# Profile of a Faceted Macrostep Caused by Anomalous Surface Tension

Authors: Noriko Akutsu
Journal: Advances in Condensed Matter Physics
 (2017)

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## Abstract

The height profile of a macrostep on a vicinal surface near equilibrium is studied numerically using a restricted solid-on-solid model with a point-contact-type step-step attraction (p-RSOS model). We calculate the surface tension of vicinal surfaces around the (001) surface inclined towards the111 direction using the density-matrix-renormalization group method. We also calculate the height profiles of vicinal surfaces using the Monte Carlo method and study the connection between the height profile of the macrostep near equilibrium and the discontinuous surface tension. We find that the height profile of a macrostep on a vicinal surface near equilibrium can be classified depending on the zone in the faceting diagram where the system exists. We also find finite size effects both for the height profile and for the inhibition of the macrostep motion in the relaxation process to the equilibrium state.


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## Body

## 1. Introduction

For 4H-SiC crystals grown by a solution method, macrostep formation is considered to be a major reason for grown crystal degradation [1]. The profile of a macrostep of 4H-SiC crystal formed on a vicinal surface is shown in Figure 1. To date, the formation of faceted macrosteps near equilibrium has been considered to be due to the effects of anomalous surface tension. 4H-SiC can be used as an example for understanding the importance of controlling the dynamics of macrosteps on a vicinal surface near equilibrium. To accomplish control of the dynamics of macrosteps, we have to clarify the fundamentals of the phenomenon step by step.Figure 1
Three-dimensional profiles of macrostep growth of 4H-SiC with (a)Si0.5Cr0.5 and (b) Si0.46Cr0.5Al0.04 solvents. The lower graphs show the 2D height profiles of the macrosteps indicated by the white lines in the upper figures. Reprinted from [1], Copyright (2015), with permission from Elsevier.
(a) (b)The instabilities with respect to macrostep formation near equilibrium have been considered phenomenologically to be the result of anomalous surface tension [2, 3]. However, to date, such anomalous surface tension has not been calculated using a microscopic model. Since a crystal surface is a low-dimensional substance, it is difficult to obtain reliable results theoretically due to the large contribution from thermal fluctuations [4].In previous papers [5–10], we calculated the surface tension numerically using a restricted solid-on-solid (RSOS) model with a point-contact-type step-step attraction (p-RSOS model, Figure 2). Here, “restricted” means that the height difference between nearest neighbor sites is restricted to 0,±1. We consider that the origin of the point-contact-type step-step attraction is the orbital overlap of the dangling bonds at the meeting point of the neighboring steps. The energy gained by forming the bonding state is regarded as the attractive energy between steps.Figure 2
(a) Perspective view of the RSOS model tilted towards the direction110. (b) Top-down view of the RSOS model. The thick blue lines show the surface steps.
(a) (b)To obtain reliable results, we used the density-matrix-renormalization group (DMRG) method [11–16], and we showed numerically that the slope dependence of the surface tension is discontinuous (note that, in mathematical terms, “disconnected” is a more accurate description than “discontinuous”) at low temperatures at equilibrium.This discontinuous surface tension affects the morphology of the crystal [5–10]. The equilibrium crystal shape (ECS), which is the crystal droplet shape with the least surface free energy, can be obtained from a polar graph of the surface tension using the Wulff theorem [17–20] or obtained using the Landau-Andreev method [21–25]. The discontinuous surface tension causes a jump on the surface slope at the facet edge on the ECS. This jump of the surface slope is referred to as a first-order shape transition, after the fashion of a phase transition [26].The discontinuous surface tension also leads to the formation of a faceted macrostep on a vicinal surface, as two surfaces coexist [8] at equilibrium. Since the side surface of a faceted macrostep is smooth, the kink density of the surface becomes small. Hence, under a small driving force for crystal growth or sublimation (dissolution), the velocity of the faceted macrostep is significantly smaller than the velocity of a single step [9, 10]. In particular, at sufficiently low temperature, the macrostep moves intermittently. This inhomogeneous motion of macrosteps degrades the grown crystal.In the present work, we study how the height profile of the macrostep is classified with the profile linked to the discontinuous surface tension. In a previous paper [5], we obtained the faceting diagram for the p-RSOS model using the DMRG method. The faceting diagram is shown in Figure 3, illustrating the various zones that the model can be divided into (see [5] for details). The macrostep is formed in the step-faceting zone and in the step droplet zone. We consider three cases in detail: a typical example from the step-faceting zone (ϵint/ϵ=-0.9 and kBT/ϵ=0.6), a typical example from step droplet zone I (ϵint/ϵ=-0.9 and kBT/ϵ=0.63), and a typical example from Gruber-Mullins-Pokrovsky-Talapov (GMPT) zone I [27, 28], where ϵ and ϵint are explained in the following section.Figure 3
Faceting diagram of the p-RSOS model for a vicinal surface obtained by the DMRG method. Squares: calculatedTf,1. For T<Tf,1, the surface tension is discontinuous (disconnected, more precisely) around the (111) surface. Triangles: calculated Tf,2. For T<Tf,2, the surface tension is discontinuous around the (001) surface. Open circles: calculated roughening transition temperatures of the (001) surface. Solid line: zone boundary line calculated by the two-dimensional Ising model. For the definition and details about QI Bose solid, QI Bose liquid, and QI Bose gas, refer to [5]. This figure is taken from [5].This paper is organized as follows. In Section2, we explain the p-RSOS model. The DMRG calculated surface tensions are shown in Section 3. In Section 4, height profiles at equilibrium calculated by the Monte Carlo method for the step-faceting zone, step droplet zone, and GMPT universal zone are studied. In Section 5, we study the finite size effects both on the height profile and on the inhibition of the macrostep motion in the relaxation process to the equilibrium state. Discussions are given in Section 6 and conclusions are given in Section 7.


## 2. p-RSOS Model

The Hamiltonian of the (001) surface can be written as(1)Hp-RSOS=Nϵsurf+∑n,mϵhn+1,m-hn,m+hn,m+1-hn,m+∑n,mϵintδhn+1,m+1-hn,m,2+δhn+1,m-1-hn,m,2,where N is the total number of lattice points, ϵsurf is the surface energy per unit cell on the planar (001) surface, ϵ is the microscopic ledge energy, δ(a,b) is the Kronecker delta, and ϵint is the microscopic step-step interaction energy. The summation with respect to (n,m) is taken over all sites on the square lattice. The RSOS condition is required implicitly. When ϵint is negative, the step-step interaction becomes attractive (sticky steps).


## 3. Anomalous Surface Tension: DMRG Calculation

The surface tension is the surface free energy per normal area. In order to evaluate the surface free energy for a vicinal surface, we add the terms concerning the Andreev field [23, 24]: η=(ηx,ηy). The Hamiltonian for the grand canonical ensemble with respect to the number of steps is [29, 30] (2)Hvicinal=Hp-RSOS-ηx∑n,mhn+1,m-hn,m-ηy∑n,mhn,m+1-hn,m.The Andreev field behaves like a chemical potential with respect to a single step.From statistical mechanics, the grand partition functionZ is calculated as follows: Z=∑hm,n‍exp⁡-βHvicinal, where β=1/kBT. The summation with respect to hm,n is taken over all possible values of h(m,n). The Andreev free energy f~(η) [23, 24] is the thermodynamic grand potential calculated from the grand partition function Z, as follows [29, 30]: (3)f~η=f~ηx,ηy=-limN→∞⁡1NkBTln⁡Z,where N is the number of lattice points on the square lattice.In low-dimensional cases, more precision is required than that provided by a mean field calculation of the partition function [4]. Hence, to obtain reliable results, we adopt the DMRG method [11–16], which was developed for one-dimensional quantum spin systems. We use the transfer matrix version of the DMRG method, which is known as the product wave-function renormalization group (PWFRG) method [14–16].Examples of the Andreev free energy calculated by the DMRG method are shown in Figure4 as a function of (ηx/ϵ,ηy/ϵ) for ηx=ηy. It should be noted that the profile of the Andreev free energy f~(ηx,ηy) is similar to the profile of ECS z=z(x,y), where f~(ηx,ηy)=λz(x,y), (ηx,ηy)=-λ(x,y), and λ represents the Lagrange multiplier related to the crystal volume [23–25].Figure 4
Profile of the Andreev free energy, which is also the profile of the ECS.kBT/ϵ=0.63, ϵint/ϵ=-0.9. Z=f~(ηx,ηy)/ϵ=(λ/ϵ)z(x,y), R=±X2+Y2 with X=Y, (X,Y)=(ηx/ϵ,ηy/ϵ)=-(λx/ϵ,λy/ϵ), where z=z(x,y) represents the ECS and λ represents the Lagrange multiplier related to the crystal volume. We assume ϵsurf is equal to ϵ. The points P and Q indicate the (001) and (111) facet edges, respectively. At Q, the (111) surface and the surface with slope p1 have the same value of Andreev free energy. The pale pink line near Q indicates a metastable surface. The end point of the pale pink line shows the spinodal point approximately.Using the inverse Legendre transform with respect tof~(η), (4)fp=f~η+η·p,we obtain the surface free energy per unit x-y area, f(p). The surface tension is calculated from the surface free energy, as follows: (5)γsurfpx,py=fpx,py1+px2+py2.The polar graphs of the surface tension for several typical temperatures are shown in Figure5. In the figure, the angle θ is the tilt angle from the (001) surface towards the 111 direction. The surface gradient p is related to θ by |p|=±tan⁡θ.Figure 5
(a)kBT/ϵ=0.6, ϵint/ϵ=-0.9 (in the step-faceting zone). (b) kBT/ϵ=0.63, ϵint/ϵ=-0.9 (in step droplet zone I). (c) kBT/ϵ=0.73, ϵint/ϵ=-0.9 (in GMPT zone I). (d) kBT/ϵ=0.6, ϵint=0 (original RSOS model). Red lines: Z(R) calculated by the DMRG method. Blue lines and filled blue squares: polar graph of surface tension γsurf(θ)/ϵ (-54.74°<θ<54.74°), where θ is the tilt angle of the vicinal surface from the (001) surface towards the direction 〈111〉. The pale blue lines in (b) indicate the values for the metastable surfaces. The end points of the pale blue lines show the spinodal points approximately. Point O represents the Wulff point. We assume ϵsurf is equal to ϵ.
(a) (b) (c) (d)A characteristic of the surface tension is a discontinuity near the (001) surface and the (111) surface. There are two transition temperaturesTf,1 and Tf,2 [8]. For T<Tf,1, the surface tension is discontinuous around the (111) surface (Figure 5(b)). For T<Tf,2 (Figure 5(a)), the surface tension is discontinuous around the (001) surface. In the case of ϵint/ϵ=-0.9, kBTf,2/ϵ=0.613±0.02 and kBTf,1/ϵ=0.709±0.02 [5]. As shown in Figure 3, we call the temperature range T<Tf,2 the step-faceting zone, the range Tf,2≤T<Tf,1 the step droplet zone, and the range Tf,1≤T the GMPT universal zone. Furthermore, the step droplet zone and the GMPT zone are divided into two zones by the line of the roughening transition temperature of the (001) surface.The ECS calculated as the Andreev free energy (shown by red lines) agrees well with the ECS obtained by the Wulff theorem [17–20]. In the step-faceting zone (Figure 5(a)), the (001) surfaces directly contact the (111) surface without a curved area. In the step droplet zone (Figure 5(b)), the 001 surfaces contact the curved areas without a jump of the slopes (point P in Figure 4), whereas the 111 surfaces contact the curved areas with a jump of the slopes (point Q in Figure 4). In the GMPT zone (Figure 5(c)), both the (001) surfaces and (111) surfaces contact the curved areas without a jump of the slopes.


## 4. Profile of Macrosteps

To see the morphological effect of the discontinuous surface tension on the height profile of macrosteps, we studied the vicinal surface of the Hamiltonian equation (1) with a fixed number of steps Nstep using the Monte Carlo method.In this method, at the initial time, the steps were positioned at equal distances. Then, a lattice site to be updated is chosen randomly. The surface structure is updated nonconservatively by the Monte Carlo method with the Metropolis algorithm. With the RSOS restriction taken into consideration, the structure is updated with probability 1 whenΔE≤0 and with probability exp⁡[-ΔE/kBT] when ΔE>0, where ΔE=Ef-Ei for Ei the energy of the present configuration and Ef the energy of the updated configuration. The energy is calculated based on the Hamiltonian equation (1). In the direction along the steps, we required a periodic boundary condition. For the direction normal to the steps, we connected the lowest side of the structure to the uppermost side of the square by adding a height with a number of steps Nstep. The obtained snapshots of the surfaces and the height profiles of the vicinal surfaces at the bottom line of the top-down-view of the surface after 2×108 Monte Carlo steps per site (MCS/site) are shown in Figure 6.Figure 6
Top-down and side views of vicinal surfaces calculated by the Monte Carlo method. (a)kBT/ϵ=0.6, number of steps Nstep=60. (b) kBT/ϵ=0.63, Nstep=80. (c) kBT/ϵ=0.73, Nstep=60. ϵint/ϵ=-0.9. Size: 802×802. Time: 2×108 Monte Carlo steps per site (MCS/site). The surface height in the top-down view is represented by brightness with 10 gradations, with brighter regions being higher. Because of the finite gradation, the height of the terraces with the darkest color is higher by one than the height of the terraces with the brightest color. The line in the side views shows the surface height on the bottom side of the top-down view.
(a) (b) (c)For the vicinal surface in the step-faceting zone (Figure6(a)), there is one macrostep with an orientation of the side surface of (111). The structure of a regular train of steps is thermodynamically unstable for the direction between 001 and 111, except for the directions very close to 001 (see Figure 5(a)). In this manner, the steps self-assemble to form a faceted macrostep on the vicinal surface at equilibrium. Furthermore, the motion of the macrostep is almost inhibited. This is because the kinks in the vicinal surface are significantly small relative to the vicinal surface of the original RSOS model. It should be noted that the surface stiffness ∂2f(p)/∂px∂py for the (001) surface and for the (111) is infinite since both the (001) and (111) surfaces are smooth.For the vicinal surface in GMPT zone I (Figure6(c)), the steps appear to be dissociating and the height profile is almost linear. With a small driving force for growth or for sublimation, the mean surface height increases or decreases linearly. However, the growth or sublimation rate is smaller than that of the original RSOS model because there remain locally merged steps (step droplets).The structure of the vicinal surface in step droplet zone I (Figure6(b)) is complicated. The surface slope p1 at the point Q in Figure 4 is the key slope. For a low step density (p<p1), the steps are dissociated. No macrosteps are formed on the vicinal surface. For a high step density (p≥p1), the structure of a regular train of steps is thermodynamically unstable due to the discontinuity in the surface tension (Figure 5(b)). At that location, the vicinal surface has two surfaces, as shown at point Q in Figure 4. At point Q, the (111) surface and the surface with slope p1 have the same value of the Andreev free energy. Therefore, the (111) surface and the surface with slope p1 coexist on the vicinal surface [8]. We calculated the averaged height profiles h¯(x~) for the vicinal surface using the Monte Carlo method (Figure 7). Here, h¯(x~) is defined by(6)h¯x~=1Ny∑j=1Nyhx~,y~j,where Ny is the number of lattice points along the step direction, y~j is the location of the lattice points along the step direction, and x~=x2+y2 is the location of steps in the direction normal to the step edge.Figure 7
Averaged height profilesh¯(x~) of a vicinal surface. kBT/ϵ=0.63. Size: 1602×1602. Nstep=120. ϵint/ϵ=-0.9. Time: 2×108 MCS/site. x~=x2+y2; L~ represents the linear size of the system.Comparing Figure1 on 4H-SiC with the side views in Figure 6, the height profile in Figure 1(b) looks similar to the height profile in the step droplet zone,that is, Figure 6(b) or Figure 7. As for the case of Figure 1(a), it looks similar to Figure 6(a). According to [1], however, the terrace surface in Figure 1(a) is also slightly tilted. This similarity suggests that the surface tension of 4H-SiC around the side surface is discontinuous, whereas the surface tension around the terrace surface is continuous.


## 5. Finite Size Effect

Since the faceted step moves very slowly at lower temperatures in the step-faceting zone, it takes an extremely long time to reach global equilibrium. The time evolution of the averaged surface heightht at temperatures kBT/ϵ=0.2 and kBT/ϵ=0.4 is shown in Figure 8, where ht=(1/Nx)∑{x~}‍h¯(x~) and Nx is the number of lattice points normal to the mean running direction of the steps. In Figure 9, we show snapshots of the vicinal surface after 2×108 MCS/site. Since attachments and detachments on the crystal surface are balanced, local equilibrium is achieved. However, global equilibrium is not reached. At global equilibrium, only one macrostep should be formed.Figure 8
Time evolution of the averaged surface height in the step-faceting zone. (a)kBT/ϵ=0.2. (b) kBT/ϵ=0.4. ϵint/ϵ=-0.9. Nstep=60. Size: 802×802.
(a) (b)Figure 9
Freezing of the surface structure in the step-faceting zone. (a)kBT/ϵ=0.2. (b) kBT/ϵ=0.4. ϵint/ϵ=-0.9. Nstep=60. Size: 802×802. Time: 2×108 MCS/site.
(a) (b)For the temperaturekBT/ϵ=0.2, the surface structure changed during the initial few thousands of MCS/site but did not change for the rest of the period 2×108 MCS/site (Figure 8(a)). The surface froze due to the faceted macrosteps. The steps moved more frequently for kBT/ϵ=0.4 than for kBT/ϵ=0.2. From Figure 8(b), we can see that a step that detaches from the edge of the side surface of a macrostep due to thermal fluctuations can be readily captured by a neighboring macrostep. In this manner, the structure of the surface changes intermittently and takes far more time than 2×108 MCS/site to reach global equilibrium.Two questions arise at this point: whether the system always reaches global equilibrium forkBT/ϵ=0.6 and why the edge of the macrostep looks smeared in Figure 6 when it should be sharp according to the surface tension of Figure 5(a). In the following, we present the Monte Carlo results, which demonstrate that local equilibrium is always achieved, while global equilibrium is not reached for a large system size, and that the edge of the macrostep converges to a sharp edge in the large size limit.In order to study the size dependence of the height profile of macrosteps, we calculated the averaged height profileh¯(x~) of vicinal surfaces. We show h¯(x~) in Figure 10. As can be seen from the figures, even a time of 2×108 MCS/site is not sufficient to reach global equilibrium for sizes of 1602×1602 and 2402×2402.Figure 10
Size effect on the averaged height profilesh¯(x~) of the vicinal surface. (a) Size: 802×802, Nstep=60. (b) Size: 1602×1602, Nstep=120. (c) Size: 2402×2402, Nstep=180. kBT/ϵ=0.6. ϵint/ϵ=-0.9. Time: 2×108 MCS/site.
(a) (b) (c)However, the edges of the macrosteps become sharper as the size increases. If we express the width of the curved area byΔ(L~), Δ(L~) can be estimated by the following inequality relationship: Wv2(ρ,L~)ρ=1<Δ2(L~)<W2(L~), where Wv2(ρ,L~) is the fluctuation width of a single step on the vicinal surface with step density ρ [31, 32] and W2(L~) is the fluctuation width of a single step on the (001) planar surface [33]. Then, we obtain the inequality expression:(7)1π2ln⁡L~<Δ2L~<kBTγ~ϕL~,where γ~(ϕ) is the step stiffness defined by γ~(ϕ)=γ(ϕ)+∂2γ(ϕ)/∂ϕ2, γ(ϕ) is the step tension, and ϕ is the in-plane tilt angle relative to the [010] direction. Hence, Δ(L~)/L~→0 in the thermodynamic limit L~→∞.This result is consistent with a discontinuous surface tension (Figure5). From the thermodynamics of two-surface coexistence [8], the surface slope jumps at the cross sectioned line of the two surfaces on the vicinal surface. Hence, in the thermodynamic limit, the curved area near the edge of the macrosteps converges to zero. Thus, we can conclude that the curved areas at the edge of a macrostep appear to be due to the finite size effect at equilibrium.In the step droplet zone, global equilibrium is achieved more quickly. Though the faceted (111) side surface is difficult to move, the tilted surface (surface slope =p1) which contacts the (111) surface has many kinks. The thermal fluctuations on the tilted surface promote the achievement of global equilibrium.


## 6. Discussions

It would be helpful if real numbers for surface tension and temperatures were known for 4H-SiC. However, in order to determine these, we need to know the values ofϵ and ϵint, and to determine ϵ and ϵint, we need real values for the kink energies and temperature dependence of the step stiffness of a single step for the interface of 4H-SiC. To date, these real values are not accurately known. Therefore, more experimental observations on ϵ and ϵint are required.Fortunately, for step-faceting the atomic composition of the substance is less relevant because it is a first-order phase transition. From thermodynamics, the free energyF is expressed as F=E-TS, where E is an internal energy and S is entropy. At T=0, F=E, and E depends on the atomic composition of the substance. In the phase transition (T>0), however, entropy plays a dominant role. The entropy is hardly affected by the atomic species of the substance but depends on the symmetry, the interacting force range, the number of components, and the dimensionality (as seen in Trouton’s rule for molar entropy of vaporization, e.g., the molar entropy of vaporization for nonpolar materials is almost the same value for various kinds of liquids).In the present work, we have shown that the characteristics of the height profile of a macrostep can be classified by the zone in the faceting diagram where the vicinal surface exists. In the step-faceting zone, the terrace for a macrostep is (001), whereas, in the step droplet zone, the surface in contact with a macrostep has the slopep1≠0. This classification is a direct result of the connectivity of the surface tension. Since the p-RSOS model is constructed on a slightly coarse grained vicinal surface [8] with a square mesh, this result is applicable to a wide variety of materials to qualitatively explain the effect of a discontinuous surface tension.From the classification scheme, we can find a method of dissociation of macrosteps for 4H-SiC at equilibrium. As mentioned at the end of Section4, 4H-SiC in Figure 1 is thought to be in the step droplet zone. To dissociate the macrostep, the temperature should be increased so that the state of the system moves to the GMPT zone. When the step-step attraction is strong (a large ϵint), the area of the step droplet zone is large. Hence, we have to raise the temperature substantially to reach the GMPT zone. In that case, the introduction of a small amount of “surfactant” like Al in 4H-SiC (Figure 1(b)) may make it easier to dissociate the macrostep [34–36], more so than increasing the temperature.Comparing the height profiles of macrosteps observed for 4H-SiC (Figure1) with the height profiles simulated by the p-RSOS model in detail, we find that the shape of the profile at the upper edge of a macrostep differs slightly from that at the lower edge of a macrostep for a real system, whereas, for the p-RSOS model at equilibrium, the shapes of the height profiles at the upper and lower edges are the same. Since the height profile of 4H-SiC (Figure 1) is obtained during the crystal growth, some nonequilibrium effects may apply. In the case of vapor growth, surface diffusion and the Ehrlich-Schwoebel barrier effect [37–40] are known to cause step bunching or step meandering.It is important to distinguish between the relaxation process from an initial configuration to the equilibrium state and the steady-state motion of a vicinal surface with a driving force for crystal growth or crystal dissolution. For the relaxation to the equilibrium state near equilibrium on the vicinal surface, the faceted step is not able to move due to the divergence of the surface stiffness [9, 10, 41]. We can see this from the time-dependent Ginzburg-Landau (TDGL) theory for the surface (interface) motion [42, 43]. Applying the finite size effect, the results in Section 5 are consistent with the expectation from the TDGL equations. Macrosteps with a small number of steps are easier to move than macrosteps with a larger number of steps. In real observations near equilibrium, the same phenomenon will be encountered.The dynamics of the height profile of a macrostep for steady-state motion will be studied elsewhere.Furthermore, in a real system there exists a step-step repulsion originating from the elastic interaction among the steps [44–46], which is expressed by the equation ∑i≠j‍∑y~‍g0/(x~i(y~)-x~j(y~))2, where g0 is the coupling constant of the elastic repulsion, x~i is the location of the ith step on a vicinal surface normal to the mean running direction of steps, and y~ is the location along the steps.The elastic step-step repulsion causes three effects: (1) an increase of the (111) surface energy, (2) a weakening of the step-step attraction [5], and (3) the development of a regular array of macrosteps [47]. Regarding (1), the step-step repulsion increases the surface energy of the (111) surface, which increases the area of the (001) surface on the ECS while decreasing the area of the (111) surface. Hence, wider (001) terraces are preferred on the vicinal surface. For (2), if we consider an averaged field for interactions among the steps, the step-step repulsion reduces the step-step attraction overall. Then, the step-faceting zone shrinks and the zone boundary between the step-faceting zone and the step droplet zone shifts to a lower temperature [5]. For (3), due to the difference in the force range between the step-step attraction (point-contact type) and the step-step repulsion (long range), the balance between the accumulated interactions depends on the distance between steps. For short distances between steps, the attractive interaction dominates, whereas for long distances between steps, the repulsive interaction dominates. Hence, a regular array of n-merged steps develops [47], where n is selected as the least free energy of the system.We studied the height profile of macrosteps for 4H-SiC because faceted macrosteps for 4H-SiC can be explained by the anomaly of the surface (interface) tension [1]. No other explanation for the formation of the faceted macrostep has yet been found. For 6H-SiC or 3C-SiC the classification of the height profile of a macrostep in the present work is applicable to explain the effect of the surface tension qualitatively. If faceted macrosteps in the mesoscopic scale for 6H-SiC or 3C-SiC are observedat equilibrium, the surface tension is suggested to be discontinuous. When the slope of a terrace p1 is zero (a smooth flat surface), the vicinal surface is in the step-faceting zone, whereas when the slope of a terrace is not zero, the vicinal surface is in the step droplet zone.Finally, we note that, formally, the Andreev free energy equals the negative pressure-P [5]. We can see this from the analogy between (3) and the grand potential of a gas-liquid system. This pressure represents the tendency of the surface area to become wider. On the mesoscopic scale, the surfaces on a crystal determine their locations so that the pressure of neighboring planar surfaces balances out. Since this pressure is a kind of force, it propagates rapidly around the crystal surface.


## 7. Conclusions

The ECS and surface tension of the vicinal surface for the p-RSOS model withϵint/ϵ=-0.9 were calculated (Figures 4 and 5). To ensure the reliability of the calculations, the DMRG method was used. The microscopic structure of vicinal surfaces was studied using the Monte Carlo method.We showed that the characteristics of the height profile for a macrostep can be classified by the zone in the faceting diagram where the vicinal surface exists. In thestep-faceting zone, the terrace for a macrostep is (001), whereas in thestep droplet zone, the surface which contacts a macrostep has the slope p1≠0. This classification is a direct result of the connectivity of the surface tension.Furthermore, in the step-faceting zone, the macrostep is so stiff that it takes an infinitely long time to reach global equilibrium in the thermodynamic (large system size) limit. In the Monte Carlo simulation, global equilibrium is reached because of the finite size. The relaxation time increases as the temperature decreases (Figures8 and 9). Smearing at the edge of the macrostep is observed, which is also caused by the finite size effect (Figure 10).The observed height profile of a macrostep of 4H-SiC [1] has the characteristics of the height profile in the step droplet zone. This suggests the surface tension is discontinuous around the side surface of the macrostep.



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