Graphene Helicoids: Mechanics, Thermal & Electronic Effects

• Graphene helicoids are graphene structures arranged in a helical topology that yield enhanced mechanical flexibility, tunable thermal transport, and unique electronic phenomena.
• They exhibit nanospring behavior with stage-resolved tensile response, including delamination and an elastic-limit force around 6 nN, enabling stretchability beyond 1000% strain.
• The helical geometry induces pseudo-electric fields and modulates Dirac-carrier scattering, paving the way for topological flat bands in systems like helical trilayer graphene.

Searching arXiv for relevant papers on graphene helicoids and helical graphene systems. Graphene helicoids are graphene-based structures in which the material is organized with helicoidal or helical screw symmetry rather than as a flat sheet or a simple multilayer stack. In the literature surveyed here, the term covers at least two closely related regimes: a graphene nanoribbon wrapped into a screw-dislocation-like helicoid, often denoted GH, and a helical multilayer moiré system exemplified by helical trilayer graphene (HTG) with twist configuration $(\theta_1,\theta_2,\theta_3)=(\theta,0,-\theta)$ (Zhan et al., 2017, Zhan et al., 2018, Devakul et al., 2023). Across these regimes, the helicoidal geometry is not merely morphological. It governs tensile response, heat transport, and Dirac-carrier dynamics, and in the multilayer moiré setting it can self-organize into locally periodic $C_{2z}$-broken domains with topological flat bands and near-ideal quantum geometry (Atanasov et al., 2016, Atanasov et al., 2016, Devakul et al., 2023).

1. Structural definitions and geometric realizations

In nanoscale mechanical and thermal studies, a graphene helicoid is constructed from a graphene nanoribbon arranged into a helicoid topology via a screw-dislocation geometry (Zhan et al., 2017). One formulation uses a single screw dislocation with Burgers vector magnitude $|b| = 3.4\ \text{Å}$, together with two graphene monolayers at the two ends to reduce edge effects (Zhan et al., 2017). The geometry is described by the outer radius $R$, inner radius $r$, and turn number $N$, with width

$w = R - r$

and total height

$h_{\text{tot} = (N+2)|b|$

in the molecular-dynamics nanospring model; the deformable region has

$N_{\text{eff} = N - 2$

turns (Zhan et al., 2017).

A closely related thermal-transport construction also treats GH as a graphene nanoribbon wrapped into a continuous helical, screw-dislocation-like shape, with

$w = R - r, \qquad L_{\text{tot} = N \times |b|$

and a Burgers vector spacing of $C_{2z}$0 (Zhan et al., 2018). In that setting, GH is contrasted with multilayer graphene (MLG): instead of discrete sheets held together only by van der Waals interactions, the helicoid is a continuous spiral with strong geometric continuity along its axis (Zhan et al., 2018).

In continuum electronic treatments, the helicoid is a minimal surface. One helicoidal nanoribbon parameterization is

$C_{2z}$1

with twist rate

$C_{2z}$2

and induced metric

$C_{2z}$3

(Atanasov et al., 2016). A related helicoid parameterization used for massless Dirac particles is

$C_{2z}$4

with induced metric

$C_{2z}$5

(Watanabe et al., 2015). In both formulations the helicoid is emphasized as a minimal surface, so the electronic effects are attributed to geometry itself rather than to ordinary strain (Atanasov et al., 2016, Watanabe et al., 2015).

In moiré graphene, the helical concept is extended to a three-layer stack with distributed twist. HTG is defined by

$C_{2z}$6

with a special angle

$C_{2z}$7

(Devakul et al., 2023). Unlike ordinary bilayer moiré systems, this unrelaxed structure contains two coupled moiré patterns and is therefore a supermoiré, or moiré-of-moiré, structure (Devakul et al., 2023).

2. Mechanical response and nanospring behavior

The GH nanospring literature identifies a tensile response that is governed by delamination and subsequent ribbon stretching rather than by the mechanics of a conventional helical wire spring (Zhan et al., 2017). The deformation comprises four stages overall: initial delamination, stable delamination, elastic deformation, and failure (Zhan et al., 2017). The three elastic stages are central to its characterization as a nanospring.

During initial delamination, the force and strain energy increase sharply as adjacent turns begin to separate, overcoming interlayer van der Waals attraction. The force reaches a local maximum

$C_{2z}$8

which is the critical force for initiating stable delamination (Zhan et al., 2017). In the stable-delamination stage, the force drops to

$C_{2z}$9

and remains nearly constant with fluctuations, while the strain energy increases approximately linearly with strain (Zhan et al., 2017). After full delamination, the system behaves like a stretched graphene nanoribbon. The force rises again nonlinearly, the strain energy follows a parabolic trend, and the elastic-limit force is

$|b| = 3.4\ \text{Å}$0

marking the onset of failure (Zhan et al., 2017).

The reported tensile deformation capability is unusually large. The paper states that the yield strain can exceed $|b| = 3.4\ \text{Å}$1, with a representative case approaching $|b| = 3.4\ \text{Å}$2 (Zhan et al., 2017). The geometry dependence is explicit. With initial turn height

$|b| = 3.4\ \text{Å}$3

and fully delaminated height

$|b| = 3.4\ \text{Å}$4

the delamination strain is estimated as

$|b| = 3.4\ \text{Å}$5

For a GH with eight effective turns and $|b| = 3.4\ \text{Å}$6, the predicted value is

$|b| = 3.4\ \text{Å}$7

in agreement with simulation (Zhan et al., 2017). The delamination strain is independent of turn number $|b| = 3.4\ \text{Å}$8 and outer radius $|b| = 3.4\ \text{Å}$9, and increases with increasing inner radius $R$0 (Zhan et al., 2017).

A central finding is that the elastic-limit force remains nearly constant across geometries, around $R$1 (Zhan et al., 2017). The paper attributes this to highly non-uniform strain localization: inner edge atoms experience the largest tensile strain, atoms in the ribbon middle may experience slight compressive strain, and failure is governed by fracture of the graphene nanoribbon rather than by failure of the helicoid topology itself (Zhan et al., 2017). Crack initiation occurs at the armchair edge location, propagation follows the zigzag direction, and monoatomic chains and pentagon carbon rings form near fracture (Zhan et al., 2017).

Hydrogen saturation changes these behaviors only modestly in force but more strongly in strain. The reported changes are a decrease of about $R$2 in $R$3, a decrease of about $R$4 in $R$5, and a slight increase of $R$6 from $R$7 to $R$8; the strains decrease from about $R$9 to $r$0 for $r$1, from about $r$2 to $r$3 for $r$4, and from about $r$5 to $r$6 for $r$7 (Zhan et al., 2017). This suggests that edge chemistry modifies stretchability more strongly than the characteristic force scales.

3. Thermal transport and strain-dependent heat conduction

Graphene helicoids have also been studied as thermal-transport systems whose axial conduction differs qualitatively from the through-plane response of multilayer graphene (Zhan et al., 2018). The key structural distinction is continuity: GH is a single continuous helical graphene ribbon, so axial heat flow contains both interlayer van der Waals transport and projected in-plane graphene conduction along the helical axis (Zhan et al., 2018). By contrast, MLG is composed of separate graphene sheets whose through-plane transport is governed mainly by weak van der Waals coupling, and whose cross-plane thermal conductivity saturates with increasing thickness (Zhan et al., 2018).

The simulation framework adopts the Fourier-law-style definition

$r$8

with

$r$9

and $N$0 (Zhan et al., 2018). Over the simulated range, GH exhibits thickness scaling

$N$1

whereas MLG follows

$N$2

and converges to a thickness-independent limit (Zhan et al., 2018). The GH power law is attributed to the divergent in-plane thermal conductivity of 2D graphene, projected onto the helical axis (Zhan et al., 2018).

Interlayer overlap and alignment are quantitatively important. The paper reports larger interlayer centroid misfit in MLG, around $N$3 Å, and smaller misfit in GH, around $N$4 Å (Zhan et al., 2018). The smaller misfit indicates better alignment between adjacent turns, which increases effective contact area for van der Waals-mediated transfer and reduces sliding or misalignment losses (Zhan et al., 2018).

The strain response is nonstandard. In the small-strain regime, below roughly $N$5, compressive strain enhances thermal conductivity in GH because adjacent turns are pushed closer and van der Waals coupling becomes stronger (Zhan et al., 2018). Under ultra-large tensile strain, on the order of

$N$6

and even approaching the broader elastic regime near $N$7, the heat current does not collapse as in generic solid-state materials (Zhan et al., 2018). After initial delamination, the structure contains a delaminated region behaving like a curved nanoribbon and an undelaminated region where van der Waals interactions remain. The reported heat current drops substantially only in the early stretching stage and then converges to a nearly constant value around

$N$8

in the large-strain regime (Zhan et al., 2018).

The phonon interpretation is expressed through a phonon population variation ratio,

$N$9

with the conclusion that GH consistently has a larger out-of-plane phonon population than MLG (Zhan et al., 2018). This supports the view that GH thermal transport is a mixed in-plane/out-of-plane phonon transport problem rather than simple interlayer conduction.

4. Curved-surface Dirac physics and helicoidal nanoribbons

The electronic structure of graphene helicoids has been analyzed by treating graphene carriers as Dirac fermions constrained to a curved surface with screw symmetry (Watanabe et al., 2015). In the general isothermal-coordinate construction, the curved-space Dirac equation is written using zweibeins and covariant derivatives,

$w = R - r$0

with

$w = R - r$1

(Watanabe et al., 2015). For the helicoid, the metric factor is $w = R - r$2, and the Dirac equation becomes

$w = R - r$3

(Watanabe et al., 2015).

Because the coordinate $w = R - r$4 is cyclic, one separates variables as

$w = R - r$5

where $w = R - r$6 is the conserved momentum along the helical direction (Watanabe et al., 2015). After the transformation

$w = R - r$7

the problem reduces to a one-dimensional Schrödinger-type equation,

$w = R - r$8

with effective potential

$w = R - r$9

(Watanabe et al., 2015).

A central conclusion is the absence of bound states. Since

$h_{\text{tot} = (N+2)|b|$0

the asymptotic region is free and the spectrum consists of scattering states rather than localized bound states (Watanabe et al., 2015). The paper concludes that massless Dirac electrons do not form bound states on the helicoid, and further shows that bound states remain absent even after adding a mass term (Watanabe et al., 2015).

Scattering therefore becomes the dominant phenomenon. Reflection and transmission amplitudes are written as

$h_{\text{tot} = (N+2)|b|$1

and the potential obeys the symmetry

$h_{\text{tot} = (N+2)|b|$2

(Watanabe et al., 2015). The local density of states is defined by

$h_{\text{tot} = (N+2)|b|$3

and decomposed as

$h_{\text{tot} = (N+2)|b|$4

The reported numerical behavior is that the LDoS has large oscillations near the helicoid axis and becomes more uniform far from the axis (Watanabe et al., 2015). The helicoid thus acts as a smooth curved scattering region rather than a trap.

5. Geometry-induced pseudo-electric fields and chiraltronics

A distinct continuum treatment of helicoidal graphene nanoribbons emphasizes an iso-spin-dependent separation mechanism generated by the twist itself (Atanasov et al., 2016). In that formulation the stationary curved-surface Dirac equation is

$h_{\text{tot} = (N+2)|b|$5

with separated spinor components

$h_{\text{tot} = (N+2)|b|$6

and quantized longitudinal momentum

$h_{\text{tot} = (N+2)|b|$7

(Atanasov et al., 2016).

The two iso-spin components satisfy effective Schrödinger-like equations,

$h_{\text{tot} = (N+2)|b|$8

$h_{\text{tot} = (N+2)|b|$9

with

$N_{\text{eff} = N - 2$0

(Atanasov et al., 2016). The second term is the twist-induced contribution whose sign depends on iso-spin and chirality.

In the thin-strip regime,

$N_{\text{eff} = N - 2$1

the potentials reduce to linear form,

$N_{\text{eff} = N - 2$2

which the authors interpret as opposite effective potentials from a transverse electric field,

$N_{\text{eff} = N - 2$3

(Atanasov et al., 2016). This is described as reminiscent of the Hall effect, except that the field is geometry-induced rather than externally applied (Atanasov et al., 2016).

The sign difference is the basis for iso-spin separation onto opposing rims of the ribbon. For one chirality convention and $N_{\text{eff} = N - 2$4, iso-spin $N_{\text{eff} = N - 2$5 and $N_{\text{eff} = N - 2$6 are driven toward opposite edges, with the assignment reversed by changing the sign of $N_{\text{eff} = N - 2$7 or the helicoid chirality $N_{\text{eff} = N - 2$8 (Atanasov et al., 2016). The effective potential also hinders rim-to-rim transport; in the Born approximation the backward-scattering probability is

$N_{\text{eff} = N - 2$9

(Atanasov et al., 2016).

The same framework predicts iso-spin transitions. The transition frequency is estimated as

$w = R - r, \qquad L_{\text{tot} = N \times |b|$0

and, using the thin-strip condition,

$w = R - r, \qquad L_{\text{tot} = N \times |b|$1

For a micron-sized ribbon with $w = R - r, \qquad L_{\text{tot} = N \times |b|$2 and $w = R - r, \qquad L_{\text{tot} = N \times |b|$3, this gives

$w = R - r, \qquad L_{\text{tot} = N \times |b|$4

that is, the THz range (Atanasov et al., 2016). The authors describe this program as “chiraltronics,” in which geometry replaces an external field and the relevant degree of freedom is graphene iso-spin or chirality (Atanasov et al., 2016).

6. Helical trilayer graphene, local reconstruction, and topological flat bands

Helical trilayer graphene extends the helicoid concept from a single twisted ribbon to a multilayer moiré structure with distributed twist (Devakul et al., 2023). The proposed configuration,

$w = R - r, \qquad L_{\text{tot} = N \times |b|$5

contains two coupled moiré patterns and nominally forms a supermoiré structure (Devakul et al., 2023). The central structural result is that lattice relaxation reconstructs this supermoiré into large periodic single-moiré domains.

The local aperiodicity measure is defined by

$w = R - r, \qquad L_{\text{tot} = N \times |b|$6

with

$w = R - r, \qquad L_{\text{tot} = N \times |b|$7

in the bulk of a relaxed domain (Devakul et al., 2023). The relaxed state forms a triangular tiling of large domains separated by domain walls. The locally periodic domains are denoted h-HTG and $w = R - r, \qquad L_{\text{tot} = N \times |b|$8, related by $w = R - r, \qquad L_{\text{tot} = N \times |b|$9 (Devakul et al., 2023).

A defining feature of h-HTG is a finite lateral offset

$C_{2z}$00

between the two moiré superlattices (Devakul et al., 2023). This shift breaks $C_{2z}$01 symmetry locally, and that symmetry breaking is identified as crucial for nontrivial valley topology (Devakul et al., 2023). At $C_{2z}$02, the relaxed h-HTG domain hosts a pair of nearly flat isolated bands with valley-contrasting Chern numbers

$C_{2z}$03

(Devakul et al., 2023). In the chiral limit $C_{2z}$04, the model becomes analytically tractable and yields exactly flat bands with ideal quantum geometry; at the first magic angle the exact zero-mode bands have

$C_{2z}$05

(Devakul et al., 2023).

The bands are isolated from remote bands by a large gap. The abstract states a scale

$C_{2z}$06

and the detailed estimate in the continuum model is around $C_{2z}$07 meV (Devakul et al., 2023). This is large compared with the flat-band bandwidth

$C_{2z}$08

(Devakul et al., 2023). The geometric diagnostic for ideality is the trace-condition violation

$C_{2z}$09

with ideal quantum geometry corresponding to

$C_{2z}$10

(Devakul et al., 2023). The paper reports that HTG is close to this ideal limit, with remarkably uniform Berry curvature and charge density (Devakul et al., 2023).

The correlated and topological implications follow directly from this band structure. The paper highlights the prospect of integer and fractional quantum anomalous Hall states in $C_{2z}$11 and $C_{2z}$12 bands, generalized quantum Hall ferromagnets at integer filling, and fractional Chern insulators at fractional filling (Devakul et al., 2023). It also notes that the large domains, on the scale of hundreds of nanometers, are amenable to local probes such as scanning SETs or scanning SQUIDs, and that a very small uniform heterostrain of order $C_{2z}$13 could relax the entire device into a single h-HTG domain (Devakul et al., 2023).

When multiple domains remain, adjacent h-HTG and $C_{2z}$14 regions carry opposite valley-Chern responses, and the domain walls host gapless counterpropagating edge modes, producing a Chalker–Coddington-like network on the supermoiré scale (Devakul et al., 2023). This suggests a real-space network realization of helical edge transport tied directly to the reconstructed helicoidal moiré texture.

7. Conceptual scope, common distinctions, and research significance

The literature does not use “graphene helicoid” in a single narrow sense. One well-established usage refers to a graphene nanoribbon-based helicoid produced by a screw dislocation and studied for its mechanical and thermal properties (Zhan et al., 2017, Zhan et al., 2018). Another usage concerns helicoidal or screw-symmetric curved graphene surfaces analyzed within Dirac theory, where geometry-induced potentials modify scattering and iso-spin transport (Watanabe et al., 2015, Atanasov et al., 2016). A broader helical-graphene extension appears in HTG, where the twist is distributed across three consecutive layers and relaxation produces locally periodic topological domains (Devakul et al., 2023).

A common misconception is that the helicoid effects are reducible to ordinary strain engineering. The continuum electronic papers explicitly stress that the helicoid is a minimal surface and that the dominant effects arise from geometry itself, not from the usual strain-induced pseudo-magnetic-field mechanism (Atanasov et al., 2016, Watanabe et al., 2015). Another possible misconception is that helicoidal curvature should confine Dirac carriers. The curved-surface Dirac analysis reaches the opposite conclusion: neither massless nor massive Dirac electrons form bound states on the helicoid, and the principal signatures are scattering, phase shifts, and local density variations near the axis (Watanabe et al., 2015).

The combined body of work shows that helicoidal graphene architectures couple geometry, van der Waals interactions, and low-dimensional transport in ways unavailable to flat graphene or ordinary multilayers. In GH nanosprings, that coupling yields very large tensile deformation capability, stage-resolved delamination mechanics, and a nearly constant elastic-limit force set by inner-edge bond loading (Zhan et al., 2017). In GH thermal transport, the same continuity and overlap produce axial conductivity that scales as a power law with thickness and remains functional under extreme strain (Zhan et al., 2018). In helicoidal nanoribbon electronics, twist generates iso-spin-dependent effective potentials and THz-scale transition estimates (Atanasov et al., 2016). In HTG, a helical multilayer geometry reconstructs into $C_{2z}$15-broken periodic moiré domains with flat topological bands, valley Chern numbers $C_{2z}$16, near-ideal quantum geometry, and a large remote-band gap (Devakul et al., 2023).

Taken together, these results establish graphene helicoids as a structural and theoretical class in which screw symmetry or helical stacking is the organizing principle. Depending on the realization, that principle yields nanospring mechanics, anomalous axial heat transport, geometry-controlled Dirac scattering, iso-spin separation, or correlated topological moiré bands.