Scaling laws for band gaps of phosphorene nanoribbons: A tight-binding calculation
Esmaeil Taghizadeh Sisakht, Mohammad H. Zare, Farhad Fazileh
SScaling laws for band gaps of phosphorene nanoribbons: A tight-binding calculation
Esmaeil Taghizadeh Sisakht, Mohammad H. Zare, and Farhad Fazileh ∗ Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran (Dated: August 17, 2015)In this study, we analyze the band structure, the state characterization, and electronic transport ofmonolayer black phosphorus (phosphorene) zigzag nanoribbons (zPNRs) and armchair nanoribbons(aPNRs), using five-parameter tight-binding (TB) approximation. In zPNRs, the ratio of the twodominant hopping parameters indicates the possibility of a relativistic dispersion relation and theexistence of a pair of separate quasi-flat bands at the Fermi level. Moreover, the corresponding statesare edge localized if their bands are well separated from the valence and conduction bands. We alsoinvestigated the scaling laws of the band gaps versus ribbon widths for the armchair and zigzagphosphorene nanoribbons. In aPNRs, the transverse electric field along the ribbon width enhancesthe band gap closure by shifting the energy of the valence and conduction band edge states. ForzPNRs, a gap occurs at the middle of the relatively degenerate quasi-flat bands; thus, these ribbonsare a promising candidate for future field-effect transistors.
I. INTRODUCTION
Two-dimensional (2D) structures that are inspired bygraphene such as hexagonal boron nitride (BN) and tran-sition metal dichalcognides (TMDs) have attracted con-siderable attentions owing to their remarkable electronicproperties . Graphene is known to have novel elec-tronic and mechanical properties such as high carrier mo-bility; however, its zero band gap limits its performance.As a TMD, molybdenum disulphide (MoS ) has a di-rect band gap of ∼ and a relatively high on/offratio . However, the carrier mobility of MoS is muchless than that of graphene. These layered structures canbe etched or patterned as quasi-one-dimensional (1D)strips referred to as nanoribbons. Graphene nanoribbons(GNRs) and MoS nanoribbons are examples of these 1Dstrips. These 1D nanoribbons can offer better tunabilityin electronic structures because of quantum confinementand edge effects .Monolayer black phosphorus, referred to as phospho-rene, has attracted much attention recently because ofits potential applications in nano-electronics, thermo-electronics and opto-electronics . Phosphorene hasa finite band gap and greater mobility as compared withMoS . Similar to bulk graphite, black phosphorus is alsoa layered structure in which the layers are held togetherby Van der Waals interactions . Each layer consists ofphosphorus atoms that are covalently bonded to three ad-jacent phosphorus atoms, thus forming a puckered honey-comb structure because of sp hybridization, as shown inFig. 1. As can be seen, the phosphorus sites are groupedin two zigzag layers. The upper and lower sites are shownwith darker and lighter colors, respectively. Phosphorenehas been successfully fabricated in the laboratory by nu-merous researchers . Graphene can be isolated bypeeling; similarly, phosphorene can also be isolated fromblack phosphorus via mechanical exfoliation. Phosphorushas a direct band gap of 0.3 eV . Phosphorene lay-ers can be mechanically exfoliated from bulk phosphorus,and the band gap of phosphorene thus obtained rangesfrom 2.0 eV (monolayer) to 0.6 eV (five-layers) . Al- A o t b a o A oo A xy FIG. 1. (a) Crystal structure and hopping integrals t i of singlelayer phosphorene for the TB model. (b) Top view. Notethat the dark (gray) balls represent the phosphorus atomsin the upper (lower) layer. The dotted rectangle indicates aprimitive unit cell containing four atoms. The parameters forthe bond angles and unit cell lengths are taken from . though phosphorene nanoribbons (PNRs) have not yetbeen fabricated, experience from graphene and other 2Dmaterials suggests the electronic structure and opticalproperties of PNRs must be studied for future researchon phosphorene-based nanoelectronics. Numerous stud-ies have focused on first-principle calculations . Re-cently, a TB model has been proposed by introducinghopping integrals ( t i ) over five neighbouring sites , asshown in Fig. 1(a).Our goal is to apply the above mentioned TB modelto zigzag and armchair phosphorene nanoribbons to ana-lyze their band structure and quantum conductance andcompare the results with other more sophisticated calcu-lations. Thereafter, we examine the effect of transverseelectric field on the band structure and quantum conduc-tance of both zigzag and armchair nanoribbons.In section II, the TB model is introduced. In sectionIII, the band structure and effective masses of the mono-layer phosphorene near the gap are presented based onthe TB model and it is shown that the dispersion is rel-ativistic along the y direction and the Fermi velocitiesalong this direction are calculated. In section IV, the nu-merical data for this model is presented for zPNRs andaPNRs, and the emergence of edge states and the grad- a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug ual emergence of flat bands in zPNRs when | t /t | ratiois increased is discussed. The scaling behavior of bandgap with ribbon width is presented and the obtained re-sults are compared with those of the other methods. Inaddition, the effect of transverse electric field on the bandgap in aPNRs and the transistor effect in zPNRs are in-vestigated. II. MODEL HAMILTONIAN
The TB Hamiltonian recently proposed for this systemis given by H = (cid:88) i,j t ij c † i c j (1)where the summation is over the lattice sites, and t ij are the hopping integrals between the i th and j th sites.Further, c † i and c j represent the creation and annihilationoperators of electrons in sites i and j , respectively. Thesehopping integrals between a site and its neighbours areshown in Fig. 1(a).The connections in the upper or lower layers in eachzigzag chain are represented by t hopping integrals, andthe connections between a pair of upper and lower zigzagchains are represented by t hopping integrals. Further, t denotes the hopping integrals between the nearest sitesof a pair of zigzag chains in the upper or lower layer, and t denotes the hopping integrals between the next nearestneighbor sites of a pair of upper and lower zigzag chains.Finally, t is the hopping integrals between two atomson the upper and lower zigzag chains that are farthestfrom each other. The specific values of these hopping in-tegrals as suggested in are as follows: t = − .
220 eV, t = 3 .
665 eV, t = − .
205 eV, t = − .
105 eV, and t = − .
055 eV. The special characteristic of this modelis that the second hopping integral is positive. This im-plies that the zigzag chains have negative t hopping in-tegrals along the chains and positive t hopping integralsconnecting these chains. For zPNRs, the eigenstates ofthe transverse modes, which characterize the behavior ofthe states as edge or bulk states, are along both t and t connections. The role of this behavior in creation of arelativistic band dispersion along the Γ-X direction willbe discussed in the next section. III. MONOLAYER PHOSPHORENE
In this section the band structure and effective massesof the electron and hole states of the bulk monolayerphosphorene is calculated based on the above mentionedTB model and the results are compared with ab-inito cal-culations. Since each unit cell of a single layer phospho-rene contains four phosphorus atoms [Fig. 1(b)], initially,a four band model is created. The band dispersion alongthe two periodic directions of Γ-X and Γ-Y are compared -404 Γ XY E n e r gy ( e V ) Γ XY S SS FIG. 2. Tight-binding energy band structure for bulk phos-phorene. and the electron and hole effective masses are comparedalong the two directions. In the next subsection it isargued that the unit cell for the electronic model onlycontains two phosphorus atoms resulting in a two bandmodel. Finally, the band gap at Γ point is derived as afunction of the hopping parameters.
A. Four-band tight-binding model
As shown in Fig. 1(b) the unit cell of the monolayerphosphorene is a rectangle containing four phosphorusatoms. Fourier transforming, the general Hamiltonian inmomentum space is given by: H = (cid:88) k ψ † k H [4] k ψ k (2)where ψ † k = ( a † k b † k c † k d † k ) and H [4] k is a 4 × H [4] k = A k B k C k A ∗ k D k B k B ∗ k D ∗ k A k C ∗ k B ∗ k A ∗ k (3)whose elements are given by A k = t + t e − ik a B k = 4 t e − i ( k a − k b ) / cos( k a /
2) cos( k b / C k = 2 e ik b / cos( k b / t e − ik a + t ) D k = 2 e ik b / cos( k b / t + t e − ik a ) . (4)Here k a = k · a and k b = k · b , where a = a ˆ x and b = b ˆ y are the primitive translational vectors of the structure TABLE I. Fermi velocities and effective masses of electron andhole states near the CBM and VBM along the two directionsof Γ-X and Γ-Y.Band v F ( × m/s ) m/m Γ-X (e) 9.71 0.164Γ-X (h) 8.26 0.179Γ-Y (e) – 0.873Γ-Y (h) – 1.175 displayed in Fig. 1(b). Bulk energy bands for the mono-layer phosphorene are shown in Fig. 2. The band dis-persion is relativistic along the x direction whereas it isnonrelativistic along the y direction. Considering a rela-tivistic band dispersion, E = (cid:112) m v F + p v F , along theΓ-X direction and a parabolic form along the Γ-Y di-rection near the conduction band minimum (CBM) andvalence band maximum (VBM) the effective masses andthe Fermi velocities are calculated and presented in Ta-ble I. It can be deduced from Table I that electrons andholes moving along the zigzag direction are more thansix times heavier than those moving along the armchairdirection.There is a simple explanation for the reason why thisspecial combination for the dominant hopping parame-ters ( t = − .
220 eV and t = 3 .
665 eV) creates a nearlyrelativistic dispersion near Γ point along the x direction.We introduce a lattice model [Fig. 3(a)], which is equiv-alent to the monolayer phosphorene within the two pa-rameter TB approximation. For this model the disper-sion along the y direction for large wavelengths along x ( | k a | ∼ x direction) canbe modeled by TB on a linear chain shown in Fig. 3(b).Similarly, the dispersion along the x direction for | k b | ∼ t cos( k b ), which near k b (cid:39) −| t | + | t | k b , and it isparabolic. This dispersion gives rise to an effective massof m = 1 . m , which is consistent with the data in Ta-ble I. The dispersion for the linear chain of Fig. 3(c) alongthe x direction is given by ± (cid:112) (2 t ) + t + 4 t t cos( k a ).In terms of the absolute values of the hopping param-eters and near the k a (cid:39)
0, this relation is reducedto ± (cid:112) ( | t | − | t | ) + 2 | t t | k a . When | t | is close to | t | , we can ignore the first term under the squareroot and the dispersion will be linear ± (cid:112) | t t | k a andthe constant of proportionality gives a Fermi velocity of2 π (cid:112) . . × (4 . / / (12400eV˚A) × c ∼ m/s which is consistent with the data of Table I. Forthe model of Eq.1, | t | (cid:39) | t | which does not give an ex-actly linear dispersion but it gives a massive relativisticdispersion, and for larger values of k a it is nearly linear. t1t1t1t1 t2t2t1t1 t2 t1 t1t1t2t1t1t2t2 t1 t1t1t2t1t1t2 t2 t22t 1 2t 1 t1t1t1t1 (c)(a) (b) FIG. 3. (a) Topologically equivalent structure to monolayerphosphorene within two parameter TB model. (b) Equivalentlinear chain model along the zigzag direction. (c) Equivalentlinear chain model along the armchair direction.
B. Two-band tight-binding model
In the TB Hamiltonian of Eq. 1, if we project the posi-tions of the upper and lower zigzag chains on a horizontalplane and keep the previous hopping integrals, the unitcells of the electronic system is reduced to two phospho-rus atoms per unit cell. The Fourier transform of theresulting two band model is given by H = (cid:88) k φ † k H [2] k φ k (5)where φ † k = ( a † k b † k ) and H [2] k is a 2 × H [2] k = (cid:18) B k e i ( k a − k b ) / A k + C k e i ( k a − k b ) / A ∗ k + C ∗ k e − i ( k a − k b ) / B k e i ( k a − k b ) / (cid:19) (6)Diagonalizing the above matrix, the energy spectrumis E k = | B k | ± | A k + C k e i ( k a − k b ) / | (7)The band gap in the Γ point is E g = 4 t + 2 t + 4 t + 2 t = 1 .
52 eV . (8) IV. ELECTRONIC AND TRANSPORTPROPERTIES OF PHOSPHORENENANORIBBONS
In the following numerical analysis, the commonly usedmethod for determining the width of graphene nanorib-bons is employed to determine the PNR structures. Ac-cording to this method, the structure of aPNR is definedby the number of dimmer lines across the ribbon width( N a -aPNRs), whereas that of zPNR is defined by thenumber of zigzag chains across the ribbon width ( N z -zPNRs) . To calculate the band structure and eigen-states of the nanoribbons, we obtain the eigenvalues andeigenvectors of the following matrix, which is the crystalHamiltonian between Bloch sums: M αβ ( k ) = − (cid:88) ij t iα ; jβ e i k · R ij (9)where i and j denote different unit cells, α and β denotethe basis sites in a unit cell. Further, k is the wave vector,and R ij represents a bravais lattice vector. Moreover, t iα ; jβ are the hopping integrals between the basis site α of unit cell i and the basis site β of unit cell j , andwill be substituted by the five hopping parameters of themodel, accordingly. For nanoribbons, the periodicity isonly along the ribbon length; therefore, the number ofbasis sites in each unit cell is proportional to the ribbonwidth. A. Edge modes in zPNRs
In order to understand the physics of this model, westudy the influence of the ratio of the two dominant hop-ping parameters on the behavior of the electronic struc-ture for zPNRs. We first study the dependence of quasi-flat bands and their corresponding edge states in zPNRson the ratio of | t /t | . The band structure and proba-bility amplitude of the upper valence band eigenstates of100-zPNRs for | t /t | =1, 2, and 3 for k = 0 are shownin Fig. 4. As can be seen in Figs. 4(a), (b), and (c), asthe | t /t | ratio increases, the two middle bands (shownwith grey lines) are detached from the bulk bands. Thecritical value of the ratio for the emergence of edge statesat k = 0 is 2, namely, at this ratio, the average ampli-tude of | Ψ i | becomes nearly homogeneous in the bulk.It should be noted that the states corresponding to thequasi-flat bands that are outside the middle region in-cluding between Dirac-like points and k = π or k = − π are always localized on the edges. Fig. 4(c) shows theband structure for | t /t | =3. In this case, the edge bandsare isolated from the bulk states, and are two-fold de-generate. This degeneracy is lifted in zPNRs with smallwidths ( N z <
40) for wave vectors near k = 0. Thisbehavior can be explained by considering the effect offinite electron tunneling between two opposite edges ofzPNRs with small widths . Fig. 4(d) shows the proba-bility amplitude of the upper valence band eigenstate for k = 0 as a function of the position of phosphorus atoms.As can be clearly seen, for small values of this ratio, theprobability amplitude is large for the bulk sites, whereasfor the edge sites, it is minimal or zero. The probabilityamplitude of the bulk sites decreases as this ratio is in-creased. For | t /t | =3, only the probability amplitude ofthe sites near the edges are non-zero, thus indicating theimportance of the | t /t | ratio in the creation of the edgestates.If we refer to the model that was introduced in Fig. 3,we can explain the above mentioned behavior as follows.The structure shown in Fig. 3 is a bipartite lattice, andeach site is connected to three sites of the other sublat-tice with two t links and one t link. If we separatethe total wave function to two components, each having E n e r gy ( e V ) t =|t | t =2|t | t =3|t | −π π −π ππ−π kkk (a) (b) (c) x i t =3|t |t =2|t |t =|t | |ψ i |
40 60 80 100 (d) FIG. 4. Top: Band structure of 100-zPNRs ( w ∼
22 nm)for | t /t | ratio values of (a) | t /t | =1, (b) | t /t | =2, and (c) | t /t | =3 for t = − . t = − . t = − . k = 0 of a zigzag phosphorene nanoribbon for differ-ent ratios of | t /t | . Note that the horizontal axis representsa unit cell in the width of the ribbon. -12 -8 -4 0 4 8 12 x i | ψ i | FIG. 5. Probability amplitude of the edge band eigenstatesof 6-zPNR ( w ∼ w ∼ amplitudes only on one sublattice, the local energy con-tribution of a wave function is proportional to the localamplitudes of the two component wave functions times∆ ≡ t + t . In the case of ∆ < | t /t | < > | t /t | > B. Scaling laws of band gaps for PNRs
Fig. 6 shows the variation in band gap with ribbonwidth for zPNRs and aPNRs owing to the quantum con-finement effect . In contrast to boron nitride nanorib-bons (BNNRs) , graphene nanoribbons (GNRs) , and α -graphdiyne nanoribbons , the band gap of PNRs de-creases monotonically as the ribbon width increases.Fig. 6 shows that the bang gap is larger in zPNRs forthe same ribbon width, indicating that the energy con-tribution from quantum confinement is higher in zPNRs,thus resulting in a stronger quantum confinement effectin zPNRs. The scaling behavior of band gap with in- B n a d G a p ( e V ) zPNRaPNR FIG. 6. Variation in band gap of zPNRs and aPNRs withribbon width. E ff ec ti v e M a sss ( m ) (e)-ZPNR(h)-ZPNR(e)-APNR(h)-APNR FIG. 7. Variation in effective masses of zPNRs and aPNRswith ribbon width. creasing ribbon width for both types of PNRs has beencalculated using DFT calculations . They suggesteda scaling behavior of ∼ /w for aPNRs whereas a ∼ /w for zPNRs. We argue that the scaling law for the zPNRsis not 1 /w . In fact, since the electrons along the con-finement direction of zPNRs, which is the armchair di-rection, behave like massive-relativistic particles, we fitour data for zPNRs with E gap = (cid:112) A /N zα + B + C ( w (cid:39) . N z − .
08 nm). The fitted values for the pa-rameters are A = 22 . α = 2 . B = 1 .
10 eV, and C = 0 .
42 eV. In this formula, we expect a parabolicscaling law as long as the second term under the squareroot is much larger than the first term. This conditionfor the above fitted values occurs for w (cid:29) . w (cid:28) . /w scalinglaw for zPNRs with ribbon widths larger than 3 . E gap = A (cid:48) /N aβ + C (cid:48) ( w (cid:39) . N a −
1) nm), and the fitted values for the pa-rameters are A (cid:48) = 20 . β = 1 .
92, and C (cid:48) = 1 .
52 eV,in agreement with previous results .We have also calculated the effective masses of the elec-tron and hole states near the VBM and CBM of PNRswith different ribbon widths. The results are shown ifFig. 7. The effective masses of zPNRs are more thansix times larger than aPNRs and for small widths theireffective masses increase even to higher values.
C. Response of aPNRs to E ext
Next, we analyze the relationship between the elec-tronic properties of aPNRs (periodicity along the x -direction) and the external electric field ( E ext ) along theribbon width. The band structure for E ext = 0 is shownin Fig. 8(a), in which the CBM and VBM determine theband gap. The electronic states associated with the VBMand CBM are located in the bulk of the ribbon [Fig. 8(b)].Also, all aPNRs are semiconductors independent of theirribbon width. When a transverse E ext is applied alongthe width, the states corresponding to the CBM, whichhave a positive band curvature (electron states), will shiftto lower energies owing to Stark effect, whereas the statescorresponding to the VBM (hole states) shift to higherenergies. Therefore, the CBM and VBM states will lo-calize on the ribbon edges [Fig. 8(d)]. By further in-creasing the field strength, the two bands approach oneanother because of the electrostatic potential differencebetween the opposite edges, and the band gap decreasesand eventually closes at a critical transverse field, E c [Fig. 8(c)]. This trend in band gap variation with E ext has already been observed in other materials such asGNRs , carbon nanotubes , MoS nanoribbons , andBNNRs . It should be noted that in contrast toother compounds such as BN , that the structure havea polarization along the width, the gap closure does notchange if we reverse the direction of the transverse E ext along the width.We also calculated the variation in band gap of aPNRswith E ext for four different widths (Fig. 9). As the trans-verse E ext increases, the band gap decreases uniformly.Similar behavior has been observed in the nanoribbonsof BN and MoS .As the aPNR width increases, the band gap decreasesrapidly with increasing transverse field E c , and the gap closure occurs for smaller fields because the electrostaticpotential difference is proportional to the ribbon width.The variation in band gap with ribbon width and trans-verse E ext has been calculated recently using DFT .For aPNRs with large widths, the results obtained withthe TB approach are in good agreement with the DFT-calculations. As the transverse E ext increases, the gapcloses directly at k = 0 for E c =0.339 V/˚A, and theedge band states corresponding to the VBM and CBMstates are localized on the opposite edges of the aPNRs[Fig. 8(d)].As shown in Fig. 9, the gap closure of aPNRs withsmall widths exhibits an interesting trend. For instance,for the 8-aPNR, the band gap varies slowly under a strong -6-4-20246 k E ext =0 E n e r gy ( e V ) (a) (b) -6-4-20246 k E ext =0.339 (V/Å) E n e r gy ( e V ) −π π (c) (d) FIG. 8. Top: (a) Band structure, and (b) probability ampli-tudes of 8-aPNR for zero transverse electric field. Bottom:The same graph for E ext = 0 .
339 V/˚A. Note that the eigen-states correspond to k = 0. Electric Field (V/Å) B a nd G a p ( e V ) ● -- ● FIG. 9. Variation in band gap of aPNRs with transverseexternal electric field for five different ribbon widths. E ext = 0.339 (V/Å) E n e r gy ( e V ) E ext = 0.406 (V/Å)E ext = 0.527 (V/Å) k π−π k=1.47k=0.708 FIG. 10. Conduction and valance bands of 8-aPNR for E ext =0 . . .
527 V/˚A. E ext , and the band gap closes for E c =0.339, it opensagain and closes at 0.527 V/˚A. Fig. 10 shows the valenceand conduction bands for E ext = 0.339, 0.406, and 0.527V/˚A. The opening up of the band gap after its closurefor very small ribbon widths is related to the finite hop-ping integrals between the two opposite edges and themechanism for a similar behavior in MoS nanoribbonshas been explained elsewhere . D. Transistor effect in zPNRs
A recent study based on the TB model has investigatedthe effect of an external in-plane ( E ext ) electric field onthe edge modes of zPNRs and the effect of an externalelectric field ( E z ) perpendicular to the ribbon surface onzPNRs . The results show that the band gap increasesin accordance with ( lE z ) where l is the separation dis-tance between the upper and lower layers of phosphorene.Moreover, for E ext greater than a critical strength ( E c ), -6-4-20246 k E ext =0 π E n e r gy ( e V ) conductance (e /h) (a) (b) (c) -6-4-20246 k E ext =0.016 (V/Å) E n e r gy ( e V ) π (d) conductance (e) (f) (e /h) FIG. 11. Top: (a) Conductance, (b) band structure, and(c) probability amplitudes of the band gap edge states of a10-zPNR under zero transverse electric field. Bottom: Thesame graph for E ext = 0 .
016 V/˚A. Note that the eigenstatescorrespond to k = 0. the degeneracy of the edge bands in Fig. 4c is lifted for thequasi-flat bands, and a transistor effect can be observed.Further, E c is inversely proportional to the ribbon width( ∝ /w ).In this study, we investigated the transistor effect inzPNRs using the Landauer formalism . In this for-malism, the conductance σ ( E ) for nanoscale devices atFermi energy ( E F ) between a pair of leads p and q isgiven by σ ( E ) = ( e h ) T r [Γ p ( E ) G RD ( E )Γ q ( E ) G AD ( E )] (10)where G RD ( E ) is the retarded Green’s function of the de-vice and G AD ( E ) = G RD † ( E ). In this equation, Γ p ( q ) = i [Σ p ( q ) ( E ) − Σ p ( q ) † ( E )] where Σ p ( q ) ( E ) is the self energyrelated to lead p ( q ). The retarded Green’s function ofthe device ( G RD ( E )) is given by G RD ( E ) = [ E − H D − Σ Rp ( E ) − Σ Rq ( E )] − (11)
20 40 60 80
Ribbon Width (N) C r iti ca l E l ec t r i c F i e l d ( V / Å ) E cr ~N z-1.05 -0.4-0.20 E n e r gy ( e V ) -0.8-0.400-0.4-0.20 E n e r gy ( e V ) kk −π −ππ π (a)(b)(c) E cr =0.007 (V/Å) E ext =0 E ext =0 E cr =0.012 (V/Å) FIG. 12. (a) Variation in critical transverse electric field withribbon width of zPNRs. Quasi-flat bands for (b) 20-zPNR for E ext = 0 and E ext = 0 .
007 V/˚Aand (c) 6-zPNR for E ext = 0and E ext = 0 .
012 V/˚A.
We now analyze the conditions under which the tran-sistor effect can be observed in zPNRs. The conductance,band structure, and wave functions of a 10-zPNRs for E ext =0 and 0.016 V/˚A are shown in Fig. 11. As canbe seen in Fig. 11(b), the degeneracy between the twoedge modes at zPNRs is slightly lifted close to k = 0.Therefore, the conductance is slightly asymmetric near k = 0. As shown in Fig. 11(c), the wave functions of theupper and lower quasi-flat bands are localized on boththe edges. As the external electric field is increased upto a critical field, the overlap between these two bandsvanishes. What we have is a conductance controlled bythe external electric field at Fermi energy, which is a field-effect transistor behavior. In this case, the wave functionsof the upper and lower edge bands are localized on theopposite edges [Fig. 11(f)].The relationship between E c and ribbon width is shown in Fig. 12(a). For zPNRs with widths greater than N z = 14, E c scales as 1 /N z . , which is in good agree-ment with the results previously reported by Ezawa .However, for ribbons with widths smaller than N z = 14,we found a completely different behavior. To explain thisportion of the graph, we considered the behavior of theedge bands of zPNRs with different widths. Figs. 12(b)and 12(c) show the quasi-flat bands for 20-zPNR and 6-zPNR, respectively. For E ext = 0, the quasi-flat bandsare different for these two widths. The VBM and CBMof 20-zPNR are located at k = π and k = 0, respectively.The VBM of 6-zPNR is also located at k = π whereasthe CBM is located at a k between 0 and π . This dis-placement of the CBM in 6-zPNR is caused by the finiteinteraction between the two edge modes. Therefore, alower external electric field is needed for observing thetransistor effect. V. CONCLUSION
In summary, we presented the numerical results forthe band structure and quantum conductance of zPNRsand aPNRs based on a five parameter TB model. It wasshown that the general form of the electronic structureis controlled by the two dominant hopping parameters.It was discussed that the opposite sign of these two hop-ping integrals is the origin of the creation of a relativisticband dispersion along the armchair direction. Our nu-merical results for zPNRs predicts a pair of degeneratequasi-flat bands at the Fermi level that are localized onthe ribbon edges, and this degeneracy is lifted for smallribbon widths owing to finite interactions between theedge states. Additionally, our calculations provide scal-ing laws of the band gap for PNRs as a function of ribbonwidth. We discussed that the band gap scaling law forboth nanoribbons with widths much larger than 3 . /w . For aPNRs, a semiconducting behavioris predicted, and an insulator-metal transition can be ex-pected when a transverse electric field is applied. In zP-NRs, an external transverse electric field can remove theoverlap between quasi-flat bands. The anisotropy in themobility , tunability of the band gap with ribbon width,and the field dependent conductance make this system apromising candidate for the future of field-effect transis-tor technologies. ∗ [email protected] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A.Firsov, Science , 666 (2004). A. K. Geim and K. S. Novoselov, Nature Materials , 183(2007). A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov,and A. K. Geim, Rev. Mod. Phys. , 109 (2009). A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C. Y.Chim, G. Galli, and F. Wang, Nano Lett. , 1271 (2010). K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz,Phys. Rev. Lett. , 136805 (2010). D. Xiao, G. B. Liu, W. Feng, X. Xu, and W. Yao, Phys.Rev. Lett. , 196802 (2012). X. Blase, A. Rubio, S. G. Louie, and M. L. Cohen, Phys.Rev. B , 6868 (1995). K. Watanabe, T. Taniguchi, and H. Kanda, Nature Ma-terials , 404 (2004). A. Kuc, N. Zibouche, and T. Heine, Phys. Rev. B ,245213 (2011). B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti,and A. Kis, Nature Nanotechnology , 147 (2011). Y. W. Son, M. L. Cohen, , and S. G. Louie, Nature (Lon-don) , 347 (2006). L. Yang, M. L. Cohen, and S. G. Louie, Nano Letters ,3112 (2007). X. Wang, Y. Ouyang, X. Li, H. Wang, J. Guo, and H. Dai,Phys. Rev. Lett , 206803 (2008). L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng,X. H. Chen, and Y. Zhang, Nature Nanotechnology ,372 (2014). H. Liu, A. T. Neal, Z. Zhu, X. Xu, D. Tomanek, and P. D.Ye, ACS Nano , 4033 (2014). F. Xia, H. Wang, and Y. Jia, Nature Communications ,4458 (2014). A. Castellanos-Gomez, L. Vicarelli, E. Prada, J. O. Island,K. L. Narasimha-Acharya, S. I. Blanter, D. J. Groenendijk,M. Buscema, G. A. Steele, J. V. Alvarez, H. W. Zandber-gen, J. J. Palacios, and H. S. J. van der Zant, 2D Materials , 025001 (2014). S. P. Koenig, R. A. Doganov, H. Schmidt, A. H. C. Neto,and B. Oezyilmaz, Appl. Phys. Lett. , 103106 (2014). A. Morita, Appl. Phys. A , 227 (1986). D. Warschauer, J. Appl. Phys. , 1853 (1963). S. Narita, Y. Akahama, Y. Tsukiyamaa, K. Muroa, S. Mo-ria, S. Endo, M. Taniguchi, M. Seki, S. Suga, A. Mikuni,and H. Kanzaki, Physica B+C , 422 (1983). Y. Maruyama, S. Suzuki, K. Kobayashi, and S. Tanuma,Physica B+C , 99 (1981). L. Liang, J. Wang, W. Lin, B. G. Sumpter, V. Meunier,and M. Pan, Nano Lett. , 6400 (2014). V. Tran, R. Soklaski, Y. Liang, and L. Yang, Phys. Rev.B , 235319 (2014). J. Qiao, X. Kong, Z.-X. Hu, F. Yang, and W. Ji, NatureCommunications , 4475 (2014). Q. Wei and X. Peng, Appl. Phys. Lett , 251915 (2014). J. Zhang, H. Liu, L. Cheng, J. Wei, J. Liang, D. Fan, J. Shi,X. Tang, and Q. J. Zhang, cond-mat/arXiv:1405.3348(2014). H. Y. Lv, W. J. Lu, D. F. Shao, and Y. P. Sun, cond-mat/arXiv:1404.5171 (2014). K. Gong, L. Zhang, W. Ji, and H. Guo, cond-mat/arXiv:1404.7207 (2014). A. N. Rudenko and M. I. Katsnelson, Phys. Rev. B ,201408 (2014). V. Tran and L. Yang, Phys. Rev. B , 245407 (2014). K. Dolui, C. D. Pemmaraj, and S. Sanvito, ACS Nano ,4823 (2012). Y.-W. Son, M. L. Cohen, and S. G. Louie, Phys. Rev.Lett. , 216803 (2006). X. Y. Zhao, C. M. Wei, L. Yang, and M. Y. Chou, Phys.Rev. Lett. , 236805 (2004). L. Yang, C.-H. Park, Y.-W. Son, M. L. Cohen, and S. G.Louie, Phys. Rev. Lett. , 186801 (2007). C.-H. Park and S. G. Louie, Nano Lett. , 2200 (2008). Z. Zhang and W. Guo, Phys. Rev. B , 075403 (2008). X. N. Niu, D. Z. Yang, M. S. Si, and D. S. Xue, J. AppliedPhysics , 143706 (2014). L. S. Q Wu, M. Yang, Z. Huang, and Y. P. Feng, cond-mat/arxiv.org:1405.3077 (2014). J. ´OKeeffe, C. Y. Wei, and K. J. Cho, Appl. Phys. Lett. , 676 (2002). V. Barone and J. E. Peralta, Nano Letters , 2210 (2008). M. Ezawa, New J. Phys. , 115004 (2014). S. Datta,
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