Interlayer Sliding in Bilayer Graphene

• Interlayer sliding in bilayer graphene is defined as the lateral displacement between atomic layers that modulates friction, domain textures, and electronic band topology.
• Recent DFT and continuum modeling studies quantify ultra-low energy barriers and reveal mechanisms such as superlubricity, strain-bound solitons, and sliding-induced ferroelectricity.
• Advances in experimental techniques demonstrate sliding-induced phonon splitting, Berry curvature reversals, and quantized charge pumping, pointing to applications in nanomechanics and memory devices.

Interlayer sliding in bilayer graphene refers to the lateral displacement of one graphene layer relative to another, fundamentally modulating the registry-dependent physical properties of the system. The phenomenon is central to the tribological, electronic, and topological behaviors observed throughout commensurate, twisted, and defected bilayer systems. Recent density functional theory (DFT), continuum modeling, and experimental investigations reveal vanishing friction ("superlubricity") for pristine twisted bilayers, robust static and dynamic barriers induced by atomic-scale defects, strain-bound soliton domain-wall formation, symmetry-driven ferroelectric switching, and topological quantum pumping tied to sliding-induced Berry curvature reversals.

1. Microscopic Origin and Quantitative Characterization

A rigorous DFT approach employing the vdW-DF3 functional quantifies the potential energy surface (PES) governing interlayer sliding for commensurate moiré patterns (notably (2,1) and (3,1)). The first spatial Fourier harmonics capture the PES:

$$\delta U(x', y') = U_1 [2\cos(k'_y y')\cos(k'_x x') + \cos(2k'_y y')]$$

with $N_c = n_1^2 + n_1 n_2 + n_2^2$, $k'_x = 2\pi \sqrt{N_c} / a$, $k'_y = 2\pi \sqrt{N_c/3} / a$, $a = 2.4660$ Å. For the (2,1) pattern, DFT yields a corrugation amplitude $\Delta U_\mathrm{max} = 0.4$ to $0.8$\,$\mu$eV per atom (relaxation doubles the value), while the (3,1) cell shows $\Delta U_\mathrm{max} \sim 0.03$\,$\mu$eV per atom (Lebedeva et al., 3 Nov 2025). These barriers are orders of magnitude below those for aligned Bernal bilayer systems.

Critical interlayer distance variation can reverse the PES minima and maxima, fundamentally altering sliding energetics. The finite barrier for rotation to an incommensurate state for (2,1), $\Delta U_\mathrm{rot} \sim 0.25$\,$\mu$eV/atom, stabilizes the superlubric regime.

Static friction force, shear mode frequency, and modulus derive from PES derivatives; e.g., for the (2,1) relaxed cell: $f = 0.59$ cm$^{-1}$, $C_{44} = 1.6 \times 10^6$ Pa, $F_\mathrm{max}/w = 1.3 \times 10^{-2}$ N/m (Lebedeva et al., 3 Nov 2025).

2. Structural Superlubricity and Its Limiting Factors

Interlayer superlubricity—vanishing measurable friction—is robust for defect-free twisted bilayer graphene, with computed PES corrugation $<6 \times 10^{-6}$ meV/atom, below numerical noise (Minkin et al., 2021). However, atomic-scale defects (vacancies) break local registry symmetry and introduce finite barriers: $U_\mathrm{max} = 28$ meV/vacancy, sliding barriers $\Delta E \sim 7–8$ meV/vacancy, static friction $F_\mathrm{stat} \sim 12–16$ pN/vacancy. Defect density as low as $<1\%$ produces macroscopic friction scaling as $F_d \sim N_d^{1/2} \sim A^{1/2}$ (Minkin et al., 2021).

Semiempirical potentials (Lebedeva, Popov) fitted to DFT capture defect-limited superlubricity with minor deviations in energy barriers, adequate for dynamic friction modeling but less so for static friction directionality. Computationally efficient first-harmonic approximations match DFT corrugation magnitudes within 6% (Minkin et al., 2021).

3. Domain Walls and Stacking Textures

Large overlaps between commensurate moiré domains favor strain relief through solitonic domain-wall (stacking-dislocation) formation. The Frenkel–Kontorova model yields wall width $l_D \sim 0.7\,\mu$m and formation energy $W_D \sim 0.3$ meV/Å for (2,1) (Lebedeva et al., 3 Nov 2025). These 1D solitons connect AB–BA stacking, propagate as strain-minimizing smooth textures, and can be constructed analytically or as neutral multi-pole configurations (Gong et al., 2013):

$$\Delta^>(z) = \Delta_\alpha + (\Delta_\beta - \Delta_\alpha)\, \bar z_0/(\bar z_0 - \bar z)$$

$$\Delta^<(z) = \Delta_\alpha + (\Delta_\beta - \Delta_\alpha)\, (z_0 - z)/z_0$$

These stacking defects localize the commensuration energy and their formation can be directly imaged in large ($>10\,\mu$m) twisted flakes.

4. Electronic and Topological Response to Sliding

Sliding modifies the band topology via phase factors $e^{-i G_j \cdot u}$ in the interlayer coupling Hamiltonian (Pan et al., 16 Nov 2025), mediating Berry curvature reversals and valley Chern number jumps:

$$\Omega_\tau(k;u) \simeq -\tau \frac{v_F^2\,\Delta_\mathrm{eff}(u)}{2[\Delta_\mathrm{eff}(u)^2 + (v_F k)^2]^{3/2}}$$

$$C_\tau(u) = \frac{1}{2\pi} \int_{|k|<\Lambda} \Omega_{-,\tau}(k;u)\, d^2k$$

Experimental realization involves bending bilayer graphene across a nanoridge, inducing a domain-wall profile where the band gap closes/reopens as stacking evolves AB → AA′ → BA. Transport measurements yield quantized conductance $G \approx 8 e^2/h$, consistent with eight topological valley channels (Pan et al., 16 Nov 2025). This effect generalizes across 2D materials with nontrivial stacking landscapes.

5. Topological Charge Pumps and Quantum Transport

Interlayer sliding in twisted bilayer graphene implements quantized topological pumping: the sliding Chern number $C_s$ equals the net edge states transferred per mechanical cycle (Fujimoto et al., 2020):

$$C_{ij}^{(l)} = \sum_{n=1}^N \frac{1}{2\pi}\int_0^1 d k_j \int_0^1 d\lambda_i\, \Omega_{\lambda_i k_j}^{(n)}(\mathbf k, \lambda)$$

$I \approx e\, C_s\, \dot\lambda\, L_y / L_M$

Experimental signatures include stepwise dc current proportional to sliding rate, periodic in-gap edge-mode appearance in STM, and detectable nonlocal voltage via counterpropagating edge states. The bulk-edge correspondence directly links $C_s$ and quantized charge transfer (Fujimoto et al., 2020).

6. Phonon and Plasmon Mode Modulation by Sliding

Sliding induces sensitive and symmetry-dependent optical phonon splittings: the IR-active $E_u$ LO/TO pair splits linearly with sliding vector, $\Delta \omega_{E_u} \sim \pm \kappa\, |\vec\delta|$ ($\kappa \sim 20–40$ cm$^{-1}$ per bond length), while Raman-active $E_g$ frequency remains almost unchanged but develops strong polarization anisotropy (Choi et al., 2013). High-resolution IR and polarized Raman spectroscopy can thus resolve interlayer misalignment to sub-Ångström scales.

The electron-plasmon spectrum also exhibits sliding-dependent transitions: acoustic and optical plasmon mode dispersions, damping thresholds, and intensity are sensitive to stacking vector, doping, and interlayer registry (Lin et al., 2018). Momentum-frequency phase diagrams provide precise experimental targets for EELS and IR absorption measurements.

7. Sliding-Induced Ferroelectricity in Heterostacks

Bilayer graphene encapsulated between h-BN layers exhibits "across-layer sliding ferroelectricity" (ALSF) due to next-neighbor interlayer coupling asymmetry, breaking inversion symmetry and enabling switchable out-of-plane polarization:

$$U(\mathbf{u}) = \sum_{m=1}^3 [V^{\mathrm{NN}} \cos(\mathbf{G}_m \cdot \mathbf{u}) + V^{\mathrm{NNN}} \sin(\mathbf{G}_m \cdot \mathbf{u})] + \mathrm{const}$$

$$P_z(\mathbf{u}) \simeq P_0 \sin \left( \frac{2\pi u_x}{a} \right)$$

DFT/NEB calculations yield sliding barriers $\Delta U \sim 3.1$ meV/unit cell, remnant polarization $P_0 \sim 0.48$ pC/m, and coercive fields $E_c \sim 30$ mV/nm, matching experimental hysteresis (Yang et al., 2022). Multilayer and molecule-decorated stacks realize nonvolatile, ultra-high-density switchable memory via local sliding.

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In summary, interlayer sliding in bilayer graphene establishes an intricate interdependence between structural registry, frictional and electronic properties, topological transport, phonon and plasmon spectra, and symmetry-breaking ferroelectric switching. Advanced first-principles calculations, continuum models, and well-controlled experiments underpin the understanding and engineering of these phenomena, with vibrant prospects for nanomechanics, topological electronics, and nonvolatile atomic memory.