Chirality dependent spin polarization of carbon nanotubes

The spin polarization of carbon nanotubes (CNTs) offers a tunable building block for spintronic devices and is also crucial for realizing carbon-based electronics. However, the effect of chiral CNTs is still unclear. In this paper, we use the density functional theory (DFT) method to investigate the spin polarization of a series of typical finite-length chiral CNTs (9, m). The results show that the spin density of chiral CNTs (9, m) decreases gradually with the increase in m and vanishes altogether when m is larger than or equal to 6. The armchair edge units on both ends of the (9, m) CNTs exhibit a clear inhibition of spin polarization, allowing control of the spin density of (9, m) CNTs by adjusting the number of armchair edge units on the tube end. Furthermore, analysis of the orbitals shows that the spin of the ground state for (9, m) CNTs mainly comes from the contributions of the frontier molecular orbitals (MOs), and the energy gap decreases gradually with the spin density for chiral CNTs. Our work further develops the study of the spin polarization of CNTs and provides a strategy for controlling the spin polarization of functional molecular devices through chiral vector adjustment.


Introduction
Low-dimensional carbon materials show great potential in realizing spin polarized injectors [1,2], spin transport [3,4], spin valves [5,6], and similar molecular devices due to their capabilities to present of spin polarized electronic structures. Studies show that for the ground state of finite-length zigzag edge carbon nanotubes (CNTs), the spin polarization is ferromagnetically coupled among the atoms at one end and antiferromagnetically coupled between the two zigzag edges [7][8][9], similar to the coupling seen in some graphene fragments [10,11]. Further research on defects in low-dimensional carbon materials shows that vacancies and doping at the zigzag edges suppresses their spin polarization [11]. It is clear that the magnetism involved in these carbon-based materials arises mainly from the spin polarizations of local structural defects [12][13][14][15][16][17][18] rather than perfect structures. In particular, the local defect structure dependence of the spin polarization can be seen clearly in the different characteristics of CNTs with pure zigzag or pure armchair edges. The former demonstrates considerable spin polarization, as shown in both theoretical and experimental studies [7,[19][20][21][22][23], while the latter shows none [10]. A recent study of graphene nanoflakes (GNFs) shows that eliminating the armchairs region between the zigzag edges significantly improves their spin-polarized properties, such as magnetic coupling [24].
As both zigzag and armchair edges are present in chiral CNTs (n, m) (n≠m≠0) at each end, side by side [25,26], their spin polarization can be very complicated. It is hoped that it can be controlled by tuning the values of the chiral vectors m and/or n. The study of such tunable spin polarization will be significant in molecular device applications, but unfortunately no work has yet been done on this. In related work, a study of graphene nanoribbons (GNRs) has identified spin-polarized phenomena for at least three consecutive zigzag units at the edge [11], highlighting the importance of studying the spin polarization of chiral CNTs when the chiral vector (n, m) changes. Previous studies of nanotubes show that changes of edge have a significant impact on their electronic properties [27], such as becoming metals or semiconductors depending on the chiral index (n, m) [28,29]. Moreover, the thermal conductivity of a CNT is dependent on its chirality [30]. These characteristics are a further reason to consider engineering the spin polarization through changing (n, m) and thus the chirality of CNTs.
(9, 0) CNTs are typical structures for studying CNTs, which can be synthesized experimentally [31,32] and are also widely studied theoretically [7]. Meanwhile, (9, m) tubes offer sufficient variances of chiral vectors with acceptable calculation cost, providing effective models to study the chirality dependent properties of carbon nanotubes. In this paper, using the density functional theory (DFT) method, we investigate for the first time in detail the spin polarization of chiral CNTs (9, m, L) with m varying from 1 to 8. The typical (9, 0), (9,9) CNTs are also used as references. For this purpose, we will analyze their spin electronic structures and geometric stability. We expect this work to be helpful in engineering the spin polarization by changing the chiral vectors of CNTs.

Methods
DFT [33][34][35] methods have been shown to provide reliable results with moderate computational costs and thus are widely accepted as a suitable tool for investigating the characteristics of geometric and electronic structures [11,20,[36][37][38][39]. DFT methods are able to deal not only with the spin polarizations resulting from defects and the unsaturated edge structures of carbon material systems [10,11,20,40,41], but can also describe the interactions between the sp valence electrons of carbon materials and the d or f shell electrons of other elements, including the magnetism arising from them [42][43][44].
In this work, we use the screened exchange hybrid density functional HSE06 proposed by Heyd et al [45,46]. This method gives more reliable maximum Mulliken spin density values than those obtained using the local spin density approximation (LSDA) or the semi-local gradient corrected functional of Perdew, Burke and Ernzerhof (PBE) [10]. It also generates band gaps that are in good agreement with the experimental data and hence more suitable for dealing with systems involving electronic localization [37]. To confirm the reliability of the results, we study CNTs with four lengths (L) for the different chiral indices (9, m, L) (L=2, 3, 4, and 5 denotes the number of carbon rings, and m=0-9) and obtain consistent results across all cases. We performed all calculations using the 6-31G ** basis set with the Gaussian 09 program [47] and analyze the electron density difference and other data using Multiwfn [48]. We examine the obtained structures by calculating the frequencies at the same level of theory and then analyze their electronic structures.

Geometrical structures
The geometrical structures of the spin-polarized chiral CNTs of (9, m) with m=1-5 and zigzag CNTs studied in this paper are shown in figure 1. Other structures with m=6-9 can be seen in the supporting information (SI) in figure S1. To find the ground state of each structure, we compare the relative energies of the different multiplicities, which can be found in SI figure S2. The chiral CNTs with m=1-5 for four lengths obtained show that the ground states are open shell singlets, and that the ground states of those chiral CNTs with m=6-8 and the armchair CNTs with m=9 are nonmagnetic. For these chiral CNTs in open shell singlet ground states, the spin polarization is antiferromagnetically coupled between both ends. This result is similar to that obtained for zigzag CNTs and GNRs [7,10]. The spin density distributions of the ground state can be found in figure S3 of the SI. In order to investigate the dependence of spin polarization on chirality, we also compare their spin density distributions with respect to the different chiral vectors. The characteristics of these distributions at the tube ends for the chiral CNTs are shown in figure 2.

Spin polarization
Comparing the results for the (9, 1) and (9, 2) CNTs in figure 2, it can be seen that the spin density of the former is larger. The analysis of (9, 3), (9, 4), and (9,5) shows that the spin density of the chiral CNTs gradually decreases as m increases. For (9, 1) CNTs with one armchair edge unit, the spin density of the carbon atoms gradually increases and then gradually decreases along the direction of rotation. The location closer to the carbon atom on the armchair edge, which demonstrates net spin-down electrons, involves fewer net spin-up electrons on the zigzag edge atom. The result for (9, 2) CNTs is similar to that of (9, 1) which has the largest spin density for the carbon atom between the armchair edge units, such as the spin density of the four carbon atoms on the zigzag edges between the armchair edge units is bigger than for the three carbon atoms. This location preference for spin distribution is also found in CNTs with other chiral vectors. When m is larger than or equal to 6, the spin of  the carbon atoms localized at the ends vanishes. These results show that armchair edge atoms can inhibit spin polarization at the ends, which is consistent with the findings for graphene nanobelts [24,49]. We reach the same conclusion for these four lengths of chiral nanotubes.
Based on the optimized structures, we also calculate the electron density distribution and spin density along the tube. We obtain the local integral curves of the electron density distribution along the tube direction according to the equation where ρ (x, y, z) is the total electron density. Interestingly, we find that the armchair edge atoms not only suppress the spin density at the ends but also inhibit the spin density and change the spin distribution along the tube axis. The electron density distribution of (9, m, 4) shows clear change as m increases, as shown in figure 3. In terms of the variation of electron density, the bigger the value of m, the closer the peaks. This is because the carbon atoms of CNTs are gradually dispersed along the tube axis direction with the change of chiral vector. It can also be seen that the average electron density for α and β gradually increases with m, owing to the higher number of carbon atoms. The result of the electron density difference between α and β, which is also termed the spin density along the tube axis (denoted by a black line), shows that the spin density is larger at both ends and smaller in the middle. However, with the increase in the number of armchair edge unit, the spin density decreases all along the tube and the spin distribution is different from that found in (9, 0) CNTs. These results show that a change in the chiral vectors can be used to control the spin distribution and spin density along the tube axis by adjusting the locations of the anti-spin carbon atoms.

Electronic structures
In order to evaluate which orbitals are contributing to the spin density, we analyze the frontier molecular orbitals (MOs) of the CNTs as shown in figure 4. It can be seen that the highest occupied MO (HOMO) for the α electron (denoted as HOMO-α) and for the β electron (HOMO-β) mainly occupy both ends of the nanotubes and are complementary to one another in space. It can also be seen that the morphology of HOMO-α is similar to that of the lowest unoccupied molecular orbital (LUMO) for the β electron, and LUMO-α is similar to HOMO-β. From the local integral curve of the spin density (denoted by the black line on figure 3), we know that the spin density is at its maximum at both ends of the nanotubes. Therefore, their spin density originates mainly from the contributions of the frontier MOs. Then, we investigate the HOMO-LUMO gap for the nanotubes and obtain the maximum value of about 1.86 eV for the (9, 0) nanotube and the smaller value of about 1.41 eV for the (9, 1) nanotube. The HOMO-LUMO gaps for (9, 2), (9, 3), (9, 4), (9, 5), (9, 6), (9, 7), (9,8), and (9, 9) are 1.28 eV, 1.06 eV, 0.97 eV, 0.69 eV, 1.38 eV, 1.18 eV, 1.61 eV, and 1.63 eV, respectively. These results show that the HOMO-LUMO gap decreases gradually for the chiral vector pointing from (9, 0) to (9,5), but the HOMO-LUMO gaps for (9, 6) to (9,9) do not show a clear trend. The above study shows that HOMO-LUMO gap and spin density follows the same trend respect to different chiral vectors. They both decrease from (9, 0) to (9,5), and when the spin density vanishes from (9, 6) to (9,9), the gap also lose its variance monotonicity. This suggests the spin density could directly characterize the HOMO-LUMO gap of finite nanoscale chiral CNTs. Spin density reasonably reflects the change in electronic structure which is the actual reason of the chirality dependency of HOMO-LUMO gap in these systems.

Raman activity
To facilitate comparison with possible experimental observations, we analyze the Raman activity of the ground state and closed shell chiral CNTs (9, m, 4). The results are shown in figure 5. For the infrared (IR) spectra, the vibration modes of the highest peak come mainly from hydrogen atoms, and thus are not discussed here. Among the three typical modes for CNTs (the breathing, tangential, and radial modes), the tangential and radial modes are seriously distorted with the change in the chiral vectors (9, m) and thus are difficult to determine. In figure 5, the blue arrows indicate the breathing mode for the ground state, and the pink arrows represent the breathing mode for the closed shell of the CNTs. The breathing mode for the ground state of (9, 0, 4) has a frequency of 360.96 cm −1 , and 361.40 cm −1 for the closed shell. For the (9, 1, 4) CNTs, the frequencies of the breathing mode for the ground state and closed shell are 346.86 cm −1 and 347.06 cm −1 , respectively. The frequencies at 321.33 cm −1 and 321.58 cm −1 denote the breathing modes for the ground state and closed shell of (9, 2, 4). For the (9, 3, 4) CNTs, the breathing modes of the ground state and closed shell are 312.59 cm −1 and 312.73 cm −1 , respectively. The frequencies of the breathing modes for the ground state and closed shell are 306.49 cm −1 and 307.16 cm −1 for the (9,4,4) CNTs. For the (9,5,4) CNTs, the breathing modes of the ground state and closed shell are both found at 302.20 cm −1 . These results show that the breathing modes of the CNTs red shift successively compared with the (9, 0) CNTs by an order of magnitude of 10 0 -10 1 cm −1 . Furthermore, we compare the ground state and closed shell and show that the breathing modes of the closed shell blue shift slightly in an order of magnitude 10 −1 -10 0 cm −1 .

Summary and conclusions
In this work, we calculate the electronic structures and related properties of chiral CNTs (9, m) using the firstprinciples DFT method. Our results are the first to show that the spin density of chiral CNTs decreases gradually as the armchair edge unit increases, and the spin vanishes when the number of armchair edge units is larger than or equal to 6. The armchair edge atom suppresses the spin density of CNTs, which is similar to the situation for GNFs [24]. The ground state of these chiral CNTs is a spin antiferromagnetically coupled singlet state when m is 1-5, which is similar to the result for zigzag nanotubes. For other chiral CNTs with m of 6-8, the ground state is nonmagnetic. The result of the local integral curve for the α and β electron densities indicates that they increase gradually on average. Furthermore, the analysis of MOs reveals that the spin density of chiral CNTs originates mainly from the contributions of the frontier MOs. The change of spin density well characterizes the variance of HOMO-LUMO gap respect to chirality, connecting the response of electronic-structure and the change of geometric chiral vector. Finally, an analysis of the Raman activity shows that the breathing modes of the ground state of the chiral CNTs red shift successively compared with the (9, 0) CNTs by an order of magnitude of 10 0 -10 1 cm −1 , and the breathing modes of the closed shell states do not shift, or blue shift slightly, in the order of magnitude 10 −1 cm −1 . The results of the series of chiral CNTs we have investigated in this work show that we can control the spin polarization by changing the chiral vectors. This work may be helpful in molecular device design. Besides, the spin transport in nanoscale carbon systems is highly focused, with predictions and even some observations of quantum Hall effect [50,51], spin Hall effect [52], chiral spin state and spin current [53] in graphene-based systems have been reported. Here, through the change of spin polarization, we actually demonstrated, in real space, the evolution of spin polarized electronic structure of CNTs respect to chirality, especially that of spin polarized edge state which is the key of chiral spin current in CNTs under quantum Hall effect [54]. We hope our results could also contribute to further understanding the fundamental properties and novel applications of corresponding nanoscale systems.