Cao Z. (Ed.) Thin Film Growth: Physics, materials science and applications

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56 Thin ﬁlm growth

substrate, ordered arrays can be formed. In Yu and Voigt (2009) such a strain

eld is used to nucleate islands in an ordered array which leads to perfectly

ordered trenches with tunable size in the phase-eld simulation.

3.3.2 Spiral and mound growth

The large-scale shape of spirals and mounds are similar. However, it is

clear that the growth mechanism near the top of the structure is completely

different for spiral mounds and so-called wedding cakes. The growth of

wedding cakes is governed by the rate of two-dimensional nucleation of

islands on the top terrace, which is enhanced by the connement of adatoms

due to the Ehrlich–Schwoebel barrier. In contrast, atoms landing near the

top of the spiral mound can avoid the Ehrlich–Schwoebel barrier by moving

around the core which should reduce the local adatom concentration and

hence the growth rate. On the other hand the indelible presence of the step

emanating from the screw dislocation obviates the need for two-dimensional

nucleation, which should increase the rate of vertical growth. We use a

setup in which both growth modes are possible and are interested in their

competition. Two-dimensional nucleation is thereby modeled as an intrinsic

feature of the phase-eld approximation. If the adatom density becomes

sufciently large a new terrace nucleates as a result of the changing barrier

in the multiwell potential. Figure 3.2 shows a typical surface conguration

after deposition of 200 monolayers. Under these conditions in which the

adatom concentration is allowed to become sufciently large so that islands

nucleate deterministically, spiral mounds were generally seen to grow higher

than regular wedding cakes. The results nicely coincide with the experiments

on Pt(111) surfaces in Redinger et al. (2008).

3.3.3 Anomalous spiral growth

The approach discussed in the previous section neglects the inuence of

the strain eld resulting from the screw dislocation. Taking this effect into

account might lead to anomalous spiral step motion as found on a Si(001)

surface by Hannon et al. (2006) and Voorhees (2006). The anomalous

behavior is attributed to interaction between the surface structure and the

strain eld of the screw dislocation. The screw dislocation thereby breaks

two single atomic height steps. At each step, the surface phase switches

between (1 ¥ 2) and (2 ¥ 1). The strain eld of the screw dislocation causes

an asymmetry between the two phases, with one phase more energetically

favored than the other. The step growth is coupled with the asymmetric

phase switching between (1 ¥ 2) and (2 ¥ 1), causing the anomalous spiral

prole. The related dynamics for this growth mode is analysed by Yu et al.

(2009). In order to account for the elastic effects we use alternating surface

ThinFilm-Zexian-03.indd 56 7/1/11 9:40:03 AM

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57Phase-ﬁeld modeling of thin ﬁlm growth

phases of (1 ¥ 2) and (2 ¥ 1) with different surface energies. We dene

Dg = g

2¥1

– g

1¥2

corresponding to the strain eld of the screw dislocation,

which enters the denition of the multiwell potential. Figure 3.3 shows

typical congurations under the inuence of Dg.

3.4 Conclusion

Phase-eld models have been used to approximate BCF-type models for step

ow growth. The phase-eld variable thereby models the discrete height

of the lm. The novelty of such models is the ability to approximate not

only the jump in the uxes across the step edges but also the jump in the

density eld which results from the kinetic boundary conditions. This is

realized through a degenerate mobility function in the phase-eld equation,

which can be interpreted as a degenerate Cahn–Hilliard-type equation. The

phase-eld approach allows the nucleation of new islands, it allows for

coalescence and also for vanishing of islands, without special treatment.

The derived set of equations can be solved with standard tools; however, an

adaptive mesh renement along the step edges is desirable. Using matched

asymptotic analysis the connection between these phase-eld models and

the BCF model can be shown.

The approach is applied to understand various interesting phenomena

in thin lm growth. Among them are the formation of trenches as a result

3.2 Growth spirals and mounds obtained from phase-ﬁeld simulation

with D = 10, F = 0.2 ML/s, g = 1, r* = 0.1, k

= 10 and k

–

= 1. The

surface is shown after deposition of 200 ML showing spiral mounds

and wedding cakes and demonstrating the faster growth of the spiral

mounds. The simulations were performed by A. Rätz.

ThinFilm-Zexian-03.indd 57 7/1/11 9:40:03 AM

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58 Thin ﬁlm growth

of the Ehrlich–Schwoebel barrier and anisotropic kinetic coefcients, the

competition between mound and spiral growth, which for typical material

parameters will always be dominated by spirals, and an anomalous spiral

growth mode, which results from the interaction of the strain eld associated

with the screw dislocation and the surface structure.

3.5 References

Borca B, Fruchart O, David P, Rousseau A, Meyer C (2007) ‘Kinetic self-organization

of trenched templates for the fabrication of versatile ferromagnetic nanowires’, Appl.

Phys. Lett. 90 142507.

Burton W K, Cabrera N, Frank F C (1951) ‘The growth of crystals and the equilibrium

of their surfaces’, Phil. Trans. R Soc. A 243 299–358.

Hannon J B, Shenoy V B, Schwarz K W (2006) ‘Anomalous spiral motion of steps near

dislocations on silicon surfaces’, Science 313 1266–1269.

Hausser F, Voigt A (2008) ‘Step meandering in epitaxial growth’, J. Cryst. Growth

303, 80–84.

(a)

(d) (e)

(b) (c)

(f)

3.3 Images of the spiral growth at the steady-state regime simulated

using F = 0.005 ML/s (left), 0.02 ML/s (middle) and 0.04 ML/s (right)

for non-zero in (a)–(c) and zero in (d)–(f). The lightest gray color

corresponds to the highest surface height, being around 109, 193,

231, 111, 200 and 233 atomic layers in (a)–(f), respectively. The sharp

change in gray indicates the position of the step. Inset (enlarged):

the corresponding simulated image of the surface termination,

where the (2 ¥ 1) and (1 ¥ 2) phases are denoted by bright and dark,

respectively, and the position of the step is indicated by switching

between bright and dark. The simulations were performed by Y.-M.

Yu.

ThinFilm-Zexian-03.indd 58 7/1/11 9:40:03 AM

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59Phase-ﬁeld modeling of thin ﬁlm growth

Hu Z Z, Lowengrub J S, Wise S M, Voigt A (2011) ‘Phase-eld modeling of epitaxial

growth: applications to step trains and island dynamics’, Physica D (to appear)

Karma A, Plapp M (1998) ‘Spiral surface growth without desorption’, Phys. Rev. Lett.

81 4444–4447.

Liu F, Metiu H (1997) ‘Stability and kinetics of step motion on crystal surfaces’, Phys.

Rev. E 49 2601–2616.

Otto F, Penzler P, Rätz A, Rump T, Voigt A (2004) ‘A diffuse interface approximation

for step ow in epitaxial growth’, Nonlin. 17 477–491.

Redinger A, Ricken O, Kuhn P, Rätz A, Voigt A, Krug J, Michely T (2008) ‘Spiral

growth and step edge barriers’, Phys. Rev. Lett. 100 035506.

Torabi S, Lowengrub J, Voigt A, Wise S (2009) ‘A new phase-eld model for strongly

anisotropic systems’, Proc. Roy. Soc. A 465 1337–1359.

Voorhees P W (2006) ‘Materials science – step dances on surfaces’, Science 313

1247–1248.

Yeon D H, Cha P R, Chung S I, Yonn J K (2003) ‘Phase-eld model for the dynamics

of steps and islands on crystal surfaces’, Metals Mater. Int. 9 67–76.

Yeon D H, Cha P R, Lowengrub J S, Voigt A, Thornton K (2007) ‘Linear stability analysis

for step meandering instabilities with elastic interactions and Ehrlich–Schwoebel

barriers’, Phys. Rev. E 76 011601.

Yu Y-M, Voigt A (2009) ‘Directed self-organization of trenched templates for nanowire

growth’, Appl. Phys. Lett. 94 043108.

Yu Y-M, Backofen R, Voigt A (2008) ‘Phase-eld simulation of stripe arrays on metal

bcc(110) surfaces’, Phys. Rev. E 77 051605.

Yu Y-M, Liu B G, Voigt A (2009) ‘Phase-eld modeling of anomalous spiral step growth

on Si(001) surfaces’, Phys. Rev. B 79 235317.

ThinFilm-Zexian-03.indd 59 7/1/11 9:40:03 AM

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Analysing surface roughness evolution

in thin films

Y. KajiKawa, The University of Tokyo, japan

Abstract: Surface roughness is a key factor in determining lm quality and

the physical properties of materials deposited by chemical vapor deposition

(CVD). However, there have been few attempts to comprehensively

overview the mechanism underlying the evolution of lm roughness

during deposition, from atomic to macroscopic scale. In this chapter, the

mechanisms causing roughness evolution during CVD are surveyed from

a phenomenological viewpoint. Firstly, roughness formation processes

at atomic scale are reviewed, focusing on homo-epitaxial growth. Next,

roughness generation during hetero- or non-epitaxial growth is discussed,

as well as the process of roughness amplication. In the growth stage of a

lm, roughness evolution can be classied into two types: positive feedback

growth and non-positive feedback growth, and these are discussed in the

nal part of the chapter.

Key words: surface roughness, nucleation and growth, mechanism,

modeling.

4.1 Introduction

Chemical vapor deposition (CVD) involves many chemical and physical

phenomena: in the gas phase with a ow of a mixture of reactant and carrier

gas, in the solid phase of deposited materials, and at the surface where a

variety of elementary processes occur to determine the structure and physical

properties of deposited materials. By controlling operating conditions and

reactor geometry, highly uniform coatings can be obtained over a broad

range of scale levels, from atomic to reactor. However, CVD often produces

a variety of surface roughnesses which in most cases is not suitable for

application. Because currently the roughness is suppressed by trial-and-error

adjustments of operating conditions, at factory level, many wafers are wasted

to nd a good recipe for the process. Although a better understanding of the

underlying phenomena enables better design of CVD equipment and efcient

adjustment of operating conditions, there are few comprehensive surveys on

the underlying mechanism of surface roughness evolution.

To date, a number of reviews have been published on roughness evolution

during vapor deposition, owing to its importance in application and also to

scientic research interests (Meakin, 1993; Krug, 1997; Politi et al., 2000).

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61Analysing surface roughness evolution in thin ﬁlms

These reviews usually focus only on the specic phenomena causing roughness

evolution, such as anisotropic diffusion and shadowing instability, and its

strict mathematical expressions. They can provide scholars with guidelines

for simulating roughness evolution under virtual conditions. However, from a

pragmatic view, such approaches do not meet the common demand to restrain

the roughness evolution. The aim of this chapter is to review roughness

evolution processes during CVD from a phenomenological viewpoint, rather

than focusing on mathematical details.

4.2 Roughness during homo-epitaxial growth

4.2.1 Thermodynamics

The surface of a lm can become rough during homo-epitaxial growth through

both thermodynamic and kinetic factors. The surface of a single crystal

under thermodynamic equilibrium has an atomically smooth structure at zero

temperature. However, at high temperatures, the surface of a single crystal

is atomically rough even under thermodynamic equilibrium. The transition

temperature at which the surface of a single crystal in thermodynamic

equilibrium becomes rough is called the roughening temperature, and the

transition is called the roughening transition (Burton et al., 1951). This can

be understood by the increase in entropy with increasing temperature. In

this case, the surface with atomic-scale roughness is the most stable. At

temperatures higher than the roughening temperature, facets disappear and

isolated crystals have a curved morphology which reects a rough surface

on the facets (Heyraud and Métois, 1987).

4.2.2 Kinetics

Below the roughening temperature, surface roughness can appear due to

kinetic factors when there is not sufcient relaxation to reach thermodynamic

equilibrium. This process is called kinetic roughening. In kinetic roughening,

adatoms do not have sufcient time to migrate on the surface to a stable

position such as kinks and steps. One of the factors inuencing roughness

evolution in kinetic roughening is the nucleation and growth of islands.

Another is surface diffusion governed by the Ehrlich–Schwoebel barrier

(Ehrlich and Hudda, 1966; Schwoebel and Shipsey, 1966).

Figure 4.1 schematically illustrates elementary processes during homo-

epitaxial growth. Nucleation on the two-dimensional layer involves the

arrival and adsorption of reactants on the surface (deposition), dissociation of

coordinated molecules to become adatoms, their migration along the surface

(diffusion), and aggregation of adatoms to create islands. These processes

(adsorption, diffusion, and aggregation) are the causes of roughness evolution.

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62 Thin ﬁlm growth

During epitaxial growth, adatoms arriving on the substrate surface migrate

to nucleate 2D islands, and these islands expand laterally by capturing other

migrating adatoms. When the nucleation rate on each layer is slow, islands

grow laterally by step-ow growth, and as a result the surface becomes

smooth because adatoms bond preferentially to steps and kinks in the existing

islands. When islands are created at a fast nucleation rate, the surface formed

is rough. Therefore, controlling nucleation rate is crucial to suppress surface

roughness, which is in turn governed by a variety of processes illustrated

later.

Another factor we must consider is surface diffusion between different

layers. Atoms arriving on the islands also contribute to this lateral layer-by-

layer growth when the diffusion barrier at the edge of the island is small. This

barrier is called the Ehrlich–Schwoebel barrier. When the barrier becomes

too large to allow interlayer transportation, 3D island growth occurs. This

is called the Schwoebel effect. At sufciently high temperatures, this effect

is negligible. An off-oriented substrate is also used to enhance the step-ow

growth and realize the smooth surface, minimizing the chance of 3D growth.

However, in this vicinal surface, existing steps are sometimes observed to

form bunches, giving a rough surface (Matsunami and Kimoto, 1997). This

step bunching is caused by the larger deposition ux on larger terraces, by

the step edge barrier and by different levels of diffusion on the different

terraces (Saito and Uwaha, 1994).

Predicting island density and size distribution from an atomistic view is

the primary task of nucleation theories. Figure 4.1 shows the elementary

processes taking place on the surface that contribute to surface roughness at

the atomic level. Because it is difcult to simultaneously account for all of

the elementary processes in theoretical models, such as kinetic Monte Carlo

Adsorption

Desorption

on terraces

Desorption

at steps

4.1 Elementary processes on the growing surface.

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63Analysing surface roughness evolution in thin ﬁlms

simulations (KMC), usually only a subset of the complete set of processes are

accounted for in a given model. There are a number of papers on nucleation

models simulating these processes. One goal of nucleation models is to

provide information on the maximum island density, N

max

, such as N

max

(F/D)

, where F is the ux of the growth species, D is the surface diffusivity

of adatoms, and c is a scaling exponent.

Determining c for various processes is one goal of most nucleation studies.

Control of nucleation density is an important key factor in restraining roughness

during homo-epitaxial growth. In order to maintain step-ow growth, the

island density must be kept as small as possible, which requires low F and

large D depending on c. However, when 3D growth of islands becomes

dominant, such a low nucleation density becomes the cause of rough surfaces.

This is because small island density allows large spaces where islands can

grow and results in a rough surface caused by agglomeration of large islands.

We can use nucleation theory to deduce not only island density but also the

dominant processes (Kajikawa et al., 2004a).

The simplest nucleation theory is the deposition-diffusion-aggregation

(DDA) model, which, as its name suggests, includes the processes of

deposition, diffusion, and aggregation. Because review papers describing

nucleation theory have already been published (Venables et al., 1984;

Stoyanov and Kaschiev, 1981; Ratsch and Venables, 2003), in this chapter

we review theoretical efforts to model deposition, diffusion, aggregation

and other complex elementary processes such as island migration and defect

trapping. Although understanding of such elementary processes can be used

as a basis for understanding the root cause of roughness evolution during

homo-epitaxial growth, it is not possible with current nucleation models

to simultaneously include all the physical processes that can take place.

Therefore, we review indices for determining which processes are active in

a given system and therefore must be included in a corresponding model.

The following models focus mainly on homo-epitaxial growth. However, it is

possible to obtain a fundamental understanding of the elementary processes

relating to other types of lm growth (e.g. hetero-epitaxial and non-epitaxial

growth) which determine the morphology of deposited lms.

Deposition-diffusion-aggregation (DDA) model

The DDA model assumes that growing species adsorb onto a surface,

diffuse, and then collide with other adatoms to create stable islands. A

phenomenological version of the DDA model has already been described

in the literature (Villain et al., 1992; Jensen et al., 1997, 1998), and is

summarized in the following.

The adatom density, r, is determined by a balance between deposition

and capture by islands, so that

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64 Thin ﬁ lm growth

r ~ F/DN [4.1]

where F is the deposition rate per surface site per unit of time, D is the surface

diffusion rate of adatoms, and N is the island density. If the contribution of

direct impingement to nucleation is negligible, the nucleation rate of islands

can be estimated as

j ~ D r

~ N

max

[4.2]

where N

max

is the maximum island density obtained and t

is the time when

islands come into contact. N

max

is obtained at time t

. In Eq. 4.2, we assume

the nucleation rate is constant. For islands with a 3D shape, the growth rate

of islands can be expressed as

d(R

)/dt ~ D r [4.3a]

If the rate of edge diffusion and deformation is not suf ciently fast to maintain

a 3D structure, then Eq. 4.3a is expressed as

()

dR()dR

dt/ dt

()

[4.3b]

where d

is the fractal dimension of the islands. At the limit of slow edge

diffusion and deformation, the island shape is determined by diffusion-limited-

aggregation (DLa), with d

= 1.7 (Tang, 1993). For fast edge diffusion and

no deformation, islands become 2D disks, and d

= 2. at t = t

R ~ N

–1/2

[4.4]

so that N

max

becomes (Villain et al., 1992; Tang, 1993):

2/4+

NF ~NF

(

NF (NF

NF ~NF (NF ~NF

D/)D

[4.5]

Therefore, N

max

~ (F/D)

2/7

for 3D islands, and N

max

~ (F/D)

1/3

for 2D islands,

which agrees well with Kinetic Monte Calo (KMC) simulations (Villain

et al., 1992).

There are some variations on the DDA model. In the DDA model,

atoms are assumed to deposit at random positions on a surface at  ux F. in

some cases temporal  uctuations of the deposition  ux, like the situation

in modulated CVD, are treated by both the rate equation (RE) and KMC

models (Jensen and Niemeyer, 1997; Combe and Jensen, 1998). Another

model includes anisotropic diffusion. In most deposition processes, surface

diffusion is expected to occur as 2D Brownian diffusion. Other modes of

diffusion such as anisotropic and long-range diffusion can modify c from

1/3 to 1/4 for 2D islands (Villain et al., 1992; Amar et al., 1998).

Evaporation

Isolated adatoms can evaporate from the surface, and models including

evaporation, deposition, diffusion, and aggregation have been developed.

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65Analysing surface roughness evolution in thin ﬁlms

Using the phenomenological approach, Jensen et al. (1997, 1998) derived

max

as follows. If evaporation exists, the adatom density quickly approaches

a stable value after deposition starts, i.e.,

r ~ Ft

[4.6]

where t

is the mean time of evaporation for an adatom. when direct

impingement exists, this increases the number of islands linearly according

j ~ (F + D r) r [4.7]

For 3D islands, island growth is governed by

d(R

)/dt ~ F(X

+ R)

[4.8]

where X

is the mean diffusion length of adatoms before desorption, expressed

as X

= (Dt

)

1/2

. Combining these equations, the maximum island density

can be expressed as

max

~ (Ft

)

2/3

(1 + X

)

2/3

[4.9]

From the RE approach, Jensen et al. (1997, 1998) also derived island growth

rates and deposition rates when coalescence starts. The scaling exponent

was checked using KMC simulations, which showed good agreement with

Eq. 4.9.

Edge diffusion/deformation

A monomer reaching an existing island sticks to the edge of it. If it cannot

move, diffusion limited aggregation (DLA) occurs. When islands are small,

islands remain compact to minimize the free energy. The fractal dimension

of islands formed during DLA is about 1.7 (Tang, 1993). Therefore, if edge

diffusion and deformation are not included in the model, it is possible to

overestimate the island size, which decreases the accuracy of the simulated

island density and size distribution (see Eq. 4.5). Detailed models including

edge diffusion and deformation have been presented by Amar and Family

(1996), Jensen et al. (1999) and Krug (2000).

Direct impingement

Direct impingement onto existing islands can also change the scaling behavior.

Most theoretical models neglect this effect, and assume that monomers

deposited on top of islands has no effect on the system. This implies that

either the Ehrlich–Schwoebel barrier prevents monomers from falling onto

the bottom layer or that the diffusion rate onto the second layer is lower

than that onto the bottom layer. Direct impingement is important when either

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