We report on temperature-dependent (10 K – 250 K) spectral and dynamical properties of free exciton–polariton and bound exciton emission in copper iodide (CuI) bulk single crystals analyzed by means of time-resolved photoluminescence spectroscopy. The characteristic line shape of the polariton emission at low temperatures is interpreted in terms of the “k-linear term effect” on the degenerate *Z*_{1,2} excitons in CuI. For free exciton–polaritons, an increase in the decay time with increasing temperature up to 360 ps at 160 K is observed. For bound exciton emission, decay times between 180 ps and 380 ps are observed at low temperatures, revealing the expected $EB3/2$ dependence of radiative lifetime on the localization energy. Based on the observed rise times of bound excitons at low temperatures, a defect density of shallow acceptors of 1 × 10^{17} cm^{−3} was estimated, in agreement with measured room temperature free hole density.

## I. INTRODUCTION

Copper iodide in the thermodynamically stable zincblende phase (*γ*-CuI) is a promising wide-bandgap material^{1} (*E*_{G} ≈ 3.1 eV) for transparent optoelectronic and thermoelectric applications due to its large exciton binding energy^{2} $(EXB\u224862meV)$, intrinsic p-type conductivity with a hole mobility of up to 43 cm^{2}/Vs in bulk crystals,^{3} and its high thermoelectric figure of merit.^{4} Until now, CuI has been successfully applied in transparent p–n heterojunctions,^{5–8} thin film transistors,^{9–11} solar cells,^{12–14} and scintillators.^{15} Published lasing emission in CuI-based microstructures^{16} substantiates the suitability of CuI as a p-type component in compact integrated optoelectronic circuits and light-emitting devices. Furthermore, large exciton oscillator strengths were obtained for CuI,^{17} making this material interesting not only for the investigation of properties of bulk polaritons but also for the investigation of polariton-related effects in CuI-based cavities.

In the last few years, CuI crystals and films have been studied in terms of structural,^{18–20} electrical,^{21,22} and optical properties.^{16,23–25} In particular, the excitonic emission lines of the near-band edge emission of CuI were investigated extensively at low temperatures.^{2,16,17,26–28} The properties of the degenerate excitonic states were investigated by uniaxial-stress and magneto-optical measurements and explained in context of the polariton picture.^{17,28}

Although many optical properties of CuI have been investigated, detailed studies of temporal emission dynamics in CuI are still lacking. For thin films, averaged lifetimes of excitonic emission of around 900 ps were reported at room temperature.^{29} However, the interpretation of different recombination dynamics of free and bound states was strongly limited by significant broadening of the observed emission band. The excitonic decay times at low temperatures were reported for high quality CuI microwires.^{16} However, the discussion of different observed decay components and the interpretation of the decay times as a function of photon energy and temperature are still an open research question.

The aim of this work is to provide a detailed discussion of the recombination dynamics of exciton–polaritons in CuI for temperatures between 10 K and 250 K. First, we present and recall the static emission properties of CuI single crystals as a function of temperature and assign the observed near-band edge emission peaks to free polariton emission arising from different polariton branches and bound exciton states. The subsequent part of this paper deals with the time evolution of the observed emission peaks. The energy relaxation of the polariton population after pulsed excitation at low temperatures is discussed based on the transient line shape of the polariton emission peaks. Subsequently, the different recombination dynamics of free and localized states are discussed as a function of temperature.

## II. EXCITON–POLARITONS

The large oscillator strength of excitons in CuI^{17} leads to a pronounced coupling between the excitons and the electromagnetic field (photons), resulting in a mixed state called exciton–polaritons. The dispersion relation between the energy *E* and the wave vector *k* of the polariton modes is given by the so-called polariton equation,^{31}

where *ɛ*(*E*) denotes the dielectric function in the vicinity of the excitonic resonance. The dispersion of the lower polariton branch (LPB) and upper polariton branch (UPB) resulting from Eq. (1) in the most simple case of a single exciton resonance is schematically shown in Fig. 1(a). The dashed and dotted lines indicate the dispersion relations of the uncoupled excitons and photons, respectively. In the case of a non-vanishing coupling, the crossing is avoided and the solid lines describe the dispersion relation of the coupled polariton state. The resulting energy difference between both branches is given by the Rabi splitting Ω. The LPB dispersion starts at *E* = 0 and *k* = 0 (off scale in Fig. 1) as a straight line and flattens by approaching the resonance energy *E*_{T}. The UPB dispersion, on the other hand, begins at *k* = 0 and *E* ≈ *E*_{L} and approaches the photon dispersion relation in the material $hck/\epsilon b$ at larger wave vectors. We note that although the UPB energy at *k* = 0 coincides with *E*_{L}, the longitudinal exciton branch usually does not couple to the electromagnetic field. In the case where the Rabi splitting [see Fig. 1(a)] is smaller than the broadening due to the low oscillator strength, polariton effects can usually be neglected.

Due to the mixed nature of exciton–polaritons, their physical properties, such as, e.g., lifetime, depend on their characteristics, i.e., whether they are more exciton-like or photon-like. The corresponding contributions that determine the overall character of the coupled polariton state can be illustrated by the square of the Hopfield coefficients *H*_{exc} and *H*_{ph}.^{30} For the coefficients of each polariton branch, |*H*_{exc}|^{2} + |*H*_{ph}|^{2} = 1 holds. For a small wave vector and energies well below the exciton resonance, the dispersion of the LPB is similar to those of the bare photon and thus $Hph2\u22481$. For large wave vectors and energies well above the excitonic resonance, the polariton dispersion is mainly determined by the exciton dispersion $(Hexc2\u22481)$. An opposite behavior is observed for the UPB. In the vicinity where the dispersion of the bare photon and the exciton crosses, the exciton and photon equally contribute to the polariton state, i.e., |*H*_{exc/ph}|^{2} = 0.5.

## III. EXPERIMENTAL

### A. Crystal growth

Copper iodide was synthesized from copper acetate monohydrate (Sigma-Aldrich, 98%) and acetone and iodine (Sigma-Aldrich, 99.5%) in hot acetic acid, similar to the procedure reported by Hardt and Bollig.^{32} The solubility of copper iodide in acetonitrile is around 30 g/l at 20 °C.^{33} The raw product was purified by crystallization of an acetonitrile copper iodide complex. As identified by x-ray diffraction (XRD), at low temperature (below −20 °C), the main part of copper iodide crystallizes as colorless crystals of [(CH_{3}CN)_{2}(CuI)_{2}]_{n}.^{34} This complex that is stable only in an acetonitrile atmosphere or at low temperature^{35} was completely transformed to copper iodide by treatment in vacuum at 120 °C for 10 h. According to x-ray fluorescence analysis and powder XRD, the resulting white powder is highly pure (≥99.993%) copper iodide.

Large crystals of copper iodide can be grown using solution methods.^{33,35–38} Based on the decreasing solubility of copper iodide in acetonitrile with increasing temperature in the range of 20 °C–60 °C, crystals of *γ*-CuI with varying optical quality^{33,35} were grown by slow heating of a saturated solution. The partially brown color of the crystals is due to oxidation processes. Several methods to trap the oxidation products are known.^{33,35,38} The best way to yield colorless crystals in high optical quality is the combination of suppressing oxidation and trapping of iodine. Trapping agents should be liquid in order to maintain high purity and to suppress inclusion of impurities in the copper iodide crystals. A good option is the use of ketones, which react with iodine by formation of HI and the corresponding iodine substituted ketones. Additionally, the crystal growth should be carried out in an inert atmosphere. We realized the growth of crystals in autoclaves filled with copper iodide acetonitrile solution saturated at room temperature and penta–2,4–dione in the ratio 10:1. All compounds were purified before use. The temperature was increased from 25 °C to 46 °C within two weeks. The size of the resulting colorless and clear crystals is in the range of (1–4) mm [see Fig. 2(c)].

### B. Structural and electronic characterization

X-ray diffraction (XRD) was carried out for the structural investigation of CuI. The XRD setup used for the investigation of CuI crystals consists of a PANalytical X’Pert Pro diffractometer with a Cu K_{α} radiation source, a parabolic mirror, and a PIXcel^{3D} detector. For the investigation of CuI powder, a Stoe STADI P diffractometer equipped with a DECTRIS MYTHEN 1 K detector was employed. X-ray fluorescence analysis was measured using a Bruker S2 Picofox TRFA spectrometer.

The electronic properties of the crystals were investigated using temperature-dependent van der Pauw (VDP) measurements. For this purpose, a shadow mask and DC magnetron sputtering were employed in order to fabricate Ohmic gold contacts onto the crystal surface. The sample was then mounted on a LakeShore 8425 probe station for temperature-dependent resistivity and Hall effect measurements. The maximum magnetic field applied during the Hall effect measurements was 2 T.

### C. Optical characterization

For temperature-dependent optical measurements, the substrate with the CuI crystals was mounted in a helium-flow cryostat (Janis ST-500). The sample temperature was measured with a silicon-diode-based temperature sensor mounted on the cryostat cold finger. A helium–cadmium laser (Kimmon HeCd, 325 nm, cw) was used for time-integrated measurements. For time-resolved measurements, a frequency-doubled mode-locked titanium–sapphire laser (Coherent Mira HP TiSa: 383 nm wavelength, 200 fs pulse duration, and 76 MHz repetition rate) was used. The laser beam was focused by using a Mitutoyo 50× NUV objective (NA = 0.42) to a spot size of about 4 *µ*m. A variable attenuator was used to adjust the excitation density. The emitted photoluminescence (PL) was collected confocally by using the same objective, dispersed by using a spectrometer (Horiba Jobin Yvon iHR320) with either 2400 grooves/mm or 600 grooves/mm grating, and guided to a CCD camera (Horiba Jobin Yvon Symphony Open STE) or to a streak camera (Hamamatsu C5680), respectively. The spectral resolution of both detection systems in combination with the corresponding gratings is 0.05 nm for time-integrated measurements and 0.3 nm for time-resolved measurements. The choice of the grating for time-resolved measurements represents a compromise between the required spectral and temporal resolution, where the latter is affected by the path-length difference through the spectrometer.^{39} The best obtained temporal resolution in the chosen time range of the measurements was ∼15 ps. The selected excitation density was about 6 nJ/cm^{2}, which is about five orders of magnitude smaller than the previously reported laser thresholds in CuI microwires.^{16} Furthermore, the absence of saturation of bound exciton emission and non-linear effects associated with exciton–exciton scattering^{40,41} was verified by excitation-dependent measurements with excitation energy densities *D*_{exc} in the range (1–1000) nJ/cm^{2}.

## IV. RESULTS AND DISCUSSION

### A. Structural properties

Figure 2(a) shows the XRD 2*θ*–*ω* scan of a solution-grown CuI single crystal. The scan shows that the orientation presented in Fig. 2(c) corresponds to the (111) direction of the *γ*-phase. A splitting of *K*_{α,1}/*K*_{α,2} can be observed for all reflections. The cubic lattice constant, extracted from powder diffraction, and the scan in Fig. 2(a) are $a=6.0526(3)A\u030a$, being in agreement with the literature.^{42,43} The *ϕ* scan in Fig. 2(b) shows threefold symmetry with 120° spacing and a 60° shift between the (200) and (220) peaks. This is to be expected for a (111) orientation; therefore, we can deduce that the crystal is single crystalline without twinning or rotational domains.

### B. Steady-state low temperature PL properties

Figure 3(a) shows a typical time-integrated photoluminescence (PL) spectrum of a CuI single crystal at 10 K. The observed emission lines are attributed to free and localized excitonic transitions, which are discussed individually below.

#### 1. Free exciton–polariton photoluminescence

Although the emission lines between 3.055 eV and 3.12 eV were often attributed to the free transversal (*FX*_{T}) and longitudinal (*FX*_{L}) exciton ground states^{16,45} (*n* = 1) and the corresponding excited states (*n* > 1), this classification is, strictly speaking, inaccurate since *FX*_{L} does not couple to the electromagnetic field.^{31} Instead, the observed emission features must be discussed in the exciton–polariton framework since the large oscillator strength of free excitons leads to a remarkable splitting of the polariton branches that is larger than the observed emission linewidth, in contrast to, e.g., GaAs, where polariton effects can be neglected due to the small exciton oscillator strength of 7 × 10^{−5}.^{46} Here, we start the discussion based on the findings presented by Suga *et al.*^{17,28}

Since the valence (conduction) band in CuI has a Γ_{6} (Γ_{8}) symmetry, the lowest exciton states are represented by Γ_{6} × Γ_{8} = Γ_{3} + Γ_{4} + Γ_{5} states.^{47} Due to the dipole–dipole interaction, the Γ_{5} singlet state splits into longitudinal (Γ_{5L}) and transversal (Γ_{5T}) states. The anisotropic spin-exchange splitting between the Γ_{3} and Γ_{4} triplet states is neglected in this work similar to Refs. 17 and 48.

The dispersion of the pure excitonic states was calculated according to Ref. 17 taking into account the electron–hole exchange interaction and energy terms linear and quadratic in wave vector *k*. According to Eq. (1), by taking into account only excitonic contributions to the dielectric function, the polariton equation is given by^{48}

with the oscillator strength *f*_{i}, the energy $Eiexc$ of the corresponding exciton state depending on the wave vector, and the background dielectric constant $\epsilon bUV$ in the UV spectral region where excitonic contributions to the dielectric function are not considered. In this context, CuI is an interesting material where not only the transverse Γ_{5} state but also the Γ_{3,4} triplet states couple with photons due to the finite oscillator strength at *k* > 0.^{17} In comparison to the case, where only one exciton is involved [see Fig. 1(a)], the dispersion relation becomes more complicated in this case and exhibits four distinct polariton branches.

The polariton dispersion was calculated considering only ground state excitons (*n* = 1) using the parameters reported by Suga *et al.*^{17,28} [see Fig. 3(b)]. We find very good agreement between the calculated polariton dispersion and our experimental data and assign the emission peaks as follows: the emission around 3.057 eV appears to originate from the bottleneck-region, i.e., the region where the dispersion curvature changes its sign, of the lower polariton branch (LPB). At low temperatures, this peak energy coincides with the transverse exciton energy *E*_{T} of the degenerate triplet states Γ_{3,4}.^{49} Due to the scattering of the polaritons at the LPB dispersion toward smaller *k* values, a low-energy tail is expected for the LPB emission. However, this low-energy tail spectrally overlaps with the emission of weakly bound excitons and is therefore difficult to observe experimentally, especially under cw excitation. As will be shown later, such a low-energy tail can be clearly seen a few ps after pulsed excitation, when bound excitons are not yet formed. The peak at 3.058 eV is assumed to arise from the overlap of the emission from the lower parts of the middle polariton branches (MPB_{1} and MPB_{2}). The transition energy of 3.065 eV corresponds to the energy of the longitudinal exciton (*FX*_{L}) of the singlet Γ_{5} state. However, emission from this longitudinal state is not expected due to a weak coupling to the electromagnetic field,^{31} as mentioned before. Instead, this emission is assigned to the upper polariton branch (UPB), which results from the polariton effect on the transverse Γ_{5T} exciton and also starts at an energy close to those of Γ_{5L} at *k* = 0. The thermal distribution of the polaritons in the vicinity of the UPB leads to the characteristic asymmetric line shape of the corresponding emission peak with a steep slope on the low-energy side and an exponential tail on the high-energy side.^{28}

The weak emission peaks around 3.105 eV and 3.115 eV (see Fig. 3) are assigned to excited exciton states with *n* = 2 and *n* = 3, respectively. The polariton splitting for excited states decreases with *n*^{−3/2} and becomes smaller than the peak broadening for *n* > 2 transitions. Thus, the polariton effects can be neglected in this case.^{50} The exciton binding energy is determined to be (62 ± 2) meV in a hydrogenic model [see the inset of Fig. 3(a)] in excellent agreement with previous reports.^{51} Using this value, the bandgap energy at 10 K yields (3.120 ± 0.002) eV, similarly to literature values of *E*_{G} = 3.1 eV.^{1}

#### 2. Bound exciton photoluminescence

The observed peaks at energies between 2.98 eV and 3.05 eV are assigned to the emission of bound excitons and are labeled from B to H according to previous publication.^{16} The emission in the energy range of the lines C–E is probably caused by recombination of excitons bound to shallow neutral acceptors.^{26} It is known that the dissociation energy and thus the existence of excitons bound to ionized acceptors strongly depend on the ratio of the effective masses of the charge carriers *σ* = *m*_{e}/*m*_{h}.^{54} Excitons bound to ionized acceptors are expected to be stable only for *σ* > 2.33. In CuI, the values for *σ* are, however, significantly smaller than the critical mass ratio *σ*_{c} such that excitons localized to such ionized acceptors are unstable and only neutral acceptors are expected to trap free excitons. The activation energy of electrically active shallow acceptors *E*_{A,1} = 108 meV, determined from temperature-dependent VDP measurements [see Fig. 4(a)], is in reasonable agreement with literature values.^{26,55} Assuming that the acceptors contributing to free carrier density are associated with the bound exciton transitions labeled C–E, the factor of Haynes’s rule describing the ratio of the binding energy of the exciton to the impurity to the binding energy of the hole to the impurity $EB/EAb$^{56} can be determined to be 0.08–0.17. However, these values are only a rough estimation and have to be verified, e.g., by further investigations of intentionally doped CuI samples. Furthermore, we note that a second activation energy of around *E*_{A,2} = 48 meV was observed in Hall effect measurements. Using the determined Haynes rule factor, the corresponding shallow acceptor could be assigned to the weak transition peak B occurring at 3.054 eV. However, it is difficult to assign the corresponding emission unambiguously since it could also result from the recombination of excitons bound to shallow donors.^{26}

The exact origin of transitions F and G, which were also observed in previous experiments,^{16} is still unclear. In terms of their spectral position, these peaks could be 1-LO-phonon replicas of the C and D transitions, although such assignment could only be partially confirmed by means of time-resolved photoluminescence, as discussed below.

The emission around 3.016 eV (labeled H) was ascribed to recombination of excitons bound to deep neutral acceptors by Sauder and Certier.^{2} We note that the reason for the occurrence of a doublet structure in comparison to a single peak observed for CuI microwires^{16} is unclear until now. However, the activation energy of this deep defect could not be investigated by Hall effect measurements since such defects are assumed to contribute to the carrier density at temperatures far above the room temperature. The weakly pronounced doublet structure at around 3 eV is assigned to phonon replica of the transitions H with a LO-phonon energy of ∼19 meV, which is in excellent agreement with our Raman measurements performed on the same sample and with literature values.^{57,58}

#### 3. Temperature-dependent polariton emission

Temperature-dependent PL measurements were performed for temperatures between 10 K and 250 K. Figure 5(a) shows the evolution of the time-integrated emission near the band edge of CuI with increasing temperature. It can be clearly seen that with increasing temperature, the relative intensity of the emission from the free polaritons increases compared to the emission of bound exciton states due to the dissociation of bound excitons at higher temperatures. Furthermore, the peaks related to the free exciton–polaritons shift to lower energies mainly due to bandgap shrinkage. While the UPB emission peak can be tracked up to about 200 K, LPB- and MPB-related emission peaks are difficult to follow at higher temperatures due to increasing broadening.

Moreover, it has already been shown for other copper halides that the LPB bottleneck-region responsible for the emission peak is not located at a fixed wavenumber along the polariton dispersion but also depends on the temperature via polariton–phonon scattering processes.^{49} Therefore, we focus in the following on the temperature-dependent peak position of the UPB emission [see Fig. 5(b)].

The temperature-dependent energy shift of the bandgap can be described by^{45,60}

Here, *E*_{0} is the bandgap energy at 0 K. The second term on the right-hand side describes the energy shift due to renormalization by the electron–phonon interaction (Δ*E*_{ph}), where *α* and *k*_{B}Θ_{B} denote the coupling strength and the effective phonon energy, respectively. The last term (Δ*E*_{th}) takes into account the contribution of the thermal lattice expansion *a*(*T*) − *a*_{0} via the deformation potential *γ*. Since the latter nearly takes the same form [see Fig. 5(c)], it is often not discussed separately and is often included in the effective phonon energy Θ_{B}.^{61} However, this would lead to an effective phonon energy of 26 meV, which is significantly larger than the LO-phonon energies reported for CuI^{57,58} and thus unphysical. Considering the effects of lattice expansion and electron–phonon interaction individually according to Eq. (3), we obtain an effective phonon energy *k*_{B}Θ_{B} = (20 ± 2) meV [see Fig. 5(b)], which is in reasonable agreement with experimental values for the LO-phonon energy, determined by Raman spectroscopy. The coupling constant was determined to be *α* = 0.11 meV/K, in accordance with previous results.^{24}

The expression describing the electron–phonon interaction (Δ*E*_{ph}) used in this work also corresponds to the model presented by Fan,^{62,63} where *α*Θ_{B} can be identified with Fan’s parameter *A*. The analytic expression for *A* is given by^{64}

where *E*_{ph} is the effective phonon energy; $\epsilon s(\epsilon bVIS)$ is the static (background) dielectric constant; *m*_{e}(*m*_{h}) is the effective electron (hole) mass; and *e*, *ℏ*, *ɛ*_{0}, and *m*_{0} are the elementary charge, the reduced Planck constant, the vacuum permittivity, and the rest mass of electron, respectively. Using spectroscopic ellipsometry, the background dielectric constant in the visible spectral range below the excitonic resonance is determined to be $\epsilon bVIS=4.86$, and by using the Lydanne–Sachs–Teller (LST) relation,^{65} we obtain for the static dielectric constant *ɛ*_{s} = 6.43, in good agreement with the literature.^{57} The effective masses of electrons and holes *m*_{e} = 0.3 and *m*_{lh} = 0.3 are taken from Ref. 66. Using the above-mentioned values and the experimental value for the effective phonon energy *k*_{B}Θ_{B} = 20.6 meV, we obtain *A* = 25.8 meV, which is in excellent agreement with the experimentally obtained value of *α*Θ_{B} and therefore supports the distinction between the electron–phonon interaction and the lattice expansion. The corresponding individual contributions to the bandgap are presented in Fig. 5(c). For calculation of the temperature-dependent lattice constant [see the inset of Fig. 5(c)], we use the temperature-dependent thermal expansion coefficients reported by Berger^{59} for CuI bulk crystals and the room temperature lattice constant of 6.055 nm, determined by XRD measurements [see Fig. 2(a)]. Unfortunately, to the best of the authors’ knowledge, no experimental data are available for temperatures below 80 K, so we assume that the absolute value of the thermal expansion coefficient is small at low temperatures and therefore set Δ_{th}*E*_{0}(*T*) = 0 for *T* < 80 K. We note that a detailed determination of the contributing phonon branches would require more precise knowledge of the thermal lattice expansion at low temperatures.

### F. Polariton dynamics

#### 1. Cooling of polaritons

In the case of non-resonant optical excitation, as used in the present work, excitons are generated with high excess kinetic energies. Since the exciton–polariton emission is observed a few ps after excitation, we assume that the initial carrier thermalization process and the corresponding generation of excitons are completed on timescales far below the temporal resolution of our setup, in agreement with the literature.^{67} The subsequent cooling of the exciton–polaritons, mainly due to interaction with phonons and, in particular, LO-phonon scattering, leads to a reduction in their excess kinetic energy.^{68} Assuming that the polaritons are in quasi-thermal equilibrium with each other, the distribution of UPB polaritons can be described by the Maxwell–Boltzmann statistics,^{68} which is in good agreement with the observed line shape of the corresponding emission peak (see, e.g., Fig. 3).

The initial polariton distribution is expected to cool by emission of phonons. Interestingly, we observe a significant change in the slope of the high-energy tail of the UPB emission within the first 100 ps [see Fig. 6(a)] after excitation, indicating a change in the polariton momentum-space distribution due to the energy relaxation processes at low temperatures. The effective temperature of the experimentally observed distribution of UPB polaritons was extracted by using the Maxwell–Boltzmann statistics, i.e., *I*(*E*) ∝ exp(−*E*/*k*_{B}*T*) to describe the high-energy tail of the corresponding emission peak at $\u223c3.065eV$ [see Fig. 6(b)]. We note that no reasonable fit of the line shape could be obtained at later delay times due to the insufficient signal-to-noise ratio. The transient decrease in the obtained polariton temperature from an initial value of about 60 K to a value of 30 K is well described by simple exponential cooling,

where *κ* and *T*_{0} are the cooling constant and final polariton temperature, respectively. We obtain a cooling constant of ∼20 ps for employed experimental conditions, revealing fast energy relaxation of polaritons.

However, the entire thermal distribution of the total polariton population is still barely accessible in a photoluminescence experiment, as explained in the following. Although the wave vector is a correct quantum number to describe the polariton state in the bulk, only the in-plane component of the wave vector is preserved upon recombination at the surface.^{31} Therefore, only polaritons with *k*_{∥} ≤ *k*_{vac} = *E*/(*ℏc*) can be transmitted through the crystal surface [case of the total inner reflection in Fig. 6(c)]. Thus, at larger *k* values, only a small fraction of the polaritons can escape from the crystal and contribute to PL emission. Furthermore, the fraction of the experimentally observable polariton population is further limited by the maximum aperture angle of Θ_{NA} ≈ 25° [see Fig. 6(c)]. Thus, the fraction of the experimentally accessible polariton population decreases dramatically for *k* ≥ 6 × 10^{4} cm^{−1}. Therefore, the effective temperature values extracted from the line shape of the high-energy tail of the UPB emission systematically underestimate the temperature describing the overall thermal distribution of UPB polaritons.

To estimate the mentioned deviations, the measured PL signal was corrected by a factor of [1 − cos(*φ*_{max}(*E*))], which considers the fraction of the polaritons with $k\u2225\u2264k\u2225max$ for a given wave vector $|k\u20d7|$ [see Fig. 6(c)]. The angle *φ*_{max} describes the half-cone angle of the directional distribution of polaritons and is determined by the numerical aperture of the objective and the polariton dispersion. Thus, the energy-dependent fraction of polaritons contributing to the measured signal is taken into account. Figure 6(d) shows an example of the corrected line shape of the UPB emission about 100 ps after excitation at 10 K [see Fig. 6(a)] in comparison with the raw PL data. We indeed find a slightly higher temperature of about 40 K compared to the previously determined value of 30 K, as expected from the previous discussion. However, although the absolute values for the temperature in Fig. 6(b) are underestimated by about 10%–20%, the relative cooling behavior of the polaritons is still well described in Eq. (5).

Interestingly, the finally observed polariton temperature after 100 ps still exceeds the nominal lattice temperature, which is assumed to be equal to the experimentally measured cryostat cold finger temperature. In general, an increase in the lattice temperature after non-resonant pulsed excitation can be expected for CuI due to an energy transfer of the excess carrier energy to phonons. This, in turn, leads to delayed carrier relaxation as a consequence of re-absorption of phonons.^{25} However, for the experimental conditions given in the present work, the maximum increase in lattice temperature is expected to be below 10 K and cannot fully explain such large polariton temperatures. An explanation of this behavior might be that the depletion of the polariton population in the radiation region with *k*_{∥} ≤ *k*_{vac} prevents a complete thermalization of the entire population.^{68,69}

Consequently, a time-averaged effective temperature of the distribution of UPB polaritons in the case of cw excitation at low temperatures would also exceed the nominal lattice temperature. In fact, for cw excitation at lattice temperatures below ∼45 K [see Fig. 3(a)], the experimentally determined temperature of the UPB polariton distribution remains constant around 40 K, taking into account the previously discussed correction due to the limited numerical aperture. This shows that the polaritons are indeed not in thermal equilibrium with the lattice, in good agreement with time-resolved measurements. For higher lattice temperatures, the effective polariton temperature linearly follows the nominal lattice temperature. We also note that similar overheating of charge carriers has also been reported for CdS,^{70} GaAs,^{71} and GaN.^{72}

Finally, a slight blue shift of the polariton emission is visible in the transient spectra [see Fig. 6(a)]. Such an energy shift could indicate a renormalization of the bandgap as a consequence of the decreasing lattice temperature,^{25} although the expected changes in the bandgap energy cannot fully explain the observed shift. Another possible reason could be the previously discussed change in the polariton distribution with time. However, a clear discussion of the lattice dynamics in CuI after pulsed excitation based on time-resolved PL spectroscopy is extremely difficult and is beyond the aim of this work.

#### 2. Free exciton–polariton dynamics

As mentioned above, the lifetime of polaritons is strongly influenced by their character, which can be represented by the Hopfield coefficients,^{50} i.e., for a strong photonic (excitonic) character, the lifetime of the photon (exciton) dominates the polariton lifetime. Interestingly, it turns out that in the region of the dispersion from which the observed emission originates, the polaritons on the LPB and UPB branches reveal predominantly exciton-like character. The same is true for the polaritons on the middle polariton branches. Therefore, we claim that the decay characteristics and lifetimes discussed further are mainly determined by the exciton properties. Nevertheless, we note that the description in terms of polaritons is necessary to describe the observed emission properties.

Figure 7(a) shows the time-resolved PL spectra of the UPB emission (red symbols), which qualitatively represent the transient spectra of all observed free polariton peaks. It can be clearly seen that the emission exhibits a non-mono-exponential decay, which can be well described by a sum of two-exponential decays,

where *A*_{1} and *A*_{2} express the contributions of the fast and slow decay, with decay times *τ*_{1} and *τ*_{2}, respectively. In the literature, different explanations for such biexponential behavior can be found. The biexponential behavior of the polariton luminescence in CdS for energies above the bottleneck-region (*E* > *E*_{T}) was associated by Askary and Yu with fast relaxation of nonequilibrium polariton population via acoustic phonons.^{73,74} However, such a scattering process should result in refilling of the polariton population in the bottleneck-region and thus a rise time of the luminescence emission of the LPB peak. Similar observation of slower rise time of the LPB emission in comparison to the UPB emission was also reported for ZnO.^{75} However, for CuI, no rise time could be observed within the temporal resolution for the emission of free polaritons. One possible reason could be the low probability of interbranch scattering from the UPB to the LPB branch compared to intrabranch scattering within the UPB. This would lead to a quasi-thermal equilibrium of polaritons around the energy minimum of the UPB branch,^{49} which is in good agreement with the observed Maxwellian line shape of the corresponding PL peak.

Instead, we associate the fast decay component with decay times in the range of 10 ps…40 ps with the trapping of polaritons by defects corresponding to their strong exciton-like character. The implied slight decrease in *τ*_{1} decay time with increasing photon energy along the UPB emission, which is shown in Fig. 7(b), reflects the cooling of the polariton distribution and the associated changes in the UPB line shape on short timescales [see Fig. 6(a)]. This assumption is further supported by the observation of similar rise times for the emission of bound excitons, suggesting a refilling of the bound exciton population by the above-mentioned trapping of free polaritons. We discuss the slow decay component and its relation to the lifetime of the polariton population in more detail below.

Assuming that polaritons are in quasi-thermal equilibrium among each other, an energy-independent decay time dominated by the excitonic component of the polaritons is expected for all free polariton emission peaks.^{74} At 10 K, we find polariton decay times *τ*_{2} in the range of 140 ps…165 ps [see Fig. 7(b)]. The fact that the observed decay time of the UPB emission is almost independent of the photon energy supports the previous statement that polaritons around the energetic minimum of the UPB are in quasi-thermal equilibrium among each other and reveal, therefore, similar transient characteristics. For the LPB emission, slightly larger decay times of around 160 ps were observed. A similar increase in the decay time in the vicinity of the bottleneck-region was reported for polaritons in CdS.^{70,74} In CuI, however, the large spectral overlap of the low-energy tail of the LPB emission and the weakly bound excitons (B and C) makes a clear distinction of the spectral dependence of the decay time along the LPB branch difficult.

Finally, very short decay times in the range of 5 ps…15 ps were found for the recombination of the excited states (*n* ≥ 2), revealing fast relaxation of excited exciton states with *n* = 1 to the ground state. We note that these values are below the temporal resolution of our setup and thus the decay times might be even smaller.

With increasing lattice temperature, the spectral overlap of the LPB and MPB emission and bound exciton emission becomes larger due to increasing broadening, making the discussion of decay characteristics of the LPB or MPB emission as a function of temperature difficult. For this reason, we focus here on the decay dynamics of the UPB polaritons and use the corresponding *τ*_{2} decay time as a measure of the exciton lifetime in CuI. The temperature dependence of *τ*_{2} is shown for the UPB emission in Fig. 7(c). It is clearly visible that the slow decay component increases with increasing temperatures up to ∼350 ps at 160 K. Such a behavior is expected for the radiative decay time of free excitons in bulk material^{31} *τ*_{r} ∝ *T*^{3/2}. The increase in the decay times can be understood qualitatively in terms of conservation of the in-plane wave vector component *k*_{∥} at the surface. The broadening of the thermal distribution with increasing temperature leads to occupation of the polariton branches at higher wave numbers *k* and thus to an increasing fraction of polaritons, which cannot be transmitted through the surface. We note that at low temperatures below 45 K, the decay time remains nearly constant, which is not expected from the *T*^{3/2} dependence of the decay time. However, such a behavior can be explained by the fact that for low temperatures, thermal equilibrium of polariton distribution with the lattice is not achieved, as mentioned before. The final effective temperature of the polariton distribution was shown to be nearly constant for nominal lattice temperatures below ∼30 K, which also explains the constant decay times in this temperature range. In contrast to *τ*_{2}, the fast decay time *τ*_{1} fluctuates statistically in the range of 25…50 ps independent of the sample temperature [see Fig. 7(c)], supporting the interpretation of this decay component as involving non-radiative decay processes, as, e.g., trapping by defects.

#### 3. Bound exciton dynamics

The generated exciton–polaritons can be effectively trapped by defects due to their exciton-like character. This capture process couples, therefore, the dynamics of free and localized states, where the rise time of bound excitons is strongly connected to the fast initial decay of free exciton–polaritons.^{76} In general, these bound excitons can be either thermally activated back into free exciton state or recombine to the ground state.^{72} However, the thermal activation of bound excitons is unfavorable at low temperatures due to relatively high localization energies (E_{B} $>10meV$) of the defects.

The transient behavior of weakly bound excitons in CuI is exemplary shown for the D transition in Fig. 7(a) and can be modeled by a simple monoexponential decay with a rise time at the beginning,

where *τ*_{0} and *τ* describe the formation time and decay time of bound excitons, respectively.

The obtained rise times of weakly bound excitons (C–E) are around 30 ps and thus comparable with the fast decay component of free polaritons, supporting our assignment of this rise time to a formation of the bound exciton population. We note that although similar transient behavior is also expected for deeper bound excitons (labeled H), the rise time could not be reasonably determined for this transition due to the overlap with a broad donor–acceptor pair recombination band at lower energies.^{18} In general, the formation time is expected to depend in on the impurity concentration *N* first approximation via the following equation:^{54,77}

where *σ*_{FX} is the trapping cross section and *v*_{th} is the thermal velocity of the exciton diffusion through the lattice. Although the corresponding trapping cross sections of the defects in CuI are unknown to the best of the authors’ knowledge, we use $\sigma FX\u2248\pi aFX2\u22486.1\xd710\u221214cm2$ as a first approximation with the exciton Bohr radius in CuI *a*_{FX} = 1.4 nm.^{78} The average kinetic energy of free excitons can be determined via *E*_{kin} = 3/2*k*_{B}*T*, leading to an averaged thermal exciton velocity of around 2.4 × 10^{6} cm/s for a mean polariton temperature of ∼40 K. Thus, the estimated defect density is in the order of 2 × 10^{17} cm^{−3}, which is in reasonable agreement with measured hole density at room temperature (see Fig. 4).

The observed decay times of bound excitons (B-H) are shown in Fig. 7(b). It can be clearly seen that the decay times increase with increasing binding energy. Such a behavior is expected for the radiative lifetime of localized states, which are inversely proportional to the corresponding transition oscillator.^{46,79,80} According to the theory of Rashba and Gurgenishvili,^{81} the oscillator strength of bound excitons is proportional to the volume occupied by the corresponding wave function, leading to a $EB3/2$ dependence of the lifetime. The determined decay times of the bound excitons are plotted in the inset of Fig. 7(b) vs $EB3/2$. The observed proportionality of lifetimes for the emission peaks (B–E) is in well qualitative agreement with the theory of Rashba and Gurgenishvili.^{81} However, it has to be considered that the experimentally determined lifetimes include contributions of radiative (*τ*_{r}) and non-radiative (*τ*_{nr}) times via 1/*τ* = 1/*τ*_{r} + 1/*τ*_{nr}. Therefore, non-radiative recombination processes, e.g., Shockley–Read–Hall (SRH) or Auger recombination, may reduce the total lifetime of bound excitons, especially at high (unintentional) doping concentrations.^{76} Nevertheless, we would like to emphasize that if non-radiative Auger recombination would dominate, the recombination lifetimes should follow the fourth power of the binding energy,^{82} which is clearly not the case, such that radiative recombination still seems to dominate in our samples.

The fact that the decay times of the transitions F and G differ strongly from the increasing trend observed for the transitions C–E could indicate a different origin of the emission lines. Based on the energetic distance, these two transitions could be the 1-LO replicas of transitions C and D, respectively, as mentioned above. Although the decay times of the transitions F and G differ slightly from the corresponding zero-phonon lines, such an assignment could explain the observed deviations from the $EB3/2$ trend.

The reason for the excessively short decay times of the transition H is not entirely clear. Since it is assumed that the corresponding transition arises from the recombination of excitons bound to near-surface Cu vacancies, we speculate that the contribution of the non-radiative recombination processes could reduce the corresponding lifetime and thus lead to the observed behavior. Still, the previous identification of the 1-LO-phonon replica of the H-doublet by means of the energy position is confirmed by similar dynamics to the respective zero-phonon lines.

Although, in general, no temperature dependence of the radiative decay of localized excitons is expected,^{83} we also observe a slight increase in the decay times with increasing temperature for bound exciton transitions. However, an unambiguous determination of the corresponding decay times becomes more difficult with increasing temperature due to the increasing spectral overlap of the bound exciton transitions with the low-energy tail of the LPB emission and the dissociation of the corresponding bound excitons.

## V. SUMMARY

Temperature-dependent static and dynamical properties of the near-band edge emission in CuI single crystals have been investigated. The exciton binding energy was determined to be about 62 meV based on the energy position of the exciton states (*n* = 1, 2, and 3). By means of temperature-dependent PL measurements, the electron–phonon coupling strength of about *α* ≈ 0.11 meV/K and an average phonon energy of 20.6 meV were determined. From a comparison of PL spectra with temperature-dependent Hall effect measurements, Haynes’s rule factor for neutral shallow acceptors was estimated to be in the range of 0.08–0.17.

The different decay characteristics of free and localized exciton states were explained by their coupled interaction: trapping of free exciton–polaritons by defects leads to the rapid decay of free polariton luminescence with *τ*_{1} ≈ 30 ps and simultaneously to formation time of bound excitons, which exhibit similar capture time constants. Under this assumption, the defect density of the shallow neutral acceptors was estimated to be about 10^{17} cm^{−3} in agreement with room temperature hole density. The slower decay component *τ*_{2} was assigned to the intrinsic lifetime. For the bound excitons, this decay time at 10 K ranges between 180 ps…360 ps and thus shows the expected $EB3/2$ proportionality. This indicates that recombination is dominated mainly by radiative recombination processes. Decay times around 150 ps were determined for polariton emission at low temperatures. It was shown that the decay time of UPB polaritons increases with increasing temperature between 10 K and 160 K to about 350 ps and that the expected *T*^{3/2} proportionality for radiative recombination of non-localized states in the bulk is fulfilled up to that temperature. Above 160 K, the decay time appears to decrease, presumably due to more nonradiative recombination channels at higher temperatures.

## ACKNOWLEDGMENTS

We gratefully acknowledge funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through FOR 2857 (Projects P02, P05, and P06) - 403159832. Furthermore, we would like to thank Bernd Rheinländer, Daniel Splith, Philipp Storm, and Sebastian Henn (all Universität Leipzig) for fruitful discussions. E.K., M.S.B., L.T., R.H., A.M., and O.H. acknowledge the Leipzig School of Natural Sciences BuildMoNa.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to declare.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.