Theory of weak localization in graphene with spin-orbit interaction

The Gist

The Big Picture: A Traffic Jam of Electrons

Imagine a crowded city street (the graphene sheet) where people (electrons) are trying to walk from point A to point B. Usually, they bump into obstacles (impurities) and scatter randomly. However, because electrons act like waves, they can also interfere with each other, much like ripples in a pond.

Weak Localization is a phenomenon where these "ripples" accidentally line up perfectly when a person turns around and walks back the way they came. This constructive interference makes it harder for them to move forward, effectively creating a traffic jam. In normal metals, this makes the material slightly more resistant to electricity.

However, in graphene, things get weird. Because of the unique way electrons move in this material, they usually get a "twist" (called a Berry phase) that makes them interfere destructively when they turn around. This usually helps them move, reducing resistance. This is called Weak Antilocalization.

The New Ingredient: The "Spin" Twist

The paper focuses on what happens when we add Spin-Orbit Interaction to graphene. Think of "spin" as the electron's internal compass needle. When an electron moves, the "Rashba effect" (caused by nearby materials) acts like a strong wind that forces these compass needles to spin and change direction as the electron travels.

The Old Map vs. The New Map

For a long time, scientists used a standard formula (the Hikami-Larkin-Nagaoka or HLN formula) to predict how this "wind" affects the traffic jam. They assumed the wind just made the compass needles spin so fast that they lost their memory of direction (dephasing).

The Paper's Discovery:
The author, L. E. Golub, argues that the old map is wrong for this specific type of graphene.

• The Old View: The wind just scrambles the compass needles (spin dephasing).
• The New View: The wind doesn't just scramble the needles; it acts like a magnetic steering wheel (a "spin-orbit vector potential") that actively pushes the electrons in specific directions depending on which way their compass is pointing.

Because of this "steering wheel" effect, the math changes completely. The old formula (HLN) is like trying to navigate a city with a map that only shows potholes, ignoring the fact that there are also one-way streets and traffic lights.

What the New Theory Says

The author developed a new, more complex mathematical expression to describe this behavior.

1. It's not just about losing memory: The effect isn't just that electrons forget their spin; it's that the spin actively changes how they interfere with themselves.
2. The Result: The new formula predicts a different pattern of electrical resistance when you apply a magnetic field. It shows that the "anti-traffic jam" effect (Weak Antilocalization) happens much faster and more strongly than the old formula predicted, even with a moderate amount of "wind" (spin-orbit coupling).
3. Why it matters: If scientists use the old formula to analyze experiments on graphene, they will get the wrong numbers for how strong the spin-orbit interaction actually is. The new formula is the correct tool to measure these properties accurately.

A Simple Analogy: The Spinning Top

Imagine two spinning tops (electrons) trying to walk in a circle and meet back at the start.

• Without the wind: They spin in sync and meet perfectly.
• With the old theory: The wind makes them wobble so much they forget which way they were spinning, so they don't meet well.
• With the new theory (this paper): The wind doesn't just make them wobble; it tilts their spinning axis in a specific way as they move. This tilt changes the path they take, making them meet in a completely different pattern than the "wobble" theory predicted.

Who is this for?

The paper specifically mentions that this theory is designed for graphene heterostructures, particularly those stacked with materials called transition metal dichalcogenides (TMDCs). These are the specific setups where this "steering wheel" effect is strong enough to matter.

Summary

This paper fixes a broken tool. Scientists have been using an old formula to measure how electrons behave in special graphene setups. The author shows that the old formula misses a crucial "steering" effect caused by the electrons' spin. By using the new, more complex formula, researchers can finally get the correct measurements of how these materials work.

Technical Summary

Technical Summary: Theory of Weak Localization in Graphene with Spin-Orbit Interaction

Problem Statement
Weak localization (WL) in graphene is a quantum interference phenomenon sensitive to Berry phases, intervalley scattering, and spin-orbit coupling (SOC). While the anomalous magnetoresistance in standard two-dimensional semiconductors with Rashba SOC is often described by the Hikami-Larkin-Nagaoka (HLN) formula, previous studies in those systems indicate that the HLN expression fails to capture the physics when Rashba splitting is present. In graphene, the situation is further complicated by the unique Berry phase ($\pi$) of Dirac fermions and the potential for strong intervalley scattering. Existing theoretical frameworks for graphene heterostructures (e.g., graphene on transition metal dichalcogenides) frequently rely on the HLN function to extract SOC parameters from experimental magnetoresistance data. However, it is unclear whether the standard HLN approach remains valid for graphene when Rashba splitting induces a spin-orbit vector potential that cannot be reduced to simple spin dephasing.

Methodology
The paper develops a microscopic theory of WL in graphene with Rashba SOC by constructing a Hamiltonian for electrons on two sublattices in two valleys with spin projections. The Hamiltonian includes Dirac kinetic energy, Rashba SOC, Kane-Mele SOC, a staggered sublattice potential (orbital gap), and trigonal warping. Disorder is modeled via spin-independent and spin-dependent scattering terms, distinguishing between symmetric and asymmetric perturbations.

The authors transform the problem into a basis of conduction-band states characterized by momentum $p$ and valley index $K_{\pm}$. In this basis, the Hamiltonian resembles that of spin-split massive electrons with angle-dependent scattering, modified by a phase factor $e^{-i\theta/2}$ in the scattering amplitude due to the Berry phase. The core of the calculation involves deriving the equation for the Cooperon $C(q)$, which describes the interference of time-reversed electron paths. Crucially, the Cooperon equation includes a term linear in the total spin operator $S$ and the magnetic field momentum $q$, representing a spin-orbit vector potential ($A_{SO}$).

The authors solve the Cooperon equation in a magnetic field normal to the graphene layer. They project the dephasing operator onto valley and spin singlet ($s$) and triplet ($t_0, t_1$) interference channels, calculating nine distinct dephasing rates ($\Gamma^l_j$) that account for intervalley, intravalley, spin-orbit, and spin-valley scattering processes. The resulting magnetoconductivity is expressed as a sum of contributions from these channels, utilizing a generalized function $F_t$ for the triplet sector and the standard HLN function $F$ for the singlet sector.

Key Contributions and Results

1. Derivation of a New Magnetoconductivity Formula: The paper derives an analytical expression for the WL-induced anomalous magnetoconductivity that differs fundamentally from the traditional HLN formula. The derived expression (Eq. 6) incorporates a four-parametric function $F_t$ for the triplet contribution, which arises because the Rashba splitting generates a spin-orbit vector potential that mixes different spin interference channels.
2. Role of the Spin-Orbit Vector Potential: The theory demonstrates that the Rashba effect in graphene is not merely a spin dephasing mechanism. Instead, it introduces a vector potential term in the Cooperon equation that is linear in spin and momentum. This term prevents the reduction of the problem to simple dephasing rates, necessitating the more complex functional form derived in the paper.
3. Comparison with Existing Models:
   • McCann-Fal'ko (MF) Theory: The paper compares its results with the MF expression (Ref. [21]), which treats Rashba SOC solely as a dephasing rate. The authors show that the MF formula is only accurate in the limits of zero Rashba splitting or extremely large splitting where triplet contributions are negligible. At intermediate Rashba strengths ($B_R \gtrsim 3B_\phi$), the MF theory significantly deviates from the results of the present work, underestimating the weak antilocalization (WAL) effect.
   • Limiting Cases: In the absence of Rashba splitting ($B_R=0$), the derived formula reduces to an HLN-like expression. In the limit of fast valley-triplet relaxation, the formula simplifies to a form dependent on three dephasing rates ($\Gamma_\phi, \Gamma_{SO}, \Gamma_{asy}$), recovering the structure of Ref. [21] only when spin-orbit scattering is absent.
4. Impact of Scattering Types: The theory distinguishes between symmetric and asymmetric spin-orbit scattering. Results indicate that asymmetric scattering processes induce a stronger WAL effect than symmetric ones for a given Rashba splitting magnitude.
5. Magnetic Field Orientation: The paper briefly addresses in-plane magnetic fields, noting that the Zeeman term suppresses the singlet channel. In the limit of large Zeeman splitting, the theory reverts to the form with vanishing Rashba-induced vector potential effects.

Significance
The paper claims that the developed theory is essential for the correct interpretation of experimental data in graphene heterostructures with strong spin-orbit coupling, such as those involving transition metal dichalcogenides (TMDCs). By demonstrating that the Rashba interaction acts via a spin-orbit vector potential rather than simple dephasing, the authors argue that using the traditional HLN formula to extract spin-orbit and dephasing parameters from experimental magnetoresistance data is inadequate. The new expression allows for the adequate determination of these parameters, correcting potential errors in previous analyses that relied on the HLN approximation. The theory is specifically applicable to monolayer graphene systems where linear momentum terms in the Hamiltonian generate the spin-orbit vector potential; it notes that this specific effect is absent in bilayer graphene and TMDC layers, where HLN-like theories remain valid.