LaNiO3 superlattices

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Journal of Crystal Growth 279 (2005) 114–121 www.elsevier.com/locate/jcrysgro

Real-time X-ray study of roughness scaling in the initial growth of epitaxial BaTiO3/LaNiO3 superlattices Yuan-Chang Lianga, Hsin-Yi Leeb,, Heng-Jui Liub, Chun-Kai Huanga, Tai-Bor Wua a

Department of Materials Science and Engineering, National Tsing Hua University, Hsinchu, Taiwan, Republic of China National Synchrotron Radiation Research Center, 101 Hsin-Ann Road, Hsinchu Science Park, Hsinchu 30076, Taiwan, Republic of China

b

Received 8 November 2004; accepted 11 February 2005 Available online 7 April 2005 Communicated by M. Schieber

Abstract The evolution of surface and interface roughness during heteroepitaxial growth of strained BaTiO3/LaNiO3(BTO/ LNO) superlattices on SrTiO3 (0 0 1) substrates was investigated using real-time X-ray reflectivity measurements in situ with synchrotron radiation. The roughness scaling of the growth front and interface of superlattices with modulation length below (2 and 6 nm) and beyond (20 nm) the critical thickness was studied against the bilayer number. The fitted results of in situ specular X-ray reflectivity curves reveal a two-dimensional growth of BTO and LNO sublayers on the SrTiO3 substrate for superlattices with a modulation length below the critical thickness. Moreover, a larger root-meansquare roughness of BTO/LNO interface was obtained in the superlattice with modulation length beyond the critical thickness indicating lattice relaxation in the superlattice structure. Fitted results of in situ specular X-ray reflectivity curves provide the first evidence for power-law scaling of the root-mean-square roughness of an interface and a surface in the superlattice structure. Observation of such a roughness scaling behavior indicates that strain plays an important role in the determination of microstructure in the growth of epitaxial BTO/LNO superlattices. r 2005 Elsevier B.V. All rights reserved. PACS: 68.10.Nz; 68.10.Kw; 68.55.Jk Keywords: A1. Surface structure; A1. X-ray diffraction; A3. Superlattices

1. Introduction Corresponding author. Tel.: +886 35780281x7120; Fax:

+886 35783813. E-mail address: [email protected] (H.-Y. Lee).

Clarifying the evolution of morphology of surface and interface is a challenging problem in

0022-0248/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2005.02.047

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the study of superlattice materials. The presence of interfacial roughness and strain are inevitable during the formation of a boundary between two media. The formation of interfaces and surface is influenced by many factors, which prove almost impossible to distinguish. It is highly desirable to find a functional form of the surface and interface roughness in terms of epilayer thickness, which might allow the prediction of roughness parameters under various processing conditions. Important progress has been made in a dynamic scaling approach; it is generally recognized that the morphology and formation dynamics of the film surface and interface exhibit a simple dynamic scaling behavior despite the complicated nature of the growth process. In the scaling regime, the change of roughness ðsÞ follows time (t) according to a power law stb ; and the duration of deposition is directly proportional to the epilayer thickness or the number of bilayers in the superlattice [1,2]. Theoretical calculation of the exponent b depends on the modeling of the dynamics that would capture the basic physics of epitaxial growth. Hence, experimental determination of this exponent is important, not only for a practical interest of characterizing the surface and interface roughness but also to assess the validity of theoretical models. In previous experiments on roughness scaling [3–5], the diversity of these measured values of b indicates that growth dynamics are complicated for various material systems, varied surface diffusion, or surface relaxation mechanisms [6]. The development of interface structure is, in general, a function of the method and conditions of deposition [3,7], but apart from the effect of the deposition parameters another important parameter that might influence the surface and interface structure is the epilayer thickness. The mechanism of multilayer growth is associated with evolution of roughness at the interface and its nature along the whole stack of the multilayer. The roughness is strongly dependent upon both the number of bilayers (N) [8] and the value of modulation length (d). Because of the non-equilibrium situation, measurement in situ is advantageous to diminish the influence from kinetic effects. On investigating the scaling behavior of surface and

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interface roughness in the initial growth of BaTiO3/LaNiO3 (BTO/LNO) superlattices as a function of d and N, we here report the results of experiments on real-time X-ray scattering in situ with synchrotron radiation. The BTO/LNO superlattice was chosen for this purpose because of its remarkable dielectric properties compared to the single-layer BTO film of the same effective thickness, and also because strain between constituent compounds in the initial several N has a large influence in determining the evolution of surface and interface roughness in a superlattice, further influencing the dielectric performance of superlattice [9].

2. Experiments The BTO/LNO superlattice films were epitaxially grown on SrTiO3 (0 0 1) single-crystal substrates. The designed thickness of each sublayer was fixed at 1, 3, and 10 nm. We adopted a symmetric sublayer structure, i.e. (BTOm/ LNOm)N, in which m is the thickness of a sublayer with unit nanometer and N is the number of bilayers. The deposition was performed at sputtering power densities 3.95 and 2.96 W/cm2 for BTO and LNO sublayers, respectively. During deposition, the substrate temperature was maintained at 500 1C, and the working pressure of deposition was fixed at 2 Pa, with an Ar/O2 ratio 4:1. The rate of deposition was 0.5 nm/min. Specular X-ray reflectivity was measured in situ and in real time through y–2y scans, using synchrotron radiation performed on wiggler beamline BL-17B1 at National Synchrotron Radiation Research Center, Hsinchu, Taiwan. The incident X-rays were focused vertically with a mirror and made monochromatic to an energy 10 keV with a Si (1 1 1) double-crystal monochromator. With two pairs of slits between sample and detector, the typical wave vector resolution in the vertical scattering plane was 0.005 nm1 in this experiment. The surface morphology of the films was investigated ex situ with an atomic-force microscope (AFM). These observations were conducted with a contact mode on an area 1 mm  1 mm.

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3. Results and discussion

Intensity (arb. units.)

0.12

0.08

unit:nm (BTO-1/LNO-1)N/STO

1010 108 106 104

5bilayers 4bilayers 3bilayers 2bilayers 1bilayer

102 100 10-2 0

1

2

(a)

3 qz (nm-1)

5

4

6

1013

Intensity (arb. units)

1011

(BTO-3/LNO-3)N/STO

109 107 105

5bilayers

103

4bilayers

101

3bilayers 2bilayers 1bilayer

10-1 10-3 0

1

2

(b)

3 qz (nm-1)

4

5

6

1012 (BTO-10/LNO-10)N/STO

1010 Intensity (arb. units)

The epitaxial nature of BTO and LNO layers in a superlattice is demonstrated by the in-plane orientation ((1 0 1) reflection) relation with respect to the major axes of the SrTiO3 (STO) substrate. The azimuthal diffraction patterns of the (B3/L3)10 superlattice film in the vicinity of a main peak are shown in Fig. 1. The clean four peaks at 90o intervals to each other have nearly the same intensity. No other peaks are observed between the intervals of four peaks, indicating the alignment of a and c axes of BTO and LNO strained unit cells along those of the STO substrate to be perfect. Figs. 2(a)–(c) show the specular component of the X-ray reflectivity in situ against the momentum transfer (qz) in reciprocal space along the substrate surface normal direction obtained from a superlattice with varying d and N. In each curve, the diffuse scattering was subtracted first, and the data points (open circles) represent only the specular component. The intensity oscillation originated from the interference of X-rays reflected from the growing surface and those from the interfaces between constituent compounds and that of film/ STO substrate [10]. If the surface of the film were perfectly flat, the intensity of the X-rays reflected from it would be comparable to the intensity of the X-ray beam reflected from the interface between

Intensity (arb. units)

1012

108 106 104

5bilayers 4bilayers 3bilayers 2bilayers 1bilayer

102 100 10-2 10-4

0.04

0 (c)

0.00 -50

0

50

100 150 200 Phi (Degree)

250

300

Fig. 1. X-ray phi-scan of (1 0 1) main peak for (B3/L3)10 superlattice.

1

2

3 qz (nm-1)

4

5

6

Fig. 2. In situ specular reflectivity curves (open circle) and its best fit (solid line) from superlattice with varing modulation length (d) and bilayer number (N): (a) d ¼ 2 nm (b) d ¼ 6 nm (c) d ¼ 20 nm:

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1.2

Roughness (nm)

1.0

Surface

0.8

Sublayer thickness (nm) 1 3 10

0.6 0.4 0.2 1

2

(a)

3 4 Number of bilayers

5

1.4 1.2 Roughness (nm)

the film and the STO substrate. If, however, the surface of the film is too rough, the intensity of Xrays reflected from the surface is negligible and only that from the STO substrate is detectable. In this situation, no interference between these two interfaces occurs and we observe no oscillation in the X-ray intensity. The oscillation amplitude and the overall X-ray intensity depend strongly on the roughness of the surface and of the interface [11]. Well-defined oscillations (Kiessig fringes) of reflected intensity are clearly visible in Figs. 2(a)–(c), indicating the presence of a well-ordered layer structure independent of the variation in d or N. The thickness of the film is given as 2p=Dqz [12], in which Dqz is the period of the oscillation. Hence, the period of oscillations decreases as the thickness of the stacking period or the number of bilayers increases. In the case of a BTO/LNO superlattice, the lattice mismatch in this system is estimated to be 3%; this large misfit makes lateral coherent growth of constituent compounds occur to only a short modulation length. According to the estimation from Matthews–Blakeslee theory [13], the value d ¼ 20 nm is larger than the critical value (16 nm) for the generation of misfit dislocations [9]. Figs. 3(a) and (b) summarize the evolution of surface and interface roughness in situ from superlattices with d ¼ 2; 6, and 20 nm against N. From the multilayer stack with the dp6 nm and N ¼ 125; the surface and interface roughness are maintained at a value about 0.28 nm (0.25 nm for a bare STO substrate); this condition might be a consequence of the repetition of two-dimensional nucleation and growth of BTO and LNO sublayers on the flat terrace of the STO substrate, as proposed by Visinoiu et al. [14] and Terashima et al. [15]. As d reaches 20 nm hence larger than the critical thickness, the surface and interface roughness increase largely relative to that with dp6 nm; due to strain relief in the multilayer stack. The interface roughness seems notably to be larger than the surface roughness for d ¼ 20 nm with N ¼ 125: According to the mechanism of strain relief in a heteroepitaxy proposed by Sun et al. [16] and by Matthews [17], the dislocation half-loop generally nucleates at a film surface resulting in surface undulation and then extends toward the

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Sublayer thckness (nm) 1 3 10

Interface

1.0 0.8 0.6 0.4 0.2 0.0 1

(b)

2

3 4 Number of bilayers

5

Fig. 3. Evolution of surface and interface roughness of superlattice with varying sublayer thickness and bilayer number obtained from the fitted data shown in Fig. 2: (a) surface roughness; (b) interface roughness.

interface. More dislocation half-loops might meet and combine to form a long straight edge dislocation line at the interface, which helps to relax the misfit strain. Hence, a large density of misfit dislocations appears at the BTO/LNO interface in the superlattice with d ¼ 20 nm: Moreover, oxygen vacancies that are the most common and mobile defects in perovskite ferroelectrics tend to form planar clusters associated with interfacial misfit dislocations during film growth [18,19], which roughen the interface structure at the BTO/LNO interface. This phenomenon was ob-

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served in other ferroelectric superlattices by Catalan et al. [20]. Figs. 4(a) and (b) exhibit the surface morphology of superlattices with varied d and N observed from AFM. The morphology analysis reveals important clues for the growth mechanism in superlattices with varied d and N. Fig. 4(a) exhibits the surface morphology of a superlattice with d ¼ 2 nm and N ¼ 5: At this stage, the film has small nuclei or grains that grow uniformly across the substrate. Moreover, in Fig. 4(b) corresponding to d ¼ 20 nm and N ¼ 5; distinct and widely spaced three-dimensional features that develop on the surface are observed, indicating that strain relaxation in multilayer stack is related to a Stranski–Krastanov nucleation mode [21,22]. The observation of AFM images agrees with the results from Figs. 3(a) and (b), i.e. the morphology of the surface has undergone a

Fig. 4. AFM surface images: (a) multilayer film with d ¼ 2 nm and N ¼ 5; (b) multilayer film with d ¼ 20 nm and N ¼ 5:

significant modification when the critical thickness becomes exceeded. A difficulty in applying a kinetic roughening model to explain the experimental results arises from the substrate effect. The substrate morphology plays an important role in the evolution of surface roughness during deposition of ultra-thin films [23]. Especially in heteroepitaxial deposition, with a thickness less than the critical value, the substrate effects are more problematic because strain is present from the lattice-mismatch between the substrate and the overlayer films. The ubiquitous layer-by-layer growth, i.e. two-dimensional growth, of perovskite oxide films on latticemismatched substrates might be also considered a result of the substrate effect. The roughness scaling in the growth of superlattices is illustrated in Figs. 5(a)–(c), in which the change of roughness against the total thickness of superlattice is plotted on a log–log scale [24,25]. It seems difficult to determine exactly the growth exponent b of surface roughness from these figures because of the substrate roughness contribution [26]. When fitting all root-meansquare (rms) roughness data of a surface to tb ; the obtained values of exponent b for d ¼ 2 and 6 nm are about 0.377 and 0.3, respectively. The evolution of surface roughness for coherently strained sublayers on a STO substrate can be discussed in terms also of a competition between the substrate roughness contribution and growth-induced roughening. The roughness of a growth front, ss ; has contributions from substrate roughness sSTO and growthinduced roughness sg ; i.e. ss ¼ ðs2STO þ s2g Þ1=2 : From Fig. 5(a), the surface roughness of a multilayer stack with d ¼ 2 nm and N ¼ 1 is ss ¼ 0:245  0:014 and that for the same modulation length but N ¼ 2 is ss ¼ 0:25  0:02: The above result shows that surface roughness exhibits rms roughness slightly less than or equal to the bare STO substrate ss 0:25 nm: In contrast, the surface roughness of a multilayer stack with d ¼ 6 nm for N ¼ 1 ðss ¼ 0:232  0:018Þ and N ¼ 2 ðss ¼ 0:242  0:019Þ shows the same tendency (Fig. 5(b)), i.e. a smoothing effect is observable in the initial growth of two bilayers on the substrate. The result is similar to observations by Ballestad et al. [27] and of Gyure et al. [28]. Considering sg values extracted from surface roughness, ss ; and taking

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Surface Interface

Roughness (nm)

β~0.377+/-0.09 β~0.13+/-0.03 1

Roughness σg (nm)

Y.-C. Liang et al. / Journal of Crystal Growth 279 (2005) 114–121 1

β~0.381+/-0.03

0.1 10 Thickness of superlattice (nm)

0.1 1 (a)

10 Thickness of superlattice (nm)

Interface

Roughness (nm)

β~0.3+/-0.05 1

β~0.12+/-0.03

Roughness σg (nm)

1 Surface

β~0.6+/-0.12

0.1 10 100 Thickness of superlattice (nm)

0.1 10 Thickness of superlattice (nm)

(b)

100

Surface

Roughness (nm)

Interface β~0.243+/-0.04

1

β~0.18+/-0.01

100 (c)

Thickness of superlattice (nm)

Fig. 5. Evolution of surface and interface roughness during the in situ growth of multilayer stack obtained from the fits of the data shown in Fig. 2. The evolution of roughness as a function of total thickness of superlattice was plotted on a log–log scale. The slope of the curves represent the dynamic roughness exponent b: (a) d ¼ 2 nm (b) d ¼ 6 nm (c) d ¼ 20 nm: The insets of Figs. 5(a) and (b) show the growth-induced roughness versus total thickness of superlattice on a log–log scale for d ¼ 2 nm and d ¼ 6 nm; respectively.

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into consideration only the points after the roughness transition from smoothing to roughening, the growth-induced exponent b ¼ 0:381  0:03 and 0.670.12 is obtained for d ¼ 2 and 6 nm, respectively (illustrated in the insets of Figs. 5(a), (b)). On the basis of the dynamical scaling approach, the above results are taken to indicate the occurrence of kinetic roughening in the growth front of superlattices. A Schwoebel barrier, i.e. asymmetry in the vertical and horizontal diffusion of adatoms, is reported to cause instability during film growth [29]; this effect also exhibits a dynamic roughening proportional to time [30]. However, the roughening effect generated from a Schwoebel barrier results in pyramids in the epitaxial growth with exponent b1 [31]. This large exponent b value seems unusable to interpret the evolution of surface roughness in the current case. Under the conditions of a multilayer stack for d ¼ 2 or 6 nm along with the initial several bilayers, a coherent strain between constituent sublayers is expected [9]. The presence of coherent strain in the heteroepitaxial superlattice structure might also provide a pinning force during epitaxial growth so that constituent atoms are not allowed to relax or to diffuse freely [32]. This mechanism is not taken into consideration in derivation of most theoretical models, which possibly explains that observed b values are larger than predicted according to Kardar–Parisi–Zhang theory [33]. The surface exponent b for d ¼ 20 nm is smaller than that for d ¼ 2 and 6 nm. In this case, strain relief at the interface between consecutive BTO and LNO layers is realized because the sublayer thickness exceeds the critical value for maintenance of misfit strain at the BTO/LNO interface [9]. The exponent b in this case is about 0.234, which implies that surface roughness increases at a smaller rate with increasing total thickness of the multilayer structure than that with d value below the critical thickness. Moreover, this value is near the predicted value from theory, i.e. 0.25 [34,35]. The evolution of surface roughness in the growth stage with layer thickness beyond the critical thickness in other heteroepitaxies shows a similar result. The Fe/Si heteroepitaxy shows an exponent b value 0.24 and that in InN/Al2O3 system is 0.236 [24,25].

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The evolution of interface roughness with increasing total thickness of superlattice is also shown in Figs. 5(a)–(c). The interface roughness is almost constant at 0.2570.03 nm with b0.13 for superlattices having d ¼ 2 or 6 nm. Hence, the interface roughness is not describable with a power law of scaling the same as that for surface roughness, i.e. the surface exponent b is not equal to interface exponent b: In contrast, for d ¼ 20 nm; the interface roughness shows a visible cumulative characteristic with increasing total thickness of the superlattice compared to that having d ¼ 2 or 6 nm, and a maximum accumulation of roughness between N ¼ 1 and N ¼ 2 was observed irrespective of any other adjacent bilayer number. From comparison of interface and surface roughness in superlattices with a modulation length below the critical value (d ¼ 2 and 6 nm), a trend of slightly smaller rms roughness of interface was observed with respect to that of the surface. In other heteroepitaxial systems, the surface exhibits larger rms roughness than that of the interface in a regime of the layer thickness less than a critical value [32]. In other superlattice systems, the multilayer structure suppresses an increase of interface roughness by providing a periodic restarting layer [36,37]; this suppression of the interface roughness would not be observed if a single layer of similar thickness was deposited [10]. However, only the interface itself is insufficient to explain completely the roughening process at the interface in our case. The evolution of roughness within multilayers depends on the thermodynamic properties of the constituent layers, intrinsic defects in the multilayer stack, and the kinetic properties of deposited atoms in a way that is incompletely understood [38]. Nevertheless, the interface still exhibits a simple dynamic scaling behavior with sublayer thickness beyond or under a critical value despite the complicated nature of the growth process.

4. Conclusion We have investigated in situ the roughness scaling behavior in initial growth of a BTO/LNO superlattice by real-time X-ray reflectivity mea-

surements with synchrotron radiation. These Xray reflectivity curves confirm that a well-ordered layered structure was formed irrespective of the variation in stacking period or bilayer number. Moreover, fitted results of specular X-ray reflectivity curves provide first evidence for power-law scaling of the rms roughness of interface and surface in the superlattice structure. The evolution of surface roughness in the superlattice with d ¼ 2 and 6 nm is describable with a dynamic scaling exponent b0.377 and 0.3, respectively. The presence of coherent strain in the superlattice structure with modulation length below the critical thickness might play an important factor in the observed b values. However, the evolution of surface roughness for a superlattice with d ¼ 20 nm (larger than the critical value) shows a smaller exponent b0.234 consistent with reported values in other heteroepitaxial materials. All observed roughness scaling behavior in the BTO/ LNO superlattice clearly exhibits that the strain condition in the superlattice has a profound effect on the microstructure development of surface and interface in BTO/LNO superlattices.

Acknowledgments We thank for support the National Science Council of the Republic of China under Contract no. NSC 92-2216-E-213-001, and the Ministry of Education of Republic of China under Contract no. A-92-E-FA04-1-4.

References [1] H. Yan, Phys. Rev. Lett. 68 (1992) 3048. [2] F. Family, Physica A 168 (1990) 561. [3] Y.L. He, N.H. Yang, T.M. Lu, G.C. Wang, Phys. Rev. Lett. 69 (1992) 3770. [4] H. You, R.P. Chiarello, H.K. Kim, K.G. Vandervoort, Phys. Rev. Lett. 70 (1993) 2900. [5] E.J. Ernst, F. Fabre, R. Folkerts, J. Lapujoulade, Phys. Rev. Lett. 72 (1994) 112. [6] J. Krug, M. Plischke, M. Siegert, Phys. Rev. Lett. 70 (1993) 3271. [7] E.E. Fullerton, J. Pearsion, C.H. Sowers, S.D. Bader, Phys. Rev. B 48 (1993) 17432.

ARTICLE IN PRESS Y.-C. Liang et al. / Journal of Crystal Growth 279 (2005) 114–121 [8] E.E. Fullerton, D.M. Kelly, J. Guimpel, I. Schuller, Phys. Rev. Lett. 68 (1992) 859. [9] Y.C. Liang, T.B. Wu, H.Y. Lee, Y.W. Hsieh, J. Appl. Phys. 96 (2004) 584. [10] A. Paul, G.S. Lodha, Phys. Rev. B 65 (2002) 245416. [11] D.Y. Noh, Y. Hwu, H.K. Kim, M. Hong, Phys. Rev. B 51 (1995) 4441. [12] A. van der Lee, Solid State Sci. 2 (2000) 257. [13] W. Matthews, A.E. Blakeslee, J. Crystal Growth 27 (1974) 118. [14] A. Visinoiu, M. Alexe, H.N. Lee, D.N. Zakharov, A. Pignolet, D. Hesse, U. Gosele, J. Appl. Phys. 91 (2002) 10157. [15] T. Terashima, Y. Bando, K. Ijima, K. Yamamoto, K. Hirata, K. Hayashi, K. Kamigaki, H. Terauchi, Phys. Rev. Lett. 65 (1990) 2684. [16] H.P. Sun, W. Tian, X.Q. Pan, J.H. Haeni, D.G. Schlom, Appl. Phys. Lett. 84 (2004) 3298. [17] J.W. Matthews, J. Vac. Sci. Technol. 12 (1975) 126. [18] L. He, D. Vanderbilt, Phys. Rev. B 68 (2003) 134103. [19] K. Szot, W. Speier, R. Carius, U. Zastrow, W. Beyer, Phys. Rev. Lett. 88 (2002) 075508. [20] G. Catalan, D. O’Neill, J.M. Gregg, Appl. Phys. Lett. 77 (2000) 3078. [21] B.P. Rodriguez, J.M. Millunchick, J. Crystal Growth 264 (2004) 64. [22] Y.C. Liang, T.B. Wu, H.Y. Lee, H.J. Liu, Thin Solid Films 469–470C (2004) 500.

121

[23] T. Hasegawa, M. Kohno, S. Hosaka, S. Hosoki, Phys. Rev. B 48 (1993) 1943. [24] I.J. Lee, J.W. Kim, T.B. Hur, Y.H. Hwang, H.K. Kim, Appl. Phys. Lett. 81 (2002) 475. [25] H.J. Kim, D.Y. Doh, J.H. Je, Y. Hwu, Phys. Rev. B 59 (1999) 4650. [26] Z.J. Liu, Y.G. Shen, L.P. He, T. Fu, Appl. Surf. Sci. 226 (2004) 371. [27] A. Ballestad, B.J. Ruck, M. Admcyk, T. Pinnimgton, T. Tiejie, Phys. Rev. Lett. 86 (2001) 2377. [28] M.F. Gyure, J.J. Zinck, C. Ratsch, D.D. Vvedensky, Phys. Rev. Lett. 81 (1998) 4931. [29] R.L. Schwoebel, J. Appl. Phys. 40 (1969) 614. [30] M. Siegert, M. Plischke, Phys. Rev. Lett. 73 (1994) 1517. [31] I.K. Zuo, J.F. Wendelken, J. Lapujoulade, Phys. Rev. Lett. 78 (1997) 2791. [32] Z.H. Ming, S. Huang, Y.L. Soo, Y.H. Kao, T. Carns, K.L. Wang, Appl. Phys. Lett. 67 (1995) 629. [33] M. Kardar, G. Parisi, Y. Zhang, Phys. Rev. Lett. 56 (1986) 889. [34] J. Cherrier, V.L. Thanh, R. Buys, J. Derrien, Europhys. Lett. 16 (1991) 737. [35] S.D. Sarma, P. Tamborenea, Phys. Rev. Lett. 66 (1991) 325. [36] J.M. Freitag, B.M. Clemens, J. Appl. Phys. 89 (2001) 1101. [37] D.E. Savage, N. Schimke, Y.H. Phang, M.G. Lagally, J. Appl. Phys. 71 (1992) 3283. [38] P. Houdy, P. Boher, C. Schiller, P. Luzeau, R. Barchewitz, N. Alehyane, M. Quahabi, Proc. SPIE 984 (1988) 95.