Quantitative characterization of the mesoscopic surface roughness in a growing island film

Computational Materials Science 33 (2005) 369–374 www.elsevier.com/locate/commatsci

Quantitative characterization of the mesoscopic surface roughness in a growing island film Vladimir I. Trofimov b

a,*

, Ilya V. Trofimov a, Jong-Il Kim

b

a Institute of Radioengineering and Electronics of RAS, 11/7 Mokhovaya Street, 125009 Moscow, Russia Cheonan Valley, CN-Regional Innovation Agency 43-5, Sameun-Ri, Jiksan-Eup, Cheonan-Si, Chungnam-Do 330-708, South Korea

Abstract A quantitative description of mesoscopic surface roughness generated in thin film growing via 3D island mechanism is presented. Analysis is based on the statistical model of a random nucleation and growth of hemispherical islands accounting for their collisions at late stages. Analytical expressions for a number of surface relief parameters: the rms roughness, the roughness coefficient, the surface height (depth) distribution and the package density factor providing a rather complete quantitative description of the evolving surface morphology during growth process in different condensation regimes are derived. It is shown that the surface height distribution is a non-Gaussian with a negative skewness and that the rms roughness and the roughness coefficient kinetics can be represented as a universal (independent of a condensation regime) unimodal function of either coverage or film thickness with a maximum just prior the completed film formation. The non-monotonic surface roughness dynamics in a growth process predicted by the model is consistent with experimental data. Ó 2004 Elsevier B.V. All rights reserved.

1. Introduction Interface roughness is one of the central features in many film technologies, since it directly controls many physical film properties and plays a key role in the so-called surface enhanced phenomena, e.g., the giant Raman scattering, so a

*

Corresponding author. Tel.: +7 095 203 3689; fax: +7 095 203 8414. E-mail address: [email protected] (V.I. Trofimov).

quantitative characterization of the surface morphology is of great importance for materials science. A huge number of theoretical studies had been devoted to a noise-induced surface roughening in stochastic growth models (ballistic and random deposition, Eden, Kardar–Parisi–Zhang, solid-on-solid), which gives rise to an interesting phenomenon of dynamic scaling [1–3]. However, the surface morphology in ‘‘classical’’ Volmer– Weber (VW) 3D island growth model had been studied a little [4–8]. In this article, we present a quantitative description of the mesoscopic surface

0927-0256/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2004.12.004

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V.I. Trofimov et al. / Computational Materials Science 33 (2005) 369–374

roughness generated in thin film growing via VW mechanism. The treatment is based on the statistical growth models approach [9] developed by an extension of a familiar theory of Kolmogorov [10], Johnson and Mehl [11], and Avrami [12] (KJMA) taking into account finite-size effects in thin films.

2. Growth model In this section, we briefly recall our growth model [13] that allows describing film growth in a self-consistent manner at all deposition stages. Initially, free islands grow isotropically in a hemisphere form. After collisions there are two limiting cases: coalescence, when colliding islands merge each other and impingement, when no redistribution of matter among colliding islands occurs. We suppose here that the film growth is governed by the impingement mechanism, i.e. after collisions islands simply cease growth at the contact boundaries and continue to grow in all other available directions with a growth law, which will be found self-consistently below. The main concept of the model is a feeding zone (FZ) that is introduced as a join of substrate regions covered by islands and a band of width Xa adjoining to the islands perimeter; Xa = (Dsa)1/2 is a mean diffusion length of adatom before desorption, D the surface diffusion coefficient of adatom, sa a mean residence time of adatom on a substrate before desorption. It is supposed that all the atoms incoming onto a FZ are incorporated into a film, whereas those landing outside a FZ reevaporate from a substrate. Designate a specific FZ area at time moment t as nf(t), and a specific film volume as V(t), which coincides obviously with a mean film thickness h. Then, a simple kinetic equation may be written down dV ðtÞ=dt ¼ dh=dt ¼ J Xnf ðtÞ;

ð1Þ

where J is a deposition rate and X an atomic volume. Consider a film at some time moment t (Fig. 1). Let n(z, t) be a coverage of a film section at height z from a substrate; itÕs clear that n(z, t) is ever decreasing function of z: n(0, t) = n(t)  substrate

Fig. 1. Sketch of a film at some time moment t.

coverage and n(z, t) = 0 at z P Rm, Rm is the largest island radius. Then, a specific film volume may be written down as Z 1 V ðtÞ ¼ nðz; tÞ dz: ð2Þ 0

To find a function n(z, t) one needs to specify a nucleation mode. We assume that all the islands nucleated at time t = 0 in random points of substrate with density N (cm2). This assumption of instant nucleation serves a good approximation of a nucleation process in thin film [9]. Let A is some point at a height z from a substrate. The probability q(z, t) that it will survive by time t, i.e. not be captured by a growing film, coincides evidently with a probability of nuclei absence in a circle of a radius r = [R2(t)  z2]1/2, which due to the Poisson distribution of nuclei over a substrate is given by q(z, t) = exp[pN(R2(t)  z2)], i.e.   ð3Þ nðz; tÞ ¼ 1  exp pN ðR2 ðtÞ  z2 Þ : Insert of Eq. (3) into Eq. (2) yields Z Rm     dz: V ðtÞ ¼ 1  exp pN R2 ðtÞ  z2

ð4Þ

0

Noticing that a specific FZ coincides with substrate coverage n by an imaginary film generated from a real film by an instant increment of all the islands by DR = Xa, from Eq. (1) one gets a second necessary equation n h io dV 2 ¼ J X 1  exp pN ðRðtÞ þ X a Þ : ð5Þ dt Coupled Eqs. (4) and (5) determine self-consistently a desired growth law R(t) Z q 0 R q0 2q 0 exp ½ðq02  y 2 Þ dy 0  ð6Þ pffiffi 2  dq ¼ s 0 1  exp ðq0 þ zÞ

V.I. Trofimov et al. / Computational Materials Science 33 (2005) 369–374

in dimensionless variables q = R/a and s = t/t0 with natural scales of the model a = (pN)1/2, t0 = 2 a/(JX); here, z  ðX a =aÞ ¼ pNX 2a is a single model parameter, characterizing condensation regime: at z 1 (high substrate temperatures) a so called extremely incomplete condensation (EIC) takes place; with increasing z a crossover to incomplete (z < 1) and in the limit z ! 1–to complete condensation (CC) occurs. Calculation results with Eq. (6) for various z are shown in Fig. 2 by solid curves. There exist two limiting curves at z = 1 and 0 corresponding to CC and EIC regime, respectively. All the curves tend with time towards the limiting curve for z = 1, since with increasing substrate coverage any initial condensation regime determined by the parameter z asymptotically tends towards CC. As is seen the curve with z = 1 nearly coincides with a limiting curve forz = 1, it means that a CC regime is practically realized at z P 1. It is also seen that growth accelerates at late stages because of the reduction of the growth front perimeter at island collisions. To detect an effect of collisions in explicit form consider growth of a free island. The latter occurs due to capture of atoms from its feeding zone, a circle of a radius R + Xa, so that accounting for their overlapping, we can write down

3

4

3

2

h' 2

5 1

1

0

1

2

3

τ

Fig. 3. The time dependence of the mean film thickness at different values of the parameter z: 104 (1), 102 (2), 101 (3), 1 (4) and 1 (5).

Calculation results with Eq. (7) confirm unambiguously (Fig. 2, broken curves) the growth acceleration at late stages caused by island collisions. This effect is naturally stronger manifested in CC regime because of constancy of an average atomic flux onto each island during growth process. The knowledge of the growth law allows calculating kinetics of a mean film thickness and substrate coverage Z s 2 h0 ¼ s  exp½ðqðs0 Þ þ z1=2 Þ ds0 h0  h=a; 0

ð8Þ nðsÞ ¼ 1  expðq2 ðsÞÞ

ð9Þ

2

2

2pR dR=dt ¼ J Xf1  expbpN ðR þ X a Þ cg; whence Z q 0

371

1  exp



2q0

2

ðq0

0 pffiffi 2  dq ¼ s: þ zÞ

ð7Þ

ρ ρ ∝ τ 1/3

2 1

needed for analysis the surface roughness evolution in growing film. Calculation results with Eq. (8) show that like the growth law with increasing z kinetic curves h0 ðsÞ tend asymptotically towards a limiting law h0 ¼ s for a CC and at z = 1 almost merge it, and any kinetic curve h0 ðsÞ tends with time to a linear law h0 / s (Fig. 3).

3. Surface morphology z=∝

1 0.1

z =0 10 -2

0.001

10

0.01

-4

0.1

1

τ

Fig. 2. Growth law in different condensation regimes.

Now we can describe quantitatively the mesoscopic roughness of a film surface composed of hemispherical growth hillocks,shown schematically in Fig. 1. Designate as n(h, t) a coverage of a film section at height h from a substrate. It coincides obviously with a probability that a surface height at a randomly chosen substrate point at

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V.I. Trofimov et al. / Computational Materials Science 33 (2005) 369–374

~ðsÞ moves from one limtemperature) the curve r iting curve (z = 1) corresponding to CC to another one (z = 0) corresponding to EIC, because of the growth rate decrease caused by desorption of adatoms. However, a maximum height ~m ffi 0:457 does not depend on a condensation rer gime, and to all sm values the same coverage value corresponds. Indeed, both quantities q and h0 are universal functions of coverage: q(n) is given by Eq. (9), and h0 given by Eq. (8) may be rewritten as Z ð ln j1njÞ1=2 h i 2 h0 ðnÞ ¼ 1  ð1  nÞex dx ð12Þ

time t is not less than h, hence a probability distribution density of a surface height may be written down as  onðh; tÞ=oh; 0 6 h 6 R; ft ðhÞ ¼ ð1  nðtÞÞdðhÞ þ 0; R 6 h; ð10Þ where the first term with a delta-function accounts for the contribution of the uncovered parts of a substrate and n(h, t) is given by Eq. (3). Now we can calculate the variety of the surface morphology characteristics. One of its main is the R1 2 rms roughness r2  0 ðh  hÞ ft ðhÞ dh for which using Eq. (10) one gets 2

~2 ðsÞ ¼ q2 ðsÞ  h0 ðsÞ  nðsÞ; r

~ ¼ r=a: r

0

~ given by Eq. (12) can also be repand therefore r resented as a universal function of either coverage ~ðnÞ is depicted or film thickness. The first of them, r by a curve (Fig. 5, left) with a maximum ~m ffi 0:457 just prior completed film formation, r ~ðh0 Þ that may be nm ffi 0.856. The second one, r more useful in applications, especially when one deals with a continuous film, is shown in Fig. 5 (right) where as an example the absolute values of r and h for typical interisland distance a = 25 nm are indicated. At n ffi 1 Eq. (12) simplifies: h0 ffi q  1=2q, i.e. q ffi h0 þ 1=2h0 , so from Eq. (9) it follows that in continuous film ðh0 > 3Þ the rms roughness drops with film thickness in a ~  1=2h0 , i.e. rather slowly so that in simple law r film of finite thickness a finite surface relief is formed, which depends on a scale a = (pN)1/2, i.e., on the island density. Thus, at typical for metallic films a = 25 nm in 60-nm film r ffi 3 nm in consistence with experimentally measured roughness (e.g., [14]).

ð11Þ

Eq. (11) combined with Eqs. (7)–(9) yields curve ~ðsÞ with a maximum (Fig. 4) what has a simple r explanation: initially the nucleation and growth of islands increase the roughness but at later stages due to islands impingement and filling of interisland voids the roughness decreases with time. With decreasing z (i.e. increasing substrate

z =∝

0.5

1 10

-2

10

-4

z=0

σ~

0

2

1

3

τ

4

Fig. 4. The time evolution of the rms surface roughness in different condensation regimes.

σ~, k

50

σ~

k

1

100

200

h , nm σ , nm

0.5

10

σ~

5 0

ξ

1

0

5

h'

10

Fig. 5. The rms roughness and the roughness coefficient as a function of coverage (left) and the rms roughness as a function of average film thickness (right).

V.I. Trofimov et al. / Computational Materials Science 33 (2005) 369–374

h'd 1

10

-2

z = 10

373

h'd

-4

0.5

0 .5

2

1

0

3

4

τ

ξ

0

1

Fig. 6. The average surface depth kinetics at various values of z (left plot) and function of coverage (right).

Another important surface relief characteristic is the mean surface depth h0d ¼ q  h0 . Its kinetics behaviour depending on the condensation regime is like to that of the rms roughness (Fig. 6, left). This quantity is also a universal function of either coverage or film thickness. The first of them, h0d ðnÞ is also depicted with a unimodal curve with an earlier (nm ffi 0.6) maximum h0d m ffi 0:54 (Fig. 6, right). The next important surface morphology characteristic (e.g., for adsorption film properties) is a roughness coefficient k defined as a specific actual film surface. It coincides obviously with a specific growth front surface, i.e. equals to a derivative of a specific film volume (or average film thickness) with respect to a radius: k ¼ dV =dR ¼ dh=dR ¼ dh0 =dq, whence k ¼ 2qðq  h0 Þ:

ð13Þ

This quantity is also a universal function of either n or h0 and the former is depicted by a curve (Fig. ~ðnÞ with a slightly later 5, left) similar to that of r (nm = 0.895) maximum km = 1.285.

~ f (h )

In continuous film, throwing off the first term in Eq. (10) for the probability distribution density of a deviation of a surface height from its average, ~h ¼ h0  h0 one gets f ð~ hÞ ¼

8 h  < 2ð~ h þ h0 Þexp  1 þ

2 1 ~ h  2~ hh0 4h02

i

: 0;

; ~ h6

1 ; 2h0

~ h>

1 : 2h0

ð14Þ Fig. 7 shows how the surface width narrows down with film thickness and that the surface height distribution is asymmetric, non-Gaussian, with a negative skewedness. In optical thin film applications, for characterizing a film microstructure is used the so-called packing density factor (PDF) defined as a ratio of a volume of the solid part of the film to the total volume occupied by a film [15] P ¼ h=hg ¼ h0 =h0g ;

ð15Þ

where the geometric film thickness hg in our model coincides with a maximum island radius hg = Rm(t). Calculations with Eq. (15) show (Fig. 8) how the PDF tends to unity with film thickness because of the surface roughness decrease. This plot may be used for estimation of a contribution

h' = 3 k , P ,σ

5 h' = 2

k

1 P

h' = 1

- 0.5

0

0 .5

~ h'

Fig. 7. The time evolution of the surface height distribution.

σ

0

1

2

3

4

h'

Fig. 8. The roughness coefficient, packing density factor and rms roughness as a function of a mean film thickness.

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V.I. Trofimov et al. / Computational Materials Science 33 (2005) 369–374

of surface roughness associated porosity to an overall film porosity. 4. Conclusions A quantitative description of the surface morphology evolution in thin film growing via 3D hemispherical island mechanism is presented. It is found that while the time evolution of the surface roughness parameters depends on growth regime, each of them can be represented as universal function of either coverage or average film thickness, e.g. the rms surface roughness and the roughness coefficient are unimodal function of either coverage or average film thickness with a maximum just prior the completed film formation. The surface height distribution is non-Gaussian with negative skewness. Acknowledgement This work was partially supported by the Russian Foundation for Basis Research under grant number 04-02-17681.

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