Probing valley population imbalance in transition metal dichalcogenides via temperature-dependent second harmonic generation imaging

Degenerate minima in momentum space - valleys - provide an additional degree of freedom that can be used for information transport and storage. Notably, such minima naturally exist in the band structure of transition metal dichalcogenides (TMDs). When these atomically thin crystals interact with intense laser light, the second harmonic generated (SHG) field inherits special characteristics that reflect not only the broken inversion symmetry in real space, but also the valley anisotropy in reciprocal space. The latter is present whenever there exists a valley population imbalance (VPI) between the two valleys. In this work, it is shown that the temperature-induced changes of the SHG intensity dependence on the excitation fieldpolarization, is a unique fingerprint of VPI in TMDs. Analysis of such changes, in particular, enables the calculation of the valley-induced to intrinsic second order susceptibilities ratio. Unlike temperature-dependent photoluminescence (PL) measurements of valley polarization and coherence, the proposed polarization resolved SHG (PSHG) methodology is insensitive to the excitation field wavelength, an advantage that renders it ideal for monitoring VPI in large crystalline or stacked areas comprising different TMDs.


Introduction
Conventional optoelectronics is based on the manipulation of electronic charge with light, for information transport, storage and readout. In electronic systems with degenerate minima in their band structure -valleys-an additional degree of freedom that labels those minima, i.e., the valley index, can serve as the information carrier.
Such a possibility has opened a new field of electronics, namely valleytronics, which enables the processing of additional information within the same physical space 1, 2, 3 .
Specifically, the valley index can be mapped to a pseudospin that, similar to the response of a spin to magnetic fields, is affected by the Berry curvature of the bands 4,5 .
In TMDs, the valley pseudospin is coupled to the electron spin 6 to give rise to selection rules that allow the excitation of carriers in specific valleys only when light of suitable helicity is used 7,8,9 . More specifically, the excitation and control of carriers is achieved by circularly polarized light that populates only one of the two valleys. Interestingly, the polarization is transferred to and subsequently measured in the emitted (one-and two-photon) PL 10,11,12 . Such measurements are used to indirectly probe the population imbalance in different valleys.
Several other phenomena, associated with the existence of valleys and related with time reversal symmetry, have been reported in atomically thin crystals with hexagonal structure and corresponding degenerate (but inequivalent) K and K' valleys in the hexagonal Brillouin zone. For example, in the presence of external fields, this imbalance may generate valley currents that give rise to valley Hall conductivity 13,14 .
In addition, when linearly polarized light is used for excitation, superpositions of excitons in different valleys are created and one can measure valley coherence 15,16 .
Valley injection and transport may be used to create a valley filter, i.e., a means to populate a single valley and therefore induce valley polarization. Serial combination of two such filters acts as a valley valve that can be externally controlled 18 . Moreover, although intervalley scattering is suppressed in ideal crystals, there are recent reports of intervalley collective modes in the presence of unequal valley populations 19 . 4 In all the aforementioned cases, the common underlying physical principle is the population imbalance between different valleys. This is reasonable considering that polarization and transport effects are associated with charge separation and local variations in the chemical potential, respectively. Hence, they directly reflect the crystal symmetries in both real and momentum space. In real space, the charge is locally accumulated around the atomic positions, whereas in momentum space, the carriers occupy states in the vicinity of high symmetry points within the hexagonal Brillouin zone. Interestingly, both polarization and transport effects can be examined by analyzing the second order nonlinear optical response of atomically thin crystals.
Recently, SHG spectroscopy has been highly appreciated as a powerful tool to study two-dimensional (2D) TMDs 20,21,22,23,24,25,26,27 . Owing to the vanishing thickness of these thin crystals, phase matching conditions are readily satisfied and thus the second order nonlinear susceptibility, (!) , is large 28 . Under the D 3h point group symmetry of TMD crystals, the (!) tensor shows non-vanishing elements for odd-layer TMDs in 2H configuration and for arbitrary-layered in 3R stacking geometry 28 . These elements are not independent but are inter-related through: (2) = − (2) = − (2) = − (2) = (2) , where x denotes the crystal mirror symmetry axis, i.e., the armchair direction and !"# (!) is the nonzero element of the intrinsic second order susceptibility tensor.
In addition, both SHG intensity and polarization have been found to reveal information about the main crystallographic axis 20 Indeed, the ability to control the polarization state of the SHG signal enables the extraction of additional information from a single measurement, since the second order response is determined by a third rank susceptibility tensor 31 ; therefore 5 measurements at higher order of response enable access to a larger number of independent quantities of a system 32,33 .
Recent theoretical studies suggest that the second order optical response is also a useful tool to probe the electronic configuration of 2D crystals 32,34,35 . This is possible due to the symmetry characterizing the hexagonal Brillouin zone in momentum space.
Similarly to the alternating atoms at the corners of the hexagon in real space, characterized by the D 3h symmetry of the trigonal prismatic structure with Bernal stacking (Fig. 1a), in momentum space the alternating K and K' points also result in D 3h symmetry, reflecting the trigonal warping of electrons in the vicinity of high symmetry points (Fig.1b) 36 . Besides, the conduction and valence band states at the corners of the hexagonal Brillouin zone are formed by hybridization of the transition metal d-orbitals with the chalcogen p-orbitals and therefore are strongly localized in the metal atom plane 37,38,39 . A direct consequence of this effect is the possibility for such crystals to produce valley-induced SHG, additionally to the intrinsic second order response. As a result, in the presence of population imbalance between the two valleys, additional elements in the second order nonlinear optical susceptibility tensor become nonzero 32,34,35 . Therefore, we have: !!!
where !"# (!) is the nonzero element of the VPI second order susceptibility tensor.
VPI can be either induced 10 or emerge spontaneously 40 and depending on the way it is created, it reflects different aspects of the electronic system. For instance, when circularly polarized light is used for excitation, only one of the two valleys is populated and the spin-valley polarization is transferred to the detected one-or twophoton PL 15 . In contrast, for linearly polarized light, a superposition of excitons in the K and K΄ valleys is created and one can measure valley coherence 16 . Even at equilibrium, i.e., valley population balance, there is a small, but finite probability that 6 an electron will be transferred to the adjacent valley, while at the same time flipping its spin, thus perturbing the balance. In this last case, access to the degree of VPI would reveal information about the intrinsic valley relaxation time 40 .
In this work, we take advantage of the temperature dependence of PSHG to account for the VPI in atomically thin TMDs. Notably, since the VPI defines the difference between the valley populations, = ! − ! ! , it also reflects the chemical potential difference, , between the two valleys ( Fig. 1c). Hence, in the presence of imbalance, the additional valley-induced contribution to the SHG that is intrinsically generated by the TMD crystal, can be estimated as with the corresponding contribution to the second order nonlinear susceptibility being proportional to the chemical potential difference, i.e., !"# (!)~34, 35,40 . Accordingly, local variations in the chemical potential affect the SHG induced by the VPI and thus can be probed by nonlinear optical experiments 41 . More importantly, and in contrast to the intrinsic nonlinear optical response of 2D TMDs, the valley-induced SHG is sensitive to temperature variations. Based on this, we here vary the temperature of a 2D TMD crystal and the corresponding changes in the SHG intensity are used to probe the inter-valley chemical potential difference and therefore the VPI. Unlike temperature-dependent PL measurements of valley polarization and coherence, the proposed SHG methodology is insensitive to the excitation field wavelength, an advantage that renders it suitable for monitoring VPI in TMDs.
We consider an electromagnetic field that is normally incident to a 2D TMD sample with polarization parallel to the sample plane, at an angle φ (Fig. 2a); the crystal !"# !"#~!~! armchair direction is oriented at an angle θ. Here we implement the experiment proposed by Hipolito and Pereira 32 , in which a quarter-wave plate is placed before the sample with its fast axis at an angle α. Using a half-wave plate we control the orientation φ of the fundamental linear polarization, while a linear polarizer placed before the detector at an angle ζ, selects suitable SHG components. All angles are measured with respect to the laboratory x-axis.
PSHG imaging was performed in 78K-300K temperature range using a fs laserscanning microscope coupled with a liquid nitrogen cryogenic system ( Fig. 2b; see also Methods). The infrared laser beam is guided into an inverted microscope, while its polarization is controlled by properly rotating half-and quarter-wave plates. A pair of galvanometric mirrors enables raster-scanning of the sample, which is placed inside a continuous flow cryostat. The SHG signal from the sample passes through a rotating linear polarizer and is collected in reflection geometry. Different rotation speeds of the optical elements allows control over the angles φ, α and ζ. The armchair direction θ can be determined with the same experimental setup 26 .
For the PL measurements, a micro-PL setup was used to collect PL in a backscattering geometry (see also Methods). Emitted light was dispersed by a single monochromator equipped with a multichannel CCD detector. Following the excitation, the emitted PL spectra were analyzed as σ + and σusing a combination of quarter-wave plate and linear polarizer placed in front of the spectrometer entrance slit. Α cryogenic system was coupled with the optical setups to perform temperaturedependent second harmonic and spin-valley polarization measurements in a range of temperatures from 78K up to 300K. 8 Application of nonlinear optics for a crystal with D 3h symmetry yields the SHG field emerging from the crystal, as 42, 43 Here !"# (!) and !"# (!) correspond to the intrinsic and induced due to VPI contributions to the second order response, respectively. This means that the SHG intensity reaching the detector depends on four angles, namely φ, θ, α, ζ, corresponding to the effects of excitation linear polarization, crystal orientation, quarter wave plate and linear polarizer, respectively (Fig. 2). In this case, the detected SHG intensity is given by: where κ denotes the absolute value of the valley-induced to intrinsic susceptibility ratio.
The ratio κ can be extracted upon fitting of the experimentally measured SHG intensity with equation (2) and reflects the degree of VPI. In our experiment the optical elements are controlled using motorized stages synchronized to rotate in phase and therefore φ = α = ζ, giving rise to a six-petal pattern for the PSHG intensity. With this experimental configuration, the normalized SHG intensity recorded at the detector reads

Results and Discussion
WS 2 samples were prepared with mechanical exfoliation and characterized using Raman mapping (see Methods). Fig. 3(a) shows an optical image of the sample where the monolayer (1L) region is indicated. In order to quantify the VPI, we fit the experimentally measured SHG intensity at each temperature with equation (4) to extract the dimensionless parameter κ. For 300 K, in particular, we assume that the valley-induced SHG is negligible (κ=0) and use the same equation to determine the armchair orientation θ. As shown in Figures 3(b) and (c), the SHG intensity, as well as the VPI mapping of WS 2 at 78K appear to be relatively uniform across the sample area yielding a value of κ=0.1. However, several points of the flake boundaries correspond to increased or decreased κ values, most probably originating from local field effects that affect the electron distribution in the valleys, hence the VPI.
In Figures 3(d)-(i) we present polar plots of the SHG emerging from the same monolayer region, as a function of temperature, ranging from 78 K to 300 K. The effect of low temperature is to preserve VPI by hindering the relaxation processes due to scarcity of phonons 44 ; this effect is readily imprinted onto the PSHG patterns. As a consequence, as the temperature rises the PSHG intensity becomes progressively lower (see also Fig. 4), indicating the suppression of VPI.
The experimental data (red spheres) -obtained from representative monolayer regions of interest-are fitted with equation (4)  The monotonic decrease of the SHG with temperature (Fig.4b) is reflected in the temperature dependence of κ-ratio (Fig.4c). For comparison, we present in Fig. 4d  In order to further validate our method, we examine another flake of exfoliated WS 2 , comprising monolayer, few-layer and bulk regions (Fig. 5a). The SHG intensity from the monolayer region at 78K is higher than the multilayer one, as it is clearly shown in Fig. 5b. In contrast, the VPI, depicted in Fig. 5c in terms of the parameter κ, is higher in the few-layer region. This is actually expected, since there is an additional In contrast to PL measurements used to measure valley polarization and coherence 10,11,12,15,16,17 , VPI imaging by means of PSHG is insensitive to the excitation wavelength due to the coherent character of the underlying process. Whereas for both one-and two-photon PL imaging, access to the electronic states has to be achieved by tuning the excitation wavelength in the vicinity of the (real) excitonic resonances, such a limitation does not apply for PSHG as the scattering of two photons occurs in virtual states and no absorption is required. This advantage renders the PSHG method universally applicable for imaging VPI for the various atomically thin crystals. For instance, within the same field of view one can image the VPI of adjacent crystalline areas consisting of different 2D TMD flakes as demonstrated in Fig. 6, showing the WS 2 flake presented above together with a WSe 2 monolayer in the same field of view (Fig. 6a). Comparing the SHG from the WS 2 and WSe 2 monolayers we observe that the latter exhibits increased intensity (Fig. 6b), reflecting the larger intrinsic nonlinear susceptibility characterizing the selenides. On the contrary, the VPI imaging (Fig. 6c) and histogram (Fig. 6d)

Optical spectroscopy measurements.
We used a micro-PL setup with a 50x objective and appropriate filters and Temperature-dependent measurements.
Α cryogenic system has been coupled with the optical setup to perform temperature-  Supplementary Information, Fig. S4c). Raman mapping revealed a difference of 11 cm -1 , which also indicates monolayer thickness for WSe 2 . Finally, Raman characterization of the WS 2 sample comprising monolayer, few-layer and bulk areas ( Supplementary Information, Fig. S5a) was performed at liquid nitrogen conditions, with an excitation energy of 543nm. The corresponding Raman spectra for the three areas, ( Supplementary Information, Figs. S5d-f), suggest that they comprise one, two to three and above six layers, respectively.

Polarization-resolved second harmonic generation imaging.
PSHG imaging was performed using a fs laser raster-scanning microscope coupled with a liquid nitrogen cryogenic system (ST500, Janis, USA) (Fig. 2b) Coordination of PMT recordings with the galvanometric mirrors for the image formation, as well as the movements of all motors, are carried out using LabView (National Instruments, USA).

Data availability statement
The data that support the findings of this study are available from the corresponding authors upon reasonable request.      28 Figure 6