Orbital and Spin Nernst Effects in Monolayers of Transition Metal Dichalcogenides

Abstract: In recent years, orbitronic effects have attracted growing attention as complementary counterparts to the well-established spintronic phenomena. In this work, we demonstrate that monolayers of transition metal dichalcogenides provide an excellent platform for the observation of the orbital Nernst effect, a relatively less explored phenomenon describing the generation of a transverse orbital current in response to an applied temperature gradient. We show that, similar to its electrical counterpart, viz., the orbital Hall effect, the orbital Nernst effect does not require the presence of spin-orbit coupling. Analytical results based on a low-energy valley model offer key insights into the underlying mechanisms, highlighting in particular the crucial role of electronic states at the Fermi energy for the emergence of this effect. The inclusion of spin-orbit coupling further gives rise to a spin Nernst effect, which scales with the strength of spin-orbit coupling and vanishes in its absence. We substantiate our analytical findings with full Brillouin-zone tight-binding results for two representative systems, monolayer 2H MoS$_2$ and 2H NbS$_2$. Our results show that while both orbital and spin Nernst conductivities in MoS$_2$ require electron or hole doping, both effects are intrinsically present in metallic NbS$_2$. Our work reveals the central role of orbital and spin Berry curvatures, identifies doping as an effective route for tuning orbital and spin Nernst responses, and proposes a possible experimental setup for detecting these effects in monolayer transition metal dichalcogenides.

• The paper demonstrates that monolayer TMDCs can exhibit tunable orbital and spin Nernst effects through carrier doping and spin-orbit coupling modulation.
• It employs both a low-energy valley model and a full tight-binding approach to quantitatively estimate Nernst conductivities in insulating and metallic regimes.
• The study finds that metallic TMDCs like NbS₂ show robust orbital responses with conductivities significantly exceeding the spin-related effects.

Orbital and Spin Nernst Effects in Monolayer Transition Metal Dichalcogenides

Introduction and Context

This paper provides a comprehensive theoretical investigation of the orbital Nernst effect (ONE) and spin Nernst effect (SNE) in monolayer 2H-phase transition metal dichalcogenides (TMDCs), focusing on the representative systems MoS₂ (insulating) and NbS₂ (metallic) (2605.10033). It demonstrates that TMDC monolayers constitute a highly tunable platform for realizing both effects, with distinct mechanisms for the orbital and spin channels, and offers predictions for their experimental detection and future utilization in caloritronic and orbitronic devices.

Theoretical Framework: Valley Model and Tight-Binding Approach

The authors employ two complementary frameworks:

1. Valley Model: A low-energy analytical model near the $K$/$K'$ valleys, capturing the essential Berry curvature physics relevant for low doping or near the band extrema.
2. Full-Brillouin-Zone Tight-Binding Model: Material-specific tight-binding Hamiltonians—parametrized from first-principles—describe MoS₂ and NbS₂ across the Brillouin zone, enabling quantitative estimates of Nernst conductivities in both insulating and metallic regimes.

The valley model Hamiltonian explicitly incorporates the absence/presence of spin-orbit coupling (SOC) and provides closed-form analytical expressions for OHC, SHC, ONC, and SNC, highlighting their dependence on the Fermi level and SOC strength.

Figure 1: Crystal structure and valley band structure of monolayer MX₂ TMDCs, showing inversion symmetry breaking and valley-contrasting physics.

Physical Mechanisms and Analytical Results

Orbital and Spin Nernst Effects: These phenomena correspond to transverse charge-neutral flows of orbital and spin angular momentum, respectively, induced by a longitudinal temperature gradient. The key findings are:

• ONE persists without SOC, in contrast to the SNE.
• The ONE vanishes in intrinsic (undoped) insulators but is induced with electron or hole doping. In contrast, ONE is intrinsic in metallic TMDCs. The ONC exhibits a strong Fermi energy dependence, $\propto 1/\epsilon_F^2$, enabling sensitive control.
• SNE and SHC are strictly proportional to SOC strength, vanishing as $\zeta \rightarrow 0$. This is a consequence of mutually cancelling spin subbands in the absence of SOC.
• Berry Curvature Distribution: Both effects are traceable to the Berry curvature (orbital or spin) localized near $K$/$K'$ for valence states and around $\Gamma$ for conduction states (in electron-doped cases).

  Figure 2: Schematic depiction of (a) orbital Nernst effect and (b) spin Nernst effect; electrons with different angular momenta are driven transversely by a thermal gradient.

Numerical Results for MoS₂ and NbS₂

The tight-binding calculations quantitatively establish several strong claims:

• MoS₂ (Insulating TMDC):
  • OHC is nonzero throughout the band gap, but ONC is strictly zero at intrinsic (undoped) Fermi energy and only arises upon substantial doping.
  • The ONC magnitude is larger upon electron doping (due to Berry curvature at $\Gamma$), and tightly linked to band structure details.
• NbS₂ (Metallic TMDC):
  • Both OHC and ONC are finite at the Fermi energy without any doping. The ONC, in particular, can reach values of order $10^2$ in normalized units, much larger than the SNC.
  • The magnitude of ONC generally exceeds SNC by an order of magnitude or more.
  • A sign change in ONC as a function of energy (doping) is observed, directly corresponding to the sign of underlying orbital Berry curvature.
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  • Figure 3: Orbital and spin Hall and Nernst conductivities in MoS₂ as a function of energy. ONC is zero at the Fermi energy but becomes sizeable outside the gap.

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  • Figure 4: Orbital and spin Hall and Nernst conductivities in NbS₂; metallicity ensures finite ONC and SNC at the Fermi level, with strong dependence on the SOC.

Berry Curvature Analysis

The Berry curvature distribution at the Fermi level is the microscopic origin for both Nernst responses:

• MoS₂: Berry curvature is sharply peaked at $K$, $K'$0 (for holes) and $K'$1 (for electrons), with the orbital component much stronger than the spin component.
• NbS₂: While orbital Berry curvature peaks around valleys, the immediate $K'$2, $K'$3 regions are unoccupied due to the Fermi level position; nevertheless, regions near the Fermi surface exhibit substantial orbital Berry curvature.

  Figure 5: $K'$4-space distribution of orbital and spin Berry curvature for MoS₂ (top) and NbS₂ (bottom); Berry curvature hotspots determine the sign and magnitude of conductivities.

  

  Figure 6: Linear dependence of SHC and SNC on SOC strength $K'$5 in NbS₂, confirming analytical scaling predictions.

Experimental Proposal

A Hall-bar experimental geometry is proposed for direct detection via MOKE (magneto-optic Kerr effect). The setup leverages the edge accumulation of spin (SNE) and orbital (ONE) magnetization, with the following measurement scheme:

• Scanning a polarized laser across the sample detects edge-accumulated angular momentum via Kerr rotation;
• Reversing the thermal gradient flips the sign of the accumulated moments;
• SNE and ONE can be differentiated by analyzing the sign and the response to in-plane magnetic fields.

  Figure 7: Schematic of the proposed experimental device structure for MOKE-based detection of SNE and ONE in monolayer TMDCs.

Implications and Outlook

Theoretical Significance

• Establishes a clear separation of orbital- and spin-driven thermal transport mechanisms in 2D TMDCs.
• Demonstrates that orbital Nernst responses can be engineered even in systems with negligible SOC, provided metallicity is present, which dramatically widens material design space versus spin-based devices.
• Highlights the tuning capability of doping/gate voltage for ONC/SNC, predicting practical pathways for device integration.

Practical Relevance

• The possibility to harness both orbital and spin Nernst effects for energy harvesting, i.e., conversion of waste heat to information-carrying orbital/spin currents.
• TMDCs, with their easily accessible 2D structure, chemical tunability, and strong Berry curvature effects, are identified as ideal candidate materials.
• The MOKE-based detection scheme should be broadly applicable for mapping edge magnetizations in 2D materials.

Future Directions

• Exploration of other metallic TMDCs (e.g., TaS₂, NbSe₂) with heavier transition metals and stronger SOC for larger SNCs.
• Investigation of extrinsic effects (e.g., disorder, temperature dependencies) on ONC/SNC in realistic environments.
• Integration of ONC/SNC effects into functional devices for spin-orbitronic and caloritronic applications, potentially in combination with optical or gate-driven control.

Conclusion

This study presents a detailed, predictive account of the orbital and spin Nernst effects in 2D TMDCs, displaying strong evidence that both effects can be selectively engineered via SOC and carrier doping. The findings underscore the centrality of orbital Berry effects in future 2D electronics and caloritronics, motivating further theoretical and experimental efforts in orbitronics and beyond.