One-dimensional magnetism and Rashba-like effects in zigzag bismuth nanoribbons

Discoveries of low-dimensional quantum materials have renewed interest in some of the fundamental phenomena, such as magnetism and Rashba-type spin orbit coupling. In particular, exploring these phenomena by themselves and/or in combination within one-dimensional (1D) systems is of interest for fields as disparate as spintronics and biology. For example, a better understanding of Rashba-type spin orbit coupling in 1D systems may be used to explain spin-selective electron transport in long helical molecules. In this paper, using first principle calculations, we show that each edge of a zigzag nanoribbon composed of a bismuth (Bi) bilayer is a truly 1D structure that naturally combines both 1D magnetism and Rashba-type spin orbit coupling in a single system. In particular, we study the combined effects of exchange and spin-orbit coupling in nanoribbons that are: (i) ideal freestanding, (ii) placed on a hexagonal boron nitride substrate, and (iii) decorated with N atoms. The edges of the Bi zigzag NRs can display ferromagnetic, antiferromagnetic, or noncollinear ordering, resulting in a broken quantum spin Hall state. The interplay of Rashba and exchange effects in different magnetic phases can result in different spin-dependent transport regimes.

Figure 1

(a) Side and top views of the Bi (111) NRs with the zigzag relaxed edges (REs). (b) The top view of the structure with the shear distorted edges (SEs). The black vertical arrows indicate the shear distortions. (c) The side and top views of the edge structure with the dimerized Klein edges (DKEs).

Figure 2

Band structures of the nonmagnetic (NM) Bi (111) zigzag bilayer NR. (a) The edge bands localized on the left and right edges of the NRs are indicated by the green and orange circles, respectively. The bulk states are represented by the black curves. The edge states from the opposite sides are degenerate. Note that the gap between the edge bands and the bulk conduction bands is due to the finite width of the considered NR, which contains 16 zigzag chains. This gap should ultimately close for wider NRs, according to bulk-surface correspondence in 2D topological insulators. [(b)–(e)] Spin-resolved band structures for the edge bands localized on the left [(b),(c)] and right [(d),(e)] sides. The panels (b) and (d) correspond to the $\pm S_y$, while panels (c) and (e) to $\pm S_z$ spin projections; the $\pm S_x$ projections being small. Colors code the expectation values of the spin projections.

Figure 3

Spin densities of the NR with the open edges corresponding to AFM (left panel) and FM (right panel) solutions. Blue (yellow) color indicates the negative (positive) isovalues in the isosurface plots.

Figure 4

[(a),(b)] Band structures of the magnetic Bi (111) zigzag bilayer NR for the AFM stable self-consistent spin configurations. The edge bands localized on the left and right edges of the NRs are indicated by the orange and green circles, respectively. The bulk states are represented by the black curves. Two different band structures (a) and (b) correspond to two equivalent AFM solutions with opposite spin moments. Similar to the NM case, for the AFM structure, the edge states from the opposite sides are degenerate. To emphasize this, the orange circles in panels (a) and (b) are made larger to stand out. The oblique black arrows indicate the shift of the bands along the $\mathit{k}$ axis relative to those in the nonmagnetic phase [see Fig. 2]. (c) Schematic diagram showing Rashba effect in Bi zigzag NRs acting locally at each edge, where the potential gradients or electric fields develop. This is in contrast with the ordinary 1D systems, where such fields are usually associated with the nonequivalent edges (d).

Figure 5

Band structure for the Bi (111) zigzag bilayer NR with the FM configuration, without (a) and with [(b),(c),(d)] spin projections. In the panel (a) the edge bands localized on the left and right edges of the NRs are indicated by the orange and green circles, respectively. The panels (b), (c), and (d) correspond to the $\pm S_x, \pm S_y$, and $\pm S_z$ spin projections, respectively. Colors code the expectation values of the spin projections.

Figure 6

Separations between the neighboring chains of Bi atoms, straight and parallel to $x$, in the $y$ direction in the relaxed Bi/BN(N) heterostructures obtained without (a) and with (b) the inclusion of SOC at $a$ = 4.340 Å. The pairs of chains are counted in the direction from the left edge to the right. The horizontal dash lines are guides for the eye showing the distances in the middle of the NRs. (c) Spin density (its dominant $y$ component) for the Bi/hBN(N) heterostructure with the FM configuration and corresponding to panel (a). Blue (yellow) color indicates the negative (positive) isovalues in the isosurface plots. The total magnetic moments of the heterostructure is [0.00,$-0.60$,0.46] ${\mu }_B$. (d) Top view of the heterostructure. The grey circles indicate N atoms and green–B atoms.

Figure 7

Band structure for the Bi/hBN(N) heterostructure with the FM configuration, without (a) and with [(b),(c)] spin projections. In the panel (a) the edge bands localized on the left and right edges of the NRs are indicated by the orange and green circles, respectively. Within the narrow $\mathit{k}$ interval centered on $\mathrm{\Gamma }$, where the orange and green circles are replaced by black ones, the states are not well localized and spread over the entire NR width as $\mathit{k}\to \mathrm{\Gamma }$ (see the main text). The panels (b) and (c) correspond to the $\pm S_y$ and $\pm S_z$ spin projections, respectively. (d) Charge density plot for the highest-energy state at the $\mathrm{\Gamma }$ point, which is results from hybridization between the edge states and is delocalized over the entire region between the two edges.

Figure 8

Band structure for the Bi (111) zigzag bilayer NR decorated with nitrogen atoms, for the (a) NM and (b) FM states. In (b), the bands drawn with orange (green) circles are localized at the left (right) edge of the decorated NR.

Figure 9

Spin densities of a NR with the edges decorated with nitrogen, for a FM solution. Yellow (blue) colors in the spin density plots correspond to positive (negative) isovalues of the plotted quantity. In the cases of $x$ and $z$ magnetizations, an expanded view of the NR edges are shown on the right.

Figure 10

Band structure for the NR decorated with nitrogen atoms with FM spin configuration, showing spin projections. The panels (a), (b), and (c) correspond to the $\pm S_x, \pm S_y$, and $\pm S_z$ spin projections, respectively. Colors code the expectation values of the spin projections.

Figure 11

Schematic of transport in the FM and AFM Bi zigzag NRs under $n$ doping. The edge states carrying opposite spins are denoted by red and blue.