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Applied Surface Science 298 (2014) 171–175

Contents lists available at ScienceDirect

Applied Surface Science

jou rn al h om ep age: www.elsev ier .com/ locate /apsusc

elf-assembly of condensates with advanced surface by means of theompeting field selectivity and Gibbs–Thomson effect

yacheslav Perekrestov, Yuliya Kosminska, Alexander Mokrenko, Taras Davydenko ∗

epartment of Nanoelectronics, Faculty of Electronics and Informational Technologies, Sumy State University, 2, Rymskogo-Korsakova St.,0007 Sumy, Ukraine

r t i c l e i n f o

rticle history:eceived 13 November 2013eceived in revised form 24 January 2014ccepted 24 January 2014

a b s t r a c t

Copper and silicon layers were deposited using the accumulative plasma-condensate system. Their sur-face was found to possess the complex developed morphology using SEM technique. Competing processesof the field selectivity and Gibbs–Thomson effect are considered to describe the formation of the surface.The mathematical model is created on the basis of these effects which describes self-assembly of the

vailable online 31 January 2014

eywords:elf-assemblyuasi-equilibrium condensationurface morphologyibbs–Thomson effect

surface at the form of adjoining elements of an elliptic section. The comparative analyses of theoreticaland experimental results are given.

© 2014 Elsevier B.V. All rights reserved.

. Introduction

Currently, self-assembly of low-dimensional structures byeans of deposition under conditions of proximity to thermo-

ynamical equilibrium is of high research interest [1–3]. Untilow, only few studies of the structure formation principles haveeen focused namely on the quasi-equilibrium conditions andheir consequences. Therefore, there are still uninvestigated mech-nisms of self-organization of morphologically complex growingurface which are based not only on near-equilibrium condensa-ion requirements, but also on focusing of ion fluxes onto projectingarts of the growth surface. We consider using the term “field selec-ivity” to determine the lastly named factor in further. Therefore,he field selectivity means the idea that ionized deposited atomseing are selectively fixed on projecting parts of the growth sur-ace at the presence of the electric field above it. Our experimentsave shown that structures obtained under field selectivity condi-ions can correspond to coupled columnar elements (an exampleill be given further in the Section 2 of the present study). Thin film

olumnar structures known from literature [4,5] do not conform to

urs either by morphological characteristics or growth conditions.

So, in the present stage, we set the problem to determinehe central mechanisms of structure formation under the field

∗ Corresponding author. Tel.: +380 542 335513/996828072; fax: +380 542 334049.E-mail addresses: regiomontan 4020@mail.ru, uss-celvin@mail.ru

T. Davydenko).

169-4332/$ – see front matter © 2014 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.apsusc.2014.01.153

selectivity. The next section of the article covers the peculiaritiesof the experiment with an explanation of the physical principlesof film growth. The third section is devoted to the creation of themathematical model which describes self-assembly of the devel-oped surface of condensates under the field selectivity. Possiblevariants of self-assembled surface morphology are discussed. Thefinal section briefly concludes the study.

It should be noted that the obtained structures are of certainlyapplied interest due to their properties which, however, do not lieat a focus of the present work. Some characteristics of the structuressuch as electroconductivity are determined by contact phenomenabetween the columnar elements to a large extent. Therefore, thetechnological approach offered in the present work can be useful forproducing sensors operating in various chemically active medium.Besides, the preliminary tests have shown that the copper conden-sates change their electrical resistance approximately to 10% duringbending the substrate that is due to contact loss between columnarelements and can be used in tensometry.

2. Experimental device and microstructures obtained

In order to perform corresponding experiments, the accumula-tive plasma-condensate system (APCS) was used [6]. Such devicesconsist of the magnetron sputterer and the hollow cathode 5 placed

in the middle of the magnetron (see Fig. 1).

The material was condensed on the glass substrate 6 situatedin the hollow cathode 5 on the water-cooled holder 7. The oper-ational principles of the APCS are based on self-organization of

172 V. Perekrestov et al. / Applied Surface

Fig. 1. Cross-section of the axisymmetrical accumulative plasma-condensate sys-th

cmsttct

acurtpvdponT

Fdr

em (APCS) (1 – anode; 2 – substrate; 3 – erosion zone; 4 – magnet system; 5 –ollow cathode volume; 6 – substrate; 7 – chiller).

ondensation conditions near thermodynamical equilibrium. Theathematical model of self-organization of critically low steady-

tate supersaturation and physics of the operation principles ofhe APCS are discussed in detail in Ref. [6]. Due to the fact thathe deposited substance localized inside the hollow cathode 5 isonsiderably ionized [7], and also the negative potential is suppliedo the growth surface, there are prerequisites for field selectivity.

The necessity to create the mathematical model describing self-ssembly of the developed surface of condensates comes from ourorresponding experiments concerning deposition of Cu and Sising the APCS. Argon was used as the working gas during the mate-ial deposition. Argon was undergoing deep cleaning directly insidehe vacuum chamber during all the technological process by therocedure given in Ref. [8]. We used non-running gas inflow, i.e. theacuum chamber was cut off from the pumping system after pump-own. After that the chamber was filled with argon to its working

ressure PAr = 6–15 Pa. Then two additional magnetron sputterersf titanium shielded from the APCS were turned on. During tita-ium sputtering residual chemically active gases were absorbed.hus, their partial pressure was minimized to the level of about

ig. 2. SEM images of surface morphology of Cu and Si condensates obtained ex-situ oneposition time is 6 h, PAr = 15 Pa, Pw = 16.8 W; (c) Si, deposition time is 2 h, PAr = 6 Pa, Pw

epresent enlarged or lateral view of the samples.

Science 298 (2014) 171–175

10−7–8 × 10−8 Pa [8]. The discharge power input Pw was 6–21.9 Wand the ratio of the cathode inlet hole area to the cathode innersurface area (see [6]) was 0.5.

The examples of self-assembled developed surfaces of Cu and Siat the form of adjoining convex and concave ellipsoids of revolutionare shown in Fig. 2. XRD investigations of the condensates revealedwell defined texture of nucleation and growth for Cu ((1 1 1) beingparallel to the substrate surface). Meanwhile, the Si condensatespossessed amorphous structure. We have assumed this to be causedby primary origination of Si condensate in amorphous form onstructurally isotropic glass substrate. The atom-by-atom attachingto the base amorphous layer occurs near thermodynamical equilib-rium. That growth process complicates transition to the crystallinestate. Vicinity to the phase equilibrium promotes the maximal fill-ing of chemical bonds that are the prerequisite for quasi-crystallinestructure formation.

As it can be seen from the given examples of the devel-oped surface (see Fig. 2a and c), only individual projections andhollows exist on the growth surface during the first stage ofself-assembly. However, sufficiently long condensation leads tostatistically homogeneous morphology (see Fig. 2b and d) thatindicates on the features of self-organization. It should be also men-tioned that the field selectivity is possible if the growth surfacepossesses initial local curvature. Such deviations from perfectly flatsurface exist on a substrate and can be also formed during conden-sate growth.

3. Mathematical model of self-organization of theadvanced condensate surface

3.1. Physical basics of the surface growth process

Our aim is not to study kinetics of atomic processes but tofollow the transformations of structural elements on macro level.We suppose that adatoms diffusion processes are not crucial in

glass substrates. (a) Cu, deposition time is 30 min, PAr = 15 Pa, Pw = 16.8 W; (b) Cu,= 3.4–6 W; (d) Si, deposition time is 8 h, PAr = 6 Pa, Pw = 3.4–6 W. Small insertions

urface Science 298 (2014) 171–175 173

stlc

tttstm

atb

�

w

p

�

teae

t≈fcTowHabp

vp(

(Gd

�

r

3

gs�

AcItat

V. Perekrestov et al. / Applied S

tructure formation of advanced surface. This is due to the fact thathe residence time of an adatom on the surface � ∝ exp

(Ed/kT

)is

ow because of low desorption energy Ed during quasi-equilibriumondensation [9].

The samples shown in Fig. 2 do not reveal crystal faceting. Hence,he growth surface can be considered as structurally isotropic. So,he field selectivity and the Gibbs–Thomson effect [10,11] can beaken as two key factors determining the processes of the surfaceelf-assembly. These factors are in essence competing and showhemselves at the formation of convex and concave surface frag-

ents.Low values of the chemical potential difference �� of vapour

nd condensed phase one can take as a criterion of proximity tohermodynamical equilibrium. In vicinity to equilibrium �� cane presented in the form of the two constituents:

� = ��s + ��r, (1)

here ��s and ��r are to be found separately below.The first constituent ��s depends on the growth surface tem-

erature T and the deposited vapour pressure P as follows [10]:

�s = (˝v − ˝c)(P − Pe) + (sv − sc)(Te − T). (2)

Here ˝v and ˝c are the particular volumes per a single atom inhe vapour and the condensate correspondingly; sv and sc are thentropies of the vapour and the condensate respectively; Pe and Te

re the pressure and the temperature determining a point on thequilibrium curve of the vapour-condensate system.

Considering both summands in Eq. (2) separately, let us notehat the absolute supersaturation �P = P − Pe near equilibrium is0 above every point of the structurally isotropic flat growth sur-

ace. The presence of positive curvature at local surface parts canause focusing of deposited ion fluxes through the electric field.his means that the limited fluxes will be redistributed and directednto positive curvature parts, while parts with negative curvatureill lack for deposited substance thus reducing �P above them.ence, under conditions of extremely low supersaturations the rel-tive differences �P/Pe above the curved growth surface possessoth positive and negative values, and that is the most significantrerequisite for field selectivity.

In our case the temperature deviation from its equilibriumalue is always negligible quantity (�T = Te − T ≈ 0) and inde-endent from surface curvature. Then in Eq. (2) the summandsv − sc)(Te − T) can be excluded.

For structurally isotropic surface the second summand in Eq.1) ��r is defined by surface curvature corresponding to theibbs–Thomson relation [10,11], valid for quasi-equilibrium con-itions:

�r = ˝c˛(

1R1

− 1R2

). (3)

Here is the particular free surface energy, R1 and R2 are theadii of negative and positive curvature correspondingly.

.2. Model equations

Fig. 3 presents the cross-section of the simplified model of therowth surface at the initial period of condensation. The cross-ection includes half-ellipses with the central semiaxis a, b, c, and

− b (2� being the period of the relief, assumed to be constant).Besides, the surface morphology does not change in z direction.

ccording to Fig. 3, the parameters a, b, and c are determined by theoordinates of the A, B, C points in the coordinate system (x, y, and z).

t should be mentioned that this coordinate system is floating andhe point of origin 0 is specified by the position of the maximum And the inflection point B. Obviously, it is possible to create variousypes of the growth surface relief by varying the initial values of a,

Fig. 3. Growth surface model representation.

b and c. Therefore, our aim is to obtain kinetic equations describingtime dependence of the relief geometrical parameters a, b, c duringcondensation.

First, using the geometrical parameters of the growth surfacemodel, we derive the chemical potential difference at each of A,B, and C points (see Fig. 3). In order to obtain the constituent��s, using Eq. (2), the absolute supersaturation �P = P − Pe shouldbe expressed through the parameters, which determine the fieldselectivity. According to the physical meaning of the field selectiv-ity, described in previous sections, the electric field strength abovea surface point directly influences both local values of the depositedion flux and the supersaturation. It should be also emphasized thatit is not the total deposited flux but a local flux value that is ofessential importance, since the system is near equilibrium and thetotal deposited flux is considerably low. Therefore, let us assumethe absolute supersaturation

�P = P − Pe to be in direct proportion to the electric field strengthE, i.e.

�Pi = ωEi, (4)

where i means A, B, or C point (here and hereinafter), ω is theproportionality coefficient.

So, we should obtain the value of the electric field strength abovea surface point. In the simplest case, the Gauss theorem gives us thefollowing expression at the A, B and C points:

Ei = E0

(1 ± d

Ri

), (5)

where, E0 is the electric field strength above a flat condensate sur-face, d is the length of the cathode fall, Ri is the curvature radiusat the i point, the plus sign should be taken for the positive sur-face curvature, and the minus should be for the negative surfacecurvature.

Taking into account the half-elliptic cross-section, the curva-ture radii are related to the geometrical parameters of the growthsurface model by the following relations:

1RA

= a

b2, (6)

1RB

= b

a2− (� − b)

c2, (7)

1RC

= c

(� − b)2. (8)

According to Eqs. (4)–(8), the absolute supersaturation �P atthe A, B, C points can be expressed as

�PA = ωE0

(1 + da

b2

), (9)

�P = ωE[

1 + d(

b − (� − b))]

, (10)

B 0a2 c2

�PC = ωE0

[1 + dc

(� − b)2

]. (11)

1 urface

dt

�

�

�

naf

ut(joeiiuwweOept

a

w

dTa

a

b

c

eatoti[dicsr

74 V. Perekrestov et al. / Applied S

Thereby we can get the expressions for the chemical potentialifference �� at the A, B, and C points, if substituting Eqs. (9)–(11)o Eq. (2), Eqs. (6)–(8) to Eq. (3), and Eqs. (2)–(3) to Eq. (1):

�A = ω(˝r − ˝c)E0

(1 + da

b2

)− ˛˝ca

b2, (12)

�B = ω(˝r − ˝c)E0

[1 + d

(b

a2− (� − b)

c2

)]

− ˛˝c

(b

a2− (� − b)

c2

), (13)

�C = ω(˝r − ˝c)E0

[1 + dc

(� − b)2

]− ˛˝cc

(� − b)2. (14)

Since the choice of the reference system (see Fig. 3) determinesegative sign of the c parameter, the sign before c has been invertednd c should be considered as its absolute value in Eq. (14) andollowing expressions.

So, the final task is to derive the sought-for kinetic equations. Lets choose an element of the growth surface such that it is limited byhe relief period 2� along y axis and by a unit dimension along z axissee the high-lighted central area in Fig. 3). The time-constant fluxc condense onto the chosen element. Then consider the incrementf the condensate volume per unit time within the chosen surfacelement from the two points of view. On the one hand, the volumencrement dV can be found by summarizing changes of the follow-ng two areas: convex and concave half-ellipse areas, multiplied bynit length along z axis. Thus, dV = d(�ab/2) + d(�c(� − b)/2). Heree assume a to be positive and c to be negative in accordanceith the chosen coordinate system. It should be noted that if, for

xample, a becomes negative, the convex ellipse becomes concave.n the other hand, the volume increment per unit time can bexpressed through a simple product of the incident flux jc and thearticular volume per a single atom ˝c in condensate. So, we obtainhe equality:

˙ b + ba + c� − bc − cb = (2˝cjc)�

, (15)

here the point above a symbol means time derivative.Further, from Eq. (15) we can derive the system of equations

etermining the rates of time change of the a, b and c parameters.his system describes selective self-assembly of advanced surfacend includes the following expressions:

˙ =[

2˝cjc�

− ���B(a − c) − ��C(� − b)

]× 1

b, (16)

˙ =

[2˝cjc

�− ���Ab − ���C(� − b)

]× 1

(a − c), (17)

˙ =[

2˝cjc�

− ���Ab − ���B(a − c)

]× 1

(� − b). (18)

The time derivatives a, b and c in the left-hand part of thequations are determined directly from Eq. (15). Any derivativeppearing in the right-hand part of each equation are expressed inerms of corresponding chemical potential difference ��A, ��Br ��C for the two following reasons. Firstly, in the case of struc-urally isotropic growth surface the speed of normal crystal growths observed to be proportional to the chemical potential difference10]. Therefore, � is the proportionality factor. Secondly, the con-ensate growth rates at different points of the growth surface are of

nterdependent character. So, for example, if the surface curvaturehanges at B and C points in a manner, and at A point it remainstable, then the condensation rate can change at A point because ofedistribution of the limited flux jc.

Science 298 (2014) 171–175

The final and complete form of the equations can be obtained byeasy substitution of Eqs. (12)–(14) to the system of Eqs. (16)–(18).This step is omitted here in order to exclude unnecessary cumber-some formulas.

3.3. Analysis of the model

Let us analyze meaningful solutions of the system of Eqs.(16)–(18) using the method of phase portraits. It should be statedthat the presented solutions correspond to the region of posi-tive values of a and b < � . The first example should be consideredconcerns high supersaturations. That means that the sufficientlyintensive flux jc falls onto the singled surface element and the2 ˝cjc/� parameter in Eqs. (16)–(18) is considerably higher thanthe sum of the other summands. For this example the phase por-traits drawn on the basis of Eqs. (16)–(18) have no peculiaritiescorresponding to self-organization, and this occurs independentlyfrom initial surface relief. Hence, during condensation the initialmorphology does not undergo changes that can take place if thegrowth rate of condensates is both the same at each point of thegrowth surface and stable in time. It confirms in this way that theself-organization is impossible at high supersaturations.

Further let us consider the situation of low supersaturations. Foreach particular case discussed below Fig. 4 represents correspond-ing phase portraits plotted on the basis of Eqs. (16)–(18). Each phasepath in the phase portraits relates to different initial conditions ofgeometrical characteristics of the surface, i.e. a, b, and c values. Inpoint of fact, a family of phase paths determines physical processesof condensation or surface morphology alteration which depend oninitial surface configuration. A singular point indicates formation ofsuch relief that does not change during further condensation. Thiscorresponds to the fact that the chemical potential is the same ineach point of the growth surface. And hence, the growth rate ineach surface point will be the same too.

Under condition of E0 ≈ 0 independently from the initial param-eters a, b and c the growth surface gradually becomes absolutelyflat in the course of condensation, i.e. the singular point has coor-dinates a ≈ 0 and c ≈ 0 (see Fig. 4a). Such smoothing of the growthsurface is quite explainable as in absence of the field selectivity it isthe Gibbs–Thomson effect that flattens the surface. For this reasoncondensation onto the substrate 2 (see Fig. 1) at rather high tem-peratures (∼800 K) did not result in self-assembly of any developedsurface.

The phase portraits drawn at increased values of E0 and weakcontribution of ��r constituent to self-assembly are characterizedby a part of the general path where self-assembly evolves for a longtime (Fig. 4b). Wherein, b approaches zero value, a and c possessfinite values that indicates self-organization of the growth surfacein the form of hollows with elliptic cross-sections. Such type of thegrowth surface has been realized by condensation of Si onto glasssubstrate (Fig. 2c). Apparently, low values of E0 are caused by weakconductivity of Si.

It is possible to obtain the phase portraits corresponding to thestructures with the identical elliptic cross-sections (Fig. 4c andd) by varying the deposition parameters in Eqs. (16)–(18) suchas E0, jc, and d. Obviously, the singular points on these portraitsindicate such self-organization of surface morphology that makesthe growth rate to be independent from the surface coordinates.Experimentally such variants of self-assembly are obtained duringcondensation of Cu (see Fig. 2b). Besides, it should be mentionedthat different sizes of projections are produced by different dis-tances between them or by 2� parameter (see Fig. 3).

The well-defined damped cyclicity in Fig. 4d is of particularinterest since it evidently shows the periodical changes of ��A,��B and ��C around their constant low values under the devel-opment of the growth surface curvature.

V. Perekrestov et al. / Applied Surface Science 298 (2014) 171–175 175

F (a) E0

4 t (stabs e para

4

tscomppe

R

ig. 4. Phase portraits obtained at different parameters of the technological system

.22, 0.00); (b) E0 = 1.0, jc = 1.0, d = 2.1; (c) E0 = 0.24, jc = 2.0, d = 1.5, the singular poiningular point (stable node) coordinates (a; b; c) – (0.48, 2.67,−11.11); the rest of th

. Conclusion

In the present study we managed to show experimentally andhrough the mathematical model that condensation of ionized sub-tance in vicinity to equilibrium and in presence of electric fieldan cause self-assembly of the developed surface, the morphol-gy of which can be constructed on ellipsoids of revolution. Theain cause of that condensation is the balancing of the chemical

otentials at different points of the growth surface due to the com-eting macroscopic effects of field selectivity and Gibbs–Thomsonffect.

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[

[

= 0.001, jc = 0.01, d = 4.1, the singular point (stable node) coordinates (a; b; c) – (0.00,le node) coordinates (a; b; c) – (0.95, 3.40,−0.63); (d) E0 = 0.24, jc = 2.0, d = 4.1, themeters were taken as follows: ω(˝r − ˝c) = 1.0, = 1.0, � = 1.0, � = 5.0, ˝c = 1.0.

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[7] A.G. Zhyglinsky, V.V. Kuchinsky, Mass-Transfer at Plasma-Surface Interaction,Energoatomizdat, Moscow, 1991 (in Russian).

[8] V.I. Perekrestov, S.N. Kravchenko, Change in the composition of residual gasesin a vacuum chamber during Ti film deposition, Instrum. Exp. Tech. 45 (2002)404–407.

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