The antichiral edge states can be obtained based on the modified Haldane model in zGNRs. Considering the side potentials *U*1,2, *E**Z*1,2, and *M*1,*2*, the corresponding Hamiltonian with the tight-binding model is described as:

where are the electronic creation (annihilation) operator with the spin \(\sigma \;(\sigma = \uparrow \downarrow )\) at site *i*; \(⟨i,j⟩\) and \(⟨⟨i,j⟩⟩\) run over all the nearest and the next-nearest-neighbor hopping sites. The first term describes the nearest-neighbor coupling of electrons with *t* = 2.7 eV. The second term denotes the modified Haldane model resulting in the antichiral edge state, which has been experimentally demonstrated [20, 21]. The \({t_2}\) and \(\phi\) is set as 0.03 eV and \(- \pi /2\), respectively. For the modified Haldane model, \({v_{ij}}=1( - 1)\) represents the counterclockwise (clockwise) hopping between the sublattice A, while \({v_{ij}}= - 1(1)\) for the sublattice B. The third and last terms are the side potential, including the potential field *U*1,2, the staggered electric potential *E**Z*1,2 with \({\mu _i}= \pm 1\) for A or B sublattice, and the exchange field *M*1,2, which are applied along the boundaries of the nanoribbon. \({\sigma _z}\) are the *z* components of \(2 \times 2\) Pauli matrix for the electron spin. As shown in Fig. 1, the side potential is applied along two boundaries of the zGNRs with the same width *W = 8*. Here we note that the *U*, *E*Z and *M* are applied separately or jointly along two boundaries of zGNRs according to the different modulation requirements.

For the three-terminal devices, the transmission coefficients (*T**ij*) from lead *i* to lead *j* is calculated by the Non-equilibrium Green’s function (NEGF) formalism. In the spin-resolved case, it is expressed as

$$T_{{ij}}^{\sigma }(E)=Tr[{\mathbf{\Gamma }}_{j}^{\sigma }(E){{\mathbf{G}}^{R,\sigma }}(E){\mathbf{\Gamma }}_{i}^{\sigma }(E){{\mathbf{G}}^{A,\sigma }}(E)]$$

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In Eq. (2), \({{\mathbf{G}}^{R,\sigma }}(E)\) and\({{\mathbf{G}}^{A,\sigma }}(E)\) are the retarded and advanced Green’s function with the spin \(\sigma\); \({\mathbf{\Gamma }}_{i}^{\sigma }(E)\;(i=1,2,3)\) is the spin-resolved linewidth function of lead *i*, which describes the coupling between the conductor region and lead *i*. The retarded (advanced) Green’s function is calculated by the formula below.

$${{\mathbf{G}}^{R(A),\sigma }}(E)={[{E_{+( - )}}{\mathbf{I}} - {\mathbf{H}}_{D}^{\sigma } - \sum\limits_{i} {{\mathbf{\Sigma }}_{i}^{{R(A),\sigma }}(E)} ]^{ - 1}}$$

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In Eq. (3), \({E_+}=E+i\eta ={\left[ {{E_ - }} \right]^*}\), where and \(\eta\) are the incoming electron energy and an infinitesimal positive number, respectively; \({\mathbf{I}}\)is the identity matrix;\({\mathbf{\Sigma }}_{i}^{{R,\sigma }}(E)={{\mathbf{{\rm H}}}_{D,i}}{\mathbf{g}}_{i}^{{R,\sigma }}{{\mathbf{H}}_{i,D}}\) is the retarded self-energy matrix with \({{\mathbf{{\rm H}}}_{D,i}}\) and \({{\mathbf{H}}_{i,D}}\) being the coupling matrix between the conductor and the lead *i*; \({\mathbf{g}}_{i}^{{R,\sigma }}\)is the retarded surface Green’s function of lead *i*, which can be calculated by using the routine of Lopez-Sancho’s iterative method [32].

To investigate the antichiral edge states in zGNRs and understand the electron transport details in the three-terminal device, we plot the local bond current in the leads and conductor region. The energy-dependent local bond current formula between site *i* and *j* reads [33, 34]:

$$J_{{ij}}^{\sigma }(E)=H_{{ji}}^{\sigma }G_{{ij}}^{{<,\sigma }}(E) - G_{{ji}}^{{<,\sigma }}(E)H_{{ij}}^{\sigma }$$

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where \({{\mathbf{G}}^{<,\sigma }}(E)\) is the lesser Green’s function in the energy domain expressed as:

$${{\mathbf{G}}^{<,\sigma }}(E)= - i{{\mathbf{G}}^{R,\sigma }}(E){\mathbf{\Gamma }}_{\alpha }^{\sigma }(E){{\mathbf{G}}^{A,\sigma }}(E)$$

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and \(H_{{ij}}^{\sigma }\) is the relevant matrix element of the conductor’s Hamiltonian. It is noted that this formula is related to the local bond current from the incidence of lead \(\alpha\).