A microscopic kinetic theory of crystal growth

A microscopic kinetic theory of crystal growth

Journal of Crystal Growth 97 (1989) 319—323 North-Holland, Amsterdam 319 A MICROSCOPIC KINETIC THEORY OF CRYSTAL GROWTH S. HARRIS College of Enginee...

426KB Sizes 35 Downloads 174 Views

Journal of Crystal Growth 97 (1989) 319—323 North-Holland, Amsterdam

319

A MICROSCOPIC KINETIC THEORY OF CRYSTAL GROWTH S. HARRIS College of Engineering and Applied Sciences, SUNY, Stony Brook, New York 11794, USA

Received 30 January 1989; manuscript received in final form 5 May 1989

We re-examine the rate-determining kinetic process that is used in the Burton—Cabrera—Frank (BCF) theory of crystal growth using the Fokker—Planck equation to describe the diffusion of molecules along terraces. We show that the use of the diffusion equation with Fick’s law to relate the surface concentration and flux requires the use of a modified surface diffusion coefficient which has the form D,’ = D,(1 + o(/3)). Here D~is the surface diffusion coefficient in the absence of adsorption and desorption from and to the bulk, and /3 = I/f TD, the ratio of relaxation times characterizing surface diffusion (1/f) and desorption (Ta) and is generally a small quantity. We also present explicit results for the molecular flux entering and exiting the steps. The latter results cannot be determined in the diffusion equation description on which the BCF theory is based.

1. Introduction The basic theory for the growth of vicinal crystal surfaces has been developed by Burton, Cabrera,

ered, Fick’s law is only valid if a modified diffusion coefficient is used. However the correction to the usual surface diffusion coefficient is of order /3 l/~T0, where 1/~and TD are relaxation times

and Frank [1] (BCF). The original theory has since been generalized in a number of directions through the relaxation of a number of the original assumptions [2,3], but in all of this work the description of the fundamental rate-determining process is that of surface diffusion along the terraces to stationary steps. Diffusion to the terraces from the bulk and to the kinks in a given step are considered to be fast processes not requiring theoretical elaboration for a clear understanding of the overall growth mechanism. An essential ingredient in these theories is the use of a macroscopic diffusion equation (DE) incorporating Fick’s law relating the surface flux and density. In what follows we re-examine this surface diffusion process from a microscopic viewpoint. There are two distinct reasons motivating out interest. The first is to test the validity of Fick’s law in the present circumstances where adsorption and desorption occur and the diffusion process takes place in a bounded region. Second, as has been noted in a number of other situations [4-6], the DE does not provide a detailed description of the boundary behavior. We find that, in the order of approximation consid-

characterizing surface diffusion and desorption, and in general [3] /3 < 1 or /3 ~ 1 so that our result does does not have major quantitative implications. We will make use of the Fokker— Planck equation [7,8] (FPE) to provide the basic microscopic description. This has a fundamental advantage over the DE in that the correct macroscopic description is embedded in it and emerges as a natural outcome of the solution process. Also, with this level of description we are able to more precisely describe the boundary than with the DE [5]. For purposes of comparison we briefly describe the BCF theory in section 2. In section 3 we reformulate the surface diffusion process in terms of the FPE. In general it is only possible to find approximate solutions for the latter and in section 4 we make use of standard methods to find such a solution. Without reference to the explicit boundary condition at the steps we are able to show the validity of Fick’s law with modified diffusion coefficient D~’> D~,and we present an argument to justify this result. In the concluding section 5, we consider the separate boundary fluxes into and

=

0022-0248/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

320

S. Harris

/

Microscopic kinetic theory of crystal growth

out of the steps, and we determine their dependence on the surface and bulk properties. We wish to emphasize that our purpose here is not to detract from the remarkable achievement that the BCF theory represents but rather to view this theory in a manner consistent with the original conceptualization and from a somewhat different perspective and using different “tools”.

2. BCF theory of surface diffusion The basic kinetic process in the BCF theory is steady surface diffusion. Following adsorption from the bulk, the molecules diffuse one-dimensionally along a terrace until they are either desorbed or reach one of the two steps that terminate the terrace surface. The density of adsorbed molecules at the steps is prescribed as the equilibrium value, so that the steps are not perfect absorbers (i.e. they do not retain all incident molecules) but density.only retain The sufficient exchange molecules of molecules to with maintain the bulk this is described through separate adsorption and desorption terms that are included in the steady-state DE, D~-~—~+ CB —

dx

-~-

=

0,



~

~ ~L.

(1)

3.1. The FPE formulation

The FPE provides a microscopic description as opposed to the DE which is limited to the macroscopic level. The former is fundamental in that the macroscopic description can be, at least in principlc, extracted from it [7,8], e.g. by averaging and a closure approximation (see below). An additional advantage of the FPE is that it permits a more detailed description at boundaries where incident and emergent fluxes can be separately determined. For a one-dimensional system without sinks and sources the steady FPE is

(

a2

a

kT av +1+v—

)f Qf

(3)

where f(x, v) is the density in position—velocity space and ~ is the friction coefficient which is related to the diffusion coefficient through the Einstein relationship D~ kT/~m (in what follows we set the molecule mass m 1). The quantity 1/~which has units time, is the relaxation time characterizing the surface diffusion process in the FPE description [8]. Moments of f are defined as =

KN

KN°(1 + a).

(2a)

In equilibrium the adsorption and desorption balance so that C~= KN° n°/TD, 0 the= equilibrium surface concentration.(2b) Eq. withis na flux balance and directly incorporates the (1)

Fick’s law relationship j D~dn/dx between the surface flux j and concentration n. In what follows we will examine two aspects of this de=



M,(x)

ci~v7(x, v),

=

(4)



The moments M 0 and M1 are, respectively, the molecule concentration and flux and therefore the

the equilibrium bulk concentration [2], so that =

3. The Fokker—Planck equation

=

Here D~is the surface diffusion coefficient, n(x) is the density of adsorbed molecules on the terrace which is of length L, and CB accounts for adsorption from the bulk and is proportional to the bulk concentration N. If the relative supersaturation a is small we can express N in terms of a and N°,

CB

scription, the use of Fick’s law and the details of the flux at the steps where the DE can only determine the net flux.

FPE can be used to generate equations of change for these quantities. Unfortunately these equations are not closed because the gradient term on the left-hand side of eq. (3) introduces a term Mr±i into the equation for Mr found by multiplying the FPE by (Technically this is an the operation andf~ notdour. a multiplication.) To solve r moment equations for M 0, M1,... Mr it is then necessary to approximate Mr+ which appears in the last equation, in terms of the former quantities thus “closing” the equations by reducing the num~,

S. Harris

/

321

Microscopic kinetic theory of crystal growth

ber of unknowns to the number of equations. In most applications a closure is effected by approximating f and a variety of methods have been developed for this purpose [9].

4. Approximate solution of the Fokker—Planck equation

3.2. FPE theory of surface diffusion

f= ~

We look for a solution of the form [9]

s=0

The FPE must be modified to account for the adsorption and desorption processes before being applied to the surface diffusion problem described above. Assuming that adsorbed molecules have an equilibrium velocity distribution (the bulk is supposed to be near equilibrium) we can write a modified steady FPE as

Qf+ c~, (2~kT)1~2 2/kT)

_~

exp(—v

(5)

TD’

2/2kT) a~(x)H~(v)exp(—u (2~kT)”~2

(7)

where the a are related to the moments of f and H~(v) are Hermite polynomials, the eigenfunctions of Q [7]. In general we are more interested in the moments of f than f itself. A standard approach [9] to determine these quantities is to truncate the above expansion, i.e. set a~ 0 for ~ ~ Here we choose s * 4 and use eqs. (6) to =

=

determine the four unknowns a 0, a1, a2, convenience we re-write the truncated f asa3. For

(~~)~+~

u / v2

where CB is again determined by the equilibrium conditions on the moments of f. A more complete treatment would include the derivation of a modified FPE from first principles. We justify the use of eq. (5) here on the same basis that eq. (1) is used in the BCF theory. In both cases the

f=

+

+



x exp(

— v2/2kT) (2~rkT)1~2

where

n

=



3)~J (8)

,

M

0, j = M1, q = M2/2kT— ~n, and M3/6kT—j/2kT. These identities follow from

surface—bulk interaction is of secondary importance compared to the rate-determining surface diffusion process. If we integrate eq. (5) over v we will obtain an equation relating the concentration n M0 and flux I M1: dM1/dx CB Mo/TD, (6a)

p eq. (8) by explicit calculation as does the closure relationship

and we see that if Fick’s law holds we have derived eq. (1). We will also make use of additional moment equations by multiplying 3 and found then integrating over eq. (5) by u, v2, and u v:

qualitatively describe any departure from the behavior predicted by the macroscopic description, here eq.explicit (1). The boundary condition at the steps will be used in the next section. For our present purposes we will only have to make use of two

=

=

=



=

2M M4 = 6kTM2 + 3(kT)

0.

(9)

Based on experience with the solution of Boltzmann’s equation [9], we expect that the approximation made in using eq. (8) will allow us to

dM 2/dx

=

0,

(6b)

dM3/dx + ~2M2



2~kTM0= kTCB,

(6c)

dM4/dx



6.EkTM1

+ E1M1

+ ~3M3

=

0,

(6d)

with ~ 5~+ l/TD. In the FPE description the individual moments follow from the solution for f, and to proceed we consider a standard approximation scheme. =

general conditions which the solutions must satisfy, symmetry, i.e. the even moments must be even functions of x while the odd moments must be odd functions, and correct limiting behavior as l/~D 0 (i.e. the surface—volume interaction becomes negligible) which in the present situation requires that the even moments remain independent of x and the odd moments identically vanish.

/3

=

—~

322

5. Harris

/

Microscopic kinetic theory of crystal growth

The above system of moment equations together with the closure condition on M 4 leads to the following fourth order differential equation for

where again a~2 is the positive root of a~. Explicitly calculating the latter we find 2/3/kT)(1 /3) + o(/3~), a (~ -

(17)

=

d 4M2 + ~ —

=

6~E0+ 2~~)d2M2

+

The relationship between M1 and the gradient of M0 can now be determined from eqs. (14)—(16). D~’ Denoting we findthe Fick’s law diffusion coefficient as

~O~rE2~3

2 3(kT) — (CB/3kT)~o~l,~

3(l + ~ (10) 1. The solution of the homogewhere equation d,1 d/dx’ neous is determined by the four distinct roots of the algebraic equation d4 + A d2 B, where A and B are the coefficients of the d 2M2 and M2 terms in eq. (10). Writing out these coefficients explicitly,

-

= =

(k7-’,,/fl[l +

/3

(18)

+ O(~~)],

=

A

(~2/3kT)[6

=

+

14/3 + 6/32],

2I[6 + 11$

+ 6/32 +

[~~/3(kT)

B=

o($~)].

(ha) (lib)

If we define a1, a2 2 as4AB)~2j, a1 _~[A (A —

=

2

a 2

(12)





4AB)1/2],

~[A + (A

=

then the roots of the characteristic equation are ±aV2,±aV2.The second pair is of o(1) and are pure imaginary, while the first pair is of o(/3) and are real. Therefore =

a

2x + a 1 cosh aV

2x

+

2 cos aV

i.e. Fick’s law holds, but with a dressed diffusion coefficient. For /3 << 1, D~’ approximates the surface diffusion coefficient D~ kT/~, but in =

general this is not the case although we expect that this section with hold. a short discusin We mostconclude circumstances /3 < 1 does sion of the above result and consider the explicit solution for M 0 and M1 in the next section. What we ahave shown is that the surface flux is enhanced as result of the adsorption—desorption process, and we believe that this can be explained by fairly straightforward reasoning. For a steady state to exist there must be a net flux onto the surface from the bulk and this must be exactly cornpensated by a net flux into the steps. Thus, e.g. in the region 0 ~ x ~ L along the terrace there are fewer molecules moving to the left (u < 0) than

a 3,

(13)

would occur in the case of one-dimensional diffu-

where a3 is the particular solution of eq. (10). Since a2 is o(1), we require that a2 0 in order to satisfy the condition that for /3 0, M2 is inde-

sion on the terrace with no net adsorption—dcsorption flux and compensating exit flux at the boundary. This results in a increased flux, moving to the right, as described by eq. (18).

=

—~

pendent of x. Then, from eqs. (6a) and (6c), we have M0

=

(h/~~~~) d2M2

+ CB/~O,

(14) 5. The boundary fluxes

or

1~2x+ C~/~

M0 = a cosh a

(15)

In concluding this study we consider the entry

where we can now drop the subscript on a1 and on a1, here, and in what follows. We also have from eq. (6b) 2 sinh a1~’2x, M1 —(1/~~)d1M2 —a~0a”~ (16)

and exit molecular fluxes at the step. This is a level of detail that cannot be obtained with the DE which only describes the net flux. The explicit boundary condition is M~ 1(~L) n°, the step equilibrium concentration [2], from which it follows directly from the general solutions for M0, M1

=

0,

=

=

S. Harris

/ Microscopic kinetic theory of crystal growth

and M2 found earlier, eqs. (15), (16) and (13), together with M3 as given by eq. (6d) that 2 M cosh ~La1”2’ xa~ cosh 0 n°(1 + a) (19) =



D~’n°aa’~2 sinh xa1~2 cosh ~La”2’

=

(20)

dent molecules retained in the steps is also small. In the case where > we have ~-,

j

2[1 _nb(kT/2~)l/

where 0.46




a(2~$)1”2E’



(26)

1.00 and the same conclusion

holds. A more detailed theory of the step absorp-

k Tn°(1 + a) (1 + 2

=

323

tion efficiency would have to include the dynamics of the step boundary growth in determin-

~o~i cosh xa1~’2 —n 0a— a cosh ~La1~2’

(21)

ing ratherinthan ing the the boundary boundary conditions fixed as is done bothconsiderhe BCF theory and our microscopic treatment of this

M 3

~1

=

[6kT~M,



problem.

3kTd1(2M2 + kTM0)].

(22) In the above equations we have used. eqs. (2) to replace GB in favor of the relative supersaturation 0. a and concentration n now be found The equilibrium entry flux into the steps can by explicit calculation from eq. (8) with ,~, j, q and p given by the above M~.The flux into the step at x ~L from the terrace, j~’is

Acknowledgement This work was supported in part by the Research Foundation of the State University of New York.

=

i±=I

dvuf(~L, v)

References

no(kT/2~)1/2[1 + a(2~/3)~”2(1 + ~)

=

xtanh ~La1’~2 /3a + Similarly, the exit flux at x —

o(s5/2)1 =

.

~

(23)

~L onto the terrace,

j, is

j

=

fo

dvvf(~L, v)

— =

no(kT/2~.)1/2[$a + (2~$)1/2(1 + 1/3)

Xtanh ~La~’2



(24)

o($5/2)].

If the step were a perfect absorber, i would be identically zero. For the conditions that apply here, where a and /3 are small, we can approximate j when ~ is also small as —

_n0(kT/2~r)lz”2[1



/3a(1

+ (2~)1/2f],

(25) and we see that in this case the fraction of

mci-

W.K. Burton, N. Cabrera and F.C. Frank, Phil. Trans.

Soc. London A243 299.Problems (Wiley, New [2] Roy. R. Ghez, A Primer of (1951) Diffusion York, 1988) section 2.6. [3] P. Bennema and G.H. Gilmer, in: Crystal Growth; An Introduction, Ed. P. Hartman (North-Holland, Amsterdam, 1973) p. 263. [4] S. Harris, Phys. Rev. A, to be published. [5] S. Harris, J. Chem. Phys. 75 (1981) 3103. [6] S. Harris, Phys. Rev. A36 (1987) 3392; see also references cited therein. [7] C. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, 1983) section 5.3.6. Referred to here and in the following reference as Kramers equation. [8] N.G. van Kampen, Stochastic Problems in Physics and Chemistry (North-Holland, Amsterdam. 1981) section VIII.7. [9] The methods used are identical to those used in connection with Boltzmann’s equation. See, e.g., M. Kogan, Rarefied Gas Dynamics (Plenum, New York, 1969) sections 3.3, 3.4; S. Harris, Introduction to the Theory of the Boltzmann Equation (Holt, Rinehart and Winston. New York, 1971) ch. 7.