Dynamics and control of the shape and size of a sitting drop-like meniscus, occurring in single-crystal growth

https://doi.org/10.1016/j.jfranklin.2009.09.002Get rights and content

Abstract

This paper concerns the following topics: dependence of shape and size of a static meniscus sitting like a drop on the controllable part of pressure difference across the free surface; stability of static menisci; procedure for creation of a stable static meniscus, appropriate for the growth of a rod with constant radius; temperature gradients and crystallization front displacement; equation for crystal dimension change rate; evolution of crystal radius and of the level of crystallization front; and control of cross-section during growth. This kind of problems appears in semiconductor single-crystal rod growth from a melt by edge-defined film-fed growth (EFG) technique. The novel case where the upper radius of the meniscus is larger than the shaper radius and the meniscus is sitting like a drop is analyzed. For some of the topics numerical illustrations are given. The obtained results can be useful in experiment planning and technology design of single-crystal rod growth by the EFG method. With this aim this study was undertaken.

Introduction

The free surface of a static meniscus, sitting like a drop, occurring during single-crystal rod growth by edge-defined film-fed growth (EFG) method, is described in the hydrostatic approximation by the Young–Laplace capillary equation [1], [2]γ(1R1+1R2)+ρgz=-pHere γ is the melt surface tension; ρ the melt density; g the gravity acceleration; 1/R1,1/R2 are the main normal curvatures of the free surface at a point M of the free surface of the meniscus; z is the coordinate of M with respect to the Oz-axis, directed vertically upwards; and −[ρgz+p] the pressure difference across the free surface, whose controllable part isp=pg-(H+h)ρg-pm-pMHere pm is the hydrodynamic pressure in the melt due to thermal convection, pM the pressure due to Marangoni convection; pg the pressure at the free surface of the gas flow introduced in the furnace to release the heat from the wall of the rod and the free surface of the meniscus; H the melt column “height” between the horizontal crucible melt level and the shaper top level and h the meniscus height (Fig. 1). When the crucible melt level is above the shaper top level, then H>0 and when the shaper top level is above the crucible melt level, H<0.

To calculate the meniscus free surface shape, it is convenient to employ the Young–Laplace equation (1) in its differential form. This form of Eq. (1) can be obtained as a necessary condition for the minimum of the energy of the meniscus melt column, limited by the free surface.

For the growth of a single-crystal rod of radius r1, 0<r0<r1, the differential equation for axisymmetric surface is given byz=-ρgz+pγ[1+(z)2]3/2-1r[1+(z)2]zfor0<r0rr1which is the Euler equation for the energy functionalI(z)=r0r1{γ[1+(z)2]1/2-12ρgz2-pz}rdrz(r1)=h>0,z(r0)=0Here r0 is the shaper radius (capillary channel radius) and h the static meniscus height.

For physical requirements, solutions of Eq. (3) should satisfy the following boundary conditions:(a)z(r0)=0andz(r)isstrictlyincreasingin[r0,r1](b)z(r0)=tan(π-θ)(c)z(r1)=h>0(d)z(r1)=tan(π/2-αg)

  • (a)

    expresses that at the point (r0, 0) the free surface is attached to the shaper edge and the free surface of the meniscus is relatively simple (i.e. z(r) is strictly increasing);

  • (b)

    expresses that at the point (r0, 0) the angle between the tangent to the meniscus and the horizontal is equal to the wetting angle θ;

  • (c)

    expresses that the meniscus height is equal to h>0 and

  • (d)

    expresses that at the point (r1, h) (the right end of the free surface where the solidification condition has to be realized) the angle between the tangent line to the free surface and the vertical is equal to the growth angle αg, i.e. the tangent to the crystal wall is vertical. When this condition is satisfied during the growth, then the rod radius is constant, equal to r1.

The growth angle αg and wetting angle θ, which appear in the above relations, are material constants and in this paper it is assumed that they satisfyπ/2<θ<π;0<αg<π/2;π-θ<π/2-αgSuch conditions have to be fulfilled when the radius r1 of the crystal being to grow is higher than the radius ro of the shaper (Fig. 1). As concerns the type of materials that fulfill them: the crystal material has to be such that the growth angle αg satisfies 0<αg<π/2 (e.g. for Si αg=0.2967 rad); the shaper material has to be non-wetted by the melt, i.e. θ>π/2; the shaper edge has to be sharp and the condition π−θ<π/2−αg has to be satisfied. For more details see Ref. [2].

An important problem in semiconductor crystal growth from melts by the EFG technique is the location of the range where the controllable part of pressure difference across the free surface, p, has to be or can be chosen, when ρ, γ, θ, αg and r0, r1, h are given a priori.

The state of the art during 1993–1994 concerning the dependence of the shape and size of the meniscus free surface on pressure p, for small and large Bond numbers, for the growth of single-crystal rod by the EFG technique, is summarized in [2]. According to [2], for the general differential equation (3), describing the free surface of the meniscus, there is no complete analysis and solution. For the general equation only numerical integrations were carried out for a number of process parameter values that were of practical interest at the moment. In [3] the authors investigate the influence of p on the meniscus shape for rods, in the case of middle-range Bond numbers (i.e. B0=1), which most frequently occurs in practice and has been left out of the regular study in [2]. The authors use a numerical approach in this case to solve the free surface equation written in terms of the arc length of the curve. The stability of the static free surface of the meniscus is analyzed by means of the Jacobi equation. The result of this investigation is that a large number of static menisci, having drop-like shapes, are unstable.

In [4], [5] automated crystal growth processes, based on weight sensors and computers, are analyzed. An expression for the weight of the meniscus, contacted with crystal and shaper of arbitrary shape, in which there are terms related to the hydrodynamic factor, is given. In [6] it is shown that the hydrodynamic factor is too small to be considered in the automated crystal growth.

In the present paper we locate the range where p has to be, or can be, chosen in order to obtain solutions for the nonlinear boundary value problem (NLBVP) (3) and (5). We also establish conditions that assure that the obtained solution of the NLBVP is stable or is not stable. We give a procedure for the creation of a stable static meniscus, for the growth of a rod with constant radius. We establish the temperature gradients in the meniscus and in the crystal; equation of the crystallization front displacement and equation of the crystal dimension change rate. The dynamical system governing the crystal radius and the crystallization front level evolution is analyzed. A control procedure of the cross-section of the crystal is presented.

Section snippets

Location of the range of the controllable part p of the pressure difference across the free surface

Consider the NLBVP{z=-ρgz+pγ[1+(z)2]3/2-1r[1+(z)2]z,r[r0,r1]z(r0)=0,z(r0)=tan(π-θ)z(r1)=h,z(r1)=tan(π/2-αg)z(r)isstrictlyincreasingin[r0,r1]where r0, r1, h, ρ, γ, g, p, θ and αg are real numbers having the following properties:0<r0<r1;h>0;ρ>0;γ>0;g0;π/2<θ<π0<αg<π/2;π-θ<π/2-αg

Remark 2.1

If the solution z(r) of the initial value problem (IVP):z=-ρgz+pγ[1+(z)2]3/2-1r[1+(z)2]z,rr0z(r0)=0,z(r0)=tan(π-θ)is concave (z″(r)<0) for rr0, then there is no r1>r0 and h>0 such that z(r) is a solution of

Stability analysis of static menisci

Definition 3.1

A static meniscus is stable if the function z(r), describing the free surface of the meniscus, minimizes the energy functional of the melt column:I(z)=r0r1{γ[1+(z)2]1/2-12ρgz2-pz}rdrz(r0)=0,z(r1)=h,h>0

Theorem 3.2

If z(r) is a convex solution of NLBVP (7) in [r0, mr0] (m>0) and the following inequality holds:m1/2(m-1)<πγ1/2sin3/2αgr0(ρg)1/2the static meniscus is stable.

Proof

Since Eq. (7)1 is the Euler equation of Eq. (32), it is sufficient to prove that the Legendre and Jacobi conditions are satisfied. For this

Procedure for the creation of a stable static meniscus, appropriate for the growth of a rod with constant radius

In this section first it will be shown in which way some of the explicit formulas, presented in the previous section, can be used for the determination of pressure p1 (which has to be used) for the creation of a stable static sitting drop-like meniscus with a given upper radius r1, when θ, αg, ρ, γ and r0 are given a priori and π-θ<π/2-αg. After this it will be shown in which way the melt column height H (which has to be used) has to be found, when the pressure of the gas flow (on the meniscus),

Temperature gradients at the crystallization front and crystallization front displacement

We assume that the temperature in the furnace is given byTen(z)=Ten(0)-kzwhere Ten(0) is the temperature at z=0 and k>0 is the temperature gradient in the furnace [2].

We assume also that the temperature distribution in the meniscus melt (i=1) and in the crystal (i=2) is given by a function Ti(z), i=1, 2, which satisfies the one-dimensional stationary heat transport equation [2]d2Tedz2-vχidTidz-2μiλi1r(Ti-Ten)=0where v is the pulling rate; χi the thermal diffusivity λi/ρici; λi the thermal

Equation of crystal dimension change rate

When a crystal grows in the vertical direction, the condition of constant radius r1 is that the angle made by the line tangent to the meniscus at the three-phase point and the vertical is equal to the growth angle αg. This means that the angles made by the line tangent to the meniscus at the three-phase point and the horizontal are equal to π/2-αg and α=π/2-αg, respectively.

If απ/2-αg, the crystal changes in dimension according to the law [2]drdt=-Vctan[α-(π2-αg)]ordrdt=-1Λρ1[λ1G1(r,h)-λ2G2(r,

Evolution of the crystal radius and crystallization front level

The system that governs the evolution of the crystal radius r(t) and crystallization front level h(t), according to [2] and the above considerations, is{drdt=-1Λρ1[λ1G1(r,h)-λ2G2(r,h)]tan[α-(π2-αg)]dhdt=v-1Λρ1[λ1G1(r,h)-λ2G2(r,h)]Using this system it is possible to predict the evolution of r(t) and h(t) if at t=0, when the growth process starts, r(0)=the seed radius and h(0)=the level of the crystallization front are known.

In order to grow a crystal with constant cross-section we have to

Control of cross-section during growth

In the above considerations the pulling rate v, the temperature T0 of the melt at the meniscus basis, the melt column height H between the shaper top level and the crucible melt level, the temperature gradient k in the furnace and the pressure pg of the gas flow introduced in the furnace, in order to release the heat, were considered constant. If during the growth we can change these parameters, then we can use them as control parameters.

It is clear that the steady-state solution coordinates

Conclusions

It is possible to build up a coherent mathematical model that permits explanation of the hypothetical (physical) mechanism and qualitative description of the dynamics and control of shape and size of a meniscus sitting like a drop, occurring in single-crystal growth based on the following laws:

  • 1.

    Young–Laplace law concerning the equilibrium capillary surfaces;

  • 2.

    one-dimensional stationary heat transport equation by conduction, convection and radiation; and

  • 3.

    liquid/solid phase field equation.

Acknowledgements

The authors thank the anonymous referees’ precious comments, which improved the quality of the paper.

The authors thank the Romanian National Authority for Research for supporting the research under the Grant ID 354 no. 7/2007.

References (22)

  • R. Finn

    Equilibrium Capillary Surfaces

    (1986)
  • Cited by (1)

    For Special Issue on Dynamics and Control (Firdaus E. Udwadia).

    View full text