Introduction

Borophene, a monolayer of boron atoms, has attracted extensive attention as a prototype for exploring two-dimensional (2D) systems1,2,3 since its successful synthesis by two independent groups4,5 following theoretical predictions6,7,8,9,10,11,12. In contrast to other 2D materials13,14,15, multiple borophene polymorphs including freestanding ones have been realized experimentally16,17,18,19,20,21,22, which exhibit in-plane anisotropy and are tunable by ambient growth conditions. Of particular interest are the planar \(\nu _{1/5}\) and \(\nu _{1/6}\) boron sheets, which consist of (2,2) and (2,3) chains, respectively. Here, the indices an and aw of (an,aw) represent, respectively, the number of atoms in the narrow and wide rows of a single boron chain23, as illustrated in Fig. 1a. These two sheets possess many intriguing attributes, such as superior mechanical strength and flexibility24,25, conventional superconductivity26,27,28, Dirac fermions29,30,31, and ultrahigh thermal conductance3,32.

Fig. 1: Structure of a disordered BNR and electron transport along pure BNRs.
figure 1

a Schematics of a two-terminal disordered BNR coupled to left and right semi-infinite \(\nu _{1/5}\) BNRs (yellow rectangles). This disordered BNR is assembled from random arrangement of (2,2) and (2,3) chains (gray rectangles) in the CSR, where a (2,2)j chain is followed randomly by a \(\left( {2,2} \right)_{\bar j}\) or \(\left( {2,3} \right)_{\bar j}\) one and a (2,3)j chain by a (2,2)j or (2,3)j one, with \(\bar j\) = 1 (2) for j = 2 (1). This leads to structural disorder induced by randomly distributed LDs and the disorder is characterized by the ratio p of (2,3) chains to all of the chains (length L). Here, the length is L = 10 and the width defined as the number of rows is N = 9. b The top panel shows the unit cell of \(\nu _{1/6}\) borophene, which contains five boron atoms called as a, b, c, d, and e. The bottom panel plots the unit cell of \(\nu _{1/5}\) one, which includes eight atoms labeled by u, v, w, and x. Energy-dependent conductance G for c a \(\nu _{1/5}\) BNR with p = 0 and d a \(\nu _{1/6}\) BNR with p = 1.

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On the other hand, line defects (LDs) exist in diverse 2D materials33,34,35,36,37,38 and strongly affect their electronic, magnetic, optical, mechanical, and thermal properties39,40,58, owing to higher structural stability of borophene in the presence of LDs54,59. Specifically, several distinct phases are synthesized from periodic self-assembly of LDs52,53,54, implying that (2,2) and (2,3) chains function as building blocks to construct intriguing boron sheets. These LDs could considerably modulate the electronic properties of borophene52,55,60 and improve its mechanical response59, thus playing an important role in the discovery of exotic quantum phenomena and in device applications. Notice that the LDs in realistic borophene lack long-range order and structural disorder emerges simultaneously52,54. Until now, the LD-induced structural disorder has not yet been discussed and understanding this disorder may facilitate the elucidation of structure-property relationships.

In this paper, we study theoretically the electron transport through two-terminal borophene nanoribbons (BNRs) with random distribution of LDs by connecting to left and right semi-infinite \(\nu _{1/5}\) BNRs. These BNRs, termed as disordered BNRs, are assembled from random arrangement of (2,2) and (2,3) chains in the central scattering region (CSR) and leads to LD-induced structural disorder, as shown in Fig. 1a. We find that both \(\nu _{1/5}\) and \(\nu _{1/6}\) BNRs exhibit metallic behavior, consistent with experiments21,22,23,52. Remarkably, although the overall electron transport ability is dramatically declined in the presence of LDs, some resonant peaks of conductance quantum appear in the transmission spectra of various disordered BNRs, regardless of inhomogeneous potential energies and hop** integrals, nanoribbon length, and LD distribution. By changing nanoribbon width, the disordered BNRs could present metal-insulator transition with tunable transmission gap in the insulating regime. Furthermore, two evolution phenomena of resonant peaks are revealed for disordered BNRs with different widths, showing that (i) all of the resonant peaks for odd width overlap perfectly with all those of two narrow BNRs and (ii) a specific resonant peak for width Ni reappears at various BNRs with width \(N = \alpha (N_i + 1) - 1\) and α an integer.

Results

Tight-binding model

Electron transport through two-terminal disordered BNRs is simulated by the Hamiltonian29,30: \({{{\mathcal{H}}}} = \mathop {\sum}\nolimits_i {\varepsilon _ic_i^{\dagger} c_i} - \mathop {\sum}\nolimits_{\langle i,j\rangle } {t_{ij}c_i^{\dagger} c_j}\). Here, \(c_i^{\dagger}\) (ci) creates (annihilates) an electron at site i, εi is the potential energy, and tij is the nearest-neighbor hop** integral. In the numerical calculations, the parameters are set to \(\varepsilon _i = 0\), \(t_{ij} =t= 1\) (energy unit), and the nanoribbon length counted by all of the chains in the CSR to L = 2000. The most disordered BNRs are considered in which half of the chains are (2,3) ones with p = 0.5 and the results are averaged over 2500 disordered samples.

Pure BNRs

We first study the electron transport through pure BNRs with the CSR being a \(\nu _{1/5}\) or \(\nu _{1/6}\) BNR, as shown in Fig. 1c and d. It clearly appears that both \(\nu _{1/5}\) and \(\nu _{1/6}\) BNRs exhibit metallic behavior which is consistent with experiments21,22,23,52, and their conductances are asymmetric about the line of E = 0 owing to the electron-hole symmetry breaking. For the \(\nu _{1/5}\) BNR, its transmission spectrum is characterized by many conductance plateaus quantized at integer multiples of G0 as expected (Fig. 1c), because of the translational symmetry. By contrast, when the \(\nu _{1/5}\) BNR is replaced by a \(\nu _{1/6}\) one, the conductance declines and presents dramatic oscillating behavior instead of quantized plateaus (Fig. 1d), in accordance with first-principles calculations60, because of LD-induced scattering at the CSR-lead interfaces.

Transmission peaks and important transport phenomena in disordered BNRs

Figure 2a and b show energy-dependent averaged conductance 〈G〉 and standard deviation δG, respectively, for disordered BNRs with different widths N. Here, \(\delta G \equiv \sqrt {\langle G^2\rangle - \langle G\rangle ^2}\). One can see that the electron transport through disordered BNRs is strongly suppressed as expected, due to Anderson localization caused by successive scattering from randomly distributed LDs. However, it is surprising that some transmission peaks of \(\langle G\rangle = G_0\) are found in all these disordered BNRs and the peak number increases with N (Figs. 2a and 6b). In particular, the standard deviation at transmission-peak positions satisfies \(\delta G\sim 0\) except for \(E\sim 0.329t\) of N = 28 and E ~ 0.217t and 0.656t of N = 33 (Fig. 2b), because the conductance contributed from neighboring transmission peak(s) fluctuates at these energies due to finite-size effects. We emphasize that δG ~ 0 at all these transmission-peak positions for longer disordered BNRs. All these transmission peaks are very robust, implying that they could be observed in any disordered BNR with arbitrary arrangement of LDs, as further discussed below. This result explains a recent experiment that delocalized states are measured in BNRs with LDs55.

Fig. 2: Electron transport along disordered BNRs with different widths.
figure 2

Energy dependence of a averaged conductance 〈G〉 and b standard deviation δG for typical N. c 〈G〉 vs. E for three N. The gray lines in a and b denote the conductance quantum G0 and \(\delta G = 0.28G_0\), respectively.

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Besides, one can identify other important features. (i) The transport property of disordered BNRs strongly depends on N, ranging from metallic (see the red-dashed line in Fig. 2a) to insulating behavior (see the other lines in Fig. 2a). This indicates the width-driven metal-insulator transition in disordered BNRs. And the transmission gap is tunable in the insulating regime by varying N, which is similar to graphene nanoribbons61,62,63 and facilitates band-gap engineering of BNRs. (ii) The transmission peaks, characterized by full width at half maximum (FWHM), are mainly categorized into narrow and wide peaks separated by about 0.06t FWHM (Figs. 2a and 6b). (iii) Every transmission peak corresponds to two peaks in the curve of δG-E (Fig. 2b). Remarkably, the maximum of all these peaks satisfies \(\delta G\sim 0.28G_0\) (see the gray-solid line in Fig. 2b), which is close to \(\sqrt {1/12} G_0\) reported in disordered graphene nanoribbons64,65,66. (iv) Some transmission peaks can even achieve 2G0 (see the black-solid line in Fig. 2a), because of almost perfect superposition of two neighboring transmission peaks, as seen from the four-peak structure around E ~ −0.226t in Fig. 2b.

Robustness of transmission peaks in disordered BNRs

To demonstrate the robustness of transmission peaks, we study the electron transport along disordered BNRs by considering inhomogeneous potential energies εi and hop** integrals tij, different nanoribbon lengths L, and ratios p. Figure 3a shows 〈G〉 vs. E for disordered BNRs with inhomogeneous εi and tij, the values of which are determined by the number of neighboring boron atoms29,30. One can see that several transmission peaks of \(\langle G\rangle = G_0\) are observed in the energy spectra of all these disordered BNRs with different N and the peak number increases with N.

Fig. 3: Electron transport along disordered BNRs by considering inhomogeneous potential energies εi and hop** integrals tij, different lengths L, and ratios p.
figure 3

a \(\langle G\rangle\) vs E for different N with inhomogeneous εi and tij. All these inhomogeneous model parameters are determined by the number of adjacent atoms29,30, i.e., \(\varepsilon _{{{\mathrm{a}}}} = \varepsilon _{{{\mathrm{e}}}} = \varepsilon _{{{\mathrm{u}}}} = \varepsilon _{{{\mathrm{x}}}} = - 0.098t\), \(\varepsilon _{{{\mathrm{b}}}} = \varepsilon _{{{\mathrm{d}}}} = \varepsilon _{{{\mathrm{v}}}} = \varepsilon _{{{\mathrm{w}}}} = 0.029t\), \(\varepsilon _{{{\mathrm{c}}}} = 0.4225t\), \(t_{{{{\mathrm{ab}}}}} = t_{{{{\mathrm{de}}}}} = t_{{{{\mathrm{uv}}}}} = t_{{{{\mathrm{uw}}}}} = t_{{{{\mathrm{vx}}}}} = t_{{{{\mathrm{wx}}}}} = 1.02t\), \(t_{{{{\mathrm{ac}}}}} = t_{{{{\mathrm{ce}}}}} = 0.895t\), \(t_{{{{\mathrm{ae}}}}} = t_{{{{\mathrm{ax}}}}} = t_{{{{\mathrm{eu}}}}} = t_{{{{\mathrm{ux}}}}} = 1.06t\), \(t_{{{{\mathrm{bc}}}}} = t_{{{{\mathrm{cd}}}}} = 0.92t\), and \(t_{{{{\mathrm{bd}}}}} = t_{{{{\mathrm{vw}}}}} = 0.955t\). \(\langle G\rangle\) vs E for b different L with p = 0.5 and for c typical p with L = 2000. d 〈G〉 vs L and e 〈G〉 vs p at the electron energies marked by the stars in b. The other model parameters in b, c are the same as Fig. 2a and N = 33.

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Figure 3b plots 〈G〉 vs E with the length ranging from L = 500 to 104. Here, the width is N = 33 and the results are calculated from 5 × 106/L disordered samples. It is clear that several transmission peaks are found in all these disordered BNRs and their FWHM depends on L. The FWHM is large for short disordered BNRs and will be progressively reduced by increasing L as expected, because the number of randomly distributed LDs is increased and gives rise to the enhancement of Anderson localization caused by continuous scattering from these LDs. However, all these peaks are very robust against the number and distribution of LDs, as further illustrated in Fig. 3d which shows 〈G〉 vs L at typical transmission-peak positions marked by the stars in Fig. 3b. With increasing L, 〈G〉 at the transmission-peak positions of E ~ 0.217t and 0.384t declines firstly and then saturates at G0 in the large length limit (see the red-dashed and blue-dotted lines in Fig. 3d). Similar behavior can also be observed at the peak position of E ~ −0.226t, but \(\langle G\rangle\) achieves \(2G_0\) for \(10^2 < L < 10^5\) and is slightly decreased by further increasing L (see the black-solid line in Fig. 3d). This indicates that two transmission peaks almost overlap at \(E\sim - 0.226t\) and will be gradually separated from each other in the large length limit. In sharp contrast, \(\langle G\rangle\) drops to zero when E deviates from the peak positions (see the cyan-dash-dotted line in Fig. 3d).

Figure 3c and e plot \(\langle G\rangle\) vs E for typical p and \(\langle G\rangle\) vs p at several E, respectively, by fixing N = 33 and L = 2000. One can see from Fig. 3c that all of the transmission peaks remain and the FWHM also depends on p. In the relatively low p regime, the FWHM decreases with p (see the black-solid, red-dashed, and blue-dotted lines in Fig. 3c); whereas in the high p regime, the FWHM usually increases with p (see the blue-dotted, cyan-dash-dotted, and violet-dash-dot-dotted lines in Fig. 3c). This nonmonotonic phenomenon can be understood from the aforementioned LD-induced Anderson localization. Notice that the number of randomly distributed LDs increases with p for p < 0.5 and decreases with p for p > 0.5. In other words, the situation of p = 0.5 corresponds to the most disordered BNRs and the disorder strength characterized by p becomes larger when p is closer to 0.5, giving rise to nonmonotonic dependence of the FWHM on p. Similar behavior can also be observed in the curves of \(\langle G\rangle\)-p that \(\langle G\rangle\) decreases (increases) quickly with p in the extremely low (high) p regime (Fig. 3e). This can be traced back to the fact that the wave functions are very sensitive to LDs once they are introduced in the CSR and leads to dramatic modulation of the electron transport ability. However, all these peaks are very robust against LDs in this situation. For E ~ 0.217t (0.384t), \(\langle G\rangle\) reaches G0 in a wide range of \(0.17 \,<\, p \,<\, 0.97\) (\(0.04 \,<\, p \,<\, 0.98\)) (see the red-dashed and blue-dotted lines in Fig. 3e), while for E ~ −0.226t \(\langle G\rangle\) saturates at \(2G_0\) in the range of \(0.03\, <\, p\, <\, 0.99\) (see the black-solid line in Fig. 3e). By contrast, \(\langle G\rangle\) is decreased to zero when E situates away from the peak positions (see the cyan-dash-dotted line in Fig. 3e). Therefore, we conclude that the transmission peaks are very robust and could be detected in any disordered BNR with arbitrary arrangement of LDs, thus allowing for direct accessibility in experiments.

Physical origin of transmission peaks in disordered BNRs

To elucidate the physical origin of transmission peaks in disordered BNRs, we calculate bond currents of both periodic and disordered BNRs, and consider N = 2 as an example. The bond currents flowing from the lth chain to the l + 1th one read67,68,69: \({{{\mathbf{I}}}}_l = ( - 2e/h){\int} {[{{{\mathbf{H}}}}_{l,l + 1}{{{\mathbf{G}}}}_{l + 1,l}^ < (\xi ) - {{{\mathbf{H}}}}_{l + 1,l}{{{\mathbf{G}}}}_{l,l + 1}^ < (\xi )]d\xi }\), where \({{{\mathbf{G}}}}_{l,l + 1}^ <\) is the interchain lesser Green’s function. Figure 4a and b plot the spatial distributions of bond currents at the peak position of E ~ 0 for \(\nu _{1/5}\) and \(\nu _{1/6}\) BNRs, respectively, with L = 20. For the \(\nu _{1/5}\) BNR, the current always flows along the outer edges of each (2,2) chain and its spatial distributions remain unchanged for the lth chain (Fig. 4a). While for the \(\nu _{1/6}\) BNR, the current flows along the armchair edges within each (2,3) chain and the spatial distributions are independent of l as well (Fig. 4b). We then calculate the spatial distributions of bond currents at E ~ 0 for various BNRs with diverse arrangement of (2,2) and (2,3) chains in the CSR, as illustrated in Fig. 4c–e. We find that both spatial distributions and magnitude of bond currents within any (2,2) or (2,3) chain of all these BNRs, in the presence of LDs, are the same as those of \(\nu _{1/5}\) and \(\nu _{1/6}\) BNRs. In other words, the electrons at the peak positions will not be reflected by LDs and can thus propagate through any disordered BNR dissipationlessly.

Fig. 4: Spatial distributions of bond currents at the resonant energy of E ~ 0 for different BNRs with various arrangement of (2,2) and (2,3) chains in the CSR.
figure 4

The top two panels refer to (a) a \(\nu _{1/5}\) BNR and (b) a \(\nu _{1/6}\) one, the middle panel (c) to a periodic BNR with p = 0.5, and the bottom panels (d) and (e) to two disordered samples with p = 0.5. Here, the width is N = 2, the length is L = 20, and the size of the arrows is proportional to the magnitude of bond currents.

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To further understand the mechanism of transmission peaks, we study the electron transport through \(\nu _{1/5}\) and \(\nu _{1/6}\) BNRs with different L, and also take N = 2 as an example (Supplementary Fig. 1). It is clear that these transmission peaks originate from resonant tunneling, where some resonant energies remain unchanged for \(\nu _{1/6}\) BNRs with different L and the others change with L. Therefore, the electrons with invariant resonant energies cannot be reflected by LDs, leading to resonant peaks in disordered BNRs.

Evolution phenomenon (EP) I

We then focus on the evolution of resonant peaks in disordered BNRs with various N. Figure 2c shows \(\langle G\rangle\) vs E for disordered BNRs with three N. It clearly appears that some resonant peaks for N = 33 are superimposed on all those for N = 16, while the remaining ones will overlap with all those for N = 17 by properly moving peak positions. This phenomenon can also be observed in other disordered BNRs with, e.g., N = 5, 11, 13, and 27 (Supplementary Fig. 2). Therefore, we conclude that all of the resonant peaks of disordered BNRs with odd width No could be assembled from the ones with \(N = (N_{{{\mathrm{o}}}} - 1)/2\) and \((N_{{{\mathrm{o}}}} + 1)/2\), namely, EP I.

To elucidate the underlying physics of EP I, Fig. 5 plots the spatial distributions of averaged interchain currents \(\langle I_l(n)\rangle\) at typical resonant energies shown in Fig. 2c. It is clear that all of the \(\langle I_l(n)\rangle\) exhibit rather uniform fringe patterns and are independent of l albeit the existence of randomly distributed LDs, manifesting delocalized states in various disordered BNRs. And green fringes denote transmission channels with finite \(\langle I_l(n)\rangle\), whose number and location depend on E.

Fig. 5: Spatial distributions of averaged interchain currents 〈Il(n)〉 of disordered BNRs at the resonant energies marked by the stars in Fig. 2c.
figure 5

2D plots of 〈Il(n)〉 vs chain index l and row index n for N = 33 at (a) \(E\sim 0.384t\), (d) \(E\sim 0.656t\), (e) \(E\sim 0.217t\), and (h) \(E\sim 0.826t\), for N = 16 at (b) \(E\sim 0.384t\) and (c) \(E\sim 0.656t\), and for N = 17 at (f) \(E\sim 0.268t\) and (g) \(E\sim 0.903t\).

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Since the spatial inversion symmetry with respect to the row of n = (N+1)/2 is preserved for disordered BNRs with odd N (Fig. 1a), the corresponding \(\langle I_l(n)\rangle\) possess mirror symmetry (Fig. 5a and d–h) and can be divided into two cases according to the parity of wave functions. (i) The \(\langle I_l(n)\rangle\) at n = 17 are zero for N = 33 (Fig. 5a and d), corresponding to odd wave functions whose electron densities are zero along the symmetric row of n = 17. In this situation, the top segment from n = 1 to 16 and the bottom one from n = 18 to 33, separated by this symmetric row, are independent of each other. Consequently, the transmission channels of the top/bottom segment for N = 33 are the same as those for N = 16 (Fig. 5a–d), giving rise to completely identical resonant peaks at the same E of these disordered BNRs. (ii) Contrarily, the \(\langle I_l(n)\rangle\) are finite at n = 17 (Fig. 5e and h), which refers to even wave functions with nonzero electron densities along the symmetric row. Interestingly, the transmission channels of the top/bottom segment for N = 33 are also identical to those for N = 17 albeit distinct magnitude of \(\langle I_l(n)\rangle\) at the nth row (Fig. 5e–h). Correspondingly, the disordered BNRs with N = 17 and 33 possess identical resonant peaks but at different resonant energies, owing to the interference effect at the symmetric row. By contrast, the spatial inversion symmetry is destroyed for disordered BNRs with even N and the parity of wave functions disappears simultaneously, leading to the absence of EP I for even N.

EP II

Figure 6a shows \(\langle G\rangle\) vs. E around a transmission peak for three disordered BNRs, where the magenta-dashed line represents the sum of \(\langle G\rangle\) for N = 2 and 6. One can see that the peak at E ~ 0.021t for N = 20 overlaps perfectly with the superposition of two neighboring peaks for N = 2 and 6, as seen from the black-solid and magenta-dashed lines in Fig. 6a. This indicates that a transmission peak of wide disordered BNRs could be evolved from narrow ones.

Fig. 6: Evolution of resonant peaks in disordered BNRs with different widths.
figure 6

a 〈G〉 vs. E around a peak for three N. The magenta-dashed line represents the sum of 〈G〉 for N = 2 and 6. b A 2D plot of FWHM for all of the peaks vs N and E. The cyan-solid, blue-dashed, and red-dotted lines display the evolution of three peaks initially appeared in disordered BNRs with width Ni = 2, 4, and 6, respectively.

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To further elucidate the above phenomenon, Fig. 6b displays a 2D plot of FWHM for all of the peaks as functions of N and E. It is clear that the number of squares increases with N and some disordered BNRs possess peculiar resonant peaks with FWHM exceeding 0.12t (see the leftmost peak for N = 15 in Fig. 2a and the black squares in Fig. 6b). In particular, completely identical resonant peaks are observed at E ~ 0 for N = 2, 5, 8,..., at E ~ 1.017t for N = 4, 9, 14,..., and at E ~ 0.021t for N = 6, 13,... (see the cyan-solid, blue-dashed, and red-dotted lines in Fig. 6b). Therefore, we infer that a resonant peak, firstly emerged in a disordered BNR with \(N = N_i\), will reappear at the same E of various BNRs with \(N = \alpha (N_i + 1) - 1\), where α is an integer and \(N_i + 1\) the period. This characteristic is named as EP II, which compensates EP I. Notice that the peak at E ~ 0.021t for \(N = 3 \times 7 - 1 = 20\) deviates from EP II (see the surrounded blue square in Fig. 6b), because of finite-size effects. By increasing L, this transmission peak is split into two peaks at E ~ 0 and 0.021t (Fig. 6a). Consequently, this surrounded square will be evolved into a cyan (red) square on the cyan-solid (red-dashed) line and EP II holds.

To gain insights into EP II, we consider a disordered BNR with \(N = \alpha (N_i + 1) - 1\). According to its spatial structure, this disordered sample could be divided into α basic BNRs with width Ni, where the mth BNR includes the rows from \(n = m_i - N_i\) to \(m_i - 1\) and is separated from the m+1th one by the mith row, with \(0 \,<\, m \,<\, \alpha\) and \(m_i = m(N_i + 1)\). For instance, the BNR with N = 9 can be divided into two basic BNRs with \(N_i = 4\), which are separated by the fifth row (Fig. 1a). The Hamiltonian \({{{\mathbf{H}}}}_{{{\mathrm{c}}}}\) of this disordered BNR with width N can then be partitioned as:

$$\begin{array}{*{20}{l}} {{{{\mathbf{H}}}}_{{{\mathrm{c}}}} = \left[ {\begin{array}{*{20}{c}} {{{{\mathbf{H}}}}_1} & {{{{\mathbf{A}}}}_{11}} & 0 & \cdots & 0 \\ {{{{\mathbf{A}}}}_{11}^{\dagger} } & {{{{\mathbf{R}}}}_1} & {{{{\mathbf{A}}}}_{21}} & \ddots & \vdots \\ 0 & {{{{\mathbf{A}}}}_{21}^{\dagger} } & {{{{\mathbf{H}}}}_2} & \ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & {{{{\mathbf{A}}}}_{\alpha ,\alpha - 1}} \\ 0 & \cdots & 0 & {{{{\mathbf{A}}}}_{\alpha ,\alpha - 1}^{\dagger} } & {{{{\mathbf{H}}}}_\alpha } \end{array}} \right]} \hfill \end{array}$$
(1)

where \({{{\mathbf{H}}}}_m\) and \({{{\mathbf{R}}}}_m\) are the sub-Hamiltonians of the mth basic BNR and the mith row, respectively, and \({{{\mathbf{A}}}}_{mn}\) the hop** matrix from the mith row to the mth BNR when n = m and from the mth BNR to the \(m_i - 1\)th row when n = m − 1. Since the mth and m + 1th BNRs are mirror images about the mith row, the eigenstates of \({{{\mathbf{H}}}}_m\) and \({{{\mathbf{H}}}}_{m + 1}\) are the same with identical eigenenergies, and the hop** matrices satisfy \({{{\mathbf{A}}}}_{m + 1,m} = {{{\mathbf{A}}}}_{m,m}^{\dagger}\). Assuming the resonant state of the mth basic BNR is described by the Schrödinger equation of \({{{\mathbf{H}}}}_m|{{{\mathbf{{\Phi}}}}}_0\rangle = E_{{{\mathrm{r}}}}|{{{\mathbf{{\Phi}}}}}_0\rangle\), with \(|{{{\mathbf{{\Phi}}}}}_0\rangle\) the wave function and Er the resonant energy, the wave function \(|{{{\mathbf{{\Psi}}}}}\rangle\) of \({{{\mathbf{H}}}}_{{{\mathrm{c}}}}\) can then be constructed as:

$$|{{{\mathbf{{\Psi}}}}}\rangle = \sqrt \alpha /\alpha (|{{{\mathbf{{\Phi}}}}}_0\rangle ,0, - |{{{\mathbf{{\Phi}}}}}_0\rangle ,...,( - 1)^{\alpha - 1}|{{{\mathbf{{\Phi}}}}}_0\rangle )^{{{\mathrm{T}}}}.$$
(2)

Substituting Equations (1) and (2) into the Schrödinger equation, one obtains \({{{\mathbf{H}}}}_{{{\mathrm{c}}}}|{{{\mathbf{{\Psi}}}}}\rangle = E_{{{\mathrm{r}}}}|{{{\mathbf{{\Psi}}}}}\rangle\) straightforwardly. Thus, disordered BNRs with \(N = \alpha (N_i + 1) - 1\) possess a completely identical resonant peak for different α. These results are further confirmed by the numerical results of bond currents (Supplementary Fig. 3).

Discussion

In summary, the electron transport along disordered BNRs with random distribution of LDs is investigated. Our results indicate that although the overall electron transport efficiency is strongly declined, some resonant peaks are observed in the transmission spectra and leads to delocalized states in various disordered BNRs. These disordered BNRs could exhibit metal-insulator transition by varying nanoribbon width with tunable transmission gap in the insulating regime. Besides, the evolution of resonant peaks in disordered BNRs with different widths is explored, which is related to the spatial inversion symmetry.

Methods

According to the Landauer-Büttiker formula, the conductance is expressed as \(G = G_0{{{\mathrm{Tr}}}}[{{{\mathbf{{\Gamma}}}}}_{{{\mathrm{L}}}}{{{\mathbf{G}}}}^{{{\mathrm{r}}}}{{{\mathbf{{\Gamma}}}}}_{{{\mathrm{R}}}}{{{\mathbf{G}}}}^{{{\mathrm{a}}}}]\). The conductance quantum \(G_0 = 2e^2/h\), the Green’s function \({{{\mathbf{G}}}}^{{{\mathrm{r}}}}(E) = [{{{\mathbf{G}}}}^{{{\mathrm{a}}}}(E)]^{\dagger} = [E{{{\mathbf{I}}}} - {{{\mathbf{H}}}}_{{{\mathrm{c}}}} - {{{\mathbf{{\Sigma}}}}}_{{{\mathrm{L}}}}^{{{\mathrm{r}}}} - {{{\mathbf{{\Sigma}}}}}_{{{\mathrm{R}}}}^{{{\mathrm{r}}}}]^{ - 1}\), and the linewidth function \({{{\mathbf{{\Gamma}}}}}_{{{{\mathrm{L}}}}/{{{\mathrm{R}}}}} = i({{{\mathbf{{\Sigma}}}}}_{{{{\mathrm{L}}}}/{{{\mathrm{R}}}}}^{{{\mathrm{r}}}} - {{{\mathbf{{\Sigma}}}}}_{{{{\mathrm{L}}}}/{{{\mathrm{R}}}}}^{{{\mathrm{a}}}})\), with E the electron energy, \({{{\mathbf{H}}}}_{{{\mathrm{c}}}}\) the CSR Hamiltonian, and \({{{\mathbf{{\Sigma}}}}}_{{{{\mathrm{L}}}}/{{{\mathrm{R}}}}}^{{{\mathrm{r}}}}\) the retarded self-energy due to the coupling to the left/right semi-infinite \(\nu _{1/5}\) BNR70,71.

As seen from Fig. 1a, a disordered BNR with width N can alternatively be divided into N/2 (\((N - 1)/2\)) basic BNRs for even (odd) N. The width of this basic BNR is Ni = 2, and the mth basic BNR includes the 2m−1th and 2mth rows. Then, the Hamiltonian \({{{\mathbf{H}}}}_{{{\mathrm{c}}}}\) of this disordered BNR with width N can be partitioned as:

$$\begin{array}{*{20}{l}} {{{{\mathbf{H}}}}_{{{\mathrm{c}}}} = \left[ {\begin{array}{*{20}{c}} {{{{\mathbf{H}}}}_1} & {{{{\mathbf{A}}}}_{11}} & 0 & \cdots \\ {{{{\mathbf{A}}}}_{11}^{\dagger} } & {{{{\mathbf{H}}}}_1} & {{{{\mathbf{A}}}}_{11}} & \ddots \\ 0 & {{{{\mathbf{A}}}}_{11}^{\dagger} } & {{{{\mathbf{H}}}}_1} & \ddots \\ \vdots & \ddots & \ddots & \ddots \end{array}} \right],} \hfill \end{array}$$
(3)

where \({{{\mathbf{H}}}}_1\) is the sub-Hamiltonian of the basic BNR and \({{{\mathbf{A}}}}_{11}\) is the hop** matrix from the m + 1th basic BNR to the mth one. We emphasize that \({{{\mathbf{H}}}}_{{{\mathrm{c}}}}\) in Equations (1) and (3) is identical to each other but with different block matrices.

As the basic BNR can be further divided into (2,2) and (2,3) chains with width Ni = 2, \({{{\mathbf{H}}}}_1\) can be expressed as:

$$\begin{array}{*{20}{l}} {{{{\mathbf{H}}}}_1 = \left[ {\begin{array}{*{20}{c}} {{{{\mathbf{h}}}}_1} & {{{{\mathbf{h}}}}_{12}} & 0 & \cdots & 0 \\ {{{{\mathbf{h}}}}_{12}^{\dagger} } & {{{{\mathbf{h}}}}_2} & {{{{\mathbf{h}}}}_{23}} & \ddots & \vdots \\ 0 & {{{{\mathbf{h}}}}_{23}^{\dagger} } & {{{{\mathbf{h}}}}_3} & \ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & {{{{\mathbf{h}}}}_{L - 1,L}} \\ 0 & \cdots & 0 & {{{{\mathbf{h}}}}_{L - 1,L}^{\dagger} } & {{{{\mathbf{h}}}}_L} \end{array}} \right].} \hfill \end{array}$$
(4)

Here, \({{{\mathbf{h}}}}_l\) is the sub-Hamiltonian of the lth isolated chain and \({{{\mathbf{h}}}}_{l,l + 1}\) is the hop** matrix between two neighboring chains. When the lth chain is (2,2) one, \({{{\mathbf{h}}}}_l\) reads

$$\begin{array}{*{20}{l}} {{{{\mathbf{h}}}}_l = {{{\mathbf{h}}}}_{(2,2)} = \left[ {\begin{array}{*{20}{l}} {\varepsilon _{{{\mathrm{u}}}}} \hfill & {t_{{{{\mathrm{uv}}}}}} \hfill & {t_{{{{\mathrm{uw}}}}}} \hfill & 0 \hfill \\ {t_{{{{\mathrm{uv}}}}}^ \ast } \hfill & {\varepsilon _{{{\mathrm{v}}}}} \hfill & {t_{{{{\mathrm{vw}}}}}} \hfill & {t_{{{{\mathrm{vx}}}}}} \hfill \\ {t_{{{{\mathrm{uw}}}}}^ \ast } \hfill & {t_{{{{\mathrm{vw}}}}}^ \ast } \hfill & {\varepsilon _{{{\mathrm{w}}}}} \hfill & {t_{{{{\mathrm{wx}}}}}} \hfill \\ 0 \hfill & {t_{{{{\mathrm{vx}}}}}^ \ast } \hfill & {t_{{{{\mathrm{wx}}}}}^ \ast } \hfill & {\varepsilon _{{{\mathrm{x}}}}} \hfill \end{array}} \right].} \hfill \end{array}$$
(5)

When the lth chain is (2,3) one, \({{{\mathbf{h}}}}_l\) reads

$$\begin{array}{*{20}{l}} {{{{\mathbf{h}}}}_l = {{{\mathbf{h}}}}_{(2,3)} = \left[ {\begin{array}{*{20}{c}} {\varepsilon _{{{\mathrm{a}}}}} & {t_{{{{\mathrm{ab}}}}}} & {t_{{{{\mathrm{ac}}}}}} & 0 & 0 \\ {t_{{{{\mathrm{ab}}}}}^ \ast } & {\varepsilon _{{{\mathrm{b}}}}} & {t_{{{{\mathrm{bc}}}}}} & {t_{{{{\mathrm{bd}}}}}} & 0 \\ {t_{{{{\mathrm{ac}}}}}^ \ast } & {t_{{{{\mathrm{bc}}}}}^ \ast } & {\varepsilon _{{{\mathrm{c}}}}} & {t_{{{{\mathrm{cd}}}}}} & {t_{{{{\mathrm{ce}}}}}} \\ 0 & {t_{{{{\mathrm{bd}}}}}^ \ast } & {t_{{{{\mathrm{cd}}}}}^ \ast } & {\varepsilon _{{{\mathrm{d}}}}} & {t_{{{{\mathrm{de}}}}}} \\ 0 & 0 & {t_{{{{\mathrm{ce}}}}}^ \ast } & {t_{{{{\mathrm{de}}}}}^ \ast } & {\varepsilon _{{{\mathrm{e}}}}} \end{array}} \right].} \hfill \end{array}$$
(6)

The sub-Hamiltonian \({{{\mathbf{h}}}}_{(2,2)}\) corresponds to half of the unit cell of \(\nu _{1/5}\) borophene (see the bottom panel of Fig. 1b), and \({{{\mathbf{h}}}}_{(2,3)}\) to the unit cell of \(\nu _{1/6}\) borophene (see the top panel of Fig. 1b).