Bright-dark exciton splitting in lead halide perovskite crystals accessed via quantum beats in photon echoes

Abstract

Understanding the fine structure of excitons is crucial for optoelectronic and quantum photonic applications of lead halide perovskites. It is demonstrated that polarization-sensitive photon echo spectroscopy in magnetic field provides a powerful method to access coherent exciton dynamics and reveal their energy level structure, which is hidden by inhomogeneous broadening. Exciton quantum beats observed in both Faraday and Voigt geometries offer a precise probe of the energy splittings among the four 1$s$ exciton states, enabling determination of the fine structure and bright-dark splittings. Application of this technique to bulk mixed halide perovskite crystals FA$_{0.9}$Cs$_{0.1}$PbI$_{2.8}$Br$_{0.2}$ reveals a bright-dark exciton splitting of $Δ_\mathrm{X}=0.46~$meV, along with electron and hole Landé $g$ factors $g_\mathrm{e}=3.38$ and $g_\mathrm{h}=-1.14$, respectively. The quantum beats persist on timescales of 20--50$~$ps, demonstrating remarkably robust spin and optical coherences at cryogenic temperature of 2$~$K. The decay of the quantum beats of the outer doublet is governed by dephasing due to dispersion of the bright-dark splitting of $\sim0.06~$meV caused by localization potential fluctuations, while dephasing in the bright exciton inner doublet originates from a small zero field splitting of $\sim0.035~$meV due to anisotropic potentials.

Figures (11)

• Figure 1: a) Orientation of the excitation and magnetic field geometry relative to the sample surface. The laser propagates along the $z$-axis ($\mathbf{k} \parallel \mathbf{z}$), which is normal to the sample plane. The Faraday geometry corresponds to a magnetic field applied along the optical axis ($\mathbf{B}_\text{F} \parallel \mathbf{z}$), whereas in Voigt geometry, the field is applied in-plane along the $x$-axis ($\mathbf{B}_\text{V} \parallel \mathbf{x}$). A coordinate system is included for clarity. b) Energy level diagram and optical transitions for the 1$s$ exciton in Faraday (left) and Voigt (right) geometries. Optical transitions between the crystal ground state $|{\rm G}\rangle$ and bright exciton states (green) are shown with red/blue arrows for light propagating along the $z$-axis. The dark state $\ket{0,0}$ is separated from the bright states $\ket{1,(0,\pm1)}$ (the indices reflect the direction of the dipole moment) by an energy splitting $\Delta_{\rm X}$. In an external magnetic field, the exciton states undergo splitting and mixing. In Faraday geometry (left), mixed states remain inaccessible, and only $\ket{1,(\pm1)}$ are addressed optically with $\sigma^{\pm}$ polarized light, respectively. In Voigt geometry (right), the mixed states become accessible via linearly polarized light. In strong magnetic field, the inner and outer doublets are polarized perpendicular ($V$ polarization) and parallel ($H$ polarization) to the magnetic field direction $\mathbf{B}_\text{V}$, respectively. The level arrangement assumes $g_\text{e} > 0$ and $g_\text{h} < 0$, following Ref. 24KoptevaAdvSci, to provide a consistent representation of the observed splitting behavior of the exciton states.
• Figure 2: a) Four-wave-mixing (FWM, green) and photoluminescence (PL, blue) spectra of bulk mixed halide perovskite FA_0.9Cs_0.1PbI_2.8Br_0.2 sample, as previously reported in Ref. 23GrisardNano. The PL spectrum is measured under excitation with photon energy of 2.33 eV. The FWM spectrum was recorded using ps laser pulses. The excitation laser spectrum (shaded area) used in the PE experiments has a full width at half maximum of 18 meV and is centered at 1.51 eV. b) Schematic representation of the non-collinear transient FWM arrangement used for PE studies.
• Figure 3: Decay of the photon echo (PE) amplitude at zero magnetic field ($B=0$) measured in lineraly co-polarized configuration. Solid red line is a fit with Equation \ref{['eq:expexp']}. Inset: Polar plot of PE amplitude as a function of angle $\theta$ between linear polarizations of the 1st and 2nd pulse measured at $\tau_{12}=4$ ps. Solid red line is a fit with $\cos^2{\theta}$ which corresponds to typical exciton behavior19PoltavtsevSR.
• Figure 4: Photon echo oscillations in magnetic field $B$ for different polarization configurations. a)-d): Time-resolved photon echo amplitude (arb. units) for $3T$ and $5T$. Each legend indicates the horizontal ($H\parallel\mathbf{x}$) and vertical ($V\parallel\mathbf{y}$) polarization of the first and second excitation pulse and detection. Solid curves represent the results of the modeling described by Equations \ref{['eq:PEeqs_F_HHH']}-\ref{['eq:PEeqs_V_HHH']}. Full set of data and the details of analysis procedure can be found in the Supplementary Information \ref{['SI-fit']}. The evaluated parameters are summarized in Table \ref{['tab:fit']}. e) Magnetic field dependence of the energy splittings between the inner $\hbar\Omega_{\rm I}$ and outer $\hbar\Omega_{\rm O}$ excitonic doublets. Plus and circle symbols correspond to the data obtained in Faraday and Voigt geometries, respectively. Solid blue/yellow line is a linear fit with $\hbar\Omega_{\rm I}+\delta$ with $\delta\approx35~\mu$eV. Red line represents the fit with $\hbar\Omega_{\rm O} = \hbar\sqrt{\Omega_{-}^2+\Omega_{\rm X}^2}$. e) Solid red line is the fit with $\Omega^2_{-}/(\Omega_{-}+\Omega_{\rm X})^2$ where $\Delta_{\rm X}=\hbar\Omega_{\rm X}=0.46$ meV. The arrow marks the point where the Zeeman splitting $\hbar\Omega_{-}$ matches the bright-dark splitting $\Delta_{\rm X}$. f) Magnetic field dependence of the oscillations amplitude $C=(\Omega_{-}/\Omega_{\rm O})^2$. The circles are extracted from the fit of $HHH$ transients in Voigt geometry using Equation \ref{['eq:PEeqs_V_HHH']} with fixed value of $g_\text{e}-g_\text{h}=4.52$ which is evaluated from the fit of $\hbar\Omega_{\rm O}(B)$ in panel.
• Figure S1: Scheme of coordinate systems aligned with magnetic field $x'y'z'$ and the coordinate system with respect to pump/probe pulse direction $xyz$.