Helical Nanotube Structures of MoS2 with Intrinsic Twisting: An ObjectiveMolecular Dynamics Study

D.-B. Zhang, T. Dumitrica∗Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA

G. SeifertPhysikalische Chemie, Technische Universitaet Dresden, D-01062 Dresden, Germany

(Dated: January 2, 2010)

Objective molecular dynamics combined with density-functional-based tight-binding makes pos-sible to compute chiral nanotubes as axial-screw dislocations. This enables the surprising revelationof a large catalog of MoS2 nanotubes that lack the prescribed translational symmetry and exhibitchirality-dependent electronic band-gaps and elastic constants. Helical symmetry emerges as thenatural property to rely on when studying quasi-one dimensional nanomaterials formally derived orgrown via screw dislocations.

PACS numbers: 61.46.Np, 62.20.de

Due to the enormous interest in modulating propertiesat the nansocale, understanding the fundamental inter-play between the ionic and electronic degrees of freedomin chiral nanostructures is of great importance. Many ofthe extraordinary properties of single-wall carbon nan-otubes (CNTs) [1] originate in their intrinsic symmetry.Unfortunately, little is known about other chiral NTswith more complex walls. On the theoretical side, this ispartially due to the inconvenience of handling chiralitywithin the standard treatments of crystalline solids.

A CNT is usually conceptualized by the pure rollingof a graphene ribbon into a seamless cylinder. Duringthis process the circumference vector C0 is rolled into acircle while the translation vector T0 stays straight [2].The expectation is that the translational symmetry iswell preserved even though the bond lengths and anglesbetween atoms will slightly change upon rolling due tofinite curvature effects. Next, microscopic treatmentsformulated under periodic boundary conditions (PBC)are applied in order to determine the optimal periodicityand the atomic locations inside one PBC cell. Symmetry-constrained microscopic calculations confirmed [3–6] thatroll-up gives good predictions for the CNTs’ structuralparameters. This also indicates that the linear elas-tic response is largely maintained upon rolling. Dueto the isotropy of graphene, the elastic constants ofCNTs should not dependent on chirality. This is whatmicroscopic quantum-mechanical calculations have con-sistently found [5–8], sometimes at odds with classicalpotential-based studies proposing non-negligible chiral-ity dependences.

The makeup of a chiral CNT was originally depictedby Iijima [1] in a different way, namely by rolling agraphene ribbon followed by gliding of the edges pastone another, essentially via an axial screw dislocation.Microscopic modeling based on this description was onlyrecently achieved due to the recent development of ob-jective molecular dynamics (MD) [9, 10]. Our method

allows computing chiral NT structures in a nanomechan-ical way [4], as the result of the Eshelby’s twist [11] in-troduced by the axial screw dislocation.

New findings often come by using new investigationmethods. While the two (PBC and objective) MD ap-proaches are intuitively expected to give identical results,here we show that for a large catalog of chiral molybde-num disulphide (MoS2) NTs, only the new treatment isapplicable. Below ∼ 7 nm in diameter, rolling-up theMoS2 layer becomes nonlinear. The twist introduced bythe axial screw dislocation breaks the prescribed transla-tion, giving rise to a NT for which standard PBC treat-ments are no longer suited. Fortunately, the helical sym-metry introduced by the Eshelby’s twist combined withobjective MD provide a general framework to performmicroscopic studies. Using this treatment, we demon-strate electronic band-gap variations with chirality, andanisotropy in the elastic constant of chiral MoS2 NTs.

Synthesized not long ago [12], MoS2 NTs are inter-esting structures to understand chirality. They are cur-rently investigated experimentally, especially in relationwith their mechanical properties [13], and are very welldescribed with atomistic approaches [14]. The MoS2 NTwall is less stiff than graphene although still isotropic inthe linear-elastic regime. It has a rippled hexagonal ap-pearance, Fig. 1(a), consisting of a triangular Mo layerlocated between two triangular S layers, such that eachMo atom bonds with six S atoms. To generate the flatsheet, the primitive cell delineated by vectors a1 and a2

connecting neighboring Mo atoms and the positions ofthe MoS2 molecule are needed.

Fig. 1(a) (left) shows a (n, n) NT formed by rolling-up a MoS2 ribbon with zig-zag Mo and S edges. Theinfinitely long tubule can be mathematically generatedby repeated axial translations with Burgers vector b3 =a1−a2 applied to the atoms located in the translationalring below. Fig. 1(a)(right) shows a (n + 1, n − 1) NTobtained by rolling-up the same ribbon, followed by an

2

FIG. 1: (Color online) (a) Objective computational cells (be-low) for (10,10) (left) and (11,9) (right) MoS2 NTs. The NTstructures (above) were obtained by the helical replication ofthe objective cells along vector b3. (b) Strain energy vs. θfor the (10,10) ... (20,0) NT family with nearly-equal diam-eters. The energy minima correspond to the stress-free NTsof indexes indicated under each curve.

additional axial glide with b3. This glide creates an axialscrew dislocation with respect to the armchair NT, withBurgers vector ib3, where i is an integer. The glide in-duces a torque that produces an atomic-scale Eshelby’stwist, and formation of a (n+ i,n− i) chiral pattern [4].

In our objective MD, we describe a chiral (n+ i, n− i)NT from the sameN atoms belonging to the translationalcell of the (n, n) MoS2 NT. Let Xj be their atomic po-sitions after axial glide with ib3. We call this collectionof N atoms objective cell. Positions Xj,ζ of the atomslocated in the objective cell replica indexed by integer ζare obtained with

Xj,ζ = RζXj + ζb, j = 1, ..., N. (1)

Axial vector b combined with rotational matrix R ofangle θ characterize the helical transformation. Thesestructural parameters are analogous to the lattice vec-tors in PBC MD.

Special consideration was needed [10] to accommodatethe electronic subsystem to eq. (1). The one-electronstates were represented in terms of symmetry-adaptedBloch sums

|αj, κ〉 =1√Ns

Ns−1∑ζ=0

eiκζ |αj, ζ〉. (2)

Here, Ns is the number of helical operations (typically∞) over which the cyclic boundary conditions are im-posed. The Bloch factors are eigenvalues of the helicaloperators and −π ≤ κ < π is the helical quantum num-ber. |αj, ζ〉 is the orbital with symmetry α located on

FIG. 2: Scaling of the intrinsic (a) axial (ε), (b) radial (ε∗)strains with NT diameter. (c) Torsional shear strain and (d)axial strain (both multiplied by R1.55) for the chiral MoS2

NTs, as obtained from the DFTB model.

atom j, all in the objective cell indexed by ζ orientedsuch as the invariance (1) is satisfied. The valence shellbasis set used here comprises sp and spd basis functionsfor S and Mo, respectively. To describe the interatomicinteractions we rely on density functional theory-basednon-orthogonal two-centre parameterizations [14] imple-mented in the code Trocadero [15]. The linear combina-tions of atomic orbitals were sampled for 5 κ values ofthe helical Bloch phase and the forces on each N atomswere computed from the band energy. In analogy withPBC MD, the objective MD solution represents the map-ping with (1) of the time-dependent displacements of theN atoms onto the full NT structure. This way [9], theunconstrained set of coupled differential equations of themolecular dynamics for the full structure is satisfied un-der initial conditions invariant to the group operationsassociated with eq. (1).

In this theoretical framework we have performed calcu-lations on five families (n = 10, 14, 16, 18) generated byintroducing axial screw dislocations in (n,n) NT struc-tures. Additionally, a large collection of armchair andzig-zag NTs of 1 − 9 nm diameter was considered. Theinitial structural information for a (n,m) NT is adoptedfrom the rolled MoS2 layer, for which the free parame-ters of eq. (1) can be obtained with the simple expres-sions |b0| =

√3a(n + m)/2

√n2 + nm+m2 and θ0 =

π(n−m)/(n2 + nm+m2). Parameter a = 3.27 A is thelength of the primitive vectors of the flat layer. Addition-ally, the radius of this tubule writes [9] R0 = |C0|/2π =

3

a√n2 + nm+m2/2π. The stress-free atomic positions

and the Eshelby’s twist parameters (|bE | and θE) areidentified by applying an objective modeling procedure,objective MD within conjugate gradient minimization ofthe total potential energy.

Most unusually, we find that for a large catalog of NTsthe structural parameter predictions given by the roll-upare not adequate. This is exemplified in Fig. 1(b) forthe (10, 10)...(20, 0) NT family. Indeed, axial relaxationunder fixed angle θ0 only is not sufficient, and further an-gular relaxation in the vicinity of relaxed |bE | lowers theenergy. The angle values of the stress-free chiral struc-tures θE deviate from the predicted θ0 indicated by downarrows. Thus, the chiral NTs possess an intrinsic twistwith respect to roll-up configuration.

More insight into the structural parameters of stress-free NTs is obtained by analyzing the variations shownin Fig. 2(a) and (b). The axial prestrain ε = (|bE | −|b0|)/|b0| and the radial prestrain ε∗ = (R−R0)/R0 areessentially zero in the large diameter region. No intrin-sic twist was found at these diameters and we concludethat pure roll-up is achieved for diameters larger than∼ 7 nm. At lower diameters however, the behavior is non-linear. The rolling of the MoS2 layer couples both withthe in-plane strain and the rolling curvature. Fig. 2(a)shows that while armchair NTs slightly elongate with in-creasing curvature, the zig-zag ones shrink. As a result,in chiral NTs the axial prestrain values are spread be-tween these two curves. Concurrently, NTs undergo adecompression with respect to the rolled-up configura-tion, Fig. 2(b), with practically no chirality dependence.

For a compact characterization, we combined simplesymmetry arguments with the extensive numerical com-putations and derived simple functional forms for allprestrains. Ignoring the small chirality effect, the ra-dial prestrain can be described by a simple fitting ofthe atomistic data, as ε∗ = 5.4(R/A)−1.55. We fur-ther assume the same radial scaling for the other pre-strains. At constant R, the developed anisotropy be-tween special armchair and zig-zag directions implies thatboth ε∗ and the shear prestrain γ = (θE − θ0)R/|bE |,must have a 60o period in their chirality angle depen-dence. Fig. 2(c) and (d) both show a nearly lineardependence of γ and ε (both R1.55 augmented) to thesymmetry-allowed lowest-order in chiral angle χ. Thus,to a good approximation γ = 0.52(R/A)−1.55sin6χ andε = −(R/A)−1.55(0.18 + 0.24cos6χ).

To summarize, a (n,m) NT obtained by pure rolling hastranslational symmetry, with T0 =

√3|C0|/d, where d is

the greatest common divisor of 2n + m and 2m + n [2].However, the nonlinear roll-up effect introduces an in-trinsic twist. For example, a (14, 6) MoS2 NT locksa 0.87 deg/nm twist rate. How can we describe theatomic order in the chiral NTs without the translationproperty? Our comprehensive investigation of the fullyrelaxed structures found that the helical invariance is

FIG. 3: (a) Band structure for a (19,1) MoS2 NT. The Fermilevel, located at zero, is marked by a horizontal line. Thehorizontal axis represents the helical quantum numbers. (b)Calculated band-gap energies as a function of chirality.

present to the extent that the NT structures can be de-scribed by applying successive commuting helical trans-formations to the MoS2 molecule, as

Xj,(ζ,ζ′) = R′ζ(RζXj + ζb) + ζ ′T, j = 1, 2, 3. (3)

Of course, the structural parameters for the old helicaloperator are |bE | and θE . The new helical transforma-tion is characterized by a translation T = T0(1+ε) and anaxial rotation matrix R′ of angle θ′ = γT . Note that, ifγ is vanishingly small, eq. (3) becomes the translational-helical representation [2]. The helical polymers [6] gen-erated with (3), regardless of the positions of the atomsin the molecule, belong to the larger class of recently de-fined objective molecular structures [16]. Informally, inan objective molecular structure corresponding atoms indifferent molecules see exactly the same environment upto orthogonal transformation (translation and rotation).

The helical symmetry introduced by the Eshelby’stwist combined with objective MD is a suitable frame-work to study the chiral MoS2 NTs. For example, inPBC calculations the common way to analyze the elec-tronic energy is by plotting the electronic bands as afunction of the linear momentum. Analogously, in objec-tive calculations electronic states are plotted as a func-tion of the helical quantum number k, as exemplified inFig. 3(a) for the tight-binding states of a (19, 1) NT. Ouranalysis found that all computed chiral NTs have indirectband-gaps. Within the same family, band-gap decreasessmoothly with increasing χ, Fig. 3(b). The chirality de-pendence is pronounced at smaller diameters.

If the structural parameters |bE | and θE are variedaround their optimized values, the NT’s property of beingan objective molecular structure is retained but helicalstrain states are introduced [6]. Using this strategy westudied pure tensile and torsional deformations [3] withinthe adiabatic approximation, i.e. when forces on atomsare derived from the electronic ground state at each strainconfiguration. Evaluation of the elastic constants was

4

FIG. 4: (a) Young’s and (b) shear modulus vs. NT diameter.The wall thickness was taken to be 6.15 A.

then accomplished with simple second-order polynomialfits to the ground-state energy’s dependence on strain.

When the nonlinear elasticity under rolling-up arises,this should alter the isotropic elastic behavior of theMoS2 NTs. Indeed, for diameters under ∼ 7 nm there is adiscernible diameter- and chirality-dependent anisotropyin the elastic response, as can be seen in the obtainedYoung’s Y and shear G elastic constants displayed inFig. 4. Microscopically, this originates in the signif-icant distortion of the hexagonal lattice symmetry, asquantified by the ε, ε∗, and γ prestrains. The similar-ity between the elastic quantities of NTs and the flatMoS2 layer holds only at diameters larger than ∼ 7 nm,consistent with the prestrain analysis. In this region,typical values Y = 230 GPa, G = 88 GPa, and Pois-son ratio ν = 0.3 (obtained from the isotropic elas-ticity relation G = Y/2(1 + ν)), and bending rigidityD = 34.4 eV/molecule (obtained from the strain energyof NTs) compare favorably with experiment [17], demon-strating the reliability of the employed treatment.

In conclusion, adopting the screw-dislocation NT con-struction view and the recent theoretical advances, wereveal a large collection of MoS2 NTs that lack the stan-dard translational symmetry. Thus, the usual roll-upconstruction needs to be amended: while the circumfer-ence vector is being rolled, the translation vector shouldturn into a helix. Deriving the parameters for this helixis complex and requires both the ability to describe thenonlinear microscopic response of the MoS2 to rolling andan accurate description of the interatomic interactions.It is likely that many other NTs have a locked twist,including very small diameter CNTs. The methodologyintroduced here, objective MD relying on the helical sym-metry of the Eshelby’s twist, is general and can predictstructure, properties and dynamical behavior for other

chiral nanostructures, including lead sulphide [18] andlead selenide [19] nanowires grown via a screw dislocationmechanism, polymers and certain biomolecules. Here weshowed that helical MoS2 NTs with intrinsic twist have χ-dependent fundamental band-gaps and elastic responses.Although of practical interest [13], the elastic torsionalresponse of MoS2 NTs was not studied extensively untilnow due to the incompatibility of PBC with this type ofdeformation. The knowledge of the band-gap and elastic-constant variations with χ opens the possibility to designnew NT-pedal [13] electromechanical devices and exper-iments using MoS2 NT components.

We thank R.D. James and M.S. Dresselhaus for valu-able discussions. D.-B.Z. and T.D. thank NSF CA-REER Grant No. CMMI-0747684 and AFOSR GrantNo. FA9550-09-1-0339. G.S acknowledges support fromERC INTIF226639.

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