Edge and sublayer degrees of freedom for phosphorene nanoribbons with twofold-degenerate edge bands via electric field
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Edge and sublayer degrees of freedom for phosphorene nanoribbons with twofold-degenerate edgebands via electric field
Yi Ren, Xiaoying Zhou, and Guanghui Zhou ∗ Department of Physics, Key Laboratory for Low-Dimensional Quantum Structures and Quantum Control (Ministry of Education),and Synergetic Innovation Center for Quantum E ff ects and Applications, Hunan Normal University, Changsha 410081, China For the pristine phosphorene nanoribbons (PNRs) with edge states, there exist two categories of edge bandsnear the Fermi energy ( E F ), i.e., the shuttle-shaped twofold-degenerate and the near-flat simple degenerateedge bands. However, the usual experimental measurement may not distinguish the di ff erence between thetwo categories of edge bands. Here we study the varying rule for the edge bands of PNRs under an externalelectrostatic field. By using the KWANT code based on the tight-binding approach, we find that the twofold-degenerate edge bands can be divided into two separated shuttles until the degeneracy is completely removedand a gap near E F is opened under a su ffi ciently strong in-plane electric field. Importantly, each shuttle from theribbon upper or lower edge outmost atoms is identified according to the local density of states. However, undera small o ff -plane field the shuttle-shaped bands are easily induced into two near-flat bands contributed from theedge atoms of the top and bottom sublayers, respectively. The evidence provides the edge and sublayer degreesof freedom (DOF) for the PNRs with shuttle-shaped edge bands, of which is obviously di ff erent from anothercategory PNRs intrinsically with near-flat edge bands. This is because that the former category of ribbons solelyhave four zigzag-like atomic configurations at the edges in each unit cell, which also results in that the propertyis robust against the point defect in the ribbon center area. As an application, furthermore, based on this issuewe propose a homogenous junction of a shuttle-edge-band PNR attached by two electric gates. Interestingly,the transport property of the junction with field manipulation well reflects the characteristics of the two DOFs.These findings may provide a further understanding on PNRs and initiate new developments in PNR-basedelectronics. I. INTRODUCTION
The internal degree of freedom (DOF) of electrons innanostructures is an important issue of modern condensedmatter physics. In addition to the charge and spin DOFs,other ones have also been discussed. For example, inmultilayer graphene [1] and transition-metal dichalcogenides(TMDs) [2], both the layer and valley DOFs are presented[3, 4]. In addition, new DOF may appear upon tailoringa two-dimensional (2D) material into a nanoribbon. Theedge and layer, for instance, have been regarded as tunableequivalents of the spin-one-half DOF in bilayer phosphorenenanoribbons (PNRs) with zigzag-edge [5]. However, here wetreat the edge and sublayer as two DOFs in a recently revealedmonolayer slope-edged PNR (sPNR) with twofold-degenerateedge bands [6].Phosphorene is a few- or mono-layer black phosphorus(BP) where P atoms arranged in the top and bottom sublayersof a puckered honeycomb lattice [7, 8]. Inside phosphorene,each P atom is covalently bonded with three adjacent atomsto form a puckered honeycomb structure due to the sp hybridization [9, 15]. This promising new 2D material, in thesense of applications in nano-electronics, can be exfoliatedfrom bulk BP due to the weak interlayer Van der Waalsinteraction and possesses a direct band gap of 0.3 eV [7,9]. This direct gap increases up to ∼ ff ect transistor (FET) [7, 11–13] basedon phosphorene is found to have an on / o ff ratio up to 10 and a ∗ Electronic address: [email protected] high hole carrier mobility to 800 cm / Vs [14]. Further, arisingfrom the low symmetric and highly anisotropic structure,phosphorene owns strongly anisotropic electrical, thermaland optical properties, which may open up possibilities forconceptually new devices [9, 16–20].On the other hand, nanoribbons can o ff er better tunability inelectronic structures because of the quantum confinement andedge morphological influence. Tailoring a 2D phosphorenesheet along the conventional zigzag and armchair directionshave been experimentally realized [21, 22]. Hence the zigzagPNRs (zPNRs) with significant edge states and armchairPNRs (aPNRs) with a direct band-gap have been extensivelystudied [15, 23–26], and their skewed or beard counterpartshave also been further reported [24, 27]. In general, theedge states projecting to the outermost atoms of a ribbon inreal space are near the Fermi level E F [24, 28]. They havebeen extensively studied for graphene and MoS nanoribbons[29, 30]. However, a zPNR has two near-degenerate edgebands closing to E F , which are respectively contributed bythe atoms of the two edges. And the properties of theseimportant edge states have been recognized [5, 28, 31–34].Moreover, we may cut a phosphorene sheet so that the zigzag(armchair) direction intersects the puckered ridges under anchiral angle other than 0º (90º), resulting in PNRs withother possible edge geometries [6]. These ribbons can beclassified into two types, one type with edge states includingzPNR and the other without edge states including aPNR. Inspecification, from our previous definition of the chiral vector T = m a / + n a / m , n ), the cases of m + n = even integer have defined the all possible planar crystaldirections [6]. Hence a ribbon can be generally denotedas ( m , n )PNR. Furthermore, according to the edge atomicarrangement (morphology), PNRs with edge states can befurther divided into two categories. When both m and n are odd, the outermost edge atoms of a PNR are alternatelylocated at the two sublayers, resulting in the shuttle-shapedtwofold-degenerate edge bands near E F , such as (1,3)PNRand (3,1)PNR [6, 27, 35, 36]. While for both even m and n , the outermost atoms are located at the same sublayerand the ribbon only has a near-flat degenerate edge band,such as (2,4)PNR and (4,2)PNR [6]. And some of thetypical ribbons with these two categories of edge states havealready been observed in experiments [37, 38]. However, theelectronic property may be significantly di ff erent from eachother between the two categories. And there are few studieson the PNRs with shuttle-shaped twofold-degenerate edgebands [27]. Therefore, it is essentially demanded to explorethe microscopic origin of the twofold-degenerate edge bands,especially the transport property. Meanwhile, the defects inPNR samples, such as monatomic vacancies, are inevitable inexperiments [39]. It is also important to understand the defecte ff ect on the electronic and transport properties [40].In this paper we select two sPNRs, (1,3)PNR and (2,4)PNR,as the exemplary ribbons belonging to the categories withtwofold-degenerate and near-flat degenerate edge bands,respectively. By using the KWANT software within theframework of tight-binding method, we find that the twoshuttle-shaped twofold-degenerate edge bands of (1,3)PNRare separated until the degeneracy is removed and a gap near E F is opened under a su ffi ciently strong in-plane electric field.And each shuttle contributed from the outmost atoms of theribbon upper or lower edge are identified according to thelocal density of states (LDOS). However, under a small o ff -plane field the shuttle-shape bands are easily separated intotwo degenerated near-flat bands contributed from the edgeatoms of the top or bottom sublayer. The edge band variationwith external field for this category is completely di ff erentfrom that of (2,4)PNR belonging to the previously reportedzPNR category. This is because a (1,3)PNR has four zigzagatomic configurations on the upper and lower edges and thedegenerate bands are from the outermost atoms in the samesublayer or di ff erent upper-lower edge. This allows the twoDOFs to be distinguished and regulated by applying electricfield along di ff erent directions. Further, based on this issue wepropose a (1,3)PNR homogenous junction attached by electricgates. Interestingly, the transport property of the junction withfield manipulation well reflects the characteristics of the twoDOFs for (1,3)PNR category. In addition, the defect e ff ect onthe transport property is also discussed. The conclusion is thatboth DOFs are robust against the defect in the center area ofthe ribbon, but the sublayer DOF is more e ff ective to resist theedge vacancy than the edge DOF. These results may provide afurther understanding on PNRs and initiate new developmentsin PNR-based electronics.The organization of the paper is as follows. First, weclassify the categories of the PNRs with edge states, and thenanalyze their edge atomic arrangements and unit cell choiceIn Sec. II, we present the model description and the detailson the calculations. Then in Sec. III, we demonstrate thethe edge state bands of the two exemplary sPNRs blong to (c) t t t t t a a a (2,4) T x ... N21 (b)
N2 0.53 ¯0.9 ¯1 ... (a) y (1,3) T a o yz FIG. 1: The schematic illustrations of (a) (1,3)PNR and (b)(2,4)PNR, where the red dashed line parallelogram in each ribbonindicates its minimum periodical supercell, the chiral vector T isillustrated by the red solid arrow along the edge, a and a denote theprimitive vectors, and t - t are the five e ff ective hoping parameters.The distance between atomic rows for the two sPNRs is 0.9 Å and0.53 Å, respectively. And the ribbon end side views in (a) and (b)indicate the top / bottom sublayer by the red / blue color. (c) The sketchof the p - n junction composed of a (1,3)PNR where the di ff erent colorshade in two sides implies the top and bottom back-gates whichadjust the Fermi level, and the green thick arrows indicate the in-plane transverse and o ff -plane vertical electric fields, respectively. di ff erent categories and the edge band variations under in- ando ff -plane electric fields, respectively. Hence according to thecoexistence of two (upper or lower edge and sublayer) DOFsin monolayer (1,3)PNR, with which a p - n junction is proposedand the the important transport property is discussed. Finally,the conclusion is briefly drawn in Sec. IV. II. MODEL DESCRIPTION AND METHOD
The schematic illustrations of the considered (1,3)PNR and(2,4)PNR are shown in Figs. 1(a) and 1(b), respectively. Thered dashed parallelograms are the minimum supercells, wherethe red solid arrows are the chiral vectors indicating the crystaldirection of the ribbon edges [6]. The integer N denotesthe number of atomic chains (blue dashed lines) across theribbon width. Further, for (1,3)PNR shown in Fig. 1(a), thered and blue outmost edge atoms are arranged alternately onthe two sublayers (see the side view), as a consequence thesupercell width must be the twice of | T | . In contrast, theoutmost edge atoms for (2,4)PNR are always located at onesublayer and the supercell width is equal to | T | . Moreover, asshown in Fig. 1(c), we propose a homogenous p - n junctionbased on a (1,3)PNR, where the blue and orange backgroundimply the positive and negative potentials provided by back-gate electric gates, respectively. And the thick green arrowsin upper and lower panels indicate the in-plane transverseelectric field along the y -direction and the o ff -plane verticalone along the z -direction, respectively. The stability of PNRshave been analyzed in our previous work [6]. Since the | T | of the two selected PNRs here are much less than 20Å, hencethey are considered relatively stable and do not appear obviousedge reconstruction.We use the KWANT code based on tight-binding (TB)Hamiltonian [41] to calculate the electronic energy bandfor the selected PNRs, and the atomistic quantum-transportsimulations for the proposed junction are based on thescattering-matrix method from matching wavefunctions [42–44]. In comparison with the first-principles calculation [33],this approach can treat large nanostructures matching theusual experimentally reachable sample size up to sub-100nm scales with better precision without large computationalconsumption.The TB model Hamiltonian for phosphorene [45] in thepresence of an electric field can be described as H = X < i , j > t i j c † i c j + ( el ε y / z + U i ) X i c † i c i , (1)where c † i ( c j ) is the creation (annihilation) operator of electronat site i ( j ), the summation h i , j i means over all the neighboringatomic sites with hopping integrals t i j , l is the component ofthe atomic position from the selected origin along the electricfield direction, ε y / z is the strength of the field along the y / z direction as shown in Fig. 1(c), and U i is the impurity-induced potential if available. It has been shown that fivehopping parameters [see Fig. 1(b)] are enough to describethe electronic band structure of phosphorene. The valuesof these hopping integrals suggested in the previous studies[45] are t = -1.220 eV, t = t = -0.205 eV, t = -0.105eV and t = -0.055 eV. Therefore, by solving the discreteSchr¨odinger equation corresponding to Hamiltonian (1) onthe proper basis for the supercell drawn by the red dashedparallelograms in Fig. 1 and applying the Bloch theorem,the k -dependent Hamiltonian for a PNR can be written as H ( k ) = H , + H , e ika + H † , e − ika in the form of ( N ′ × N ′ )dimensional matrix. Here N ′ is the number of atoms inthe supercell, H , is the matrix of the central cell, H , thecoupling matrix with the right-hand adjacent cell, and a is thelength between two nearest-neighbor cells. Diagonalizing this k -dependent Hamiltonian, we can obtain the band spectrumand the corresponding eigen-wavefunctions. Then we cancalculate the LDOS using the following formulaLDOS( E , r ) = c √ π X n | Ψ n ( r ) | e − ( En − E )22 c , (2)where c is broadening parameter, Ψ n ( r ) and E n are the eigen-wavefunction and eigenvalue, respectively, in which n denotesthe energy band index and r the atom position.In addition, we mention that the description of the TBmodel has been e ff ectively used for PNRs with di ff erent edgegeometries [5, 6, 20, 25, 27, 28]. We have verified the validityof TB approach by the first-principles calculation using theAtomistix Toolkit code [46]. The result shows that the edgebands can also be near the Fermi energy and the di ff erencebetween them is quantitative. As for the influence of spinpolarization, it is found that the total energy of (1,3)PNR for the spin polarization case is 8.82 meV lower than that of non-polarized case. And for the (2,4)PNR, the energy di ff erenceis only 0.04 meV. This is well conceivable that the few-meVenergy di ff erence will be mitigated by a reasonable finitetemperature (few tens of Kelvin), i.e., the spin-polarized stateswill transit into paramagnetic states in actual experiments. Sothat we do not consider the e ff ect of spin polarization on theresults for the present work.In calculating the conductance for the junction under anexternal electric field, it is divided into the left electrode, theright electrode, and the middle scattering region. The S matrixcan be obtained by matching the wavefunctions at the twointerfaces of electrode / scattering-region. Once the S matrix isobtained, the conductance of the system at zero temperaturecan be calculated by using the Landauer formula [47] G LR = e h T LR = e h X n ∈ L , m ∈ R | S mn | , (3)where L / R labels the left / right leads, T LR is the transmissioncoe ffi cient from lead L to lead R, and S mn gives the scatteringamplitude from an incoming mode n to an outgoing mode m ,both of which are the elements of the scattering matrix. It isclear that the conductance depends on the number of availabletransport modes through the junction. III. RESULTS AND DISCUSSIONSA. edge bands under in-plane electric field
The calculated energy spectra for the selected exemplary(1,3)PNR ( N =
35) and (2,4)PNR ( N =
49) with nearly the samewidth ∼ ff erentstrength are shown in Figs. 2(a-c) and 2(d-f), respectively,where the Fermi level E F is set to zero. The left and rightinserts in (b, c and f) indicate the LDOS correspondingto energies marked respectively by the red and blue pointscontributed by the ribbon upper or lower edge outermostatoms, where the side view for each supercell is embodied.First, as shown in Fig. 2(a), the band structure of the pristine(1,3)PNR has a shuttle-shaped edge band near E F , whichimplies the ribbon is metallic. In fact, the shuttle-shapededge bands are twofold-degenerate di ff erentiating by the reddashed and blue solid lines. However, the degeneracy can beeliminated by applying an in-plane electric field along the y -direction. As a small field with strength ε y = / Å is appliedshown in Fig. 2(b), the twofold-degenerated edge bandsobviously separate into two partially overlapped shuttles.Interestingly, as shown in Fig. 2(c), the two shuttles arecompletely separated as the filed strength is further increasedup to 5 mV / Å, which means the degeneracy is completelyremoved. In contrast, for the (2,4)PNR as the example ofthe another category, it has two near-flat degenerate edge statebands in the original band structure which are drawn by thered dashed and blue solid lines shown in Fig. 2(d). Theyalso pass through E F exhibiting metallicity, which is verysimilar to that of a conventional zPNR except for a little -1.5-1.0-0.50.00.51.0 (2,4)PNR N=49 - E n e r gy ( e V ) - (a) (1,3)PNR N=35 y =0 mV/¯ y =2 mV/¯ (1,3)PNR N=35 (b) y =5 mV/¯ (1,3)PNR N=35 (c) y =0 mV/¯ (d) y =2 mV/¯ (2,4)PNR N=49(2,4)PNR N=49 (e) y =5 mV/¯ kkkkk ----- (f) - k FIG. 2: The band structure for (a-c) (1,3)PNR with N =
35 and (d-f) (2,4)PNR with N =
49 under an in-plane electric field with di ff erent strengths,(a and d) ε y =
0, (b and e) 2 mV / Å and (c and f) 5 mV / Å, respectively. The Fermi energy E F is set to zero, and the left and right inserts in(b), (c) and (f) indicate the LDOS distributions corresponding to energies near E F marked by the red and blue points, respectively, where theside view for each supercell LDOS is embodied. The right-side attached color bar indicates the electron density from the lowest (blue) to thehighest (red) value. di ff erence in the curvature of the edge bands [15, 24, 48].However, much di ff erent from the twofold-degenerated bands,the degeneracy for this category of ribbon can be easilyremoved by a small electric field as shown in Figs. 2(e) and2(f). The characteristic of edge band under an electric fieldfor (2,4)PNR is very similar to the extensively studied zPNR[24, 28, 49], and hence we attribute them to the same category.An in-plane electric field removing the degeneracy of theedge bands can be understood from Hamiltonian (1). Theadditional diagonal terms increase (accumulate) linearly withthe y -coordinate. From the analysis of the calculated results,it can be seen that the degenerate edge state bands for bothcategories of sPNRs would be broken by the application of anin-plane electric field. As the field strength increases, a bandgap is opened and the transition from metal to semiconductoris occurred. The variation of (2,4)PNR edge bands withelectric field is basically the same as zPNR [24, 28, 49].This is because that they belong to the same category ofsPNRs, where the outermost atoms of the ribbon are all atthe same sublayer. However, for the (1,3)PNR with twofold-degenerate edge bands, the E F is embedded in a mirroredshuttle-shape bands. In order to further know which atoms inthe ribbon contribute the edge state bands, we have calculatedthe LDOS at the energies near E F marked by the red and bluepoints, respectively. When the two shuttle-shape bands are notcompletely separated shown in Fig. 2(b) for ε y = / Å, fromthe inserts we see the di ff erence in LDOS between energieson the two shuttles. The red / blue dashed / solid shuttle bandsare mainly contributed the outermost atoms at the upper / loweredge.In addition, the left and right inserts in Figs. 2(c)and 2(f) depict the real-space electronic distributions at theenergies of conduction band minimum (CBM) and valence band maximum (VBM) marked by the red and blue points,respectively. The right-side attached color bar indicates thedensity from the lowest (blue) to highest (red) value. Fromthe LDOS inserts we can identify whether an edge band iscompletely contributed by the outmost atoms of the upperor lower edge of the ribbon, which provides a (upper-lower)edge DOF. However, we cannot distinguish them from whichsublayers since their responses to an in-plane field are thesame. Therefore, we need to apply an o ff -plane verticalelectric field along the z -direction respectively to the twocategories of sPNRs. B. edge bands under o ff -plane electric field The di ff erence in the layer atomic arrangement on the edgeof a sPNR with di ff erent parity rows can be reflected by theresponse to an o ff -plane electric field, and then a ff ects thecorresponding band structure. This e ff ect has been revealedand discussed for other 2D material nanoribbons, such asgraphene, silicene and phosphorene ones [28, 50, 51]. Here,in Fig. 3 we present the result of the edge band response toa vertical electric field with strengths ε z = / Å forthe two exemplary ribbons, respectively. For the considerationof the even-odd parity [20, 28], we consider the two widthcases of N =
35, 36 and N =
49, 50 for (1,3)PNR and (2,4)PNR,respectively. The LDOS left / right insert corresponding to theenergies marked by the red / blue point. From Fig. 3(a) for(1,3)PNR with N =
35, the twofold-degenerate edge bands inshuttle-shape are almost unchanged under small ε z = / Å.However, when ε z is increased up to 0.4 V / Å, interestingly, thetwo shuttle-shape edge bands are separated into two nearlyflat bands as shown in Fig. 3(b). This results implying a -1.5-1.0-0.50.00.51.0 z =0.1 V/¯ (1,3)PNR N=35(1,3)PNR N=35 E n e r gy ( e V ) (a) z =0.1 V/¯ z =0.5 V/¯ kk (d)(c)(b) (1,3)PNR N=36 z =0.5 V/¯ (1,3)PNR N=36 k z =0.1 V/¯ z =0.5 V/¯ (2,4)PNR N=50 z =0.1 V/¯ z =0.5 V/¯ (e) (h)(g) (2,4)PNR N=49 k ---- (f) (2,4)PNR N=50(2,4)PNR N=49 - k - k - k - k FIG. 3: The band structures for (a-d) (1,3)PNR with odd ( N =
35) and even ( N =
36) parities under an o ff -plane vertical electric field withdi ff erent strengths, respectively, where (a and c) ε z = / Å and (b and d) 0.5 V / Å. The band structures for odd-even ( N = transition from metal to semiconductor phase. But the two-fold degenerate is still not removed. Further, from Figs. 3(a-d) it seems that the number of atom rows (parity) does nota ff ect the band structure of (1,3)PNR. In addition, the sideview in (b) shows that the left / right inserted LDOS comefrom the edge outermost atoms of the top / bottom sublayer.We can identify a band from the top or bottom sublayer,which provides a sublayer DOF. This is because that the edgeoutmost atoms are alternately located at the top and bottomsublayers. In the contrast, for (2,4)PNR belongs to anothercategory with two near-degenerate edge bands, the responseof the edge bands to the field is sensitive on the ribbon width(parity), which is the same as that for the conventional zPNRs[28]. As shown in Figs. 3(e) and 3(f), in specification, with theincrease of the field strength the edge band degeneracy of theodd-numbered ( N =
49) ribbon can not be broken but enterallymove upward a little. Since the degenerate edge bands arecontributed by the outermost atoms of both edges. However,for the even-numbered ( N =
50) ribbon shown in Figs. 3(g)and 3(h), the outermost atoms of the two edges come from thesame sublayer. Hence as the field strength increasing, the edgeband degeneracy is removed and the transition from metal tosemiconductor occurs.
C. (1,3)PNR homogenous junction
Next, from the above results for the two exemplary sPNRsbelonging to di ff erent categories, we know that a monolayer(1,3)PNR owns two DOFs of edge and sublayer, which isparticularly similar to a bilayer zPNR with two DOFs ofedge and layer [5]. Using this similarity, we may constructa homogenous p - n junction using a (1,3)PNR. The two sidesof a (1,3)PNR are attached by the near top and bottom back-gates which adjust the E F , leading to the lifting-up or -downof the edge bands for two ends. Further, the external in-plane and o ff -plane electric fields applied on the junction shown inFig. 1(c) can be realized by using a side-gate and anotherback-gate electrodes, respectively. The side-gate techniquehas been proven to be experimentally feasible in grapheneas a channel material for other applications [52, 53]. Theback-gates can arouse a potential di ff erence (electric field)across the whole monolayer. This kind of setup has beenrealized for bulk phosphorene transistors [54, 55]. And apseudospin field e ff ect transistor has also been proposed andcharacterized based on a bilayer zPNR-based junction, inwhich a pseudospin-polarized current is generated [5]. Thesimilar interesting e ff ect may also be realized in the p - n junction based on a monolayer (1,3)PNR, which can generatean edge- or sublayer-polarized current by properly adjustingthe gate electrodes.In Fig. 4(a), we show the calculated conductance spectrumfor the proposed junction based on (1,3)PNR with a width ∼ ff erentstrengths ε y = / Å (red),respectively. The solid / dashed line indicates the alignmentfor the edge bands from the same / di ff erent upper-lower edge.The separated edge bands from the two edges of the terminalswith / without alignment by back-gates is shown in Fig. 4(c),where the upper / lower panel corresponds to the case of thesolid / dashed line in 4(a), and the green / red arrows impliesthe switch on / o ff state. First, we find a conductance plateauof 2 e / h around E F for the junction without external fieldshown by the grey solid line, which is in accord with theband structure for a pristine (1,3)PNR shown in Fig. 2(a).When a small field 0.5 mV / Å is applied, by adjusting thepositive and negative voltages of the two back-gate as to theedge bands from the same upper or lower edge be aligned, theconductance still maintains a 2 e / h plateau (blue solid line)near E F , whereas the conductance is decreased to 0 . e / h from di ff erent edge as shown by the blue dot-dashed line.This is because that the band degeneracy is broken not enough RightLeft off on Right on off Left
Energy (eV)
0 0.5 mV/¯ 0.5 mV/¯ 5 mV/¯ 5 mV/¯ C ondu c t a n ce ( e / h ) (a) in-plane electric field off-plane electric field (d)(c)
0 0.1 V/¯ 0.1 V/¯ 0.5 V/¯ 0.5 V/¯ (b)
FIG. 4: The conductance spectrum for the (1,3)PNR-based p - n junction with a width about 3 nm under (a) in-plane transverse and(b) o ff -plane vertical electric fields, respectively. The gray solidline implies the conductance without applied field. The blue andred solid (dashed) lines indicate the edge bands from the same(di ff erent) (a) upper-lower edge with field strength 0.5 and 0.5 mV / Å,(b) top-bottom sublayer with field strength 0.1 and 0.5 V / Å arealigned, respectively. (c) and (d) The alignment of the edge bandsof the two semi-infinitive ribbons under transverse and vertical fields,respectively, where the green / red arrow implies the on / o ff state of thejunction. so that the two shuttle-shaped edge bands are not completelyseparated. Further, as the field strength increases up to5 mV / Å, the degenerate shuttle-shaped edge bands is fullyremoved and a significant band gap is opened around E F [seeFig. 2(c)]. As a consequence, when the same edge bands arealigned as shown in the upper panel in 4(c), a conductancegap correspondingly appears with two 2 e / h plateaus besideit (red solid line). Otherwise the conductance is almost zeroeven though there are two shuttle-shaped edge bands from thedi ff erent upper or lower edge are aligned as shown in the lowerpanel in 4(c), which shows a transport o ff state.Since the speciality of the edge atom arrangement for themonolayer (1,3)PNR, sublayer is also an important DOF inits transport property. In Fig. 4(b), we show the conductancespectrum for the junction under an o ff -plane vertical electricfield with strengths, 0 (gray), 0.1 (blue) and 0.5 V / Å (red),respectively. Here the solid / dashed lines indicates the twoedge bands from the same / di ff erent sublayers are aligned, ofwhich the diagrammatic sketch for the edge band alignmentis shown in 4(d). First, we also find a conductance plateau of2 e / h around E F for the junction without external field shownby the grey solid line. In the presence of applied field ε z = / Å, there is a gap opened within the shuttle-shaped edge in-plane electric field -12 -12 -12 y (mV/¯) (a) Left Right -12 L * ( n , E ) R ( n ’ , E ) RightLeft (b) z (V/¯) off-plane electric field FIG. 5: The wavefunction overlap, < Ψ ∗ L ( n , E ) | Ψ R ( n ′ , E ) > , as afunction of (a) ε y and (b) ε z for the two terminals of the junction,where n ( n ′ ) is the band index number for the junction left (right)side. And the LDOS diagrams with certain energies at twodi ff erent field strengths pointed by black arrows are inserted withthe embodied side view of supercell. bands. Therefore, as the edge bands from the same upperor lower edge are adjusted (by back-gate) to be aligned, theconductance exhibits two steps of 2 e / h besides E F (see theblue solid line), which is in accord with the band structureshown in Fig. 3(b) or 3(d). In this case the increase of the fieldstrength (e.g., ε z = / Å) only result in the two narrowersteps and more far away from E F (see the res solid line). Onthe contrary, when the edge bands from the same upper orlower edge are not aligned, the conductance is nearly zeroexcept for very small peaks at E F as shown by the blue andred dashed lines. The reason for the results is that the edgeoutermost atoms of (1,3)PNR are alternately located at thetop and bottom sublayers. Therefore, a small vertical electricfield can results in wave function superposition in di ff erentsublayers.In order to further understand the rule of the two DOFs intransport for the junction, in Fig. 5 we plot the dependenceof wavefunction overlap, < Ψ ∗ L ( n , E ) Ψ R ( n ′ , E ) > , as a functionof the field strength (a) ε y and (b) ε z . Here Ψ L / R ( n , E ) is thewavefunction in the left / right (L / R) side of the junction. TheLDOS with certain energies at two di ff erent field strengths arealso inserted with the embodied side view of supercell. FromFig. 5, we first note that the overlap value drops rapidly as thefield strength increasing due to the degeneracy elimination,with which the edge bands from the di ff erent upper-loweredge or top-bottom sublayer are separated. Therefore, withoutapplied field the wavefunction overlap is equal to 1. In thiscase the edge bands are completely superposed, which refersto its pristine state shown in Fig. 2(a). Meanwhile, byadjusting the two back-gates as to the two side edge bandsfrom the same edge or sublayer aligned as shown in upperpanels of Fig. 4(c) or 4(d), the overlap degree is also equal to1 regardless of the electric field strength.Further, as shown in Fig. 5(a), we find that the two shuttle-shaped edge bands from the upper or lower edge are alignedby adjusting the back-gates as to the potential di ff erencebetween the junction two sides is equal to the electric fieldstrength [see the lower panel in Fig. 4(c)], the wavefunctionoverlap for two terminals is greatly reduced. When the twoshuttle-shape edge bands are just separated a little by a smallfield 0.5 mV / Å, the wavefunction overlap is greatly reducedto a small value, which corresponds to the blue dashed peaksnear E F in Fig. 4(a). This is because that the overalldegeneracy of the two shuttle bands would be almost removedeven under a small electric field, but some of the degeneracy isstill remained. Therefore, at certain energies the superpositionof the wavefunctions in two sides still has a definitive value.In this case, from the LDOS shown in left upper panel ofFig. 5(a)], we can see it distributes at the upper and loweredge outermost atoms. Specifically, the upper edge atomsin the left side account for the major contribution (bright redspots), while the reddish ones at lower edge indicate a partialcontribution. On the contrast, the LDOS distributed are theupper-lower edge for the right side is opposite to the leftone. As the field strength is increased up to 5 mV / Å, thewavefunctions overlap for two side approaches zero (as reddashed line shown in Fig. 4(a)). This is because that the twoshuttle edge bands are completely separated, and electrons arelocalized only at the outmost atoms of one edge. This can alsobe seen from the LDOS inserts with the embodied side viewof supercell. On the other hand, since the edge mostout atomsof (1,3)PNR are alternately arranged on the top and bottomsublayers, the wavefunctions of the top and bottom sublayeratoms still mix together when a large vertical electric fieldis applied as shown in the lower insert of Fig. 5(b). Hencethe amplitude scale of y -axis in Fig. 5(b) is much larger thanthat in 5(a). Therefore, when ε z = / Å the wavefunctionoverlap for two sides is still large, which corresponds to theblue dashed plateau near E F in Fig. 4(b). When the verticalfield becomes larger, the wavefunction overlap of the topand bottom sublayers tends to zero due to the localization ofelectrons, as shown by the lower insert on Fig. 5(b) for LDOSwith 0.5 V / Å.Finally, we demonstrate the influence of defect on thetransport, which is also very important for device applicationby using the proposed DOFS since defects are inevitable onPNR samples in experiments [39]. For the sake of argumenton the two DOFs in a (1,3)PNR, here we mainly consider twokinds of monatomic vacancies. As shown in the top of Fig.6 for a defective (1,3)PNR, there are two monatomic vacancypositions indicated by the black dotted circles, one at the edge(EV) and the other in the center (CV) of the ribbon. Figure 6gives the conductance spectrum, where (a, c) and (b, d) for CVand EV cases with ε y = / Å and ε z = / Å, respectively.As a reference, the red dashed lines in (a, c) indicate theconductances for a pristine (1,3)PNR without defect but within- and o ff -plane fields, respectively, which can be refer tothe corresponding energy spectrum shown in Figs. 2 and 3.As is seen from the conductance spectra shown by the bluesolid lines, a CV does not a ff ect the edge bands regardlessthe field direction, resulting in an unchanged conductance in Energy (eV) y =5 mV/¯ (CV) y =5 mV/¯ (pristine) (a) EV (1,3)PNR with defect (d)(c)(b) y =5 mV/ ¯ (EV) z =0.1 V/¯ (CV) z =0.1 V/¯ (pristine) C ondu c t a n ce ( e / h ) z =0.1 V/¯ (EV) FIG. 6: The conductance spectrum of a (1,3)PNR with monatomicvacancy respectively located at the (a, c) center and (b, d) edge ofthe ribbon, where (a, c) with ε y = / Å and (c, d) ε z = / Å. Thered dashed line in (a) and (c) corresponds to the pristine case withoutdefect. comparison the pristine case shown in (a, c). But the situationfor EV is much di ff erent. Comparing with the conductancespectrum shown by the red solid line in Fig. 4(a) for the sameupper-lower edge are aligned (also shown by red dashed linein Fig. 6(a)). As shown in Fig. 6(b), we find that the edge stateon one of edges with an EV is severely weakened, which resultin that the conductance plateau on the right side of E F almostdisappears. The original conductance plateau is maintained.In addition, for a EV with ε z = / Å shown by the bluesolid line in Fig. 6(d), the conductance steps are retained asthe ideal case [see the blue solid line in Fig. 4(b) and the reddashed line in Fig. 6(c)]. But the conductance peak on the leftside of E F is reduced by half, while the right side peak is stillclose to 2 e / h . This importantly implies that the CV has noe ff ect on the edge state. The same result is expected for the in-plane field case. Moreover, due to the particularity of the twoDOFs, the e ff ect of EV on edge-state contributed conductanceis di ff erent under the regulation of electric field. As shownby the LDOS in Fig. 5(a), a very small in-plane field canmake the outer electrons of the edge atoms localized, whichprovides the transport channel corresponding to the edge withan EV almost forbidden. On the other hand, a large o ff -planefield would cause a certain degree of wavefunction overlapbetween the sublayer atoms on the same side [as shown inFig. 5(b)], so that an EV in one sublayer only destroysthe edge state of the corresponding sublayer, while the othersublayer on the same side still maintains a channel. In general,the sPNRs with two-DOFs can reduce the influence of low-concentration edge monatomic defects on the edge states to acertain extent.At last but not least, besides the atomic defects the edgepassivation of nanoribbons is also an important issue becausethat the edges are usually passivated in real experimentalsamples. When edge hydrogen(H)-passivation is consideredin the TB calculation, usually an extra potential field is appliedon the edge atoms of the ribbons. This may lead to that theoriginal edge bands are disappeared or modified (see, e.g.,[20]). Here we have also tested the H- and O-passivationcases by using a DFT calculation, respectively. The resultshows that the H-passivated edges do not exhibit edge states,while O-passivated ones remain qualitatively similar to the thecase of bare edges. And the result of the edge bands is thequalitatively the same as those by the TB approach, whichconfirms that the device assumption based on (1,3)PNR isscientifically valuable [6]. IV. SUMMARY AND CONCLUSION
In summary, we have studied the di ff erence in electronicstructures between the two categories of sPNRs with edgestates and the variation under an external electrostatic field.Taking (1,3)PNR and (2,4)PNR as the examples, we firstidentify the distinction of edge morphology and the unit cell selection for these two ribbons. And then, by usingthe KWANT code based on TB approach, we find that theshuttle-shaped twofold-degenerate edge bands of (1,3)PNRcan be became two separated shuttles until the degeneracyis removed and a gap near E F is opened under a su ffi cientstrong in-plane electric field. And each shuttle contributedfrom the outmost atoms of the ribbon upper or lower edge isverified according to the LDOS. However, under a small o ff -plane field the shuttle-shape bands are easily separated intotwo degenerated near-flat bands contributed from the edgeatoms of the top or bottom sublayer. The edge band variationwith external field for this category is completely di ff erentfrom that of (2,4)PNR belonging to previously reported zPNRcategory. This is because a (1,3)PNR has four zigzagatomic configurations on the upper and lower edges and thedegenerate bands are from the outermost atoms in the samesublayer or di ff erent upper-lower edge. This allows the twoDOFs to be distinguished and regulated by applying electricfield along di ff erent directions. Further, based on this issue wepropose a (1,3)PNR homogenous junction attached by electricgates. Interestingly, the transport property of the junctionwith field manipulation well reflects the characteristics of thetwo DOFs for (1,3)PNR category. In addition, the defecte ff ect from the vacancies in edge and bulk on the transportproperty is also discussed. These results may provide a furtherunderstanding on PNRs and initiate new developments inPNR-based electronics. Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (Grant Nos. 11774085, 11804092, and11664019), the Project Funded by China Postdoctoral ScienceFoundation (Grant Nos. BX20180097, 2019M652777), andHunan Provincial Natural Science Foundation of China (GrantNo. 2019JJ40187). [1] 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).[2] K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, Phys. Rev.Lett. , 136805 (2010).[3] P. San-Jose, E. Prada, E. McCann, and H. Schomerus, Phys.Rev. Lett. , 247204 (2009).[4] X. Xu, W. Yao, D. Xiao, and T. F. Heinz, Nat. Phys. , 343(2014).[5] S. Soleimanikahnoj, and I. Knezevic, Phys. Rev. Appl. ,064021 (2017).[6] Yi Ren, Pu Liu, Benliang Zhou, Xiaoying Zhou, and GuanghuiZhou, Phys. Rev. Appl. , 064025 (2019).[7] Likai Li, Yijun Yu, Guo Jun Ye, Qingqin Ge, Xuedong Ou, HuaWu, Donglai Feng, Xian Hui Chen, and Yuanbo Zhang, Nat.Nanotech. , 372 (2014).[8] Xi Ling, Han Wang, Shengxi Huang, Fengnian Xia, andMildred S. Dresselhaus, PNAS , 4523 (2015). [9] C. Q. Han, M. Y. Yao, X. X. Bai, Lin Miao, Fengfeng Zhu, D.D. Guan, Shun Wang, C. L. Gao, Canhua Liu, Dong Qian, Y.Liu, and Jin-feng Jia, Phys. Rev. B , 085101 (2014).[10] Likai Li, Jonghwan Kim, Chenhao Jin, Guojun Ye, Diana Y.Qiu, Felipe H. da Jornada, Zhiwen Shi, Long Chen, ZuochengZhang, Fangyuan Yang, Kenji Watanabe, Takashi Taniguchi,Wencai Ren, Steven G. Louie, Xianhui Chen, Yuanbo Zhang,and Feng Wang, Nat. Nanotech. , 21 (2017).[11] Steven P. Koenig, Rostislav A. Doganov, Hennrik Schmidt, A.H. Castro Neto, and Barbaros ¨Ozyilmaz, Appl. Phys. Lett. ,103106 (2014).[12] Michele Buscema, Dirk J. Groenendijk, Sofya I. Blanter, GaryA. Steele, Herre S. J. van der Zant, and Andres Castellanos-Gomez, Nano Lett. , 3347 (2014).[13] Fengnian Xia, Han Wang, and Yichen Jia, Nat. Commun. ,4458 (2014).[14] Sherman Jun Rong Tan, Ibrahim Abdelwahab, Leiqiang Chu,Sock Mui Poh, Yanpeng Liu, Jiong Lu, Wei Chen, and Kian Ping Loh, Adv. Mater. , 1704619 (2018).[15] A. Carvalho, A. S. Rodin, and A. H. Castro Neto, Europhys.Lett. , 47005 (2014).[16] Han Liu, Adam T. Neal, Zhen Zhu, Zhe Luo, Xianfan Xu,David Tomanek, and Peide D. Ye, ACS Nano , 4033 (2014).[17] Pengke Li and Ian Appelbaum, Phys. Rev. B , 115439 (2014).[18] Xiaoying Zhou, Wen-Kai Lou, Dong Zhang, Fang Cheng,Guanghui Zhou, and Kai Chang, Phys. Rev. B , 045408(2017).[19] X. Y. Zhou, R. Zhang, J. P. Sun, Y. L. Zou, D. Zhang, W. K.Lou, F. Cheng, G. H. Zhou, F. Zhai, and Kai Chang, Sci. Rep. , 12295 (2015).[20] Pu Liu, Xianzhe Zhu, Xiaoying Zhou, Guanghui Zhou, and KaiChang, Sci. China-Phys. Mech. Astron. 64, 217811 (2021).[21] Paul Masih Das, Gopinath Danda, Andrew Cupo, William M.Parkin, Liangbo Liang, Neerav Kharche, Xi Ling, ShengxiHuang, Mildred S. Dresselhaus, Vincent Meunier, and MarijaDrndic, ACS Nano , 5687, (2016).[22] Mitchell C. Watts, Loren Picco, Freddie S. Russell-Pavier,Patrick L. Cullen1, thomas S. Miller, Szymon P. Bartu´s, OliverD. Payton, Neal T. Skipper, Vasiliki tileli, and Christopher A.Howard, Nature , 216 (2019).[23] Vy Tran and Li Yang, Phys. Rev. B , 245407 (2014).[24] M. Ezawa, New J. Phys. , 115004 (2014).[25] Rui Zhang, X. Y. Zhou, D. Zhang, W. K. Lou, F Zhai, and KaiChang, 2D Mater. , 045012 (2015).[26] Ajanta Maity, Akansha Singh, Prasenjit Sen, Aniruddha Kibey,Anjali Kshirsagar, and Dilip G. Kanhere, Phys. Rev. B ,075422 (2016).[27] Marko M. Gruji´c, Motohiko Ezawa, Milan ˇZ. Tadi´c, andFrancois M. Peeters, Phys. Rev. B , 245413 (2016)[28] Benliang Zhou, Benhu Zhou, Xiaoying Zhou, and GuanghuiZhou, J. Phys. D: Appl. Phys. , 045106 (2017).[29] K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus,Phys. Rev. B , 17954 (1996).[30] M. V. Bollinger, J. V. Lauritsen, K. W. Jacobsen, J. K. Nørskov,S. Helveg, and F. Besenbacher, Phys. Rev. Lett. , 196803(2001).[31] D. J. P. de Sousa, L. V. de Castro, D. R. da Costa, and J. M.Pereira, Jr. Phys. Rev. B , 235415 (2016).[32] SKFiroz Islam, Paramita Dutta, A. M. Jayannavar, and ArijitSaha, Phys. Rev. B , 235424 (2018).[33] Yi Ren, Fang Cheng, Z. H. Zhang, and Guanghui Zhou, Sci.Rep. , 2932 (2018).[34] E. Taghizadeh Sisakht, F. Fazileh, M. H. Zare, M. Zarenia, andF. M. Peeters, Phys. Rev. B , 085417 (2016). [35] Ashwin Ramasubramaniam and Andre R. Muniz, Phys. Rev. B , 085424 (2014).[36] Yuanyue Liu, Fangbo Xu, Ziang Zhang, Evgeni S. Penev, andBoris I. Yakobson, Nano Lett. , 6782 (2014).[37] Yangjin Lee, Sol Lee, Jun-Yeong Yoon, Jinwoo Cheon, HuYoung Jeong, and Kwanpyo Kim, Nano Lett. , 559 (2020).[38] Liangbo Liang, Jun Wang, Wenzhi Lin, Bobby G. Sumpter,Vincent Meunier, and Minghu Pan, Nano Lett. , 6400 (2014).[39] B. Kiraly, N. Hauptmann, A. N. Rudenko, M. I. Katsnelson,and A. A. Khajetoorians, Nano Lett. , 3607 (2017).[40] L. L. Li, and F. M. Peeters, Phys. Rev. B , 075414 (2018).[41] C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal,New J. Phys. , 063065 (2014).[42] Yongjin Jiang and Liangbin Hu, Phys. Rev. B , 195343(2007).[43] M. Zwierzycki, P. A. Khomyakov, A. A. Starikov, K. Xia, M.Talanana, P. X. Xu, V. M. Karpan, I. Marushchenko, I. Turek,G. E. W. Bauer, G. Brocks, and P. J. Kelly, Phys. Stat. Sol. B , 623 (2008).[44] Tatiane P. Santos, Leandro R. F. Lima, Caio H. Lewenkopf, J.Comput. Phys. , 440 (2019).[45] A. N. Rudenko and M. I. Katsnelson, Phys. Rev. B ,201408(R) (2014).[46] S. Smidstrup, T. Markussen, P. Vancraeyveld, J. Wellendor ff , J.Schneider, T. Gunst, B. Verstichel, D. Stradi, P. A. Khomyakov,and U. G. Vej-Hansen, J. Phys.: Condens. Matter 32, 015901(2020).[47] S. Datta, Electronic transport in mesoscopic systems (Cam-bridge University Press, Cambridge, 1995).[48] Hongyan Guo, Ning Lu, Jun Dai, Xiaojun Wu, and Xiao ChengZeng, J. Phys. Chem. C, , 14051 (2014).[49] Dong Zhang, Wenkai Lou, Maosheng Miao, Shou-chengZhang, and Kai Chang, Phys. Rev. Lett. , 156402 (2013).[50] Z. Li, H. Qian, J. Wu, B. L. Gu, and W. Duan, Phys. Rev. Lett. , 206802 (2008).[51] Jun Kang, Fengmin Wu, and Jingbo Li, Appl. Phys. Lett. ,233122 (2012).[52] B. H¨ahnlein, B. H¨undel, J. Pezoldt, H. T¨opfer, R. Granzner, andF. Schwierz, Appl. Phys. Lett. , 093504 (2012).[53] F. Molitor, J. Guttinger, C. Stampfer, D. Graf, T. Ihn, and K.Ensslin, Phys. Rev. B , 245426 (2007).[54] V. Tayari, N. Hemsworth, O. Cyr-Choini`ere, W. Dickerson, G.Gervais, and T. Szkopek, Phys. Rev. Appl. , 064004 (2016).[55] J. S. Kim, P. J. Jeon, J. Lee, K. Choi, H. S. Lee, Y. Cho, Y. T.Lee, D. K. Hwang, and S. Im, Nano Lett.15