# Simple Model for Corrugation in Surface Alloys Based on First-Principles Calculations

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

_{0}, where E

_{0}is 13.605 eV, and used 8 × 8 × 1 k-point mesh. The numerical pseudo atomic orbitals [19] were utilized as follows: For most models, the numbers of the s-, p-, and d-character orbitals were three, two, and two, respectively—especially three, three, and two for Pb, Sb, and Bi. The cut off radii of Ag, B, Al, Ga, In, Tl, C, Si, Ge, Sn, Pb, N, P, As, Sb, and Bi were 7.0, 7.0, 7.0, 7.0, 7.0, 8.0, 5.0, 6.0, 7.0, 7.0, 8.0, 5.0, 7.0, 7.0, 7.0, and 8.0, respectively, in units of a

_{0}, where a

_{0}is the Bohr radius, 0.529177 Å. These choices of parameters for pseudo atomic orbitals are well tested for bulk systems [17].

_{Ag}$=$ 4.1 Å for Ag substrates [20]. Figure 1b also shows the rhombus which represents the surface unit cell. The in-plane cell length is $\sqrt{3}$ a

_{Ag(111)}(5.021 Å), where a

_{Ag(111)}is the in-plane lattice constant for an Ag(111) surface, which is given as a

_{Ag}/$\sqrt{2}$ (2.899 Å). To treat the surface alloys with the surface slab model, the atomic positions are fixed for the bottom three Ag layers and relaxed the atomic positions for the top three layers. The convergence of corrugation parameters is checked for the thickness of slab models up to 10 layers with half of the atomic layers fixed. For example, the difference for the corrugation parameter d of Bi/Ag (111) is less than 0.03 Å.

## 3. Results and Discussions

#### 3.1. Corrugation Parameter of M/Ag(111)–$\left(\sqrt{3}\times \sqrt{3}\right)R30\xb0$

#### 3.2. Simple Hard Spherical Atomic Model for Surface Corrugations

_{M}are larger than those of Ag atoms r

_{Ag}. From Figure 3a, we obtained the following equation (Equation (1)) to relate corrugation d and atomic radii.

_{M}dependence of surface corrugations d calculated by using Equations (1)–(3), where the atomic radius of silver is evaluated as r

_{Ag}= 1.45 Å from the SHSAM with an experimental lattice constant a

_{Ag}= 4.1 Å. The dotted, solid, and dashed lines represent Equations (1)–(3), respectively. For the negative case, the solid line for Equation (2) is absolutely linear, while the dashed line for Equation (3) is almost linear. For the positive case, the dotted line for Equation (1) is nonlinear.

_{M}dependence of corrugation parameters with atomic radii of Clementi et al., where the atomic radii are computed with self-consistent field functions based on a minimal basis-set atomic functions for the ground-state atoms [25]. We can clearly see the same tendency; the r

_{M}dependence of corrugation parameters change the gradient at larger atomic radii, as Figure 4 given by the SHSAM, if we plot for the groups III, IV, and V separately. The r

_{M}dependence is classified into three behaviours of Group III, IV, and V, since elements of each group have similar properties based on the same number of valence electrons.

#### 3.3. Determination of Surface Alloy Atomic Radii

_{M}dependence of calculated corrugation parameters of surface alloys is qualitatively similar to that from the SHSAM plotted by Equations (1)–(3) in Figure 4. However, from Clementi’s atomic radii and Equation (1) derived from the SHSAM, we cannot explain the positive corrugations, since the atomic radius of an Ag atom is larger than that of any M atom: for example, r

_{Ag}= 1.65 Å and r

_{Bi}= 1.43 Å. If we use Clementi’s atomic radii and the SHSAM, all the corrugation parameters should be negative by Equation (2) or Equation (3). Though we also try to explain the surface corrugations using the other type of atomic radii: for example, metallic radii and covalent radii [26] with the SHSAM, it works partly and we cannot reproduce all density functional calculations.

_{M}are obtained from Equations (1)–(3) by entering the DFT corrugation values from our calculations in Table 1, assuming r

_{Ag}= 1.45 Å as mentioned above. Here, we call the atomic radii determined from corrugations in surface alloys r

_{M}surface alloy atomic radii (SAAR). The calculated SAAR are shown in Table 2. The SAAR are compared with the atomic radii of theoretical calculations by Clementi et al. [25], metallic radii and covalent radii by A. F. Well’s book [26].

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Atomic structure of an M/Ag(111)–$\left(\sqrt{3}\times \sqrt{3}\right)R30\xb0$ surface alloy. (

**a**) Side view. d denotes the corrugation parameter; (

**b**) Top view. The rhombus represents the unit cell. The in-plane cell length is $\left(\sqrt{3}\right)$a

_{Ag}, where a

_{Ag}is the lattice constant for Ag(111).

**Figure 2.**Summary of calculated corrugation parameter d of M atoms that consist of group III, IV, and V atoms, sorted by atomic number Z. Red and blue bars correspond to negative and positive values, respectively.

**Figure 3.**Simple hard spherical atomic model (SHSAM) for an M/Ag(111)–$\left(\sqrt{3}\times \sqrt{3}\right)R30\xb0$ surface alloy. (

**a**) Positive corrugation. (

**b**) Side view of the negative corrugation case in the “monolayer model”. (

**c**) Side view of the negative corrugation case in the “bilayer model”. The corrugation parameter d is given as the difference between the heights of M and Ag atoms: d = h

_{M}– h

_{Ag}, where ${h}_{M}=\sqrt{{\left({r}_{\mathrm{Ag}}+{r}_{M}\right)}^{2}-{(\frac{2}{\sqrt{3}}{r}_{\mathrm{Ag}})}^{2}}$ and ${h}_{\mathrm{Ag}}=2\sqrt{\frac{2}{3}}{r}_{\mathrm{Ag}}$.

**Figure 4.**Corrugation parameter d calculated with Equations (1)–(3) in the main text, where r

_{Ag}= 1.45 Å. The dotted line shows positive corrugation given by Equation (1). The solid line and dashed line show negative corrugation given by Equations (2) and (3) with “monolayer model” and “bilayer model”, respectively.

**Figure 5.**Corrugation parameters d in M/Ag(111)–$\left(\sqrt{3}\times \sqrt{3}\right)R30\xb0$ calculated based on the density functional theory (DFT) versus Clementi’s atomic radii of M [25]. The lines are a guide to the eye for group III (B, Al, Ga, In, Tl), IV (C, Si, Ge, Sn, Pb), and V (N, P, As, Sb, Bi) atoms.

**Figure 6.**Summary of calculated atomic radii of surface alloy atomic radii (SAAR), Clementi’s atomic radii, metallic radii, and covalent radii, sorted by atomic number Z.

**Table 1.**Comparison of corrugation parameters from our calculation and prior works of calculations and experiments, sorted by atomic number Z. d denotes the corrugation parameter.

Z | Atom | d (Corrugation Parameter) (Å) | Reference (Å) | |
---|---|---|---|---|

(Theoretical) | (Experimental) | |||

5 | B | −0.654 | - | |

6 | C | −0.664 | - | |

7 | N | −0.712 | - | |

13 | Al | −0.115 | - | |

14 | Si | −0.199 | - | |

15 | P | −0.202 | - | |

31 | Ga | −0.002 | - | |

32 | Ge | −0.055 | - | 0.3 [21] |

33 | As | −0.040 | - | |

49 | In | 0.190 | - | |

50 | Sn | 0.184 | - | |

51 | Sb | 0.178 | - | |

81 | Tl | 0.306 | - | |

82 | Pb | 0.556 | 0.59 [22] | 0.42 [22] |

83 | Bi | 0.690 | 0.69 [23]; 0.61 [9]; 0.65 [24]; 0.8 [14] | 0.57 [9] |

Atoms | Surface Alloy Atomic radii (Å) | Clementi’s Atomic Radii (Å) [25] | Metallic Radii (Å) [26] | Covalent Radii(Å) [26] |
---|---|---|---|---|

Ag | 1.45 | 1.65 | 1.44 | - |

B | 0.95 | 0.87 | - | - |

C | 0.94 | 0.67 | - | 0.77 |

N | 0.9 | 0.56 | - | 0.74 |

Al | 1.36 | 1.18 | 1.43 | - |

Si | 1.29 | 1.11 | - | 1.17 |

P | 1.29 | 0.98 | - | 1.1 |

Ga | 1.45 | 1.36 | 1.53 | - |

Ge | 1.40 | 1.25 | 1.39 | 1.22 |

As | 1.42 | 1.14 | - | 1.21 |

In | 1.46 | 1.56 | 1.67 | - |

Sn | 1.46 | 1.45 | 1.58 | 1.40 |

Sb | 1.46 | 1.33 | 1.61 | 1.41 |

Tl | 1.47 | 1.56 | 1.71 | - |

Pb | 1.5 | 1.54 | 1.75 | - |

Bi | 1.53 | 1.43 | 1.82 | - |

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**MDPI and ACS Style**

Nur, M.; Yamaguchi, N.; Ishii, F.
Simple Model for Corrugation in Surface Alloys Based on First-Principles Calculations. *Materials* **2020**, *13*, 4444.
https://doi.org/10.3390/ma13194444

**AMA Style**

Nur M, Yamaguchi N, Ishii F.
Simple Model for Corrugation in Surface Alloys Based on First-Principles Calculations. *Materials*. 2020; 13(19):4444.
https://doi.org/10.3390/ma13194444

**Chicago/Turabian Style**

Nur, Monika, Naoya Yamaguchi, and Fumiyuki Ishii.
2020. "Simple Model for Corrugation in Surface Alloys Based on First-Principles Calculations" *Materials* 13, no. 19: 4444.
https://doi.org/10.3390/ma13194444