#
First Principles Computation of New Topological B_{2}X_{2}Zn (X = Ir, Rh, Co) Compounds

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Computational Method

## 3. Results and Discussion

#### 3.1. Crystal Structure

#### 3.2. Volume Optimization

#### 3.3. Formation Energy

#### 3.4. Elastic Properties

#### 3.5. Phonon Frequencies

#### 3.6. Electronic Band Structure and DOS Properties

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(Color online) Crystal structure and BZ of the B${}_{2}{X}_{2}$Zn compound. (

**a**) Shows a tetragonal crystal structure of a layered pattern. The blue, green, and red solid spheres denote the Zn, B, and X atoms, respectively. The arrangement of atoms for each layer (Zn1, X1, and B1) is noted as top–down views. (

**b**) Shows the first BZ by displaying high symmetry points of the BZ (black dots) with labeling $\Gamma $(0, 0, 0), X(−0.5, −0.5, 0), P(0.75, −0.25, −0.25), and Z(0.5, 0.5, −0.5). The (001) surface BZ shows in labeling with $\overline{\Gamma}$, $\overline{M}$, and $\overline{X}$. Paths labeled in green display the k-path selection for the comparison of bulk and surface states calculations.

**Figure 2.**(Color online) Calculated phonon spectra of B${}_{2}{X}_{2}$Zn (X = Ir, Rh, Co) compounds. All three phonon spectra were calculated without the SOC effect and are shown on the k-path $\Gamma $-X-Z-P. (

**a**) The phonon spectrum of the Ir sample, (

**b**) the same for the Rh sample, and (

**c**) the same for the Co sample.

**Figure 3.**(Color online) Calculated electronic band structure of B${}_{2}$Ir${}_{2}$Zn compound with and without SOC interaction along the high symmetry lines on the BZ k-path $\Gamma $-M-X-Z-P-$\Gamma $. (

**a**) Calculated bulk band structure with SOC effect. (

**b**) Calculated atom-projected DOS for B${}_{2}$Ir${}_{2}$Zn with SOC and total DOS with SOC effect. (

**c**) An irreducible representation of the calculated bulk band structure without SOC. Colors have the meaning of band symmetries discussed in the text. The solid black line at zero in (

**a**–

**c**) indicates the Fermi level. (

**d**) The Fermi surface of B${}_{2}$Ir${}_{2}$Zn bands.

**Figure 4.**(Color online) Calculated surface states of (001) plane. (

**a**) The (001) surface BZ shown in labeling with $\overline{\Gamma}$, $\overline{M}$, and $\overline{X}$. Paths labeled in green display the k-path selection. (

**b-i**) Shows the arrangement of atoms for the B1 layer and (

**b-ii**) shows the surface states of the B-termination layer with 25 layers of atoms. (

**c-i**) Shows the arrangement of atoms for the Ir1 layer and (

**c-ii**) shows the surface states of the Ir-termination layer with 21 layers of atoms. Similarly, (

**d-i**) shows the arrangement of atoms for the Zn1 layer, and (

**d-ii**) shows the surface states of the B-termination layer with 23 layers of atoms. The arrangement of atoms for each layer (Zn1, X1, and B1) is noted in Figure 1. Surface bands are denoted by red solid lines and bulk bands are shown in blue solid lines as a comparison. All surface and bulk band calculations were performed including the SOC effect. The solid black horizontal line at zero indicates the Fermi level.

**Figure 5.**(Color online) The change in band energy gap around the Fermi level of the B${}_{2}$Ir${}_{2}$Zn sample with SOC effect by changing the volume of the lattice. Top panel (

**a**–

**d**): shows the effect of gap changes due to the compression and stretch of the lattice for two crossings in X-$\Gamma $ and X-Z paths for −10%, −5%, 0%, and 5% shown from left to right, respectively. Bottom panel (

**e**–

**i**): The plots on the left show two crossings in the Z-P-$\Gamma $ k-path with their related stretch percentages in volume. The graph on the right compares both crossings’ energy gap to the percentage of lattice parameters in P-Z and P-$\Gamma $ paths on the left. The dotted back vertical line at 0% represents the optimized lattice. The solid black horizontal line at zero indicates the Fermi level in all plots.

**Figure 6.**(Color online) Calculated band structure of B${}_{2}{X}_{2}$Zn (X = Rh, Co) without SOC, and DOS of B${}_{2}{X}_{2}$Zn (X = Ir, Rh, Co) with and without the SOC effect. (

**a**,

**b**) Display of the band structure of Rh and Co samples respectively. (

**c**) Shows total DOS for all three samples with solid and dotted lines representing calculations with and without the SOC effect, respectively. The solid black line at zero indicates the Fermi level in all plots.

**Table 1.**The calculated lattice constants a, c, and formation energies, E, of B${}_{2}{X}_{2}$Zn compounds.

a (Å) | c (Å) | E (eV/atom) | |
---|---|---|---|

${B}_{2}I{r}_{2}Zn$ | 2.9912 | 12.7118 | −0.3005 |

${B}_{2}R{h}_{2}Zn$ | 2.9377 | 12.9377 | −0.3245 |

${B}_{2}C{o}_{2}Zn$ | 2.7634 | 12.1513 | −0.2279 |

**Table 2.**The calculated elastic constants (${C}_{ij}$), bulk modulus (B), shear modulus (G), Young’s modulus (E) in units of GPa, and Poisson’s ratio ($\nu $) for the B${}_{2}{X}_{2}$Zn compounds.

${\mathit{C}}_{11}$ | ${\mathit{C}}_{12}$ | ${\mathit{C}}_{13}$ | ${\mathit{C}}_{33}$ | ${\mathit{C}}_{44}$ | ${\mathit{C}}_{66}$ | B | G | E | $\mathit{\nu}$ | |
---|---|---|---|---|---|---|---|---|---|---|

${B}_{2}I{r}_{2}Zn$ | 354.4 | 142.2 | 199.8 | 496.5 | 183.3 | 100.0 | 247.92 | 132.07 | 336.47 | 0.27 |

${B}_{2}R{h}_{2}Zn$ | 338.4 | 102.6 | 169.2 | 420.1 | 157.9 | 80.4 | 215.81 | 118.47 | 300.43 | 0.27 |

${B}_{2}C{o}_{2}Zn$ | 342.1 | 133.8 | 160.7 | 432.3 | 205.8 | 120.3 | 223.03 | 143.92 | 355.32 | 0.23 |

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**MDPI and ACS Style**

Howard, J.; Rodriguez, A.; Haldolaarachchige, N.; Hettiarachchilage, K.
First Principles Computation of New Topological B_{2}*X*_{2}Zn (*X* = Ir, Rh, Co) Compounds. *J* **2023**, *6*, 152-163.
https://doi.org/10.3390/j6010011

**AMA Style**

Howard J, Rodriguez A, Haldolaarachchige N, Hettiarachchilage K.
First Principles Computation of New Topological B_{2}*X*_{2}Zn (*X* = Ir, Rh, Co) Compounds. *J*. 2023; 6(1):152-163.
https://doi.org/10.3390/j6010011

**Chicago/Turabian Style**

Howard, Jack, Alexander Rodriguez, Neel Haldolaarachchige, and Kalani Hettiarachchilage.
2023. "First Principles Computation of New Topological B_{2}*X*_{2}Zn (*X* = Ir, Rh, Co) Compounds" *J* 6, no. 1: 152-163.
https://doi.org/10.3390/j6010011