Tunable topological Nernst effect in 2D transition metal dichalcogenides
Girish Sharma





Department of Physics, Virginia Tech, Blacksburg, VA 24061, U.S.A


Two dimensional semiconducting transition metal dichalcogenides (TMDs) exhibit an intrinsic Ising spin orbit coupling (SOC) along with a valley contrasting Berry curvature, which can generate a purely anomalous spin and valley Nernst signal driven by a thermal gradient. We show that a small Bychkov-Rashba coupling, which is present in gated TMDs, can enhance the valley Nernst signal by at least 1-2 orders of magnitude. We find that the Nernst signal in these materials is dominated by the anomalous geometrical contribution, and the conventional contribution is much weaker. Importantly, the Nernst signal is also highly tunable by external gating. Although the total Nernst signal vanishes due to time reversal (TR) symmetry,  a small magnetic coupling lifts the valley degeneracy and generates an amplified Nernst response. Additionally, we also discuss the Nernst response of bilayer TMDs, and show a similar enhancement and modulation of the Nernst signal due to Rashba SOC.
Our predictions are highly pertinent to ongoing experimental studies in TMDs. The generated large anomalous Nernst signal can directly probe the presence of a large Berry curvature in these materials, and may serve as a promising tunable platform for caloritronics applications.




Tunable topological Nernst effect in 2D transition metal dichalcogenides
    Girish Sharma
    December 30, 2023
========================================================================



§ INTRODUCTION

Recently two dimensional semiconducting transition metal dichalcogenides (TMDs) of the form MX_2, where M = Mo, W, and X = S, Se, have attracted enormous interest because of the possibility of manipulating the valley degree of freedom in addition to spin and charge <cit.>, making them highly promising for low-power electronics and several valleytronics applications. The valley degrees of freedom refer to local extrema in the band structures of these materials, a feature unique to the class of these materials. The valley degree of freedom may also be manipulated as a qubit which opens up new arenas to explore quantum computation with TMDs <cit.>.
Further, the in-plane mirror symmetry is broken and results in an unusual intrinsic spin-orbit coupling (SOC), called the Ising SOC <cit.>, which originates from the d-orbitals of the transition metal. Ising SOC aligns the electron spins near opposite valleys to opposite out of plane directions, thus behaving like a valley contrasting out of plane Zeeman field. Such a spin-orbit field can be sharply contrasted to the Rashba SOC <cit.> which aligns the electron spins along the in-plane direction and produces a helical liquid. As a consequence of Ising SOC the valley bands in TMDs are spin-locked even though overall time-reversal symmetry is preserved. This distinctive feature has spurred theoretical and experimental studies in semiconducting TMDs as well as their superconducting counterparts <cit.>. Moreover the broken spatial inversion symmetry also generates a large Berry curvature near the valleys giving rise to unconventional electronic and optical properties <cit.>, which have been recently studied in the context of TMDs <cit.>.

In TMDs, due to spin-valley locking the electron spin remains a good quantum number. However, spin may not be conserved if the out of plane mirror symmetry is broken, which can result from a number of processes such as external gating or strain, that are typically present in realistic experimental situations. A Bychkov-Rashba spin orbit coupling <cit.> can thereby naturally arise in TMDs <cit.> which aligns the electron spins away from the out-of plane direction. The non-trivial role of Rashba SOC has been extensively studied in several materials such as 2D semiconductors <cit.>, quantum dots <cit.>, more recently in  producing topological superconductivity <cit.>. Despite the fact that Rashba SOC exists in gated TMDs, its role in TMDs has received only limited attention <cit.>. Very recently the role of Rashba SOC in generating strong Berry curvature effects and anomalous Hall effect has been studied <cit.>. 

The Nernst effect has been used as a very important experimental probe in a number of physical systems such as high-temperature cuprate superconductors <cit.>, charge density waves <cit.>, and more recently as a probe of strong Berry curvature in Weyl and Dirac semimetals <cit.>. In conventional Fermi liquids, the Nernst effect is expected to be small due to Sondheimer's cancellation <cit.>. However, the Sondheimer's cancellation breakdowns for Berry mediated Nernst effect <cit.>, giving rise to a measurable anomalous Nernst current even in the absence of magnetic field. The generation of a Nernst signal in TMDs has been investigated where the intrinsic Ising SOC and Berry curvature effects can generate a pure spin and valley Nernst current <cit.>. 

In this work we show that extrinsic Rashba SOC and intrinsic Ising SOC in gated TMDs can jointly amplify the Nernst signal, which is highly tunable in nature by external gating. The generated Nernst signal is shown to enhance by at least 1-2 orders of magnitude compared to the Nernst signal generated from a purely intrinsic Ising SOC. 
We first calculate the Berry curvature due to Rashba SOC, and show that the Berry curvature is strongly enhanced and sharply peaked in the momentum space around the valley points, resembling a Dirac-delta distribution. We then calculate the Nernst response and show that even in the absence of Rashba SOC the Nernst signal at a valley point is shown to be purely dominated by the anomalous geometrical contribution, and the conventional B-dependent Nernst signal is significantly weaker for typical material parameters. 
The tunable anomalous valley Nernst signal in the presence of Rashba coupling is shown increase by at least one order of magnitude compared to the Nernst signal generated from a purely intrinsic Ising SOC. Although the total Nernst signal, i.e. summing the contributions from both valleys, vanishes due to preservation of time reversal (TR) symmetry, a small external B-field lifts the valley degeneracy and further amplifies the Nernst signal by almost two orders of magnitude. 

The Nernst effect in bilayer TMDs also has received limited attention so far. Additionally, here we briefly discuss the band structure, Berry curvature of bilayer TMDs <cit.>, and theoretically examine the generated Nernst signal due to large Berry curvature in these materials. Particularly, we calculate the Nernst response of a 3R and 2H stacked bilayer MoS_2. Including the intrinsic SOC effects, the 3R stacked bilayer MoS_2 breaks inversion symmetry, unlike the 2H stacked bilayer which remains spin-degenerate. A finite Rashba SOC can enhance and tune the valley Berry curvature of a 3R stacked bilayer. On the other hand a small Rashba SOC generates (and can also tune) a giant Berry curvature in a 2H stacked bilayer due to lifting of spin-degeneracy. We calculate the anomalous Nernst response and show that Rashba SOC can also enhance and tune the Nernst signal in bilayer TMDs.
Our predictions can be directly tested in experiments, and can be employed for several promising caloritronics applications. 

This paper is organized as follows: in Section II we discuss the Hamiltonian and the Berry curvature of monolayer TMDs in the presence of Rashba spin orbit coupling, and show the tunability of the Berry curvature with the Rasba parameter. In Section III we discuss the Nernst effect in monolayer TMDs, showing the enhancement and modulation of the anomalous Nernst coefficient with Rashba SOC. In Section IV, we discuss the band structure, Berry curvature and Nernst effect in bilayer TMDs focusing on 3R stacked bilayer MoS_2 and 2H stacked bilayer MoS_2. We then also discuss the enhancement and tunability of the anomalous Nernst coefficient in both these cases. 
Finally we summarize and conclude in Section V. 





§ HAMILTONIAN AND BERRY CURVATURE

We begin with the low energy 𝐤·𝐩 Hamiltonian for a generic monolayer TMD, which is given by <cit.> 

    H_0 = at (τ k_x σ_x+ k_yσ_y) + Δ/2σ_z

where a is the lattice constant, t is the effective hopping integral, Δ is the bare energy bandgap, 𝐤 is the momentum measured from the valley point 𝐊, and σ_i's represent Pauli matrices for the electron bands. The intrinsic spin orbit coupling can be represented by the following Hamiltonian <cit.> 

    H_SO = λ_c τσ_0+σ_z/2 s_z + λ_vτσ_0 - σ_z/2 s_z

where λ_c (λ_v) is the intrinsic Ising spin-orbit coupling in the conduction (valence) band, s_i^s represent Pauli matrices for the electron spin, and τ∈{+1,-1} is the binary valley index. This intrinsic SOC lifts the spin-degeneracy and the bands become spin-polarized.
In addition to the intrinsic spin orbit coupling, a Bychkov-Rashba coupling can also arise in TMDs broken out-of-plane mirror symmetry due to external gating. In the context of the 𝐤·𝐩 Hamiltonian this term can be written as <cit.> 

    H_R = α_R (τσ_x s_y - σ_y s_x)

where α_R is the strength of the Bychkov-Rashba SOC. The total Hamiltonian is thus given by

    H = H_0 + H_SO + H_R

The energy eigenvalues of the Hamiltonian H specifically at the valley points (i.e. 𝐤=0) are 

    E_nτ = n λ_c τ/2 - nλ_v τ/2
       ±1/2√((λ_c + λ_v)^2 + Δ^2 + 2nΔτ (λ_c+λ_v) + 8α_R^2 (1-nτ))

where n∈{+1,-1}. 
The spin degeneracy in the conduction bands is broken as a result of λ_c and α_R. Unlike λ_c which is an intrinsic SOC, the Rashba SOC allows us to tune the energy bands by external electrostatic gating. 

The presence of Berry curvature in TMDs can result in anomalous transport responses such as the anomalous Hall, spin Hall and the Nernst effects <cit.>. 
The Berry curvature for a generic m^th band is given by the expression

    Ω_ab^m = i∑_m≠ m'⟨ m|∂ H/∂ k_a|m'⟩⟨ m'|∂ H/∂ k_b|m⟩ -(a↔ b )/(E_m' - E_m)^2

The presence of a finite Rashba SOC can thus allow us to manipulate the Berry curvature of the electron bands. Even though the Berry curvature for the Hamiltonian H in Eq. <ref> can be evaluated analytically, the exact expressions are cumbersome and we do not provide them here. Instead, we directly numerically calculate the Berry curvature and related quantities. 
Fig. <ref> shows the calculated Berry curvatures of the conduction bands as a result of tuning the Rashba SOC parameter α_R. We note that the Berry curvature of the conduction bands can increase up to atleast 1 order of magnitude greater than the valence bands for typical values of α_R. On the other hand, the valence band Berry curvature stays close to the orbital (intrinsic) Berry curvature (Ω_𝐊 = -2a^2t^2 / (Δ-λ_v)^2) as the intrinsic SOC λ_v ≫α_R typically for all TMDs. The spin splitting λ_c∼α_R and the Berry curvature in conduction bands is strongly affected by both parameters, λ_v and α_R. Interestingly, we also note that Rashba SOC not only enhances the Berry curvature contribution for the conduction band but also sharpens the peak at the valley point as demonstrated in Fig. <ref>. The intrinsic Berry curvature scales as ∼ 1/(1+k^2) around 𝐊 but the Rashba mediated Berry curvature has a much sharper distribution, closely resembling a Dirac-delta function. This is expected because the Chern number of the band is a topological invariant, and therefore a sharp peak in the Berry curvature must be compensated by reducing contributions from  the neighboring points in momentum space.



§ NERNST EFFECT IN MONOLAYER TMDS

 Nernst effect refers to the generation of a transverse electric field in the presence of a longitudinal temperature gradient. Conventionally, the Nernst effect should occur only in the presence of an external magnetic field, which provides a transverse velocity to the electrons by the Lorentz force. However, a non-trivial Berry curvature Ω_𝐤 of the bands can also give rise to a Nernst response as a result of an anomalous velocity term <cit.> genrated by Ω_𝐤. 
A non-zero Nernst response is expected only when overall time-reversal symmetry is violated. In doped TMDs the contribution from the spin-up and spin-down bands exactly cancels out each other as the system preserves time-reversal symmetry. 
The anomalous Nernst coefficient is given by the product of the anomalous velocity and the entropy density <cit.> 

    α_xy = k_Be/ħ∑_m=1^4∫ d^2𝐤 Ω^m_𝐤 s^m_𝐤

where m is the band index, s_𝐤 = -n_f log n_f - (1-n_f) log (1-n_f) is the entropy density, n_f being the Fermi distribution function. It is important to point out that the anomalous Hall conductivity σ_xy directly depends on the Berry curvature of the filled bands, but on the other hand α_xy is a Fermi surface quantity, because s_𝐤 is zero for completely filled and completely empty bands. 


It is for this reason, that an insulator can give rise to an anomalous Hall response (σ_xy), but not an anomalous Nernst response α_xy. 
Typically in Fermi liquids the conventional Nernst effect is expected to be small due to Sondheimer's cancellation <cit.>, which breakdowns of for Berry mediated Nernst effects, as shown recently <cit.>.



The anomalous Nernst coefficient is calculated from Eq. <ref> by integrating the Berry curvature up to the chemical potential. In Fig. <ref> we plot the anomalous Nernst coefficient for various TMDs for α_R =0. The anomalous Nernst coefficient picks up a non-zero signal for partially filled conduction bands and its magnitude is proportional to the Berry curvature of the respective bands. In the presence of a finite magnetic field TMDs also will exhibit a conventional Nernst response which is given by <cit.> 

    α_xy = e^3 τ_s^2 B/Tħ∫d^2𝐤 (E - μ)(-∂ n_f/∂ E) ( v_x^2∂^2 E/∂^2 k_y^2 - v_xv_y ∂^2E/∂ k_x∂ k_y)
  
where τ_s is the scattering time and v_i is the semiclassical velocity. In order to compare the magnitude of the anomalous part of the Nernst coefficient in TMDs with the conventional one we also calculate the latter within the relaxation time approximation, which is valid in the limit ω_cτ_s≪ 1, ω_c being the cyclotron frequency. For typical values <cit.> of τ_s and relevant B- fields we find that this condition is satisfied. Fig. <ref> shows a comparison of the anomalous and conventional Nernst coefficients. Even when α_R=0 the Berry curvature mediated Nernst response is at least 3 orders of magnitude bigger than the conventional Nernst coefficient. 

In Fig. <ref> we plot the anomalous Nernst coefficient at valley τ=+1 as a function of the chemical potential μ calculated for WS_2. For α_R>0 the Nernst coefficient shows a significantly enhanced signal compared (by at least one order of magnitude) to the Nernst coefficient contribution arising just from intrinsic SOC (α_R=0). The figure also shows the density plot for the anomalous Nernst coefficient for various values of α_R and μ. The sharp peaks in the Nernst signal when μ is tuned to partially fill up one of the conduction bands is a striking feature which emerges in these systems from a finite Rashba SOC. As the chemical potential is varied from the bottom of the lower conduction band to the upper conduction band, the Nernst signal changes sign because the sign of the Berry curvature is opposite in the two neighboring bands.

Our discussion so far was focused on the Nernst coefficient contribution from a single valley in TMDs. We noted that the anomalous Nernst contribution is significantly larger than the conventional Nernst coefficient. Further the anomalous contribution can be enhanced by at least one order in magnitude as a result of tuning the Rashba SOC α_R, but the conventional Nernst isn't much altered with a finite α_R. However, as a result of preservation of TR symmetry the Berry curvature satisfies Ω^m_𝐊 = -Ω^m_-𝐊, and thus the total (summing contributions from both valleys) Berry curvature, and hence the total Nernst coefficient exactly vanishes. Nevertheless, breaking the valley degeneracy can result in a finite flux of the Berry curvature and therefore a finite total Nernst effect. One way to generate such asymmetry is by applying a small external magnetic field perpendicular to the 2D plane. The magnetic field acts on the valley and spin sectors as H^s_z = b_z s_z, and H^v_z = ∑_a m^a b_z, where b_z is the magnetic field, s_z is the spin-Pauli matrix, and valley orbital magnetic moments m^a. The orbital magnetic moment is given by 

    m^a = ie/2ħ∑_a≠ a'⟨ a|∂ H/∂ k_x|a'⟩⟨ a'|∂ H/∂ k_y|a⟩ +c.c/E_a - E_a'


The terms H_z^s and H^v_z can then be added to the total Hamiltonian in Eq. <ref>. The role of external magnetic field is to break time reversal symmetry and lift the band degeneracy between +𝐊 and -𝐊. As a result the system now possesses an net flux of the Berry curvature. A conventional Nernst coefficient will also be generated in the process of applying an external B-field, however it is of a much smaller magnitude compared with the anomalous contributions. 
In Fig. <ref> we plot the magnitude of the sum of Berry curvatures in the two valleys for a particular conduction band in MoS_2 for a finite magnetic field. When B=0 we expect a vanishing net Berry curvature everywhere. 
We note that for α_R=0, we need a large B-field in order to generate even a small net Berry curvature. However a small finite B-field in conjunction with a finite α_R produces a large net Berry curvature (at least 2 orders of magnitude larger) with a very sharp distribution (resembling a Dirac-delta distribution) around the valley point 𝐊, clearly distinct from the ∼ 1/(1+k^2) distribution for intrinsic Berry curvature. 

In Fig. <ref> we plot the total Nernst coefficient for MoS_2 in the presence of a finite magnetic field. In plotting this we integrated the Berry curvature of both valleys up to the chemical potential.
The sharply peaked net Berry curvature distribution around the 𝐊 point and a finite Rashba SOC produce an amplified anomalous Nernst coefficient, which can be easily tuned by external gating. It is important to emphasize the role of an external magnetic field, which by itself cannot produce the Nernst signal of this magnitude but is crucial in generating Berry curvature asymmetry between the two valleys. Interestingly, similar behavior for the Nernst response has been recently proposed <cit.> and observed <cit.> in Dirac semimetals. Even though Dirac semimetals are rather quite distinct from TMDs, the close similarity in their Nernst response is an universal manifestation of underlying non-trivial band topology.



§ NERNST EFFECT IN BILAYER TMDS

So far our discussion was focused on Nernst response in monolayer TMDs. The Nernst response in bilayer TMDs has received limited attention so far. The anomalous Nernst effect in bilayer TMDs is directly related to its Berry curvature properties, which in bilayer TMDs has both inter-layer and intra-layer contributions <cit.>. Further, the stacking of monolayers also effects the band spectrum and the Berry curvature in bilayer TMDs <cit.>. Like monolayer TMDs, bilayer TMDs also exhibit a finite Berry curvature, which leads to anomalous transport and optical properties, such as the valley and spin Hall effects. As a prototype, we will discuss (i) 3R stacked bilayer MoS_2, and (ii) 2H-stacked bilayer MoS_2. As we will examine, both of these materials can exhibit a large Nernst effect of purely anomalous origin, tunable with Rashba spin orbit coupling. Since, we already showed in the previous section that the conventional (B- field dependent) contribution to the Nernst signal is small compared to the anomalous contribution, we will neglect this conventional contribution in our analysis and just focus on the anomalous contribution.



 §.§ 3R stacked bilayer
 
We first discuss 3R stacked bilayer MoS_2, which break inversion symmetry <cit.>. Due to inversion symmetry breaking Berry curvature is expected to play an important role in transport properties. Near the 𝐊 points, the low energy Hamiltonian up to the lowest order in k can be written as 

    H^3R=( [   ϵ_cb^b  γ_3 k^+ γ_cc k^-        0;  γ_3 k^-   ϵ_cb^v        0 γ_vv k^-; γ_cc k^+        0   ϵ_cb^t  γ_3 k^+;        0 γ_vv k^+  γ_3 k^-   ϵ_vb^t ])

where the momentum k^± = (τ k_x ± i k_y) is measured from the valley points. The subscripts vb and cb refer to valence and conduction bands respectively, while t and b refer to the top and bottom layer respectively. γ_3 is the intra-layer coupling between the conduction and the valence band (similar to at in the monolayer case), while γ_cc and γ_vv are the inter-layer couplings between top and the bottom layers. From Eq. <ref> it is clear that the Berry curvature will have both intra-layer (γ_3 dependent) contributions, and inter-layer (γ_cc and γ_vv dependent) contributions. For example, at the 𝐊 points, the Berry curvature for the conduction band is given by 

    Ω_𝐊 = 1/2(2γ_3/ϵ_cb^b - ϵ_vb^b)^2 ±1/2(ϵ_cb^b-ϵ_cb^v)^2γ_cc^2
 
for the top/bottom later, and similarly one may derive an expression of the Berry curvature for the valence band. For typical material parameters, we find that the contributions of inter-layer and intra-layer on the Berry curvature are of comparable order of magnitude. The above Hamiltonian does not  contain the spin-orbit coupling effects, which may be accounted for by adding the terms Δ_cb s_z τ_z and Δ_vb s_z τ_z to the conduction and the valence bands respectively, where the Pauli matrices s_z and τ_z act on spin and valley space. Note that Δ_vb≫Δ_cb, as is the case in monolayer TMDs. Spin-orbit coupling splits the spin-degenerate bands in both the conduction and the valence bands, though its effect on the conduction bands is rather small. 





Let us now discuss the tunability of the Berry curvature in the presence of Rashba spin orbit coupling. Fig. <ref> shows the peak Berry curvature at the 𝐊 point as a function of the Rashba spin orbit coupling parameter (α_R) for four of the total eight bands of a 3R stacked bilayer MoS_2. The Berry curvature is easily tunable with Rashba spin orbit coupling and can be enhanced up to two orders of magnitude for typical α_R values. Similar to the case of a monolayer TMD, we can expect a large anomalous valley Nernst signal, which is easily tunable via Rashba spin-orbit coupling. Again, due to preservation of a TR symmetry, the net Nernst signal must vanish. A small perturbative magnetic field, can however break valley symmetry. The magnetic field acts on the valley and spin sectors i.e. H^s_z = b_z s_z, and H^v_z = ∑_i m^i b_z, for magnetic field b_z and valley orbital magnetic moments m^i. Although, the magnetic field has a small effect on the band-structures of these materials, its role in generating a valley asymmetry and thus a large overall Berry curvature is crucial. Fig. <ref> shows the total anomalous Nernst coefficient for a magnetic field of 5T, which is tunable with the chemical potential and α_R, showing sharp peaks at certain values of chemical potential. 



 §.§ 2H stacked bilayer

The low energy 𝐤·𝐩 Hamiltonian for 2H stacked bilayer MoS_2 is given by 

    H^2H=( [     ϵ_cb  γ_3 k^+ γ_cc k^-        0;  γ_3 k^-     ϵ_vb        0      t_⊥; γ_cc k^+        0     ϵ_cb  γ_3 k^-;        0      t_⊥  γ_3 k^+     ϵ_vb ])

where t_⊥ is the tunneling between the holes of top and bottom layer. Unlike 3R stacked bilayer, the above Hamiltonian is inversion symmetric. Therefore the Berry curvature vanishes identically everywhere. The spin-orbit coupling effects can be accounted for by  Δ_cb s_z τ_z σ_z and Δ_vb s_z τ_z σ_z to the conduction and the valence bands respectively. However, even in the presence of SOC, the energy bands in 2H stacked bilayer remain degenerate (see Fig. <ref>), leading to vanishing Berry curvature. A finite Berry curvature can be generated via: (i) a finite inter-layer potential U_g which breaks symmetry between the top and the bottom layers, or (ii) gate-tunable Rashba spin-orbit coupling. The effect of inter-layer coupling has been explored in Ref. Kormanyos:2018, and we will not discuss that here. Here we will explore the effect of Rashba spin-orbit coupling on the band-structure, Berry curvature and the Nernst effect in 2H stacked bilayer TMDs. 

The evolution of band spectrum due to Rashba SOC is shown in Fig. <ref>. Even in the presence of intrinsic SOC (but absence of Rashba SOC), the 2H bilayer spectrum is spin-degenerate, with vanishing  Berry curvature near a valley point. A finite Rashba SOC lifts the spin degeneracy  at and away from the 𝐊 point. We note that due to an infinitesimal Rashba SOC, the band degeneracy is lifted and a giant peak in the Berry curvature is generated. This behavior can be contrasted to the smooth Berry curvature enhancement in 3R stacked bilayer MoS_2 (see Fig. <ref>) when α_R is increased from zero. Consequently, an anomalous valley Nernst coefficient, tunable with Rashba SOC, is generated (also shown in Fig. <ref>), displaying peaks when the chemical potential touches the energy bands. Another interesting behavior occurs for large values of Rashba SOC. When the Rashba SOC parameter is increased further, at a certain critical α_R the bands cross each other (see Fig. <ref>). This band crossing also generates a peak in the Berry curvature along with a distinct sign change (Fig. <ref>). Thus, even though the 2H stacked bilayer by itself does not display peculiar Berry curvature effects, a small finite Rashba coupling can lead to striking modulation of the Berry curvature. 
Now, we can apply a small magnetic field which breaks the valley symmetry, and thereby generates a large overall anomalous Nernst effect. Fig. <ref> shows the total anomalous Nernst coefficient (α_xy) of a 2H stacked bilayer MoS_2 for a magnetic field of B=1T, tunable with Rashba SOC.



§ CONCLUSION

 In this work we analyzed the Nernst response of monolayer and bilayer TMDs. The Nernst response typically has two electronic contributions - the conventional B-field dependent contribution, where the anomalous velocity to the electrons is provided by the Lorentz force, and secondly the the anomalous  contribution where the anomalous velocity is provided by the Berry curvature. 
 We first showed that the Nernst signal at a valley point in TMDs is purely dominated by the anomalous geometrical contribution and the conventional B-dependent Nernst signal is significantly weaker for typical material parameters. 
 Further, we showed that the interplay of gate controlled Rashba SOC and intrinsic Ising SOC in gated monolayer TMDs can produce a large valley Nernst signal of purely geometrical origin in monolayer TMDs, which is highly tunable in nature. 
The tunable anomalous valley Nernst signal in the presence of Rashba coupling can be enhanced by at least one order of magnitude. Although the total Nernst signal vanishes due to preservation of TR symmetry, a small magnetic coupling can lift the valley degeneracy, and an amplified net Nernst signal of almost two orders of magnitude larger can be achieved.
 
Additionally, we also analyzed the band structure, Berry curvature, and Nernst effect in bilayer TMDs, focusing on 3R and 2H stacked bilayer MoS_2 as a prototype. Including just the intrinsic SOC effects, the 3R stacked bilayer MoS_2 breaks inversion symmetry, unlike the 2H stacked bilayer which remains spin-degenerate. We showed that a finite Rashba SOC can enhance and tune the valley Berry curvature of a 3R stacked bilayer. On the other hand a small but finite Rashba SOC generates (and can also modulate) a giant Berry curvature in a 2H stacked bilayer due to lifting of spin-degeneracy. When the Rashba SOC is increased further, then at a certain critical α_R the bands cross each other and another peak in the Berry curvature is generated along with a distinct sign change. Consequently, we also showed that Rashba SOC can enhance and tune the anomalous Nernst effect in bilayer TMDs.
Our predictions can be directly tested in experiments, and may be employed for several promising tunable caloritronics applications. 

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