Detection of spin- and valley-polarized states in van der Waals materials via thermoelectric and non-reciprocal transport

This paper predicts that thermoelectric and current rectification effects in hybrid junctions of Ising superconductors and valley-polarized van der Waals materials can serve as experimentally accessible probes for detecting spin- and valley-polarized states.

The Gist

Imagine you have a tiny, microscopic city built from layers of ultra-thin materials (like graphene or special crystals). In this city, electrons don't just move around; they have "identities" based on two things: their spin (which way they are spinning, like a top) and their valley (which neighborhood of the city they live in).

Usually, these electrons are a mixed crowd: some spin up, some spin down, living in different neighborhoods. But in certain special materials, the electrons get "polarized." This means they all agree to spin the same way and live in the same neighborhood. Scientists call this a spin- and valley-polarized state.

The big problem? It's very hard to "see" or measure this agreement. You can't just take a photo of an electron's spin. You need a special detector.

The Detective's Toolkit: The "Ising Superconductor"

The authors of this paper propose a new, clever detective tool: a hybrid junction. Think of it as a bridge connecting two different worlds:

1. World A (The Polarized Material): The place where the electrons are all in agreement (the "polarized" crowd).
2. World B (The Ising Superconductor): A special material where electrons are already "locked" in a specific way by the material's own internal rules (spin-orbit coupling).

The bridge between them is a tiny tunnel. Electrons have to jump across this tunnel to get from one side to the other.

The Magic Trick: How the Bridge Works

Here is the magic: The bridge is picky. It only lets electrons jump if their "identity" (spin and valley) matches a specific pattern.

The authors discovered that if you apply a magnetic field to the Ising Superconductor, it creates a unique "filter" inside the bridge. This filter interacts with the polarized electrons on the other side.

They found two main ways to detect the hidden agreement of the electrons:

1. The "Thermoelectric" Effect (The Heat-to-Electricity Trick)

Imagine you have a hot side and a cold side of the bridge.

• Normal situation: If you heat one side, electrons jiggle and move, creating a tiny voltage. This is standard.
• The special situation: Because the Ising Superconductor has this unique "valley-locking" rule, and the other side has "valley-polarized" electrons, the heat causes a massive, unexpected surge of electricity.
• The Analogy: Imagine a crowd of people (electrons) trying to leave a stadium. Usually, they just walk out the doors. But if the stadium has a special rule where only people wearing red shirts can leave through the North door, and you heat up the North side, suddenly everyone rushes out that door in a specific direction, creating a huge flow. The paper predicts this "surge" happens specifically because of the valley-polarized state.

2. The "Rectification" Effect (The One-Way Street)

This is like a diode in a flashlight, but for electrons.

• Normal situation: If you push electrons from Left to Right, they flow. If you push them Right to Left, they flow back the same amount. It's a two-way street.
• The special situation: With this special bridge, the electrons flow easily one way but get stuck going the other way.
• The Analogy: Think of a turnstile at a subway station. Usually, you can push through it in either direction with equal effort. But in this setup, the turnstile is rigged so that if you push from the "Valley A" side, it swings open easily. If you push from the "Valley B" side, it jams. This "one-way street" behavior is a dead giveaway that the electrons on the other side are polarized.

Why Does This Matter?

For a long time, scientists have used light (lasers) or very strong magnets to try to see these "valley" states. But those methods are clunky and don't work well with the new, tiny electronic devices we are building.

This paper says: "Stop using lasers! Just build this bridge and measure the electricity."

• It's direct: You just measure voltage or current.
• It's compatible: It works with the superconducting materials used in future quantum computers.
• It's a fingerprint: The way the electricity behaves (the heat effect or the one-way street) is a unique fingerprint that proves the electrons are polarized.

The Bottom Line

The authors have designed a theoretical "trap" for electrons. By connecting a special superconductor to a polarized material, they created a system where the electrons' hidden "valley" identity forces them to behave strangely with heat and electricity. If we build this in a lab, we can finally "see" these invisible quantum states using simple electrical wires, paving the way for a new type of electronics called valleytronics (using valleys instead of just charge to store information).

Technical Summary

1. Problem Statement

While valley physics (the use of momentum-space valleys as an electronic degree of freedom) is central to emerging quantum phenomena in van der Waals (vdW) materials like twisted bilayer graphene and transition metal dichalcogenides (TMDs), experimental detection remains challenging.

• Current Limitations: Existing methods rely heavily on optical probes, high-field magnetotransport (Quantum Hall effects), or non-local transport.
• The Gap: There is a lack of straightforward electrical transport probes capable of detecting valley-polarized states, particularly in hybrid structures involving superconductors.
• Specific Challenge: Detecting the interplay between Ising superconductivity (found in few-layer TMDs with strong spin-orbit coupling and out-of-plane spin locking) and spin-valley polarized states (found in rhombohedral graphene or twisted bilayer graphene) using standard electrical measurements.

2. Methodology

The authors propose a theoretical model for a hybrid junction consisting of:

1. Ising Superconductor (ISC): A few-layer TMD (e.g., MoS$_2$, WS$_2$) subjected to an in-plane Zeeman field ($h$).
2. Spin-Valley Polarized Material (X): A material hosting polarized states, such as rhombohedral (ABC-stacked) graphene in a quarter-metal phase or twisted bilayer graphene.
3. Tunnel Barrier: A barrier allowing momentum- and valley-conserving tunneling (e.g., an STM tip or a planar contact).

Theoretical Framework:

• Hamiltonians:
  • The ISC is modeled using a Bogoliubov-de-Gennes Hamiltonian including Ising spin-orbit coupling ($\Delta_{so}$), an in-plane exchange field ($h$), and superconducting order parameters ($\Delta$ for singlet, $\psi$ for triplet).
  • Material X is modeled with a Hamiltonian including valley polarization ($m_v$) and spin magnetization ($m_z, m_x$).
• Transport Calculation:
  • The authors calculate charge and heat currents using the standard tunneling approach.
  • They analyze the spectral function of the ISC, identifying three key density of states (DOS) components:
    • $N_0$: Conventional spin-averaged DOS (even in energy).
    • $N_x$: Spin-odd component due to Zeeman splitting (odd in energy).
    • $N_z$: Ising DOS, a unique component arising from the interplay of Ising SOC and the Zeeman field. It is odd in both energy and valley index.
• Key Mechanism: The tunneling current couples the valley-odd Ising DOS ($N_z$) of the superconductor with the spin-valley polarized states of material X. This coupling breaks electron-hole symmetry, leading to non-linear transport effects.

3. Key Contributions

The paper predicts two distinct transport signatures that serve as direct electrical probes for valley-polarized states:

A. Ising Thermopower (Thermoelectric Effect)

• Mechanism: A temperature gradient ($\delta T$) across the junction drives a current. Due to the coupling between the valley-odd $N_z$ and the spin-valley polarization of X, a net thermoelectric voltage is generated even in the absence of a bias voltage.
• Distinctive Feature: The thermoelectric coefficient $\alpha_z$ (associated with $N_z$) is even with respect to the applied magnetic field ($h$), i.e., $\alpha_z(-h) = \alpha_z(h)$. This contrasts with the conventional spin-splitting thermopower ($\alpha_x$), which is odd in $h$.
• Significance: This symmetry difference allows experimentalists to distinguish the valley-polarized signal from conventional spin-splitting effects.

B. Current Rectification (Non-Reciprocal Transport)

• Mechanism: In the non-linear regime (finite voltage bias), the breaking of electron-hole symmetry leads to current rectification, where $I(V) \neq -I(-V)$. The system acts as a diode.
• Distinctive Feature: Similar to the thermoelectric effect, the rectification coefficient $R_z$ (driven by $N_z$) is even in the magnetic field ($R_z(-h) = R_z(h)$), whereas the conventional rectification $R_x$ is odd.
• Advantage: Rectification is a purely electrical probe that does not require maintaining a temperature gradient, making it potentially easier to implement in AC measurements (generating DC or second-harmonic responses).

4. Results

• Analytical Expressions: The authors derived analytical formulas for the thermoelectric coefficient ($\alpha_z$) and rectification factor ($R_z$) in the limit of large SOC and low temperature.
  • $\alpha_z \propto \frac{h^2}{\Delta_{so}(\Delta_{so}^2 + h^2)}$
  • $R_z \propto \tanh(\frac{eV}{2k_B T})$
• Parameter Dependence:
  • SOC Strength: The effects are maximized when the Ising SOC ($\Delta_{so}$) is comparable to the superconducting gap ($\Delta$). Very strong SOC suppresses the effect (power-law decay), suggesting that materials like MoS$_2$ (moderate SOC) or multilayer TMDs (reduced effective SOC) are better candidates than WS$_2$ or NbSe$_2$ (very strong SOC).
  • Temperature: Finite temperature is required to activate quasiparticles for transport, but the effect persists at low $T$.
  • Disorder: The effects are robust against intra-valley scattering but sensitive to inter-valley scattering. However, as long as the inter-valley scattering rate is below the superconducting gap, the spectral features (like "mirage gaps") and the predicted transport signatures remain observable.
• Simulation: Numerical plots (Figs. 3–5) demonstrate that both thermopower and rectification reach significant magnitudes under realistic experimental conditions.

5. Significance and Experimental Outlook

• Direct Valley Probe: This work provides the first proposal for a direct electrical transport probe of valley-polarized states in hybrid superconducting structures, bypassing the need for complex optical or high-field setups.
• Platform Recommendations:
  • Optimal Junction: A few-layer TMD (e.g., MoS$_2$) coupled to rhombohedral graphene or twisted bilayer graphene.
  • Alternative: If material X only has valley polarization (no spin), a spin-filtering barrier (e.g., a ferromagnetic insulator) can be used to introduce the necessary spin selectivity.
• Feasibility: The proposed setup is compatible with existing technologies such as Scanning Tunneling Microscopy (STM) with quantum twisting capabilities or gate-tunable van der Waals heterostructures.
• Impact: The ability to detect and manipulate spin-valley polarized states via simple electrical measurements opens new avenues for valleytronics and the study of interaction-driven symmetry-broken states in 2D materials.

In summary, the paper establishes that the unique interplay between Ising superconductivity and spin-valley polarization creates a "valley-odd" spectral signature. This signature manifests as a magnetic-field-even thermoelectric voltage and current rectification, offering a robust and accessible method to experimentally verify valley-polarized quantum states.