Introduction↩︎

Two-dimensional semiconducting transition metal dichalcogenides (2D-TMDs) possess unique optoelectronic properties, which have propelled them in the spotlight of research in photonics and material science during more than a decade. [1]–[4] Among them are a strong direct-bandgap photoluminescence (PL) [5], [6] and a high second-order nonlinear susceptibility in the monolayer phase, [7], [8] as well as a pronounced excitonic response at room temperature. [9], [10] Furthermore, the valley pseudospin in 2D-TMDs introduces a new binary degree of freedom for electrons that may be utilized to encode information, paving the way for novel approaches in information processing and storaging. [11]–[21] The valley pseudospin arises from multiple energetically degenerate but spin-selective band extrema, the so-called valleys, in the conduction and valence bands of a crystal. These valleys form at the direct bandgaps located at the corners of the Brillouin zone, where carriers occupy one of the two subsets (K or K’ valleys) depending on their spin state. Additionally, the optical selection rules become valley-dependent, allowing spin-polarized valleys to be selectively addressed and read out using circularly polarized light. [22], [23] The degree of valley polarization (DOVP) reflects the contrast between exciton densities in different valleys and is typically measured through the circular polarization of the emitted PL. While the instantaneous DOVP can reach values close to \(\pm 1\), strong intervalley electron-hole exchange interaction leads to a fast valley depolarization, thus limiting the time available for logical processing, transporting and detecting the valley information even at cryogenic temperatures. [24]–[26]
Photonic nanostructures offer intriguing opportunities for interfacing 2D-TMDs with light at the nanoscale. [27]–[29] In particular, they have proven their potential to contribute to solutions for reducing the valley depolarization via various mechanisms. One strategy is to enhance the circular polarization contrast by using chiral nanoparticles, [30] chiral assemblies of metallic nanoparticles, [31] chiral [32]–[34] or achiral [35] metasurfaces, and other tailored designs. [29] Another approach uses engineered nanostructures to achieve valley-selective directional coupling of valley-polarized excitons or of their emitted light. [14], [18], [36]–[41]
However, most experimentally studied structures for valley routing    whether based on propagating surface plasmon polaritons [38] and guided modes, [14], [18], [36] or extended modes in metasurfaces [37], [39] and photonic crystals [40]–[42]    typically have large footprints of several square microns. The large size of the suggested structures is problematic considering the high integration densities that would be ultimately required for valleytronic devices. A solution to this problem is provided by plasmonic nanoantennas, which are well known for their ability to shape the emission patterns of localized sources. [31], [43]–[46] Plasmonic nanoantennas have also been demonstrated to facilitate free-space emission routing for rotating electric dipole sources, scattering light into different angular directions depending on the rotation direction of the nearfield source. [31], [47] Importantly, this approach directly applies to the concept of valley routing, as rotating electric dipoles accurately model the emission from valley-polarized excitons in 2D-TMDs. [14], [35], [37], [39], [48], [49]
An ideal valley-routing device should scatter PL from valley-polarized excitons into distinct directions based on the dipole’s rotation direction, while preserving circular polarization in the farfield to faithfully reflect the underlying DOVP. A natural approach to assess such functionality is through angle- and polarization-resolved measurements of the emitted light. However, interpreting these measurements can be misleading due to the interplay between spin- and valley-dependent effects. Spin-momentum coupling, for instance, can produce directional emission patterns linked to circular polarization regardless of the emitter’s internal valley state. [45], [46] As a result, such routing effects do not reliably indicate the DOVP. Additionally, the scattering process itself can alter the polarization state of the emitted light in a complex manner, [48], [50], [51] further complicating interpretation. In this study, we address these challenges by establishing a direct connection between the intrinsic DOVP of the material and the farfield emission characteristics, enabling an accurate assessment of valley-selective emission routing.
This connection is established by profound experimental and numerical analysis of valley-selective directional light emission from monolayer tungsten diselenide (1L-WSe\(_2\)) placed on top of a plasmonic nanoantenna array as sketched in [subfig:sketch95a].

The nanoantennas consist of two parallel nanobars with different sizes, which have been designed to support electric dipolar and electric quadrupolar resonances, respectively, at the operation wavelength. Chen et al. [47] numerically demonstrated that the multipolar interference between these resonances leads to emission routing when both nanobars are simultaneously excited by a rotating electric dipole source, with the direction of emission being linked to the rotation direction of the dipole.
As a first step, we show that the subwavelength-volume nanoantennas produce distinct angular scattering patterns upon illumination with circularly polarized white light of opposite handedness. We quantify this difference using a normalized contrast metric referred to as angular circular dichroism (CD). Unlike traditional CD measurements involving chiral molecules or structures, the observed effects here are extrinsic, as the nanoantennas themselves are achiral and the asymmetry arises purely from directional scattering.
Next, we exploit the valley-dependent selection rules in 1L-WSe\(_2\) to generate a pronounced DOVP using circularly polarized excitation as shown in [subfig:sketch95b]. We then analyze the directionality of the resulting PL mediated by the nanoantennas, focusing on the angular CD under circularly polarized and unpolarized detection schemes. By comparing both detection modes, we distinguish between purely electromagnetic scattering effects and those that genuinely reflect the underlying DOVP in the material.
We further support our experimental analysis with numerical calculations of the angular CD in PL. To accurately calculate the emission of valley-polarized excitons located within the periodic nanoantenna array, we employed a reciprocity-principle-based emission model. Importantly, our calculations show that valley-selective directional scattering is critically linked to the interplay of several nearfield coupling effects within the joint system. On the one hand, by systematically varying the extent of the monolayer in our simulations, we find that valley-selective emitters yield larger magnitudes of the calculated angular CD when positioned in closer proximity to the nanoantenna. On the other hand, we show that the observed directional effects result from extrinsic chirality mediated by the asymmetric nanobar dimer, as no such directional scattering is present in periodic arrays composed of single nanobars.
Our numerical analysis reveals how the nearfield polarization around resonant nanoantennas governs the angle- and polarization-dependent farfield PL of nearby valley-selective emitters, providing fundamental insights for the design and optimization of valleytronic devices.

Nanoantenna fabrication and optical characterization↩︎

Following the design proposed by Chen et al., [47] we fabricated hybrid structures consisting of 1L-WSe\(_2\) placed on an array of gold nanoantennas. The nanoantennas, each consisting of two parallel nanobars of different sizes, were fabricated on an oxidized silicon wafer (300 nm oxide layer) using standard electron-beam lithography in combination with gold evaporation and a lift-off process (see Methods for details on the fabrication process). [subfig:sample95a] shows a scanning electron micrograph of a typical fabricated gold nanoantenna array, where the nanoantennas have a fixed height \(H = \SI{40}{\nano\metre}\) and are arranged in a square lattice with a lattice constant \(\require{physics} \Lambda = \qty{1}{\micro\metre}\). [subfig:sample95b] shows a close-up view of the area indicated by the white box.

The small and large nanobar have dimensions of \(l\times w = \SI{100}{\nano\metre}\times\SI{40}{\nano\metre}\) and \(L\times W = \SI{295}{\nano\metre}\times\SI{65}{\nano\metre}\), respectively, and are separated by a \(g = \SI{50}{\nano\metre}\) gap. Additionally, we have fabricated several nanoantenna arrays with varying lengths \(l\) and \(L\), allowing us to sweep the resonance wavelengths of the short and long nanobars, respectively. To minimize potential changes in the carrier relaxation dynamics within the subsequently transferred 1L-WSe\(_2\), which may result from charge-transfer, dipole-dipole interaction, or plasmonic quenching, [52], [53] we coated the fabricated nanoantenna arrays with a \(\SI{15}{\nano\metre}\) thick layer of silicon dioxide using plasma-enhanced chemical vapor deposition.
We then investigated the optical nearfield response of the coated nanoantennas using cathodoluminescence (CL) imaging. By exciting the nanostructure with a focused electron-beam, thus generating emitted photons via cascaded processes, this technique provides information on the optical nearfields with super-optical resolution down to few tens of nanometres. [54], [55] In [subfig:sample95c], we show the corresponding CL scan of a single nanoantenna from an array similar to the one depicted in [subfig:sample95a], with the CL signal integrated over a wavelength range from 700 nm–720 nm, matching the expected trion emission band in 1L-WSe\(_2\) at cryogenic temperatures. The CL image reveals a distinct nearfield mode profile for each nanobar. For the small nanobar, we observe a nearfield hotspot at each end of the nanobar. This resembles an electric dipole mode profile, with the dipole moment \(p_x\) oriented along the nanobar and centered within it. For the large nanobar, we observe two additional nearfield hotspots near its center, resembling the mode profile of a linear quadrupolar mode, with the quadrupole moment \(q_{xx}\) oriented along the nanobar and also centered within it.
Exciting both multipolar modes within the nanobars with a rotating dipole leads to directional scattering, with light being predominantly emitted into different halfspaces (\(x<0\) and \(x>0\)) depending on the spin-orientation of the rotating dipole. [47] In our system, the rotating dipole source is realized by valley-selective excitonic emitters in 1L-WSe\(_2\) created upon circularly polarized excitation. Note that the nanobars each align along the \(x\)-direction (see [subfig:sample95a]).
Next, we investigated the circular-polarization-dependent directional scattering of the fabricated nanoantenna array by angle-resolved white light (WL) spectroscopy. For this, we prepared circularly polarized light using a stabilized tungsten-halogen source combined with a linear polarizer and a superachromatic quarter wave plate. We illuminated the sample and collected the reflected light with a 100x/0.88NA objective. Subsequently, by imaging the back-focal plane of the objective onto the slit of an imaging spectrometer, we measured the angular spectra \(I^{\sigma^\pm \text{inc.}}_{\text{WL}}(\sin\theta\cos\varphi,\lambda)\) of the light back-scattered from the sample for \(\sigma^{\pm}\) polarized illumination, respectively. Here, \((\theta,\varphi)\in[0,\pi/2]\times[0,\pi]\) are the spherical coordinates describing points on a unit sphere and \(\tilde{k}_x =\sin\theta\cos\varphi\) is their corresponding projection along the \(x\)-direction. A detailed discussion on the setup geometry and the influence of the optical components on the polarization state of the excitation and detection is provided in Sec. S.1 of the Supporting Information.
In [subfig:sample95d], we show the respective angular CD defined as the normalized contrast \(\text{CD}_{\text{WL}}=(I^{\sigma^+ \text{inc.}}_{\text{WL}}-I^{\sigma^- \text{inc.}}_{\text{WL}})/(I^{\sigma^+ \text{inc.}}_{\text{WL}}+I^{\sigma^- \text{inc.}}_{\text{WL}})\). Between wavelengths of 700 and 720 nm (see thick dotted lines), we observe a pronounced antisymmetric feature with respect to the two halfspaces \(\tilde{k}_x<0\) and \(\tilde{k}_x>0\), clearly indicating circular-polarization-dependent directional scattering mediated by the nanoantenna array. In particular, we demonstrate a maximum CD of 6.5% at a wavelength of 705 nm, matching the cryogenic peak emission wavelength of the PL spectrum measured for 1L-WSe\(_2\) on the bare substrate at 3.8 K, as shown in [subfig:sample95e]. We also observe several additional antisymmetric modes between wavelengths of 700 nm and nearly 800 nm. Since the mode dispersion closely follows that of the lattice modes (highlighted by dashed lines), we attribute these features to the interplay between directional scattering by individual nanoantennas and the diffractive grating orders arising from their periodic arrangement. For details on the dispersion of the grating orders, see Sec. S.2 of the Supporting Information.
It is worth to note that both the single nanoantennas and their periodic arrangement within the array have an achiral geometry such that the total CD of the array is expected to be zero. We have verified this by plotting the measured CD as retrieved from the angle-integrated back-scattered intensities \(I^{\sigma^\pm \text{inc.}}_{\text{WL}}(\lambda) = \int_{\text{NA}} I^{\sigma^\pm \text{inc.}}_{\text{WL}}(\tilde{k}_x,\lambda)\operatorname{d}\!\tilde{k}_x\). [subfig:sample95f] shows the resulting CD, which is close to zero across the entire spectral range of our measurements (note the scale of 0.05). The small oscillating deviations are likely caused by the dispersion of the polarization optics used in our experiments. Note further that in this experiment no polarization control was used in detection and the measured non-zero CD is purely a result of changing the illumination polarization which is a necessary condition to enable valley routing.
Finally, we synthesized 1L-WSe\(_2\) on an oxidized silicon wafer by a chemical vapour deposition process using an Knudsen-type effusion cell as described previously by George et al. [56]. This scalable process allows for a dense coverage of the growth substrate with single crystalline monolayers, which we subsequently transferred from the growth substrate onto the host substrate with the fabricated nanoantenna array using a poly(methyl methacrylate) assisted wet-transfer scheme [57].

Photoluminescence study of the hybrid system↩︎

[subfig:PL95RT95a] shows a true-color optical microscope image of a fabricated nanoantenna array after we transferred a sample of 1L-WSe\(_2\) to cover parts of the nanoantenna array and of the bare substrate. Note that apart from regions with 1L-WSe\(_2\), the crystal includes smaller regions of 2L-WSe₂, as well as triangular holes (visible as small regions with different contrast). Initially, we investigated the PL emission from the hybrid system at room temperature without employing any polarization control. [subfig:PL95RT95b] shows a measured confocal PL scan of the same sample area as shown on the left. Further details on the experimental parameters are provided in Methods.

We observe a bright and uniform PL signal from the 1L-WSe\(_2\) on the bare substrate region associated to the A-excitonic PL. The region of 2L-WSe\(_2\) appears darker, due to the drastically reduced PL quantum yield of the indirect bandgap semiconductor. The measured PL spectra of the 1L- and 2L-WSe\(_2\) are provided in Sec. S.3 of the Supporting Information. Note that the PL signal measured from the regions of the triangular holes is indistinguishable from that of the bare substrate. Further, a regular square pattern of small regions with reduced PL intensity coincides with the locations of individual nanoantennas. The reduced PL intensity measured in the farfield is attributed to the interplay of several effects, including the scattering by individual nanoantennas, diffractive grating modes of the array, and Fabry-Pérot modes within the multi-layer substrate. Hence, the interference of multiple contributing modes may lead to a modulation of the farfield PL intensity depending on the geometrical parameters of the sample and the wavelength of emission. For example, we observe enhancement of the farfield PL intensity from 1L-WSe\(_2\) placed on different nanoantenna arrays with spectrally shifted resonances (see Sec. S.3 of the Supporting Information).
We further investigated the influence of the resonant nanoantennas on the emission decay dynamics of 1L-WSe\(_2\) using time-resolved PL measurements (see Sec. S.3 of the Supporting Information). We observe a two-component decay with PL lifetimes of \((2.62\pm 0.05)\) ns and \((0.37\pm 0.01)\) ns on the bare substrate, that reduce to \((1.90\pm 0.06)\) ns and \((0.32\pm 0.01)\) ns on the nanoantenna array, showing a certain nearfield coupling between the nanoantennas and excitons in 1L-WSe\(_2\). Importantly, the nearfield coupling at cryogenic temperatures is expected to be further enhanced, as the exciton energy shifts to higher energies, coinciding with the operational bandwidth of the resonant nanoantenna.
Next, we investigated whether the fabricated nanoantenna arrays can directionally route the PL emission from valley-selective excitons in the 1L-WSe\(_2\). The induced valley contrast is typically characterized in emission by the degree of circular polarization (DOCP) of PL, defined as \(\text{DOCP}=(I_{\text{PL}}^{\sigma^+ \text{det.}}-I_{\text{PL}}^{\sigma^- \text{det.}})/(I_{\text{PL}}^{\sigma^+ \text{det.}}+I_{\text{PL}}^{\sigma^- \text{det.}})\). At room temperature, valley-selective excitation of 1L-WSe\(_2\) typically results in a negligible DOCP due to phonon-assisted ultrafast intervalley scattering. Cooling the sample to cryogenic temperatures significantly suppresses the intervalley scattering rate, thereby preserving the circular polarization contrast of the excitation field at the emission level and yielding a pronounced DOCP in PL.
Importantly, upon radiative decay, these excitons act as rotating dipolar nearfield sources that drive the nanoantenna at the exciton emission wavelength, with the dipole’s rotation sense being determined by the circular polarization of the excitation. [48] By design, the nanobar dimer exhibits valley-selective directional farfield interference, [47] linking the emission direction to the exciton’s valley-index and thereby mediating valley-momentum coupling.
We performed circular-polarization resolved cryogenic measurements (\(\require{physics} T=\qty{3.8}{\kelvin}\)) using a commercially available closed-loop liquid helium cryostat (s50, Montana Instruments) equipped with a custom-built back-focal plane imaging setup, as depicted in [subfig:PL95cryo95a].

For excitation, we focused a 633 nm continuous-wave helium-neon laser with an average power of \(\SI{100}{\micro\watt}\) on the sample using a 100\(\times\)/0.88NA objective and collected the PL signal in reflection geometry with the same objective. For polarization control, we used a combination of a linear polarizer (LP) and a quarter-wave plate (QWP) to prepare a linear polarized beam after reflection from the dichroic beam splitter (DBS). Here, the QWP is utilized to compensate the phase shift introduced by the DBS upon reflection. After the DBS, we employed a super-achromatic QWP to prepare a \(\sigma^{\pm}\) polarized excitation beam before entering the objective. In detection, the same super-achromatic QWP was used to project the circular-polarized signal into a linear-polarized basis, which is then analyzed by another LP after reflection from a mirror. For a detailed discussion on the polarizing properties of the optical components, see Sec. S.1 of the Supporting Information.
In [subfig:PL95cryo95b], we show the experimental PL spectra of 1L-WSe\(_2\) for \(\sigma^+\) polarized excitation and detection measured on the bare substrate (blue curve) and on top of the nanoantenna array (orange curve). In both cases, we observe a multi-peak spectrum that is typical for tungsten-based 1L-TMDs. [58]–[63] In ascending wavelength order, the first peak corresponds to the neutral bright exciton (\(X^0\)), whereas the second peak results from several exciton complexes including negatively charged trions (\(X^-\)), as well as grey/dark excitons (\(D^0\)) and trions (\(D^-\)). Thus, we label this peak as \(X^-/D\). Two additional peaks at longer wavelengths correspond to localized defect states, while the third peak also spectrally overlaps with the expected energy of the bi-exciton (\(XX\)). Thus, we label the third and fourth peak as \(L_1/XX\) and \(L_2\). A detailed discussion on the excitonic contributions based on multi-Voigt-line fitting is provided in Sec. S.4 of the Supporting Information.
On the nanoantenna array, we observe a higher PL intensity across the entire emission spectrum of 1L-WSe\(_2\), likely resulting from excitation and emission enhancement mediated by the nanoantennas (see also Sec. S.3 of the Supporting Information for time-resolved measurements at room temperature). Additionally, the PL spectrum of 1L-WSe\(_2\) on the nanoantennas exhibits a redshift by several meV. While such spectral shifts naturally arise from variations across the monolayer sample (compare Fig. S4b), we find a larger shift for the \(X^-/D\) peak, hinting at a potential brightening of dark excitons mediated by the nearfield interaction with the nanoantennas (see Sec. S.4 of the Supporting Information).
Next, we measured the DOCP in PL from 1L-WSe\(_2\) (see [subfig:PL95cryo95c]) on the bare substrate (blue curve) and on the nanoantenna array (orange curve), under \(\sigma^+\) (solid curves) and \(\sigma^-\) (dashed curves) polarized excitation. In each case, we find the highest DOCP associated with the \(X^-/D\) peak, reaching values of 0.65 for the bare substrate and 0.43 for the nanoantenna array. As this reduction in DOCP for 1L-WSe\(_2\) on the nanoatenna array occurs across the entire emission spectrum, it likely results from scattering by the plasmonic nanoantennas, [48] as further analyzed by numerical emission modeling in Sec. S.9 of the Supporting Information. Note that the slightly asymmetric DOCP in PL on the nanoantenna array is not expected for an achiral structure and may have resulted from small differences in the measurement position between the measurements with different excitation polarization.
To quantify the valley-dependence of the emission patterns, we performed angle-resolved spectroscopy of the PL from 1L-WSe\(_2\) placed on top of the gold nanoantenna array. For these measurements, we used the same detection scheme as in our angle-resolved WL spectroscopy, with the excitation beam widened to cover the same area as for the focused white light illumination (see Methods and Sec. S.1 of the Supporting Information). [subfig:PL95cryo95d] shows the corresponding angular CD in PL, defined as \(\text{CD}_{\text{PL}}=(I^{\sigma^+ \text{inc.}}_{\text{PL}}-I^{\sigma^- \text{inc.}}_{\text{PL}})/(I^{\sigma^+ \text{inc.}}_{\text{PL}}+I^{\sigma^- \text{inc.}}_{\text{PL}})\), retrieved from the angular PL spectra \(I^{\sigma^\pm \text{inc.}}_{\text{PL}}(\tilde{k}_x,\lambda)\). For comparison, [subfig:PL95cryo95e] and [subfig:PL95cryo95f] show, respectively, the integrated PL intensity spectrum \(I_{\text{PL}}(\lambda) = I^{\sigma^+ \text{inc.}}_{\text{PL}}(\lambda) + I^{\sigma^- \text{inc.}}_{\text{PL}}(\lambda)\), where \(I^{\sigma^\pm \text{inc.}}_{\text{PL}}(\lambda) = \int_{\text{NA}} I^{\sigma^\pm \text{inc.}}_{\text{PL}}(\tilde{k}_x,\lambda)\operatorname{d}\!\tilde{k}_x\), and the CD\(_{\text{PL}}\) calculated from \(I^{\sigma^\pm \text{inc.}}_{\text{PL}}(\lambda)\). We find that, in contrast to the case of WL scattering, there is no pronounced directional pattern for the CD\(_{\text{PL}}\). Instead, we observe only a minor antisymmetric feature near the DOCP maximum (see dotted line and circles). For clarity, [subfig:PL95cryo95g] shows the cross-section of the measured CD\(_{\text{PL}}\) at a wavelength of 710 nm (orange dots), as well as the cross-section of the CD\(_{\text{WL}}\) measured on the nanoantenna array without the monolayer, using white light spectroscopy (green triangles). Compared to the pronounced antisymmetric features observed for CD\(_{\text{WL}}\), the distribution for CD\(_{\text{PL}}\) is less systematic. Nevertheless, within the angular range discussed previously, as highlighted by the shaded regions, we observe a significant antisymmetric feature in the CD\(_{\text{PL}}\), with a magnitude of \(2\%\) and its sign opposite to that of the white light result.
Note that although the integrated \(\mathrm{CD_{PL}}\) must vanish for an achiral nanoantenna array, we observe subtle spectral modulations: the \(X^-/X^D\) peak near 709 nm diminishes, while the \(X^0\) peak and a broad defect band increase in intensity over successive measurements. These changes are likely caused by slow, photo-induced charge trapping at defect sites and appear to occur on a timescale comparable to the interval between measurements. Importantly, they do not affect the directional \(\mathrm{CD_{PL}}\) at any fixed wavelength.

Valley-selective directional scattering↩︎

Next, to obtain deeper insight into the observed valley-selective directional asymmetry, we performed polarization-resolved back-focal plane imaging (see Methods for details). [subfig:cryoPL-BFP95a] shows the measured angular PL intensity, obtained with a \((710\pm 10)\) nm bandpass filter, and for the same measurement location on the nanoantenna array as discussed previously. Specifically, we focused on changes in the angular PL intensity upon switching the circular polarization of the excitation (top and bottom row), and analyzed these emission patterns using a circular-polarized detection basis (left and middle column) and for unpolarized detection (right column).

For simplicity, we introduce a Jones notation to distinguish between these cases as follows: \(\sigma^{+}_{\text{det.}}\,|\,\sigma^{-}_{\text{exc.}}\equiv{+-}\), with analogous definitions for all other polarization combinations. These emission patterns show distinct arc patterns resulting from diffractive grating orders that arise due to the periodic nanoantenna arrangement, as discussed in Sec. S.2 of the Supporting Information. For comparison, we overlaid the calculated diffraction patterns in the bottom left panel in [subfig:cryoPL-BFP95a].
In these emission patterns, not all diffractive modes seam to appear with equal intensity, with favored emission directions depending on the circular polarization of detection. For example, for the cases \(--\) and \(-+\) (i.e. \(\sigma^-\) polarized detection), the diffractive modes are pronounced in diagonal \(k\)-space directions (\(k_x\cdot k_y>0\)), while for \(++\) and \(+-\) (i.e. \(\sigma^+\) polarized detection) anti-diagonal directions (\(k_x\cdot k_y<0\)) are favored. This polarization dependence is a signature of spin-momentum coupling mediated by the nanoantenna array. Additionally, these patterns are linked by the system’s symmetry in \(k\)-space, as depicted in [subfig:cryoPL-BFP95b]. The emission patterns for opposite detection polarizations appear as mirror images with respect to the \(k_y\)-axis due to the respective mirror symmetry of the nanoantenna array and the handedness reversal of circularly polarized light under mirror operations. Note that the symmetry of the system is reduced when a valley-specific excitation is introduced. For a given detection polarization, changing the excitation polarization alters the intensity but not the shape of the emission pattern. This intensity difference stems from the finite DOCP in emission from valley-polarized excitons.
These observations demonstrate that the angular emission distribution in our system is sensitive to the valley polarization. For unpolarized detection (right column in [subfig:cryoPL-BFP95a]), this intensity contrast results in visibly different emission patterns. The respective angular CD distribution in [subfig:cryoPL-BFP95c] provides a quantitative measure for these differences, showing a significant contrast across several emission directions. We find angular CD magnitudes up to \(15\%\) for specific emission directions, and antisymmetric features with peak values up to \(\pm6\%\) along the \(k_x\) axis. For a comparison with the results in [subfig:PL95cryo95g], see Sec. S.5 of the Supporting Information. Importantly, this finite angular CD is a signature of the valley-selectivity in emission with respect to these angular directions. By utilizing valley-selective excitation and unpolarized detection, we demonstrate directional effects arising purely from valley-momentum coupling, distinguishing our results from previous demonstrations of spin-momentum locked emission using emitters without internal (pseudo-)spin.
We further performed numerical simulations of the emission from the studied system using a reciprocity-principle-based model. [64], [65] This approach bases on the equivalence of placing a single dipolar emitter in the nearfield region of a suitable nanophotonic system and studying its emission behavior in the farfield, with the situation of illuminating the system from the farfield and studying the system’s nearfield response. For a detailed description of the numerical model, see Methods. This approach allows us to calculate the radiated power and farfield polarization from incoherent dipolar emitters placed in periodic systems. To model the emission response from 1L-WSe\(_2\) on the nanoantenna array, we have calculated the in-plane nearfield distribution on top of the embedding layer upon plane wave illumination. Within this framework, we identify the contribution from a valley-selective emitter in K/K’ with the \(\sigma^+/\sigma^-\) polarized in-plane nearfield intensity, and associate the polarization of the emitted light with that of the incident plane wave driving the system.
By varying the angle of incidence, we obtain the calculated angular emission intensity distributions for unpolarized detection, as depicted in [subfig:cryoPL-BFP95d] for \(\sigma^+\) polarized (top) and \(\sigma^-\) polarized (bottom) nearfield intensities, and the respective CD shown in [subfig:cryoPL-BFP95e]. Our calculations show diffractive mode patterns as for our experimental results, giving rise to similar symmetric features as discussed previously. The calculated emission patterns for circular detection polarization are available in Sec. S.6 of the Supporting Information. The calculated angular CD distribution further confirms the experimentally observed order of magnitude in contrast values for the nanoantenna array homogeneously covered by 1L-WSe\(_2\). For small emission angles, we find similar antisymmetric features as previously observed in experiments with an angular CD magnitude of \(1\%\) (see also Sec. S.5 of the Supporting Information). At high emission angles, our model seems to overestimate the emission intensities, and hence contrast values, as such are clearly not observed in experiments. We note that the predictive accuracy of this model is limited for these grazing emission angles, as we did not radiometrically normalize the power radiated into different angles.

Emitter distribution↩︎

We recall that both the experimentally and numerically observed angular CD remain systematically below values predicted for valley-emitters placed in the vicinity of a single nanoantenna. [47] This deviation likely stems from contributions of emitters located farther away from the nanoantenna, experiencing weaker nearfield interactions. A natural way of testing this hypothesis is to adjust the monolayer area in the numerical simulations, effectively reducing the contributions of these remote emitters. Here, we numerically explore the potential of varying the monolayer extent within the unitcell, as a free design parameter, leaving other sample parameters intact.
Exemplary, we discuss several cases of differently sized monolayers placed on top of the nanoantenna within each unitcell. [subfig:numerics-nearfieldarea95a] and [subfig:numerics-nearfieldarea95b] show the monolayer areas and, respectively, the calculated angular CD distributions for three different cases (A-C).

For case A, i.e. the monolayer filling the gap region between the nanobars, the angular CD takes magnitudes up to \(53\%\). As the monolayer area is further increased (cases B and C), the range of the angular CD drops significantly due to the emission of mostly uncoupled emitters, exhibiting no valley-selective emission patterns. Additionally, this results in a qualitative change of the angular CD pattern, as clearly visible when comparing cases A and C. The pattern in A mostly reflects the nanoantenna’s antisymmetry with respect to the \(k_y\) axis (see also [subfig:cryoPL-BFP95b]), due to the efficient nearfield interaction of emitters located within the nanoantenna’s gap region. In C, most of the emitters are located further apart from the nanoantenna, and we find a two-fold antisymmetric distribution that originates from the square array arrangement.
Based on these considerations, we propose the following metric to quantitatively characterize the strength of the valley-routing effect. We consider two farfield detectors measuring the intensities, \(I_\mathrm{left}\) and \(I_\mathrm{right}\), emitted into two fixed farfield directions within the left and right halfspaces, respectively. Initially, we chose the two farfield directions indicated by the dashed circles in [subfig:numerics-nearfieldarea95b]. We then define a measurable farfield quantity as the intensity contrast \((I_\mathrm{left}-I_\mathrm{right})/(I_\mathrm{left}+I_\mathrm{right})\) between both directions. In the left panel in [subfig:numerics-nearfieldarea95c], we analyze this directional intensity contrast obtained for different filling factors of the monolayer flake covering the nanoantenna array’s unitcell (red and blue circles). For a fixed valley state or nearfield polarization, this contrast yields a finite quantity, with larger contrast values being obtained for smaller filling factors. Importantly, the sign of the contrast follows the valley-state, allowing the efficient readout of the valley-information from the farfield even in the presence of a resonant nanoscatterer, often obscuring this link due to its scattering response. [48]
For comparison, we also calculated the intensity contrast obtained for arrays of individual nanobars, where we have adopted the same geometric parameters of the small nanobar (red and blue squares) and the large nanobar (magenta and cyan triangles). For small filling factors, we observe a clear difference in their directional scattering behavior compared to the nanobar dimer antenna. Hence, we conclude that the large directional contrast values for filling factors below \(\sim 3\%\) result from the resonant interaction between both nanobars. For larger filling factors, however, the difference between dimer and monomer arrays becomes negligible, as the directional scattering is mostly influenced by diffractive modes.
Alternatively, in the right panel in [subfig:numerics-nearfieldarea95c] we calculated the directional contrast for two different directions as indicated by the dashed squares in [subfig:numerics-nearfieldarea95b]. Different to the previous case, we now find a clear difference in the directional contrast exhibited by the nanobar dimer array compared to the monomer arrays for all filling factors. Thus, we conclude that the directional contrast for this pair of farfield directions purely results from the valley-dependent directional scattering of the nanoantenna, with no additional contributions from diffractive modes.
It is important to note that the two previous cases reveal two different symmetries within the system, namely that of the lattice and that of the dimer nanoantenna. Importantly, only for the dimer nanoantenna this directional intensity contrast between left and right halfspaces is a symmetry-protected property, owing to the extrinsically induced chirality. [66]–[69] For a detailed symmetry analysis, see Sec. S.7 of the Supporting Information. Consequently, when integrating over all farfield directions within each of the halfspaces separately, the directional intensity contrast mediated by the nanoantenna dimer is expected to be robust, while that for the arrays with single nanobars is expected to vanish. We have numerically verified this behavior as discussed in Sec. S.8 of the Supporting Information. This symmetry-protected valley-dependent behavior therefore provides a robust platform for valleytronic processing owing to certain tolerances with respect to deviations in farfield directions and emitter distribution.

Conclusion↩︎

We have experimentally investigated valley-momentum coupled emission from hybrid nanophotonic structures consisting of CVD-grown 1L-WSe\(_2\) placed on top of circular-polarization selective directional nanoantennas. Our hybrid nanostructures exhibit an antisymmetric angular CD with a magnitude of \(6\%\). Crucially, no polarization analyzer is employed, clearly indicating the pivotal role of angle-selective coupling of the nanoantennas to valley-polarized exciton populations. Further, we have discussed the importance of experimental conditions able to distinguish valley-momentum locking from farfield-polarization sensitive directionality in the context of harnessing nanophotonics for valleytronics. These findings are supported by a novel numerical approach, allowing to model incoherent valley-selective emitters within periodic nanostructures. Our numerical results agree well with the experimental observation that the nanoantenna array exhibits a systematically lower angular CD than predicted for a single nanoantenna. By varying the coverage of the unitcell area by the monolayer, we are able to show that this discrepancy results from a reduced coupling of emitters distant to the nanoantenna. We demonstrate numerically that angular CD values above \(50\%\) can be retained when limiting the monolayer area to the nanoantenna’s gap region. Importantly, such capabilities may be realized in experiment using already existing techniques for nanopatterning of monolayer TMDs. [70], [71] Ultimately, we show that this directional intensity contrast can be harnessed robustly, owing to the extrinsically chiral properties of the dimer nanoantenna, providing a novel platform for valleytronic processing based on resonant nanoantennas.
Generally, the performance of nanoscopic valleytronic devices based on the non-vanishing DOVP will be challenged by two aspects: (1) The efficient optical addressing and reading out the valley degree of freedom in the presence of resonant nanostructures due to complex nearfield effects both at the excitation and emission level. Importantly, our scheme for valley-routing does not depend on the polarization state of emission in the farfield but relies on the valley-selective nearfield excitation alone. This provides a practical solution to overcome the limitation of an otherwise partly or fully obscured valley-information as measured in the farfield by encoding the valley-information in the PL emission direction. (2) Operating those devices at room temperature due to ultrafast spin-relaxation processes. Promisingly, several works demonstrate a non-zero DOVP at room temperature involving charge capturing [72], alloying [73], multilayer [14] and heterostructure [74] materials paving the way for room temperature plasmonic and dielectric nanoantenna based valleytronic devices.

Methods↩︎

Fabrication and nearfield characterization of gold nanoantennas↩︎

Gold nanoantennas were fabricated on a thermally oxidized silicon wafer (300 nm silicon dioxide) using electron-beam lithography followed by a lift-off process. Initially, the electron-beam resist (ZEP520A) was spin-coated on the pre-cleaned silicon wafer at 4000 rpm for 40 s and baked at 180   for 120 s. Next, the nanoantenna arrays were defined by electron-beam lithography using a base dose of 100  C cm⁻² and dose factor variations between 1.2 and 1.4 for the large and small nanobars, respectively. After electron beam exposure, the resist was developed in N-amyl acetate for 90 s and rinsed by isopropanol to remove the exposed areas of the resist. Subsequently, 40 nm of gold were deposited using electron-beam evaporation (>99.5 % target purity, 9 × 10⁻⁶ Torr chamber pressure). For lift-off, the coated substrate was left for several hours in a N-N dimethyl acetamide bath followed by 10 s ultrasonication and rinsing with acetone and isopropanol.

Cathodoluminescence imaging was performed using a FEI Verios scanning electron microscope equipped with a Gatan MonoCL4 Elite detection system.

Optical experiments↩︎

Angle-resolved white light spectroscopy was performed using a stabilized tungsten-halogen light source. The white light was prepared under circular polarization using a linear polarizer and a super-achromatic quarter wave plate and focused onto the sample by a 100x/0.88NA objective resulting in an illumination spot diameter of about \(\SI{5.5}{\micro\metre}\). In reflection geometry, the back-scattered light was collected by the same objective, passed through a 30(R):70(T) plate beam splitter, and analyzed in wavelength-momentum space by imaging the back-focal plane of the objective onto the entrance slit of an imaging spectrometer (Andor Shamrock 750). This resulted in a spectral and momentum resolution of \(\Delta\lambda=\SI{0.23}{\nano\metre}\) and \(\Delta k_x/k_0 = 0.016\). The momentum-axis was defined by orienting the sample with respect to the orientation of the spectrometer slit. A sketch of the optical setup and a detailed discussion of the polarization control are provided in Sec. S.1 of the Supporting Information.

Room temperature photoluminescence measurements were conducted using a commercial confocal fluorescence lifetime imaging microscope (MicroTime 200, PicoQuant). A pulsed laser source (\(\SI{100}{\pico\second}\) pulse length, \(\SI{40}{\mega\hertz}\) repetition rate, \(\SI{30}{\micro\watt}\) average power) at \(\SI{532}{\nano\metre}\) wavelength was used for excitation and focused on the sample by a 100x/0.95NA objective resulting in an estimated spot diameter of \(2r = 2\lambda/(\mathrm{NA}\cdot\pi)\approx\SI{0.36}{\micro\metre}\). The signal was collected by the same objective in reflection geometry. The reflected laser light was blocked using a 550 nm longpass filter.

Polarization-resolved cryogenic photoluminescence measurements were conducted at a temperature of 3.8 K using a closed-loop helium cryostat (s50, Montana Instruments). A 633 nm wavelength continuous-wave helium-neon laser (\(\SI{100}{\micro\watt}\) average power) was used for excitation. The laser light was prepared under circular polarization using a linear polarizer and a quarter wave plate and focused on the sample by a 100x/0.88NA objective resulting in an estimated spot diameter of \(2r = 2\lambda/(\mathrm{NA}\cdot\pi)\approx\SI{0.46}{\micro\metre}\). The signal was collected in reflection geometry by the same objective, passed through a dichroic beam splitter and analyzed in a helical polarization basis by a super-achromatic quarter wave plate and a linear polarizer. For angle-resolved PL imaging, the back-focal plane of the objective was directly imaged onto an electron-multiplying charge-coupled device (EMCCD, iXon897 Ultra, Andor) or onto the entrance slit of an imaging spectrometer as described above. On the EMCCD, this resulted in a momentum resolution of \(\Delta k_x/k_0 = 0.011\). A sketch of the optical setup and a detailed discussion of the influence of the optical components on the detection polarization are provided in Sec. S.1 of the Supporting Information.

Numerical simulations↩︎

For modeling incoherent emission from periodic nanostructures, we employ an Averaged Reciprocal Modal Analysis (ARMA), exploiting Lorentz reciprocity. This allows us to infer the emission pattern of a two-dimensional semiconductor from its plane wave excitation response. [64], [65]
We model this by scanning a plane wave over a regular grid of incident wavevectors \(\mathbf{k}=(k_x,k_y)\), bound by the numerical aperture \(|k|\leq NA = 0.9\), and computing the respective nearfield intensity distribution of the nanoantenna array at the emission wavelength and within the monolayer plane. By reciprocity, we approximate the polarization-dependent angular PL intensities measured in the experiment by the obtained spatially-averaged nearfield intensity.
Applying this concept to valley-selective emission, we additionally identify the circularly polarized components of the nearfield to emission contributions obtained from respective valley-excitons. Ultimately, we obtain a polarization-dependent spatially-averaged nearfield intensity in \(k\)-space, allowing qualitative analysis of emission directionality features connected to polarization and nearfield asymmetries.

Fourier modal method↩︎

For the full-wave calculations, we employed a custom three-dimensional Fourier modal method (also known as rigorous coupled‐wave analysis) with two-dimensional in-plane periodicity. [75]–[77] The simulation domain is a \(\require{upgreek} 1\upmu\mathrm{m}\times 1\upmu\mathrm{m}\) unitcell containing the nanobar dimer and the layered dielectric stack (compare 1 and 2). The angular space of the incident plane waves is defined in \(k\)-space via \[\begin{align} \begin{matrix} k_x &=& \sin(\theta)\cos(\phi),\\ k_y &=& \sin(\theta)\sin(\phi) \end{matrix} \quad \Rightarrow \quad \begin{matrix} \theta &=& \arcsin(\sqrt{k_x^2 + k_y^2}),\\ \phi &=& \arctan2(k_y,k_x), \end{matrix} \end{align}\] where \(\sqrt{k_x^2 + k_y^2}\le 0.9\). The numerical solutions obtained by the Fourier modal approach yield the scattering matrix of the layered system, whose entries are the complex amplitudes of all reflected and transmitted diffraction orders at the monolayer plane. The evanescent mode coefficients then comprise the nearfield components originating from the nanoantenna that interact with the adjacent layers. [69], [78]
The electric field components in a linear polarization basis then follow from the Rayleigh-expansion, [75] as \[E_n(x,y,z) = \sum_{l,m=-N}^{N} \mathcal{E}_{n,lm} \, \exp\left(i \left[\gamma_{lm} z + k_{x,l} x + k_{y,m} y\right]\right), \label{eq:rayleighseries}\tag{1}\] where \(\mathcal{E}_{n,lm}\) are the Rayleigh coefficients of the \(n-\)th field component, with \(n\in \{x,y,z\}\), obtained from the scattering matrix coefficients \((l,m)\). The propagation constant in \(z\)-direction is given by \(\gamma_{lm} = \sqrt{\left(\tilde{n}k_0\right)^2 - k_{x,l}^2 - k_{y,m}^2}\), with the vacuum wavenumber \(k_0\) and the refractive index of the surrounding medium \(\tilde{n}\).

Helical basis projection↩︎

To impose circular polarization, we first project the incident field from its \(s\)-\(p\) basis [75] in spherical coordinates onto the in-plane Cartesian coordinate system of the monolayer. For a given emission direction \(\mathbf{k} = (\sin\theta\cos\phi,\, \sin\theta\sin\phi,\, \cos\theta)\), we find: \[\begin{align} \mathbf{p} &= (\cos\theta\cos\phi,\; \cos\theta\sin\phi,\; -\sin\theta), \\ \mathbf{s} &= (-\sin\phi,\; \cos\phi,\; 0), \end{align}\] with the respective transformation matrix: \[U = \begin{bmatrix} \mathbf{s}\\ \mathbf{p} \end{bmatrix}\cdot [\mathbf{x} \, \mathbf{y}] = \begin{pmatrix} \cos\theta\cos\phi & \cos\theta\sin\phi \\ -\sin\phi & \cos\phi \end{pmatrix}.\] For oblique emission angles, any circularly polarized farfield component yields an elliptical projection onto the monolayer plane, with the ellipticity and orientation of the polarization ellipse being determined by the emission angle. Hence, we find the in-plane nearfield distribution \((E_{\pm,x}^\prime, E_{\pm,y}^\prime)^T\) upon a \(\sigma^\pm\) polarized plane wave, as: \[\begin{pmatrix} E_{\pm,x}^\prime\\ E_{\pm,y}^\prime \end{pmatrix} = \frac{1}{\sqrt{2}}\, U \begin{pmatrix} E_x \\ \pm i\cdot E_y \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} \cos\theta\, e^{i\phi}\cdot E_x \\ (\mp\sin\phi \pm i\,\cos\phi)\cdot E_y \end{pmatrix}.\] Finally, we transform the obtained in-plane nearfield distribution from a linear polarization basis to a helical basis, [78] as: \[\begin{pmatrix} E_{+,\pm} \\ E_{-,\pm} \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ i & -1 \end{pmatrix} \begin{pmatrix} E_{\pm,x}^\prime \\ E_{\pm,y}^\prime \end{pmatrix},\] with the Jones notation \(\sigma^+_{\mathrm{farfield}}|\sigma^-_{\mathrm{nearfield}}\equiv+-\). In this helical bases, we are able to compare the emerging patterns in \(k\)-space from the ARMA approach to the measured valley-selective angular PL intensities.

Field averaging and box-size dependence↩︎

In order to obtain the contribution of the nearfield intensities, reciprocally yielding emission contributions in different angular directions, we calculate the averaged nearfield intensity related to that plane wave direction \[M^{\pm}(k_x,k_y;\{x,y \in \text{box}\}) = \frac{1}{N_{\text{box}}}\sum_{\{x,y \in \text{box}\}} |E_{\sigma^{\pm}}(k_x,k_y;x,y)|^2.\] We mimic the experimental conditions by choosing an integration box covering the entire unitcell area.
Additionally, by varying the averaging box size, we are able to investigate both spatially- and polarization-dependent directional coupling effects between valley-selective emitters and the nanobar dimer antennas.

References↩︎