Topological origin of quasi-flat edge band in phosphorene

Phosphorene, a honeycomb structure of black phosphorus, was isolated recently. The band structure is highly anisotropic, where the kx direction is Dirac-like and the ky direction is Schrödinger-like. A prominent feature is the presence of a quasi-flat edge band entirely detached from the bulk band in phosphorene nanoribbons. We explore the mechanism of the emergence of the quasi-flat band by employing the topological argument invented to explain successfully the flat band familiar in graphene. The quasi-flat band can be controlled by applying in-plane electric field perpendicular to the ribbon direction. The conductance is switched off above a critical electric field, which acts as a field-effect transistor.


Introduction
Graphene is one of the most fascinating materials found in this decade [1,2]. The low-energy theory is described by massless Dirac fermions, which leads to various remarkable electrical properties. In practical applications to current semiconductor technology, however, we need a finite band gap in which electrons cannot exist freely. For instance, armchair graphene nanoribbons have a finite gap depending on their width [3][4][5], while bilayer graphene under a perpendicular electric field also has a gap [6]. It is desirable to find an atomic monolayer bulk sample that has a finite gap. Silicene is a promising candidate for post graphene materials, which is predicted to be a quantum spin-Hall insulator [7]. Nevertheless, silicene has so far been synthesized only on metallic surfaces [8][9][10]. Another promising candidate is a transition metal dichalcogenides such as molybdenite [11][12][13].
A newcomer challenges the race of the post-graphene materials. That is phosphorene, a honeycomb structure of black phosphorus. It has been successfully generated in the laboratory [14][15][16][17][18] and has revealed a great potential in applications to electronics. Black phosphorus is a layered material where individual atomic layers are stacked together by Van der Waals interactions. Just as graphene can be isolated by peeling graphite, phosphorene can be similarly isolated from black phosphorus by the mechanical exfoliation method. As a key structure it is not planer but puckered due to the sp 3 hybridization, as shown in figure 1. There are already several works based on first-principle calculations on phosphorene [19][20][21][22] and its nanoribbon [23][24][25][26][27]. The edge states detouched from the bulk band have been found in pristine zigzag phosphorene nanoribbons [24][25][26][27]. The tight-binding model was proposed [28] very recently by including the transfer energy t i over the five neighbor hopping sites ( = ⋯ i 1, 2, , 5), as illustrated in figure 1.
A striking property of phosphorene nanoribbon is the presence of a quasi-flat edge band [24][25][26][27] that is entirely detached from the bulk band. We explore the band structure of phosphorene nanoribbon analytically and numerically as a continuous deformation of the honeycomb lattice by changing the transfer energy parameters t i . The graphene is well explained in terms of electron hopping between the first neighbor sites with = ≠ t t 0 , where the origin of the flat band has been argued to be topological [29]. We are able to employ the same argument to show the topological origin of the quasi-flat edge band in phosphorene. The essential roles are played by the ratio t t | | 2 1 and t 4 : The flat bands are detached from the bulk band when the ratio is 2, while the flat bands are bent due to the transfer energy t 4 .  where a x and a y are the length of the unit cell into the x and y directions. We show the Brillouin zone and the energy spectrum of the tight-binding model in figures 2(a) and (c), respectively.

Two-band tight-binding model
We are able to reduce the four-band model to the two-band model due to the D h 2 point group invariance. We focus on a blue point (atom in upper layer) and view other lattice points in the crystal structure (figure 1). We also focus on a red point (atom in lower layer) and view other lattice points. As far as the transfer energy is concerned, the two views are identical. Namely, we may ignore the color of each point. Hence, instead of the unit cell containing four points, it is enough to consider the unit cell containing only two points. This reduction makes our analytical study considerably simple.
The two-band model is given by The rectangular Brillouin zone of the four-band model is constructed by folding the hexagonal Brillouin zone of the two-band model, as illustrated in figure 2(a). The equivalence between the two models (2) and (4) is verified as follows. We have explicitly shown the energy spectra of the four-band model (2) and the two-band model (4) in figures 2(c) and (d), respectively. It is demonstrated that the energy spectrum of the four-band model is constructed from that of the two-band model: The two bands are precisely common between the two models, while the extra two bands in the four-band model are obtained simply by shifting the two bands of the two-band model, as dictated by the folding of the Brillouin zone.
By diagonalizing the Hamiltonian, the energy spectrum reads The band gap is given by The asymmetry between the positive and negative energies arises from the f 4 term.

Low-energy theory
In the vicinity of the Γ point, we make a Tayler expansion of f i in (3) and obtain The Hamiltonian (4) reads with the Pauli matrices τ, where α β γ Hence, the dispersion is linear in the k y direction, but parabolic in the k y direction. The energy spectrum is given by The low-energy Hamiltonian agrees with the previous result [30] with a rotation of the Pauli matrices τ τ ↦ x z and τ τ ↦ y x .

Phosphorene nanoribbons
We investigate the band structure of a phosphorene nanoribbon placed along the y direction (figure 1). There are two ways of cutting a honeycomb lattice along the y direction, yielding a zigzag edge and a beard edge. Accordingly, there are three types of nanoribbons, whose edges are (a) both zigzag, (b) zigzag and beard, (c) both beard. We show their band structures in figures 3(a), (b) and (c), respectively.

Quasi-flat bands
A prominent feature of a phosphorene nanoribbon is the presence of the quasi-flat edge modes isolated from the bulk modes found in figures 3(a) and (b). They are doubly degenerate for a zigzag-zigzag nanoribbon, and nondegenerate for a zigzag-beard nanoribbon, while they are absent in a beard-beard nanoribbon [figure 3(c)].
We explore the mechanism of how such an isolated quasi-flat band emerges in phosphorene. As far as the energy spectrum is concerned, instead of H k k ( , ) x y 2 as defined by (4), it is enough to analyze the two-band Hamiltonian x y x y x y 2 2 4 The term f k k ( , ) , where the quasi-flat edge modes are found to become perfectly flat. It is a good approximation to set = = t t 0 3 5 , since the transfer energies t 1 and t 2 are much larger than the others. Indeed, we have checked numerically that almost no difference is induced by this approximation. The two-band model ′ H k k ( , ) This Hamiltonian has been studied in the context of strained graphene and optical lattice [31][32][33][34]. The energy spectrum reads for < t t 2 2 1 , where we have set = a a y .

Flat bands in anisotropic honeycomb lattice
We explore the origin of the flat band in the anisotropic honeycomb-lattice model (13). It is instructive to study the change of the band structure of nanoribbon by changing the parameter t 2 continuously, with t 1 being fixed. We show the band structure with (a) the zigzag-zigzag edges, (b) the zigzag-beard edges, and (c) the beard-beard edges in figure 4 for typical values of t 1 and t 2 .
(i) We start with the case = t t 2 1 , where the energy spectrum (14) becomes that of graphene with two Dirac cones at the K and ′ K points. The perfect flat band connects the K and ′ K points, that is, it lies for (a) π π − ⩽ ⩽ − ak 2 3 and π π ⩽ ⩽ ak (ii) As we increase t 2 but keep t 1 fixed, the two Dirac points move towards k = 0, as is clear from (15). The flat band is kept be present and connects the two Dirac points. See figures 4 (d)-(f). (iii) At = t t 2 2 1 , the two Dirac points merge into one Dirac point at k = 0, as implied by (15). The flat band touches the bulk band at k = 0 for the zigzag-zigzag nanoribbon and the zigzag-beard nanoribbon, but disappears from the beard-beard nanoribbon. See figures 4(g)-(i). (iv) For > t t 2 2 1 , the bulk band shifts away from the Fermi level, as follows from (14 ). The flat band is disconnected from the bulk band for the zigzag-zigzag nanoribbon and the zigzag-beard nanoribbon, where it extends over all of the region π π − ⩽ ⩽ k . On the other hand, the edge band becomes a part of the bulk band and disappears from the Fermi level for the beard-beard nanoribbon. See figures 4(j)-(l).

Topological origin of flat bands
The topological origin of the flat band has been discussed in graphene [29]. It is straightforward to apply the reasoning to the anisotropic honeycomb-lattice model (13). We consider onedimensional (1D) Hamiltonian H k ( ) k x in the k x space, which is essentially given by the twoband model (13) at a fixed value of ≡ k k y . We analyze the topological property of this 1D Hamiltonian. Because the k x space is a circle due to the periodic condition, the homotopy class is  π = S ( ) 1 1 .

We write the Hamiltonian as
It is important to remark that the gauge degree of freedom is present in this Hamiltonian [35]. Indeed the phase of the term F k ( ) k x is irrelevant for the energy spectrum of the bulk system. However, this is not the case for the analysis of a nanoribbon, since the way of taking the unit cell is inherent to the type of nanoribbon, as illustrated in figure 5. It is necessary to make a gauge fixing so that the hopping between the two atoms in the unit cell becomes real, namely, t 1 for the zigzag edge and t 2 for the beard edge. For the zigzag edge we thus make the gauge fixing such that where the parameter k corresponds to the momentum of zigzag nanoribbons. Here, t 1 for the link in the unit cell and − e t iak 1 for the neighboring link [ figure 5(a)]. The topological number of the 1D system is given by a a x k k x wind x By an explicit evaluation, N k ( ) wind is found to take only two values; We may interpret this as the winding number as follows. We consider the complex plane for counts how many times the complex number F k ( ) k x winds around the origin as k x moves from 0 to π a 2 for a fixed value of k. Note that the origin implies where the Hamiltonian is ill defined. We have shown such a loop for typical values of k in figure 5(b)  We have demonstrated that the system is topological for > k k | | D and trivial for < k k | | D . We next appeal to the bulk-edge correspondence to the topological system. When we cut the bulk along the y direction, the 1D system has an edge. Since the edge separates a topological insulator and the trivial state (i.e., the vacuum state), the gap must close at the edge, namely, there must appear a gapless edge mode. The gapless edge mode appears for all < k k | | D , implying the emergence of a flat band connecting the two Dirac points given by (15).
For the beard edge, we make the gauge fixing such that and carry out an analogous argument (see figure 5). We reach at the conclusion that a flat band appears for > k k | | D and connects the two Dirac points given by (15) but in an opposite way to the case of the zigzag edge.

Wave function and energy spectrum of edge states
We have explained how the flat band appears in the anisotropic honeycomb-lattice model. The flat band corresponds to the quasi-flat band in the original Hamiltonian Ĥ 2 .
We construct an analytic form of the wave function at the zero-energy state in the anisotropic honeycomb-lattice model (13) as follows. We label the wave function of the atom on the outer most cite as ψ 1 , and that of the atom next to it as ψ 2 , and as so on. The total wave function is ψ ψ ψ ψ = ⋯   . On the other hand, we numerically obtain = − E (0) 0.301 eV. The agreement is excellent.

Nanoribbons with in-plane electric field
It is possible to make a significant change of the quasi-flat edge band by applying external electric field parallel to the phosphorene sheet as in the case of graphene nanoribbon [4]. Let us apply electric field E x into the x direction of a nanoribbon with zigzag-zigzag edges. A large potential difference (∝ WE x ) is possible between the two edges if the width W of the nanoribbon is large enough. This potential difference resolves the degeneracy of the two edge modes, shifting one edge mode upwardly and the other downwardly without changing their shapes. We present the resultant band structures in figure 6 for typical values of E x .
The energy shift due to the in-plane electric field is given by ⎛ n n x 0 2 2 2 This is well approximated by Δ = E k WE ( ) x for wide nanoribbons. Electric current may flow along the edge. We derive the conductance. In terms of singleparticle Greenʼs functions, the low-bias conductance σ E ( ) at the Fermi energy E is given by [37] ⎡