# A tight-binding investigation of biaxial strain induced topological phase transition in GeCH

###### Abstract

We propose a tight-binding (TB) model, that includes spin-orbit coupling (SOC), to describe the electronic properties of methyl-substituted germanane (GeCH). This model gives an electronic spectrum in agreement with first principle results close to the Fermi level. Using the formalism, we show that a topological phase transition from a normal insulator (NI) to a quantum spin Hall (QSH) phase occurs at 11.6% biaxial tensile strain. The sensitivity of the electronic properties of this system on strain, in particular its transition to the topological insulating phase, makes it very attractive for applications in strain sensors and other microelectronic applications.

###### pacs:

73.22.-f,71.70.Ej,73.63.-b^{†}

^{†}thanks: These two authors contributed equally.

^{†}

^{†}thanks: These two authors contributed equally.

## I Introduction

Topological insulators (TIs) are a new state of matter that have attracted a lot of interest within the condensed matter physics community Kane and Mele (2005); Hasan and Kane (2010); Qi and Zhang (2011); Moore (2010); Fu et al. (2007); Bernevig et al. (2006). It is now well established that TIs are promising candidates for future advanced electronic devices. They possess a bulk insulating gap and conducting edge states. The edge states are protected by time-reversal symmetry (TRS) against backscattering and this property makes them robust against disorder and nonmagnetic defects. Consequently, the edge channels normally possess very high carrier mobility.

Among TI materials two-dimensional (2D) van der Waals systems have attracted a lot of attention during the past decade Mannix et al. (2017). The interest in these systems originates from the discovery of graphene, which has a very high carrier mobility (200 000 cm/(V s)), thermal conductivity, and mechanical strength Neto et al. (2009); Katsnelson (2012); however, its zero electronic band gap has severely limited its applicability in electronic devices. Also, the proposal for the existence of a topological insulating phase in graphene by Kane and Mele was shown to be unrealistic, because of its extremely small SOC strength Yao et al. (2007); Huertas-Hernando et al. (2006). Hence, extensive efforts have been devoted to open a band gap and increase the effective SOC in graphene or find other 2D systems with favorable SOC, carrier mobility, and appropriate band gap.

Other 2D materials such as single- or few-layer transition metal dichalcogenides (TMDs), boron nitride, silicene, germanene, phosphorene, stanene and MXene, have been extensively explored Ren et al. (2016); Mannix et al. (2017).

Another important issue for applications in electronic industry is the compatibility of the material with current silicon-based electronic technology. Therefore, the group IV elements with honeycomb structure are more favorable for this purpose.

One method for tuning the electronic band structure of 2D systems is the use of surface functionalization. Functionalization of graphene with hydrogen, the so-called hydrogen-terminated graphene or graphane, opens a sizeable band gap, but its carrier mobility decreases dramatically to 10 cm/(Vs) Elias et al. (2009). Silicene and germanene the other analogues of graphene have also attracted much attention. However, the small band gap of these systems and mobility issues have limited their application for electronics. Functionalized germanene provide enhanced stability and tunable properties Jiang et al. (2014a). Compared with bulk Ge, surface functionalized germanene possess a direct and large band gap depending on the surface ligand. These materials can be synthesized via the topotactic deintercalation of layered Zintl phase precursors Jiang et al. (2014a, b). In contrast to TMDs, the weaker interlayer interaction allows for direct band gap single layer properties such as strong photoluminescence that are readily present without the need to exfoliate down to a single layer. Bianco et al. Bianco et al. (2013) produced experimentally hydrogen-terminated germanene, GeH (also called, germanane). Recently the new material GeCH was synthesized Jiang et al. (2014a), that exhibit an enhanced thermal stability. GeCH is thermally stable up to which compares to for GeH. The electronic structure of GeCH has been shown to be very sensitive to strain, which makes it very attractive for strain sensor applications Ma et al. (2014); Jing et al. (2015); Ma et al. (2016). It has also a high carrier mobility and pronounced light absorption which makes it attractive for light harvesting applications Jing et al. (2015); Ma et al. (2016).

At present there exist already a few first-principle studies of GeCH that also include the effect of SOC Jiang et al. (2014a); Ma et al. (2014); Jing et al. (2015); Ma et al. (2016). To fully understand the physics behind the electronic band structure close to the Fermi level, we propose a TB model. Our TB model is fitted to the density functional theory (DFT) results both for the case with and without SOC. In the next part of this work we applied biaxial tensile strain to examine the effect of strain on the electronic properties of this system and compare our results with DFT calculations. The possibility of a topological phase transition in GeCH under biaxial tensile strain is also examined. Our finding that there is a transition to the QSH phase is further corroborated by the fact that we find TRS protected edge states in nanoribbons made out of GeCH.

This paper is organized as follows. In Sec. II, we introduce the crystal structure and lattice constants of monolayer GeCH. Our TB model with and without SOC is introduced in Sec. III, and the effect of strain on the electronic properties of monolayer GeCH is examined. In Sec. IV, using the formalism we demonstrate the existence of a topological phase transition in the electronic properties of monolayer GeCH when biaxial tensile strain is applied. The paper is summarized in Sec.V.

## Ii lattice structure of monolayer GeCH

The hexagonal atomic structure of monolayer GeCH and its geometrical parameters are shown in Figs. 1(a-c). As shown in Figs. 1(a,b) it consists of three atomic layers where a buckled honeycomb sheet of Ge atoms is sandwiched between two outer methyl group layers. Each unit cell of monolayer GeCH consists of two Ge atoms and two CH groups. Previous DFT calculations gave for the lattice constant Å, and the Ge-Ge and Ge-C bond lengths are 2.415Å and 1.972Å, respectivelyMa et al. (2014). The buckling height, , indicating the distance between two different Ge sublattices, is 0.788 Å.

We have chosen the and axes along the armchair and zigzag directions, respectively. The axis is in the normal direction to the plane of the monolayer GeCH. With this definition of coordinates, the lattice vectors are written as , where the corresponding hexagonal Brillouin zone of the structure (see Fig. 1(d)) is determined by the reciprocal vectors .

## Iii Tight-Binding Model Hamiltonian

Electronic structure of monolayer GeCH has been obtained by using DFT calculations in Ref. Ma et al. (2014). It is shown that the low-energy electronic properties of this system are dominated by , and atomic orbitals of Ge atoms. DFT calculations including SOC interaction have shown that applying an in-plane biaxial tensile strain induces a topological phase transition in the electronic properties of monolayer GeCH Ma et al. (2014). Although such a DFT approach, provides valuable information regarding the electronic properties of such system, it is limited to small computational unit cells. For example, large nanoribbons consisting of hundreds of atoms and including disorder require very large super-cells which go beyond present day computational DFT capability. This motivated us to derive a TB model for monolayer GeCH that is sufficiently accurate to describe the low-energy spectrum and the electronic properties of this system.

In the following we will propose a low-energy TB model Hamiltonian that includes SOC for monolayer GeCH. We show that our model is able to predict accurately the effect of strain on the electronic properties of the system.

### iii.1 Model Hamiltonian without SOC

We propose a TB model including , and atomic orbitals with principal quantum number of Ge atoms to describe the low-energy spectrum of this system. The nearest-neighbor effective TB Hamiltonian without SOC in the basis of and in the second quantized representation is given by

(1) |

where and represent the creation and annihilation operators for an electron in the -th orbital of the -th atom, is the onsite energy of -th orbital of the -th atom and is the nearest-neighbor hopping amplitude between -th orbital of -th atom and -th orbital of -th atom. We will show that this effective model is sufficiently accurate to describe the low-energy spectrum of this system.

Note that the above Hamiltonian is quite different from the effective Hamiltonian that describes the electronic properties of pristine germanene Kaloni and Schwingenschlögl (2013). In the pristine honeycomb structures of the group IV elements, the effective low-energy spectrum is described by the outer atomic orbitals. However, in monolayer GeCH, the orbitals mainly contribute to the -bonding between Ge and C atoms to form the energy bands that are far from the Fermi level. Therefore, we will neglect the contribution of the orbitals of the Ge atoms and the other orbitals of the CH molecule in our TB model.

Hopping parameters | Without strain | With biaxial strain |
---|---|---|

With the above description, the hopping parameters of Eq. (1) can be expressed in terms of the standard
Slater-Koster parameters as listed in the middle column of Table 1,
where and are, respectively, function of the cosine of
the angles between the bond connecting two neighboring
atoms with respect to and axes.

Using the Fourier transform of Eq. (1), and numerically diagonalizing the resulting Hamiltonian in space,
one can fit to the ab-initio results
in order to obtain the numerical values of the mentioned Slater-Koster parameters. The density functional calculation
results ma2 () including the Heyd-Scuseria-Ernzerhof (HSE)
functional approximation Heyd et al. (2003) are used to parametrize the TB model given by Eq. (1).
We have listed the obtained numerical values
of these parameters in Table 2. The numerically calculated TB energy bands of monolayer GeCH
in the absence of strain, as shown in Fig. 2(a), are
in excellent agreement with the ab-initio results. The direct band gap of monolayer GeCH at the point is
1.82 eV.

-2.20 | 2.62 | 2.85 | -0.85 | -5.09 | 2.1 |

### iii.2 Strain effects

Applying strain to a system modifies its electronic properties Bir and Pikus (1974). This is due to the fact that it changes both the bond lengths and bond angles leading to a modulation of the hopping parameters that determine the electronic properties of the system.

An accurate prediction of the electronic properties of the system in the presence of different types of strain, is a stringent test of the accuracy of our TB model. To this end, we now first calculate the modification of the hopping parameters when biaxial tensile strain is applied to the plane of monolayer GeCH. Then we will study the modification of the energy spectrum in the presence of such a strain to show that our results agree very well with the DFT calculations. This particular type of strain noticeably simplifies our calculations.

When biaxial tensile strain is applied in the plane of monolayer GeCH leaves the honeycomb nature of its lattice intact and the initial lattice vectors and evolve to the deformed ones and . Therefore, the vector , in the presence of in-plane strain is deformed into , where and are the strain in the direction of the and axes, respectively. In the following, for simplicity we assume that the strengths of the applied biaxial strains in the two directions are equal, i.e., . In the linear deformation regime, one can perform an expansion of the norm of to first order in an which results in

(2) |

where and are coefficients related to the geometrical structure of GeCH. For the three nearest neighbor Ge atoms, one can write , where is the initial buckling angle. We note that in the presence of biaxial strain, the bond lengths and buckling angles are both altered. Thus, we consider their effects on the modification of the hopping parameters, simultaneously. Based on elasticity theory, we know that the main features of the mechanical properties in a covalent material are determined by the structure of the system and the strength of the covalent bonds. Therefore, one can expect that the change of the buckling angle in germanene Kaloni and Schwingenschlögl (2013) and GeCH be akin. The variation of the buckling angle Kaloni and Schwingenschlögl (2013) as a function of biaxial strain can be fit to the linear form (see Fig. 3), where .

According to the Harrison rule Harrison (1999), the standard Slater-Koster parameters related to and orbitals are proportional to the bond length as . Using Eq. (2), the modified parameters are given by

(3) |

One can then use the change of the buckling angle and the Slater-Koster parameters to obtain the modified hopping parameters as listed in the last column of Table 1, where represents the unstrained hopping parameters. For instance, the new hopping parameter can be approximated by

(4) |

Substituting and into the above equation gives

(5) |

In a similar way, one can obtain the other modified hopping parameters in order to study the evolution of the energy spectrum of monolayer GeCH as a function of applied biaxial tensile strain.

Straightforward substitution of the new hopping parameters in Hamiltonian, Eq. (1), gives the Hamiltonian for the strained system. The calculated TB energy spectrum in the presence of biaxial tensile strain with strengths of , , and are shown in Figs. 2(b), (c) and (d), which are in excellent agreement with the DFT results Ma et al. (2014); ma2 ().

We show in Fig. 4 the dependence of the band gap of GeCH as function of biaxial tensile strain. Notice the good agreement between both DFT and TB approaches demonstrating the validity of our proposed TB model.

### iii.3 Spin-Orbit coupling

Spin-orbit interaction is a relativistic correction to the Schrödinger equation. It can significantly affect the electronic properties of systems that consists of heavier elements. In such systems, the major part of SOC originates from the orbital motion of electrons close to the atomic nuclei. In the Slater-Koster approximation, one can assume an effective spherical atomic potential , at least in the region near the nucleus. Therefore, one can substitute and into the general form for the SOC term Li et al. (2015); Zhao et al. (2015)

(6) |

to obtain the SOC in the form of

(7) |

where is a radial function whose value depends on the type of atomic species. In the above equations, and , are Plank constant, free mass of electron, speed of light, and momentum, respectively; and and represent the Pauli matrices, angular momentum operator and electron spin operator, respectively.

Using the well known ladder operators and , one can obtain the matrix representation of the SOC Hamiltonian in the basis set of for monolayer GeCH with matrix elements

(8) |

where and represent the atomic orbitals of -th atom. Note that since the two atom basis in the unit cell of the monolayer GeCH are the same, we have .

Thus, the representation of the SOC Hamiltonian in the above mentioned basis is

(9) |

whose elements are matrices with , and

(10) |

The value of the strength of the SOC should be chosen either in agreement with experiment or by fitting the TB bands to the ab-initio results near some points such that it gives the correct band gap. In order to evaluate the strength of the SOC for Ge atoms in monolayer GeCH, we fitted the spectrum obtained from our multi-orbital TB model to the one from density functional calculations within the local density approximation (LDA) for the exchange correlation in Ref. Ma et al. (2014). As shown in Fig. 5, there is excellent agreement between the TB spectrum and the DFT results for the SOC strength eV. We adopt this SOC strength in the following calculations of the TB spectrum when we use the hopping parameters from Table 2.

The TB energy spectrum of monolayer GeCH are shown in Figs. 6 (a) and (b) for 0% and 12.5% strain, respectively.

Note that due to the presence of time reversal and inversion symmetry, each band in the energy spectrum of monolayer GeCH is doubly degenerate. As shown in Fig. 7, by applying biaxial tensile strain, the global band gap located at gradually decreases and eventually a band inversion occurs at 11.6% strain. By further increasing strain, the induced band gap due to SOC, (see Figs. 6(b), and (c)) becomes indirect, and at a reasonable strength of 12.8% reaches the value of 115 meV.

One can use the TB spectrum of Figs. 6, to calculate the effective masses of electrons and holes near the conduction band minimum (CBM) and the valence band maximum (VBM). The results, in unit of free electron mass , are listed in Table 3 for 0%, 6%, 9%, and 12.5% biaxial tensile strain. Note that, the electron and hole effective masses near the CBM and VBM along the two directions of -K and -M are the same.

Strain ()\Effective mass () | Electron | Hole |
---|---|---|

0% | 0.135 | 0.157 |

6% | 0.074 | 0.105 |

9% | 0.045 | 0.058 |

12.5% | 0.033 | 0.316 |

Another way to test the validity of our TB model, is its ability to predict a possible topological phase transition in the electronic properties of monolayer GeCH. In the next section we will study the strain-induced topological phase in monolayer GeCH using our TB model.

## Iv topological phase transition of monolayer GeCH under strain

In the previous section, using the TB model including SOC, we showed that monolayer GeCH is a NI. We also showed that one can manipulate its electronic properties by applying in-plane biaxial strain. It is clear from Eq. (8) that SOC preserves the TRS. Thus, the monolayer GeCH can exhibit a QSH phase when its energy spectrum is manipulated by an external parameter that does not break TRS. The classification is a well known approach to distinguish between the two different NI and TI phases Kane and Mele (2005); Hasan and Kane (2010). In the following, we briefly introduce the lattice version of the Fu-Kane formula Fu and Kane (2006), to calculate the invariant. Then, we show numerically that by applying biaxial tensile strain, a change in the bulk topology of monolayer GeCH occurs.

### iv.1 Calculation of the invariant

The Fu-Kane formula Fu and Kane (2006), for the calculation of the invariant is given by

(11) |

where the integral is taken over half the Brillouin zone as denoted by HBZ.
Here, the Berry gauge potential , and the Berry field
strength are
given by , and
, respectively; where represents
the periodic part of the Bloch wave function with band index , and the summation in
runs over all occupied states.

Note that, in this approach one has to do some gauge fixing procedure Fukui and Hatsugai (2007)
to fulfill the TRS constraints and the periodicity of the
points which are related by a reciprocal lattice vector .
Moreover, due to the TRS and the inversion symmetry in monolayer GeCH, each band
is at least doubly degenerate. Therefore, one needs to generalize the definition of
and to non-Abelian gauge field analogies Hatsugai (2004)
constructed from the 2M dimensional ground state multiplet ,
associated to the Hamiltonian Fukui and Hatsugai (2007); Hatsugai (2004).

In order to compute the invariant, a lattice version of Eq. (11) is more favorable for numerical calculations. To this end, one can simply convert the equivalent rhombus shape of the honeycomb Brillouin zone in space as shown in Figs. 8(a) and (b), into a unit square in space by the following change of variables

(12) |

This, allows us to use the more simple lattice version of Eq. (11) Fukui and Hatsugai (2007)

(13) |

where the lattice sites of the Brillouin zone are labeled by . Thus the above mentioned gauge fixing procedure and TRS constraints are applied on the equivalent points. Using the so-called unimodular link variable Fukui and Hatsugai (2007)

(14) |

where denotes a unit vector in the - plane, one can define the Berry potential and Berry field in Eq. (13) as

(15) | |||||

(16) |

Note that both the Berry potential and the Berry field strength are defined within the
branch of and .

The numerical results of the invariant are shown in Fig. 9.
As seen, for %, monolayer GeCH is a NI and at the critical value of
%, the invariant jumps from 0 to 1, indicating a strain-induced TI phase transition in
the electronic properties of the system. The topologically
protected global bulk gap for a strain of 12.8% is 115 meV, which is much larger than the thermal
energy at room temperature and therefore the monolayer GeCH is an excellent candidate for strain related applications.

In the next subsection we examine the formation of topologically protected edge states
in a typical nanoribbon with zigzag edges when the system is driven into the TI phase by applying
biaxial tensile strain.

### iv.2 Electronic properties of GeCH nanoribbons under strain

The appearance of helical gapless states at the edge of a 2D topological insulator, is a crucial consequence of its nontrivial bulk topology. In the previous section, we showed that a jump from 0 to 1 in the invariant for biaxial strain at takes place, demonstrating a topological phase transition in the electronic properties of monolayer GeCH. As an example, in this subsection, we study the 1D energy bands of GeCH nanoribbons with zigzag edges in the presence of biaxial tensile strain. Our TB model predicts the appearance of topologically protected edge states with increasing strain when the invariant becomes 1. We denote the width of the zigzag GeCH nanoribbon (z-GeCH-NR) by N, which is the number of zigzag chains across the ribbon width. To calculate the energy spectrum of a z-GeCH-NR with width N, we construct its supercell Hamiltonian () in the basis of where , , and represent the , , and orbitals of Ge atoms along the nanoribbon width.

and represent the atomic orbitals of H atoms that are introduced to passivate the Ge atoms on each edge, respectively. We assume that the width of the nanoribbon is large enough that the interaction between the two edges is negligible, and one can safely neglect the tiny change of the hopping parameters due to the passivation procedure. Therefore, one can write the matrix elements of the nanoribbon Hamiltonian as

(17) |

where are the basis site indices in a supercell; denote the atomic orbitals; denote the spin degrees of freedom; and is the translational vector of the -th supercell. The corresponding onsite energy of Ge atoms and the hopping parameters pertinent to the Ge-Ge bonds are substituted from Tables 1 and 2. Moreover, one has to define the onsite energy , and the hopping parameters and in the above equation corresponding to the matrix elements related to the H-Ge bond. We adopt from the fitting procedure the numerical values eV, eV, and with eV where +(-) denotes the lower (upper) H-Ge edge bonds. One can diagonalize the corresponding TB Hamiltonian, Eq. (IV.2), in order to obtain the energy spectrum. By applying biaxial tensile strain we found that the band gap of the nanoribbon gradually decreases and eventually the metallic edge states protected by TRS appear for a strain value where a band inversion takes place in the TB energy spectrum of bulk monolayer GeCH. The numerically calculated energy bands of z-GeCH-NR with in the presence of 9%, 11%, and 13% biaxial tensile strain are shown in Figs. 10(a), (b), and (c), respectively. This demonstrates a topological phase transition from the NI to the QSH phase in the electronic properties of monolayer GeCH.

## V Conclusions

To conclude, we have proposed an effective TB model with and without SOC for monolayer GeCH including , , and orbitals per atomic site. Our model reproduces the low-energy spectrum of monolayer GeCH in excellent agreement with ab-initio results. It also predicts accurately the evolution of the band gap in the presence of biaxial tensile strain. By including the SOC, this band gap manipulation leads to a band inversion in the electronic properties of monolayer GeCH, giving rise to a topological phase transition from NI to QSH. Our model predicts that this phase transition takes place for 11.6% biaxial tensile strain as verified by the formalism. The topologically protected global bulk gap at a strain of 12.8% is 115 meV, which is much larger than the thermal energy at room temperature and makes monolayer GeCH a promising candidate for future applications. We also showed the emergence of topologically protected edge states in a typical z-GeCH-NR in the presence of biaxial strain larger than 11.6%. This is an additional confirmation of the existence of the TI phase in the electronic properties of monolayer GeCH.

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