Resonant free-carrier absorption in 2D hybrid organic-inorganic perovskites: The Rashba effect or small polarons?

The unprecedented solar-cell performance attained in hybrid organic-inorganic perovskites (HOIPs)^1–3 originates from their fortuitous confluence of several desirable optoelectronic properties. Understanding these properties requires detailed knowledge of HOIPs’ electronic structures and microscopic interactions involving carriers, such as electron-electron and electron-phonon interactions. Because of the concurrent strong spin-orbit coupling (SOC)^4,5 of their heavy elements and lack of inversion symmetry in their crystalline structures,^6–8 HOIPs can exhibit the Rashba effect^9,10 in the conduction and valence bands. The conduction band of HOIPs is characterized by the total angular momentum j_e = l + s (l = 1 for Pb’s 6p orbitals), (j_e, j_ez) = (1/2, ±1/2), and its Hamiltonian, with the Rashba effect included, reads

where k_q (q = x, y, z) is the q-component of wavevector k, m^* is the effective mass, α_c is the Rashba strength of the conduction band, and $σ$ are the Pauli matrices of the 2 × 2 pseudospin space spanned by j_ez = ±1/2. The Rashba effect destroys the pseudospin degeneracy and transforms the conduction band into upper and lower subbands. The energy minimum of the lower subband occurs at a ring in the k-space, $kx2+ky2=k0=αcm*/ℏ2$⁠, and is lower than the energy at |k| = 0 by $ER=ℏ2k02/(2m*)$⁠. The Rashba effect and associated unusual band dispersions have been cited to rationalize the outstanding optoelectronic properties of HOIPs, including slow electron-hole recombination and its temperature dependence,^11–13 moderate carrier mobilities,¹⁴ and bright exciton luminescence in perovskite nanocrystals (NCs).¹⁵ These conjectures are being scrutinized experimentally.^16–18 

It should be noted that breaking of (pseudo) spin degeneracy is fairly common in non-centrosymmetric semiconductors, such as zinc blende and wurtzite structures, where the k-dependent SOC in a given band is generally written as H_so = $σ$ · g(k), with g(−k) = −g(k).¹⁹ g(k) in the vicinity of k = 0 can be expanded in terms of odd powers of k, and the Rashba (Dresselhaus⁹) effect emerges when the first nonzero term is k (k³). The prototypical direct-gap semiconductors GaAs, GaN, and CdSe all have spin splitting at nonzero k due to either the Rashba effect (∼α_ck) or the Dresselhaus effect (∼α₃k³ and the band extremum still locates at k = 0). However, the spin splittings in those semiconductors (α₃ = 27.58 eV ų for GaAs,²⁰ α_c = 0.009 eV Å for GaN,²¹ and 0.06 eV Å for CdSe²²) are too small to influence optoelectronic properties. In order for the Rashba effect to alter optoelectronic and transport behaviors, the carrier distribution should deviate significantly from that of a (pseudo) spin-degenerate band. This requires that the energy gain E_R is larger than or at least comparable to the thermal energy k_BT, with k_B and T being the Boltzmann constant and temperature. If, on the other hand, E_R ≪ k_BT as in GaN and CdSe (E_R < 10⁻⁴ eV), the material’s optical and carrier transport properties would differ little from those of a direct-gap semiconductor. Thus, in discussing the Rashba effect in HOIPs, it would be more productive to give its strength range than to simply debate its presence or absence.

Despite its critical importance, experimental quantifications of the Rashba effect in HOIPs are scarce due largely to the challenging angle-resolved photoelectron spectroscopy (ARPES), a common experimental tool for measuring band structures.²³ Even the powerful ARPES has its limitations, as the signal from bulk interior may be masked by that from the surface. Moreover, for HOIPs with a large band gap, only the valence-band dispersion can be easily accessed by ARPES, although the Rashba effect in the conduction band is expected to be much stronger than in the valence band. Because of the experimental constraints and the paucity of available data, estimation of α_c in HOIPs relies on density-functional-theory (DFT) computations, which, as shown in a recent study,²⁴ may be susceptible to numerical artifacts.

Since the Rashba effect, or the underlying inversion asymmetry, can manifest itself in second harmonic generation (SHG), electron-hole recombination, and spin relaxation, the Rashba effect may be assessed via these (non)linear optical properties and spin dynamics in HOIPs. Recently, the measured angle dependence of SHG signals was found incompatible with the strong Rashba effect predicted by DFT calculations.²⁴ According to latest studies of electron-hole recombination in 3D HOIPs, the observed temperature dependence^16,17 is more consistent with that of a direct-gap semiconductor than that of an indirect-gap one.^11–13 Based on the newly developed SHG²⁵ and spin relaxation²⁶ theories of HOIPs, as well as the measured values of SHG²⁷ and spin relaxation lifetimes,²⁸ we have estimated the Rashba strength in 3D HOIPs to be small, E_R < 10⁻³ eV,^25,26 i.e., E_R ≪ k_BT. Thus, 3D HOIPs are essentially direct-gap semiconductors as far as optoelectronic properties are concerned. For perovskites NCs, the strong Rashba effect was invoked to argue that the lowest exciton state is bright,¹⁵ which, however, was challenged by a recent magneto-photoluminescence study that directly shows that the lowest exciton in perovskite NCs is dark.¹⁸ Thus, a strong Rashba effect in perovskite NCs is also in doubt.

Compared to bulk HOIPs and NCs, 2D HOIPs, because of their layered structure and a possible strong electric field associated with asymmetric organic ligand arrangement, are believed to have a large Rashba effect. The long organic ligands in 2D HOIPs introduce a large distance between layers of PbX₆ (X = I or Br) octahedra and impede interlayer electron motion. Recently, free-carrier absorption (FCA) in 2D HOIPs was shown to contain a resonant peak, which has been interpreted as a direct optical transition from the lower to the upper Rashba subbands at |k| = k₀ with frequency ℏω = 4E_R.²⁹ This observation has since been substantiated by a systematic model.³⁰ In particular, a magnetic field is found to lead to oscillatory magnetic circular dichroism, which can completely determine the Rashba subbands’ dispersion.³⁰ From the observed resonance energy, the Rashba strength is estimated as α = 1.45 eV Å if the effective mass m^* = 0.29m³¹ is used.

While the Rashba effect is a plausible explanation of the observed resonant FCA in 2D HOIPs, the Rashba strength estimated from a recent time-resolved circular-dichroism experiment, α_c = 0.08 eV Å,³² is considerably weaker than that from Ref. 29. More important, 2D HOIPs also exhibit another intriguing optoelectronic property, broadband white luminescence,³³ which is not seen in their 3D counterpart. The broadband luminescence stems from self-trapped excitons (STEs),^33–35 namely, excitons trapped by a highly distorted lattice, as revealed by experimental and theoretical studies.^36–40 In particular, it is suggested that the STEs in 2D HOIPs are induced by a short-range optical deformation potential (ODP) associated with fluctuations in the thickness of Pb-X layers.⁴⁰ Besides excitons, this ODP can also trap carriers, forming small polarons localized in individual Pb-X layers. Formation of small polarons qualitatively alters carrier transport behavior and therefore would have a major impact in the optoelectronic performance of layered perovskites.

Here, we show that in 2D HOIPs, and more generally, in layered perovskites,⁴¹ the small polarons would give rise to a deceivingly similar FCA resonance as that from the Rashba effect, at ℏω = 4E_b, where E_b is the binding energy of the small polaron. This resonant FCA originates from the Franck-Condon optical transition between the potential energy surfaces (PESs) belong to adjacent layers. The similarity between the FCA spectra in the two different scenarios indicates that the resonant FCA alone is not a sufficient proof of the Rashba effect. Fortunately, the two scenarios can be readily distinguished via an electric field modulation of the resonant FCA. Hence, electric-field-modulated FCA would help prove the Rashba effect and/or small polarons, both of which can significantly influence optoelectronic properties in layered perovskites. This paper is organized as follows: in Sec. II, we will apply the Kubo formula to the Rashba effect and small polarons. Then, in Sec. III, we will discuss the electric-field effects on their resonant FCA spectra. Finally, we conclude this paper in Sec. IV by enumerating similarities and differences between the Rashba effect and small polarons and their implications.

FCA describes carrier optical transition within the same band and is the dominant infrared absorption process in doped semiconductors.⁴² It is a convenient optical tool to probe carrier properties in a given band. For a typical parabolic band well separated from other bands, direct optical transition within the band is not possible and FCA must involve phonons to conserve both energy and momentum, resulting in a power-law frequency dependence, ω⁻².⁴² As we will see later, the unusual electronic structures arising from the Rashba effect and from small polarons make phonon-less intraband optical transition possible. For the optical transition, we consider only the electric-dipole transition with the electron-photon interaction being

where A is the vector potential of the electromagnetic wave, e is the electron charge, c is the speed of light in vacuum, and $v$ is the velocity operator, which is related to the current operator j and momentum operator p via $v$ = j/em = p/m.

To facilitate a direct and systematic comparison, we apply the Kubo formula to both the Rashba effect and small polarons and juxtapose the results for the two scenarios. The FCA spectrum is proportional to the real part of the frequency-dependent conductivity, which, according to the Kubo formula, can be expressed in terms of the current-current, or equivalently, velocity-velocity correlation function.⁴³ Thus, the optical absorption spectrum is

where β = 1/k_BT, ⟨⋯⟩ represents the ensemble average, $μ$⁠, $ν$ = x, y, z are Cartesian components, $V$ is the volume, and n is the refractive index.

In the absence of the Rashba effect, the velocity operator v is proportional to the carrier’s momentum, $v$ = ℏk/m^*, and conserves (pseudo)spin. Consequently, the optical transition is possible only between states with the same spin. However, in the presence of the Rashba effect, the velocity operator must be rederived from the Hamiltonian in Eq. (1),^6,44

where r is the location operator, $e±=12(ex±iey)$ (e_q is the unit vector along the q-axis) and $σ±=σx±iσy$⁠. Besides the usual spin-conserving term ℏk/2m^*, the velocity operator contains a term involving $σ±$⁠, i.e., spin-flip processes, stemming from the coupling between the momentum and the pseudospin operator in Eq. (1). This spin-flip term is the origin of the electric-dipole spin resonance pointed out by Rashba,⁶ as opposed to the conventional magnetic-dipole induced electron spin resonance.

To use the Kubo formula, we write Hamiltonian (1) in terms of creation and annihilation operators,

where ↑ (↓) represents pseudospin j_ez = +(−)1/2, k_± = ke^±iϕ, and $k≡|k|=kx2+ky2$⁠. The off-diagonal elements can be eliminated by introducing new operators c_k+(−),

and the Hamiltonian becomes

with energy dispersions

as plotted in Fig. 1(a). Thus, $ck+(−)†$ creates an electron with wavevector k in the upper (lower) conduction band. The velocity operator can be expressed in terms of the upper- and lower-band operators as

For light polarized along the μ direction, the FCA spectrum is

where $fk+(−)≡⟨ck+(−)†ck+(−)⟩$ is the carrier distribution in the upper (lower) subband, which, for a low carrier density, follows the Maxwell distribution,

Here, $N$ is the carrier density, and $I=∫0∞xdxe−u(x−1)2/π/u=1+O(u−1)$⁠, with u ≡ E_R/k_BT. The new band dispersions of Eq. (9) alter the distribution of free electrons from the Maxwell distribution for a parabolic band $Ek0=ℏ2k2/(2m*)$⁠, $fk0=Nπβℏ2m*e−βℏ2k2/(2m*)$⁠, when E_R ≥ k_BT.

The integration over time in Eq. (12) gives energy conservation condition $ℏω=±(Ek+−Ek−)$⁠, which fixes k for a given $ω$⁠, and we have

Figure 1(b) shows the calculated FCA spectra from inter-subband transitions at different temperatures. We see a resonance in the FCA spectrum, whose position does not depend on temperature. The resonance can be traced to the carrier distribution concentrated at k = k₀, where the photon energy between the lower and upper subbands is $ℏω=Ek0+−Ek0−=4ER$⁠. At a finite temperature, the carrier distribution spreads around k₀, |k − k₀| ≤ $δ$k with ℏ²(⁠$δ$k)²/2m^* ∼ k_BT, where their transition energy deviates from 4E_R, as described by the dashed lines in Fig. 1(a), leading to a finite width of the resonance peak. At low temperatures with smaller thermal energies, the resonant peak becomes narrower, which is shown in Fig. 1(b). The width of the resonant peak, according to Eq. (14), is $8ERkBT$⁠, which decreases with temperature slowly, T^1/2. As shown in Fig. 1(c), the experimentally observed resonant FCA in Ref. 29 can be accounted for with E_R = 0.045 eV, which corresponds to α_c = 1.45 eV Å if m^* = 0.29m³¹ is used.

In 2D HOIPs, individual inorganic Pb-X layers are separated by long organic-ligand layers. One unique optoelectronic property in 2D HOIPs, which is not seen in 3D HOIPs, is their broadband white emission³³ due to prevalent STEs-localized excitons surrounded by a highly distorted lattice,^34,35 as indicated by recent experimental and theoretical studies.^36–40 

A Pb-X layer in 2D HOIPs can be regarded as a quantum well (QW) with its thickness determined by the distance between two apical X atoms in a PbX₆ octahedron.^45,46 In such a QW, the electron’s motion normal to the QW (z-axis) is confined, resulting in quantized energies for both conduction electrons and valence holes. The conduction-band electron energy is quantized with the ground-state energy being

where l_a is the QW width. Here, we have neglected the possible Rashba effect for clarity. The quantum confinement of electron motion depends on the composition and structure,⁴¹ which can be incorporated via a change in effective mass m^* and/or QW width l_a.

Now, consider the three-atom X-Pb-X chain in a PbX₆ octahedron normal to the Pb-X layer. Of the three eigenmodes of the X-Pb-X chain, one has the central Pb stationary and the two X atoms moving out-of-phase, which corresponds to the B_1g mode in 3D HOIPs,⁴⁷ and is called the B_g mode in the rest of the paper. Its frequency is $Ω0=K*/M$⁠, with K^* being the effective elastic constant of the Pb-X bond and M being the mass of the X atom. The B_g vibration will change the QW’s thickness from l_a to l_a + 2Q_i, where Q_i is the X-atom displacement from its equilibrium position, and the electron’s energy in the QW becomes^40,48

Hence, there exists a strong coupling between electrons and the B_g phonons,

where $ai†$ creates an electron in layer i and the coupling strength A can be estimated by

Since the ODP is caused by the lattice deformation,⁴⁹ it is of short-range as opposed to the long-range polar (Fröhlich) coupling, with the latter responsible for the formation of large polarons.⁵⁰ Using typical values of m^* and l_a, l_a = 6 Å, m^* = 0.2 − 0.4m,^27,31 we estimate the ODP strength A ∼ 0.5–1 eV/Å.

The Hamiltonian of carriers, with the ODP and the lattice energy included, can be written as

where Ω₀ is the frequency of the B_g mode. The second term describes electron hopping between adjacent layers, denoted by ⟨ij⟩. To simplify our discussions without losing any essential physics, we consider two adjacent layers.⁵¹ By separating the average and relative lattice distortions $Q¯=(Q1+Q2)/2$ and Q = Q₁ − Q₂, and noticing that only the latter affects the electron motion, the small-polaron Hamiltonian becomes

If we temporarily neglect the hopping term, by minimizing the potential energy with respect to Q, ⟨∂H_P/∂Q⟩ = 0, the potential energy minimum would occur at

which is $Q=−Q0≡−A/MΩ02$ when the electron is in layer 1 and Q = Q₀ when the electron is in layer 2. The potential energy surface (PES) associated with a polaron at layer 1 or 2 would be two parabola centered at Q = ±Q₀,

where $Eb=A2/(4MΩ02)$ is the polaron binding energy, as shown in Fig. 2(a).

The hopping term would couple the polaron states in adjacent layers and facilitate polaron’s transition across layers. We follow Holstein’s classical work and solve the small-polaron Hamiltonian via the canonical transformation,^43,52

The purpose of this canonical transformation is to eliminate the strong electron-lattice coupling term, which cannot be adequately treated by perturbation. If we express the lattice distortion Q by the creation and destruction operators of the B_g phonon,

the lattice energy in Eq. (20), $M4(Q̇2+Ω02Q2)$⁠, reduces to ℏ$Ω0$b^†b. The suitable S is found to be

which results in an Hamiltonian

in which

Through the canonical transformation, electrons are dressed by a phonon cloud

with j = 1, 2, and the lattice vibration shifts its equilibrium away from zero,

The velocity operator, again, can be obtained from the location operator, defined as $r=∑jjdaj†aj$⁠, with d being the interlayer distance,

whose direction is perpendicular to Pb-X layers. The interlayer distance, d = l_a + l_b, with l_b being the wall thickness between adjacent Pb-X QWs, can be systematically tuned by using different ligands. For example, l_b = 7.5 Å in (C₄H₉NH₃)₂PbI₄, 10.2 Å in (C₆H₅(CH₂)₂NH₃)₂PbI₄, and 12.4 Å in (C₈H₁₇NH₃)₂PbI₄.⁵³ The velocity operator in Eq. (30) describes hopping of the phonon-dressed electron (small polaron) between adjacent Pb-X layers and gives rise to a hopping mobility,⁵¹ 

which is a thermal activation process and qualitatively different from band-like transport of large polarons⁵⁰ in 3D HOIPs. Hence, 2D HOIPs are expected to have distinct carrier transport properties, in addition to disparate luminescence features, from 3D HOIPs.

By using the aforementioned values of ODP strength A and typical Pb-X stretching frequencies ℏΩ₀ = 11–16 meV,⁴⁷ the binding energy E_b in layered perovskites is estimated to be 0.02–0.06 eV. This value range, which is consistent with a recent experimentally estimation for layered perovskites (∼35 meV),^38,39 is comparable to the thermal energy at room temperature, suggesting that carrier transport across Pb-X layers is possible. It is, however, much smaller than the value of E_b ∼ 0.5–1 eV predicted for several 3D HOIPs in Ref. 54. Such a large E_b, according to Eq. (31) with d replaced by the 3D HOIP’s lattice constant, would result in a negligible carrier mobility, which seems at odds with the band-conduction transport in 3D HOIPs as demonstrated from cyclotron resonance.⁵⁵ 

The optical absorption of small polarons was systematically studied by Reik and Heese⁵⁶ and by Bogomolov et al.⁵⁷ in 1960s, and we will apply their results to 2D HOIPs. According to the Kubo formula, the conductivity is

The time integral in Eq. (32) can be evaluated via the steepest descent method as shown in  Appendix A, and we have

where η and γ are a function of the phonon number n_p, $np=[eβℏΩ0−1]−1$⁠,

The resonance frequency and the peak width, according to Eq. (33), are

which are 4E_b and $8EbkBT$ when k_BT > ℏΩ₀/2.

Figure 2(b) shows the calculated FCA spectra from small polarons at different temperatures. We see that the resonant frequency shifts toward a higher energy with temperature and then stays at ℏω = 4E_b after ℏΩ₀/2 > k_BT. The broadening is caused by carrier distribution around Q = ±Q₀, $MΩ02(δQ)2/8∼kBT$⁠. With increasing temperature, the width becomes larger, δω ∼ T^1/2.

We see that both the Rashba effect and small polarons in 2D HOIPs can give rise to a resonant FCA, at ℏω = 4E_R in the former and at ℏω = 4E_b in the latter. Their widths have a similar temperature dependence ∼T^1/2. Thus, the observed resonant FCA can be well described by either of the two situations. The obtained small-polaron binding energy is also very reasonable. Hence, a resonant FCA is necessary but not sufficient evidence of the Rashba effect and/or small polarons. Is it possible to definitely distinguish these two scenarios?

While the band dispersions of the Rashba effect in Fig. 1(a) and the PESs of small polarons in Fig. 2(a) have similar double parabolic shapes, the Rashba subbands describe electrons within a PbX layer, whereas the PESs of small polarons describe polarons in different layers. Thus, an applied field normal to the layers would have different effects on the electronic structures.

The Rashba strength α_c for the conduction band, according to the effective-mass model recently developed for 3D HOIPs,⁵⁸ which was later extended to 2D HOIPs,³⁰ is related to the inversion asymmetry parameter ζ, ζ ≡ ⟨S|H_cr|Z⟩, with H_cr being the crystal-field potential,

Here, E_c and E_v are the conduction and valence band edges, respectively, P is the Kane parameter, and ξ is related to spin-orbit coupling λ and crystal-field splitting Δ_cr via $tan⁡2⁡ξ=22λ/(λ−3Δcr)$⁠. An applied electric field $E$ normal to the layer would alter the crystal-field, $Hcr→Hcr+eEz$⁠, and accordingly, the degree of the inversion asymmetry $ζ′=ζ+(dζ/dE)0E$⁠. Consequently, the effective Rashba strength would change

which, in turn, leads to a modified k₀ and E_R, denoted as $k0′$ and $ER′$⁠. Thus, we expect that the FCA resonance would move to ℏω = 4$ER′$ from 4E_R, as shown in Fig. 3(a). When the field direction is reversed, the resonance would move in the opposite direction, i.e., the resonant FCA in the Rashba effect depends on the sign of the electric field.

On the other hand, an applied electric field would result in an energy difference of $Δ≡eEd$ for polarons localized in adjacent layers. The small-polaron Hamiltonian becomes⁵¹ 

In the absence of hopping, the PESs of polarons in adjacent layers would be

While the minimum of $E1(2)′$(Q) still occurs at Q = −(+)Q₀, the energy differences between the two PESs at ∓Q₀ are no longer identical but become $4Eb∓eEd$⁠, as shown in Fig. 3(b). Accordingly, the resonant peak would split into two, located at $ℏω=4Eb∓eEd$⁠, which was first recognized by Austin.⁵⁹ When the electric field flips its direction, polaron PESs in Fig. 3(b) become their mirror image with respect to Q = 0, i.e., the right parabola is higher than the left parabola, the absorption, however, will not change. Applying the canonical transformation Eq. (25) to Hamiltonian (39), we obtain the FCA spectrum

where $ℏω±=ℏω±eEd$⁠. The extrema of the two terms in Eq. (41) happen at

which are $ℏω1(2)=4Eb−(+)eEd$ for k_BT > ℏΩ/2. The splitting between the two peaks are $2eEd$⁠.

Figures 4(a) and 4(b) show the calculated FCA spectra in the presence of an electric field. For the Rashba effect, the field shifts the resonance to a higher or lower energy depending on whether it increases or decreases the Rashba strength $αc$⁠, with the line shape largely intact. For small polarons, the field broadens FCA spectrum and two peaks would emerge, which becomes clearer at low temperatures, where the broadening in FCA resonance is smaller. The reduced amplitude of FCA at low temperatures is due to the quantum nature of phonons with occupation n_p ≪ 1, which makes the Franck-Condon principle less accurate.

The difference in the electric-field effect is more pronounced if we examine the change in FCA, $Δκ(ω)≡κ(ω;E)−κ(ω;0)$⁠, which has distinct line shapes in the two scenarios, as shown in Figs. 4(c) and 4(d). For the Rashba effect, Δ$κ$(ω) reaches zero at ℏω₀ = 4E_R, around which, Δ$κ$(ω) is an odd function of $ω−ω0$ and has a maximum (minimum) at $ℏω≃4ER−(+)4ERkBT$⁠. For small polarons, on the other hand, Δ$κ(ω)$ is approximately symmetric about $ω=ω0$⁠: Δ$κ$(ω) reaches the minimum at ℏω = 4E_b around which there are two maxima, separated by an energy of $2eEd$⁠, when it is larger than the broadening of FCA resonance. The distinct line shapes in Δ$κ$ allow an easy determination of the origin of resonant FCA, being the Rashba effect or small polarons.

The Rashba effect and small polarons would have profound implications to optoelectronic properties in layered perovskites. They both can lead to a resonant FCA with similar line shapes and temperature dependences. This similarity stems from the resembling double parabolic electron energies shown in Figs. 1(a) and 2(a): the band dispersion E −k for the Rashba effect and the PES E − Q for small polarons, where the energy minimum is located at k₀ ≠ 0 or ±Q₀ ≠ 0. For the Rashba effect, the optical transition is between the two Rashba subbands at the same k as required by momentum conservation. For small polarons, the transition is between two PESs at the same lattice distortion Q according to the Franck-Condon principle. The finite Q₀ in small polarons is closely related to the Jahn-Teller effect⁶⁰ commonly found in ionic compounds and organic complexes, where the electron’s orbital degeneracy would be destroyed by the electron-lattice coupling. In fact, small polarons and associated lattice distortions are referred to as the pseudo-Jahn-Teller or second-order Jahn-Teller effect.⁶¹ The resemblance of the resonant FCA spectra from the two scenarios indicates that the observed resonant FCA in 2D HOIPs is not a sufficient proof of the Rashba effect.

There also exist significant differences between the two cases. First, since the FCA resonance frequency for the Rashba effect is determined by the band dispersions, the resonant frequency is independent of temperature. For small polarons, however, the PES is the adiabatic energy and the quantum nature of lattice vibrations can influence the polaron’s binding energy. Consequently, the resonant frequency becomes independent of temperature only in the high-temperature regime, k_BT > ℏΩ₀/2. Second, the energy minimum at |k| = k₀ in the Rashba effect is in fact a ring in the k-space, within which electrons can freely move. Whereas in the small-polaron case, there exists an energy barrier of E_b between energy minima at Q₀ and −Q₀. Thus, distinct electrical transport behaviors are expected in the two cases. Third, for the Rashba effect, spin orientation in an eigenstate at a given k is always perpendicular to k, but in the small-polaron case, spin is not explicitly involved, which suggests different spin dynamics for the Rashba effect and for small polarons. Finally, the FCA would have a different polarization. For the Rashba effect, the optical selection rule is that polarization is in the layer plane, whereas the polarization in the small-polaron case is normal to the layer. This polarization difference would be particularly useful to discern the FCA origin in single-crystal 2D HOIPs.

The differences between the Rashba effect and small polarons can be exploited to distinguish the two scenarios. At low temperatures with k_BT < ℏΩ₀/2, the FCA resonance frequency is fixed in the former but shifts with temperature in the latter. A more revealing way is to use the electric-field modulation of FCA. The electric field would alter the Rashba strength and shifts the FCA peak but would destroy the energy degeneracy between polarons localized in adjacent Pb-X layers, resulting in the two resonance peaks. The resulted FCA change for the Rashba effect is antisymmetric around ℏω = 4E_R but is approximately symmetric around ℏω = 4E_b. Such disparate electric-field modulations allow a definitive proof and quantification of the Rashba effect and/or small polarons, both of which have great implications in transport, optical, and optoelectronic properties in layered perovskites.

This work was partly supported by the U.S. Army Research Office under Contract No. W911NF-17-1-0511.

Here, we provide details on how the time integration in Eq. (33) was evaluated. Since an electron localized at site 1 or 2 has the same energy, $⟨a1†(t)a2(t)a2†a1⟩=⟨a2†(t)a1(t)a1†a2⟩=1$⁠. By using the fact $X1=X2†$⁠, the time integration in Eq. (33) becomes

where $ρ=(A/ℏΩ0)(ℏ/MΩ0)1/2$⁠. To evaluate the exponential of noncommuting phonon operators, we employ the Baker-Hausdorff theorem,

provided that [O₁, O₂] commutes with O₁ and with O₂ and obtain

For phonon operators, there is a useful identity to transform the thermodynamic average of an exponential of operators into an exponential of the thermodynamic average of operators,⁶² 

and we have

Thus, the time integral becomes

where we have changed the integration variable, z = t + iβ/2. Using the steepest descent method,

where z₀ is the saddle point of W(z), W′(z₀) = 0 and U(z) only weakly depends on z, we obtain

and Eq. (33).

To derive Eq. (41), we notice that according to Eq. (39),

Consequently, the time integral of Eq. (32) is

Applying the same procedure as described in  Appendix A to the two terms, we obtain