Abstract
Hybrid inorganicorganic perovskites have proven to be a revolutionary material for lowcost photovoltaic applications. They also exhibit many other interesting properties, including giant Rashba splitting, largeradius Wannier excitons and novel magnetooptical 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 magnetooptical properties can be efficiently developed. Here we construct an effectivemass model for the hybrid perovskites based on the group theory, experiment and firstprinciples calculations. Using this model, we relate the Rashba splitting with the inversionasymmetry parameter in the tetragonal perovskites, evaluate anisotropic gfactors for both conduction and valence bands and elucidate the magneticfield 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 organicinorganic perovskites such as CH_{3}NH_{3}PbI_{3} represents a revolutionary breakthrough for lowcost 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 spinorbit coupling^{7} and the associated Rashba effect^{8,9}, largeradius Wannier excitons^{10} and novel magneticfield effect (MFE) in photoluminescence (PL) and photoconduction^{11,12} and show promise in lightemitting^{13} and thermoelectric^{14} 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 densityfunctional calculations^{15,16,17}. While such firstprinciples 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 effectivemass 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 gfactors, are usually obtained from magnetooptical studies, which turn out to be a major experimental means of validating the principle of the band theory of semiconductors.
Exciton, an electronhole 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 lightemitting efficiencies and has been a subject of intense debate^{21,22}. This controversy can be resolved via a definitive measurement of magnetooptical 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 CH_{3}NH_{3}PbI_{3}^{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 gfactors) mechanism, frequently encountered in organic radical pairs^{24}. However, the lack of knowledge on the gfactors in CH_{3}NH_{3}PbI_{3} hampers the development of a clear understanding of the MFE. In addition, the MFE is sensitive to the intensity of photoexcitation^{12}, which is not well understood.
Here we construct an effectivemass model of CH_{3}NH_{3}PbI_{3} based on information available in literature. This model, which can be extended to other hybrid perovskites by using suitable parameters, reveals connections among the gfactors, 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 gfactors) 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 magneticfield, 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 effectivemass model in understanding magnetooptical properties and suggest it a foundation for systematically studying many other transport, optical and spintronic processes in the hybrid perovskites.
Results
Model
Crystalline CH_{3}NH_{3}PbI_{3} can have the hightemperature αphase with the pseudo cubic (O_{h}) symmetry, the intermediatetemperature βphase with the tetragonal (C_{4v}) symmetry and the lowtemperature orthorhombic γ phase. Phase transitions from high to low temperatures, being of groupsubgroup type, take place at 333 K and 150 K, respectively^{25,26}. We focus in this paper on the βphase, which is also a good description of the approximately uniaxial γphase^{25}. In the βphase, PbI_{6} octahedra are misalign with the Caxis (symmetry axis of C_{4v}) and the structure is noncentrosymmetric.
Firstprinciple 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 zincblende semiconductors is not suitable for the tetragonal perovskites^{15}. In the absence of magnetic field, the Hamiltonian can be written as H = H_{0} + H_{SO}, where H_{0} = p^{2}/2 m + V(r) is spinindependent part and is the spinorbit 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 βphase, 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 crystalfield splitting between Z and X (Y), 〈XH_{0}X〉 = 〈YH_{0}Y〉 = −〈ZH_{0}Z〉/2 = δ/3. And c_{5} originates from the lack of inversion asymmetry in C_{4v}, giving rise to^{28} 〈SH_{0}Z〉 = 〈ZH_{0}S〉^{*} = c_{5}〈SzZ〉 ≡ ζ.
The Hamiltonian H_{0}, up to the second order of k, can be written as^{19,20}
Here E_{v} is the valenceband 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 valenceband and conductionband orbitals^{20}.
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 = 0^{20}, 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 CH_{3}NH_{3}PbI_{3} (Fig. 1), the first three peaks, located at 1.6 eV, 2.8 eV and 3.4 eV^{29}, 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.
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 quasidegenerate conductionband 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
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 quasicubic symmetry, L_{1} = L_{2}, M_{1} = M_{2} = M_{3} and κ_{1} = κ_{2}.
gfactors, 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 theory^{31},
Here E_{m}(k) is the energy of mth band, which, for small k, can be approximated by the bandedge value. Taking into account the noncommutative 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 Caxis 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 Caxis 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 firstprinciples approaches^{16,17}. One of the most accurate values are given in ref. 17, which are also consistent with recent cyclotron resonance measurements^{23}, , 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}/ħ^{2} = −25.90 and mM_{i}/ħ^{2} = 23.39.
A free electron possesses a magnetic moment of its spin and has a gfactor of g_{0} = 2.0023. The SOC enables the electron orbital motion to contribute to the magnetic moment and, consequently, the effective gfactor deviates from g_{0}. In Eq. (9), the gfactors of the valenceband edge along and perpendicular to the Caxis are
We see that the gfactors depend on the energies of conductionband 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 2HPbI_{2}, g_{h} = −0.4^{32}. The negative gfactor means that the up spin has a lower energy than the down spin.
The conductionband gfactors along and perpendicular to the Caxis are
which depend on the antisymmetric Luttinger parameters κ_{1} and κ_{2}, in addition to the . Thus the values of g_{e} and g_{e⊥} can be used to determine κ_{1} and κ_{2}. Experimentally, the exciton gfactor are measured from the energy splitting between left and rightcircularly 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 2HPbI_{2}, g_{e} = 1.4 and g_{e⊥} = 2.4^{32}.
Since both the gfactors 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 Caxis,
The derivations can be found in the Methods section. We plot in Fig. 2 the gfactors 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 .
Rashba splitting
The Rashba term, E_{c(v)r}(k) = α_{c(v)r}(k_{y}σ_{x} − k_{x}σ_{y}) in Eqs (9) and (10), destroys the spin degeneracy, giving rise to energymomentum 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 ζparameter 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 firstprinciples calculations vary significantly and direct measurements, such as spinpolarized photoemission and spinflip Raman scattering, of CH_{3}NH_{3}PbI_{3} 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 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} = ±, with being the timereversal 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 ψ_{1}, (J, J_{z}) = (1, 0) for ψ_{2} and (J, J_{z} = 1, ±1) for . The absorption and emission of these states are proportional to the modular square of their electricdipole elements,
with . Hence ψ_{2} can absorb and emit light polarized along the zaxis and can absorb and emit light circularly polarized in the xy plane. ψ_{1}, however, is dark, for it contains only spin triplets and is therefore dipoleforbidden.
The above selection rules in the hybrid perovskites is in stark contrast to those in πconjugated 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 , 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 gfactor, 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 Caxis, B = (0, 0, B), will split in energy, and the splitting in absorption and luminescence peaks of left and rightcircularly polarized light, would be ^{11,33,34}.
The magnetic field B will also mix ψ_{1} and ψ_{2}, 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.
In the Vogit configuration with B perpendicular to the Caxis, 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 (ψ_{1}, ψ_{5L}) as well as pair (ψ_{2}, ψ_{5T}) are coupled via the magnetic field, with
and their energies are and . 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.
Magneticfield 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 magneticfieldinduced 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,
where is a 2 × 2 density matrix spanned by ψ_{1} and ψ_{2}, 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 and spin relaxation , , for ^{5,6,11}. In the steady state, , the densities at ψ_{1} and ψ_{2} can be written as and , where and . The intensity change in PL is due to the change in ρ_{22},
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 systems^{24}. 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 ψ_{1} with the dipoleallowed ψ_{5L} and ψ_{2} with ψ_{5T}, which have different oscillator strengths and polarizations, and along the zaxis and and along the yaxis for ψ_{5T}, respectively. Using the Bloch equation, we express the PL change in the ψ_{1} and ψ_{5L} manifold as
where Δg_{⊥} = g_{e⊥} − g_{h⊥} and the PL change in the manifold of ψ_{2} and ψ_{5T} as
where Σg_{⊥} = g_{e⊥} + g_{h⊥}. The MFE in the ψ_{1} and ψ_{5L} manifold, ΔI_{2}, depends on the difference in the gfactors along the xaxis, the direction of the magnetic field, in a very similar fashion as ΔI_{1}.
The MFE in the ψ_{2} and ψ_{5T} manifold, ΔI_{3}, however, depends on the summation of the electron and hole gfactors along the xaxis. 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), ΔI_{3} is then completely determined by the exciton relaxation time τ, whereas ΔI_{1} and ΔI_{2} 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 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 shortrange 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,
where a_{0} is the effective Bohr radius of the exciton, a_{0} = ħ^{2}ε/e^{2}μ with being the effective mass of exciton.
A highintensity photoexcitation creates many free electronhole 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, s and ħω is the photon energy. These free electronhole pairs will screen the Coulomb interaction, which can be modeled by the DebyeHückel 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 groundstate 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 , 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 and the line shape in ΔI_{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 . 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 electronhole 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 hydrogenlike exciton wavefunctions. Such a perturbation must fail when the magnetic length becomes much smaller than the orbital radius of exciton, a_{0} = ħ^{2}ε/(e^{2}μ). In this highmagnetic 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) magneticfield regimes.
The Hamiltonian for the envelop function Φ(r_{e} − r_{h}) at K = 0 (K being the centerofmass 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 sets^{38}. In low magnetic fields, by using a_{0} (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 H_{0} = −∇^{2} + r^{2}
with are chosen to be another basis set. These basis sets, with correct characteristics of wave functions in the low and highfield 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 conductionband 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 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 and , according to the selection rule, are electricdipole 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 electronhole absorption, can completely determine both m_{e} and m_{h}. It is noted that the interband cyclotron resonance of free electronhole pairs can measure only μ, not individual m_{e} and m_{h}. Thus cyclotron resonance of excitons reveals more information.
Discussion
The hybrid inorganicorganic perovskites have shown great promise in photovoltaic and many other important applications because of their outstanding transport, optical and magnetooptical properties. To understand these properties, in this paper, we have constructed a reliable and accessible effectivemass model of the hybrid perovskites, which connects effective masses, Rashba splittings and anisotropic gfactors of conduction and valence bands. Using this effectivemass 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 effectivemass model is a foundation on which systematic models of electronphonon coupling, carrier mobility and other transport properties can be developed. Because of the concise expressions of SOC, Rashba effect and gfactor, this model also facilitates studies of spin relaxation^{39}, spin Hall effect^{40} 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 Caxis with an angle θ, we can define the new zaxis (denoted z′) along B and assume that the tilting is in the xz plane with the new xaxis 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 gfactor 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 angledependent term in the Hamiltonian can be calculated via the integral
where the Wigner 3j 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 _{2}F_{1}(α, β, γ; z is the Gauss’ hypergeometric function.
The matrix element of r^{s} between basis functions and , neglecting the normalization factor, is
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How to cite this article: Yu, Z. G. Effectivemass model and magnetooptical properties in hybrid perovskites. Sci. Rep. 6, 28576; doi: 10.1038/srep28576 (2016).
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Acknowledgements
I thank B. Hu, Y. Li, D. Sun and Z. V. Vardeny for useful discussions. This work was partly supported by the US Army Research Office under Contract No. W911NF1510117.
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Yu, Z. Effectivemass model and magnetooptical properties in hybrid perovskites. Sci Rep 6, 28576 (2016). https://doi.org/10.1038/srep28576
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DOI: https://doi.org/10.1038/srep28576
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