Transport Induced Dimer State from Topological Corner States
TTransport Induced Dimer State from Topological Corner States
Kai-Tong Wang,
1, 2
Yafei Ren, Fuming Xu, ∗ Yadong Wei, and Jian Wang
4, 5, † College of Physics and Energy, Shenzhen University, Shenzhen 518060, China Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province,College of Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China Department of Physics, University of Hong Kong, Pokfulam Road, Hong Kong
Recently, a new type of second-order topological insulator has been theoretically proposed byintroducing an in-plane Zeeman field into the Kane-Mele model in the two-dimensional honeycomblattice. A pair of topological corner states arise at the corners with obtuse angles of an isolateddiamond-shaped flake. To probe the corner states, we study their transport properties by attachingtwo leads to the system. Dressed by incoming electrons, the dynamic corner state is very differentfrom its static counterpart. Resonant tunneling through the dressed corner state can occur bytuning the in-plane Zeeman field. At the resonance, the pair of spatially well separated and highlylocalized corner states can form a dimer state, whose wavefunction extends almost the entire bulkof the diamond-shaped flake. By varying the Zeeman field strength, multiple resonant tunnelingevents are mediated by the same dimer state. This re-entrance effect can be understood by a simplemodel. These findings extend our understanding of dynamic aspects of the second-order topologicalcorner states.
I. INTRODUCTION
Two-dimensional insulators with nontrivial bandtopology have been widely studied in the past decades ,which include quantum anomalous Hall effect , quan-tum spin Hall effect , quantum valley Hall effect ,and topological crystalline insulators . These topo-logical phases are characterized by nontrivial band topo-logical indices as well as gapless edge states, which arerobust against disorders and exhibit quantized conduc-tivity .Recently, higher-order topological insulators (TIs)have been theoretically proposed in various systems ,which are characterized by hinge modes in three-dimensional (3D) materials or corner states in two-dimensional (2D) systems . The existence of one-dimensional hinge states has been experimentally con-firmed in bismuth and multi-layer WTe . In 2Dhigher-order TIs, one-dimensional edges are insulatingwhereas the corners between different edges can hostzero-dimensional in-gap states that are isolated fromboth edge and bulk bands by an energy gap .In contrast to the gapless edge states in conventionaltopological insulators, higher-order topological cornerstates are not conducting and behave like localized boundstates. Therefore, it is challenging to detect these cornerstates by transport measurement.In this work, we numerically study the transport prop-erties of topological corner states in the two-dimensionalhoneycomb lattice. Based on the modified Kane-Melemodel with an in-plane Zeeman field, the second-ordertopological insulator is realized and zero-energy cornerstates are localized at the intersections of different zigzagboundaries of an isolated diamond-shaped flake . Byconnecting two leads to the diamond-shaped flake whilekeeping the corner states intact, dynamic features of cor- ner states are revealed. Different from the static cornerstate, incident electrons dwelling in the corner state cancause its ”delocalization” that mediates the transport.By tuning the Zeeman field, corner-state-mediated reso-nant tunneling occurs. For a single corner-state setup, asthe Zeeman field is increased, resonant tunneling via thecorner state or its precursor persists, until a thresholdfield strength closes the resonant channel. For a setupwith two corner states, a dimer state is formed at theresonance with its wavefunction spanning almost the en-tire bulk of the flake. As the Zeeman field is scanned,it is found that after the corner states emerge, multipleresonant peaks mediated by the dimer state can arisein the transmission spectrum, which is counter-intuitivesince there is only one dimer state. A simple model isconstructed, which explains that this re-entrance effectis indeed caused by the same dimer state. II. MODEL AND FORMALISM
In the conventional Kane-Mele model , intrinsic spin-orbit coupling gives rise to quantum spin Hall effect in thetwo-dimensional honeycomb lattice. By applying an in-plane Zeeman field, the time-reversal symmetry is brokenand higher-order topological states emerge . This mod-ified Kane-Mele model has the following tight-bindingHamiltonian : H D = − t (cid:88)
3. Eigenfunction dis-tributions of the corner states are shown in the inset, and thecorresponding eigenenergies are highlighted in red. C and C are the intersections of zigzag boundaries where corner statesemerge. which only involves the next-nearest-neighbor hoppingand ν ij = +1 / − j tosite i by taking a left/right turn. The last term orig-inates from the in-plane Zeeman field, whose directiondepends on the magnetic field. s is the Pauli matrixfor spin and λ denotes the Zeeman field strength. We set t so = 0 . t and choose the magnetic field along y -directionas B = (0 , B y , t is set as the energyunit.We consider the diamond-shaped honeycomb-latticeflake surrounded by zigzag boundaries as shown inFig.1(a). In the absence of the Zeeman field, gapless edgestates distribute continuously along zigzag boundariesof the flake. When the Zeeman field is turned on, theedge states start to shrink towards the corners with ob-tuse angles while delocalizing away from the flake bound-aries. Meanwhile, the zero-energy levels stay unchangedin the energy spectrum while non-zero energy levels arerepelled from E = 0. As the Zeeman field is increased to λ ≥ . , a large energy gap opens up accompanied bya pair of zero-energy states, which are highlighted in redin Fig.1(b). The corresponding eigenfunctions of thesezero-energy states are localized around C and C cor-ners of the diamond-shaped flake as illustrated in the in-set. These zero-energy corner states manifest the second-order topological phase characterized by nonzero windingnumbers of the bulk band .To investigate transport behaviors of the corner states,the diamond-shaped flake is connected to two conductingleads to calculate the conductance or transmission of thesystem. The full Hamiltonian of the transport system isgiven by H = H L + H D + H T , (2)where H L is the lead Hamiltonian following the Kane-Mele model H L = − t (cid:88)
Several transport setups are proposed to probe the lo-calized corner states and reveal their transport character-istics, by connecting the diamond-shaped flake to zigzaghoneycomb-lattice leads in different ways. The size of thisflake is fixed as L x × L y = 30 a × a with a being the lat-tice constant. Unless specified otherwise, eigenfunctionfor the isolated system, partial LDOS and transmissionfor the open system, are all evaluated at zero energy.Four possible setups are investigated. Our numericalresults show that, corner states are destroyed once theleads are directly connected to C and C regions of theflake. Two setups shown in Fig.5(a) and Fig.6(a) belongto this category, where detailed results and analysis arepresented in the supplemental material. In the following,we consider the other two setups. In the first setup, onlyone corner state survives, which can be probed by res-onant tunneling. In the second setup, the leads are faraway from C and C so that both corner states remainintact. In this case, a counter-intuitive resonant tunnel-ing phenomenon mediated by both corner states occurs. FIG. 2: (Color online) (a) Schematic of the single corner-state setup coupled with left and top leads, together with theeigenfunction distribution shown in blue. (b) Transmissionversus the Zeeman field strength for different energies E . (c)Partial LDOS for E = 10 − , i.e., the α ( λ = 0 .
2) point for T =2 in panel (b). (d) Partial LDOS for E = 10 − , which isthe α point for T =1. During this tunneling process, the incident electron tra-verses almost the entire region of the flake, which is theclassically forbidden region. The dressing of incomingelectrons leads to the formation of a dimer state, whichis symmetrically distributed among both corner states.
A. Single Corner-State Setup
In the single corner-state setup shown in Fig.2(a),left and top leads are attached to the central region.Edge states along zigzag boundaries of the λ =0 re-gions are shown in blue. Single corner state arises atthe C corner with an exponentially decaying wavefunc-tion. The dependence of the transmission T on the Zee-man field strength for different energies are plotted inFig.2(b) showing two transmission plateaus with T =2and T =1. For a particular incident energy, there is a crit-ical strength of the Zeeman field beyond which the trans-mission drops. When the incident energy is further awayfrom zero, the T = 2 plateau is more easily destroyedby the Zeeman field. These transmission plateaus usu-ally manifest the signature of quantized transport. How-ever, the T =2 transmission plateau is contributed fromtwo distinct conducting channels, one from the edge-statechannel and the other from the corner-state-mediatedresonant tunneling channel. To understand the physi-cal origin of transmission plateaus, we plot the partialLDOS for open systems in Fig.2(c) and Fig.2(d). Given λ =0.2 and E = 10 − for T = 2, the incoming electrontunnels through the flake via the corner state at C . Due to the dressing of incoming electrons, the corner statebecomes delocalized and its wavefunction expands intothe classically forbidden bulk region. Two conductingchannels are clearly seen in Fig.2(c): one connects edgestates along zigzag boundaries of both leads as labeled bythe red arrow near C , the other is formed through thecorner state at C . When the Zeeman field is increasedfurther, the corner-state wavefunction shrinks towardsthe C corner until it is completely localized and the cor-responding conducting channel is closed. Thus, only oneedge-state channel survives in Fig.2(d) resulting in the T =1 plateau.Now we study the typical resonant feature. The trans-mission vs the incident energy is shown in Fig.S3(a) of thesupplemental material, which exhibits extremely shapepeaks near E = 0. The T = 2 transmission peak is eas-ily destroyed by either changing the incident energy orincreasing the Zeeman field strength, showing its reso-nant nature. Meanwhile, the T = 1 plateau is robustagainst both λ and E , since it originates from the edgestate connecting left and top leads at the C corner inFig.2(c). To further confirm the resonant nature, we de-fine ∆ E as the half-width energy at half-maximum of theresonant peak ( T = 1 . E . A stronger Zeemanfield leads to more localized corner states, and it is easierto close the resonant channel through the C corner. ∆ E can also be obtained by solving the eigenvalue problemof the effective Hamiltonian including the effect of bothleads. The imaginary part of the eigenvalue is identifiedas the lifetime of the resonant state, which is equivalentto ∆ E . Numerical results presented in the supplementalmaterial show that this is indeed the case. B. Double Corner-State Setup
As shown in Fig.3(a), the diamond-shaped flake is con-nected to narrower leads around the C corner. Bothleads are far away from C and C to ensure weak im-pact on the corner states. The zero-energy eigenfunctionof this isolated system with λ = 0 . C and C . Transmission asa function of the Zeeman field is displayed in Fig.3(b). At λ = 0, there are two conducting channels due to the edgestates giving rise to T = 2. When the Zeeman field is ap-plied and increased, T decreases from T = 2 and quicklyreaches the T =1 plateau, which indicates the closing ofone conducting channel. Remarkably, with further in-creasing of λ , multiple resonant peaks of T = 2 emerge.As discussed in the supplemental material, the secondsharp peak is actually double peaks and the influence oflead width is also evaluated. In view of resonant trans-port, these integer transmission peaks of T = 2 manifestthe existence of two perfect propagating channels in thesystem. In the following, we will examine the nature ofthis resonant tunneling and explain why multiple reso-nant peaks appear. FIG. 3: (Color online) (a) The double corner-state setup,where the central region is connected to narrower left andbottom leads around the C corner. (b) Transmission withlead width W L =11. Four typical λ are shown in red arrows,which are 0 . .
2, 0 .
28, and 0 .
31, respectively. Panels (c)-(f): Partial LDOS for λ - λ . We focus on the setup with lead width W L =11 andplot partial LDOS of the open system in Fig.3(c-f), whichcorrespond to four typical Zeeman field strengths labeledas λ to λ in Fig.3(b). When λ =0, two edge states ofquantum spin-Hall nature dominate the transport, whichleads to T = 2. For a small λ =0.024, two conductingchannels are still visible in Fig.3(c). But the channelalong boundary 2 is not transmissive enough so that itcan only sustain a T = 1 . λ =0.2 at the T =1 plateau, only the channelalong boundary 1 survives and the other channel is de-stroyed (Fig.3(d)). Nevertheless, electrons incident fromthe left lead can still tunnel through the barrier U la-beled in Fig.3(a) and reach the upper corner C withlarge partial LDOS, which shows the dynamic signatureof the dressed corner state. However, it can not reachthe lower corner C leaving the static corner state com-pletely isolated. Fig.3(d) shows the distinction betweenstatic and dynamic corner states located at C and C ,respectively. When λ is increased to λ , Fig.3(e) showsthe dynamic feature of corner states at both C and C ,which expand and grow inside the bulk due to the dress-ing of incoming electrons. Counter-intuitive phenomenonhappens at λ . In Fig.3(f), dynamic corner states at C and C bridge the spatial gap inside the system and forma dimer state, which leads to a new resonant channel in-side the bulk instead of along the boundaries of the flake.As a result, a resonant peak with T =2 appears, which issharp and sensitive to both λ and E . Numerical results FIG. 4: (Color online) (a) The triple-barrier model in one-dimensional tight-binding chain . U / denotes the strengthof potential barriers with ∆ U the relative variation. (b)Transmission spectrum for U / = 1. (c) Transmission withrespect to the relative variation ∆ U for U = 1 + ∆ U and U = 1 − ∆ U at the resonant level E res = 0 . E r of the complex eigenenergies versus ∆ U . Red dashline corresponds to E res = 0 . verify that, all the resonant peaks in Fig.3(b) originatefrom the same mechanism: the dimer-state-mediated res-onant tunneling. The nature of the dimer state has beenstudied in detail in the supplemental material. To iden-tify the dimer state, the following observations are inorder: (1) the dressing of incoming electrons leads to thebinding of two corner states; (2) the distribution of thedimer wavefunction should be symmetrical in order tomediate the resonant tunneling; (3) comparing Fig.3(f)with Fig.S5(b) the binding of two corner states is closelyrelated to the sharpness of the resonant peak.The existence of multiple resonant peaks with respectto the Zeeman field strength, instead of single one, canbe understood with a simple model. We note that in thepresence of the Zeeman field, there are three spatial bar-riers in Fig.3(a) for an incident electron to overcome inresonant tunneling. U is the effective potential betweenthe leads and the corner states, and U is that in the bulkseparating C and C . The evolution of U and U followsthat of the corner state. As the Zeeman field is increased,the edge state shrinks along the boundary and delocalizesaway from it so that the strength of U increases while U decreases. In another word, the Zeeman field strengthcontrols the barrier variation in this system, which is con-firmed by partial LDOS shown in Fig.3. We constructa triple-barrier model based on one-dimensional tight-binding chain , where barrier strengths U and U aresimultaneously varied with a relative amount ∆ U (seeFig.4(a)). The tight-binding chain has 35 lattice sitesand all three barriers with width 5 sites are evenly spaced.When U / = 1, the resonant level for this triple-barriersystem is found to be E res =0.1828 as shown in Fig.4(b).By fixing E res and tuning the effective potentials with U = 1 + ∆ U and U = 1 − ∆ U , the transmission withrespect to ∆ U shows two resonant peaks with T = 1 asshown in Fig.4(c). These peaks are similar to the multi-ple resonant peaks in Fig.3(b). Here the relative varia-tion ∆ U plays a similar role as the Zeeman field strength λ in Fig.3(b). The evolution of the resonant level E res of the triple-barrier system is plotted in Fig.4(d). Theprocedure of calculating E r , which is the real part ofthe complex eigenenergy, is demonstrated in the supple-mental material. We see that E r crosses the incidentenergy twice, which gives rise to two resonant peaks inFig.4(c). Hence these two resonant peaks are mediatedby the same resonant state showing the re-entrance ef-fect. Similarly, by varying the Zeeman field strength inthe double corner-state setup, multiple resonant tunnel-ing processes are mediated by the same dimer state. IV. CONCLUSION
In summary, by connecting two leads to the systemhosting topological corner states, we studied dynamicfeatures of the corner state. Dressed by incoming elec- trons, the localized static corner state becomes extended,which can give rise to corner-state-mediated resonanttunneling. For a setup with a single corner state, bytuning the Zeeman field, we observed resonant tunnelingmediated by a state that evolves from the edge state,the precursor of a corner state, and the corner state,and finally the closing of this resonant channel with alarge enough Zeeman field. When two corner states arepresent, the resonant tunneling can also occur. At theresonance, the dressing of incoming electrons can bindtwo corner states to form a dimer state, whose wavefunc-tion extends to most of the bulk region that is classi-cally forbidden. 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FIG. 5: (a) Schematic of the setup connected to left and right leads. The system consists of zigzag ribbons without Zeemanfields and the diamond-shaped flake with λ = 0 . V. SUPPLEMENTAL MATERIALSS1. Setup I: the central region coupled with left and right leads
In the setup of Fig.5(a), the diamond-shaped central flake with nonzero Zeeman field is connected to left andright leads. The lead Hamiltonian follows the conventional Kane-Mele model defined in Eq.(3), and spin-helical edgestates propagate in both zigzag leads when the electron energy is inside the bulk gap. In Fig.5(a), we plot thezero-energy eigenfunction distribution for the isolated system, and blue dots along zigzag boundaries of those λ = 0regions illustrate the topological edge states. The edge states disappear in the λ = 0 . C and C of the central flake, inside the λ (cid:54) = 0 region there are no corners or boundaryintersections where corner states could reside; (2) in the presence of leads, corner states will leak out. We haveconfirmed that, when a narrow lead of width W L = 1 is directly connected to the corner, the corresponding cornerstate will be destroyed. Besides, the eigenfunction distribution in the λ = 0 . T = 2 to zero with the increasing of λ , and T is hardly affected by the variation of electron energies. T = 2 corresponds to the propagation of two in-gapedge modes, and they are quickly destroyed by the Zeeman field inside the central flake. In Fig.5(c), we plot thezero-energy partial LDOS for electrons incident from the left lead, which shows only nonzero density of states in theleft lead due to the spin-helical edge states. S2. Setup II: the central region coupled with left and bottom leads
Two setups are depicted in panels (a) and (d) of Fig.6, where left and bottom leads are connected to the centralregion. The zero-energy eigenfunction shows no sign of corner states in Fig.6(a), which is similar to Setup I. InFig.6(b), the transmission of this system rapidly declines from T = 2 to T = 1 when the Zeeman field is applied,and slightly decays upon further increasing of λ . Single conducting channel is found in Fig.6(c) labeled by the redarrow, which connects the left and down leads near the C corner. In the presence of strong Zeeman fields, the C FIG. 6: Panels (a) and (d): Two setups coupled with left and bottom leads for lead widths W L = 30 and W L = 25, respectively,accompanied by zero-energy eigenfunction distributions. Panels (b) and (c): Transmission and partial LDOS for the setup inpanel (a). Panels (e) and (f): Transmission and partial LDOS for the setup in panel(d). corner with an acute angle does not host electronic states, which is responsible for the slow decay of transmission from T = 1 in this setup. Compared with the single corner-state setup in Fig.2, though no corner state is found at the C corner, it can still support certain electronic states and result in a robust T =1 plateau against the Zeeman field asshown in Fig.2(b). The single conducting channel in Fig.6(c) can be easily destroyed by considering narrower leadsand exposing the C corner in vacuum. Such a setup is presented in Fig.6(d), where the eigenfunction distribution atthe C corner is depleted by the in-plane Zeeman field. In Fig.6(e) and (f), it is found that the transmission of thissetup directly drops to zero at large Zeeman fields, which is consistent with the partial LDOS that the propagatingchannel is closed. S3. Single Corner-State Setup: features of resonant tunneling
To demonstrate the resonant feature of the T = 2 transmission plateau for the single corner-state setup shown inFig.2, we plot the dependence of transmission of this setup on the incident energy in Fig.7(a). Note that the energyis on the logarithmic scale in order to illustrate the extremely shape peak near E = 0. The T = 2 transmissionpeak is easily destroyed by either changing the incident energy or increasing the Zeeman field strength, showing itsresonant nature. Meanwhile, the T = 1 plateau is robust against both λ and E , since it originates from the edgestate connecting left and top leads at the C corner in Fig.2(c). To further confirm the resonant nature, we define∆ E as the half-width energy at half-maximum of the resonant peak ( T = 1 . E . A stronger Zeeman field leads to more localized corner states, and it is easier to close theresonant channel through the C corner. It is well known that in resonant tunneling, electrons can tunnel throughthe system if the incident energy E is in line with the resonant level E res . E res can be calculated by solving theeigenvalue problem of an effective Hamiltonian H eff = H + Σ r for the open system, which satisfies [ H + Σ r ] ψ γ = (cid:15) γ ψ γ . (7)The eigenenergies of this effective Hamiltonian are complex, i.e., (cid:15) γ = E r + iE i . The real part E r correspondsto the resonant level E res , which is zero in the zero-energy corner-state-mediated resonant tunneling process. The FIG. 7: (a) Transmission versus the incident energy E for several Zeeman field strengths of the single corner-state setup. Theblack arrow corresponds to the half-width energy ∆ E at half-maximum T = 1 .
5. (b) ∆ E and the imaginary part E i of (cid:15) γ inEq.(7) with respect to λ for the T = 2 resonant peak.FIG. 8: (a) The double corner-state setup connected to narrower left and bottom leads around the C corner. (b) Transmissionfor different lead widths W L . The inset highlights the region in the dash rectangle. corresponding imaginary part E i is equivalent to ∆ E , the half-width energy at half-maximum of the resonant peak.This well-established theory has been verified in various tunneling systems . In Fig.7(b), we plot ∆ E as well as E i against the Zeeman field strength for the T = 2 peak in Fig.7(a). ∆ E is measured at the transmission peak and E i is obtained through diagonalizing the effective Hamiltonian H eff . Clearly, ∆ E perfectly matches with E i in a widerange of λ , which further consolidates the resonant tunneling characteristic of the T = 2 transmission peak. S4. Double Corner-State Setup: the influence of lead width on transmission
For the double corner-state setup shown in Fig.8(a), narrower left and bottom leads are connected to the diamond-shaped flake around the C corner to ensure weak impact on the corner states, which are clearly seen at C and0 FIG. 9: (a) The double transmission peaks for lead width W L = 11. λ to λ correspond to two peaks and one dip in thetransmission spectrum. Panels (b)-(d): Partial LDOS for λ to λ . C . We evaluate the influence of lead width on the transmission of this setup. Transmission as a function of theZeeman field strength for two lead widths are displayed in Fig.8(b). These two systems share similar features: thetransmission drops from T = 2 in the presence of a Zeeman field, and reaches the T = 1 plateau with the increasing of λ , and finally exhibits multiple T =2 resonant peaks when the Zeeman field is further enhanced. For a narrower leadof width W L = 7, the multiple resonant peaks are wide and close to each other. While for the lead width W L = 11,single peak is separated from double peaks shown in the inset. A larger Zeeman field is required to detect the first T =2 resonant peak for W L = 11. Dynamic details of the double resonant peaks are revealed in the following.The double transmission peaks with respect to the Zeeman field strength is highlighted in Fig.9(a), where two sharppeaks is separated by a dip. The transmission is T =2 for λ and λ , and T drastically drops to exactly T =1 at λ forming the dip. The partial LDOS in Fig.9(b) shows that, at λ , a symmetrical dimer state mediates a resonanttunneling channel, which crosses almost the entire bulk of the diamond-shaped flake. In contrast, the precursor of thedimer state has asymmetrical distributions, which is exemplified at λ where this delicate dimer state is ”destroyed”by slight increasing of the Zeeman field strength. In Fig.9(c), it is obvious that the resonant channel is closed at λ and single edge-state channel (red arrow) is conducting. Remarkably, the dressing of incoming electrons (albeitasymmetrical) around both corner states is still present, which is preparing for the next resonance. However, thewavefunction becomes asymmetrical, which is typical for the precursor of the dimer state. When the Zeeman field islightly enhanced to λ , the dimer state emerges again and leads to the second T = 2 resonant peak. These numericalresults vividly demonstrate the dynamic nature of the dimer state induced by electron-dressed corner states. The1 FIG. 10: (a) Transmission versus the Zeeman field strength for the system size 20 a × a . (b) Partial LDOS corresponding to λ =0.44. (c) Transmission as a function of λ for the system size 50 a × a . (d) Partial LDOS corresponding to λ =0.247. Thelead width is fixed at W L = 11. same dimer state repeatedly emerges in the resonant window when the system is tuned by the Zeeman field, whichgives rise to multiple resonant peaks in the transmission spectrum. Finally, we note that the total DOS characterizingthe bonding of the dimer state is just the dwell time of incoming electrons, which is proportional to the sharpnessof the resonance. A sharper resonance leads to tighter bonding of the dimer state. To summarize, three featuresof the dimer state are identified: (1) the dressing of incoming electrons bridges two corner states; (2) the scatteringwavefunction symmetrically distributes around two corner states; (3) a sharper resonance gives tighter bonding of thedimer state. S5. Size effect of the diamond-shaped flake
To evaluate the size effect of the diamond-shaped flake, we consider two system sizes in the double corner-statesetup, which are 20 a × a and 50 a × a , respectively. The lead width is fixed at W L = 11. The correspondingtransmission and partial LDOS results are shown in Fig.10. Combined with numerical results for the system size30 a × aa