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

The calculated energy spectra for the selected exemplary (1,3)PNR ($N$=35) and (2,4)PNR ($N$=49) with nearly the same width $\sim$3 nm under an in-plane electric field with different strength are shown in Figs. 2(a-c) and 2(d-f), respectively, where the Fermi level $E_{F}$ is set to zero. The left and right inserts in (b, c and f) indicate the LDOS corresponding to energies marked respectively by the red and blue points contributed by the ribbon upper or lower edge outermost atoms, where the side view for each supercell is embodied. First, as shown in Fig. 2(a), the band structure of the pristine (1,3)PNR has a shuttle-shaped edge band near $E_{F}$, which implies the ribbon is metallic. In fact, the shuttle-shaped edge bands are twofold-degenerate differentiating by the red dashed and blue solid lines. However, the degeneracy can be eliminated by applying an in-plane electric field along the $y$-direction. As a small field with strength $\varepsilon_{y}$=2 mV/Å is applied shown in Fig. 2(b), the twofold-degenerated edge bands obviously separate into two partially overlapped shuttles. Interestingly, as shown in Fig. 2(c), the two shuttles are completely separated as the filed strength is further increased up to 5 mV/Å, which means the degeneracy is completely removed. In contrast, for the (2,4)PNR as the example of the another category, it has two near-flat degenerate edge state bands in the original band structure which are drawn by the red dashed and blue solid lines shown in Fig. 2(d). They also pass through $E_{F}$ exhibiting metallicity, which is very similar to that of a conventional zPNR except for a little difference in the curvature of the edge bands Carvalho ; Ezawa ; Guo . However, much different from the twofold-degenerated bands, the degeneracy for this category of ribbon can be easily removed by a small electric field as shown in Figs. 2(e) and 2(f). The characteristic of edge band under an electric field for (2,4)PNR is very similar to the extensively studied zPNR Ezawa ; ZhouBL ; Zhang , and hence we attribute them to the same category.

An in-plane electric field removing the degeneracy of the edge bands can be understood from Hamiltonian (1). The additional diagonal terms increase (accumulate) linearly with the $y$-coordinate. From the analysis of the calculated results, it can be seen that the degenerate edge state bands for both categories of sPNRs would be broken by the application of an in-plane electric field. As the field strength increases, a band gap is opened and the transition from metal to semiconductor is occurred. The variation of (2,4)PNR edge bands with electric field is basically the same as zPNR Ezawa ; ZhouBL ; Zhang . This is because that they belong to the same category of sPNRs, where the outermost atoms of the ribbon are all at the same sublayer. However, for the (1,3)PNR with twofold-degenerate edge bands, the $E_{F}$ is embedded in a mirrored shuttle-shape bands. In order to further know which atoms in the ribbon contribute the edge state bands, we have calculated the LDOS at the energies near $E_{F}$ marked by the red and blue points, respectively. When the two shuttle-shape bands are not completely separated shown in Fig. 2(b) for $\varepsilon_{y}$=2 mV/Å, from the inserts we see the difference in LDOS between energies on the two shuttles. The red/blue dashed/solid shuttle bands are mainly contributed the outermost atoms at the upper/lower edge.

In addition, the left and right inserts in Figs. 2(c) and 2(f) depict the real-space electronic distributions at the energies of conduction band minimum (CBM) and valence band maximum (VBM) marked by the red and blue points, respectively. The right-side attached color bar indicates the density from the lowest (blue) to highest (red) value. From the LDOS inserts we can identify whether an edge band is completely contributed by the outmost atoms of the upper or lower edge of the ribbon, which provides a (upper-lower) edge DOF. However, we cannot distinguish them from which sublayers since their responses to an in-plane field are the same. Therefore, we need to apply an off-plane vertical electric field along the $z$-direction respectively to the two categories of sPNRs.

The difference in the layer atomic arrangement on the edge of a sPNR with different parity rows can be reflected by the response to an off-plane electric field, and then affects the corresponding band structure. This effect has been revealed and discussed for other 2D material nanoribbons, such as graphene, silicene and phosphorene ones ZhouBL ; LiZ ; Kang . Here, in Fig. 3 we present the result of the edge band response to a vertical electric field with strengths $\varepsilon_{z}$=0.1 and 0.4 V/Å for the two exemplary ribbons, respectively. For the consideration of the even-odd parity LiuP ; ZhouBL , we consider the two width cases of $N$=35, 36 and $N$=49, 50 for (1,3)PNR and (2,4)PNR, respectively. The LDOS left/right insert corresponding to the energies marked by the red/blue point. From Fig. 3(a) for (1,3)PNR with $N$=35, the twofold-degenerate edge bands in shuttle-shape are almost unchanged under small $\varepsilon_{z}$=0.1 V/Å. However, when $\varepsilon_{z}$ is increased up to 0.4 V/Å, interestingly, the two shuttle-shape edge bands are separated into two nearly flat bands as shown in Fig. 3(b). This results implying a transition from metal to semiconductor phase. But the two-fold degenerate is still not removed. Further, from Figs. 3(a-d) it seems that the number of atom rows (parity) does not affect the band structure of (1,3)PNR. In addition, the side view in (b) shows that the left/right inserted LDOS come from the edge outermost atoms of the top/bottom sublayer. We can identify a band from the top or bottom sublayer, which provides a sublayer DOF. This is because that the edge outmost atoms are alternately located at the top and bottom sublayers. In the contrast, for (2,4)PNR belongs to another category with two near-degenerate edge bands, the response of the edge bands to the field is sensitive on the ribbon width (parity), which is the same as that for the conventional zPNRs ZhouBL . As shown in Figs. 3(e) and 3(f), in specification, with the increase of the field strength the edge band degeneracy of the odd-numbered ($N$=49) ribbon can not be broken but enterally move upward a little. Since the degenerate edge bands are contributed by the outermost atoms of both edges. However, for the even-numbered ($N$=50) ribbon shown in Figs. 3(g) and 3(h), the outermost atoms of the two edges come from the same sublayer. Hence as the field strength increasing, the edge band degeneracy is removed and the transition from metal to semiconductor occurs.

Next, from the above results for the two exemplary sPNRs belonging to different categories, we know that a monolayer (1,3)PNR owns two DOFs of edge and sublayer, which is particularly similar to a bilayer zPNR with two DOFs of edge and layer Soleimanikahnoj . Using this similarity, we may construct a homogenous $p$-$n$ junction using a (1,3)PNR. The two sides of a (1,3)PNR are attached by the near top and bottom back-gates which adjust the $E_{F}$, leading to the lifting-up or -down of the edge bands for two ends. Further, the external in-plane and off-plane electric fields applied on the junction shown in Fig. 1(c) can be realized by using a side-gate and another back-gate electrodes, respectively. The side-gate technique has been proven to be experimentally feasible in graphene as a channel material for other applications Hahnlein ; Molitor . The back-gates can arouse a potential difference (electric field) across the whole monolayer. This kind of setup has been realized for bulk phosphorene transistors Tayari ; Kim . And a pseudospin field effect transistor has also been proposed and characterized based on a bilayer zPNR-based junction, in which a pseudospin-polarized current is generated Soleimanikahnoj . The similar interesting effect may also be realized in the $p$-$n$ junction based on a monolayer (1,3)PNR, which can generate an edge- or sublayer-polarized current by properly adjusting the gate electrodes.

In Fig. 4(a), we show the calculated conductance spectrum for the proposed junction based on (1,3)PNR with a width $\sim$3 nm under a transverse in-plane field with different strengths $\varepsilon_{y}$=0 (gray line), 0.5 (blue) and 5 mV/Å (red), respectively. The solid/dashed line indicates the alignment for the edge bands from the same/different upper-lower edge. The separated edge bands from the two edges of the terminals with/without alignment by back-gates is shown in Fig. 4(c), where the upper/lower panel corresponds to the case of the solid/dashed line in 4(a), and the green/red arrows implies the switch on/off state. First, we find a conductance plateau of $2e^{2}/h$ around $E_{F}$ for the junction without external field shown by the grey solid line, which is in accord with the band structure for a pristine (1,3)PNR shown in Fig. 2(a). When a small field 0.5 mV/Å is applied, by adjusting the positive and negative voltages of the two back-gate as to the edge bands from the same upper or lower edge be aligned, the conductance still maintains a $2e^{2}/h$ plateau (blue solid line) near $E_{F}$, whereas the conductance is decreased to $0.3e^{2}/h$ from different edge as shown by the blue dot-dashed line. This is because that the band degeneracy is broken not enough so that the two shuttle-shaped edge bands are not completely separated. Further, as the field strength increases up to 5 mV/Å, the degenerate shuttle-shaped edge bands is fully removed and a significant band gap is opened around $E_{F}$ [see Fig. 2(c)]. As a consequence, when the same edge bands are aligned as shown in the upper panel in 4(c), a conductance gap correspondingly appears with two $2e^{2}/h$ plateaus beside it (red solid line). Otherwise the conductance is almost zero even though there are two shuttle-shaped edge bands from the different upper or lower edge are aligned as shown in the lower panel in 4(c), which shows a transport off state.

Since the speciality of the edge atom arrangement for the monolayer (1,3)PNR, sublayer is also an important DOF in its transport property. In Fig. 4(b), we show the conductance spectrum for the junction under an off-plane vertical electric field with strengths, 0 (gray), 0.1 (blue) and 0.5 V/Å (red), respectively. Here the solid/dashed lines indicates the two edge bands from the same/different sublayers are aligned, of which the diagrammatic sketch for the edge band alignment is shown in 4(d). First, we also find a conductance plateau of $2e^{2}/h$ around $E_{F}$ for the junction without external field shown by the grey solid line. In the presence of applied field $\varepsilon_{z}$=0.1 V/Å, there is a gap opened within the shuttle-shaped edge bands. Therefore, as the edge bands from the same upper or lower edge are adjusted (by back-gate) to be aligned, the conductance exhibits two steps of $2e^{2}/h$ besides $E_{F}$ (see the blue solid line), which is in accord with the band structure shown in Fig. 3(b) or 3(d). In this case the increase of the field strength (e.g., $\varepsilon_{z}$=0.5 V/Å) only result in the two narrower steps and more far away from $E_{F}$ (see the res solid line). On the contrary, when the edge bands from the same upper or lower edge are not aligned, the conductance is nearly zero except for very small peaks at $E_{F}$ as shown by the blue and red dashed lines. The reason for the results is that the edge outermost atoms of (1,3)PNR are alternately located at the top and bottom sublayers. Therefore, a small vertical electric field can results in wave function superposition in different sublayers.

In order to further understand the rule of the two DOFs in transport for the junction, in Fig. 5 we plot the dependence of wavefunction overlap, $<$$\Psi^{*}_{L}(n,E)\Psi_{R}(n^{\prime},E)$$>$, as a function of the field strength (a) $\varepsilon_{y}$ and (b) $\varepsilon_{z}$. Here $\Psi_{L/R}(n,E)$ is the wavefunction in the left/right (L/R) side of the junction. The LDOS with certain energies at two different field strengths are also inserted with the embodied side view of supercell. From Fig. 5, we first note that the overlap value drops rapidly as the field strength increasing due to the degeneracy elimination, with which the edge bands from the different upper-lower edge or top-bottom sublayer are separated. Therefore, without applied field the wavefunction overlap is equal to 1. In this case the edge bands are completely superposed, which refers to its pristine state shown in Fig. 2(a). Meanwhile, by adjusting the two back-gates as to the two side edge bands from the same edge or sublayer aligned as shown in upper panels of Fig. 4(c) or 4(d), the overlap degree is also equal to 1 regardless of the electric field strength.

Further, as shown in Fig. 5(a), we find that the two shuttle-shaped edge bands from the upper or lower edge are aligned by adjusting the back-gates as to the potential difference between the junction two sides is equal to the electric field strength [see the lower panel in Fig. 4(c)], the wavefunction overlap for two terminals is greatly reduced. When the two shuttle-shape edge bands are just separated a little by a small field 0.5 mV/Å, the wavefunction overlap is greatly reduced to a small value, which corresponds to the blue dashed peaks near $E_{F}$ in Fig. 4(a). This is because that the overall degeneracy of the two shuttle bands would be almost removed even under a small electric field, but some of the degeneracy is still remained. Therefore, at certain energies the superposition of the wavefunctions in two sides still has a definitive value. In this case, from the LDOS shown in left upper panel of Fig. 5(a)], we can see it distributes at the upper and lower edge outermost atoms. Specifically, the upper edge atoms in the left side account for the major contribution (bright red spots), while the reddish ones at lower edge indicate a partial contribution. On the contrast, the LDOS distributed are the upper-lower edge for the right side is opposite to the left one. As the field strength is increased up to 5 mV/Å, the wavefunctions overlap for two side approaches zero (as red dashed line shown in Fig. 4(a)). This is because that the two shuttle edge bands are completely separated, and electrons are localized only at the outmost atoms of one edge. This can also be seen from the LDOS inserts with the embodied side view of supercell. On the other hand, since the edge mostout atoms of (1,3)PNR are alternately arranged on the top and bottom sublayers, the wavefunctions of the top and bottom sublayer atoms still mix together when a large vertical electric field is applied as shown in the lower insert of Fig. 5(b). Hence the amplitude scale of $y$-axis in Fig. 5(b) is much larger than that in 5(a). Therefore, when $\varepsilon_{z}$=0.1 V/Å the wavefunction overlap for two sides is still large, which corresponds to the blue dashed plateau near $E_{F}$ in Fig. 4(b). When the vertical field becomes larger, the wavefunction overlap of the top and bottom sublayers tends to zero due to the localization of electrons, as shown by the lower insert on Fig. 5(b) for LDOS with 0.5 V/Å.

Finally, we demonstrate the influence of defect on the transport, which is also very important for device application by using the proposed DOFS since defects are inevitable on PNR samples in experiments Kiraly . For the sake of argument on the two DOFs in a (1,3)PNR, here we mainly consider two kinds of monatomic vacancies. As shown in the top of Fig. 6 for a defective (1,3)PNR, there are two monatomic vacancy positions indicated by the black dotted circles, one at the edge (EV) and the other in the center (CV) of the ribbon. Figure 6 gives the conductance spectrum, where (a, c) and (b, d) for CV and EV cases with $\varepsilon_{y}$=5 mV/Å and $\varepsilon_{z}$=0.1 V/Å, respectively. As a reference, the red dashed lines in (a, c) indicate the conductances for a pristine (1,3)PNR without defect but with in- and off-plane fields, respectively, which can be refer to the corresponding energy spectrum shown in Figs. 2 and 3. As is seen from the conductance spectra shown by the blue solid lines, a CV does not affect the edge bands regardless the field direction, resulting in an unchanged conductance in comparison the pristine case shown in (a, c). But the situation for EV is much different. Comparing with the conductance spectrum shown by the red solid line in Fig. 4(a) for the same upper-lower edge are aligned (also shown by red dashed line in Fig. 6(a)). As shown in Fig. 6(b), we find that the edge state on one of edges with an EV is severely weakened, which result in that the conductance plateau on the right side of $E_{F}$ almost disappears. The original conductance plateau is maintained. In addition, for a EV with $\varepsilon_{z}$=0.1 V/Å shown by the blue solid line in Fig. 6(d), the conductance steps are retained as the ideal case [see the blue solid line in Fig. 4(b) and the red dashed line in Fig. 6(c)]. But the conductance peak on the left side of $E_{F}$ is reduced by half, while the right side peak is still close to $2e^{2}/h$. This importantly implies that the CV has no effect on the edge state. The same result is expected for the in-plane field case. Moreover, due to the particularity of the two DOFs, the effect of EV on edge-state contributed conductance is different under the regulation of electric field. As shown by the LDOS in Fig. 5(a), a very small in-plane field can make the outer electrons of the edge atoms localized, which provides the transport channel corresponding to the edge with an EV almost forbidden. On the other hand, a large off-plane field would cause a certain degree of wavefunction overlap between the sublayer atoms on the same side [as shown in Fig. 5(b)], so that an EV in one sublayer only destroys the edge state of the corresponding sublayer, while the other sublayer on the same side still maintains a channel. In general, the sPNRs with two-DOFs can reduce the influence of low-concentration edge monatomic defects on the edge states to a certain extent.

At last but not least, besides the atomic defects the edge passivation of nanoribbons is also an important issue because that the edges are usually passivated in real experimental samples. When edge hydrogen(H)-passivation is considered in the TB calculation, usually an extra potential field is applied on the edge atoms of the ribbons. This may lead to that the original edge bands are disappeared or modified (see, e.g., LiuP ). Here we have also tested the H- and O-passivation cases by using a DFT calculation, respectively. The result shows that the H-passivated edges do not exhibit edge states, while O-passivated ones remain qualitatively similar to the the case of bare edges. And the result of the edge bands is the qualitatively the same as those by the TB approach, which confirms that the device assumption based on (1,3)PNR is scientifically valuable Ren .