Band alignment of one-dimensional transition-metal dichalcogenide heterotubes†

Mei Ge^a, Fanmin Zeng^b, Zixuan Wang^b, Jiang-Jiang Ma*^b and Junfeng Zhang*^a
aCollege of Physics and Electronic Engineering, Hainan Normal University, Haikou, 571158, China. E-mail: zhangjf@hainnu.edu.cn
^bSchool of Physics and Information Engineering, Shanxi Normal University, Taiyuan 030031, China. E-mail: majiangjiang@sxnu.edu.cn

First published on 28th August 2024

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Introduction

Two-dimensional (2D) heterostructures, whether van der Waals (vdW)¹ or lateral,² have attracted significant attention due to their unique physics not observed in conventional 2D materials and devices of various applications.^3–9 Despite considerations of symmetry and lattice mismatch, both the transfer approach¹⁰ and chemical vapor deposition (CVD) methods^5,11,12 have proven successful in obtaining 2D heterostructures with atomic sharp interfaces and controllable domain sizes. These 2D heterostructures exhibit novel properties that enable the realization of diverse electronic devices, including tunnelling transistors, flexible electronics, photodetectors, photovoltaics, and light-emitting devices.¹³ Furthermore, considerable efforts have been stirred up for the new class of devices based on 2D–1D^14,15 or 1D–1D heterostructures.^16–19 The quantum confinement effects within heterotubes make the 1D physics more intriguing.²⁰ As a cylinder rolled up from a 2D sheet, a 1D nanotube can serve as the blocking component for 1D heterotubes. In 1D nanotubes, the curvature breaks the mirror symmetry of the 2D sheet, rendering both the chiral vector and radial size crucial. Therefore, compared with 2D heterostructures, 1D heterotubes offer greater flexibility in tuning the properties and designing new 1D devices.^21–25

The most prominent among 1D materials is the carbon nanotube (CNT),²⁶ known for its self-assembling capabilities. Additionally, boron nitride nanotube (BNNT),^27–29 TMD NTs,^30,31 lead iodide nanotubes (PbI₂ NTs)³² and covalent organic frameworks nanotubes (COF NTs),³³ are noteworthy carbon-like nanotubes. Particularly interesting thing is the phenomenon observed as the diameter of 1D heterotubes increases: a transition from a straddling to staggered band gap between the inner and outer layer CNTs of 1D heterotubes takes place. This transition has been verified in charge separation in photovoltaic devices.²⁶ By using in situ electron irradiation methods, Arenal et al.²⁸ experimentally synthesized 1D CNT@BNNT heterotubes (where “@” represents “encapsulated in”), providing a material platform for realizing topological, low dissipation CNT nanoelectronics.²⁹ Another investigation by Cao et al.³¹ demonstrated that 1D CNT-MoS₂ heterotubes can function effectively as battery electrodes, exhibiting high specific capacity and low degradation. In the case of PbI₂@multiwall CNTs (MWCNTs), both MWCNTs and PbI₂ NT demonstrated enhanced conductivity.³² Additionally, COF@CNTs heterotubes exhibited dramatically enhanced catalytic activity, attributed to charge transfer from CNTs, which decreases COF's bandgap and work function.³³ Moreover, heterotubes containing more than two components can be fabricated with controllable layer number and diameter size. In these heterotubes, intertube Coulomb interactions play prominent roles in carries transport.^34–36 While the experimental synthesis of numerous 2D materials with a wide spectrum of electronic properties has made remarkable progress, the investigation of coaxially stacked 1D heterotubes, composed of different types of 1D counterparts, is still in its early stages. A significant amount of related research remains to be undertaken.

2D TMD vdW heterostructures have been fabricated with controllable domain size, layer number and atomic sharp interface.^5,37–39 As a representative example, the 2D MoS₂/WS₂ vdW heterostructure exhibits a type II band alignment,⁵ where the conduction band minimum (CBM) of MoS₂ aligns with the valence band maximum (VBM) of WS₂. This arrangement results in efficient charge separation,⁶ which makes it highly promising for applications in photovoltaics and optoelectronics.⁴⁰ Recent efforts have focused on understanding the ultrafast interlayer charge transfer processes in these vdW heterostructures.^6,41–46 Additionally, novel physical phenomena such as interlayer excitons, Bose–Einstein condensation, superfluidity, and K–K′ valley dynamics, have garnered significant attention.⁴⁷ Although TMD tubes have been successfully synthesized,⁴⁸ and their mechanical and electronic properties have been extensively investigated.^49–53 1D TMD heterotubes have been largely overlooked based on our limited knowledge. To fully comprehend the novel physical phenomena and unlock the potential practical applications associated with 1D TMD heterotubes, further research is imperative.

In this work, we investigate the structural and electronic properties of 1D MoS₂@WS₂ (WS₂@MoS₂) heterotubes through first-principles calculations. Armchair (ac), zigzag (zz) and chiral heterotubes are considered, with diameters ranging up to ∼4 nm. Initially, the stability of the MoS₂ (WS₂) NTs and MoS₂@WS₂ (WS₂@MoS₂) heterotubes is explored. Subsequently, the band alignment between the inner and outer layer of 1D TMD heterotubes is examined. Finally, the effect of the interlayer electric field between the outer and inner layers is analyzed using an electrostatics model.

Computational details

First-principles calculations were performed using the Vienna ab initio simulation package (VASP),⁵⁴ based on density functional theory (DFT).⁵⁵ We used the Perdew–Burke–Ernzerhof (PBE) functional and the projector augmented wave (PAW)⁵⁶ approach for the exchange–correlation potential and energy. A plane-wave energy cutoff was set to 500 eV. A larger supercell of 60 Å (perpendicular to the NT) was chosen to separate the periodic images. By employing Grimme's semiempirical DFT-D3⁵⁷ scheme, the vdW interactions between the inner and outer layer were included. Atomic structures were fully relaxed until the Feynman–Hellman force on each atom was less than 0.01 eV Å⁻¹. The electronic self-consistent convergence criterion was set at 10⁻⁶ eV, and the Brillouin zone was sampled with a K-point mesh density of 0.02 Å⁵⁸ to ensure accurate calculations. Note that band alignment of the heterotube concerns mainly with the alignment of the electrostatic potentials across the interface, specifically, the long-range electrostatic potential which depends only on the charge distribution. Since DFT is known to yield rather reasonable charge distribution, the difference between PBE and more accurate hybrid functional like HSE06 or G₀W₀ calculation is expected to be small. Therefore, our current conclusion derived from PBE calculations should be reasonable.

Results and discussion

Thermodynamic stability and electronic properties of single-layer nanotubes

We begin by examining single-layer MoS₂ and WS₂ NTs. Fig. 1 illustrates the structure of MoS₂ NTs with ac and zz edges. MS₂ (M = Mo, W) NTs with diameters (D) ranging from 8 to 40 Å were considered. To assess the stability of the NTs relative to their 2D counterparts, we introduce the strain energy (E_s) defined as E_s (eV per atom) = (E_tot − E_MS2)/N. Here, E_tot is the total energy of a MS₂ NT, E_MS2 is the total energy of the corresponding 2D MS₂ sheet with the same number of atoms as the NT, and N donates the total number of atoms in the NT. Fig. 2 shows the plot of E_s as a function of D. For both ac and zz MS₂ NTs, E_s decreases with increasing D. However, for MS₂ NTs with D < 12 Å, the large curvature breaks the triple S–M–S structure of the NT, leading to unpredictable E_s values. On the other hand, for MS₂ NTs with D > 12 Å, E_s presents a clear relationship with D, for both MoS₂ and WS₂ NT. This dependency can be described by E_s = a/D² + b, where “a” is constant related to the tube's curvature and “b” is a constant representing the strain energy caused by other factors, such as stretching or compression. The observed E_s behavior of either MoS₂ or WS₂ NTs follows the a/D² trend, which is consistent with previous studies.^34,59

The electronic properties of MoS₂ also vary with D. Fig. 3a and b depict the band structure of zz and ac MoS₂ NT with D of 39.08 Å and 38.44 Å, respectively. The energy gap (E_g) for all the considered MS₂ NTs is summarized in Fig. 3c and d. Generally, WS₂ exhibits a larger E_g compared to MoS₂ for the same D. MS₂ (zz) with D < 12 Å exhibits semiconducting behavior with E_g approximately 0.7 eV. However, ac MS₂ NTs with D < 14.5 Å exhibits metallic properties. For these NTs with larger D, E_g strongly depends on D, consistent with previous reports.^59–61 For TMD tube with D smaller than 12 Å, the geometric profile of the tube can hardly be constrained, and the linear dependence between the band gap and diameter does not hold. This behavior is similar to that of E_s. In the case of larger D, the E_g of zz NTs exhibits a linear dependence on D, with a periodic fluctuation of (3n) observed for both MoS₂ and WS₂ NT. This behavior is consistent with similar finding in Carbon NTs and Carbon nanoribbons, owing to Clar's rule.⁶² With the increasing of D, the band structure of ac MoS₂ NTs undergoes an indirect (I) to direct (D) transition. However, ac WS₂ NTs are consistently direct, as shown in Fig. 3d. Note that, only direct E_g values at the G point are plotted in Fig. 3.

Band alignment of heterotubes

We now explore 1D heterotubes composed of MoS₂ and WS₂ NTs. As shown in Fig. 4a, the heterotube can be either MoS₂@WS₂ or WS₂@MoS₂ with ac or zz edges. We tested the combinations by fixing D of the outer tube. We fixed the chiral vectors of ac and zz outer tubes as integer indices (20, 20) and (30, 0), corresponding to D of 35.01 Å and 30.4 Å, respectively. We can determine the most favorable energic combinations by selecting the inner tube with varies dimeters. Hereby, we define the energy difference (ΔE) as:

ΔE = (Ehetero − Eouter − Einner)/NMS2,	(1)

where E_hetero, E_outer and E_inner are the total energy of the heterotube, isolated outer and inner MS₂ tube, N_MS2 is the (MS₂) number of the outer tube. Fig. 4b and c summarize ΔE of WS₂@MoS₂ and MoS₂@WS₂ heterotubes, respectively. Note that smaller inner tubes may be located off-center. However, for larger inner tubes, the preferred location tends to shift towards the center of the outer tube. Therefore, only heterotubes with an inner tube positioned at the center have been included for ΔE in Fig. 4. With the increase of D of the inner tube, ΔE decreases and reaches the minimum when the intertube distance is 6.08 Å for aa and 6.14 Å for AC, respectively. These correspond to the combination with chiral vectors (17, 0)@(30, 0) for zz heterotube, and (13, 13)@(20, 20) for the ac one. Therefore, we fix the intertube distance by selecting the same chiral vector difference for different heterotubes henceforward.

We consider the heterotubes with the outer layer of either MoS₂ or WS₂, and the outer and inner layer share the same chiral vector type. About 33 heterotubes have been built with the diameters ranging between 20 Å to 40 Å. The thermodynamic stability is determined by the formation energy (E_form), which is defined as:

Eform = (Eouter + Einner − Ehetero)/N,	(2)

where E_hetero, E_outer and E_inner are the same as in eqn (1), N is the total number of atoms in the heterotube supercell. As shown in Fig. 5, E_form of either WS₂@MoS₂ or MoS₂@WS₂ is positive and increase with D of the heterotube. Meanwhile, ac heterotubes have lower E_form (with the averaged E_form difference of ∼1.5 meV), indicating that they are generally less stable than zz ones. For the same D, WS₂@MoS₂ has slightly higher E_form (∼3 meV) than that of MoS₂@WS₂ heterotube, which is consistent with the nearly identical mechanical properties between MoS₂ and WS₂ sheet.⁶³ All the diameters of the inner and outer tube can be found in Tables S1 and S2 in the ESI.† Because of the computation ability limitation, it is impractical for us to investigate heterotubes with larger D.

Fig. 6 plots the band alignments of 1D heterotubes for both MoS₂@WS₂ (Fig. 6a, c and e) and WS₂@MoS₂ (Fig. 6b, d and f) as a function of D ranging from 26 Å to 39 Å. All band structures for the inner and outer tube of the heterotubes can be found in Fig. S1–S4 in the ESI.† Notably, for small D, the inner tube has significant distortion, resulting in a metallic heterotube. Additionally, an indirect-to-direct band transition is observed in ac heterotubes as D increases. Consequently, band alignments caused by both indirect (Fig. 6a and b) and direct (Fig. 6c and d) band structure are considered. The difference is that, the direct band gap in 1D heterotubes is larger than the indirect one. In summary, the transition of band alignment from type I to type II can be achieved through increase the size of both ac and zz WS₂@MoS₂ (MoS₂@WS₂) heterotubes.

We also investigated 1D chiral MoS₂@WS₂ (WS₂@MoS₂) heterotubes. Typically, chiral nanotubes result in a larger supercell size compared to the ac or zz types. Considering the limitations of the computational capabilities, we have selected chiral MoS₂ and WS₂ tubes with the chiral index of (41) as representative components for constructing 1D heterotubes, with varying D of 26.83 Å, 31.34 Å, 35.88 Å and 40.42 Å. Fig. S5 and S6† plotted the band structures of (41) chiral WS₂@MoS₂ and MoS₂@WS₂ heterotubes. Accordingly, we can obtain the band positions near the Fermi lever of the inner and outer layer components for (41, 41) TMD heterotubes with varying D. As illustrated in Fig. 7, type I band alignment is observed for both the chiral WS₂@MoS₂ and MoS₂@WS₂ heterotubes with a D of 26.83 Å. With the increase in D, the difference in both CBM and VBM band position between the inner and outer tubes decrease. For heterotubes with a D of 40.42 Å, the inner and outer tube contribute almost equally to the VBM. Meanwhile, the difference in CBM band position between the inner and outer tube for heterotube with D of 40.42 Å is smaller than for the heterotube with D of 26.83 Å. Type II band alignment has not been observed in either WS₂@MoS₂ or MoS₂@WS₂ heterotubes; however, it may be anticipated in heterotubes with larger D, according to the observed tends in band position difference.

The transition in band alignment between MoS₂@WS₂ and WS₂@MoS₂ heterotubes arises from different origins. As shown in Fig. 6, the transition of band alignment type in MoS₂@WS₂ heterotubes results from the exchange of the VBM between the inner MoS₂ tube and the outer WS₂ tube. Conversely, in WS₂@MoS₂ heterotubes, the exchange of the CBM between the inner WS₂ tube and the outer MoS₂ tube leads to the band alignment type transition. Taking the example of 1D zz WS₂@MoS₂ heterotube, Fig. 8 illustrates the band structures for various D values, i.e. curvature of the tube. With an increase in D, a significant elevation of the WS₂ CBM is observed, indicating the onset of a band alignment type transition. The dependence of band alignment on diameter, as demonstrated here, is consistent with previous findings in CNTs 1D heterotubes.²⁶ In the 1D heterotube, the curvature of the inner and outer tube decreases with different rates. This is one reason for the transition of band alignment in heterotubes with the increase of D. In heterotubes with a larger D, as shown in Fig. 6, the WS₂ tube possesses a higher CBM and VBM compared to MoS₂. Consequently, both electron and hole carriers are localized within the WS₂ tube, which corresponds to the type II band alignment. Thus, the band alignment type in 1D MoS₂@WS₂ (or WS₂@MoS₂) heterotube with a large D is similar to that in 2D vdW MoS₂/WS₂ heterosturcture.⁵

Electrostatic model of heterotubes

When 2D TMD sheets are bent to form 1D nanotubes, there is a significant charge redistribution that remold the inner electric field. Consequently, the positions of the VBM or CBM shift. To gain further insights into the impact of the inner electric field, we construct an electrostatic model. As depicted in Fig. 9a, the charged atoms within the 1D TMD heterotube can be conceptualized as the charged cylinders with different radii. In accordance with Gauss’ law, the electric field E is zero inside and 2Kρ/r outside the charged cylinder, where K is the Coulomb constant and r is the line charge density of the cylinder. Therefore, inside the 1D TMD heterotube (Fig. 9a):

	(3)

where r_S1 (r_S2, r_S3, or r_S4) denotes the radius of the first (second, third, or fourth) S atom layer, r_M1 (r_M2) presents the radius of the first (second) metal atom layer (as labeled in Fig. 9a), ρ_S1 (ρ_S3, ρ_M1, ρ_M2) presents the line charge density of the first S atom (third S atom, first and second metal atom) layer. We assume that the charge transfer between the TMD tubes can be neglected, such that ρ_M1 + ρ_S1 + ρ_S2 = 0 and ρ_M2 + ρ_S3 + ρ_S4 = 0. Consequently, we can calculate the electrostatic potential difference between the Mo and W layers as:

	(4)

where e is the elementary charge. From Bader analysis, we can obtain ρ_S1 (ρ_S3, ρ_M1) of the 1D TMD heterotube with various D. Taking the 1D zz WS₂@MoS₂ heterotube as an example, we have calculated the eΔU as shown in Fig. 9b. In general, except for those with small D (<13 Å), eΔU decreases with increasing D, and converges to 0.25 eV with infinite D. This kind of D dependent eΔU is another reason for the transition of band alignment from type I to type II. The 1D zz WS₂@MoS₂ heterotube with infinite D can be seen as a 2D WS₂/MoS₂ heterostucture, with eΔU arising from the parallel plane electric field. Similar behavior can be expected for other 1D heterotubes.

Conclusion

In summary, by using first-principles calculations, we performed a systematic investigation on the thermodynamic stability and electronic properties of the 1D WS₂@MoS₂ (MoS₂@WS₂) heterotubes. Our calculations demonstrate that the heterotube is more stable when the intertube distance is approximately 6.08 Å for zz and 6.14 Å for ac edges, respectively. Additionally, we revealed that both 1D MoS₂@WS₂ and WS₂@MoS₂ heterotubes undergo a band alignment type transition from type I to type II. This transition is attributed to the exchange of the VBM in 1D MoS₂@WS₂ heterotubes, but the exchange of the CBM in WS₂@MoS₂ heterotubes. Furthermore, employing an electrostatic model, we demonstrated a reduction in the electrostatic potential difference between the inner and outer tubes, as the heterotube diameter increases.

Author contributions

M. G. conceptualization, writing—review & editing, funding acquisition. F. Z. writing—original draft preparation, designing and performing the DFT calculations. Z. W. investigation, software. J. M. conceptualization, software. J. Z. conceptualization, project administration, funding acquisition.

Data availability

The data that support the findings of this study are available from the corresponding author, [J. Z.], upon reasonable request.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (grant number: 12074235) and the Natural Science Foundation of Shanxi province (grant number: 202103021223252).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4nr03384a

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