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

Abstract

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.

Introduction

Hybrid organic-inorganic perovskites such as CH₃NH₃PbI₃ represents a revolutionary breakthrough for low-cost solar cells^1,2,3,4 because of their desirable optical and carrier transport properties^5,6. The materials also exhibit many intriguing features, including strong spin-orbit coupling⁷ and the associated Rashba effect^8,9, large-radius Wannier excitons¹⁰ and novel magnetic-field effect (MFE) in photoluminescence (PL) and photoconduction^11,12 and show promise in light-emitting¹³ and thermoelectric¹⁴ applications.

These outstanding properties are interconnected and are ultimately determined by the material’s unusual electronic structure, which has been intensively studied by a variety of density-functional calculations^15,16,17. While such first-principles calculations are indispensable in predicting the crystal structure, carrier effective mass and band gap, they become increasingly unwieldy in studying processes involving excited states and under external fields. An alternative is to develop an effective-mass Hamiltonian^18,19,20, which is both tractable and transparent in physics with parameters determined by experimentally measured properties. These properties, such as effective masses and g-factors, are usually obtained from magneto-optical studies, which turn out to be a major experimental means of validating the principle of the band theory of semiconductors.

Exciton, an electron-hole pair bounded by the Coulomb interaction, is a fundamental excitation in semiconductors. The exciton binding energy in the hybrid perovskites is critical to photovoltaic and light-emitting efficiencies and has been a subject of intense debate^21,22. This controversy can be resolved via a definitive measurement of magneto-optical absorption (cyclotron resonance)²³, which reveals characters of exciton as well as constituent electron and hole. To take advantage of the wealth of information, a detailed analysis of cyclotron resonance is needed. While diamagnetic response of an exciton is similar to that of a hydrogen, a key difference is that the electron and hole in an exciton have a comparable effective mass, particularly in the hybrid perovskites.

A moderate magnetic field less than 1 T is found to be able to influence exciton PL in CH₃NH₃PbI₃^11,12, indicating that magnetic field is a versatile tool for studying excitons and free carriers. This MFE has been attributed to the Δg (the difference between electron and hole g-factors) mechanism, frequently encountered in organic radical pairs²⁴. However, the lack of knowledge on the g-factors in CH₃NH₃PbI₃ hampers the development of a clear understanding of the MFE. In addition, the MFE is sensitive to the intensity of photoexcitation¹², which is not well understood.

Here we construct an effective-mass model of CH₃NH₃PbI₃ based on information available in literature. This model, which can be extended to other hybrid perovskites by using suitable parameters, reveals connections among the g-factors, effective masses and Rashba spin splittings. Using this model, we examine the MFE on exciton PL and find that the MFE is controlled by the interplay of exchange energy, exciton (spin) relaxation time and the Zeeman energy. Besides the Δg mechanism, a Σg (summation of the electron and hole g-factors) mechanism can manifest itself in the MFE. The dependence of MFE on the intensity of photoexcitations is quantitatively explained in terms of the screening effect by the photogenerated carriers, which greatly reduces the exchange coupling of excitons. The diamagnetic effect on excitons under an arbitrarily large magnetic field is reliably calculated and in an excellent agreement with recent cyclotron measurements. The experimentally observed pronounced excitonic absorption peak, induced by the magnetic-field, can be attributed to the 3D_±2 states, whose energy splitting can be used to determine the difference in electron and hole effective masses. Our results demonstrate the efficacy of the effective-mass model in understanding magneto-optical properties and suggest it a foundation for systematically studying many other transport, optical and spintronic processes in the hybrid perovskites.

Results

Model

Crystalline CH₃NH₃PbI₃ 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-phase²⁵. In the β-phase, PbI₆ 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 point⁷, 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·ru_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 perovskites¹⁵. In the absence of magnetic field, the Hamiltonian can be written as H = H₀ + H_SO, where H₀ = p²/2 m + V(r) is spin-independent part and 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, Γ₁ ⊕ Γ₁₂ ⊕ Γ₁₅. Neglecting the trivial Γ₁ representation, we write V(r) = ∑_jc_jd_j, where d₁ = 2z² − x² − y² and d₂ = x² − y² are the basis functions of Γ₁₂ and d₃ = x, d₄ = y and d₅ = z are the basis functions of Γ₁₅. By requiring D(G_i)V(r) = V(r) with G_i being the symmetry operators in C_4v, c₂ = c₃ = c₄ = 0. The nonzero c₁ reflects a crystal-field splitting between Z and X (Y), 〈X|H₀|X〉 = 〈Y|H₀|Y〉 = −〈Z|H₀|Z〉/2 = δ/3. And c₅ originates from the lack of inversion asymmetry in C_4v, giving rise to²⁸ 〈S|H₀|Z〉 = 〈Z|H₀|S〉^* = c₅〈S|z|Z〉 ≡ ζ.

The Hamiltonian H₀, up to the second order of k, can be written as^19,20

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 and are the Kane parameters that connect the valence-band and conduction-band orbitals²⁰.

The SOC among the p orbitals mixes up and down spins, λ = i〈X↑|H_SO|Y↓〉 = i〈Y↑|H_SO|Z↓〉 = i〈Z↑|H_SO|X↓〉, which is particularly strong in the hybrid perovskites due to the heavy element Pb, λ = 1.2–1.5 eV^7,8,9 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 = 0²⁰, which can be achieved by choosing the following basis functions u_n(r),

with . The angular momentum is s = 1/2 for the valence band v_± and j = 1/2 (j = l + s with l = 1 and s = 1/2) for the first conduction band c_±. The two upper conduction bands, and , have j = 3/2, with j_z = ±3/2 for and j_z = ±1/2 for . The diagonal elements at k = 0 in these basis functions are E_v, , E_c′ = 0, . Here we temporally neglect ζ, which is small as compared to other parameters. In the measured absorption spectra of CH₃NH₃PbI₃ (Fig. 1), the first three peaks, located at 1.6 eV, 2.8 eV and 3.4 eV²⁹, can be attributed to electron transitions from the valence band to the three conduction bands. Thus we obtain the values E_v = −2.8 eV, E_c = −1.2 eV and E_c′ = 0.6 eV, which fix the parameter values, λ = 1.4 eV, δ = −0.7 eV and sin ξ = 0.411.

Figure 1

Schematic diagram of band edges and their angular momenta in CH₃NH₃PbI₃.

The optical absorption spectrum is adapted from ref. 29.

An applied magnetic field B have two effects on a charge carrier: paramagnetic magnetism due to the carrier’s spin and diamagnetic orbital magnetism due to the lack of commutation among momentum components, ³⁰, where 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 κ₁ and κ₂ are the Luttinger antisymmetric parameters for tetragonal structures.

The total Hamiltonian, in the presence of magnetic field B, can now be written as

where 4 × 4 matrices H_A, H_B and H_C are displayed in the Methods section. In this Hamiltonian, we have eleven parameters, L_i and κ_i (i = 1, 2), M_i and N_i (i = 1, 2, 3), and P_⊥. If the information at hand is insufficient to fix all these parameters, one can resort to the first principles calculations, or via the quasi-cubic symmetry, L₁ = L₂, M₁ = M₂ = M₃ and κ₁ = κ₂.

g-factors, effective masses and cyclotron frequencies

The most important transport and optical processes occur in the conduction and valence bands, which are both nondegenerate (excluding spin). We map the above 8 × 8 Hamiltonian H_mn into two effective 2 × 2 Hamiltonians in spin space for these two bands by employing the Löwdin method for degenerate perturbation theory³¹,

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),

where μ_B ≡ eħ/(2 mc) is the Bohr magneton, and , which can be approximated by E_v and E_c.

The effective masses of valence band along and perpendicular to the C-axis in Eq. (9) are expressed as

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

which are influenced by the interaction with the valence band, as well as by the symmetric parameters L_i and M_i, stemming from the interaction with far bands. The anisotropic effective masses for the valence and conduction bands have been extensively calculated by several first-principles approaches^16,17. One of the most accurate values are given in ref. 17, which are also consistent with recent cyclotron resonance measurements²³, , m_h⊥ = 0.23 m, m_e|| = 0.15 m and m_e⊥ = 0.21 m. From these effective masses, we obtain the Kane parameters, P_|| = 7.64 eV Å and P_⊥ = 6.95 eV Å, as well as the symmetric parameters mL_i/ħ² = −25.90 and mM_i/ħ² = 23.39.

A free electron possesses a magnetic moment of its spin and has a g-factor of g₀ = 2.0023. The SOC enables the electron orbital motion to contribute to the magnetic moment and, consequently, the effective g-factor deviates from g₀. 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-PbI₂, g_h = −0.4³². 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 κ₁ and κ₂, in addition to the . Thus the values of g_e|| and g_e⊥ can be used to determine κ₁ and κ₂. Experimentally, the exciton g-factor are measured from the energy splitting between left- and right-circularly polarized absorption^33,34 and PL¹¹, 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 κ₁ = 0.269. If we further assume κ₂ = κ₁, we have g_e⊥ = 2.281. These values are also similar to those of electrons in 2H-PbI₂, g_e|| = 1.4 and g_e⊥ = 2.4³².

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, as a function of θ. At θ = 0, and m_ce(h)(θ) = m_e(h)⊥. At θ = π/2, g_e(h)(θ) = g_e(h)⊥ and .

Figure 2

g-factors and cyclotron masses of conduction and valence bands.

Left panels describe the electron (g_e) and hole (g_h) g-factors as a function of angle θ between the C-axis and the applied magnetic field. Right panels describe the electron (m_ce) and hole (m_ch) cyclotron masses.

Rashba splitting

The Rashba term, E_c(v)r(k) = α_c(v)r(k_yσ_x − k_xσ_y) in Eqs (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.

Figure 3

Rashba effect in the conduction and valence bands near the R point.

Left (right) panel is for k along the [0, 1, 1] ([0, 1, 1]) direction. ζ = 0.5 eV.

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,

which indicate that the Rashba splittings in the valence and conduction bands are correlated. Currently the Rashba splittings obtained from different first-principles calculations vary significantly and direct measurements, such as spin-polarized photoemission and spin-flip Raman scattering, of CH₃NH₃PbI₃ are not yet available. For ζ = 0.5 eV, we have α_vr = 0.565 and α_cr =1.088 eVÅ.

Exciton wavefunctions

Hybrid perovskite CH₃NH₃PbI₃ has a large dielectric constant ε and the excitons are of the Wannier type, whose wave functions can be written as³⁵

where j_e, j_h = ±, with being the time-reversal operator. is the envelop function describing the relative motion of electron and hole and may have S, P, or D characteristics in low magnetic fields, which gradually transforms to that of Landau wave functions with increase of magnetic field.

The hole (electron) state follows the representation of C_4v. With 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 ψ₁, (J, J_z) = (1, 0) for ψ₂ and (J, J_z = 1, ±1) for . The absorption and emission of these states are proportional to the modular square of their electric-dipole elements,

with . Hence ψ₂ can absorb and emit light polarized along the z-axis and can absorb and emit light circularly polarized in the x-y plane. ψ₁, 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 SOCs³⁶, 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, ψ₁, ψ₂ and , in general, are not degenerate in energy because of possible exchange interaction between spins , where σ^e/2 = j_e and σ^h/2 = s_h. Consequently the energies of these excitons .

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 B² and shift states equally in energy.

In the Faraday configuration with B along the C-axis, B = (0, 0, B), will split in energy, and the splitting in absorption and luminescence peaks of left- and right-circularly polarized light, would be ^11,33,34.

The magnetic field B will also mix ψ₁ and ψ₂, with

and the energies of eigenstates become and their wave functions are (i = 1, 2). Hence, with increase of the magnetic field, will gain oscillator strength, and flare up, while will lose oscillator strength, , as illustrated in Fig. 4.

Figure 4

Exciton energies and oscillator strengths vs magnetic field.

Panels (a,b) delineate exciton energies as a function of magnetic field in the Faraday and Voigt configurations, respectively. Panels (c,d) are the oscillator strengths of these exciton states along the z-axis and the x-axis, respectively. Panel (e) illustrates the electric-dipole selection rule of the exciton states. Black and red lines correspond to ψ₁ and ψ₂ excitons. Green and blue lines correspond to and excitons in (a,c) and ψ_5L and ψ_5T in (c,d). Green and blue lines in (c) as well as black and blue lines in (d) are on top of each other. .

In the Vogit configuration with B perpendicular to the C-axis, B = (B, 0, 0), we can construct transverse and longitudinal states out of states, and , which have polarization along the y and x axis, respectively. It is readily to verify that in this configuration, pair (ψ₁, ψ_5L) as well as pair (ψ₂, ψ_5T) are coupled via the magnetic field, with

and their energies are and . Figure 4 also plots the exciton energies E₁, E₂, E_5L and E_5T, as well as their corresponding oscillator strengths |〈e_xp_x〉|² as a function of the magnetic field.

Magnetic-field effect on photoluminescence

The PL intensity in CH₃NH₃PbI₃ is found to be susceptible to a magnetic field at low temperatures¹¹. In the Faraday configuration, the magnetic field couples ψ₁ and ψ₂ excitons. Since only the recombination of ψ₂ can give rise to luminescence, the magnetic-field-induced change in populations of ψ₂ and ψ₁ would lead to an MFE. We employ the Bloch equation of the density matrix to systematically describe the population dynamics,

where is a 2 × 2 density matrix spanned by ψ₁ and ψ₂, 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_mδ_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 and spin relaxation , , for ^5,6,11. In the steady state, , the densities at ψ₁ and ψ₂ can be written as and , where and . The intensity change in PL is due to the change in ρ₂₂,

When the exchange is significant with , 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 , the denominator in Eq. (24) becomes , which represents the so called Δg mechanism of MFE. The Δg mechanism has been found responsible for many MFE phenomena involving radical pairs in organic systems²⁴. If the 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 ψ₁ with the dipole-allowed ψ_5L and ψ₂ with ψ_5T, which have different oscillator strengths and polarizations, and along the z-axis and and along the y-axis for ψ_5T, respectively. Using the Bloch equation, we express the PL change in the ψ₁ and ψ_5L manifold as

where Δg_⊥ = g_e⊥ − g_h⊥ and the PL change in the manifold of ψ₂ and ψ_5T as

where Σg_⊥ = g_e⊥ + g_h⊥. The MFE in the ψ₁ and ψ_5L manifold, ΔI₂, depends on the difference in the g-factors along the x-axis, the direction of the magnetic field, in a very similar fashion as ΔI₁.

The MFE in the ψ₂ and ψ_5T manifold, ΔI₃, however, depends on the summation of the electron and hole g-factors along the x-axis. Thus in addition to the Δg mechanism, a Σg 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 Σg mechanism is particularly important if the exchange is approximately isotropic, , where the J = 1 triplet states are degenerate in energy and according Eq. (26), ΔI₃ is then completely determined by the exciton relaxation time τ, whereas ΔI₁ and ΔI₂ in Eqs (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, ΔI_i (i = 1, 2, 3), suggesting that multiple Lorentzen functions may be required to describe experimental data, as shown experimentally¹¹.

Photoexcitation intensity dependence of MFE

It is observed that the MFE in PL in CH₃NH₃PbI₃ is also dependent on the photoexcitation intensity¹². 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 ψ₁ and ψ₂ 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,

Figure 5

Magnetic field effect in exciton photoluminescence.

Panel (a) describes ΔI₁ for exchange splitting between ψ₂ and ψ₁ being 4J_⊥ = 1 (black line), 0.1 (red line), 0.01 (green line) and 0.001(blue line) meV. Panel (b) describes ΔI₁ (black line), ΔI₂ (red line) and ΔI₃ (green line) for meV. Panel (c) describes the carrier density and the reduction of the exchange as a function of the intensity of excitation light. Panel (d) plots ΔI₁ as a function of magnetic field with 4J_⊥ = 1 meV for intensities of 0.34 (black line) and 0.85 W/cm² (red line). Black and red dots are the corresponding experimental values in ref. 12. Panels (e,f) illustrate the Δg and Σg mechanisms. The exciton relaxation time is fixed at 10⁻¹⁰ s.

where a₀ is the effective Bohr radius of the exciton, a₀ = ħ²ε/e²μ with being the effective mass of exciton.

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 ~ 10⁵ cm^−1 5 for the CH₃NH₃PbI₃, τ_l is the carrier recombination lifetime,  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 gases³⁷. The Coulomb potential −e²/εr, in the presence of charged particles, is replaced by the potential U that satisfies the Poisson equation, (∇² − Q²)U = 0 with Q² = 8πe²/k_BTN. The solution U(r) is of the Yukawa type U(r) = −e²e^−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)β²Q²ħ²/μ − 4β³e²/[εQ(4β² + 4β + 1)]. The wave function , 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 10¹⁸ cm⁻³, the exchange is so small that and the line shape in ΔI₁ 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 mass²³. 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 . Since the anisotropy in effective mass for both electron and hole are relatively small, as shown in Fig. 2, we use isotropic effective masses, , m_h = and , 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 becomes much smaller than the orbital radius of exciton, a₀ = ħ²ε/(e²μ). 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 = μe⁴/(2ε²ħ²), 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

where L = −ir × ∇ is the orbital angular momentum. While this Hamiltonian of exciton is similar as that of hydrogen atom in a magnetic field, the key difference is that the third term contains , 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 sets³⁸. In low magnetic fields, by using a₀ (R_y) as the length (energy) scale, , where η = m/m_e − m/m_h and Y_lm is spherical harmonic function. We choose the basis set as

which are the eigenstates of with is the generalized Laguerre polynomials.

In high magnetic fields, by using and as the length and energy scales and the eigenfunctions of a spherical harmonic oscillator H₀ = −∇² + r²

with 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. and . The optical selection rule for the transition from valence- to conduction-band Landau levels is N_h = N_e = N and the absorption peaks are located at , where E_g is the band gap.

In Fig. 6, we compare the theoretical results with recent cyclotron resonance experiment²³, 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 2P₀ state, as assigned in ref. 23 for the absorption peak, is almost degenerate in energy with 2S state. Indeed, transitions to exciton states and , 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, , 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.

Figure 6

Cyclotron resonance of excitons in CH₃NH₃PbI₃.

Solid lines are energies of different exciton states. Dashed lines are the energy difference between the Landau levels of free electron and holes with (N, N) being the Landau-level index. Dots are the experimental data from ref. 23. The absorption peak between 2S and (1, 1) coincides with the energies of 3D_±2 states.

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 Σg 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 relaxation³⁹, spin Hall effect⁴⁰ and other magneto optical and spintronic phenomena.

Methods

Expressions of H_A, H_B and H_C

The three 4 × 4 matrices in Eq. (7) are displayed as

where

where

where

Derivations of Eq. (15)

When the applied magnetic field B tilts away from the crystal C-axis with an angle θ, we can define the new z-axis (denoted z′) along B and assume that the tilting is in the x-z plane with the new x-axis denoted as x′. The transformations of coordinates between the two references are

The Zeeman energy in Eqs (9) and (10) then becomes

and the effective g-factor g(θ) in Eq. (15) can be obtained by diagnonalizing this Hamiltonian.

In the new reference system, and the kinetic energy in Eqs (9) and (10) reads

with , , . We can define the ladder operators b and b⁺ of the Landau levels as

with , which satisfy [b, b⁺] = 1. The kinetic energy is then expressed as

with , which gives m_c(θ) in Eq. (15).

Matrix elements of H_D in basis sets of Φ_nlm and Ψ_nlm

The matrix elements of H_D among Φ_nlm and among Ψ_nlm can be evaluated analytically, which greatly simplifies the eigenstate calculations.

The matrix element of angle-dependent term in the Hamiltonian can be calculated via the integral

where the Wigner 3-j symbols are used.

The only type of matrix elements to be evaluated is r^s between the basis functions. Between basis functions and ,

The integral can be worked out analytically,

where γ = l_i + l_j + s + 2, , α = 2l_i + 1, β = 2l_j + 1, m = n_i − l_i − 1, n = n_j − l_j − 1, λ = 2/n_j, μ = 2/n_j and ₂F₁(α, β, γ; z is the Gauss’ hypergeometric function.

The matrix element of r^s between basis functions and , neglecting the normalization factor, is