Abstract
Utilizing the spin degree of freedom of photoexcitations in hybrid organic inorganic perovskites for quantum information science applications has been recently proposed and explored. However, it is still unclear whether the stable photoexcitations in these compounds correspond to excitons, free/trapped electronhole pairs, or charged exciton complexes such as trions. Here we investigate quantum beating oscillations in the picosecond timeresolved circularly polarized photoinduced reflection of single crystal methylammonium triiodine perovskite (MAPbI_{3}) measured at cryogenic temperatures. We observe two quantum beating oscillations (fast and slow) whose frequencies increase linearly with B with slopes that depend on the crystal orientation with respect to the applied magnetic field. We assign the quantum beatings to positive and negative trions whose Landé gfactors are determined by those of the electron and hole, respectively, or by the carriers left behind after trion recombination. These are \({g}_{[001]}^{e}\) = 2.52 and \({g}_{[1\bar{1}0]}^{e}\,\)= 2.63 for electrons, whereas \(\big{g}_{[001]}^{h}\big\,\)= 0.28 and \(\big{g}_{[1\bar{1}0]}^{h}\big\,\)= 0.57 for holes. The obtained gvalues are in excellent agreement with an 8band K.P calculation for orthorhombic MAPbI_{3}. Using the technique of resonant spin amplification of the quantum beatings we measure a relatively long spin coherence time of ~ 11 (6) nanoseconds for electrons (holes) at 4 K.
Introduction
The hybrid organic–inorganic perovskite (HOIP) semiconductors have attracted intensive research interest due to their remarkable optoelectronic properties such as large absorption coefficients, strong photoluminescence emission, high carrier photogeneration efficiency and large carrier diffusion length^{1,2,3,4}. Photovoltaic powerconversion efficiency of solar cells based on the HOIPs has reached 25.6%^{5,6}. Moreover, the HOIPs have also been used as active materials in other optoelectronic devices, including lightemitting diodes^{7,8} and phototransistors^{9}. In addition, HOIP spintronic properties have recently attracted growing attention due to the relatively long spinrelaxation time in these materials^{10,11,12}. In particular, Rashba spin splitting^{13,14}, optical spinselection rules^{15} and magneticfield effects on photocurrent, electroluminescence and photoluminescence in HOIP devices and films, respectively, have been observed^{16}. Noteworthy potential applications of HOIPs in quantuminformation science (QIS) such as quantum logic and quantum communication have been recently demonstrated^{1,17,18}.
One of the key requirements for an active semiconductor in QIS is utilizing the spin degree of freedom of its lowestlying photoexcitations. In this regard the photoexcitations in the orthorhombic HOIP phase at cryogenic temperatures are particularly important. However, the stable photoexcitations in HOIP single crystals have not been sufficiently investigated, and it remains unclear whether they correspond to free excitons, bound excitons, free/trapped electron–hole (e–h) pairs, or chargedexciton complexes such as trions^{19,20,21,22}. The reason for this debate is the relatively small excitonbinding energy in these compounds, which makes it difficult to decide whether or not the excitons dissociate immediately upon light excitation, and then fall into shallow traps known to exist in HOIPs. For proper spectroscopic analysis, it is noteworthy that the exciton species in these compounds have four different quantum substates (namely exciton fine structure, EFS), comprising one ‘dark singlet’ and three ‘bright triplet’ levels that may be split in energy by few 100 μeV^{23,24,25,26,27,28}. Consequently, it is conceivable that quantum beatings (QBs) among the four excitonic states would reveal up to six different oscillation frequencies when applying an external magnetic field, which further splits these states and renders all four states optically active^{10,29}. However, only two QB frequencies have been observed^{10}, with negligible zerofield frequencies^{10} that imply no zerofield finestructure splitting, this casts doubt on the exciton involvement in the QB phenomenon. Another issue with the exciton model for explaining the QBs is the photoluminescence (PL) spectrum from HOIPs that rarely shows free exciton emission or emission from e to h pairs associated with the continuum at the band edge. In fact, most of the steadystate PL emission in these compounds originates from subbandgap states^{30,31,32,33}, whose nature has not yet been clearly established.
Here we have used ultrafast spectroscopy at liquid He temperature (4 K) for measuring transient QBs in a highquality single crystal of the prototypical HOIP, namely the methylammonium lead iodine, MAPbI_{3} (Fig. 1a) in the orthorhombic phase. Specifically, we measured transient spin relaxation and QBs using circularly polarized photoinduced reflectivity (PPR) that are generated due to the Zeeman splitting associated with an applied magnetic field. We found that the spinrelaxation time in the highquality singlecrystalline MAPbI_{3} at 4 K is of the order of ten ns, which enables the observation of several oscillations associated with two different QBs with slow and fast frequencies. Importantly, we found that the QB frequencies at zero field are miniature (i.e., <0.1 GHz), this is in contrast to the exciton explanation for the QB, since the lack of QB at zero field indicates ultrasmall EFS that is not the case in HOIPs^{24,25,26,27}. From this and the different excitation spectra of the slow and fast oscillations, we assign the QBs in MAPbI_{3} to positive and negative trions, for which the Zeeman energy is determined by the lone charge in the trio of particles^{34}, as explained in Supplementary Note 3. Consequently from the linear QBfrequency responses with B, we obtained the anisotropic gvalues of the electron and hole (e&h) in the MAPbI_{3} crystal. The measured anisotropy is in excellent agreement with an 8band K.P calculation for orthorhombic MAPbI_{3}, presented in Supplementary Note 1, and allows us to determine for the key bandstructure parameters by fitting to the measured gfactor anisotropy.
Results
The MAPbI_{3} single crystals in our study were synthesized by a ligandassisted growth method, which has been demonstrated to tune the facets of the crystals^{35}. The grown crystals also have a recordlow chargetrap density due to the modulation of ion addition during the solutiongrowth process. Our experimental setup for measuring the transient PPR is shown in Fig. 1c. It is a derivative of the wellknown degenerate pump/probe technique, where the polarization of the pump beam is modulated by a photoelastic modulator between left (δ^{+}) and right (δ^{}) circular polarization (LCP or RCP) for circular PPR (cPPR). In this scheme the probe beam is circularly polarized (either LCP or RCP) by a quarterwave plate. The transient change in the probe reflection, \({\triangle R}_{{\delta }^{}{\delta }^{+}}^{{\delta }^{+}{\delta }^{+}}\,\) = \({R}_{{\delta }^{+}{\delta }^{+}}\) − \({R}_{{\delta }^{}{\delta }^{+}}\) is recorded (see ‘Methods’ for more details). In contrast to the more traditional pump/probe technique in which the pump intensity is modulated and the measured photoreflectivity (or photoabsorption) is proportional to the photogenerated exciton density, N; the cPPR is proportional to the population difference \(({N}_{{\delta }^{+}}{N}_{{\delta }^{}})\) between LCP and RCPpump excitations.
For our tPPR measurements, we have used a Ti: sapphire pulse laser having ~150femtosecond pulse duration at ~80 MHz repetition rate, which could be tuned from 730 nm to 830 nm. The fundamental beam was split into two beams by an 80/20 beam splitter for pump and probe in the degenerate configuration. The pump and probe beams were aligned onto the MAPbI_{3} crystal that was placed inside a cryostat with a builtin electromagnet that delivered a field with strength, B up to 700 mT that was applied parallel to the crystal surface (i.e., Voigt geometry), at temperature between 4 K and 300 K. Since the pulsedlaser linewidth (~8 meV) is relatively broad, at resonance condition with the exciton levels in MAPbI_{3}, the pump pulse may simultaneously excite spin sublevels of various excitations such as free and trapped excitons and e&h pairs that may lead to transient QBs among their spin sublevels. When measuring QBs at B = 0, the transient PPR technique is a unique method for resolving a small ZFS, which may be in the µeV energy range (resulting in few ns QB oscillatory period). At B > 2 mT, we found that the QB frequencies change linearly with B, from which we obtained the associated gfactors and their anisotropy in the MAPbI_{3} crystal.
The transient cPPR measurements were conducted on a MAPbI_{3} single crystal at cryogenic temperature, where the beams were aligned along two crystal orientations, namely [100] and [110] (Fig. 1b), with an applied magnetic field in the Voigt configuration, in which the field direction is parallel to the crystal surface directed along [001] and \([1\bar{1}0]\), respectively. Figures 2 and 3 show the cPPR(t) dynamics and their corresponding fast Fourier transform (FFT) spectra measured for different crystal orientations and magneticfield strengths at ~758 nm (~1.64 eV) pump excitation, which is in resonance with the exciton feature in MAPbI_{3}. As seen from the FFT spectra (Fig. 2a–f and Fig. 3d–f), at B > 50 mT, we clearly observe two QB frequencies, namely slow and fast, which increase linearly with B.
In order to study the ZFS of the QBrelated photoexcitations, we measured the cPPR dynamics at zero magnetic field up to 8 ns time delay, as shown in Fig. 4a for measurements on the (100) facet, with magnetic field directed along [001]. It is seen that the cPPR(t) at B = 0 lasts beyond 8 ns with estimated spin lifetime, τ > 10 ns. Applying a small B = 2 mT causes the cPPR(t) to cross the zero line at ~5 ns, indicating that there is a very long QB oscillation whose period is comparable to the pulse laser repetition time (t_{rep} ~ 12.5 ns here). At increasing B, the QB period decreases further (Fig. 4a).
For better studying the ultrasmall QB frequency at B = 0, we used a different approach, where we take into account the remaining response from the preceding pulses due to the relatively short repetition time, t_{rep}. In this approach, known in the literature as resonant spin amplification (RSA)^{36} instead of scanning the delay time, t, in measuring the cPPR(t) response, we scanned the B field at a constant delay time, t and measured the magnetic PPR response; we name this cPPR version ‘magnetic field circular PPR’, or McPPR(B,t), as seen in Fig. 4b. The observed oscillations in the McPPR(B) can be analyzed by the equation^{36}
where the index n runs over the present and preceding laserpump pulses that contribute to the McPPR response. The index i represents the fast and slow contributions for which \({\Gamma }_{f,s}\) and g_{f,s} are their spinrelaxation rate and Landé gfactors, respectively. The important parameter in Eq. (1) that we measure is ΔE, which is the ZFS energy of the related photoexcitations. The obtained McPPR(B) responses were fitted by adjusting the parameters A_{i} and ΔE, while fixing g_{f/s} values as obtained from the cPPR(t) dynamics at various B fields (Fig. 2), as well as the Bdependent spinrelaxation rate Γ_{i}(B), which were measured separately (see below). A good fit to the experimental McPPR(B) response was obtained using ΔE_{(100)} = 40 neV (nano eV). This value is in good agreement with the ultrasmall ZFS energy that is obtained from the extrapolation to B = 0 of the QB frequencies vs. B in the cPPR(t) (Fig. 2g).
The fact that the ZFS is negligibly small indicates that excitons are not the photoexcitations underlying the observed QBs^{37}. In Supplementary Note 2, we show the calculated magnetoexciton fine structure for MAPbI_{3} and the resulting simulated cPPR signatures; the participation of excitons in the QB response is shown in the S.I. to lead to two distinct beat frequencies at zeroapplied magnetic field with two additional beat frequencies emerging with increased magnetic field due to magnetic activation of the dark exciton. This behavior stands in stark contrast to the experimental observations here. We thus conclude that the QBs in MAPbI_{3} are due either to electrons and holes or to positive and negative trions, where the excitons are trapped by photogenerated holes and electrons, respectively, or to separately localized resident carriers remaining after trion recombination^{38,39,40}.
MAPbI_{3} is known to have photogenerated charges with long lifetime even at room temperature, let alone at 4 K. Therefore, there is a large background density of photogenerated electrons and holes that are due to the pump excitation that can trap the photogenerated excitons. Note that trions comprise a bound state of three particles. A negative trion, T^{−} is composed of one hole and two electrons (hee), whereas a positive trion, T^{+} is composed of one electron and two holes (ehh), as schematically shown in Fig. 4c and d, respectively. It is known that the overall exchange interaction of trions among the three bound particles, which determines the trion ZFS energy, in fact vanishes in the trion ground state. This was shown in ref. ^{41} and discussed in Supplementary Note 3. Moreover, when trions are photogenerated, the spins of the two likecharge particles need to align antiparallel to each other (see Supplementary Fig. 5 and related detailed discussion in Supplementary Note 3). Therefore, the Zeeman splitting of a trion is determined by the lone charge in the particles’ trio^{34}. In particular, the Zeeman splitting of T^{+} is governed by the electron gfactor, \({g}_{e}\), since \({H}_{Z}={g}_{e}{\mu }_{B}B\); whereas the Zeeman splitting of T^{−} is governed by the hole gfactor, \({g}_{h}\). Consequently, in either case (trions or residual carriers left behind after trion decay), the gfactors extracted from the QB frequencies vs. B are those of the electron and hole. It is not possible to assign the particular gvalue to either particle, but when comparing with calculations^{10, 29}, it is likely that the larger gvalue (fast oscillation) is due to the electron in the T^{+}; whereas, the smaller gvalue (slower oscillation) is due to the hole in T^{−}. From the QBs measured along different crystal orientations, we thus obtain the anisotropic gfactors in MAPbI_{3} to be \(\big{g}_{[001]}^{e}\big\) = 2.52 and \(\big{g}_{[1\bar{1}0]}^{e}\big\) = 2.63 for the electron; whereas \(\big{g}_{[001]}^{h}\big\) = 0.28 and \(\big{g}_{[1\bar{1}0]}^{h}\big\) = 0.57 for the hole; where the subscripts refer to the direction of the applied magnetic field. The relation \(\big{g}_{e}\big \; > \; \big{g}_{h}\big\) observed here is consistent with the K.P model calculation of ref. ^{29}, which was performed for tetragonal MAPbI_{3}. However, our experiments are performed at liquid helium temperature at which MAPbI_{3} is known to be in the orthorhombic phase, so that the model of ref. ^{29} is not quantitatively applicable.
We therefore developed an 8band K.P model for orthorhombic HOIPs such as MAPbI_{3}. This model is outlined in the Methods section and described in detail in Supplementary Note S1. In short: Employing the quasicubic approach of ref. ^{27}, we determine the bandedge Bloch functions of the conduction band by diagonalization of the K.P Hamiltonian, that also includes the spinorbit interaction and the crystal field contributions^{27,29} which break the symmetry of the [100], [010] and [001] directions in the orthorhombic structure. Writing the full 8band K.P Hamiltonian that includes kdependent terms, see Supplementary Eq. S17–S18 in SI, and adding the magnetic Hamiltonian^{42} given in Supplementary Eq. S20 as a perturbation (see Methods), we evaluate the effective Zeeman Hamiltonian for the lowest conduction band and the valence band using Löwdin’s partition method^{43} and thereby determine the electron and hole gfactors. While the gfactors are isotropic in the cubicphase HOIPs, in tetragonal or orthorhombic perovskites, the symmetry between the orthogonal [100], [010] and [001] directions is broken, leading to anisotropic gfactor tensor components as shown in Fig. 5. Note that the six principal gfactor components (x, y and z values for the electron and the hole, respectively, where x, y and z align to [100], [010] and [001]) are fully determined by six parameters: the bandgap, \({E}_{g}\); the Kane energy^{27}, \({E}_{p}\) the spin–orbit splitoff parameter, \(\Delta\); the tetragonal^{29} and orthorhombic^{27} crystal field parameters \(\delta ,\zeta\), respectively, and Luttinger’s magnetic parameter^{42}, \(\kappa\). The bandgap is determined from the PL spectra (see Fig. 6), while the splitoff parameter is known from independent measurements and calculations^{29}, enabling determination of the remaining four parameters \({E}_{p}\), \(\delta ,\zeta\) and \(\kappa\) by measuring the electron and hole gfactors along two axes of the crystal as described above. In Table 1, we show the values of these parameters calculated by best fit to the measured gfactors; the resulting calculated gfactors match the experimental values.
The resulting values for the Kane energy and Luttinger’s magnetic parameter are close to the corresponding values for tetragonal MAPbI_{3}^{29} as expected in a quasicubic system. From the fit, we find that the tetragonal crystal field is positive, in agreement with densityfunctional theory (DFT) calculations of band structure of tetragonal MAPbI_{3} reported in refs. ^{44,45}, the hybrid DFT calculations for both tetragonal and orthorhombic MAPbI_{3} reported in ref. ^{27} and the 16band K.P model for tetragonal MAPbI_{3} reported in ref. ^{24}, but contrary to the analysis in ref. ^{29} (see the illuminating discussion in ref. ^{26}). The magnitude, δ = + 349.8 meV, is in line with the range of the values calculated using DFT or hybrid DFT ~ + 100–240 meV^{24,27}. The orthorhombic crystal field is found to be \(\zeta\) = +147.7 meV, which agrees in sign but is larger in magnitude than the value (+82 meV) calculated using hybrid DFT in ref. ^{27}. The bandedge energies and the exciton fine structure that result from these parameters are shown in Supplementary Figs. 1, 2.
In order to test the trion scenario, we measured the excitation dependence of the two QBs. We changed the pump–probe wavelength and measured the QBs at fixed B = 400 mT on (100) crystal facet with Bfield direction along [001]. Subsequently, we performed a Fast Fourier transform of the periodic cPPR(t) response to obtain the FFT component of the slow and fast QB oscillations. The excitation dependences of the QB FFT components are shown in Fig. 6a. It is seen that the excitation spectrum of the slow (hole) and fast (electron) is different. In particular, they peak at different energies; the fast QB component peaks at ~1.63 eV, whereas the slow QB component peaks at ~1.64 eV. Both energies are below the MAPbI_{3} exciton energy, E_{x} at this temperature (E_{x} ~ 1.65 eV) as deduced from the free exciton PL band (Fig. 6c)^{32}. From the different excitation spectra for the slow and fast oscillations we conclude that the QBs cannot be due to free e–h pairs, which should show overlapping excitation spectra; and this supports the trion interpretation. Consequently, we calibrate the positive and negative trion optical transition in MAPbI_{3} crystal to be at 1.63 eV and 1.64 eV, respectively.
In further support for this assignment we note that the sample is constantly illuminated by the pulsed excitation at low temperature, where the resulting photoexcitations do not decay inbetween adjacent pump pulses. Under these conditions steady state photocarrier density may be formed in the illuminated area of the crystal that subsequently leads to trion photogeneration, especially at the resonance excitation condition. To check this assumption, we performed steady state photoinduced absorption (PA) measurement in MAPbI_{3} film using a cwlaser illumination and probed by a FTIR spectrometer (see Methods). In this experiment, the laser light was slowly modulated and the change, ΔT in the transmission spectrum, T was monitored, where PA = −ΔT/T. Figure 6b shows the PA spectrum of MAPbI_{3} measured at low temperature. It clearly shows the PA spectrum of free carriers, namely free carrier absorption (FCA) that increases towards small photon energy, ħω, as ω^{−2}. In the PA experiment it can be easily shown that −ΔT/T = Δαd; where d is the film’s thickness, Δα is the induced absorption that is given by the relation Δα = Nσ, where N is the photocarriers density and σ is the optical cross section. Since we know the film thickness d = 100 nm, we estimate Δα from the PA band to be ~100 cm^{−1} at ħω = 1000 cm^{−1}. Consequently, from the previous FCA crosssection value at 1000 cm^{−1}, σ ~ 10^{−16} cm^{2}, we estimate the steadystate photocarrier density in MAPbI_{3} at liquid He temperature to be N ~ 2 × 10^{17} cm^{−3}. Taking into account that the average power in the pulsed experiment is about an order of magnitude larger than that used in the cw measurements, we estimate a background steady state photocarriers density of ~10^{18}/cm^{3}, which is much larger than the carrier density in equilibrium at 4 K, or the density of impurities. This high carrier density is sufficiently large to capture most photoexcited excitons in the sample^{46}. As a result, the photoluminescence spectrum in the MAPbI_{3} crystal at low temperatures is dominated by trion emission, as seen in Fig. 6d and Supplementary Fig. 7, which is further redshifted possibly due to the photonrecycling process^{47}. The PL emission of the MAPbI_{3} crystal at 10 K as a function of the laser excitation intensity (I_{L}) shows a dominant trion band that grows as (I_{L})^{1.5} as shown in Supplementary Fig. 7b. In addition, the PL emission band blueshifts with increasing temperature and changes abruptly at the tetragonaltoorthorhombic phasetransition temperature^{32}. Taken together, these measurements support our assignment of photogenerated trions in MAPbI_{3}.
To provide additional experimental evidence for trions, we measured the QBs of a pristine MAPbI_{3} film and compared these to those of a seemingly ntype doped film; the doping was achieved by soaking the MAPbI_{3} film in benzyl viologen (BV) solution in toluene for ~one minute, where BV is an electrondonating molecule (see ref. ^{48}). It is known that MAPbI_{3} can selfdope according to the precursor ratio of MAI to PbI_{2} (see ref. ^{49}). Our ‘as grown’ MAPbI_{3} films fall in the category of pdoped, so that the light doping achieved by soaking in BV actually compensates the ptype dopants that are originally in the film. As shown in Supplementary Fig. 8, both QB amplitudes increase upon compensation, because the steadystate photocarrier density background increases in ‘intrinsic’ semiconductors. The mere fact that the QB amplitude changes upon ‘doping’ supports our assignment for the trion quantum beatings, otherwise, there should not be any dependence of the QB amplitude on the Fermi level in the film.
We further use the QB spectroscopy to investigate the spin dynamics of the trions in MAPbI_{3}. First, we fit the two QB oscillations using the equation
where f_{1} and f_{2} are the two QB frequencies (namely T^{+} and T^{−}) that were extracted from the FFT of the transient dynamics, and τ_{1} and τ_{2} are the QBdecay lifetimes. This was done at different magnetic fields at 4 K (Fig. 7a and Supplementary Figs. 9–11) and at various temperatures at a fixed field of 400 mT (Fig. 7b and Supplementary Fig. 12). As depicted in Fig. 7a, the T^{+} and T^{−} spinrelaxation times, \({\tau }_{e}^{+}\) and \({\tau }_{h}^{}\) measured along [100] crystal orientation, are nearly constant at field B < 400 mT, but steeply decrease as 1/B at B ≧ 400 mT. We note that the \({\tau }_{e}^{+}\) saturation at small magnetic fields could be due to the limit of our pulserepetition time (t_{rep} ~ 12.5 ns). Therefore, for obtaining the correct T^{+} spin lifetime, we measured the McPPR(B,t) response (i.e., the RSA method discussed above) at negatively fixed delay time t = −400 picoseconds (ps) (see Fig. 7c). This RSA method has been used before to extract \({\tau }_{2}^{* }\) for electrons in GaAs (\({\tau }_{2}^{* }\) ≈ 130 ns, ref. ^{36}) and CdTe quantum wells (\({\tau }_{2}^{*}\)≈ 30 ns, ref. ^{50}) even though that the pulsetopulse time period, t_{rep} was much smaller than the extracted \({\tau }_{2}^{* }\) in both cases. Given that t_{rep} is actively stabilized to 1.5 ps, and the magnetic field is measured to 0.1 Gauss, the RSA method is able to determine the spin lifetime as long as 5 μs, even that t_{rep} is ~2 orders of magnitude shorter (see ref. ^{36}. and ref. ^{40}).
As seen in Fig. 7c, the RSA response at t = −400 ps is comprised of a sequence of RSA resonances having full width at half maximum (FWHM) as narrow as ~1 mT. We note that \({\tau }_{2}^{* }\) may be directly obtained from the width of the RSA resonances, where longer \({\tau }_{2}^{* }\) corresponds to narrower width. For example, for electrons in GaAs with \({g}_{e}\,\) = 0.44, the FWHM of ~0.3 mT corresponds to \({\tau }_{2}^{* }\) of 130 ns. Similarly, in CdTe QWs with \({g}_{e}\,\) =1.64, the FWHM of ~0.6 mT results in \({\tau }_{2}^{* }\) ~ 30 ns. In the MAPbI_{3} crystal with \({g}_{[001]}^{e}\,\) = 2.52, the FWHM at B = 2 mT is ~ 1mT, which gives \({\tau }_{2}^{* }\) of ~10–15 ns. In order to get a precise \({\tau }_{2}^{* }\) for T^{+} (\({\tau }_{2}^{* }={\tau }_{e}^{+}\)) we fitted the RSA response at B = 2 mT using Eq. (1) and obtain \({\tau }_{e}^{+}\) = 11 ± 1 ns, which is consistent with the \({\tau }_{e}^{+}\) value extracted from the timescan experiments. In addition, we also explored the RSA resonances at larger B. We observed that the FWHM of the RSA resonances at B < 400 mT is the same as for B = 2mT, confirming that \({\tau }_{e}^{+}\) saturation at small fields extracted from the timescan experiment is correct.
The steep decrease of τ(B) was previously observed in CdTe quantum wells^{40,50} and attributed to an inhomogeneous dephasing of an ensemble of carrier spins caused by a variation of Larmor precession frequencies due to dispersion of the carriers’ gfactor, Δg. This can be written as
Here \({\tau }_{2}^{* }\) = \({\tau }_{e}^{+}\) or \({\tau }_{h}^{}\), \({\tau }_{0}\) is the spin lifetime at zero and small B fields, and \(\Theta \left({B}_{0}\right)\)is a step function that describes the saturation of \({\tau }_{2}^{* }\) for B_{0} < 400 mT. From the 1/B fitting of experimental data for B ≥ 400 mT (Fig. 7a), we obtain \(\triangle {{{{{{\rm{g}}}}}}}_{{{{{{\rm{e}}}}}}}^{+}\) = 0.005 and \(\triangle {{{{{{\rm{g}}}}}}}_{{{{{{\rm{h}}}}}}}^{}\) = 0.011 for B along [001] crystal orientation. We note that the observed saturation of \({\tau }_{e}^{+} \sim\) 11 ns and \({\tau }_{h}^{}\) ~ 6 ns sets a lower limit of \({\tau }_{0}\) for T^{+} and accordingly for T^{−}.
Furthermore, we found that the spinrelaxation rates for both T^{+} and T^{−} increase with the temperature T up to 60 K (Fig. 7b and Supplementary Fig. 12). For T > 60 K, we do not observe any QB signals, which could be due to fast disintegration of trions at higher temperatures. In this case, we estimate the trionbinding energy to be of the order of 5 meV. The τ(T) response supports the Elliot–Yafet (EY) spinrelaxation mechanism, which arises from spin–orbitrelated scattering collisions with phonons^{51}. In this case, the spinrelaxation rate is proportional to the phononscattering rate Γ_{p}. Since the occupancy, 〈n〉, of optical phonon increases with T, the carrier–phononscattering rate increases, leading to an increase in the spinrelaxation rate Γ (=1/τ). We fitted the temperaturedependent rates using the function \({\Gamma }^{+,()}\) = \({\Gamma }_{0}^{+,()}+{\Gamma }_{\omega }^{+,()}\frac{1}{{e}^{\frac{\hslash \omega 0}{{K}_{B}T}}1}\), where \({\Gamma }_{0}^{+,()}\) is the temperatureindependent scattering rate from defects and impurities, \({\Gamma }_{\omega }^{+,()}\) is the scattering rate from phonons and ω_{0} is a typical optical phonon frequency. Good fits for \({\Gamma }_{e}^{+}\)(T) and \({\Gamma }_{h}^{}\)(T) have been obtained with \({\Gamma }_{0}^{+()}\) = 0.09 (0.2) ± 0.005 (0.02) (GHz), \({\Gamma }_{{\omega }}^{+()}\) = 13 (10) ± 2 (1) (GHz) and \({\hslash \omega }_{0}\) = 5 ± 3 meV (see Fig. 7b). That Γ_{0} ≪ Γ_{ω} indicates that carrier scattering with defects/impurities is negligibly small in this highquality MAPbI_{3} single crystal. The fitting also shows that the optical phonon modes having energy <10 meV are the dominant scatterers that influence the trion spinrelaxation kinetics.
In conclusion, the quantum beatings observed in a highquality MAPbI_{3} single crystal at cryogenic temperature originate from positive and negative trions for which the beating frequencies increase linearly with the applied magnetic field. This may be interpreted as the Larmor precession frequencies of electron (hole) in the positive T^{+} (negative T^{−}) trions or to the residual carriers left after trion decay. The obtained Landé gfactors of both electrons and holes show a significant anisotropy when the magnetic field was directed along two different crystal axes [001] and [\(1\bar{1}0\)], respectively. In addition, the unexpected long spinrelaxation time τ of T^{+} trions ~11 ns at liquid He temperature could make the MAPbI_{3} single crystal a potential candidate for quantum information science.
Methods
Sample preparation
In our studies, we have used both single crystals and thin films of MAPbI_{3}. The single crystals were grown by typical ITC method^{35} in which we mixed CH_{3}NH_{3}I and PbI_{2} in a 1:1 molar ratio in gammabutyrolactone (GBA) to form the precursor solution with a concentration of 1.23 mol/ml. The precursor solution was stirred overnight at 80 °C. Subsequently, the solution was filtered using PTFE filter with 0.22μm pore size. Finally, the precursor solution was kept in a large beaker, which was halffilled with mineral oil and placed on a hot plate at 95 °C for about 2 days. Small MAPbI_{3} particulates were harvested as seed crystals. By placing a seed crystal in 10 mL of the same precursor solution and keeping it at 95 °C for another 2 days, the seed grew into a large MAPbI_{3} crystal of mm size. For larger crystals, the latter step was repeated with the large crystals used as the new seeds.
Thin films were prepared using the standard procedure in which a precursor solution of CH_{3}NH_{3}I_{3} and PbCl_{2} in a molar ratio of 3:1 in N,Ndimethylformamide was used to form a concentration of 0.8 mol/ml. The precursor solution was stirred on the hotplate at 50 °C overnight. The solution was cooled to room temperature and spincoated on glass substrates at 3000 rpm for 60 sec. The resulting film was then annealed at 100 °C for 2 h.
Characterization of the crystal structure
XRD measurements of singlecrystal MAPbI_{3} (Fig. 1) were performed using a Bruker D2 Phaser Xray diffractometer in 2theta configuration (see S.I.). The flat face of the MAPbI_{3} crystal was placed facing down on a flat surface of the sampleholder ring, while the remainder of the ring was covered with amorphous clay to form a mold around the crystal, such that only the intended face of the crystal remained uncovered. The Xray source was the K\(\alpha\) emission from a Cu target. A divergence slit of 0.2 mm was placed in the inputbeam path and an antiscatter screen above the sample provided efficient measurements. The measurements were done with step resolution of 0.022˚ over the continuous 2theta range of 10˚–70˚. The obtained spectrum was compared with that in the literature to determine the crystalline orientation of the measured face. Similar procedure was repeated along other flat faces of the crystal, and the corresponding orientation axis was determined.
Transient circularPPR spectroscopy
The transient circularPPR technique is a derivative of wellknown optical pump/probe correlation spectroscopy in which only the polarization of the pump beam is modulated with a PEM (i.e., photoelastic modulator) at 41 kHz between left and right circular polarizations. The probe beam was circularly (linearly) polarized for circular (linear) PPR. The pump and probe beams were split from the output of a Ti: sapphire laser (SpectraPhysics) with pulse duration of 150 fs and 80 MHz repetition rate that could be continuously tuned from 730 nm to 810 nm. In order to minimize the large scattering intensity from the stronger pump beam, we used a double lockin technique in which the probe beam was also modulated by a mechanical chopper at 1.2 kHz. The pump and probe beams having average intensity of ~20 Wcm^{−2} and ~3 Wcm^{−2}, respectively, were aligned through various optical components to spatially and temporally overlap onto a small area of the samples with spot size of ~100 µm. The probebeam path length was extended by a mechanically delayed stage up to ~8 ns. For the probebeam intensity detection, we used a silicon photodiode connected to the first lockin amplifier that was externally synchronized with the PEM at 41 kHz. The second lockin amplifier was externally synchronized with the chopper frequency at 1.2 kHz. The crystal was placed in a cryostat that controlled the temperature between 4K and 300K. A magnetic field, B from an electromagnet with strength, B up to 700 mT, was applied in the direction parallel to the crystal chosen surface (i.e., Voight configuration).
Photoinducedabsorption spectroscopy
Photoinduced absorption (PA) in the midIR spectral range was measured using a Fouriertransform infrared spectrometer (FTIR, Thermo Scientific) having an external detecting system as a probe in the spectral range of 500–4000 cm^{−1}. The film was excited at ħω ≈ 2.77eV using a diode laser of which beam was modulated at 50 mHz using a shutter controlled by a function generator. The change, ΔT in the IR transmission spectrum, T induced by the pumpbeam excitation was signalaveraged over 5000 scans. The PA spectrum was subsequently calculated as −ΔT/T.
Calculation of the gfactors in orthorhombic MAPbI_{3}
To describe the anisotropic electron and hole gfactors in orthorhombic MAPbI_{3}, we employ an 8band K.P model based on the quasicubic approach of ref. ^{27}. The bandedge valenceband function Bloch functions are represented as the 2fold degenerate states with Sorbital symmetry, while for the conduction bands, the Bloch functions have orbital Psymmetry^{27,44}. To find the bandedge Bloch functions, we diagonalize the conductionband Hamiltonian, including the spin–orbit interaction, \({H}_{{LS}}=\frac{2}{3}\Delta {{{{{\bf{L}}}}}}{{{{{\boldsymbol{\cdot }}}}}}{{{{{\bf{S}}}}}}\), whose strength is given by \(\Delta\), the spin–orbit splitoff parameter that separates the upper\(J=3/2\) derived conduction bands from the lower \(J=\) ½derived conduction bands in the cubic crystal structure. We include as well the crystal field Hamiltonian^{27,29}, \({H}_{{CF}}\), which breaks the symmetry of the x,y,z directions in the orthorhombic structure. This is given by^{27}
where \({L}_{x},{L}_{y},{L}_{z}\) are the x, y, z components of the orbital angular momentum operator, and the tetragonal^{29} and orthorhombic^{27} crystalfield parameters \(\delta\), \(\zeta\) reflect symmetry breaking relative to the cubic phase in the \({{{{{\bf{z}}}}}}\) and in the \({{{{{\bf{x}}}}}},{{{{{\bf{y}}}}}}\) directions, respectively. Diagonalization of \({H}_{{LS}}+{H}_{{CF}}\) results in Bloch eigenfunctions for the lower conduction bands that can be represented in the general form^{27}
where the symbols \(X,Y,Z\) denote orbital functions that transform like x, y, z under rotations, while \({{{{{{\mathcal{C}}}}}}}_{X},{{{{{{\mathcal{C}}}}}}}_{Y},{{{{{{\mathcal{C}}}}}}}_{{{{{{\rm{Z}}}}}}}\) are realvalued cnumbers determined by numerical diagonalization and subsequent Gram–Schmidt orthogonalization; these coefficients are functions of the crystalfield parameters \(\delta ,\zeta\) as well as the spin–orbit splitoff parameter, \(\Delta\). Similarly, the energies of the upper heavy and lightelectronband edges are found by numerical diagonalization of \({H}_{{LS}}+{H}_{{CF}}\). The corresponding Bloch functions can be shown to have the following general forms: for the heavyelectron^{52} band,
where the realvalued coefficients \({{{{{{\mathcal{H}}}}}}}_{X},{{{{{{\mathcal{H}}}}}}}_{Y},{{{{{{\mathcal{H}}}}}}}_{{{{{{\rm{Z}}}}}}}\) are found numerically; and for the lightelectron^{52} band,
Here again, the realvalued coefficients \({{{{{{\mathcal{L}}}}}}}_{X},{{{{{{\mathcal{L}}}}}}}_{Y},{{{{{{\mathcal{L}}}}}}}_{{{{{{\rm{Z}}}}}}}\) are found by numerical diagonalization of \({H}_{{LS}}+{H}_{{CF}}\). Writing the full 8band K.P Hamiltonian including the kdependent terms, which give rise to dispersion of the band energies away from the zone center (see Eq S19 in Supplementary Note 1), adding the magnetic Hamiltonian given by^{42}
where \({\mu }_{B}=e\hslash /2{m}_{0}\) is the Bohr magneton, \({g}_{0}=\big{g}_{e}\big\) is the free electronspin gfactor ≈ \(2.0023\) and \(\kappa\) is Luttinger’s magnetic parameter^{42}, we evaluate the effective Zeeman Hamiltonian for the lowest conduction band and the valence band using L\(\ddot{{{{{{\rm{o}}}}}}}\)wdin’s partition method [43], and utilizing the commutators \([{k}_{x},{k}_{y}]=i\frac{e}{\hslash }{B}_{z}\) and their cyclic permutations [42]. While the gfactors are isotropic in the cubicphase HOIPs (see Supplementary Note 1, Eq. S26), in tetragonal or orthorhombic perovskites, the symmetry between the orthogonal x, y and z directions is broken, leading to anisotropic gfactors, see the expressions in Supplementary Table 1, and the calculated gfactors shown in Fig. 5. We note that the six principal gfactors (x, y, z values for the electron and the hole, respectively) are fully determined by six parameters: these are the bandgap, \({E}_{g}\); the Kane energy, \({E}_{p}\); the spin orbit splitoff parameter, \(\Delta\); the tetragonal and orthorhombic crystalfield parameters \(\delta ,\zeta\) and Luttinger’s magnetic parameter, \(\kappa\).
Data availability
All data are available in the main text or the Supplementary Information. Additional data related to the findings of this study may be requested from the authors.
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Acknowledgements
We thank Sarah Li (UofU) and ZhiGang Yu (WSU) for fruitful discussions, and Rikard Bodin and Xin Pan for help with MAPbI_{3}film preparation. The spectroscopic measurements, the singlecrystal growth and the theoretical modeling were supported by the Center for Hybrid OrganicInorganic Semiconductors for Energy (CHOISE), an Energy Frontier Research Center funded by the Office of Basic Energy Sciences, Office of Science within the US Department of Energy through contract number DEAC3608G028308. The filmgrowth facility and the steadystate measurements were supported by the Department of Energy Office of Science, Grant DESC0014579.
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Z.V.V. planned the project. J.H. and Y.L. grew the single crystals. A.C. characterized the crystals. U.N.H. and D.R.K. conducted all the transient and steadystate measurements. Z.V.V., U.N.H. and P.C.S. analyzed the data. P.C.S. proposed and wrote the theory. Z.V.V., U.N.H. and P.C.S. wrote the paper and all authors reviewed it.
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Huynh, U.N., Liu, Y., Chanana, A. et al. Transient quantum beatings of trions in hybrid organic triiodine perovskite single crystal. Nat Commun 13, 1428 (2022). https://doi.org/10.1038/s41467022290536
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DOI: https://doi.org/10.1038/s41467022290536
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