NiS ultrafine nanorod with translational and rotational symmetry

ABSTRACT Anisotropy is a significant and prevalent characteristic of materials, conferring orientation-dependent properties, meaning that the creation of original symmetry enables key functionality that is not found in nature. Even with the advancements in atomic machining, synthesis of separated symmetry in different directions within a single structure remains an extraordinary challenge. Here, we successfully fabricate NiS ultrafine nanorods with separated symmetry along two directions. The atomic structure of the nanorod exhibits rotational symmetry in the radial direction, while its axial direction is characterized by divergent translational symmetry, surpassing the conventional crystalline structures known to date. It does not fit the traditional description of the space group and the point group in three dimensions, so we define it as a new structure in which translational symmetry and rotational symmetry are separated. Further corroborating the atomic symmetric separation in the electronic structure, we observed the combination of stripe and vortex magnetic domains in a single nanorod with different directions, in accordance with the atomic structure. The manipulation of nanostructure at the atomic level introduces a novel approach to regulate new properties finely, leading to the proposal of new nanotechnology mechanisms.


Materials
Nickel acetylacetonate (Ni(acac) 2 ), Diphenyl oxide (C 12 H 10 O) were purchased from J&K Chemicals.1-Dodecanethiol (C 12 H 26 S) was purchased from Alfa Aesar.Ethanol(C 2 H 5 OH) and n-hexane(C 6 H 14 ) were purchased from Beijing Chemical Works.All chemical reagents used in this experiment were analytical grade and used without further purification.

Characterization
Transmission electron microscopy (TEM) images were carried out on JEOL JEM-2100 microscopes.X-ray powder diffraction (XRD) patterns were obtained by a Shimadzu LabX-6000 X-ray diffractometer and the Data were collected in Bragg-Brentano mode with a scan rate of 1° min -1 .LAADF STEM imaging experiments were carried out using a 200 kV STEM with a sub-Å resolution (ARM200FC, JEOL) operated at 200 kV.For STEM observations, we adopted a probe size of approximately 0.78 Å, a probe convergence angle of approximately 25 mrad, and collection semi-angles for LAADF imaging of 30-120 mrad.

Synthesis of NiS Ultrafine Nanorod
In a typical synthesis, 0.0321 g (0.125 mmol) Nickel acetylacetonate was dissolved into 5 mL diphenyl oxide at room temperature in a three-neck flask.After ultrasonicated for 5 min, the flask was transferred into a water-bath at 40℃ and kept for 10 min.Then 5 mL 1-Dodecanethiol was added dropwise into the above solution and maintained heating 40℃ for 30 min.Subsequently, the flask was heated to 220℃ for 1 h under magnetic stirring.After that, the mixture was centrifugated at 13000 rpm for 5 min and washed with ethanol for the first time and n-hexane for following two times.

TEM electron holography
The TEM electron holography assists in revealing the magnetic moment orientation of our sample, based on the division of the two beams from the field emission electron gun by the interference of the prism.One beam is propagated into the vacuum, providing the information of the reference, while the other beam is transmitted through the sample.The information encoded inside the sample can be exfoliated by the extraction of the vacuum image from the sample overlapping and vacuum part to cultivate the magnetism electron holography images with the pure magnetism information in the sample based on the equation of: In which the Φ  indicates the magnetic flux inside the sample overlapping area with the vacuum part, surface integrated by the magnetism.

Computational method
The calculations were performed using first-principles density functional theory (DFT), as implemented in the Vienna Ab Initio Simulation Package (VASP) (1,2).The general gradient approximation of Perdew-Burke-Ernzerhof (GGA-PBE) functional was used to describe the exchange-correlation (3).The DFT+U was employed with the value of 6.2 eV for the 3d orbitals of Ni.A plane wave cutoff of 500 eV was applied to expand the electron wave functions.The vacuum layers were set to 10 Å to avoid periodic interaction.The supercell with a = 30.00Å, b = 9.46 Å and c = 30.00Å was employed for NiS nanorods, and the reciprocal space was sampled using a 1 × 2 × 1 mesh grid by using Monkhorst-Pack k-points scheme (4).The structures were relaxed until the variation of the total energy was smaller than 10 -6 eV and all force on each atom was less than 0.01 eV/Å.The Becke-Johnson damping function (5,6) was used to describe the van der Waals (vdW) interaction.For the structural optimization of NiS, linear magnetism was uniformly adopted in our calculations.Furthermore, the most stable electronic structure of the NiS nanorod configuration was determined by considering the different initial noncollinear magnetisms.
To explore the atomic structures of the large NiS nanorods for 4 or 5 layers, we employed the CP2K/Quickstep package, which is more efficient to simulate the large supercell (7).The similar parameters and criteria were employed in these two packages.The exchange correlation energy was carried out within the GGA-PBE (8).The norm-conserving Goedecker-Teter-Hutter (GTH) pseudopotentials was used to describe the core electrons (9).The correlation energy (U) for the 3d orbitals of Ni also set as 6.2 eV.The NiS nanorod was modeled using a cell with dimensions of 30.00 Å along a axis, 19.50 Å along b axis and 45.00 Å along c axis.Gaussian functions with molecularly optimized double-zeta polarized basis sets (MOLOPT-DZVP) were adopted for expanding the wave function of Ni 3p 6 4s 2 3d 8 and S 3s 2 3p 4 electrons (10).

Fig. S2 .
Fig. S2.The interlayer spacing of the lamellar structure form side view of NiS ultrafine nanorod is measured to be ~0.48nm.

Fig. S3 .
Fig. S3.ADF-STEM image of the precursors collected when the temperature was just raised to

Fig. S4 .
Fig. S4.ADF-STEM image and the corresponding electron energy loss spectroscopy (EELS) mapping images of NiS nanorod.

Fig. S5 .
Fig. S5.(a) The corresponding FFT pattern of Fig. 1f, the thicker NiS nanorod in radial direction.(b) The corresponding FFT pattern of the inner part of the thicker NiS nanorod in radial direction.

Fig. S6 .
Fig. S6.The interlayer spacing of the lamellar structure form side view of the thicker NiS nanorod is measured to be ~0.46 nm.