Manipulating topological intravalley and intervalley scatterings of inner edge states in hybrid nanoribbons

We investigate the formation of inner edge states and their transport properties in hybrid nanoribbons. Some new inner edge states, such as spin-polarized, spin-valley-polarized and valley-polarized antichiral inner edge states, are obtained, different from the current existence of valley- and spin-valley-momentum locked inner edge states. We also obtain general formula of local bond current with the wave-function matching technique and use it to discover three interesting transport phenomena of the intravalley and intervalley scatterings that depend on the propagating direction, propagating path, spin mode and wave-vector mismatches between inner edge states. In particular, these transport phenomena are further used to design topological spin, spin–valley and valley filters and be representative for graphene, silicene, germanene and stanene, supporting a potential application of inner edge states, which are robust against random vacancies.

Inspired by the transport applications of topological edge states distributed in outer boundary of system, inner edge states have been rapidly recently investigated and developed for different types of valleytronics. In particular, the inner states with spin and valley degrees of freedom generally exist in hybrid nanoribbon linking two or more subsystems with different topological phases, which are characterized by spin-and valley-dependent Chern numbers [26,27]. In transport applications, there are robust against short-and long-range and magnetic long-range disorders [28,29], breeding the field of valleytronics and spin-valleytronics in addition to spintronics. Similar to spintronics, generating and manipulating the valley degree of freedom is the central control of valleytronics.
So far, the transport investigations of inner edge states are mainly relied on the valley-locking and spin-valley-locking inner edge states [29,30], and some works are proposed to break the valley degenerate for the valley polarization, important for applications for designing device. For instance, Xu and Jin [26] used the single valley and spin-valley inner edge states in hybrid silicene nanoribbon, where two half-nanoribbons are modulated independently by different external fields, to design the valley electronics and thermoelectronics. Han et al [31] proposed a gate-tunable device in multichannel zero-line system to modulate the switch between the current partitions for designing multifunctional valley-based electronics. Zhou et al [29] proposed the quantum spin-valley Hall kink states, protected by the valley-inversion and time-reversal symmetries, which can be used for designing robust spin-valley filter. In our recent work [32], based on the gap effect, propagating direction and spin mode mismatch of inner edge states, we proposed two types of spin-valley filters in three-terminal honeycomb lattice system. Although abundant applications of inner edge states for spin-valleytronics and valleytronics have been proposed, some new inner edge states and their transport phenomena of the intravalley and intervalley scatterings and applications are still less investigated.
In this work, we focus on proposing new types of inner edge states charactered by spin-and valley-dependent Chern numbers, and then further investigate their transport phenomena and applications in topological junction made of hybrid zigzag nanoribbon. Based on original Haldane model, we present some types of spin-polarized and spin-valley-polarized and valley-polarized antichiral inner edge states.
With tight-binding model and wave-function matching technique, we obtain the local bond current that can calculate the contribution of each propagating mode and the sum of each propagating mode. On this basis, we use these inner edge states to construct some topological junction with regions P and N as a base for finding out three underlying mechanisms of the intravalley and intervalley scatterings, where the factors of the spin mode, propagating direction and wave-vector mismatch play an important role in transport phenomena. In addition, these results offer a diverse platform for designing abundantly robust valleytronics and spin-valleytronics against random vacancies.

Model and theory
In our proposed topological junction with inner edge states, the upper and lower half-nanoribbons are modulated by different external fields in regions P and N denoted by the green and yellow, respectively. The corresponding tight-binding Hamiltonian model reads: where c † iα (c iα ) denotes the creation (annihilation) operator at site i with spin α; < i, j > and << i, j >> run over all the nearest-neighbor and next-nearest-neighbor sites, respectively; υ = +1(−1) as the next-nearest-neighbor hopping is anti-clockwise (clockwise) with respective to positive z-axis; µ i = +1(−1) corresponds to the site A (B) and S z αβ is the z-component Pauli matrix with spin indices α and β. The first term denotes the nearest-neighbor hopping with t = 1 eV throughout this paper. The second term denotes the spin-orbit coupling neglected here. The superscripts I and II in the last three terms stand for that the corresponding external field are applied on the upper and lower half-nanoribbons respectively, and Θ(y) is the Heaviside function. The third term denotes the off-resonant circularly polarized (ORCP) light, where the opposite (negative) value of λ Ω for the right (left) circulation [33][34][35]. The fourth term denotes the antiferromagnetic exchange field λ AF , which can be induced by the coupling to a honeycomb-lattice antiferromagnet [36,37], and the last term denotes the staggered electric field λ E .
In the low-energy Dirac approximation, the Hamiltonian (1) in the momentum space can be expressed as: where η = ±1 for the valleys K and K ′ respectively, σ i (i = x, y, z) is the i-component Pauli matrix for the sublattice pseudospin, and: where s = ±1 for the spin-up and spin-down modes respectively. By diagonalizing equation (3), one can get the gap of the energy spectrum with a spin s and valley η as 2 |∆|,where ∆ I(II) = sλ is the Dirac mass term for the upper (lower) half-nanoribbon. Based on the Dirac mass term, the corresponding Chern numbers are given as C [19,26]. In order to character the properties of inner edge states between the regions I and II, one can use the corresponding spin-and valley-dependent Chern numbers of inner edge states , which is consistent with the method of using the band structure. In addition, these Chern numbers of inner edge states stand for different inner edge states with spin and valley mode, where the sign of Chern numbers denotes its propagating direction.
With the tight-binding model, we use the wave-function matching technique [38,39] instead of the non-equilibrium Green's function to investigate the intervalley and intravalley scatterings of inner edge states, the transmission coefficient from the lead L to the lead R reads: where the transmission amplitude from the incoming mode m to the outgoing mode n is expressed as: its detailed derivation is listed in the appendix A. In order to further understand the transport properties of inner edge states, one needs to master the transport details by the local bond current. With the Schrödinger equation and wave-function matching technique, the local bond current flowing from atoms i to j with an incoming mode m from the lead L is expressed as: where C m is the wave function of the conductor and H ij denotes the matrix element of the conductor's Hamiltonian. Also, the detailed derivation of equation (6) is presented in the appendix A. It is noted that we assume the local bond current is from the lead L.
In our calculation throughout this paper, the external fields in equation (1) are all chosen as relatively large values ±0.2t. Compared to the values of external fields, graphene, silicene, germanene and stanene have small intrinsic spin-orbit coupling (SOC) values of 10 −3 meV, 3.9 meV, 40 meV and 100 meV respectively in 2D materials with honeycomb lattice. And these values of intrinsic SOC have very small effect on outer and inner edge states induced by relatively large values of external fields, discussed in appendix C. Therefore, the following results and discussions are consistent with graphene, silicene and germanene.

Results and discussions
We investigate different inner edge states in hybrid nanoribbon, where the upper and lower half-nanoribbons are in different topological phases, and then the corresponding transport properties and applications in topological junction with inner edge states. Throughout the paper, the fixed size parameters for calculating the band structures and transport phenomena are set as N y = 20 containing 80 atoms along the armchair boundary and N x = 14 for the length of the conductor, as shown in figure 1.
The red (blue) denotes the spin-up (spin-down) mode, the black denotes the spin degeneracy, the direction of the arrow denotes the propagating direction of the edge state and the width along armchair boundary contains 80 atoms as shown in figure 1.

Different types of inner edge state
In figure 2(e), it shows the antichiral edge state for the outer and inner boundaries, which is induced by the left and right circulation of the ORCP light applied on the upper and lower half-nanoribbons respectively. The outer edge states propagate to the left, while the inner edge states with the valleys K ′ and K propagate to the right, which can be obtained by the corresponding band structure in figure 2(a). The antiferromagnetic exchange field and staggered electric field in addition to the ORCP light can also induce the spin-polarized antichiral edge states for outer and inner boundaries as shown in figures 2(f)-(h) (we chose the case in figure 2(b) as an example to illustrate the edge states by the wave function distributions in appendix B), and the corresponding band structures are presented in figures 2(b)-(d) respectively. Actually, the edge states distributed in outer and inner boundaries in figures 2(e)-(g) have reverse cases with opposite propagating direction, we just show the reverse case in figure 2(h) for the one in figure 2(g). In particular, these spin-polarized edge states are a platform for designing a spin filter with spin-up or spin-down mode. One  can use ∆C sη = (∆C ↑K ′ , ∆C ↓K ′ , ∆C ↑K , ∆C ↓K ) to character different types of inner edge states. We just regard the case in figure 2(f) as an example to explain its formation. According to the spin-and valley-dependent Chern numbers C sη and the parameters of external fields, one can obtain two Chern numbers C I sη = (−1/2, −1/2, 1/2, −1/2) and C II sη = (−1/2, 1/2, 1/2, 1/2). Therefore, the Chern numbers of inner edge state is ∆C sη = (0, 1, 0, 1), meaning the existence of inner edge states with spin-valley modes K ′ ↓ and K ↓ in positive propagating direction. Moreover, in hybrid nanoribbon, the initial edge states in outer boundaries remain unchanged.
After introducing the spin-polarized edge states for the outer and inner boundaries. Here, we introduce valley and spin-valley polarized edge states. In figures 3(a)-(c), one can see that these band structures have the identical gapless bands in the bulk gap, in which they also share the same edge states for the outer and inner boundaries with the spin-degeneracy, spin-down and spin-up modes respectively as shown in

The intravalley and intervalley scatterings
In the following, we focus on the transport properties in topological junction with different edge states. In figure 4(a), one can see that the spin-valley polarized inner edge states between regions P and N are consistent for the spin-up mode, propagating direction and path, except for the valleys. Along the inner boundary, we expect that the transmission from the left to the right of the intervalley scattering is 100% although the large wave-vector distance (wave-vector mismatch) between the valleys K ′ and K. In figures 4(b) and (c), the corresponding transmission is 100% and local bond current propagates along the inner boundary, which indicates that when the factors of the spin mode, propagating direction and path of inner edge state are consistent between regions P and N, the transport phenomenon is independent on the wave-vector mismatch. Moreover, when the condition that the former three factors are both consistent is not satisfied, the latter factor has an influence on the transport phenomenon discussed in figure 5. It is also noted that the local bond current from the left passes through the system with slight mess due to the wave-vector mismatch.
In this section, we investigate other transport phenomenon of co-existing intravalley and intervalley scattering. In figure 5(a), it shows that the inner edge states with the valleys K ′ and K propagate from the left  in the region P while the inner edge state with the valley K propagates along the same direction in the region N. Besides, the spin mode, propagating direction, propagating path and wave vector are consistent for the inner edge states with the valley K between the regions P and N, but for the inner edge states with the valleys K ′ and K, there exists a wave-vector mismatch. Therefore, we expect that the transmission of the intravalley scattering is 100%, and the transmission of the intervalley scattering is zero due to its propagating channel is occupied by the intravalley scattering. In figures 5(b) and (c), the corresponding transmissions and local bond currents indicate the above results. In figure 5(c), one can see that when the former three factors are consistent, the local bond currents with the intravalley scattering preferentially occupy the propagating channel due to the wave-vector match, while the local bond currents from the left are absolutely reflected back along the lower boundary and not upper boundary. Because in the reflected process, the reflected current need to the buffer area in the region N and then is fully reflected back, and the upper buffer area in Here, we investigate the transport phenomenon dependent on the wave-vector mismatch. In figure 6(a), it shows that the topological junction is constructed by the spin-polarized antichiral edge states with opposite propagating direction for outer and inner boundaries. Obviously, because of the opposite propagating direction of the inner edge states between regions P and N, the corresponding local bond currents will not choose the inner boundary channel but the outer ones in the region N. In figure 6(b), it shows that the transmission T k←K ′ gradually decreases from 1 to 0 with the increasing energy, while the transmission T k←K has opposite tendency compared to T k←K ′ . Through the analysis of the band structures in regions P and N, one can find that these tendencies are caused by the wave-vector mismatch. For instance, from the band structures in figures 2(c) and (d) for the regions P and N respectively, one can see that the distance between the valley K ′ in figure 2(c) and the non-valley k (outer edge state) increases with increasing the energy in the bulk gap, while the distance between the valley K in figure 2(c) and the non-valley k decreases with increasing the energy. These results indicate that the greater the wave-vector mismatch, the less the transmission. It is noted that the summation of the transmission T k←K ′ and T k←K is always 100%. In figures 6(c) and (d), we also investigate the local bond current. In figure 6(c), at E = −0.15 eV, the corresponding inner local bond current chooses to the outer boundaries to propagate with transmission 100%, caused by the propagating direction mismatch between the inner edge states. Based on this mismatch, at E = 0, half of local bond current propagates along the outer boundary and the other half is reflected along the outer boundary due to the wave-vector mismatch. In figure 6(e), due to the large wave-vector mismatch, the inner local bond current is fully reflected back along the outer boundary. In overall effect, the local bond currents with the valleys K ′ and K are reflected back and propagate along the outer boundaries respectively with transmission 100% as shown in figure 6(f).

Applications for different filters
Based on the analysis of the transport phenomena above, we construct some types of topological junction with the edge states for applications in transport. In figures 7(a)-(c), there are spin-degenerated antichiral edge states in all regions P for the outer and inner boundaries, and in all regions N for spin-polarized antichiral, spin-valley polarized and valley-polarized cases for the outer and inner boundaries respectively. In figure 7(a), one can expect that the inner edge states with the spin-up mode and the valleys K ′ and K occupy their respective propagating channels, while the inner edge states with the spin-down mode are fully reflected due to the gap effect, which induced the spin filter with transmission 2 as shown in figure 7(d). In figure 7(b), we can find that the spin-up edge state with the valley K occupies the propagating channel, while the other inner edge state is reflected, which induces the spin-valley filter with transmission 1 as shown in

Conclusions
In summary, we have investigated some types of inner edge states in hybrid nanoribbon and obtained local bond current with the wave-function matching technique to investigate their transport phenomena and applications in topological junction with regions P and N. These results show that different combination of the ORCP light, antiferromagnetic exchange field and staggered electric field can facilitate spin-polarized, spin-valley-polarized and valley-polarized antichiral inner edge states with the inner spin-and valley-dependent Chern numbers. Based on these inner edge states, we further find three transport phenomena by local bond current. (a) When the spin mode, propagating direction and path of the inner edge states are consistent between regions P and N, the transmission of the intervalley scattering is 1 with slight mess of local bond current due to the wave-vector mismatch. (b) Based on these consistent factors, when the inner edge states with the valleys K ′ and K simultaneously propagate from the left, the current of the intravalley scattering preferentially occupies the inner propagating channel due to the wave-vector match, while the other type of the current is fully reflected along the outer boundaries. (c) When the spin mode and propagating direction are consistent between the inner and outer edge states without the condition of the consistent propagating path, the transport phenomenon is dependent on the wave-vector mismatch. The transmission will decrease from 1 to 0 as the degree of the wave-vector mismatch increases, and the local bond current propagates from the inner boundary to outer boundaries. In addition, the transmission summation of the intravalley and intervalley scatterings always remains 1. Based on these transport phenomena, we also design the spin-polarized, spin-valley-polarized and valley polarized filters, which supports a platform for the future nanodevice.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

Appendix A. Transmission and local bond current
In this section, we first discuss the Bloch matrices to find the solutions of the right-and left-going modes for the leads, and then the transmission amplitude and local bond current with the wave-function matching technique. In figure A1, the Schrödinger equation for the lead L (R) is expressed as: Where H L(R) and B L(R) are the unit-cell Hamiltonian and the coupling Hamiltonian between nearest unit cells respectively in the lead L (R), I denotes the unit matrix with dimension M and ϕ i is the M-dimension eigenvectors on all sites of the unit cell i. According to the Bloch theorem ϕ i = λ i−j ϕ j with λ = e ikxa , equation (A1) can be transformed as: where 0 is M × M zero matrix. It can be known that equation (A2) has 2M solutions for eigenvalue λ, which can be classified into M right-going modes and M left-going modes [40,41]. In addition, these going modes contain the corresponding propagating modes and evanescent modes. For the evanescent modes, |λ| < 1 corresponds to the right-going evanescent mode and |λ| > 1 corresponds to the left-going evanescent mode.
In particular, |λ| = 1 corresponds to both right-and left-going propagating mode, one has to use the sign of the Bloch velocity to distinguish the propagating direction. The Bloch velocities are given by this expression: In figure A1, the red dotted box indicates the Hamiltonian including the conductor and two unit-cells in both lead L and lead R, the corresponding tight-binding equations can be expressed as: Specially, we assume that the accident wave is from the lead L and then transmits to the lead R or reflects into the lead L. Therefore, the wave function at site j in the lead L can be written as the combination of incident and reflected wave: where [λ n (±)] j ϕ n (±)α n (±) with the corresponding amplitude coefficient α n of ϕ n ; while the wave function in the lead R is written as the transmitted wave: Moreover, it is noted that the eigenvectors are nonorthogonal in general, the dual vectorsφ n (±) are defined as:φ † n (±)ϕ m (±) = δ nm , ϕ † n (±)φ m (±) = δ nm . (A7) Figure A1. Schematic view of Hamiltonian matrix of a system divided into slices. The yellow and green regions denote the leads and conductor respectively, and the red dotted box region is used to derive and calculate the transmission coefficient and local bond current.
From the above equations, one can find the expressions in the following: where: Combining equations (A8) and (A9), equation (A4) can be rewritten as: In the matrix form with the Green's function, equation (A10) leads to this expression in the following: and the Green's function in equation (A11) reads: where the G 1 and G 3 are the M × M matrices and G 2 is the (N × M) × M matrix, the Hamiltonian reads: Combining equations (A11) and (A12), one can get this expression: Actually, we can choose an incoming mode ϕ L,m (+) from the lead L as a start to calculate the transmission element. And the wave function in the last equation in equation (A14) can be expressed in the other form: Compared with the last one in equation (A14), the transmission element is induced as: Then, the corresponding normalizing transmission element can be induced as: Besides, when ϕ L,m (+) is chosen for all possible incoming modes m = 1 · · · M, all the corresponding normalizing transmission elements can be found. Therefore, the transmission coefficient from the lead L to lead R is naturally written as: By using the Schrödinger equation, the local bond current propagating from the atoms i to j with an incoming mode m from the lead L is expressed as: with the first one in equation (A14), the wave function C D,i at site i can be solved by:

Appendix B. The wave function distributions
Here, we choose a case in figure 2

Appendix C. Transmission and band structure under different spin-orbit coupling
In this section, we discuss the effect of different spin-orbit coupling λ so on the transmission and band. The case in figure 6 is just chose as an example to illustrate the validity of our results, other cases are not shown here. In figures A3(a) and (b), one can find that weak λ so has on influence on the band structure in the region P in figure 6(a). In figures A3(c) and (d), the stronger λ so decreases the energy range (red dotted box) for outer and inner edge states, while the bands for outer edge state are split. In addition, compared to the case in figure A3(c), the energy range in figure A3(d) almost keep unchanged by the wider width. It is noted that the types of outer and inner edge states are independent on the λ so , the corresponding wave function distributions are not shown here. Actually, we can also use the spin-and valley-dependent (inner) Chern numbers C (∆C sη = (∆C ↑K ′ , ∆C ↓K ′ , ∆C ↑K , ∆C ↓K ))to explain the outer (inner) edge states. According to the values of parameters chose as 0.2t, the range 0 ⩽ λ so < 0.2t will not change the sign of C I(II) sη as well as ∆C sη . Therefore, the considered values of the spin-orbit coupling have no influence on the types of outer and inner edge states. Besides, the transmissions in, (a1)-(d1) share the structures with the case in figure 6(b), controlled by the wave-vector mismatch. In figure A4, one can see that when ten vacancies are randomly distributed in the conductor region consisting of 1120 atoms, the transmission (blue line) slightly decreases compared to the red line. Subsequently, the transmissions are gradually decrease with the increasing number of vacancies. It is noted that large number of vacancies can dramatically destroy the transport properties shown in the cyan and magenta lines, while moderate number of vacancies have litter impact on the transport properties. Moreover, other cases also share same tendency that vacancies decrease the transmission, not shown here. Therefore, we come to the conclusion that the results are moderately robust against random vacancies.