Measurement of microscopic coupling constants between atoms on a surface: Combination of LEEM observation with lattice model analysis

Surface Science 630 (2014) 109–115

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Measurement of microscopic coupling constants between atoms on a surface: Combination of LEEM observation with lattice model analysis Noriko Akutsu Faculty of Engineering, Osaka Electro-Communication University, Neyagawa, Osaka 572-8530, Japan

a r t i c l e

i n f o

Article history: Received 14 June 2014 Accepted 15 July 2014 Available online 22 July 2014 Keywords: Step tension Step stiffness Two-dimensional Ising model LEEM

a b s t r a c t We present a method combining low-energy electron microscopy (LEEM) and lattice model analysis for measuring the microscopic lateral coupling constants between atoms on a surface. The calculated step (interface) stiffness in a honeycomb lattice Ising model with the nearest neighbor and the second nearest neighbor interactions (J1 = 93.8 meV and J2 = 9.38 meV) matched the experimental step quantity values on an Si(111)(1 × 1) surface reported by Pang et al. and Bartelt et al. based on LEEM measurements. The experimental value of step tension obtained by Williams et al. lies on the calculated step tension curve. The polar graphs of the step tension and a two-dimensional island shape at the temperature T = 1163 K also agree well with the experimental graphs reported by Métois and Müller. The close agreement between the LEEM observations and the lattice model calculations on a Si(111) surface suggests that our method is also suitable for measuring microscopic lateral coupling constants on the surface of other materials that are less well-studied than Si. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Recent developments have made first-principles calculations a powerful tool for describing surface phenomena quantitatively on a microscopic scale. However, the step energies and kink energies on a surface are too small to be calculated reliably by the firstprinciples method. If the kink energy could be calculated, it would be important to confirm the values of these microscopic quantities experimentally. Si is one of the most extensively studied materials, and Si surface energetics is currently being investigated. Understanding Si energetics provides insights into the quantitative understanding of the crystal surfaces of other materials that are less intensively studied than Si. These insights will help overcome difficulties in fabricating more complex nanoscale devices from other materials. Step stiffness measurements using direct imaging techniques with low-energy electron microscopy (LEEM) [1] were established using steps on Si(001) and Si(111) surfaces [2,3]. In 1993, Bartelt et al. [2] used LEEM to measure the step stiffness of an Si(111) surface at 1173 K, and obtained a value of 30 meV/Å. Because a step is a onedimensional (1D) entity, the zigzag structures in a step are easily excited by thermal fluctuations. The variation in the step structure makes an entropic contribution to the step free energy. Hence, calculations based on statistical mechanics are required to link measured surface thermodynamic quantities, such as step stiffness, with the microscopic coupling constants between atoms. Akutsu and Akutsu [4] calculated the step tensions and step stiffness for Si(111), assuming surface structures of (7 × 7), (1 × 1), (2 × 2), and

http://dx.doi.org/10.1016/j.susc.2014.07.017 0039-6028/© 2014 Elsevier B.V. All rights reserved.

pffiffiffi pffiffiffi 3  3 . They tried to determine the microscopic coupling constants so that the calculated step tension and step stiffness reproduced the observed values. However, the number of observed values was small. Moreover, the effect of the metastable structures, such as (2 × 2) pffiffiffi pffiffiffi [5] and 3  3 [6], could not be excluded at temperatures above (7 × 7)–(1 × 1), the phase transition temperature, Tc. Recently, more sophisticated LEEM observations of Si(111) with a (1 × 1) structure were reported by Pang et al. [7] and what they call the “linear temperature coefficient” of the observed values was −0.14 meV/(ÅK) between 1145 and 1233 K, where the linear temperature coefficient represents the temperature gradient of their observed quantities [7]. In the LEEM method, the two-time correlation function G(t1, t2) for the fluctuations of a step position is digitally analyzed to obtain the step quantities, and G(t1, t2) is originally expressed by the step stiffness [2]. They assumed that the line tension is sufficiently isotropic to replace the step stiffness with the line tension. However, their obtained values were almost 1.7 times larger than the known step tension value [8,9]. In this study, we intended to reproduce the LEEM values using the imaginary path-weight (IPW) random walk method [10,11] with the 2D Ising model on a honeycomb lattice with nearest neighbor (nn) and second nn interactions based on Ref. [4]. We also aimed to obtain the microscopic coupling constants of the nn and the second nn. The consistency of the experimental and calculated values means that combining LEEM observations with lattice model analysis can be used to measure the microscopic lateral coupling constants between atoms on a surface.

N. Akutsu / Surface Science 630 (2014) 109–115

This paper is organized as follows. In Section 2, we give a brief explanation of the model for the calculations. Thereafter, the two microscopic lateral coupling constants between atoms on a surface are provided in Section 3. The equilibrium island shape and the polar graphs of the step tension and the step stiffness are also presented in this section. In Section 4, we compare our results with previous experimental results. Conclusions are given in Section 5.

[112]

2

Y kBT/ J1

110

H LG ¼ −ϕ1

X

C Ai C B j −ϕA2

bi; jN

X

C Ai C A j −ϕB2

bi; jN

X

C Bi C B j

bi; jN

N N h iX h iX − μ A;2D −μ A;gas ðP A ; T Þ C Ai − μ B;2D −μ B;gas ðP B ; T Þ C Bi ; i¼1

ð1Þ

i¼1

where Ci = 0, 1 is the lattice gas variable, N is the number of unit cells in the 2D lattice, and μA,gas(PA, T) = μB,gas(PB, T) = μgas(P, T) is the chemical potential of the atoms in the ambient gas phase of Si at pressure P and temperature T. The chemical potentials on the 2D lattice can be approximated as [4]:   ϕ μ A;2D ¼ − z3 1 þ z6 ϕ2 þ ΔE þ μ solid ðT Þ; 2

ð2Þ

  ϕ μ B;2D ¼ − z3 1 þ z6 ϕ2 −ΔE þ μ solid ðT Þ; 2

ð3Þ

where μsolid(T) is the chemical potential of the atoms in the bulk solid at temperature T, and z3 = 3 and z6 = 6 are the coordination numbers of the first and second nn atoms on the honeycomb lattice, respectively. ΔE is half the energy of the bond normal to the 2D lattice. The vertical bonds of the A-sublattice connect the 2D lattice to the substrate, and those of the B-sublattice form dangling bonds. We assume ΔE = ϕ1/2. At equilibrium, the energy of a lattice gas with {CAi, CBi} = (0, 0) for all sites should equal the energy of a lattice gas with {CAi, CBi} = (1, 1) for all sites. From this condition, after some calculations, we obtain μ solid ðT Þ−μ gas ðP e ; T Þ ¼ 0;

b

0

-2

--2 [112]

c

40

1

50

0

-1

2. The model In this section, we describe the model for estimating the step free energy per length, which equals step tension [4]. The model is a 2D interacting lattice gas model on a honeycomb lattice, which is the same as the Si(111) (1 × 1) surface model in Ref. [4]. It should be noted that the lattice gas is a device for quantifying the entropy caused by the zigzag structure of a step. Using the lattice gas model with the IPW method, the overhang structures in a step are taken into consideration for calculating the step tension and stiffness. The honeycomb lattice is divided into two triangular sublattices called the A sublattice and the B sublattice. The first nn coupling constant is ϕ1 and the second nn coupling constant between atoms in the A sublattice and the second nn coupling constant between atoms in the B sublattice are ϕA2 and ϕB2, respectively. For Si(111), we assume that ϕA2 = ϕB2 = ϕ2. The chemical potentials on the lattice are μA,2D for the A sublattice and μB,2D for the B sublattice. The energy of the lattice gas is given by the following lattice gas Hamiltonian HLG:

a

0

-40 -1 0 1 X kBT/ J1

2

-40

0

40

-50

[meV/A]

-50

0

50

[meV/A]

Fig. 1. (a) Calculated two-dimensional (2D) island shape on a Si(111) 1 × 1 surface. (b) Polar graph of step tension. (c) Polar graph of step stiffness. T = 1163 K. ϕ1 = 375 meV, ϕ2 = 37.5 meV, and ΔE = 188 meV.

3. Step tension and step stiffness 3.1. Calculation method In this subsection, we explain the definition of step tension1 and the method used to calculate the step tension and step stiffness. The equilibrium island shape (e.g. Fig. 1(a)) can be obtained using a Wulff construction from a Wulff figure, which is a polar graph of the step tension γ(θ) (e.g. Fig. 1(b)). Here, θ is the tilt angle of the mean direction of a step from a crystal axis (Fig. 2(a)). At T = 0 K the island shape becomes a truncated triangle and the calculated Wulff figure shows a raspberry shape with cusps. A step is defined as a phase separation line in the lattice gas model (Fig. 2(a)) resulting from the anti-phase boundary condition (Fig. 2(a)). Here, we explain the step notation. In the figure, a horizontal h i line, which is labeled θ = 0, is normal to 112 . A step running in a horh i izontal mean direction is a 112 step. The mean direction of the step connecting O and P is tilted by angle θ. It should be noted that the lattice does not satisfy mirror symmetry with respect to the horizontal line. A h i h i 112 step is not the same as a 1 12 step. Therefore, we define the direction of the normal vector of a step such that the vector is directed into the empty phase from the filled phase (Fig. 2(b) and (c)). The h i angle of a 1 12 step becomes θ = 180°. The step tension is defined as the free energy per unit length of a phase separation line in the large size limit. The free energy of the phase separation line is defined as the free energy difference between the bulk free energy with an anti-phase boundary condition and the bulk free energy with a homogeneous boundary condition. Hence, the free energy of the bulk is canceled out. This definition is met by calculations using the IPW method [10,11]. The IPW method is an extension of Vdovichenko–Feynman's method [12] which can be used to obtain the exact free energy of a 2D nn Ising model. Although the IPW method is approximate for complex models, it provides exact results for known 2D nn Ising models. In the IPW method [10,11], a  2D equilibrium crystal (island) shape is   ! !  obtained from the D-function, D iω with ω ¼ ωx ; ωy , as   ! D iω ¼ 0; ωx ¼ λy=kB T;

ð5Þ ωy ¼ λx=kB T;

ð6Þ

ð4Þ

where Pe is the pressure of Si in the gas phase with crystal–gas coexistence. The lattice gas model and the equilibrium conditions can be converted to a 2D antiferromagnetic (AF) Ising model (Eq. (2.7) in Ref [4], A.1). Therefore, in the lattice gas model of Eq. (1) the unknown parameters are the microscopic coupling constants ϕ1 and ϕ2.

where λ is the Lagrange multiplier associated with the volume-fixing constraint in the Wulff construction, x and y are the Cartesian coordinates describing the 2D island shape, and kB is the Boltzmann constant. The explicit form of the D-function is given by Eqs. (B.1)–(B.8).

1

In the rigorous sense, the interface tension for Ising model.

N. Akutsu / Surface Science 630 (2014) 109–115

+

+

+ −

+ + P

+ − O

a

+ −

Step tension [meV/A]

+

b

− + Q

− + −+ −+ −+ − +−

a

c

blue) filled circles and gray (light blue) open circles: {CA, CB} = {0, 0}. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

! The step tension is obtained using ω as     ! ! ∂D iω ∂D iω = ; tan θ ¼ ∂ωy ∂ωx

ð7Þ

  γ ðθ; T Þ ¼ kB T ωx cos θ þ ωy sin θ :

ð8Þ

e ðθÞ is defined by γ e ðθÞ ¼ γðθÞ þ The step stiffness (e.g. Fig. 1(c)) γ 2 ∂ γðθÞ=∂θ2 , which relates to the Gibbs–Thomson effect (Appendix D.12 in Ref. [13]) for an anisotropic step. In the IPW method, the step stiffness is given by the following equation [14,15]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i 2 2 −1 D2x þ D2y  −Dxx sin θ þ Dxy sin2θ−Dyy cos θ ;

ð9Þ

where Dx ¼

∂D ; ∂ωx

Dyy ¼

2

∂ D ; ∂ω2y

Dy ¼

∂D ; ∂ωx

Dxy ¼

Dxx ¼

∂2 D ; ∂ω2x

2

∂ D : ∂ωx ωy

60 40

-[112] [110] [112]

20

0

Fig. 2. (a) A step as a phase separation line connecting O and P. The angle POQ is labeled θ. h i   h i (b) A kink in a 1 12 step with kink energy ϵK 1 12 . (c) A kink in a 112 step with kink   energy ϵK 112 . Black filled circles and black open circles: {CA, CB} = {1, 1}. Gray (light

e ðθÞ ¼ kB T γ

80

Tc,2D =2101 K

ð10Þ

Using the IPW method, we obtain explicit analytic equations – Eqs. (6)–(10) with Eqs. (B.1)–(B.8) – for step tension and step stiffness with full anisotropy. Hence, these equations can be used in the LEEM equipment. 3.2. Microscopic coupling constants In this subsection, we show that the calculated step stiffness agrees with the experimentally observed values obtained by Pang et al. [7] if we choose ϕ1 and ϕ2 appropriately. We were unable to fit the step tension curve, γ(θ), to the observed values using any set of (ϕ1, ϕ2). As the temperature increased the observed values (Fig. 3(a), filled circles) decreased too steeply to be fitted by the Ising model. h i However, we were able to obtain the step stiffness curve for a 112 (θ = 0) step that described the experimental values (Fig. 3(b), filled circles) well. Moreover, we could not fit the step stiffness curves for the h i h i 112 orientation (θ = 180°) or for the 110 orientation (θ = 90°) to the observed values. In addition, the experimental value obtained by Bartelt et al. [2] (Fig. 3(b), filled triangle) lies on the curve for a h i 112 (θ = 180°) step. This value was obtained by analyzing a LEEM

500

b Step stiffness [meV/A]

+

111

1000

1500

T [K]

2000

80 60

[110]

40

-[112]

20

0

[112]

500

1000

1500

T [K]

2000

Fig. 3. (a) Temperature dependence of step tension. (b) Temperature dependence of h i h i step stiffness. Solid (red) line: 112 . Broken (blue) line: 1 12 . Dotted (green) line: h i 110 . ϕ1 = 375 meV, ϕ2 = 37.5 meV, and ΔE = 188 meV. Filled circles (black): Ref. [7]. Filled triangle (black): Ref. [2]). Filled diamond: Ref. [8]. Diagonal cross: Ref. [16] Open squares: Ref. [17]. Open diamond: Ref. [18]. Crosses: Ref. [19] Open reversed triangles: Ref. [20]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

image. Therefore, the discrepancies between the two values can be understood as a result of anisotropy. The steep decrease in the step stiffness results from the small coordination number z3 = 3 of the honeycomb lattice. Because only three bonds surround each atom in a honeycomb lattice, the bonds cannot hold an atom on a step edge in the solid phase. Hence, as the temperature increases, thermal agitation destabilizes the atoms on the step edge at a relatively low temperature. This causes the phase transition temperature Tc,2Dhoneycomb of the nn honeycomb lattice to be kBTc,2Dhoneycomb/J = 2/cos h−1(2) ≈ 1.519 [21], where J is the nn coupling constant of the 2D Ising model. However, the second nn bonds in the present model hold the atoms in the solid phase more effectively. The phase transition temperature, Tc,2Dtriangle, of an nn triangular lattice is kBTc,2Dtriangle/J = 4/ln(3) ≈ 3.641 [21]. Hence, if the value of J2/J1 is larger in the present model, the step stiffness should decrease more gently as temperature increases, and Tc,2D should increase. The calculated step tension curves are different from the values obtained by Pang et al. However, the step tension of 40 meV/Å at h i 1073 K obtained by Williams et al. [8] almost lies on the 112 step curve, whereas the step tension obtained by Bermond et al. [16] at 1323 K is off the curves. Bermond obtained the step tension from the contour shape of the facet on the equilibrium crystal shape (ECS) of Si. Because the absolute value of the Lagrange multiplier for the ECS is difficult to determine, the absolute value of the step tension is also difficult to determine from the ECS.

112

N. Akutsu / Surface Science 630 (2014) 109–115

Table 1 Microscopic quantities (a = 3.84 Å). Quantity

Equation

Value

ϕ1 ϕ2 ΔE   ϵK 112   ϵK 1 12

4J1 4J2 = 0.1ϕ1 2H = ϕ1/2 ϕ1/2 + ϕ2 + (1/3)ΔE

375 meV 37.5 meV 188 meV 288 meV

ϕ1/2 + ϕ2 − (1/3)ΔE

163 meV

[ϕ1/2 + 2ϕ2 − (1/3)ΔE]/a

52.1 meV/Å

[ϕ1/2 + 2ϕ2 + (1/3)ΔE]/a

84.7 meV/Å

T=0 γ(0°) γ(180°)

h

i 112 h i 1 12

|r m |/ |r3 | [%]

30

20

10

0 The microscopic values for the calculations, the energies of the kinks shown in Fig. 2(b) and (c), and the step tensions in the T → 0 limit are listed in Table 1. All these values are generally consistent with those reported by Williams et al. [8,9] and other researchers [22]. The phase transition temperature, Tc,2D, is obtained from the temperature where the step tension of the Ising model is zero. We obtain Tc,2D as 2101 K. It should be noted that Tc,2D suggests a relative height of the roughening transition temperature TR of planar Si(111). In general, TR is higher than Tc,2D [23,24]. Tc,2D provides the lower limit [25] for TR. The zigzag structures of a step are pinned by the multi-height islands; the multi-height islands on the terraces are formed by thermal fluctuations. This pinning of the fluctuations on the step-configuration reduces the step entropy. Hence, the step tension 2 on a surface should be zero at TR, which is higher than Tc,2D. The relationship Tc,2D b TR has been shown directly for the Si(001) (2 × 1) surface [15,26]. 3.3. Anisotropy In this subsection, we consider the anisotropy of the equilibrium ! island shape. By solving Eq. (5) with respect to ω at 1163 K, we obtain the equilibrium island shape on an Si(111) surface (Fig. 1(a)). Polar graphs of the step tension and step stiffness at 1163 K are calculated from Eqs. (8) and (9), respectively (Fig. 1(b) and (c)). Because the temperature is high, the island shape and the polar graph of the step tension are almost circular. However, the step stiffness shows a strong anisotropy. We can see that ∂2γ(θ)/∂θ2 is explicitly from the difference between the step h i stiffness and the step tension in Fig. 1(b) and (c). Around 112 , h i ∂2γ(θ)/∂θ2 N 0; and around 112 , ∂2γ(θ)/∂θ2 b 0. The calculated equilibrium island shape is similar to the observed shape shown in Fig. 7(b) in the paper published by Métois and Müller [20]. They measured the shape using reflection electron microscopy (REM) at 1373 K. The LEEM image obtained by Pang et al. at 1163 K (Fig. 1a in their work) looks circular. To study the anisotropy in the equilibrium island shape using a different approach, we calculate the coefficient of the Fourier series expansion as in the work of Pang et al. [7]. We parameterize the contour of the calculated shape (Fig. 1(a)) as s, and the distance between the center of the shape and point s is described by r(s). The perimeter length S is the maximum value of s. r(s) is a periodic function with period S, and r(s) can be expanded in a Fourier series as r(s) = ∑ m rm exp[iqms], where qm = 2πm/S. The coefficient rm is obtained by Z 1 S r ðsÞexp½−iqm sds: ð11Þ rm ¼ S 0 Eq. (11) is numerically calculated using 9600 points on the perimeter, and the ratio |rm|/|r3| is shown in Fig. 4. 2 Step tension equals step free energy per unit of length. Note that F = E − TS, where F is the free energy, E is the internal energy, and S is the entropy. Thus, the temperature at which F = 0 increases as S decreases.

0

5

m

10

Fig. 4. Coefficients of the Fourier series of r(s), where s is a parameter on the perimeter of the island shape (Fig. 1(a)), and r(s) is the distance between an original point in the island shape and a point on the perimeter at s. Wave number qm = 2πm/S, where S is the length of the perimeter.

As in previous experiments [7], a small but finite value of |r6| is obtained (Fig. 4). The values are 2.68 × 10−2 for |r6|/|r3|, 6.92 × 10−4 for |r9|/|r3|, and 5.13 × 10−6 for |r12|/|r3|. Other |rm|/|r3| values are less than 3 × 10− 7. These results and Fig. 4 show slightly more threefold symmetric components than the experimentally observed island shape [7]. The island in Fig. 1 reported by Pang et al. may have evaporated quite slowly. The REM observations suggested [20] that the island shape becomes more isotropic due to the curvature-driven decay [20,27] known as the Gibbs–Thomson effect. In addition, according to the linear response theory [28], the mass flow normal to a step edge vn(t) caused by the change of an external driving force δ(Δμ) is described by a relaxation function, α(t), such as vn(t) = α(t)δ(Δμ), sufficiently near equilibrium. From the fluctuation–dissipation theorem, the two-time correlation function G(t2, t1) is expressed by the relaxation function as G(t2, t1) = kBTα(t2 − t1) [28]. The expressions of G(t2, t1) are known for several systems, such as Eqs. (2) and (4) in Ref. [2]. In both cases, G(t2, t1) is inversely proportional to the step stiffness. Hence, vn(t) for h i a 1 12 step should be more sensitive to slow, small changes of Δμ h i than a 112 step. Therefore, the island shape reported by Pang et al. could become circular.

3.4. Si(001) surface Bartelt et al. [2,3] showed that the digital Fourier transformation of the time correlation function of the step fluctuations obtained from a LEEM image can provide the step stiffness of a step on an Si(001) (2 × 1) surface. In previous work, Swartzentruber et al. [29] observed step fluctuations on an Si(001) (2 × 1) surface using STM. They statistically analyzed the data to obtain the microscopic values of kink energy. Using these microscopic values, Akutsu and Akutsu [15,26] reproduced the values of the step stiffness on Si(001) (2 × 1) without adjustable parameters. The model used in Ref. [15,26] was the 2D lattice gas model with the nn and second nn interactions on a square lattice. For the step stiffness on the Si(001) (2 × 1) surface, the combination of the LEEM results [3] and the IPW calculations based on the 2D lattice gas model [15,26] provides three 3 reliable microscopic coupling constants.

3 In the case of the Si(001) (2 × 1) surface, the microscopic constants are ϕ1x and ϕ1y for the nn sites in the x- and y-directions, respectively, in addition to ϕ2. for the second nn sites.

N. Akutsu / Surface Science 630 (2014) 109–115

4. Discussion Most of the previous experimental values of step stiffness, except for the value obtained by Alfonso et al. [17], fall below the curve obtained from the values reported by Pang et al. [7] (Fig. 3(b)) and contain a great deal of scatter. In the previous measurements, the step stiffness values were estimated from the mean square fluctuations of a step using the following equation, e ¼ Ce γ

kB TL ; bx2 N

ð12Þ

where b x2 N is the averaged squared step fluctuations over time, L is the length of a step, and Ce is a coefficient. In the experiments, Ce was assumed to be 1/6. It should be noted that Ce depends on the boundary conditions of a step and the averaging method for the large L limit. The step fluctuation is characterized by the width, w, defined by w2 = b x(y)2 N, where y is the position along a line corresponding to the mean direction that the step runs in, x(y) is the local normal deviation of a step from the line at y, and b ⋅ N is the thermal average [30]. Under periodic boundary conditions, w2 can be calculated using a Fourier series expansion, such as x(y) = ∑ q x(q)eiqy: 2

w ¼

XD

2

jxðqÞj

E

q

¼

kB TL : e ðθÞ 12γ

ð13Þ

For fixed boundary conditions, care should be taken to measure w2. 2 According to a similar

manipulation of Eq. (13), w was calculated by 2 e [30] w ¼ kB TL= 6γðθÞ . However, when both ends of a step are fixed, the fluctuations near the end points are smaller than the fluctuations near the center of a step. Then, on average and for the long length limit, the fluctuation width at any point depends on its position on the step. If we parameterize the position on a step using u = y/L, considering the Markovian character of the probability with respect to deformations, the fluctuation width, w(u), becomes [31] 2

w ðuÞ ¼

kB TL uð1−uÞ e ðθÞ γ

2

kB TL : e ðθÞ 4γ

h i become smaller than the maximum values of a 112 step because of h i the averaging procedure. Therefore, the true step stiffness of a 112 step may be larger than the curve in the present paper, and this produces the larger microscopic quantities.

5. Conclusions We obtained the values of two microscopic lateral coupling constants, ϕ1 and ϕ2, on a Si(111) (1 × 1) surface by fitting the calculated h i step stiffness for a 112 step to the low-energy electron microscopy (LEEM) results obtained by Pang et al. [7]. We used a 2D lattice gas model on the honeycomb lattice with the nn and the second nn coupling constants [4]. Using the IPW method, we obtained explicit analytic equations – Eqs. (6)–(10) with Eqs. (B.1)–(B.8) – to obtain the step tension and step stiffness with full anisotropy. These equations can be used with the LEEM equipment to measure ϕ1 and ϕ2. The step stiffness of the earlier LEEM results [2] lies on the calculated h i curve for a 112 step. The discrepancies between the values reported by Pang et al. and those reported by Bartelt et al. arise from anisotropy. The step tension observed at 1073 K [8] almost lies on the calculated h i curve for a 112 step. The calculated 2D equilibrium island shape agrees with that observed by Métois and Müller [20] by using reflection electron microscopy (REM). For the Si(001) (2 × 1) surface, the step stiffness based on the 2D lattice gas model on a square lattice with nn and second nn interactions [15,26] reproduced the LEEM results [3], and the microscopic coupling constants agree with the constant obtained by STM [29]. Therefore, reliable values of microscopic coupling constants ϕ1 and ϕ2 can be obtained from the LEEM results of the step stiffness combined with statistical mechanical calculations of the step stiffness based on the 2D lattice gas model with nn and second nn interactions. This method can be applied to other materials that are less intensively studied than silicon.

ð14Þ Acknowledgments

for the long length (L → ∞) limit. w2 is obtained as a simple average with

e ðθÞ respect to u, such as w2 = ∫ 10w2(u)du. Then, we have w2 ¼ kB TL= 6γ again. From Eq. (14), the fluctuation width of a step at the middle of the step becomes w ð1=2Þ ¼

113

This work was supported by a Japan Society for the Promotion of Science (JSPS) KAKENHI Grant number 25400413. The author thanks Professor M. S. Altman for helpful discussion of the island shape and its anisotropy. The author also thanks Professor Takanori Koshikawa for his continuous encouragement.

ð15Þ

This means that Ce = 1/4 [31,32]. This value was confirmed by Monte Carlo calculations [33,34] by comparing the step stiffness obtained from Eq. (14) with u = 1/2 to the interface stiffness calculated using a 2D Ising model. If we multiply the step stiffness values obtained by Métois and Müller [20] (reverse triangles in Fig. 3(b)) by 1.5, we obtain 34.2–47.8 meV/Å. 47.8 meV/Å lies on the calculated step stiffness curve h i for a 112 step, and 34.2 meV/Å lies on the calculated step stiffness h i curve for a 110 step. In addition, the value obtained by Bermond et al. [18] (diamond in Fig. 3(b)) at 1173 K is 68 meV/Å after multiplying h i by 1.5. The value agrees well with the present calculations for a 112 step. The true values of the microscopic quantities may be slightly larger than the ones in Table 1. Most of the points in Fig. 3 published by Pang et al. are averaged with respect to the steps of all orientations. When there is anisotropy in the step stiffness, the observed values

Appendix A. Hamiltonian of the Ising antiferromagnet The lattice gas model (Eq. (1)) and the equilibrium conditions (Eqs. (2)–(4)) are converted to a 2D AF Ising model (Eq. (2.7) in Ref [4]). In the conversion, the lattice gas variable Ci = 0, 1 on site i is transferred to the Ising spin variable σi = ± 1 as σAi = 2CAi − 1 for the A sublattice and as σBi = 1 − 2CBi for the B sublattice. The Ising AF Hamiltonian H is

H ¼ J1

X bi; jN

−H

σ Ai σ B j − J A2

X bi; jN

σ Ai σ A j − J B2

X

σ Bi σ B j

bi; jN

N X ðσ Ai þ σ Bi Þ þ Nz3 J 1 þ Nz6 ð J A2 þ J B2 Þ=2; H ¼ ΔE=2;

ðA:1Þ

i¼1

where H is the external uniform field. ϕ1, ϕ2, and ΔE are converted to 4J1, 4JA2 = 4JB2 = 4J2, and 2H, respectively.

114

N. Akutsu / Surface Science 630 (2014) 109–115

Appendix B. Explicit form of the D-function

3

2

s1 ¼ −½W ð−1 þ W H Þð1 þ W H ÞðW−W þ W H þ WW H

    D ikx ; iky ¼ M þ c1 coshðkx Þ þ c1 cosh kx =2−cy ky   þ c1 cosh kx =2 þ cy ky þ c2 coshð2kx Þ     þ c2 cosh kx −2cy ky þ c2 cosh kx þ 2cy ky     þ c3 cosh 2cy ky þ c3 cosh 3kx =2−cy ky   þ c3 cosh 3kx =2 þ cy ky þ s1 sinhðkx Þ   þ s2 sinhð2kx Þ−s2 sinh kx −2cy ky     þ s4 sinh kx =2−cy ky þ s4 sinh kx =2 þ cy ky   − s2 sinh kx þ 2cy ky ;

3

2

3

3

3

3

2

3

2

3

2

2

− W WH þ W Wa þ W WHWa þ W Wa þ W WHWa

The D-function for the 2D Ising model with the Hamiltonian (A.1) is expressed as follows [4]

3

2

3

3

3

2

− W W a −W W H W a þ W W b þ W W H W b −WW a W b 3

2

4

− W W a W b −2W W H W a W b þ 2W W H W a W b 2

3

2

4

2

− WW H W a W b −W W H W a W b −3W W H W a W b 4

4

3

2

3

2

2

2

ðB:6Þ

2

þ W WHWaWb þ W Wb þ W WHWb 4

2

4

3

3

− 3W W H W a W b þ 3W W H W a W b −W W b

ðB:1Þ

3

2

3

4

4

2

− W W H W b þ W W H W a W b Þ=W H ; h i 3 s2 ¼ − W ð−1 þ W H Þð1 þ W H Þð−1 þ W a Þð−1 þ W b Þ =W H ;

3

ðB:7Þ

2

s4 ¼ ½W ð−1 þ W H Þð1 þ W H ÞðW−W þ W H þ WW H

where

3

2

3

2

3

2

3

3

2

3

2

− W WH þ W Wa þ W WHWa þ W Wa 2

6

3

3

3

3

6

M ¼ 1 þ 3W þ 4W −W =W H −W W H −12W W a   6 2 6 3 3 3 3 3 3 3 þ 9W W a þ 4W W a þ W W a =W H þ W W H W a 6

4

6

6

6

2

6

6

3

6

þ 9W

2 6 2 W b −18W W a W b

4

3

3

6

4

6

4

2

6

2 2 WaWb

3

2

2

6

2

2

2

4

2

4

2

4

4

3

ðB:8Þ

2

2

3

3

4

2

4

2

2

3

2

3

4

4

2

where

2

  h i 3 2 c2 ¼ W 1 þ W H ð−1 þ W a Þð−1 þ W b Þ =W H ;

c3 ¼ 2W ð−1 þ W a Þð−1 þ W b Þð−1 þ W a þ W b þ W a W b Þ;

pffiffiffi 3=2; W ¼ exp½−2ð J 1 þ 2 J A2 þ 2 J B2 Þ=ðkB T Þ;

W H ¼ exp½−2H=ð3kB T Þ; W a ¼ exp½4J A2 =ðkB T Þ;

2

− 2W W H W a W b −W W H W a W b −W W H W a W b   5 2 5 2 þ 3W W a W b =W H þ 3W W H W a W b     5 4 5 4 4 2 2 − W W a W b =W H −W W H W a W b þ W W b =W H   4 2 2 5 2 5 2 þ W W H W b þ 3W W a W b =W H þ 3W W H W a W b     5 2 2 5 2 2 4 3 2 − 3W W a W b =W H −3W W H W a W b − W W b =W H   4 2 3 5 4 5 4 − W W H W b − W W a W b =W H −W W H W a W b ;

4

3

6

c1 ¼ W =W H −W =W H −W=W H −WW H þ W W H −W W H     4 2 4 2 4 2 2 þ W W a =W H þ W W H W a þ W W a =W H   4 2 2 4 3 2 4 2 3 þ W W H W a − W W a =W H −W W H W a     4 2 4 2 2 2 þ W W b =W H þ W W H W b − W W a W b =W H     4 2 3 − W W a W b =W H þ 2W W a W b =W H   5 3 − 2W W a W b =W H þ 2W W H W a W b 5

2

− W W b −W W H W b þ W W H W a W b Þ=W H :

2

4

3

þ W W H W b −3W W H W a W b þ 3W W H W a W b

2 2 WaWb

i ¼ −1; cy ¼ 2

3

3

2

þ 2W W a W b −6W W b þ 6W W a W b þ W W b ;

2

2

− 3W W H W a W b þ W W H W a W b þ W W b

ðB:2Þ

4

4

3

4

þ 3W þ 9W   6 3 3 3 3 3 3 3 6 3 þ 4W W b þ W W b =W H þ W W H W b −6W W a W b 6

3

þ 2W W H W a W b −WW H W a W b −W W H W a W b

− 18W W a W b −6W W a W b þ 6W W a W b 6

3

þ W W H W b −WW a W b −W W a W b −2W W H W a W b

− 6W W a þ W W a −12W W b þ 30W W a W b   3 3 3 3 þ 3W W a W b =W H þ 3W W H W a W b 6

2

þ W W H W a −W W a −W W H W a þ W W b

W b ¼ exp½4 J B2 =ðkB T Þ; r p ¼ expðiπ=6Þ and r m ¼ expð−iπ=6Þ:

References

ðB:3Þ

ðB:4Þ

ðB:5Þ

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