Effective-mass model and magneto-optical properties in hybrid perovskites

Hybrid inorganic-organic perovskites have proven to be a revolutionary material for low-cost photovoltaic applications. They also exhibit many other interesting properties, including giant Rashba splitting, large-radius Wannier excitons, and novel magneto-optical effects. Understanding these properties as well as the detailed mechanism of photovoltaics requires a reliable and accessible electronic structure, on which models of transport, excitonic, and magneto-optical properties can be efficiently developed. Here we construct an effective-mass model for the hybrid perovskites based on the group theory, experiment, and first-principles calculations. Using this model, we relate the Rashba splitting with the inversion-asymmetry parameter in the tetragonal perovskites, evaluate anisotropic g-factors for both conduction and valence bands, and elucidate the magnetic-field effect on photoluminescence and its dependence on the intensity of photoexcitation. The diamagnetic effect of exciton is calculated for an arbitrarily strong magnetic field. The pronounced excitonic peak emerged at intermediate magnetic fields in cyclotron resonance is assigned to the 3D±2 states, whose splitting can be used to estimate the difference in the effective masses of electron and hole.


Results
Model. Crystalline CH 3 NH 3 PbI 3 can have the high-temperature α-phase with the pseudo cubic (O h ) symmetry, the intermediate-temperature β-phase with the tetragonal (C 4v ) symmetry, and the low-temperature orthorhombic γ phase. Phase transitions from high to low temperatures, being of group-subgroup type, take place at 333 K and 150 K, respectively 25,26 . We focus in this paper on the β-phase, which is also a good description of the approximately uniaxial γ-phase 25 . In the β-phase, PbI 6 octahedra are misalign with the C-axis (symmetry axis of C 4v ), and the structure is noncentrosymmetric.
First-principle calculations indicate that the valence and conduction bands are mainly associated with cationic (Pb) s and p orbitals, respectively [15][16][17]27 , denoted as S, X, Y, and Z. The direct band gap is located at R point 7 , which has the same C 4v symmetry as the crystal structure. Since physically relevant states are those close to the band extremes, we derive band structure in the neighborhood of R point, via the k · p method, where k is the wave vector away from the R point. In this method, the wave function at k is expressed as ψ nk = e ik·r u n (r) with u n (r) being the basis function of nth band at the R point. We note that the k · p Hamiltonian for zinc-blende semiconductors is not suitable for the tetragonal perovskites 15 . In the absence of magnetic field, the Hamiltonian can be written as is the spin-orbit coupling (SOC). Here V(r) is periodic potential, p is momentum, m is free electron mass, c is the speed of light, and σ are the Pauli matrices.
In the β-phase, the potential should be an identical representation of group C 4v , a subgroup of the cubic group O h , and therefore can be expressed in terms of the O h representations, in particular, its first three irreducible representations, Γ 1 ⊕ Γ 12 ⊕ Γ 15 . Neglecting the trivial Γ 1 representation, we write V(r) = ∑ j c j d j , where d 1 = 2z 2 − x 2 − y 2 and d 2 = x 2 − y 2 are the basis functions of Γ 12 , and d 3 = x, d 4 = y, and d 5 = z are the basis functions of Γ 15 . By requiring D(G i )V(r) = V(r) with G i being the symmetry operators in C 4v , c 2 = c 3 = c 4 = 0. The nonzero c 1 reflects a crystal-field splitting between Z and X (Y), 〈 X|H 0 |X〉 = 〈 Y|H 0 |Y〉 = − 〈 Z|H 0 |Z〉 /2 = δ/3. And c 5 originates from the lack of inversion asymmetry in C 4v , giving rise to 28 〈 S|H 0 |Z〉 = 〈 Z|H 0 |S〉 * = c 5 〈 S|z|Z〉 ≡ ζ.
The Hamiltonian H 0 , up to the second order of k, can be written as 19 Here E v is the valence-band maximum, L i , M i , and N i are parameters due to the interaction between the conduction bands with far bands other than the valence band, and are the Kane parameters that connect the valence-band and conduction-band orbitals 20 . The SOC among the p orbitals mixes up and down spins, which is particularly strong in the hybrid perovskites due to the heavy element Pb, λ = 1.2-1.5 eV 7-9 , and cannot be treated as a perturbation.
The crux of the k · p method is that the basis functions of u n (r) should be the eigenstates of H at k = 0 20 , which can be achieved by choosing the following basis functions u n (r), Scientific RepoRts | 6:28576 | DOI: 10.1038/srep28576 , where  αβγ is the antisymmetric tensor of rank three. Because of the latter, the effective Zeeman energy of quasi-degenerate conduction-band orbitals can be written as where l (l = 1) is the angular momentum operator, and κ 1 and κ 2 are the Luttinger antisymmetric parameters for tetragonal structures. The total Hamiltonian, in the presence of magnetic field B, can now be written as Here E m (k) is the energy of mth band, which, for small k, can be approximated by the band-edge value. Taking into account the non-commutative relations among k components, we obtain the effective Hamiltonians, up to the second order of k, for the valence and conduction bands, with basis functions of v ± and c ± in Eqs (2) and (3), It is interesting to note that m h|| (m h⊥ ) depends on the interaction with the conduction bands via the Kane parameter P || (P ⊥ ). Similarly, the effective masses of the conduction band along and perpendicular to the C-axis are A free electron possesses a magnetic moment of its spin and has a g-factor of g 0 = 2.0023. The SOC enables the electron orbital motion to contribute to the magnetic moment and, consequently, the effective g-factor deviates from g 0 . In Eq. (9), the g-factors of the valence-band edge along and perpendicular to the C-axis are We see that the g-factors depend on the energies of conduction-band edges as well as the Kane parameters. In contrast to the effective mass, g h|| is connected to the conduction bands only via P ⊥ . This is understandable because a magnetic field B affects the electron orbital motion perpendicular to the field. For the same reason, g h⊥ is connected to the conduction bands via both P || and P ⊥ . Using the values of P || and P ⊥ , we obtain the g h|| = − 0.472 and g h⊥ = − 0.354, which are similar to the value in 2H-PbI 2 , g h = − 0.4 32 . The negative g-factor means that the up spin has a lower energy than the down spin.
The conduction-band g-factors along and perpendicular to the C-axis are which depend on the antisymmetric Luttinger parameters κ 1 and κ 2 , in addition to the ⊥ P ( ) . Thus the values of g e|| and g e⊥ can be used to determine κ 1 and κ 2 . Experimentally, the exciton g-factor are measured from the energy splitting between left-and right-circularly polarized absorption 33,34 and PL 11 , which, as we will discuss below, is g e|| + g h|| . If we use g e|| + g h|| = 1.2, as in ref. 33, we find g e|| = 1.672 and κ 1 = 0.269. If we further assume κ 2 = κ 1 , we have g e⊥ = 2.281. These values are also similar to those of electrons in 2H-PbI 2 , g e|| = 1.4 and g e⊥ = 2.4 32 .
Since both the g-factors and the effective masses are anisotropic, the effective spin splitting and the cyclotron frequency depend on the angle θ between the magnetic field and the C-axis, The derivations can be found in the Methods section. We plot in Fig. 2 the g-factors and the effective cyclotron mass, (9) and (10), destroys the spin degeneracy, giving rise to energy-momentum dispersions, for the conduction band and for the valence band, as plotted in Fig. 3.
The Rashba strengths α c(v)r are directly related to the C 4v potential ζ-parameter that characterizes the inversion asymmetry of the structure, Exciton wavefunctions. Hybrid perovskite CH 3 NH 3 PbI 3 has a large dielectric constant ε, and the excitons are of the Wannier type, whose wave functions can be written as 35 where j e , j h = ± ,  The being the 1S state, the exciton wavefuctions can be characterized by the C 4v representations, The total angular momentum J = j e + j h is J = 0 for ψ 1 , (J, J z ) = (1, 0) for ψ 2 , and (J, J z = 1, ± 1) for ψ ± 5 . The absorption and emission of these states are proportional to the modular square of their electric-dipole elements, . Hence ψ 2 can absorb and emit light polarized along the z-axis, and ψ ± 5 can absorb and emit light circularly polarized in the x-y plane. ψ 1 , however, is dark, for it contains only spin triplets and is therefore dipole-forbidden.
The above selection rules in the hybrid perovskites is in stark contrast to those in π-conjugated organic materials with weak SOCs 36 , where the electron (hole) has zero angular momentum (l z = 0 for π orbitals) and J = 1 is dipole forbidden whereas J = 0 is dipole allowed. As we will see below, this difference gives rise to a far richer physics of MFE in the hybrid perovskites.
Paramagnetic effects on excitons. The four 1S exciton states, ψ 1 , ψ 2 , and ψ ± . An applied magnetic field can modify the exciton energy via the Zeeman energy,  It should be noted that from the time reversal symmetry, the hole's g-factor, including the sign, is identical to that of the valence electron. Since we are concerned with relatively weak magnetic field, we temporarily neglect the diamagnetic effect, which is proportional to B 2 and shift states equally in energy.
In the Faraday configuration with B along the C-axis, B = (0, 0, B), ψ ± 5 will split in energy, , which have polarization along the y and x axis, respectively. It is readily to verify that in this configuration, pair (ψ 1 , ψ 5L ) as well as pair (ψ 2 , ψ 5T ) are coupled via the magnetic field, with  Figure 4 also plots the exciton energies E 1 , E 2 , E 5L , and E 5T , as well as their corresponding oscillator strengths |〈 e x p x 〉 | 2 as a function of the magnetic field.
Magnetic-field effect on photoluminescence. The PL intensity in CH 3 NH 3 PbI 3 is found to be susceptible to a magnetic field at low temperatures 11 . In the Faraday configuration, the magnetic field couples ψ 1 and ψ 2 excitons. Since only the recombination of ψ 2 can give rise to luminescence, the magnetic-field-induced change in populations of ψ 2 and ψ 1 would lead to an MFE. We employ the Bloch equation of the density matrix to systematically describe the population dynamics, 1 g where ρ is a 2 × 2 density matrix spanned by ψ 1 and ψ 2 , ρ ρ ψ ψ = ∑ m n mn m n , with m, n = 1, 2. (∂ ρ/∂ t) g represents the generation of the exciton states, which is finite only for diagonal terms, (∂ ρ mn /∂ t) g = F m δ mn , because the PL in the MFET measurements 11,12 is not resonantly excited. τ is the relaxation time of these exciton states, which includes both recombination τ −  When the exchange is significant with . In this regime, the MFE is suppressed because the magnetic field cannot overcome the exchange to effectively alter the populations on the exciton states for H < 1T. When , which represents the so called Δ g mechanism of MFE. The Δ g mechanism has been found responsible for many MFE phenomena involving radical pairs in organic systems 24 . If the ∆g is known, the Lorentz line shape in Eq. (24) can be used to measure the exciton relaxation time τ, or equivalently, its spin relaxation time.
In the Vogit configuration, the MFE in PL is also expected because the magnetic field, as shown in Eqs (21) and (22), mixes the dark ψ 1 with the dipole-allowed ψ 5L , and ψ 2 with ψ 5T , which have different oscillator strengths and polarizations, ξP sin /4 2 2 and along the z-axis and ξ ⊥ P cos /2 2 2 and along the y-axis for ψ 5T , respectively. Using the Bloch equation, we express the PL change in the ψ 1 and ψ 5L manifold as where Δ g ⊥ = g e⊥ − g h⊥ , and the PL change in the manifold of ψ 2 and ψ 5T as where Σ g ⊥ = g e⊥ + g h⊥ . The MFE in the ψ 1 and ψ 5L manifold, Δ I 2 , depends on the difference in the g-factors along the x-axis, the direction of the magnetic field, in a very similar fashion as Δ I 1 .
The MFE in the ψ 2 and ψ 5T manifold, Δ I 3 , however, depends on the summation of the electron and hole g-factors along the x-axis. Thus in addition to the Δ g mechanism, a Σ g mechanism is taking effect in the hybrid perovskites. In the former, the magnetic field modulates the populations between states J = 0 and (J, J z(x) ) = (1, 0), which can be visualized as the electron and hole spins precess along the magnetic field in the opposite directions. In the latter, the magnetic field modulates populations between (J, J z ) = (1, 0) and (1, ± 1), which can be visualized as the electron and hole spins precess along a transverse magnetic field in the same direction. The Σ g mechanism is particularly important if the exchange is approximately isotropic,  (24) and (25) are suppressed by the exchange splitting 4J ⊥ between the singlet and triplet. For polycrystalline materials, the intensity change should be the combination of the three processes, Δ I i (i = 1, 2, 3), suggesting that multiple Lorentzen functions may be required to describe experimental data, as shown experimentally 11 .

Photoexcitation intensity dependence of MFE. It is observed that the MFE in PL in CH 3 NH 3 PbI 3 is
also dependent on the photoexcitation intensity 12 . Only when the intensity reaches a certain threshold, does the MFE become significant. The line shape of MFE shrinks with the increase of the photoexcitation intensity and is eventually stabilized. To explain this unusual intensity dependence, we notice that the MFE, as shown in Fig. 5, is pronounced only when the Zeeman energy dominates over the exchange splitting. As a specific example, we consider the Faraday configuration, where the energy splitting between ψ 1 and ψ 2 is 4J ⊥ . This exchange is of short-range and related to the exciton envelop function Φ (r) at r = 0, i.e., when the electron and hole are at the same location. For the 1S state, A high-intensity photoexcitation creates many free electron-hole pairs, whose density can be estimated as N = αIτ l /ħω, where α is the absorption coefficient, α ~ 10 5 cm −1 5 for the CH 3 NH 3 PbI 3 , τ l is the carrier recombination lifetime, τ − 10 l 5 s, and ħω is the photon energy. These free electron-hole pairs will screen the Coulomb interaction, which can be modeled by the Debye-Hückel theory of ion gases 37 . The Coulomb potential − e 2 /εr, in the presence of charged particles, is replaced by the potential U that satisfies the Poisson equation, (∇ 2 − Q 2 )U = 0 with Q 2 = 8πe 2 /k B TN. The solution U(r) is of the Yukawa type U(r) = − e 2 e −Qr /(εr), and the ground-state wave function in such a potential can be written as , where the trial parameter β can be obtained by minimizing the ground state energy of Hamiltonian − ∇ 2 /2μ + U(r), E = (1)/(2)β 2 Q 2 ħ 2 /μ − 4β 3 e 2 / [εQ(4β 2 + 4β + 1)]. The wave function Φ ∼ (r), as compared to Φ (r), is more spreaded in space, and the exchange will be reduced by a factor Figure 5 illustrates the screening effect. We see that as the intensity of photoexcitation increases, the exchange is greatly reduced. The MFE, meanwhile, becomes significant. After the carrier density reaches 10 18 cm −3 , the exchange is so small that τ ⊥  J Q 4 ( ) 1, and the line shape in Δ I 1 is independent of exchange, and therefore the photoexcitation intensity.
Diamagnetic effect on excitons. So far we have considered only the 1S exciton states and neglected the diamagnetic effect on excitons. The diamagnetic effect originates from the orbital motions of electron and hole, and has been used to directly measure the exciton's binding energy and effective mass 23 . For free electrons and holes, an applied magnetic field can localized their orbital wave function normal to the magnetic field, forming the Landau levels with the magnetic length of c eB /  . Since the anisotropy in effective mass for both electron and hole are relatively small, as shown in Fig. 2 , to study the diamagnetic effect. The diamagnetic response of excitons is far more complex than that of free electron-hole pairs, but offers more valuable information on excitons as well as constituent electrons and holes. Under a small magnetic field, the Coulomb interaction in an exciton is predominant and the diamagnetic effect can be studied by applying the perturbation theory to the hydrogen-like exciton wavefunctions. Such a perturbation must fail when the magnetic length c eH /  becomes much smaller than the orbital radius of exciton, a 0 = ħ 2 ε/(e 2 μ). In this high-magnetic field regime, it is more appropriate to use the Landau levels as the starting point. A ratio, γ = ħω c /2R y , between the exciton cyclotron energy ħω c = eB/(μc) and the exciton binding energy, R y = μe 4 /(2ε 2 ħ 2 ), can be used to distinguish the weak (γ < 1) and strong (γ > 1) magnetic-field regimes.
The Hamiltonian for the envelop function Φ (r e − r h ) at K = 0 (K being the center-of-mass momentum of exciton) reads , the difference between electron and hole effective masses, which will reduce to μ −1 in the hydrogen case.
To reliably solve the Hamiltonian for an arbitrary magnetic field, we employ two different basis sets 38 . In low magnetic fields, by using a 0 (R y ) as the length (energy) scale, ηγ γ θφ , where η = m/m e − m/m h and Y lm is spherical harmonic function. We choose the basis set as In high magnetic fields, by using γ a ( 6/ ) 1/2 0 and γ R / 6 y as the length and energy scales =  1/2 are chosen to be another basis set. These basis sets, with correct characteristics of wave functions in the low-and high-field regimes, facilitate an analytical evaluation of the Hamiltonian matrix elements (see the Methods section). Moreover the basis sets are eigenstates of parity and L z = m, which are good quantum numbers of the Hamiltonian. We use large basis sets in both regimes, n, l ≤ 20 for Φ nlm and n, l ≤ 29 for Ψ nlm , and diagonalize the Hamiltonian to obtain the eigenstates. The large basis sets allow us to approach the intermediate γ ~ 1 from both γ < 1 and γ > 1 regimes so that the solutions from both ends are smoothly connected. In the limit of γ → ∞ , the eigenstates become the the Landau levels of free electron and hole. ω , where E g is the band gap. In Fig. 6, we compare the theoretical results with recent cyclotron resonance experiment 23 , using the effective masses of m h = 0.223 m and m e = 0.188 m, and the binding energy of 16 meV. The agreement between theory and experiment is excellent. In addition, the pronounced absorption peaks above the 2S state, induced by the magnetic field, are very close to the state 3D ±2 , whereas the 2P 0 state, as assigned in ref. 23 for the absorption peak, is almost degenerate in energy with 2S state. Indeed, transitions to exciton states ψ + − D 5 2 and ψ − + D 5 2 , according to the selection rule, are electric-dipole allowed and the only reason that these the state is dark at zero field is Φ (r = 0) = 0, which become finite in large magnetic fields. Thus we believe that the observed excitonic absorption peak is due to the 3D ±2 states. 3D ±2 states are important in that from their energy splitting, D  e  h  3  3  1  1   2  2 , we can obtain the difference in the effective masses of electron and hole, which, together with μ measured from the Landau levels of free electron-hole absorption, can completely determine both m e and m h . It is noted that the inter-band cyclotron resonance of free electron-hole pairs can measure only μ, not individual m e and m h . Thus cyclotron resonance of excitons reveals more information.

Discussion
The hybrid inorganic-organic perovskites have shown great promise in photovoltaic and many other important applications because of their outstanding transport, optical, and magneto-optical properties. To understand these properties, in this paper, we have constructed a reliable and accessible effective-mass model of the hybrid perovskites, which connects effective masses, Rashba splittings, and anisotropic g-factors of conduction and valence bands. Using this effective-mass model, we have elucidated the observed MFE in exciton PL and its dependence on photoexcitations and identified a new Σ g mechanism of MFE. We have also calculated the cyclotron resonance of excitons for arbitrarily strong magnetic fields and pointed out that excitonic states such as 3D ±2 provide information on the difference in effective masses of electron and hole.
This effective-mass model is a foundation on which systematic models of electron-phonon coupling, carrier mobility, and other transport properties can be developed. Because of the concise expressions of SOC, Rashba effect, and g-factor, this model also facilitates studies of spin relaxation 39 , spin Hall effect 40 , and other magneto optical and spintronic phenomena. A , H B , and H