Magnetic-Field-Induced Darkening of Bright Excitons in a MoWSe$_2$ Monolayer Alloy

Introduction

The optical activation (brightening) of dark excitons, electron-hole states whose direct radiative recombination is suppressed by spin-selection rules, is a well-established phenomenon in semiconductor nanostructures. Dark excitonic states and their coupling to optically active counterparts have been extensively investigated in conventional quantum wells [1] and semiconductor quantum dots, where dark-exciton brightening can be induced by magnetic fields, structural symmetry breaking, or coupling to magnetic impurities [2, 3, 4, 5]. In such systems, dark excitons provide access to otherwise hidden excitonic fine structure and play an important role in spin relaxation, exchange interactions, and the formation of long-lived excitonic reservoirs. More recently, dark-exciton physics has gained increasing attention in layered semiconductors, including layered metal-halide perovskites [6, 7, 8] and atomically thin van der Waals (vdW) materials, such as InSe [9], magnetic CrSBr [10], and, most prominently, semiconducting transition-metal dichalcogenides (S-TMDs) [11, 12, 13, 14, 15, 16, 17, 18, 19, 20].

Monolayers (MLs) of Mo- and W-based S-TMDs provide a particularly rich platform for exploring bright- and dark-exciton physics. Owing to the absence of inversion symmetry and strong spin-orbit coupling, the electronic bands at the inequivalent K$^{\pm}$ valleys are spin split. Consequently, the sign of the conduction-band (CB) spin splitting determines whether the lowest intravalley excitonic transition is spin allowed or spin forbidden [11, 12, 13, 14, 16, 18, 21, 22, 23, 24]. Accordingly, S-TMD MLs are commonly classified as darkish or bright: in MoS$_2$, WS$_2$, and WSe$_2$, the lowest spin-forbidden excitons lie below their bright counterparts, whereas in MoSe$_2$ and MoTe$_2$ the lowest excitonic state is optically allowed [13, 14, 17, 18, 24]. This energetic ordering strongly affects the optical response, as relaxation into low-lying dark states suppresses photoluminescence (PL) and modifies exciton and trion recombination dynamics, whereas the reduced radiative decay of dark states enables the formation of long-lived excitonic and valley reservoirs [16, 18, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]. In S-TMD MLs, an in-plane magnetic field mixes opposite-spin conduction-band states and redistributes oscillator strength between bright and spin-forbidden excitons, thereby enabling optical access to dark excitonic states and providing insight into the dark-bright splitting and conduction-band spin structure [13, 14, 17, 18, 42, 43]. Related brightening mechanisms based on symmetry breaking, strain, proximity effects, and nanophotonic coupling have also been explored [4, 44, 45, 46, 47]. By contrast, the inverse process, the loss of optical activity of an initially bright excitonic state, here referred to as darkening, has remained essentially unexplored in S-TMD MLs. Although strain- and composition-induced bright-to-dark crossovers have been reported in selected systems [44, 48], and enhanced in-plane magnetic responses have recently been observed in MoWSe$_2$ alloys [49], these studies did not demonstrate a controlled magnetic-field-induced darkening of a bright exciton accompanied by activation of its dark counterpart. Here we establish this missing complementary regime: magnetic brightening of the dark exciton occurs together with pronounced magnetic darkening of the initially bright state. Alloying provides the key control parameter because, in Mo$_x$W$_{1-x}$Se$_2$, composition strongly modifies the band-edge spin structure [49, 50] and, through the resulting changes in the low-energy electronic states, can also enhance the effective magnetic coupling between bright and dark excitons.

In this Letter, we investigate the interplay between bright and dark excitons in an hBN-encapsulated Mo$_{0.68}$W$_{0.32}$Se$_2$ ML alloy using low-temperature PL spectroscopy in in-plane magnetic fields $B_{\parallel}$ up to 25 T, complemented by density-functional-theory calculations including $GW$ corrections and a theoretical $\mathbf{k}\cdot\mathbf{p}$ analysis. The $\mathbf{k}\cdot\mathbf{p}$ model describes the coupled evolution of the bright and dark excitons under an in-plane magnetic field, while complementary DFT-$GW$ calculations support a spin-forbidden lowest-energy transition for an alloy composition close to that investigated experimentally. The alloyed ML exhibits a bright, MoSe$_2$-like emission at zero field. Surprisingly, an additional emission line emerges below the bright exciton for $B_{\parallel}\gtrsim15$ T, with an intensity that increases quadratically with $B_{\parallel}$, as expected for the magnetic brightening of a spin-forbidden dark exciton. Remarkably, the emergence of the dark-exciton emission is accompanied by a pronounced suppression of the initially bright exciton, revealing the simultaneous magnetic brightening of the dark state and darkening of its bright counterpart. The magnetic-field evolution of the exciton energies and intensities is consistently captured by the bright-dark exciton model, yielding a zero-field bright-dark splitting of $\Delta=14.2\pm1.3$ meV and an effective in-plane $g$ factor of $\lvert g_{\parallel}\rvert=5.2\pm0.2$.

Theoretical background— To describe the magnetic-field-induced mixing between bright and dark excitons in S-TMD MLs, we employ a $\mathbf{k}\cdot\mathbf{p}$ approach formulated in the basis of intravalley A-exciton states [12, 15, 24]. At zero magnetic field, each excitonic state is specified by (i) the valley $K^{\pm}$ occupied by the electron and hole and (ii) the spin indices $s_v,s_c=\uparrow,\downarrow$ of the valence- and conduction-band states forming the exciton.

The configurations $s_v=s_c=\uparrow$ in the $K^+$ valley and $s_v=s_c=\downarrow$ in the $K^-$ valley correspond to the bright A excitons, denoted by $\lvert K^{\pm},b\rangle$, respectively. The opposite-spin configurations, $s_v=\uparrow$, $s_c=\downarrow$ in $K^+$ and $s_v=\downarrow$, $s_c=\uparrow$ in $K^-$, form the dark A-exciton states $\lvert K^{\pm},d\rangle$.

The bright states possess in-plane transition dipole moments [12, 51], $\langle0|\mathbf{d}|K^{\pm},b\rangle=-id_{\parallel}(\mp\mathbf{e}_x+i\mathbf{e}_y)$, and couple to $\sigma^{\pm}$ circularly polarized light at normal incidence. In contrast, the dark states possess out-of-plane transition dipole moments [12, 51, 52, 53], $\langle0|\mathbf{d}|K^{\pm},d\rangle=id_{\perp}\mathbf{e}_z$, and couple to $z$-polarized light.

An in-plane magnetic field $\mathbf{B}_{\parallel}=(B_x,B_y)$ applied parallel to the ML’s plane mixes opposite-spin CB states within the same valley through the Zeeman interaction $g_{\parallel}\mu_BB_{\pm}$, where $B_{\pm}=B_x\pm iB_y$. In the excitonic basis $\lvert K^+,b\rangle$, $\lvert K^-,b\rangle$, $\lvert K^+,d\rangle$, and $\lvert K^-,d\rangle$, the corresponding Hamiltonian reads [13, 16]

where $\Delta$ denotes the zero-field bright-dark exciton splitting. The short-range electron-hole exchange interaction, described by $\delta$, mixes the dark excitons from opposite valleys and splits them into gray and strictly dark states [16, 26]. Since this splitting is typically much smaller than the bright-dark separation ($\delta\ll\Delta$) [16, 18, 26, 54, 55], we set $\delta=0$ in the following. The Hamiltonian then separates into two $2\times2$ blocks describing bright-dark mixing independently within each valley:

Writing $B_x=B_{\parallel}\cos\phi$ and $B_y=B_{\parallel}\sin\phi$, such that $B_{\pm}=B_{\parallel}e^{\pm i\phi}$, and introducing the mixing angle $\tan\vartheta=\frac{g_{\parallel}\mu_BB_{\parallel}}{\Delta}$, we obtain

The resulting bright-like and dark-like branches remain valley degenerate and have energies

The corresponding eigenstates are

The bright-state admixture in the dark-like exciton is therefore [13]

Since $\tan\vartheta=g_{\parallel}\mu_BB_{\parallel}/\Delta$, the leading contribution to $w$ scales as $B_{\parallel}^2$. Within the standard description of magnetic brightening of dark excitons [13], the intensity of ્ર 红鼎 the dark-exciton emission is expected to be proportional to the bright-exciton admixture $w$, the dark-exciton population $n_{\mathrm{D}}$, and the zero-field bright-exciton intensity $I_{\mathrm{B}}$:

The same mixing also modifies the bright-like exciton. As discussed in Sec. S1 of the Supplemental Material (SM), the change in the collected emission arising solely from the field-induced modification of the bright-exciton transition dipole is small under our experimental conditions and cannot account for the pronounced quenching of the A-exciton emission discussed below. We therefore consider the magnetic-field dependence of the bright-exciton population.

In a minimal model, we assume thermal equilibrium at temperature $T$ and take the population of the bright-like branch to follow the Boltzmann distribution, $n_{\mathrm{B}}(B_{\parallel}) \propto\exp\left[-\frac{E_{\mathrm{B}}(B_{\parallel})}{k_{\mathrm{B}}T}\right]$. Using the field dependence of the bright-like exciton energy in Eq. (4), the population normalized to its zero-field value becomes

This expression qualitatively describes the magnetic-field-induced evolution of the bright-exciton population $n_{\mathrm{B}}$. To illustrate the roles of the bright-dark exciton splitting $\Delta$ and the in-plane coupling parameter $g_{\parallel}$, Figure 1 shows the calculated $n_{\mathrm{B}}(B_{\parallel})/n_{\mathrm{B}}(0)$ at $T = 4.2\ \mathrm{K}$ and $B_{\parallel} = 25\ \mathrm{T}$ using Eq. (9). The strongest field-induced depletion of the bright-exciton population is expected for small $\Delta$ and large $g_{\parallel}$.

For WS$_2$ and WSe$_2$ MLs, where the bright-dark exciton splitting is typically $\Delta\approx40\ \mathrm{meV}$ [16, 18, 53, 54, 56] and $g_{\parallel} \approx2$ [16, 18], the predicted reduction of the bright-exciton population at 25 T is only about 10%. Such a weak effect may explain why a pronounced magnetic-field-induced depletion of the bright-exciton emission has not been reported in previous studies of dark-exciton brightening [13, 14, 18, 54]. This establishes a simple criterion for strong magnetic darkening: a small bright-dark splitting combined with strong in-plane coupling.

Magnetic brightening of the dark exciton—To test this prediction, we performed low-temperature PL measurements on an hBN-encapsulated Mo$_{0.68}$W$_{0.32}$Se$_2$ ML in the Voigt geometry, with an in-plane magnetic field $B_{\parallel}$ up to 25 T. Details of the sample fabrication and magneto-optical measurements are provided in the Methods section of the SM [57]. The resulting magnetic-field evolution of the PL spectra is shown in Figure 2(a). The same data plotted on a logarithmic intensity scale are presented in Sec. S2 of the SM [57], where the weak $X^{\mathrm{D}}$ emission is more clearly visible.

At zero magnetic field, the spectrum closely resembles that of bright MoSe$_2$ [17, 18, 26, 58, 59, 60]. Following the assignment established for MoWSe$_2$ alloy MLs, and in particular for Mo$_{0.68}$W$_{0.32}$Se$_2$ [50], the $X^{\mathrm{B}}$ and T lines are attributed to the bright neutral and charged excitons, respectively.

For $B_{\parallel} > 15\ \mathrm{T}$, an additional emission line, labeled $X^{\mathrm{D}}$, emerges on the low-energy side of $X^{\mathrm{B}}$, separated from it by approximately 15 meV [inset of Figure 2(a)]. Its integrated intensity, extracted from Lorentzian fits, increases with $B_{\parallel}$ [Figure 2(b)]. The field dependence of the $X^{\mathrm{D}}$ intensity is well described by the quadratic relation $I_{X^{\mathrm{D}}}(B_{\parallel}) = \alpha B_{\parallel}^{2}$, yielding $\alpha= (7.51 \pm0.15) \times10^{-3}\ \mathrm{T}^{-2}$. This quadratic dependence is consistent with Eq. (8), according to which the dark-exciton emission intensity is governed by the bright-state admixture induced by the in-plane magnetic field. Such a $B_{\parallel}^{2}$ scaling is a characteristic signature of the magnetic brightening of spin-forbidden dark excitons and has been previously observed in S-TMD MLs [13, 14, 16, 18, 54].

The observed $X^{\mathrm{B}}$-$X^{\mathrm{D}}$ separation of approximately 15 meV is close to the bright-dark splitting reported for MoS$_2$ MLs (around 15 meV) [18], and substantially smaller than the values of about 40 meV typical of WS$_2$ and WSe$_2$ [16, 36, 37, 40, 42, 52, 54]. The assignment of $X^{\mathrm{D}}$ to the neutral dark exciton is further supported by spin-resolved band-structure calculations, which are consistent with a spin-forbidden lowest-energy transition at the K valleys; see Sec. S3 of the SM [57].

Magnetic-field evolution of bright and dark excitons—Additionally, a striking feature of the spectra is the simultaneous suppression of the $X^{\mathrm{B}}$ emission with increasing $B_{\parallel}$, while the T intensity remains nearly unchanged over the entire magnetic-field range [see Figure 3(b)]. This behavior goes beyond the conventional magnetic brightening of dark excitons observed in binary S-TMD MLs, where the $X^{\mathrm{B}}$ emission remains essentially unaffected by the applied in-plane magnetic field [13, 14, 16, 18, 54]. Thus, in the studied Mo$_{0.68}$W$_{0.32}$Se$_2$ alloy ML, the magnetic brightening of $X^{\mathrm{D}}$ is accompanied by a pronounced magnetic-field-induced darkening of $X^{\mathrm{B}}$.

To analyze this effect quantitatively, we first extract the magnetic-field evolution of the $X^{\mathrm{B}}$ and $X^{\mathrm{D}}$ energies from the measured PL spectra, Figure 3(a). With increasing $B_{\parallel}$, the $X^{\mathrm{B}}$ and $X^{\mathrm{D}}$ lines exhibit characteristic blue- and redshifts, respectively, resulting from the bright-dark exciton mixing.

Following the two-level bright-dark mixing model introduced above, the field evolution of the bright-like and dark-like exciton branches is described by

where the + and − signs correspond to the $X^{\mathrm{B}}$ and $X^{\mathrm{D}}$ branches, respectively. Here, $E_{0}$ denotes the mean energy of the bright and dark exciton states, whereas $\Delta$ is their zero-field energy splitting. These quantities are defined as

where $E_{\mathrm{B}}(B_{\parallel})$ and $E_{\mathrm{D}}(B_{\parallel})$ denote the energies of the bright-like and dark-like exciton branches at a given magnetic field, respectively.

Since the dark exciton is not optically resolved at zero magnetic field, $E_{\mathrm{D}}(0)$ cannot be determined directly from the PL spectrum. We therefore adopt the following procedure. First, $E_{0}$ is determined by averaging the quantity $[E_{\mathrm{B}}(B_{\parallel}) + E_{\mathrm{D}}(B_{\parallel})]/2$ over the experimental data in the field range 15-25 T, where both excitonic transitions are resolved. The zero-field bright-dark splitting is then obtained from $\Delta= 2[E_{\mathrm{B}}(0) - E_{0}]$. The field dependence of the $X^{\mathrm{B}}$ and $X^{\mathrm{D}}$ energies [Figure 3(a)] is then fitted using Eq. (10), with $E_{0} = 1.63350 \pm0.00040$ eV and $\Delta= 14.2 \pm1.3$ meV fixed to the values determined above.

Thus, $\lvert g_{\parallel}\rvert$ is the only free fitting parameter. The resulting curves, shown as solid lines in Figure 3(a), reproduce the experimental data very well. The extracted value, $\lvert g_{\parallel}\rvert= 5.2 \pm0.2$, is substantially larger than the value of approximately 2 reported for binary S-TMD MLs [16, 18]. This enhanced magnetic response can be qualitatively understood as a consequence of alloy-induced modifications of the band-edge electronic structure, consistent with the strong composition dependence of the out-of-plane excitonic $g_{\perp}$ factor recently observed in Mo$_x$W$_{1-x}$Se$_2$ alloys [50]. Changes in the relative positions of the bands modify the contributions of remote bands to the low-energy states and can therefore substantially renormalize the effective in-plane coupling $g_{\parallel}$ governing the bright-dark mixing. A larger $\lvert g_{\parallel}\rvert$, together with the reduced bright-dark splitting near the conduction-band spin-orbit crossover, favors particularly strong magnetic mixing in the alloy. Interestingly, an effective in-plane excitonic $g_{\parallel}$ factor of $6.4 \pm0.3$ was recently reported for a Mo$_{0.7}$W$_{0.3}$Se$_2$ ML [49], further highlighting the unusually strong in-plane magnetic response of Mo$_x$W$_{1-x}$Se$_2$ alloys in this composition range. A more detailed effective-medium description is provided in § 3.8 of the SM [57].

We note that the extracted value of $|g_{\parallel}|$ is sensitive to the uncertainties in $E_0$ and $\Delta$. In particular, the determination of $|g_{\parallel}|$ strongly depends on the assumed value of $\Delta$. A detailed analysis of this sensitivity is provided in Sec. S5 of the SM [57].

We now turn to the magnetic-field dependence of the $X^{\mathrm{B}}$ emission intensity. A pronounced suppression of $X^{\mathrm{B}}$ is already evident in the PL spectra shown in Figure 2(a). To quantify this effect, we extract the integrated intensities of the $X^{\mathrm{B}}$ and T emission lines from Lorentzian fits and normalize them to their zero-field values [Fig. 3(b)].

While the T intensity remains nearly constant over the entire magnetic-field range, the $X^{\mathrm{B}}$ emission is strongly suppressed, decreasing by approximately 85% at $B_{\parallel}=25$ T. The distinct field evolution of the two transitions rules out a trivial loss of the PL signal, for example, due to a drift of the excitation or collection spot across an inhomogeneous sample. Instead, it indicates that the in-plane magnetic field selectively suppresses the neutral bright-exciton emission.

This magnetic-field-induced darkening of $X^{\mathrm{B}}$ can be understood within the population model introduced above. Assuming thermal equilibrium at $T=4.2$ K, the relative population of the bright-like exciton branch is given by Eq. (9). Using the experimentally determined parameters $\Delta=14.2\pm1.3$ meV and $|g_{\parallel}|=5.2\pm0.2$, we obtain $n_{\mathrm{B}}(25\ \mathrm{T})/n_{\mathrm{B}}(0)\simeq0.08\pm0.02$, corresponding to a 92% reduction of the bright-exciton population. This value is close to the experimentally observed suppression of the $X^{\mathrm{B}}$ emission by approximately 85%.

Using the same values of $\Delta$ and $|g_{\parallel}|$, we calculate the full magnetic-field dependence of $n_{\mathrm{B}}(B_{\parallel})/n_{\mathrm{B}}(0)$ from Eq. (9). The resulting curve, shown as the solid line in Figure 3(b), closely reproduces the measured evolution of the normalized $X^{\mathrm{B}}$ intensity. This agreement supports the interpretation that the pronounced magnetic darkening of the bright exciton is dominated by the field-induced increase in the energy of the bright-like branch and the consequent depletion of its thermal population.

Summary– We have demonstrated that an in-plane magnetic field can simultaneously brighten a spin-forbidden dark exciton and strongly suppress its bright counterpart in a $\mathrm{Mo}_{0.68}\mathrm{W}_{0.32}\mathrm{Se}_2$ ML alloy. Although the zero-field PL resembles that of a bright $\mathrm{MoSe}_2$-like ML, the high-field response reveals a dark neutral exciton below the bright state. The coupled evolution of the two exciton branches is captured by a bright-dark mixing model, while the pronounced quenching of the bright-exciton emission is accounted for by the field-induced increase in the energy of the bright-like branch and the consequent depletion of its thermal population. The relatively small bright-dark splitting and enhanced in-plane coupling place the alloy in a regime where magnetic darkening becomes directly observable. More generally, because the effect is controlled by the competition between $\Delta$ and $g_{\parallel}\mu_{\mathrm{B}}B_{\parallel}$, it should be enhanced in excitonic systems with near-degenerate bright and dark states. These findings establish composition as a route to engineering the bright-dark exciton hierarchy and to magnetic control of exciton populations and light emission in two-dimensional semiconductors.

Acknowledgements

The work was supported by the National Science Centre, Poland (grants no. 2022/45/N/ST3/01887 and 2025/57/B/ST5/03288). The publication was created as part of a project co-financed by the Polish Ministry of Science and Higher Education under contract no. 2025/WK/01. We acknowledge the support of the LNCMI-CNRS, a member of the European Magnetic Field Laboratory (EMFL). A.O.S. acknowledges the support of the Czech Science Foundation (project GAČR 26-21965S). A.O.S. and M.P. acknowledge support from the CENTERA2, FENG.02.01-IP.05-5 T004/23 project funded within the IRA program of the FNP Poland, co-financed by the EU FENG Programme. A.P. and E.B. acknowledge funding by the European Union’s EIC Pathfinder Open 2024 under grant agreement 101186642 (VALERIA). E.B. acknowledges support from the Italian Science Fund (Fondo Italiano per la Scienza, FIS), FIS 3 Call (Project INCOGNITO, ID: FIS-2024-07158). K.W. and T.T. acknowledge support from the CREST (JPMJCR24A5), JST and World Premier International Research Center Initiative (WPI), MEXT, Japan.

References

Supplemental Material: Magnetic-Field-Induced Darkening of Bright Excitons in a MoWSe$_2$ Monolayer Alloy

Katarzyna Olkowska-Pucko; Artur O. Slobodeniuk; Mehdi Arfaoui; Elena Blundo; Natalia Zawadzka; Łucja Kipczak; Jan Szpakowski; Diana Vaclavkova; Dipankar Jana; Piotr Kapuściński; Kenji Watanabe; Takashi Taniguchi; Clement Faugeras; Marek Potemski; Adam Babiński; Antonio Polimeni; Maciej R. Molas

This Supplemental Material provides: Methods - Fabrication of the hBN-encapsulated Mo$_{0.68}$W$_{0.32}$Se$_2$ monolayer, the magneto-photoluminescence measurements performed in high in-plane magnetic fields, and computational details. S1 - Emission intensity of field-mixed bright and dark excitons. S2 - Logarithmic-scale representation of the magneto-photoluminescence spectra highlighting the weak X$^\mathrm{D}$ emission. S3 - Band structure and spin-resolved unfolding of monolayer Mo$_{0.75}$W$_{0.25}$Se$_2$. S4 - Effective-medium description of magnetic darkening in Mo$_{0.68}$W$_{0.32}$Se$_2$. S5 - Sensitivity of the extracted $g_{\parallel}$ factor to the bright-dark exciton splitting.

Methods

Fabrication of hBN encapsulated sample

Mo$_{0.68}$W$_{0.32}$Se$_2$ was mechanically exfoliated from a bulk crystal purchased from HQ Graphene. The monolayer (ML) was exfoliated onto polydimethylsiloxane (PDMS) using scotch tape. Relatively thick hBN flakes were mechanically exfoliated onto SiO$_2$/Si substrates consisting of a 300 nm-thick SiO$_2$ layer on a 500 $\mu$m-thick Si wafer. The Mo$_{0.68}$W$_{0.32}$Se$_2$ ML was then deterministically transferred from the PDMS onto an hBN flake. A thin hBN flake was exfoliated onto PDMS and subsequently transferred on top of the ML to complete the encapsulation. After each transfer step, the sample was annealed under high vacuum ($10^{-6}$ mbar) at approximately 120°C for several hours to promote the coalescence of contaminants and improve the adhesion between the layers.

Magneto-optical experiments

Low-temperature micro-magneto-photoluminescence investigations were performed in the Voigt geometry, i.e., with the magnetic field oriented parallel to the ML plane. Measurements with a spatial resolution of approximately 1 $\mu$m were carried out using a resistive magnet capable of producing magnetic fields up to 30 T and a free-space optical setup. The sample was mounted on an $x$-$y$-$z$ piezoelectric stage and maintained at $T = 4.2$ K. The sample was excited using a continuous-wave (CW) laser diode with a wavelength of 515 nm (photon energy of 2.41 eV). The emitted light was dispersed by a monochromator with a focal length of 0.5 m and detected using a CCD camera.

Computational details

First-principles calculations were performed within (density-functional-theory) DFT using the plane-wave pseudopotential approach as implemented in the Quantum ESPRESSO package [61, 62, 63]. The exchange-correlation interaction was described within the generalized-gradient approximation using the Perdew-Burke-Ernzerhof functional [64]. Optimized norm-conserving Vanderbilt pseudopotentials were employed for Mo, W, and Se. The Mo$_{0.75}$W$_{0.25}$Se$_2$ alloy was represented by a $2 \times2 \times1$ supercell (SC) containing four transition-metal sites (three Mo and one W) and eight Se atoms [65]. The alloy configuration was constructed by replacing one of the four Mo atoms in the $2 \times2$ MoSe$_2$ SC by W. The atomic positions and in-plane lattice parameters were optimized within DFT using Broyden-Fletcher-Goldfarb-Shanno algorithm, while the out-of-plane lattice parameter was kept fixed at 20 Å. Structural relaxation was performed until the total-energy and force convergence thresholds of $10^{-4}$ Ry and $10^{-3}$ Ry/Bohr, respectively. The optimized in-plane SC lattice constant was 6.64 Å. The Brillouin zone (BZ) was sampled with a $12 \times12 \times1$ k-point grid during the structural relaxation. For the subsequent electronic-structure calculations, the Kohn-Sham wave functions were expanded in a plane-wave basis with a kinetic-energy cutoff of 60 Ry. The BZ was sampled using an $18 \times18 \times1$ k-point grid. The self-consistent-field calculations were performed with a convergence threshold of $10^{-8}$ Ry. Spin-orbit coupling (SOC) was explicitly included in the electronic-structure calculations. A two-dimensional isolated-system treatment was employed along the out-of-plane direction.

Quasiparticle corrections were calculated within the $G_0W_0$ approximation using Yambo [66, 67]. The dynamically screened Coulomb interaction was described within the plasmon-pole approximation. The independent-particle polarizability included 400 bands, and the dielectric matrix was expanded using a cutoff of 10 Ry. The convergence of the self-energy with respect to high-energy unoccupied states was accelerated using the Bruneval-Gonze terminator [68]. A slab Coulomb cutoff was applied along the out-of-plane direction to suppress spurious interactions between periodically repeated MLs.

Emission intensity of field-mixed bright and dark excitons

In this section, we examine how the magnetic-field-induced mixing of bright and dark excitons modifies the transition dipole moment and, consequently, the light intensity collected in the experimental geometry. In particular, we derive the dependence of the collected emission intensity on the mixing angle and show that the modification of the transition dipole alone produces only a weak reduction of the bright-exciton emission under our experimental conditions.

The field-mixed bright-like state introduced in the main text is characterized by the transition dipole moment $\mathbf{d}_{\pm}=\langle0|\hat{\mathbf{d}}|\mathrm{B}_{\pm}\rangle$, which has the form

The dipole vectors associated with the two valleys can generally be written as $\mathbf{d}_{\alpha}=a_{\alpha}[\mathbf{e}_x+i\mathbf{e}_y]+b_{\alpha}[-\mathbf{e}_x+i\mathbf{e}_y]+c_{\alpha}\mathbf{e}_z$, with $\alpha=\pm$.

We now derive the intensity of circularly polarized emission as a function of the collection angle and the orientation of the dipole moments. The derivation is based on a transformation to the spherical basis and takes into account the directional character of optical-dipole emission. The resulting expressions provide a quantitative description of the influence of the finite collection angle of the detection system on the measured emission intensity.

Here $\mathbf{e}_{\pm}(\theta,\phi)$ are the polarization vectors of $\sigma_{\pm}$-polarized photons propagating in the direction defined by the angles $(\theta,\phi)$. To write $\mathbf{e}_{\pm}$ explicitly, we introduce the local orthogonal unit vectors in the directions of increasing $r$, $\theta$, and $\phi$, respectively [see Figure S1.1]:

The spherical basis $\{\mathbf{e}_{\theta},\mathbf{e}_{\phi},\mathbf{e}_r\}$ has the same relative orientation as $\{\mathbf{e}_x,\mathbf{e}_y,\mathbf{e}_z\}$. Hence, we define $\mathbf{e}_{\pm}(\theta,\phi)\equiv(\pm\mathbf{e}_{\theta}+i\mathbf{e}_{\phi})/\sqrt{2}$. Taking the corresponding scalar products gives

For the two valleys, we obtain

and

Here, $\phi$ denotes the azimuthal orientation of the in-plane magnetic field, whereas $\varphi$ denotes the azimuthal direction of photon emission.

Squaring these amplitudes gives

These expressions determine the intensity of the $\sigma_{\pm}$-polarized light emitted by excitons from the $\mathbf{K}_{\pm}$ valleys. The total intensity of the $\sigma_{\pm}$-polarized light collected within the solid angle of the objective is therefore

Evaluating the sums over $\alpha$, we obtain

Integration over $\varphi$ eliminates the interference term in both expressions and demonstrates that the total $\sigma_{+}$ and $\sigma_{-}$ intensities are equal:

Here, $\mathcal{N}$ is a coefficient independent of $\theta_0$ and $\vartheta$ and proportional to the bright-exciton population, $\mathcal{N} \propto n_{\mathrm{B}}$. In the limit $d_{\parallel} \gg d_{\perp}$, the collected emission intensity as a function of the in-plane magnetic field $B_{\parallel}$ is given by

The reduction in the collected emission intensity predicted by this expression originates from the field-induced rotation of the transition dipole moment of the bright-like exciton toward the out-of-plane direction. However, even at high magnetic fields ($B_{\parallel} \sim30$ T), this effect produces only a minor decrease in the collected $A$-exciton emission intensity:

Magnetic brightening of the dark excition

To highlight the weak $X^{\mathrm{D}}$ emission, Figure S2.2 presents the same PL spectra as in the main text on a logarithmic intensity scale. This representation more clearly reveals the low-intensity $X^{\mathrm{D}}$ line emerging below $X^{\mathrm{B}}$ for $B_{\parallel} \gtrsim15$ T.

BAND STRUCTURE AND SPIN-RESOLVED UNFOLDING OF MONOLAYER Mo$_{0.75}$W$_{0.25}$Se$_2$

The emergence of $X^{\mathrm{D}}$ below $X^{\mathrm{B}}$ in an in-plane magnetic field suggests that the lowest-energy excitonic state in the investigated Mo$_{0.68}$W$_{0.32}$Se$_2$ monolayer (ML) is optically forbidden. In particular, at $B_{\parallel}=25$ T, $X^{\mathrm{D}}$ lies approximately 15 meV below $X^{\mathrm{B}}$ (see Figure 2 of the main text), while its intensity increases quadratically with $B_{\parallel}$, as expected for the magnetic brightening of a spin-forbidden dark exciton. To further substantiate this assignment, we performed spin-resolved electronic-structure calculations to determine the band-edge spin ordering at the $K^{\pm}$ valleys. The calculations were performed for ML Mo$_{0.75}$W$_{0.25}$Se$_2$, represented by a $2 \times2 \times1$ alloy SC, which provides a commensurate approximation to the experimentally investigated Mo$_{0.68}$W$_{0.32}$Se$_2$ composition. Our calculations are intended to identify the relevant band-edge electronic structure rather than to reproduce quantitatively the experimentally determined bright-dark exciton splitting.

$G_{0}W_{0}$ quasiparticle band structure and spin ordering The spin-resolved $G_{0}W_{0}$ quasiparticle band structure of ML Mo$_{0.75}$W$_{0.25}$Se$_2$ is shown in Figure S3.3. Details of the computational method are provided in the Methods section. Because the calculations are performed using a $2 \times2 \times1$ alloy SC, the quasiparticle bands are folded into the reduced SC Brillouin zone. We denote the primitive-cell (PC) reciprocal-lattice basis vectors by $\mathbf{b}_{i}$ and the corresponding SC reciprocal-lattice basis vectors by $\mathbf{B}_{i}$, where $i=1,2,3$. The real-space PC and SC lattice vectors are related by the transformation matrix $\mathbf{M}=\operatorname{diag}(2,2,1)$. Accordingly, in reciprocal space, $\mathbf{B}_{1}=\mathbf{b}_{1}/2$, $\mathbf{B}_{2}=\mathbf{b}_{2}/2$, and $\mathbf{B}_{3}=\mathbf{b}_{3}$. The quasiparticle band structure is evaluated along the SC path $\Gamma_{\mathrm{SC}}M_{\mathrm{SC}}K_{\mathrm{SC}}\Gamma_{\mathrm{SC}}$, where $\Gamma_{\mathrm{SC}}=(0,0,0)$, $M_{\mathrm{SC}}=(1/2,0,0)$, and $K_{\mathrm{SC}}=(1/3,2/3,0)$ are given in fractional coordinates of the SC reciprocal basis $\{\mathbf{B}_{i}\}$. The inset of Figure S3.3 compares the first SC BZ with the first PC BZ. The DFT calculation including SOC yields a direct fundamental band gap at $K$ of 1.33 eV. The $G_{0}W_{0}$ correction increases the direct quasiparticle gap at $K$ to 2.29 eV.

Spin-resolved band unfolding The SC folds states originating from different PC momenta into the reduced SC BZ, complicating the identification of their valley and spin character. To overcome this, we employ a spin-resolved band-unfolding procedure applied to fully relativistic DFT+SOC wave functions. The implementation follows the plane-wave spectral-weight formulation of SC band unfolding [69], generalized to two-component spinor wave functions as in Medeiros et al. [70]. For an arbitrary SC wave vector $\mathbf{K}$, the corresponding PC momenta satisfy $\mathbf{K}+\mathbf{G}=\mathbf{k}+\mathbf{g}$, where $\mathbf{G}=\sum_{i}g_{i}\mathbf{B}_{i}$ and $\mathbf{g}=\sum_{i}m_{i}\mathbf{b}_{i}$ are SC and PC reciprocal-lattice vectors, respectively, with $g_{i},m_{i}\in\mathbb{Z}$. Equivalently, a given SC momentum $\mathbf{K}$ unfolds onto the PC momenta $\mathbf{k}_{j}=\mathbf{K}+\mathbf{G}_{j}$ modulo a PC reciprocal-lattice vector, where $\mathbf{G}_{j}$ denotes one of the distinct SC reciprocal-lattice translations connecting $\mathbf{K}$ to the PC BZ. A fully relativistic SC eigenstate is expressed as the two-component spinor $\boldsymbol{\Psi}_{n\mathbf{K}}(\mathbf{r})=(\psi^{\uparrow}_{n\mathbf{K}}(\mathbf{r}),\psi^{\downarrow}_{n\mathbf{K}}(\mathbf{r}))^{\mathrm{T}}$, where $n$ denotes the SC band index. Each spinor component is expanded in a plane-wave basis as $\psi^{\sigma}_{n\mathbf{K}}(\mathbf{r})=\sum_{\mathbf{G}}C^{\sigma}_{n\mathbf{K}}(\mathbf{G})e^{i(\mathbf{K}+\mathbf{G})\cdot\mathbf{r}}$, with $\sigma=\uparrow,\downarrow$. The spin-resolved unfolded spectral weight at a given PC momentum $\mathbf{k}$ is obtained by summing the plane-wave contributions over the subset $\mathcal{U}_{\mathbf{k}}$ of SC reciprocal-lattice vectors satisfying the unfolding condition,

The total unfolded spectral weight is

For the present $2 \times2 \times1$ transformation, $\det\mathbf{M}=4$, giving four distinct unfolding sectors for each SC wave vector $\mathbf{K}$. In our plane-wave implementation, the subset $\mathcal{U}_{\mathbf{k}}$ entering Eq. (27) is identified from the integer coefficients $(g_1,g_2,g_3)$ of the SC reciprocal-lattice vectors. Because the SC is doubled along the first two lattice directions, the unfolding sectors are determined by the parities of $g_1$ and $g_2$. We therefore label each sector by $(s_1,s_2)\in\{(0,0),(1,0),(0,1),(1,1)\}$, such that the selected plane-wave components satisfy $g_1-s_1=0\pmod{2}$ and $g_2-s_2=0\pmod{2}$. The normalized out-of-plane spin polarization of each unfolded state

Here, $P_{z,n\mathbf{K}}(\mathbf{k})\in[-1,1]$, with positive and negative values corresponding to opposite out-of-plane spin projections. Within the convention used here, $P_z=\langle\sigma_z\rangle=2\langle S_z\rangle/\hbar$. For visualization, the discrete unfolded states are broadened using a Gaussian function $g_\eta(E-E_{n\mathbf{K}})$ with broadening $\eta$. The total unfolded spectral function is

while the spin-weighted spectral function is defined as

The energy-resolved spin polarization used to color the unfolded band structure is

The ratio is evaluated only where $A_{\mathrm{tot}}(\mathbf{k},E)$ exceeds a small numerical threshold.

Unfolded band structure and spin-orbit splitting

The spin-resolved unfolded band structure of ML Mo$_{0.75}$W$_{0.25}$Se$_2$ is shown in Figure S3.4. Unfolding onto the PC BZ allows the band-edge states and their spin character at the $\mathbf{K}^{+}$ valley to be identified directly. We denote the two uppermost valence branches by $v_1$ and $v_2$, with energies $E_{v_1}(\mathbf{k})$ and $E_{v_2}(\mathbf{k})$, and the two lowest conduction branches by $c_1$ and $c_2$, with energies $E_{c_1}(\mathbf{k})$ and $E_{c_2}(\mathbf{k})$, respectively. At the $\mathbf{K}^{+}$ valley, $v_1$ corresponds to the valence-band maximum (VBM) and has an out-of-plane spin polarization $P_z=-1$, whereas $c_1$ corresponds to the conduction-band minimum (CBM) and has $P_z=+1$. The next conduction-band state $c_2$, has $P_z=-1$, i.e., same spin character as $v_1$. For the momentum-resolved maps shown in the right panels of Figure S3.4, the conduction- and valence-band spin-orbit splittings are defined as $\Delta_{\mathrm{SO}}^{c}(\mathbf{k})=E_{c_2}(\mathbf{k})-E_{c_1}(\mathbf{k})$ and $\Delta_{\mathrm{SO}}^{v}(\mathbf{k})=E_{v_1}(\mathbf{k})-E_{v_2}(\mathbf{k})$. At the $\mathbf{K}^{+}$ valley of the PC BZ, the corresponding conduction- and valence-band spin-orbit splittings are $\Delta_{\mathrm{SO}}^{c}(\mathbf{K})=18\ \mathrm{meV}$ and $\Delta_{\mathrm{SO}}^{v}(\mathbf{K})=200\ \mathrm{meV}$, respectively. At the $\mathbf{K}^{+}$ valley, the lowest-energy interband transition $(v_1,c_1)$ is a spin-forbidden, whereas the higher-energy $(v_1,c_2)$ transition is a spin-allowed. The $G_0W_0$ quasiparticle calculation and DFT+SOC unfolding provide complementary descriptions of the alloy electronic structure. The resulting band-edge spin ordering is consistent with the assignment of the experimentally observed X$^{D}$ emission in Mo$_{0.68}$W$_{0.32}$Se$_2$ to a spin-forbidden dark exciton.

Effective-medium description of magnetic darkening in Mo$_{0.68}$W$_{0.32}$Se$_2$

The theoretical analysis is based on an effective-medium description of the monolayer alloy Mo$_{0.68}$W$_{0.32}$Se$_2$, in which its low-energy electronic structure is assumed to retain the main features of the parent MoSe$_2$ and WSe$_2$ monolayers. In particular, we assume that well-defined Bloch states can be introduced at the $\mathrm{K}^{\pm}$ valleys, where the relevant excitonic transitions occur. We further neglect local structural distortions and differences in the spatial extent of the relevant Mo and W orbitals, and describe the alloy by an effective hexagonal lattice with a common lattice constant. Within this approximation, the electronic states near the $\mathrm{K}^{\pm}$ valleys can be treated using the same symmetry-based framework as in binary S-TMD monolayers [23, 71].

A $\boldsymbol{k}\cdot\boldsymbol{p}$ model for monolayer Mo$_{0.68}$W$_{0.32}$Se$_2$ at the $\mathrm{K}^{\pm}$ points can therefore be constructed analogously to that of the parent S-TMD monolayers. The simplest four-band model is characterized by: (i) the spin-resolved valence- and conduction-band extrema; (ii) momentum matrix elements between valence- and conduction-band Bloch states of the same spin in the $\mathrm{K}^{\pm}$ valleys, which determine the optical activity of the excitonic transitions; and (iii) matrix elements coupling these bands to higher conduction and lower valence bands, which provide higher-order corrections within the $\boldsymbol{k}\cdot\boldsymbol{p}$ approximation. We next discuss how these parameters are expected to evolve in Mo$_{0.68}$W$_{0.32}$Se$_2$ compared with those in MoSe$_2$ and WSe$_2$ monolayers.

As a lowest-order approximation, the energy of the $n$th band of the Mo$_x$W$_{1-x}$Se$_2$ monolayer at the $\mathrm{K}^{\pm}$ points can be estimated by linear interpolation between the corresponding band energies of the parent compounds:

Such a linear interpolation provides a simple qualitative description of the evolution of the band-edge energies with composition, although deviations from linearity and band-gap bowing are known to occur in Mo$_x$W$_{1-x}$Se$_2$ alloys [72, 73].

Within this approximation, the conduction-band spin splitting in the alloy for $x=0.68$ at the $\mathrm{K}^{\pm}$ points becomes substantially smaller in absolute value than the corresponding splittings in pure MoSe$_2$ and WSe$_2$ monolayers. The reduced conduction-band spin splitting enhances the mixing between bright and dark excitons induced by an in-plane magnetic field. Consequently, a larger fraction of the optical activity can be transferred from the bright to the dark exciton than in the parent MoSe$_2$ and WSe$_2$ monolayers. The resulting loss of optical activity of the bright exciton increases with increasing magnetic field.

The momentum matrix elements between valence- and conduction-band Bloch states of the same spin in the $\mathrm{K}^{\pm}$ valleys can likewise be approximated by interpolation between the corresponding values for MoSe$_2$ and WSe$_2$. These matrix elements determine the optical activity of the bright excitons. However, within the present model, they do not depend explicitly on the applied in-plane magnetic field and therefore do not themselves account for the magnetic-field-induced darkening.

Finally, we consider higher-order corrections to the four-band $\boldsymbol{k}\cdot\boldsymbol{p}$ model. Changes in the relative positions of the bands modify the contributions of remote bands to the low-energy electronic states and, consequently, their magnetic response [23, 74]. The effective in-plane coupling parameter $g_{\parallel}$ entering the bright-dark mixing can therefore differ substantially from the corresponding values in the parent MoSe$_2$ and WSe$_2$ monolayers. A larger $\lvert g_{\parallel}\rvert$ enhances the magnetic-field-induced mixing of the bright and dark excitons and thus increases the transfer of oscillator strength between the two states.

In addition to this redistribution of optical activity, the bright-like exciton shifts to higher energy with increasing $B_{\parallel}$. At finite temperature, this energy shift reduces the thermal population of the bright-like branch and thereby suppresses its photoluminescence intensity. As shown in the main text, this population effect provides the dominant contribution to the pronounced magnetic darkening observed experimentally.

Interestingly, a large effective in-plane excitonic $g$ factor has recently been reported for a Mo$_{0.7}$W$_{0.3}$Se$_2$ monolayer [49]. In that work, however, the enhanced photoluminescence $g$ factor was attributed to many-body interactions rather than directly to a single-particle bright-dark mixing parameter. Nevertheless, the observation highlights the unusually strong in-plane magnetic response of Mo$_x$W$_{1-x}$Se$_2$ alloys close to the conduction-band spin-orbit crossover.

The present model is restricted to neutral bright and dark excitons. A quantitative description of trions, which are three-particle complexes and involve additional Coulomb and exchange interactions, is beyond the scope of this work.

Sensitivity of the extracted $g_{\parallel}$ factor to the bright-dark exciton splitting

The extracted value of $|g_{\parallel}|$ is sensitive to the assumed value of the zero-field bright-dark exciton splitting $\Delta$. To quantify this sensitivity, we performed three fits using different constraints on $\Delta$ and $|g_{\parallel}|$. The corresponding fits are shown in Figure S5.5. The uncertainty in the extracted transition energies is primarily determined by the experimental energy resolution and spectral sampling. The energy spacing between adjacent data points in the recorded PL spectra was approximately 0.28 meV.

For the main fit, the mean exciton energy $E_0$ was determined by averaging $[E_{\mathrm{B}}(B_{\parallel}) + E_{\mathrm{D}}(B_{\parallel})]/2$ over the field range 15-25 T, where both transitions are experimentally resolved. Together with the zero-field energy of $X^{\mathrm{B}}$, this yields $\Delta_1 = 14.2 \pm1.3$ meV, as described in the main text. Fixing this value of $\Delta$ in the fit gives $|g_1| = 5.2 \pm0.2$.

As a second fitting procedure, $\Delta$ was treated as a free parameter, yielding $\Delta_2 = 14.48 \pm0.12$ meV and $|g_2| = 4.6 \pm0.2$. Conversely, fixing the in-plane coupling parameter to the literature value $|g_3| = 6.4$ reported for a Mo$_{0.7}$W$_{0.3}$Se$_2$ monolayer [49] gives $\Delta_3 = 13.51 \pm0.24$ meV.

The values of $\Delta_2$ and $\Delta_3$ are consistent with $\Delta_1$ within its uncertainty. At the same time, the corresponding values of $|g_{\parallel}|$ differ substantially, demonstrating that the extracted in-plane coupling parameter is highly sensitive to relatively small variations in the assumed zero-field bright-dark exciton splitting.

Overall, the three fitting procedures yield bright-dark splittings in the range of approximately 13.5-14.5 meV, while the corresponding values of $|g_{\parallel}|$ span the range 4.6-6.4. Despite this sensitivity of $|g_{\parallel}|$, all fits consistently place the zero-field dark-exciton energy at approximately 1.62 eV, about 14 meV below the bright exciton. Thus, the conclusion that the dark exciton forms the lowest-energy excitonic state is robust with respect to the fitting procedure.