Exciton–polaritons in a 2D hybrid organic–inorganic perovskite microcavity with the presence of optical Stark effect

Kenneth Coker, Chuyuan Zheng(郑楚媛), Joseph Roger Arhin,Kwame Opuni-Boachie Obour Agyekum, and Weili Zhang(张伟利)

1School of Information and Communication Engineering,University of Electronic Science and Technology of China,Chengdu 611731,China

2Department of Electrical and Electronic Engineering,Ho Technical University,Ho 00233,Ghana

3Telecommunication Engineering Department,Kwame Nkrumah University of Science and Technology,Kumasi 00233,Ghana

Keywords: exciton-polaritons,perovskite,microcavity,optical Stark effect

1.Introduction

The layered material class known as two-dimensional(2D) hybrid organic-inorganic perovskites (2D HOIPs) is quickly gaining popularity as a promising semiconductor for optoelectronic and energy-harvesting applications.[1-5]There are several causes for this rise in interest.One of them is the fact that the structure and composition of organic and inorganic components can be changed to alter the physical properties of 2D HOIPs.[6]Furthermore, at ambient temperature,the 2D HOIPs exhibit a high quantum confinement charge that has significant excitonic effects.[7,8]

Whenever an extrinsic trigger, like excitons, collides with light intensively, hybrid particles called polaritons are formed.[9,10]These polaritons have also played a vital role in the development of optoelectronic devices over the last few years.[11,12]Moreover,the behavior of polaritons in 2D HOIPs exhibits remarkable characteristics when confined within a highly reflective structure, such as a distributed Bragg reflector(DBR)mirror.[13]In comparison to more compact designs, the DBR mirror effectively confines light with minimal loss, rendering it highly advantageous for the advancement of polaritonic devices.[14]Leveraging the impact of the optical Stark effect (OSE) on these exciton-polaritons confined between two DBR mirrors becomes particularly promising,given the successful control of polaritons.[15-17]In recent times, researchers have explored the influence of the OSE in 2D WS2[18]monolayers at room temperature and developed an all-optical method to realize topological insulators using OSE within exciton-polariton systems.[19]These studies provide ways to manipulate polaritons through OSE with a large degree of control freedom.

Alternatively, this study introduces 2D HOIP into a microcavity to form exciton-polaritons and focuses on the influence of OSE on the exciton-polaritons.Using both steady and dynamic state analyses,strong coupling between excitons and polaritons is revealed through the observation of Rabi splitting, and the evaluation of polariton branches in the momentum space is analyzed at different strengths of OSE.

2.Methodology

This study presents a setup aimed at analyzing the influence of the OSE on exciton-polaritons within a 2D HOIP active layer.The schematic structure of the setup is illustrated in Fig.1, featuring exciton-polaritons confined within two highly reflective DBR mirrors.These DBR mirrors are made up of numerous, alternate layers of materials with low and high refractive indices that have been precisely structured to provide high reflectivity at a wavelength band that can confine photons with the same energy to the excitons of the embodied 2D HOIP.The 2D HOIP with a thickness of 0.8 nm, serves as the active material, with the organic layer sandwiched between two inorganic layers,forming the hybrid perovskite.[20]The incorporation of the OSE is applied by illuminating an optical pulse with lower photonic energy to the excitons.

A complex numberm+i×nis used to characterize the refractive index of 2D HOIP.[21]The real partmof the complex number describes how much light is slowed down as it passes through the material.The imaginary componentnrepresents the extinction coefficient,which quantifies the amount of light absorbed by the 2D HOIP.The complex refractive index of 2D HOIPs is dependent on various factors such as the chemical composition,crystal structure,temperature,and the wavelength of light that passes through it.The complex refractive index can also be influenced by the presence of defects and impurities in the material,which can result in significant changes in the optical properties.The refractive index can be calculated through the complex permittivity,εr.[22]As illustrated in Fig.2, the ability of the 2D HOIP to store electric charges is characterized by its real componentεm, which represents the permittivity.On the other hand,the energy dissipated due to the absorption of electromagnetic waves is denoted by the imaginary componentεn, signifying the dielectric loss.The relatively higher permittivity observed in Fig.2 imparts the 2D HOIP with a distinctive advantage in efficient electric charge storage,making it highly suitable for photovoltaic applications to store electric charges generated from absorbed light.

Fig.2.Complex permittivity of 2D HOIP against wavelength.The blue curve represents the real component (permittivity) whereas the orange curve represents the imaginary component(dielectric loss).

Moreover,the 2D HOIP exhibits a favorable characteristic of relatively lower dielectric loss, signifying minimal energy dissipation during electromagnetic wave absorption.The focus will be at the wavelength close to where the first excitonic peak occurs.

3.Results and analysis

In the proposed setup, exciton-polaritons are introduced into the microcavity using a quasi-resonant continuous-wave(CW)laser.Once the injecting laser’s parameters,i.e.,angle of incidence,frequency,and intensity,have been properly set,the injected photons and the excitons in the active layer can couple strongly,forming Rabi splitting and two dips in the reflection spectrum.The reflectivity of the proposed setup against wavelength is plotted for different values of the incident angleθi(i.e.,0°,10°,20°,and 30°respectively)as shown in Fig.3(a).The presence of the 2D HOIP yields two dips that correspond to the phenomenon of Rabi splitting, indicating strong coupling between the confined cavity photons and the excitons in the 2D HOIP.The leftmost dips represent the branch of upper polaritons(UP),while the rightmost dips represent the branch of lower polaritons (LP).From Fig.3(b), it is realized that the Rabi splitting increases as the incident angle is increased.With an increase in the incident angle,the horizontal-direction momentum of input photons changes,leading to a modified interaction between the excitons and photons in the active layer.This interaction causes a widening of the separation between the branches,thereby increasing the Rabi splitting.In the presence of OSE,the excitonic energy is expressed asωx=ω0x+δwhereω0xis the excitonic energy that is free to flow andδis the OSE-induced energy shift.δis directly proportional to the intensity of the Stark optical pulse(square of the field,|Eo|2),and inversely proportional to the photon-exciton detuning,Δ.In the presence of the OSE,expanding the expression for the excitonic energy being[23]

whereMdmis the dipole moment matrix.The extended Gross-Pitaevskii equation[24]in Eq.(1)is used to model the system’s dynamics and characterize the system’s interactions.

whereγxis the excitonic decay rate,andγcis the photonic decay rate.ψxis the excitonic field,ψcis the photonic field,gxis the exciton-exciton interaction strength,ΩRis the Rabi frequency,ωcis the photonic energy,and ˆHodenotes the Hamiltonian matrix.For brevity, the decay rates and the nonlinear terms have been excluded from the simulation.As a result of the Rabi frequencyΩR,the LP and UP branches become the two main eigenmodes of ˆHoas shown below:

Fig.3.Reflectivity of the microcavity with 2D HOIP present.(a)Reflectivity against wavelength.(b)Rabi splitting for the different incident angles.The plots represent different incident angles θi (i.e., 0°, 10°,20°,and 30°)from left to right respectively.The left dips represent the UP and the right dips represent the LP.

Fig.4.Dispersion of the UP and LP branches as a function of the inplane momentum.The Stark-induced energy levels,ELP1,UP1,ELP2,UP2,ELP3,UP3, and ELP4,UP4 correspond to |Eo|2 values of 0, 50, 100, and 150 MV respectively.

Fig.5.Changes in photonic and excitonic fractions. |C(k)1|2|X(k)1|2,|C(k)2|2|X(k)2|2, |C(k)3|2|X(k)3|2, and |C(k)4|2|X(k)4|2, correspond to|Eo|2 values of 0,50,100,and 150 MV respectively.

The OSE can happen in a fast transient, paving the way for the dynamic control of the polaritonic field as seen in Fig.6.In the dynamic analysis, equation (1) is numerically solved using the Runge-Kutta method to the fourthorder.[18]The analysis is done for two cases: when the optical Stark pulse is absent and when the optical Stark pulse is present, considering the excitonic fraction as an example.A transient optical Stark pulse which is of the structurewith a Gaussian shape is applied to the field.Ais the amplitude of the pulse,tcis the center of the pulse (time at which the pulse reaches the maximum value),andβis the pulse width.In the analysis,A=3 V,tc=0.46 ps,andβ=0.125 ps.The time domain waveform clearly showing the energy shift in the polaritonic field as a result of the introduction of the optical Stark pulse is seen in Fig.6(a).The highlighted area shows a peak value at the said transient and this is as a result of the influence of the OSE.The fast Fourier transform (FFT) is then employed to plot the dispersion of the polaritons in the frequency domain of the polaritonic field,shown in Fig.6(b).The blue curve in Fig.6(b)represents the case when the OSE is absent.The leftmost peak represents the UP branch whereas the rightmost peak represents the LP branch.In the presence of the OSE(area highlighted with red),an energy shift and numerous frequency points are seen for the polariton branches.Since the optical Stark pulse with Gaussian shape is a superposition of different frequency components,when the pulse interacts with the field,different energy transitions are excited.This excitation leads to the generation of new frequency components.The field’s response to the introduction of the Gaussian-shaped optical Stark pulse exhibits resonance with the polariton branches.This resonance results in different frequency components corresponding to the energy differences associated with the polariton branches.These oscillations show different frequencies that correlate to the energy differences.Depending on the amplitude and timing of the pulse,different energy shifts and states can be realized offering opportunities for controlling and manipulating polaritonic properties in a highly tunable manner.For clarity, the values of certain key parameters used in this study have been presented in Table 1.

Fig.6.Dynamic analysis of the OSE on the polaritonic field.(a) Time domain waveform showing the influence of the OSE.(b) Frequency domain showing the dispersion of polaritons and energy shift of the UP(leftmost peaks)and LP(rightmost peaks)branches in the absence and presence of the OSE.

Table 1.Key parameters and their values in the simulation.

4.Conclusion

This study has explored the optical properties of 2D HOIP materials and investigated the impact of the OSE on excitonpolaritons within a 2D HOIP active layer.Through steady and dynamic state analyses,the study demonstrated that an external optical Stark pulse effectively modifies various polariton characteristics, including dispersion, excitonic and photonic fractions,and energy levels.The results revealed that the OSE induces energy shifts in excitons proportional to the pulse intensity, leading to significant changes in the polaritonic field.The dynamic state analysis showed a transient fluctuation of the polaritonic field as a result of the influence of the OSE in the time domain and a distinctive energy shift in the Fourier domain.These analyses indicate the nonlinear and complex nature of the field’s response to the OSE which is essential for studying the dynamics and behavior of polaritons in optoelectronic devices,microcavities,and other related systems.These findings have significant bearings for the development of optoelectronic devices utilizing 2D HOIP materials.The results provide valuable insights for future research in this rapidly evolving field.

Acknowledgements

The authors would like to express their gratitude to the Key Laboratory of Optical Fiber Sensing and Communication,University of Electronic Science and Technology of China,Ministry of Education of China.

Project supported by the National Natural Science Foundation of China (Grant Nos.11974071 and 62375040)and the Sichuan Science and Technology Program (Grant Nos.2022ZYD0108 and 2023JDRC0030).