# Magnetic field control of light-induced spin accumulation in monolayer MoSe2

Rafael R. Rojas-Lopez<EMAIL_ADDRESS>Departamento de Física,
Universidade Federal de Minas Gerais, 31270-901, Belo Horizonte, Brazil.
Zernike Institute for Advanced Materials, University of Groningen, 9747 AG
Groningen, The Netherlands.  Freddie Hendriks  Zernike Institute for Advanced
Materials, University of Groningen, 9747 AG Groningen, The Netherlands.
Caspar H. van der Wal  Zernike Institute for Advanced Materials, University of
Groningen, 9747 AG Groningen, The Netherlands.  Paulo S. S. Guimarães
Departamento de Física, Universidade Federal de Minas Gerais, 31270-901, Belo
Horizonte, Brazil.  Marcos H. D. Guimarães<EMAIL_ADDRESS>Zernike
Institute for Advanced Materials, University of Groningen, 9747 AG Groningen,
The Netherlands.

###### Abstract

Semiconductor transition metal dichalcogenides (TMDs) have equivalent dynamics
for their two spin/valley species. This arises from their energy-degenerated
spin states, connected via time-reversal symmetry. When an out-of-plane
magnetic field is applied, time-reversal symmetry is broken and the energies
of the spin-polarized bands shift, resulting in different bandgaps and
dynamics in the K+ and K- valleys. Here, we use time-resolved Kerr rotation to
study the magnetic field dependence of the spin dynamics in monolayer MoSe2.
We show that the magnetic field can control the light-induced spin
accumulation of the two valley states, with a small effect on the
recombination lifetimes. We unveil that the magnetic field-dependent spin
accumulation is in agreement with hole spin dynamics at the longer timescales,
indicating that the electron spins have faster relaxation rates. We propose a
rate equation model that suggests that lifting the energy-degeneracy of the
valleys induces an ultrafast spin-flip toward the stabilization of the valley
with the higher valence band energy. Our results provide an experimental
insight into the ultrafast charge and spin dynamics in TMDs and a way to
control it, which will be useful for the development of new spintronic and
valleytronic applications.

Atomically-thin transition metal dichalcogenides (TMDs) offer the possibility
to optically address the spin and valley degrees-of-freedom of charge
carriers. Through circularly polarized light, $\sigma_{+}$ or $\sigma_{-}$,
one can excite electron-hole pairs at opposite points at the edges of the
Brillouin zone, known as $K_{+}$ and $K_{-}$ valleys [1, 2, 3]. The high spin-
orbit coupling in these materials further causes a spin splitting of the
levels, connecting the spin and valley degrees-of-freedom and their dynamics.
For this reason, TMDs are very attractive for (opto)valleytronic and
(opto)spintronic applications [4, 5, 6, 7, 8]. One of the main bottlenecks for
such applications is the control of these degrees-of-freedom. Magnetic fields
have been shown to strongly affect the valley polarization. This has been
demonstrated by photoluminescence (PL) measurements of TMD monolayers under
high out-of-plane magnetic fields, showing a linear shift of the emission
peaks reflecting the reduction of the band gap for one valley and the increase
of the other [9, 10, 11, 12, 13]. This effect became known as the valley-
Zeeman effect. These experiments extract an exciton and trion g-factor of
$\sim$ $-$4, which is in agreement with theoretical expectations [14, 15, 16].
Different works have reported the dynamics of electrons, excitons, and trions
by performing time-resolved photoluminescence (TRPL) [17, 18, 19], time-
resolved differential reflectivity (TRDR) [20, 21, 22], and time-resolved Kerr
rotation (TRKR) [23, 24, 25, 26, 27, 28], with most works focusing on either
zero or in-plane external magnetic fields. More recently, Zhang et al. [29]
showed the effect of out-of-plane magnetic fields on polarized TRPL, but
without any clear difference between the results obtained at different
circularly polarized excitation and high magnetic fields. Nonetheless, PL is
solely sensitive to the valley polarization and radiative decay of excitons
and trions. Therefore, PL is insensitive to light-induced spin polarization of
resident carriers which can sustain the spin information over much longer
times. The understanding of the decay processes and possibilities of control
of resident carriers are of huge importance for the engineering of a new
generation of opto-spintronic devices [30, 31, 8, 32]. However, the use of an
out-of-plane magnetic field to control the long-lived spin dynamics in
monolayer TMDs – beyond the radiative recombination times – remains largely
unexplored.

Figure 1: (a) Schematics of the TRKR measurements with an external magnetic
field. (b) Valley-Zeeman effect on the K+ and K- spin states of monolayer
MoSe2. Dashed lines indicates the zero magnetic field spin states while solid
lines represent the magnetic field shifted states. The Valley-Zeeman effect
results in larger energy shifting in the valence band $\Delta E_{v}$ than in
the conduction band $\Delta E_{c}$. (c) TRKR signal for $\sigma_{+}$ (red) and
$\sigma_{-}$ (blue) polarized pump, at different magnetic fields: B = -5 T, 0
T (d), +5 T (e), for $\lambda$= 755 nm and T = 6 K.

Here we show that the spin information of long-lived carriers in monolayer
MoSe2 can be effectively controlled via the valley-Zeeman effect. Using time-
resolved magneto-optic Kerr effect, we show that magnetic fields can induce an
ultrafast spin-valley scattering and enhancement of a light-induced spin
accumulation with a preference for the valley with higher valence band energy.
While we find that the spin scattering rates do not show a clear trend for the
external magnetic field strength, the magneto-optical signal strength shows a
clear linear behavior. Particularly, we observe a nearly field-independent
long spin lifetime of $\sim$2 ns. The magnetic field dependence of the light-
induced spin accumulation we measure agrees with hole spin dynamics and
resident carrier recombination at longer timescales, while we argue that
electron spins may have faster relaxation rates. Our results are well
described by a simple rate-equation model, which takes into consideration a
field-dependent carrier recombination and intervalley scattering. Our model
indicates that the magnetic field has a stronger effect on the hole transfer
between valleys than in electrons, resulting in a spin imbalance with a larger
population of carriers from the valley with the smaller bandgap. In this way,
we demonstrate here that an applied magnetic field can be used to effectively
control the optically-generated spin accumulation in TMDs.

Our samples were prepared by mechanical exfoliation of a bulk MoSe2 crystal
(HQ Graphene) onto a polydimethylsiloxane (PDMS - Gel Pack) and then
transferred to a 285 nm SiO2/Si substrate. The spin dynamics in our samples
was measured by TRKR, which presents a higher sensitivity to spin-related
phenomena, compared with TRPL or TRDR. We perform a single-color (degenerated)
pump-probe technique in a dual-frequency modulation (see supplementary
information) using a similar experimental setup as described elsewhere [25].
When a high intensity circularly polarized ($\sigma_{\pm}$) laser pulse (pump)
excites the monolayer, electrons of a single valley ($K_{\pm}$) are excited to
the conduction band and can generate a spin imbalance. By shining a linearly
polarized (probe) pulse at a certain time-delay after the pump pulse, the
spin/valley imbalance is measured by a change on the polarization axis of the
reflected beam, which is rotated by an angle $\theta_{K}$ (Fig. 1a). Finally,
the time evolution of the spin imbalance in the sample is measured by changing
the delay time (d$t$) between the pump and probe pulses. All measurements were
performed at low-temperature T = 6 K.

Figure 1d shows the TRKR at zero magnetic field and how the sign of the Kerr
rotation depends on which valley we are exciting: a positive (negative) Kerr
rotation is observed upon shining $\sigma_{+}$ ($\sigma_{-}$) polarized pump
pulses. This behavior is a fingerprint of the symmetry and spin selectivity of
the two valleys of the TMD, when excited with circularly polarized light [23,
24, 33, 25, 28]. Two decay time constants are visible in our measurements, a
faster ($\tau_{1}\sim$ 1.5 ps) and a longer one ($\tau_{2}\sim$ 2 ns) that
goes beyond the measurement range.

We observe a drastic change on the TRKR dynamics upon an applied magnetic
field. Figure 1c and e show the TRKR signal of our sample for an applied
magnetic field of -5 T and +5 T, respectively. Within the first 3 ps after
excitation we observe a reversal of the TRKR signal for one pump polarization,
while a stabilization for the other. This can be explained by an out-of-plane
magnetic field lifting the valley degeneracy causing an opposite shift on the
TMD bands of opposite valleys [9, 10, 11, 12] (Fig 1b). Due to the different
net angular momenta of the valence and conduction bands, the energy shifts for
the conduction ($\Delta E_{c}$) and valence bands ($\Delta E_{v}$) are also
different. The resulting effect is that the lowest energy state for holes lies
within one valley, while for electrons lies in the other.

These observations reveal that our TRKR signal is dominated by hole
relaxation. Upon populating the $K_{-}$ valley through a
$\sigma_{-}$-polarized pump, we observe a reversal of the signal from negative
to positive for positive magnetic fields. For this case, the electron ground
state lies in the conduction band of the $K_{-}$ valley, while the ground
state of holes is located at the $K_{+}$ valley. The fact that we observe a
positive TRKR signal after $\sim$ 3 ps, indicates a higher spin population of
the $K_{+}$ valley, implying a fast intervalley scattering and that the spin
accumulation measured in our experiments arises from the hole spin. When the
direction of the magnetic field is reversed, similar behavior is observed but
with an opposite TRKR signal.

The long decay times we observe do not show any clear dependence with the
magnetic field. The two decay time constants are obtained through a
biexponential decay of the form
$\theta_{K}=A_{1}e^{-t/\tau_{1}}+A_{2}e^{-t/\tau_{2}}$, where $A_{n}$ are the
Kerr rotation angles at a delay time d$t$ = 0 and $\tau_{n}$ the decay times
at the fast and slow decay processes $n=1,2$, respectively. This is done for
measurements using both directions of pump polarization and various values of
applied magnetic fields, from B = -5 T to +5 T. Figures 2a and 2b show the
TRKR signals at B = -5 T to +5 T over a long d$t$ range, up to 1.55 ns. The
TRKR signals are still clearly measurable even at these long-time delays. The
magnetic field dependence for $\tau_{2}$ is shown in Figure 2c (see the
supplementary material for $\tau_{1}$). For zero, and also high magnetic
fields, we find non-zero pump-induced TRKR signals. However, for 0$<|B|<$ 3 T,
the signal for one of the pump polarizations is within our experimental noise,
and therefore only measurements for one pump polarization were used within
this range. Nonetheless, we do not observe any striking features on the TRKR
decay times for the whole range of magnetic fields studied, even though
additional spin scattering channels are predicted to arise from the breaking
of time-reversal symmetry [34]. This indicates that, while the intervalley
scattering rate could be modified by the magnetic field, the main source of
spin scattering remains unaffected for our samples.

Figure 2: (a) TRKR signal for $\sigma_{+}$ (red) and $\sigma_{-}$ (blue)
polarized pump, at B = -5 T (a) and +5 T (b) at long time scales. The orange
dashed lines show the fit that extracts the slow relaxation times. (c)
Extracted slow decay times ($\tau_{2}$) and (d) amplitudes (A2) at different
magnetic fields for the two excitation polarizations.

The amplitude of our TRKR signals shows a clear linear behavior with the
magnetic field, Fig. 2d. Strikingly, the slopes for the two pump polarizations
are slightly different, resulting in a non-symmetric response with the
magnetic field direction at long time delays. Previous measurement runs of the
same sample also presented a linear trend with the magnetic field, with
different slopes for each excitation polarization, see supplementary material
for details. Although small experimental artifacts are not discarded, a
phenomenological origin of these results would be inconsistent with a simple
description involving solely a Zeeman energy shift of the energy states. We
currently do not have a clear understanding of the origin of such an effect,
which should be explored in more detail in later studies.

Figure 3: (a) TRKR Polarization at different wavelengths and zero external
magnetic fields. Measurements were shifted in the vertical axis for clarity.
(b) Polarized photoluminescence spectra of our sample at B = 0 T. Inset:
Intensity profile of the TRKR Polarization at different delay times: 2, 4, 6,
10, 100, and 200 ps. (c) TRKR for $\sigma_{+}$ (red) and $\sigma_{-}$ (blue)
polarized pump at $\lambda$ = 765 nm for B = -5 T (c) and +5 T (d) at T = 6 K.
Figure 4: (a) Experimental (left) and theoretical (right) results for B = 0 T
with $\tau_{+}=\tau_{-}=$ 20 ps, $\tau_{c_{+-}}=\tau_{c_{-+}}=$ 1.8 ps and
$\tau_{v_{+-}}=\tau_{v_{-+}}=$ 5 ns. (b) Experimental (left) and theoretical
(right) results for B = +5 T with an asymmetric hole scattering time
$\tau_{v_{+-}}=5$ ps, $\tau_{v_{-+}}=$ 5 ns. (c) Representation of the
dynamics of the spin population under positive magnetic fields: the ground
state of the system with an initial hole population is photo-excited
($\sigma_{-}$) at the valley $K_{-}$. Later, electrons are scattered between
valleys in the conduction band followed by the scattering of holes from
$K_{-}$ to $K_{+}$. Subsequently, excited states recombine radiatively.
Finally, resident holes in $K_{+}$ reach thermal equilibrium with $K_{-}$. The
represented transfer times indicate the dominant process at each panel.

By studying the dependence of the TRKR signal on the excitation wavelength, we
can probe the contribution of excitons and trions to the spin signal. To
reduce spurious effects which are independent of the valley/spin polarization,
here we use the polarization of the Kerr rotation, namely the difference of
the TRKR measurements for the two polarizations of excitation
$(\theta_{K}(\sigma_{+})-\theta_{K}(\sigma_{-}))/2$ [25]. As expected, our
signal is strongly modulated upon a change of the wavelength when going over
the exciton and trion resonances (Fig. 3a). We observe that when exciting with
$\lambda\leqslant$ 745 nm, KR polarization decays in the first couple of
picoseconds. On the other hand, when exciting at $\lambda\geqslant$ 750 nm,
the spin imbalance remains finite, within the time range of our measurements.
The largest signal is seen at $\lambda=$ 755 nm, see inset of Figure 3b. A
comparison to photoluminescence measurements (Fig. 3b) reveals that our
findings are consistent with short-lived signals coming from excitations at
and below the wavelength of the exciton resonance (X), while the long-lived
signals come from energies close to the peak of trion emission (T). Our
observations are in agreement with previous reports in literature [35, 18, 24,
17] showing that trions dominate the spin signal in TMDs monolayers.

We observe an interesting effect arising from the magnetic field-induced
reduction of the bandgap for one valley with respect to the other. For high
magnetic fields (B = $\pm$5 T) and with the lowest excitation energy resulting
on a measurable signal ($\lambda$ = 765 nm), we only observe a TRKR signal
above our noise level for one pump polarization, Figure 3c and d. For the
positive magnetic field, the bandgap of the K+ valley is reduced and produces
a spin accumulation when excited with $\sigma_{+}$. Nevertheless, as the
bandgap at K- is increased, the energy of the pumping photons is not enough to
excite the electrons and generate any spin imbalance, resulting in no TRKR
signal for $\sigma_{-}$ excitation. A similar behavior is observed for
negative magnetic fields, but with opposite spin accumulation in the K- valley
and no signal for $\sigma_{+}$ excitation. Additional measurements of
wavelength dependence with the magnetic field can be found in the
supplementary material.

To quantitatively describe the spin dynamics and to elucidate the relaxation
processes behind our measurements, we use a rate-equation model similar to
what has been proposed before [24], but including the effect of an external
magnetic field. We consider three main carrier scattering processes: direct
radiative recombination at the same valley, the spin-valley flip of electrons
in the conduction band, and the spin-valley flip of holes in the valence band.
The breaking of the energy degeneracy of the $K_{+}$ and $K_{-}$ valleys is
represented by different intervalley scattering rates. Therefore, considering
the more general case, where conduction and valence band transfer rates, and
also radiative recombination of both valleys, are different, we get to the set
of equations

$\displaystyle\frac{dn_{+}}{dt}$ $\displaystyle=$
$\displaystyle-\frac{n_{+}p_{+}}{\tau_{+}}-\frac{n_{+}}{\tau_{c_{+-}}}+\frac{n_{-}}{\tau_{c_{-+}}}$
(1) $\displaystyle\frac{dn_{-}}{dt}$ $\displaystyle=$
$\displaystyle-\frac{n_{-}p_{-}}{\tau_{-}}-\frac{n_{-}}{\tau_{c_{-+}}}+\frac{n_{+}}{\tau_{c_{+-}}}$
(2) $\displaystyle\frac{dp_{+}}{dt}$ $\displaystyle=$
$\displaystyle-\frac{n_{+}p_{+}}{\tau_{+}}-\frac{p_{+}}{\tau_{v_{+-}}}+\frac{p_{-}}{\tau_{v_{-+}}}$
(3) $\displaystyle\frac{dp_{-}}{dt}$ $\displaystyle=$
$\displaystyle-\frac{n_{-}p_{-}}{\tau_{-}}-\frac{p_{-}}{\tau_{v_{-+}}}+\frac{p_{+}}{\tau_{v_{+-}}}$
(4)

where $n_{i}$ and $p_{i}$ are the photo-excited populations at the $K_{i}$
valley for electrons and holes, respectively. The first term at the right side
of each equation represents the radiative recombination at a specific valley
with recombination time $\tau_{\pm}$. The second term represents the
population of carriers that is transferred to the opposite valley in the
conduction or valence bands with an intervalley scattering time
$\tau_{c,v_{\pm\mp}}$. Finally, the last term relates to the carrier transfer
coming from the opposite valley (source term). As Kerr rotation measures the
spin/valley imbalance of the system, we model our measurements as the
difference between the valley populations
$\theta_{K}=(n_{+}+p_{+})-(n_{-}+p_{-})$. The initial conditions allow us to
consider the two types of excitation, for instance, $n_{+}(0)=N/2$,
$p_{+}(0)=N/2$, $n_{-}(0)=0$, and $p_{-}(0)=0$ for $\sigma_{+}$ excitation,
where $N/2$ is the number of excited carriers.

The results obtained by solving our rate-equation system are in good agreement
with the experimental measurements with and without magnetic fields. In Figure
4a we compare our experimental and theoretical results at zero magnetic field,
where valley degeneracy is still present. We noticed that to properly
reproduce our measurements, the radiative recombination rate could not be
either the shorter nor the longer decay process, and instead, good results
were achieved by using decay times ($\tau$) of tens of picoseconds. The latter
agrees with previous reports of the trion lifetime for MoSe2 [17, 18, 26, 29].
Therefore, as our results point to a spin accumulation of holes at the long
time range ($\tau_{v}$ = 5 $ns$), the fast decay ($\tau_{c}$ = 1.8 $ps$) is
assigned to electron transfer at the conduction band. This is consistent with
previous reports pointing out to the fast depolarization of electrons in the
conduction band when compared to the holes, and even radiative recombination
of trions [24, 36].

To simulate our results at high magnetic fields (Figure 4b) we introduce an
asymmetry in the scattering times of the different considered processes (see
supplementary information for details). Our model indicates that the main
process responsible for the fast switch of spin polarization is a strong
reduction of the hole inter-valley scattering through the valley with higher
valence band energy. In the right panel of Figure 4b we show the result of the
TRKR calculation just with the effect of an asymmetric scattering process at
the valence band, by considering $\tau_{v_{-+}}$ = 5 ps, three orders of
magnitude smaller than $\tau_{v_{+-}}$. Meanwhile, the introduction of
different scattering times for the electrons in the conduction band and
different radiative recombination times at the the two valleys can help to
refine the result at the short lifetimes but they have no major impact on the
total decay profile. We point out that in both calculations in Figures 4a and
4b, we did not obtain a long-lived decay process, such as the ones observed in
the experimental results, even considering larger scattering lifetimes for
holes $\tau_{v}$. Therefore, we associate this long-lived component to
resident charge carriers in agreement with previous reports [35, 27, 26, 24].
In Figure 4c, we illustrate the obtained spin dynamics for positive magnetic
field and an optical excitation of the $K_{-}$ valley.

Our results show that optically-generated spins in TMDs can be very
effectively controlled by magnetic fields, which is explained by a strong
field-induced asymmetry on the intervalley scattering rates. We envision that
the external magnetic field used in our experiments could be replaced by
magnetic proximity to van der Waals magnets. These heterostructures have
already shown to result on a significant control over valley polarization in
photoluminescence measurements [30], and our work implies that a significant
polarization of long-lived resident carriers can potentially be efficiently
obtained and controlled via proximity effects, with lifetimes of several ns
opposed to a few ps for excitons/trions. We believe that this will open new
directions for opto-spintronic devices based on TMDs which do not depend on
the stabilization of bound electron-hole pairs, but make use of the full
potential of the long spin lifetimes offered by resident carriers in these
materials.

See Supplementary Information for additional details regarding the
experimental setup, photoluminescence measurements with magnetic field, TRKR
data at different wavelengths with an external magnetic field, TRDR, and
further discussion of the rate-equation model.

We thank J. G. Holstein, H. de Vries, F. van der Velde, H. Adema, and A.
Joshua for their technical support. This work was supported by the Dutch
Research Council (NWO—STU.019.014), the Zernike Institute for Advanced
Materials, and the Brazilian funding agencies CNPq, FAPEMIG and the
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) -
Project code 88887.476316/2020-00.

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## Supplementary Information:
\-
$\large\text{Magnetic field control of light-induced spin accumulation in
monolayer MoSe}_{2}$

##  I Details on the experimental setup

Figure S5: Experimental setup used for our time-resolved measurements.

For measuring time-resolved data we used a single-color (degenerated) pump-
probe technique in a double modulation configuration. Figure S5 pictures our
experimental setup with the key elements represented on it. For excitation, we
used a tunable Ti:sapphire laser (Mai Tai - Spectra Physics) with a repetition
rate of 80 MHz and a pulse width $<$ 300 fs. With a beam-splitter and a set of
neutral density filters (not shown in Figure S5), we set the pump and probe
beams to a 4:1 fluence ratio with a $\sim$ 6 $\mu$J/cm2 probe fluence. The
pump is set to circularly polarized light ($\sigma_{\pm}$) with a linear
polarizer (P1) and a quarter wave plate (QWP) after passing through a chopper
with a frequency of 70 Hz. For blocking the pump beam after the reflection on
the sample we use the set of lenses L1, L3, and L4 that help focus the beam in
a different place between L3 and L4, on its way back from the cryostat. The
lens L1 also ensures a larger spot size for the pump beam at the sample, due
to a different focal point. The probe beam is linearly polarized (P2) and its
polarization is modulated by a photoelastic modulator (PEM) at a frequency
$\Omega\ =$ 50 kHz. The beam is detected on the way back from the sample by an
amplified photodetector (PD) after passing through a linear polarizer (A), set
cross-polarized with the probe for detecting the Kerr rotation signal.

The presence of the chopper in the pump path and the PEM in the probe path has
the purpose of improving the signal-to-noise ratio, also known as the double
modulation technique. The PEM is set with its fast axis at 45$\degree$ to the
axis of polarization of the incident beam that makes the polarization of the
probe oscillate between circularly and linearly polarized with a frequency
$2\Omega$. The difference in the modulation frequencies allows us to separate
the signal coming from each beam. With this configuration, the reflected
signal is sent to one lock-in amplifier with the reference to the PEM
oscillation frequency, which helps clean the signal coming from the probe
beam. The output of the filtered signal is sent to a second lock-in amplifier
that has its reference on the chopper frequency and helps to improve the
signal from the probe which is affected by the pump. Using the Jones matrix
formalism, it is possible to determine that the intensity of the signal at the
second harmonic of the PEM oscillation frequency is related to the Kerr
rotation as $I_{2\Omega}=\theta_{K}RJ_{2}(\rho)\cos(2\Omega t)$ [37, 25],
where $\theta_{K}$ the Kerr rotation, $R$ is the average reflected intensity,
$J_{2}$ the spherical Bessel function of the first kind and $\rho$ = 3.041 is
the retardation given by the PEM to optimize the signal. Therefore, the first
lock-in amplifier is set to detect and amplify the second harmonic which is
proportional to the Kerr rotation signal.

##  II Photoluminescence at high magnetic fields

Figure S6: Photoluminescence spectra of 1L MoSe2 at different magnetic fields
for the two polarization excitations. (b) Valley-Zeeman splitting for the
trion peak, calculated as the difference of the energies
$E_{\sigma_{+}}-E_{\sigma_{-}}$. (c) Degree of circular polarization (Pc) of
excitons and trions at different magnetic fields.

Photoluminescence measurements were performed with $\lambda=$ 690 nm laser
diode with a power of 100 $\mu$W at a temperature of 6 K. For controlling the
polarization, we set the fast axis of a quarter wave plate (QWP) aligned to
the polarization of the excitation light. When the emitted light from the
sample, $E_{+}\hat{\sigma}_{+}+E_{-}\hat{\sigma}_{-}$, with $\hat{\sigma}_{+}$
the Jones matrix associated to the circularly polarized light, passes again
through the waveplate, the fast axis is no longer aligned with the
polarization of the light. The latter will decompose most of the signal of the
circularly polarized states at $\pm\pi/4$ with respect to the axis of the QWP.
Therefore, by placing an analyzer (polarizer) at those angles, we can
independently measure the intensity $|E_{+}|^{2}$ and $|E_{-}|^{2}$.

Figure S6a shows the photoluminescence spectra of our sample at different
magnetic fields B = 0, $\pm$5 T. Each spectrum was measured at the two
polarizations, $\sigma_{+}$ (red) and $\sigma_{-}$ (blue). The two
characteristic peaks of MoSe2 are observed: one coming from exciton
recombination (1.664 eV) and the other from the emission of trions (1.630 eV).
For analyzing the valley-Zeeman shifting we fitted the trion peak for both
polarizations and plotted the difference of the peak centers
$E_{\sigma_{+}}$-$E_{\sigma_{-}}$ as a function of the magnetic field (Figure
S6b). We obtained a linear behavior with a slope of ($-0.26\pm 0.01$) meV/T,
indicating a $g$-factor of -4.5, which is in good agreement with previous
reports in the literature [9, 13, 16]. Figure S6c shows the degree of circular
polarization of the exciton and trion peaks as a function of the magnetic
field calculated as the difference of the integrated intensity at the two
different polarizations
$P_{c}=(I_{\sigma_{+}}-I_{\sigma_{-}})/(I_{\sigma_{+}}+I_{\sigma_{-}})$.

##  III Fast decay process in TRKR vs. Magnetic field

Figure S7 shows the fast decay times $\tau_{1}$ and corresponding amplitudes
$A_{1}$ obtained from the biexponential decay fits of the Kerr rotation angles
measured at different magnetic fields.

Figure S7: Extracted fast decay times $\tau_{1}$ and (b) amplitudes A1 from
the fits of TRKR presented in the main manuscript.

##  IV Spin dynamics: wavelength dependence

For keeping track of the wavelength dependence of the Kerr rotation with the
magnetic field, we also measured the TRKR at the two polarizations
($\sigma_{\pm}$) for different wavelengths at B = 0 T, B = $\pm$5 T. To reduce
spurious effects and simplify the analysis, we calculate the Kerr rotation
polarization defined as ($\theta_{K}(\sigma_{+})-\theta_{K}(\sigma_{-})$)/2.
The results are shown in Figure S8. In all the cases, we observe that the
stronger and long-lived signals are associated with the wavelengths between
$\lambda$ = 755 nm and 760 nm. Therefore, the magnetic field does not induce
clear changes in the KR signal close to exciton resonances nor change
(significantly) the resonance close to the trion energy peak.

Figure S8: Kerr rotation polarization of our sample at different wavelengths
studied with an out-of-plane magnetic field B=$\pm$5 T and at B = 0 T.
Extracted lifetimes are shown at each plot.

##  V Rate equation Model -Different parameters

Figure S9a shows all the different decay paths considered in our model. When a
magnetic field is applied, valley degeneracy breaking can affect the
recombination and scattering times of excited spins at each valley. To start
exploring the effect of such asymmetry, we first pursued the reproduction of
our results at zero magnetic field, Figure S9b. In Figure S9(c-h) we present
the TRKR calculated for different sets of parameters. As for zero magnetic
field valley degeneracy is still present, equal scattering and recombination
rates are considered for both valleys: $\tau_{+}=\tau_{-}=\tau_{r}$,
$\tau_{v_{+-}}=\tau_{v_{-+}}=\tau_{v}$, and
$\tau_{c_{+-}}=\tau_{c_{-+}}=\tau_{c}$.

Figure S9: (a) Representation of the valley-Zemman shifting and carrier
transfer process of the bands at the vallleys $K_{\pm}$ under a positive
magnetic field. (b) Experimental result of TRKR in MoSe2 at zero magnetic
field for the two pump polarizations $\sigma_{\pm}$ at T = 6 K. (c-h)
Simulation of the TRKR using the model presented in the main text. The
parameters for the calculation are presented in each figure. When two sets of
parameters are presented on the same figure, it indicates that the same result
is obtained for both cases.

With the obtained parameters at zero magnetic field (Figure S9h), we proceed
to study the effect of the magnetic field introducing the assymetry in the
decay times. Figure S10 show the results of TRKR calculted with different
parameters for reproducing our experimental results at B = $\pm$ 5 T.

Figure S10: (a) Experimental TRKR of MoSe2 at B = 5 T. (b-c) Calculated TRKR
for modeling (a). (d) Experimental result at B = -5 T. (e-f) Simulated TRKR
for modeling the results shown in (d)

##  VI TRKR vs. B - Different set of measurements

Figure S11: (a-b) TRKR decay times and (c-d) amplitudes extracted at a
different point of the sample, in a different cool down of the same sample,
with a different alignment with $\lambda$ = 755 nm, $F_{pump}$ = 100 $\mu
J/cm^{2}$, $F_{probe}$ = 10 $\mu J/cm^{2}$, T = 6 K.

##  VII Further discussion on fitting the decay process

For fitting the spin dynamics in the main document we considered a simple
approach of two exponential decay, as it is within the scope of our paper. For
the fits, we use the data measured in two ranges with different acquisition
parameters. One for short-time scale analysis up to 20 ps, acquired at 0.1
ps/s, and a long one for the slow decay process up to 1550 ps, with a velocity
of acquisition of 3 ps/s. For each data set is possible to accurately fit
either the fast or the slow decay process. Nevertheless, fits for intermediate
decay times (tens of ps) can fail for some curves (see Figure S12a). Here, we
present an alternative three-decay process fit
($A_{1}e^{-t/\tau_{1}}+A_{2}e^{-t/\tau_{2}}+A_{3}e^{-t/\tau_{3}}$) over the
long-range data set that helps to elucidate the dynamics of the intermediate
process and its effect on the results of the two-exponential decay.

Figure S12: (a) TRKR at different magnetic fields for a $\sigma_{+}$
excitation fitted with a bi-exponential decay and (b) a three-exponential
decay (black dashed line). Figure S13: Fitting parameters extracted from a
three exponential decay fit.

Figure S12b presents the fits of the TRKR when exciting with polarization
$\sigma_{+}$ at three different magnetic fields. In Figure S13 we present the
extracted lifetimes and amplitudes of each decay process. We observe a
consistent result for the fast and slow decay processes when compared with the
data presented with the two-exponential decay case. For $\tau_{2}$ we observed
lifetimes of tens to a hundred picoseconds, which is consistent with the trion
recombination lifetime reported in the literature [35, 18, 24, 17]. Also, for
the amplitude $A_{2}$, we observe a linear behavior with the magnetic field as
observed for $A_{3}$.

##  VIII Time-Resolved Differential Reflectivity at B = 0 T

Figure S14: (a) Time-resolved differential reflectivity studied at different
wavelengths and their fits (red dashed lines). Inset: extracted amplitudes
$A_{i}$ and; (b) extracted lifetimes $\tau_{i}$ of the bi-exponential fit.