Nanoscale-Resolved Surface-to-Bulk Electron Transport in CsPbBr3 Perovskite.

Describing the nanoscale charge carrier transport at surfaces and interfaces is fundamental for designing high-performance optoelectronic devices. To achieve this, we employ time- and angle-resolved photoelectron spectroscopy with ultraviolet pump and extreme ultraviolet probe pulses. The resulting high surface sensitivity reveals an ultrafast carrier population decay associated with surface-to-bulk transport, which was tracked with a sub-nanometer spatial resolution normal to the surface, and on a femtosecond time scale, in the case of the inorganic CsPbBr3 lead halide perovskite. The decay time exhibits a pronounced carrier density dependence, which is attributed via modeling to enhanced diffusive transport and concurrent recombination. The transport is found to approach an ordinary diffusive regime, limited by electron-hole scattering, at the highest excitation fluences. This approach constitutes an important milestone in our capability to probe hot-carrier transport at solid interfaces with sub-nanometer resolution in a theoretically and experimentally challenging, yet technologically relevant, high-carrier-density regime.

A profound understanding and a reliable description of charge carrier transport at the nanometer length scale are crucial for a multitude of optoelectronic applications. In bulk materials with large absorption coefficients, such as lead halide perovskites (LHPs),[1] strong gradients of carrier density occur on the nanometer scale and transport phenomena are expected to take place in a femtosecond regime. These surface or interface transport processes are central for the operation of single-junction and tandem solar cells,[2−4] nanolasers,[5−7] light-emitting diodes (LEDs),[8] and photocatalytic systems.[9,10] From a spatial perspective, while recent advances in ultrafast optical imaging have made outstanding contributions in characterizing the lateral transport of charge carriers,[11−14] these studies lack the ability to monitor the surface-to-bulk transport, which as a consequence has remained elusive up to now.[15] In addition, these techniques do not have a sensitivity to the electron momentum, which is crucial in discerning properties of band gap edge carriers.[16−18] From a temporal perspective, the hundreds of femtoseconds time scales are intricate due to a crossover between the ballistic[12] and the diffusive[11,13,14,19] transport regimes, making either analytical or numerical description difficult. In this work, we overcome these limitations by monitoring the ultrafast dynamics of photoexcited electrons using time- and angle-resolved photoelectron spectroscopy (TR-ARPES),[16,20,21] supported by theoretical modeling. We apply this approach to the case of the inorganic lead halide perovskite CsPbBr3 upon ultraviolet (UV) excitation, creating a strong charge gradient at the surface. Vital to this strategy is the use of extreme ultraviolet (EUV) probe pulses. Indeed, in solids containing heavy elements (such as Pb and Br), the inelastic mean free path (MFP) of photoelectrons at 20–30 eV photon energies is ∼5 Å.[22] Unlike the case of bulk-integrating techniques, this extreme surface sensitivity is used to extract information about the carrier transport on a sub-nanometer scale in the direction orthogonal to the surface (along the z axis in Figure ). This unique capability, combined with femtosecond temporal and momentum resolution, enables us to selectively follow in real space and real time the ultrafast evolution of the electron population in the conduction band (CB).

Figure 1

Schematic view of the experiment. The numbered arrows correspond to (1) carrier transport into the bulk and (2) lateral transport across the surface.

LHPs earned their place among the most studied semiconductors due to their outstanding optoelectronic properties,[23,24] which can be engineered by small modifications in composition and structure,[25,26] adapting them for a wide variety of applications.[27−29] LHPs possess some remarkable transport properties, such as long carrier lifetimes,[30−32] carrier effective mass renormalization,[17,33,34] and defect tolerance,[35,36] which were partially attributed to polaron formation. The complexity in describing carrier transport in LHPs is due to a delicate interplay of electronic and lattice dynamics with disorder and defects.[37−41] Furthermore, their transport properties can be strongly influenced by the operating conditions of a photonic device.[42,43] Many applications, such as LEDs, lasers, and solar concentrators,[44] typically operate under high carrier densities, where, in view of an enhanced long-term stability,[45−47] fully inorganic LHPs are better suited in comparison to their hybrid counterparts. At charge carrier densities above ∼1018 cm–3, complex many-body phenomena can be observed in LHPs,[48−50] such as hot-phonon bottleneck[51−54] and band-filling[55] effects, Auger processes become an important recombination channel for photocarriers,[56,57] an excitonic Mott transition occurs,[5] and the polaronic transport picture breaks down.[48] Photophysical studies of LHPs in this convoluted yet technologically relevant regime are theoretically and experimentally challenging.[58−63]

Here, we monitor the charge carrier dynamics of CsPbBr3 single crystals by TR-ARPES as a function of carrier density. The observed evolution of the photoemission signal reflects the migration of near-surface carriers into the bulk on a time scale that depends significantly on the carrier density. With the support of modeling based on diffusion and recombination rate equations, our study of strongly photoexcited samples (∼1019–1020 cm–3) reveals that at lower densities the carrier diffusivity is strongly enhanced, possibly due to high velocities of hot carriers, sustained by band-filling, hot-phonon bottleneck, and Auger heating[56] effects, resulting in a fast decay of surface population. However, at higher densities, the transport is predominantly limited by carrier–carrier scattering and can be described by conventional diffusivity already at ultrafast time scales.

The experimental scheme is depicted in Figure . The UV pump pulse initially creates a depth distribution of photocarriers with an exponentially decaying profile inside the crystal (dashed line). The EUV probe pulse then photoemits primary electrons from a small volume, indicated in green. A change in the carrier density (solid line) in the probed volume is reflected in the photoemission signal strength, which provides a direct access to the evolution of the carrier population.

Figure a shows the ARPES intensity plotted as a function of the k and k in-plane components of the electron momentum, acquired at 2.6 eV above the valence band (VB) maximum immediately after (t = 100 fs) the pump excitation. The measured constant energy map is overlapped with density functional theory (DFT) calculations and shows localized electron pockets at the corners (M̅ point) of the surface Brillouin zone (SBZ) corresponding to the (001) lattice plane.[17] When the probe energy is tuned, the maximum CB intensity is observed at 24.3 eV, close to the R point of the three-dimensional BZ.[64] In what follows we exploit the energy tunability and the momentum resolution of ARPES to observe the electron dynamics at the CB minimum.

Figure 2

Electronic structure of CsPbBr3: (a) experimentally measured constant energy map as a function of the in-plane electron momentum of the lowest CB (shown in the top right corner) overlapped with a DFT simulation; (b) Dispersions of the VB and the CB measured at t = 100 fs (the color scale of the upper part of the image was multiplied by a factor of 5) overlapped with the DFT bands at k = π/a (red) and k = 0 (blue). Black dashed lines indicate the integration range used to obtain Figure . Violet arrows indicate the allowed pump excitation. Note: the experimental surface sensitivity leads to an additional partial integration along the k, as explained in ref (17).

Figure 3

TR-ARPES data: (a) time evolution of the integrated photoemission intensity maps (the integration range is indicated by dashed lines in Figure b); (b) Gaussian fits of the integrated maps at the two time delays indicated by vertical lines in (a); (c) temporal dynamics of the CB population with a biexponential fit to the data. The time delay of 0 fs is fixed at the half-rise of the signal. The error bars indicate standard deviations of the Gaussian fit of the peak areas. The blue Gaussian represents the cross-correlation of the pump and probe pulses.

Figure b shows the dispersion of the bands forming the band gap, measured using 24.3 eV probe energy, at t = 100 fs after the excitation (indicated by violet arrows). For the first time, the energy-momentum dispersions of both the occupied and the unoccupied states are resolved simultaneously. Red and blue solid lines depict the theoretical dispersion of band gap edges as a function of the electron momentum orthogonal to the surface for the two extreme values k = π/a and k = 0, respectively. The band gap magnitude was matched to the value determined via optical absorption (∼2.29 eV, section 2.3 in the Supporting Information). It agrees with other optical data from the literature (2.2–2.4 eV)[1,65,66] and with combined direct and inverse photoemission studies (2.3 eV).[25,67] This ensures that the experiment probes the electron dynamics at the minimum of the CB.

Experimental (for the VB) and theoretical investigations indicate that the states forming the band gap of CsPbBr3 have a nearly parabolic dispersion isotropic in all three directions of the momentum space.[16−18,68] Possible direct electronic transitions for the 3.2 eV pump energy are indicated by violet arrows in Figure b. The maximum photocarrier excess energy of ∼0.9 eV is insufficient to promote electrons to higher energy valleys (see Figure S3). Therefore, the electron population is restricted exclusively to the lowest energy valley of the BZ.

Figure a shows photoemission intensity maps integrated in the momentum range of ±0.23 Å–1 around the M̅ point (between black dashed lines in Figure b), plotted vs time delay, and shows the evolution of the CB population. The integrated maps were fitted with a Gaussian, as shown in Figure b. The peak area is proportional to the photoexcited electron population and is shown in Figure c, as a function of time delay. The CB population increases upon UV excitation within our temporal resolution (∼70 fs), and it decays on a picosecond time scale.

The decay is well fitted with a double-exponential function, pointing to at least two underlying processes. To elucidate the role of the carrier density on the CB dynamics, we investigated the dependence of the extracted decay time constants as a function of the incident pump fluence in the range of (25–325) ± 5 μJ/cm2. The experimental traces and the corresponding fits are shown in Figure a. The maximum electron population scales linearly with the excitation fluence, indicating that the experiments are conducted within a linear absorption regime (section 2.7 in the Supporting Information). Furthermore, a progressively slower population decay at higher fluences is visually apparent.

Figure 4

Temporal evolution of the CB population (a) Biexponential fits (lines) to the experimental data (circles). The legend shows the incident pump fluence of the corresponding scan. The traces are offset by 100 arb.u. for better visibility. (b) The time scale of the fast decay component τfast as a function of electron–hole pair density in the probed volume. The vertical error bars indicate standard deviations of the plotted fit parameters. The horizontal error bars indicate estimated uncertainties originating from pump power fluctuations.

The time constant of the fast-decay component is denoted as τfast and that of the slower contribution as τslow. The fast time scale τfast is weakly affected by the choice of τslow for all of the pump fluences (section 2.9 in the Supporting Information), and a good agreement with the experimental traces is retained when they are fitted globally with a single τslow, yielding a value of ∼20 ps.

The increase of τfast as a function of photogenerated electron–hole pair density N in the probed volume (section 2.6 in the Supporting Information) is shown in Figure b. It increases 3-fold over the entire range of probed densities.

Taking into account the extreme surface sensitivity and the picosecond time scale of the population decay, we consider carrier recombination within and transport out of the probed volume (indicated as (1) in Figure ) to be the most likely pathways. In particular, our analysis, detailed below, reveals that in the high-N regime the carrier transport is limited by electron–hole scattering and can be accounted for by steady-state diffusion with a strong contribution of Auger recombination (AR). However, at low N, we observe a fast transport characterized by anomalously high values of diffusivity, reflecting a significant contribution from carriers that have undergone few or no scattering events (quasi-ballistic transport).

In order to provide a quantitative insight into the transport and recombination properties of CsPbBr3, we modeled them numerically, considering the carrier diffusion and the trap-assisted surface recombination, along with various electron–hole recombination pathways within the bulk, as being potentially responsible for the population dynamics in the probed volume. Subsequently, we model the depth–temporal profile N(z,t) of the carrier density inside the crystal to compare it with the near-surface density evolution tracked experimentally. We proceed by solving the differential equationwhich is often adopted in photoemission[69,70] and optical studies.[71−74] Here, D is the diffusion coefficient, S is the surface recombination velocity,[45] and k1, k2, and k3 are the mono-,[66] bi-, and trimolecular (AR) bulk recombination coefficients,[71,75,76] respectively. The term G(z,t) represents carrier generation and follows the temporal and spatial profiles of the pump. Various other processes, including a photo-Dember effect,[77] surface photovoltage, and lateral transport (indicated as (2) in Figure ), act on longer time scales (section 2.11 in the Supporting Information) and therefore are not included in the model.

At photocarrier densities above the onset of many-body effects in LHPs, the density dependence of the bimolecular k2(N), the AR k3(N), and the diffusion D(N) coefficients[71,72] must be considered. The last coefficient stems from numerous mechanisms, such as the saturation of trap states at low densities or the increased effect of carrier degeneracy and of electron–hole scattering at high densities.[13,71,72] The diffusivity D is expected to increase with N due to degeneracy according to the generalized Einstein relation.[72] The resulting dependence, calculated using optical data from ref (71), is shown by the blue line in Figure a. Concurrently, a more efficient electron–hole scattering at higher N results in an asymptotic ∼1/N dependence,[13] shown by the orange line in Figure a, which was calculated using reported values of the static permittivity[78] and of the reduced carrier effective mass.[79] The value combined via Matthiessen’s rule, as 1/Dtheor(N) = 1/Ddeg(N) + 1/Deh(N), is plotted as the red line in Figure a.

Figure 5

Results of modeling charge carrier density evolution in the probed volume: (a) carrier density dependence of the diffusion coefficient originating from carrier degeneracy (Ddeg, blue) and electron–hole scattering (Deh, orange), overlapped with the combined dependence calculated according to Matthiessen’s rule (Dtheor, red) and with the effective diffusivity extracted from the global fit of the solutions of eq to the experimental data under the assumption of its linear variation at low N (Deff, green); (b) comparison of selected simulated traces (with normalized amplitude) extracted by solving eq , incorporating the diffusivity dependences D(N) depicted in (a) by red and green, correspondingly; (c) comparison of density dependences of τfast extracted from global biexponential fits to the experimental data (blue) and to the green and red traces in (b), correspondingly. The error bars indicate standard deviations of the plotted fit parameters.

The bi- and trimolecular recombination coefficients are expected to reduce and saturate at high carrier densities due to an increased Coulomb screening[80] and state filling.[72] The corresponding approximation for k2(N) was adopted from ref (71). To model the saturation of k3(N), we used the heuristic form k3 = k30/(1 + N/Ng)3/2, proposed in the literature,[72] and the experimental values k30 = 3.6 × 10–27 cm6 s–1 and Ng = 7 × 1018 cm–3.[71,72] This implies a gradual decrease in k3 by 1 order of magnitude within our range of N (section 2.12 in the Supporting Information). According to our band structure calculations, CsPbBr3 possesses a so-called coincidental resonance between the band gap and an interband transition in the VB (unlike the CB) (section 2.2 in the Supporting Information), also reported for the CB of MAPbI3.[81] Therefore, the hhe AR process (involving two holes and one electron) is expected to dominate over the eeh process. The latter is, nonetheless, possible in the form of phonon-assisted AR, which is usually dominant in degenerate wide-gap semiconductors.[80,82]

After incorporation of the density dependence of various coefficients into eq , the experiment was matched only at high carrier densities, as shown by the red traces in Figure b. Conversely, the model strongly underestimates the decay at lower carrier densities, as shown in Figure c, which compares the values of τfast extracted from the global fit to the experimental data (in blue) and to the simulated traces (in red) displayed in Figure b, pointing toward additional mechanisms at low densities. In fact, an accelerated nonequilibrium transport persisting for tens of picoseconds is expected to occur in LHPs when the photocarriers have sufficient excess energy, as demonstrated by a recent experimental study.[42] Under such conditions, the effective diffusivity can exceed the equilibrium value by 2 orders of magnitude. To quantify the possible deviation from the conventional transport model, we extract an effective diffusivity Deff(N) by globally minimizing the mismatch between the experiment and the simulation using a modified model, in which the diffusivity can vary under a certain N, assuming a linear dependence for simplicity. The resulting green traces in Figure b exhibit a significantly improved agreement with the experiment. This is also seen when the corresponding τfast values in Figure c (in green) are compared with the experimental values (in blue).

In a striking manner, Figure a shows that the extracted Deff(N) (green line) surpasses the equilibrium values (red line), which describe the steady-state transport,[71,72] by almost 1 order of magnitude at low densities. When the ultrashort time scale accessible by our experiments is considered, the enhanced transport of hot carriers can explain the fast population decay. In line with our findings, even higher nonequilibrium diffusion coefficients (∼100 cm2/s), in comparison to ours (cm2/s), were observed in the optical microscopy study of lateral transport on a picosecond time scale at a lower density (4 × 10–17 cm–3) and at a higher excess energy (1.49 eV) of the photocarriers.[42]

Another similar study observed carriers in LHPs propagating ballistically over 150 nm within the first 20 fs upon light absorption.[12] Moreover, the transport length was found to be reduced with increasing N, due to enhanced carrier–carrier scattering, reaching a value of 66 ± 10 nm at N = 0.25 × 1019 cm–3. This suggests that the ballistic transport lengths should be even shorter at our carrier densities.

According to diffusion theory, D is proportional to the photocarrier effective transport MFP λeff and to the average carrier velocity v: D ≈ vλeff.[83,84] According to Matthiessen’s rule, 1/λeff = ∑1/λ, where λ values are the individual MFPs related to the corresponding microscopic scattering mechanisms. In view of this, the shape of Deff(N) in Figure a can be qualitatively interpreted as follows: At low N, λeff is long and is limited by scattering with longitudinal optical (LO) phonons.[71] When λeff is comparable[84] to the probing depth, the electrons appear to leave the probed volume quasi-ballistically. This, together with the high velocities of hot carriers, leads to the observed enhanced diffusive transport. Three additional effects can contribute to the increased values of D: the hot-phonon bottleneck and Auger heating can maintain high carrier velocities by slowing down their cooling, whereas the polaron formation can increase λeff by screening the carrier electrostatic potential.[85]

The impact of carrier–carrier scattering increases at high N and results in λeff becoming shorter than the probing depth. This explains why the diffusivity in this regime agrees with studies of steady-state transport properties and asymptotically approaches the 1/N dependence. Ultimately, this reflects the dominance of carrier–carrier scattering as the main transport-limiting mechanism at high N. According to Figure c, this regime sets in above ∼5 × 1019 cm–3, where the conventional diffusion model starts to adequately describe the carrier transport at interfaces, even on a sub-picosecond time scale.

To summarize, the dominant mechanisms of carrier transport and recombination depend on a multitude of factors during the operation of an optoelectronic device, such as the surface contact (as in photocatalysis), the carrier density (as in solar cells vs solar concentrators), the time scale (as in pulsed lasers), and the length scale (as in nanoscale devices). In the case of CsPbBr3, where the absorption depth is only ∼60 nm, the carrier cooling, recombination, scattering, and transport develop a prominent depth dependence, following the photocarrier density distribution. A fundamental understanding of ultrafast processes driving carriers from interfaces toward the bulk (or e.g. the electrical contacts) is essential for improving device efficiencies. The tools capable of directly accessing these material properties are in great demand, as they are indispensable in developing qualitative and quantitative guidelines for mitigating functional bottlenecks and optimizing the performance. The EUV-based TR-ARPES provides a unique view of charge carrier dynamics, as it can selectively follow photocarriers at true band gap edges to extract the most precise and relevant photophysical characteristics. The ability to control the surface and bulk properties (such as doping, orientation, structure, etc.) of LHPs[25,27,28] and to choose the best operating conditions to take advantage of the density dependence of various many-body effects make LHPs an exciting material to be custom-tailored for next-generation applications: whether we want, for example, to enhance the radiative recombination, to maximize the transport length, or to minimize the carrier cooling rate, depending on the requirements. This approach is generally applicable to other technologically relevant semiconductors by selecting suitable pump and probe energies, whereas recent advances in nano-ARPES[86] expand its applicability even to nanosized samples. Future studies with energy-tunable excitation could further elucidate the role of carrier excess energy in the nanoscale transport.

From a theoretical perspective, our simulations highlight the limitations in applying the conventional diffusion theory to the nanoscale transport, prompting the community to further advance our capability of calculating nonequilibrium transport properties ab initio, upon unveiling their crucial role on the ultrafast time scale.

In conclusion, we utilized the surface sensitivity, temporal, and momentum resolution of EUV-based TR-ARPES to investigate the dynamics of electrons photoexcited into the conduction band of CsPbBr3 single crystals. This allowed us to observe ultrafast electron transport into the bulk with sub-nanometer resolution, which we studied as a function of pump fluence. By combining our experimental results with simulations of the spatiotemporal evolution of the hot-electron population, we clarify the microscopic mechanisms governing the ultrafast surface-to-bulk carrier transport and recombination. This allows us to distinguish the carrier-density-dependent crossover from the enhanced to the ordinary diffusive transport regime at ∼5 × 1019 cm–3, with the former being limited by LO phonon scattering and the latter by carrier–carrier scattering. The results are consistent with the saturation of carrier recombination coefficients due to phase-space filling and Coulomb screening mechanisms enhanced at high carrier densities. Our results constitute a new milestone in understanding and describing the ultrafast nanoscale surface-to-bulk hot-carrier transport in LHPs in the high-carrier-density regime, which represents the operational conditions of various photonic and nano-optoelectronic devices. This approach can be generally implemented for studying nanoscale transport in all highly absorbing photonic materials, even beyond LHPs.