Quantum transport in defective phosphorene nanoribbons: Effects of atomic vacancies

Defects are almost inevitably present in realistic materials and defective materials are expected to exhibit very different properties than their nondefective (perfect) counterparts. Here, using a combination of the tight-binding approach and the scattering matrix formalism, we investigate the electronic transport properties of defective phosphorene nanoribbons (PNRs) containing atomic vacancies. We find that for both armchair PNRs (APNRs) and zigzag PNRs (ZPNRs), single vacancies can create quasilocalized states, which can affect their conductance. With increasing vacancy concentration, three different transport regimes are identified: ballistic, diffusive, and Anderson localized ones. In particular, ZPNRs that are known to be metallic due to the presence of edge states become semiconducting: edge conductance vanishes and transport gap appears due to Anderson localization. Moreover, we find that for a fixed vacancy concentration, both APNRs and ZPNRs of narrower width and/or longer length are more sensitive to vacancy disorder than their wider and/or shorter counterparts, and that for the same ribbon length and width, ZPNRs are more sensitive to vacancy disorder than APNRs.

Figure 1

Model system under consideration: (a) armchair and (b) zigzag PNRs. The system is composed of two semi-infinite leads of width $W$ (light-red shaded region) and a finite scattering region of length $L$ and width $W$ (dark-gray shaded region). Here, three types of atomic vacancies are taken into account, i.e., monovacancy (MV), divacancy of type I (DV1), and divacancy of type II (DV2), which can be located at the edges or in the bulk of the scattering region. Two types of divacancies (DV1 and DV2) are distinguished where DV1 (DV2) keeps (breaks) the sublattice symmetry of the PNR.

Figure 2

Band structure, DOS, and conductance of perfect PNRs (without vacancies): (a)–(c) for APNR and (d)–(f) for ZPNR. Here, the ribbon width is set to $W=4$ nm for both APNR and ZPNR.

Figure 3

(a)–(d) DOS and conductance of defective APNR and ZPNR for different types of a single vacancy placed in the center of the PNR, together with the results when no vacancy (NV) is present; (e) and (f) Spatial LDOS of edge propagating modes in the ZPNR without and with a single vacancy. Here, NV, MV, DV1, and DV2 denote the cases of no vacancy, monovacancy, divacancy of type I, and divacancy of type II, respectively. In panel (d), the black arrow indicates the energy of those edge propagating modes at which their spatial LDOS is calculated. In panel (f), the black circle indicates the single vacancy and its position.

Figure 4

(a)–(d) DOS and conductance of defective APNR and ZPNR for different positions $\left(d\right)$ of the monovacancy (MV) from the center of the PNR, together with the results of their perfect counterparts with no vacancy (NV). (e)–(h) Spatial LDOS of bulk propagating modes in the ZPNR for the cases of NV and MV. In panel (d), the black arrow indicates the energy of those bulk propagating modes at which their spatial LDOS is calculated. In panels (f)–(h), the black circles indicate the different vacancy positions.

Figure 5

DOS and conductance of defective PNRs with random vacancies for different vacancy concentrations $P$ as indicated, together with the results of their perfect counterparts with no vacancy $\left(P=0\right)$: (a), (c) for the APNR case and (b), (d) for the ZPNR case. Here, each panel contains the individual results for all the simulated PNRs with vacancies (gray lines), the averaged results of them (color lines), and the result for perfect PNR (black line). The inset of panels (c) and (d) show schematically the APNR and ZPNR containing a random distribution of vacancies.

Figure 6

Ratios of the change in the conductance of defective PNRs with random vacancies relative to that of their perfect counterparts, for different ribbon widths $W$ and vacancy concentrations $P$ as indicated: (a), (c), (e) for the APNR case and (b), (d), (f) for the ZPNR case. Here, the ribbon length is set to be $L=10$ nm for all the simulated PNRs.

Figure 7

Conductance of defective PNRs with random vacancies for different ribbon length $L$ and vacancy concentrations $P$ as indicated: (a), (c), (e) for the APNR case and (b), (d), (f) for the ZPNR case. Here, the ribbon width is set to $W=4$ nm for all the simulated PNRs. In panels (a)–(d), black lines are the results of perfect PNRs with no vacancy $\left(P=0\right)$, which are presented for reference purposes. In panels (e) and (f), red (blue) dots are the numerical results of bulk (edge) conductance, and corresponding lines are the fitted ones.