Design of graphene waveguide: Effect of edge orientation and waveguide configuration
Nayyar Abbas Shah, Vahid Mosallanejad, Kuei-Lin Chiu, Guo-ping Guo
DDesign of graphene waveguide: Effect of edge orientation andwaveguide configuration
Nayyar Abbas Shah, Vahid Mosallanejad, ∗ Kuei-Lin Chiu, † and Guo-ping Guo ‡ CAS Key Laboratory of Quantum Information,and Synergetic Innovation Center of Quantum Information and Quantum Physics,University of Science and Technology of China,Chinese Academy of Sciences, Hefei 230026, China Shenzhen Institute for Quantum Science and Engineering,Southern University of Science and Technology, Shenzhen 518055, China (Dated: June 13, 2019)Electron transport in a graphene quantum well can be analogous to photon trans-mission in an optical fiber. In this work, we present a detailed theoretical analysisto study the transport characteristics of graphene waveguides under the influence ofdifferent edge orientations. Non-equilibrium Green’s function approach in combina-tion with tight-binding Hamiltonian has been utilized to investigate the conductanceproperties of straight armchair and zigzag oriented graphene waveguides. Conduc-tance plateaus at integer steps of 4 e /h have been observed in both orientations whilethe zigzag oriented waveguides present a wider first quantized plateau compared tothat in the armchair oriented ones. Using various geometric and physical parameters,including side-barrier and waveguide width, and the metallic properties of terminals,we investigate the conductance profile of waveguides. In addition to the observationof valley-symmetry in both edge orientations, this article explores the critical influ-ence of drain contacts on waveguide conductance. Furthermore, we extended ourtransport study to three different highly bent waveguide configurations, such as U-shape, L-shape and split-shape waveguides, in order to explore their applications ingraphene-based ballistic integrated circuit devices. In the end, we also calculated theconductance of larger graphene waveguides using the scalable tight-binding model,in order to compare the results obtained from the original model. a r X i v : . [ c ond - m a t . m e s - h a ll ] J un I. INTRODUCTION
Ballistic transport and coherent conductance quantization are the key elements for en-gineering sophisticated nanoelectronic devices in new classes of materials [1–7]. Physicallytailored graphene channels with widths less than 50 nm, often noted as graphene nanoribbons(GNRs), provide an opportunity to manipulate the electrical properties of the intrinsicallygapless crystal [8–11]. Electronic properties and stability of GNRs have been investigatedfor realistic applications such as transistors, filters and polarizers [12–16]. The two well-known edge configurations, i.e., armchair and zigzag, result in two distinct forms of GNRs(commonly abbreviated by AGNRs and ZGNRs) [17, 18]. Transport properties in these twostructures are different in many aspects, such as the spacing between conductance plateaus.Although ideal GNRs should possess the quantization of conductance, unavoidable disorderson the edges have become dominant sources of incoherent scattering, making the quanti-zation of conductance hardly visible in plasma-etched GNRs [19–23]. To date, only fewinvestigations into conductance quantization in GNRs fabricated using shadow mask oxy-gen plasma etching exist [24, 25]. Further improvement is now incorporated into the designof graphene point contacts and GNRs by using hexagonal-born-nitride as bottom and topdielectrics to reduce substrate disorders [26–28]. However, the pronounced quantization ofconductance (mostly appearing as kinks) is not easily accessible due to the hypersensitivityof the system to edge disorders [29, 30]. On the other hand, charge carriers in graphenerevealed phenomena such as refraction, reflection and Fabry-P´erot interference that canbe analogous to electromagnetic phenomena [31–33]. It has also recently been shown thatthe long phase coherence length in graphene embedded in van der Waals heterostructuresprovides unique opportunities to observe electron interference and other peculiar electrontransmission states such as the snake states [34–36]. The optics-like phenomena of electronsin graphene enables the design of all graphene electronic devices resembling an optical fiber,which effectively works as an electron waveguide [37–39]. When a uniform potential wellis imposed across a graphene flake, the induced 1D quantum confinement in 2D electrongas results in straight graphene waveguides which have been explored both theoreticallyand experimentally with middle-scale (sub-micron size) and large-scale (micron size) ge-ometries [40–45]. In line with the aforementioned theoretical studies, we have previouslydemonstrated that the quantization of conductance can be achieved in straight and bentarmchair graphene waveguides by using Non-equilibrium Green’s function (NEGF) calcula-tion and proper design of contacts [46–48]. Recent work in the field studies also suggeststhat the connection between the external electrodes and the ribbon scattering area playsan important role in the conductance of GNRs [49, 50]. Since AGNRs and ZGNRs havevery different transport properties, we aim to address the question: what are the differencesin transport between armchair-oriented and zigzag-oriented graphene waveguides (abbrevi-ated as AO-GWs and ZO-GWs, respectively) with similar sizes? Our study includes twomain parts. Firstly, we present a theoretical comparison between transport in straight AO-GWs and ZO-GWs. Secondly, we investigate the transmission characteristics of graphenewaveguides with different geometries (L-shape, U-shape and split-shape), which had beenpreviously studied in tailored graphene systems [51, 52]. We organize this article in the fol-lowing way: the geometry of AO-GW and ZO-GW and the details of our model are presentedin section II. Conductance and local density of state are compared for straight AO-GW andZO-GW in the first part of section III, where the corresponding quasi-one dimensional bandstructures for slices of waveguides are calculated for reference. Furthermore, the effect ofgeometrical parameters such as the widths of side-barriers, waveguide (potential well) andterminals were investigated. Similar transport studies were also carried out for L-shape,U-shape and split graphene waveguides. The results are presented in the second part ofsection III. In addition, the scalable tight-binding method has been utilized to examine thequantization of conductance for larger graphene waveguides in the last part of section III.Finally, we will provide conclusive remarks about all waveguide configurations in section IV.
II. DEVICE DESCRIPTION AND METHODOLOGY
Fig. 1 illustrates the geometry of our devices. Middle-size strips of graphene with widthW and length L are considered as the scattering area, where the armchair and zigzag edgesare distributed along the horizontal (x-axis) and vertical (y-axis) directions, respectively.We introduce an external rectangular gate to induce a spatially varied atomic on-site energyin the graphene strip, which divides the scattering area into a centrally located region ofwaveguide and two side-barriers. In this way, two distinct edge orientations for graphenewaveguide (AO-GW and ZO-GW) can be created as shown in Figs. 1(a) and 1(b), respec-tively. W G (W SB ) represents the width of waveguide (side-barrier) with fixed on-site energy SBW LW S W D W S W D (a) (b)xy -1 -0.8 -0.6 -0.4 -0.2 0 W G W SB W SB U SB U WG U(eV) U N ch =3 - A G N R st , 2 nd , 3 rd NN (c) (d) FWHM Δ W U SB U WG AO-GWSBSB Z O - G W S B S B S D
FIG. 1: (Color online) Schematic diagrams of graphene waveguides. (a) Armchair oriented waveg-uide (AO-GW). (b) Zigzag oriented waveguide (ZO-GW). SB indicates the side-barrier. (c) Thecross section of ZO-GW showing the smooth variation of the on-site potential energy. The scale ofon-site potential at each atomic site is indicated by different color. The potential profile (U) acrossthe x-axis is shown underneath, which ranged from U
W G on the bottom of the waveguide to U SB on the side-barriers. (d) An example of N A -GNR with N A = 9 together with a small scattering areawith N ch = 3 to show the different tight-binding approximations with 1st, 2nd, and 3rd nearestneighbors. U W G (U SB ), in which we have considered the full width at half maximum (FWHM) account-ing for the smoothed on-site energy as shown in Fig. 1(c). Note that the potential energyon the atomic sites is indicated by color in Figs. 1(a) and 1(b), and can be referred to thecolor bar shown in Fig. 1(c). Each graphene waveguide contains two fundamental parts:the scattering area and leads (the areas that stick out from the scattering area). We use thenotation N A -AGNR to label the central scattering area. N A stands for the number of dimerlines and is defined as N A = 1 + (cid:98) W/ (0 . √ a cc ) (cid:99) , in which W is the width of AGNR anda cc = 0 .
142 nm is the carbon-carbon bond length. The length of the scattering area (L) isrelated with the chain number ( N ch ) via N ch = (cid:98) L/ (3 a cc ) (cid:99) (note that each chain contains2N A atoms). Parameters N A and N ch are two essential inputs to build the scattering area.The second part of the device is contacts (source and drain) which are also made ofcarbon and are in fact finite-width GNRs attached to the scattering area, as illustrated bythe extended GNRs sticking out of the rectangular region of W × L in Figs. 1(a) and 1(b).The width of source (drain) in both orientations is labeled by W S (W D ) and is also related approx. (cid:15) ( eV ) t ( eV ) t ( eV ) t ( eV ) s ( eV ) s ( eV ) s ( eV )1st 0 -2.74 0 0 0 0 03rd -0.36 -2.78 -0.12 -0.068 0.106 0.001 0.003TABLE I: Hopping energies and overlap integral values for the 1st (first row) and the 3rd (secondrow) nearest neighbor tight-binding approximations [54, 55]. with the number of dimer lines in source (drain) by N eS (N eD ), where the first index (e = a,z) stands for the edge orientation. The orientation of scattering area is kept unchanged,whereas the position of the leads and the edge orientation of the waveguide are differentfor ZO-GW and AO-GW (see Figs. 1(a) and 1(b)). It can be assumed that wider leads(as compare to W G ) provide denser subbands and consequently higher density of state forcarriers to get in and out of the waveguide. On the other hand, wider leads may also provideextra paths for carriers to go through the side-barriers instead of the waveguide and thus theinterference may demolish the coherent transmission from source to drain [48, 53]. Thus, inmost configurations discussed in this report W S, D is equal to W G unless otherwise stated.Moreover, our previous studies have shown that a metallic AGNR is a better choice tomake an ideal contact to armchair oriented graphene waveguide [48, 53] . Indeed, the zero-energy modes in metallic AGNRs permit the low energy electrons from the source to beinjected into the waveguide region. The advantage of using metallic GNRs as leads reflectsitself as an early onset of the first conductance plateau around the Dirac point. Thus, wemay modify N aS, aD by 1 or 2 to yield a number of dimer lines of N aS, aD = 3 m +2 ( m isan integer), which is the condition for building metallic AGNRs. On the other hand, idealZGNR leads (with an even number of atoms in the unit cell) connecting to ZO-GWs donot need any modification, because they naturally have zero-energy modes. Source leadshave the same on-site energy as in the guiding region while the drain leads are grounded(zero on-site energy) in all examples. Tight-binding Hamiltonian of a graphene device canbe expressed as: H = (cid:88) i µ i c † i c i + (cid:88) i,j t i,j ( c † i c j ) , (1)where c † i ( c i ) is the creation (annihilation) operator and µ i , indicates the on-site energy at thei-th atomic site. The on-site energy can be tuned through the external gate potentials and isdescribed by U as depicted in Fig. 1(c). Hopping between the nearest neighbors (e.g., i andj sites) is the origin of second term where t i,j denotes a fixed energy value based on tight-binding approximations, as in Table I [54, 55]. A small size scattering area with N ch = 3is shown in Fig. 1(d) in which the 1st, 2nd, and 3rd order tight-binding approximationsare indicated by green, blue and red circles, respectively. Following the Landauer-B¨uttikerformalism, conductance of a two-terminal device in low-temperature and low-bias can beexpressed as G = G T , where G = 2 e /h represents the quanta of conductance and T isthe transmission coefficient. Spin degree of freedom is included by the factor 2 in G while e and h are the electron charge and Planck’s constant [56]. The source-to-drain transmissioncoefficient T can be calculated using the Caroli’s formula [57]: T = trace (Γ s G r Γ d G a ) , (2)where Γ s (Γ d ) is the broadening matrix of the source (drain) lead. G r ( G a = G r † ) representsretarded (advanced) Green’s function given by G r ( E ) = [( E + iη ) S − H − Σ s ( E ) − Σ d ( E )] − , (3)where η is a small infinitesimal number usually about 10 − . Here, S is the overlap matrix built in a similar way to the second term in Eq. (1), and takes the form S = (cid:88) i,j s i,j ( c † i c j ) , (4)where s i,j represents the overlap integral between atomic orbitals (p z ) located at i and j .It is worth noting that orbitals at two different atomic sites are not necessarily orthogonalto each other. Therefore, non-zero values exist on the S matrix if the third (3 rd ) nearestapproximation is considered (see Table I). However, these values are small due to the long-distance interactions between atomic orbitals. The open boundary condition at the sourceand drain is incorporated into the transport study via the last two terms in Eq. (3), whichare the so called self-energy terms. Self-energy matrices are calculated via Σ s = A † s g s A s and Σ d = A d g d A † d , in which A s, d are given by A s, d ( E ) = [( E + iη ) S s S , S d − H s S , S d ] . (5)Here, H s S and S s S are the interaction Hamiltonian and interaction overlap matrices betweenthe source and the first super cell in the scattering area, while H S d and S S d are the interactionHamiltonian and interaction overlap matrices between the last supercell in the scatteringarea and drain lead (index S refers to the scattering area whereas s and d denote thesource and drain). In the process of building H s S ( S s S ), the i -th index in Eq. (1) (Eq. (4))goes over the atomic sites in the source lead while the j -th index goes over the atomicsites in the first super-cells of the central scattering area. H S d and S S d are constructedsimilarly. We employed the Sancho-Rubio iterative scheme to calculate the retarded surfaceGreen’s functions , g s, d [58, 59], from which one can easily obtain the broadening matrices via Γ s, d = i (Σ s, d − Σ † s, d ). Another important parameter is the local density of state (LDOS)given by LDOS ( E ) = ( i/π ) diag ( G r ( E ) − G a ( E )) , (6)where diag refers to the diagonal elements of the matrix. We can also evaluate LDOS byextracting the real part of the diagonal elements of the spectral function ( G r Γ s, d G a ). Thisparameter determines the spatial distribution of wave function at a specific Fermi energy.Inversion of the large matrix in Eq. (3), which is associated with the large number of atomsin the scattering area, is a massive task. For many of the physical quantities such as thetransmission function and LDOS, only part of the full Greens function is required. Therecursive scheme, explained in detail in Ref. [60], allows us to obtain the essential parts ofthe Green’s function to perform the necessary calculations.In tight-binding theory, expansion of free electron wave function in terms of the Block’swavefunction together with the minimization of energy converts the Schr¨odinger equationinto an eigenvalue matrix equation, H( k )-E( k )S( k )=0, where k is the two dimensionalwavevector whose range is determined by high symmetry points in graphene’s reciprocallattice [61]. In systems with a physical confinement in the transverse direction, it is possibleto further simplify the 2D bandstructure calculation by assuming a plane-wave wavefunctionin the longitudinal direction: e ik (cid:107) x (cid:107) , where the index (cid:107) denotes the longitudinal (transport)direction. Physical confinement in the transverse direction leads to H( k ⊥ )-E( k ⊥ )S( k ⊥ )=0,where the index ⊥ denotes the transverse direction. The eigenvalues E( k ⊥ ) of the followingcharacteristic equation (the so-called secular equation ), det ( H ( k ⊥ ) − E ( k ⊥ ) S ( k ⊥ )) = 0 , (7)give rise to the quasi-one dimensional band structure. Note that H ( k ⊥ ) is given by H ( k ⊥ ) ≡ H lc e ( − ik ⊥ a c ) + H cc + H cr e ( ik ⊥ a c ) , (8)where a c is the distance between the neighbor super-cells. H cc denotes the interaction Hamil-tonian between all atoms in the central super-cell, while H lc ( cr ) represents the interactionHamiltonian between atoms in the left (central) super-cell with atoms in the central (right)super-cell. One can use Eq. (1) to build each of the Hamiltonian matrices in Eq. (8). S ( k ⊥ )has a similar form to H ( k ⊥ ) in which S lc , S cc and S cr (constructed via Eq. (4)) replacing theequivalent Hamiltonian terms in Eq. (8). Altogether, Eq. (7) can be constructed to solvethe eigenvalue problem. III. RESULTS AND DISCUSSIONSA. Straight Waveguides
We begin our study by considering straight graphene waveguides in both edge orientations(AO-GWs and ZO-GWs), exploring three different side-barrier widths (W SB ) and investi-gating the effect of W SB on the conductance. The length of graphene waveguide (L) and thewidth of the guiding region (W G ) are fixed at 100 nm and 20 nm, respectively. The 20 nmwide guiding region is equivalent to the number of dimer lines N A -GW = 163 in AO-GWand N Z -GW = 188 in ZO-GW. The total width of scattering area W is 40, 60 and 80 nmwhich corresponds to W SB = 10, 20 and 30 nm, respectively. At the same time, leads withthe number of dimer lines N aS, aD = 161 (metallic armchair leads; a stands for armchairand S ( D ) stands for source (drain)) and N zS, zD = 188 (symmetric zigzag leads; z standsfor zigzag) have been considered for AO-GW and ZO-GW, respectively. The on-site po-tential energy in the scattering area is smoothly varied within ∆W = 44a cc ≈ SB = 0 eV at the side-barriers to U W G = -0.3 eV at the guiding area for all devices [53].As mentioned earlier, source leads and waveguide areas set to possess the same potentialenergy (U
W G ) while drain leads are grounded in all samples. We conducted a transportstudy for these six samples by considering both the first (1NN) and the third (3NN) nearesttight-binding approximations. The results are shown separately on the left and right panelsin Fig. 2. In Fig. 2 both the 20 nm ZO-GW and AO-GW exhibit a quantization of conduc-tance G = 1, 3, 5 G in each configuration (see the green curve and red curve in each panel).The first plateau of ZO-GW is clearly wider in energy axis than that of AO-GW. The firstconductance plateaus for both ZO-GW and AO-GW are flat, whereas other higher plateaus S B = 1 0 n m ( d )
G (G0) ( b )W
S B = 2 0 n m ( e )
E ( e V ) ( c )W
S B = 3 0 n m ( f )
FIG. 2: Conductance of 20 nm-wide AO-GW (red-dot lines) and ZO-GW (green-solid lines) withdifferent side-barrier width, W SB . (a)-(c) with the 1NN approximation and (d)-(f) with the 3NNapproximation. are not, and show a gradual losing flatness toward more positive energies. Importantly, theeffect of side-barrier widths (W SB ) seems negligible for both orientations. This suggests aminimum influence of edge disorders on conductance of a gate-defined graphene waveguideas long as the edges (the border between side-barriers and vacuum) are far enough from thewaveguide area. When the 3NN approximation is employed, noticeable dips in the conduc-tance of AO-GWs (red-dot lines) appeared around E = 0 eV, as can be seen in Figs. 2(d)-(f).This can be understood by the fact that the 3NN approximation tends to yield a small bandgap in an AGNR (i.e., terminals) [62]. Both 1NN and 3NN approximations give rise tothe noisy conductance features at E < < -0.2 eV (notshown in Figs. 2) [48]. We attribute these noises to the increase of current passing throughside-barriers in this range of energy. At higher energies, the plateaus gradually disappearbecause there are only a few confined wavefunctions localized in the waveguide area.In addition, with the 3NN approximation, the difference in conductance between twoorientations became more visible. For example, the conductance of ZO-GW exhibits largervalues at E < - 0 . 1 0 0 . 1 0 . 2 0 . 31591 3 0 0 . 1 0 . 2 0 . 3 W G = 2 0 n m G (G ) D Z G N R S E ( e V )
FIG. 3: Conductance of source (blue-dot line), drain (red-dashed-dot line) and ZO-GW (green-solidline) considering (a) 1NN and (b) 3NN approximations. drain electrode and ZO-GW under the 3NN approximation, as shown in Fig. 3. Note that,the conductance of the drain electrode refers to the conductance of the semi-infinite GNRthat is used as a drain lead in our structure. The correspondence between the red-dashed-dot line and green-solid line in Fig. 3 suggests that the conductance of the waveguide followsthe conductance behavior of the drain terminal.We further explore the effect of leads on waveguide transport properties. Here, we mod-ified the number of dimer lines of leads by 1 or 2 to make them either metallic or semi-conducting (nonmetallic). In contrast to the insensitivity of conductance to the widths ofside-barriers, conductance of waveguide for both orientations shows a clear dependence onthe metallic (m) or nonmetallic (n) nature of leads, as shown in Figs. 4(a) and 4(b). Onethird of the AGNRs and an ideal ZGNR have metallic behavior because their band structurespossess zero-energy mode.Different combinations of metallic and non-metallic leads are considered for a previouslystudied configuration, i.e., W SB = 20 nm and W G = 20 nm. Non-metallic drain in AO-GW yields a finite gap on conductance around E = 0 eV (gray-solid and green-dot lines inFig. 4(a)) while the conductance of a configuration with non-metallic source and metallicdrain is identical to that with both metallic leads (i.e., blue-solid line is identical to red-dot line in Fig. 4(a)). Moreover, the conductance of AO-GW with non-metallic drain (m-nand n-n) shows shorter spacing between plateaus with quantization steps at multiple ofG compared to that with metallic drain (n-m and m-m) which shows quantization steps atmultiple of 2G . On the other hand, an ideal ZGNR (with closed hexagonal crystal structure)represented by an even number of dimer lines is indeed metallic. However, a ZGNR with an1 G (G0) m - n n - n n - m m - mS - D ( a )A O - G W ( b )Z O - G W 1 6 n m 2 0 n m 2 4 n m 4 0 n mW
S / D = ( c )E ( e V ) ( d )
FIG. 4: The conductance through straight graphene waveguides considering W SB = 20 nm andW G = 20 nm with different combination of leads nature. (a) Conductance of AO-GW for dif-ferent combination of metallic (m) and nonmetallic (n) leads. (b) Same as (a) for ZO-GW. (c)Conductance of AO-GW for various widths of metallic leads. (d) Same as (c) for ZO-GW. odd number of dimer lines results in breaking the crystal symmetry and is non-metallic dueto the absence of the zero-energy mode. As a result, the gap in conductance is even wider inthe case of ZO-GW with disordered (non-metallic) drain (green-dot line in Fig. 4 (b)). Here,we refer a ZGNR lead with an odd number of dimer lines as a disordered lead. Also, likeAO-GW, the conductance of ZO-GW with a non-metallic source and a metallic drain (n-m)is identical to that with both metallic leads (m-m), as shown by the blue-solid and red-dotlines in Fig. 4 (b). Configurations with non-metallic drain (m-n and n-n) in ZO-GW do notchange the quantization step (in contrast to AO-GW) but it has shifted the conductanceboth vertically and horizontally, as shown in Fig. 4 (b). This result again indicates that thenature of the drain plays a significant role on the conductance of graphene waveguide forboth orientations. Therefore, we adopted metallic leads for the rest of our studies becausethey yield early onset of non-zero conductance plateau for both edge orientations. Alteringthe width of leads at nanometer scale also influences the conduction of graphene waveguide,as shown in Figs. 4(c) and 4(d) for both edge orientations. Wider conductance plateaus arepresented for short leads and vice versa. Note that the situation W D, S (cid:54) = W G has added avisible level of noise to the conductance plateaus in the cases of much shorter (16 nm) andmuch wider (40 nm) leads as compared to the primary case of W D, S = W G .2 - 0 . 10 . 00 . 10 . 20 . 3 1 5 9 - 0 . 2 0 0 . 2- 0 . 2 0 . 2- 0 . 2 0 0 . 2- 0 . 10 . 00 . 10 . 20 . 3 - 0 . 10 . 00 . 10 . 20 . 3 1 5 9 E (eV)
A O - G W W G = ( a ) ( d ) k ( n m - 1 ) ( c ) ( b )G ( G ) E (eV)
Z O - G W ( e )W
S / D = W G ( h ) k ( n m - 1 ) ( g )( f ) FIG. 5: (Color online) (a) Conductance of AO-GWs for various W G = 20, 30 and 40 nm. (b)-(d)Band structure plotted for different W G employed in (a). The color of the bands in each panelcorrespond to the color used in (a). (e) Conductance of ZO-GWs for various W G used in (a).(f)-(h) band structure plotted for corresponding W G of (e). Solid gray lines in the band structuresdenote the bands corresponding to the wavefunctions that are not confined in the waveguide. In further study of the effect of parameter W G on the conductance of graphene waveguidefor both edge orientations, we evaluated three values of W G (20, 30 and 40 nm), with leadssatisfying the condition W D, S = W G . For these tests, length L = 100 nm and side-barriersW SB = 20 nm are kept fixed. Conductance and quasi-one dimensional band structures forsupercells corresponding to each W G are plotted in Fig. 5 with the same color schemes. Forboth edge orientations, as W G decreases from 40 nm to 20 nm, conductance plateaus getlonger. This is a result of larger spacing between the energy bands, as visible in Figs. 5(b)-5(d) and 5(f)-5(h). Note that the subbands of AO-GW are two-fold degenerate (see Fig. 5(b)-5(d)) while the subbands of ZO-GW are not degenerate (see Figs. 5(f)-5(h)). For a specificW G , one can deduce that the first plateau on conductance for E > G = 20 nm.In Figs. 6(a)-6(d), normalized LDOS for both orientations of the 20 nm wide waveguideare presented at two Fermi energies (E = 0.05 eV and E = 0.15 eV), which correspond tothe conductance plateau at G and 3G , respectively. Right (lower) panels of Figs. 6(a)-6(b)(Figs. 6(c)-6(d)) plot the average of the unnormalized LDOS ( < LDOS > ) within the black-dashed lines shown in Figs. 6(a)-6(b) (Figs. 6(c)-6(d)). Reasonable localization of LDOS isapparent within the waveguide area at E for both orientations, as shown in Figs. 6(a) and6(c). The four peaks visible in the right panel of Fig. 6(a) correspond to the fourth mode inthe band structure of AO-GW (see Fig. 5(d)), which contributes to the first plateau in theW G = 20 nm waveguide. Similar analysis can be performed for other graphene waveguideswith different widths and at different energies. Comparison between Fig. 6(a) and Fig. 6(b)(or Figs. 6(c) and Fig. 6(d)) shows stronger confinement of wavefunction at E as comparedto E . Nevertheless, < LDOS > shows that leakage of wavefunction toward side-barriers isstill negligible at E for both edge orientations. B. U-, L-Shape and Split Waveguides
In this section, we further study the transport properties of waveguides with the ge-ometries that can be potentially used in nanoelectronic devices. Three types of curvedwaveguides, U-shape, L-shape and split-shape, have been taken into account to investigatethe conductance profile and the ability to confine the charge carriers in these highly bentstructures. In a U-shape graphene waveguide, both the source and drain leads are connectedto the same edge orientation (either armchair or zigzag edge). In the following, we use thenotation U-AO-GW (U-ZO-GW) to represent the U-shape AO-GW (ZO-GW). A U-AO-GW(U-ZO-GW) can be constructed by bending a straight AO-GW (ZO-GW) by 180 ◦ as shownin Fig. 7(a) (Fig. 7(b)). Dimension of the scattering area is W × L = 120 nm ×
80 nm forU-AO-GW and W × L = 80 nm ×
120 nm for U-ZO-GW. Here, we consider the waveguides4 y ( n m ) x (nm) y ( n m ) E E y ( n m ) < L D O S > L D O S ( a . u . ) < L D O S > E E (a)(b)(c) (d) AO-GWAO-GW Z O - G W Z O - G W FIG. 6: (Color online) LDOS for a AO-GW (a) at E = 0.05 eV and (b) at E = 0.15 eV. Rightpanels of (a) and (b) show the averages of the unnormalized LDOSs ( < LDOS > ) in the selectedregion between the black-dashed lines shown in (a) and (b). (c) and (d) The same as (a) and (b)but for ZO-GW. with two different widths ( W G = 20 nm and 30 nm) in each orientation. The width of themiddle-barrier between the source and drain (i.e., 2R in Fig. 7(a) and Fig. 7(b)) is set to40 nm (30 nm) when W G = 20 nm (30 nm), while W SB = 20 nm was consistent across allstructures. The on-site potential energy of the U-shape waveguides with W G = 20 nm isconstructed by a combination of three segments: two AO(ZO)-GWs with L = 20 nm which5 D r a i n S ou r c e U-AO-GW U-ZO-GWSource
Drain R R W SB R R (a) (b)W SB W G S S S S S S C C W SB W SB - 0 . 1 0 0 . 1 0 . 2 0 . 31357 0 0 . 1 0 . 2 0 . 3 2 0 n m 3 0 n m G (G0) W G = U - A O - G W ( c )E ( e V ) U - Z O - G W ( d ) x (nm) y ( n m ) x (nm) (e) (f) E (eV)
FIG. 7: (Color online) (a) and (b) show the schematic diagram for U-AO-GW and U-ZO-GW.(c) and (d) show the conductance of U-AO-GW and U-ZO-GW, with W G = 20 nm (red-dot line)and 30 nm (green-solid line), respectively. (e) and (f) show LDOS calculated for U-AO-GW andU-ZO-GW with W G = 20 nm and at E = 0.03 eV. are parallel to each other, and half of a circular waveguide with inner (outer) radius of 20 nm(40 nm) which provides a smooth bending around the center of the circular part (i.e., pointC in Fig. 7(a) and Fig. 7(b)). Conductance of the U-AO-GWs and the U-ZO-GWs bothresemble that of their counterparts (straight AO-GWs and ZO-GWs), as can be observed bycomparing Fig. 7 (c) with Fig. 5(a) and Fig. 7(d) with Fig. 5(e). In the U-shape case, thegeneral form of quantized conductance is preserved, but the second plateau is modulated bya visible oscillation as highlighted by a dashed ellipse in Fig. 7(d). This oscillation is morepronounced in the W G = 20 nm case and becomes less visible when W G is 30 nm. Thenormalized LDOS for U-shape waveguide with W G = 20 nm in both orientations at a givenenergy of E = 0.03 eV (which locates within the first plateau) is plotted in Figs. 7(e) and67(f), respectively. Both LDOS again show reasonable confinement at given Fermi energywhich corresponds to the conductance plateau.Next, we studied the L-shape graphene waveguide to investigate the effect of 90 ◦ bendingon their transport properties. Here, we considered two configurations of L-shape waveguidein a fixed-size scattering area (W = L = 100 nm), as shown in Fig. 8(a) and Fig. 8(b).First, a AO-GW bent to become a ZO-GW, with source on the zigzag interface and drainon the armchair interface, as labeled as L-AZ-GW. Secondly, a ZO-GW bent to become aAO-GW, with source on the armchair interface and drain on the zigzag interface, as labeledas L-ZA-GW. Note that the edge orientation of the scattering area is fixed while the locationof source and drain leads is different for each case, as visible in Fig. 8(a) and Fig. 8(b). Thewaveguide (equivalently the on-site potential energy) is constructed using a combination ofAO-GW and ZO-GW (both with L = 50 nm) perpendicular to each other, and a quarter ofa circular waveguide with inner (outer) radius of 10 nm (30 nm), which provides a smooth90 ◦ bending around the center of the system (i.e., point C in Figs. 8(a) and 8(b)). Tocalculate the conductance of the aforementioned configurations, one only needs to switchthe on-site potential energy between source and drain, and the relative positions of Γ s and Γ d in Eq. (2). Conductance of the L-shape waveguide in each configuration, with W G = 20 nmand 30 nm, is plotted in Figs. 8(c) and 8(d), respectively. Consistent with the previousresults of straight waveguides, conductance of the L-shape graphene waveguides (both ZAand AZ) show dependence on the nature of the drain, as can be seen by comparing Fig. 8(c)with Fig. 5(a) and Fig. 8(d) with Fig. 5(e) for W G = 20 nm and 30 nm. Conductanceof a 20 nm L-ZA-GW also shows a visible oscillation at the second conductance plateau,which is similar to the case of U-shape graphene waveguide. This phenomenon could beattributed to the bending-induced scattering between K and K (cid:48) sub-lattices. Similarly, wecalculated the LDOS of L-shape graphene waveguides with W G = 20 nm and at E = 0.05 eV(within the first conductance plateau). Both L-shape graphene waveguides present a decentconfinement of wave function along the straight parts and around the bending area, as shownin Figs. 8(e) and 8(f). As an extension to the L-shape graphene waveguide, we subsequentlystudied the split waveguides, which could be viewed as the counterpart of an optical beamsplitter. The on-site energy of a split graphene waveguide can be constructed by combiningthat of two adjacent L-shape waveguides bent in opposite directions. The split waveguidebuilt in the scattering area consists of two parts: a stem part and two split parts. In our7 L-AZ-GW R R S S S (a) SB S ou r c e Drain
L-ZA-GWR R S S S (b) SB Source D r a i n C C - 0 . 1 0 0 . 1 0 . 2 0 . 31357 - 0 . 1 0 0 . 1 0 . 2 0 . 3 G (G0) W G = L - A Z - G WE ( e V ) E ( e V )( c ) L - Z A - G W ( d ) x (nm) y ( n m ) x (nm) (e) (f) FIG. 8: (Color online) Schematic diagram for (a) L-AZ-GW and (b) L-ZA-GW. (c) and (d) showthe conductance of L-AZ-GW and L-ZA-GW, with W G = 20 nm (red-dot line) and 30 nm (green-solid line), respectively. (e) and (f) show LDOS calculated for L-AZ-GW and L-ZA-GW withW G = 20 nm and at E = 0.05 eV. example, the stem part is 40 nm wide and it splits equally into two 20 nm wide bent graphenewaveguides. We also considered two configurations for the split waveguide, labeled by SP-AZ-GW and SP-ZA-GW, in which SP-AZ-GW (SP-ZA-GW) refers to a split waveguidewhere the orientation of stem is armchair (zigzag), while that of the branches is zigzag(armchair). Like the case of the L-shape waveguide, drain leads at the end of branches areconnected to different interfaces, which are opposite to the interface between source lead andthe stem, due to the 90 ◦ bending of each L-shape waveguide. The calculated conductancethrough different paths (G12 and G13) is shown in Figs. 9(a) and 9(b), in which the firstsubindex (i.e., 1) refer to the stem while the second subindex (i.e., 2 or 3) refers to eachbranch. Conductance for both paths in the three-terminal SP-AZ-GW show similar trendto that of the 20 nm straight ZO-GW.8 - 0 . 1 0 0 . 1 0 . 2 0 . 30246 - 0 . 1 0 0 . 1 0 . 2 0 . 30246 G (G0)
S P - A Z - G W S P - Z A - G W G G G G E ( e V ) E ( e V ) x (nm) y ( n m ) L D O S ( a . u . ) x (nm) (a) (b)(c) (d) FIG. 9: (Color online) (a) and (b) show the conductance of SP-AZ-GW and SP-ZA-GW withW G = 40 nm for stem and W G = 20 nm for branches. LDOSs of SP-AZ-GW and SP-ZA-GW arepresented in (c) and (d) at E = 0.05 eV. The conductance of SP-ZA-GW also follows a similar pattern to the 20 nm straightAO-GW, which can be recognized by the small dip in conductance around E = 0 eV (seeFig. 9(b)). Together with the small dip observed in other armchair drain-based waveguides,we concluded that the nature of drain leads (metallic or nonmetallic, and width) significantlydetermines the conductance profile of various types of graphene waveguides, regardless oftheir bending geometries [53]. Again, we plotted the normalized LDOS of split waveguidesfor each configuration in Figs. 9(c) and 9(d) to depict the confinement at E = 0.05 eVcorresponding to the first conductance plateau. In addition, quasi-one dimensional bandstructures for selected supercells around the splitting point, indicated by dashed rectanglesin Figs. 9(c) and 9(d), are plotted in Figs. 10(a) and 10(b), respectively. We have chosenthese segments of the scattering area, because they give us the information of the energybands at the beginning of two independent branches. The calculated energy bands showthe two-fold (Fig. 10(b)) and four-fold (Fig. 10(a)) degeneracy for supercells with zigzag(Fig. 9(d)) and armchair (Fig. 9(c)) edges. The number of energy bands in the presence ofbranches has doubled compared to the band structures of the straight graphene waveguides(see Figs. 5(b)-5(d) and Figs. 5(f)-5(h)). Each of the two-fold energy bands in Fig. 10(b) canbe attributed to a non-degenerate energy band belongs to each branches. Similarly, one can9 - 0 . 2 0 0 . 2- 0 . 10 . 10 . 3 1 . 9 5 2 . 1 2 . 2 5- 0 . 10 . 10 . 3 k ( n m - 1 ) ( a ) S P - A Z - G W
E (eV) k ( n m - 1 ) S P - Z A - G W ( b )
E (eV)
FIG. 10: Energy band structures for the split waveguides around the splitting area. (a) Bandstructure calculated for the supercell indicated by the dashed rectangle (an armchair supercell)in Fig. 9(c). (b) Band structure calculated for the supercell indicated by the dashed rectangle (azigzag supercell) in Fig. 9(d). Different color lines (except for the gray lines) are used to distinguishthe four-fold and two-fold degenerated subbands in (a) and in (b), respectively. Upper solid graylines denote the bands corresponding to the wavefunctions that are not confined in the waveguide. divide the four-fold degenerate energy bands of SP-AZ-GW in Fig. 10(a) into two two-folddegenerate bands resulting from each branch. Moreover, the symmetry of system along thetransport direction in the stem part assures the spatial continuity of energy channels alongeach branch segments. Therefore, the incoming wave has equal probability to scatter intoeach branch at the splitting point and results in ballistic transport from splitting point todrains. This justifies the similarity of conductance between two branches, as can be observedin G and G (see Figs. 9(a) and 9(b)).0 C. Upscaling Graphene Waveguides
Although the recursive NEGF enable us to perform transport calculations on all theaforementioned examples, the large amounts of memory required by the algorithm rendersit incapable of handling structures longer than 200 nm in common computing machines. Onesolution to this hurdle is to employ a scalable tight-binding approach to examine the quanti-zation of conductance on much larger graphene waveguides. A scalable tight-binding modelrefers to upscaling the real carbon-carbon bond length (a cc ) in graphene via a Scale = S f a cc ,with the scaling factor S f > mustbe modified to t /S f to keep the energy band structure unchanged in the low energy regime.First, we performed the transport study on 20 nm waveguides (i.e., our early example witharmchair and zigzag edge orientations) with two different scaling factors 2 and 4. Note thatthe size of waveguide is fixed, so the increase of the scaling factor actually reduces the num-ber of carbon atoms in the calculation. Conductance of the scaled graphene waveguides withboth orientations along with conductance of the non-scaled devices (S f = 1 as a reference)have been shown in Figs. 11(a)-11(b). Conductance calculated by the scalable model showsreasonable consistency with that calculated using the real model. However, we detectedtwo minor differences. First, the resulting conductance of the scaled model in the case ofAO-GW delivered noisier conductance in the upper range of Fermi energy. Secondly, theconductance of the scalable model with a larger scaling factor tended to lower the originalspacing between plateaus in the case of ZO-GW. Furthermore, we performed a transportstudy for 80 nm waveguides with L = 400 nm and L = 600 nm in both orientations usingthe scaling factor S f = 4. The results are plotted in Fig. 11(c). In general, conductancein both types of large-scale graphene waveguides showed reduced spacing between plateaus(less than 2G = 4 e /h ) and became more fractional with respect to nG (n = 1, 3, 5,. . . seeinset in Fig. 11(c)). Spacing between plateaus in zigzag oriented waveguides is more uniformthan in armchair oriented waveguides, which has presented a series of hardly distinguishableplateaus for E > ), when the scaled model is applied. Twoexamples of normalized LDOS, for 80 nm-wide graphene waveguides in both orientations, are1 - 0 . 1 0 . 0 0 . 1 0 . 2 0 . 30246 0 0 . 1 0 . 2 0 . 3 - 0 . 1 0 . 0 0 . 1 0 . 2 0 . 3061 21 8 s f = 1 s f = 2 s f = 4 G (G ) ( a )A O - G W ( b )Z O - G W · · · · G (G ) E ( e V )
A O - G W Z O - G W ( c ) W ( n m ) · L ( n m ) W ( n m ) · L ( n m ) G (G ) E ( e V ) x (nm) y ( n m ) x (nm) FIG. 11: (Color online) (a) and (b) show the conductance of a 20 nm wide AO-GW and ZO-GWcalculated by the scalable tight-binding model with a scaling factor S f = 1, 2, and 4, respectively.Note that S f = 1 corresponds to the original tight binding model (green-solid line). (c) Conductancecalculated by the scalable tight-binding model with S f = 4 for longer graphene waveguides in bothorientations. (d) and (e) LDOSs of AO-GW and ZO-GW. LDOSs are extracted at the energy valuethat is indicated by an arrow in (c). plotted in Figs. 11(d)-11(e). These show the effect of confinement achieved by the quantumwell in the scalable tight-binding model. In summary, our results show that a small-widthgraphene waveguide is capable of delivering quantized conductance with the scalable modelas long as the well potential is deep enough, which is in contrast to the shallower quantumwells used in Ref. [43].2 IV. CONCLUSION
To conclude, by applying the Non-equilibrium Greens function, we have investigatedthe transport property of straight and various bent graphene waveguides with two typesof edge orientations, i.e., armchair and zigzag configurations. For the straight waveguides,we have shown that the width of side-barrier has little effect on the conductance, while thenature (metallic or non-metallic) and width of the source/drain leads plays an importantrole in waveguide conductance profiles. In particular, the conductance of waveguides isfound to primarily follow the conductance property of the drain terminal in the case ofZO-GW under the 3NN approximation. The conductance in both armchair and zigzagoriented waveguides can be quantized by steps of 4 e /h in a similar manner, but the zigzagoriented waveguide shows a longer first plateau in cases where its drain terminal possesseszero energy modes. From a series of analyses into conductance characteristics, we haveobserved that the conductance of bent graphene waveguides is similar to that of their straightcounterparts, regardless of the bending degree of the guide region for different geometricconfigurations. LDOS maps for all configurations have shown a good capacity to confinecharged particles at the Fermi energies corresponding to the first few conductance plateaus.Moreover, we have employed the scalable tight-binding model to effectively capture theconductance of large-scale straight graphene waveguides. The conductance profile of large-scale graphene waveguides with both orientations exhibits quantized steps close to 4 e /h ,while the spacing between plateaus is sensitive to the employed scaling factor. . Altogether,this study has demonstrated that coherent transport can be achieved in various electricallygated graphene waveguides with different edge orientations. The conductance quantizationrealized in straight and highly bent graphene waveguides is promising for application ofgraphene in modern nanoelectronic devices and thus making all-graphene integrated circuitspossible in the future. ACKNOWLEDGMENTS
This work was financially supported by the National Key Research and Development Pro-gram of China (Grant No. 2016YFA0301700), NNSFC (Grant No. 11625419 ), the StrategicPriority Research Program of the CAS (Grant Nos. XDB24030601 and XDB30000000), the3Anhui initiative in Quantum information Technologies (Grants No. AHY080000). This wasalso supported by Chinese Academy of Sciences and The World Academy of Science for theadvancement of science in developing countries. ∗ Correspondence author: [email protected] † Correspondence author: [email protected] ‡ Correspondence author: [email protected][1] B. Van Wees, H. Van Houten, C. Beenakker, J. G. Williamson, L. Kouwenhoven, D. Van derMarel, and C. Foxon, Phys. Rev. Lett. 60, 848 (1988).[2] D. Wharam, T. J. Thornton, R. Newbury, M. Pepper, H. Ahmed, J. Frost, D. Hasko, D.Peacock, D. Ritchie, and G. Jones, J. Phys. C: Solid State Phys. 21, L209 (1988).[3] S. Somanchi, B. Terr´es, J. Peiro, M. Staggenborg, K. Watanabe, T. Taniguchi, B. 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