Multidimensional coherent spectroscopy reveals triplet state coherences in cesium lead-halide perovskite nanocrystals

Multidimensional spectroscopy suggests that perovskite nanocrystals have a dark state interleaved with the triplet states.

Abstract

Advances in optoelectronics require materials with novel and engineered characteristics. A class of materials that has garnered tremendous interest is metal-halide perovskites, stimulated by meteoric increases in photovoltaic efficiencies of perovskite solar cells. In addition, recent advances have applied perovskite nanocrystals (NCs) in light-emitting devices. It was found recently that, for cesium lead-halide perovskite NCs, their unusually efficient light emission may be due to a unique excitonic fine structure composed of three bright triplet states that minimally interact with a proximal dark singlet state. To study this fine structure without isolating single NCs, we use multidimensional coherent spectroscopy at cryogenic temperatures to reveal coherences involving triplet states of a CsPbI₃ NC ensemble. Picosecond time scale dephasing times are measured for both triplet and inter-triplet coherences, from which we infer a unique exciton fine structure level ordering composed of a dark state energetically positioned within the bright triplet manifold.

INTRODUCTION

Cesium lead-halide perovskites were first synthesized over a century ago with a general chemical formula CsPbX₃ (where X = Cl, Br, or I). Recently, synthesis of CsPbX₃ nanocrystals (NCs) was achieved (1, 2), which combines the advantages of perovskites (e.g., efficient luminescence and long carrier diffusion length) with those of colloidal NC materials (e.g., surface engineering and size-tunable emission). Perovskite NCs exhibit luminescence with quantum yields reaching nearly unity (3), in contrast to the optimized 80% quantum yield achieved by chalcogenide NCs coated with a gradient shell (4). Although all other colloidal materials suffer inhibited emission from lower-energy dark states (5), the unusual brightness of perovskite NCs is now believed to originate from an optically active, nondegenerate triplet state that emits efficiently despite the presence of a dark singlet state (6, 7).

The unique exciton fine structure of perovskite NCs has substantially extended the potential applications of colloidal NCs. In particular, the three nondegenerate bright triplet states and their orthogonally oriented dipole moments (arising from atomic orbital mixing along the lattice symmetry axes) have generated much excitement for potential applications in quantum information processing (8, 9). However, engineering exciton superposition states as information carriers will require an intimate knowledge of their coherent dynamics, which are still not well understood. The exciton fine structure of perovskite NCs has thus far only been studied via single-NC photoluminescence (6, 10, 11) and transient absorption (12, 13) techniques, which have provided information only about their incoherent population dynamics. Furthermore, inhomogeneous spectral broadening due to NC size dispersion limits the utility of linear spectroscopic techniques in studying NC ensembles. More sophisticated methods are thus required (14–17) to extract the desired ensemble-averaged coherent properties of perovskite NCs.

Here, we extract crucial figures of merit for quantum information processing, the ensemble-averaged coherence times, and reveal coherence times of both optical triplet coherences and terahertz inter-triplet coherences for an ensemble of CsPbI₃ NCs (see Fig. 1, A and B) at cryogenic temperatures. We also present evidence for a mixed bright-dark level ordering (see Fig. 1C) that renders the triplet state excitons only partially bright. These measurements are enabled by using multidimensional coherent spectroscopy (MDCS) (18) to circumvent the inhomogeneous broadening that obscures spectral signatures of the exciton fine structure and resolve coherent coupling involving different triplet states. The extracted coherence times for CsPbI₃ NCs are an order of magnitude longer compared to candidate materials for valleytronics (14, 15, 19), which positions perovskite NCs as a potential material platform for quantum information applications via bottom-up assembly.

Fig. 1. CsPbI₃ perovskite NCs studied via MDCS.

(A) Perovskite NC absorption spectra as a function of temperature (34). (B) Energy level diagram of the nondegenerate bright triplet states {Ψ_{i = x},_y,_z} that comprise the band-edge transitions to and from |Ψ_g⟩. The dark singlet state |Ψ_d⟩ is shown to lie between states |Ψ_y⟩ and |Ψ_z⟩, which is argued in the main text. (C) Schematic of the MDCS experiment. Three pulses {A,B,C} are focused onto the sample with varying time delays. Inset shows a transmission electron micrograph of representative CsPbI₃ NCs. (D) Excitation pulse sequence and excitation polarization schemes used to acquire one-quantum and zero-quantum spectra, in which double-sided arrows in circles denote the polarization of each pulse. Pulses A and C are horizontally polarized, and pulse B is either horizontally or vertically polarized, which corresponds to an emitted signal of either horizontal or vertical polarization respectively. a.u., arbitrary units.

RESULTS

Multidimensional coherent spectroscopy

To perform MDCS, we use a multidimensional optical nonlinear spectrometer (20), which focuses three laser pulses onto the perovskite NC ensemble sample as a function of three time delays τ, T, and t (schematically shown in Fig. 1, C and D). By Fourier transforming the emitted four-wave mixing (FWM) signal as a function of two or all three time delays, the coherences and populations induced by each pulse are correlated in a multidimensional spectrum. In order, the first pulse (A) generates a coherent superposition between the ground and an optically active excited state, which oscillates at the energy difference ħω_τ. The second pulse (B) then converts this optical coherence into either a population state that decays according to its relaxation rate or a coherence between two neighboring states within a ground or excited state manifold that oscillates at ħω_T. The last pulse (C) then generates a final optical coherence that oscillates at ħω_t and radiates an FWM signal. These sequences of light-matter interactions are commonly referred to as quantum pathways [(21) and the Supplementary Materials]. Fourier transforming along the variables τ and t returns one-quantum spectra (which correlate the absorption energy ħω_τ with the emission energy ħω_t) and has been applied in recent studies (22–24) to reveal novel electronic properties of various perovskite materials. In this study, we also transform along the variables T and t to obtain zero-quantum spectra [which correlate the intraband mixing energy (25) ħω_T and the emission energy ħω_t]. Furthermore, the polarization of the second pulse is chosen to align either parallel or orthogonal to the colinear polarizations of the other two pulses to probe different quantum pathways. We denote the two polarization schemes as colinear excitation and cross-linear excitation, respectively, as shown in Fig. 1D.

One-quantum spectra probe optical triplet coherences

One-quantum spectra were acquired at a temperature of 4.6 K with colinear and cross-linear excitation (shown in Fig. 2, A and B). Both spectra reveal numerous peaks elongated in the diagonal direction (|ħω_τ| = |ħω_t|), reflecting inhomogeneous spectral broadening (26). Cross-slices taken in the orthogonal direction, which reflect the ensemble-averaged homogeneous response at a certain resonance energy (26), are plotted in Fig. 2 (C and D). In the full slices (inset), asymmetric peaks are observed for |ΔE| > 4 meV, which could arise from either electron-phonon coupling (27) or biexciton transitions (28). Here, we focus on features related to the triplet state fine structure and defer discussion of vibrational features and biexciton transitions [which appear at the biexciton binding energy ΔE = −E_XX, which is well over 10 meV (10), and do not influence the low-energy features studied here] to future reports. The main plots of each slice section (highlighted by the yellow boxes inset) show symmetric peaks that, due to their polarization dependence, we attribute to absorption and emission involving different triplet state coherences. As described above, the origin of the observed peaks is interpreted as changes in the system density matrix induced by each pulse that form accessible quantum pathways (21). First, peaks 1 and 3 in Fig. 2C are generated by absorption and emission of coherences involving |g⟩ and both triplet states |Ψ_x⟩ and |Ψ_y⟩. We note that our measurements do not inform the ordering of states |Ψ_x/y⟩, so we assume the ordering shown in Fig. 1B for labeling the dephasing rates discussed below. The central peak 2 is likewise generated by quantum pathways involving absorption and emission by coherences of identical resonance energy |g⟩⟨Ψ_i| and |Ψ_i⟩⟨g|, respectively. In Fig. 2D, peaks 1 and 3 are visible as shoulders on two new peaks 4 and 5, which are generated by absorption and emission of coherences involving |g⟩ and triplet states |Ψ_y⟩ and |Ψ_z⟩.

Fig. 2. Optical triplet coherences in one-quantum spectra.

(A and B) Magnitude one-quantum spectrum at 4.6 K with (A) colinear and (B) cross-linear excitation. The white/red dashed arrows and solid black lines indicate the cross-slice locations and laser pulse spectra, respectively. (C and D) Bottom plots show normalized cross-slices centered at |ħω_τ| = |ħω_t| = 1900 meV of the (C) colinear and (D) cross-linear excitation one-quantum spectrum. Top plots show theoretically calculated relative peak strengths in the NC reference frame (see the Supplementary Materials). Numbers label peaks arising from electronic interband coherences and populations.

The polarization dependences of the five observed peaks is not immediately intuitive. To explain the peak structure for both excitation polarization schemes, we theoretically calculate the relative peak strengths for varying dipole matrix elements and vector orientations of each triplet state transition (see the Supplementary Materials). From these calculations, we draw two important conclusions: (i) To observe sidebands with colinear excitation, the two triplet states involved must have transition dipole moments that are not perpendicular, and (ii) for peaks 4 and 5 to each be primarily composed of a single peak located at ΔE = ±Ω₁, as suggested by our fitting attempts, the dipole moment of |Ψ_y⟩ must be substantially stronger than those of |Ψ_x⟩ and |Ψ_z⟩. We achieve good agreement with experiment for a 45° angle between the dipole moments of |Ψ_x⟩ and |Ψ_y⟩ and a dipole matrix element of |Ψ_y⟩ that is 1.5 times the magnitude of the dipole matrix elements of |Ψ_x⟩ and |Ψ_z⟩ (see the Supplementary Materials).

Fitting the cross-diagonal line shapes also extracts the homogeneous linewidths γ_i (25) of triplet state transitions between |g⟩ and |Ψ_i⟩. In this context, one-quantum spectra are particularly useful when compared to integrated FWM techniques (29), since the cross-diagonal slice position |ħω_τ| = |ħω_t| reflects an effective NC size (see the Supplementary Materials). Although peaks 1 and 4 and peaks 3 and 5 would ideally be mirror images, vibrational coupling distorts the line shapes of peaks 1 and 4. We thus fit only the ΔE ≥ 0 side (details are in the Supplementary Materials). Fit of the colinear slice line shape (Fig. 2C) gives a sideband (peaks 1 and 3) dephasing rate ħ$\frac{{\mathrm{\gamma }}_x+{\mathrm{\gamma }}_y}{2}$ = 0.120 ± 0.023 meV (5.49 ps) and a zero-phonon line (peak 2) dephasing rate of 0.124 ± 0.008 meV (5.32 ps). Fit of peaks 4 and 5 in the cross-linear slice line shape (Fig. 2D) gives a dephasing rate ħ$\frac{{\mathrm{\gamma }}_y+{\mathrm{\gamma }}_z}{2}$ = 0.496 ± 0.017 meV (1.33 ps). The fitted triplet state energy splittings are likewise Ω₁ = 1.82 ± 0.02 meV and Ω₂ = 0.24 ± 0.016 meV. The provided parameter uncertainties reflect the accuracy of the fitting procedure rather than statistical error. If the dipole moment of state |Ψ_y⟩ is indeed much larger than those of |Ψ_x⟩ and |Ψ_z⟩, the zero-phonon line dephasing rate will approximately equal ħγ_y, which, in turn, determines the individual triplet state dephasing rates γ_x = 0.116 meV ($T_2^x$ = 5.68 ps), γ_y = 0.124 meV ($T_2^y$ = 5.32 ps), and γ_z = 0.868 meV ($T_2^z$ = 0.76 ps). To test the above claim, fits assuming additional peaks at ΔE = ±(Ω₁ + Ω₂) were attempted but did not yield reasonable line shapes.

Zero-quantum spectra probe terahertz inter-triplet coherences

Many of the quantum pathways that generate the sidebands in Fig. 2 (A and B) involve inter-triplet coherences, which are quantum coherences between triplet states that are not necessarily dipole-coupled (30). Of both fundamental and practical importance is the inter-triplet coherence time, which defines the time scale during which the superposition states involved may be coherently manipulated. Inter-triplet coherences are those density matrix elements of the form |Ψ_i⟩⟨Ψ_j| where i, j = {x,y,z} and i ≠ j. To directly measure and characterize these coherences, we take zero-quantum spectra at varying temperature and delay τ. For colinear excitation, no inter-triplet coherences between |Ψ_x⟩ and |Ψ_y⟩ are observed (see the Supplementary Materials). It is ambiguous whether their corresponding peaks are weak or are simply obscured by the linewidth of a central ħω_T = 0 peak. For cross-linear excitation, we further isolate the inter-triplet coherence pathways by passing the measured FWM signal through a vertical polarizer. We plot a resultant cross-linear zero-quantum spectrum at 20 K in Fig. 3A. Sidebands are observed at mixing energies identical to the positions of peaks 4 and 5 in Fig. 2D, which we attribute to inter-triplet coherences between |Ψ_y⟩ and |Ψ_z⟩. An inter-triplet coherence between |Ψ_x⟩ and |Ψ_z⟩ is observed in neither the colinear nor cross-linear zero-quantum spectra, which is consistent with a dominant transition dipole of state |Ψ_y⟩ as argued above.

Fig. 3. Terahertz inter-triplet coherences in zero-quantum spectra.

(A) Magnitude zero-quantum spectrum taken at τ = 0 fs and 20 K. Two sidebands due to inter-triplet coherences are observed. (B) Evolution of normalized slices taken along the dashed red line in (A) at ħω_τ = 1890 meV as a function of delay τ (at 20 K) and temperatures [10, 15, 20, 25, 30, 40] K (at τ = 0 fs). (C) Fit of normalized cross-slice taken at τ = 0 fs, in which the complex Lorentzians of peaks 6 and 8 are shifted by phases −π/2 relative to peak 7. The shaded curves represent the real quadratures of each Lorentzian in the fit line shape, and the top plot is the phase of the fitted complex line shape.

In Fig. 3B, the evolutions of normalized slices (at ħω_T = 1890 meV) as a function of delay τ and temperature are shown. The FWM signal dephases rapidly with increasing τ and results in an equally rapid decrease of sideband visibility (see the Supplementary Materials), in contrast to the opposite behavior of vibrational intraband coherences [see (16) and the Supplementary Materials], which confirms its electronic nature. No change in the amplitude ratio between sidebands 6 and 8 is observed as temperature increases, confirming that the state splitting observed indeed belongs to the bright triplet excited state rather than from thermal filling of higher-lying ground states. We note that the triplet state coherences in one-quantum spectra broaden substantially with increasing temperature and are not resolved at temperatures above 15 K. In contrast, minimal broadening of the inter-triplet coherence linewidth is observed up to 40 K (see Fig. 3B), indicating that inter-triplet coherences are robust against thermal dephasing (14).

A slice at τ = 0 fs is plotted in Fig. 3C, from which we can extract the inter-triplet coherence time. However, the quantum pathways that generate peaks 6 and 8 involve identical dipole moments${\mathrm{\mu }}_y^2{\mathrm{\mu }}_z^2$, from which we expect equal peak amplitudes contrary to the uneven peaks observed. This difference is due to interference between the three complex Lorentzian line shapes underlying the overall amplitude line shape (see the Supplementary Materials), and the fit to experiment is performed by shifting the phase of each sideband Lorentzian line shape by identical factors of −π/2 relative to the central ħω_T = 0 peak. These phase factors arise from a −π/4 phase difference between the complex transition dipole moments μ_y and μ_z, which may be directly related to structural information about the initial/final states of an optical transition and their associated wave functions (31). From our fit, we extract an energy splitting Ω₁ = 1.61 meV and an inter-triplet coherence time $T_2^{\mathit{yz}}$ = 1.36 ps at 20 K.

DISCUSSION

It is quite unexpected that the optical dephasing rate γ_z is so much faster than those of the other two triplet states γ_x and γ_y. Although this disparity suggests a fundamentally different dephasing mechanism for coherences involving state |Ψ_z⟩, photoluminescence of similar orthorhombic perovskite NCs that exhibit triplet state structure reveals similar emission linewidths for all three states of the manifold (6, 7). We resolve this discrepancy by proposing a unique exciton fine structure composed of a dark singlet state |Ψ_d⟩ that lies above the states |Ψ_x⟩ and |Ψ_y⟩, which form the band edge, while remaining below the third triplet state |Ψ_z⟩ (shown in Fig. 1B). Camargo et al. (24) have similarly applied MDCS to reveal dark states in a perovskite thin film, although the sub-gap defect states they observe are fundamentally different from the dark state discussed here, which arises from the lead-halide atomic orbitals and would exist in a perfect lattice. For our proposed level ordering, rapid relaxation from |Ψ_z⟩ to |Ψ_d⟩ then significantly decreases the population lifetime $T_1^z$ and consequently $T_2^z$ as well (21). Such a fine structure has been theoretically predicted in certain ranges of NC size due to competition between the Rashba effect and electron-hole exchange interaction (32). Although we cannot rule out alternative sources of line broadening, such as the imperfect correlation between the two inhomogeneously broadened transitions between the ground state and {|Ψ_y⟩, |Ψ_z⟩} that would selectively broaden sidebands 4 and 5 in Fig. 2 (see the Supplementary Materials), our hypothesis is further supported by previous photoluminescence studies of CsPbI₃ NCs of nearly identical size (10). These incoherent spectra, which are not affected by inhomogeneous broadening correlation, revealed polarized doublets corresponding to |Ψ_x⟩ and |Ψ_y⟩ but did not detect the third triplet state |Ψ_z⟩ (whose emission would be quenched by nonradiative relaxation to |Ψ_d⟩ according to our hypothesis of a partially bright triplet exciton band edge). In accordance with the predicted size dependence of the relative dark state energy (32), we also observe an abrupt increase in $T_2^y$with increasing slice position (see the Supplementary Materials), which results from a crossing in energy of |Ψ_y⟩ and |Ψ_d⟩. These results stand in stark contrast to recent studies of CsPbBr₂Cl (6) and FAPbBr₃ (7) NCs, which have reported the highest- and lowest-lying dark states, respectively. Evidently, a change in even a single constituent atom of the perovskite lattice may drastically alter the exchange and Rashba interaction coefficient ratio that determines the fine structure level ordering (32), which warrants future investigation.

To conclude, we have measured and characterized both optical frequency triplet coherences and terahertz frequency inter-triplet coherences. We have also presented evidence of an exciton band edge whose emission is partially quenched by an intermediate dark state, which contributes important insight into the controversial nature of exciton ground states in different perovskite NC materials (6, 7). As a material still in its infancy, perovskite NCs show promise for applications in optoelectronic devices. In particular, the minimal thermal broadening of inter-triplet coherences observed here motivates study of applications above cryogenic temperatures. For example, in a triplet state analog of valleytronics in two-dimensional materials (19), superpositions of triplet states could be initialized and read out with linearly polarized light and coherently manipulated via terahertz radiation as information carriers (33).

MATERIALS AND METHODS

The NCs studied are cube-shaped, with side lengths of 8.7 ± 2.6 nm measured from 100 NCs in transmission electron microscopy measurements. These sizes are comparable to the CsPbI₃ exciton Bohr diameter (12 nm) and correspond to a room temperature 1S exciton absorption peak centered around 1900 meV (shown in Fig. 1B). The synthesis method is described in the Supplementary Materials. To study these NCs using MDCS at cryogenic temperatures, we disperse them in heptamethylnonane and load the colloidal suspension in a copper sample holder with 0.5-mm-thick sapphire windows. The sample optical density is measured to be 0.3 at room temperature 1S exciton absorption peak. Upon cooling of the sample below ~100 K, heptamethylnonane freezes into a transparent glass, resulting in a stationary, randomly oriented NC ensemble.

To perform MDCS, we use a multidimensional optical nonlinear spectrometer (20), which focuses three laser pulses onto the sample. The excitation pulses are of 90 fs duration at a 250-kHz repetition rate, and the excitation intensity of 5 W/cm² generates a predominately third-order response as verified by a power-dependence measurement of the generated transient FWM signal. The emitted signal is then heterodyne-detected with a co-propagating local-oscillator pulse as a function of time delays τ, T, and/or t with sub-wavelength stability.

SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/7/1/eabb3594/DC1