Valley selectivity of soft x-ray excitations of core electrons in two-dimensional transition metal dichalcogenides

Optical properties of semiconducting monolayer transition metal dichalcogenides have received a lot of attention in recent years, following the discovery of the valley selective optical population of either K + or K − valleys at the direct band gap, depending on the polarization of the incoming light. We use group theoretical selection rules, as well as ab initio DFT calculations, to investigate whether this valley selectivity effect is also present in x-ray optical transitions from the ﬂat core level of the transition metal atom to the valence and conduction band K valleys. Valley selectivity is predicted for s , p 1 / 2 , and p 3 / 2 edges in transitions to and from the valence band edges with circularly polarized radiation. Possible novel applications to the diagnostics of valleytronic properties and intervalley dynamics are investigated and the feasibility of ultrafast pump-probe and Kerr rotations experiments with suitable soft-x-ray free-electron laser sources is discussed.


I. INTRODUCTION
Two-dimensional transition metal dichalcogenides (2D-TMD) provide a very interesting platform for the investigation of two-dimensional solids, with some specific features that differentiate them from either graphene or hexagonal boron nitride (hBN) [1]. In particular, the semiconducting members of this family (MX 2 compounds, with M = Mo, W and X = S, Se, Te) display interesting optical properties in the fundamental gap region; the monolayer gap is direct, and located at two distinct opposite corners of the Brillouin zone, denoted in the following by K + , K − , referred to as "valleys," where pronounced exciton effects are observed and where selective optical population of either valley, depending on the sign of the circularly polarized incoming light [2,3], is observed. This effect has led to investigate the potential use of these "valleytronics" effects for advanced technological applications. The origin of the effect in 2D-TMD and other compounds * massimo.altarelli@mpsd.mpg.de with the same symmetry is ascribed to the lack of inversion symmetry and to the mirroring of the K + , K − band extrema under time-reversal symmetry [4]. The role of the spin-orbit interaction, frequently emphasized in the literature, however, is not essential (see Appendix A for a numerical example).
One of the hotly discussed topics is the understanding and possible control of the decay time of the difference in occupation of the two valleys, related to the dynamics of electrons, holes and excitons in interaction with the other elementary excitations of the two-dimensional crystal. In this context, it may be of interest to analyze alternative excitation paths, that for example do not simultaneously generate valence band holes and conduction electrons in the selected band valley.
Here we consider the possibility to excite electrons from core levels by absorption of x-ray photons, or to fill core holes by valence electrons in x-ray emission processes, investigating possible valley selectivity effects of the photon polarization. Given that the conduction and valence band valleys have more than 75% metal d-states character [5], it is to be expected that the most intense dipole allowed transitions occur from the p-like core levels of the transition metal atoms, for example, at the L 2,3 , M 2,3 , or N 2,3 edges, corresponding to 2p, 3p, 4p core levels (although the low, parity-breaking lattice symmetry allows other possibilities, as we shall soon see).
Very recently, ultrafast far-UV and soft-x-ray spectroscopy investigations of transition metal dichalcogenides films, after excitation with a pump laser (with duration <5 fs), have been performed using high harmonic generation sources [6,7], and a free-electron laser [8], with a pump duration 50 fs; the samples, however, were multilayer films with several tens of nm thickness, and the photons were not circularly polarized. These pioneering experiments were therefore addressing interesting issues of carrier dynamics, but, because of the sample and photon source properties, were not accessing valleytronic properties, which are the focus of the present work.
Practical reasons suggest to concentrate attention on rather shallow core levels, with ionisation energies restricted to the far UV or soft-x-ray regions. Spectroscopy with photon energies of several keV can only be performed with limited energy resolution, unsuitable to differentiate band features such as primary and secondary valleys; in addition, the photon wave vector at such energies cannot be considered as negligible with respect to the size of the Brillouin zone, and therefore optical transitions cannot be described in the dipole approximation, with initial and final single-electron states with the same quasi momentum in the zone.
This paper is organized as follows. In Sec. II standard group-theoretical techniques are applied to 2D-TMD, recovering the known symmetry characters of valence and conduction band edges [9] but also identifying the irreducible representations of the s-, p-, and d-like core-level edges and deriving the selection rules for optical transitions, for both core and valence excitations. The results, pointing to the valley selectivity of circularly polarized x-ray transitions to the valence bands from the s and p core-level edges, but not from the d edges, are briefly discussed. Transitions into valence band states, of course, require also that the Fermi level is below the valence band top, which can occur by electrostatic gating, by optical pumping or by p-type doping. In Sec. III, ab initio DFT calculations of the x-ray absorption spectra are presented, that confirm and complement the group theoretical results. In Sec. IV, taking inspiration from existing experimental results for multilayer TMD, we discuss possible contributions of laser pump, soft-x-ray probe spectroscopy to the study of the electronic properties and dynamics of valley-pumped systems. The requirements on the soft-x-ray source (high intensity and monochromaticity, short pulses, and variable polarization) point in the direction of seeded free-electron lasers [10].

II. GROUP-THEORETICAL SELECTION RULES FOR CORE ELECTRON TRANSITIONS
Van der Waals multilayer stackings of transition metal dichalcogenides, MX 2 , have been investigated for many years. The progress in the manipulation and exfoliation of twodimensional structures lead more recently to the fabrication of monolayers of the 2H structure that, unlike the multilayer versions, have a direct band gap [1]. In Figs. 1(a) and 1(b) a schematic side and top view of the monolayer structure are shown. It consists of three planes of a triangular arrangement of atoms, with the metal M atoms at the central plane, sandwiched between the two chalcogen X planes, with a stacking that locates metal and chalcogen atoms, when projected on a single plane, at alternating vertices of regular hexagons. If we denote by a the distance of two consecutive vertices of a hexagon [ Fig. 1(b)], then a is, e.g., approximately 0.182 nm and an orthogonal cartesian frame to describe all vectors in real and reciprocal space. (c) Brillouin zone for the two-dimensional lattice structure, with the unit reciprocal lattice vectors b 1 , b 2 and with the two wave vectors of the band structure "valleys," K + , K − . A vector equivalent to K + is shown for illustration purposes.
for MoS 2 and 0.189 nm for WSe 2 , as can be derived from the crystallographic data reproduced by Kormányos et al. [11]. In terms of this a length, the cartesian coordinates of the lattice unit vectors a 1 , a 2 in terms of the x, y axes of Fig. 1 In Fig. 1(c) the corresponding Brillouin zone (BZ) is shown, with the unit vectors of the reciprocal lattice b 1 , b 2 and with K + , K − as the two opposite, nonequivalent points where the direct gap is located. It is important to note that, in the same cartesian frame, and the K + and K − points are not connected by a reciprocal lattice vector, Also, the star of K + only contains (besides itself) K − because all other vertices of the hexagonal BZ are connected to either K + or K − by a reciprocal lattice vector [an example is shown in Fig. 1(c)]. It is important to remember that, since in the case of the 2D-TMD inversion is absent, Kramers' theorem [12] does not apply, and nondegenerate spin bands occur throughout the zone. Furthermore, the point group of the structure shown in Fig. 1 is the dihedral group D 3h . To determine the symmetry properties and the selection rules for optical transitions of the Bloch functions at the K + or at the K − point, we have to consider the "small group" or "wave vector group" at this point, the subgroup of D 3h containing the operations that leave this point invariant [13,14]. It easy to see that the C 2 rotations and the corresponding σ v mirror reflections of D 3h interchange the two (nonequivalent) vectors, and are therefore not part of the small group. The remaining operations form the point group C 3h . This is a rather simple Abelian group, with six operations and six classes, each comprising one element, where E is the identity, and the 2π/3 rotations C 3 and the rotation-reflections S 3 (about an axis perpendicular to the layer and intersecting the center of a hexagon) are labeled + or −, depending on whether they are counterclockwise or clockwise. To obtain all the required information about the representations of the point group C 3h and of other groups we shall need later, such as the full rotation-reflection group in three-dimensional space, O(3), many textbooks can be consulted. In the following the notations and conventions of the excellent compilation by Koster, Dimmock, Wheeler, and Statz [15] (KDWS) shall be adopted.
To classify the symmetry of electronic states at the relevant wave vectors K + or K − we must first determine the action of the six C 3h operations on the corresponding Bloch functions. We shall suitably adapt the treatment, as given in Refs. [9,11,16], for the lowest conduction and upmost valence bands to include the case of bands derived from core levels.
A. Symmetry character of conduction and valence levels at To establish the basic methods, notations and conventions, we sketch here the derivation of the symmetry properties and optical selection rules for the valence to conduction band transitions (for more details, see Appendix B).
Following the results of ab initio and semiempirical band structure calculations, the dominant orbital character of lowest conduction band valleys at K + and K − is from the m = 0 d orbital of the metal atom (with a weight, e.g., for MoS 2 estimated [5] at 82%). The remaining fraction is ascribed to p orbitals of the chalcogen, but for symmetry considerations this adds no relevant information. The Bloch function at K ± can be written in the general form [17] c,K ± = where (see Fig. 2) the metal atomic sites are identified by the integers j 1 , j 2 , according to having introduced the vectors and having shown in Fig. 2, next to a few atoms, the ( j 1 , j 2 ) integer values as examples.
In each term of Eq. (5), the first factor is the Bloch phase factor, 2,0 is a Wannier function with l = 2, m = 0 character and χ (σ z ) denotes the spinor wave function, distinguishing the spin up or down conduction valley. In adopting this form of the Bloch function to discuss the symmetry properties, we implicitly acknowledge that, in the fundamental gap region, the crystal field effects, that lift the fivefold degeneracy of the d orbitals of the metal atoms by several eV's, are more important than the spin orbit interaction, that mixes different m and different σ z values. To proceed, one must consider the effect of the C 3h operations on each factor. In doing so, we follow the standard group-theoretical procedure [13][14][15] to describe the effect of a space symmetry operationĈ on a generic function f of the space coordinates r ≡ (x, y, z) As detailed in Appendix B, the Bloch phase factor at K + transforms according to the 3 irreducible representation, and at K − according to 2 ; the d-like Wannier function, however, acquires a factor e −imφ for all rotations about the z axis; so for m = 0 it is invariant for all rotations and also for the perpendicular plane reflection, and therefore belongs to 1 [18]. As to the spin wave functions χ , they are by definition a basis for the two-dimensional D + 1/2 representation of the full rotation-reflection group O(3), where the + superscript acknowledges that the spin pseudovector is parity even. It is important to recall that when dealing with half-integer spin systems we must consider the "double" groups, resulting from introducing an additional operation E , that reverses the sign of the half-integer spin functions, but does not affect the space coordinates (equivalently, the additional elements of the group are often described by adding 2π to the angle of every rotation of the group) [13][14][15]; and consider only the so-called additional representations, odd under E . When restricted to the operations of C 3h , D + 1/2 is reducible (the fact that C 3h has 6 group elements (12 for the double group) and 6 group classes (12 for the double group) implies that all irreducible representations are one-dimensional, corresponding to the absence of spin degeneracy). We can reduce D + 1/2 (as C 3h representation) as follows [19]: Therefore, from Ref. [20] at K + : Similarly, at K − : with the two representations corresponding to the two spin directions. Note that 9,11 are respective complex conjugates of the 10,12 . This is a reformulation of the time reversal symmetry condition E n (k, ↑) = E n (−k, ↓) implied by the discussion immediately following Eq. (3).
Consider now the top of the valence bands. The corresponding dominant orbital is the metal d orbital with m = ±2 [17], schematically behaving as (x ± iy) 2 . We can write the Bloch functions as In either case, namely, K + , m = +2 or K − , m = −2 it is readily seen that the product of the Bloch factor and the d-like Wannier function is invariant under all C 3h operations [2], belonging to the 1 identical representation. The overall Bloch function with the spin part is a basis for the D + 1/2 , and we already know that D + 1/2 = 7 + 8 . On the basis of this symmetry assignment, it is possible to establish the selection rules for optical transitions. The group theoretical necessary but not sufficient prescription for a nonvanishing transition matrix element from an initial state ψ i to a final state ψ f under the transition operator (T α ), can be expressed as [21] The fact that the "small" group of the wave vectors K + , K − has only one-dimensional irreducible representations makes our life much easier (do not need to specify and discuss individual rows of irreducible representations). For the valence and conduction edges we determined the representations; for the transition operator A · p, corresponding to dipole optical transitions, we have the vector potential where denotes the polarization vector of mode λ = ±1, 0 for positive circular, negative circular and linear z polarization, respectively, and b + , b are photon creation and annihilation operators. Including also the photon states, we can precisely define the transition operator in Eq. (15) for a photon absorption process as with n λ , n λ − 1 denoting the number of photons of the λ polarization in the initial and final state, respectively. The transition operator (in the k −→ 0 limit) contains therefore λ · p (for photon emission) and * λ · p (for photon absorption), therefore transforms like the components of the p vector, i.e., like the l = 1 spherical harmonics, or the D − and from a look at the character table of C 3h , keeping in mind the convention embodied in Eq. (8) and applying it to l = 1 spherical harmonics [23], it is easy to establish that , thus assigning the proper symmetry to positive and negative circular polarization for propagation perpendicular to the plane, and to linear polarization perpendicular to the plane. We are especially concerned with the first two, that determine the chirality of optical properties of MTMD. Taking Eq. (16) into account, we identify 2 , (T α ∼ * +1 ) as the "+" circular polarization, and 3 (T α ∼ * −1 ) as the "−" one, following the convention implicitly assumed in Ref. [2].
A straightforward perusal of the C 3h table of multiplication [22] and character table [24], yields the following allowed transitions from the valence band state 7 , 8  This recovers the remarkable effect of the valley selectivity for circularly polarized photons. It is important to observe that although the excitonic character of the observed optical transitions at the fundamental gap is very pronounced [1], this does not alter the selection rules derived from one-electron Bloch functions at the relevant edges, as the excitonic wave function satisfies those symmetries.
B. Symmetry character of core levels at K + , K − As already mentioned in the Introduction, the core electron levels have very distinctive features when compared to valence states: one is their very localized character, so that, when described in terms of bands in momentum space, they have negligible dispersion and very large effective masses. In addition, the spin-orbit splitting of core levels (except the s-like ones) is much larger than for valence states and constitutes an energy scale exceeding the fundamental band gap and in most cases the typical bandwidth of valence levels (see a compilation of atomic binding energies of Mo and W in Appendix E). Before dealing with the discussion of strongly spin-orbit-split p and d core levels, let us briefly address the s-like core levels. This can in point of fact proceed very quickly, as the discussion parallels completely that of the conduction band bottom. From the point of view of the C 3h group, the conduction band Wannier function 2,0 and the s-like function 0,0 have exactly the same symmetry properties of invariance under all group operations. Therefore, we obtain the symmetry classification of 10 , 12 for the two spin orientations at K + and 9 , 11 at K − . This is enough to conclude that, in p-type samples, transitions to the valence band top shall display the valley selectivity of polarized photons. Transitions to the conduction band minima, corresponding to states with symmetry identical to the s core levels, are forbidden.
Going back to the strongly spin-orbit split p and d levels, it appears appropriate to modify the description of Bloch functions as given in Eqs. (5) or (12) by diagonalizing at the very start the spin-orbit interactions and replacing the eigenfunctions of l, m, and σ z with those of j and j z . Given the very nearly dispersionless nature of the core bands, all the 2 j + 1 values of j z are degenerate, to a very good approximation, throughout the Brillouin zone and constitute a basis for the D ± j representation of O(3); the ± sign ambiguity depends on the parity of the spatial wave functions, e.g., odd for p 1/2 or p 3/2 levels, even for d 3/2 or d 5/2 , etc.
Thus, the general form of a core wave function is From the discussion of the valence levels, we learned that the Bloch factor in each addendum transforms like 3 for K + and like 2 for K − . Therefore, the complete Bloch function transforms like the product representation with the odd D − j representation for p core levels, and the even D + j for the d-like levels. From tabulated results [22] we have the following reductions: which allow us to identify the reduction of the complete Bloch functions for p 1/2 and p 3/2 functions, i.e., for .. core levels. Starting with the p 1/2 levels at the K + point, from Eq. (21) they belong to the following representations: and at the K − point, to the representations We are then in a position to derive the selection rules for optical transitions to the conduction band minima, using again Eqs. (13)- (16). The only allowed transitions are at K + : − In this case of p-type samples, therefore, the valley selectivity should be in principle observable at the L 2 , M 2 , N 2 absorption edges. Also, x-ray emission deriving from transitions from the full valence band states to core holes should obey related selection rules (interchanging initial and final states and the sign of the polarization).
We now consider the p 3/2 levels. From Eq. (22) one obtains that, at the K + point they correspond to the following representations: and at K − : Notice that 7 and 8 levels are present at both points, and we know from the discussion of the valence band transitions that these symmetries imply the valley selectivity of optical absorption to the conduction bands. It remains to investigate which transitions are allowed from the 9 and 11 levels to the 10 and 12 levels at K + , and from the 10 and 12 levels 9 and 11 levels at K − . The analysis based on Eq. (14) gives the following results for allowed transitions at K + : We thus see that the sixfold degenerate d 5/2 state contains all six additional irreducible representations of C 3h . They shall also all appear both at K + (K − ), because multiplication by 2 ( 3 ) of the whole list of representations shall deliver a permutation of the same list. This tells us without further calculations that the valley selectivity does not occur at d 5/2 edges, where all possible symmetries and transitions occur at both K ± points. An explicit calculation shows that the selectivity is lost also at the d 3/2 edges. In fact, the fourfold degeneracy of this state is split into four representations, namely, at K + : and at K − : This combination of symmetries is not verifying valley selectivity. For example, we find at K + : Initial Final Polarization 9 12 − so that transitions with both polarizations are allowed in both valleys. It is also easily verified that no selectivity is present in p-type samples either. In the following section we will use first-principle calculations to confirm all the selection rules obtained for high symmetry points, and extend them through large regions of the Brillouin zone.

III. NUMERICAL CALCULATIONS OF TRANSITION RATES
To confirm the findings of the group-theoretical analysis of the selection rules for the K + and K − band edges, DFT ab initio calculations were performed throughout the Brillouin zone for the prototypical MoS 2 MTMD, using the all-electron exciting code [25] with linearized augmented plane-wave and local orbitals (LAPW + lo) basis set [26]. In this way, the rapid oscillation of the localized wave functions around the nuclear position is handled in the local atomic orbital basis within the sphere of the "muffin-tin" radii and the rest of the solid (interstitial region) in the plane-wave basis. Due to the dominance of the relativistic effects for the core states, we obtained core-state wave functions by solving the set of coupled Dirac equations in a spherically symmetric potential as implemented in exciting [25,28]. In contrast, the conduction and valence states were obtained in the scalar-relativistic approximation. The latter also implies that the small components of the four-spinor wave function were neglected when calculating the transition matrix elements. We employ the generalized gradient approximation (GGA) PBE [27] for exchange-correlation functional, a momentum space grid of 21 × 21 × 1 points, and a muffin-tin radius of 2 a.u. for both species. Excitonic effects were not included at this stage, as they should not alter the main results and conclusions, for the reasons outlined later, at point 2 in Sec. V.
The selected core level is the Mo 3p 3/2 , or M 3 level, which (see Table I) is located about 390 eV below the Fermi energy. The quantity that is computed and plotted is the normalized "degree of optical polarization" η as defined in Ref. [2], for the transitions from the four degenerate p 3/2 levels to the band b, where D jb in the independent particle limit in vicinity of K ± points, Here the integration area (k max ) is defined by the parameter k max as |k − K ± | < k max . The latter is chosen to be the half of the distance between the K + and K − points. The transition matrix elements M jb (k) between core and valence (conduction) states were calculated in the dipole approximation and  (Fig. 4). The dominant contribution from the corresponding valley (K + for right circular polarization and K − for the left), however, highlights strong selectivity (see Fig. 3).
The calculations, besides confirming the opposite value of η at the K + , K − points for valence band transitions, also reveal that the strong polarization character persists in very large portions of the Brillouin zone, as one moves away from these points; a fact that would not be easily predictable by group theory, but that is in complete analogy to the findings of Cao et al. [2] for transitions at the fundamental gap. This is an aspect that is important for soft-x-ray spectroscopy, because it implies that the valley selectivity is not going to be washed out by the relatively larger bandwidth of x-ray pulses. The DFT calculations also underline very different behavior of transitions to the conduction bands, where, besides confirming the zero selectivity at the K ± points, they reveal a very different distribution of η in the two regions surrounding the points. To study the role of the orbital character in the selection rules for the whole Brillouin we performed a projection of the Kohn-Sham wave functions onto the basis of the local orbitals of the molybdenum atom and results are presented in Fig. 5. Qualitatively, instead of almost uniform distribution, which we observed for the core to valence band transition, the η distribution for the core to conduction band transitions shows two types of subvalleys with the opposite signs. The first type comprises the valleys around the K ± point, and the second the small triangular valleys in between K ± and . The latter corresponds to the local minimum of the conduction band (Fig. 5). The opposite selectivity of the two subvalleys is related to the d orbital character of the conduction bands where d 2 and d −2 interchange each other around local and global minima. From Fig. 5 we could also see that for the conduction band in the close vicinity of the K ± the d 0 type of orbitals have the most substantial contribution and correlate with the absence of dichroism predicted from group theory. Furthermore, for the valence band, orbital characters d ±2 are dominant in K ± valleys, respectively, which expand almost up to point and lead to the more isotropic distribution of η. In line with the conclusion of the previous section our ab initio calculations highlight that the symmetry of the local orbitals involved in the transition lies at the very origin of the valley selectivity.

IV. SOFT-X-RAY CORE-LEVEL SPECTROSCOPY: APPLICATIONS TO "VALLEYTRONICS"
The main result of the previous sections is the valley selectivity of transitions from the metal s and p core levels to the valence bands. To have empty states near the valence band top, the available options are gated devices, typically with a MTDM sandwiched between hBN layers with electrodes at the bottom and the top [29], or optical pumping, as in Refs. [6][7][8] and possibly p-type doping [30,31]. If we postpone for the time being the discussion on the feasibility of XUV and soft-x-ray transmission measurements in atomically thin samples, then novel diagnostic tools for valleytronics can be envisaged. For example, taking inspiration from the pumpprobe transient absorption experiments reported by Chang et al. [7] at the W 5p 3/2 edge of multilayer WS 2 samples, consider the pump-probe experiments for the monolayer case schematically displayed in Figs. 6(c) and 6(d). Here we propose to use circularly polarized pump and probe pulses; we argue that a circularly polarized pump pulse should induce a pronounced circular dichroism in the pump absorption. If the pump and the probe polarizations are the same [ Fig. 6(c)], then the transient absorption of the probe should be observed at very small delay, because the transitions from the core levels access the valley in which a hole population was created by the pump. As the delay of the two pulses increases the transient absorption should be attenuated, with characteristic time τ , given by because the hole population is reduced by recombination (rec) and by transfer(trans) of holes to other valleys. If the probe polarization is inverted [ Fig. 6(d)], then the probe can now promote core electrons only to the valley opposite to the one populated with holes by the pump: the absorption signal is therefore much smaller than in the previous case at short delay times, and can possibly build up in time, if sufficient transfer to the opposite valley takes place, with characteristic time (where the prime acknowledges that only transfers to the opposite valley are included, and not, e.g., to the valley). At any delay time, therefore, the absorption of the pump pulse should display circular dichroism and, according to Eqs. (33) and (34), should deliver information on the characteristic decay times of the valence valley polarization.
To consider specifics related to WS 2 monolayer, we performed calculations of the normalized polarization degree for transitions first from the core levels (5p 3/2 of W) to the uppermost of the valence bands [ Fig. 6(a)] and then from the uppermost of the valence bands to the lowest of the conduction bands [ Fig. 6(b)]. As a result, both core and optical transitions show pronounced valley selectivity with the same sign in K + and K − valleys. The latter means that if circular polarization of the optical and x-ray photons are the same, we can probe information about the single valley; and if, however, they are different, contributions from cross valleys processes will be dominant. In contrast to the previous section, calculations for WS 2 include relativistic effects for core and valence electrons and consequently reproduce spin-orbit splitting of ∼8 eV for 5p states of tungsten, and ∼0.3 eV for the valence d states. The results are presented in Fig. 6 and agrees with the group theoretical predictions. It is also worth noticing that the two valleys are extended through the most part of the Brillouin zone and manifest strong difference in partial absorption. In Figs. 6(a), 6(e), and 6(i) we are showing only characteristic contributions from those 5p 3/2 fourfold degenerate levels that participate in nonzero transitions at the K ± points, contribution from the rest of 5p 3/2 levels favorably support the conclusion in the extended valley region but lead to the divergence of the polarization degree due to the zero denominator. As expected from the group theoretical analysis the core to the lowest conduction band transitions are not valley selective at K ± , but have certain selectivity for the small parts of the Brillouin zone (see also Appendix C). Following the approach stated in the previous section we obtained partial absorption cross-sections for the both types of transitions which confirms strong dichroism through the entire energy range of the highest (lowest) valence (conduction) bands.
It is generally presumed that intervalley transfer occurs for excitons, via exciton-exciton or exciton-phonon scattering, with much higher probability than for single carriers (electrons or holes) as the latter case implies either a spin flip or a sizable energy difference [1]. Nonetheless, it is worthwhile to notice that the core electron circular polarization transitions are strictly a probe of the hole population, unlike, e.g., photoluminescence. In special cases, such as the presence of nonradiative recombination or trapping centers for either carrier, the distinction may become relevant.
The actual feasibility of core-level absorption spectroscopy on single layer samples deserves discussion, in particular in the pump-probe mode as discussed above, although recent XAS measurements of monolayer MoS 2 at various core thresholds, performed by electron yield at the SSRL synchrotron in Stanford, are encouraging [32].
With respect to the results reported in Ref. [7], there is, on the one hand, the reduction from several tens of layers to one, and the requirement of circularly polarized photons, on the 115433-9 other hand. To the former challenge, the possible strategies could involve the growth of samples in which single layers are stacked with a hBN spacer in between; and also the selection of a more intense source. By replacing the HHG source used in Ref. [7] with a free-electron laser such as the FERMI facility [10], the intensity increases by about 2 to 3 orders of magnitude; and one can ripe the additional benefit of satisfying the requirement of circularly polarized photons.
Another technique to explore the intervalley relaxation of carriers is time-resolved Kerr rotation (TRKR) [33,34] which has been used in samples where tuning the Fermi level by the gate potential produces, independent of optical pumping, an equilibrium population of electrons or one of holes. In presence of these so-called resident carriers, optical pumping with circularly polarized photons near the band gap produces, in one of the valleys, a rapidly decaying population of charged excitons, or trions, which results in an upset of the balance in the population of the two valleys. The imbalance corresponds to a nonzero value of the orbital angular momentum. The exact role of the trions decay and the role of non radiative traps due to impurities is not completely clear [34]. In any event, the measurement of the decay time of the Kerr rotation allows access to the decay times of the difference in valley population of electrons in n-type samples and of holes in p-type samples. The latter case is especially interesting as decay times as long as several μs are observed. In these samples, generation of the holes imbalance between the valleys could be produced by transitions from the core levels with circularly polarized x-ray photons to the valence bands, without intermediate stages in which excitons or trions are present. The core hole would be very rapidly filled by Auger processes or by x-ray fluorescence from electron levels from the whole valence bands; such core hole annihilation should not involve any valley selectivity and should therefore not influence the Kerr rotation signal. This would therefore simplify the understanding of the imbalance relaxation paths and of the corresponding timescales.

V. CONCLUSIONS
In summary the conclusions of the present analysis of the selection rules for transitions in core-level spectroscopy in MTMD are that: (1) The valley selectivity of circular polarization transitions to the conduction band minima is not to be expected for s-, p-, or d-like metal core levels, as shown by group-theoretical arguments and by ab initio calculations, in contrast to the well-known top of the valence to conduction valleytronic selectivity.
(2) Valley selectivity is, however, expected for transitions from s-and p-like core levels (but not for d-like levels) to valence states (in contrast to the well known top of the valence-to-conduction selectivity, valleytronics), and conversely for x-ray emission. This conclusion assumes of course that the valence bands are not completely full, as can be obtained in a static way by electrostatic biasing, or in a static or time-dependent way by optical pumping.
(3) We argued that these selection rules open the way to a new tool for the study and time resolved monitoring of valleytronics in 2D-TMD, and a specific example was discussed.
The potential applications, however, are not limited to the idea of using an x-ray pulse as a probe. In fact, another possibility to profit from derived selection rules is to use the core-to-valence transition under circularly polarized light to create an imbalanced population in the two valleys for the initially hole-doped TMD sample. That would allow more direct access to dynamics of the hole/electron carriers without creating multiparticle excitations (e.g., excitonis, trions), which accompany transitions in the optical region.
Before further considering possible experimental implications of the results derived here for core-level transitions, the following points should be taken into account: (1) Feasibility of specific experiments, depending on the actual cross sections, considering the extremely diluted character of monolayer samples.
(2) Excitonic effects. The valence to conduction spectroscopy is strongly affected by exciton effects [1], as expected from considerations on screening of the electronhole interaction in low-dimensional systems. The same arguments should apply to electron-core hole interactions, and core excitons can be expected to modify the transition energies and intensities, but not necessarily the symmetry considerations discussed here. In general, if a pair of bands are connected by an allowed dipole transition, then so is the corresponding ground exciton state.
(3) The just mentioned dipole approximation, i.e., the assumption that the photon momentum is negligible in comparison to the size of the Brillouin zone. This is reasonable for the shallowest core levels, like the M edges of Mo or the N edges of W, but certainly not for the deeper ones.
(4) The lifetime broadening of core levels (especially the deeper ones) is a limitation of the achievable energy resolution. In some cases it can prevent a clear distinction of transitions to the K ± valleys from nearby secondary band extrema, requiring a specific discussion of each compound.
Finally, we considered a material with close structural symmetry but without strong spin-orbit coupling such as hBN. We confirmed valley selectivity for excitations in the optical region and derived corresponding selection rules. However, there are very few dipole allowed transitions from core levels to the lowest conduction and valence band and in total they do not show the valley selectivity.

APPENDIX A: VALLEY SELECTIVITY WITHOUT SPIN-ORBIT COUPLING: hBN
The purpose of this Appendix is to briefly consider the valley-selective effects in monolayer hexagonal boron nitride, hBN, with a simple extension of the discussion given in Sec. II A. In fact, the lattice symmetry for a monolayer of hBN is identical to that of 2D-TMD, with D 3h as point group and C 3h as the wave vector group at the K ± points [35]. According to the prevailing results of DFT theoretical work, the valence band top of monolayer hBN is located at the K ± points; whether the conduction band minimum is at or at K ± is a very close call [36][37][38][39] and the computed gap is extremely sensitive to the chosen functional and to details of the numerical procedure [40]. Nonetheless, even if the lowest gap is indirect, optical properties are very likely dominated by the larger oscillator strength of the direct transition, and experimental evidence [41] points in this direction. We are going to consider transitions at these direct gaps to provide an example of nonvanishing valley selectivity in a system in which, with good approximation, the spin-orbit interaction can be neglected (see Fig. 7).
The group theoretical analysis can follow the pattern of Eq. (5) and following of Sec. II A, the only difference being the orbital character of the Wannier-like functions for hBN. According to the band structure calculations, see for example Ref. [36], the dominant character of the conduction band minima is derived from the B 2p z states, whereas the valence band top is ascribed to 2p z orbitals of the N atoms. Given the low atomic number of these elements, spin-orbit interactions are negligible. Starting with the conduction band minimum at K + , we observe that the 2p z orbitals are invariant for all proper rotations about the z axis (m = 0), but are odd under all operations reversing the z axis directions, i.e., σ z , S ± 3 . A look at the character table of C 3h associates this behavior to 4 representation. The symmetry character of the exponential Bloch factor for the metal sites is identical to the MTMD case and belongs to the 3 irreducible representation. Finally, the spin wave function belongs to the reduction of the D + 1/2 representation that gives 7 + 8 . Putting all together, the conduction band minimum at K + for the two spin directions is associated to Consider now the top valence bands, corresponding to similar 2p z orbitals of N, centered however on the "empty" vertices of the lattice sketched in Fig. 2. These sites can be indexed in analogy to Eq. (6) as Note that with this indexing, since K + · δ 2 = 0, both j 1 , j 2 -indexed anion and cation sites belong to the same wavefront and have a common phase, Repeating the procedure outlined in Appendix B, and noting that C − 3 δ 2 = δ 3 we find that after a C + 3 rotation, the phase factor of the N sites becomes and, in general, C ± 3 , S ± 3 produce a factor e ±(i2π/3) , which allows us to associate the Bloch factor at the N sites to the 2 representation.
The valence bands analogous of Eq. (A1) becomes then It is readily seen that if we now consider the same bands at K − we must interchange 2  showing as valley selectivity effect at the K ± valleys of hBN. Our numerical calculations (Fig. 7) confirm such analysis; however, they also suggest that the valley are smaller then in case of MTMD and elongate toward the point. In considering the transitions from the 1s core levels, however, we find no valley selective circular dichroism probing the highest valence and the lowest conduction bands. This is however not attributable to the negligible spin-orbit interaction, but rather to 2p z , m = 0 character of both band edges, resulting in forbidden transitions for circular polarization.

APPENDIX B: SYMMETRY CHARACTER OF THE SPACE PART OF THE BLOCH FUNCTIONS AT K +
For completeness, we enumerate the steps leading to the conclusion that the space part of the Bloch function (i.e., the part that does not include spin variables) at K + transforms as the 3 representation of C 3h . The first observation is that the σ h leaves all factors unchanged, because atomic position vectors lie in the plane, 2,0 ∼ (3z 2 − r 2 ) is even and the spin is a pseudovector. Consider now the counterclockwise rotation by 2π/3, C + 3 . To determine its effect on the spatial part of the wave function, i.e., on following Eq. (8) we have to evaluate This can be done by noticing that the sum is unchanged if each R j 1 , j 2 is replaced by C − 3 (R j 1 , j 2 ), i.e., This is because, when j 1 , j 2 run over all integers, both sums on the right-and on the left-hand side run once over each and every metal site. To make further progress, we determine the lines of constant phase of the unrotated Bloch wave [the Bloch wavefronts, shown as dotted lines in Fig. 2, the integer near the top of each line being (2 j 1 + j 2 mod 3)]; the phase factor being given by e iK + ·( j 1 a 1 + j 2 a 2 +δ 3 ) = e i(2 j 1 + j 2 −1) 2π 3 .
To see the effect of the C − 3 rotation, we notice that its effect is to transport the atom located at j 1 a 1 + j 2 a 2 + δ 3 to the new location −( j 1 + j 2 )a 1 + j 1 a 2 + δ 1 . This is readily seen by inspection of Fig. 2 The phase factor associated to the j 1 , j 2 term after the rotation becomes where the right-hand side results from inserting factors e i3 j i 2π 3 = 1, i = 1, 2; comparing to Eq. (12) it is immediately apparent that the action of C + 3 multiplies each addendum of the Bloch sum by e (i4π/3) ≡ e (−i2π/3) , and likewise in general the elements C ± 3 , S ± 3 produce a factor e ∓(i2π/3) . Now a simple look at the character table of C 3h [15] reveals that these complex factors reproduce exactly the characters of the 3 representation.
The m = 0 d-like Wannier function, however, is unaffected by all rotations about the z axis, and it does not affect the overall symmetry of the Bloch function.

APPENDIX C: CORRESPONDENCE BETWEEN SPIN-ORBIT PARTNERS AND IRREDUCIBLE REPRESENTATIONS
The purpose of this Appendix is to deepen the symmetry assignments of the electronic states at K ± ; for example, it was established, Eqs. (23) and (24), that the doublet of p 1/2 states with j z = ±1/2 belong to the 8 and 11 representations at K + ; which one corresponds to the j z = +1/2 state? This should allow us to identify the wave functions for a specific representation and to obtain a direct, intuitive picture of the optical selection rules.
The procedure to associate a specific wave function to an irreducible representation, in the somewhat subtle case of a double group, can be derived from Ref. [44].
Let us start from the simple case of the conduction band, where we can basically ignore the spin-orbit interaction. Going back to Eq. (5) and considering one addendum of the Bloch sum, we can write the wave function for spin-up and -down as e iK + ·R 2,0 (r − R) There are therefore three parts of the wave function: Bloch sum, a spherical harmonic type orbital, and a spinor. When we act with a C 3h operation on it, the wave function is multiplied by a factor that is the product of factors related to the Bloch exponential, to the orbital and to the spinor. Evaluating this product gives us the character of the operation in the corresponding irreducible representation, and comparing to the character table we can identify the symmetry type of the wave function. The fact that all representations are onedimensional is of course making this particularly simple. Let us start with the spin-up case and with C + 3 . Looking back to Appendix B, we know already that the Bloch term contributes a factor e −i2π/3 , and the m = 0 function a factor 1. The spinors transform according to the D + 1/2 representation of O(3), and following Eqs. (3.10)-(3.16) of KDWS, a rotation by an angle α about the z axis corresponds to the matrix Therefore, for a −2π/3 rotation, there is a factor e −iπ/3 for the spin-up case, e iπ/3 for the spin down. Putting it all together, for C + 3 , we get an overall factor −1 for spin up, and e −iπ/3 for spin down. Taking a look at the character table (Table 57 on page 59 of KDWS) this is already sufficient to establish that spin up is 12 and spin down is 10 , as Eq. (10) has already restricted the selection to these two representations. For future reference and to appreciate the care needed for the double groups in the general case, it is instructive to obtain the same result again, but this time choosing the σ h operation. Since Eq. (8), or the equivalent Eqs. (3)(4)(5)(6)(7)(8)(9)(10) and (3)(4)(5)(6)(7)(8)(9)(10)(11) of KDWS, lead us to apply the inverse of σ h , we must avoid the pitfall of identifying σ h as its own inverse: in fact the inverse of σ h is σ h , which is σ h plus a 2π rotation (that for a spinor does not coincide with the identity E , because it changes all signs). In fact [43], So we know that the effect of σ h on the Bloch factor and on the m = 0 orbital is a factor of 1, to find the action on the spinors we must remember that σ h = C 2Î , a rotation by π times the space inversion. We therefore obtain where we used the fact that D + 1/2 is even under the inversion, and therefore the action of σ h is equivalent to that of C 2 . We therefore conclude that the effect of σ h for the spin-up state is to multiply it by a factor −i, and for the spin-down by i, and a look at the character table confirms the 12 assignment for spin up, 10 for spin down. At K − the states transforming as 9 and 11 correspond, respectively, to spin up and spin down as they are identifiable as the time-reversal partners of, respectively, 10 and 12 , as already pointed out in the discussion following Eq. (11).
Following the reasoning already adopted at the beginning of Sec. II B, the same assignments are valid also for s-like core levels.
Next we consider the top valence bands. They are unique in their symmetry properties, because as described in Sec. II A, the Bloch factor for K + and the m = 2 d orbital combine to form a C 3h invariant function, and the same happens at K − with the m = −2 orbital. Therefore, only the behavior of the up and down spinors characterizes the symmetry behavior of the top valence bands, and on the basis of Eq. (42) we can assign 8 to spin up and 7 to spin down. Again the time reversal invariance imposes E (K + , 8 Let us now consider the p 1/2 levels. The corresponding Bloch functions at K + can be written, in spinor notation, as j 1 , j 2 e iK + ·R j 1 , j 2 1/2, j z r − R j 1 , j 2 , where, for j z = +1/2 and, for j z = −1/2, To compute the character of an operationĈ, following the usual procedure, we must evaluate The effect on the exponential Bloch factor can be evaluated as in the previous sections, and for the spinor we apply the procedure described by KDWS on p. 10 ff., Eqs. (3-13)- (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16). For example, for the C + 3 operation, the exponential acquires a factor e −i2π/3 , for the D matrix we can use Eq. (C3) (the ± parity sign intervening for improper rotations only) and for the spinor component we simply apply C − 3 (z) = z, C − 3 (x ± iy) = e i∓2π/3 (x ± iy), to obtain Therefore, the 1/2, 1/2 partner of the p 1/2 doublet is multiplied by −1 when acted upon by C + 3 , which associates it to the 11 representation [see Eq. (23)], while, necessarily, the 1/2, −1/2 partner belongs to 8 .

APPENDIX D: WS 2 : CORE-TO-CONDUCTION BAND TRANSITIONS
For the experiment described in Sec. IV, it is essential to have valley selectivity for the transitions from and to the top of the valence band. However, the valley selectivity is not expected for transitions from p core levels to the conduction band minima at the K ± points, according to the findings of the group theoretical analysis of Sec. II. It is interesting to explore the first principle calculations for the WS 2 monolayer in some detail (see Fig. 9). In fact, transitions from each individual W 5p 3/2,m j level are valley selective, but transitions with positive and negative m j have an equal and opposite valley sensitivity and cancel each other completely around the K ± points. Similar to what is found also in MoS 2 , there are small secondary valleys in between and K ± points which show a partial degree of dichroism [see Fig. 9(b)]. The latter could be relevant for the future studies involving relaxation and intervalley scattering.

APPENDIX E: X-RAY EDGES FOR Mo AND W
We reproduce here the atomic binding energies of some of the core levels of Mo and W, for convenience of the reader (Table I).