Next Article in Journal
Facile Fabrication of Fluorescent Inorganic Nanoparticles with Diverse Shapes for Cell Imaging
Previous Article in Journal
A Comparative Study of Particle Size Distribution of Graphene Nanosheets Synthesized by an Ultrasound-Assisted Method
Previous Article in Special Issue
Engineering Planar Transverse Domain Walls in Biaxial Magnetic Nanostrips by Tailoring Transverse Magnetic Fields with Uniform Orientation
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

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

1
Department of Physics, Faculty of Science, Zagazig University, 44519 Zagazig, Egypt
2
Department of Physics, Pukyong National University, Busan 608-737, Korea
*
Author to whom correspondence should be addressed.
Nanomaterials 2019, 9(2), 153; https://doi.org/10.3390/nano9020153
Submission received: 20 December 2018 / Revised: 15 January 2019 / Accepted: 21 January 2019 / Published: 26 January 2019
(This article belongs to the Special Issue Ferromagnetic and Magnetic Properties of Nanostructures)

Abstract

:
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.

1. 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 (CrI3) monolayer has a finite band gap of 1.2 eV with an FM ground state. It was found that the CrI3 monolayer had a band gap of 1.2 eV with a critical Curie temperature of Tc = 45 K and strong magnetic anisotropy [1,2,3]. The pristine CrI3 layer consists of three monoatomic planes: one plane of chromium (Cr) atoms sandwiched between two atomic planes of iodine (I). Each Cr3+ 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 3d54s1, and the largest atomic magnetic moment among all elements in the 3d transition metal series (6 µB). Thus, the Cr in the pristine CrI3 layer is expected to have a +3 state with an electron configuration of 3d34s0. This is consistent with the observed saturation magnetization of CrI3, which has a magnetic moment of around 3 µB 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 (a1 and a2) along a certain direction Ch = na1 + ma2, 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 Ch and T vectors define the unit cell of a SWNT. Alternatively, one can use θ, the chiral angle between a1 and Ch, 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 + n m + m 2 .
Since CrI3 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 CrI3 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 CrI3 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 CrI3 NT, and propose that it can be utilized for potential multifunctional applications.

2. 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 CrI3 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 CrI3 NT systems with a moderate value of Ueff = 2.65 eV, which is adequate for obtaining a reasonable correction, as shown in the CrI3 monolayer system [22]. The optical properties were calculated from the frequency-dependent dielectric function ε(ω) = ε1(ω) + 2(ω). The imaginary part of this function is determined by the following equation:
ε 2 α β ( ω ) =   4 π 2 e 2 Ω l i m q 0 1 q 2 c , v , k 2 ω k δ ( ϵ c k ϵ v k ω ) × u c k + e α q | u v k u c k + e β q | u v k *
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:
ε 1 α β ( ω ) = 1 + 2 π P 0 ε 2 α β ( ω ) ω ω 2 ω 2 + i η d ω
The real and imaginary parts of the dielectric function are correlated by the well-known Kramers–Kronig relations [23,24].

3. Results and Discussion

3.1. Structural Characterization

Before we discuss the structure of the CrI3 SWNT, it is helpful to review the monolayer geometry. Figure 1a 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 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 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 , π 6 ]   [25,26] as presented in Figure 1a. Figure 1b,c displays two examples of typical zigzag (ZZ) and armchair (AC) CrI3 SWNTs. The CrI3 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.
To check the dynamic stability of CrI3 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 CrI3 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.

3.2. Electronic Band Structures

We now present the electronic band structures and band gaps. Figure 2 and Figure 3 show the electronic band structures and the orbital projected partial density of states (PDOS). Note the overlap of the I-px and I-py states and the Cr-dxz and dyz 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 CrI3 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 CrI3 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-pz 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-pz 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-pz orbital in the majority spin band formed the VBM edge. In contrast, for the (10,10) tube, the I-pz 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 CrI3 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 CrI3 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.

3.3. 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 CrI3 SWNT systems. One of the most important physical properties of a ferromagnetic material is the Curie temperature (Tc). Using Metropolis Monte Carlo (MC) simulations [28] based on the Ising model, we calculated the Curie temperature for CrI3 NTs (AC and ZZ). According to the Ising model, the Hamiltonian equation can be written as H ^ = i , j J m ^ i   m ^ j , where m ^ i   and m ^ j are the magnetic moments (in µB) 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 e x N m 2 where Eex, N, and m2 represent the exchange energy defined by E e x = ( E A F M E F M ) , 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 CrI3 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 CrI3 layer to be 51 K. Note that the experimentally measured value was 45 K [1]. For the (5,0)-ZZ CrI3 NT, the Curie temperature (Tc) was 33 K, while the Tc of (5,5)-AC CrI3 NT was increased almost twice when the diameter is doubled, and this Tc 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 CrI3-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 Tc 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 Tc 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 Tc, and this suggests that the Tc 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 Tc 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 (Eg) 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 Tc obtained using the Monte Carlo simulation for (5,0)-ZZ, (7,0)-ZZ, (10,0)-ZZ, and (12,0)-ZZ NTs, we estimated the Tc by assuming a linear relationship. Figure 5 shows the calculated result. As mentioned above, the Tc should start to drop after a certain size of NT. Nonetheless, if we can assume that the Tc 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.

3.4. 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 ) + i ε 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-pz to Cr-dz2 transition, while transition from both I-pz to Cr-dz2 and from I-px,y to Cr-dxz,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-dyz,zx to I-px,y, and Cr-dz2 to I-pz 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:
N ( ω ) =   ε 1 ( ω ) + i ε 2 ( ω ) = n ( ω ) + i k ( ω )
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 CrI3 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:
R ( ω ) = | 1   ε 1 ( ω ) + i ε 2 ( ω ) 1 +   ε 1 ( ω ) + i ε 2 ( ω ) | 2
Figure 5d shows the calculated results. For CrI3 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 CrI3 NTs display rather weak reflectivity in the visible range for both directions of polarization, and all CrI3 NT systems would be optically transparent in a wide range of visible frequencies.

4. Conclusions

In summary, we investigated the geometry, electronic band structure, magnetic state, and optical properties of CrI3 SWNTs with different sets of chiral indices. Unlike the carbon nanotubes, we found that all CrI3 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 Tc displayed the same feature. In the zigzag NTs, the Tc was enhanced with increasing tube diameter, while the Tc of armchair NT decreased with increasing diameter. The (10,0) ZZ NT had a Tc of 86 K, which is almost twice of that found in the 2D CrI3 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 CrI3 NTs showed a large reflectivity near the ultraviolet regime, which can be used for UV protection. Overall, we propose that the CrI3 NTs have multifunctional physical properties for spintronics and optical applications.

Supplementary Materials

The following are available online at https://www.mdpi.com/2079-4991/9/2/153/s1, Figure S1: Snapshots of MD at 5 ps with top and side views at 300 K for (a) (5,0)-ZZ NT (b) (5,5)-AC NT, Figure S2: Atomic structure of chromium(III) iodide (CrI3) NTs: (a) illustration of (10,0)-ZZ NT (b) (10,10)-AC NT with side views, Table S1: Energy difference (in meV) between the FM, NM and AFM for (5,0)-ZZ, (5,5)-AC, (7,0)-ZZ, (10,0)-ZZ, (10,10)-AC, and (12,0)-ZZ CrI3 SWNT, Table S2: Structural information of CrI3 NTs such as lattice parameters (L), the diameter (d), the calculated diameter (dcalculated), Cr-I bond length (dCr-I), the interlayer vertical distance between two iodine cylinders (δ), and energy gap (Eg), Figure S3:The energy gap of CrI3 ZZ NTs with respect to diameter reciprocals, Table S3: Calculated local magnetic moments on the Cr site (MCr), local magnetic moments on the I site (MI), total magnetic moments in a unit cell (Mcell), exchange energy (Eex), exchange magnetic coupling (J), and Curie temperature (Tc), Figure S4: Temperature dependent magnetization curve for CrI3 (a) (7,0)-ZZ NT, and (b) (12,0)-ZZ NT.

Author Contributions

J.H. conceived the idea of this study. M.M. performed DFT, the geometry, electronic band structure, magnetic state, and optical properties of CrI3 SWNTs. J.H. wrote the manuscript and all the authors commented on it.

Funding

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2016R1A2B4006406) and by the Supercomputing Center/Korea Institute of Science and Technology Information with supercomputing resources including technical support (KSC-2018-C2-0014).

Acknowledgments

The authors acknowledge the Basic Science Research Program through the National Research Foundation of Korea (NRF) and the Supercomputing Center/Korea Institute of Science and Technology Information.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Huang, B.; Clark, G.; Navarro-Moratalla, E.; Klein, D.R.; Cheng, R.; Seyler, K.L.; Zhong, D.; Schmidgall, E.; McGuire, M.A.; Cobden, D.H.; et al. Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit. Nature 2017, 546, 270–273. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  2. McGuire, M.A.; Dixit, H.; Cooper, V.R.; Sales, B.C. Coupling of crystal structure and magnetism in the layered, ferromagnetic insulator CrI3. Chem. Mater. 2015, 27, 612–620. [Google Scholar] [CrossRef]
  3. Lado, J.L.; Fernández-Rossier, J. On the origin of magnetic anisotropy in two dimensional CrI 3. 2D Mater. 2017, 4, 035002. [Google Scholar] [CrossRef] [Green Version]
  4. Iijima, S. Helical microtubules of graphitic carbon. Nature 1991, 354, 56–58. [Google Scholar] [CrossRef]
  5. Iijima, S.; Ichihashi, T. Single-shell carbon nanotubes of 1-nm diameter. Nature 1993, 363, 603–605. [Google Scholar] [CrossRef]
  6. Laird, E.A.; Kuemmeth, F.; Steele, G.A.; Grove-Rasmussen, K.; Nygård, J.; Flensberg, K.; Kouwenhoven, L.P. Quantum transport in carbon nanotubes. Rev. Mod. Phys. 2015, 87, 703–764. [Google Scholar] [CrossRef] [Green Version]
  7. Lu, X.; Chen, Z. curved pi-conjugation, aromaticity, and the related chemistry of small fullerenes. Chem. Rev. 2005, 105, 3643–3696. [Google Scholar]
  8. Dresselhaus, M.S.; Dresselhaus, G.; Saito, R. Physics of carbon nanotubes. Carbon 1995, 33, 883–891. [Google Scholar] [CrossRef] [Green Version]
  9. Eklund, P.C.; Holden, J.M.; Jishi, R.A. Vibrational modes of carbon nanotubes; Spectroscopy and theory. Carbon 1995, 33, 959–972. [Google Scholar] [CrossRef]
  10. Sinnott, S.B.; Andrews, R. Carbon nanotubes: Synthesis, properties, and applications. Crit. Rev. Solid State Mater. Sci. 2001, 26, 145–249. [Google Scholar] [CrossRef]
  11. Hohenberg, P.; Kohn, W. Inhomogeneous electron gas. Phys. Rev. 1964, 136, B864–B871. [Google Scholar] [CrossRef]
  12. Kohn, W.; Sham, L.J. Self-consistent equations including exchange and correlation effects. Phys. Rev. 1965, 140, A1133–A1138. [Google Scholar] [CrossRef]
  13. Kresse, G.; Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B 1993, 47, 558–561. [Google Scholar] [CrossRef]
  14. Kresse, G.; Hafner, J. Ab initio molecular-dynamics simulation of the liquid-metal–amorphous-semiconductor transition in germanium. Phys. Rev. B 1994, 49, 14251–14269. [Google Scholar] [CrossRef]
  15. Kresse, G.; Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 1996, 6, 15–50. [Google Scholar] [CrossRef]
  16. Kresse, G.; Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 1996, 54, 11169–11186. [Google Scholar] [CrossRef]
  17. Perdew, J.P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996, 77, 3865–3868. [Google Scholar] [CrossRef] [PubMed]
  18. Perdew, J.P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1997, 78, 1396. [Google Scholar] [CrossRef]
  19. Blöchl, P.E. Projector augmented-wave method. Phys. Rev. B 1994, 50, 17953–17979. [Google Scholar] [CrossRef] [Green Version]
  20. Kresse, G.; Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 1999, 59, 1758–1775. [Google Scholar] [CrossRef]
  21. Dudarev, S.L.; Botton, G.A.; Savrasov, S.Y.; Humphreys, C.J.; Sutton, A.P. Electron-energy-loss spectra and the structural stability of nickel oxide: An LSDA+U study. Phys. Rev. B 1998, 57, 1505–1509. [Google Scholar] [CrossRef]
  22. Liu, J.; Sun, Q.; Kawazoe, Y.; Jena, P. Exfoliating biocompatible ferromagnetic Cr-trihalide monolayers. Phys. Chem. Chem. Phys. 2016, 18, 8777–8784. [Google Scholar] [CrossRef] [PubMed]
  23. Kronig, R.D.L. On the theory of dispersion of X-Rays. JOSA 1926, 12, 547–557. [Google Scholar] [CrossRef]
  24. Kramers, H.A. La diffusion de la lumière par les atomes. Atti del Congresso Internationale dei Fisici. Como 1927, 2, 545–557. [Google Scholar]
  25. Dobardžić, E.; Milošević, I.; Nikolić, B.; Vuković, T.; Damnjanović, M. Single-wall carbon nanotubes phonon spectra: Symmetry-based calculations. Phys. Rev. B 2003, 68, 045408. [Google Scholar] [CrossRef]
  26. Lee, R.K.F.; Cox, B.J.; Hill, J.M. The geometric structure of single-walled nanotubes. Nanoscale 2010, 2, 859–872. [Google Scholar] [CrossRef]
  27. Dresselhaus, M.S.; Dresselhaus, G.; Eklund, P.C. Science of fullerenes and carbon nanotubes. Adv. Mater. 1997, 9, 1193. [Google Scholar]
  28. Moaied, M.; Alvarez, J.V.; Palacios, J.J. Hydrogenation-induced ferromagnetism on graphite surfaces. Phys. Rev. B 2014, 90, 115441. [Google Scholar] [CrossRef]
Figure 1. Atomic structure of chromium(III) iodide (CrI3): (a) side and top views of pristine 2D layer (b) illustration of (10,0) zizag (ZZ) nanotubes (NT) (c) (5,5) armchair (AC) NT with side and top views.
Figure 1. Atomic structure of chromium(III) iodide (CrI3): (a) side and top views of pristine 2D layer (b) illustration of (10,0) zizag (ZZ) nanotubes (NT) (c) (5,5) armchair (AC) NT with side and top views.
Nanomaterials 09 00153 g001
Figure 2. Electronic band structures of the: (a) (5,0) ZZ NT (b) (5,5) AC NT (c) (10,0) ZZ NT, and (d) (10,10) AC CrI3 NT.
Figure 2. Electronic band structures of the: (a) (5,0) ZZ NT (b) (5,5) AC NT (c) (10,0) ZZ NT, and (d) (10,10) AC CrI3 NT.
Nanomaterials 09 00153 g002
Figure 3. Orbital projected partial density of state (PDOS) of (a) (5,0) ZZ NT (b) (5,5) AC NT (c) (10,0) ZZ NT, and (d) (10,10) AC NT.
Figure 3. Orbital projected partial density of state (PDOS) of (a) (5,0) ZZ NT (b) (5,5) AC NT (c) (10,0) ZZ NT, and (d) (10,10) AC NT.
Nanomaterials 09 00153 g003
Figure 4. Temperature-dependent magnetization curve for (a) (5,0)-ZZ NT (b) (5,5)-AC NT (c) (10,0)-ZZ NT, and (d) (10,10)-AC NT.
Figure 4. Temperature-dependent magnetization curve for (a) (5,0)-ZZ NT (b) (5,5)-AC NT (c) (10,0)-ZZ NT, and (d) (10,10)-AC NT.
Nanomaterials 09 00153 g004
Figure 5. Diameter dependent Curie temperature of zigzag nanotubes.
Figure 5. Diameter dependent Curie temperature of zigzag nanotubes.
Nanomaterials 09 00153 g005
Figure 6. (a) Real and (b) imaginary parts of the frequency-dependent dielectric function, (c) real and imaginary parts of the refractive index, and (d) reflectivity for parallel (E||) and perpendicular (E) electric field polarization with respect to the tube axis. Four tubes are presented: the (5,0)-ZZ, (5,5) AC, (10,0)-ZZ, and (10,10)-AC CrI3 SWNT.
Figure 6. (a) Real and (b) imaginary parts of the frequency-dependent dielectric function, (c) real and imaginary parts of the refractive index, and (d) reflectivity for parallel (E||) and perpendicular (E) electric field polarization with respect to the tube axis. Four tubes are presented: the (5,0)-ZZ, (5,5) AC, (10,0)-ZZ, and (10,10)-AC CrI3 SWNT.
Nanomaterials 09 00153 g006
Table 1. Structural information of CrI3 NTs such as lattice parameters (L), the diameter (d), the calculated diameter (dcalculated), Cr–I bond length (dCr–I), the interlayer vertical distance between two iodine cylinders (δ), and energy gap (Eg).
Table 1. Structural information of CrI3 NTs such as lattice parameters (L), the diameter (d), the calculated diameter (dcalculated), Cr–I bond length (dCr–I), the interlayer vertical distance between two iodine cylinders (δ), and energy gap (Eg).
CompoundL
(Å)
d
(Å)
dcalculated
(Å)
dCr–I
(Å)
δ
(Å)
Eg
(eV)
(5,0)-ZZ11.82112.0110.872.712.830.435
(5,5)-AC6.82519.3018.822.682.930.695
(10,0)-ZZ11.82121.8121.732.682.950.809
(10,10)-AC6.82535.9337.652.692.960.427
Table 2. Calculated local magnetic moments on the Cr site (MCr), local magnetic moments on the I site (MI), total magnetic moments in a unit cell (Mcell), exchange energy (Eex), exchange magnetic coupling (J), and Curie temperature (Tc).
Table 2. Calculated local magnetic moments on the Cr site (MCr), local magnetic moments on the I site (MI), total magnetic moments in a unit cell (Mcell), exchange energy (Eex), exchange magnetic coupling (J), and Curie temperature (Tc).
CompoundMCr
B)
MI
B)
Mcell
B)
Eex
(×10−3 eV)
J
(×10−3 eV)
Tc
(K)
(5,0)-ZZ3.55−0.1860.36351.811.9533
(5,5)-AC3.47−0.1660.09675.813.7563
(10,0)-ZZ3.42−0.14119.661859.975.1786
(10,10)-AC3.390.14119.301044.062.9048

Share and Cite

MDPI and ACS Style

Moaied, M.; Hong, J. Size-Dependent Critical Temperature and Anomalous Optical Dispersion in Ferromagnetic CrI3 Nanotubes. Nanomaterials 2019, 9, 153. https://doi.org/10.3390/nano9020153

AMA Style

Moaied M, Hong J. Size-Dependent Critical Temperature and Anomalous Optical Dispersion in Ferromagnetic CrI3 Nanotubes. Nanomaterials. 2019; 9(2):153. https://doi.org/10.3390/nano9020153

Chicago/Turabian Style

Moaied, Mohammed, and Jisang Hong. 2019. "Size-Dependent Critical Temperature and Anomalous Optical Dispersion in Ferromagnetic CrI3 Nanotubes" Nanomaterials 9, no. 2: 153. https://doi.org/10.3390/nano9020153

APA Style

Moaied, M., & Hong, J. (2019). Size-Dependent Critical Temperature and Anomalous Optical Dispersion in Ferromagnetic CrI3 Nanotubes. Nanomaterials, 9(2), 153. https://doi.org/10.3390/nano9020153

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop