Spin-Polarizing Electron Beam Splitter from Crossed Graphene Nanoribbons

Junctions composed of two crossed graphene nanoribbons (GNRs) have been theoretically proposed as electron beam splitters where incoming electron waves in one GNR can be split coherently into propagating waves in two outgoing terminals with nearly equal amplitude and zero back-scattering. Here we scrutinize this effect for devices composed of narrow zigzag GNRs taking explicitly into account the role of Coulomb repulsion that leads to spin-polarized edge states within mean-field theory. We show that the beam-splitting effect survives the opening of the well-known correlation gap and, more strikingly, that a spin-dependent scattering potential emerges which spin polarizes the transmitted electrons in the two outputs. By studying different ribbons and intersection angles we provide evidence that this is a general feature with edge-polarized nanoribbons. A near-perfect polarization can be achieved by joining several junctions in series. Our findings suggest that GNRs are interesting building blocks in spintronics and quantum technologies with applications for interferometry and entanglement.

Figure 1

Transport setup and spin-dependent properties for $AB$-stacked 8-ZGNR devices. (a),(b) Two different self-consistent solutions for the spin-density distribution in the device region, labeled $\boxed{\uparrow \downarrow }$ and $\boxed{\uparrow \uparrow }$, respectively, defined by the spin orientation of the lower edge of each GNR. The up (down) spin density is shown in red (blue). The lower, horizontal ribbon is plotted in black, while the upper, intersecting at an angle of 60°, is depicted in gray. Electrodes 1–4 are indicated. The ribbons are separated by a distance $d=3.34\AA$ along the $z$ axis, as displayed in the side view [lower part of (b)]. The dashed lines in each configuration indicate a symmetry axis that maps the device geometry to itself through mirror operations, where the red (black) color of the axis further indicates that the spin index is inverted (conserved) by the symmetry operation. (c),(d) Spin-resolved density of states of scattering states incoming from electrode 1 for the $\boxed{\uparrow \downarrow }$ and $\boxed{\uparrow \uparrow }$ spin configuration, respectively, computed at $E-E_F=0.5\mathrm{eV}$. The dominant spin on each site at this energy is shown in red for up spins and in blue for down spins. (e),(f) Sketch of incoming and outgoing waves through the scattering center (represented by the circled cross) and the corresponding transmission probabilities from calculations.

Figure 2

Spin- and energy-resolved transmission probabilities $T_{12}$, $T_{13}$, $T_{14}$ and reflection $R_1$ for (a)–(d) the $\boxed{\uparrow \downarrow }$ and (e)–(h) $\boxed{\uparrow \uparrow }$ configurations of Fig. 1. Electrons are injected from electrode 1. The red (blue) curves correspond to the up (down) spin components with $U=3\mathrm{eV}$. For comparison, the corresponding calculations for the unpolarized case ($U=0$) are indicated by dashed gray lines.

Figure 3

Spin polarization $P_{12}$ of the current from electrode 1 to 2 as a function of in plane translations of one ribbon with respect to the other for (a) $\boxed{\uparrow \downarrow }$ and (b) $\boxed{\uparrow \uparrow }$ configurations introduced in Fig. 1. The electron energy is in the conduction band at $E=0.5\mathrm{eV}$. The in plane unit cell (dashed lines) has lattice vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$, where $a_0=2.46\AA$ is the graphene lattice constant. The red crosses (green disks) indicate the high-symmetry configurations with $AB$ ($AA$) stacking.

Figure 4

(a) Sketch of an array of three consecutive $AB$-stacked 8-ZGNR crossings to enhance the spin-polarized current at the output electrode 2. (b) Spin polarization $\overline{P_{12}}(N)$ (filled symbols) and majority-spin transmission $\overline{T_{12}^{\mathrm{maj}}}(N)$ (open symbols) as a function of the number of crossings $N$ between terminals 1 and 2 in the conduction band at $E=0.5\mathrm{eV}$. Two different scenarios are considered: an ideal arrangement of identical $\boxed{\uparrow \uparrow }$ $AB$ crossings (blue circles) as well as a random sampling (orange squares) over ${10}^7$ different spin, intersection angle (within 55–65°), and translation configurations drawn from the data in the Supplemental Material, Sec. S11 [41], assuming equal weights. The average polarization $⟨|\overline{P_{12}}(N)|⟩$ follows an analytic expression (black line, Supplemental Material, Sec. S12 [41]) approaching 1 exponentially in $\sqrt{N}$. The gray lines indicate the best (1st, 5th, 10th, 25th, 50th) percentiles of the random distribution.