Ilaria Maccari of ETH Zurich, Louk Rademaker of the University of Geneva and Leiden University, and Giulia Venditti of the University of Geneva demonstrate that spatially correlated disorder in two-dimensional superconductors can obscure a hallmark transition known as the BKT transition, replacing it with a “smeared percolative transition”. Their work reveals that twist angle inhomogeneities, naturally occurring in moiré systems like twisted multilayer graphene, can effectively hide the ability to distinguish between different types of superconducting order at low temperatures. The researchers propose using the real part of conductivity as a diagnostic tool to assess the impact of this correlated disorder.

Twist Angle Inhomogeneity in 2D Superconductors

Elasticity theory reveals that twist angle inhomogeneities, inherent to moiré structures, exhibit logarithmic long-range correlations, a finding that fundamentally alters expectations for two-dimensional superconductivity. These correlations arise from the solid’s response to atomic-scale defects, creating a spatially fluctuating landscape of local superconducting properties. The research demonstrates that these correlated imperfections are not merely incidental but are an unavoidable consequence of the material’s physical properties. Numerical modeling of the elastic equations governing these twisted layers shows that the resulting twist angle disorder significantly impacts the expected behavior of 2D superconductors.

The team’s analysis, based on a random impedance network, reveals that this disorder effectively hides the ability to differentiate between nodal and fully gapped superconducting states at low temperatures. The researchers determined the correlations of the twist angle inhomogeneity by inverting the relationship between displacement and local twist angle, allowing them to calculate the statistical properties of the disorder.

This calculation relies on the premise that a local twist angle is determined by a summation over force distributions, expressed as an equation. This detailed modeling provides a framework for understanding how even small, correlated variations in twist angle can dramatically alter the macroscopic superconducting behavior. The EMT, a common analytical approach, underestimates the superfluid stiffness and its onset temperature in samples with significant disorder.

The team’s supplemental material details EMT results across a range of parameters, demonstrating the limitations of this method in the presence of correlated disorder. This suggests that standard analytical techniques may provide an incomplete picture of superconductivity in these materials, necessitating more sophisticated modeling approaches. The implications extend to experimental verification, as the researchers propose that the real part of conductivity is a key diagnostic observable. By carefully measuring this property, scientists can assess the relevance of correlated disorder, specifically, twist angle inhomogeneity, and determine whether it is masking the true superconducting behavior.

BKT Transition Smearing by Random Impedance Networks

Modeling reveals that disorder arising from variations in twist angle within layered superconductors fundamentally alters expected behavior at the transition to superconductivity, replacing a sharp transition with a gradual change in conductivity. Specifically, the work demonstrates that finite-frequency conductance measurements reveal a “smeared percolative transition” rather than the expected BKT transition, a hallmark of two-dimensional superconductivity. This shift occurs because of spatially correlated disorder, where local variations in the critical temperature, T_c, are not random but linked by the material’s elasticity.

The team’s approach involved constructing a random impedance network to simulate the disordered landscape of local T_c values, acknowledging that even seemingly uncorrelated random forces on the material’s structure generate long-range correlated displacements. This is due to the non-local nature of the elastic Green’s tensor, a mathematical description of how the material deforms under stress.

The calculations, based on elasticity theory, demonstrate that these correlated displacements significantly impact the superconducting properties. The researchers used Poisson’s ratio, Young’s modulus and medium density in their model to account for the material’s mechanical response to these forces. Understanding the order parameter is essential for understanding the underlying mechanism driving superconductivity. The team’s random impedance network models the resistive and superfluid responses, with each bond representing a mesoscopic superconducting island characterized by its local BKT transition temperature.

The network’s impedance incorporates both the amplitude and phase fluctuations of the local superconducting order parameter, providing a more realistic representation of the system. Analysis of the effective impedance, z_em(ω,T), extracted from the network reveals how the disorder affects the DC resistance, the real part of the optical conductivity, *σ₁* and the superfluid stiffness.

Elasticity Theory Modeling of Disorder Correlations

This finding challenges the conventional understanding of how superconductivity emerges in these materials, particularly in twisted multilayer graphene where such angles are inherent. Researchers found that the finite-frequency conductance, when analyzed through a random impedance network, reveals this altered transition rather than the expected sharp change in behavior. The team’s modeling, based on elasticity theory, demonstrates that the spatial arrangement of these twist angle variations is critical; it’s not simply the amount of disorder, but how it’s correlated that dictates the observed superconducting properties.

This correlation manifests as long-range connections, forming filamentary structures that deviate from predictions made by simpler analytical techniques. Specifically, the analysis reveals that the effectiveness of Effective Medium Theory, a common approach, diminishes when these organized disorder patterns are present, as the theory assumes a uniform medium.

The work goes beyond simply identifying the altered transition; it proposes a specific diagnostic tool for detecting the influence of correlated disorder. The real part of the conductivity, σ₁, is presented as a practical experimental probe for the percolative nature of the transition, offering a way to distinguish between disorder driven by random variations and that driven by correlated twist angles.

Disagreement between the experimentally measured σ₁ and predictions from Effective Medium Theory is a strong indicator of the presence of these long-range correlations. The study also details how the elasticity of the solid material leads to logarithmic correlations in the twist angle inhomogeneity.

Superfluid Stiffness as an Order Parameter Probe

The expected relationship between temperature and superfluid stiffness in two-dimensional superconductors becomes obscured by spatially correlated disorder arising from twist angle inhomogeneities, according to new modeling work. Researchers demonstrate that this disorder can effectively mask the distinction between superconducting order parameters, specifically, whether they are nodal or fully gapped, at low temperatures, complicating efforts to understand the underlying mechanisms driving superconductivity in materials like twisted multilayer graphene.

This finding challenges the conventional use of superfluid stiffness as a reliable indicator of order parameter symmetry in these systems. The team’s analysis centers on twisted bilayer graphene, a model moiré system, but the principles extend to other twisted multilayer materials. Using a random impedance network based on realistic disorder in the local T_c, they show that the finite-frequency conductance reveals a smeared percolative transition instead of a BKT transition.

The resulting simulations reveal that the expected sharp transition characteristic of a BKT transition is replaced by a significant departure from predicted behavior. Their model predicts that the low-temperature behavior of the superfluid stiffness is no longer a clear indicator of the order parameter symmetry.

The team’s calculations used specific parameters, a resonance frequency of ω₀ = 2× 10⁹ s^-1, a reference inductance of L_s(0) = 1 nH, a mean inductance of L_m = 10nH, a normal resistance of R_N = 2 kΩ, and a parameter α = 0. 2, to directly compare their results with existing microwave resonance transport experiments. These parameters define the characteristics of the simulated network and allow for a quantitative assessment of the disorder’s impact on the observed superconducting properties.

The work also highlights the potential for using multiple probes to disentangle intrinsic superconducting properties from disorder effects. The researchers propose that the real part of the conductivity can be used as a key diagnostic observable to probe the relevance of correlated disorder.

Disorder-Induced Masking of Pairing Symmetry

Twist angle inhomogeneities within layered superconductors can mimic the characteristics of nodal pairing, even when a fully gapped superconducting state exists, according to new modeling. This effect arises from the creation of superconducting islands within the material, altering the expected low-temperature behavior of the superfluid stiffness. The research reveals that correlated disorder, specifically stemming from variations in twist angle, suppresses superfluid stiffness at low temperatures in a manner that closely resembles nodal superconductivity, regardless of the actual gap symmetry.

This poses a significant challenge to interpreting experimental data, as the standard method of using low-temperature superfluid stiffness to identify order-parameter symmetry becomes unreliable. The team’s model shows that these large-scale inhomogeneities drive the system toward a percolative transition, where identifying a clear critical point for global phase coherence becomes impossible.

This is the first central result of their work, demonstrating that the observed low-temperature stiffness signature of nodal pairing can occur even when a fully gapped state is present. The team’s calculations show that the peak of $σ_{1} (ω_{0}, T)$ broadens and develops a finite residual value at low temperatures, indicating a growing non-superconducting fraction within the network. The researchers propose that the real part of the conductivity can be used as a key diagnostic observable to probe the relevance of correlated disorder, offering a new way to understand the complex behavior of these materials.

👉 More information
🗞 How twist angle inhomogeneity masks the superconducting order parameter symmetry
✍️ Ilaria Maccari, Louk Rademaker and Giulia Venditti
🧠 DOI: http://link.aps.org/doi/10.1103/djwx-lqsx

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