Edge and sublayer degrees of freedom for phosphorene nanoribbons with twofold-degenerate edge bands via electric field

For the pristine phosphorene nanoribbons (PNRs) with edge states, there exist two categories of edge bands near the Fermi energy ($E_F$), i.e., the shuttle-shaped twofold-degenerate and the near-flat simple degenerate edge bands. However, the usual experimental measurement may not distinguish the difference between the two categories of edge bands. Here we study the varying rule for the edge bands of PNRs under an external electrostatic field. By using the KWANT code based on the tight-binding approach, we find that the twofold-degenerate edge bands can be divided into two separated shuttles until the degeneracy is completely removed and a gap near $E_F$ is opened under a sufficiently strong in-plane electric field. Importantly, each shuttle from the ribbon upper or lower edge outmost atoms is identified according to the local density of states. However, under a small off-plane field the shuttle-shaped bands are easily induced into two near-flat bands contributed from the edge atoms of the top and bottom sublayers, respectively. The evidence provides the edge and sublayer degrees of freedom (DOF) for the PNRs with shuttle-shaped edge bands, which is obviously different from another category of PNRs intrinsically with near-flat edge bands. This is because the former category of ribbons solely have four zigzaglike atomic configurations at the edges in each unit cell, which also results in that the property is robust against the point defect in the ribbon center area. As an application, furthermore, based on this issue we propose a homogenous junction of a shuttle-edge-band PNR attached by two electric gates. Interestingly, the transport property of the junction with field manipulation well reflects the characteristics of the two DOFs. These findings may provide a further understanding on PNRs and initiate new developments in PNR-based electronics.

Figure 1

The schematic illustrations of (a) (1,3)PNR and (b) (2,4)PNR, where the red dashed line parallelogram in each ribbon indicates its minimum periodical supercell, the chiral vector T is illustrated by the red solid arrow along the edge, ${\mathit{a}}_1$ and ${\mathit{a}}_2$ denote the primitive vectors, and $t_1$–$t_5$ are the five effective hoping parameters. The distance between atomic rows for the two sPNRs is 0.9 Å and 0.53 Å, respectively. The ribbon end side views in (a) and (b) indicate the top/bottom sublayer by the red/blue color. (c) The sketch of the $p\text{−}n$ junction composed of a (1,3)PNR where the different color shade in two sides implies the top and bottom back gates which adjust the Fermi level, and the green thick arrows indicate the in-plane transverse and off-plane vertical electric fields, respectively.

Figure 2

The band structure for (a)–(c) (1,3)PNR with $N=35$ and (d)–(f) (2,4)PNR with $N=49$ under an in-plane electric field with different strengths, (a) and (d) ${\varepsilon }_y=0$, (b) and (e) 2 mV/Å, and (c) and (f) 5 mV/Å, respectively. The Fermi energy $E_F$ is set to zero, and the left and right insets in (b), (c), and (f) indicate the LDOS distributions corresponding to energies near $E_F$ marked by the red and blue points, respectively, where the side view for each supercell LDOS is embodied. The right-side attached color bar indicates the electron density from the lowest (blue) to the highest (red) value.

Figure 3

The band structures for (a)–(d) (1,3)PNR with odd ($N=35$) and even ($N=36$) parities under an off-plane vertical electric field with different strengths, respectively, where (a) and (c) ${\varepsilon }_z=0.1$ V/Å and (b) and (d) 0.5 V/Å. The band structures for odd-even ($N=49–50$) (2,4)PNR under the same field are correspondingly shown in (e)–(h). The insets in (b), (d), (f), and (h) depict the LDOS distributions corresponding to the edge states at the certain energies pointed by the red and blue points, where the side view for each supercell LDOS is embodied. The color bar indicating the density is neglected here since it is the same as that in Fig.  2.

Figure 4

The conductance spectrum for the (1,3)PNR-based $p\text{−}n$ junction with a width about 3 nm under (a) in-plane transverse and (b) off-plane vertical electric fields, respectively. The gray solid line implies the conductance without applied field. The blue and red solid (dashed) lines indicate the edge bands from the same (different) (a) upper-lower edge with field strength 0.5 and 0.5 mV/Å, (b) top-bottom sublayer with field strength 0.1 and 0.5 V/Å are aligned, respectively. (c) and (d) The alignment of the edge bands of the two semi-infinitive ribbons under transverse and vertical fields, respectively, where the green/red arrow implies the on/off state of the junction.

Figure 5

The wave-function overlap, $<{\mathrm{\Psi }}_L^{*}(n,E)|{\mathrm{\Psi }}_R(n^{\prime },E)>$, as a function of (a) ${\varepsilon }_y$ and (b) ${\varepsilon }_z$ for the two terminals of the junction, where $n$ ($n^{\prime }$) is the band index number for the junction left (right) side. The LDOS diagrams with certain energies at two different field strengths pointed by black arrows are inserted with the embodied side view of the supercell.

Figure 6

The conductance spectrum of a (1,3)PNR with monatomic vacancy, respectively, located at the (a),(c) center and (b),(d) edge of the ribbon, where (a),(c) with ${\varepsilon }_y=5$ mV/Å and (c),(d) ${\varepsilon }_z=0.1$ V/Å. The red dashed line in (a) and (c) corresponds to the pristine case without defect.