The impact of the halide cage on the electronic properties of fully inorganic caesium lead halide perovskites

Perovskite solar cells with record power conversion efficiency are fabricated by alloying both hybrid and fully inorganic compounds. While the basic electronic properties of the hybrid perovskites are now well understood, key electronic parameters for solar cell performance, such as the exciton binding energy of fully inorganic perovskites, are still unknown. By performing magneto transmission measurements, we determine with high accuracy the exciton binding energy and reduced mass of fully inorganic CsPbX$_3$ perovskites (X=I, Br, and an alloy of these). The well behaved (continuous) evolution of the band gap with temperature in the range $4-270$\,K suggests that fully inorganic perovskites do not undergo structural phase transitions like their hybrid counterparts. The experimentally determined dielectric constants indicate that at low temperature, when the motion of the organic cation is frozen, the dielectric screening mechanism is essentially the same both for hybrid and inorganic perovskites, and is dominated by the relative motion of atoms within the lead-halide cage.

Rapid developments in the field of hybrid organic-inorganic perovskites have led to a dramatic increase of power conversion efficiencies in perovskite-based solar cells, which currently exceed 22%. [1][2][3][4] Hybrid organic-inorganic perovskites combine low-cost fabrication processes 5,6 with strong light absorption, 7 efficient photoluminescence, 8,9 together with long carrier lifetimes and diffusion lengths. 10,11 The combination of these properties has led to numerous applications of this class of materials in optoelectronic devices beyond solar cells, including light emitting diodes, 12 lasers 13 and photodetectors. 14 Hybrid organic-inorganic perovskites are characterized by a general chemical formula ABX 3 , where A is an organic ammonium cation (Methylammonium (MA), or formamidinium, FA), B=Pb 2+ or Sn 2+ and X is a halide anion (Cl − , Br − , I − or an alloyed combination of these). Initially, the fabrication of perovskite-based solar cells was based on mono halide material. 15 In this case, the power conversion efficiency is usually limited to less than 20% for conventional MAPbI 3 based devices, 16 which are additionally plagued by poor resistance to moisture or high temperatures, 17,18 as well as by the formation of trap states induced by exposure to light. 19 The synthesis of FAPbI 3 provides, in principle, an attractive alternative, as it has a band gap smaller than its MA counterpart, 20 being closer to the optimal value for a single junction solar cell, which influences favorably its conversion efficiency. 1,21 However, the large radius of the FA cation favors the formation of a photoinactive polymorph at room temperature. 22 Alternatively, fully inorganic Caesium-based CsPbX 3 perovskite compounds 23 with an excellent thermal stability up to 450 • C 24 Interestingly, ε eff is comparable for all the iodide compounds, but decreases significantly for the bromides. This suggests that at low temperature, when the motion of the organic cations is frozen, 34 the dielectric screening mechanism is essentially the same for both the inorganic and hybrid perovskites and is controlled by the lead-halide cage.
Typical transmission spectra of CsPbBr 3 , CsPbI 2 Br and CsPbI 3 , measured over a wide range of temperatures (4.2 K -270 K), are presented in Fig. 1(a-c). We observe a consistent blue shift of the band edge absorption energy of CsPbI 3 through mixed CsPbI 2 Br to CsPbBr 3 , highlighting the good tuneability of the band gap via the introduction of a heavier halide in the crystal. 35 The transmission spectrum for each compound exhibits a single minimum at all temperatures, which blue shifts and broadens with increasing temperature. The detailed evolution of the band gap absorption energy with temperature is presented in Fig. 1(d). We note that the band gaps of all the investigated samples exhibit a well behaved monotonic dependence on the temperature. This is in stark contrast with organic-inorganic halide perovskites, where an increase in the band gap is observed at temperatures corresponding to the phase transitions to a lower symmetry crystalline structure. 32,33,[36][37][38][39] This is a particularly significant result in the case of CsPbI 3 , which suggests that our sample preparation procedure (see Experimental Methods) preserves the photoactive perovskite phase of this compound even at cryogenic temperatures. Importantly, we have monitored the absorption spectra of all the samples during the cool down. This point was critical especially for the CsPbI 3 , which is known to be unstable at ambient conditions. In the absorption spectra we did not see any signs of dramatic change of the band gap, which would suggest the transition into yellow phase. This supports our finding that we freeze the sample in the cubic phase. In the case of CsPbBr 3 , we have investigated both as prepared samples (orthorhombic phase) and samples, which have been annealed at 250 • C, which is above the transition to the cubic phase. 27,40 The identical transmission data and evolution of the gap with temperature shown in Fig. 1(a,d) demonstrate that annealing does not influence the electronic structure of CsPbBr 3 under the measurements conditions used. This suggests that independently of the thermal processing, CsPbBr 3 will always transform to the orthorhombic phase below the phase transition point at 88 • C and the continuous evolution of its band gap with temperature indicates the absence of any further phase transitions and to a good stability of the investigated samples.
Low temperature magneto transmission spectroscopy is a powerful technique which has previously been used to precisely determine the binding energy and reduced mass of the exciton in organic-inorganic perovskites. 32,33,39 In Fig. 2 a weak minimum can be resolved for magnetic fields larger than 35 T. It is more clearly seen in differential transmission spectra obtained by dividing the transmission spectra by the spectrum measured at zero magnetic field (see Fig. 2    resonances are related to free carrier inter Landau level transitions with energies where E g is the band gap, n = 0, 1, 2, . . . is the orbital quantum number of the Landau levels in the conduction and valence bands. For dipole allowed transitions (∆n = 0), and for a well-defined value of the band gap E g , the only fitting parameter in Eq.( 1) is the reduced mass µ. Fitting the observed resonances to Eq.( 1) (gray lines in Fig. 3) allows us to determine the exciton reduced mass. The excitonic like transitions close to the band edge are well described with a numerical model for a hydrogen atom in high magnetic field. 41 The eigenenergies of an excitonic system in zero field are given by where E N is the energy of N th excitonic level, R * = R 0 µ/m 0 ε 2 eff , R 0 is the atomic Rydberg, m 0 is the free electron mass and ε eff is the relative dielectric constant. The fit of the inter Landau level transitions provides an accurate estimation of µ, which in this second step is taken as a fixed parameter. This provides strong limits on the value of R * , which is further constrained by the observation of the 2s state, well resolved only in high magnetic field both for CsPbI 3 and CsPbBr 3 . Contrary to early magneto optical estimates of exciton binding energies of hybrid organic-inorganic perovskite, 42,43 our approach does not require us to assume a value for the effective dielectric constant, which we can actually determine from Eq. ( 2). The values of the effective mass and R * obtained for all three compounds are summarized in table 1 and Fig 5(a-b).
In the case of organic-inorganic perovskites, the phase transition to a higher symmetry crystalline structure allows the rotational motion of the organic cation, which enhances the dielectric screening and reduces the exciton binding energy. [32][33][34]39 The lack of an abrupt change of the band gap observed here implies that inorganic perovskites do not undergo phase transitions up to room temperature. The absence of a structural change suggests that the exciton binding energy does not vary over the investigated temperature interval. To support this conclusion, we have performed magneto-transmission measurements of CsPbBr 3 at 180 K. We compare the magnetic field dependence of the 1s transition energy at high and low temperatures in Fig. 4. The temperature induced change in the band gap has been removed by plotting the high temperature data on a different scale (right axis) but over the same range (10 meV in both cases). The excellent overlap of the two sets of the data demonstrates that the exciton binding energy does not change within experimental accuracy.
In Fig. 5(b) we plot the experimentally determined values of the exciton reduced mass µ of Caesium compounds. Our results are close to theoretical prediction of µ for CsPbI 3 , which range from ∼ 0.07m 0 44,45 to ∼ 0.18m 0 46 and in excellent agreement with density functional theory of organic-inorganic perovskites adjusted to fit the experimental band gap. 47 We compare our results on fully inorganic compounds to the reduced masses determined on organic-inorganic perovskites. 32,33,39 We observe an increase of the effective mass with  increasing value of the band gap. This trend can be understood in the frame of a simple two-band k · p model, 48 assuming the same effective mass for the valence and conduction band, the reduced mass of the exciton can be written as where P = Ψ VB |p x |Ψ CB is the momentum matrix element which couples states in the conduction and valence bands and 2|P | 2 /m 0 is the Kane energy. 33 The measured evolution of the band gap with the reduced mass is well fitted with a Kane energy of 8.3 eV, only slightly larger than theoretical predictions. 49 Knowing the values of exciton binding energy and the reduced effective mass we can calculate the effective dielectric constant ε eff . In Fig. 5(c), we compare the dielectric constant for Caesium-based compounds (stars) with the corresponding quantities for the hybrid organic-inorganic materials from our previous work 32, 33,39 (blue circles). The dielectric con- Pb-X stretching and Pb-X-Pb rocking modes harden with decreasing halide mass, with the derived dielectric function exhibiting the same trend as observed in Fig. 5(a). 54 This confirms the qualitative picture that the dielectric screening properties are mainly determined by the lead-halide cage in the low temperature, when the organic cation motions are frozen.
The exciton binding energies we have determined are significantly smaller than early estimates obtained from magneto optical measurements of the 1s excitonic states. 42 In the absence of any observed phase transitions (see Fig. 1), we argue that the exciton binding energy will depend only very weakly on the temperature, as demonstrated by the similar diamagnetic shift of the 1s state at 2 K and 170 K in Fig. 4. This suggests that the photocreated carriers in the compounds investigated exhibit essentially a free-carrier behavior at temperatures corresponding to the normal operating conditions of solar cells.

Experimental Methods
Magneto transmission spectra have been acquired by combining long pulse magnetic field measurements for magnetic fields up to 66 T and short duration pulsed magnets (for magnetic fields up to 150 T). For the long pulse measurements (typical pulse duration ∼ 100 ms), the sample was mounted in a liquid helium cryostat. White light from a halogen lamp was used as the excitation source. The light emitted from the lamp was coupled in a 200 µm diameter multimode fiber, used to illuminate the sample. The transmitted light was coupled in a 400 µm diameter multimode fiber and guided to spectrometer equipped with a liquid nitrogen cooled CCD camera. The typical exposure time was 2ms, which ensured that the transmission spectra were acquired at essentially constant magnetic field values. For the very high magnetic field measurements (B < 150 T), magnetic field pulses with a typical duration of 5 µs were generated by a single turn coil system with a bore diameter of 10 mm.
A helium-flow cryostat with a kapton tail was located in the single turn coil. The sample was kept at 5 K. Magneto-transmission measurements were conducted by using a tunable optical parametric oscillator (OPO) pumped by a Ti:sapphire laser as the light source, a fast (100 MHz) silicon detector and a high speed digital oscilloscope.
All samples were prepared on glass microscope slides, which were cleaned by sonication sequentially in acetone and isopropanol, and then treated with oxygen plasma for 10 minutes. The CsPbI 3 and CsPbI 2 Br perovskite layers were deposited in a nitrogen glovebox by spincoating (at 1500 rpm) a solution of the appropriate ratios of CsI (Alfa Aesar, 99.9%), PbI 2 (Sigma Aldrich, 99%) and PbBr 2 (Sigma Aldrich, >98%) dissolved at 0.43 M in N,N-Dimethylformamide following a previously reported procedure. 35 The CsPbBr 3 perovskite was deposited by sequential evaporation of layers of PbBr 2 (107 nm) and CsBr (93 nm), each deposited at 1-2Å/s onto room-temperature substrates, at pressures below 6 × 10 −6 mbar in a BOC Edwards Auto 306 evaporator. Owing to their limited stability in air at room temperature, CsPbX 3 samples were systematically annealed in an oven before loading them in the cryostat in order to restore the cubic phase. CsPbI 3 was annealed at 350 • C for 10 minutes. CsPbI 2 Br and CsPbBr 3 were annealed at 250 • C for 5-7 minutes. After the annealing, the samples were placed in a liquid helium cryostat within 4 minutes, which ensured that the sample remains in the cubic phase. The sample stage was not temperature-controlled.
All samples were annealed after deposition and again before measurement.
from the Chinese Scholarship Council (CSC), R. J. S. is a Commonwealth Scholar, funded by the UK government. This work was supported by EPSRC (UK) via its membership to the EMFL (grant no. EP/N01085X/1).

Supporting Information Available
The following files are available free of charge. Supporting information: Low temperature transmission spectra in magnetic field for CsPbI 2 Br and CsPbI 3 , XRD, Absorbance and PL for CsPbI 3 CsPbI 2 Br and CsPbBr 3 .