are assumed small. At the growth ( F1* ) and dissolution ( F2* ) interfaces the heat and mass balances are expressed as

Sf F *  L  L F *  S  S F * =  L {( + )  ( + )}  Pr Ta R R Z R R Z F * 1 F * C L C L (C L  CS ) = ( + ) ScTa  R R Z

(7.4.20) (7.4.21)

where we have used Eq. (7.4.14). The dimensional P concentration is approximated from the In-P phase diagram as

cLph  9.148  1023 T 8 + 6.578  1019 T 7  2.108  1015 T 6 + 3.454  1012 T 5  3.606  109 T 4 + 2.351 106 T 3

(7.4.22)

 9.354  104 T 2 + 2.076  101T  1.970 Matsumoto et al. [1997] solved the transformed field equations under the boundary and interface conditions described above using the finite difference technique combined with the boundary fitting method. Varying grids were used: 11 (radial direction) x 33 (vertical direction), 21 x 83, and 31 x 123. They found that the maximum difference in the values of the stream function, temperature, concentration, and position of the interface was less than 3.2%. The Prandtl number and the Schmidt number were, respectively, 7x10-3’ and 3.7 which were estimated by the physical properties of InP given by Danilewski et al. [1992] and the Taylor number was fixed at 150, corresponding to 5 rpm for the ampoule of 5 mm radius. The initial aspect ratios of the solution, the seed and the feed were, respectively, 2, 2 and 4. The initial temperature of the system was set at the melting point of indium and the initial concentration of P in the In solvent was zero. The ratio of the heat flux q1/q2 was 2. The thermal Rayleigh

Sadik Dost and Brian Lent

388

Interface position, F*1

2.4 RaT=103

2.2

RaT=102 RaT=10

2.0

melt back

1.8 conduction 1.6 0

1000

2000 3000 4000 Dimensionless time, 

5000

Fig. 7.4.2. Time evolution of the growth interface (after Matsumoto et al. [1997).

number was estimated using the physical properties given in Danilewski et al. [1992] as: 0, 10, 102, 103. The solutal Rayleigh number, Ra C, was 5105. Simulation Results The time evolution of the position of the growth interface at the centre is shown in Fig. 7.4.2, for which the initial position of the interface was located at Z = 2. In the figure, the evolution of the interface was also presented under conduction for comparison. In the case of buoyancy convection for RaT = 10 and 100, noticeable melt-back occurs and the crystal starts growing after a positive P concentration gradient has been established near the growth interface. In the case of strong convection (RaT = 1000), although meltback occurs in the early stage, the negative P concentration gradient is not high, and a positive P gradient establishes very quickly. The issue of constitutional supercooling was also examined by Matsumoto et al. [1997] for the THM growth of InP. The degree of the constitutional supercooling was defined as

S

c  csat csat

(7.4.23)

where csat is the saturation temperature in the In-P solution corresponding to the local temperature. Accordingly, the region will be supercooled when S is positive. Their simulation showed that the supercooling occurs only when the Rayleigh number is about 1000. The time evolution of the supercooled area is shown in Fig. 7.4.3. As seen, the simulated model indicates that strong supercooling occurs near the growth

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389

5.0

Dimensionless position, Z

solid II 4.0 solvent

3.0

S = 0.05

S = 0.05

S = 0.05

S = 0.05

Smax = 0.09

Smax = 0.36

Smax = 0.19

Smax = 0.09

2.0 solid I 0

1

0

1

0

1

0

1

Dimensionless position, R =6

 = 10

 = 50

 = 250

Fig. 7.4.3. Computed supercooling at RaT = 1000, RaC = 50000, Ta = 150 (redrawn from Matsumoto et al. [1997]).

interface in the early stage. Simulations show that although the solutal Rayleigh number is larger than the thermal Rayleigh number, the solutal natural convection is not strong even compared to the force convection induced by rotation. This is due to the fact that in the present simulations the Schmidt number is much larger than the Prandtl number, and the concentration gradient is not so large. Especially, the effect of solutal convection diminishes in the steady state since the concentration gradient becomes zero. At RaT = 10, small cells remain in the upper part of the solution which is induced by rotation, while thermal convection becomes dominant in the case of RaT = 100. At both RaT = 10 and 100, the concentration of P becomes very small in the centre of the solution zone in the early stages. This may explain the melt-back seen in Fig. 7.4.3. The temperature at the interface is lower than the melting point of InP and the process reaches its steady state when the concentration of P becomes uniform in the solution. In the case of strong convection (RaT = 1000) a positive phosphorus gradient is established very quickly at the interface. For details on the computed flow, temperature and concentration structures under various growth conditions, we refer the reader to Matsumoto et al. [1997]. 7.4.3. Growth of GaSb A numerical simulation study for the THM growth of GaSb from a Gasolution was given by Okano et al. [2002]. The effects of growth temperature,

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Sadik Dost and Brian Lent

crucible rotation, and crucible material on the growth interface shape were investigated. The numerical simulation results showed that the interface curvature increases with increasing growth temperature. At the high growth temperature, the use of a crucible rotation was found beneficial in suppressing natural convection in the melt. It also reduces the interface curvature. At the lower growth temperature scheme (with the same temperature gradient), the interface shape was slightly convex towards the crystal, and the effect of the crucible rotation on the interface shape was not significant. Finally, it was found that a pBN crucible requires a higher growth temperature scheme compared with that for SiO2 and carbon crucibles. Fig. 7.4.4 shows a schematic view of the computational domain (a), and the measured temperature profile along the crucible wall (b). In the model, the solution is considered an incompressible Newtonian viscous liquid with the assumption of the Boussinesq approximation. The system is assumed axisymmetric, and the convective flow in the solution is laminar. Under these conditions, the governing equations in the liquid phase are in the same form as those given in Eqs. (7.3.8)-(7.3.13), and (7.4.1)-(7.4.4) in the absence of magnetic effects. However, here the liquid concentration C is given in volume fraction. In the solid phase that represents the crystal, the feed, and the crucible wall, the heat conduction equation is the only governing equation. The heat conduction equation was solved in each domain. The boundary conditions for the THM system simulated are the same as those given in Section 7.3. In order to calculate the shape and the position of the growth and dissolution

Fig. 7.4.4. A schematic view of the model geometry (a), and the measured temperature profile along the crucible wall (b) (after Okano et al. [2002]).

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391

interfaces, the following heat and mass balance equations were solved together with the solid/solution phase diagram:

s H C

T f Ts T f f TL f = {ks ( s  )  kL ( L  )}{1+ ( )2 } t x y x x y x x

f C f TL f =  DC (  ){1+ ( )2 } t y x x x

(7.4.24)

where H represents the latent heat of fusion, and f(x,y) is the interface shape (growth or dissolution). The governing equations and the associated boundary conditions were transformed into dimensionless form, and the boundary fitted coordinate method was used to treat curved interfaces. The dimensionless field equations were discretized by the finite difference method, and were solved numerically by the successive over relaxation method. After examining the mesh dependency of the system, the solution region was divided into 49 (radially)79

Fig. 7.4.5. Transient behavior of the fluid flow (left half in the solution:  = 0.0002 (–)), temperature distribution (left half in the solids: T = 5.6K) and the concentration variation (right half in the solution: C = 0.02 (–)) when only the thermal (upper column) or solutal (lower column) natural convection was considered. (a), (b) and (c) correspond, respectively, to the cases where t = 30; 60 and 100 s (after Okano et al. [2002]).

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Sadik Dost and Brian Lent

(axially) meshes with uniform spacing. The transient process was simulated from the selected initial condition with no heater movement. Then, the result obtained by the conductive heat transfer analysis was used as the initial conditions for the next step. The thermal and solutal Grashof numbers, GrT and Grc, are calculated as 103 and104. 7.4.4. Simulation Results Fig. 7.4.5 shows the transient behavior of the fluid flow (left half in the solution), the temperature distribution (left half in the solids), and the concentration variation (right half in the solution) when only thermal (upper column) or solutal (lower column) natural convection was considered. The crucible was assumed to be made of SiO2. The thermal conductivity of SiO2 is 1.62 W/(mK). It was found that the thermal natural convection induces an upward flow along the crucible wall and a downward flow along the center axis. The flow structure of the solutal natural convection is complex. However, it

Fig. 7.4.6. The effects of a steady crucible rotation and the growth temperature when both the thermal and solutal natural convections were present in the solution. The upper and lower columns show, respectively, the cases corresponding to the lower and higher (30 K) growth temperatures: (a) 0 rpm, (b) 3 rpm, (c) 5 rpm and (d ) 7 rpm (after Okano et al. [2002]).

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becomes simpler as the time proceeds. This is because of the fact that the solution was agitated by the convective flow, and consequently, the concentration gradients in the solution, which are the driving force for the solutal natural convection, became smaller. Fig. 7.4.6 presents the effects of the crucible rotation and the growth temperature scheme on the fluid flow, temperature and concentration distributions, and the shape of the interfaces when both the thermal and solutal natural convections are present in the solution. The upper and lower columns represent, respectively, the cases where the growth temperatures are selected as the lower temperature scheme shown in Fig. 7.4.4(b), and as 30K higher (while keeping the same temperature gradient) than that shown in Fig. 7.4.4(b). The shape of the growth interface becomes convex towards the crystal, but the shape strongly depends on the growth temperature. The interface curvature becomes larger with the increasing crucible temperature. In such a case, crucible rotation was very beneficial in suppressing natural convection in the solution, and in obtaining a growth interface of smaller curvature. In the case of the lower growth temperature scheme, the interface shape is slightly convex towards the crystal, and the effect of crucible rotation on the interface shape is not significant. Two different crucibles were considered in the simulation: a carbon crucible (with a thermal conductivity of 80.1W/mK) and a pBN crucible. Due to the layer structure of the pBN crucible, its thermal conductivity shows anisotropy (Simpson and Struckes [1976], Lan and Ting [1995], Okano et al. [1998]). In this study, the thermal conductivity values of 2.51 and 62.8 W/mK are used along the c-direction (perpendicular to the growth direction) and the a-direction (parallel to the growth direction), respectively. The simulation results for the fluid flow, temperature and concentration distributions, and the shape of the

Fig. 7.4.7. Effect of crucible material at lower and higher crucible temperatures (after Okano et al. [2002]).

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solids/solution interfaces are presented in Fig. 7.4.7. For the pBN crucible (at the lower crucible temperature), no crystal growth was achieved. This was due to the anisotropy in the thermal conductivity of the pBN crucible that slowed down the radial heat conduction into the solution. Consequently, the solution temperature became lower than those found in the cases of carbon and SiO2 crucibles. One can then conclude that the use of a pBN crucible requires a higher crucible temperature with respect to the SiO2 and carbon crucibles. 7.5. The Use of Magnetic Fields in 2-D and Axisymmetric Models As mentioned earlier, Lan and Yang [1995] used a pseudo-steady-state model to simulate fluid flow, heat and mass transfer, and interface shapes during the THM (called the Traveling Solvent Method in Lan and Yang [1995]) growth of a CdTe/Te system under various gravity intensity levels. The effects of some growth parameters such as the growth rate, initial solvent volume, and heater temperature were studied for a crucible of 1.5 cm diameter. In the simulation of larger size crystals, computational difficulties may be encountered due to strong convection in the solution. Ghaddar et al. [1999] presented an axisymmetric steady-state numerical simulation model to investigate the influence of a rotating magnetic field (RMF) on the flow patterns and compositional uniformity in the solution zone of a THM system used for the growth of CdTe crystals. They have found that, under low gravitational field conditions, RMF can be useful to suppress residual natural convection, and at high gravity levels, RMF may result in complex flow structures and enhance the compositional non-uniformity at the growth interface. They have encountered computational difficulties under normal gravity (Earth’s gravitational field) in which the solutal transport becomes too intensive to be computationally resolvable by their model, due to the strong convection in the solution. The computations could then only be carried out up to one-tenth of the normal gravity level. At higher gravity levels, the computations were unstable. Dost et al. [2003] also carried out axisymmetric numerical simulations for the THM growth of CdTe crystals to examine the effects of applied stationary and rotating magnetic fields. This study also considered small non-uniformities in the stationary magnetic field. A strong stationary field is applied to suppress the natural convection in the solution zone, and a small but rotating magnetic field is considered for better mixing in the horizontal plane. The spatial distribution of the stationary field is almost uniform before the furnace and the growth system are lowered into the magnet opening. However, the field uniformity may be altered by the presence of the growth system. In order to determine the effect of such field non-uniformities, small magnetic body force components are considered in the model. The simulation results showed that higher stationary field levels are better in suppressing convection in the solution, but enhance the

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compositional non-uniformity in the solution. Small but unintentional nonuniformities in the stationary magnetic field also enhance the compositional non-uniformity. A rotating magnetic field, on the other hand, is beneficial for mixing in the horizontal plane, and reduces the compositional non-uniformity in the solution. The mixing is enhanced at higher rotating magnetic field frequencies. 7.5.1. Magnetic Body Force Components The field equations of the liquid and solid phases for a binary system of CdTe are the same as those given in the previous sections. In the model the liquid phase represents the dilute Te-rich solution zone, and the solid phases are the feed and seed materials, and the quartz crucible wall. The only governing equation in the solid phase is the energy balance given in Eq. (7.3.6). The continuity and energy equations of the liquid phase are the same as those given in Eqs. (7.3.8) and (7.3.13), and the momentum and mass transport equations as those given in Eqs. (7.4.1)-(7.4.4). The derivation of the magnetic body force components were given in Chapter 4 for a general case, and in Chapters 6 and 8 for LPEE and LPD systems. However, since we consider here some magnetic field non-uniformities, we present them explicitly for the benefit of the reader. Stationary Magnetic Field As described in Chapter 3, the superconducting magnet at the CGL at the University of Victoria was designed to have the externally applied magnetic field aligned with the vertical axis of the growth system (Fig. 3.3.2). The measured magnetic field is almost uniform in both the radial and vertical directions in the liquid zone region (Fig. 3.3.4). The field measurements were made when the furnace was not in the magnet opening. However, when the furnace and the growth ampoule are placed into the magnet opening, the field distribution may be affected, and there may be some small deviations from uniformity. It is not possible to measure the magnetic field distribution when the growth system is in the magnet opening. In addition, it is also possible that the experimental set up is not aligned perfectly with the magnetic field. Such a small non-uniformity in the applied magnetic field, and also small misalignments may lead to small magnetic body force components in all three coordinate directions. In order to examine the effects of such small field non-uniformities and misalignments on the THM growth process, small magnetic field components are introduced in the magnetic field vector as follows:

B = B( r e r +  e + e z )

(7.5.1)

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where B is the magnitude of the applied magnetic field B, and r and  are small values introduced in the r- and - directions. It must be noted that the system is still axisymmetric, thus, all the field variables are independent of the -coordinate. By neglecting the induced electrical field and also higher-order terms, the magnetic body force components due to the external applied magnetic field B are obtained from  E (v  B)  B as

Frms =  E B 2 (u +  r w +  r  v  2 u)   E B 2 (u +  r w) Fms =  E B 2 (v +  w +  r  u   r2 v)   E B 2 (v +  w) Fzms =  E B 2 ( r u +  v   r2 w  2 w)   E B 2 ( r u +  v)

(7.5.2)

Rotating Magnetic Field For a rotating magnetic field vector B rot , since it is not stationary, the Maxwell equations must be satisfied:

E= 

B rot , t

  E = 0,   B rot = iB J,   B rot = 0,   J = 0

(7.5.3)

With the introduction of a scalar potential  and a vector potential A,

B rot =   A , and E = ( +

A ) t

(7.5.4)

Eqs. (7.5.3) reduce to

  (E +

A )=0 t

(7.5.5)

The rotating field applied in the horizontal plane can be defined in general by

B rot = B rot {e r sin(   t) + e cos(   t)

(7.5.6)

where Brot and  are respectively the magnitude and frequency of the applied rotating field. The field defined in Eq. (7.5.6) satisfies Eq. (7.5.4)1 identically, and the solution of Eq. (7.5.5) and the electric field in Eq. (7.5.4)2 yield, respectively

A =  B rot r cos(   t)e z , and E =  +  rB rot sin(   t)e z

(7.5.7)

Substitution of the induced electric field E and the magnetic field Brot into the magnetic body force, i.e.,  E (E + v  B rot )  B rot , yields a complex expression for the magnetic body force. Under certain assumptions, this expression can take a simplified form. For instance, when the skin depth is much larger than

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the radius R,   (μBE)1/2 >> R, (for instance in the THM systems considered in Ghaddar et al. [1999] and Dost et al.[2003]  =23 cm), the magnetic field distribution can be assumed to be unaffected by the fluid flow, that is, the field penetrates the fluid unchanged. In this case, assuming that

 (r, , z,t)  1 (r, z)sin(   t) + 2 (r, z)cos(   t)

(7.5.8)

the mean magnetic body force components are obtained as

  1 1 Frmrot =  E B rot { 2  uB rot } , Fmrot =  E B rot { 1 + (r  v)B rot } 2 2 z z 1 (r2 ) (7.5.9) Fzmrot =  E B rot {  wB rot } 2r r Further assumptions, such as the amplitude of the oscillating component of the magnetic body force being negligible, the fluid rotation being considerably slower than the magnetic field frequency, the ampoule and solid CdTe being electrical insulators, and the magnetic field being continuous at all boundaries (see Ghaddar et al. [1999] for details), the mean magnetic body force components in Eqs. (7.5.9) become

Frmrot = 0,

1 Fmrot =  E r (B rot )2 , 2

Fzmrot = 0

(7.5.10)

7.5.2. Simulations Under RMF at Various Gravity Levels Ghaddar et al. [1997] carried out a numerical simulation study to examine the effect of applied rotating magnetic fields in the THM growth of CdTe under normal and microgravity conditions. The numerical analysis focused on the influence of a rotating magnetic field on flow patterns and concentration distribution in the solution zone. Gravity levels were selected to represent the ground (Earth) and space (Microgravity) processing conditions. Simulation results under microgravity conditions have shown that RMF can be used to minimize the residual natural convection (g-jitter effect) and to control the uniformity of the solution composition near the growth interface without altering the shape of the growth interface. However, as the gravity level increases, the effectiveness of RMF in controlling natural convection reduces. At high gravity levels, for the range of field strengths studied, the application of RMF led to complex flow structures and enhanced compositional nonuniformity (near the growth interface) in the solution, and increased the convexity of the growth interface. The computational domain of the THM system of Ghaddar et al. [1997] is shown in Fig. 7.5.1. The Te-solution and the CdTe feed and crystal are contained in the inner fused silica crucible of diameter 2R. The inner crucible is

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placed within a stationary outer sleeve and is pulled down at a constant velocity Vg. A resistive ring heater of constant temperature TH and length LH is used to heat the Te-solution zone which is about 20 mm long and has an average CdTe concentration of 12%. A vacuum is maintained between the sleeve and the crucible. The relevant system information and the material properties can be found in Ghaddar et al. [1997]. The governing equations of the model are the steady-state continuity, (Eq. (7.3.8)), energy (steady-state version of Eq. (7.3.6)), mass transport (Eq. (7.3.19)), and momentum (steady-state version of Eqs. (7.4.1)-(7.4.3) in the absence of a stationary field). The molar concentration of CdTe in the Tesolution, C, is used in the model. Since the model was steady-state (timeindependent), in order to be able to take the crucible translation into account, it was assumed that the growth and dissolution rates are equal to the translation rate Vg. Furthermore, to account for the translation of the crucible, a uniform downward axial velocity field of magnitude Vg was imposed such that in the absence of melt convection, the diffusion-controlled solute transport is correctly predicted. The thermal boundary conditions of the system are written as follows. Heat is

Graphite

Outer silica wall Inner silica wall

CdTe feed

Resistive heater

Dissolution interface Te solution

Growth interface

CdTe crystal

Rotating field generator

CdTe seed Graphite

Inner crucible translating at Vg

Fig. 7.5.1. Computational domain of the THM system of Ghaddar et al. [1997] (redrawn schematically).

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radiated from the heater at TH through the silica sleeve and the crucible. A twoband one-dimensional model which incorporates the semi-transparent behavior of the silica ampoules is used. Heat is also radiated from all boundaries back to the surrounding environment which is assumed to be at room temperature (300 K). At the growth and dissolution interfaces, temperature is continuous, and the balance of energy is written, including the latent heat of fusion, as

qS  qL =  V g H , at the dissolution interface

(7.5.11)

qS  qL = + V g H , at the growth interface

(7.5.12)

No slip conditions on the velocity field were imposed at all solid boundaries, and the continuity of the axial component, w = Vg at solid boundaries. For the solutal boundary conditions a constant concentration of 12% CdTe is imposed at the dissolution interface, while at the growth interface the mass flux is set proportional to the concentration at the interface

DC (C  n) = CV g

(7.5.13)

At the ampoule wall, the solutal mass flux is zero. Based on the assumptions discussed in Section 7.5.1, Eqs. (7.5.10) is used to represent the magnetic body force components. The field equations together with the boundary conditions are solved by the Galerkin finite element method and Newton-Raphson iteration using the commercial CFD package CapeSim developed by Ghaddar et al. [1997]. The overall system shown in Fig. 7.5.1 was used to obtain the temperature distribution in the system by neglecting convection in the liquid-zone. The computed temperature field was then used as thermal boundary conditions along the new top and bottom boundaries of the computational domain containing only the feed/solvent/crystal zones and inner ampoule which was used for all reported convection calculations; the radial heat loss for the new system is based on radiation through the outer crucible to the heater and environment. Based on the assumptions in the model, Ghaddar et al. [1997] stated that the new boundaries are sufficiently far away from the interfaces where the system temperature has cooled down significantly and therefore, have little effect on the solution in the Te-zone. Their steady-state calculations showed that the relative difference in the temperature at these locations in the overall THM system with and without convection is less than 1%. 9-node tensor product elements were used in the liquid phase, and 8-node elements in the solid. A typical mesh comprises 1200 convection elements and 600 conduction elements resulting in about 26000 degrees of freedom. A typical calculation is started from non-deformed interfaces. After each iteration the computational mesh is deformed according to the following criterion: using the new concentration values at the current interface locations, the new interface locations are calculated from the phase-diagram and the new temperature field.

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The mesh is then deformed so that the new interfaces lie on the element faces (see Ghaddar et al. [1997] for details). As mentioned earlier, due to the assumptions employed, the Te-solution volume is not conserved in this approach. Ghaddar et al. [1997] reported that the maximum relative change in volume among all cases is within 10%. It was suggested that one way to impose the volume constraint is to adjust the imposed value of the concentration at the dissolution interface during the iterations so as to obtain the desired Te-solution volume. The CdTe-Te phase diagram and the steep axial temperature gradients in the system, however, result in a minimal deviation from the imposed 12% concentration. Simulation Results The computed flow patterns in the absence of RMF at the 10-6 and 10-1g0 gravity levels are shown in Fig.7.5.2. In both cases, the thermal buoyancy associated with heat input into the solvent zone results in a counter-clockwise recirculating cell that transports the solute from the dissolution interface to the growth interface. The mass transfer Peclet number, which can be used as a diagnostic indicator of the strength of solutal mixing, is less than unity at 10-6g0 and increases to about 6000 at 10-1g0. At about 10-4g0, the solute transport becomes convection-dominated. The magnetic body force components in the solution tend to oppose the buoyancy-induced convection near the growth interface, and reinforce the same near the dissolution interface.

(a) at g/g0 = 10-6

(b) at g/g0 = 10-1

Fig. 7.5.2. The computed flow field at two gravity levels in the absence of an RMF: (a) B = 0, umax = 6.810-6, vmax = 1.910-5, w = 0 (cm/s); (b) B = 0, umax = 0.31, vmax = 0.35, w = 0 (cm/s) (after Ghaddar et al. [1997]).

The influence of RMF is shown for the low- and high-gravity levels in Figs. 7.5.3a,c , and 7.5.3b,d, respectively. At low-gravity, the application of a moderate magnetic field strength results in two flow cells that completely

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occupy the solution zone and overwhelm the buoyancy-induced convection, Fig. 7.5.3a. With the increasing field strength, these two cells become stronger and thinner, resulting in boundary-layer flows close to the two surfaces and a quiescent region in the solution core, Fig. 7.5.3c. These flow patterns, however, do not repeat at the high gravity level (Fig. 7.5.3b,d). At moderate magnetic field strengths, a flow cell appears at the growth interface and the buoyancy-induced cell is pushed upwards. With the increasing field strength, the flow structure becomes more complex, consisting of a strong cell near the dissolution interface and two weaker cells in the vicinity of the growth interface.

(a) at g/g0 = 10-6

(b) at g/g0 = 10-1

(c) at g/g0 = 10-6

(d) at g/g0 = 10-1

Fig. 7.5.3. The computed flow field at two gravity levels: (a) B = 1 mT, umax = 0.029, vmax = 0.0034, w = 0.026 (cm/s); (b) B = 2.8 mT, umax = 0.29, vmax = 0.63, w = 0.37 (cm/s), (c) B = 5.0 mT, umax = 0.099, vmax = 0.069, w = 0.34 (cm/s), and (d) B = 4 mT, umax = 0.31, vmax = 0.33, w = 0.44 (cm/s) (after Ghaddar et al. [1997]).

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402 -74

g/g0=10-6 5 mT 3 mT

-76 g/g0=1, B=0 mT

z (mm)

1 mT

0 mT -78

-80 0

5 r (mm)

10

Fig. 7.5.4. Growth interface shapes for different RMF strengths at g/g0=10-6 (redrawn from Ghaddar et al. [1997]).

The numerical instabilities encountered by Ghaddar et al. [1997] did not allow them the use of higher magnetic field levels and also the 1g0 level. At 1g0 and B = 14 mT field intensity, the numerical simulations could have been carried out only in the absence of the concentration field. The computed growth interface shapes at 10-6g0 under various field intensity levels are shown in Fig. 7.5.4. The interface shape at 0.1g0 and no field is also shown for comparison. The comparison of the results shows that RMF can be used in space experiments to control the radial uniformity of CdTe concentration near the growth interface. The required RMF strengths for this purpose are small enough not to alter the temperature distribution in the solution zone. Thus, the conclusion was that the growth and dissolution interface shapes in space will not be influenced by RMF and are distinctly different from those calculated for the case of 0.1g0. 7.5.3. Modeling the THM Growth of CdTe Under Strong Magnetic Fields The applied vertical magnetic field may have some unintentional misalignment with respect to the vertical axis of the growth system. Such a misalignment would influence the level of the convective flow in the vertical plane, which adversely affects the stability of the growth interface, and the level of the convective flow in the horizontal plane that gives rise to mixing in the solution. In addition, the magnetic field distribution may no longer be uniform when the growth furnace and the ampoule are lowered into the magnet

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opening. This non-uniformity in the magnetic field may give rise to small additional body force components (Dost et al. [2002]). With this objective in mind, Dost et al. [2003] numerically examined the effects of such a magnetic non-uniformity and misalignment on the growth process of THM. In THM, the shape of the growth interface is influenced by the convective flow in the solution as well as the heat and mass transfer in the vicinity of the growth and dissolution interfaces. The shapes of the interfaces are therefore a priori unknown, and are solved simultaneously with the governing equations of the model. Dost et al. [2003] used an automatic mesh generation equation to generate the adaptive mesh in the whole computational domain. The effects of both fixed and rotating magnetic fields are considered. Computational Model The schematic view of the THM growth system used in Amistar Research and Development Inc. was given in Fig.7.3.5. The computational domain that includes the axisymmetric THM growth cell configuration, the applied furnace temperature profile, and the coordinate system used are presented in Fig. 7.5.5a.

(a) Fig. 7.5.5. (a) Computational domain and the applied temperature profile, and (b) the temperature field computed under B = 10 kG at t = 8 h. The dashed lines represent the growth and dissolution interfaces (after Dost et al. [2003]).

(b)

The axisymmetric field equations of the liquid and solid phases for a binary system of CdTe are the same those given in the previous sections. In the model

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the liquid phase represents the dilute Te-rich solution zone, and the solid phases are the feed and seed materials, and the quartz crucible wall. The only governing equation in the solid phase is the energy balance given in Eq. (7.3.6). The continuity and energy equations of the liquid phase are the same as those given respectively in Eqs. (7.3.8) and (7.3.13), and the momentum and mass transport equations given in Eqs. (7.4.1)-(7.4.4). The following boundary and interface conditions are used. At the axis of symmetry in the liquid zone

u = 0, v = 0,

w C = 0, =0 r r

(7.5.14)

At the vertical wall(s)

u = 0, v = 0, w = 0,

C =0 r

(7.5.15)

At the growth interface

u = 0, v = 0, w = 0,  DC kS

C = (1 C)Vn , C = C1 , r

T T  kL =  LVn H n n

(7.5.16)

At the dissolution interface

u = 0, v = 0, w = 0, T = T2 , C = C2 , kS

T T  kL =   LVn H n n

(7.5.17)

where Vn is the normal velocity of the interface, T2 is the equilibrium temperature at the dissolution interface, and the interface concentrations in the solution, C1 , and C2 , are determined by the interfacial equilibrium condition:

C = ae(b c /T )

(7.5.18)

where temperature T is given in Kelvin (K). The interface concentration in the solid is set to unity since the solubility of Te in the solid CdTe is neglected (Lan and Yang [1995], and Ghaddar et al. [1997]). The mass balance along the growth interface, namely Eq. (7.5.16)4, is used to evaluate the shape of the growth interface. Thermal boundary conditions along the vertical wall, top surface, bottom surface, and the symmetry axis of the crucible are given, respectively, as:

kS

T = h[T  T f (z,t)] +  E [T 4  T f4 (z,t)] n

(7.5.19)

Single Crystal Growth of Semiconductors from Metallic Solutions

T = h[T  T f ,t (z,t)] +  E [T 4  T f4,t (z,t)] n T kS = h[T  T f ,b (z,t)] +  E [T 4  T f4,b (z,t)] n T =0 n kS

405

(7.5.20) (7.5.21) (7.5.22)

The initial conditions are taken as the equilibrium state of the system. At this equilibrium state under the static furnace temperature profile, the shape and location of the growth and dissolution interfaces, their equilibrium temperatures, and the temperature, concentration and velocity distributions in the system are taken as initial conditions. The selection of the furnace temperature profile is very important in establishing the growth and dissolution interfaces at desired interface equilibrium temperatures. It is also important to select an appropriate speed for the furnace movement not to lead to either an undersaturated or over supersaturated solution in the vicinity of the growth interface. The growth process and the quality of grown crystals are very sensitive to these parameters. Numerical Technique The Galerkin finite element method, based on the penalty function formulation given by Reddy [1982], is employed in the solution of the governing equations. The advantage of the penalty function formulation is that the pressure term does not enter the formulation as a primary unknown variable, and the incompressibility equation does not require an independent solution. The boundary conformal mapping technique (Christodoulou and Scriven [1992] is applied to obtain the growth interface shape which is a priori unknown, as well as the deformed finite element mesh in the whole domain. The mesh generation equations are given as (Christodoulou and Scriven [1992] and Liu et al. [2000])

 [S +  s ] 

1  ln[(r2 + z2 ) f ( )] = 0, J 

 [S 1 +  s ] 

2  ln[(r2 + z2 ) f ( )] = 0 J 

where

S = (r2 + z2 )(r2 + z2 )1

(7.5.23) (7.5.24)

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Table 7.5.1. Physical Parameters of the CdTe System (Ghaddar et al. [1999], Meric et al. [1999]). Parameter

Symbol

Values

Density of liquid

l

5.62 g/cm3

Density of solid

s

5.68 g/cm3

Density of Silica

a

2.20 g/cm3

Thermal conductivity of liquid

kl

0.057 W/(cm.K)

Thermal conductivity of solid

ks

0.038 W/(cm.K)

Thermal conductivity of silica

ka

0.031 W/(cm.K)

Specific heat of liquid

cl

0.372 J/(g.K)

Specific heat of solid

cs

0.16 J/(g.K)

Specific heat of silica

ca

0.77 J/(g.K)

Kinematic viscosity of liquid



0.008992 g/(cm.s)

Electrical conductivity of liquid

E

20000 1/(.cm)

Thermal expansion coefficient

T

8.010-5 1/K

Solute expansion coefficient

C

0.056

Diffusion coefficient of CdTe in Te solution

D

4.210-5 cm2/s

Latent heat

H

209.0 J/g

Furnace temperature traveling rate

Vg

0.2 cm/day

Crucible inner radius

R

1.0 cm

Crucible thickness Initial solution height

0.2 cm L

2.3 cm

Initial substrate thickness

0.7 cm

Initial source thickness

3.0 cm

Phase diagram coefficient

a

0.209

Phase diagram coefficient

b

8.2

Phase diagram coefficient

c

8607.29

and (r,z) and ( ,  ) are respectively coordinates of the physical and computational domains. J is the Jacobian matrix of the coordinate transformation: J = r z  r z , with the subscripts  and  denoting partial differentiation with respect to indices as  /  and  /  . There are three terms in each of Eqs. (7.5.23) and (7.5.24). The first two terms are the result of optimizing the orthogonality and smoothness of the mesh, while the third term is used to control the spacing of the mesh. According to Christodoulou and

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Scriven [1992], the parameters are chosen as  s = 0.2 , 1 =  2 = 0.1 , and f ( ) = f ( ) = 1 . The shape of the growth interface is determined by writing the mass balance across the growth interface. Note that the temperature at the growth interface should be kept in accordance with the temperature field distribution in the system, and is not a constant anymore. Correspondingly, the concentration along the interface will change with temperature according to the phase diagram relation, Eq. (7.5.18). On the other hand, the interface shape at the dissolution interface is controlled by the isotherm condition (T=T2) along the interface, which is obtained assuming a constant concentration distribution at the dissolution interface. The computation-mesh in the solution is 5656 (86247 unknowns) with a very fine meshing in the vicinity of the computational boundaries (namely, the crucible walls, symmetry axis, and growth interfaces) in the liquid phase. Simulations have shown that the selected mesh is sufficient for an accurate and stable solution. The unknown variables, including temperature, velocity, concentration, and coordinates of the physical domain, are approximated by two-dimensional, nine-node, biquadratic basis functions. The Newton-Raphson method is used to iteratively solve the set of residual equations formulated after discretization, until a selected criterion is reached. Simulation Results Physical properties and system parameters are given in Table 7.5.1. In the table, the liquid represents the Te-rich solution, and the solid stands for the grown CdTe crystal and the CdTe feed. A typical computed temperature distribution for half of the growth system is shown in Fig. 7.5.5b (under B = 10 kG and at t = 8 hours). The temperature gradients near the growth interface and the dissolution interface are, respectively about 120 K/cm and 50 K/cm. The maximum temperature difference in the solution is about 25 K. Fig. 7.5.6 shows the simulation results for concentration and flow fields in the solution under three different levels of the applied stationary magnetic field: 5 kG, 10 kG, and 20 kG. In all three cases, the concentration gradients are confined near the growth interface due to strong convection, and are very low in the rest of the solution domain (which is a reflection of the low growth rate of THM). The intensity of convection in the solution decreases with increasing levels of the applied magnetic field. The single convection cell moves toward the center of the solution region and eventually splits into two cells circulating in opposite directions. Fig. 7.5.7a presents the evolution of the growth interface under two different magnetic field levels. Dashed lines represent the growth under a 5 kG field, and the solid lines are for the growth under a 10 kG field. The effect of higher

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magnetic field levels on the shape of growth interface can be seen clearly. This is due to the significant effect of the applied magnetic field on the convective flow in the solution. The concentration distribution in the solution near the growth interface under three magnetic field levels is given in Fig. 7.5.7b. The compositional nonuniformity (the variation of concentration) in this region, , can be measured

Fig. 7.5.6. Concentration contours and stream function contours (at t = 8 h). For concentration contours under (a) B = 5 kG, (b) B = 10 kG, and (c) B = 20 kG, contour spacing is 0.0009 mole fraction, and the minimum and maximum concentrations are 0.10503 and 0.11151 respectively. For stream function contours under (d) B = 5 kG, (e) B = 10 kG, and (f) B = 20 kG, contour spacing is 0.0015 cm2/s, and the minimum and maximum values of stream functions for each case are,  min = 0.0 and  max = 0.060756 ,  min = 0.0 and  max = 0.031466 , and  min = 0.001976 and  max = 0.010024 (after Dost et al. [2003]).

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according to Alexander et al. [1989]:

 = {(cmax  cmin ) / cave }%

(7.5.25)

where cave is the average concentration in the solution near the growth interface. Under B = 5 kG, B = 10 kG and B = 20 kG, we obtain  = 2.87%,  = 2.94%, and  = 4.80%, according to Fig. 7.5.8 and Eq. (7.5.25). This shows that at higher magnetic field levels, the compositional uniformity in this region becomes worse. Results indicate that the application of higher stationary magnetic field levels can reduce the flow strength further, but increase the radial segregation near the growth interface leading to higher levels of compositional nonuniformity in the solution. Lower magnetic field levels are therefore more beneficial. The flow patterns shown in Fig. 7.5.6 support this conclusion.

(a)

(b)

Fig. 7.5.7. (a) Evolution of the growth interface under different magnetic field levels. Solid lines represent the results under B = 10 kG, and the dashed lines are for those under B = 5 kG. The time increment between two curves is 4 h, and (b) Concentration distribution near the growth interface under various magnetic fields (t = 8 h) (after Dost et al. [2003]).

At higher magnetic field levels, the convection cell (although with a weaker strength) moves towards the center of the solution zone, and reduces the strength of concentration gradients further in the vicinity of the growth interface in the region near the symmetry axis. Consequently, weaker concentration gradients in this region result in a slower growth, which, in turn, lead to a higher concentration non-uniformity in the solution. Indeed, similar results were presented for various gravity levels in Ghaddar et al. [1999], Alexander et al. [1989], and Liu et al. [2000]. It was shown that the reduction of convection strength would lead to better compositional uniformity at low gravity levels, and much worse under high gravity levels. These results would be particularly useful for the THM growth of ternary crystals. Characterization of the grown

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ternary crystals for the crystal compositional uniformity can provide insight for the compositional uniformity (mixing) in the solution. The effect of magnetic field nonuniformity and misalignment was examined under two levels of magnetic field, B = 10 kG and B = 20 kG, and for three levels of non-uniformity, 0.05, 0.02, and 0.00. The simulation results for the concentration near the growth interface are shown in Fig. 7.5.8. As can be seen, the nonuniformity in the concentration distribution increases at higher field nonuniformity levels. In addition, it becomes worse with increasing magnetic field levels (see Fig. 7.5.8b). The conclusion is that the more uniform the applied stationary magnetic field, the better the compositional uniformity in the solution.

Fig. 7.5.8. Effect of small field non-uniformities on concentration near the growth interface at (t = 8 h): (a) B = 10 kG; (b) B = 20 kG. The solid lines represent concentrations for the case with r =  = 0.05, the equal-dashed lines for r =  = 0.02, and unequal-dashed lines for r =  = 0.0 (after Dost et al. [2003]).

The flow patterns (Fig. 7.5.9) exhibit different structures under different magnetic field and uniformity levels. As can be seen from Figs. 7.5.9a,b,c, under the effect of the same magnetic field level (10 kG), the flow cell moves towards the center with increasing non-uniformity levels, and in turn, leads to nonuniform interface shapes. This becomes more pronounced at higher magnetic field levels. At the 20 kG level, as seen in Fig. 7.5.9d,e, the flow cell splits, and the concentration level in the vicinity of the growth interface decreases. The strength of convection becomes weaker. The variation of the circumferential velocity component, v, in the horizontal plane is related to the level of mixing in the solution. Fig. 7.5.10 shows its variation along the r-direction at the middle of the solution zone (a), and near the growth interface (b). Results are presented for a 10 kG magnetic field level, and two nonuniformities, 0.02 and 0.05. As expected, at the higher nonuniformity levels, the velocity component in the horizontal plane becomes

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Fig. 7.5.9. Stream function contours under various magnetic fields and non-uniformities (t = 8 h). (a) B = 10 kG, r =  = 0.0, with min = 0.0 and max = 0.031466 (b) B = 10 kG, r =  = 0.02, with min = 0.0 and max = 0.029165; (c) B = 10 kG, r =  = 0.05; with min = 0.0 and max = 0.023287; (d) B = 20 kG, r =  = 0.0, with min = 0.001981 and max = 0.010024; (e) B = 20 kG, r =  = 0.05; with min = 0.003672 and max = 0.005951. Contour spacing is 0.0015 cm2/s in all cases (after Dost et al. [2003]).

larger, implying a better mixing in the horizontal plane. The solid line (0.05) is higher everywhere in the middle of the solution zone (see Fig. 7.5.10a). The difference is more pronounced in the vicinity of the growth interface (see Fig. 7.5.10b), although the velocity distribution is slightly different in the center of the solution zone. The maximum value of v under B = 10 KG is 0.502 cm/s. In order to see the effect of a small but rotating magnetic field, numerical simulations were repeated under 5 kG and 10 kG stationary magnetic fields for a 50 G rotating field at two different frequencies: 5 Hz and 30 Hz. Fig. 7.5.11

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shows the simulation results for the velocity component v at the middle of the solution and near the growth interface. The velocity becomes higher at the higher frequency levels, contributing further to the mixing in the solution. The velocity profile near the growth interface (Fig. 7.5.11b) is almost the same as that in the middle of the liquid zone (Fig. 7.5.11a), indicating that the rotating magnetic field may significantly enhance the mixing in the vicinity of the growth interface.

Fig. 7.5.10. Variation of the circumferential velocity component v as a function of the radial coordinate (the r-direction) (B = 10 kG, t = 8 h): (a) at the middle of the solution zone, (b) near the growth interface. Solid lines represent the case of r =   = 0.05, and dashed lines for r =  = 0.02 (after Dost et al. [2003]).

Fig. 7.5.11. Variation of the circumferential velocity component v as a function of the radial coordinate (the r-direction) under two sets of magnetic field and frequency levels (t = 8 h, r =  = 0.0): (a) at the middle of the solution zone, (b) near the substrate-solution interface. Equaldashed lines represent the case of B = 10 kG, B rot = 50 G, and  = 5 Hz; thick solid lines for B = 5 kG, Brot = 50 G, and  = 5 Hz; unequal-dashed lines for B = 10 kG, Brot = 50 G, and  = 30 Hz, and light solid lines for B = 5 kG, Brot = 50 G, and  = 30 Hz (after Dost et al. [2003]).

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Results have shown that for the THM system considered in Dost et al. [2003] higher stationary magnetic field levels are better in suppressing convection, but enhance the compositional non-uniformity in the solution. Small but unintentional non-uniformities in the stationary magnetic field do not significantly affect the convective flow in the solution, but enhance the compositional nonuniformity in the solution near the growth interface. A rotating magnetic field, on the other hand, is beneficial for mixing in the horizontal plane, and reduces compositional nonuniformity. Mixing is enhanced at higher rotating magnetic field frequencies. 7.6. The Use of Magnetic Fields in 3-D Models As mentioned earlier, the Travelling Heater Method is a technologically important solution growth technique for growing binary and ternary single crystals of alloy semiconductors such as GaSb, CdTe, CdZnTe, and InGaSb. However, due to high temperature gradients used in THM, strong convective flows develop in the liquid zone. Such flows may adversely affect the quality of the grown crystals. To minimize this effect by suppressing the natural convection in the liquid solution, the use of applied magnetic fields is a feasible option. In this section we present the recent 3-D numerical simulation models developed for the THM growth of CdTe. The issues involved in the 3-D THM modeling are addressed. Numerical results under strong magnetic fields are still conflicting, and it seems will continue to be so, until solid experimental verifications have been achieved. Depending on the purpose and the experimental set up, the stationary applied magnetic field can be selected as horizontal, or vertical, or both. In any case, as discussed in Chapter 4, the applied magnetic field gives rise to a magnetic body force that acts on the points of the liquid zone. This body force affects the structure of the fluid flow in the solution. For instance, in the growth systems considered in Okano et al. [2002], Dost et al. [2003], Liu et al. [2003], Abidi et al. [2005], and Kumar et al. [2006]), the applied stationary magnetic field aligned perfectly with the axis of the growth cell gives rise to two magnetic body force components in the horizontal plane: one in the radial direction inward, and the other one in the circumferential direction. These two horizontal force components together with the vertical gravitational body force affect the structure of the fluid flow in the solution, and consequently suppress it. The flow field can be suppressed further using higher magnetic field levels. Consideration of high magnetic fields in crystal growth may become necessary for a few reasons. For instance, in order to suppress convection significantly, down to a desired level, such as corresponding to the one in the microgravity environment, or due to the high electric resistivity of the liquid solution involved, it may become necessary to use high fields. In addition, in modeling, it may be desirable to run numerical simulations up to the highest possible magnetic field levels in order to shed light on some aspects of the

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growth process and also to determine the limitations of the use of magnetic fields. However, the use of high magnetic fields in numerical simulations exhibits great challenges in terms of numerical stability and convergence. For instance, in the THM growth of CdTe single crystals given in Liu et al. [2003] the numerical model, which used a commercial finite volume–based package (CFX), led to an interesting phenomenon. The maximum flow intensity (magnitude of the flow velocity, i.e., U max = (u 2 + v 2 + w 2 )1/2 , was given as a function of the Hartmann number ( Ha = B rc ( / μ )1/ 2 with rc the radius of the growth cell), and similar to what was observed in Bed Hadid and Henry [1996a,b], Davoust et al. [1999], and Lan et al. [2003], it was shown that the maximum velocity obeys a logarithmic law. In addition, the flow field exhibited three, distinct behaviors: a stable region up to the critical value of 8.0 kG (Ha less than 160), a transitional region with B from 8.0 to some value above 12.0 kG (Ha from approximately 160 to 250), and an unstable region with higher field levels (Ha>250). The flow velocity decreases with the increasing magnetic field in the stable region, but increases in the intermediate and unstable regions. Within the stable region, the relation between the flow intensity (Umax) and the Hartmann number obeys a power law: U max  Ha 5/ 4 . The optimum magnetic field level in this system was about 8.0 kG (Ha = 160), for the flattest interface. Such results are very significant for experimentalists for designing their experimental setups properly. The question of whether such behavior of the velocity field in crystal growth under magnetic fields is physical or just numerical still remains to be proven experimentally. In the absence of any supporting experimental data, it appears to be due to the way the field equations are solved numerically. A similar behavior in the velocity field was also observed numerically in the liquid phase electroepitaxy (LPEE) of GaAs in Liu et al. [2002]. However, in this case the LPEE experiments conducted by Sheibani et al. [2003a,b] show that there is an upper limit (about 4.5 kG) of the applied field intensity above which either the growth interface looses its stability or uneven and unsuccessful growths are observed due to strong convection in the liquid solution (see Sheibani et al. [2003a,b] for details). The computations of Liu et al. [2002], Sheibani et al. [2003b], and Liu et al. [2003] could not be carried out for magnetic fields higher than 12.0 kG (Ha = 250), due to numerical instability and convergence problems in iterations. Even for the region of Ha = 160-250, many numerical innovative techniques were used to keep the system numerically stable, and it is possible that such a transient and unstable behavior at this level (Ha = 160-250) was of numerical nature. A similar problem, the THM growth of a SiGe system, was considered by Abidi et al. [2005], and the field equations were solved using a finite elementbased commercial package (FIDAP). Computations could only be carried out up to a 2.0 kG level (about Ha = 20 with respect to diameter), and at higher magnetic field levels numerical difficulties were encountered, leading to a

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strong and unstable fluid flow, and non-convergence in the numerical iterations. Results presented in Abidi et al. [2005] predicted a similar behavior of the fluid velocity given in Liu et al. [2003] since the simulation was in the stable region of Liu et al. [2003], i.e., Ha = 20 less than 160. However, the power law obtained in Abidi et al. [2005] for the stable region was different than that of Liu et al. [2003], i.e. U max  Ha 7/4 . The optimum magnetic field level for their system was about 2.0 kG (Ha = 20) for a flatter interface. As seen in the related literature, when a flow problem is simulated under strong magnetic fields, numerical instability and problems in convergence of iterations may occur. This may lead to either conflicting or incomplete results. This is mainly due to the following. The magnetic body force components are not only functions of the square of the magnetic field intensity but they also depend on the fluid velocity components. These magnetic force terms, which are on the right-hand side of the momentum equations, are usually treated as source terms in computer codes. This is usually the source of such nonconvergence or instability. Such terms must be treated carefully. Kumar et al. [2006] have introduced a numerical technique to handle such numerical difficulties. As an application, the Traveling Heater Method was selected in this work for the growth of CdTe crystals under a static vertical magnetic field. The governing equations of the THM model, which also include the electric charge balance equation in terms of the induced electric potential and the applied magnetic field intensity, are solved numerically. Evolution of the growth interface is simulated with the help of a moving grid algorithm in a block-structured finite-volume code. The convergence rate of iterations is significantly improved by treating a part of the magnetic body force terms implicitly in the iteration loop. Consequently, the magnetic field as high as 15.0 kG (Ha = 290) did not lead to any convergence problems and unstable flows. Simulation results are presented for the flow, concentration, temperature, and electric potential fields in the liquid solution. Transport structures were stable all the way up to Ha = 290. Results also show that, in spite of the initially assumed-axisymmetric boundary conditions, three dimensional transport structures develop as soon as the growth process begins. The application of magnetic field suppresses convection in the solution. The behavior of the velocity field given in Liu et al. [2003], and also implied in Abidi et al. [2005], was not observed up to a 15.0 kG (Ha = 290) magnetic field level. Instead, the velocity field continued to decrease monotonically with the increasing magnetic field level. The code was verified by solving a benchmark problem of Ben Hadid and Henry [1996]. 7.6.1. A Model for the THM Growth of CdTe We first consider the 3-D model developed by Liu et al. [2003] for the THM growth system presented in Fig. 7.5.5a. The governing equations of the liquid

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phase (the Te-rich solution) under a stationary vertical magnetic field are given in cylindrical coordinates by Continuity

1 v w 1  ru + + =0 r  z r r

( )

(7.6.1)

Momentum

u v u u v 2 u 2 v 1 p  E 2 u +u + +w  = { 2u  2  2 }  uB r r  z r t  L r  L r r 

(7.6.2)

v v v v uv v 2 v 1 p  E 2 v +u + +w + = { 2 v  2 + 2 }  vB (7.6.3) r r  z r t  L r   L r r 

w v w w 1 p w +u + +w =  2 w  -g{t (T - T0 )   c (C - C0 )} r r  z t  L z

(7.6.4)

Mass transport

C v C C C +u + +w = DC  2C r r  z t

(7.6.5)

Energy

kL T v T T T +u + +w =  2T r r  z  L c pL t

(7.6.6)

where

2 

1    1 2 2 + r + r r  r  r 2  2 z 2

(7.6.7)

In the solid phase (the CdTe seed and grown crystal, the CdTe feed, and the quartz ampoule) the energy equation is the only governing equation, given in the form

ks T =  2T t s c ps The following boundary interface conditions are assumed. At the vertical wall

(7.6.8)

Single Crystal Growth of Semiconductors from Metallic Solutions

u = 0,

v = 0,

w = 0,

C =0 r

417

(7.6.9)

At the growth interface

u = 0,

v = 0,

w = 0,

T = T1 ,

C = C1

(7.6.10)

T = T2 ,

C = C2

(7.6.11)

At the dissolution interface

u = 0,

v = 0,

w = 0,

In the above equations, T1 and T2 are respectively the solidification temperature at the growth interface and dissolving temperature at the dissolution interface. C1 and C2 are respectively the solute concentrations at the growth and dissolution interfaces, which are determined by the interfacial equilibrium conditions given in Eq. (7.5.18). Note that isothermal conditions along the growth and dissolution interfaces are assumed because of the slow growth rate in THM, which indicates that the interface undercoolings are neglected, and interface shapes are determined by the isotherms. The thermal boundary conditions surrounding the outer ampoule (vertical wall, top surface, and bottom surface) for Eq. (7.6.8) are taken as:

ks

T = h T  T f z,t  n

( )

(7.6.12)

where h is the overall heat transfer coefficient which combines the effects of heat conduction and radiation, and is estimated by considering the measured values of the temperature difference between the furnace temperature and the temperature at the inner wall of the ampoule when it is empty; Tf(z,t) represents the furnace temperature along the ampoule wall, or at the top and bottom surfaces. Initial conditions are taken as the equilibrium state of the system under the static furnace thermal profile. The growth and dissolution interface shapes and positions are obtained iteratively by adjusting the interfaces to fit the isotherms at their equilibrium temperatures (T1 and T2). Therefore, the selection of the furnace temperature profile is very important to obtain desired interface shapes at desired equilibrium temperatures. The system is left under a constant temperature profile to reach the equilibrium state. After a certain time (long enough), state variables such as temperature, concentration and flow velocity reach a steady state at which their values are taken as the initial values. It is important to mention that, during growth, the interface shapes do not deviate significantly from their initial shapes since the contribution of latent heat was neglected due to the very slow growth rate in THM. This may not be the case for some other growth techniques (such as Bridgman) of much faster growth rates where the changes in the interface shapes may become significant

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(Liu et al. [1999]) due to the strong effect of latent heat during solidification. In the numerical simulations presented here for the THM growth of CdTe crystals, the interface shapes are kept unchanged, and only the interface locations are assumed to be changing with the movement of the furnace temperature. Such a simplification was extremely helpful in handling the difficulties associated with convergence, particularly at higher magnetic field levels. The governing equations presented above are solved numerically by CFX. Several user-defined fortran subroutines were developed and used to deal with the moving grid, and complex temperature boundary conditions. The time derivatives were calculated by the backward finite difference algorithm. Simulation Results Physical properties and system parameters used in the simulations were given in Table 7.5.1. In the table, the liquid represents the Te-rich solution, and the solid the grown CdTe crystal and the CdTe feed. Unless otherwise stated, all simulation results are presented at t = 2.0 h. The simulation results without an applied magnetic field are given in this section. These results given for the fluid flow, concentration and temperature fields in the solution constitute a basis for the three-dimensional model of the THM growth system. The computed initial temperature distribution for the whole growth system is shown in Fig. 7.6.1. The temperature gradients near the

Fig. 7.6.1. Initial temperature distribution in the entire growth crucible. Contour spacing T is 8 K, and the maximum and minimum temperatures are Tmax = 1041.9 K and Tmin = 869.1 K. The lines pointed by arrows are growth and dissolution interfaces (after Liu et al. [2003]).

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growth interface and near the dissolution interface are, respectively, about 120 and 55 K/cm. The maximum temperature difference in the solution is about 35 K. Since the interface shape is determined by the isotherm at the equilibrium temperature, the growth interface would exhibit a double-humped pattern, which agrees qualitatively with experiments, as seen in Fig. 3.4.3.

(a)

(b)

(c)

Fig.7.6.2. Flow, concentration, and temperature fields in the solution (no magnetic field) in the vertical plane at  =  (left column) and in the horizontal plane at the middle of the growth cell (right column). (a) flow field, with the maximum and minimum flow velocities 2 2 2 1/2 ( U = (u + v + w ) ) Umax = 0.0123 m/s and Umin = 0.0 m/s, and contour spacing is U = 0.0006 m/s; (b) concentration field, with maximum and minimum concentrations Cmax = 0.1306 and Cmin = 0.1198, and contour spacing is C = 0.0007; (c) temperature field, with maximum and minimum temperatures Tmax = 1020 K and Tmin = 984 K, and contour spacing is T = 5 K (after Liu et al. [2003]).

A good match between the equilibrium solidification temperature and the furnace temperature profile is very important to achieve a flatter interface. Otherwise, the growth interface may become more curved, which is undesirable. The computed shapes of the isotherms in the solution are different from those obtained from the two-dimensional simulations presented in the previous section (see Fig. 7.5.7, Dost et al. [2003]). The observed difference can be attributed to the fact that a larger ampoule diameter used in the present simulation (the diameter was changed from 2.0 to 2.6 cm) leads to a stronger convection in the solution, and influences the temperature field as well (Brandon et al. [2002]). In fact, to verify this observation, 3-D simulations were also carried out for an ampoule of smaller diameter (diameter less than 2.0 cm),

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and the results exhibited a growth interface of single-humped shape (still slightly convex to the solution) similar to that of 2-D simulations. For the sake of space, the results are not presented here. Fig. 7.6.2 shows the flow, concentration, and temperature fields in the liquid solution zone computed under no magnetic field. The flow structures are more complex than those of the two-dimensional simulations presented in Fig. 7.5.8, and similar to those, for instance, observed in Liu et al. [2002], Ben Hadid and Henry [1996], Okano et al. [2002], and others (compare with the 2-D flow structures presented in Sections 7.3., 7.4, and 7.5). The strongest flow is seen in the middle part of the crucible cell and near the symmetry axis, similar to what was observed in Fig. 7.5.8. The flow is nearly symmetric but not homocentric (Fig. 7.6.2a left column). The concentration boundary layers are formed in the vicinity of the dissolution and growth interfaces (Fig. 7.6.2b) with high concentration gradients along the z-direction inside the boundary layers due to the strong convection in the solution. The concentration distribution in the rest of the solution, outside these two boundary layers, on the other hand, is nearly uniform. In this figure, it is also noted that the concentration field shows a slight asymmetry (three-dimensional effect), indicating the sensitivity of the concentration field to varying growth conditions. The difference between the temperatures of the dissolution and growth interfaces is about 10 K. Effect of Magnetic Field The growth of CdTe crystals by THM was considered under various levels of static (vertical) applied magnetic field. The static magnetic field levels of B = 2.0, 4.0, 8.0, and 12.0 kG were selected. Higher magnetic field levels were not considered for two reasons (Liu at el. [2003]): a) the highest magnetic field level that can be applied in the experimental set-up was 12.5 kG, and b) the transport structures at higher magnetic levels become computationally unstable or do not converge. The computed flow fields in the solution under B = 2.0, 4.0, 8.0 and 12.0 kG field levels are presented in Fig. 7.6.3. Similar to the computational observations made in Liu et al. [2002] for the LPEE growth of a binary system, there is also a critical value of the applied magnetic field intensity below which the flow is suppressed and stable. This critical magnetic field intensity value in THM is higher than that of LPEE (about 2.0–3.0 kG in Liu et al. [2002]), and is about 8.0 kG. When the field intensity is less than this critical value, i.e., 8.0 kG, which was called the ‘‘stable region’’, (see Figs. 7.6.3a,b,c), an increase in the applied magnetic field intensity results in significant reduction in the flow intensity, which is, in general, desirable for a stable and controlled crystal growth. The flow in the horizontal planes is nearly homocentric. With the increase in the magnetic field, the intensity of the flow in the middle region of the growth cell along the r-direction decreases significantly (suppressed), but the flow intensity near the ampoule wall does not decrease as much, which

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results in more uniform flow across the horizontal plane. This is also desirable for the THM growth in terms of the flatness and stability of the growth interface. Flow patterns in the vertical planes show visible changes with increasing magnetic field. A large flow cell in the middle region along the radial direction splits into two smaller cells. This observation agrees qualitatively with the 2-D simulation results presented in Fig.7.5.8. It must be mentioned that the Hartmann layer observed in the 3-D simulation of the LPEE growth configuration in Liu et al. [2002] and the horizontal Bridgman configuration in Ben Hadid and Henry [1996] at high magnetic field levels is not visible here. This may be attributed partially to the strong convection observed in THM.

(a)

(b)

(c)

(d)

Fig. 7.6.3. Flow field under various magnetic field levels in the vertical plane at  =  (left column) and in the horizontal plane at the middle of the growth cell (right column): (a) B = 2.0 kG, with Umax = 0.00774 m/s and Umin = 0.0 m/s; (b) B = 4.0 kG, with Umax = 0.00351 m/s and Umin = 0.0 m/s; (c) B = 8.0 kG, with Umax = 0.00153 m/s and Umin = 0.0 m/s; and (d) B = 12.0 kG, with Umax = 0.0565 m/s and Umin = 0.0 m/s. Contour spacing is U = 0.0006 m/s (after Liu et al. [2003]).

At higher magnetic field levels, above the critical value, which we have selected the level of B = 12.0 kG (Fig. 7.6.3d), the flow intensity increases dramatically similar to that observed in Liu et al. [2002] (in the LPEE growth of GaAs) in the transitional region where the flow structures have tendency towards instability. In fact, above a certain magnetic field level, they become unstable. This behavior of the fluid flow in THM could be numerical, as will be discussed later in this section. However, this was explained in Liu et al. [2002] as follows. At lower magnetic field intensity levels, the two magnetic body force components (radial and circumferential) and the vertical buoyancy force act on the points of the liquid solution, and in turn the combined effect of these body

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force components (two magnetic and one gravitational) suppresses the fluid flow to provide a prolonged and stable growth. However, when the magnetic field level exceeds a critical value, which is about 8.0 kG in this case, the two magnetic body force components become larger and consequently the delicate balance between the magnetic and gravitational body force components is altered in favor of a very strong convection in the solution. This strong convective flow may also adversely affect the concentration distribution, and lead to an unsatisfactory growth and an unstable growth interface. As can be seen from Fig. 7.6.3d, at the 12.0 kG level, the flow field in the horizontal planes is no longer homocentric and becomes very non-uniform. Very strong flow cells appear in four locations, near the ampoule wall while the flow in the rest of the solution remains weaker. Note that the flow is still stable under this condition, implying that the magnetic field level of 12.0 kG is about the border between the transitional and the unstable regions. The computations of Liu et al. [2003] for magnetic field levels higher than 12.0 kG did not converge. As we will discuss this issue later in this section, the difficulty may be just numerical, in other words, it can be fixed numerically as shown in Kumar et al. [2006], but it may also be experimental as is the case for the LPEE growth in which experiments become unstable above 4.5 kG (Sheibani et al. [2003a,b]) due to the strong interaction of the applied magnetic and electric fields. The issue in the THM growth of CdTe crystals still remains to be verified experimentally.

(a)

(b)

(c)

Fig. 7.6.4. Change of temperature distribution with time at B = 20 kG: (a) t = 1 h; (b) t = 2 h; and (c) t = 3 h. In this figure, Tmax = 1025 K, Tmin = 979 K, and T = 5 K. The computation was carried out under fixed interfaces without considering the concentration field (after Liu et al. [2003]).

The computational difficulty of Liu et al. [2003], which used CFX, may also be attributed to the extreme sensitivity of the concentration field under strong convection (Ghaddar et al. [1999] and Liu et al. [2002]) and to the coupling of the free moving interfaces with possible unsteady flows in the solution. However, for a computational run at the magnetic field strength of B = 20 kG under the conditions of fixed interfaces and without considering the concentration field, the computations were also unstable, and consequently no results were obtained for the concentration field (Liu et al. [2003]). It was

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possible to compute only the flow and temperature fields in the solution. Here, only the computed temperature field is presented (Fig. 7.6.4) for the purpose of providing a feeling for the flow and concentration fields. The temperature field shows significant changes over time, and implies the possibility of strong and unstable behavior of the flow and concentration fields. The influence of the magnetic field on flow intensity is summarized in Fig. 7.6.5. The maximum intensity of the flow field is given as a function of the Hartmann number ( Ha = B rc ( / μ )1/ 2 with rc being the radius of the growth cell, Ben Hadid and Henry [1996a,b]). Similar to what was observed in Davoust et al. [1999], the maximum velocity obeys a logarithmic law, and the flow field exhibits three distinct behaviors: a stable region up to the critical value of 8.0 kG (Ha < 160), a transitional region with B from 8.0 to some value above 12.0 kG (Ha from 160 to 250 (approximately)), and a unstable region with higher field levels (Ha > 250). The flow velocity decreases with an increase in the magnetic field intensity in the stable region, but in the intermediate and unstable regions the flow velocity increases. Within the stable region, the relation of the flow intensity (Umax) and the Hartmann number obeys a power law of

U max  Ha 5/4

(7.6.13)

which has been demonstrated by Ben Hadid and Henry [1996a,b], and Davoust et al. [1999] with a slightly different index of the power law, and by and Liu et al. [2002]. It must be mentioned that the curve given in Fig. 7.6.5 resembles the curve

Fig. 7.6.5. The variation of the maximum velocity (Umax) of the flow field as a function of the Hartmann number (Ha) (after Liu et al. [2003]).

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given in Liu et al. [2002] for the LPEE growth of GaAs in terms of its overall behavior and also in terms of the Hartmann number (although slightly different in Ha). However, these two figures are very different quantitatively in terms of the applied magnetic field and flow intensity levels since they represent the growth of two different materials by two different techniques. Also, the definition of the Hartmann number in each case is different, namely Ha is defined with respect to the radius of the crucible in this work while it was defined with respect to the solution height in Liu et al. [2002] for convenience. However, the overall behavior of the flow field under magnetic field is the same in both LPEE and THM for the systems considered. Fig. 7.6.6 shows the influence of magnetic field on the concentration field in the solution. When the field is less than 8.0 kG (within the stable region), with the increase of the magnetic field intensity, the boundary layers appearing in the vicinity of the interfaces become thinner, and the concentration cells in the middle region of the solution domain are formed and become intensive (see Fig. 7.6.2b and Figs. 7.6.6a–c) due to the suppressed flow field. The maximum concentration difference in the horizontal plane in the vicinity of the growth interface does not show significant change with varying magnetic field, but the concentration distribution exhibits notable changes, and the non-uniformity of the concentration distribution increases with the magnetic fields (see the left column of Fig. 7.6.6a–c). These results are in agreement with the changes in the flow field, and also with the results of Dost et al. [2003] presented in Section 7.5. Under the B = 12.0 kG (Fig. 7.6.6d) field, strong convection causes very large concentration differences along the radial direction in the vicinity of the growth interface, which will likely lead to a serious radial segregation along the growth interface. The non-smooth waggles appearing in Fig. 7.6.6d suggest computational instability at this magnetic field intensity level. The distribution of concentration gradients (computed according to dC/dz) in the vicinity of the growth interface were computed at various magnetic field levels, and the results are shown in Fig. 7.6.7a. As can be seen, the concentration gradient distribution is the most uniform at B = 8.0 kG, the level for which the convection is most suppressed (Fig. 7.6.5). If one considers other concentration gradients, both shallower and steeper ones, it is easy to see that these cases will cause a mismatch between the travelling heater rate and the growth rate. It appears that for the THM growth system considered here for the growth of CdTe crystals, the magnetic field level of B = 8.0 kG is the optimum. Since the interface undercooling is assumed to be negligible, because of the low growth rate in THM, the growth interface would be along the isotherm at the equilibrium solidification temperature. Therefore, by computing the maximum differences in the z-direction (dz) along the isotherms at the growth interface, one can measure the interface deflection (Kim et al. [1988]). These differences were plotted in Fig. 7.6.7b at various field intensity levels. As can be seen, the interface is the flattest at B = 8.0 kG, which indicates once more

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that the optimum magnetic field intensity is about B = 8.0 kG for the THM setup considered here.

(a)

(b)

(c)

(d)

Fig. 7.6.6. Concentration field under various magnetic fields in the vertical plane at  =  (left column) and in the horizontal plane near the growth interface (right column) (Cmax = 0.1307; C min = 0.1190; and contour spacing C = 0.0007): (a) B = 2.0 kG; (b) B = 4.0 kG; (c) B = 8.0 kG; and (d) B = 12.0 kG (after Liu et al. [2003]).

Fig. 7.6.7. (a) Concentration gradient distribution in the immediate vicinity of the growth interface under various magnetic field intensities, and (b) The deflection of the growth interface with the change of the applied magnetic field strength (after Liu et al. [2003]).

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(a)

(b)

(c)

(d)

Fig. 7.6.8. Thermal field under various magnetic fields in the vertical plane at  =  (Tmin = 983 K, Tmax = 1020 K, and contour spacing T = 8 K): (a) B = 2.0 kG; (b) B = 4.0 kG; (c) B = 8.0 kG; and (d) B = 12.0 kG (after Liu et al. [2003]).

The influence of magnetic field on the temperature field is given in Fig. 7.6.8. Note the changes in isotherms along the growth interface. At the field levels less than 8.0 kG (Figs. 7.6.8a–c), the isotherms maintain the double-humped patterns, but they become flatter with the increasing magnetic field intensity. At B = 12.0 kG, the isotherm along the growth interface becomes a single-humped, and convex to the solution, indicating that the shape of the growth interface is greatly influenced by the applied magnetic field. Kim et al. [1988] and Brandon et al. [2002] suggest that changes in isotherms are mainly due to the changes in the flow field being in agreement with the simulation results presented in Liu et al. [2003]. 7.6.2. Control of the Growth Interface Shape In THM, the control of the growth interface shape is very important. Generally a flat or slightly concave interface towards the solid is desirable, to prevent solvent inclusions and polycrystalline growth. However, the control of the shape of the growth interface in THM is very difficult since the evolution of the growth interface is influenced by convection in the solution and also by the heat and mass transfer in the vicinity of the growth and dissolution interfaces. In order to investigate this issue, a three dimensional numerical simulation study was carried out to determine the conditions for an optimum interface shape using various thermal boundary conditions and also by introducing a crucible rotation. The model equations and the associated interface and boundary conditions are the same as those given in the previous section (Eqs. (7.6.1)-(7.6.12)), except for the conditions related to crucible rotation and the heat removal at the bottom of the THM ampoule. The velocity field at the vertical wall is given by

u = 0,

v = 2  r,

w=0

(7.6.14)

where  is the crucible rotation rate, and the thermal boundary condition at the bottom of the crucible is taken as

Single Crystal Growth of Semiconductors from Metallic Solutions

ks

T = hb h T  T f z,t  Tb  n

( )

427

(7.6.15)

where hb and Tb are the constants that determine the degree of heat removal. Two cases are considered: a) a relatively low level of heat removal (hb = 1 and Tb = 110), and b) a high level heat removal hb = 50 and Tb = 110. The initial conditions are taken as the equilibrium state of the system when the furnace thermal profile is stationary. The growth and dissolution interface shapes, and the positions of the interfaces are obtained iteratively by adjusting the interfaces to fit to the isotherms at their equilibrium temperatures. Therefore, the selection of the furnace temperature profile is very important to obtain desired interface shapes at desired equilibrium temperatures. After a certain time, the temperature, concentration, and velocity fields become steady which are taken as the initial values in the computations. Simulation Results The flow, concentration, and temperature fields computed for a stationary system (at the beginning of the growth process) have shown that the growth interface is of a double-humped shape (Figs. 7.6.1-7.6.3). A slight asymmetry was observed in the concentration field. These transport structure values are used in our computations as the initial conditions. As mentioned earlier, a flat (or slightly concave interface) is desirable. In order to achieve this objective, two options were considered. The first one is to change the heat transfer conditions at the bottom surface of the ampoule (as done in experiments, Fig. 3.4.3). In this case, the computed interface shapes and thermal fields are presented in Fig. 7.6.9. As seen, with the removal of heat at the bottom of the ampoule, the shapes of the isotherms near the growth interface change. Indeed, simulations show that by controlling the heat removal at the bottom, the growth interface can be made flatter or slightly convex to the solid. As the growth proceeds and the grown crystal gets thicker, the heat removal would become less effective, unless the thermal conditions are adjusted. However, as we know from the simulations, in the absence of such a heat removal (Fig. 7.6.10) the growth interface becomes flatter as the growth proceeds. Therefore, by controlling the heat removal at the bottom of the crucible, the desired growth interface shape can be maintained throughout the growth process. Fig. 7.6.10 presents the time evolution of the growth interface. As seen, the simulation results show that as the growth interface evolves. the thermal field of the system changes and the growth interface becomes flatter. This implies that by changing the thermal characteristics of the system, the growth interface can be made slightly concave (towards the crystal) starting almost at the beginning of the growth process. We note that the interface shape at the beginning of the growth is almost the same as that given in Fig. 7.6.1, indicating that the simple

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isotherm treatment used at the growth interface in Liu et al. [2003] led to sufficiently accurate predictions.

(b)

(a)

(c)

Fig. 7.6.9. Evolution of the thermal field (at t = 10 min growth time): (a) no heat removal, (b) low heat removal (hb = 1 and Tb = 110 in Eq. (14)), (c) large heat removal (hb = 50 and Tb = 110 in Eq. (14)). Contour spacing is T =10 K, and the maximum and minimum temperatures are: (a) Tmax = 1041 K and Tmin = 859 K, (b) Tmax = 1041 K and Tmin = 838 K, and (c) Tmax = 1041 K and Tmin = 750 K.

(a)

(b)

Fig. 7.6.10. Evolution of the growth interface: (a) with no rotation, and (b) at the rotation rates of (1)  = 3 rpm, (2)  = 5 rpm, and (3)  = 7 rpm.

This is because the growth velocity in THM is very small; consequently we observe small concentration changes in the solution, making the effect of the constitutional supercooling negligible (Lan and Yang [1995]).

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The effect of convection was also numerically examined by Lan and Yang

Fig. 7.6.11. The effect of convection in the THM growth of CdTe with a small volume of solvent under three levels of gravity: (a) and (d) at g = 0, (b) and (e) at g = 10-2 g0, and (c) and (f) at g = g0 (U0 = - 7.510-6 cm/s) (after Lan and Yang [1995]).

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[1995] for a CdTe system. It was found that in the system considered supercooling is reduced by convection. Results were presented for three gravity levels (Fig. 7.6.11). An optimum growth interface is obtained under the rotation rate of 5 rpm, which agrees with the results of the two-dimensional simulations of Okano et al. [2002].

(a)

(c)

(b)

(d)

(e)

Fig. 7.6.12. The computed flow (left column) and concentration (right column) fields in the solution at (a)  = 3 rpm, (b)  = 5 rpm, and (c)  = 7 rpm. In (a) the maximum and minimum flow strengths are Umax = 0.0119 m/s and Umin = 0.0 m/s, and the maximum and minimum concentrations are Cmax = 0.1286 and Cmin = 0.1199. In (b) U max = 0.0117 m/s and Umin = 0.0 m/s, and Cmax = 0.1298 and Cmin = 0.1219. In (c) Umax = 0.0126 m/s and Umin = 0.0 m/s, and Cmax = 0.1303 and Cmin = 0.1222. Contour spacing is 0.001 m/s for the flow field and 0.001 for the concentration field. The computed flow (d) and concentration (e) fields in the solution at  = 15 rpm. In (d) Umax = 0.0198 m/s and Umin = 0.0 m/s, with contour spacing 0.001 m/s; and in (e) Cmax = 0.1271 and Cmin = 0.1209, with contour spacing 0.001.

The computed flow and concentration fields under three rotation rates are shown in Fig. 7.6.12. With the increase in the rotation rate, the flow field becomes stronger near the vertical wall. The concentration field becomes uniform everywhere in the solution except in the region near the dissolution interface. This could be beneficial for growing crystals with low point defect concentrations, but might be not favorable at higher growth rates due to the shallow concentration gradient near the growth interface, which may lead to solvent inclusions in the grown crystal.

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In order to make the study more complete, the thermal and concentration fields were also computed under a high rotation rate: 15 rpm. In this case, the tangential flow velocity component near the vertical wall of the crucible is nearly two times that without rotation. Flow instability may also occur since the flow structures clearly become asymmetric. The concentration field in the solution exhibits layer-like distributions at the whole region, which reflects the domination of the circumferential flow. 7.6.3. A Finite Element Model for the THM Growth of SiGe Abidi et al. [2005] carried out 3-D steady-state finite-element numerical simulations to study the effect of an axial magnetic field on the fluid flow, heat and mass transfer in the Ge0.98Si0.02 solution of a THM system of SiGe. In the model, in the THM crucible the Ge0.85Si0.15 polycrystalline source is placed at the top, a solid Ge0.98Si0.02 rod was melted to act as the solvent in the middle, and a Ge seed is located at the bottom (Fig. 7.6.13). In the liquid phase, the Ge-rich solution, the steady-state governing equations, namely the continuity, momentum, energy, and mass transport equations, were solved numerically using the finite element technique (Eqs.(7.6.1) -(7.6.6) in the absence of time dependency). The phase diagram is taken into account in order to determine the silicon concentration at the growth interface. A mesh sensitivity analysis was performed to find the optimum mesh size for accurate

Fig. 7.6.13. Schematic view of (a) the applied temperature profile and (b) the THM growth system of SiGe (after Abidi et al. [2005]).

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Fig. 7.6.14. The finite element mesh (after Abidi et al. [2005]).

results with a minimum computational power storage. An external vertical magnetic field was considered. By increasing the magnetic field intensity. it was observed that the intensity of the flow at the center of the solution zone reduces at a faster rate than near the wall. This phenomenon creates a stable and flatter growth interface, and the silicon distribution in the horizontal plane becomes relatively homocentric. The maximum flow intensity was found to obey a logarithmic law with respect to the Hartmann number ( U max  Ha 7/4 ) in this SiGe THM system. The following boundary and interface conditions are used. At the ampoule wall, and the growth and dissolution interfaces no-slip boundary (flow field) and zero-concentration gradient conditions are imposed. At the dissolution and growth interfaces, equilibrium concentrations are assumed, i.e., respectively

C = C1 = 0.015 , and C = C2 = 0.02

(7.6.16)

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where C1 and C2 are the concentrations of Si at the dissolution and growth interface, obtained from the Ge-Si phase diagram given in Olesinski and Abbaschian [1948]. The governing equations were solved numerically using the Galerkin finite element technique with eight-node hexahedral elements. The unknown radial, angular, and axial velocity components, and temperature and concentration are evaluated at each node of the element. The finite element mesh of the model is shown in Fig.7.6.14. In order to handle the complex flow in the solution, a finer mesh is used in this region. The segregated algorithm has been used to solve the nonlinear set of algebraic equations. A linear approximation for the pressure using the penalty method is adopted. The field equations are solved simultaneously. Convergence was reached when the difference between two consecutive iterations is less than 10-4. Different meshes were tested, and the ideal mesh was chosen according to an average Nusselt number across the solution (see Abidi et al. [2005] for details). Table 7.6.1. Physical parameters of the SiGe system (after Abidi et al. [2005]). Ge0.85Si0.15

Ge0.98Si0.02

Ge

cp

0.04008 J g  k

cp

0.04008 J g  k

cp

0.0390 J g  k

Lf

39 cal/g

Tm

944.7 °C

Tm

935 °C

0.52  10 cm s

c

1.0  10 cm s

2.6  10 cm /s

2

c

0.005 1/at%Si

c

0.005 1/at%Si

0.005 1/at%Si

t

1.110 1/°C

t

1.010-4 1/°C



0.2995 W cm  k



0.25 W cm  k

μ

8.5  10 g cm  s



1.5424  10 cm s



5.51g cm

1100 °C

c

4

4

t

1.1  10 1/°C



0.2905 W cm  k

μ

7.4  10 g cm  s



1.46  10 cm s



4

c

Tm

c

4

3

3

2

5.06785 g cm

3

2

-4

3

μ

8.3496  10 g cm  s



1.53192  10 cm s



5.4504 g cm



2.5  10 S cm

3 3

2

2

3

3

2

3

4

Three different cases were examined for the THM process of Ge0.98Si0.02 using the values for the properties presented in Table 7.6.1. First, the effect of natural convection is studied at 1 g0 (Earth’s gravitational field). In the second case, the gravity level is taken as g =10-4g0. Finally, a static vertical magnetic field is considered at 1 g0. The silicon distribution in the solution is examined at three different planes: in the vertical plane (r, = 0, z), near the growth interface (r,, z = 1.575 cm), and at the middle region (r, , z = 2.0 cm).

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Fig. 7.6.15. Silicon composition and flow velocity contours in the solution in the vertical plane (top) and in the horizontal plane (middle) and their variation along the r-direction (bottom) (after Abidi et al. [2005]).

Simulation Results Under Normal and Microgravity Levels Fig. 7.6.15a shows the computed silicon distribution in the vertical plane near the growth interface. The application of the temperature profile in the solution zone results in two flow cells circulating counter-clockwise, forcing the flow upward along the crucible wall where the temperature is the highest, and downward along the central axis where the temperature is lower. As shown, the

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silicon is following the flow pattern by moving downward from the silicon rich concentration near the dissolution interface towards the growth interface. In the microgravity case, due to the lack of natural convection, one observes a linear distribution of silicon, and the process is diffusion limited. The silicon distribution in the horizontal plane in the middle of the solution is uniform, as seen. The distribution is non-linear under 1 g0, due to convection. If the solute distribution near the interface under the 10-4g0 level is examined, one may see that it is improved, but still nonuniform since this gravity level was still not low enough. However, at this level, the convective flow has almost vanished. Two boundary layers are formed near the dissolution and growth interfaces. Under normal gravity, there are relatively high concentration gradients along the wall and the center of the solution zone, due to convection (Fig. 7.6.15a). Fig. 7.6.15b shows the flow field in both cases. The flow is nearly axisymmetric, and homocentric for both cases. A relatively strong flow cell is formed in the middle region near the centre and two weaker cells near the wall. Under normal gravity, the strongest flow cell has the magnitude of 0.4796 cm/s. For the microgravity condition, the same behavior is observed but with less intensity, at 0.000168 cm/s.

Fig. 7.6.16. Silicon distributions under different magnetic field intensities in the vertical plane (top), and at the horizontal plane near the growth interface (bottom) at z = 1.575 cm (C = 1.3%) (after Abidi et al. [2005]).

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Results Under Magnetic Field Fig. 7.6.16 presents the computed silicon distribution in the liquid for different magnetic field intensity levels. At B = 0.2 kG, the concentration distribution is more axisymmetric, and the concentration contours are spread more uniformly, compared to the case with no magnetic field. The concentration gradient near the wall is also less than that in the case of no magnetic field. With an increase in the magnetic intensity, the concentration distribution becomes smoother, and the concentration contours become flatter (top). This represents a more uniform growth. By examining the concentration distribution in the horizontal plane near the growth interface, the contours indicate that the distribution becomes more homocentric with an increase in the magnetic field intensity level. Fig. 7.6.16d shows a complete homocentric silicon distribution near the growth interface, which is desirable for crystal growth.

Fig. 7.6.17. Flow patterns under different magnetic field intensities along the vertical plane at centre axis of the sample (top) and the variation of the velocity along the radial direction (bottom) in the middle of the solvent(at z = 2.0 cm) (after Abidi et al. [2005]).

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As seen in Fig.7.6.16b, the silicon concentration near the growth interface is maximum in the middle (in horizontal plane) being 9.15% at B = 0.2 kG, and decreases with increasing magnetic field intensity to 6.55% at B = 0.6 kG. When the magnetic field intensity is increased to 1.0 kG, the silicon distribution does not show a significant change but the concentration contours become homocentric. Fig. 7.6.17 presents the computed flow patterns in the solution at B = 0.2, 0.6, 1.0, and 2.0 kG, in the vertical plane at the center of the growth cell (top) and along the radial direction at the middle of the growth cell (bottom). As mentioned earlier, three cells are observed in the solution: a relatively strong one located in the middle of the growth cell (in the vertical plane), and two weaker cells near the ampoule wall for the case of weak magnetic field intensities (see Fig. 7.6.17a). The strongest flow cell is located close to the growth interface rather than near the dissolution interface. This is because of the influence of gravity. This implies that the flow was not suppressed significantly at this magnetic field level. The magnitude of the maximum flow velocity at B = 0.2 kG is 0.19635 cm/s, which is less than the case with no magnetic field. The other two cells close to the wall have less flow intensity than the middle cell (Fig. 7.6.17a-bottom). At B = 0.6 kG, the strong cell in the middle becomes weaker but the flow intensity of the two cells close to the wall does not drop as significantly as the middle cell. Therefore the behavior of the flow changes along the radial direction (Fig. 7.6.17b-bottom). The magnitude of the maximum flow velocity

Fig. 7.6.18. Silicon concentration distribution near the growth interface under various gravity and magnetic field intensity levels (after Abidi et al. [2005]).

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Fig. 7.6.19. Variation of flow intensity with the Hartmann number (after Abidi et al. [2005]).

in the two cells close to the wall becomes higher than the cell located in the middle. The two cells close to the wall move towards to the wall, and form the Hartmann layer. As B increases beyond 0.6 kG, the Hartmann layer becomes thinner. This phenomenon confirms the observations of Liu et al. [2002], and Okano et al. [2002]. This behavior in flow intensity is desired since it gives rise to a stable and flatter growth interface. In Fig. 7.617d, the cells still are closer to the growth interface rather than the dissolution interface, which shows that the fluid flow is still not completely suppressed. Fig. 7.6.18 shows the computed silicon distribution near the growth interface along the radial direction at different levels of magnetic field and gravity. The simulation results confirm that, as expected, the silicon distribution is the most uniform under the reduced gravity condition. The variation of the maximum flow velocity with the Hartmann number is given in Fig. 7.6.19. The maximum velocity is found to obey a logarithmic law as U max  Ha 7/4 where Ha is defined with respect to the diameter of the growth ampoule. Computations could only be carried out up to about Ha = 20, and at higher magnetic field levels numerical difficulties were encountered, leading to a strong and unstable fluid flow, and non-convergence in iterations. In drawing conclusions from the results of Liu et al. [2003] and Abidi et al. [2005], one must examine the differences between these two models. First, the diameter of the growth ampoule used in the model of Abidi et al. [2005] is about the half size of the ampoule used in Liu et al. [2003]. As mentioned in the previous section, in comparison of the results of the 2-D model of Dost et al. [2003] with that of the 3-D model of Liu et al. [2003], significant differences in flow structures were observed. This was due to the smaller size of the growth

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ampoule used in Dost et al. [2003] which is almost the same as that of Abidi et al. [2005]. When Liu et al. [2003] ran 3-D simulations for the small size ampoule for the same material (CdTe), the flow structures were almost the same as those given in Dost et al. [2003]. Therefore, results show that the crucible size is an important factor, and one must be careful in comparing results from the simulations of different crucible sizes. The behavior of the maximum flow velocity with the magnetic field intensity of Abidi et al. [2005] is similar to that of Liu et al. [2003]. This is because in the simulation of Abidi et al. [2005] the maximum value of the Hartmann number is 20. This value is much less than 160, which is the maximum Hartmann number of the stable region of Liu et al. [2003]. However, the power law obtained by Abidi et al. [2005] for the stable region was different than that of Liu et al. [2003]. The optimum magnetic field level for their system was about 2.0 kG (Ha = 20) for a flatter interface, in comparison with a 0.8 kG field level in Liu et al. [2003]. One may also argue that the difference in the optimum magnetic field levels is due to the difference in the sizes of the growth ampoules. Finally, we must also mention that the physical parameters are different in the CdTe and SiGe systems used in the simulations. 7.6.4. Numerical Simulations Under High Magnetic Field Levels As mentioned in the beginning of this section, when a flow problem is simulated under strong magnetic fields, numerical instability and problems in iteration convergence may occur. This is particularly true in 3-D simulations using commercial packages of which the source codes are not available to the user. The magnetic body force components are not only functions of the square of the magnetic field intensity, but they also depend on the fluid velocity components. These magnetic force terms, which are on the right-hand side of the momentum equations, are usually treated as source terms in commercial computer codes. This is usually the source of such non-convergence or instability. Kumar et al. [2006] have introduced a numerical technique to handle such numerical difficulties. The THM system simulated in Liu et al. [2003] is considered for comparison, and an in-house computer code (FASTEST-3D) was used. The material and physical properties are the same as those given in Table 7.5.1. The governing equations of the model, which also include the electric charge balance equation in terms of the induced electric potential and the applied magnetic field intensity, are written in integral forms, as follows Continuity

V

 L dV +   LU i dSi = 0 S t

(7.6.17)

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Momentum



V t (LU j )dV + S LUiU j dSi = S μ +

V

 L g j T (T  Tref )dV + 

V

U j xi

dSi  

p dV V x j

 L g j C (C  Cref )dV

 +  { E  jqr ( +  qmnU m Bn ) Br }dV V xq  

(7.6.18)

Jq

Energy



T

V t (L c pLT )dV + S LUi c pLTdSi = S kL xi dSi

(7.6.19)

Mass transport



C

V t (L C)dV + S LUiCdSi = S L DC xi dSi

(7.6.20)

Electric charge balance



S  E xi dSi = S  E  ijkU j Bk dSi

(7.6.21)

where  ijk denotes the permutation symbol which is equal to 1 for the positive permutation, to –1 for the negative permutation, and to zero if any two indices are equal. In the case of moving grids, the volume and the surface area of the control volume are not constant in time, and hence the first terms of the left hand side can be modified according to the Leibniz rule



d

V t (L )dV = dt V L dV  S LUi dSi g

(7.6.22)

where U ig is the grid velocity and  represents any transport variables (i.e., Ui, or T, or C). By the application of the Leibniz rule, the integral of rate of change of a quantity in a fixed shape control volume is changed into two parts: the rate of change of the quantity with the deformation of the control volume, and the convective fluxes arising due to the movement of the control volume boundaries. By using the Leibniz rule in Eqs. (7.6.17)-(7.6.20), a generic transport equation for a transport variable  can be written as:

d dt



V L dV + S  L (Ui  Ui )dSi = S   xi dSi + V Q dV g

(7.6.23)

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where   and Q represent respectively a material coefficient and the source term for the transport quantity  (their values are given in Table 7.6.2). In the solid phase, the energy balance and electric charge balance are the governing equations written respectively as

d dt

T

V s c psTdV  S sUi c psTdSi = S ks xi dSi g

(7.6.24)

and



S  E xi dSi = S  E  ijkU j Bk dSi

(7.6.25)

with the appropriate material coefficients. The boundary and the interface conditions are the same those given in the previous section for the problem of Liu et al. [2003]. For the electric potential boundary condition, the continuity of the electric current is employed. Table 7.6.2. Variables in the generic transport equation, Eq.(7.6.23)

Conserved quantity





Q

Mass

1

0

0

Momentum

Uj

μ



P +  g j T (T  Tref ) x j

+  g j C (C  Cref ) +  jqr J q Br Energy

c pL T

kL

0

Solute mass

C

DC

0

In order to estimate the fluxes due to grid movement at an interface, the interfacial undercooling is neglected, and therefore the interfacial energy balance or Stefan-condition is applied (see Alexiades and Solomon [1993] for details):

1  T T  L HU iI +  L (1  L )2 (U iI )3 = ks  kL 2 xi s xi s

(7.6.26) L

where H is the latent heat of fusion and U iI are the liquid-solid interface velocity components. Since the liquid and solid densities at the interface temperature are close to each other, the influence of density change on interfacial velocity is negligible and therefore the second term on the left hand side of the above equation can be neglected. It should be emphasized that the

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interface velocity computed from the Stefan condition should not be directly utilized in the above mentioned conservation equations (for U ig ) otherwise mass, momentum, species and energy sources or sinks will be generated in the respective equations (see Demirdzic and Peric [1988, 1990] and Ferziger and Peric [1999]). Due to the discretization errors, for instance in the case of incompressible flows, the grid flux term may not cancel out of the unsteady term to yield a divergence free velocity, i.e.

S LUi dSi = 0 This inconsistency arising from the discretization of the equations introduces constraints on the size of the time step of integration and can be avoided by taking a so-called space conservation law (SCL) into account. This was pointed out initially by Thomas and Lombard [1979], and later Demirdzic and Peric [1998,1990] who have shown the necessity of solving the space conservation law for arbitrary shaped domains in the case of moving boundaries. The space conservation law has the following form:

d dt

V dV  S Ui dSi = 0 g

(7.6.27)

The interfacial velocity from the Stefan balance ( U I ,i ) is only utilized to recreate numerical grids for the domain. For the crucible outer boundaries (top, bottom and side wall) the thermal boundary condition given in Eq. (7.6.12) is applied, where h = 220 W/m2 K is the heat transfer coefficient measured from experiments, and the furnace temperature profile is the same as given in Fig. 7.5.5a. Numerical Methodology The balance equations are presented in the Cartesian form, however, solution of the equations is sought on curvilinear body-fitted grids. Hence, the terms involving gradients and divergence operators were transformed from the Cartesian to the curvilinear coordinates. The coupling between the velocities and mass in the momentum field equations was treated with the SIMPLE algorithm of Patankar and Spalding [1972] where a pressure-correction equation is solved to correct both pressure and velocity fields. The discretized continuity equation needs the velocity of the fluid at the cell faces, and if this is carried out by a linear interpolation of velocity in the case of collocated grids, non-physical checkerboard velocity and pressure fields may occur. Therefore, a velocity interpolation proposed by Rhie and Chow [1983] was taken into account (Fig. 7.6.20). The term involving the grid velocities in the transport equations needs special care. This is due to the fact that the treatment of change of volume due to the grid movement may easily give rise to numerical source or sink terms in the continuity equation, which may then cause a convergence problem in the

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443

pressure correction equation. Furthermore, in the case of high magnetic fields, the convergence rate may deteriorate significantly due to the strong negative source terms. Therefore, the source terms in the momentum equations requires a special treatment, and this issue is discussed below. The magnetic body force components were given in Chapter 4 in Section 5, Eqs. (4.5.7) (we write here again for convenience) as

F1 =  E (1 + B2 v2 + B3v3 )B1   E (B22 + B32 )v1 F2 =  E ( 2 + B1v1 + B3v3 )B2   E (B12 + B32 )v2

F3 =  E ( 3 + B1v1 + B2 v2 )B3   E (B12 + B22 )v3     exp licit

(7.6.28)

implicit

Here, by implicit one means that the coefficient of the extreme right is taken to the left hand side of the discretized momentum equations. These coefficients on the right hand side will always be negative, and therefore if kept in the source terms, they are likely to create tremendous convergence problems or the solution may not converge at all in some cases. In order to avoid possible convergence problems, the implicit part should be moved to the left hand side of the algebraic system of equations. The discretized momentum equations without the magnetic body force terms for an arbitrary control volume P may be written as

Fig. 7.6.20. The 3-D grid system used (after Kumar et al. [2006]).

Sadik Dost and Brian Lent

444

aP  P +  anb  nb = QP

(7.6.29)

nb

where  represents the unknown components U1 , U2, and U3, and a stand for the coefficient of the unknown, nb refers to the neighbors, and QP is the source term or the explicit part of the equation in concern. After adding the magnetic body force terms into the momentum equations, the coefficient and source terms can be updated according to Eq. (7.6.28). For instance, the first components of the discretized equations are modified as:

aP  aP +  E (B22 + B32 ) , and QP  QP +  E (B2U 2 + B3U 3 + A1 )B1

(7.6.30)

By following the above-described procedure, the convergence rate was improved significantly in the present numerical simulation. Nevertheless, one should bear in mind that in case the explicit part of the magnetic body force dominates, and moreover becomes negative, the convergence problem may still arise. In such situations, one has to heavily under-relax the solution to get a converged solution. However, this may increase the computational time considerably. This can be noticed in Fig. 7.6.21a,b which presents the comparison between the convergence rate of iterations at arbitrary time steps for the case with a vertical magnetic field of 6.0 kG. A complete treatment of the magnetic body force components explicitly increases the total number of iterations by a factor of three. Kumar et al. [2006] did not obtained a converged solution at all for a magnetic field above 8.0 kG without treating the abovementioned part of the magnetic force implicitly. This level of magnetic field

Fig. 7.6.21a. The comparison of rate of convergence between the fully explicit treatment of the magnetic body force components (after Kumar et al. [2006]).

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Fig. 7.6.21b. The comparison of rate of convergence between the fully implicit treatment of the magnetic body force components (after Kumar et al. [2006]).

(8.0 kG) is in agreement with that of Liu et al. [2003] where above this magnetic field level strong and unstable solutions were observed. This shows that the strong and unstable convective flow observed in Liu et al. [2003] and Abidi et al. [2005] at higher magnetic field levels might have been of numerical origin. Kumar et al. [2006] has also observed that the contribution of the gradient of electric potential in the magnetic body force is smaller than the other terms, and therefore hardly influences the convergence rate. Approximation of Fluxes The space conservation law is utilized in order to evaluate the fluxes appearing in the conservation equations due to the movement of grids. It has already been shown in several studies, e.g. Demirdzic and Peric [1988, 1990] and Thomas and Lambard [1979] that the space conservation law (SCL) must be fulfilled within a numerical computation with moving grids so that the problem of mass sink or mass source can be avoided in the discretized equation of mass. The space conservation law can be discretized for a first-order time integration scheme

 (Uig Si ) f = f

V n +1  V n t

and for a second order time integration scheme

(7.6.31)

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 (Uig Si ) f = f

3V n +1  4V n + V n 1 2t

(7.6.32)

where U ig is the grid velocity needed in Eq. (7.6.23) and f denotes a face of a control-volume and the superscript n is for the n-th time step. The interface movement resulting from the heat balance at the interface is utilized to create the new grids in the computational domain. This movement of the grids between two time-steps is responsible for the displacements of the CV faces. A schematic view of the displacements of all faces of an arbitrary control volume is shown in Fig. 7.6.22. Once the grids are moved, the displacements of the CV faces correspond to a net convective flux which (appearing in the conservation equations) needs to be accounted for. Thus, with the inclusion of fluxes due to grid movement, within the iterations in a time-step, the grid remains fixed. The flux arising from the grid movement is equal to the sum of the rate of volume swept by an individual face. The applied SCL yields a net volume change of a CV; however in the discretized equations, the convective fluxes for each face of the CV due to the grid movement need to be evaluated. This can be achieved by computing the swept volume of individual face of the CV. For instance, fluxes due to the movement of the east face shown in Fig. 7.6.22 can be calculated for the first-order time scheme as

(  LU ig )e dSi  e  e

( V )ne +1 t

(7.6.33)

Fig. 7.6.22. A schematic view of the change in control volume due to movement of the grid (after Kumar et al. [2006]).

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447

and for the second-order scheme as

(  LU ig )e dSi  e  e

3( V )ne +1  ( V )ne 2t

(7.6.34)

where ( V )e is the volume swept by face e (cuboid 1-2-3-4-1'-2'-3'-4') and can be computed easily once both vertices of the cell at old and new time steps are known. The swept volume is computed by the method suggested by Kordula and Vinkour [1983], where the volume is decomposed into six tetrahedra, and all containing a common diagonal. For the east face shown in Fig. 7.6.22, the swept volume is

1  Ve = {(44  41 + 43  44 + 41  41 + 43  43 6 + 42  41 + 43  42 )i42}

(7.6.35)

where, for instance, 44 is a vector extending from point 4 to point 4 . By computing the fluxes due to the grid movement, the space conservation law is fulfilled and the problem of artificial mass sink or source can thus be avoided Treatment of Interfacial Conditions In order to compute the interface velocity introduced in the last section, first, temperature gradients in the liquid and solid phases are calculated. For this purpose, the Stefan energy balance Eq. (7.6.26) is utilized. The liquid and solid phases are separate domains in the computational program. Hence, the heat fluxes across the growth and dissolution interfaces must be exchanged. Once the exchange has been carried out, the grid velocity can be computed for one of the domains (either solid or liquid) and copied back to the other one. By carrying out this treatment, the common faces between two different domains remain in contact with each other. This interface velocity is computed at nodes that lie on the liquid-solid interface. However, in order to recreate the grids, the interface velocities or grid displacements are needed at the grid nodes. This is achieved with the help of a bi-linear interpolation in space. Once the grid velocities are known, the new position of the grid can be computed easily for both first and second-order time integration schemes. A similar procedure is followed to achieve the continuity of the electric current at the growth and dissolution interfaces, where both the electric current and electric potential remain continuous across the interfaces. The numerical code is programmed to transfer information at the boundaries of two contiguous domains for any kind of domain orientations. Solution Approach At first, only the steady-state energy equation is solved in order to obtain a suitable initial condition for the temperature field. With the temperature field

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Sadik Dost and Brian Lent

obtained from the steady-state simulations, the time-dependent momentum, energy and mass transport equations are solved. The second-order time implicit scheme is adopted for the time integration. The implicit time schemes are inherently stable, even for large values of the Courant number, and therefore the scheme allows selection of larger time steps compared with explicit schemes. The computations were performed for various strengths of the magnetic field varying from 2.0 to 15.0 kG. In each case the simulations were performed for approximately 3500 time steps to obtain initial flow and thermal fields. Once the flow field is computed, the tracking of the growth and dissolution interfaces is started with the help of the moving grid algorithm for about 500-600 time steps. Within each time step, the residuals for each equation were brought down to 5-6 orders of magnitude. The SIMPLE algorithm was adopted for the pressure-velocity coupling in the momentum equations. After every time step, the computational grids are recreated with the help of an algebraic grid generator. The movement of a block influences all its immediate neighboring blocks, and therefore the grid management becomes relatively complex.

Fig. 7.6.23. Computed temperature (a-d), flow (e-h), and concentration (i-l) fields in the vertical plane at 2.0, 4.0, 6.0, and 10.0 kG field levels (after Kumar et al. [2006]).

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449

Simulation Results The temperature, flow and concentration fields computed at four magnetic field levels (2.0, 4.0, 6.0 and 10.0 kG) are presented in the vertical plane of the liquid zone in Fig. 7.6.23, and in the horizontal plane of the liquid zone in Fig. 7.6.24. As expected, the common characteristic of all these transport structures is the development of three-dimensional structures (non-axisymmetric) at the onset of the growth process, in spite of the initially imposed axisymmetric boundary conditions. This shows the importance of using three-dimensional models for accurate predictions. Due to the transient nature of the problem (due to the constant movement of the heater) the transport structures are always unsteady, but the isotherms shown in Fig. 7.6.23a-d may still provide an insight about the evolution of the growth interface. It can be seen, as the applied magnetic field intensity increases, the isotherms become more concave towards the liquid zone, giving rise to a favorable condition for growth of crystals with uniform composition, and less

Fig. 7.6.24. Computed temperature (a-d), flow (e-h), and concentration (i-l) fields in the horizontal plane at 2.0, 4.0, 6.0, and 10.0 kG field levels (after Kumar et al. [2006]).

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Sadik Dost and Brian Lent

inclusions. It appears that the magnetic intensity level somewhere between 6.0 to 8.0 kG is optimum, above which the growth interface becomes more concave, and is unfavorable for high-quality crystal growth. This will be seen clearly later from the computed interface shapes. This level of magnetic field is called “optimum” since it leads to the most favorable growth interface (slightly concave towards the liquid zone) for higher quality crystals. The computed optimum magnetic field level, which is about 6.0-8.0 kG, agrees with those computed in Liu et al. [2003 and Abidi et al. [2005]. However, as can be seen from Fig. 7.6.23a-l and Fig. 7.6.24a-l, all the fields (temperature, concentration and flow) become smoother and more axisymmetric with the increasing magnetic field level. There is no unstable or strong flow development above the optimum magnetic field intensity level, contrary to what was observed in both Liu et al. [2003] and Abidi et al. [2005]. This shows that, within the limits of the assumed simplifications and assumptions in the model for the THM system considered here, such unstable and strong transport structures observed in Liu et al. [2003] and Abidi et al. [2005] are perhaps numerical. The numerical treatment presented in the present study eliminated such instabilities observed earlier and allowed us to carry out computations up to 15.0 kG (Ha = 290) level, without any convergence problems. However, to the best of our knowledge, the physical existence of such unstable and strong transport structures in such a THM system remains to be proven experimentally. For instance, in the case of the LPEE growth of GaAs, the experimental limit is about 4.5 kG, above which the growth interface lost its stability, and uneven and unsuccessful growths were observed in Sheibani et al. [2003a,b].

Fig. 7.6.25. The maximum flow velocity as function of the applied magnetic field (after Kumar et al. [2006]).

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451

Fig. 7.6.26. The maximum flow velocity as function of the applied magnetic field (re-computed from Ben Hadid and Henry [1996]) (after Kumar et al. [2006]).

As seen from Fig. 7.6.23e-h and Fig. 7.6.24e-h, the flow field continues to be suppressed with the increasing magnetic field intensity. In order to show this quantitatively, the maximum velocity in the liquid zone was computed and plotted versus the applied magnetic field in Fig. 7.6.25. As expected the maximum velocity obeys a logarithmic laws as also predicted in Liu et al.

Fig. 7.6.27. The shape of the growth interface at different magnetic field levels (after Kumar et al. [2006]).

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[2003] and Abidi et al. [2005], but contrary to that observed in Liu et al. [2003] and Abidi et al. [2005], the flow field continues to be suppressed (up to a Ha = 290 magnetic field level). In order to check the validity of the present code, a benchmark problem of Ben Hadid and Henry [1996] was also simulated. The simulation results are presented in Fig. 7.6.26, and as seen, are in good agreement with those of Ben Hadid and Henry [1996]. The evolution of the growth interface is presented in Fig. 7.6.27 for five levels of the magnetic field. As expected from the computed temperature field, the magnetic field level of about from 6.0 to 8.0 kG leads to the most favorable growth interface. This is sufficiently close to the results of Liu et al. [2003], considering the absence of the effect of the induced electric field in Liu et al. [2003]. The contribution of the induced electric potential was not taken into account in both Liu et al. [2003] and Abidi et al. [2005]. This effect is included in Kumar et al. [2006]. However, in order to see the effect of including the induced electric field on the iteration convergence, Kumar et al. [2006] carried out the same computations without including the induced electric field as done in Liu et al. [2003] and Abidi et al. [2005]. It was observed that the omission of the electric potential in the analysis has no effect on the convergence. This shows that the stability of the computations in Kumar et al. [2006] is due to the numerical treatment used, but not due to the inclusion of the induced electric field in the analysis.

453

Chapter 8

LIQUID PHASE DIFFUSION

In this chapter we present the numerical simulations carried out for the liquid phase diffusion (LPD) growth of SixGe1-x crystals with and without the application of applied magnetic fields. In the models, the application of a static vertical magnetic field was considered to suppress the strong convection observed in the solution in the first a few hours of the LPD growth process, and a horizontal rotating magnetic field to obtain a uniform mixing and a flatter growth interface.

8.1. Modeling the LPD Growth of SixGe1-x We first present a computational model based on the thermomechanical balance laws of a binary continuum mixture and the related irreversible thermodynamics of the transport phenomena given in Chapter 4. The model equations and the related interface and boundary conditions are presented for the LPD system considered in Fig. 3.5.19. The effects of electric and magnetic fields are included in the model. 8.1.1. The Model In the model, the liquid phase is the Ge-rich Si-Ge solution zone of the LPD growth system, and the solid phase represents the Ge single crystal seed substrate, the Si polycrystalline feed, and the crucible wall. The basic equations of each phase and the related assumptions are presented below. The liquid phase is considered as a mixture of two viscous and heat conducting incompressible Newtonian fluids, and is assumed to be a dilute binary solution (Ge-rich) of the solute (Si) and the solvent (Ge). The

Sadik Dost and Brian Lent

454 z Si source Insulator

Ge-rich solution R

n n

Temperature profile at the wall

Growth interface

t Quartz crucible

Ge seed r

T(z)

Fig. 8.1.1. The computational domain used for the LPD growth system with a representative thermal profile (after Yildiz and Dost [2005]).

contributions of the Soret and Dufour effects are neglected since there is no physical evidence for their significance in the LPD growth process considered here. The liquid phase has relatively low viscosity, thus the contributions of concentration and temperature to the dissipative stress tensor are also neglected. As a result, the constitutive equations given in Eqs. (5.1.1) and (5.1.3) are adopted. As we have discussed in Chapters 5, 6, and 7, the well-known Boussinesq approximation is also adopted here in order to include the effect of density variations arising from the temperature and concentration gradients across the liquid phase. In the present LPD setup, the density of the Si-Ge solution decreases with the increasing temperature (positive thermal expansion) and also with the addition of lighter silicon (positive solutal expansion). The mixture density is expressed as

   L [1 T (T  T0 )  C (C  C0 )]

(8.1.1)

where, as mentioned in earlier chapters, T and C represent the thermal and solutal expansion coefficients evaluated at the reference state, respectively, C is the solute (Si) mass concentration in the liquid solution, and  L is the density of the reference (starting) liquid solution (which is the molten Ge). Governing Equations Fig. 8.1.1 shows schematically the vertical cross section of the computational domain of the LPD growth system. Under the above assumptions, the threedimensional governing equations of the liquid phase are written in cylindrical

Single Crystal Growth of Semiconductors from Metallic Solutions

455

coordinates ( r,  , z ). The field equations of the two-dimensional simulation model are not given for the sake of brevity. They can easily be written from those given in Chapter 7. It is also straightforward to write them by simply dropping the dependency on the azimuthal coordinate (  ) in the 3-D equations. For the 2-D simulations, the growth cell is assumed axisymmetric, as shown in Fig. 8.1.1. At the growing interface, the contribution of the latent heat is neglected due to a very small growth rate. The 3-D field equations of the model are in the same form as those given in Chapter 7, however, for convenience, we write them again for the LPD system considered here. Continuity

1 v w 1  ru + + =0 r  z r r

( )

(8.1.2)

Momentum

Frem u v u u v 2 1 p u 2 v u 2 (8.1.3) +u + +w  = +  ( u  2  2 )+ r r  z r t  L r L r r 

Fem v v v v uv 1 p v 2 u v 2 (8.1.4) +u + +w + = +  ( v  2 + 2 )+ r r  z r r  L  t L r r  w v w w 1 p w +u + +w = +  2 w r r  z t  L z + T (T  To )g + C (C  Co )g +

Fzem

L

(8.1.5)

Energy

k T v T T T +u + +w = L  2T r r  z  L L t

(8.1.6)

Mass transport

C v C C C +u + +w = DC  2C r r  z t

(8.1.7)

The magnetic body force components in Eqs. (8.1.3)-(8.1.5) will be considered in the next section. The solid phase is modeled as a rigid, heat conducting solid. Therefore, the energy equation is the only balance law, given as

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456

k T = s  2T t s s

(8.1.8)

Boundary and Interface Conditions At the vertical wall, we assume zero-flow velocity and no mass flow conditions, i.e.,

u = 0, v = 0, w = 0,

C =0 r

(8.1.9)

At the axis of symmetry (for only the two-dimensional model), we use

w T C = 0, = 0, = 0, u = 0 r r r

(8.1.10)

At the growth and dissolution interfaces, the condition of local thermodynamic equilibrium is assumed. Therefore, the concentration at the interface is determined from the Si-Ge binary phase diagram (Ge-rich side) as

(

)

L s = a LT 3 + b LT 2 + c LT + d L , and ceq ceq = T  a s / bs

(8.1.11)

The numerical values of the phase diagram coefficients are given in Table 8.1.1. Table 8.1.1. Si-Ge phase diagram coefficients (Yildiz and Dost [2005]). L

3

L

L

L

a (1/K )

b (1/K2)

c (1/K)

d

2.598410-9

-8.718910-6

0.0099041

-3.822

s

s

a (K)

b (K)

1211.45

639.63

Since the growth rate in LPD is approximately five orders of magnitude smaller than the flow velocity in the solution, the radial and axial velocity components are taken as zero at the growth interface. In addition, since the diffusion of solute in the crystal is far smaller than the one in the solution, the solid phase diffusion is not considered. Due to the slow growth rate, the latent heat is also neglected. In addition, due to a relatively small interface curvature, the effect of surface tension in the energy balance is also excluded. The surface tension effect in the interface energy balance, however, becomes important in the presence of an interface with large curvature such as in the case of a dendritic growth or lateral overgrowth (Nishinaga [2002], Liu et al. [2004]). Then, the conditions at the growth interface are written as

C = CgiL = CeqL at Tgi ,

ur = 0,

v = 0,

wz = 0,

Single Crystal Growth of Semiconductors from Metallic Solutions

sV g (Cgis  CgiL ) =  L DCL,Si

C , n

ks

T T  kL n s n

457

=0

(8.1.12)

L

where CgiL and Cgis is

is the equilibrium mass fraction of the solute at the growth interface, the equilibrium mass fraction of silicon in the solidified crystal. The evolution of the growth interface is computed by solving the mass balance equation at the interface. Since the experimentally observed rate of dissolution of the silicon source is very small (approximately 0.3 mm/day), the mass balance at the dissolution interface is excluded, and at the dissolution interface, we write

C = CdiL = CeqL at Tdi , u = 0, v = 0, w = 0, ks

T T  kL n s n

=0

(8.1.13)

L

CdiL

where is the equilibrium mass fraction of the solute at the dissolution interface. The thermal boundary conditions for the quartz crucible (vertical wall, top and bottom surfaces) and the symmetry axis of the crucible are given as

ks

T  = h{T  T f (z)}, n

T =0 n

(8.1.14)

where T f (z) represents the ambient temperature inside the furnace along the quartz ampoule wall and along the top and bottom surfaces, and h is the modified heat transfer coefficient, which includes the contributions of both the convective and radiation heat losses, written as  h = h +  {T + T f (z)}{T 2 + T f2 (z)} (8.1.15) where  is the thermal emissivity of the quartz ampoule, and is the Stefan Boltzmann constant. The modified heat transfer coefficient is approximated using experimental data (such as the measured thermal profile, solute distribution in a grown crystal, and the position of the initial growth interface). In addition, at all interfaces in the crucible (the melt-ampoule, crystal–ampoule, and inner-outer crucible interfaces) we assume a continuous heat flux (perfect thermal contact). The initial conditions (at t = 0) are

C = Co , u = 0,

v = 0, w = 0.

(8.1.16)

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Table 8.1.2. Design parameters for the LPD growth system (Yildiz and Dost [2005]). Total growth zone length

40 mm

Initial source height

3 mm

Initial substrate thickness

10 mm

Crystal diameter

25 mm

Total wall thickness of quartz (including the crucible and tube)

4 mm

Table 8.1.3. Physical parameters of growth charge materials*(Yildiz and Dost [2005]). Parameter

Source (Si)

Substrate (Ge)

Crystal (SixGe1-x)

Solution (Si-Ge)

Specific heat,  (J/kg.K)

967 at 1300K 1037 (liquid)

396.1 at 1210K 380 (liquid)

396.1 - 487

380 - 406

Thermal conductivity, k (W/m.K) Diffusion coefficient, DC (m2/s) Mass density, (kg/m3)

23.7 at 1273K

10.60 at 1210K

10.60

42.8

N/A

N/A

1.010-20

2.510-8

2301.6 at 1300K

5323 at 1210K

4839-5323

5633

N/A

N/A

N/A

1.210-4

N/A

N/A

N/A

0.0053

1807.9

466.5

-

-

N/A

N/A

N/A

7.3510-4

Thermal expansion T (1/K) Solutal expansion C (1/mol%Si) Enthalpy of fusion HF (kJ/kg) Viscosity μ (kg/m.s)

Specific heat (quartz crucible) (J/kg.K)

1200 at 1300K

Thermal conductivity (quartz crucible) (W/m.K)

2 at 1300K

3

Density (quartz crucible) (kg/m )

2200 at 1300K th

*Compiled from: Metals Handbook, Volume 2, 10 edition. ASM international, The Materials Information Society, 1990. E. Yamasue, et al. [2002], Palankovski et al. [2001], Nakanishi et al. [1998], Yesilyurt et al. [1999], Supplier:www.gequartz.com/en/fig11.htm.

8.1.2. Physical Parameters The physical and system parameters used in the computations are given in Tables 8.1.2 and 8.1.3. Since the liquid solution is dilute (maximum silicon molar fraction is 0.04), the properties of the Ge liquid were used when the required physical properties of the Si-Ge solution were not available. The specific heat capacity of the solid and liquid phases is approximated by a linear expression, at a given temperature, as

Single Crystal Growth of Semiconductors from Metallic Solutions

 Si Ge x

1x

459

=  Si x +  Ge (1 x)

(8.1.17)

The composition dependence of the density of the growing crystal is also calculated similarly. The reported diffusion coefficient for Si in a Ge-melt is highly diverse, varying between 110-8 to 510-8 m2/s, so 2.510-8 m2/s was used in the computations.

8.2. Numerical Simulation of the LPD Growth of SixGe1-x In this section, we present the results of the numerical simulations carried out in the absence of an applied magnetic field. 3-D simulations were performed for the initial period of the growth process only, due to the high computational power requirement (a finer mesh is used). Table 8.2.1. Dimensionless parameters (Yildiz and Dost [2005]). Thermal Grashof, GrT

(TgL3T)/2

5.09107

Solutal Grashof, GrC

(CgL3T)/2

10.7107

Prandtl, Pr

/

0.0075

Schmidt, Sc

/DC

7.6743

8.2.1. Dimensionless Parameters Dimensionless parameters shed light on the relative contributions of various physical and transport parameters. Some of these parameters are given in Table 8.2.1, where the characteristic length (L) is taken as the diameter of the liquid zone (25 mm), and C = C1 – C0 and T = T1 – T0 are the characteristic concentration and temperature, respectively. C1, C0, and T1, T0 are the reference concentration (in terms of silicon mass fraction) and temperature (in Kelvin) values at the top and the bottom of the liquid zone. For an accurate scaling analysis, the dimensionless number must be defined with respect to reliable characteristic parameters of the system. For instance, the thermal and solutal Grashof numbers are usually defined with respect to the radial length and the thermal and solutal gradients in this direction. However, in the LPD system, this would not be appropriate since the radial temperature and solutal gradients vary even over a small length scale. Due to this difficulty, in this section, the solutal and thermal Grashof numbers are defined with respect to the axial concentration and temperature gradients. The computed values show that in the present LPD system the solutal Grashof number is approximately two times greater than the thermal Grashof number, which implies that the contribution of the solutal gradient to natural convection is larger than that of the thermal gradient.

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8.2.2. Computational Procedure The field equations, Eqs. (8.1.2)-(8.1.8), are solved using CFX-4.4. Several Fortran subroutines were developed to deal with the moving grid and complex temperature boundary conditions. For the discretization of time derivatives, a fully implicit backward time difference stepping procedure was implemented. Time iterations were initiated to calculate the time dependent concentration and flow field in the liquid, and the temperature in the liquid and the solid phases, by which the changes in these field variables and also the change in the substrate thickness due to the movement of the growth interface can be tracked. The computational process starts with the selection of appropriate initial guesses and boundary conditions. Then, the time dependent flow fields are computed. The computed flow field is used to calculate the thermal and concentration profiles in the liquid zone. The thermal field has to be updated at each time step to include the influence of the growing crystal. The interface concentrations were determined from the Si-Ge binary phase diagram using the computed temperature field. Then, finally the concentration field is computed in the liquid domain. The accuracy of the computations was checked by refining the mesh until a mesh-independent solution was obtained. Equidistant finite volume meshes were used. The mesh sizes used for the substrate, liquid domain, source, and quartz ampoule were, respectively, 1500, 4000, 500 and 1200 in 2-D, and were 18000, 54000, 7200 and 42240 in the 3-D simulations. 8.2.3. Two-Dimensional Simulations Temperature Field Fig. 8.2.1 shows the computed temperature distributions in the substrate, solution, source, and quartz crucible, for various hours of growth time. The initial temperature difference between the top and bottom of the liquid domain is about 45 K. As can be seen from Fig. 8.2.1a, the initial growth interface follows a reference isothermal line and is concave to the solid. The formation of a concave interface may be attributed to the thermal characteristics of the computational domain. Large differences in the thermal conductivities of the subdomains (substrate, solution, and surrounding quartz crucible) affect the isotherms in the vicinity of the interface. In addition, the presence of a ring-shaped insulator distorts the isotherms around the boundary between the insulated and uninsulated regions. In the model, its effect is incorporated using an estimate for the heat transfer coefficient to include the contributions of both the radiative and convective heat transfer. Furthermore, the convective heat transfer from the bottom of the crucible to the surrounding furnace atmosphere is smaller than that from the periphery of the ampoule due to the presence of a hollow quartz pedestal underneath the crucible. Away from the substrate and source material, thermal lines are almost

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flat due to being inside the insulated region. In the source, isotherms are convex since the top of the source material is in contact with the vacuum environment; therefore, heat transfer from the surroundings to the source material through its surface is less than that from its periphery through the quartz crucible.

(a) Growth time: 0.5 hours

(b) Growth time: 7.5 hours

Fig. 8.2.1. Computed temperature contours within the entire computational domain (2-D) at various growth times (in Kelvin) (after Yildiz and Dost [2005]).

The isotherm near the dissolution interface is convex with a large thermal gradient. Its convexity is due to the difference in the thermal boundary conditions between the insulated and uninsulated regions. The comparison of the simulated isotherms of the liquid zone for different growth times suggests that the flow field does not have a strong influence on the thermal field. This is expected since the Prandtl number for the Ge-rich SixGe1-x solution is far smaller than unity. Therefore, heat transport within the liquid domain is mainly due to conduction with very little contribution of convective flow on the temperature field. It is interesting to note that as the growth proceeds, the growth interface is crossed by isotherms, and hence a non-isothermal interface is developed. This situation gives rise to different levels of local saturations at the growth interface, and in turn, variations in the growth velocity across the interface. Flow and Concentration Fields Fig. 8.2.2 presents the computed flow (left column) and solute concentration (right column) structures. The flow field is given in terms of the magnitude of the flow velocity, and the concentration field in silicon mass fraction. As can be seen, the computed flow field is of one main eye-shaped convective cell, which consists of several radially-stacked small cells. The flow cells are located near the growth interface. In the rest of the liquid domain, convection is very weak,

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and is not notable graphically. The main convective flow cell circulates clockwise, forcing the liquid upward along the symmetry axis, and downwards along the vertical wall. The fastest circulation is at the centre of the main cell.

Fig. 8.2.2. (a) Growth time: 0.5 hours

Fig. 8.2.2. (b) Growth time: 5 hours

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Fig. 8.2.2. (c) Growth time: 6.5 hours Fig. 8.2.2. Flow (left column) and concentration (right column) fields for different growth 2 2 2 times. The flow field is given in terms of the magnitude of the flow velocity, U = vr + v z , and the concentration in silicon mass fraction (after Yildiz and Dost [2005]).

At the early stages of the growth process, convection in the solution zone is very strong. During growth, however, the silicon source dissolves continuously into the Ge-rich solution, and decreases the solution density. This gives rise to an axial density gradient in the solution, and makes the solution heavier at the bottom, near the growth interface. This axial density gradient acts as a stabilizer

Fig. 8.2.3. Magnitude of the flow velocity as a function of silicon mass fraction (after Yildiz and Dost [2005]).

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on the flow, and suppresses it in the upper region of the solution. The flow cells shrink continuously, and disappear completely after 5.2 hours of growth. This growth mechanism makes the LPD growth processes of SiGe diffusiondominant. Therefore, concentration profiles are naturally very stable, leading to a stable growth. The formation of an axial, stabilizing density gradient was also reported in the 2-D numerical simulation of the vertical Bridgman growth of SixGe1-x and GaInSb (Adornato and Brawn [1987] and Steliana and Dufarb [2004]). The evolution of concentration in the liquid is seen in Fig. 8.2.2. Fig. 8.2.3 shows the magnitude of the flow velocity near the growth interface as a function of silicon mass fraction. In order to verify the above-mentioned role of the axial density gradient, 2-D simulations were also performed by neglecting the contribution of solutal gradients, i.e. taking  C = 0 in the momentum equation. In this case, the interface shape was assumed to remain in its initial shape for simplicity, and it was moved at a constant speed of 0.9 mm/h. Results are shown in Fig. 8.2.4. The computed flow field is presented on the left, and the concentration field on the right. As seen, the flow has two convective cells, one at the top near the dissolution interface circulates anticlockwise, and the other one at the bottom near the growth interface circulates clockwise. The top cell moves the liquid upward along the vertical wall and downward along the symmetry axis. The bottom cell moves the liquid in opposite directions. Flow intensity depends on the steepness of the radial temperature gradient while the flow direction is a function of the shape of radial isotherms. In the present growth configuration, the radial thermal gradient near the dissolution interface is steeper than that near the growth interface, so the flow intensity of the top cell is greater than that of the bottom cell. No convective cell is present in the middle region (between the top and bottom cells) because the radial temperature gradient is small in this region. The results presented in Fig. 8.2.4 indicate that the presence of convection is mainly due to the presence of radial thermal gradients. In addition, the concentration contours in this case are different than those presented in Fig. 8.2.2. Each convection cell creates a concentration vortex. Since the flow is strongest at the center of the cells, the concentration gradients at the corresponding regions are the smallest. Convection moves the solute species towards the boundary, leading to relatively large solutal gradients in the middle. Growth Rate The growth rate is computed by solving the mass balance equation together with the SiGe binary phase diagram at each time step. The computed growth interface at the beginning of the growth process is concave. The computed and measured growth rates are plotted in Fig. 8.2.5 at the axis (r = 0) and periphery (r = r) of the crystal. At the onset of growth, for several hours, the growth rate

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at the periphery of the grown crystal is higher than that at the center. As growth progresses, the central region grows faster than the periphery, and the interface becomes flatter. As the interface advances further, the curvature of the interface changes, and becomes convex.

(a)

(b) Fig. 8.2.4. Flow (left) and concentration (right) fields at (a) t = 0.5 h and (b) t = 3.5 h when  C = 0 (after Yildiz and Dost [2005]).

The time variation of the growth rate can be best visualized by examining the mass balance at the growth interface, which gives the growth rate as

Vg =

 L DSiL C (Cgis  CgiL )  s n 1

(8.2.1)

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s

where Cgi and Cgi are the interfacial equilibrium concentrations in the solution and the grown crystal, computed from the phase diagram. According to the Sis L Ge phase diagram, the difference (Cgi  Cgi ) increases with the increasing interface temperature as the growth interface moves up. Since the concentration gradient along the growth direction remains almost constant during growth, the s L growth rate can be assumed to be inversely proportional to (Cgi  Cgi ) . The growth rate must then decrease as the interface moves to the higher temperature regions. This concept relies on the fact that at the onset of growth, less solute is needed to saturate the solution for solidification. However, at the later stages of growth, since more silicon atoms are needed for supersaturation, the growth rate is naturally expected to be slower than before. Fig. 8.2.5 presents the measured and computed interface displacements at the center (a) and near the wall (b) over 29 hours of growth. The computed averaged growth rate at the center is 0.63 mm/h while the experimental value is 0.67 mm/h. The computed and measured values near the wall are 0.59 mm/h, and 0.55 mm/h, respectively. The difference between the experimental and simulation growth rates is about 6%.

(a)

(b)

Fig. 8.2.5. Interface position as a function of growth time, at (a) r = 0, and (b) r = r (after Yildiz and Dost [2005]).

As mentioned earlier in Section 3.5, the computed time evolution of the growth interface (on the left) and the cross-section of an LPD grown single crystal (on the right) were shown in Fig. 3.5.21. Comparison of the growth striations (circular mark) with the computed growth interfaces shows that the numerical model developed successfully simulates the tendency of the time evolution of the growth interface. In experiments, the interface starts becoming flatter after 16 hours of growth, while numerically, the flat interface is observed around the 24th hour of growth.

Single Crystal Growth of Semiconductors from Metallic Solutions

(a) Growth time: 0.5 hours

(b) Growth time: 3.5 hours

(c) Growth time: 5.5. hours Fig. 8.2.6. Flow (left) and concentration (right) fields at (a) t = 0.5 h, (b) t = 3.5 h, and (c) t = 5.5 h (after Yildiz and Dost [2005]).

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(a) Growth time: 0.5 hours

(b) Growth time: 3.5 hours

(c) Growth time: 4.5 hours

(d) Growth time: 5.5. hours

Fig. 8.2.7. Flow fields at different growth times: 0.5, 3.5, 4.5 and 5.5 hours. The horizontal plane is at a distance of 14 mm from the bottom of the substrate (after Yildiz and Dost [2005]).

8.2.4. Three-Dimensional Simulations The computed 3-D temperature distributions in the vertical section (at  =  and at time t =1 hour) show the same patterns as those obtained from 2-D simulations (Fig. 8.2.1). The computed flow and concentration fields are presented in Fig. 8.2.6 in the vertical plane (where  =  ) of the computational domain where the flow field is given in terms of the magnitude of the velocity vector and the concentration field in silicon concentration. As seen in Fig. 8.2.6a, similar to that of Fig. 8.2.2a, the flow field has two main cells approaching each other at the symmetry axis (at 0.5 hours). The cells circulate in the opposite directions, as described earlier. The computed flow and concentration fields from the 2-D and 3-D simulations are similar quantitatively. This is an expected result since there are no external effects, such as an applied magnetic field or a crucible rotation.

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However, in terms of flow patterns, the 3-D simulations clearly show the development of three-dimensional structures in the liquid zone in spite of the initially assumed axisymmetric boundary conditions. This is particularly obvious in the structures presented in the horizontal plane (Fig. 8.2.7) at a distance of 14 mm from the bottom of the substrate. The development of 3-D flow structures is known to be of a numerical nature ( see for instance Nikfetrat et al [1996]). Although LPD is mainly a diffusion dominated process, the presence of strong convection during the early stages of the growth gives rise to nonuniform transport of species to the growth interface. This may result in local variations in the growth rate, and lead to the formation of low angle grain boundaries, or in some cases, of polycrystalline structure. The suppression of convection, indeed, can be helpful for increasing the likelihood of a single crystalline growth. The use of an applied magnetic field may be a feasible option to achieve such a goal. This is the subject of next section.

8.3. Simulation of the LPD Growth Process Under Magnetic Fields The numerical simulations carried out for the LPD growth of SiGe crystals under applied static and rotating magnetic fields are presented in this section. The LPD set-up presented Fig. 3.5.19 is simulated using the field equations, and the interface and boundary conditions presented in the previous section. 8.3.1. Magnetic Body Force Components For convenience, the magnetic body force components due to the applied static and rotating fields are presented separately. Vertical Magnetic Field We assume that the LPD system is subject to an applied static, vertical magnetic field which is perfectly aligned with the symmetry axis of the growth system, i.e., B = Bez. We also assume that the Ge-rich Si-Ge solution is an electrically conducting, nonpolarizable and nonmagnetizable binary liquid mixture. In addition, we consider that the magnetic field in the solution is uniform (Sheibani et al. [2003a,b]). Indeed, the magnetic Reynolds number, Rem = μ0 EU0 R010-5, calculated using U0 = 10-3 m/s (the maximum magnitude of the flow velocity) and R0 = 12.510-3 m (radius of the solution), is smaller than unity, indicating that the assumption of a uniform static field is reasonable. The magnetic force components in the r- and - directions are then obtained from Eqs. (4.5.7) respectively as em Fst,r =  E ( B

1   B 2u), r 

Fst,em =  E (B

  B 2 v) r

(8.3.1)

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The electric potential is obtained from the conservation of charge, i.e.,   J =  E   ( + v  B) = 0 as

 2 = B(

v v 1 u +  ) r r r 

(8.3.2)

Since the electrical conductivity of the liquid (  E, L = 1.7  106 1/.m) is greater than those of the substrate (  E,Ge = 6.06  105 1/.m), the source (  E,si = 4.25  104 1/.m ), and the crucible (10-7 1/.m), all boundaries surrounding the liquid melt are assumed electrically insulated; leading to a boundary condition of  / n = 0 for Eq. (8.3.1). Rotating Magnetic Field The application of a rotating magnetic field was briefly discussed in Section 4.5. A rotating magnetic field can be imposed with pairs of coils by switching the power on and off. This is was done in the CGL THM furnaces using three pairs of coils. However, here we assume, for simplicity, that the generated field consists of two sinusoidally varying magnetic field components. The magnetic field components are orthogonal and have phase differences. The resultant magnetic field rotates in the horizontal plane with an angular frequency of B, and is written from Eq. (4.5.15) as

B rot = B rot {e r sin(   B t) + e cos(   B t)}

(8.3.3)

which is governed by the induction equation, i.e.,

(

)

1 B rot =  2B rot +   v  B rot . t μ E E

(8.3.4)

Since the magnetic Reynolds number is very small (about 10 5 ), the last term on the right hand side of Eq. (8.3.4) can be neglected (Mossner and Gerberth [1999]). We can then write

1 B rot   2B rot t μ E E

(8.3.5)

A solution in the form

B rot (t,x) = B 0rot (x)e

i B t

(8.3.6)

gives

i B μ E  E B 0rot e

i B t

=  2B 0rot e

i B t

.

(8.3.7)

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from which we obtain

k 2B 0rot =  2B 0rot

(8.3.8)

In Eq. (8.3.8),

k 2 =  E μ E i B or

k = (1+ i)

 E μ E B . 2

(8.3.9)

The term

=

2  E μ E B

(8.3.10)

is called the skin or penetration depth, which gives the distance at which magnetic fields travel through a conducting liquid without being changed or modified. If the characteristic velocity, U0, is replaced by BR0, the so-called shielding parameter is obtained as

S = μ E  E B Ro2

or

S = 2(Ro2 /  2 ) .

(8.3.11)

The skin depth and the shielding parameter are usually used interchangeably to characterize the interaction of the magnetic field with an electrically conducting medium.  >> R0 (or S << 1) corresponds to a condition where the magnetic field distribution is not affected by the conducting liquid, in other words, the magnetic field penetrates into the liquid solution without experiencing any change.  << R0 (or S >> 1) on the other hand implies that the magnetic field penetrates slightly into the liquid because it is expelled due to the high conductivity of the melt, large angular frequency of the magnetic field, or combination of both (Vizman et al. [2001]). In the present simulation, the low frequency approximation is adopted:  >> R0 or S << 1. For the present LPD growth system of SiGe, the penetration depth  is approximately 120 mm, which is calculated using R0 = 12.5 mm (radius of the liquid zone) and a magnetic field frequency of 10 Hz. The low frequency assumption makes the present problem simpler since the computation for the magnetic field distribution within the liquid mixture is not required. As shown in Section 4.5, the magnetic field Brot and the induced electric field ind E can be expressed in terms of a vector potential A and a scalar potential , respectively, and satisfaction of Eqs. (4.513.) and (4.5.14) yields

A =  Bre z cos(   B t) , and Eind = - +  B e z Br sin(   t)

(8.3.12)

Since the applied magnetic field rotates at B, the electric field Eind, the current density J, and the scalar electric potential  should also change with the

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same frequency. Therefore, in order to put  in the same form as the rotating magnetic field, it is split into two parts as (Barz et al. [1997], Mossner and Gerberth [1999])

 (r,  , z,t) = 1 (r, z)sin(   B t) + 2 (r, z)cos(   B t)

(8.3.13)

The magnetic body force components are obtained from Eq. (4.5.11). The force components have both time-independent and time-dependent terms. The time dependent terms oscillate at 2 B. It is reasonable to assume that the liquid solution cannot follow these oscillating components due to its inertia. We then neglect the oscillating force components, and confine the analysis to the timeindependent, mean forces (as done in Ghaddar et al. [1999], and Ben Hadid et al. [2001]). Using

Firot =

1 2 rot  F d B t , 2 0 i

(8.3.14)

the time averaged force components are calculated as

  1 rot 1 B  E ( 2  uB), Frot =  E B rot [ 1 + ( B r  v)B rot ], 2 2 z z 1 (r2 ) (8.3.15) =  E B rot (  wB rot ) 2r r

Frrot = Fzrot

where we have omited the bar above the averaged components for simplicity. For the LPD growth system under consideration, the computed characteristic values of GrT and Re are about 5.41105 and 1.2104, respectively. Since the value of (GrT)1/2/2Re is 3.0710-2, the azimuthal force component is dominant compared with the other two components. This justifies the omission of the radial and axial force components in most cases (Ghaddar et al. [1999], Ben Hadid et al. [1999], Dost et al. [2002], Liu et al. [2003]). Yildiz et al. [2006a] included all three force components. The conservation of charge,   J = 0 determines the electric potentials 2 2 1 1  1  1 1 v B rot w (  2  2 + 2 1  B rot + )sin(   B t) r r r z r  z r

+(

2 2 v v 1 2  2  2 1  2  2 + 2 2  B z + r B)cos(   B t) = 0 r r r z r z r

(8.3.16)

Note that the functions sin(   B t) and cos(   B t) cannot be zero at the same time. Therefore, in order for Eq. (8.3.16) to be satisfied for all possible functions, their respective coefficients must vanish, in other words,

Single Crystal Growth of Semiconductors from Metallic Solutions

( 2 

1 r

2

)1 = (

1 w v rot 1 u w  )B , and ( 2  2 )2 = (  )B rot r  z z r r

473

(8.3.17)

which are subject to the following boundary conditions: at the vertical wall

1 / r = 0 ,

2 / r = 0 ,

(8.3.18)

and at the dissolution and growth interfaces

2 / z = 0 ,

1 / z =  B B rot r .

(8.3.19)

8.3.2. Numerical Method The governing equations were solved by CFX. Several user-defined subroutines were developed to move the grid in time, to implement complex thermal boundary conditions and magnetic body force terms in the momentum equations, and to solve additional scalar field equations for the electric potentials. A body fitted finite volume grid was generated. Comment files were created to define physical, transient and solver parameters. The solution of the field equations starts with defining the initial guesses and the boundary conditions. First, the momentum equations were solved for velocities. The computed velocity fields were then used to calculate temperature and concentration contours in the liquid zone, which are coupled with the linear momentum equations through the flow field. The field equations for the electric potentials in both cases (vertical and rotating magnetic fields) have the form of a diffusion equation with a source term being the spatial derivatives of velocity components. It was therefore possible to describe these equations in CFX as a scalar transport equation under the steady state and no convection conditions. The source term in this scalar transport equation was treated by developing a user defined subroutine. The magnetic body force terms were included through source term subroutines. The mesh sizes for the substrate, the liquid domain, the source and the quartz ampoule, were 18,000, 54,000, 7200, and 42,240, respectively. For the discretization of time derivatives, a fully implicit backward time difference stepping procedure was used. The initial shape of the growth interface was computed by deforming the initial grid according to the isotherm at the melting temperature of the substrate. For the vertical stationary field, simulations were carried out with and without including the contributions of the electric potentials, for the purpose of comparison. In both cases, the computed flow patterns and flow strengths are the same for all practical purposes. This suggests that for the system considered here, the potential built up in the solution due to convection is negligible. The computed temperature distribution in the solution for different magnetic field

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strengths (for both vertical and rotating magnetic fields) indicate that the effect of convection on thermal field is small. This can be attributed to the small Prandtl number (0.0075) of the Si-Ge solution.

(a) B = 0.0 T

(b) B = 0.05 T

(c) B = 0.1 T

(d) B = 0.3. T

Fig. 8.3.1. Computed flow strength in the vertical plane after one hour of growth at various magnetic field intensity levels (after Yildiz et al. [2006a]).

8.3.3. Effect of the Vertical Stationary Magnetic Field Flow Field Fig. 8.3.1a presents the computed velocity profiles (with no magnetic field) at the 1st hour of growth in the vertical plane (i.e.,  =  ) of the computational domain. The flow field is given in terms of the magnitude (flow strength) of the velocity vector.

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(b) (a) Fig. 8.3.2. (a) Computed radial-meridional (vertical) velocity vector (left) and azimuthal velocity contours (right, strongest being darkest) at B = 0.2 T vertical field intensity and at the  =  plane, and (b) the computed flow field in the vertical plane, after t = 1 h growth (after Yildiz et al. [2006a]).

As seen from the figure, similar to that observed in the previous section, the fluid flow has two main convection cells in the vicinity of the growth interface. In the rest of the liquid zone, convection is very weak and not noticeable in the figure. The convective cell on the right circulates in the clockwise direction, pushing the fluid downward along the crucible wall and upward along the axis of symmetry, while the other cell on the left rotates anti-clockwise. The computed flow strengths under three magnetic field levels are presented in Figs. 8.3.1b-d. As seen, the magnetic field strength of 0.05 tesla (T) (relatively weak) decreases the strength of convection, yet the cellular form of the convective cells is not broken down completely. As the magnetic field strength increases, the convection cells expand along the radial direction while shrinking in the vertical direction. Simulation results show that a field level approximately between 0.2 T to 0.3 T is sufficient to suppress the convection significantly. Fig. 8.3.2a presents the computed velocity field at the 0.2 T field level. In order to see the effect of the applied magnetic field on flow structures, the radial and meridional velocity vector components are plotted on the left side and the azimuthal velocity component (contours) on the right. Fig. 8.3.2b shows the computed flow field in the vertical plane. In Fig. 8.3.3, the variation of the maximum flow strength (the maximum magnitude of the flow velocity, Umax) is plotted versus the Hartmann number. Results are presented with and without the contribution of the electric potential. The maximum flow strength in the liquid decreases with the increasing value of

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Fig. 8.3.3. Variation of maximum flow strength (U max) with the Hartmann number (after Yildiz et al. [2006a]).

the Hartmann number. The flow was stable up to Ha = 60. As the Hartmann number is increased further, the flow becomes unstable; however, the maximum flow strength still decreases. The flow strength varies according to Ha 2 ,which agrees with the boundary layer approximation (Kim et al. [1988]). Concentration Field The computed silicon concentration distribution is presented in Fig. 8.3.4. The concentration contours with no magnetic field are presented in Fig. 8.3.4a at t = 1 h. As seen, in the region where the convection is strong (the lower region) we observe a strong mixing, and in the region with weak convection (upper section) a diffusion-like solute distribution. With the application of a vertical magnetic field, we see a diffusion-like solute distribution near the growth interface due to the suppression of convection. The concentration contours approach the growth interface (Fig. 8.3.4b), showing that the growth process becomes diffusion dominant. Interface Shape The variation of the concentration gradient along the growth interface is presented in Fig. 8.3.5. As seen, under the effect of a static, vertical magnetic field, the interface shape is flattened, but not significantly. The growth interface preserves its initial concave shape. This suggests that the application of a static magnetic field in the present LPD system is not beneficial for flattening the growth interface, contrary to that observed in THM (Ma and Walker [2000], Dost et al. [2003], Liu et al. [2004], Wang et al. [2005]), or in LPEE (Sheibani et al. [2003a,b], Liu et al. [2003], Dost et al. [2005]). This is due to the

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following features of the LPD growth process of SiGe: i) the strong convection observed near the growth interface in the early stages of growth, and ii ) the large density difference between the solvent (Ge) and the solute (Si), and iii) the thermal character of the system.

(a) B = 0.0 T

(b) B = 0. 2 T

Fig. 8.3.4. Silicon concentration distribution at (a) B = 0.0 T, (b) at B = 0.2 T at t = 1 h (after Yildiz et al. [2006a]).

Fig. 8.3.5. Interface shape deflection at various vertical magnetic field intensities (after Yildiz et al. [2006a]).

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(a) B = 0.5 mT

(b) B = 1.0 mT

(c) B = 2.0 mT

(d) B = 3.0 mT

Fig. 8.3.6. Flow field at t = 2 h, f = 10 Hz, and (a) B = 0.0 mT. (b) B = 0.5 mT, (c) B = 1.0 mT, (d) B = 2.0 mT, (e) B = 3.0 mT, and (f) B = 4.0 mT (after Yildiz et al. [2006a]).

8.3.4. Effect of a Rotating Magnetic Field (RMF) Flow Field Fig. 8.3.6 presents the effect of RMF on the fluid flow at t = 2 h under various magnetic field strengths. The flow field under no magnetic field is the same shown in Fig. 8.3.1a. Under a 0.5 mT (millitesla) RMF (Fig. 8.3.6b), two new large convection cells develop in the upper part of the liquid zone. These cells circulate in opposite directions; the one on the left circulates anti-clockwise while the one right clockwise. The cells gain strength with increasing magnetic field levels, and come closer to the cells at the bottom (buoyancy induced cells) as seen in Figs. 8.3.6a-d. As the field intensity increases further, the cells merge, and become two new large cells that span across the entire domain. Such a flow structure due to the effect of RMF improves mixing in the solution zone. The

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flow field was numerically stable up to about B = 3 mT. However, above 3 mT, a series of local vortices form near the vertical crucible wall. Above 4 mT, the flow becomes numerically unstable.

(b)

(a) Fig. 8.3.7. At B = 2.0 mT and f = 10 Hz rotating magnetic field level: (a) radial/meridional velocity vectors (left) and azimuthal velocity contours (right) (at  =  plane), and (a) the computed flow field at t = 2 h in the horizontal plane at a distance of 6 mm from the growth interface (after Yildiz et al. [2006a]).

(a) (b) Fig. 8.3.8. Computed flow velocity with and without the contribution of electric potential at various magnetic field levels (at 10 Hz): (a) the maximum azimuthal velocity component, and (b) the maximum flow strength. (after Yildiz et al. [2006a]).

The computed velocity field under a field of 2.0 mT intensity and 10 Hz frequency is shown in Fig. 8.3.7. The flow strength in a horizontal plane near

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the interface is given in Fig. 8.3.7b. The radial and meridional velocity vector components are plotted on the left of Fig. 8.6.7a while the azimuthal velocity contours are presented on the right. The improvement in mixing in the liquid zone is clearly seen. The strong fluid flow near the interface leads to a further flattening in the interface. The present simulation results suggest that a rotating magnetic field level about 3.0 mT appears to be optimum for the LPD growth system of SiGe presented in Fig. 3.5.19. The field levels up to this value may provide the desired mixing in the solution. Above this value, however, the application of RMF may not be so beneficial for the LPD system considered here.

(a)

(b)

Fig. 8.3.9. (a) Si concentration distribution at t = 2 h of growth (at B = 3.0 mT and 10 Hz, and at  =  plane), and (b) radial segregation as a function of the RMF intensity (at 10 Hz). (after Yildiz et al. [2006a]).

In order to see the effect of the induced electric potential, the velocity field was also computed by neglecting the contribution of the electric potential. The results for the azimuthal velocity component are presented in Fig. 8.3.8a, and for the flow strength in Fig. 8.3.8b. As can be seen, the contribution of the electric potential is significant, contrary to that observed in the case of a static vertical magnetic field. We also note that the variation of the azimuthal velocity component is almost identical to that of the flow strength. This implies that the azimuthal velocity component is the dominant. Concentration Field The computed concentration profiles of Si in the Si-Ge solution at t = 2 h of growth are presented in Fig. 8.3.9a. Results show that in the absence of a rotating magnetic field, the concentration profiles exhibit diffusion-like patterns with a distribution varying linearly from the dissolution interface to the growth

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interface (Fig. 8.3.1a). However, under RMF the concentration profiles become affected by the induced convection, and the concentration distribution become mostly convection dominant, as the field level increases. The concentration profiles at the B = 3.0 mT level at t = 2 h of growth are presented in Fig. 8.3.9a. As seen, at this level, the concentration distribution is almost convection dominant, providing a good mixing in the solution. The large Schmidt number (about 5.22) indicates that the flow structure under the effect of RMF has a significant influence on the transport of the solute in the solution. The radial segregation is calculated by

 c = (cmax  cmin ) / cav

(8.3.20)

for different levels of RMF intensities, and plotted in Fig. 8.3.9b. As seen, the computed value of the radial segregation decreases with increasing RMF intensity levels, supporting the earlier conclusion that the application of RMF may give rise to a good mixing in the solution. Interface Shape Experimental and numerical studies have shown that RMF is beneficial for flattening the growth interface (for instance, Salk et al. [1994], Fiederle et al. [1996], Barz et al. [1997], Senchenkov et al. [1999], Ghaddar et al. [1999], Gelfgat [1999], Mossner and Gerberth [1999], Ben Hadid et al. [2001], Dost et

Fig. 8.6.10. Interface shape deflection with various rotating magnetic field intensities (after Yildiz et al. [2006a]).

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al. [2003], Voltz et al. [2005], Yildiz et al. [2006a]). The application of RMF in the LPD growth of SiGe is effective even at very low RMF levels. Computed results are presented in Fig. 8.3.10. As seen, the application of a field intensity of 3 mT is sufficient to make the interface almost flat. However, above the 3 mT field level, the growth interface starts becoming concave again. Numerical simulations show that the use of a static, vertical magnetic field in the LPD growth setup (for SiGe) is effective in suppressing natural convection in the solution. A stationary, vertical field intensity of 0.3 tesla is sufficient to provide significant suppression. Above the 0.3 tesla level, however, the flow in the solution becomes numerically unstable. Results show that the stationary vertical magnetic field does not provide the expected flattening in the growth interface. However, the use of RMF is effective in providing sufficient mixing in the solution, leading to more homogeneous SiGe crystals. In addition, RMF is also very beneficial for flattening the growth interface. At the 3.0 mT RMF level, the growth interface changes from concave to almost flat. Over this critical value, unstable flow is observed. One can conclude that for the LPD system considered here, the use of a rotating magnetic field is more beneficial than a stationary magnetic field.

483

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