Perspective on the physics of two-dimensional perovskites in high magnetic field

The term perovskite designates any solid which shares the same crystal structure as calcium titanate,¹ and is generically described by the formula ABX₃. The corresponding crystal structure, shown schematically in Fig. 1(a), comprises corner-sharing octahedra BX₆, which compose the inorganic backbone of the crystal, and an A cation, which fills the voids between the octahedra. Metal halide perovskites are direct bandgap semiconductors in which the A site is occupied by an organic or inorganic cation and the B site is generally occupied by Pb or Sn. Finally, the octahedra vertices are occupied by halide anions, such as Cl, Br, or I. Even though the burgeoning research activity on this class of materials is relatively recent,² the first synthesis of a metal halide perovskite compound was reported in 1893,³ followed almost a century later by the first hybrid organic–inorganic semiconducting perovskite.⁴ The cation included in the A site has a strong impact on the crystal structure of the resulting material. If its size is sufficiently small to be accommodated between the octahedra, as is the case for small organic [methylammonium (MA), formamidinium (FA)] or inorganic [cesium (Cs)] cations, then the crystal structure will be a periodic repetition in the three dimensions of the inorganic backbone represented in Fig. 1(a). These materials are often referred to as three-dimensional (3D) or bulk perovskites. If the A site cation is substantially larger than the space between the octahedra (as in the case of large organic molecules consisting of aliphatic or aromatic chains), it will violate the Goldschmidt tolerance factor,⁵ which leads to the formation of crystals consisting of thin slabs of lead-halide octahedra, surrounded by large organic molecules which act as spacers, as illustrated schematically in Fig. 1(b). These materials are informally referred to as two-dimensional (2D) or layered perovskites and belong to the Ruddlesden–Popper (RP)^6,7 or Dion–Jacobson (DJ)⁸ phase. The general formula describing RP 2D perovskites is $( A ) 2 A ′ n − 1 MX 3 n + 1$⁠, where A represents a large organic monovalent cation (ended with an amine group, NH₃), which acts as a spacer between inorganic octahedral slabs.⁹ In the case of DJ phase,⁸ the organic spacer is a divalent cation ended from both sides by amine groups: $( A ) A ′ n − 1 MX 3 n + 1$⁠, where A′ is a small cation (MA, FA, Cs).

Initially, 2D perovskites have been intensively investigated for their fundamental electronic properties as they can be considered as an “ideal and natural quantum well,” where charge carriers are confined in the central lead-halide slab, while the organic spacers act as barriers¹⁰ and they are not plagued by interface roughness or the intermixing characteristic of epitaxially grown quantum wells. Moreover, a significant mismatch between the dielectric constant of the organic and inorganic moieties¹¹ leads to a considerable reduction of the dielectric screening of the Coulomb interaction between the photocreated electron-hole pairs. 2D perovskites have been the first system where theoretical studies on Coulomb correlation in ultrathin films^12,13 could be tested experimentally.¹⁴ The strong reduction of the dielectric screening leads to tightly confined excitonic complexes, with binding energies on the order of hundreds of millielectronvolts.^14–16 Subsequently, layered perovskites were used as the charge transport material in transistor devices.^17,18 The rise of bulk perovksites as one of the most promising materials for photovoltaic applications² and the simultaneous wide scientific interest in layered materials such as graphene,¹⁹ transition metal dichalcogenides,²⁰ hexagonal boron nitride,²¹ and their heterostructures,^22,23 has led to a rediscovery of 2D perovskites. Their better environmental stability^24–27 as compared to their 3D counterparts^28,29 has been used to protect 3D perovskites in a heterojunction geometry.³⁰ However, 2D perovskites are now intensively investigated as materials in their own right, with applications in light emitting diodes,^31,32 lasers,^33,34 white light emitters,^35,36 high efficiency solar cells,^25,37 photodetectors,^38,39 and polaritonic devices.⁴⁰ Their electronic and optical properties can be tuned by acting on a variety of knobs, such as their thickness,⁶ the organic spacers,^14,41,42 which also enable a certain degree of control over the dielectric confinement,¹⁶ the composition,^43,44 and the thickness of the metal-halide slab.^6,15 Importantly, the choice of the organic spacer can influence the distortion of the inorganic sublattice,⁴⁵ which, in turn, has a considerable impact on the optical properties of the 2D perovskites by changing the coupling between charge carriers and lattice vibrations.^41,46–50 Thus, 2D perovskites offer a vast playground to explore extremely rich physics where the electronic and lattice properties are intertwined.

The investigation of the optical response of 2D perovskites under high magnetic field represents a powerful experimental tool to achieve a thorough understanding of their electronic properties and to disentangle the effects of the different degrees of freedom listed above. Optical spectroscopy in high magnetic field can be considered a conceptually simple experimental technique, which yields straightforward information about charge carrier effective masses, their g-factors, the spatial extension of the exciton wave function, and the energy structure of the exciton levels. These quantities are in turn of high importance for practical applications or are fed into theoretical models, which makes magneto-optical spectroscopy one of the workhorses of semiconductor physics.

In this Perspective, we summarize the progress in the understanding of the fundamental electronic properties of 2D perovskites revealed by magneto-optical spectroscopy. Magneto-optical measurements are experimentally challenging in the case of layered perovskites due to their very high exciton binding energies¹⁵ and low carrier mobility.^51–53 In order for the effects of magnetic field to be experimentally detectable, usually very large magnetic fields (up to $≳ 100 T$⁠) are required. In the first part of this Perspective, we describe how magnetic field is the only direct and straightforward experimental technique which has allowed a reliable and accurate determination of the exciton reduced mass in metal halide perovskites. Later, we comment shortly on how the use of high magnetic field can help study the coupling between charge carriers and phonons. Subsequently, we describe how magnetic field can be used to unveil the energy structure of excitonic complexes. In final part, we highlight research directions which can potentially benefit from the use of high magnetic fields.

Unfortunately, the knowledge of the diamagnetic coefficient alone does not allow us to determine the reduced mass of exciton. This is because the spatial extension of the wave function depends not only on the effective mass, but also on the dielectric screening and quantum confinement. To derive a quantitative estimate of the effective mass from the diamagnetic coefficient, a model of the exciton has to be assumed. In the case of 2D perovskites, characterized by a strong difference between the dielectric constant of the inorganic wells and the organic barriers,⁷⁵ the 2D hydrogenic model,⁶⁹ the Keldysh potential,⁷⁶ image charge effect,⁷⁴ or more complex models based on first principles calculations^15,77 can be found in the literature.

Nevertheless, some important conclusions can be drawn from the measurements of the exciton diamagnetic shift. Based on an advanced modeling of the exciton wave function, it has been shown that the exciton reduced mass can be substantially modified by increasing the thickness of the inorganic quantum well.¹⁵ For the series of (BA)₂(MA)_n−1 Pb_nI_3n+1 structures with $n = 1 , 2 , 3 …$ (where BA stands for buthylammonium, MA for methylammonium cations and n indicates the number of inorganic layers), the diamagnetic coefficient was reported to systematically increase with increasing n (see Table I). Correspondingly, the authors observed a decrease in the effective mass from 0.221 to 0.186 going from n = 1 to n = 5, as summarized in Fig. 2(f). Interestingly, even for n = 5 the reduced mass of exciton μ is much larger than the reduced mass of the exciton in bulk MAPbI₃ (⁠ $μ = 0.104$^59,61,62). These measurements have been performed at 4 K, thus addressing only the problem of the exciton reduced mass of the low temperature phase of (BA)₂(MA)_n−1 Pb_nI_3n+1.⁷⁸ In another work, the direct measurement of the diamagnetic coefficient enabled to unveil the change of the exciton reduced mass of 2D perovskites with spacers consisting of aliphatic chains of increasing length [(C_mH_2m+1NH₃)₂PbI₄ with varying m] upon structural phase transition.⁴¹ In this work, it has been demonstrated that the length of the organic spacer has a vanishingly small effect on the diamagnetic coefficient, which instead increases by an almost a factor three between the low and the high temperature crystal phases, as seen from Table I. This considerable enhancement has been attributed to a change of the exciton reduced mass upon structural phase transition, as also suggested by band structure calculations.⁴¹ By inspecting Table I, one realizes also that 2D perovskites with an aromatic spacer, such as phenylethylammonium (PEA), have consistently larger diamagnetic coefficients than the layered perovskites of the same thickness with an aliphatic spacer. This can be ascribed to the larger spatial extension of the exciton wavefunction in layered perovskites with an aromatic spacer due to a smaller dielectric mismatch between the well and the barrier than that of 2D perovskites with aliphatic spacers.¹⁶ Moreover, the aromatic spacers introduces smaller structural distortions of the inorganic sublattice, which leads to consistently smaller effective masses in 2D perovskites with aromatic spacers than those with aliphatic spacers.^66,67

The observation of the Landau level dispersion allows also to precisely determine the single particle bandgap $E g$ because all Landau levels extrapolate to $E g$ at zero magnetic field.^63,66 The direct determination of the bandgap from linear absorption spectra might be highly inaccurate if the excited exciton states merge with the bandgap transition into a broad spectral feature, which results in an underestimation of the bandgap value and of the exciton binding energy.⁶⁶ 

The investigation of the exciton states in high magnetic field clarifies also the origin of the complex line shape of the absorption spectrum of 2D perovskites.^46,49,69 The multiple peak structure has been so far attributed to vibronic progression^41,46,69 or distinct excitonic (polaritonic) states.^49,50,83 The results of magnetoabsorption studies of (PEA)₂(CH₃NH₃)_n−1Pb_nI_3n+1 (for n = 1, 2, 3)⁸⁴ and (C_mH_2m+1NH₃)₂PbI₄ (for $m = 4 , 6 , 8 , 10 , 12$⁠)⁴¹ support the vibronic progression scenario.^41,46,69 All the absorption spectral features are separated by around $∼ 40 meV$ in (PEA)₂(CH₃NH₃)_n−1Pb_nI_3n+1 and $∼ 15 meV$ in (C_mH_2m+1NH₃)₂PbI₄. The identical magnetic field induced shifts of the side peaks, as shown in Fig. 3,⁶⁶ strongly support their vibronic progression origin. This is the expected behavior of phonon replicas given that, if the lattice vibrations are unaffected by the magnetic field, all phonon assisted transitions should follow the evolution of the zero phonon line. If the side peaks originated from distinct excitonic states due to the spin–orbit coupling, or exchange interaction, they should exhibit different shifts in magnetic field, as exemplified by the opposite shift of exchange-interaction split bright excitonic states induced by the magnetic field.⁸⁵ 

Excitons—electron–hole pairs correlated by Coulomb interaction—possess a rich energy level structure, referred to as an exciton fine structure, which depends on the individual spins of the electron and the hole. The fine structure is of paramount importance to understand the interaction of excitonic complexes with light and reflects the underlying symmetry of the crystal and of the quantum confinement. The energy structure of band edge excitons in 2D perovskites⁸⁶ is very similar to that of their 3D counterparts.^87,88 Metal halide perovskites display an unusual, “inverted” band structure, which is strongly influenced by a considerably strong spin–orbit coupling in the conduction band and crystal field.⁸⁹ The states of band edge excitons are built from s-like hole states and p-like electron states with total angular momentum J = 1/2.^86,87 The presence of electron-hole exchange interaction results in an exciton fine structure where the ground state is a J = 0 optically dark (⁠ $ϕ 1$⁠) state and a threefold degenerate, optically active state with total angular momentum J = 1, which lies at higher energy.⁹⁰ In 2D perovkites, the lack of the symmetry in the z direction further lifts the degeneracy of J = 1 states, which yields two degenerate states with $J z = ± 1$ (⁠ $ϕ 3 ±$⁠) and one with J_z = 0 (⁠ $ϕ 2$⁠). The pair of degenerate states has a dipole moment in the xy plane of the crystal and couples to left- or right-handed circularly polarized light, while the latter state has a dipole moment perpendicular to the metal-halide octahedra plane (z). In the case of broken in-plane symmetry, the degeneracy of the $J z = ± 1$ state is completely lifted. This results in two excitonic states with a perpendicular in plane dipole orientation (⁠ $ϕ X / Y$⁠), which couple to linearly polarized light. A schematic representation of the fine structure of the exciton is illustrated in Fig. 4(a).⁸⁶ The exciton fine structure in the case of 2D perovskites is analogous to that observed in 3D perovskites in tetragonal and orthorhombic phases,^85,88,91 with the notable difference that the enhanced Coulomb interaction is expected to lead to a much larger splitting of the states.⁹² The splitting between the bright and dark states is expected to be in the range of a few to few tens of millielectronvolts,^93,94 one or two orders of magnitude larger than in bulk semiconductors.^85,88

Quantifying the splitting between dark and bright states is important not only from a fundamental viewpoint, but also for the practical implementation of perovskite-based light sources. A large bright-dark exciton splitting might be detrimental for lighting applications. It can prevent efficient thermal mixing of both states even at room temperature, thus reducing the efficiency of the light emitting device. Recent experiments suggest however that this negative aspect can be partially mitigated by the soft nature of perovskites, which determines a slow exciton relaxation to the low energy dark exciton, due to the weak coupling of exciton with acoustic phonons.^95–97 A detailed knowledge of exciton and phonon energy spectrum is crucial to understand the fundamental mechanisms which may limit the performance of 2D perovskites in light emitting applications.

Considering that half the exciton states are optically inactive in the standard experimental back scattering geometry, accessing the full exciton fine structure can be an experimentally challenging task. The $ϕ 1$ and $ϕ 2$ excitons are dipole forbidden under the light propagation in the direction perpendicular to the metal-halide plane. The $ϕ 2$ state can be optically accessible by moving the optical axis of the detected signal off the direction perpendicular to the quantum well⁹⁸ or by using a high numerical aperture objective,⁹⁹ but the lowest dark state remains inaccessible by the means of simplest optical measurements.

The magnetic field induces a mixing of the dark and the bright states, thereby providing a comprehensive picture of the exciton fine structure.¹⁰⁰ This magnetic-field mediated transfer of oscillator strength from the dipole allowed to the dipole forbidden transitions gives a direct optical access to dark states. In the same experiment, the magnetic field dependence of the exciton transition energies enables a direct measurement of carriers' g-factors, spin–orbit splitting, or crystal field.^69,90,91,100

In spite of these experimental challenges, there have been some reports which studied quantitatively the fine structure of exciton states of 2D perovskites. A splitting of the order of 30 meV has been measured from the PL spectrum of (C₄H₉NH₃)₂PbBr_4.⁹³ This observation has been attributed to the concomitant breakdown of selection rules in the presence of crystal distortion, higher-order transition moment processes, and phonon-assisted transitions. More recently, a 2 meV fine-structure splitting of bright exciton states in (PEA)₂PbI₄ has been reported, as shown in Fig. 4(d).⁹⁴ This splitting has been attributed to a broken in-plane symmetry in this 2D perovskite. So far, this is the largest splitting between bright exciton states reported for two-dimensional semiconducting systems. Interestingly, the authors of this work report four bright exciton states. This observation cannot be explained in the excitonic picture discussed above and presented in Fig. 4(a). A direct observation of dark states of A₂PbI₄ or A₂SnI₄ has not yet been reported. An indirect approach to accessing the energy difference between optically active and inactive levels relies on the influence of the thermal activation of the dark exciton population to the bright states on the PL spectrum and dynamics.¹⁰² By making use of this method, a 10 meV splitting has been estimated for (PEA)₂Pb(Sn)I_4.^103,104 There is only one report where a high magnetic field in the Voigt configuration (up to 40 T) has been applied to study 2D perovskites.⁶⁹ The authors of this work observe the appearance of new features in the absorption spectra for magnetic fields larger than 30 T, as shown in Fig. 4(b). The energy shift analysis of π and σ polarized states with the equations presented in Table II reveals a bright dark splitting of the order of only $∼ 1 meV$ in (C₆H₁₃NH₃)₂PbI₄⁶⁹ (see Fig. 4(c) ⁶⁹). This value is significantly lower than what recently reported for other 2D perovskites,⁹⁴ as discussed above. The complex line shape of the absorption spectrum of (C₆H₁₃NH₃)₂PbI₄ could hinder a precise determination of the fine structure of the exciton manifold. This motivates additional studies to unveil the exciton fine structure in 2D perovskites.

The use of high magnetic field has enabled to elucidate some of the fundamental electronic properties of 2D perovskites. Many open questions could benefit from the application of high magnetic field. A question which could be addressed is the accurate measurement of the effective mass of electron and holes, which could be achieved by THz spectroscopy in magnetic fields. However, for this method to be viable, free carriers should be injected in 2D perovskites, for example, by performing above bandgap illumination.¹⁰⁵ Another aspect which should be clarified is whether magneto-optical experiments allow us to access the bare carrier mass or whether they can probe the polaron mass.⁷³ Because of the ionic character of the perovskite crystal, a strong coupling of charge carriers to optical phonons is expected.^106,107 At relatively low magnetic fields, the effective mass is in fact the polaron mass, which is larger than the bare carrier mass by a factor of $1 + α / 6$⁠,¹⁰⁸ where α is the Frölich coupling constant. In high field limit, instead, when the phonon frequency is much lower than the cyclotron frequency, the motion of the carrier decouples from the lattice vibrations and the quantity probed by magneto-optical techniques is expected to be the bare carrier mass.⁷³ In the intermediate range, when $ℏ ω c$ approaches energy of the longitudinal optical (LO) phonon, the interaction between electrons and LO phonons resonantly increases, which results in an anti-crossing between the N = 0 Landau levels with one LO phonon and the N = 1 Landau level with no LO phonon.^109,110 The experimental observation of this effect would be a very elegant proof of the polaronic nature of carriers in perovskite compounds. A potentially interesting candidate for this experiment would be (PEA)₂SnI₄ due to its low effective mass.⁶⁶ 

Another interesting problem, which might be addressed by magneto-optical spectroscopy, is the anisotropy of exciton wave function related to the in-plane and out of plane directions. So far, only the exciton size in the in-plane direction has been determined for 2D perovskites.⁶⁷ However, there is a lack of information concerning the penetration depth of the exciton wave function into the organic barriers. This information might be helpful to explain why in some 2D perovskites the excitons seem to couple to vibration modes localized in the organic spacers,^46,84,111 while in other compounds this does not occur.^41,112 We can imagine at least two approaches to reveal the out-of-plane exciton size. One requires the use of the Faraday geometry. In this case, the plane of the inorganic layers should be parallel to the magnetic field and the light emitted/transmitted/reflected from the edge of structure should be analyzed. This could be challenging from the experimental point of view; however, the analysis should be relatively simple by making use of Eq. (1). Alternatively, the Voigt geometry could be used, which might be easier experimentally. However, the more complex dependence of the transition energies (see Table II) can make the precise determination of diamagnetic coefficient, which should be added to the Zeeman splitting, difficult to achieve.

Finally the successful synthesis of phase pure 2D perovskites with n > 1^113,114 would enable the investigation of the evolution of the FSS in 2D perovskites with different inorganic slab thickness. For this purpose, the magnetic field represents the ideal experimental tool. The recent progress made in optical spectroscopy in extreme magnetic fields^115–117 together with the continuous improvement in the crystal synthesis represents a fertile ground for successful studies and to deepen our understanding of the fundamental physics of 2D perovskites.

The authors appreciate support from the National Science Centre Poland within the MAESTRO program (Grant No. 2020/38/A/ST3/00214).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.