Spin-Electric Coupling in Lead Halide Perovskites

Lead halide perovskites enjoy a number of remarkable optoelectronic properties. To explain their origin, it is necessary to study how electromagnetic fields interact with these systems. We address this problem here by studying two classical quantities: Faraday rotation and the complex refractive index in a paradigmatic perovskite ${\mathrm{CH}}_3{\mathrm{NH}}_3{\mathrm{PbBr}}_3$ in a broad wavelength range. We find that the minimal coupling of electromagnetic fields to the $\mathbf{k}·\mathbf{p}$ Hamiltonian is insufficient to describe the observed data even on the qualitative level. To amend this, we demonstrate that there exists a relevant atomic-level coupling between electromagnetic fields and the spin degree of freedom. This spin-electric coupling allows for quantitative description of a number of previous as well as present experimental data. In particular, we use it here to show that the Faraday effect in lead halide perovskites is dominated by the Zeeman splitting of the energy levels and has a substantial beyond-Becquerel contribution. Finally, we present general symmetry-based phenomenological arguments that in the low-energy limit our effective model includes all basis coupling terms to the electromagnetic field in the linear order.

Figure 1

(a) Faraday rotation ${\mathrm{\Theta }}_F$ as a function of magnetic field $B$ for two wavelengths. (b) Normalized Verdet constant $\tilde{V}(\omega )$ extracted from the data in (a) as a function of the photon energy of the incident beam (blue circles); red curve, fit according to Eq. (1) (see text); purple crosses, high-frequency data of Ref. [15] shown here for comparison; gray curve, normalized Verdet constant $\tilde{V}(\omega )$ computed from the Becquerel formula $\gamma (e/2mc)\omega \partial n/\partial \omega$, with $n(\omega )$ extracted from the data in Fig. 3. Note that the fit according to Eq. (1) differs from the Becquerel curve only by a constant shift (red dashed curve) at low frequencies.

Figure 2

Basic elements of the effective Hamiltonian. Left: conduction band made of spin-orbit coupled $p$-type orbitals ($J=1/2$) and valence band made of $s$-type orbitals ($J=1/2$). Zeeman interaction projected on this basis acts only on the valence band. Middle: intersite hopping between $s$ and $p$ orbitals; $t$ and $t_3$ correspond to the inter- and intraorbital hoppings. respectively. Right: On-site electric-field-induced hybridization between $s$ and $p$ orbitals. See Eq. (2) for the definition of the basis states.

Figure 3

(a) Reflectivity ($p$ polarization) as a function of the angle of incidence for two different wavelengths with Fresnel fits to extract the complex refractive index. Note that the reflectivity has a nonzero minimum value for $\hslash \omega >\mathrm{\Delta }$, implying nonzero imaginary part of $n(\omega )$. Inset: Brewster angle measurement. (b) Blue, experimental values of $\mathrm{Re}(n)$ (dots) plotted together with a Sellmeyer fit (solid curve); red, experimental values of $\mathrm{Im}(\alpha )$ (dots) extracted from the imaginary part of the refractive index plotted together with our fit (solid curve). The purple curve is a fit based on our effective model (without excitons). The gray curve shows the excitonic peak. Note that both $t$ and $\mu$ terms are necessary for a faithful fit, see the Supplemental Material [18].

Figure 4

(a) Rashba splitting that follows from the interplay of the hopping and spin-electric terms. (b) Axion-type ($\overrightarrow{E}·\overrightarrow{B}$) term as a result of the spin-electric and regular Zeeman terms. (c) Photoinduced momentum shift under circularly polarized irradiation as a result of the spin-electric and orbital-selective Zeeman terms.