Resonant tunneling in disordered borophene nanoribbons with line defects

Very recently, borophene has been attracting extensive and ongoing interest as the new wonder material with structural polymorphism and superior attributes, showing that the structural imperfection of line defects (LDs) occurs widely at the interface between $\nu_{1/5}$ ($\chi_3$) and $\nu_{1/6}$ ($\beta_{12}$) boron sheets. Motivated by these experiments, here we present a theoretical study of electron transport through two-terminal disordered borophene nanoribbons (BNRs) with random distribution of LDs. Our results indicate that LDs could strongly affect the electron transport properties of BNRs. In the absence of LDs, both $\nu_{1/5}$ and $\nu_{1/6}$ BNRs exhibit metallic behavior, in agreement with experiments. While in the presence of LDs, the overall electron transport ability is dramatically decreased, but some resonant peaks of conductance quantum can be found in the transmission spectrum of any disordered BNR with arbitrary arrangement of LDs. These disordered BNRs exhibit metal-insulator transition by varying nanoribbon width with tunable transmission gap in the insulating regime. Furthermore, the bond currents present fringe patterns and two evolution phenomena of resonant peaks are revealed for disordered BNRs with different widths. These results may help for understanding structure-property relationships and designing LD-based nanodevices.

Very recently, borophene has been attracting extensive and ongoing interest as the new wonder material with structural polymorphism and superior attributes, showing that the structural imperfection of line defects (LDs) occurs widely at the interface between ν 1/5 (χ3) and ν 1/6 (β12) boron sheets. Motivated by these experiments, here we present a theoretical study of electron transport through two-terminal disordered borophene nanoribbons (BNRs) with random distribution of LDs.
Our results indicate that LDs could strongly affect the electron transport properties of BNRs. In the absence of LDs, both ν 1/5 and ν 1/6 BNRs exhibit metallic behavior, in agreement with experiments. While in the presence of LDs, the overall electron transport ability is dramatically decreased, but some resonant peaks of conductance quantum can be found in the transmission spectrum of any disordered BNR with arbitrary arrangement of LDs. These disordered BNRs exhibit metalinsulator transition by varying nanoribbon width with tunable transmission gap in the insulating regime. Furthermore, the bond currents present fringe patterns and two evolution phenomena of resonant peaks are revealed for disordered BNRs with different widths. These results may help for understanding structure-property relationships and designing LD-based nanodevices.
In this Letter, 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 ν 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. 1(a). We find that both ν 1/5 and ν 1/6 BNRs exhibit metallic behavior, consistent with experiments [21][22][23]50]. Remarkably, although the overall electron transport efficiency 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 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 . 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 (2,2)j or (2,3)j one and a (2,3)j chain by a (2,2)j or (2,3)j one, with 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. Energy-dependent conductance G for (b) a ν 1/5 BNR with p = 0 and (c) a ν 1/6 BNR with p = 1.
overlap perfectly with all those of two narrow BNRs and (ii) a specific resonant peak for width N i reappears at various BNRs with width N = α(N i + 1) − 1 and α an integer.
Model.-Electron transport through two-terminal disordered BNRs is simulated by the Hamiltonian [29,30]: Here, a † i (a i ) creates (annihilates) an electron at site i, ǫ i is the potential energy, and t is the nearest-neighbor hopping integral. According to the Landauer-Büttiker formula, the conductance is expressed as , with E the electron energy, H c the CSR Hamiltonian, and Σ r L/R the retarded self-energy due to the coupling to the left/right semi-infinite ν 1/5 BNR [59,60].
In the numerical calculations, the parameters are fixed to ǫ i = 0, t = 1 (energy unit), and the length counted by all of the chains in the CSR is taken as L = 2000, unless stated otherwise. We consider the most disordered BNRs with half of the chains being (2,3) ones (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 ν 1/5 or ν 1/6 BNR, as shown in Figs. 1(b) and 1(c). It clearly appears that both ν 1/5 and ν 1/6 BNRs exhibit metallic behavior which is consistent with experiments [21][22][23]50], and their conductances are asymmetric about the line of E = 0 owing to the electron-hole symmetry breaking.  For the ν 1/5 BNR, its transmission spectrum is characterized by many conductance plateaus quantized at integer multiples of G 0 as expected [ Fig. 1(b)], because of the translational symmetry. By contrast, when the ν 1/5 BNR is replaced by a ν 1/6 one, the conductance declines and presents dramatic oscillating behavior instead of quantized plateaus [ Fig. 1(c)], in accordance with firstprinciples calculations [58], because of LD-induced scattering at the CSR-lead interfaces. Disordered BNRs. -Figures 2(a) and 2(b) show energydependent averaged conductance G and standard deviation δG, respectively, for disordered BNRs with different widths N . Here, δG ≡ G 2 − G 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 [61]. However, it is surprising that some transmission peaks of G = G 0 are found in all these disordered BNRs and the peak number usually increases with N [ Figs. 2(a) and 4(b)]. In particular, the standard deviation at peak positions satisfies δG ∼ 0 [62] and these peaks are robust against L and p [61], implying that the transmission peaks could be observed in any disordered BNR with arbitrary arrangement of LDs. This result explains a recent experiment that delocalized states are measured in BNRs with LDs [53]. These transmission peaks originate from resonant tunneling [61], where some resonant energies remain unchanged for ν 1/6 BNRs with different L and the others change with L. Therefore, the electrons with invariant resonant energies cannot be reflected by LDs, as further demonstrated from spatial distributions of bond currents [61], leading to resonant peaks in disordered BNRs.
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. 2(a)] to insulating behavior [see the other lines in Fig. 2(a)]. This indicates the width-driven metalinsulator transition in disordered BNRs. And the transmission gap is tunable in the insulating regime by varying N , which is similar to graphene nanoribbons [63][64][65] and facilitates band-gap engineering of BNRs. (ii) The resonant peaks, characterized by full width at half maximum (FWHM), are mainly categorized into narrow and wide peaks separated by about 0.06t FWHM [Figs. 2(a) and 4(b)]. (iii) Each resonant conductance peak corresponds to two peaks in the curve of δG-E [ Fig. 2(b)]. Remarkably, the maximum of all these peaks satisfies δG ∼ 0.28G 0 which is close to 1/12G 0 reported in disordered graphene nanoribbons [66,67]. (iv) Some transmission peaks can even achieve 2G 0 [see the black-solid line in Fig. 2(a)], because of almost perfect superposition of two neighboring resonant peaks, as seen from the four-peak structure around E ∼ −0.226t in Fig. 2

(b).
Evolution phenomenon (EP) I.-We then focus on the evolution of resonant peaks in disordered BNRs with various N . Figure 2(c) shows G 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 [61]. Therefore, we conclude that all of the resonant peaks of disordered BNRs with odd width N o could be assembled from the ones with N = (N o − 1)/2 and (N o + 1)/2, namely, EP I.
To elucidate the underlying physics of EP I, Fig. 3 plots spatial distributions of averaged interchain currents I l (n) at typical resonant energies shown in Fig. 2(c). The currents flowing from the lth chain to the l + 1th one read [68][69][70]: is the interchain lesser Green's function. It is clear that all of the I l (n) 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 I l (n) , whose number and location depend on E.
Since the spatial inversion symmetry with respect to the row of n = (N + 1)/2 is preserved for disordered BNRs with odd N [ Fig. 1(a)  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 4(a) shows G vs E around a transmission peak for three disordered BNRs, where the magentadashed line represents the sum of G 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. 4(a). This indicates that a transmission peak of wide disordered BNRs could be evolved from narrow ones.
To further elucidate the above phenomenon, Fig. 4 Fig. 4(b)]. 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 = α(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 × 7 − 1 = 20 deviates from EP II [see the surrounded blue square in Fig. 4(b)], because of finite-size effects. By increasing L, this transmission peak is split into two peaks at E ∼ 0 and 0.021t [ Fig. 4(a)]. 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 = α(N i + 1) − 1. According to its unique structure, this disordered sample could be divided into α basic BNRs with width N i , 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 m i th row, with 0 < m < α 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. 1(a)]. The Hamiltonian H c of this disordered BNR with width N can then be partitioned as: where H m and R m are the sub-Hamiltonians of the mth basic BNR and the m i th row, respectively, and A mn the hopping matrix from the m i th row to the mth BNR when n = m and from the mth BNR to the m i − 1th row when n = m − 1. Since the mth and m + 1th BNRs are mirror images about the m i th row, the eigenstates of H m and H m+1 are the same with identical eigenenergies, and the hopping matrices satisfy A m+1,m = A † m,m . Assuming the resonant state of the mth basic BNR is described by the Schrödinger equation of H m |Φ 0 = E r |Φ 0 , with |Φ 0 the wave function and E r the resonant energy, the wave function |Ψ of H c can then be constructed as: Substituting Eqs. (1) and (2) into the Schrödinger equation, one obtains H c |Ψ = E r |Ψ straightforwardly. Thus, disordered BNRs with N = α(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 [61].
Conclusion.-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 metalinsulator 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.
Pei-Jia Hu thanks Jing-Ting Ding for useful discussions. This work is supported by the National Natural Science Foundation of China (Grants No. 11874428, No. 11874187, and No. 11921005), the National Key Research and Development Program of China (Grant No. 2017YFA0303301), and the High Performance Computing Center of Central South University.