Electrically Tunable Quasi-Flat Bands, Conductance and Field Effect Transistor in Phosphorene
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Electrically Tunable Quasi-Flat Bands, Conductance and Field Effect Transistor in Phosphorene
Motohiko Ezawa
Department of Applied Physics, University of Tokyo, Hongo 7-3-1, 113-8656, Japan
Phosphorene, a honeycomb structure of black phosphorus, was isolated recently. We investigate electricproperties of phosphorene nanoribbons based on the tight-binding model. A prominent feature is the presenceof quasi-flat edge bands entirely detached from the bulk band. We explore the mechanism of the emergence ofthe quasi-flat bands analytically and numerically from the flat bands well known in graphene by a continuousdeformation of a honeycomb lattice. The quasi-flat bands can be controlled by applying in-plane electric fieldperpendicular to the ribbon direction. The conductance is switched off above a critical electric field, whichacts as a field-effect transistor. The critical electric field is anti-proportional to the width of a nanoribbon. Thisresults will pave a way toward nanoelectronics based on phosphorene.
Graphene is one of the most fascinating material found inthis decade . The low-energy theory is described by mass-less Dirac fermions, which leads to various remarkable elec-trical properties. In practical applications to current semi-conductor technology, however, we need a finite band gapin which electrons cannot exist freely. For instance, arm-chair graphene nanoribbons have a finite gap depending on thewidth , while bilayer graphene under perpendicular electricfield also has a gap . It is desirable to find an atomic mono-layer bulk sample which has a finite gap. Silicene is one ofthe promising candidates of post graphene materials, whichis predicted to be a quantum spin-Hall insulator . A strik-ing property of silicene is that a topological phase transitionis induced by applying electric field . Nevertheless, silicenehas so far been synthesized only on metallic surfaces .Another promising candidate is a transition metal dichalco-genides such as molybdenite .A new comer challenges the race of the post-graphenematerials. That is phosphorene, a honeycomb structureof phosphorus. It has been successfully obtained in thelaboratory and revealed a great potential in applica-tions to electronics. Black phosphorus is a layered materialwhere individual atomic layers are stacked together by Vander Waals interactions. Just as graphene can be isolated bypeeling graphite, phosphorene can be similarly isolated fromblack phosphorus by the mechanical exfoliation method. Asa key structure it is not planer but puckered due to the sp hybridization, as shown in Fig.1. There are already sev-eral works based on first-principle calculations . Thetight-binding model was proposed only recently by includ-ing the transfer energy t i over the 5 neighbor hopping sites( i = 1 , , · · · , ), as illustrated in Fig.1.The aim of this paper is to investigate the physics of phos-phorene nanoribbons based on the tight-binding model. Thetight-binding model is essential to make a deeper understand-ing of the system, which is not attained by first-principle cal-culations and low-energy effective theory. Aa a striking prop-erty of phosphorene nanoribbon, we demonstrate the pres-ence of a quasi-flat edge band which is entirely detachedfrom the bulk band. We explore the band structure of phos-phorene nanoribbon numerically and analytically from thatof graphene nanoribbon as a continuous deformation of thehoneycomb lattice by changing the transfer energy parame-ters t i . The graphene is well explained in terms of electron FIG. 1: Illustration of the structure and the transfer energy t i of phos-phorene. (a) Bird eye’s view. (b) Side view. (c) Top view. The leftedge is zigzag, while the right edge is beard. Red (blue) balls rep-resent phosphorus atoms in the upper (lower) layer. A dotted (solid)rectangular denote the unit cell of the 4-band (2-band) model. Theparameters of the unit cell length and angles of bonds are taken fromReference . hopping between the first neighbor sites with t = t = 0 and t = t = t = 0 , where the presence of flat bands is wellknown . We follow the fate of the flat bands by changingthese parameters. The essential roles is played by the ratio t /t : The flat bands are detached from the bulk band whenthe ratio is , while the flat bands are bent by the term t .It is an exciting problem to control the band structure ex-ternally. We may change the band gap by applying externalelectric field E z perpendicular to the phosphorene sheet withthe use of the puckered structure. Although we can controlthe band gap, the amount of the controlled gap is very tinycompared to the large band gap, that is, ∆ = 1 . eV sincethe height of the puckered structure ℓ is the order of nm. Onthe other hand, the quasi-flat edge band is found to be sensibleto external electric field E x parallel to the phosphorene sheet.This is because a large potential difference ( ∝ W E x ) is pos-sible between the two edges if the width W of the nanoribbonis large enough. We propose a field-effect transistor driven byin-plane electric field with the use of the quasi-flat edge bands.The conductance is shown to be either or e /h with respectto the in-plane electric field. If the nanoribbon width is µm ,the critical electric field is given by E cr = 0 . meV/nm. Thisis experimentally feasible. The unit cell of phospho-rene contains 4 phosphorus atoms, where two phosphorus ex-
FIG. 2: Brillouin zones and energy spectra of the 4-band and 2-bandmodels of phosphorene. (a) The Brillouin zone is a rectangular in the4-band model, which is constructed from two copies of the hexagonalBrillouin zone of the 2-band model. A magenta oval denotes a Diraccone present at the Γ point. (b) The Brillouin zone is a hexagonalin the 2-band model. A magenta oval denotes a Dirac cone presentat the M point. (c) The band structure of the 4-band model, whichis constructed from two copies of that of the 2-band model. (d) Theband structure of the 2-band model. ist in the upper layer and the other two phosphorus exist inthe lower layer. The tight-binding model of phosphorene wasrecently proposed and is given by H = X h i,j i t ij c † i c j , (1)where summation runs over the lattice sites, t ij is the trans-fer energy between i th and j th sites, and c † i ( c j ) is the cre-ation (annihilation) operator of electrons at site i ( j ). It hasbeen shown that it is enough to take hopping links, as illus-trated in Fig.1. The transfer energy explicitly reads as t = − . eV, t = 3 . eV, t = − . eV, t = − . eV, t = − . eV for these links.In the momentum representation the 4-band Hamiltonianreads as H = P k c † ( k ) ˆ H ( k ) c ( k ) with ˆ H = f + f f f + f f ∗ + f ∗ f f f ∗ f ∗ f + f f ∗ + f ∗ f ∗ f ∗ + f ∗ , (2)where f =2 t e ik x / cos √ k y , f = t e − ik x ,f =2 t e − ik x / cos √ k y ,f =4 t cos 32 k x cos √ k y , f = t e ik x . (3)We show the Brillouin zone and the energy spectrum of thetight-binding model in Fig.2(a) and (c), respectively. FIG. 3: Band structure of phosphorene nanoribbons when the trans-fer energy t is nonzero and zero. (a,d) Both edges are zigzag. (b,e)One edge is zigzag and the other edge is beard. (c,f) Both edges arebeard. The quasi-flat edge mode emerges for (a) and (b). When weset t = 0 , the quasi-flat band becomes perfectly flat band. Flat andquasi-flat Edge states are marked in magenta. We are able to reduce the 4-band model to the 2-band model due to the C h point groupinvariance. We focus on a blue point (atom in upper layer)and view other lattice points in the crystal structure (Fig.1).We also focus on a red point (atom in lower layer) and viewother lattice points. As far as the transfer energy is concerned,the two views are identical. Namely, we may ignore the colorof each point. Hence, instead of the unit cell containing 4points, it is enough to consider the unit cell containing only2 points. This reduction makes our analytical study consider-ably simple.The 2-band model is given by H = P k c † ( k ) ˆ H ( k ) c ( k ) with ˆ H = (cid:18) f f + f + f + f f ∗ + f ∗ + f ∗ + f ∗ f (cid:19) . (4)The rectangular Brillouin zone of the 4-band model is con-structed by folding the hexagonal Brillouin zone of the 2-bandmodel, as illustrated in Fig.2(a).The equivalence between the two models (2) and (4) is ver-ified as follows. We have explicitly shown the energy spectraof the 4-band model (2) and the 2-band model (4) in Fig.2(c)and (d), respectively. It is demonstrated that the energy spec-trum of the 4-band model is constructed from that of the 2-band model: The two bands are precisely common betweenthe two models, while the extra two bands in the 4-band modelare obtained simply by shifting the two bands of the 2-bandmodel, as dictated by the folding of the Brillouin zone. Phosphorene nanoribbons.
We investigate the bandstructure of a phosphorene nanoribbon placed along the y di-rection (Fig.1). We study zigzag and beard edges. There arethree types of nanoribbons, whose edges are (a) both zigzag,(b) zigzag and beard, (c) both beard. We show their bandstructures in Fig.3(a), (b) and (c), respectively.A prominent feature is the presence of the edge modes iso-lated from the bulk modes found in Fig.3(a) and (b). Theycomprise a quasi-flat band. It is doubly degenerate for azigzag-zigzag nanoribbon, and nondegenerate for a zigzag-beard nanoribbon, while it is absent in a beard-beard nanorib-bon [Fig.3(c)].Let us explore the mechanism how such an isolated quasi-flat band appears in phosphorene. By diagonalizing theHamiltonian, the energy spectrum reads E ( k ) = f ± | f + f + f + f | . (5)The band gap is given by ∆ = 4 t + 2 t + 4 t + 2 t = 1 . eV . (6)The asymmetry between the positive and negative energiesarises from the f term. Then it is interesting to see whatwould happen when we set t = 0 . We show the band struc-tures in Fig.3(d), (e) and (f) for the three types of nanoribbons,where the quasi-flat edge modes are found to become perfectlyflat. Consequently it is enough to show the emergence of theflat band by studying the model with t = 0 . Furthermore itis a good approximation to set t = t = 0 , since the trans-fer energies t and t are much larger than the others. Indeedwe have checked numerically that no qualitative difference isinduced by this approximation. Flat bands in anisotropic honeycomb lattice.
To explorethe origin of the isolated quasi-flat band we analyze theanisotropic honeycomb-lattice model, which is described bythe Hamiltonian (4) by setting f = f = f = 0 . ThisHamiltonian is well studied in the context of graphene andoptical lattice . The energy spectrum reads E = s t + 4 (cid:18) t + t t cos 32 k x (cid:19) cos √ k y , (7)which implies the existence of two Dirac cones at k x = ± arctan( q t − t /t ) , k y = 0 , (8)for | t | < | t | .We now study the change of the band structure of nanorib-bon by a continuous deformation of the honeycomb lattice,starting that of graphene. We show the band structure with (a)the zigzag-zigzag edges, (b) the zigzag-beard edges, and (c)the beard-beard edges in Fig.4 for typical values of t and t .(i) We start with the isotropic case t = t , where the en-ergy spectrum (7) becomes that of graphene with two Diraccones at the K and K ′ points. The perfect flat band connectsthe K and K ′ points, that is, it lies for (a) − π ≤ k ≤ − π and π ≤ k ≤ π ; (b) − π ≤ k ≤ π ; (c) − π ≤ k ≤ π .It is attached to the bulk band. See Fig.4(a),(b),(c). The topo-logical origin of flat bands in graphene has been thoroughlydiscussed .(ii) As we increase t but keeping t fixed, the two Diracpoints move towards the Γ point ( k = 0 ), as is clear from(8). The flat band keeps to be present between the two Diracpoints. See Fig.4(d),(e),(f).(iii) At | t | = 2 | t | , the two Dirac points merge into oneDirac point at the Γ point, as implied by (8). The flat bandtouches the bulk band at the Γ point for the zigzag-zigzagnanoribbon and the zigzag-beard nanoribbon, but disappearsfrom the beard-beard nanoribbon. See Fig.4(g),(h),(i). FIG. 4: Band structure anisotropic honeycomb nanoribbons. Theflat edge states are marked in magenta. We have set t = − and t = t = t = 0 . We have also set (a,b,c) t = 1 , (d,e,f) t = 1 . ,(g,h,i) t = 2 , (j,k,l) t = 2 . . The unit cell contains atoms. (iv) For | t | > | t | , the bulk band shifts away from theFermi level, as follows from (7). The flat band is discon-nected from the bulk band for the zigzag-zigzag nanoribbonand the zigzag-beard nanoribbon, where it extends over all re-gion − π ≤ k ≤ π . On the other hand, the edge band becomesa part of the bulk band and disappears from the Fermi levelfor the beard-beard nanoribbon. See Fig.4(j),(k),(l). Energy spectrum of quasi-flat bands.
We have explainedhow the flat band appears in the anisotropic honeycomb-latticemodel. The flat band is bent into the quasi-flat band by switch-ing on the transfer interaction t .We can derive the energy spectrum of the quasi-flatband perturbatively. We construct an analytic form of thewave function at the zero-energy state in the anisotropichoneycomb-lattice model ( t = t = t = 0 ). By solvingthe Hamiltonian matrix recursively from the outer most cite,we obtain the analytic form of the local density of states of thewave function for odd cite j , | ψ ( j ) | = α j p − α , (9)with α = 2 | t | (cos k ) / | t | . The wave function is zero foreven cite. It is perfectly localized at the outer most cite when k = π , and describes the flat band. With the use of this wavefunction, the energy spectrum E qf ( k ) of the quasi-flat band isestimated perturbatively by taking the expectation value of the t term as E qf ( k ) = − t t t (1 + cos k ) eV , (10)where t t /t = 0 . . On the other hand we numericallyobtain E (0) = − . eV. The agreement is excellent. FIG. 5: Band structure and conductance in unit of e /h of phospho-rene nanoribbons with the both edges being zigzag in the presenceof in-plane electric field E x . (a,b) E x = 0 , (c,d) E x = 0 . meV/nm,(e,f) E x = 1 . meV/nm, (g,h) E x = 2 . meV/nm. Magenta curvesrepresent the quasi-flat bands, while cyan lines represent the Fermienergy. The unit cell contains atoms. Phosphorene nanoribbons with in-plane electric field.
Itis an intriguing problem to control the band structure exter-nally. We may try to change the band gap by applying externalelectric field E z perpendicular to the phosphorene sheet. Wefind the band gap to behave as ∆ = 1 .
52 + 0 .
28 ( ℓE z ) eV , (11)where ℓ is the separation between the upper and lower lay-ers. The gap becomes simply larger when we apply E z . Fur-thermore, in practical applications, it needs very large electricfield since ℓ is the order of nm.On the other hand, it is possible to make a significantchange of the quasi-flat edge band by applying external elec-tric field parallel to the phosphorene sheet. Let us apply elec-tric field E x into the x direction of a nanoribbon with zigzag-zigzag edges. A large potential difference ( ∝ W E x ) is possi-ble between the two edges if the width W of the nanoribbon islarge enough. This potential difference resolves the degener-acy of the two edge modes, shifting one edge mode upwardlyand the other downwardly without changing their shapes. We present the resultant band structures in Fig.5 for typical valuesof E x . Field-effect transistor.
Electric current may flow alongthe edge. We show the conductance in Fig.5. Without in-planeelectric field, the conductance at the Fermi energy is e /h since there is a two-fold degenerate quasi-flat band. Abovethe critical electric field, the conductance becomes since thequasi-flat band splits perfectly. This acts as a field-effect tran-sistor driven by in-plane electric field. The critical electricfield is anti-proportional to the width W .We derive the critical electric field E cr . The energy shift at k = π is given by ∆ E ( π ) = ± W E x since the wave functionis perfectly localized at the outer most edge cite. In generalthe energy shift is given by ∆ E ( k ) = ± (cid:18) W − α − α (cid:19) E x . (12)It is well approximated by ∆ E ( k ) = ± W E x for widenanoribbons. The conductance is written as e h [ θ ( E − ε − | ∆ E (0) | ) − θ ( E − | ∆ E ( π ) | )+ θ ( E − ε + | ∆ E (0) | ) − θ ( E + | ∆ E ( π ) | )] , (13)where θ ( x ) is the step function θ = 1 for x > and θ = 0 for x < . The critical electric field is determined as E cr = | ε | / (2 W ) (14)If the nanoribbon width is µm , the critical electric field isgiven by E cr = 0 . meV/nm. This is experimentally feasible. Discussion.
We have analyzed the band structure of phos-phorene nanoribbons based on the tight-binding model, anddemonstrated the presence of quasi-flat edge modes entirelydetached from the bulk band. Starting from the well-knownstructure of graphene, we have explained the mechanism howsuch edge modes emerge by a continuous deformation of thehoneycomb lattice. The conductance due to the quasi-flat edgemodes is quantized to be either or e /h with respect tothe in-plane electric field E x . The critical electric field is E cr = 0 . meV/nm for a nanoribbon with width µm . Afield-effect transistor is possible with the use of this property.We may also expect a similar structure made of arsenic andantimony, which should be called as "arsenene" and "anti-monene". The electronic properties will be explained simplyby setting the transfer energies appropriately. Acknowledgements
The author thanks the support by theGrants-in-Aid for Scientific Research from the Ministry ofEducation, Science, Sports and Culture No. 25400317. M.E. is very much grateful to N. Nagaosa for many helpful dis-cussions on the subject.
Supporting Materials
Wave function of edge state.
We construct an analyticform of the wave function at the zero-energy state in theanisotropic honeycomb-lattice model ( t = t = t = 0 ) asfollows. We label the wave function of the atom on the outermost cite as ψ , and that of the atom next to it as ψ , and asso on. The total wave function is ψ = { ψ , ψ , · · · , ψ N } ifthere are N atoms across the nanoribbon. The Hamiltonian isexplicitly written as H = t g ∗ · · · t g t · · · t t g ∗ · · · t g · · ·· · · · · · · · · · · · · · · , (15)with g = 1 + e ik . The eigenvalue problem Hψ = 0 is triviallysolved, yielding ψ n = 0 and ψ n +1 = [ t (cid:0) e ik (cid:1) /t ] n ψ .The dispersion of the quasi-flat band is determined as E qf ( k ) = ∞ X n =0 t (1 + e − ik ) ψ ∗ n ψ n +2 + t (1 + e ik ) (cid:0) ψ ∗ n +2 ψ n (cid:1) = − t t t (1 + cos k ) , (16)which is (10) in the text. The energy shift due to the in-planeelectric field is given by ∞ X n =0 ( W − n ) | ψ n | = (cid:18) W − α − α (cid:19) E x , (17)which is (12) in the text. Electric field.
We apply electric field E z perpendicularto the sheet. It is necessary to use the 4-band tight-bindingmodel, since the electric field breaks the C h point groupinvariance. Namely, the upper and lower layers are distin-guished by the electric field. We introduce E z into the Hamil-tonian (2), ˆ H ( E z ) = ˆ H + diag. ( ℓE z , ℓE z , − ℓE z , − ℓE z ) . (18)By diagonalizing this Hamiltonian at k = 0 , the band gap isfound to be ∆ = X s = ± q ( t + s t + t ) + ( ℓE z ) + 4 ( t + t ) , (19)which yields numerically (11) in the text. Low-energy theory.
In the vicinity of the Γ point, wemake a Tayler expansion and obtain f = t (cid:18) ik x − k x − k y (cid:19) ,f = t (cid:18) − ik x − k x (cid:19) ,f = t (cid:18) − ik x − k x − k y (cid:19) ,f = t (cid:18) − k x − k y (cid:19) ,f = t (cid:0) ik x − k x (cid:1) . (20)The Hamiltonian (4) reads ˆ H = f + (cid:0) ε + αk x + βk y (cid:1) τ x + γk x τ y , (21)with the Pauli matrices τ , where ε =2 t + t + 2 t + t = 0 . eV, α = − t − t − t − t − t = 0 . eV, β = − t + 34 t + 32 t = 0 . eV, γ = − t + t + 5 t − t = 3 . eV. (22)Hence the dispersion is linear in the k y direction but parabolicin the k y direction. The low-energy Hamiltonian agrees withthe previous result with a rotation of the Pauli matrices τ x τ z and τ y τ x . Conductance.
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