Incorporation of surface tension to interface energy balance in crystal growth

Transcription

1 Cryst. Res. Techno. 42, No. 9, (2007) / OI /crat Incorporation of surface tension to interface energy baance in crysta growth M. Yidiz and. ost* Crysta Growth aboratory, epartment of Mechanica Engineering, University of Victoria, Victoria, BC, V8W 3P6, Canada Received 27 February 2007, revised 25 March 2007, accepted 6 Apri 2007 Pubished onine 12 June 2007 Key words crysta growth from soution, jump conditions, Gibbs-Thomson effect. PAC n, ab, dm Effect of surface tension across a growth interface is nown as the Gibbs-Thomson effect, and the associated energy baance is widey referred to as the tefan condition in the iterature, which is derived from thermodynamics. In this artice, the interface energy baance that accounts for the effect of surface tension is derived by writing the jump condition for the energy baance on a surface of discontinuity which represents in crysta growth the evoving growth interface (soidification front) between the iquid and soid phases. To the best of our nowedge, the derivation of energy baance by writing jump conditions on a surface of discontinuity (interface) is new. 1 Introduction In crysta growth when the growth interface has a arge curvature, the surface tension may pay a significant roe in the evoution of growth interface shape, affecting the concentration distribution in the met/soution near the interface. In such a case, the incusion of the effect of interface curvature (nown as the Gibbs-Thomson effect) in a mode may become necessary for accurate predictions. In addition, if a free surface exists in the system, the therma and souta gradients aong the growth interface give rise to an additiona convective fow in the met/soution near the interface, which is nown as the Marangoni convection. In crysta growth modeing invoving an interface, either a free surface or an interface between the iquid and soids phases, writing accurate expressions for interface conditions is a chaenge. In thermomechanics of a continuum, the associated boundary and interface conditions are obtained by writing the jump conditions for mass, momentum, and energy baances on a moving surface of discontinuity on which certain quantities may suffer jumps. This approach provides confidence in obtaining accurate boundary and interface conditions for a seected mode. The derivation of the jump conditions reated to mass, momentum, and energy baances through this approach is we-nown (see for instance [1]). urface tension appears in the momentum baance since it contributes to the force baance aong the boundary of the domain. However, when the effect of surface tension needs to be incuded in the energy baance on a soidifying interface, referred to as the tefan condition in the iterature, the derivation is done through thermodynamics considerations, by introducing the effect of surface tension in temperature [2,3]. imiary, the effect of surface tension on growth rate in crysta growth is incuded through its incorporation into concentration [4]. The contribution of surface tension may aso be incorporated into growth rate by modifying the mass transport equation writing the jump condition for the species mass baance on a surface of discontinuity (interface) [5,6]. In this artice, we present the derivation of the interface energy baance that accounts for the effect of surface tension, by writing the jump condition for the energy baance on a surface of discontinuity [7]. Here, a * Corresponding author: e-mai: sdost@me.uvic.ca

2 Cryst. Res. Techno. 42, No. 9 (2007) 915 discontinuity surface represents the evoving growth interface (soidification front) between the iquid and soid phases. Mass, momentum, and heat transport are considered in the iquid phase, and ony heat transport in the soid phase. For simpicity, the species mass transport is not incuded in the iquid phase. Beow, we aso present briefy the mass and momentum baances since the associated jump conditions are needed in the derivation of the energy baance at the interface. 2 erivation of jump conditions For a materia body of voume V encosed by a surface (voume boundary) which contains a discontinuity surface moving with a veocity u (we use Cartesian components, see fig. 1), we obtain the mass and momentum baances [1,7] as t ρdv = 0 and ρvdv = ρ fdv + tnda + tbd t V and the energy baance as t (1) V V ρ{(1/ 2) vv + ε} dv = ρbvdv + ρhdv + tvnda qnda + tv bd (2) V V V where /t is the materia derivative (or denoted by a dot on a variabe) defined by / t / t+ v( / x), ρ the density of the phase, v the fow veocity components, f the body force components, ε the interna energy per unit mass, h the interna heat generation rate per unit mass within the voume, q the heat fux vector, v the components of veocity of surface partices, t the stress tensor, and t the symmetric interface stress tensor which is defined as t ( b ) = t b [7]. Here, t ( b ) is the interface stress vector defined as a contact force per unit ength on and is a function of position on. dv, da, and d represent the voume, area, and ength eements, respectivey, V - and - denote the voume and surface excuding points ying on the discontinuity surface (t), is the curved ine formed by the intersection of the discontinuity surface (t) with the cosed surface bounding the voume V as iustrated in figure 1b. The vector b = h n is the binorma vector which is tangent to the discontinuity surface (t), norma to, and outwardy directed with respected to V. h is the tangentia unit norma, and n is the exterior unit norma vector. Note that a fied variabes are functions of time and space coordinates, and throughout the artice, the summation convention is empoyed. Fig. 1 a) a surface of discontinuity, and b) representation of norma, binorma and tangent unit vectors, and the ength eement d. We assume that the surface of discontinuity is a materia surface, and thus, the norma component of the veocity of a surface partice is equa to the norma component of the veocity of the discontinuity surface. The veocity of a surface partice is referred to as surface veocity (denoted by v ) and is defined as the time rate of change of the position of a partice. It is aso assumed that the tangentia components of the surface veocity are continuous across the discontinuity surface, which means that the discontinuity surface does not suffer a ine of discontinuity. In the derivation of the oca baance aws and the associated jump conditions on the surface of discontinuity, we use the Green-Gauss and transport theorems

3 916 M. Yidiz and. ost: Incorporation of surface tension into energy baance φ, dv + φnda = φnda and ϕ = { + ( ϕv ), } + ϕ( ) V t V ϕ dv dv v u n da (3) t V () t where the symbo indicates the jump of the encosed quantities across the discontinuity surface, i.e. + ϕ ϕ ϕ where ϕ + and ϕ are the vaues on the positive and negative sides of the discontinuity surface. The use of the above equations in eq. (1) yieds the we-nown expressions for mass baance [1]: ρ + ( ρv ), = 0 in V t, m m ρ( v u ) n = 0 on (4) and In order to write the momentum baance, we first examine the ast term in eq. (1) which represents the contribution of the interface stress t to the momentum baance. ince the interface stress tensor can be decomposed into two parts as t = + γ P [2] where is the viscous portion of the interface stress tensor, and γ is the thermodynamic surface tension due to the surface stress generated by the deformation of the interface. Ignoring the viscous effect at the interface, we may write t = γ P where P = δ n n ( δ is the Kronecer symbo). Here, P is nown as the projection tensor and when it is operated on a vector fied, it extracts the tangentia component of the vector fied. Using the surface divergence theorem for a tensor fied, one may write [7] t b d = ( t P + 2 Ω t n ) da (5) where Ω is the mean curvature. Upon repacing t by γ P, and noting that b = P b and Pn = 0, and using 2Ω nmm, and the foowing ( γ ) γ γ γ γ { δ δ = γ P γ nn = γ P + 2Ωγ n P, m Pm =, mp Pm + P, mpm =, mpm n, mn m n, mn nmn + nn, m m nn mn, mn}, m m mm,, m m Eq. (5) then becomes γ bd = ( γ P + 2 Ωγ n ) da (7), m m Using Eqs. (3) and (7) in eq. (1), we obtain the momentum baance as [2] v ρ t = t + ρb in V. ρv ( v u ) t n = γ P + 2Ω γ n on (8) and, m m If we assume surface tension is independent of position, then the term γ P on the right-hand side vanishes., m m We now obtain the energy baance. Using the surface divergence theorem for a vector fied, the ast term in Eq. (2) is written as γ v Pbd = { ( γ v P), mpm + 2 Ωγ v P } n da = 0 (9) = ( P γ v + v P + γ v P P ) da= ( P γ v + v P + 2Ωγ v n ) da m, m, m m, m m m, m, m m and by the use of Eqs. (3), (7), (8) and (9) in Eq. (2), we obtain the energy baance as ρε = t v q + ρh in V,, ρ{ (1/ 2) vv + ε}( v u) + q tv n= Pmγ, mv + v, mpm+ 2Ωγ v n on (10) The right-hand side of the jump condition in Eq. (10) represents the contribution of surface tension to the energy baance on the interface. If we impose incompressibiity on the interface, and assume surface tension is (6)

4 Cryst. Res. Techno. 42, No. 9 (2007) 917 independent of position, then the terms vm, P Eq. (10), then becomes 1 ρ( vv + ε)( v u ) + q t v n = 2Ωγ v n 2 m and P γ,mv vanish, and the energy baance on the interface, m on (11) 2 Interface conditions for a moving soidification front We now write the above jump conditions expicity for specific interface conditions in crysta growth. First, the condition for the overa mass baance in Eq. (4) becomes since the veocity of the soid phase is zero: ρ ( v u ) n = ρ u n (12) where the capita supercripts and represent the quantities of the iquid and soid phases respectivey. imiary, the jump condition corresponding to the momentum baance can be written as v ρ ( v u ) n t n = 2γ Ω n, or ρ unv t n = 2γ Ω n (13) By the use of Eq. (12), Eq. (13) becomes ( t t ) n = ρ vu n + 2γ Ω n (14) The condition corresponding to the energy baance in Eq. (12) now becomes {(1/ 2) }( ) ( ) ( ρ v v + ε v u n + q n t t vn ρ ε u n + q n ) = γ v 2Ω n (15) or using Eqs. (12) and (13), we write / } ρ {(1/ 2) v v + ε }{ ( ρ ρ ) u n + q n ( t t ) vn ( ρ ε u n + q n ) = γ v 2Ω n {(1/ 2) } ( 2 ) ρ vv + ε un + qn + ρ vun + γ Ω n v + ρ ε un qn = γ v 2Ω n ρ {(1/ 2) vv + ε ε } un + qn qn + 2 γ Ω( v v ) n = 0 H or using ( ρ / ρ ) un = ( v v ) n, we arrive at the interface condition corresponding to the energy baance as { qn qn = ρ H + 2 ( γ / ρ ) Ω } un (16) where H is the atent heat or enthapy of fusion. Note the ast term on the right-hand-side, the contribution of interface curvature, wi be positive for a positive curvature. 3 iscussion The interface conditions in Eqs. (12), (14) and (16) represents the overa mass conservation, momentum, and energy baances at the interface, respectivey. Note that the contribution of surface tension in the momentum baance, Eq. (14), depends ony on the interface curvature, but is independent of the interface veocity. In other words, it is not couped with the growth veocity. When the curvature is zero or negigibe, the contribution of surface tension vanishes, and Eq. (14) becomes ( t t ) n = ρ vu n. When the growth veocity is sma, in crysta growth techniques such as eectroepitaxy and the traveing heater method for instance, its contribution to the momentum baance may be negected. In such a case, the momentum baance taes the form of ( t t ) n = 0 which is simpy the static force baance at the interface. In the energy baance in Eq. (16), the contributions of the atent heat of fusion and the interface curvature are both couped with the interface veocity u. It appears that in sow soidification processes, such as eectroepitaxy or the traveing heater method, their contributions may be negected. However, even such cases, a care must be taen, and the physica phenomena must be examined on physica grounds whether such a

5 918 M. Yidiz and. ost: Incorporation of surface tension into energy baance simpification can be made. In some crysta processes such as atera overgrowth of semiconductors, even the growth veocity may be sma, the contribution of interface curvature to the growth process may be significant [4-6]. In the absence of the contributions of interface curvature and atent heat, the condition for the energy baance in Eq. (16) taes the commony used form of qn = qn which is simpy the continuity of heat fux through the interface. In this short artice, as mentioned earier we have not incuded the species mass baance for a mixture, for simpicity. For a binary mixture, for instance, there wi be one more interface condition to be written from the mass baance of one of the two constituents. epending on the growth technique, this condition may be used to determine the growth rate. For instance, in the soution growth of semiconductors, this is written as [8] u V ρ C n g C ( ) = ρ C C n (17) where u n is the norma veocity of the growth interface. In the case of an appied eectric current, for instance in eectroepitaxy, the contribution of eectrotransport to the growth rate wi be incuded (see [6,9]). imiary, the contributions of surface inetics and interface curvature can aso be incuded in the species mass baance at the interface [6]. In addition, the interfacia concentrations must aso be updated through the satisfaction of the phase diagram equation(s) [9]. We now consider an exampe from crysta growth. We write the energy baance condition on the growth interface of the P (iquid phase diffusion) growth system of ige given in [10] (fig 2a). For this crysta growth system, upon assuming the Fourier aw of heat fux for both the iquid (the iquid Ge-i soution) and soid (the grown ige crysta) phases, i.e., q = T, and q = T, where and are the corresponding heat conductivity coefficients of the iquid and soid phases, the energy baance given in Eq. (16) taes the foowing expicit form T T 2 γ = ρ ( H + Ω ) u n n ρ n The contribution of interface curvature on the right hand side of the above equation must be taen into account in cases where the curvature is arge. For instance, the effect of interface curvature was very significant in the atera growth of GaAs simuated in [4] and [6] since the interface curvature remains arge during entire growth period. In the P growth of ige crystas [10, 11], the interface curvature remains arge ony during the first five hours of about a 40-hour growth process (see the evoution of the growth interface in figure 3a). The interface is then graduay becoming fatter, and the contribution of the ast term on the right-hand-side of Eq. (18) diminishes. The contribution of the interface curvature in the simuations of [11] was not significant since the interface curvature was not so arge compared with that of [4, 6]. In the simuation resuts presented in figure 3b, the contribution of atent heat was not taen into account. (18) Fig. 2 The mode domain of the P growth system of ige [10,11].

6 Cryst. Res. Techno. 42, No. 9 (2007) 919 Fig. 3 The evoution of P growth interface: a) numericay predicted, and b) experimentay observed growth striations that represent the evoution of the growth interface [10,11]. 4 Concusions In the artice, we derived the interface energy baance that accounts for the effect of surface tension by writing the jump condition for energy baance on a surface of discontinuity which represents in crysta growth the evoving growth interface (soidification front) between the iquid and soid phases. To the best of our nowedge, the derivation of energy baance by writing jump conditions on a surface of discontinuity (interface) is new. Acnowedgments The financia support provided by the Canadian pace Agency, and by the Canada Research Chairs Program is gratefuy acnowedged. References [1] A. C. Eringen, Mechanics of Continua, Robert E. Krieger Pubishing Company, New Yor (1980). [2] J. C. attery, Interfacia Transport Phenomena, pringer-verag, New Yor (1990). [3] V. Aexiades and A.. oomon, Mathematica Modeing of Meting and Freezing Processes, Hemisphere Pubishing Corporation, Chapter 2, Washington (1988). [4] Y. C. iu, Z. R. Zytiewicz, and. ost, J. Cryst. Growth 275, E959 (2005). [5] M. Khener, R. J. Braun, and M.G. Mau, J. Cryst. Growth 235, 425 (2002). [6] Y. C. iu, Z. R. Zytiewicz, and. ost, J. Cryst. Growth 265, 341 (2004). [7] M. Yidiz, A Combined Experimenta and Modeing tudy for the Growth of i x Ge 1 x inge Crystas by iquid Phase iffusion (P), Ph.. Thesis, University of Victoria, Victoria, BC, Canada (2005). [8] R. W. Wicox, J. Cryst. Growth 65, 133 (1983). [9]. ost and B. ent, inge Crysta Growth of emiconductors from Metaic outions, Esevier, Amsterdam (2007). [10] M. Yidiz,. ost, and B. ent, J. Cryst. Growth 280, 151 (2005). [11] M. Yidiz and. ost, Int. J. Engng. ci. 43, 1059 (2005).