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Vicinal silicon surfaces: from step density wave to faceting

This paper investigates faceting mechanisms induced by electromigration in the regime where atomic steps are transparent. For this purpose we study several vicinal orientations by means of in-situ (optical diffraction, electronic microscopy) as well as ex-situ (AFM, microprofilometry) visualization techniques. The data show that faceting proceeds in two stages. The first stage is short and leads to the appearance of a step density wave, with a wavelength roughly independent of the surface orientation. The second stage is much slower, and leads to the formation of a hill-and-valley structure, the period of which depends on the initial surface orientation. A simple continuum model enables us to point out why the wavelength of the step density wave does not depend on the microscale details of the surface. The final wavelength is controlled by the competition between elastic step-step interaction and facet edge energy cost. Finally, the surface stress angular dependence is shown to emerge as a coarsed-grained picture from the step model.


Introduction
Due to its scientific and technological interest, faceting of stepped surfaces has been a long standing subject of intensive research . Indeed, from a fundamental viewpoint the underlying mechanisms are still matter of debate. Furthermore, facetted systems appear to be promising templates for the "bottom-up" design of nanostructures.
One of the most important mechanisms for faceting is current-induced step bunching. While the instability of the surface is driven by electromigration [5,[13][14][15][16], the resulting pattern arises from the interplay between electromigration-induced mass transport and the minimization of the elastic energy variations resulting from the changes in the surface morphology. As we shall see in the following, step bunching also appear as a promising way to study fundamental aspects of step-step elastic interactions as well to control the surface morphology at the micro or nano scale. At the nanoscale, considerable amount of research has been devoted to the understanding of the role of surface steps in the morphological evolution of vicinal Si(111) surfaces during sublimation . These phenomena depend both on temperature and on the direction of the heating current. Stoyanov [16] was the first to propose a step model based on the Burton-Cabrera-Frank (BCF) model [26], in which electromigration is introduced as a bias in the Brownian motion of the adatoms on the surface [14][15][16]. At the microscale, Marchenko [27,28] then Alerhand [29] proposed a simple theory, based on elastic minimisation, to explain the micrometric periods which appear by annealing unstable surfaces.
In this paper we study the influence of the surface orientation on the instability, as well as the link between nanoscopic and microscale models. To do so, we have performed a systematic study of the surface morphology from the first stages (where the vicinal surface is described as a step pattern) towards the microscale state (where the facetted surface is described as a hill-and-valley structure formed by microscale facets) for various vicinal orientations. In all cases the morphological evolution proceeds in two stages: a short one based on the formation of a step density wave (the period of which, roughly does not depend upon the surface orientation) followed by a much slower one, where periodic microscale facets form via a step bunching mechanism. The final faceting period seems to depend on the elastic properties of the so-formed microscale facets.
The paper is divided in four parts. In the first part, the first section is devoted to the Vicinal surfaces can be described as stairs-like surfaces, where monatomic steps separate microscopically flat terraces. Since the atoms belonging to the step edges have a different number of nearest neighbors than the atoms in the underlying bulk, steps give rise to a lattice distortion that mediates an elastic interaction between them. The elastic description of the steps depends upon the state of the surface (surface of a stress-free body or surface of a stressed body for example [30]) as well as upon its structure. As a preamble, we thus would like to provide the reader with a detailed description of the geometry and the elastic properties of the surfaces that will be analyzed in the subsequent sections.
As shown in Fig. 1a, the selected vicinal orientations -(118), (223) (443) and (105) surfaces -form a closed cycle in the stereographic projection. More precisely, in Fig. 1a, are shown: in red the normal to the selected vicinal faces, and in blue, the normal to the (001), (113), (111), (110) and (100) surfaces which appear on the Si equilibrium shape [31]. In figure 1b are also reported the morphologies of some crystal surfaces with zone axis [ ] 10 1 .We may see, e.g., that the (118) surface is a vicinal of the (001) surface and thus is constituted of (001) terraces separated by monoatomic steps forming (111) micro-facets while  (101) surfaces can be considered to be flat at the second neighbor scale (at least for the fcc model).
In such a terrace/step model, important differences exist between the different vicinal surfaces under study. Let us thus consider separately vicinals of Si(111) and vicinals of Si(001).
(1) Since the (111) surface is isotropic, vicinals of Si(111) exhibit equivalent (111) terraces characterized by isotropic surface stresses (see figure 2a). In other words, the surface stress is a scalar. From an elastic viewpoint, steps, separating the (111) terraces, can be modeled by rows of elastic dipoles distributed along the step edge [30]. The elastic interaction between steps per unit length then scales as 2 − ℓ where ℓ is the inter-step distance (see appendix A).
(2) Si(001) is not an isotropic surface. Indeed, its number of dangling bonds is reduced by the formation of dimer pairs aligned along the 10  in the direction normal to the step gives birth to a net force across the step. The action of these steps on the underlying crystal can be modeled by a line of elastic monopoles perpendicular to the steps and distributed along the step edge [30]. The elastic interaction between steps thus scales as ℓ ln where ℓ is the inter-step distance (see [30] and appendix A).
(ii) For vicinal faces with 100 zone axis, (case of (510) surface) the steps are parallel to the 100 direction (see figure 2c) giving rise to a shear stress discontinuity at the step edge. This discontinuity can be described by a row of monopoles parallel to the step 1 . The monopoles of two neighboring steps are antiparallel. We show in appendix A that the elastic interaction between such monopoles also scales as ℓ ln .
As a partial conclusion, from an elastic viewpoint, the action of the steps of such vicinal faces on the underlying crystal can be modeled by a line of elastic monopoles parallel to the step and distributed along the step edge to the usual dipolar contribution should be added.
In   The experiments have been performed with an ascending step current direction (regime II of [35]), for which a surface instability occurs. The current direction used for the various vicinal surfaces is reported in Table 1. Heating duration varies from 15 minutes to more than 100 hours in order to observe the whole kinetic behaviour of the faceting process. The residual pressure during heating was less than 1x10 -9 mbar. Notice that thank to the evaporation regime, a clean surface is periodically regenerated so that the surface remains clean during all the process. The samples are observed in situ by optical diffraction (see the experimental setup in Fig: 3a), TEM and ex situ by AFM and optical microscopy. The TEM apparatus is a JEOL 100C microscope modified for UHV in-situ experiments [36], the AFM is a Nanoscope III from Digital Instruments used in the non-contact mode. Optical diffraction experiments were performed with a laser beam (λ=0.53 nm) at an incident angle closed to the normal incidence. The scattered light is observed with a CCD camera (see fig 3a). For longer annealing duration (several hours) the period slowly grows towards an asymptotic stae, and the diffracted pattern is slowly blurred because of the appearance of numerous defects.

TEM experiments:
TEM has been used to follow the early stages of the instability. More precisely the silicon surface is illuminated in grazing incidence, so that the shadow of the edge of the sample can be observed. The amplitude of the surface corrugation is enhanced by rotating the screen in the microscope and observing the image also in grazing conditions (as described in [36]). The results, given in figure 4, show that the surface morphology exhibits a sinusoidal shape at the early beginning of the process. The time evolution of the amplitude of the corrugation is also reported in figure 4. This evolution can be perfectly fitted by an exponential law.
In other words, in-situ experiments point out that the early stages of annealing are characterized by the appearance of a characteristic wavelength with an exponential "explosion" of the amplitude, which are the characteristic feature of linear instability with a unique most unstable mode.

II.2/ Ex-situ experiments
Ex situ experiments essentially consist in « post-mortem » examination of the samples. . In-situ optical diffraction measurements as well as TEM measurements clearly show that no smaller periodicity is observed at shorter times. This result highlights the fact that simple mechanisms based on step-pairing then double-steps pairing and so on… (e.g. zipping mechanisms.) as described in [8,37] are not appropriate to describe the underlined mechanism. Our opinion, reinforced by ex-situ AFM images is that the instability proceeds by a collective motion of the steps, giving birth to a step density wave. Curiously the value of the wavelength is roughly the same whatever the initial vicinal surface (see Fig 7). In other words, at first order, this value does not depend upon the initial distance between the monoatomic steps on the initial vicinal surface. Some other authors have yet noticed that the initial wavelength roughly does not change with the vicinality angles [19][20] . In Fig 7 we report our results (stars). We can thus define a domain (the upper dotted segment in Fig. 7) in which the wavelength roughly does not depend upon the vicinality. Some authors have also studied the wavelength change versus the vicinality, so that we can report in Fig 7 two other domains where the wavelength seems to be constant whatever the initial inter-step distance. These domains are also drawn as dotted segments in Fig. 7. The three dotted segments do not merge in a single dotted line because the experiments have not been performed at the same temperature while the wavelength depends on the sample temperature [19][20][21][22][23][24][25]. For completeness we also report some other values "gleaned" in literature [21][22][23][24][25]  (ii) At latter stages, the kinetics of faceting becomes slow, and a hill-and-valley structure forms. The bunches then start to form facets which crystallographic orientation can be easily obtained from angle measurements. It is found that the angle (α ) of one of the microscale facet remains constant while the other ( β ) increases with time.
(iii) After a long time, a stationary state is reached. It is formed by the 1 F and 2 F facets which crystallographic indexes are reported in table 2 for each initial vicinal face. The crystallographic nature of the facets shows that the bunches evolve towards the closest densely packed crystallographic planes surrounding the initial vicinal surface in the equilibrium crystal shape [31]. Notice that the 2 F facets are not flat at the atomic scale because it is easier to reach a stepped face than a flat one for which supplementary activation energy is needed for step coalescence. At the end of the first regime there is a unique wavelength but the step density still depends upon the initial vicinality. Notice that further annealing by an alternative current of the so-facetted structure restores the flatness of the nominal vicinal surface as it should be for electromigration-induced faceting.

III/ Discussion:
To sum up, all these experimental results are compatible with a quick step density wave mechanism followed by a much slower step bunching mechanism as mentioned in a previous paper [39], and as proposed in the case of non-transparent steps in Ref. [40]. During step bunching, the angle β (defined in Fig. 6) of the microscale facet increases with time while the terraces of the initial vicinal surface remain flat (α is constant). The final state is a stationary state formed with the two closest facets in the equilibrium shape surrounding the initial vicinal face (see Fig. 1a) and thus is fixed by crystallography. A sketch of the mechanism of kinetic faceting (with t the time) is reported in figure 8.
In the following, we will focus on the final and initial stages of the process.

III.1/ Analysis of the final state: towards a Marchenko-Alerhand description
Let us consider the usual faceting transition: an unstable surface (with thus negative surface stiffness) decomposes into a periodic sequence of facets with orientations 1 θ and 2 θ having different surface stresses [41]. The unstability originates in the decrease of the total surface energy from the planar to the facetted state. The slopes of the facets are given, but the period of the sequence is fixed by elasticity [27,29]. The surface stress discontinuities at the boundaries can be modelled by rows of monopoles perpendicular to the discontinuities [27,29]. The elastic relaxation induced by these forces diverges logarithmically [27,29] so that the elastic relaxation overcomes the energy of the domain boundaries. This results in the spontaneous formation of periodic facets with period L [27,29]. More precisely, the total energy change from the flat towards the facetted state classically reads [30]:  [30]. Other mechanisms can also lead to a selection of an average distance between bunches [42].
In the case under study, annealing the facetted structure without electromigration restores the nominal flat surface. In other words, in absence of electromigration, the final state is unstable The faceting thus is no more caused by the surface energy reduction but by a driving force due to the electrical field. Here we assume, as in Refs. [8,11,43] that the selection of the period remains based on the elastic relaxation whatever the origin of the destabilisation (thermodynamic or kinetics). It should mean that the electromigration field role is equivalent to define an effective surface energy change does not play a role in the selection of the period.
Furthermore, in order to have a general picture -based on atomic steps-which applies at all times, we describe the final state as an elastic interaction between steps characterised by dipoles or monopoles rather than an interaction between microscale facets characterized by their own surface stress tensor.
The usual approach to calculate step-induced elastic field is (i) to describe the step in terms of localized forces distributions applied at the step edge, (ii) to model the action of these forces on the underlying crystal by point forces acting on a semi-infinite flat crystal, and (iii) to use the Green function to calculate the strain field and then the stored elastic energy [44]. The result is well known for the surface of a stress-free (resp: stressed) body (for a review see [30]) modeling the vicinal surface as a periodic array of 1D rows of elastic dipoles (resp: monopoles) perpendicular to the step edge. In our case, the description of the elastic interactions between the steps is more complex for two reasons: (i) as shown in section I the vicinal initial surfaces may be described by various configurations (alterned monopoles and/or dipoles), (ii) in the final state these rows rearrange to form a hill-and-valley structure characterized by two lengths: the step-step distance in a bunch and the distance in between two neighboring bunches. Thus the elastic description of the final state depends on the type of monoatomic steps (that means upon the initial vicinal surface) and on the characteristic lengths. However, even if electromigration is known to induce kinetics instability, the elastic energy we calculate is that one of the final facetted structure consisting in large (001) terraces separated by step bunches. This final state, reached for a maximum of the step density, is driven by energetic and not kinetics. Electromigration thus does not modify the interstep distance in a dense bunch.
To estimate the stored elastic energy modification arising from the faceting we will proceed in three steps: (i) calculation of the elastic interaction between two steps, (ii) calculation of the elastic energy of the facetted state, (iii) calculation of the elastic energy difference between the initial vicinal face and the facetted final state. Finally, we will compare our results to the usual Alerhand-Marchenko microscale approach [27,29]. We will see that the comparison will give access to the surface stress change close to a high index surface.
Notice that in the following, we will use isotropic linear elasticity. Indeed while isotropic elasticity fails to reproduce the displacement field induced by the steps, it is now well known that isotropic elasticity can be used for determining the elastic energy with a good accuracy [45].

III.1.1/ Elastic interaction between steps:
As recalled in appendix A, the elastic interaction energy per unit length between twoparallel steps separated by a distance ℓ is well known (for a review see [30]). For elastic dipoles perpendicular to the step edge it scales as The elastic energy in the facetted final configuration can be easily obtained by adequate summations of the elastic energy interactions between two parallel rows. For the sake of simplicity we will calculate separately the elastic energy due to the interaction of steps in a bunch (containing N steps) and the elastic interaction between the bunches (see figure 9 for the geometrical definitions). The first term will be called intra-bunch energy, the second the inter-bunch energy. The analytical expressions of these energies are reported in tables 3 where for the sake of simplicity we separate the dipolar and the monopolar contributions.
Thus in the following we consider the step-step interaction as described by dipole-dipole interaction or monopole-monopole interaction but never consider the dipole-monopole interaction.
The exact expressions can be expressed as a summation of the elastic energy between two steps over the considered configuration (intra or inter bunch). Approximated analytical expressions are obtained by (i) transforming the summation to an integral then by (ii) In tables 3 are reported the expressions for elastic dipoles (Table 3a) and for elastic alterned monopoles (Table 3b).

Intra-bunch
Interaction between two bunches (inter-bunch) Interaction energy for an infinite periodic surface.
Exact expression   (see appendix A). the + and -sign arises respectively for N even and N odd. Moreover, the expressions are given per unit step-length, thus the unity is an energy over surface area.
In figure 10 are plotted the elastic interactions calculated numerically by performing the exact summations but without any monopole-dipole interaction. They are in good agreement with the approximated analytical expressions calculated by integration so that in the following we will use the approximated expressions. The main results are (i) the intra-bunch contributions depend linearly on the number of steps in a bunch (for monopoles or dipoles), (ii) for monopoles the intra-bunch elastic energy is smaller for N even than for N odd so that bunches prefer to be formed by an even number of steps, (iii) the inter-bunch analytical expressions are similar to the expressions given by Marchenko [27] and Alerhand [29] who

III.1.3/ Elastic energy change due to faceting
Let us consider the energy change due to the faceting process that means the energy change due to the transformation from a vicinal surface towards a facetted system. This energy change per unit length reads: is the relative coverage of one phase with respect to the other (see figure 9) and 0 p and 1 p the slopes of the . The term τ has been introduced to take into account the boundary energy between both domains. It does not appear naturally in the simple sketch given in figure 8 but should appear when considering that because of the symmetry breaking, the steps located at the edges of the bunch cannot have the same energy as the steps inside the bunch. Finally, the last term describes the inter-bunch elastic interaction. It does not depend upon the nature of the step interaction excepted the prefactor.
When considering that the surface occupation of each domain is, at least for the final state, fixed by crystallography (since the facets in the final state correspond to cusps of the gammaplot [31]), the energy change per unit length is a simple function of L , so that its minimum value is reached for . The equilibrium period thus reads: This expression can be compared to that obtained by Marchenko [27] and Alerhand [29].
They considered the final state as formed by microscale facets characterized by their own surface stress whose components perpendicular to the facet edges are drawn in Fig. 8 (2) and (3)  For weak values of θ one obtains from comparison of the two previous relations: This expression is analogous to the one found by Salanon for stressed solids [47] where the steps are described by the sum of rows of dipoles and monopoles (both perpendicular to the step) and the surface stress expression is developed up to second order in θ .
Equation (7) means that since the presence of steps leads to surface stress relaxation, the surface stress is maximum for a low index surface and thus decreases with θ . On the contrary, the energy cost to create surface steps implies that the surface energy increases with θ . In other words, local minima (cusps) of the surface energy plot (gamma-plot) correspond to local maxima (anticusps) of the surface stress plots [30,39,47].
Beyond this approach it is also possible to use our experimental results to obtain absolute values of surface stress. Indeed, using the microscale model of Marchenko-Alerhand [27,29], the measurement of the final period gives the difference ) ( for very transparent steps [40]. In both expressions 0 a is an atomic distance unit, A is an elastic quantity describing the dipolar forces at the steps, ξ the reduced electromigration force and Q a characteristic length varying from a tenth of an atomic distance up to some atomic distances [40]. Both expressions can be put in the generic form of a characteristic lengthscale times a "geometric factor", which is a dimensionless combination of atomic scales. Indeed, the interstep distance ℓ is of the order of some atomic distances in the experiments presented above. We here show that this generic form can be derived within the frame of a continuous model, which does not refer to microscale details.
For this purpose, we consider a model in which the initial surface is rough since the vicinal surfaces under study have high slopes. We write a continuum model based on macroscopic quantities having smooth orientation dependence. We use a 1D model, along the variable x and we neglect sublimation or growth.
From the mass conservation equation: with h is the local height and j is the surface flux.
We then consider the diffusion process driven by the variations of the chemical potential µ and the electromigration force: where M , f and c are respectively the orientation-dependent mobility, migration force, and concentration of adatoms at the surface.
We write the free energy of the surface as: where 0 γ and 1 γ are constants, and 3 γ is a function of the local slope The chemical potential is defined as is the number of atoms of the solid.
A variational calculation then leads to the usual Herring expression [48] of the surface energy variation: so that for a positive slope one obtains: Using Eq. (13) into Eqs. (8) and (9), we obtain the time evolution equation of the surface height as:

•
For the monopolar case monop A can be roughly estimated from the surface stress of the Si(001) surface. Since Nm s s [29,53], and using formula of appendix A2 one obtains Amonop .
The fact that the wavelengths are comparable for monopoles and dipoles can be easily understood, since in elasticity the only dimensional constant is the Young modulus E , and the only specific length-scale is the atomic distance 0 a . As (ii) At latter stages, the kinetics of faceting becomes slow, and a hill-and-valley structure form by a process in which the terrace orientation is conserved but the facet orientation increases with time. We have not studied in detail the kinetics of the mechanism, which will be reported in a fore-coming paper. as described by Marchenko [27] then Alerhand [29] by using directly a microscale model. The comparison between the analytical expressions issued from the two approaches: step models and the microscale approaches gives access to the angular dependence of the surface stress.
This can be used to study the surface stress anisotropy as first reported in [39].
Last but no least, our results show that it is possible to tune the period of the faceting in the micrometric range. The goal now is to be able to tune the faceting at the nanoscale. It could be possible by using growth instability [55] or externally applied stress [56].

Acknowledgements:
This work has been supported by the ANR PNANO grant: Nano-morphogénèse. Appendix:

Elastic interaction between steps
The elastic energy stored in an elastic body is simply half the work done by the surface force distribution ) (x P (characterized by its components ( ) x P α ) against the surface displacement. It can be written (for a review see [30]): ) is the Green tensor that means the displacement field ( ) x u associated to a point force of amplitude unity located at ' x [44].
For two (parallel or antiparallel) monopoles located in ( ) with the sign + for parallel monopoles and -for antiparallel monopoles 2 where α F has the dimension of a force For two (parallel or antiparallel) dipoles perpendicular to the y -direction (parallel to the step), the force distribution reads [30]: ( ) A has the dimensions of a mechanical torque. Here again, the sign + is for parallel dipoles and -for antiparallel dipoles. For a step dividing the surface in two equivalent terraces, the surface stresses 11 s of the two neighboring terraces exert a mechanical torque per unit length of moment y h s1 1 ( h being the step height) which has to be equilibrated by the torque of the force dipolar distribution so that [57,58,30] h s A 11 3 = . On the contrary, the 1 A component can only be calculated by means of inter-atomic potentials (see for example [59]).
Let us now consider two ℓ -apart steps parallel to the y direction bearing identical dipoles parallel to the x direction or antiparallel monopoles in the y direction Using the properties of the Dirac "function" there is for the monopoles Performing thus the integral there is, when defining the density of elastic energy of interaction per unit length of the step L W w L ∞ → =lim : where the quantity 0 a is an atomic unit introduced as a cutoff in order to avoid local divergences in the calculation of the integrals. In both cases the rows repulse each other.
Notice that monop w diverges while dip w converges.
Let us note that when performing the same summation for parallel monopoles in the x direction one recovers the well-known result: In the following, equations (A2) will be written:     According to the terrace structure, the steps can be described as rows of elastic dipoles or by the sum of elastic monopoles and elastic dipoles. Furthermore for the vicinal of the (001) surface one step is rougher than the other [32] as drawn in Fig. 2b . The couples of dots represent the dimers and the lines on the terraces the rows of dimmers.    : In thus figure are reported the wavelengths that appear at the very beginning of the process. More precisely we report our results (stars) as well as the results obtained by other authors in other contexts. The dotted lines correspond to domains in which no wavelength change (see text for more details). White circle [21], black down-triangles [25], white squares [19][20], diamonds [60], white up-triangle [23], white down-triangles [22], black cross [24]. Figure 8: Sketch of the faceting mechanisms. At t=0 is the initial vicinal surface. At t=t 1 , a step wave density forms by collective motion of the steps. At t=t 2 the step bunching mechanism starts so that the initial terrace (F a ) grows at constant angle while a facet (F b ) forms with the angle β(t 2 ). At the end of the process (t=∞) there is a stationary state formed by the flat facet F 1 and the stepped facet F 2 characterised by their own surface stress tensors. The surface stress component perpendicular to the edge are 1 s and 2 s .   [001] [223] [111] [443] [110] [100] [510] [113]     Initial terrace width (nm)