A Perturbative Solution of Sine-Gorden Equation

Appendix F A PERTURBATIVE SOLUTION OF SINE-GORDEN EQUATION

In this appendix, we give a perturbation solution of Equation (14.3) for a small pinning force. Letting V - 6Vo, where ~ is small and h - ho + Chl -~~2h2 + ..., Equation (14.3) can be written as Oho Ot Oh1 0t

:

R q- v V 2 h o - t~74ho -+- r/,

(F.1)

=

v~72hi - t ~ 4hi - Vo sin( 2rhOc )"

(F.2)

Letting ho(r, t) - ho,o(t) + ho,1 (r, t), we have ho,o(t)

=

h o , l ( k , t)

-

(F.3)

Rt,

/o

O(k, t)e -(tck4+vk2)(t-T)dT,

(F.4)

N

where O(k, t) and ho,1 (k, t) are the spatial Fourier transform of rl(r, t) and ho,1 (r, t), respectively, as defined in Equations (A.9) and (A.10). Clearly ho,o is the average height (or thickness) of the zeroth-order perturbation, and ho,1 is the height fluctuation of the zeroth-order perturbation. Usually, 2 >, therefore Equation (F.2) can be further approximated ho,o >> ~ / < h o,1 V by Oh1 . . .- 27rVo . . ho. 1 cos( 27rRt ). Ot = v V2 hi - t~74h1 - Vo sin(27rRt) C C ' C

(F.5)

Similarly, letting hi (r, t) - hl,o(t) + h1,1 (r, t), we obtain hl,o(t)

=

h1,1 (k, t)

=

N

Voc [cos(2~Rt) _ 1]

2~R

c

2~VOc L t c~

(F.6)

' )no,1 (k, T)e -('~k4+vk2)(t-r)dT.

(F.T) Therefore, the average thickness of a film grows as Voc [cos(27rRt < h > ~ R t + ~-~-~ c )-

1],

and the power spectrum due to surface roughness evolves as

-

D

1 - e -2('~k4+vk2)t

~ k 4 + ~,k 2 399

(F.8)

400

A PERTURBATIVE

SOLUTION OF SINE-GORDEN EQUATION

120 90 60 30 0 3 2 1

(b)

0 1.2 0.8 0.4 0.0

5

10

15

20

25

30

35

40

Rt ( M L ) FIG. F.1

The average thickness < h >, the growth rate d~h>,

and the interface width

w a s a f u n c t i o n of g r o w t h t i m e a t a f i x e d r a t e R f o r u - 0, ~ -- 2, D -- 0.5, R -- 3, c -- 3, a n d V0 -- 0.2.

+

DVoc sin(27rRt)e_2(~k4+vk2)t 7rR(t~k 4 -4- vk 2) c -

-

4D VoTrR sin ( ~ c( k4 +

+ 4(

) k4 +

(F.9)

Figure F.1 shows the change of the average thickness < h >, the growth rate d and the interface width w as a function of growth time for u - 0, = 2, D = 0.5, R = 3, c = 3, and V0 = 0.2. As V0 << R, the average thickness increases almost linearly with time, but the actual growth rate oscillates with a small amplitude depending on the pinning force V0. The interface width increases and the oscillation amplitude becomes larger and larger as the time increases. The above treatment does not account for the discrete nature of the lattice, especially at the very initial stages. For example, when O - 0.5, the interface width w should not be less than 0.5. However, from Figure F.1 we can see that the calculated w is less than 0.4. This inconsistency arises when we neglect the discrete lattice effect in the Langevin equation, Equation (14.3). As long as the O is not an integer, no matter what smoothening mechanism dominates the growth,

A PERTURBATIVE

S O L U T I O N OF S I N E - G O R D E N E Q U A T I O N

401

1.8 1.5 1.2 0.9 0.6 0.3 0.0

-

0

,

2;

,

After lattice effect correction

-

4;

,

6;

i

8;

|

100

Rt (ML)

FIG. F.2 The interface width w as a function of growth time t before and after the discrete lattice effect correction. the crystalline surface has to be rough. This is the intrinsic nature of a crystalline surface. A continuous equation washes out this nature. In order to incorporate the discrete lattice effect, we make the following simple assumption: if the surface is not totally covered by adatoms, i.e., R t / c is not an integer, we assume that the surface has two basic growth modes: one is the 2-D growth and the other one is dynamic roughening. Those two modes are independent of each other. Therefore, the total interface width should be wt = v / w 2 + O ( 1 - O). Figure F.2 shows the interface width w as a function of growth time t before and after the discrete lattice effect correction. At the very initial growth stage, the discrete lattice effect contributes quite a lot to the w value, while for long times the Langevin equation dominates the roughening.