Phason-driven temperature-dependent transport in moir'e graphene

The gist The experimental and theoretical investigation points to phasons as the dominant source of electron scattering in mTBG, revealing their central role in shaping temperature-dependent resistivity in the absence of electronic correlations.

Moiré System and Phason Modes

The formation of a moiré superlattice dramatically alters the electronic and vibrational properties of 2D materials, where the lowest-energy phonon modes are two acoustic branches called phasons that describe sliding motion between layers. These moiré phonons describe fluctuations of the stacking order in the reconstructed superlattice and include two branches of acoustic-like modes known as phasons. In realistic systems, interlayer interactions, electronic coupling, and structural disorder introduce damping, rendering phasons overdamped and diffusive <ref:2511.01691#pg4> This dissipative character is key to their influence on electronic transport, where phason scattering has been predicted to extend the linear-in-T behavior of ρ(T) down to temperatures lower than the Bloch-Grüneisen scale <ref:2511.01691#pg4>.

Temperature Dependence of Resistivity

The study investigates temperature-dependent electrical transport in minimally twisted bilayer graphene (mTBG) at a minimal twist angle (θ = 0.36◦), which permits ruling out electronic correlation influence and considering a reconstructed lattice structure with a triangular soliton network <ref:2511.01691#pg5>. The experimental results show a linear-in-temperature resistivity across the band manyfold above T ∼ 10 K, preceded by a quadratic temperature dependence While the linear-in-temperature resistivity is up to two orders of magnitude larger than in monolayer graphene, it is reduced (approximately by a factor of three) with respect to magic-angle twisted bilayer graphene. This linear-in-T behavior is compatible with scattering processes mediated by longitudinal phasons, which dominate the resistivity over the contribution from conventional acoustic phonons of the monolayer.

Electron-Phason Coupling and Scattering Mechanism

The electron-phason coupling is modeled using a continuum model of TBG, where the coupling term Hˆe-ph contains intra-layer terms coupled to dynamical heterostrain fluctuations and inter-layer terms representing phonon-assisted tunneling events <ref:3pg21,inter-layer terms representing phonon-assisted tunneling events, as recently observed with the quantum twisting microscope [17]>. The contribution of phasons to the resistivity dominates over the phonon contribution by at least one order of magnitude <ref:2511.01691#pg10>. The low-temperature part of the resistivity follows a quadratic trend which is a characteristic feature of phason scattering, since those processes are dominated by long-wavelength incoherent fluctuations <ref:2511.01691#pg12>. This T squared behavior is found even at n = 0, where the charge carrier density vanishes, showing that the T squared behavior cannot be attributed to this mechanism <ref:2511.01691#pg13>.

Phason vs. Acoustic Phonon Contributions

The spectral function of a phason mode α entering in the scattering rates of electrons is given by χ''α(k, ω) = 2ϱγω[ω2 − ω2α(k)]2 + γ2ω2, where γ is a phenomenological damping constant <ref:2511.01691#pg12>. The Bloch-Grüneisen temperature scale saturates for fillings n/ns ≥ 0.15 for both phasons and in-phase phonons <ref:2511.01691#pg22>. Both types of collective modes give a linear-in-T resistivity for temperatures above the Bloch-Grüneisen scale <ref:2511.01691#pg24>. The quadratic in T resistivity at the lowest temperatures is a characteristic feature of phason scattering, since those processes are dominated by longwavelength incoherent fluctuations <ref:2511.01691#pg12>.

Experimental Validation and Conclusions

The results are validated by comparing experimental data with theoretical calculations, where the computed ρe-ph terms follow the same behavior with band filling as measured experimentally <ref:2511.01691#pg23>. The analysis confirms that the driving mechanism of the large linear-in-T resistivity is restricted to the lower-energy moir´e bands, as minima in dρ/dT map approximately to integer fillings <ref:2511.01691#pg12>. In conclusion, phasons are identified as the dominant source of electron scattering in mTBG, revealing their central role in shaping temperature-dependent resistivity in the absence of electronic correlations <ref:2511.01691#pg13>. Beyond graphene, this work contributes to a growing understanding of phasons as a general mechanism in moir´e materials <ref:2511.01691#pg13>.

How it works

The lattice relaxation and moir´e phonons are studied using continuum models where the free energy includes an adhesion potential term that introduces the moiré superlattice effects. The electronic Hamiltonian is formulated in second quantization, incorporating Dirac Hamiltonians and interlayer tunneling events described by Tˆ(ξ)[ϕ(r)] <ref:3pg20,The electronic Hamiltonian in second quantization is given by Hˆ [ϕ(r)] = X∑ξ=±1Zdrψ†ξHˆ(ξ)[ϕ(r)]ψ ξ>.

Resistivity Calculation

The resistivity ρe-ph is calculated semiclassically within Boltzmann transport theory, where the transition rate P k'k(α) is given by Fermi’s Golden rule <ref:3pg23,P k'k(α) = 2ħgα(k, k') 2nF(k)[1 − nF(k')]Imχ ph(k' - k, εk' - εk)nB(εk' - εk)>. The decay of the Bose distribution and phonon response function allows one to calculate it over the whole real line, instead of over a small interval around ω = 0 <ref:2511.01691#pg24>. The variational ansatz for resistivity is found by minimizing a functional, leading to ρe-ph = 1/4kBT∫X⊤P-1X <ref:2511.01691#pg25>.

Numerical Implementation

The numerical implementation involves using 55 stars in codes for phonon dispersions and 15 stars to calculate electron bands and matrix elements of the electron-phonon coupling <ref:2511.01691#pg25>. The density of states is calculated with a factor of 2 to account for spin degeneracy and a broadening ζ = 0.3 meV <ref:2511.01691#pg25>. For resistivity estimation, variational ansatz expanded in a basis of cylindrical harmonics is used for connected Fermi surfaces <ref:2511.01691#pg26>. The calculation introduces an approximation g(kθ, k′θ) ≈ g(kθ−k′θ') which is exact only for spherical Fermi surfaces <ref:2511.01691#pg26>.

Key Findings Summary

1. The linear-in-T resistivity is up to two orders of magnitude larger than in monolayer graphene but is reduced by a factor of three compared to magic-angle twisted bilayer graphene.

2. The experimental trends are compatible with scattering processes mediated by longitudinal phasons, which dominate the resistivity over conventional acoustic phonons.

3. The low-temperature part of the resistivity follows a quadratic trend, characteristic of phason scattering, while intermediate temperatures show linear-in-T behavior <ref:2511.01691#pg12>.

4. The saturation of ρ(T) determines a threshold up to which transport can be considered limited to individual low-energy bands <ref:2511.01691#pg23>.

5. The quadratic T squared behavior in resistivity is found even at n = 0, ruling out electron-electron scattering combined with umklapp processes <ref:2511.01691#pg12>.

How it works

The electronic band Hamiltonian Hˆ(ξ)[ϕ(r)] is a block-matrix Hamiltonian containing Dirac Hamiltonians and interlayer tunneling events encoded in the off-diagonal blocks Tˆ(ξ)[ϕ(r)] <ref:3pg20,Hˆ 0 ≡ Hˆ [ϕ0(r)] with translational symmetry in the moir´e superlattice>. The electron-phason coupling term Hˆe-ph contains intra-layer terms coupled to dynamical heterostrain fluctuations and inter-layer terms representing phonon-assisted tunneling events <ref:3pg22,Hˆ e-ph = 1/sqrt(A)X∑ξ=±1Xn1,n2Xk1,k2∈mBZgαξ,n1,n2(k1, k2)ˆc†ξ,n2,k2cˆξ,n1,k1ϕα k2-k1 <ref:2511.01691#pg31>. The resistivity is calculated by minimizing a functional derived from the linearized Boltzmann equation <ref:2511.01691#pg25>.

1. textbfIncreased predictive accuracy for electronic transport in moiré systems using phason scattering models. This system can now predict temperature-dependent resistivity trends by explicitly incorporating scattering processes mediated by longitudinal phasons, as shown in the abstract, and quantitatively compare experimental data with theoretical predictions derived from the Boltzmann equation where longitudinal modes dominate the resistivity over the contribution from conventional acoustic phonons.

2. textbf Enhanced material design capabilities for 2D heterostructures. The improved system can now guide experimentalists in engineering moiré patterns by predicting how structural changes, such as twist angle variations, affect transport, allowing for better control over phenomena like isospin Pomeranchuk effect and the identification of regions where the linear-in-T resistivity is up to two orders of magnitude larger than in monolayer graphene.

3. textbf Development of a robust methodology for discerning scattering mechanisms. The AI can now definitively distinguish between different vibrational modes by analyzing the temperature dependence of their contributions, as detailed in Figure 4, specifically identifying that the quadratic in T resistivity at the lowest temperatures is a characteristic feature of phason scattering versus other processes.

The electronic and vibrational properties of 2D materials are dramatically altered by the formation of a moiré superlattice. The lowest-energy phonon modes of the superlattice are two acoustic branches (called phasons) that describe the sliding motion of one layer with respect to the other. Considering their low-energy dispersion and damping, these modes may act as a significant source of scattering for electrons in moiré materials. Here, we investigate temperature-dependent electrical transport in minimally twisted bilayer graphene, a moiré system developing multiple weakly-dispersive electronic bands and a reconstructed lattice structure. We measure a linear-in-temperature resistivity across the band manyfold above T about 10 K, preceded by a quadratic temperature dependence. While the linear-in-temperature resistivity is up to two orders of magnitude larger than in monolayer graphene, it is reduced (approximately by a factor of three) with respect to magic-angle twisted bilayer graphene. Moreover, it is modulated by the recursive band filling, with minima located close to the full filling of each band. Comparing our results with a semiclassical transport calculation, we show that the experimental trends are compatible with scattering processes mediated by longitudinal phasons, which dominate the resistivity over the contribution from conventional acoustic phonons of the monolayer. Our findings highlight the close relation between vibrational modes unique to moiré materials and carrier transport therein.