Topological insulators have unique properties that make them promising materials for future implementation in next-generation electronic devices. However, topological insulators like stanene nanoribbons need to be passivated before they can be used in devices. We calculate the electronic band structure of stanene nanoribbons (SNRs) that are passivated by hydrogen (H), fluorine (F), chlorine (Cl), bromine (Br), iodine (I), or sodium (Na). We show that the difference between the electronegativity of the passivation material and the tin atoms defines the position of the Dirac cone of the topological insulator edge states. We develop a four-parameter tight-binding model based on the Kane–Mele model [Kane and Mele, Phys. Rev. Lett. **95**, 226801 (2005); Kane and Mele, Phys. Rev. Lett. **95**, 146802 (2005)]. The hopping parameters of the TB model are obtained by fitting the tight-binding model to the density functional theory (DFT) calculations. Finally, we demonstrate that the DFT band structures and the tight-binding model band structures are in good agreement with each other at low energies around the Dirac point, thereby capturing the complete physics of the passivated edge bands.

## I. INTRODUCTION

Topological insulators express many useful properties that can be employed in future electronic and spintronic devices.^{3} One of the most promising properties is the existence of one-dimensional (1D) spin channels, at the interface between $Z2$ two-dimensional (2D) topological insulators and conventional insulators, called edge states.^{2} For example, using 2D topological insulators as active channels in devices, such as topological insulator field-effect transistors (TIFETs), enables high electron mobilities.^{4}

Since topological insulators possess a non-trivial band structure with band inversion, edge states are guaranteed to exist at the boundary between a topological insulator and a conventional insulator.^{1,5} Furthermore, the edge states carry electrons that are spin-momentum-locked, meaning that electrons moving in a certain direction are spin-polarized. Consequently, electrons are resistant to backscattering due to weak perturbations, e.g., defects in the bulk or at the edges, as long as the perturbations are not magnetic causing a spin-flip or do not scatter the electron to the other edge of the ribbon.^{6,7}

Theoretically, graphene would be capable to exhibit topologically insulating behavior because of its hexagonal lattice arrangement.^{1} However, graphene’s spin–orbit interaction is too small^{8,9} to open a significant bandgap. On the other hand, stanene consists of a 2D monolayer of tin arranged in a buckled honeycomb lattice.^{10,11} Stanene has a bandgap of about $Egap=100$ meV,^{11} which is induced by spin–orbit interactions of the tin atoms.^{1} Stanene has the largest bandgap compared to its other group IV counterparts like silicene and germanene,^{12} because Sn has larger spin–orbit interactions due to being a higher atomic number element. As a consequence of the topologically different nature of the bandgap compared to conventional insulators, stanene shows topologically insulating behavior at interfaces with conventional insulators.^{1,5} Other groups have been able to epitaxially synthesize stanene^{13} and indirectly observe the edge state currents.^{14}

When implementing topological insulators in devices, it is important that we can access the Dirac cone so that most carrier transport occurs through edge states. However, often the Dirac cone does not lie at the Fermi level of the topological insulator. In other works, the position of the Dirac cones or the Fermi level was manipulated by functionalizing the surface^{10,11,15} or by doping^{16} the topological insulator. We propose a different approach to manipulating the Dirac cone of a 2D topological insulator in which we only passivate the dangling bonds at the edges of the 2D topological insulator. Unlike functionalizing the entire ribbon, the bulk band structure would remain unchanged, while the edge states forming the Dirac cone can be brought to the Fermi level.

In this work, we theoretically investigate the impact of edge passivation on the electronic properties of stanene nanoribbons (SNRs). We analyze the band structures obtained by density functional theory (DFT) of stanene nanoribbons passivated with different passivations. We show that passivation influences the shape of the edge bands. Interestingly, we find that the edge states are bent either upwards or downwards with different magnitudes, depending on the electronegativity of the passivants relative to the tin atoms. Finally, we construct a tight-binding model that can reproduce the results obtained by DFT. The development of the effective tight-binding model would allow for further transport calculations of SNRs with larger length scales, thereby capturing the complete physics of the passivated edge states.

## II. METHODOLOGY

First, we calculate and analyze the electronic band structures of stanene nanoribbons (SNRs) passivated by different elements with the help of first-principles calculations. Next, we fit the band structure of the passivated SNRs to a simple low effective energy model and further analyze the consequences of passivation.

The simulated SNRs are 2D nanoribbons of tin arranged in a hexagonal lattice as illustrated schematically in Fig. 1(a). We choose to do band structure calculations of zig–zag-oriented SNRs that are 16 atoms wide. For smaller ribbons, a bandgap opens due to edge state interactions,^{17} while wider ribbons would require more computational resources. Six passivation elements are studied: hydrogen, fluorine, chlorine, bromine, iodine, and sodium. One atom of passivation is attached to the zig–zag edge of the SNR [shown in Fig. 1(b)].

### A. First-principles calculations

We calculate the electronic band structure of the SNRs using first-principles density functional theory (DFT) calculations. The DFT calculations are performed with the help of the DFT software package *Quantum Espresso*.^{18,19} While performing the DFT calculations, we use the exchange-correlation energies through the generalized gradient approximation parameterized by the Perdew–Burke–Ernzerhof (PBE).^{20} We use fully relativistic pseudopotentials generated by the projector augmented wave (PAW) method^{21} to account for spin–orbit interactions.

Throughout the band structure calculations of the SNRs with hydrogen and halogen passivation, the kinetic energy cutoffs for the wavefunctions and the charge density is set to 100 and 400 Ry, respectively. These values are chosen so that the self-consistent calculation converges. For the simulations with sodium passivation, we increase the energy cutoff for the wavefunction energies to 120 Ry and for the energy density to 550 Ry to reach convergence for Na energy levels.

Since the SNR is a one-dimensional system, we choose a $16\xd71\xd71$ k-grid as generated by the Monkhorst–Pack scheme.^{22} The vacuum inserted in both the *y*-direction and *z*-direction of the 1D system is about 20 Å. Before calculating the band structures of the differently passivated ribbons, we perform relaxations and proceed by calculating the band structure.

We also perform self-consistent field calculations of each passivation element in its radical form. The calculations use the same pseudopotentials and energy cutoffs as before and are performed within a $1\xd71\xd71$ k-grid with a vacuum of 20 Å. We compute the work functions of the six passivation radicals (H, F, Cl, Br, I, and Na) from the self-consistent calculations.

Orbital projection calculations are performed to gauge the density of states (DOS) contribution from each atom and orbital to the band structure. We use the capability of Quantum Espresso to project the wavefunctions obtained from the self-consistent calculations on the orthogonalized atomic $s,p,d,f$-orbitals. From the orbital wave function projections, the projected DOS are obtained.

### B. Tight-binding model and fitting

#### 1. Kane–Mele model

We utilize the tight-binding (TB) model proposed by Kane and Mele^{1,2} to model the electronic structure of our SNRs. The Kane–Mele Hamiltonian incorporates the contribution of two electronic states per atom, which interact through nearest-neighbor hopping and spin–orbit interaction between next-nearest neighbors,

where $h.c.$ stands for the Hermitian conjugate. The first term describes the nearest neighbor hopping, with an interaction strength $t1$. The nearest neighbor hopping is described by the sum over all nearest neighbors of the $i$th atom and spin states $\alpha \u2208{\u2191,\u2193}$, wherein $cj,\alpha $ is the annihilation of an electron at the $j$th atom site with spin state $\alpha $ and $ci,\alpha \u2020$ the creation of an electron at the $i$th atom site with spin state $\alpha $, adjacent to the annihilation site. The second term in Eq. (1) relates to the spin–orbit interaction of next-nearest neighbors. More explicitly, the spin–orbit interaction strength is given by $\Delta SO=33t2$.^{1} The sum goes over all next-nearest neighbors of $i$th site and spin indices $\alpha $ and $\beta \u2208{\u2191,\u2193}$. $s\alpha \beta z$ is the third Pauli matrix, while $\nu i,j$ is either $+1$ or $\u22121$ depending on whether the hopping path to a next-nearest neighbor over a nearest neighbor site makes a clockwise turn or a counterclockwise turn, respectively. In our case, we neglect the Rashba term since we want to simulate a freestanding SNR.

Solving for the eigenvalues of the Kane–Mele Hamiltonian at different points in k-space, we obtain the band structure of non-passivated hexagonal 2D materials as shown in Fig. 2. The bulk states are presented as black lines, while the edge states are highlighted in blue. In the Kane–Mele model, the edge states of a non-passivated ribbon demonstrate a linear dispersion relationship around $k=\pi $.

#### 2. Passivation

We model passivated SNRs by adding one atom of passivation to both ends of each unit cell as shown in Fig. 1(b). Therefore, we add terms to the Hamiltonian accounting for the passivation,

Here, $HKM(t1,t2)$ represents the original Kane–Mele Hamiltonian. The first added term is the sum of each on-site potential at the passivation atom sites. The sum goes over all the passivation atoms $l$ and each spin state $\alpha $. The second added term accounts for the interaction strength between the passivation atoms and the ribbon atoms by going over each passivation atom and its nearest neighbors, which is only one ribbon atom in the case of a zig–zag ribbon and each of the spin states $\alpha $. We ignore spin–orbit interaction terms between the passivation atoms and the ribbon atoms.

The Hamiltonian with passivation introduced in Eq. (2) has four parameters: $t1$, $t2$, $t3$, and $\Delta V$ to describe the electronic properties of topological insulators. The on-site potential $\Delta V$ is calculated from first principles, unlike the other three parameters that are obtained by fitting the TB model to the band structures obtained by DFT.

The on-site potential $\Delta V$ of the passivation atoms is defined as the difference between the work functions of a passivation atom radical and the non-passivated ribbon: $\Delta V=Wpas\u2212WSn$.^{23} The on-site potential of the Sn atoms is set to zero.

Considering that the TB model only accounts for two bands per atom while the DFT calculation uses multiple orbitals per atom, only the Dirac cones consisting of the edge bands within the bandgap are fitted. The TB edge bands are fitted to the edge bands acquired from DFT by the method of least squares where we minimize

Here, $EDFT$ and $ETB$ are the DFT and TB energy bands, respectively. The sum goes over every data point at the associated wave vector $ki$. Additionally, when fitting the TB model to the DFT results, we ensure that the Dirac point of the TB band structure overlaps with the Dirac point of the DFT band structure. Once $\Delta V$ is determined from DFT and $t1$, $t2$, $t3$ are obtained through fitting, we have all parameters for the TB Hamiltonian.

## III. RESULTS AND DISCUSSION

Figure 3 illustrates the relaxed structure of a hydrogen passivated (H-passivated) stanene nanoribbon (SNR) where (a) shows a top view and (b) shows a side view. In the top view, we identify that the ribbon is arranged in a hexagonal lattice, and in this case, the edges are passivated by hydrogen atoms. The side view indicates that the stanene naturally forms a low buckled structure as already reported in other publications.^{11,24}

In the next sections, the ribbons are 16 atoms wide unless noted otherwise and have either no passivation or are passivated with one edge atom.

### A. DFT band structure

Figures 4(a)–4(d) show the DFT band structures in the first Brillouin zone of a non-, H- , F- , and Na-passivated SNR, respectively. The Fermi level is marked by the red dotted line in each of the plots, and in the case of the passivated SNRs, the edge bands are highlighted in blue. The DFT band structure plots for Cl-, Br-, and I-passivated SNRs can be found in the supplementary material.

The band structure of a non-passivated SNR is demonstrated in Fig. 4(a). The non-passivated SNR band structure is in stark contrast with the non-passivated SNR band structure calculated using the TB model (shown in Fig. 2). Since the edge atoms are not passivated, there is an excessive number of dangling bonds moving the Fermi level away from the Dirac cone. As a result, the ribbon does not behave like a topological insulator.

Figure 4(b) shows the band structure of a H-passivated SNR. In this case, the band structure exhibits a bandgap, which is traversed by edge states (marked in blue).

Compared to the theoretical non-passivated case (Fig. 2), we observe that the edge bands are slightly bent below the Fermi level. The bending effect is due to the interaction between the passivation atoms and the edge atoms of the SNR. In contrast, the edge states in the TB model (Fig. 2) are not interacting with any passivation or dangling bonds, thus the edge state possesses linear dispersion relations.

The band structure of a fluorine passivated (F-passivated) SNR is shown in Fig. 4(c). It shows similar features as the H-passivated SNR. There is a clear bandgap between bulk bands which is crossed by edge bands around the Fermi level at $k=\pi $, and the overall shape of most of the bulk bands is preserved. However, comparing the edge bands of the F-passivated to the H-passivated case, we find that the edge bands are more bent downwards in the F-passivated case. Since fluorine is more electronegative than hydrogen, fluorine makes a much stronger bond with the tin atoms. As a result of the increase in the bond energy, the energies of the participating electrons are decreased compared to the vacuum energy. Therefore, the Dirac cone is lower compared to the case of the H-passivated atoms.

Figure 4(d) demonstrates the band structure of a sodium passivated (Na-passivated) SNR. Here, the band structure looks significantly different compared to the band structure of H-passivated SNRs. In particular, the edge bands bend upwards as opposed to downwards, which is due to sodium being more electropositive compared to tin. Furthermore, we can discern that in this case, we do not have a well-defined bandgap since the Fermi level overlaps with bulk bands at around $k=0$.

In Fig. 5, we plot the band structure of the H-passivated SNR where the size of the points indicates the projected density of states (DOS) of different orbitals of a tin atom. We project the DOS for edge and center atoms of the ribbon to see their impact on the band structure. Figure 5(a) shows the projected DOS of a tin atom at the edge of a H-passivated SNR. In the case of an edge atom, the DOS of the edge bands is large compared to the other regions in the band structure. In contrast, when analyzing the projected band structure due to an atom in the middle of the ribbon [see Fig. 5(b)], the DOS at the edge bands vanish. We conclude that the wavefunctions associated with the Dirac cone are mostly located at the edge, while minimally located in the bulk.

### B. Tight-binding model

We found that non-passivated ribbons do not have a Dirac cone, H- and F-passivated ribbons exhibit a Dirac cone below the Fermi level, and Na-passivated ribbons have a Dirac cone above the Fermi level. The simplest TB Kane–Mele model however, simply shows a Dirac cone exactly in the middle of the bandgap. We proceed using the modified Kane–Mele introduced in Eq. (2) to determine the parameter set that describes the shifting Dirac cone.

#### 1. On-site potentials

The on-site potential for each of the halogens, hydrogen, and sodium is plotted in Fig. 6, together with the difference between Pauling electronegativities^{25} of the passivation elements and tin atoms, $\Delta EN=ENpas\u2212ENSn$. The on-site potentials are calculated from the difference in the work function of the passivation radical and the work function of the non-passivated ribbon. We observe an increase in the on-site potentials when moving from hydrogen passivation to fluorine passivation. However, passivating the ribbon with heavier halogens decreases the on-site potential. We recognize the same progression in the difference between the difference in electronegativity of the passivation elements and tin atoms. Thus, we conclude that the on-site potential of the passivation elements depends on their electronegativities.

The on-site potential of the passivation element is positive when the passivation element is more electronegative than the ribbon element. As a consequence of the positive on-site potential, the Dirac cones are located below the Fermi level as shown in Fig. 4. Conversely, the on-site potential of the passivation element is negative when the passivation element is less electronegative than the ribbon element, and thus, the Dirac cones are found above the Fermi level.

#### 2. Fitting

The interaction parameters, $t1$, $t2$, and $t3$, of the TB model are obtained from the DFT band structure (Fig. 4) by fitting to minimize Eq. (3). We fit the edge bands of the SNRs within an energy range $E=EF\xb10.3$ eV, where $EF$ signifies the Fermi energy.

Figure 7(a) shows the fit of the TB model to the DFT band structure of a H-passivated SNR. The TB model fits the DFT results well at low energies around the Fermi level. Remarkably, our TB model can reproduce the accurate physics of the edge states using only four parameters.

Figures 7(b)–7(d) illustrate the values of $t1$, $t2$, and $t3$ parameter for each of the differently passivated SNRs, respectively. The values of the fitting parameters are also listed in Table I. From Fig. 7(b), we observe that the $t1$ parameter stays within a range between 0.70 and 0.80 eV. The parameter $t1$ remains similar for all the differently passivated SNRs, which is attributed to the ribbon atoms being unchanged.

. | Tight-binding parameter (eV) . | |||
---|---|---|---|---|

Passivation . | t_{1}
. | t_{2}
. | t_{3}
. | ΔV
. |

H | 0.78 | 0.009 | 0.57 | 2.30 |

F | 0.75 | 0.009 | 1.35 | 6.88 |

Cl | 0.74 | 0.009 | 0.96 | 4.34 |

Br | 0.71 | 0.009 | 0.82 | 3.47 |

I | 0.72 | 0.009 | 0.60 | 2.52 |

Na | 0.77 | 0.003 | 0.72 | −1.43 |

. | Tight-binding parameter (eV) . | |||
---|---|---|---|---|

Passivation . | t_{1}
. | t_{2}
. | t_{3}
. | ΔV
. |

H | 0.78 | 0.009 | 0.57 | 2.30 |

F | 0.75 | 0.009 | 1.35 | 6.88 |

Cl | 0.74 | 0.009 | 0.96 | 4.34 |

Br | 0.71 | 0.009 | 0.82 | 3.47 |

I | 0.72 | 0.009 | 0.60 | 2.52 |

Na | 0.77 | 0.003 | 0.72 | −1.43 |

Figure 7(c) plots the spin–orbit interaction parameter $t2$. For H-passivated and halogen-passivated SNRs, the value of the parameter $t2$ remains within a fitting error at about $t2=0.009$ eV. We use the $t2$-value to evaluate the bandgap energy of stanene^{1} as $Eg=2\Delta SO=63\u22c5t2=0.10$ eV, which is precisely the theoretical bandgap predicted for stanene.^{11} However, for the Na-passivated SNR, we calculate a $t2\u223c0.003$ eV which is significantly smaller than for the halogen-passivated SNRs. Since $Eg=63\u22c5t2$, we conclude that there is less of a bandgap opening in the case of a Na-passivated ribbon compared to a halogen-passivated ribbon. Of course, we already observed that the Na-passivated ribbon has metallic characteristics, as the Fermi level crosses into the conduction bands, which masks the topological properties of the Na-passivated ribbon.

Figure 7(d) shows the values of the $t3$-parameters describing the interaction strength between the ribbon atoms and the passivation atoms. $t3$ varies significantly for each of the differently passivated SNRs and follows a progression that mimics the amount of bending of the edge states. $t3$ seems to be dependent on both magnitudes of the on-site potential and the amount of bending in the edge states. We conclude that $t3$ depends on the magnitude of the relative electronegativity difference between the passivation atoms and the tin atoms of the ribbon.

The results of the TB model are shown in Figs. 8(a) and 8(b) for a 32 atom-wide, H-passivated ribbon and a 32 atom-wide, F-passivated ribbon, respectively. The band structures represent the eigenvalues of the Kane–Mele Hamiltonians built with the acquired fitting parameters. The model correctly predicts that the edge bands of the F-passivated ribbon bend further down than the bands of the H-passivated ribbon. The bending is mostly dependent on the $t3$ parameter and the on-site potential $\Delta V$.

## IV. CONCLUSION

We have presented the impact of edge passivation on the electronic band structure of 2D topological insulators such as stanene. We have conducted DFT band structure calculations (shown in Fig. 4) of stanene nanoribbons that were either not passivated or passivated by hydrogen, fluorine, chlorine, bromine, iodine, or sodium. We conclude that passivation is necessary for the edge states to appear in SNRs. In non-passivated SNRs, the edge bands are pushed up into the conduction band, because the unpassivated edge of the ribbon has an excessive number of dangling bonds. Passivation neutralizes these bonds, lowering the energies of the edge states which leads to recognizable Dirac cones. The passivation material properties determine the shape and position of the Dirac cones with respect to the Fermi level which will influence the charge transport properties.

We have constructed a tight-binding model based on the Kane–Mele Hamiltonian that accurately captures the electronic properties of the stanene nanoribbons around the Fermi level. The parameters of the tight-binding model ($t1$, $t2$, $t3$, $\Delta V$) have been obtained from DFT.

We have shown that edge bands bend either upward or downward depending on the sign of the on-site potential. The magnitude of the bending is proportional to the interaction parameter $t3$ and the magnitude of the on-site potential. Both the on-site potential and the $t3$ parameter values of different passivation materials are connected to their respective electronegativity. We believe that the tight-binding model can be used for studying the charge transport of wide SNRs, which is impossible using full atomistic methods.

## SUPPLEMENTARY MATERIAL

See the supplementary material for the DFT band structures, figures of fitting the tight-binding model to the DFT band structures, and tight-binding model band structures of all differently passivated stanene nanoribbons.

## ACKNOWLEDGMENTS

This work was supported by imec’s Industrial Affiliation-Program. This material is based upon work supported by the National Science Foundation (NSF) under Grant No. 1802166. We acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for, in part, providing HPC resources that have contributed to the research results reported within this work.

The computational resources and services used in this work were, in part, provided by VSC (Flemish supercomputer Center), funded by the Research Foundation—Flanders (FWO) and the Flemish Government—Department EWI.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Emeric Deylgat:** Conceptualization (equal); data curation (equal); formal analysis (equal); investigation (equal); methodology (equal); software (equal); validation (equal); visualization (equal); writing – original draft (equal); writing – review & editing (equal). **Sabyasachi Tiwari:** Conceptualization (equal); data curation (equal); formal analysis (equal); investigation (equal); methodology (equal); project administration (equal); software (equal); supervision (equal); validation (equal); writing – review & editing (equal). **William G. Vandenberghe:** Conceptualization (equal); data curation (equal); formal analysis (equal); funding acquisition (equal); project administration (equal); resources (equal); supervision (equal); writing – review & editing (equal). **Bart Sorée:** Conceptualization (equal); data curation (equal); formal analysis (equal); funding acquisition (equal); project administration (equal); resources (equal); Supervision (equal); writing - review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.