Valley Hall Transport of Photon-Dressed Quasiparticles in 2D Dirac Semiconductors

We present a theory of the photovoltaic valley-dependent Hall effect in a two-dimensional Dirac semiconductor subject to an intense near-resonant electromagnetic field. Our theory captures and elucidates the influence of both the field-induced resonant interband transitions and the nonequilibrium carrier kinetics on the resulting valley Hall transport in terms of photon-dressed quasiparticles. The non-perturbative renormalization effect of the pump field manifests itself in the dynamics of the photon-dressed quasiparticles, with a quasienergy spectrum characterized by {dynamical gaps $\delta_\eta$ ($\eta$ is the valley index)} that strongly depend on field amplitude and polarization. Nonequilibrium carrier distribution functions are determined by the pump field frequency $\omega$ as well as the ratio of intraband relaxation time $\tau$ and interband recombination time $\tau_{\mathrm{rec}}$. We obtain analytic results in three regimes, when (I) all relaxation processes are negligible, (II) $\tau \ll \tau_{\mathrm{rec}}$, and (III) $\tau \gg \tau_{\mathrm{rec}}$, and display corresponding asymptotic dependences on $\delta_\eta$ and $\omega$. We then apply our theory to two-dimensional transition-metal dichalcogenides, and find a strong enhancement of valley-dependent Hall conductivity as the pump field frequency approaches the transition energies between the pair of spin-resolved conduction and valence bands at the two valleys.


Introduction
Low-dimensional quantum systems subject to an externally applied large power high frequency electromagnetic field (EMF) display a great variety of interesting phenomena, such as multi-photon induced macroscopic quantum tunneling [1], multi-photon Rabi oscillations and the dynamic Stark effect in superconducting or hybrid qubits [2,3], dissipationless electron transport [4], polaritons and condensates [5,6], and Floquet nonequilibrium states [7,8]. In many cases of interest, the quantum dynamics of systems strongly interacting with an EMF can be described in terms of nonequilibrium quasiparticles called photon-dressed quasiparticles (PDQs) [2,9]. They are characterized by a specific quasienergy spectrum and nonequilibrium steady state distribution functions. Such a quasiparticle description is particularly useful for near-resonant excitation, i.e. when the frequency of the EMF is close to the difference of the intrinsic energy levels. The quasienergy spectrum of such PDQs shows a dynamical gap [10,11] that is proportional to the amplitude of the EMF, and the nonequilibrium steady state of the PDQs is determined by interplay between different time scales: the inverse dynamical gap, inverse frequency, and relaxation times [12].
As we turn to spatially extended systems, PDQs naturally appear in two-band semiconductors in the presence of EMF-induced interband transitions. The quasiclassical dynamics of PDQs in a spatially dependent potential, for example, leads to a ballistic photocurrent in graphene-based nanostructures [13][14][15][16]. The dependences of the photocurrent on the gate voltage, amplitude, frequency, and polarization of the EMF are mostly determined by the energy spectrum of the PDQs and, in particular, by the dynamical gap. However, a nonequilibrium steady state of PDQs cannot be achieved under these conditions, and thus has not been observed in such experiments. Dynamical gaps have been extensively studied in originally gapless materials [17,18] under the high-power EMF, where rather complicated spectra of quasienergies with multiple dynamical gaps have been found.
In this paper we theoretically study the valley Hall transport of PDQs in homogeneous two-dimensional (2D) Dirac semiconductors under irradiation of circularly polarized light, or in other words, a photovoltaic valley-dependent Hall effect. It is well known that, in addition to momentum and spin, 2D materials with a hexagonal lattice (such as graphene [19,20]) host valley degrees of freedom, which are quantum numbers describing corners K and K′ of their hexagonal Brillouin zone. The presence of valleys gives rise to new valleyresolved physics [21] that has been much heralded as valleytronics [22]. 2D Dirac semiconductors are gapped materials characterized by low-energy massive Dirac electrons in the vicinity of the two valleys. As an example of Dirac semiconductors, 2D transition-metal dichalcogenides (TMDs) [23,24] provide a much sought-after platform to realize valley-resolved physics [25,26] due to a particularly large band gap, Δ, advantageously occurring within the optical frequency range (e.g., MoS 2 has a band gap at 1.66 eV [27]).
An important property underlying many valleytronic phenomena is the valley selection rule: the low-energy electrons at each valley couple predominantly to one particular state of optical polarization (left or right circular polarization), enabling valley-selective interband transitions. Under a DC probe field, there will be an excess population of majority-valley electrons driven in the transverse direction, leading to an anomalous Hall effect. While the linear response optical conductivity of TMDs has been extensively studied in a number of works (e.g., [28][29][30]), nonlinear optical phenomena [31] remain largely unexplored despite attracting increasing attention [32,33].

Model, Hamiltonian and energy spectrum of PDQs
Let us consider the electron dynamics in a 2D Dirac semiconductor subjected to an externally applied strong pump EMF (figure 1), characterized by the vector potential t where ω is the frequency of the applied field. The total Hamiltonian Ĥ of this system consists of two parts: the equilibrium Hamiltonian, , , x y x y 0 hs s = = (ˆˆ) (ˆˆ)are the momentum and the single-particle velocity operator, respectively. The equilibrium Hamiltonian describes a pair of gapped Dirac cones (with the energy gap Δ) at the two corners K and K′ of the hexagonal Brillouin zone (labeled by the valley index η=±1), and x y z , , ŝ are Pauli matrices describing the pseudospin degrees of freedom. To apply our results to TMD materials, e.g. MoS 2 , we take into account the spin-orbit interaction λ so in the last term of the Hamiltonian in equation (1), with s=±1 being the electron spin. Equation (1) is the minimal model for TMD that captures valley Hall transport. Since the pump field is illuminated at near-resonant frequencies, effects from the conduction band edge spin splitting (∼1 meV) and trigonal warping further from the band edge [34,35] are expected to be quantitatively small and can be neglected. In the absence of an external EMF, the electron energy spectrum of a TMD near K and K′ consists of conduction (+) and valence (−) bands that are spin and valley-dependent, We note that δ η (p) is proportional to the amplitude A | |, and it strongly depends on pump EMF polarization. In the vicinity of the valley centers, where v p ), capturing the seminal valley-dependent selection rule [27]. Therefore, while a dynamical gap opens in each of the four copies of the gapped Dirac dispersions in the TMD band structure, the valley selection rule causes a dynamical gap in one of the valleys to dominate. In what follows, we will write δ η (p) instead of p d h | ( )|thus dropping the irrelevant phase factor.

Hall transport of PDQs
Hall transport in the presence of a strong pump EMF can be obtained as the linear response to a weak probe field of frequency Ω (see figure 1), characterized by the vector potential . The resulting current density is given by the expectation value j e v G t t i Tr , =a a < [ˆ( )], where α=x, y and G < is the lesser Green's function. In the linear regime over the probe field we obtain where the times t, t′ are taken on the Keldysh contour C. The contour-ordered Green's functions G t t , ¢ ( )in(5), which are 2×2 matrices due to the pseudospin structure of the Hamiltonian (1), are calculated by treating the pump field non-perturbatively within the RWA.
The time-averaged Hall current is expressed via the Hall conductivity σ xy as j x xy y  s = , and t t y c t y is the probe electric field taken along the y axis. Following calculation given in appendix A, we find in the limit of a static probe field ( 0 W  , thus where the coefficients u p and v p satisfy the following conditions (we will use p instead of p in the indices in what by particle number conservation, the above equation implies the conservation of PDQs with n p n p 1 . The valley-dependent Hall conductivity in equation (6) depends on the population difference of the PDQs, which is given by n p n p n p f p p p p 1 2 1 2 c )calculated in the absence of EMF [40]. These small corrections cannot be elaborated precisely in the framework of RWA but their typical value

Kinetics of PDQs
We consider an insulating Dirac semiconductor in equilibrium, where the Fermi level is located in the middle of the band gap. The temperature is taken to be much smaller than the band gap so that thermally excited carriers can be ignored. In the presence of a strong pump EMF, the nonequilibrium distribution function of electrons depends on the ratio of the intraband relaxation time τ and the interband recombination time τ rec [41,42] 8 . In the absence of any intraband relaxation and interband recombination, i.e. the ballistic regime (later referred to as regime I), the difference in the distribution functions of the PDQs is given by n p 1 2 s i g n , corresponding to the distribution function of nonequilibrium electrons in the conduction band, (red); regime II, τ=τ rec (green); and regime III, τ?τ rec (blue). The momentum p c is the value at resonance and it is found from the relation: ξ(p c )=0.
. Here, our results coincide with a kinetic equation analysis based on the density matrix approach [43,44].
Under a strong pump field with large Rabi frequency In the opposite case, when the interband recombination time is . Note here that the nonequilibrium state of the PDQs in regime III is analogous to a nonequilibrium steady state of a two-level system subject to a strong resonant EMF [45]. See also table 1 for the summary of three regimes.

Nonequilibrium valley-resolved Hall conductivity
In order to focus on the essential valley-resolved physics, we will first disregard the spin-orbit interaction.
In the limit of vanishing pump EMF, i.e. δ η → 0 (and for arbitrary frequency), the distribution functions reduce to those in equilibrium, f c (p)=0 and n p 1 2 1 , so that equation (8) Further analytic progress can be made if we disregard the dependence d h on p. Indeed, δ η (p) is a smooth function. In the mean time, the main contribution to the nonequilibrium part of the valley Hall conductivity in equation (10) comes from the vicinity of the resonant points. Thus we substitute the dependence δ η (p) by the value δ η =δ η (p c ). (It should be noted that we keep the dependence of δ η on frequency ω.) Furthermore let us focus on the frequency range w If the pump EMF frequency is below the gap, Δ−ÿω?δ η , only virtual transitions between the conduction and valence bands occur, resulting in a renormalization of band energies, i.e. the dynamic Stark effect, as described by the quasienergies ε 1,2 (p) of the PDQs. This scenario corresponds to regime I. Calculating the integral over ξ in equation (10) In the opposite limit (ω>Δ/ÿ), interband transitions occur and all three regimes can be established. Calculating the integral over ξ in equation (10), we arrive at the following results: In the presence of the pump field, the signs of the Berry curvatures of the renormalized bands should remain the same. Therefore, the Hall conductivity contribution from the conduction (valence) electrons will be negative (positive) at valley K and positive (negative) at valley K′. The negative sign of σ xy then follows from the larger population of excited carriers at valley K in comparison with valley K′ due to the valley selection rule.
Using equation (8) and taking the sum over η-valley-dependent contributions, we can numerically calculate the total Hall conductivity as a function of pump frequency ω (see solid curves in figure 3). We notice a similar behavior in regimes I and III, namely, an abrupt increase in the absolute value of conductivity as the frequency approaches Δ/ÿand a further smooth decrease of σ xy .
The most significant feature is that regime II shows a completely different behavior (inset of figure 3) by revealing a dramatically enhanced Hall conductivity. Indeed, the ratio II I s s ( ) ( ) for similar parameters is on the order of 10 5 . Further, it saturates at large frequencies to a value independent of applied EMF power. This is a direct consequence of the inversion of electron population in regime II.
Next, we take into account the full momentum dependence of the light-matter coupling at both valleys K and K′ by using the exact relations, equations (4) and (6), for a left circularly polarized pump field (σ=1). Now, the light couples strongly to the K valley and weakly to the K′ valley, inducing an enhanced dynamical gap at the K valley with δ 1 >δ −1 (see dashed lines in figure 3). A crucial assumption of these calculations is that both the valleys are described by the same type of steady state distribution functions, regardless of different values of their dynamical gaps.
Accounting for the small dynamical gap in the K′ valley leads to minute changes of σ xy (ω) in regimes I and III (compare the dashed and solid red (blue) curves in figure 3). However, the results obtained for regime II are drastically different. They show a small, sharp peak when ÿω=Δ (compare the dashed green curve in main plot and solid green curve in Inset of figure 3). We explain this behavior as a consequence of the crucial assumption that nonequilibrium distributions are realized in both valleys. Observation of the frequency dependence of σ xy enables us to distinguish between the different nonequilibrium steady states under an optical pump field.

Spin-orbit coupling effects in TMDs
Finally, using the full k·p Hamiltonian(1) of TMDs, we include spin-orbit coupling effects in our analysis. Typical parameters of MoS 2 monolayer [27] are employed: Δ=1.66 eV and λ so =75 meV. Calculation results for the three regimes are presented in figures 4(a)-(c). As expected, SOI results in the appearance of a second threshold in the conductivity of regimes I and III (figures 4(a) and (c)) and two sharp peaks in regime II ( figure 4(b)), once the EF frequency reaches the band gap values Δ±λ so (at ÿω=1.585 eV and ÿω=1.735 eV) for the two spin-split bands 9 . The plots also demonstrate the dependence of σ xy on the value of the gap, δ η (0). It is important to note, that with account of the SOI, there opens a possibility to established spinpolarized Hall conductivity if ω is in the narrow frequency interval (Δ−λ so , Δ+λ so ). Indeed, at the first threshold (see figure 4) due to the energy conservation, there will be Hall current of electrons and holes with a predefined projection of spin [46].

Conclusions
We have developed a theory for the photovoltaic valley-dependent Hall effect in a 2D Dirac semiconductor driven by a strong EMF. We have found that the valley-dependent Hall conductivity is strongly enhanced when the pump field frequency is close to the transition energies of the two spin-split bands at K and K′ valleys. We have also shown that the conductivity is highly sensitive to nonequilibrium carrier distribution functions due to the joint influence of the pump field and the intraband relaxation and interband recombination processes.
Here we present a detailed discussion of the RWA used; properties of the gapped dispersion of PDQs; calculation of the general expression for the conductivity using a nonequilibrium Keldysh approach.
Assuming that the probe field is weak, the current can be calculated as a linear response to this field: , w h e r e , i T r , , , where C is the Keldysh contour. The Green's functions in equation (A1) should be calculated in the presence of the pump field accounting for it in unperturbed manner. Thus, in contrast to standard linear response technique, Green's functions in(A1) are principally nonequilibrium and in general case, they depend on both the times t and t¢ separately. Thus, the Green's function satisfies the following equation: It is written in a pseudospin representation of the operators s â . This representation is not very convenient. Therefore using a unitary transformation, we switch to another representation using the conduction and valence bands: . After the transformation into cv-basis, we find Important to note, that v cṽ and v vc depend on the momentum, p. In order to transform into a rotating reference frame, we utilize the operator

Note, this equation is translational invariant in time.
The resulting quasienergy spectrum is given by As indicated in figure A1, hj hj hj hj hj hj hj hj and Green's function in band representation accounting for the strong pump field reads G t t S t g t t S t g t t g t t e e e e . A 1 3 Substituting equations (A13) in (A12) and performing the Fourier transform in time, we find that non-diagonal terms of Q t t , All the Green's functions here depend on the absolute value of particle momentum p thus, the term sin 2f ( )in F (p) does not play the role due to the angle integration. The structure of gg g g g g R A = + < < < [ ] contains the lesser g < and retarded/advanced g R, A functions which can be easily found from expression (A9).
The time-averaged Hall current is expressed via the Hall conductivity σ xy as j x xy y  s = , and t t y c t y 1   = - ¶ ( ) ( ) is the probe electric field taken to be along the y axis. The Hall conductivity σ xy contains nonlinear effects due to the presence of a strong pump EMF. Taking the integration over ε in(A14), we find (in the limit of a static probe field 0 W  ) a generic expression for the photovoltaic valley-dependent Hall conductivity(6).    . Parameters used are the same as in figure A2.