Interplay of Rashba and valley-Zeeman splittings in weak localization of spin-orbit coupled graphene

This paper develops a weak localization theory for graphene heterostructures with strong Rashba and valley-Zeeman spin-splittings, demonstrating that while valley-Zeeman splitting alone does not affect weak localization, its interplay with Rashba coupling and intervalley scattering can reverse the sign of the anomalous magnetoconductivity.

The Gist

Imagine a sheet of graphene (a material made of a single layer of carbon atoms) as a vast, two-dimensional dance floor. On this floor, electrons are the dancers. In a perfect world, these dancers would move in perfect sync, creating a beautiful, constructive interference pattern that makes the dance floor conduct electricity very well. This is the concept of Weak Localization: a quantum effect where electrons act like waves that reinforce each other, making it easier for current to flow.

However, in the real world, things get messy. The paper by L. E. Golub explores what happens when we introduce two specific types of "noise" or "rules" to this dance floor, which change how the electrons dance and, consequently, how electricity flows.

Here is the breakdown of the paper's findings using simple analogies:

The Two New Rules of the Dance Floor

The paper looks at graphene placed next to special materials (like topological insulators) that impose two new rules on the electron dancers:

1. The Rashba Splitting (The "Spin-Flip" Rule): Imagine a rule that forces the dancers to spin their bodies as they move. If they spin one way, they get pushed left; if they spin the other way, they get pushed right. This is the Rashba effect.
2. The Valley-Zeeman Splitting (The "Valley-Specific" Rule): The dance floor has two distinct zones (called "valleys"). This rule says that dancers in Zone A must spin clockwise, while dancers in Zone B must spin counter-clockwise. This is the Valley-Zeeman effect.

There is also a third factor: Inter-valley Scattering. This is like a bouncer who occasionally kicks a dancer from Zone A over to Zone B, or vice versa, disrupting their rhythm.

The Main Discovery: A Tug-of-War

The core of the paper is a tug-of-war between these rules and how they affect the "Weak Localization" (the helpful interference).

1. The Rashba Effect Alone:
If you only have the "Spin-Flip" rule (Rashba) and no bouncer (no inter-valley scattering), the dancers get so confused by their spinning that they stop reinforcing each other. Instead of helping the current flow, they start fighting it. This flips the sign of the effect: the material goes from helping electricity flow to resisting it. In physics terms, this is a switch from "Weak Antilocalization" (resistance) to "Weak Localization" (conductivity).

2. The Valley-Zeeman Effect Alone:
If you only have the "Valley-Specific" rule (Valley-Zeeman) but no Rashba effect, nothing changes. The dancers in Zone A and Zone B are just doing their own thing, but since they aren't spinning wildly, the interference pattern remains the same. The paper confirms that without the Rashba rule, the Valley-Zeeman rule is invisible to this specific quantum effect.

3. The Tug-of-War (Rashba vs. Valley-Zeeman):
This is where it gets interesting. When you have both rules active:

• The Rashba rule tries to make the dancers spin wildly and mess up the interference (causing resistance).
• The Valley-Zeeman rule tries to lock the dancers into specific zones with specific spins.
• The Result: If the Valley-Zeeman rule is strong enough, it actually "calms down" the Rashba chaos. It forces the dancers into a state where they stop interfering with each other in a way that causes resistance. The paper shows that a strong Valley-Zeeman effect can flip the sign back again, restoring the original behavior (or even reversing it further), effectively canceling out the Rashba effect's influence.

The Role of the "Bouncer" (Inter-valley Scattering)

The paper also introduces the "bouncer" (inter-valley scattering).

• Without the Valley-Zeeman rule: If the bouncer kicks dancers between zones frequently, it disrupts the rhythm enough to flip the sign of the effect, turning resistance back into conductivity.
• With a strong Valley-Zeeman rule: If the Valley-Zeeman rule is already strong, adding the bouncer flips the sign again, reversing the previous outcome.

The "Sign Reversal" Analogy

Think of the electrical conductivity correction as a volume knob on a speaker.

• Normal state: The volume is low (positive magnetoconductivity).
• Rashba effect: Turns the volume knob the other way (negative magnetoconductivity).
• Valley-Zeeman effect: If Rashba is on, a strong Valley-Zeeman effect turns the knob back toward the original position.
• Inter-valley scattering: Acts like a second hand that can also turn the knob, but its direction depends on whether the Valley-Zeeman rule is present or not.

The Bottom Line

The paper provides a mathematical "recipe" (analytical expressions) to predict exactly what will happen to the electrical flow in these graphene sheets. It tells us that:

1. Valley-Zeeman splitting does nothing on its own, but it is a powerful counter-force to Rashba splitting.
2. Inter-valley scattering (dancers jumping zones) always changes the outcome, but the direction of that change depends on how strong the Valley-Zeeman rule is.

By understanding this delicate balance, scientists can use these formulas to figure out exactly how strong the spin-orbit interactions are in real graphene devices just by looking at how they conduct electricity in a magnetic field. It's like being able to tell how strong the wind is just by watching how a specific type of leaf dances on the ground.

Technical Summary

Technical Summary: Interplay of Rashba and Valley-Zeeman Splittings in Weak Localization of Spin-Orbit Coupled Graphene

Problem Statement
Graphene heterostructures proximitized by strongly spin-orbit coupled materials, such as transition metal dichalcogenides and topological insulators, exhibit significant Rashba spin splitting ($\lambda_R$) and valley-Zeeman splitting ($\lambda_{VZ}$). While weak localization (WL) and weak antilocalization (WAL) are well-understood in ordinary systems and pristine graphene, the quantum transport properties of these heterostructures are complicated by the simultaneous presence of three factors: Rashba splitting, valley-Zeeman splitting, and inter-valley scattering. Specifically, the Rashba splitting and inter-valley scattering are known to induce a sign reversal of the conductivity correction (transitioning between WAL and WL), but the specific role of the valley-Zeeman splitting in this interplay, particularly when Rashba coupling is present, had not been fully analytically characterized.

Methodology
The author develops a theoretical framework for weak localization in graphene based on the Hamiltonian of spin-orbit coupled graphene, which includes Dirac fermion velocity, Rashba coupling, and valley-Zeeman terms. The quantum correction to conductivity is calculated via the Cooperon, representing the interference amplitude of time-inversion coupled particle loops.

Key methodological steps include:

1. Operator Formalism: The Cooperon is treated as the inverse of an operator $L$. In the absence of valley-Zeeman splitting, this operator acts on spin triplet and singlet channels. The inclusion of valley-Zeeman splitting introduces a term linear in the spin difference operator $L$, distinct from the total spin operator $S$, leading to a mixing of singlet and triplet interference channels.
2. Matrix Inversion: The problem is reduced to inverting specific matrices representing the interference channels.
   • In the absence of inter-valley scattering, a 4-rank matrix is inverted to account for the mixing of spin channels ($t_1, t_0, t_{-1}, s$) induced by $\lambda_{VZ}$.
   • In the presence of inter-valley scattering, the theory is generalized to a 16-channel problem (combining valley and spin degrees of freedom). This reduces to inverting a 6-rank matrix for the coupled channels, while other channels are treated as decoupled or independent.
3. Analytical Derivation: The magnetoconductivity $\Delta\sigma$ is derived analytically for arbitrary relations between the Rashba splitting, valley-Zeeman splitting, and inter-valley scattering rates. The results are expressed in terms of the digamma function $\psi$ and specific polynomial roots derived from the matrix traces.

Key Results
The paper derives analytical expressions for the anomalous magnetoconductivity under various regimes:

• Absence of Rashba Splitting: The valley-Zeeman splitting has no effect on the weak localization correction. The system behaves according to the standard Hikami-Larkin-Nagaoka (HLN) formula with a factor of 4 (due to two valleys and two spin channels), resulting in negative magnetoconductivity (WAL).
• Presence of Rashba Splitting (No Inter-valley Scattering):
  • At $\lambda_{VZ} = 0$, strong Rashba coupling leads to WL (positive magnetoconductivity at low fields).
  • As $\lambda_{VZ}$ increases, it competes with the Rashba effect. For large $\lambda_{VZ}$ (specifically $\Delta_\phi \gtrsim 30$), the system undergoes a transition back to WAL (negative magnetoconductivity). This occurs because the valley-Zeeman splitting aligns spins along the $\pm z$ direction, effectively decoupling the spin subsystems and suppressing the Rashba-induced WL.
• Effect of Inter-valley Scattering:
  • Without Valley-Zeeman Splitting: Inter-valley scattering reverses the situation observed in pure Rashba systems. It drives a transition from WAL to WL, resulting in negative magnetoconductivity at low fields.
  • With Large Valley-Zeeman Splitting: Inter-valley scattering induces a reciprocal transition (WL to WAL). In the presence of large $\lambda_{VZ}$, the magnetoconductivity is negative (WAL-like) without scattering; however, introducing inter-valley scattering creates a minimum in the magnetoconductivity curve and causes a sign reversal to positive values at high fields.

Significance and Claims
The paper claims to provide the first analytical expressions for quantum corrections to magnetoconductivity in graphene heterostructures where Rashba splitting, valley-Zeeman splitting, and inter-valley scattering coexist with arbitrary relative strengths.

The primary significance lies in demonstrating that the valley-Zeeman splitting acts as a control parameter that can reverse the sign of the conductivity correction in Rashba-coupled graphene, effectively suppressing the Rashba-induced WL. Furthermore, the work clarifies that inter-valley scattering and valley-Zeeman splitting can induce reciprocal transitions between WL and WAL depending on the presence or absence of the other. The developed theory is presented as a tool to adequately determine spin- and valley-dependent parameters of graphene heterostructures from experimental magnetoconductivity data, particularly in regimes beyond the motional-narrowing approximation.