Electronic structures and magnetism of MTe₂ (M = Cr, V, and Fe) monolayer nanoribbons

As is widely acknowledged, one-dimensional nanoribbons derived from the majority of two-dimensional materials manifest physical characteristics entirely disparate from their bulk counterparts, attributable to the emergence of edge states and the quantum confinement effects.^1–10 For instance, graphene is both theoretically and experimentally established to be a non-magnetic metal. However, zigzag graphene nanoribbons exhibit semiconducting characteristics and their edge states display ferromagnetic behavior.^1–3 Similarly, MoS₂, a non-magnetic direct bandgap semiconductor, has been predicted through first-principles calculations to exhibit non-magnetic, semiconducting behavior with a fluctuating bandgap for armchair nanoribbons, while its zigzag counterparts display metallic ferromagnetism, independent of width.⁵ Furthermore, monolayer nanoribbons of materials such as silicene, phosphorene, TiS₃, and ZnO manifest electronic structures and magnetism that relate to the widths and edge features.^6–9 Importantly, some of these 2D monolayered nanoribbons have already been experimentally synthesized with results that align remarkably well with theoretical predictions. For instance, zigzag graphene nanoribbons with widths of 11 Å are prepared by employing a bottom-up approach, exhibiting bandgaps corresponding with calculated values.³ Notably, recent reports have shown examples of transition metal dichalcogenide (TMDC) nanoribbons with well-defined edge atoms, including the successful synthesis of zigzag MoS₂ nanoribbons with molybdenum-edged atoms through a high-temperature annealing technique,¹¹ and zigzag MoSe₂ with selenium-edged atoms by using chemical vapor deposition (CVD) methods.¹² These theoretical and experimental achievements provide effective means to manipulate the electronic structures and magnetism of 2D materials.

Lately, significant attention has been directed toward stratified transition metal ditellurides (MTe₂) owing to their distinctive attributes. These materials are known for exhibiting a vast range of unsaturated magnetoresistance,^13,14 along with phenomena such as superconductivity,^15–17 charge density waves,^18,19 and Dirac and Weyl semimetal characteristics,^{14,15,20–22} with a particular emphasis on magnetic properties.^23–30 Inspired by the TMDC nanoribbons, we focus on three types of H-phase monolayer MTe₂ (M = Cr, V, and Fe) nanoribbons. As previously studied, monolayer CrTe₂ is a non-magnetic direct bandgap semiconductor with a bandgap range of 0.53–0.60 eV,^23,31–33 whose excellent stability has been theoretically validated, suggesting its potential for experimental synthesis. Monolayer VTe₂ is a ferromagnetic semiconductor with an indirect bandgap and a Curie temperature exceeding room temperature.³⁴ Monolayer FeTe₂ is metallic and has been predicted by using the HSE method to potentially exhibit half-metal ferromagnetism.^35,36 Recent studies indicate that these three monolayer films hold promise in the field of spintronics.^33–37 Given that monolayer MTe₂ shares the same crystal structure with MoX₂ (X = S, Se), and considering that monolayer CrTe₂ and MoS₂ are both non-magnetic direct bandgap semiconductors, it is natural to consider that the methods applied to obtain precise edge structures in MoX₂ may be equally viable for these monolayer films as well. Therefore, it is imperative to investigate the electronic structures and magnetism of MTe₂ monolayer nanoribbons.

In this paper, we explore the structural stability, electronic structures, and magnetism of zigzag and armchair MTe₂ monolayer nanoribbons, probing how the width of the nanoribbons influences these properties for both types of edge configurations.

All calculations are based on density functional theory (DFT)^38,39 by employing the Vienna Ab initio Simulation Package (VASP). The interactions between the ionic cores and the valence electrons are described by the projector-augmented wave (PAW) method,⁴⁰ and the exchange-correlation potential is treated within the Perdew–Burke–Ernzerhof (PBE)⁴¹ formalism of the generalized gradient approximation (GGA).⁴⁰ The density of k-points in the 2D Brillouin zone is set to (1 × 15 × 1) for the structural relaxation and self-consistent iterations of the MTe₂ monolayer nanoribbons. Periodic boundary conditions are applied, and a vacuum layer of 20 Å thickness is introduced to prevent interactions between neighboring periodic images of the nanoribbons. First, the monolayer MTe₂ cell structure is optimized; then, using the optimized lattice constants, the nanoribbons are constructed by expanding and cutting the cell. All nanoribbon structures undergo thorough relaxation calculations, with force and energy convergence criteria set at 0.01 eV/Å and 10⁻⁵ eV, respectively. The cutoff energy for the plane-wave basis set is chosen as 500 eV.

Figure 1(a) depicts the crystal structure of monolayer MTe₂ (M = Cr, V, and Fe).^30,33,42 Viewing from the top, each M atom is situated at the center of a trigonal prismatic configuration and is coordinately bonded to six Te atoms. From the side perspective, the M atom layer is sandwiched between two layers of Te atoms. Depending on the direction of slicing, MTe₂ monolayer nanoribbons can be categorized into two types: zigzag and armchair. For ease of discussion, nanoribbons cut along these two directions are designated as N-MTe₂-ZNR and N-MTe₂-ANR, respectively, where N indicates the number of transition metal M atoms in the nanoribbon, representing the nanoribbon’s width. The two types of MTe₂ nanoribbons exhibit different edge atoms. The MTe₂-ZNR has edges composed of either M or Te atoms, while the edges of MTe₂-ANR contain both M and Te atoms. Figures 1(b) and 1(c) illustrate the structures of 7-MTe₂-ZNR and 9-MTe₂-ANR, respectively. Since we only consider armchair MTe₂ nanoribbons with symmetrical edges, N is taken as odd-numbered values (N = 5, 7, 9, 11, and 13).

Full relaxation calculations have been performed on the geometric structures of MTe₂-ZNR and MTe₂-ANR with varying widths. The results revealed that MTe₂-ZNR retains its original tri-layer geometry with minimal internal alterations, primarily evident by a modest adjustment in the M-Te bond distance (d_M-Te), in addition to the more pronounced lateral displacement of its edge atoms. Figure S1 in the supplementary material presents the zigzag monolayer nanoribbons of CrTe₂, VTe₂, and FeTe₂ with widths N = 5 and 11. It is evident that, due to edge effects, the M atoms at the boundaries are noticeably displaced inward toward the center of the nanoribbon, while the Te atoms at the edges exhibit a slight outward movement. The relaxed configurations of MTe₂-ANR have showcased some surprising structural changes. Figure S2 in the supplementary material portrays nanoribbons of two widths, N = 7 and 11, for each type of compound. The results demonstrate a pronounced inward relocation of boundary Cr atoms in the CrTe₂-ANR. For the VTe₂-ANR, boundary V atoms exhibit an inward movement and bond formation with second-nearest Te atoms when N ≤ 9. However, the optimized structures of 11- and 13-VTe₂-ANR are similar to those observed in CrTe₂ monolayer nanoribbons. The FeTe₂-ANR experiences more drastic structural changes upon relaxation. With N ≤ 9, boundary Fe atoms migrate inward and establish bonds with second-nearest Te atoms, accompanied by disruption of internal Fe–Te bonds, thereby causing significant deformations of the structure. For widths of N = 11 and 13, boundary Fe atoms continue the inward movement to bond with second-nearest Te atoms. Apart from the 5-, 7-, and 9-FeTe₂-ANR, whose structures undergo considerable changes following relaxation, the remaining armchair nanoribbons retain their intrinsic three-layer geometric structure with minimal internal variation. Due to the extensive internal structural changes in 5-, 7-, and 9-FeTe₂-ANR, which no longer pertain to the armchair nanoribbon configuration, we will not discuss their electronic structures and magnetic properties.

For clarity, we separately address the electronic structures and magnetic properties of CrTe₂ monolayer nanoribbons with two distinct edge configurations. The computational results for VTe₂ and FeTe₂ monolayer nanoribbons are discussed using a similar methodology.

The monolayer CrTe₂ is a non-magnetic semiconductor. To determine the ground state of CrTe₂-ZNR, we have calculated the energies for both the spin-polarized and non-spin-polarized states during structure optimization. The energy difference is defined as $ΔE=ESP−ENSP/N$⁠, where n is the number of Cr atoms in the nanoribbon unit cell. The calculated results are shown in Table I. It is evident that the energy of the spin-polarized state is lower than that of the non-spin-polarized state, indicating that CrTe₂-ZNR has a magnetic ground state. Table I also lists the total magnetic moment, average magnetic moment, local magnetic moments of the boundary Cr1 atoms and boundary Te4 atoms, and the local magnetic moments of adjacent Cr and Te atoms for different widths N of the CrTe₂-ZNR unit cell, with atomic numbering shown in Fig. 1(b). Table I reveals that when N = (5–13), the total magnetic moment of the nanoribbons tends to decrease with increasing N. For instance, when N ≤ 11, the magnetic moment is about 3 μ_B; when N = 12, it drops to about 2 μ_B; and when N = 13, it is ∼1.5 μ_B. The average magnetic moment of the nanoribbons also shows a decreasing trend with increasing N.

Next, we examine the magnetic moment distribution of each atom. The boundary Cr1 atom of the CrTe₂-ZNR has two dangling bonds, while the boundary Te4 atom has one dangling bond, leading to partially filled d orbitals in the Cr1 atom and p orbitals in the Te4 atom. Due to the strong interatomic exchange interaction between the electrons in the d orbital of the Cr1 atom and the p orbital of the Te4 atom, both atoms are in a spin-polarized state. The larger local magnetic moment of the Cr1 atom is around 3.2 μ_B, while the magnetic moment of the Te4 atom varies slightly with N—for N < 12, the magnetic moment of Te4 is ∼0.07 μ_B, and for N ≥ 12, it is about 0.3 μ_B. Assessing the fully relaxed structures, we find that for N < 12, the boundary Te4 atom moves slightly outward from the nanoribbon, leading to an increase in the Te4–Cr3 bond length to 2.66 Å. In contrast, when N ≥ 12, the positional change of the boundary Te4 atom is minimal, and the Te4–Cr3 bond length is 2.62 Å. The shortening of the Te4–Cr3 bond causes an increase in Te4’s magnetic moment due to this edge deformation. Interestingly, the Cr2 atom has a negative magnetic moment of approximately −2.20 μ_B, resulting in antiferromagnetic coupling between Cr1 and Cr2 atoms. To understand the distribution of the magnetic moments in nanoribbons, we calculated the spin-polarized charge densities of CrTe₂-ZNR. Here, we take the 7-CrTe₂-ZNR as an example, as shown in Fig. 3(a). It is clear that due to the influence of the edge states, all Cr atoms are in a spin-polarized state (some in spin antipolarized state), and the magnetism of 7-CrTe₂-ZNR is almost entirely distributed on Cr atoms, with only a minimal amount on Te atoms (too small to be shown, such as the magnetic moments of Te4 and Te5 being less than 0.10 μ_B). The calculations also indicate that the magnetic moments of internal Cr atoms are smaller than those of boundary Cr1 atoms. Overall, 7-CrTe₂-ZNR exhibits ferrimagnetism with a total magnetic moment of 3.02 μ_B. The magnetic moments distribution for other N–CrTe₂-ZNR are similar to that of 7-CrTe₂-ZNR and also show ferrimagnetism.

Figure 4 shows the band structures for 7- and 13-CrTe₂-ZNR unit cells, along with the corresponding DOS and PDOS for atomic orbitals. Examination of the band diagrams reveals that the majority and minority spin channels are asymmetric and partially filled, with both crossing the Fermi level concurrently, indicative of metallic character. Calculations demonstrate that CrTe₂-ZNR of various widths all exhibit metallic behavior, showing that this metallic nature is independent of width N. The bonding and antibonding states near the Fermi level in monolayer CrTe₂ are mainly composed of the d states of Cr atoms. The formation of dangling bonds at the edges of the CrTe₂ monolayer nanoribbon disrupts the bands formed by the bonding and antibonding d orbitals of edge Cr atoms. Correspondingly, these orbitals approach and cross the Fermi level, rendering CrTe₂-ZNR metallic. For further insight into the source of magnetism and the metallic bands, the DOS is examined. The DOS of 7-CrTe₂-ZNR reveals that Cr atoms throughout the nanoribbon are spin-polarized, indicating that internal Cr atoms transition from non-magnetic to magnetic due to the influence of edge states. The d states of Cr atoms contribute substantially to the spin states near the Fermi level, highlighting the pivotal role of Cr atoms across the entire band structure of the nanoribbon. By contrast, the contribution from Te atoms is negligible. In addition, the majority spin states of the edge Cr1 atom’s d orbitals are almost completely occupied, presenting the largest magnetic moment. Internal Cr2 atoms exhibit antiferromagnetic coupling, while the minority spin states of the Cr3 atom’s d orbitals are partially filled, resulting in smaller magnetic moments relative to edge Cr1. The DOS of 13-CrTe₂-ZNR differs from that of 7-CrTe₂-ZNR, particularly regarding the p states of the boundary Te4 atom. As previously mentioned, due to the structural relaxation, the bond length of Cr3–Te4 in 13-CrTe₂-ZNR is shortened by 0.04 Å relative to that in 7-CrTe₂-ZNR, leading to an increased magnetic moment for the edge Te4 atom. Consequently, p orbitals of Te atoms also contribute to the spin states near the Fermi level.

Similarly, the calculated energy differences between spin-polarized and non-spin-polarized states suggest that CrTe₂-ANR nanoribbons possess a magnetic ground state. Table I lists the energy differences between spin-polarized and non-spin-polarized states for CrTe₂-ANR. It can be seen that the total magnetic moments of CrTe₂-ANR do not follow a clear trend with N (with the total magnetic moment of 15-CrTe₂-ANR computed to be 7.68 μ_B); however, within the range of N = (5–13), the average magnetic moment tends to decrease with increasing N. Moving to the distribution of individual atomic local magnetic moments, the edge Cr1 and Te4 atoms, due to the presence of dangling bonds along with strong interatomic exchange interactions, are all in a spin-polarized state, with their magnetic moments ∼3.2 and 0.1 μ_B, respectively. Interestingly, there is antiferromagnetic coupling between the edge Cr1 atom and the adjacent Cr2 atom, while there is ferromagnetic coupling with the nearby Cr3 atom. Upon inspecting the fully relaxed geometric structures, we discovered that the distance between Cr1 and Cr2 (2.74 Å) is smaller by 0.21 Å than the distance between Cr1 and Cr3 (2.97 Å), with edge deformation inducing different coupling modes between neighboring Cr–Cr atoms. To understand the distribution of magnetic moments in the nanoribbons, we calculated the spin-polarized charge densities for CrTe₂-ANR, exemplified here by 9-CrTe₂-ANR as shown in Fig. 3(b). It is observed that due to the influence of edge states, all Cr atoms display spin polarization (some with spin antipolarization), and nearly the entire magnetism of the nanoribbon is distributed on Cr atoms, with only a tiny fraction on boundary Te atoms (the magnetism is too small to show up; e.g., magnetic moments of Te4 and Te5 atoms are about 0.10 μ_B). Calculations also indicate that the magnetic moments of the internal Cr atoms are smaller than those of the edge Cr1 atom. Overall, 9-CrTe₂-ANR displays a ferrimagnetic nature with a total magnetic moment of 8.25 μ_B. The distribution of magnetic moments in other widths of armchair CrTe₂ monolayer nanoribbons is similar to that of 9-CrTe₂-ANR, exhibiting ferrimagnetism.

The results of the band structures reveal that the electronic behavior of armchair CrTe₂ monolayer nanoribbons differs from that of zigzag nanoribbons, as demonstrated in Fig. 5(a). The 5-CrTe₂-ANR and 7-CrTe₂-ANR exhibit semiconducting properties, with band gaps (E_g) of 0.23 and 0.18 eV, respectively, which are smaller than that of the monolayer CrTe₂ (E_g = 0.53 eV). When N ≥ 9, two spin bands cross the Fermi level, indicating metallic behavior in the nanoribbon. This suggests that the electronic structure of armchair CrTe₂ monolayer nanoribbons strongly depends on N. The transition from semiconducting to metallic behavior with increasing N in CrTe₂-ANR has the same physical mechanism as CrTe₂-ZNR. For more insight into the magnetic origin and the semiconductor-to-metal transition in CrTe₂-ANR, we plotted the DOS and PDOS charts for 5-CrTe₂-ANR and 11-CrTe₂-ANR, as shown in Figs. 5(b) and 5(c). It is apparent that Cr atoms are significantly influenced by edge states and are in a spin-polarized state. The spin states near the Fermi level are predominantly contributed by Cr atoms, while the contribution from edge Te atoms to the Fermi level is minimal. As N increases, the filling of Cr atomic spin states across the Fermi level leads to a semiconductor-to-metal transition in the CrTe₂ monolayer nanoribbons. In addition, the DOS indicates that the majority of spin states of the boundary Cr1 atom in CrTe₂-ANR are almost fully occupied. The Cr1 atom also has two dangling bonds, so it has a magnetic moment of ∼3.20 μ_B, which is the same as the edge Cr1 atom in CrTe₂-ZNR. The Cr2 atom shows antiferromagnetic coupling, and the minority spin states of the Cr3 atom are half-filled, resulting in a smaller magnetic moment compared to the boundary Cr1 atom.

The magnetic moments for VTe₂-ZNR are listed in Table S1 in the supplementary material. It is clear that the total magnetic moments of VTe₂-ZNR tend to increase with increasing N. This trend is readily comprehensible, as when N is sufficiently large, the average magnetic moment will approach that of a single VTe₂ monolayer cell (1 μ_B).^23,42 Upon evaluating the local magnetic moments of individual atoms, we note that the magnetic moments of the edge V1 atom and adjacent V2 atom significantly vary. This phenomenon differs from that observed in CrTe₂-ZNR, where the magnetic moments of the edge Cr1 atom and the neighboring Cr2 atom remain almost constant. This could be because monolayer CrTe₂ itself is a non-magnetic semiconductor. Due to edge effects in the nanoribbon, edge atoms first exhibit spin polarization, which then induces spin polarization in internal atoms. In contrast, monolayer VTe₂ is inherently a ferromagnetic semiconductor, and the magnetism of its nanoribbons is not caused by edge states.

The spin-polarized charge densities (see Fig. S3 in the supplementary material) show that the magnetic moments are almost entirely distributed on the V atoms, and the contribution from Te atoms can be considered negligible. In addition, due to edge effects, some outer V atoms exhibit spin antipolarization, resulting in all VTe₂-ZNR displaying ferrimagnetism. Simultaneously, it is observed that the average magnetic moment of the V atoms at the outer edge of the nanoribbons (labeled V1, V2, and V3) is smaller than that of the inner V atoms, with a range between 0.82 and 1.1 μ_B, which is consistent with the average moment of a monolayer VTe₂. This can be attributed to the weakening magnetism of the outer atoms as a result of edge deformation following complete structural relaxation. This also provides another perspective for explaining why the average magnetic moment of VTe₂-ZNR increases with N, as the larger N becomes, the smaller the contribution of outer V atoms to the average magnetic moment.

Figure 6(a) presents the band structures for 7-VTe₂-ZNR, showing that both majority and minority spin bands are asymmetrically filled and partially cross the Fermi level, indicating metallic characteristics. Calculations show that VTe₂-ZNR all exhibit metallic behavior, suggesting that this metallic nature is independent of N. Figure 6(b) shows the DOS of 7-VTe₂-ZNR. The DOS near the Fermi level of the nanoribbon for both spin channels is asymmetric, primarily derived from the contribution of V atoms’ d orbitals with a minor contribution from Te atoms’ p orbitals, indicating that the magnetism of the nanoribbon is mainly due to V atoms. Furthermore, the total DOS, d states of V atoms, and p states of Te atoms all cross the Fermi level, indicating metallic characteristics. This is understandable because, in the monolayer VTe₂, the bonding and antibonding bands near the Fermi level are composed of V atoms’ d states and Te atoms’ p states. With the formation of dangling bonds at the edges of VTe₂ monolayer nanoribbons, the bands formed by the bonding and antibonding d orbitals of V atoms and p orbitals of Te atoms are disrupted. Consequently, these orbitals approach and cross the Fermi level, resulting in metallic behavior in VTe₂-ZNR.

As mentioned earlier, with armchair VTe₂ monolayer nanoribbons when N ≤ 9, the bonding between edge V atoms and sub-nearest Te atoms significantly affects the electronic structure and magnetism of the nanoribbons. The magnetic moments of VTe₂-ANR are listed in Table S1 in the supplementary material. It indicates no clear pattern in the total magnetic moment of VTe₂-ANR within N = (5–13), with the moments being very small for N = 5 and 7, and the maximum observed at N = 11. Upon examining the local magnetic moments of atoms, it is noticed that at N = 11, all V atoms in the nanoribbon have positive magnetic moments, while for other VTe₂-ANR, the edge V1 atom generally exhibits a negative magnetic moment. The spin-polarized charge densities (see Fig. S4 in the supplementary material) show that all V atoms in 11-VTe₂-ANR demonstrate spin-up polarization, exhibiting ferromagnetic behavior, whereas other VTe₂-ANR exhibit ferrimagnetic behavior.

Band structures indicate that VTe₂-ANR all exhibit metallic behavior, as shown in Fig. S5(a) in the supplementary material. Figures S5(b) and S5(c) show the DOS of 9- and 11-VTe₂-ANR, respectively. It is clear that there is an asymmetric DOS for both spin channels near the Fermi level, primarily resulting from contributions by the d orbitals of V atoms, with a minor part from the p orbitals of Te atoms, indicating that the magnetism of the nanoribbons is mainly due to V atoms. The mechanism of the metallic behavior for VTe₂-ANR is the same as that for VTe₂-ZNR, arising from the hybridization of V atom d orbitals and Te atom p orbitals.

The magnetic moments FeTe₂-ZNR are listed in Table S2 in the supplementary material. It is apparent that the total magnetic moment of FeTe₂-ZNR increases with increasing N, while the average magnetic moment decreases with increasing N, a trend that differs from VTe₂-ZNR. However, it can be anticipated that as N becomes sufficiently large, the average magnetic moment will approach the magnetic moment of a single unit cell of monolayer FeTe₂ (1.48 μ_B).^23,42 Examining the local magnetic moments of individual atoms, both edge and internal Fe atoms display positive magnetic moments, without any showing spin antipolarization. We calculated the spin-polarized charge density for FeTe₂-ZNR and found that all nanoribbons in this series exhibit ferromagnetism, which differs from the ferrimagnetism shown by zigzag CrTe₂ and VTe₂ monolayer nanoribbons. Taking 7-FeTe₂-ZNR as an example, the charge densities (see Fig. S6 in the supplementary material) reveal that all Fe atoms are in a spin-up polarized state, with the magnetism almost entirely distributed on Fe atoms, and the contribution from Te atoms being negligible, indicating ferromagnetic behavior in the nanoribbon. In addition, the magnetic moment distribution in FeTe₂-ZNR differs from that of zigzag VTe₂ monolayer nanoribbons. Apart from the central layer Fe atoms, which possess a magnetic moment (∼2.5 μ_B) larger than that of other Fe atoms, there are no other notable patterns. The band structures and DOS (taking 7-FeTe₂-ZNR as an example, as shown in Fig. S7 in the supplementary material) indicate that FeTe₂-ZNR all exhibit metallic behavior, suggesting that creating nanoribbons does not fundamentally alter the electronic structure of monolayer FeTe₂. The DOS reveals that both spin channels near the Fermi level are asymmetric, primarily resulting from the contribution of Fe atom d orbitals, with a minor part from Te atom p orbitals, signifying that the magnetism of the nanoribbon is mainly due to Fe atoms.

As previously mentioned, due to the complete change in the internal structure and bonding states of 5-, 7-, and 9-FeTe₂-ANR, the electronic structure and magnetism will no longer be discussed. Calculation results indicate that 11- and 13-FeTe₂-ANR are both ferromagnetic metals, with total magnetic moments of 19.15 and 23.23 μ_B, respectively. Furthermore, calculations were also carried out for 15-FeTe₂-ANR, which also exhibits ferromagnetic metallic behavior, with a magnetic moment of ∼26.45 μ_B. The spin-polarized charge densities (see Fig. S8 in the supplementary material) indicate that all Fe atoms are in a spin-up polarized state, demonstrating their ferromagnetism. The band structures and DOS (see Fig. S9 in the supplementary material) indicate that FeTe₂-ANR all exhibit metallic behavior. The mechanism of ferromagnetism in FeTe₂-ANR is the same as that for FeTe₂-ZNR.

We have investigated the influence of edge configurations and nanoribbon widths on the electronic structures and magnetism of MTe₂ (M = Cr, V, and Fe) monolayer nanoribbons via first-principles calculations. The results indicate that all three monolayers can form structurally stable zigzag and armchair nanoribbons, as shown in Table II. For the zigzag nanoribbons, CrTe₂-ZNR transitions from a non-magnetic semiconductor monolayer to a ferrimagnetic metal, with metallic properties independent of the width. The metallic properties arise from the d orbitals of the edge Cr atoms. VTe₂-ZNR evolves from a ferromagnetic semiconductor monolayer to a ferrimagnetic metal, and the metallic nature is also width-independent and is a result of hybridization between d orbitals of edge V atoms and p orbitals of edge Te atoms. FeTe₂-ZNR maintains the attributes of its monolayer counterpart (ferromagnetic metal). For armchair nanoribbons, CrTe₂-ANR exhibits ferrimagnetic properties and its conductivity is dependent on width. 5- and 7-CrTe₂-ANR are semiconductors, while wider CrTe₂-ANR become ferrimagnetic metals. VTe₂-ANR shows variable behavior between ferrimagnetic and ferromagnetic metals with changes in the width. FeTe₂-ANR with widths N ≥ 11 remains a ferromagnetic metal. The calculated spin-polarized charge densities of states indicate that the magnetic moments of all monolayer nanoribbons are primarily contributed by the 3d transition metal atoms. Our results suggest that MTe₂ nanoribbons have potential applications in nanoelectronic devices.

In the supplementary material, further computational details, as well as some intermediate results and figures, are presented.

This work was supported by the Hunan Provincial Natural Science Foundation of China (Grant No. 2023JJ40074), the Hunan Provincial Department of Education Outstanding Youth Project (Grant Nos. 21B0757 and 22B0821), Yunnan Province Natural Science Foundation (Grant No. 00900206020616034), and the National Natural Science Foundation of China (Grant No. 42471384).

The authors have no conflicts to disclose.

Wei Chen: Investigation (equal); Writing – original draft (equal). Qi Chen: Data curation (equal); Writing – review & editing (equal). Jianming Zhang: Methodology (equal); Writing – review & editing (equal). Yu Zheng: Funding acquisition (equal). Ying Long: Funding acquisition (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.