Ferromagnetic semiconductor nanotubes with room Curie temperatures

Abstract

Realizing ferromagnetic semiconductors with high Curie temperature T_C remains a challenge in spintronics. Recent experiments have obtained some two-dimensional (2D) room temperature ferromagnetic metals, such as monolayers CrS₂ and VSe₂. Inspired by the recent experimental progress on the nanotubes based on 2D van der Waals non-magnetic transition-metal dichalcogenides, magnetic nanotubes based on monolayer ferromagnetic materials are highly possible. Here, by the density functional theory calculations, we proposed a way to obtain a high T_C ferromagnetic semiconductor in magnetic nanotubes. Some high T_C ferromagnetic semiconductors are predicted in the MX₂ nanotubes (M = V, Cr, Mn, Fe, Co, Ni; X = S, Se, Te), including CrS₂ and CrTe₂ zigzag nanotubes with a diameter of 18 unit cells, showing T_C above 300 K. In addition, due to the strain gradient in the walls of nanotubes, an electrical polarization at the level of 0.1 eV/Å inward of the radial direction is obtained. Our results propose a way to obtain high-temperature ferromagnetic semiconducting nanotubes based on experimentally obtained 2D high T_C ferromagnetic metals.

Introduction

Ferromagnetic semiconductors integrate magnetic functionality into conventional semiconductor platforms, enabling novel spintronic devices capable of simultaneous spin and charge manipulation^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19}. Within standard semiconductor architectures, these materials uniquely generate, control, and detect spin-polarized currents through engineered spin-orbit coupling and magnetic anisotropy. This synergy unlocks transformative applications, including spin injection masers for coherent terahertz emission^18,19; circularly polarized light-emitting diodes with chirality-tunable output¹⁰; magnetic diodes and p-n junctions exhibiting rectification tunable by external fields^11,12,13,14; magnetic tunnel junctions with enhanced spin-dependent tunneling ratios^15,16; and spin valve structures for nonvolatile memory integration¹⁷. For practical implementation, such devices require ferromagnetic semiconductors with Curie temperature T_C above 300 K to ensure room-temperature operation. However, most ferromagnetic semiconductors exhibit T_C values below 200 K due to magnetic dilution and weak exchange interactions. This critical limitation impedes their technological viability.

In 2017, the successful synthesis of two-dimensional (2D) van der Waals ferromagnetic semiconductors CrI₃²⁰ and Cr₂Ge₂Te₆²¹ in experiments has attracted extensive attention to 2D ferromagnetic semiconductors. According to Mermin–Wagner theorem²², the magnetic anisotropy is essential to produce long-range magnetic order in 2D systems. Recently, with great progress of 2D magnetic materials in experiments, more 2D ferromagnetic materials have been obtained, such as some ferromagnetic semiconductors with T_C far below room temperature^{23,24,25,26,27}, and some ferromagnetic metals with high T_C above room temperature^{28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43}. In addition, some ferromagnetic semiconductors with T_C above room temperature have been predicted based on theoretical calculations^{44,45,46,47,48,49,50,51,52,53}, while their synthesis remains a challenge.

Since the first report of carbon nanotubes (CNTs)⁵⁴, one-dimensional materials based on 2D sheets have attracted attention because of their unique mechanical and electrical properties due to their low-dimensional structures. The structures of CNTs consist of nanodimensions made up of rolled sheets of 2D graphite. There are many ways a sheet of graphite is rolled up to form a tube, such as zigzag, chiral, and armchair; these are the names given to different types and could be controlled in experiments^55,56. In addition, the diameters of nanotubes could be as small as sub-nm level^{57,58,59,60,61}. There are some applications of CNT^{55,62,63,64,65,66,67,68,69,70,71,72,73}, such as chemical sensors⁶⁴, biomedical^65,66,67, biosensors^71,73 and digital electronics^68,72, etc.

There are some studies of magnetic nanotubes^{74,75,76,77,78,79,80,81,82,83,84,85,86}. They were prepared by adding magnetic particles to non-magnetic nanotubes such as CNT doped with magnetic atoms, or ferromagnetic materials such as Fe, Co, and alloys, etc^{74,75,76,77,78,79,80}. The nonreciprocal spin waves were observed in magnetic nanotubes under magnetic field⁷⁹. Ni-doped silicon nanotubes were reported to be room temperature ferromagnetic semiconductors with tiny saturation magnetization⁸⁰. It has some interesting applications, such as a platform for studying the magnetochiral effect, drug delivery, sorbent, catalysis, sensor, and basic building blocks for future memory elements^{76,77,78,79,86}. Theoretical studies have demonstrated that magnetic nanotubes exhibit unique magnetic properties, including distinct spin-wave dynamics, domain wall propagation, unidirectional magnonic waveguiding, etc^{81,82,83,84,85}.

With the fast development of 2D van der Waals transition-metal dichalcogenides (TMD) materials, there are many experiments of TMD nanotubes^{87,88,89,90,91,92,93,94,95,96}, including WS₂^92,93,94,96, WSe₂⁹¹, and MoS₂^89,97. They show interesting properties, such as superconductivity^92,93 and photovoltaic effect^94,96. In addition, some density functional theory (DFT) studies have investigated the properties of TMD nanotubes^{98,99,100,101,102,103,104,105,106,107}. In this article, we theoretically predict high T_C ferromagnetic semiconducting nanotubes based on experimentally known 2D high T_C ferromagnetic metals. Considering the high T_C in experiments, ferromagnetic monolayers 1T-MX₂ (M = V, Cr, Mn, Fe, Co, Ni; X = S, Se, Te) were chosen as the initial 2D material to roll up. The DFT calculations were performed to investigate the properties of armchair and zigzag nanotubes with different diameters. We predicted some ferromagnetic semiconducting nanotubes with T_C above 200 K, including zigzag CrS₂ nanotube with 18 CrS₂ units (Z-18-CrS₂) showing T_C of 364 K and band gap of 0.53 eV, and Z-18-CrTe₂ showing T_C of 441 K and band gap of 0.16 eV. From monolayer cases to nanotube cases, they transition from high T_C ferromagnetic metals to high T_C ferromagnetic semiconductors. As shown in Fig. 1, monolayer CrS₂ is a ferromagnetic metal, whereas Z-18-CrS₂ is a ferromagnetic semiconductor. The stability of nanotubes is confirmed by the calculations of strain energy and molecular dynamics simulations. The strain energies of predicted nanotubes are lower than the experimentally prepared narrow MoS₂ nanotube, CNT, and BN nanotubes, suggesting the feasibility of preparation. Radial electric polarization at 0.1 eV/Å level is observed in nanotubes due to the strain gradient in the wall. As three representative results, the properties of CrS₂, CrTe₂, and VSe₂ nanotubes are discussed in detail in the paper. The properties of nanotubes are closely related to their diameter and structural configuration. Our theoretical results propose a way to obtain high T_C ferromagnetic semiconducting nanotubes derived from high T_C 2D ferromagnetic metals in experiments.

Fig. 1: Metal-semiconductor transition in CrS₂ from monolayer to zigzag nanotube.

a Crystal structure of monolayer CrS₂. b Spin-polarized band structure of monolayer CrS₂, demonstrating metallic behavior. c Crystal structure of zigzag CrS₂ nanotube. d Spin-polarized band structure of the zigzag CrS₂ nanotube, demonstrating semiconducting behavior.

Results

Structures and stability of nanotubes

The crystal structure of monolayer CrS₂ is shown in Fig. 1a, with space group P$\overline{3}$ m1 (164). The calculated in-plane lattice constants are ${a}_{0}/\sqrt{3}={b}_{0}=3.40$ Å, in agreement with previous calculation results of 3.28 Ź⁰⁸. The metallic band structure of monolayer CrS₂ is obtained by the DFT calculation, as shown in Fig. 1b. DFT calculation and Monte Carlo simulation results show that monolayer CrS₂ is a ferromagnetic metal with T_C of 295 K, close to the experimental results of T_C = 300 K⁴⁰. Detailed calculation processes are given in Supplementary section 1 in Supplementary Material.

As shown in Fig. 1, zigzag and armchair nanotubes are obtained by rolling up monolayers along different directions⁵⁵. The repeating MX₂ units in a unit cell of a nanotube are 12, 14, 16, 18, 20, 22, and 24, respectively. The diameter of nanotubes is defined according to the outer anions. Due to the different lattice constants of monolayer 1T-MX₂ along zigzag and armchair directions, armchair nanotubes exhibit larger diameters than their zigzag counterparts with the same number of atoms.

To analyze the stability of nanotubes, the strain energy E_strain is calculated by refs. ^{89,97,98,100,104,107}

$${E}_{{\rm{strain}}}=\frac{{E}_{{\rm{nanotube}}}}{{N}_{{\rm{nanotube}}}}-\frac{{E}_{{\rm{monolayer}}}}{{N}_{{\rm{monolayer}}}},$$

(1)

where E_nanotube, E_monolayer, N_nanotube, and N_monolayer are the energies and total number of atoms in the nanotube and monolayer cases, respectively. Calculation results of E_strain of CrS₂, CrTe₂, and VSe₂ are shown in Fig. 2a. The positive E_strain indicates that the atoms in the nanotube possess higher energy than those in the monolayer, due to the strain induced by rolling up the monolayer. E_strain decreases with increasing diameter due to the decrease in strain. The detailed results of E_strain of zigzag and armchair nanotubes are listed in Table 1. The larger diameters of armchair nanotubes result in lower E_strain compared to zigzag nanotubes containing the same number of atoms. Armchair VSe₂ nanotubes with diameters exceeding 2.5 nm exhibit negative E_strain, suggesting the possibility of spontaneous curling. Similar positive results and behavior of E_strain are obtained in previous calculations^{89,97,98,100,104,107}. CNTs with diameters <1 nm have been experimentally synthesized, exhibiting an E_strain exceeding 0.1 eV/atom^56,57,58. Narrow h-BN nanotubes with a diameter of 0.45 nm were prepared and exhibited an E_strain of 0.25 eV/atom^61,109. In addition, the single-walled MoS₂ nanotubes with a diameter of ~3.9 nm and E_strain of ~0.05 eV/atom were synthesized⁹⁷. These experimental results support the stability of the nanotubes predicted in our calculations. It is worth noting that, even with different E_strain, the configuration and diameter of nanotubes can be controlled in experiments⁵⁵.

Fig. 2: Calculated properties of zigzag CrS₂, CrTe₂ and VSe₂ nanotubes.

a The results of strain energies E_strain. The E_strain of MoS₂ nanotubes with diameter of ~3.9 nm in experiment is ~0.05 eV/atom⁹⁷, of BN nanotubes with diameter of 0.45 nm is ~0.25 eV/atom⁶¹, of CNT with diameter of 1 nm is ~0.10 eV/atom^57,58. These values are shown for reference. b The results of band gaps. c Results of Curie temperatures T_C obtained by DFT calculation and Monte Carlo simulation. d The results of effective radial electrical field E_r inward along the radius. The results of room temperature ferromagnetic semiconductors of zigzag CrS₂ and CrTe₂ nanotubes are emphasized by circles.

Table 1 The band gap, diameter, Curie temperature T_C, strain energy E_strain and radial electric field E_r inward along the radius for zigzag and armchair nanotubes

To further assess their stability, we performed molecular dynamics simulations on the Z-18-CrS₂ and Z-18-CrTe₂ nanotubes at 300 K using an NVT ensemble. Simulations were performed for 6 ps. The results presented in Fig. 3 provide key evidence for their dynamical stability. First, the total energy exhibits only minor oscillations around a stable baseline, indicating that the nanotubes are in thermal equilibrium. Moreover, the final atomic configurations, appearing distorted yet preserving the nanotube topology, confirm that the structures do not unravel, validating their room-temperature stability.

Fig. 3: Molecular dynamics simulation results for Z-18-CrS₂ and Z-18-CrTe₂.

The simulation was run at 300 K for 6 ps of molecular dynamics simulation results, energy and structure. Inside are the side views of crystal structures after 6 ps.

Band gap of nanotubes

The results of band gaps of zigzag CrS₂, CrTe₂, and VSe₂ nanotubes are shown in Fig. 2b. In monolayer cases, they are room temperature magnetic metals^40,42,43. Non-zero band gaps were observed in zigzag nanotubes with some diameters. Zigzag CrS₂ nanotubes with diameters of 1.05 nm and 1.28 nm, zigzag VSe₂ nanotubes with diameters of 0.97 nm, 1.07 nm, 1.24 nm, and 1.69 nm, and zigzag CrTe₂ nanotubes with a diameter of 1.45 nm exhibited non-zero band gaps. The detailed results of band gaps of zigzag and armchair nanotubes are listed in Table 1. Nearly all armchair nanotubes display metallic behavior, except the armchair VSe₂ nanotube with a diameter of 1.46 nm, showing a band gap of 0.11 eV. Their band structure is given in Supplementary Figs. S3–S8 in the Supplementary Material. This configuration and perimeter-dependent electronic behavior indicate a metal-to-semiconductor transition influenced by structural deformation.

Similar to CNTs, TMD nanotubes exhibit a configuration and diameter-dependent metal-to-semiconductor transition^56,110. For CNTs, some zigzag CNTs show a band gap while all armchair CNTs remain metallic; this behavior is attributed to the folding of the Brillouin zone, which depends on both configuration and diameter^56,110. The configuration and perimeter-dependent metal-to-semiconductor transition in TMD nanotubes arises from two key effects: Brillouin zone folding and uniaxial strain. The top view and band structure of monolayer CrS₂ are shown in Fig. 4a, b, respectively. Its Brillouin zone is shown in Fig. 4e. Figure 4f depicts the Fermi contour of monolayer CrS₂, defined as the locus of K points satisfying E(k) = E_F. The features of the Fermi contour agree with the band structure in Fig. 4b. The existence of a band gap along a specific K-path can be determined by examining whether there exist intersections of the Fermi contour with the corresponding path. Intersections are observed with the Γ-X, Γ-Y, and S-X paths, indicating no band gap along these directions. Conversely, the absence of intersections with the S-Y path indicates the presence of a band gap along this path.

Fig. 4: Analysis of band structure evolution in CrS₂ from monolayer to nanotubes.

a Orthorhombic unit cell of monolayer CrS₂. b–d Band structures without strain (b), under 34% tensile strain along x direction (c), and under 12% tensile strain along y direction (d). e The 2D Brillouin zone of monolayer CrS₂. f–h Fermi contours at E(k) = E_F without strain (f), under 34% tensile strain along x direction (g), and under 12% tensile strain along y direction (h). Schematic diagram of a zigzag nanotube (z1) and the corresponding 1D Brillouin zone (z2). (z3) The Fermi contour of monolayer CrS₂ under 34% tensile strain along x direction, lines parallel to Γ-Y represent the 1D Brillouin zone of zigzag nanotubes. (z4) The band structure corresponds to (z3). Schematic diagram of an armchair nanotube (a1) and the corresponding 1D Brillouin zone (a2). (a3) The Fermi contour of monolayer CrS₂ under 12% tensile strain along y direction, lines parallel to Γ-X represent the 1D Brillouin zone of armchair nanotubes. (a4) The band structure corresponds to (a3).

The electronic properties of the monolayer are significantly affected by applied strain. To quantify strain, we define the strain ratio by ϵ = (a − a₀)/a₀ × 100%, where a and a₀ are the lattice constants with and without strain, respectively. The band structure and Fermi contour of monolayer CrS₂ under 34% tensile strain along x direction are shown in Fig. 4b, g, respectively. Under this strain, no intersections are observed between the Fermi contour and the Γ-Y or S-X paths, implying the presence of energy gaps along both directions. Similarly, the band structure and Fermi contour of monolayer CrS₂ under 12% tensile strain along y direction are shown in Fig. 4d, h, respectively.

Extending to nanotubes, zigzag nanotubes experience uniaxial strain along the perimeter, corresponding to the x direction in the monolayer structure (Fig. 4(z1)). As shown in Fig. 4(z2), when the structure transforms from monolayer to 1D zigzag nanotube, the Brillouin zone folds from its 2D planar form into lines parallel to the Γ-Y direction. Fig. 4(z3) show the Fermi contour in the Brillouin zone of monolayer CrS₂, where the 1D Brillouin zone of zigzag nanotube is schematically indicated by lines. The Brillouin zone and the Fermi contour exhibit no intersections, thus demonstrating a semiconducting band structure as shown in Fig. 4(z4).

In contrast, armchair nanotubes experience uniaxial strain along their perimeter, corresponding to y direction in the monolayer structure (Fig. 4(a1)). Upon transformation from monolayer to 1D armchair nanotube, the Brillouin zone folds from a 2D plane into lines parallel to the Γ-X direction. Fig. 4(a3) depicts the Fermi contour for this folded zone, indicated schematically by lines. The presence of intersections between the Fermi contour and the folded zone demonstrates a metallic band structure, as shown in Fig. 4(a4).

The Brillouin zone of zigzag and armchair nanotubes always contains the Γ-Y and Γ-X paths, respectively. The evolution of band gap along Γ-Y/Γ-X line in monolayer CrS₂ under uniaxial strain along x/y direction are presented in Fig. 5a. Without strain, no gap exists along either path; under strain along x direction, a gap opens along Γ-Y for ϵ = −32% ~−20% and ϵ > 28%, whereas strain along y direction induces no gap along Γ-X. As depicted in Fig. 5b, the external S atoms experience tensile strain, which decreases from 40% to 15% with increasing nanotube diameter from 14 to 24 CrS₂ units. As Fig. 5a shows, semiconducting behavior in zigzag nanotubes requires a large tensile strain (>28%), limiting it to narrow tubes with big strain. For example, in semiconducting Z-18-CrS₂, outer S, Cr, and inner S layers exhibit large strains of −23%, 13%, and 30%, respectively. The DFT results for TMD nanotubes are not entirely consistent with those from the 2D band structure, primarily due to their complex structure comprising multiple atomic layers subjected to different strains. Together, Brillouin zone folding and uniaxial strain govern the diameter- and configuration-dependent metal-to-semiconductor transition in TMD nanotubes. The strained band structures of monolayer CrS₂ are provided in Supplementary Figs. S1, S2 and Supplementary Material.

Fig. 5: Strain-induced band gap changes in monolayer CrS₂ and strain in nanotubes.

a Band gap of monolayer CrS₂ along Γ-Y/Γ-X with uniaxial strain along x/y direction. b Uniaxial strain along x/y direction of zigzag/armchair CrS₂ nanotubes with different perimeters.

Calculation of Curie temperature T _C of the nanotube

To study the magnetic properties of the zigzag CrS₂ nanotube with 18 CrS₂ units (Z-18-CrS₂), we consider a Heisenberg-type Hamiltonian:

$$H=J\sum _{ < i,j > }\overrightarrow{{S}_{i}}\cdot \overrightarrow{{S}_{j}}+\sum _{i}{A}_{\parallel }{S}_{i\parallel }^{2},$$

(2)

where $\overrightarrow{{S}_{i}}$ and $\overrightarrow{{S}_{j}}$ are spin operators of Cr atoms at site i and j, respectively, J is the exchange coupling constant between the nearest-neighboring Cr atoms. A_∥ is the single-ion magnetic anisotropy parameter defined as A_∥S² = (E_∥−E_⊥)/N_Cr, where E_∥ and E_⊥ are the DFT results of energies of CrS₂ nanotube with magnetization parallel and perpendicular to the extension direction of nanotube, respectively. N_Cr is the number of Cr atoms in the considered cell. The calculated result of A_∥S² is −0.82 meV/Cr, indicating that the magnetization is parallel to the extension direction of the nanotube. Energies of Z-18-CrS₂ nanotube with an FM and an AFM spin configurations, as shown in Fig. 6a, b, respectively, can be expressed as

$$\begin{array}{rcl}{E}_{{\rm{FM}}}\,=\,3{N}_{{\rm{Cr}}}J{S}^{2}+{E}_{0},\\ {E}_{{\rm{AFM}}}\,=\,-{N}_{{\rm{Cr}}}J{S}^{2}+{E}_{0}.\end{array}$$

(3)

E₀ is the energy part independent of spin configurations, which is included in the total energy of DFT results for Z-18-CrS₂. N_Cr is the number of Cr atoms in the unit cell of the nanotube, i.e., 18 for Z-18-CrS₂. The exchange coupling parameter JS² can be calculated by $J{S}^{2}=({E}_{{\rm{FM}}}-{E}_{{\rm{AFM}}})/\left(4{N}_{{\rm{Cr}}}\right)$. The DFT results of the relative total energy of Z-18-CrS₂ in FM and AFM states are 0 and 2132.8 meV, respectively, which gives JS² = −29.62 meV. By the Monte Carlo simulation based on the Heisenberg model in Eq. (3), the temperature-dependent magnetization and susceptibility of Z-18-CrS₂ were calculated, as shown in Fig. 6c, giving a T_C of 364 K.

Fig. 6: Spin configurations and Monte Carlo results of Z-18-CrS₂.

A FM (a) and an AFM (b) spin configuration are considered to get the exchange coupling parameter JS². c Magnetization and susceptibility of Z-18-CrS₂ as a function of temperature, obtained by the Monte Carlo simulation, giving a T_C of 364 K.

Due to their similar structures, all nanotubes are calculated in this way. The T_C of zigzag type CrS₂, CrTe₂, and VSe₂ nanotubes are obtained, as shown in Fig. 2c. T_C ~ 300 K for zigzag CrS₂ nanotubes, ~450 K for zigzag CrTe₂ nanotubes and ~100 K for zigzag VSe₂ nanotubes are observed. From monolayer ferromagnetic metals with high T_C to nanotubes, they maintain strong ferromagnetic coupling. The data are presented in Table 1. The T_C in armchair configuration shows similar values of T_C in zigzag configuration.

Among these materials, Z-18-CrS₂ and Z-18-CrTe₂ are room temperature ferromagnetic semiconductors, corresponding to diameters of 1.28 nm and 1.45 nm, respectively. Z-18-CrS₂ is an FM semiconductor with T_C of 364 K and a band gap of 0.53 eV. Z-18-CrTe₂ shows T_C of 431 K and a band gap of 0.16 eV. To obtain their accurate band gaps, the HSE functional is applied in DFT calculation¹¹¹. Metal-to-semiconductor transition happens in CrS₂ and CrTe₂ from monolayer to zigzag nanotubes. E_strain of 0.05 eV/atom and 0.09 eV/atom are observed in Z-18-CrS₂ and Z-18-CrTe₂, respectively, indicating their feasibility of preparation. Detailed results concerning the band gap, diameter, Curie temperature T_C, and strain energy E_strain of zigzag and armchair CrS₂ and CrTe₂ nanotubes are summarized in Table 1.

Discussion on T _C of monolayer CrS₂ and zigzag CrS₂ nanotubes

The exchange couplings of monolayer CrS₂ and zigzag CrS₂ nanotubes exhibit distinct characteristics. Using DFT calculations, we determined the exchange coupling constants for the first (J₁), second (J₂), and third (J₃) nearest-neighbor interactions, as summarized in Table 2. Detailed calculation process are given in the Method section. For monolayer CrS₂, our DFT results reveal ferromagnetic coupling with J₁S² = −10.78 meV, J₂S² = −4.53 meV, and J₃S² = −3.20 meV. Monte Carlo simulations demonstrate that including only J₁ yields a T_C of 132 K. In contrast, zigzag CrS₂ nanotubes exhibit more complex magnetic interactions. While J₁ remains ferromagnetic and dominant in magnitude, J₂ also favors ferromagnetism while J₃ becomes antiferromagnetic. This competition results in T_C being primarily governed by J₁. For Z-18-CrS₂ nanotubes: J₁S² = −19.38 meV, J₂S² = −14.30 meV, and J₃S² =10.08 meV. Monte Carlo simulations predict T_C = 238 K when considering only J₁, rising to 309 K when all three couplings are included. Similarly, for Z-12-CrS₂ nanotubes: J₁S² = −16.96 meV, J₂S² = −6.75 meV, and J₃S² = 4.92 meV. The corresponding T_C values increase from 208 K (J₁ only) to 230 K (all couplings). These findings indicate that while monolayer CrS₂’s T_C is determined by the combined effects of J₁, J₂, and J₃, the T_C of zigzag CrS₂ nanotubes is predominantly controlled by the dominant ferromagnetic J₁ interaction.

Table 2 Coupling constants and Curie temperatures for CrS₂ nanotubes and monolayer CrS₂

Since T_C of zigzag CrS₂ nanotubes is predominantly governed by the ferromagnetic J₁ interaction, the observed decrease in T_C with decreasing perimeter can be reasonably attributed to variations in J₁. Recent studies^{47,51,112,113} have proposed that ferromagnetic J₁ in two-dimensional magnetic semiconductors and metals can be effectively described by the superexchange model. The superexchange coupling ${J}_{ij}^{super}$ between Cr atom at site i and Cr atom at site j can be obtained as ref. ⁵¹:

$${J}_{ij}^{super}=\left(\frac{1}{{E}_{\uparrow \downarrow }^{2}}-\frac{1}{{E}_{\uparrow \uparrow }^{2}}\right)\sum _{k,p,d}| {V}_{ik}{| }^{2}{J}_{kj}^{pd}=\frac{1}{A}\sum _{k,p,d}| {V}_{ik}{| }^{2}{J}_{kj}^{pd}.$$

(4)

The indirect exchange coupling ${J}_{ij}^{super}$ consist of two processes. One is the direct exchange process between the d-electrons of Cr at site j (Cr_j) and the p-electrons of S at site k (S_k), represented by ${J}_{kj}^{pd}$. Another is the electron hopping process between the p-electrons of S_k and the d-electrons of Cr_i, presented by ∣V_ik∣²/A. V_ik is the hopping parameter between d electrons of the Cr_i and p electrons of S_k. $A=1/(1/{E}_{\uparrow \downarrow }^{2}-1/{E}_{\uparrow \uparrow }^{2})$ is taken as a pending parameter, where E_↑↑ and E_↑↓ are energies of two d electrons in Cr atoms with parallel and antiparallel spins, respectively. Eq. (4) represents the summation of contributions from all possible superexchange paths between Cr_i and Cr_j via all p-orbitals of all intermediate S_k atoms. The direct exchange coupling ${J}_{kj}^{pd}$ can be expressed as ${J}_{kj}^{pd}=2| {V}_{kj}{| }^{2}/| {E}_{k}^{p}-{E}_{j}^{d}|$^{47,51,112,113}. V_kj is the hopping parameter between p electrons of the S_k and d electrons of Cr_j, ${E}_{k}^{p}$ is the energy of p electrons of the S_k, and ${E}_{j}^{d}$ is the energy of d electrons of Cr_j.

The parameters V_ik, V_kj, ${E}_{k}^{p}$, and ${E}_{j}^{d}$ in Eq. (4) are obtained from DFT and Wannier function calculations. The exchange term ${J}_{ij}^{super}A$ is computed by summing over all relevant p orbitals at all possible S sites and all d orbitals at Cr sites. The parameter ${J}_{ij}^{super}A$ is an analytical construct, not a fundamental physical observable. Composed of the superexchange constant ${J}_{ij}^{super}$ (unit: eV) and parameter A (unit: eV²), its derived dimension is energy cubed (eV³). This unconventional unit arises mathematically from our deliberate construction: ${J}_{ij}^{super}A$ serves as a theoretical tool to isolate hopping-dependent trends in superexchange interactions by explicitly factoring out the A term from the exchange coupling framework. The nearest coupling constant J^superS² is obtained by averaging couplings:

$${J}^{{\rm{super}}}{S}^{2}=\frac{1}{z}\sum _{ < i,j > }{J}_{ij}^{super}{S}^{2},$$

(5)

where z is the number of nearest couplings, which is 6 in nanotubes. Table 3 presents the calculated values for zigzag CrS₂ nanotubes with perimeters containing 12 and 18 CrS₂ units, demonstrating an increase in superexchange strength with tube diameter. This trend of J^superA is consistent with the J_DFTS² values obtained from DFT, where the same parameter A is taken for nanotubes with different diameters.

Table 3 Coupling constants of zigzag CrS₂ nanotubes containing 12 and 18 CrS₂ units, obtained by DFT calculation and Eq. (4)

As an example, we examine the superexchange coupling between Cr atom at site 2 and Cr atom at site 4 in Z-12-CrS₂ mediated by S atoms at sites 4 and 15. The structure is shown in Fig. 7. The coupling strength ${J}_{24}^{{\rm{super}}}A$ increases from 9.9 to 32.8 eV³ as the nanotube diameter increases from 12 to 18 CrS₂ units, as shown in Table 3. To understand this behavior, we analyze the 3d orbital energies E_d of Cr atom at site 2 and Cr atom at site 4, the 5p orbital energy E_p of S atoms at sites 4 and 15, and the hopping matrix element ∣V_pd∣ between them. The superexchange coupling strength is represented by the factor ∣V_pd∣⁴/∣E_p-E_d∣, according to Eq. (4). As shown in Table 4, the energy difference ∣E_p-E_d∣ decreases markedly, and the hopping term ∣V_pd∣² increases with perimeter, resulting in a stronger superexchange coupling.

Fig. 7: Structure of Z-12-CrS₂.

The zigzag CrS₂ nanotube with a perimeter of 12 CrS₂ units (Z-12-CrS₂), highlighting the superexchange path between Cr₂ and Cr₄.

Table 4 The parameters of superexchange coupling in Eq. (4) for the zigzag CrS₂ nanotubes with perimeters of 12 and 18 CrS₂ units, obtained by DFT

Calculation of electric polarization E _r of nanotube

According to previous studies of flexoelectricity, an inhomogeneous strain will break the inversion symmetry of 2D materials, and electric polarization is permitted^114,115,116. Flexoelectricity holds immense application potential in sensors, actuators, and next-generation electronic devices^117,118,119, which has been reported to tune magnetic properties in some materials^120,121. When a nanotube is formed by rolling a 2D layer of finite thickness, radial electric polarization is possible due to the difference in strain between the inside and outside of the wall^105,106. We calculated the flexoelectric-like radial polarization of nanotubes. The radial electrostatic polarization is obtained by E_r = (V_R(outside)–V_R(inside))/h, where V_R(inside) and V_R(outside) are the radial electrostatic potential inside and outside of the nanotube, and h is the thickness of the tube wall¹⁰⁵. The radial electrostatic potential V_R(r) is obtained by converting the DFT results of electrostatic potential V(r, θ, z) to a function of radius by V_R(r) = ∫dθ∫V(r, θ, z)/(2πz₀)dz, where r, θ, and z are the three parameters of position in cylindrical coordinates, and z₀ is the length of unit cell. The radial electrostatic potential of Z-18-CrS₂ outside of tube is 0.51 eV higher than inside, as shown in Fig. 8c. Considering a thickness h of 2.6 Å, the effective electrical polarization is 0.19 eV/Å inward along the radius. All electrical polarization of nanotubes was obtained in similar way. As shown in Fig. 2d), E_r of nanotubes decreases with increasing diameters. The data are presented in Table 1. Both zigzag and armchair nanotubes show radial electrical polarization at 0.1 eV/Å level inward along the radius.

Fig. 8: Radial polarization driven by radial strain gradient in nanotubes.

a Cross-sectional view of a zigzag CrS₂ nanotube showing compressive strain in the inner wall and tensile strain in the outer wall. b Dependence of inner and outer wall strain on the nanotube perimeter. c The radial electrostatic potential V_R of Z-18-CrS₂. V_R inside of tube is 0.51 eV lower than outside.

The radial polarization correlates positively with the strain difference between the inner and outer layers of the nanotube (Fig. 8a). As the tube circumference expands from 12 to 24 CrS₂ units, the outer layer relaxes from 40 % to 15 % strain while the inner layer holds at −25% (Fig. 8b), diminishing the overall strain differential and reducing polarization. The geometric parameters also dominate the Curie temperature (Fig. 2c). Because both E_r and T_C derive from nanotube geometry, independent adjustment of each one and analyzing their direct coupling is challenging.

The properties of zigzag and armchair magnetic nanotubes family MX₂

In a similar way, MX₂ nanotubes (M = V, Cr, Mn, Fe, Co, Ni; X = S, Se, Te) nanotubes were calculated in a similar way. Some high T_C ferromagnetic semiconductors are predicted, such as Z-12-FeTe₂ and A-12-VTe₂ with T_C higher than 200 K. They show E_strain lower than 0.06 eV/atom, indicating their stability. The detailed results of zigzag nanotubes and armchair nanotubes are shown in Tables 5 and 6, respectively.

Table 5 For zigzag magnetic nanotubes family MX₂, the band gap, diameter d, Curie temperature T_C, and strain energy E_strain

Table 6 For armchair magnetic nanotubes family MX₂, the band gap, diameter d, Curie temperature T_C, and strain energy E_strain

Discussion

By rolling the 2D high T_C ferromagnetic metals MX₂ (M = V, Cr, Mn, Fe, Co, Ni; X = S, Se, Te) into nanotubes, we theoretically predicted some high T_C ferromagnetic semiconductors, including Z-18-CrS₂ with T_C of 364 K and band gap of 0.53 eV, Z-18-CrTe₂ with T_C of 441 K and band gap of 0.16 eV. The predicted nanotubes show strain energies lower than experimental nanotubes, such as narrow CNTs and BN nanotubes, suggesting the feasibility of preparation. In addition, an electrical polarization on the order of 0.1 eV/Å inward of the radial direction is observed due to the strain gradient in the tube wall. Our theoretical results demonstrate a way to obtain high T_C ferromagnetic semiconducting nanotubes derived from experimentally obtained 2D high T_C ferromagnetic metals.

Methods

Density functional theory calculations

All calculations were based on the DFT as implemented in the Vienna ab initio simulation package (VASP)¹²². The exchange-correlation potential is described by the Perdew-Burke-Ernzerhof (PBE) form with the generalized gradient approximation (GGA)¹²³. The electron-ion potential is described by the projector-augmented wave (PAW) method¹²⁴. We carried out the calculation of PBE + U with U = 4 eV for 3d elements. The band structures were calculated in Heyd-Scuseria-Ernzerhof (HSE) hybrid functional¹¹¹. The plane-wave cutoff energy is set to be 600 eV. 7 × 1 × 1 Γ center K-points were used for the Brillouin zone (BZ) sampling in nanotubes. The structures of all materialÅ, respectively. The Wannier90 code was used to construct a tight-binding Hamiltonian^125,126 to calculate the magnetic coupling constants.

Monte Carlo program

The Heisenberg-type Monte Carlo simulation of monolayer CrS₂ and Z-18-CrS₂ nanotubes was performed on 25 × 25 × 1 and 50 × 1 × 1 lattices, respectively. 1 × 10⁵ steps were carried out for each temperature, and the last one-thirds steps were used to calculate the temperature-dependent physical quantities. The Curie temperature calculation algorithm has been validated in previous work on some 2D and 3D ferromagnetic and antiferromagnetic systems, as shown in Table 7. Results of monolayers MnSe₂^39,127, CrTe₂²⁹, Cr₃Te₆^31,51, Cr₂Ge₂Te₆^21,128 and CrS₂⁴⁰, and bulk LaFeO₃^129,130 are listed. The computed T_C and T_N agree well with experimental measurements, confirming the validity of our calculation.

Table 7 Comparison between calculated and experimental T_C and T_N values for some 2D and 3D magnetic materials

Calculation of J ₁, J ₂, J ₃ and Curie temperatures of monolayer CrS₂ and monolayer CrTe₂

To study the magnetic properties of monolayer CrS₂, we consider the Heisenberg-type Hamiltonian:

$$\begin{array}{rcl}H&=&{J}_{1}\mathop{\sum}\limits _{\left\langle i,j\right\rangle }\overrightarrow{{S}_{i}}\cdot \overrightarrow{{S}_{j}}+{J}_{2}\mathop{\sum}\limits _{\left\langle \left\langle i,j\right\rangle \right\rangle }\overrightarrow{{S}_{i}}\cdot \overrightarrow{{S}_{j}}+{J}_{3}\mathop{\sum}\limits _{\left\langle \left\langle \left\langle i,j\right\rangle \right\rangle \right\rangle }\overrightarrow{{S}_{i}}\cdot \overrightarrow{{S}_{j}}+A\mathop{\sum}\limits _{i}{S}_{iz}^{2}+{E}_{0},\end{array}$$

(6)

where $\overrightarrow{{S}_{i}}$ and $\overrightarrow{{S}_{j}}$ are spin operators of Cr atoms at sites i and j, respectively. J₁, J₂, and J₃ are the nearest, second-nearest, and third-nearest exchange coupling constants between Cr atoms, as shown in Fig. 9a. A is the single-ion magnetic anisotropy parameter defined as A = (E_⊥ − E_∥)/N_Cr, where E_⊥ and E_∥ are energies of monolayer CrS₂ with out-of-plane and in-plane magnetization, respectively. N_Cr is the number of Cr atoms in the considered cell. The results are A = −0.23 meV/Cr. Energies of monolayer CrS₂ with a ferromagnetic and three antiferromagnetic spin configurations, as shown in Fig. 9a–d, respectively, can be expressed as

$$\begin{array}{lll}{E}_{{\rm{FM}}}&=&24{J}_{1}{S}^{2}+24{J}_{2}{S}^{2}+24{J}_{3}{S}^{2}+{E}_{0},\\ {E}_{{\rm{AFM1}}}&=&8{J}_{1}{S}^{2}-8{J}_{2}{S}^{2}-8{J}_{3}{S}^{2}+{E}_{0},\\ {E}_{{\rm{AFM2}}}&=&-8{J}_{1}{S}^{2}-8{J}_{2}{S}^{2}+24{J}_{3}{S}^{2}+{E}_{0},\\ {E}_{{\rm{AFM3}}}&=&-8{J}_{1}{S}^{2}+8{J}_{2}{S}^{2}-8{J}_{3}{S}^{2}+{E}_{0}.\end{array}$$

(7)

E₀ is the energy part independent of spin configurations, which is included in the total energy of DFT results for monolayer CrS₂. The exchange coupling parameters can be calculated by solving the equations. The DFT results of the relative total energy of monolayer CrS₂ in FM, AFM1, AFM2, and AFM3 states are 0, 423, 490, and 520 meV, respectively, which gives J₁S² = −10.78 meV, J₂S² = − 4.53 meV and J₃S² = −3.20 meV. HSE hybrid functional approach was used to obtain the energies with different magnetic configurations¹¹¹. 7 × 4 × 1 Γ center K-points were used. As shown in Fig. 9e, the Monte Carlo simulation result gives T_C of 295 K, close to the experimental results of T_C = 300 K⁴⁰.

Fig. 9: Spin configurations and Monte Carlo results of monolayers CrS₂ and CrTe₂.

A ferromagnetic (FM) (a) and three antiferromagnetic (AFM) (b–d) spin configurations are considered to get the exchange coupling parameters. e The magnetization and susceptibility of monolayer CrS₂ as a function of temperature obtained by the Monte Carlo simulation, give a T_C of 295 K. f The energy and specific heat of monolayer CrTe₂ as a function of temperature obtained by the Monte Carlo simulation, give a T_N of 352 K.

The magnetic properties of monolayer CrTe₂ were calculated in a similar way. The lattice constant of monolayer CrTe₂ is chosen as the experimental value of $a=b/\sqrt{3}=6.8$ Ų⁹. The relative energies of FM, AFM1, AFM2, and AFM3 states are 1350.15, 490.65, 705.26, and 0.00 meV, respectively, showing an AFM3 ground state, agree with previous experiment²⁹ and DFT studies¹³¹. The results of coupling constants and MAE are J₁S² = 24.41 meV, J₂S² = −5.26 meV, J₃S² = 19.41 meV, and A = −3.00 meV/Cr, respectively. As shown in Fig. 9f, the Monte Carlo simulation result gives T_N of 352 K, which agrees with the experimental results above room temperature²⁹.

Fig. 10: Spin configurations considered for CrS₂ nanotubes.

A ferromagnetic (FM) (a), two antiferromagnetic (AFM) (b, c), and one ferrimagnetic (FIM) (d) spin configurations are considered to get the exchange coupling parameters.

Calculation of J ₁, J ₂ and J ₃ for Zigzag CrS₂ nanotubes

To calculate the magnetic coupling constants of zigzag CrS₂ nanotubes, we consider the Heisenberg-type Hamiltonian similar to Eq. (6). One FM, two AFM, and a ferrimagnetic configuration are considered, as shown in Fig. 10a–d, respectively. Their energies can be expressed as

$$\begin{array}{lll}{E}_{{\rm{FM}}}/{N}_{{\rm{Cr}}}&=&12{J}_{1}{S}^{2}+12{J}_{2}{S}^{2}+12{J}_{3}{S}^{2}+{E}_{0}/{N}_{{\rm{Cr}}},\\ {E}_{{\rm{AFM1}}}/{N}_{{\rm{Cr}}}&=&-2{J}_{1}{S}^{2}-2{J}_{2}{S}^{2}+12{J}_{3}{S}^{2}+{E}_{0}/{N}_{{\rm{Cr}}},\\ {E}_{{\rm{AFM2}}}/{N}_{{\rm{Cr}}}&=&2{J}_{1}{S}^{2}-2{J}_{2}{S}^{2}-2{J}_{3}{S}^{2}+{E}_{0}/{N}_{{\rm{Cr}}},\\ {E}_{{\rm{AFM3}}}/{N}_{{\rm{Cr}}}&=&4{J}_{1}{S}^{2}+4{J}_{3}{S}^{2}+{E}_{0}/{N}_{{\rm{Cr}}}.\end{array}$$

(8)

N_Cr is the number of Cr atoms in the considered cell. The DFT results of the relative total energy of Z-18-CrS₂ in FM, AFM1, AFM2, and FIM states are −582.22 meV, −577.98, −579.94 meV, and −580.01 meV, respectively. The calculated coupling constants are J₁S² = −19.38 meV, J₂S² = −14.30 meV, and J₃S² = 10.08 meV, respectively.