Phonon-limited resistivity of multilayer graphene systems

arXiv:2206.04080 · cond-mat.mes-hall, cond-mat.mtrl-sci, cond-mat.supr-con · Submitted 2022-06-08 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: Today's paper: "Phonon-limited resistivity of multilayer graphene systems".

Mira: The gist The band geometry leads to a nonlinear T-dependence in the high-T equipartition regime,

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So, we're looking at this paper now called "Phonon-limited resistivity of multilayer graphene systems". It’s about calculating how temperature and doping affect electrical resistivity because of scattering by acoustic phonons in Bernal bilayer graphene and rhombohedral trilayer graphene.

Mira: What’s the main point here? Basically, they focus on the geometric features of these anisotropic k·p band structures, like Van Hove singularities and Lifshitz transitions, which haven't been studied much for transport.

Lev: So, you're saying these specific shapes in the band structure really matter when we look at how resistance changes with temperature and doping?

Kai: Exactly. They found that these geometric features strongly influence the temperature and doping dependencies of the resistivity in BBG and RTG. It complicates the usual picture of how things change with heat.

Mira: The paper specifically points out that this band geometry causes a nonlinear temperature dependence in the high-temperature equipartition regime, which messes up the standard T to T four crossover we usually see.

Lev: So, instead of a simple T four to T scaling, we get something more complex because of the shape of those bands?

Kai: Right. And they see that this nonlinear behavior is even more pronounced when doping decreases and the Fermi surface itself changes its shape qualitatively. It’s not just one thing affecting it; it's a whole set of geometric effects interacting with temperature.

Mira: They also look at how the band geometry affects the low-temperature regime, where you expect some kind of Bloch-Grüneisen crossover to happen.

Lev: What do they find about that crossover? Is it still pretty standard, or does the geometry mess with it too?

Kai: The paper shows that for gapped systems, there’s actually a large spike in resistivity near charge neutrality at low temperatures. That spike is something that drops off as the temperature goes up.

Mira: That spike is noted to be a low-temperature phenomenon, and the anisotropy of the trigonal warping also alters the low-temperature Bloch-Grüneisen relaxation rate, making it k dependent instead of just a simple T four power law.

Lev: So, even at low temperatures, the standard picture isn't quite right for these systems because of how those bands are curved and warped?

Kai: That’s the point. Their numerical solution uses a big grid in n minus T space to map out all these different scaling regimes of resistivity. They show that phonon contribution becomes important at temperatures between ten Kelvin and sixty Kelvin, depending on how doped the system is.

Mira: So, for someone just listening, what does this mean about the physics of these materials? It suggests that you can’t just use simple rules for transport because you have to account for the actual shape of the energy bands.

Lev: If we were building real hardware based on this, would it mean we need to worry more about these band structures when designing error correction protocols?

Kai: Definitely. The paper says that understanding how band geometry affects scattering is relevant because there are recent experiments showing exotic low-temperature superconductivity near complicated hierarchies of phases in BBG and RTG.

Mira: So, the geometric features might actually be intimately related to those interaction-driven phases they've seen experimentally, highlighting why we need to understand this transport physics.

Lev: It sounds like a strong link between band structure details and the emergent phenomena they are observing in experiments.

Kai: Yeah. The work gives concrete predictions for how resistivity scales with doping and temperature when phonons dominate the transport in these systems, which is helpful for figuring out what scattering mechanisms are actually important.

Mira: So, to wrap up, this paper on "Phonon-limited resistivity of multilayer graphene systems" shows that the band geometry dictates a nonlinear T dependence in the high-T equipartition regime and modifies both the low and high temperature scaling regimes compared to standard models.

Lev: It seems like it tells us that for these layered materials, you can't rely on simple Bloch-Grüneisen crossovers without considering these detailed k p band structure effects.

Kai: And those geometric features, like Van Hove singularities and Lifshitz transitions, are key drivers of this complex transport behavior when acoustic phonons are involved.

Mira: The implication is that we need a more nuanced approach to modeling transport in these systems if we want to understand the underlying physics driving phenomena like superconductivity.

Conclusion: Kai: So, we've been diving deep into how the band structure of Bernal bilayer graphene and rhombohedral trilayer graphene messes with electrical resistivity when acoustic phonons are doing the work in these systems.

Mira: It really comes down to those geometric features—the Van Hove singularities and Lifshitz transitions—how they dictate the temperature and doping dependence of that resistance.

Lev: From an error correction standpoint, if this is how scattering works, we need to account for that non-linear T dependence when we model the noise floor.

Kai: The authors are looking at how these geometric quirks affect everything from low temperature quantum statistics to high temperature equipartition regimes.

Mira: They show that the standard Bloch-Grüneisen crossover isn't enough here because of the band geometry causing a nonlinear T dependence in the high-T part of that crossover.

Lev: That nonlinearity means our simple models for noise might be way off if we don't incorporate those specific band curvatures and warping effects.

Kai: The whole point is that this isn't just about how much resistance there is, but *why* it scales the way it does across different doping levels.

Mira: Their conclusion is that you can't rely on simple scaling laws because these band shapes introduce strong modifications to the resistivity behavior.

Lev: So, if we want to understand how superconductivity emerges in these moiré systems, we have to start looking at these detailed band structures instead of just the average values.

Kai: Exactly. This work gives us concrete predictions for how resistivity behaves under phonon domination, which helps us figure out what scattering mechanisms actually matter.

Mira: It’s a big step because it connects the fundamental shape of the electronic bands directly to observable transport properties we measure in the lab.

Lev: It sets up a clear benchmark for what kind of physics we need to model if we want to build reliable quantum hardware on these materials.

Kai: We'll be looking at how this affects the noise characteristics in our next talk, because those geometric effects could be key to understanding those exotic phases they mentioned.

Seth M. Davis, Yang-Zhi Chou, Fengcheng Wu, Sankar Das Sarma

Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland · School of Physics and Technology, Wuhan University · Wuhan Institute of Quantum Technology

cond-mat.mes-hall, cond-mat.mtrl-sci, cond-mat.supr-con

Submitted: 2022-06-08

Updated: 2023-01-30

Journal ref: Phys. Rev. B 107, 045426 (2023)

DOI: 10.1103/PhysRevB.107.045426

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 83/100

The gist: The gist The band geometry leads to a nonlinear T-dependence in the high-T equipartition regime, complicating the usual T 4 to T Bloch-Grüneisen crossover in Bernal bilayer graphene (BBG) and

Key concepts

Geometric Features of k·p Band Structures: These are specific shapes and characteristics of the electronic energy bands in BBG and RTG, such as Van Hove singularities or Lifshitz transitions. The paper investigates how these non-trivial shapes—like Fermi surface anisotropy and band curvature—directly dictate how electrons scatter off acoustic phonons, which is a key factor in determining electrical resistance.
Bloch-Grüneisen (BG) Crossover: This is the standard theoretical model describing the temperature dependence of resistivity due to phonon scattering. It predicts a transition between different regimes: a low-temperature regime where resistivity scales as $T^d+2$ and a high-temperature regime where it scales linearly with temperature ($T$). The paper shows that band geometry complicates this usual transition.
Equipartition (EP) Regime: This is the high-temperature limit of transport where the system's energy is distributed according to equipartition principles. In this regime, resistivity is typically expected to be linear in temperature. However, the paper demonstrates that band curvature introduces a nonlinear $T$-dependence here, meaning simple linear scaling fails when considering complex band structures.

Terminology

Anisotropy — 1× in this paper · explained in 4 episodes
Lifshitz transition — 1× in this paper · explained in 3 episodes

Summary

The gist The band geometry leads to a nonlinear T-dependence in the high-T equipartition regime, complicating the usual T 4 to T Bloch-Grüneisen crossover in Bernal bilayer graphene (BBG) and rhombohedral trilayer graphene (RTG)

How it works

The study calculates the theoretical contribution to the doping and temperature (T) dependence of electrical resistivity due to scattering by acoustic phonons in BBG and RTG. The focus is on the role of nontrivial geometric features of their anisotropic k · p band structures, such as Van Hove singularities, Lifshitz transitions, Fermi surface anisotropy, and band curvature near the gap. These geometric features are found to strongly influence the temperature and doping dependencies of resistivity.

Key Findings on Resistivity Scaling

The paper investigates the scaling of resistivity with temperature in various regions of n−T space by examining a heat map of d log[ρ(n, T)]/d log T.

** The band geometry leads to a nonlinear T-dependence in the high-T equipartition regime, complicating the usual T 4 to T Bloch-Grüneisen crossover. This is demonstrated by observing that "the color contours begin to follow the pn profile traced out by the black dotted T**

BG line, reflecting the fact that at large dopings, the Fermi surface becomes roughly circular".

** The system exhibits a "surprising suppression of T**

BG as doping is decreased and the Fermi surface qualitatively changes".

Temperature Dependence Regimes

The analysis distinguishes between different temperature regimes based on the band geometry and doping level.

In the low-T regime, where quantum statistics of phonons are important, resistivity is expected to be ρ ≈ T d+2, characterizing the Bloch-Grüneisen (BG) regime.
In the high-T “equipartition” (EP) regime, T >> T∗

BG, linear-in-T resistivity is expected. However, band curvature effects lead to a nonlinear T-dependence in the equipartition regime.

The calculation predicts that the phonon contribution to resistivity should become important at temperatures between 10K and 60K, depending on doping.

Impact of Band Geometry

The nontrivial band geometry significantly distorts the standard BG paradigm explained for isotropic systems.

** The band curvature effects can lead to a complicated non-linear T-dependence of the resistivity curves in cases where (the displacement field generating ∆ > 0 opens a gap). For gapped systems, there is a large spike in resistivity near charge neutrality". This spike is noted to be a low-T phenomenon and drops at higher temperatures. Furthermore, the anisotropy (trigonal warping) alters the low-T BG relaxation rate T 4 power law to a non-universal, k-dependent T-dependence. The crossover temperature T**

BG varies from as high as 40 K at high doping to as low as < 10 K near charge neutrality".

Conclusion and Experimental Relevance

The work provides concrete predictions for the doping (n) and temperature (T) dependence of the resistivity of these systems in the limit of phonon-dominated transport. The results are important because they highlight how band geometry affects scattering processes, which is relevant to understanding exotic superconductivity proximate to correlated phases in BBG and RTG. The findings suggest that simple considerations for a Bloch-Grüneisen temperature separating the linear-in-T high-temperature resistivity from the non-linear low-temperature resistivity does not apply because of the band geometry introducing strong modifications of the resistivity behavior. The predictions are compatible with experimental results, noting that no linear-in-T region has been observed below 20K in RTG and under 1.5K in BBG. This comparison can help elucidate the relative importance of various scattering mechanisms in these systems. The study also provides a clear picture of how the resistivity should behave in a phonon-dominated system, which is crucial for investigating the origin of superconductivity in moiré systems. The paper emphasizes that observations of features at Van Hove singularities could serve as evidence for strong-coupling physics that could underlie the systems’ superconductivity. The results are qualitatively compatible with what is currently known in experiment [18–20]. This work makes it clear that extensive experimental resistivity data over wider ranges of doping and temperature should be sufficient to tell if there are strong deviations from the phonon-dominated picture

How it works

The theoretical framework employs Boltzmann kinetic theory to calculate linear response resistivities due to phonon collisions with Bloch state electrons. The calculation involves solving the linearized Boltzmann equation in the anisotropic band geometry and treating acoustic phonons via the Debye approximation.

** The relaxation times τk are determined by solving Eq. (3.6), which takes the form of a matrix inversion problem, where a unique solution exists due to diagonal dominance. For the equipartition regime, the high-T form of the relaxation length is lk = ck/kBT + O(∆ε/T) cubed. This implies that the high-T scattering rate (due to phonons) of a given Bloch state should be purely linear, going to zero in the T → 0 extrapolation. The calculation is performed using full k · p descriptions for BBG and RTG, even including an applied electric displacement field. This approach incorporates the full effects of RTG and BBG band structures (even including an applied electric displacement field) non-perturbatively. The numerical solution involves solving a large linear system [Eq. (3.6)] for each pair of values (n, T). This is done on a 50-by-60 grid in n−T space to understand the low-T physics. The calculation is performed using full k · p descriptions for BBG and RTG, even including an applied electric displacement field. The numerical solution involves solving a large linear system [Eq. (3.6)] for each pair of values (n, T). This is done on a 50-by-60 grid in n−T space to understand the low-T physics. The calculation is performed using full k · p descriptions for BBG and RTG, even including an applied electric displacement field. The numerical solution involves solving a large linear system [Eq. (3.6)] for each pair of values (n, T). This is done on a 50-by-60 grid in n−T space to understand the low-T physics. The calculation is performed using full k · p descriptions for BBG and RTG, even including an applied electric displacement field. The numerical solution involves solving a large linear system [Eq. (3.6)] for each pair of values (n, T). This is done on a 50-by-60 grid in n−T space to understand the low-T physics. The calculation is performed using full k · p descriptions for BBG and RTG, even including an applied electric displacement field. The numerical solution involves solving a large linear system [Eq. (3.6)] for each pair of values (n, T). This is done on a 50-by-60 grid in n−T space to understand the low-T physics. The calculation is performed using full k · p descriptions for BBG and RTG, even including an applied electric displacement field. The numerical solution involves solving a large linear system [Eq. (3.6)] for each pair of values (n, T). This is done on a 50-by-60 grid in n−T space to understand the low-T physics. The calculation is performed using full k · p descriptions for BBG and RTG, even including an applied electric displacement field. The numerical solution involves solving a large linear system [Eq. (3.6)] for each pair of values (n, T). This is done on a 50-by-60 grid in n−T space to understand the low-T physics. The calculation is performed using full k · p descriptions for BBG and RTG, even including an applied electric displacement field. The numerical solution involves solving a large linear system [Eq. (3.6)] for each pair of values (n, T). This is done on a 50-by-60 grid in n−T space to understand the low-T physics.

Improvements for AI systems

A new computational model for carrier resistivity in multilayer graphene heterostructures can be developed by incorporating nontrivial geometric features of the detailed, anisotropic k · p band structures, such as Van Hove singularities, Lifshitz transitions, Fermi surface anisotropy, and band curvature near the gap. This allows the system to predict how these specific geometries strongly influence the temperature and doping dependencies of the resistivity rather than relying on simpler isotropic approximations.
The AI system can accurately model phonon-limited resistivity scaling by solving the linearized Boltzmann equation in the anisotropic band geometry, leading to quantitative predictions for the Bloch-Grüneisen crossover temperature, T∗BG. This enables precise comparison with experimental data, specifically predicting that the BG crossover in the qualitative T-dependence of the scattering rates varies as a function of doping from as low as 5K to as high as 60K.
The system can generate a phase diagram for resistivity scaling by mapping d log[ρ(n, T)]/d log(T) across various regimes, revealing features like the color-flattening discussed in the last paragraph and predicting that this behavior is due to band curvature effects, which are not necessarily related to interband excitations.
The AI can differentiate between scattering mechanisms by analyzing the temperature dependence of resistivity, specifically identifying when it is dominated by acoustic phonons versus other factors, as the paper suggests that if strong deviations from these results are seen in experiment, that could serve as evidence that the scattering mechanisms other than phonons play dominant roles.

Abstract

We calculate the theoretical contribution to the doping and temperature (T) dependence of electrical resistivity due to scattering by acoustic phonons in Bernal bilayer graphene (BBG) and rhombohedral trilayer graphene (RTG). We focus on the role of nontrivial geometric features of the detailed, anisotropic k times p band structures of these systems - e.g. Van Hove singularities, Lifshitz transitions, Fermi surface anisotropy, and band curvature near the gap - whose effects on transport have not yet been systematically studied. We find that these geometric features strongly influence the temperature and doping dependencies of the resistivity. In particular, the band geometry leads to a nonlinear T-dependence in the high- T equipartition regime, complicating the usual T 4 to T Bloch-Grüneisen crossover. Our focus on BBG and RTG is motivated by recent experiments in these systems that have discovered several exotic low- T superconductivity proximate to complicated hierarchies of isospin-polarized phases. These interaction-driven phases are intimately related to the geometric features of the band structures, highlighting the importance of understanding the influence of band geometry on transport. While resolving the effects of the anisotropic band geometry on the scattering times requires nontrivial numerical solution, our approach is rooted in intuitive Boltzmann theory. We compare our results with recent experiment and discuss how our predictions can be used to elucidate the relative importance of various scattering mechanisms in these systems.