Orbital and Spin Nernst Effects in Monolayers of Transition Metal Dichalcogenides

This paper demonstrates that monolayer transition metal dichalcogenides serve as an ideal platform for observing both orbital and spin Nernst effects, revealing that the orbital Nernst effect arises intrinsically without spin-orbit coupling while the spin Nernst effect scales with it, with both phenomena being tunable via doping in semiconducting MoS$_2$ and intrinsically present in metallic NbS$_2$.

The Gist

Imagine a tiny, ultra-thin sheet of material, just one atom thick, made of a sandwich of metal atoms and sulfur (or selenium) atoms. Scientists call these "Transition Metal Dichalcogenides" (TMDCs), but let's just call them super-thin metal sandwiches.

This paper is about discovering how heat can make invisible "spins" and "orbits" of electrons move sideways in these sandwiches, creating new ways to potentially harvest energy.

Here is the breakdown of their discovery using simple analogies:

1. The Setup: A Crowded Dance Floor

Imagine the electrons in this material are dancers on a crowded floor. Usually, when you push them (with electricity), they move forward. But sometimes, if the floor has a specific texture, they might get pushed sideways instead.

• The "Spin" Dancers: Some dancers have a natural "spin" (like a top spinning).
• The "Orbit" Dancers: Other dancers are moving in specific circular paths around the center of the atom (like planets orbiting the sun). This is their "orbital" motion.

For a long time, scientists thought the "orbit" part of the dance was frozen and useless in solid materials. This paper says: "No, it's actually very active!"

2. The Main Discovery: The "Thermal Sidewalk"

The researchers found that if you heat up one side of this super-thin sandwich and keep the other side cool, the electrons don't just flow from hot to cold. Instead, they start flowing sideways (perpendicular to the heat).

They call this the Nernst Effect.

• Spin Nernst Effect: The "spinning" dancers drift to the right.
• Orbital Nernst Effect: The "orbiting" dancers drift to the left.

The Big Surprise:
Usually, to make these dancers move sideways, you need a special ingredient called "Spin-Orbit Coupling" (a heavy, complex interaction).

• The Paper's Claim: The Orbital effect (the orbiting dancers) does not need this heavy ingredient at all. It happens naturally just because of the shape of the dance floor. This means it can happen in lighter, simpler materials, not just heavy ones.

3. The Two Types of Sandwiches

The team tested two specific types of these metal sandwiches:

• The "Insulator" Sandwich (MoS2):

  • Think of this as a dance floor where the dancers are stuck in their seats. They can't move freely unless you give them a ticket (add extra electrons or remove some to "dope" the material).
  • Result: If you don't add tickets, the sideways flow stops. But if you add the right amount of "doping," the sideways flow turns on.

• The "Metal" Sandwich (NbS2):

  • Think of this as a dance floor where the dancers are already running wild and free.
  • Result: The sideways flow happens naturally, without needing any extra tickets or doping. It's always on.

4. How to See It (The Experiment)

Since you can't see these tiny electron flows with your eyes, the paper proposes a way to detect them using a "magnetic camera" (called MOKE).

• The Setup: Imagine a long, thin strip of the material. You heat one side of the strip.
• The Effect: The "spin" and "orbit" dancers rush to the edges of the strip.
• The Detection: Because these dancers carry a tiny magnetic personality, they create a weak magnetic field at the edges. The researchers suggest shining a laser at the edges; the laser light will twist slightly (like a steering wheel turning) if these magnetic fields are there.
• The Trick: The "spin" dancers and "orbit" dancers twist the laser in opposite directions. This allows scientists to tell them apart, like seeing a left-turning car and a right-turning car in the same lane.

5. Why Does This Matter?

The paper suggests this is a new way to harvest energy.

• Imagine your computer getting hot. Instead of that heat just wasting away, this effect suggests we could turn that waste heat into a useful "current" of orbital or spin information.
• It opens the door to "Orbitronics," a new field where we use the "orbit" of electrons (instead of just their charge or spin) to build faster, cooler, and more efficient devices.

Summary in One Sentence

This paper proves that in ultra-thin metal sheets, heat can naturally push electrons sideways based on their "orbit" (without needing heavy atoms), and this effect is strongest in metallic versions of these sheets, offering a new way to turn waste heat into useful electronic signals.

Technical Summary

Technical Summary: Orbital and Spin Nernst Effects in Monolayers of Transition Metal Dichalcogenides

Problem Statement
While spintronic phenomena such as the spin Hall and spin Nernst effects are well-established, their reliance on spin-orbit coupling (SOC) limits their magnitude and restricts applicability primarily to materials containing heavy elements. In contrast, orbital-driven effects (orbitronics) offer a pathway to larger conductivities without requiring SOC. Although the orbital Hall effect (OHE) has been extensively studied and experimentally observed, its thermal analogue, the orbital Nernst effect (ONE), remains underexplored. Existing proposals for the ONE have largely been limited to inversion-symmetric systems. This work addresses the gap in understanding the ONE and the spin Nernst effect (SNE) in non-centrosymmetric monolayer transition metal dichalcogenides (TMDCs), specifically investigating whether these effects can exist intrinsically in metallic systems or require doping in insulating ones.

Methodology
The authors employ a multi-scale theoretical approach combining analytical low-energy models with full Brillouin-zone (BZ) tight-binding calculations:

1. Valley Model Analysis: A low-energy effective Hamiltonian is constructed around the $K$ and $K'$ valley points of the BZ for $MX_2$ monolayers. This model incorporates electronic hopping parameters, energy gaps, and SOC strength ($\zeta$). The authors derive analytical expressions for orbital and spin Berry curvatures, orbital and spin Hall conductivities (OHC, SHC), and subsequently the Nernst conductivities (ONC, SNC) using the Mott relation (energy derivatives of Hall conductivities at the Fermi energy).
2. Tight-Binding Calculations: To capture realistic material features beyond the valley approximation, the authors construct a $d$-orbital tight-binding Hamiltonian for two representative systems: insulating monolayer $2H$-MoS$_2$ and metallic $2H$-NbS$_2$. Parameters are extracted using the $N$-th order muffin-tin orbital (NMTO) method, including onsite energies and hopping up to the 4th nearest neighbor. SOC is included explicitly.
3. Numerical Computation: The full BZ is sampled (100 $\times$ 100 k-mesh) to compute the distribution of orbital and spin Berry curvatures. OHC and SHC are calculated by summing these curvatures over occupied states. ONC and SNC are then derived via numerical differentiation with respect to energy.
4. Experimental Proposal: A device geometry is proposed utilizing the magneto-optic Kerr effect (MOKE) to detect the accumulation of out-of-plane spin and orbital moments at sample edges induced by an in-plane temperature gradient.

Key Contributions and Results

• Existence of ONE without SOC: The study demonstrates that the orbital Nernst effect does not require SOC, distinguishing it from the spin Nernst effect. Analytical results show that the orbital Berry curvature exists even when $\zeta = 0$, leading to a non-zero OHC. However, the ONE (the thermal derivative) vanishes in undoped insulating systems where the Fermi energy lies within a band gap.
• Role of Doping and Metallicity:
  • Insulating MoS$_2$: The ONE and SNE are absent in the intrinsic insulating state. They can be induced only through electron or hole doping, which shifts the Fermi energy into the bands, creating a non-zero density of states at the Fermi level. The paper notes that electron-doped MoS$_2$ exhibits significantly larger ONC than hole-doped MoS$_2$, driven by inter-orbital $d_{xz}$-$d_{yz}$ hopping at the $\Gamma$ point.
  • Metallic NbS$_2$: In contrast to MoS$_2$, both ONE and SNE are intrinsically present in metallic NbS$_2$ without the need for doping.
• Dependence on SOC: While the ONE is independent of SOC, the SNE scales linearly with the SOC strength ($\zeta$) and vanishes in its absence. The magnitude of the spin-driven effects is consistently smaller than the orbital-driven effects due to the relatively weak SOC in these materials.
• Berry Curvature Origins:
  • In hole-doped systems (valence bands), the effects are dominated by Berry curvature contributions near the $K$ and $K'$ valley points.
  • In electron-doped systems (conduction bands), particularly for MoS$_2$, the $\Gamma$ point plays a crucial role, driven by specific inter-orbital hopping parameters.
• Sign Reversal: The calculations reveal a sign reversal in OHC and ONC for NbS$_2$ at specific Fermi energies (around 2.43 eV), attributed to the sign change in the orbital Berry curvature distribution.

Significance and Claims
The paper claims to extend the study of the ONE from centrosymmetric systems to non-centrosymmetric TMDCs, identifying this material family as a promising platform for observing these effects. The work highlights that:

1. Metallicity is a key criterion: Metallic TMDCs are intrinsic hosts for the ONE, whereas insulating TMDCs require extrinsic doping.
2. Tunability: Gate-voltage control and electronic doping serve as effective mechanisms to tune the magnitude of both ONE and SNE.
3. Energy Harvesting: As thermally driven effects, ONE and SNE offer potential for energy harvesting by converting thermal gradients into orbital and spin currents.
4. Experimental Feasibility: The authors propose a specific experimental setup using MOKE to detect the out-of-plane accumulation of moments, noting that the opposite polarity of spin and orbital moments allows for distinguishing between SNE and ONE via sign reversal in the Kerr signal.

The authors maintain a modest tone regarding applications, framing the work as a theoretical demonstration and a proposal for detection rather than a report of experimental realization. They suggest that future studies could explore materials with heavier elements (e.g., TaS$_2$, NbSe$_2$) to enhance the SNE.