The quantum-confined Stark effect in layered hybrid perovskites mediated by orientational polarizability of confined dipoles

The quantum-confined Stark effect (QCSE) is an established optical modulation mechanism, yet top-performing modulators harnessing it rely on costly fabrication processes. Here, we present large modulation amplitudes for solution-processed layered hybrid perovskites and a modulation mechanism related to the orientational polarizability of dipolar cations confined within these self-assembled quantum wells. We report an anomalous (blue-shifting) QCSE for layers that contain methylammonium cations, in contrast with cesium-containing layers that show normal (red-shifting) behavior. We attribute the blue-shifts to an extraordinary diminution in the exciton binding energy that arises from an augmented separation of the electron and hole wavefunctions caused by the orientational response of the dipolar cations. The absorption coefficient changes, realized by either the red- or blue-shifts, are the strongest among solution-processed materials at room temperature and are comparable to those exhibited in the highest-performing epitaxial compound semiconductor heterostructures.

The total fit is given as the solid black trace. Unlike, for the layered hybrid perovskites, the fit here has been produced using Elliott's formula modified to account for variation from the parabolic-band regime. 1-3 b, Third-derivative of the optical absorption. c, EA spectra at various applied voltages (Vpk). A strong correspondence exists with the third-derivative spectrum, indicating the low-field Franz-Keldysh-Aspnes (FKA) effect 4 , a result in agreement with a thorough work from Ziffer et al. 5 . d, Quadratic dependence of the EA on applied field, in agreement with the FKA effect 4 . Figure 11. EMA Calculated Stark Shifts for Layered Perovskites. a, Cumulative Stark shifts to electron and hole energy levels as a function of effective well width for an infinite well potential subjected to an electric field of 20 kV·cm -1 . Consistency between the calculated shifts and the theoretical relation for shifts in the quantum regime is shown. In these calculations, the effective well width is greater than the actual well width due to penetration of the wave function into the real finite well. b, Cumulative Stark shifts to electron and hole energy levels of n = 3 perovskites with different cations as a function of applied field. The finite well model has been used. In our electroabsorption studies, the field applied to our n = 3 perovskites is 56 kV·cm -1 for methylammonium films and 53 kV·cm -1 for cesium films. The EMA calculations indicate similar energy level shifts of ~100 µeV. This value is close to our experimentally found 300 µeV.

Supplementary Tables
Supplementary Table 1. Internal Electric Field. Electric fields (kV·cm -1 ) generated across the modulator device, within the perovskite platelets, and within the quantum wells and barriers of the platelets. Applied bias is 5 V.

Supplementary Discussion
Second-Harmonic Modulation. The use of a lock-in amplifier referenced to the second harmonic of the modulation frequency is standard practice for electroabsorption measurements as it avoids any spurious effects that may result from linear dependences on the modulation field.
Within the low-field regime electroabsorptive effects vary quadratically with field strength, whereas linear electro-optic or electromechanical effects such as the Pockels effect or piezoelectric effect vary linearly with field strength. 4,28 These effects may produce changes to the reflectance spectra of electroabsorption modulators and therefore appear as absorption changes. If these undesired effects are grouped under A1 and the electroabsorptive effects under A2, then, The electric field used in modulation spectroscopy can be written as, where ω is the modulation frequency. The square of this is then, which is equivalent to, 2 = App 2 [ 1−cos (2 )   2 ] .
The total absorption/reflection changes, ∆ , can then be written as: By locking into the second harmonic of the modulation frequency, only changes related to the electroabsorptive effects will be detected.
Similarly, if the device had a built in field, , the total field would be, Again considering the quadratic dependence of electroabsorptive changes on field, only electroabsorptive changes related to the modulation bias will be detected when locking into the second harmonic.