Dr Yaowei Hu

Prof. F Matthieu Picher

Dr Ngoc Minh Tran

Dr Marlène Palluel

Dr Laurentiu Stoleriu

Dr Nathalie Daro

Dr Stephane Mornet

Dr Cristian Enachescu

Dr Eric Freysz
email: eric.freysz@u-bordeaux.fr

Florian Banhart
email: florian.banhart@ipcms.unistra.fr

Dr Guillaume Chastanet
email: guillaume.chastanet@icmcb.cnrs.fr

Banhart

Photo-Thermal Switching of Individual Plasmonically Activated Spin Crossover Nanoparticle Imaged by Ultrafast Transmission Electron Microscopy

Keywords: ultrafast electron microscopy, single nanoparticle detection, spin crossover, timeresolved spectroscopy, nanohybrids

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Introduction

Materials that enable mechanical actuation or sensing under stimulation are highly demanded for many applications. Among such materials, spin crossover (SCO) systems (mainly metallic coordination complexes) [ 1 , 2 , 3 ] are investigated because the stimulus-induced electronic configuration change between low spin (LS) and high spin (HS) states is accompanied by an average local volume change of the coordination polyhedron of the metal site experiencing the spin crossover of 25 %. In the crystals, this local volume change at the metal site propagates through the whole material by intermolecular interactions, producing up to around 10 % expansion or contraction of the whole crystal [START_REF] Guionneau | [END_REF] , thus making them of interest as actuators [5,6,7] but also in electronic devices [8,9,10] or in nanothermometry [11] . The ultrafast spin state switching deserves investigations on the impact of the mechanical changes at the nanoscale especially in opto-electronic [ 12 ,13 ] or memristance [ 14 , 15 ,16 ] devices involving SCO. Indeed, this volume change is likely to create internal strains and therefore alter the SCO through structural fatigability [ 17 , 18 , 19 ] , even if mechanical resilience [ 20 ] has been evidenced. This point is of importance for device durability upon intensive use since a change in the microstructural features can affect the switching properties and consequently the reliability of the device.

Spin crossover can be induced by several means like temperature, pressure and light irradiation. Optical switching of spin crossover materials is of great interest because of the large prospects for optical memories. The optical switching of these materials may be achieved through several processes: light irradiation can promote a photo-reaction on ligands, populate excited states, or heat the sample [21] . Previous studies on the LS→HS (respectively HS→LS) photo-excitation, also referred as Light-Induced Excited Spin State Trapping (LIESST, respectively reverse-LIESST), showed this is a multistep process that allows the population at low temperature of the metastable HS* state [22,23] : (i) The LS to HS* state photoswitching occurs in few femtoseconds, with a local increase of the coordination sphere volume, but at constant unit-cell volume. (ii) These newly formed HS* molecules generate an internal pressure on the LS lattice while the energy deposited by the laser dissipates through lattice heating. These stresses are relaxed through dilation of the lattice mediated by acoustic phonons at time scales determined by the size of the crystals and the speed of sound in the material. (iii) This lattice expansion, which takes place few nanoseconds after the excitation, can promote the switching of additional molecules toward the HS* state, and increases the average crystal temperature as well as its volume. This increase of temperature gets faster as the crystal size is reduced. The whole phenomenon is then highly non-linear since one photon can convert several sites [24] .

As mentioned above, the temperature increase induced by the laser excitation can also be used to trigger the LS to HS spin crossover. Here we study the photo-thermal case, where the process is mainly limited by the diffusion of heat within the sample. This process strongly depends on the size of the particles [24] and therefore only the steps occurring above the nanosecond time scale are at play. It is of strong interest when applied to cooperative SCO materials. In such materials, the strong intermolecular interactions [25] involve an additional energy to overcome, leading to the occurrence of hysteretic SCO features. As a consequence, memory effects are accessible, around room temperature, for such systems. When the irradiation is applied onto the LS state, at a temperature inside the hysteresis, the laser pulse induces a quantitative LS → HS irreversible photoswitching due to the stable character of the HS state in the hysteresis [ 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 ] . This can be applied to generate room temperature opto-electronic devices [13] . The efficiency and speed of such photothermal effect can be highly improved by reducing the size of the particles [32,[START_REF] Salmon | [END_REF] and incorporating nanoheaters into the SCO particles, like gold nanoparticles that produce heat when excited in their surface plasmon resonance (SPR) band [ 37 ] . Indeed, the absorption coefficient of plasmonic nanoparticles is much higher than of the SCO compound. Therefore, excitation of the SPR band should heat the sample more efficiently compared to direct absorption of light by the SCO component. Strong reductions of the photothermal LS to HS switching threshold were observed while incorporating gold NPs onto (SCO@Au architectures) or inside (Au@SCO architectures) the SCO particles [38,39,40,41,42,43] .

For the application of SCO materials in opto-mechanical devices, it is indispensable to know the shape changes of individual nanoparticles and, in particular, the time-dependent response on photon pulses. In most previous studies, the response of large ensembles of SCO particles to laser pulses was studied at a macroscopic scale. However, these measurements did not reveal the behaviour of individual SCO particles and thus didn't allow to detect size changes of the nanocrystals. Indeed, the knowledge of the exact volume change of a single SCO particle, its time scale upon photo-switching and its repeatability is of importance to understand the transport mechanisms, the occurrence of percolation effect (when the increase of volume induces nanoparticles to be in contact) and the fatigue of the devices. A recent study has shown the thermal hysteresis loop and LS to HS switching kinetic at the second time scale on individual nanoparticles through optical methods without providing information on the volume change 44 . In a previous investigation [45,46] , ultrafast TEM has already shown shape changes of SCO particles after laser irradiation, but details of this transformation and the influence of embedded plasmonically active particles remain unexplored. The knowledge of the photoswitching dynamic as function of the laser fluence, the efficiency of the heat transfer, the influence of the substrate bearing the nanoparticles, and other parameters are still to be investigated and remain a challenge. In this paper, time-resolved measurements of the switching dynamics are carried out on individual pure and gold-filled SCO nanoparticles by ultrafast electron microscopy. The expansion is measured under laser pulses of different energies at the nanosecond to the millisecond timescale. This study is performed on single nanoparticles and compared with an optical characterization of an assembly of particles. Such an investigation is of importance as it can be extended to breathing or mechanically active materials (photochromic materials, batteries upon charge and discharge, actuators, nanomachines…).

Results and Discussion

Studied systems

In this study, we focus our attention on triazole-based SCO nanoparticles which are widely studied for implementation in devices [5,8,13] . They are built from a 1D coordination polymer made of Fe(II) metal ionic switchable sites linked together by bridging triazole ligands to form linear chains separated by counter-anions [47] . The [Fe(Htrz)2trz](BF4) is probably the most studied compound since large hysteresis loops can be observed above room temperature [48,49] and at the nanoscale [START_REF] Salmon | [END_REF]50] . The photo-thermal effect on 115 nm nanoparticles of this compound has been reported to occur within 12 ns by optical measurements performed on an assembly of particles [32] .

Gold nanorods were embedded into these SCO nanoparticles to promote local heating through light-activation of the surface plasmon resonance [41] (Figure 1a and supporting information for the protocol). An assembly of such nanohybrids exhibits a wide thermal hysteresis of 34 K, centered at 360 K with switching temperatures in the heating mode T1/2↑ = 378 K and in cooling mode T1/2↓ = 344 K (Figure 1b). Moreover, on the same synthetic batch, the SCO nanoparticles can host one or several gold nanorods. (black, for an ensemble of particles) and length of a single pure SCO particle along its long axis (orange/blue). The length measurements were taken in the heating stage of a conventional TEM with continuous electron beam. The accuracy of the length measurements is approximately 1 nm. A similar hysteresis taken from another SCO particle and a Au@SCO particle is shown in fig. SI2.

Static temperature effects

In previous studies, the crystal structure of the one-dimensional coordination polymer [Fe(Htrz)2trz](BF4) showed changes in the unit-cell volume by 11 % with a pronounced elongation of the b axis (axis along which the one dimensional polymer grows) of 0.46 Å from LS to HS, corresponding to an expansion of 6 % [51] . For a 400 nm nanoparticle, this should correspond to almost 24 nm of stretching.

In the first set of experiments, the static lengths of pure SCO particles are measured in a wide temperature range. This is carried out at thermal equilibrium by continuous heating of SCO particles in a high-temperature TEM specimen stage. The length changes are measured by taking conventional images with a continuous electron beam. Figure 1b shows TEM measurements of the expansion of a single pure SCO particle along its long axis together with the magnetic susceptibility of a macroscopic amount of SCO particles as a function of temperature. The same measurements for a Au@SCO particle are shown in Figure SI2. Both measurements show a hysteresis upon heating and cooling. While the hysteresis of the first cycle taken from a new specimen is always approximately 10 K wider, all following curves were precisely reproducible upon repeated heating-cooling cycles, in agreement with the known run-in process for this compound [17,48,49,50] . The hysteresis is approximately 37 K wide for both measurements; the switching temperatures of the length changes are in the warming mode T1/2↑ = 371 K and in the cooling mode T1/2↓ = 334 K. The switching temperatures measured through the length change are slightly shifted towards low temperature compared to the susceptibility measurements. This may arise from various sources, the most probable ones being the different temperature calibrations (+/-5° in the UTEM heating stage) and a slight length difference (330 nm for the magnetic measurement versus 400 nm for UTEM measurements for about the same width). The hysteresis from a Au@SCO particle (Fig. SI2) is qualitatively similar but slightly narrower, most likely due to the non-expanding Au rods that replace some SCO material. The slightly varying length within the hysteresis appears in all measurements but isn't understood in detail yet. This measurement clearly shows that upon warming, the nanoparticle expands from 400 nm to 416 nm. This corresponds to 5 % of elongation, very close to the 6 % increase in lattice parameter along the c-axis recorded by X-ray crystallography. The TEM observation shows that the widening is less pronounced than the lengthening, in accordance with the expected anisotropic expansion (which is larger for the b, than for a (+ 0.9%) and c (+ 4%) unit-cell parameters). These results confirm the observation of the bistable effect within the hysteresis loop on one individual nanoparticle of 330 nm, in agreement with the report on 200 nm particles of the same compound [46] recorded by surface plasmon resonance microscopy and on 600 nm particles of another SCO compound [45] recorded by electron diffraction.

Time-resolved TEM studies on individual nanoparticles

The optical absorption of SCO at the wavelength of the laser pulses in this study (1064 nm) is very low and so allows us to study the heating effect of the embedded Au rods. Therefore, pure SCO particles are mainly heated from the carbon substrate in our setup [46] and Au@SCO from the embedded Au rods [41] . In the first case, the copper grid holding the graphene support film on which the SCO particles are deposited absorbs the heat which then diffuses through the graphene layer to the SCO particles. In the second case, the encapsulated Au nanorods heat the SCO particles much more efficiently from inside. No static specimen heating was applied in the time-resolved studies.

Figure 2 shows UTEM images of a SCO particle (see Methods section and figure SI1) taken with nanosecond electron pulses before and 500 ns after an IR pulse (1064 nm, 7 ns, 45 mJ cm - ²). As seen from the contrast profiles along the long axis of the particle, the length has increased by 20 nm which corresponds to 5% of its total length. A width change of 3 nm corresponding to 2% of the total width was measured. This result is in accordance with the expected expansion anisotropy of the SCO particles. Here the b axis corresponds to the long axis of the particles. Figure 3a shows the expansion of a pure (unfilled) SCO particle (LS length 390 nm) as a function of time after excitation with laser pulses of different energies. At 8 µJ/pulse (45 mJ cm -²/pulse), the particle reaches a maximum expansion of 1.7% after approximately 15 ns and then contracts again to its original length within 100 ns. At 15 and 20 µJ/pulse (85 and 115 mJ cm -²/pulse, respectively), the particle starts expanding at the same time scale (15 ns) and with the same expansion speed as with 8 µJ pulses but the expansion process continues until at 100 ns the maximum length is reached. With 20 µJ pulses, a saturation expansion of almost 5 % is reached. At high pulse energies (here visible at 15 and 20 µJ/pulse), the particle appears unstable with oscillating length which stabilizes after a few hundred nanoseconds. The length then remains temporarily unchanged until the particle contracts after 5 µs (15 µJ) or 100 µs (20 µJ). The initial length is reached again after 0.1 -1 ms. The overall behavior is reproducible from one nanoparticle to another, with minimal variations in the expansion process due to different substrate-particle interfaces. No effects of fatigue were observed when the measurements were repeated (each length measurement consists of 1200 expansion-contraction cycles). 370x 160 nm) filled with one gold nanorod and, a SCO particle (390 x 160 nm) with three gold nanorods. The laser power was always 45 mJ cm -²/pulse. The insets show the respective particles where the measurements were undertaken. Another example is shown in Figure SI3.

Figure 3b shows the behaviour of a SCO particle encapsulating three Au nanorods. For pulse energies where no saturation in pure SCO is reached, less laser power is needed to achieve the same length changes as in unfilled SCO. For example, 8 µJ pulses lead to 1.7% expansion in pure SCO and 3.5% expansion when three Au rods are encapsulated. Furthermore, the maximum expansion at higher pulse energies is reached earlier than in pure SCO. Whereas the maximum expansion at 20 µJ pulses in pure SCO appears after approximately 100 ns (Figure 3a), the same expansion is already reached after 25 ns if the SCO particle encapsulates three gold nanorods (Figure 3b).

The effect of different numbers of Au particles inside one SCO particle on the expansion is compared in Figure 3c for a pure SCO particle, a SCO particle with one Au nanorod and a particle with three Au nanorods, all exposed to the same pulse energy (8 µJ). The more Au rods are embedded, the higher and faster the expansion is up to saturation.

Time-resolved study on assembly of nanoparticles by optical spectroscopy

The optical absorbance of a macroscopic sample in the 400 to 800 nm range (Figure SI4) is steadily decreasing as the wavelength increases and it is higher in the LS state. The absorbance of the Au@SCO exhibits two absorption peaks. The first one around 520 nm is related to the transverse plasmon resonance of the Au nanorods. The second marked one appears in the 650 to 850 nm spectral range. It is associated to the longitudinal plasmon resonance of the Au nanorods. The maximum of the absorption of Au@SCO is centered around 740 nm when the SCO compound is in the LS state and is shifted towards 705 nm when the SCO compound is brought to the HS state. We have previously shown that this shift is related to change of the index of refraction of the SCO compound upon the LS to HS transition [41] .

Figure 4a displays the typical increase of the transmission of an Au@SCO at T = 333K upon excitation by a nanosecond laser pulse at 520 nm and with a fluence of ∼34 mJ cm -2 (green dots). At this temperature the sample is just below the descending branch of the thermal hysteresis loop. Upon excitation, the transmission rapidly increases within few tens of nanoseconds, remains almost constant during almost 20 µs and then decreases and recovers its initial state after about 1 ms. The increase of the transmission T(t) is fitted using the following function T(t) = T (1-exp(-t/ts)) where T and ts are the change in transmission and switching time, respectively. Similarly after the plateau, the relaxation of the transmission of the sample towards its initial value is fitted using the following decay law T(t) = T exp(-t/tr) where tr is The same experiment was repeated with a different wavelength of 470 nm, with and without embedded gold nanoparticles (Figure 4b). The difference between SCO and Au@SCO is marked. The change in HS ~ 5% for the SCO sample is lesser compared to the change recorded for Au@SCO (HS ~10 %). Hence, the relaxation time recorded for the SCO (tr = 12.5 ± 1 µs) is faster than the one for the Au@SCO (tr = 65 ± 5 µs). Let us notice that upon irradiation in the SPR band we readily damaged the sample after few nanosecond laser shots, even with the lowest accessible photo-switching fluence (~5 mJ cm -2 ), due to a too intense local heating of the SCO at the interface with the gold nanorods.

Figure 4: (a) Photoswitching kinetics of Au@SCO upon excitation by nanosecond laser pulses at 520 nm for various laser fluences. (b) Photoswitching kinetics of Au@SCO and SCO upon

excitation by nanosecond laser pulse centered at 470 nm with a 10 mJ cm -2 laser fluence. The solid lines are fits of HS(t)using the equations described in the text.

Discussion

We have previously proposed a possible mechanism accounting for the LS to HS state photoswitching of a macroscopic SCO sample, displaying a broad thermal hysteresis loop, upon excitation by nanosecond laser pulses [32,30] . The proposed scenario makes it possible to describe the evolution of the sample in the (T, HS) phase diagram upon excitation as displayed in Figure 5. The sample is initially in the LS state (HS = 0) near the ascending branch of the thermal hysteresis loop (point A). Upon excitation and absorption of the laser light by the sample or the substrate, the temperature rapidly increases within the sample while most of the SCO remains in the LS state (Point B). When the fluence is high enough, the SCO sample is over heated and the HS fraction rapidly growths. Then as heat diffuses, the average temperature lowers due to the endothermic nature of the LS to HS transition process. This process stops when the temperature of the sample reached the ascending branch of the thermal hysteresis loop (Point C). Beyond this temperature, the HS state fraction remains constant whereas the temperature further decreases. This lasts as long as the temperature of the sample has not reached the descending branch of the thermal hysteresis loop (Point D). Below that latter temperature both the LS fraction and temperature relax back to their initial position within the (T, LS) phase diagram.

Figure 5: Evolution of the sample within the thermal hysteresis loop under a heat pulse.

The time-resolved optical measurements follow this model quite well. In such experiments, light is directly absorbed by the sample (made of an ensemble of nanoparticles). The temperature increase inside such sample, T, can be written as:

T = (lp) Ep /(e  Cp) (1)
where , (lp ), Ep , e, Cp are the density of SCO particles, absorption at the pump wavelength lp, the laser fluence, the thickness, and the heat capacity of the sample, respectively. The latter physical parameters are known for SCO samples. However, it is noteworthy that due to the sample preparation and the random packing of the nanoparticles, the actual density  of our sample is somehow uncertain. We also suppose that, apart from the absorption, the physical parameters of SCO and Au@SCO are almost similar. Finally, we consider that 60 % of laser fluence Ep is absorbed by the 4 µm thick sample (composed of SCO nanoparticles, see experimental section) with  = 0.9 g cm -3 , Cp =1.0 J g -1 K -1 and e = 4 µm. As a general consequence, the speed at which HS fraction grows and its amount depend on the amplitude of the overheating T: the higher the laser fluence, the higher T and the faster the switching time.

For Ep = 34 mJ cm -2 pump fluence, one readily finds that T~56 K. According to this estimation and knowing the initial temperature of the sample T0 = 333 K, the temperature of the sample upon excitation is T0 + T = 389 K. Thus, based on the (T, HS) phase diagram (Figure 5), the sample remains in the HS state when it cools down right after the laser pulse. Hence, upon excitation, the HS fraction increases. After this rapid increase HS remains almost constant for ~20 µs and then the sample relaxes back to its initial LS state. If the laser fluence is reduced to 20 mJ cm -2 , T ~ 33 K (from ( 1)) and the sample is just brought about the ascending branch of the thermal hysteresis loop. The increase in the HS fraction HS is lesser and happens on a longer time scale. In Figure 4a for laser fluence of 9.5 mJ cm -2 the increase of HS is small (~0.05 %) but, according to the calculated increase in temperature, it is higher than expected. This may result from the fact that our simple computation neglects firstly that, due to the high absorption of the sample (thickness 4 µm), particles at the top of the samples absorb more strongly than those at the bottom and secondly, within the excited area part of the Au@SCO nanohybrids may contain more than one gold nanorods (as seen on the TEM pictures, Figure 2).

Equation 1 also indicates that T and therefore HS depends on the absorption coefficient of the sample. If one tunes the laser wavelength towards a spectral region where the absorption of the samples is higher, the increase in temperature should be higher and HS should be increased.

This is indeed what we observe as we tune the laser wavelength from 520 nm to 470 nm, the laser fluence for which a sensible change in HS is largely reduced (see Figure SI4). Moreover, as the Au@SCO NPs have a stronger absorption than SCO NPs, due to the inclusion of Au nanorods, HS is higher in Au@SCO NPs upon excitation compared to SCO NPs. All these conclusions are in agreement with the data presented in Figure 4.

Compared to the optical measurements, the heating processes during the UTEM measurements are different. Indeed, based on an earlier analysis of TEM measurements [45,46] ,

we have to assume that heating of the pure SCO particles occurs mainly through the substrate.

This is due to the low absorption of SCO at the used excitation wavelength (1064 nm) for these experiments. Heating of the substrate occurs by absorption of the Cu grid and to a smaller extent by the graphene film. The high thermal conductivity of the graphene film ensures rapid heat conduction to the particles. The heating of the pure SCO particles is therefore mainly limited by the interfacial area with the graphene which is difficult to control and varies slightly between different particles. This finally determines the timescale of heating and expansion of SCO.

Indeed, a variation in the onset of heating between different samples was observed in some cases. However, SCO particles encapsulating Au rods always show rapid heating at all laser powers within approximately 20 ns (Figures 3b andc). This is understandable due to the more efficient heat transfer from the Au rods that are embedded in the particles and share a large interface with SCO. This demonstrates that the plasmonic heating effect is indispensable if rapid switching of devices at low energies of the laser pulses is desired.

The temporal development of the length of pure (unfilled) SCO particles after laser pulses shows that the onset of expansion occurs approximately 10 ns after the laser pulse, irrespective of the pulse energy. From the left slope of the curves, an averaged expansion speed for these particles (390 nm) of approximately 0.4 m/s is obtained. Towards the maximum expansion of 4.5-5%, which is reached at higher pulse energies, the average expansion speed is slightly less.

The comparison with gold-filled particles (Figure 3b) shows that the expansion for lowenergy pulses (8 or 15 µJ) is higher at the same pulse energies if gold is encapsulated. Since the major contribution from the Au particles is plasmonic heating, the influence on the expansion of SCO has to be of purely thermal nature. It is obvious that an increasing number of Au particles has more influence on heating (Figure 3c). Due to the more efficient heat transfer, the expansion is faster than if heat has to be transported via the substrate. However, once the particles expand, the expansion speed is almost the same in pure SCO and Au@SCO in the lower expansion regime. The maximum expansion of Au@SCO is reached here after 20-30 ns (Figure 3), while more than 100 ns are needed for pure SCO.

For high excitation energies, the photoswitching time scale, the plateau reached after the expansion and the relaxation rates to the initial LS state agree well with the model proposed from the time-resolved optical measurements (equation 1 and Figure 5). However, this is not the case for lower excitation energies and this probably comes from different heating mechanisms.

Equation 1 holds for an ensemble or a single SCO or Au@SCO NPs exhibiting a spin state transition associated with a thermal hysteresis loop. In order to discuss the thermal evolution of the system, we use a mechanoelastic model, which was previously applied to simulate the dynamic phenomena after ultrafast photoexcitation [24] . In this model, the molecules, represented as rigid spheres, are situated in open boundary lattices and the elastic interactions are mimicked by springs linking a molecule to its closest neighbours. When a molecule changes its state, its volume varies, which results in an elongation or a compression first of its closest springs, which determines the change in position of molecules and a propagation of the initial perturbation.

The initial evolution of the system was simulated (a) by randomly creating an additional given number of HS molecules (50% in our case), whose temperature is increased by T = 300 K to mimic a direct absorption of light by the compound; (b) by the heat diffusion from a hot substrate towards the sample, or (c) by heat diffusion from 5 hot gold particles inside the lattice.

The dynamics of the "temperatures" of the system is then phenomenologically described for every molecule i by the following differential equation ( 2).

( ) (

)

i ii L ij edge B dT T T T T dt  = - - - - (2)
where i T is the temperature corresponding to the i molecule, ij T is the average temperature of all neighbors of the i molecule, is the diffusion coefficient between molecules, B T is the thermal bath temperature, i edge T is the temperature of molecules in contact with the thermal bath, and  is the heat transfer coefficient to the bath.

In Figure 6 we present the evolution of the temperatures in the system for all three cases presented above, using equation ( 2), for a system composed of 10,000 molecules. In all these temperature maps, one can clearly see the inhomogeneity of the temperature inside the sample.

For initially randomly switched molecules (Figure 6 left panel), as the nucleation centers are close to each other, the temperatures of all molecules in the system vary immediately after the first step following the switching as the heat propagates in all directions (from a to c). A hot substrate (Figure 6 middle panel) divides the sample in two parts, separated by a moving interface (from d to e): one hot and the other one maintaining its initial temperature for a longer time. Finally, the five gold particles (Figure 6 right panel) act as nucleation centers, changing step by step the temperatures of neighboring sites, while further sites keep their temperature for a while (from f to g). This creates large variety of behaviors based on out-of-equilibrium inhomogeneous heating. The lower the switched HS fraction, the higher the inhomogeneity of temperatures inside the lattice. This is most likely at the origin of the difference between the optical measurements where a plateau in the photoswitching profile is observed and the UTEM measurements where this plateau only appears above 60 % of photoswitching. It indicates that uniform heating throughout the whole particle is necessary to lead to the appearance of a hysteresis. Short and localized heating, as it appears when heating occurs from the substrate or from few localized heating centers such as the metal particles, does not lead to collaborative expansion of the whole particle and therefore not to a hysteresis when the saturation expansion is not reached.

Conclusion

In this study, we have investigated the behavior of spin crossover nanoparticles, embedding from zero to seven plasmonically active gold nanorods. TEM measurements performed on single nanoparticles have evidenced the cooperative character of the SCO of such nanocrystals by probing the length change upon temperature variations. This strongly confirms that the breathing of the crystallographic unit cell is reflected at the nanoscopic scale. Moreover, by measuring this length change using UTEM allowed us to probe the photoswitching dynamics of a single nanoparticle and how it is influenced by the presence of gold nanorods. An increasing number of gold nanorods fastens the photoswitching rate from more than 100 ns without any gold particle to 10 ns with five gold nanorods. This study confirms that heating by laser pulses and heat diffusion inside the particle are the key factors controlling this photoswitching. We evidenced here the efficient plasmonic heating at the single nanoparticle level. We went a step further in comparison to most of the current studies on nanoparticles by studying the behavior of individual nano-objects. Moreover, this study also shows that the time-resolved optical study performed on an assembly of nanoparticles is in accordance with the UTEM measurements on individual nanoparticles, leading to similar photoswitching time scales, despite the differences in the heat absorption processes. This is crucial when fast opto-mechanical switches have to be realized with several nanoparticles integrated into a device. It is obvious that plasmonic heating by embedded gold particles leads to the most efficient and fastest switching behaviour and would therefore be adequate for building an optimized and reliable device. A characterization technique combining high spatial and temporal resolution such as UTEM is indispensable when the behaviour of individual SCO nanocrystals needs to be monitored. The combination with optical measurements that average over many particles, gives a comprehensive picture of the processes at different scales. Such an investigation is not only of importance for the understanding of spin crossover phenomena but also for applications of breathing or mechanically active materials (photochromic materials, batteries upon charge and discharge, actuators, nanomachines…).

Experimental Section/Methods

Ultrafast transmission electron microscopy (UTEM, see Figure SI1) allowed us to observe the behavior of individual particles. In a pump-probe approach [52,53] , a pulsed nanosecond laser (1064 nm, 7 ns pulse duration) is aligned onto a spot (150 µm) on the specimen grid in the column of the UTEM. Although the absorption of SCO at 1064 nm is very low, length changes of pure SCO particles up to saturation could be achieved through heat conduction from the graphene substrate [46] by pulse energies in the range 4 -40 µJ/pulse, corresponding to 20-200 mJ/cm 2 per pulse. Each IR pulse is followed after an adjustable delay by a 7 ns electron pulse [52] that served for imaging the particles so that their length could be measured as a function of time. The temporal resolution in this configuration is approximately 10 ns; however, relative length changes can be recorded with slightly smaller steps. Due to repulsive electron-electron interactions within the intense pulses which cause a loss of coherence and increased energy width, the intensity of the electron pulses had to be limited [52,54] . The pump-probe delay is adjusted by synchronizing the two lasers with an electronic delay unit. Since the length changes of the SCO particles are reversible, the experiment is carried out repeatedly for each measurement in a stroboscopic approach with a repetition frequency of 20 Hz. For recording an image, 1200 pulses are summed up within the typical acquisition time of 60 s. All the timeresolved measurements were performed at room temperature.

The SCO particles are deposited on monolayer graphene grids (Agar Ltd.) for TEM studies.

To ensure that no undesired radiation damage of SCO by the electron beam occurred, some particles were exposed deliberately to electron irradiation. After a dose exceeding approximately 1000 electrons/nm 2 , the thermally induced expansion isn't observable anymore.

Therefore, all pump-probe experiments are carried out at much lower electron dose (typically 0.02 electrons/nm 2 per pulse, that is 24 electrons/nm 2 for the recording of an image), where electron beam effects are negligible. The probe beam transmitted by the sample is focused on the entrance slit of a monochromator to transmit the probe wavelength. The signal transmitted by the monochromator is further filtered by a laser-line filter and focused on a high-speed silicon photodiode connected to a 1

GHz numerical oscilloscope. The oscilloscope is triggered by the nanosecond laser pulse. It is worth mentioning the central wavelength of the laser diode is centered about the maximum absorption of the Au@SCO sample in the LS state. There is a sensible change in absorption when both SCO and Au@SCO samples are switched from the LS to the HS state 41 .

The way we estimated γHS in Figure 4 is as follows. We increase the temperature of the oven in which the sample is inserted and we recorded the amplitude of the signal recorded by the photodiode placed after the spectrometer when the sample is set in the LS and HS state. We consider the change in amplitude ∆V is associated to HS = 1. Then we consider that upon laser excitation the change in γHS(t) scales with respect to the signal V(t) recorded by the photodiode: γHS(t) =∆V(t)/∆V.

Figure 1 :

 1 Figure 1: a) TEM image of the Au@SCO nanohybrids with various amounts of gold nanorods

Figure 2 :

 2 Figure 2: SCO particle encapsulating two Au nanorods before (a) and 500 ns after (b) an IR

Figure 3 :

 3 Figure 3: Expansion of an SCO particle as a function of time after an IR pulse. (a) shows a

  the relaxation time. As displayed in Figures 4a the data are well fitted considering ts = 18 ± 2 ns and tr = 410 ±50 µs. Besides, the change in amplitude of the transmission indicates that about 45% (i.e. the HS fraction HS ≈ 45%) of the sample is switched from the LS to HS state (see experimental section). The same experiment was performed at 9.5 mJ cm -2 and 19 mJ cm -2 laser fluences, as displayed in Figure 4a (blue and red dots). For the lowest laser fluence the switching time increases to ~ 250 ± 50 ns, the plateau lasts almost the same while the relaxation rate decreases: tr = 120 ± 20 µs and 70 ± 10 µs for 19 mJ cm -2 and 9.5 mJ cm -2 laser fluences, respectively.

Figure 6 :

 6 Figure 6: Evolution of temperatures inside the system upon heat diffusion from half of the

  Time-resolved optical measurements: For the time resolved absorption measurements of an assembly of particles, the samples are prepared as follows. A powder of either SCO or Au@SCO nanoparticles is dispersed in ethanol, stirred, drop-casted and dried on a microscope glass slide. Afterwards, the deposited layer with thickness about 4.0±2.0μm is covered by another microscope glass slide. The set-up we used to measure the time resolved absorption of the sample is displayed in FigureSI5. The SCO and Au@SCO samples are inserted in an oven whose temperature can be controlled (±0.1•C) in the range from 300 K to 440 K. It is then excited by the nanosecond pulse slightly focused on beam spot Φ of ∼1.5 mm in diameter. The nanosecond pulses are provided by an optical parametric oscillator (Continuum OPO Panther EX). The latter delivers nanosecond pulses tunable from 410 nm to 2700 nm. The pulse duration and energy are about ∼6 ns and 40 mJ, respectively. The transmission of the sample is probed using a continuous wave (CW) laser diode delivering few milliwatts at 785 nm wavelength.

Acknowledgements

CNRS, University of Bordeaux, Nouvelle Aquitaine Region and the ANR program (HEROES ANR-17-CE09-0010-01) are acknowledged for fundings. Funding by the EQUIPEX program of the Agence Nationale de Recherche, contract ANR-11-EQPX-0041 (project UTEM) and the network METSA (Réseau national de plateformes en microscopie électronique en transmission et sonde atomique) is gratefully acknowledged. Y. Hu acknowledges support from the China Scholarship Council. G.C. would like to thank Guillaume Rogez to have envisioned the possibility of UTEM in the study of SCO nanohybrids.

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Supporting Information ((delete if not applicable))

Supporting Information is available from the Wiley Online Library or from the author.


J Klein
email: jpklein@mit.edu

A Hötger

M Florian

A Steinhoff

A Delhomme

T Taniguchi

K Watanabe

F Jahnke

A W Holleitner

M Potemski

C Faugeras

J J Finley
email: finley@wsi.tum.de

A V Stier

Controlling exciton many-body states by the electric-field effect in monolayer MoS 2

We report magneto-optical spectroscopy of gated monolayer MoS2 in high magnetic fields up to 28 T and obtain new insights on the many-body interaction of neutral and charged excitons with the resident charges of distinct spin and valley texture. For neutral excitons at low electron doping, we observe a nonlinear valley Zeeman shift due to dipolar spin-interactions that depends sensitively on the local carrier concentration. As the Fermi energy increases to dominate over the other relevant energy scales in the system, the magneto-optical response depends on the occupation of the fully spin-polarized Landau levels in both K/K valleys. This manifests itself in a many-body state. Our experiments demonstrate that the exciton in monolayer semiconductors is only a single particle boson close to charge neutrality. We find that away from charge neutrality it smoothly transitions into polaronic states with a distinct spin-valley flavour that is defined by the Landau level quantized spin and valley texture.

A bosonic exciton immersed in a Fermi gas is an example of the impurity-model that can be understood in terms of a polaronic picture -dressing of the exciton by collective excitations of the many-body environment [1][2][3][4][5][6][7][8]. The role of electron-electron interactions on the properties of two-dimensional (2D) electron and hole gases has been extensively studied in GaAs [9][10][11], AlAs [12,13] and recently also in graphene [14,[START_REF] Young | Spin and valley quantum hall ferromagnetism in graphene[END_REF]. Typically the spin-and valley susceptibility of the system is found to be enhanced by interactions. Signatures of collective phenomena are particularly pronounced in 2D materials [START_REF] Huang | Layer-dependent ferromagnetism in a van der waals crystal down to the monolayer limit[END_REF][START_REF] Tang | Simulation of hubbard model physics in WSe2/WS2 moiré superlattices[END_REF][START_REF] Shimazaki | Strongly correlated electrons and hybrid excitons in a moiré heterostructure[END_REF][START_REF] Regan | Mott and generalized wigner crystal states in WSe2/WS2 moiré superlattices[END_REF] such as semiconducting transition metal dichalcogenide (TMDC) MoS 2 owing to strong Coulomb interaction [START_REF] Chernikov | Exciton binding energy and nonhydrogenic rydberg series in monolayer WS2[END_REF][START_REF] Stier | Magnetooptics of exciton rydberg states in a monolayer semiconductor[END_REF]. Elegant control knobs, such as static electric-field effects have been used for tuning manybody interactions for the study of Fermi-polarons [START_REF] Sidler | Fermi polaron-polaritons in chargetunable atomically thin semiconductors[END_REF][START_REF] Ravets | Polaron polaritons in the integer and fractional quantum hall regimes[END_REF][START_REF] Wang | Strongly interactionenhanced valley magnetic response in monolayer WSe2[END_REF][START_REF] Smoleński | Interaction-induced shubnikov-de haas oscillations in optical conductivity of monolayer MoSe2[END_REF][START_REF] Tan | Interacting polaron-polaritons[END_REF][START_REF] Liu | Landau-quantized excitonic absorption and luminescence in a monolayer valley semiconductor[END_REF]. Specifically in the very low density regime, valley dichroism in the presence of a magnetic field generates a unique local spin and valley texture of excess charge carriers. Here, only few spin-polarized Landau levels (LLs) with low filling factors are involved in shaping the many-body environment. Therefore, the interaction of excitons immersed in the fermionic bath depends on those LLs. Quantum transport studies are intrinsically restricted to the high density regime where LLs with high filling factors are typically probed [START_REF] Pisoni | Interactions and magnetotransport through spin-valley coupled landau levels in monolayer MoS2[END_REF][START_REF] Movva | Density-dependent quantum hall states and zeeman splitting in monolayer and bilayer WSe2[END_REF][START_REF] Xu | Odd-integer quantum hall states and giant spin susceptibility in p-type few-layer WSe2[END_REF][START_REF] Gustafsson | Ambipolar landau levels and strong band-selective carrier interactions in monolayer WSe2[END_REF][START_REF] Larentis | Large effective mass and interactionenhanced zeeman splitting of k-valley electrons in MoSe2[END_REF][START_REF] Fallahazad | Shubnikov-de haas oscillations of high-mobility holes in monolayer and bilayer WSe2: Landau level degeneracy, effective mass, and negative compressibility[END_REF][START_REF] Lin | Determining interaction enhanced valley susceptibility in spin-valley-locked MoS2[END_REF][START_REF] Li | Spontaneous valley polarization of interacting carriers in a monolayer semiconductor[END_REF] whereas optical measurements facilitate access to the entire density regime [START_REF] Smoleński | Interaction-induced shubnikov-de haas oscillations in optical conductivity of monolayer MoSe2[END_REF][START_REF] Liu | Landau-quantized excitonic absorption and luminescence in a monolayer valley semiconductor[END_REF]. The large variation of reported gfactors for excitons in MoS 2 [START_REF] Stier | Exciton diamagnetic shifts and valley zeeman effects in monolayer WS2 and MoS2 to 65 tesla[END_REF][START_REF] Mitioglu | Magnetoexcitons in large area CVD-grown monolayer MoS2 and MoSe2 on sapphire[END_REF][START_REF] Cadiz | Excitonic linewidth approaching the homogeneous limit in MoS2-based van der waals heterostructures[END_REF][START_REF] Goryca | Revealing exciton masses and dielectric properties of monolayer semiconductors with high magnetic fields[END_REF][START_REF] Jadczak | Fine structure of negatively charged and neutral excitons in monolayer MoS[END_REF] and other 2D materials [START_REF] Smoleński | Interaction-induced shubnikov-de haas oscillations in optical conductivity of monolayer MoSe2[END_REF][START_REF] Liu | Landau-quantized excitonic absorption and luminescence in a monolayer valley semiconductor[END_REF][START_REF] Stier | Exciton diamagnetic shifts and valley zeeman effects in monolayer WS2 and MoS2 to 65 tesla[END_REF][START_REF] Goryca | Revealing exciton masses and dielectric properties of monolayer semiconductors with high magnetic fields[END_REF][START_REF] Aivazian | Magnetic control of valley pseudospin in monolayer WSe2[END_REF][START_REF] Srivastava | Valley zeeman effect in elementary optical excitations of monolayer WSe2[END_REF][START_REF] Macneill | Breaking of valley degeneracy by magnetic field in monolayer MoSe2[END_REF][START_REF] Li | Valley splitting and polarization by the zeeman effect in monolayer MoSe2[END_REF][START_REF] Wang | Valley-and spinpolarized landau levels in monolayer WSe2[END_REF][START_REF] Stier | Probing the influence of dielectric environment on excitons in monolayer WSe2: Insight from high magnetic fields[END_REF][START_REF] Lyons | The valley zeeman effect in inter-and intra-valley trions in monolayer WSe2[END_REF][START_REF] Wang | Observation of quantized exciton energies in monolayer WSe2 under a strong magnetic field[END_REF] with arbitrary carrier concentration further highlights the necessity for studying valley Zeeman shifts with tunable and controlled carrier densities in 2D semiconductors.

In this Letter, we fully address the variation of gfactors in the literature which is a direct consequence of the many-body interaction with the fermionic bath. We control the carrier concentration n in a dual-gate fieldeffect device (Fig. 1(a)) and study the magneto-optical response in high magnetic fields up to B = 28 T. Simultaneous control of n and B allows us to prepare a unique global spin texture originating from the quantization of excess carriers in fully spin-polarized LLs. We study the magnetic field dependence of neutral and charged excitons (Fermi-polarons) which encode the evolution of the total magnetic energy including carrier spin, valley magnetic moment (Berry phase) and cyclotron phenomena arising from quantization of electrons and/or holes into discrete LLs [START_REF] Smoleński | Interaction-induced shubnikov-de haas oscillations in optical conductivity of monolayer MoSe2[END_REF][START_REF] Liu | Landau-quantized excitonic absorption and luminescence in a monolayer valley semiconductor[END_REF][START_REF] Aivazian | Magnetic control of valley pseudospin in monolayer WSe2[END_REF][START_REF] Srivastava | Valley zeeman effect in elementary optical excitations of monolayer WSe2[END_REF][START_REF] Macneill | Breaking of valley degeneracy by magnetic field in monolayer MoSe2[END_REF][START_REF] Li | Valley splitting and polarization by the zeeman effect in monolayer MoSe2[END_REF][START_REF] Wang | Valley-and spinpolarized landau levels in monolayer WSe2[END_REF][START_REF] Wang | Observation of quantized exciton energies in monolayer WSe2 under a strong magnetic field[END_REF]. From our measurements, we directly observe that shape and magnitude of the valley Zeeman shift ∆E V Z of excitons very sensitively depend on the spin and valley texture. Our results suggest that the interaction of the exciton with the Fermi-bath at low densities is driven by dipolar spin-interaction which markedly differs from previous observations that have not taken into account the unique LL quantization in 2D TMDCs [START_REF] Sidler | Fermi polaron-polaritons in chargetunable atomically thin semiconductors[END_REF][START_REF] Ravets | Polaron polaritons in the integer and fractional quantum hall regimes[END_REF][START_REF] Wang | Strongly interactionenhanced valley magnetic response in monolayer WSe2[END_REF][START_REF] Smoleński | Interaction-induced shubnikov-de haas oscillations in optical conductivity of monolayer MoSe2[END_REF][START_REF] Tan | Interacting polaron-polaritons[END_REF][START_REF] Liu | Landau-quantized excitonic absorption and luminescence in a monolayer valley semiconductor[END_REF].

We excite the sample with unpolarized light at λ = 514 nm and an excitation power of 30 µW and detect σ -polarized PL at T = 5 K. Due to the robust optical selection rules in monolayer TMDCs, we only probe the emission from excitonic recombination in the K valley for positive and negative polarities of the B-field [START_REF] Xiao | Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides[END_REF]. rically tune the voltage applied to the top and bottom gates, which effectively counteracts Fermi-level pinning due to the reduction of band tail states [START_REF] Volmer | How photoinduced gate screening and leakage currents dynamically change the fermi level in 2d materials[END_REF][START_REF] Klein | Impact of substrate induced band tail states on the electronic and optical properties of MoS2[END_REF]. This allows us to observe the X + transition in the p-charged regime I for the first time in MoS 2 . The data reveal narrow emission lines of the X 0 due to the hBN encapsulation [START_REF] Cadiz | Excitonic linewidth approaching the homogeneous limit in MoS2-based van der waals heterostructures[END_REF][START_REF] Wierzbowski | Direct exciton emission from atomically thin transition metal dichalcogenide heterostructures near the lifetime limit[END_REF][START_REF] Florian | The dielectric impact of layer distances on exciton and trion binding energies in van der waals heterostructures[END_REF] and LL oscillations in the intravalley X -at higher magnetic field. Importantly, another PL feature, labelled X -, appears red shifted from X -at n 2 ∼ 4 • 10 12 cm -2 . This peak quickly gains oscillator strength with increasing n to dominate the spectrum at the highest value of n studied. X -is commonly observed for n-doping in MoS 2 [START_REF] Pisoni | Interactions and magnetotransport through spin-valley coupled landau levels in monolayer MoS2[END_REF][START_REF] Roch | First-order magnetic phase transition of mobile electrons in monolayer MoS2[END_REF] and WSe 2 [START_REF] Barbone | Charge-tuneable biexciton complexes in monolayer WSe2[END_REF]. It has recently been attributed to a Mahan-like exciton [START_REF] Roch | First-order magnetic phase transition of mobile electrons in monolayer MoS2[END_REF], or an exciton-plasmon-like excitation [START_REF] Tuan | Marrying excitons and plasmons in monolayer transition-metal dichalcogenides[END_REF], but its precise origin is not yet fully understood. We repeated the measurement for various static magnetic fields ranging from -28 T to 28 T.

K' K B = 0T (II) c 2 c 1 (c) (I) N = N ↑ ↑ B = -28T PL (σ -) (b) (a) (II) (III) (IV) n 2 n 1 𝜂 𝑠 1 -1 FLG V g V g hBN MoS 2 - + + - B X 0 X + X - X' - (f) ∆ db +1K +1K' +1K +0K +0K c 1 c 2 ∆ db +1K' +0K' +1K +1K +0K' +1K' E +1K' (d) B < 0T X 0 B > 0T (II) K' K' σ- N ≠ N ↑ ↑ (
Figure 1(c) depicts the spin and valley band sequence of monolayer MoS 2 in an excitonic picture. Electron-hole exchange leads to an optically dark alignment of the spinorbit split conduction bands c 1 and c 2 (∆ db ∼ 14 meV), which has been theoretically proposed [START_REF] Deilmann | Dark excitations in monolayer transition metal dichalcogenides[END_REF] and experimentally verified [START_REF] Pisoni | Interactions and magnetotransport through spin-valley coupled landau levels in monolayer MoS2[END_REF][START_REF] Robert | Measurement of the spin-forbidden dark excitons in MoS2 and MoSe2 monolayers[END_REF]. Throughout this Letter, we describe all observed effects on the basis of this picture. For B = 0 and E F situated slightly above the lower conduction band minima (c 1 ), an equal number of spin-up (↑) and spin-down (↓) electrons occupy the bands at K and K (N ↑ = N ↓ ) resulting in zero net spin-polarization (Fig. 1(c)). However, an applied magnetic field breaks time-reversal symmetry, shifts the conduction band minima, and the resident electrons condense into LLs resulting in an occupation imbalance between the valleys. A direct consequence is the emergence of a spin-polarized Fermi sea (N ↑ = N ↓ ) in either K or K for positive and negative magnetic fields (Fig. 1(d)). Therefore, the resulting spin-valley texture of the electrons depends on the carrier concentration (E F ) and the direction and magnitude of the applied magnetic field as visualized for monolayer MoS 2 in Fig. 1(f). For other monolayer TMDCs, this depends on the details of the respective band structure. In general, the global degree of spin-polarization η s = (N ↑ -N ↓ )/(N ↑ + N ↓ ) enters different regimes ranging from |η s | = ±1 at low carrier concentration and high magnetic fields (regime II), through intermediate densities with |η s | < 1 (regime III) to the absence of η s at highest n where E F resides well above the minimum of the upper conduction band c 2 (regime IV). For our consideration, we model only the relevant LLs in the conduction bands with a finite inhomogeneous broadening (4 meV) and determine the number of spin-↑ and spin-↓ electrons from integrating over the corresponding density of states (see Fig. 1(e) and further details in the Supplemental Material [59]).

An applied magnetic field lifts the K/K valley degeneracy by shifting time-reversed pairs of states in opposite directions in accord with the Zeeman energy -µ B •B [START_REF] Xu | Spin and pseudospins in layered transition metal dichalcogenides[END_REF]. This effect will shift the exciton energy when the magnetic moment of conduction and valence bands are not equal,

∆E V Z = -(µ c -µ v ) • B = 1 2 gµ B B. Figures 2(a)
and 2(b) show the magnetic field dependent X 0 PL close to charge neutrality (n ∼ 0 cm -2 ) and for low electron densities (n ∼ 1.45 • 10 12 cm -2 ). While ∆E V Z is completely linear at charge neutrality, strikingly it becomes nonlinear with electron doping. The observation is summarized in Fig. 2(c), where the B-field dependent peak positions for a sequence of charge concentrations are shown.

At charge neutrality, we measure a g-factor of X 0 , g X 0 = -1.27 ± 0.09. This value is consistent with recent reports in high quality and hBN encapsulated MoS 2 [START_REF] Cadiz | Excitonic linewidth approaching the homogeneous limit in MoS2-based van der waals heterostructures[END_REF][START_REF] Jadczak | Fine structure of negatively charged and neutral excitons in monolayer MoS[END_REF] but contrasts earlier work on non-encapsulated samples [START_REF] Stier | Exciton diamagnetic shifts and valley zeeman effects in monolayer WS2 and MoS2 to 65 tesla[END_REF][START_REF] Mitioglu | Magnetoexcitons in large area CVD-grown monolayer MoS2 and MoSe2 on sapphire[END_REF]. Our measured g-factor appears to be unusually small compared to other TMDCs where a linear valley Zeeman shift for X 0 near g ∼ -4 is reported [START_REF] Stier | Magnetooptics of exciton rydberg states in a monolayer semiconductor[END_REF][START_REF] Stier | Exciton diamagnetic shifts and valley zeeman effects in monolayer WS2 and MoS2 to 65 tesla[END_REF][START_REF] Goryca | Revealing exciton masses and dielectric properties of monolayer semiconductors with high magnetic fields[END_REF][START_REF] Aivazian | Magnetic control of valley pseudospin in monolayer WSe2[END_REF][START_REF] Srivastava | Valley zeeman effect in elementary optical excitations of monolayer WSe2[END_REF][START_REF] Macneill | Breaking of valley degeneracy by magnetic field in monolayer MoSe2[END_REF][START_REF] Li | Valley splitting and polarization by the zeeman effect in monolayer MoSe2[END_REF][START_REF] Wang | Magneto-optics in transition metal diselenide monolayers[END_REF]. In general, the total exciton magnetic moment has contributions from an orbital and a spin component µ = µ orb + µ s , where µ orb = -e 2me l z and µ sege 2me s z . Due to the hybridization of the d x 2 -y 2 and d xy orbitals in the valence band, the Bloch electrons at K/K provide an angular momentum of l z = ±2h such that an estimate for the valley Zeeman splitting in a simple, non-interacting model is µ = -4µ B . [START_REF] Srivastava | Valley zeeman effect in elementary optical excitations of monolayer WSe2[END_REF] Such g-factors are reproduced well by recent first-principle studies including excitonic effects [START_REF] Deilmann | Ab initio studies of exciton g factors: Monolayer transition metal dichalcogenides in magnetic fields[END_REF][START_REF] Woźniak | Exciton g factors of van der waals heterostructures from first-principles calculations[END_REF]. However, as pointed out in Ref. [START_REF] Wang | Magneto-optics in transition metal diselenide monolayers[END_REF] and demonstrated in the Supplemental Material [59], the decomposition of µ orb in a contribution from inter -cellular hopping, captured by a lattice model, and atom-like intra-cellular corrections is a delicate problem. Localized orbitals that do not allow for atom-like dipole transitions (e.g. d z 2 -and d x 2 -y 2 /d xy -orbitals in MoS 2 ) result in a vanishing intra-cellular correction rendering µ orb = 0. Breaking of the electron-hole symmetry is expected due to transitions from each band to energetically higher and lower lying bands resulting in a finite inter -cellular contribution. However, even the most sophisticated theoretical approaches to date ignore the many-body nature of the electron-hole pair which we demonstrate here. We further note that the exciton g-factor results from couplings of energetically remote bands [START_REF] Jovanov | Observation and explanation of strong electrically tunable excitongfactors in composition engineered In(Ga)As quantum dots[END_REF], which may be perturbed by free carriers as well as impurities [START_REF] Klein | Impact of substrate induced band tail states on the electronic and optical properties of MoS2[END_REF].

The nonlinear valley Zeeman shift in the n-charging regime directly correlates with η s and is a direct consequence of a dipolar spin-interaction of the exciton with the local spin-polarized fermionic bath. An enhancement in the context of conduction band filling has been observed in WSe 2 at high densities [START_REF] Wang | Strongly interactionenhanced valley magnetic response in monolayer WSe2[END_REF]. However, our reported results clearly show that the interaction is driven by few electrons occupying fully spinpolarized LLs. Similar physics can be found for electrons interacting with local magnetic moments in dilute magnetic semiconductors [START_REF] Furdyna | Diluted magnetic semiconductors[END_REF]. In our case, the effective magnetic moment originates from a magnetic field induced net spin-polarization of excess carriers as depicted schematically in Fig 1(d). Qualitatively, this can be pictured as a paramagnetic spin ordering effect of conduction band electrons where the MoS 2 effectively undergoes a ferri magnetic-to-ferromagnetic phase transition [START_REF] Lin | Determining interaction enhanced valley susceptibility in spin-valley-locked MoS2[END_REF]. When a sufficiently large magnetic field is applied, the valley degeneracy is lifted, which leads to a repopulation of electrons into the lowest conduction band LL. As such a net spin imbalance, S Z (B) = -1/2 • B J (x), between the valleys occurs at higher B. Here, the Brillouin function [START_REF] Furdyna | Diluted magnetic semiconductors[END_REF] is

B J (x) = 2sz+1 2sz coth 2sz+1 2sz x - 1 2sz coth 1 2sz
x where s z = 1/2 is the electron spin in c 1 and x = gµ B s z B/k B T . The net magnetization is then M ∼ ∆N •S z (B) with the total number of polarized electron spins ∆N = |N ↓ -N ↑ |. Therefore, the total valley Zeeman g-factor for the exciton in regime II can be expressed as the addition of spin and valley net magnetization effects as

g * (n, B) = g X 0 + g m (n) • S z (B) (1)
such that the nonlinear valley Zeeman shift is

∆E X 0 V Z (n, B) = 1/2g * (n, B)µ B B.
As shown by the fits to the data in Fig. 2(c), this simple model captures the nonlinear valley Zeeman shift surprisingly well. Note that we do not observe a nonlinearity in the p-charging regime I. This may be expected as interaction effects are more prominent in the conduction band [START_REF] Pisoni | Interactions and magnetotransport through spin-valley coupled landau levels in monolayer MoS2[END_REF].

Finally, we summarize the valley susceptibility of all features in our accessible density range in Fig. 3(a).

Regime I.-The X 0 g-factor monotonically increases from ∼ -1.27 at charge neutrality to -2.1 at the highest hole density while the X + g-factor exhibits a value of -(3.52 ± 0.35) suggesting that the extra hole adds ∼ 1.5µ B to the valley susceptibility of the exciton. Most strikingly, we find that g * monotonically increases from the value observed close to charge neutrality to the trion g-factor, showing the direct dependence of the exciton g-factor on the carrier density and strongly pointing towards the exciton having the character of a spin-polarized magnetic polaron at finite electron density.

(II) (III) (a) n 1 n 2 (IV) n 0 X 0 X + X -(s -) X -(s + ) n 2 n 3 n 3 MB. n > n 3 (IV) n 2 < n < n 3 K K' (b) (c) E B = E F B = 0T X' X' - K' / K n 1 (II) (III) (
Regime II.-Introducing electrons to the system and using Eq. 1 to obtain g * (n, B), the magnetic moment of X 0 (see Fig. 2(c)) increases monotonically with carrier concentration, peaking at g * ∼ -5.5 for a density of n 1 ∼ 2 • 10 12 cm -2 . This confirms our expectation that for n < n 1 , the increased magnetic susceptibility of X 0 arises from the density of spin-polarized electrons in the lower K valley (see Fig. 1(f)) and the magnitude of g * (n, B) is highly sensitive to the local electron concentration. This conclusion may explain the large variation in the literature of reported g-factors for excitons in MoS 2 [START_REF] Stier | Exciton diamagnetic shifts and valley zeeman effects in monolayer WS2 and MoS2 to 65 tesla[END_REF][START_REF] Mitioglu | Magnetoexcitons in large area CVD-grown monolayer MoS2 and MoSe2 on sapphire[END_REF][START_REF] Cadiz | Excitonic linewidth approaching the homogeneous limit in MoS2-based van der waals heterostructures[END_REF][START_REF] Goryca | Revealing exciton masses and dielectric properties of monolayer semiconductors with high magnetic fields[END_REF][START_REF] Jadczak | Fine structure of negatively charged and neutral excitons in monolayer MoS[END_REF]. Crucially, g * (n, B) smoothly approaches the g-factor observed for X -at higher electron concentrations, indicating that both states have a similar spin and valley structure. The X 0 interacts with the spinpolarized Fermi sea while the strength of the interaction is tuned by the Fermi energy, very similar to the Kondo problem of an isolated impurity spin interacting with a spin-polarized Fermi sea [START_REF] Latta | Quantum quench of kondo correlations in optical absorption[END_REF]. For high densities, the spin structure of X 0 increasingly resembles that of X -. Indeed, at the carrier density of n 1 , X 0 sees on average ∼ 0.1 electrons within its wave function.

Regime III.-Increasing the electron concentration from moderate (n 1 ) to high (n 2 ) densities, we find an asymmetry of the X -valley Zeeman shift in positive (X -(s + )) and negative (X -(s -)) magnetic fields (see Fig. S5 in the Supplemental Material [59]). We attribute this to LL occupation differences in the K/K valley in the lower conduction bands c 1 (see Fig. 1(e)). The gfactor for negative B increases to ∼ -10. Similar to X 0 , X -also interacts with electrons in the Fermi sea. However, unlike X 0 the local spin valley texture admits electrons residing in both valleys since E F is located in the lower conduction bands (regime III). This picture is also supported by the spin texture, shown in Fig. 3(c).

Regime IV.-For n > n 2 , the PL is quickly dominated by the feature X -which emerges precisely at the carrier density when E F shifts into the upper conduction bands c 2 (n 2 ∼ 4 • 10 12 cm -2 in our device) [START_REF] Pisoni | Interactions and magnetotransport through spin-valley coupled landau levels in monolayer MoS2[END_REF]. We find that ∆E V Z of X -is symmetric and linear in B for all accessible densities (see inset Fig. 3(a)). The g-factor of X - equals g X -(s -) at n 2 , which suggests a qualitatively similar magnetic moment. Indeed, a bound complex with the excess electron occupying the ν = +0K LL in the c 2 is such a configuration (see inset Fig. 3(b)). This is further substantiated by the very strong valley polarization of this feature (see Fig. S6 and S7 in the Supplemental Material [59]). Both, η s and the g-factor of X -simultaneously diminish above n 3 with η s ∼ 0 and a minimum gfactor of ∼ -5. Strikingly, at exactly this concentration, a change in the dependence of the X -binding energy E B = E(X -) -E(X -) occurs from a linear to a sublinear dependence on E F (see Fig. 3(b)). For n > n 3 , all valleys and electron spin species are available since E F is situated well within c 2 . The electron concentration approaches the Mott density [START_REF] Steinhoff | Exciton fission in monolayer transition metal dichalcogenide semiconductors[END_REF] where strong many-body effects start to dominate the interaction of X -and electrons of all spins forming a strongly dressed many-body state (see inset Fig. 3(b)). Here the precise local spin structure becomes less relevant, and mean field-"like" theories become applicable. A detailed description of this state would call for dedicated many-body calculations of the spin susceptibility.

Our results show that all excitons in 2D materials are many-body correlated states that have a magneto-optical response that is sensitive to the local carrier density and related spin and valley textures. We explain the large variation of g-factors observed in the literature as arising from lack of control of local doping. The findings of our study represent an important step towards studying and engineering many-body related phases and novel interaction phenomena in atomically thin materials. For the gate-dependent magneto-photoluminescence measurements, we apply a magnetic field and vary the gate voltage in steps of 100 mV while collecting PL spectra for every gate voltage step. We perform the same gate biasing sequence for every magnetic field, thus ensuring that the voltage sweeps at different magnetic fields are directly comparable. We first apply a static magnetic field and then we tune the bias voltage from max. V + to max.

V -. Typical current voltage characteristics are presented in Fig. S2(a) and (b). We apply equal voltage to top-and bottom-gate (V = V bg = V tg ). From the transfer characteristics of both devices, we find that our biasing scheme is highly reproducible for all magnetic fields applied in the experiment. The reproducibility is due to the graphite contacts to the MoS 2 which is known for low contact resistance and small Schottky barrier heights. [9,10] The leakage currents are in the noise floor for most of the range and negligible leakage currents of < 1.5 nA, that correspond to current densities of < 10 -2 A cm -2 , at the highest bias voltages.

The data are collected for a laser excitation power of 30 µW with a laser excitation energy of 2.41 eV.

III. CARRIER DENSITY DEPENDENT MAGNETO-PHOTOLUMINESCENCE OF MONOLAYER MOS 2

The σ -circularly polarized, charge carrier density dependent magneto-photoluminescence is shown in Fig. S3. The above described biasing sequence is used to maintain sample stability throughout the individual voltage sweeps for static magnetic fields ranging from B = -28 T to B = 28 T. ranging from -28 T to 28 T. The X 0 , X -, X + and X -PL features are marked.

28T 15T 10T 5T 20T -28T -15T -10T -5T -20T X 0 X + X - X' -

IV. QUANTIZED SPIN-VALLEY TEXTURE

The Zeeman shift of electrons in the spin-orbit split conduction band valleys in monolayer MoS 2 manifests from spin, the Berry phase and due to quantization of electrons in Landau levels in each valley. The shift for the lower conduction band c 1 is given by

E c 1 = τ s s z 2µ B B + τ i m e µ B B + ν heB m e , (1)
while the shift of the upper conduction band c 2 is

E c 2 = ∆ db + τ s s z 2µ B B + τ i m e µ B B + ν heB m e . (2)
Here, the valley and spin indices are τ s = ±1 (K = +1K and K = -1K) and τ i = ±1

(+1 spin-↑ and -1 spin-↓). We used an electron mass of m e = 0.44m 0 . Moreover, ν is the filling factor of the LL and ∆ db the energy splitting between c 1 and c 2 for no magnetic field applied. We model two LLs for each valley with ν = +0 and ν = +1 in K . [11] After quantifying the energy shift of each LL of each spin in every valley, we can further calculate the density of states (DOS) as a function of the Fermi level E F (applied gate voltage). From this quantity, we can then infer the number of electrons populating the spin-↓ and spin-↑ LLs to deduce the degree of spin polarization for a given E F . We model each LL by using a Gaussian function

DOS LL = e|B| h 1 σ √ 2π exp - (E -E LL ) 2 2σ 2 , (3)
with eB/h as the degeneracy per unit area and the energetic position of the LL E LL as defined in Eq. 1 and 2 and a FWHM of each LL of Γ = 2 2ln(2)σ = 4 meV accounting for the experimentally observed inhomogeneous broadening. The total DOS for all spin-↑ electrons is given through

DOS ↑ LL = ν=1 i=0 DOS ↑,K LL + ν=2 i=1 DOS ↑,K LL , (4)
and the DOS for all spin-↓ electrons are given by

DOS ↓ LL = ν=2 i=1 DOS ↓,K LL + ν=1 i=0 DOS ↓,K LL . (5)
By integrating the DOS to E F we obtain the number of electrons populating each LL with the total number of electrons with spin-↑

N ↑ = E F 0 DOS ↑ LL dE (6)
and spin-↓

N ↓ = E F 0 DOS ↓ LL dE . (7)
We can now compute the global degree of spin polarization all magnetic fields and

E F η s (B, E F ) = N ↓ -N ↑ N ↓ + N ↑ . (8)
V. MAGNETIC MOMENT OF BLOCH ELECTRONS: LATTICE HAMILTONIAN

VS. ATOMIC CONTRIBUTIONS

The magnetic moment is comprised of a contribution due to the orbital motion of a Bloch electron and a contribution due to the electron spin. The z-component of the orbital magnetic moment is given by:

µ orb,n zk = - e 2m e Φ n k | lz |Φ n k = - e 2m e Φ n k |xp y -ŷ px |Φ n k . (9)
Consider the general expression Φ n k |x i pj |Φ n k and insert a complete set of Bloch states:

Φ n k |x i pj |Φ n k = n k Φ n k |x i |Φ n k Φ n k |p j |Φ n k . (10)
The momentum matrix elements are diagonal in k due to translational invariance:

Φ n k |p j |Φ n k = Φ n k |p j |Φ n k δ k,k . (11)
The position matrix element can be transformed using the Schrödinger equation of Bloch states,

H Φ n k = ε n k Φ n k , (12)
and the commutator relation [12] 1

m e p = i h H, r , (13)
which holds in case of a local one-electron potential. It is still valid in the presence of spinorbit interaction, as long as the latter can be approximately treated as an on-site potential.

Using Eqs. ( 11), ( 12) and ( 13), we find:

Φ n k |x i pj |Φ n k = n h im e Φ n k |p i |Φ n k 1 ε n k -ε n k Φ n k |p j |Φ n k . (14)
The crystal wave functions can be constructed as a linear combination of localized orbitals in the following way such that they fulfill Bloch's theorem:

Φ n k = α c n α (k) kα , kα = 1 √ N R e ik•R Rα (15)
with the orthonormality relations Rα R α = δ RR δ αα [START_REF] Huang | Layer-dependent ferromagnetism in a van der waals crystal down to the monolayer limit[END_REF] and

kα k α = δ kk δ αα , (17)
where N is the number of lattice sites. We can formulate the crystal (lattice) Hamiltonian in terms of the localized orbitals:

H = RR αα t αα RR Rα R α . (18)
Inserting the ansatz (15) into Eq. ( 14), we obtain:

Φ n k |x i pj |Φ n k = n h im e 1 ε n k -ε n k αα (c n α (k)) * c n α (k) kα|p i |kα αα (c n α (k)) * c n α (k) kα |p j |kα . (19)
Following [13], we analyze the momentum matrix element by using the commutator relation (13) again, transforming the momentum states according to [START_REF] Young | Spin and valley quantum hall ferromagnetism in graphene[END_REF] and inserting a complete set of position states:

kα |p j |kα = - im e h 1 N R,R e ik•(R-R ) dr R α |r j |r r|H|Rα -R α |H|r r|r j |Rα . (20)
One has to distinguish between the continuous space variable r and the discrete unit-cell label R. The position operator acts as rj |r = r j |r . The momentum matrix element contains contributions that can be directly related to the discrete lattice as well as contributions that arise due to the spatial extension of orbitals. We separate these contributions by shifting r → r + R in the first term and r → r + R in the second term:

kα |p j |kα = - im e h 1 N R,R e ik•(R-R ) dr R α |r j + R j |r + R r + R |H|Rα -R α |H|r + R r + R|r j + R j |Rα = - im e h 1 N R,R e ik•(R-R ) (R j -R j ) R α |H|Rα - im e h 1 N R,R e ik•(R-R ) drr j R α |r + R r + R |H|Rα -R α |H|r + R r + R|Rα , (21)
where we made use of the completeness of position states again to derive the first term of the second line. This so-called Peierls contribution given by inter -site hopping can be written as a generalized Fermi velocity:

- im e h 1 N R,R e ik•(R-R ) (R j -R j ) R α |H|Rα = m e h ∂ ∂k j 1 N R,R e ik•(R-R ) R α |H|Rα = m e h ∂ ∂k j Hα α k . (22)
It follows from the Schrödinger equation ( 12) that the Hamiltonian Hαα

k defines the tight- binding-like eigenvalue problem α Hαα k c n α (k) = ε n k c n α (k). (23)
The second term in Eq. ( 21) contains continuum contributions due to the finite extension of the electron wave functions. It can be written as

- im e h 1 N R,R e ik•(R-R ) drr j R ,α R α |r + R r + R |R α R α |H|Rα -R α |H|R α R α |r + R r + R|Rα = - im e h 1 N R,R e ik•(R-R ) drr j R ,α (χ R α (r + R )) * χ R α (r + R ) R α |H|Rα -R α |H|R α (χ R α (r + R)) * χ Rα (r + R) (24)
with the wave functions χ Rα (r) = r|Rα . This term accounts on the one hand for all atomic or intra-site (R = R ) processes and on the other hand for corrections to the intersite processes contained in the Peierls term. Hence, we can split the momentum matrix element into three contributions:

kα |p j |kα = m e h ∂ ∂k j Hα α k + p α α k,j inter-site corr. + p α α k,j intra-site corr. . (25)
In the limit of well-localized orbitals, the dominant correction is given by the intra-site term [13]. It is obtained by using the lattice periodicity, χ Rα (r + R) = χ 0α (r), to shift the origins of all wave functions to the same unit cell, which is labeled 0. We then identify those terms where the wave function arguments also lie within the same unit cell (R = R in the first term, R = R in the second term):

p α α k,j intra-site corr. = - im e h 1 N R,R e ik•(R-R ) drr j α (χ 0α (r)) * χ 0α (r) R α |H|Rα -R α |H|Rα (χ 0α (r)) * χ 0α (r) = - im e h drr j α (χ 0α (r)) * χ 0α (r) Hα α k -Hα α k (χ 0α (r)) * χ 0α (r) = - im e h α r α α j Hα α k -Hα α k r α α j ( 26
)
with the atom-like dipole matrix elements

r αα j = drr j (χ 0α (r)) * χ 0α (r) (27)
forcing the usual optical selection rules ∆l = ±1, ∆m = 0, ±1. Note that the intra-site term (26) needs to take into account the same set of localized orbitals that is used to set up the lattice Hamiltonian [START_REF] Shimazaki | Strongly correlated electrons and hybrid excitons in a moiré heterostructure[END_REF]. The fact that the coupling to the magnetic field via the vector potential has to be two-fold in a tight-binding or lattice approach has also been discussed in Ref. [14]. There are always contributions that can not be captured by the socalled Peierls substitution [13] in the lattice Hamiltonian leading to the first term in [START_REF] Smoleński | Interaction-induced shubnikov-de haas oscillations in optical conductivity of monolayer MoSe2[END_REF].

If we nevertheless use the Peierls contribution alone, we end up with the following lattice formulation of the orbital magnetic moment:

In a simple two-band model, the structure of this expression leads to equal orbital magnetic moments for conduction and valence electrons. Corrections are expected due to transitions from each band, respectively, to energetically higher and lower bands. Moreover, if the two fundamental bands are composed of localized orbitals that do not allow for atom-like dipole transitions (e.g. d z 2 -and d x 2 -y 2 /d xy -orbitals in transition metal dichalcogenide monolayers), the intra-site correction will vanish. The exciton g-factor, calculated directly from the magnetic moments of conduction and valence bands at the K-point as g X = 2(µ c zKµ v zK )/µ B , will therefore be zero since the net spin of the exciton is zero. This is consistent with the k • p-picture discussed in Ref. [14].

  Figure 1(b) shows a carrier concentration dependent false colour PL map recorded at B = -28 T. We symmet-

  FIG.| 1. (a) Schematic of the dual-gate monolayer MoS2 vdW device in Faraday geometry. A monolayer MoS2 is sandwiched between hexagonal boron nitride (hBN) and contacted with thin graphite (FLG) electrodes. (b) Carrier concentration dependent low-temperature (T = 5 K) µ-PL at B = -28 T. Dashed lines show the region of charge neutrality (n ∼ 0 cm -2 ). Arrows for B = -28 T highlight the +0K and +1K LLs in the X -. (c) Spin and valley exciton band structure at the K and K points of monolayer MoS2 for restored time-reversal symmetry (B = 0 T). (d) A finite magnetic field (B = 0 T) breaks time-reversal symmetry inducing an electron spin imbalance (N ↑ = N ↓ ). For positive magnetic field, the dipolar-spin interaction is of intravalley-type, while for negative magnetic field of inter valley-type. (e) Calculated LL fan diagram. (f) Calculated degree of spin-polarization ηs showing the density and magnetic field dependent spin and valley texture in the conduction bands.

  FIG.| 2. (a) and (b) Normalized µ-PL spectra of X 0 for applied magnetic fields at n = 0 cm -2 (n = 1.45 • 10 12 cm -2 ) from -28 T to 28 T. The solid lines are a guide to the eye. (c) Carrier density dependent valley Zeeman shift ∆EV Z of the X 0 in the electron and hole charged regime.

  FIG.| 3. (a)Carrier density dependent g-factors of X 0 (black and gray data), the negatively (red and blue data) and positively (green data) charged excitons X -and X + , and the X -feature (yellow data). (b) Carrier density dependence of the X - binding energy EB. For n2 < n < n3 EB is linear with EF while it becomes sub-linear for n > n3. For the highest carrier densities (regime IV), the quasi-particle excitation is strongly interacting and dissolving into a many-body state X M B. . (c) Electron concentration dependence of the degree of global spin-polarization ηs for B = -20 T and 20 T.

Figure S 1 .

 1 Figure S 1. Schematic of the field-effect van der Waals device. Monolayer MoS 2 is encapsulated between hBN. The device is a dual-gate device where the carrier density is controlled with a topand bottom-gate V tg and V bg . The same voltage with same polarity is applied to both gates for enhancing the gating effect.

Figure S 3 .

 3 Figure S 3. Carrier density dependent low-temperature (T = 5 K) µ-PL for static magnetic fields

Figure S 4 .Figure S 7 .Figure S 8 .

 478 Figure S 4. (a) Carrier density dependent Zeeman shift of the positively charged exciton X + . (b) Carrier density dependent Zeeman shift of the strongly dressed high density feature X -.

Supported by Deutsche Forschungsgemeinschaft (DFG) through the TUM International Graduate School of Science and Engineering (IGSSE) and the German Excellence Cluster-MCQST and e-conversion. We gratefully acknowledge financial support by the PhD program ExQM of the Elite Network of Bavaria, financial support from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 820423 (S2QUIP) the German Federal Ministry of Education and Research via the funding program Photonics Research Germany (contracts number 13N14846) and the Bavarian Academy of Sciences and Humanities. J.K. acknowledges support by the Alexander von Humboldt foundation. M.F., A.S. and F.J. were supported by the Deutsche Forschungsgemeinschaft (DFG) within RTG 2247 and through a grant for CPU time at the HLRN (Berlin/Göttingen). J.J.F and A.H. acknowledge support from the Technical University of Munich -Institute for Advanced Study, funded by the German Excellence Initiative and the European Union FP7 under grant agreement 291763 and the German Excellence Strategy Munich Center for Quantum Science and Technology (MCQST). K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, Grant Number JPMXP0112101001, JSPS KAKENHI Grant Number JP20H00354 and the CREST(JPMJCR15F3), JST. The work has been partially supported by the EC Graphene Flagship project (no. 604391), by the ANR projects ANR-17-CE24-0030 and ANR-19-CE09-0026. We further thank Scott Crooker and Mark Goerbig for insightful and stimulating discussions.

CONTENTS I. Field-effect device for carrier density control in monolayer MoS 2 II. Transfer characteristics in high magnetic fields III. Carrier density dependent magneto-photoluminescence of monolayer MoS 2 IV. Quantized spin-valley texture V. Magnetic moment of Bloch electrons: lattice Hamiltonian vs. atomic contributions VI. Valley Zeeman shift of X -, X + and X - VII. Magnetic field and density dependent valley dichroism References

We make use of field-effect devices to control the carrier density in monolayer MoS 2 . [1] Figure S1 shows the two device geometries used in this manuscript. In our device, monolayer MoS 2 is encapsulated in hBN. We use the hBN for two main reasons: (i) Encapsulation reduces inhomogeneous linewidth broadening of excitons [2] and (ii) as gate dielectric that withstands high breakdown fields, [3] thus preventing leakage currents. We use few-layer graphite as the gate-electrode and to directly contact the MoS 2 . The heterstacks are assembled by the dry viscoelastic transfer technique iteratively stacking the individual layers using PDMS stamping. [1] The dual-gate device with top and bottom gates as shown in Fig. S1. We apply equal gate voltages with the same polarity to the gates V bg = V tg with respect to the monolayer MoS 2 for controlling the carrier density. In this device, we are also able to tune from the n-into the p-doped regime. The observation that only the n-doped regime is accessible with a single-gate is common in the literature for monolayer MoS 2 . [4,5] It is likely that the dual-gate device allows to access the p-doped regime since it overcomes Fermi-level pinning effects by the symmetric device geometry. Furthermore, dual gates allow larger applied effective fields for tuning the carrier density. We determine the carrier density by using a simple plate capacitor model where the device capacitance is C = 0 hBN /d with the dielectric constant of multilayer hBN hBN = 2.5 [3,[6][7][8] and the hBN layer thickness d = 14 nm which is determined by atomic force microscopy (AFM). Since top and bottom hBN thickness are very similar for the dual-gate device we relate, the carrier density with the gate voltage through n = C(V tg + V bg )/e = 2CV /e. black line. The X -and X -reveal enhanced degree of polarization for higher magnetic field.