Size-Dependent Critical Temperature and Anomalous Optical Dispersion in Ferromagnetic CrI3 Nanotubes

Using first principles calculations, we explored the magnetic and optical properties of chromium(III) iodide (CrI3) nanotubes (NTs) by changing their chirality and diameter. Here, we considered six types of NTs: (5,0), (5,5), (7,0), (10,0), (10,10), and (12,0) NTs. We found that both zigzag and armchair NTs had a ferromagnetic ground with a direct band gap, although the band gap was dependent on the chirality and diameter. Using the Monte Carlo simulation, we found that the Curie temperatures (Tc) exhibited chirality and diameter dependence. In zigzag NTs, the larger the tube diameter, the larger the Tc, while it decreased with increasing diameter in the armchair tube. We found that the Tc was almost doubled when the diameter increased two-fold. This finding may guide development of room temperature ferromagnetism in zigzag NTs. We also found that the CrI3 NTs displayed anisotropic optical properties and anomalous optical dispersion in the visible range. Specifically, the (10,0) zigzag NT had a large refractive index of 2 near the infrared region, while it became about 1.4 near blue light wavelengths. We also obtained large reflectivity in the ultraviolet region, which can be utilized for UV protection. Overall, we propose that the CrI3 NTs have multifunctional physical properties for spintronics and optical applications.


Introduction
Extensive studies have focused on two-dimensional (2D) materials because they display many peculiar physical properties not found in bulk or macroscopic structures. So far, numerous types of 2D materials have been fabricated. Most of these are non-magnetic, although few theoretical works have investigated the physical properties of magnetic 2D structures. Despite the difficulty of synthesizing intrinsic materials, it would be an interesting finding if a 2D ferromagnetic (FM) material with a finite band gap could be synthesized, because this would offer potential applications for spintronic or opto-spintronic devices. It was recently reported that the chromium(III) iodide (CrI 3 ) monolayer has a finite band gap of 1.2 eV with an FM ground state. It was found that the CrI 3 monolayer had a band gap of 1.2 eV with a critical Curie temperature of T c = 45 K and strong magnetic anisotropy [1][2][3]. The pristine CrI 3 layer consists of three monoatomic planes: one plane of chromium (Cr) atoms sandwiched between two atomic planes of iodine (I). Each Cr 3+ ion is arranged in a honeycomb network of an edge-sharing octahedral coordinated by six I − ions, and each I − ion is bonded to two Cr ions. The isolated Cr atom has an electron configuration of 3d 5 4s 1 , and the largest atomic magnetic moment among all elements in the 3d transition metal series (6 µ B ). Thus, the Cr in the pristine CrI 3 layer is expected to have a +3 state with an electron configuration of 3d 3 4s 0 . This is consistent with the observed saturation magnetization of CrI 3 , which has a magnetic moment of around 3 µ B per Cr atom [2].
After the discovery of carbon nanotubes (CNTs) in 1991 [4,5], both theoretical and experimental investigations have revealed that the unique structure of the CNT produces remarkable chiral dependent physical properties. This motivated the scientific community to search for another nanotube (NT) material with desirable features: boron nitride or phosphorene NT. From a structural point of view, a single-walled nanotube (SWNT) can be imagined as a cylinder with a diameter of only a few nanometers. The SWNTs may be rolled into one another to form 'Russian dolls' known as multi-walled carbon nanotubes (MWNTs) [4]. Unlike ordinary materials, it has been shown that the carbon based SWNT could be either metallic or semiconducting [6,7] according to its chirality. SWNTs can be constructed by rolling up the sheet within unit vectors (a 1 and a 2 ) along a certain direction C h = na 1 + ma 2 , also known as a chiral vector [8][9][10]. Chiral indices (n,m) are commonly used to label SWNTs. The chiral vector is perpendicular to a translational vector T pointing to the long axis of the SWNT. Both C h and T vectors define the unit cell of a SWNT. Alternatively, one can use θ, the chiral angle between a 1 and C h , to classify such materials. Due to the symmetry of the hexagonal lattice, |θ| ≤ 30 • . SWNTs with chiral indices n = m (θ = 30 • ) or m = 0 (θ = 0 • ) have the highest symmetry; they are known as armchair (AC) or zigzag (ZZ) NTs, with a diameter, d, related to the chiral indices (m and n) by the equation d = a π √ n 2 +nm + m 2 . Since CrI 3 has a band gap with a magnetic state, it might be useful to explore the electron structure and magnetic and optical properties of the NTs by changing the chirality. Moreover, the critical temperature is one of the main issues in the magnetic materials, and the 2D CrI 3 has a rather low Curie temperature of 45 K. Thus, it is necessary to increase the critical temperature for future device applications. So far, no studies on these issues in the CrI 3 nanotube system are available. Thus, in our report, we aim to investigate the chirality and size-dependent magnetic state, band gap tuning, enhancement of critical temperature, and optical properties of the CrI 3 NT, and propose that it can be utilized for potential multifunctional applications.

Numerical Method
We performed the ab initio simulations within the density functional theory [11,12] framework using the Vienna Ab initio Simulation Package (VASP) [13][14][15][16]. Exchange correlation interactions were treated with the generalized gradient approximation (GGA) within the Perdew-Burke-Ernzerhof (PBE) formulation [17,18]. The full-potential projected plane-wave framework [19,20] was used with an energy cutoff of 500 eV for the plane-wave basis set. We fully optimized the structure until the force on each atom was smaller than 0.01 eV Å −1 and the energy convergence reached up to 10 −5 eV/atom, using the conjugate gradient method. For the one dimensional NTs, a unit cell with periodic boundary conditions was adopted to simulate the infinite z direction, and the vacuum distances of 15 Å along the x and y axes were applied to avoid interaction between two neighboring images. The Brillouin zone was sampled using the Monkhorst-Pack scheme using a 1 × 1 × 13 k point mesh for atomic and electronic relaxations. The electronic band structure was obtained from energy eigenvalues of 50 points along the Γ-X line in the Brillouin zone. Due to the localized 3D electrons of Cr, the correlation effect may alter the magnetic properties of CrI 3 systems. It is therefore necessary to take the onsite Coulomb repulsion interaction of Cr 3D electrons into account, using the generalized gradient approximation and the Hubbard U term (GGA + U) method [21] to check the magnetic state of CrI 3 NT systems with a moderate value of U eff = 2.65 eV, which is adequate for obtaining a reasonable correction, as shown in the CrI 3 monolayer system [22]. The optical properties were calculated from the frequency-dependent dielectric function ε(ω) = ε 1 (ω) + iε 2 (ω). The imaginary part of this function is determined by the following equation: where c and v represent the conduction and valence band states, k is the wave vector, and u ck represents the wave function with a lattice constant periodicity. The real part is determined by the following equation: The real and imaginary parts of the dielectric function are correlated by the well-known Kramers-Kronig relations [23,24].

Structural Characterization
Before we discuss the structure of the CrI 3 SWNT, it is helpful to review the monolayer geometry. Figure 1a illustrates the atomic structure of a pristine CrI 3 layer, which consists of three monoatomic planes X-M-X (X = I and M = Cr). All three planes belong to the same trigonal lattice with the basis vectors → a and → b of equal length, a = 6.8259 Å. The unit cell contains 6 I and 2 Cr atoms. The iodine planes are separated by distance δ = 2.88 Å, and the Cr atom is located at the center between two triangular iodine planes at a relative height of δ 2 . Similar to the single-walled CNTs, a CrI 3 SWNT can be constructed by folding a monolayer into a tube form. The CrI 3 SWNT with chiral index (n,m) is obtained by rolling up the helical vector in a given direction, where the chiral angle is also in the range θ = 0, π 6 [25,26] as presented in Figure 1a. Figure 1b,c displays two examples of typical zigzag (ZZ) and armchair (AC) CrI 3 SWNTs. The CrI 3 tube single wall consists of three coaxial cylinders (X-M-X) of thickness δ, and it endures different distortions when the monolayer is folded into a tube. Due to the advanced experimental technique, it is possible to fabricate CNTs with diameters of 1~2 nm [27]. Thus, in our work, we consider four types of NTs which have diameters of~1-2 nm. Table 1 shows the structural information, such as lattice parameters (L), diameter (d), and Cr-I bond length in each tube in the ground state. The tube radius (d/2) corresponds to the distance between the axis of the tube and the cylinder of Cr atoms, and the interior and exterior iodine atoms are shrunken and stretched. A comparison of the calculated diameter d calculated = a π √ n 2 + nm + m 2 with the optimized value (d) yields an agreement. where c and v represent the conduction and valence band states, k is the wave vector, and uck represents the wave function with a lattice constant periodicity. The real part is determined by the following equation: The real and imaginary parts of the dielectric function are correlated by the well-known Kramers-Kronig relations [23,24].

Structural Characterization
Before we discuss the structure of the CrI3 SWNT, it is helpful to review the monolayer geometry. Figure 1(a) illustrates the atomic structure of a pristine CrI3 layer, which consists of three monoatomic planes X-M-X (X = I and M = Cr). All three planes belong to the same trigonal lattice with the basis vectors ⃗ and ⃗ of equal length, a = 6.8259 Å. The unit cell contains 6 I and 2 Cr atoms. The iodine planes are separated by distance δ = 2.88 Å, and the Cr atom is located at the center between two triangular iodine planes at a relative height of . Similar to the single-walled CNTs, a CrI3 SWNT can be constructed by folding a monolayer into a tube form. The CrI3 SWNT with chiral index (n,m) is obtained by rolling up the helical vector in a given direction, where the chiral angle is also in the range θ = 0, [25,26] as presented in Figure 1(a). Figure 1 [27]. Thus, in our work, we consider four types of NTs which have diameters of ~1-2 nm. Table 1 shows the structural information, such as lattice parameters (L), diameter (d), and Cr-I bond length in each tube in the ground state. The tube radius (d/2) corresponds to the distance between the axis of the tube and the cylinder of Cr atoms, and the interior and exterior iodine atoms are shrunken and stretched. A comparison of the calculated diameter d = √n + nm + m with the optimized value (d) yields an agreement.   To check the dynamic stability of CrI 3 SWNT systems, we calculated the molecular dynamics, because it is a reliable tool to check whether a virtual structure is stable. The calculated dynamic stability and mechanical properties suggest that the 1D of CrI 3 SWNT systems could be synthesized by appropriate experimental techniques on a suitable substrate, and that these materials can exist at ambient conditions. To show this, as an illustration, we present the snapshots of molecular dynamics at 5 ps with top and side views at 300 K for (5,0)-zigzag and (5,5)-armchair NTs in Supplementary Figure S1.

Electronic Band Structures
We now present the electronic band structures and band gaps. Figures 2 and 3 show the electronic band structures and the orbital projected partial density of states (PDOS). Note the overlap of the I-p x and I-p y states and the Cr-d xz and d yz states. This is due to the symmetry of the NT. It is well-known that CNTs can have either metallic or semiconducting band gaps according to the chirality, even if they have the same diameters [27]. However, all the CrI 3 NTs had a direct band gap at the Γ-point, regardless of their chirality. We also found that these NTs displayed diameter-and chirality-dependent band gaps. For instance, in the ZZ CrI 3 NT, both conduction band minimum (CBM) and valence band maximum (VBM) appeared in the majority spin state, and the band gaps increased from 0.435 (for (5,0) ZZ NT) to 0.909 eV (for (12,0) ZZ NT) with increasing diameter. As shown in the PDOS for the (5,0)-ZZ NT, we found strong hybridization between I-p and Cr-d states in the majority spin state. With increasing diameter, the repulsive interaction between Cr-d and I-p states in the majority spin state was further enhanced. In particular, the downshift of the I-p z orbital in the majority spin valance band was clearly observed, and this resulted in an increase of the band gap. Additionally, both valence and conduction band edges appeared in the majority spin state. We also found that the I-p z state contributed to the valence band edge, while the conduction band edge originated strongly hybridized states. However, the AC tube displayed different behavior. In the (5,5) configuration, both the CBM and VBM appeared in the majority spin state with a band gap of 0.695 eV, and the I-p z orbital in the majority spin band formed the VBM edge. In contrast, for the (10,10) tube, the I-p z orbital in the valence band with majority spin state was pushed down below the Fermi level. The I-p (x,y) orbital in the minority spin band formed the VBM, and the I-Cr repulsion was suppressed. Consequently, we observed a reduced band gap of 0.427 eV. As described, the size-dependent band gap appeared differently in the zigzag and armchair NTs, and this can be understood from a structural point of view. For instance, the zigzag NTs will be identical to the monolayer when the tube diameter becomes infinite [27]. Thus, the increasing band gap of zigzag CrI 3 NTs with diameter is reasonable. In contrast, the armchair NT showed a distorted structure, and the geometric feature was still far away from the 2D characteristic for the armchair NTs considered in our calculations. Thus, the band gap of armchair NTs showed a different tendency from that found in zigzag NTs. However, when the tube diameter becomes infinite, the armchair NT will be identical to the 2D monolayer. To show this, as an illustration, we present the structures of (10,0)-zigzag and (10,10)-armchair NTs in Figure S2. Despite the chirality-and diameter-dependent band gap, all the CrI 3 NTs have a semiconductor band structure with a band gap. The robustness of semiconducting half-metallic materials will be advantageous for spintronics applications, because no extra process is required to sort which NTs have either metallic or semiconducting properties with the FM state.

Curie Temperature
We now explore the magnetic ground state. To determine the magnetic ground state, we considered three possible spin configurations: ferromagnetic (FM), nonmagnetic (NM), and antiferromagnetic (AFM) coupling between chromium atoms. In Table S1, we present the energy difference between the FM, NM, and AFM for CrI 3 SWNT systems. One of the most important physical properties of a ferromagnetic material is the Curie temperature (T c ). Using Metropolis Monte Carlo (MC) simulations [28] based on the Ising model, we calculated the Curie temperature for CrI 3 NTs (AC and ZZ). According to the Ising model, the Hamiltonian equation can be written asĤ = − ∑ i,j Jm imj , wherem i andm j are the magnetic moments (in µ B ) at sites i and j, and J is the exchange parameter. For simplicity, only the nearest neighboring exchange interaction is taken into account, and J = E ex Nm 2 where E ex , N, and m 2 represent the exchange energy defined by E ex = (E AFM − E FM ), number of Cr atoms per unit cell, and square magnetic moments. In Table 2, we present calculated results. During the MC simulations, a 50 supercell was used to mimic the CrI 3 NTs, and this was large enough to minimize the periodic constraints. The MC simulation allowed us to calculate variations in the average magnetic moment per unit cell at a given temperature. Figure 4 displays the temperature dependent magnetization curve. The magnetic moment retains a high spin state in the low temperature range, and then drops to near zero at the critical temperature.   To check the reliability of the MC calculations, we also calculated the Curie temperature of the 2D CrI 3 layer to be 51 K. Note that the experimentally measured value was 45 K [1]. For the (5,0)-ZZ CrI 3 NT, the Curie temperature (T c ) was 33 K, while the T c of (5,5)-AC CrI 3 NT was increased almost twice when the diameter is doubled, and this T c was slightly larger than that of the 2D layer. In (10,0)-ZZ NT, it was significantly enhanced to 86 K. Unlike the zigzag NTs, the Curie temperature of armchair CrI 3 -NTs decreased from 63 K to 48 K with increasing diameter. Indeed, the Curie temperature is closely related to the strength of exchange interaction, and we expect a higher T c if the J is larger. As shown in Table 2, the J value of zigzag NT is increased in a bigger NT, while it is suppressed in armchair NTs. This can nicely account for the chiral-and size-dependent critical temperature found in our work. One of the main issues in the low dimensional magnetic materials is achieving a room temperature ferromagnetism. It is obvious that the T c will finally converge to that of the 2D sheet when the diameter of ZZ NT becomes infinite. Nonetheless, as shown in Table 2, the larger ZZ NT shows a larger T c , and this suggests that the T c will increase until the diameter reaches a certain critical value. Due to our limited computing power, we cannot fully investigate this issue with very big ZZ NT. As an alternative approach, we performed an estimation for the T c based on rough approximation. After that, we considered (7,0)-ZZ and (12,0)-ZZ NTs. We first checked the diameter-dependent band gap of ZZ NTs. Table S2 presents the structural information and energy gap for (7,0)-ZZ NT and (12,0)-ZZ NT. The energy gaps (E g ) for (7,0)-ZZ NT and (12,0)-ZZ NT are 0.702 and 0.909 eV. Based on our diameter dependent band gap, we extrapolated the band gap for (5,0)-ZZ, (7,0)-ZZ, (10,0)-ZZ, and (12,0)-ZZ NTs, and Figure S3 shows the calculated result. By extrapolation, we find that the band gap of 2D layer will be about 1.30 eV, and this is in good agreement with the previously reported value of 1.2 eV [1]. Table S3, and Figure S4 shows the magnetic information for (7,0)-ZZ NT and (12,0)-ZZ NT. The calculated Curie temperatures were 71 K and 134 K for (7,0)-ZZ NT and (12,0)-ZZ NT, respectively. From the T c obtained using the Monte Carlo simulation for (5,0)-ZZ, (7,0)-ZZ, (10,0)-ZZ, and (12,0)-ZZ NTs, we estimated the T c by assuming a linear relationship. Figure 5 shows the calculated result. As mentioned above, the T c should start to drop after a certain size of NT. Nonetheless, if we can assume that the T c progresses linearly until the diameter reaches about 5 nm, then our calculation suggests that the room temperature ferromagnetic structure can be possible if the diameter reaches about~5-6 nm.

Optical Properties
We now discuss the optical properties. The most important physical quantity for optical properties is the frequency dependent dielectric function ε(ω) = ε 1 (ω) + iε 2 (ω), because all the optical quantities can be extracted from this frequency dependent dielectric function. The dielectric function ε(ω) is dependent on the polarization of the incident light. In this report, we consider two types of incident light propagation: electromagnetic waves perpendicular (or parallel electric polarization to the tube axis E || ) and parallel (or perpendicular electric polarization to the tube axis E ⊥ ) to the tube axis. Figure 6a,b shows the real and imaginary parts of the frequency-dependent dielectric function ε(ω). It is clear, especially in the long wavelength case, that the ε(ω) strongly depends on the electric polarization and chirality of the NTs. Subsequently, we can expect the anisotropic optical properties to be more noticeable in long wavelength conditions. Since the ε 2 (ω) is related to the optical transition by the incident light, it is important to understand this function. If we consider the conventional experimental situation, it will be more realistic to explore the electromagnetic wave propagation perpendicular to the tube axis. Thus, we focus on the parallel electric field polarization (E || ) in the following discussion. From the calculated ε 2 (ω), we found that the first peaks in the low energy region (between~1.3-1.7 eV) were rather broad. Nonetheless, the essential characteristic can be understood from the band structure and DOS. Considering the optical transition selection rule, in the (5,0) and (5,5) NTs we found that the major contribution to the absorption occurred from I-p z to Cr-d z 2 transition, while transition from both I-p z to Cr-d z 2 and from I-p x,y to Cr-d xz,yz contributed to the absorption in (10,0) and (10,10) NTs. The second absorption peaks (between~2.7-3.2 eV) were much broader than the first peaks. In the second peaks, in both zigzag and armchairs tubes, the transitions from Cr-d yz,zx to I-p x,y, and Cr-d z 2 to I-p z were responsible for the absorption. For optical device applications, both refractivity and reflectivity are key properties which can be extracted from the frequency-dependent dielectric function. Figure 5c presents the refractive index given by: where n(ω) and k(ω) represent real and imaginary parts of the refractive index. The real part of N(ω) is called the index of refraction, and it is related to the speed of the electromagnetic wave, while the imaginary part is related to the wave absorption. We found that the refractive index is strongly sensitive to the wavelength and diameter of the NT, and such a dependency was clearly observed in long-wavelength cases. For instance, in the one edge of visible light (~380 nm = 3.3 eV), the (5,0)-ZZ NT (smallest diameter NT) had an n-value of 1.6, while the (10,10)-AC NT (largest diameter NT) had n = 1.30. By increasing the wavelength, we also found an interesting optical property. In many materials, the refractive index increases at shorter wavelengths, which is known as normal dispersion. For CrI 3 NTs, however, the n(ω) becomes larger with increasing wavelength (anomalous dispersion). A large value of n was also observed near the infrared edge (~700-800 nm =~1.8-1.6 eV). For instance, we obtained a refractive index of 2 for light propagating perpendicular to the tube axis (E || polarization), and this value may have potential applications for infrared optical devices. As presented in Table 2, the (10,0)-ZZ tube had the highest critical temperature-almost twice that of the 2D sheet-as well as the largest refractive index. The refractive index varied from 1.4 to 2 in the visible range at different wavelengths. Thus, we find that the (10,0)-ZZ NT displays the most promising magnetic and optical properties for potential device applications. We also calculated the reflectivity given by: (4) Figure 5d shows the calculated results. For CrI 3 NTs, both ZZ and AC tubes have a large reflectivity in the ultraviolet (UV) wavelength region, and this characteristic can be utilized for UV protection. On the other hand, the CrI 3 NTs display rather weak reflectivity in the visible range for both directions of polarization, and all CrI 3 NT systems would be optically transparent in a wide range of visible frequencies.

Conclusions
In summary, we investigated the geometry, electronic band structure, magnetic state, and optical properties of CrI 3 SWNTs with different sets of chiral indices. Unlike the carbon nanotubes, we found that all CrI 3 NT systems had a direct band gap with a ferromagnetic ground state, and the band gap was dependent on the chirality and the diameter. The band gap of zigzag NTs was increased with larger diameter, while the armchair NT showed the opposite trend. This can be understood from the geometric feature. We also calculated the Curie temperature using the Monte Carlo simulation, and the T c displayed the same feature. In the zigzag NTs, the T c was enhanced with increasing tube diameter, while the T c of armchair NT decreased with increasing diameter. The (10,0) ZZ NT had a T c of 86 K, which is almost twice of that found in the 2D CrI 3 layer. We also explored the frequency dependent optical properties. We found that the optical properties were strongly anisotropic according to the electric polarization direction, and also dependent on the chirality and diameter of NTs. The anisotropic behavior was more noticeable in the long wavelength regime, and the anomalous optical dispersion was observed in the visible range because the refractive index was also increased in the long wavelength. In particular, the (10,0) ZZ NT, which had the largest Curie temperature in our systems, displayed a large refractive index of 2 near the infrared regime, while it became about 1.4 near the blue light wavelength. This wide variation may suggest that the ZZ NTs have potential infrared optical device applications. Additionally, the CrI 3 NTs showed a large reflectivity near the ultraviolet regime, which can be used for UV protection. Overall, we propose that the CrI 3 NTs have multifunctional physical properties for spintronics and optical applications.