Strain engineering of Zeeman and Rashba effects in transition metal dichalcogenide nanotubes and their Janus variants: an ab initio study

We study the influence of mechanical deformations on the Zeeman and Rashba effects in transition metal dichalcogenide nanotubes and their Janus variants from first principles. In particular, we perform symmetry-adapted density functional theory simulations with spin–orbit coupling to determine the variation in the electronic band structure splittings with axial and torsional deformations. We find significant effects in molybdenum and tungsten nanotubes, for which the Zeeman splitting decreases with increase in strain, going to zero for large enough tensile/shear strains, while the Rashba splitting coefficient increases linearly with shear strain, while being zero for all tensile strains, a consequence of the inversion symmetry remaining unbroken. In addition, the Zeeman splitting is relatively unaffected by nanotube diameter, whereas the Rashba coefficient decreases with increase in diameter. Overall, mechanical deformations represent a powerful tool for spintronics in nanotubes.

Spintronics or spin electronics refers to the exploitation of both spin and the electronic charge in solid state devices 43 .In this context, the Zeeman and Rashba effects are of particular interest, both being relativistic effects arising from spin-orbit coupling (SOC).In particular, the Zeeman and Rashba effects result in splitting of the electronic bands along the energy and wavevector axes, respectively, of particular importance being those at the valence band maximum (VBM) and the conduction band minimum (CBM).These effects have been studied in TMD monolayers and their Janus variants not only experimentally [44][45][46] , but also theoretically using ab initio Kohn-Sham density functional theory (DFT) calculations 43,47,48 .
In addition, the effect of strain on the Zeeman and Rashba splittings has been studied in Janus TMD bilayers 47 and their heterostructures 49 using DFT.However, there have been no such studies for TMD and Janus TMD nanotubes (and nanotubes in general), which provides the motivation for the current investigation.
In this work, we study the influence of mechanical deformations on the Zeeman and Rashba effects in the synthesized TMD nanotubes and their Janus variants using Kohn-Sham DFT calculations.In particular, we perform symmetry-adapted DFT simulations with SOC to determine the variation in the Zeeman and Rashba splittings with axial and torsional deformations.We find significant splitting for the nanotubes having the transition metal as either molybdenum or tungsten.In particular, axial and torsional deformations can be used to vary the Zeeman splitting, while torsional deformations can be used to introduce and vary the Rashba splitting, making the nanotubes particularly well-suited for spintronics applications.
The remainder of this manuscript is organized as follows.In Section II, we list the standard and Janus TMD nanotubes studied and describe the symmetry-adapted Kohn-Sham DFT simulations for the calculation of the Zeeman and Rashba splittings.Next, we present and discuss the results of the simulations in Section III.Finally, we provide concluding remarks in Section IV.

II. SYSTEMS AND METHODS
We start by considering the TMD nanotubes that have been synthesized 1-3 : {MoS 2 , MoSe 2 , MoTe 2 , WS 2 , WSe 2 , WTe 2 , NbS 2 , NbSe 2 , TaS 2 , TiS 2 , TiSe 2 , HfS 2 , and ZrS 2 }.Since we have found that spin-orbit coupling (SOC) does not cause any splitting in the nanotubes: {NbS 2 , NbSe 2 , TaS 2 , TiS 2 , TiSe 2 , HfS 2 , and ZrS 2 }, we henceforth consider the remaining TMD nanotubes: {MoS 2 , MoSe 2 , MoTe 2 , WS 2 , WSe 2 , WTe 2 }, as well as their Janus variants with the heavier chalcogen on the outside: {MoSSe, MoSTe, MoSeTe, WSSe, WSTe, WSeTe}, all with 2H-t symmetry.We consider their armchair configurations, since the results remain unchanged for the zigzag configuration, in agreement with previous observations for SOC in the MoS 2 nanotube 50 .The diameters of the TMD nanotubes are chosen to be commensurate with those synthesized, and the diameters of the Janus TMD nanotubes are set to DFT-calculated equilibrium values (Table I).The axial and torsional deformations considered are also commensurate with those in experiments 19,33,[51][52][53] .Indeed, through phonon calculations using ABINIT 54 , we have verified that the monolayer counterparts are stable at the largest tensile/shear strains (Supplementary Material), which suggests the stability of the nanotubes at the chosen strains, as curvature effects on the phonon spectrum are expected to be minor at the relatively large diameters of the nanotubes.
Table I: Diameters of the TMD nanotubes 55,56 and their Janus variants 10,57 .We perform Kohn-Sham DFT simulations using the Cyclix-DFT 58 feature in the state-ofthe-art real-space code SPARC [59][60][61] .In particular, we perform symmetry-adapted calculations that exploit the cyclic and/or helical symmetry in the system to reduce the Kohn-Sham problem to the unit cell/fundamental domain with minimal number of atoms 58,62 , e.g., the fundamental domain for the chosen nanotubes contains only 3 atoms, i.e., 1 metal and 2 chalcogen atoms (Fig. 1).This reduction due to symmetry can be exploited even on the application of axial and/or torsional deformations, tremendously lowering the computational expense, given that DFT calculations scale cubically with system size, making otherwise impractical calculations routine, e.g., a 8.5 nm diameter MoSSe nanotube with an external twist of 6 × 10 -4 rad/bohr has 219, 888 atoms in the simulation domain when employing periodic boundary conditions, a system size that is impractical even with state-of-the-art approaches 63 .Cyclix-DFT is now a mature open source feature in SPARC, having been verified by comparisons with established DFT codes 58 , and ability to make accurate predictions in diverse physical applications 10, [55][56][57][64][65][66][67] .

Material
In all simulations, we employ the Perdew-Burke-Ernzerhof (PBE) 69   geometry of the nanotubes (Supplementary Material) is in very good agreement with previous DFT calculations 8,16,72-75 , and the equilibrium geometry of the corresponding monolayers (Supplementary Material) is in very good agreement with experiments [76][77][78][79] as well as DFT calculations 16,17,[80][81][82] , verifying the accuracy of the chosen pseudopotential and exchangecorrelation functional.Though more advanced and expensive exchange-correlation functionals such as hybrid generally provide better spectral properties, this is not always the case, e.g., Janus TMD monolayers 57 , motivating the choice of PBE exchange-correlation here, as done in previous works for such systems 8,72,74,75,81 .The numerical parameters in the Cyclix-DFT simulations, including grid spacing for real-space discretization, grid spacing for Brillouin zone integration, vacuum in the radial direction, and structural relaxation tolerances, are chosen such that the Zeeman splitting values and Rashba coefficients are converged to within 0.01 eV and 0.01 eV Å, respectively.This translates to an accuracy of 10 −4 Ha/atom in the ground state energy.

III. RESULTS AND DISCUSSION
We now use the aforedescribed framework to study the effect of axial and torsional deformations on the Zeeman and Rashba splittings in the molybdenum and tungsten TMD nanotubes and their Janus variants.In the results presented here, the axial strain (ε) is defined as the change in nanotube length divided by its original length; the shear strain (γ) is defined as the product of the nanotube radius with the applied twist per unit length; the Zeeman splitting (λ VBM ) corresponds to the valence band maximum (VBM), where the effect is significantly more pronounced (∼5x larger) than the conduction band minimum (CBM) (Supplementary Material); and the Rashba splitting coefficient (α) corresponds to the VBM, calculated at the zero wavevector in the axial direction.All the data can be found in the Supplementary Material.
In Fig. 2, we present the variation in the Zeeman splitting with tensile and shear strains.
We observe that the Zeeman splitting in the undeformed state is significant, being comparable to the monolayer counterparts 43 , with the WTe 2 and MoS 2 /MoSTe nanotubes having the largest and smallest values of λ VBM = 489 and ∼146 meV, respectively.In addition, the splitting decreases with increase in tensile strains, going to zero for large enough strains.A similar behavior is also observed for shear strains, other than for the MoSe 2 , WS 2 , WSSe, and WSTe nanotubes, where the splitting remains unaffected by the torsional deformations.Such a decrease in the Zeeman splitting values upon the application of biaxial strains has been observed for Janus TMD bilayers 47 .Note that the values for the Janus TMD nanotubes are generally in between their parent TMDs.Note also that the sudden jumps in the Zeeman splitting values are a consequence of the VBM location shifting to a different wavevector.
In Fig 3, we present the variation in the Rashba coefficient with shear strain.Unlike torsional deformations, axial deformations do not break the inversion symmetry of the nanotube, and therefore the Rashba effect remains absent 83 .We observe that the Rashba coefficient increases linearly with shear strain -average coefficient of determination of linear regression over all the materials is 0.97 -reaching significantly large values that are comparable to those for Janus TMD monolayers 47 , systems where we have found the Rashba effect to be insensitive to shear strains.Indeed, the Rashba effect is not observed in TMD monolayers and MoS 2 nanotubes, respectively, whose undeformed configurations also have the largest and smallest Zeeman splitting, respectively.However, this correlation is not generally true, e.g., MoTe 2 has one of the smallest Zeeman splitting of λ VBM = 213 meV for the undeformed nanotube, whereas it has one of the largest Rashba coefficient of α = 0.65 eV Å for the maximum shear strained tube (γ = 0.15).Note that the values for the Janus TMD nanotubes are generally in between their parent TMDs.
To understand the effect of the nanotube diameter on the results obtained, we now consider the nanotubes that demonstrate the largest Zeeman and Rashba effects, i.e., WSe 2 , WSeTe, and WTe 2 , with diameters spanning the range ∼ 2 − 10 nm.In Fig. 4, we present the variation in the Zeeman splitting and Rashba coefficient with the diameter, while considering the unstrained and largest shear strain (γ = 0.15) configurations, respectively.We observe that the Zeeman splitting values remain relatively unchanged, increasing ever so slightly with diameter -around 1% over the entire diameter range -approaching the flat sheet values of λ VBM = 463, 473, and 493 meV for WSe 2 , WSeTe, and WTe 2 , respectively 43 .
In addition, the Rashba coefficient decreases significantly with increase in diameter, e.g., the value for WTe 2 reduces from α = 0.83 eV Å at a diameter of 2 nm to α = 0.73 eV Å at a diameter of 9 nm, expectedly heading towards the zero value for the flat sheet configuration.
The results presented here clearly demonstrate that mechanical deformations can be used to engineer the Zeeman and Rashba splittings in molybdenum and tungsten TMD nanotubes as well as their Janus variants, making them a powerful tool for spintronics applications.In particular, the Zeeman effect is especially significant for the undeformed nanotubes, becoming progressively smaller and even disappearing with increase in axial/shear strains, and the Rashba effect can be introduced through torsional deformations -break the inversion symmetry of the system -becoming especially significant as the shear strain increases.

IV. CONCLUDING REMARKS
In this work, we have studied the strain engineering of Zeeman and Rashba effects in synthesized TMD nanotubes and their Janus variants using first principles DFT simulations.In particular, we have performed symmetry-adapted Kohn-Sham calculations with spin-orbit coupling to determine the effect of axial and torsional deformations on the Zeeman and Rashba splittings in the electronic band structure.We have found that there is significant splitting in the molybdenum and tungsten nanotubes, for which the Zeeman splitting decreases with increase in tensile/shear strain, reaching zero for large enough strains, while the Rashba splitting coefficient increases linearly with shear strain, while being zero for all axial deformations, a consequence of the inversion symmetry remaining unbroken.In addition, the Zeeman splitting is relatively unaffected by the nanotube diameter, whereas the Rashba coefficient decreases with increase in diameter.Though the current study has been restricted to TMD nanotubes and their Janus variants, other nanotubes are expected to demonstrate similar behavior, particularly those with heavy chemical elements.Overall, mechanical deformations represent a powerful tool for spintronics applications using nanotubes.

Figure 1 :
Figure 1: Left: Illustration of a Janus TMD nanotube subject to both axial and torsional deformations, with the three fundamental domain atoms colored blue, orange, and dark green (structural model generated using VESTA 68 ).Right: Illustration of the Zeeman and Rashba splittings in the electronic band structure, with λ VBM and λ CBM denoting the Zeeman split at the valence band maximum (VBM) and conduction band minimum (CBM), respectively, and α representing the Rashba splitting coefficient.

Figure 2 :
Figure 2: Variation in the Zeeman splitting (λ VBM ) for the molybdenum and tungsten TMD nanotubes and their Janus variants with (a) tensile strain (ε) and (b) shear strain (γ).The tensile and shear strains result from the application of axial and torsional deformations, respectively.
Figure 3: Variation in the Rashba splitting coefficient (α) for the molybdenum and tungsten TMD nanotubes and their Janus variants with shear strain (γ).The shear strain results from the application of torsional deformations.