Tunable thermoelectricity in monolayers of MoS$_{2}$ and other group-VI dichalcogenides

We study the thermoelectric properties of monolayers of MoS$_{2}$ and other group-VI dichalcogenides under circularly polarized off-resonant light. Analytical expressions are derived for the Berry phase mediated magnetic moment, orbital magnetization, as well as thermal and Nernst conductivities. Tuning of the band gap by {\it off-resonant} light enhances the spin splitting in both the valence and conduction bands and, thus, leads to a dramatic improvement of the spin and valley thermoelectric properties.


I. INTRODUCTION
Being the first truly two dimensional material [1], graphene has attracted remarkable attention, both due to its exotic transport behavior and technological applications in various fields [2]. Still, fundamental problems restrict its applicability, in particular the negligible band gap and weak spin orbit coupling (SOC). These limitations could be overcome by monolayer MoS 2 , which therefore is interesting for next generation nanoelectronics [3][4][5][6][7].
MoS 2 combines the honeycomb structure of graphene with a large intrinsic direct band gap of 2∆ = 1.66 eV and a large SOC of λ = 74 meV, providing mass to the Dirac fermions [8][9][10]. As a consequence, preliminary results indicate potential in valleytronics, because the dispersion can be manipulated in a flexible manner for optoelectronic applications [10][11][12][13].
Spin and valley Hall effects have been predicted in an experimentally accessible temperature regime [9], the former arising from the strong SOC and the latter from the broken inversion symmetry.
In addition to the electrical and optical properties, Berry phase mediated thermoelectric effects due to a temperature gradient have been proposed for two-dimensional systems [14]. Orbital magnetic moments, orbital magnetizations [15], as well as thermal and Nernst conductivities have been addressed in Refs. [17][18][19] and theoretical models for the thermoelectric transport have been presented for graphene in Ref. [20] and for topological insulators in Ref. [21]. Of particular interest is the tuning of the spin and valley thermoelectric properties of MoS 2 and other group-VI dichalcogenides, where a temperature gradient gives rise to transverse spin/valley accumulation and spin/valley current. In graphene this is difficult to realize due to the negligible band gap and weak SOC.
In the present work we quantify the Berry phase mediated thermoelectric properties of MoS 2 and other group-VI dichalcogenides by deriving analytical expressions for the key thermoelectric quantities in the presence of circularly polarized off-resonant light. Gap opening by such light has been predicted for graphene and for the surface states of topological insulators [22], and has been confirmed experimentally for the latter [23]. For graphene the chiralities for different values of the frequency have been given in Ref. [24]. Moreover, opening of a trivial gap has been reported under high-frequency linearly polarized light [25].
Going beyond these findings, we demonstrate in the following that by off-resonant light large spin and valley thermoelectric effects can be achieved.

II. MODEL FORMULATION
Extending the approach of Ref. [9] by introducing time dependence, we start from the effective Hamiltonian in the xy-plane in the presence of circularly polarized light, where η = ±1 represents the K-and K ′ -valleys, respectively, ∆ is the mass term that breaks the inversion symmetry, σ x , σ y , and σ z are the Pauli matrices, λ is the SOC with real spin index s z , and v = at 0 / is the Fermi velocity of the Dirac fermions (with t 0 being the nearest neighbour hopping amplitude and a the lattice constant). We use the gauge in the two-dimensional canonical momentum Π(t) = p + eA(t) with vector potential where Ω is the frequency of the light and A = E 0 /Ω (E 0 is the amplitude of the electric field, E(t) = −∂A(t)/∂t). We have A(t + T 0 ) = A(t) for T 0 = 2π/Ω. The plus/minus sign in Eq. (2) refers to right/left-handed circular polarization of the light.
The effect of off-resonant light can be described by a static Floquet Hamiltonian [22], which yields excellent agreement with experiments [23]. A static approach is satisfied for low intensity (evA ≪ ℏΩ) and high frequency (t 0 ≪ ℏΩ) light, which does not directly excite electrons but effectively modifies the band structure through virtual photon absorption and emission processes. We arrive at the effective Hamiltonian (see the Appendix) where ∆ Ω = e 2 v 2 ℏ 2 A 2 /ℏ 3 Ω 3 is an effective energy term representing the circularly polarized off-resonant light, which essentially renormalizes the mass of the Dirac fermions. Similar approaches have been used for describing gapped systems such as silicene [26] and disordered topological insulators [27]. Diagonalization of the Hamiltonian leads to the eigenvalues and eigenfunctions Here ζ = ±1 denotes electron/hole bands, s = ±1 stands for spin up/down, and ϕ = tan −1 (k y /k x ) with k x = k cos ϕ, k y = k sin ϕ, and k = k 2 x + k 2 y . The energy eigenvalues E s,η ζ are illustrated in Fig [23]. For ∆ Ω = 0.8 eV we observe that the spin splitting in the conduction band is increased and the band gap is reduced to 0.06 eV in the K ′ -valley, so that only the K ′ -valley is relevant, whereas for the K-valley the band gap becomes 3.26 eV. In the following we restrict the discussion to the K ′ -valley (η = −1).

III. ORBITAL MAGNETIC MOMENT AND TEMPERATURE DEPENDENT ORBITAL MAGNETIZATION
In order to study the Berry phase mediated thermoelectric transport, we consider the free energy, which, for a weak magnetic field B, is given by [14] F s,−1 where β = 1/k B T , k B = 8.62 × 10 −5 eV/K is the Boltzmann constant, µ is the Fermi energy, and T is the temperature. Eq. (6) can be simplified by converting the summation into an integral, where is the Berry curvature. The energy E s,−1 The orbital magnetization then is obtained as where f is the Fermi distribution function. From Eqs. (4) and (5) we obtain for the z- and correspondingly for the z-component of the orbital magnetic moment For finite ∆ − ∆ Ω + sλ the orbital magnetic moment has a peak at k = 0. For λ = 0 we obtain for ∆ − ∆ Ω = 30 meV for a single valley and ∆ Ω = 0.8 eV an orbital magnetic moment of 35 Bohr magnetons. High magnetic moments have been predicted for systems involving orbital degrees of freedom, such as graphene [15].
Using Eqs. (11) and (12) in Eq. (10), we obtain for T → 0 for the conduction band the which again can be enhanced by reducing the band gap by off-resonant light. Eq. (13) reduces to a previous result for gapped graphene in Ref. [15] in the limit of λ = 0 and ∆ Ω = 0. As an example, for µ = 0.2 eV we obtain an orbital magnetization (by dividing the results of Eq. (13)  As compared to Fig. 2, without off-resonant light we obtain about half the value for M v , while M s is 100 times smaller (with opposite sign), since it is dominated by the band gap (the system is pinned to the valley transport regime). For T = 160 K we observe spin effects close to the Dirac point, whereas they are suppressed for T = 360 K. These effects will become clear in the next section, since the orbital magnetization is proportional to the Hall conductivity [15]. Susceptibility measurements, electron paramagnetic resonance, x-ray magnetic circular dichroism, and neutron diffraction can be used to probe the orbital magnetization [28][29][30].

IV. SPIN/VALLEY THERMAL AND NERNST CONDUCTIVITIES
Eq. (10) contains conventional (first term) and Berry phase mediated (second term) contributions. It has been demonstrated in Refs. [14,15] that the conventional part does not contribute to the transport, while the Berry term directly modifies the intrinsic Hall current (which is obtained by integrating the Berry curvature over the two-dimensional Brillouin zone). In contrast to the Hall conductivity, the Nernst conductivity is determined not only by the Berry curvature but also by entropy generation around the Fermi surface [20,21] and therefore is sensitive to changes of the Fermi energy and temperature. For a weak electric field E, the Hall current is given by j x = α s,−1 xy (−∇ y T ) and the Nernst conductivity by [20] α s,−1 being the entropy density. Recent experiments found that Eq. (14) describes graphene very well [31]. We obtain the transverse thermal conductivity where Li 2 is the polylogarithm. Eqs. (14) and (15) can be simplified in the limit of low temperature, using Mott relations [14], to where with f ± representing the distribution function of the electron/hole bands. According to Eq.
(16), the Nernst conductivity is proportional to the derivative of the thermal conductivity.
We solve Eq. (18) for T = 0 by performing the integral and obtain in the case that the Fermi level is in the conduction band Eqs. (14) and (19) show that the spin (α s xy = α +1 Experiments indicate that the thermoelectric properties can be understood by Mott relations, which agree with experimental data for low temperature [31][32][33][34]. In these experiments the thermoelectric properties, in particular the Nernst conductivity, have been measured for gapless graphene in a transverse magnetic field. The Nernst effect discussed in our work exists even without external magnetic field, being solely driven by the effective magnetic field due to the Berry curvature. Note that Eq. (14) is more general than Eq. (16), because it goes beyond the linear temperature dependence.
It has been found experimentally that the dependence of the thermoelectric transport on the gate voltage (Fermi energy) can be tuned by controlling the band gap in monolayer [17][18][19][31][32][33] and bilayer [34] graphene. Being the electrical response to the thermodynamic perturbation, a giant thermoelectric transport is achieved when the bands come close to the Dirac point. In Fig. 3  VI dichalcogenides has been desired for thermoelectric applications since the discovery of graphene. Band gap opening by off-resonant light has been achieved in Ref. [23] for the surface states of topological insulators and can also be used for monolayer MoS 2 , since transistors [5] and amplifiers [7] already have been realized.  = v(ησ x p x + σ y p y ) + ∆σ z − ληs z σ z + ληs z , which describes a static honeycomb lattice with hopping t 0 (in a standard tight binding notation) and a band gap of 2∆. When