Philipp S Grigoryev

Vasilii V Belykh

Dmitri R Yakovlev

Emmanuel Lhuillier

Manfred Bayer

Coherent Spin Dynamics of Electrons and Holes in CsPbBr 3 Colloidal Nanocrystals

Keywords: Perovskite nanocrystals, CsPbBr 3, coherent spin

The spin dynamics in CsPbBr 3 lead halide perovskite nanocrystals are studied by picosecond pump-probe Faraday rotation in an external magnetic field. Coherent Larmor precession of electrons and holes with spin dephasing times of ∼600 ps is detected in a transversal magnetic field. The longitudinal spin relaxation time in weak magnetic fields reaches 80 ns at a temperature of 5 K. In this regime, the carrier spin dynamics is governed by nuclear spin fluctuations characterized by an effective hyperfine field strength of 25 mT. The Landéfactors determining the carrier Zeeman splittings are g e = +1.73 for electrons and g h = +0.83 for holes. A comparison with a CsPbBr 3 polycrystalline film and bulk single crystals evidences that the spatial confinement of electrons and holes in the nanocrystals only slightly affects their g factors and spin dynamics.

In this Letter, we investigate the coherent spin dynamics of electrons and holes in CsPbBr 3 NCs using time-resolved pump-probe Faraday rotation (ppFR). We measure the electron and hole g-factors, their spread within the NCs ensemble, and the temperature dependence of the transverse and longitudinal spin relaxation times. The carrier interaction with the nuclear spins and its role in the spin dynamics are studied. The spin dynamic parameters in CsPbBr 3 NCs are compared with those in a polycrystalline film and a bulk single crystal.

■ OPTICAL PROPERTIES OF NANOCRYSTALS

We study CsPbBr 3 NCs dispersed in polymethyl methacrylate (PMMA) and spin-coated on a glass coverslip. (For synthesis details, see Section S2 in the Supporting Information.) The NCs have a cubic shape with a typical size of 10 nm, which slightly exceeds the exciton Bohr diameter of 7 nm. Therefore, the excitons experience a rather weak size quantization. Transmission electron microscopy images and optical spectra of similar CsPbBr 3 NCs can be found in ref 13. It should be noted that whereas CsPbBr 3 NCs are known to crystallize in the cubic phase, 3 a recent report shows that they can exhibit tetragonal and orthorhombic phases at cryogenic temperatures. [START_REF] Fu | Neutral and charged exciton fine structure in single lead halide perovskite nanocrystals revealed by magneto-optical spectroscopy[END_REF] Lower crystal symmetry will likely result in an anisotropy of the charge-carrier g-factors.

To characterize the optical properties of the studied CsPbBr 3 NCs, we measured the photoluminescence (PL) spectrum and recombination dynamics at a temperature of T = f1 5 K; see Figure 1a,b. The results are typical for CsPbBr 3 NCs, 96 as reported in literature; see, for example, ref 13 and references 97 therein. The PL line has maximum at 2.317 eV and a full width 98 at half-maximum of 29 meV. It shows a fast recombination, 99 two-component dynamics with a decay of 0.07 and 0.2 ns 100 (Figure 1b), in line with the 0.2 ns decay measured in single 101 NC experiments. [START_REF] Fu | Neutral and charged exciton fine structure in single lead halide perovskite nanocrystals revealed by magneto-optical spectroscopy[END_REF] We have checked that the application of 102 the magnetic field up to 6 T only slightly elongates the PL 103 decay. These decay times are considerably shorter than the 104 exciton decay time of 0.9 ns in CsPbBr 3 bulk crystals. [START_REF] Belykh | Coherent spin dynamics of electrons and holes in CsPbBr 3 perovskite crystals[END_REF] In 105 general, the emission of an NC ensemble can be contributed 106 by neutral and charged NCs. Therefore, the PL can contain the 107 emission of neutral, negatively charged, and positively charged 108 excitons. The ratio of their populations depends on the 109 synthesis protocol, the sample preparation, and the optical 110 excitation conditions. In our previous study, [START_REF] Canneson | Negatively charged and dark excitons in CsPbBr 3 perovskite nanocrystals revealed by high magnetic fields[END_REF] we measured 111 polarized PL and recombination dynamics in high magnetic 112 fields up to 30 T of similar CsPbBr 3 NCs and concluded that at 113 liquid helium temperatures, the emission is mainly contributed 114 by negatively charged excitons with some contribution from 115 neutral excitons. The present study using the ppFR 116 technique [START_REF] Yakovlev | Chapter 6. Coherent Spin Dynamics of Carriers[END_REF] allows us to gain more information on the NC 117 charging, [START_REF] Feng | Dynamic evolution from negative to positive photocharging in colloidal CdS quantum dots[END_REF] as will be discussed as follows. polarization, while the linearly polarized probe pulses detect its time evolution by varying their delay relative to the pump.

Figure 1c shows the spin dynamics measured for different magnetic fields applied in the Voigt geometry (field orientation perpendicular to the collinear pump and probe beams). The Faraday rotation amplitude oscillates with time, reflecting the coherent spin precession at the Larmor frequency about the magnetic field. The precession frequency increases with magnetic field strength, B V , growing from 0.125 to 1 T. Simultaneously, the signal decay accelerates. The modulations in the oscillatory signal arise from the contribution of two components with different precession frequencies, as confirmed in the fast Fourier transform (FFT) spectrum of the dynamics in Figure 1d. Indeed, the Faraday rotation dynamics can be well-fitted by the sum of two exponentially decaying oscillations

ω ω = - * + - * A S t
Here S e(h) is the amplitude of the electron (hole) spin signal, ω L,e(h) is the Larmor precession frequency of electrons (holes), We attribute the high-frequency component with the gfactor of 1.73 to the electron and the low-frequency component with the g-factor of 0.83 to the hole by analogy to bulk CsPbBr 3 perovskites. [START_REF] Belykh | Coherent spin dynamics of electrons and holes in CsPbBr 3 perovskite crystals[END_REF] Note that our experimental technique is not sensitive to the g-factor sign, which should be determined using an alternate method. We use here the knowledge of the exciton g-factor, g X = +2.4, that we measured in CsPbBr 3 NCs, [START_REF] Canneson | Negatively charged and dark excitons in CsPbBr 3 perovskite nanocrystals revealed by high magnetic fields[END_REF] and the fact that in lead halide perovskites, g X = g e + g h . This allows us to conclude that in the studied NCs, both electron and hole g-factors are positive, similar to CsPbBr 3 bulk crystals; [START_REF] Belykh | Coherent spin dynamics of electrons and holes in CsPbBr 3 perovskite crystals[END_REF] for details see Section S4 in the Supporting Information.

Let us clarify now the important question of whether neutral or singly charged NCs are responsible for the ppFR signal. In the case of neutral NCs, the signal should originate from the photogenerated electrons and holes bound to neutral excitons. There are two arguments that let us exclude this neutral exciton scenario. First, the measured spin dynamics should then be limited by the exciton recombination time, which is 0.2 ns for the studied NCs (Figure 1b). The measured spin dephasing times of electrons (0.6 ns) and holes (0.5 ns) are much longer than the exciton recombination time. The second argument is related to the exciton fine structure. Magnetooptical studies of emission from single CsPbBr 3 NCs show that the zero-field exciton exchange splitting of neutral excitons is ∼0.6 meV [START_REF] Fu | Neutral and charged exciton fine structure in single lead halide perovskite nanocrystals revealed by magneto-optical spectroscopy[END_REF] and that most NCs at low temperatures have either tetragonal or orthorhombic crystal symmetry. Note that in rather similar FAPbBr 3 NCs the splitting is found to be in the range from 0.3 to 1.7 meV. [START_REF] Tamarat | The ground exciton state of formamidinium lead bromide perovskite nanocrystals is a singlet dark state[END_REF] The exciton exchange interaction couples electron and hole spin so that the spins of

* = * + Δ ℏ T B T g B 1 ( ) 1 (0) 2,e(h) V 2,e(h) e(h) B V

226

(2)

Here Δg e(h) is the dispersion of electron (hole) g-factors.

228

* T (0) 2,e(h)
is the spin dephasing time at zero magnetic field, [START_REF] Belykh | Coherent spin dynamics of electrons and holes in CsPbBr 3 perovskite crystals[END_REF] This may be related to the spread of 238 NC sizes and shapes, which determines the g factor and its 239 anisotropy. The magnetic field dependence of the Faraday rotation signal measured at a negative time delay of -1 ns is shown in f3 Figure 3a. The finite signal amplitude implies that the spin relaxation time of carriers exceeds the repetition period of the laser pulses, T R = 13.1 ns. In fact, as we will show, it exceeds several T R periods and corresponds to a cumulative effect due to repeated pulse application. In this experiment, the laser helicity is kept constant, but the laser intensity is modulated with frequencies in the f = 0.3-6 MHz range. One can see that at the low modulation frequency of 0.3 MHz (black line), the signal amplitude has a minimum at zero magnetic field and increases with the field increase. It thus corresponds to a socalled polarization recovery curve (PRC). [START_REF] Zhukov | Spin inertia of resident and photoexcited carriers in singly charged quantum dots[END_REF] The half width at half-maximum of the PRC equals 25 mT, which is the characteristic field of nuclear spin fluctuations acting on the carriers via the hyperfine interaction. In our recent study of CsPbBr 3 crystals, we showed experimentally and theoretically that in lead halide perovskites, the holes in the valence band have a considerably stronger hyperfine interaction compared with the conduction band electrons. [START_REF] Belykh | Coherent spin dynamics of electrons and holes in CsPbBr 3 perovskite crystals[END_REF] Therefore, we suggest that the PRC signal in CsPbBr 3 NCs is mainly contributed by the hole-nuclear hyperfine interaction.

Nano

When the pump modulation frequency is increased to 6

MHz, the PRC amplitude decreases; see Figure 3a. This is evidence that the modulation period, 1/f, is comparable to T 1 in this case, and the spin accumulation loses its efficiency.

Therefore, we used the spin inertia method to measure the spin relaxation time, T 1 , in the longitudinal magnetic field. [START_REF] Belykh | Coherent spin dynamics of electrons and holes in CsPbBr 3 perovskite crystals[END_REF][START_REF] Zhukov | Spin inertia of resident and photoexcited carriers in singly charged quantum dots[END_REF][START_REF] Heisterkamp | Longitudinal and transverse spin dynamics of donor-bound electrons in fluorine-doped ZnSe: Spin inertia versus Hanle effect[END_REF] In addition, the detected signal is retarded by the phase, ϕ, which we also measure; see the Methods in the Supporting Information for details. Figure 3b shows the PRC amplitude and tan ϕ as a function of the modulation frequency, which gives us two ways to evaluate T 1 . Fitting the frequency dependence of the PRC amplitude with the spin inertia equation (Section S3 in the Supporting Information), we obtain T 1 = 83 ± 9 ns. The fit of the tan ϕ dependence on f with a linear equation (Section S4 in the Supporting Information) gives T 1 = 56 ± 5 ns. This difference in T 1 is caused by the nonmonoexponential spin relaxation dynamics:

The amplitude of the spin polarization is more sensitive to the slow component in the dynamics, whereas the retardation phase is more dependent on the fast component. We note that 323 We also studied the coherent spin dynamics in a CsPbBr 3 f4 324 polycrystalline film. Its PL spectrum (Figure 4a) has an 325 asymmetric shape with a low-energy tail, similar to the low-326 temperature emission of bulk CsPbBr 3 . [START_REF] Belykh | Coherent spin dynamics of electrons and holes in CsPbBr 3 perovskite crystals[END_REF] The spin precession 327 monitored by ppFR is given in Figure 4b at B V = 0.25 T and T 328 = 5 K. Similar to the NCs, it contains two Larmor precession 329 frequencies, where a fit with eq 1 gives the following ns. It is reasonable to suggest that similar to 332 CsPbBr 3 bulk and NCs, g e > 0 and g h > 0 in the film. 338 1). We note a small but monotonic variation of the g-factors 339 when going from NCs to film and further to bulk. The rather 340 close g-factor values are related to the weak quantum 341 confinement in the studied NCs, which is also confirmed by their close energy positions of the PL emission. The behavior is in line with the well-known relation between the optical transition energy and the electron g-factor for semiconductors. [START_REF] Roth | Theory of optical magnetoabsorption effects in semiconductors[END_REF][START_REF] Yugova | Universal behavior of the electron g-factor in GaAs/AlGaAs quantum 590 wells[END_REF][43] The small effect of confinement on the spin properties of the compared samples is further confirmed by the similar range of spin relaxation times in NCs and bulk perovskites. [START_REF] Belykh | Coherent spin dynamics of electrons and holes in CsPbBr 3 perovskite crystals[END_REF] In conclusion, we performed a detailed study of the electron * T 2 values in NCs and bulk are given for B V = 0.125 T, and those in film are given for B V = 0.25 T.

118■

  COHERENT SPIN DYNAMICS OF ELECTRONS AND 119 HOLES IN NANOCRYSTALS 120 To address the carrier spin dynamics, we use ppFR; see the 121 Methods in the Supporting Information. Here circularly 122 polarized pump pulses optically induce a carrier spin

Figure 1 .

 1 Figure 1. (a) Photoluminescence spectrum of CsPbBr 3 NCs. (b) Time-resolved recombination dynamics measured at the maximum of the PL line. The black dotted line shows the instrumental response function, whereas the red dashed line shows the double-exponential fit. (c) Coherent spin dynamics measured at 2.330 eV in different magnetic fields applied in Voigt geometry. The curves are shifted vertically and horizontally for clarity. The thick red line is a fit to the data at B V = 0.125 T with eq 1 and S e /S h = 0.5. In the bottom, the black and red lines are the two components of the fit describing the electron and hole spin precession. (d) Fast Fourier transform (FFT) spectrum of the Faraday rotation dynamics at B V = 1 T. (e) Magnetic field dependencies of the Larmor precession frequencies (corresponding Zeeman splittings are shown on the right axis) for electrons and holes. The error bars are within the data points. Solid lines are linear fits, giving |g e | = 1.73 and |g h | = 0.83. (f) Magnetic field dependencies of the spin dephasing times for electrons and holes. Lines are fits according to eq 2, with parameters given in the text.

  and * T 2,e(h) is the electron (hole) spin dephasing time. The contributions of the two dynamics components at B V = 0.125 T are shown separately in the bottom of Figure 1c. The fit electron and hole signals, S e /S h ≈ 0.5. The Larmor precession frequencies of both components increase linearly with magnetic field (Figure 1e). By fitting them with ω L,e(h) = |g e(h) |μ B B V /ℏ, where μ B is the Bohr magneton and g e(h) is the electron (hole) g factor, we evaluate |g e | = 1.73 and |g h | = 0.83.

  photoexcitation of a charge exciton (trion) state, as studied in

229 which is limited

  by the spin coherence time, T 2,e(h) , of the 230 individual carriers and the spin relaxation mechanisms not 231 related to the g-factor dispersion, for example, the hyperfine 232 interaction with nuclear spin fluctuations. This equation gives 233 reasonable fits to the experimental dependencies of * T B ( ) 2,e(h) V with * = T (0) 1.2 2,e ns and * = T (0) 1.0 2,h ns and g-factor 235 spreads of Δg e = 0.07 and Δg h = 0.08. Note that the measured 236 g-factor spreads are about two times larger than those in bulk 237 CsPbBr 3 single crystals.

  260 ■ LONGITUDINAL SPIN RELAXATION OF CARRIERS 261 IN NANOCRYSTALS 262 Let us turn to experiments in the Faraday geometry where the 263 external magnetic field, B F , is applied along the light wave 264 vectors. Consequently, the optically induced spin polarization of carriers along the field direction does not show Larmor precession. This allows one to measure the longitudinal spin relaxation time, T 1 , and gather information on the nuclear spin fluctuations.

Figure 2 .

 2 Figure 2. Dynamics of Faraday rotation signal in CsPbBr 3 NCs measured at 2.330 eV for different temperatures in B V = 0.125 T. Data are shifted vertically for clarity. The inset shows the temperature dependence of the spin dephasing time for electrons and holes.

Figure 3 .

 3 Figure 3. Spin dynamics of CsPbBr 3 NCs measured in longitudinal magnetic fields (Faraday geometry). (a) Polarization recovery curves measured at different pump modulation frequencies. Data are shifted vertically for clarity. T = 5 K. (b) PRC amplitude (red squares) and tangent of the retardation phase (green circles) as a function of the modulation frequency. Lines show fits with eqs S3 and S4 in the Supporting Information to the experimental data. T = 5 K. (c) Temperature dependence of the longitudinal spin relaxation time, T 1 . Inset shows the Faraday rotation signal dynamics in the longitudinal magnetic field, B F = 0.15 T, at T = 22 K.

  330 parameters: |g e | = 1.78, |g h | = 0.77,

  333 ■ SUMMARY OF g-FACTORS AND SPIN 334 RELAXATION TIMES IN CsPbBr 3 PEROVSKITE 335 STRUCTURES 336 It is instructive to compare the g-factor values for CsPbBr 3 337 perovskite NCs, polycrystalline film, and bulk crystals (Table

  and hole spin dynamics in CsPbBr 3 perovskite NCs. Electron and hole g-factors of +1.73 and +0.83, respectively, have been measured from spin beats in the transverse magnetic field. The spin dephasing times * T 2 of ∼1 ns considerably exceed the exciton recombination time, which has allowed us to attribute the measured spin dynamics to NCs singly charged with an electron or a hole. In the longitudinal magnetic field, a spin relaxation time T 1 exceeding 80 ns has been measured. The influence of the carrier-nuclear hyperfine interaction, which controls the carrier spin dynamics at zero and weak magnetic fields, has been identified. Our study underlines that perovskite NCs have promising spin-dependent properties, very much comparable to the comprehensively studied epitaxially grown III-V quantum dots and colloidally synthesized II-VI NCs.

Figure 4 .

 4 Figure 4. (a) Photoluminescence spectrum of the CsPbBr 3 polycrystalline film. (b) Coherent spin dynamics measured in the film at 2.330 eV for B V = 0.25 T. The panel gives experimental data (green line) and fit to the data (red line) with eq 1 and S e /S h = 0.75.

  The amplitudes of both the electron and hole spin signals 243 gradually decrease with increasing temperature. That means 244 that the decrease in the signal amplitude is provided by a 245 decreasing efficiency of the optical spin polarization. On the 246 contrary, the spin dephasing times, shown in the inset of Figure2472, change only weakly in this temperature range. The hole

			Letters	pubs.acs.org/NanoLett	Letter
		240	Oscillating signals in the Faraday rotation dynamics are
	f2	241 detectable up to temperatures of 35 K, as shown in Figure 2.
		248 dephasing time remains almost constant at * = T 2,h	0.5	ns,
		249 whereas the electron time, * T 2,e , slightly decreases from 0.7 to
		250 0.4 ns. This is in contrast with the strong temperature
		251 dependence of * T 2,e in bulk CsPbBr 3 , 32 where temperature
		252 induces the delocalization of carriers and the activation of spin
		253 relaxation mechanisms related to the spin-orbit coupling. On
		254 the contrary, in NCs, carriers are localized at any temperature.
		255 The other difference in NCs from bulk CsPbBr 3 , presumably
		256 related to the localization, is the much more homogeneous
		257 spin dynamics for NCs (dephasing times and ratio between
		258 electron and hole contributions) for different positions on the
		259 sample.
				C

https://doi.org/10.1021/acs.nanolett.1c03292 Nano Lett. XXXX, XXX, XXX-XXX 242

  The longitudinal spin relaxation time, T 1 , determined from 315 the frequency dependence of the PRC amplitude is shown in 316 Figure3cas a function of temperature. It rapidly decreases 317 with temperature and saturates at ∼2 ns above 20 K. At 318 temperatures higher than 20 K, T 1 was measured by fitting the 319 decay of the Faraday rotation signal with a single exponential 320 function; see the inset of Figure3c.

		Nano Letters	pubs.acs.org/NanoLett	Letter
		311 the evaluated T 1 values for NCs are a few tens of nanoseconds 312 longer than T 1 for the corresponding bulk perovskites; 32 see
	t1	313 Table 1.
		321 ■ COHERENT SPIN DYNAMICS IN 322 POLYCRYSTALLINE FILM
			D

https://doi.org/10.1021/acs.nanolett.1c03292 Nano Lett. XXXX, XXX, XXX-XXX 314

Table 1 .

 1 Spin Parameters Measured for Different CsPbBr 3 Perovskite Samples a Results for bulk CsPbBr 3 (single crystal) are taken from ref 32.

		nanocrystals	polycrystalline film	bulk 32
	electron g-factor, g e	+1.73	+1.78	+1.96
	hole g-factor, g h	+0.83	+0.77	+0.75
	exciton g-factor, g X	+2.4 13		+2.35
	electron spin dephasing time, T 2,e *	0.6-0.7 ns	0.25 ns	0.9-5.2 ns
	hole spin dephasing time, T 2,h *	0.5 ns	0.4 ns	0.7-1.9 ns
	longitudinal spin relaxation time, T 1	56-83 ns		32-53 ns
			■ ASSOCIATED CONTENT * sı Supporting Information
			The Supporting Information is available free of charge at
			https://pubs.acs.org/doi/10.1021/acs.nanolett.1c03292.
			Details of experimental methods, synthesis and charac-
			terization of CsPbBr 3 NCs and polycrystalline film, and
			Zeeman Hamiltonian and g-factors of charge carriers and
			excitons (PDF)
			■ AUTHOR INFORMATION Corresponding Author
			Dmitri R. Yakovlev -Experimentelle Physik 2, Technische
			Universität Dortmund, 44221 Dortmund, Germany; P.N.
			Lebedev Physical Institute, Russian Academy of Sciences,
			119991 Moscow, Russia; Ioffe Institute, Russian Academy of
			Sciences, 194021 St. Petersburg, Russia; orcid.org/0000-
			0001-7349-2745; Email: dmitri.yakovlev@tu-dortmund.de

Authors

Philipp S.

Grigoryev

-Spin Optics Laboratory, St. Petersburg State University, 199034 St. Petersburg, Russia; orcid.org/0000-0002-3349-062X Vasilii V. Belykh -Experimentelle Physik 2, Technische Universität Dortmund, 44221 Dortmund, Germany; P.N. Lebedev Physical Institute, Russian Academy of Sciences, 119991 Moscow, Russia; orcid.org/0000-0002-0032-748X a