Dynamics of epitaxial graphene growth and adsorptions of cobalt 2

2 Concepts of nucleation and growth on a surface This chapter describes and reviews the nucleation and island growth concepts that will be employed as an approach for the third and fourth objectives of this thesis (i.e. probing the adsorption behaviour of Co on graphene, graphite and carbon-rich 63). These concepts or collectively known as atomistic nucleation theory are based on diffusion of adatoms and their atomistic activities on the surface such as nucleation, island growth and desorption. The equations derived from this concept link the experimental observations (island density and size distribution) to the microscopic terms related to nucleation and growth dynamics of the system (for example diffusion barrier, nucleation barrier and critical nucleus size, i*). The last section of this chapter ends with review of size scaling model which is widely used to deduce the critical nucleus size, i* for nucleation of stable island. Concepts of nucleation and growth on a surface 2.1 Vapour deposition As atoms from an atomic vapour (gas phase) approaches a surface, they condense on this surface due to supersaturation of this vapour stream above the surface. These adsorbed atoms (adatoms) grow into various microscopic structures via one of these growth modes as depicted in Fig. 2.1 i.e. Frank-van der Merwe (FM), Volmer-Weber (VW) and Stranski-Krastanov (SK), all named after its founder [1]. In FM mode, films grow in layer-by-layer fashion i.e. a preceding layer is fully completed before the next layer forms. Consequently, this FM mode produces 2-dimensional (2D) structure. In VW mode, the second, third and even higher layers start to form although the first monolayer has yet to be completed. Hence this mode forms 3-dimensional (3D) islands. Finally, the SK mode is an intermediate between FM and VW mode. The system initially grows 2D but the subsequent layer growth is unfavourable and grows 3D. (a) FM growth (b) VW growth γs-f γs > γs-f + γf γf γs < γs-f + γf (c) SK growth γs γs + γs-f < γf Fig. 2.1 The three growth modes of thin film i.e. (a) layer-by-layer or FM growth, (b) 3D or VW growth and (c) layer plus 3D or SK growth. The surface energy, γ conditions that decide the growth mode is also presented. Readers are referred to text for more detailed description. The growth mode of a system depends on the balance between the surface energy of substrate, γs and the newly created energy i.e. the interface energy between substrate 36 Chapter and film, γs-f and the surface energy of film, γf. The choice is always driven by the desire of the system to keep the total surface energy as low as possible. To satisfy FM growth, formation of new surface energies must be lower than the surface energy of substrate where Δ = γf + γs-f  γs < 0. Under this condition, the film will wet the surface in layer-bylayer fashion. When the sum of γs-f and γf is larger than γs (Δ > 0), the system will adopt the 3D or VW growth mode to minimise both γs-f and γf on the surface and to keep the substrate’s surface expose as much as possible. This often happens when the adatoms are more strongly bound to each other than to the substrate. For the SK mode, there are many possible reasons for a system to switch from 2D to 3D growth and one of them is the presence of strain due to the mismatch in the lattice parameter between the layer and substrate structure at the interface. In this situation, during the early stage where 2D growth is still taking place, the film is constrained to adopt the surface structure of the substrate. This built-up in the strain energy increases with layer thicknesses and hence γs-f value increases with coverage. Initially, as coverage is low, the strain energy is small and the condition γs > γs-f + γf for the layer growth is not perturbed. Continuous increase of γs-f as the film thickness increases will eventually results in γs < γs-f + γf. In this instance, a change in growth mode i.e. from 2D to 3D occurs to relieve the strain and hence minimise γs-f to reduce the total surface energy [2,3]. 37 Concepts of nucleation and growth on a surface 2.2 Atomistic Approach to Nucleation and Growth on Surfaces In addition to consideration of surface energy being the thermodynamic component in determining the morphology of the surface, growth also depends on kinetics factors such as adsorption, desorption, nucleation and attachment dynamics on the surface. The atomistic nucleation process leading to growth can be envisaged to begin with the adsorbed adatoms thermally accommodated to the surface, after which they diffuse randomly on the surface. Because of their mobility, adatoms can undergo various atomistic pathways during the initial stage of growth as shown in Fig. 2.2 [4]. Nucleation of an island takes place when a mobile adatom meets another adatom and if the configuration is allowed, they aggregate as a pair. Growth occurs when (i) further addition of adatom onto this new nucleus happens or (ii) instead of meeting another mobile adatom, the adatom meets an existing island on the surface and attaches to it. Adatoms may also detach from an island and reduces its size. Besides activities above, desorption (re-evaporation) of adatoms into the gas phase is also possible. All of the atomistic processes above obey the Arrhenius’s law and require surface diffusion of adatoms over an intermittent potential separated by an energy barrier, Ed. This potential arises from the periodicity of the surface lattice, a. The adatom diffusion rate, D (cm2s-1) is given as [5,6,7,8]: D  E a  o exp   d  k T  B    (2.1) where a can be approximated as No -1/2 , No is the density of adsorption sites on substrate (cm-2), vo is the attempt frequency which often associated to the lateral vibration mode of 38 Chapter adatom,* kB is the Boltzmann constant and T is the surface temperature. Apart from Ed, adatoms are also subjected to other energy barriers depending on the process involved, such as binding barrier surmounted by two adatoms to form bonding (Eb, binding energy), desorption barrier (Edes) and detachment barrier (nEb, n = number of bonds to break) (see Fig. 2.2). Deposition flux, F Desorption (Edes) Surface diffusion (Ed) Island growth Smallest stable nucleus, i*+1 Nucleation (Eb , Ed) Dissociation (nEb, Ed) Surface lattice Fig. 2.2 Key processes undertaken by adsorbed atoms (adatoms) on the surface leading to formation of a stable island. These processes include surface diffusion, nucleation of islands, capturing of adatoms by a growing island, regeneration of free adatoms via dissociation of an unstable island and loss of adsorbed adatoms via desorption. The corresponding energy barriers associated with each process are given in bracket. Although adatoms on the surface undergo the atomistic process individually, their activities can be expressed collectively since they are exposed to the same conditions i.e. they are thermally accommodated to the same temperature, their diffusion is subjected to the same surface potential and they are subjected to the same energy barriers. They were first treated collectively by Frenkel [9] using steady-state rate equations. The theory has since been improved by Zinsmeister [10], Rhodin and Walton [11], Lewis [12] and others [4,13] and has been successfully applied to experimental observations [14,15,16,17]. This * vo= kBT/h where h is Planck’s constant. From 273K to 1073 K, vo range from x 1012 to x 1013 s-1. 39 Concepts of nucleation and growth on a surface atomistic nucleation model centered on an idea called critical nucleus size, i*. i* is defined as an island size (number of atoms) that upon additional of an extra adatom, (i*+1) turns into the smallest stable island on the surface. The reason for this is that adatoms on the surface are less stable than those bonded together in an island. A smallest stable island often has higher propensity towards growth than decay i.e. rate of adatom attachment always larger than rate of adatom detach from it. Under an arrival of a constant flux, F, the average time-dependence of adatoms density, n1 (cm-2s-1) on substrate is assumed to be of first-order† and is given as [13]:   dn1 n  F   21n1   i ni  22 n2   i ni dt  i 2 i 3 (2.2) where ni is the density of island size i,  (s) is the lifetime of an adatom on the surface before desorption, i (s-1) is the attachment frequency of adatom to island size i and i (s-1) is the detachment frequency of adatom from island size i. Equation (2.2) can be comprehended by dividing the terms into two groups: (a) supply of adatom (all terms in Eq. (2.2) with “+” sign) i.e.: (i) supply from flux of adatoms, F, (ii) dissociation of a dimer that generates two free adatoms, 22 n2 , (iii) detachment of adatoms from a island, i ni , (b) depletion of adatom (all terms with “–” sign) i.e.: (i) desorption of adatoms from surface, n1  , (ii) dimer formation by adatoms, 21n1 , and (iii) attachment of single adatoms to existing islands, i ni . † Rate is proportional to the population of adatoms. 40 Chapter The time-dependence of island densities, ni (cm-2s-1) is controlled by: (a) growth of island size i via attachment of single atom to island size i1 and detachment of single atom from island size i+1 (these processes have “+” sign), (b) decay of island size i via attachment of single adatom to island size i and detachment of single adatom from island size i (these processes have “”sign). Hence time-dependence of island densities ni (cm-2) can be written as: dni  i 1ni 1  i1ni 1  (i  i )ni dt (2.3) The adatom residence time on the surface before desorption,  (s-1) in Eq. (2.2) is defined as:  E  exp  des  vo  k BT  (2.4) where Edes is the activation energy for adatom desorption. The attachment frequency, ω* in Eqs. (2.2) and (2.3) is proportional to the density of adatoms on the surface, n1, the diffusion rate of adatoms on the surface, D which decide the likelihood of adatom meeting an island and capture number, σ* that gives the probability an adatom successfully attach to the island when they meet one. Hence the equation for ω* has the form: *   * Dn1 (2.5). 41 Concepts of nucleation and growth on a surface The rate equation in Eq. (2.3) can be simplified assuming nucleation rate is limited to island size i > i* [10-13]. Hence Eq. (2.3) becomes: dN   *n* dt (2.6) where *  i* , n*  ni* . The population of ni* can be found in Walton’s relation [11] and has the form: i*  n   E  ni*  N o   exp  i   k BT   No  (2.7) where Ei is dissociation energy of island i* and equals to nEb, n= number of bonds in the island and Eb= binding energy between two adatoms in the island. E1 = for i*= 1. When the surface temperature is sufficiently low for complete condensation, desorption of adatom is slow compared to the time frame for adatom to migrate and attach to an island, hence desorption can be comfortably neglected. Time-independent adatom density can be set. When surface temperature is sufficiently high for desorption, incomplete condensation occurs. Density of adatoms, n1 is limited by the average time adatoms spent on the surface before desorption, i.e. n1= Fτ. Combining Eqs. (2.1), (2.4) and (2.7), solving Eq. (2.6) has the following form:  F    E  N    exp      i *  k BT   vo  (2.8) where E is the effective energy barrier for nucleation which includes Ei, Ed and Edes. χ is an exponent parameter and a function of i*. Both E and χ depends on the condensation conditions mentioned above as tabulated in Table 2.1 below. 42 Chapter Table 2.1: The exponent, χ and energy barrier, E for various geometry-dependant condensation processes [4,13]. Conditions χ E i*/(i*+2) i*/(i*+2.5) Ei  i * Ed Ei  i * Ed Initially incomplete condensation - 2-dimensional - 3-dimensional i*/2 2i*/5 Ei  i * Edes Ei  i * Edes Extreme incomplete - 2-dimensional - 3-dimensional i* 2i*/3 Ei  (i * 1) Edes  Ed Ei  (i * 1) Edes  Ed Complete condensation - 2-dimensional - 3-dimensional For a complete condensation system, Eq. (2.8) can be written in another form that illustrates the competition between arrival rate, F and the diffusion rate of adatom, D on the surface as follow:     E  F N    exp   i  D   i *  k BT  (2.8a). The diffusion rate, D can be extracted out based on Eq. (2.1). Equation (2.8a) relates the morphology (N and size) of the system to F and D as follow. For a system held at a constant temperature, T, increasing the arrival rate of adatoms, F, will increase the probability of adatoms to meet another adatoms and this leads to higher density of island, N. On contrary, reducing F will create lower adatom population on the surface. This permits adatoms to diffuse on the surface at a longer period of time before meeting another mobile adatom and increases their probability of meeting an island and attached to it. Hence high F increases the system propensity towards nucleation but the average size of the islands is also smaller. Smaller F however increases the system propensity towards island growth but reduces the island density. If surface temperature is increased, the diffusion rate of adatoms is also increased and hence they effectively cover a larger area. This will increase their probability to meet an existing island than nucleating a fresh 43 Concepts of nucleation and growth on a surface island. Hence higher T (higher D) will produce a distribution of islands with larger island size but lower density. The above nucleation dynamics is limited to the pre-coalescence regime i.e. before saturation of island density occurs. Under a constant arrival flux and growth temperature, the coverage- or deposition time-dependence of island density is shown in Fig. 2.3. As depicted the density profile can be divided into four main regimes. In regime I, which occurs during the initial stage of growth, is dominated by nucleation of new islands. In this regime, density of island increases linearly at a constant nucleation rate, J. As the density of islands increases with coverage/ time, the island-island separation becomes comparable to the diffusion length of adatoms. Hence, most of the adatoms are captured by existing island and nucleation rate of fresh islands decreases. This regime, which is shown as regime II in Fig. 2.3, is dominated by growth. As the size of the islands continues to increase, coalescence begins where the islands growth zone overlaps with the neighbouring islands causes the island density to reduce despite nucleation may still occur at a much lower rate than regime II. This competition between nucleation and coalescence creates a saturation regime (Regime III) where the maximum island density is observed when the nucleation rate is comparable to coalescence rate. When the coalescence rate continually increase with coverage and eventually faster than nucleation rate, the density of island begins to decrease as observed in regime IV. This density profile is also a function of temperature. As the growth temperature is increased from T1 to T2, the diffusion length of adatom increases and the system propends towards growth than nucleation (Eq. (2.8a)). Hence, slower nucleation rate is observed for J2. Because the density of island also increases much slower, the time taken to reach 44 Chapter saturation regime (regime III) is longer i.e. t2 > t1. The maximum cluster density for higher growth temperature also lower since the average island size is bigger. Their growth zone is hence also larger than islands at lower temperature. This translates to lower saturation density. II Island density (/cm2) I IV III Nsat,1 J1 T1 < T2 T1 J2 Nsat,2 T2 II I t1 III t2 IV Coverage/ deposition time Regime I: Nucleation dominated regime Regime II: Growth dominated regime Regime III: Onset of coalescence Regime IV: Coalescence dominated regime Fig. 2.3 Typical coverage/ deposition time-dependant island density for nucleation on a surface under a fixed arrival rate and constant growth temperature conditions. The atomistic model of nucleation in Eq. (2.8) can be applied to understand dynamics of surface nucleation and growth. According to Eqs. (2.8) and (2.8a), i* (or E) can be extracted from a series of experiments that carried out under different F (or T). Invention of high resolution imaging techniques such as scanning tunneling microscopy (STM) allows island density to be extracted [16,18]. Figure 2.4 shows an example of this [16]. In Fig. 2.4a, the log-log plot of island density vs. flux, F for 0.1ML Cu deposited on 45 Concepts of nucleation and growth on a surface Ni(001) at 215K is a linear function as predicted by Eq. (2.8). i* = is extracted from the slope. For higher T i.e. 345K (not shown here), i*=1 becomes unstable and i*=3 is extracted. One the other hand, carrying out the growth at different temperatures but maintaining same F allow us to extract the energy barriers in Eq. (2.8). An example is given in Fig. 2.4b where the energy barrier, E is given by the slope of the Arrhenius plot of a series of 0.1ML Cu deposited on Ni(001) at different T. (b) (a) Fig. 2.4 (a) Double-logarithmic plot of island density, nx vs deposition flux of 0.1 ML Cu deposited on Ni(001) at 215K and (b) Arrhenius plot of island density of 0.1 ML Cu on Ni(100) deposited at flux 1.34x10-3 ML/s. Figures adapted from Ref. [16]. Besides performing experiments under different incident flux to produce Fig. 2.4a, there is also another alternative to extract i* from experimental observations i.e. via scaling of island size that developed by Evans et al. [19]. This scaling model will be described in the following section. 46 Chapter 2.3 Scaling of island size and critical nucleus size, i* 2.3.1 Introduction As discussed above, an island is required to achieve a certain critical size, i* before it is stable. i* is affected by temperature and a bigger i* is anticipated when the growth temperature increases. At any given temperature, an isotropic system with homogenous surface potential has only one i*. Hence all stable islands formed on an isotropic surface originated from a same i*. For a given coverage, θ, the distribution of the island size, f(s) is anticipated to have a single peak. When this distribution is divided with average island size, for that coverage, the scaled distribution would peak at s/= 1. According to Evans et al. [19], if this scaled distribution is further normalised with the density of island with size s, (Ns) and total island density (NT), this distribution together with distributions from other coverage,  will collapse onto a common curve. This scaling relation has the following form: N s    s   Here    fi  s  s    dxf ( x)   dx.xf ( x)  , and (2.9).   dx.x f ( x)    where σ2 denotes the variance of f. According to this model, as long as the system has only one i*, the size distributions from different coverage should collapse onto a single curve after being scaled according to Eq. (2.9). The shape of this curve defines the f(s/) that related to i*. Hence fitting the curve allows one to extract i* [20]. Figure 2.5a shows the f(s/) curve for i*  according to equation given in Ref. 20. As illustrated, the f(s/) is increasingly sharper 47 Concepts of nucleation and growth on a surface and peaks at a bigger maximum as i* increases. For i*= 0, since the smallest stable island is single adatom, the distributions of island size during the pre-coalescence regime will always has the form depicted in Fig. 2.5b where f(s/) has a maximum at s/= 0. 2.5 10 f (s /) 2.0 1.5 1.0 0.5 0.0 0.0 0.5 1.0 1.5 s/ 2.0 2.5 (b) 0.8 i*=0 f(s/) (a) 0.6 0.4 0.2 i s/ Fig. 2.5 The shape of f(s/) for (a) i* = to 10 and (b) i*=0, as predicted by Family et al. [20]. The scaling model is based on the atomistic nucleation theory and in the present form it is derived with the following assumption. First, the model requires the ratio of D/F to be sufficiently high so that adatoms are able to diffuse randomly on the surface and that the density and size of the islands are obtained in the pre-coalescence regime. Second, the surface diffusion is isotropic i.e. the energy barrier for adatom diffusion for all possible directions is equal. Third, islands should ideally be free of any strain energy that may affect its growth. Therefore in reality the scaling function work best for isotropic epitaxy systems such as Fe/Fe(001) [15] and Rh/Rh(111) [21]. Deviations from the scaling function have also been observed and these are described in the next section. 48 Chapter 2.3.2 The deviation from scaling function, f(s/) (i) f(s/) of certain island dimension does not fall onto a same curve This has been observed on an anisotropic surface such as GaAs(001)-2x4 [22-24] where the scaling of InAs island width along the [110] not fall onto a same scaling curve (see Fig. 2.6a). Other island dimension i.e. the length and volume have been successfully scaled based on scaling function in Eq. (2.9). The reason for this is that the growth of island width is constrained by anisotropy of GaAs(001) surface along the [110]. Weinberg et al. [23] show that if the growth along certain dimension is constantly restricted even as the coverage increases, then that particular dimension will not be defined by a single scaling function. In contrast, the island length along the [ 10] direction, which grows unobstructed, falls onto a same scaling curve. (ii) f(s/) skews towards smaller regime i.e. s/ less than For island growth with strain on an anisotropic system, scaling is found skewed toward the left side of the x-axis i.e. peak maximum is found in the regime of s/ less than 1. One example is shown in Fig. 2.6a where strained InAs islands on GaAs(001) often skewed to the left side [ 23 , 24 ]. This behaviour is not observed for relaxed dislocated islands such as 3D InN islands on GaN(0001) as depicted in Fig. 2.6b where scaling of volume, basal area and height obey f(s/) in Eq. (2.9) and have a maximum that peaks at s/= [25]. (iii) scaling requires mass conservation f(s/) for volume and other reduced dimension such as area, width and length 49 Concepts of nucleation and growth on a surface are always found not have same f(s/) (or i*). Scaling of reduced dimension often shows bigger i* than the scaling of volume (for 3D islands system) or area (for 2D islands system). For example, length distributions of InN/GaAs often scales with i higher than basal area and volume distributions as shown in Fig. 2.6b [25]. The reason for this observation at present is still not well understood. (b) Density, Ns (a) S (x1000 nm3) Fig. 2.6 (a) Scaling of island volume (stotal) and island length along [110] of 2D InAs islands grown on As-GaAs(001) surface skew towards left. Scaling of island width along [110] does not fall onto a same curve. Figures adapted from Ref. 23; (b) scaling of island volume (top), height (bottom) and basal area (bottom) of 3D InN islands on GaAs(0001) surface under various flux and coverage at 370oC. Scaling of height shows bigger i* than scaling of island basal area. Figures adapted from Ref. 25. Equations 2.8 and 2.9 will thus be used to examine the relationship of density of islands as a function of temperature when growth studies as performed in the precoalescence regime. 50 [...]... (x1000 nm3) Fig 2. 6 (a) Scaling of island volume (stotal) and island length along [110] of 2D InAs islands grown on As-GaAs(001) surface skew towards left Scaling of island width along [110] does not fall onto a same curve Figures adapted from Ref 23 ; (b) scaling of island volume (top), height (bottom) and basal area (bottom) of 3D InN islands on GaAs(0001) surface under various flux and coverage at... Fe/Fe(001) [15] and Rh/Rh(111) [21 ] Deviations from the scaling function have also been observed and these are described in the next section 48 Chapter 2 2.3 .2 The deviation from scaling function, f(s/) (i) f(s/) of certain island dimension does not fall onto a same curve This has been observed on an anisotropic surface such as GaAs(001)-2x4 [22 -24 ] where the scaling of InAs island width along... and length 49 Concepts of nucleation and growth on a surface are always found do not have same f(s/) (or i*) Scaling of reduced dimension often shows bigger i* than the scaling of volume (for 3D islands system) or area (for 2D islands system) For example, length distributions of InN/GaAs often scales with i higher than basal area and volume distributions as shown in Fig 2. 6b [25 ] The reason for this... II: Growth dominated regime Regime III: Onset of coalescence Regime IV: Coalescence dominated regime Fig 2. 3 Typical coverage/ deposition time-dependant island density for nucleation on a surface under a fixed arrival rate and constant growth temperature conditions The atomistic model of nucleation in Eq (2. 8) can be applied to understand dynamics of surface nucleation and growth According to Eqs (2. 8)... GaAs(001) often skewed to the left side [ 23 , 24 ] This behaviour is not observed for relaxed dislocated islands such as 3D InN islands on GaN(0001) as depicted in Fig 2. 6b where scaling of volume, basal area and height obey f(s/) in Eq (2. 9) and have a maximum that peaks at s/= 1 [25 ] (iii) scaling requires mass conservation f(s/) for volume and other reduced dimension such as area, width and. .. the form depicted in Fig 2. 5b where f(s/) has a maximum at s/= 0 2. 5 1 2 3 4 5 6 7 8 9 10 f (s /) 2. 0 1.5 1.0 0.5 0.0 0.0 0.5 1.0 1.5 s/ 2. 0 2. 5 (b) 0.8 i*=0 f(s/) (a) 0.6 0.4 0 .2 i 0 1 2 3 4 s/ Fig 2. 5 The shape of f(s/) for (a) i* = 1 to 10 and (b) i*=0, as predicted by Family et al [20 ] The scaling model is based on the atomistic nucleation theory and in the present form it...Chapter 2 saturation regime (regime III) is longer i.e t2 > t1 The maximum cluster density for higher growth temperature also lower since the average island size is bigger Their growth zone is hence also larger than islands at lower temperature This translates to lower saturation density II Island density (/cm2) I IV III Nsat,1 J1 T1 < T2 T1 J2 Nsat ,2 T2 II I t1 III t2 IV Coverage/ deposition... example is given in Fig 2. 4b where the energy barrier, E is given by the slope of the Arrhenius plot of a series of 0.1ML Cu deposited on Ni(001) at different T (b) (a) Fig 2. 4 (a) Double-logarithmic plot of island density, nx vs deposition flux of 0.1 ML Cu deposited on Ni(001) at 21 5K and (b) Arrhenius plot of island density of 0.1 ML Cu on Ni(100) deposited at flux 1.34x10-3 ML/s Figures adapted from... and (2. 8a), i* (or E) can be extracted from a series of experiments that carried out under different F (or T) Invention of high resolution imaging techniques such as scanning tunneling microscopy (STM) allows island density to be extracted [16,18] Figure 2. 4 shows an example of this [16] In Fig 2. 4a, the log-log plot of island density vs flux, F for 0.1ML Cu deposited on 45 Concepts of nucleation and. .. (bottom) of 3D InN islands on GaAs(0001) surface under various flux and coverage at 370oC Scaling of height shows bigger i* than scaling of island basal area Figures adapted from Ref 25 Equations 2. 8 and 2. 9 will thus be used to examine the relationship of density of islands as a function of temperature when growth studies as performed in the precoalescence regime 50 . time-dependence of adatoms density, n 1 (cm -2 s -1 ) on substrate is assumed to be of first-order † and is given as [13]: 11 11 2 2 23 22 ii i i ii dn n Fnnnn dt            (2. 2). of adatoms. Chapter 2 41 The time-dependence of island densities, n i (cm -2 s -1 ) is controlled by: (a) growth of island size i via attachment of single atom to island size i1 and. density (/cm 2 ) I II III Coverage/ deposition time IV J 1 N sat,1 I II III IV N sat ,2 J 2 T 1 T 2 T 1 < T 2 t 1 t 2 Concepts of nucleation and growth on a surface 46 Ni(001) at 21 5K is a linear