Nanotube Slidetronics

One-dimensional slidetronics is predicted for double-walled boron-nitride nanotubes. Local electrostatic polarization patterns along the body of the nanotube are found to be determined by the nature of the two nanotube walls, their relative configuration, and circumferential faceting modulation during coaxial interwall sliding. By careful choice of chiral indices, chiral polarization patterns can emerge that spiral around the nanotube circumference. The potential usage of the discovered slidetronic effect for low-dimensional nanogenerators is briefly discussed.


S1. Polarization registry index (PRI)
The polarization registry index (PRI) is a geometric measure that correlates interlayer registry in layered material interfaces and their electric polarization characteristics.A detailed description of the PRI, its global (GPRI) and local (LPRI) definitions, and their performance are provided in Ref. 1. Originally developed for planar bilayer systems, such as hexagonal boron-nitride (h-BN), it can be generalized to describe curved structures, e.g.double walled nanotubes (DWNT).The generalization follows the spirit of the PRI for planar systems consisting of non-trivial sublayer structure, such as transition metal dichalcogenides.Here, the projected overlap between two-dimensional Gaussian functions (the standard deviations of which are marked by ) associated with atoms residing on adjacent layers is scaled by a factor that depends exponentially (with an exponent ) on the vertical distance between the two.Specifically, for the case of h-BN, Eq. ( 4) (rather than Eq. ( 1)) of Ref. 1 is used in Eq. ( 2) of the same reference for the PRI calculation.The vertical distance between the two atoms is defined according to their local normal vectors following the procedure described in Ref. 2. Parameter refitting (see Table S1) following the fitting procedure described in Ref. 1 for the planar bilayer system, yields excellent agreement between the  ℎ−BN and density functional theory (DFT) calculated polarization profiles (see Fig. S1).  ) is given in units of /√3, where  is the lattice constant ( ℎ− = 2.51 Å).Average (̅ ) difference between the normalized reference DFT potential drop profile and the  ℎ− results, and the corresponding standard deviation () obtained using the optimal parameter sets are also provided.

h-BN
Parameter

S2. Quasi-static simulations
Quasi-static axial inter-wall sliding simulations were performed in steps of Δ = 0.1 Å using classical force-fields as described in the methods section.For each inter-wall shift the atomic coordinates of the double-walled boron-nitride nanotube (DWBNNT) structure were first optimized (keeping their axial, , coordinate fixed) using the Fire algorithm 3 with fixed supercell size.This was followed by an optimization step of the nanotube supercell length using the conjugate gradient (CG) algorithm. 4This two-step energy minimization procedure was repeated five times, which is sufficient to obtain well converged results (see Fig. S2a).In each repetition, both minimization stages were terminated when the forces acting on each degree of freedom reduced below 10 −6 eV/Å .Finally, the atomic coordinates were further relaxed using the Fire algorithm with a force tolerance of 10 −6 eV/Å .A typical diagram presenting the total energy minimization during the relaxation procedure is shown in Fig. S2a for the (70,70)@(77,74) DWBNNT.
The corresponding energy profiles, calculated using single-point DFT calculations (see Methods section) on the ILP relaxed structures of the ZZ (55,0)@(63,0) and (31,31)@(36,36)AC DWBNNTs, are depicted in Figs.S2b-c.The sliding energy corrugations of ~1.1 and ~0.4 meV/atom, respectively, are lower than those obtained using the ILP (3.4 and 0.9 meV/atom, respectively).Nonetheless, considering the overall very small energy landscape corrugation, these differences can be considered to be minor.We note that these sliding energy barriers compare well to those obtained for 2D bilayers of h-BN (2.3 meV/atom) 5 and MoS2 (0.2 meV/atom) 6 .
The ILP energy barriers per unit-cell for DWBNNT coaxial sliding are 0.5 eV, 0.1 eV, and 0.2 meV for ZZ, AC, and bichiral DWBNNTs, respectively, calculated based on the above reported values and on the number of atoms per unit cell.We therefore conclude that room temperature fluctuations are insufficient to induce coaxial sliding of the achiral DWBNNTs.

S4. Additional details regarding the calculations of AC@AC DWBNNTs
S4.1.Charge redistribution map of parallelly-stacked (31,31)@(36,36) DWBNNT In the discussion of Fig. 2 provided in the main text, we attributed the emerging electrostatic potential energy patterns to charge redistribution between the two AC (31,31)@(36,36) tube walls.To demonstrate this, we present in Fig. S7 charge density difference maps (with respect to the individual tubes) obtained at different interlayer displacements, corresponding to the structures discussed in the main text.We find that for the facetted structures, the charge density redistribution becomes more pronounced (see Fig. S7a and c).This can be attributed to the increased inter-wall lattice commensurability at the facet regions.Notably, the AB and BA stacking regions in each facet show opposite charge difference polarity, and locally asymmetric patterns due to circumferential frustration.These results are in line with the polarization maps presented in the main text.

Figure S1 .
Figure S1.Polarization profiles calculated using DFT (adapted from Ref. 1, open circles) and the generalized GPRI ℎ−BN (full lines) along the armchair direction.The x-axis is normalized to the intralayer lattice constant of h-BN ( = 2.51Å).

Figure S4 .
Figure S4.(a) Cross-sectional view of the ZZ (30,0)@(38,0) DWBNNT.(b) DFT calculated electrostatic potential difference maps (with respect to the same individual walls) plotted along the (001) face of the DWBNNT presented in panel a.(c) Polar diagrams presenting the radial electrostatic potential energy drop (in meV) across the DWBNNT presented in panel a.

Figure S5 .
Figure S5.(a) Cross-sectional view of anti-parallelly stacked (55,0)@(63,0) DWBNNT.(b) DFT calculated electrostatic potential difference maps (with respect to the same individual walls) plotted along the (001) face of the DWBNNT presented in panel a.(c) Polar diagrams presenting the radial electrostatic potential energy drop (in meV) across the DWBNNT presented in panel a.

Figure S6 .
Figure S6.(a) DFT calculated (see Methods section) bandgap of parallelly-stacked ZZ (55,0)@(63,0) DWBNNTs as a function of coaxial interwall displacement.The horizontal dashed line represents the band gap of AB stacked h-BN bilayer calculated at the same level of theory.(b) The corresponding band structure obtained at various coaxial interwall displacements.The top of the valence band is set to zero in all panels.(c) Work function variations as a function of coaxial interwall displacement.The horizontal dashed line represents the work function of AB stacked h-BN bilayer calculated with the same level of theory.

Figure S9 .
Figure S9.Axial interwall shift induced superstructure and radial polarization variations calculated for the anti-parallelly stacked AC (31,31)@(36,36) DWBNNT.(a)-(d) Crosssectional views of the relaxed DWBNNT for several coaxial shifts,  = 0, 0.6, 1.2, and 1.8 Å. (e)-(h) DFT calculated electrostatic potential difference maps (with respect to the same individual walls) plotted along the (001) face of the DWBNNTs presented in panels a-d, respectively.The color bar appearing in panel e is common to panels f-h.(i)-(l) Polar diagrams presenting the radial electrostatic potential energy drop (in meV) across the DWBNNTs presented in panels a-d, respectively.

Figure S10 .
Figure S10.(a) DFT calculated (see Methods section) bandgaps of parallelly-stacked (black) and anti-parallelly-stacked (red) AC (31,31)@(36,36) DWBNNTs.The horizontal dashed line represents the bandgap of AB stacked h-BN bilayer calculated at the same level of theory.(b) and (c) show the band structures of parallelly-stacked and anti-parallellystacked DWBNNTs at various interwall coaxial displacements, respectively.The top of the valence band is set to zero in all panels.(d) Work function of parallelly-stacked (black) and anti-parallelly-stacked (red) AC (31,31)@(36,36) DWBNNTs at various coaxial interwall displacements.The horizontal dashed line represents the work function of AB stacked h-BN bilayer calculated at the same level of theory.