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Morphological instability of Cu nanolines induced by Ga+-ion bombardment: In situ scanning electron microscopy and theoretical model Qiangmin Wei, Weixing Li, Kai Sun, Jie Lian, and Lumin Wang Citation: J. Appl. Phys. 103, 074306 (2008); doi: 10.1063/1.2903881 View online: http://dx.doi.org/10.1063/1.2903881 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v103/i7 Published by the AIP Publishing LLC.

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JOURNAL OF APPLIED PHYSICS 103, 074306 共2008兲

Morphological instability of Cu nanolines induced by Ga+-ion bombardment: In situ scanning electron microscopy and theoretical model Qiangmin Wei,1,a兲 Weixing Li,1 Kai Sun,1 Jie Lian,2 and Lumin Wang3,b兲 1

Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA 2 Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180, USA 3 Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA and Department of Nuclear Engineering and Radiological Science, University of Michigan, Ann Arbor, Michigan 48109, USA

共Received 29 November 2007; accepted 6 February 2008; published online 8 April 2008兲 The morphological evolution of copper nanolines induced by focused ion beam at normal bombardment has been investigated by in situ scanning electron microscopy. A periodic array of particles is observed when the width of lines reaches a certain value. The stability of a nanoline is studied in terms of a model based on Nichols and Mullins 关Trans. Metall. Soc. AIME 233, 1840 共1965兲兴 instability and curvature-dependent sputtering yield. A critical line width is found by linear analysis. When the line width is below this value, unstable mode whose wave vector is parallel to the line axis develops and a chain of periodic particles forms. When the width is above this critical value, the sputtering etching only leads to the decrease of width. The flux and temperature dependence of wavelength is measured and explained based on this model. The predictions of the model are in good agreement with the experimental results. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2903881兴 I. INTRODUCTION

Bombardment of solid surfaces by ions with intermediate energy can generate a rich variety of interesting patterns. Ripples and quantum dots are two examples which have been intensively studied both experimentally and theoretically.1–10 This self-organized structure has received particular interests recently as a promising candidate for an easy, inexpensive, and large area fabrication of patterns.11 A common feature of most studies is the formation of patterns induced on two-dimensional surface. A systematic treatment of the evolution of morphologies for one dimensional materials subjected to ion bombardment, such as lines, has been lacking so far. Recently, it has been reported that ion sputtered nanolines can self-assemble into chains of ordered nanoparticles. Lian et al.12 demonstrated that focused ion beam 共FIB兲 induced Co nanolines can form nanoparticles with the wavelength of 256 nm and diameter of 58 nm. Similarly, Zhao et al.13 reported that Pt and Au nanolines can also form linear arrays of dots induced by focused 30 keV Ga+ ion beam and 1 MeV Kr ion beam. They employed Rayleigh instability14 to explain the formation of nanoparticles. However, the classical Rayleigh instability describes the break up of the cylindrical liquid jet under surface tension in which the volume is conservative. In their cases, due to the sputtering yield, the volume of lines decreases with the bombardment. For a solid rod with conservative volume at high temperature, the stability was originally discussed by Nichols and Mullins.15 By linear analysis of surface diffua兲

Electronic mail: [email protected]. Corresponding author: Electronic mail: [email protected].

b兲

0021-8979/2008/103共7兲/074306/9/$23.00

sion, they found that solid cylinder was unstable for wavelength exceeding the circumference of the cylinder, in agreement with Rayleigh instability. In this paper, we report the experimental observation and theoretical modeling on morphological instability of Cu nanolines induced by FIB. A detailed description of evolution of lines under ion beam etching at normal incidence with various flux and temperature is shown. Following the pioneering work for ripple formation given by Bradly and Harper 共BH model兲,9 we derived a partial differential equation based on the Sigmund theory16,17 and Nichols and Mullins approach.15 By linear analysis, a critical radius is found for the instability of nanolines. When the line width is below this critical value, a wave vector along the longitudinal direction develops, giving rise to the formation of periodic dots. II. EXPERIMENT AND RESULTS

The nanolines used in our work were created via FIB direct writing on Cu thin films 共thickness ⬃80 and ⬃120 nm兲 with different widths. This thin film was fabricated by deposition of Cu on Si 共100兲 substrates with a native oxide at room temperature in a high vacuum magnetron sputtering chamber 共10−9 Torr兲. The ion bombardment experiments were performed in a scanning electron microscope 共SEM兲 equipped with FIB instrument 共FEI Nova 200 Nanolab兲 at the vacuum of 10−7 Torr. Focused Ga+ ions beam with energy 10 keV at normal incidence were used to sputter the thin film surface for all ion beam experiments. The scanned area 15⫻ 5 ␮m2 was kept constant. Wavelength and size of nanoparticles were monitored in situ by SEM. The current of ion beam and substrate temperature were varied.

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FIG. 1. 共Color online兲 SEM images of chains of nanoparticle formation. 共a兲 Cu lines with four different initial widths cut from a continuous 80 nm thick film using FIB before irradiation. 共b兲 Bombardment up to fluence of 5 ⫻ 1016 cm−2. Ion beam energy ␧ = 10 keV, flux f = 4.2⫻ 1014 cm−2 s−1, and temperature T = 300 K. 关共c兲 and 共d兲兴 AFM image and cross-section profile showing the morphology variation for Cu lines. The scanning area of 共c兲 is 7 ⫻ 7 ␮m2. Scale bar is 2 ␮m.

The spot size of 10 keV Ga+ ion beam was 50 nm with the overlap of 50%. Each spot size was bombarded during 1 ␮s with repetition time of 100 ms. The topography of patterned structures was also measured using atomic force microscopy 共AFM兲 operated at a taping mode. The evolution of nanolines subjected to ion bombardment has been reported by Lian et al. and Zhao et al.12,13 Following their experimental procedure, four Cu lines with different widths 共600, 400, 200, 100 nm兲 were created by FIB direct writing 关Fig. 1共a兲兴. After exposure time of about 100 s, the lines self-assemble into chains of periodic nanodots 关Fig. 1共b兲兴. Combining with previous studies, the main experimental features can be briefly summarized as 共1兲 line widths decrease linearly at the beginning of bombardment; 共2兲 when line widths reach a critical value, periodic patterns start to form; 共3兲 the line with larger width can develop into two lines and these two lines can evolve into periodic nanodots individually; 共4兲 the pattern formation is within several minutes; and 共5兲 wavelength is nearly the same for different width lines. A detailed study of the morphological evolution of Cu lines subjected to FIB bombardment at room temperature for different exposure time is shown in Fig. 2. A Cu line with cross section of 120⫻ 120 nm2 was created via FIB direct writing 关Fig. 2共a兲兴. At the beginning of bombardment, no periodic pattern is found while the width of line decreases linearly. After a sputtering time of ⬃60 s, corresponding to a fluence of ⬃2.92⫻ 1016 cm−2, a periodic morphology with the wavelength of ⬃320 nm starts to appear 关Fig. 2共b兲兴, in which the average width of nanoline reduces to ⬃100 nm. With increasing bombardment time, the chain of nanoparticles begins to form at time ⬃90 s 关Fig. 2共c兲兴. These nanoparticles become smaller and eventually are sputtered away upon further milling 关Fig. 2共d兲兴. Due to redeposition and surface tension, the doughnut-shaped rims on the edges of line are generated during the ion bombardment process 关Figs.

FIG. 2. SEM images showing the temporal development of instability of a copper line with cross section of 120⫻ 120 nm2 at 共a兲 t = 0 s, 共b兲 t = 60 s, 共c兲 t = 90 s, and 共d兲 t = 120 s. 共e兲 High magnification image of 共b兲. Flux f = 4.2 ⫻ 1014 cm−2 s−1 and energy ␧ = 10 keV. Scale bar= 300 nm.

2共a兲 and 2共e兲 and AFM image in Figs. 1共c兲 and 1共d兲兴. This is a characteristic feature of the dewetting process.12,13,18 With the increase of width of lines, this rim would grow up to a curved cylinder and would therefore break up into a chain of nanoparticles 共Fig. 1 for 400 and 600 nm lines兲. This morphology is similar to the shape on the edge of plate discussed by Nichols and Mullins.15 In order to investigate the development of rim, we made another experiment on the line with the cross section of 500 共width兲 ⫻ 80 共thickness兲 nm2 共Fig. 3兲. After ion etching with energy of 10 keV and flux of 4.2 ⫻ 1014 cm−2 s−1 for ⬃180 s, the craters or holes form and randomly distribute along the line 关Fig. 3共b兲兴. As the sputtering proceeds, this randomly distributed dots develops into two chains of regularly arranged patterns, parallel to the axis of wire 关Fig. 3共c兲兴. The degree of ordering is improved for further bombardment 关Fig. 3共d兲兴. Sinusoidlike structure with the nanoparticles on the vertex can be identified by tilting the sample 关Figs. 3共e兲 and 3共f兲兴. The evolution of wavelength of nanopartilces as a function of flux is shown in Fig. 4 at fixed energy, ion fluence, bombarded area, and temperature. We found that wavelength is independent of the ion flux at temperatures 300 and 350 K 共Fig. 4兲. The same behavior has been reported on the Cu共100兲 surface for ripple formation.19–23 III. MODEL

The experimental results can be schematically described in Fig. 5. For small width lines, the cross section is ellipse 关Fig. 1共c兲 and 1共d兲兴. When the width reaches a critical value 关Fig. 5共a兲兴, periodic patterns start to develop until nanoparticles are formed. For larger width lines, due to dewetting effect,12,13,18 two rims on both edges of the line are generated 关Figs. 5共b兲, 1共c兲, and 1共d兲兴. According to Sigmund theory,16 the sputtering is higher in the middle of the line than that on the rims. Thus original line breaks down to two lines, each of which evolves into periodic nanodots 关Fig. 1共a兲 and 1共b兲兴.


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FIG. 3. SEM images of morphological evolution of a copper line with rectangle cross section of 500⫻ 80 nm2 at different bombardment time: 共a兲 t = 0 s, 共b兲 t = 180 s, 共c兲 t = 240 s, and 共d兲 t = 300 s. 共a兲–共d兲 are viewed from surface normal, while 共e兲 and 共f兲 are viewed from 52° relative to the surface normal of 共c兲 and 共d兲, respectively. 共h兲 High magnification image of 共e兲. Flux f = 4.2⫻ 1014 cm−2 s−1 and energy ␧ = 10 keV. Scale bar= 500 nm.

Next we will establish a linear partial differential equation based on the roughening and smoothing mechanisms to show that there exists a critical width for pattern formation. This section is organized as follows: we first apply Sigmund theory to an arbitrary line structure to derive a partial differential equation describing the ion-induced roughening mechanism, and then another roughening mechanism due to Ehrlich–Schwoebel barrier is discussed. Part III C contains a brief decription of thermally induced smoothing mechanism. In part III D, ion-induced smoothing mechanism is proposed. A linear partial differential equation is given by cooperating all these effects in the last part.

FIG. 5. Schematic representation of particle formation. 共a兲 Instability for a small size line and 共b兲 instability for a large size line. R0 is original size of line, and Rc is critical size at the beginning of pattern formation. Rim structure can be seen for line with larger size 共b兲.

A. Ion-induced roughening

Theoretical approaches to surface erosion by ion bombardment typically used Sigmund’s transport theory to calculate the sputtering yield at a point on an arbitrary surface function under the assumption of Gaussian distribution of deposited energy of ions.9,10,16,17,24 According to Sigmund’s theory,16 when the ion penetrates an average distance inside the solid it spreads their kinetic energy to the neighboring sites following Gaussian distribution. The sputtering yield at which material is ejected from a point is proportional to the energy accumulated there by all ions. For a homogeneous flux, the distribution of the total number of sputtered atoms per unit area per unit time along the outward normal surface is given by N共r兲 = ⌳

冕冕

␾共r⬘兲F共r − r⬘兲dA⬘ ,

共1兲

S

FIG. 4. Wavelength as a function of flux at various temperatures. Within the limit of experimental error, the wavelength with respect to flux is constant.

where ⌳ is a constant characterizing the target material,16 the integral is evaluated over the area S, ␾共r⬘兲 is a correction to the uniform flux f, ␾共r⬘兲dA⬘ is the number of ions hitting on an area dA⬘, F共r − r⬘兲 denotes the sputtered atoms at position r generated by an ion hitting the surface in a point r⬘. To evaluate Eq. 共1兲, we need to know the surface profile. In our experiments, the cross sections of nanolines are square or rectangular before ion bombardment. However, as shown in our experiments 关Fig. 1共d兲兴 and other experimental data,12,13 when we reduce the sample less than 100 ⫻ 100 nm2 from larger cross section by ion bombardment, due to redeposition, surface diffusion, and dewetting, such small cross section can be approximated as ellipse. Gener-


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⳵Rz = − ⍀N共Ry,Rz兲冑1 + ⵜ2y x + ⵜz2x ⬇ − ⍀N共Ry,Rz兲, ⳵t

共6兲

where ⍀ is atomic volume. Substituting Eq. 共5兲 into 共6兲 gives

⳵Rz = − v0 + ␩K, ⳵t FIG. 6. Schematic illustration of line and coordinate frame used to calculate the sputtering yield at point O. The ion beam direction is perpendicular to the wire axis. Two principle curvatures have opposite signs 共we assume curvature is positive for convex surface兲.

ally, the surface height x = h共y , z兲 can be expanded into powers of derivatives of h共y , z兲. For BH model,9 it was assumed that surface height can be expended as h共y , z兲 = −y 2 / 2Ry − z2 / 2Rz, where Ry and Rz are the two principle radii of curvature. If expending surface profile into fourth power of height, Barabási and co-workers proposed nonlinear equation to explain the morphology evolution of ripple for long time bombardment, such as roughening, coarsening, and rotation of ripple.10,24 In our case, due to shadow effects, only half of line can be bombarded by the ion beam 共Fig. 6兲. Thus we can assume that the surface function is y2 z2 − , x= 2Rz 2Ry

共2兲

where Rz and Ry are two principal radii of curvature of surface at point O. According to Sigmund’s theory, the deposited energy distribution is given by F共r兲 =

⑀ 共2␲兲 ␣␤ 3/2

2

冉

exp −

共x − a兲 y +z − 2 2␣ 2␤2 2

2

2

冊

共3兲

,

where the ion beam is along the positive x axis, ⑀ is the ion energy, a is the ion penetration depth, ␣ and ␤ are the cascading sizes in transverse 共␣兲 and longitudinal 共␤兲 directions. For normal bombardment, we have

␾共r⬘兲 =

冑

f

共4兲

冑1 + ⵜ2y⬘x⬘ + ⵜz2⬘x⬘

and dA⬘ = 1 + ⵜ2y⬘x⬘ + ⵜz2⬘x⬘dy ⬘dz⬘. Let r = 0 and the integral be over the range of −⬁ ⬍ y , z ⬍ + ⬁, we obtain the sputtered atoms at point O contributed from all slowing down of ions inside materials. Following ripple formation,9,10,24 we assume that the initial surface varies slowly enough that Rz and Ry are much larger than average penetration depth. Thus we can expand the exponential terms which include 1 / Ry , 1 / Rz in Eq. 共1兲 to the first order and neglect second and higher orders. Combination of Eqs. 共1兲–共4兲 gives N共Ry,Rz兲 =

⌳⑀ f

冑2␲␣

冉冊冋

exp −

a2 2␣2

1−

冉

1 a␤2 1 − 2␣2 Rz R y

冊册

. 共5兲

This can be understood as the flux of atoms outgoing from the wire along the surface normal as a result of ion bombardment. For small slope approximation, we have

where v0 =

␩=

K=

⍀⌳⑀ f

冑2␲␣

冉冊冉冊

exp −

⍀⌳⑀ fa␤2 2冑2␲␣3

共7兲

a2 , 2␣2

exp −

a2 , 2␣2

共8兲

1 1 − , Rz R y

where the ion beam is along the positive x axis 共Fig. 1兲, and K is the mean curvature of surface. B. Ehrlich–Schwoebel roughening

For a metal surface under its roughening temperature, there is a barrier at step edges that inhibits diffusing atoms from coming down toward lower levels 关Ehrlich–Schwoebel 共ES兲 barrier25,26兴. This barrier leads to a surface flux in the uphill direction and to an increase of slope. Following Refs. 11, 21, 23, and 27–30 we can introduce in Eq. 共7兲 the term f slsl2d⍀ ⳵h = K = SK ⳵t 2共ls + ld兲a0

共9兲

for ES effect, where S is roughening coefficient describing the effects of ES barrier to interlayer diffusion, f s is the flux of mobile species on the surface, ls and ld are Schwoebel length and diffusion length, respectively, and a0 is the lattice constant. The Schwoebel length is defined as ls = a0关exp共EES / kT兲 − 1兴, where EES is ES energy. The diffusion length is determined by the competition between flux of mobile species and diffusion. Since both length ls and ld are exponential temperature dependence, Eq. 共9兲 shows that S has a maximum value at a certain temperature and dramatically decreases below or beyond this temperature. Thus ES barrier on the roughening plays an important role in a narrow range of temperature.23 In principle, we can compare the value of two roughening coefficients S in Eq. 共9兲 and ␩ in Eq. 共7兲 to determine the controlling mechanism. Because ␩ increases linearly with the increase of energy 关a / ␣ and ␤ / ␣ in Eq. 共8兲 are approximately constant兴 while S is ion energy independent, and ␩ increases faster than S with increasing flux, for high flux and energy, curvature dependent roughening becomes dominant.19,23 For Cu共001兲 system bombarded by 0.8 keV Ar+ ion with 2.13⫻ 1014 cm−2 s−1 flux and temperature from 393 to 473 K, Chason and co-workers20–22 showed that morphological evolution of Cu in the early stages can be well described by the continuum theory in which ES barrier is neglected. In our experiments, as shown in Fig. 7 for planar surface, no crystallographic feature is induced by ion beam at normal bombardment, which is not


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formation.20,21,37,38 In our case, by expanding Eq. 共2兲 into the fourth order, we obtain ion-induced diffusion term

⳵h a␤⌳⑀ f⍀ exp共− a2/2␣2兲ⵜs2K = ␨ⵜs2K = ⳵t 8冑2␲

FIG. 7. Evolution of thin film subjected to ion beam at normal incidence with various fluence: 共a兲 8.5⫻ 1016 cm−2 and 共b兲 1.6⫻ 1017 cm−2. Inset is fast fourier transformation 共FFT兲 showing the isotropic property of pattern. Flux f = 4.2⫻ 1014 cm−2 s−1, energy ␧ = 10 keV, and scale bar= 1 ␮m.

consistent with the observations controlled by ES barrier.31–34 Here we assume the mass transport do not face ES barrier at step edges. This simplifying assumption enables us to make quantitative comparison of our model with experimental observations. As shown below this assumption does not change the general results of our model. C. Thermally induced smoothing

The mobile species on the surface, biased by changes in surface energy, can smooth the surface by thermally induced diffusion. According to the classical work of Herring and Mullins,35,36 for amorphous surfaces, a chemical potential energy is proportional to the surface curvature

␮ = K␥⍀,

D sc ⵜ s␮ , kT

共11兲

⳵Rz = − v0 + ⌫K + Dⵜs2K, ⳵t

共14兲

where ⌫ = ␩ + S and D = B + ␨. If we substitute the equations ⳵Rz / ⳵t = −⳵x / ⳵t, 1 / Ry = −⳵2x / ⳵y 2, and 1 / Rz = ⳵2x / ⳵z2 into Eq. 共14兲, we obtain a linear partial different equation which shows that the line is unstable with arbitrary wave vector due to the curvature dependent sputtering yield. It should be noted that the similar equations are derived in Refs. 9 and 39 for the ripple formation induced by the oblique ion beam on the two-dimensional surface. IV. ANALYSIS OF STABILITY

For an infinite long line and small amplitude perturbations, using cylinder coordination, the curvature can be expressed by

冉冊冋冉冊册

⳵ 2R z ⳵Rz 2 − Rz 2 ⳵␪ ⳵␪ ⳵ 2R z − 2. 2 3/2 ⳵Rz ⳵y Rz2 + ⳵␪

Rz2 + 2

where Ds is surface self-diffusion coefficient, c is the number of diffusing atoms per unit surface area, and kT has its usual meaning. Divergence of −J gives rise to the outward normal growth velocity15,36 D s␥ ⍀ 2c 2 ⵜ K ⬅ Bⵜs2K. vn = kT

at normal bombardment. It can be seen that our expression is different from that derived in Refs. 24 and 37 by atomic volume ⍀ used in the Eq. 共6兲. This difference results in the small value compared with previous studies. Similar to smoothing mechanism, we can make comparison of ␨ with B to determine which mechanism dominates. Combining Eqs. 共7兲, 共10兲, 共12兲, and 共13兲, we have linear equation for evolution of lines subjected to normal bombardment,

共10兲

where ␥ is surface tension. According to Nernst–Einstein relation, a current of species on the surface is proportional to a gradient of the chemical potential J=−

共13兲

K=

If we introduce an infinitesimal perturbation given by Rz = R0 + ␦共t兲exp共iky + in␪兲,

共12兲

This equation does not take into account of anisotropic movement of species on a single crystal metal surface. As shown by Rusponi et al.,27,28 the diffusion coefficient DsK can be replaced by two terms Ds,xKx + Ds,Y Ky to give the different rate along two axes. D. Ion-induced smoothing

By expanding surface height into the fourth order, Makeev et al.24,37 used Eq. 共1兲 to obtain a term which is proportional to the fourth derivative of height. Since this term is fully determined by the process of surface erosion and has a smoothing effect similar to the thermal diffusion, it is called ion-induced surface diffusion. It has been reported that this term plays an important role at the low temperature in which the thermal diffusion is small.20,21,37,38 Combination of this term with thermal diffusion, the relationship between wavelength and flux is successfully explained for ripple

共15兲

共16兲

where R0 is initial radius of wire, n = 0 , 1 , 2 , 3 , . . ., substituting Eqs. 共15兲 and 共16兲 into Eq. 共14兲 and retaining only first order term in ␦共t兲 yield

⳵R0 ␩ = − v0 + ⳵t R0 and

冋冉

1 ⳵␦共t兲 n2 = − + k2 ⳵t␦共t兲 R20 R20

共17兲

冊册冋冉 D

冊册

n2 + k2 − ⌫ . R20

共18兲

Equation 共17兲 expresses the shrinkage rate of the mean radius subjected to the ion beam. Since R0 Ⰷ a and ␣ ⬇ ␤, we can neglect last term in Eq. 共17兲 and obtain the evolution of radius with time Rz = R0 − v0t.

共19兲

This result is different from that given by Nichols and Mullins15 in which the variance of the mean radius with time


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is zero. If we assume small slope, Eq. 共19兲 is the direct result of bombardment. Equation 共18兲 shows the dispersion relation of the fractional growth rates for various frequencies. If ⌫ = 0, the pure capillary effect is realized. It can be seen that any mode for n 艌 1 is stable. The instability occurs with wave vector along the axis of cylinder and wavelength of ␭ = 8.89R0, the identical results given by Nichols and Mullins.15 If D = 0, it reduces to the pure irradiation effect. The surface is unstable with arbitrary wave vector, consistent with ripple formation for normal bombardment on the planar surface.9,10 When both irradiation and capillary are present, the unstable modes can be determined under two conditions: first, if two terms on the right hand of Eq. 共18兲 are all positive, we have R0 ⬍

冑

D , ⌫ 共20兲

␭=

8.89R0

冑

⌫ 1 + R20 D

,

with the wave vector parallel to the axis of line. Second, if two terms on the right hand of Eq. 共18兲 are all negative, it shows that when R0 ⬎ 冑D / ⌫, the arbitrary wave vector with n2 / R20 + k2 ⬎ 1 / R20 forms, which means we can not observe periodic structures on the surface of wire. Since Eq. 共19兲 is still valid, this process can be understood as only ion milling phenomenon. Once the radius of line satisfies Eq. 共20兲, the periodic wave parallel to line axis forms and then the line breaks into particles. Under this condition, we have R0 ⬎

冑

␭ = 2␲

D , ⌫

冑

共21兲 D . ⌫

Thus, we obtain a critical width of line given by Rc = 冑D/⌫.

共22兲

If the width of line exceeds the critical value, although the unstable modes can occur, no ordered patterns can develop. Sputter etching gives rise to the decrease of width, leading to the maximum wavelength given by Eq. 共21兲. Compared with wavelength, ␭ = 8.89R0, given by Nichols and Mullins15 for solid cylinder 共or Rayleigh instability14 for an inviscid liquid兲 under the condition of conservation of volume, Eqs. 共20兲 and 共21兲 show that ion bombardment tends to suppress the instability induced by surface diffusion and leads to the decrease of wavelength. The ratio of coefficient ⌫ describing roughening mechanism to coefficient D describing smoothing mechanism determines the correction to the Rayleigh instability. This is reasonable because the ion beam bombardment can reduce the radius of wire, thus leading to the decrease of wavelength. The relationship between the wavelength and radius of wire is not linear for ion bombardment. Equation 共21兲 shows if R0 → ⬁, wavelength is independent of initial radius, and if the radius

FIG. 8. Dependence of the wavelength on the wire width from Eqs. 共20兲 and 共21兲. At T = 300 K, Rc = 冑D / ⌫ = 48 nm, and at T = 320 K, Rc = 冑D / ⌫ = 82 nm. Rayleigh instability is given for comparison.

of wire is small and temperature is high enough Eq. 共20兲 reduces to classical Rayleigh instability of a cylindrical rod. Therefore, for small radius and intermediate energy ions as shown by Lian et al.12 and Zhao et al.,13 Rayleigh instability can be used to approximately predict the experimental observations. V. COMPARISON AND DISCUSSION A. Comparison with Rayleigh instability and BH model

Figure 8 gives the comparison of our model with Rayleigh instability by assuming thermally induced smoothing mechanism and ion-induced roughening mechanism dominate. These two dominant mechanisms are always assumed in the analysis of ripple formation.9,10,21,22,24 Under these assumptions the temperature dependence of wavelength is a simple Arrhenius equation, the same behavior as BH model.9 The parameters we used in Fig. 8 are energy ␧ = 10 keV, flux f = 4.2⫻ 1014 cm−2 s−1, kT = 4.14⫻ 10−21 J, atomic volume material factor ⌳ = 1.4⫻ 10−9 ⍀ = 1.2⫻ 10−23 cm3, cm eV−1,16 penetration depth a = 3.9 nm,40,41 straggling ␣ = 2 nm,40,41 and straggling ␤ = 1.2 nm.40,41 Since the surface tension, concentration of mobile defects, and diffusion coefficient are dependent on irradiation conditions and surface properties, for simplicity, we assume surface tension ␥ = 1.78⫻ 10−4 J cm−2,42 diffusion coefficient Ds = 2.74 ⫻ 10−12 cm2 s−1,43 and concentration of mobile defects c = surface atomic density= 1.9⫻ 1015 cm−2. Substituting these values into Eqs. 共8兲 and 共12兲 gives B = 3.3⫻ 10−26 cm4 s−1, ␩ = 1.4⫻ 10−15 cm2 s−1, and v0 = 0.21 nm s−1. Rough theoretical estimate of the critical radius is Rc = 48 nm, wavelength and ␭ is ␭ = 300 nm for any R0 ⬎ Rc −4 2 冑 = 8.89R0 / 1 + 4.4⫻ 10 R0 共in unit of nm兲 for R0 ⬍ Rc. It should be noted that Arrhenius behavior is very sensitive to temperature and energy, a small temperature or energy change can result in a significant different outcome, which can differ by an order of magnitude or more. For example, if temperature changes from 300 to 320 K, the critical radius calculated from above assuming values can increase from 48 to 82 nm as shown in Fig. 8. However, it is well known that the temperature dependent wavelength cannot be de-


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scribed by a simple Arrhenius behavior. Surface tension and concentration of mobile defects are not only dependent of temperature but also dependent of slope.23 Fragmentation of Cu nanowire driven by Rayleigh instability at various temperatures has been reported.44 Since the breakup time is proportional to R40 according to Nichols and Mullins instability,15 nanowire with 55 nm radius exhibits no change after annealing at 400 ° C during 20 min. In our experiments, however, within 1 min, the same size nanolines can undergo significant changes at room temperature 共Fig. 1兲. A conclusion can be drawn that the interaction between ion and nanowire is the main reason for the fragmentation of nanoline. According to the linear equation for ripple formation,9 with normal bombardment on the amorphous surface, no ordered ripple forms although the surface is unstable. For single crystal surface, the ripple can be induced by the surface anisotropy diffusion at normal bombardment. In our case, however, the ordered patterns can be formed at normal bombardment although we show that surface effects can be negligible. This is because of a relationship between two principle curvatures as shown in Eq. 共15兲. The critical radius seems to provide a criterion for distinguishing between plane and cylinder. When the radius exceeds the critical value, the behavior of cylinder is similar to the planar surface described by BH model.9 For BH model,9 the smoothing terms induced by sputtering and surface tension are competitive with the roughening terms induced by sputtering etching and ES barrier, while for the wire, both smoothing and roughening terms make the wire unstable. The role of smoothing is to select one unstable mode for the radius less than critical value. B. Flux dependence of wavelength

Flux dependence of ripple wavelength has been extensively studied.21–23,39,45 According to BH model,9 the wavelength is inversely proportional to the square root of flux. This relationship is derived under the assumption of fixed concentration of mobile species on the surface. However, the concentration of mobile species is ion flux and temperature dependent as shown by Chan et al.21–23 and Erlebacher et al.39 Using creation rate equations, they derived variation of ripple wavelength with flux and temperature under different conditions. Their approach can be extended to our model. If we neglect cluster formation, the concentration of mobile defects on the surface, depending on the production and annihilation rates, satisfies the following mass conservation equations:46–48

⳵Cv = D vⵜ 2C v + K d − K ivC iC v − 兺 K svC sC v , ⳵t s

共23a兲

⳵Ci = Diⵜ2Ci + Kd − KivCiCv − 兺 KsiCsCi − Y f , ⳵t s

共23b兲

where Di and Dv are the defect diffusion coefficients, Kd is the defect generation rate, Ci and Cv are the defect concentration, Cs are the sink densities, Ksi and Ksv are the sink reaction rates, Y is sputtering yield, Kd = f ␴d / N, where ␴ is

displacement cross section in units of displacements/ion/Å, and N is atomic density of target. Because two equations are coupled in a nonlinear form, they have no analytical solution, and are normally solved numerically. In the cases where the spatial distribution of defects is important, Eq. 共23兲 can be solved in their spatially dependent form 关third and fourth terms in Eq. 共23a兲 are zero兴.21 In the cases where the defect separation is larger than the distance between extended sinks, this equation is used in its spatially independent form 关first terms in Eq. 共23兲 are zero兴.22,39,45 Depending on the irradiation temperature, the microstructure of materials, and the damage rate, two types of processes can be identified in spatially independent form: for high mobility of defects, high sink density and low displacement rate, defects absorption at sinks limits the accumulation, and thus we have sinkdominated regime, whereas for low temperature, high displacement rate and low sink density microstructures, recombination limits accumulation, giving rising to a recombination-dominated regime. At steady state, in sink-dominated regime, the first and third terms on the right hand of Eq. 共23兲 can be assumed to be zero, thus Cv ⬀ f and Ci ⬀ f, and then B ⬃ f. From Eqs. 共20兲 and 共21兲, we know that wavelength is independent of flux. In recombination-dominated regime, the first and fourth terms on the right hand of Eq. 共23兲 are zero. We have Cv ⬀ f 1/2 and Ci ⬀ f 1/2, and then B ⬃ f 1/2. Therefore, under this condition wavelength is flux dependent. If we further assume that ␨ Ⰶ B, we have ␭ ⬀ f −1/4. In our experiments, ion flux is high 共f = 共4.2⫻ 1014兲 – 共2 ⫻ 1016兲 cm−2 s−1兲. We should consider the sinks induced by the implanted Ga atoms on the surface. It is well known that, with the receding of the sample surface, the initial Gaussian shaped distribution of implanted atoms overlaps and the concentration of implanted atoms at the surface increases. The Ga atom fraction on the surface at steady state can be approximated as CGa = 1 / 共1 + Y兲,49 where Y is the sputtering yield per implanted ion. For 10 keV Ga ion on Cu surface, Y ⬇ 4,40 and then CGa ⬇ 20%. These implanted Ga atoms on the Cu surface can create high sinks for mobile species. Considering the mobilities of species at the temperature in experiments, we believe our experimental conditions belong to sink-dominated regime in which the wavelength is flux independent. This result is in good agreement with experimental observations shown in Fig. 4. C. Temperature dependence of wavelength

The thermal equilibrium concentration of defects is small compared with the ion-induced defect concentration in our experimental conditions. We neglect this effect in Eq. 共23兲. But at high temperature, the defects have high mobility and thus the thermal equilibrium defects are dominant. We need to use thermal defect creation rate to replace Kd in Eq. 共23兲. As a result, we have ␭ ⬀ f −1/2.21 At intermediate temperatures where kd dominates over diffusion process, wavelength is independent of flux, corresponding to the sinkdominated regime. At still lower temperature and lower sink densities, the concentration of defects induced by the beam


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becomes high enough that recombination starts to dominate. This situation corresponds to a recombination-dominated regime in which ␭ ⬀ f −1/4. During bombardment, the slowing down of the incident ions in the target materials initiates displacement cascades. Except a small fraction of kinetic energy is emitted by sputtered atoms or stored as defects, almost all of the kinetic energy is converted to heat. Under conditions that sample well contacts to a heat reservoir, a finite steady-state temperature increase is given by Ti = P / 共␲ri␬i兲,50,51 where P is beam power, ri is beam size, ␬i is sample thermal conductivity. In our case, Cu line on the surface of Si substrate, T ⬍ 1 ° C. Therefore, in our experiment, it is reasonable to assume that temperature is the heat reservoir temperature. If ion-induced smooth mechanism and ion-induced roughening mechanism dominate at low temperature,21,22,24,38 from Eqs. 共7兲, 共13兲, and 共21兲, we have ␭ = 2␲冑␨ / ␩ ⬇ ␣ / 2 for ␣ = ␤, independence of flux and temperature. Because ␣ is around several nanometers for the ion energy less than 10 keV, the wavelength derived from these two mechanisms is two orders of magnitude smaller than the experimental data. Similar results have been reported for the ripple formation on silicon surface by low-energy noble gas bombardment at room temperature.2 On the other hand, if thermally induced smoothing mechanism and ion-induced roughening mechanism dominate,9,10,21,22,24 from Eqs. 共7兲, 共12兲, and 共21兲 we obtain the wavelength which is a function of diffusivity, concentration of mobile species on the surface, and temperature. The simple relationship between wavelength and flux or temperature can be derived under special cases from rate equations similar to Eq. 共23兲. In our approximation, we did not consider the effects of redepositon and viscous flow on the instability of nanolines. The first effect can lead to the nonlinear terms which have little effects on the growth of a small disturbance for linear stability analysis.52 Viscous flow is another important relaxation mechanism in amorphous systems as pointed out by a number of groups.45,53–55 In our case, because the surface remains crystalline during bombardment, this effect is neglected in our model.

VI. CONCLUSIONS

In conclusion, we have studied the morphological evolution of ion-induced nanoparticle formation on Cu lines both experimentally and theoretically. The experimental data show that when the line width reaches a certain value, it breaks up to particles with periodic spacing. We combine the Sigmund theory16,17 and Nichols and Mullins15 approach to develop a new model to interpret the experimental observations. Linear-stability analysis shows that there exists a critical radius for particle formation. For the line with width less than this value, a longitudinal mode can develop until periodic particles form. The line with larger width will eventually reach this critical value by ion etching and start to form periodic particles.

ACKNOWLEDGMENTS

This work was supported by the Office of Basic Energy Science of the U.S. Department of Energy through Grant No. DE-FG02-02ER46005. 1

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