Massive Dirac-Pauli physics in lead-halide perovskites (2025)

Here, we first illustrate that the non-standard matter-field interaction in LHPs is actually the Pauli coupling with the (effective) atomic dipole polarizability in place of the anomalous moment, \(\mu ^{\prime}\). Further, to facilitate future theoretical and experimental studies, we outline a few effects that Dirac-Pauli physics can have on the properties of LHPs. We consider bound states of charge carriers to local defects, such as vacancies and interstitials, and also electron-hole binding. Finally, we discuss the physics of scattering in the presence of the Pauli term.

Pauli coupling in LHPs

The Hilbert space for the low-energy description of LHPs is determined by the hybridization of orbitals of lead and halide atoms²³. In the vicinity of the chemical potential, the states have predominantly s- (valence band) and p-type (conduction band) characters^19,20. Strong spin-orbit coupling splits the p-type states. Therefore, both the top of the valence band and the bottom of the conduction band have J = 1/2, and low-energy optoelectronic properties of LHPs are shaped by four states that can be conveniently written as {⇑↑, ⇓↑, ⇑↓, ⇓↓}, where

The right-hand-side of these expressions follows the standard notation^24,25 for the spin structure of the state [\(|\downarrow\!\!\left.\right\rangle\), \(|\uparrow\!\!\left.\right\rangle\)] and for its spatial component [\(| s\left.\right\rangle\), \(| {p}_{x}\left.\right\rangle\), \(| {p}_{y}\left.\right\rangle\), \(| {p}_{z}\left.\right\rangle\)]. The Hermitian operators that act in the Hilbert space of Eq. (2) can be readily written as the direct product τ_i ⊗ σ_j, where τ_i and σ_j are the Pauli matrices that operate correspondingly in the orbital ({⇑, ⇓}) and the quasi-spin ({↑, ↓}) subspaces defined via the left-hand-side of the identities in Eq. (2).

In the absence of external fields, the low-energy spectrum of LHPs is determined by diagonalizing the Hamiltonian (ℏ = c = 1)¹⁹:

where σ is a vector of Pauli matrices so that σ ⋅ k = ∑_iσ_ik_i. \(\Delta ({\boldsymbol{k}})=\frac{1}{2}(\Delta +\frac{{t}_{3}{a}^{2}| {\boldsymbol{k}}{| }^{2}}{2})\), k is the quasi-momentum counted from the high-symmetry R-point relevant for the low-energy physics of LHPs, Δ is the bandgap at the R-point, a is the lattice constant, t₃ and t are further k ⋅ p parameters of the Hamiltonian. They can be estimated by fitting to numerical calculations or experimental data. For example, for methylammonium lead bromide (MAPbBr₃), these parameters are found to be: t ≃ 0.6eV, t₃ ≃ 0.9eV, a ≃ 0.586nm, Δ ≃ 2.3eV^20,22.

It was recently shown both experimentally and theoretically that in order to account for the full effect of the external electric field E on LHPs, one should employ not only the minimal coupling \(\widetilde{{\boldsymbol{k}}}={\boldsymbol{k}}-q{\boldsymbol{A}}\) (where q is the charge of an electron) in Eq. (3) but also an additional term μτ₁ ⊗ σ ⋅ E, dubbed “spin-electric” in refs. ^21,22 because it couples the quasi-spin degree of freedom to the external electric field. The non-minimal coupling term accounts for the local effects of the electric field on the lattice, in particular, on the highly polarizable Pb²⁺. Therefore, we interpret μ as coming from the effective atomic polarizability of lead ions. For MAPbBr₃, the experiment demonstrates that the magnitude of this effect corresponds to μ ≃ −0.3∣q∣a²².

A close examination reveals that the operator μτ₁ ⊗ σ ⋅ E is nothing else but the electric part of the Pauli term in Eq. (1). Indeed, the Hamiltonian in Eq. (3) amended with the spin-electric term can be written as:

where \(\beta ={\tau }_{3}\otimes {\mathbb{I}}\), α = τ₂ ⊗ σ, and γ = βα. As the matrices γ^ν = {β, γ} satisfy the Dirac algebra {γ^μ, γ^ρ} = diag(2, − 2, − 2, − 2), the first two terms in Eq. (4) lead to the Dirac equation in Eq. (1) with \(\mu ^{\prime} =0\) and t₃ = 0. [The “standard (Dirac) representation” of the gamma matrices⁹ is obtained by the unitary transformation U^†αU where \(U={e}^{-i{\tau }_{3}\pi /4}\), which corresponds to a π/2-rotation in the τ-subspace.] Note that t₃ ≠ 0 is beyond the Dirac equation, as is common for gapped Dirac materials²⁶. We note that while this term does not break the algebraic structure of the Dirac equation, its presence leads to measurable consequences, e.g., in the determination of the energy of a bound state in an impurity potential (see below).

Finally, in the absence of magnetic fields [γ^μ, γ^ν]F^μν = 4γ ⋅ E, establishing an equivalence between Eqs. (1) and (4) with \(\mu ^{\prime} =\mu\). Note that the magnetic part of the Pauli term [γ^μ, γ^ν]F^μν leads to an anomalous τ₃ ⊗ σ ⋅ B coupling, which also enters the low-energy description of LHPs placed in the magnetic field B^21,22. However, the precise magnitude of this magnetic term in MAPbBr₃ is currently undetermined, and therefore, we leave the corresponding investigation to future studies.

In what follows, we illustrate the physical implications of the Pauli term for LHPs. To this end, we solve the stationary Dirac-Pauli equation

where Ψ is the 4-component Dirac spinor, \({\Psi }^{T}=({\psi }_{{\mathcal{\Uparrow }}\uparrow },{\psi }_{{\mathcal{\Uparrow }}\downarrow },{\psi }_{{\mathcal{\Downarrow }}\uparrow },{\psi }_{{\mathcal{\Downarrow }}\downarrow })\), and \({\mathcal{E}}\) is the energy. [Here, the first (last) two elements of the spinor are typically referred to as electron (hole) components.] Specifically, we solve Eq. (5) with the Dirac-Pauli Hamiltonian for two standard physical problems: bound states and scattering. Specifically, we discuss the binding of a charge carrier by an attractive potential and scattering of free charge carriers off a potential step (Klein paradox; see Fig. 2a). To guide experimental studies of the Pauli term, we further discuss spin effects in scattering and formation of excitons. The coupling between spin and the electric field is a hallmark feature of the Pauli coupling, providing a clear pathway for its detection in a laboratory.

Bound states near static impurities in LHPs

In this subsection, we investigate how the bound states of an electron in the vicinity of a spherically symmetric local impurity potential, ϕ, are modified by the Pauli term. To this end, we focus on the effective behavior of electrons by integrating out the hole degrees of freedom following the procedure of ref. ²⁷ (see Supplementary Note 1). This approach resembles the Foldy-Wouthuysen transformation in QED²⁸, which is a perturbative study with the expansion parameter ϵ/Δ (ϵ is the energy of the bound state). A perturbative approach to bound states is natural in LHPs due to its “defect tolerance”, i.e., only shallow states are observed in the badgap²⁹.

By doing so, we frame the problem in the form of the Schrödinger equation for electrons that, in the leading (in the powers of 1/Δ) order reads

where \({\psi }^{T}=({\psi }_{{\mathcal{\Uparrow }}\uparrow },{\psi }_{{\mathcal{\Uparrow }}\downarrow })\) is the electronic wavefunction; \(\epsilon ={\mathcal{E}}-\Delta /2\) is the corresponding energy; \({m}_{e}^{* }={(8{a}^{2}{t}^{2}/\Delta +{t}_{3}{a}^{2}/2)}^{-1}\) is the effective mass, note that the parameter t₃ enters only in the effective mass, otherwise its presence does not affect our results. The terms V₀ and H_P originate from the coupling between the conduction and valence bands. \({V}_{0}=-\frac{{q}^{2}\phi {(r)}^{2}}{\Delta }+\frac{2\epsilon }{\Delta }q\phi (r)\) is the beyond-Coulomb potential energy of an electron in the electric field generated by the impurity for μ = 0²⁷; H_P is the sum of terms arising from Pauli coupling:

The first term in H_P mimics the effect of the polarizability of quasi-electrons. This term is non-vanishing even if t = 0, implying that this polarizability of electrons inside lead-halide perovskites can be traced back to the atomic polarizability of the underlying lattice ions (primarily Pb²⁺), see also the “Discussion” section below. The ∇ ⋅ E(r) term is referred to as the Darwin term; in QED, this term is interpreted as coming from the jitter motion (Zitterbewegung) of a relativistic electron in the presence of a non-uniform electric field²⁸. The last term in H_P describes the spin-orbit coupling (SOC).

If μ ≠ 0, SOC appears as \(\sim \mu | {\boldsymbol{E}}({\boldsymbol{r}})| {\boldsymbol{S}}\cdot {\boldsymbol{L}}/(r\sqrt{{m}_{e}^{* }\Delta })\) in the effective Schrödinger equation (6) of the Dirac-Pauli system. Hence, its influence on the states and energies is different when compared to a Dirac system of an electron in a given electric field. For example, in a hydrogen atom SOC has the paradigmatic form ~q∣E(r)∣S ⋅ L/(rm_eΔ)²⁸. We see that the above-presented terms have identical functional dependence. However, their strengths are different with the ratio \(\mu \sqrt{{m}_{e}^{* }\Delta }\). A similar conclusion can also be reached for the Darwin-like term.

To illustrate the effect of H_P on the properties of bound states, we investigate the ground state of an electron interacting with a model Gaussian defect whose charge density is chosen as

where Q is the charge of the defect. The width of the Gaussian is determined by the lattice constant, a, to model a single-site defect of a semiconductor. We consider a defect with Q > 0, which binds electrons. However, the hole-electron symmetry allows us to also easily understand what happens for Q < 0 from our calculations.

The charge density produces a radial electric field given by

where ε is the (dynamic) dielectric constant of the medium (we take ε = 5 in our numerical calculations³⁰) and \({\rm{erf}}\) is the standard error function. Once the electric field is known, we solve the Schrödinger equation (6) numerically by diagonalizing the corresponding matrix (see Methods). Specifically, we calculate the ground state energy of the system with and without H_P, and study the difference introduced by H_P.

The ground state of Eq. (6) has L = 0, hence, the spin-orbit coupling term does not play a role in Eq. (7) and can be ignored. In other words, the effects of SOC can only be seen in the states with L ≠ 0. A study of the ground state allows us to demonstrate the effect of the first two terms in H_P: the polarizability and the Darwin terms. Note that in LHPs, the parameter μ is negative, which implies that the matrix H_P is positive-definite, and hence, that the ground state energy is increased by the presence of the Pauli coupling.

We show our findings in Fig. 1. First, we plot the electronic density in Fig. 1a. We see that the wavefunction becomes spatially broader as the Pauli term repels the electron. This change in the electronic density affects other observables as well, for example, the binding energy ϵ_B > 0 (ϵ_B = −ϵ) of the electron-defect system. Indeed, Fig. 1b confirms that the Pauli term leads to a noticeable reduction of ϵ_B (see also Supplementary Fig. 1). In the figure, we plot the relative decrease in binding energy:

We conclude that the Pauli coupling in lead-halide perovskite can have a measurable effect on the energies of electrons in the vicinity of defects. This effect scales with the parameter μ, which depends on the chemical structure of the material. As we expect that μ can be affected by one’s choice of the halogen atom, further studies are needed to determine the lead-halide perovskite with the most pronounced effect of Pauli coupling for tabletop simulations of the corresponding physics.

Spin structure of excitons in LHPs

LHPs are known for their anomalously fast photon emission at cryogenic as well as at room temperatures^20,31. It has been suggested that this is because excitons in LHPs have a unique fine structure with an optically active (“bright”) ground state. Such level ordering can be obtained by introducing a Rashba-type spin-orbit interaction^20,32. In general, there is a consensus that Rashba coupling is featured prominently in LHPs, however it is usually introduced phenomenologically to account for experimental observations. The microscopic origin of Rashba coupling is a subject of debate, especially at high temperatures where LHPs have a nominally cubic structure^33,34,35.

With the Pauli term, Rashba coupling follows naturally, with a coefficient determined by the parameters of the perovskite structure (cf. ref. ²²):

where E is the electric field explicitly breaking inversion symmetry. The Rashba energy that corresponds to H_Rashba is

According to ref. ²⁰, the lowering of the “bright” state is given by δE_X ≃ − E_Rashba. To compensate for the standard singlet-triplet splitting (≃0.3 meV²⁰) due to the electron-hole exchange interaction³⁶, one should have δE_X ≃ 1 meV. Under the hypothesis of local polar fluctuations³⁷, this implies the fields ∣E∣ ~ 0.6 V/nm.

While internal fields of this strength are reasonable on the atomic scales³⁸, they are quite considerable and are yet to be experimentally confirmed. This requirement for the strength of these fields is high because of the factor 1/Δ² in Eq. (12), which follows from the use of second-order perturbation theory. Surprisingly, it turns out that the Pauli term can lower the optically active exciton state at a lower order of perturbation theory, thus producing the desired level splitting with lower electric fields (∣E∣ ~ 0.3 V/nm). Indeed, the effective polarizability of a charge carrier due to Pauli coupling implies a van-der-Waals-like lowering of the energy by

which enters in the first order in 1/Δ, in contrast to the result in Eq. (12).

We conclude that the Pauli coupling may play a role in the observed fast photon emission in LHPs. Note that all derivations in this work apply to a cubic phase of LHPs. However, the polarizability of lead ions is a local property of the material, thus, we expect that our discussion applies also at lower temperatures, where LHPs crystals are tetragonal or orthorhombic.

Klein problem in LHPs

Klein paradox refers to a counterintuitive behavior of scattering amplitudes off a step potential whose height is larger than twice the particle’s rest mass. In the first quantization picture, the paradox manifests itself in the seemingly unphysical values for the reflection and transmission coefficients. In particular, Klein’s solution suggests that the transmitted current does not vanish even when the height of the barrier approaches infinity^39,40 (for more details, see refs. ^41,42,43).

Direct experimental observation of this physics in quantum electrodynamics would require access to extremely high electric fields not available in modern laboratories (the exception is electron beams in aligned crystals, which, however, can only be used for indirect studies of Klein paradox)⁴⁴. This motivates searches for tabletop set-ups that would realize Dirac’s equation, such as gapless Dirac materials^45,46 or photonic superlattices⁴⁷. The most direct analogy to the Klein paradox in its original formulation, however, requires working with gapped Dirac semiconductors such as LHPs. The corresponding bandgap Δ then would play the role of a pair-creation energy 2mc² in QED, where m is the mass of the particle that is being created. As the gap Δ can be about six orders of magnitude lower than 2mc², the corresponding effects in Dirac semiconductors are much more accessible as compared to QED. Nevertheless, it should be noted that the very existence of an energy gap of a few eV may already result in a considerable experimental obstacle for the observation of the Klein paradox in LHPs. Still, we find it fitting to start the discussion of the effect of the Pauli term by considering the corresponding solution of the Dirac equation.

For simplicity, we assume that t₃ = 0 here (otherwise, we should explicitly consider other electronic bands whose presence leads to t₃ ≠ 0). Similar to the original formulation of the problem by Klein, we consider the one-dimensional potential

where Θ(x) is the Heaviside step function, and V₀ is the strength of the potential. For μ ≠ 0, the associated electric field E_x = −∂ϕ/∂x = −V₀δ(x) enters the Dirac equation via the Pauli term. This leads to a term with a δ-function in Eq. (4), giving rise to a discontinuity in the spinor components at x = 0, which can be treated as a specific boundary condition, see Supplementary Note 2 and ref. ⁴⁸.

We solve the Dirac-Pauli equation for each spinor component and arrive at the following expressions for the reflection and transmission coefficients (see Supplementary Note 2):

where \(\lambda =4iatk/(\Delta +2{\mathcal{E}})\) and \({{\Lambda }}\,=\,4iatK/(\Delta +2{\mathcal{E}}-2q{V}_{0})\), \(r=\lambda {e}^{-\frac{\mu {V}_{0}}{at}}/{{\Lambda }}\) and k and K are the momenta of the charge carrier for x < 0 and x > 0. For μ → 0, these expressions turn into the original Klein solution⁴³. Non-vanishing values of μ lead to an exponential dependence on V₀ in Eq. (15). Subject to the sign of μ, the exponent \({e}^{-\frac{\mu {V}_{0}}{at}}\) that appears in the parameter r is either increasing or decreasing, leaving marked imprints on the transmission and reflection coefficients.

When μ < 0, as in LHPs, the Klein paradox is modified in comparison to the case when there is no Pauli coupling. Curiously, the introduction of the Pauli term suppresses the anomalous (R > 1) Klein reflectivity in the asymptotic limit of large barrier heights, qV₀ → ∞. Figure 2b illustrates this for the parameters of MAPbBr₃. The behavior of the reflection coefficient for μ > 0 is even more intriguing as it features a resonance-like divergent behavior, which can be interpreted as a resonant enhancement of particle–antiparticle creation within the potential barrier. This might be indicative of a novel vacuum breakdown channel that has previously been overlooked (see Supplementary Note 3 for additional details). We conclude that for either sign of μ, Pauli coupling strongly affects Klein’s physics.

Spin effects in scattering

Without doubt, modifications to Klein’s physics is of general interest, however, their observation appears challenging due to the energy gap in LHPs. As we discuss below, the effects of Pauli coupling on scattering are more readily studied through the spin degree of freedom. To illustrate this, let us assume that a sample of LHPs has an energy barrier described by the potential ϕ(x) = v₀f(x), where v₀ (f) is the strength (shape) of the barrier. By assumption, the function f(x) has a finite support parametrized by d, i.e., f(x) = 0 if ∣x∣ > d/2. One realization of the potential ϕ(x) is an n-p-n junction⁴⁹. Unlike the Klein scenario, the potential barrier here is weak, v₀ ≪ Δ, allowing us to implement Foldy-Wouthuysen-like transformation and derive the equation for electron propagation, cf. Eq. (6):

where we chose the z-axis perpendicular to the scattering plane; W(x) is the spin-independent part of the potential (see Supplementary Note 1); k_y is the quantum number that parametrizes the momentum along the y-direction.

The Pauli coupling in Eq. (16) turns the energy barrier into a magnetic impurity, which affects spin-up and spin-down electrons differently, allowing one to manipulate the spin of an electron. For example, given a spin-unpolarized current for x < 0, one can generate a spin-polarized current for x > d. Indeed, let us consider the transmission coefficient (see, e.g., ref. ⁵⁰)

Here, ± corresponds to the spin of the incoming electron; \({W}_{\pm }=W\pm 4at{v}_{0}\mu f^{\prime} {k}_{y}/\Delta\). The function ψ₁ is the solution to Eq. (16) that satisfies the integral equation

where the momentum k_x parametrizes the incoming flux. As W₊ ≠ W₋, we conclude that T₊ ≠ T₋, which leads to the spin polarization of the outgoing current: ∣T₊ − T₋∣/(T₊ + T₋). To measure it in LHPs, one can use the standard methodology established for other semiconductors⁵¹.

The simplest estimate for T_± is obtained in the quantum tunneling regime using the WKB approximation, \({T}_{\pm }^{{\rm{WKB}}} \sim \exp (-2\mathop{\int}\nolimits_{{x}_{1}}^{{x}_{2}}\sqrt{2{m}_{e}^{* }({W}_{\pm }-\epsilon )})\), where x₁ and x₂ are classical turning points. This expression allows one to optimize f(x) for the problem at hand. It is worth noting that \({T}_{+}^{{\rm{WKB}}}\ne {T}_{-}^{{\rm{WKB}}}\) only for potentials that break mirror symmetry, i.e., if f(− x) ≠ f(x). This is in line with the recent observation of spin-polarized currents in chiral LHPs at room temperature in the absence of external magnetic fields⁵². Our results demonstrate that similar observations can be made in non-chiral LHPs in the presence of an energy barrier with broken mirror symmetry.