1 Introduction

Van der Waals (vdW) heterostructures have been recently found to possess many exotic physical properties that can be widely used in advanced physical mechanical devices [1,2,3,4,5,6,7,8,9,10]. The key to such success is the interaction between different layers of such heterostructures. For example, the breaking of symmetry between bilayer graphene was shown to open up a small band gap such that it can be used in semiconducting devices [11]. Dean et al. demonstrated that by putting graphene on boron nitride (BN) substrate, the performance of graphene-based electronic device was substantially improved [12]. Ever since such advances, fabrication of two-dimensional vdW heterostructures has become one of the central research topics, numerous layered material candidates have been widely used and extensively explored, including but not limited to BN, MoS2, WS2, black phosphorus (BlackP), blue phosphorus (BlueP), borophene, etc. [13,14,15,16,17,18,19,20] A new flavor recently added to this area is the magic-angle twisted bilayer graphene [21,22,23] and trilayer graphene [24, 25], where new exciting electronic states include the correlated electronic states and superconducting phase were found. Combining with the structurally induced incommensurate tribological states and the moire pattern, these structures are expected to bring a lot more intriguing properties, promising for future advanced smart devices [26,27,28,29,30].

While from the mechanical point of view, although rich physics has been discovered in the above-mentioned low-dimensional devices, a fundamental challenge remains, which is the scaled production of such vdW heterostructures in laboratory. Owing to the weak vdW interaction, small size and complicated environments are required for such structures to operate in devices [1, 31, 32]. Although a few strategies have been proposed and demonstrated, such as the AFM manipulation approach [33], there is still a long way toward industrial applications.

It is interesting to mention that, prior to the discovery of layered graphene [34] and other 2D materials, carbon nanotubes (CNTs) [35, 36] and other 1D materials have already been synthesized and proven to host rich mechanical and physical properties. These nanotubes can also form vdW structures such as multi-walled CNTs (MWCNTs) or other coaxial nanotubes. For example, by melting fullerenes in BN nanotubes, one can produce coaxial CNT/BNNT vdW structures [37]. Zhang et al. demonstrated that such structures can be used as nanocables due to the conducting and insulating nature of the CNT and BNNT components [38, 39]. Therefore, it is also of high importance to explore the possibility of fabricating similar 1D heterostructures composed of different materials as they possess the advantages of well-defined geometry, ease of assembly, etc. [40] Efforts have been directed toward producing more of such heterostructures, for example, Xiang et al. reported the successful synthesis of BN and MoS2 crystals on single-walled CNTs (SWCNTs) and even BN/MoS2/C multi-walled nanotubes, which are useful for many applications [41].

In this work, we show via molecular dynamics (MD) simulations that the hollow space inside mechanically stable CNTs can be used as a template to produce 1D vdW heterostructures. The confined geometry of the nanotube may also enhance the mechanical stability of encapsulated nano-species. We use the recently synthesized BlueP [42] as an example to validate such a scenario because BlueP is a promising layered semiconductor, but its stability is not comparable to that of graphene nanoribbons. The confined BlueP/CNT structure may lead to interesting dynamics, improved structural stability and new physical properties suitable toward new devices. Our results show that driven by vdW interaction, 1D BlueP nanoribbons (BluePNR) can be spontaneously encapsulated into the inner side of SWCNTs while the structural transition is determined by the interplay between vdW interaction and bending energy. When the vdW interaction is sufficiently strong, BluePNR will be driven to form curved helical structure, self-healing occurs at bare edges of the nanoribbon and further lowers the potential energy of the system. Based on a simple energetic model, we draw a phase diagram for the final assembled structure. We expect that our results can be used for understanding a variety of nanomaterials and guiding self-assembly of 1D vdW heterostructures with desired functionality.

2 Methods

Molecular dynamics was performed via the LAMMPS [43] package. Carbon nanotubes of 200.0 Å in length were built in all the simulations with non-periodic boundary conditions. The diameters of the CNTs range from 20.3 to 74.6 Å. Armchair, zigzag and chiral CNTs were all considered in the simulations to examine the chirality effects on the dynamics and the final structure. The BluePNRs built in this work mostly have the length of 350.0 Å and the width of 8.2 to 50.8 Å, and both armchair- and zigzag-edged ribbon structures were considered. BluePNRs were placed at the open end of the CNTs to examine how they reacted with the environment at a low temperature of 5 K using the NVT thermostat [44], with the timestep of 1 fs in the simulations. CNTs were not allowed to move during the simulations in order to simplify the data collection and track the dynamic behavior. The interatomic potential used in this study includes the AIREBO potential [45] for carbon, the Stillinger–Weber potential [46] for P, and the Lennard–Jones (L-J) potential [47] for vdW interactions between P and C. The L-J parameters are: \(\varepsilon =0.00776\; {\rm eV}, \;\mathrm{and}\;\sigma =3.5627 \)Å. The potential energy and vdW energy between CNTs and BluePNRs were calculated, and the structures were visualized using OVITO to examine the underlying physics responsible for the assembly behavior.

3 Results and Discussion

Figure 1 shows a typical example of encapsulating BluePNR into SWCNTs and the evolution of the potential energy of the system in time. The BluePNR in this case is 11.5 Å in width and the (15,15) SWCNT has a diameter of 20.3 Å. It is seen that at the beginning of the simulation, the potential energy of the system decreases sharply by 39.7 eV within 90 ps, correspondingly, the BluePNR moves translationally from the right end of the SWCNT until they are in complete contact, as can be seen in snapshots e1 and e2. Afterwards, the contact area between BluePNR and SWCNT does not decrease anymore. Therefore, the potential energy reaches the minimum value and gradually increases with further translational motion of the BluePNR toward the left end of the SWCNT, leading to slightly decreased contact area, and the BluePNR is also seen bent somewhat, which raises the bending energy (see e4 in Fig. 1). As a result, a potential energy bump is observed in Fig. 1a. Afterwards, the trajectory of BluePNR converges toward the center of the SWCNT, and the potential energy starts to approach the saturation value. The saturation value of potential energy is higher than that at e2. This can be attributed to the bending deformation of BluePNR that introduces strain energy to the system, as can be seen in snapshots e5, e6 and e7 shown in Fig. 1. This type of 1D vdW nanoribbon/nanotube structure is similar to that observed in graphene nanoribbon (GNR) encapsulated in SWCNT, where exotic transport properties were found owing to the hybridization [48].

Fig. 1
figure 1

Self-assembly of a BluePNR of 11.5 Å in width into a (15,15) SWCNT of 20.3 Å in diameter. The potential energy variation with respect to simulation time is displayed in a and typical snapshots are shown in b, the labels (from e1 to e7) in the snapshots correspond to each energy state shown in a

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Interestingly, when the (25,25) SWCNT of a larger diameter is used for the same simulation, a distinct behavior is observed. As can be seen in Fig. 2, during the early stage of the encapsulation process, the structural dynamics and the potential energy change are very similar to those in Fig. 1. However, the potential energy drop is stronger as compared with that encapsulated in the (15,15) SWCNT, reaching around 67 eV at 100 ps. This is because the (25,25) SWCNT has larger diameter, meaning smaller curvature; therefore, the contact quality between SWCNT and the BluePNR is improved (i.e., shorter distance between P and C atoms on average), leading to improved vdW interaction. As a result, it is found that after 310 ps of simulation (n6 of Fig. 2), the potential energy suddenly decreases, in dramatic contrast to the slight increasing behavior observed in Fig. 1. A close look at the snapshots during this stage shows that the BluePNR starts to curve up after around 240 ps (n5 of Fig. 2) to make closer contact with the SWCNT. Although this process induces extra bending energy to the BluePNR, the decrease in vdW interaction energy (of negative values) within the BluePNR nullifies this cost to favor the increase in curvature. Further evolving of such process enables a full contact between SWCNT and BluePNR until the potential energy reaches the global minimum of −158 eV at 480 ps. The helical tubular-like BlueP structure is thus obtained.

Fig. 2
figure 2

Self-assembly of a BluePNR of 11.5 Å in width into a (25,25) SWCNT of 17 Å in radius. The potential energy variation of the system is shown in a and typical snapshots are shown in b

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As mentioned above, BlueP is a newly synthesized semiconductor, but similar to black phosphorus, its stability under ambient condition is not comparable to typical 2D materials such as graphene and BN. To retain good stability in the presence of bare edges in these structures is essential to its performance in semiconducting devices. Fortunately, within the confined space in SWCNT, little room is provided for BluePNR to break into pieces or its structural form as observed in the red phosphorus case. Instead, after curving into helical structures, the bare edges of nanoribbon vibrate around their neighboring edges, as can be seen in Fig. 3. At certain moments, the dynamic motion of the ribbon enables unsaturated edge atoms to approach the bonding distance, which is 2.26 Å in the BlueP case. The bonding of two unsaturated P atoms can reduce the potential energy of the system by about 0.399 eV, and further pulls the next pair of P atoms closer to the bonding distance. As a result, a cascading self-healing process takes place that drives the helical structure to form BlueP nanotube (BluePNT), as observed in Fig. 3d − f. In this simulation, 120 pairs of P-P edge atoms are presented, the corresponding total amount of energy drop can be estimated to 120*0.399 = 47.88 eV. As can be seen in Fig. 3a, after the assembly process is completed, the vdW energy is about 48.01 eV above the potential energy of the final structure, indicating fully self-healing of the BlueP structure. A previous study by Kit et al. showed that twisting bare-edged graphene nanoribbons can produce SWCNTs of desired diameter [49], where precise external manipulation is needed for the nanotube formation. In contrast, the driving force for the formation of the BluePNT here is pure vdW interaction and the process is spontaneous, these give dramatic advantage of our approach over the previous ones in ease of implementation.

Fig. 3
figure 3

a Variation of potential energy and vdW energy with respect to time during the self-assembly of BluePNR into the (25,25) SWCNT. Typical snapshots of the BluePNR from 345 to 410 ps are shown in d–f, where local unsaturated P atoms pair with one another to form new P − P bonds in order to lower the potential energy of the system. With two representative P atoms (P1 and P2) highlighted to display the scenario, the potential energy and distance between P1 and P2 are shown in b and c, the rebonding event occurs at 367.4 ps

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As discussed above, the main driving forces for the encapsulation process are the vdW energy and bending energy. We next proceed to discuss in detail the competition between them that governs the final configurations. Considering the segments of BluePNR inserted into the SWCNT, the maximum vdW energy that can be applied to the BluePNR is:

$${{{E}}}_{{\text{v}}{\text{d}}{\text{W}}}={{a}}{{l}}{{w}}$$
(1)

where a is a constant equalling 270.5 eV/Å2, l and w are the length and width of the BluePNR, respectively. If the BluePNR is bent to make full contact with the inner wall of SWCNT, the bending energy required is:

$$E_\text{bend}={\int }_{0}^{l}{D}{w}{k}^{2}\left({{x}}\right) \text{d}{x}$$
(2)

where k is the curvature of the bent BluePNR and \(\frac{1}{k}=R-0.47\). Substitute this relation into Eq. (2), one therefore finds:

$${{{E}}}_\text{bend}=\frac{{{D}}{{w}}{{l}}}{{({{R}}-0.47)}^{2}}$$
(3)

where D is the bending stiffness of BluePNR and D = 0.84 eV from our calculations. It is straightforward that if the BluePNRs curve up to form nanotube structures, the following condition must meet:

$$\left|\frac{{{{E}}}_{{\text{v}}{\text{d}}{\text{W}}}}{{{{E}}}_{{\text{b}}{\text{e}}{\text{n}}{\text{d}}}}\right|=\left|\frac{{{a}}}{{{D}}}{({{R}}-0.47)}^{2}\right|>1$$
(4)

One therefore obtains the necessary condition for the formation of BluePNT in SWCNT (BluePNT/SWCNT):

$${{R}}>10.3{\mathop {\rm{A}}\limits^ \circ}$$
(5)

This explains why in our simulations, the BluePNR cannot form tubular structures in the (15,15) SWCNT, the radius of which is 10.17 Å. Because it is smaller than 10.3 Å, only ribbon structures in SWCNT are obtained. While for the (25,25) SWCNT case in Fig. 2, the radius of 16.95 Å is larger than 10.3 Å, we thus obtain the 1D BluePNT/SWCNT vdW structure.

It should be noted that for larger SWCNTs, an additional condition is also needed for the formation of BluePNT/SWCNT structure: the width of the ribbon must be sufficiently small to enter the SWCNTs at the early stage, this leads to:

$${{R}}>\frac{{{w}}}{2}+0.47$$
(6)

This scenario is demonstrated by our simulations on the size effect of self-assembly, as shown in Figs. 4 and 5. A (30,30) SWCNT of 20.3 Å in radius is utilized in Fig. 4, and BluePNRs with different widths ranging from 8.2 to 41 Å are considered in the simulations to examine the effect of width. Since 20.3 Å is greater than 10.3 Å, BluePNRs all curve up to tubular structures as long as they can be inserted to the SWCNT, i.e., when the width is not too wide as Eq. (6) requires, as clearly seen in Fig. 4. We also plot the vdW energy change and potential energy change during the assembly process, as can be seen in Fig. 4b. It is seen within the narrow BluePNR width region that the vdW energy change per atom remains at the same level of − 0.12 eV/atom, while the potential energy change is larger, owing to the contribution of self-healing process. Interestingly, the potential energy change increases with width. This is because wider BluePNRs have lower proportion of edge atoms. As a result, the self-healing-induced energy drop makes less contribution to the average potential energy.

Fig. 4
figure 4

Effect of BluePNR width on the self-assembly process in a (30,30) SWCNT. The vdW energy variation is shown in a, comparison of potential energy change and vdW energy change is shown in b, and the obtained final configurations are shown in c

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Fig. 5
figure 5

Effect of SWCNT radius on the self-assembly process. The vdW energy variation is shown in a, comparison of potential energy change and vdW energy change is plotted in b, and the obtained final configurations are shown in c

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When the BluePNR width approaches the critical value, the structural evolution starts to vary. Taking the BluePNR with w = 31.16 Å as an example, although the final configuration is similar to other narrower BluePNRs, the decrease rate of the vdW energy is smaller (see Fig. 4a), meaning it takes longer for the BluePNR to enter the SWCNT and then curve up to tubular structures. When the width further increases to 34.4 Å, which breaks the condition in Eq. (6), the simulation shows that the vdW energy stays at high level throughout the simulation and no tubular structures form. As a result, both the vdW energy change and the potential energy change remain at very low levels owing to the trivial contact. We further study BluePNRs of a larger width of 41 Å. Interestingly, due to its large size, the BluePNR stays at the end of the SWCNT for a long time of 250 ps, although the vdW interaction is stronger than other cases simulated above. The BluePNR then manages to find a way to contact the outside of the SWCNT and eventually form a tubular structure outside the SWCNT. This wrapping behavior is similar to the self-assembly of core–shell all-carbon photovoltaic heterostructures [40, 50]. Similarly, one can expect the assembled 1D BluePNR/SWCNT heterostructures to possess intriguing physical properties suitable for advanced electronics.

The effect of SWCNT radius on the assembly is summarized in Fig. 5, where it can be seen that when the SWCNT is too small, the BluePNR enters the inner side of the SWCNT but does not curve up, the vdW energy therefore stays at a high value. When the radius increases to larger than 10.3 Å, tubular structures form for the BluePNRs of the same width, as clearly seen in Fig. 5, and the vdW energy and potential energy both drop to lower values, indicating higher stability, as shown in Fig. 5b. Note that the variation trend of potential energy here is not monotonic. This is because the contact quality of edge P-P pairs is not consistent, indicating that the final tubular BluePNT structure is defective in some cases.

Based on the simulation results and our analysis described above, we finally present a phase diagram for the self-assembly behavior, as shown in Fig. 6. Namely, we identify three phases for the assembly of BluePNR to the SWCNT to form the 1D vdW heterostructure, the BluePNT in SWCNT heterostructure (phase I), the BluePNR in SWCNT heterostructure (phase II), and the BluePNR wrapping SWCNT structure (phase III). The corresponding side and axial views of the structures are shown in Fig. 6a, and the three phases are clearly separated in Fig. 6b with respect to the SWCNT radius and BluePNR width. It is worth noticing that the above conclusions are not sensitive to edge passivations of the SWCNTs. We have run similar simulations with the SWCNT edges passivated with hydrogen atoms, and found that the trends are very consistent, with only negligible differences in vdW energy. Recent experiments have also observed encapsulation of 1D structures within the inner wall of CNTs, e.g., red phosphorus and carbon chains [48, 51]. We therefore expect future experimental efforts to fabricate such 1D vdW configurations for desirable applications with high performance, such as the recent work by Rybkovskiy et al. [51]

Fig. 6
figure 6

a Three typical final configurations obtained via self-assembly; b the phase diagram with respect to SWCNT radius and BluePNR width, based on the combination of analysis and simulation data

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4 Conclusions

In conclusion, we found by molecular dynamics simulations that vdW interaction can drive BluePNRs of narrow width to be encapsulated into larger-diameter SWCNTs and form 1D vdW heterostructures. When the vdW energy is sufficiently strong to overcome the bending energy of BluePNR, coaxial tubular structure is formed and self-healing process takes place for bare-edged BluePNR. While for wide BluePNRs or 2D BlueP sheets, wrapping behavior is observed for the same simulation conditions. We derived simple analysis model to predict such behavior and drew a phase diagram for the assembly of 1D vdW heterostructures. As the 1D vdW heterostructure is an exciting complement to the 2D vdW heterostructure family and the fabrication of 2D vdW heterostructure is still a great challenge to the research community, we expect that our results can offer an alternative approach and guideline for addressing the urgent issue of identifying high-throughput fabrication methods for low-dimensional functional structures and devices with desired performance.