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Topological origin of quasi-flat edge band in phosphorene

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2014 New J. Phys. 16 115004

(http://iopscience.iop.org/1367-2630/16/11/115004)

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Topological origin of quasi-at edge band inphosphorene

Motohiko EzawaDepartment of Applied Physics, University of Tokyo, Hongo 7-3-1, 113-8656, JapanE-mail: ezawa@ap.t.u-tokyo.ac.jp

Received 4 July 2014Accepted for publication 24 September 2014Published 31 October 2014

New Journal of Physics 16 (2014) 115004

doi:10.1088/1367-2630/16/11/115004

AbstractPhosphorene, a honeycomb structure of black phosphorus, was isolated recently.The band structure is highly anisotropic, where the kx direction is Dirac-like andthe ky direction is Schrdinger-like. A prominent feature is the presence of aquasi-at edge band entirely detached from the bulk band in phosphorenenanoribbons. We explore the mechanism of the emergence of the quasi-at bandby employing the topological argument invented to explain successfully the atband familiar in graphene. The quasi-at band can be controlled by applying in-plane electric eld perpendicular to the ribbon direction. The conductance isswitched off above a critical electric eld, which acts as a eld-effect transistor.

Keywords: topological insulator, bulk-edge correspondence, topology, phos-phorene, black phosphorus

1. Introduction

Graphene is one of the most fascinating materials found in this decade [1, 2]. The low-energytheory is described by massless Dirac fermions, which leads to various remarkable electricalproperties. In practical applications to current semiconductor technology, however, we need anite band gap in which electrons cannot exist freely. For instance, armchair graphenenanoribbons have a nite gap depending on their width [35], while bilayer graphene under aperpendicular electric eld also has a gap [6]. It is desirable to nd an atomic monolayer bulk

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New Journal of Physics 16 (2014) 1150041367-2630/14/115004+13$33.00 2014 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

sample that has a nite 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 beensynthesized only on metallic surfaces [810]. Another promising candidate is a transition metaldichalcogenides such as molybdenite [1113].

A newcomer challenges the race of the post-graphene materials. That is phosphorene, ahoneycomb structure of black phosphorus. It has been successfully generated in thelaboratory [1418] and has revealed a great potential in applications to electronics. Blackphosphorus is a layered material where individual atomic layers are stacked together by Vander Waals interactions. Just as graphene can be isolated by peeling graphite, phosphorene canbe similarly isolated from black phosphorus by the mechanical exfoliation method. As a keystructure it is not planer but puckered due to the sp3 hybridization, as shown in gure 1.There are already several works based on rst-principle calculations on phosphorene [1922]and its nanoribbon [2327]. The edge states detouched from the bulk band have been foundin pristine zigzag phosphorene nanoribbons [2427]. The tight-binding model was proposed[28] very recently by including the transfer energy ti over the ve neighbor hopping sites( = i 1, 2, , 5), as illustrated in gure 1.

A striking property of phosphorene nanoribbon is the presence of a quasi-at edge band [2427] that is entirely detached from the bulk band. We explore the band structure of phosphorenenanoribbon analytically and numerically as a continuous deformation of the honeycomb lattice bychanging the transfer energy parameters ti. The graphene is well explained in terms of electronhopping between the rst neighbor sites with = t t 01 2 and = = =t t t 03 4 5 , where the origin ofthe at band has been argued to be topological [29]. We are able to employ the same argument toshow the topological origin of the quasi-at edge band in phosphorene. The essential roles areplayed by the ratio t t| |2 1 and t4: The at bands are detached from the bulk band when the ratio is 2,while the at bands are bent due to the transfer energy t4.

Figure 1. Illustration of the structure and the transfer energy ti of phosphorene. (a) Birdeyes view. (b) Side view. (c) Top view. The left edge is zigzag while the right edge isbeard as a nanoribbon. Red (blue) balls represent phosphorus atoms in the upper (lower)layer. A dotted (solid) rectangular denote the unit cell of the four-band (two-band)model. The parameters of the unit cell length and angles of bonds are taken from [17].

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New J. Phys. 16 (2014) 115004 M Ezawa

2. Model hamiltonian

The tight-binding model is given by

=H t c c , (1)i j

ij i j4

,

where summation runs over the lattice sites, tij is the transfer energy between ith and jth sites,and ci

(cj) is the creation (annihilation) operator of electrons at site i (j). It has been shown [28]that it is enough to take ve hopping links to describe phosphorene, as illustrated in gure 1.The transfer energy explicitly reads as = t 1.2201 eV, =t 3.6652 eV, = t 0.2053 eV,

= t 0.1054 eV, = t 0.0555 eV for these links.

2.1. Four-band tight-binding model

The unit cell of phosphorene contains four phosphorus atoms, where two phosphorus atomsexist in the upper layer and the other two phosphorus atoms exist in the lower layer. In themomentum representation we obtain the four-band Hamiltonian = k k kH c H c( ) ( ) ( )k4 4 with

=

+ ++

++ +

H

f f f f f

f f f f

f f f f

f f f f f

0

0

0

0

, (2)4

1 3 4 2 5

1*

3*

2 4

4*

2*

1 3

2*

5*

4*

1*

3*

where

= = =

= =

f t e a k f t e f t e a k

f t a k a k f t e

2 cos12

, , 2 cos12

,

4 cos32

cos12

, , (3)

i a ky y

i a k i a ky y

x x y yi a k

1 1 2 2 3 3

4 4 5 52

x x x x x x

x x

12 3

13

52 3

13

where ax and ay are the length of the unit cell into the x and y directions. We show the Brillouinzone and the energy spectrum of the tight-binding model in gures 2(a) and (c), respectively.

2.2. Two-band tight-binding model

We are able to reduce the four-band model to the two-band model due to the D h2 point groupinvariance. We focus on a blue point (atom in upper layer) and view other lattice points in thecrystal structure (gure 1). We also focus on a red point (atom in lower layer) and view otherlattice 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, itis enough to consider the unit cell containing only two points. This reduction makes ouranalytical study considerably simple.

The two-band model is given by = k k kH c H c( ) ( ) ( )k2 2 with

=

+ + ++ + +

Hf f f f f

f f f f f . (4)2

4 1 2 3 5

1*

2*

3*

5*

4

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New J. Phys. 16 (2014) 115004 M Ezawa

The rectangular Brillouin zone of the four-band model is constructed by folding the hexagonalBrillouin zone of the two-band model, as illustrated in gure 2(a).

The equivalence between the two models (2) and (4) is veried as follows. We haveexplicitly shown the energy spectra of the four-band model (2) and the two-band model (4) ingures 2(c) and (d), respectively. It is demonstrated that the energy spectrum of the four-bandmodel is constructed from that of the two-band model: The two bands are precisely commonbetween the two models, while the extra two bands in the four-band model are obtained simplyby shifting the two bands of the two-band model, as dictated by the folding of theBrillouin zone.

By diagonalizing the Hamiltonian, the energy spectrum reads

= + + +kE f f f f f( ) . (5)4 1 2 3 5The band gap is given by

= + + + =t t t t4 2 4 2 1.52 eV. (6)1 2 3 5The asymmetry between the positive and negative energies arises from the f4 term.

Figure 2. Brillouin zones and energy spectra of the four-band and two-band models ofphosphorene. (a) The Brillouin zone is a rectangular in the four-band model, which isconstructed from two copies of the hexagonal Brillouin zone of the two-band model. Amagenta oval denotes a Dirac cone present at the point. (b) The Brillouin zone is ahexagonal in the two-band model. A magenta oval denotes a Dirac cone present at theM point. (c) The band structure of the four-band model, which is constructed from twocopies of that of the two-band model. (d) The band structure of the two-band model.

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New J. Phys. 16 (2014) 115004 M Ezawa

2.3. Low-energy theory

In the vicinity of the point, we make a Tayler expansion of fi in (3) and obtain

= + ( )( )f t

ia k a k a k2

3

112

14

,x x x x y y1 12 2

=

=

=

= +

( )

( )

( )

( )

( )

( )

f tia k a k

f tia k a k a k

f t a k a k

f tia k a k

13

16

,

25

3

2512

14

,

432

12

,

12

3

23

. (7)

x x x x

x x x x y y

x x y y

x x x x

2 22

3 32 2

4 42 2

5 52

The Hamiltonian (4) reads

= + + + +( )( )H f a k a k a k , (8)x x y y x x x y2 4

2 2

with the Pauli matrices , where

= + + + = = =

= + + = = + + =( )

m t t t t t t t t t

t t t t t t t

2 2 0.76 eV,112

16

2512

32

23

0.112 eV,

14

14

12

0.201 eV,1

35 2 2.29 eV. (9)

1 2 3 5 1 2 3 4 5

1 3 4 1 2 3 5

Hence, the dispersion is linear in the ky direction, but parabolic in the ky direction. The energyspectrum is given by

= E f E , (10)4 0with

= + + + ( )( )E k m a k a k . (11)x x x y y0 2

2 22

The low-energy Hamiltonian agrees with the previous result [30] with a rotation of the Paulimatrices x z and y x.

3. Phosphorene nanoribbons

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

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New J. Phys. 16 (2014) 115004 M Ezawa

3.1. Quasi-flat bands

A prominent feature of a phosphorene nanoribbon is the presence of the quasi-at edge modesisolated from the bulk modes found in gures 3(a) and (b). They are doubly degenerate for azigzagzigzag nanoribbon, and nondegenerate for a zigzag-beard nanoribbon, while they areabsent in a beardbeard nanoribbon [gure 3(c)].

We explore the mechanism of how such an isolated quasi-at band emerges inphosphorene. As far as the energy spectrum is concerned, instead of H k k ( , )x y2 as dened by(4), it is enough to analyze the two-band Hamiltonian

= ( ) ( ) ( )H k k H k k f k k , , , . (12)x y x y x y2 2 4The term f k k( , )x y4 only shifts the energy spectrum. The energy spectrum of H k k ( , )x y2 issymmetric between the positive and negative energy spectra, as we have noticed in (5). Weshow the band structures of the three types of nanoribbons in gures 3(d), (e) and (f) for theHamiltonian H k k ( , )x y2 , where the quasi-at edge modes are found to become perfectly at.

It is a good approximation to set = =t t 03 5 , since the transfer energies t1 and t2 are muchlarger than the others. Indeed, we have checked numerically that almost no difference is inducedby this approximation. The two-band model H k k ( , )x y2 is well approximated by the anisotropichoneycomb model,

=

++

Hf f

f f

0

0. (13)2

1 2

1*

2*

This Hamiltonian has been studied in the context of strained graphene and optical lattice [3134]. The energy spectrum reads

Figure 3. Band structure of phosphorene nanoribbons when the transfer energy t4 isnonzero and zero. (a, d) Both edges are zigzag. (b, e) One edge is zigzag and the otheredge is beard. (c, f) Both edges are beard. The quasi-at edge mode emerges for (a) and(b). When we set =t 04 , the quasi-at band becomes perfectly the at band. Flat andquasi-at edge states are marked in magenta.

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New J. Phys. 16 (2014) 115004 M Ezawa

= + +E t t t t a k a k4 cos

32

cos12

, (14)x x y y22

12

1 2

which implies the existence of two Dirac cones at =k 0x and =k ky D with

= ( )ak t t t2 arctan 4 , (15)D 12 22 2for t t22 1 , the bulk band shifts away from the Fermi level, as follows from (14 ). Theat band is disconnected from the bulk band for the zigzagzigzag nanoribbon and thezigzag-beard nanoribbon, where it extends over all of the region k . On the otherhand, the edge band becomes a part of the bulk band and disappears from the Fermi levelfor the beardbeard nanoribbon. See gures 4(j)(l).

3.3. Topological origin of flat bands

The topological origin of the at band has been discussed in graphene [29]. It is straightforwardto apply the reasoning to the anisotropic honeycomb-lattice model (13). We consider one-dimensional (1D) Hamiltonian H k( )k x in the kx space, which is essentially given by the two-band model (13) at a xed value of k ky. We analyze the topological property of this 1DHamiltonian. Because the kx space is a circle due to the periodic condition, the homotopy classis =S( )1 1 .

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New J. Phys. 16 (2014) 115004 M Ezawa

We write the Hamiltonian as

=( )

( )( )

H kF k

F k

0

0. (16)k x

k x

k x*

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 unitcell is inherent to the type of nanoribbon, as illustrated in gure 5. It is necessary to make agauge xing so that the hopping between the two atoms in the unit cell becomes real, namely, t1for the zigzag edge and t2 for the beard edge.

Figure 4. Band structure of anisotropic honeycomb nanoribbons. The at edge states aremarked in magenta. We have set = t 11 and = = =t t t 03 4 5 . We have also set (a, b, c)

=t 12 , (d, e, f) =t 1.72 , (g, h, i) =t 22 , (j, k, l) =t 2.52 . The unit cell contains 144atoms.

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New J. Phys. 16 (2014) 115004 M Ezawa

For the zigzag edge we thus make the gauge xing such that

= + = + + + ( )( )( )( )F k e f f t e t e1 ,k x i a k ak iak i a k ak12 1 2 1 2 ( 3 )x x x x12 3 12where the parameter k corresponds to the momentum of zigzag nanoribbons. Here, t1 for thelink in the unit cell and e tiak 1 for the neighboring link [gure 5(a)]. The topological number ofthe 1D system is given by

=

( )N k dk F k( ) log . (17)

a

a

x k k xwind x

By an explicit evaluation, N k( )wind is found to take only two values; =N 1wind for k k| | D. We may interpret this as the winding number as follows. We

consider the complex plane for =F k F k i k( ) | ( ) | exp [ ( )]k x k x k x . The quantity N k( )wind countshow many times the complex number F k( )k x winds around the origin as kx moves from 0 to a2 for a xed value of k. Note that the origin implies =F k( ) 0k x , where the Hamiltonian is illdened. We have shown such a loop for typical values of k in gure 5(b), where the horizontalaxis is for Re F k[ ( )]k x and the vertical axis is for Im F k[ ( )]k x ,

= + + ( )( )F k t ak t a k ak aRe (1 cos ) cos 12

3 , (18 )k x x x1 2

= + ( )( )F k t ak t a k ak bIm sin sin 12

3 . (18 )k x x x1 2

Figure 5.Unit cells and winding numbers for the zigzag and beard edges. It is necessaryto make a gauge xing in the Hamiltonian so that the hopping between the two atoms inthe unit cell becomes real, namely, (a) t1 for the zigzag edge and (b) t2 for the beardedge. (c, d) The winding number reads =N k( ) 1wind if the loop encircles the origin (redcircle), and =N k( ) 0wind if not (blue circle). The origin is represented by a dot, wherethe Hamiltonian is ill dened.

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New J. Phys. 16 (2014) 115004 M Ezawa

The loop surrounds the origin and =N k( ) 1wind when < k kD and < k kD , while itdoes not and =N k( ) 0wind when k k| | D and trivial for

where =t t t4 0.141 4 2 . On the other hand, we numerically obtain = E (0) 0.301eV. Theagreement is excellent.

4. Nanoribbons with in-plane electric eld

It is possible to make a signicant change of the quasi-at edge band by applying externalelectric eld parallel to the phosphorene sheet as in the case of graphene nanoribbon [4]. Let usapply electric eld Ex into the x direction of a nanoribbon with zigzagzigzag edges. A largepotential difference ( WEx) is possible between the two edges if the widthW of the nanoribbonis 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. Wepresent the resultant band structures in gure 6 for typical values of Ex.

The energy shift due to the in-plane electric eld is given by

= = =

E k W n W E( ) ( )

1. (23)

nn x

0

2 2

2

This is well approximated by =E k WE( ) x for wide nanoribbons.Electric current may ow along the edge. We derive the conductance. In terms of single-

particle Greens functions, the low-bias conductance E( ) at the Fermi energy E is given by[37]

= ( )E e h E G E E G E( ) Tr ( ) ( ) ( ) ( ) , (24)2 L D R Dwhere = E i E E( ) [ ( ) ( )]R(L) R(L) R(L) with the self-energies E( )L and E( )R , and

= [ ]G E E H E E( ) ( ) ( ) , (25)D D L R 1

Figure 6. Band structure and conductance in unit of e h2 of phosphorene nanoribbonswith the both edges being zigzag in the presence of in-plane electric eld Ex. (a, b)

=E 0x , (c, d) =E 0.5x meV/nm, (e, f) =E 1.5x meV/nm, (g, h) =E 2.0x meV/nm.Magenta curves represent the quasi-at bands, while cyan lines represent the Fermienergy. The unit cell contains 144 atoms.

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New J. Phys. 16 (2014) 115004 M Ezawa

with the Hamiltonian HD for the device region. The self-energy E( )L(R) describes the effect ofthe electrode on the electronic structure of the device, whose the real part results in a shift of thedevice levels, whereas the imaginary part provides a life time. It is to be calculated numerically[3841].

We use this formula to derive the conductance,

+ + +( ) ( )eh

E E E E E E E E(0) ( ( ) ) (0) ( ( ) ) ,2

0 0

where x( ) is the step function = 1 for >x 0 and = 0 for

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New J. Phys. 16 (2014) 115004 M Ezawa

1. Introduction2. Model hamiltonian2.1. Four-band tight-binding model2.2. Two-band tight-binding model2.3. Low-energy theory

3. Phosphorene nanoribbons3.1. Quasi-flat bands3.2. Flat bands in anisotropic honeycomb lattice3.3. Topological origin of flat bands3.4. Wave function and energy spectrum of edge states

4. Nanoribbons with in-plane electric field5. ConclusionAcknowledgementsReferences