# Polaronic Mass Enhancement and Polaronic Excitons in Metal Halide Perovskites

Michal Baranowski, Andrzej Nowok, Krzysztof Galkowski, Mateusz Dyksik, Alessandro Surrente, Duncan Maude, Marios Zacharias, George Volonakis, Samuel Stranks, Jacky Even, et al.

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Michal Baranowski, Andrzej Nowok, Krzysztof Galkowski, Mateusz Dyksik, Alessandro Surrente, et al.: Polaronic Mass Enhancement and Polaronic Excitons in Metal Halide Perovskites. ACS Energy Letters, 2024, 9 (6), pp.2696-2702. 10.1021/acsenergylett.4c00905. hal-04596394

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Polaronic Mass Enhancement and Polaronic Excitons in Metal Halide Perovskites

Michal Baranowski,¹, * Andrzej Nowok,¹, ² Krzysztof Galkowski,¹ Mateusz Dyksik,¹

Alessandro Surrente,¹ Duncan Maude,² Marios Zacharias,³ George Volonakis,⁴ Samuel D.

Stranks,⁵,⁶ Jacky Even,³ Miroslaw Maczka,⁷ Robin Nicholas,⁸ and Paulina Plochocka¹, ², †

¹Department of Experimental Physics, Faculty of Fundamental Problems of Technology, 
Wroclaw University of Science and Technology, 50-370 Wroclaw, Poland

²Laboratoire National des Champs Magnétiques Intenses, EMFL, 
CNRS UPR 3228, Université Grenoble Alpes, Université Toulouse, 
Université Toulouse 3, INSA-T, 38042 Grenoble and 31400 Toulouse, France

³Univ Rennes, INSA Rennes, CNRS, Institut FOTON - UMR 6082, F-35000 Rennes, France

⁴Univ Rennes, ENSCR, INSA Rennes, CNRS, ISCR - UMR 6226, F-35000 Rennes, France

⁵Cavendish Laboratory, University of Cambridge,
JJ Thomson Avenue, Cambridge CB3 0HE, UK.

⁶Department of Chemical Engineering and Biotechnology,
University of Cambridge, Philippa Fawcett Drive, Cambridge CB3 0AS, UK

⁷Institute of Low Temperature and Structure Research,
Polish Academy of Sciences, ul. Okolna 2, 50-422 Wroclaw, Poland

⁸Department of Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford, OX1 3PU, UK

(Dated: April 26, 2024)

**Abstract:** In metal halide perovskites the complex dielectric screening together with low energy of phonon modes leads to non-negligible Fröhlich coupling. While this feature of perovskites has been already used to explain some of the puzzling aspects of carrier transport in these materials, the possible impact of polaronic effects on the optical response, especially excitonic properties, is much less explored. Here with the use of magneto-optical spectroscopy, we revealed the non-hydrogenic character of the excitons in metal halide perovskites, resulting from the pronounced Fröhlich coupling. Our results can be well described by the polaronic-exciton picture where electron and hole interaction are no longer described by a Coulomb potential. Furthermore, we show experimental evidence that the carrier-phonon interaction leads to the enhancement of the carrier's effective mass. Notably, our measurements reveal a pronounced temperature dependence of the carrier's effective mass, which we attribute to a band structure renormalization induced by the population of low-energy phonon modes. This interpretation finds support in our first-principles calculations.

Interaction between charge carriers and lattice excita-
tion (phonons) have a profound impact on the optical
response of semiconductor materials, determining tran-
sition linewidth or fulfilling energy-momentum conserva-
tion requirements in indirect light absorption and emis-
sion processes[1, 2]. The coupling of the carriers to opti-
cal lattice vibrations can also significantly affect the spec-
trum of the excitonic transitions[3–6] leading to a devi-
ation from hydrogenic exciton model[2] (commonly used
for many semiconductors) or enhancing carrier mass[7, 8].
This is especially important for ionic semiconductors,
where carriers-optical phonon coupling is strong. For a
long time, ionic semiconductors have stayed in the shade
of covalent semiconductors (III-V, II-VI), where the im-
pact of the phonons on the exciton energy structure could
be neglected (the hydrogen model works very well). How-
ever, the recent boom in the field of metal-halide per-
ovskites has brought the discussion of the impact of pola-
ronic effects on the excitonic properties into the spotlight
of scientific interest[4, 8–14].

Tremendous interest in metal-halide perovskites has
been triggered by skyrocketing progress in energy har-
vesting application [15–18], followed by promising results
in the field of light emitters and detectors [19, 20]. At

the same time, perovskite semiconductors constitute a
fascinating and challenging material system for funda-
mental understanding as they bridge inorganic and or-
ganic semiconductors [10]. Despite extensive research on
the physics of these materials, a complete and unified un-
derstanding of their electronic and optical properties is
still missing [8–11]. The crucial features of perovskites,
which distinguish them from regular inorganic semicon-
ductors (GaAs, Si, GaN), are the richness of their lattice
dynamics [21–25] and the related complex dielectric re-
sponse [23, 26]. Indeed, the soft ionic and anharmonic
lattice makes perovskites an intriguing system for the
investigation of carrier-lattice coupling [8, 10, 27]. The
coupling of carriers to anharmonic vibrations plays a key
role around room temperature and above and is essential
to explain the mild bandgap change across-phase tran-
sitions [27–31], the temperature dependence of carriers
mobility[10, 32] or thermal conductivity[30]. At very low
temperatures, anharmonicity-related effects are expected
to be weak; however, the carriers-phonon coupling still
significantly influences the optoelectronic properties of
these materials.

In metal halide perovskites the non-negligible coupling
of carriers with optical phonons and formation of po-
larons is expected due to significant contrast between high $\epsilon_\infty$ and low $\epsilon_s$ frequency part of dielectric function [12, 25, 26] and the low energy of some phonon modes [25]. This interaction quantified by Fröhlich coupling constant ($\alpha = 1.7 - 2.2$) [25] is orders of magnitude larger in perovskites than in GaAs ($\alpha = 0.068$) or other III-V or II-VI semiconductors [33]. Therefore it should have a profound impact on the electronic and optical properties of perovskites and cannot be ignored as is often the case with other semiconductor materials. For example, Fröhlich coupling is known to enhance the effective masses [7, 34], $m^* \approx m(1 + \alpha/6)$, which is one of the most crucial parameters for device operation and should therefore be thoroughly understood.

The concept of large polarons (i.e., carriers dressed with the lattice interaction) has already been used to explain some of the puzzling aspects of carrier transport in perovskites [8, 10, 25, 35-40]. Yet, the possible impact of polaronic effects was mostly neglected in the analysis of the optical response, leading to confusing reports [4, 9]. For instance, the excited excitonic states have been hardly observed even for the highest quality crystals [41-43], clearly deviating from the hydrogen-like exciton model [4] (where the oscillator strength of the 2s states is the 1/8 of 1s state). More importantly, there are evident discrepancies in the reported carrier mass values determined through magneto-optical studies performed in the low [41, 44] and the high field limits [9, 45, 46]. The higher effective mass is consistently observed in the former case, providing another indication of the importance of carrier-phonon coupling in perovskites for their optical response.

Here, we employ magneto-optical spectroscopy and exciton-polaron modelling, to show that the above discrepancy can be reconciled by taking into account the impact of the Fröhlich coupling on the optical response of metal halide perovskites. By solving the exciton-polaron model [3-6] in the magnetic field we prove that the contrasting values of the effective masses observed in the low and high magnetic field, regime stem from the polaronic nature of excitonic transitions. In this way, we provide direct evidence of the importance of Fröhlich coupling for understanding the optical response and carriers' effective mass in perovskite crystals. Moreover, we highlight the importance of carrier-phonon coupling for band structure renormalization, which leads to strong temperature dependence of the carrier's effective mass.

We perform our investigation for three high-quality perovskite single crystals: MAPbI$_3$, MAPbBr$_3$ and CsPbBr$_3$ (for synthesis details see methods section). To reveal the impact of polaronic effects on carriers' mass we measured the optical response in a pulsed magnetic field up to 90 T. In the low field limit when $\hbar\omega_c < \hbar\omega_{LO}$ (where $\omega_c = eB/\mu$ is a cyclotron frequency, $\hbar\omega_{LO}$ is the LO phonon energy and $\mu$ is a reduced carrier mass probed by interband transitions $1/\mu = 1/m_h + 1/m_e$) the

optical properties are determined by the polaron mass as the lattice follows the cyclotron motion of the carriers. In the high field limit when $\hbar\omega_c \gg \hbar\omega_{LO}$ the bare carrier mass $\mu$ is probed because lattice deformation cannot follow fast cyclotron movement of the carrier [47]. Assuming reduced carrier mass $\mu$ in the range of $0.1 - 0.15 m_0$ [45] (where $m_0$ is free electron mass) and an effective LO phonon energy $\sim 15$ meV [25, 48] the crossover between the two magnetic field regimes should occur around $\sim 20$ T. Therefore with the available magnetic field range, we can probe optical response related to the polaron mass and bare carrier mass limit [47].

Fig. 1 (a) shows the reflection spectrum of MAPbBr$_3$ measured at 2 K (blue curve) together with its derivative (yellow curve), whose minima correspond to the energy of the excitonic transitions [42] (for other samples see Fig. S1 and S2). The optical response is dominated by the strong 1s excitonic transition at 2.249 eV which on the high energy side (2.262 eV) is followed by a much weaker 2s exciton state. The 2s transition oscillator strength is much lower than 1/8 of 1s strength ($\sim 2-3\%$ instead of $\sim 12\%$) which is a first hallmark of the polaronic character of the observed excitonic transition [4].

To probe the reduced effective mass we investigated exciton and Landau level transitions evolution in the presence of high magnetic fields. The evolution of the reflection spectrum of MAPbBr$_3$ is presented in Fig. 1 (b) (for remaining crystals see SI). Both excitonic transitions blueshift and at higher magnetic fields Zeeman splitting of the excitonic states can be observed. To reveal the transitions between interband Landau levels in Fig. 1 (c), we plot spectra obtained at a nonzero magnetic field divided by the spectrum measured at zero magnetic field. Such ratioed spectra exhibit equally spaced features (on the high energy part of the spectrum), which we ascribe to the interband Landau levels transitions. The evolution of these interband Landau level transitions is summarized in panels (d)-(f) of Fig. 1. This dependence can be well fitted with a standard formula for parabolic band dispersion (since the excitonic corrections to the higher Landau levels are very small) [45, 47, 49]:

$$E_{LLn} = E_g + (n + 1/2)\hbar\omega_c \quad (1)$$

which provides a very direct measure of carriers' reduced mass $\mu$ which are 0.096, 0.106 and 0.117 mass of electrons for MAPbI$_3$, MAPbBr$_3$ and CsPbBr$_3$ respectively (see also Tab. I). It is important to note that the Landau levels could be observed only in the field above 40-50 T, therefore, we can safely assume that the extracted values correspond to the masses of bare carriers'.

To measure the effective reduced mass at the low magnetic field limit we make use of excitonic transitions which can be observed regardless of field strength. Thanks to the superior quality of our crystals both 1s and 2s excitonic transitions can be observed. Their evolution in the magnetic field (for all three crystals) is summarized
<figure><img src="image_2.png" /><figcaption>FIG. 1. (a) MAPbBr$_3$ reflectance spectrum (blue) taken at 2 K and its derivative (yellow) showing resonance features related to 1s and 2s excitonic transitions. (b) Evolution of reflectance spectrum in magnetic field. (c) Rationed spectra of reflectance for different magnetic field values, black lines indicated Landau levels (LL) transitions. (d)-(f) Fan charts summarizing interband Landau levels transition energies as a function of the magnetic field for MAPbI$_3$, MAPbBr$_3$ and CsPbBr$_3$. Efectice reduced mass $\mu$ with the use of eq.1.</figcaption></figure>

<table><caption>TABLE I. Summary of extracted parameters. $\mu$ - reduced effective mass determined based on Landau level transitions, $\mu_X$ - effective mass extracted from hydrogenic exciton model, $\mu^*$ polaron mass calculated according to eq. 5 and eq. 6 with parameters $\epsilon_\infty$, $\epsilon_s$ and $E_{LO}$ obtained through modelling of excitonic transition with the use of Bajaj potential.</caption><thead><tr><th></th><th>μ</th><th>μ<sub>x</sub></th><th>μ<sup>*</sup></th><th>ε<sub>∞</sub></th><th>ε<sub>s</sub></th><th>E<sub>LO</sub></th><th>α</th></tr></thead><tbody><tr><td>MAPbI<sub>3</sub></td><td>0.096</td><td>0.102</td><td>0.114</td><td>7</td><td>15</td><td>12 meV</td><td>1.12</td></tr><tr><td>MAPbBr<sub>3</sub></td><td>0.106</td><td>0.151</td><td>0.137</td><td>5.2</td><td>16</td><td>15 meV</td><td>1.75</td></tr><tr><td>CsPbBr<sub>3</sub></td><td>0.117</td><td>0.163</td><td>0.158</td><td>4.5</td><td>16</td><td>18 meV</td><td>2.1</td></tr></tbody></table>

in Fig.2 as diamond points (the contribution of Zeeman splitting is cancelled out by averaging the energy of split states). At first, we model the behaviour of 1s and 2s exciton states in a magnetic field using the numerical solutions of the hydrogen atom in a magnetic field proposed by Makado and McGill [50]. In this approach, the energy shift induced by the magnetic field depends (only) on the dimensionless parameter $\gamma = \frac{1}{2}\hbar\omega_c/E_B$ which involves the exciton binding energy $E_B$ and the reduced effective mass used for the cyclotron energy. The simultaneous observation of 1s and 2s exciton states delivers strong constraints for the exciton binding energy there-

fore effectively $\mu$ becomes the only fitting parameter. As is shown by the dashed red lines magnetic field-induced shifts of the 1s and 2s states as well as their energy distance can be reasonably reproduced with the hydrogenic model. However, significantly higher values of $\mu$ (higher than the one extracted from Landau levels observed at high magnetic field values see Tab.I) have to be used to fit experimental data well. In other words, the excitons behave if they were composed of heavier carriers than probed by Landau level analysis (which is sensitive to the bare carrier mass). This is a smoking gun for polaronic mass enhancement of the carriers in metal-halide
<figure><img src="image_3.png" /><figcaption>FIG. 2. Dependence of 1s and 2s excitonic transition energy as a function of magnetic field (open points) for MAPbI$_3$, MAPbBr$_3$ and CsPbBr$_3$ crystals at 2 K. The dashed red line represents the best fitting with the hydrogen model while the short-dash red line shows a prediction of the hydrogen model with the $\mu$ extracted from Landau levels spectroscopy. The black line shows modelling with the use of Bajaj potential and parameters summarized in Tab.I</figcaption></figure>

perovskites. The usage of bare carrier masses (in the hydrogen exciton model) fails to fit the data (especially in Br-containing compounds) as shown by short-dashed red lines.

At this point, it is important to emphasize that in ionic crystals the hydrogenic exciton model should be treated as a simplified picture of bound electron-hole pairs [3–6, 51]. More precise descriptions should take into account the contrasting low and high-frequency parts of the dielectric function which cause the electron-hole interaction to no longer be described by a simple Coulomb

potential and that the carrier’s mass can be enhanced [3–7]. Therefore, to further support the exciton-polaron character of the observed transition, we use the effective Hamiltonian for excitons in polar semiconductors proposed by Bajaj [6] to model the energies of exciton states energy in the magnetic field. The Bajaj potential is a phenomenological modification of the Haken potential [3], which remains in quantitative agreement with more advanced models [5, 51] (as we show in Tab. S1) and can be easily extended for the case of a magnetic field (see detailed description SI). It has the following form:

$$V(R) = -\frac{1}{\varepsilon_s R^2} - \left(\frac{\varepsilon_\infty}{\varepsilon_s}\right)^\gamma \left(\frac{1}{\varepsilon_\infty} - \frac{1}{\varepsilon_s}\right) \left(e^{-R/l_e} + e^{-R/l_h}\right) \frac{e^2}{2R} \quad (2)$$

where $l_{e/h}$ is an electron and hole polaron radius defined as:

$$l_{e/h} = \sqrt{\frac{\hbar^2}{2m_{e/h}E_{LO}}} \quad (3)$$

The extra factor $(\varepsilon_\infty/\varepsilon_s)^\gamma$ (which is not present in the Haken potential) provides an improved agreement with the experimentally measured values of exciton binding energy. For a range of polar crystals, including the high $\alpha$ value halides, it was shown that $\gamma = 3/5$ is the optimal choice [6]. We solve numerically the Schrödinger equation for the exciton-polaron problem in a magnetic field,

which for the optically active s-states take the following form (see SI for detailed description):

$$\left(\frac{\mathbf{p}^2}{2\mu^*} + V(R) + \frac{e^2 B^2}{8\mu^*} (x^2 + y^2)\right) \Psi(\mathbf{R}) = E \Psi(\mathbf{R}) \quad (4)$$

where $\mathbf{p}$ is the momentum operator, $e$ is the elementary charge and $x, y$ are spatial coordinates in the plane perpendicular to the magnetic field direction. The $\mu^*$ is the reduced polaron mass $1/\mu^* = 1/m_e^* + 1/m_h^*$ where:

$$m_{e/h}^* = m_{e/h} \left( 1 + \frac{\alpha}{6} \right) \quad (5)$$
<figure><img src="image_4.png" /><figcaption>FIG. 3. (a) Comparison of the Landau levels shifts at 2 K (blue) and 60 K (red) for MAPbBr$_3$, the different slope of two fan charts is highlighted by the yellow filling. (b) Summary of the bare reduced effective mass $\mu$ for all three compounds at different temperatures. Diamonds correspond to experimental results while circle-points are DFT calculations. (c) Experimentally measured temperature-induced relative increase of the bandgap and $\mu$. (d) Calculated temperature-induced relative increase of the bandgap and $\mu$.</figcaption></figure>

and $\alpha$ is a Fröhlich coupling constant:

$$ \alpha = \frac{e^2}{\hbar} \left( \frac{1}{\varepsilon_{\infty}} - \frac{1}{\varepsilon_s} \right) \sqrt{\frac{m_e/\hbar}{2E_{LO}}} \quad (6) $$

In the modelling, we used the bare carrier reduced mass $\mu$ extracted from the Landau level analysis (assuming the same mass for the electron and the hole) as the input parameter and polaron mass was calculated based on equations,6 and 5. The results of the simulation are presented as black lines in Fig.2, which shows very good agreement with experimental data for the reasonable values (summarized in Tab. I) of $\varepsilon_{\infty}$ and $E_{LO}$ [25, 26, 48, 52, 53]. It is important to note that parameters used in the modelling should be treated as effective values as the model used (and all Haken model developments) assumes coupling to only one optical phonon mode. This “effective” character might be the reason for somehow lower values of $\varepsilon_s$ than the ones typically reported [25, 26]. Nevertheless, our model allows us to very well describe 1s and 2s excitonic transition behaviour in metal halide perovskites under the presence of the magnetic field. Therefore we show that the effective mass probed in the low and high field limits can be reconciled by taking into account Fröhlich coupling and its impact on the excitonic properties of the material. In contrast to the simple hydrogenic model, our approach relates the clear mass enhancement to the dielectric properties of the material providing very direct evidence for the polaronic nature of the excitons in metal halide perovskites and the importance of Fröhlich

coupling for the optical response of these materials.

The measured Fröhlich coupling scales with the bandgap or halide anion as can be seen from Tab. I and the deviation from the hydrogen model is more evident for Br compounds. This can be understood as an effect of stronger ionic bonding of the metal with the halide, when I is exchanged to Br or Cl, which leads to a lower value for the effective screening [14, 46]. This scaling of the Fröhlich coupling also indicates that the high effective mass of exciton observed in CsPbCl$_3$[54] might partially stem from a stronger polaronic enhancement. At the same time, the impact of the A-site cation is not very pronounced (see Tab. I) showing that polaronic properties of perovskites are dominated by LO phonon modes of the inorganic metal-halide sublattice, in agreement with previous works [14, 21].

We have also explore the evolution of the bare carriers’ mass with temperature. Generally, in semiconductors, the temperature dependence of the effective carrier mass is expected due to the variation of the band gap with temperature[47, 55, 56]. In perovskites, unusually for semiconductors, the bandgap opens with increasing temperature therefore an increase of the effective mass is expected as temperature increases. Indeed this effect can be observed via Landau level spectroscopy, as shown in Fig. 3. Evidently, the slope of the Landau level fan chart decreases as temperature rises from 2 K to 60 K (see blue and red lines) pointing to an increasing reduced effective mass $\mu$. The same behaviour can be observed by
analysing the excitonic transitions as shown in Fig. S3. A similar dependence of $\mu$ vs $E_g$ is consistently observed for all investigated compounds which is summarized in panel (b) of Fig. 3.

In metal halide perovskites the relation between the effective mass and the bandgap can be well described with the simple linear dependence[45, 49] (which comes from 2 bands $k \cdot p$ model (see SI for details):

$$ \frac{1}{\mu} \approx \frac{17.3}{E_g} \qquad (7) $$

This simple model successfully predicts the value of effective mass, measured at 2 K for a broad range of metal-halide perovskite compounds characterized by different bandgaps [49]. However, at the same time, it fails to explain the magnitude of effective mass increase induced by temperature. It is straightforward to show from eq. 7 that the relative increase of mass should be the same as the bandgap: $\mu(T)/\mu(2K) = E_g(T)/E_g(2K)$. As shown in Fig. 3(c) this is not the case and mass increases much faster. More precise 4-band $kp$ with generally accepted band structure parameters [57] gives the same result as shown in SI.

This strong temperature dependence is another indication of carrier-phonon interaction in metal halide perovskites which leads to the renormalization of band dispersion when low-energy phonon modes become populated with increasing temperature. To illustrate this phenomenon, we conducted first-principles calculations on the orthorhombic CsPbBr₃ band structure, explicitly considering the influence of electron-phonon coupling. This analysis was performed using the special displacement method (SDM) [58, 59] within the harmonic approximation as implemented in the EPW/ZG module [60]. Special displacements were generated for a 4 × 2 × 2 supercell containing 320 atoms, using the phonons that incorporate the effect of long-range interactions. The calculated phonon band dispersion is shown in Fig. S5 in SI. The impact of spin-orbit coupling was included in all SDM supercell calculations. The effective masses were determined using finite differences and taking the average of the diagonal elements of the effective mass tensor. The calculated values of the bandgap and effective masses as a function of temperature are summarized in Tab. S2. The corresponding reduced mass $\mu$ of carriers is also shown in Fig. 3 (b) as a circle-point. The calculated values for $\mu$ lie within 0.095 - 0.105 in the temperature range 0 - 75K and underestimate the experimental data by around 20% but still follow the experimental trend. However, as shown in panel (d) of Fig. 3, the temperature-induced relative increase of the effective masses and the bandgap show very good agreement with the measured values illustrating the importance of electron-phonon interaction for temperature evolution of the effective mass. The absolute values from DFT could be adjusted by considering quasiparticle corrections

within the GW approximation which increase the effective mass by ~ 25% [61], but they are not expected to change qualitatively the whole picture, inducing only small quantitative changes [34] in the relative increase of the effective masses.

In summary, our study highlights the polaronic nature of charge carriers in metal halide perovskites, revealing the crucial role of the Fröhlich coupling in shaping their optical response. The observed differences in the effective carrier mass between low and high magnetic field limits highlight the importance of this interaction for the optical properties of these materials. With the use of the Bajaj exciton-polaron model, accounting for the dielectric properties, we have shown that the contrasting values of the effective mass can be reconciled corroborating the polaronic and non-hydrogenic character of the excitons in perovskites. Moreover, our research uncovers a unique temperature-dependent increase in carriers' effective mass, which results from the band structure renormalization when the low-energy phonons became populated as shown by first-principles calculations. The highlighted impact of carrier-phonon coupling on the carrier mass enhancement and the optical properties shows that polaronic effects cannot be neglected in the intended optoelectronic applications.

## METHODS

Reflection spectra as a function of the magnetic field were measured in a pulsed field magnet with maximum field B=90 T and the whole pulse duration of ≈ 100 ms. Broad-band white light was provided by a tungsten halogen lamp and directed on the sample inside the magnet with the use of optical fibre. The magnetic field measurements were performed in the Faraday configuration, with the light propagating parallel to the magnetic field. The reflected signal was collected by a fibre bundle, which surrounds the excitation fibre (back-scattering geometry). The signal was spectrally analysed with the use of a grating monochromator and equipped with a liquid-nitrogen-cooled CCD camera. The sample was placed in a liquid helium cryostat inside the magnet.

Single crystals of all perovskites were grown using the antisolvent vapour-assisted crystallization method, in which the appropriate antisolvent is slowly diffused into a solution containing the crystal precursors. In the case of MAPbBr₃, 1 mL of 2M solution of methylamine in methanol (2mmol, Sigma-Aldrich) and 2 mmol of HBr (48 wt % in H₂O, Sigma-Aldrich) was mixed with 5 mL of acetonitrile (Sigma-Aldrich). Then 2 mmol of PbBr₂ (98%, Sigma-Aldrich) was added to the prepared above amine hydrobromide solution under stirring. In the next step, dimethylsulfoxide (DMSO, Sigma-Aldrich) was added drop by drop until the orange precipitate, which appeared in the solution, disappeared completely.
The clear solution was transferred into a glass vial and this vial was placed in a second larger glass vial containing methyl acetate (99.5%, Sigma-Aldrich). The lid of the outer vial was thoroughly sealed but the lid of the inner vial was loosened to allow diffusion of the methyl acetate into the precursor solution. Orange crystals, which grew at the bottom of the vial, were separated from the mother liquid and dried at room temperature. A precursor of MAPbI₃ was prepared by dissolving 2 mmol of PbI₂ (99%, Sigma-Aldrich) in 0.5 mL of HI (57 wt% in H₂O, stabilized with H₃PO₄, Sigma-Aldrich). Then 2 mmol of methylamine was added leading to the precipitation of a black powder. In the next step, acetonitrile was added drop by drop under constant stirring until the solution became clear. This solution was transferred into a glass vial and this vial was placed in a second larger glass vial containing methyl acetate. The grown black crystals with dimensions up to 5 mm were separated from the liquid after one week and dried at room temperature. In order to grow single crystals of CsPbBr₃, 5 mmol of CsBr (99.9%, Sigma-Aldrich) and 6 mmol of PbBr₂ were added to 10 mL of DMSO. The mixture was stirred at room temperature for 24 h. The solution was then titrated using methanol (Sigma-Aldrich) until an orange precipitate started to form. Then a few drops of DMSO were added to dissolve the precipitate, the solution was filtrated and placed in a small glass vial. This vial was placed in a second larger glass vial containing methanol. The orange crystals were harvested after one week and dried at room temperature.

## SUPPORTING INFORMATION

Magetoptical spectroscopy results for MAPbI₃ and CsPbBr₃, Derivation of the exciton-polaron model in a magnetic field. Description of 2 bands and 4 bands *kp* model. Comparison of Bajaj model prediction with other exciton-polaron models. Details of *ab initio* band structure calculation.

## ACKNOWLEDGEMENTS

This work has been partially funded by the National Science Centre Poland within the MAESTRO program Grant No. 2020/38/A/ST3/00214 and the OPUS program (no. 2019/33/B/ST3/01915(P.P.) and Sonata program No. (2020/39/D/ST3/03000) (A.S.). This study has been partially supported through the EUR grant NanoX no. ANR-17-EURE-0009 in the framework of the "Programme des Investissements d'Avenir". (M.D.) acknowledges support from the National Science Centre Poland, Grant No. 2021/43/D/ST3/01444. S.D.S. acknowledges funding from the Royal Society and the Tata Group (UF150033). The work was supported by a

Royal Society International Exchanges Cost Share award (IECnR2n170108) and the Hubert Curien Alliance Programme run by the British Council. M.Z. acknowledges funding from the European Union (project ULTRA-2DPK / HORIZON-MSCA-2022-PF-01 / Grant Agreement No. 101106654). G.V. acknowledges support from the Agence Nationale pour la Recherche through the CPJ program. This work was granted access to the HPC resources of TGCC/CINES under the allocation A0140911434 made by GENCI R.N. acknowledge support from the Engineering and Physical Sciences Research Council via its membership to the European Magnetic Field Laboratory (grant number EP/N01085X/1 and NS/ A000060/1)

## BIBLIOGRAPHY

* Corresponding author:michal.baranowski@pwr.edu.pl

† Corresponding author:paulina.plochocka@lncmi.cnrs.fr

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<figure><img src="image_5.png" /><figcaption>FIG. 4. TOC FIGURE</figcaption></figure>