Mode locking of hole spin coherences in CsPb(Cl, Br)3 perovskite nanocrystals

The spin physics of perovskite nanocrystals with confined electrons or holes is attracting increasing attention, both for fundamental studies and spintronic applications. Here, stable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{{{\rm{CsPb}}}}}}}}{({{{{{{{{\rm{Cl}}}}}}}}}_{0.56}{{{{{{{{\rm{Br}}}}}}}}}_{0.44})}_{3}$$\end{document}CsPb(Cl0.56Br0.44)3 lead halide perovskite nanocrystals embedded in a fluorophosphate glass matrix are studied by time-resolved optical spectroscopy to unravel the coherent spin dynamics of holes and their interaction with nuclear spins of the 207Pb isotope. We demonstrate the spin mode locking effect provided by the synchronization of the Larmor precession of single hole spins in each nanocrystal in the ensemble that are excited periodically by a laser in an external magnetic field. The mode locking is enhanced by nuclei-induced frequency focusing. An ensemble spin dephasing time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T}_{2}^{ * }$$\end{document}T2* of a nanosecond and a single hole spin coherence time of T2 = 13 ns are measured. The developed theoretical model accounting for the mode locking and nuclear focusing for randomly oriented nanocrystals with perovskite band structure describes the experimental data very well.

The spin physics of perovskite nanocrystals with confined electrons or holes is attracting increasing attention, both for fundamental studies and spintronic applications. Here, stable CsPbðCl 0:56 Br 0:44 Þ 3 lead halide perovskite nanocrystals embedded in a fluorophosphate glass matrix are studied by timeresolved optical spectroscopy to unravel the coherent spin dynamics of holes and their interaction with nuclear spins of the 207 Pb isotope. We demonstrate the spin mode locking effect provided by the synchronization of the Larmor precession of single hole spins in each nanocrystal in the ensemble that are excited periodically by a laser in an external magnetic field. The mode locking is enhanced by nuclei-induced frequency focusing. An ensemble spin dephasing time T Ã 2 of a nanosecond and a single hole spin coherence time of T 2 = 13 ns are measured. The developed theoretical model accounting for the mode locking and nuclear focusing for randomly oriented nanocrystals with perovskite band structure describes the experimental data very well.
Lead halide perovskite semiconductors are highly attractive due to their remarkable photovoltaic efficiency 1,2 , and are promising for optoelectronic 3,4 and spintronic [4][5][6][7] applications. Perovskite nanocrystals (NCs) have recently extended the wide class of semiconductor NCs grown by colloidal synthesis [8][9][10][11][12][13] . They show a remarkable quantum yield of up to 90% even for bare NCs, as surface states do not act detrimentally on the exciton emission efficiency. For inorganic CsPbX 3 (X = I, Br, or Cl) NCs, the band gap can be tuned from the infrared up to the ultraviolet by mixing the halogen composition and by changing NC size, varying the quantum confinement of charge carriers. Additionally, the exciton fine structure can be adjusted by the NC shape [14][15][16][17] .
In addition to the high quantum yield and tunable optical properties, the simple fabrication makes lead halide perovskite NCs interesting for applications. As far as spintronics is concerned, only quite a few studies have been performed so far. Neutral and charged excitons in single NCs were identified by their Zeeman splitting in magnetic field 18,19 . For NC ensembles, negatively charged excitons (trions) and dark excitons were identified in strong magnetic fields of 30 T 20 .
Optical orientation, optical alignment and anisotropic exciton Zeeman splitting were observed 21 . The coherent spin dynamics of electrons and holes in CsPbBr 3 NCs 22,23 were explored, and the picosecond spin dynamics of carriers in CsPbI 3 NCs were reported 24 . Despite the progress in crystal growth, perovskite NCs still suffer from insufficient long-term stability of optical properties. A promising approach here is to synthesize NCs embedded in glass, providing protection by encapsulation 25,26 .
As a result, perovskite NCs in a glass matrix may be suitable for quantum technologies. For reference, one may compare NCs with the established system of self-assembled (In,Ga)As/GaAs quantum dots (QDs) singly charged with an electron or a hole. Using all-optical approaches one can orient and manipulate the spins of the charge carriers on the picosecond timescale at high operation frequencies 27,28 . There are two complementary concepts: the first one uses a single spin in a QD for establishing a quantum bit, the second one exploits a QD ensemble which may be suited for a quantum memory with sufficiently strong light-matter interaction. Here, we focus on a NC ensemble, that provides information about the average spin properties and their dispersion due to inhomogeneity. The inhomogeneity obstacle, however, can be overcome by applying a periodic laser excitation that synchronizes the electron spin Larmor precession in different NCs subject to a transverse magnetic field. This spin mode locking (SML) effect, initially discovered in singly charged (In,Ga)As QDs, allows one to uncover the spin coherence of individual carriers, which is typically prevented due to the faster spin dephasing in the ensemble [29][30][31] . The interaction of resident carrier spins with the surrounding nuclear bath provides further flexibility with potential memory times up to hours and hyperfine interaction fields up to a few Tesla 32,33 . This collective phenomenon of spin dynamics homogenization has been demonstrated so far only for (In,Ga)As QDs.
In this paper we demonstrate the SML effect for holes in a totally different class of QDs, namely perovskite CsPbðCl 0:56 Br 0:44 Þ 3 NCs synthesized in a glass matrix. We provide detailed information on the spin dynamics of confined holes correlated with the appearance of spin mode locking. We measure the g-factor, spread of g-factors, longitudinal spin relaxation time T 1 , transversal spin coherence time T 2 , and inhomogeneous spin dephasing time T Ã 2 . Exploiting the hyperfine interaction with nuclear spins, we implement dynamic nuclear polarization and optically-detected nuclear magnetic resonance (ODNMR) to identify the involved nuclear isotopes. A theoretical model of the SML in perovskite structures is developed to account for the inverted band structure and for the dominating role of the holenuclear interaction, compared to self-assembled QDs. As an additional challenge, the random orientation of NCs in the ensemble is considered.

Results
The optical properties of CsPbðCl 0:56 Br 0:44 Þ 3 nanocrystals embedded in a fluorophosphate glass are shown in Fig. 1a. At a cryogenic temperature of T = 5 K the transmission spectrum of the NCs, with about 8 nm size, shows a pronounced exciton resonance at 2.743 eV, which is broadened due to NC size dispersion. The exciton resonance can be also traced by the spectral dependence of the time-resolved Faraday ellipticity (TRFE) amplitude. The exciton population dynamics is measured by time-resolved differential transmission (ΔT/T), showing a decay time of 160 ps, which corresponds to the exciton lifetime (inset in Fig. 1a).
To study the coherent spin dynamics of resident carriers we use the pump-probe technique with spin-sensitive detection of the Faraday ellipticity, whose potential has been approved for lead halide perovskite crystals [34][35][36] , polycrystalline films 23,37,38 , and CsPbBr 3 NCs 22,23 . The pump laser pulses are circularly polarized and, according to the selection rules, generate spin polarization of electrons and holes along the light k vector. The oriented carrier spins precess at the Larmor frequency around the magnetic field applied in the Voigt geometry B V (B⊥k). The coherent spin dynamics is then detected through the Faraday ellipticity of the linearly-polarized probe pulses.
A typical TRFE signal, measured at T = 1.6 K in B V = 1 T, is shown in the upper part of Fig. 1b. It shows oscillations with the Larmor frequency ω L which is a linear function of B V : ω L = |g|μ B B V /_. This allows us to evaluate the g-factor of |g| = 1.20. ω L has no offset at zero magnetic field (Fig. S4a), which is a strong argument in favor of the presence of resident carriers in the NCs rather than of carriers bound in an exciton 23 . In the latter case, an offset equal to the electron-hole exchange would be expected. General theoretical arguments and the experiments on dynamic nuclear polarization (Fig. 2c, see details below) indicate that the g-factor has a positive sign. From its value and the strong changes of ω L in presence of polarized nuclei we conclude that the signal is dominated by resident holes with g h = +1.20 (see Supplementary Notes 1 and 2). Note, that the stronger hole-nuclear interaction compared to the electron-nuclear one is specific for lead halide perovskite semiconductors 35 , due to their "inverted" band structure in comparison to semiconductors like GaAs or CdTe.
The TRFE amplitude decays within a nanosecond due to spin dephasing in the ensemble of NCs showing a dispersion of Larmor frequencies. The spin dephasing time at B V = 1 T is T Ã 2 = 0:26 ns, and it increases up to 0.5 ns at low magnetic fields. There it becomes limited by the hole interaction with the nuclear spin fluctuations. In stronger fields, T Ã 2 decreases as 1/B V due to the g h dispersion Δg = 0.03 (Fig. S4b). By using the spin inertia technique 39 we measured the longitudinal spin relaxation time T 1 = 7.6 μs in zero longitudinal magnetic field, which gives an estimate of the upper limit of T 2 in the studied NCs (Supplementary Note 4).

Spin mode locking in nanocrystal ensembles
The most striking feature of the TRFE signal (blue) in Fig. 1b is the presence of an oscillating signal at negative time delays, before the pump arrival. Its amplitude has a maximum at small negative delays and decays with increasing negative delay with the same dephasing time as at positive delays. For such a signal to occur, optically-induced spin coherence should be present in the NCs for times exceeding the repetition period of the laser pulses T R = 13.2 ns, and can therefore be obtained for spin coherence times T 2 ≥ T R . The same effect was observed in ensembles of singly charged (In,Ga)As/GaAs QDs with both negative and positive charging 29,31,40 . It is caused by synchronization of the Larmor spin precession in the ensemble of NC carriers with the periodic laser pumping, termed the spin mode locking effect 41 , an effect similar to the resonant spin amplification 42 .
The SML effect results from the efficient accumulation of spin polarization from carrier spins which Larmor frequencies ω L are commensurate with the laser repetition period T R . These spin precession modes fulfill the phase synchronization condition (PSC): ω L = Nω R = 2πN/T R , with N being an integer. If the single spin coherence time exceeds T R , the spin polarization of these modes accumulates. Modes fulfilling the PSC are shown schematically in Fig. 1c, assuming a Gaussian distribution of the Larmor frequencies centered around N = 3 for the weight of individual contributions, as shown Fig. 1d by the green line with the spread of Larmor frequencies Δω. Δω is contributed by the g-factor dispersion and by the nuclear spin fluctuations (Supplementary Note 4). Within this distribution, the five essential PSC modes (red needles) are located.
The signal resulting from the sum of the five PSC modes is shown at the bottom of Fig. 1c. The maxima of the PSC mode signal coincide with the pump pulse arrival times, in between the signal is reduced with a symmetry relative to the center between the pump pulses: the ensemble spin polarization dephases after a pump pulse, but then revives before the next pump. The pump pulse excites each time all modes within the Gaussian distribution, but only the PSC modes are amplified from pulse to pulse and contribute to the SML signal at negative delays. Remarkably, the spin coherence time T 2 of the holes in individual NCs can be determined from the ratio of the signal amplitudes at negative and positive delays. This requires a modeling approach, that we will present in the following.
It was shown for (In,Ga)As QDs that the spectrum of PSC modes and the shape of TRFE signal can be tailored by using a two-pump protocol, where an additional second pump pulse is applied with a time delay T D with respect to the first pump pulse. In this case, an additional synchronization condition is introduced, as the amplified Article https://doi.org/10.1038/s41467-023-36165-0 PSC modes have to satisfy two commensurability conditions: ω L = 2πN/T R and ω L = 2πM/T D , with M being an integer 29,31 . In the experiment this leads to the emergence of signal bursts at multiple times of T D . This behaviour is indeed also found in the studied perovskite NCs, as one can see from the red colored signal in Fig. 1b and its zoom in Fig. 1e, measured with T D = 0.1 T R = 1.32 ns. Two bursts at −1.32 ns and +2.64 ns are seen very prominently. A scheme of the PSC mode spectrum modification for the two-pump case is given in Fig. 2a.

Nuclei-induced frequency focusing
In semiconductors, the spin dynamics of electrons and holes are strongly influenced by their interaction with the nuclear spin system in which they are embedded 43 . Polarized nuclei provide the Overhauser field B N , which shifts the carrier Larmor precession frequency to ω L = |g|μ B (B V + B N )/_. Thereby, spin precession modes in Fig. 1d, that do not match the PSC, can be shifted in their frequency such that they satisfy the PSC. This effect was found experimentally in (In,Ga)As QDs and identified as nuclei-induced frequency focusing (NIFF) of the carrier spin coherence 30 . The NIFF mechanism represents a positive feedback loop, that provides the required nuclear polarization in each QD, so that a large dot fraction in the ensemble is pushed into PSC modes. Due to the very slow nuclear spin relaxation at cryogenic temperatures, the NIFF configuration can be kept for hours.
In an ideal, very efficient, system as is the case for (In,Ga)As QDs with strong carrier-nuclei interaction, NIFF results in identical amplitudes of the TRFE signal at negative and positive time delays, meaning that all precession frequencies are focused on the PSC modes. This is obviously not the case for the studied perovskite NCs, see Fig. 1b. Still, NIFF is present as can be evidenced by the relaxation dynamics of the bunch amplitude at −1.32 ns delay when the second pump is blocked 30 . In Fig. 2b we demonstrate such a measurement. We find the decay time of τ = 0.5 s, which is related to the repolarization of the nuclear spin system from the two-pump to the one-pump protocol, sketched in Fig. 2a. Without nuclear involvement the amplitude of the signal would decay within the carrier spin coherence time T 2 after switching off the second pump, as was the case for hole spins in (In,Ga)As QDs 40 .

Dynamic nuclear polarization and ODNMR
To further justify the involvement of nuclear spins, we apply the method of dynamic nuclear polarization (DNP) in a titled magnetic field 44 . The spin polarization of optically-oriented carriers is transferred via the hyperfine interaction to the nuclear spin system. The Overhauser field of the polarized nuclei B N acts back on the carriers and shifts their Larmor precession frequency, the detailed scheme can be found in Supplementary Note 2 and ref. 35 . The shift of the Larmor frequency allows one to measure the nuclear spin polarization.
One can see in Fig. 2c, that for σ + pump the Larmor precession becomes faster compared to the σ − case. The relative frequency shift is considerable, which is typical for the strong hole-nuclei spin interaction in perovskites. It allows us to confirm that the TRFE signal is related to holes and to evaluate the Overhauser field of B N = 5.8 mT. The sign of the observed Larmor frequency change corresponds to a positive hole g-factor.
The variation of the TRFE signals by the DNP effect allows us to use it for optical detection of nuclear magnetic resonance (ODNMR) 35 . For that, the time delay is fixed and the TRFE amplitude is measured as a function of the radio frequency of additional radiation with f RF = 0.1−10 MHz. The nuclear spin system is depolarized when the energy hf RF matches to the Zeeman splitting of a nuclear isotope μ N g N B, i.e. for a NMR. Here, μ N is the nuclear magneton and g N is the nuclear g-factor (Supplementary Table 1). In the experiment it is detected as a resonant decrease of the TRFE amplitude, as shown in the inset of Fig. 2d at B = 0.2 T. From a linear fit of the resonance frequency dependence on the magnetic field f NMR RF ðBÞ, we evaluate the gyromagnetic ratio γ = μ N g N /_ = 8.80 MHz/T, see Fig. 2d. One can conclude from the Supplementary Table 1 that among the nuclear isotopes present in the CsPb(Cl, Br) 3 NCs only the 207 Pb isotope matches this value. The dominant role of 207 Pb on the carrier spin dynamics was identified earlier in FA 0.9 Cs 0.1 PbI 2.8 Br 0.2 crystals and the theoretical analysis establishes it as common feature for lead halide perovskites 35 .

Theory of spin mode locking in perovskite NCs
The theory of spin mode locking was previously developed for selfassembled quantum dots 45 and successfully applied to explain the experimental results 40,41,46 . However, the discovery of SML in perovskite nanocrystals provides new theoretical challenges. First, the generation of spin coherence in perovskite structures with an inverted band structure occurs on the basis of other selection rules 43 . Circularly polarized light generates an exciton with an electron and a hole, both having spin 1/2, instead of an exciton with an electron (1/2) and a heavy hole (3/2). Second, perovskite colloidal nanocrystals in an ensemble are randomly oriented. For a complete theoretical description, the anisotropy of the g-factor for a single NC and the possibility of generating spin coherence through positively and negatively charged trions have to be considered. Taking all these facts into account requires a significant adjustment of the theoretical model.
The model, which details are given in Supplementary Notes 6-10, has two key parts: the first one considers the spin coherence generation and its dynamics in magnetic field for a single nanocrystal with arbitrary oriented crystallographic axes. The second part accounts for the inhomogeneity of the NC ensemble, namely the NCs random orientation and the g-factor spread.
Without loss of generality we consider resident holes, that can be created by photo-charging, as in the experiment the SML of holes is observed. The resident carrier spin polarization, providing the pumpprobe signal, is generated by a mechanism involving trions. The oriented spin polarization components are tilted relative to the direction of the magnetic field, precess about it in time, and decay with the spin coherence time T 2 . For T 2 ≥ T R the spin polarization accumulates by pumping with a train of optical pulses. The SML is an ensemble effect resulting from a large number of oscillating signals with frequencies commensurable with T R , therefore, the model takes into account the spread of the g-factors (Supplementary Note 8).
The experimentally observed spin dynamics, measured at B V = 0.35 T (black dots in Fig. 3a), are modeled numerically using Supplementary Eqs. 28-61. The signal decay, at negative and positive time delays, is described by the T Ã 2 , that is given by the g-factor spread, Δg, and nuclear spin fluctuations, δB N . The ratio S b /S a of the amplitudes before pulse arrival S b and after pulse arrival S a is determined by T 2 , the optical pulse area Θ and the NIFF effect. Technically, we calculate the spin polarization distribution S a as a function of the Larmor frequency parameterized by the hole g-factor ω L = g h μ B B V /_ (Fig. 3b). The distribution width is defined by 2Δg with Δg = 0.1. The multiple peaks correspond to synchronized precession modes, with the width of each peak determined by 1=T 2 Þ is related to the nuclear spin fluctuations. The modeled SML signal, corresponding to this distribution, is shown by the red line in Fig. 3a. It reproduces well all features of the experimental data. Finally we note, that the presented theoretical approach for the description of the SML effect is universal and applicable for different crystal phases of the NCs.
In order to highlight the role of the hole-nuclei interaction for the measured SML, we perform modeling without and with account for the NIFF. Without NIFF, we get a hole spin coherence time T 2 = 28 ns for Θ = π as best fit parameter. The nuclear polarization changes the spin polarization distribution by increasing T Ã 2 and narrowing the precession modes. This results in an increase of the ratio S b /S a , and to match its experimental value of 0.1 a shorter spin coherence time should be taken, to keep the mode width the same (insert of Fig. 3b). With NIFF we get T 2 = 13 ns, which is at the lower limit of the times required for the SML to occur. It is remarkable that due to the involvement of NIFF we are able to observe the SML effect for such a short-lived hole spin coherence.

Discussion
In summary, we have demonstrated the outstanding potential of perovskite NCs embedded in a glass matrix, using their spin properties as exemplary feature. We find a very long hole spin lifetime up to the microsecond range. We present an additional class of materials, in which the effects of spin mode locking and nuclei-induced frequency focusing are observed. This allows one to measure the hole spin coherence time T 2 , disregarding the random NC orientation and variation of NC sizes and shapes. It opens the pathway for implementation of multi-pulse protocols for manipulating the spin coherence, also with involvement of strong hole-nuclei interaction provided by the inverted band structure of the lead halide perovskites.
Despite our main goal here to report on the SML effect in perovskite NCs, we would like to compare our discovery with other systems. In the studies of colloidal CsPbBr 3 NCs (grown in solution) the SML effect was not observed 22,23 . It demonstrates, that more investigations are needed to clarify whether it is related to some principal difference between colloidal NCs and NCs in glass, e.g. due to their very different surface conditions, and how this difference affects the spin coherence of carriers. However, the choice of a glass matrix or NC synthesis should not limit the observed effects fundamentally. We are convinced that they can be observed in different lead halide perovskite NCs, but also in lead-free NCs.

Samples
The studied CsPb(Cl, Br) 3 nanocrystals embedded in a fluorophosphate BaðPO 3 Þ 2 glass matrix were synthesized by rapid cooling of a glass melt enriched with the components needed for the perovskite crystallization. Samples of fluorophosphates (FP) glasses of 60BaðPO 3 Þ 2 À 15NaPO 3 À 12AlF 3 À 1Ga 2 O 3 À 4Cs 2 O À 8PbF 2 (mol. %) composition doped with 16 mol.% NaCl, 3.4 mol.% BaBr 2 were prepared using the melt-quench technique. The glass synthesis was performed in a closed glassy carbon crucible at temperatures of 1000-1050°C. About 50 g of the mixed powder was melted in the crucible during 20 min. Then the glass melt was cast on a glassy carbon plate and pressed to form a plate with a thickness of about 2 mm. The formation of the crystalline phase in glass can be carried out directly in the process of cooling the melt. During the pouring out of the transparent melt spontaneous precipitation of NCs with sizes smaller than the Bohr exciton radius occurs. By a subsequent heat treatment at a temperature just above the glass transition temperature (400-429°C), it is possible to sequentially grow NCs to large sizes. The details of the method are given in ref. 26 .
The NC size of 8 ± 1 nm is evaluated from a scanning transmission electron microscopy technique, see Supplementary Note 11. The concentration of NCs, of about 6 × 10 15 cm −3 , is evaluated from optical absorption measurements. The composition of the CsPbðCl 0:56 Br 0:44 Þ 3 NCs is estimated from the XRD data, see Supplementary Note 11. At T = 5 K the exciton resonance, mainly broadened due to the NC size dispersion, is at 2.743 eV. The band gap lies between that of bulk CsPbBr 3 and bulk CsPbCl 3 . Additional for NCs, compared to the bulk case, the optical transition is shifted due to the quantum confinement 47,48 . For optical experiments, the sample with a thickness of 500 μm was polished from both sides. The remarkable advantage of NCs in a glass matrix versus wet-chemistry synthesized NCs is the encapsulation resulting in long-term stability. Also the possibility to have optically flat sample surfaces with diminished light scattering is important for the polarization sensitive optical techniques used in this study. The effect of phase separation due to a laser excitation is excluded by low temperatures and relatively low laser powers applied resonantly to the NCs transitions. An additional long-term stability test is provided, in the Supplementary Note 11, to prove our claims.

Magneto-optical measurements
For low-temperature optical measurements we use a liquid helium cryostat with a variable temperature from 1.6 K up to 300 K. At T = 1.6 K the sample is immersed in superfluid helium, while at 4.2-300 K it is held in helium gas. A superconducting vector magnet equipped with three orthogonal pairs of split coils to orient the magnetic field up to 3 T along any arbitrary direction is used. The magnetic field parallel to k is denoted as B F (Faraday geometry), perpendicular to k as B V (Voigt geometry). If not stated otherwise, the B V is oriented horizontally. The angle α is defined as the angle between B and the light wave vector k, where α = 0°corresponds to B F .

Transmission spectra
A white-light halogen lamp is used to measure the optical transmission. The sample was thinned to 43 μm in order to avoid saturation of the light absorption in the spectral range of the exciton resonances. The spectra are detected by a 0.5 m monochromator equipped with a silicon charge-coupled-device camera.

Time-resolved Faraday ellipticity (TRFE)
The coherent spin dynamics are measured using a degenerate pumpprobe setup 31 . A titanium-sapphire (Ti:Sa) laser generates 1.5 ps long pulses in the spectral range of 700-980 nm (1.265-1.771 eV), which are frequency doubled with a beta barium borate crystal to the range of 350-490 nm (2.530-3.542 eV). The laser spectral width is about 1 nm (about 2 meV), and the pulse repetition rate is 76 MHz (repetition period T R = 13.2 ns). The laser output is split into two beams, pump and probe, which pulses can be delayed with respect to one another by a mechanical delay line. The laser photon energy is tuned to be in resonance with the exciton transition, e.g. at 2.737 eV at T = 1.6 and 5 K. The pump and probe beams are modulated using photo-elastic modulators (PEM). The probe beam is linearly polarized and its amplitude is modulated at 84 kHz, while the pump beam is either helicity modulated at 50 kHz between σ + /σ − circular polarizations or amplitude modulated at 100 kHz with the circular polarization fixed at either σ + or σ − . The signal was measured by a lock-in amplifier locked to the difference frequency of the pump and probe modulation. For measuring by spin inertia technique, the modulation frequency of the pump helicity f m was varied from 1 kHz to 5 MHz by an electro-optical modulator (EOM). For measuring the spin dynamics induced by the pump pulses, the polarization of the transmitted probe beam was analyzed with respect to its ellipticity (Faraday ellipticity) using balanced photodiodes and a lock-in amplifier. In B V ≠ 0, the Faraday ellipticity amplitude oscillates in time reflecting the Larmor precession of the charge carrier spins. The dynamics of the Faraday ellipticity signal can be described by a decaying oscillatory function: Here S is the amplitude at zero delay, ω L is the Larmor precession frequency, and T Ã 2 the spin dephasing time. For application of a twopump pulse scheme, after the delay line the so far not modulated linearly polarized pump beam was passed through a 50/50 nonpolarizing beam splitter. The deflected beam was sent via two mirrors over a certain distance and merged with the transmitted pump beam via a second nonpolarizing 50/50 beam splitter. By careful adjustment both pump beams were brought aligned parallel to each other, to excite the same sample spot with 300 μm diameter.

Pump-probe time-resolved differential transmission
Here, for excitation by the pump beam, the same scheme as in TRFE is used, but with linear polarization. The pump is amplitude modulated by a photo-elastic modulator at 100 kHz. The linearly polarized, unmodulated probe beam is separated into a reference beam, which is routed around the cryostat and the test beam, which is transmitted through the sample. The signal is recorded with respect to the intensity difference between the transmitted and the reference beams, using a balanced photodetector.

Optically-detected nuclear magnetic resonance (ODNMR) with TRFE detection
In this technique nuclear magnetic resonances are measured via resonant decrease of dynamic nuclear polarization by radio frequency (RF) radiation. For that optical detection of the TRFE amplitude at fixed magnetic field is used. A small RF coil of about 5 mm diameter having 5 turns is placed close to the sample surface, similar to ref. 49 . The coil is mounted flat at the sample surface and the laser beam is transmitted through the bore of the coil. The RF induced magnetic field direction is perpendicular to its surface and parallel to k. The RF in the frequency range f RF from 100 Hz up to 10 MHz is driven by a frequency generator, with an applied voltage of 10 V leading to an oscillating field amplitude of about 0.1 mT. The RF is terminated by internal 50Ω resistors, but not frequency matched to the circuit. In the low frequency range up to 5 MHz the current is nearly frequency independent, as the inductive resistance of the coil is small compared to the internal termination. For observation of DNP it is essential to tilt the external magnetic field away from the Voigt geometry, to have a nonzero scalar product B ⋅ S and apply a constant pump helicity.

Data availability
The data on which the plots in this paper are based and other findings of this study are available from the corresponding authors upon justified request.

Code availability
The code on which the calculations within this paper are based and other findings of this study are available from the corresponding author upon justified request.