Electron-Phonon Coupling in Kekulé Graphene

• Electron-phonon coupling strength is defined as the measure of interactions between electrons and lattice vibrations, crucial for understanding resistivity and superconductivity.
• The methodology uses frozen-phonon and deformation potential approximations to compute local variations in the hopping parameter based on bond-length distortions in Kekulé-ordered graphene.
• Spatial inhomogeneity from Kekulé distortion produces two distinct bond types with up to a 40% variation in coupling, enabling targeted engineering of superconducting phases.

Electron-phonon coupling strength quantifies the interaction between electronic states and lattice vibrations in solids, governing phenomena such as electrical resistivity, carrier relaxation, and phonon-mediated superconductivity. In graphene and related low-dimensional materials, the modulation of electronic structure by lattice distortions—especially in the presence of bond-ordering instabilities like the Kekulé distortion—gives rise to pronounced spatial variation and tunability of electron-phonon interactions. The recent investigation of Kekulé-ordered graphene explicitly demonstrates how bond-dependent modifications yield locally resolved, highly inhomogeneous electron-phonon coupling, profoundly affecting pairing phenomena and potentially enabling engineered superconducting phases (Szczȩśniak, 20 Jun 2025).

1. Distance-Dependent Electronic Coupling and Its Role in Electron–Phonon Interactions

The essential microscopic ingredient determining local electron–phonon coupling in Kekulé-ordered graphene is the dependence of the electronic (hopping) parameter $t$ on the interatomic distance $l$. This variation is captured by an exponential form: $$t = t_0 \exp\left[ \eta \left( \frac{l}{a_0} - 1 \right) \right]$$ where $t_0$ is the hopping integral at equilibrium bond length $a_0$, and $\eta$ encodes the decay proportion (empirically fit to ab initio calculations). As lattice vibrations propagate (e.g. E$_{2g}$-type in-plane phonons), $l$ fluctuates about $a_0$, directly modulating $t$.

The electron–phonon coupling constant on a given bond, $\alpha$, is obtained as the derivative of $t$ with respect to $l$: $\alpha = \frac{dt}{dl} = C t$ with $C \approx 1.49817$ Å$^{-1}$ in Kekulé-ordered graphene [Eq. 10 in (Szczȩśniak, 20 Jun 2025)]. This linear proportionality (termed the "bond-coupling rule," Editor's term) establishes that the electron–phonon interaction on each bond is directly tied to the locally renormalized hopping parameter.

2. Spatial Inhomogeneity Induced by Kekulé Bond Order

A Kekulé distortion induces two distinct bond lengths in graphene:

• Shortened bonds ($a_0'$): 2/3 of all bonds; enhanced $t$ and $\alpha$
• Lengthened bonds ($a_0''$): 1/3 of all bonds; reduced $t$ and $\alpha$

Consequently, the local electron–phonon coupling varies periodically across the lattice:

Bond Type	Fraction	Relative $\alpha$ ($\alpha/\alpha_0$)
Short ($a_0'$)	2/3	$>1$, up to $1.4$+
Long ($a_0''$)	1/3	$<1$

The difference in electron–phonon coupling between these two bond types can exceed 40% for realistic band gap magnitudes ($\Delta$), with the percentage enhancement for short bonds tracking $\Delta$ linearly (see inset Fig. 4 in (Szczȩśniak, 20 Jun 2025)). This periodic spatial modulation produces domains of locally enhanced electron–phonon interaction, which may nucleate as "pairing islands" in a superconducting regime.

3. Methodological Framework: Frozen-Phonon and Deformation Potential Approximations

The calculation proceeds via the frozen-phonon approach: a static, symmetry-adapted lattice displacement representing the E$_{2g}$ phonon mode is imposed, and the variation in local $t$ is computed ab initio (or fitted using the exponential model). The deformation potential emerges as the key descriptor: $$\alpha = \frac{dt}{dl} = t_0 \eta \exp\left[ \eta \left( \frac{l}{a_0} - 1 \right) \right]$$ By evaluating $\alpha$ at distorted bond lengths, a detailed non-uniform map of the electron–phonon coupling (and thus local pairing strength) is constructed (Szczȩśniak, 20 Jun 2025).

4. Implications for Superconductivity: Local Pairing and Percolative Phases

The domains of enhanced electron–phonon coupling establish a spatially modulated pairing landscape. In the superconducting context, electron pairs would preferentially localize in these "islands" of strong coupling, as opposed to forming a homogeneous condensate. For global superconductivity to emerge, these domains must connect percolatively—analogous to granular superconductors.

Moreover, the competition between local coupling enhancement (via increased $t$ and $\alpha$ on short bonds) and phonon stiffening (higher phonon energies on altered lattices) sets the magnitude of the effective coupling constant and the superconducting gap. Enhanced local coupling may support higher critical temperatures or facilitate non-adiabatic pairing mechanisms, especially as the nonuniformity disrupts standard mean-field descriptions.

5. Pathways to Engineering and Control

The robust, local, bond-dependent formulation of electron–phonon coupling presents several avenues for engineered quantum phases in graphene and related materials:

• Strain Engineering: Local or patterned strain modifies bond lengths, enabling spatial control over $t$ and $\alpha$.
• Substrate and Adatom Decoration: Surface effects or chemical functionalization (e.g., Li, H) can introduce targeted Kekulé distortions or bond inhomogeneities, breaking chiral symmetry and modulating local coupling.
• Electrical Gating/Doping: Altering carrier concentration may further enhance or suppress electron–phonon interactions in specific regions, tuning the global response.
• Computational Design: The linear relation $\alpha=C t$ allows machine learning models or data-driven frameworks to predict and optimize structures for desired local electron–phonon characteristics.

6. Physical Context and Broader Significance

The understanding that bond-dependent electron–phonon coupling can be highly nonuniform in Kekulé-ordered or otherwise distorted graphene reframes the approach to phonon-mediated phenomena, including superconductivity and potentially charge-ordered or density-wave states. It rationalizes the experimental observation of unusual or "granular" superconducting behavior in structurally modulated low-dimensional materials. The explicit "bond-coupling rule" and the associated bond-order engineering strategies offer a generalizable approach to control quantum phases by tuning real-space lattice–electronic correlations.

7. Outlook: Experimental Probes and Theoretical Extensions

The prediction of bond-resolved variation in electron–phonon coupling can be tested directly via spatially resolved spectroscopies, such as scanning tunneling spectroscopy (STS) or resonant inelastic x-ray scattering (RIXS) with sublattice sensitivity. The framework also opens paths for theoretical developments in strongly correlated and topologically nontrivial systems, where similar local coupling rules may inform the emergence of unconventional superconductivity, charge order, or non-adiabatic phenomena in broader classes of two-dimensional and interface materials.