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Interplay between spin proximity effect and charge-dependent exciton dynamics in MoSe2/CrBr3 van der Waals heterostructures

Semiconducting ferromagnet-nonmagnet interfaces in van der Waals heterostructures present a unique opportunity to investigate magnetic proximity interactions dependent upon a multitude of phenomena including valley and layer pseudospins, moiré periodicity, or exceptionally strong Coulomb binding. Here, we report a charge-state dependency of the magnetic proximity effects between MoSe2 and CrBr3 in photoluminescence, whereby the valley polarization of the MoSe2 trion state conforms closely to the local CrBr3 magnetization, while the neutral exciton state remains insensitive to the ferromagnet. We attribute this to spin-dependent interlayer charge transfer occurring on timescales between the exciton and trion radiative lifetimes. Going further, we uncover by both the magneto-optical Kerr effect and photoluminescence a domain-like spatial topography of contrasting valley polarization, which we infer to be labyrinthine or otherwise highly intricate, with features smaller than 400 nm corresponding to our optical resolution. Our findings offer a unique insight into the interplay between short-lived valley excitons and spin-dependent interlayer tunneling, while also highlighting MoSe2 as a promising candidate to optically interface with exotic spin textures in van der Waals structures.


Supplementary Note 1: Details of the DFT calculations
The ab initio calculations of the electronic structure of the heterostructure MoSe 2 /CrBr 3 have been performed using density-functional theory (DFT) at the level of the local density approximation (LDA), as implemented in Quantum Espresso package [1]. In addition we have applied the on-site Hubbard correction with values U = 1.5 eV and Hund's exchange interaction J = 0.5 eV [2]. We have included spin-orbit interaction with spinorial wave functions, using norm-conserving full relativistic pseudopotentials. The pseudopotentials of Cr and Mo include semi-core valence electrons and have been generated with ONCVPSP and PSEUDOJO [3,4]. The electronic density converges with an energy cutoff of 80 Ry and a k-grid of 12 × 12 × 1. We use a slab model with a 17Å vacuum thickness to avoid interactions between periodic images.
In order to make the interpretation of the band structure easier we have followed the unfolding procedure formulated by Popescu and Zunger [5]. This method maps the energy eigenvalues obtained in supercell calculations into an effective band structure (EBS). The reference band structure is that of monolayer MoSe 2 .
We have performed two sets of ab initio calculations: (i) imposing the experimental lattice constant of MoSe 2 and relaxing the atomic positions of the whole heterostructure, and (ii) relaxing the lattice parameter of the supercell. In this way we assure that no artifact from the simulation alters the conclusions. Figure 1a-b shows the EBS materialprojected and spin-projected, respectively, for the case of MoSe 2 experimental lattice constant. We find the strongest hybridization of the EBS at the intermediate Q point in the conduction band. Moreover, the conduction band is immersed within the t 2g bands of CrBr 3 . Figure 1c shows the detail of the conduction band at the −K and +K points of the EBS, which reveals a valley splitting of 2.2 meV. In free-standing MoSe 2 and without magnetic field, conduction band states at −K and +K have the same energy. Our calculations indicate no valley splitting in the valence band.  Figure 2 shows the relevant wavefunctions at important points in the EBS at the K-point. There is evidence of interlayer band hybridization between the materials' conduction bands, corresponding to box 5 in Figure 2. The orbital content of the MoSe 2 valence band is localized primarily in the molybdenum plane (box 1), and so experiences a far weaker spin proximity effect as compared to the MoSe 2 conduction band, in which the orbital spread is more prominent out of the plane (boxes 3 and 4). This explains the breaking of valley degeneracy in the conduction band (Figure 1c), absent for the valence band. We have also evaluated the EBS for an optimized heterostructure, as shown in Figure 3. In this case we find an indirect bandgap for MoSe 2 , and a stronger hybridization of the conduction band at the Q-point. This is a well-known consequence of the dependency of the bandgap directness on the lattice parameter in transition metal dichalcogenides [6]. In this case a conduction band valley splitting is also expected, as shown in Figure 3c We have confirmed the lack of significant contribution to the proximity effects from underlying CrBr 3 layers by comparing the electronic structure of monolayer MoSe 2 on top of single-layer and double-layer CrBr 3 . Essentially we find that adding more layers results in the increasing of states from CrBr 3 . The thicker CrBr3 adds more bands to the EBS as a whole, and the absolute energies of some of the bands shift, but the eg bands remain at several hundred millielectron Volts below the t2g bands, and so the spin-dependent interlayer tunnelling discussed in the Main Text is not expected to change substantially.

Supplementary Note 3: Measurements on additional samples
We have fabricated and measured the DOCP in photoluminescence of 2 additional MoSe 2 / CrBr 3 samples. The results are presented below. As can be seen, both samples reproduce the core finding from the sample presented in the Main Text, that is, exciton insensitivity to the proximity effects, alongside trion DOCP switching (evidenced by the sharp discontinuity, present only in the trion state, in the colourmaps below) arising from spin-dependent interlayer charge transfer.
We note some minor differences attributable to sample variation. This is to be expected, especially considering that these additional samples were not fabricated using the same bulk CrBr 3 crystal as Sample 1 presented in the Main Text. For instance, the exact shape of the trion DOCP vs B is not identical to Sample 1. This reflects a difference in ferromagnetic domain dynamics. As discussed in the Main Text, the domain formation patterns and sizes depend heavily on a host of factors which will inevitably vary from one flake to the next. We also note that CrBr 3 is very unstable and sensitive to degradation. We cannot be sure how many layers within each flake are still magnetically active, and how that may influence the domain dynamics of the flake as a whole.
The other noticeable difference is that the trion DOCP in Sample 3 does not cross DOCP = 0. This is a linear offset in DOCP which does not influence the hysteresis behaviour. We expect that it arises from a portion of the CrBr 3 flake which is not responsive to B-field, as the magnetization may be pinned by disorder or degradation. In all 3 samples, only the trion state is sensitive to CrBr 3 magnetization fluctuations, while the exciton displays only a shallow gradient owing to the conventional valley Zeeman effect, rather than any proximity interactions. We heat the sample from the base temperature of 4.2 K up to 60 K, under a constant applied magnetic field of B = +200 mT, in order to saturate the CrBr 3 magnetization. The polarization degree is observed to decrease with increasing temperature, and to tend towards zero above the reported Curie temperature of CrBr 3 of ∼ 37 K [7]. We note that the CrBr 3 becomes paramagnetic above the Curie temperature, and so some spin polarization of the electronic bands is likely to persist above ∼ 37 K owing to the influence of the external saturation field. Therefore, no sharp drop in DOCP is necessarily expected exactly at the Curie point. The laser remains in σ + circular polarization for all data presented in the Main Text. Nominally, this polarization state addresses only the +K valley of MoSe 2 , rather than the −K valley, which in principle may introduce a finite valley polarization of photogenerated carriers, excitons or trions, leading to a non-zero circular polarization degree in the eventual photoluminescence. However, the lack of significant retention of non-resonantly optically injected valley polarization in photoluminescence from monolayer MoSe 2 is well known [8,9], most commonly attributed to extremely efficient and rapid depolarization owing to long-range electron-hole exchange interactions, which effectively couple excitons of opposite valley index [10,11].
To confirm that the polarization state of the laser in our experiments has no effect on the results, we repeat our experiments with σ − laser polarization, and measure polarization resolved PL intensity, as presented in the Main Text. The result for both laser polarizations are shown in Suppl. Fig. 7, where it is clear that the PL response is essentially identical regardless of laser polarization, confirming that any non-zero polarization in emission is a result of interaction with the ferromagnet or external B-field, and not the laser itself.  Fig. 2e. The identical result displayed here confirms that the polarization degree dependence is due to interactions with the CrBr3 and external B-field, rather than the laser polarization, which has no effect owing to extremely rapid valley depolarization. This ensures that all optically generated valley polarization is lost before luminescence, if the excitation is sufficiently non-resonant, as is the case here (laser energy is ∼ 300 meV above the exciton energy).

Supplementary Note 6: Peak fitting for extraction of linewidth and valley splitting
In order to extract the linewidths of the exciton and trion states as a function of applied external B-field (as shown in Main Text Fig. 3c), the polarization resolved photoluminescence spectra at each B-field increment were fitted to two Gaussian peak functions, corresponding to X and T. From the fitting we also extract a small valley splitting in both states, attributed to conduction band splitting as discussed in Supplementary Note 1. While the DFT calculations predict a larger proximity induced CB valley splitting of ∼ 2 meV, the smaller experimental value may be due to the fact that interlayer exchange couplings are notoriously difficult to compute accurately, the fact that lattice alignment may be different from the one assumed in the calculation, and the fact that excitonic effects have not been included in the calculation. We also fit the same spectra with 2 peaks: a higher energy symmetric Gaussian function for the neutral exciton, and a lower energy peak for the trion, which is the convolution between a Gaussian and a low energy exponential function, in order to attempt to account for the trion tail. Examples of the alternative fitting are shown below, along with the extracted standard deviation (a direct measure of FWHM in the exponentially modified Gaussian is not a well defined parameter, and so we use standard deviation instead to display how the peak width varies with B) and valley splitting vs B-field. As can be seen, the results are qualitatively in agreement with the double Gaussian peak fitting method shown above. The trion width switches between two values while the exciton remains almost constant (in agreement with Main Text Fig. 3c), while both peaks show a small valley splitting (in agreement with the valley splitting shown on the previous page).  Throughout the Main Text we do not exceed 200 mT applied external B-field strength. This is to ensure that the valley Zeeman effect in MoSe 2 can be neglected, and so any observations must result purely from a magnetic proximity effect. The weak B-field serves only to control the domain dynamics in CrBr 3 , and is too weak to have any appreciable effect directly on band energies in MoSe 2 .
However, in order to investigate whether the CrBr 3 substrate modifies the valley Zeeman response of MoSe 2 , we measure the valley splitting from the sample in applied fields up to B = 8 T. The result is shown below, where we additionally perform a linear fitting for both exciton and trion lineshifts to extract an average rate of shift. We find the exciton to shift at (−0.28 ± 0.01) meV / T while the trion shifts at (−0.11 ± 0.01) meV / T. Variation in rates of shift in MoSe 2 have been reported to be a consequence of doping level [12]. We also expect that CrBr 3 , although saturated above B = 200 mT, may also modify the absolute rates of shift. In general, the exact nature of interplay between band shifts due to the valley Zeeman effect, and those due to interfacial exchange field, are not well understood. In addition to the magneto-photoluminescence measurements presented in the Main Text, the sample was studied using a low temperature wide-field Kerr microscope. Details of the wide field Kerr microscopy are given in the methods section. Despite storage under high vacuum and dark surroundings, significant degradation was observed to have occurred in the CrBr 3 flake in the period between completing the PL measurements and commencing the Kerr measurements. This most likely happened during sample transit in low vacuum, and during periods of sample mounting and unmounting from cryostats in air. Figure 13a shows the sample before the PL measurements, with the CrBr 3 flake intact. Figures 13b and c show the sample before Kerr measurements, after degradation. Figures 13d, e, and f show images of the sample from the Kerr microscope at out of plane magnetic field strengths of B = −80, 0, +80 mT, and a sample temperature of 11 K. A background image, taken at zero applied field, is subtracted from the live CMOS feed, such that any magnetization manifests as brighter or darker regions of the image relative to the non-magnetic surroundings. This occurs as light reflected from magnetized material displays a small rotation of the linear polarization plane, and so is transmitted through the analyzer by a greater or lesser extent. In this way, the spatially resolved brightness of the image denotes the polar Kerr signal.
The brightness change of the remaining CrBr 3 is very clear, becoming darker than the surroundings at negative field and brighter at positive field. The change in brightness of the surrounding substrate is due to Faraday rotation in the microscope objective and cryostat window. Crucially, no domain structures could be resolved in the Kerr microscope at any field strength, rather, the entire flake appears to brighten and darken at the same rate. This indicates that the domains are smaller than the optical resolution of the microscope, ∼ 300 nm. If they were larger, a mixed pattern of bright and dark regions would be visible in Figure 13e at zero field, but no such pattern is observed.