Electron beam-splitting effect with crossed zigzag graphene nanoribbons in high-spin metallic states

The increasing interest in graphene nanoribbons (GNRs) for molecular-scale electronic and spintronic devices has emerged because it is well known that they inherit some of the exceptional properties of graphene while having tunable electronic properties, such as the dependence of the bandgap on their width and edge topology,¹ and the appearance of π-magnetism,² absent in pure two-dimensional (2D) graphene. Moreover, these systems are a remarkable platform for electron quantum optics, where the electrons propagating coherently in these ballistic conductors resemble photons propagating in optical waveguides.³ On the one hand, it has been shown that electrons can propagate without scattering over large distances of the order of ∼100 nm in GNRs.^4–6 On the other hand, ballistic transport in ZGNRs can be fairly insusceptible to edge defects as a consequence of the prevailing Dirac-like behavior, which makes the electronic current flow maximally through the central region of the ribbon.⁷ Furthermore, with the advent of bottom-up fabrication techniques, long samples of GNRs free of defects can now be chemically realized via on-surface synthesis, as demonstrated in the seminal works by Cai et al. for armchair GNRs⁸ and by Ruffieux et al. for ZGNRs.⁹ 

It is known that the ground state of ZGNRs corresponds to a ferromagnetic (FM) ordering of spins along the edges and antiferromagnetic (AFM) ordering between the edges,^10,11 i.e., with total spin projection per unit cell equal to zero, S_z = 0. In this configuration, the magnetic instabilities of the states localized at the edges coming from the flat bands of ZGNRs open a bandgap due to the Coulomb repulsion in the otherwise metallic ribbons.¹² The opening of the bandgap and the edge states associated with the AFM coupling in ZGNRs have been confirmed by experiments, where the magnetic order has been shown to be stable up to room temperature.^13,14 The spin-polarized states localized at the edges are coupled such that there is an energy penalty to excite the AFM ground state to the FM state (exchange interaction). In the case of wider ZGNRs, the AFM and FM solutions are close in energy (small exchange interaction) due to the decoupling of the localized edge states, as they decay exponentially toward the center of the ribbon.^10,15–18 In this case, the FM solution can be favored due to the Zeeman energy under a relatively small magnetic field. The presence of a net spin-polarization, in the absence of transition metals or heavy atoms, makes these structures privileged for spintronics due to the weak spin scattering in pure carbon-based systems.^2,19 For instance, the intrinsically weak spin-orbit and hyperfine couplings in graphene lead to long spin coherence and relaxation times²⁰ and a large spin-diffusion length that is expected to reach $∼10μ$m even at room temperature.²¹ 

Recently, devices formed of crossed GNRs have been predicted to behave as perfect beam splitters, where the injected electron beam is divided into two of the four arms with near 50–50 probability and zero backscattering.^22–24 Furthermore, the particular case of devices formed of crossed ZGNRs is even more interesting, since they can create a spin-polarizing scattering potential²⁵ where the device can work as a spin-polarizing beam splitter. Following these ideas for electron quantum optics applications, a Mach–Zehnder-like interferometer in a GNRs network has recently been proposed.²⁶ In terms of their feasibility, manipulation of GNRs in STMs^27,28 has opened the possibility of building 2D multi-terminal GNR-based electronic circuits.²⁹ The spin properties of such devices can be addressed by measuring with spin-polarized STMs^30,31 and probed by shot-noise measurements.³² For instance, a device formed of two crossed ZGNRs has been experimentally realized with the control over the crossing angle reaching a precision of 5°.³³ 

While, in previous studies, only the AFM regime has been explored, other spin configurations can appear and show interesting spin-polarized transport properties. For instance, in contrast to the AFM case, the FM band structure of periodic ZGNRs does not show a bandgap around the Fermi level, which makes this regime interesting since there is conduction of electrons at the Fermi level. Given the metallic character of the FM-like spin configuration, one can envision to generate a minimal excitation in the device with only one particle and no hole (a leviton)^34–37 by applying a Lorentzian-like voltage pulse of specific amplitude and duration, enabling the generation of a single-electron excitation.³⁸ 

Here, we analyze the functioning of an electronic beam splitter built with two crossed ZGNRs (of width 30 carbon atoms across) in an FM-like configuration, i.e., where the total magnetization of the device is different from zero. To describe the spin physics of the system, we employ the Hubbard Hamiltonian in the mean-field approximation (MFH).³⁹ The main complexity of the modeling lies in the description of the coupling between ZGNRs at the crossing, for which we use a Slater–Koster parameterization⁴⁰ that has shown to be in good agreement with other more accurate descriptions, such as density functional theory.²⁴ By employing this simple, yet powerful description based on single-electron physics, we can explore large systems composed of $∼8000$ atoms.

The manuscript is structured as follows: In Sec. II, we explain in detail the theoretical methods employed in this work (MFH Hamiltonian and NEGF formalism). In Sec. III, we present the obtained results for a device formed of two crossed-wide ZGNRs in its FM-like configuration, and finally, the conclusions are provided in Sec. IV.

The system of study is composed of two infinite crossed ZGNRs placed one on top of the other separated by an inter-ribbon distance d, with a relative crossing angle of around θ = 60°, as shown in Fig. 1. Here, the semi-infinite electrodes are indicated by red squares numbered 1–4.

As the junction between the ribbons breaks the translational invariance of the perfect ZGNRs, we use Green’s function^43,44 formalism to solve the Schrödinger equation for the open quantum system. Details of the implemented MFH model with open boundary conditions⁴² can be found in the supplementary material of Ref. 25.

In this section, we analyze the transport properties for a device formed of two crossed ZGNRs of W = 30 carbon atoms across (30-ZGNR) as a function of the inter-layer separation d for values close to the typical distance between layers in graphite (d = 3.34 Å), and the intersecting angle θ for values close to the commensurate case where θ = 60°. To understand the spin states of ZGNRs, we performed different spin-polarized calculations changing the total mean value of the spin operator $Ŝz$ per unit cell, $⟨Ŝz⟩=12∑i⟨ni↑⟩−⟨ni↓⟩≡Sz$⁠, where the summation goes over the sites i within the unit cell of the periodic ZGNR.

In Fig. 2(a), we show the total energy per unit cell as a function of S_z relative to the case of S_z = 0 (the AFM case) for a periodic ZGNR of W = 30 carbons across. As can be seen here, there is a local minimum at S_z = 0.317, corresponding to the solution of the lowest energy for S_z ≠ 0. The fact that the solution of minimum energy appears at such total S_z can be understood from the fact that, in the AFM case, the local spin projection summed over the bottom (or top) half of the unit cell of the ZGNR is $|Szhalf|=0.159$⁠. This means that the total S_z per unit cell in the FM case needs to reach twice this value to flip the local magnetic moment at one edge. Note that the magnetic moment associated with S_z is μ = g_Sμ_BS_z, where g_S ≈ 2 is the electron spin g-factor and μ_B is the Bohr magneton. To see to what extent the ribbon width affects these results, we compare E(S_z) for W = 10, 20, 30, 40-ZGNRs in Fig. S1 in the supplementary material, where we observe two main features: While the qualitative behavior is the same for all of them, the value of S_z at which the minimum of energy appears is larger for wider ribbons, and, as expected, the minimum value of E(S_z > 0) diminishes with the width.

For each S_z, we plot the energy corresponding to the spin configuration of the lowest energy in Fig. 2(a). Here, we distinguish between two phases depending on S_z: AFM character (for S_z < 0.15), where the spin polarization shows opposite spin majorities at the edges, and FM character (for S_z > 0.15), where the spin polarization shows the spin majority of equal spin index. The two insets to Fig. 2(a) show the spin polarization for a 30-ZGNR: one in the AFM-like spin configuration (calculated with S_z = 0.08), where it can be seen that the colors at the edges are different (red and blue), and another one in the FM-like spin configuration (calculated with S_z = 0.21), where it can be seen that the same color appears at both edges (red). In the case of the AFM-like spin configuration for S_z ≠ 0, not only the sign of the local magnetic moments at the bottom and top edges of the unit cell is different but also the magnitude, as a consequence of the existing spin imbalance. Whereas when the FM character is achieved, both the magnitude and sign of the local magnetic moments at the bottom and top edges of the unit cell are equal.

In Fig. 2(b), we plot the band structure for the FM solution of the lowest energy for the 30-ZGNR, obtained with S_z = 0.317, for spin σ = ↑ (red lines) and σ = ↓ (blue lines). Here, we can observe the metallic character of the FM configuration for the ZGNR, as there are states available at the Fermi level, E_F, for both up and down spins.

As mentioned above, although the ground state corresponds to the configuration with S_z = 0, the presence of a magnetic field B in the z-direction (cf. Fig. 1) can stabilize a high-spin configuration due to the Zeeman energy ΔE = μB = g_Sμ_BS_zB. For instance, the corresponding electronic energy E(S_z) for the FM-like configuration of the lowest energy is E(S_z = 0.317) = 0.97 meV/cell above the ground state, implying that a critical magnetic field of the order B_c = 26.6 T (parallel to the z-axis in this case) is needed to make the two spin states degenerate. In Fig. 3(a), we study the zero-bias conductance G_αβ(V) with (α, β) ∈ {(1, 2), (1, 3)} (black and green lines, respectively) for a device formed of two crossed 30-ZGNRs as a function of the inter-layer separation d. Here, V represents a rigid shift of the Fermi level E_F. We consider inter-layer distances close to the typical van der Waals distance between graphene layers in graphite (d = 3.34 Å).^23,49,50 In the first place, we can infer that the total spin-averaged conductance (sum of intra- and inter-layer conductances) is 1 since the values for $Ḡ12$ and $Ḡ13$ are symmetric with respect to 0.5G₀, which means that there is no backscattering for an incoming electron at the Fermi level in these devices at least for these ranges of d and θ. In the second place, we observe an oscillating behavior of $Ḡαβ$ with respect to this varying parameter. For instance, the inter-/intra-layer conductance ratio reaches its maximum for d = 3.34 Å. While one would expect that for smaller inter-layer distances d the interlayer $(Ḡ13)$ conductance would increase, as the interlayer hopping integral depends exponentially on the distance between the ribbons, we observe a decrease (and increase in $Ḡ12$⁠) for smaller d in Fig. 3(a), as a consequence of an interference process due to the scattering potential created by the crossing. We also observe that, for d between 3.44 and 3.49 Å, there is a crossing between $Ḡ12$ and $Ḡ13$⁠, implying that, for that inter-layer separation, the device behaves as a perfect 50:50 beam splitter where the incoming electron beam is equally separated in the two possible outgoing directions with $Ḡαβ=0.5G0$ for low gate voltages V.

Similarly, in Fig. 3(b), we study G_αβ(V) for different crossing angles close to the commensurate configuration with θ = 60°. We apply the rotation around the center of the scattering region (crossing) that is obtained for the case with θ = 60° and accounts for the effect of different possible stackings by averaging over the in-plane translations of one ribbon with respect to the other. By doing so, we aim to provide a comprehensive overview of the results, accounting for the variability in stacking configurations that might occur in practical scenarios. The in-plane unit cell is determined by the graphene lattice vectors. We obtain the conductance for a mesh of four points along each lattice vector within the unit cell. The error bars are calculated as the standard deviation of the spin-averaged conductance $Ḡαβ$ at each point, averaging over the in-plane translations. The observed variance of approximately ∼10%–20% reflects the variations across different translational configurations, showing the inherent differences sampled by these translations. However, not all the stackings are equivalent. For instance, the most energetically favorable (and therefore most likely) configuration is the AB-stacking (see the supplementary material of Ref. 25). By analyzing the transport properties relative to this varying parameter in Fig. 3(b), we observe, on the one hand, that the inter-/intra-layer conductance ratio reaches its maximum for θ = 55°, 65°. On the other hand, the sum of the total spin-averaged conductance is 1 as in panel (a), since the values for $Ḡ12$ and $Ḡ13$ are symmetric with respect to 0.5G₀ as well, meaning that the variation of θ does not introduce backscattering. We can see that the oscillatory dependence of the conductance on the crossing angle is less smooth than the one seen in Fig. 3(a). This occurs due to a more complicated dependence of the σ- and π-type hopping integrals on θ.

To see the effect of the width on the transport properties as a function of these two varying parameters, we performed a similar analysis for a 20-ZGNR device in the supplementary material (see Fig. S2), where we observe that, qualitatively, the behavior is maintained. For further detail, we plot the energy-resolved transmission probabilities for the 30-ZGNR device as a function of d and θ in the supplementary material (see Figs. S3 and S4).

Finally, we note that it has been previously shown that the symmetries associated with the spatial distribution of the spin densities are crucial for the transport properties of the device.^24,25 In this case, since the FM character implies that ⟨n_↑⟩ ≠ ⟨n_↓⟩, there will not be a symmetric behavior for the existing spin channels. However, the spin-density distribution possesses a symmetry axis at y = sin (−60°)x that maps the device geometry to itself through mirror operations, and applies to each spin component individually (conserves the spin index). As it has been shown in Refs. 24 and 25, certain symmetrical combinations of electrodes lead to equal transmission probabilities $Tαβσ=Tγδσ$⁠. In this case, the symmetrical electrode mapping corresponds to (1, 2, 3, 4) ↔ (4, 3, 2, 1).

We have analyzed the electron transport properties for a device formed of two crossed infinite ZGNRs of W = 30 carbon atoms across (30-ZGNRs) as a function of the spin configuration by fixing different values for the total spin per unit cell S_z. In the first place, by computing the total energy associated with these configurations E(S_z), we have shown that there is a local minimum for the solution with S_z > 0, with E(S_z > 0) close to 1 meV/cell above the ground state [E(0)]. We have also seen that, depending on S_z, there are two possible phases: AFM-character, where the edges of the ZGNR unit cell are populated by opposite spin majorities, and FM-character, where the two edges of the ribbon are populated by the same spin majority. These two phases appear for S_z < 0.15 and S_z > 0.15, respectively. We also computed the band structure for the FM-like configuration of the lowest energy, where we observe that this system in such a spin state shows a metallic character. We estimate that the critical magnetic field needed to make this FM-like solution degenerate with the AFM ground state is B_c = 26.6 T for this particular case, although this value will further decrease for wider ribbons.

We have also calculated the inter- and intra-layer electrical conductances for different gatings varying the inter-layer distances, for distances close to the van der Waals distance between graphene layers in graphite (d = 3.34 Å), and crossing angles close to the commensurate stacking where θ = 60° for this four-terminal device. We have shown that the (spin- and displacement-averaged) electrical conductance displays an oscillatory behavior with respect to these varying parameters at low gate voltages (−10 meV ≤ V ≤ 10 meV) while maintaining the sum $Ḡ12+Ḡ13=1$⁠, which means that there is no backscattering for the devices for different values of d and θ within the shown ranges nor conductance into terminal 4. The maximum value for the inter-/intra-layer spin-averaged conductance ratio $(Ḡ13/Ḡ12)$ for this device is found for d = 3.34 Å and θ = 55°, 65°. In addition, to show that these results are not exclusive to the chosen ZGNR width, we performed a similar analysis for a 20-ZGNR device (see supplementary material), where we show that it possesses similar qualitative behavior.

The results presented here add to the vision of using GNR-based devices for spintronics and quantum technologies. On top of the already discussed properties and applications of spin-polarized GNR-based beam splitters for electron quantum optics,^24–26 this device in its FM-like spin configuration can be a promising candidate due to its metallic nature, which facilitates electron injection through the generation of a minimal excitation. This can be achieved by applying a Lorentzian-like voltage pulse with a specific amplitude and duration to produce a single-electron excitation within the device.^34–37 In fact, performing time-dependent quantum transport calculations for levitonic excitations^34–37 could offer critical insights into the nonequilibrium dynamics of the proposed devices, and further elucidate the role of minimal excitation states in transport phenomena of the charges injected by the pulse.^51–53 

See the supplementary material for additional calculations, including transport calculations for devices with other ribbon widths and transmission curves as a function of electronic energy for the device discussed in the main text.

This work was funded by the Spanish Grant No. MCIN/AEI/10.13039/501100011033 (Grant Nos. PID2020-115406GB-I00, TED2021-132388B-C44, PID2022-140845OB-C66, and JDC2022-048665-I), the Basque Department of Education (Grant No. PIBA-2023-1-0021), the University of the Basque Country (UPV/EHU) through Grant No. IT-1569-22, and the European Union’s Horizon 2020 (FET-Open project SPRING Grant No. 863098).

The authors have no conflicts to disclose.

Sofia Sanz: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Géza Giedke: Formal analysis (supporting); Funding acquisition (equal); Investigation (equal); Supervision (supporting); Writing – original draft (supporting); Writing – review & editing (equal). Daniel Sánchez-Portal: Formal analysis (supporting); Funding acquisition (equal); Investigation (equal); Supervision (supporting); Writing – original draft (supporting); Writing – review & editing (equal). Thomas Frederiksen: Data curation (supporting); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Resources (equal); Software (equal); Supervision (equal); Visualization (equal); Writing – original draft (supporting); Writing – review & editing (equal).

The dataset that supports the findings of this study is publicly available on Zenodo, https://doi.org/10.5281/zenodo.14224593.